Citation
Evaluation of fiber reinforced polymer tube bracing as a seismic retrofit method for existing building

Material Information

Title:
Evaluation of fiber reinforced polymer tube bracing as a seismic retrofit method for existing building
Creator:
Hessek, Christopher J. ( author )
Language:
English
Physical Description:
1 electronic file (197 pages). : ;

Subjects

Subjects / Keywords:
Concrete-filled tubes ( lcsh )
Loads (Mechanics) ( lcsh )
Tubes ( lcsh )
Concrete-filled tubes ( fast )
Loads (Mechanics) ( fast )
Tubes ( fast )
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Review:
Thousands of buildings throughout the United States were designed and constructed prior to the implementation of formal seismic design standards. These buildings have an increased risk of collapse if an earthquake event were to occur. Many seismic retrofit techniques are currently used to improve the structural performance of these buildings during earthquake events. The aim of this study is to determine if the use of Fiber Reinforced Polymer FRP tubes as diagonal bracing members is a viable option for the seismic retrofit of existing buildings. ( ,, )
Review:
A parametric analytical study has been conducted to predict the performance of axially loaded FRP bracing members within a representative building structure subjected to simulated seismic loads. 23 unique models were developed and tested using the structural analysis software SAP2000. Various combinations of structural parameters such as bracing configuration material properties and loading type were evaluated to obtain an understanding of their effect on the structure s ability to resist seismic load. Bracing materials used include hollow structural steel tubes hollow FRP tubes and FPR confined concrete tubes with the hollow structural steel tubes being included as a control case.
Review:
Hollow FRP tubes were found to perform poorly under seismic loading primarily due to their very brittle failure mode remaining linear elastic until ultimate rupture. The ductility ratios of the FRP confined concrete tubes were found to be 40 to 50 percent less than those of the hollow structural steel tubes. However FRP confined concrete tubes performed sufficiently enough to warrant continuing research efforts regarding their use for seismic bracing applications.
Thesis:
thesis (M.S.)--University of Colorado Denver.
Bibliography:
Includes bibliographic references.
System Details:
System requirements: Adobe Reader.
General Note:
Department of Civil Engineering
Statement of Responsibility:
by Christopher J. Hessek.

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Source Institution:
|University of Colorado Denver
Holding Location:
Auraria Library
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
930181130 ( OCLC )
ocn930181130

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Full Text
EVALUATION OF FIBER REINFORCED POLYMER TUBE BRACING AS A
SEISMIC RETROFIT METHOD FOR EXISTING BUILDINGS
by
CHRISTOPHER J. HESSEK
B.S., Colorado State University, 2005
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillment
of the requirements for the degree of
Master of Science
Civil Engineering
2015


This thesis for the Master of Science degree by
Christopher J. Hessek
has been approved for the
Civil Engineering Program
by
Yail Jimmy Kim, Chair
Chengyu Li
Frederick R. Rutz
July 24, 2015
n


Hessek, Christopher, J. (M.S., Civil Engineering)
Evaluation of Fiber Reinforced Polymer Tube Bracing as a Seismic Retrofit Method for
Existing Buildings
Thesis directed by Associate Professor Dr. Yail Jimmy Kim
ABSTRACT
Thousands of buildings throughout the United States were designed and constructed prior
to the implementation of formal seismic design standards. These buildings have an
increased risk of collapse if an earthquake event were to occur. Many seismic retrofit
techniques are currently used to improve the structural performance of these buildings
during earthquake events. The aim of this study is to determine if the use of Fiber
Reinforced Polymer (FRP) tubes as diagonal bracing members is a viable option for the
seismic retrofit of existing buildings.
A parametric analytical study has been conducted to predict the performance of
axially loaded FRP bracing members within a representative building structure, subjected
to simulated seismic loads. 23 unique models were developed and tested using the
structural analysis software SAP2000. Various combinations of structural parameters
such as bracing configuration, material properties, and loading type were evaluated to
obtain an understanding of their effect on the structures ability to resist seismic load.
Bracing materials used include hollow structural steel tubes, hollow FRP tubes, and FPR
confined concrete tubes, with the hollow structural steel tubes being included as a control
m
case.


Hollow FRP tubes were found to perform poorly under seismic loading, primarily
due to their very brittle failure mode remaining linear elastic until ultimate rupture. The
ductility ratios of the FRP confined concrete tubes were found to be 40 to 50 percent less
than those of the hollow structural steel tubes. However, FRP confined concrete tubes
performed sufficiently enough to warrant continuing research efforts regarding their use
for seismic bracing applications.
The form and content of this abstract are approved. I recommend its publication.
Approved: Yail Jimmy Kim
IV


ACKNOWLEDGMENTS
First and foremost, I would like to acknowledge the sacrifice of my wife and
children while I have pursued a Masters of Science degree. Their willingness to allow
my time away from the family for the countless nights and weekends of schoolwork has
not gone unnoticed.
I would also like to thank my committee members Dr. Jimmy Kim, Dr. Chengyu
Li, and Dr. Fredrick Rutz for their time in reviewing my work.
Finally, I would like to express my gratitude to my thesis advisor, Dr. Jimmy
Kim, for his encouragement and guidance throughout my work. His expertise and wealth
of knowledge in my area of study was a valuable resource that was essential to the
success of this project. Dr. Kims positive, pleasant demeanor was refreshing and
uplifting.
v


TABLE OF CONTENTS
List of Tables..............................................................x
List of Figures.............................................................xi
Notation................................................................... xxiii
Chapter
1. Introduction............................................................. 1
1.1 Introduction......................................................... 1
1.2 Research Significance............................................... 1
1.3 Objectives.......................................................... 2
1.4 Scope............................................................... 3
1.5 Thesis Outline...................................................... 4
2. Literature Review........................................................ 5
2.1 Introduction........................................................5
2.2 Code-Based Seismic Analysis and Design.............................. 5
2.2.1 Seismic Design Criteria ASCE 7-10............................ 6
2.2.2 Seismic Design Requirements for Building Structures ASCE 7-10. 8
2.2.3 Seismic Response History Procedures ASCE 7-10................ 11
2.3 Seismic Retrofit of Existing Buildings.............................. 14
2.3.1 S ei smi c Retrofit B ackground................................. 15
2.3.2 Common Retrofit Techniques Currently in Use.................... 17
2.3.3 Seismic Retrofit Process ASCE 41-13.......................... 19
2.3.4 Performance Objective for Existing Buildings ASCE 41-13......20
2.4 Fiber Reinforced Polymer Overview................................... 22
vi


2.4.1 Material Properties of FRP Tubes
23
2.5 Hollow FRP Tubes as Axially Loaded Structural Members............. 24
2.5.1 Axial Tensile Properties..........................................24
2.5.2 Axial Compressive Properties......................................26
2.5.3 Stress-Strain Behavior of Hollow FRP Tubes........................28
2.6 FRP Confined Concrete Tubes as Axially Loaded Structural Members.....29
2.6.1 Confinement Characteristics...................................... 29
2.6.2 Linear Elastic Behavior Confinement Model........................ 31
2.6.3 Nonlinear Behavior Confinement Model..............................35
2.6.4 Axial Tensile Properties..........................................36
2.6.5 Axial Compressive Properties Cyclic Loading.................... 37
2.7 Current Use of FRP in Seismic Applications...........................38
3. Analytical Modeling....................................................... 54
3.1 Introduction.......................................................... 54
3.2 Model of Three-Story Steel Framed Building Using SAP2000............. 54
3.2.1 Description of the Analytical Model...............................54
3.2.2 Structural Analysis Approach..................................... 56
3.2.3 Modal Analysis....................................................57
3.2.4 Non-Linear Static Pushover Analysis...............................57
3.2.5 Non-Linear Dynamic Analysis Using Time-History Acceleration
Function........................................................ 59
3.2.6 Dynamic Loading of Model Using El Centro Earthquake Ground
Motion...........................................................60
vii


3.3 Parametric Study Parameters
60
3.3.1 Matrix of Parameters Evaluated..................................61
3.3.2 Stress-Strain Curve of ASTM A500 Grade B Steel Tubes............62
3.3.3 Stress-Strain Curves of Hollow FRP Tubes....................... 62
3.3.4 Stress-Strain Curves of FRP Confined Concrete...................63
3.3.5 Assumed Failure Mode......................................... 64
3.4 Model Validation................................................... 65
3.4.1 Validation of Strength and Stiffness Properties................ 65
3.4.2 Validation of Static Pushover Analysis......................... 66
3.4.3 Validation of Modal Response Properties.........................67
4. Analytical Results and Discussion.........................................83
4.1 Introduction.........................................................83
4.2 Results of Modal Participation and Frequency Analysis............. 83
4.3 Results of Nonlinear Static Pushover Analysis..................... 84
4.4 Results of Nonlinear Time History Analysis........................ 86
4.5 Discussion and Comparison of Results.................................90
5. Summary and Conclusions..................................................152
5.1 Summary............................................................. 152
5.2 Conclusions........................................................ 152
5.3 Design Recommendations..............................................153
5.4 Recommendations for Future Work.................................... 154
viii
References
156


Appendix
A El Centro Earthquake Ground Acceleration Record........................158
B SAP2000 Modeling Procedure.............................................162
IX


LIST OF TABLES
Table
3-1: Matrix of Parameters Evaluated......................................... 68
3-2: Axial Properties of Hollow FRP Tubes Evaluated....................... 69
3- 3: Axial Properties of FRP Confined Concrete Evaluated.................. 69
4- 1: Summary of Modal Response of Each Bracing Configuration...............94
4-2: Summary of Non-Linear Dynamic Analysis Results........................95
x


LIST OF FIGURES
Figure
2-1: Map of MCER Spectral Response Parameters United States
(FEMA P-750, 2009)................................................... 40
2-2: Design Response Spectrum Curve (FEMA P-750, 2009).................... 40
2-3: Equal Displacement Approximation Theorem (FEMA, P-750, 2009)......... 41
2-4: Various Nonlinear Force-Deformation Curves (FEMA 440, 2005).......... 41
2-5a: Ductile Hysteresis Loops (FEMA P-750, 2009)...........................42
2-5b: Pinched Hysteresis Loops (FEMA P-750, 2009)...........................42
2-6: Fiber Reinforced Polymer General Makeup (ISIS Candada, 2006)......... 43
2-7: Stress-Strain Curves of Various Fibers (ISIS Candada, 2006).......... 43
2-8: Pultrusion Process Schematic Diagram (Strongwell, 2007).............. 44
2-9: Filament Winding Process for Fabricating FRP Tubes (Warner, 2000).....44
2-10: Failure Strain of Matrix Less than that of Fiber (ISIS Canada, 2006). 45
2-11: Failure Strain of Matrix More than that of Fiber (ISIS Canada, 2006). 45
2-12: Fiber Buckling from Axial Compressive Loads (Jones, 1999)............ 46
2-13: Experimental Results for Fiber Buckle Wavelength versus Fiber Diameter
(Jones, 1999).........................................................46
2-14: Buckling of a Discretely Supported Euler Column (Jones, 1999).........47
2-15: Effectively Confined Core for Rectangular Hoop Reinforcement
(Mander, 1988)....................................................... 47
2-16: Confining Action of FRP Shell on Concrete Core (Ozbakkaloglu
et. al., 2012)........................................................48
xi


2-17: Representative Stress-Strain Curves for Confined versus Unconfined
Concrete (Fam and Rizkalla, 2001).....................................48
2-18: Solid Cylinder and Thin Shell under Different Stresses (Fam and
Rizkalla, 2001)........................................................49
2-19a: Varying Modulus of Elasticity of Concrete (Fam and Rizkalla, 2001)....49
2-19b: Varying Poissons Ratio of Concrete (Fam and Rizkalla, 2001).......... 50
2-19c: Nonlinear Stress-Strain Curve for Confined Concrete (Fam and Rizkalla,
2001)................................................................. 50
2-20: Example Test Setup Diagram Axially-Loaded FRP Confined Concrete
Cylinder (Ozbakkaloglu & Akin, 2012)...................................51
2-21: Experimental Stress-Strain Curve for Normal-Strength Concrete Confined
with Two Plies of CFRP (Lam et. al., 2006)............................ 51
2-22: Experimental Stress-Strain Curve for High-Strength Concrete Confined
with Six Plies of CFRP (Ozbakkaloglu and Akin, 2012)..................52
2-23: Experimental Stress-Strain Curve for Normal-Strength Concrete Confined
with Two Plies of AFRP (Ozbakkaloglu and Akin, 2012)..................52
2- 24: Experimental Stress-Strain Curve for High-Strength Concrete Confined
with Six Plies of AFRP (Ozbakkaloglu and Akin, 2012)...................53
3- 1: Plan View Typical Floor and Roof Framing.............................70
3-2: Isometric Model View - Single-Bay X-Bracing - Extruded Sections....71
3-3: Isometric Model View - Single-Bay X-Bracing - Joint Labels......... 71
3-4: Isometric Model View - Single-Bay X-Bracing - Member Labels.........72
xii


3-5: Isometric Model View Single-Bay X-Bracing Individual Joint
Restraints............................................................ 72
3-6: Elevation Model View Single-Bay X-Bracing Member Labels..........73
3-7: Elevation Model View Single-Bay X-Bracing Mode Shape 1........ 73
3-8: Elevation Model View Single-Bay X-Bracing Mode Shape 2.........74
3-9: Elevation Model View Single-Bay X-Bracing Mode Shape 3.........74
3-10: Isometric Model View Super X-Bracing Extruded Sections..............75
3-11: Isometric Model View Super X-Bracing Joint Labels.................. 75
3-12: Isometric Model View Super X-Bracing Member Labels................. 76
3-13: Elevation Model View Super X-Bracing Member Labels..................75
3-14: Isometric Model View Chevron Bracing Extruded Sections..............77
3-15: Isometric Model View Chevron Bracing Joint Labels...................77
3-16: Isometric Model View Chevron Bracing Member Labels..................78
3-17: Elevation Model View Chevron Bracing Member Labels..................78
3-18: Nonlinear Static Pushover Load Case Parameters........................79
3-19: El Centro Time-History Load Case Parameters...........................79
3-20: El Centro Time-History Plot.............................................80
3-21: Material Stress-Strain Plot Steel ASTM A500 Grade B.................80
3-22: Material Stress-Strain Plot Hollow GFRP Tube Strongwell Extren
500/525............................................................... 81
3-23: Material Stress-Strain Plot Hollow CFRP Tube GW Composites..........81
3-24: Material Stress-Strain Plot Normal Strength Concrete Confined by Two
Plies of CFRP..........................................................81
xiii


3-25: Material Stress-Strain Plot High Strength Concrete Confined by Six
Plies of CFRP....................................................... 82
3-26: Material Stress-Strain Plot Normal Strength Concrete Confined by Two
Plies of AFRP........................................................82
3- 27: Material Stress-Strain Plot High Strength Concrete Confined by Six
Plies of AFRP........................................................82
4- la: Static Pushover Curve Base Shear vs. Roof Displ. Model #1.........96
4-lb: Non-Linear Time-History - Base Shear vs. Time Model #1..............96
4-lc: Non-Linear Time-History - Roof Displacement. Vs. Time Model #1......96
4-ld: Non-Linear Time-History - Roof Acceleration vs. Time Model #1.......97
4-le: Non-Linear Time-History - M2 Axial Force vs. Time Model #1..........97
4-lf: Non-Linear Time-History - Kinetic Energy vs. Time Model #1..........97
4-2a: Static Pushover Curve Base Shear vs. Roof Displ. Model #2..........98
4-2b: Non-Linear Time-History - Base Shear vs. Time Model #2..............98
4-2c: Non-Linear Time-History - Roof Displacement vs. Time Model #2...... 98
4-2d: Non-Linear Time-History - Roof Acceleration vs. Time Model #2...... 99
4-2e: Non-Linear Time-History - M2 Axial Force vs. Time Model #2..........99
4-2f: Non-Linear Time-History - Kinetic Energy vs. Time Model #2..........99
4-3a: Static Pushover Curve Base Shear vs. Roof Displ. Model #3..........100
4-3b: Non-Linear Time-History - Base Shear vs. Time Model #3.............100
4-3c: Non-Linear Time-History - Roof Displacement vs. Time Model #3......100
4-3d: Non-Linear Time-History - Roof Acceleration vs. Time Model #3..... 101
4-3e: Non-Linear Time-History M2 Axial Force vs. Time Model #3...........101
xiv


101
102
102
102
103
103
103
104
104
104
105
105
105
106
106
106
107
107
107
108
108
108
109
Non-Linear Time-History Kinetic Energy vs. Time Model #3..
Static Pushover Curve Base Shear vs. Roof Displ. Model #4.
Non-Linear Time-History Base Shear vs. Time Model #4......
Non-Linear Time-History Roof Displacement vs. Time Model #4
Non-Linear Time-History Roof Acceleration vs. Time Model #4..
Non-Linear Time-History M2 Axial Force vs. Time Model #4....
Non-Linear Time-History Kinetic Energy vs. Time Model #4..
Static Pushover Curve Base Shear vs. Roof Displ. Model #5.
Non-Linear Time-History Base Shear vs. Time Model #5......
Non-Linear Time-History Roof Displacement vs. Time Model #5
Non-Linear Time-History Roof Acceleration vs. Time Model #5..
Non-Linear Time-History M2 Axial Force vs. Time Model #5....
Non-Linear Time-History Kinetic Energy vs. Time Model #5..
Static Pushover Curve Base Shear vs. Roof Displ. Model #6.
Non-Linear Time-History Base Shear vs. Time Model #6......
Non-Linear Time-History Roof Displacement vs. Time Model #6
Non-Linear Time-History Roof Acceleration vs. Time Model #6..
Non-Linear Time-History M2 Axial Force vs. Time Model #6....
Non-Linear Time-History Kinetic Energy vs. Time Model #6..
Static Pushover Curve Base Shear vs. Roof Displ. Model #7.
Non-Linear Time-History Base Shear vs. Time Model #7......
Non-Linear Time-History Roof Displacement vs. Time Model #7
Non-Linear Time-History Roof Acceleration vs. Time Model #7..
xv


4-7e: Non-Linear Time-History M2 Axial Force vs. Time Model #7.........109
4-7f: Non-Linear Time-History Kinetic Energy vs. Time Model #7.........109
4-8a: Static Pushover Curve Base Shear vs. Roof Displ. Model #8........110
4-8b: Non-Linear Time-History Base Shear vs. Time Model #8..............110
4-8c: Non-Linear Time-History Roof Displacement vs. Time Model #8.......110
4-8d: Non-Linear Time-History Roof Acceleration vs. Time Model #8...... Ill
4-8e: Non-Linear Time-History M2 Axial Force vs. Time Model #8.........Ill
4-8f: Non-Linear Time-History Kinetic Energy vs. Time Model #8.........Ill
4-9a: Static Pushover Curve Base Shear vs. Roof Displ. Model #9........112
4-9b: Non-Linear Time-History Base Shear vs. Time Model #9..............112
4-9c: Non-Linear Time-History Roof Displacement vs. Time Model #9.......112
4-9d: Non-Linear Time-History Roof Acceleration vs. Time Model #9...... 113
4-9e: Non-Linear Time-History M2 Axial Force vs. Time Model #9.........113
4-9f: Non-Linear Time-History Kinetic Energy vs. Time Model #9.........113
4-10a: Static Pushover Curve Base Shear vs. Roof Displ. Model #10......114
4-10b: Non-Linear Time-History Base Shear vs. Time Model #10............114
4-10c: Non-Linear Time-History Roof Displacement vs. Time Model #10.... 114
4-10d: Non-Linear Time-History Roof Acceleration vs. Time Model #10.....115
4-10e: Non-Linear Time-History M2 Axial Force vs. Time Model #10........115
4-10f: Non-Linear Time-History Kinetic Energy vs. Time Model #10........115
4-1 la: Static Pushover Curve Base Shear vs. Roof Displ. Model #11.....116
4-1 lb: Non-Linear Time-History Base Shear vs. Time Model #11...........116
4-1 lc: Non-Linear Time-History Roof Displacement vs. Time Model #11... 116
xvi


