
Citation 
 Permanent Link:
 http://digital.auraria.edu/AA00001476/00001
Material Information
 Title:
 Anisotropic elastic modeling of reinforced soils
 Creator:
 AbuHassan, Mohammad
 Place of Publication:
 Denver, Colo.
 Publisher:
 University of Colorado Denver
 Publication Date:
 2006
 Language:
 English
 Physical Description:
 xxxv, 545 leaves : ; 28 cm.
Thesis/Dissertation Information
 Degree:
 Doctorate ( Doctor of Philosophy)
 Degree Grantor:
 University of Colorado Denver
 Degree Divisions:
 Department of Civil Engineering, CU Denver
 Degree Disciplines:
 Civil Engineering
 Committee Chair:
 Chang, NienYin
 Committee Members:
 Ko, HonYim
Xi, Yunping Brady, Brian T. Wang, ShingChun Trever Rorrer, Ronald A. L.
Subjects
 Subjects / Keywords:
 Soil mechanics  Mathematical models ( lcsh )
Soil mechanics  Mathematical models ( fast )
 Genre:
 bibliography ( marcgt )
theses ( marcgt ) nonfiction ( marcgt )
Notes
 Thesis:
 Thesis (Ph. D.)University of Colorado Denver, 2006. Civil engineering
 Bibliography:
 Includes bibliographical references (leaves 540545).
 General Note:
 Department of Civil Engineering
 Statement of Responsibility:
 by Mohammad AbuHassan.
Record Information
 Source Institution:
 University of Colorado Denver
 Holding Location:
 Auraria Library
 Rights Management:
 All applicable rights reserved by the source institution and holding location.
 Resource Identifier:
 263685156 ( OCLC )
ocn263685156

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ANISOTROPIC ELASTIC MODELING OF REINFORCED SOILS
by
Mohammad AbuHassan
B.S,5 University of Jordan, 1999
M.S., University of Colorado at Denver, 2002
A thesis submitted to the
University of Colorado at Denver and Health Science Center
in partial fulfillment
of the requirements for the degree of
Doctoral of Philosophy
Civil Engineering
2006
This thesis for the Doctor of Philosophy
degree by
Mohammad AbuHassan
has been approved
by
ShingChun Trever Wang
Ronald A. L. Rorrer
Mohammad AbuHassan (PhD,, Civil Engineering)
Anisotropic Elastic Modeling of Reinforced Soils
Thesis directed by Professor NienYin Chang
ABSTRACT
Numerical modeling is a powerful tool for studying the static and the dynamic
responses of different types of geostructures such as MSE foundations^ walls,
and bridge abutments. It imposes no size limitation; it is speedy; and it is cost
effective. However, numerical modeling must be calibrated and validated by
determining the correct material parameters and comparing the results of the
numerical model with those of carefully performed physical modeling. When
proven effective, numerical modeling can be relied on heavily for further
explorations of geostrucutre behaviors. An accurate numerical analysis of
reinforced soil structures must account for frictional sliding between the soils and
the reinforcing elements. The tensile stresses that are built up in reinforcing
layers, due to friction, are transferred to soil's layers, which compensates for their
lack of tensile resistance. After all, it is a composite material that exhibits the best
qualities of both of its constituents, the compressive resistance from soil and the
tensile resistance from geosynthetics.
When modeling MSE structures, using finite element method, difficulties in
forming the finite element mesh and time consuming while preprocessing and
CPU processing cannot be avoided, especially for a large problem. To alleviate
the interface convergence problems and speed up the computation one can treat
the geocomposite as an anisotropic homogeneous material, instead. This
homogeneous material must account for constituents^ properties, which includes
their interface frictional interaction. Therefore, the main goal of this research is to
develop a constitutive model that predicts the mechanical properties of reinforced
soil, while taking into account the soil properties, the reinforcing properties, the
vertical spacing between reinforcing layers, the confining pressure, and the
frictional interface. The constitutive model which can best account for the above
said characteristic is identified as ''transversely isotropic linear elastic modeI,\
The transversely isotropic models were formulated through statistical modeling of
the results from finite element modeling of the behavior of the composite cube.
These transversely isotropic models were found to be effective in simulating the
linear elastic resposnses of laboratory samples and full geostrucutres, including
full scale MSE foundations, MSE walls, and MSE abutments.
This abstract accurately represents the content of the candidate^ thesis. I
recommend its publication.
Signei
DEDICATION
This thesis is dedicated to my loving parents, Dr. Eng. Abdelfattah AbuHassan
and Fatek AbuHassan, who continuously give me the unlimited support in
achieving all my goals.
ACKNOWLEDGMENT
Thank you GOD
I would like to begin by expressing my gratitude to my Professor, NienYin
Chang, chair of this thesis committee who gave me full support and help. Prof.
Chang provided all the possibilities to complete this project including this final
thesis. Working with Prof* Chang was an experience that helped me gain broader
perspective on many areas. I also wish to thank the members of my committee,
Prof. HonYim Ko3 Prof. Yunpin Xi, Prof. Trever Wang, Prof Brian Brady, and
Prof. Ronald Rorrer for tneir helpful comments and suggestions.
I want to also thank ZehZon Lee for his vast knowledge in MSE walls and their
behavior Thanks also to Jan Chang for his guidance in preparing the triaxial
tests, for Russel Cox for his assistance in creating the nmte element mesh of the
MSE bridge abutment, Cesar Gonzalez for conducting the tensile strength test,
Myron Lacome for his experience with AutoCad, Khaled AbuNameh and
Dawood Oqlah for editing this thesis, and also like to thank Paola Mera for the
incessant encouraging and support.
Last but not least, I would like to thank my family for their support even though
they were thousand of miles away in Jordan.
To all, I say thank you for making this overall education experience possible.
CONTENTS
Figures..................................................................xvi
Tables..........................................*........................xxxii
Chapter
1. Introduction.......................*...........................1
1.1 Problem Statement.............................................1
1.2 Research Goals and Objectives.................................2
1.3 Research Methodology and Tasks................................3
1.4 Significance of Research......................................4
2. Literature Review.............................................5
3. Theoretical Backgroimd.......................................11
3.1 Introduction.................................................11
32 Finite Element Theory for Linear Materials...................11
3.3 Finite Element Theory for Nonlinear Materials................16
3.3.1 Tangent Stif&iess Method.....................................16
33.2 Viscoplastic Method.........................................17
3.3.3 Modified NewtonRaphson Method...............................17
33.3.1 Explicit Algorithm...........................................18
3.3.3.2 Implicit Algorithm...........................................19
3.4 Finite Element Commercial Codes..............................19
3.4.1 NIKE3D, TRUGRID, and GRIZ....................................20
3 A2 LSDYNAFEMBPCand LSPRE/POST...............................21
vii
3.5 Constitutive Modeling of Soil................................23
3.5.1 Hyperbola Model............................................ 24
3.5.2 RambcrgOsgood Model.........................................26
3.5.3 MohrCoulomb Failure Criterion...............................27
3.5.4 DruckerPrager Criterion.....................................28
3.5.5 Cap Models...................................................30
3.6 Contact in rmite Element.....................................34
3.7 Summary......................................................36
4. Anisotropic Properties of GeoComposite......................37
4.1 Introduction.................................................37
4.2 Composite Materials.........................................38
4.2.1 Orthotropic (Three Planes of Symmetry).......................43
4.2.2 Transversely Isotropic (A plane of Isotropy).................44
4.2.3 Isotropic (Complete Symmetry)................................46
4.3 Evaluation of Composite Material Properties................ 47
4.3.1 Finite Element Approach in Determining the Mechanical
Properties of a Geocomposite...............................48
4.3.2 Analytical Method in Determining the Mechanical Properties
of a GeoComposite(Jones 1975)..............................59
4.3.2.1 Isostrain Model.............................................60
4.3.2_2 Isostress Model.............................................62
4.4 Stress Distribution of Reinforcement.........................68
4.4.1 Weak Reinforcement...........................................68
4.4.2 Stiff Reinforcement..........................................74
4.5 Summary And Conclusion.......................................79
5. Statistical Modeling of Transversely Isotropic Geo
Composite(Michael H. Kutner 2005)...........................80
5.1 Theory of Multiple Regression Analysis.......................80
viii
5.1*1 Analysis of Variance (ANOVA) Table...............................83
5.1.1.1 Test for Regression Relation.....................................85
5.1.2 Inferences about Regression Parameters...........................86
5.1.2.1 Interval Estimation of pic.......................................86
5.1.2.2 Test for pk......................................................86
5.1.3 Estimation of Mean Response and Prediction of New
Observation................................................. 88
5.1.3.1 Interval Estimation of E{ Yh}....................................88
5.1.3.2 Prediction of New Observation E{ Yh}........................... 88
5.1.4 Diagnostic and Remedial Measures.................................89
5.1.5 Building the Regression Model....................................90
5.1.6 Automatic Search Procedures for Model Selection..................90
5.1*6.1 Best Subset Algorithms____.......................................91
51.6.1.1 Coefficient of Determination (R2P)...............................91
5.L6.1.2 Mallows9 Cp Criterion.......................................... 92
5.1.6.1.3 Akaikes Information Criterion (AICP) and Schwarzs Bayesian
Criterion (SBCP).............................................. 92
5.L6.1.4 Prediction Sum of Squares (PRESSp) Criterion.................... 93
5.1.6.2 Stepwise Regression Methods......................................93
5.1.7 Multicollinearity and Its Effect.................................94
5.2 Application of Multiple Regression Analysis on Geo
Composite ......................................................94
5.2.1 Plane Modulus of Elasticity (Eh).................................96
5.2.1.1 Model Selection for Eh...........................................99
5.2.1.2 Diagnostic and Remedial Measure of Eh...........................102
5.2.1.3 Inference about Regression Parameters of Eh.....................104
5.2.1.4 Interval Estimation of E{Yh}..............................*......107
5.2.1.5 Interval Prediction for New Observation Yh(new).................108
ix
5.2,1.6 Alternative Model of Eh........................................109
5.3 Summary and Conclusions:.......................................109
6. Drained Triaxial Tests on GeoComposite Samples................Ill
6.1 Introduction...................................................Ill
6.2 Test Materials.................................................112
6.2.1 Ottawa Sand....................................................112
6.2.2 Polypropylene Geotextiles......................................113
6.3 Laboratory Tests......................*.......................113
6.3.1 Tensile Geotextile............................................113
63.1.1 Equipments and Test Specimen...................................114
63.1.2 Preparation...................................................114
63.1.3 Results and Discussion.............*..........................414
6.3.2 Hydrostatic Compression........................................117
6*3.2.1 Sample Preparation.............................................118
63.2.2 Results Discussion of Hydrostatic Compression Test.............134
6.3.3 Conventional1 naxial Compression Tests.........................136
6.3.3.1 Samples Preparation............................................137
63.3.2 Results and Discussion of Drained Triaxial Tests...............138
6.4 Finite Element Calibration.....................................146
6.4.1 Ottawa Sand....................................................147
6.4.2 Reinforcement Properties (Geotextile)..........................156
6.4.3 Bottom and Top Cap (Steel).....................................159
6.5 Finite Element Simulation and Validation.......................159
6.5.1 Tensile Test on Geotextile................................... 160
6*5.2 Drained Triaxial Test on Ottawa Sand......................... 161
6.5.3 Drained 1 naxial Test on Reinforced Samples with Discrete
Approach....................................................lt>^
x
6.5.4 Drained Triaxial Test on Reinforced Samples with
Homogeneous Material Approach...............................171
6.5.5 Results Summary of Discrete and Homogeneous Models...........178
6.6 Summary and Conclusion.......................................183
7. Validation of Finite Element Method on GeoComposite.........185
7.1 Introduction.................................................185
7.2 Triaxial Test (Liu 1987).....................................186
7.2.1 Laboratory Test..............................................186
7.2.1.1 Materials Preparation........................................186
7.2.1.1.1 Soil....................................................... 186
7.2.1.1.2 Reinforcement............................................... 187
7.2.1.2 Samples Patterns.............................................189
7.21.3 Test Results.................................................191
7.2.2 Finite Element Analysis of UnReinforced and Reinforced
Samples using Discrete Models...............................193
7.2.2.1 Geometries of Specimens......................................194
1222 Test Simulation..............................................195
7,2*2.3 Materials Parameters....................*......................195
1223A Sandy Soil Parameters........................................196
12232 Reinforcement Parameters.....................................198
7,2.2.4 Results of the Finite Element Analysis using Discrete Models.199
7.2.3 Finite Element Analysis of Reinforced Samples using
Homogeneous Model...........................................208
7.2.3.1 Equivalent Properties........................................208
72.3.2 Results of Finite Element Model using Homogeneous Models.....209
7.3 Plane Strain Testing (Kanop Ketchart 2001)...................216
7.3.1 Background...................................................216
7J2 Conventional Compression Triaxial Tests......................217
xi
7.3.2.1 Laboratory Results...........................................217
1322 Finite Element Results.......................................219
73.3 SGP Tests....................................................224
7.3.3.3 Finite Element Results using Homogeneous Model...............231
7.4 Summary and Conclusions......................................234
8. ThreeDimensional Foundation with Soil Reinforcement.........238
8.1 Introduction.................................................238
8.2 Concept and Design Consideration of Foundation(McCarthy
1988).......................................................239
8.2.1 Unreinforced Foundation Soil................................239
8.2.2 Foundation with Soil Reinforcement...........................247
83 Finite Element Analysis of Square Footing Supported by Un
Reinforced and Reinforced Soil..............................254
8.3.1 Materia] Properties........................................ 257
8.3.1.1 Ottawa Sand..................................................258
8.3.1.2 Geotextile................................................. 260
8.3.1.3 Concrete.....................................................261
83*2 Finite Element Modeling......................................261
8.4 Results and Discussion.......................................262
84.1 Results of Foundation without Soil Reinforcement.............262
8.4,2 Results of Foundation with Soil Reinforcement................265
8.4.2.1 Effect of Spacing............................*...............266
8.4.2.2 Effect of Geosynthetic Stiffiiess............................269
8.4.23 Summary of Spacing and Stiffness Effects.....................271
8.4.2.4 Stress Distribution on Reinforcement Layers..................277
8*4*3 Results of Foundations with Reinforced Soil Modeled as
Homogeneous Material........................................288
8.5 Summary and Conclusions......................................294
xii
9, Static and Dynamic Analysis of MSE Wall with Rigid Facing........*.297
9.1 Introduction.....................................................297
9.2 Concept and Design Consideration of Retaining Walls...............298
9.2.1 Retaining Walls with Geosynthetic Reinforcement (Vector
Elias 2001).......................................................299
9.2. LI External Stability of Vertical MSE Walls and Horizontal
Backfill..........................................................302
9.2. L 1.1 Sliding Stability.................................................305
9.2. LL2 Overturning Stability.............................................305
9.2* 1.1.3 Bearing Capacity Failure..........................................306
9.2.1.2 Internal Stability of MSE walls..................................309
9.2.2 Hybrid Retaining Walls...........................................313
9.3 Finite Element Analysis on MSEWall with Rigid Facing.............314
9.3.1 Loading...........................................................317
9.3.2 Material Properties............................................. 318
9.3.2.1 Backfill and Foundation Soil......................................318
93.2.2 Inclusion...................................................... ,320
9.3.2.3 Concrete Wall and Footing.........................................320
9.33 Sliding Interface.................................................322
9.3*4 Boundary Conditions...............................................322
93.5 External and Internal Stability of MSE wall.......................323
9.3.5.1 External Stability due to Static Loading..........................324
93.5.2 Internal Stability due to Static Loading..........................326
9.3.5.3 External Stability due to Seismic Loading.........................329
9.3.5.4 Internal Stability due to Seismic Loading.....................*...332
9*4 Results and Discussion............................................336
9.4.1 Results of Discrete Model.........................................337
9.4.1.1 Lateral Earth Pressure (ax) behind the Hybrid Wall................337
xiii
9.4.1.2 Lateral Wall Displacement (8X).................................339
9.4.1.3 Bearing Pressure (az) and Settlement (8Z) of Foundation........341
9.44 Inclusion Tensile Stresses.....................................343
9.4.2 Results of Homogeneous Model...................................350
9.4*2.1 Lateral Earth Pressure (ax) behind the Hybrid Wall.............352
94,2.2 Lateral Wall Displacement (6X).................................354
9A2.3 Bearing Pressure (cjz) and Settlement (5Z) of Foundation.......356
9,5 Summary and Conclusions.................................... 360
10. Static and Dynamic Analysis of MSE Bridge Abutments............364
10.1 Introduction......................................... 364
10.2 Design Consideration of Bridge Abutment (FHWA, 2001).:........367
10.3 Finite Element Analysis on Bridge Span Supported by MSE
Abutment.......................................................368
10.3.1 Loading...................................................... 372
10.3.2 Material Properties............................................374
10.3.2.1 Backfill and Reinforced Soil..,,...............................374
10.3.2.2 Inclusion......................................................376
10.3.2.3 Concrete.......................................................376
10.3.3 Sliding Interface..............................................377
10.3.4 Boundary Conditions ......................................... 377
10.4 Results and Discussion....................................... 378
10.4.1 Results of Discrete Mode!.................................... 378
10A1.1 Lateral Pressure or Bndge Abutment.............................379
10.4.L2 Bearing Pressure and Settlement Distribution of MSE Backfill
beneath the Spread Footing.....................................386
10.4.1.3 Longitudinal and Transverse Earth Pressure behind the MSE
Wall......................................................... 392
xiv
10.4.1.4 Longitudinal and Transverse Displacements of MSEWall.....400
10.4.1.5 Inclusion Tensile Stresses in the MSEWall................407
10.4.2 Results of Homogeneous Model............................ 415
10.4.2.1 Lateral Pressure of Bridge Abutment.......................418
10.4.2.2 Bearing Pressure and Settlement Distribution of MSE Backfill
beneath the Spread footing................................425
10.4.2.3 Longitudinal and Transverse Earth Pressure behind the MSE
Wall......................................................430
10.4*2.4 Longitudinal and Transverse Displacements of MSEWall.....437
10.5 Summary and Conclusions...................................440
11. Summary, Conclusions, and Recommendations.................443
111 Summary...................................................443
11.2 Conclusions....................................... ....450
11.3 Recommendations for Future Studies........................458
Appendix
A. Equivalent Transversely isotropic Properties of Geo
Composites................................................460
B. Regression Analysis of Independent Parameters (Ev> Vh, vV5 Gv,
andGh)................................................... 471
REFERENCES.........................................................540
xv
LIST OF FIGURES
Figure
3.1"Parameters of MohrCoulomb model11.......................................27
3.2 lfDruckerPrager criterion''............................................29
3.3 Tap model"...............................................................31
3.4 "Interpretation of parameters of fi".....................................32
4.1"Deviator stresses on cube element1'......................................42
4.2 "Transversely isotropic material (23) plane of isotropy)................45
4.3 "Geocomposite cube element (one, three, and five Layers)11.............49
4.4 "Es (MPa) distribution under different a"..............................51
4.5 "Theoretical frame of all numerical tests11.............................52
4.6 "Load curve distributed for all 3 loads; gravity, confining pressure, and
deviator stress".........................................................53
4.7 Parallel (Isostrain) Model".................................... 60
4.8 "Serial (Isostress) Model1.............................................62
4.9 "Xstress distribution for geosynthetic with E = 22 MPa............... 70
4.10 TTYstress distribution for a geosynthetic with E = 22 MPaT,............70
4.11"Zstress distribution (Pa) for a geosynthetic with E = 22 MPan............71
4.12 "Plane shear stress, XY, distribution (Pa) for a geosynthetic with E 
22 MPa"..................................................................71
4.13 11 Vertical shear stress, YZ, distribution (Pa) for a geosynthetic with E
=22 MPa.................................................................72
4.14 "Vertical shear stress, XZ, distribution (Pa) for a geosynthetic with E
=22 MPa.............................................................. 72
xvi
4.15 "Xstress distribution (Pa) for a geosynthetic with E =1100 MPa"......75
4.16 nYstress distribution (Pa) for a geosynthetic with E =1100 MPa'1.....75
4.17 MZstress distribution (Pa) for a geosynthetic with E =1100 MPa"......76
4.18 ^Plane shear stress, XY, distribution (Pa) for a geosynthetic with E =
1100 MPa.....................'........................................76
4.19 MVertical shear stress, YZ, distribution (Pa) for a geosynthetic with E
=1100 MPa"..............................................................77
4.20 "Vertical shear stress, XZ, distribution (Pa) for a geosynthetic with E
=1100 MPa"..............................................................77
5.1"The regression analysis framework"......................................95
5.2 "The matrix plot of Eh with respect to all X variables...............97
5.3 'The residual versus the fitted values of Ehn.........................102
5.4 "Normal probability plot of the residuals of Eh.......................103
61"Geotextile Tensile Tests"..............................................117
6.2 "Triaxial Cell".......................................................120
6.3 ^Essential materials for hydrostatic and triaxial tests1'..............120
6.4 "Measuring devices for hydrostatic and triaxial Tests................121
6.5 ^Ottawa Sand, raining device and iunnel'1.............................121
6.6 "Grease, rubber cement, screw driver and brush".......................122
6.7 Vacuum Device"......................................................122
6.8 ''Initial height measurement"........................................ 123
6.9 "Thickness of membrane measurement"...................................124
6.10 Pulling the membrane over the bottom cap of the sample1'.............125
6.11"Assembling the split mold around the bottom cap and wrapping the
membrane around the mold"..............................................126
6.12 ''Applying vacuum between the split mold and the membrane'1...........127
6.13 H Adding the soil the using a raining device and a funnel*1...........127
6.14 "Lifting the raining device slowly"..................................128
XVII
6.15 "Tamping the sample gently"..........................................128
6.16 "Placing and leveling the top cap"..................................129
6.17 "Applying vacuum pressure to the top of the sampleN..................129
6*18 "Measurement of samples height.....................................131
6.19 Measurement of samples Diameter.................................131
6.20 Adding rubber cement around the porous stone area"..................132
6.21 "Placing a greased Oring"..........................................132
6.22 "Adding water through the confining valven...........................133
6.23 "Confining pressure and volume change"..............................133
6.24 "Isotropic consolidation test results due to Ottawa sand and the
triaxial ceUM.........................................................135
6.25 "Hydrostatic compression result of Ottawa sand only".................135
6.26 "Drained triaxial test"..............................................138
6.27 StressStrain relation at or]=103 kPa.............................141
6*28 "StressStrain relation at = 207 kPaM.................................141
6.29 ,rStressStrain relation at <73 = 310 kPaF,..........................142
6.30 nPeak stress difference as a fucntion of number of reinforcing Layer
and confining pressure"...............................................142
631 initial Youngfs Modulus as a function of number of Reinforcing
Layer"................................................................143
6.32 "Volumetric strain Axial strain relation of Ottawa sand"............143
6.33 HMohrCoulomb envelope for samples with different reinforcing
pattemsrT.............................................................145
6.34 "'Relationship between the logarithm of equivalent cohesion and
spacing between layers"...............................................146
6.35 "Geologic Cap model"............................................... 147
6.36 "DruckerPrager criterion"..........................................148
xviii
637 "Relation between normalized deviator stress and axial strain11....151
6.38 "The shape of first yield surface, fl,for Ottawa sand according to
Cap model11.........................................................152
6.39 "Observed and predicted Vol. strain vs. Mean pressure".........*.155
6.40 "Isotropic Elastic Plastic material"..............................157
6*41 'Tensile force vs, axial strain of the heat bounded geotextile"....161
6.42 "Finite element mode) of Ottawa sand specimen1'....................162
6.43 nFESimulation vs. measured triaxial results of Ottawa Sand @ Â¢73 =
103 kPa".............................................................163
6.44 MFESimulation vs. measurea iriaxial results of Ottawa Sand @ a3 =
207 kPa"...............................................;............164
6.45 "FESimulation vs. measiired triaxial results of Ottawa Sand @^2
310 kPa.............................................................164
6.46 "Patterns of reinforced samples'.................................. 166
6.47 ^FESimulation vs. measured triaxial results of reinforced Sand and 2
layers of reinforcement @ CT3 =103 kPa,T............................167
6.48 'TESimulation vs. measured triaxial results of reinforced Sand and 2
layers of reinforcement @03 = 207 kPa"............................. lo/
6*49 "FESimulation vs. measured triaxial results of reinforced Sand and 2
layers of reinforcement @ CT3 = 310 kPa............................168
6.50 MFESimulation vs. measured triaxial results of reinforced Sand and 4
layers of reinforcement @ 03 =103 kPa"..............................168
6.51 T,FESimulation vs. measured triaxial results of reinforced Sand and 4
layers of reinforcement @03 207 kPa1'............r.................169
6.52 'TESimulation vs. measured triaxial results of reinforced Sand and 4
layers of reinforcement @ a3 = 310 KPa".............................169
xix
6.53"FESimulation vs. measured triaxial results of reinforced Sand and 6
layers of reinforcement @ <73 =103 kPa".................................170
6.54 "FESimulation vs. measured triaxial results of reinforced Sand and 6
layers of reinforcement @ Â¢13 = 207 kPaM................................170
6.55 "FESimulation vs. measured triaxial results of reinforced Sand and 6
layers of reinforcement @ a3 = 310 kPa".................................171
6.56 f,Triaxial test results of transversely isotropic material vs. the lab and
the discrete model of 2 layers sample @ Â¢73 =103 kPa,T..................174
6.57 "Triaxial test results of transversely isotropic material vs. the lab and
the discrete model of 2 layers SMiple @ =207 kPa"......................174
6.58 "Triaxial test results of transversely isotropic material vs. the lab and
the discrete model of 2 layers sample @ Â¢13 = 310 kPa"..................175
6.59 rtTnaxiai test results of transversely isotropic material vs. the lab and
the discrete model of 4 layers sample @ Â¢13 =103 kPa r,.................175
6.60 "Tnaxial test results of transversely isotropic material vs. the lab and
the discrete model of 4 layers sample @ Â¢53 = 207 kPa"....................176
6*61 ,rTnaxial test results of transversely isotropic material vs, the lab and
the discrete model of 4 layers sample @ CT3 = 310kPaM...................176
6*62 tflnaxial test results of transversely isotropic material vs. the lab and
the discrete model of 0 layers sample @ 03=103 kPaM......................177
6.63 11 Triaxial test results of transversely isotropic material vs. the lab and
the discrete model of 6 layers sample @03 = 207 kPaM...........*........177
6.64 M Triaxial test results of transversely isotropic material vs. the lab and
the discrete model of 6 layers samples @ CT3 = 310 kPa1'................,,178
7.1'The non woven geotextile material used in Liu s testn....................187
7.2 "Thickness versus pressure relationship of Biddim C34 geotextile11.....189
7.3 "Reinforcing Ppattems for triaxial test"................................190
xx
7.4"StressStrain relationship for samples tested at 103 kPa confining
pressure1...............................................................192
7.5 nStressStrain relationship for samples tested at 310 kPa confining
Pressure1'..............................................................192
7.6 '^MohrCoulomb envelope for samples with different reinforcing
patterns".................................................................193
7.7 "Models of unreinforced and reinforced soil cylinders using
LSDYNA...................................................................194
7.8 "StressStrain relation of Ottawa sand @103 kPa confining pressure1'......201
7.9 "StressStrain relation of Ottawa sand @310 kPa confining pressure'1......202
7*10 11 StressStrain relation of Ottawa sand and 1 layer of reinforcement @
103 kPa confining pressure"...............................................202
7.11,1 StressStrain relation of Ottawa sand and 1layer of reinforcement @
310 kPa confining pressure"...............................................203
7.12 riStressStram relation of soil and 4 layers of reinforcement @ 103
kPa confining pressure................................................ 203
7.13 M StressStrain relation of Ottawa sand and 4 layers of reinforcement
@310 kPa confining pressure'1.............................................204
7.14 MStressStrain relation of Ottawa sand and 6 layers of reinforcement
@103 kPa confining pressure11.............................................204
7.15 "StressStrain relation of Ottawa sand and 6 layers of reinforcement
@310 kPa confining pressure"..............................................205
7.16 TTriaxial test results of transversely isotropic material vs. the lab and
the discrete mbdel of 1layer sample @ <73 =103 kPa"......................211
7t 17 'Triaxial test results of transversely isotropic material vs. the lab and
the discrete model of 1layer sample @ <73 = 310 kPa......................211
xxi
7.18 "Triaxial test results of transversely isotropic material vs. the lab and
the discrete model of 4 layers sample @ =103 kPa11...................212
7.19 M Inaxial test results of transversely isotropic material vs* the lab and
the discrete model of 4 layers sample @ a3 310 kPa'*..................212
7*20 "Tnaxial test results of transversely isotropic material vs. the lab and
the discrete model of 6 layers sample @ =103 kPa............*.........213
7.21 M Tnaxial test results of transversely isotropic material vs. the lab and
the discrete model of 6 layers sample @ a3 ~ 310 kPa1'..................213
7.22 StressStrain results of CTC test on Ottawa sand...........:.......218
7.23 "Volumetric strain versus axial strain results of CTC tests on Ottawa
sand11..................................................................218
7.24 "StressStrain relation of soil @ 69 kPa confining pressure measured
(Ketchart2001)versus results of FEdiscrete model".....................221
7.25 "StressStrain relation of soil @ 207 kPa cormmng pressure
measured (Ketchart, 2001) versus results of FEdiscrete model"..........221
7.26 "StressStrain relation of soil @ 345 kPa confining pressure measured
(Ketchart2001)versus resulte of FEdiscrete model.....................222
7.27 "Specimen dimensions of SGP,!.........................................225
/.28 irVertical load versus vertical displacement of unreinforced sample
and reinforced sample with Amoco2044...................................226
7.29 "Layout of unreinforced and reinforced samples using the finite
element method.........................................................227
7.30 ^Vertical loaddisplacement relation of unreinforced sample
measured by Ketchart versus discrete finite element model'r.............228
7.31 f, Vertical loaddisplacement relation of reinforced sample measured
by Ketchart versus discrete unite element model"........................228
XXII
7.32 MVertical loaddisplacement relation of reinforced sample measured
by Ketchart versus discrete finite element model and homogeneous
model"............................................................. 232
8.1"Types of shallow spread footing a) square footing; b) strip footing; o)
rectangular footing; d) trapezoidal footing....................... 240
8.2 "Failure modes when reaching bearing capacity".......................242
8.3 "Modes of bearing failure of reinforced earth".......................248
8.4 "Non dimensional quantities (AlA2, and A3) function of z/B1........250
8.5 "Effective length of reinforcement (L) as function of /TB"..........251
8.6 H3Dimenioanl foundation modelrr................................... 256
8.7 "Variation of vertical stress beneath a footing based on Boussinesq
analysis and Westergaard analysisrespectively"......................2^7
8.8 "Depth versus G for Ottawa sand, y =1719 kg/m3r,.....................259
8*9 "Overburden pressure of footing on top of foundation soil,FE vs.
analytical1'.........................................................263
8.10 ^Bearing pressure of soil foundation resulted from finite element
analysis.............................................................265
8.11 ftBiearing pressure of foundation due different spacing when Eg =160
MPa"............................................................. 266
8.12 " Bearing pressure of foundation due different spacing when Eg =
320 MPa........................................................... 267
8.13 h Bearing pressure of foundation due different spacing when Eg =
640 MPa".............................................................267
8.14 ''Bearing pressure of foundation due different geosynthetic stiffness
when S =1000 mm".....................................................270
8.15 f Bearing pressure of foundation due different geosynthetic stiffiiess
when S =500 mmM......................................................270
xxiii
8.16 11 Bearing pressure of foundation due different geosynthetic stiffness
when S = 250 mm1'....................................................271
8.17 Ma2 increase due different reinforcement patterns'1................272
8.18"Performance based bearing capacity at different spacing when E
=160 MPa".............................................................274
8.19 "Performance based bearing capacity at different spacing when E
=320 MPa..............................................................274
8*20 "Performance based bearing capacity at different spacing when E
=640 MPa..............................................................275
8.21 "Contour and 3Dimensional surface plots of horizontal stress
distribution for geo textile at depth of 500 mm from the footing..279
8.22 "Contour and 3Dimensional surface plots of horizontal stress
distribution for geotextile at depth of 0.58 from the footing".......280
8.23 "Contour and 3'Dimensional surface plots of horizontal stress
distribution for geotextile at depth of IB from the footing".........281
8.24 f'Contour and 3Dimensional surface plots of horizontal stress
distribution for geo textile at depth of L5B from the footing".......282
8.25 "Contour and 3Dimensional surface plots of horizontal stress
distribution for geotextile at depth of 2B from the footing"..........283
8.26 "Contour and 3Dimensional surface plots of horizontal stress
distribution for geotextile at depth of 0.5B from the footing".......284
8.27 ^Horizontal stress distribution of reinforcement layers at distance
away from the footing^ centerline, S = 250 mm, Eg 320 MPa"...........285
8.28 NHorizontal stress distribution of geosynthetic layers at different
depths for foundation on top of reinforced soil,S =1000 mmM..........286
8.29 n Horizontal stress distribution of geosynthetic layers at different
depths for foundation on top of reinforced soilS =500 mm..........287
xxiv
8.30 "Horizontal stress distribution of geosynthetic layers at different
depths for foundation on top of reinforced soil,S =250mmM..................287
8.31 "Bearing pressure of foundation with homogeneous reinforced soil
for S=1000 mm"......................................................293
8*32 "Bearing pressure of foundation with homogeneous reinforced soil
for S = 500 mm.....................................................293
8.33 "Bearing pressure of foundation with homogeneous reinforced soil
for S = 250 mmrr...........................................................294
9.1"Retaining wall external stability; a) overturning, b) sliding, c) bearing
capacity d) deep seated.............................................299
92 "External stability of MSE walln.............................................303
9.3 "External loads on vertical MSE wall with horizontal backfill due to
weight and surcharge"...................................:..................304
9.4 "External loads on vertical MSE wall due to seismic loading"........309
9.5 "Inclusion stress distribution along the height of the wall"........311
9.6 "Side view of MSEWall".............................................316
9.7 MElCentro Earthquake acceleration time history11...................318
9.8 "Extemal load on MSE Wall model................................... 324
9.9 "Lateral earth pressure (ax) of backfill along the hybrid retaining wallIT.339
9.10 "Lateral displacements of hybrid retaining wall due static and .
dynamic loads.................................................... 340
9*11 MBearing pressure (az) of foundation soil due to static and dynamic
loadf,..............................................................342
9.12 ''Settlement (6z) of foundation soil due to static and dynamic load11.....343
9.13 "Tensile stress of reinforcements at distance 0 m flrom the wall due
static and dynamic loadings11.......................................346
9*14 'Tensile stress of reinforcements at distance 1.5 m from the wall due
static and dynamic loadings'1........................................346
9.15 'Tensile stress of reinforcements at distance 3.75 m from the wall
due static and dynamic loadings.....................................347
9.16 "Tensile stress of reinforcements at distance 7 m from the wall due
static and dynamic loadings'*........................................347
9.17 "Summary of tensile stresses of reinforcements due to static
loadings11...........................................................349
9* 18 n Summary of tensile stresses of reinforcements due to dynamic
loadingsf,...........................................................349
9.19 ''Lateral earth pressure (ax) of homogeneous model due to static
loading".............................................................353
9.20 "Earth pressure (ax) of homogeneous model due to dynamic loading"...353
9.21 "Lateral wall displacements (5X) of homogeneous model due static
loading"...........................................................355
9.22 "Lateral wall displacements (5X) of homogeneous model due dynamic
loading"....................................................... 355
9.23 "Bearing pressure (cjz) of toundation soil in homogeneous model due
to static load......................................................357
9.24 "Bearing pressure (az) of loundation soil in homogeneous model due
to dynamic load.................................................. 358
9.25 "Settlement (5Z) of foundation soil in homogeneous model due to
static load'................................................. 359
9.26 "Settlement (5a) of foundation soil in homogeneous model due to
dynamic load".......................,.......................r.......359
10.1 ,TSide view of MSE abutments"..................................... 366
xxvi
10.2 ,fMSE wall facing; a) geosynthetics warp, b) segment concrete block,
c) full height panel".................................................366
10.3 "Isometric view of MSE bridge abutment'1.............................371
10.4 "Northridge Transverse (Y) horizontal acceleration time history11
(Berkeley).........................*..................................373
10.5 "Northridge Longitudinal(X) horizontal acceleration time history"
(Berkeley).......:....................................................373
10.6 rtLongitudinal earth pressure of abutment along the centerline1'.....380
10.7 ^Contour and 3Dimenensional surface plots of longitudinal earth
pressure of the bridge abutment on the Stemwall due static loading1'.381
10.8 ,fContour and 3Dimensional surface plots of longitudinal earth
pressure of the bridge abutment on the Stemwall due dynamic
loading".............................................................382
10.9 "Lateral earth pressure of abutment against the wing wall along the
edge"........................,........................................383
10.10 ''Contour plot of transverse earth pressure of the bridge abutment on
the wingwall due static loading"....................................384
10.11"Contour plot of transverse earth pressure of the bridge abutment on
the wingwall due dynamic loading11..................................385
10.12 ^Contour and 3Demensional plots of MSE backfill bearing pressure
due static loading"..................................................388
10.13 11 Contour and 3Demensional plots of MSE backfill bearing
pressure due static loading due dynamic loading11....................389
10.14 '^Contour and 3Demensional plots of MSE backfill settlement due
static loading".................................................... 390
10.15 f,Contour and 3Demensional plots of MSE backfill settlement due
dynamic loading".....................................................391
10.16 "Longitudinal horizontal stress (ax) of soil behind the MSE wall"...394
XXVll
10.17 'Transverse horizontal stress (ay) of soil behind the MSE wall".....395
10.18 r, Contour and 3Dimensional plots of longitudinal earth pressure on
the longitudinal side of the MSEWall due static loading^............396
10.19 "Contour and 3Dimensional plots of longitudinal earth pressure on
the longitudinal side of the MSEWall due dynamic loading"...........397
10.20 T, Contour and 3Dimensional plots of transverse earth pressure on
the transverse side of the MSEWall due static loading"..............398
10.21 n Contour and 3Dimensional plots of transverse earth pressure on
the transverse side of the MSEWall due dynamic loading".............399
10.22 "Longitudinal displacements of MSE wall due to static and dynamic
loadings11...........................................................401
10.23 T,Transverse displacements of MSE wall due to static and dynamic
loadings............................................................401
10.24 MContour and 3Dimensional surface plots of longitudinal wall
displacement due static loading".....................................403
10.25 "Contour and 3Dimensional surface plots of longitudinal wall
displacement due dynamic loading"....................................404
10.26 "Contour and 3Dimensional surface plots of transverse wall
displacement due static loading".....................................405
10.27 ,rContour and 3Dimensional surface plots of transverse wall
displacement due dynamic loading'1...................................406
10.28 "Contour plot of longitudinal stresses of bottom reinforcing layer
due to static loading"...............................................409
10.29 "Contour plot of longitudinal stresses of bottom reinforcing layer
due to dynamic loading1'.............................................409
10.30 "Contour plot of transverse stresses of bottom reinforcing layer due
to static loading.....................................................410
XXVIII
10.31''Contour plot of transverse stresses of bottom reinforcing layer due
to dynamic loading^.................................................410
10.32 "Contour plot of longitudinal stresses of middle reinforcing layer,
H=2.4mdue to static loading".......................................411
10.33 "Contour plot of longitudinal stresses of middle reinforcing layer,
H=2.4m5 due to dynamic loading1'....................................411
10.34 ^Contour plot of transverse stresses of middle reinforcing layer,
H=2_4m, due to static loading.....................................412
10.3b ^'Contour plot of transverse stresses of middle reinforcing layer,
H=2.4m? due to dynamic loading^.....................................412
10.30 ^Contour plot of longitudinal stresses of top reinforcing layer,
H==43mdue to static loading1.....................................413
10.37 ^Contour plot of longitudinal stresses of top reinforcing layer,
H=4*3m, due to dynamic loading".....................................413
1038 nContour plot of transverse stresses of top reinforcing layer,
H=4.3mdue to static loading.......................................414
10.39 "Contour plot of transverse stresses of top reinforcing layer,
H=4.3m? due to dynamic loading".....................................414
10.40 "Longitudinal earth pressure of abutment along the centerline of
homogeneous model vs. discrete model1*..............................420
10.41 11 Contour plot of longitudinal earth pressure of the bridge abutment
on the Stemwall of homogeneous model due static loading"...........421
10.42 "Contour plot of longitudinal earth pressure of the bridge abutment
on the Stemwall of homogeneous model due dynamic loading"..........422
10.43 "Contour plot of transverse earth pressure of the bridge abutment on
the wingwall of homogeneous mode due static loading'1..............423
10.44 T,Contour plot of transverse earth pressure of the bridge abutment on
the wingwall of homogeneous mode due dynamic loading"..............424
xxix
10.45 "Contour and 3Demensional plots of MSE backfill bearing pressure
of homogeneous mode due static loading"...........................426
10.46 "Contour and 3Demensional plots of MSE backfill bearing pressure
of homogeneous mode due dynamic loading..........................427
10.47Contour plot of MSE backfill settlement of homogeneous model
due static loading'1............................................429
10.48 "Contour plot of MSE backfill settlement of homogeneous model
due dynamic loading.......................................... 429
10.49 "Longitudinal horizontal stress (ax) along the centerline of the MSE
wall of discrete and homogeneous model due to static loading'1....433
10.50 ^Longitudinal horizontal stress (ax) along the centerline of the MSE
wall of discrete and homogeneous model due to dynamic loading"....433
10.51,f Longitudinal horizontal stress (ax) of soil along the edge of the
MSE wall of discrete and homogeneous model due static loading1*....434
10.52 ^Longitudinal horizontal stress (crx) of soil along.tbe edge of the
MSE wall of discrete and homogeneous model due to dynamic
loading...................................................... 434
10.53 'Transverse horizontal stress (ay) of soil along the centerline of the
MSE wall of discrete and homogeneous model due to static loading".435
10.54 "Transverse horizontal stress (ay) of soil along the centerline of the
MSE wall of discrete and homogeneos model due to dynamic
loading".........................................................435
10.55 "Transverse horizontal stress (ay) of soil along the edge of the MSE
wall of discrete and homogeneous model due to static loading".....436
10.56 "Transverse horizontal stress (ay) of soil along the edge of the MSE
wall of discrete and homogeneous model due to dynamic loading"....436
XXX
10.57 ''Longitudinal displacement (8x) of MSE wall along the centerline
of discrete and homogeneous model due static loading"....................439
10*58 'Transverse displacement (5y) of MSE wall along the centerline of
discrete and homogeneous model due static loading".......................439
B.l"The matrix plot of vh with respect to all X variables"....................473
B.2 ,rThe residual versus the fitted values of Vh"............................485
B.3 '"Normal probability plot of the residuals of VhM.........................486
B.4 ,fThe matrix plot of Ev with respect to all X variables"..................488
B5 "Residual versus the fitted values of Ey.................................494
B.6 11 The residual plots of Ev against each variable'"'......................495
B.7 '"Normal probability plot of the residuals of Ev,f .......................496
B.8 'The matrix plot of vv with respect to all X variables11.................,504
B.9 "The residual versus the fitted values of vv".............................513
B.10 "Normal probability plot of the residuals of vv".........................514
B.l1"Matrix plot of Gv with respect to all X variables......................516
B.12 Residuals versus the fitted value of Gv........................:.....525
B.l 3 "Normal probability plot of the residuals of Gv1'.......................526
B.l4 "Matrix plot of Gh with respect to all X variables"......................528
B.15 "The residual versus the fitted values of Gh............................538
B.16 "Normal probability plot of the residuals of Gh".........................539
XXXI
LISTS OF TABLES
Table
4_1 Compliance matrix components [C]_...................................40
4.2 "Soil and geosynthetic properties"....................................48
4.3 "Strain components of geocomposite with 1layer inclusion............54
4.4 11[C] for geocomposite with 1layer inclusion"........................54
4.5 "Strain components for geocomposite with 3 layers inclusions11.......56
4.6 "[C] for geocomposite with 3 layers inclusion.......................56
4.7 "Strain components of geocomposite with 5 layers inclusion"..........57
4.8 M[C] for geocomposite with 5 layers inclusion''......................58
4.9 "E and E* using finite element and analytical method".................67
4.10 ^Properties and applied loadings of the cube element"................68
4.11,fStress components of weak reinforcement11............................73
4.12 "Stress components of strong reinforcement"............................78
5.1"ANOVA Table for general linear regression model".......................85
5.2 "The correlation matrix of Eh and all X variables^....................98
5.3 "MINITAB output for Best two subsets of Eh model for each subset
sizen.........................................................!.......99
5.4 "MINITAB Forward/backward stepwise regression output of Eh
variable"................... .........................................100
5*5 MANOVA Table for Eh regression model"..................................101
5.6 MANOVA Table for alternative mode of Eh"..............................109
5.7 ''Summary of constitutive models equations for transversely isotropic
geocomposite"........................................................110
xxxii
6.1"Summary of the geotextile tensile test results....................116
6*2 "A summary of Ottawa sand shear failure dataM........................152
6.3 "'Parameters of Cap model for Ottawa Sandn...........................156
6.4 "Summary of the geotextile properties"..............................159
6.5 "Summary of steel caps properties"..................................159
6*6 nThe Transversely Isotropic Properties of Reinforced Samplesn........173
6.7 M Young's modulus and normalized Young's modulus of soil specimens
with 2 layers of reinfrocement^EF...................................180
6.8 "Youngs modulus and normalized Youngs modulus of soil specimens
with 4 layers of reinforcement".....................................181
6.9 "Youngs modulus and normalized Youngs modulus of soil specimens
with 6 layers of reinforcement".....................................182
7.1"Typical Physical Properties of Bidim C34 Engineering Fabric"........188
7.2 ^The Cap model properties of Ottawa sand,r...........................198
7.3 Youngs modulus and normalized Youngs modulus of soil
specimens......................................................... 206
7.4 "Young's modulus and normalized Young's modulus of soil specimens
with 1 layer of reinforcement"......................................206
7.5 "Young's modulus and normalized Young's modulus of soil specimens
with 4 layers of reinforcement'1....................................207
7.6 ,lYoung,s modulus and normalized Young's modulus of soil specimens
with 6 layers of reinforcement1'.....................................207
7.7 rtThe Transversely Isotropic Properties of Reinforced Samples"......209
7.8 nYoungrs modulus and normalized Young's modulus of soil specimens
with 1layer of reinforcement........................................214
7.9 ^Young's modulus and normalized Youngfs modulus of soil specimens
with 4 layers of reinforcement'1................................... 214
xxxiii
7,10 11 Young's modulus and normalized Youngfs modulus of soil
specimens with 6 layers of reinforcement11..........................215
7.11"The Cap model properties of Ottawa sandKetchart samples11.........220
7*12 "Normalized Young's Modulus of Finite element method compared
with Experiment1...................................................223
7.13 "Normalized Young's Modulus of Finite element method compared
with Experiment, plane strain test on unreinforced sample''........230
7A4 "Equivalent properties of SGP consisting of Ottawa sand and 3 layers
of Amoco 2044 geotextileT,..........................................231
7.15 "Youngs modulus and normalized Youngs modulus of
homogeneous mode from simulating plane strain test".................233
8.1"Ottawa sand properties for Cap model^...............................260
82 "Geosynthetic properties for Elasticplastic".......................261
83 "Concrete properties, Elastic n (Boresi 2003)........................261
8.4 uTerzaghirs corresponding settlement of reinforced soil models'*....268
8.5 "Bearing capacity of reinforced soil models corresponding to
Terzaghi's settlement"..............................................269
8.6 MBCR due to different reinforcement spacing and geosynthetic
stiffness"..........................................................273
8.7 "Homogeneous properties for lm spacing and Eg 320 MPa".............289
8.8 "Homogeneous properties for 0.5m spacing and Eg =320 MPar,..........290
8.9 "Homogeneous properties for 0.25m spacing and Eg =320 MPa"..........291
8.10 'Terzaghi's corresponding settlement of reinforced soil models;
homogeneous vs. discrete...........................................292
8.11!tBearing capacity of reinforced soil models corresponding to
Terzaghis settlement; homogeneous vs. discrete....................292
9.1"Load curves summary11............................................... ,317
9.2 "Properties of soil for MSE wall".................................. 321
9*3 "Properties of inclusion for MSE wall"........................................321
9.4 Properties of Concrete wall of MSE wall"....................................321
9.5 ^'Factor of safety against reinforcement rupture at different elevation
due to static loading"........................................................327
9.6 "Internal stability with respect to pullout failxire due to static loading11.329
9.7 ^Factor of safety against reinforcement rupture at different elevation
due to dynamic loading'1......................................................335
9.8 "Factor of safety against reinforcement pullout at different elevation
due to dynamic loading^...............................................336
9.9 "Mechanical properties of the reinforced soil composite".....................351
9.10 ^'Maximum bearing pressure of soil beneath footing".................357
10.1 "Dimensions of MSE abutment".........................................370
10.2 "Northridge Earthquake information".................................374
10.3 "Properties of Ottawa sand along different elevationsT,.............375
10.4 "Reinforcement properties in MSE abutment and backfill soil"........376
10.5 "Concrete properties'1..............................................376
10.6 ^Equivalent properties of lower zone of the MSEWalln...............416
10.7 "Equivalent properties of middle zone of the MSEWall"............**416
10.8 "Equivalent properties of top zone of the MSEWall".................417
10.9 "Equivalent properties of lower zone of the abutment"...............417
10.10 "Equivalent properties of top zone of the abutment..........................418
10.11"Range of lateral earth pressure (kPa) using discrete and
homogeneous approaches"...............................................432
10.12 "The regression equations of lateral earth pressure"........................437
B,1"The correlation matrix of Eh and all X variables".............................474
B+2 "output for Best two subsets for each subset size of vj/.....................475
B.3 nMINITAB forward/backward stepwise regression output of VhT..................477
XXXV
B.4 "ANOVA Table for Vh regression model".................................478
B.5 MANOVA Table for alternative model"...................................487
B.6 The correlation matrix of Ey and all X variables"...................489
B.7 nMINITAB output for Best two subset results for each subset size of
EvM...................................................................490
B.8 !'MINITAB forward/backward stepwise regression output of EvT,.........492
B.9 "ANOVA Table for Ey regression model".................................493
B.10 nANOVA Table for Ev regression model when dropping the S termn.......502
B.l1 nANOVA Table for alternative model of Vh"............................503
B.12 MThe correlation matrix of vv with all X variables*'.................505
B.l3 "MESflTAB output for best two subsets for each subset size of Vy1'...507
B.l4 ,1MINITAB forward/backward stepwise regression output of vvM.........507
B.15 ANOVA Table for vv regression mode"................................509
B.l6 "ANOVA Table for alternative model of vv"............................515
B.17 "The correlation matrix of Gv and all X variables...................517
B.l 8r fTMINITAB out put for best two subsets for each subset size of Gvn.518
8.19 ''MINITAN forward/backward stepwise regression output of Gy11........519
B.20 "ANOVA Table for Gv regression model"...........................520
B.21 "ANOVA Table for alternative model of Gvn........................526
B.22 "Correlation matrix of Gh and all X variables".......................529
B.23 "MINITAB output for best two subsets for each size of Gh"..........530
B.24 "MINITAB forward/backward stepwise regression output of Gh"..........532
B.25 MANOVA Table for Gh regression piodd................................533
B.26 "ANOVA Table for alternative model of Gh............................539
xxxvi
Introduction
1.1 Problem Statement
In recent years, more and more geosynthetic materials have been introduced as
engineering materials and widely applied in earthquake and geotechnical
engineering. These materials cause significant modification and improvement in
the engineering behavior of soil such as strength, stiffness, and corrosion
resistance. The resultant reinforced soils (geocomposites) are in general
composite materials that result from combination and optimization of individual
constituent materials. They are constructed in a manner that produces a structure
of alternating layers of soil and reinforcing elements (Vector Elias 2001).
Reinforced soils are generally an anisotropic and inhomogeneous. In order to
determine the physical and the mechanical properties of this anisotropic material,
field or laboratory tests need to be conducted. There are several factors that
directly affect the behavior of reinforced soil: properties of soil, properties of
reinforcing material (inclusion), confining pressure, spacing between Inclusion
layers, and frictional interface (NienYia Chang 2006)*
Similar to concrete, soil is relatively weak in tension as compared to its strength
in compression. The effect of soil reinforcing is due to tensile stress builtup in
horizontal reinforcing layers transferred to the soil through sliding friction as
compressive stresses. An accurate analysis of this geocomposite should consider
the sliding interaction between the two components, soil and inclusions.
If reinforced soil, such as in Mechanically Stabilized Earth (MSB) walls, is
modeled using a finite element code, difficulties and time consuming models can
not be avoided when defining the sliding interface properties between the two
materials. Therefore, there is a need for defining a constitutive model of
reinforced soil that will eliminate the use of interface friction between
reinforcement and soil layers.
1.2 Research Goals and Objectives
Reinforced soil is considered as geocomposite material that consists of soil and
reinforcing material with a slippage interface and subjected to a confining
pressure. This composite is an anisotropic. The main goal of this research is to
develop a constitutive model of this geocomposite and use it in predicting the
behavior of earth structures such as in MSE walls.
The main objectives of this research are the following:
1 Find material type and properties of soil reinforced with horizontal layers
of reinforcing element using a finite element approach*
2 Develop a constitutive model of reinforced soil that predicts the material
properties under different applied conditions.
...
3 Obtain material properties from sets of laboratory tests.
4 Validate the results of finite element analysis.
5 Investigate the effectiveness of this model by building a reinforced
foundation soil, a MSE wall, and MSE bridge abutment. Discrete and
homogeneous models will be used for the analysis.'
1.3 Research Methodology and Tasks
The goal of this research is to develop a constitutive model of reinforced soil.
This goal will be achieved using numerical and physical testing methodologies
following the subsequent tasks:
1 Perform a finite element analysis using NIKE3D code on cubes of
reinforced soil with lmxlmxlm soil dimensions, and equally spaced
inclusion elements. In this analysis, all cubes will be subjected to their
own gravity load followed by confining pressure and deviator stresses
under different applied conditions of spacing, properties of soil, properties
of inclusion, and friction coefficient. For each case, a total of six tests will
be applied on each specimen; three normal and three shear stresses, to
determine the compliance matrix. From the resultant compliance matrix,
the material type and properties can be determined. The analysis will be
completed using geosynthetic materials as reinforcement.
2 Perform a statistical analysis on the extracted data from the finite element
analysis. The statistical analysis will correlate the strongly related
parameters in linear equations forming the constitutive model of
reinforced soil.
3 In the laboratory, perform sets of triaxial tests on unreinforced and
reinforced soil samples, determine the material properties, and compare
them to those resulted from finite element analysis. Once calibration and
validation are achieved, further studies on different applications will be
preceded.
4 Perform further validations by comparing the results from finite element
method with those obtained from different tests completed on reinforced
soil specimens.
5 Investigate the efficiency of the constitutive model by numerically
analyzing three geocomposite structures: reinforced foundation soil3 MSE
wall, and MSE bndge abutment. These structures will be built and
anatyzed with the aid of LSDYNA code using two different approaches:
a. Considering the foil interface between the soil and geosynthetic
material, briefed as discrete model, and
b. Considering the constitutive model of reinforced soil, briefed as
homogeneous model.
1.4 Significance of Research
MSE walls have exhibited the ability of withstanding earthquake forces without
being severely damaged. For that reason, several tests have been completed on
these walls under seismic conditions. These test improved the understanding of
the mechanics of reinforced soil method. Among all approaches, finite element is
the most powerful tool* Depending on the degree of sophistication, finite element
method is capable of describing almost any kind of reinforced system, and gives
results that closely resemble real conditions. But because of the difficulties in
modeling these walls due to the sliding interfaces between all soil and inclusion
layers, this research becomes pertinent. Finding an elastic constitutive model of
reinforced soil will facilitate the analysis of MSE walls and foundation supported
by reinforced soil. This constitutive model will also reduce the amount of time
required to model these walls, as well as eliminate the sources of error.
2. Literature Review
Despite their initial cost, composite materials have higher specific strength and
stiffness than traditional materials. These properties greatly increase the demand
for the use of composite materials in new civil engineering constructions and also
in the rehabilitation and strengthening of existing structures. They were first used
in aeronautical and aerospace applications, and then expanded to other
applications.
Composite structures involve the use of wood, steel, or fabric reinforced elements.
Although steel is highly susceptible to corrosion, it was found that even in an
aggressive environment, the galvanized strip used in earth reinforcing has a useful
life of more than 120 years (Liu 1987). Fabric elements are potentially suitable
alternatives for use in many harsh environments due to their chemical and
corrosion resistance properties. Furthermore, fabric elements are light and easy to
construct. Brimahet et al.(1998) suggested to totally eliminate the steel from
bridge slabs by combining carbon fiber reinforced plastic tendons with
polypropylene fiber reinforced concrete (A. Braihmah 1998). In their tests, a one
fourthscale model of a bridge deck was subjected to a concentrated load with
thickness 20% less than the minimum recommended in the Ontario highway
bridge design code (OHBC) of 1991,They found that the deck slabs under static
loading offer a minimum safety factor of 6.8 against punching shear. Fiber
reinforced plastics are also used as jackets to enhance both strength and ductility
of concrete columns by providing confining pressure.
5
Previous studies on these walls have shown the effect of concrete strength, types
of fibers, fiber volume fraction, fiber orientation, jacket thickness, and the
interface between the core and the jacket on the confinement effectiveness.
Additional investigations were completed using more than 100 specimens to
investigate the effect of cross section shape^ column length/diameter ratio, and the
interface bond (A. Mirmiah 1998). First, they investigated the shape effect by
conducting a series of uniaxial compression tests on cubical and cylindrical
specimens. Effectiveness of circular sections was reported, this was due to the
uniformity of the confining pressure provided by the plastic jackets on circular
cross sections. Then they investigated the effectiveness of length to diameter
(L/D) ratio. Specimens with different L/D ratios were instrumented at different
locations with sets of vertical and horizontal strain gauges followed by a set of
uniaxial compression test. It was found that L/D has insignificant effect on
confinement effectiveness. After that, they investigated bond effect, between the
fabric reinforced plastic jackets and the concrete core. Both, adhesive and
mechanical bond were investigated. Only mechanical bond (shear connectors)
was found to improve the performance of the section by distributing the
confinement pressure around the circumference of the tube.
.r
For the past two decades, fiber reinforced polymers, such as geogrid used in
retaining walls, have been increasingly used in soil reinforcement to obtain an
improved geotechnical material. Ever since, reinforced soil has been a subject of
research. Many of the earlier test results proved the effectiveness of fiber
reinfpreement and the interaction between soil and reinforcement material he
following paragraphs point at different research that was conducted on reinforced
soil.
A comprehensive study was conducted to investigate the static strength and the
dynamic properties of horizontally reinforced soil subjected to vertical cycling
loading using triaxial testing (Liu 1987). In his static analysis, three different
reinforcing patterns plus one imreinforced pattern were used to investigate the
effectiveness of reinforcement with different numbers of reinforcing layers. The
reinforced sample consisted of Ottawa sand and needle punched non woven
geotextile. Each sample was 12 (in) height and six (in) diameter. It was found
that the ultimate strength in reinforced samples increases with increasing the
number of reinforcing layers. This is accompanied by an increase in the axial
strain.
Another study was completed to investigate the static response of sand reinforced
with randomly distributed fibers (Maher 1990), In this study, a laboratory triaxial
compression tests were performed to determine the stressstrain response of sand
reinforced with randomly distributed fibers* Different fibers (Buna, Reed, and
Palmyra)^ and different sand (Mortar sand, Ottawa, Muskegon, and Glass sphere)
were investigated. Experiments were followed by statistical analysis to develop a
model. This model predicts the fiber contribution to strength under static loading.
It was found that an increase in fiber aspect ratio, except for fibers with very low
modulus, resulted in a lower critical confining stress. Also, shear strength
increases linearly with increasing amounts of fiber.
In 1999? Frost et al.conducted an experimental study using shear tests to
investigate the behavior of sandreinforcement interfaces (J. Frost 1999). It was
found that the peak interface friction coefficient decreases with the increase of
normal stress, and increases linearly with the relative roughness. Frost et al.
reported a minor effect of the preparation method, the rate of shearing, and the
thickness of the soil specimens on the interface, friction. However, tests have
shown that the friction coefficient resulted from direct shear test is much smaller
than that of pullout tests, where pullout test is being used lately for calculating
friction coefficient (Z. Wang 2002). Other shear tests were conducted to
investigate the effect of fly ashsoil mixture (S. Kaniraji 2001). It was shown that
the fiber inclusions increase the strength of raw fly ashsoil specimens as well as
that of the cementstabilized specimens. It was also noticed that the brittle
behavior changed to ductile one after adding fiber inclusions.
In 2002, experimental investigations and modeling of non linear elasticity of fiber
reinforced soil under cyclic loading at small strain were conducted (J. Li 2002).
Cycle shear tests were conducted on 27 cylindrical specimens using triaxial
apparatus, each specimen was 2.8 in diameter and 5.6 in height Different fiber
contents under both different confining pressure and incremental loading were
investigated. The results indicated that the shear modulus of reinforced soil is
affected by all variant factors. Also, the elastic modulus of reinforced soil is
directly proportional to both fiber content and confining pressure, and inversely
proportional to the incremental loading repetition. After that, a linear regression
with multiple variables analysis was completed to calibrate the related parameters.
Another investigation on the interface between soil and reinforcement was
conducted by Ensan and Shahrour (M. Ensan 2002). They presented an elasto
plastic microscopic constitutive model for the multi layer materials with the
imperfect interfaces and implemented this model in a finite element program
* 4
(PECLAS). From their analysis, it was determined that the presence of an
imperfect interface reduces the resistance of the reinforced soil by about 55%.
In 2002, a study using fibershaped waste material, such as polyethylene
terephthalate (PET), in soil reinforcing was conducted (N. Consoli 2002). In this
study, emphasis was placed on the influence of the fiber length and fiber content
on the basic aspect of soil behavior, such as initial stiffness, peak ultimate
strength, ductility, and energy absorbance capacity. The stressstrainstrength
response was evaluated using unconfmed compression tests, splitting tests, and
drained triaxial tests. Experimental tests were followed by multiple regression
analysis to obtain and interpret a representative experimental data base. From
their results, they indicated the efficiency of the fiber reinforcement was
dependent on the fiber length, where the greatest improvements in triaxial
strength and energy absorption capacity were observed for the longer, 36 mm,
fiber. They also observed strength and stiffness increase for the uncemented soil
by fiber reinforcing. For cemented soils the effect was more pronounced for the
lowest cement content,
A year after, a total of 14 series of triaxial tests were performed on a reinforced
element with different types of sand and different fiber geometry (R. Michalowski
2003). All specimens we cylinders of a specific height and diameter equal to 94.5
mm. The results indicated that up to 70% increase in the strength of un
reinforced soil can be gained with fiber concentration of 2%, This percentage will
change by changing the aspect ratio of reinforcing fibers. Furthermore, small
fiber concentration will strongly affect fine sand. However5 for large
concentration of fibers, the coarse sand becomes much stronger.
Reinforced soil is a practical solution to improving geotechnical properties such
as bearing capacity, this is best for reinforcement in weak soil foundations
(Michalowski 2004). Michalowski proposed a numerical method supported with
experimental data for calculating limit loads on strip footing over foundation soils
reinforced with horizontal layers of geosynthetics. He found that the
reinforcement length of four times of the footing width is recommended to get all
the benefit of reinforcing.
Although sufficient testing has been conducted at the coupon level on reinforced
soil, no constitutive model of reinforced soil has been conducted taking in
consideration all variations in soil and geosynthetic properties under different
confining pressure. Therefore, this research is an attempt at producing such a
model of reinforced soil followed by mathematical equations to describe the
behavior of it. This model predicts the material properties of reinforced soil by
knowing the spacing between reinforcements, soil properties, reinforcement
properties, friction coefficient, and confining pressure.
10
3.
Theoretical Background
3.1 Introduction
This chapter provides a brief background on finite element methodology,
including some of the commercial codes (NIKE3D and LSDYNA) used in this
research. Additionally, emphasis is placed on using the finite element method on
geotechnical applications. It is fruitful to refer to this chapter, from time to time,
in conjunction with the study of topics in other chapters.
3.2 Finite Element Theory for Linear Materials1
The concept Finite Element Method (FEM) has a wide range of engineering
applications. It was initially developed by A. Hrennikoff (1941) and R Courant
(1942). According to Courant, the domain is divided into finite triangular sub
regions for solution of second order elliptical partial differential equations. In the
1950s, the development of FEM began for airframe and structural analysis. Since
then, it has been generalized into a branch of applied mathematics for numerical
1110deling of physical systems in an extensive variety of engineering disciplines
such as electromagnetic and fluid dynamics.
1 The theory of linear and nonlinear finite element is obtained from Potts and Zdravkovic, 1999.
11
FEM is based on an energy principle which is a fundamental concept used in
physics and engineering. This principle expresses the relationships between
stresses, strain, material properties, and external work done by internal and
external forces. Virtual work is an example of this energy principle. Virtual
work is a mathematical product of forces and displacement. It is the work done
on a particle due to real or imaginary force producing a real or imaginary
displacement in the direction of the applied force. MathematicallyFEM method
is used for finding approximate solution of partial differential equations. The
solution approach is based on either eliminating the differential equations
completely in steady state problem, or rendering the partial differential equation
into an equivalent ordinary differential equation, which is then, solved using
standard techniques such as finite differences.
The FEM is a good choice for solving partial differential equations over complex
domains Por this kind of computationfinite element analysis (PEA) is the most
effective and efficient computer simulations. In FEA, the system is represented
by a model consisting of multiple linked discrete regions. At each element, the
concepts of continuum mechanic approach (equilibrium, compatibility, and
constitutive relations) are applied. As a result, a system of simultaneous
equations is constructed and solved for the unknown values.
In finite element, the geometry is replaced by an equivalent finite element mesh
which is composed of small regions called finite elements. In twodimensional
analyses, the finite elements are usually triangular or quadrilateral in shape
composed of key points called nodes (ZdravkovicT 1999). In order to obtain
accurate solution^ the zones of attention should contain finer mesh with proper
aspect ratios. For each element, based on the energy principle, the domain is
presented by Equation 3.1.
12
(3.1)
[KE^dE)=\ARE\
Where:
[Ke] = element stiffness matrix,
{Adg} = the vector of incremental dement nodal displacements, and
{ARh} = vector of incremental element nodal forces*
In finite element formulation, the element displacements and geometry are
expressed in terms of interpolation functions or shape ftinctions (Nj) using natural
coordinate system that varies from 1 to L Each node has one shape function and
is equal to 1 at that node and 0 at other nodes. For example Ni is equal to 1 at
node 1 and N2 is equal to 1 at node 2 and so on. Corresponding to these shape
function and based on the minimum potential energy principle for linear elastic
material, the global finite element equation of a body has the following form,
In Equation 3.2, n is the number of element, and [Kc] is the element stifi5iess
matrix. For plane strain condition, [Ke] becomes
ke
Where:
f ]t[Bj[D\B\j\dSdT
11
(3.3)
t = unity thickness,
'I j = Jacobian determinant that is responsible for the mapping1 between the
global and natural element, and is shown In Equation 3.4,
[DI =constitutive matrix
(3.2)
Â£
nA
X
1
\l/t
Â£
rF
nz 
13
[B] = Niatnx that contains the derivatives of the shape iunctions, and is
shown m Equation 3.5
S and T = natural coordinates that are related to the global coordinate
using the jacobianas shown in Equation 3.0, and
{ARc} = righthand side load vector, and is shown in Equation 3.7.
dx dy dy dx
dS dT dS dT
W =
dN,
dx
0
0
dN{
3y
dN,
2
8N
n
dx
0
dN.
97V,
2
2
dy dx dy
9r
0
8x
0
dNn
Sy
o
8N
Sy
5N
dx
dN. dN.
T
dT
dN. dN.
dx dy
T
\[nY {AF}dvol+ \[Nf{AT}dSrf
surface
volume
(34)
(3.5)
(3.6)
Â¢3.7)
After forming the equilibrium equations of all separate elements, the set of global
equations is formed by assembling the separate equations, as shown in Equation
3.8, In this equation, the global stiffiiess matrix is obtained jfrom assembling the
separate elements stiffness matrices using the direct stiffness method. Also, the
boundary conditions including concentrated load and surcharge pressure are
added to the rightside of the global equation, {AR}
KW}nG^RG)
Where:
Â¢3.8)
55
lacl^larl^
14
[Kg] = global stiffness matrix,
{Adfi} = vector that contains the unknown degree of freedoms for the
entire finite element meshs and
{ARh} = global righthand side load vector.
Once the global equilibrium equation is established, the nodal displacements
values {Me} are solved. The solution of these nodal displacements is obtained
using different techniques, such as Gaussian elimination or iterative methods.
From there, secondary quantities such as strains and stress can be computed, as
shown in Equations 3.9 and 3.10, respectively.
As
x
As
y
Ay
4 xy
Afiz
dx
dN2 dx 0 dN n dx
0 dNl 0
dN2 uy 8Zi dN n
sy dx dy
0 0 o
dN
dy
dN
dx
Au]
Avi
Au.
Av
Aw
r
Av^
(3.9)
} =[}
Where:
M:
T
Arxy Aand
(3.10)
[D] = constitutive matrix that only depends on Youngs modulusEand
Poisson's ratio, v5 for isotropic elastic materials.
15
3.3 Finite Element Theory for Nonlinear Materials
Most of the material models, especially, in geotechnical applications behave non
linearly. For that reason, applying the theory of linear finite element would result
in inaccurate results. In elastic materials, the constitutive matrix [D] is considered
a constant. This is not the case for nonlinear material, where [D] varies with
stress and strain during a finite element analysis. Therefore, the governing finite
element equations are reduced from Equation 3,8 following incremental form as
shown in Equation 3.11
W.(3i1)
Where:
[K]1 = is the incremental global system stiffiiess matrix,
{ Ad}1is the vector of incremental nodal displacements,'
{AR}' = is the vector of incremental nodal forces, and
i = is the increment number.
Since [D] is not constant for non linear materials, [K]1 will also be dependent on
the current stress and strain level and will vary over an increment.
The three main popular solutions for nonlinear materials are tangent stiffness,
viscoplastic, and the modified NewtonRaphson.
3.3.1 Tangent Stiffness Method 1
In this method, \K]1 of Equation 3.11 is assumed to be constant over each
increment. Since the stiffness is constant over each loading increment, the load
displacement curve will be a straight line. In reality, the stiffness of nonlinear _
16
material will not be constant during any loading increment. Therefore, there will
be error of the predicted displacement. This error increases with increments away
from starting point and the predicted displacement further deviates from the true
solution. To obtain more accurate solutions, smaller load increments are required,
3.3.2 Viscoplastic Method
This method was originally developed for linear elastic viscoplastic material*
This kind of material is time dependent and is represented by a network that
consists of elastic and viscoplastic components connected in series. When using
this method, the system is assumed to behave elastically. If the resulting stress .
lies below the yield surface, the incremental performance is elastic and the
calculated displacements are correct. If the resulting stress violates yielding
condition, the stress state can only be sustained momentarily,and viscoplastic
straining occurs. The viscoplastic method is simple and has been widely used.
Complication rises when solving problems involving non viscous materials such
as elastoplastic materials.
3.3.3 Modified NewtonRaphson Method
i
The modified NewtonRaphson method uses an iterative technique to solve
Equation 3.11 The first step is similar to the tangent stiffness method but the
error is recognized and the predicted incremental displacements are used to
calculate the residual load, which is a measure of the error in the analysis.
Equation 3,11 is then solved with the residual load forming the incremental right
hand side vector as in Equation 3.12
17
K1
(3.12)
Where:
j =is the iteration number, and
=is the residual load.
The procedure is repeated until the residual load is small, and the incremental
aisplacements are then the sum of the iterative displacements. r determine the
residual load vector, an estimate of the incremental aisplacements is calculated at
the end of any iteration. From there, the incremental strains are evaluated, and the
constitutive model is then integrated along the incremental strain paths to obtain
an estimate of the stress changes. Theses stress changes are added to the stresses
at the beginning of the increment. Knowing these stresses, the consistent
equivalent nodal forces are evaluated. The difference between these forces and
the externally applied loads give the residual load vector. The integration
methods are termed stress point algorithms and can be computed using both
explicit and implicit approaches known as substepping algorithm and return
algorithm, respectively.
3.33.1 Explicit Algorithm
In this approach, the incremental strains are divided into number of substeps,
hence name substepping algorithm. In each substep, the strains are a proportion
of the incremental strains, as shown in Equation 3.13. The constitutive equations
are then integrated numerically over each substep.
(3.13)
Where:
18
{Asss}= are the strains in each substep,
{Aejnc}= are the incremental strains
AT ~ is the proportion
3.3.3,2 Implicit Algorithm
In this approach, the plastic strains over the increment are calculated from the
stress conditions corresponding to the end of the increment. These stresses are
not known, therefore initial estimate based on elasticity are completed to predict
the stress changes. Further iterations are completed to insure convergence where
the constitutive behavior is fulfilled and the final stress that Is within the yield
surface is computed at the end of that increment. From the stress state, the plastic
strains over the increment can be calculated.
Both algorithms give accurate result. However, explicit algorithm is more
vigorous in case of nonlinear constitutive model that contain 2 or more
concurrently active yield surfaces (Zdravkovic1 1999)s which is the case for most
of geotechnical engineering applications.
3_4 Finite Element Commercial Codes
In this research two main codes were heavily used. These were NIKE3D and
LSDYNA. With each code a complete package including the preprocessor and
post processors were also utilized. With NIKE3D, TRUGRID and GRIZ were
used as prq>rocessor and postprocessor, respectively. On the other hand, FEMB
and LSTCPrePost were used as preprocessor and post processor, respectively.
19
3.4.1 NIKE3D, TRUGRH), and GRIZ
N1KE3E is a nonlinear, implicit, and 3Dimensional finite element code for solid
and structure mechanics(Maker 1995), It was originally developed and has been
used by the Lawrence Livermore Nation Laboratory, LLNL for over 20 years. It
is used to study the static and dynamic response of structures undergoing finite
deformations. The main elements used in this program are the 8node solid
element5 2node truss element, and 4node shell element. The 8node solid
elements are integrated with a 2 x 2 x 2 point Gauss quadratic rule. The 2 node
beam elements use 1 integration point along the length. And the 4 node shell
elements use 2x2 Gauss integration in the plane.
There are more than 20 material models implemented in this code to simulate a
wide range of materia behavior including elastoplasticityanisotropycreep, and
rate dependence. Between independent bodies an arbitrary contact is handled by
variety of sliding algorithms such as modeling the sliding along matenal
interfaces including frictional interface.
NIKE3D is based on an updated Lagrangian formulation. Throughout each load
step? nodal displacement increments which produce a geometry that satisfies
equilibrium at the end of the step are computed. Once the displacement
increments are updated, the displacement, energy, and residual norms are
computed. After that, the equilibrium convergence is tested using user defined
tolerances and the analysis is preceded to the next load step. If convergence is not
achieved within the user specified iteration limitSj the optional automatic time
step controller will adjust the time step size and try again.
20
A preprocessing program, TRUGRID, was developed for NIKE3D and other
finite element and difference codes? such as ADINA, ANSYS, MARC, LS
DYNA, etc., to generate such finite mesh. TRUGRID is a 3Dimensional mesh
generator developed by LLNL as well TRUGRID is also used to specify loads,
element typessliding surfacesboundary conditions, and material models. The
output file from TRUGRID would then become the input file for N1KE3D or
other finite element programs. It uses a special projection method for mapping a
block mesh onto one or more surfaces. Therefore, a complex a looking mesh can
be built from a simple block very easily.
After running the analysis using the main processor, NIKE3D, a post processor,
GRIZ, was used to extract the results. GRIZ was also developed by Lawrence
Livermore Nation Laboratory. The output file from NIKE3D becomes the input
file for GRIZ. With GRIZ, the analysis results could be visualized and animated
passing through specified loading increments. The results from GRIZ could be
also printed into text file for plotting and analyzing.
3.4.2 LSDYNA, FEMBPC, and LSPRE/POST
LSDYNA is an allpurpose finite element code developed at Livennore Software
Technology Corporation, LSTC for analyzing deformation and dynamic response
of structures (LSDYNA 2003). The main solution method is based on explicit
time integration. Nevertheless an implicit solver is currently available. The
origin of LSDYNA dates back to the public domain software, DYNA3D, which
was developed more than 30 years ago at the LLNL. Throughout the past decade,
21
considerable progress has been made by Hallquist2 in developing the LSDYNA
code. This code was first initiated in 1989 for automotive applications. In this
version,1989, many enhanced capabilities such as a one way treatment of slide
surfaces with voids and friction and unique penalty specifications for each side
surface were brought in. Since then, many developments have occurred to this
program. Currently, LSDYNA contains approximately 100 constitutive models
representing wide range of material behaviors including linear and non linear,
elastic and inelastic, homogeneous and composite materials, and so on.
Furthermore, difficult contact problems such as contactimpact are easily treated.
This code helped vastly in the automotive industry because of the specialized
capabilities such as airbags, sensors, and seat belts that have been adapted in
LSDYNA
In LSDYNA, elements can be solids, shells, beams, or discrete elements. The
solids composed of 8 node elements, the shell composed of 3 or 4 node elements,
the beam of 2 node elements, and the discrete elements are springs or dashpots. It
is operational on a large number of mainframes, workstations, and PC^s,
A powerful preprocessing program for use with LSDYNAPC is FEMBPC
Version 27. It was developed by Engineering Technology Associates, Inc. in
2001,FEMBPC contains a swarm of automated, easy to use model building
functions. It supports a complete LSDYAN interface including material types,
contact types, boundary conditions, and output data base,
LSPRE/POST is a preprocessor and post processor. Both functions are still in
the developing stages. In this research, this program was heavily used as a post
2 In 1988 J, 0. Hallquist worked half time at LLNL to devote more time to the development of
LSDYNA. In 1989 he resigned from LLNL to continue the development of LSDYNA at LSTC.
22
processor. The current edition, Vl? was sufficient in producing the required
results in both graphical and text versions. The LSPRE/POST is very friendly
software, where the analysis results could be visualized and animated through
loading increments.
3.5 Constitutive Modeling of Soil
In general, the constitutive models for geomaterials such as rocks, concrete, and
soil are based on the same mathematical plasticity theory used to mode! common
metals. However, geomaterials slightly differ from the other metals due to their
compressibility and low tensile strength* Their behavior is stress path dependent
and the total deformation is composed of recoverable and irrecoverable part (C.S.
Desai 1984).
As for any other material, the only lucid way to determine the parameters that
define the constitutive model of soil is to conduct appropriate laboratory and/or
field tests. The parameters obtained from the laboratory or field test must be
verified by reproducing the observed data using the constitutive model. Have
satisfaction obtained, further usage of this model can be utilized to implement
different projects or applications. Element testing that characterizes the
mechanical behavior of soil is the most appropriate investigation for deriving the
parameters of the constitutive models. Of these tests are odometer, triaxial, true
triaxial, direct and simple shear, hollow cylinder, and standard penetration test
known as SPT. When simulating the element testing, it is very important to
closely simulate the field conditions including initial stress and drainage
conditions.
23
Soil is a non linear material3. When subjected to loaa, the physical characteristics
including the moduli continue to change. These moduli define the material
stiffriess and hence the constitutive matrix of the material. Throughout loading,
the components of constitutive matrix keep changing resulting in non linear
behavior, Quasilinear model based on piecewise linear behavior can be used as
an approximation of nonlinear behavior. In Quasilinear model,a given nonlinear
behavior is divided into pieces of linear elastic behaviorin which Hookes law
can be used to solve for Modulus of Elasticity, E5 and Poisson5s ratio, v, at each
loading increment. In this approach it is important to determine the variable
parameters as the state of stress changes during the incremental loading. Hence
these models are also referred to by variable parameter (VP) or variable moduli
(VM) model. Both elastic parameters can vary during the step loading or one of
them may vary while the other is kept constant. Of these VM models, hyperbolic
and RambergOsgood model are considered and briefly discussed.
3.5.1 Hyperbola Model
The stress strain curves of soil maybe simulated using a hyperbola or parabola
function. The hyperbola model for representing the stress strain curves for soil
was first proposed by Konder in 1963 and is given in Equation 3*14.
e
Where:
aiCT3 = is the deviator stress at any load increment,
e ^ is the axial strain at any load increment,
3 Non linearity renders the load displacement behavior.
(3.14)
24
E = initial tangent modulus, and
(cti
Since the response of soil or any geologic material is a fimction of confining
pressure, it becomes necessary to express the hyperbola model in conjunction
with the relation between initial modulus and confining pressure. Therefore, the
stress strain curve of the hyperbola has the form of:
2c cos $ + sin
/ \n
K7P
nap
\ a )
(3.15)
Where:
Et = is the tangent modulus at a point,
Rf = is the ratio of ultimate deviator stress to the failure deviator stress
c = is the cohesive strength
(()=is the angle of friction,
Pa = is the atmospheric pressure,
Kh and n = are material parameters.
1'his model is simple, and material parameters can be determined from laboratory
test results such as conventional triaxial compression, CTC, test. Results obtained
from this model are reliable especially when subjected to monotonic loading.
This is not the case when loadingunloading conditions are involved because
these loadingunloading conditions include wide range of stress paths where as
the hyperbola mode contains only one stress path. Furthermore, the Hyperbola
model does not account for dilatancy effects which become a deficiency when
modeling dense sand or overconsolidated clay.
25
3.5.2 RambergOsgood Model
The RambergOsgood model is an analytical model and it is often used to
represent the hydrostatic behavior of soil materials subjected to cyclic loading
(Oncul 2001). For monotonic loading, the stress strain relation of the Ramberg
Osgood can be expressed by:
y = is shear strain,
yy is reference shear strain^
t = is shear stress,
xy = is reference shear stress,
a = is constant factor, larger than 03 that adjust the position of the curve
along the strain axis, and
R = is constant factor, larger than 1,that controls the curvatures of the
graph.
The above VM models can account for behavior of limited class of materials
loadings, and stress paths, and therefore may not treated as general models.
Furthermore, geologic materials exhibits frictional resistance that is proportional
to the normal loading. The above mentioned models do not account for such
behavior.
The following models, Mohr Coulomb, DruckerPrager and Cap model, do take
into account for complex loading paths and frictional resistance. They are based
(3.16)
26
on theory of plasticity that was developed for metals taking in consideration the
dependency on hydrostatic stress.
3.5.3 MohrCoulomb Failure Criterion
MohrCoulomb is the most commonly used failure criterion in engineering
practice. According to this criterion^ the shear strength increases with increasing
normal stress on the failure plane, as shown in Figure 3.1, and can be expressed
as:
r + tan^
(cTj + ct^ jsin^ + 2ccos^
Where:
t = is the shear stress on failure planes
 is the angle of internal friction,
a = is the normal effective stress on the failure surface,
c = is the cohesion of the soil, and
G\ and CT3 = are the major and minor principal stress, respectively*
(3.17)
(3.18)
27
As seen in Equations 3.17 and 3.18, and Figure 3.1this model is expressed in
terms of maximum and minimum principal stress. Furthermore, Equation 3.18
indicates that MohrCoulomb failure criterion is represented by irregular
hexagonal pyramid m the stress space. According to this shape, the yield strength
in compression is higher than in extension.
In MohrCoulomb failure criterion, only two parameters are needed. These are c
and <(). They can be easily obtained from conducting two or more CTC tests on
cylindrical soil sample. For frictionless material, where ^ is equal to 0, the Mohr
Coulomb criterion reduces to Tresca criterion.
3.5.4 DruckerPrager Criterion
The DruckerPrager yield criterion is a generalization of the Von Mises criterion
that includes the influence of hydrostatic stress (Boresi 2003). The yield function
can be expressed as shown in Equation 3.19. This failure criterion can be further
explained using Figure 3.20, where constant parameters, a and k? can be
determined from the slope and intercept of failure envelope plotted in first
invariant of principal stresses5 Ji and, square root of second invariant of deviatoric
principal stresses,J2D space, as shown in Equations 3,20 and 3.21.
(319)
Jl=
(3.20)
(3.21)
28
Figure 3.2 "DruckerPrager criterion"
In order to establish this failure envelope or determine the model constants, it is
necessary to perform some laboratory tests, such as CTC test or plane strain test.
