SEISMIC INTERFACE STABILITY OF COMPOSITE DAMS
by
Fatih Onciil
B.S., Middle East Technical University, 1992
M.S., University of Colorado at Denver, 1995
A thesis submitted to the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Civil Engineering
2001
This thesis for the Doctor of Philosophy
degree by
Fatih Onciil
has been approved
by
John R. Mays
Brian T. Brady
v/ / cx /
Date
Onciil, Fatih (Ph.D., Civil Engineering)
SEISMIC INTERFACE STABILITY OF COMPOSITE DAMS
Thesis directed by Professor Nien-Yin Chang
ABSTRACT
Many earth darns failed or suffer great distress in the past earthquakes. The potential
grave consequence of dam failures has led research scientists to pay great attention to
the investigation of the potential causes of dam failures and the formulation of the
mechanism for their prevention. Because of the site condition and the need for the
power generation, many dam are of composite type. This means the body of the dam is
composed of a concrete dam, embankment wing dams and the transitional (or
wrapped-around) section. There are over one hundred composite dams in the world that
are higher than 100 feet and some of them are located in seismically active areas. Thus,
it is important to study the seismic safety of composite dams.
Though much research effort has been devoted to the seismic dam safety in the last
three decades, the seismic soil-concrete interface stability in a composite dam has
received insignificant attention. The late Professor H. B. Seed pointed out the need for
the investigating the seismic interface stability in one of its research memorandum.
Then, in early 1980s, the Earthquake Engineering and Geoscience Division.
Geotechnical and Structural Laboratory, the Waterways Experiment Station, the U.S.
Army Corps of Engineers began the study of the seismic stability of the soil-concrete
interface stability. The research was hindered by the lack of appropriate computer code.
The University of Colorado at Denver (UCD) was involved in this initial effort. Initially,
FLUSH developed at the UC Berkeley was used in the study and the potential for the
interface separation was confirmed. However, the evaluation of the size and depth of
separation was not possible, until the availability of NIKE3D computer code developed
at the Lawrence Livermore National Laboratory (LLNL) through the collaborative
agreement between LLNL and UCD.
NIKE3D computer code has the interface formulation needed for the assessment of the
interface behavior under seismic load and it has been used extensively in this study. Its
effectiveness in evaluating the interface behavior was calibrated using the dynamic
centrifuge test on a retaining wall model with a dry sand backfill. The centrifuge test
was performed at the University of Colorado at, Boulder (Stadler, 1996). The excellent
agreement between the centrifuge model test and the numerical analysis using NIKE3D
confirms the validity of the NIKE3D as a study tool for the soil-structure interface
behavior under seismic shaking.
Extensive numerical analyses were performed to study the seismic interface stability of
composite dams. Numerous plane-strain analyses were performed to assess the effect of
the slope of embankment and concrete dam on the size and depth of interface separation
of composite dams with height ranging from 100 to 400 feet during strong seismic
shaking. The results show the interface behavior is strongly affected by the slope of the
respective side of the dam and the height of the dam. Six 3-dimensional analyses were
also performed. The significant difference in 2-D and 3-D results indicate the need for
3-D analysis for accurate assessment of the performance of the interface in a composite
dam.
IV
Both elastic model and Ramberg-Osgood model were used in the investigation of the
soil model effect. It was determined that an appropriate soil model is critical to the
effective assessment of the interface behavior.
The effect of the imposition of all three components of earthquake ground motion on the
interface performance was also studied. It was found that while the effect on the
transverse interface was pronounced, its effect on the upstream and downstream
interfaces was not.
While the study is extensive, it only touches the surface of the problem. Much research
work is needed to critically examine the seismic stability of interface and the overall
stability of composite dams.
i
This abstract accurately represents the content of the candidates thesis. I recommend
its publication.
Signed
v
DEDICATION
Dedicated
to my wife EBRU ONCUL and, my son OMER
ACKNOWLEDGMENTS
I would like to take this opportunity to express my gratitude to those who have
contributed to the development of this thesis.
Firstly, I am greatly indebted to Prof. Nien-Yin Chang for his guidance, support,
friendship, and valuable comments on each part of the thesis. Having the opportunity to
work with Prof. N. Y. Chang was an experience that helped me gain broader
perspective on many areas.
I also wish to thank the members of the committee, Prof. H.Y. Ko. Prof. Benson Shing,
Prof. John Mays, and Prof. Brian Brady for their helpful comments and suggestions.
The guidance and support from Dr. Mike Puso is greatly acknowledged. Phone
conversations during the numerical analysis part of the thesis was very helpful.
The financial support of the Dumlupinar University in Kutahva, Turkey is gratefully
acknowledged. Special thanks to Colorado Department of Transportation (CDOT) for
providing financial support during the final years of my doctoral study.
I also would like to thank Dr. Liu Jiang who helped me on using NIKE3D, and
numerous occasions. Special thanks to Jan Chang for his encouragement, and valuable
friendship during my study. My thanks are extended to Mr. Cengiz Alkan, and Mr.
Kevin Lee for their support.
I profoundly thank my parents, sister, brother, and their families for providing me moral
support and for their patience. Finally, but not the least, I am grateful to my dear wife
Ebru and my son Orner for their continuous love and patience during my Ph.D. study.
CONTENTS
Figures ....................................................................... xviii
Tables.......................................................................... xliv
Chapter
1. Introduction.................................................................. 1
1.1 Problem Statement............................................................. 1
1.2 Significance of Research ..................................................... 3
1.3 Research Objectives .......................................................... 5
1.4 Research Approach............................................................. 6
2. Literature Survey............................................................. 9
2.1 Introduction.................................................................. 9
2.2 Existing Composite Dams....................................................... 9
2.3 Literature Review on Analysis of Composite Dams ............................. 10
2.4 Soil-Concrete Interface Models............................................... 14
2.4.1 Stiffness Approach........................................................ 15
IX
2.4.2 Constrained Approach
20
2.5 A Brief Review of Computer Codes...................................... 24
2.5.1 Interface formulation in FLAC ....................................... 27
2.6 Constitutive Models .................................................... 29
2.6.1 Equivalent Linear Elastic Model...................................... 29
2.6.2 Hyperbolic Model..................................................... 30
2.6.3 Mohr-Coulomb Model................................................... 32
2.6.4 Ramberg-Osgood Model................................................. 32
2.6.5 Other Models......................................................... 38
2.7 Pore Pressure Generation Models......................................... 38
2.7.1 Uncoupled Seeds Method.............................................. 39
2.7.2 Partially Coupled Model of Finn...................................... 40
2.7.3 Fully Coupled Method: Biot Theory.................................. 42
2.8 Summary and Conclusion ................................................. 44
3. Finite Element Analysis Codes........................................... 45
3.1 Introduction............................................................ 45
3.2 TRUEGRID................................................................ 45
x
3.3 NIKE3D................................................................... 47
3.3.1 Element Library ......................................................... 48
3.3.2 Solution Strategy........................................................ 48
3.3.3 Element Formulation...................................................... 49
3.3.4 Material Models ......................................................... 53
3.3.5 Interface models ........................................................ 54
3.3.6 Penalty Formulation in NIKE3D ........................................... 54
3.3.7 Damping Methods.......................................................... 56
3.3.8 Algorithmic Damping ..................................................... 57
3.3.9 Rayleigh Damping......................................................... 59
3.4 GRIZ .................................................................... 61
3.5 Summary and Conclusions.................................................. 61
4. Centrifuge Testing and A Retaining Wall Simulation with NIKE3D .......... 63
4.1 Introduction............................................................. 63
4.2 Dynamic Centrifuge Testing............................................... 64
4.2.1 Geometry Considerations ................................................. 67
4.2.2 Material Considerations.................................................. 67
xi
4.2.3 Concrete................................................................. 67
4.2.4 Soil and water .......................................................... 69
4.2.5 Boundary Effects......................................................... 70
4.2.6 Foundation-composite dam interaction..................................... 70
4.2.7 Box walls and the model.................................................. 70
4.2.8 Input ground motion...................................................... 71
4.2.9 Instrumentation ......................................................... 71
4.2.10 A Recommended Dynamic Composite Dam Centrifuge Test..................... 72
4.3 Stadlers Centrifuge Testing............................................. 73
4.3.1 Loading Sequence......................................................... 73
4.3.2 Wall Geometry and Material Properties ................................... 75
4.4 Finite Element Model of the Retaining Wall............................... 76
4.5 Input Motion............................................................. 82
4.6 Centrifuge Test Results.................................................. 83
4.7 NIKE3D Simulation and Comparisons........................................ 88
4.8 Conclusions............................................................. 104
5. Selection of Numerical Analysis Parameters .............................. 106
xii
5.1 Introduction
106
5.2 Input Motion........................................................... 106
5.3 Finite Element Models.................................................. 109
5.3.1 Two-Dimensional Model.................................................. Ill
5.3.2 Three-Dimensional Model................................................ 113
5.4 Material Parameters.................................................... 117
5.4.1 Ramberg-Osgood Model Parameters........................................ 120
5.4.2 Linear Elastic Model Parameters........................................ 124
5.5 Interface Treatment.................................................... 129
5.6 Rayleigh Damping Parameters............................................ 130
5.7 Summary and Conclusions................................................ 134
6. Two-Dimensional Parametric Study........................................ 136
6.1 Introduction........................................................... 136
6.2 Natural Vibration Characteristics...................................... 137
6.3 2-D Parametric Finite Element Analysis................................. 150
6.3.1 FE Analysis Procedure.................................................. 150
6.3.2 Separation Calculations................................................ 152
xi n
6.3.3 Acceleration Calculations
155
6.3.4 Interface Pressure Calculations........................................... 155
6.3.5 FE Analysis Results....................................................... 156
6.4 Effect of Vertical Component of Ground Motion........................... 192
6.5 Interpretation of 2-D FE Results ....................................... 200
6.5.1 Maximum Separation ....................................................... 204
6.5.2 Maximum Acceleration and RMS Acceleration Ratios........................ 206
6.5.3 Interface Pressure........................................................ 208
6.6 Summary and Conclusions................................................... 212
7. Elastic Three-Dimensional Finite Element Analysis.......................... 214
7.1 Introduction.............................................................. 214
7.2 Natural Vibration Characteristics......................................... 215
7.3 3-D Finite Element Analysis Results..................................... 216
7.4 Effect of Nonlinear Soil Model............................................ 235
7.5 Effect of Vertical and Longitudinal Components of Ground Motion ........ 239
7.5.1 Case I: Linear Elastic Soil Model....................................... 239
7.5.2 Case II: Ramberg-Osgood Nonlinear Soil Model............................ 243
7.6 Interpretation of Results
247
7.6.1 Maximum Acceleration........................................................ 247
7.6.2 Maximum Separation ......................................................... 247
7.6.3 Separation Depth............................................................ 248
7.7 Summary and Conclusions..................................................... 248
8. A Case Study.................................................................. 251
8.1 Introduction................................................................ 251
8.2 Folsom Dam.................................................................. 251
8.2.1 FLUSH Analyses.............................................................. 251
8.2.2 NIKE2D Analyses............................................................. 252
8.3 Comparisons ................................................................ 255
8.4 Summary and Conclusions..................................................... 258
9. Statistical Assessment of Parametric Effects................................ 259
9.1 Introduction................................................................ 259
9.2 Basic Statistics and Polynomial Models...................................... 260
9.3 Univariate Statistical Assessment........................................... 262
9.3.1 Variation and Trend......................................................... 262
xv
9.3.2 Effects of 6 Variation................................................. 263
9.3.3 Effects of 4> Variation................................................ 264
9.3.4 Effects of mu Variation ............................................... 265
9.3.5 Effects of md Variation................................................ 266
9.3.6 Polynomial Best Fits................................................... 275
9.4 Multivariate Statistical Assessment.................................... 283
9.5 Summary and Conclusions................................................ 290
10. Summary, Conclusions and Recommendations................................. 292
10.1 Summary................................................................ 292
10.2 Conclusions............................................................ 295
10.3 Recommendations for Further Study...................................... 297
Notation Index............................................................... 299
Appendix
Appendix..................................................................... 301
A. Figures................................................................... 301
A.l Free Vibrational Behavior................................................ 301
A.2 2-D FE Results........................................................... 304