4-1 Id: Non-Linear Time-History Roof Acceleration vs. Time Model #11... 117
4-1 le: Non-Linear Time-History M2 Axial Force vs. Time Model #11.......117
4-1 If: Non-Linear Time-History Kinetic Energy vs. Time Model #11.......117
4-12a: Static Pushover Curve Base Shear vs. Roof Displ. Model #12......118
4-12b: Non-Linear Time-History - Base Shear vs. Time Model #12...........118
4-12c: Non-Linear Time-History - Roof Displacement vs. Time Model #12....118
4-12d: Non-Linear Time-History - Roof Acceleration vs. Time Model #12....119
4-12e: Non-Linear Time-History M2 Axial Force vs. Time Model #12.......119
4-12f: Non-Linear Time-History Kinetic Energy vs. Time Model #12.......119
4-13a: Static Pushover Curve Base Shear vs. Roof Displ. Model #13......120
4-13b: Non-Linear Time-History - Base Shear vs. Time Model #13...........120
4-13c: Non-Linear Time-History - Roof Displacement vs. Time Model #13... 120
4-13d: Non-Linear Time-History - Roof Acceleration vs. Time Model #13... 121
4-13e: Non-Linear Time-History M2 Axial Force vs. Time Model #13.......121
4-13f: Non-Linear Time-History Kinetic Energy vs. Time Model #13.......121
4-14a: Static Pushover Curve Base Shear vs. Roof Displ. Model #14......122
4-14b: Non-Linear Time-History - Base Shear vs. Time Model #14...........122
4-14c: Non-Linear Time-History - Roof Displacement vs. Time Model #14... 122
4-14d: Non-Linear Time-History - Roof Acceleration vs. Time Model #14....123
4-14e: Non-Linear Time-History M2 Axial Force vs. Time Model #14.......123
4-14f: Non-Linear Time-History Kinetic Energy vs. Time Model #14.......123
4-15a: Static Pushover Curve Base Shear vs. Roof Displ. Model #15......124
4-15b: Non-Linear Time-History Base Shear vs. Time Model #15...........124
xvii


4-15c: Non-Linear Time-History Roof Displacement vs. Time Model #15.... 124
4-15d: Non-Linear Time-History Roof Acceleration vs. Time Model #15......................................125
4-15e: Non-Linear Time-History - M2 Axial Force vs. Time Model #15. 125
4-15f: Non-Linear Time-History - Kinetic Energy vs. Time Model #15..125
4-16a: Static Pushover Curve Base Shear vs. Roof Displ. Model #16......126
4-16b: Non-Linear Time-History Base Shear vs. Time Model #16......................................126
4-16c: Non-Linear Time-History Roof Displacement vs. Time Model #16..................................... 126
4-16d: Non-Linear Time-History Roof Acceleration vs. Time Model #16......................................127
4-16e: Non-Linear Time-History - M2 Axial Force vs. Time Model #16..127
4-16f: Non-Linear Time-History - Kinetic Energy vs. Time Model #16..127
4-17a: Static Pushover Curve Base Shear vs. Roof Displ. Model #17......128
4-17b: Non-Linear Time-History Base Shear vs. Time Model #17......................................128
4-17c: Non-Linear Time-History Roof Displacement vs. Time Model #17......................................128
4-17d: Non-Linear Time-History Roof Acceleration vs. Time Model #17......................................129
4-17e: Non-Linear Time-History - M2 Axial Force vs. Time Model #17..129
4-17f: Non-Linear Time-History - Kinetic Energy vs. Time Model #17..129
4-18a: Static Pushover Curve Base Shear vs. Roof Displ. Model #18......130
4-18b: Non-Linear Time-History Base Shear vs. Time Model #18......................................130
4-18c: Non-Linear Time-History Roof Displacement vs. Time Model #18..................................... 130
4-18d: Non-Linear Time-History Roof Acceleration vs. Time Model #18..................................... 131
4-18e: Non-Linear Time-History - M2 Axial Force vs. Time Model #18..131
4-18f: Non-Linear Time-History - Kinetic Energy vs. Time Model #18..131
4-19a: Static Pushover Curve Base Shear vs. Roof Displ. Model #19......132
xviii


4-19b: Non-Linear Time-History Base Shear vs. Time Model #19............132
4-19c: Non-Linear Time-History Roof Displacement vs. Time Model #19... 132
4-19d: Non-Linear Time-History Roof Acceleration vs. Time Model #19....133
4-19e: Non-Linear Time-History M2 Axial Force vs. Time Model #19.......133
4-19f: Non-Linear Time-History Kinetic Energy vs. Time Model #19.......133
4-20a: Static Pushover Curve Base Shear vs. Roof Displ. Model #20......134
4-20b: Non-Linear Time-History Base Shear vs. Time Model #20...........134
4-20c: Non-Linear Time-History Roof Displacement vs. Time Model #20... 134
4-20d: Non-Linear Time-History Roof Acceleration vs. Time Model #20....135
4-20e: Non-Linear Time-History M2 Axial Force vs. Time Model #20.......135
4-20f: Non-Linear Time-History Kinetic Energy vs. Time Model #20.......135
4-21a: Static Pushover Curve Base Shear vs. Roof Displ. Model #21......136
4-21b: Non-Linear Time-History Base Shear vs. Time Model #21...........136
4-21c: Non-Linear Time-History Roof Displacement vs. Time Model #21... 136
4-21d: Non-Linear Time-History Roof Acceleration vs. Time Model #21... 137
4-21e: Non-Linear Time-History M2 Axial Force vs. Time Model #21.......137
4-21f: Non-Linear Time-History Kinetic Energy vs. Time Model #21.......137
4-22a: Static Pushover Curve Base Shear vs. Roof Displ. Model #22......138
4-22b: Non-Linear Time-History Base Shear vs. Time Model #22...........138
4-22c: Non-Linear Time-History Roof Displacement vs. Time Model #22... 138
4-22d: Non-Linear Time-History Roof Acceleration vs. Time Model #22....139
4-22e: Non-Linear Time-History M2 Axial Force vs. Time Model #22.......139
4-22f: Non-Linear Time-History Kinetic Energy vs. Time Model #22........139
xix


4-23a: Static Pushover Curve Base Shear vs. Roof Displ. Model #23.....140
4-23b: Non-Linear Time-History Base Shear vs. Time Model #23............140
4-23c: Non-Linear Time-History Roof Displacement vs. Time Model #23.....140
4-23d: Non-Linear Time-History Roof Acceleration vs. Time Model #23.....141
4-23e: Non-Linear Time-History M2 Axial Force vs. Time Model #23........141
4-23f: Non-Linear Time-History Kinetic Energy vs. Time Model #23........141
4-24a: Static Pushover Curve Comparison - Single-Bay X-Brace Models.......142
4-24b: Static Pushover Curve Comparison - Single-Bay X-Brace Models.......142
4-25a: Static Pushover Curve Comparison - Super X-Brace Models............143
4-25b: Static Pushover Curve Comparison - Super X-Brace Models............143
4-26: Static Pushover Curve Comparison - Chevron Brace Models............144
4-27: Comparison of Maximum Elastic Displacement vs. Maximum Total
Displacement Static Pushover Analysis............................144
4-28: Ratio between Maximum Total Displacement and Maximum Elastic
Displacement Static Pushover Analysis........................... 145
4-29a: Graphical Representation of Ductility Ratio Typical Steel Brace. 145
4-29b: Graphical Representation of Ductility Ratio Typical Hollow FRP Brace.. 146
4-29c: Graphical Representation of Ductility Ratio Typical FRP Confined
Concrete Brace.................................................... 146
4-30: Comparison of Maximum Base Shear Nonlinear Dynamic Earthquake
Loading............................................................147
4-31: Comparison of Maximum Roof Displacement Nonlinear Dynamic
Earthquake Loading.................................................147
xx


4-32: Comparison of Maximum Roof Acceleration Nonlinear Dynamic
Earthquake Loading...................................................148
4-33: Comparison of Maximum Axial Force in Level 1 Bracing Member -
Nonlinear Dynamic Earthquake Loading.................................148
4-34: Comparison of Kinetic Energy vs. Time Non-Linear Dynamic Earthquake
Loading..............................................................149
4-35: Correlation between Natural Period of Structure and Base Shear from
Dynamic Earthquake Loading...........................................149
4-36: Correlation between Maximum Static Pushover Displacement and
Maximum Displacement from Dynamic Earthquake Loading.................150
4-37: Correlation between Maximum Base Shear and Maximum Roof
Acceleration from Dynamic Earthquake Loading.........................150
4-38: Correlation between Maximum Base Shear and Maximum Axial Force in
Level 1 Bracing Member from Dynamic Earthquake Loading...............151
B-l: New Model Startup Window..............................................162
B-2: 3D Frames Window..................................................... 162
B-3: Basic Material Properties Window......................................163
B-4: Advanced Material Properties Window.................................. 163
B-5: Nonlinear Material Data Window....................................... 164
B-6: Frame Member Properties Window....................................... 165
B-7: Section Properties Window.............................................165
B-8: Area Sections Window..................................................166
B-9: Shell Section Data Window............................................ 166
xxi


167
168
168
169
169
170
170
171
171
172
Mass Source Data Window
Define Constraints Window......................
Diaphragm Constraints Window...................
Define Time History Functions Window...........
Time History Function Definition Window........
Define Load Cases Window.......................
Load Case Data Modal Window..................
Load Case Data Nonlinear Static Window.......
Load Case Data Nonlinear Time History Window
Set Load Cases to Run Window...................
XXII


NOTATION
C damping matrix of structure
Cs seismic response coefficient (dimensionless)
Cm vertical distribution factor
E modulus of elasticity
Ec secant modulus of concrete
Ef modulus of elasticity of the reinforcing fibers
Efrp modulus of elasticity of FRP
Em modulus of elasticity of the polymer matrix
Es elastic modulus of tube in hoop direction
fee axial stress of confined concrete at general point
fc compressive strength of unconfined concrete
fee peak strength of confined concrete
Fx portion of the seismic base shear, V, induced at Level x
hx the height above the base to Level x
/ moment of inertia
Ie seismic importance factor
k distribution exponent
K stiffness matrix of structure
L column length
m number of discrete lateral supports along the length of the column
M diagonal mass matrix of structure
P column axial load
r constant in Manders equation relates initial tangential modulus to secant modulus
of concrete, applied load
xxm


R response modification coefficient, radius of confining tube
Ss mapped MCER, 5 percent damped, spectral response acceleration parameter at
short periods
Si mapped MCER, 5 percent damped, spectral response acceleration parameter at a
period of 1 second
Sds design, 5 percent damped, spectral response acceleration parameter at short
periods
Sdi design, 5 percent damped, spectral response acceleration parameter at a period of
1 second
Sms the MCER, 5 percent damped, spectral response acceleration parameter at short
periods adjusted for site class effects
Smi the MCEr, 5 percent damped, spectral response acceleration parameter at a period
of 1 second adjusted for site class effects
t thickness of confining tube, time
T fundamental period of the building
Tl long period transition period
u displacement of structure
ii velocity of structure
u acceleration of structure
uR radial displacement of point at interface surface between concrete core and outer
tube
V total design lateral force or shear at the base
Vf volume fraction of reinforcing fibers
Vm volume fraction of polymer matrix
W effective seismic weight of the building
wx portion of Wthat is located at or assigned to Level x
x ratio between any axial strain level of confined concrete and strain corresponding
to peak strength under constant confining pressure
£cc general axial strain level of confined concrete
XXIV


£ c strain at peak strength of unconfined concrete
'cc strain at peak strength of confined concrete under constant confining pressure
£y strain level in confining jacket in axial direction
Ofrp,uit ultimate tensile strength of FRP
(£m,uit ultimate tensile strength of polymer matrix
(JR confining pressure at interface between concrete core and jacket
as tensile stress in jacket in hoop direction due to given confining pressure
/V Poissons ratio
uc Poissons ratio of concrete at given load level
XXV


1.0 Introduction
1.1 Introduction
The seismic analysis and design procedures in the building codes currently in use
in the United States are very robust and will generally produce safe, reliable
structures. However, the development of these design requirements for new buildings is
relatively recent, meaning that there are many thousands of existing buildings throughout
the country that were designed and constructed prior to the implementation of sound
design standards. Public safety is understandably a concern within these buildings. If an
earthquake were to occur, there is an increased risk of collapse since they were not
specifically designed and detailed to resist the imposed forces.
Fiber-reinforced polymer structural members are increasingly being used as an
alternative to traditional building materials such as steel, concrete, and wood in certain
applications. FRP members can have distinct advantages over the traditional materials
such as being lightweight, corrosion-resistant, and having high tensile strengths. FRP
tubes are a potentially cost-effective and reliable method to seismically retrofit existing
buildings.
1.2 Research Significance
New innovative methods of seismic bracing are needed to address the large
number of seismically deficient buildings in the United States. These bracing techniques
should ideally be easy to install and have minimal impact on the operation of existing
1


buildings, since they will predominately be installed as retrofit measures while existing
buildings are still in service.
Recent research by Nakahara et al. in 2013 has investigated the use of concrete
filled steel tubes as diagonal members for seismic retrofit. While this system may be
prudent in certain applications, it may be advantageous to instead use FRP tubes in place
of mild steel tubes. Material properties of FRP tubes can be configured such that its
tensile strength can far exceed that of steel. The author is not aware of any published
research that investigates the use of concrete filled FRP tubes specifically as diagonal
bracing members in a buildings lateral force-resisting system.
Lam et al. in 2006 and Ozbakkaloglu and Akin in 2012 studied the behavior of
FRP confined concrete cylinders subjected to simulated seismic loads in the axial
direction. Experimental testing was performed in each of these studies, with a somewhat
arbitrary reversible load applied to the FRP confined concrete cylinder system. The
current study evaluates the behavior of both hollow FRP tubes and FRP confined
concrete tubes subjected to real ground acceleration records from actual past earthquake
events. Similarly, the author is not aware of any published research that studies the FRP
confined concrete tube system under loading from true earthquake ground acceleration
records.
1.3 Objectives
The objective of this study is to determine if the use of FRP tubes as diagonal
bracing members is a viable option for the seismic retrofit of existing buildings. The
behavior of hollow FRP and FRP confined concrete tubes under a variety of static
2


loading conditions (tension, compression, bending) has been extensively studied and is
well known. However, a limited amount of research has been conducted to determine
how these FRP members behave when subjected to dynamic loading such as that
resulting from earthquake ground motions.
Seismic forces from earthquakes create a very unique type of loading and structural
behavior. The applied force is cyclical and reversible due to the ground moving back and
forth during a seismic event. This cyclical force creates repetitive loading and unloading
of the structure. It is not economically feasible to design structures to behave elastically
and remain essentially undamaged in the event of a Maximum Considered Earthquake,
which is expected to occur at a given site approximately once every 2500 years.
Therefore, buildings are typically designed to behave nonlinearly and experience
permanent damage under design seismic loads. Through analysis, testing, and
performance results from actual buildings, traditional building materials such as steel and
reinforced concrete have been proven to be safe and reliable when this certain amount of
permanent damage is allowed to occur. This study investigates whether this same type of
predictable nonlinear behavior can be expected in FRP tube bracing, which will
determine if FRP tubes can be used for retrofit diagonal bracing in existing buildings.
1.4 Scope
The scope of the research consists of a parametric study of a sample three-story
building with FRP tubes inserted as bracing elements. A structural analytical model was
created for the sample building and subjected to nonlinear static and nonlinear dynamic
lateral loads to simulate seismic ground motion events. Various bracing configurations
3


were evaluated for the two loading scenarios. The individual brace and overall building
response were studied for each of the bracing configurations.
1.5 Thesis Outline
The contents of the thesis are briefly outlined below:
Chapter 2: presents a literature review related to current seismic analysis and design
standards currently used in the United States. Current seismic retrofit methods and
adopted standards are also investigated. The material properties and structural behavior
of axially loaded FRP tubes are reviewed.
Chapter 3: provides a detailed description of the structural analytical model that was
created to test the various bracing configurations.
Chapter 4: presents the results of the nonlinear static and nonlinear dynamic loading of
the analytical model for each of the bracing configurations evaluated. A discussion and
comparison of the results is also presented.
Chapter 5: presents the summary and conclusions of the study as well as
recommendations for future research regarding the use of FRP tubes for seismic bracing
applications.
References.
Appendix.
4


2.
Literature Review
2.1 Introduction
This chapter provides a summary of background information that has been
gathered and evaluated in order to perform the technical analysis portion of the study.
The general scope of the study is to investigate the use of FRP tubes as diagonal bracing
members for seismic retrofit applications. Therefore, detailed knowledge of both seismic
analysis procedures as well as material properties and behavior of FRP tubes is necessary.
The first part of this chapter examines the seismic analysis and design standards that are
currently in use in the United States. The use of these standards is typically mandatory
through the legal adoption of a model building code by the governing jurisdiction where
the building in question is to be located. The second section of this chapter discusses
seismic retrofit procedures that are currently used for existing buildings, along with the
specific building codes that govern the retrofit requirements and techniques. The third
section studies the structural properties and behavior of hollow FRP tubes. Finally, FRP
confined concrete solid cylinders are investigated as axially loaded structural members.
2.2 Code-Based Seismic Analysis and Design
The building code that has been predominately adopted by jurisdictions
throughout the United States is the International Building Code either the 2012 IBC or
earlier editions. There are numerous secondary codes and standards that are adopted by
reference in the 2012 IBC, one of which is ASCE 7-10 Minimum Design Loads for
Buildings and Other Structures, published by the American Society of Civil Engineers.
5


Chapters 11 through 23 of the ASCE 7-10 standard provide specific requirements for the
seismic analysis and design of building structures.
2.2.1 Seismic Design Criteria ASCE 7-10
The general seismic design criteria for building structures is presented in Chapter
11 of ASCE 7-10. The objective of the criteria given in the standard is to provide a
consistent and measurable margin of safety against collapse for any given structure. This
is accomplished through the use of the seismic design parameters Ss and Si. The
parameter Ss represents the maximum considered earthquake acceleration response
(MCEr) for a building with a period of 0.2 seconds, and the Si parameter represents the
MCEr for a building with a period of 1.0 seconds.
The MCE typically equates to a magnitude of ground shaking that has a 2 percent
probability of exceedance in 50 years, or an annual return period of once every 2500
years. Although its technically possible for a magnitude of ground shaking larger than
the MCE to occur for any given site, it is extremely unlikely from a probabilistic
standpoint. Therefore, the MCE magnitude has been determined by the standard
developers to provide a structure that is relatively economical to build, while still having
an acceptable level of safety for its occupants. The Ss and Si values may be obtained
from the U.S. Geological Survey (USGS) website http://earthquake.usgs.gov/designmaps
for any given building site. Figure 2-1 shows a map of typical MCEr 1-second spectral
response acceleration values throughout the United States, expressed in terms of percent
gravity. The figure illustrates that several regions of the United States are classified as
having high seismic risk.
6