For CTC test, the minor and the intermediate principal stress, G3 and aj, are equal
to the confining stress. On the other hand, ai is equal to the confining stress in
addition to the deviatoric stress, Aa, as shown in Equation 3,22. Applying two or
more CTC tests at dififerent confining pressure, allows the failure curve to be
established by finding Ji and ^]J2d for each test. Doing so, and using the least
square method, a and k can be determined. The values of a and k in case of CTC
tests can be expressed in terms of <)> and c as shown in Equations 3.23 and 3.24,
respectively (C.S. Desai 1984). The values of ^ and c are first determined using
MohrCoulomb failure criterion obtained from conducting two or more CTC tests
under different confining pressure.
29
<7^ C7^ + Act Â¢3.22)
2 x sin ^ x (3sin (3.23)
^ 6xcxcos sin (3.24)
According to this criterion and as shown in Figure 3,2 there is always a negative
plastic strain component indicating a volume increase or dilation. For dense sand,
during shearing this phenomenon occurs at even very small strains. However,
loose sands experience compressive deformations and volume reduction. This ^
Indicates that n loose sand is used and tested, the results obtained from this
method are not reliable.
3.5.5 Cap Models
Cap models are based on yielding of soil They are expressed in terms of 3
Dimesnicnal state of stress. The elliptical yield surfaces looking like caps
resulted in the name of such models. The cap models use two intersecting
surfaces, fixed or ultimate yield surface, f]? and yield cap, fa, as shown in Figure
3.3.
30
I
Figure 3*3 **Cap model"
Based on DiMaggio and Sandler^ the ultimate yield surface that represents the
shear failure surface has the form of:
Where:
a5 p, y = are material parameters
GJ] = is convenient for fitting shear failure data that has fairly linear
straight line representation, which allows this surface to be model using
the DruckerPrager Model,
The constant parameters for the shear failure surface can be obtained from
experimental data such as CTC test results* Once obtaining the results at different
confining pressure and plotting J] versus the best curve fit according to
Equation 3.25 is obtained, and therefore the constant parameters can be
determined. If DruckerPrager model was assumed to represent the ultimate
(3.25)
31
yielding surface as in Figure 3.4, where a straight line can fit these data points,
then the constant parameters of this ultimate yield surface can be solved using the
following equations
6c cos
V3(3sin^)
0 2smÂ¢
W(3 sin
Where a in this equation represents the Yaxis intercept and G represents the
slope of the failure surface. For simplicity, both DruckerPrager failure lines can
be represented by one linewhich makes the value of equal to zero. In case of
cohesionless soil,a is also zero, and therefore, the shear failure line starts from
the ongin (0, 0). The only parameter to determine is p. p can be determined by
solving Equation 3.25 at a point on the transition curve considering all other
constants (C.S. Desai 1984).
Figure 3.4 "Interpretation of parameters of fin
(3.26)
(3.27)
32
The other surface is the moving yield surface and is referred to by cap. The cap
surface represents an ellipse with its long axis along the mean pressure, Jl? axis,
and can be expressed in the form of:
/2=jR2^2Z)+(/1c)2=(XC)2 (3.28)
Where:
R = is the ratio of the major to minor axis of the ellipse, and lies in the
range of L67 lo 2 (C.S. Desai 1984),
X = is hardening parameter that depends on the plastic volumetric strain as
shown in Equation 3*29 and can be written in terms of mean pressure
as in Equation 330 Graphically, it is the position on J] axis where the
cap surface intersects, and
C = is the value of Ji at the center of the ellipse.
/
X
In
D
sl
W
+z
(3,29)
X = 3p (3.30)
Where:
Ds Z, and W are hardening parameters
p = is the mean pressure, which is equal to the confining pressure in case
of triaxial test
In case of very small initial yielding, Z? the size of the initial cap is almost zero.
Therefore after rearranging Equation 3.29 and plugging in Equation 3,30,
considering the elastic, 8ve? and the plastic, evp ? volumetric strains components of
the total volumetric strain, ev, the W and D parameters can be determined using
Equation 3.31.In Equation 3.31 the elastic volumetric strain is substituted by the
value of p/K, where K is the bulk modulus and is defined as the slope of the
unloading curve in a hydrostatic compression test (C.S. Desai 1984). Several
33
trials must be conducted to obtain the most accurate values of W and D that
reproduce the hydrostatic curve (volmnetric strain versus pressure). Ideally, when
approaches zero5 and therefore W is the volumetric strain measured at that
pressure (C,S. Desai 1984),
As a summary, defining cap model parameters several tests must be completed.
Hydrostatic compression and CTC tests can be sufficient. The hydrostatic
compression test is subjected to loading and unloading. The loading portion helps
in determining the hardening parameters, W and D, while the slope of unloading
is necessary in determining the elastic parameter K. Performing sets of the CTC
tests under two or more confining pressure provides an excellent prediction of the
parameters for fi surface and initial Yoxang's Modulus, Ei, for each confining
pressure* Once all the parameters constants are determined, it becomes necessary
to reproduce the hydrostatic curve and CTC results xmder different confining
pressures in order to validate the model.
3.6 Contact in Finite Element
In finite element there are three methods to define contact between different parts.
These are kinematic constraint for tied interfaces and called Lagmngemultipliers,
distributed parameter where sliding occurs without separation, and penalty
method which includes sliding, separation, and friction* Most of the applications
in soil structural interaction involve separation and sliding between adjacent
particles. From there, a brief background and description of penalty method is
presented.
applying large pressure during hydrostatic compression test the term e
3PD
(3.31)
34
In the penalty method, a normal interface springs are placed between all
penetrating nodes and contact surfaces. For each slave node the closest master
node and closest master segments are found and a check is completed to
determine if any penetration. Then penetration is then reduced by applying forces
through the existing massless spring that has a stiffness k and is called penalty
stiffness. It is unique for each segment and is based on the bulk modulus of the
penetrated material. For 2dimensioaal shell elements and 3Dimensional brick
elements, k is defined as shown in Equations 3.32 and 3.33, respectively.
Where:
a = is the penalty stiffiiess scale factor. In LSDYNA default is 0.1,
B = is the bulk modulus,
A
t = is the nominal shell thickness, t =
max shell diagonal
A = is the segment area, and
V is the brick element volume.
Due to relative movement between contacted parts, friction is generated. Friction
is based on a Coulomb formulation, and the friction coefficient, ]<, It is a function
of coefficient static friction, [is, coefficient of dynamic friction, decay
coefficient, dc3 and relative velocity between the slave node aiid the master
segment, v3 as shown in Equation 3.34.
ax fixt
(3.32)
(333)
(3.34)
35
For many geotechnical applications, especially including reinforcement layers
within the soil mass, the 4 node shell element is included in the model. Even
though it is a 2 dimensional element, thickness must be properly accounted for in
contact. Therefore, surface or nodes contact must have an offset to account for
shell thickness. When contact is detected on both sides of the shell, half of the
thickness must be considered as an offset on each side of the element.
During contact, kinetic energy is developed. This energy is directly proportional
to the spring stiffness and the penetration distance. Preferably this energy should
be very small to indicate a minimal penetration compared to other physical
energies in the system. In order to obtain small contact energy, l.e. small
penetration, it becomes necessary to avoid initial penetrations and tangled meshes
at any cost. This can be done by adequately offsetting the contacted surfaces and
use consistent fine meshes on adjacent parts.
3.7 Summary
In this chapter a brief back ground is presented to cover some aspects of finite
element method and analysis especially for non linear materials such as soil. This
was followed by considering some of the constitutive models such as Rambo
Osgood and Cap models for geological materials. Since soil is frictional material
it was significant to define contact properties between soil particles themselves
and between soil and other adjacent parts such as reinforcement layers and
concrete foundations.
36
4. Anisotropic Properties of GeoComposite
4.1 Introduction
Over the last three decades, the use of geosnthetic reinforced soil, GRS, in geo
structures like mechanically stabilized earth, MSE? walls had steadily increased.
Reinforced soil structures are constructed in a manner that produces a composite
structure of alternating layers of soil and reinforcing elements (Vector Elias
2001), Combining the compression strength of Ottawa sand and the tensile
strength of geosynthetic produces a geocomposite that is strong in tension and
compression (NienYin Chang 2006).
Performing finite element (or difference) analyses on geostructures requires both
the material models and soilgeosynthetic interface model that allows realistic
behavior like interface slippage and separation, which increases the computation
time especially for a large problem. It would be much easier and less time
consuming if the geocomposite can be treated as a homogeneous material instead
of adjacent parts with frictional interfaces. This would alleviate the interface
convergence problem and speed up the computation when using the finite element
method.
In this chapter finite element analyses were completed on cubic samples of this
geocomposite to evaluate its equivalent properties as function of its constituent
materials and other factors. These factors were spacing between inclusion layers,
friction coefficient, and confining pressure.
37
The tests were performed onlraxlmxlm geocomposite cube with linear
elastic soil and geosynthetics. The cubes were subjected to a combination of
confining pressures and deviator stresses in six directions successively. The
directions were XYZXYYZand X Once these forces were applied, the
resulted strains in all six directions were calculated. Based on Hooke5s law? the
compliance matrices were determined, which allowed in observing the material
type and properties for each configuration case. Also, the Young's moduli of the
composite evaluated from the finite element analyses were compared with the
values from the analytical formulation.
The purposes of these tests were to determine the material type of the geo
composite and to develop a data base of equivalent properties that could be used
in creating a constitutive model. This model would be obtained (Chapter five)
based on linear regression analysis as function of all input parameters (Young^
modulus of soilPoissons ratio of soilYoungs modulus of geosynthetic
Poissons ratio of geosyntheticSpacing between reinforcement layersrnction
coefficient, and confining pressure)
4.2 Composite Materials
The composite material term signifies that two or more materials are combined on
a macroscopic scale to form a useful material (Jones 1975). The advantage of
composites is that they usually exhibit the best qualities of their constituent and
often some qualities that neither constituent posses. Composite materials are
widely employed in both or in the combination of structural and soil fields* These
applications are expected to continue due to increasing requirements for light
weight, nigh stiffness and/or strength, nonconducting, and noncorroding
38
materials. Geosynthetic material fits these requirements and has served as subject
of research since 1940 when it was utilized in the military and aerospace
industries (J. Frost 1999).
Composite materials are generally anisotropic4 and inhomogeneous5, so that the
strength of these composites cannot be computed by identifying a single stress
level Stress state is nearly always complex, even when only one stress is applied
at the global level. To obtain the mechanical properties of anisotropic materials,
several test must be conducted such as multiaxial test to cubical composite (Hon
Yim Ko 1974). In this approach all the material properties of composite material
is obtained by applying sets of deviator stresses on a confined specimen. From
these applied stresses, all components of compliance matrix [C] are evaluated
using Hooked law, in which all strain components are linearly related to stress
components as shown in Equations 4. La and 4.1.b.
Â£ = [c]*o
Â£ X cn C12 C13 C14 C15 C16 a X
y C21 C22 C23 C24 C25 C26 a y
s z C31 C32 c33 C34 C35 C36 a z
8 w C41 C42 C43 C44 C45 C46 r
s C51 C52 C53 C54 C55 C56 r 2
Â£ xz _C61 C62 C63 C64 C65 C66 T 1 xz A
A complete anisotropic elastic material has 36 material parameters in its [C] as
shown in Table 4.1. However, less than 36 of the constants are actually
4 An isotropic body has material properties that are the same in every direction at point in the
body,
5 A homogeneous body has uniform properties throughout.
(4.1.a)
(4.1.b)
39
independent parameters for elastic materials when the strain energy is considered.
Due to symmetry, where CyCji, only 21 of the parameters are independent.
Where:
E E E E E E ax
yy yy yy yy yy yy
V xz V l Tj 2 TJ 7 fzx,z
E E E E E E z
zz ZZ ZZ ZZ ZZ zz z
Tj x.xy Tf 7 1 fi y^xy fi zx3xy V
G G G G G G
xy xy xy xy xy xy r yz
77 Â¥ 7] yyz 7 jl xy,yz i u zx,yz r
G G G G G G X2
Z yz Z
^x9zx 77 zzx ^xy,zx ^yz,zx 1
G G G G G G
xz XZ xz XZ XZ XZ
ExxEyyEzzYoungs moduli (Tension compression)
Table 4.1 ''Compliance matrix components IC]M
Cii Cl2 C13 C14 C5 Cl6
C21 C22 C23 C24 C25 C26
C31 C32 C33 C34 C35 C36
C41 C42 C43 C44 C45 C46
C51 C52 C53 C54 C55 C56
C6I 62 C63 C64 C65 ^66
According to the generalized Hooked law, the general shape of the resulting [C]
for an anisotropic material is as follow (Lekhnitskii 1963):
, u v 7] rf n
1 yx y2x *xyyx fyz3x fzx,x
Exx Exx Exx Exx Exx EXX
V A V 71 77 71
xy 1 zy fxy,y fyz,y lzx,y *
Fxz
40
Gxy, Gyz, Gxz Shear moduli
Vyx, v^y, v^, vxy> Vyz, Vjtt are the Poisson coefficients which characterize the
transverse compression for tension in the direction of the axis of the
coordinate (thusVyx is a coefficient which characterizes the decrease in
the x direction for tension in the y direction; Vxy is the coefficient which
characterizes the decrease in the y direction for the tension along the x
axis and so forth)
ji is the coefficient of Chentsov. These constants characterize the shear in
the planes, parallel to the coordinates and also induce tangential stresses
parallel to other coordinate planes. For example characterizes the
shear plane parallel to the yz plane which induces the stress
r is the coefficient of mutual influence of two characteristics. The first
characteristic is the elongation in the positive directions parallel to the
axes. These are induced by tangential stresses. The second characteristic
is shear in the planes parallel to the coordinates under the influence of
normal stress.
From Equation 4^2, it is clear that the material properties can be directly
determined by finding all the components of [C]. Components of [C] can be
determined by applying stresses, one at a time on a cube element of the composite
material as shown in Figure 4.1, The stresses are normal stresses on the negative
and positive X faces of the cube element (ax)5 normal stresses on the negative and
positive Y faces of the cube element (ay)3 nonnal stresses on the negative and
positive Z face of the cube element (az)s shear stresses on the negative and
positive X faces in the Y direction or on the negative and positive Y face in the X
direction (Txy), shear stresses on the negative and positive Y face in the Z
direction or on the negative and positive Z faces in the Y direction (Tyz), and shear
41
stresses on the negative and positive X faces in the Z direction or on the negative
and positive Z faces in the X directions ().
z
X
Figure 4*1 T, Deviator stresses on cube element11
Loading the sample in erne direction produces six strain components which are
linear functions of unknown compliances (HonYim Ko 1974). From each stress,
all strain components can be evaluated by either the use of strain gages in physical
testing or finite element approach. Using Equation 4.1, all columns of [C] can be
calculated. For examplewhen cyx is applied and all other stress (CTycrzTxyTyz
and Txz) arc zero, all six strain component can be determined. Therefore, from
this test the first column of [C] can be calculated as follow:
42
s X C11 c12 c13 C14 C15 a
s y C21 c22 C23 C24 c25 C26 X 0
Â£ Z C31 C32 C33 C34 C35 C36 0
xy C41 C42 C43 C44 C45 C46 0
Â£ yz C51 C52 C53 C54 C55 c56 0 _0
s L xz. _C61 C62 C63 c64 C65 C66
X :cii
8 = y C21CTx
s = z C31CTx
s CA.cr 41 x
s C^a 51 x
Â£ XZ = Ca^ct 61 x
The same procedure can be followed to determine the second} the third, the
fourth, the fifth, and the sixth columns of [C] by applying ay, az> txy, xyZj and xxz,
respectively. Once all the [C] components are calculated, the material type and
properties can be determined. The types that are widely used here will be briefly
discussed, which are orthotropic, transversely isotropic, and isotropic.
4.2.1 Orthotropic (Three Planes of Symmetry)
In orthotropic material, there are three principal directions of elasticity that are
mutually orthogonal which pass through each point of the body (Jones 1975). An
orthotropic material has no coupling between any normal stress and shear strain,
and no coupling between any two distinct shear strains*
The principal constants are Ei,E^, E3 (the Young moduli), Gnt G13 (the shear
moduli) and vu, v2!V13, v31, V23, V32 (Poissons ratio coefficients). The
generalized Hooke's law equation can be written in the following matrix form:
43
y,
12
x
y
V.
e
i
13
s
Â£
xz
El
o
o
Where:
v12 21
121
E
23
E1
0
0
0
v,
31
v
32
0
(T
X
a
y
E.
2
E,
23
0 0
E3 0 1 0 1
.2G12 0
0
2G23
0 0 0
32 ^13 V31
0
cr
0
1
xy
2
xz
2G.
13
(4.3)
The above material resulted with 12 constants. Due symmetry, 9 of them are
independent.
4.2.2 Transversely Isotropic (A plane of Isotropy)
In transversely isotropic materials there is one plane in which the mechanical
properties are equal in all directions (Lekhnitskii 1963). A transversely isotropic
material is an orthotropic in which two of the directions affect the third direction
in the same manner. For example, illustrated in Figure 4.2, is a fibrous element in
which directions 2 and 3 behave the same with respect to direction 1,this is 4t23
plane of isotropy.
44
The generalized Hookers law can be written in the following matrix form:
E and E? are Young moduli (for tensioncompression) with respect to directions
lying in the plane of isotropy and perpendicular to it. v is the Poisson coefficient
which characterizes the transverse reduction in the plane of isotropy for tension in
^lv
4
CTxb bclv
o o o 001
2G
0 0
2G0
0
0 0
IS
0 0
V
Â£ o 0 0
Â£11Â£V Â£ 0 0 0
Â£Â£
000
s^
45
the same plane, v9 is the Poisson coefficient which characterizes the transverse
reduction in the plane of isotropy for tension in a direction normal to it. G? and G
are the shear moduli for the plane normal and parallel to the plane of Isotropy,
respectively. From Equation 4.4, six constants can be observed. Only five of
them are independent. These are E, E\ v, v\ and G5. G is dependent on both E
and v as shown in Equation 4.5.
~2{\+v)
4.2.3 Isotropic (Complete Symmetry)
In isotropic bodies, any plane being a plane of elastic symmetry and all points are
equivalents in all directions (Lekhnitskii 1963). A three dimension (3D)
isotropic material is an orthotropic material that has no direction preference.
The equation of isotropic material which expresses the generalized Hookers Law
can be written in the flowing matrix form:
1 1 V 1 V 0 0 0 a
s V V X
X 8 y ~~E Â£ ~~E 1 0 0 0 a y
Â£ 0 0 0 a
Z _  E E ~E _ 2
e xy 0 0 0 1 2G 0 0 T xy
yz 1 T
E xz 0 0 0 0 0 Z
2G T
0 0 0 0 0 1 XZ
2G
(4.5)
(4*6)
46
Where E is Young's modulus, G is shear modulus, and v is the Poisson
Coefficient. These three elastic constants are identical in all directions. Only two
of them are independent and can be explained using Equation 4.5.
In order to determine the material type of a composite, it is required determining
all components of [C], where the general shape of [C] indicates the type of the
material. If Ciand C44=C55=C66, the
material is considered an elastic isotropic material. If C2=C2i; Cn=C3i; and
C23C32, the material is considered an orthotropic material. If Cn=C22 C\2=C2\
C i3==0231^032 and Css^C^e, the material is considered a transversely isotropic
material.
In order to determine the equivalent properties of a composite, several approaches
can be utilized, such as laboratory testing, unite element analysis, and analytical
methods. In this chapter, the finite element and the analytical methods will be
used. The results from both methods will be compared and used in statistical
analysis for future predictions.
4.3 Evaluation of Composite Material Properties
The geocomposite properties were determined in two approaches, the finite
element and the analytical methods. The results obtained from both methods
were compared and evaluated.
47
4.3.1 Finite Element Approach in Determining the
Mechanical Properties of a Geocomposite
Numerical study using NIKE3D was completed on cubical reinforced soil
specimens with lm x lm x lm dimensions each. The purpose of this study was to
find the type and properties of the reinforced soil. The geocomposite specimens
consisted of medium dense and dense Ottawa sandy soil, and geosynthetic
inclusion. Different types of inclusions were considered in this study, including
geotextile, geomembrane, and geogrid. Both, soil and reinforcement were
assumed to be isotropic elastic materials. The sand and geosynthetic properties
used are shown in Table 4,2.
Table 4.2 f1Soil and geosynthetic properties"
Soil (Das 1993)
Void ratio Poissons ratio Density Density/g
(V) (Mg/m3) (Kg.s2/m4)
Medium Dense to Dense sand 0.630.53 0.20.3 1.9 193.68
Gcosynthetic
Thickness Poissons ratio Density Youngs Modulus
(mm) (V) (g/cm3) (MPa)
Geosynthetic 2.5 0.2 0.4 0.94 221100
For the geocomposite specimens, the bedding plane was on the horizontal(X, Y)
plane parallel to the bottom side of the cubic specimen and Z was in the vertical
direction perpendicular to XY plane. The geosynthetic inclusion was allowed to
slide relative to the sand due to the frictional interface between the two materials.
In this study, tests were completed on geocomposite with 15 3, and 5 layers of
48
equally spaced inclusion as shown in Figure 4.3, Under each spacing (S)
category, tests were completed with three different values of mean pressure (a).
These values were 68947.5 N/m2 (10 psi)5 172368.9 N/m2 (25 psi)3 and 344737.8
N/m2 (50 psi).
49
In 1968, Hardin and Black suggested the following two equations, 4.7 and 4.8, to
determine the shear modulus (G) of sandy soil based in the shear wave velocity
for low amplitudes of vibrations (Das 1993). In these equations, G of round
grained and angular sandy soil primarily depends on a, and soil's void ratio (e).
By means of the former equations 4.7 and 4.8, average shear modulus (G) of
sandy soil for the two cases, round and angular grained soil was calculated. Soil
modulus of elasticity (Es) was then calculated using Equation 4.5, presented in the
following format:
Â£2(lhv)G
For the first a value, Es was found to vary from 70 MPa for medium dense sand
to 300 MPa for dense sand. For the second c value, Es was found to vary from
100 MPa for medium dense sand to 450 MPa for dense sand. For the third a
value, Es was found to vary from 150 MPa for medium dense sand to 650 MPa for
dense sand. In each range of Es, two supplementary inner points were selected to
recognize its effect on the geocomposite properties as shown in Figure 4.4.
6908(2.17e)2 .g
\ + e
(4.7)
(4.8)
50
Figure 4.4 nÂ£s (MPa) distribution under different a,r
Under all Es categories for geocomposite specimens with 1layer inclusion,1100
MPa? 550 MPa, and 270 MPa were used for modulus of elasticity of geosynthetic
(Eg), This range of Eg represented the geogrid, geomembrane, or geotextile
materials that have strong stiffiiess. Values of 0.3, 0.25, and 0.2 were sued for
Poisson ratio of soil (vs). Values of 0.4, 03 and 0.2 were used for Poisson ratio of
geosynthetic (vg). Values of 1.0, 0.5, and 0.25 were used for friction coefficient
(f) between soil and inclusion. For geocomposite with 3 and 5 layers inclusion,
1100 MPa and 270 MPa were used for Eg, 0.3 and 0.2 were used for vs? 0,4 and
0.2 were used for vS51,0.5, and 0.25 were used for f. Further investigations were
completed on this cube to analyze its performance when weak geotextile were
used. The elastic moduli of such material were in between 22 MPa and 100 MPa.
For that reason, tests of specimen with different Eg luider the variation of a, S,
Es, vs? vg> and f were completed. The frame in Figure 4.5 was provided to show
a summary of all the performed tests.
51
Figure 4.5 "Theoretical frame of all numerical tests'1
A total of 468 combination tests were applied on the geocomposite specimens.
For each combination, the deviator stresses (ox, ay Txy, iyz, and tXi) were
applied separately on the geocomposite specimens with each magnitude of
689475 N/m2 (100 psi). The gravity load due to the specimen's own weight and
a were also applied. This resulted in a grand total of 2808 tests. Each test lasted
30 seconds with 1 second increment, starting with gravity load where it reached
an acceleration of 9.81 m/s2 after 10 seconds and stayed constant for the rest of
the test period, a was applied after 10 seconds, it reached its ultimate load at 20
seconds and stayed constant for the rest of the testing period After 20 seconds
52
the deviator stress was applied and reached its ultimate load at 30 seconds. These
three loads distributions were shown in Figure 4.6.
Loads Increments
1.2
1
0.8
 0,6
0.4
0.2
0
0 5 10 15 20 25 30 35
Time (sec)
Gra^ty / Confining / D^iator /
/ pressure / / / stress
/ / /
/ / / / .
/ / /
T
Figure 4.6 f,Load curve distributed for aU 3 loads; gravity, confining
pressureand deviator stress"
After running the numerical analysis using NIKE3D on the geocomposite
specimensthe strain components (sx, ey ExyÂ£yzExj) due to the application of
each individual stress of ctx, ay> az, txy, Tyz? Txz were determined. These results
were then used to calculate the components of [C], From [C], the material type
and properties were determined. In this study, only three set of tests are
explained and the rest of all results are tabulated in Appendix A. These sets are 1)
geocomposite with 1layer inclusion, 2) geocomposite with 3 layers inclusion,
and 3) geocomposite with 5 layers inclusion. For the three mentioned sets, =
68947.5 N/m2 (10 psi), Es= 70 MPa, Eg=1100 MPa, vs = 0.3, vg = 0.4, and
Deviator stresses = 689475 (100 psi).
53
1)Geocomposite specimen with 1layer inclusion
The resulted strain components due to all applied stresses are shown in Table
4.3. To calculate the components of [C], shown in Table 4.4, these strain
components were divided by the deviator stress.
Table 4.3 "Strain components of geocomposite with 1layer inclusion"
Â£x Sy Sz ^yz
0.0084 0.0023 0.0026 0.0000 0.0000 0.0000
crY 0.0023 0.0085 0.0027 0.0000 0.0000 0.0000
0.0027 0.0027 0.0095 0.0000 0.0000 0.0000
XV 0.0000 0.0000 0.0000 0.0106 0.0000 0.0000
tvz 0.0000 0.0001 0.0000 0.0000 0.0128 0.0000
xz 0.0000 0.0000 0.0000 0.0000 0.0000 0.0128
Table 4.4 "[C] for geocomposite with 1 layer inclusion"
0.0122 0.0034 0.0039 0.0000 0.0000 0.0000
0.0034 0.0123 0.0039 0.0000 0.0000 0.0000
0.0037 0.0039 0.0138 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 54 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0186 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0186
From the results in Table 4.4, it was observed that Cn=C22 C^^Cii
Ci 3=C23=C3 032; and C55=C66. This indicated that geocomposite material
acted as transversely isotropic. From this observation, with the aid of
Equations 4.4, the mechanical properties of geocomposite with 1 layer of
inclusion under the specified mentioned conditions of r, EsEgvsVgand
Deviator stress were as follow:
54
S
y
"z
xy
s
y^
s
E
0
0
V V E 0
1 1 t V E 0
V E 1 E 0
0 0 1 2G
0 0 0
0 0 0
2G
0
0
0
1
2G
a
x
y
xy
G
E
E
v
~E
2((l+v)
^(verage(0.0122,0.0123) = 0.01225 => E = 81.648(JWPa)
^vemge(0.0034,0.0034) = 0.0034 ^v = 0.2755
P = 0.0138 =>Â£ =12.S{MPa)
E
P
7 =  verage(0.00390.00390,00390.0037) = 0.0039
=> '=0.280
=dvera(0.0186,0.0186) = 0.0186 => G = 32_4209(M?a)
2G
2G
0.0154 => G = 32.4209(MPa)
Using Equation 5, G was found to be;
E 81.648
G
2((l + v) 2(1+ 0.2755)
32.0066(MPa)
55
2) Geocomposite specimen with 3 layers inclusion
The resulted strain components due to all applied stresses are shown in Table
4.5. To calculate the components of [C]shown in Table 4.6, these strain
components were divided by the deviator stress.
Table 4.5 "Strain components for geocomposite with 3 layers inclusions1'
Ex Â£y Â£z Ejcy yz
0.0091 0.0024 0.0029 0.0000 0.0000 0.0000
aY 0.0024 0.0089 0.0029 0.0000 0.0000 0.0000
0.0027 0.0027 0.0093 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0111 0.0000 0.0000
Tvz 0.0000 0.0000 0.0000 0.0000 0.0124 0.0000
xz 0.0000 0.0000 0.0000 0.0000 0.0000 0.0124
Table 4,6 for geocomposite with 3 layers inclusion
0.0034 0.0039 0.0000 0.0000 0.0000
0.0035 0.0129 0.0039 0.0000 0.0000 0.0000
0.0042 0.0042 0.0135 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0161 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0179 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0179
Again, the shape of [C] indicated that the geocomposite sample with 3 layers
of inclusion was transversely isotropic. Using Equations 4,4 and 4.5, the
mechanical properties under the same specified conditions were
56
=y4verage(0.0132,0.0129) = 0.0130 = 76.7667(MPa)
E
=Aver age(0.003 A0.0035) = 0.00345 ^>v~ 0.2667
E
^ = 0.0135 =>Â£' ^n.m(MPa)
E
^ = jvera(0.0039,0.0039,0.004290.0042) = 0.0040
=> = 0.2970
t = jverage(0.0179,0.0179) = 0.0179 a G = 27.8614(Ma)
2G
=0.0161=> G = 30.9921(MPa)
2G
Using Equation 5, G was found to be
G
E 76.7667
2((l + v) 2(1 + 0.2667)
= 30.3015(MPa)
3) Geocomposite specimen with 5 layers inclusion
The resulted strain components due to all applied stresses are shown in Table
4,1.To calculate the components of [C], shown in Table 4.8, these strain
components were divided by the deviator stress.
Table 4.7 "Strain components of geocomposite with S layers inclusion"
ex Â£y exv Eyz 8xz
0.0087 0.0023 0.0027 0.0000 0.0000 0.0000
0.0023 0.0086 0.0027 0.0000 0.0000 0.0000
CTz 0.0027 0.0027 0.0096 0.0000 0.0000 0.0000
Txv 0.0000 0.0000 0.0000 0.0110 0.0000 0.0000
vz 0.0000 0.0000 0.0000 0.0000 0.0127 0.0000
Txz 0.0000 0.0000 0.0000 0.0000 0.0000 0.0128
57
Table 4.8 f1[Cj for geocomposite with 5 layers inclusion"
^0.0126 : 0.0033 0.0031 0.0000 0.0000 0.0000
.0.0034 ;0.0124 0.0000 0.0000 0.0000
^.0040 = .0.0040 0.0139 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0160 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0185 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0186
Again, the shape of [C] Indicated that the geocomposite sample with five
layers inclusion was transversely isotropic. Using Equations 4.4 and 4.5, the
mechanical properties under the same specified conditions were
=dverage(0.0126,0.0124) = 0.0130 =79.9425(MPa)
E
=^verage(0.00330.0034) = 0.00335 => v = 0.2686
E
^ = 0.0139 => Â£* = 72.0054(MPa)
E
^7 = ^verage(0.00390.0039,0.0040,0.0040) = 0.00395 => k' = 0.2842
E
= v4verage(0.0185,0.0186) = 0.01855 G* = 27.9892(MPo)
2G
=0.0160 G = 31.3239(MPa)
Lising equation 4.5, G was found to be
^ E 79.9425 ,
= 7 == 31.5074(MPa)
2((l + v/) 2(1 + 0.2686) v
As a result from all sets, geocomposite material was determined to be a
transversely isotropic in which the mechanical properties were equal in the
horizontal X, Y plane. Both values of G, from [C] and Equation 4.5 were very
58
close, which proved that G is consistently dependent variable and only five
independent constants represented this type of material.
4.3,2 Analytical Method in Determining the Mechanical
Properties of a GeoComposite (James C. Gerdeen
A 3D composite consist of two materials, the matrix which is denoted as M, and
the inclusion or fiber which is denoted as f. In general, both materials have
different properties which are also different &om the equivalent properties of the
composite.
In the analytical method, both materials are assumed to be glued together. For
that reason, the effective properties do not depend on the frictional interface.
Instead, the composite material has properties based on the amount of each
materialthe shape of the inclusions, the orientation of the inclusionsand the
material properties of both materials.
Using this method, the effective modulus of elasticity (Ec) of the composite can
be determined. This can be completed using one or both of the following models:
isostiain model and isostress model.
59
4.3.2.1 Isostrain Model
In this model, the matrix and the inclusions are separated into parallel
^compartments95 in the composite as shown in Figure 4.7. The strain in each of
the two materials is then equal.
Figure 4.7 "Parallel (Isostrain) Model11
F =F ^Fr
c m f
Where:
Fc = Total force acting on the composite element.
Fm = Force in the matrix.
Ff= Force in the fiber.
(49)
60
F F
c m
A A
c c
Where:
Am = Area of matrix.
Af = Area of fiber
A
V ^
A
c
Where:
Vm = Volume fraction of matrix.
Vf = Volume fraction of fiber.
(4.10)
(4.11)
(4.12)
Substituting Equations 4.11 and 4.12 in Equation 4.10, the following stresses can
be reorganized:
c
=am
Where:
(4,13)
ac = the applied stress on top of the composite = Fc/A.
= the applied stress on top of the matrix = Fm/Am.
CTf = the applied stress on top of the fiber = Ff/Af.
Since both materials are assumed to be glued together, the strains on both
materials, em and 8f, are equal. These strains are also equal to the composite
strain, sc as shown in Equation 4*14,
G S ~ ^ Â£
(4.14)
61
Dividing Equation 4.13 by the composites strainsthe modulus of elasticity of
the composite can be determined as follow;
E =E V r
c mm j j
Where:
Ec = Equivalent modulus of elasticity = cr/s.
Em = Modulus of elasticity of the matrix = ojz.
Ef = Modulus of elasticity of fiber = Gf/e.
4.3.2.2 Isostress Model
In this model, the two materials axe lumped in a serial arrangement, which makes
the stress the same in each material, however the strains are different.
Figure 4.8 "Serial (Isostress) ModeI,1
(4.15)
62
L
V =^
L
c
V
L
(4.16)
(417)
For this model, the change in length ALC is due to the contribution of both
materials as shown in Equation 4.18.
AL = AL + AL r
c m f
(4.18)
Dividing Equation 4.18 by Lc will give the resulted strain of the composite (ec)
and the strain contribution of each material (8m, 8f).
ec
A7, / A L AZ ^ L
L ^l = + a = s V
L L L L L Lr L rn m f f
c c c me f c
(4.19)
As mentioned above, the stresses in both materials are equal, and these stresses
are also equal to the stress in the composite material, shown in Equation 4.20,
"  cy " cx
m f
(4.20)
Substituting Â£c by Â£m by am/Em, and Â£f by a/Ef, and considering the
stresses equality in Equation 4.20, the following formula can be used to determine
the effective modulus of elasticity
Ec Em
+
E, ^
1
Vm.+ VJ
(4.21)
Em E
63
The three examples used in the finite element sections (1layer, 3 layers, and 5
layers) are repeated and analyzed using the analytical method considering both
models. The isostrain model will give the plane modulus of elasticity and the
isostress model will give the vertical modulus of elasticity.
L 1 layer
For 1layer, the following properties were given
Em = 70 MPa
Ef=1100 MPa
Isostrain model was used to determine the plane modulus of elasticity (E) of
the composite:
t of reinforcement =2.5 mm
Ac =1000 mm x 1,000 mm =1,000,000 mm2
Af = 2.5 mm x 1000 mm = 2,500 mm2
Am =1000,000 2,500 = 997,500 mm2
Vm 997,500/1,000,000 0.9975
Vf= 2,500/1,000,000 = 0,0025
Ec=E V + E^Vj. = 70(0.9975) + 1100(0.0025)=12.575MPa
E = 72.575 MPa
Isostress model was used to determine the vertical modulus of elasticity (EJ)
of the composite:
Lc =1000 mm.
64

Full Text 
PAGE 1
ANISOTROPIC ELASTIC MODELING OF REINFORCED SOILS by \ Mohammad AbuHassan B.S., University of Jordan, 1999 M.S., University of Colorado at Denver, 2002 A thesis submitted to the University of Colorado at Denver and Health Science Center in partial fulfillment of the requirements for the degree of Doctoral ofPhilosophy Civil Engineering 2006 j.. I /
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This thesis for the Doctor of Philosophy degree by Mohammad AbuHassan has been approved by HonYimKo Brian T. Brady ShingChun Trever Wang Ronald A L. Rorrer Date
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Mohammad AbuHassan (Ph.D., Civil Engineering) Anisotropic Elastic Modeling of Reinforced Soils Thesis directed by Professor NienYin Chang ABSTRACT Numerical modeling is a powerful tool for studying the static and the dynamic responses of different types of geostructures such as MSE foundations, walls, and bridge abutments. It imposes no size limitation; it is speedy; and it is cost effective. However, numerical modeling must be calibrated and validated by determining the correct material parameters and comparing the results of the numerical model with those of carefully performed physical modeling. When proven effective, numerical modeling can be relied on heavily for further explorations of geostrucutre behaviors. An accurate numerical analysis of reinforced soil structures must account for frictional sliding between the soils and the reinforcing elements. The tensile stresses that are built up in reinforcing layers, due to friction, are transferred to soil's layers, which compensates for their lack of tensile resistance. After all, it is a composite material that exhibits the best qualities of both of its constituents, the compressive resistance from soil and the tensile resistance from geosynthetics. When modeling MSE structures, using finite element method, difficulties. in forming the finite element mesh and time consuming while preprocessing and CPU processing cannot be avoided, especially for a large problem. To alleviate
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the interface convergence problems and speed up the computation one can treat the geocomposite as an anisotropic homogeneous material, instead. This homogeneous material must account for constituents' properties, which includes their interface frictional interaction. Therefore the main goal of this research is to develop a constitutive model that predicts the mechanical properties of reinforced soil, while taking into account the soil properties the reinforcing properties the vertical spacing between reinforcing layers the confining pressure, and the frictional interface. The constitutive model which can best account for the above said characteristic is identified as "transversely isotropic linear elastic model". The transversely isotropic models were formulated through statistical modeling of the results from finite element modeling of the behavior of the composite cube. These transversely isotropic models were found to be effective in simulating the linear elastic resposnses of laboratory samples and full geostrucutres, including full scale MSE foundations, MSE walls, and MSE abutments. This abstract accurately represents the content of the candidate's thesis. I recommend its publication.
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DEDICATION This thesis is dedicated to my loving parents, Dr. Eng. Abdelfattah AbuHassan and Fatek Abu: Hassan, who continuously give me the unlimited support in achieving all my goals.
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ACKNOWLEDGMENT Thank you GOD ... I would like to begin by expressing my gratitude to my Professor, NienYin Chang, chair of this thesis committee who gave me full support and help. Prof. Chang provided all the possibilities to complete this project including this fmal thesis. Working with Prof. Chang was an experience that helped me gain broader perspective on many areas. I also wish to thank the members of my committee, Prof. HonYim Ko, Prof. Yunpin Xi, Prof. Trever Wang, Prof. Brian Brady, and Prof. Ronald Rorrer for their helpful comments and suggestions. I want to also thank ZehZon Lee for his vast knowledge in MSE walls and their behavior. Thanks also to Jan Chang for his guidance in preparing the triaxial tests, for Russel Cox for his assistance in creating the finite element mesh of the MSE bridge abutment, Cesar Gonzalez for conducting the tensile strength test, Myron Lacome for his experience with AutoCad, Khaled AbuNameh and Dawood Oqlah for editing this thesis, and also like to thank Paola Mera for the incessant encouraging and support Last but not least, I would like to thank my family for their support even though they were thousand of miles away in Jordan. To all, I say thank you for making this overall education experience possible.
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CONTENTS Figures ........................................................................................................................ xvi Tables ......................................................................................................................... xxxii Chapter 1. 1.1 1.2 1.3 1.4 2. 3. 3.1 3.2 3.3 3.3.1 3.3.2 3.3.3 3.3.3.1 3.3.3.2 3.4 3.4.1 3.4.2 Introduction ............................................................................................ } Problem Statement ................................................................................. 1 Research Goals and Objectives .............................................................. 2 Research Methodology and Tasks ......................................................... 3 Significance ofResearch ........................................................................ 4 Literature Review ................................................................................... 5 Theoretical Background ....................................................................... II Introduction .......................................................................................... 11 Finite Element Theory for Linear Materials ........................................ 11 Finite Element Theory for Nonlinear Materials ...... ............................ .16 Tangent Stiffness Method .................................................................... 16 Viscoplastic Method ........................................................................... 17 Modified NewtonRaphson Method .................................................... 17 Explicit Algorithm ............................................................................... 18 Implicit Algorithm ............................................................................... 19 Finite Element Commercial Codes ...................................................... 19 NIKE3D, TRUGRID, and GRIZ ......................................................... 20 LSDYNA, FEMBPC, and LSPRE/POST ......................................... 21 Vll
PAGE 8
3.5 3.5.1 3.5.2 3.5.3 3.5.4 3.5.5 3.6 3.7 4. 4.1 4.2 4.2.1 4.2.2 4.2.3 4.3 4.3.1 4.3.2 4.3.2.1 4.3.2.2 4.4 4.4.1 4.4.2 4.5 5. 5.1 Constitutive Modeling of Soil.. ............................................................ 23 Hyperbola Model ................................................................................. 24 RambergOsgood Model ...................................................................... 26 MohrCoulomb Failure Criterion ........................................................ .27 DruckerPrager Criterion ..................................................................... 28 Cap Models .......................................................................................... 30 Contact in Finite Element .................................................................... 34 Summary .............................................................................................. 36 Anisotropic Properties of GeoComposite ........................................... 3 7 Introduction .......................................................................................... 3 7 Composite Materials ......................................................... : .................. 38 Orthotropic (Three Planes of Symmetry) ........................................... .43 Transversely Isotropic (A plane oflsotropy) ...................................... .44 Isotropic (Complete Symmetry) ......................................................... .46 Evaluation of Composite Material Properties ....................... _. ............. .47 Finite Element Approach in. Determining the Mechanical Properties of a Geocomposite ............................................................ .48 Analytical Method in Determining the Mechanical Properties of a GeoComposite(Jones 1975) ......................................................... 59 !sostrain Model ................................................................................... 60 I sostress Model ................................................................................... 62 Stress Distribution ofReinforcement.. ................................................. 68 Weak Reinforcement ........................................................................... 68 Stiff Reinforcement .............................................................................. 7 4 Summary And Conclusion ........... ........................................................ 79 Statistical Modeling of Transversely Isotropic GeoComposite(Michael H. Kutner 2005) .................................................. 80 Theory of Multiple Regression Analysis ............................. ............... 80 viii
PAGE 9
5.1.1 Analysis of Variance (ANOVA) Table ................................................ 83 5.1.1.1 Test for Regression Relation ................................................................ 85 5 .1.2 Inferences about Regression Parameters ............................................. 86 5.1.2.1 Interval Estimation ofpk ...................................................................... 86 5.1.2.2 Test for pk ............................................................................................ 86 5.1.3 Estimation ofMean Response and Prediction ofNew Observation .................... ; ..................................................................... 88 5.1.3.1 Interval Estimation ofE{Yh} ............................................................ .. 88 5.1.3.2 Prediction ofNew Observation E{Yh} ................................................ 88 5.1.4 Diagnostic and Remedial Measures ..................................................... 89 5.1.5 Building the Regression Model ........................................................... 90 5.1.6 Automatic Search Procedures for Model Selection ............................. 9Q 5 .1.6.1 Best Subset Algorithms ........................................................................ 91 5.1.6.1.1 Coefficient ofDetermination (R2p) ...................................................... 91 5 1.6.1.2 Mallows' Cp Criterion .......................................................................... 92 5.1.6.1.3 Akaike's Information Criterion (AICp) and Schwarz's Bayesian Criterion (SBCp) ...................... ............................................................ 92 5 .1.6.1.4 Prediction Sum of Squares (PRESSp) Criterion .................................. 93 5.1.6.2 Stepwise Regression Methods ........................................................ .... 93 5.1.7 Multicollinearity and Its Effect.. .......................................................... 94 5.2 5.2.1 5.2.1.1 5.2.1.2 5.2.1.3 5.2.1.4 5.2.1.5 Application of Multiple Regression Analysis on GeoComposite .................... : ....................................................................... 94 Plane Modulus of Elasticity (Eh) ........... ; .............................................. 96 Model Selection for Eh ......................................... 99 Diagnostic and Remedial Measure of Eh ........................................... 1 02 Inference about Regression Parameters of Eh .................................... 1 04 Interval Estimation ofE{Yh} ............................................................. 107 Interval Prediction for New Observation Y h(new) ............................... 1 08 lX
PAGE 10
5.2.1.6 5.3 6. 6.1 6.2 6.2.1 6.2.2 6.3 6.3.1 6.3.1.1 6.3.1.2 6.3.1.3 6.3.2 6.3.2.1 6.3.2.2 6.3.3 6.3.3.1 6.3.3.2 6.4 6.4.1 6.4.2 6.4.3 6.5 6.5.1 6.5.2 6.5.3 Alternative Model of Eh ..................................................................... 1 09 Summary and Conclusions : .................... ........................................... 1 09 Drained Triaxial Tests on GeoComposite Samples .......................... 111 Introduction .. ........................................................................... ...... .... 111 Test Materials .............. ...................................................................... 112 Ottawa Sand ................ ...................................................................... 112 Polypropylene Geotextiles ................................................................ 113 Laboratory Tests ......................... ....................................................... 113 Tensile Geotextile ..................... ; ........................................................ 113 Equipments and Test Specimen ............. ... ......................................... 114 Preparation ......... .... ............................................................................ 114 Results and Discussion ...................................................................... l14 Hydrostatic Compression ............................................................ ...... 117 Sample Preparation ............................................................................ 118 Results Discussion of Hydrostatic Compression Test ......... ... ... ....... 134 Conventional Triaxial Compression Tests .. ....................................... l36 Samples Preparation ........ ..................... ................................ ............. 13 7 Results and Discussion of Drained Triaxial Tests ............................. 138 Finite Element Calibration ................................................................. 146 Ottawa Sand ....................................................................................... 14 7 Reinforcement Properties ( Geotextile) ...... ... .......... .......................... 156 Bottom and Top Cap (Steel) ............................................................. .159 Finite Element Simulation and Validation ........................................ .! 59 Tensiie Test on Geotextile .............. ................................................... 160 Drained Triaxial Test on Ottawa Sand ............................................... 161 Drained Triaxial Test on Reinforced Samples with Discrete Approach ............................. .. ...... ........ ..... .... .. .... .. . ... .......... ............ l65 X
PAGE 11
6.5.4 6.5.5 6.6 7. 7.1 7.2 7.2.1 7.2.1.1 7.2.1.1.1 Drained Triaxial Test on Reinforced Samples with Homogeneous Material Approach ..................................................... 171 Results Summary ofDiscrete and Homogeneous Models ................. 178 Summary and Conclusion .................................................................. 183 Validation of Finite Element Method on GeoComposite ................ .185 Introduction ........................................................................................ 185 Triaxial Test (Liu 1987) ..... ........................ ....................................... 186 Laboratory Test .................................................................................. 186 Materials Preparation ........................................................ .' ................ 186 Soil ............ .......................................................................... .............. 186 7 .2.1.1.2 Reinforcement ... .................. ............... . ........ ................................... 187 7 .2.1.2 Samples Patterns ........ ....................................................................... 189 7.2.1.3 Test Results ........................................... ............................................. 191 7.2.2 7 2.2.1 7.2.2.2 7.2.2 3 7.2.2.3.1 Finite Element Analysis ofUnReinforced and Reinforced Samples using Discrete Models ... ............................................. ........ 193 Geometries of Specimens ............. ................. .......................... ....... 194 Test Simulation ........................................... ." ..................... ................ 195 Materials Parameters ............................ ............................................. 195 Sandy Soil Parameters ....................................................................... 196 7 .2.2 3 .2 Reinforcement Parameters . ....... ... . . ... ....... . ........ . ...... ......... ....... 198 7.2.2.4 7.2.3 7.2.3 1 7.2.3.2 7.3 7.3.1 7.3.2 Results of the Finite Element Analysis using Discrete Models ........ 199 Finite Element Analysis of Reinforced Samples using Homogeneous Model ................... ........................... ........ ................ 208 Equivalent Properties . ... ..... .... ................... .... ..... . . . ... .... ....... ... .... ... 208 Results of Finite Element Model using Homogeneous Models ......... Plane Strain Testing (Kanop Ketchart 2001) ..................................... 216 Background .................................. ............................................. ....... 216 Conventional Compression Triaxial Tests ................................ ... ... ... 217 Xl
PAGE 12
7.3.2.1 7.3.2.2 7.3.3 7.3.3.3 7.4 8. 8.1 8.2 8.2.1 8.2.2 8.3 8.3.1 8.3.1.1 8.3.1.2 8.3.1.3 8.3.2 8.4 8.4.1 8.4.2 8.4.2.1 8.4.2.2 8.4.2.3 8.4.2.4 8.4.3 Laboratory Results ............................................................................. 217 Finite Element Results ....................................................................... 219 SGP Tests ........................................................................................... 224 Finite Element Results using Homogeneous Model .......................... 231 Summary and Conclusions ............................................................... .234 ThreeDimensional Foundation with Soil Reinforcement.. ............... 238 Introduction ........................................................................................ 23 8 Concept and Design Consideration ofF oundation(McCarthy 1988) .................................................................................................. 239 Unreinforced Foundation Soil .......................................................... 239 Foundation with Soil Reinforcement.. .... ..... .................................... .247 Finite Element Analysis of Square Footing Supported by UnReinforced and Reinforced Soil ......................................................... 254 Material Properties ............................................................................. 257 Ottawa Sand ....................................................................................... 25 8 Geotextile ........................................................................................... 260 Concrete ............................................................................................. 261 Finite Element Modeling .................................................................. .261 Results and Discussion ...................................................................... 262 Results ofFoundation without Soil Reinforcement .......................... .262 Results of Foundation with Soil Reinforcement.. .............................. 265 Effect of Spacing ....... ......................................................................... 266 Effect of Geosynthetic Stiffness ........................................................ 269 Summary of Spacing and Stiffness Effects ........................................ 271 Stress Distribution on Reinforcement Layers .................................... 277 Results ofF oundations with Reinforced Soil Modeled as Homogeneous Material ...................................................................... 288 8.5 Summary and Conclusions ................................................................ 294 Xll
PAGE 13
9. Static and Dynamic Analysis ofMSE Wall with Rigid Facing ........ .297 9.1 Introduction ................................................ .' ....................................... 297 9.2 Concept and Design Consideration of Retaining Walls .................... 298 9.2.1 Retaining Walls with Geosynthetic Reinforcement (Vector Elias 2001) ......................................................................................... 299 9.2.1.1 External Stability of Vertical MSE Walls and Horizontal Backfill ............................................................................................... 3 02 9 .2.1.1.1 Sliding Stability ................................................................................. 305 9.2.1.1.2 Overturning Stability ......................................................................... 305 9.2.1.1.3 Bearing Capacity Failure ................................................................... 306 9.2.1.2 Internal Stability ofMSE walls .......................................................... 309 9.2.2 Hybrid Retaining Walls ..................................................................... 313 9.3 9.3.1 9.3.2 9.3.2.1 9.3.2.2 9.3.2.3 9.3.3 9.3.4 9.3.5 9.3.5.2 9.3.5.3 9.3.5.4 9.4 9.4.1 9.4.1.1 Finite Element Analysis on MSEWall with Rigid Facing ................ 314 Loading .............................................................................................. 31 7 Material Properties ............................................................................. 318 Backfill and Foundation Soil ............................................ ................ 318 Inclusion ............................................................................................. 320 Concrete Wall and Footing ............................................................... .320 Sliding Interface ................................................................................. 322 Boundary Conditions ....................................... : External and Internal Stability ofMSE wall ...................................... 323 External Stability due to Static Loading ............................................ 324 Internal Stability due to Static Loading ............................................. 326 External Stability due to Seismic Loading ......................................... 329 Internal Stability due to Seismic Loading .......................................... 332 Results and Discussion ...................................................................... 336 Results of Discrete Model.. ................................................................ 337 Lateral Earth Pressure (crx) behind the Hybrid Wall .......................... 337 Xlll
PAGE 14
9.4.1.2 9.4.1.3 9.4.1.4 9.4.2 9.4.2.1 9.4.2.2 9.4.2.3 9.5 10. 10.1 10.2 10.3 10.3.1 10.3.2 10.3.2.1 10.3.2.2 10.3.2.3 10.3 3 10.3.4 10.4 Lateral Wall Displacement (ox) ................................................ : ........ 339 Bearing Pressure (crz) and Settlement (oz) ofFoundation ................ . 341 Inclusion Tensile Stresses .................................................................. 343 Results ofHomogeneous Model ........................................................ 350 Lateral Earth Pressure (crx) behind the Hybrid Wall ......................... .352 Lateral Wall Displacement (ox) ......................................................... 354 Bearing Pressure (crz) and Settlement (oz) ofFoundation .................. 356 Summary and Conclusions ................................................................ 360 Static and Dynamic Analysis ofMSE Bridge Abutments ................. 364 Introduction .............................. ..................................................... .... 364 Design Consideration of Bridge Abutment (FHWA,2001) Finite Element Analysis on Bridge Span Supported by MSE Abutment. ........................................................................................... 368 Loading ::::372 Material Properties .................................................. ........................... 374 Backfill and Reinforced Soil.. ............................................................ 374 Inclusion ............................................................................................. 376 Concrete ..................................... : ....................................................... 376 Sliding Interface ............................................................... ,. ................. 377 Boundary Conditions ......................................................................... 3 77 Results and Discussion ...................................................................... 378 10.4.1 Results ofDiscrete Model ................................................. ................. 378 10.4.1.1 Lateral Pressure of Bridge Abutment ................................................ 379 10.4.1.2 Bearing Pressure and Settlement Distribution ofMSE Backfill beneath the Spread Footing ................................................................ 386 1 0.4.1.3 Longitudinal and Transverse Earth Pressure behind the MSEWall ........... ........................................................................................ 392 XIV
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10.4.1.4 10.4.1.5 10.4.2 10.4.2.1 10.4 2.2 1 0.4.2.3 10.4.2.4 10.5 11. 11.1 11.2 11.3 Appendix A. Longitudinal and Transverse Displacements ofMSEWall ............. .400 Inclusion Tensile Stresses in the MSEWall.. .............................. .... .407 Results of Homogeneous Model ...................................................... .415 Lateral Pressure of Bridge Abutment .............................................. .418 Bearing Pressure and Settlement Distribution of MSE Backfill beneath the Spread footing ................................................................ .425 Longitudinal and Transverse Earth Pressure behind the MSEWall .................. ..................... ..... ....................................................... 430 Longitudinal and Transverse Displacements ofMSEWall ............. .437 Summary and Conclusions ................................................................ 440 Summary, Conclusions and Recommendations ............................... 443 Summary ............................................................................................ 443 Conclusions .................................................................................... : ... 450 Recommendations for Future Studies ............................................... .458 Equivalent Transversely Isotropic Properties of GeoComposites ............................................................. ........................... 460 B. Regression Analysis of Independent Parameters (Ev, vh, Vv, Gv, and Gh) ................................................................ ............................... 471 REFERENCES .................................................................................................... 540 XV
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LIST OF FIGURES Figure 3.1 "Parameters ofMohrCoulomb model" ........................................................... 27 3.2 "DruckerPrager criterion" ............................................................................... 29 3.3 "Cap model" ..................................................................................................... 31 3.4 "Interpretation ofparameters off1 32 4.1 "Deviator stresses on cube element" .............................................................. .42 4.2 "Transversely isotropic material (23) plane of isotropy)" ............................. .45 4.3 "Geocomposite cube element (one, three, and five Layers)" ........................ .49 4.4 "Es (MPa) distribution under different cro" ...................................................... 51 4.5 "Theoretical frame of all numerical tests" ... ................................................... 52 4.6 "Load curve distributed for all3 loads; gravity, confining pressure, and deviator stress" ....................................................... ....................................... 53 4.7 "Parallel (!sostrain) Model" ............................................................................ 60 4.8 "Serial (!sostress) Model ................................................................................. 62 4.9 "Xstress distribution for geosynthetic withE= 22 MPa" .............................. 70 4.10 uy stress distribution for a geosynthetic with E = 22 MPa11 70 4.11 "Zstress distribution (Pa) for a geosynthetic withE= 22 MPa" .................. 71 4.12 "Plane shear stress, XY, distribution (Pa) for a geosynthetic withE= 22 MPa" .... ......................................................................... ........................... 71 4.13 ''Vertical shear stress, YZ, distribution (Pa) for a geosynthetic withE = 22 MPa11 72 4.14 "Vertical shear stress, XZ, distribution (Pa) for a geosynthetic withE = 22 MPa" ...................................................................................................... 72 XVI
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4.15 "Xstress distribution (Pa) for a geosynthetic withE= 1100 MPa" ................. 75 4.16 "Ystress distribution (Pa) for a geosynthetic withE= 1100 MPa" .............. 75 4.17 "Zstress distribution (Pa) for a geosynthetic withE= 1100 MPa" .............. 76 4.18 "Plane shear stress, XY, distribution (Pa) for a geosynthetic withE = 1100 MPa" ..................................................................................................... 76 4.19 "Vertical shear stress, YZ, distribution (Pa) for a geosynthetic withE = 1100 MPa" .................................................................................................. 77 4.20 "Vertical shear stress, XZ, distribution (Pa) for a geosynthetic withE = 1100 MPa" ............ ...................................................................................... 77 5.1 "The regression analysis framework" ............................................................. 95 5.2 "The matrix plot ofEh with respect to all X variables" ................................... 97 5.3 "The residual versus the fitted values ofEh" .................................................. 102 5.4 "N oimal probability plot of the residuals of Eh" ............................................ 1 03 6.1 "Geotextile Tensile Tests" ................................................. : .................... ...... 117 6.2 "Triaxial Cell" ................................................................................................ 120 6.3 "Essential materials for hydrostatic and triaxial tests" .................................. 120 6.4 "Measuring devices for hydrostatic and triaxial Tests" ................................. 121 6.5 "Ottawa Sand, raining device and funnel" ..................................................... 121 6.6 "Grease, rubber cement, screw driver and brush" .......................................... 122 6.7 "Vacuum Device" .......................................................................................... 122 6.8 "Initial height measurement" ................. ................................. : .................... .123 6.9 "Thickness of membrane measurement" ....................................................... 124 6.10 "Pulling the membrane over the bottom cap of the sample" ....................... 125 6.11 "Assembling the split mold around the bottom cap and wrapping the membrane around the mold" ........................................................................ 126 6.12 "Applying vacuum between the split mold and the membrane" .................. 127 6.13 "Ad<;ling the soil the using a raining device and a funnel" ........................... 127 6.14 "Lifting the raining device slowly" .............................................................. 128 xvii
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6.15 "Tamping the sample gently" ....................................................................... 128 6.16 "Placing and leveling the top cap" .............................................................. .129 6.17 "Applying vacuum pressure to the top of the sample" ................................. 129 6.18 "Measurement of sample's height" .............................................................. 131 6.19 "Measurement of sample's Diameter" ......................................................... 131 6.20 "Adding rubber cement around the porous stone area" ............................... 132 6.21 "Placing a greased 0ring" ........................................................................... 132 6.22 "Adding water through the confining valve" ............................................... 133 6.23 "Confining pressure and volume change" ........................... ........................ 133 6.24 "Isotropic consolidation test results due to Ottawa sand and the triaxial cell" .................................................................................................. 135 6.25 "Hydrostatic compression result of Ottawa sand only" ............................... 13 5 6.26 "Drained triaxial test" ....................... ........................................................... 13 8 6.27 "StressStrain relation at cr3 = 103 kPa" ...................................................... .141 6.28 "StressStrain relation at cr3 = 207 kPa" ...................................................... .141 6.29 "StressStrain relation at cr3 = 310 kPa" ....................................................... 142 6.30 "Peak stress difference as a fucntion of number of reinforcing Layer and confining pressure" ............................................................................... 142 6.31 "Initial Young's Modulus as a function of number of Reinforcing Layer .......................................................................................................... 143 6.32 "Volumetric strainAxial strain relation of Ottawa sand" ........................... 143 6.33 "MohrCoulomb envelope for samples with different reinforcing patterns" ....................................................................................................... 145 6.34 "Relationship between the logarithm of equivalent cohesion and spacing between layers" ............................................................................... 146 6.35 "Geologic Cap model" ............................................ .................................... 6.36 "DruckerPrager criterion" ........................................................................... 148 XVlll
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6.3 7 "Relation between normalized deviator stress and axial strain" .................. 151 6.38 "The shape of first yield surface, fl, for Ottawa sand according to Cap model" .................................................................................................. 152 6.39 "Observed and predicted Vol. strain vs. Mean pressure" ........................... .155 6.40 "Isotropic Elastic Plastic material" ............................................................ 157 6.41 "Tensile force vs. axial strain of the heat bounded geotextile" .................... 161 6.42 "Finite element model of Ottawa sand specimen" ...................................... .162 6.43 "FESimulation vs. measured triaxial results of Ottawa Sand @ cr3 = 103 k:Pa" .............................. ... ........................................................... ... ........ 163 6.44 "FESimulation vs. measured triaxial results of Ottawa Sand @ cr3 = 207 k:Pa" ....................................................................................................... 164 6.45 "FESimulation vs. measured triaxial results of Ottawa Sand @ cr3 = 310 k:Pa" ....................................................................................................... l64 6.46 "Patterns of reinforced samples" : 6.4 7 "FESimulation vs. measured triaxial results of reinforced Sand and 2 layersofreinforcement@ cr3 = 103 k:Pa" .... .............................................. .167 6.48 "FESimulation vs. measured triaxial results of reinforced Sand and 2 layers of reinforcement @ cr3 = 207 k:Pa" ................................................... .167 6.49 "FESimulation vs. measured triaxial results of reinforced Sand and 2 layers of reinforcement @ cr 3 = 31 0 k:Pa" ................................................... .168 6.50 "FESimulation vs. measured triaxial results of reinforced Sand and 4 layers of reinforcement@ cr3 = 103 k:Pa" 6.51 "FESimulation vs. measured triaxial results of reinforced Sand and 4 layers of reinforcement@ cr3 = 207 k:Pa" 6.52 "FESimulation vs. measured triaxial results of reinforced Sand and 4 layers of reinforcement@ cr3 = 310 kPa" .... ... ............ .... .. ,., ............... ...... 169 XlX
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6.53"FESimulation vs. measured triaxial results of reinforced Sand and 6 layers of reinforcement @ cr3 = 103 kPa" ......... : .......................................... 170 6.54 "FESimulation vs. measured triaxial results of reinforced Sand and 6 layers of reinforcement @ cr3 = 207 kPa" .................................................... 170 6.55 "FESimulation vs. measured triaxial results of reinforced Sand and 6 layers of reinforcement @ cr3 = 31 0 kPa" ................................................... .1 71 .6.56 "Triaxial test results of transversely isotropic material vs. the lab and the discrete model of2layers sample@ cr3 = 103 kPa" .............................. 174 6.57 "Triaxial test results of transversely isotropic material vs. the lab and the discrete model of 2 layers sample @ cr3 = 207 kPa" .............................. 17 4 6.58 "Triaxial test results of transversely isotropic material vs. the lab and the discrete model of 2 layers sample @ cr3 = 310 kPa" .............................. 175 6.59 "Triaxial test results of transversely isotropic material vs. the lab and the discrete model of 4 layers sample @ cr3 = 1 03. kPa ...... ....... .............. 17 5 6.60 "Triaxial test results of transversely isotropic material vs. the lab and the discrete model of 4 layers sample @ cr3 = 207 kPa" .............................. 176 6.61 "Triaxial test results of transversely isotropic material vs. the lab and the discrete model of 4 layers sample @ cr3 = 31 OkPa" ...... ........................ 176 6.62 "Triaxial test results of transversely isotropic material vs. the and the discrete model of 6 layers sample @ cr3 = 1 03 kPa" ......... , .............. .. 1 77 .6.63 "Triaxial test results of transversely isotropic material vs. the .lab and the discrete mo
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7.4"StressStrain relationship for samples tested at 103 kPa confining pressure 11 192 7.5 11StressStrain relationship for samples tested at 310 kPa confining Pressure11 : 192 7.6 11MohrCoulomb envelope for samples with different reinforcing pattems11 ....................................................................................................... 193 7. 7 "Models of unreinforced and reinforced soil cylinders using LSDYNA11 194 7.8 relation of Ottawa sand@ 103 kPa confining pressure11 ........ 201 7.9 "StressStrain relation of Ottawa sand @ 310 kPa confining pressure" ........ 202 7.10 "StressStrain relation of Ottawa sand and 1layer of reinforcement@ 103 kPa confming pressure" ........................................................................ 202 7.11 "StressStrain relation of Ottawa sand and 1 layer of reinforcement@ 310 kPa confiDing pressure" ........................................................................ 7.12 "StressStrain relation of soil and 4layers of reinforcement@ 103 kPa confining pressure" ............................................................................... 7.13 "StressStrain relation of Ottawa sand and 4 layers of reinforcement @ 310 kPa confining pressure" ................................................................... .204 7.14 "StressStrain relation of Ottawa sand and 6 layers of reinforcement @ 103 kPa confiDing pressure" .................................................................... 204 7.15 "StressStrain relation of Ottawa sand and 6 layers of @ 310 kPa confining pressure" .................................................................... 205 7.16 "Triaxial test results of transversely isotropic material vs. the lab and the discrete model of 1 layer sample @ 0'3 = 103 kPa" ................................ 211 7.17 "Triaxial test results of transversely isotropic material vs. the lab .. and the discrete model of 1 layer sample @ 0'3 = 31 0 kPa" ................................ 211 XXI
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7.18 "Triaxial test results of transversely isotropic material vs. the lab and the discrete model of 4layers sample@ cr3 = 103 kPa" .............................. 212 7.19 Triaxial test results of transversely isotropic material vs. the lab and the discrete model of 4 layers sample @ cr3 = 31 0 kPa" .............................. 212 7.20 "Triaxial test results of transversely isotropic material vs. the lab and the discrete model of 6 layers sample @ cr3 = 103 kPa" .............................. 213 7.21 "Triaxial test results of transversely isotropic material vs. the lab and the discrete model of 6 layers sample @ cr3 = 310 kPa" .............................. 213 7.22 "StressStrain results of CTC test on Ottawa sand" ........................ ........... .218 7.23 "Volumetric strain versus axial strain results of CTC tests on Ottawa sand" ............................................................................................................. 218 7.24 "StressStrain relation of soil @ 69 kPa confining pressure measured (Ketchart, 2001) versus results ofFEdiscrete model" ........................ ...... 221 7.25 "StressStrain relation of soil @ 207 kPa confining pressure measured (Ketchart, 2001) versus results of FEdiscrete model" ................ 221 7.26 "StressStrain relation of soil @ 345 kPa confining pressure measured (Ketchart, 2001) versus results of FEdiscrete model" ............................... 222 7.27 "Specimen dimensions of SGP" ................................................................... 225 7.28 "Vertical load versus vertical displacement of unreinforced sample and reinforced sample with Amoco2044" ................................................... 226 7.29 "Layout of unreinforced and reinforced samples using the finite . . element method" .......................................................................................... 227 7.30 "Vertical loaddisplacement relation ofunreinforced sample measured by Ketchart versus discrete finite element model" ..................... .228 7.31 "Vertical loaddisplacement relation of reinforced sample measured by Ketchart versus discrete finite element model" ...................................... 228 xxn
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7.32 "Vertical loaddisplacement relation of reinforced sample measured by Ketchart versus discrete finite element model and homogeneous model" ...... ........................ .................................... ........ ... .................... ....... 232 8.1 "Types of shallow spread footing a) square footing; b) strip footing ; c) rectangular footing; d) trapezoidal footing" .................. .... ............. ... ... .... 240 8.2 "Failure modes when reaching bearing capacity" ..................... ................... .242 8.3 "Modes of bearing failure of reinforced earth" ........ ...................................... 248 8.4 "Non dimensional quantities (A1, A2, and A3) function ofz/B" ............. .... .250 8.5 "Effective length of reinforcement (L0 ) as function ofz/B" .......... .......... ..... 251 8 6 "3Dimenioanl foundation model"................ ................................................ 256 8.7 "Variation of vertical stress beneath a footing based on Boussinesq analysis and Westergaard analysis, respectively" ... .............. ..... ................ 257 8.8 "Depth versus G for Ottawa sand y = 1719 kg/m3 ........ ......................... 259 8.9 "Overburden pressure of footing on top of foundation soil, FE vs. analytical" .................................... .................................. ..... ... ........ .... . .. .. 263 8.10 "Bearing pressure of soil foundation resulted from finite element analysis" .................................................. .... .......... ... ............................ .... 265 8.11 "Bearing pressure of foundation due different spacing when Eg = 160 MPa" .... .... ........... ............................................ .......................... ....... ....... .... 266 8.12" Bearing pressure of foundation due different spacing when Eg = 320 MPa" . ...... ....... ..................... ........ ....... ... .... .......... .............. ................ 267 8.13 "Bearing pressure of foundation due different spacing when Eg = 640 MPa" ... ... ..... ........ ..... ... ........... ........ ........... ........... .......... ................ 267 8.14 "Bearing pressure of foundation due different geosynthetic when S = 1000 mm" ....... .......... ...... .............. ........... ....... .... ... ... ................ 27Q 8.15 "Bearing pressure of foundation due different geosynthetic stiffness when S =500 mm" .......... .......... .......... .......... ...................... ........... ........... 270 xxm
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8.16 Bearing pressure of foundation due different geosynthetic stiffness when S = 250 mm" ....................................................................................... 271 8.17 "crz increase due different reinforcement patterns" ...................................... 272 8.18"Performance based bearing capacity at different spacing when E =160 MPa" ............................................................... ................................... 274 8.19 "Performance based bearing capacity at different spacing when E =320 MPa ................................................................. ................................... 274 8.20 "Performance based bearing capacity at different spacing when E =640 MPa ...................................................... .............................................. 275 8.21 "Contour and 3Dimensional surface plots of horizontal stress distribution for geotextile at depth of 500 mm from the footing" ......... ..... 279 8.22 "Contour and 3Dimensional surface plots of horizontal stress distribution for geotextile at depth of 0.5B from the footing" ..................... 280 8.23 and 3Dimensional surface plots of horizontal stress distribution for geotextile at depth of IB from the footing" ....................... .281 8.24 "Contour and 3Dimensional plots of horizontal distribution for geotextile at depth of 1.5B from the footing" ..................... 282 8.25 "Contour and 3Dimensional surface plots of horizontal stress distribution for geotextile at depth of 2B from the footing" ........................ 283 8.26 "Contour and 3Dimensional surface plots of horizontal stress distribution for geotextile at depth of 0.5B from the footing" ..................... 284 8.27 "Horizontal stress distribution of reinforcement layers at distance away from the footing's centerline, S = 250 mm, Eg 320 MPa" ................. 285 8.28 "Horizontal stress distribution of geosynthetic layers at different depths for foundation on top of reinforced soil, S =1000 mm" ................... 286 8.29 Horizontal stress distribution of geosynthetic layers at different depths for foundation on top of reinforced soil, S =500 mm" ..................... 287 XXlV
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8.30 "Horizontal stress distribution of geosynthetic layers at different depths for foundation on top of reinforced soil, S =250mm" ...................... 287 8.31 "Bearing pressure of foundation with homogeneous reinforced soil for S = 1000 mm" ......................................................................................... 293 8.32 "Bearing pressure of foundation with homogeneous reinforced soil for S = 500 mm" ........................................................................................... 293 8.33 "Bearing pressure of foundation with homogeneous reinforced soil for S = 250 mm" ........................................................................................... 294 9.1 "Retaining wall external stability; a) overturning, b) sliding, c) bearing capacity d) deep seated ................................................................................ 299 9.2 "External stability ofMSE wall" .................................................................. .303 9.3 "External loads on vertical MSE wall with horizontal backfill due to weight and surcharge" .......... : ................................................... ; ..................... 304 9.4 "External loads on vertical MSE wall due to seismic loading" .................... .309 9.5 "Inclusion stress distribution along the height of the wall" ........................... 311 9.6 "Side view ofMSEWall" .............................................................................. 316 9.7 "ElCentro Earthquake acceleration time history" .......................................... 318 9.8 "External load on MSE Wall model" ............................................................. 324 9.9 "Lateral earth pressure (crx) ofbackfill along the hybrid retaining wall" ...... 339 r 9.10 "Lateral displacements of hybrid retaining wall due static and dynamic loads" ............................................................................................. 340 9.11 "Bearing pressure (crz) of foundation soil due to static and dynamic load" .............................................................. .............................................. 342 9.12 (8z) of foundation soil due to static and dynamic load" ........... 343 9.13 "Tensile stress of reinforcements at distance 0 m from the wall due static and dynamic loadings'' ............ ............. ..................... : ...................... 346 XXV
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9.14 "Tensile stress of reinforcements at distance 1.5 m from the wall due static and dynamic loadings" ...................................................................... .346 9.15 "Tensile stress of reinforcements at distance 3.75 m from the wall due static and dynamic loadings" ................................................................. 347 9.16 "Tensile stress of reinforcements at distance 7 m from the wall due static and dynamic loadings" ....................................................................... 347 9.17 "Summary of tensile stresses of reinforcements due to static loadings" ...................................................................................................... 349 9.18 "Summary of tensile stresses of reinforcements due to dynamic loadings" ...................................................................................................... 349 9.19 "Lateral earth pressure ( crx) of homogeneous model due to static loading" ........................................................................................................ 353 9.20 pressure (crx) ofhomogeneous model due to dynamicloading" ....... 353 9.21 "Lateral wall displacements (8x) of homogeneous model due static loading" ........................................................................................ ; ............... 355 9.22 wall displacements (8x) of homogeneous model due dynamic ... 9.23 "Bearing pressure ( crz) of foundation soil in homogeneous model due to static load" ............................................................................................... 357 9.24 "Bearing pressure ( crz) of foundation soil in homogeneous model due to dynamic load" .......................................................................................... 358 9.25 "Settlement (8z) of foundation soil in homogeneous model due to static load". ................. ........................................ .......................................... 359 9.26 "Settlement (8z) of foundation soil in homogeneous model due to dynamic load" ........... .......................... ................................... .... .............. 359 10.1 "Side view ofMSE abutments" ................................................ ................. 366 XXVl
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10.2 "MSE wall facing; a) geosynthetics warp, b) segment concrete block, c) full height panel" ............. : ... ..................................................................... 366 10.3 "Isometric view ofMSE bridge abutment" .................................................. 371 10.4 "Northridge Transverse (Y) horizontal acceleration time history" (Berkeley) ....................................... ........ .......... ........................................ 373 10.5 "Northridge Longitudinal (X) horizontal acceleration time history" (Berkeley) : 10.6 "Longitudinal earth pressure of abutment along the centerline" ....... .......... 380 10.7 "Contour and 3Dimenensional surface plots of longitudinal earth pressure of the bridge abutment on the Stemwall due static loading" ........ 381 10.8 and 3Dimensional surface plots of longitudinal earth pressure of the bridge abutment on the Stemwall due dynamic loading" ... ..................................................................................................... 382 10.9 "Lateral earth pressure of abutment against the wing wall along the edge ......................................... ................................................................... 383 10.10 "Contour plot of transverse earth pressure of the bridge abutment. on the wingwall due static loading" ................................................................. 1 0.11 "Contour plot of transverse earth pressure of the bridge abutment on the wingwall due dynamic loading" ........................................................... 385 10.12 "Contour and 3Demensional plots of MSE backfill bearing pressure due static loading" ...................................... ................................................. 388 10.13 "Contour and 3Demensional plots ofMSE backfill bearing pressure due static loading due dynamic loading" ....................................... 389 10.14 "Contour and 3bemensional plots ofMSE backfill settlement due static loading" ...................... : ....................................................................... 390 10.15 "Contour and 3Demensional plots ofMSE backfill settlement due dynamic loading" ......................................................................................... 391 10.16 "Longitudinal horizontal stress (ox) of soil behind the MSE wall" .......... 3 94 xxvn
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10.17 "Transverse horizontal stress (cry) of soil behind the MSE wall" .............. 395 10.18 Contour and 3Dimensional plots of longitudinal earth pressure on the longitudinal side ofthe MSEWall due static loading" ........................ .396 10.19 "Contour and 3Dimensional plots of longitudinal earth pressure on the longitudinal side ofthe MSEWall due dynamic loading" .................... 397 10.20" Contour and 3Dimensional plots of transverse earth pressure on the transverse side of the MSEWall due static loading" ............................. 398 10.21 "Contour and 3Dimensional plots of transverse earth pressure on the transverse side ofthe MSEWall due dynamic loading" ....................... 399 10.22 "Longitudinal displacements ofMSE wall due to static and dynamic loadings" ...................................................................................................... 401 10.23 "Transverse displacements ofMSE wall due to static and dynamic loadings" ....................................................................... ,. .............................. 401 10.24 "Contour and 3Dimensional surface plots oflongitudinal wall displacement due static loading" ............................................ ,. .................... .403 10.25 "Contour and surface plots oflongitudinal wall displacement due dynamic loading" ........................................................... .404 10.26 "Contour and 3Dimensional surface plots of transverse wall displacement due static loading" ................................................................. .405 10.27 "Contour and 3Dimensional surface plots of transverse wall displacement due dynamic loading" ........................................................... .406 10.28 "Contour plot of longitudinal stresses ofbottom reinforcing layer due to static loading" .................................................................................... 409 10.29 "Contour plot oflongitudinal stresses ofbottom reinforcing layer due to dynamic loading" .............................................................................. 409 10.30 "Contour plot of transverse stresses of bottom reinforcing layer due to static loading" .......................................................................................... 410 xxvm
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10.31 "Contour plot of transverse stresses of bottom reinforcing layer due to dynamic loading" ............... ........ . ......... .............. . ...... ........ .... . ........... 410 10.32 "Contour plot oflongitudinal stresses of middle reinforcing layer H=2.4m, due to static loading" ...... . ........................... .............................. .411 10.33 "Contour plot of longitudinal stresses of middle reinforcing layer, H=2.4m, due to dynamic loading" ............................. .......... ..................... .411 10.34 "Contour plot oftransverse stresses of middle reinforcing layer, H=2.4m, due to static loading" ....... ... : ......................... : .............................. .412 10.35 "Contour plot of transverse stresses of middle reinforcing layer, H = 2.4m, due to dynamic loading" ............................. ........ ....................... .412 10.36 "Contour plot of longitudinal stresses of top reinforcing layer, H=4.3m, due to static loading" ................................ .... .......... .......... ......... .413 10. 37 "Contour plot oflongitudinal stresses oftop reinforcing laye,r, H=4.3m, due to dynamic loading" ......................... ... ........ ............ ... ...... .413 '10. 38 "Contour plot of transverse stresses oftop reinforcing layer, H = 4.3m, due to static loading" ... . . ......................... ... ... ...... .................... .414 10.39 "Contour plot oftransverse stresses oftop reinforcing layer, H=4.3m, due to dynamic loading" .............................................................. .414 10.40 "Longitudinal earth pressure of abutment along the centerline of homogeneous model vs. discrete model" ..................... ....... ..................... .420 10.41 "Contour plot of longitudinal earth pressure of the bridge abutment on the Stemwall of homogeneous model due static loading" .................... .421 10.42 "Contour plot earth pressure of the bridge abutment on the Stemwall of homogeneous model due dynamic loading" .............. .422 10.43 "Contour plot of transverse earth pressure of the bridge abutment on the wingwall ofhomogeneous mode due static loading" .......................... .423 10.44 "Contour plot of transverse earth pressure of the bridge abutment on the wingwall of homogeneous mode due dynamic loading" .................... .424 XXIX