xvi
A.2.1 H=100ft
304
A.2.2 H=200ft............................................................... 325
A. 2.3 H=300ft.............................................................. 344
A.2.4 H400ft............................................................... 363
A.3 Polynomial Best Fit Curves.............................................. 382
A.4 Bar Charts ............................................................. 397
A. 5 Pressure Profiles .................................................... 410
B. Tables.................................................................. 418
References
423
FIGURES
Figure
1.1 Folsom Dam, California........................................................ 2
1.2 A typical soil-concrete interface of a composite dam.......................... 3
1.3 The U.S earthquake hazard map and the number of composite darns by
state (circled numbers)....................................................... 4
2.1 Relative displacement element [17]........................................... 16
2.2 Interface element representation in separated state [39]..................... 18
2.3 A typical interface and zone dimensions of FLAC interface.................... 27
2.4 Deviatoric stress vs strain relation for hyperbolic model.................... 31
2.5 Mohr Coulomb yield surface................................................... 33
2.6 Typical loading and unloading curve for Ramberg-Osgood Model................. 35
3.1 Analysis sequence............................................................ 46
3.2 A simple block part and cylindrical projection surfaces [83]................. 47
3.3 Elements available in NIKE3D [47]............................................ 49
3.4 Contact of node m with segment of jk [21].................................... 54
xviii
4.1 A representative composite dam cross-section in a centrifuge test container.
(Not drawn to scale.)...................................................... 66
4.2 Instrumentation plan for centrifuge testing................................... 73
4.3 Simplified composite dam section.............................................. 74
4.4 Sequence of static and dynamic events......................................... 75
4.5 Centrifuge test box configuration. [73]....................................... 76
4.6 Accelerometer and LVDT locations [73]......................................... 77
4.7 Earth pressure transducer (EP#), and strain gage (SG#) instrumentation
on inboard face of the model wall. [73] ................................... 77
4.8 Wall mesh and selected nodes for data presentation............................ 79
4.9 Input Motion for Static Loading............................................... 82
4.10 Typical Prescribed and Measured Horizontal Input Motions [73]................. 83
4.11 Input Motion for NIKE3D Analysis. Digitized from measured input motion
in Figure 4.10............................................................. 84
4.12 Measured Input Motion and step numbers for data gathering [73]................ 85
4.13 Dynamic Profiles at Step 249 [73]............................................. 86
4.14 Dynamic Profiles at Step 637 [73]............................................. 87
4.15 Representation of deflection, v, and depth, z................................. 89
xix
4.16 Wall deflection comparison for Â£ = 0%.......................................... 93
4.17 Wall deflection comparison for Â£ = 10%......................................... 94
4.18 Wall deflection comparison for Â£ = 20%......................................... 95
4.19 Earth pressure comparison tor Â£ = 0%........................................... 96
4.20 Earth pressure comparison for Â£ = 10%.......................................... 97
4.21 Earth pressure comparison for Â£ = 20%.......................................... 98
4.22 RMS acceleration ratio comparison for Â£ = 0%................................. 99
4.23 RMS acceleration ratio comparison for Â£ = 10%............................... 100
4.24 RMS acceleration ratio comparison for Â£ = 20%............................... 101
4.25 Location of soil columns for RMS comparisons.................................. 102
4.26 RMS acceleration ratio in soil along the interface, 1.3 and 6 inches away
from the interface for Â£ = 0. 10, and 20%................................. 103
4.27 Deflection of wall top at different time steps for different damping ratios. . 105
5.1 (a) Transverse, (b)Longitudinal, and (c)Vertical components of Koyna Dam
Earthquake Record, 1967...................................................... 108
5.2 Horizontal component of Koyna Dam Earthquake Record and its accelera-
tion response spectrum...................................................... 110
5.3 Hypothetical composite dam.................................................... Ill
xx
5.4 The mesh for plane strain analysis......................................... 112
5.5 A complete 3-D model of a composite dam.................................... 114
5.6 3-D Mesh of the composite dam and dimensions............................... 115
5.7 Representations of upstream, transverse, and downstream soil-concrete in-
terface areas............................................................ 116
5.8 FE Mesh of upstream embankment and clay core............................... 117
5.9 FE Mesh of downstream embankment and clay core............................. 118
5.10 FE Mesh of concrete monolith and upstream interface....................... 118
5.11 FE Mesh of concrete monolith and upstream interface....................... 119
5.12 Idealized cross-section of a composite dam with soil layers................ 119
5.13 fc2max effect on maximum U/S interface acceleration........................ 121
5.14 k2max effect on maximum U/S interface separation........................... 121
5.15 Damping ratio, Â£ and G/Gmax vs shear strain curves for clay.............. 125
5.16 Damping ratio, Â£ and GjGmax vs shear strain curves for sand.............. 125
5.17 Comparison of interface pressures obtained from the interface element ap-
proach................................................................... 131
5.18 Comparison of interface pressures obtained from the nodal force approach. . 131
5.19 Damping ratio effec t on maximum accelerations............................. 133
5.20 Damping ratio effect on maximum separation................................... 133
5.21 k2max and damping ratio effects on maximum separation and separation
depth....................................................................... 135
6.1 Natural frequencies for all cases of 8,4),mu, and md (H = 100)............. 138
6.2 Natural frequencies for all cases of 8,4>,mu, and md. (H=200).............. 139
6.3 Natural frequencies for all cases of 6,
6.4 Natural frequencies for all cases of 9,,mu, and md (H=400)................ 140
6.5 Natural frequencies for all cases of 8, (p,mu,md and all heights........... 140
6.6 First five mode shapes of three different configurations. H=100 ft........... 142
6.7 First five mode shapes of three different configurations, H=400 ft........... 142
6.8 Unit base acceleration impulse............................................... 143
6.9 Crest horizontal displacement (a, b, and c) and acceleration (d, e, and f)
time histories of concrete monolith without U/S and D/S soil embankments
for 0%, 5%, and 10% damping ratios respectively........................... 146
6.10 Horizontal displacement and acceleration time histories of soil and concrete
nodes at the crest, Â£ = 0% (H=100)........................................ 147
6.11 Horizontal displacement and acceleration time histories of soil and concrete
nodes at the crest, Â£ = 5% (H=100)........................................ 147
6.12 Horizontal displacement and acceleration time histories of soil and concrete
nodes at the crest, Â£ = 10% (H=100)....................................... 148
xxii
6.13 Horizontal displacement and acceleration time histories of soil and concrete
nodes at the crest, Â£ = 0% (H=200)........................................ 148
6.14 Horizontal displacement and acceleration time histories of soil and concrete
nodes at the crest, Â£ = 5% (H=200)........................................ 149
6.15 Horizontal displacement and acceleration time histories of soil and concrete
nodes at the crest, Â£ = 10% (H=200)....................................... 149
6.16 Sequence of static and dynamic events used in NIKE3D Analysis............... 151
6.17 Location of selected nodes for data presentation............................ 153
6.18 Illustration of SEPARATION for soil-concrete interface areas................ 154
6.19 Sample time histories of soil node located at the crest of the dam (H=100 ft). 158
6.20 Sample time histories of soil node located at the crest of the dam (H=400 ft). 159
6.21 A shared legend for figures showing interface performance................... 161
6.22 Representation of height, z/H............................................... 161
6.23 Representation of separation depth, d/H..................................... 162
6.24 Max. Separation, Max. Acceleration, and Nodal Interface Stresses (Max.,
Static, Min.) along the UPSTREAM Interface due to change of 8 (in de-
grees) (H=100) .................................................... 164
6.25 (Continued from previous figure) Max. Separation. Max. Acceleration,
and Nodal Interface Stresses (Max., Static, Min.) along the UPSTREAM
Interface due to change of 8 (in degrees) (H=1()0) .................. 165
xxiii
6.26 Collective graphs of Max. Separation, Acceleration, and interface pressures
along the UPSTREAM Interface at all 9's (H=100) .................... 166
6.27 Max. Separation, Max. Acceleration, and Nodal Interface Stresses (Max.,
Static, Min.) along the DOWNSTREAM Interface due to change of 9 (in
degrees) (H=100) .............................................. 167
6.28 (Continued from previous figure) Max. Separation. Max. Acceleration, and
Nodal Interface Stresses (Max., Static, Min.) along the DOWNSTREAM
Interface due to change of 9 (in degrees) (H=100) ............. 168
6.29 Collective graphs of Max. Separation, Acceleration, and interface pressures
along the DOWNSTREAM Interface at all O's (H = 100)................. 169
6.30 Max. Separation, Max. Acceleration, and Nodal Interface Stresses (Max.,
Static, Min.) along the UPSTREAM Interface due to change of 9 (in de-
grees) (H=200) ................................................ 170
6.31 (Continued from previous figure) Max. Separation, Max. Acceleration,
and Nodal Interface Stresses (Max., Static, Min.) along the UPSTREAM
Interface due to change of 9 (in degrees)(H=200) .............. 171
6.32 Collective graphs of Max. Separation, Acceleration, and interface pressures
along the UPSTREAM Interface at all 0s (H = 200) .................. 172
6.33 Max. Separation. Max. Acceleration, and Nodal Interface Stresses (Max.,
Static, Min.) along the DOWNSTREAM Interface due to change of 9 (in
degrees)(H=200) ............................................... 173
xxiv
6.34 (Continued from previous figure) Max. Separation, Max. Acceleration, and
Nodal Interface Stresses (Max., Static, Min.) along the DOWNSTREAM
Interface due to change of 9 (in degrees) (H=200) ............ 174
6.35 Collective graphs of Max. Separation, Acceleration, and interface pressures
along the DO WNSTREAM Interface at all 9's (H=200)................. 175
6.36 Max. Separation, Max. Acceleration, and Nodal Interface Stresses (Max.,
Static, Min.) along the UPSTREAM Interface due to change of 6 (in de-
grees) (H300) ............................................... 176
6.37 (Continued from previous figure) Max. Separation, Max. Acceleration,
and Nodal Interface Stresses (Max., Static, Min.) along the UPSTREAM
Interface due to change of 9 (in degrees) (H=300) ............ 177
6.38 Collective graphs of Max. Separation, Acceleration, and interface pressures
along the UPSTREAM Interface at all 9s (H=300) ................... 178
6.39 Max. Separation, Max. Acceleration, and Nodal Interface Stresses (Max.,
Static, Min.) along the DOWNSTREAM Interface due to change of 9 (in
degrees) (H=300).............................................. 179
6.40 (Continued from previous figure) Max. Separation, Max. Acceleration, and
Nodal Interface Stresses (Max., Static, Min.) along the DOWNSTREAM
Interface due to change of 9 (in degrees) (H=300) ............ 180
6.41 Collective graphs of Max. Separation, Acceleration, and interface pressures
along the DOWNSTREAM Interface at all 9's (H=300).................. 181
XXV
6.42 Max. Separation. Max. Acceleration, and Nodal Interface Stresses (Max.,
Static, Min.) along the UPSTREAM Interface due to change of 8 (in de-
grees) (H=400) ................................................ 182
6.43 (Continued from previous figure) Max. Separation, Max. Acceleration,
and Nodal Interface Stresses (Max., Static, Min.) along the UPSTREAM
Interface due to change of 6 (in degrees) (H=400) .............. 183
6.44 Collective graphs of Max. Separation, Acceleration, and interface pressures
along the UPSTREAM Interface at all 8's (H=400) .................... 184
6.45 Max. Separation, Max. Acceleration, and Nodal Interface Stresses (Max.,
Static, Min.) along the DOWNSTREAM Interface due to change of 8 (in
degrees) (H=400)...................................................... 185
6.46 (Continued from previous figure) Max. Separation, Max. Acceleration, and
Nodal Interface Stresses (Max., Static, Min.) along the DOWNSTREAM
Interface due to change of 8 (in degrees) (H=400) .............. 186
6.47 Collective graphs of Max. Separation, Acceleration, and interface pressures
along the DOWNSTREAM Interface at all 0s (H=400)................... 187
6.48 Max. Separation and Separation Depth of U/S interface vs 8 and Height. . 188
6.49 Max. Acceleration Ratio and Max. RMS Acceleration Ratio of U/S inter-
face vs 8 and Height.................................................. 189
6.50 Max. Separation and Separation Depth of D/S interface vs 8 and Height. . 190
6.51 Max. Acceleration Ratio and Max. RMS Acceleration Ratio of D/S inter-
face vs 8 and Height.................................................. 191
xxvi
6.52 The effect of 6 on minimum and maximum interface pressures along the
U/S interface.................................................................... 192
6.53 The effect of 6 on minimum and maximum interface pressures along the
D/S interface.................................................................... 193
6.54 Collective graphs of minimum an maximum pressures versus 6 for both U/S
and D/S interfaces............................................................... 194
6.55 Maximum separation, acceleration, and interface stress distribution for both
upstream and downstream interfaces under ground shaking with vertical
and transverse components (H=100)................................................ 196
6.56 Maximum separation, acceleration, and interface stress distribution for both
upstream and downstream interfaces under ground shaking with vertical
arid transverse components (H=200)............................................... 197
6.57 Maximum separation, acceleration, and interface stress distribution for both
upstream and downstream interfaces under ground shaking with vertical
and transverse components (H=300)................................................ 198
6.58 Maximum separation, acceleration, and interface stress distribution for both