The ASCE 7-10 standard recognizes that seismic shear waves can be amplified as
they travel through certain types of soil. The Ss and Si acceleration response parameters
are therefore modified to account for the soil characteristics at the building site. The
design spectral response acceleration parameters are defined as Sds and Sdi, and are
calculated as follows:
Ss and Sr. obtained from USGS for specific building site
Sms ~ SiSs (2-1)
SmI SVS1 (2-2)
2 Sds ^ Sms (2-3)
2 Sdi (2-4)
Once the design response acceleration parameters, Sds and Sdi, have been
calculated, a seismic design response spectrum is developed. The design response
spectrum represents that maximum magnitude of acceleration that will occur in any given
structure that is constructed at the particular building site in question. The structures
magnitude of acceleration in response to the ground shaking is heavily dependent on its
degree of stiffness. Therefore, the acceleration magnitude in the design response
spectrum varies based on the natural period of the particular structure. A typical design
response spectrum is shown in Figure 2-2.
The final notable objective of Chapter 11 of the ASCE 7-10 is to determine the
Seismic Design Category for the structure based on the previously defined parameters.
The standard defines 6 possible Seismic Design Categories A through F. The Seismic
7


Design Categories are used to trigger progressively stringent requirements for the design,
construction, and inspection of the structure as the magnitude of seismic hazard increases.
The standard prescriptively groups the various requirements in the SDC A through F
since the requirements are often on or off parameters i.e. parameters that are not
scalable.
2.2.2 Seismic Design Requirements for Building Structures ASCE 7-10
Chapter 12 of the ASCE 7-10 standard provides the required seismic analysis and
design procedures specifically related to building structures. The standard states that a
mathematical model shall be constructed and evaluated for the structure to demonstrate
that it is capable of resisting the internal forces and deformations that will result from the
applied seismic design forces.
Three types of seismic analysis procedures are presented in the standard -
Equivalent Lateral Force Analysis, Modal Response Spectrum Analysis, and Seismic
Response History Procedures. With all three of these procedures, the structure is
expected to behave inelastically, meaning that buckling, yielding, and permanent
deformation of the structural components is expected during the design earthquake event.
By allowing a certain level of damage and permanent deformation to occur in the
structure, a significantly more economical building can be provided than if the structure
were designed to remain entirely elastic and undamaged after the design earthquake
event.
A brief description of the Equivalent Lateral Force Analysis (ELF) and the Modal
Response Spectrum Analysis procedures will be provided here. A more detailed
8


description of the Seismic Response History Procedure will be provided in the following
section, since this is the method that will be employed in the projects parametric study.
The ELF analysis method is the least rigorous of the three available methods, yet
it provides sufficiently accurate results for most structures and is therefore very widely
used. As previously stated, it is recognized that the seismic load is dynamic in nature and
that the structure will behave inelastically under this applied seismic load. However, the
ELF method utilizes two key idealizations that make the analysis much simpler the
dynamic load is converted to an equivalent static load, and the structure is assumed to
behave linear elastically. The inelastic effects are accounted for through the use of a
response modification factor, R, and a deflection amplification factor, Cd. The previously
calculated Sds and Sdi values are modified by the R factor and the seismic importance
factor, Ie, to obtain the seismic response coefficient, Cs. Cs represents the
pseudoacceleration of the structure, and is expressed in terms of percent gravity units.
The equations for Cs given in the ASCE 7-10 standard are dependent on the structures
fundamental period and are as follows:
r $ds S~ (t) (2-5)
V/g/
SD1 = for T < Tl (2-6)
= for T >Tl Hr) (2-7)
: > 0.044SD5/e > 0.01 (2-8)
9


For structures located where Si is equal to or greater than 0.6g, Cs shall not be less than:
_ 0.55!/
KR/le)
(2-9)
The seismic response coefficient is then multiplied by the effective seismic weight of the
structure to obtain the seismic base shear, V:
V = CSW (2-10)
The total seismic base shear, V, is vertically distributed to each of the discrete levels of
the structure by use of the following equations:
F = C V 1 X vxv (2-11)
wxh*
Cvx ~ 7 k (2-12)
The vertical distribution factor Cm uses the idealization that 100 percent of the structures
mass participates in the first mode the structures natural mode. All of the higher
modes of the structure are neglected and the structure is reduced to one single-degree-of-
freedom system, which greatly simplifies the analysis. The structure is analyzed for the
series of static lateral seismic forces applied at each level, and the individual structural
members are designed for the resulting forces.
Modal Response Spectrum Analysis (MRS), similar to ELF analysis, uses a linear
elastic idealization to simplify the analysis. The same response modification factor, R, is
used to account for the true inelastic behavior of the structure. Whereas the ELF
procedure is only interested in the first mode of vibration, MRS requires a more rigorous
10


analysis to determine several of the natural modes of vibration of the structure. For each
direction under consideration, the analysis requires that a sufficient number of modes are
obtained to account for a combined modal mass participation of at least 90 percent of the
structures actual mass. With a sufficient number of modes obtained, the true multi -
degree-of-freedom structure is converted into a series of single-degree-of-freedom
systems, each of which have a unique mode shape and period of vibration. Each of the
single-degree-of-freedom systems is then separately analyzed to determine the story
forces, member forces, and displacements due to the design seismic loads. It should be
noted that although the MRS procedure utilizes certain dynamic properties of the
structure, it is not a full-blown dynamic analysis method. The MRS results provide only
the magnitude of maximum acceleration for each of the modes. Neither the sign (positive
or negative), nor the time at which this maximum acceleration occurs are known.
Therefore, the individual modal responses cannot be recombined exactly since their
maximum values each occur at a different point in time. A statistical combination of the
individual modes is instead used the results from the series of uncoupled analyses are
then combined through the use of either the square root sum of squares (SRSS) or
complete quadratic equation (CQC) method to obtain the response of the true structure.
2.2.3 Seismic Response History Procedures ASCE 7-10
The Seismic Response History Procedures of ASCE 7-10 are presented in Chapter
16 of the standard. The procedure consists of a full dynamic analysis of the structure,
wherein the analytical model is subjected to a series of time-dependent ground motions.
Although the use of simulated ground motions is permitted and appropriate in certain
11


instances, the ground motions used in the analysis will often be from actual past seismic
events recorded in the vicinity of the structure.
Two subcategories exist within the Chapter 16 Seismic Response History
Procedures the Linear Response History Procedure and the Nonlinear Response History
Procedure. As is evident by their titles, the Linear Procedure utilizes the idealization that
the structure will behave linear-elastically throughout the course of the dynamic seismic
loading event. This is the same idealization that is made in both the Equivalent Lateral
Force Procedure and the Modal Response Spectrum Analysis procedure, and ultimately
results in a much simpler and manageable analysis. This idealization can be made due to
the Equal Displacement Approximation Theorem, which states that the displacement of a
structure undergoing nonlinear deformation is nearly equal to the displacement of the
same structure behaving linear elastically. The linear force-deformation curve can
therefore be extrapolated out to determine an equivalent linear design force for any give
deformation magnitude. Figure 2-3 depicts a graphical representation of the Equal
Displacement Approximation Theorem concept.
To the contrary, the Nonlinear Procedure assumes a nonlinear hysteretic behavior
of the structure, which is in fact how it will behave when subjected to a real-life seismic
event. The Nonlinear Procedure is obviously the more rigorous of the two analysis
methods to implement, but it should be expected to yield more accurate results since
more of the true characteristics of the structure are being captured in the mathematical
model.
As mentioned earlier, the response history analysis uses a series of time-
dependent ground motions as the loading parameters. Ideally, the ground motions that
12


are selected will be similar in nature to any future ground motions that the structure could
be subjected to during an actual earthquake event. Therefore, the selected records should
have magnitudes, fault distances, source mechanisms, and soil conditions that are all
similar to what will be encountered at the project site. (Federal Emergency Management
Agency, 2009). The Pacific Earthquake Engineering Research Center (PEER) website,
www.peer.berkeley.edu, provides a large suite of past ground acceleration records that
can be downloaded and used for a response history analysis. Records from actual past
earthquakes have been downloaded from the PEER website and used in the parametric
study section of this report.
Once a sufficient number (typically 3 to 7) of ground motions records has been
selected for use in the analysis, each of the acceleration records is scaled using the
procedures of either Section 16.1.3.1 or 16.1.3.2. Scaling of the ground motions is
necessary because although the type of ground motion in the record may be appropriate,
its likely that the magnitude of acceleration is either higher or lower than the
acceleration that should be used for the structure being designed. The more general
Design Response Spectrum that has previously been developed using the procedures of
ASCE 7-10 Chapter 11 is used as the benchmark for the acceleration magnitude that the
more detailed time-dependent ground motions must be scaled to. Specifically, each
ground motion must be scaled so that the average of each time-dependent spectra is not
less than the Chapter 11 Design Response Spectrum. The magnitude of acceleration is
dependent on the natural period of the structure, so this requirement must be satisfied for
the entire range of 20 percent to 150 percent of the natural period of the structure under
consideration. The 0.2T lower bound of the range is meant to account for the higher
13


modes of the structure, and the 1.5T upper bound of the range is meant to account for
inelastic response, which will likely have a higher period than the purely linear-elastic
system.
Due to the complexity and certain subjective characteristics of the Seismic
Response History Procedures, the ASCE 7-10 standard requires that a third-party design
review is conducted on the seismic analysis and design if this procedure is used. The
standard states that the peer review must be conducted by an independent team of
registered design professionals who have demonstrated experience in the Response
History Procedure. This is the only such strict requirement of a peer review found in the
ASCE 7-10 standard, which illustrates the complexity of the Seismic Response History
Procedure.
2.3 Seismic Retrofit of Existing Buildings
Modem building codes contain well defined seismic design parameters that, when
properly applied, produce safe and predictable building performance during earthquakes.
Seismic design provisions began making their way into building codes in the 1960s,
though they were somewhat primitive and flawed in their infancy. Seismic analysis and
design standards have progressively evolved over the past 40 plus years from research
and industry knowledge, into what is currently in use today. Therefore, there are
thousands of buildings throughout the United States that were designed and constructed
prior to the implementation of modem seismic design standards. These buildings pose a
significant risk to their occupants in the event that a large earthquake should occur.
14


Seismic retrofit is used as a means to upgrade these substandard buildings to reduce their
risk of collapse and ensure the safety of their occupants.
2.3.1 Seismic Retrofit Background
The Long Beach, California earthquake of 1933 triggered the development of the
first formal seismic building standards in the United States. Members of the California
state legislature personally witnessed the collapse of buildings from the earthquake, and
consequently enacted the Field Act of 1933. Through the Field Act, a statewide building
code was developed with provisions to safeguard new buildings from earthquakes.
(Department of General Services, 2002).
Although the Field Act was instrumental in providing seismic safety for newly
built structures, it did nothing to address the considerable number of buildings
constructed prior to its enactment. Recognizing this issue, the California legislature
passed the Garrison Act in 1939. (Department of General Services, 2002). The Garrison
Act specifically addressed California public school buildings that were built prior to
1933, requiring mandatory seismic evaluation of these structures. If the buildings were
found to be substandard for seismic performance, they were required to be either
retrofitted or abandoned.
Seismic design standards continued to develop at the state and local levels until
the National Earthquake Hazards Reduction Program (NEHRP) was established by
Congress in 1977, with the intent to reduce the risks of life and property from future
earthquakes in the United States through the establishment and maintenance of an
effective earthquake hazards reduction program. This significance of NEHRP is that it
15


brought the issues to the national level, providing a consistent and unified voice for
earthquake risk mitigation in the United States.
There are four primary Federal agencies that coordinate the activities of NEHRP
and implement its programs the Federal Emergency Management Agency (FEMA), the
National Institute of Standards and Technology (NIST), the National Science Foundation
(NSF), and the U.S. Geological Survey (USGS). The work of FEMA is of particular
interest regarding building seismic design, because FEMA developed many of the
fountainhead documents that lead to the current seismic design and rehabilitation
provisions that are currently included in model building codes today. These include
FEMA 547 Techniques for the Seismic Rehabilitation of Existing Buildings and FEMA
P-750 NEHRP Recommended Seismic Provisions for New Buildings and Other
Structures, to name a few.
Seismic retrofit initiatives can generally be classified into two categories -
mandatory programs and voluntary programs. Several local jurisdictions have
implemented mandatory seismic retrofit programs for existing buildings that are deemed
to be unsafe. These mandatory requirements are clearly meant to protect the safety of the
public, but can come with serious political, economic, and societal side effects, and
should therefore not be taken lightly. Voluntary seismic rehabilitation carried out
proactively by building owners, without set requirements from the governing building
jurisdiction make up a large percentage of the retrofit projects carried out. Motives that
can drive a building owner to voluntarily retrofit their property vary widely, the most
obvious being to provide basic life safety to their occupants. Other motivations include
protecting historic or landmark structures, minimizing disruptions and downtimes to their
16


business, minimizing administrative efforts required to file insurance claims and apply
for disaster relief assistance. Regardless of what a particular building owners motive is
to seismically retrofit an existing structure, there are numerous approaches for how it can
be executed, which will be described in the following section.
2.3.2 Common Retrofit Techniques Currently In Use
This study examines the applicability of a single proposed seismic retrofit
technique, using very specific materials and strengthening configurations. The author
recognizes that numerous seismic retrofit techniques are available as alternates to the
method investigated in this study, many of which have proven to be practical,
economically viable, and effective through performance during actual earthquake events.
Therefore, it is appropriate to review these techniques currently in use a basis of
comparison.
Once it is recognized that a seismic deficiency exists, there are different
distinguishable classes of measures that can be taken to retrofit a building:
Adding elements to increase strength or stiffness
Enhancing performance of existing elements by increasing strength or
deformation capacity
Improving connections between components
Reducing demand
Removing selected components from the lateral system
Increasing ductility
(Federal Emergency Management Agency, 2006)
17


Adding new structural elements such as shear walls, braced frames, or moment
frames is the most commonly used retrofit technique. The new elements are typically
used in combination with any existing lateral force resisting elements to create a new
global strength and global stiffness of the structure. When adding new structural
elements, consideration must be given to the existing components such as diaphragms,
chords, and collectors to ensure that they are capable of delivering the loads attracted by
the newly introduced elements.
Rather than adding new elements to the structure to improve its performance, the
already existing elements of the lateral force resisting system can be altered for enhanced
performance. Examples of these methods include wrapping concrete columns with steel
or composite materials to provide increased confinement and shear strength, and layering
concrete and masonry shear walls with plate steel or composite materials to increase
ductility and in-plane shear capacity. For this class of retrofit techniques, the vertical and
horizontal distribution of seismic forces is not significantly changed since no new lateral
elements are introduced to the seismic force resisting system.
Sometimes the seismic evaluation process will reveal that the existing lateral
force resisting elements themselves have adequate strength and stiffness to resist the
imposed earthquake forces, but the connections between these components are deficient.
In these cases, retrofit techniques are available to improve the connections between
components so that a complete load path exists and assumed force distributions can
occur. This class of techniques can be classified as targeting load path deficiencies.
Reducing the overall seismic demand is a technique that can be used when the
structure contains a complete and well-connected seismic force-resisting system, but it is
18


weak relative to the design earthquake forces. One specific method within this class is
reducing seismic mass, usually accomplished by removing one or more top floors, which
in turn reduces lateral load to the lower levels. This class of seismic demand reduction
techniques also includes modification of dynamic response of the structure most
commonly through the use of localized damping devices or base isolation.
The final class of retrofit techniques is removal of select components from the
structure. This is typically done to enhance the structures deformation capacity by
removing or uncoupling brittle elements from the lateral system. An example of this
technique is introducing vertical sawcut joints in unreinforced masonry shear walls to
change their failure mechanism from shear to a bending or rocking mode. (Federal
Emergency Management Agency, 2006)
2.3.3 Seismic Retrofit Design Process ASCE 41-13
ASCE 41-13: Seismic Evaluation and Retrofit of Existing Buildings is intended to
be the single standard addressing the seismic performance of existing buildings in the
United States. It was developed on the heels of two fountainhead documents -A TC-14:
Evaluating the Seismic Resistance of Existing Buildings by the Applied Technology
Council (1987) and I'EM A 273: NEHRP Guidelines for the Seismic Rehabilitation of
Buildings by the Federal Emergency Management Agency (1997). Prior to these two
documents, no formal standards existed for seismic evaluation and retrofit. Instead,
design engineers were forced to use their individual judgment in analysis and design,
usually attempting to apply standards meant for new buildings to evaluate and seismically
retrofit existing structures.
19


Performance-based seismic design is the main approach of the provisions
contained in the ASCE 41-13 document. The concept of performance-based design and
the specific parameters related to ASCE 41-13 are described in the following section.
2.3.4 Performance Objective for Existing Buildings ASCE 41-13
The seismic design provisions of ASCE 7-10 can generally be categorized as
being prescriptive. That is, the criteria specifies a minimum level of strength and
stiffness of the structure, but little information is actually known regarding how the
structure will perform during an earthquake event (i.e. does the building experience
substantial structural damage, limited cosmetic damage, or remain fully operational with
no damage).
ASCE 41-13 on the other hand utilizes a performance-based approach to seismic
design. Performance-based seismic design explicitly evaluates a structures anticipated
performance for a specific seismic event it is likely to experience. Since more insight is
gained regarding the buildings actual performance, the seismic force resisting system
can either be made more economical while still meeting minimum performance
standards, or higher levels of performance can be confirmed beyond what is assumed
using the code-based prescriptive methods.
ASCE 41-13 contains four separate sets of performance objectives that can be
used for seismic analysis and design of existing buildings:
1. Basic Performance Objective for Existing Buildings (BPOE)
2. Enhanced Performance Objectives
3. Limited Performance Obj ectives
20


4. Basic Performance Objective Equivalent to New Building Standards (BPON)
The engineer, owner, building official, or a combination of the three determine which of
the four performance objectives will be utilized based on the acceptable level of seismic
hazard for the particular building in question. For example, some buildings may be
required to remain fully operational after a seismic event, while other buildings may
allow a substantial amount of structural damage to be acceptable, only requiring total
collapse prevention so that building occupants can safely exit the structure after a seismic
event.
Specific factors that are used to quantify the performance criteria listed above
include the post-yield ductility of the structure, the post-yield residual strength of the
structure, and the failure type of the structure. Generally, a structure is considered to be
seismically superior if it can withstand large plastic deformations beyond the
displacement at which it first begins to yield, and it can exhibit a post-yield ultimate
strength that is significantly greater that the magnitude of the initial yield strength.
Figure 2-4 depicts examples of various nonlinear static force-deformation curves,
exhibiting both good and poor post-yield seismic characteristics. Similarly, the degree of
hysteretic behavior displayed by the structures lateral force resisting system is
considered when quantifying its seismic performance. Ductile hysteresis loops, as shown
in Figure 2-5a, are desired since seismic loading is cyclical by nature, with the structure
likely undergoing many reversible cycles of plastic deformation during an earthquake
event. A pinched hysteresis loop, as shown in Figure 2-5b, performs poorly during a
seismic event due to its inability to undergo large post-yielding deformations and
dissipate the kinematic energy within the structure from the dynamically applied load.
21