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10.45 "Contour and 3Demensional plots ofMSE backfill bearing pressure of homogeneous mode due static loading" ................................................. .426 10.46 "Contour and 3Demensional plots ofMSE backfill bearing pressure ofhomogeneous mode due dynamic loading" ............................................ .427 10.47"Contour plot ofMSE backfill settlement ofhomogeneous model due static loading" ........................................................................................ 429 10.48 "Contour plot ofMSE backfill settlement of homogeneous model due dynamic loading" .................................................................................. 429 10.49 "Longitudinal horizontal stress (crx) along the centerline of the MSE wall of discrete and homogeneous model due to static loading" ................ .433 10.50 "Longitudinal horizontal stress (crx) along the centerline ofthe MSE wall of discrete and homogeneous model due to dynamic loading" ........... .433 10.51 Longitudinal horizontal stress ( crx) of soil along the edge of the MSE wall of discrete and homogeneous model due static loading" ........... .434 10.52 "Longitudinal horizontal stress ( crx) of soil along .the edge of the MSE wall of discrete and homogeneous model due to dynamic loading" ............................ .......... 10.53 "Transverse horizontal stress (cry) of soil along the ofthe. MSE wall of discrete and homogeneous model due to static loading" ..... . .435 10.54 "Transverse horizontal stress (cry) of soil.along the of the MSE wall of discrete and homogeneous model to dynamic loading" .......................................................................... .......... ;.:' 10.55 "Transverse horizontal stress (cry) of soil along the e dge of the MSE wall of discrete and homogeneous model due to static loading" .. :.436 1 p.56 "Transverse horizontal stress (cry) .of soil along the edge of the MSE .. ' .. . .. wall of discrete and homogeneous model due to dynamic loading" ........... .436 ' XXX
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10.57 "Longitudinal displacement (8x) ofMSE wall along the centerline of discrete and homogeneous model due static loading" ............................ .439 10.58 "Transverse displacement (8y) ofMSE wall along the centerline of discrete and homogeneous model due static loading" ................................ .439 B.1 "The matrix plot of vh with respect to all X variables" ................................ 4 73 B.2 ''The residual versus the fitted values of vh" ................................................ .485 B.3 "Normal probability plot of the residuals ofvh" .......................................... .486 B.4 "The matrix plot ofEv with respect to all X variables" ................................ .488 B.5 "Residual versus the fitted values ofEv" ...................................................... .494 B.6 "The residual plots ofEv against each variable"" ........................................ 495 B.7 "Normal probability plot of the residuals ofEv" ...................... : .................... 496 B.8 "The matrix plot ofvv with respect to all X ............. ............... B.9 "The residual versus the fitted values ofvv" ........ ......................................... 513 B.1 0 "Normal probability plot of the residuals of vv" ................. ....................... 514 B.11 "Matrix plot of Gv with respect to all X variables" ............. ........................ 516 B.l2 versus the fitted value ofGv" ........................ .. .... : ......... ....... 525 B.l3 "Normal probability plot ofthe residuals .. : ............... : ............... : .... 526 B.14 "Matrix plot of Gh with respect to all X variables" ..................................... 528 . . . . B.l5 "The residual versus the fitted values of Gh" .............................................. 538 B.16 "Normal probability plot of the residuals ofGh" ......................................... 539 ' . XXXI
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LISTS OF TABLES Table 4.1 "Compliance matrix components [C]" ............................................................ .40 4.2 "Soil and geosynthetic properties" .................................................................. .48 4.3 '"Strain components of geocomposite with 1 layer inclusion" ........................ 54 4.4 "[C] for geocomposite with I layer inclusion" ............................................... 54 4.5 "Strain components for geocomposite with 3 layers inclusions" ................... 56 4.6 "[C] for geocomposite with 3 layers inclusion" ............................................. 56 4. 7 "Strain components of geocomposite with 5 layers inclusion" ..................... 57 4.8 "[C] for geocomposite with 5 layers inclusion" ............................................. 58 4.9 "E and E' using finite element and analytical method" .................................... 67 4.10 "Properties and applied loadings of the cube element" .................................. 68 4.11 "Stress components ofweak reinforcement" ................................................. 73 . 4.12 "Stress components of strong reinforcement" ................................................ 78 5.1 "ANOVA Table for general linear regression model" ..................................... 85 5.2 "The correlation matrix ofEh and all X variables" .......................................... 98 5.3 "MINITAB output for Best two subsets ofEh model for each subset size" ........................ .......... .............................................................................. 99 5.4 "MINIT AB F orwardlbackward stepwise regression output o f Eh variable" ....................................................................................................... 100 5.5 OVA Table for Eh regression model" .................................................... 1 01 5.6 "ANOVA Table for alternative mode ofEh" ................................................. 109 5.7 "Summary of constitutive models equations for transversely isotropic ............................................................................................ 11.0 xxxii
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6.1 11Summary of the geotextile tensile test results11 ................... ..................... 116 6.2 "A summary of Ottawa sand shear failure data" ........................................... .152 6.3 11Parameters of Cap model for Ottawa Sand" ...... .................................... ... 156 6.4 11Summary of the geotextile properties" ......................... ............................... 159 6.5 "Summary of steel caps ............................................................... 159 6.6 "The Transversely Isotropic Properties ofReinforced Samples11 .173 6. 7 "Young's modulus and normalized Young's modulus of soil specimens with 2 layers of reinfrocement"EF .................... ............... .................... ..... 180 6.8 "Young's modulus and normalized Young's modulus of soil specimens with 4 layers of reinforcement" ....................... ........................................... 181 6.9 11Young's modulus and normalized Young's modulus of soil specimens with 6 layers of reinforcement" ........................................................ .... ..... 182 7.1 "Typical Physical Properties ofBidim C34 Engineering Fabric11 .188 7.2 "The Cap model properties of Ottawa sand" ............... ........ .......... .... ..... . .198 7.3 "Young's modulus and normalized Young's modulus of soil specimens" ...................................... ................ ....... ................................... 206 7.4 "Young's modulus and normalized Young's modulus of soil specimens with 1 layer of reinforcement" ........ ..................................... ...................... 206 7.5 "Young's modulus and normalized Young's modulus of soil specimens . with 4 layers of .......................................... ........... .... . ..... 207 7.6 "Young's modulus and normalized Young's modulus of soil specimens r with 6 layers of reinforcement11 207 7.7 "The Transversely Isotropic Properties ofReinforced Samples11 209 7.8 "Young's modulus and normalized Young's modulus of soil specimens with 1 layer of reinforcement" ... ...... .............. ... . ..... ................................. 214 7.9 "Young's modulus and normalized Young's modulus of soil specimens with 4 layers of reinforcement11 214 XXXlll
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7.10 "Young's modulus and normalized Young's modulus of soil specimens with 6 layers of reinforcement" .................................................. 215 7.11 "The Cap model properties of Ottawa sandKetchart samples" .................. 220 7.12 "Normalized Young's Modulus of Finite element method compared with Experiment" ......................................................................................... 223 7.13 "Normalized Young's Modulus of Finite element method compared with Experiment, plane strain test on unreinforced sample" ...................... 230 7.14 "Equivalent properties ofSGP consisting of Ottawa sand and 3 layers of Amoco 2044 geotextile" .......................................................................... 231 7.15 "Young's modulus and normalized Young's modulus of homogeneous mode from simulating plane strain test" ............................... 233 8.1 "Ottawa sand properties for Cap model" ....................................................... 260 8.2 "Geosynthetic properties for Elasticplastic" ................................................. 261 . "Concrete properties, Elastic" (Boresi 2003) ................................................. 261 8.4 "Terzaghi's corresponding settlement of reinforced soil models" ................. 268 8.5 "Bearing capacity of reinforced soil models corresponding to Terzaghi's settlement" ....................................... _. ............ ............................ 269 8.6 "BCR due to different reinforcement spacing and geosynthetic stiffness" .......................... ............................................................................ 273 8.7 "Homogeneous properties for 1m spacing and Eg =320 MPa" ..................... 289 8.8 "Homogeneous properties for 0.5m spacing and Eg =320 MPa" .................. 290 8.9 "Homogeneous properties for 0.25m spacing and Eg =320 MPa" ............... .291 8.10 "Terzaghi's corresponding settlement of reinforced soil models; homogeneous vs. discrete" ........................................................................... 292 8.11 "Bearing capacity of reinforced soil models corresponding to Terzaghi's settlement; homogeneous vs. discrete" ..................... ................ 292 9.1 "Load curves summary" ................................................................................. 317 9.2 "Properties of soil for MSE wall" ................................ ... ............................... 321 XXXIV
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9.3 "Properties of inclusion for MSE wall" ......................................................... 321 9.4 "Properties of Concrete wall ofMSE wall" ........................ ............. ........... .321 9.5 "Factor of safety against reinforcement rupture at different elevation due to static loading" .................................................................................... 327 9.6 "Internal stability with respect to pullout failure due to static loading" ........ 329 9.7 "Factor of safety against reinforcement rupture at different elevation due to dynamic loading" .............................................................................. 335 9.8 "Factor of safety against reinforcement pullout at different elevation due to dynamic loading" ... ... .......... ...... .................. ...................... .... ......... 336 9.9 "Mechanical properties of the reinforced soil composite" ............................. 351 9.10 "Maximum bearing pressure of soil beneath footing" ................................ 10,1 "Dimensions ofMSE abutment" .................................................................. 370 10.2 "Northridge Earthquake information" .......................................................... 374 10.3 "Properties of Ottawa sand along different elevations" ............................... 375 1 0.4 "Reinforcement properties in MSE abutment and backfi ll soil" .................. 3 7 6 10.5 "Concrete properties" ................................................................................... 376 10.6 "Equivalent properties oflower zone of the MSEWall" ........................... .416 10.7 "Equivalent properties of middle zone of the MSEWall" ........................ :.416 10.8 "Equivalent properties oftop zone of the MSEWall" ............................... .417 10.9 "Equivalent properties of lower zone of the abutment" .............................. .417 10.10 "Equivalent properties of top zone of the abutment" ................................ .418 10.11 "Range oflateral earth pressure (kPa) using discrete and homogeneous approaches" ........................................................................... 432 1 0. q "The regression equations oflateral earth pressure" ................................. .4 3 7 B.1 "The correlation matrix ofEh and all X variables" ...................................... .474 B.2 "output for Best two subsets for each subset size ofvh" .............................. .475 B.3 "MINITAB forward/backward stepwise regression output ofvh" ............... .477 XXXV
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B.4 "ANOVA Table for vh regression model" ....... ............................................ .478 B.5 "ANOV A Table for vh alternative model" ................................................... .487 B.6 "The correlation matrix ofEv and all X variables" ...................................... .489 B. 7 "MINIT AB output for Best two subset results for each subset size of Ev" .................. ............................................................................................. 490 B.8 "MINITAB forward/backward stepwise regression output ofEv" ............... .492 B.9 "ANOV A Table for Ev regression model" ................................................... .493 B.lO "ANOV A Table for Ev regression model when dropping the S term" ........ 502 B.11 "ANOV A Table for alternative model of vh" .............................................. 503 B.12 "The correlation matrix ofvv with all X variables" .................................... 505 B.lJ "MINITAB output for best two subsets for each subset size ofvv" B.14 "MINITAB forward/backward stepwise regression output ofyv" .............. 507 B.15 "ANOV A Table for Vv regression mode" B.16 "ANOVA Table for alternative model ofvv" .............................................. 515 B.17 "The correlation matrix of Gv and all X variables" .................................... .517 B.18 "MINIT AB out put for best two subsets for each subset size of Gv" .......... 518 B.19 "MINIT AN forward/backward stepwise regression output of Gv" ...... : ...... ? 19 B.20 "ANOV A Table for Gv regression model" ...................... ; .. ;: .......... .;;.: ......... 520 B.21 "ANOVA Table for alternative model ofGv" .......................... ,.,., ...... ,., .... 526 B.22 ''Correlation matrix ofGh and allX variables" .. : }3.23 "MINIT AB output for best two subsets for each size of Gh" :: B.24 "MINIT AB forward/backward stepwise regression output of Gh" ............. 532 . . . . . B.25 "ANOV A Table for Gh regression ITiodel" : .............................. 533 I ' B.26 "AN OVA Table for alternative model of Gh" ............................................. 539 j:_ :.' 'I,', : . i t xxxvi
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1. Introduction 1.1 Problem Statement In recent years, more and more geosynthetic materials have been introduced as engineering materials and widely applied in earthquake and geotechnical engineering. These materials cause significant modification and improvement in the engineering behavior of soil such as strength, stiffness, and corrosion resistance. The resultant reinforced soils (geocomposites) are in general composite materials that result from combination and optimization of individual constituent materials. They are constructed in a manner that produces a structure of alternating layers of soil and reinforcing elements (Vector Elias 2001 ). Reinforced soils are generally an anisotropic and inhomogeneous. In order to determine the physical and the mechanical properties of this anisotropic material, or laboratory tests need to be conducted. There are several factors that directly affect the behavior of reinforced soil: properties of soil, properties of reinforcing material confining pressure, spacing between inclusion layers, and frictional interface (NienYin Chang 2006). Similar to concrete, soil is relatively weak in tension as compared tci its strength in compression. The effect of soil reinforcing is due to tensile stress builtup in horizontal reinforcing layers transferred to the soil through sliding friction as compressive stresses. An accurate analysis of this geocomposite should consider the sliding interaction between the two components, soil and mclusions. I 1
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If reinforced soil such as in Mechanically Stabilized Earth (MSE) walls is modeled using a finite element code, difficulties and time consuming models can not be avoided when defining the sliding interface properties between the two materials. Therefore, there is a need for defining a constitutive model of reinforced soil that will eliminate the use of interface friction between reinforcement and soil layers. 1.2 Research Goals and Objectives Reinforced soil is considered as geecomposite material that consists of soil and reinforcing material with a slippage interface and subjected to a confil).ing . pressure. This composite is an anisotropic. The main goal of this research is to develop a constitutive model of this geocomposite anq use it in predicting the behavi9r of earth structures such as in MSE walls The main objectives of this research are the following: 1Find material type and properties of soil reinforced with horizontal layers of reinforcing element using a finite element approach. 2Develop a constitutive model of reinforced soil that predicts the material properties under different applied conditions . 3Obtain material properties from sets of laboratory tests. 4Validate the results of finite element analysis 5Investigate the effectiveness of this model by building a reinforced foundation soil, a MSE wall, and MSE bridge abutment. Discrete and homogeneous models will be used for the analysis. ( 2
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1.3 Research Methodology and Tasks The goal of this research is to develop a constitutive model of reinforced soil. This goal will be achieved using numerical and physical testing methodologies following the subsequent tasks: 1Perform a finite element analysis using NIKE3D code on cubes of reinforced soil with 1 m x 1 m x 1 m soil dimensions, and equally spaced inclusion elements. In this analysis, all cubes will be subjected to their own gravity load followed by confining pressure and deviator stresses under different applied conditions of spacing, properties of soil, properties of inclusion, and friction coefficient. For each case, a total of six tests will be applied on each specimen; three normal and three shear stresses, to determine the compliance matrix. From tlle resultant compliance matrix, the material type and properties can be determined. The analysis will be completed using geosynthetic materials as reinforcement. 2a statistical analysis on the extracted data from the finite element analysis. The statistical analysis will correlate the strongly related parameters in linear equations forming the constitutive model of reinforced soil. 3In the laboratory, perform sets of triaxial tests on unreinforced and reinforced soil samples, determine the material properties, and compare them to those resulted from finite element analysis. Once .calibration and validation are achieved, further studies on different applications will be preceded. 4Perform further validations by comparing the results from finite element method with those obtained from different tests reinforced soil specimens. 3
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5Investigate the efficiency of the constitutive model by numerically analyzing three geocomposite structures: reinforced foundation soil, MSE wall, and MSE bridge abutment. These structures will be built and analyzed with the aid ofLSDYNA code using two different approaches: a. Considering the full interface between the soil and geosynthetic material, briefed as discrete model, and b. Considering the constitutive model of reinforced soil, briefed as homogeneous model. 1.4 Significance of Research MSE wallshave exhibited the ability ofwithstanding earthquake withqut being severely damaged. For that reason, several tests have been completed on these walls under seismic conditions. These test improved the understanding of the mechanics of reinforced soil method. Among all approaches, finite element is the most powerfut tpol. Depending on the degree of sophistication, finite element method is capable of describing almost any kind of reinforced system, l:!lld gives results that closely resemble real conditions. But because of the difficulties in modeling these walls due to the sliding interfaces between all soil and inclusion layers, this research becomes pertinent. Finding an elastic constitutive model of reinforced soil will facilitate the analysis ofMSE walls and foundation supported by reinforced soil. This constitutive model will also reduce the amount of time ' required to model these walls, as well as eliminate the sources of error. 4 I
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2. Literature Review Despite their initial cost, composite materials have higher specific strength and stiffness than traditional materials. These properties greatly increase the demand for the use of composite materials in new civil engineering constructions and also in the rehabilitation and strengthening of existing structures. They were first used in aeronautical and aerospace applications, and then expanded to other applications Composite structures involve the use of wood, steel, or fabric reinforced elements. Although steel is highly susceptible to corrosion, it was found that even in an aggressive environment, the galvanized strip used in earth reinforcing has a useful life of more than 120 years (Liu 1987). Fabric elements are pottmtially suitable alternatives for use in many harsh environments due to their chemical and corrosion resistance properties. Furthermore, fabric elements are light and easy to construct. Brimahet et al. (1998) suggested to totally eliminate the steel from bridge slabs by combining carbon fiber reinforced plastic tendons with polypropylene fiber reinforced concrete (A. Braihmah 1998). In their tests a one fourthscale model of a bridge deck was subjected to a concentrated load with thickness 20% less than the minimum recommended in the Ontario highway ' . bridge design code (OHBC) of 1991. They found that the deck slabs under static loading offer a minimum safety factor of 6.8 against punching shear Fiber . reinforced plastics are also used as jackets to enhance both strength and ductility of concrete columns by providing confining pressure. 5
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Previous studies on these walls have shown the effect of concrete strength, types of fibers, fiber volume fraction, fiber orientation, jacket thickness, and the interface between the core and the jacket on the confinement effectiveness. Additional investigations were completed using more than 100 specimens to investigate the effect of cross section shape, column length/diameter ratio, and the interface bond (A. Mirmiah 1998). First, they investigated the shape effect by conducting a series of uniaxial compression tests on cubical and cylindrical specimens. Effectiveness of circular sections was reported, this was due to the uniformity of the confining pressure provided by the plastic jackets on circular cross sections. Then they investigated the effectiveness of length to diameter (LID) ratio. Specimens with different.LID ratios were instrumented at different locations with sets of vertical and horizontal strain gauges followed by a set of ' I . uniaxial compression test. It was found that LID has insignific ant on confinement After that they investigated bond effect, between the fabric reinforced plastic jackets and the concrete core. Both adhesive and mechanical bond were investigated. Only mechanical bond (shear connectors) was found to improve the performance of the section by distributing the . ' I confinement pressure around the circumference of the tube. For the past two decades fiber reinforced polymers, such as geogrid used in retaining walls, have been increasingly used in soil reinforcement to obtain an improved geotechnical material. Ever since, reinforced soil has been a subject of . . research. Many of the earlier test proved the effectiveness of fiber reinforcement and the interaction between soil and reinforcement material. The following paragraphs point at different research that was conducted on reinforced soil. 6
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A comprehensive study was conducted to investigate the static strength and the dynamic properties of horizontally reinforced soil subjected to vertical cycling loading using triaxial testing (Liu 1987). In his static analysis, three different reinforcing patterns plus one unreinforced pattern were used to investigate the effectiveness of reinforcement with different numbers of reinforcing layers The reinforced sample consisted of Ottawa sand and needle punched non woven geotextile. Each sample was 12 (in) height and six (in) diameter. It was found that the ultimate strength in reinforced samples increases with increasing the number of reinforcing layers. This is accompanied by an increase in the axial strain. Another study was completed to investigate the static response of reinforced . ' with randomly distributed fibers (Maher 1990). In this study, alaboratory triaxial compression tests were performed to determine the stressstrain response of sand I I reinforced with randomly distributed fibers. Different fibers (Buna, Reed, and Palmyra), and different sand (Mortar sand, Ottawa, Muskegon, and Glass sphere) were investigated. Experiments were followed by statistical analysis to develop a model. This model predicts the fiber contribution to strength under static loading. It was found that an increase in fiber aspect ratio, except for fibers with very low . . ; modulus, resulted in a lower critical confining stress. Also, shear strength increases linearly with increasing amounts of fiber. In 1999, Frost et al. conducted an experimental study using shear tests to investigate the behavior of sandreinforcement interfaces (J. Frost 1999). It was found that the peak interface friction coefficient decreases with the increase of normal stress, and increases linearly with the relative roughness. Frost et al. reported a minor effect of the preparation method, the rate of shearing, and the thickness of the soil specimens on the interface friction. However, tests have 7
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shown that the friction coefficient resulted from direct shear test is much smaller than that of pullout tests where pullout test is being used lately for calculating friction coefficient (Z. Wang 2002). Other shear tests were conducted to investigate the effect of fly ashsoil mixture (S. Kaniraji 2001). It was shown that the fiber inclusions increase the strength of raw fly ashsoil specimens as well as that of the cementstabilized specimens. It was also noticed that the brittle behavior changed to ductile one after adding fiber inclusions. In 2002, experimental investigations and modeling of non linear elasticity of fiber reinforced soil under cyclic loading at small strain were conducted (J. Li 2002). Cycle shear tests were conducted on 27 cylindrical specimens using triaxial apparatus, each specimen was 2.8 in diameter and 5.6 in height. Different fiber contents under both different confining pressure and were investigated. The results indicated that the shear modulus of reinforced soil is affected by all variant factors. Also, the elastic modulus of reinforced soil is directly proportional to both fiber content and confining pressure, and inversely proportional to the incremental loading repetition. After that, a linear regression with multiple variables analysis was completed to calibrate the related parameters. Another investigation on the interface between soil and reinforcement was .. conducted by Ensan and Shahrour (M. Ensan 2002). They presented cm elasto plastic microscopic constitutive model for the multi layer materials with the imperfect interfaces and implemented this model in a finite element program . . \ (PECLAS). From their analysis, it was qetermined that the presence of an.. imperfect interface reduces the resistance of the reinforced soil by about 55%. In 2002, a study using fibershaped waste material, such as polyethylene terephthalate (PET), in soil reinforcing was conducted (N 2002). In this 8
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study, emphasis was placed on the influence of the fiber length and fiber content on the basic aspect of soil behavior, such as initial stiffness, peak ultimate strength, ductility, and energy absorbance capacity. The stressstrainstrength response was evaluated using unconfined compression tests, splitting tests, and drained triaxial tests. Experimental tests were followed by multiple regression analysis to obtain and interpret a representative experimental data base From their results, they indicated the efficiency of the fiber reinforcement was dependent on the fiber length, where the greatest improvements in triaxial strength and energy absorption capacity were observed for the longer, 36 mm, fiber. They also observed strength and stiffness increase for the uncemented soil by fiber reinforcing. For cemented soil, the effect was more pronounced for the lowest cement content. A year after, a total of 14 series of triaxial tests were performed on a reinforced element with different types of sand and different fiber geometry (R. Michalowski 2003). All we cylinders of a specific height and diameter 94.5 mm. The results indicated that up to 70% increase in the strength of un reinforced soil can be gained with fiber concentration of2%. This percentage will change by changing the aspect ratio of reinforcing fibers. Furthermore, small fiber concentration will strongly affect fine sand. However, for large concentration of fibers, the coarse sand becomes much stronger. Reinforced soil is a practical solution to improving geotechnical properties such as bearing capacity, this is best for reinforcement in weak soil foundations (Michalowski 2004). Michalowski proposed a numerical method l?Upported with experimental data for calculating limit loads on strip over foundation reinforced with horizontal layers of geosynthetics. He found that the 9
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reinforcement length of four times of the footing width is recommended to get all the benefit of reinforcing Although sufficient testing has been conducted at the coupon level on reinforced soil, no constitutive model of reinforced soil has been conducted taking in consideration all variations in soil and geosynthetic properties under different confining pressure. Therefore, this is an attempt at producing such a model of reinforced soil followed by mathematical equations to describe the behavior of it. This model predicts the material properties of reinforced soil by knowing the spacil).g between reinforcements, soil properties, reinforcement properties friction coefficient, and confining pressure. 10
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3. Theoretical Background 3.1 Introduction This chapter provides a brief background on finite element methodology, including some of the commercial codes (NIKE3D and LSDYNA) used in this research. Additionally, emphasis is placed on using the finite element method on geotechnical applications. It is fruitful to refer to this chapter, from time to time, in conjunction with the study of topics in other chapters. 3.2 Finite Element Theory for Linear Materials1 The concept Finite Element Method (FEM) has a wide range of engineering applications. It was initially developed by A. Hrennikoff(1941) and R. Courant (1942). According to Courant, the domain is divided. into finite triangular sub regjons for solution of second order elliptical partial differential equations. In the 1950s, the development ofFEM began for airframe and structural analysis . Since then, it has been generalized into a branch of applied mathematics for numerical modeling of physical systems in an extensive variety of engineering disciplines such as electromagnetic and fluid dynamics. 1 The theory of linear and nonlinear finite element is obtained from Potts and Zdravkovic, 1999. 11
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FEM is based on an energy principle which is a fundamental concept used in physics and engineering. This principle expresses the relationships between stresses, strain, material properties, and external work done by internal and external forces. Virtual work is an example of this energy principle. Virtual work is a mathematical product of forces and displacement. It is the work done on a particle due to real or imaginary force producing a real or imaginary displacement in the direction of the applied force. Mathematically, FEM method is used for finding approximate solution of partial differential equations. The solution approach is based on either eliminating the differential equations completely in steady state problem, or rendering the partial differential equation into an equivalent ordinary differential equation, which is then, solved using standard such as finite differences. The FEM is a good choice for solving partial differential equations over complex . . . .. ' domains. For this kind of computation, finite element analysis (FEA) is the most effective and efficient computer simulations. In FEA, the system is represented by a model consisting of multiple linked discrete regions. At each element, the concepts of continuum mechanic approach (equilibrium, compatibility, and constitutive relations) are applied. As a result, a system of simultaneous equations is constructed and solved for the unknown values. In finite element, the geometry is replaced by an equivalent finite element mesh which is composed of small regions called finite elements. In twodimensional ai).alyses, the finite elements are usually triangular or quadrilateral in shape composed of key points called nodes (Zdravkovic' 1999). In order to obtain accurate solution, the zones of attention should contain finer mesh with proper . aspect ratios. For each element, based on the energy principle, the domain is presented by Equation 3.1. 12
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lK E jlL1d E J= E J Where: [KE] = element stiffness matrix, { i1dE} = the vector of incremental element nodal displacements, and { i1RE} = vector of incremental element nodal forces. In finite element formulation, the element displacements and geometry are expressed in terms of interpolation functions or shape functions (Ni) using natural coordinate system that varies from 1 to 1. Each node has one shape function and is equal to 1 at that node and 0 at other nodes. For example N1 is equal to 1 at node 1 and N2 is equal to 1 at node 2 and so on. Corresponding to these shape function and based on the minimum potential energy principle for linear elastic material, the global finite element equation of a body has the following form, In Equation 3 .2, n is the number of element, and [KE] is the element stiffness matrix. For plane strain condition, [KE] becomes [icE]= f ft[B]T[DiB]JidSdT Where: t = unity thickness, jJj == detertninant that is responsible for the mapping between the global and natural element, and is shown in Equation 3.4, [D] =constitutive matrix ; ' I : . ' 13 (3.1) (3.2) (3.3)
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[B) = Matrix that contains the derivatives of the shape functions, and is shown in Equation 3.5, S and T = natural coordinates that are related to the global coordinate using the jacobian, as shown in Equation 3.6 and {ARE}= righthand side load vector, and is shown in Equation 3.7. ax ay ay ax III = as aT as aT aNI aN2 aN 0 0 n 0 ax ax ax aNI aN2 aN 0 0 0 n [B)= 8y 8y 8y aNI aN1 aN2 aN2 aN aN n n 8y ax 8y ax 8y ax 0 0 0 0 0 0 {aNi aNi}T [!; as aT ax aT : ]{aNi aNi }T ay ax. ay aT f[N]T{LIF}dvol+ J[N]T.{LIT}dSrf volume surface After forming the equilibrium equations of all separate elements, the set of global equations is formed by asserp.bli!lg the separate equations as shown in Equation 3.8. In this equation the global stiffness matrix is obtained from assembling the separate elements stiffness matrices using the direct stiffness method. Also, the boundary including concentrated load and surcharge pressure are to the rightside of the global equation, lKG = J Where: '. 14 (3.4) (3.5) (3.6) (3.7) (3.8)
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[Ko] = global stiffness matrix, = vector that contains the unknown degree of freedoms for the entire finite element mesh, and =global righthand side load vector. Once the global equilibrium equation is established, the nodal displacements values are solved. The solution of these nodal displacements is obtained using different techniques, such as Gaussian elimination or iterative methods. From there, secondary quantities such as strains and stress can be computed as shown in Equations 3.9 and 3.10 respectively. aN1 aN2 aN 0 0 n o ax ax ax X aN1 aN2 aN 0 0 0 n y = ay ay ay xy aNI aNI aN2 aN2 aN aN n n ay ax ay ax ay ax 0 0 0 0 0 0 n {L1u} = [D ]{Lie} Where : = [ x J and y xy [D] =constitutive matrix that only depends on Young's modulus E and Poisson s ratio, v, for isotropic elastic materials. 15 (3.9) (3.10)
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3.3 Finite Element Theory for Nonlinear Materials Most of the material models, especially, in geotechnical applications behave non linearly. For that reason, applying the theory of linear finite element would result in inaccurate results. In elastic materials, the constitutive matrix [D] is considered a constant. This is not the case for nonlinear material, where [D] varies with stress and strain during a finite element analysis. Therefore, the governing finite element equations are reduced from Equation 3.8 following incremental form as shown in Equation 3.11 Where: [Ka]i =is the incremental global system stiffness matrix, ='is the vector of incremental nodal di splacements, { 8Ra} i = is the vector of incremental forces: and i = 1s the increment number. Since [D] is not constant for non linear materials, [KG]i will also be on the durrent stress and strain level and will vary over an increnieht. The three main popular solutions for nonlinear materials are tangent stiffness, viscoplastic, and the modified NewtonRaphson. 3.3.1 Tangent Stiffness Method : In this method, [KG]i ofEquation 3.11 is assumed to be constant over each increment. Since the stiffness is constant over each loading increment, the load displacement curve will be a straight line. In reality, the stiffness ofnonlinea,r . . ' . 16 (3.11)
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material will not be constant during any loading increment. Therefore, there will be error of the predicted displacement. This error increases with increments away from starting point and the predicted displacement further deviates from the true solution. To obtain more accurate solutions, smaller load increments are required. 3.3.2 Viscoplastic Method This method was originally developed for linear elastic viscoplastic material. This kind of material is time dependent and is represented by a network that consists of elastic and viscoplastic components connected in series. When using this method, the system is assumed to behave elastically . If the : resulting lies below; the yield surface, the incremental is the . . . . . . '.. . calculated displacements are correct. If the resulting .yi,elping. cpridition stress state can only be sustained momentarily and viscoplastic straining occurs. The viscoplastic method is simple and has been widely used Complication rises when solving problems involving non viscous materials such as elastopla:stic 3.3.3 Modified NewtonRaphson Method The modified NewtonRaphson method uses an iterative technique to solve Equation 3 .11. The first step is similar to the tangent stiffness method but the \ \ ' ,' I error is recognized and the predicted incremental used to calculate the residual load which is a measure of the error in the analysis. . . . . Equation 3.11 is then solved with the residual load forming the right hand side vector 1:!S in Equation 3.12 . \ 17
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(3.12) Where: j = is the iteration number, and \}' = is the residual load. The procedure is repeated until the residual load is small, and the incremental displacements are then the sum of the iterative displacements. To determine the residual load vector, an estimate ofthe incremental displacements is calculated at the end of any iteration. From there, the incremental strains are evaluated, and the constitutive model is then integrated along the incremental strain paths to obtain an estimate of the stress changes. Theses stress changes are added to the stresses at the beginning of the increment. Knowing these stresses, the consistent equivalent nodal forces are evaluated. The difference between these forces and the externally applied loads giye the residual load vector. The integration methods are termed stress point algorithms and can be computed using both e:x:plicit and implicit approaches known as substepping algorithm and return algorithm respectively 3.3.3.1 Explicit Algorithm this approach, the incremental strains are into number of substeps, hence name substepping algorithm. In each substep, the are a proportion ' . ' ofthe strains, as shown in Eqmt.tion 3.13. The equations are tl:len integrated numerically over each substep. 8 = 11T 8 ss J me J Where : 18 (3.13)
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{L\Ess}= are the strains in each substep, { L\Einc}= are the incremental strains L\ T = is the proportion 3.3.3.2 Implicit Algorithm In this approach, the plastic strains over the increment are calculated from the stress conditions corresponding to the end of the increment. These stresses are not known therefore initial estimate based on elasticity are completed to predict the stress changes. Further iterations are completed to insure convergence where the constitutive behavior is fulfilled and the final stress that is within the yield surface is computed at the end of that increment. From the stress state, the plastic strains over the increment can be calculated. Both algorithms give accurate result. However, explicit algorithm is more vigorous in case of nonlinear constitutive model that contain 2 or more concurrently active yield surfaces (Zdravkovic' 1999), which is the case for most of geotechnical engineering applications. 3.4 Finite Element Commercial Codes In this research two main codes were heavily used. These were NIKE3D and LSDYNA. With each code a complete package including the preprocessor and post processors were also utilized. With NIKE3D, TRUGRJD and GRJZ were used as preprocessor and postprocessor, respectively. On the other hand, FEMB and LSTCPrePost were used as preprocessor and post processor, respectively. 19
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3.4.1 NIKE3D, TRUGRID, and GRIZ NIKE3E is a nonlinear, implicit, and 3Dimensional finite element code for solid and structure mechanics(Maker 1995). It was originally developed and has been used by the Lawrence Livermore Nation Laboratory, LLNL for over 20 years. It is used to study the static and dynamic response of structures undergoing finite deformations. The main elements used in this program are the 8node solid element, 2node truss element, and 4node shell element. The 8node solid elements are integrated with a 2 x 2 x 2 point Gauss quadratic rule. The 2 node beam elements use 1 integration point along the length. And the 4 node shell elements use 2 x 2 Gauss integration in the plane. There are more than 20 material models implemented in this code to simulate a wide range of material behavior including elastoplasticity, anisotropy, creep, and rate dependence. Between independent bodies an arbitrary contact is handled by vaiiety of sliding algorithms such as modeling the sliding along material interfaces including frictional interface. NIKE3D is based on an updated Lagrangian formulation. Throughout each load step, nodal displacement increments which produce a geometry that satisfies equilibrium at the end of the step are computed. Once the displacement increments are updated, the displacement, energy, and residual norms are computed. After that, the equilibrium convergence is tested using user defined tolerances and the analysis is preceded to the next load step. If convergence is not achieved within the user specified iteration limits, the optional automatic time step controller will adjust the time step size and try again. 20
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A preprocessing program, TRUGRlD, was developed for NIKE3D and other finite element and difference codes, such as ADINA, ANSYS, MARC, LS DYNA, etc., to generate such finite mesh. TRUGRlD is a 3Dimensional mesh generator developed by LLNL as well. TRUGRlD is also used to specify loads, element types, sliding surfaces, boundary conditions, and material models. The output file from TRUGRID would then become the input file for NIKE3D or other finite element programs. It uses a special projection method for mapping a block mesh onto one or more surfaces. Therefore, a complex a looking mesh can be built from a simple block very easily. After running the analysis using the main processor, NIKE3D, a post processor, GRlZ, was used to extract the results. GRlZ was also developed by Lawrence Livermore Nation Laboratory. The output file from NIKE3D becomes the input file for GRIZ. With GRlZ, the analysis results could be visualized and animated passing through loading increments. The results from GRIZ could be also printed into text file for plotting and analyzing. 3.4.2 LSDYNA, FEMBPC, and LSPREIPOST LSDYNA is an allpurpose finite element code developed at Livermore Software Technology Corporation, LSTC for analyzing deformation and dynamic response of structures (LSDYNA 2003). The main solution method is based on explicit time integration. Nevertheless an implicit solver is currently available. The origin ofLSDYNA dates back to the public domain software, DYNA3D, which was developed more than 30 years ago at the LLNL. Throughout the past decade, 21
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considerable progress has been made by Hallquist2 in developing the LSDYNA code. This code was first initiated in 1989 for automotive applications. In this version, 1989, many enhanced capabilities such as a one way treatment of slide surfaces with voids and friction and unique penalty specifications for each side surface were brought in. Since then, many developments have occurred to this program. Currently, LSDYNA contains approximately 100 constitutive models representing wide range of material behaviors including linear and non linear, elastic and inelastic, homogeneous and composite materials, and so on. Furthermore, difficult contact problems such as contactimpact are easily treated. This code helped vastly in the automotive industry because of the specialized capabilities such as airbags, sensors, and seat belts that have been adapted in LSDYNA In LSDYNA, elements can be solids, shells, beams, or discrete elements. The solids composed of 8 node elements, the shell composed of 3 or 4 node elements, the beam of 2 node elements, and the discrete elements are springs or dashpots. It is operational on a large number of mainframes, workstations, and PC's. A powerful preprocessing program for use with LSDYNAPC is FEMBPC Version 27. It was developed by Engineering Technology Associates, Inc. in 2001. FEMBPC contains a swarm of automated, easy to use model building functions. It supports a complete LSDYAN interface including material types, contact types, boundary conditions, and output data base. LSPRE/POST is a preprocessor and post processor. Both functions are still in the developing stages. In this research, this program was heavily used as a post 2 ln 1988 J. 0. Hallquist worked halftime at LLNL to devote more time to the development of LSDYNA. In 1989 he resigned from LLNL to continue the development ofLSDYNA at LSTC. 22
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processor. The current edition, Vl, was sufficient in producing the required results in both graphical and text versions. The LSPRE/POST is very friendly software, where the analysis results could be visualized and animated through loading increments. 3.5 Constitutive Modeling of Soil In general, the constitutive models for geematerials such as rocks, concrete, and soil are based on the same mathematical plasticity theory used to model common metals. However, geematerials slightly differ from the other metals due to their compressibility and low tensile strength. Their behavior is stress path dependent and the total deformation is composed of recoverable and irrecoverable part (C.S. Desai 1984). As for any other material, the only lucid way to determine the parameters that define the constitutive model of soil is to conduct appropriate laboratory and/or field tests. The parameters obtained from the laboratory or field test must be verified by teproducing the observed data using the constitutive model. Have satisfaction obtained, further usage of this model can be utilized to implement different projects or applications. Element testing that characterizes the mechanical behavior of soil is the most appropriate investigation for deriving the parameters of the constitutive models. Of these tests are odometer, triaxial, true triaxial, direct and simple shear, hollow cylinder, and standard penetration test known as SPT. When simulating the element testing, it is very important to closely simulate the field conditions including initial stress and drainage conditions. 23
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Soil is a non linear material3 When subjected to load, the physical characteristics including the moduli continue to change. These moduli define the material stiffness and hence the constitutive matrix of the material. Throughout loading, the components of constitutive matrix keep changing resulting in non linear behavior. Quasilinear model based on piecewise linear behavior can be used as an approximation of nonlinear behavior. In Quasilinear model, a given nonlinear behavior is divided into pieces of linear elastic behavior, in which Hooke's law can be used to solve for Modulus of Elasticity, E, and Poisson's ratio, v, at each loading increment. In this approach it is important to determine the variable parameters as the state of stress changes during the incremental loading. Hence these models are also referred to by variable parameter (VP) or variable moduli (VM) model. Both elastic parameters can vary during the step loading or one of them may vary while the other is kept constant. Of these VM models, hyperbolic and RambergOsgood model are considered and briefly discussed. 3.5.1 Hyperbola Model The stress strain curves of soil maybe simulated using a hyperbola or parabola function. The hyperbola model for representing the stress strain curves for soil was first proposed by Konder in 1963 and is given in Equation 3.14. Where: cr1 cr3 = is the deviator stress at any load increment, 8 = is the axial strain at any load increment, 3 Non linearity renders the load displacement behavior. 24 (3.14)
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E = initial tangent modulus, and ( cr1 cr3)ult = is the ultimate deviator stress. Since the response of soil or any geologic material is a function of confining pressure, cr3 it becomes necessary to express the hyperbola model in conjunction with the relation between initial modulus and confining pressure. Therefore, the stress strain curve of the hyperbola has the form of: [ R /(1sinXa1 a312 (a3 Jn E = 1K P t 2ccos + 2a3 sin h a Pa Where: Et = is the tangent modulus at a point, Rr = is the ratio of ultimate deviator stress to the failure deviator stress c = is the cohesive strength, = is the angle of friction, p a = is the atmospheric pressure, Kh and n = are material parameters. This model is simple, and material parameters can be determined from laboratory test results such as conventional triaxial compression, CTC, test. Results obtained from this model are reliable especially when subjected to monotonic loading. This is not the case when loadingunloading conditions are involved because these loadingunloading conditions include wide range of stress paths where as the hyperbola mode contains only one stress path. Furthermore, the Hyperbola model does not account for dilatancy effects which become a deficiency when modeling dense sand or overconsolidated clay. 25 (3.15)
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3.5.2 RambergOsgood Model The RambergOsgood model is an analytical model and it is often used to represent the hydrostatic behavior of soil materials subjected to cyclic loading (Oncul2001). For monotonic loading, the stress strain relation of the Ramberg Osgood can be expressed by: r r r [ r 1] Yy = 'y 1+a 'y Where: y = is shear strain, yy =is reference shear strain, 't = is shear stress, 'ty = is reference shear stress, a= is constant factor, larger than 0, that adjust the position of the curve along the strain axis, and R =is constant factor, larger than 1, that controls the curvatures of the graph. The above VM models can account for behavior of limited class of materials, loadings, and stress paths, and therefore may not treated as general models. Furthermore, geologic materials exhibits frictional resistance that is proportional to the normal loading. The above mentioned models do not account for such behavior. The following models, Mohr Coulomb, DruckerPrager and Cap model, do take into account for complex loading paths and frictional resistance. They are based 26 (3.16)
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on theory of plasticity that was developed for metals taking in consideration the dependency on hydrostatic stress 3.5.3 MohrCoulomb Failure Criterion MohrCoulomb is the most commonly used failure criterion in engineering practice. According to this criterion, the shear strength increases with increasing normal stress on the failure plane as shown in Figure 3.1, and can be expressed as: r=c+O"tan O" 1 O" 3 = lO" 1 + O" 3 )sin + 2c cos Where: 't = is the shear stress on failure plane, = is the angle of internal friction, cr = is the normal effective stress on the failure surface, c = is the cohesion of the soil, and crt and cr3 =are the major and minor principal stress respectively. o,J "Jf 11 Figure 3.1 "Parameters of MohrCoulomb model" 27 (3.17) (3. 18)
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As seen in Equations 3.17 and 3 .18, and Figure 3.1, this model is expressed in terms of maximum and minimum principal stress. Furthermore Equation 3.18 indicates that MohrCoulomb failure criterion is represented by irregular hexagonal pyrainid in the stress space. According to this shape, the yield strength in compression is higher than in extension. In MohrCoulomb failure criterion, only two parameters are needed. These are c and They can be easily obtained from conducting two or more CTC tests on cylindrical soil sample. For frictionless material, is equal to 0, the Mohr Coulomb criterion reduces to Tresca criterion. 3.5.4 DruckerPrager Criterion The DruckerPrager yield criterion is a generalization of the Von Mises criterion that includes the influence ofhydrostatic stress (Boresi 2003). The yield function can be expressed as shown in Equation 3.19. This failure criterion can be further explained using Figure 3.20, where constant parameters a and k, can be determined from the slope and intercept of failure envelope plotted in first invariant of principal stresses, J 1 and, square root of second invariant of deviatoric principal stresses, J zD space, as shown in Equations 3.20 and 3 .21. f = ai1 + .J[JJk 28 (3.19) (3.20) (3.21)
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{,., 17""';. "'J n !.., i V "l"!t ,u, 'i .t.t.7 r : Figure 3.2 "DruckerPrager criterion" In order to establish this failure envelope or determine the model constants, it is necessary to perform some laboratory tests, such as CTC test or plane strain test. For CTC test, the minor and the intermediate principal stress, cr3 and cr2 are equal to the confining stress. On the other hand, cr1 is equal to the confining stress in addition to the deviatoric stress, as shown in Equation 3.22. Applying two or more CTC tests at different confining pressure, allows the failure curve to be established by finding J1 and for each test. Doing so, and using the least square method, a and k can be determined. The values of a and k in case of CTC tests can be expressed in terms and cas shown in Equations 3.23 and 3.24, respectively (C.S. Desai 1984). The values and care first determined using MohrCoulomb failure criterion obtained from conducting two or more CTC tests under different confining pressure. 29
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2x a = ;:::::':.fj x (3k = .J3 x (3 According to this criterion and as shown in Figure 3.2 there is always a negative plastic strain component indicating a volume increase or dilation. For dense sand, during shearing this phenomenon occurs at even very small strains. However, loose sands experience compressive deformations and volume reduction. This indicates that if loose sand is used and tested, the results obtained from this method are not reliable. 3.5.5 Cap Models Cap models are based on yielding of soil. They are expressed in terms of 3Dimesnional state of stress. The elliptical yield surfaces looking like caps resulted in the name of such models. The cap models use two intersecting surfaces, fixed or ultimate yield surface, and yield cap, f2 as shown in Figure 3.3. 30 (3.22) (3.23) (3.24)
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Figure 3.3 "Cap model" Based on DiMaggio and Sandler, the ultimate yield surface that represents the shear failure surface has the form of: J 2 D = a r exp(,811 )+ W 1 Where: a, y = are material parameters 8J 1 = is convenient for fitting shear failure data that has fairly linear straight line representation, which allows this surface to be model using the DruckerPrager Model. The constant parameters for the shear failure surface can be obtained from experimental data such as CTC test results. Once obtaining the results at different confining pressure and plotting J 1 versus J 2D the best curve fit acc?rding to Equation 3.25 is obtained, and therefore the constant parameters can be determined. If DruckerPrager model was assumed to represent the ultimate 31 (3.25)
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yielding surface as in Figure 3.4, where a straight line can fit these data points, then the constant parameters of this ultimate yield surface can be solved using the following equations 6ccos a = =='v'3(3sin) () = 2sin v'3(3 sin) Where a in this equation represents the Y axis intercept and 8 represents the slope of the failure surface. For simplicity, both DruckerPrager failure lines can be represented by one line, which makes the value of y equal to zero. In case of cohesionless soil, a is also zero, and therefore, the shear failure line starts from the origin (0, 0). The only parameter determine be determined by solving Equation 3.25 at a point on the transition curve considering all other constants (C.S. Desai 1984). Figure 3.4 "Interpretation of parameters of f1 32 (3.26) (3.27)
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The other surface is the moving yield surface and is referred to by cap. The cap surface represents an ellipse with its long axis along the mean pressure, J 1 axis, and can be expressed in the form of: !2 =R2J2D +{J1cf =(X c)2 Where: R = is the ratio of the major to minor axis of the ellipse, and lies in the range of 1.67 to 2 (C.S. Desai 1984), X = is hardening parameter that depends on the plastic volumetric strain as shown in Equation 3.29 and can be written in terms of mean pressure as in Equation 3.30 Graphically it is the position on J1 axis where the cap surface intersects and C = is the value of J 1 at the center of the ellipse, X=3p Where: D, Z, and W are hardening parameters p = is the mean pressure, which is equal to the confining pressure in case of triaxial test In case of very small initial yielding, Z, the size of the initial cap is almost zero. Therefore after rearranging Equation 3.29 and plugging in Equation 3.30, considering the elastic, Eve, and the plastic, e/, volumetric strains components of the total volumetric strain, Ev, the W and D parameters can be determined using Equation 3 .31. In Equation 3.31 the elastic volumetric strain is substituted by the value of p/K, where K is the bulk modulus and is defined as the slope of the unloading curve in a hydrostatic compression test (C.S. Desai 1984). Several 33 (3.28) (3. 29) (3.30)
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trials must be conducted to obtain the most accurate values of W and D that reproduce the hydrostatic curve (volumetric strain versus pressure). Ideally, when applying large pressure during hydrostatic compression test the term e3 P 0 approaches zero, and therefore W is the volumetric strain measured at that pressure (C.S. Desai 1984). &v =W(le 3pD)+ As a summary, defining cap model parameters several tests must be completed. Hydrostatic compression and CTC tests can be sufficient. The hydrostatic compression test is subjected to loading and unloading. The loading portion helps in determining the hardening parameters, W and D, while the slope of unloading is necessary in determining the elastic parameter K. Performing sets of the CTC tests under two or more confining pressure provides an excellent prediction of the parameters for f1 surface and initial Young's Modulus, Ei, for each confining pressure. Once all the parameters constants are determined, it becomes necessary to reproduce the hydrostatic curve and CTC results under different confining pressures in order to validate the model. 3.6 Contact in Finite Element In finite element there are three methods to define contact between different parts. These are kinematic constraint for tied interfaces and called Lagrangemultipliers, distributed parameter where sliding occurs without separation, and penalty method which includes sliding, separation, and friction. Most of the applications in soil structural interaction involve separation and sliding between adjacent particles. From there, a briefbackground and description of penalty method is presented. 34 (3.31)
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In the penalty method, a normal interface springs are placed between all penetrating nodes and contact surfaces. For each slave node the closest master node and closest master segments are found and a check is completed to determine if any penetration. Then penetration is then reduced by applying forces through the existing massless spring that has a stiffness k and is called penalty stiffness. It is unique for each segment and is based on the bulk modulus of the penetrated material. For 2dimensional shell elements and 3Dimensional brick elements, k is defined as shown in Equations 3.32 and 3.33, respectively. k shell = a x fJ x t A2 kb. k =ax{Jxnc v Where: a = is the penalty stiffness scale factor. In LSDYNA default is 0 .1, B = is the bulk modulus, A t = is the nominal shell thickness, t = segment max shell diagonal A = is the segment area and V = is the brick element volume. Due to relative movement between contacted parts, friction is generated. Friction is based on a Coulomb formulation, and the friction coefficient, IJ.c It is a function of coefficient static friction, !J.s, coefficient of dynamic friction, !J.d, decay coefficient, de, and relative velocity between the slave node ahd the master segment, v, as shown in Equation 3.34. Jl =Jl +( Jl \ dc v c d lfs dp 35 (3.32) (3.33) (3.34)
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For many geotechnical applications, especially including reinforcement layers within the soil mass, the 4 node shell element is included in the model. Even though it is a 2 dimensional element, thickness must be properly accounted for in contact. Therefore, surface or nodes contact must have an offset to account for shell thickness. When contact is detected on both sides of the shell, half of the thickness must be considered as an offset on each side of the element. During contact, kinetic energy is developed. This energy is directly proportional to the spring stiffness and the penetration distance. Preferably this energy should be very small to indicate a minimal penetration compared to other physical energies in the system. In order to obtain small energy, i.e. small penetration, it becomes necessary to avoid initial penetrations and tangled meshes at any cost. This can be done by adequately offsetting the contacted surfaces and use consistent fine meshes on adjacent parts. 3.7 Summary In this chapter a brief back ground is presented to cover some aspects of finite elemept method and analysis especially for non linear materials such as soil. This was followed by considering some of the constitutive models such as Rambo Osgood and Cap models for geological materials. Since soil is frictional material it was significant to define contact properties between soil particles themselves and between soil and other adjacent parts such as reinforcement layers and concrete foundations. 36
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4. Anisotropic Properties of Geo.;Composite 4.1 Introduction Over the last three decades, the use of geosnthetic reinforced soil, GRS, in geo structures like mechanically stabilized earth, MSE, walls had steadily increased. Reinforced soil structures are constructed in a manner that produces a composite structure of alternating layers of soil and reinforcing elements (Vector Elias 2001). Combining the compression strength of Ottawa sand and the tensile strength of geosynthetic produces a geocomposite that is strong in tension and compression (NienYin Chang 2006). Performing finite element (or difference) analyses on geostructures requires both the material models and soilgeosynthetic interface model that allows realistic behavior like interface slippage and separation, which increases the computation time especially for a large problem. It would be much easier and less time consuming if the geocomposite can be treated as a homogeneous material instead of adjacent parts with frictional interfaces. This would alleviate the interface convergence problem and speed up the computation when using the finite dement method. In this chapter finite element analyses were completed on cubic samples of this geocomposite to evaluate its equivalent properties as function of its constituent materials and other factors. These factors were spacing between inclusion layers, friction coefficient, and confining pressure. 37
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The tests were performed on 1 m x 1 m x 1 m geocomposite cube with linear elastic soil and geosynthetics. The cubes were subjected to a combination of confining pressures and deviator stresses in six directions successively. The directions were X, Y, Z, XY, YZ, and XZ. Once these forces were applied, the resulted strains in all six directions were calculated. Based on Hooke's law, the compliance matrices were determined, which allowed in observing the material type and properties for each configuration case. Also, the Young's moduli of the composite evaluated from the finite element analyses were compared with the values from the analytical formulation. The purposes of these tests were to determine the material type of the geo composite and to develop a data base of equivalent properties that could be used in creating a constitutive model. This model would be obtained (Chapter five) based on linear regression analysis as function of all input parameters (Young's modulus of soil, Poisson's ratio of soil, Young's modulus of geosynthetic, Poisson's ratio of geosynthetic, Spacing between reinforcement layers, friction coefficient, and confining pressure) 4.2 Composite Materials The composite material term signifies that two or more materials are combined on a macroscopic scale to form a useful material (Jones 1975) .The advantage of composites is that they usually exhibit the best qualities of their constituent and often some qualities that neither constituent posses. Composite materials are widely employed in both or in the combination of structural and soil fields. These applications are expected to continue due to increasing requirements for light weight, high stiffness and/or strength, nonconducting, and noncorroding 38
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materials. Geosynthetic material fits these requirements and has served as subject of research since 1940 when it was utilized iri the military and aerospace industries (J. Frost 1999). Composite materials are generally anisotropic4 and inhomogeneous5 so that the strength of these composites cannot be computed by identifying a single stress level. Stress state is nearly always complex, even when only one stress is applied at the global level. To obtain the mechanical properties of anisotropic materials, several test must be conducted such as multiaxial test to cubical composite (Hon Yim Ko 197 4 ). In this approach all the material properties of composite material is obtained by applying sets of deviator stresses on a confined specimen. From these applied stresses, all components of compliance matrix [C] are evaluated using Hooke's law, in which all strain components are linearly related to stress components as shown in Equations 4.1.a and 4.1.b. E = [C] a & ell c12 cl3 c14 cl5 c16 (j X X & c21 c22 c23 c24 c25 c26 (j y y 8 c31 c32 c33 c34 c35 c36 (j z z & c41 c42 c43 c44 c45 c46 "&' xy xy & c51 c52 c53 c54 c55 c56 "&' yz yz 8 c61 c62 c63 c64 c65 c66 "&' xz xz A complete anisotropic elastic material has 36 material parameters in its [C] as shown in Table 4.1. However, less than 36 ofthe constants are actually 4 An isotropic body has material properties that are the same in every direction at point in the body. 5 A homogeneous body has uniform properties throughout. 39 (4.l.a) (4.l.b)
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independent parameters for elastic materials when the strain energy is considered. Due to symmetry, where Cij=Cjb only 21 of the parameters are independent. Table 4.1 "Compliance matrix components [C]" c11 cl2 cl3 C14 C1s C16 c21 C22 C23 C24 C2s C26 C31 C32 c33 c34 c3s c36 C41 C42 c43 c44 C4s c46 Cst Cs2 Cs3 Cs4 Css Cs6 c61 c62 c63 c64 C6s c66 According to the generalized Hooke's law, the general shape of the resulting [C] for an anisotropic material is as follow (Lekhnitskii 1963): 1 v v 1lxy,x 17yz,x 17zx x yx E E E E E E XX XX XX XX XX XX v 1 v 1Jxy,y 1J yz,y 1Jzx,y xy E E E E E E (j & yy yy yy yy yy yy X X v v 1lxy,z 17 yz,z 17zx z (j & 1 y y xz yz & E E E E E E (j z zz zz zz zz zz zz z = rxy 17x,xy 1J y,xy 1J z,xy 1 flyz,xy flzx,xy xy ryz G G G G G G xy xy xy xy xy xy yz rxz 11x,yz 1ly,yz 17 z,yz flxy,yz 1 flzx,yz xz G G G G G G yz yz yz yz yz yz 11x zx 1ly,zx 17z zx flxy,zx flyz,zx 1 _,_ G G G G G G xz xz xz xz xz xz Where: Exx, Eyy, Young's moduli (Tension compression) 40 (4.2)
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Gxy, Gyz, Gxz Shear moduli Vyx, Yzy, Vzx, Yxy, Yyz, Vxz are the Poisson coefficients which characterize the transverse compression for tension in the direction of the axis of the coordinate (thus, Vyx is a coefficient which characterizes the decrease in the x direction for tension in the y direction; Yxy is the coefficient which characterizes the decrease in they direction for the tension along the x axis and so forth) ).1 is the coefficient of Chentsov. These constants characterize the shear in the planes, parallel to the coordinates and also induce tangential stresses parallel to other coordinate planes. For example ).lzx,xy characterizes the shear plane parallel to the yz plane which induces the stress 'txz 11 is the coefficient of mutual influence of two characteristics. The first characteristic is the elongation in the positive directions parallel to the axes. These are induced by tangential stresses. The second characteristic is shear in the planes parallel to the coordinates under the influence of normal stress. From Equation 4 : 2, it is clear that the material properties can be directly determined by finding all the components of [C]. Components of [C] can be determined by applying stresses, one at a time on a cube element of the composite material as shown in Figure 4.1. The stresses are normal stresses on the negative and positive X fa,ces of the cube element ( cr x), normal stresses on the negative and positive Y faces of the cube element (cry), normal stresses on the negative and positive Z face of the cube element (crz), shear stresses on the negative and positive X faces in the Y direction or on the negative and positive y face in X direction ('txy), shear stresses on the negative and positive Y face in the Z direction or on the negative and positive Z faces in theY direction ('tyz), and shear 41
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stresses on the negative and positive X faces in the Z direction or on the negative and positive Z faces in the X directions ( 'txz). Y l j. i I (J I yyl I Figure 4.1 Deviator stresses on cube element" Loading the sample in one direction produces six strain components which are linear functions of unknoWn. compliances (HonYim Ko 1974). From each stress, all strain components can be evaluated by either the use of strain gages in physical testing or finite element approach. Using Equation 4.1, all columns of [C] can be calculated. For example, when crx is applied and all other stress (cry, crz, 'txy, 'tyz, and 'txz) are zero, all six strain component can be determined. Therefore, froJTI this test the first column of [C] can be calculated as follow: 42
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& ell c12 c13 c14 c15 c16 8x = CllCT X X (]' & c21 c22 c23 c24 c25 c26 X 8y =C21ax y 0 & c31 c32 c33 c34 c35 c36 0 s z = c31a x z = => & c41 c42 c43 c44 c45 c46 0 8xy =C41ax xy & c51 c52 c53 c54 c55 c56 0 8 =C51a yz 0 yz x & c61 c62 c63 c64 c65 c66 s = c61a xz XZ X The same procedure can be followed to determine the second, the third, the fourth, the fifth, and the sixth columns of [C] by applying cry, O"z, 'txy, 'tyz, and 'txz, respectively. Once all the [C] components are calculated, the material type and properties can be determined. The types that are widely used here will be briefly discussed, which are orthotropic, transversely isotropic, and isotropic. 4.2.1 Orthotropic (Three Planes of Symmetry) In orthotropic material, there are three principal directions of elasticity that are mutually orthogonal which pass through each point of the body (Jones 1975). An orthotropic material has no coupling between any normal stress and shear strain, and no coupling between any two distinct shear strains. The principal constants are E1, E 2 E3 (the Young moduli), 012, 023, 013 (the shear moduli) and v12, v21, v13, V31, V23, v32 (Poisson's ratio coefficients). The generalized Hooke's law equation can be written in the following matrix form: 43
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1 v21 v31 0 0 0 E1 E2 E3 v12 1 v32 0 0 (}" e 0 X X E1 E2 E3 (}" e y y v13 v23 1 e 0 0 0 (}" z E1 E2 E3 z (4.3) = e r xy 0 0 0 1 0 0 xy e 2G12 r yz yz e 1 x z 0 0 0 0 0 r 2G23 x z 0 0 0 0 0 1 2G13 Where: v12 v21 v23 v32 v13 v31 ==, =E2 E1 E3 E2 E3 E1 The above material resulted with 12 constants. Due symmetry, 9 ofthem are independent. Transversely Isotropic (A plane of Isotropy) In transversely isotropic materials there is one plane in which the mechanical properties are equal in all directions (Lekhnitskii 1963). A transversely isotropic material is an orthotropic in which two of the directions affect the third direction in the same manner. For example, illustrated in Figure 4.2, is a fibrous element in which directions 2 and 3 behave the same with respect to direction 1, this is "23 plane of isotropy" 44
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3 2 1 Figure 4.2 "Transversely isotropic material (23) plane of isotropy)" The generalized Hooke's law can be written in the following matrix form: 1 v v 0 0 0 E E E a E: v 1 v X X 0 o 0 E E a E: E y y E: v v 1 0 0 0 a z ' z = (4.4) E: E E E T xy 1 xy E: 0 0 0 0 0 yz 2G T E: 1 yz xz 0 0 0 0 0 T 2G xz 0 0 0 0 0 1 2G E and E' are Young moduli (for tensioncompression) with respect to directions lying in the plane of isotropy and perpendicular to it. v is the Poisson coefficient which characterizes the transverse reduction in the plane of isotropy for tension in 45
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the same plane. v' is the Poisson coefficient which characterizes the transverse reduction in the plane of isotropy for tension in a direction normal to it. G' and G are the shear moduli for the plane normal and parallel to the plane of isotropy, respectively. From Equation 4.4, six constants can be observed. Only five of them are independent. These are E, E', v, v', and G'. G is dependent on both E and vas shown in Equation 4.5. G= E 2(1 + v) 4.2.3 Isotropic (Complete Symmetry) In isotropic bodies, any plane being a plane of elastic symmetry and all points are equivalents in all directions (Lekhnitskii 1963). A three dimension (3D) isotropic material is an orthotropic material that has no direction preference. The equation of isotropic material which expresses the generalized Hooke's Law can be written in the flowing matrix form: & X & y & z & xy & yz & xz = 1 E v E v E 0 0 0 v E 1 E v E 0 0 0 v E v E 1 E 0 0 0 0 0 0 1 2G 0 0 0 0 0 0 1 2G 0 0 0 0 0 0 1 2G 46 CY X CY y CY z T xy T yz T xz (4.5) (4.6)
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Where E is Young's modulus, G is shear modulus, and vis the Poisson Coefficient. These three elastic constants are identical in all directions. Only two of them are independent and can be explained using Equation 4.5. In order to determine the material type of a composite, it is required determining all components of [C], where the general shape of [C] indicates the type of the material. IfC11=C22=C33; C12=C13=C21=C23=C31=C32; and C44=Css=C66, the material is considered an elastic isotropic material. IfC12=C21; C13=C31; and C23=C32, the material is considered an orthotropic material. IfC11=C22; C12=C21; C13=C23=C31=C32; and Css=C66, the material is considered a transversely isotropic material. In order to determine the equivalent properties of a composite, several approaches can be utilized, such as laboratory testing, finite element analysis, and analytical methods. In this chapter, the finite element and the analytical methods will be used. The results from both methods will be compared and used in statistical analysis for future predictions. 4.3 Evaluation of Composite Material Properties The geecomposite properties were determined in two approaches, the finite element and the analytical methods. The results obtained from both methods were compared and evaluated. 47
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4.3.1 Finite Element Approach in Determining the Mechanical Properties of a Geocomposite Numerical study using NIKE3D was completed on cubical reinforced soil specimens with lm x lm x lm dimensions each. The purpose of this study was to find the type and properties of the reinforced soil. The geocomposite specimens consisted of medium dense and dense Ottawa sandy soil, and geosynthetic inclusion. Different types of inclusions were considered in this study, including geotextile, geomembrane, and geogrid. Both, soil and reinforcement were assumed to be isotropic elastic materials. The sand and geosynthetic properties used are shown in Table 4.2. Table 4.2 "Soil and geosynthetic properties" Soil (Das 1993) Void ratio Poisson's ratio Density Density/g (e) (v) (Mglm3 ) (Kg.s2/m4 ) Medium Dense 0.630.53 0.20.3 1.9 193.68 to Dense sand Geosynthetic Thickness Poisson's ratio Density Young's Modulus (mm) (v) (glcm3 ) (MPa) Geosynthetic 2.5 0.20.4 0.94 221100 For the geocomposite specimens, the bedding plane was on the horizontal (X, Y) plane parallel to the bottom side of the cubic specimen and Z was in the vertical direction perpendicular to XY plane. The geosynthetic inclusion was allowed to slide relative to the sand due to the frictional interface between the two materials. In this study, tests were completed on geocomposite with I, 3, and s layers of 48
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equally spaced inclusion as shown in Figure 4.3. Under each spacing (S) category, tests were completed with three different values of mean pressure ( cr0). These values were 68947.5 N/m2 (10 psi) 172368.9 N/m2 (25 psi) and 344737.8 N/m2 (50 psi) m :: : _.,. ;:""'_..., r J Figure 4.3 "Geocomposite cube element(one, three, and five Layers)" 49
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In 1968, Hardin and Black suggested the following two equations, 4.7 and 4.8, to determine the shear modulus (G) of sandy soil based in the shear wave velocity for low amplitudes ofvibrations (Das 1993). In these equations, G of round grained and angular sandy soil primarily depends on cr0 and soil's void ratio (e). Ground(N lm2) = 6908(2.17 e)2 0"0.5 max 1+e 0 Gangular (N 1m2)= 3230(2.97e )2 0"0.5 max 1+e 0 By means of the former equations 4.7 and 4.8, average shear modulus (G) of sandy soil for the two cases, round and angular grained soil was calculated. Soil modulus of elasticity (E5 ) was then calculated using Equation 4.5, presented in the following format: E=2(1+v)G For the first cro value, Es was found to vary from 70 MPa for medium dense sand to 300 MPa for dense sand. For the second cr0 value, Es was found to vary from 100 MPa for medium dense sand to 450 MPa for dense sand. For the third cr0 value, E5 was found to vary from 150 MPa for medium dense sand to 650 MPa for dense sand. In each range ofE5 two supplementary inner points were selected to recognize its effect on the geecomposite properties as shown in Figure 4.4. ,. 50 (4.7) (4.8)
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70 150 I I I 220 3 3 0 450 1 t I I r.,..,.....;.......,....,._.,.,..,.,..1.....,... ..,..;. I apt12368.9 w.m:... 2s psi Figure 4.4 "Es (MPa) distribution under different cr0 Under all Es categories for geocomposite specimens with 1 layer inclusion, 1100 MPa, 550 MPa, and 270 MPa were used for modulus of elasticity of geosynthetic (E8). This range ofE8 represented the geogrid, geomembrane, or geotextile materials that have strong stiffness. Values of0.3, 0.25 and 0.2 were sued for Poisson ratio of soil (vs). Values of0.4, 0.3 and 0.2 were used for Poisson ratio of geosynthetic (vg) Values of 1.0, 0.5, and 0.25 were used for friction coefficient (f) between soil and inclusion. For geecomposite with 3 and 5 layers inclusion, 1100 MPa and 270 MPa were used for E8 0.3 and 0.2 were used for Ys, 0.4 and 0 2 were used for Ys, 1, 0.5, and 0.25 were used for f. Further investigations were completed on this cube to analyze its performance when weak geotextile were used The elastic moduli of such material were in between 22 MPa and 100 MPa. For that reason, tests of specimen with different E8 under the variation of cr0 S, E5 v5 V g, and fwere completed. The frame in Figure 4.5 was provided to show a summary of all the performed tests. 51
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1 Layer, S .500 mm Figure 4.5 "Theoretical frame of all numerical tests" A total of 468 combination tests were applied on the geocomposite specimens. For each combination, the deviator stresses ( O"x, O"y, O"z, 'txy, 'tyz, and 'txz) were applied separately on the geocomposite specimens with each magnitude of 689475 N/m2 (100 psi). The gravity load due to the specimen's own weight and o0 were also applied. This resulted in a grand total of 2808 tests. Each test lasted 30 seconds with 1 second increment, starting with gravity load where it reached an acceleration of9.81 m/s2 after 10 seconds and stayed constant for the rest of the test period. cro was applied after 10 seconds, it reached its ultimate load at 20 seconds and stayed constant for the rest of the testing period. After 20 seconds, 52
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the deviator stress was applied and reached its ultimate load at 30 seconds. These three loads distributions were shown in Figure 4.6. 1 0 _. Loads Increments 1.2 ""I 1 0.8 0.6 0.4 0,2 Gravfty I COnfinfng /1 1 / I 1'ressu'l stress < .r L I i / I j / .i / .. / k< 1 / /, I 0 0 10 15 20 25 TJme (sec) Figure 4.6 "Load curve distributed for all 3 loads; gravity, confining pressure, and deviator stress" I f 35 After running the numerical analysis using NIKE3D on the geocomposite specimens, the strain components (Ex, Ey, Ez, Exy, Eyz, Exz) due to the application of each individual stress of crx, cry, crz, 'txy, 'tyz, 'txz were determined. These results were then used to calculate the components of [C]. From [C], the material type and properties were determined. In this study, only three set of tests are explained and the rest of all results are tabulated in Appendix A. These sets are 1) geocomposite with 1 layer inclusion, 2) geocomposite with 3 layers inclusion, and 3) geocomposite with 5 layers inclusion. For the three mentioned sets, cr 0 = 68947.5 N/m2 (10 psi), Es= 70 MPa, Eg= 1100 MPa, v5= 0 3, Yg = 0.4, and Deviator stresses= 689475 (100 psi). 53
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1) Geecomposite specimen with 1 layer inclusion The resulted strain components due to all applied stresses are shown in Table 4.3. To calculate the components of [C], shown in Table 4.4, these strain components were divided by the deviator stress. Table 4.3 "Strain components of geocomposite with 1 layer inclusion" Ex By Bz Bxy Byz Bxz crx 0.0084 0.0023 0.0026 0.0000 0.0000 0.0000 crv 0.0023 0.0085 0.0027 0.0000 0.0000 0.0000 crz 0.0027 0.0027 0.0095 0.0000 0.0000 0.0000 'txv 0.0000 0.0000 0.0000 0.0106 0.0000 0.0000 'tvz 0.0000 0.0001 0.0000 0.0000 0.0128 0.0000 'txz 0.0000 0.0000 0.0000 0.0000 0.0000 0.0128 Table 4.4 "[C) for geocomposite with I layer inclusion" 1 : 0.0000 0.0000 0.0000 o.oooo o.oooo o.oooo .. o.oo@' 7 o.oooo o.oooo o.oooo 0.0000 0.0000 0.0000 !';(])J Q l54': 0.0000 0 0000 o.oooo o.oooo o.oooo o.oooo :o.ot&,' 6 o.oooo 0.0000 0 0000 0.0000 0.0000 0.0000 From the results in Table 4.4, it was observed that Cu=C22; C12=C21; C13=C23=C31:::;:C32; and Css=C66 This indicated that geecomposite material acteci as transversely isotropic From this observation, with the aid of Equations 4.4, the mechanical properties of geecomposite with 1 layer of inclusion under the specified mentioned conditions of cr0 E5 Eg, Vs, vg, and Deviator stress were as follow: 54
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1 v v 0 0 0 I E E E (J' & v 1 v X X 0 0 0 E E I (J' & E y y & v v 1 0 0 0 (J' z I I z = E E E & r xy 1 xy & 0 0 0 0 0 yz 2G r & yz xz 0 0 0 0 I 0 r 2G xz 0 0 0 0 0 1 I 2G G= E 2((1+v) 1 = Average(0.0122,0.0123) = 0.01225 =:> E = 81.648(MPa) E v E =Average( 0.0034,0.0034) = 0.0034 => v = 0.2755 1 = 0.013 8 => E = 72.5(MPa) E v 1 =Average( 0.0039,0. 0039,0.0039 0.0037) = 0.0039 E I => v =0.280 1 I = Average(0.0186,0.0186) = 0.0186 => G = 32.4209(MPa) 2G 1 = 0.0154 => G = 32.4209(MPa) 2G Using Equation 5, G was found to be: G = E = 81.648 = 32.0066(MPa) 2((1 + v) 2(1 + 0.2755) 55
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2) Geecomposite specimen with 3 layers inclusion The resulted strain components due to all applied stresses eire shown in Table 4.5. To calculate the components of [C], shown in Table 4.6, these strain components were divided by the deviator stress. Table 4.5 "Strain components for geocomposite with 3 layers inclusions" O'x O'v O'z 'txv 'tvz 'txz Ex Ey Ez Exy Eyz Exz 0.0091 0.0024 0.0029 0.0000 0.0000 0.0000 0.0024 0.0089 0.0029 0.0000 0.0000 0.0000 0.0027 0.0027 0.0093 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0111 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0124 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0124 Table 4.6 "[C) for geocomposite with 3 layers inclusion" 0.0000 0.0000 0.0000 ' . ,.,, '"'.?c . :.>"< 'i ,_j ': 0;0()35 j)'; 0129 : ; 0.0000 0.0000 0.0000 o.oooo o.oooo o.oooo o.oooo o.oooo o.oooo o.oooo o.oooo o.oooo o.oooo o.oooo o.oooo o.oooo . ;i j o.oooo o .oooo o.oooo o.oooo o.oooo Again, the shape of [C] indicated that the geecomposite sample with 3 layers of inclusion was transversely isotropic. Using Equations 4.4 and 4.5, the mechanical properties under the same specified conditions were 56
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1 = Average(O.Ol32,0.0129) = 0.0130 => E = 76.7667(MPa) E v =Average( 0.0034,0.0035) = 0.00345 => v = 0.2667 E 1 , = 0.0135 => E = 73.883(MPa) E v , =Average( 0.0039,0.0039,0.00429 ,0.0042) = 0.0040 E => v = 0.2970 1 , = Average(0.0179,0.0179) = 0.0179 => G = 27.8614(MPa) 2G l =0.0161=>G=30.992l(MPa) 2G Using Equation 5 G was found to be G = E = 76 7667 = 30.3015(MPa) 2((1 + v) 2(1 + 0.2667) 3) Geocomposite specimen with Slayers inclusion The resulted strain components due to all applied stresses are shown in Table 4.7. To calculate the components of [C], shown in Table 4.8, these strain components were divided by the deviator stress. Table 4.7 "Strain components of geocomposite with Slayers inclusion" Ex Ey Ez Exy Eyz Exz O'x 0.0087 0.0023 0.0027 0.0000 0.0000 0.0000 O'v 0.0023 0.0086 0.0027 0.0000 0.0000 0.0000 O'z 0.0027 0.0027 0.0096 0.0000 0.0000 0.0000 'txy 0.0000 0.0000 0.0000 0.0110 0.0000 0.0000 'tyz 0.0000 0.0000 0.0000 0.0000 0.0127 0.0000 'txz 0.0000 0.0000 0.0000 0.0000 0.0000 0.0128 57
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Table 4.8 "[C] for geocomposite with Slayers inclusimi" 0.0000 0.0000 0.0000 0.0000 0.0 )1.85 0.0000 o.oooo o.oooo o.oooo o.oooo o.oooo Again, the shape of [C] indicated that the geocomposite sample with five layers inclusion was transversely isotropic. Using Equations 4.4 and 4.5, the mechanical properties under the same specified conditions were 1 = Average(0.0126,0.0124) = 0.0130 E = 79.9425(MPa) E v =Average( 0.0033,0.0034) = 0.00335 v = 0.2686 E 1 I = 0.0139 E = 72.0054(MPa) E V I 1 =Average( 0.0039,0.0039,0.0040,0.0040) = 0.00395 v = 0.2842 E 1 I 1 = Average(0 0185,0.0186) = 0.01855 G = 27 9892(MPa) 2G _l_ = 0.0160 G = 31.3239(MPa) 2G Using Equation 4.5, G was found to be G = E = 79 9425 = 31.5074(MPa) 2((1 + v) 2(1 + 0.2686) As a result from all sets, geocomposite material was determined to be a transversely isotropic in which the mechanical properties were equal in the horizontal X, Y plane. Both values ofG, from [C] and Equation 4.5 were very 58
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close, which proved that G is consistently dependent variable and only five independent constants represented this type of material. 4.3.2 Analytical Method in Determining the Mechanical Properties of a GeoComposite (James C. Gerdeen 2006) A 3D composite consist of two materials, the matrix which is denoted as M, and the inclusion or fiber which is denoted as f. In general, both materials have different properties which are also different from the equivalent properties of the composite. In the analytical method, both materials are assumed to be glued together. For that reason, the effective properties do not depend on the frictional interface. Instead, the composite material has properties based on the amount of each material, the shape ofthe inclusions, the orientation of the inclusions, and the material properties of both materials. Using this method, the effective modulus of elasticity (Ec) of the composite can be determined. This can be completed using one or both of the following models: isostrain model and isostress model. 59
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4.3.2.1 Isostrain Model In this model, the matrix and the inclusions are separated into parallel "compartments" in the composite as shown in Figure 4.7. The strain in each of the two materials is then equal. r==> Figure 4.7 "Parallel (Isostrain) Model" F =F +Ff c m Where: Fe= Total force acting on the composite element. F m = Force in the matrix. Fr= Force in the fiber. 60 (4.9)
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F F Ff F A FJAJ A A A A A Af A c c c m c c Where: Am = Area of matrix. Ar = Area of fiber A v = __!!1_ m A c Af v =! A Where: c V m = Volume fraction of matrix V r = Volume fraction of fiber. Substituting Equations 4.11 and 4.12 in Equation 4.10, the following stresses can be reorganized: (Y = (Y v + (Yfvf c m m Where: crc =the applied stress on top of the composite= FcfAc. crm =the applied stress on top of the matrix= Fm!Am. crr= the applied stress on top of the fiber= FriAr Since both materials are assumed to be glued together, the strains on both materials, Em and Er, are equal. These strains are also equal to the composite strain Ec as shown in Equation 4.14. & = & = & =& c m f 61 (4.10) (4.11) (4.12) (4.13) (4.14)
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Dividing Equation 4.13 by the composite's strain, the modulus of elasticity of the composite can be determined as follow: E =E V +EJVJ c m m Where: Ec = Equivalent modulus of elasticity = crJE. Em= Modulus of elasticity of the matrix = crm/. Er = Modulus of elasticity of fiber = crr/E. 4.3.2.2 Isostress Model In this model, the two materials are lumped in a serial arrangement, which makes the stress the same in each material, however the strains are different. Figure 4.8 "Serial (Isostress) Model" 62 (4.15)
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L v = m m L c Lf Vf=L c For this model, the change in length is due to the contribution of both materials as shown in Equation 4.18. M =M +Mf c m Dividing Equation 4.18 by Lc will give the resulted strain of the composite (Ec) and the strain contribution of each material (Em, Er). MMMMLML & = _c_ = _!!1_ + _1_ = _!!1__!!1_ + _1_ __[_ = & V + & V c L L L L L L L mm ff c c c m c f c As mentioned above, the stresses in both materials are equal, and these stresses are also to the stress in the composite material, shown in Equation 4 20. CY = CY = CYf = CY c m Substituting Ec by crJE c Em by crm!Em, and E r by crtffir, and considering the . stresses equality in Equation 4.20, the following formula can be used to determine the effective modulus of elasticity v v 1 _1_ = __!!i_ + _j__ E = E E E c Vm VI c m f +Em Ef 63 (4.16) (4.17) (4.18) (4.19) (4.20) (4.21) ... ...