upstream and downstream interfaces under ground shaking with vertical
and transverse components (H=400)................................................ 199
6.59 Representation of typical maximum separation response of U/S and D/S
interfaces....................................................................... 201
6.60 Representation of typical maximum acceleration response of U/S and D/S
interfaces....................................................................... 202
xxvii
6.61 Representation of typical interface pressure response of U/S and D/S inter-
faces.......................................................................... 203
6.62 Comparison of maximum acceleration ratio with pure soil embankment and
composite section.............................................................. 209
6.63 Comparison of RMS acceleration ratio with pure soil embankment and com-
posite section........................................................................ 210
7.1 First mode shape of 300 ft high composite dam................................ 217
7.2 Max. Separation along (a) U/S and (b) D/S interfaces at maximum cross-
section for all heights (in ft)................................................ 219
7.3 Normalized Max. Separation along (a) U/S and (b) D/S interfaces at max-
imum cross-section for all heights............................................. 220
7.4 Max. Acceleration ratio along (a) U/S and (b) D/S interfaces at maximum
cross-section for all heights.................................................. 221
7.5 Surface and contour plots of maximum separation (in ft) for upstream soil-
concrete interface area with elastic soil model (H=100ft)................... 223
7.6 Surface and contour plots of maximum separation (in ft) for transverse soil-
concrete interface area with elastic soil model (H=100ft)................... 224
7.7 Surface and contour plots of maximum separation (in ft) for downstream
soil-concrete interface area with elastic soil model (H=100ft).............. 225
7.8 Surface and contour plots of maximum separation (in ft) for upstream soil-
concrete interface area with elastic soil model (H=200ft)...................... 226
xxviii
7.9 Surface and contour plots of maximum separation (in ft) for transverse soil-
concrete interface area with elastic soil model (H=200ft)..................... 227
7.10 Surface and contour plots of maximum separation (in ft) for downstream
soil-concrete interface area with elastic soil model (H=200ft)................ 228
7.11 Surface and contour plots of maximum separation (in ft) for upstream soil-
concrete interface area with elastic soil model (H=300ft)..................... 229
7.12 Surface and contour plots of maximum separation (in ft) for transverse soil-
concrete interface area with elastic soil model (H=300ft)..................... 230
7.13 Surface and contour plots of maximum separation (in ft) for downstream
soil-concrete interface area with elastic soil model (H=300ft)................ 231
7.14 Surface and contour plots of maximum separation (in ft) for upstream soil-
concrete interface area with elastic soil model (H=400ft)..................... 232
7.15 Surface and contour plots of maximum separation (in ft) for transverse soil-
concrete interface area with elastic soil model (H=400ft,).................... 233
7.16 Surface and contour plots of maximum separation (in ft) for downstream
soil-concrete interface area with elastic soil model (H=400ft)................ 234
7.17 Surface and contour plots of maximum separation (in ft) for upstream soil-
concrete interface area with R-0 soil model (H=400ft)......................... 236
7.18 Surface and contour plots of maximum separation (in ft) for transverse soil-
concrete interface area with R-O soil model (H=400ft)......................... 237
xxix
7.19 Surface and contour plots of maximum separation (in ft) for downstream
soil-concrete interface area with R-0 soil model (H=400ft)................. 238
7.20 Surface and contour plots of maximum separation (in ft) for upstream soil-
concrete interface area with elastic soil model and all three ground motion
components (H=400ft)....................................................... 240
7.21 Surface and contour plots of maximum separation (in ft) for transverse soil-
concrete interface area with elastic soil model and all three ground motion
components (H=400ft)....................................................... 241
7.22 Surface and contour plots of maximum separation (in ft) for downstream
soil-concrete interface area with elastic soil model and all three ground
motion components (H=400ft)................................................ 242
7.23 Surface and contour plots of maximum separation (in ft) for upstream soil-
concrete interface area with R-0 soil model and all three ground motion
components (H=400ft)....................................................... 244
7.24 Surface and contour plots of maximum separation (in ft) for transverse soil-
concrete interface area with R-0 soil model and all three ground motion
components (H=400ft)................................................. 245
7.25 Surface and contour plots of maximum separation (in ft) for downstream
soil-concrete interface area with R-0 soil model and all three ground motion
components (H=400ft)................................................. 246
8.1 FLUSH output of superimposed max. & min. normal stresses along the
upstream interface of Left Wing Dam of Folsom Dam for hinge k roller
conditions.(Tension is negative) ................................................. 253
xxx
8.2 Top: Finite element mesh of NIKE2D at the time of maximum separation,
Middle: Separation time history of node 444 for original and scaled ground
motions, Bottom: Maximum separation versus depth for original and scaled
ground motions........................................................... 256
8.3 Maximum Normal Stress Distribution along the interface.................... 257
9.1 Effect of 0 on U/S response, H=100 ft..................................... 275
9.2 Effect of 6 on D/S response, H=100 ft..................................... 277
9.3 Effect of
9.4 Effect of 4> on D/S response, H=100 ft..................................... 278
9.5 Effect of mu on U/S response, H=100 ft.................................. 278
9.6 Effect of mu on D/S response, H=100 ft.................................. 279
9.7 Effect of md on U/S response, H=100 ft.................................. 279
9.8 Effect of md on D/S response, H=100 ft.................................. 280
A.l Horizontal displacement and acceleration time histories of soil and concrete
nodes at the crest, Â£ = 0% (H=300)....................................... 302
A.2 Hor izontal displacement and acceleration time histories of soil and concrete
nodes at the crest, f = 5% (H=300)....................................... 302
A.3 Horizontal displacement and acceleration time histories of soil and concrete
nodes at the crest, Â£ = 10% (H=300)...................................... 303
xxxi
A.4 Horizontal displacement and acceleration time histories of soil and concrete
nodes at the crest, Â£ = 0% (H=400)................................... 305
A.5 Horizontal displacement and acceleration time histories of soil and concrete
nodes at the crest, Â£ = 5% (H=400)................................... 305
A.6 Horizontal displacement and acceleration time histories of soil and concrete
nodes at the crest, Â£ = 10% (H=400).................................. 306
A.7 Max. Separation, Max. Acceleration, and Nodal Interface Stresses (Max.,
Static, Min.) along the UPSTREAM Interface due to change of 0 (in de-
grees) (11 ---100)............................................... 307
A.8 (Continued from previous figure) Max. Separation. Max. Acceleration,
and Nodal Interface Stresses (Max., Static, Min.) along the UPSTREAM
Interface due to change of 0 (in degrees) (H=100).......................... 308
A.9 Collective graphs of Max. Separation, Acceleration, and interface pressures
along the UPSTREAM Interface at all 0s (H=100)...................... 309
A.10 Max. Separation, Max. Acceleration, and Nodal Interface Stresses (Max.,
Static, Min.) along the DOWNSTREAM Interface due to change of 0 (in
degrees) (H=100)................................................. 310
A.11 (Continued from previous figure) Max. Separation, Max. Acceleration, and
Nodal Interface Stresses (Max., Static, Min.) along the DOWNSTREAM
Interface due to change of 0 (in degrees) (H=100).......................... 311
A. 12 Collective graphs of Max. Separation, Acceleration, and interface pressures
along the DOWNSTREAM Interface at all 0s (H=100)................... 312
xxxii
A. 13 Max. Separation, Max. Acce., and Nodal Interface Stresses (Max., Static,
Min.) along the UPSTREAM Interface due to change of mu (H = 100). . . 313
A. 14 (Continued from previous figure) Max. Separation, Max. Acceleration,
and Nodal Interface Stresses (Max., Static, Min.) along the UPSTREAM
Interface due to change of m,u (H=100)....................................... 314
A. 15 Collective graphs of Max. Separation, Acceleration, and interface pressures
along the UPSTREAM Interface at all mus (H=100)....................... 315
A.16 Max. Separation, Max. Acce., and Nodal Interface Stresses (Max., Static,
Min.) along the DOWNSTREAM Interface due to change of mu (H=100) . 316
A. 17 (Continued from previous figure) Max. Separation, Max. Acceleration, and
Nodal Interface Stresses (Max., Static, Min.) along the DOWNSTREAM
Interface due to change of mu (H=100)........................................ 317
A. 18 Collective graphs of Max. Separation, Acceleration, and interface pressures
along the DOWNSTREAM Interface at all mus (H=100)..................... 318
A. 19 Max. Separation, Max. Acce., and Nodal Interface Stresses (Max., Static,
Min.) along the UPSTREAM Interface due to change of md (H=100). . . 319
A.20 (Continued from previous figure) Max. Separation, Max. Acceleration,
and Nodal Interface Stresses (Max., Static, Min.) along the UPSTREAM
Interface due to change of md (H=100)........................................ 320
A.21 Collective graphs of Max. Separation, Acceleration, and interface pressures
along the UPSTREAM Interface at all mds (H=100)....................... 321
xxxiii
A.22 Max. Separation, Max. Acceleration, and Nodal Interface Stresses (Max.,
Static, Min.) along the DOWNSTREAM Interface due to change of md
(H=100)...................................................... 322
A.23 (Continued from previous figure) Max. Separation, Max. Acceleration, and
Nodal Interface Stresses (Max., Static, Min.) along the DOWNSTREAM
Interface due to change of rnd (H = 100)............................... 323
A.24 Collective graphs of Max. Separation, Acceleration, and interface pressures
along the DOWNSTREAM Interface at all mds (H=100)............... 324
A.25 Max. Separation, Max. Acceleration, and Nodal Interface Stresses (Max.,
Static, Min.) along the UPSTREAM Interface due to change of 4> (in de-
grees) (H=200)............................................... 326
A.26 (Continued from previous figure) Max. Separation, Max. Acceleration,
and Nodal Interface Stresses (Max., Static, Min.) along the UPSTREAM
Interface due to change of (in degrees) (H=200).................... 327
A.27 Collective graphs of Max. Separation, Acceleration, and interface pressures
along the UPSTREAM Interface at all <^s (H=200)................... 328
A.28 Max. Separation, Max. Acceleration, and Nodal Interface Stresses (Max.,
Static, Min.) along the DOWNSTREAM Interface due to change of (in
degrees) (H=200)............................................. 329
A.29 (Continued from previous figure) Max. Separation, Max. Acceleration, and
Nodal Interface Stresses (Max., Static, Min.) along the DOWNSTREAM
Interface due to change of o (in degrees) (H200)...................... 330
xxxiv
A.30 Collective graphs of Max. Separation, Acceleration, and interface pressures
along the DOWNSTREAM Interface at all d>s (H=200).................. 331
A.31 Max. Separation, Max. Acceleration, and Nodal Interface Stresses (Max.,
Static, Min.) along the UPSTREAM Interface due to change of mu (H=200). 332
A.32 (Continued from previous figure) Max. Separation, Max. Acceleration,
and Nodal Interface Stresses (Max., Static, Min.) along the UPSTREAM
Interface due to change of mu (H=200)..................................... 333
A.33 Collective graphs of Max. Separation, Acceleration, and interface pressures
along the UPSTREAM Interface at all mu's (H=200).................... 334
A.34 Max. Separation, Max. Acceleration, and Nodal Interface Stresses (Max.,
Static, Min.) along the DOWNSTREAM Interface due to change of mu
(H=200)......................................................... 335
A.35 (Continued from previous figure) Max. Separation, Max. Acceleration, and
Nodal Interface Stresses (Max., Static, Min.) along the DOWNSTREAM
Interface due to change of rrm (H=200).................................... 336
A.36 Collective graphs of Max. Separation, Acceleration, and interface pressures
along the DOWNSTREAM Interface at all mus (H=200).................. 337
A.37 Max. Separation, Max. Acceleration, and Nodal Interface Stresses (Max.,
Static, Min.) along the UPSTREAMlnXexia.ee due to change of md (H=200). 338
A.38 (Continued from previous figure) Max. Separation. Max. Acceleration,
and Nodal Interface Stresses (Max., Static, Min.) along the UPSTREAM
Interface due to change of md (H=200)..................................... 339
xxxv
A.39 Collective graphs of Max. Separation, Acceleration, and interface pressures
along the UPSTREAM Interface at all mds (H=200).................... 340
A.40 Max. Separation, Max. Acceleration, and Nodal Interface Stresses (Max.,
Static, Min.) along the DOWNSTREAM Interface due to change of md
(11=200)....................................................... 341
A.41 (Continued from previous figure) Max. Separation, Max. Acceleration, and
Nodal Interface Stresses (Max., Static, Min.) along the DOWNSTREAM
Interface due to change of md (H=200)..................................... 342
A.42 Collective graphs of Max. Separation, Acceleration, and interface pressures
along the DOWNSTREAM Interface at all mds (H=200).................. 343
A.43 Max. Separation, Max. Acceleration, and Nodal Interface Stresses (Max.,
Static, Min.) along the UPSTREAM Interface due to change of
grees) (H=300)........................................................ 345
A.44 (Continued from previous figure) Max. Separation, Max. Acceleration,
and Nodal Interface Stresses (Max., Static, Min.) along the UPSTREAM
Interface due to change of
A.45 Collective graphs of Max. Separation, Acceleration, and interface pressures
along the UPSTREAM Interface at all (p's (H=300).................... 347
A.46 Max. Separation. Max. Acceleration, and Nodal Interface Stresses (Max.,
Static, Min.) along the DOWNSTREAM Interface due to change of (p (in
degrees) (H=300)............................................... 348
xxxvi
A.47 (Continued from previous figure) Max. Separation, Max. Acceleration, and
Nodal Interface Stresses (Max., Static, Min.) along the DOWNSTREAM
Interface due to change of 0 (in degrees) (11=300)......................... 349
A.48 Collective graphs of Max. Separation, Acceleration, and interface pressures
along the DOWNSTREAM Interface at all 0s (H=300)................... 350