The ductility and post-elastic characteristics of various structural materials and
bracing configurations have been evaluated in the parametric study portion of the report.
2.4 Fiber Reinforced Polymer Overview
The general family of fiber reinforced polymers, or fiber reinforced plastics, are
made up of two homogeneous materials with different properties that are mechanically
bonded to create a composite material with certain advantageous characteristics. The two
materials combined are a polymer resin matrix and reinforcing fibers. Figure 2-6 shows
the general makeup of an FRP composite. Common materials used for fiber reinforcing
are glass, carbon (graphite), basalt, and aramid (Kevlar). The various available fiber
materials have advantages and disadvantages for specific applications, including cost,
strength, stiffness, and durability. Figure 2-7 shows the typical stress-strain curves for
commonly used fiber materials. The curves represent the behavior of the pure fibers
only, without effects of the polymer matrix.
Glass fiber reinforced polymer (GFRP) commonly referred to simply as
fiberglass specifically uses glass fibers as its reinforcement material. The material is
manufactured into various shapes typically using one of two methods the pultrusion
process or the filament winding process. In the pultrusion process, the raw materials of
resin and glass fibers are pulled (rather than pushed, as in the case of extrusion) through a
heated stationary forming die to form a constant cross-section member of any desired
length. See Figure 2-8 for a schematic diagram of the basic pultrusion process. In the
filament-winding process, the fiber and resin materials are wound around a forming
shape, called a mandrel. The filament winding process is typically used to manufacture
22


cylindrical or tube shaped members. Figure 2-9 shows a schematic diagram of the basic
filament winding process.
FRP is extremely versatile in its applications, spanning a wide range of industries
including marine vessels, aircraft, automobiles, storage tanks, piping, and structural
members. The materials use in structural building applications is of interest in this
study, and will therefore be discussed in the following sections.
2.4.1 Material Properties of FRP Tubes
Glass-fiber reinforced polymers can be made up of a virtually infinite number of
different component combinations. This can include variations in fiber type, polymer
matrix type, fiber concentration, fiber cross-sectional area, and fiber orientation. This
versatility in material makeup also leads to a wide range of structural material properties,
such as modulus of elasticity, yield stress, rupture stress, and ductility. It is therefore
very difficult to make generalizations about the structural behavior of FRP. (ISIS
Canada, 2006) One generalization that can be made is that FRP is an orthotropic
material, meaning that it has different properties and strengths in different directions.
Therefore, the direction of stress relative to the material orientation must be considered
when evaluating material properties. FRP exhibits the greatest strength and stiffness
when the direction of all of the fibers are oriented parallel to the axis of the member and
the loading direction this is referred to as a unidirectional composite. Unidirectional
fiber orientation is used in most all structural engineering applications, so the discussion
of mechanical properties of FRP will be limited to this type.
23


2.5 Hollow FRP Tubes as Axially Loaded Structural Members
Since FRPs are made up of two components the polymer matrix and the fiber
reinforcing it is intuitive that the modulus of elasticity of the composite material is
proportional to the modulus of the two individual components. An equation known as
the rule of mixtures is commonly used to approximate the elastic modulus of the FRP
composite, which is expressed as follows:
Efrp = EmVm + EfVf = (Ef Em)Vf + Em (2-13)
The rule of mixtures is only applicable for unidirectional composites, for either a tension
or compression load parallel to the fibers.
Axial strength of FRP composites is dependent on whether the applied load is
tensile or compressive. The material generally exhibits greatest strength when the normal
force is tensile. The compressive strength is typically about 50 to 60 percent of the axial
tensile strength. (ISIS Canada, 2006) However, certain pultruded shapes can exhibit
compressive strengths and stiffness equal to the tensile properties.
2.5.1 Axial Tensile Properties
For axial tensile loads, two types of failure modes are possible for FRP members
- failure of the reinforcing fibers or failure of the polymer matrix. Due to deformation
compatibility, the axial strain in the fiber and in the matrix must always be equal. If the
ultimate strain of the matrix is smaller than the ultimate strain of the fibers, then the
strength of the FRP composite will likely be controlled by the matrix. This type of
failure is graphically depicted in Figure 2-10. Conversely, if the ultimate strain of the
24


fibers is smaller than the ultimate strain of the matrix, then the FRP strength will likely be
controlled by the fibers. This type of failure mode is graphically represented in Figure 2-
11.
While the relative ultimate strains of the matrix and fibers are important, the fiber
volume concentration is also influential on whether the FRP strength is matrix-controlled
or fiber-controlled. Therefore, four possible scenarios are possible when evaluating
potential failure modes of axially-loaded FRP, which are as follows:
1. High fiber volume, failure strain of matrix less than failure strain of fibers.
2. High fiber volume, failure strain of matrix greater than failure strain of fibers.
3. Low fiber volume, failure strain of matrix less than failure strain of fibers.
4. Low fiber volume, failure strain of matrix greater than failure strain of fibers.
A unique equation is required to describe each of these possible failure modes.
When the fiber volume concentration is high and the relative ultimate strain of the
matrix is low, the fibers will resist the majority of the axial load, and the failure of the
matrix component is irrelevant. This type of scenario is expressed by the following
equation:
frp,ult ~ f,ultVf (2-14)
When the fiber volume concentration is high and the relative ultimate strain of the
matrix is higher than that of the fibers, then a large percentage of the axial force is
required to be transferred from the fibers to the matrix as the fibers rupture. The ultimate
strength of the composite for this condition is expressed as follows:
frp,ult ~ &f,ultVf "b bjp) (2-15)
25


For the case of a low fiber volume concentration and a matrix ultimate strain that
is less than the fiber ultimate strain, the axial strength of the FRP will be governed by the
matrix failure. That is, when the ultimate strain of the matrix is reached, the entire
composite section will fail. This condition is represented by the following equation:
frp,ult GfVf &m,ult(^- ~ ^/) (2-16)
Finally, when the fiber volume concentration is low and the ultimate strain of the
matrix is greater than the fiber ultimate strain, then the axial load that was being carried
by the reinforcing fibers will be transferred to the matrix when the fibers rupture, and the
composite will continue to support load until the matrix reaches its higher ultimate strain.
This scenario is expressed as follows:
frp,ult ~ m,ult(^- ~ ^/) (2-17)
All of these failure modes described refer to a large fiber volume concentration
versus a small fiber volume concentration. The balance point between a large
concentration classification and a small concentration classification is typically a fiber
reinforcing fraction in the range of 10 percent of the total composite volume.
2.5.2 Axial Compressive Properties
While the tensile failure mode can be attributed to rupture of the matrix and the
fiber materials, the axial compressive failure mode is typically due to buckling of the
reinforcing fibers, within the restraint of the polymer matrix material. (Jones, 1999)
26


Figure 2-12 shows a schematic representation of various buckled shapes of reinforcing
fibers.
The buckling behavior of the individual fibers is somewhat analogous to a column
on elastic foundation model, wherein the buckling wavelength is directly proportional to
the fiber diameter. Figure 2-13 shows a plot of buckle wavelengths for various glass-
epoxy fiber diameters that have been observed through previous experimental testing.
(Jones, 1999)
It is recognized that an individual glass fiber of any given diameter will have a
specific critical length that causes buckling under a constant axial load. It can also be
seen that in a composite material, the polymer matrix that surrounds the fibers acts as a
lateral support or brace, resisting the tendency of the fiber to buckle. The critical
buckling load for a glass fiber embedded in a polymer matrix is therefore much higher
than if the fiber were not surrounded by the polymer. The traditional Euler buckling load
of a column is given by:
El
P = m2n2 (2-18)
Lz
where m-1 is the number of lateral supports along the length of the column. Figure 2-14
shows buckling configurations of Euler columns with various numbers of discrete lateral
supports. The Euler column equation presented above assumes that each discrete support
is infinitely rigid. However, the polymer matrix that surrounds a glass reinforcing fiber is
certainly not rigid in the transverse direction. Instead, it behaves more like a spring,
providing a certain degree of lateral stability to the fiber, yet deforming laterally so that
the buckling susceptibility of the fiber is increased. Mathematical models much more
27


sophisticated than the Euler column equation are required to represent the buckling
behavior of an axially loaded FRP material, and literature is available that presents these
models. The critical concept to be aware of is that the compressive strength of a FRP will
virtually always be smaller than the tensile strength, because axial buckling of the glass
fibers will occur before the rupture strength of the matrix or the fibers can be reached.
2.5.3 Stress-Strain Behavior of Hollow FRP Tubes
Theoretical and experimental test results have demonstrated that uniaxial
pultruded FRP will typically exhibit a linear stress-strain behavior from zero load until
the ultimate load that causes failure/rupture. (Jones, 1999) Glass reinforcing fibers are
very brittle by nature, and therefore possess little to no ductility under axial load. The
matrix material by itself will generally exhibit nonlinear structural properties since it is a
plastic or polymer material, and plastic tends to have ductile deformation ability.
However, since the glass reinforcing fibers typically account for the majority of the
strength and stiffness in the axial load direction, any nonlinear characteristics of the
polymer matrix are negated in the behavior of the composite system as a whole. (Jones,
1999)
Many theoretical equations and models have been developed through research to
predict the behavior of FRP composites subjected to structural loads. Elementary
models, such as the basic Rules of Mixtures, have been discussed in the previous
sections. The complexity and sophistication of these theoretical models can grow, and
consequently so can their accuracy regarding correlation with real measured results.
However, no matter how accurate theoretical models may be, measured data from actual
28


experimental testing is obviously more reliable and indicative of expected behavior from
future structural loading. Therefore, experimentally measured not theoretically
calculated structural properties such as strength and stiffness are typically what is used
in structural design.
Further description of the stress-strain curves that have been used in this
analytical study is provided in Chapter 3.
2.6 FRP Confined Concrete Tubes as Axially Loaded Structural Members
Previous sections of the report have demonstrated that the axial compressive
properties of FRP are considerably inferior to the axial tensile properties. The
generalization can be made that the ideal loading condition for uniaxial FRP is in axial
tension. The opposite can generally be said for plain concrete its structural
performance is far superior in axial compression than in axial tension.
The use of FRP confined concrete tubes as axially loaded members has certain
advantages in that it can limit the tensile stresses in the concrete while also limiting the
compressive stresses in the FRP composite material. The study of this structural member
type has been given significant attention through a large number of experimental and
analytical studies over the past twenty plus years. The structural behavior and
characteristics of this type of member will be reviewed in the following sections.
2.6.1 Confinement Characteristics
As plain concrete or any material for that matter is axially compressed in one
direction, it has a tendency to expand laterally in the other two directions. The magnitude
29


of lateral expansion relative to the magnitude of axial compression is expressed by the
materials Poissons Ratio, v.
d£ trans
d^axial
~dey
dex
~dez
dex
(2-19)
As concrete under axial load laterally expands, it begins to develop micro cracks and
tensile splitting, since it has a relatively low tensile strength in this transverse direction.
The lateral splitting is what ultimately leads to failure of unconfined plain concrete
loaded in axial compression.
Providing some sort of confinement mechanism to control the lateral expansion of
concrete is the key to increasing its strength and ductility. (Fam and Rizkalla, 2001). The
most traditional method of providing this confinement is the use of transverse steel
reinforcing ties or spirals embedded within the perimeter of the concrete member. Figure
2-15 shows the confining pressure provided by traditional steel reinforcing ties.
Research has shown that casting concrete members within FRP tubes is also an
excellent method to provide the necessary concrete confinement. When the concrete
member is loaded in axial compression, its lateral expansion is restrained by the FRP
tube, resulting in a radial pressure in the tube. (Fam and Rizkalla, 2001). This type of
radial confinement action is graphically shown in Figure 2-16.
The magnitude of radial pressure applied by the tube is a function of the stiffness
of the tube itself. As the axial load in the concrete increases, the concretes tendency for
lateral expansion increases, which in turn causes the confining pressure in the tube to
increase.
30


For the case of a steel tube surrounding a concrete core, the steel material will
deform linear elastically up to its yield point, after which it can be idealized as being
perfectly plastic the strain increases as the magnitude of stress remains constant. In this
steel tube case, the confining pressure provided by the steel will linearly increase up to
the yield point, and then will remain constant, even as the magnitude of compressive load
applied to the concrete continues to increase.
FRP on the other hand typically behaves linearly-elastically until failure, not
exhibiting a defined yield point. Therefore, when concrete is confined by FRP tubes, the
radial pressure in the circumferential direction will be variable for all magnitudes of axial
load from the point of zero load until the failure of the system when the FRP shell
ruptures.
2.6.2 Linear Elastic Behavior Confinement Model
Research by Mander et al. in 1988 provided equations to define the axial stress-
strain relationships of confined concrete for the specific case of a constant confining
pressure, the strain at which this peak strength occurs as £cc. Their model shows that the axial
stress of the confined concrete is related to the peak strength by the following equation:
J CC A'1
r 1 + xr
(2-20)
where
(2-21)
31


and
r =
(2-22)
Eco is the tangent modulus of elasticity for the case of unconfined concrete, as can be seen
in Figure 2-17. Esec is the secant modulus of elasticity for the case of confined concrete,
which can also be seen from the plot in Figure 2-17. The maximum stress that can be
achieved in the axially compressed concrete, fcc, is a function of the concretes
unconfined strength, fc, and the lateral confining pressure (which is a constant magnitude
for this particular case).
From the Mander et al. set of equations, a nonlinear stress-strain curve can be developed
if two parameters are known the unconfined compressive strength of the concrete, fc,
and the constant lateral confining pressure, As noted throughout the above description, the Mander et al. equations only apply
to the case when the lateral confining pressure is of a constant magnitude. However,
when the confining material is a FRP tube, the confining pressure will not be constant. It
will continuously increase for all magnitudes of axial load since the FRP remains linear
elastic until rupture, never experiencing a plastic yield point.
(2-23)
The axial strain that occurs at this maximum axial stress is defined as:
(2-24)
32


In 2001, Fam and Rizkalla presented a model for predicting the behavior of
axially loaded concrete that is confined by a material with a variable confining pressure,
such as a FRP tube.
Fam and Rizkalla first sought to establish a relationship between the axial strain,
cc, and the radial displacement, ur, of an axially loaded concrete cylinder. For an
unconfined concrete cylinder, they noted that the radial displacement is:
Ur vcRscc (2-25)
This relationship is graphically shown in Figure 2-18a.
However, when an external radial stress is applied to the circumference of the
cylinder, the radial displacement is:
1 vr
uR =-Rgr (2-26)
This relationship is graphically shown in Figure 2-18b.
The external radial stress applied to the concrete is also the same as the internal
radial stress applied by the FRP tube. Therefore, the hoop stress, as, and the radial
displacement of the tube can be calculated based on the internal radial stress as follows:
Uc
Ur
orR t (2-27)
orR2 Eqt (2-28)
33


The radial displacement is depicted in Figure 2-18c.
Similar to Equation 2-28 above, the radial displacement of the tube can also be
calculated based on the axial strain of the system:
uR = vsRscc (2-29)
The relationship between axial strain and radial displacement is shown in Figure 2-18d.
Two possible axial loading scenarios of the system exist:
1. The concentric axial load is applied to the concrete core only.
2. The concentric axial load is applied to both the concrete core and the FRP tube.
For Case 1, Fam and Rizkalla recognized that the outward radial displacement of
the concrete core and the outward radial displacement of the tube must be equal.
Both axial strain, £cc, and radial pressure, 'u, influence the radial expansion of
the core, but radial pressure alone influences the radial expansion of the tube
(since it is not subjected to any externally applied axial force). The radial stress at
any magnitude of loading for Case 1 can be expressed as:
vc
R R 1~vc£cc (2-30)
Est+ Ec
For Case 2, both the concrete core and the FRP tube will tend to expand outward
when subjected to the axially compressive load. However, their relative rate of lateral
expansion will likely be different since each of the two materials will likely have a
different Poissons Ratio. In fact, Fam and Rizkalla noted that if the Poissons Ratio of
34


the tube is greater than the Poissons Ratio of the concrete, the confining tube will be
expanding laterally at a faster rate than the concrete, and the radial pressure, actually be negative. Since a negative radial pressure is not physically practical, this
scenario leads to a separation between the concrete core and confining tube. Several
researchers have observed this type of behavior in their experimental results, such as
Fardis and Khalili in 1981, and Wei et al. in 1995. Steel tubes used for confinement are
more susceptible to this negative radial stress separation since the Poissons Ratio of steel
(0.3) is greater than the Poissons Ratio of concrete in early stages of loading (0.15 to
0.2). Conversely, FRP tubes can be designed to have a lower Poissons Ratio to avoid
this separation scenario. For Case 2, the radial pressure for any given magnitude of
loading is given by:
(Vc ~ vs)
R ~ R l-vc£cc (2-31)
Est+ Ec
From the equations presented for both Case 1 and Case 2 above, it is apparent that
the confining pressure can be increased by either stiffening the tube in the radial
direction, or increasing the Poissons Ratio of the concrete, or both.
2.6.3 Nonlinear Behavior Confinement Model
The preceding section describes the case where both the concrete core and the
surrounding FRP tube are behaving linear elastically throughout the axial loading history.
This assumption is usually valid for the FRP material, which essentially behaves linear
elastically until rupture. However, it is expected that concrete modulus of elasticity, Ec,
35


and Poissons Ration, jyc, will not be a constant, but will instead vary for each different
load magnitude. Recognizing this, Fam and Rizkalla developed a confinement model to
account for the nonlinear properties of the concrete. Their model is essentially an
incremental step function procedure, calculating a new Ec and c value for each
sequential magnitude of axial strain. A nonlinear stress strain curve is then developed by
connecting the dots from each of the individual points calculated along the curve.
Figure 2-19a shows a representation of varying moduli of elasticity that may be used
based on different magnitudes of axial strain. Figure 2-19b shows a representation of
varying Poissons Ratios that may be used based on different magnitudes of axial strain.
Figure 2-19c shows a nonlinear stress-strain diagram that can be constructed using the
method presented by Fam and Rizkalla.
2.6.4 Axial Tensile Properties
The analytical portion of this study utilizes the nonlinear stress-strain curves
developed by other researchers through their experimental test results. In nearly all of the
literature found regarding testing of FRP confined concrete tubes subjected to cyclical
axial loading, the samples were tested with compression-only loading. Since the system
of concrete tubes is very anisotropic in the axial loading direction, the tensile nonlinear
stress-strain curve will vary significantly from the compressive stress-strain curve. A
lack of experimental results on the behavior of concrete-filled FRP tubes exists, so the
analytical study will be limited to compression-only members for this particular system.
36


2.6.5 Axial Compressive Properties Cyclic Loading
Many experimental studies have been conducted over the past 20-plus years
investigating the axial stress-strain behavior of FRP-confined concrete under monotonic
loading. This research has led to the development of over 80 stress-strain models for
monotonic axial loading. A 2012 study by Ozbakkalogu, Lim, and Vincent presented a
review and assessment of the various experimental results and stress-strain models for the
structural system that have been presented in published literature. The monotonic
behavior of FRP-confined concrete is therefore well understood due to the large and
reliable database of past test results.
On the other hand, research investigating the behavior of FRP-confined concrete
under cyclic axial loading has been relatively limited up to this point. Three previous
experimental studies that exist and were considered as part of this report are as follows:
FRP-Confined Concrete Under Axial Cyclic Compression (2006), by Lam,
L., Teng, J.G., Cheung, C.H., and Xiao, Y.
FRP-Confined High-Strength Concrete Under Axial Cyclic Compression
(2008), by Ozbakkaloglu, T., Lim, J.C., and Griffith, M.C.
Behavior of FRP-Confined Normal- and High-Strength Concrete under
Cyclic Axial Compression (2012), by Ozbakkaloglu, T., and Akin, E.
In each of these studies, concrete cylinders were jacketed with various
configurations of FRP material for confinement, and then loaded in cyclic axial
compression until failure. Figure 2-20 shows an example test setup for the experiments.
One common observance in these studies has been that the envelope stress-strain curves
of cyclically-loaded FRP-confined concrete closely follow the stress-strain curves of the
composite members under monotonic loading. (Ozbakkaloglu et. al., 2008). Figure 2-21
through 2-24 show example test results of cyclic compression-only load cycles (red)
37


overlain with the monotonic loading stress-strain curve (blue) for the same specimen. It
can be seen from the figures that the monotonic stress-strain curve is also essentially an
envelope curve for the cyclic loading case. This correlation between the monotonic and
cyclic loading matches the hypothesis that was originally proposed by Karsan and Jirsa
(1969), which states there is a unique envelope curve for any given cyclically loaded
concrete member which is identical to the stress-strain curve of the same concrete
member under monotonic loading.
The experimental studies noted above also observed that the rupture of the FRP
jacket is always what defined the failure condition of the FRP-confined concrete system.
The ultimate strength and maximum axial strain of the concrete were observed to occur
simultaneously with FRP rupture. Further description of the stress-strain curves that have
been used in this analytical study is provided in Chapter 3.
2.7 Current Use of FRP in Seismic Applications
Although the use of both hollow FRP tubes and FRP confined concrete tubes as
diagonal bracing members for seismic retrofit of buildings is a new concept being
explored, FRP is already commonly used in practice for many other seismic load
resistance applications. The first to note is the use of FRP in industrial, marine, and off-
shore structures. For these structures, the entire structural frame will commonly be
constructed out of FRP sections, instead of traditional steel or concrete. This is primarily
because the FRP material can be engineered for superior durability in these harsh
environments, enhancing its resistance to corrosion, freeze-thaw effects, and fatigue due
to load cycling. When the entire structural frame is constructed with FRP sections, this
38


includes the diagonal bracing members that resist lateral loads. Therefore, there are
many built structures in existence where hollow FRP tubes are used as diagonal bracing
members.
Secondly, FRP is commonly used to retrofit existing concrete buildings by
installing FRP confinement strips to the existing concrete columns. The FRP
confinement strips can significantly increase the strength and ductility of the concrete
columns, which often have an inadequate amount of spiral or tie reinforcing. The
concept of applying FRP to existing concrete columns is similar to the FRP confined
concrete tubes that have been evaluated in this study. For both instances, the concrete
component is resisting the applied axial load, while the FRP is enhancing its ability to do
so by providing a radial confinement pressure.
39