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The three examples used in the finite element sections (1 layer, 3 layers, and 5 layers) are repeated and analyzed using the analytical method considering both models. The isostrain model will give the plane modulus of elasticity and the isostress model will give the vertical modulus of elasticity. 1. 1/ayer For 1layer, the following properties were given Em =70 MPa Er= 1,100 MPa !sostrain model was used to determine the plane modulus of elasticity (E) of the composite: t of reinforcement = 2.5 mm Ac = 1,000 mm x 1,000 mm = 1,000,000 mm2 Ar = 2.5 mm x 1000 mm = 2,500 mm2 Am = 1 ,000,000 2,500 = 997,500 mm2 Vm = 997,500/1,000,000 = 0.9975 V r= 2,500/1,000,000 = 0.0025 E = E V + EJVJ = 70(0.9975) + 1100(0 0025) = 72.575MPa c m m E = 72.575 MPa !sostress model was used to determine the vertical modulus of elasticity (E') of the composite: Lc= 1000 mm. 64
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Lm = 997.5 mm Lr= 2 5 mm V m = 997.5/1,000 = 0.9975 v f = 2.5/1,000 = 0.0025 E = 1 c vm VI +Em Ef = 1 = 70.164MPa 0.9975 0.0025 +70 1100 E' = 70.164 MPa 2. 3/ayers !sostrain model was used to determine the plane modulus of elasticity (E) of the composite: t of reinforcement = 2 5 mm Ac = 1,000 mm x 1,000 mm = 1,000,000 mm2 Ar=3 x 2.5 mm x 1000 mm = 7,500 mm2 Am = 1 ,000,000 7,500 = 992,500 mm2 Vm = 992 500/1,000,000 = 0.9925 Vr= 7,500/1,000,000 = 0.005 E = E V + EJVJ = 70(0.9925) + 11 00(0.0075) = 77. 725MPa c m m E = 77.725 MPa !sostress model to determine the vertical modulus of elasticity (E') of the composite: Lc= lOOOmm. 65
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. Lm = 992.5 mm Lr= 2.5 x 3 =7.5 mm Vm = 992.5/1,000 = 0.9925 Vr= 7.5/1,000 = 0.0075 E = 1 c vm VI +Em Ef 1 = 0.9925 0.0075 = 10.495MPa +70 1100 E' = 70.495 MPa 3 5 layers !sostrain model to determine the plane modulus of elasticity (E') of the composite: t of reinforcement = 2.5 mm Ac = 1,000 mm x 1,000 mm = 1,000,000 mm2 Ar =5 x 2.5 mm x 1000 mm = 12,500 mm2 Am= 1,000,00012,500 = 987,500 mm2 Vm = 987,500/1,000,000 = 0.9875 Vr= 12,500/1,000,000 = 0.0125 E = E V + E/Vf = 70(0.9875) + 1100(0.0125) = 82.875MPa c m m E' = 82.875 MPa !sostress model was used to determine the vertical modulus of elasticity (E') of the composite: Lc= 1000mm. 66
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Lm=987.5 mm Lr= 2.5 x 5 =12.5 mm Vm = 987.5/1,000 = 0.9875 Vr= 12.5/1,000 = 0.0125 1 1 E c = _v_m __ v_f_ = 0.9875 0 0125 = 70.829MPa ++70 llOO Em Ef E' = 70.829 MPa The results obtained from the finite element and the analytical methods (Table 4.9) in good agreement. Table 4.9 "E and E' using finite element and analytical method" !layer 31ayers 5 Layers E E' E E' E E' (MPa) (MPa) (MPa) (MPa) (MPa) (MPa) Finite 81.648 72.5 76.76 73.883 79.425 72.001 element Analytical 77.725 70.164 77.725 70.495 82 875 70.829 Error 5.05 3.33 1.24 4.81 4.16 1.65 The analytical approach was simple and had results that were close to those obtained from finite element approach. However, this approach didn't consider the frictional interface between the reinforcement elements and the soil surface, where both materials were assumed to be glued together. In real applications, the soil and reinforcement layers are not tied to each other which cause the relative sliding between both parts. Also, using analytical method, only the modulus of 67
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elasticity in both directions can be determined which is not sufficient to determine all the other properties or the material type. 4.4 Stress Distribution of Reinforcement To investigate the stress distribution of geosynthetic, two extreme cases of the cube element with 1 layer were studied. In the first case, a very weak reinforcement that was weaker than the soil. In the second case, the reinforcement was very stiff. For both cases, the soil was considered to be soft sandy soil, with a low modulus of elasticity. The properties and the applied loadings were as summarized in Table 4.1 0. Table 4.10 "Properties and applied loadings ofthe cube element" Es cro O'z #of layers Eg Comment (MPa) (Pa) (Pa) (MPa) Case 1 70 68947.5728 689475.728 1 22 Weak Case2 70 68947.5728 689475.728 1 llOO Stiff 4.4.1 Weak Reinforcement In the first case, the cube element of 1 layer reinforcement was subjected to confining pressure and vertical pressure on the top surface (crz) of magnitudes 68947.5 MPa (10 psi) and 689475.7 MPa (100 psi), respectively. The soft soil with Es = 70 MPa was considered. However, the soil stiffness was more than three times of the reinforcement stiffness, where the Eg was only 22 MPa. For case 1, the resultant stress distributions. of each stress component were as shown in Figure 4.9 through Figure 4.14. Also, these components were summarized in Table 4.11. 68
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By observing the next Figures 4.9 through 4.14, and Table 4.11, following was found: 1When applying vertical pressure ( O'z), the maximum resultant stresses on the shell element were the horizontal normal stresses (crx and cry) of magnitude 0.14 MPa. 2The maximum normal stresses (crx, cry and crz) occurred near the middle of the reinforcement element and decreased at further elements from the center. 3The minimum stresses were the vertical shear stresses ('t yz, 'txz) with average magnitude of 65.0 Pa. 4The absolute vertical shear stresses increased at regions that were far from the center. 5All the horizontal normal stresses ( crx and cry) were positive. This meant that due to the applied vertical pressure, the normal horizontal stresses in the geosynthetic were tensile. 6The vertical normal stresses ( O'z) were negative. This indicated that due to the applied vertical pressure the normal vertical in the geosynthetic were compressive. 69
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Figure 4.9 "Xstress distribution for geosynthetic with E = 22 MPa" Figure 4.10 "Ystress distribution for a geosynthetic withE= 22 MPa" 70
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Figure 4.11 "Zstress distribution (Pa) for a geosynthetic withE= 22 MPa" Figure 4.12 "Plane shear stress, XY, distribution (Pa) for a geosynthetic withE= 22 MPa" 71
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Figure 4.13 "Vertical shear stress, YZ, distribution (Pa) for a geosynthetic withE= 22 MPa" Figure 4.14 "Vertical shear stress, XZ, distribution (Pa) for a geosynthetic withE= 22 MPa" 72
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......:1 UJ Node#. 301 302 303 304 297 298 299 300 293 294 295 296 289 290 291 292 Absolute Max X y 0 0 0.3 0 0.6 0 0.9 0 0 0.3 0.3 0.3 0.6 0.3 0.9 0.3 0 0.6 0.3 0.6 0.6 0.6 0.9 0.6 0 0.9 0.3 0.9 0.6 0.9 0.9 0.9 Table 4.11 "Stress components of weak reinforcement" O"x O"y O"z (Pressure) 'txy 'tyz 'txz (Pa) (Pa) (Pa) (Pa) (Pa) (Pa) 1.24E+05 1.24E+05 8.27E+04 3.16E+03 4.83E+Ol 4.01E+Ol 1.28E+05 1.34E+05 8.74E+04 1.56E+03 2.44E+Ol 5.63E+:01 1.28E+05 1.34E+05 8.74E+04 1.60E+03 2.20E+Ol 5.82E+Ol 1.24E+05 1.24E+05 8.26E+04 3.15E+03 4.45E+Ol 4.39E+Ol 1.34E+05 1.28E+05 8.75E+04 1.74E+03 6.30E+Ol 1.92E+Ol 1.39E+05 1.39E+05 9.29E+04 8.99E+02 3.26E+Ol 2.71E+Ol 1.39E+05 1.39E+05 9.28E+04 7.24E+02 2.90E+Ol 2.78E+Ol 1.34E+05 1.28E+05 8.73E+04 1.51E+03 6.03E+Ol 2.07E+Ol 1.35E+05 1.28E+05 8.76E+04 1.02E+03 6.51E+Ol 2.92E+Ol 1.40E+05 1.39E+05 9.29E+04 3.63E+02 3.55E+Ol 3.44E+Ol 1.39E+05 1.39E+05 9.26E+04 1.02E+03 2.87E+Ol 3.22E+Ol 1.33E+05 1.28E+05 8.70E+04 1.75E+03 6.32E+Ol 2.48E+Ol 1.25E+05 1.23E+05 8.29E+04 2.35E+03 5.26E+Ol 5.68E+Ol 1.29E+05 1.34E+05 8.76E+04 9.69E+02 3.02E+Ol 6.68E+Ol 1.28E+05 1.33E+05 8.71E+04 1.90E+03 2.13E+Ol 6.19E+Ol 1.23E+05 1.23E+05 8.19E+04 3.38E+03 5.02E+Ol 4.71E+Ol 1.40E+05 1.39E+05 9.29E+04 3.38E+03 63 66.8
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4.4.2 Stiff Reinforcement In the second case, the cube element of 1 layer with the same soft sandy soil was subjected to the same loading of the first case. However, the reinforcement was much stiffer than that in first case, where Eg was 11 00 MPa. For case 2, the resultant stress distributions of each stress component were as shown in Figure 4.15 through Figure 4.20. Also, these components were summarized in Table 4.12. By observing the preceding graphs, 4.15 through 4.20, and Table 4.12, the following was found: 7When applying vertical pressure (crz), the maximum resultant stresses on the shell element were the horizontal normal stresses (ax and cry) of average magnitude 2.15 MPa. 8The maximum normal stresses (ax, cry and crz) occurred near the middle of the reinforcing element and decreased at further nodes from the center. 9The minimum stresses were the vertical shear stresses ( 'tyz, 'txz) of average magnitude 80.85 Pa. 10The absolute vertical shear stresses increased at regions that were far from the center. 11All the horizontal normal stresses ( crx and a y) were positive. This meant that due to the applied vertical pressure, the normal horizontal stresses in the geosynthetic were tensile. 12The vertical normal stresses (crz) were negative. This indicated that due to the applied vertical pressure, the normal vertical stresses in the geosynthetic were compressive. 74
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Figure 4.15 "Xstress distribution (Pa) for a geosynthetic withE= 1100 MPa" Figure 4.16 "Ystress distribution (Pa) for a geosynthetic withE= 1100 MPa" 75
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Figure 4.17 "Zstress distribution (Pa) for a geosynthetic withE= 1100 MPa" Figure 4.18 "Plane shear stress, XY, distribution (Pa) for a geosynthetic withE= 1100 MPa" 76
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Figure 4.19 "Vertical shear stress, YZ, distribution (Pa) for a geosynthetic withE= 1100 MPa" Figure 4.20 "Vertical shear stress, XZ, distribution (Pa) for a geosynthetic withE= 1100 MPa" 77
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....} 00 Node# 301 302 303 304 297 298 299 300 293 294 295 296 289 290 291 292 Absolute Max X y 0 0 0.3 0 0.6 0 0.9 0 0 0.3 0.3 0.3 0.6 0.3 0.9 0.3 0 0.6 0.3 0.6 0.6 0.6 0.9 0.6 0 0.9 0.3 0.9 0.6 0.9 0.9 0.9 Table 4.12 "Stress components ofstrong reinforcement" O'x cry crz (Pressure) 'txy 'tyz 'txz (Pa) (Pa) (Pa) (Pa) (Pa) (Pa) 1.31E+06 1.29E+06 8.66E+05 5.99E+05 6.34E+02 5.96E+02 1.19E+06 1.96E+06 1.05E+06 3.00E+05 3.22E+02 5.92E+02 1.18E+06 1.96E+06 1.05E+06 2.99E+05 3.05E+02 6.01E+02 1.30E+06 1.30E+06 8.66E+05 5 99E+05 6.19E+02 6.14E+02 1.98E+06 1.19E+06 1.06E+06 2.83E+05 6.37E+02 2.78E+02 2.15E+06 2.13E+06 1.43E+06 1.41E+05 3.43E+02 2.72E+02 2.14E+06 2.13E+06 1.42E+06 1.51E+05 2.85E+02 2.75E+02 1.97E+06 1.18E+06 1.05E+06 3.01E+05 6.19E+02 2.84E+02 2.00E+06 1.20E+06 1.07E+06 3.16E+05 7.18E+02 4.32E+02 2.16E+06 2.14E+06 L46E+06 1.58E+05 4.15E+02 3.92E+02 2.16E+06 2.13E+06 1.43E+06 1.44E+05 2.74E+02 3.62E+02 1.98E+06 1.18E+06 1.05E+06 2.88E+05 6.61E+02 3.73E+02 1.34E+06 1.32E+06 8.85E+05 6.01E+05 7.95E+02 8.22E+02 1.23E+06 1.98E+06 1.07E+06 2.98E+05 4.66E+02 7.35E+02 1.22E+06 1.96E+06 1.06E+06 2.92E+05 2.84E+02 6.74E+02 1.33E+06 1.29E+06 8.71E+05 5.79E+05 7.04E+02 7.00E+02 2.16E+06 2.14E+06 1.46E+06 6.01E+05 79.5 82.2
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In both cases, the cube element acted similarly, where the center of reinforcement element experienced the most normal stresses and the edges experienced the most shear stresses. The vertical shear stresses were very small and didn't significantly change by increasing the stiffness of the reinforcement, but all the other stress components increased significantly. Also, for both cases the plane normal stresses ( crx and cry) were the maximum. For that reason, one of each should be always considered when analyzing the stresses of reinforcements. In other words, the tensile stresses in reinforcing elements must be considered in designing reinforced structures rather than any other stresses. 4.5 Summary And Conclusion Geocomposite model behaved as transversely isotropic material. It had six constants that would represent the material. The variation in elastic properties of constituent materials and other factors such as spacing and confining pressure affected the general outcome of the composite. The developed model was based on Hooke's law that linearly relates the stress to the strain components in 6 x 6 compliance matrix. No matter the configurations of either the cubes or the constituents' properties, the shape of this matrix was always indicating that the geocomposite was as transversely isotropic. All the results due to the finite element tests on the reinforced cube are included in Appendix A. Furthermore the Young's moduli obtained from finite element method were comparable, within a range of 5 % differences, with those obtained from the analytical method which proved the accuracy of the finite element method. 79
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5. Statistical Modeling of Transversely Isotropic GeoComposite (Michael H. Kutner 2005) 5.1 Theory of Multiple Regression Analysis The general linear regression model, with normal error terms, simply in terms of X variables. Y. =Po+ p1x.I + p2x.2 + ... + P 1x. +e. z z z pzp z Where: 13o, 13t, . j3p are parameters Xit, ... Xi,p are known constants Ei are independent that are normally distributed with mean value of zero and constant variance N(O, cr2 ) i = 1, ... n The mean response function for regression model ( 5.1) is E{Y} =Po+ fJ1X1 + fJ2X2 + + PPXP Thus, the general linear regression model with normal error terms implies that the observation Yi are independent normal variabl es, with mean E(HonYim Ko) as given by Equation 5.2 and constant variance fi. 80 (5.1) (5.2)
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To express general linear regression model in matrix terms, the following matrices must be defined: 1 xll x12 x1p 1 x21 x22 x2p &1 /31 y2 &2 /32 Y= ,X= 1 x31 x32 x3p &= ,/3= Yn & f3 1 XI x2 X n np The row subscript for each element Xik in the X matrix identifies the trial or case, and the column subscript identifies the X variable. In matrix terms, the general regression model ( 5.1) is Where: Y is a vector of responses (n x 1), X is a vector of parameters (n x (p)), J3 is a matrix of constants ((p) x 1), and E is a vector of errors. These errors are independent normal random variables (nx1), haze zero mean, E{E} = 0, and constant variancecovariance matrix (n x n): o2 0 o2 o2{s}= 0 o2 0 = o21 0 0 o2 Consequently, the random vector Y has expectation E{Y}=XP and the variancecovariance matrix (n x n) ofY is the same as that of E o2 {Y} = o21 81 (5.3) (5.4) (5.5) (5.6)
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The regression function represents a hyperplane through a cloud of points. In this model, is where the hyperplane intercepts the Y axis. In other words, if the scope of the model includes the point XI= 0, ... Xp = 0, then represents the E{Y} at XI= 0, ... Xp = 0. The parameters fori :f:. Ogives the change in mean response E{Y} per unit increase in Xi when all other variables are held constant. When estimating the regression coefficients, the least squares criterion for general linear regression model regression model is generalized as follow: Q =I (Y; Po fJ!Xil . f3 pxip )2 = (Yxp )' (Yxp) i=l The least square estimators are those values of ... that minimize Q, and the vector of the least square estimated regression coefficients bo, bi, ... bp as b. bo The normal equations suggest that the estimator b must satisfy X'Xb = X'Y which gives b=(xxtxv The fitted values are expressed by 'and the vector of fitted values yequal to y=Xb =X(xxtxv =HY H=x(xxtx 82 (5.7) (5.8) (5.9) (5.10)
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Where Y is the vector of the response values, and H (the hat matrix) is a projection operator. The residuals term, which is the difference between the response and the fitted values for the same Xi becomes e=Yy=YHY = (1H)Y and the covariance matrix ofthe residuals is given by 5.1.1 Analysis ofVariance (ANOVA) Table The analysis of variance involves examining sources of variation and the contribution of the regression to the variation of the Yi. This approach is based on partitioning of sums of squares and degree of freedom associated with the response variable Y. There are three deviations to be examined. These are: 1Total deviation: Yi Y mean, where Y mean is average value of the responses. 2Regression deviation: Deviation of line (simple regression) or plane (multiple regression) aroUnd mean: yi Y mean 3Error deviation: Deviation of observation around line for simple or plane for multiple: Yi Y'i. Hence, the resulted sum of squares are the total sum of squares (SSTO) which has n1 degrees offreedom associated with it, the regression sum of squares (SSR) which has p1 degrees of freedom associated with it, and the errors sum of 83 (5.11) (5.12)
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squares (SSE) which has np degrees of freedom. The values of these sums of squares are given by SSTO= 'L(Yi Ymeanf =Y'Y (:)Y'JY =SSR+SSE where J is the matrix of 1 s. SSE= l =Y'Yb'X'Y SSR = SSTOSSE = b'X'Y _!_ Y'JY n where n is the number of responses, and J is an n x n matrix of 1 s. The mean squares (MSR and MSE) can be calculated by dividing the sum of square by its own degree of freedom, which gives MSR= SSR p1 MSE= SSE np The above values including the sum of squares (SS), degree of freedoms ( df), and mean squares (MS) are summarized in Table 5.1. (5.13) (5.14) (5.15) (5.16) (5.17)
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Table 5.1 "ANOVA Table for general linear regression model" Source of ss df MS variation SSR = SSTOSSE MSR= SSR Regression = b'X'Y _!_ Y'JY p1 p1 n Error SSE= :Le{ np MSE= SSE =Y'Yb'X'Y np SSTO = L (; Ymean t Total = Y'Y (:)Y'JY n1 =SSR+SSE 5.1.1.1 Test for Regression Relation The analysis of variance is used to examine the hypotheses for testing the significance of the regression, i.e. whether there is a linear relation between the response and the set of predictor variables. These hypotheses are given by Eqution 5.18. Ho :/31 =/32 = ... =[JP =0 Ha: some [Jk O,k = 1, ... ,p The test static is constructed from the ratio shown in Equation 5.19. F*= MSR MSE The decision rule to control is If F* F(la; p 1, n p ), Conclude H0 IfF*> F(la;p 1,np), Conclude Ha The existence of a regression relation by itself does not, of course, ensure that useful predictions can be made by using it. 85 (5.18) (5.19) (5.20)
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Another factor used to measure the proportionate of total variation in Y associated with the use of the set of variables Xr, ... Xp is the coefficient of multiple determination R2 as shown in Equation 5.21. R2 = SSR =1 SSE SSTO SSTO The value ofR2 varies between 0 and 1. When it is equal to 0, all bk = 0 (k = 1, ... ,p), and when it is equal to 1, allY observations fall directly on the fitted regression surface. 5.1.2 Inferences about Regression Parameters 5.1.2.1 Interval Estimation of The confidence limits for with 1a. confidence coefficient are Where: 5.1.2.2 Test for Test for are conducted to determine whether or not = 0 in multiple regression model and set up with the following alternatives: Ho:f3k=O Ha:f3k;t:O 86 (5.21) (5.22) (5.23) (5.24)
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The test statistic, Equation 5.25, and the decision rule, Equation 5.26, are as follow: srkf iflt *I { 1 ; n p). conclude H 0 Otherwise conclude H a Furthermore, Partial F tests can be completed to test whether pkxk can be dropped from the model. The alternatives and decision rule of this test are shown in Equations 5.24 and 5.20, respectively. In this kind of test, two models are examined, full and reduced model. For both models SSE are calculated and donated by SSE(F) and SSE(R), respectively. The general linear test statistic is as shown in Equation 5.27, where df= np. F* = SSE(R)SSE(F) 1 (SSE(F)J dfR dfF dfF: Similarly, to test whether several pk = 0 can be dropped from the model, similar test can be constructed. The alternatives of this model are as indicated in Equation 5.28 Ho :/3 =/3 1 = ... =/3 1 =O q q+ pH a :not all of the /Jk in H 0 equal to zero The test statistic, where the first pq coefficients are the ones to be tested, is F*= ssR( xq, ... ,x pllx1 ... ,x q _1 )jssE( x 1 ... ,x P _1 ) pq np Usually, larger values ofF* lead to conclusion Ha 87 (5.25) (5.26) (5.27) (5.28) (5.29)
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5.1.3 Estimation of Mean Response and Prediction of New Observation 5.1.3.1 Interval Estimation of E{Y h} The estimated mean response yh corresponding to given values ofX1 ... Xp is yh =Xhb Where: xh =[1 xhl ... xhp l The confidence interval for the prediction is given by Where: 5.1.3.2 Prediction of New Observation E{Y h} The 1a precdition interval for a new observation Y h(new) corresponding to Xh is given by Equation 5.33. Y,1 {1;np }{pred} Where: s2 {pred} = MSE + s2 {rh} 88 (5.30) (5.31) (5.32) (5.33) (5.34)
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5.1.4 Diagnostic and Remedial Measures Diagnostics play an important role in the development and evaluation of multiple regression models. Each predictor should be examined with appropriate box plots, histograms, and dot plots. These plots can provide helpful, preliminary univariate information. Scatter plots of the response variable against each predictor variable can aid in determining the nature and strength of the bivariate relationships between each of the predictor variables and the response variable and in identifying gaps in the data points as well as outlying data points. The basic scatter plot can be facilitated by the use of the scatter plot matrix. A numerical representation of the scatter plot matrix is the correlation matrix shown in Equation 5.35. This matrix is symmetric and contains the coefficients of simple correlations rv 1 rY2, ... rvp between Y and each of the predictor variables, as well as all of the coefficients of simple correlation among the predictor variables such as r12 between X1 and Xz. 1 rYI rY2 rYp rYI 1 r12 rip p= rYp rlp r2p 1 Another important tool is residual plots. The residuals plotted against the fitted values can still be useful to assess the adequacy of the fit, the assumption of constant variance, outliers, etc. This can be completed by observing if any systematic deviations from the response plan, and if the varies of error terms varies with the level ofY. Residuals should also be plotted against each other of the predictor variables. Each of these plots can provide further information about 89 (5.35)
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adequacy of the regression function with respect to that variable. For assessing normality QQplots of the residuals are useful. 5.1.5 Building the Regression Model In many regression problems there will be a fixed sample size to work with and a moderate to large number of potential predictor variables. Generally, adding additional variables to a regression problem that already contains a small number of variables will improve predictive accuracy. Continuing to add variables (without adding more samples) will often lead to deterioration in predictive accuracy ( overfitting). From there, it is important to find the best subset of variables (variable section) considering that many subsets of variables are likely to do well. The model building process consists of the following major steps: 1Data collection and preparation: In this step, the data are examined through sample histograms, box plot, scatter plots, etc. 2Reduction of explanatory variables (variable selection) 3Model refinement and selection: This is completed by the examination of residuals ad other diagnostics 4Model validation: This is the last step where the best model is picked. 5.1.6 Automatic Search Procedures for Model Selection For any set ofp predictors, there are 2P alternative models can be constructed The calculation is based on the fact that each predictor can be either included or excluded from the model. It is very difficult and time consuming to make a detailed examination of all possible regression models. For that reason, automatic 90
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computersearch procedures have been developed to examine every combination of possible predictors for small to moderate sized numbers of candidate variables. The limited number of good subsets allows the investigator to choose the best model. The 2 most common approaches in model selection, known as variables selections, are best subset regression and stepwise regression. 5.1.6.1 Best Subset Algorithms The best subsets according to a specified criterion such as R2 Cp, and PRESSp are identified without requiring the fitting of all possible subset regression models. 5.1.6.1.1 Coefficient of Determination (R2p) According to this criterion, subsets for which R2 is high are identified as good subset. It is equivalent to using the error sum of squares SSEp following the form: SSE R 2 =1P SSTO This criterion is very effective used in many regression program such as MINIT AB and R, but it could be misleading, where adding more variables to the model will always increase R2 However, this method becomes more effective when looking for point at which adding more variables lead to small increase in R2. 91 (5.36)
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5.1.6.1.2 Mallows' Cp Criterion This criterion is concerned with the total mean squared error of the fitted values for each subset regression mode. The model which includes P potential X variables is assumed to have been carefully chosen so that . Xp) is unbiased estimator of (i. The solution of this criterion is: SSE C = ( p ) (n 2P) P MSE x1 ... ,X p Where, SSEp is the error sum of square for the fitted subset regression model with P parameters, and n is the number of cases. In order to identify a good subset of X variables, Cp value should be small and near to the number of variables (P). For example, if3 X variables are investigated, then Cp value should be somewhere near 3, and so on. 5.1.6.1.3 Akaike's Information Criterion (AICp) and Schwarz's Bayesian Criterion (SBCp) These criteria are given by AICP = nlnSSEP n1nn+2p SBC p = n In SSE p n Inn + [ln n }p The investigator should search for models that have small value of AICp or SBCp 92 (5.37) (5.38) (5.39)
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5.1.6.1.4 Prediction Sum of Squares (PRESSp) Criterion The PRESSp criterion is the sum of the squared prediction errors over all n cases, and is given by: PRESS p = :t (Y; Y;(i) Y (5.40) i=l The models with small PRESSp values fit well because of having small prediction 'errors. For that reason models with small PRESSp values are considered good candidates. 5.1.6.2 Stepwise Regression Methods This method becomes more useful when considering more potential variables for building the model. There are two main procedures, forward stepwise and backward stepwise. The forward stepwise, most widely used, starts by choosing the best single variable that has the largest t* statistics, Equation 5.41, then adding the best second variable with the first in the model, then adding the best third variable with the first two variables are still in the model, and so on until no additional variables have t* statistics above the present criterion. The purpose here oft* statistics is to test whether or not the slope is zero. Also, when doing so, the Pvalue at each step must be considered. For instance, if the Pvalue is less than 0.15 for that variable, then that variable is retained in the model. bk The backwards stepwise starts with all variable in the model and then drop the worst variables. 93 (5.41)
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5.1.7 Multicollinearity and its Effect One of the goals of regression is to understand the impact of a particular variable on the response. For uncorrelated predictors, identifying the impact of particular variable is straight forward. On the other hand, correlated predictors are offering the same information making it hard to decouple the impact of each variable. Good news, predictions are not often terribly affected by correlated variables, but standard errors may be affected. Multicollinearity can be detected with several ways, such as if large changes in the coefficients when certain variables are added or deleted, and large correlations between predictor variables. A more formal measure is the variance inflation factor (VIF) given by: V!Fj =(1Rlt Where, R/ is coefficient of multiple determination when Xj is regressed on the other p2 variables in the model. A roleofthumb is that a maximum VIF above 10 indicates that multicollinearity may be unduly influencing the leastsquares estimates. 5.2 Application of Multiple Regression Analysis on GeoComposite (5.42) The results, of 468 tests, from the finite element model have shown that there were two types of parameters. These parameters were input data (independents) and output results (dependents) as shown in Figure 5 .1. The independent parameters were cro, E5 Eg, v5 Vg, f, and S. The dependents parameters were E, E', v, v', G', and G. After completing all sets oftests, it was difficult to inform which parameters were strongly correlated, except for G. According to the 94
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definition of transversely isotropic materials, G is related to both E and v. From here, it was necessary to apply statistical analysis to correlate the strongly related parameters in linear equations. These equations would be used in the future for prediction purposes. Input Finite Element jjfft 1 Figure 5.1 The regression analysis framework" The regression analysis was completed for each of the six dependent variables separately using the statistical package, MINITAB Release 14. But before doing so, the predictor variables should be examined for multicollinearity using VIFs test. VIF1 = (1R t1 For S, R12= 0.016 VIF1= 1.0162. For cro, Rl= 0.283 VIF2= 1.3947. For E5 R/= 0.283 VIF3= 1.3947. For v5 R/= 0.246 VIF4= 1.32626. For Eg, Rs2 = 0.007 VIFs= 1.007. 95
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For vg, Rl= 0.252 VIF6= 1.3368. From these results, there is some impact of collinearity, but the VIFs are all below 10, rule of thumb, so collinearity should not be taken into consideration. In this chapter, a complete procedure for developing the plane modulus of elasticity, Eh, is described. The procedures of developing the other dependent variables are presented in Appendix B. At the end of this Chapter, a summary of all constitutive equations are provided. 5.2.1 Plane Modulus of Elasticity (Eh) The matrix plot and the correlation matrix were obtain and shown in Figure 5.2 and Table 5.2. A very strong linear relation was observed from the matrix plot between Eh (Y) and the Es, X3. This result agreed with that obtained by the correlation matrix, where rYX3 was equal to 0.9980. Also, the correlation matrix showed that Y is also correlated to the confining pressure, X2 and Spacing, X1 with less correlation factors. 96
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\0 ....] Figure 5.2 "The matrix plot of Eh with respect to all X variables"
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Table 5.2 "The correlation matrix of Eh and all X variables" Spacing Confining Soil.mod. Soilpois. Geo.mod. Geo.pois. I Fric.coef. XI x2 XJ X. Xs x6 Ehor I Spacing 0.045122 1 0.000889 1 0.001783 0.531605 1 XJ Soilpois. 0.00946 0.00257 I 0.00516 I 1 \0 p 00 Geomod. 0.01513 I 0.010852 I 0.004626 I 0.009296 1 0.03349 I 1 Xs Geppois. 0.008891 0.00345 0.004275 0.008591 0.24936 0.00258 1 Fric.coef. 0.00142 0.109655 0.0034 0.00683 0.31729 0.04435 0.33021 I 1
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5.2.1.1 Model Selection for Eh Using only the first order terms in the model with the aid of the best subset algorithms, the following model alternatives were obtained and were shown in Table 5.3. This method was based on R2 criterion, but also showed for each of the best subsets the Cp and (MSE)0 5 The best 2 subsets for each number of variables were identified. The mostright column of the tabulation shows the X variables in the subset. Table 5.3 "MINITAB output for Best two subsets ofEh model for each subset size" # of Variable R:z Co (MSE)u.s Predictors 1 99.7 630.7 10.419 Es 1 28.1 238382.0 153.9 cro 2 99.9 8.4 6.8369 S, Es 2 99.9 621.3 10.375 Es Eg 3 99.9 0.8 6.7740 S, Es......Eg 3 99.9 10.3 6.8433 S, Es, f 4 99.9 2.5 6.7793 S, Es, E.,.. f 4 99.9 2.7 6.7807 S, Es, Eg, Vo 5 99.9 4.3 6.7850 S, E8 f 5 99.9 4.4 6.7862 S, cro, Es, Eg, f 6 99.9 6.1 6.7907 S, Es, V8 Eg, Vg, f 6 99.9 6.2 6.7919 S, cro, Es, Eg, Vg, f 7 99.9 8.0 6'.7976 S, cro, Es, Vs, Eg, Vg, f From Table 5.3, it was seen that the best subset, according to the R2 criterion, is the ones with 2 variables and more. The R2 criterion value for all these models is 0.999, which is relatively high. This value stayed constant even when adding more variables. For that reason, it is not recommended to look at models with more than 2 or 3 variables at the most. To limit the choices, it is helpful to look at the Cp value. This value should be small and close to the number of variables. For that reason, the models of the 2 variable S, Es, and the model of the 3 99
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variables were good candidates since the Cp values were 0.8 and 10.3, respectively. Using the forward and backward stepwise regression, 3 steps were concluded and were as shown in Table 5.4. In the first step of stepwise regression method, the variable E5 X3 was chosen. This variable has the largest t* value, t = Xtb k} and smallest Pvalue of 0.00. The R2 resulted from the E, variable was 99.7% which is relatively high. However, Cp was very high and was very far from the number of variables In the second step, the S, variable with the second highest t* value of24.84 was added to the model. The addition of the S variable increased the t* value ofEs to 572.17 and R2 to 99.9%, and significantly reduced the Cp value to 8.4. In the third step, Eg with the third larges t* value of3.11 was added to the model. By adding the Eg variable, the t* value ofE5 slightly increased to 577.44. As a result, the R2 value didn't increase and stayed constant, but Cp decreased to 0.8. Further addition of any variable didn't affect t* value ofEs nor the R2 hence the analysis was terminated. Table 5.4 "MINIT AB Forward/backward stepwise regression output of Eh variable" Ste).l_ 1 2 3 Constant 1.405 16.178 17.134 Es (X3) 1.1081 1.108 1.108 t* 375.51 572.17 577.44 Pvalue 0.00 0.00 0.00 S (Xl) 0.0533 0.0532 t* 24.84 25.04 Pvalue 0.00 0.00 Eg (XS) 0.00189 t* 3.11 Pvalue 0.002 (MSE)us 10.4 6.84 6.77 R2 99.6 99.78 99.78 Mallow Cp 6307 8.4 0.8 100
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The results obtained from the stepwise regression method agreed with those obtained from best subset methodology. As a result, the regression function of Eh was as shown in Equation 5.43. The analysis of variance for this regression analysis was as shown in Table 5.5. E11 =17.l+l.llxEs +0.0532xS+0.00189xEg Where: (5.43) Eh = is the plane modulus of elasticity of the geocomposite material (MPa) Es = is the modulus of elasticity of soil (MPa) S =is the spacing between reinforcement layers (mm) Eg = is the modulus of elasticity of reinforcements (MPa) Table 5.5 "ANOV A Table for Eh regression model" Analysis of Variance Source Df ss MS F p Regression 3 15334843 5111614 111396.61 0.00 Regression 464 21291 46 Error Total 467 15356134 In this regression model, b1=1.11, b2 = 0.0532, and b3 = 0.00189. This means that 1 MPa increase in Es while the S and Eg are staying constant would increase the Eh by 1.11 MPa. Also, the 1 mm increase in S while Es and Eg stayed constant would increase the Eh by 0.0532 MPa. And, the 1 MPa increase in Eg while Es and S stayed constant would increase the Eh by 0.00189 MPa. 101
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5.2.1.2 Diagnostic and Remedial Measure of Eh The best model for Eh included E5 S, and, Eg. To asses the adequacy of the fit, and the assumption of constant variance, the residual plots must be prepared to suggest any modification if needed. The first graph to be considered was the residual versus the fitted values (Y) as shown in Figure 5.3. This plot didn't suggest any systematic deviation from response plane, which indicated that this linear relation was valid. .. '" I I ' ' I : Figure 5.3 "The residual versus the fitted values of Eh" To check if the residuals were normally distributed, the normal probability plot of the residual was plotted as shown in Figure 5.4. This plot showed that the residuals were normally distributed because the pattern was linear with some departure from linearity. 102
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... :,. t 1 .... .. { .... ........... .... ....... .,.. .... ... ""<.......... .t> ,,_.... .......... ,,,_ ""' < ;, '"t. ' '<' \ rt j r ;"' 7"' + = , w,,;.. :....0 .J......... .:.... ,: .......... ........... ..:.. .... ............ ""'"' ..... ..... ,.., ... !....;. .;: ... ._. J l i l I ... !..__ ,)._ "'J: ' "' .t __ .;! j 1 l Figure 5.4 "Normal probability plot of the residuals ofEh" To test whether there was a regression relation between the response Y h and the predictor variables Es, S, and Eg, the overall F test statistic is constructed: F* = MSR = 111396.61 MSE The alternatives of this test were Ho: /31 = /32 = = f3 p = 0 H a: some f3k :t: O,k = 1, ... ,p And the decision rule was IfF*:::;; F(1a;p 1,n p), ConcludeH0 IfF*> F(1a;p 1,np), Conclude Ha Considering ex= 0.01, and knowing that n = 468 and p = 3, 103
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F(l0.01; 31,4683) = F(0 99; 2, 465)= 2.30 Since F* = 111396.61 >> F = 2.6, Ha was concluded that Eh was related to E5 S, andEg. 5.2.1.3 Inference about Regression Parameters ofEh The confidence limits for (31 with 1a = 0.99 confidence coefficient are b 1 {1;np Since b1 = 1.11, s{bi}= 0.00192, and t(10.01/2; 4683) = t(0.995;465) = 2.576, the confidence limit for (31 was 1.11 2.576(0.00192) 1.10505 1.114945. With confidence coefficient 0.99, it was estimated that confidence interval for b1 was somewhere between 1.10505 and 1.114945 After that it was needed to test if Es term should be dropped from the model (partial test). The general linear test statistic is F* = SSE(R)SSE(F) 1(SSE(F)) dfR dfF dfF Where: SSE (R) = 15321577, SSE(F) = 21291 dfR = n3 = 4683 = 465 dfp = n4 =464 Then F* = 333442 For this test the alternatives are 104
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Ho: /31 = 0 Ha : /31 ;t: 0 and the decision rule is F(1a;p 1,np), Conclude H 0 IfF*> F(1a;p 1,np), Conclude H a Considering the decision rule, Ha was concluded because of the very large value ofF*. This indicated that J3t, Es should note be dropped from the model. Sterm The confidence limits for J3z with 1a. = 0.99 confidence coefficient are b2 { 1:; np }{b2 } Since bz = 0.0532, s{bz}= 0.002125, and t(10.01/2; 4683) = t(0.995;465) = 2.576, the confidence limit for J3z was 0.0532 2.576(0.002125) => 0.047726 /32 0.058674 With confidence coefficient 0.99, it was estimated that confidence interval for bz was somewhere between 0.047726 and 0.058674. After that it was needed to test if S term should be dropped from the model (partial test). The general linear test statistic is F* = SSE(R)SSE(F) /(SSE(F)) dfR dfF dfF 105
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Where: SSE (R) = 50057, SSE(F) = 21291, dfR = n3 = 4683 = 465, dfF = n4 = 464 Then F* = 626.9 For this test the alternatives are Ho: /31 = 0 Ha : fJI =f. 0 and the decision rule is F(la;p 1,n p), Conclude H0 IfF*> F(la;p 1,np), Conclude Ha Since F* = 626.9 > F(0.99;2,465) = 4.65, Ha was concluded. This indicated that S shouldn't be dropped from the model. The confidence limits for with 1a = 0.99 confidence coefficient are h1 {1;np}{b1 } Since b3 = 0.0019, s{b3}= 0.0006087, and t(l0.0112; 4683) = t(0.995;465) = 2.576, the confidence limit for is 0.0019 2.576(0.0006087) => 0.00033198 /31 0.003468 With confidence coefficient 0.99, it was estimated that confidence interval for b3 was somewhere between 0.00033198 and 0.003468. 106
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After that, it was needed to test if Eg term should be dropped from the model (partial test) The general linear test statistic is F* = SSE(R)SSE(F) 1(SSE(F)J djR djF dfF Where: SSE (R) = 21736, SSE(F) = 21291 dfR = n3 = 4683 = 465, dfp = n4 = 464 Then F* = 9.697 For this test the alternatives are H0 : {31 = 0 H0 :{31 :;:.0 and the decision rule is F(1a;p 1,np), Conclude H0 IfF*> F(1a;p l,np), Conclude Ha Since F* = 9.69799 > F(0.99;2,465) = 4.65, Ha was concluded. This indicated that B3, Eg shouldn t be dropped from the model. 5.2.1.4 Interval Estimation of E{Y h} The estimated mean response yh corresponding to given values ofEs, X1 = 220 MPa, S, Xz = 250 mm, and Eg, X3 = 1100 MPa is yh= = 242.0159 MPa Where, X h = [1 200 250 1100 J The confidence interval for the prediction is given by yh{1;np }{r;.} 107
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Where: s2 {yh}= {b}Xh and s2 {b} = MSE(X'Xt'. From the analysis ofVariance MSE = 46.00, and therefore using MATLAB software, s2{Yh}= 0.2679 With a confidence of0.99, t(l0.01/2;4683) = 2.576. From there, the 0.99 confidence limits for E{Yh} were 242.0159 2.576(0.2679t.5 => 240.6825 yh243.3492 Hence, with a confidence coefficient of0. 99 the estimated mean level ofEh with Es of220 MPa, S of250 mm, and Eg of 1100 MPa were somewhere between 240 6825 and 243.3492 MPa. This estimated interval was small which indicated the accuracy of the chosen model. 5.2.1.5 Interval Prediction for New Observation Y h(new) The Ia prediction limits for a new observation Y h(new) corresponding to Xh are YJ; {1;np )s{pred} Where Considering the previous given values ofXh, Xh =[I 200 250 II 00 ]', and knowing that both MSE and s2 {Y h }equal to 46.00 and 0.2679, respectively, therefore s2{pred}= 46.2679. From there, the prediction interval for new observation using a 99 percent confidence coefficient was 108
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242.0159 2.576(46.2679) 0 5 =:> 224.494:::; Yh :::; 259.538 5.2.1.6 Alternative Model of Eh In this section, several alternatives were considered in addition to the one that was obtained. The choice of these alternatives was based on the best subset method, and consequently the best alternatives and its analysis of variance were as shown in Equation 5.44, and Table 5.6, respectively. Eh = 17.5 + l.llEs + 0.05318 + 0.00191Eg + 0 .55/ Table 5.6 "ANOV A Table for alternative mode of Eh" Analysis of Variance Source Df ss MS F p Regression 4 15334855 3833714 83416.13 0.00 Regression Error 463 21279 45 Total 467 15356134 5.3 Summary and Conclusions Table 5.7 summarizes all the best subset and best alternative regression equation from the analyses that were performed to formulate the functional relationships between dependent and independents variables with high correlation coefficient using MINITAB Release 14. Using these equations one can effectively evaluate all parameters of the transversely isotropic model with given properties of soil, geosynthetic, interface friction coefficient, geosynthetic layer spacing, and confining pressure 109 (5.44)
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Table 5.7 "Summary of constitutive models equations for transversely isotropic geo composite" GeoComposite Rz Equations % Eh = 17.1 + 1.11x Es + 0.0532x S +0.00189x Eg 99.7 Eh = 17.5 + L11x E + 0.0531x S + 0.00191x E + 0.55 x f s g V h = 0000641 + 0.944745 XV S 000000425 X E 0.00001358x S g + 0 00000665x E8 0.008614x a 0 98.9 v h = 0.00230 + 0.946 X v 0.00000425 X E 0.00001358x S + s g 0.00000665 X E 0.008614 X (j 0 + 0 00326 X v s g E = 6.32 + 1.02 X E + 24.6 X v + 0.000953 X E v s s g 99 9 E = 5.62 + 1.02 X E + 23.4 X v + 0.000937 X E 0.448 X f v s s g v = 0.001350 + 0.989925 XV 0.0000308 X s0 00000222 X E v s g v = 0.00317 + 0.987x v 0.000031 X s0.000002 X E 97.2 v s g 0.00123xf G = 27.7608 + 0.39485 X E 92.4579 X v 0.0052xS v s s 99.8 G =27.8+0.395xE 92.5xv 0.00517xS0.034xf v s s Gh =21.6+0.449xE 10l.Oxv +0.0202xS+0.00132xE s s g 99.8 Gh = 22.3 + 0.449* E 101x v 0.0202x S + 0.113 x f s s 110
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6. Drained Triaxial Tests on GeoComposite Samples 6.1 Introduction In this chapter, three main tests were completed. They were: Geosynthetic Tensile, Isotropic Compression, and Conventional Compression Triaxial Test. Each test was essential in providing the materials properties of the Ottawa sand and the Polypropylene geotextiles, and was further simulated via the finite element method. Eventually, in case of reinforced samples, the transversely isotropic properties were determined based on the stress strain state and the spacing between reinforcing layers, where the constructive equations developed in Chapters 5 were applied to determine the homogeneous properties of the reinforced cylindrical samples. The main purpose of this chapter was to perform drained triaxial test on dry reinforced soil specimens of diameter 71 mm (2.8 in) and height 152.4 mm (6 in). Four different patterns were selected in this study to investigate the effect of spacing between reinforcement layers. These were soil without any reinforcement (unreinforced samples), with 2layers, with 4layers, and with 6 layers of reinforcement. In case of reinforced samples, the geotextiles were equally spaced at 76.2 mm, 38.1 mm, and 25.4 mm for the 2layers, 4layers, and 6 layers samples, respectively. 111
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For all patterns, three different confining pressures (cr3) were investigated to see the effect of stress path on these samples behavior. These three stresses were 103 kPa (15 psi), 207 kPa (30 psi), and 310 kPa (45 psi), and were essential in determining the soil properties that further needed in the finite element simulations. Before conducting the triaxial test, isotropic compression test was completed on the sandy soil alone to obtain the bulk modulus (K) and hardening parameters, and the geosynthetic tensile test was completed to obtain the elastoplasitc properties of the nonwoven geotextiles used in this project. 6.2 Test Materials 6.2.1 Ottawa Sand Dry Ottawa 3040 sand was used throughout the test program. Based on the results obtain at the Soil Laboratory at the Bureau of Reclamation located in the Denver Federal Center, the maximum and minimum unit weights of the sand were reported as Ymax = 112.19 pcf and Ymin = 97.52 pcf. In this study, a relative density (Dr) of70% was chosen. Using Equation (6.1), the density (y7o) was calculated as 1719 kg/m3 (107 pet). 1 1 r min r70 Dr = 7 0% = '==o=____;__;:__ 1 1 rmin rmax 112 (6.1)
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Since the desired height (H) ofthe soil samples was 152 mm and the desired diameter (D) was 71 mm, the amount of soil required for a sample, Ws, was estimated to be 1. 041 kg, where Ws = Vs xr70 And, V = 1C x(D)2 xH 4 s 6.2.2 Polypropylene Geotextiles A propylene, continuous filament, heat bounded nonwoven geotextile was used in the study. The mass per Unit area was 135 gm, and the average measured thickness was 0.4 rrtm, which resulted in a mass density of0.3 kg/m3 The geotextile was cut into circular discs with diameter slightly smaller than the overall sample diameter to ease the sample perpetration. 6.3 Laboratory Tests As mentioned before, three main tests (tensile geotextile, isotropic compression, and conventional triaxial tests) were completed to obtain the reinforcement and the soil propert:ies. For each test, the preparation process and results were in the following subsections. 63.1 Tensile Geotextile The tensile properties of geotextiles were obtained using a widewidth strip specimen tensile method as specified by ASTM D4595 (D459586 1986). This 113 (6.2) (6.3)
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test method is applicable to most geotextiles that include woven fabrics, and non woven fabrics. Through this test, the tensile strength, elongation, initial modulus, offset modulus, secant modulus, and breaking toughness can be calculated. 6.3.1.1 Equipments and Test Specimen An Instron, constant rate extension (CRE) testing machine was used. This machine must have adequate pen response to properly record the forceelongation curve. Each sample was 200 mm (8 in) wide and at least 200 mm (8 in) long with the length dimension being parallel to the direction in which the tensile strength is being measured. The entire width of each sample was sufficiently gripped by clamps in a way to prevent slippage or crushing. 6.3.1.2 Preparation Following the procedure stated in the ASTM D459586, the machine was set to a strain rate of 10 3% I minute. Also, the specimen must be mounted centrally in the clamps. The specimen length in the machine direction and the crossmachine direction test, respectively, must be parallel to the direction if application force. After that the tensile testing machine was started until rupture occurred. 6.3.1.3 Results and Discussion In the tensile geotextile test, two specimens were tested in the machine direction and another two were tested in the cross machine direction. All these tests were completed by Cesar Gonzales6 as a term project for Application of reinforced soil 6Graduate student of Civil Engineering at the University of Colorado at Denver. 114
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at the University of Colorado at Denver. For each specimen, the tensile strength ( ar), elongation (E), and the initial tensile modulus, were measured based on Equations 6.4, 6.5, and 6.6, respectively. The results of the four tests were as shown in Figure 6."1 and were summarized in Table 6.1. In Figure 6.1, an average curve of strength versus axial strain was included. The slope and the ultimate strength of this curve was obtained, where the it was eventually compared with that obtained from the finite element simulation. Where: ar = tensile strength which is the maximum force per unit width to cause the specimen to rupture, N/m (lbf/in) of width, F r = observed breaking force, N (lbt), and Ws =specified specimen width, m (in) (Ex Rx 100) & = '
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Where: Ji =initial tensile modulus which is the tangent to the first straight portion ofthe forceelongation curve, N/m (lbf/in) of width, F = determined from on the drawn tangent line, N (lbf), Ep = corresponding elongation with respect to the drawn tangent line and determined for,%, and Ws =specimen width, m (in) Table 6.1 "Summary of the geotextile tensile test results" Specimen# ac Ji (N/m) E (N/m) Machine 1 5801 0.625 57266.5 direction 2 6304.6 0.625 57266.5 Crossmachine 3 6282.7 0.75 63571.1 direction 4 5253.8 0625 63571.1 116 (6.6)
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1311 1aDD z eiDD .e ;.., c + 1 Test 2 ... Test3 Test4 Average a 10 2D 30 4a Ia 70 Axial Strain (%) Figure 6.1 "Geotextile Tensile Tests" 6.3.2 Hydrostatic Compression For this test, the state of stress is isotropic indicating that all principal stresses are equal, i.e. cr1 = cr2 = cr3 Since no shear stresses are included within an isotropic material in hydrostatic compression test, the bulk modulus, K, can be determined from this test using the theory of elasticity indicated in Equations 6. 7 through 6.10. 1 8 v = Kp K= E 3(12v) 8 = v v 0 117 (6.7) (6.8) (6.9)
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Where: Ev = is the volumetric strain, p = is the mean pressure, E =is the initial Young's modulus of soil, v =is the Poisson's ratio, 11 V = is the change in soil's volume, and V0 =is the initial volume of soil's sample. The hydrostatic compression test was conducted on a specimen of Ottawa sand only. The relative density of the Ottawa sand was 70%. In this test, a confining pressure was applied to the sample, and was adjusted in step increments. The corresponding volume change was measured using a burette filled with water. The change of water level in the burette would indicate the volume change of the specimen. The sample was subjected to a loading and unloading, as the unloading portion was necessary to determine the bulk modulus and the hardening parameters used for the cap model in the finite element simulations. 6.3.2.1 Sample Preparation Jo perform the hydrostatic compression test, a triaxial cell must be used to contain the sample and provide the required confining pressure, cr3 The main components of the cell were a base pedestal, top cap, a top plate, and locking ring as shown in Figure 6.2. The base have three holes with appropriate opening for connections. The three in the front, from left to right, have connectors to the confining pressure, the back pressure to the bottom of the sample, and the back 118 (6.10)
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pressure to the top of the sample. The fourth hole located in the back had a quick connector for the pressure transducer. Before starting the sample preparation, several things were checked and completed. These were: 1The bottom and the caps corresponding to the sample to be prepared were selected. For this test, a sample with a diameter 71 mm (2.8 in) was used. 2The confining pressure burette tubes were filled with water, and the tubes were then inspected to make sure no air bubbles were present. 3All the needed materials (sample mold, ring clamps, rubber membrane, a rings, filter papers, porous stone, level), measuring devices (tap measure, standing ruler, pi tape, weighing scale, calipers), Ottawa sand, raining device, funnel, spatula, rubber cement, vacuum grease, squeeze bottle, brush and screwdriver set were obtained as shown in Figure 6.2 through Figure 6.6. 4The vacuum device, Figure 6.7, was inspected to make S'!lfe that the oil level was appropriate. 119
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Figure 6.2 "Triaxial Cell" Figure 6.3 "Essential materials for hydrostatic and triaxial tests" 120
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Figure 6.4 "Measuring devices for hydrostatic and triaxial Tests" Figure 6.5 "Ottawa Sand, raining device and funnel" 121
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Figure 6.6 "Grease, rubber cement, screw driver and brush" Figure 6.7 "Vacuum Device" 122
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The first important step was cleaning and flushing the triaxial cell, where a compressor was used to blow out each exit opening. Then initial height including the base plate, bottom cap, top cap, two filter papers, and two porous stone was measured in 3locations, Figure 6.8, using a standing ruler, and the average value was recorded. Figure 6.8 "Initial height measurement" The thickness of the membrane was then measured at the four corners, Figure 6.9, using a caliper, and the average value was recorded. After the initial measurements, the sample was ready for preparation. This started with placing one porous stone and one piece of filter paper on the bottom cap followed by pulling the membrane over the cap. Two small 0rings were fitted over the membrane and bottom cap as shown in Figure 6.1 0. The main objectives of membrane were to enclose the specimen in the triaxial cell proving a cylindrical shape and preventing any water leaking from the surrounding confining pressure into the sample. This was followed by carefully assembling a split mold around 123
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the bottom cap. The top end of the rubber membrane was folded back and wrapped around the top end ofthe split mold as shown in Figure 6.11. A vacuum pressure was then applied through the mold to hold the membrane tightly in place against the inner wall of the split mold as shown in Figure 6.12. Figure 6.9 "Thickness of membrane measurement" 124
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Figure 6.10 "Pulling the membrane over the bottom cap of the sample" Since the required Dr was 70 2%, and the sample volume was determined using a diameter of 71 mm and a height of 152 mm, the amount of sand was determined based on Equations 6.1 through 6.3 to be 1041 gm. This amount was added in three layers using a raining device and a funnel to obtain uniform sample. Adding the soil was completed by placing the raining device inside the mold before pouring the sand. The raining device was filled with the soil and was lifted up slowly. This was followed by taping the mold very gently. This was repeated for the other two soil layers and shown in Figures 6 .13 through 6.15. After placing all the soil layers, the remaining filter paper, porous stone, and the top cap were then place on top of the sand. At this point, the sample was leveled, as in Figure 6.16, using a small leveling device. Then the upper end of the rubber membrane was released from the split mold and was wrapped around the top cap and sealed using two 0rings. The vacuum was then removed from the mold and applied to the top of the sample producing an 125
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effective stress of 13.8 kPa (2 psi) to prevent the sample from collapsing upon the removal of the split mold. The reason behind applying a vacuum to the sample was because sandy soil has zero cohesion and this stress was allowing the sample to stand without the split mold support. Figure 6.11 "Assembling the split mold around the bottom cap and wrapping the membrane around the mold" 126
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Figure 6.12 "Applying vacuum between the split mold and the membrane" Figure 6.13 "Adding the soil the using a raining device and a funnel" 127
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Figure 6.14 "Lifting the raining device slowly" Figure 6.15 "Tamping the sample gently" 128
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Figure 6.16 "Placing and leveling the top cap" Figure 6.17 "Applying vacuum pressure to the top of the sample" 129
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After 10 to 15 minutes of applying the vacuum, the split mold was then removed and the sample dimensions were measured. The height of the sample was measured again at three locations using a standing ruler and the diameter of the samples was measured at three locations along the height of the sample as shoWn in Figures 6.18 and 6.19. An average value for both height and diameter were recorded and the Dr% was checked again to make sure that the value was within the 70 %. Leaks were checked using a squeeze bottle by observing if any of the water was sucked into the sample. Holes were patched using rubber cement as in Figure 6.20. The ball was then placed on the top cap. This step, especially when conducting the triaxial test is a must. Without it loading the sample was impossible. The outer ring of the base plate was then greased and a greased large 0ring was placed on it, as shown in Figure 6.21. The top chamber was then set on the bottom plate and rotated to get a good seal followed by placing the locking ring on the cell and open ended tube to the air bleed of the top cap. The triaxial chamber was slowly filled with confining water, as shown in Figure 6.22, through the confining valve until the water flowed out of the open tube or a big bubble was generated at the top of the chamber. Slowly the vacuum was released and the confining pressure was increased equally and simultaneously to maintain the effective stress inside the sample at 13.8 kPa. The confining pressure was increased to 34.5 kPa (5 psi) and the volume change in the burette reading was recorded, as shown in Figure 6.23, since the drop of the water level reflected the contraction of the sample volume. Afte:r that the pressure 130
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was increased with 34.5 kPa increment up to 655 kPa (95 psi). The sample was then unloaded with the same increment. Figure 6.18 "Measurement of sample's height" Figure 6.19 "Measurement of sample's Diameter" 131
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Figure 6.20 "Adding rubber cement around the porous stone area" Figure 6.21 "Placing a greased 0ring" 132
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Figure 6.22 "Adding water through the confining valve" Figure 6.23 "Confining pressure and volume change" 133
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6.3.2.2 Results Discussion of Hydrostatic Compression Test After the completion of the isotropic consolidation test, the volumetric strain, Ev versus the mean pressure, p, was plotted as shown in Figure 6.24. In Figure 6.24, both unloading and loading results were calculated. Ev was determined using Equation 6.9, and p was simply the confining pressure considering all principal stresses were equal in the case of hydrostatic compression test. This curve reflected the volume change ofboth the sample and the system (the triaxial cell, porous stones, filter papers and the membrane). For that reason another hydrostatic compression test was conducted without including the soil sample. The same increment, 34.5 kPa, of confining pressure was chosen for this test in both loading and unloading paths. Considering both tests, the hydrostatic compression behavior of Ottawa sand only was obtained as shown in Figure 6.25 The bulk modulus, K, was determined to be the slope of the unloading portion of the hydrostatic compression test (C.S. Desai 1984), and was equal to 209 MPa. 134
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D.7l 0.1 :. o l i i !. OA l lljll lljll f! 0.3 IL IC .. 0.2 0.1 I I l l l. a I 0 1 2 3 4 I I 7 Volumetric Str1ln (%) Figure 6.24 "Isotropic consolidation test results due to Ottawa sand and the triaxial Cell" 0.7 Bulk Modulus { K):: Slope ot unloading portion 0.1 1\. = 209 MPa :_ 0.1 !. OA .. .. 0.3 IL Unlo3ding : 0.2 0.1 0 0 0.2 0.4 0.1 D.l 1 Valum .tric 1traln (%) Figure 6.25 "Hydrostatic compression result of Ottawa sand only" 135
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6.3.3 Conventional Triaxial Compression Tests The triaxial test is most reliable shear test and the apex of soil investigation. It is used for major projects such as earth dams. In this test a cylindrical soil sample with a length of at least twice its diameter is placed in a triaxial chamber. A specific confining pressure is applied by means of water, followed by a vertical load. This vertical load is applied at the top of the sample and gradually increased until the sample fails in shear along a diagonal plane (Chen 2000). A series of triaxial tests results, using different values of confining pressure contributes in creating Mohr's circles of failure stresses. Aseries of Mohr's circles are an excellent tool in determining some of the soil's strength parameters, such as cohesion, c, and internal friction The information about a soil's behavior that is obtained in a triaxial test is similar to that obtained in a direct shear test. However, this should not be taken to mean that the direct shear test and the triaxial test are identical, they are not. The triaxial test provides much more reliable results and the extent of application is superior to the direct shear test. In triaxial test, drainage can be controlled to simulate actual conditions in the field, the soil is free to fail on the weakest surface (Chen 2000), and more important, there is a controlled rotation of the principal planes. Thus, plots of a stress versus strain can be obtained, unlike the direct shear test. In both triaxial and direct shear tests, stress concentrations exist, but are less severe in the triaxial test. The better results obtained are not without cost; triaxial test a more expensive, more labor intensive, more time consuming, more intensive, and more complex than the direct shear test. In this chapter, sets of drained triaxial tests were conducted on two types of samples. The first set consisted of Ottawa sand and was referred to as un reinfoiced or unreinforced samples. The other set consisted of both Ottawa sand 136
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and geotextile reinforcements. For the reinforced samples, the reinforcement layers were equally spaced at 76.2 mm, 38.1 mm and 25.4 mm for the 2layers, 4 layers, and 6layers samples, respectively. For all four patterns, specimens were subjected to three confining pressure, 0'3. These were 107 kPa (15 psi), 207 kPa (30 psi) and 310 kPa(45 psi). Since the tests were completed in drained condition manner, the volume change was recorded and plotted against the axial strain. However, the most important outcome of this test was the plot of the deviator stress = cr1cr3) versus the axial strain, Ea. Eventually, these stress strain curves with the aid of Mohr circles were successful guides for determining most of the soil properties, such as initial elastic modulus (Ei), Shear modulus (G), and friction From these sets, the effect of reinforcement patterns and the confining pressure on the soil strength was observed. 6.3.3.1 Samples Preparation The sample preparation was similar to the hydrostatic compression test, considering the following factors: 1When preparing reinforced samples, the thickness of each reinforcing layer must be included in determining the final volume the specimen. 2The metal ball was very crucial. Without it, loading the sample would be impossible. 3The sample was consolidated up to the required 0'3 without any attempt to unloading. 4The Dr of the sandy soil was consistent for all samples and was 70 2%. While consolidating the sample, the MTS machine was allowed to warm up for at least 1/2 an hour. After consolidating the sample, the cell was carefully moved to 137
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the MTS machine in order to shear the sample. The tests were completed using strain control, where a displacement of up to 10% of the initial height was applied to the sample. Shearing each sample was completed in 1 hour. After that, sample was unloaded and the cell was cleaned to be prepared for the next test. Figure 6.26 "Drained triaxial test" 6.3.3.2 Results and Discussion of Drained Triaxial Tests The vertical deformation, the resulting force, and the burette reading were all recorded simultaneously. From there, the axial straiq (8a), deviator stress and the volumetric strain (8v) were measured at each deformation reading using the following Equations 6.11, 6.12, 6.13, and 6.9, respectively. HH c = 0 a H 0 Where: 138 (6.11)
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H = is the height of the piston at specific deformation, and Ho = is the initial height of the piston. F /J..(j=A Ao A==1& Where: a F = is the resulting force at a specific deformation, A = is the area at a specific deformation, and Ao =is the initial area. (6.12) (6.13) The stressstrain relationships for different combination of cr3 and spacing between reinforcement were as shown in Figures 6276.29. For all specimens, the deviator stress increased with the axial strain until failure occurred. Also, it was indicated that under the same confining pressure, the ultimate strength of reinforced samples exceeded that ofthe nonreinforced samples. For instance, the peak strength of a sample was 0.3 MPa under a confining pressure of 103 kPa. This strength increased to; 0.38 MPa when embedding 2layers of geotextile, 0.5 when embedding 4layers of geotextile, and 0.65 MPa when embedding 6 layers of geotextile. A significant increase of the peak strength was also observed when increasing the confining pressure. For a unreinforced sample, the peak strength increased from 0.3 MPa under 103 k.Pa confining pressure to 0.64 MPa and 0.88 MPa under confining pressures of 207 kPa and 310 kPa, receptively. Figure 6.30 was provided to summarize the peak deviator stresses as function of confining pressure and the number of reinforcing layers. In this figure, linear trends and almost parallel, were observed indicating the increase 139
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of soil strength as a result of adding more reinforcing layers and increasing the confining pressure. Furthermore, a plot of initial Young's Modulus versus the number of layers was plotted as shown in Figure 6.31. Figure indicated that under large confining pressure, the increase in the peak strength when adding more reinforcing layers is associated with a precipitous drop in initial stiffness. On the other hand, at a smaller confining pressure, the initial stiffness decreases when adding a small number of reinforcing layers, and adding more reinforcing layers doesn't significantly change the initial stiffness. The largest initial stiffness of 272 occurred for a unreinforced sample that was subjected to a confining pressure of 310 kPa. The initial stiffness of the specimen significantly dropped to 142 MPa when adding 2 layers of the geotextile, and further dropped to 92 MPa when embedding 6layers of the geotextile. In 1977, Broms concluded from his triaxial tests that at small strains, the stiffness of a reinforced soil is somewhat smaller than that in the unreinforced soil (Kanop Ketchart 2001). Wu, in 1989, indicated that this loss of stiffness at the small strain is negligible in field construction because the ratios of the reinforcement spacing to the reinforcement thickness are much greater than those in the triaxial tests (Kanop Ketchart 2001 ). In addition, the samples initially contracted during shear and started to dilate at axial strain of less than 1 %. This observable fact was noticed while measuring the volume change from the water level in the burette. Figure 6.32 indicated this phenomenon for all the three cr3. The reason behind this dilation was due to the relatively high density of the specimen. The sand particles were incompressible and with small gaps between them. Increasing the deviator stress forces the particles to slid resulting in an expansion in the sample volume. 140
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0.7 ..,........,._, 0.1 IL 0.1 1Soi1+4 layers i 0.4 So it+ 2 I er$' Q 0.3 Soli .. .. 0.2 0.1 D 2 4 I I 10 Axial Strain Figure 6.27 "StressStrain relation at a3 = 103 kPa" 1.2 1 IL l.u ., Solt+2 lwers !a.a Soil a :u ... Ill D.2 D D 2 4 I I 10 12 Axllil Sl:rwin rAt Figure 6.28 "StressStrain relation at a3 = 207 kPa" 141
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1.4 1.2 ... l!u iu Q I a.4 en D.2 a 1AOCII ';"1.20CD 11. !1.11111 u D.IOOD !! Q G.IOOD !a..m .X l D.ZIGD o.oaaa Soii+G l:.yers Soi1+4 l;t y ers Soil 2 4 I I 1a 12 Axial Stnln C'i) Figure 6.29 "StressStrain relation at 0'3 = 310 kPa" a 1 2 3 4 I 7 Figure 6.30 "Peak Stress Difference as function of number of reinforcing Layer and confining pressure" 142
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300 11' 1.210 iii :1 "9 J,., 0 UJO .. 10 !! .!: 0 0 2 :1(3. 1 OkPa) "'5,6003{.Loyera #): 6 1 .J(LO)e!d} + 2134 52 R:: .. o .9m E1 #)+ OOJ3S .. o..a.;s 207 kP" 103 kP3 3 4 I I 1 Num br of lainfarci ng Lay.,. Figure 6.31. "Initial Young's Modulus as a function of number of reinforcing layer" D 2 4 I 10 O.B 0 .CJ.I _, Dilati on i!. _,.I c .a ...: .1 D > .a 3.5 ... 4.S Aldllltraln C%) Figure 6.32 "Volumetric strainAxial strain relation of Ottawa sand" 143
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Since three different confining pressures were used in the triaxial test, Mohr's circles for each test at its peak vertical stress together with failure envelopes for each of four different reinforcing patterns were plotted in Figure 6.33. In this figure, the sample without reinforcement exhibited frictional resistance with zero cohesion. The other three sets of reinforced samples exhibited a near identical friction of36 and cohesion intercept (equivalent cohesion or C') which increased with increasing the number of reinforcing layers, i.e. less spacing, as shown in Figure 6.34 The equivalent cohesions for reinforced sandy soil were 22.3 kPa, 64.5 kPa, and 108.5 kPa for the 2 layers, 4 layers, and 6 layers samples, respectively. By observing the data in Figure 6.34, a linear model was obtained to represent the relation between C' and the spacing between reinfo rcement layers, and was as follow: log(c'(kPa))= 0.0235(S(mm))+ 2.5383 144 (6.14)
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c (kPa) Mohr's Circle for 0Layer, 2Layers, 4Layers, & 6Layers 2Layti'S Soil \ JEr 3\j\3\ \ cr(kPa) Figure 6.33 "MohrCoulomb envelope for samples with different reinforcing patterns" 145
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2.5 , I 2 I I \ D D.DD 10.111 2D.DD 3DJID S [mm) 40.00 10.00 Figure 6 34 "Relationship between the logarithm of equivalent cohesion and spacing between layers" 6.4 Finite Element Calibration Calibration was completed by obtaining the materials properties resulted from the laboratory tests. LSDYNA code was adopted to simulate the laboratory results. Different models were chosen to represent each material. For instance Cap model was best assigned for the Ottawa sand, elasticperfect plasticity for the geotextile and linear elastic for the bottom and the top cap. The properties of each material were determined based on the previous mentioned tests. 146
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6.4.1 Ottawa Sand A cap modee was used to simulate the sandy soil. This model as shown in Figure 6.35 consists of two yield surfaces, f1 and f2. In this model, f1 is the fixed yield surface and can be expressed as shown in Equation 6.15, in is function of J1 (C.S. Desai 1984) Where: J 2D = [ (o1 o2 f + (a 2 o3 f + (o1 o3 f] ..JJ;;. I _:.Z_I I I I I I I \b I I I I I I = Rb c Figure 6.35 "Geologic Cap model" 7 Detailed description of this model is presented in Chapter 3, THEORETICAL BACKGROUND. 147 (6.15) (6.16) (6.17)
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Based on triaxial test results, at different confining pressure, the first yielding surface can be plotted by fitting the data of J 20 versus J I From a triaxial test, crz is equal to cr3, and is equal to the confining pressure, where cr1 is the sum of the deviator stress, and the confining pressure, 0"3. If the trend of versus J1, due to different confining pressures, follows a straight line, further simplification to the first yielding surface by using DruckerPrager model as shown in Figure 6.36. lt Figure 6.36 "DruckerPrager criterion" For the case of Conventional triaxial test, a and k can be determined using the following formulas (C.S. Desai 1984). 2x sin a = ==__:_.J3 x (3 sin) k = 6xcxcos .J3x(3sin) 148 (6.18) (6.19)
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Where: = is the frictional angle, and c = is the cohesion. On the other hand, f2 is represented the elliptical yield caps and is expressed as shown in Equations 6.20 and 6.21 (C.S. Desai 1984). /2 =R2J2o +{JlCY =R2b 2 X=;)+Z Where: R =is the ratio of the major to minor axis of the ellipse, and lies in the range of 1.67 to 2 (Desai and Siriwardae, 1984), X = is hardening parameter that depends on the plastic vohunetric strain and can be written in terms of mean pressure, such as X = 3 x p. Graphically, it is the position on J1 axis where the cap surface intersects, C = is the value of J 1 at the center of the ellipse, El = is the plastic volumetric strain and is obtained from subtracting the elastic volumetric strain from the total volumetric strain, such as ep =8 ee v v v Z = is the size of initial cap D and W = are hardening parameters, and p = is the mean pressure. In general, the following parameters were to be determining to describe the cap model: 1Elastic parameters: bulk modulus (K), and shear Modulus (G), 149 (6.20) (6.21)
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2Failure envelope (f1 ): a,y, 8, and B 3Yield cap (f2): D, W, Z and R The bulk modulus, K, thus measures the response in pressure due to a change in relative volume. Therefore, K was computed from the hydrostatic test data of the sample. It was defmed as the slope of the unloading curve (C.S. Desai 1984) and was determined to be K=209MPa Other moduli describe the material's response (strain) to other kinds of stress are the shear modulus and Young's modulus. The shear modulus describes the material's response to shearing strains, and Young's modulus describes the material's response to linear strain. It was possible to normalize the stressstrain curves of unreinforced specimens by plotting llcr/cr3 against Ea as shown in Figure (6.37). This curve suggested that all curves could collaps into one curve giving an initial slope value of 6.876. Multiplying this value by cr3 as shown in Equation (6.22) resulted in an initial Young's Modulus (Ei) for each of the three cr3 E; =6.876x100xa3 (6.22) From there, Ei was determined to be as E 103 kPa = 70 MPa, E2o7 kPa = 100 MPa, and E3 1 o kPa = 260 MPa Knowing that the Poisson's ratio (v) of sandy soil was between 0 2 and 0.3 (Das 1999), v was assumed to be 0.25. From there and using Equations 6.22 and 6.23, G was determined to be function of cr3 as follow: G = Ei = _6_.8_76_x:I_o_o _a.::..3 2 X (1 + V) 2 X (1 + V) 150 (6.23)
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3.1 3 2.1 rr 2 'a <11.1 D.l D G103 kPa= 28 MPa, G2o1 kPa= 56 MPa, and G31o kPa= 84 MPa D 2 3 4 A:Kial Strain(%) 103kPa ...... 207 kPa ""*'"310kPOl I Figure 6.37 "Relation between normalized deviator stress and axial strain" I The failure envelope, f1 is determined by plotting the data points of the square root of ho versus J 1 that were resulted from the triaxial test on unreinforced samples under different confining pressure. J1 is the sum of the three principal stresses, such as J 1 = a 1 +a 2 +a 3 In the J 1 expression, cr2 = cr3 = the confining pressure, and a 1 = a 3 + 11a where 11cr is the peak stress difference deviator stress. On the other hand ho, the second invariant of the deviatoric stress tensor has a formJ 2 D = a 2 f + (a2 a 3 f + (a1 a 3 f J. Under three confining pressure, J1 and J2o o s were calculated and summarized in Table 6.2. 151
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Table 6.2 "A summary of Ottawa sand shear failure data" 0'2 = 0'3 /!cr O'I Jl J 0.5 20 (MPa) (MPa) (MPa) (MPa) (MPa) 0.103 0.2995 0.4025 0 6085 0.1729 0.207 0.6393 0.8463 1.2603 0.3691 0.31 0.8842 1.1942 1.8142 0.5105 Plotting and fitting these data, as shown in Figure 6.38, indicated that for Ottawa sand, the f1 sUrface could be simplified to a linear surface as indicated by DruckerPrager criterion unlike the elliptical surface shown in the cap model. 0.4 S' w tG.3 .,. eft 0,2 D., D DJID a UJD J1 1.1D 2.DO Figure 6.38 "The shape of first yield surface, fl, for Ottawa sand according to Cap model" 152
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Based on Mohr Coulomb envelop for Ottawa sand, as shown in Figure 6.33, c were determined to be 36 and zero, respectively. Applying Drucker criterion, the parameters a and e of the fixed yield surface, ft, were determined to be: a= 6x ex cos = 6x Ox'cos36 = 0 and J3 x (3 sin) J3 x (3 sin36) B = 2x sin = 2x sin36 = 0 2815 J3 x (3 sin) J3 x (3 sin36) Furthermore, it was assumed that y is zero, so that the curvedtransition part, in Figure 6.36, coincides with the linear part of the fixed yield surface. Due to the linear behavior of fixed yield surface, it was also assumed that B is zero. From there: a=O, 8 = 0.2815, y = 0, and B =0 It is important to notice that a in the cap model becomes the J 2 D intercept of the fixed yield surface, which is the constant K in the Drucker model, and 8 in the cap model is the slope of the fixed yield surface, which is the parameter a in the Drucker model. The parameters y and B were assumed to be zero for simplicity. Further parameters, X, D, W, and Z can be computed using Equation 6.17 from hydrostatic test data. According to Zaman el al. (1982), most soil exhibit plastic deformation at very low stress level, hence it is reasonable to assume that Z, the initial yielding cap, is negligibly small and equal to zero (C.S. Desai 1984). The value of X can be expressed in terms of mean pressure, as X = 3 x p Rearranging Equation 6.21 and considering the elastic volumetric strain, Eve as 153
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function ofK as in Equation 6.7, the total volumetric strain was written in the following form: &v = w(1e 3pD )+ Where: Equation 6.23 can be used to determine the values of W and D based on the mean pressure versus the total volumetric strain, Ev curve obtained in the hydrostatic compression test. In the hydrostatic compression test, the soil specimen was subjected to high value of mean pressure, p, of 655 kPa (95 psi). This high pressure value forced the expression eJpD in Equation 6.23 to approach zero. Therefore, the value of W was equal to the measured volumetric strain at the large confining pressure (C.S. Desai 1984), and was equal to: W=0.7907% By observing Equation 6.23, p, Ev, K, and W were known, and the only unknown was D. Several trials were conducted to computeD that would give the best correlation between the predicted total volumetric strain using Equation 6.23 and observed total volumetric strain data from the hydrostatic compression test as shown in Figure 6.39. The value ofD that gives a good prediction ofthe mean pressure was found to be: D=0.6 The last parameter to determine in this model was R, which is the ratio of the major to the minor axis of the ellipse. The recommended values of R were in the range of 1.6 to 2 (C.S. Desai 1984). For this Ottawa sand and after several finite 154 (6.24)
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element simulations of the triaxial test on a unreinforced sample, the value of R was chosen to be R= 1.6 As a summary, the required parameters of Ottawa sand for cap model were as shown in the Table 6.3. u lu I. !u :I .. !u IlL c: ID.2 0.1 a a . aA a.a "' I Obserevt!cl u I I I : 1 Volumtrlc Stnln (%) Figure 6.39 "Observed and predicted Vol. strain vs. Mean pressure" 155
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T bl 6 3 "P a e arameters o fC d H 0 ap mo e or ttawa s d" an O'J = 103 O'J = 207 0'3 = 310 kPa kPa kPa p (kg/mJ) 1719 1719 1719 K(MPa) 209 209 209 G(MPa) 28 56 84 a 0 0 0 e 0.2815 0.2815 0.2815 'Y 0 0 0 (3 0 0 0 Xo 0 0 0 D 0.6 0.6 0.6 w 0.07907 0.07907 0.07907 R 1.6 1.6 1.6 6.4.2 Reinforcement Properties (Geotextile) By observing, the behavior of reinforcement, Figures 4.1 when subjected to tensile loading, it initially behaved linearly elastic and then plasticity was initiated after 1 0% of straining. LSDYNA does not have the model that exactly represents this behavior. The model that was chosen to represent the reinforcement was Isotropic ElasticPlastic as shown in Figure 6.40. This was chosen due to the fact it exhibits similar behavior as geotextile used in these tests. For Isotropic ElasticPlastic materials, five constants must be defines. Theses are mass density (p ), shear modulus (G), Yield stress (Y), Plastic straining hardening (Etan), and Bulk modulus (K). 156
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Figure 6.40 "Isotropic Elastic Plastic material" The thickness of the geosynthetic element was measured at 3 locations using a caliper and an average value was recorded The average thickness was found to be: t=0.4mm The mass per unit area was measured also and was equal to 135 gm/m2 From measurements, t and mass per unit area, the density was determined to be: p = 0.3 kg/m3 The results obtained from the test results showed that the average initial secant modulus was 60.8 kN/m. Initial Young's Modulus ofthis geosynthetic, Egis equal to the initial secant modulus divided by its thickness. Therefore, an average Eg value was equal to: Eg= 159 MPa 157
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E Assuming a Poisson's ratio (v) of0.3, knowing that K = ( g ) and 3x l2v E G = ( g ) therefore, K and G were equal to: 2x l+v K= 132.5 MPa G = 61.5 MPa Because of the difficulties of simulating the exact behavior or reinforcement when subjected to tensile loading, Y was assumed to be halfway between the stresses ofthe initiation of nonlinearity and ultimate strength which was 15.5 MPa, therefore, Y was assumed to be: Y= 13 MPa Finally, the hardening parameter, E1an, was computed as the slope of the line between the yielding strength and ultimate strength reached in the stress strain curve and was equal to: Etan = 10 MPa As a summary the required parameters of the heat bounded nonwoven geotextile used in the following drained triaxial tests and according to isotropic elastic plastic model were as shown in the next table. 158
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Table 6.4 "Summary of the geotextile properties" p (kglm3 ) K G y Etan (MPa) (MPa) (MPa) (MPa) 0.3 132.5 61.5 13 10 6.4.3 Bottom and Top Cap (Steel) Caps, top and bottom, were made of Stainless Steel material that has a very large value of yield strength compared to the Ottawa sand and geotextile reinforcements. For that reason, Isotropic Elastic model was assumed for both caps. In this model, three constants must be defined. They were the mass density (p), Young's Modulus (E), and Poisson's ratio (v). For stainless steel, the elastic properties were as shown in the next table (Beer, et. al, 2003). Table 6.5 "Summary of steel caps properties" 7920 E (MPa) 132.5 v 61.5 6.5 Finite Element Simulation and Validation Finite element analyses, using the finite element code LSDYNA, were completed to simulate the measured test results that were obtained from laboratory experiments. The properties of the utilized materials in fmite element simulations were implemented from the laboratory test. First, the tensile test on geosynthetic material was completed, and then the triaxial tests under three different confining pressures (103, 207, and 310 kPa) were completed on Ottawa sand only. After 159
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that, reinforced samples with all three patterns (2, 4, and 6layers) with each subjected to the three confining pressures were subjected to a triaxial loading. 6.5.1 Tensile Test on Geotextile From Section 6.4.2, it was assumed that an Isotropic ElasticPlastic material model was assigned for the nonwoven geotextile used in this part of this project. This material was subjected to tensile loading following the ASTM D459586 standards. As for the laboratory test, the sample was 200 mm wide by 200 mm length. This geosynthetic was modeled as fournod shell element with a constant thickness of 0.4 mm. The reinforcement sample was subjected to prescribed vertical deformation of 120 mm which was equal to 65% axial strain The resulted forces were recorded and plotted against the axial strain. The finite element results were compared with those obtained from the laboratory test as shown in Figure 6.41. From Figure 6.41, it was noticed the suitability of isotropic elastic plastic model in simulating the Continuous Filament NonWoven HeatBounded Geotextile material. This model and the properties of the geotextile were investigated when simulating the triaxial test on the reinforced cylindrical samples. 160
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1.40Et03 1.2DE+03 I.OOEt02 .e I.ODE+02 c t4.DDE+02 2.GOEt02 .* .... .... lli. ... ""'!+ .... ,,._ _ ;,_.;. "' M&asured O.DDE400 +,a 10 70 Figure 6.41 "Tensile force vs. axial strain of the heat bounded geotextile" 6.5.2 Drained Triaxial Test on Ottawa Sand In this section, a mesh of Ottawa sand with properties determined in Section 6.4.1 was built using FEMB computer code as shown in Figure 6.42. The soil specimen was 71 mm diameter and 152.4 mm height. Two caps bottom and top (loading cap), that have the same diameter of the soil specimen were also added to the model. The caps provided boundary conditions, where the bottom cap was constrained from any movement. On the other hand, the top cap was allowed to move only in the direction of the applied deformation (Zdirection). Both, the soil and the caps were modeled as eight node brick elements. Loading the sample was completed in two stages. The first stage represented the confining pressure and the second one was the deviator stress applied in strain control condition. The confining pressure was applied in a way that all the element were subjected to that 161
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pressure that increased linearly and stayed constant during the application of the deviator load. In the second stage, the top nodes of the loading cap were subjected to vertical deformation that increased linearly up to 15.24 mm, which was equal to 10 % of axial deformation. Loading cap Base Plate Figure 6.42 "Finite element model of Ottawa sand specimen" Three finite element were completed considering the three confming pressures (103, 207, and 310 kPa). The average vertical stresses on the top ofthe soil were computed and plotted against the axial strain of the soil specimen. The 162
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results were compared to those obtain from laboratory test for the three confining pressures as shown in Figures 6.43, 6.44, and 6.45. These figures indicated that the cap model is capable in representing Ottawa sand, and further tests on this kind of soil using the cap model is promising. 0.3 Mea.sured fESim ulation 0 1 I I Figure 6.43 "FESimulation vs;Measured triaxial results of Ottawa sand@ cr3 = 103 kPa" 163
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0.7 D.l lo.s u 0.4 lu a D.2 en D.t 0 0 t 2 "'Measured FESimul a tion 3 4 I Axil Strain (%) I I Figure 6.44 "FESimulation vs. Measured triaxial results of Ottawa sand @ cr3 = 207 kPa" 0.1 ... "' ................ .... ,... ..... .. ,._ ....... ,..., . .,,. . "" .. ... ,..., ..... .,.,.. . 'iir 0.1 lo.T :: 0.8 I o.s !E a 0.4 Measured I 0.3 0.2 FESimul.uion 0.1 0 0 2 4 I I 10 Axial Strain (%) Figure 6.45 "FESimulation vs. Measured triaxial results of Ottawa sand@ cr3 = 310 kPa" 164
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6.5.3 Drained Triaxial Test on Reinforced Samples with Discrete Approach Once the properties of Ottawa sand and geotextile reinforcement were confirmed through triaxial and axial tests, respectively, the triaxial simulation of reinforced sample with different patterns were completed next. The three different patterns of reinforcement under the same three confining pressures were simulated. The procedure of simulating triaxial reinforced sample was similar to that of un reinforced sample . However, when generating a mesh of solid and shell elements for the discrete model, a gap must be taken into consideration that accounted for the reinforcement thickness. The thickness was distributed equally above and underneath the shell element. For all patterns, the reinforcing layers were equally spaced. The spacing between reinforcement layers was 76.2 mm, 38.1 mm, and 25.4 mm for the 2 layers, the 4 layers, and the 6layers specimens, respectively. The side view of these patterns was as shown in Figure 6.46. Furthermore, a contact friction was allowed between the soil surface and the geotextiles. The friction coefficient, J.l., was 0.5 corresponding to 36 internal friction angle $, where J.l. can be determined using the following equation (Lee 2000). 2 ,u = 3tan 165 (6.25)
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2ayers 4layers 64ayers Figure 6.46 "Patterns of reinforced samples" As for the unreinforced samples, the plots of axial strains versus the stress difference were plotted and compared to those obtained from the laboratory tests and shown in Figures 6.47 through 6.55. There were several observations that were noticed while simulating the laboratory test results using the finite element method. One of the most important observations was the capability of Geological Cap model in representing the behavior of sandy soil, where the results of both methods were comparable even under large strain deformations. Not only the unreinforced samples were well simulated, also the simulation of reinforced soil specimens agreed with the results that were obtained in the laboratory tests. Both finite element simulations and laboratory tests on the fact that adding more reinforcement and increasing the confining pressure can significantly increase the soil strength. 166
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OAI a A 'ii'UI l a.3 .. I a.2 a Measured FE Sim ul3tion = 0.11 i a.1 G.DI a a 1 2 3 4 5 I Axt.a Stnin Figure 6.47 "FESimulation vs. Measured triaxial results of reinforced sand and 2 layers of reinforcement@ cr3 = 103 kPa" 0.1 0.1 0.1 ................ ,. ... ""' .. .... .. .. "" ... "'! . .__;.... M e;u ured FESim ! 0 1 2 3 4 I I 7 I Axial StRin (%) Figure 6.48 "FESimulation vs. Measured triaxial results of reinforced sand and 21ayers of reinforcement@ cr3 = 207 kPa" 167
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1A 12 t 1 t a M&"sured JoA FESim 02 0 a 2 4 I I 1a 12 Axial Stnln (%) Figure 6.49 "FESimulation vs. Measured triaxial results of reinforced sand and 2 layers of reinforcement@ cr3 = 310 kPa" 0.1 0.1 ';' loA u c: 10.3 Q Me