A.49 Max. Separation, Max. Acceleration, and Nodal Interface Stresses (Max.,
Static, Min.) along the UPSTREAM Interface due to change of mu (H=300). 351
A.50 (Continued from previous figure) Max. Separation. Max. Acceleration,
and Nodal Interface Stresses (Max., Static, Min.) along the UPSTREAM
Interface due to change of mu (H=300)...................................... 352
A.51 Collective graphs of Max. Separation, Acceleration, and interface pressures
along the UPSTREAM Interface at all mus (H=300).................... 353
A.52 Max. Separation, Max. Acceleration, and Nodal Interface Stresses (Max.,
Static, Min.) along the DOWNSTREAM Interface due to change of mu
(H=300)......................................................... 354
A.53 (Continued from previous figure) Max. Separation, Max. Acceleration, and
Nodal Interface Stresses (Max., Static, Min.) along the DOWNSTREAM
Interface due to change of mu (H=300)...................................... 355
A.54 Collective graphs of Max. Separation, Acceleration, and interface pressures
along the DOWNSTREAM Interface at all mus (H=300).................. 356
A.55 Max. Separation. Max. Acceleration, and Nodal Interface Stresses (Max.,
Static, Min.) along the UPSTREAM Interface due to c hange of mat (H=300). 357
xxxvii
A.56 (Continued from previous figure) Max. Separation, Max. Acceleration,
and Nodal Interface Stresses (Max., Static, Min.) along the UPSTREAM
Interface due to change of md (H=300)........................................ 358
A.57 Collective graphs of Max. Separation, Acceleration, and interface pressures
along the UPSTREAM Interface at all md's (H=300)....................... 359
A.58 Max. Separation, Max. Acceleration, and Nodal Interface Stresses (Max.,
Static, Min.) along the DOWNSTREAM Interface due to change of md
(H-300).................................................................. 360
A.59 (Continued from previous figure) Max. Separation, Max. Acceleration, and
Nodal Interface Stresses (Max., Static, Min.) along the DOWNSTREAM
Interface due to change of md (H=300)........................................ 361
A.60 Collective graphs of Max. Separation, Acceleration, and interface pressures
along the DOWNSTREAM Interface at all mils (H=300).................... 362
A.61 Max. Separation, Max. Acceleration, and Nodal Interface Stresses (Max.,
Static, Min.) along the UPSTREAM Interface due to change of 4> (in de-
grees) (H=400).................................................... 364
A.62 (Continued from previous figure) Max. Separation, Max. Acceleration,
and Nodal Interface Stresses (Max., Static, Min.) along the UPSTREAM
Interface due to change of 0 (in degrees) (H=400)............................ 365
A.63 Collective graphs of Max. Separation, Acceleration, and interface pressures
along the UPSTREAM Interface at all 4>'s (H=400)...................... 366
xxxviii
A.64 Max. Separation, Max. Acceleration, and Nodal Interface Stresses (Max.,
Static, Min.) along the DOWNSTREAM Interface due to change of 0 (in
degrees) (H=400)............................................. 367
A.65 (Continued from previous figure) Max. Separation, Max. Acceleration, and
Nodal Interface Stresses (Max., Static, Min.) along the DOWNSTREAM
Interface due to change of 0 (in degrees) (H=400)...................... 368
A.66 Collective graphs of Max. Separation, Acceleration, and interface pressures
along the DOWNSTREAM Interface at all 0s (H=400)................ 369
A.67 Max. Separation, Max. Acceleration, and Nodal Interface Stresses (Max.,
Static, Min.) along the UPSTREAM Interface due to change of mu (H=400). 370
A.68 (Continued from previous figure) Max. Separation, Max. Acceleration,
and Nodal Interface Stresses (Max., Static, Min.) along the UPSTREAM
Interface due to change of mu (H=400).................................. 371
A.69 Collective graphs of Max. Separation, Acceleration, and interface pressures
along the UPSTREAM Interface at all mu's (H=400)................. 372
A.70 Max. Separation, Max. Acceleration, and Nodal Interface Stresses (Max.,
Static, Min.) along the DOWNSTREAM Interface due to change of mu
(H=400)...................................................... 373
A.71 (Continued from previous figure) Max. Separation, Max. Acceleration, and
Nodal Interface Stresses (Max., Static, Min.) along the DOWNSTREAM
Interface due to change of mu (H=400).................................. 374
xxxix
A.72 Collective graphs of Max. Separation, Acceleration, and interface pressures
along the DO WNSTREAM Interface at all mus (H=400).................... 375
A.73 Max. Separation, Max. Acceleration, and Nodal Interface Stresses (Max.,
Static, Min.) along the UPSTREAM Interface due to change of md (H400). 376
A.74 (Continued from previous figure) Max. Separation, Max. Acceleration,
and Nodal Interface Stresses (Max., Static. Min.) along the UPSTREAM
Interface due to change of md (H=400)......................................... 377
A.75 Collective graphs of Max. Separation, Acceleration, and interface pressures
along the UPSTREAM Interface at all rnd's (H=400)...................... 378
A.76 Max. Separation. Max. Acceleration, and Nodal Interface Stresses (Max.,
Static, Min.) along the DOWNSTREAM Interface due to change of md
(11=400)........................................................... 379
A.77 (Continued from previous figure) Max. Separation, Max. Acceleration, and
Nodal Interface Stresses (Max., Static, Min.) along the DOWNSTREAM
Interface due to change of md (H=400).......................................... 380
A.78 Collective graphs of Max. Separation, Acceleration, and interface pressures
along the DOWNSTREAM Interface at all md's (H=400)..................... 381
A.79 Effect of (9 on U/S response, H=200 ft.................................. 383
A.80 Effect of 6 on D/S response, H=200 ft.................................. 384
A.81 Effect of
A.82 Effect of 4> on D/S response, H=200 ft.................................. 385
xl
A.83 Effect of mu on U/S response, H=200 ft...................................... 385
A.84 Effect of mu on D/S response, H=200 ft...................................... 386
A.85 Effect of md on U/S response, H=200 ft...................................... 386
A.86 Effect of md on D/S response, H=200 ft...................................... 387
A.87 Effect of 6 on U/S response, H=300 ft...................................... 387
A.88 Effect, of 6 on D/S response, H=300 ft...................................... 388
A.89 Effect of 0 on U/S response, H=300 ft...................................... 388
A.90 Effect of 0 on D/S response, H=300 ft...................................... 389
A.91 Effect of mu on U/S response, H=300 ft...................................... 389
A.92 Effect of mu on D/S response, H300 ft...................................... 390
A.93 Effect of md on U/S response, H=300 ft...................................... 390
A.94 Effect of md on D/S response, H=300 ft...................................... 391
A.95 Effect of 9 on U/S response, H=400 ft...................................... 392
A.96 Effect of 9 on D/S response, H=400 ft...................................... 393
A.97 Effect of 4> on U/S response, H=400 ft...................................... 393
A.98 Effect of 0 on D/S response, H=400 ft...................................... 394
A.99 Effect of mu on U/S response, H=400 ft...................................... 394
xli
A. 100 Effect of mu on D/'S response, H=400 ft..................................... 395
A.101 Effect of md on U/S response, H=400 ft................................... 395
A.102 Effect of md on D/S response, H=400 ft................................... 396
A. 103 Max. Separation and Separation Depth of U/S interface vs 0 and Height. . 398
A. 104 Max. Acceleration Ratio and Max. RMS Acceleration Ratio of U/S inter-
face vs (p and Height....................................................... 399
A.105 Max. Separation and Separation Depth of D/S interface vs and Height. . 400
A.106 Max. Acceleration Ratio and Max. RMS Acceleration Ratio of D/S inter-
face vs
A.107 Max. Separation and Separation Depth of U/S interface vs mu and Height. 402
A. 108 Max. Acceleration Ratio and Max. RMS Acceleration Ratio of U/S inter-
face vs mu and Height....................................................... 403
A. 109 Max. Separation and Separation Depth of D/S interface vs mu and Height. 404
A. 110 Max. Acceleration Ratio and Max. RMS Acceleration Ratio of D/S inter-
face vs mu and Height....................................................... 405
A.lll Max. Separation and Separation Depth of U/S interface vs md and Height. 406
A.112 Max. Acceleration Ratio and Max. RMS Acceleration Ratio of U/S inter-
face vs md and Height....................................................... 407
A. 113 Max. Separation and Separation Depth of D/S interface vs md and Height. 408
xlii
A. 114 Max. Acceleration Ratio and Max. RMS Acceleration Ratio of D/S inter-
face vs md and Height............................................................... 409
A. 115 The effect of
U/S interface....................................................................... 411
A. 116 The effect of
D/S interface....................................................................... 412
A. 117 Collective graphs of minimum an maximum pressures versus for both U/S
and D/S interfaces.................................................................. 413
A. 118 The effect of md on minimum and maximum interface pressures along the
U/S interface....................................................................... 414
A. 119 The effect of md on minimum and maximum interface pressures along the
D/S interface....................................................................... 414
A. 120 Collective graphs of minimum an maximum pressures versus md for both
U/S and D/S interfaces.............................................................. 415
A. 121 The effect of mu on minimum and maximum interface pressures along the
U/S interface....................................................................... 416
A. 122 The effect of mu on minimum and maximum interface pressures along the
D/S interface....................................................................... 416
A. 123 Collective graphs of minimum an maximum pressures versus mu for troth
U/S and D/S interfaces.............................................................. 417
xliii
TABLES
Table
2.1 List of Dams with Embankment Wing Dams...................................... 11
2.2 Number of dams, higher than 150 meters, classified as concrete & earth
outside the US [53]........................................................ 12
2.3 Codes for Dynamic Analysis.................................................. 25
4.1 Scaling relations [41, 15]. It is assumed that same soil and fluids are used
in model and prototype..................................................... 65
4.2 Scaling relations for retaining wall interpretations [19]................... 68
4.3 Ramberg-Osgood and Elastic Model Parameters used in NIKE3D Analysis. 78
4.4 Ramberg-Osgood material properties for Seeds Average Sand with 7=100
pcf........................................................................ 81
4.5 Numbers of figures illustrating the comparison of measured and calculated
dat a...................................................................... 88
5.1 Unit Weight Distribution................................................... 118
5.2 Shear wave (Us), and maximum shear modulus (Gmax) distributions............ 122
xliv
5.3 Ramberg-Osgood material properties for Seeds [68] average sand with 7 =
125 pcf........................................................... 126
5.4 Ramberg-Osgood material properties for Seeds [68] average clay with 7 =
135 pcf........................................................... 127
5.5 Youngs modulus (E), and Poissons ratio (/i) used in FE analyses. 128
6.1 Case numbers for all parameters................................... 138
6.2 Numbers of figures illustrating 2-D response ..................... 160
7.1 Natural Frequencies (in Hz) of Dam Models......................... 216
7.2 Comparison of 3-D and 2-D results................................. 218
7.3 Numbers of figures illustrating 3-D response...................... 222
9.1 UPSTREAM Response due to 9........................................ 267
9.2 DOWNSTREAM Response due to 9..................................... 268
9.3 UPSTREAM Response due to (b....................................... 269
9.4 DOWNSTREAM Response due to Q..................................... 270
9.5 UPSTREAM Response due to mu....................................... 271
9.6 DOWNSTREAM Response due to mu .................................... 272
9.7 UPSTREAM Response due to rnd...................................... 273
9.8 DOWNSTREAM Response due to rnd.................................... 274
xlv
9.9 Numbers of Figures showing variation of independent variables and poly-
nomial best-fit curves................................................... 276
9.10 Best Fit Parameters and correlation coefficient for heights 100 and 200 ft. . 284
9.11 Best Fit Parameters and correlation coefficient for heights 300 and 400 ft. . 285
9.12 Correlation matrix of normalized variables for U/S interface............................... 286
9.13 Correlation matrix of un-normalized variables for U/S interface............................ 287
9.14 Correlation matrix of normalized variables for D/S interface............................... 287
9.15 Correlation matrix of un-normalized variables for D/S interface............................ 287
9.16 R2 values of regression models in the form of y = a-rbH + c9 + d(j> + e{mu) +
f(md)................................................................. 288
9.17 R2 values of linear regression models with selected independent variables. . 288
9.18 R2 values of second-degree linear regression models in the form of y =
a +bx + ex2 + dH...................................................... 289
9.19 Best multivariate regression models selected for U/S interface performance
parameters............................................................ 289
9.20 Best multivariate regression models selected for D/S interface performance
parameters............................................................ 290
B.l Ramberg-Osgood material properties for Seeds [68] average sand with 7
130 pcf............................................................... 419
xlvi
B.2 Ramberg-Osgood material properties for Seeds [68] average sand with 7 =
135 pcf.......................................................... 420
B.3 Ramberg-Osgood material properties for Seeds [68] average sand with 7 =
140 pcf.......................................................... 421
B.4 Ramberg-Osgood material properties for Seeds [68] average sand with 7 =
145 pcf.......................................................... 422
xlvii
1. Introduction
1.1 Problem Statement
Many earth dams either failed or suffered severe distress in past earthquakes. Thus, it is
critical to analyze their performance during strong motion earthquakes to provide
sufficient information for dam- safety assessment. Though many dynamic analysis
methods have been developed in the past two and a half decades, the complicated
Interface Behavior of Composite Dams (IBCD) has never received significant attention
because of the lack of more realistic soil models and soil-concrete interface models.