Spectral Response Acceleration, Sa (g)
SDC A SDC B SDC C SDC D SDC D SDC D SDC E
Figure 2-1: Map of MCEr Spectral Response Parameters for United States
(FEMA P-750, 2009)
Figure 2-2: Typical Design Response Spectrum (FEMA P-750, 2009)
40


Figure 2-3: Equal Displacement Approximation Theorem (FEMA P-750, 2009)
Backbone
(deformation controlled) (force contolled)
Figure 2-4: Various Nonlinear Force-Deformation Curves (FEMA 440, 2005)
41


Force
(b)
Figure 2-5: (a) Ductile Hysteresis Loops (b) Pinched Hysteresis Loops
(FEMA P-750, 2009)
42


Stress (MPa)
FIBRES
^-------
POLYMER
MATRIX
I D !
I D I
i a i
I D I
i a i
I 9 I
l J-7
^""*->1 *...
FRP
Figure 2-6: Fiber Reinforced Polymer General Makeup (ISIS Canada, 2006)
Figure 2-7: Stress-Strain Curve of Various Reinforcing Fibers (ISIS Canada, 2006)
43


SURFACING
Figure 2-8: Pultrusion Process Schematic Diagram (Strongwell, 2007)
Resin
Rolls of Fiber Application
Reinforcement System Fiber Delivery
Figure 2-9: Filament-Winding Process for Fabricating FRP Tubes (Warner, 2000)
44


Figure 2-10: Matrix Failure Strain Smaller than Fiber Failure Strain
(ISIS Canada, 2006)
Figure 2-11: Matrix Failure Strain Larger than Fiber Failure Strain
(ISIS Canada, 2006)
45


1 \ \
from Axial Compressive Load (Jones, 1999)
BUCKLE
WAVELENGTH
in
FIBER DIAMETER, min
BUCKLE
WAVELENGTH
mm
Figure 2-13: Experimental Results for Fiber Buckle Wavelength vs. Fiber Diameter
(Jones, 1999)
46


m a high
Figure 2-14: Buckling of a Discretely Supported Euler Column (Jones, 1999)
Figure 2-15: Effectively Confined Core for Rectangular Hoop Reinforcement
(Mander, 1988)
47


fc
Figure 2-16: Confining Action of FRP Shell on Concrete Core
(Ozbakkaloglu et. al., 2012)
Figure 2-17: Representative Stress-Strain Curves for Confined vs. Unconfined
Concrete (Fam and Rizkalla, 2001)
48


Figure 2-18: Solid Cylinder and Thin Shell under Different Stresses (Fam and
Rizkalla, 2001)
Figure 2-19a: Varying Modulus of Elasticity of Concrete (Fam and Rizkalla, 2001)
49


Lateral strain
Axial strain
Figure 2-19b: Varying Poissons Ratio of Concrete (Fam and Rizkalla, 2001)
Figure 2-19c: Nonlinear Stress-Strain Curve for Confined Concrete (Fam and
Rizkalla, 2001)
50


/ 500 mm
Axial SG
Figure 2-20: Example Test Setup Diagram Axially-Loaded FRP Confined
Concrete Cylinder (Ozbakkaloglu & Akin, 2012)
Lateral strain s. Axial strain ec
Figure 2-21: Experimental Stress-Strain Curve for Normal-Strength Concrete
Confined with Two Plies of CFRP (Lam et. al., 2006)
51


140
03
CL
120
100
b 80
<1)
L.
W 40
5
'x
<
20
0
0


m
/ji/j/i j
iiiin/
iiiiiiim -H -H -C-6L-M1 -C-6L-M2
H-C-6L-C2 Envelope (H-C-6L-C2)
0.005 0.01
Axial strain, sc
0.015
Figure 2-22: Experimental Stress-Strain Curve for High-Strength Concrete
Confined with Six Plies of CFRP (Ozbakkaloglu and Akin, 2012)
Figure 2-23: Experimental Stress-Strain Curve for Normal-Strength Concrete
Confined with Two Plies of AFRP (Ozbakkaloglu and Akin, 2012)
52


Figure 2-24: Experimental Stress-Strain Curve for High-Strength Concrete
Confined with Six Plies of AFRP (Ozbakkaloglu and Akin, 2012)
53


3.
Analytical Modeling
3.1 Introduction
A parametric analytical study has been conducted to predict the performance of
axially loaded FRP bracing members within a representative building structure, subjected
to simulated seismic loads. Various combinations of structural parameters such as
bracing configuration, material properties, and loading type have been evaluated in the
model to obtain an understanding of their effect on the structures ability to resist seismic
load.
3.2 Model of Three-Story Steel Framed Building Using SAP2000
The software package SAP2000 has been used to develop and test a structural
analytical model as part of this study. SAP2000 is a three-dimensional structural analysis
program developed by Computers & Structures Inc. The program is capable of analyzing
a wide variety of structure types including stadiums, towers, piping systems, dams, and
buildings. Both linear-elastic and non-linear structural behavior can be defined and
evaluated with the model. The program is also capable of performing both static and
dynamic analysis. A description of the specific model definition, various bracing
configurations, and analysis procedures are described in the following sections.
3.2.1 Description of the Analytical Model
An analytical model has been created in SAP2000 to represent a three-story steel
framed building. The building has been modeled with a floor-to-floor height of 12 feet
54


for each of the three stories, for a total height of 36 feet. A two bay by three bay floor
framing layout has been used, with a typical column spacing of 30 feet, yielding overall
plan dimensions of 60 feet by 90 feet. A plan view of the typical floor and roof framing
layout is shown in Figure 3-1.
Each of the twelve columns has been pinned at its base, meaning that they can
support horizontal or vertical reactions, but cannot support any rotational moment
reactions (in any direction). All individual framing members (beams and columns) have
also been modeled as pinned at each end, meaning that there is no rotational restraint at
member ends, and no moment is transferred between elements framing into each other.
This type of all-pinned connection idealization is typical for a steel framed building
structure.
At each of the three elevated framing levels, a rigid diaphragm constraint has been
defined in the model. The diaphragm constraint causes all joints to move together as a
planar diaphragm that is idealized as being infinitely rigid against in-plane membrane
deformation. The assumption of an infinitely rigid diaphragm is typical for a building
structure with floors made up of either a solid concrete slab or a concrete-filled slab on
metal deck, since the relative in-plane stiffness is so much larger than the stiffness of the
vertical bracing elements.
With the base structural model defined, three different lateral bracing layouts
were added to the structure as follows:
Single-bay X-brace. Diagonal bracing members arranged in an X shape
are located in a single bay at all three stories. Figures 3-2 through 3-9
show the typical single-bay X-brace model configuration.
55


Super X-brace. Diagonal bracing members were added along each of the
perimeter walls to form a single, large X shape. Instead of a series of
single X-braces being contained within lxl bays, the Super X-brace has
an overall extent of 3x3 bays. Figures 3-10 through 3-13 show the typical
super X-brace model configuration.
Chevron brace. Diagonal bracing members arranged in an Inverted V
shape are located in a single bay at all three stories. This type of bracing
configuration relies on each pair of bracing members having an equal and
opposite magnitude of axial load, so that the net vertical reaction at the
midspan of the beam they frame into is zero. Figures 3-14 through 3-17
show the typical Chevron brace model configuration.
The single-bay X-brace, super X-brace, and chevron brace are all braced frame
configurations that are typically used in real life building applications. The comparative
results from the parametric study provides useful insight into the relative performance of
each bracing system.
3.2.2 Structural Analysis Approach
Four different types of analysis procedures are typically used for structural
seismic design, which are as follows:
Linear static analysis
Non-linear static analysis
Linear dynamic analysis
Non-linear dynamic analysis
56


Both non-linear static analysis (pushover analysis) and non-linear dynamic analysis
(time-history response analysis) have been implemented in this study. A modal analysis
has also been performed for each structure. Each of the analysis types and the specific
approaches used are described in the following sections.
3.2.3 Modal Analysis
A modal analysis was performed in SAP2000 for each structure to determine its
vibration behavior. The vibration characteristics are inherent properties of the structure
itself, and are independent of any externally applied loads. Linear elastic behavior of the
entire structure is assumed for modal analysis.
The frequency of vibration and the corresponding modal participation factor was
recorded for the first three modes of each structure. The modal participation factor is a
measure of the modes relevance for computing the response of the structure to an
externally applied acceleration in the direction in question. The sum of the participation
factors for all modes theoretically equals one, since the entire dynamic response of a
structure must be captured by all of its mode shapes.
3.2.4 Non-Linear Static Pushover Analysis
A nonlinear static pushover analysis has been performed in SAP2000 for each of
the 23 structure configurations. In the pushover analysis, a lateral load was applied
quasi-statically from zero magnitude to the full magnitude that caused collapse of the
structure. The lateral load applied consisted of a discrete lateral load at each of the three
levels of the structure. While the magnitude of the lateral loads was steadily increasing,
57


they were always proportional to the mode shape of the structures first mode, only the
scaling of load was varied. As demonstrated in Chapter 4 of the report, the first mode
typically captures around 90 percent of the mass participation in the translational
direction in question. For all models, the pushover analysis has been limited to the X-
direction. Both the applied loading and the stiffness of the structure are symmetric in the
X-direction, so the structure experiences translational-only displacements in this
direction. No translation in the Y-direction or rotation about the Z-axis was present or
considered in the analysis due to the symmetric conditions.
A plot of the total base shear versus the lateral displacement of the roof structure
has been obtained for each structure analyzed. The pushover curve approximates the
nonlinear material behavior the structure will be subjected to from earthquake induced
forces. It also demonstrates the level of post-yield ductility that the structure can provide,
which is an important component for seismic performance.
The cross-sectional area of bracing members was calibrated in each model so that
every structure evaluated had an identical magnitude of base shear strength that caused
collapse. This can be seen in the various static pushover curves plotted in Chapter 4, in
which the base shear for every curve peaks at the same magnitude. By giving each of the
23 structures an equal lateral strength, meaningful comparisons could be made between
the performances of one structure to another.
The typical non-linear static load case parameters that were used in the SAP2000
models are shown in Figure 3-18.
58


3.2.5 Non-Linear Dynamic Analysis Using Time-History Acceleration Function
A non-linear time-history response analysis has been performed for each of the 23
structure configurations in SAP2000. In a time-history analysis, the structure is subjected
to a specified loading that varies with time. In this particular study, the applied loading is
a ground acceleration, which is meant to simulate seismic loading of a structure. The
dynamic response of the structure due to this time-dependent loading is evaluated. The
dynamic equilibrium equation of motion is solved for each time step, which is given by:
K u(t) + C ii(t) + M u(t) = r(t) (3-1)
where u, u, and u are the displacements, velocities, and accelerations of the structure,
respectively. Since the applied load is a ground acceleration as stated above, these
displacements, velocities, and accelerations are relative to the ground motion. K
represents the stiffness matrix, C represents the damping matrix, and M represents the
diagonal mass matrix. R is the applied load, with respect to time. The material non-
linearity (i.e. the nonlinear stress-strain behavior) of the members is included in the
equation of motion that is solved in the analysis.
Similar to the static pushover analysis, the dynamic loading was always applied in
the X-direction for the time-history portion of the study. Since the strength and stiffness
of the structure are symmetric in the X-direction, the structure experiences translational
displacements in the X-direction only. No translation in the Y-direction or rotation about
the Z-axis was present or considered in the time-history analysis. Limiting the loading
and structure response to a single direction simplified the analysis and yielded structural
59


analysis results that were more meaningful to compare for the purpose of the parametric
study.
3.2.6 Dynamic Loading of Model Using El Centro Earthquake Ground Motion
The time-history function that was used for the non-linear dynamic analysis is
from the recorded ground acceleration of an actual past earthquake the El Centro
earthquake of 1940. The earthquake took place in Southern California, near the border
between the United States and Mexico. The earthquake had a moment magnitude of 6.9
and was the first major seismic event where ground accelerations were able to be
recorded by a seismograph at the fault line. The ground acceleration data from the El
Centro earthquake is commonly used in the seismic design of structures today.
The record consists of 1559 ground acceleration data points, each recorded at an
equal time interval of 0.02 seconds. The total time of the ground acceleration is therefore
31.18 seconds. The units of acceleration in the record are g. Each g value has been
multiplied by the constant 9.81 m/s to give a ground acceleration with units of m/s .
The typical non-linear time-history load case parameters that were used in the
SAP2000 models are shown in Figure 3-19. A plot of the recorded ground acceleration
versus time from the El Centro earthquake is shown in Figure 3-20.
3.3 Parametric Study Parameters
23 unique structural models were evaluated in the analytical study. Brace
material properties and bracing configuration were parametrically changed to create the
23 different structures.
60


3.3.1 Matrix of Parameters Evaluated
Table 3-1 itemizes the parameters of the 23 different structural models that were
evaluated. Models #1 through #5 used HSS steel tubes as their bracing members, models
#6 through #15 used hollow FRP tubes as their bracing members, and models #16
through #23 used FRP-confined concrete tubes as their bracing members.
For both the HSS steel tube bracing and the hollow FRP tube bracing, all three
bracing configurations single-bay X-brace, super X-brace, and chevron brace were
evaluated. For these materials, the single-bay X-brace and the super X-brace models
were evaluated for both the case where bracing members are subjected to tension and
compression axial loading, as well as tension-only axial loading. The chevron brace was
only evaluated for the case where bracing members are subjects to tension and
compression loading, since tension-only loading would result in a net vertical reaction on
the supporting beam.
For the FRP-confined concrete tube bracing models, all bracing members were
subjected to compression-only axial loading. As can be seen in the stress-strain curves
presented in the following sections, the FRP-confined concrete tubes exhibit very little
axial tensile capacity, but have considerable strength when loaded in compression. If
subjected to axial tension, these members would exhibit a very brittle failure do to
concrete tensile rupture, and the ability to resist future compression loads from the
seismic load reversal would be compromised. Therefore, it is prudent to configure the
FRP-confined bracing members so they resist compression-only loading. As can be seen
in Table 3-1, chevron bracing was not used in any of the FRP-confined concrete tube
61


models since this type of bracing relies on the brace members being able to resist both
tension and compression loads.
3.3.2 Stress-Strain Curve of ASTM A500 Grade B Steel Tubes
Hollow structural steel (HSS) tubes were evaluated for the control portion of this
study. The most commonly used material for AISC standard shapes of HSS square and
rectangular members is ASTM A500 Grade B. This material has a specified yield
strength of 315 Mpa (46 ksi), and a specified ultimate tensile strength of 400 Mpa (58
ksi). The material has a modulus of elasticity of 200 Gpa (29000 ksi), which is typical
for most structural steel materials. A symmetric stress-strain curve for both tensile and
compressive axial loading is typically assumed for this material. The stress-strain curve
that has been used in the analysis for the ASTM A500 Grade B bracing members is
shown in Figure 3-21.
3.3.3 Stress-Strain Curves of Hollow FRP Tubes
Published material properties from current FRP manufacturers have been obtained
for use in the analytical portion of this study. Material properties used are for currently
market-available hollow structural tube shapes. The two materials selected for use in the
analytical study are as follows:
1. Extren 500/525 Series Shapes, manufactured by Strongwell Corporation.
Material is a pultruded glass fiber reinforced polymer. Fibers are unidirectionally
oriented. Isophthalic polyester and vinyl ester resins are used for the matrix
compound. Detailed product literature can be found on the manufacturers
62


website, at www.strongwell.com. Figure 3-22 shows the stress-strain curve for
this bracing material that has been used in the analysis.
2. Carbon Fiber Tube, manufactured by Good Winds Composites. Material is a
pultruded carbon fiber reinforced polymer. Fibers are unidirectionally oriented.
Bisphenol epoxy vinyl ester is resins are used for the matrix compound. Detailed
product literature can be found on the manufacturers website, at
www.gwcomposites.com. Figure 3-23 shows the stress-strain curve for this
bracing material that has been used in the analysis.
Table 3-2 lists the relevant structural properties of the two specific materials described
above. From Table 3-2, Figure 3-22, and Figure 3-23, it can be seen that although the
glass fiber tube and the carbon fiber tube have very different magnitudes of ultimate
stress and moduli of elasticity, the magnitude of ultimate axial strain for these two
materials is nearly identical. Therefore, both of these materials will reach ultimate failure
or rupture at approximately the same magnitude of elongation, even though the
magnitude of stress that causes this failure to occur will be very different.
3.3.4 Stress-Strain Curves of FRP Confined Concrete
Four non-linear stress-strain curves for axially-loaded FRP confined concrete are
used in the analytical portion of this study. As noted in the previous section, all four of
the curves result from actual experimental test results from past research. A brief
description of each of the four axial members follows.
1. Normal-strength concrete confined by two plies of carbon FRP jackets. This
system was experimentally tested by Lam et. al. in 2006. Specific material
63


properties observed from test results are shown in Table 3-3. The stress-strain
curve published in the previous study is shown in Figure 3-24.
2. High-strength concrete confined by six plies of carbon FRP jackets. This system
was experimentally tested by Ozbakkaloglu and Akin in 2012. Specific material
properties observed from test results are shown in Table 3-3. The stress-strain
curve published in the previous study is shown in Figure 3-25.
3. Normal-strength concrete confined by two plies of aramid FRP jackets. This
system was experimentally tested by Ozbakkaloglu and Akin in 2012. Specific
material properties observed from test results are shown in Table 3-3. The stress-
strain curve published in the previous study is shown in Figure 3-26.
4. High-strength concrete confined by six plies of aramid FRP jackets. This system
was experimentally tested by Ozbakkaloglu and Akin in 2012. Specific material
properties observed from test results are shown in Table 3-3. The stress-strain
curve published in the previous study is shown in Figure 3-27.
3.3.5 Assumed Failure Mode
For both the nonlinear static pushover analysis and the nonlinear time history
analysis, the assumed failure mode of the structure is a nonlinear mechanism developing
in the diagonal bracing members. For HSS steel braces, this consists of ductile yielding
of the bracing member due to axial load. For the FRP confined concrete braces, this
consists of nonlinear plastic crushing of the confined concrete, until the FRP jacket
ruptures due to radial strain, causing ultimate failure of the brace.
64