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1.2 1 ... "' a. ...... !.a .a .. .. c Me .. sured aA I FESimulation !I CtJ a.2 a a 2 4 I I 1a 12 Axi.t Strain(%) Figure 6 .51 "FESimulation vs. Measured triaxial results of reinforced sand and 4 layers of reinforcement@ cr3 = 207 kPa" 1.2 ... 1 a. &a .a .. !a.l !E 0 Measured aA I FESimulation Itt a.2 a a 1 2 3 4 I I 1 I Axial Stnln ('1.) Figure 6.52 "FESimulation vs. Measured triaxial results of reinforced sand and 4layers of reinforcement@ cr3 = 310 kPa" 169
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D.l D.l ... 0.1 a. .. l.o.a u c Q.l D.4 c Me.1sured :0,3 FESimulation D.2 ell 0.1 a 0 2 4 I I 10 Axial Strain(%) Figure 6 53"FESimulation vs. Measured triaxial results of reinforced sand and .6layers of reinforcement@ cr3 = 103 kPa" 1 : l.o.l u IE c oA I CIJ 0.2 . .. Measured FESimuladon a 2 4 I 10 Axial Stnln (%) Figure 6.54 "FESimulation vs. Measured triaxial results of reinforced sand and 6 layers of reinforcement@ cr3 = 207 kPa" 170
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1A :. 1.2 !. u ju au .. .. ; 0.4 CIJ 0.2 .. ,. .. ..... ... Measured a 2 4 I I 10 Axial Strain fl.) Figure 6.55 "FESimuJation vs. Measured triaxial results of reinforced sand and 6 layers of reinforcement@ cr3 = 310 kPa" 6.5.4 Drained Triaxial Test on Reinforced Samples with Homogeneous Materia.l Approach In this section the transversely isotropic properties of reinforced soil samples were determined based on the constitutive equations developed in Chapter 5 and were as summarized in Table 6.6. The transversely isotropic properties varied from spacing to another and from confining pressure to another. The homogeneous models were also subjected to the same boundary conditions and applied loads of the discrete models. The results ofthis homogeneous approach were compared with those obtained from the laboratory test results and the results obtained from the finite element analyses on the discrete models. In case of transversely isotropic samples, the behavior acted, as expected, linearly. For that reason, only 0.5% of the strain was taking into consideration, and was as 171
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shown in Figures 6.56 through 6.64. These figures indicated the homogeneous approach gave reasonable results under small loading compared to the results obtained from laboratory tests and those of the discrete model. At larger strain, more than 0.5 %, the results obtained from the homogeneous models significantly deviated from the results of other two approaches. Further comparisons and evaluations of the homogeneous and discrete modes are presented in the next section with the aid of Young's moduli. 172
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Table 6.6 "The Transversely Isotropic Properties of Reinforced Samples" 2layers S =50.8 mm Given _properties cr3 Es Vs Eg Vg s MPa MPa MPa mm 0.103 70 0.25 160 0.3 50.8 0.207 100 0.25 160 0.3 50.8 0.310 260 0.25 160 0.3 50.8 Equivalent properties cr3 Eh vh Ev Vv Gv Gh MPa MPa MPa MPa MPa 0.103 63.60 0.23 71.38 0.25 32.02 28.99 0.207 96.90 0.23 101.98 0.25 43.87 42.46 0.310 274.50 0.22 265.18 0.25 107.04 114.30 4layers S =30.48 mm cr3 Es Vs Eg Vg s MPa MPa MPa mm Equivalent properties cr3 Eh vh Ev Vv Gv Gh MPa MPa MPa MPa MPa 0.103 62.52 0.23 71.38 0.25 32.13 28.58 0.207 95.82 0.23 101.98 0.25 43.97 42.05 0.310 273.42 0 23 265.18 0.25 107.15 113.89 S=21.7mm Equivalent properties cr3 Eb Vh Ev Vv Gv Gh MPa MPa MPa MPa MPa 0.103 62.06 0.23 71.38 0.25 32.17 28.40 0.207 95.36 0.23 101.98 0.25 44.02 41.87 0.310 272.96 0.23 265.18 0.25 107.19 113.71 173
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0.21 02 D.. !. i0.11 .. 0.1 a ... 0 0 ,/ / / / ,I / ... / .. / / .. ,I / / . Measured Homo_ Tr;'4nslsotr<,pic; 0.4 0.1 AKII Stnin (%) 1 Figure 6.56 "Triaxial test results of transversely isotropic material vs. the lab and the discrete modelof2layers sample@ cr3 = 103 kPa" 0.3 ... i_o.211 i 0.2 0.1 &; D.GII Mea:Sured FESim ulation Homo_Tnmslsotropic D 0.1 D.2 0.3 0.4 0.1 0.1 Axial Stnln (%) Figure 6.57 "Triaxial test results of transversely isotropic material vs. the lab and the discrete model of 2 layers sample @ cr3 = 207 kPa" 174
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0.1 0.1 lo.s I. c ;oA co.a la.2 en 0.1 0 a Measured FESimult)tion Hom_Translsotropic . . .. ... ... .. . . . . . . ... "' ..... 0.1 0.3 DA Axial Stnaln (IS) ... 0.1 0.1 Figure 6.58 "Triaxial test results of transversely isotropic material vs. the lab and the discrete model of21ayers sample@ o3= 310 kPa" CL I. :!0.11 Q a.1 cno.as Meuured FE.Sim ulation a 0 0.1 G.2 0.3 0.4 0.1 0.1 Axial Strain (%) Figure 6.59 "Triaxial test results of transversely isotropic material vs. the lab and the discrete model of 4 layers sample@ o3 = 103 kPa 175
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OA Measure.d Homo ... TransIsotropic 0.1 0.2 0.3 0.4 D.l 0.1 Axial Stnln (%) Figure 6.60 "Triaxial test results of transversely isotropic material vs. the lab and the discrete model of 4 layers sample@ cr3 = 207 kPa" 0.7 . . # Measure.d 0.1 .. FESimuJ.tion ,_,. Homo Trmsi$otro ie a 0.1 0.2 0.3 DA D.l D.l Axial Stntn fl.) Figure 6.61 "Triaxial test results of transversely isotropic material vs. the lab and the discrete model of 4 layers sample@ o3 = 310kPa" 176
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... 0.2 A. !. c 0.1 I cno.as Measured. FESimul3tion Homo_Translsotropic 0 0 0.1 D.2 0.3 0.4 0.1 0.1 Axial Snln (%) Figure 6.62 "Triaxial test results of transversely isotropic material vs. the lab and the discrete model of 61ayers sample@ cr3 = 103 kPa" 0.3 .... lo.21 u 0.2 1 o., ltJ O.DII 0 0.1 D.2 0.3 0.4 0.1 0.1 Axial Snln (%) Figure 6.63 Triaxial test results of transversely isotropic material vs. the lab and the discrete model of 6 layers sample@ cr3 = 207 kPa" 177
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0.1 o.a Measured :,o:r !.o.a *FESim uJ:ttion Homo TransIsotropic 0 '0.4 c o.a "0.2 , 0.1 0 a 0.1 0.2 0.3 DA D.l 0.1 hill Str.ln f%) Figure 6.64 Triaxial test results of transversely isotropic material vs. the lab and the discrete model of 6 layers samples @ cr3 = 310 kPa" 6.5.5 Results Summary of Discrete and Homogeneous Models In order to evaluate the efficiency of the homogeneous models and the discrete models in simulating the laboratory test results, values of Young's moduli from all approaches, laboratory experiments, finite element with discrete models, and fmite element with homogeneous model were obtained. For discrete models, the Young's and secant moduli were obtained at 0% strain (E1an), 1 %strain (Esec 1%), and 2% strain (Esec2%). At each strain level, the Young's moduli of the discrete models were normalized through dividing the Young's modulus of finite element approach by the Young's modulus of experiment result. For instance the initial Young's modulus (Etan) resulted from the discrete model was divided by Etan from the experiment, and so on for Esec 1% and Esec 2% A value of 1 or around 1 178
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indicates a good representation of such a model. In case of the homogeneous models, only the initial Young's moduli were determined and normalized with respect to those of the experiment results. The Young's moduli and the normalized Young's moduli were obtained and summarized in Tables 6.7, 6.8, and 6.9 for2layers, 4layers and 6layers. From Tables 6.7 through 6.9, it was observed that Esecl% and Esec2% for the discrete model gave better normalized values than that ofEtanThis indicated that the initial performance under small loading, of discrete model was a little off, but it started to converge with the laboratory test results after further and larger axial strains. On the other hand, the normalized Young's moduli were close to unity at small confining pressure, but started to deviate when more pressure was applied. 179
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...... 00 0 Table 6.T"Young's modulus and normalized Young's modulus ofsoil specimens with 21ayers ofreinfrocement" Young's modulus Normalized Young's modulus Method 0'3 Etan Eto;. E2o;. (EIEe xp)tan (EIEexp)t% (EIEexp)2% (kPa) (MPa) (MPa) (MPa) 103 40.4 32.20 21.10 1.00 1.00 1.00 Experiment 207 53.8 48.80 34.40 1.00 1.00 1.00 310 170.2 83.40 55.30 1.00 1.00 1.00 103 28 5 23.20 18 90 0.71 0.72 0.90 Discrete 207 47 5 52.60 37.70 0.88 1.08 1.10 310 50.4 57.9 48.5 0 3 0.69 0.88 103 39.5 NA NA 0.98 NA NA I Homogeneous 207 57.5 NA NA 1.07 NA NA I 310 148.2 NA NA 0.87 NA NA I
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....... 00 ....... Table 6;8 "Young's modulus and normalized Young's modulus of soil specimens with 4 layers of reinforcement" Young's modulus Normalized Young's modulus I Method 0'3 Etan Et% E2% (EIEexp)tan (EfEexp)t% I (kPa) (MPa) (MPa) (MPa) (EfEe xp)2% 103 42 00 30.60 22.50 1.00 1.00 1.00 Experiment 207 146.30 63.50 41.70 1.00 1.00 1.00 310 133.30 77.90 53 50 1.00 1.00 1.00 103 32.40 27.80 20.30 0.77 0.91 0.90 Discrete 207 83.60 62.90 40.00 0.57 1.00 0.96 310 84.04 73. 20 51.60 0 .63 0.94 0.96 103 38.60 NA NA 0.92 NA NA Homogeneous 207 54.2 NA NA 0.37 NA NA 310 146.06 NA NA '1.1 NA NA
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00 N Table 6.9 "Young's modulus and normalized Young's modulus of soil specimens with 61ayers of reinforcement" Youf!g's modulus Normalized Young's modulus Method O'J Etan Et% E2% (EIEexp)tan (EIEexp)I% (EIE exp)2% (kPa) (MPa) (MPa) (MPa) 103 87.90 36.90 25.90 1.00 1.00 1.00 Experiment 207 115.90 54.40 40.40 1.00 1.00 1.00 310 83.90 74.20 53.10 1.00 1.00 1.00 103 42.30 34.40 23.20 0.48 0.93 0.89 Discrete 207 62.00 56 00 38.30 0.53 1.03 0.95 310 69 80 53.80 42.50 0.83 0.73 0.80 103 37.90 NA :NA 0.43 NA NA I I Homogeneous 207 55.30 NA NA 0.48 NA NA I 310 157.00 NA NA 1.87 NA NA 
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6.6 Summary and Conclusion Laboratory and numerical tests were conducted on cylindrical reinforced soil specimen. The specimens were 71 mm diameter and 152 mm height. Some of these samples didn't include any reinforcement, and were refereed to by un reinforced samples. These unreinforced samples were utilized to obtain the strength properties of the used soil, Ottawa sand, used in the study with the aid of Mohr Coulomb envelope. Internal friction of36, the initial Young's Modulus between 70 MPa and 260 MPa for 1 03kPa and 310 kPa confining pressures, respectively, and bulk modulus of209 MPa were all determined from hydrostatic compression test and triaxial tests. Tensile tests were completed on the reinforcement inclusion. A 200 x 200 mm inclusion sample, nonwoven heat bounded geotextile with a thickness of0.4 mm, was subjected to axial tensile test. These tests provided its elastic and plastic parameters, Young's Modulus of 159 MPa and yielding stress of 13 MPa. Once the properties were evaluated, a conventional drained triaxial test was conducted on the reinforced cylinders under different confining pressures. The reinforced samples were built using two approaches, one with the discrete model, and the other using the homogeneous model using the transversely isotropic constitutive equations, that were developed in Chapter 5. The results of both models were compared to each other and to the results obtained from the laboratory experiment. The discrete mode yielded results very comparable to the laboratory results even at large strain deformations. On the other hand, the transversely isotropic homogeneous model gave a good agreement to the results of laboratory experiments and the numerical simulations at strains lower than the elastic limit or 183
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small strains. Beyond 1 % of strain, the homogeneous response deviated significantly from those of other two approaches. The transversely isotropic model is a linear model and the constitutive equations were based on the linear properties of the soil and geosynthetic, therefore linear behavior of the geo composite was expected. However, when evaluating the normalized initial Young's modulus, it was found that homogeneous model gave good representation of the laboratory test results, especially when subjected to small confining pressure, where the normalized Young's moduli in most cases were not far away from unity. One of the advantages when using the homogeneous approach compared to the discrete model was the computation time. For example when processing a model with 6 layers using discrete model 9 hours were need to complete each run, but when using the transversely isotropic model only 34 minutes were needed to complete the analysis. Furthermore, when building the homogeneous model, time and effort were saved a primarily. To build a discrete model, especially with so many layers, it is required to define the interface friction between each soil layer and the adjacent reinforcement layers. This was not straight forward, and requires a lot of repetitions to obtain a normal termination without any penetration problems. In homogeneous model, building shell elements and their frictional interaction with the soil layers were avoided, but they were accounted for through the equivalent properties. 184
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7. Validation of Finite Element Method on Geo Composite 7.1 Introduction In this chapter, the efficiency of the finite element approach and the regression model when analyzing reinforced soil were investigated. This process named validation was completed by comparing the results obtained from reinforced soil model using both finite element method and field/laboratory tests. Tests used for validations: 1. Triaxial tests on cylindrical reinforced soil samples with 6inch diameter and 12inch height. The physical tests were conducted in 1987 by H.C. Liu at the University of Colorado at Denver, 2. Planestrain tests conducted on the geosyntheticreinforced cube samples with equally spaced geotextile. Tests were conducted in 2001 by K. Ketchart at the University of Colorado at Denver. After the comparison between the results of finite element numerical experiments and laboratory tests, numerical experiments were conducted on homogeneous geocomposite samples with the equivalent transversely isotropic material properties obtained from the regression equations formulated in Chapter 5. 185
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7.2 Triaxial Test (Liu 1987) 7.2.1 Laboratory Test Triaxial test were conducted to investigate the effectiveness of reinforcement with different number of reinforcing layer. RC. Liu completed these tests in 1987 as part of his master thesis. In his tests, Ottawa 3040 sand prepared at a relative density of70% and a geotextile with commercial name, Bidim C34, were used. From these tests, the stress strain relationship and the equivalent Poisson's ratios of the geecomposite at different combinations of confining pressure and reinforcing pattern were obtained. 7.2.1.1 Materials Preparation 7.2.1.1.1 Soil Commercially available white clean sand (Ottawa 3040 sand) was used throughout the study. The maximum and the minimum unit weight of the sand, Ymax is 112.19 pcf and the minimum, Ymin is 97.52 pcf. These values were determined at the US Bureau of Reclamation located in the Denver Federal Center. In this study, the unit weight of the sand at the relative density (Dr)= 70 % was used. Therefore based on Equation 7.1, Y7o was calculated to be 107.3 5 pcf. 186
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1 1 rmrn" r70 Dr = 70% = =1 :::=..:1,...:::rmin rmax 7.2.1.1.2 Reinforcement Bidim C34, a needle punched nonwoven geotextile (Figure 7.1 with physical properties shown in Table 7.1) was used as reinforcing material. It was cut into circular discs with diameters slightly smaller than the overall sample diameter. In his study, Liu observed that the geotextile becomes thinner as the normal pressure increase (Figure 7 .2). Figure 7.1 "The non woven geotextile material used in Liu's test" (Liu 1987) 187 (7 .1)
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Table 7.1 "Typical Physical Properties of Bidim C34 Engineering Fabric" (Liu 1987) Bidim Product Code No. us C34 Bidim Product Code No. International U 34 Oz/yd2 Mass per unit area Nominal 8.0 gm/M2 270 Nominal Porosity (calculated) 0.005 bar% 91 2.000 bar% 81 Surface thickness (Mils) 0.005 bar 90 2.000 bar 41 Surface thickness (mm) 0 005 bar 2.3 2.000 bar 1.05 Nominal Permeability 10:J (M/sec) 3 Planar Permeability 103 (M/sec) 0.6 Grab Tensile( lbs) ASTMD1117 255 Grab elongation (%) ASTMD1117 75 Trapezoid Tear Strength (lbs) ASTMD2263 125 Mullen Burst Strength (psi) 400 Restrained Tensile Test (lbs/in) 120 Elon2ation (%) 35 E.O.S Ds 70 Dso 100 Abrasion Resistance 135 Heat Resistance F 50 psi loadin2 480 Puncture Stren2th ASTMD751 125 188
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. .. c: ... a "' 'o .. .... 4 . . lG2 !', :: .... : t O ; ltfl . . . : ..... : .. : .. .. Figure 7.2 "Thickness versus pressure relationship of Biddim C34 geotextile" (Liu 1987) 7 .2.1.2 Sample Patterns All samples subjected to triaxial test were 304.8 mm (12 inches) in height and 152.4 mm (6 inches) in diameter. A number of reinforcing patterns were used to investigate the effect of geotextile spacing (or the number of reinforcing layers) on the ultimate strength of soil samples. There were four patterns included in this study (Figure 7.3): 1) no reinforcement, 2) I layer of reinforcement, which was placed at the mid height of the sample 3) 4 layers of reinforcement, which were 189
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placed at the spacing of76.2 mm (3 inches), and 4) 6layers of reinforcement, which were placed at the spacing of 50.8 mm (2 inches). l==:.,;:""" c .. """ _,.. ... ;' ,. ._ ... . l : ... .. ' ... . : '!t' ,.._.,'"
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When reaching the ultimate strength or 15% axial strain, the test was terminated and the stressstrain relation was computed. The purpose from this load application was to determine the stress strain relationship of reinforced and un reinforced samples. 7.2.1.3 Test Results For each sample with different patterns, the stress strain relationship under 103 kPa and 310 kPa confining pressures was developed, as shown in Figure 7.4 and Figure 7.5, respectively. In these figttres, it was observed that linder the same confining pressure, the ultimate strength of the reinforced samples exceeded that of unreinforced samples Also, the ultimate strength increased with increasing the number of reinforcing layers and the confining pressure. One of the major findings from his project was the friction which was determined to be approximately 3 7, and was obtained by plotting Mohr Coulomb envelope (Figure 7.6). For unreinforced soil, the cohesion, c, was equal to zero. However, it was noticed that an equivalent cohesion, c', was initiated due to adding reinforcing layers. The value of c' increased with increasing the number of reinforcing layers. For instance, an equivalent cohesion value was 22 kPa when embedding 1 layer of reinforcement, and it increased to 66 kPa and 120 kPa due to having 4 layers and 6 layers of reinforcement, respectively. 191
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OJD O.ID ._o.1D !,O.ID : o.m 10.40 ao.aa E o.m .. en o.,o ers ___ j _O.O ___ ___ j.O Axial Stnln, Ba (%) Figure 7.4"StressStrain relationship for samples Tested at 103 kPa confining pressure" (Liu 1987) 1.1D .... 6 layers ;,.20 i 1JD .. 11ayer IC !a.ID IE Q layer aa.aa laAD en 0.20 DJIJ o.o 2.0 4.0 1.0 1.0 10.0 ,2.0 ,4.0 .a Axial StnJn, .l%) Figure 7.5 "StressStrain Relationship for samples tested at 310 kPa confining pressure" (Liu 1987) 192
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uo ..... .! .... .. tO I "' 40 0 .0. llo I W !1141'1Mt S trut+ o Figure 7.6 "MohrCoulomb envelope for samples with different reinforcing patterns" (Liu 1987) 7.2.2 Finite Element Analysis ofUnReinforced and Reinforced Samples using Discrete Models The finite element method was used to simulate the triaxial tests on the reinforced and unreinforced soil specimens. These cylindrical specimens had different patterns a,nd similar to those used in the laboratory tests. The finite element code LSDYNA, and the preprocessor FEMB were used to in analysis of both un reinforced and reinforced samples. The specimens were first subjected to confining pressure 103 kPa and 310 kPa and then followed by vertical load application until failure or 10 % strain. 193
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7.2.2.1 Geometries of Specimens For all patterns, the specimens were 152.4 mm in diameter and 304.8 mm in height. In case of reinforced samples, reinforcement layers were equally spaced as shown in Figure 7. 7. In the 1 layer specimens, the reinforcement was placed in the middle of the specimen, at a distance of 152.4 mm from the bottom. In the 4 layers specimens, the reinforcement layers were placed at a spacing of76.2 mm. The first and the last layers were placed at a distance of 38.1 mm (1.5 inch) from the bottom and the top. In the 6 layers specimens, the reinforcement layers were placed at a spacing of 50 mm (2 inches). For each model, a bottom cap and top cap were added to the model. These caps were utilized in order to correctly simulate the boundary conditions of the laboratory tests. Figure 7.7 "Models ofunreinforced and r:einforced soil cylinders using LSDYNA" 194
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7.2.2.2 Test Simulation These models were all subjected to triaxial tests similar to the ones completed in the laboratory. In the analyses, the samples were first subjected to the confining pressure that increased linearly reaching its ultimate value at 1 second, and stayed constant until termination, at 2 seconds. After 1 second of the analysis, the top of these samples were subjected to vertical prescribed motion that increased linearly. The maximum allowed deformation was 10% of the total sample height, and was equal to 30.48 mm. As in the two plates were used, bottom and top. The bottom one was there to provide suitable boundary conditions, where it was fixed in all global directions, X, Y, and Z. On the other hand, the upper plate was used to transmit the deformations and stresses to the soil specimen, and also, it was constrained from any lateral movement. Furthermore, the sand was modeled as blocks consisting of 8 nodes and the reinforcement was modeled as shell elements consisting of 4 nodes with a constant thickness. 7.2.2.3 Materials Parameters In all these models, the sandy soil was assumed to be a nonlinear material with Cap model of 1 0 input parameters. The 1 0 parameters were divided into three categories: the elastic parameters; the failure parameters, and the hardening parameters. The elastic parameters included the initial shear modulus (G) and initial bulk modulus (K). The failure parameters included the failure envelope parameter (a.), failure envelope linear coefficient (8), failure envelope exponential coefficient (y), and failure envelope exponent (!3). Finally, the hardening parameters included the surface axis ratio (R), hardening law exponent (D), hardening law coefficient (W), and hardening law exponent (X0). The 195
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reinforcement was assumed to be elastic isotropic with two inputted parameters: Modulus of Elasticity (E), and Poison's Ratio (v). 7 .2.2.3.1 Sandy Soil Parameters In his tests, Liu used Ottawa sand. The properties of the Ottawa sand were developed in Chapter 6, therefore there was no need to calculate these properties; however, these properties were reviewed. The only thing that would change in these parameters is the shear modulus, G, of soil. G is function the of overburden pressure, and therefore with different confining pressure, G was expected to be also different. To determine the values ofG, it was required to determine the initial slopes, which was obtained from triaxial test results on the unreinforced samples under different confining pressures. The slope of each curve represented the Young's modulus, E, and was equal to 50 MPa and 70 MPa when subjected to 103 kPa and 307 kPa confining pressures, respectively. For the Ottawa sand, the Poisson's ratio, v, was assumed to be 0.25. Therefore, G was determined using Equation 7 2 to be 20 and 28 MPa, when subjected to 103 kPa and 310 kPa confining pressures, respectively. Also, K, the bulk modulus of Ottawa sand was determined in the last chapter from the hydrostatic compression test, and was equal to 209 MPa. Liu presented was around 37, which resulted from two Mohr circles However, the drained triaxial tests that were completed in Chapter 6 indicated was 36, which resulted from applying three confining pressures. Therefore, it was decided to use a value of 37 when completing the validation. G = ,E. 2(1 +v) 196 (7.2)
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After determining elastic parameters, the failure envelope parameters were determined. These parameters were determined based on the DruckerPrager model criterion. In this model, the parameters and y were assumed to be zero. Since the cohesion of the Ottawa sand was zero, the failure envelope parameter, a., was equal to zero (Equation 7.3). a= 6xcxcos = 6x0xcos36 =O. J3 x (3sin) J3 x (3sin36) The other important parameter was e. It was influenced by the value of friction as shown in Equation 7.4. Applying a value of37 for the friction angle resulted in a value of0.2815 for e. e = 2x sin = 2 X Sin36 = 0.2815 J3x(3sin) .J3x(3sin36) The hardening parameters, R, D, W, and Xo were 1.6, 0.6, 0.007907;and 0, respectively. The hardening parameters, D and W were calculated, as described in chapter 6, from the hydrostatic compression test. The value ofR was concluded after several trials of simulations, which resulted in better validation of the laboratory test results. On the other hand, a value of Xo of zero was assumed due to the assumption that plasticity would initiate at very small loading. The soil properties are summarized in Table 7.2. 197 (7.3) (7.4)
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Table 7.2 "The Cap model properties of Ottawa sand" K 209.0 MPa Elastic parameters G @ 0.10342 MPa cr3 20 .0MPa G 0.31026 MPa cr3 28.0 MPa a 0.0 Failure envelope e 0.2815 parameters 0.0 y 0.0 R 1.6 Hardening D 0.6mm2/N parameters w 0.007907 Xo 0.0 7.2.2.3.2 Reinforcement Parameters This geosynthetic material was assumed to be linear elastic; and it was very difficult to obtain the exact properties of elasticity (E and v) from the given data in Table 7.1. For that reason, E and v were assumed and investigated using a parametric study. The thickness, t, was assumed to be constant and equal to 1 mm. Several trials were completed on samples with 1 layer under both confining pressure to determine E and v. It was determined that 350 MPa forE and 0.2 for v were best values that could represent the model. The same values of E and v were further applied to the 4 and 6 layers samples. In reality, geotextiles are not elastic. A better model to represent this kind of materials would be elastic perfectly plastic, which was used in the Ghapter 6. But, since the properties were not completely available, the elastic model was sufficient to represent general behavior of these specimens when subjected to small deformations of up to 7% strain. 198
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7.2.2.4 Results of the Finite Element Analysis using Discrete Models After obtaining the model parameters of the sandy soil and the nonwoven geotextile, all the analyses were completed and compared with those obtained from laboratory tests. The results under the two confining pressures, 103 kPa and 31 0 kPa were as shown in Figures 7. 8 through 7.15. At the end of each simulation the average vertical stress, crz, at the top of the soil samples were measured and subtracted from the confining pressure to obtain the deviator stress, llcr. From there, the stressstrain relationship resulted from finite element analysis, under each confining pressure was plotted and compared to the results obtained from laboratory (or measured). In spite of not having the exact material properties of reinforcement layers, the fmite analysis with the aid of Cap model was capable of reproducing the stress strain plots, particularly within small displacements. When observing the soil alone, as shown in Figures 7.8 and 7.9, there was a good agreement between the results obtained from the physical test and the finite element analysis. This can be further explained by determining the normalized Young's modulus (Table 7.3). The normalized Young's modulus is the ratio between the Young's modulus using finite element method and Young's modulus obtained from experiment. A value of 1 or close to 1 indicates compatibility between both approaches. Therefore, by observing Table 7.3, it was noticed that the initial Young's modulus that resulted from the finite element method was very close to that of experimental data. The normalized Young's modulus was 0.94 under a confining pressure of 103 kPa, and 1.01 under a confining pressure of 310 kPa. When observing the normalized Young's modulus at 1% strain and 2% 199
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strain, it was noticed that these values were also close to unity. From there, the numerical behavior of Ottawa sand under triaxial test was similar to that of physical test even when subjected to large axial load. When adding reinforcement layers, the strength that resulted from finite element analysis kept on increasing, unlike the measured results, where constant stress level was observed after certain amount of displacement. The properties of these inclusion layers were assumed to be elastic. Also, these elastic properties were not provided, and several trials were conducted on the 1 layer sample to obtain approximate values of the geotextiles properties. So, the strength increase in these geocomposite samples was expected. But with that, the initial trends of all these graphs, except the one containing 6 layers of reinforcement and under a confining pressure of 103 kPa, resulted with good validation using the finite element method. In case of the peculiar sample, the measured strength was initially flat, unlike the strength of the finite element method. The flat portion could have resulted from the fact that this geotextile was not fully stretched, and therefore, was not contributing any resistance at the early stages of loading. If this portion is to be neglected, the initial stressstrain behavior of both data experimental and finite element would be similar, and the normalized Young's modulus would be closer 1. To check for the finite element method accuracy in modeling reinforced soil with the aid of discrete model, the Young's moduli and the normalized Young's moduli for the 1 layer samples, 4 layer samples, and 6 layers samples were obtained and are summarized in Tables 7.4, 7.5, and 7.6, respectively. In these tables, the values of normalized Young's modulus were close to 1, except for the 6layers sample when subjected 103 kPa confining pressure. Not only the initial Young's moduli of finite element was close to the experiment ones, also the 200
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secant moduli at 1 % and 2 % strain were close to those of experiment stress strain results. CUI 0.30 D.2li .D.2D t .8 0.11 Meot$Ured 0.10 FE Slm ulo.tlon D.DI a.oo D.GDDD D.IDDD 1 .DODO 1 .IDDD 2.DDCJD 2JIGCID a.aaaa Axial Strain. a.(%] Figure 7.8 "StressStrain relation of Ottawa sand@ 103 kPa confining pressure" 201
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1.11J O.ID 'ii"O.IO a. !, O.'JU :a.., ia.m 0.40 =0.30 !a.2D ........ Me .. Sim o.oaaa 1JXIIO 2.00013 3.CJDOO Axial Stnln, Ira (%) 4.oaaa l.aaaa Figure 7.9 "StressStrain relation of Ottawa sand@ 310 kPa confining pressure" 0.40 l.o.31 1.0.30 fi u IC 0.21 ia.2D c .0.11 sa.10 ..... ...... Me:lsured FE Simulat ion OJIS i I a.aa i'r,.,.r. ; o.aa 1.00 2.00 3.00 4.00 5.00 1.00 Axlalltntn, s. (% ) Figure 7.10 "StressStrain relation of Ottawa sand and 11ayer of reinforcement@ 103 kPa confining pressure" 202
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UD .. a. !,O.ID u c Co.m 1: Q :oAD 0.20 FE Simulation 0.00 ,.ao 2.00 3.00 4.00 I.DO Axial Strain, Sa (%} 1.00 1.00 Figure 7.11 "StressStrain relation of Ottawa sand and 11ayer of reinforcement@ 310 kPa confining pressure" 0.10 0.10 !,o :o !0.40 fj ao.30 !o.m Me:as u r ed 0.10 :. FE Simulation o.ao 0.00 2.00 4.GO 1.00 I.GD 10.GO 12.00 Axial Strain, Sa fl.) Figure 7.12 "StressStrain relation of soil and 4layers of reinforcement@ 103 kPa confining pressure" 203
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1JD,1.1D : 1.4D !.,.3) u :i UID .. fo.ID c D.ID I D.2D . I ; "'. Measured FE Sim utation D.DD 2.DD 4JO 1.00 I.DD 1D.GD 12.00 Axial Strain, (II.) Figure 7.13 "StressStrain relation of Ottawa sand and 4 layers of reinforcement@ 310 kPa confining pressure" 1.00 .. lD.IO u ro.eo f 0.40 c .. !D.20 eft , . . ; ... ., .... Me:.sured FE Simul a tion D 2.00 4 00 6,()0 s.oo 10 00 G.20 Aldal Strain, fl.) Figure 7.14 "StressStrain relation of Ottawa sand and 61ayers of reinforcement@ 103 kPa confining pressure" 204
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1.10 ... l 1.40 1.2D u I 1.111 !;:OJD c 110JD I 0.40 0.210 Measured Simulation OJII 2.00 4.DO 1.00 1.00 10.00 12.DO Aldal Strain, e.(%) Figure 7.15 "StressStrain relation of Ottawa sand and 6 layers of reinforcement@ 310 kPa confining pressure" 205
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Table 7.3 "Young's modulus and normalized Young's modulus of soil specimens" Young's modulus Normalized Young's modulus Method O'J Etan Esecl% Esec2% (EIEexp)tan I (EIEexp)I% I (EIEexp)2% (kPa) (MPa) (MPa) (MPa) Discrete 103 40.00 33.20 20.2 0.78 0.97 1.03 310 56.6 59.00 48.00 1.1 0.80 1.14 Table 7.4 "Young's modulus and normalized Young's modulus of soil specimens with 1 layer of reinforcement" Young's modulus Normalized Young's modulus Method O'J Etan Esecl% Esec2% (EIEexp)tan I (EIEexp)t% (EIEexp)2% (kPa) (MPa) (MPa) (MPa) Discrete 103 37 10 29.90 14.16 0.94 0.85 8.87 310 62.80 63.30 46.70 1.01 0.99 1.00
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Table 7.5 "Young's modulus and normalized Young's modulus of soil specimens with 41ayers of reinforcement" Young's modulus Normalized Young's modulus Method O'J Etan Esec 1% Esec2% (EfEexp)tan (EfEexp)t% (EfEexp)2% (kPa) (MPa) (MPa) (MPa) Discrete 103 23.80 29 90 20.50 0.82 1.14 1.07 310 65.6 54.9 41.7 1.52 1.056 0.85 Tabl 7 6 "Y e oungs mo d I u us an d r dY norma IZe oun1:rs mo u us o s01 specimens wa d I f 'I 'th61 f ti ayers o rem orcem ent" Young's modulus Normalized Young's modulus Method O'J Etan Esecl% Esec2% (EfEexp)tan (EfEexp)l% (EfEexp)2% (kPa) (MPa) (MPa) (MPa) Discrete 103 44 90 31.50 22.7 6.80 3.2 2 .63 310 29.50 29.20 35.50 0.86 0.83 0.86 re: = the initial slope of the stressstrain, %or2% = is the secant Young's modulus at 1% and 2 %strain.
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7.2.3 Finite Element Analysis of Reinforced Samples using Homogeneous Model 7.2.3.1 Equivalent Properties In this section, the finite element analyses were completed on homogeneous reinforced soil sample, with transversely isotropic properties. The equivalent properties were calculated according to the regression equations of Chapter 5, and are summarized in Table 7.4. In this table, the equivalent properties were determined as function ofE5 Eg, V5 Vg, and S. Respectively, Es was 50 MPa and 70 MPa when subjected to I 03 kPa and 3I 0 kPa confining pressure. For all the three spacing patterns, and under both confining pressure, v5 was constant and equal to 0.25. Eg and Vg were also constants, and were equal to 350 MPa and 0.2, respectively. The three spacing patterns were I52.0 mm for the I layer samples, 76.2 mm for the 4 layers samples, and 50.8 mm for the 6 layers samples. 208
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Table 7. 7 "The Transversely Isotropic Properties of Reinforced Samples" 1layer O'J Eh vh Ev Vv MPa MPa MPa 103.00 49.19 0.23 0.24 310.00 71.35 0.23 72.19 0.24 4layers O'J Eh Vb Ev Vv MPa MPa MPa 103.00 47.57 0.23 51.89 0.24 310.00 69.73 0.23 72.30 0.24 6layers O'J Eh Vh Ev Vv MPa MPa MPa 103.00 47.03 0.23 51.92 0.24 310.00 69.19 0.23 72.34 0.24 7.2.3.2 Results of Finite Element Model using Homogeneous Models Gv Gh MPa MPa 23.64 22.81 31.54 31.79 Gv Gh MPa MPa 23 .83 22.20 31.73 31.18 Gv Gh MPa MPa 23.89 22.00 31. 79 30.98 The results of the homogeneous models due to the triaxial test simulations were compared to those obtained from the discrete finite element model and physical test a:s shown in Figures 7.16 through 7 .21. Due to the elastic behavior of the homogeneous model, the comparisons were completed within 1 % strain only. At further displacements the homogeneous response significantly deviated from the other two approaches, and therefore large displacements were not considered. In general, the initial response of the homogeneous model was similar to the initial response of the discrete model and the physical testing. One exception occurred when comparing the results of 6 layers sample that was subjected to 103 kPa. The stressstrain relation of that sample using the homogeneous approach significantly deviated from those measured in the laboratory. However, for that case the results of homogeneous and discrete models were almost identical within 209
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the initial stages of loading. Have the measured stressstrain curve been shifted to the left, by neglecting the flat portion, the simulation of both finite element approaches would become similar to that of experiment. As mentioned before, the flat portion of the experiment stressstrain results could be obtained due to the combination of low confining pressure and the large number of reinforcing layers. In such a condition, further displacements are needed to cause the stretching of geosynthetic, so that the geosynthetic would attain its large tensile strength and, therefore, contribute to the soil strength resistance. Nevertheless, the other results due to different combination of confining pressure and spacing patterns showed acceptable symmetry. The comparisons were completed by calculating the Young's modulus and the normalized Young's modulus, for each of the finite element models, and the physical models. These comparisons were completed and are shown in Tables 7.6, 7.7, and 7.8. In general, the values of normalized Young's moduli for the homogeneous, (EHomogeneousiEExperiment)tan were close to 1. Except for the sample that had 6 layers of reinforcement and 1 03 kPa confining pressure, the range of (EHomogeneousiEExperiment)tan was varying from 0. 8 to 1. 7, which showed that homogeneous model is adequately capable in reproducing the linear portion of the stressstrain curve. As a summary, when numerically analyzing the reinforced soil sample using equivalent properties, the stress strain response was linear. That was expected since the material had transversely isotropic properties. Even though the response was linear, it was observed that under small strains it behaved similarly to the linear portion of samples tested in the laboratory or when simulated using discrete finite element model. 210
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0.31 ';' i_D.3D c So.2D co.11 !0.10 OJII ., .. / .,.. ,. ,r .. . / / / / Mus ured FE Simulat ion Tran s ver$aly l $ optrop i c D.DO D.2G 0.40 D.IG D.ID 1.DO 1.2G Axtai ltnin, s. f!') Figure 7.16 "Triaxial test results of transversely isotropic material vs. the lab and the discrete model oft layer sample@ o3= 103 kPa" 1.40 1.2D .. a. !, UIJ u o.m IC i o.m Q Me:asured DAD s FE SIM U l .ltion 0.2111 Transversely Isotrop i c o.aa O.DO 1.GO 1.!0 2.GO 2.!0 Axial Stnln, Sa(%] Figure 7.17 "Triaxial test results of transversely isotropic material vs. the lab and the discrete model of 1 layer sam pie @ o3 = 310 kPa" 211
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OAI 0.40 lo.31 !.0.31 u fo.m Q i0.11 ,; 0.10 O.GI Measured FE Simulation 0.20 0.40 CliO 0.10 1.00 1.20 Axll Strain, Ira ('1.) Figure 7.18 "Triaxial test results of transversely isotropic material vs. the lab and the discrete model of 4 layers sample@ cr3 = 103 kPa" 0 .1D o.m CL !.O.tl3 u IE o.3D c .. lo.m ell 0.10 o.aa o.ao 0.20 Measured FE Simul>ltion Transversely Isotropic 0.40 D.IO 0.10 UID Axll Strain, a.. ('1.) 1.20 Figure 7.19" Triaxial test results of transversely isotropic material vs. the lab and the discrete model of 4 layers sample@ cr3 = 310 kPa" 212
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Measured F E Simulation Transversely Isotropic: 0 o.eo . "' ... ... 1 .00 Axial Strain. a. f%] Figure 7.20 "Triaxial test results of transversely isotropic material vs. the lab and the discrete model of6Iayers sample@ cr3= 103 kPa" 0.41 'il D.4D la.a UQ.3D la.21 .. D.11 a a.1a . .. ,_ it .. FE Simul 3 tion Trilnsversely Isotropic DJII am a.e a.&a a.m 1.DD 1.20 AxJ al Strain. e. f%) Figure 7.21 "Triax ial test results of transversely isotropic material vs. the lab and the discrete model of 61ayers sample@ cr3= 310 kPa" 213
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Table 7.8 "Young's modulus and normalized Young's modulus of soil specimens with 1 layer of reinforcement" Young's modulus Normalized Young's modulus Method O"J Etan El% E2% (EIEexp)tan (EIEexp)I% (EIEexp)2% (kPa) (MPa) (MPa) (MPa) Discrete 103 37.10 14.16 0.94 0.85 8.87 310 62.80 63.30 46.70 1.01 0.99 1.00 Homogeneous 103 39.90 NA NA 1.01 NA NA 310 58.10 NA NA 0.94 NA NA I Table 7.9 "Young's modulus and normalized Young's modulus of soil specimens with 41ayers of reinforcement" Young's modulus Normalized Young's modulus Method O"J Etan E1% E2% (EIEexp)tan (EIEexp)I% (EIEexp)2% (kPa) (MPa) (MPa) (MPa) Discrete 103 23.80 29 90 20.50 0.82 1.14 1.07 310 65.6 54.9 41.7 1.52 1.056 0.85 Homogeneous 103 42.60 NA NA 1.47 NA NA 310 51.90 NA NA 1.12 NA NA
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Table 7.10 "Young's modulus and normalized Young's modulus of soil specimens with 61ayers of reinforcement" modulus Normalized Young's modulus Method 0'3 Etan Et% E2% (EIEexp}tan (EIEexp}t% (EfEexp}2% (kPa) (MPa) (MPa) (MPa) Discrete 103 44.90 31.50 22.7 6.80 3.2 2.63 310 29.50 29.20 35.50 0.86 0.83 0.86 Homogeneous 103 40.9 NA NA 6.19 NA NA 310 57.2 NA NA 1.67 NA NA
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7.3 Plane Strain Testing (Kanop Ketchart 2001) 7.3.1 Background In 2001, K Ketchart8 investigated the behavior of geosynthetic reinforced soil mass under various loading conditions with emphasis on the effect of preloading. In his study, laboratory tests were completed on soil, geosynthetic, and the soil geosynthetic composites. Ottawa sand and a road base soil were used in the study, while the reinforcements were Amoco 2044 and Typar 3301. Soil was subjected to conventional compression triaxial test under three different confining pressures, cr3 : 69 kPa, 207 kPa, and 345 kPa. Also, a direct shear test was completed under different normal forces to determine the friction angle of the soil. The geosynthetics were subjected to inisolation load extension tests to obtain linear parameters of reinforcement. These tests were followed by laboratory test known as SoilGeosynthetic Performance (SGP) test, which was conducted in a plane strain condition. In the following sections, validations were completed on samples that were subjected to the monotonic loading condition, using Ottawa sand and Amoco 2044 reinforcement. From there, the conventional triaxial test on Ottawa sand the SGP test on reinforced samples were completed numerically using the discrete and homogeneous model approaches, and their results were compared with those obtained from laboratory testing. 8 PhD, Civil Engineering, Department of Civil Engineering, University of Colorado, Denver Colorado 216
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7 .3.2 Conventional Compression Triaxial Tests A series of conventional compression triaxial (CTC) tests were completed on Ottawa sand. The tests were initially performed by Ketchart in 2001 and were checked here numerically using the finite element code LSDYNA. 7.3.2.1 Laboratory Results The CTC test was performed on an unsaturated soil specimen of Ottawa sand. The sample was 158 mm high and 71 mm in diameter and the unit weight of each specimen was16.85 kg/m3 corresponding to Dr 70%. The procedure of sample preparation is similar to the one shown in Chapter 6. Three confining pressures were applied to the sample: 69 kPa (10 psi), 207 kPa (30 psi), and 345 psi (60 psi). The results of these triaxial tests were as shown in Figure 7 .22. As expected, the strength of these samples was directly proportional to the confining pressure. Also, it was observed that while shearing the sample, initial contraction followed by dilation at axial strain of less than 0.8 % occurred (Figure 7.23). 217
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Figure 7.22 "StressStrain results ofCTC test on Ottawa sand" (Kanop Ketchart 2001) 45 4 0 35 ;i'lO .. 25 J 20 )ts ::::. 05 0 0 .QS 0 2 4 6 8 1) AxbSIJ*t(%) Figure 7.23 "Volumetric strain versus axial strain results ofCTC tests on Ottawa sand" (Kanop Ketchart 2001) 218
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7.3.2.2 Finite Element Results The finite element models were built in a similar manner to the ones utilized in the laboratory, where the loads were applied consequently, cr3 and ilcr, in a 2 secondsrun. The cr3 was applied to all segments of the specimen within the first second of the analysis, and stayed constant until termination. This was followed by the application of ilcr. The deviator stress was then applied in straincontrolled condition, and was referred to by prescribed motion. Only 10 % deformation of sample height, 15.24 mm, was applied to the top of the sample using a rigid plate. The segmental pressure and the prescribed motions were applied incrementally. As in the experiment, two plates were used, bottom and top. The bottom one was there to provide suitable boundary conditions, where it was fixed in all global directions, X, Y, and Z. On the other hand, the upper plate was used to transmit the deformations and stresses to the soil specimen, and also, it was constrained from any lateral movement. Since the material used here was Ottawa sand, the same properties obtained in Chapter 6 were used, considering Cap model. The properties are summarized in Table 7 .11. Even though, the frictional angle obtained by Kethcart in 2004 from the direct shear test was 3 7, a value of 36 was used to complete this validation. The reason of using the later and not the former angle was due to the confidence in the results obtained from triaxial tests on Ottawa sand that was completed in Chapter 6. The value of G was dependent on the overburden pressure. So, for each cr3, the Young's modulus, which is the slope of each stresstrain curve, was calculated. Assuming that v for Ottawa sand is equal to 0.25 and using Equation 7.2, then under the three confining pressure (69 kPa, 207 kPa, and 345 kPa), G was equal to 39 MPa, 68 MPa, and 94 MPa, respectively. 219
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Table 7.11 "The Cap model properties of Ottawa sandKetchart samples" K 209.0MPa Elastic G@ 69 kPa 0"3 39.0 MPa parameters G @ 207 kPa cr3 68.0MPa G _@ 345 kPa cr3 94MPa a 0.0 Failure envelope e @36cj) 0.2815 parameters 0.0 'Y 0.0 R 1.6 Hardening D 0.6mm.l!N parameters w 0.007907 Xo 0.0 After completing the analysis, the average vertical stress, O"z, was calculated. As a result, the stress strain curve that corresponds to each confining pressure was plotted (Figures 7.24 through 7.26). In each of these figures, a comparison between the finite element simulation and the laboratory test result was provided. From these figures, it was noticed that the finite element method with the aid of Cap model was able to simulate the behavior of the triaxial tests to good extent under different confining pressures. Further verification was provided in Table 7.12. In trustable, the Young's moduli and the normalized Yom;tg's' moduli at the three strain level (tangent, secant at 1 %, and secant at 2 %) was provided. Initially, the values of normalized Young's moduli were not close to 1, but they were close to 1 at 1 % and 2 %, indicating the ability of finite element method with the aid of cap model in reproducing the stress strain relation, even under large deformations. 220
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:a.m 1.. eft eft In a: aia.os c FESin1 ul ation a.aa 2JD 4.00 a.aa 1.00 12.00 AXIAL SmAIN (%) Figure 7.24 "StressStrain relation of soil @ 69 kPa confining pressure measured (Ketchart, 2001) versus results of FEdiscrete model" ... 0.10 a. l..a.m eft In 0.40 teft a: a.aa e 0.32 Bi c 0.10 MU$Ured FESlm ulation OlD D.DIIJ 2.DDD 4JIJD IJIJD I.DDD 1a.aaa 12.DOO AXIAL STRAIN (%) Figure 7.25 "StressStrain relation of soil@ 207 kPa confining pressure measured (Ketchart, 2001) versus results of FEdiscrete model" 221
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._.,..., IL !. en a en en a: 0 taAO it Q 020 Musured FESihlula tion a.oa 2.00 4.Ga I.GD I.GD 1D.GD 12.Ga AXIAL STRAIN (%) Figure 7.26 "StressStrain relation of soil @ 345 kPa confining pressure measured (Ketchart, 2001) versus results ofFEdiscrete model" 222
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Table 7.12 "Normalized Young's Modulus of Finite element method compared with Experiment" Young's Modulus Normalized Young's Modulus Method cr3 2% 69.00 Discrete 207.00 345.00
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7.3.3 SGP Tests The SGP test apparatus was developed at the University of Colorado at Denver. The main purpose of this apparatus was to investigate the behavior of a generic geosynthetic reinforced soil mass when subjected to monotonic loading and unloading reloading cycles. It consisted of a rigid container that is able to deform only laterally representing the plane strain condition. As shown in Figure 7.27, the dimensions of the specimen inserted in the rigid container were 61 0 mm high, 254 mm wide, and 565 mm deep. The apparatus was prepared twice: once without any reinforcement known as unreinforced sample, and then with 3 layers of reinforcement considering the strong reinforcement, Amoco2044. The 3 layers were inserted at the bottom at the center, and at the top of sample. The spacing between these inclusion layers was 305 mm. The apparatus consisted of base and top rigid plates to ensure that the vertical load from the loading rod was applied to the center of the test specimen, and to provide the desirable boundary conditions The thickness of each plate was 25 mm. 224
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Figure 7.27 "Specimen dimensions of SGP" (Kanop Ketchart 2001) 7.3.3.1 Laboratory Results As mentioned before, the laboratory tests were completed on unreinforced and reinforced samples. Both specimens were subjected to vacuum pressure that would maintain the specimen shape, and equal to 69 kPa. This was followed by vertical force, applied in strain control manner under a constant strain rate of 0.5 % per minute. Once the tests were completed, a plot of vertical load versus the vertical deformations was obtained for both specimens (Figure 7.28). It was clear that reinforcing layers resulted in stronger composite, where the failure load reached 10 and 17 kN for the unreinforced and reinforced specimens, respectively. Furthermore, the failure load was reached at around 10 mm vertical displacement that corresponded to 2 % of sample height. 225
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P.J.Jl:.S ....... 10 0 5 10 15 20 25 30 Veltical Disptacement (mm) Figure 7.28 "Vertical load versus vertical displacement of noreinforced sample and reinforced sample with Amoco2044" (Kanop Ketchart 2001) 7 .3.3.2 Finite Element Results using Discrete Models The finite element analyses were completed on the unreinforced and reinforced specimens considering all the loads and boundary conditions that were applied to the experiment specimens The unreinforced sample consisted of OttawC!: sand and two plates, bottom and top. The reinforced sample consisted of 3 layers of geotextiles embedded horizontally at spacing of307 mm. Based on the extensional test of geosynthetic completed by Kethcart in 2001, the secant stiffness of the Amoco2044 was 10 kN/m. For this geosynthetic it was assumed that thickness was 1 mm and v was 0.2. Therefore, the Elastic modulus E, was determined to be 1000 MPa. The soil and rigid plate were modeled as brick 226
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elements of 8 nodes, while the reinforcements were modeled as shell elements of 4 nodes. The layouts of unreinforced and reinforced specimens were as shown in Figure 7.29. In this experiment, the boundary conditions were applied at the constrained sides, where deformations perpendicular to that side were not allowed. On the other two sides, segmental pressure of 69 kPa was applied. The bottom nodes of the base plates were constrained in all directions, while the top nodes of the loading plates were constrained from any plane deformation. The deviator stress was applied on the top nodes of loading plate in strain controlled manner. A prescribed vertical motion in the Z direction was applied at these nodes. The ultimate deformation reached at the end of the analysis was 20 mm, and the vertical forces at the top of the soil elements were computed and plotted against the vertical displacements for the unreiforced and the reinforced specimens, as shown in Figures 7.30 and 7.31, respectively. Deformed side Constrained side Reinforced sample Virgin sample Figure 7.29 "Layout of onreinforced and reinforced samples using the finite element method" 227
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50000 45000 401100 35900 g lOOOO 1 251100 a _, 201100 15900 10000 5000 .. ...... . Measured Sond 0 o.aaa I.DDO 10.aaD 11.DDO 2D.oaD 2&.000 30.oaD Vrtlnl Dls,lcmnt [m m) Figure 7.30 "Vertical loaddisplacement relation of onreinforced sample measured by Ketchart versus discrete finite element model" Meas ured_reinforeed sand F'E_reinforced s:.,.nd D.DIIJ e.aaa 1 a.aaa 11.aaa 20.aaa 21.DOD V.nicl Dl,lcmnt(mm] Figure 7.31 "Vertical loaddisplacement relation of reinforced sample measured by Ketchart versus discrete finite element model" 228
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From Figures 7.30 and 7.31, the initial behaviors obtained from fmite element were similar to the ones obtained from the experiments that were completed by Ketchart. The results from experiments showed a failure of the loaddeformation curve after 10 to 15 mm. That was not the case for the results obtained from finite element analysis, especially for the reinforced sample case. For the unrelnforced sample, in finite element, the specimen observed nonlinearity after 10 mm of deformation, which indicated that the sample yielded and became in the plastic range, but did not fail On the other hand, the reinforced sample behaved linearly, where the strength of sample kept increasing. The increase in the resulted load was directly proportional to the vertical displacement. This could be explained by the fact that geosynthetic was modeled using linear elastic model. Furthermore, Table 7 6 indicates the efficiency of finite element, especially at the early stages. In this table the tangent Young's Modulus, E1an, and the secant Young's Modulus at 1 %and 2 %, Et% and E2%, were normalized with respect to the ones from experimental results. In this table, a:ll results showed that the normalized Young's Moduli were near unity, which again, proved the suitability of finite element analysis in simulating the initial results of plane strain test on un reinforced and reinforced soil specimens. 229
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Table 7.13 "Normalized Young's Modulus of Finite element method compared with Experiment, plane strain test on onreinforced samp .. e" Young's Modulus Normalized Young's Modulus Method O"J Etan Et% E2% (E/Eexp )tan (E/Eexp)t% (E/Eexp)z% _(kPa) (MPa) (MPa) (MPa) Discrete 69.00 21.10 19.50 15.40 1.00 0.99 1.01
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7.3.3.3 Finite Element Results using Homogeneous Model The first step in conducting finite element analysis on homogeneous model was to evaluate the equivalent properties. The equivalent properties were function of the elastic properties of the constituents, Ottawa sand and Amoco 2044 geotextile, spacing between reinforcement layers, which was constant and equal to 305 mm, and cr3 which was equal to 69 kPa. The resulted properties were estimated using the regression equations developed in Chapter 5, shown in Table 7.14 Table 7.14 "Equivalent properties of SGP consisting of Ottawa sand and 3 layers of Amoco2044 geotextile" Given ro s 0.25 0.23 The homogeneous specimen was subjected to loadings and condition similar to that on the discrete model. After running the analysis, the vertical load was plotted against vertical displacements and compared to the results obtained from the experiment and discrete finite element model (Figure 7.32). Since the homogeneous model was essentially established based on the elastic properties of the constituents, the response of this model was compared to the linear responses of the two other responses, and was up to 10 mm of displacement. 231
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1GDDGD 1;::= === === = == = = I IDOCIJ Reinforced s:unple J IDOOD FEOiscrete / lUOOD FEHom ogeneous // / / / giDDUD / /.( 1.., .. / a ..J 40000 "' I // IIi /.; I 2DCIIJ I 1aaaD i l l O.DO 2.00 4.00 1.DD I.GO 10.00 12.00 14.GO 11.00 Vrtlcal Di111lac.n .nt (m m) Figure 7.32 "Vertical loaddisplacement relation of reinforced sample measured by Ketchart versus discrete finite element model and homogeneous model" Evidently, the homogeneous model behaved similarly to the other two models at the very early stages. After 8 mm of displacement, the response of homogeneous model started to deviate from the other two responses and started to over predict the strength of the model. Table 7.8 indicates the efficiency of finite element using homogeneous model, with transversely isotropic properties, up to 2 % of axial strain deformation. In this table the Etan, Er% and E2 were normalized with respect to the ones from experimental results. All results showed that the normalized Young's Moduli were close to unity. Therefore the discrete finite element and the transversely isotropic models are able to simulate the initial behavior of reinforced soil under plane strain condition. 232
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Table 7.15 "Young's modulus normalized Young's modulus of homogeneous mode from simulating plane strain test" Young's Modulus Normalized Young's Modulus Method O'J Etan (EfEexp)tan (kPa) (MPa) Homogeneous 69.00 18.25 0.92
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7.4 Summary and Conclusions The main purpose of this chapter was to validate different experiment results with the aid of finite element method. First, the validation was completed considering a discrete model, in which all adjacent parts were built separately with full frictional interface. Second, the validation was completed considering a homogeneous model, in which the reinforced soil is built as one homogeneous material with transversely isotropic equivalent properties. The validation was then completed by comparing the results of both approaches to those obtained from the experimental tests. Once the validation is satisfied, further use of the discrete model and the homogeneous model can be applied on different reinforced soil structures, as will be discussed in the following three chapters. To perform this validation, two sets of laboratory tests were selected: conventional triaxial tests on cylindrical reinforced soil samples that were conducted by H. C. Liu in 1987, and plane strain test on the geosynthetic reinforced soil that was conducted by K. Ketchart in 2001. These laboratory tests were numerically simulated with the aid of the finite element code, LSDYNA. Similar loading and boundary conditions of experiment were applied to the numerical models. Before staring the simulation, the materials properties must be obtained. Since Ottawa sand with a relative density of 70 ro was used in both experiments, the soil properties obtained from previous calibration, in Chapter 6, were used here for validation purposes. The only paramter that would vary here was the shear modulus, G. G is a function of the overburden pressure, and therefore, different confining pressure result in different values of G. The triaxial tests completed by Liu were subjected to two confining pressure 103 kPa and 310 kPa. Under each confinement, the samples, with diameter of 152 mm and height of 304.8 mm, were unreinforced once, and reinforced three times with a needle punched non234
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woven geotextile following different reinforcing patterns. The reinforcing patterns were: 1 layer located in the middle of the sample, 4 equally spaced layers, and 6 equally spaced layers. When comparing the results from the finite element methods, discrete and homogeneous, to those of laboratory, several things were observed. In the case of the discrete model, the numerical results of unreinforced samples were very similar to those resulted from the experiment. The initial Young's modulus and the secant Young's moduli at 1% and 2% strain were similar to those from the experiment. This proved the effectiveness of using Cap model in simulating the Ottawa sand. The numerical validation of reinforced soil was also satisfactory, especially at small displacement. In these discrete reinforced samples, an incessant increase in the strength was observed, unlike the ones from the experiment. The reason for the strength increase was due to assuming a lineal elastic model for the reinforcing material. The major difference of the results occurred when analyzing the 6 layer reinforced sample that was subjected to 103 kPa confining pressure. In this particular case, the results of the discrete finite element model significantly deviated from the experimental results. It is believed that the low confining pressure and the large number of reinforcing layers permitted some initial weakness in the reinforced sample, and the flat portion in the stressstrain relation. Further vertical load assisted in developing the tensile strength of these geotextile, and therefore, a strong geocomposite was the result. If this flat portion had not to been considered, then the initial results of both experiment and discrete finite element model would have been very similar. Furthermore, the results of homogeneous model were also satisfactory to some extent. The homogeneous model had transversely isotropic properties, and it was developed based on the linear properties of the constituents, soil and geosynthetic. Therefore, when comparing the results of this approach to those of experiment and discrete finite element model, only the initial data points, within the elastic range, of the stressstrain curve were taken into consideration. By doing so, it 235
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was concluded that the homogeneous model was capable in reproducing the initial potion of stressstrain curve, up to 1 %, that resulted from triaxial tests. For the homogeneous model, the initial Young's modulus was calculated and divided by the corresponding one of the experiment. The resulted values named normalized Young's modulus were not far from unity. These values showed again that the homogeneous model is capable in simulating the initial response of the triaxial test on reinforced soil samples. The other simulation was the plane strain test of reinforced soil that was completed by K. Ketchart in 2001. This test was referred to by soil geosynthetic performance, or SGP. In this test the geosynthetic reinforced soil mass was subjected to monotonic load. Furthermore, in this test the sample was placed in a rigid container that was able to deform only laterally to represent the plane strain condition. In the SGP test, two samples were prepared: unreinforced and reinforced. They were subjected to a total vertical displacement of 25 rinn. The reinforced sample consisted of 3 geotextile layers. When simulating the behavior of this test using discrete and homogeneous finite element models, several observations were noticed. When observing the unreinforced soil sample, the discrete model was successfully able to predict the initial portion, up to 15 mm vertical displacement, of the loaddisplacement curve. In the experiment, the sample failed after the 15 mm displacement. The discrete model, on the other hand, showed that the sample yielded without failur. However, before that, the responses of both approaches were identical. When observing the reinforced soil sample, the discrete model was able to predict the experiment behavior to some extent. The strength of the reinforced discrete model kept on increasing, unlike the physical specimen which failed after 15 mm of displacement. The main reason for that strength increase was the use of a linear elastic model in representing the geosynthetic material. But within the first 15 mm deformation, 236
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the responses of the discrete reinforced model and the experiment were alike. This was indicated through determining the normalized Young's modulus and the normalized secant modulus at 1 % and 2 % strain, and they were all very close to 1. On the other hand, when analyzing the homogeneous model, the comparison was within 10 mm displacement. Due to its elasticity, at large displacements the response of the homogeneous model significantly deviated from the other two approaches. But the initial response of discrete model and the physical tests was similar to that of the homogeneous model. This was also shown through determining the Young's modulus and the normalized Young's modulus ofthe homogeneous model, where a value of0.92 for the normalized Young's modulus was observed. To sum up, the discrete model with the aid of Cap model is a powerful tool in representing the soil structure and reinforced soil structure. The homogeneous model is also capable to some extent in representing the reinforced soil structure. When modeling using the discrete model, difficulties were met when setting up the mesh, and several trials were conducted to eliminate the potential of initial penetration between the soil and the reinforcement. Also, while running this kind of analysis using an x86 processor, the simulations of the discrete model lasted much loner time than for the homogeneous model. For instance, when simulating the plane strain test of reinforced sample, the analysis lasted more than 12 minutes in the case for discrete and less than 10 second in the case of homogeneous model. When simulating the triaxial test, the analysis lasted more than 6 hours for the case of 6 layers sample, and less than 10 minutes when simulating the homogeneous model. Of course, due to not embedding inclusion layers, the homogeneous model did not have the problem of penetration, and therefore, building the mesh and processing the homogeneous model was much easier. 237
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8. ThreeDimensional Foundation with Soil Reinforcement 8.1 Introduction In this chapter, 3D finite element analyses were performed on a 3m square footing resting on a soil and reinforced soil of different spacing. First, a case of footing on soil was analyzed with its result serving as the baseline for comparison. After that, the cases of footing resting on reinforced soil with different spacing and Young's moduli were considered to investigate the influence of reillforcement spacing and stiffness on the footing's bearing capacity. The spacing patterns were 1000 mm; 500 mm, and 250 mm, while the geosynthetic Young's moduli were 160 MPa, 320 MPa, and 640 MPa. To minimize the boundary effect on the results of the analysis a 3m x 3 footing, the following foundation soil dimensions were selected: 21m x 21m x 15m high. The hypothetical model involves Ottawa sand as foundation soil and nonwoven geotextile as reinforcement. The reinforced zone extended to a depth of 9 m, 3 times the width of the footing. Each reinforcement layer extended 4.5 m horizontally in both directions from the centerline of the footing covering 9 times of the footing area. After completing the analyses mentioned above with full geosyntheticsoil interface interaction, briefed as discrete model, another model briefed as homogeneous model was analyzed. The homogeneous model consisted ofhomogeneous composite foundation soil with transversely isotropic properties. 238
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The model properties were then evaluated using the regression equations formulated in Chapter 5 and the given properties of soil, geosynthetic reinforcement, and the reinforcing spacing. Furthermore, in the discrete models, the tensile stress distribution of geotextile layers was evaluated to investigate the variation of the reinforcement tensile stresses with depth and distance from the footing's centerline. 8.2 8.2.1 Concept and Design Consideration of Foundation(McCarthy 1988) Unreinforced Foundation Soil For many structures, it is the earth underlying the structures that provides the ultimate support. For that reason, the soil at a building location becomes the material affecting the stability of structure on top of it. A load transfer ,devices, named foundations, are required to impart the load from structure members into the soil without stressing the soil beyond its strength limit or causing excessive settlement of the supported structure. Foundations can be classified based on the depth of the foundation and the soil providing the support. The two main classes of foundation are shallow foundations such as mat and spread footing, and deep foundations such as piles and caissons. Spread footings are indeed less expensive and reduce the overall cost of several applications such as bridges by up to 20% resulting on very similar settlements ofbridges on deep foundation (JeanLouis Briaud 1999). Therefore, spread footings are always considered as a: foundation alternative and eliminated only on the basis of calculations. They are typically of plain concrete or reinforced concrete pad used to spread out building column and wall loads over a 239
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sufficiently large soil area. They have the shape of squares, rectangles, trapezoids, and long strips as in Figure 8.1. On the other hand, mat foundations are considered large footings existing over a great area, entire building area. Mat foundations are utilized to distribute building loads in order to reduce differential settlement betweenadjacent areas. In this research, emphasis was placed on spread footing in general and square footing in particular. Q HI> per foot otkN per rn Figure 8.1 "types of shallow spread footing a) square footing; b) strip footing; c) rectangular footing; d) trapezoidal footing" (McCarthy 1988) The pressure that a foundation unit imposes onto the supporting earth mass is the soil's bearing pressure. The ultimate pressure causing the shear failure in the supporting soil is the bearing capacity Bearing capacity is related to the properties of soil and the characteristics of the foundation such as size depth, and shape. When ultimate bearing capacity of foundation is reached, the soil fail in different mode depending on the density. The three main modes of soil failure are 240
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general shear, local shear, and punching shear as shown in Figure 8.2. To investigate which failure is occurring, rigidity index, Ir and critical rigidity index, Ircriticat, must be obtained and compared. The rigidity index is a relative measure of soil deformability (McCarthy 1988), where the critical rigidity index is a reference value. Both Ir and Ircriticai are shown in Equations 8.1 and 8.2. A larger_ value of rigidity index compared to the critical value indicates the general shear failure mode indicating the rigidity of soil when developing a foundation bearing capacity G I=r c + otan v I .1 1 = lexp[(3.30.45 B)cot(45)] rcrz zca 2 L 2 Where: G = soil shearing modulus, c = soil cohesion = angle of internal friction crv = effective soil overburden pressure at a depth corresponding to B/2 below the foundation base B = width of foundation L = Length ofF oundation 241 (8.1) (8.2)
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.: '7; .. : F ;:;:: . ...... '! ; ,. ,_. Figure 8.2 "Failure modes when reaching bearing capacity" (McCarthy 1988) ; ':' Based on the classical theory of plasticity, a solution of the foundation bearing capacity for the case of a general shear failure was developed. It is assumed in this theory that no deformations occur prior to the point of shear failure but the plastic flow occurs at constant stress after shearing failure. The developed bearing capacity equation combines the effects of soil cohesion, internal friction, size and depth of foundation, and soil weight. This general equation, Equation 8.3, was developed by Terzaghi for strip footings and applies superposition for the factors Nc, Ny, and Nq whose values are a function of the shear possessed by the supporting soils. The bearing capacity factors Nq and Nc are based on the rigid242
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perfectly plastic theory, where as the Ny is a semiempirical bearing capacity factor (Steven W. Perkins 2000). 1 quit =cNc + 2Br1Nr +r2D fNq Where: Quit= ultimate gross bearing capacity or soil bearing pressure (ksf, kN/m2 ) c = cohesion of the soil below the foundation level (ksf, kN/m2 ) Dr= depth of footing below the lowest adjacent soil surface (ft, m) YI = effective unit weight of soil below foundation level (pcf, kN/m3 ) y2 = effective unit weight of soil above foundation level (pcf, kN/m3 ) Nc, Ng, Nq =soil bearing capacity factors, dimensionless terms, whose values relate to the angle of internal friction, f, whereby Nq N c = ( N q 1) cot Nr =2( Nq +l}an Equation 8.3 is mainly used for the condition of long footing. In order to take in consideration different shapes such as square, rectangular, circular, or foundation inclination, etc., different modification factors were introduced to Terzaghi's equation. To account for these modification factors, the extended bearing capacity equation becomes, q t = eN s d i b )+ _!_ Br1N (r s d i b ) ne c c c c c c 2 r r r r r r + r 2 D 1 ( Nq 1)(rqs qd /qb q) 243 (8.3) (8.4)
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Where: re, ry,rq =reduction factors to compensate for the punchinglocal shear condition, Se, Sy,Sq = shape factors for square, rectangular, and circular foundations, de, dy,dq = depth factors, ie, iy,iq = load inclination factors, be, by,bq =base tilt facotrs The shape factors were developed from empirical data depending on foundation dimensions and internal friction angle as in Equations 8.5 through 8.7. For strip footings, the B/L ratio approaches zero and the shape factors become unity s1 =10.4(%) s q = l+(o/z,)tan Equations 8.3 and 8.4 provide ultimate values of bearing capacity. For designing purposes, a factor of safety of 2.5 or 3 is included to the value of qnet accounting for variations in soil conditions and environmental factors. qnet qdesign = SF An alternative methodology for determining bearing capacity was proposed by Bolton in 1986 (Steven W. Perkins 2000). In his approach, Bolton proposed an empirical equation of dilatancy angle at peak strength, which was a 244 (8.5) (8.6) (8.7) (8.8)
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function of the soil relative density, Dr, the mean normal effective stress confinement with unit ofkPa, p', strain condition, and other empirical constants as shown in equations 8.9 and 8.10. According to Bolton's approach, the parameter IR can be used to describe the effect of progressive failure on foundation capacity. In his approach, it is only required the determination of Dr and which eliminate the need of conducting sets of triaxial tests. Dr is determined from standard penetration test and cone penetration test, whereas can be determined from conducting a shear test. The limitation of this approach, it is only valid for sand with moderate to high relative densities. ' "' ="' +AI "'peak "'cv R IR =Dr(Qlnp')R Where: cv = is the friction angle, A = is an empirical constant that is equal to 3 and 5 for triaxial strain condition and plane strain condition, respectively, IR = is a relative dilatancy index, and Q and R = are empirical material constants with values of 1 0 and 1, respectively. Another approach for predicting bearing capacity of shallow foundations on sand was proposed by Perkins and Madson (Steven W. Perkins 2000) and is referred to by relative density approach. This approach accounts for scale effects due to nonlinear strength behavior through the use of Bolton's equation. In addition to the soil's relative density and constant volume friction angle, the unit weight is necessary for the iterative design procedure of this approach. Such approach has shown an improvement over the classical approaches. However, this approach 245 (8.9) (8.1 0)
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does not account for strength anisotropy which can be a significant influence on the bearing capacity. Results obtained from such approach give a conservative bearing capacity when applied to flexible solution since it is based mainly on rigid foundations. The predicted bearing capacity can be determined using Equations 8.11 through 8.13 that provide the best fit line when plotting the progressive failure index, lpF, versus IR. quitpredicted= quitpeak I PF( quitpeak qultcv) q =q'N +_!_r'BN ultpeak q 2 r I pp=0.044IR +0.65 Where: q' = is effective surcharge existing at the footing base prior to any loading and subsequent settlement of the footing into underlying soil, Nq and Ng =are bearing capacity factors that depend only on the peak friction angle, and (8.11) (8.12) (8.13) qultcv = is calculated using Equation 8.12 with friction angle corresponding to fcv The finite element solution, presented in the following sections, is compared to only Terzaghi's equation because of its wide range use in design and analysis purposes. 246
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8.2.2 Foundation with Soil Reinforcement When artificial fill (soil) is used for construction purposes, it is typically placed on compacted layers to improve the engineering properties of soil and to develop a final elevation and shape. The soil should have adequate strength and satisfactory compressibility to maintain the structural ability and minimize the post construction settlement. When the natural soil materials at planned construction site are too weak or deemed inadequate for the needs of the project such as inadequate shear strength, method of basic site improvement may be possible and practical. The addition of reinforcement layers such as strong metallic strips, geotextile, and geogrid in the soil supporting the foundations can improve the ultimate bearing capacity (Das 1999). The deformation properties of reinforced soil are totally different from those of unreinforced because of the tensile properties of the reinforcing materials. It depends on the conditions of the reinforcement such as the number of layers, the embedded width, the depth, and the stiffness of reinforcing materials. Guido et al. (1985) realized from their laboratory experiments on square footing supported by reinforced loose sand that geotextile layers placed within a depth equal to the width of the foundation would increase the bearing capacity of foundation after obtaining measurable settlement; more than 0.02 of the foundation width (Das 1999). Furthermore, the modes of bearing capacity failure in reinforced earth are different from failure modes of shallow foundations resting on a compact and homogeneous soil. In 1975, Binquet and Lee suggested that the bearing capacity failure in reinforced earth with strong strip reinforcements primarily depends on the depth of the first reinforcing layer, d, compared to the width of the footing, B, and the number of reinforcing layers, N, as shown in Figure 8.3 (Das 1999). By observing these graphs, case (a), where the ratio d/B is greater than 2/3 gives the 247
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lowest effect of reinforcement. In this case the full shear failure plane is developed above the first reinforcement layer. On the other hand, case (c), where the ratio d/b is less than 2/3 and N is larger than 4, has the most beneficial effect. T d . I Figure 8.3 "Modes of bearing failure of reinforced earth" (Das 1999) The shear stress distribution, 'txz, caused by footing loading at depth z below the footing at distance x measured from the center line of the footing can be obtained using Equation 8.14. 248
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Where: b = halfwidth of the footing = B/2, B = width of footing, qR = load per unit area on the foundation. The tensile force developed in reinforcement (TN) can be calculated .from equilibrium equations and is shown in Equation 8. 15. In this equation, the tensile force is calculated for the case ofN layers of reinforcements under the footing with centertocenter spacing of T =_!_[q [qR N N 0 q 2 p Where: qo = load pre unit area on the foundation of unreinforced soil, qR = load pre unit area on the foundation of reinforced soil, N = number of reinforcement layers, = spacing between reinforcement layers, and Al, A2, and B =non dimensional quantities f(z/B) after Binquet and Lee, 1975 as shown in Figure 8.4. 249 (8.14) (8.15)
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2 '::/ 8 .: Figure 8.4 "Non dimensional quantities (Al, A2, and A3) function ofz!B" (Das 1999) An engineer must determine whether the tie at any depth z will fail either by breaking or pullout. For that reason and for designing purposes, factors of safety against both failure mechanisms must be included as shown in Equations 8.16 and 8.17. For breaking mechanism, the most important .factors to be taken in considerations are related to the reinforcement properties such as dimensions, number of layers and breaking strength. On the other hand, for pullout mechanism, the resistance against the tie being pulled out derives from the frictional resistance between the soil and the reinforcement at any depth. wtnf FS y (B)T(N) Where: FS(B) = factor of safety against reinforcement breaking, w = width of a single reinforcement, t = thickness of each reinforcement, 250 (8.16)
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n = number of reinforcement per unit length of foundation, and fy = yield or breaking strength of the reinforcement materials. FS 2tan,(wn{ A,Bq,( ;,)+r(L, x,Xz+D1 )] (p)T(N) Where: FS(p) = factor of safety against reinforcement pullout, y = unit weight of soil, Dr= depth of foundation, =reinforcementsoil friction angle, L0 =effective length of reinforcement, f (z/B) after Binquet and Lee, 1975 as shown in Figure 8.5, and A3 = f(z!B); Figure 8.4. ,_ / a.o .................. ..... Ito. +f..... i. : : / . ; / l f . I t l. O 7 ';:. '.' #1!. .. : Figure 8.5 "Effective length of reinforcement (L0 ) as function of z/B" (Das 1999) 251 (8.17)
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It is obvious that due to reinforcement, the bearing capacity of foundation resting on reinforced soil significantly increases. The effect of reinforcement for increasing the ultimate bearing capacity has been generally expressed in terms of bearing capacity ratio, BCR, as shown in Equation 8.18. Alawaji conducted Odometer tests on circular foundations models supported by unreinforced and reinforced soil (Alawaji, 2000). He concluded that up to 95% reduction in settlement and up to 320% increase in bearing capacity of foundation over collapsible sand. According to Alawaji's results from testing a thin layer of sand reinforced with Tensar SS2 Biaxial geogrid on top of crushed and collapsible soil, the maximum efficiency of sand geogrid system is achieved when increasing the geogrid width and decreasing the geogrid depth. BCR = q u'( R') Where: qu(R) = is the ultimate bearing capacity of reinforced soil, and qu =is the ultimate bearing capacity ofunreinforced soil. Determining the ultimate bearing capacity of foundation supported by un reinforced and reinforced soil has been intensively investigated. For foundation without soil reinforcement, Terzaghi equation (classical solution) as shown in Equation 8.3, has been widely used for practical and design purposes. However, for the foundation with reinforced soil, the bearing capacity is still to be determined. In 1997, Huang and Meng have provided an uncertain relationship, Equation 8.19, to determine the ultimate bearing capacity of strip surface footing on reinforced sand based on the web slab mechanism (C. R. Patra 2005). The results obtained from such approach provide a conservative prediction of the 252 (8.18)
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ultimate bearing capacity when compared with laboratory model results (C. R. Patra 2005). q R =0.5x(B+M)xyxN +yxdxN u r q Where: M3 = 2dtanf3 tanfJ = 0.68+ 0.743(CR)+ o.o{!) w B = is the width of footing, h = is the vertical spacing between consecutive layers, b = is the width of the geogrid reinforcements under the foundation, d= is the depth of reinforcement below the bottom of the foundation, CR = is the cover ratio, w = is the width of longitudinal ribs, and W = is the center to center spacing of the longitudinal ribs. Based on the upperbound theorem that results in a compatible plastic deformation, the bearing capacity for foundations with and without reinforcement is obtained using Equation 8.20 (Kentaro Yamamoto 2001). This equation provides a reasonable agreement of bearing capacity and failure mechanism with the observations. The failure mechanism is determined by a parameter of the wedge angle of soil expressed in radians. The wedge angle of soil increases with the increase of deformed area, therefore for foundation with reinforcement is larger than that of foundation with unreinforced soil. 253 (8.19)
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tanq cos(;) + 2 2 cos 2 q 2( 1 + tan 2 2) q = yB N = yB ))exp{(;rq %)tan2}} ( 8 20 ) 2 r 2 + tan2cos(; +)+sin(;+) cos(% )cos(; ) f{ rr !) } + expl\;rq tan2 2 cos 2 q cos 2 8.3 Finite Element Analysis of Square Footing Supported by UnReinforced and Reinforced Soil Several hypothetical 3D foundation models were constructed and analyzed using FEMB and LSDYNA finite element codes, respectively. Each model consisted of square footing with dimensions of 3 m wide x 3 long x 1 m thick. The footing was placed on top of the soil. In order to eliminate the boundary effect on the analysis results as shown in Figure 8.6 the foundation soil was assumed to be 21 m wide x 21 m long x 15 m thick. The analyzed models were divided into two groups: unreinforced and reinforced foundation soil. While the foundation with unreinforced soil consisted ofhomogeneous soil layers, foundations with reinforced soil consisted of soil with embedded inclusion layers at the 3 vertical spacing: 1m, 0.5 m, and 0.25 m, respectively. For each case of foundation with reinforced soil, the reinforcing layers extended horizontally in both direction, X andY, to a distance 4.5 m from the centerline of the footing and depth, Z direction, of9 m (or 3B) from the base of the footing, where B is the width of the square footing. Guido et al. (1986) have reported that in order to obtain the maximum benefit from the reinforcement with respect to the bearing capacity, the width of reinforcement should be between 2 to 3 times of the plate's width (Alawaji 2001). Furthermore, based on Buissnesq and Westgaard analyses, the 254
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vertical stress increment due to the footing's load application becomes insignificant at a depth of3B from the base of the footing, as shown in Figure 8.7 (McCarthy 1988). A similar conclusion was attained from tests completed by Briaud and his colleges when investigating the behavior of 5 large spread footings on sand (JeanLouis Briaud 1999). They noticed that 97% of the settlement takes place within a depth of 2B below the footing. As a result, the reinforcing layers in the current foundation were extended to 3B horizontally and vertically. In addition to the spacing effect on the bearing capacity of the foundation, the reinforcement stiffness was also investigated. For each spacing category, the Young's modulus of reinforcement was allowed to vary representing weak strength, moderate strength, and high strength geotextile; they were 160 MPa, 320 MPa, and 640 MPa, respectively. 255
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Figure 8.6 "3D foundation model"
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Figure 8.7 "Variation of vertical stress beneath a footing based on Boussinesq analysis and Westergaard analysis, respectively" (McCarthy 1988) 8.3.1 Material Properties The three main materials used in this model were Ottawa sand foundation soil, geotextile reinforcement, and concrete footing. The involved constitutive models representing these materials were the cap model for Ottawa sand, elasticplastic (or himodulus) model for geotextile, and isotropic linear elastic model for concrete. 257
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8.3.1.1 Ottawa Sand Since Ottawa sand was used in this model, the parameters were obtained from the triaxial test results presented in Chapter 6. All the parameters except for the shear modulus, G, did not change. G, represented in Pascal unit, was allowed to vary with depth as a function of the mean pressure, cro (or overburden pressure), and void ratio, e, as seen in Equation 8.2l.a and 8.2l.b (Das 1993). Ground = 6908(2.17 e )2 if Ji max l+e 0 Gangular = 3230(2.97 e)2 ah max l+e 0 The typical void ratio for uniform sandy soil varies from 0.8 to 0.45 for both loose and dense sand (Das 1999). In this project Dr of the Ottawa sand was 70 %, medium dense. Therefore, an average e value of 0.6 was assumed for the foundation soil. cro is the mean stress defined in Equation 8.22, where crv is the vertical (overburden) stress and crh is the horizontal stress at a depth z. a 0 = v + 2a h) In homogeneous soil with a constant unit weight, y, crv at a depth Z below the ground surface is the following: a =rxZ v and the horizontal stress at the same depth Z is ah =K xa 0 v where k0 is the coefficient of earth pressure at rest and can be determined using the following empirical equation (McCarthy 1988). 258 (8.21.a) (8.21.b) (8.22) (8.23) (8.24)
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. K =1sin 0 Using a friction angle of 36 as determined in Chapter 6, Ko was found to be 0.4. Also, considering 1719 kg/m3 as the unit weight of Ottawa sand corresponding to Dr of 70%, and by using Equations 8.2l.a and 8.2l.b, G was plotted versus the Depth Z in Figure 8.8. 0.00 20.00 40.00 60.00 $0.00 100.00 120.00 1e0.00 2 ""' t Q 10 12 (MP:a) Figure 8.8 versus G for Ottawa sand, y = 1719 kg/m3 The soils beneath the footing were divided into three zones. The first zone, top soil layers, extended to 2 m below the ground surface, and the average corresponding G value was 26.5 MPa. The second zone, middle soil layers, was extended from 2 m to 9 m below the ground surface, and the average corresponding G value was 82.9 MPa. The third, bottom soil layers, was extended from 9 m down to the base of the model, and the average corresponding 259 (8.25)
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G value was 126.0 MPa. The rest of the soilparameters were obtained from chapter 6 for Ottawa sand and used directly in the Cap model. The soil properties are summarized in Table 8.1. Table 8.1 "Ottawa sand properties for Cap model" K G 9 R D w Xo (MPa) (MPa) a. 'Y 209 26.5 0 0.2815 0 0 1.6 0.6 0.007907 0 209 82.9 0 0.2815 0 0 1.6 0.6 0.007907 0 209 126.0 0 0.2815 0 0 1.6 0.6 0.007907 0 8.3.1.2 Geotextile Different reinforcement properties were investigated to see the effect of reinforcement stiffness on the foundation's bearing capacity. Initially, the same material properties of the nonwoven heat bounded geotextile used in Chapter 6 were used here. Then, the Young's modulus was increased by factors of2 and 4, respectively. The model used to represent the geotextile was elasticplastic and the properties are summarized as in Table 8.2. In this table, pis the density, G is the shear modulus, Y is the Yielding strength, Etan, is the hardening slope after yielding, and K is the bulk modulus. G= E 2x(1+v)' And K= E 3 X (12 X v) 260 (8.26) (8.27)
