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EVALUATION OF BRIDGE STRUCTURES
SUBJECTED TO SEVERE EARTHQUAKES
by
VOSSI REICHMAN
September 1996
A dissertation submitted to the
Faculty of the Graduate School of the
State University of New York
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
ABSTRACT
Current state-of-practice methods used for seismic analysis and design of bridge
structures are based on modified elastic spectral methods. These methods are unable to
address certain critical issues that may be peculiar to bridge structures, namely: cyclic
plastic behavior of structural elements, soil-structure interaction, differential ground
motion, the nonlinear modeling of base isolators and supplemental damping devices. To
address these deficiencies, this research develops two evaluation methods based on (i)
nonlinear time history analysis including the effects of the transient development of dam
age in structural components, and (ii) pushover procedures based on static and dynamic
lateral loading approaches. Using the latter, it is possible to assess the global seismic
capacity of an entire bridge structure which can be used in conjunction with seismic
demand determined from inelastic spectral techniques. Classical pushover analysis uses
lateral loads that are increased in a monotonic fashion until the bridge reaches its failure
limit state, this defines displacement capacity. This approach assumes a first-mode distri
bution of lateral forces and this distribution of forces is apt to miss higher mode effects
which may be important, particularly for long bridges. Therefore, several procedures are
considered to address this issue including: adaptive modal distribution, acceleration
ramps, and acceleration pulses. The latter two, being dynamic procedures, are generally
capable of capturing multi-modal and total dynamic effects.
Several new modeling features are addressed in this research by the development of a
3D computational platform (IDARC-BRIDGE). First, a new tri-axial model is developed
that is capable of accurately developing the behavior of sliding isolators including the
influence of the changing vertical force and velocity on the friction coefficients. Second,
modeling of the connection between the bridge deck and the substructure is advanced.
This includes a macro-element representation of the deck seat with the effect of the spatial
layout of bearings and diaphragms by using features of elastic elements, hinge, spring, end
releases and rigid end block transformations.
Finally, the proposed computational procedures are verified by comparing predictive
results with a combination of closed-form analytical solutions, laboratory and field exper
imental results, as well as existing computational software. Several case studies are then
used to explore the utility of the proposed computational framework with a particular
emphasis on capturing observed failure modes.
ii
Acknowledgments
This dissertation could not have been completed without the help of many people
who have contributed in different ways to make this work possible.
I wish to thank my committee members for helping me finish my dissertation by
contributing from their valuable time to edit it. I would like to thank my major advisor,
Prof. A.M. Reinhorn, for introducing me to an exciting field of research, and escorting me
along the way. I have special regards for Prof. M.C. Constantinou. His remarkable dedica
tion to teaching and research introduced me to the field of base isolation and passive con
trol and inspired me in my work. I would like to thank Prof. J.B. Mander for introducing
me to the subject of capacity design, which was an important part of my work. Special
thanks to Prof. M.P. Gaus, for his help and support in hard times, and for providing me
with a "fair deal".
I am sure I could have never been able to perform research at this level without the
encouragement and support from Prof!. Sheinman from the Technion, ISRAEL, and other
Professors at the Technion, who provided me the foundational of body of my professional
knowledge.
It is obvious that without true friends who have offered their support through this
roller coaster experience, success would not have been possible. Dr. Roy F. Lobo and Dr.
Chen Li are such friends. I would like to mention also all the other friends I acquired
through the years in Ketter Hall.
My arrival to V.B. would not have been possible without the initial support and the
on going patience of my superiors at the Israeli Electric Corporation, especially Mr.
iii
Moshe Lasry. Mrs. Efrat Eshcol, Mr. Dan Levi, and Mr. Shlomo Abramovich who have
demonstrated considerable understanding in waiting for me to finish this study.
Last but not least, I would like to thank my family for encouraging me on this dif
ficult but rewarding path: my parents, Ester and Aharon Reichman, ISRAEL, my brother
Hillel Reichman and his lovely family, and especially my dear wife, Roxana Reichman,
and our three lovely children, Inbal, Ran and Yael.
iv
Dedication
To my loving, understanding and supporting wife,
Dr. Roxana Reichman,
who believed in me and supported me
on the long and hard way,
and to our wonderful three children,
Inbal (12), Ran (9) and Yael (5),
who have put up with two part time parents
and were a constant source of happiness and consolation.
v
Table of Contents
1.0 INTRODUCTION .................................................................................................. 1
1.1 Bridges as Unique Structures .............................................................................................. .4
1.2 The Use of 3D vs. 2D Models ............................................................................................ 7
1.3 Evaluation of Bridges with Protective Systems .................................................................. 8
1.3.1 Models of Isolation Components .......................................................................... 9
1.3.2 Modeling of Damping Devices .......................................................................... 11
1.4 Approaches to Modeling and Solution of Nonlinear Systems ......................................... .14
1.4.1 Macro-Modeling Approach ................................................................................ 15
1.5 Soil Structure Interaction .................................................................................................. 16
1.6 Damage Evaluation in the Inelastic Response of Structures ............................................ 17
1.7 Modeling of Reinforced Concrete .................................................................................... 20
1.7.1 Modeling of Shear Walls .................................................................................... 22
1.8 Evaluation Methods of Bridges ........................................................................................ 23
1.9 Comparison of Evaluation Procedures for Bridges .............•............................................. 24
1.9.1 Nonlinear Time History with Damage Index ...................................................... 24
1.9.2 Nonlinear Time History ....................................................................................... 26
1.9.3 Linear Time History ............................................................................................ 27
1.9.4 Modal - Response Spectrum Analysis ................................................................. 29
1.9.5 Capacity - Push over analysis I Demand - Spectral.. ........................................... 29
1.10 Summary of Evaluation Procedures ................................................................................... 31
2.0 ANALYTICAL TOOLS FOR EVALUATION OF BRIDGES - STATE-OF-THE-
ART ........................................................................................................................ 32
2.1 Introduction ...................................................................................................................... .32
2.2 Methods of Analysis ......................................................................................................... 32
vi
2.3 Damage Analysis ............................................................................................................. .36
2.4 Research Needs ................................................................................................................. 37
3.0 DEVELOPMENT OF NEW ANALYTICAL PLATFORM FOR SEISMIC
RELIABILITY EVALUATION ............................................................................ .39
3.1 Bridge Structure and Structural Components - Descriptive Presentations ....................... .40
3.1.1 Components and System Definition ................................................................... .41
3.1.2 Comparison of Buildings and Bridges / Specific Issues ..................................... .43
3.2 Structural Component Modeling ....................................................................................... .43
3.2.1 Basic Stick Element Formulations ...................................................................... 44
3.2.1.1 Rigid arms ........................................................................................ 45
3.2.1.2 End releases ...................................................................................... 50
3.2.1.3 End springs ....................................................................................... 52
3.2.2 Inelastic Hysteretic Beam Columns with Nonlinear Shear and Flexure Behavior
in Beam/Column elements .................................................................................. 55
3.2.2.1 .Bending ............................................................................................ 56
3.2.2.2 Combined Bending and Shear .......................................................... 59
3.2.3 Shear and Bending Wall Panel Model (for modeling Abutment) ....................... 65
3.2.4 Supports and Base Isolation Models ................................................................... 66
3.2.4.1 Bouc-Wen's Model for Seismic Isolators ......................................... 67
3.2.4.2 Triaxial Isolator ................................................................................ 73
3.2.5 Three Dimensional Spring - Connection Element Model ................................... 82
3.2.6 Modeling of Damping in Bridges ........................................................................ 84
3.2.6.1 Global Damping ............................................................................... 84
3.2.6.2 Linear Damping Element ................................................................. 85
3.2.6.3 Nonlinear Damping .......................................................................... 86
3.2.6.4 Damping Devices ............................................................................. 86
3.2.7 Nonlinear, Yielding in Tension, Stiffening in Compression Element ................. 87
vii
3.3 Modeling of Bridge Subsystems ........................................................................................ 89
3.3.1 Modeling of Foundation and Soil System ........................................................... 89
3.3.1.1
3.3.1.2
Elastic Linear Models for Soil Structure Interaction ........................ 90
Frequency Dependant Modeling ...................................................... 90
3.3.2 Abutment Modeling ............................................................................................. 90
3.3.3 Bents modeling .................................................................................................... 91
3.3.4 Deck modeling ..................................................................................................... 92
3.3.5 Expansion Joints and Special Connections ......................................................... 95
3.4 Solution Procedures for Coupled motions ......................................................................... 97
3.5 Evaluation of Bridges via Approximated Methods ......................................................... 103
3.5.1 Methods for Determining Seismic Demand ...................................................... 105
3.5.1.1 Modal (Spectral) Analysis by Seismic Codes ............................... 105
3.5.1.2 Static Equivalent Procedure ........................................................... 108
3.5.2 Time History Analysis ...................................................................................... 109
3.5.2.1 Bridges Under Seismic Excitation -Uniform and Differential Excita-
tions .................................................................................................... 110
3.5.3 Numerical Solution for Bridge Systems ............................................................ 116
3.5.3.1
3.5.3.2
3.5.3.3
Identification of Special Issues in Solution Procedures ................. 116
Sparse Matrix Formulations of Structures Equations ..................... 117
Conjugate Gradient Iterative Solver (Kincaid et at. 1990) ............. 118
3.6 Methods of Analysis to Evaluate Bridge Capacity .: ........................................................ 123
3.6.1 Direct Section Evaluation .................................................................................. 124
3.6.1.1 Mechanical Properties Modeling .................................................... 125
3.6.1.2 Fiber Analysis ................................................................................. 125
3.6.2 Incremental Monotonic Static Analysis ........................................................... .125
3.6.2.1
3.6.2.2
Arbitrary Force Loading ................................................................. 128
Arbitrary Displacement Input ......................................................... 129
viii
3.6.2.3 Adaptable (Modal) Loading ........................................................... 129
3.6.2.4 Acceleration Ramp Load ................................................................ 130
3.6.2.5 Acceleration Pulse Input ................................................................. 131
3.6.3 Comparing Demand to Capacity ....................................................................... 132
3.6.3.1 Demand/Capacity comparison in Inelastic structures (Damage Index-
ing) ..................................................................................................... 132
3.6.3.2 Evaluation of Bridges in ModerateILow Seismicity Zones - Proce-
dure ..................................................................................................... 134
3.6.3.3 Evaluation of Bridges in Severe Seismicity Zones ........................ 135
4.0 CASE STUDIES FOR BRIDGE EVALUATIONS ............................................. 140
4.1 Evaluation of Retrofit of Existing Bridges ....................................................................... 140
4.1.1 Case Study #1: Analytical Evaluation of Small Scale Bridge System .............. 140
4.1.2 Case Study #2: East Aurora Bridge - Field Test.. .............................................. 149
4.1.3 Case Study #3: Bridge in Los Angeles - Retrofit Solution ................................ 159
4.1.4 Case Study #4: Bridge over Los Angeles River -Retrofit Solution with Base Iso-
lation and Restrained Expansion Joints ............................................................. 180
4.1.5 Case Study 5: MiyagawaBridge ....................................................................... 196
4.1.6 Case Study 6: Evaluation of Bridge Capacity Using Different Push - Over Proce-
dures and Dynamic Analysis ............................................................................. 206
5.0 DISCUSSIONS AND CONCLUDING REMARKS .......................................... 227
5.1 Further Research Suggestions .......................................................................................... 231
6.0 RFERENCES ....................................................................................................... 233
A. APPENDIX A ...................................................................................................... 248
A.I IDARClBridge - Inelastic Damage Analysis of Reinforced Construction =
Platform for Bridges ............................................................................................ 248
ix
List of Figures
Fig # Title Page #
1.1 Structures Idealization (a) building (b) Bridge 4
3.1. Bridge Assembly and Major Components 40
3.2. Connection Between Deck and Bent Using Rigid Arm and End Springs (a) 46
Physical Description (b) Regular Modeling (c) Modeling with Rigid Arm
and End Springs
3.3. Rigid Body Transformation from Joint P to Joint J 48
3.4. Element End Releases 51
3.5. Elements End Springs 53
3.6. Trilinear Hysteretic Model, a) General Model, b) Slip Definition 57
3.7. Stiffness Calculation of Damaged Elements a)MomentDistribution b)Flexi- 58
bility Distribution c)Damaged Zones
3.8. Combined Flexure and Shear Stiffness Derivation 59
3.9. Example Frame, for Illustration of Combined Shear and Flexure Model 63
3.10. Constitutive Relations in Shear and Bending of the Frame's Columns 64
3.11. Influence of Shear on Frame's Capacity 65
3.12. Shear Wall Definition (a) Shear Wall (b) Shear Wall Modeling for Analysis 67
3.13. Elastomeric Isolator. a) side view, b) force displacements relations, c) top 68
view of deformed isolator
3.14. Triaxial Sliding Isolator, a) side view, b) sliding force - displacements rela- 74
tion. c) dependancy of coeficient of friction on velocity
3.15. Graph of Theoretical Input Motion of 8 Shaped Test 79
x
3.16. Analytical Loops of Frictional Force and Displacements in 8 Shaped Motion 80
Test
3.17. Recorded Loops of Frictional Force and Displacements in 8-Shaped Motion 81
Tests
3.18. Spring Element Representation 82
3.19. Damper Element Representation 85
3.20. Typical Expansion Joint 88
3.21. Force Displacements Relations of Expansion Joint 88
3.22. Typical Arrangement of Bridge Abutment. The Soil Stiffness Modeled By 91
Springs.