In the evaluation of the seismic stability of a composite dam, besides others, the main
problem is the dynamic interaction between concrete gravity dam and soil embankment.
The wrap-around sections are the transitional sections of a dam where it changes from
concrete dam to embankment wing dams. As an example; Folsom Dam is shown in
Figure 1.1. Soil-concrete interface areas between concrete dam and both upstream and
downstream embankment wing dams are also illustrated in Figure 1.1. Figure 1.2 shows
a typical two dimensional composite section in the wrap around region. The disastrous
consequences of a dam failure provides incentives for a precise analysis of the problem.
During strong earthquake shaking, the soil may slip and/or separate (debond) from
concrete and, upon the reversal of the direction of motion, the soil may reattach
1
(rebond) itself to the concrete surface. Debonding along the upstream surface allows
water to enter the gap created during the process and water is expelled upon rebonding
from the gap. The repeated debonding-rebonding can result in a permanent gap due to
plastic embankment deformation, internal erosion due to the water pumping action and
further the dam failure. Due to the lack of necessary analysis tools it has been difficult
to perform the rational analysis of the interface performance. This thesis aims to
provide more insight into I BCD.
Figure 1.1 Folsom Dam, California.
The study covers both 2-D and 3-D modelling of interface area, elastic-plastic material
model, effect of different geometric configurations of the interface on the separation,
separation depth, horizontal acceleration, and earthquake induced stresses along the
both upstream and downstream interfaces.
2
Figure 1.2 A typical soil-concrete interface of a composite dam.
1.2 Significance of Research
Because of the growing public concern for the safety of dams and reservoir and the
oversimplifications of the analysis methods used in designing dams it is urgently needed
to reevaluate the original designs and current procedures using new and more realistic
computation tools.
So far the soil-concrete interface has not been reported as the major reason for the
earthquake induced darn failure or distress. However, this does not mean that the
seismic safety of IBCD is not of concern. In fact, the analysis results indicate the
possibility of interface separation, acceleration amplification, and high pressure which
could lead to the dam distress or failure and the problem requires immediate research
attention. In the list of The World's Major Dams and Hyroplants [53], it is reported
that there are 36 concrete dams having wing dams in the countries other than US. In
the US alone, there are more than 40 dams, higher than 100 ft, having composite
sections. Figure 1.3 shows the U.S. earthquake hazard map and the circled numbers
represent the number of composite dams by state.
3
DAMAGE
0- None
1- Minor
lil 2-Moderate
i 3-Major
Figure 1.3 The U.S earthquake hazard map and the number of composite dams by state
(circled numbers).
4
The ability to evaluate the earthquake effect on the behavior of the composite dam is
critical to the assessment of the overall seismic safety of dams, to the design of remedial
measures, and to the new design strategy combating this potential problem.
1.3 Research Objectives
This study uses the finite element analysis to study the composite dam interface
behavior in lieu of the many other research tools available, mainly because of its cost
effectiveness over physical model tests. An extensive survey was carried out to find
computer codes appropriate for the study, particularly the code with a suitable interface
formulation for simulating the soil-concrete interface effect in a composite dams.
NIKE3D developed at the Lawrence Livermore National Laboratory (LLNL) was
selected for use in this study and beyond for the availability of its source code and
technical consultation as long as the LLNL and the University has an appropriate
collaborative agreement. The effectiveness of the code in simulating the interface
behavior was calibrated using the centrifuge model test results. An excellent agreement
was achieved. This provides a great confidence in the ability of NIKE3D in simulating
the interface problems in composite dams under seismic shaking. The objectives of this
study are multiple:
To develop a better understanding of the behavior of the soil-concrete interface
in a composite dam under strong seismic shaking in terms of the potential
5
repetitive debonding and rebonding, the depth of such action, and the extend of
such problem.
To clearly delineate the critical problems for future research on the interface
behavior in composite dams.
To investigate the effect of the slope of the concrete dam and embankment in the
wrapped-around section on the magnitude of separation and separation depth.
To assess the distribution of the dynamic earth pressure along the interface, and
the motion amplification.
To assess the three-dimensional effect by performing three-dimensional
modelling of the wrapped-around section.
To assess the effect of soil models on the interface behavior.
To assess the effect of imposing all three components (N/S, E/W and Vertical)
of ground motion on the interface behavior.
1.4 Research Approach
To achieve the research objectives a systematic approach was adopted. It should also be
noted that no other work has been found on IBCD in the literature, except some
numerical analyses. Therefore this study is the first detailed research effort on IBCD.
The steps of the research approach followed in this study can be itemized as follows;
Selection of numerical analysis computer code, and verification of its reliability,
Building hypothetical 2-D, and 3-D finite element, model,
6
Determining controlling parameters, and interface performance parameters,
Performing 2-D, and 3-D parametric finite element analyses,
Comparing 2-D, and 3-D finite element analysis results,
Statistical assessment of finite element results.
Offering recommendations on remedial measures for existing dams, and new
design strategies.
A proper analysis of the dynamic interface behavior of a composite dam requires a
numerical analysis computer code(s) with a nonlinear soil model and soil-structure
interface characteristics. The interface characteristics must include the capability of
modelling the interface slippage, debonding and rebonding. NIKE3D was selected for
this study with Ramberg-Osgood non-linear material model and interface algorithm that
allows separation and frictional sliding.
Hypothetical FE models were created using the mesh generator code TrueGrid. The
effect of overburden pressure on the material properties was considered by dividing
upstream and downstream embankments into different layers.
Geometric cross-sectional parameters were selected as the controlling parameters; they
are the upstream soil slope, upstream interface angle, downstream soil slope,
downstream interface angle, and height. As the interface performance parameters the
separation, separation depth, maximum acceleration, maximum RMS acceleration, and
minimum, static, and maximum pressures along the interface were selected.
i
Numerous 2-D FE analyses have been performed. Due to the time consuming nature of
3-D FE analysis, limited number of 3-D FE analyses were performed. Results are
compared and interpreted.
To investigate the degree of dependency of interface performance parameters on the
controlling parameters, statistical assessment has been performed.
8
2. Literature Survey
2.1 Introduction
In this chapter, a survey has been carried out on existing concrete dams with
embankment abutment or wing dams in the US and in the world. Besides the existing
dams, an extensive literature survey has been conducted about interface behavior study
of composite dams. Moreover, literature review on dynamic interface effects, available
computer codes to analyze IBCD and constitutive soil models are included.
2.2 Existing Composite Dams
In order to locate the existing concrete dam with soil in its cross-section, numerous
databases have been searched. Bureau of Reclamation online database is one of them.
USACEs (US Army Corpse of Engineers) database called NID (National Inventory of
Dams) has also been used. NID has around 75,000 records and 59 information fields.
NID database does not have any specific field entry to search composite dams, but dams
classified as 'concrete gravity & earth or rock fill can be searched. This is the closest
definition for composite dams. Findings of all database research are tabulated in Table
2.1. Additionally, The Worlds Major Dams and Hydroplant [53] lists 36 concrete &
9
earthfill dams outside the US. Table 2.2 shows the distribution of these dams by
country.
2.3 Literature Review on Analysis of Composite Dams
An extensive literature survey has been conducted about interface behavior study of
composite dams. Because of the lack of necessary analysis tools only a few calculations
have been made that account for the nonlinear effects with rebonding and debonding
behavior of the soil-concrete interface of composite dams. In the literature it is possible
to find extensive research reports about concrete gravity and embankment dams
studying the effect of dam-water and dam-water-foundation interactions, water
compressibility, base sliding of concrete monoliths, reservoir bottom absorption,
cavitation flexibility of concrete dams etc., but none of these addresses the
soil-concrete interface separation problem. In the literature, the interface separation
phenomenon was first mentioned in Seismic Analysis of Wing Dams of Folsom Dam
reports [80, 74]. Because of the lack of necessary tools, no numerical analysis results,
related to separation, were presented in these reports.
The interface behavior of composite dams was first studied by Chang [10] and Oncul
[57]. Chang [10] conducted analyses using the code FLUSH with nonlinear shear
modulus and damping ratio versus shear strain. Oncul performed analyses on IBCD
using computer code NIKE2D [21] with Ramberg-Osgood (R-O) non-linear material
model. The NIKE2D code also contains the interface algorithm that allows debonding,
10
Table 2.1 List of Dams with Embankment Wing Dams
Some Geometric and Dynamic Properties of Composite Dams
State Height Crest Width Base Width Dynamic Properties
(/<) (/o (ft) MCE(M) Peak Accc. (3) f(hz)
Concrete Gravity Dams
American Falls ID 104 43 59 7.5 0.67 10.3
Angostura SD 193 10 230 6.7 0.19 -
Jackson Lake WY 66 24 72 - - -
Keswick CA 157 20 111 - - -
Marshall Ford TX 278 20 286 5.3 0.09 2.70
Olympus CO 70 10 50 5.0 0.17 -
Shosta CA 602 30 543 6.25 0.46 2.90
Brantley NM - - - - - -
Folsom CA 170 32 139 6.5 0.35 -
Lower Monumental WA 155 - - - - -
Little Goose WA - - - - -
Ice Harbor WA 123 - - - - -
Lower Granite WA - - - - -
Butress Dams
Lake Tahoe CA 18 11 19 6.0 0.58 -
Minidoka ID 86 - - - - -
Pueblo CO 250 30 147 5.0 0.13 -
Thin Arch Dams
Nambe Falls NM 150 5 15 6.5 0.69 6.67
Dams Classified as Concrete Gravity & Earth in NID Database
Thomaston CT 137 - - - - -
Colebrook CT 215 - - - - -
Black Rock CT 130 - - - -
Carters Main GA 464 - - - - -
Carters Reregulation GA 105 - . - - -
Borden Broook MA 110 - - - - -
Knightville MA 150 - - - - -
Stockton MO 161 - - - - -
Everett NH 110 - - - - -
Franklin Falls NH 116 - . - - -
Kinzua PA 177 - - - - -
Ball Mountain VT 247 - - - - -
North Hartland VT 182 - . - - -
Townshend VT 126 - - - - -
Dams Classified as Concrete Gravity & Rockfill in NID Database
Kentucky KY 206 - - - - -
Hoover OH 124 - - - -
Mcnarv OR 220 - - - - -
Gainer Memorial RI 109 - - - - -
Norris TN 265 - - - - -
Normandy TN 110 - - - - -
Chickamauga TN 129 - - - - -
Pickwick Landing TN 113 - - - -
Fort Loudoun TN 125 . - - -
Tellico TN 129 - - - - .
Columbia TN 105 - - . - -
Watts Bar TN 112 - . - - -
Efoone TN 168 - - - - -
11
Table 2.2 Number of dams, higher than 150 meters, classified as concrete & earth'
outside the US [53].
Country # of Dams Country # of Dams
Argentina 7 India 5
Brazil 13 Netherlands 1
Canada 3 Nigeria 1
Czechoslovakia Paraguay 3
Hungary Turkey 1
rebonding and frictional sliding. Analysis results are compared to assess the
effectiveness of these two computer codes in evaluating the phenomenon of interface
separation [57]. The Left Wing Dam of the Folsom Dam, 180 ft high, is chosen in the
analyses. Kovna Dam Earthquake Record with amai 0.87g was used in both NIKE2D
and FLUSH analyses.
In summary, the NIKE2D analysis confirmed the suspicion of the interface separation
using FLUSH with the following observations: i) the soil-concrete interface can separate
under a strong motion earthquake, ii) the separation is significant and repetitive, and
iii) the separation can reach a significant depth.
To get a better understanding of IBCD parametric studies should be performed to see
the effects of different ground motions, interface geometry and material models. Chang
et al. [11] performed preliminary parametric analysis on a 100 ft high hypothetical
composite dam using NIKE3D. Results indicate that angle of upstream interface has
signific ant effect on separation and separation depth.
Due to the fact that hydraulic fracture is very possible in the interface area because of
its debonding and rebonding nature, it is necessary to mention the literature on
12
hydraulic fracture. As the soil and concrete separate the minor principal reduces to zero
at the soil face and that causes hydraulic fracture to occur. Hydraulic fracture is
promoted by the presence of the discontinuity, such as borehole, an existing crack, or
loose soil adjacent to rock joint. Hydraulic separation may occur at an interface between
soil and adjacent dissimilar material such as concrete or rock as soon as the water
pressure reaches the same magnitude as the normal stress across the interface [38].
Hydraulic separation is more likely to occur than hydraulic fracturing because there is
no tensile strength to resist separation. Jaworski et al. [38] showed in experimental
studies that initial hydraulic fracturing requires, in general, hydraulic pressure in the
range of 1.5 to 1.8 times larger than the minor total principal stress. However they also
show that existing crack would require hydraulic pressure equal to minor total principal
stress. Sherard [70] states that very steep rock abutments and near vertical concrete
structures produce zones susceptible to cracking and leakage, because heavy compaction
equipment can not operate close to concrete structures due to limited space, and
compac tion is done with hand operated small machines.