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Table 8.2 "Geosynthetic properties for Elasticplastic" p G y Etan K (tons/mm3 ) (MPa) (MPa) (MPa) (MPa) easel 3.38E10 61.5 13 10 132.5 Case2 3.38E10 123 13 10 266.5 Case3 3.38E10 246 13 10 533.3 8.3.1.3 Concrete Concrete was assumed to be an elastic material. Medium strength qoncrete has the following parameters. Table 8.3 "Concrete properties, Elastic (Boresi 2003) p E v (tons/mm3 ) (MPa) 2.32E9 25000 0.15 8.3.2 Finite Element Modeling As mentioned earlier, LSDYNA was the major code to analyze the shallow foundations with both unreinforced and reinforced supporting soil. Initially, a model without any reinforcement was assembled. A sliding frictional contact was allowed between the foundation soil and the concrete footing. The coefficient of friction, 11, was dependent on the internal friction angle of soil and was equal to (Lee 2000). From there, Jl was equal to 0.5. Additionally, the foundation model was subjected to a total of two loads in 2 seconds duration. The first load was gravity and was constant until termination with a gravitational acceleration of 9.81m/s2 The second load was representing the compression loading on top of 261
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the footing applied in displacementcontrolled fashion. A maximum deformation of 300 mm was applied incrementally starting after 1 second from the analysis. The foundation of reinforced soil cases were subjected to the same load as for the unreinforced case. Here a frictional contact was allowed between soil layers and the adjacent reinforcement. The contact between the adjacent materials was based on surface to surface algorithm, with a frictional coefficient of 0.5 as well. The reinforcement layers were 9 x 9m2 They were modeled as 2:D shell elements rather than eight nodded brick elements which were used in the soil and concrete footing. 8.4 Results and Discussion Results from finite element analyses were computed for the footing resting on top of the unreinforced and reinforced soil. The results obtained from the foundation with unreinforced soil were compared with analytical solution. Furthermore, the foundation models with reinforced soil were assembled using homogeneous material with the equivalent transversely isotropic material properties using the regression equations formulated in Chapter 5. 8.4.1 Results of Foundation without Soil Reinforcement After completing the analysis, the overburden pressure was computed at different elevations and compared to those obtained from the analytical solution using Equation 8.23, O"z = y.z, as shown in Figure 8.9. 262
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o.oaoo 0.0810 0.10110 0.1100 G.2a 0.2fiCJD G.300D 0 2 4 Analytical .. FESimulatlon . !. .. N . ..,0 ... 12 ... 11 Q., (MPa) Figure 8.9 "Overburden pressure of footing on top of foundation soil, FE vs. analytical" The next step was to determine the bearing pressure of the foundation and was compared with Terzaghi's bearing pressure, Equation 8.4. Since Terzaghi's equation is based on general shear failure criteria, it was essential to check if this model was meeting this mechanism. So the rigidity index, Ir, was measured and compared to the critical rigidity index, Ircritical In this model G was determined for the top layers to be 26.5 MPa, the overburden pressure, crv at depth 1.5m, B/2 was 0.0153 MPa, and the soil' friction was 36. Therefore, the following apply. I = G = 26.5(MPa) = 2433.4 r c+atan 0+0.0153(MPa)tan36 v 263
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I .t. 1 =.!.exp[(3.30.45B)cot(45)] rcrz zca 2 L 2 ex{(33 0.45)co{ 453;)] 100 By comparing the results Ir was much larger than Ircrticah hence a general shear failure mechanism was concluded. Froni there, Terzaghi's equation was used including the correction factor for the square footing. Since the cohesion of soil was zero and the foundation was lying on the ground surface, the first and third terms were eliminated. All modification factors were set to 1 except for the shape factor because of the square footing, which resulted in a bearing capacity of0.854 MPa, as shown below. sr = 10.4(%)= 10.4 = 0.6 q =.!.Br N (r s d i b ) net 2 1 r r r r r r l 1719.s(k%3 ) (NJ =x3(m}x m x9.81x56.3x(1x0.6xlxlxl} 2 1000 kg => q net = 854. 7(kN I m2 ) = 0.854(MPa} After determining the bearing pressure from Terzaghi's equation, it was measured from the finite element solution. The bearing pressure was measured by taking an average value of O"z underneath the footing. The vertical stress was measured at different time increments. Each increment represented a 15 mm settlement as shown in Figure 8.10. The corresponding deformation from Terzaghi's equation was 53.5 mm. From Figure 8.10, the finite element analysis results using cap model to represent the sand showed that the Terzaghi bearing capacity is much too conservative. This phenomenon was also observed by Atlar and Patra when 264
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conducting laboratory tests on strip foundation supported by sandy soil (C. R. Patra 2005). 6, (mm) Figure 8.10 "Bearing pressure of soil foundation resulted from finite element analysis" 8.4.2 Results of Foundation with Soil Reinforcement In this section, the results of foundation models with reinforced soil were compared to those obtained from unreinforced model. The purpose was to confirm the efficiency of reducing spacing between reinforcing layers and increasing their stiffness. After that the tensile stress distribution along reinforcing layers at different elevations was investigated to determine the optimum reinforcement dimensions. 265
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8.4.2.1 Effect of Spacing For constant E, three spacing patterns between the reinforcing layers were investigated. The spacing was 1000 mm, then 500 mm and 250 mm. The results were as shown in Figure 8.11, 8.12, and 8.13 forE 160MPa, 320MPa, and 640 MPa, respectively. 4.1 'ii 4 D.. I. 3.1 3 .. 2.1 .. 2 IC 1.1 ID 1 D.l ... o .5m S=lm. Soil 10D 110 2ICI 8, (mm) Figure 8.11 "Bearing pressure offoundation due different spacing when Eg = 160 MPa" 266
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I I "il a. !.4 :I S ltn SoU & 1:2 1i Ill 1 a a 10 100 1110 2DD 210 6y(mm) Figure 8.12 11 Bearing pressure of foundation due different spacing when Eg = 320 MPa11 I SO.l5m I "il a. 1.4 :I .. & c 2 li Ill 1 a a Ill 1111 210 Figure 8.13 11 Bearing pressure of foundation due different spacing when Eg = 640 MPa 11 267
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The corresponding deformation from Terzaghi's equation for model containing sandy soil only was 53.5 mm. For all reinforced models, the corresponding settlement for Terzaghi's bearing capacity was less and further decreased with less spacing. For instance, at reinforcement stiffness of 160 MPa, the corresponding settlement was 50.03 mm, 48.38 mm, and 38.14 mm when S = 1000 mm, 500 mm, and 250 mm, respectively. The corresponding settlement at different spacing and stiffness were as seen in Table 8.4. The decrease in the settlement when reducing the spacing indicated a stronger foundation with larger bearing capacity. This can be further explained by evaluating the bearing capacity of all reinforced models associated with the settlement limit resulted from Terzaghi's bearing capacity, as shown in Table 8.5. In this table, it was evident that for any reinforcement stiffness, the bearing capacity at settlement associated with Terzaghi's bearing capacity, 53.5 mm, would increase due decreasing the spacing between reinforcement layers. Table 8.4 "Terzaghi's corresponding settlement of reinforced soil models" Spacing Settlement (mm) (mm) 1000 50.03 E=160MPa 500 48.38 250 38.14 1000 50.82 E=320MPa 500 46.12 250 36.04 1000 50.33 E=640MPa 500 45.54 250 35.57 268
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Table 8.5 "Bearing capacity of reinforced soil models corresponding to Terzaghi's settlement" Spacing Bearing pressure (mm) (MPa) 1000 0.905 E=160MPa 500 0.928 250 1.077 1000 0.897 E=320MPa 500 0.973 250 1.136 1000 0.905 E=640MPa 500 0.993 250 1.136 8.4.2.2 Effect of Geosynthetic Stiffness For constant S, three stiffness categories of reinforcement with E varying from 160 MPa, 320 MPa to 640MPa, were analyzed as shown in Figure 8.14, 8.15, and 8.16 for spacing of 1000 mm, 500 mm, and 250 mm, respectively. From these figures, and Tables 8.4 and 8.5, it was noticed that the Terzaghi's corresponding settlements of foundation with soil reinforcement models were not far off from the one of unreinforced foundation. Also, the bearing capacity of reinforced models associated with Terzaghi's settlement was noticed to have slight increase due increasing the stiffness. This indicated that the efficiency of increasing reinforcement stiffness was very minor especially at the early stages of loading. 269
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4.1 T4 ii 3.1 a. I. 3 ;: 2.1 .. t 2 .. 1.1 .. 0.1 a 1DD 111) 2GO G., (mm) Figure 8.14 "Bearing pressure of foundation due different geosynthetic stiffness when S = 1000 mm" 4.1 ii 4 a. !,U 3 :I !H & 2 IC 11.1 ID 1 0.1 0 0 10 10D 110 Figure 8.15 Bearing pressure of foundation due different geosynthetic stiffness when S =500 mm" 270
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Soil 1110 110 2DO a, (rn m) Figure 8.16 "Bearing pressure offoundation due different geosynthetic stiffness when S = 250 mm" 8.4.2.3 Summary of Spacing and Stiffness Effects It was essential to combine the effect of spacing and stiffness in order to locate where the maximum strength occurred. This was completed by calculating the strength increase compared to the case of foundation with unreinforced soil. As expected, Figure 8 .17 shows that the maximum increase, 40 %, occurred when the spacing between reinforcement layers was 0.25m and E was 640MPa. However, the increase due to increasing E was not as significant as when reducing the I spacing. Yamamto and Kusodo observed the same conclusion in 2001 when investigating the failure mechanism and bearing capacity of foundations resting on reinforced soil (Kentaro Yamamoto 2001). They found that the reinforcing effects are more likely influenced by the width of the reinforcement and the number of layers than the stiffness of reinforcement. Determining the beating 271
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capacity ratio, BCR, as shown in Table 8.6, gave the same conclusion, in which the largest value ofBCR was 143% that occurred when using Eg of640 and spacing between reinforcement layers of250 mm. The value ofBCR was most affected by the spacing between reinforcing layers, and was less affected by the stiffness of geosynthetic. The value of BCR is the ratio of the ultimate bearing capacity of reinforced soil, qu(R) divided by the ultimate baring capacity ofunreinforced soil, qu, as shown in Equation 8.28. aa 1!. ... . I I :20 u .5 1D .c t 0 50 100 150 200 250 300 en 1D ..,._E160 S=1m rE=160 S=0.25m ..... E=320 S=1m En320MPa S=0.5m +E=320MPa S=0.25m E=640MPaS=1m ... E=640MPa:S=o.sm E=640MPaS=0.25m &.., (mm) Figure 8.17 "
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Table 8.6 "BCR due to different reinforcement spacing and geosynthetic stiffness" Spacing Eg BCR (mm) (MPa) % 160 109.2 1000 320 112.4 640 112.3 160 114.9 500 320 121.2 640 126.6 160 133."9 250 320 140.0 640 143. 0 A very useful tool that would combine the effect of foundation settlement, stiffness of reinforcing layers, and spacing between reinforcing layers on bearing capacity was the performance based bearing capacity, Rb, for all data. Rb is defined as the ratio of bearing capacity of numerical analysis divided by the Terzaghi's bearing capacity. Figures 8.18, 8 19, and 8.20 were presented to graphically summarize Rb at different settlements when E was 160 MPa, 320 MPa, and 640 MPa, respectively. 273
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I 4 2 1 l I l I !. D 1CII 1fi0 200 2fll 3DD 3fi0 Figure 8.18"Performance based bearing capacity at different spacing when E =160 MPa" I I I 4 3 2 1 1CII 110 200 Sc::O.lSl s = o . S. n S=hn Soil Figure 8.19 "Performance based bearing capacity at different spacing when E =320 MPa 274
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il 7 I I 4 3 2 1 l o2sJ l s .. o s m l S a lm. j Soil } J I i ,. a 10 100 110 2GD &.,(mmJ Figure 8.20 "Performance based bearing capacity at different spacing when E =640 MPa Again, Figures 8.18 through 8.20 indicated that bearing capacity was most enhanced because of the spacing effect rather than geosynthetic stiffness. In order to support this observation, a linear regression analysis was completed between the independent variables: settlement (bv), Spacing (S), and geosynthetic Young's modulus (Eg), and the dependent parameter, Rb. While conducting the statistical analysis, the correlation matrix between the independent and dependent variables indicated that Rb was most influenced by bv then S with p values of 0.976 and 0.145, respectively. The p value between Rb and Eg was very small and was equal to 0.038. Although adding Eg to the regression analysis containing bv and S slightly increased R2 from 97.45 %to 97.55 %, the addition was necessary to include this expression to the regression analysis due to its effect on the value of mallowsCp. The MallowsCp criterion is concerned with the total mean squared error of the n fitted values for each regression model. A small Cp value that is 275
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close to the number of independent variables would identify the subsets of independent variables (Michael H. Kutner 2005). Before adding Eg to the regression, Cp was 13 when using 8v and S. The value ofCp dropped to 4 when adding Eg to the equation. Therefore, the equation for performance bearing capacity of reinforced sandy soil was as shown in Equation 8.29. In this equation, the positive sign indicates the increase in Rb quantity due to increasing the independent variable, and the negative sign indicates the increase in Rb quantity due to reducing the independent variable. Therefore an increase in bearing pressure can be achieved by increasing the settlement and geotextile stiffness and by reducing the spacing between adjacent reinforcing layers. When using this equation, the user must be aware of its limitations. Its only valid for the following conditions: 1) Ottawa sand with relative density of70 %, 2) The Young's modulus of geotextile must be in between 160 MPa and 640 MPa, 3) The spacing between geotextile is not more than 1m and not less that 250 mm, and 4) The maximum settlement of the foundation is 300 mm. Rb = 0.619 + 0.0168 X 0 (mm)0.000727 X s(mm)+ 0.0003 X E (MPa) v g As an example, assuming a constant Eg of 640 MPa, and Spacing of 250 mm, then with different settlements of25 mm, 50 mm, and 75 mm, Rb would be equal to 1.049, 1.469, and 1.889, respectively. The values of these Rb at different settlements were obtained from Equation 8.29 and were in agreedment with those obtained from Figure 8.20. 276 (8.29)
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8.4.2.4 Stress Distribution on Reinforcement Layers When modeling the foundation with reinforced soli, geotextile layers were extended to 9 m below the ground surface, covering 9 times the footing area. The following questions needed to be addressed: Do we need all the reinforcement layers? Do we need to cover all this area? To answer these questions, contours of horizontal stresses, crx and cry, were obtained at different depths from the base of footing: 500 mm, 0.5B (1.5 m), 1B (3m), 1.5B (4.5 m), 2B (6 m), and 2.5B (7.5 m), as shown in Figures 8.21, 8.22, 8.23, 8.24,8.25, and 8.26, respectively. The first general observation was that the maximum horizontal stresses were located underneath the footing. These stresses reduced gradually to very small values. This trend was observed at all elevations. When comparing the contours at different depths, it was noticed that significant reduction in these stresses were obtained at much deeper depths. For instance, at a depth of 0.5B (1.5 m) below the footing the maximum horizontal stress was around 12 MPa. On the other hand, at a depth of2.5B (7 m) below the footing, the maximum horizontal stress was around 0.6 MPa. The other contour plots for depths between 1B and 2.5B showed that tensile stresses had values falling between the 2 extreme cases, where the values started to reduce from top to bottom. The only different trend among these contour plots was shown at a depth of 500 mm, in Figure 8.21. At 500 mm, a volcano shape was observed underneath the footing, indicating slight reduction in the tensile stresses at the centerline with maximum value close to the edge of the footing, which then decreased following the same trends of other elevations. These contour plots were further investigated when combining all these contour plots, as shown in Figure 8.27. In Figure 8.27, it was shown that at each elevation, the horizontal stresses had maximum values beneath the footing, which 277
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then decreased drastically away from the centerline. These stresses also decreased with depth to very negligible values near the base of the reinforced soil, reaching negative values near the edges of these deep layers. Also the same volcano distribution phenomenon was observed at the reinforcing layers located at shallow depths; 250 mm, 500 mm, 750, and 1000 mm, where the tensile stresses increased from the centerline toward the edge of the footing and reduced again at segments located far away from the footing. The peak value of 13.7 MPa occurred beneath the footing at depth of750 mm from the ground surface. Comparing the maximum tensile stress values of these reinforcement layers to the ultimate stress value of 15.5 MPa, provided by the tensile testing ori the geotextile in Chapter 6, it was shown along each elevation the factor of safety against the maximum tensile stress was ranging from 1.13 at shallow depth to 3 8.2 near the bottom reinforcing layer (3B = 9 m). Similar observation was determined when considering different depths. It was found that the factor of safety against maximum tensile stress was ranging from 1.13 underneath the footing to 98 beside the end zone of reinforcement (1.5B = 4.5 m). From there, it was clear that tensile stresses became very small at segments far away from the centerline and at deeper distances downward. Therefore, covering 9 times of the footing area with geotextile was not very beneficial. In fact 2 times, 4.5 m X 4.5 m, of the footing area would be enough since beyond that very small tensile stresses were generated and sometimes negative value representing a compression behavior. Geotextile members are only capable of supporting tensile stresses, where all the compressive stresses are carried by soil elements. 278
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Figure 8.21 "Contour and 3Dimensional surface plots of horizontal stress distribution for geotextile at depth of 500 mm from the footing" 279
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Figure 8.22 "Contour and 3Dimensional surface plots of horizontal stress distribution for geotextile at depth of O.SB from the footing" 280
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Figure 8.23 "Contour and 3Dimensional surface plots of horizontal stress distribution for geotextile at depth of lB from the footing" 281
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," ;,. ooo 't, < :. . .. . .... . Figure 8.24 "Contour and 3Dimensional surface plots of horizontal stress distribution for geotextile at depth of l.SB from the footing" 282
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' Figure 8.25 "Contour and 3Dimensional surface plots of horizontal stress distribution for geotextile at depth of2B from the footing" 283
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. .. ontat: stresses at !, Figure 8.26 "Contour and 3Dimensional surface plots of horizontal stress distribution for geotextile at depth of 0.5B from the footing" 284
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11.0 14.0 'iii12.D D. !.10.0 .. .. 1.0 .. I 1.0 I 4.0 '1: 0 z: 2.0 a .a 4.0 Faating 1000 2000 adepth= 250 mm ..... depth= 500 mm ..... depth= 750 hHn ....,_depth= 1000 mm ..... depth= 1500 m m ..... depth= 3000 mm depth= 4500 mrn depth = 6000 m m "'1!1depth = 7500 m m =8750 mm Distance from centerline (mm) Figure 8.27 "Horizontal stress distribution of reinforcement layers at distance away from the footing's centerline, S = 250 mm, Eg 320 MPa" In the Figures 8.21 through 8 27, the distributions of stresses in both horizontal and vertical directions were considered. But from these graph, the effect of spacing on the stress distribution was not indicated. For this reason, Figures 8.28 through 8.30 were provided to show stress distributions for the foundation supported by reinforced soil with spacing equal to 1000 mm, 500 mm, and 250 mm. In the following figures, the horizontal stresses were calculated along the soil's depth at six locations with respect to the centerline of the footing: centerline, 0.25B (750 mm), 0.5B (1500 mm), 0.75B (2250), 1B (3000 mm), and 1.5B (4500 mm) Again, these figures indicated that most of the tensile stresses occurred beneath the footing. The stresses decreased significantly in both directions, away from the footing and at deeper depth, where after 6 m depth stresses at all the locations were very small. At locations of lB and 1.5B from the centerline, the stresses were very small at all the elevations, which indicated that 285
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at these locations, 1B and further from the centerline reinforcement became inconsequential. Also, from these figures, it was noticed that when larger spacing was considered, the maximum tensile stresses were observed to occur near the top at distance of 1 m, which was similar to the cases with less spacing. However, when observing the case of250 mm spacing for instance, it was noticed that the tensile stress on the top layers were not as large as the one depth of 1 m, but these values were still very large reaching 6 MPa. After a depth of 1 m, the trends of all cases were the same, a significant decrease with further depth. This indicated that it would be beneficial and more economical to have more reinforcement layers underneath the footing at shallow depths, and to allow the increase in the spacing at further depths. 2 D 2 I lD 12 14 Figure 8.28 "Horizontal stress distribution of geosynthetic layers at different depths for foundation on top of reinforced soil, S =1000 mm" 286
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E !. 5 t 0 8000 .eooo a 2 4 I I 10 12 14 Figure 8.29 Horizontal stress distribution of geosynthetic layers at different depths for foundation on top of reinforced soil, S =500 mm" 14 11 .eooo Tensile stres .. s, ax (MPa) Figure 8.30 "Horizontal stress distribution of geosynthetic layers at different depths for foundation on top of reinforced soil, S =250mm" 287
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8.4.3 Results of Foundations with Reinforced Soil Modeled as Homogeneous Material In this section, a comparison between results obtained from the discrete model and homogeneous was provided. In the homogeneous model, the equivalent properties of reinforced foundation so' il were estimated using the regression equations formulated in Chapter 5. As explained earlier, this geocomposite had transversely isotropic properties with six constants: E v, E', v', G, and G', which were determined as function of the elastic properties of soil and geosynthetic, spacing between geosynthetic layers, confining pressure, and friction coefficient between soil and the geosynthetic. The model here had the same dimensions and boundary conditions of the discrete with the same loading increments. Once the model was assembled, the analyses were completed considering the spacing 1 000 mm, 500 mm, and 250 mm. However, only a geosynthetic with Young's modulus, Eg, of 320 MPa was considered for the entire spacing patterns. To obtain the homogeneous properties of this geocomposite one must be aware of the effect of confining pressure that varies with depth Accordingly, the shear modulus, Gs, and Young's modulus, Es, of soil will be affected. When modeling the discrete element, the foundation soil was divided into three zones: un reinforced base, middle, and top. The middle and top were reinforced beneath the footing and extended horizontally to 4.5 m from the footing centerline. Therefore, when adopting the homogeneous approach, only the reinforced zones were considered to be geocomposites with isotropic properties, i.e. the base soil and the upper unreinforced soil layers were modeled using the same properties they had in the discrete model, using the cap model. Using Figure 8.8, the values of G were calculated for each of the two reinforced zones and were equal to 82.8 MPa and 26.5 MPa, respectively. Assuming a v for Ottawa sandy 288
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soil of 0.25, then based on Equation 8.26, G = ( E ) the corresponding Es for 21+v each zone was 207 MPa and 65 MPa, respectively. From there, the homogeneous properties of the reinforced foundation soil for the and top reinforced zones considering a geotextile with Eg of320 MPa and spacing of 1000 mm, 500 mm, and 250 mm were as summarized in Tables 8.7, 8.8, and 8.9, respectively. Table 8.7 "Homogeneous properties for lm spacing and Eg =320 MPa" Toplayers S = 1000 mm Given properties O'J Es Vs Eg Vg s MPa MPa MPa mm 0.0076 66.25 0.25 319.8 0.3 1000 Eguivalent O'J Eh Vh Ev Vv Gv Gh. MPa MPa MPa MPa MPa 0.0076 85.10 0.23 67.09 0.23 27.94 36.85 Middlelayers S = 1000 mm Given _l)_rop_erties O'J Es Vs Ee Ve s MPa MPa MPa mm 0.056 207 0.25 319.8 0.3 1000 Equivalent properties O'J Eh Vh Ev Vv Gv Gh MPa MPa MPa MPa MPa 0.056 241.05 0.23 210.75 0.23 83.50 100.04 289
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Table 8.8 "Homogeneous properties for 0.5m spacing and Eg =320 MPa" toplayers S = 500 mm Given properties CfJ Es Vs Eg Vg s MPa MPa MPa mm 0.0076 66.25 0.25 319.8 0.3 500 Equivalent properties CfJ Eh Vb Ev Vv Gv Gh MPa MPa MPa MPa MPa 0.0076 74.50 0.23 67:82 0.24 29.19 32.85 Middlelayers S= 500 mm Given properties cr3 Es Vs Eg Vg s MPa MPa MPa mm 0.056 207 0.25 319.8 0.3 500 Equivalent properties cr3 Eh Vb Ev Vv Gv Gh MPa MPa MPa MPa MPa 0.056 230.45 0.23 211.48 0.24 84 75 96.04 290
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Table 8.9 "Homogeneous properties for 0.25m spacing and Eg =320 MPa" Toplayers S= 250 mm Given properties O'J Es Vs Eg Vg s MPa MPa MPa mm 0.0076 66.25 0.25 319.8 0.3 250 Equivalent properties O'J Eh Vb Ev Vv Gv Gh MPa MPa MPa MPa MPa 0.0076 69.20 0.23 68.19 0.24 29.81 30.85 Middlelayers S = 250 mm Given properties O'J Es Vs V_g_ s MPa MPa MPa mm 0.056 207 0.25 319.8 0.3 250 J!I"O_perties O'J Eh Vb Ev Vv Gv Gh MPa MPa MPa MPa MPa 0.056 225.15 0.23 211.85 0.24 85.38 94.04 With the properties provided, the homogeneous models were then subjected to the same loads and boundary conditions of discrete models. The results of this approach were compared with those of discrete models as shown in Figures 8.31 through 8.33. In these figures, only the initial portion, up to 25 mm settlement, was taken in consideration. At larger settlement the homogeneous models behavior significantly deviated from the discrete models and hence results of large settlement were not considered in these comparisons. These figures indicated that the responses of the homogeneous models at small displacement were very comparable with those obtained from the discrete models. However, in the homogeneous model, larger bearing pressure and smaller settlements than the discrete and soil models were obtained. Also, the behaviors of homogeneous models were not much different from each other at different spacing. In fact, 291
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small increase of bearing capacity was observed when increasing the spacing This is of course just the opposite of what the discrete model indicated. But this odd phenomenon was a result of the regression equations developed in Chapter 5. In these equations, the plane modulus of elasticity was weakly correlated to the spacing between reinforcement with positive sign. However and because of the weak correlation, the effect of reducing the spacing had not major differences on the overall outcome. Tables 8.10 and 8.11 provide a summary of the corresponding Terzaghi's settlement and bearing capacity compared with those of discrete models Both tables pointed at the differences obtained from this approach compared with the discrete approach, indicating that a stiffer foundation would result from using the homogeneous approach. Table 8.10 "Terzaghi's corresponding settlement of reinforced soil models; homogeneous vs. discrete" Eg Spacing OvDiscrete OvHomogeneous (MPa) (mm) (mm) (mm) 1000 50 82 18.16 320 500 46.12 19.06 250 36.04 22 69 Table 8.11 "Bearing capacity of reinforced soil models corresponding to Terzaghi's settlement; homogeneous vs. discrete" Eg Spacing 0' zDiscrete cr zhomogeneous (MPa) (mm) (MPa) (MPa) 1000 0.897 2.215 320 500 0.973 2.078 250 1.136 1.940 292
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1.2 ... 1 a. !.a.l .. 0.8 & COA 1i a:JD.2 +Sol! 1m_E320 Terzaghr ...,_hom .1m 0 I 10 5v(mm) 2D 21 Figure 8.31 "Bearing pressure of foundation with homogeneous reinforced soil for S = 1000 mm" 1.2 1 'il IL .. .. CQA a ID D.2 0 a 10 115 (mm) +Soil ...... + TerzQgh i homo 0 .5m 20 21 Figure 8.32 "Bearing pressure of foundation with homogeneous reinforced soil for S = 500 mm" 293
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, 0.1 0.1 1l' k 0.7 ; 0.1 1 o.s .. !' 0.4 I o3 0.2 0.1 0 0 I 10 11 soil o .25m_E320 ...._Terzaghi rhomo 0.25m 21 30 Figure 8.33 "Bearing pressure of foundation with homogeneous reinforced soil for S = 250 rom" 8.5 Summary and Conclusions In summary, a vertical compressive load was applied on a square footing supported by unreinforced and reinforced soils in finite element simulations. Up to 300 mm vertical settlement was incrementally applied to the top of the footing. The resultant vertical stresses corresponding to the settlements were calculated. As expected, the bearing capacity of the footing, especially under large deformations, increases with an increasing number of reinforcing layers, where the contribution of geosynthetic tensile strength became more effective. Applying regression analysis to the concept of performance based bearing capacity, it was determined that bearing capacity was directly proportional to the settlement and stiffness of reinforcement, and inversely proportional to the spacing between 294
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inclusion layers. The bearing capacity was much more dependent on settlement and spacing than inclusion's stiffness. The analysis also indicated that the tensile stress of inclusions was much larger under the footing than away from it, and decreased rapidly with the distance from the footing. For models with smaller spacing between inclusion layers, beneath the footing, the tensile stress increase reached its maximum value at a depth of 1 000 mm and then decreased drastically with the depth of reinforcement layer. The soil beneath the footing was reinforced with several layers of nonwoven geotextile. Each layer was extended horizontally by a distance of 1.5B (B, width of footing) from the centerline of the footing covering an area 9 times that of the footing area. Also, the last layer of geotextile was located at a depth of 3B. From Figures 8.21 to 27, it is concluded that, to reap the maximum efficiency, the maximum lateral extent of the reinforcement needs not to be greater than lB from the footing centerline and the vertical extent of 2B from the base of footing. The last part of this chapter was emphasized with treating the reinforced soil beneath the foundation as a homogeneous material that had equivalent transversely isotropic material properties. The results of this approach had larger bearing capacity and smaller corresponding settlements than those of the discrete model. Therefore, such an approach resulted with stiff foundation that over predict the bearing capacity of this foundation. It would be beneficial to compare the results of both approaches with large scale tests on reinforced foundation soil and make some necessary revisions to enhance the prediction. Despite the differences, which were small at the early age of loading, the use of homogeneous model would simplify the analysis and save a lot of CPU time. In this approach the time required to assemble the model was significantly reduced 295
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due to the fact of not including the reinforcing shell elements and their contact with the soil layers. Both, the shell elements properties and the interface between soil and reinforcement layers were included in the regression equations of the transversely isotropic geecomposite. Also, for example, when running the analysis of the discrete model containing geosynthetic with 250 mm spacing, the analysis lasted more than 7 hours using an x869 processor. On the other hand, when completing the analysis on the homogeneous model using the same processor, the analysis lasted for 1 hour only. 9 Family 6 Model 13 Stepping Genuinelntel800 MHz 296
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9. Static and Dynamic Analysis ofMSE Wall with Rigid Facing 9.1 Introduction Mechanically Stabilized Earth (MSE) walls are one of the major types of retaining structures. Unlike gravity walls, MSE walls are considered as internally stabilized structures due to internal reinforcement layers. Generally speaking, MSE walls consist of reinforcing layers which are embedded in the soil's back fill and facing system. The facing system is either segmental blocks or precast concrete panel, both having their own advantages. MSE segmental, or modular block retaining wall, is very cost effective, saving up to 40 % of the total cost, compared to conventional concrete retaining walls, and it is easy to construct (Sam M. B. Helwany 2001 ). In fact, geosynthetic reinforced walls are found to be the least expensive of all wall categories at all wall heights. In these kinds of walls, the geosynthetic layers are sandwiched between stacked concrete blocks at regular vertical spacing (Hoe I. Ling 2005). Externally stabilized walls, such as gravity and semi gravity, have the benefit of continuous rigid facet. It is of a great advantage to combine the benefits of both systems. Hybrid retaining walls inherit the continuous rigid facet from externally stabilized walls, and geosynthetic reinforcements from internally stabilized walls. In order to enhance the structural properties without adding cost, the volume of the facing section be less than the volume of externally stabilized walls, as well as the amount of geosynthetic reinforcements must be less than the one in internally stabilized wall. 297
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In this chapter, static and dynamic analyses were completed on a Hbrid retaining wall. The reinforced backfill in the MSE wall was constructed first using a discrete finite element model and then using homogeneous composite model. The results of both models were calculated and compared to investigate the effectiveness of the constitutive equations, developed in Chapter 5, under both static and dynamic loadings. Additionally, horizontal stress distribution of geosynthetic layers was computed at different elevations along the wall. The finite element code, LSDYNA, was the major code for completing these analyses. 9.2 Concept and Design Consideration of Retaining Walls The proper design of retaining structures requires an estimation of lateral earth pressure, which is a function of the type and the amount of wall movement, shear strength parameters of soil, and the unit weight of soil (Das 2004). Also it is important to define the failure envelope which shows how the wall can fail. Under static conditions, retaining walls are acted upon by body forces related to the mass of the wall, by soil pressure, and by external forces. Achieving equilibrium of these forces without inducing the shear stresses that approach the shear strength of the soil indicates a proper design of the retaining structure. Under seismic loading, permanent deformation of the wall may occur due to inertial forces that could cause a violation in force equilibrium and excessive dynamic lateral earth pressure on retaining structures. For all retaining walls, after determining the lateral earth pressure, the structure as a whole is checked for stability against overturning, sliding, bearing capacity, and deepseated failures as shown in Figure 9.1. For MSE walls, in addition to 298
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external stability, internal stability is checked against the tension and the pullout of reinforcements at different elevations, Section 9 .2.1. (c) r I I t I I I I l I I I I I I I I (b) (d) Figure 9.1 "Retaining wall external stability; a) overturning, b) sliding, c) bearing capacity d) deep seated. 9.2.1 (McCarthy 1988) Retaining Walls with Geosynthetic Reinforcement (Vector Elias 2001) Recently, soil reinforcement has been widely used in the construction and the design of foundations, retaining walls, slopes, and other structures. Adding reinforcement materials such as metallic strips, geotextiles, and geogrid to the backfill of retaining walls will increase the tensile strength of the back fill and develop shear resistance from the friction at the soil reinforcement interfaces. Not to mention the simplicity of the construction that does not require large 299
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construction equipment or experienced craftsmen, and most importantly, the flexibility that allows them to withstand differential settlement in seismic shaking. The concept of reinforced soil is similar to that of reinforced concrete, where the mechanical properties are enhanced by reinforcement sited parallel to the principal strain direction to make up for soil's lack of tensile resistance(V ector Elias 2001). Based on Zonberg and Leshchinsky, the reinforcement must resist lateral earth pressure, thus mobilizing some of its tensile strength, which is enabled by the interaction between the soil and the reinforcement along common interfaces (Dov Leshchinsky 2004). The principal role of the reinforcement is to control soil deformations. The frictional interaction, due to relative shear displacement and corresponding shear stress between soil and reinforcement surface, allows the stress transfer along the reinforcement. Once the transfer occurs, all tensile stresses are carried on by reinforcement layers, resulting in a stronger composite material. In most cases, a relatively high tensile resistance is provided by reinforcement layers. However, if tensile forces in the inclusions become so large that the geosynthetic elongate excessively (breakage of reinforcement), or if the tensile forces become larger than the pullout resistance that the shear stresses in the surrounding soil increase. significantly (pullout failure), internal failure of MSE wall can occur. In order to avoid such failure, tensile strength of geosynthetic must be provided by the manufacturer. The strength capacity of reinforcement must be larger than the developed tensile stresses in the structure at different elevations. The behavior of geosynthetic material is more complex than that of steel reinforcement. Its behavior is affected by a number of different factors such as creep, installation damage, temperature, and confining stress. Therefore, reduction factors to the tensile strength of geosynthetic are applied when deign. 300
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The reduction factors vary from one material to another depending on its strength and sensitivity to those factors. For example, the creep reduction factor for Polyester varies from 2.5 to 1.6, while High Density Polyethylene varies from 5 to 2.6. In addition to the reduction factor, a global factor of safety is included to account for uncertainties in externally applied loads, structure geometry, fill properties, and load nonuniformity. A traditional value of 1.5 has been used as the global factor of safety, and a value of 7 has been used to take into account all reduction factors. From there, the design longterm reinforcement tension load, Ta, for the limit state is as shown in Equation 9.1. T = Tult = Tult a RFFS 1FS (9.1) Having a strong reinforcement material in the backfill of a retaining wall provides a stronger structure due to stress transfer. For this to happen, granular soils with less than 15% fines passing the No. 200 sieve should be used that would allow more frictional resistance(Vector Elias 2001). On the other hand, Koerner recommends that zero particles pass the No. 200 sieve (Robert M. Koerner 2001). The cohesionless backfill shall be reasonably free from organic materials and shall be well compacted resulting in a minimum friction angle of34, except for areas near the wall's face to prevent facing panel movement. Adequate draining of the backfill must be used in the reinforcement zone, where low permeability is a major reason for serviceability and actual failure problems (Robert M. Koerner 2001). When designing an MSE wall using the Limit equilibrium method10, it is important to first satisfy the external stability, as is the case for any retaining wall, and then satisfy the internal stability. 10 A limit equilibrium method analysis consists of a check of the overall stability of the structure. 301
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9.2.1.1 External Stability of Vertical MSE Walls and Horizontal Backfill External stability analysis for MSE walls treats the reinforced section as a rigid soil mass. The evaluation is similar to the ones obtained using conventional retaining wall systems. Due to the flexibility of MSE walls, the factors of safety for external failure are in some cases lower than those used for gravity walls. These factors of safety account for uncertainties in the soil properties, reinforcement properties, and loads, as well as inadequacies in the stability model (Tanit Chalermyanont 2005). The evaluation of external stability includes sliding, overturning, bearing capacity, and deep seated stability as shown in Figure 9.2. After the static analysis, a check for seismic stability in regions must be also completed. 302
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Figure 9.2 "External stability of MSE wall" (Vector Elias 2001) For walls with a vertical face, the MSE wall mass acts as a rigid body. The developed earth pressure rises from the back end of the reinforcements, as shown in Figure 9.3. This pressure is only a function of the reinforced soil mass and surcharge, if it was accounted for . The weight of any wall facing is neglected. For a vertical wall and horizontal backslope, the active coefficient of earth pressure (Ka) depends on the internal friction angle, and has the form of K0 = tan2 ( 45303 (9.2)
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The resultant horizontal thrust, FT, located at H/3, is simply the area of the resultant triangle of the horizontal pressure, and is equal to Where: y = is the unit weight of reinforced soil H = is the height of the wall .,_ I J J j I 1 1,1 Iii J I I 4 : . . mm ' m m f I! il ! l1ifdldlf. '1! I ! . .. f. } L 8 .... Figure 9.3 "External loads on vertical MSE wall with horizontal backfill due to weight and surcharge" (Vector Elias 2001) 304 (9.3)
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9.2.1.1.1 Sliding Stability When checking against sliding at the base of the MSE wall, a minimum factor of safety of 1 5 must be satisfied. The resisting force,. PR, which is caused by the shear resistance along the base of the wall, must be at least 1.5 times the driving forces due lateral stresses, Po, as shown in Equation 9.4. L,PR FS zd = ;:;: 1.5 s z zng L,P d Where: PR =rxHxLxtan PD=FT L = is the reinforcement length If the required factor of safety against sliding was not satisfied, then the reinforcement length, L, must be increased. 9.2.1.1.2 Overturning Stability When checking against overturning of the MSE wall a minimum factor of safety of2.0 must be satisfied. The resisting moment due to the mass of the reinforced soil, MR, must be at least twice the overturning moment, M0 as shown in Equation 9.5. MR FSOT =:?:2 Mo 305 (9.4) (9 5)
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Where: MR Mo_= xyxH2 xKa x( Again, in case of not achieving the required factor of safety, it is recommended to increase the reinforcement length, L. 9.2.1.1.3 Bearing Capacity Failure When checking against the bearing capacity of the MSE wall a factor of safety of 2.5 must be satisfied. The allowable bearing capacity of the foundation soil, quit. must be 2.5 or more times the vertical stress at the base of the MSE wall, crv, as shown in Equation 9.6. FS =quit ;:: 2.5 beanng a Where: v q ult = c I X N c +: 0.5 XL X r X N r yxHxL (]' ='v L2e FT X IJj e==yxHxL Again, in case of not achieving the required factor of safety, it is recommended to increase the reinforcement length, L. 306 (9.6)
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Uncertainties of soil properties can also be accounted for using probabilistic analysis. Based on that, reliability based design method (RBD) for external stability of MSE wall considering sliding, overturning, and bearing modes was developed (Tanit Chalermyanont 2005). Based on Chalermyanont and Benson "A RBD method provides the designer with means to determine the reinforcing length (L) required, ensuring that a specified probability of failure, Pr, will be achieved for any particular external failure mode and set of statistical characteristics of the soil properties". The maximum accepted target probability of failure for such geotechnical structures is 0.01. Therefore, RBD charts corresponding to Prof0.01, 0.001, and 0.0001 were developed for each failure mode by determining the required reinforcement length. By studying these charts, it was found that the mean and the coefficient of variation of the friction angle are significant for sliding, the mean and coefficient of variation of the friction angle of the backfill and coefficient variation of the unit weight of the backfill are significant for overturning, and the mean and coefficient of variation of both friction angle of the backfill and the foundation soil are significant for the bearing capacity. Once the static stability analysis is completed, the stability due to seismic loading must be checked, especially for retaining wall located in active seismic zones. The dynamic response of even the simplest type of retaining wall is quite complex due to the resultant inertia force on the reinforced zone, PIR. However, simplified methods were developed to estimate the earthquakeinduced loading. Okabe (1926), and Mononobe and Matsu (1929) developed the basis ofpseudostatic analysis of seismic earth pressure on retaining structures. It has become popularly known as the MononobeOkabe (M0) method. This M0 method is similar to the static Coulomb theory with additional pseudostatic acceleration applied to the Coulomb active wedge. The pseudostatic soil thrust, P AE, on the retaining wall in 307
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addition to the static thrust, P A, is exerted. Both P1R and P AE are functions of the maximum horizontal acceleration in the reinforced soil wall, Am, as shown in Equations 9.7 and 9.8, respectively (Kramer 1996). 2 P 1 R =0.5xAmxyxH where: A = (1.45 A)A m A = is the maximum ground acceleration coefficient at the centered of the wall mass, obtained from the AASHTO code (AASHTO 2004). 1 2 PAE =2_xK AE xyxH where: = is the internal friction angle, = is the backfill slope angle, 8 = is the seismic inertia angle =tan1 [ K h ] 1K v Kh, and kv = are the horizontal and vertical peak ground acceleration coefficients, respectively, and co = is the angle of wall face from the vertical. In designing the MSE wall for seismic loading, the P A must be added to P1 R and to 0.5P AE, as shown in Figure 9.4. The P AE was reduced because it is unlikely that both forces, PIR and P AE, would peak simultaneously. Once the total thrust due to 308 (9.7) (9.8)
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dynamic analysis is computed, the sliding stability and bearing capacity are evaluated. The computed factor of safety must be equal to or greater than 75% of the minimum static safety factors. Figure 9.4 "External loads on vertical MSE wall due to seismic loading" (Vector Elias 2001) 9.2.1.2 Internal Stability of MSE walls When conducting the internal stability analysis, it is important to obtain spacing between geosynthetic layers so as not to overstress them (Robert M. Koerner 309
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2001 ). Therefore, the maximum tensile force in each reinforcement layer must be calculated. Connecting the maximum tensile forces from each layer defines the critical slip surface in a reinforced soil wall. The critical slip surface inclined from the horizontal with an angle '!'a, and can be determined as a function of the soil's friction angle and other angles. For a vertical wall, \jl can be determined using Equation 9.9. Based on large number of previous experiments, the maximum tensile forces surface assumed to be approximately linear and pass through the toe of the wall as shown in Figure 9.5 (Vector Elias 2001). The maximum tensile force in each reinforcement layer per unit width, T max, is a function of the overburden pressure and the spacing between reinforcements, as shown in Equation 9.10. II'= 45 + 2 T =K xyxZxA max a t Where: Z = is the depth of reinforcement from the top of the MSE wall, At = tributary area. 310 (9.9) (9.10)
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For .wer1&cel ..U., . .,,.f' :; ..... $ cf.: ..U. d. ,.,.. '*"" tt .: . froa ... f). t.enlt1! .t;;;.J;.;;;;;,. .. oot!.,"' ..... ... floo\ c. . ,_ I CM '",. Ill tift...,,,. oot.lt+ 1..tam ...... < Figure 9.5 "Inclusion stress distribution along the height of the wall" (Vector Elias 2001) . Once the values ofT max are obtained for the reinforcing layers, the MSE wall is checked for the internal stability against reinforcement rupture, as shown in Equation 9 .11. In This equation, T a is the allowable tension force of the reinforcement, and is equal to the ultimate tensile strength from the wide width tensile tests divided by the reduction factors and an overall factor of safety, as shown in Equation 9.12 The reduction factors are: Creep (RFcR) which is equal to 1.6 for polyester, durability (RFn) with a minimum value of 1.1, and installation damage (RFm) with a minimum value of 1.5. The overall all factor of safety (FS) is assumed to have a minimum value of 1.5. T FS =a;::::1 rupture T max 311 (9.11)
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T T = ult a RFCR RFD RFID FS Then, the MSE wall is checked for the internal stability with respect to Pullout Failure, as shown in Equation 9.13. In this equation, the length of geosynthetic reinforcement within the soil mass behind the maximum te.nsile surface, Le, is calculated (Robert M. Koerner 2001 ). This length is assumed to have a minimum value of 1 m. Based on the embedment length, a factor of safety against pullout, FSpo, is produced. The minimum required value ofFSpo is 1.5 1 Pr T = xF xyxZxL xCxR xa=max FS e c FS PO pullout Where: p = is the friction bearing interaction ( ), 3 Le = is the embedment length in the resisting zone behind the failure surface ( 1m ), C = is the reinforcement effective unit perimeter (2 for geosynthetic ), Rc =is the overage ratio (1 for continuous reinforcements) a= is a scale correction factor (0.8 for geogrid, and 0.6 for geotextile), In case of not achieving the miirimum factor of safety against pullout, several alternatives can be done: Increase the total length of reinforcement, L, use a stronger reinforcement, or reduce the vertical spacing between reinforcement layers. The total length of reinforcement in each layer is the sum of the embedment length Le, and Length of active zone, La, as shown in Equation 9 14. L=L +L a e Where for vertical walls 312 (9.12) (9 13) (9.14)
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As for external stability, internal stability against rupture and pullout for seismic condition must be calculated. The seismic safety factors must be at least 75% of the minimum allowable static safety factors. The inertial force, produced by the seismic event leads to an incremental dynamic increase in the maximum tensile forces in the reinforcements. Therefore, the ultimate strength of the geosynthetic reinforcement is the sum of the reinforcement strength per unit width needed to resist the static component of the load, Srs, and the reinforcement strength needed to resist the dynamic component of the load, Srt, as shown in Equation 9.15. Tult = S rs + S rt (9.15) Due to seismic loading, the friction coefficient between soil and reinforcement maybe reduced. Hence, 80% reduction factor to the fiction coefficient p* is included when evaluating for pullout under seismic loading. Therefore the total pullout, Ttotai. load must be Cx 0.8F ( *) T xyxZxL xR xa tota 0.75 x 1.5 e c 9.2.2 Hybrid Retaining Walls A Hybrid wall lies between an externally stabilized retaining wall system internally stabilized wall system. It has reinforcement at the backfill and has rigid facing instead of the modular block. Its rigid facing should not be as : wide as that . of gravity walls due to embedded reinforcements within the backfill. The external and internal stability for MSE walls can also be used for this kind of walls. 313 (9.16)
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A detailed study on hybrid retaining walls was completed in 2000 by ZehZon Lee. In his study, Lee Numerically investigated several hybrid retaining walls under both static and dynamic loading conditions. One of his finding, that Hybrid Twall with small toe oflength equal tol.5% of the wall's height (H) and heel of length equal to 15% of the wall's height (H) behaved very well compared to other configurations when subjected to different seismic shaking (Lee 2000). 9.3 Finite Element Analysis on MSEWall with Rigid Facing A 3Dirnnesional hypothetical hybrid retaining wall model, under plane strain condition, was assembled and analyzed using FEMB and LSYDNA codes, respectively. The height of the wall was 7 m and included Ottawa sand as its backfill material, and Tensar SR2 geogrid as its inclusion material. The wall was subjected to static loading due to gravity and dynamic loading due to earthquake. The model was initially built allowing interface between soil and reinforcement, briefed as discrete model, and then as homogeneous materials using the equivalent transversely isotropic properties, that were formulated in chapter 5. The hypothetical MSE model shown in Figure 9.6 consisted of a concrete hybrid T retaining wall with small toe, foundation soil, backfill soil, and inclusion. Typically, for a conventional retaining wall the base of the wall, B, is approximately 75% of the wall's height, H. For this model, and because of including reinforcements, it was decided to reduce B to 30% of H. The resulted B was 2.4 m. Of this dimension, 113 was assigned for the toe and 2/3 was assigned 1 I ., for the heel. The wall's height was defmed as the height of the wall above the ground surface, and it had an embedment depth of 1 m. The footing and the stem 314
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thicknesses were 0.5 m, and 0.6 m, respectively. Furthermore, the inclusion layers were extended to Length L, equal to the wall's height of 7 m, with constant vertical spacing of 0.3 m. The inclusion layers were not embedded nor attached to the wall, allowing free movement of the wall. This represented the condition of weak connection between the wall and the geogrid inclusion. The extended length of foundation soil in front of the wall was 2H, 14m, and the total length of the backfill was 3H, 21m. The reason for extending the foundation and the backfill was to exclude any boundary effect on the analysis results. The thickness of the MSE model in Y direction was constant and equal to 5 m. In this 3Dimensional model, all elements but inclusions were constructed using brick element containing 8 nodes. The inclusion layers were built using the 2dimesnional 4noded shell element. 315
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Lx Hybrid retaining H=7m Reintbrced zone L1 =7m Fotmdatiori Hl,l,!fl = 6 m HFI.ight = 5 ttl 35.5 m Figure 9.6 "Side view ofMSEWall" Backfill L:! =14m
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9.3.1 Loading Both models, discrete and homogeneous, were subjected to a gravity load due to the weight of all model's components, and a shaking load due to earthquake ) motion. These loads were referred to as static and dynamic, respectively. The static was applied by constant gravitational ground acceleration of 9. 81 ml sec2 throughout the analysis. After 5 seconds of the analysis, the horizontal component of ground motion acceleration was applied for an additional30 seconds. The reason for applying the dynamic load after 5 seconds was to exclude any oscillation resulting from the applied instant gravitational acceleration. The horizontal component of 1940 ElCentro earthquake acceleration, shown in Figure 9.7, was selected and applied representing the dynamic load. The peak ground acceleration (PGA)11 ofElCentro earthquake was 0.32g or 3.12 rnlsec2 To magnify the earthquake effect, a factor 2 was applied to all the data in this accelerogram, from start till end. Once the earthquake motion was completed, another 5 seconds of the analysis was allowed under gravitational loading only until termination at 40 seconds. Both static and dynamic loadings were applied through the command ofbody load. Table 9.1 is presented to briefly summarize both load curves, static and dynamic. Table 9.1 "Load curves summary" Load curve 1 2 9.81 (constant) (Fi e9.7)x2 Termination time of 40 11 PGA is the larges absolute value of acceleration from a given time history. 317
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F UIIO I .&! D E E GDD .. Figure 9.7 "ElCentro Earthquake acceleration time history" 9.3.2 Material Properties One of the most important steps in constructing a model using finite element method is to assign correct properties of the utilized materials. The three main materials that were used in this model were Ottawa, geogrid reinforcement, and concrete section. The constitutive models representing these materials involved in this study were the cap model for Ottawa sand, and isotropic linear elastic model for inclusion layers and concrete wall. 9.3.2.1 Backfill and Foundation Soil Ottawa sandy soil with density of 1719 kg/m3 corresponding to relative density, Dr, of70% was used to represent the backfill material and the foundation soil. A 318
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geologic cap model was used to represent these elements. The 10 material constants as determined from previous calibration, Chapter 6, for Ottawa sand were as shown in Table 9.2. Among these constants, the shear modulus, G; would vary with depth depending on the mean stress, cr0 as shown in Equations 9.17 and 9.18 for round and angular grains respectively. The reinforced zone and the backfill were divided into 3 segments (Top, middle, and bottom) and the average value of G was determined to represent each segment. On the other hand, the foundation was represented by 1 segmental with 1 value of G. The G for the foundation soil was allowed to slightly increase from the calculated value to 120 MPa to 210 MPa to increase its rigidity. Rowe and Skinner reported in their numerical analysis of geosynthetic reinforced retaining wall, constructed on layered of soil foundation using finite element program AFENA, that the stiffness of the foundation can have a significant effect on the wall's behavior(R. Kerry Rowe 2001). In their analysis, they found that a weak foundation can significantly increase the wall's deformation and the strains in reinforcement layers compared to rigid foundation, but it doesn't affect the earth's pressure behind the wall face. G round 6908(2.17 e )2 112 max= CJ'72 1+e o Gangular max= 3230(2.97 e)2 1+e o Where: Gmax = maximum shear modulus in kN/m2 e = 0.58 for dense sand 1 CJ'o =((J' v + 2CJ'11) 3 319 (9.17) (9.18)
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K0 = 1sin Z = depth from top of the wall 9.3.2.2 Inclusion A commercially available geogrid named Tensar UXll OOHS was used as the inclusion material for the analysis. This geogrid is made of High Density Polyethylene, HDPE and has properties as shown in Table 9.3 (GFR 2004). The geogrid in this simulation was assumed to have elastic properties. Therefore, only the elastic modulus, Eg, and Poisson's ration, Vg, properties were of interest. Eg was taken to be the secant modulus at 5 % strain divided by the thickness, t, which was 2 mm. 9.3.2.3 Concrete Wall and Footing The hybrid retaining wall was modeled as continuous concrete. The concrete wall was assumed to have elastic as shown in Table 9.4. 320
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K (MPa) MSETop 209 MSEMdl 209 MSEBtm 209 Foundation 209 Table 9.2 "Properties of soil for MSE wall" G a cp J3 y R D (MPa) (MPa) (mm2/N) 32.6 0 0.2815 0 0 1.6 0.6 70 0 0.2815 0 0 1.6 0.6 125 0 0.2815 0 0 1.6 0.6 210 0 0.2815 0 0 1.6 0.6 Table 9.3 "Properties of inclusion for MSE wall" (Lee 2000) p Eg v (Kg/m3 ) (MPa) (assumed) 1030 290 0.3 Table 9.4 "Properties of Concrete wall of MSE wall" (Boresi 2003) 2320 E (MPa) 25000 v 0.15. w Xo (MPa) 0.0079 0 0.0079 0 0.0079 0 0.0079 0
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9.3.3 Sliding Interface In this model, a frictional interface based on penalty function was allowed between adjacent surfaces causing sliding and separation. As it was mentioned before, the tensile strength of reinforced soil would increase by stress transfer from the soil to inclusion layers. Friction interaction between soil and reinforcement would allow such a phenomenon The sliding interface was not only defined between soil and inclusions, but also between soil and concrete wall. The static friction coefficient, was determined based in the internal friction angle of soil, which was obtained in chapter 6 from drained triaxial tests and was equal to 36. The interface friction angle, 8, is defined to be (Lee 2000), could be determined using Equation 9.18. was determined to be around 0.5 ,u = tano (9.19) 9.3.4 Boundary Conditions The MSE wall model was subjected to plane stain condition due to the thickness of the wall, and therefore, the lateral displacement in Y direction would be zero. In addition to plane strain condition, roller condition at both ends and fixed condition at the base of the wall were applied. The roller condition allowed the nodes to move vertically in Z direction only, while the fixed condition didn't allow any movement. 322
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9.3.5 External and Internal Stability ofMSE wall Before subjecting this model to finite element simulations it was decided to check if this structure met the external and internal requirement by satisfying the minimum safety factors. It was assumed that no traffic surcharge was applied on top of the MSE wall. Also, the weight of the concrete wall was relatively small compared to the soil weight and therefore, was not included in the analysis. As shown in Figure 9.8, there were 2 main loads that must be taken in consideration, the weight of reinforced soil, Vl, and the lateral thrust, Fl. The locations of these forces were at L/2 and H/3, respectively, where Lis width of reinforced zone and H is height of the reinforced soil. Properties of reinforced soil, Backfill and foundations were y = 1719 (kg/m3 ) = 16.9 kN/m3 = 36o, c = 0 (kPa) K a =tan 2 ( 45 }=0.26 323
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Vl H=7m >E(L=7m ) Figure 9.8 "External load on MSE Wall model" 9.3.5.1 External Stability due to Static Loading When conducting the external stability analysis, it was required to determine the driving and resisting forces. From Figure 9.8, Vl is weight of the reinforced soil and was equal to On the other hand, F 1 was estimated by evaluating the area of the lateral earth pressure triangle and was equal to Fi= x yx H 2 x K0 =0.5 xl6.{ %3 )x 72x0.26=107.65( ko/n,) 324
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The overturning moment, M0 was resulted from the lateral thrust, F 1 multiplied by the arm, H/3, and was equal to M 0 = xyxH2 xKa = 107.65 (kN!m)x (2.33 m) = 251.2( kN.t%J And the resisting moment, MR, was resulted from the weight of the reinforced soil, V 1 multiplied by the arm, L/2, and was equal to M R = 828.4 (kN!m)x (3.5 m) = 2899.4 (kN%) Factor o[safety against sliding / From Equation 9.4, the minimum allowed factor of safety against sliding was 1.5. Applying this equation and considering that the as the friction coefficient, the factor of safetY against sliding was L P R VI x tan tjJ 828.4(kN.'%z)x tan 36 FS == = T ) =56>15 sliding z: p d Fl 107 .65\kN.% Factor o[safety against over turning From Equation 9.5, the minimum allowed factor of safety against overturning was 2. Applying this equation, the factor of safety against overturning was M R 2899.4(kN.%) FSOT =. = TkN nl) M 0 "}m 325 OK OK
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Factor of safety against bearing pressure From Equation 9.6, the minimum allowed factor of safety against bearing capacity failure was 2.5. Applying this equation, the factor of safety against bearing capacity failure was q cf X N + 0.5 X LX r X N ult c r FS bearing = CJ' = Vl v L 0+0.5x7(m)x16.{ %3 )xs6.31 > = (k% ) 28 2.5 828.4 N m 7(m) By checking all the safety factors it was determined that the MSE wall was externally stable. 9.3.5.2 Internal Stability due to Static Loading This model consisted of 7 m backfill reinforced with geogrid. The vertical spacing Sv, between inclusions was 0.3 m resulting in 23 layers of reinforcements. In this analysis the model was tested against reinforcement rupture, and then against the pullout failure Internal stability with respect to rupture failure The first step was to determine T max in each of the reinforcing layers, and was equal to T =K xyxS xZ="0.26x16.9fkN/ 3)x0.3(m)xz max a v X /m 326 OK
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Based the on the internal stability against reinforcement rupture, the allowable tension force, T 3,of each reinforcement layer must be larger than T max, therefore T FS =a;:::1 rupture T max Knowing that the ultimate tensile strength, T ult. of the geogrid used in this model was 58kN/m (GFR 2004), Tawas equal to T 58fkN I ) T = ult = \ / m =19.97(k%) a RFCR RF D RFID FS 1.6x1.1x1x 1.5 m By comparing Ta to Tmax at each layer, FSrupture was determined and was as shown in Table 9.5. From this table it was clear that factor of safety against rupture of reinforcement in all elevations was satisfied. Table 9.5 "Factor of safety against reinforcement rupture at different elevation due to static loading" z Tmax Ta FSrupture Comment (m) (kN/m) (kN/m TaiTmax 0.3 0.39546 19.97 50.50 Ok 0.9 1.18638 19.97 16.83 Ok 1.5 1.9773 19.97 10.10 Ok 2.1 2.76822 19.97 7.21 Ok 2.7 3.55914 19.97 5.61 Ok 3.3 4.35006 19.97 4.59 Ok 3.9 4.74552 19.97 3.88 Ok 4.5 5.53644 19.97 3.37 Ok 5.1 5.9319 19.97 2.97 Ok 5.7 6.72282 19.97 2.66 Ok 6.3 7.51374 19.97 2.40 Ok 6.9 8.30466 19.97 2.20 Ok 327
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Internal stability with respect to pullout failure Based on the pullout criterion, the required factor of safety is 1.5. The first step here was determining the critical slip surface which was inclined from the horizontal surface at an angle \jl. For this problem cp = 36 for Ottawa sand as determined in Chapter 6, p = oo, 3 = 2/3cp = 24. Therefore: 36 If/= 45 += 45 += 63 2 2 From there the embedded length of each layer in the resistant zone, L e was L = L(HZ)* tan(9063)) = 7(m)(7mz)x 0.5095 e Moreover when using Equation 9.13 in determining the Factor of safety against pullout, the following constants were considered: 2 2 F = 3tan = 3tan36 = 0.484 C=2, R: = l.O, and a=0.8. From there, the Pr was determined at each layer had the form Pr=F xyxZxL xCxR xa=0.484xl6.9(kN!m)xZxL x2xlx0.8 e c e Pr =13.087 x Zx L e 328
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At each layer, the value ofPr was determined and divided by Tmax to obtain the SFpo, and were as shown in Table 9.6. In this table, it was evident that the values of SF Pullout were much larger than 1.5, which indicated that due to the static loading, the wall was internally stabilized with respect to the pullout failure. Table 9 6 "Internal stability with respect to pullout failure due to static loading" z Le Pr Tmax FSpullout Comment (m) (m) (kN/m) (kN/m) Pr!Tmax 0.3 3.59 14.08 0.4.0 35.61 Ok 0.9 3.89 45.84 1.19 38.64 Ok 1.5 4.20 82.41 1.98 41.68 Ok 2.1 4.50 123.77 2.77 44.71 Ok 2.7 4 .81 169.94 3.56 47.75 Ok 3.3 5.11 220.90 4.35 50.78 Ok 3.9 5.42 276.67 5.14 53.82 Ok 4.5 5.73 337.24 5.93 56.85 Ok 5.1 6.03 402.61 6.72 59.89 Ok 5.7 6.34 472.78 7.51 62.92 Ok 6.3 6.64 547.75 8.30 65.96 Ok 6.9 6.95 627.52 9.10 68.99 Ok 9.3.5.3 External Stability due to Seismic Loading In external stability analysis due to seismic loading, the horizontal inertia force, P1R and the seismic thrust, P AE were determined. P1R is a function of the maximum acceleration developed within the reinforced soil IJlass, AM. For a peak acceleration coefficient, A, equal to 0.4, Am was equal to A = (1.45A)A = (1.450.4)x 0.4 = 0.42 m From there, PIR was equal to 329
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PIR O.Sx Am x yx H 2 0.5x(0.42)x16.9( %3 )x (7(m)f l73.9kN I m In this model = 36 = oo, e I[ 1 I[ A;]= tan and As a result, using the MononobeOkabe active earth pressure equation, Therefore, It is unlikely that P AE and P1R are to peak simultaneously. Therefore, P AE was reduced to 50% of its original value, resulting with (so%)P AE = (o.s)x (24t.s(ko/nJ)= 120.9(kN/m) 330
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Factor o(sa(ety against sliding The minimum allowed factor of safety against sliding when subjected to seismic loading was 75 %of the one when subjected to static loading, 1.5. Therefore, the factor of safety against sliding was L,PR V1x tan FS = = ;___!._ _______ sliding 'I p d Fl + PIR +(50% )P AE 828.4(ko/n )x tan 36 = r J r J r ) = 1 5 > 1 25 107.65\k% + 173.9\k% + 120.9\k% . Factor o(safety against over turning The minimum allowed factor of safety against overturning, when subjected to seismic loading was 75% of the one when subjected to static loading, 2. Therefore, the factor of safety against overturning moment was L MR Vx1 2 FS OT = = =::MO F1x(I%)+(50%)PAE x(0.6H)+PIR x( 828.4(ko/,)x I 07.65 x G)+ 120.9(kN/m)x (0.6 x 7(m ))+ 173.9ft l) = 2899.4(kN) = 2.12 l.S 1367.6(kN) 331 OK OK
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Factor o(safety against bearing pressure The minimum allowed factor of safety against bearing capacity failure when subjected to seismic loading was 75% of the one when subjected to static loading, 2.5. Therefore, the factor of safety against bearing capacity failure was cf xN +0.5xLxyxN c r FS bearing =a= V1 v L 0+0.5x7x16.{ %3 )x56.31 > = 828.4(ko/.J 28 _1.875 7(m) By checking all the safety factors, it was determined that the MSE wall was externally stable under seismic loading. 9.3.5.4 Internal Stability due to Seismic Loading The first step in evaluating the internal stability due to seismic loading was determining the dynamic tensile load, T md, on each reinforcement layer induced by the inertia force, P1. In which L. T = p x ez md I n And, L: L. 1 ez l= PI =A xW m a 332 OK (9.20) (9.21)
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Where: W a = is weight of active zone To do so, it was important to determine the angle of critical slip surface, \jf, from horizontal. It was assumed that the location and slope of the critical surface remain the same, then Therefore, Le = L(HZ)* tan(9063))= 7(m)(7(m)Z(m))x 0.5095, aJ?.d 7 L L = 63.36(m) 1 ez l= From there, w = r x !__ x H 2 x tan(90 If/) a 2 = 16.9(ko/nJx (7(m)) 2 x tan(9067.83)= 168.69(kN/m) PI =Am xWa =0.42x168.69(k%J=70. 86(kN/m) L. ( %) L. T = P x ez = 70.86 kN x ez md I n m 63.6(m) L L. 1 ez l= The total maximum tensile load at each reinforcement layer, Ttotat, is T =T +T total max md 333 (9.22)
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Internal stability with respect to rupture failure When conducting the internal stability with respect to rupture failure, the reinforcement strength needed to resist the static load, Srs, must be larger than T max, and the reinforcement strength needed to resist the dynamic load, Srt must be larger thanTmd As in the static analysis reduction factors and global factor of safety are taken into consideration as shown in Equation 9.23 and 9.24. s s T < rs = rs max (75% )x RF CR x RF D x RFID x FS (75% )x RF x FS s T < rt md (75%)x RF D x RFID x FS Assuming that Srs is 40 % of Tult and that Srt is 60 % of Tult, therefore Srs =(40%)xTult =0.4x5s(ko/m)=23.2(ko/nJ Srt =(60%)xTult =0.6x5s(k%)=34.s(k%) From there, S 23 2(kNI) rs = l m = (75%)xRFCR xRFD xRFID xFS (75%)x1.6xl.lxl.lxl.5 m' and S 25.56(kN I ) rt = I m = 25 56(kN I ) (75%)xRFDxRF1DxFS (75%)xl.lxl.lxl.5 lm Comparing and Tmd to the reduced strength in each layer, as shown in Table 9.7, indicated that the reinforcement layers were stabilized against rupture failure under seismic loading. 334 (9.23) (9.24)
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Table 9.7 "Factor of safety against reinforcement rupture at different elevation due to dynamic loading" z Tmax Srs/(0.75.RF.FS) Tmd Srt/(0. 75.RF .FS) Comment (m) (kN/m) (kN/m) (kN/m) (kN/m) 0.3 0.40 10.65 4.04 25.56 Ok 0.9 1.19 10.65 4.38 25.56 Ok 1.5 1.98 10.65 4.72 25 56 Ok 2.1 2.77 10.65 5.06 25.56 Ok 2.7 3.56 10.65 5.40 25.56 Ok 3.3 4.35 10.65 5.74 25.56 Ok 3.9 5.14 10.65 6.07 25.56 Ok 4.5 5.93 10.65 6.41 25.56 Ok 5.1 6.72 10.65 6.75 25.56 Ok 5.7 7.51 10.65 7.09 25.56 Ok 6.3 8.30 10.65 7.43 25.56 Ok 6.9 9.10 10.65 7.77 25.56 Ok Internal stability with respect to pullout failure Based on the seismic stability with respect to pullout failure, 80 % reduction to the static pullout resistance was applied. The factor of safety against pullout was reduced by 75% of the static one. The result shown in Table 9.8 indicated that all the layers were internally stabilized against pullout due to seismic loading (80%)xF* xyxZxL xCxR xa (80o/c)Pr FS PO = T e c = T o 1.125 total total 335
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Table 9.8 "Factor of safety against reinforcement pullout at different elevation due to dynamic loading" z Le 80xPr Tmd Tmax Ttotal FSrupture Comment (m) (m) (kN/m) (kN/m) (kN/m) (kN/m) 0.3 3.61 11.34 4.04 0.40 4.43 2.56 Ok 0.9 3.91 36.88 4.38 1.19 5.56 6.63 Ok 1.5 4.22 66.24 4.72 1.98 6.69 9.89 Ok 2.1 4.52 99.40 5.06 2.77 7.82 12.70 Ok 2.7 4.82 136.39 5.40 3.56 8.95 15.23 Ok 3.3 5.13 177.18 5.74 4.35 10.09 17.57 Ok 3.9 5.43 221.79 6.07 5.14 11. 22 19.77 Ok 4.5 5.74 270.21 6.41 5.93 12.35 21.89 Ok 5.1 6.04 322.45 6.75 6.72 13.48 23.93 Ok 5.7 6.34 378.50 7.09 7.51 14.61 25.91 Ok 6.3 6.65 438.36 7.43 8.30 15.74 27.86 Ok 6.9 6.95 502 04 7.77 9.10 16.87 29.76 Ok 9.4 Results and Discussion In this section and subsequent sections the results of finite element simulations of the MSE wall models are presented. Both models, discrete with full frictional interface and homogeneous with transversely isotropic properties were subjected to gravity and earthquake shaking under plane strain condition. At 5 second of analysis the were gathered to represent the static response. After that, within the dynamic simulation, the maximum absolute results were gathered to represent the dynamic response. The results obtained from discrete model were eventually compared with those obtained from homogeneous model. The study items, due to static and dynamic loadings were: 1Lateral earth pressure ( crx), 2Lateral wall displacement (Bx), 336
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3Bearing pressure (crz) and settlement (8z) of foundation, and 4Inclusion tensile stresses in reinforcement layers (for the case of discrete mode). 9.4.1 Results of Discrete Model For this discrete model and homogeneous model, the plane strain condition was imposed constraining any lateral strain in Y direction, along the thickness of the retaining wall. Therefore, only the global directions X and Z were of interest in this chapter. In real applications of MSE walls, geosynthetic layers are attached to the wall to prevent excessive wall displacement and rotation. In this model, the reinforcement layers were not attached to the concrete wall and therefore more displacements of the wall was expected; the MSE model was a hypothetical model and was completed mainly for comparison reasons with the homogeneous one. 9.4.1.1 Lateral Earth Pressure (crx) behind the Hybrid Wall Due to the application of both loads, static and dynamic, the lateral earth pressure on the hybrid retaining wall was obtained and plotted along the centerline of the MSE backfill as shown in Figure 9.9. In this figure, the data of both static and dynamic lateral pressure were smoothly fitted to recognize the trends of both loadings. The equations of these fits were shown in the graph. The earth pressure increases with depth, with range of 0 kPa to 20 kPa under gravitational load, and 40 to 140 kPa dynamic loads. Without a doubt, there was a significant effect of 337
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the earthquake motion on lateral pressure along the hybrid wall. The increase of lateral earth pressure was more than 6 times at most of the elevation points. Also, in this figure, the static and dynamic lateral earth pressure of finite element method was compared to those obtained from analytical solution. In which,a t. =K xyxZ=0.26xl6.9(kN/3)xZ,and xs at1c a I m a d =a + KAE x r x z = 0.26 x 16.9(kN/ 3 ) x z(m) x rynam1c x statlc I m + 0.583 x 16., %3 )x z(m) Where, Z is the depth from top. The finite element response slightly varied from the analytical solution, but the trends were the same. Previous study on hybrid T walls, when subjected to dynamic load, indicated that the distributions oflateral earth pressure in case of detached geosynthetic didn't significantly differ from the cases of attached geosynthetic, except in the upper 1/3 portion for the wall. In the detached condition, due to a stronger mutual wall backfill slippage, the upper 1/3 portion of the wall the lateral earth pressure was larger than that in the attached condition (NienYin Chang 2004 ). 338
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... .. .. .. .. 4 Sttlt!c .. ftft 11. .. ... l}+l:J t,., .... 7JIJ .... ... !s.aa .. 4 ,, l: 3.00 2JIJ 1JIJ '. ' OJIJG 20.000 40.000 IO.DDD IO.OaD 1CIUIHI12DJIJG 140.CIKI110.000 Lu.,.l mh ,,..u,._ax (kPa) Figure 9.9 "Lateral earth pressure (ax) of backfill along the hybrid retaining wall" 9.4.1.2 Lateral Wall Displacement (Bx) The lateral displacements of the hybrid retaining wall were obtained at the centerline along the wall's height, and were as shown in Figure 9.10. In this figure, it was obvious that static load was already a cause of large displacement which significantly increased when imposing the dynamic load Due to static and dynamic loadings, the maximum displacement occurred near the top of the wall, with absolute magnitudes of23 mm and 240 mm, respectively, where the negative values of displacements indicated that the wall was moving away from the backfill forward displacements. In general, the average values of displacement resulted from dynamic shaking were 13 times more than those resulted from static simulation. These displacements especially due to earthquake would be much less if the reinforcing layers were to be attached to the concrete wall by merging 339
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the coincident nodes of concrete wall and inclusion layers. Chang and his colleagues noticed that the deformed shapes and the maximum forward displacements are very different between the two different connection conditions. In which, the detached cases show a free wall vibration with only the wall base restraint during a strong seismic shaking. On the other hand, the displacements of the attached cases had a bulb shape and were much smaller than those of detached cases (NienYin Chang 2004). Despite the large lateral displacement, the geosynthetic reinforced soil supported by hybrid concrete wall did not fail. This is an indication of the ability of geosynthetic reinforced soil to accommodate large movement without failure. This observation was also noticed when monitoring a 6yearold geosynthetic reinforced retaining wall in South Korea, built in 1994 (Y oo 2004). This wall revealed excessive lateral movement during construction of more than 40 mm, without structural failure. 110 &c (mm) 100 a Figure 9.10 "Lateral displacements of hybrid retaining wall due static and dynamic loads" 340
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9.4.1.3 Bearing Pressure ( crz) and Settlement (oz) of Foundation In this section the analysis was completed to report the weight effect of: the hybrid retaining wall, the reinforced backfill, and unreinforced backfill on the foundation soil. The results presented here focused on obtaining the vertical pressure ( crz) and vertical settlement (8z) due to gravity and earthquake motion, respectively. Since the earthquake motion was imposed horizontally, a little increase of the vertical bearing pressure and settlement were expected. Ling and his colleagues in 2005 indicated that vertical earth pressure developed at foundation was quite uniform but the distribution became non uniform as it gets nearer to the blocks. This observation was completed when conducting large scale shaking table tests on 3 modularblock reinforced soil retaining walls (Hoe I. Ling 2005). Similar observation was noticed when conducting the numerical analysis here on the MSE hybrid retaining wall. Figure 9.11 indicated that most of resulted vertical stresses were located beneath the reinforced and unreinforced zones. As expected, most of the segment of foundation soil didn't behaved differently when subjected to dynamic load, except beneath the footing. Due to both loading, a sudden increase of bearing pressure was observed beneath the footing exceeding 200 kPa as a result of the gravity and 1100 kPa as a result of the earthquake. This indicated that even though the acceleration due to ground motion was horizontally, it had significant effect on foundation's bearing pressure underneath the footing. This increase in crz was mainly related to the excessive lateral displacement of the wall. The wall tilted terribly away from the backfill causing more pressure to the foundation beneath the toe of the footing. Due to static load, the top of the wall also tilted causing an increase in crz beneath the 341
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footing. That increase was not as large as due to earthquake where the wall displaced much more laterally. Beyond the reinforced zone, the vertical pressure due static loading was constant and equal to 140 kPa. In general this value is constant and equal to Y s H+q (Kianoosh Hatami 2006), where q is the vertical surcharge and H is the depth of foundation. For this model, considering q =0, H = 8 m and ys =16.9 kN/m3 therefore, the analytical crz would equal to 135.5 kPa which is very comparable with the value obtained by finite element analysis. R elnfr actid Not supprting 1 2110 anyla:ul 81. zane t&s .,aaa l .. 1110 I .. iii IDO u i! .... > 2DO a D.DD I \ =Sta ti .t.laa ding I.DD 1D.GD 11.DD 2D.DD 21.00 30.DD 31.DD 40.DD X (m) Figure 9.11 "Bearing pressure (crz) of foundation soil due to static and dynamic load" Similar to bearing pressure, most of the settlement of foundation appeared to happen beneath the reinforced and unreinforced soil, as shown in Figure 9.12. Dynamic load didn t cause more settlement than gravity, except beneath the 342
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footing. As explained earlier, the excessive wall displacement caused the wall to be tilted away from the backfill. Therefore, this wall tilting caused larger bearing pressure under the wall's footing of the wall which resulted in larger settlement. The maximum settlements of foundation due static and dynamic loadings were 17 mm and 29 mm, respectively. DDD I.DD 1D.OD 11.DD 2D.DD 21.DD 30.00 31.DO 40.DD 10 E !. t: _,. Ia I Nat mpprting anyln.cl X (m) UnReinfruced zane Static Figure 9.12 "settlement (5z) of foundation soil due to static and dynamic load" 9.4.1.4 Inclusion Tensile Stresses The reinforced soil behind the retaining wall had 23 equally geogrid layers at 300 mm spacing with relatively small stiffuess of290 MPa. A study completed by Hatami and his colleagues concluded that using a larger number of lower stiffuess reinforcement layers at relatively smaller spacing is more effective in reducing the wall's deformations than stiffer reinforcement layers with larger 343
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spacing (K. Hatami 2001). A same conclusion was obtained in Chapter 8 when analyzing the foundation on top of reinforced soil of different stiffness and different spacing. Siddharthan et al. (2004) reported from centrifuge tests on bar mat mechanically stabilized earth walls that walls with longer reinforcement deformed less than walls with shorter reinforcements(Raj V. Siddharthan 2004). And current design of geosynthetic reinforced retaining wall considers a length of reinforcement of0.5 to 0.7 times the height of the wall (Dov Leshchinsky 2004). In this project, the length of these inclusions was assumed to be equal to the wall's height of7 m. They were extended to the wall but not attached with its nodes letting the wall to bulge freely. Furthermore, these layers had a thickness of 2 mm and were modeled as 2dimensional shell elements. The tensile stresses due static and dynamic loading of these geogrid layers were described at different distances from the wall, and along the wall height as shown in Figures 9.13 through 9 .16. After smoothing the tensile stresses by eliminating compressive stresses and drawing a trend line that best fitted the available data, the effects of distance from the wall and elevation were combined in 2 graphs, due static and dynamic loads, and were as shown in Figures 9.17 and 9.18, respectively. When observing the tensile strength in Figures 9.13 though 9.16, it was noticed that there was an increase of these stresses with depth. In Figure 9 .16, different phenomenon was observed due to dynamic load. In here, the stresses were maxima near the top and decreased toward the bottom. Based on FHW A publication 2001 and other laboratory and filed tests on reinforced soil walls, the largest axial loads are observed in the top geogrid layer and decreased toward the bottom (Taesoon Park 2005), but that didn't happen here in most ofthe sections of inclusion layers. In real life applications the inclusions are attached to the wall preventing lateral displacements. Since most of the displacements are expected to happen near the top, large stresses would develop on the inclusion layers that are 344
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located near the top. That was not the case in the current numerical simulation, where, again, the nodes of inclusion were not merged with the coincident nodes of the hybrid wall, hence zero connection strength of the wall face to reinforced soil mass through reinforcing layers was developed. Therefore, the stresses would depend on the frictional forces only that increased with increasing the normal forces. Previous study on a failed segmental retaining wall completed by Collin showed that poor connection between the soil reinforcement and the wall unit was the main cause of failure, and therefore connection between the reinforcement and the facing of retaining wall is crucial component of the system that must be well thoughtout in both the design and construction ofMSE retaining walls (Collin 2001). Furthermore, in those figures, significant increase of stresses was developed when subjected to dynamic loading. The developed tensile stresses followed the same trend of most of geosynthetic segments when subjected to gravity load; where the stresses increased with depth. Ling and his colleagues also noticed from their shaking table tests that bottom geogrid layers experienced largest peak force during shaking (Hoe I. Ling 2005). At segments that were far from the wall, 7 m in particular, the increase of stresses happened to near the top as opposed to other segments. Near there, the stresses of geosynthetic were larger than near the bottom, which were developed to prevent the excessive soil sliding that was generated as result of the bulky wall tilting. 345
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I 1 6 5 E l!4 :z 1 "f/1' .. ,. ..... ' ,, ... . "'... .... Static Static..;average Dynamic:avrage c:l: .... a . ...... ... ., . . 0 0 D 400 600 100 1000 1200 1400 1600 T81'1islaltNSsn_CIC (kPa) Figure 9.13 "Tensile stress of reinforcements at distance 0 m from the wall due static and dynamic loadings" I 1 8 s .!. l:4 "i 2:3 2 1 ..... .. .. t .. Static Dynamic Staticaverage D D Tansila strwssn_ax (kPa) Figure 9.14 "Tensile stress of reinforcements at distance 1.5 m from the wall due static and dynamic loadings" 346
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7 I ',, Dynamic .. Staticaverage 2 . : ', . .. . .. .. 1 Dynamicaverage ":, .. .. . .. '. ,,. ., a 10111 ,., T.nslla strnsas_ux Figure 9.15 "Tensile stress of reinforcements at distance 3.75 m from the wall due static and dynamic loadings" E .1: .. :c 7 5 4 3 2 1 .. / / Dynamic _, '1 / . .. .. Dynam lcaverage zoo 400 600 100 1000 Ten1ila ltnsn_ax (kPa) 1200 Figure 9.16 "Tensile stress of reinforcements at distance 7 m from the wall due static and dynamic loadings" 347
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The summary of Figures 9.13 through 9.16 was provided in Figures 9.17 and 9.18 due to static and dynamic loading, respectively. When observing the stresses developed because of a gravity load, it was noticed that most of the stresses were generated at distance of 1.5 m from the wall. Along that section, the increase with depth was from 0 kPa to more than 550 kPa. At a distance of7 m the stresses were much less where the maximum stress developed near the base was around 100 kPa. Between the wall's side and a distance of 3. 7 5 m, the static tensile stresses were almost similar, except for the small increase along the 3.75 m section. The same thing happened when subjected to the dynamic load, most of the increase happened at distance of 1.5 m and the lowest values occurred near the right edge of reinforcement layers, 7 m. At 1.5 m section, the stresses varied from 100 kPa near the top to 3 500 kPa near the base of the wall. Compared to the tensile stresses at distance of 7m, ranging from 500 kPa near the top to 100 kPa near the bottom, it was evident the reinforcements element located far away from the wall didn t much resist tensile stresses. Along the sections ofO m and 3.75 m, middle of the reinforcement length, the tensile stresses ranged from 1 00 kPa to 1000 kPa and from 500 kPa to 2000 kPa. In general, both figures showed that most of the stresses occurred at small distance of 1.5 m. The decreased of segments toward the right end, with minimal values around the right edge. At a mid distance from the wall, the stresses noticeably decreased from the maximum section, however, that decrease was not prevailing when subjected to dynamic loading. 348
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7 I I .!.c l: '53 %: 2 1 0 0 \ \ \ X :== 1.5 m x==3.75 m 100 2111 :soo 4GO sao Tensile stNsses_ax (kPJ Figure 9.17 "Summary of tensile stresses of reinforcements due to static loadings" I 7 I E i.4 3 2 I I 1 I 0 0 I I .... . X=Om X=1. Sm "* l( = 3.75 hl 11110 1100 2aDO 2all T.,ll stnsss_ax (kP) Figure 9.18 "Summary of tensile stresses of reinforcements due to dynamic loadings" 349
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9.4.2 Results of Homogeneous Model In this section a simplified model was developed and analyzed. This model contained elements with transversely isotropic homogeneous properties that accounted for soil and reinforcement properties using the regression equations developed in Chapter 5. The homogeneous model had the same geometry and boundary conditions of the discrete model. It was subjected to the same loading curves; load curve number 1 due gravity, and load cure number 2 due earthquake shaking. The reinforced and unreinforced backfill were divided into 3 zones: Bottom, middle, and top. The properties of unreinforced zone stayed the same as for discrete model where cap model was used. On the other hand, the reinforced zone properties were affected by the overburden pressure, and the transversely isotropic properties of the 3 reinforced zones were as shown in Table 9.9. The results due gravity and earthquake loadings were compared to those obtained from the discrete model. Therefore, the study items of homogeneous model were: 1Lateral earth pressure ( crx) behind the hybrid wall, 2Lateral displacement (8x) of the hybrid wall, and 3Bearing pressure ( crz) and vertical settlement (8z) of foundation Each item was investigated individually by observing the static results first and then the dynamic results. 350