3.23. Modeling of Connection Between Deck and Bearings 94
3.24. Typical Expansion Joints in Bridge 95
3.25. Expansion Joint Modeling 96
3.26. Rigid Floor Displacements 99
3.27. Deck Section Modeled by Several Beams 102
3.28. Evaluation of Inelastic Response Using Composite Spectrum 104
3.29. Verification Frame Model 114
3.30. Frame Response to Displacement and Acceleration Input. 115
3.31. Ramp Acceleration Loading 131
3.32. Force Displacements Relations for "Push Over" Analysis 137
4.1 Experiment Set Up of the Bridge on the Shaking Table. 143
4.2 Input Ground Motion (Japanese code, level 2, ground condition I) 143
4.3 Construction of Friction Pendulum System Bearing 144
xi
4.4 Experimental Derivation of The Sliding Bearings Frictional Characteristic 145
(Coefficient of Friction as Function of Velocity)
4.5 Base Isolated Bridge Model 146
4.6 Displacements at the Top of the Pier (point 3 and 4) 147
4.7 Displacements of deck (point 5 and 6) 148
4.8 Top View of the Bridges 151
4.9 Typical Bent 152
4.10 Typical Abutment 153
4.11 Side View of the Bents 154
4.12 Input Force History, Measured During the Test. 155
4.13 Analytical Model of the Bridge, Used for Analysis 156
4.14 Modeling of Bridge Deck Connection to Abutment (a) Modeling with Stiff 157
Elements (b) Modeling with Rigid Body Transformation
4.15 Comparison of Analytical and Experimental Results for Bridge on Steel 158
Bearing at Abutment.
4.16 Comparison of Analytical and Experimental Results for Bridge on Rubber 158
Bearing at Abutment.
4.17 Bridge Top View and Typical Bent Cross Section 164
4.18 Deck and Bent Cross Section 165
4.19 Structural Model of Original Bridge - Before Retrofit 166
4.20 Structural Model of Isolated Bridge 167
4.21 Ground Displacements on Stiff Soil 168
4.22 Ground Displacements on Soft Soils 169
xii
4.23 Relative Defonnations in X Direction, Between Ground a9d Deck Disp., at 170
Piers Centerline for Unifonn Ground Motion
4.24 Relative Defonnations in X Direction, Between Ground and Deck Disp., at 171
Piers Centerline for Variable Ground Motion
4.25 Shear Forces in The Pier's X Direction, for Piers Bl and B5, for Isolated 172
Bridge, at Different Heights Along the Pier
4.26 Shear Forces in X Direction, for Nonisolated Bridge, at Base of piers 173
4.27 Comparison of Shear Forces in X Direction, for Piers B I and B5 174
4.28 Relative Defonnations in Z Direction, Between Ground and Deck Disp., at 175
Piers Centerline for Unifonn Ground Motion
4.29 Relative Defonnations in Z Direction, Between Ground and Deck Disp., at 176
Piers Centerline for Variable Ground Motion
4.30 Shear Forces in The Pier's Z Direction, for Piers B I and B5, for Isolated 177
Bridge, at Different Points Along the Pier
4.31 Shear Forces in The Pier's Z Direction, for Nonisolated Bridge, at Their 178
Base
4.32 Shear Forces in The Pier's Z Direction, for Piers Bland B5 179
4.33 Original Bridge Before and After First Retrofit 185
4.34 Analytical Model of the Bridge 186
4.35 Base Isolated Bridge with and without Expansion Joints 187
4.36 Displaced Shape of (a)Original Bridge, (b) Isolated Bridge 188
4.37 Displaced Shape of Typical Frame in the (a) Original Bridge, (b) Isolated 189
Bridge
xiii
4.38 Comparison of Shear Force in Non Isolated Bridge, and Bridge Without 190
Gap
4.39 Comparison of Shear Forces in Original and Retrofitted (Alt.1) Bridge 191
4.40 Comparison of Shear Force in Isolated Bridge, and Bridge Without Gap 192
4.41 Comparison of Deck Displacements in Isolated Bridge, and Bridge Without 193
Gap
4.42 Comparison of Deck Displacements in Isolated Bridge, and Not Isolated 194
Bridge
4.43 Comparison of Deck Displacements in Non Isolated Bridge, and Bridge 195
without Gap
4.44 Completed Miyagawa Bridge 199
4.45 Bridge Geometry 200
4.46 Lead Rubber Bearings for Bridge Abutment 201
4.47 Lead Rubber Bearings for Bridge Pier 201
4.48 Cross Section of Piers 202
4.49 Load Displacements Curves 202
4.50 Structural Model of the Bridge 203
4.51 Moment at Bottom of Pier and Foundation 204
4.52 Displacements at Pier Centerline, Under and Above Isolator 205
4.53 SR141I-5 Separation and Overhead (Southbound) - General Plan and Eleva- 208
tion
4.54 SR141I-5 Separation and Overhead (Southbound) - Typical Deck Section 209
4.55 SR141I-5 Separation and Overhead (Southbound) -Pier Sections and Rein- 210
forcement.
xiv
4.56 SRI4/I-5 Separation and Overhead (Southbound) -Structural Model for
Analysis
211
4.57 Distribution of Shear Forces in Piers 2-7 for Different Analysis Procedures 217
4.58 Response of the Bridge to Adaptable Proportional to Instantaneous Mode 219
Shape Distributed Load (relative to mass). (a) Displaced Shape at Failure,
(b) Loading History, (c) Shear History for Piers 2-7
4.59 Response of the Bridge to Non Adaptable Uniform Distributed Load (rela- 220
tive to mass). (a) Displaced Shape at Failure, (b) Loading History, (c) Shear
History for Piers 2-7
4.60
4.61
Response of the Bridge to Ramp Loading. Acceleration Rate: 0.25m!
sec3 (a) Displaced Shape at Failure, (b) Loading History, (c) Shear
History for Piers 2-7
Response of the Bridge to Ramp Loading. Acceleration Rate: 0.5 m!sec 3
(a) Displaced Shape at Failure, (b) Loading History, (c) Shear History for
Piers 2-7
221
222
4.62 Response of the Bridge to Ramp Loading. Acceleration Rate: 5m!sec3. (a) 223
Displaced Shape at Failure, (b) Loading History, (c) Shear History for Piers
4.63
4.64
4.65
2-7
Response of the Bridge to Pulse Loading 0.1 sec. (a) Displaced Shape at
Failure, (b) Loading History, (c) Shear History for Piers 2-7
Response of the Bridge to Pulse Loading 0.05 sec. (a) Displaced Shape at
Failure, (b) Loading History, (c) Shear History for Piers 2-7
Response of the Bridge to Dynamic Impulsive Loading. (a) Displaced
Shape at Failure, (b) Northridge Earthquake Record From Santa Monica
City Hall (input motion), (c) Shear History for Piers 2-7
224
225
226
xv
A.I
A.2
Program General Structure
Processing Routines Arrangements
248
251
xvi
LIST OF TABLES
Table # Title Page #
1.1. Evaluation Methods for Bridges 24
1.2. Comparison of the Different Evaluation Procedures 31
4.1. Hysteretic Pier Nonlinear Properties 197
4.2. Geometry and Structural Elastic Properties of the Bridge Elements 207
4.3. Transversal Moment Curvature Relations and Shear Capacity of Piers 207
4.4. Comparison of Shear Force Distribution Between the Piers at the Time of 216
Failure of Critical Pier in Shear
4.5. Moment at Bridge Piers due to Different Push-over Procedures at Shear 218
Failure
4.6. Comparison of Performances of The SR14115 Bridge Subjected to Different 218
Push-Over Procedures
A.l Input Routines Description 248
A.2 Preprocessing Routines 250
A.3 Update Elements Properties 251
A.4 End Conditions Description 252
A.5 Global Matrices Building Description 252
A.6 Load Vector Description 253
A.7 Solution Description 254
A.8 Results Processing Description 254
A.9 Modal Analysis Description 254
xvii
1.0 INTRODUCTION
Bridges are an important element in modem urban societies and are vital to normal
economic functioning as well as in earthquake hazard events. In the case of an earthquake
event, bridges can pose a significant life hazard if failure should take place and the loss of
bridges may cause severe difficulties in emergency response to the earthquake as well as
great difficulties in subsequent recovery activities. Thus it is important to develop reliable
methods for the analysis and prediction of performance of designs for new bridges and the
evaluation of existing bridges.
The objectives of this dissertation are to evaluate previous developments in the
evaluation procedures of bridges and to develop an improved computer capability to carry
out reliable evaluation of highway bridges.