For all cases, the diagonal braces where assumed to be adequately braced or have
adequate slenderness characteristics such that buckling of the member due to
compression loads would never occur. Buckling of compression members is a failure
mode that must be always be considered in practice. Since the focus of the study was
specifically related to the plastic deformation characteristics of the various materials
evaluated, buckling was idealized as not being a concern.
3.4 Model Validation
The SAP2000 analytical models described in the previous sections required
verification of structural properties and behavior in order to validate the results of the
analysis that has been performed. Critical structural parameters such as mass
distribution, strength, stiffness, modal shapes, and seismic design forces have been
included in the verification process. The results of these parameters produced using
SAP2000 were compared to independent sources, such as hand calculations and results
obtained from other structural finite element analysis programs.
3.4.1 Validation of Strength and Stiffness Properties
The strength and stiffness properties of the SAP2000 models were validated by
comparison to results obtained from hand calculations. A static lateral load of 100 kips
was applied at each level of the structure. The model was then analyzed in SAP2000 to
determine the forces within each brace member, and the lateral deflections of joints at
Levels 1, 2, and 3. The hand calculations were performed under the idealization that 100
percent of the lateral deflection was attributed to axial deformation of the diagonal
65


bracing members (i.e. the axial deformation of vertical columns and horizontal beams
could be neglected). A comparison between the SAP2000 results and the hand
calculation results showed a variation of less than 1 percent between the two.
3.4.2 Validation of Static Pushover Analysis
It can be readily recognized that the rupture of the Level 1 diagonal bracing
members is always the failure mechanism that causes collapse under nonlinear static
pushover loading. From the pushover analysis results presented in Chapter 4, each of the
23 models evaluated has a certain magnitude of lateral load and lateral deflection when
the system failure occurs. Knowing that the total base shear of the building accumulates
from the top of the building down, and is ultimately resisted by the Level 1 diagonal
braces, it is possible to verify the results of the nonlinear static pushover analysis through
relatively simple hand calculations. Using hand calculations the ultimate axial strength
of bracing members was determined based on their cross-sectional area and the ultimate
stress of the bracing material. The total base shear resisted by the structure was then
calculated from static analysis of the geometric bracing configuration, using the axial
brace force previously obtained. This base shear obtained through hand calculations was
then compared to the results from the SAP2000 models used in the analytical study. The
base shear from the SAP2000 analysis and the hand calculation verification were
typically within one to three percent of each other.
66


3.4.3 Validation of Modal Response Properties
Three independent structural models a typical single-bay X-brace, a typical
super X-brace, and a typical chevron brace were built in the structural analysis program
RISA 3D for the purpose of validating the modal response properties that have been
obtained from SAP2000 in this study. Each RISA 3D structures mode shapes,
frequencies of vibration, and modal participation factor for each respective frequency
were compared to the corresponding SAP2000 results that were obtained from the study,
as presented in Chapter 4. The modal response properties were consistent between the
SAP2000 models and the RISA 3D verification models. Variations ranging from one to
three percent can be attributed to different finite analysis techniques that are employed by
each of the two software platforms.
67


Table 3-1: Matrix of Parameters Evaluated
Model No. I Brace Material I Specimen No. I Brace Configuration! Axial Load Resistance I Load Type
1 Steel HSS (control) ASTM A500 Gr. B Single-Bay X-Brace Tension & Compression Static Pushover El Centro (l.Ox)
2 Tension-Only Static Pushover El Centro (l.Ox)
3 Super X-Brace Tension & Compression Static Pushover El Centro (l.Ox)
4 Tension-Only Static Pushover El Centro (l.Ox)
5 Chevron Brace Tension & Compression Static Pushover El Centro (l.Ox)
6 Hollow FRP Tube Strongwel 1 Extren Series 500/525 Pultruded GFRP Si ngle-Bay X-Brace Tension & Compression Static Pushover El Centro (l.Ox)
7 Tension-Only Static Pushover El Centro (l.Ox)
8 Super X-Brace Tension & Compression Static Pushover El Centro (l.Ox)
9 Tension-Only Static Pushover El Centro (l.Ox)
10 Chevron Brace Tension & Compression Static Pushover El Centro (l.Ox)
11 GW Composites Pultruded CFRP Si ngle-Bay X-Brace Tension & Compression Static Pushover El Centro (l.Ox)
12 Tension-Only Static Pushover El Centro (l.Ox)
13 Super X-Brace Tension & Compression Static Pushover El Centro (l.Ox)
14 Tension-Only Static Pushover El Centro (l.Ox)
15 Chevron Brace Tension & Compression Static Pushover El Centro (l.Ox)
16 FRP Confined Concrete CII-SC1, CM- SC2, CII-RC (average) Si ngle-Bay X-Brace Compression-Only Static Pushover El Centro (l.Ox)
17 Super X-Brace Compression-Only Static Pushover El Centro (l.Ox)
18 H-C-6L-C1, H-C-6L-C2 (average) Si ngle-Bay X-Brace Compression-Only Static Pushover El Centro (l.Ox)
19 Super X-Brace Compression-Only Static Pushover El Centro (l.Ox)
20 N-A-2L-C1, N-A-2L-C2 (average) Si ngle-Bay X-Brace Compression-Only Static Pushover El Centro (l.Ox)
21 Super X-Brace Compression-Only Static Pushover El Centro (l.Ox)
22 H-A-6L-C1, H-CA6L-C2 (average) Si ngle-Bay X-Brace Compression-Only Static Pushover El Centro (l.Ox)
23 Super X-Brace Compression-Only Static Pushover El Centro (l.Ox)
68


Table 3-2: Axial Properties of Hollow FRP Tubes Evaluated
Axial Properties of Hollow FRP Tubes
Manufacturer Strongwell GW Composites
Product Series Extren Series 500/525 Extren Series 500/526
Ultimate Tensile Stress 30,000 psi 240,000 psi
Tensile Modulus of Elasticity 2,500,000 psi 19,500,000 psi
Ultimate Compressive Stress 30,000 psi 202,000 psi
Compressive Modulus of Elasticity 2,500,000 psi 19,000,000 psi
Poisson's Ratio 0.33 Not Listed
Table 3-3: Axial Properties of FRP Confined Concrete Evaluated
Axial Properties of FRP Confined Concrete
Specimen Source Lam et. al. (2006) Ozbakkaloglu & Akin (2012) Ozbakkaloglu & Akin (2012) Ozbakkaloglu & Akin (2012)
Specimen Name Cl l-SCl/CI ISC2/CI l-RC H-C-6L-C1/H-C-6L-C2 N-A-2L-C1/N-A-2L-C2 H-A-6L-C1/H-A-6L-C2
Concrete Type Normal Strength High Strength Normal Strength High Strength
fco 5.6 ksi 15.5 ksi 5.6 ksi 15.2 ksi
38.9 Mpa 107.0 Mpa 38.5 Mpa 105.0 Mpa
£co(/o) 0.25 0.35 0.21 0.35
Confinement Material CFRP CFRP AFRP AFRP
No. of Layers 2 6 2 6
foe 11.3 ksi 18.4 ksi 9.3 ksi 23.8 ksi
77.8 Mpa 126.8 Mpa 64.3 Mpa 164.1 Mpa
Ecu (96) 1.99 1.15 2.25 2.00
£h,rup (%) 1.08 0.73 1.53 1.31
69


| 1Q'-0" ^ 1Q'-0" ^ IQ'-O" | 1Q'-Q" ^
10-0 10-0' 10*-0" 10'-0" 10'-0"
-------------Jf------------------/ vf-----------------f---------------
6" NW CONCRETE SLAB
SELF-WT = 75 PSF
\ 50 PSF UNIFORM
SUPERIMPOSED
I OEADLOAD I
/
W 16x36 STEEL
JOIST-TYP.
V
H
H
W 12x65 COLUMN TYP. W24x68 STEEL
GIRDER TYP.
Figure 3-1: Plan View Typical Floor and Roof Framing
70


Figure 3-2: Isometric Model View Single-Bay X-Bracing Extruded Sections
Figure 3-3: Isometric Model View Single-Bay X-Bracing Joint Labels
71


Figure 3-4: Isometric Model View Single-Bay X-Bracing Member Labels
Figure 3-5: Isometric Model View Single-Bay X-Bracing Individual Joint
Restraints
72


50-1
Figure 3-6: Elevation Model View Single-Bay X-Bracing Member Labels
Figure 3-7: Elevation Model View Single-Bay X-Bracing Mode Shape 1
73
55-1 . 104-


Figure 3-8: Elevation Model View Single-Bay X-Bracing Mode Shape 2
74


Figure 3-11: Isometric Model View Super X-Bracing Joint Labels
75


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PAGE 13

!'%=)r9r;r&r'(n+E'(nnA rn/!'1=.rn9r;r&r'(n+E'(n9r)rnr/! !'/=.rn9r;r&r'(n+E'(n9r nr/! !'3=.rn9r;r&r'(n+E'(n9r nr/$ !'8=.rn9r;r&r'(n+E'(n9r nr!/$ !'0=)r9r;r&rE'(n-. r r/% !'=)r9r;r&rE'(n-A nr/% !'=)r9r;r&rE'(n-9r) rnr/1 !'!=.rn9r;r&rE'(n-9r) rnr/% !'$=)r9r;r&-r(n-. r r// !'%=)r9r;r&-r(n-A nr// !'1=)r9r;r&-r(n-9r) rnr/3 !'/=.rn9r;r&-r(n-9r) rnr/3 !'3=rn n5r-nnr5nn)r r/8 !'8=.r)r'4+-nnr5nn)rr /8 !'0=.r)r'4+5 30 !'=9nrn r' n5rr-* 9*% 00:nr(30 !'=9nrn r' n5-4&:5 r&r.r %00F%%3 !'!=9nrn r' n5-4&5 r-:<)r3 !'$=9nrn r' n5-)n r rrr+& 5r53

PAGE 14

!'%=9nrn r' n5-4 r rrr+ 5r53!'1=9nrn r' n5-)n r rrr+& 5r*53!'/=9nrn r' n5-4 r rrr+ 5r*53$'n= n5rr-(nr rn, 9rG81 $'='rn)r'4+-(nr rn)r 9rG81 $'='rn)r'4+-,nr)r ;)r9rG81 $'='rn)r'4+-*rrn )r9rG8/ $'r='rn)r'4+-9*nr )r9rG8/ $'='rn)r'4+-Hr.r+ )r9rG8/ $'n= n5rr-(nr rn, 9rG83 $'='rn)r'4+-(nr rn)r 9rG83 $'='rn)r'4+-,nr)r )r9rG83 $'='rn)r'4+-*rrn )r9rG88 $'r='rn)r'4+-9*nr )r9rG88 $'='rn)r'4+-Hr.r+ )r9rG88 $'!n= n5rr-(nr rn, 9rG!00 $'!='rn)r'4+-(nr rn)r 9rG!00 $'!='rn)r'4+-,nr)r )r9rG!00 $'!='rn)r'4+-*rrn )r9rG!0 $'!r='rn)r'4+-9*nr )r9rG!0

PAGE 15

$'!='rn)r'4+-Hr.r+ )r9rG!0 $'$n= n5rr-(nr rn, 9rG$0 $'$='rn)r'4+-(nr rn)r 9rG$0 $'$='rn)r'4+-,nr)r )r9rG$0 $'$='rn)r'4+-*rrn )r9rG$0! $'$r='rn)r'4+-9*nr )r9rG$0! $'$='rn)r'4+-Hr.r+ )r9rG$0! $'%n= n5rr-(nr rn, 9rG%0$ $'%='rn)r'4+-(nr rn)r 9rG%0$ $'%='rn)r'4+-,nr)r )r9rG%0$ $'%='rn)r'4+-*rrn )r9rG%0% $'%r='rn)r'4+-9*nr )r9rG%0% $'%='rn)r'4+-Hr.r+ )r9rG%0% $'1n= n5rr-(nr rn, 9rG101 $'1='rn)r'4+-(nr rn)r 9rG101 $'1='rn)r'4+-,nr)r )r9rG101 $'1='rn)r'4+-*rrn )r9rG10/ $'1r='rn)r'4+-9*nr )r9rG10/ $'1='rn)r'4+-Hr.r+ )r9rG10/ $'/n= n5rr-(nr rn, 9rG/03 $'/='rn)r'4+-(nr rn)r 9rG/03 $'/='rn)r'4+-,nr)r )r9rG/03 $'/='rn)r'4+-*rrn )r9rG/08

PAGE 16

$'/r='rn)r'4+-9*nr )r9rG/08 $'/='rn)r'4+-Hr.r+ )r9rG/08 $'3n= n5rr-(nr rn, 9rG30 $'3='rn)r'4+-(nr rn)r 9rG30 $'3='rn)r'4+-,nr)r )r9rG30 $'3='rn)r'4+-*rrn )r9rG3 $'3r='rn)r'4+-9*nr )r9rG3 $'3='rn)r'4+-Hr.r+ )r9rG3 $'8n= n5rr-(nr rn, 9rG8 $'8='rn)r'4+-(nr rn)r 9rG8 $'8='rn)r'4+-,nr)r )r9rG8 $'8='rn)r'4+-*rrn )r9rG8! $'8r='rn)r'4+-9*nr )r9rG8! $'8='rn)r'4+-Hr.r+ )r9rG8! $'0n= n5rr-(nr rn ,9rG0$ $'0='rn)r'4+-(nr rn) r9rG0$ $'0='rn)r'4+-,nr)r )r9rG0$ $'0='rn)r'4+-*rrn )r9rG0% $'0r='rn)r'4+-9*nr )r9rG0% $'0='rn)r'4+-Hr.r+ )r9rG0% $'n= n5rr-(nr rn ,9rG1 $'='rn)r'4+-(nr rn) r9rG1 $'='rn)r'4+-,nr)r )r9rG1

PAGE 17

$'='rn)r'4+-*rrn )r9rG/ $'r='rn)r'4+-9*nr )r9rG/ $'='rn)r'4+-Hr.r+ )r9rG/ $'n= n5rr-(nr rn ,9rG3 $'='rn)r'4+-(nr rn) r9rG3 $'='rn)r'4+-,nr)r )r9rG3 $'='rn)r'4+-*rrn )r9rG8 $'r='rn)r'4+-9*nr )r9rG8 $'='rn)r'4+-Hr.r+ )r9rG8 $'!n= n5rr-(nr rn ,9rG!0 $'!='rn)r'4+-(nr rn) r9rG!0 $'!='rn)r'4+-,nr)r )r9rG!0 $'!='rn)r'4+-*rrn )r9rG! $'!r='rn)r'4+-9*nr )r9rG! $'!='rn)r'4+-Hr.r+ )r9rG! $'$n= n5rr-(nr rn ,9rG$ $'$='rn)r'4+-(nr rn) r9rG$ $'$='rn)r'4+-,nr)r )r9rG$ $'$='rn)r'4+-*rrn )r9rG$! $'$r='rn)r'4+-9*nr )r9rG$! $'$='rn)r'4+-Hr.r+ )r9rG$! $'%n= n5rr-(nr rn ,9rG%$ $'%='rn)r'4+-(nr rn) r9rG%$

PAGE 18

$'%='rn)r'4+-,nr)r )r9rG%$ $'%='rn)r'4+-*rrn )r9rG%% $'%r='rn)r'4+-9*nr )r9rG%% $'%='rn)r'4+-Hr.r+ )r9rG%% $'1n= n5rr-(nr rn ,9rG11 $'1='rn)r'4+-(nr rn) r9rG11 $'1='rn)r'4+-,nr)r )r9rG11 $'1='rn)r'4+-*rrn )r9rG1/ $'1r='rn)r'4+-9*nr )r9rG1/ $'1='rn)r'4+-Hr.r+ )r9rG1/ $'/n= n5rr-(nr rn ,9rG/3 $'/='rn)r'4+-(nr rn) r9rG/3 $'/='rn)r'4+-,nr)r )r9rG/3 $'/='rn)r'4+-*rrn )r9rG/8 $'/r='rn)r'4+-9*nr )r9rG/8 $'/='rn)r'4+-Hr.r+ )r9rG/8 $'3n= n5rr-(nr rn ,9rG3!0 $'3='rn)r'4+-(nr rn) r9rG3!0 $'3='rn)r'4+-,nr)r )r9rG3!0 $'3='rn)r'4+-*rrn )r9rG3! $'3r='rn)r'4+-9*nr )r9rG3! $'3='rn)r'4+-Hr.r+ )r9rG3! $'8n= n5rr-(nr rn ,9rG8!

PAGE 19

$'8='rn)r'4+-(nr rn) r9rG8! $'8='rn)r'4+-,nr)r )r9rG8! $'8='rn)r'4+-*rrn )r9rG8!! $'8r='rn)r'4+-9*nr )r9rG8!! $'8='rn)r'4+-Hr.r+ )r9rG8!! $'0n= n5rr-(nr rn ,9rG0!$ $'0='rn)r'4+-(nr rn) r9rG0!$ $'0='rn)r'4+-,nr)r )r9rG0!$ $'0='rn)r'4+-*rrn )r9rG0!% $'0r='rn)r'4+-9*nr )r9rG0!% $'0='rn)r'4+-Hr.r+ )r9rG0!% $'n= n5rr-(nr rn ,9rG!1 $'='rn)r'4+-(nr rn) r9rG!1 $'='rn)r'4+-,nr)r )r9rG!1 $'='rn)r'4+-*rrn )r9rG!/ $'r='rn)r'4+-9*nr )r9rG!/ $'='rn)r'4+-Hr.r+ )r9rG!/ $'n= n5rr-(nr rn ,9rG!3 $'='rn)r'4+-(nr rn) r9rG!3 $'='rn)r'4+-,nr)r )r9rG!3 $'='rn)r'4+-*rrn )r9rG!8 $'r='rn)r'4+-9*nr )r9rG!8 $'='rn)r'4+-Hr.r+ )r9rG!8

PAGE 20

$'!n= n5rr-(nr rn ,9rG!$0 $'!='rn)r'4+-(nr rn) r9rG!$0 $'!='rn)r'4+-,nr)r )r9rG!$0 $'!='rn)r'4+-*rrn )r9rG!$ $'!r='rn)r'4+-9*nr )r9rG!$ $'!='rn)r'4+-Hr.r+ )r9rG!$ $'$n= n5rr)nr'(n +E'(nr9r$ $'$= n5rr)nr'(n +E'(nr9r$ $'%n= n5rr)nrE'( nr9r$! $'%= n5rr)nrE'( nr9r$! $'1= n5rr)n-r( nr9r$$ $'/=)n9n)).n,nr)r 9n))n ,nr)rn5r*n+ $$ $'3=nr&rr9n))n,nr)rn 9n)).n ,nr)rn5r*n+ $% $'8n=:nnrrrn,+n -+n rr(nr$% $'8=:nnrrrn,+n -+n4&5(nr$1 $'8=:nnrrrn,+n -+n5r rr(nr$1$'!0=)n9n))(nr rn-rn ,+n).n2n6r n$/ $'!=)n9n)),nr)rrn,+n) .n2n6rn$/

PAGE 21

$'!=)n9n))*rrnrn,+n) .n2n6rn$3$'!!=)n9n))*nrrr (n9r)rrn,+n).n2n6rn $3 $'!$=)nHr.r+)r-' rn,+n).n2n6r n$8 $'!%=rnr&rrnn5r rn(nr rn) ,+n).n2n6rn$8 $'!1=rnr&rr9n)) n5r, nr)rn 9n)),nr)r),+n).n2n6rn %0 $'!/=rnr&rr9n))(nr rnn9n )) *rrn),+n).n2n6rn %0 $'!3=rnr&rr9n))(nr rnn9n ))*nr rr(n9r)r),+n).n2n6rn % ('=r&9r n<&1 ('=!,n)r<&1('!=(n9nrn5rr<& 1! ('$=*nr9nrn5rr<& 1! ('%=rn9nrn,nn<& 1$ ('1=n)r9r)r5rr<& 1% ('/= r5rr<& 1% ('3=*rn r<&11 ('8= r r,nn<& 11