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Table 9.9 "Mechanical properties of the reinforced soil composite" Bottom Given J:!rOJ:!erties O'J Es Vs Eg Vg s MPa MPa MPa mm 0.0091062 312.75 0.25 290 0.3 300 Eguivalent J:!rOJ:!erties O'J Eh Vb Ev Vv Gv Gh MPa MPa MPa MPa MPa 0.0091062 343.33 0.23 319.69 0.24 127.00 141.89 Middle Given J:!rOJ:!erties O'J Es Vs Eg Vg s MPa MPa MPa mm 0.0349072 175 0.25 290 0.3 300 Eguivalent J:!rOJ:!erties O'J Eh Vb Ev Vv Gv Gh MPa MPa MPa MPa MPa 0.0349072 190.70 0.23 179.09 0.24 72.62 80.04 TOJ:! Given J:!rOI!erties O'J Es Vs Eg Vg s MPa MPa MPa mm 0.066687 81.5 0.25 290 0.3 300 Eguivalent J:!rOJ:!erties O'J Eh Vb Ev Vv Gv Gh MPa MPa MPa MPa MPa 0.066687 87.10 0.23 83.65 0.24 35.71 38.06 351
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9.4.2.1 Lateral Earth Pressure (crx) behind the Hybrid Wall The lateral earth pressure of the backfill was obtained along the hybrid wall. The results were first gathered due static and then due dynamic. As for the discrete model, the gathered data were smoothly fitted and the negative, tensile, values were omitted due to the lack of soil in resisting tensile pressure. Due static and dynamic loadings, the lateral stresses were compared with those obtained of discrete model as shown in Figures 9.19 and 9.20, respectively. Due to static loading, in Figure 9.19, both models had similar lateral earth pressure that varied from 0 kPa to 20 kPa that increased from top to bottom. When subjected to dynamic loading, the lateral earth pressure trends of both models were similar, increasing with depth. However, in the homogeneous model, the stresses were larger than that of the discrete. In homogeneous model, the stresses increased from 0 kPa near the surface to 200 kPa near the base of the wall. On the other hand, the lateral earth pressure of the discrete model varied from 40 kPa, near the top, to 130 kPa, near the base of the wall. The difference in the lateral pressure of this model was an outcome of the differences in the wall's lateral displacement, as descried in the next section. 352
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7 Dj$crete _. .... H : omogeneou$ !. :1:3 2 1 0 0 I 10 11 20 21 Labtral arth ,,. u,._ax (kPa) Figure 9.19 "Lateral earth pressure (crJ of homogeneous model due to static loading" 1 .!. l!4 'ii :z:::s 2 1 0 D Discrete: ' '', Homogeneous ..... "' 1112 11D ' 2GD Lamr.lrth p,.11urw_m (kPJ 210 Figure 9.20 "earth pressure (crx) of homogeneous model due to dynamic loading" 353
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9.4.2.2 Lateral Wall Displacement (8x) Similar to the discrete model, the lateral displacements of the hybrid retaining wall were obtained at the centerline along the wall's height. The lateral displacements due static and dynamic loading were obtained respectively and compared to those of discrete model as shown in Figures 9.21 and 9.22. Both figures indicated large differences of displacement between the two models. Initially, the discrete model observed very large displacements along the hybrid wall, even when subjected to gravitational load. As mentioned earlier, the wall was not prevented from lateral movement along its height, where the geosynthetic layers were not attached to the wall. The homogeneous model with transversely isotropic properties predicts the smaller displacements of a non linear model such as cap model. The difference in the lateral displacements was not very large when subjected to gravity load, and hence the better representation of lateral earth pressure in the last section. However, when subjected to the earthquake loading, the difference in the lateral displacements of both models was much larger. The homogeneous model, despite its being unattached to the wall, it was much stiffer than the discrete model, and therefore, less lateral displacements were resulted. 354
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I e i 7 1 l i l l: !: '4 X 3 2 1 a D OI$erete Hom ogeneou 6[mm) Figure 9.21 "Lateral wall displacements of homogeneous model due static loading" Discrete .. Hom ogeneou s \ . i a i[mm) D a Figure 9.22 "Lateral wall displacements of homogeneous model due dynamic loading" 355
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9.4.2.3 Bearing Pressure (az) and Settlement (Bz) of Foundation In the section, bearing pressure and settlement of foundation were determined and compared with those resulted from the discrete model. The comparisons were completed considering both loads, static and dynamic. As for the bearing pressure, shown in Figures 9.23 and 9.24, similar behavior was noticed along the foundation when comparing both approaches, except beneath the footing. The reason for so, was due to the excessive lateral displacement of the hybrid wall that occurred in the discrete model causing a sudden increase in the bearing pressure beneath the toe of the foundation, which resulted in large settlement of the foundation. For the homogeneous model, the wall didn't displace as much as in the discrete model, and therefore, there was smaller peak of foundation's bearing pressure that resulted in smaller settlement. An increase of the bearing pressure was noticed when subjected to dynamic load, which again it had maximum values beneath the toe of the footing. The maximum bearing pres sure resulted from both loading of both models were summarized and compared with Terzaghi's' bearing pressure in Table 9.10. In this table it was clear that the maximum bearing pressure increased when subjected to dynamic loading. And since the lateral wall displacement of discrete model was much larger, more bearing pressure was resulted beneath its footing. When comparing the results of both approach with Terzaghi's bearing pressure, homogeneous approach showed a closer value when subjected to static loading. 3.56
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Table 9.10 "Maximum bearing pressure ofsoil beneath footing" O'z (finite element) O'z (Terzaghi) kPa MPa Static Dynamic Static Discrete 275 166 118 Homogeneous 1150 640 Homogeneous !a :::1110 .. ,ao I: "5 ID 5 10 15 20 .., .. ........ J (m) Figure 9.23 "Bearing pressure (crz) of foundation soil in homogeneous model due to static load" 357
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Di5crete Hom ogeoe ous Figure 9.24 "Bearing pressure (crz) of foundation soil in homogeneous model due to dynamic load" Similarly, the wall lateral displacement had significant effect on the foundation settlement So with earthquake the shaking would cause more lateral displacement of the wall and hence more foundation settlement. Due static loading both models resulted in similar trend of settlement with maximum values of 18 mm. The settlement increased significantly when subjected to dynamic loading with maximum values of 30 mm and 25 mm for discrete and homogeneous models, respectively. 358
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a I 40 4 4 'E .. !. .. 1: 10 112 en 14 11 .. Homogeneous X (m) Figure 9.25 "Settlement (Sz) offoundation soil in homogeneous model due to static load" ; Discrete ;.... .. H omogeneous .as ............ .. ....... X (m) Figure 9.26 "Settlement (Bz) offoundation soil in homogeneous model due to dynamic load" 359
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9.5 Summary and Conclusions An MSE hybrid retaining wall consisted of reinforced backfill with equally spaced geogrid layers was analyzed under both static and dynamic loadings. The height of the wall was 7 m and the base of the hybrid wall was 30 % of the wall height. The dynamic load was simulating the ground motion of ElCentro earthquake with a magnification factor of 2 which was applied to the acceleration data that lasted for 30 seconds. A plane strain condition was applied to the model constraining any lateral movement in the Y direction, perpendicular to the wall plane. Furthermore, the model was investigated using two approaches, discrete with full frictional interface and homogeneous with transversely isotropic properties using the regression equations developed in Chapter 5. In both approaches, the models had the same configurations, same boundary conditions, and were subjected to the same loadings. Therefore, the results of both approaches due to static and dynamic loadings were compared. Before running . ', . the simulation, the structure was checked for and internal stabilities under both static and dynamic loadings to check for any failure criterion. The system was externally and internally stabilized with large factor of safeties. First, the discrete finite element model was investigated. In this structure, the soil layers represented by the cap model were allowed to slide relative to the elastic geogrid layers with the aid of the defined frictional interface. The reinforcement in each layer was extended horizontally to 7 m equate the wall's height. These layers were extended to the wall, but not attached, which resulted eventually in separation between the backfill and the wall upon loads application. Initially, due to gravitational load, the wall displaced in the forward direction with a maximiun displacement of 23 mm near the top, and zero at the wall base. Once the earthquake motion was applied, the wall displaced further resulting in a maximum 360
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displacement of250 mm at the wall top and a minimum displacement of 18 mm at the wall base. Furthermore, the lateral earth pressure was calculated along the wall's height. The pressure increased from 0 to 20 kPa when subjected to static loading and from 40 to 130 kPa when subjected to dynamic loading. The stresses here were increasing with depth, where the highest stresses occurred at the base of the backfill, near the footing. Due to the wall's lateral tilting away from the backfill, which was significantly increasing upon the earthquake shaking, larger bearing pressure of the footing occurred; causing larger settlement at the foundation toe. That would not be the case if the inclusion layers were attached to the hybrid wall, where the lateral movement would be minimized due to the tensile effect of reinforcing layers. Therefore, it was noticed that the tensile stresses of reinforcement had opposite trends from the expected ones in real applications, where the maximum stresses are developed near the top and decreased to minimal value at the wall base. In real application, the developed tensile stresses in the reinforcement layers are results of the wall movement and the frictional forces between soil and geosynthetic. However, in this finite element model, the developed tensile stresses in these reinforcement layers were results of the frictional forces between soil and geosynthetic that increased with depth only. The maximum tensile stresses were attained at a distance of 1.5 m from the hybrid wall with magnitudes of 500 kPa and 3500 kPa due to static and dynamic loadings, respectively. The Discrete model gave large displacements that started at the early stages of the load applications. The Homogeneous model with transversely isotropic properties is considered a linear model, and is limited to simulating the elastic performance. But because of the large displacements of discrete model, the homogeneous model was not able to do so, especially when subjected to dynamic loading, and therefore the difference in the lateral displacement profiles was observed. The 361
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differences in the lateral displacement profiles indicated a stiffer model using the homogeneous approach resulted in smaller displacements even when subjected to seismic ground motion. Displacements are constitutively related to the stresses. Therefore, upon earthquake shaking, the lateral earth pressure values of the homogeneous model were quite different from those of discrete ones. However, due to static loading, where displacements were smaller, the lateral earth pressure values of the homogeneous model were almost identical to those of the discrete model. Furthermore, bearing pressure and settlement of foundations of both models were very similar except beneath the toe of the footing under the dynamic loading, which was mainly contributed to the large wall's lateral displacements of the discrete model that caused peak bearing pressure and corresponding settlements. The question was: why do we use a homogeneous model for MSE hybrid retaining wall? The Homogeneous model gave reasonable prediction of stresses and displacements when loads are small and they behave elastically. Furthermore, the shell elements in the homogeneous model are not built, neither the friction interface between soil and reinforcement, but they are accounted for using the equivalent transversely isotropic properties, therefore preprocessing becomes easier and less time consuming. On top of that, CPU time is much shorter and less memory is required. When modeling the discrete finite element model with the aid of an x8612 processor, the analysis lasted around 120 hours to normally terminate the 40 seconds loading curves, more than 10 time of what was needed when processing the homogeneous model. As a result, simulating the MSE 12 Family 6 Modell3 Stepping MHz 362
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hybrid retaining wall using the homogeneous model saves a lot of effort and time and gives reasonable prediction when subjected to small loads. 363
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10. Static and Dynamic Analysis of MSE Bridge Abutments 10.1 Introduction Bridge abutment is a structure that is located at the ends of a bridge. It provides the basic foundation of retaining the earth underneath and adjacent to the approaching roadway. The most common types of abutments are c
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The reinforced soil mass is typically supported by facing panels, rigid or flexible, as shown in Figure 10.2. The rigid facing is a continuous reinforced concrete panel. It offers a certain degree of global bending resistance, which generates a larger restraint to the lateral earth pressure (Kevin z.z. Lee 2004). However, rigid walls are more expensive and need longer construction time compared to the flexible facing. The flexible facing, depending on the design and aesthetic requirement, takes the form of warped geosynthetic sheets or segmental concrete modular blocks. The field tests conducted on flexible segmental block facing bridge abutments have demonstrated excellent performance characteristics and a very high carrying capacity ofloads (Naser AbuHejleh 2001). Therefore, for all the advantages that they offer, flexible facing panels supporting MSE abutments became a alternative despite their narrow design background. In this chapter, the static and dynamic behaviors of a 3dimensional simple span concrete bridge supported by MSE abutments were examined using the finite element method. Two models were constructed and evaluated using the finite element code, LSDYNA: the discrete model with fully interface and the homogeneous model with the transversely isotropic material properties. The results due to static (gravity) and dynamic (Northridge earthquake), were obtained from both models and compared simultaneously. 365
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Figure 10.1 "Side view ofMSE abutments" (Jalinsky 2004) Figure 10.2 "MSE wall facing; a) geosynthetics warp, b) segment concrete block, c) full height panel" (Jalinsky 2004)
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10.2 Design Consideration of Bridge Abutment (FHWA, 2001) The FHW A sponsored research report (200 1) was heavily referenced in this section. Similar to the design of MSE retaining wall, both external and internal stability must be checked when designing an MSE bridge abutment. In bridge abutments with surcharge loads, horizontal stress crh at each depth Z from the top can be represented by ah =K{rxZ+ilaJ+ilah Where: ilcrv = is the increment of vertical stress due to the concentrated vertical surcharge Pv, assuming a 2V:1H pyramidal distribution, ilcrh = is the increment of horizontal stress due to the horizontal load Ph, y x Z = is the vertical stress at layer that has a depth of Z from the top due to overburden pressure. Bridge abutments have been designed by sustaining the bridge beams on the spread footing that is constructed directly on the reinforced soil mass. According to field studies that were completed on the actual structures, the allowable differential settlement between abutments ranges from 120 mm to 150 mm for continuous and simple spans, with a 30m span, respectively (Ellias, 2004). Based on experience, the following details should be considered when designing an MSE bridge abutment: 1A minimum offset of 1 m from the front of the facing to the center line of the bridge bearings, 2A required clearance of 150 mm between the back face of the facing panels and the front edge of the footing, 3A maximum bearing capacity of200 kPa, 367 (1 0.1)
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4A maximum horizontal force must be applied at each reinforcement level when designing the connections to the panel, 5The length of reinforcements of the abutments to wing walls should be extended for a horizontal distance of at least 50 % of the height of the abutment wall, and 6The seismic design forces should include seismic forces transferred from the bridge through bearing supports that do not slide freely. 10.3 Finite Element Analysis on Bridge Span Supported by MSE Abutment A 3Dimensional finite element mesh of a simple span bridge that is supported by MSE abutment was assembled and analyzed using FEMB and LSDYNA codes, respectively. The Abutments and MSE retaining walls were built from elements with full interface characteristics, which was named discrete model, followed by the homogeneous model with equivalent transversely isotropic properties using the linear regression equations that were developed in Chapter 5. The model consisted of four main components: Super structure girders, substructures (abutments), MSE retaining walls, and foundation soil. The super structure girders were precast concrete, with both ends seated on top of the steps of the stemwalls. The stemwall was connected monolitically to footing and wing walls. Behind the stemwall and wingswalls, the backfill of Ottawa sand was reinforced with geogrid represented by shell elements. In order to provide a more stabilized system, the reinforcing elements were attached to all surrounding walls. For each layer of these reinforcing elements, a gap was introduced in order to correctly simulate the frictional interface between the soil and the embedded geosynthetic elements. The gap was equal to the thickness of the geogrid, 2 mm. 368
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Half of that thickness was on top of the reinforcing element and the other half was underneath it. Both abutments, left and right, with their components were located on top of MSE retaining walls. Unlike the bridge abutments, MSE retaining walls consisted of flexible facing. The segmental concrete blocks were stacked on top of each other. The reinforcement length of each layer was 9.5 m which satisfied the minimum required length of0.6 times the height ofthe wall as specified by the National Concrete Masonry Association (NCMA) (1996). Furthermore, the spacing between the reinforcing layers was 400 mm, which was limited to less than twice the facing block width of 300 mm as recommended by the American Association of State Highways and Transportations Officials (AASHTO) of 1996 (Graeme D. Skinner 2005). Not only was the geogrid embedded between soil layers, but it was also embedded between the segmental blocks to provide a better connection with the backfill. Again, a half of the shell thickness was considered on the top and the bottom of each reinforcing layer. Beneath the MSE retaining wall was the foundation soil. The foundation soil consisted of Ottawa sand as well, and supported with proper boundary conditions. The isometric .view of this model was as shown in Figure 10.3, and the dimensions ofthis model were given in Table 10.1. All the components of the MSE bridge abutment, including the superstructure, were modeled using an 8noded brick element, except for the reinforcing layers, which were assembled using 2dimensional shell element. For this MSE abutment model, the vertical direction was assumed in Z, while the transverse and longitudinal directions were assumed in Y and X, respectively, following the Cartesian coordinates systems. 369
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Component Girder Foundation Soil Footing Stem Wall Wing Walls Abutment backfill MSE Wall Table 10.1 "Dimensions ofMSE abutment" (Jalinsky 2004) Length Width Depth (m) (m) (m) Lgx = 50.0 Wgy = 12.0 Dgz = 2.0 Lroundationx = 12 Wroundationy = 16.0 Droundationz = 7. 0 Lrootingx = 2.7 Wrootingy = 12.0 Drooting= 0.6 Lbasex = 0. 76 Dstemz = 3. 0 Lstemx = 0.3 Wstemy = 12.0 Hstep = 0.9 Lwingx = 7_0 W wingy = 0.3 Dwingz = 3.6 Lbackfillx = 7. 0 Wbackfilly = 11.2 Dbackfillz = 3.6 LivtsEx = 9.5 WMSEy = 13.7 DMsEz = 4.7 wblockelementy = 0.3 Dblockelementz = 0.2 Spacing (m) Sbackfill = OA SMsE = 0.4
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Stemwall Strip footinl! MSEwall Fotmdation soil y (Transverse) z Reinforced backfill X (Longitudinal) Figure 10.3 "Isometric view ofMSE bridge abutment" Wing
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10.3.1 Loading Both models, discrete and homogeneous, were subjected to static and dynamic loads. The static load was due to the bridge components' weight, and the dynamic was due to the earthquake ground motion. Furthermore, the static was applied through a constant gravitational acceleration of9.81 m/sec2 15 seconds into the analysis, the horizontal components of the ground motion acceleration was applied to the analyzed model. The reason for applying the dynamic load after 15 second was to exclude any oscillation that resulted from the application of instant gravitational acceleration. The horizontal components, transverse (Y) and longitudinal (X), ofNorthridge earthquake were selected and applied to this model, with a peak ground acceleration (PGA)13 of0.62g and 0.42g as shown in Figures 10.4 and 10.5, respectively, and are summarized in Table 10.2. These accelerograms contain all the acceleration from the time the earthquake begins until the time the motion has returned to the level of background noise. For engineering purposes, only the strong motion portion of the accelerogram is of interest, and is defined as the time between the first and last exceedances of 0.05g (Kramer 1996). For that reason, only the first 15 seconds of the earth quake acceleration components were adopted in the analysis. When applying the static load, the Body Loadcommand approach was selected. The body load option defines the force loads due to a prescribed base acceleration using global axes directions (LSDYNA 2003). This force is applied to all nodes in the model until termination. Similarly, the body load option was also used to define the dynamic load acceleration in both global directions, X andY, starting from 15 seconds until30 seconds. The analysis was terminated at 35 seconds, 5 seconds after the end of the dynamic siml!lation. 13 PGA is the largest absolute value of acceleration from a given time history. 372
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OA 0.2 F 0 E .0.2 .OA .o tim (sc) Figure 10.4 "Northridge transverse (Y) horizontal acceleration time history" (Berkeley) 0.1 0.4 0.3 0.2 0.1 r 0 E .0.1 2.5 .0.2 .0.3 .OA PGA=DAlg .0.1 Figure 10.5 "Northridge longitudinal (X) horizontal acceleration time history" (Berkeley) 373
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Table 10.2 "Northridge Earthquake information" (Berkeley) Category Magnitude7 Date January 17, 1994 Station 90014 Beverly_ Hills12520 Mulhol Magnitude 6.7 (ML) Intensity 9(MM) Depth 17.5 (km) Epicentral Distance 19 (km) PGA 0.62 (MM) = Modified Mercalli 10.3.2 Material Properties The three main materials were Ottawa sand foundation, geogrid reinforcement, and concrete section. In this study, the constitutive models representing these materials were the cap model for Ottawa sand, and isotropic linear elastic model for inclusion and the concrete. 10.3.2.1 Backfill and Reinforced Soil Ottawa sandy soil with density of 1719 kg/m3 corresponding to the relative density, Dr, of70%, was used to represent the reinforced soil in the MSE wall, the backfill material behind the stem wall, and the foundation soil. A geologic cap model was used to represent these parts. The 10 constants as determined from previous calibration, Chapter 6, for Ottawa sand were as shown in Table 10.3. Among these constants, the shear G, would vary with depth, depending on the mean stress, cro, as shown in Equations 10.2 and 10.3 for'round and angular grains, respectively. In the MSE retaining wall, the reinforced soil was divided into three segments (Top, middle, and bottom). The bridge abutment soil behind 374
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the stem wall was also divided into two segments (Top and bottom). The foundation soil was represented by one segment. Each of these five segments had a different G value. G = 6908(2.17 e)2 max l+e 0 G = 3230(2.97 e)2 crJi max l+e 0 Where: Gmax =maximum shear modulus in kN/m2 e = 0.58 for dense sand, 1 cr 0 = ( cr v + 2cr h) cr v = r X z cr h = K 0 X cr v K 0 = 1 sm and 3 Z = depth from top of the wall Table 10.3 "Properties of Ottawa sand along different elevations" K G a
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10.3.2.2 Inclusion A high density Polyethylene Geogrid was used as the inclusion for the analysis. A commercially available geogrid, named Tensar UX11 00 HS, with ultimate strength of 56 kN/m was selected (GFR 2004). An elastic model was assumed to represent this material, where the Young's modulus was taken to be the secant modulus at 5 % strain divided by the thickness, t, which was 2 mm. The elastic properties ofthe selected geogrid were as shown in Table 10.4 Table 10.4 "Reinforcement properties in MSE abutment and backfill soil" (Lee 2000) p E v Kglm3 MPa (assumed) 1030 290 0.3 10.3.2.3 Concrete Footing, wingwalls, stemwall, segmental block facing, precast concrete girder were all modeled as continuous concrete. The concrete section was assumed to have elastic properties and was as shown in Table 10.5. 2320 Table 10.5 "Concrete properties" (Boresi 2003) E MPa 25000 376 v 0.15
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10.3.3 Sliding Interface In this model, frictional interface was allowed between adjacent parts. The sliding interface was not only defmed between soil and inclusions, but also between soil and concrete walls. The static friction coefficient, Jl, was determined based on the internal friction angle of soil, which was obtained in chapter 6 from drained triaxial tests and was equal to 36. The interface friction angle,(), is defined as (Lee 2000), and Jl could then be determined using Equation 10. 4. Therefore Jl was determined and chosen to be 0.5 between all adjacent parts. .u = tano For a discrete model, a gap representing the thickness of the shell element was considered. Half of the thickness was on top of the reinforcing layer and the other half was under the reinforcing layer. This gap was considered to correctly model the frictional interface between the soil layers and the reinforcing layers on one hand, and the block elements and the embedded reinforcing layers on the other hand. 1 0.3.4 Boundary Conditions For this model, boundary conditions in the X, Y, and Z directions need to be defined to correctly simulate the structure. A fixed condition with displacement constraints in the X, Y, and Z directions was applied at the bottom nodes of the foundation soil. On the other hand a roller condition was applied along: the sides of the foundation, the back face of the MSE soil backfill, and the abutment backfill. The roller condition allowed displacement in the z direction only. 377 (10. 4)
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10.4 Results and Discussion The finite element simulation results were divided into two categories, static due to gravity load and dynamic due to earthquake. The dynamic load was applied after 15 seconds from the initiation of this simulation. Therefore, after 15 seconds from start, the static results were obtained. On the other hand, the dynamic results were obtained when the analyses were terminated. Maximum absolute values of the dynamic response during earthquake loading were obtained and taken into consideration, such as the maximum wall displacement or the maximum lateral earth pressure along the MSE wall, and so on. The results obtained from the simulation of the bridge abutment when using the discrete model were eventually compared to those obtained when using the homogeneous model. The study items, due to static and dynamic loadings, were 1Longitudinal and transverse MSE wall displacements, 2Longitudinal and transverse earth pressure distribution, 3Bearing pressure distribution on MSE backfill beneath the footing, 4Settlement of MSE backfill beneath the footing, 5Inclusion tensile stresses distribution (for the case of discrete model). The transverse results. referred to theY direction, perpendicular to the bride span, and the longitudinal results referred to the X direction, parallel to the bridge span. 10.4.1 Results of Discrete Model In this section, the results obtained from the discrete model with full frictional interface due to static and dynamic loadings were calculated and evaluated. Due to symmetry, only the results of the left abutment were obtained and summarized. 378
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10.4.1.1 Lateral Pressure of Bridge Abutment In this section, lateral earth pressure of the bridge abutment due to static and dynamic loading was calculated. The abutment was supported by concrete walls, stem wall, footing, and wing walls. The long side of the stem wall, as shown in Figure 10.3, was vertical to the bridge span, while the wing walls were parallel to the bridge span. Therefore, the pressure against the stem wall was in the global X (longitudinal direction), and the pressure against the wing walls was in the global Y (transverse direction). The lateral earth pressure was first calculated along the centerline behind the stem wall, as shown in Figure 1 0.6. In this figure, the lateral earth pressure due to both static and dynamic loads increased with depth until reaching maximum values and then decreased near the base of the wall. Due to the static load, the lateral earth pressure increased from 20 kPa at height of 3. 7 m to 25 0 kPa at height of 1. 7 m, and decreased back to 80 kPa near the base of the wall. Due to the dynamic load, the lateral earth pressure increased from 42 kPa at height of3.7 m to 290 at height of 1. 7 m and decreased to 120 kPa near the base of the wall. Furthermore, contour and 3Dimesional surface plots of static and dynamic lateral earth pressure against the stem wall were provided as shown in Figures 10.7 and 10.8, respectively. When observing the lateral pressure against the stem wall due to gravity load only, it was noticed that stresses increased from 0 kPa near the top to 470 kPa near the midheight, and then reduced to 100 kPa near the base of the abutment as shown in Figure 10.7. Following the same trend, due to the dynamic load, the lateral earth pressure against the stem wall increased with depth reaching a maximum value of 590 kPa near the mid height of the backfill as shown in Figure 10.8. In Figure 10.8, the maximum stress was moved towards the left side of the wall near the edge of the strip footing, where the highest stress 379
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concentration was observed. Evidently, the longitudinal earth pressure in the bridge abutment was significantly affected by the earthquake ground motion, where the lateral stresses had larger values behind the entire wall section and especially near the footing. 4aJD 31110 \ 3aDO ,\ '\ 'E211DD \ i \ !. \ l:2CIJO \ .. \ ,.., \ \ 11111 \ l \ \ \ D D .St>ltic Dynamic Conventionalstatic Conventionaldyn ::tm ic w. .. A.' 1DD 110 2m 210 (kPa) ... t 3111 Figure 10.6 "Longitudinal earth pressure of abutment along the centerline" 380
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Figure 10.7 "Contour and 3Dimenensional surface plots of longitudinal earth pressure of the bridge abutment on the Stemwall due static loading" 381
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F,'igure 10.8 "Contour and 3Dimensional surface plots oflongitudinal earth pressure of the bridge abutment on the Stemwall due dynamic loading" 382
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The lateral earth pressure due to static and dynamic loads was plotted against the wing wall along the edge of the wall, as shown in Figure 10.9. Due to the static load, the lateral earth pressure against the edge of the wing wall increased from 20 kPa at height of3.5 m to 160 kPa at height of2.0 m, and decreased to 55 kPa near the base. On the other hand, due to the dynamic load the lateral earth pressure increased from 20 kPa near the top to 350 kPa at height of 1 0 m, and then decreased slightly to 220 kPa near the base of the wall. Furthermore from the contour plots, shown in Figures 10.10 and 10.11, it was obvious that stresses increased with depth, but most of the increase was noticed to the right side of the abutment; i.e. near the footing. The same phenomenon occurred when dynamic loading was applied, except for the increase in the stresses due to shaking effect. The maximum lateral earth pressures due to gravity and dynamic loading were 365 and 385 kPa, respectively. 4000 .., 3DOD e2100 !. J:ZIDD .. 1000 100 a a 1aD Dy namic ............. >;. .... .... ,.. ... '"' ,. 180 . ,.. .., .. ,. ... .,. . .... 210 aaa. : j I I I l 400 Figure 10.9 "Lateral earth pressure of abutment against the wing wall along the edge" 383
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Figure 10.10 "Contour plot of transverse earth pressure of the bridge abutment on the wing wall due static loading" 384
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23))' .. :, ... r ':., . ,_ J.COO ,.,, l("m) ,..._;' ,.;. :'i>; _,_ ... '>. ; 4)00 <"JGV . ...... ":,' ., Figure 10.11 "Contour plot of transverse earth pressure of the bridge abutment on the wing wall due dynamic loading" 385
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10.4.1.2 Bearing Pressure and Settlement Distribution of MSE Backfill beneath the Spread Footing Highway bridge structures are usually supported by either shallow spread footings on bed rock or pile foundations on very stiff soil or socketed on rock to assure longterm serviceability. In recent years, spread footings on soil became an attractive substitute (Shad M. Sargand 1999). It is cheaper than pile foundations and requires less time. Based on Sargand and his colleagues, spread footing could be used successfully to support highway bridge structures as long the foundation is sufficiently strong. Therefore, the analyses in this section were completed to report the weight effect of super and sub structure components (abutment reinforced backfill, wingwalls, stemwall, and spread footing) on the MSE wall. So how much bearing pressure and settlement was imposed to the MSE backfill? And was the effect of the earthquake on these vertical components significant? In order to do so,.contour and 3Dimensional surface plots of vertical pressure (crz) and vertical settlements (8z) were obtained at the top soil layer of the MSEWall. These plots were first developed due to gravity load and then due to dynamic load. Figures 10.12 and 10.13 indicated that the resulted vertical stresses from static and dynamic loads ranged from 0 to 900 kPa and from 100 to 11 00 kPa, respectively. In fact, there was a noticeable effect on bearing pressure due to the earthquake, where the vertical stress increased on most of the segments of the MSE backfill. Due to both loadings, the largest bearing pressure values were under the spread footing, which decreased with distance away from the footing. Thus, the vertical stress was highly concentrated beneath the footing exceeding the recommended maximum bearing pressure of200 kPa 0/ector Elias 2001) by a factor of 4.5 and 5.5 due to static and dynamic loading, respectively Where, the 386
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average bearing pressure beneath the footing was 349.0 kPa due to static load, and 613 kPa due to dynamic load. This indicated that enhancement to this structure is required This can be completed by increasing the width of the strip footing, and locating the footing at further distance from the edge of the MSE wall. Where in this model the width of the footing in transverse direction was 2. 7 m, and the clearance was 0.3 m only. Similar to bearing pressure, the maximum settlement of the MSE backfill due to static and dynamic loadings occurred at the nodes that are beneath the footing, as shown in Figures 10.14 and 10. 15, respectively Indeed, the effect ofthe earthquake was remarkable on the backfill settlement, especially beneath the footing. As a result of gravitational load, the maximum and the average settlement under the footing were 170 mm and 117 mm, respectively. Due to dynamic shaking, the maximum the average settlements were 250 mm and 166 mm, respectively. The negative values in Figures and 10.15 referred to the settlement. Some positive values near the edges and at the end of the backfill were noticed due to the dynamic shaking These positive values referred to the heaving at points located away from the footing which resulted from the large settlement beneath the footing 387
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, .... ..... ,;, .r< crintcJ. plot.of on MSEabutment Q.::drection) : dUe tn statiC .(9ravity) loadng' Q. ; ... :. '.' ,,. _, .: }' .. .. ,tf' \''<. :''f 10000 8000 it:;i Figure 10.12 "Contour and 3Demensional plots ofMSE backfill bearing pressure due static loading" 388
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'h.v .. eooo::: Figure 10.13" Contour and 3Demensional plots of MSE backfill bearing pressure due static loading due dynamic loading" 389
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' '. : ,; : : ,:. . .. .=.: :: plotof . ..:. 1Qadl'9'(mri1) ;>'.:. ' . ' __ ... . ' . s. ' ..:: ,.t; eooo :. ::.. . 'if ..... ::;_; v :. .. .. .. static (g"
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Figure 10.15 "Contour and 3Demensional plots ofMSE backfill settlement due dynamic loading" 391
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10.4.1.3 Longitudinal and Transverse Earth Pressure behind the MSEWall Due to the application of both static and dynamic loads, the longitudinal and transverse lateral pressures on the longitudinal and transverse sides were obtained. The results were first calculated by plotting the distribution along the centerline and the edge of the MSE backfill. Then, the results were further explained with the aid of contour and 3Dimensional surface plots. All data points were presented and the best fit regression equations were provided, as shown in Figures 10.16 and 10.17. In these figures, it was obvious that the values of lateral pressures were increasing in both directions: horizontally, towards the edges, and vertically, towards the base of the wall. In addition to that, a significant increase in the lateral earth pressure was observed on both sides as a result of the earthquake ground motion. When considering the longitudinal side, it was noticed that the stresses due to the gravity load increased from top to bottom with values of 0 kPa to 90 kPa along the centerline. The trend was slightly different along the edges, where the stresses' increase happened with depth, from 40 kPa near the top to 130 kPa at an elevation of 1.5 m from the I wall's base, and then reduced back to 110 kPa at the wall's base. Along that side, the same trends with larger lateral earth pressures were observed when subjected to the dynamic load. The dynamic stresses along the centerline varied from 20 kPa near the top to 11 0 kPa near the base, where as the dynamic stresses along the edges increased with depth from 0 to 210, and reduced back to 150 kPa. The lateral earth pressure on the transverse side of the MSE wall had consistent trends. Due to both loads, the stresses along the centerline and the edge increased from minimal values near the top to maximum values near the base. Indeed, the stresses along the edges were larger than those along the centerline, and the 392
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stresses due to the earthquake were lager than those due to gravity. Along the centerline, the stresses increased from 0 kPa to 60 kPa and from 40 kPa to 1 00 kPa, due to gravity and earthquake, respectively. Along the edge the stresses increased from 10 kPa to 310 kPa and from 90 kPa to 360 kPa, due to gravity and dynamic and earthquake, respectively. It was clear that when the wall was subjected to earthquake loading, the lateral pressure behind the edges and around the top layers was no longer negligible. Figures 10.16 and 10.17 could be further explained with the aid of contour and 3Dimensional surface plots as shown in Figures 10.18 through 10.21: In these plots, the longitudinal and transverse earth pressure due to either static or dynamic loading increased with depth and away from the centerline, toward the edges. This increase became more noticeable when the dynamic load was applied. When observing the longitudinal pressure under static loading, the stresses ranged from 5 kPa near the top and increased with depth to up to 80 kPa. By traveling horizontally from the centerline along the global Y axis, the increase of stress became more dominant, reaching more than 150 kPa. Similar to the trend of static longitudinal earth pressure, the pressure, due to dynamic loading, increased from 10 kPa near the top to 100 kPa at the base of the wall, and up to 220 kPa near the edges. When observing the stresses on the transverse side, it was also noticed that the stresses increased with depth and away from the centerline. That trend was obvious when the model was subjected to gravity and dynamic loading. When subjected to the dynamic loading, the shaking effect was more obvious on the transverse side than on the longitudinal side, where the stresses by the edges reached more than 300 kPa. The plots of the backfill pressure distribution indicated that the stresses are more likely to have peak values around the edges of the MSEWall. Thus the effect of gravity or seismic loading generates stresses along the edges more than the centerline. 393
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50.00 0.00 50,00 100;00 150 : 00 200.00 250.00 Static_ Centerline Dynamac_Ed1t? lateral ,earth ptelc.re_ ox (kP_, Dyn.amic_;Cente:rllne ..., Stath;:_Cenlerllnedata + Static Edge ttFigure 10.16 "Longitudinal horizontal stress (ax) of soil behind the MSE wall" 301100
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4500 4000 3500 'E3000 .. 2! 2500 a 2000 1500 1000 500 ... ... 50.00 Static_ Centertine D;narnic_Edge Static_Edgedata 100 00 150.00 200.00 250.00 300.00 350.00 400 00 L.teral earth ...._ Static_Centerlinedat. a Dynamic_Edgedata Static_E<).;}e Figure 10.17 "Transverse horizontal stress (cry) of soil behind the MSE wall"
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Figure 10.18" Contour and 3Dimensional plots of longitudinal earth pressure on the longitudinal side of the MSEWall due static loading" 396
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Figure 10.19 "Contour and 3Dimensional plots of longitudinal earth pressure on the longitudinal side of the MSEWall due dynamic loading" 397
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. Yr;12.: :. ::>r: ; ; ;'\O n M.SE'JYall . . . 'due il) adfi:Ui(kPa) .. : .. ;r .. .. 150o' 1000 .: Figure 10.20" Contour and 3Dimensional plots of transverse earth pressure on the transverse side of the MSEWall due static loading" 398
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Figure 10.21 "Contour and 3Dimensional plots of transverse earth pressure on the transverse side of the MSEWall due dynamic loading" 399
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10.4.1.4 Longitudinal and Transverse Displacements of MSEWall The longitudinal and transverse displacements of the segmental blocks along the MSE wall were obtained at the centerline of the X andY sides, and were as shown in Figures 10.22 and 10.23. As it is shown in Figure 10. 22, a maximum displacement of more than 50 mm away from the backfill occurred near the mid height of the wall due to static load. It decreased significantly near the bottom of the wall and decreased slightly to 40 mm near the top. A similar trend was noticed when the dynamic load was applied. However, because of the shaking effect, a peak value of 1 00 mm away from the backfill took place at the very top of the wall. In general, the displacements due to dynamic loading were twice that of static loading. Also due to static and dynamic loading, the maximum transverse displacement in Figure 10.23 took place near the top with magnitudes of at least twice of those resulted at the longitudinal face of the wall. The larger Y displacements were attributed to the configuration of the model and the larger PGA value. 400
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4100 4DDO 31100 E 3DDO E !'21100 11100 1DOD 100 "' ., ..... . .1' .. I' ... I I Statics . o y .. n>im ic o.a 40.0 10.0 5x (mm) m.a 100.0 1211.0 Figure 10.22 "Longitudinal displacements of MSE wall due to static and dynamic loadings" E 3DDO .!.. .. 21100 .1: 'i moo l: 1100 J 1DOD liDO . ' . . .. . """ ,.. J .,.. Statics Oyanamic .j I l I 0.0 10.0 10D.O 110.0 211J.O 210.0 300.0 3&0.0 tit' (mm) Figure 10.23 "Transverse displacements ofMSE wall due to static and dynamic loadings" 401
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Furthermore, contour and 3Dimensional surface plots oflongitudinal and transverse displacements were obtained due to static and dynamic loadings as shown in Figures 10.24 through 10.27. The contour and 3Dimensional surface plot in Figure 10.24 indicated that most of the displacements occurred at the mid height due to static load. They decreased to 0 near the bottom and the sides of the wall. In the same figure, the 3Dimesnional surface plot indicated a significant tilt of the upper concrete block. A similar phenomenon occurred when dynamic load was applied, as shown in Figure 10.25. The bending occurred near the mid height and vanished near the bottom and the sides; however, the highest displacement happened at the top block, with a displacement of more than 1 00 mm away from the backfill. When comparing Figure 10.24 to Figure 10.25, it was evident that the outcome of the earthquake contributed to a much larger displacement than that resulted from gravity, more than twice. The Contour and 3Dimensional surface plots of the transverse displacement shown in Figure 10.26 . and 10.27, showed a slight difference in deformation shapes. From both figures, it was noticed that the highest displacement occurred near the top of the wall and reduced gradually to zero near the base and the sides. Due to the static loading, the maximum transverse displacement was more than 150 mm; it increased to more than 300 mm when subjected to the Northridge earthquake ground motion. These magnitudes were more than twice the longitudinal displacements; where the static and the dynamic longitudinal displacements were around 50 mm and 110 mm, respectively. The PGA in the longitudinal direction was 0.42, and the transverse PGA was 0.62. Beside the restraint from the bridge girders, the larger PGA in the transverse direction was the major reason for obtaining larger deformations in that direction. 402
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Figure 10.24 "Contour and 3Dimensional surface plots oflongitudinal wall displacement due static loading" 403
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Figure 10.25 "Contour and 3Dimensional surface plots of longitudinal wall displacement due dynamic loading" 404
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Figure 10.26 "Contour and 3Dimensional surface plots of transverse wall displacement due static loading" 405
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Figure 10.27 "Contour and 3Dimensional surface plots of transverse wall displacement due dynamic loading" 406
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10.4.1.5 Inclusion Tensile Stresses iii the MSEWall The MSE wall had a total of 11 equally spaced geogrid layers of 400 mm spacing. These layers were extended between the soil and the facing block elements. It was necessary to determine the effect of reinforcing layers on the behavior of the bridge abutment. In other words, it was important to determine how much tensile stresses were carried by these inclusions. To do so, the contour plots of horizontal stresses, crx and cry, were calculated at three different elevations due to both static and dynamic loads. The three selected elevations were near the base of the wall (Height= 0.4 m), at the mid height (Height= 2.4 m), and at the top reinforcement layer (Height= 4.2 m). At these elevations, the stresses were calculated in the longitudinal direction first, and then in the transverse direction. From Figure 1 0.28, it was clear that the longitudinal tensile stresses, crx, were relatively small at most ofthe bottom inclusion layer's segments when subjected to gravity load. In fact, most of the inclusion surface except near the edges had negative stress values, indicating a compressive stresses. A trivial increase in the longitudinal stresses ofthe bottom layer was observed on most of the segments when subjected to dynamic loading, as shown in Figure 10.29. Initially, the maximum longitudinal stress of2,000 kPa occurred near the edge when the model was subjected to the gravity load. On the same slice, a value of 4,000 kPa was observed due to seismic shaking. Additionally, the transverse stresses, cry were slightly larger than the longitudinal stresses when the model was subjected to gravity loading. Most of the tensile stresses were concentrated near the edges as shown in Figure 1 0.30. Similar behavior was also noticed when the dynamic. loading was applied with trivial increase of the transverse stresses, as shown in Figure 1 0.31. In the transverse direction, the maximum stresses occurred also near the comers with magnitudes of2,500 kPa and 2,700 kPa due to static and 407
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dynamic loading. A significant increase of tensile stresses was noticed at a reinforcing layer located at Height of2.4 from the base in the MSEWall, as shown in Figures 10.32 though 10.35. Figures 10.32 and 10.33 showed that the stresses in the layer located at height of 2.4 m were at a maximum underneath the strip footing. The longitudinal stresses then reduced gradually away from the footing region. The maximum observed values of longitudinal stresses due to static and dynamic loading in that region were around 7,000 kPa and 10,000 kPa. Despite having the same trend, the longitudinal stresses of both loadings at this elevation were considerably larger than the transverse stresses, as shown in Figures 10.34 and 10.35. The maximum static and dynamic transverse tensile stresses at the layer located at height of2.4 m were around 5,500 kPa and 7,300 kPa. A further increase of stresses occurred on the top reinforcing layer, as shown in Figures 10.36 through 10.39. Compared to other layers, the top layer had the greatest values of tensile stresses in the longitudinal and transverse directions. Similar to the layer at height of 2.4 m, the maximum stresses occurred in the longitudinal directions beneath the footing. The maximum longitudinal and transverse stresses due to the static load were similar to each other, and were equal to 7,000 kPa. The stresses increased drastically to 14,000 kPa in the longitudinal direction and 10,000 kPa in the transverse direction when dynamic load was applied. Since the movement of the bridge abutment was constrai1,1ed in the longitudinal direction, more tensile stresses were generated in that compared to the freemoving bridge abutment in the transverse direction. These stresses were larger than those in the transverse direction, where the maximum PGA occurred, resulting in a larger displacement. 408
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Figure 10.28 "Contour plot oflongitudinal stresses of bottom reinforcing layer due to static loading" ( . Figure 10.29 "Contour plot of longitudinal stresses of bottom reinforcing layer due to dynamic loading" 409
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Figure 10.30 "Contour plot of transverse stresses of bottom reinforcing layer due to static loading" Figure 10.31 "Contour plot of transverse stresses of bottom reinforcing layer due to dynamic loading" 410
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Figure 10.32 "Contour plot of longitudinal stresses of middle reinforcing layer, H=2.4m, due to static loading" Figure 10.33 "Contour plot of longitudinal stresses of middle reinforcing layer, H=2.4m, due to dynamic loading" 411
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Figure 10.34 "Contour plot of transverse stresses of middle reinforcing layer, H=2.4m, due to static loading" Figure 10.35 "Contour plot of transverse stresses of middle reinforcing layer, H=2.4m, due to dynamic loading" 412
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Figure 10.36 "Contour plot of longitudinal stresses of top reinforcing layer, H=4.3m, due to static loading" Figure 10.37 "Contour plot of longitudinal stresses of top reinforcing layer, H=4.3m, due to dynamic loading" 413
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Figure 10.38 "Contour plot of transverse stresses of top reinforcing layer, H=4.3m, due to static loading" Figure 10.39 "Contour plot of transverse stresses of top reinforcing layer, H=4.3m, due to dynamic loading" 414
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10.4.2 Results of Homogeneous Model Detailed finite element model of MSE bridge abutment suffers from complexity and requires lengthy time while pre processing, CPU processing, and post processing. For that reason, a more simplified model using homogeneous properties of the reinforced soil is desirable. When the homogeneous model was used, the transversely isotropic equivalent properties of reinforced soil were computed using the regression equations developed in Chapter 5, as shown in Tables 10.6 through 10.10. The properties of this geocomposite were functions of spacing between the inclusion layers and the properties of the constituents, the soil, and the geosynthetic. As was mentioned earlier, the MSE wall was divided into three zones: bottom, middle, and top, and the reinforced backfill in the bridge abutment was divided into two zones: bottom . and top. The variation of the equivalent properties in these zones was a function of the overburden pressure. Once the transversely isotropic properties were calculated, the mesh was generated using the same boundary conditions and loading curves of the discrete model. The reinforcements between block facing elements remained unchanged with full frictional interface because they were originally embedded between the concrete blocks in the discrete model. Also, the equivalent properties of the blocks and the reinforcement layers were not taken into consideration in this research. Due to static and dynamic loadings, the results of the transversely isotropic homogeneous model were calculated and compared with those of the discrete model. The study items of the homogeneous model were the lateral earth pressure of bridge abutment, the bearing pressure and the settlement of abutment, the . 415
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MSEbackfill earth pressure, and the MSEWall displacement. Due to symmetry, only the results of left abutment were obtained and evaluated. Table 10.6 "Equivalent properties of lower zone of the MSEWall" MSEBottom Given properties 0"3 Es Eg s Vs Vg (MPa) (MPa) (MPa) (mm) 0.076 258.00 0 25 290.00 0.30 400.00 Equivalent properties O"J Eh Ev Gv Gh Vh Vv (MPa) (MPa) (MPa) (MPa) (MPa) 0.076 284.79 0.23 264.66 0.24 105.14 118.11 Table 10.7 "Equivalent properties of middle zone of the MSEWall" MSEMiddle Given properties O"J Es Eg s (MPa) (MPa) Vs (MPa) Vg (mm) 0.06 229.00 0 25 290.00 0.30 400.00 Equivalent properties O"J Eh Ev Gv Gh Vh Vv (MPa) (MPa) (MPa) (MPa) (MPa) 0.0 6 252.65 0.23 234.06 0.24 93. 69 105 08 416
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Table 10.8 "Equivalent properties of top zone of the MSEWall" MSETop Given properties O'J Es Eg s (MPa) (MPa) Vs (MPa) Vg (mm) 0,045 197.00 0.25 290.00 0.30 400.00 Equivalent properties O'J Eh Ev Gv Gh Vb (MPa) (MPa) (MPa) Vv (MPa) (MPa) 0.045 217.20 0.23 201.39 0.24 81.06 90.72 Table 10.9 "Equivalent properties of lower zone of the abutment" AbutmentBottom Given properties O'J Es Eg s (MPa) (MPa) Vs (MPa) Vg (min) 0.028 153.00 0.25 290.00 0.30 400.00 Equivalent properties O'J Eh Ev Gv Gh Vb Vv (MPa) (MPa) (MPa) (MPa) (MPa) 0.028 168.45 0.23 165.48 0.24 63.68 70.96 417
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Table 10.10 "Equivalent properties of top zone of the abutment" AbutmentTop Given properties O"J Es Eg s (MPa) (MPa) Vs (MPa) Vg (mm) 0.009 63. 50 0.25 290 00 0.30 400.00 Equivalent properties O"J Eh Ev Gv Gh (MPa) (MPa) Vb (MPa) Vv (MPa) (MPa) 0 009 69.28 0.23 65.13 0.24 28.35 30.77 10.4.2.1 Lateral Pressure of Bridge Abutment Similar to the discrete model the bridge abutment was supported longitudinally by a stem wall, and transversely by two wing walls. In this section, the lateral earth pressures of the bridge abutment along these walls were calculated first due to static loading, then due to dynamic loading. Results obtained from this model were compared with those obtained from the discrete model. The lateral earth pressure was first calculated along the centerline behind the stem wall, as shown in Figure 1 0.40. In this figure, the lateral earth pressure due to both static and dynamic loads increased with depth until reaching maximum values and then decreased near the base of the wall. Due to both static and dynamic load the lateral earth pressures of homogeneous model were much larger than those obtained from discrete model. Furthermore, The Contour and the 3Dimensional surface plots in Figures 10.41 and 10.42 indicated that the longitudinal pressure in the bridge abutment of the homogeneous model acted in a 418
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similar manner as in the discrete model. Earth pressure increased with depth until it reached the maximum stresses near the center, and decreased to minimal values near the base of the wall. Under the dynamic load, the maximum stress shifted more toward the left side of the stem wall, near the edge of the strip footing, where the highest concentration of the stress was observed. However, the increase that occurred in the linear homogeneous model further exceeded that in the discrete model. For instance, the maximum lateral pressure in case of the discrete model, under both loadings, were 470 kPa and 590 kPa, while the homogeneous model's maximum pressures were 1,700 kPa and 2,700 kPa for static and dynamic loading, respectively. However, the sides and the edges of both abutments were carrying similar stresses ranging from 0 to 200 kPa when subjected to static loading, and from 0 to 300 kPa when subjected to dynamic loading. The lateral earth pressures of the bridge abutment against the wing walls were calculated under both static and dynamic loadings as shown Figures 10.43 and 10.44, respectively. Most of the compressive stresses were observed near the footing with maximum values of 150 kPa due to the static load, and 300 kPa due to the dynamic load. These values were slightly smaller than those obtained from the discrete model where the stresses were at 365 kPa and 385 kPa, respectively. 419
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4aaJ 3IGO 3CXD 'E2HD E l:2GGD 11CD 1DGD IRD a a 200 400 +DiscreteStatic .... Hom ..... DiscreteDy nam ie ...,.Hom eCo nventional .. stati c ..... 100 1GDO 1200 1400 1100 Figure 10.40 "Longitudinal earth pressure of abutment along the centerline of homogeneous model vs. discrete model" 420
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Figure 10.41 "Contour plot of longitudinal earth pressure of the bridge abutment on the Stemwall of homogeneous model due static loading" 421
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Figure 10 .42 "Contour plot of longitudinal earth pressure of the bridge abutment on the Stemwall of homogeneous model due dynamic loading" 422
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Figure 10.43 "Contour plot of transverse earth pressure of the bridge abutment on the wing wall of homogeneous mode due static loading" 423
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Figure 10.44 "Contour plot of transverse earth pressure of the bridge abutment on the wing wall of homogeneous mode due dynamic loading" 424
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10.4.2.2 Bearing Pressure and Settlement Distribution of MSE Backfill beneath the Spread footing In this section, the bearing pressure and the settlement of the MSE backfill of the homogeneous model are being compared to those obtained from the discrete model. Additionally, the comparison took into consideration the effect of static and dynamic loadings. In the homogeneous model, the bearing pressure in the contour and the 3Dimensional surface plots ranged from 0 to 1,300 kPa due to static loading, and from 100 to 2,500 kPa due to dynamic loading, as shown in Figures 10.45 and 10.46. For the discrete model, the ranges of bearing pressure of the due to the static and dynamic loadings were from 0 to 900 kPa and, from 1 00 and 1,1 00 kPa, respectively. Beneath the footing the average vertical stress was 400 kPa due to the static load, and 712 kPa due to the dynamic load. These values were in good agreement with those resulted from the discrete model, in which the 350 kPa and 612 kPa were resulted due to the static and dynamic loads, respectively. The contour and the 3Dimension surface plots of the homogeneous model followed the same path as in the discrete model. Under static and seismic loadings, both models appeared to have maximum stresses beneath the footing. These values decreased as it is presented away from the footing. 425
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Figure 10.45 "Contour and 3Demensional plots of MSE backfill bearing pressure of homogeneous mode due static loading" 426
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Figure 10.46 "Contour and 3Demensional plots of MSE backfill bearing pressure of homogeneous mode due dynamic loading" 427
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A maximum settlement of the abutment corresponding to a maximum bearing pressure occurred near the edge of the footing due to static and dynamic loadings. These results are shown in Figures 10.47 and 10.48, respectively. The effect of the earthquake was noticed on the backfill settlement, especially beneath the footing. However, less settlement occurred in the homogeneous model than in the discrete model. The homogeneous model had maximum settlements of 65 mm, due to the gravity, and 85 mm, due to the earthquake loading, without any heaving at the segments located away from the footing. On the other hand, the discrete model had maximum settlements of 170 mm and 250 mm due to gravity and earthquake loading, respectively. In case of the homogeneous model, the average settlement of the foundation was 30 mm due to the static load, and 3 7 mm due to dynamic load. These values were less than those obtained in the discrete model, in which, 118 mm and 166 mm were resulted from static and dynamic load, respectively. From there, it was evident that using the homogeneous model resulted in a stiffer abutment than the one obtained using the discrete model. Clearly, a good illustration of the bearing pressure was observed when the bridge abutment was analyzed using the transversely isotropic homogeneous properties. However, the settlement of the homogeneous model was considerably underestimated. 428
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Figure 10.47"Contour plot ofMSE backfill settlement of homogeneous model due static loading" Figure 10.48 "Contour plot of MSE backfill settlement of homogeneous model due dynamic loading" 429
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1 0.4.2.3 Longitudinal and Transverse Earth Pressure behind the MSEWall The lateral earth pressure of the MSE backfill was obtained along the centerline and along the edge. The horizontal static and dynamic stresses were calculated in both directions, longitudinal and transverse. Longitudinal and transverse earth pressures were computed and compared to those obtained from the discrete model as shown in Figures 10.49 through 10.56. Figures 10.49 though 10.52 were provided to show the longitudinal stresses of both models: along the centerline due to the gravity, along the centerline due to the earthquake, along the edge due to the gravity, and along the edge due to the earthquake. In Figure 10.49, it was noticed that under static loading, the longitudinal earth pressure along the centerline had a maximum value of 300 kPa near the top, which reduced to 150 kPa at height of 1.5 m, and increased again to a value of 200 kPa near the base. The maximum longitudinal earth pressure near the top was attributed to the foundation effect. That was not observed in the discrete model, in which zero longitudinal earth pressure was observed at the top of the wall. When subjected to the dynamic load, as shown in Figure 1 0.50, it was shown that the longitudinal earth pressure along the centerline had a maximum value of300 kPa also, which is attributed to the foundation effect. The longitudinal earth pressure of the homogeneous model decreased along the centerline to aminimum value of 150 kPa at height of2 m, and increased again to 200 kPa near the base of the wall. This behavior was also different from that of discrete model, where the stresses increased from 20 kPa at the top of the wall, to 100 kPa near the base of the wall. In Figure 10.51, the longitudinal earth pressure along the edges, of the homogeneous model, due to static load, had a larger pressure of 100 kPa, which increased with depth to a maximum value of 180 kPa at height of 2 m, and decreased to value of 80 kPa near the base of the wall. This 430
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trend was in good agreement with that of the discrete model, in which the pressure was 60 kPa at the top ofthe wall; increased to 130 kPa at height of 1.5 m and decreased to 110 kPa near the base of the wall. When subjected to dynamic load, as shown in Figure 10.52, the trends of both homogeneous and discrete along the MSE wall edge were similar, but with different values. In this figure, the longitudinal earth pressure had a value of 240 kPa at the top of the wall, which increased to 340 ]4>a at height of 2.5 m and decreased to 170 kPa near the base of the wall. On the other hand, the discrete model had a zero lateral pressure at the top of the wall, and increased to 220 kPa at height of 1.5 m and decreased to 145 kPa near the base of the wall. Figures 10.53 through 10.56 were provided to the show the comparisons of the transverse stresses from both models. In these figures, it was noticed that the resulted earth pressures from both models were very comparable; they followed similar paths and had values that were close to each other, with some exceptions. Along the centerline, as shown in Figure 1 0.53, the transverse earth pressure of both models, due to static load increased from 10 kPa at the top of the wall to 60 kPa neat the base wall. When subjected to dynamic load, as shown in Figure 10.54, the transverse earth pressure ofthe homogeneous model increased from 60 kPa at the top of the wall to 140 kPa near the base of the wall. These values were larger than those of discrete model, in which the lateral earth pressure increased from 40 kPa near the top 100 kPa near the base of the wall. Figure 10.55 and 10.56 indicated that due to both static and dynamic loadings, the transverse lateral pressures obtained from homogeneous model along the edges were close to those of discrete model and followed similar trends. In Figure 10.55, due to static load, the transverse pressure of homogeneous model increased with depth from 70 kPa at the top of the wall to 160 kPa near the base of the wall, where as in the discrete model, the transverse pressure increased from 71 kPa at the top ofthe wall to 300 431
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kPa near the base of the wall.. Due to dynamic load, there was a significant effect on the transverse earth pressure along the edge compared to those of static as shown in Figure 10.56. In the homogeneous model, the transverse earth pressure increased form 215 kPa at the top of the wall to 320 kPa near the wall base, where in the discrete model, the lateral earth pressure increased from 1 00 kPa at the top of the wall tp 360 kPa near the base of the wall. It was observed, from figures 10.49 through 10.56 and from Table 10.11, that the homogeneous model approach was capable for some extent in predicting the lateral earth pressure behind the MSE wall. This capability was noticed even more when predicting the transverse pressure, especially under gravity load. The observed trends in Figures 10.49 through 10.56 were smooth fits of the gathered data, and the equations that represented the homogeneous response was as shown in Table 10.12. Table 10.11 "Range of lateral earth pressure (kPa) using discrete and homogeneous approaches" Discrete Static Centerline 090 145315 50130 85180 Dynamic Centerline 20110 145290 Edge 0215 165340 Static Centerline 1060 1060 15310 70165 Dynamic Centerline 40100 60140 Edge 100360 215330 432 :
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4GDD ... 3CJDD E !.BD !2GDO .. :r: UIID 1CIDD .., j . . I .. .. Discrete {Centerline) (Centerline) 0.0 1ao.o .o a.o 40Q.D IDO.O prwnurw_c:m (kPa) Figure 10.49 "Longitudinal horizontal stress (crx) along the centerline of the MSE wall of discrete and homogeneous model due to static loading" 4DGD 3100 3DOD E .!.2100 .. 'i :1:1100 1DDD IUD # ,. .. .. .. . oisc. rete (Centerline} Hom ogen eous: .{CenterHne ) D.O. 1011.0 200.0 31JD.O 4GD.O Lmr11l uth prnsurw_c:m (kPa) Figure 10.50 "Longitudinal horizontal stress (crx) along the centerline of the MSE wall of discrete and homogeneous model due to dynamic loading" 433
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4JCD 31111 DJD E !.21GD t.mciJ .. 11GD 1aaD IOD . Dit.cret4! {Edge) Homogeneous (Edge ) a .a 1111.0 2DO.D 300.0 4111.0 8DO.O La..,.learth llr81lllr8_ax (kPaJ Figure 10.51 "Longitudinal horizontal stress (crx) of soil along the edge of the MSE wall of discrete and homogeneous model due static loading" _3DDO E !,21100 iaa 1i ::J:1100 1aaa saa "' . \ I : Oiserete ( Edge) . Homogeneous (Edge } . a.a 1111.0 2CII.O 300.0 400.0 100.0 Lat.nl earth pressu,._ax (kPa) Figure 10.52 "Longitudinal horizontal stress (crx) of soil along the edge of the MSE wall of discrete and homogeneous model due to dynamic loading" 434
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41111 ,..... ...., 4DGO 3ICII 3DGD E !.2RD i21111 {Centerliner) 'li :1:11111 ( Centerline) 1IXIJ IGD D D.D 1DO.D 2GClD 3DD.D 400.0. D.D Lawai earth prnsure_a, (kPa) Figure 10.53 "Transverse horizontal stress (ay) of soil along the centerline of the MSE wall of discrete and homogeneous model due to static loading" 3DDD E .!.2B .. 'i2GDO 'i ,.., 1CIIJ . DIHrete (Cent erline} Homogeneou s (Centerline) : 0.0 1DD.D 20D.D D.D 40D.O Lat.aleuth pressu,.._or (kPa) IDO.D Figure 10.54 "Transverse horizontal stress (ay) of soil along the centerline of the MSE wall of discrete and homogeneous model due to dynamic loading" 435
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4111J 4GDD 3IIIJ 3GIXI E !.2111J .. ::t: ,.., .., D D.D . . . . . . . Discrete (Edge) Homogeneou $ (Edge) 1DD.D 2GD.D 4GD.D Latwalrth ,rnsurw_oy (kPa) IDD.D Figure 10.55 "Transverse horizontal stress (cry) ofsoil along the edge of the MSE wall of discrete and homogeneous model due to static loading" .., 4000 \ \ 31100 \ \ \ 3DDD E \ !.2100 i.:2DDD .. =z=,.., 1DDD IDD D D.D \ \ ' Discret e (Edge) Homogeneou s (Edge) \. ..... \ ' ' ', ...... .... IDD.D Figure 10.56 "Transverse horizontal stress (cry) of soil along the edge of the MSE wall of discrete and homogeneous model due to dynamic loading" 436
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Table 10.12 "The regression equations of lateral earth pressure" O" ( ) h =2.0x105 x(Height)2 0.0516x(Height)+178.72 x cntr stat1c om 0" ( d ) h = 3.0 X 105 X (Height )2 0.1202 x (Height)+ 268.29 x cntr rynam1c om O" ( d . ) h =2.0x105 x{Height)2 +0.0877x{Height)+87.186 x e ge stat1c om O" ( d dy ) h =3.0x1o5 x(Height)2 +0.1443x(Height)+l65.39 x e ge nam1c om O" = 58253e0.0004 x Height y( cntr stat1c) _hom 0" = 136 87 e0 0002 x Height y( cntr dynamic)_ hom 0" = 164_87e0.0002x Height y( edge static)..:. hom = 326 76 0.0001 x Height 0" y( edge dynamic)_ hom e 10.4.2.4 Longitudinal and Transverse Displacements of MSEWall In this section, a comparison of the MSE wall displacements at the centerline, due to static and dynamic loadings, was completed using the homogeneous approach, and was compared to the displacements that were obtained from the discrete model. Figures 10.57 indicated that using the homogeneous approach would underestimate the longitudinal wall displacement along the wall's height when subjected to static and dynamic loadings. However, due to the static loading, the top portion of the wall was displaced by a lateral distance of 50 t:nm, which was equal to the displacement that resulted from the discrete model. Also, when the 437
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model was subjected to dynamic loading, a very large displacement occurred at the top segment of the wall, which exceeded the discrete model's displacement by a factor of 2. Furthermore, the homogeneous model underestimated the lateral displacement in the transverse direction under both loadings, as shown in Figure 10.58. The trends of the transverse displacements in the homogeneous model were similar to those obtained from the discrete model, but they were very small along the height of the wall. Because of the smaller displacement that resulted from the homogeneous model, it was concluded that the homogeneous model is stiffer than the discrete model. Also, the displacements of the discrete model, due to both load curves, were already large. The homogeneous model was developed based on the theory of elasticity which would predict the soil's deformation at small ranges of loading. Therefore, the homogeneous model was not able to correctly predict the large displacements that resulted from large forces such as seismic motion. 438
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E E .! Ill "i % 4500 20 Discrete ...;static Homoge neo us_st atic .. D iscrete_ dynamic Hom o geneous_ d ynar r k 10 120 110 6X [mm) 220 Figure 10.57 "Longitudinal displacement (ox) of MSE wall along the centerline of discrete and homogeneous model due static loading" r 500 E 000 !. .. 500 I: fooo Ill .. % 500 000 I I <' I 20 ... .= .... *11,._, rDiscrete ... st"tie Homogeneous_st.tic Ois crete_dynam ie 7D 120 170 l!f [m m) Figure 10.58 "Transverse displacement (oy) of MSE wall along the centerline of discrete and homogeneous model due static loading" 439
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10.5 Summary and Conclusions To swnmarize, static and dynamic loadings were applied on an MSE bridge abutment system with a bridge slab supported by two reinforced abutments and MSE walls. The MSE wall consisted of Ottawa sand backfill, and geogrid sheets reinforcement. Ottawa sand backfill properties were determined using triaxial tests, where the reinforcement properties were determined based on the properties supplied by the manufacturer. Furthermore, the model was built using two approaches, discrete and homogeneous. Displacements and stresses ofboth models were calculated and compared. Considering the discrete model, the static and dynamic loads resulted in large settlement, large lateral earth pressure, large bearing pressure, large lateral wall displacement, and large reinforcing tensile stresses. Even though this model was under a bearing pressure that was larger than the recommended by the AASHTO, it gave reasonable results that can be utilized for comparison reasons. To overcome the large bearing pressure problem, wider strip footing located at further distance from the edge of the MSE wall is recommended. From the results of the discrete model, it was noticed that the lateral pressure of the bridge abutment had the highest values near the strip footing. Furthermore, most of the abutment settlement and bearing pressure occurred beneath the footing. The stress distribution.ofthe MSE backfill' was increasing with depth. Also, the longitudinal and transverse tensile stresses of the reinforcing layers were large near the top, and very small near the base of the MSE wall. In fact, the reinforcement near the base of the MSE wall was carrying negative stresses indicating compressive stresses. Finally, due to dynamic loading, it was determined that all stresses and displacements of the MSE abutment increaseed. The wall displacement in the transverse direction was much larger than that in the 440
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longitudinal direction. This was contributed to the larger PGA in the Y direction and the restraint from the bridge girder in X direction. When considering the homogeneous model, the results in most cases followed similar trends of the discrete model. For instance, the earth pressure at the edges of the bridge abutment in the homogeneous approach had stress values that were similar to the ones obtained from the discrete model. Also, it was noticed that the lateral earth pressure of the MSE abutment in the homogeneous model had maximum values near the strip footing. This result was similar to the one that was observed in the discrete model. Furthermore, the settlement and the bearing pressure of the homogeneous model were similar to the ones of the discrete model, where the maximum values occurred beneath the footing. Larger lateral and bearing pressures were observed from the homogeneous model, compared to those observed in the discrete model. Additionally, smaller settlements of the foundation in the homogeneous model were observed, indicating that a stiffer abutment would be the result of using the homogeneous model. When analyzing the MSE wall beneath the abutment, it was noticed that the homogeneous model was able to predict the lateral pressure in the transverse direction, which is perpendicular to the bridge span. Along the longitudinal side, the predictiop. of the lateral earth pressure in the homogeneous model was not close to that.ofthe discrete model, but the trends were similar. Also, displacements in both directions using the homogeneous model were much lower than those obtained from the discrete model. This indicated that the wall system in a homogeneous model would be stiffer than the one in a discrete .model. The previous observations indicated that constitutive modeling of reinforced soil, where equivalent properties were used to replace a fully frictional interface model, was in agreement for some extent with the behavior of discrete model 441
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when subjected to static loadings. Such a constitutive model was developed based on elastic properties of soil; therefore, it would be able to provide better prediction under small deformations. Further deformations would cause nonlinearity in the stress straincurve. This was not supported by the developed constitutive model. It was noticed that large displacements occurred in the discrete model, even under static loading. Therefore, the differences in the responses of both models were observed. Had the walls and the footing been less displaced in the discrete model, the homogeneous model would have been able to predict, for some extent, the displacement of the wall and the settlement of the footing. Despite the large difference in the wall lateral displacement and the footing settlement from the discrete model, the general trends and the lateral earth pressure distribution of the homogeneous model were similar to those of the discrete model The question to be asked here is what were the advantages of using the homogeneous model instead of the discrete model in analyzing the MSE bridge abutment? The principal advantage of using this technique was the amount of time that was saved while constructing the model. This model considers equivalent homogeneous transversely isotropic elastic materials, and does not include geosynthetic between the soil layers, and therefore, no frictional interfaces were needed to be modeled between the soil and the reinforcement, but they were considered in the developed equations. Further time saving was obtained during the CPU processing. The amount of time needed to process the MSE bridge abutment using the discrete finite element model was more than 4 times that of the homogeneous model when using x86 processor14, 14 Family 6 Model 13 Stepping Genuinelntel800 MHz 442
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11. Summary, Conclusions, and Recommendations 11.1 Summary The main goal of this research was to develop a constitutive model that predicts the mechanical properties of reinforced soil, while taking into account the soil properties, the reinforcing properties, the vertical spacing between reinforcing layers, the confining pressure, and the frictional interface. In order to develop the constitutive model, a large database of the independent parameters was developed (Chapter 4). Medium dense to dense Ottawa sand and weak to strong stiffed geosynthetic were used in this database. Only the elastic properties of constituents were considered, i.e. the Elastic modulus, E, and Poisson's ratio, v. Additionally, different spacing patterns between inclusion layers, different friction coefficients between soil and geosynthetic, and different confining pressures were also used in the database. Once the database of independent variables was established, 468 numerical tests using NIKE3D finite element code were completed on cubic specimens of this geocomposite to evaluate its equivalent properties. Each specimen was 1 m x 1 m x 1 m, and was subjected to a combination of confining pressure and deviatoric stresses in six directions successively. Ofthe deviator stresses, three were normal (crx, cry, and crz), and three were shear (txy, 'tyZ, and 'txz). The resulted strains in all six directions were calculated using each deviatoric stress application. Using Hooke's law, the compliance matrices relating the applied stress to the resulted strain were determined. 443
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The shape of these matrices was unique, indicating a transversely isotropic material with six representative parameters (E, E', v, v', G, and G'). Thus, the material properties in the horizontal plane are identical. From there, the equivalent properties of the reinforced soil specimen were determined. Evidently, each specimen had its homogeneous transversely isotropic properties depending on its independent variables. The development of linear regression equations as a function of (E5 Eg, v5 Vg, S, and f.) was facilitated by the large database of dependent and independent parameters. The regression analysis (Chapter 5) was completed using the following steps: 1Determine the correlation matrix between the response and all the independent variables, and observe which variables have strong influence on the response. 2Using the option of automatic modeling in a statistical package, build a regression model by choosing between the "best subset" method and the "forward stepwise" method. Otherwise, it is difficult and timeconsuming to make a detailed examination of all the possible regression models. a. In the best subset criterion, observe the model that contains the least number of variables which result in a maximum value ofR2 and a minimum value of Cp that is not far from the number of variables. b. In the forward stepwise criterion the variables of highest t* statistics are added to the model sequentially while observing the vales ofR2 and Cp. When adding more variables, the first added t* statistic will usually increase. R 2 will also increase and Cp will decrease. Once the increase oft* statistics and R2 becomes 444
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insignificant, the analysis is terminated and the model becomes ready to be checked for validity. c. The variables obtained from both criteria should be the same. 3Conduct diagnostic and remedial measures of the response, where each predictor should be examined with appropriate plots to provide preliminary information. a. A plot of residual values versus fitted values to asses the adequacy of the fit, and to determine if any outlying residuals exist. b. A Normality plot of residuals to check if the model is normally distributed. c. Construct the overall F test statistic to examine whether there is a linear relation between the response and the set of predictor variables. 4An inference of the regression parameters to determine whether or not any of the predictor variables can be dropped from the model. 5Estimate the interval of the mean response corresponding to the given values of the predictor variables. 6Predict the interval for the new observation corresponding to the given values of the predictor variable. Given the elastic material properties of the soil/geosynthetic, the interface friction coefficient between soil/geosynthetic, the spacing between geosynthetic layers, and the confining pressure, one can effectively predict the mechanical properties of the reinforced soil structure using the developed regression equations. These equations (Chapter 5) treat the geocomposite as homogeneous material with equivalent transversely isotropic material properties. To check the accuracy of these constitutive models, the results obtained from the homogeneous approach ought to be compared with the results obtained from the discrete finite element 445
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model due any load application. But before completing this comparison, the results of the finite element analysis on the discrete model must be calibrated and validated. When calibrating the tensile properties of geotextile inclusions were first obtained from a widewidth strip specimen tensile test as indicated by ASTM D459586 (Chapter 6). On the other hand, the properties of Ottawa sand were determined using a combination of a hydrostatic compression test and a drained triaxial compression test. The hydrostatic test and the triaxial test were conducted on cylindrical specimens of Ottawa sand. Each specimen was 71 mm in diameter and 152 mm in height, with a relative density of 70 % 2 %. From the hydrostatic compression test, where the sample was subjected to isotropic loading and unloading, the mean pressure was plotted versus the volumetric strain of the soil specimen. The slope of the unloading portion of this curve was defined as the bulk modulus, K. The other set of tests on soil were the conventional compression triaxial tests. The triaxial tests were completed on two types of specimens, unreinforced and reinforced samples. For both types, specimens were subjected to three confining pressures ( cr3): 103 kPa, 207 kPa, and 310 kPa. For each confining pressure, the most important outcome of the triaxial test was the plots of the deviator stress versus the axial strain. The plots of stressstrain curves were used in determining some of the strength parameters, such as Young's modulus and ultimate strength (maximum deviator stress). This allowed the plotting of Mohr circles, which can be used in determining the maximum shear strength and the friction angle. After performing the laboratory tests and obtaining the material properties, it was essential to reproduce the same results using finite element method with the aid of a finite element code, validation. LSDYNA finite element code was used to simulate these triaxial tests on both unreinforced and reinforced soil specimens. LSDYNA consists of several material models. Among these models, the Isotropic 446
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ElasticPlastic model was used to simulate the geotextile shell element and the Geologic Cap model was used to simulate the Ottawa sand. When comparing the triaxial test results of the finite element simulation and the laboratory test results, it was concluded that the numerical results were satisfying and very comparable those obtained from the laboratory experiment. Furthermore, the triaxial test simulations were completed on a homogeneous model with equivalent transversely isotropic material properties. The results obtained from the homogeneous model were compared with those obtained from the laboratory test and the discrete finite element model. The homogeneous model behaved as elastic material, and therefore, only 0.5% of the strain was taken irito consideration when comparing the results of the homogeneous model with the other two approaches. Within that range, the results of homogeneous model gave reasonable results, where the values of Young's moduli were in good agreement with the other two approaches. Further validations of the finite element method, with both discrete and homogeneous approaches were completed in Chapter 7. In Chapter 7, two main laboratory tests on reinforced soil specimens were simulated. These were triaxial tests on cylindrical reinforced soil samples with 152 mm diameter and 304.8 mm height by H.C Liu in 1987 at the University of Colorado at Denver, and plane stain tests conducted on the geosyntheticreinforced cube samples with equally spaced geotextile in 2001 by K. Ketchart at the University of Colorado at Denver. The soil used in both projects was Ottawa sand with relative density of 70 %; therefore, the soil properties obtained in Chapter 6 were used to complete these validations. In Liu's tests, the cylindrical samples had four patterns of spacing between the reinforcing layers. The first set of samples did not include any inclusion, the 447
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second set had 1 layer placed at the midheight of the sample, the third and the fourth sets had 4 and 6 layers which were equally spaced. All samples were subjected to two confining pressures, 103 kPa and 310 kPa. In his tests, Liu used a needle punched nonwoven geotextile material as reinforcing elements, Bidim C34. The elastic properties of this geotextile were not provided, therefore, several numerical trials were completed on the samples with 1 layer reinforcement to obtain approximate properties ofthe inclusion layers. The results from the finite element method showed good similarities to those obtained from laboratory tests, except for 6 layers sample when subjected to 103 kPa confining pressure, it acted The equivalent transversely isotropic properties of each reinforced sample were then determined using the linear regression equations that were developed in Chapter 5. The initial response, within 2 % strain, of the homogeneous model was similar to the initial response of the discrete model and the physical testing. For both the discrete and homogeneous models, the Young's modulus of each simulation was determined and divided by the Young's modulus obtained from the experiment results by Liu, which is called normalized Young's modulus. Furthermore, in Chapter 7, the experimental results obtained from the Soil Geosynthetic Performance (SGP) test, conducted by Ketchart in 2001, were compared to those obtained from the finite element method using both the discrete and the homogeneous approaches. A plane strain condition was imposed on this model preventing any displacement in the longitudinal direction. The analysis was completed on two samples: one was unreinforced and the other was reinforced with three equally spaced geotextile layers, Amoco2044. The equivalent transversely isotropic properties of the homogeneous model were determined using the formulated equations in Chapter 5. When plotting the sample's vertical displacement versus the resulting pressure, it was found that the 448