Specific advances made in this dissertation include the development of new struc
tural element models to analyze situations which were previously difficult to model or
which could not be modeled, the development of structural element models for new tech
nologies such as base isolation and improved methods to model and analyze structures
which have loosely connected elements and the assembly of complex nonlinear structural
elements into a working computer program which provides a capability to model the
dynamic 3-D behavior of complex bridge structures and their supporting systems. One of
the unique features of the work is the integration of considerations of design and analysis.
This is accomplished by using two evaluation methods in which each of the methods pro
vides a capability for design which is not well handled by the other. The methods used are
nonlinear dynamic damage analysis combined with a push-over analysis specially devel
oped for bridges.
There were, no doubt, bridges which were destroyed or lost in different parts of the
world as a result of earthquakes over the centuries. Unfortunately, there is not much docu
mentation available for. these occurrences. In addition the design and construction of
bridges has undergone many changes in the past 50 years as a result of new materials and
equipment being available and the large growth in the use of motor vehicles. This is par
ticularly true in the U.S. where the number of motor vehicles has shown large growth and
a large program of construction was launched to accommodate these vehicles in both
urban areas and for cross-country travel and transportation of goods. Similar expansions
have recently been underway in other countries. Unfortunately, the inclusion of earth
quake loadings in the design of many bridges was not a major design consideration for
most of these bridges. Thus many bridges were built without adequate provision for earth
quake loadings nor was the state of knowledge on earthquake loadings and bridge design
and construction techniques at the level such knowledge is today.
The San Fernando Earthquake (1971) in the Los Angeles area resulted in the loss
of many major freeway bridges and the loss of $6.5M. This event helped to draw attention
to the need to utilize the highest level of analysis and design for bridges which may be
subjected to earthquakes.
A discussion of failures and damage suffered by bridges during earthquakes is
contained in a report by Imbsen and Penzien (1986). Among the failure mechanisms
which were considered to be important are:
1) Tilting of substructure
2) Supports displacements.
2
3) Settlements of abutments.
Examples of these types of failures and resulting damage can be found by examin
ing the Alaska earthquake (1964), in which most of the damage to bridges was caused by
large ground displacements, settlements and loss of the bearing capacity of the main struc
tural systems of the bridges. Dynamic inertia effects were not observed as a primary cause
of the bridge damage. During the period up to 1971 several earthquakes occurred in Japan
and California but these earthquakes caused insignificant amounts of damage to bridges.
For this reason, the design codes for bridges up to that time put very little emphasis on
provisions for dynamic seismic design of bridges, and designs were largely based on
equivalent static forces of a level up to 9% of the dead load of the bridge.
The San Fernando Earthquake (1971) initiated a series of changes in the approach
toward seismic design of bridges which is continuing today. A significant amount of
bridge damage was caused during this earthquake due to dynamic inertia effects, and the
evaluation of the bridges for design started to change to consider these dynamic effects.
This change require the consideration of the ductility of the bridge components which
could be subjected to dynamic inertia effects. Particular emphasis started to be paid to sup
porting elements such as piers and bents.
Since the 1971 San Femando event, a number of extremely destructive earth
quakes have occurred in the vicinity of highly populated cities in California and Japan.
The 1989 Loma-Priata, the Northridge (Goltz 1994) and Kobe (Aydinoglu et al. 1995)
earthquakes caused a large amount of damage to bridge structures and highlighted the
3
need to improve analysis techniques for new and existing bridges and the need to retrofit
hazardous existing bridges.
The extensive damage and the surprising levels of ground accelerations (up to 20),
presented the need for more accurate and scientific evaluation and design method of
bridges.
1.1 Bridges as Unique Structures
distance
(a) (b)
FIGURE 1.1 Structures Idealization (a) building (b) Bridge
The structural behavior of bridges involves a number of special problems which
must be considered carefully to avoid failures of the bridges during earthquake shaking.
Some of the features of bridges which require special considerations are:
a) The bridge may involve parts which are loosely connected.
b) The bridge may have many supports, not all of which are directly connected to
the deck.
4
c) The supports of the bridge may be spaced a long distance apart and are not gen
erally connected to one another. This may result in differences in the dynamic
loading of each pier due to phase differences in the ground motion waves arriv
ing at each pier.
d) The geologic conditions under each pier may have significant variations result
ing in variable ground motion amplification effects.
e) Differing heights of supporting piers or bents result in a variation of the stiffness
of supporting elements.
f) The effects of nonlinear behavior of the bents may result in variation of stiffness
along the bridge.
Thus the behavior of bridges is different from the structural behavior of most of
the other types of structures dealt with in the civil engineering profession and require spe
cial attention in analysis and design. Perhaps the most commonly analyzed class of struc
tures in civil engineering are building structures. Usually a building is an integral
structural unit, which can be characterized as a collection of shear elements, frames and
cantilever beams (Fig. 1.1 (a», all of which are well connected. A gradual variation of
stiffness and mass distribution is reflected in well defined dynamic behavior. When there
are abrupt changes in stiffness, as is the case when the first story of the building is open
without walls (soft first story) this can cause a concentration of dynamic rotations
response at the first story level and has often been observed as a cause of failure of build
ings.
5
In bridge structures (Fig. 1 (b» the dynamic, mass and stiffness properties change
along the height and the length of the bridge. In the vertical direction, the stiffness of the
bridge system is changing from the foundation stiffness to the pier, bent or end wall stiff
ness and then to the deck level which may have a higher level of stiffness. In the horizon
tal direction, the height of the piers (or bents) may vary and therefore their stiffness may
be different. This factor is even more important if nonlinear deformations occur in which
there could be a significant reduction in some of the pier stiffnesses due to plastic behav
ior. The stiffness properties of the bridge system can change during the duration of a
dynamic loading due to relative movement of the bridge sections at the thermal expansion
joints.
The ground motion under a building is generally characterized by using a single
record of ground motion acceleration and displacements. The ground motion effects on a
bridge (Fig. 1.1 (b» can be quite different between the supports of the bridge. This can
occur because of two reasons (Der Kiuereghian 1995):
1) If the bridge supports are located far away from one another, the arrival time of
the incident shear waves differs.
2) Different types of soil amplify differently the bed rock motion, and causes dif
ferent ground motions at different supports.
Both of these phenomena may result in quasistatic and additional dynamic forces
on the bridge system.
6
1.2 The Use of 3D vs. 2D Models
For purposes of analysis it is necessary to construct a mathematical model of the
physical bridge structure. The subsequent analysis is then carried out using the mathemat
ical model. If the mathematical model can adequately represent the true behavior of the
bridge, a satisfactory prediction of the true behavior of the bridge will be obtained. Thus it
is imperative that careful attention be paid to the generation of the mathematical model
and that the model represent both the topological and physical behavior of the materials of
the bridge into the range of dynamic nonlinear behavior. For complex loosely connected
structures the behavior of the structure is always 3-dimensional and the model and analy
sis system must be developed to truly represent this behavior.
Seismic demand as well as the seismic capacity evaluation are dependent on the
structural model and must be fully represented. Thus great care must be exercised in the
selection of representative degrees of freedom, stiffness matrix formulation, mass distri
bution and special connectivity rules. In bridge structures the deck and the supports form
longitudinal frames which are frequently represented in two dimensional models. Individ
ual supports, piers or bents are also frequently represented by 2d models to describe the
transverse vibration of the bridge. Analyses are then carried out in the 2-D directions for
loadings in these directions. If a loading is not in one of the 2-D directions an attempt is
made to represent the resulting response by combinations of 2-D analysis. Unfortunately,
even such combinations fail to adequately represent 3-D responses which can occur in
structures which appear to be very regular.
Irregular bridges with skewed supports to the deck, with multiple gaps skewed to
the bridge axes, curved deck bridges, etc. cannot be modeled by two dimensional models.
7
Moreover, regular bridges with symmetrical geometrical construction, but with unknown
distribution of strength or soil support conditions should be modeled by 3D models to cap
ture their inelastic influence.
Therefore a 3D representation of the bridge is essential in determining the demand
and also the capacity of the bridge. It should be noted however, that when the bridge is
represented by a 3D model, modal spectral analysis using a reduced number of modes,
requires a somewhat arbitrary and careful selection of contributory modes. The selection
can be based on base shear contributions or on deformation contribution methods of selec
tion of the modes. A discussion of this problem can be found in Clough and Penzien
(1975).
As can easily be seen there are several issues which require complex modeling of
the soil and bridge structures using a 3D approach. The spatial variation of propagating
ground motion, the soil supports (springs), expansion joints and seismic isolation systems
in bridges all are influenced by three dimensional motion and therefore three dimensional
considerations should be used. Two dimensional models can provide only limited infor
mation, which in many cases may be nonconservative in predicting the overall behavior of
the bridge.
1.3 Evaluation of Bridges with Protective Systems
In recent years modern protective systems have been introduced to reduce the vul
nerability of bridges to seismic events. These protective systems include base isolation
devices of different types, damping devices and active control devices.
8
The influence of added protective systems does influence the structural configura
tion of a bridge, such as in the case when a sliding isolation bearing changes from "stick"
to "slip". Initially the substructure and the superstructure act as a single dynamic unit but
when "slip" occurs the bridge behaves as two different dynamic entities. Turkington et. al.
(1989) observed that the equivalent damping of an isolation system is a function of the
type of the earthquake. In an earthquake in which ground shaking continues for a long
period of time the effective damping is higher than in an impulsive earthquake in which
the strong ground shaking is of relatively short duration. Therefore the only way to cap
ture the true behavior of the isolation system and it's effect on the overall response of the
bridge is to use nonlinear dynamic analysis which can take into account time-dependent
damping effects.
A discussion of the different kinds of mathematical models used to represent the
behavior of isolation elements used for nonlinear dynamic structural analysis is presented
in section 3.0.
1.3.1 Models ofIsolation Components
The essential features that need to be modeled to capture the behavior of isolation
bearings are:
(i) The appropriate shear stiffness representation in the pre and post yielding
range;
(ii) Representation of the strain dependence of shear stiffness appropriately
(iii) Representation of the loss of shear stiffness.
9
Bilinear or trilinear models can be used to model isolation elements like lead-rub
ber bearings and mild steel dampers. Lee(1980) has used the bilinear model for modeling
lead-rubber bearings. Many Japanese researchers (Yasaka et al. 1988 and others) have
used the bilinear model for modeling lead-rubber bearings, and steel dampers. Fujita et
al.(1989) have used the bilinear model with modifications for modeling high damping
elastomeric bearings. The trilinear model has been used by Miyazaki et al.(1988) for mod
eling lead -rubber bearings. The Ramberg-Osgood model (1943) has been used for model
ing high damping elastomeric bearings by Yasaka et al.(1988) and Fujita et al.(1989).1t is
difficult to capture all the essential features by these simple models.
A Coulomb model in which the transition from stick to sliding mode and vice
versa is controlled by stick-slip conditions was described by Mostaghel et al. (1988) and
Su et al. (1987). This model has been used for modeling sliding bearings. The viscoplastic
model for sliding bearings proposed by Constantinou et al. (l990b) has been used for
modeling sliding bearings. The Viscoplastic model proposed by Ozdemir (1976) has been
used for modeling steel dampers by Bhatti et al.(1980) and by Fujita et al.(1989) with
modifications for modeling high damping elastomeric bearings and lead-rubber bearings.