In terms of centrifuge testing, there is no particular composite dam centrifuge test
attempt. Arulanandan et. al. [1, 54] has performed dynamic centrifuge tests on an earth
dam that has sand-clay interfaces. The results show that failure mode on the dam is
greatlv influenced by the reduction in strength caused by the void-ratio increase at the
sand-clay interface and the corresponding decrease in effective stress.
13
Aforementioned hydraulic fracture and clay-sand interface performance indicate the
complexity of interface behavior between dissimilar materials. There has been no
complete and reliable analytical methodology to resolve the soil-concrete interface
problems. On the other hand computer programs developed recently with powerful
algorithms may serve as important tools to understand the dynamic soil-concrete
interface.
2.4 Soil-Concrete Interface Models
Besifles the non-linear stress-strain relation of soils, contact-impact, problems are among
the most difficult nonlinear problems because the response in contact-impact problems is
not smooth. The velocities normal to the contact surface are discontinuous in time when
impact occurs. For Coulomb friction models, the tangential velocities along the interface
are discontinuous when stick-slip behavior encountered, and they introduce significant
difficulties in the time integration of the discrete equations [4].
Dcsai et al. [18] performed numerous lab tests on the interface behavior and developed
an interface model using Ramberg-Osgood model without debonding and rebonding
features. In the lab environment it is very difficult to obtain accurate volume change
and stress measurements required to determine parameters for advanced interface
models such as those based on the theory of plasticity [18].
14
A realistic interface algorithm should be able to simulate frictional slippage, separation
and re-bonding behavior of the soil-concrete interface at any loading step. The ideal
interface algorithm should possess the following characteristics;
closely simulate the soil-concrete interface behavior during seismic shaking,
easily adaptable to existing FEM codes,
should maintain numerical stability.
should be verifiable with physical test results.
The literature survey reveals various interface algorithms. They can be classified as [39];
1) stiffness approach (e.g. directionally stiff elements), 2) constrained approach (e.g.
Lagrange multipliers, penalty method), 3) mixed approach.
2.4.1 Stiffness Approach
One of the commonly used interface element is based on the joint element proposed by
Goodman, Taylor and Brekke [28]. The element formulation is derived on the basis of
relative nodal displacements of the solid elements surrounding the interface element as
shown in Figure 2.1. This formulation can simulate slip conditions well, but has
penetration problems. Using high normal stiffness causes numerical problems. In two
dimensional analysis, the constitutive or stress-relative displacement relation is given in
Equation 2.1:
' \ /
On kn 0 Vr vr
" = < = [CU
T 1 J O ur ur ) l
15
*
Interface
SO LID n
SOLID
t=thickness=0
** s y
Figure 2.1 Relative displacement element [17]
where an =normal stress, r =shear stress, ks =shear stiffness, kn =normal stiffness and
vr and uT =relative normal and shear displacements, respectively, and [C], ^constitutive
matrix for the interface element. In the last two decades many researchers have
attempted to develop models to simulate the behavior in normal and tangential
directions.
Ghabussi, Wilson, Isenberg [27] proposed a formulation which is derived by considering
relative motions between surrounding solid elements as independent degrees-of-freedom.
This proposed model has less numerical errors and has the ability to simulate positive
and negative dilatancv resulting from shear deformations.
For nondilatant joints there is no volume change due to shearing strains and therefore
shear and normal components of deformation are uncoupled and Stress-strain relations
is given in Equation 2.2:
' On , = cnn 0 H
Os 0 Css 1 .
16
where Cnn and Csx are nonlinear functions. In relating stress to deformation in the
direction normal to the joint, three distinct regions are defined: (1) Separation
Cnn = Css = 0 when en > 0; (2) crushing of surface irregularities Cnn Ec{tcn < en < 0),
for smooth surfaces ecn 0; and (3) contact, Cnn = Ej(cn < Â£)
The tangential stress-strain relationship is assumed to be elastic-perfectly plastic using
Mohr-Coulomb yield criterion: Css = G for as < c +
(7,5 c T tancp (plastic, os has reached limiting shear strength).
Herrmann [34] introduced an interface algorithm considering various modes of interface
behavior such as re-bonding, de-bonding and slippage. Herrmanns interface algorithm
assumes there are tangential and normal springs along the interface to model frictional
slip and de-bonding. Although using relative movements as global unknowns eliminates
the numerical problems [81], when the spring coefficient is very large, very severe bond
stress gradients may result which in return may cause convergence problems.
Katona [39] developed a procedure for friction-contact interface which simulates
frictional slippage, separation and re-bonding by using principle of virtual work. Figure
2.2 represents the proposed element in a separated state. At the end of any load step k,
the interface responses are characterized by interface forces A(j and A*, and/or relative
movements AÂ£ and AÂ£ where subscripts n and s refer to normal and tangent directions.
In order to define relative movements, the nodes 1 and 2 in Figure 2.2 are assumed to be
in the same location prior to any loading.
17
J. v\
Figure 2.2 Interface element representation in separated state [39].
The use of thin layer elements has shown significant promise in terms of dynamic
behavior of interfaces. The idea of thin layer is based on the assumption that the
behavior near the interface involves a thin zone, rather than a zero thickness in several
investigations [84], The choice of thickness is very important for dynamic analysis where
the mass and damping properties need to be considered.
The idea of thin-layer sounds reasonable and studied by many researchers.
Pancle and Sharma [59] used 8-node iso-parametric parabolic element using relative
displacements as independent degrees of freedom. In this study, the numerical
ill-conditioning associated with the choice of thickness for thin joint elements was also
presented. If the thickness is too large then it is going to behave like a solid element. If
it is too small then numerical problems are very likely to occur.
Desai et. al. [17] proposed a thin-layer element for interfaces and joints. The element is
essentially treated like any other solid (soil, rock or structural) element. Since
penetration is not allowed at the interface, a high value, of the order of 108 1012 units,
18
is assigned for the normal stiffness kn which, in fact, has no physical explanation.
Therefore, normal stiffness of the element is determined by considering the surrounding
geological and structural materials as well as the interface itself. Interpenetration of
nodes is treated with special algorithm explained in [17].
Zarnan [84] has shown the implementation of the same thin-layer element into dynamic
soil-structure interaction problems with debonding and rebonding capabilities.
Furthermore in most problems, the formulations mentioned above can provide
satisfactory solutions for the stick and slip modes for which the normal stress remains
compressive. For other modes such as de-bonding, the solutions are often unreliable [17].
As a brief summary the advantages and disadvantages of stiffness approach can be
itemized as follows;
Advantages;
easy to implement into existing FEM codes,
complicated constitutive models can be used for interface elements,
works well for no separation case,
a wide literature is available on theory and application in soil mechanics.
Disadvantages;
not reliable in case of separation,
19
implementations are based on small strains assumptions,
selection of thickness of interface elements may cause numerical problems,
materials must be initially in contact at the discontinuities,
in most of the current implementations, mesh discontinuity at the interface is
not allowed.
2.4.2 Constrained Approach
In contact problems involving friction there are three methods called as Lagrange
Multipliers, Penalty Method and Augmented Lagrangian Method. Basically these
methods treat the interface as surfaces on each side without introducing any elements
into the interface area. Lagrange multipliers are studied by many researchers [3, 35] and
implemented in computer codes such as ADINA. Penalty formulation is studied by
[31, 32] and implemented in NIKE [21, 47] codes. Augmented lagrangian method is a
hybrid of the other two and is accepted as an improved formulation [71, 82]. It is also
implemented in NIKE codes.
In the next section each of these methods will be explained briefly together with their
shortcomings and advantages. Consider the variational formulation of a discrete
structural modal for a steady-state analysis:
n* iurKU UtR (2.3)
where II* is the total potential energy of the system, U is the displacement vector, R is
the load vector and K is t he stiffness matrix of the system and with the conditions
20
<911* = 0 for all i. Assume that the displacement at the degree of freedom Ui with
Ui U* needs to be imposed. Next subsections discuss how this constraint is handled in
each method.
Lagrange Multipliers
In the Lagrange Multiplier method we amend the right hand side of Equation2.3 to
obtain:
n* ^UtKU UtR + A (Ui U*) (2.4)
where A is an additional variable, and invoke 511* = 0, which gives:
K el u R
O i A y: _
(2.5)
where e, is a vector with all entries equal to zero except zth entry, which is equal to one.
Disadvantages of this method can be listed as;
may lead to singular stiffness matrices unless partitioning method is used,
new variables introduced which require more computational effort
Advantages are;
effective procedure,
easy to implement,
no numerical ill-conditioning,
21
Penalty Formulation
In the penalty method we also amend the right-hand side of Equation2.3 but without
introducing an additional variable. Consider the Equation 2.6:
rr* = Ju7ku utr + |(i/, u*f (2.6)
in which a is a constant of relatively large magnitude, a max(/c). The condition
911** = 0 now yields:
dUTKU <9UtR + a(Ui U*)dUr = 0
(2.7)
and
{K + aeieJ)U = R + aU-ei (2.8)
Hence using this technique, a large value is added to the ith diagonal element of K and
a corresponding force is added so that the requirement displacement Ui is approximately
equal to U*.
Advantages;
no additional variables,
has simple technique,
can be interpreted from a physical standpoint.
22
Disadvantages;
solution is sensitive to the choice of penalty number a.
numerical ill-conditioning due to a very large penalty number a,
exact satisfaction of constraint can only be achieved at a value of infinite for
penalty number.
Therefore maximum attention should be given to the choice of penalty number. The
optimum choice for penalty number is studied in [2].
In penalty formulation, any node that penetrates through its respective contact surface
causes a linear interface spring to be inserted into the stiffness matrix.
Augmented Lagrangian Method
It has been proposed to reduce or eliminate the shortcomings of the previous two
methods. Augmented Lagrangian technique has been applied to incompressible finite
deformation elasticity, frictionless contact problems and viscoplasticity [71]. Simo et.
al. [71] extended the method to apply to frictional problems.
Advantages;
convergence is achieved with finite penalty numbers,
inherits advantages of Lagrange multipliers method,
no additional variables.
23
Disadvantages;
by experience it is observed that this method needs more computation time.
2.5 A Brief Review of Computer Codes
There are various numerical analysis codes that has interface model and nonlinear soil
models. In this section a list of computer codes for dynamic analysis of dams and their
main features are given.
Table 2.3 is the summary of the survey about computer codes suitable for dynamic
analyses of dams. Besides the interface property, other features such as pore pressure
generation ability, solution method and type of elements are also included.
Since the analysis of IBCD is very complicated, it is required to have a powerful
computer code with interface model and pore pressure generation (e.g. Biot Theory)
algorithms. Unfortunately none of the computer codes has Biot theory and interface
algorithm at the same time. Among all computer codes, NIKE2D 3D and FLAC with
dynamic option are chosen to study IBCD. When appropriate other programs may be
added in the future.
NIKE2D and NIKE3D
Computer programs NIKE2D and NIKE3D developed at Lawrence Livermore National
Laboratory (LLNL) for defense program applications provide a powerful tool that can
24
Table 2.3 Codes for Dynamic Analysis
Codes for Dynamic Analysis of Dams
Software Type Element Type 2D 3D Pore Pressure Soil Model Interface
CO ST
CESAR-LCPO FE * M-C.D- P.CC.VE.H-B
DIANA FE - T.M-C.D-P
TARA-3 FE - - F -
ADINA FE - D-P
ABAQUS FE CC.D-P.M-C
NIKE2D FE,IMP - - - R-O
NIKE3D() FE?IMP - R-O
DYNA2D FE,EXP - - - R-O
DY.NA3D FE,EXP - R-O
DYNAFLOW FE.IMP (Biot) D-P.M-C
FLAC FD.EXP - M-C.DY.D-P
FLAC3D FD.EXP D-P.M-C
Sigma-2D FE - - -
Sigma-3D FE - -
FLUSH FE.FR - EQL -
QUAD4M FE - _ - EQL -
TELDYN FE - - -
RHEO-STAUB FE - - - VE.VP.AEP -
DIANA-SWANDYNE II FE - - (Biot) M-C.TSH Slip Elcm
SASSI FE.FR - -
DYSAC2 FE - - (Biot) - -
CO Continuum ST=Structural, FE=Finite Element. IMP= Implicit FR = Frequency Domain. FD=Fimlc Diffrcncc.
EXP^Explicit. M-C= Mohr-Coulomb, D- P = Druckcr-Prage r CC = Cam Clay, VN = von Mises. CC=Cam Clav. VE^Visco-
clastic VP- Visco-plastic W-W = W illiam-Wronkle. L)Y = Double yield H-B = Hoek-Brown. T=lVesca. F- 'inn s Model.
EQL-Equivalent Linear. R-0=Ramberg-Osgood. TSH = Two-surfacc hardening. LA = Lades Model. AEP = Anisotropic
clast ic-plastic ().= Modified Cam Clay and Lades model have been implemented by the of Colorado at Denver through its colloborative agreement with LLNL research group at the University
be used to analyze the response of important structures to large earthquakes. Computer
simulation of nonlinear behavior is quite complex and the nonlinear finite element
computer programs developed at the LLNL are some of the worlds most powerful
programs for performing nonlinear structural analysis.