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initial behavior of both discrete and homogeneous models were similar to the ones obtained from the laboratory experiment. The results from laboratory experiment showed a failure ofthe loaddeformation after 10 mm displacement. That was not the case for the results obtained from finite element analysis. The comparison was also completed by determining the normalized Young's modulus. Once the numerical model has been calibrated and validated, it can be further used and applied to different geocomposite structures. In this project, three hypothetical models were chosen and investigated. These were the MSE foundation (Chapter 8), the MSE retaining walls (Chapter 9), and the MSE bridge abutment (Chapter 10). All of the three models were built using the discrete model with fully frictional interface and the homogeneous model with equivalent transversely isotropic properties. The results of both approaches were evaluated and compared to each other. The soils in the unreinforced and reinforced zones were assumed to be Ottawa sand; therefore, the developed properties in Chapter 6 can be used. However, due to the overburden pressure effect, the shear modulus was allowed to vary with depth. In Chapter 8, 3D finite element analyses performed on a 3m square footing resting on a soil and reinforced soil of different spacing and Young's modulus. First, a case of footing on soil was analyzed with its results serving as the baseline for comparison. The analyses on the reinforced soil were performed using both discrete and homogeneous models, and the results were compared. With the aid of regression analysis, the performance based bearing capacity was determined. Furthermore, in the discrete models, the tensile stress distribution of geotextile layers was evaluated to investigate the variation of the reinforcement tensile stresses with depth and distance from the footing's centerline. 449
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In Chapter 9, static and dynamic analyses were completed on a hybrid retaining wall. The hybrid wall was 7 m height with a 2 m base, and was subjected to plane strain condition constraining any lateral movement in the Y direction. The reinforced backfill in the MSE wall was constructed first using a discrete fmite element model with fully interface, and then using homogeneous model with the transversely isotropic material properties. The wall displacement, the lateral earth pressure, the bearing pressure of foundation, and the settlement of foundation of both models were calculated and compared. External and internal stabilities under both static and dynamic load were also evaluated to check for any failure criterion. In chapter 10, the static and dynamic behavior of a 3Dimensional simple span concrete bridge supported by MSE abutments was examined using the finite element method. The model was built and analyzed using two approaches: discrete and homogeneous. Due to static and dynamic load, the lateral earth pressure, the lateral wall displacement, the bearing pressure of foundation, and the settlement of the foundation were calculated. In the case of discrete model, the tensile stress distribution in the longitudinal and the transverse directions of different geosynthetic layers was calculated. 11.2 Conclusions The main goal ofthis project was to develop a constitutive model of reinforced soil. This model would be able to predict the mechanical properties of a gee composite as a function of its constituents, and other factors such as the spacing between geosynthetic materials and the confining pressure. In order to develop this constitutive model, several tasks were completed. The first task was 450
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completed by conducting numerical tests on a cubical element of the reinforced soil. From there, and with the aid of Hooke's law, the mechanical properties of the geocomposite were developed accordingly. After that, regression equations were developed. With both the equations and the values of the independent variables, one can predict the transversely isotropic properties of the geo composite. The efficiency of such a model was completed by comparing the behavior of several models using both discrete model with fully frictional interface and the homogeneous model with equivalent transversely isotropic properties. These validations were completed by first conducting laboratory tests. The properties of the geosynthetic and the soil were obtained and calibrated to be used in the numerical model. Consequently, the tests were numerically conducted using both approaches, discrete and homogeneous, under the same boundary conditions. After validation, the application of the finite element method with the aid of both approaches was completed on different geostructures which are the Foundations resting on reinforced soil, the MSE wall, and the MSE bridge abutments. From this project the followings were points were concluded: 1The geocomposite model behaved as a transversely isotropic model with six representative constants. These constants are Eh, vh, Ev, Vv, Gh, and Gv. 2The Young's moduli obtained from the finite element method were comparable to those obtained from the analytical method of transversely isotropic materials. 3Using the regression equations developed in Chapter 5, and given the properties of soil, geosynthetic, interface friction coefficient, spacing between geosynthetic layers, and confining pressure, one can effectively 451
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predict the parameters of the transversely isotropic model of the geo composite. 4From conducting laboratory tests on Ottawa sand with relative density of 70 %, it was found that such type of soil had an internal friction angle of 36 with zero cohesion, and bulk modulus of209 MPa. 5From drained triaxial tests on unreinforced and reinforced specimens, it was evident that through increasing the confining pressure and by adding more reinforcing layers, the strength of the soil would increase significantly. The peak strength increased linearly under each confining pressure when increasing the number of reinforcing layers: 6An initial contraction followed by a volumetric increase (dilation) occurred when shearing the unreinforced and reinforced soil samples in the triaxial test. 7From the triaxial tests on reinforced specimens, the same internal friction angle of36 was obtained in addition to a cohesion intercept (C') which increased with the increasing number of reinforcing layers. The values of C' were 22.3 kPa, 64.5 and 108.5 kPa for the 2, 4, 6 specimen layers, respectively. 8When simulating the triaxial tests using the finite element method, a Cap model was selected to model the Ottawa sand, and an Isotropic Elastic Plastic material was selected to model the geotextile material, which was used in the reinforced soil specimen. They both showed good agreements with laboratory results. 9When plotting J 1 J 2 D which were obtained from drained triaxial test on the unreinforced specimens, a linear relation was observed. This indicated that the first yielding surface of the cap model 452
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could be simplified to a linear surface as described in the DruckerPrager criterion. 10The results obtained from triaxial tests on unreinforcedlreinforced specimens using finite element method with the aid of fully frictional interface showed high similarities to those obtained from the laboratory tests. This was observed while evaluating the normalized Young's modulus at 0 %, 1%, and 2 % strains, where all of these values for different reinforcing patterns and under different confining pressure were proximal to 1. 11The triaxial test results from the finite element method, and with the aid of the homogeneous approach, were close at the early stages of loading to those obtained from laboratory and the discrete finite element model. Beyond 1 % straining, the homogeneous response deviated from the two other tests. However, when evaluating the normalized initial Young's modulus, the homogeneous model gave good representation of the laboratory test results, especially when subjected to small confining pressure, where the normalized Young's moduli in most cases were not far away from unity. 12When validating Liu's and Kethcart triaxial test results on unreinforced soil (Chapter 7), the Cap model was found capable to represent the soil materal to great extend. 13When validating Liu's triaxial test results on reinforced soil, the initial response of numerical analysis was similar to that ofLiu's tests. However, an incessant increase in the strength of the numerical results was noticed, unlike the ones from the experiment. This increase resulted from assuming a linear model of the reinforcement material. 14A major difference from Liu's experiments was noticed when analyzing the 61ayer reinforced sample that was subjected to 103 kPa confining 453
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pressure. It is believed that the low confining pressure and the large number of reinforcing layer permitted some initial weakness in the reinforced sample, and hence the flat portion in the stressstrain relation determined from the experiment. 15The homogeneous model was capable in reproducing the initial stress strain curve from Liu's test results. 16When validating the plane strain test of reinforced soil that was completed by Ketchart in 2001, the discrete model was able to predict the experiment behavior to some extent. The strength of the discrete model kept on increasing, unlike the physical specimen where it failed after 15 mm of displacement. The reason for the strength increase resulted from the use of a linear elastic model in representing the geosynthetic material. 17When simulating the plane strain test of reinforced soil using the homogeneous model, the initial response, within 10 mm displacement, was similar to both the discrete model and the Ketchart's experiment, in which the normalized Young's moduli of the homogeneous model was 0.92. Beyond the 10 mm displacement, and due its elasticity, the response of homogeneous model significantly deviated from the other two models. 18From the analysis of the 3D square footing, it was evident that the bearing capacity of the footing, especially under large deformation increases with an increasing number of reinforcing layers, where the contribution of geosynthetic tensile strength becomes more effective. 19Applying the regression analysis to the concept of performance based bearing capacity, it was determined that bearing capacity was directly proportional to the settlement and stiffness of reinforcement and 454
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inversely proportional to the spacing between reinforcing inclusion layers. 20The bearing pressure was much more dependent on settlement and spacing than inclusion s stiffness. 21The tensile stress distribution of inclusions was much larger under the footing than away from it. 22The maximum efficiency from reinforced soil foundation can be obtained by having several reinforcement layers, in which the maximum lateral extent of the reinforcement needs not to be greater than lB from the footing centerline and the vertical extent of 2B from the base of the footing. 23When treating the reinforced soil beneath the foundation as a homogeneous material with equivalent transversely isotropic material properties, larger bearing capacity and smaller corresponding settlement than those of discrete model were observed Therefore, such approach resulted in stiff foundation that overpredicts the bearing capacity of the foundation. 24When analyzing the 7 m MSE hybrid retaining wall, the wall was externally and internally stabilized due to both static and dynamic loads. 25As a result of gravitational load, the MSE hybrid retaining wall displaced in the forward direction with a maximum displacement of 23 mm. Due to earthquake motion the wall displaced further, resulting in a maximum displacement of 250 mm at the top and a minimum displacement of 18 mm at the wall base. The large displacement mainly resulted from not attaching the geosynthetic layers to the concrete wall. 26When imposing gravitational load on the MSE hybrid retaining wall, the lateral earth pressure of the homogeneous model was similar to that of the discrete model. Also the foundation bearing pressure and the 455
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foundation settlement of the homogeneous model was similar to that of discrete model, except beneath the toe of the footing where larger peak values in the discrete model resulted from the large lateral displacement of the wall. However, the displacement of discrete model was much larger than that of homogeneous model. 27The difference in the lateral displacement profiles of the MSE hybrid retaining wall indicated a stiffer model using the homogeneous approach. 28When imposing dynamic load on the MSE hybrid retaining wall, the discrete model displaced much more than the homogeneous model. This resulted in slight differences in lateral earth pressure, bearing pressure, and settlement responses. 29When analyzing the discrete model of the MSE bridge abutment, it was noticed that due to static and dynamic loading, large settlement, larger bearing pressure, larger lateral earth pressure, large lateral wall displacements were observed. 30The bearing pressure that resulted form the MSE bridge abutment model was much larger than the recommended value by the AASHTO. 31Beside the restraint from the bridge girders; the larger PGA in the transverse direction was the major reason for obtaining larger deformations in that direction. 32The homogeneous model of the MSE abutment followed similar trends of the discrete model. 33Smaller settlements were noticed in the homogeneous model of the MSE abutment when compared to the discrete, which indicates that a stiffer abutment would result from the homogeneous approach. 456
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34The lateral earth pressure of the homogeneous MSE wall supporting the MSE bridge abutment was in good agreement with that of the discrete model, especially in the transverse direction. From this project, it was concluded that using homogeneous model with transversely isotropic was able to predict, to some extent, the behavior of discrete models within small displacements. At larger displacements, such model was significantly deviating from those of discrete models. The main reason for these deviations was due to the linearity of this constitutive model; and therefore it would only predict the linear responses ofthese' reinforced soil models. With this limitation, the homogeneous model had a great advantage of saving time while preprocessing and CPU processing. This kind of model would take into consideration the inclusion properties, their spacing, and their frictional interfaces with soil. Therefore, the time needed for modeling these layers and including their frictional interaction with adjacent elements would be eliminated. Furthermore, the CPU time while processing kind of analysis was enormously reduced by factors reaching up to 10 times in some cases. As a closing remark, such approach is an effective tool in simulating different reinforced soil applications in both laboratory and the field. It reasonably predicts the behavior of these applications within small loading and has the great benefit of saving a lot of time. 457
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11.3 Recommendations for Future Studies The developed constitutive model of reinforced soils is based on the linear properties of their constituents. The linear response was observed in all the applications throughout this project. The soil does not behave linearly unless when subjected to very small loadings. Therefore, it is recommended to develop a constitutive model that accounts for the nonlinear portion of the soil model. Also, it will be useful to include other factors such as the geosynthetic's thickness and length in developing a constitutive model of reinforced soil. In addition to that, the following studie studies can be included to strengthen this research: 1Perform a non linear regression analysis between the independent variables and the dependent variables. 2Obtain the cap material properties of Ottawa sand corresponding to different relative densities, by conducting several sets of hydrostatic compression tests and triaxial tests. 3Conduct triaxial test on reinforced soil samples under the combination of different soil relative densities and different reinforcing properties. 4Reproduce the triaxial test results of unreinforced and reinforced soil specimens numerically using different material models (Cap, Rambo Osgood ... ),different finite element codes (LSDYNA, NIKE3D .. ) and different algorithms (Implicit and Explicit). 5Conduct a large.scale test on foundation on top of reinforced soil and compare the results with those of finite element. 6Compare the test results of the MSE wall and the MSE bridge abutment due to dynamic load with those obtained from shaking table or centrifuge testing. 458
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7For the MSE bridge abutment analyze its performance with different footing size. Also, generate finer mesh of the abutment and compare the lateral stress distribution with that of the current coarse mesh. 459
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Appendix A. Equivalent Transversely Isotropic Properties of GeoComposites 460
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0\ s 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0 500 0 500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.069 0.069 0.069 0.069 0.069 0.069 0.069 0.069 0.069 0.069 0.069 0 069 0.069 0.069 0.069 0.069 0.069 0.069 0.069 0.069 0.069 0.069 0.069 0.069 0.069 0.069 0.069 70 70 70 70 70 70 70 70 70 I 50 I 50 I 50 I 50 I 50 I 50 I 50 I 50 I 50 225 225 225 225 225 225 225 225 225 Vs 0.3 0.3 0.3 0.3 0.3 0.25 0.2 0.3 0.3 0.3 0.3 0 3 0.3 0.3 0.25 0 2 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.25 0.2 0.3 0.3 IIOO 550 270 1100 1100 IIOO IIOO 1100 IIOO 1100 550 270 IIOO 1100 I100 1100 IIOO 1100 IIOO 550 270 1100 1100 1IOO 1IOO 1IOO IIOO Vg 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.3 0.2 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.3 0.2 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.3 0.2 friction I 0.5 0.25 1 1 1 0.5 0.25 1 I I 0.5 0 25 I I 81.648 81.299 81.074 Vb 0.275 0.276 0.276 8I. 512 0.276 81.348 0.276 8I.462 0.228 81.274 0.181 81. 626 0.275 81.603 0.275 173.389 0.275 172.924 0.275 172.614 0 275 173.243 0.275 I73.13I 0.275 173.221 0.227 72.500 72.563 72.576 72.444 72.432 71.996 71.489 72.476 72.55I I54.054 154.I39 I54 13I 154.049 154.020 153.042 Vv 0.280 0.280 0.281 0 280 0.281 0.230 O.I82 0 280 0.280 0.280 0.280 0.280 0.280 0.280 0.230 173. 008 0.180 I52 .171 0.182 173.397 0.286 154.III 0.286 I73.097 0.274 154.102 0.280 259.395 0.274 230.584 0.280 258 810 0.279 230 623 0.282 258.455 0.279 230.640 0.282 259.125 0.275 230.536 0.280 258.965 0.275 230.504 0.280 259 I80 0.227 229 039 0.230 259.207 O.I8I 227.813 0.182 259 I98 0.274 230.576 0.280 258.953 0.274 230.571 0.280 26.920 26.920 26.920 32.421 32.245 32.111 26 920 32.313 26.920 32.176 27.992 33.633 29.153 34.94I 26 920 32.442 26.920 32.472 57.690 69.058 57.690 68.834 57.469 68. 660 57.690 68.936 57.690 68.796 59.993 71.650 62.485 74.494 57.690 69.059 57.690 68.984 86.535 I 03.330 86. 535 I03.092 86.535 I01.806 86.535 103.205 86.536 103.100 89.991 107. 290 93.736 IIl.535 86.535 I 03.25I 86.535 103.196
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01 N s (m) 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0 500 0.500 O'o (MPa) 0.069 0.069 0.069 0.069 0.069 0.069 0.069 0.069 0.069 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 O.I72 Es (MPa) 300 300 300 300 300 300 300 300 300 IOO IOO 100 100 100. 100 100 100 IOO 220 220 220 220 220 220 220 220 220 330 Vs 0.3 0.3 0.3 0.3 0.3 0.25 0.2 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.25 0.2 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.25 0.2 0.3 0.3 0.3 Eg (MPa) I100 550 270 I 100 I IOO 1100 1100 1 IOO 1100 1100 550 270 1100 1100 1100 I100 1100 IlOO I100 550 270 I IOO 1100 1100 I100 1100 1100 1100 Vg 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.3 0.2 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.3 0.2 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.3 0.2 0.4 friction 1 Eh (Mpa) 345.363 344.586 344.320 0.5 344.955 0.25 344.809 I 345.029 345.272 344.868 344.862 II5.638 I15.560 II5.375 0.5 I Il5. 509 0.25 I 15.330 115.751 I15.538 I15.680 115.666 253.076 252.871 252.699 0.5 0.25 1 1 I 253.085 252.683 253.000 252.825 252.967 252.906 378.863 Vh 0.275 0.274 0.279 0.281 0.275 0.228 0.181 0.274 0.274 0.269 0.272 0.273 0.268 0.268 0.224 O.I77 0.269 0.269 0.272 0.273 Ev (MPa) 307.058 307.097 307.I36 307.017 306.99I 305.053 303 413 307.107 307.082 103.085 I02.977 I02.903 103.047 103.078 102.166 101.516 I03.058 103.033 225.510 225.368 0.276 225.311 0.272 225.5I9 0.27I 225.472 0.225 223.960 0.178 222.705 0.273 225.460 0.272 225.456 0.272 337.724 Vv 0.279 0.280 0.282 0.283 0.280 0.230 0.182 0.280 0.280 0.280 0.280 0.280 0.280 0.281 0.230 0.182 0.280 0.280 0.280 0.280 0.281 0.280 0.280 0.230 0.182 0.280 0.280 0.280 Gv (MPa) 1I5.379 I15.379 II5.379 II5.379 115.380 119.989 124. 984 115.379 115.379 38.4I9 38.4I9 38.419 38.419 38.419 39.951 41.610 Gh (MPa) 137.633 137.338 I37.203 137.490 I37.338 142.864 148.578 137.505 137.469 46.195 46.018 45.887 46.051 45.894 47.818 49.706 38.419 46.104 38.419 46.086 84. 248 101.031 84.57I 100.814 84.572 100.712 84.572 100.910 84.571 100.841 87.949 104.870 91.609 109.035 84.571 100.975 84 .57I I00.869 I26.876 151.284
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0\ l.;.) s (m) 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.5. 00 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 O'o (MPa) 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0:112 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.345 0.345 0.345 0 345 0.345 0.345 0.345 0.345 0.345 0.345 0.345 Es (MPa) 330 330 330 330 330 330 330 330 450 450 450 450 450 450 450 450 450 150 150 150 150 150 150 150 150 150 320 320 Vs 0.3 0.3 0.3 0.3 0.25 0.2 0.3 0.3 0 3 0.3 0.3 0.3 0.3 0.25 0.2 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.25 0.2 0.3 0.3 0.3 0.3 Eg (MPa) 550 270 IIOO IIOO IIOO IIOO llOO 1100 1100 550 270 1100 I100 1IOO IIOO llOO IIOO llOO 550 270 1100 1100 IIOO 1100 I100 1100 1IOO 550 Vg 0.4 0.4 0.4 0.4 0.4 0.4 0.3 0.2 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.3 0.2 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.3 0.2 0.4 0.4 friction I Eh (Mpa) 378.795 I 378.6I3 0.5 378.780 0.25 378.576 379.I08 I I 378.877 I 378.804 378.790 I I 5I6.11I I 5I6.023 5I6.0I6 0.5 516.I8I 0 25 5I6.0IO I 516.I87 5I6.443 516 .126 Vb 0.278 0.276 0.272 0.272 0.225 O.I79 0.272 0.272 0.273 0.273 0.281 0.277 0.273 0.226 0.179 0.273 Ev (MPa) 337.592 337 505 337.727 337.724 335.436 333.6I8 337.675 337.628 460 087 459.97I 459.905 460.169 460 I69 457.056 454.621 460.082 5I6.221 0.273 460.030 0 5 0.25 1 172.77I 0.267 I54.433 I72.699 0.276 154.2I7 172 .703 0.273 154.084 172.601 0.272 154.399 172.347 0.267 I54.434 172.364 0.2I9 I53.309 172.295 0.173 152.360 172 734 0.268 154.373 I72.688 0.269 154.304 367.175 367.201 0.271 0.273 327.862 327.603 Vv 0.282 0.281 0.280 0.280 0.230 O.I82 0.280 0 280 0.280 0.280 0.283 0.282 0.280 0.230 0.182 0.280 0.280 0.280 0.282 0.279 0 282 0.280 0.230 0.18I 0 280 0.280 0.280 0.280 Gv (MPa) 126.876 126.390 126.876 I26.876 I31.947 I37.440 I26.876 I26.877 I73 025 I73.026 I72.365 I73.025 183.6I4 I87.437 Gh (MPa) 149.974 151.284 151.092 151.051 156.993 163.311 151.160 151.102 206.039 205.909 204.320 205.962 205.887 213.980 222.635 173.027 206.015 I73 026 205.937 57.613 69.169 57.613 68.944 57.394 68.303 57. 614 69.046 57.613 68.950 59.907 71.747 62.393 74.575 57.612 69.I62 57.613 69.09I I22.995 146.844 122.995 146.614
PAGE 500
.,J::. 0'\ .,J::. s (m) 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.250 0.250 0.250 O"o (MPa) 0.345 0.345 0.345 0.345 0.345 0.345 0.345 0.345 0.345 0.345 0.345 0.345 0.345 0.345 0.345 0.345 0.345 0.345 0.345 0.345 0.345 0.345 0.345 0.345 0.345 0.069 0.069 0.069 Es (MPa) 320 320 320 320 320 320 320 480 480 480 480 480 480 480 480 480 650 650 650 650 650 650 650 650 650 70 70 70 Vs 0.3 0.3 0.3 0.25 0.2 0.3 0.3 0.3 0.3 0.3 0 3 0.3 0.25 0.2 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.25 0.2 0.3 0.3 0.3 0.3 0.3 Eg (MPa) 270 1100 1100 1100 1100 1100 1100 1100 550 270 1100 1100 1100 1100 1100 1100 1100 550 270 1100 1100 1100 1100 1100 1100 1100 270 1100 Vg 0.4 0.4 0.4 0.4 0.4 0.3 0.2 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.3 0.2 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.3 0.2 0.4 0.4 0.4 friction I Eh Vb Ev (MPa) 327.479 327 865 327.873 325.631 323.803 327.768 327.741 491.093 490.831 490.667 491.073 491.083 487.788 485.151 491. 004 490.935 (Mpa) 367.169 0.5 I 367.070 0.25 367.001 0.273 0.271 0.271 0.224 0.177 0.271 0.272 0.272 0.273 0.273 0:272 0.272 0.225 O.I78 0.277 0.272 367.066 367.079 367.180 367.217 550.'375 550.322 550.300 0.5 550.067 0.25 549.987 I 550.208 1 550.496 I I 0.5 0.25 I 1 1 1 0.5 550.261 550.356 745.I46 0 .273 664.463 744.941 0.278 664.199 744.932 0.281 664 .081 744.736 0.272 664.435 744.804 0 272 664.461 744.789 0.225 660.063 745.450 0.179 656.561 744.885 0.273 664.380 745.024 0.273 664.3I9 76.767 0.267 73.888 77.089 0.28I 73.544 76.393 0.269 73.850 Vv 0.280 0.280 0.280 0.230 0.181 0.280 0.280 0.280 0.280 0.280 0.280 0.280 0.230 0.182 0.282 0.280 Gv (MPa) 122.996 122. 997 122.995 127.905 133.218 122.994 122.995 184.529 184.528 184.529 184.529 184.529 191.902 199.889 184.528 Gh (MPa) 145.412 146.734 146.633 152.470 158.571 146.727 146.712 219.944 219.706 219.601 219.784 219.687 228.357 237.547 219.814 I84.531 2I9.828 0.280 249.907 297.535 0.282 249.912 297.389 0.283 249.909 297.289 0.280 249.908 297.452 0.280 249.908 297.384 0.230 259.895 309.022 0.182 270.716 321.436 0.280 249.906 297.512 0.280 249.908 297.507 0.297 27.86I 30.992 0.296 27.860 30.625 0.298 27.861 30.764
PAGE 501
+:. 0\ Vl s (m) 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0 250 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0 250 0 250 0.250 0 250 0.250 0.250 0.250 0.250 0.250 0.250 0.250 O"o (MPa) 0.069 0 069 0.069 0.069 0.069 0.069 0.069 0.069 0.069 0 .069 0.069 0.069 0.069 0.069 0.069 0.069 0.069 0.069 0.069 0.069 0.069 0.172 0.172 0.172 0.172 0.172 0 .172 0.172 Es (MPa) 70 70 70 150 150 150 150 150 150 225 225 225 225 225 225 300 300 300 300 300 300 100 100 100 100 100 100 220 Vs 0.3 0.2 0.3 0.3 0.3 0.3 0.3 0.2 0.3 0 3 0.3 0.3 0 3 0.2 0.3 0.3 0.3 0.3 0.3 0.2 0.3 0.3 0.3 0.3 0.3 0.2 0.3 0.3 Eg (MPa) 1100 1100 1100 1100 270 1100 1100 1100 1100 1100 270 1100 1100 1100 1100 1100 270 1100 I100 1100 1100 1100 270 1100 1100 1100 1100 1100 Vg 0.4 0.4 0.2 0.4 0.4 0.4 0.4 0.4 0.2 0.4 0.4 0.4 0.4 0.4 0.2 0.4 0.4 0.4 0.4 0.4 0.2 0.4 0.4 0.4 0.4 0.4 0.2 0.4 friction 0 25 I 1 1 0 5 0.25 1 0.5 0.25 0.5 0.25 1 I 0 5 0.25 Eh (Mpa) 77.628 76.905 77.384 164.496 162.601 I63.945 I63 970 162.483 164.223 Vb 0.273 O I77 0.267 0.275 0.279 0.274 0.274 0.177 0.273 243.804 0 273 242.758 0.279 243.285 0.272 242 .839 0 .271 243.944 0.181 243.634 0.274 327.872 0.281 Ev (MPa) 73.830 72 969 73.650 I57.355 156.I68 I57.354 157.355 155.990 156.891 235.002 233.622 235.000 235.000 233.359 234.450 312.483 Vv 0.295 0 .198 0.297 0.294 0.298 0.295 0.295 O.I96 0.296 0 297 0.298 0.297 0 298 0.196 0.298 0.295 322.914 0.279 3Il.068 0.299 327.870 0.28I 3I2.480 0.295 327.868 0.281 312.479 0.295 323.295 0.179 310.793 0.198 323.225 0 274 31I.913 0 298 109.896 0.268 104.889 0.297 109.165 0.280 104.614 0.297 109.453 0 267 104.820 0 298 108.90I 0.267 I04.782 0.299 109.791 0.180 104.336 0.192 109. 560 0 267 105.33I 0.296 238.554 0.272 229.994 0.297 Gv (MPa) 27.86I 30 .180 27.861 59 724 59.72I 59.724 59. 724 64.697 59.724 Gh (MPa) 30.603 33. 390 31.533 65.638 64.934 65.470 65.308 70.607 65.933 89.592 98 865 89.59I 97.095 89.593 98.86I 89.593 98.858 97 055 I06.520 89.592 97.737 119.462 131.046 119.458 129.255 119.462 131.043 119.462 131.041 129.413 I41.318 119.461 129.687 39 792 44.4I7 39 790 43.522 39 792 44.404 39 792 44.291 43.096 47.091 39 .791 44.046 87. 584 96.747
PAGE 502
0\ 0\ s (m) 0.250 0.250 0.250 O'o (MPa) 0.172 0.172 0.172 0.250 0.172 0.250 0.172 0.250 0.172 0.250 0.172 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.345 0.345 0.345 0.345 0.345 0.345 0.345 0.345 0.345 0.345 0.345 Es (MPa) 220 220 220 220 220 330 330 330 330 330 330 450 450 450 450 450 450 150 150 150 150 150 150 320 320 320 320 320 Vs 0.3 0.3 0.3 0.2 0.3 0.3 0.3 0.3 0.3 0.2 0.3 0.3 0.3 0.3 0.3 0.2 0.3 0.3 0.3 0.3 0.3 0.2 0.3 0.3 0.3 0.3 0.3 0.2 Eg (MPa) 270 1100 1100 1100 1100 1100 270 1100 1100 1100 1100 1100 270 1100 1100 1100 1100 1100 270 1100 1100 1100 llOO 1100 270 1100 1100 1100 Vg 0.4 0.4 0.4 0.4 0.2 0.4 0.4 0.4 0.4 0.4 0.2 0.4 0.4 0.4 0.4 0.4 0.2 0.4 0.4 0.4 0.4 0.4 0.2 0.4 0.4 0.4 0.4 0.4 friction I Eh (Mpa) 237.425 Vh 0.279 0.270 0.270 Ev (MPa) 228.539 229.993 229.993 228.244 229.448 343.632 342.141 343.631 343.632 341.841 343.066 467.550 466.060 467.550 Vv 0.298 0.297 0.298 0.196 0.298 0.297 0.299 0.297 0.297 0.198 0.299 0.296 0.299 0.296 0 5 I 238.148 0.25 237.733 0.5 0.25 1 0.5 0.25 1 0.5 0.25 1 1 1 0.5 0.25 1 238.862 0.181 238.540 0.272 357.287 0.276 354.985 0.278 357.141 0.275 356.946 0.275 355.125 0.178 355.038 0.274 488.138 0.279 483.227 0.278 488.136 0.279 488.134 0.279 467.546 0.296 488.272 0.186 465.544 0.197 481.897 0.274 466.996 0.299 164.792 0.274 157.602 0.294 162.618 0.279 156.388 0.297 164.526 0.273 157.587 0.294 164. 382 0.272 157.581 0 294 162.533 0.173 156.222 0.195 164.529 0.271 157.136 0.295 344.643 0.270 157.136 0.216 344.317 0.278 331.934 0.298 344.213 0.269 333.550 0.299 343.977 0.269 333.551 0.299 345.265 0.177 331.716 0.197 Gv (MPa) 87.580 87.584 87.584 Gh (MPa) 94.981 96.741 96.738 94.871 104.221 87.583 95.688 131.390 143.943 131.388 142.149 175.026 143.939 131.389 143.937 142.327 155.246 131.389 143.197 179.177 195.406 179.175 193.603 179.177 195.403 179.177 195.401 194.102 210.849 179.180 I 95.565 59.674 65.702 59.671 65.013 59.674 65.720 59.674 65.516 64.622 70.912 59.673 66.104 127.373 139.700 127.374 137.909 127.373 139.694 127.374 139.692 137.962 150.630
PAGE 503
0\ .....:1 s (m) 0 250 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0 250 0.250 0.250 0.250 0.250 0.167 0.167 0.167 O"o (MPa) 0 345 0.345 0.345 0.345 0.345 0.345 0.345 0.345 0.345 0.345 0.345 0.345 0.345 0.069 0.069 0.069 0.167 0.069 0.167 0.069 0.167 0.069 0.167 0.069 0.167 0.069 0.167 0.069 0.167 0.069 0.167 0.069 0.167 0.069 0.167 0.069 0.167 0.069 0.167 0.069 Es (MPa) 320 480 480 480 480 480 480 650 650 650 650 650 650 70 70 70 70 70 70 150 150 150 150 150 150 225 225 225 Vs 0.3 0.3 0.3 0.3 0.3 0.2 0.3 0.3 0.3 0.3 0.3 0.2 0 3 0.3 0.3 0.3 0.3 0.2 0.3 0.3 0.3 0.3 0.3 0.2 0.3 0.3 0.3 0.3 Eg (MPa) 1100 1100 270 1100 1100 1100 1100 1100 270 1100 1100 1100 1100 1100 270 1100 1100 1100 llOOc:\ 1100 270 1100 1100 llOO 1100 1100 270 1100 Vg 0 2 0.4 0.4 0.4 0.4 0.4 0.2 0.4 0.4 0.4 0.4 0.4 0.2 0.4 0.4 0.4 0.4 0.4 0.2 0.4 0.4 0.4 0.4 0.4 0 2 0.4 0.4 0.4 friction I Eh (Mpa) 344 758 520.053 515.311 Vb 0.272 0.279 0.278 0.279 0.279 0.177 0.273 0.279 0 278 Ev (MPa) 333.002 498.795 497.152 498.794 498.794 496.666 498.219 674.317 672.703 674.315 674.315 677.840 673 789 72.005 72.191 71.941 Vv 0.299 0.296 0.299 0.296 0 296 0.199 0.300 0.297 0 299 Gv (MPa) 127.375 191.095 191.092 191.096 191.096 206.995 191.097 258.796 Gh (MPa) 139 119 208.316 206.514 208.313 208 312 224.763 208.486 281.217 0.5 I 520.051 0.25 520 047 514 618 1 I 513.956 1 701.773 258.799 279.406 0.5 0.25 0.5 0.25 0.5 0.25 0 5 696.978 701.771 0.279 701.770 0.279 701.815 0.185 701.272 0.277 79 942 0.269 78.161 0.279 79.087 0.270 0.297 258.796 281.215 0.297 258.795 281.214 0.199 280.336 303.507 0 298 258.799 281.425 0.284 26.989 3 i .324 0.285 26.989 31.252 0.287 26.989 31.628 78 787 0 272 71.908 0.286 26 989 31.579 79.277 0.179 71.300 0.183 29.234 33.751 79.412 0.267 72.457 0.283 26.989 31.598 166. 984 0 271 153.359 0.286 57.856 67.156 166.510 0.285 153 116 0 284 57 856 65 933 166.360 0.272 153.296 0.287 57.856 66.952 164.960 0.271 152.990 0.289 57.856 66.808 167 258 0.183 152.139 0.182 62.672 72.329 166.436 0.273 152.940 0.286 57.856 66.233 251.104 0.277 230.800 0.282 86.792 99.361 248.186 0.285 228.963 0.285 86.793 99.412 248.763 0.276 230.800 0.284 86.792 98.995
PAGE 504
..j:::.. 0\ 00 s (m) 0.167 0.167 0.167 0.167 0.167 0 .167 0.167 0 .167 0.167 0.167 0.167 0.167 0.167 0.167 0.167 0.167 0.167 0.167 0.167 0 .167 0.167 0.167 0.167 0.167 0.167 0.167 0.167 0.167 O'o (MPa) 0.069 0.069 0.069 0.069 0.069 0.069 0.069 0.069 0.069 0.172 0.172 0.172 0 .172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 0.172 Es. (MPa) 225 225. 225 300 300 300 300 300 300 100 100 100 100 100 100 220 220 220 220 220 220 330 330 330 330 330 330 450 Vs 0.3 0.2 0.3 0.3 0.3 0.3 0.3 0.2 0.3 0 3 0.3 0.3 0.3 0.2 0.3 0.3 0.3 0.3 0.3 0.2 0.3 0.3 0.3 0.3 0.3 0.2 0.3 0.3 Eg (MPa) 1100 1100 1100 1100 270 1100 1100 1100 1100 1100 270 1100 1100 1100 1100 1100 270 1100 1100 1100 1100 1100 270 1100 1100 1100 1100 1100 Vg 0.4 0.4 0.2 0.4 0.4 0.4 0.4 0.4 0.2 0.4 0.4 0.4 0.4 0.4 0 2 0.4 0.4 0.4 0.4 0.4 0.2 0.4 0.4 0.4 0.4 0.4 0.2 0.4 friction I Eh Vh (Mpa) o.25 1 25o oo9 0.276 0.188 0.5 0.25 0.5 0.25 0.5 0.25 1 1 0.5 0.25 1 254.479 250 537 0 279 331.704 0.277 330.589 0.285 330 940 0 277 330.214 0.276 331.989 0.184 331.230 0.278 112.076 0 272 110.720 0 278 113.393 0.274 112.488 0 273 111.942 0.177 112.739 0.267 245.247 0 278 242.742 244.872 0.284 0.277 244.569 0.277 246 .198 0.185 245.170 0 278 364.556 0 277 362 527 0.284 363 839 .. 0.276 363 .113 0.275 363.930 0.184 364 246 0 277 495.625 0.276 Ev (MPa) 230 797 227.674 229.972 306.702 304.811 306.695 306.689 303.068 305.957 103.025 Vv 0.283 0.183 0.284 0.284 0.286 0.285 0.286 0.185 0.285 0.284 102.497 0.285 102.974 0.282 103.100 0.283 101.914 0.180 102.755 0.285 225.623 0.282 224.003 225.617 225.611 222 .725 224.990 337.244 335.245 337.243 337.246 333.022 336.494 458.597 0.285 0.282 0.282 0.183 0.283 0.285 0.286 0.285 0.286 0.185 0 285 0.286 Gv (MPa) 86. 792 94.018 Gh (MPa) 98. 737 107.017 86.793 99 793 115.727 133.681 115. 728 130.947 115.727 133.678 115.727 133.677 125. 363 144.104 115. 727 132.008 38.542 44.707 38.543 44.289 38.543 44.708 38.542 44 .141 41.742 48 093 38 542 45. 052 84. 839 97.351 84 842 84. 840 96.304 96. 986 84.840 96.705 91.897 104. 830 84.841 97.724 127. 279 146.718 127. 280 143.978 127.279 146.715 127. 279 146.713 137.871 158.201 127. 279 144. 804 173. 577 198. 742
PAGE 505
..j::. 0\ 1.0 s (m) 0.167 0.167 0.167 0.167 0.167 0.167 0.167 0.167 0.167 0.167 0.167 0.167 0.167 0.167 0.167 0.167 0.167 0.167 0.167 0.167 0.167 0.167 0.167 0.167 0.167 0.167 0.167 0.167
PAGE 506
.....:J 0 0.345 /. 650 Vs 0.3 1100 0.2 friction Vb Vv 709.982 0.278 660.474 0.287 250.697 285.809
PAGE 507
B. Regression Analysis of Independent Parameters (Ev, vh, Vv, Gv, and Gh) 471
PAGE 508
B.l Plane Poisson's Ratio (vh) The matrix plot and the correlation matrix were obtain and shown in Figure B.l and Table B. I, respectively. A very strong linear relation can be observed from the matrix plot between vh (Y) and the Vs CX4). This result agreed with that obtained by the correlation matrix, where rvx4 was equal to 0.9905. Also, the correlation matrix showed that Y was also correlated to the Poisson ratio of reinforcement (X6) and friction coefficient (X 7 ) with less correlation factors. 472
PAGE 509
473
PAGE 510
.....:1 Poisshor y Spacing Xt Confining x2 Soilmod x3 Soilpois X. Geomod Xs Geopois Xti friccoef x1 Poisshor (Y) 1.0000 0.0677 0.0141 0.0105 0 0960 Table B.l "The correlation matrix ofEh and all X variables" Spacing Confining Soilmod Soilpois Geomod Geopois friccoef Xt. x2 x3 X. Xs x1 1.0000 0.0020 1.0000 0.0022 0.5316 1.0000 0.0100 0.0026 0.0052 1.0000 0.0118 0 0046 0.0093 0.0335 1 0000 I 0.0012 0.0043 0.0086 0.2494 0.0026 1.0000 0.1088 0.0034 0.0068 0.3173 0.0444 0.3302 1.0000
PAGE 511
B.l.l Model Selection for vh Using only first order terms in the model with the aid of the best subset algorithms, the following model alternatives were obtained and shown in Table B.2. This method was based on R2 criterion, but also showed for each of the best subsets the Cp and (MSE)0 5 The best two subsets for each number of variables were identified. The rightmost columns of the tabulation show the X variables in the subset. Table B.2 "output for Best two subsets for each subset size ofvh" # of Variable Rz Co (MSEt5 Predictors 1 98.1 346.2 0.0048324 Vs 1 9.7 38229.5 0.033395 f 2 98.5 179.1 0.0043032 V5 Eg 2 98.4 204.9 0.0043889 S V5 3 98.8 41.3 0.0038094 S, V5 Eg 3 98.5 169.9 0.0042700 E5 V5 Eg 4 98.9 32.0 0.0037701 S, E5 V5 Eg 4 98.9 38.0 0.0037932 S;cro V5 Eg 5 98.9 10.6 0.0036829 S, cro....Es......Ys Eg 5 98.9 32.0 0.0037665 S, E5 V5 Eg, Vg 6 98.9 10.6 0.0036790 S, cro....Es ......Ys...E!!..>. 6 98.9 10.9 0.0036807 S, cro, E5 V5 Eg, f 7 98.9 8 0.0036648 s, cro, Es, Vs, Eg, Vg, f From Table B.2, it was seen that the best subset, according to the R2 criterion, was the 4 variables and more. The R2 criterion value for all these models is 0.989, which is relatively high. This value stayed constant even when adding more variables. To limit the choices, it is helpful to look at the Cp value. This value should be small and near to the number of variables. For that reason, the model of the 5 variables (S, cro, E5 V5 Eg) was a good candidate since the Cp values was 10.6. Using the forward and backward stepwise regression, 5 steps were 475
PAGE 512
concluded and shown in Table B.3. In the first step of stepwise regression method, the variable Vs or X.. was chosen. This variable has the largest t* value, oft;= {bk }'and smallest Pvalue ofO.OO. The R2 resulted from the Vs variable s bk was 98.11% which is relatively high. However, Cp was very high and very far from the number of variables. In the second step, Eg, X5 variable with the second highest t* value of 11.07 was added to the model. The addition ofthe Eg variable increased the t* value ofEs to 174.14 and R2 to 98.5%, and reduced the Cp value into 179 .1. In the third step, S with the third largest t* value was added to the model. By adding the Eg variable, the t* value ofEs slightly increased to 196.59. As a result, the R2 increased and Cp decreased to 41.3. After that, Es was added to the model. The addition ofEs was not very useful, where R2 increased very slightly from 98.83% into 98.86%. In the last step, cro was added to the model. Again, this addition was not very important since R2 didn't significantly increase. Further addition of any variable didn't affect the t* value ofEs nor the R2 hence the analysis was terminated. 476
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Table B.3 "MINIT AB forward/backward stepwise regression output of vh" Step 1 2 3 4 5 Constant 0 0078328 00050123 0 0004047 0.0014164 0.0006412 vs (X4) 0.947452 0.9452 0.9447 0.9447 0.944745 t* 155.48 174.14 196.59 198.65 203.36 Pvalue 0.00 0.00 0.00 0.00 0.00 Eg(XS) 0.0000 0.0000 0.0000 0.00000425 t* 11.07 12.38 12.54 12 84 Pvalue 0.00 0.00 0.00 0 00 S (Xl) 0.00001 0.00001 0.00001358 t* 11.37 11.50 11.77 Pvalue 0 00 0 00 0 00 Es (X3) 0.00 0.00000665 t* 3.28 5.40 Pvalue 0.001 0.000 cro (X2) 0.008614 t* 4 .81 Pvalue 0.00 (MSE)05 0.00483 0;0043 0.00381 0.00377 0.00368 R2 98.11 98.5 98.83 98.86 98 .91 Mallow Cp 346.2 179.1 41.3 32.0 10.6 The results obtained from the stepwise regression method agreed with those obtained from best subset methodology As a result, the regression function of vh has the form shown in Equation B.1, and the analysis of variance for this regression was as shown in Table B.4. vh =0000641+0.944745xvs 0.00000425xEg 0.00001358 X S + 0.00000665 XEs 0.008614 X CJ'0 Where: vh =is the plane Poisson ratio of the geocomposite material (MPa) v5 =is the Poisson' ratio of soil Eg = is the modulus of elasticity of reinforcement (MPa) S =is the spacing between reinforcement layers (mm) 477 (B.l)
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E5 = is the modulus of elasticity of reinforcements (MPa) cro =is the confining pressure (MPa) In this regression model, b 1 =0.944745, b2 = 0.00000425, b3 = 000001358, b4 = 0.00000665, and b6 = 0.008614. This means the 1 unit increase in V5 while the other variables stayed constant would increase the vh by 0.944745 MPa, and so on for other variables. Table B.4 "ANOV A Table for vh regression model" Anal vsis of Variance Source Df ss MS F p Reg_ression 5 0.56916 0.11383 8392.35 0.00 Error 462 0.00627 0.00001 Total 467 0.57542 B.1.2 Inferences about Regression Parameters of vh The confidence limits for with 1a. = 0.99 confidence coefficient are b1 {1;np }{b,} Since b1 = 0.944745, s{b1}= 0.004646, and t(10.01/2; 468.:.3) = t(0.995;465) = 2.576, the confidence limit for was 0.944745 2.576(0.004646) => 0.93277 /3, 0.956713 478
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With confidence coefficient 0.99, it was estimated that confidence interval for b1 was somewhere between 0.93277 and 0.95671. Now, need to test ifv5 term should be dropped from the model (partial test). The general linear test statistic is F* = SSE(R)SSE(F) t(SSE(F)J dfR dfF dfF where SSE (R) = 0.567189, SSE(F) = 0.00627, dfR = n3 = 4683 = 465, dfp = n4 =464 Then F* = 41509.79. For this test the alternatives are Ho: P1 = 0 Ha: P1 ::j:. 0 and the decision rule is F(la;p 1,np) Conclude H0 IfF*> F(1a;p l,np), ConcludeR a Considering the decision rule, Ha was concluded because of the very large value ofF*. This indicated that J31 i.e. v5 shouldn't be dropped from the model. The confidence limits for J33 with 1a = 0.99 confidence coefficient are b2 {1;np }{b2 } Since b2 = 0.00000425, s{b2}= 0.00000033 and t(10.01/2; 4683) = t(0.995;465) = 479
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2.576, the confidence limit for P2 is 0.00000425 2.576(0.00000033) => 5.1*10{j p2 3.4*10{j With confidence coefficient 0.99, it was estimated that confidence interval for b3 was somewhere between 5.1 *10"6 and 3.4*106 Now, need to test ifEg term can be dropped from the model (partial test). The general linear test statistic is F* = SSE(R)SSE(F) 1 (SSE(F)) dfR dfF dfF Where, SSE (R) = 0.0085, SSE(F) = 0.00627 dfR = n3 = 468 3 = 465 dfF = n4 = 464 Then F* = 165 For this test the alternatives are Ho :/]2 =0 Ha: /]2 :;f! 0 and the decision rule is F(la;p l,np), Conclude H0 IfF*> F(la;p l,np); Conclude Ha Since F* = 165.027 > F(0.99;2,465) = 4.65, Ha was concluded. This indicated that P2 i.e. Eg shouldn't be dropped from the model. 480
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Sterm The confidence limits for B3 with 1a = 0.99 confidence coefficient are b3 {!;np }{b,) Since b3 = 0.00001358, s{b3}= 0.00000115, and t(10.0l/2; 4683) = t(0.995;465) = 2.576, the confidence limit for B3 was 0.00000425 2.576(0.00000115) => 7.21 *10{j fl3 1.287*10{j With confidence coefficient 0.99, it was estimated that confidence interval for b3 was somewhere between 7.21 1 o6 and 1.287* 1 o6 Now, need to test if S term should be dropped from the model (partial test). The general linear test statistic is F* = SSE(R)SSE(F) 1 (SSE(F)J djR dfF djF Where, SSE (R) = 0.00814, SSE(F) = 0.00627, dfR = n3 = 4683 = 465 dfF = n4 =464 Then F* = 138.38 For this test the alternatives are Ho: fl3 = 0 Ha: fJ3 '*0 and the decision rule is IfF*:::; F(1a;p l,n p), Conclude H0 IfF*> F(la;p 1,np), ConcludeHa 481
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Since F* = 138.38 > F(0.99;2,465) = 4.65, Ha was concluded. This indicated that i.e. S shouldn't be dropped from the model. The confidence limits for with 1a. = 0.99 confidence coefficient are b4 {1;np }{b4 } Since b4 = 0.00000665, s{b 4 }= 0.00000123, and t(10.01/2; 4683) = t(0.995;465) = 2.576, the confidence limit for was 0.00000665 2.576(0.00000123) '5:{34 '5.3.48*106 With confidence coefficient 0.99, it was estimated that confidence interval for b3 was somewhere between 9.818*106 and 3.48*106 Now, need to test ifEs term should be dropped from the model (partial test). The general linear test statistic is F* = SSE(R)SSE(F) 1(SSE(F)) dfR djF dfF where SSE (R) = 0.00666, SSE(F) = 0.00627 dfR = n3 = 468 3 = 465 dfF = n4 =464 Then F* = 28.86 For this test the alternatives are Ho :{34 =0 Ha :{34 :;t:O 482
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and the decision rule is F(la;p l,n p), Conclude H0 IfF*> F(la;p l,np), Conclude Ha Since F* = 28.86 > F(0.99;2,465) = 4.65, Ha was concluded. This indicated that i.e. Es shouldn't be dropped from the model. The confidence limits for with 1a = 0.99 confidence coefficient are b5 { 1; n p )s{b5 } Since bs = 0.008614 s{bs}= 0.001789, and t(l0.0112; 4683) = t(0.995;465) = 2.576, the confidence limit for is 0.008614 2.576(0.001789) 4.00*103 {35 9.644*103 With confidence coefficient 0.99, it was estimated that confidence interval for b3 was somewhere between 4.00*103 and 9.644*103 Now, need to test if S term should be dropped from the model (partial test). The general linear test statistic is F* = SSE(R)SSE(F) t(SSE(F)) dfR dfF dfF Where, SSE (R) = 0.00658, SSE(F) = 0.00627 dfR = n3 = 468 3 = 465 dfF = n4 =464 483
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Then F* = 22.94 For this test the alternatives are Ho :fls =0 Ha :/35 :;t:Q and the decision rule is IfF* :s; F(la;p l,np), Conclude H0 IfF*> F(la;p l,np), Conclude Ha Since F* = 22.94 > F(0.99;2,465) = 4.65, Ha was concluded. This indicated that i.e. cro cannot be dropped from the model. B.1.3 Diagnostic and Remedial Measure of vh The best model for vh included vs, Eg, S, E5 and, cr0 To asses the adequacy of the fit, and the assumption of constant variance, the residual plots must be prepared to suggest any modification if needed. The first graph to be considered was the residual versus the fitted values (Y) as shown in Figure B.2. This plot showed a very small deviation from response plane, which indicated that this linear relation was valid except for some outlier. 484
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Figure B.2 "The residual versus the fitted values of vh" To check if the residuals were normally distributed, the normal probability plot of the residual was plotted as shown in Figure B.3. This plot showed that the residuals were normally distributed because the pattern was almost linear with some departure from linearity. 485
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Figure B.3 "Normal probability plot ofthe residuals ofvh" To test whether there was a regression relation between the response Vh and the predictor variables (vs, Eg, S, Es, and, cro), the overall F test statistic is constructed: F* = MSR = 8392.35 MSE The alternatives of this test were: Ho :fJI =flz = ... =fJP =0 Ha :some fJk :1:O,k = 1, ... ,p and the decision rule was: F(la;p l,np), Conclude H0 IfF*> F(1a;p l,np), Conclude Ha Considering a. = 0.01, and knowing that n = 468 and p = 3, F(l0.01; 31,4683) = F(0.99; r2, 465)= 2.30 486
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Since F* = 8392.35 >> F = Ha was concluded that vh was related to (vs, Eg, S, Es, and, cro). B.1.4 Alternative Model of vh In this section, several alternatives were considered in addition to the one that was obtained. The choice of these alternatives was based on the beset subset method, and consequently the best alternative and its analysis of variance were as shown in Equation B.2 and Table B.5. vh = 0.00230+ 0.946vs 0.00000425Eg0.000.01358S + 0.00000665Es0.008614a0 + 0.00326v g Table B.5 "ANOV A Table for vh alternative model" Analysis of Variance Source Df ss MS F Regression 6 0.569182 0.094864 7008.59 Regression Error 461 0.00624 0.000014 Total 467 0.575422 B.2 Vertical Modulus of Elasticity (Ev) p 0.00 The matrix plot and the correlation matrix were obtain and shown in Figure B.4 and table B.6, respectively. A very strong linear relation can be observed from the matrix plot between Ev (Y) and the Es, X3). This result agreed with that obtained by the correlation matrix, where rYX3 was equal to 0.9998. Also, the correlation matrix showed that Y is also correlated to the confining pressure, X2 with less correlation factor. 487 (B.2)
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488
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Table B.6 "The correlation matrix ofEv and all X variables" Confining Soilmod Soilpois Geomod Geopois friccoef Xz x3 Ever 1:0000 Spacing fl flflfiO I 1.0000 0.0036 1.0000 +:a I 0.0024 0.5320 1.0000 00 1,0 Soilpois 0.0001 0.0094 0.0040 0.0054 1.0000 Geomod 0.0115 0.0132 0.0014 0.0088 0.0347 I 1.0000 Xs Geopois 0.0073 0.0006 0.0028 0.0084 0.2500 0.0038 1.0000 friccoef I _0 0097 0.1097 0.0053 0.0071 0.3182 0.0460 0.3312 I 1.0000
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B.2.1 Model selection for Ev Using only first order terms in the model with the aid of the best subset algorithms, the following model alternatives are obtained and shown in Table B.7. This method was based on R2 criterion, but also showed for each of the best subsets the Cp and (MSE)0 5 The best two subsets for each number of variables were identified. The rightmost column of the tabulation shows the X variables in the subset. Table B.7 "MINIT AB output for Best two subset results for each subset size ofEv" # of Variable C_p (MSE)u.!' Predictors 1 100.0 52.1 3.1763 Es 1 28.2 1021800.5 141.5 cro 2 100.00 13.9 3.051 Es, Ys 2 100.00 43.4 3.1463 Es Eg 3 100.00 3.6 3.0166 E5 V5 Eg 3 100.00 13.1 3.0473 S, Es, Ys 4 100.00 2.6 3.0101 S, 4 100.00 4.8 3.0172 5 100.00 4.1 3.0117 S, E5 V5 Vg, f 5 100.00 4.5 3.0129 s, cro, Es, Ys, Eg 6 100.00 6.0 3.0145 S, cro, Es' Ys, Eg, f 6 100.00 6.1 3.015 S, E5 V5 Eg, Yg, f 7 1oo.oo 8.0 3.0178 s, cro, Es, Ys, Eg, Vg, f From Table B.7, jt was observed that the best subset, according to the R2 criterion, is the 1 variable and more except for the model that only contains cro variable. The R2 criterion value for all these models is 1.00. This value stayed constant, even when adding more variables. For that reason, it is not recommended to investigate models with more than 3 variables. To limit the choices, it is helpful to look at the Cp This value should be small and near to the number of 490
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variables. For that reason, the models of the 3 variables (Es v5 and Eg) and the model of 4 variables (Es V5 Eg, S) were good candidates since the Cp values were 3.6 and 2.6, respectively. Using the forward and backward stepwise regression, 4 steps were concluded and shown in Table B.8. In the first step of stepwise regression method, the variable Es or X3 was chosen. This variable has the largest t* value, oft; = {bk } and smallest Pvalue ofO.OO. The R2 resulted from the Es s bk variable was 99.96% which is very high. Cp was far from the number of variables. In the second step, the v5 X1 variable with the second highest t* value of 6.27 was added to the model. The addition of the v5 variable increased the t* value ofEs into 1180.23. It slightly affected the value ofR2 and significantly reduced the Cp value to 13.9. In the third and the fourth steps, Eg and S were added. The addition of these variables didn't change the value ofR2 but reduced Cp to 2.6. Further addition of any variable didn't affect the t* value of Es nor the R2 hence the analysis was terminated. 491
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Table B.8 "MINIT AB forward/backward stepwise regression output of Ev" Step 1 2 3 4 Constant 1.119 5.694 6.317 5.764 Es (X3) 1.02068 1.02071 1.069 1.02069 t* 1134.44 1180.23 1194.44 1197.02 Pvalue 0.00 0.00 0.00 0.00 Vs (X4) 24.1 24.6 24.5 t* 6.27 6.46 6.46 Pvalue 0 00 0.00 0.00 Eg (X5) 0.00095 0.00096 t* 3.51 3.54 Pvalue 0.00 0.00 s (Xl) 0.00164 t* 1.73 Pvalue 0.083 (MSE)0 3.18 3.05 3.02 3.01 R2 99.96 99.97 99.97 99.97 MallowCp 52.1 13.9 3.6 2.6 The results obtained from the stepwise regression method agreed with those obtained from best subset methodology. As a result, the regression function ofEv had the form as shown in Equation B3, and the analysis of variance as shown in Table B.9. E = 5.7638 + 1.0207 E + 24.5345v + 0.001E 0.0016S v s s g Where: Ev = is the plane modulus of elasticity of the geocomposite material (MPa) Es = is the modulus of elasticity of soil (MPa) Vs =is the Poisson's ratio of soil S =is the spacing between reinforcement layers (mm) Eg = is the modulus of elasticity of reinforcements (MPa) 492 (B.3)
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Table B.9 "ANOVA Table for Ev regression model" Analysis of Variance Source Df ss MS F p Regression 4 12984467 3246117 358265.0 0.00 Regression Error 462 4186 9 Total 466 12988653 In this regression model, br=l.0207, b2 = 24.5345, b3 = 0.001, and b4 = 0.0016. This meant the 1 MPa increase in Es while the other constant stayed constant would increase the Ev by 1.0207 MPa. Also, the 1 unit increase in Ys while the other constants stayed constant would increase the Ev by 24.5345 MPa. The 1 MPa increase in Eg while the other constants stayed constant would increase the Eh by 0.0010 MPa. And finally, the 1 mm increase in spacing while all the other constants stayed constant would decrease Ev 0.0016 MPa. B.2.2 Diagnostic and Remedial Measure The best model for Ev included Es, v5 Eg, and S To asses the adequacy of the fit, and the assumption of constant variance, the residual plots must be prepared to suggest any modification if needed. The first graph to be considered is the residual versus the fitted values (Y) as shown in Figure 5.9. This plot suggested an increase in the residual deviation from response plane with increasing the level of ywith some outlier results. To examine the reason behind this deviation, the residual plots should be plotted against each of the predictor variables as shown in Figure B.S In Figure B 6, the plot of residuals against Es and S showed that the variance of the error terms varied with the level of that variable The other plots, against v5 and Eg; in the same figure didn't have that problem, in which the variance of the error terms didn't significantly vary with the level of that variable. Due to this variation, further investigation on these terms must be completed 493
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Figure B.S "Residual versus the fitted values ofEv" 494
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.......... ...... ... ..... ..... ... lL lUI"' ... '!!!IIJ*.. llii!!!!!!_MIII. 101!1. !llill!llilll'!.i!'J"'' t "f ............... ..... 495
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To check if the residuals were normally distributed, the normal probability plot of the residual was plotted as shown in Figure B. 7. This plot showed that the residuals were normally distributed because the patter was almost linear with some departure from linearity. Figure B.7 "Normal probability plot of the residuals ofEv'' To test whether there was a regression relation between the response Y h and the predictor variables E5 vs, Eg, and S the overall F test statistic is constructed: F* = MSR = 358265.00 MSE The alternatives of this test were H o : f3r = fJ2 = = f3 p = 0 Ha: some f3k O,k = l, ... ,p and the decision rule was 496
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F(1a;p 1,np), Conclude H0 IfF*> F(1a;p 1,np), ConcludeHa Considering a= 0.01, and knowing that n = 467 where 1 test have been dropped due to some errors in extracting the data and p = 4 F(l0.01 ; 41 4674) = F(0 99; 3 463) 2.30 Since F* = 358265.00 >> F = 2.6, Ha was conCluded that Eh was related toEs, vs, Eg, and S. B.2.3 Inferences about Regression Parameters of Ev Es term The confidence limits for J31 with 1a = 0.99 confidence coefficient are h1 {1;np }{b1 } Since bt = 1.02 07 s{bt}= 0.00085, and t(l0.0112; 4674) = t(0 995 ; 463) 2 576 the confidencelimit for f3t is 1.0207 2.576(0.00085) => 1.0185 1.02288 With confidence coefficient 0.99 it was estimated that confidence interval for bt was somewhere between 1.0185 and 1.02288. Now, need to test ifEs term should be dropped from the model (partial test). The general linear test statistic is 497
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F* = SSE(R)SSE(F) /(SSE(F)) djR djF djF where, SSE (R) = 12986911, SSE(F) = 4186 dfR = n3 = 46 73 = 464 dfp = n4 = 463 Then F* = 1435977.467 For this test the alternatives are Ho: /31 = 0 Ha: /31 :;t: 0 and the decision rule is F(1a;p 1 ,np), Conclude H0 IfF*> F(1a;p 1,np), Conclude Ha Considering the decision rule, Ha was concluded because of the very large value ofF*. This indicated that p1 Es shouldn't be dropped from the model. The confidence limits for P2 with 1a. = 0.99 confidence coefficient are b2 { 1 ; np }{b2 } Since b2 = 24.5345, s{b2}= 3.798, and t(10.01/2; 4683) = t(0.995;465)::::: 2.576, the confidence limit for P2 is 1.0207 2.576(3.798) 498
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=> 8.762948 fJ2 10.8043 With confidence coefficient 0.99, it was estimated that confidence interval for b2 was somewhere between 8.7629 and 10.8043. Now, need to test ifv5 term should be dropped from the model (partial test). The general linear test statistic is F* = SSE(R)SSE(F) 1(SSE(F)) djR djF dfF Where, SSE (R) = 4564, SSE(F) = 4186 dfR = n3 = 46 73 = 464 dfp = n4 = 463 Then, F* = 42.8 For this test the alternatives are Ho: /31 = 0 Ha: /32 'if. 0 and the decision rule is F(1a;p 1,np), Conclude H0 IfF*> F(la;p l,np), Conclude Ha Since F* = 42.8 > F(0.99;2,465) = 4.65, Ha was concluded. This indicated that P2 i.e. V5 shouldn't be dropped from the model. Egterm The confidence limits for PJ with 1a. = 0.99 confidence coefficient are ;np }{b3 } 499
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Since b3 = 0.001, s{b3}= 0.0002710, and t(10.0l/2; 4683) = t(0.995;465):::: 2.576, the confidence limit for is 0.001 2.576(0.000271) => 3.019*104 fJ3 1.698*103 With confidep.ce coefficient 0.99, it is estimated that confidence interval for b 3 is somewhere between 3.019*104 and 1.698*103 Now, need to test ifEg tenn can be dropped from the model (partial test). The general linear test statistic is F* = SSE(R)SSE(F) 1 (SSE(F)J dfR dfF dfF Where SSE (R) = 4299 SSE(F) 4186, = n3 = 46 7 3 = 464, dfF = n4 = 463 Then F* = 12.498 For this test the alternatives are Ho: /31 = 0 Ha: fJI :;t: Q and the decision rule is F(1a;p 1,np), Conclude H0 IfF*> F(la;p l,np), Conclude H a Since F* = 9.69799 > F(0.99;2,465):::: 4.65, Ha was concluded. This indicated that i.e. Eg shouldn't be dropped from the model. 500
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Sterm The confidence limits for B3 with 1a = 0.99 confidence coefficient are b4 {1;np }{b4 } Since b4 = 0.0016, s{b4}= 0.0009433, and t(10.0l/2; 4683) = t(0.995;465) 2.576, the confidence limit forB3 is 0.0016 2.576(0.0009433) => 4.029 1 o3 /34 8.2994 1 o4 With confidence coefficient 0.99, it was estimated that confidence interval for b3 was somewhere between 4.029*103 and 8.2994*104. Now, need to test ifS term can be dropped from the model (partial test). The general linear test statistic is .F* = SSE(R)SSE(F) 1 (SSE(F)J dfR dfF dfF Where, SSE (R) = 4213, SSE(F) = 4186, = n3 =46 73 =464, dfp = n4 = 463 Then F* = 2.986 For this test the alternatives are Ho: fJI = 0 Ha: fJI =I' 0 and the decision rule is F(la;p l,np), Conclude H0 IfF*> F(la;p l,np), Conclude Ha 501
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Since F* = 2.9863 < F(0.99;2,464)::::: 4.65, Ho was concluded. This indicated that B4 i.e. S can be dropped from the model. The difference was small and it is for this project intention to investigate the spacing effect so it was recommended not to drop the S term. The linear equation the linear regression equation of Ev when dropping the S term would have the form shown in Equation B.4 and the analysis of variance was as shown in Table B10. E = 6.32+1.02x E + 24.6+v + 0.000953x E v s s g Table B.lO "ANOVA Table for Ev regression model when dropping the S term" Analysis of Variance Source Df ss MS F p Regression 3 12984440 428147 475622.46 0.00 Regression Error 463 4213 9 Total 466 12988653 B.2.4 Alternative Model of Ev In this section, several alternatives were considered in addition to the one that was obtained. The choice of these alternatives was based on the beset subset method, and consequently the best alternative and its analysis of variance were as shown in Equation B.S, and Table B 11. E v =5.62+1.02Es +23.4vs +0.000937E g 0.448/ 502 (B.4) (B.5)
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Table B 11 "ANOVA Table for alternative model ofvh" Analysis of Variance Source Df ss MS F p Regression 4 1298447 3246112 356571.22 0.00 Regression Error 462 4206 9 Total 466 12988653 B.3 Vertical Poisson Ratio (vv) The matrix plot and the correlation matrix is obtain and shown in Figure B.8 and table B.12, respectively. A very strong linear relation was observed from the matrix plot between Vv (Y) and the vs, )4. This result agreed with that obtained by the correlation matrix, where ry)(4 was equal to 0.9732. Also, the correlation matrix showed that Y was also correlated to the Poisson's ratio of reinforcement (X6) and friction coefficient (X7 ) with less correlation factors. 503
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504
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Vl 0 Vl Poisshor (Y) Spacing x. Confining x2 Soilmod XJ Soilpois X. Geomod Xs Geopois friccoef x1 Poishor (Y) 1.0000 0.1307 0.0111 0.0050 0.9732 0.0638 0.2459 0.3295 Table B.12 "The correlation matrix of Vv with all X variables" Spacing Confming Soilmod Soilpois Geomod Geopois friccoef Xt x2 XJ X. Xs x1 1.0000 0.0016 1.0000 0.0012 0.5315 1.0000 0.0088 0.0024 0.0047 1.0000 0.0099 0.0043 0.0086 0.0327 1.0000 I 0.0064 0.0034 0.0068 0.2484 0.0064 1.0000 0.1106 0.0031 0.0063 0.3182 0.0433 0.3290 1.0000
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B.3.1 Model Selection Using only first order terms in the model with the aid of the best subset algorithms, the following model alternatives were obtained and shown in Table B.l2. This method was based on R2 criterion, but also showed for each of the best subsets the Cp and (MSE)0 5 The best two subsets for each number of variables were identified. The rightmost column of the tabulation shows the X variables in the subset. From Table B.l3, it was observed that the best subset, according to the R2 criterion, was the 3 variables and more. The R2 criterion value for all these models was 0.963, which was relatively high. This value stayed constant even when adding more variables. To limit the choices, it was helpful to observe at the Cp value. This value should be small and near to the number of variables. For that reason, the model of the 3 variables (S, v5 Eg) and 4 variables (S, v5 Eg, f) were good cap.didates since the Cp values were 3.7 and 4.6, respectively. Using the forward and backward stepwise regression, 3 steps were concluded and were as shown in Table B.14. 506
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Table B.13 "MINITAB output for best two subsets for each subset size of Vv" # ofVariable R:z Cp (MSEt' Predictors 1 94.7 197.1 0.0086173 Ys 1 10.6 10682.9 0.035384 f 2 96.2 13.3 0.073147 S, V5 2 94.8 186.4 0 0085428 V5 Eg 3 96.3 3.7 0.007232 S, Ys..__Eg 3 96.2 14.4 0.0073154 S, cro, Ys 4 96.3 4 6 0.0072316 S, v5..__Eg,_ f 4 96.3 4.8 0.0072328 s, cro Ys, Eg 5 96.3 5.3 0.007229 S, V5 Eg, Yg, f 5 96.3 5.7 0.0072323 S, cro, Ys, Eg, f 6 96.3 6.4 0 0072298 S, cro, v5 Vg, f 6 96.3 7.3 0.0072368 S, E5 V5 Eg, Vg, f 7 96.3 8.0 0.0072347 s, cro, Es, Ys, Eg, Vg, f Table B.14 "MINITAB forward/backward stepwise regression output of v v Step 1 2 3 Constant 0.0106364 0.00008253 0.00135041 vs (X4) 0.9922 0.9910 0.9899 t* 91.33 107.46 108 .51 Pva1ue 0.00 0.00 0.00 S(X1) 0.00003 0.00003 t* 13.48 13.60 Pvalue 0.00 0.00 Eg(X5) 0.000002 t* 3.42 Pvalue 0.001 (MSE)os 0.00862 0.00731 0.00723 94:71 96.2 97.29 Mallow Cp 197.1 13.3 3.7 507
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In the first step of stepwise regression method, the variable v5 X4 was chosen. This variable has the largest t* value, t; = {bk } and smallest Pvalue of 0.00. s bk The R2 resulted from the Vs variable was 94.71 which was relatively high . However, was very high and very far from the number of variables. In the second step, the S, X1 variable with the second highest absolute t* value of 13.48 was added to the model. The addition of the S variable increased the t* value of Vs to 107.46 and R2 to 96.2%, and significantly reduced the Cp value to 13.1. In the third step, Eg with the third largest t* value was added to the model. By adding the Eg variable, the t* value of v5 slightly increased to 108.51. As a result, the R2 increased to 97.29 and Cp decreased to 3. 7. Further addition of any variable didn't affect the t* value ofEs nor the R2 hence the analysis was terminated. The results obtained from the stepwise regression method agreed with those obtained from best subset methodology. As a result, the regression function of Vv has the form as shown in Equation B.6, and the analysis of variance of this model was as shown in Table B.l5. Vv = 0.001350 + 0.989925vs0.000030880.00000222Eg Where: Vv = is the vertical Poisson ratio of the geocomposite material (MPa) Vs = is the Poisson' ratio of soil S =is the spacing between reinforcement layers (mm) Eg = is the modulus of elasticity of reinforcement (MPa) In this regression model, b1=0.989925, b2 = 0.0000308, and b3 = 0.0000222. This means the 1 unit increase in v5 while the other variables stayed constant would increase the Vv by 0.989925 MPa, and so on. 508 (B.6)
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Table B.15 "ANOVA Table for Vv regression mode" Anal vsis of Variance Source Df ss MS F p Regression 3 0.62979 0.20993 4013.82 0.00 Regression Error 464 0.02427 0.00005 Total 467 0.65406 B.3.2 Inferences about Regression Parameters ..!$term The confidence limits for with 1a == 0.99 confidence coefficient are b1 {1;np }{b1 } Since b 1 == 0.9899, s{bi}= 0.009123, and t(10.0l/2; 4683) = t(0.995;465) = 2.576, the confidence limit for is 0.9899 2.576(0.009123) ==? 0.99664 /31 1.0134 With confidence coefficient 0.99, it was estimated that confidence interval for b1 was somewhere between 0.99664 and 1.0134. Now, need to test ifvs term should be dropped from the model (partial test). The general linear test statistic is F* = SSE(R)SSE(F) /(SSE(F)J dfR dfF dfF 509
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Where: (R) = 0.00001162, SSE(F) = 0.02427, dfR = n3 = 4683 = 465 dfF = n4=464 Then F* = 463.77 For this test the alternatives are Ho :Pt =0 Ha :Pt :;t:O and the decision rule is IfF*:::; F(1a;p 1,np), Conclude H0 IfF*> F(1a;p 1,np), Conclude H3 Considering the decision rule, Ha was concluded because of the very large value ofF*. This indicated that B1 i.e. v5 should not be dropped from the model. Sterm The confidence limits for BJ with 1a = 0.99 confidence coefficient are b2 { 1 ; np }{b2 } Since b2 = 0.00003, s{b2}= 0.00000227, and t(10.01/2; 4683) = t(0.995;465) = 2.576 the confidence limit for B2 is 0.00003 2.576(0.00000227) =>3.5847*105 ::;p2 :::;2.4152*105 With confidence coefficient 0.99 it was estimated that confidence interval for b3 was somewhere between 3.5847*105 and 2.4152*10 5 510
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Now, need to test ifS term should be dropped from the model (partial test). The general linear test statistic is F* = SSE(R)SSE(F) /(SSE(F)J djR dfF djF Where: SSE (R) = 0.03394, SSE(F) = 0.02427, dfR = n3 = 4683 = 465 dfF = n4 = 464 Then F* = 184.8 For this test the alternatives are Ho: /32 =0 Ha: {32 :;t: 0 and the decision rule is IfF* s F(1a;p l,np), Conclude H0 IfF*> F(la;p 1,np), Conclude Ha Since F* = 184.8 > = 4.65, Ha was concluded. This indicated that i.e. S shouldn't be dropped from the model. The confidence limits for with 1a = 0.99 confidence coefficient are b3 {1;np }{bJ Since b3 = 0.000002, s{b3}= 0.00000065, and t(l0.01/2; 4683) = t(0.995;465) = 2.576, the confidence limit for is 0.000002 2.576(0.000000065) 3.6744*106 s /33 s 3.256*107 511
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With confidence coefficient 0.99, it was estimated that confidence interval for b3 was somewhere between 3.6744*106 and 3.256*107 Now, need to test ifS term can be dropped from the model (partial test). The general linear test statistic is F* = SSE(R)SSE(F) /(SSE(F)J dfn dfF djF Where: SSE (R) = 0.02488, SSE(F) = 0.02427, dfR = n3 = 4683 = 465 dfF = n4 = 464 Then F* = 28.86 For this test the alternatives are Ho :f34 =0 Ha :{J4 :;tO and the decision rule is F(la;p 1,n p), Conclude H0 IfF*> F(1a;p 1,np), Conclude H a Since F* = 11.66213 > F(0.99;2,465) = 4.65, Ha was concluded. This indicated that J33, Es should not be dropped from the model. B.3.3 Diagnostic and Remedial Measure of Vv The best model for Vv included vs, S, and Eg: To asses the adequacy of the fit, and the assumption of constant variance, the residual plots must be prepared to suggest any modification if needed. The first graph to be considered is the 5i2
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residual versus the fitted values (Yl as shown in Figure B.9. This plot showed a very small deviation from response plane; all within 0.075 which indicated that this linear relation was valid. Figure B.9 "The residual versus the fitted values ofvv" To check if the residuals were normally distributed, the normal probability plot of the residual was plotted as shown in Figure B.l 0. This plot showed that the residuals were normally distributed because the patter was almost linear with some departure from linearity. To test whether there was a regression relation between the response vh and the predictor variables (vs, S, and Eg), the overall F test statistic is constructed: F*=MSR =4013.82 MSE The alternatives of this test were 513
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H 0 : fJI = /]2 = ... = fJ p = 0 Ha :some {Jk :;t: O,k = l, ... ,p and the decision rule was IfF* s F(la;p l,np), Conclude H0 IfF*> F(la;p l,np), Conclude Ha Considering a= 0.01, and knowing that n = 468 and p = 3, F(l0.01; 31 4683) = F(0.99; 2, 465)= 2.30 Since F* = 4013.82 >> F = 2.6, Ha was concluded that vh was related to (vs, S, and Eg) :: i t } ; 'l .t '".. .,._,..,t .. ... .... ;}'"""A ;;;..;.:oi..... _i'"" ., . : ... lj .. f ; l l } ) "! 1 1, ..w.).,: .w.. ... f.oc ......... _ ............... .... '"""'' ,,. ........ ..._,_,.. .... &,, ....... '"''""b.:.:.<. ) 'I 1 ) i... ., .. ,.,... ..... :e ... ...... _..._..J;...,.,.,. ..., ..... .;( ,. ...... .... _.h.= ... ) l '1 l :! .,!. .. .... ,,.; .if .;J,. ..,. ...... ,;. ... .) ... ;. ...... .,. . ,, I $ t t ::.... t;. ..... ,. .,. ,. .. ,.. "'' ':1! ... <.' .q. ... :l ..... '> "'' >""?. w "'',.... :.$ ."t... __. . ..., _,,. q l .......... ____ ..., ... .,oi J :::r.., E' ....,. ___ ._,:.,,...;.. .... .., .... ...,. ... t ...... .... N.... ......... ...... ,., ....... "! .J:.... ............ ...... r.JOO ... = ..:i "" '"" "" .... :. ,.t"" ... ..., _ J !; >'>... .. '"'"'' ... "'f'"''"''""'"'"'"'"!i'"'""""''"""''"''''i"" . ; .................. 7 ,... ..... t _., ......... ... ,.,. .J"" ... ,.._._;:.. .... ,.._;,., .. ...... ...;::_ .,._ .,_,_,_,,;:.,,.,..,. ..... .. ...;. ............. ,. ..... r...., _,,.,.,,.. ....... f ...... ... ...... $" .>,, __ _, ..!_,,_ J : 1 ,J :t ., < 't ... ::> .... ""''"  = : ...... "' ,r .... 'x' ... r l 'i::""'" 't: ,;}.... .... "'' .... _ "; .......... ..,. ).. ............ .... _.,. __ }... ,.._....._ ..
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B.3.4 Alternative Model of Vv In this section, several alternatives were considered in addition to the one that was obtained. The choice of these alternatives was based on the beset subset method, and consequently the best alternative and its analysis of variance were as shown in Equation B.7, and Table B.16. Vv = 0.00317 + 0.987vs 0.00003180.000002Eg 0.00123/ Table B.16 "ANOVA Table for alternative model ofvv" Anal vsjs of Variance Source Df ss MS F p Regression 4 0.62985 0.15746 3011.00 0.00 Regression Error 463 0.02421 0.00005 Total 467 0.65406 B.4 Vertical Shear Modulus (Gv)15 The matrix plot and the correlation matrix is obtain and shown in Figure B.11 and Table B.17, respectively. A very strong linear relation was observed from the matrix plot between Gv (Y) and the Es, X3. This result agreed with that obtained by the correlation matrix, where rYX3 was equal to 0.9962. Also, the correlation matrix showed that Y was also correlated to the cr0 X2 with a less correlation factor. 15 For this variable, 2 tests were not considered due to an error in data extracting 515 (B .7)
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VI 0\ Figure B.ll "Matrix plot of Gv with respect to all X variables"
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Vl .....) Gv (Y) Spacing x. Confining Xz Soilmod x3 Soilpois X. Geomod Xs Geopois friccoef x1 Gv (Y) 1.0000 0.0061 0.5286 0.9976 0.0591 0.0121 0.0192 0.0116 Table B.17 "The correlation matrix ofGvand all X variables" Spacing Confining Soilmod Soilpois Geomod Geopois friccoef x. Xz x3 X. Xs x1 1.0000 0.0050 1.0000 0.0053 0.5302 1.0000 0.0082 0.0038 0.0067 1.0000 0.0109 0.0079 0.0112 0.0337 1.0000 0.0006 0.0030 0.0070 0.2507 0.0027 1.0000 0.1056 0.0009 0.0036 0.3160 0.0443 0.3289 1.0000
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B.4.1Model Selection Using only the first order terms in the model with the aid of the best subset algorithms, the following model alternatives were obtained and shown in Table B.18. This method was based on R2 criterion, but also showed for each of the best subsets the Cp and (MSE)0 5 The best two subsets for each number of variables were identified. The rightmost column of the tabulation shows the X variables in the subset. Table B.18 "MINITAB out put for best two subsets for each subset size of G" v # of Variable Rl Co (MSE)u.5 Predictors 1 99.5 700.5 4.4744 Es 1 27.9 175132 54.990 cro 2 99.8 32.6 2.9156 Es, Vs 2 99.5 646.1 4.3692 E5 f 3 99.8 0.5 2.8161 3 99.8 34.1 2.9173 E5 V5 f 4 99.8 2.3 2.8184 S, Es.....Y.s.J 4 99.8 2.4 2.8211 S, Es, V5 Eg 5 99.8 4.2 2.8211 S, E5 V5 Eg, Vg 5 99.8 4.2 2.8212 S, E5 V5 V_g, f 6 99.8 6.0 2.8238 S, E5 V5 Eg, Vg, f 6 99.8 6.1 2.8241 S, cro, Es, Eg, Vg 7 99.8 8.0 2.8268 s, cro, Es, Vs, Eg, Vg, f From Table B.l8, it has seen that the best subset, according to the R2 criterion, was the 3 variables and more. The R2 criterion value for all these models was 0.998, which was relatively high. This value stayed constant even when adding more variables. To limit the choices, it is helpful to look at the Cp value. This value should be small and near to the number of variables. For that reason, the model of the 3 variables (S, Eg, V5 ) and 4 variables (S, Eg, v5 f) were good 518
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candidates since the Cp values were 1.1 and 2.2, respectively. Using the forward and backward stepwise regression, 3 steps were concluded and shown in Table B.19. Table B.19 "MINITAN forward/backward stepwise regression output of Gv" Step 1 2 3 Constant 0.05455 26.00779 27.76084 Es (X3) 0.39493 0.39479 0.39481 t* 311.11 477.26 494.15 Pvalue 0.00 0.00 0.00 Vs (X4) 92.3 92.5 t* 25.09 26.03 Pvalue 0.00 0.00 S(X1) 0.00517 t* 5.86 Pvalue 0.00 4.47 2.92 2.82 Rz 99.52 99.80 99.81 Mallow Cp 700.5 32.6 0.5 In the first step of stepwise regression method, the variable Es or X3 was chosen. This variable has the largest t* value, t; = {bk } and smallest Pvalue of 0.00. s bk The R2 resulted from the Es variable was 99.52% which was very high. However, Cp was very high and very far from the number of variables. In the second step, the v5 or X. variable with the second highest absolute t* value of 25.09 was added to the model. The addition of the v5 variable increased the t* value ofEs to 144.26 and R2 to 99.80 %, and significantly reduced the Cp value to 32.6. In the third step, S with the third larges t* value was added to the model. By adding the S variable, the t* value ofEs slightly increased to 494.15. As a 519
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result, the R 2 increased to 99.81% and Cp decreased to 0.5. Further addition of any variable didn't affect the t* value ofEs nor the R 2 hence the analysis was terminated. The results obtained from the stepwise regression method agreed with those obtained from best subset methodology. As a result, the regression function ofvv has the form as shown in Equation B.8, and the analysis of variance of this model was as shown in Table B.20. Gv = 27.7608+0.39485Es 92.4579vs 0.00528 Where: Gv is the vertical shear modulus of the geocomposite material (MPa) Es is the modulus of elasticity of soil (MPa) vs is the Poisson' ratio of soil S is the spacing between reinforcement layers (mm) In this regression model, br=0.0.39485, b2 = 92.4579, and b3 = 0.0052. This meant the 1 MPa increase in E5 while the other variables stayed constant would increase the Gv by 0.39485 MPa, and so on. Table B.20 "ANOVA Table for Gv regression model" Analysis of Variance Source Df ss MS F p Regression 3 1943396 647799 81683.69 0.00 Regression Error 462 3664 8 Total 465 1947060 To test whether there was a regression relation between the response vh and the predictor variables (vs, S, and Eg), the overall F test statistic is constructed: 520 (B.8)
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F* = MSR = 81683.69 MSE The alternatives of this test were: H 0 : PI = /32 = ... = p p = 0 Ha :some f3k :;t: O,k = 1, .. ,p and the decision rule was: IfF*::::;; F(la;p 1,np), Conclude H0 IfF*> F(la;p 1,np), Conclude Ha Considering a= 0.01, and knowing that n = 468 and p = 3, F(10.01; 31 4663) = F(0.99; 2, 463) 2.30 Since F* = 4013.82 >> F = 2.6, Ha was concluded that Vh was related to (Eg, vs, and S). B.4.2 Inferences about Regression Parameters of Gv The confidence limits for P1 with 1a = 0.99 confidence coefficient are b1 {1;np }{b1 } Since b1 = 0.3948, s{b1}= 0.000799, and t(l0.01/2; 4663) = t(0.995;463) 2.576, the confidence limit for P1 is 0.3948 2.576(0.000799) => 0.39274::::;; PI ::::;; 0.39685 521
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With confidence coefficient 0.99, it was estimated that confidence interval for b1 was somewhere between 0.39274 and 0.39685. Due to the very large correlation between Es and Gv, there was no need to check ifEs term should be dropped from the model (partial test). The confidence limits for with 1a = 0.99 confidence coefficient are b2 {1;np }{b2 } Since bz = 92.4579, s{bz}= 3.552, and t(10.01/2; 4663) = t(0.995;463) 2.576, the confidence limit for is 92.4579 2.576(3.552) => 101.6078 p2 83.308 With confidence coefficient 0.99, it is estimated that confidence interval for b3 is somewhere between 101.6078 and 83.308. Now, need to test if S term should be dropped from the model (partial test). The general linear test statistic is F* = SSE(R)SSE(F) /(SSE(F)J djR dfF djF where SSE (R) = 9037, SSE(F) = 3664, dfR = n3 = 4663 = 463 dfp = n4 = 462 Then F* = 677.5 For this test the alternatives are 522
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Ho: /32 = 0 Ha: /32 =1:0 and the decision rule is IfF*::; F(1a;p 1, np), Conclude H0 IfF*> F(la;p 1,np), Conclude Ha Since F* = 677.5 > F(0.99;2,463)::::;: 4.65, Ha was concluded. This indicated that P2 i.e. vg should not be dropped from the model. Sterm The confidence limits for P3 with 1a = 0.99 confidence coefficient are b3 {1;np }{b3 } Since b3 = 0.0052, s{b3}= 0.0008836, and t(l0.01/2; 4683) = t(0.995;465) = 2.576, the confidence limit for P3 is 0.0052 2.576(0.0008836) => 7.476*103::; /33::; 2.924*103 With confidence coefficient 0 99, it was estimated that confidence interval for b3 was somewhere between 7.476*103 and 2.924*103 Now, need to test if S term should be dropped from the model (partial test). The general linear test statistic is F* = SSE(R)SSE(F)1(SSE(F)J dfR djF dfF Where: 523
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SSE (R) = 3936, SSE(F) = 3664 dfR = n 3 = 4663 = 463 dfF = n4 = 462 Then F* = 34.29 For this test the alternatives are Ho :{34 =0 Ha :/] 4 :;t:Q and the decision rule is If F* F(la; p 1, n p ) Conclude H0 IfF*> F(la; p l,np) ConcludeH a Since F* = 34 296 > F(0.99;2,463) ;:::;; 4.65, Ha was concluded. This 'indicated that J33 i.e. S should not be dropped from the model. B.4.3 Diagnostic and Remedial Measure of Gv The best model for Gv included Es, vs, and S. To asses the adequacy of the fit and the assumption of constant variance, the residual plots must be prepared to suggest any modification if needed. The first graph to be considered was the residual versus the fitted values (Yl as shown in Figure 5.16. This plot showed a very small deviation from response plane which indicated that this linear relation was valid. 524
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Figure B.l2 "Residuals versus the fitted value of Gv'' To check if the residuals were normally distributed, the normal probability plot of the residual was plotted as shown in Figure 5.17. This plot showed that the residuals were normally distributed because the patter was linear with some departure from linearity. 525
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Figure B.13 "Normal probability plot of the residuals of Gv'' B.4.4 Alternative Model of Gv In this section, several alternatives were considered in addition to the one that was obtained. The choice of these alternatives was based on the beset subset method, and consequently the best alternative and its analysis of variance were as shown in Equation 5.53, and Table 5.27. Gv =27.8+0.395Es 92.5vs 0.00517S0.034J Table B.21 "ANOV A Table for alternative model of Gv" Analysis of Variance Source Df ss MS F p Regression 4 1943396 485849 61130.88 0.00 Regression Error 461 3664 8 Total 465 1947060 526 (B.9)
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B.5 Horizontal Shear Modulus (Gh) The matrix plot and the correlation matrix is obtain and shown in Figure B.l4 and Table B.22, respectively. A very strong linear relation can be observed from the matrix plot between Gx (Y) and the E5 X3. This result agreed with that obtained by the correlation matrix, where rYX3 was equal to 0.9968. Also, the correlation matrix showed that Y was also correlated to the cr0 X2 with less correlation factor. 527
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Vl N 00 Figure B.14 "Matrix plot ofGh with respect to all X variables"
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Vl ('..) \0 Gv {Y) Spacing Xt Confining x2 Soilmod XJ Soilpois Geomod Xs Geopois friccoef x1 Gv Spacing y Xt 1.0000 0.0432 1.0000 0.5307 0.0020 0.9968 0.0022 0.0562 0 0100 0.0206 0.0118 0.0216 0.0012 0.0141 0.1088 Table B.22 "Correlation matrix of Gh and all X variables" Confining Soilmod Soilpois Geomod Geopois friccoef x2 XJ Xs x1 1.0000 0.5316 1.0000 0 0026 0.0052 1.0000 0 0046 0.0093 0.0335 1.0000 0.0043 0.0086 0.2494 0.0026 1 0000 0.0034 0.0068 0 3173 0 0444 0.3302 1.0000 I
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B.S.l Model Selection Using only the first order terms in the model with the aid of the best subset algorithms, the following model alternatives were obtained and shown in Table B.23. This method was based on R2 criterion, but also showed for each of the best subsets the Cp and (MSE)0 5 The best two subsets for each number of variables were identified. The rightmost column of the tabulation shows the X variables in the subset. Table B.23 "MINITAB output for best two subsets for each size ofGh" #of Variable R:z Cp (MSE)u.s Predictors 1 99.4 956.3 5.9079 Es 1 28.0 158623.4 62.526 cro 2 99.6 382.0 4.5592 Es, Vs 2 99.5 584.5 5.0767 S, Es 3 99.8 19.4 3.4396 S, E5 V5 3 99.6 363.4 4.508 E5 V5 Eg 4 99.8 2.7 3.3755 S, Es....Y.s.....Eg 4 99.8 21.2 3.4427 S, O'Q, E5 V5 5 99.8 4.5 3.3784 S, Es....Y.s.....E_g,._ f 5 99.8 4.5 3.3786 S, cro,Es, V5 Eg_ 6 99.8 6.2 3.3810 S, Es, Vs, Eg, Vg, f 6 99.8 6.3 3.3815 S, O'Q, E5 V5 Eg, f 7 99.8 8.0 3.3841 S, O'Q, Es, V5 Eg, Vg, f From Table B.23 it has observed that the best subset, according to the R2 criterion, contained 4 variables and more. The R2 criterion value for all these models was 0.998. This value stayed constant even when adding more variables. To limit the choices, it is helpful to look at the Cp value. This value should be small and near to the number of variables. For that reason, the model ofthe 4 variables (S, E5 V5 Eg) was a good candidate since the Cp values were 2.7. Using 530
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the forward and backward stepwise regression, 4 steps were concluded and shown in Table 30. In the first step of stepwise regression method, the variable Es or X3 was chosen. This variable has the largest t* value, oft; = {bk } and smallest Ps bk value ofO.OO. The R2 resulted from the Es variable was 99.36% which was very high. However, Cp was very high and very far from the number of variables. In the second step, the v5 or X. variable with the second highest absolute t* value of 17.82 was added to the model. The addition of the v5 variable increased the t* value ofEs into 348.07 and R2 into 99.62 %, and significantly reduced the Cp value into 382.0. In the third step, S with the third largest* value was added to the model. By adding the S variable, the t* value of Es slightly increased into 470.02. As a result, the R2 increased into 99.78 %and Cp decreased into 19.4. In the last step, the Eg variable was added, by which the R2 increased very slightly and Cp reduced all the way to 2.7. Further addition of any variable .didn't affect the t* value ofEs nor the R2 hence the analysis was terminated. 531
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Table B.24 "MINITAB forward/backward stepwise regression output ofGh" Step 1 2 3 4 Constant 0.4032 29.3229 22.4182 21.5814 Es (X3) 0.44961 0.44949 0.44946 0.44942 t* 268.68 348.07 461.32 470.02 Pvalue 0.00 0.00 0.00 0.00 Vs (X4) 102.4 101.6 101.0 t* 17.82 23.43 23.72 Pvalue 0.00 0.00 0.00 S(Xl) 0.0202 0.0202 t* 18.79 19.09 Pvalue 0.00 0.00 Eg(X5) 0.00132 t* 4.33 Pvalue 0.00 (MSE)u5 5.91 4.56 3.44 3.38 R:l 99.36 99.62 99.78 99.79 Mallow Cp 956.3 382.0 19.4 2.7 The results obtained from the stepwise regression method agreed with those obtained from best subset methodology. As a result, the regression function of Gh has the form as shown in Equation B. I 0, and the analysis of variance of this model was as shown in Table B.25. Gh = 21.6 + 0.449Es IOI.Ovs + 0.0202S + 0.00132Eg Where, Gh =is the vertical shear modulus of the geocomposite material (MPa) Es = is the modulus of elasticity of soil (MPa) v5 = is the Poisson' ratio of soil S =is the spacing between reinforcement layers (mm) Eg = is the shear modulus of reinforcement layers (MPa) 532 (B.IO)
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In this regression model, br=0.449, bz = 101.0, b3 = 0.0202, and b4 = 0.00132. This meant the 1 MPa increase in Es while the other variables stayed constant would increase the Gh by 0.449 MPa, etc Table B 25 "ANOVA Table for Gh regression model" Anal vsis of Variance Source Df ss MS F p Regression 4 2530670 632667 55525.1 0.00 Regression Error 463 5276 11 Total 467 2535945 To test whether there was a regression relation between the response vh and the predictor variables ( vs, S, and Eg), the overall F test statistic is constructed: F* = MSR = 55525.1 MSE The alternatives of this test were: H o : /31 = fJ2 = .. = f3 P = 0 Ha: some f3k '* O,k = l, ... ,p and the decision rule was: F(la;p l,np), Conclude H0 IfF*> F(la;p l,np), Conclude H3 Considering a= 0.01, and knowing that n = 468 and p = 3, F(10.01; 31,4663) = F(0.99; 2, 463) 2.30 Since F* = 55525.1 >> F = 2.6, Ha was concluded that Gh was related to (Es, vs, S, and Eg). 533
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B.5.2 Inferences about Regression Parameters of Gh The confidence limits for p, with 1a. = 0.99 confidence coefficient are b1 {1;np }{b1 } Since b, = 0.449, s{b,}= 0.000956, and t(10.01/2; 4683) = t(0.995;465) 2.576, the confidence limit for p1 is 0.449 2.576(0.000956) => 0.4465 fJ. 0.45146 With confidence coefficient 0.99, it is estimated that confidence interval for b1 is somewhere between 0.0.4465 and 0.4515. Due to the very large correlation between Es and Gh, there was no need to examine ifEs term could be dropped from the model (partial test). The confidence limits for P2 with 1a. = 0.99 confidence coefficient are b2 { 1; np }{b2 } Since b2 = 101.00, s{b2}= 4.258, and t(10.01/2; 4683) = t(0.995;465) 2.576, the confidence limit for P2 is 101.00 2.576(4.258) 534
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=> 111.969 s /32 s 90.0314 With confidence coefficient 0.99, it was estimated that confidence interval for b3 was somewhere between 111.969 and 90.0314. Now, need to test if S term can be dropped from the model (partial test). The general linear test statistic is F* = SSE(R)SSE( F) /(SSE(F)) djR djF dfF Where, SSE (R) = 11685, SSE(F) = 5276, dfR = n3 = 4683 = 465 dfp = n4 = 464 Then F* = 563.64 For this test the alternatives are Ho: flz = 0 Ha: /32 ;!:. 0 and the decision rule is IfF* s F(la;p l,np), Conclude H0 IfF*> F(la;p l,np), Conclude Ha Since F* = 563.64 > F(0.99;2,463) 4.65, Ha was concluded. This indicated that i.e. Eg shoudnot be dropped from the model. Sterm The confidence limits for !33 with 1a = 0.99 confidence coefficient are 535
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Since b3 = 0.0202, s{b3}= 0.001057, and t(l0.01/2; 4683) = t(0.995;465) = 2.576, the confidence limit for is 0.0202 2.576(0.001057) => 0.0174:::; /33 :::; 0.0229 With confidence coefficient 0.99, it was estimated that confidence interval for b3 was somewhere between 0.0174 and 0.0229. Now, need to test if S term can be dropped from the model (partial test). The general linear test statistic is F* = SSE(R)SSE(F) 1(SSE(F)J dfR djF dfF Where: SSE (R) = 9429, SSE(F) = 5276, dfR = n 3 = 4683 = 465 dfF = n4 = 464 Then F* = 365.23 For this test the alternatives are Ho: /33 = 0 Ha: /33 :1:0 and the decision rule is IfF*:::; F(la;p l,np), Conclude H0 IfF*> F(la;p l,np), ConcludeHa Since F* = 365.23 > F(0.99;2,463) 4.65, Ha was concluded. This indicated that Le. S should not be dropped from the model. 536
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The confidence limits for with 1a. = 0.99 confidence coefficient are b4 {1;np }{b4 } Since b4 = 0.00132, s{b4} = 0.0003035, and t(10.0l/2; 4683) = t(0.995;465) = 2.576, the confidence limit for is 0.00132 2.576(0.0003035) => 5.3814*104:;, /34:;, 2.1018*103 With confidence coefficient 0.99, it was estimated that confidence interval for b 4 was somewhere between 5.3814*104 and 2.1018*10"3 Now, need to test ifEg term could be dropped from the model (partial test). The general linear test statistic is F* = SSE(R)SSE(F) /(SSE(F)J djR djF djF Where, SSE (R) = 5490, SSE(F) = 5276, dfR = n3 = 468 3 = 465 dfp = n4 = 464 Then F* = 18.82 For this test the alternatives are Ho :fJ4. = 0 Ha :/34 :;t:O and the decision rule is IfF* <5, F(la;p 1,np), Conclude H0 IfF*> F(la;p l,np), Conclude Ha Since F* = 18.82 > F(0.99;2,463) 4.65, Ha was concluded. This indicated that Eg should not be dropped from the model. 537
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B.5.3 Diagnostic and Remedial Measure of Gh The best model for Gh included Es, Vs, S, and Eg. To asses the adequacy of the fit, and the assumption of constant variance, the residual plots must be prepared to suggest any modification if needed. The first graph to be considered is the residual versus the fitted values (Yl as shown in Figure B.l5. This plot didn't show any systematic deviation from response plane, which indicated that this linear relation was adequate with nearly constant error of variance. Figure 8.15 "The residual versus the fitted values of Gh" To check if the residuals were normally distributed the normal probability plot of the residual was plotted as shown in Figure B.16. This plot showed that the residuals were normally distributed because the patter was almost linear with some departure from linearity. 538
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Figure B.16 "Normal probability plot of the residuals of Gh" B.5.4 Alternative Model of Gh In this section several alternatives were considered in addition to the one that was obtained. The choice of these alternatives was based on the beset subset method and consequently the best alternative and its analysis of variance were as shown in Equation 5.11, and Table B.26. Gh =22.3+0.449Es 101vs60.0202S+0.113/ Table B.26 "ANOV A Table for alternative model of Gh" Anal "Sis of Variance Source Df ss MS F p Regression 4 2530456 632614 53360.4 0.00 Reg!ession Error 463 5489 12 Total 467 2535945 539 (B.ll)
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