The viscoplastic or the rate model captures most of the essential features of these bearings.
The differential equation model developed by Wen et al.(l976) collapses to the
visocplasticity model under certain conditions (Constantinou et al. 1990b) had captured
most of the features. Plasticity based yield surface models have been used to model lead
rubber bearings and high damping elastomeric bearings (Tarics et al. 1984).
10
Experimental evidence in tests on steel dampers and high damping bearings
(Yasaka et al. 1988) reveal the importance of biaxial interaction. Japanese researchers
(Wada et al. 1988, Yasaka et al. 1988, Nakamura et al. 1988) have used the multiple shear
spring model to account for biaxial effects in steel dampers, lead rubber bearings and high
damping elastomeric bearings. This model consists of a series of shear springs arranged in
a radial pattern. The plasticity based yield surface model has been used by Tarics et al.
(1984) for modeling lead rubber bearings and Mizukoshi et al. (1989) for modeling lami
nated rubber bearings and hysteretic dampers.
Constantinou et al. (1990b) have used a differential equation model for biaxial
interaction which was proposed by Park et al.(1986), and which is an extension of the
model proposed by Wen(1976) for uniaxial behavior. This model gives accurate modeling
in the elastic and fully plastic zone but not in the transition zone, where the change from
elastic to plastic behavior occurs. To use this model efficiently in an analysis code, the
pseudo force method must be used (Nagarajaiah et al. 1989). This kind of modeling might
cause convergence problems in the presence of additional nonlinear elements. An addi
tionalproblem is that the influence of the vertical load cannot be efficiently incorporated.
1.3.2 Modeling of Damping Devices
An analytical and experimental study on the behavior of different damping devices
was carried out by Li and Reinhorn(1995), which included several models that can be
used for modeling of complex dampers behavior. A summary of these models is presented
below:
• Linear Viscous Dampers
11
The behavior of linear dampers is velocity dependent, in which the damping force F d
is a linear function of the velocity Ii (t) such that
(EQ 1.1)
where C is a constant.
When this linear damper is subjected to a harmonic loading of frequency Q the rela-
tionship between force and displacements will follow the rule:
(~J2 + (.!!:..)2 = 1 l CQuo Uo (EQ 1.2)
where Uo is the maximum displacements in the damper.
The energy dissipated by this device Wd is:
(EQ 1.3)
• Kelvin Model
This model is used to model behavior of an element with both velocity and displace-
ment dependency. The force developed in this kind of element is:
Fd (t) = Ku (t) + Cli (t)
and under harmonic loading the force is:
F(ro) = K(ro)u(ro) +iroC(ro)u(ro)
(EQ 1.4)
(EQ 1.5)
12
• Maxwell Model
For an element with a strong frequency dependency such as occurs in fluid viscous
dampers with which incorporate accumulators (Constantinou et al. 1992) a model of
stiffness and damping in "series" can be used.
The force relation of this model are:
(EQ 1.6)
where A = C D/ KD and CD is the damping coefficient at zero frequency and KD is the
value of stiffness at large frequency.
• Wiechert Model
For dampers which experience significant stiffening in low frequencies a combined
Maxwell-Kelvin model can be used to model damping elements with bituminous fluids.
The force deformation of this element are defined as:
(EQ 1.7)
where Kg is the "glossy" and Ke is the "rubbery" stiffness.
Additional models based on Maxwell and Kelvin models with fractional deriva
tives (Bagley and Torvik 1983, Makris 1991 and Kasai et al. 1993) can be used to define
larger range of behavior of damping elements.
It should be noted that all the models presented are based on properties related to
harmonic loading, this type of modeling might be adequate only if the nature of the earth-
13
quake is vibratory. In general for nonlinear analysis, a better quantity to relate the proper
ties of the damper to is the instantaneous velocity. This can be achieved by using nonlinear
stiffness and damping properties ((C(U) and K(u, u)) to represent the behavior of the
element.
More Details of integration of these models in a platform for planner structural
analysis are presented by Valles et aI. (1996)
1.4 Approaches to Modeling and Solution of Nonlinear Systems
In order to analyze a structural system, adequate component modeling is required.
In many cases the behavior of a component is very complicated, and sophisticated, com
plex methods are required to obtain sufficiently accurate results. Those methods might be
adequate for some cases but not for others if the other models are very sensitive to the size
of the increment of force, or displacement. In these latter cases very small time steps are
required to accurately follow the force displacements curves which are coupled with Itera
tive solution. While an iterative solution is not difficult to perform when a single element
is considered, in the presence of several nonlinear elements in a structural system, the con
vergence of the entire system is not guaranteed. While the solution for one element is con
verging, the solution for another element might be diverging. This problem is especially
severe when loading and unloading curves are very different, and the search for the equi
librium point might take strange directions.
In order to overcome the difficulties presented above, the use of iterative proce
dures should be avoided as much as possible. A solution based mostly on components that
contribute directly to the global stiffness and damping matrices is preferred. A contribu-
14
tion of pseudo forces to the forcing vector in the governing equations should be avoided as
much as possible. Where necessary, a combination of stiffness damping and pseudo force
should be used.
1.4.1 Macro-Modeling Approach
Two approaches towards modeling of nonlinear behavior of concrete elements for
structural analysis dominate today: (i) Micro-modeling - Concrete elements such as beam
columns and wall, are discretize into small elements, in which the actual material proper
ties are introduced. This type of analysis requires the use of very large number of ele
ments, and cannot be applied on structures of significant size because of computational
capacity limitation. (ii) Macro-modeling - significant structural elements such as beams
and columns are modeled using hysteretic rules for an entire element. Using this approach
although the accuracy is compromised somewhat the solution is more feasible and
requires much less computational resources as well as providing better significant engi
neering insight.
Following the limitations presented above, a macro model approach is adopted in
this dissertation. By using this approach, even though the exact modeling of the compo
nents is slightly compromised, the major characteristics contributing to the overall system
behavior are captured. This way, a good understanding and modeling of system behavior
is reached, and if more accurate observation of component behavior is required, a dis
placement or force history can be extracted from the global analysis, and component
micromodel analysis can be performed.
15
1.5 Soil Structure Interaction
The correct evaluation of the dynamic properties (mass, damping, stiffness) of the .
soil foundation system of a structure is of major interest in evaluation of dynamic behavior
of structures (Wolf 1988). A rigorous incorporation of soil structure interaction methods in
the analysis requires that the ground be included as part of the structural model. At the
present time, even with the advanced computational facilities which are available, this is
simply not feasible.
Therefore simplified methods are generally used for the representation of soil
structure interaction. Some of these methods utilize a superposition approach in the fre
quency domain, and utilize nonlinear hysteretic models in the time domain.
The most simplified method to represent soil-structure interaction is to use equiva
lent linear stiffness and damping properties to represent soil properties but to use a fully
nonlinear modeling for the superstructure. The use of this simple approximation which
requires a careful evaluation of the soil conditions and rigorous engineering judgment of
the equivalent linear damping and stiffness of the soil, is the best way to model a system
which is expected to have nonlinear structural behavior at the present time for two rea
sons:
(i)Information on soil properties has a much higher uncertainty then the other com
ponents in the structural system (concrete, steel)
(ii)The capacity to contribute additional accuracy to nonlinear structural analysis
from more rigorous soil modeling is limited and would not improve the accuracy of the
results.
16
1.6 Damage Evaluation in the Inelastic Response of Structures
The design philosophy prevalent over the past 30 years for structures which may
be subjected to infrequent intense lateral loads is based on the ability of structures to
undergo inelastic response with hysteretic energy dissipation and possibly result in perma
nent residual deformations. By allowing relatively large displacements associated with the
nonlinear behavior the total energy input in a structure is reduced due to the dynamic soft
ening of the system. This results in producing smaller force (response) demands on other
elements of the structure. The benefit of the inelastic behavior on the strength require
ments of structural members, may be limited however, by the damage inflicted to the
members in terms of permanent deformations and deterioration of their subsequent load
carrying capacity during other future seismic events. Another result of this large displace
ment may be degradation in the functionality of the structure after the earthquake event.
A survey on the use of damage indexing was done by Valles et al. (1996)
The prediction and evaluation of seismic damage is a probabilistic problem (Park
et al. 1985 Powell and Allahabadi 1988) because the future occurrences of earthquakes,
their location and signature characteristics cannot be exactly determined. Nevertheless, a
deterministic analysis can provide valuable information on structural behavior. The need
for assessment of condition of structures and their performance during earthquakes led to
the development of damage indicators (surveyed by Chung et al. 1987, Manfredi 1993,
Williams and Sexsmith 1994) provide a way to quantify numerically the seismic damage
sustained by individual members, substructures (bents) or complete structural assemblies.
Although such indices incorporate many uncertainties resulting from modeling assump
tions, from incomplete knowledge of actual and current material properties, and from
17
future unknown ground motions, the relative ratio of demands and capacities can provide
reliable information on the envelope of requirements for which a structure should be
designed.
Damage indices were defined as a ratio of demands versus capacities in terms of
either strength, or maximum deformations, or maximum range of deformations, or cumu
lative displacements, or energy dissipation (Williams and Sexsmith 1994, Manfredi 1993,
Powel and Allahabadi 1988, Chung et al. 1987). Numerous indices can characterize the
physical quantities mentioned above at the level of individual members (columns, beams,
walls, etc.), at the level of substructures (bents, abutments), or at the global level.
Some suggested indices characterize damage as a function of the ratio of maxi
mum deformations, rotations, or curvatures in respect to the yielding levels (Banon et al.
1981). More recently, the permanent (unrestored) deformation demand (u-uy) related to
the ultimate capacity of permanent deformation (uu-uy) was suggested (Powel and Allaha
badi 1988; Yao et al. 1985): DI = (u - uy) 1 (uu - uy) . A similar index using the ductil
ity demand (11) versus the ductility capacity (l1u)' has been used (Darwin and Nmai 1986;
Mahin and Bertero 1981): DI = (11-1)/(l1u-l)
In the above formulations the capacity of the section, or structure, is assumed con
stant during the seismic event. Other indices, relate implicitly deformation and forces
using energy dissipated through hysteretic action. Mahin and Bertero (1981) suggested
design methods based on the determination of the ductility demand, instead of only hys
teretic energy while maintaining the ultimate capacity constant.
18
Concern for the influence of the hysteretic energy dissipation, in addition to the
deformation changes, produced two component damage models, one which is related to
deformations and other to energy (Banon et al. 1981 and 1982: Park and Ang 1985, and
Bracci et al 1989). The difference in the above models stem from the method used to com
bine the two influences. It should be noted that, while the two components of the damage
models are assumed independent (Park et al. 1985), or statistically independent (Banon et
al. 1981; Bracci et al. 1989), the influence of hysteretic dissipation on the capacity terms is
neglected.