PAGE 22

('0=9n r,nn<&1 / ('=,rrn<& 13 ('=,nn)n<& 13 ('!=,rr)r4+<& 18 ('$=)r4+,r<& 18 ('%=,rrnnr<& /0 ('1=nnr,nn-9n<& /0 ('/=nnr,nn-rn n<& / ('3=nnr,nn-rn)r4+< &/ ('8= rnnr<& /

PAGE 23

nn n))nr r)rrrr>)rr@ rnn )rn+ rn)rr )rn+rrr n)rn+5 )rn+r+)r)n rn)rr nnrrrrnrrn )rrrrrr rn6rrrr r rr)nrrn? ?rnrr rrnrrnrrr ))rrn r))nrn rr r)nr )r )rrrnrnnrr r) nn)n)nr )nnn n9nrCr2nrnrnnr n)rn) rr?nrn

PAGE 24

rr)nrr?n r )nr9.?%rrn)r?rnrrnrrn nn)rrn r )nr9.?%rrn)r?rnrrnrrn nn)rrnn rr r?%rrn)r?rnrrnrr nnn)rrn r r?%rrn)r?rnrrnrr nnn)rrnnr r r9.?%rrn)r?rnrrnrrn nn)rrn rn#rrnrr r9.?%rrn)r?rnrrnrrn nn)rrnnr rn#rrnrr 6rr?)r n)rnrr rnr nr)rr r+r nrrnr nnnr)rnrnrnr r&rrrrrnr r nrnrnrrnnrnr )rnrr )rn+)r)n rrrr)&rr nnrnnrrr nr&rrn+nnnrrr rrnnr rn6rrnrr rrnnnnrrrrr

PAGE 25

nnrn6rrrr nnrn6rrrrr nrrnrr#n6rr nrr#n6rnnr !")nrrrrrrn!")nrrrr5!")nrrrr+)r)nrrnrnrr&rrrr rn#n6rrrr#n6rrr rrr 5Cn5Cnrrnrnrr

PAGE 26

nrnrnnrn r nnrn nrnn nrnnnnr n rnn n nn nrnnrn !n nn"nn "nr nrnn #$rnn nrnn nnnnnn nnn nn#% nnnn nnnnn n$nn n#% nnnr$nn nr rrr&nnn nn nr nn nrnrnnnn nn

PAGE 27

' rnrn nn n %nr&n"nnnn'()n n nnn *rnr nnnnrnnn n#% n +nn#% nn nnn nnnnr nnn #% nrnnn nn,nn$ r -nn'((.n/0n""nn1"'( 'n #% r2n nnn 3nn nnn nnrnn#% rr rnnn #% n#% 2nnn nnnnn" nrnnnnr nn#% rn nn"nn r2r #% nnn nnn n#% n#% nnrn

PAGE 28

) n45 nrn "nnn n #% n2rn nnn nn"nn"nnr rnnn nnrnn n"n nrn nnnn !nrn nnnr nnnrnnn +n63nn" nnn nrr'7((rn nrnrn nrn nnnn nnrn nnnn nnnnn nnn nnn nnnnr nnr nnnn# % n #% nn nn nnnn rnn$r #% nn 1nnnrnn nnn2 nnnnrn nnnn 8nnn

PAGE 29

9 nnnn nnnn nn n r rnr : !" nnn nnrn nnrn6 n nnnnnnn nnnn nnrn#% n!" nnnn nrnnn nnnn !" nnn nrnn nnrnnn nnn1n nn!" nrnrn n nnn #% n nn#nr$

PAGE 30

nr nnrrr rnrnn nnrnrrnn rnrrnn rrnnnrnnn rnr rnrnnrr nrrrn rnrn rrn nnrnnn nnn nn n!nnr nnnnr nrnrnrrrr nr"nr n#nrnrrn rnrrnn nrnrnnn rnrnn rnnrnnrn #nn#n nrnnnnrn rrrrn rrnrnn rnn rnnrnr n"nr nrrnn n!nnn$nnr% &r'nn()*($%&r nrrrr nnnrn n()*($%&rr+!&,-.* )/01rr %2n!nnn+ !rnr&,

PAGE 31

3 &n**nr(4rn+!&,-.*)n r#nrn rnn !"nr nnn&n **r+!&,-.*)r"nrnn nnnrr rnnrnn rrnn rnrnrn n n nnrn# nrr 5/&,6rnrr)(r n nnn /&,rnrr*)r /&,n#nnrnrr nn(n rnr)r nrrr()) +nrn7nrr nrrn n/&,nrrrnnn rrn nrnrn/&,n nnn rnrrnnnnn rrnrn nrnrnrn rn rn !8rr!5 !8!6nnn 9::n#r: rn(.*r rn/&,*.rn rnrnrrnn n !nnnrn nnnnnr rn n!nn

PAGE 32

+!&,-.*)nr;nn nnnrnnrr nrrn nrrnrrnrnr nnnnn nrnrn nrr9 9rnr !8!rn 5(.*6 5(.(6 n 5(.46 n 5(.<6 2nrnrn nrn rr nnnnnr nrnnr nnnnrnnnnn n#nrnn7 nrnrrnrnr nrn rnrnnr nnr nrnnrrn nnn+n rnr(.(rnr"nr&n**rn+ !&,-.*)nrnn !0&nrrnnnr nrn n3r!0&n r'+nr!

PAGE 33

= 0&nrnrnr nn#nrn rnnrnrrnnnn nr; nnrnr# nn!0&+nr n#nrn>rrr? n'nnnrn #n$nn n !" &n*(rn+!&,-.*)nrn # rnnr nnnnnnn nnrrnnn rnnnnrrnn nnnrnnnr rnrnnnrn rnrr nnn' ,#n1nr+/rr! n+! r@nrrAnnrn rnnn nnrnnn n rnrrnnnrrnn nn#n %rnrn rnrnrrn nnnrrr rnnnn nrnn nnn# n+nrrn,#n1nr +5,16n/r r!n+rr +rn

PAGE 34

B nrrn!r@nrr rnrr nrnnnrnn rnr"n7nn ,1nrnnrrrn nnrn nrnnnrrn nnnr +rnnnr;nn nrn nnnnnn nr@rn ,1nrn;nr;nrnn n'n rrnnr#nnn rnnnnr nnn rnrnrnr rrnrnrnr nrnr r n!0!!0*rn nrnrn nr$nrrnnrrn nn rnrrnnn nr>nn?n #nrr n+!&,-.*)nnrn nn7 nrrr9 r n 5(.6 r n 5(.36 r n 5(.-6 r n 5(.=6

PAGE 35

*) rnnrn #nrrnn)3 rnn9 r # $ % 5(.B6 rrnnn nnnrn nnnrrnn 9 & r 5(.*)6 nrn nnnnrrnn r nnnrnrr#nr9 ( r ( & 5(.**6 r ( ) ( ( + ) + / 5(.*(6 nnnrnr nn;nrnn*))nrnn n7 nnnnr'nnn 7nr+rn rrnnnnnnn nrr..r. rnnn nn;rn rnnnrn nnn rnnr/rr!n+5/!6nr ,1 n;nrnrn rrnrnr nrrnrnnnrr nnnAn,1 rrnnnnrr nr/!#rrr

PAGE 36

** nrnrnnr rnrrnnnr nrrnrn# nnnrr rnnrrnrrrn nrrnnB)nrn nn7nAnn rrrnnnn. .r.rnnrnnr r..r.r nr#r rrnr,rn ..r.rnnn ;nrnnnr rrnnrn r$nr rnnnnrn/!rn;n rnrn nnnrn.r nr/!nrr nnrnrrrn rCnn5rn rn6rnnnn nrrr rnrrrn rnn rnnrnn +nnnrnrrn rn'nnr nrr nrnrnrnn# rrnr#5!!!6r rn#n#nr5&D&6nrnrrn nrrnnnn %&'(n !r@nrrr+!&,-. *)n&n *3rnnrrnr rnnn nnr"nnr rn.nrrnr +nrnrnrrnr nnrnn

PAGE 37

*( nnrrnrn rnrnn nrnnrnnnrnrnnn&n*3! r@nr r'n1r@nrr nCrr@nr r+nnnnn1 rn;n;nrnn nnn.nnr rnnrrn rnn;nrnn rnn,#n1n rrn/rr!n+ rnn n ;nrnr n,#0n+rnrr nnnnnnr nnrrrnr #nrnnrn nnn r.rnr nrnrnrnnrn# nrr rnrn(.4n nnrrn,# 0n+rnrrrnrnrnnCrr rnnr rnnnnrn "nnr. nCrrrrnr rrrnnr nrnrnnnrnnr rnn rrnnnnrnnn nnnn r+nrnrnr rn. nrrnrnrn $nrrnrnn

PAGE 38

*4 nnnrn rrnrnnnnnr "nnrnn#n rnnrr nnnr rrnrnn nrnrnnnr"n n5,/n +())B6,n#, &n5,,6n rnr nrnrrnn rrrrnr rrnn n#rrrn,, nnn nnrrnrn2n5n4nr-6rr rnrr nrnrn nrrn rrn!nr*3*4*r*3*4( !rnrrnr nrnnrrrnr nrrn n7nnnnrnr nrrnn nrnnrrnnn r 0r!nnnr rnrr +!&,-.*)&n**nrn nrnnnn rnn.nrrnrn nr! rrnrnrnnnr n.nnrn nn&n**0r!n nrnr nrnnrrnnn rn#nnnr nnr()nnr*)nrn nrrnnn rnr)(rrrn nnrrnrn

PAGE 39

*< rrnnnn*rr nnnrrnr nr rnn.n n0nrnrnn"n nnrn! r@nrrn+!&,-.*)n #nnn.n rnrn nr nnnnnnnr nnnr nrrrrnn nr @nrrnrnn #nrrn +!&,-.*)nnnnrn rn!r@nr r%$$ )n/rrrn nnn rrn rn# !rrn nrrn*B3) nrnrnnn nrrrn n<)r nrnrnn nrrn nrrnrrnn n!nn nnrnn rnrnnnrrr nr nnrnrnnnn nn#rr

PAGE 40

* !nrnnrn nnrn rrnnrnr n %$*n1r%&rn#r*B44n nrnrn nrnn n !nn/rn&r nnnrnnr rrnn# r#nnn+nr*B44r n+nnn rrnrrnr rn# 50nnr8!())(6+nrn+nnnr nr nnnnrnnrnr r rnnrnrnnnr;n n&rn n8r+n*B4B50nnr8 !())(68r +n&rrr nnnrnr *B44#nrnrrn nn$n rnrnrrn #nrn nrnnrr!nrnnrrnn nnrn nCnr,n#@;nrr5 C,@6n &r*B--nnnn>nrn rrnrn n#n n!nnnrnn nnr nn#;nrr? rC,@nnn

PAGE 41

*3 rnnnrnnrr rnnrr n#nnrn n!nnrnnrr nnnnrC,@ nnr'n, /n+5,/+6n Cnr$nnnr!nrr5C$! 6nCnr!rnr 5C!6n !8rr!5 !8!6 rr,/+rn nn ,/+rrn rnrnnnnrnn nnr rrnnnr rnr ,/+<-' rrn ,/+ .-)' !rr!" # r nr !nrnnn nrnrnr' nrrrnr! r"nr nnrnrnrr nnn nrnr#n nnrrnnnnrn nrnrrnrr rnn rnrrnnnErn nnr'rn rnrnrnn# nrnr "nr'nrn nrnr"nrn/rnnn rnrrnnrn nrnnrn rrnrrnnrn rn2nrnnr rnnnrrnn; nrrnnrn

PAGE 42

*;nnrn# nr rnnrn nr7rn nrnrnnnnn rrrrn nnrr nr %$+,#nnnnnr rrnrn n#nn nrnrnr r;nnrnrnn# nnnrn nrnnnnr rnrn rrnnrr nn#n rnrnnrnn #nr rr2nr;nn nnn nrnnn nrnrn9 +nnrnnrn ,rrnn nnr rnrn $rrnrnrrn rnrrnrnnn $nn5,/n+())36

PAGE 43

*= +nnn rrn nrnrrnrnn# nn rnrnnnr nnnrn rnnrnrnnn Ann nrnrnnrnn rrn rrnrnrnnn rnrnnn nnrnnnnnrnnnn rrnrn nnrnnr nnnr r,rnnr rnrnn rrrnnnrrr nnn rnrnnnr rrnnnr nn.nrn rnrnn#nn r;rnnnrrrrn nrn nnrnrnrn n !rnnnrr nnnnn rnnn#n nnnnrnn rn#rnnrnrn nrrnn $nnrnn#n rrnrnrn rrnrnnrnrnn rnnr rrn# nnrn nrn# nnn nnrnrn.rn r.nnnn

PAGE 44

*B nnrnn#r2 nrnn rr rrrnrrr nnrnrnr rnr n#rrnrr rrnnn'rn rrnrnrr; rrnr rnrnn#r rnrrnrn nnnrnrn nn7rnrn rrrnnnrn nn+rn n#nrnn"rn rrnr nrnr rrr5 ,/n+())36%%$( .% $%&'( rn nnr nnnr rnn n!nn$nrrnrn rrnrn' $&%( rrn n+rr &r5*B=-6 )*$+,'( !-r n,/n+5*BB-6 rnrnnr rnrrnnr nrnrn$n rnrn "n nnnnrnnr nrn nrnnnn

PAGE 45

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rnrrr$n nr rnr nrr rrrnr rrr nrr r r-rr nnrrr +r rr.nrrrr rB".rr r$ nrrn rn+nr rnr rr rnrn n rrr1( r!"/2r .rnrr rr rnrr r nrr+rr .rnnr rnnr rrr. rrrnrr .rnn. rrrnrrr r rrn rrrrn rrr rnnrr rn nrr $r rr nrrr rrrr rrn rrrn

PAGE 114

* nnrrrr r rrnr .rnnr rr12(r!"#r!". r=# r rrnr r. r=# r rr r r rrr rrrr -r=# r r"r r r" rrrr r rr r rnr rr12(r!"#r!"n rrr+? rrn rrr nrr ?r(r$ r rn r nrnr rr Arrr n rrrr r nn rrrr rr (r? r 3!"#4

PAGE 115

* ?rr rnrnr%r rrn?r rnr rr+n%nn&#nr rrr rr nr Arnrrnrr rrr ,nrrnr rn @rnrnr rr(r!"! r!":rr nn? rnn5r" /" r nr/" rrr r (; (;r rrrrr A rrrA rrrrr rrnr (r!"!!"6!":nrr (; r rrrA rr r(rr rr 3" /" rnr/" rr r 4<(; n:nrr A<(; rn*nrr A(r /" rnr/" rnr rrr r rr ,r nrr "nrr rr

PAGE 116

*# rrn38(; r nr r8(; r"4 r" r rr r .n r r r n r (r!"! !"6 r nrrrr (;r r Ar nrrrr $(;r rnn A r rrnrrrr rrnr )nrrrr r<;; r. rr r"r r:r<(; rn rnnr nrrr rrrnrr n rrrnr r(r!")!" rnrnrnrr( r!")n rr nn? r nnr rrr nn rrrr rrr r (r!"r rrrr rn rrr rr

PAGE 117

* rrrAr r 6 8(;<(;rr .# rrr rr rr<(;r(; rrr "rr .rr<(;r(; r 6nr rnr rr nr(r !"(; rnr r r nn(;r nrr r(; rrrA r rr r rr. (r!"*!"* !"*nrnrnr rrn r n(; r n(; r r rn(r!"6n.rr rr r%rrrr 5 rrr% rr% r%r rrrrr r rrr r rr.rrrr rn nrnr" r rnnr<>)"#nr nnrr nrnr(r"

PAGE 118

* (r!":rn rrrr r%r rrr rr rrr-? r rrr rr r r r rrrr nnr rr r rrnr (r !") (r!"rr rr r .rr. rr=# r rrr rrrr r$ nr rr% r rrr rrrrr rrn nrr5 r. rr5nr r rr .r r r rr nrnrn rrnr r r rnrnrn r$rn r

PAGE 119

*! "'r()##rr* +n ,&n,n nrn n nrn n nrn n nnnnnnnnnnnnnnn nnnnnnnnnnnnnnnnnnnnnnnnnnnn nrn n

PAGE 120

*6 "'r()##(-nr#nn r +n,&n,n n n n n !"rn n !#n n #nn n $%& ( '') nnnnnnnnnnnnnnnnnnnnnnnnnnnnn nnnnnnnnnnnnnnnnn

PAGE 121

*: n,r()(-nrn !r&!r.+ r r!n. r/ n,r(')(-nr"n#r$n.+r r! "n#r.r/ n,r()(-nr"n#r$n.n r#r!"n#r.r/

PAGE 122

*) n,r()(-nr"n#r$n.rr n!"n#r.r/ n,r(r)r#'r/0n+rr!"n#r. r/ n,r()r1nrn*r,!"n#r. r/

PAGE 123

* n,r()(-nrn !r&!r.+ r r!n. r/ n,r(')(-nr"n#r$n.+r r! "n#r.r/ n,r()(-nr"n#r$n.n r#r!"n#r.r/

PAGE 124

** n,r()(-nr"n#r$n.rr n!"n#r.r/ n,r(r)r#'r/0n+rr!"n#r. r/ n,r()r1nrn*r,!"n#r. r/

PAGE 125

# n,r()(-nrn !r&!r.+ r r!n. r/ n,r(')(-nr"n#r$n.+r r! "n#r.r/ n,r()(-nr"n#r$n.n r#r!"n#r.r/

PAGE 126

## n,r()(-nr"n#r$n.rr n!"n#r.r/ n,r(r)r#'r/0n+rr!"n#r. r/ n,r()r1nrn*r,!"n#r. r/

PAGE 127

# n,r()(-nrn !r&!r.+ r r!n. r/ n,r(')(-nr"n#r$n.+r r! "n#r.r/ n,r()(-nr"n#r$n.n r#r!"n#r.r/

PAGE 128

# n,r()(-nr"n#r$n.rr n!"n#r.r/ n,r(r)r#'r/0n+rr!"n#r. r/ n,r()r1nrn*r,!"n#r. r/

PAGE 129

#! n,r(%)(-nrn !r&!r.+ r r!n. r/% n,r(%')(-nr"n#r$n.+r r! "n#r.r/% n,r(%)(-nr"n#r$n.n r#r!"n#r.r/%

PAGE 130

#6 n,r(%)(-nr"n#r$n.rr n!"n#r.r/% n,r(%r)r#'r/0n+rr!"n#r. r/% n,r(%)r1nrn*r,!"n#r. r/%

PAGE 131

#: n,r(2)(-nrn !r&!r.+ r r!n. r/2 n,r(2')(-nr"n#r$n.+r r! "n#r.r/2 n,r(2)(-nr"n#r$n.n r#r!"n#r.r/2

PAGE 132

#) n,r(2)(-nr"n#r$n.rr n!"n#r.r/2 n,r(2r)r#'r/0n+rr!"n#r. r/2 n,r(2)r1nrn*r,!"n#r. r/2

PAGE 133

# n,r(3)(-nrn !r&!r.+ r r!n. r/3 n,r(3')(-nr"n#r$n.+r r! "n#r.r/3 n,r(3)(-nr"n#r$n.n r#r!"n#r.r/3

PAGE 134

#* n,r(3)(-nr"n#r$n.rr n!"n#r.r/3 n,r(3r)r#'r/0n+rr!"n#r. r/3 n,r(3)r1nrn*r,!"n#r. r/3