NIKE2D is an implicit finite element code for analyzing the finite deformation,
quasistatic and dynamic response of two dimensional, axisyminetric, plane stress and
plane strain solids. The finite element formulation accounts for both material and
geometric non-linearities. A number of material models are incorporated to simulate a
wide range of material behavior including, elasto-plasticity, anisotropy, creep, and rate
dependence. Arbitrary contact between independent bodies is handled by a variety of
slideline algorithms. These algorithms model gaps and sliding along material interfaces,
including frictional interface.
NIKE3D is fully implicit three dimensional finite element code for analyzing the finite
strain static and dynamic response of inelastic solids, shells and beams. There are more
than twenty material models implemented in the code. Like in NIKE2D contact-impact
algorithms permit gaps, frictional sliding and mesh discontinuities along material
interfaces.
Chapter 3 discusses theoretical background of NIKE codes in detail.
26
FLAC
FLAG [37] is an explicit 2D finite difference program. Constitutive equations are solved
incrementally, thus allow large strains, material anisotropies, sliding interfaces and other
nonlinearities. It has library of material models such as elastic, Mohr-Coulomb
plasticity, ubiquitous joint, double yield, viscous and strain softening. Some of its main
features are: several automatic grid generators, statistical distribution of any property,
groundwater flow with full coupling to mechanical calculation including negative pore
pressure for saturated soil, structural elements (including non-linear cables) with general
coupling to continuum. FLAC has built-in language called FISH to add user defined
features (new constitutive models, variables or commands).
2.5.1 Interface formulation in FLAC
Figure 2.3 A typical interface and zone dimensions of FLAC interface.
FLAC has three options to define an interface;
(1) Glued interface: No slip or opening is allowed.
(2) Coulomb Shear-Strength: The Coulomb shear-strength criterion limits the shear
force by the following relation:
27
(2.9)
max eT T tajl 4)Fn
where c =cohesion L =effective contact length and d> ^friction angle of interface,
Fs =tangential force, Fn ^normal force along the interface.
If the criterion is satisfied (i.e., if \FS\ > Fsmax), then Fs = Fsmax.
(3) Tension Bond: If the tension at the interface exceeds the tension bond interface
breaks and all forces are set to zero.
Stability and time step adjustments are strongly affected by interface stiffness
parameters kn and ks. The effect of pore pressure is included in the interface calculation
by using effective stress as the basis for the slip condition. No pressure drop normal to
the joint and no influence of normal displacement on pore pressure are calculated. Also
condition of fluid along the interface is not modelled.
In FLAC manual [37] it is recommended to choose kn and k as ten times equivalent
stiffness of the stiffest neighboring zone. The stiffness of a zone in the normal direction
is;
max
-(K+joy
AZmm
(2.10)
where K and G are the bulk and shear moduli, and AZmin is the smallest width of an
adjoining zone in the normal direction and it is shown in Figure 2.3.
28
Selection of interface stiffness parameters requires utmost attention. High values may
cause very small calculation timestep and low values may not reflect the real behavior of
the interface.
2.6 Constitutive Models
A material model idealization should possess the following necassary properties
(Prevost, 1996);
(1) The model should be complete, i.e. able to make statements about the material
behavior for all strain and stress paths, and not merely restricted to a single
class of paths;
(2) It should be possible to identify the model parameters by means of a small
number of standart or simple material tests;
(3) The model should be founded on some physical interpretation of the ways in
which the material is responding to changes in applied stress or strain (e.g., the
material should not be modeled as elastic if permanent deformations are
observed upon loading).
2.6.1 Equivalent Linear Elastic Model
Under strong shaking the soil may undergo large deformations and that might introduce
nonlinear effects. In order to take this into account equivalent elastic linear procedure
was developed by Seed and Idriss (1969). In this representation the shear modulus
29
(G/Gmax) and damping ratio of the soil are expressed as a function of shear strain.
Nonlinear stress-strain behavior of the soil is accounted for by iteratively adjusting the
modulus and damping values until a reasonable consistency is obtained between the
selected parameters and the computed strain levels throughout the analysis. Seed and
Idriss (1970) have published effective shear strain versus shear modulus reduction factor
and damping ratio data for sand and clay. Currently the computer codes such as
FLUSH and QUAD4M use this approach to reflect the nonlinear soil behavior in FEM
analyses. The drawback of equivalent linear analysis is it is not possible to obtain
permanent deformations.
2.6.2 Hyperbolic Model
Duncan and Chang [20] proposed a stress-strain relation to approximate the non-linear
soil behavior. Stress-strain curve is given by the equation:
Â£ fj Rf (1 sin) (gi ~ Qj)
2 ccos(j> + 2
where n ^modulus exponent, K ^modulus number, Ei KPa
Pa =atmospheric pressure, Rj = c ^cohesion, and 4> =angle of internal
friction.
The usefullness of Equation 2.11 lies in its simplicity with regard to two factors [20];
Ei
(2.11)
30
e
Figure 2.4 Deviatoric stress vs strain relation for hyperbolic model.
(1) Because the tangent modulus is expressed in terms of stresses only it may be
employed in analyses for problems involving any arbitrary initial stress
conditions without any additional complications,
(2) The parameters K, n, and Rf may be determined readily from series of triaxial
test tests for different confining pressure. The amount of effort required to
determine parameters K, n, and Rj is not much greater than that required to
determine the values of c and
The model mentioned above has been implemented into many FEM codes. It is being
used for static and cyclic loading with small hysteresis. For cyclic loading,
modulus-confining pressure relation is;
in which Eur =is the unloading-reloading modulus, and I\ur =is the corresponding
modulus number and lets users to obtain permanent deformation upon unloading.
Deviatoric stress and stress relation of hyperbolic model is shown in Figure 2.4.
(2.12)
31
2.6.3 Mohr-Coulomb Model
Mohr-Coulomb is the most coinmmonly used failure criterion in engineering practice.
The relation between shear strength and the normal stress can be expressed as;
r = c + an tan (j)
(ctj cr3) = sin <6 (
(2.13)
(2.14)
where (j) is the friction angle and c is the cohesion. Since Mohr-Coulomb failure criterion
relates two parameters r and an or ny and
addition Figure 2.5 shows the failure criteria in a three-dimensional stress space where
axes are principal stresses cry, cr2 and cr3. Hydrostatic axis is the line which forms equal
angles with the three axes.
Since intermadiate principle stress cr2 is not included in the failure criterion it assumed
not to have any influence on the strength. Mohr-Culomb theory is widely used in
engineering practice because it is well understood and strength parameters 4> and c can
easily be obtained from confined compression and direct shear tests.
2.6.4 Ramberg-Osgood Model
The R-O model is an analyt ic al model and it is often used to represent the hysteretic
behavior of soil materials subjected to cyclic loading. Although unloading and reloading
are included in the model, inelastic deformations are not included in the sense of theory
of plasticity.
32
Shear Stress
Mohr Coulomb yield surface
Normal Stress 0
Figure 2.5 Mohr Coulomb yield surface.
33
The backbone (monotonic loading) strain-stress relation of the Ramberg-Osgood
elasto-plastic model can be expressed by:
7 r
Ty
1 + Q ----
Tv
T
(2.15)
where 7=shear strain, r=shear stress, 7y=reference shear strain, ry=reference shear
stress, a=constant > 0, and r^constant >1. The factor a can be varied to adjust the
graph. When r=l a linear relationship between shear stress and shear strain is
described. Formulation of R-0 permits the use of integer value r which provides more
flexibility in fitting laboratory test data.
The Ramberg-Osgood equation is inherently one dimensional and is strictly applicable
to shear components. To generalized this equation to the multidimensional case, it is
assumed that each component of the deviatoric stress, and deviatoric strain is
independently related by the above Equation 2.15. The volumetric behavior is assumed
to elastic, and therefore the pressure, p, is determined using the elastic relation:
where e is the volumetric strain, and K is the bulk modulus.
Figure 2.6 shows a typical loading and unloading curve for Ramberg-Osgood model. For
unloading and reloading, according to Masings rule the relation becomes:
position of the curve along the strain axis and the value r controls the curvature of the
p = -Kev
(2.16)
(2.17)
34
Figure 2.6 Typical loading and unloading curve for Ramberg-Osgood Model.
35
where 70=shear strain at point of stress reversal, and r0 =shear stress at point of stress
reversal. The values of 7, ry, a, and r are to be determined from the laboratory
experiment results.
Ueng and Chen [77] developed an iterative procedure to obtain R-0 parameters for soils
using G/Gmax and damping ratio versus shear strain curves. Richart [61] performed
curve fitting studies by changing R-0 parameters a and r for clay and sand.
By rearranging Equation 2.15 the secant modulus for the backbone curve can be
expressed as:
/
n y
C'o ~
7 7y
1
\
1 + a
(2.18)
For very small strain, i.e., 7 >0 and r 0, since r >1,
(Go)7=0 Grnax
(2.19)
ly
Then the backbone relation can be rewritten as:
1 + a
Gmax7j/
r1 >
7y G max7y
Therefore, besides Gma* other parameters 7y, a, r should be determined for the
Ramberg-Osgood model. Substituting r = G7 and rearranging Egn. 2.20 we get:
(2.20)
Gn
1 = a
Goi
r-l
Gn iax7y
log _ 1 j = log a + (r 1) log ^
G07
G
L7max ,y
(2.21)
(2.22)
Equation 2.22 can be plotted using only GjGmax data and then the values of r and a
can be determined from slope and intercept respectively. Next step is plotting a similar
36
curve using only damping ratio curve and obtaining the same parameters one more time.
Final values of r and aare determined by taking the average, a and r are determined
using only damping ratio curve as follows: The equivalent critical damping ratio, 0, for
a hysteresis loop with the tip at ('ya, t0) can be expressed as:
a = AE = 2o'(r-l) G0 >r(7oxr-i
27tt070 7r(r + l) Gmax 7y
(2.23)
where AE energy" dissipation in one loading cycle. Substituting Equation 2.21 in
Expiation 2.23 we get;
2(n-l)(1__G^
7r(r + 1) Ginax
0ix (r + 1)
(2.24)
or
Go
= 1 -
Cmax 2(r 1)
Substitute Equation 2.25 in Equation 2.22 we obtain;
(2.25)
log
0ir (r + 1)
2(r 1) 0n(r + 1)
= log(a) + (r 1) log
/
1 -
,8tt (r + 1)
2(r 1)
7_
7y
(2.26)
Finally Equation 2.26 can be plotted like Equation 2.22. Thus, a best fit straight line,
and values of a and r can be found for data including both modulus and damping data.
If we examine Equations 2.22 and 2.26 we see that both have 7y Therefore to
determine 7y and the other parameters an iterative procedure is followed as below;
(1) Assume a value for 7y and obtain the values of a and r by plotting the data
according to Equations 2.22 and 2.26.
(2) Compute 7y according to Equation 2.23 from the given modulus and obtain an
average value of yy.
37
(3) Compare the new value of 7y with the previous value. Repeat steps 1 and 2 if
the difference is too great.
Finally a, 7y, and r are all obtained. Then ry can be calculated by using Equation 2.19
and this finishes the procedure for obtaining the Ramberg-Osgood material properties.
2.6.5 Other Models
The other widely used soil models, Modified Cam Clav and Lades Model, have been
implemented in NIKE3D by the research group at the University of Colorado at Denver
through its colloborative agreement with Lawrance Livermore National Laboratory
(LLNL). This newly enhanced NIKE3D may be used in the study of composite dams.
Theoretical discussion of these models are not included in this study.
2.7 Pore Pressure Generation Models
There are several pore pressure generation procedures developed for the last three
decades. These procedures can be gathered under three headings; l)uncoupled method,
2)partially coupled method, 3)fully coupled method.
Uncoupled method is based on an emprical relationship for pore pressure development
in uniform cyclic tests as a function of the number of cycles of loading normalized by the
number of cycles to initiate liquefaction. This method was developed by Seed [67] and
will be expalined shortly in the next section.
38
In partially coupled method [25] the change in pore pressure is related to change in
volumetric strain. Therefore pore pressure generation mechanism is linked to the
nonlinear analysis results. Finn's model falls in this category and it will be discussed
later in this chapter.
In fully coupled method, pore pressure generation and dissipation are directly connected
to the soil skeleton deformation according to Biots formulation. This is the most
rational method of analysis.
2.7.1 Uncoupled Seeds Method
In this model, actual measurements of pore water pressure build-up in cyclic loading are
used. The only criteria is to determine the number of uniform stress cycles N[ which will
produce a condition of initial liquefaction under undrained conditions. This can be
obtained from cyclic simple shear or triaxial tests on representative samples. This
method of analysis is implemented in the computer code APOLLO [51] and based on
the emprical findings that the development of pore water pressure in granular soils
under cyclic loading is of the form;
ug = ojF
(2.27)
where aj is the effective overburden pressure, N is the number of uniform cycles
undergone by the soil sample and Nt is the accumulative number cycles at the same
stress level required to reach initial liquefaction. As it has been stated in [51], for many
39
soils, the function F may be expressed as;
F
2
= arcsin
7T
(2.28)
where a is an emprical constant and has a value of 0.7 which represent the average curve
for many soils.