The influence of change in capacity during cumulative permanent deformations
has been formulated using "cumulative damage theory" (Chung et al. 1987; Powel and
Allahabadi 1988; Cosenza and Manfredi 1992). However, the cumulative formulation
requires calibrated indices and does not explicitly correlate the hysteretic energy dissipa
tion with the cumulative deformation influences.
Attempts to determine changes in strength capacity as a result of the past histories
of deformations were made by Wand and Shah (1987) and by Chung et aI. (1987). Chung
et al. introduced the notion of a failure envelope which links both deformation and
strength capacity limits, based on a fatigue model. However, the deterioration rules in the
above mentioned references need to be calibrated from statistical or experimental data,
without a relation to the work done and to the energy dissipated by the structural members.
Attempts to link changes in strength capacities using energy dissipation and
fatigue rules produced additional damage indices (McCabe and Hall 1989; Chang and
19
Mander 1994a). However, the damage indices proposed are limited to strength damage,
with little attention paid to the permanent and residual deformations.
From experimental data and from shaking table studies (Bracci et al. 1995) it can
be observed that there is a strong correlation between the deterioration of strength and
deformation capacity of structural elements. It is also observed that the damage in most
components and structures is correlated to the excessive deformations of elements beyond
the elastic limits, but collapse occurs only when those deformations reach their capacity.
In some cases, repetitive cycles at constant displacements, which dissipate hysteretic
energy, can deteriorate the capacity to the level of the applied loading, leading to total col
lapse (Bracci et al. 1989).
1.7 Modeling of Reinforced Concrete
An important part of the modeling of bridges for nonlinear analysis requires realis
tic modeling of the nonlinear behavior of its concrete components. Both macro and micro
modeling approaches can be used to model concrete behavior through the nonlinear range
up to failure. Because closed form solutions for complex structures are seldom possible, a
finite element approach in which the structure is divided into discrete (finite) elements for
the purpose of analysis is used. Each element can then be modeled and the total model of
the structure generated from the assemblage of individual elements. In some cases the ele
ments consist of discrete members of the structure but more often individual members or
structural components will have to be further broken down into subelements.
20
The plastic behavior of a concrete structural element can be assumed to be concen
trated at discrete locations or node points at the elements ends or spread along it's length.
A survey on the different methods of modeling and the spread of plasticity is given below:
Early work on the inelastic analysis of RIC structures was based on elastoplastic or
non-degrading bilinear systems (Velestos and Newmark 1960). The energy dissipation
capacity of such systems, however, did not produce results which were representative of
reinforced concrete behavior as documented in experimental studies which tested concrete
elements up to the point of failure with cyclic loading. It was noticed that the capacity for
dissipating energy by hysteresis for the bilinear models was lower than that found in
experimental studies or as predicted by newer hysteresis models which incorporate more
complex stress-strain-time considerations. This lead to the concept of stiffness degrading.
A more representative model that incorporated stiffness degradation into the elasto-plastic
model was developed by Clough and Johnston(1966). Another complex hysteretic model
based on experimental data was developed by Takeda et al.( 1970). The model is com
prised of a trilinear envelope curve designed to dissipate energy once the cracking point of
the concrete was exceeded. Thus even at low cycles of loading, energy dissipation was
incorporated. Celebi and Penzien(1973) showed that sections experiencing severe shear
deformations have marked pinching in their hysteretic loops. Thus energy dissipating
capacity of sections is also influenced by the shear deformations through the pinched hys
teresis loops. The Takeda model cannot capture this behavior. Several researchers thus
introduced modifications to this model to account for pinching of slip effect from shear or
bond slip (Takayanagi and Schnobrich 1977, Emori and Schnobrich 1978, and Saiidi and
Sozen 1979). Takayanagi and Schnobrich(1977) incorporated the effect of axial force
21
along with pinching and strength decay into the Takeda model. Strength deterioration, also
an important parameter in the performance of RC structures to repeated cycles of load
reversals was incorporated in other models by Nakata(1978), Aoyama (1968), and Park et
aI.(1987), Kunnath et aI. (1990).
1.7.1 Modeling of Shear Walls
Wall elements are defined in some finite element formulations as panels with four
nodes defining the corners of the panel. They have large stiffness in-plane, and small stiff
ness out of plane. In bridge analysis these types of elements are sometimes used to model
abutment walls, and wall type bridge bents.
One method of modeling walls for in plane loading, using a frame analogy was
developed by MacLeod (1973, 1976,1977). He suggested that the wall properties be spec
ified along the wall vertical centerline in the plane of the wall with nodes at the top and
bottom of this centerline. Rigid body transformations of the degrees of freedom are made
from the wall centerline, to the four comers of the wall. The rotational degrees of freedom
at the corners were eliminated by introducing additional shears thus giving the same
moment as existed at the wall centerline. The connections to the walls were in the form of
stiff coupling beams consisting of only shear and axial forces. This procedure however
resulted in parasitic moments on the overall wall element (Kwan, 1993). He noted that the
end rotation of shear walls does not reflect the influence of shear deformations on walls.
He therefore proposed the use of beam elements with vertical rigid arms (instead of the
conventional beam elements with horizontal rigid arms in the coupling beams). This
method eliminates the errors in beam end rotation and ensures compatibility between ver
tical rotations at the extreme end of the wall fiber and the rotations in the horizontal cou-
22
pIing beams. However this procedure cannot model walls connected to out-of-plane
members. Other procedures for modeling of walls, involved specifying the degrees of
freedom along the wall centerline with transformations of the stiffness from nodes at the
corners to the wall centerline. This procedure has difficulties in modeling interconnected
panels.
1.8 Evaluation Methods of Bridges
The evaluation of the performances of a structure during seismic event is the first
major step in the design and retrofit of the structure. The evaluation procedure is aimed at
determining and comparing the estimated seismic demands placed on the structure to the
estimated capacity of the structure. The performance and the level of safety of the struc
ture is derived from this comparison. The capacity of a structure is defined as the level of
forces and deformations that the structure can sustain without reaching a specified limit
state. The limit state is either the level at which the structure reaches collapse or a level
defined by design standards as the maximum deformations which the structure can sustain
and still provide either an acceptable level of life safety or to remain serviceable.
The demand from a structure can be defined as the forces and deformations that the
structure will be subjected to during a seismic event.
A summary of existing evaluation methods for bridges is presented in Table 1. A
detailed evaluation of each method will be given below. All the methods presented are
based on comparison of seismic loading demands for a bridge as a entire assembly or on
smaller parts of the bridge, to calculated capacities of the entire structure or its parts.
23
TABLE 1.1. Evaluation Methods for Bridges
Evaluation Capacity Evaluation method No. Demand Evaluation Method Method
I nonlinear time history + damage index (direct evaluation)
2 nonlinear time history (element demands) element (section) nonlin-ear capacity
3 linear time history analysis (element demand) element (section) + reduc-tion factor
4 Total force on the whole bridge response spec- "push over analysis" trum (frequency from elastic structure)
5 Forces in elements calculated from static analysis elements capacity + using forces from response spectrum reduction factor
1.9 Comparison of Evaluation Procedures for Bridges
1.9.1 Nonlinear Time History with Damage Index
The governing equation of motion is
(EQ 1.8)
Where ut
is the total deformations (ug + u), c (t) is the time dependent damping
and k (t) is the time dependent stiffness. EQ. 1.8 can be integrated in a step by step man-
ner to derive the deformations ut
• The stiffness of concrete element in general is modeled
using nonlinear rules. The protective systems are modeled using nonlinear rules also such
that the stiffness k (t) and the damping c (t) . are changing in time.
The nonlinear deformations of the concrete elements can be compared with the
capacity using the damage index indicator DI (IDARC -2D Valles et aI. 1996).
24
This procedure captures accurately the dynamic behavior of the all structural sys-
tern. It is very comprehensive, and gives very good insight into the dynamic behavior of
the bridge during earthquake.
The non linear behavior of the concrete elements is accurately modeled, and the
influence of the changes in elements properties on the response of the global system is
direct.
Because of the direct interaction in time, between the global and local systems, the
influence of the protective systems such as base isolation, and damping devices on the glo-
bal bridge system is direct and no approximations of the influence are required.
The influence of the difference between the ground motion records, at different
supports of the bridge can be directly accounted for.
• The damage caused to the concrete components is evaluated instantaneously dur-
ing the analysis duration, and since the damage index is related to reduced capacity due to
energy dissipated by inelastic deformations, an accurate indication about the actual reli-
ability of the element (section) can be achieved.
Although the amount of damage to a single element is derived accurately, it cannot
be related directly to the reliability of the whole bridge system, since the deformations and
forces in the elements are dependent on the change in the nonlinear properties of all the
elements in the system, and to find the actual capacity of the bridge, a series of analysis
should be performed with increasing ground motion intensity, until failure condition in the
bridge is reached. Relating the design level intensity of the ground motion to its ultimate
25
capacity will provide the reliability of the system. Or through direct evaluation of the
return period of the ultimate intensity of the ground motion the reliability of the bridge can
be evaluated.
It is obvious that this kind of procedure in not feasible today, because of the large
number of time consuming nonlinear analyses involved.
This procedure although very fine has few additional limitations:
The response of the global bridge system, is sensitive to the modeling of the non
linear behavior of the components. The analysis can be performed only on limited number
of ground motions, which might not be enough to represent accurately the future earth
quake.
1.9.2 Nonlinear Time History
This procedure is similar in nature to the procedure in the previous section. The
major difference is that the comparison between capacity and demand in the nonlinear ele
ment is not done instantaneously, and the extreme value of plastic displacement demand is
compared with capacity which is not reduced due to the energy dissipated in previous
reversals and therefore the calculated damage level is lower then the actual one, especially
when subjected to vibratory type earthquakes in which the element experiences large
number of plastic cycles. Most of the software packages used for nonlinear analysis of
reinforced concrete (DRAIN-2D Kannan and Powell etc.)can be classified under this defi
nition.
26
--_ .. _ ... -
1.9.3 Linear Time History
The bridge is modeled as a multiple degrees of freedom system representing the
movement and restrains at important locations. A stiffness matrix and a lumped mass
matrix is used to characterize the bridge properties. The equation of motion under earth-
quake loading are represented as.
Ku (t) + Mil (t) + ell (t) = -Mrilg (t»
Using the modal transformation
u (t) = Lqi (t) <l>i i
where <l>i is the i'th mode shape vector
introducing Fig. 1.10 into EQ. 1.9 gives
where
r = <l>TMr and <l>TM<I> = I for mass normalized modes.
(EQ 1.9)
(EQ 1.10)
(EQ 1.11)
(EQ 1.12)
EQ. 1.11 is a differential equation with one unknown q(t). This equation can be
integrated in time, to find q(t) for each mode. Using the solution of q(t) the maximum
forces and displacements can be calculated using EQ. 1.1 0 .
Because the analysis is linear, to consider possible nonlinear behavior of elements,
the maximum deformations are increased, and the maximum forced are decreased by
reduction factors related to the rotational ductility of the elements. It should be noticed
27
that the influence of the inelastic behavior is indirect. Thus there is no direct influence on
the analysis procedure (reduced properties such as elements stiffness are considered to
account for this effect).