PAGE 135

## n,r(4)(-nrn !r&!r.+ r r!n. r/4 n,r(4')(-nr"n#r$n.+r r! "n#r.r/4 n,r(4)(-nr"n#r$n.n r#r!"n#r.r/4

PAGE 136

### n,r(4)(-nr"n#r$n.rr n!"n#r.r/4 n,r(4r)r#'r/0n+rr!"n#r. r/4 n,r(4)r1nrn*r,!"n#r. r/4

PAGE 137

## n,r(5)(-nrn !r&!r.+ r r!n. r/5 n,r(5')(-nr"n#r$n.+r r! "n#r.r/5 n,r(5)(-nr"n#r$n.n r#r!"n#r.r/5

PAGE 138

## n,r(5)(-nr"n#r$n.rr n!"n#r.r/5 n,r(5r)r#'r/0n+rr!"n#r. r/5 n,r(5)r1nrn*r,!"n#r. r/5

PAGE 139

##! n,r(6)(-nrn !r&!r.+ r r!n. r/6 n,r(6')(-nr"n#r$n.+r r !"n#r.r/6 n,r(6)(-nr"n#r$n.n r#r!"n#r.r/6

PAGE 140

##6 n,r(6)(-nr"n#r$n.rr n!"n#r.r/6 n,r(6r)r#'r/0n+rr!"n#r .r/6 n,r(6)r1nrn*r,!"n#r. r/6

PAGE 141

##: n,r()(-nrn !r&!r.+ r r!n. r/ n,r(')(-nr"n#r$n.+r r !"n#r.r/ n,r()(-nr"n#r$n.n r#r!"n#r.r/

PAGE 142

##) n,r()(-nr"n#r$n.rr n!"n#r.r/ n,r(r)r#'r/0n+rr!"n#r .r/ n,r()r1nrn*r,!"n#r. r/

PAGE 143

## n,r()(-nrn !r&!r.+ r r!n. r/ n,r(')(-nr"n#r$n.+r r !"n#r.r/ n,r()(-nr"n#r$n.n r#r!"n#r.r/

PAGE 144

##* n,r()(-nr"n#r$n.rr n!"n#r.r/ n,r(r)r#'r/0n+rr!"n#r .r/ n,r()r1nrn*r,!"n#r. r/

PAGE 145

# n,r()(-nrn !r&!r.+ r r!n. r/ n,r(')(-nr"n#r$n.+r r !"n#r.r/ n,r()(-nr"n#r$n.n r#r!"n#r.r/

PAGE 146

## n,r()(-nr"n#r$n.rr n!"n#r.r/ n,r(r)r#'r/0n+rr!"n#r .r/ n,r()r1nrn*r,!"n#r. r/

PAGE 147

# n,r()(-nrn !r&!r.+ r r!n. r/ n,r(')(-nr"n#r$n.+r r !"n#r.r/ n,r()(-nr"n#r$n.n r#r!"n#r.r/

PAGE 148

# n,r()(-nr"n#r$n.rr n!"n#r.r/ n,r(r)r#'r/0n+rr!"n#r .r/ n,r()r1nrn*r,!"n#r. r/

PAGE 149

#! n,r(%)(-nrn !r&!r.+ r r!n. r/% n,r(%')(-nr"n#r$n.+r r !"n#r.r/% n,r(%)(-nr"n#r$n.n r#r!"n#r.r/%

PAGE 150

#6 n,r(%)(-nr"n#r$n.rr n!"n#r.r/% n,r(%r)r#'r/0n+rr!"n#r .r/% n,r(%)r1nrn*r,!"n#r. r/%

PAGE 151

#: n,r(2)(-nrn !r&!r.+ r r!n. r/2 n,r(2')(-nr"n#r$n.+r r !"n#r.r/2 n,r(2)(-nr"n#r$n.n r#r!"n#r.r/2

PAGE 152

#) n,r(2)(-nr"n#r$n.rr n!"n#r.r/2 n,r(2r)r#'r/0n+rr!"n#r .r/2 n,r(2)r1nrn*r,!"n#r. r/2

PAGE 153

# n,r(3)(-nrn !r&!r.+ r r!n. r/3 n,r(3')(-nr"n#r$n.+r r !"n#r.r/3 n,r(3)(-nr"n#r$n.n r#r!"n#r.r/3

PAGE 154

#* n,r(3)(-nr"n#r$n.rr n!"n#r.r/3 n,r(3r)r#'r/0n+rr!"n#r .r/3 n,r(3)r1nrn*r,!"n#r. r/3

PAGE 155

# n,r(4)(-nrn !r&!r.+ r r!n. r/4 n,r(4')(-nr"n#r$n.+r r !"n#r.r/4 n,r(4)(-nr"n#r$n.n r#r!"n#r.r/4

PAGE 156

## n,r(4)(-nr"n#r$n.rr n!"n#r.r/4 n,r(4r)r#'r/0n+rr!"n#r .r/4 n,r(4)r1nrn*r,!"n#r. r/4

PAGE 157

# n,r(5)(-nrn !r&!r.+ r r!n. r/5 n,r(5')(-nr"n#r$n.+r r !"n#r.r/5 n,r(5)(-nr"n#r$n.n r#r!"n#r.r/5

PAGE 158

# n,r(5)(-nr"n#r$n.rr n!"n#r.r/5 n,r(5r)r#'r/0n+rr!"n#r .r/5 n,r(5)r1nrn*r,!"n#r. r/5

PAGE 159

#! n,r(6)(-nrn !r&!r.+ r r!n. r/6 n,r(6')(-nr"n#r$n.+r r !"n#r.r/6 n,r(6)(-nr"n#r$n.n r#r!"n#r.r/6

PAGE 160

#6 n,r(6)(-nr"n#r$n.rr n!"n#r.r/6 n,r(6r)r#'r/0n+rr!"n#r .r/6 n,r(6)r1nrn*r,!"n#r. r/6

PAGE 161

#: n,r()(-nrn !r&!r.+ r r!n. r/ n,r(')(-nr"n#r$n.+r r !"n#r.r/ n,r()(-nr"n#r$n.n r#r!"n#r.r/

PAGE 162

#) n,r()(-nr"n#r$n.rr n!"n#r.r/ n,r(r)r#'r/0n+rr!"n#r .r/ n,r()r1nrn*r,!"n#r. r/

PAGE 163

# n,r()(-nrn !r&!r.+ r r!n. r/ n,r(')(-nr"n#r$n.+r r !"n#r.r/ n,r()(-nr"n#r$n.n r#r!"n#r.r/

PAGE 164

#* n,r()(-nr"n#r$n.rr n!"n#r.r/ n,r(r)r#'r/0n+rr!"n#r .r/ n,r()r1nrn*r,!"n#r. r/

PAGE 165

#! n,r()(-nrn !r&!r.+ r r!n. r/ n,r(')(-nr"n#r$n.+r r !"n#r.r/ n,r()(-nr"n#r$n.n r#r!"n#r.r/

PAGE 166

#!# n,r()(-nr"n#r$n.rr n!"n#r.r/ n,r(r)r#'r/0n+rr!"n#r .r/ n,r()r1nrn*r,!"n#r. r/

PAGE 167

#! n,r()n !r&!r&#n.n ,r(+7(+rr n,r(')n !r&!r&#n.n ,r(+7(+rr

PAGE 168

#! n,r(%)n !r&!r&#n. r7(+rr n,r(%')n !r&!r&#n. r7(+rr

PAGE 169

#!! n,r(2)n !r&!r&#n.& r !+rr n,r(3)&#n0n##*nnr #r!0n##" nr#r.n !rn

PAGE 170

#!6 n,r(4)n'r8rr0n##"nr#r 0n##*n nr#r.n !rn n,r(5)9 nrrrnnn n."nrr+r

PAGE 171

#!: n,r(5')9 nrrrnnn n."n$8 +r n,r(5)9 nrrrnnn n."n&nr &rr+r

PAGE 172

#!) n,r(6)&#n0n##rr+r r.nr#n :r-n, n,r()&#n0n##rrn r#r.nr #n* :r-n,

PAGE 173

#! n,r()&#n0n##rr rrn.nr #n* :r-n, n,r()&#n0n##rr0n rn-r!r+n, r#'r.nr#n* :r-n,

PAGE 174

#!* n,r()&#n0n##rr1nrn *r,.nr #n* :r-n, n,r(%)&rn'r8rr#rrr r 0n##+r r.nr#n* :rn,

PAGE 175

#6 n,r(2)&rn'r8rr#rrr r 0n##rrn.nr#n* :r-n, n,r(3)&rn'r8rr0n##+r r 0n## rrn.nr#n* :r-n,

PAGE 176

#6# n,r(4)&rn'r8rr0n##+r r 0n##0nr n-r!r+n,r#'r.nr#n*;n,

PAGE 177

nrnrn nrn rnrnr nr n rrn nnrnrr!! rrn nrr!rrr"n nrr#nr rnn! $%!r$%nrrnrn !nr rnnrn& 'nrrrnrrrr rnrrr rrrn(rr r (rn)$%rr nr! (* rrrnr &n$%nrr nrnr+ nrrn )(!$%nrr nrnnrr rnrrrn rnnrnr rnrn&rnrnrrr nn nrn$%nr,

PAGE 178

-n.nr!(r /nn %0nrn1n rrr nrn1 nr23!3! 3 $rrrn(r! n(n( rrnr n(nr &/nr nrnrnrn n!rrnn )$%nrr nnrnr &n4n r(" rrrr!n nnnr n"4rnr4r + $%nrrnrnr nrr nrnnr &nr rr/rr/ rrr nrrrrnn rn "r!n nrnr + nrnrn 5678r67+r nrnrrn r(rnnr $%$%nrr nrnrnrr r&678

PAGE 179

+ rrnnrrn (nr rnrnnr$ /!(%9:! ;9!r69nn nrnrnn nrr!< nnrr nrrr&(r nr (r'nr( rnnnn rrnrnr rrnn "(rnr& "(r( rnrnnr!%!(r n!;!rnrnr n!6=>nrn6(7rr!?& nr rr(rr nnr!r nrnr(r rnrrrnr nrrn@rrn rnn! rr$%$%n rrnrn nrrnrn nrrn nr/n(nr((rnrnrr &nrrn Ar n$%! nnrn& rnrr6+ $%n r'nrnnBr nn rn/ nnnn

PAGE 180

nrr (rn$% r(rnr C!nr(nrA rnrnr nnrnr)(! nn(n rn$% nr!r rr( rnrr r/> nr/$%?! nr nnrnr&nrnrrn nrrnr$% nrrnrnnnr6 r+!$%nrr nrnnr(n rrr! nrn1nr/r r&rn $%nrrnrnn nrnr rnnrrnn rrrnr rnr(rr!n nrn( nnnrrnrr'r $%nrrnrnn !'nr/ nr!r/rr *nnrrr r Dr(n5 4r Arnr'nn rrr&n rnrrnnr rnrrrn /rrnr nrrnr nr/!" (r

PAGE 181

nrn nrnr rr nrn" rrrn n rr#" $%&rn rrr %'&(' r)* rn+!,!-n(.n( %*./(/* *n*.!0 !! 1!2.3%(! +!,!(45r!6!78-2 (/)((n (.n*+.n(9:*. 0 "r#r ;<7#8= ##!+!,!(45r!6!-+/*.' nn(.n( %nn**.0 "#r$ 78=""! +!,!2(r!45r!6!-n *+3*(&+. n(9:*.+(>'n*( +/*(/)(= n2(0 "#r$ ;78="##"! +(n2n %&' r rn +2"+23'$!! +(n2n; nn'r r r +2#+23'$!! +(n2n $%rr "r +2##+23'$!! +(n2n () r rnn +2#+23'$!! +(n2n; %& %*nr +29!+23' $!!+(n2n +,,-%& .nr$ +29!+23'$!! (*n rrn !+'*n'

PAGE 182

rr(-(*n+9% *n0rr ((3%2.(?!2!;;; r$rr :@+nn! 9'(%'9))!:?!&!'*!6!AB!78 -+9C(n1( /nn%0 r$ rnn)(! !<=;#;;rr 2(+9( n0 /$nn01! rnn)(!<";! 2(?!E!92!?!D!95!;<<-: 'nrr2( (n0 "#rrnn #7<8=<#(rr2 (0 nn *#;=<< 3(%/ nrrr$2nrr E 7 >>>!> !n8

PAGE 183

n rData for El Centro 1940 North South Component (Pekn old Version) 1559 points at equal spacing of 0.02 sec Points are listed in the format of 8F10.5, i.e., 8 points across in a row with 5 decimal places The units are (g) *** Begin data *** 0.00630 0.00364 0.00099 0.00428 0.00758 0.01087 0.00682 0.00277 -0.00128 0.00368 0.00864 0.01360 0.00727 0.00094 0.00420 0.00221 0.00021 0.00444 0.00867 0.01290 0.01713 -0.00343 -0.02400 -0.00992 0.00416 0.00528 0.01653 0.02779 0.03904 0.02449 0.00995 0.00961 0.00926 0.00892 -0.00486 -0.01864 -0.03242 -0.03365 -0.05723 -0.04534 -0.03346 -0.03201 -0.03056 -0.02911 -0.02766 -0.04116 -0.05466 -0.06816 -0.08166 -0.06846 -0.05527 -0.04208 -0.04259 -0.04311 -0.02428 -0.00545 0.01338 0.03221 0.05104 0.06987 0.08870 0.04524 0.00179 -0.04167 -0.08513 -0.12858 -0.17204 -0.12908 -0.08613 -0.08902 -0.09192 -0.09482 -0.09324 -0.09166 -0.09478 -0.09789 -0.12902 -0.07652 -0.02401 0.02849 0.08099 0.13350 0.18600 0.23850 0.21993 0.20135 0.18277 0.16420 0.14562 0.16143 0.17725 0.13215 0.08705 0.04196 -0.00314 -0.04824 -0.09334 -0.13843 -0.18353 -0.22863 -0.27372 -0.31882 -0.25024 -0.18166 -0.11309 -0.04451 0.02407 0.09265 0.16123 0.22981 0.29839 0.23197 0.16554 0.09912 0.03270 -0.03372 -0.10014 -0.16656 -0.23299 -0.29941 -0.00421 0.29099 0.22380 0.15662 0.08943 0.02224 -0.04495 0.01834 0.08163 0.14491 0.20820 0.18973 0.17125 0.13759 0.10393 0.07027 0.03661 0.00295 -0.03071 -0.00561 0.01948 0.04458 0.06468 0.08478 0.10487 0.05895 0.01303 -0.03289 -0.07882 -0.03556 0.00771 0.05097 0.01013 -0.03071 -0.07156 -0.11240 -0.15324 -0.11314 -0.07304 -0.03294 0.00715 -0.06350 -0.13415 -0.20480 -0.12482 -0.04485 0.03513 0.11510 0.19508 0.12301 0.05094 -0.02113 -0.09320 -0.02663 0.03995 0.10653 0.17311 0.11283 0.05255 -0.00772 0.01064 0.02900 0.04737 0.06573 0.02021 -0.02530 -0.07081 -0.04107 -0.01133 0.00288 0.01709 0.03131 -0.02278 -0.07686 -0.13095 -0.18504 -0.14347 -0.10190 -0.06034 -0.01877 0.02280 -0.00996 -0.04272 -0.02147 -0.00021 0.02104 -0.01459 -0.05022 -0.08585 -0.12148 -0.15711 -0.19274 -0.22837 -0.18145 -0.13453 -0.08761 -0.04069 0.00623 0.05316 0.10008 0.14700 0.09754 0.04808 -0.00138 0.05141 0.10420 0.15699 0.20979 0.26258 0.16996 0.07734 -0.01527 -0.10789 -0.20051 -0.06786 0.06479 0.01671 -0.03137 -0.07945 -0.12753 -0.17561 -0.22369 -0.27177 -0.15851 -0.04525 0.06802 0.18128 0.14464 0.10800 0.07137 0.03473 0.09666 0.15860 0.22053 0.18296 0.14538 0.10780 0.07023 0.03265 0.06649 0.10033 0.13417 0.10337 0.07257 0.04177 0.01097 -0.01983 0.04438 0.10860 0.17281 0.10416 0.03551 -0.03315 -0.10180 -0.07262 -0.04344 -0.01426 0.01492 -0.02025 -0.05543 -0.09060 -0.12578 -0.16095 -0.19613 -0.14784 -0.09955 -0.05127 -0.00298 -0.01952 -0.03605 -0.05259 -0.04182 -0.03106 -0.02903 -0.02699 0.02515 0.01770 0.02213 0.02656 0.00419 -0.01819 -0.04057 -0.06294 -0.02417 0.01460 0.05337 0.02428 -0.00480 -0.03389 -0.00557 0.02274 0.00679 -0.00915 -0.02509 -0.04103 -0.05698 -0.01826 0.02046 0.00454 -0.01138 -0.00215 0.00708 0.00496 0.00285 0.00074 -0.00534 -0.01141 0.00361 0.01863 0.03365 0.04867 0.03040 0.01213 -0.00614 -0.02441 0.01375 0.01099 0.00823 0.00547 0.00812 0.01077 -0.00692 -0.02461 -0.04230 -0.05999 -0.07768 -0.09538 -0.06209 -0.02880 0.00448 0.03777 0.01773 -0.00231 -0.02235 0.01791 0.05816 0.03738 0.01660 -0.00418 -0.02496 -0.04574 -0.02071 0.00432 0.02935 0.01526 0.01806 0.02086 0.00793 -0.00501 -0.01795 -0.03089 -0.01841 -0.00593 0.00655 -0.02519 -0.05693 -0.04045 -0.02398 -0.00750 0.00897 0.00384 -0.00129 -0.00642 -0.01156 -0.02619 -0.04082 -0.05545 -0.04366 -0.03188 -0.06964 -0.05634 -0.04303 -0.02972 -0.01642 -0.00311 0.01020 0.02350 0.03681 0.05011 0.02436 -0.00139 -0.02714 -0.00309 0.02096 0.04501 0.06906 0.05773 0.04640

PAGE 184

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PAGE 186

-0.00494 -0.00488 -0.00482 -0.00475 -0.00469 -0.00463 -0.00456 -0.00450 -0.00444 -0.00437 -0.00431 -0.00425 -0.00418 -0.00412 -0.00406 -0.00399 -0.00393 -0.00387 -0.00380 -0.00374 -0.00368 -0.00361 -0.00355 -0.00349 -0.00342 -0.00336 -0.00330 -0.00323 -0.00317 -0.00311 -0.00304 -0.00298 -0.00292 -0.00285 -0.00279 -0.00273 -0.00266 -0.00260 -0.00254 -0.00247 -0.00241 -0.00235 -0.00228 -0.00222 -0.00216 -0.00209 -0.00203 -0.00197 -0.00190 -0.00184 -0.00178 -0.00171 -0.00165 -0.00158 -0.00152 -0.00146 -0.00139 -0.00133 -0.00127 -0.00120 -0.00114 -0.00108 -0.00101 -0.00095 -0.00089 -0.00082 -0.00076 -0.00070 -0.00063 -0.00057 -0.00051 -0.00044 -0.00038 -0.00032 -0.00025 -0.00019 -0.00013 -0.00006 0.00000 *** End Data ***

PAGE 187

n nr !"#$%$ !"&' ()%$

PAGE 188

nrr !&"))%$ !*"+)%$

PAGE 189

!,"#'%$

PAGE 190

nr !nr !-" ((.)%$ !/")%$

PAGE 191

nrnnr !0")%$ !1"'%$

PAGE 192

" nrn !"))'%$

PAGE 193

nr#$ !"'2r))%$ !"'(r)%$

PAGE 194

" nr"%!&' !&"'23(4)5 )%$ !*"3(4)5 '2% $

PAGE 195

" nr()$ !,"'26r))%$ !-"6r)'7%$

PAGE 196

" !/"6r)'7#% $ !0"6r)'7#3(4) 5%$

PAGE 197

" nr*'+) !1"6r))%$