Chameau [12] modified Seeds equation and obtained two parameter equation which is
found to better follow pore pressure generation characteristics observed in laboratory
tests;
where a and 0 Eire shape parameters and R is the excess pore pressure Au, divided by
initial effective confining pressure oj. Chamaeu suggested that these parameters are
cyclic stress ratio dependent.
2.7.2 Partially Coupled Model of Finn
An effective stress model based on strain controlled simple shear tests was developed by
Finn, Lee and Martin [25] to account for the nonlinear accumulation of pore pressure
during cyclic loading. The first assumption of this model is that the pore pressure
development occurs due to the potential for the volumetric deformation of the soil when
tested under drained condition. Pore water is assumed to be incompressible, in
comparison with the soil skeleton. These assumptions yield the change in pore pressure
as;
(2.29)
40
A u = Er Aet.
(2.30)
Where Au is the change in pore prssure, Er is the elastic rebound modulus of the soil
skeleton and Ae is the change in volumetric strain during a drained simple shear test.
The incremental volumetric strain is a function of the total accumulated total
volumetric strain, e, and the applied shear strain 7. The change in volumetric strain,
Ae, is calculated as;
Ae = C1(7 C2ev)
C3ev2
7 + C^v
(2.31)
where C\, C>, 63, and C\ are emprically determined constants and 7 is determined from
a hyperbolic stress-strain relationship;
T ~
(2.32)
where a and b arc emprical parameters. By knowing the stress history the development
of pore pressures can be calculated by a sequential procedure. All the emprical
parameters mentioned above are determined from strain controlled simple shear tests.
41
2.7.3 Fully Coupled Method: Biot Theory
A freely moving fluid in a porous material causes changes in the behavior of bulk
material. An increase in pore pressure causes the dilation of or contraction of solid
results in the change in pore pressure. The theory of linear 3D consolidation of soil was
formulated by Biot [5, 6]. Later this theory was extended to include various non-linear
effects, both material and geometrical. A finite element formulation of Biot's theory was
first presented by Sandhu [62]. Currently, Biot's principle is implemented in numerical
analysis codes such as DIANA-SWANDYNE. DYSAC2 and DYNAFLOW, etc.
Biots model of poroelastic materials addresses a coherent solid skeleton and a freely
moving pore fluid in which solid and liquid phases are fully connected. Biot made the
following assumptions:
soils are homogeneous and isotropic,
the strains are very small,
stress-strain relations are linear and perfectly elastic,
pore water is compressible and has no shear resistance,
the flow of water is viscous and follow Darcys law,
no capillary effects.
Then lie obtained the following equations for streses;
(Tij Zl\ f -\ICfckSjj 4 Qt^ij (2.33)
'U'ij Q^kk^ij ficSij (2.34)
42
where e is the dilatancy of the fluid and expressed by e = A wp and wp is the fluid
displacement vector, 5ij is the Kroneckers delta and is the strain tensor of soil
skeleton.
A close examination of constants N, M shows that these constants corresponds to
Lame's constants in the theory of elesticity, The coefficient N represents shear modulus
C! of the bulk material, and the coefficient R is a measure of the pore pressure required
on the fluid phase to force a certain volume of the fluid into the pores of the soil
aggregate while the total volume remains constant. The coefficient Q is of the nature of
a coupling between the volume change of the solid and that of fluid phase. Biot claimed
that one can perform experiments to measure the four elastic constants [69]. The shear
modulus can be measured directly.
In Biot's theory the assumption that the pore fluid is compressible introduces significant
improvement over previos theories and provides a more realistic explanation for the pore
pressure generation. The treatment of soil as two-phase medium, accounting for the
coupling between phases, and, therefore, determining the stress distribution in both the
fluid and the soil phases, gives a more accurate and realistic representation of actual soil
behavior.
43
2.8 Summary and Conclusion
This chapter discusses the literature survey on composite dams, soil-concrete interface
models, numerical analysis codes to analyze IBCD, constitutive soil models, and pore
pressure generation models.
The literature survey revealed that there is no concrete and reliable method to
investigate IBCD.
Two different interface treatment approach was explained; stiffness approach and
constraint approach. Stiffness approach techniques are are simple to implement but not
reliable in case of separation. Therefore to study IBCD constrained approach is more
appropriate.
Several computer codes are listed including their main features. It has been shown that
among others NIKE3D is one of most suitable FE software to analyze IBCD.
Several soil constitutive models including Ramberg-Osgood, Mohr-Coulomb, etc. are
presented. In terms of availability in NIKE3D, Ramberg-Osgood is most suitable soil
model that can be used in dynamic analysis.
Different pore pressure schemes are also presented. Biots coupled method is the most
realistic pore pressure generation algorithm. Since it is not available in NIKE3D,
analyzing IBCD using pore pressure generation models is left as a future study.
44
3. Finite Element Analysis Codes
3.1 Introduction
This chapter describes the softwares used in 2-D, and 3-D parametric FE analyses. Pre
and post processors, and NIKE3D are presented. TrueGrid [83] and GRIZ have been
used as the pre-processor and post-processor, respectively. Theoretical background of
NIKE3D, its interface algorithm, and solution strategy are discussed.
First, an input batch hie is used by TrueGrid to produce an input hie for NIKE3D.
Then NIKE3D produces series of binary plot hies to be read by GRIZ storing all nodal
and element information throughout the analysis. Finally, one can obtain time history
data using GRIZ. The analysis sequence is illustrated in Figure 3.1.
3.2 TRUEGRID
TrueGrid [83] is a powerful, easy-to-use interactive and batch mesh generator. TrueGrid
generates meshes for hnite difference and finite element simulation codes that model the
behavior of fluids and structures. However, TrueGrid can generate complete input hies
for many simulation codes, such as ADINA, ANSYS, MARC, LS-DYNA, LS-NIKE, etc.
Along with defining the mesh, you can specify physical properties on the mesh.
45
Input Batch File
I
TRUEGRID
i
Input File for NIKE30
t
NIKE3D
t
Binary Plot Files
GRIZ
POST PROCESSOR
Time History Ascii Files
Figure 3.1 Analysis sequence.
46
TrueGrid generates multi-block, structured meshes. Each block is composed of solid
hexahedral (six-sided) elements and/or structural quadrilateral shell and beam elements
arranged in rows, columns, and layers. TrueGrid uses a special projection method for
mapping a block mesh onto one or more surfaces. Therefore a complex looking mesh can
be built from a simple block very easily. An example is shown in Figure 3.2 with a
simple block and cylindrical projection surfaces.
tiifMi'S.EJH; ....
I; '
Figure 3.2 A simple block part and cylindrical projection surfaces [83].
TrueGrid have been used in all 2-D, and 3-D modelling efforts. Particularly, the
complicated 3-D FE model was created easily saving considerable time.
3.3 NIKE3D
NIKE3D developed at Lawrence Livermore National Laboratory (LLNL) for defense
program applications provide a powerful tool that can be used to analyze the response
of important structures to large earthquakes. Computer simulation of nonlinear
behavior is quite complex and the nonlinear finite element computer programs
developed at the LLNL are some of the worlds most powerful programs for performing
nonlinear structural analysis [47],
47
NIKE3D is fully implicit three dimensional finite element code for analyzing the finite
strain static and dynamic response of inelastic solids, shells and beams. A number of
material models are incorporated to simulate a wide range of material behavior
including, elasto-plasticity, anisotropy, creep and rate dependence. Arbitrary contact
between independent bodies is handled by a variety of slideline algorithms. These
algorithms model gaps and sliding along material interfaces, including frictional
interface.
3.3.1 Element Library
NIKE3D utilizes a relatively small set of elements. All elements use low order
interpolation, requiring no midside node definitions. This approach chooses highly
efficient elements over more costly higher order elements. The available elements are
solid, beam, and/or shell elements, and they are shown in Figure 3.3. Eight node solid
elements are integrated with a 2x2x2 point Gauss quadrature rule. Four node shell
elements use 2x2 Gauss integration in the plane, and one of many available schemes for
integration through the thickness. Two node beam elements use one integration point
along the length, with many options for integration of the cross section.
3.3.2 Solution Strategy
In NIKE3D, several nonlinear solution strategies are available, including Full-,
Modified-, and Quasi-New'ton method. By default, NIKE3D uses the BFGS method. An
48
Figure 3.3 Elements available in NIKE3D [47].
extensive set of diagnostic messages have been incorporated into the quasi-Newton
solvers to allow their convergence progress to be monitored.
NIKE3D is based on updated Lagrangian formulation. During each load step, nodal
displacement increments which produce a geometry that satisfies equilibrium at the end
of the step are computed. After obtaining updated displacement increments, the
displacement, energy, and residual norms are computed, and equilibrium convergence is
tested using user defined tolerances. Once convergence is obtained, displacements and
stresses are updated and proceeded 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.
3.3.3 Element Formulation
The governing equation which is called equation of motion is:
Tlj + bi = pul (3.1)
49
where r is the Cauchy total stress tensor, bt is the body force per unit volume, p is the
density, ux are the relative displacements, Q represents the continuum domain. The
boundary of continuum can be divided into two parts as the boundary Tu where
displacements are described and the boundary Fr where stresses are described and the
conditions on the boundary can be defined as:
Ui = Ui
(3.2)
and
respectively.
TijTlj = Ti
(3.3)
The initial conditions are:
u,{ 0) = Ui,
(3.4)
Ui(0) = Ui,
(3.5)
The rate deformation tensor dXJ is defined in terms of velocity u as:
dij ^ O'Cj 4~ Uj L
(3.6)
The Cauchy stress is, in general, a function of rate of deformation d!;) a set of history
variables H, and the temperature T:
Tjj Tjj (dij, H, T)
(3.
50
In NIKE codes, quadrilateral elements are used for the spatial discretization, yielding a
system of ordinary differential equations:
Mii + Fint (u, u, T) = P(u, b, t, T) (3.8)
Where M is the mass matrix, F is the internal nodal force vector, and P is the external
nodal force vector. P can be a function of nodal displacement u, body force per unit
volume b, time t, and nodal temperature T. In dynamic analysis the Equation 3.8 is
solved by Newmark-d method. For quasistatic analysis the Equation 3.8 becomes:
Fintfu,u,T) = P(u,b,t,T) (3.9)
In quasist.atic analysis the equation 3.9 is solved incrementally by an iterative strategy.
Without thermal and viscous effects the Equation 3.9 becomes:
Fint(un+1) = P+i (3.10)
where un+1 and Pn+i are the nodal displacements and external load evaluation at time
tn+1- At each time step, quantities are known at tn, and the solution involves finding the
displacement un+i that satisfies the equation below:
K'Au1 = Pn+1-Fint+1) (3.11)
This new equilibrium solution is found by iteration. To obtain the solution at tn+i, the
finite element Equations 3.9 are first linearized about the configuration at tn and the
displacements at tn+ \ are determined. Later the Equation 3.11 solved iteratively.
51
Convergence is determined by examining the both the displacement and energy norms.
For displacement norm
|| An11|
< dd
(3.12)
where umax is the maximum displacement norm obtained overall of the n steps is used,
and for energy norm:
(At^QUi
(Au)QÂ£+1 ~Cd
is used. Where e
(3.13)
For dynamic analysis the N'e\vmark-/3 method is used to integrate the semidiscretized
finite element equations 3.1 in time. The Newmark-/? family methods is given by:
un4i = u + Afun + (1/2 f3)At2un + /3At2iin+i (3.14)
u+i = u + (1 l)Atun + 7Afiin+i (3.15)
In NIKE2D the default values for 13 and 7 are 1/4 and 1/2 respectively. This choice of
parameters represents the trapezoidal rule which is second order accurate in time, is
energy conserving for linear problems and does not introduce numerical dissipation.
To obtain the dynamic solution at <+i, the finite element equations 3.1 are first
linearized about the configuration at tn and using the Expressions 3.14 and 3.15 the
nodal acceleration, velocity, and displacements at /.+1 are obtained. Finally the
52
equilibrium iterations performed with:
K*Au! = Pn+i F*(u)1+1)
(3.16)
where
(3.17)
(3.18)
where M is the damping matrix. For dynamic analysis, damping can be incorporated
material behavior or by a nonlinear adaptation of Rayleigh damping. Dissipative
mechanisms in material behavior are pointwise in nature and include viscoelasticity and
include hvsteretic damping caused by cyclic plasticity. Generalized Rayleigh damping is
more global in nature. However, it is implemented at the element level to maximize
computational efficiency.
3.3.4 Material Models
NIKE3D has a wide range of material library. Among others Ramberg-Osgood with its
hysteresis behavior is most suitable material model for soils. Ramberg-Osgood material
model is discussed in Section 2.6.4 in detail. UCD Geotechnical Engineering Center is
collaborating with Lawrence Livermore National Laboratory to make new soil models
into the model at the element level in two ways: through the specification of dissipative
53