Modeling of protective systems, can be approximated, by linearization of their
behavior using equivalent damping and stiffness. It should be noted however that those
properties are not placed in the element's specific location, and are "smeared" allover the
structure, for each mode of vibration.
The influence of expansion joints (gaps), cannot be directly modeled. To compen
sate for this shortcoming, an approximate procedure which includes interpolation between
analysis with open and with closed gap was suggested (Caltrans 1992).
The effect of differential ground motions, cannot be considered using this type of
analysis.
The evaluation of the reliability of a bridge using this method, is subjected to same
limitations, as the previous method. Furthermore the reliability of each element (section),
cannot be derived exactly, and it is unclear, what portion of the ductility is actually used.
As mentioned earlier the common limitation for all the procedures using time his
tory analysis, is the uncertainty in the ability of few ground motions to accurately repre
sent the future motion.
28
1.9.4 Modal - Response Spectrum Analysis.
The modal response spectrum analysis is actually a simplification of the aforemen
tioned, linear time history analysis. The element's maximum forces fmax,i, deformations
umax,i' and total force BSmax,,-on the bridge are derived as:
Umax,i = SRSS'!'=l [cI>5rjSd(roj'~)1
[max, i = SRSS'!'= 1 [cI>5rl a (roj' ~) 1
BSmax,i = SRSS'!'= 1 [r]Sa (roj'~) 1
(EQ 1.13)
(EQ 1.14)
(EQ 1.15)
where Sd and Sa are the displacements and acceleration response spectra usually linked by
the relation, Sa = ro2S d' It should be noted that these spectra are obtained by averaging
multiple spectra from a large number of earthquakes. Therefore this analysis is representa
tive to many episodes including the "unknown" future motions.
As compare to the previous procedure, only the maximas of each mode are
obtained, and using conventional SRSS procedure the required values of force and defor
mation are derived. The use of this procedure gives a limited insight into the dynamic
behavior of the bridge.
The calculation of the reliability of the all system, has the same limitations as the
previous procedure.
1.9.5 Capacity - Push over analysis I Demand - Spectral
The limitations presented for both the linear time history analysis, and modal spec
tral analysis, in considering the nonlinear behavior, and deriving the reliability of a bridge
as a system, were the motivation for developing the current method.
29
The method is based on a well known method of plastic analysis, or capacity anal
ysis, which was used for beams, plates, etc.
A short description of the procedure is presented (Priestley et al. 1991): The ele
ments of a concrete frame are assumed to have limited capacity to resist moment. A lateral
load is applied to the frame, until the moment capacity of a critical section is reached. The
section is replaced by a hinge. This procedure continuous until total mechanism is formed.
The maximum deformations are compared to the deformations at first yield. The ratio
between the maximum deformations and deformations at first yield is defined as displace
ment ductility, from which equivalent elastic force capacity can be derived using standard
procedures. This capacity can be compared to elastic force demand derived from response
spectrum curve, using the elastic frequency (or frequencies) of the structure to derive its
reliability. The reliability can be derived directly, by finding the level of earthquake that
this structure can resist, and the probability of occurrence of the earthquake is the reliabil
ity of the frame as a whole.
This method was extended (IDARC-2D v:4.0), instead of using failure analysis as
described earlier, an incremental nonlinear static analysis is performed on a structure, with
concrete elements modeled using trilinear hysteretic loop and spread plasticity. Failure cri
teria of either excessive deformations, ultimate curvature or shear capacity is used.
This procedure emphasis the nonlinear behavior of the concrete elements. The
dynamic response is represented using linear response spectrum, which doesn't consider,
the influence of the nonlinear behavior on the demand (shift in period). The influence of
protective systems, can be accounted for, only through response spectrum curves related
30
to different levels of damping. Expansion joints cannot be modeled accurately. The bridge
cannot be analyzed as one unit, and has to be broken into separate frames with assumed
tributary load to each bent frame. The interaction between the frames through the stiffness
of the deck is ignored. There is no possibility to consider the influence of differential
ground motion.
The big advantage of this method is the fact that the reliability of a subsystem can
be derived directly, as compare to the previously presented methods, where only reliability
of elements (sections) was derived directly.
1.10 Summary of Evaluation Procedures
The evaluation procedures presented earlier are compared in a qualitative way in
Table 1.2 below. The procedures numbered from I through 5 are presented in sections
1.9.1 through 1.9.5 respectively
TABLE 1.2. Comparison of the Different Evaluation Procedures
procedure dynamic protective differential plastic reliabllity ground # response systems motion behavior evaluation
I good good good good deficient
2 good good good deficient deficient
3 good deficient deficient weak deficient
4 deficient weak no deficient good
5 weak weak no weak weak
31
2.0 ANALYTICAL TOOLS FQR EVALUATION OF BRIDGES -STATE-OF-THE-ART
2.1 Introduction
This section describes the state-of-the-art of the analytical techniques and compu-
tational tools for the seismic analysis and evaluation of bridge structures. First, the various
methods of analysis are outlined pointing out the extent of the applicability for each of
these. This is followed by a description of research needs that highlight the purpose of this
dissertation.
2.2 Methods of Analysis
There are various analytical techniques that can be used to evaluate the seismic
resistance of bridge structures. The following lists most of the methods in approximate
order of complexity:
I) 2-D Elastic Analysis. This method is sometimes still used and has its roots in the
early computer programs that were based on frame analysis. The bridge is analyzed in
each of two orthogonal directions. With this approach it is possible to capture various
modes of vibration. The elastic demands on members can also be identified. This approach
lacks the ability to predict force and/or displacement limit states.
2) 2-D Plastic Analysis. This approach, although not commonly used in practice
(particularly in North America, but is perhaps more popular in other parts of the world) is
useful in identifying force (strength) capacity, coupled with inelastic spectra displacement
limit states can also be found.
32
3) 3-D Elastic Analysis. This method is considered by many practitioners to be the
state-of-the-art. Computer programs that are commonly used for such analysis of bridges
(for example, SAP90, GT-STRUDL and SEISAB) are capable of accurately identifying
three-dimensional responses providing nonlinear behavior is not expected. Most commer
cial software vendors of these programs are currently exploring ways to extend the pro
grams to incorporate nonlinear analysis.
4) 2-D Inelastic Analysis. Computer programs that can perform inelastic time his
tory dynamic analysis of bridge structures have been available for over 20 years (Kanaan
and Powell, 1973). Inelastic 2D analysis programs have generally been developed by the
research community and are sometimes used by practitioners who wish to do advanced
analysis. The use of such programs, however, has been somewhat limited as they have to
compete with 3-D linear analysis methods. In the early 1990s, many of the original nonlin
ear time history dynamic computer programs were extended to enable nonlinear static
(pushover) analysis to be performed. Following the Lorna Prieta earthquake of 1989, there
was heightened awareness within the bridge engineering community of the need for more
advanced analysis tools and bridge structures. As a first step, the inelastic time history pro
grams were enhanced to perform a static pushover analysis to assess the strength and
deformation capability of the bridge structures at various pier bents.
5) Capacity Demand Analysis. Capacity demand analysis of bridge structures are
recommended in the recently published FHWA Seismic Retrofitting Manual (Buckle and
Friedland, 1995). The aforementioned methods require a hybrid computational strategy,
which on the one hand evaluates demand (typically using 3-D linear programs by modal
analysis), and on the other hand assesses capacity using the equivalent lateral strength
33
pushover methods. For the latter, there are still few commercially available computer pro
grams that are capable of undertaking such nonlinear analysis, therefore the Retrofitting
Manual suggests hand methods of computation for this purpose.
6) 3-D Inelastic Analysis. The finite element method of analysis has evolved over
the last 35 years and the present state-of-the-art permits the use of inelastic time history
analysis of complex systems. There are several advanced computer codes that are capable
of undertaking such a task such as ANSYS, ADINA and ABAQUS. Most civil engineer
ing offices feel uncomfortable using these programs as the analysis requires considerable
skill and experience in utilizing some of the advanced attributes and capabilities of the
software. Although finite element analysis has its roots in civil engineering, it seems that
the mechanical engineering profession has more readily adopted the method and uses
advanced (but mostly linear) analysis techniques for the routine design of manufactured
products such as cars, engines, airplanes, etc. To use some of these state-of-the-art finite
element programs for bridge analysis requires special preparation of the input. The major
impediment in using these programs for earthquake engineering is that the nonlinear fea
tures are somewhat simplistic, particularly for analyzing structural concrete elements as
well as behavior of foundation systems. Sophisticated hysteretic models such as the
Takeda model resident in the original version of DRAIN-2D (Kanaan and Powell 1973)
are quite powerful for analyzing reinforced concrete structures. Such a macro element is
not available in commercial finite element programs. In spite of this, advanced models can
be constructed with a fine mesh of finite elements in commercial software that perhaps
could reasonably emulate reinforced concrete behavior under cyclic loading. However, the
overhead necessary for such a micro mechanics based computational modeling strategy is
34
-------- - ----
excessive. Moreover, most civil engineering design offices do not have suitable computer
hardware to run such advanced analysis.
Perhaps the state-of-the-practice in civil engineering can be summarized in a
recent paper by Wilson (1994) who outlines the use of substructuring techniques on PC
based computers using the program SADSAP to handle a limited amount of nonlinear
analysis. This program, however, is limited in its nonlinear capabilities and cannot handle
well the highly nonlinear attributes and damage associated with degrading reinforced con
crete elements.
In the research community, several computational platforms have been developed
for inelastic 3-D analysis of structural systems. Few of these programs have been devel
oped for bridges, most have been developed for buildings. The program DRAIN-3DX
(Prakash et ai., 1992) uses a fiber element formulation to model the nonlinear behavior of
the structural concrete elements. Because of the very complex nature of the modeling, it is
not realistic to use such a program for production design office use as the problems them
selves tend to be ill conditioned, particularly when a large number of nonlinear elements
are present.
One way of overcoming the computational difficulties associated with utilizing
fiber models is instead to use macro models that have bi-directional modeling capabilities.
Such an approach generally leads to improved problem tractability, but the problem
remains of how to calibrate the macro models for bi-directionalloading. An example of
such a program is IDARC 3D (Lobo, 1994). In this program macro models are calibrated
through a combination of moment curvature analysis parameters that are identified from
35
experimental studies. Alternatively, an off-line fiber element analysis of a single column
member can be analyzed using an experimental-type of load history. This is then used to
more accurately calibrate the macro model. Such an approach has been advocated by
Chang and Mander (1994a,b).
Perhaps the only computational platform that has been already dedicated to nonlin
ear 3D analysis of bridge structures is the software NEABS II (Imbsen and Penzien,
1986). Although this program has many desirable attributes for nonlinear 3D analysis, its
utility is limited by the simplistic nature of the hysteretic models that it uses. For example,
structural concrete elements are modeled by a bi-linear representation. It is difficult to uti
lize such a simplistic approach for highly irregular bi-directional loading. The program
also suffers from lack of flexibility that is given to the analyst as the program was devel
oped to model conditions being characteristic of California-type bridges. For bridges that
do not fall into this pattern of construction, the program is difficult to use. The program
also has some limited capability for handing differential ground motion input.
2.3 Damage Analysis
The concept of damage analysis coupled with nonlinear time history analysis is an
important issue. To date there are no commercially available software packages that com
bine nonlinear time history analysis with damage analysis. It is only the university origi
nated research type software that undertake nonlinear damage analysis. Programs that can
do this either in a somewhat crude or sophisticated way include DRAIN-2DX, SARCF
and IDARC-2D.
36
2.4 Research Needs
Many features that are desirable for the nonlinear dynamic analysis of bridge sys
'terns already exist in various forms and have been implemented in other areas. There is a
clear need to assemble these features together on to one software platform. For example,
much work has been done over the last few years on the 2- and 3-D behavior of seismic
isolation systems with a particular emphasis on building structures. Certain programs that
have resulted from this work (e.g., 3-D BASIS Nagarajaiah et al. 1989) have capabilities
that are also desirable in bridge analysis. As mentioned above, the seismic capacity and
demand analysis procedures are presently fragmented and these should be amalgamated
along with damage analysis concepts in such a way that it should be possible to analyze a
bridge that incorporates the primary source of energy dissipation in locations such as
joints and gap hinges in the deck, 3-D behavior in column elements and plastification of
these elements in the substructure (the piers), and the nonlinear behavior of the foundation
system including the nonlinear and radiation damping effects of soil structure interaction.
It should be emphasized that present pushover analyses are conducted on a frame
by-frame basis and then combined in some way to give the overall effect for the bridge
structure. The problem with this approach is that the combination may not conform to the
dynamic response that could be expected in an extreme earthquake. It is, therefore, desir
able that a 3-D computer program be capable of undertaking a pushover analysis of the
bridge both statically and dynamically. In this fashion, some of the uncertainties that per
tain to modal combination for large 3-D structures can be eliminated.
37
What distinguishes bridges perhaps more than anything else in comparison to
other structural systems is the spatial layout of the structure that gives a high propensity
for differential ground motion input.
38
3.0 DEVELOPMENT OF NEW ANALYTICAL PLATFORM FOR SEISMIC RELIABILITY EVALUATION
The reliability of a bridge for seismic loading is of major concern to ensure proper
operation of road and railway systems following an earthquake. The reliability R or the
probability of the bridge to survive can be defined as
D R = 1-
C (EQ 3.1)
in which D - the demand is the magnitude of forces or deformations that the bridge will
experience, during an earthquake with certain probability of occurrence during the life
time of the bridge; and the capacity C is the magnitude of forces and deformations in
which the bridge loses serviceability or stability. In this chapter several approaches to
evaluate the seismic performance using the reliability definition for bridge structure are
presented. In the first part of this chapter, the important characteristics of bridges as struc-
tural system, and each of its major components is presented, as well as the structural
behavior of the components, under static and especially dynamic loads.
The bridge is modeled as a framed structure, so that matrix analysis using the stiff-
ness method can be performed. The structural modeling of each of the structural elements
used in modeling the bridge components, and the way this elements are used to model
each component of the bridge, and subassemblies of the bridge is presented as well as how
mathematical-physical tool such as rigid body transformations are used to achieve effi-
cient modeling of the bridge.
39
In the second part of this chapter different procedures of evaluation capacity and
demand are presented, and the procedures used to evaluate the final reliability of individ-
ual elements, and of the whole bridge structural system
3.1 Bridge Structure and Structural Components - Descriptive Presentations
Bridge is a general word to describe a structure that enables traffic to pass over an
obstacle. Bridges types, vary from very small and simple once, that bridge a small creek,
to very large once, that span over large bodies of water. Altogether all of them.have a com-
mon typical characteristics and components that differ in size and complexity. A typical
set of a bridge assembly and it's major components is shown in Fig. 3.2
deck abutment backfill
FIGURE 3.2 Bridge Assembly and Major Components
40
3.1.1 Components and System Definition
A bridge has several major components, as shown in Fig. 3.1. The components
are: deck, bents, foundations, abutment, connections systems and joints.
The deck is the surface on which the traffic is moving. The deck is supported by
large bents which provide for both gravity and lateral movement. Several types of struc
tural arrangements of the deck are used: prestressed box girders, monolitically cast beams
and slabs, and concrete or steel slabs supported on steel beams etc.
The bents are the part of the bridge that elevates the road above ground level.
Mostly two types of bents are used: (i) A multiple piers bent with a top beam connecting
them together at their top. (ii) Single walls, which support the deck directly.
The foundations are the system which interface the bridge structure to the support
ing ground soil. The main purpose of the foundation system is to transfer the dead and live
load of the bridge to the ground. This system also transfers the horizontal loads and the
overturning moments resulting from horizontal loads, such as winds and earthquakes. Var
ious types of foundations are used, which are: shallow foundation systems, and deep foun
dations such as caissons and piles. Each provides larger or smaller soil-structure
interaction during seisrnic events.
The abutments are the supports at the two ends of the bridge. They support the ver
tical and lateral forces from the bridge deck as well as the soil behind them. The abutment
is made usually of a central wall and two wing walls, which support horizontally the soil
behind them. The deck is supported vertically on the central wall trough joint (see descrip
tion).
41
In long bridges, the expansion and contraction of the deck above the bents due to
temperature changes might cause unwanted stresses. Therefore expansion joints are intro
duced to various sections of the deck.
The deck sections are supported by bents vertically. To provide horizontal motion
capacity between the deck and the bent and to prevent undesirable forces due to thermal
expansion, bearings are placed between the deck and the bent. Several kinds of bearings
are used for this purpose: elastomeric bearings, teflon steel sliding bearings, steel bearings
etc. Those bearings should also provide rotational capacity, since usually a hinge type of
support is desirable.
In moderate and severe seismic zones, the sections of the bridge are suppose to
move together in the transverse direction. This is achieved by providing shear keys, which
prevent transverse movement but allow longitudinal movement.
Another important issue relevant to severe and moderate seismic zones, is the
problem of possible unseating of structural components from their supports. Restrainers
are used to prevent such instability in modem bridge. The restrainers are steel roads or
cables, anchored in the separate deck sections, which limit the relative maximum displace
ments of the deck sections in the longitudinal direction. More modern solutions use non
linear damping devices with energy or shock absorption characteristics
To reduce the forces in the structure resulting from earthquake, modern protective
systems such as: base isolation systems, damping devices, active control, etc. are also
used. Those devices are placed usually in the expansion joints, between deck sections or
between the deck and the bent.
42
3.1.2 Comparison of Buildings and Bridges I Specific Issues
Bridge structures span over large distances in space, and therefore their founda
tions are spread apart in distinct locations, creating several different interface points
between the soil medium and the structure. The ground motion, the supports stiffness and
damping, might vary greatly from one interface point to another, influencing significantly
the structural dynamic response. In most of the other structures i.e building structures, the
interface occurs usually in small area which can be characterized as one point in space.
Although large buildings are supported on a group of individual foundations, usually those
foundations are connected by tie beams, which form a single type of interface, with single
structural properties and ground motion.
An additional characteristic which is also unique to bridges is the presence of
expansion joints. Expansion joints are provided to prevent appearance of thermal stresses
due to bridge elongation. In affect these joints influence the response of the bridge to lat
eral loads by separating the bridge into several substructures loosely connected to one
another.
3.2 Structural Component Modeling
The entire bridge structure may be viewed as a multi-bay three dimensional frame
with various members and components.
In the previous section, a general description of structural components of a bridge
was presented. In this section, mathematical representation in form of stiffness, damping
etc. is given to the basic structural elements used to compose this subassemblies and then
43
a description, how these structural elements are actually used to represent the structural
behavior of the subassemblies.
3.2.1 Basic Stick Element Formulations
In three dimensional framed structures analysis, the basic element is the space
frame member (Weaver and Gear 1990), or "stick" element. This element has two general
joints in space, with each joint having translational and rotational degrees of freedom in
all three global directions X, Y, Z all together six Degrees Of Freedom (D.O.F) in each
joint. And for the whole element 12 D.O.F's. The element has it's own coordinate system,
and can be rotated in any direction with regard to the global coordinate system (Weaver
and Gear 1990). The constitutive relations of this element can be described by
(EQ3.2)
where F, M, u and 9 are respectively the three components vectors of forces moments,
translational displacements and rotational displacements. Moreover k represents the stiff-
ness matrices relating forces to displacements.
In case of dynamic analysis additional forces related to damping may develop,
which conveniently are described by a simple viscous representation:
(EQ 3.3)
44
where Fc, Mc' Ii and e are respectively the three components vectors of damping forces,
damping moments, translational velocities and rotational velocities. Moreover c repre
sents the damping matrices relating damping forces to velocities.
The element formulation described by Eq. 3.2 and Eq. 3.3 is the so called Kelvin
model formulation.
For types of analysis which include elements that have nonlinear properties (non
linear time history analysis, incremental nonlinear static analysis, push-over analysis) the
stiffness and damping matrices k and c depend on the nonlinear rules specific to elements
used.
The "stick" elements can be provided with special end conditions, which include
end releases, end springs and rigid arms.
Moreover in the dynamic analyses using macromodels the mass is lumped in con
necting joints such that the individual structural models are massless.
3.2.1.1 Rigid arms
The connections between elements may be very complicated especially in the
interface between the deck surface and the bents. Accurate modeling would require many
elements and joints, to model essentially rigid zones. Using rigid body transformations,
element end springs attached directly to elements, and releases of D.O.F's can provide a
significant simplification in the modeling, reduce unnecessary computations and eliminate
the need to evaluate and handle unnecessary results.
45
~I
deck beams
r 1 >:1.. __
(a) joints foranal~
'.---------~--------~
(b)
element defined between
bearing
theoretica1 joint
rigid arm
two joints using rigid arms and end springs
spring
rigid arm
isolator
(c)
FIGURE 3.2. Connection Between Deck and Bent Using Rigid Arm and End Springs (a) Physical Description (b) Regular Modeling (c) Modeling with Rigid Arm and End Springs
46
A possible connection between the deck and the bent is presented in Fig. 3.2 -(a).
To model correctly this connection many elements are required (Fig. 3.2 -(b». The deck
is modeled as beam and has only one connection point. The bent however has three con
nection points at the locations of the bearings. By assuming that the deck is rigid for rota
tion around Y axis, and translation in Z direction the corresponding rigid transformation
can be applied to represent the rigid zones. The flexibility of the deck beams for transla
tion in the Z direction, can be modeled by end springs. The use of end springs and rigid
arms enables accurate representation of this area with the use of only three elements and
four nodes (Fig. 3.2 -(c».
To accommodate the reduction in number of D.O.F's due to the presence of rigid
zones the rigid arm transformation is developed. The elements stiffness matrix is created
with respect to the actual joints (Fig. 3.2 -(a» and then transformed, using rigid body
transformation, to the theoretical joints (Fig. 3.2 -(c» which are used to assemble the total
structural stiffness matrix. Following the solution of the displacements in the global sys
tem, the displacements in the actual joints and the element's end forces can be found using
again rigid body transformation.
The transformation of an action Aj at the actual jointj to the theoretical joint (used
for descritization of the structure) p (Fig. 3.3) is developed as follows:
47
y
r j
x
y
r j
x FIGURE 3.3. Rigid Body Transformation from Joint P to Joint J
The action at joints p and j is defined as
(EQ 3.4)
The action at p is represented by the action at j
(EQ 3.5)
In this equation 'pj is a vectorfrom joint p to jointj so that
48
[I j k l 0 -Zpj Ypj P J
rpjxFj = Zpj Ypj Zpj = cpjFj = Zpj 0 -xpj FY (EQ 3.6) J
F~ FY F, -Ypj x. 0 H J J J PJ J
The relations presented in EQ. 3.6 are relating a transformation of a single joint of
the "stick" elements. The relations for the whole element (two joints) are:
A = TA. p J
where T, is the element transformation matrix:
with /3 being the identity matrix of the size of 3
The displacements vectors at joints p and j are defined as
which can be related by
where
ajxrpj =
D = { up }; D. = { uj } p 0 J 0.
i
ax p
Xpj
D. = J
j k
ay a z p p
Ypj Zpj
P J
0
= -Cpjaj = -Zpj
Ypj
The relation for the element with two joints is:
',j -Y'j o Xpj
-x· 0 PJ
a~ J
a~ J
a, J
(EQ 3.7)
(EQ 3.8)
(EQ 3.9)
(EQ 3.10)
(EQ 3.11)
49
(EQ 3.12)
The force displacement relation of the element at the actual joints (physical) is:
where kj is the stiffness matrix of the actual element.
Introducing EQ. 3.12 into EQ. 3.13 gives
and from EQ. 3.7
Premultipling both sides by Twill give
Ap = TkpTTDp
Finally, the stiffness matrix relation is:
3.2.1.2 End releases
(EQ 3.13)
(EQ 3.14)
(EQ 3.15)
(EQ 3.16)
(EQ3.17)
Releases of a stable combination between the "stick" element D.O.F's as presented
in Fig. 3.5 is developed to enable flexibility in modeling complicated connections
between elements. For the sake of modularity, a fixed end element stiffness matrix is gen-
erated first and the end releases are applied later to generate the released matrix. The num-
ber of releases is limited so that no mechanism is created in any direction (such as in the
case when one side fully released and the other side is hinged).
50
ux
beam end releases (rotation or translation)
uy / possible released uz D.O.Fs rx
~~--~-~~ x
element start element end
FIGURE 3.4. Element End Releases
The force displacement relations for a single element are
kd = A (EQ 3.18)
The degrees of freedom can be separated into released and non released degrees of
freedom, with the force vector on the released degrees of freedom being zero.
(EQ 3.19)
where subscripts n and r indicates non-released and released D.O.F's, respectively.
The stiffness matrix of the element can be related to the non-released degrees of
freedom only, when the stiffness of the released degrees offreedom is zero. Thus,
k * - k _kT k-1k - nn nr rr nr (EQ3.20)
The final stiffness matrix will have the form
k*e = [~*~] (EQ 3.21)
51
Which also satisfies the equation: A = k: d .
The use ofEq. 3.21 in the assembly of the global stiffness matrix can be done with
relation only to the nonzero terms in the matrix. It should be reminded, as it is well known
that while building a structural model, all the global D.O.F's, should have nonzero stiff
ness terms on the main diagonal, otherwise the stiffness matrix, might be ill conditioned.
The above formulation significantly simplifies the manipulation of the stick ele
ment stiffness matrix since it allows the combination of releases, rigid zones, and end
springs together, in any combination and order. Another important benefit is the fact that
this formulation can be used for any type of element, including inelastic beam elements,
walls, dampers, base isolators, gaps, etc.
3.2.1.3 End springs
A common way to formulate the stiffness matrix of a beam with end springs is by
using the flexibility method (Weaver and Gear 1990). This method requires different for
mulation for a beam without springs, and for every combination of springs at it's start and
end. To avoid this and achieve a more modular formulation, the stiffness matrix of the
fixed end beam is derived first and the contribution of the end springs is added later.
As shown in Fig. 3.5 four joints are defined for the basic "stick" element and the
two end spring elements. The constitutive relations of the combined system are:
52
kx ky kz
possible spring stiffness krx
kry
~~_2 _______________ 3~
/ element start end spring element end
FIGURE 3.5. Elements End Springs
[;2,31 = r:2' 3-2, 3 :2,3-1,41 r:2,31 I,J ~ 1,4-2,3 1,4-I,J ~ I,J
x
(EQ 3.22)
Fi,j are the forces and moments at joint i and j and uiJ are the displacements at
joint i and j. The matrix k is assembled in a standard manner by arranging the element
stiffness matrix K beam , and the stiffness matrices of the end springs K s1 ' Ks2 into the
related degrees of freedom.
Since joints 2 and 3 are internal joints, there is no external force acting on them,
and therefore Eq. 3.22 can be written as:
= r:2,3-2,3 :2,3-I,J r:2,31 ~ 1,4-2,3 1,4-I,J ~ I,J
(EQ 3.23)
After partitioning.
U = -k- I k u 2,3 2,3-2,3 2,3-1,4 1,4 (EQ3.24)
53
Introducing Eq. 3.24 into the second part of Eq. 3.23 will yield the final constitu-
tive relation for the external joints I and 4
(EQ 3.25)
where the final stiffness matrix is
k - k kT k-1 k final - 1,4- 2,3-1,4 2.3-2,3 2,3-1,4 (EQ 3.26)
In the case when only few of the D.O.F's have springs attached, the following con-
stitutive relation can be written:
[ke_e ke-IJ [U
J = [F
J k. k. 1 U· 0 l-e l- l
(EQ 3.27)
where ue are the displacements of the D.O.F's related to the external D.O.F's (similar to
joints 1-4 in the previous development) and ui are displacements related to the internal
D.O.F's (similar to joints 2-3 in the previous development). Employing simple static con-
densation, results in
k" I = k - kT k":-I.k. Jma e-e l-e l-l l-e (EQ 3.28)
This expression is general, without any limitations, and can be combined both
with rigid arms, and end releases. The stiffness of the end springs can be linear as well as
nonlinear. This operation requires finding the inverse of the matrix (k;J of the size of the
number of end springs.
54
------- -
3.2.2 Inelastic Hysteretic Beam Columns with Nonlinear Shear and Flexure Behavior in Beam/Column elements
The force displacement relation for hysteretic beam/column element is
F = ku (EQ 3.29)
Which is similar to the relations of the basic "stick" element. The stiffness matrix
of the basic "stick" element k as well as the stiffness matrix of the hysteretic beam I col-
umn element has the form
EA EA L L
12EI 6EI 12EI 6EI z _z z z ---;y L2 ----;y L2
12EI _6Ely 12Ely _6Ely --y L3 L2 ----;y L2
GJ GJ L L
4Ely 6Ely 2EI -y
L L2 L
4EI 6Elz 2EI z z k = L - L2 L (EQ 3.30)
EA L
12EI 6EI z z ---;y - L2
12EI 6Ely SYM --y L3 L2
GJ L
4EI -y
L 4EI z
L
For the hysteretic beam I column element the variables In Eq. 3.30 E(u) the Young
modulus, and J(u) the moment of inertia ofthe section are considered nonlinear functions
55
of curvature. For an inelastic beam, an equivalent EI(u) is used which is assumed to be a
function of the state of stress and strain in the element.
3.2.2.1 Bending
The stiffness matrix of a nonlinear concrete element in bending is derived using
flexibility fonnulation (the procedure is the same one used by Lobo 1994). A linear distri
bution of flexibility over the length ofthe beam(Fig. 3.7-(b» is used. The size of the pen
etration of the damage into the element indicated by xl, x2 at Fig. 3.7-(b) and designated
as cracked length in Fig. 3.7-(a). The size of the cracked (damaged) length is derived
from linear moment distribution over the element length (Fig. 3.7-(a». In the process of
the incremental analysis the damaged zones cannot recover after they were damaged pre
viously, so the maximum value reached during the analysis history is used to define the
instantaneous flexibility distribution. The maximum penetration is monitored in four
zones at the ends of element (Fig. 3.7-(c», for each local beam direction (x-z, x-y). The
elastic flexibility at the center of the elementfiexO is lIEI of the elastic (undamaged) sec
tion of the beam. The flexibility at the ends of the element defined as d<j>1 dM is obtained
from the trilinear hysteretic rule (Fig. 3.6) which includes, stiffness and strength degrada
tion and slip (Park et al. 1987). Currently the behavior of the element in bending in its two
major directions is modeled independently (no interaction).
56
4>nultimate
/ ,
MPultimate
MPyield
MPcrack
/ /
/ ' / / ,
/ , /,
/ , ,
N ,
M
, 'n ... M crack ,
Mnyield
Mnultimate
Mnyield*U,S,
(a)
first unloading with slip
M
unloading ~, ~ with slip '\ ,..-M'Crack *S.F
unloading slop
Mtarg.,*S.F \.. ~ >-............... yo \. •.•
~..... ..... L first unloading . ~ without slip
_--f~-=-:
(b)
<I> P ultimate
FIGURE 3.6. Trilinear Hysteretic Model, a) General Model, b) Slip Definition
57
cracked length(zone_number(1)) ,...:l+,
cracking moment left
cracking moment right
a)
cracked_length(zone_number(2))
x2 xl
b) up
I 2
3 4
down
c)
FIGURE 3.7. Stiffness Calculation of Damaged Elements a)Moment Distribution b)F1exlbility Distribution c)Damaged Zones
58
3.2.2.2 Combined Bending and Shear
The general behavior of bridge piers under lateral loading is dominated by bending
and shear. The shear behavior is especially important in piers with aspect ratio, less then
4.0, in which shear failure was observed in recent earthquakes.
To be able to predict the failure mechanism, a combined shear and flexure model is
developed. For the flexure model the moment curvature relation is described by the trilin-
ear model, from which the element stiffness is calculated using distributed plasticity (see
section 3.2.2.1) in terms of end forces and displacements. A trilinear shear model is used
also to define the shear force to shear deformations relationship
The final stiffness of the element is derived by combining the flexural and shear
stiffnesses into the element stiffness matrix. First the flexural stiffness matrix is derived
and shear influence is added later for the sake of modularity.
.J
k22 + klJ
~ __ ~=LO J
FIGURE 3.8. Combined Flexure and Shear Stiffness Derivation
The stiffness matrix of joint J of the beam in Fig. 3.8 which is related to flexure
only can be written as
(EQ 3.31)
59
The flexibility matrix due to flexure can be derived by inverting the stiffness matrix
F = k- I = 1 [k22 -klJ det (k) -k k
21 II
(EQ3.32)
where
(EQ3.33)
The influence of shear can be added directly to the term F II such that
(EQ 3.34)
where F II is the flexibility term including the shear deformations, and ks is the shear stiff-
ness defined as
k = fL s GA (EQ 3.35)
where f is the shape factor, L the length of the beam, G the shear modulus, and A the
effective area. It should be noted that the parameters in this equation if, A, G) can depend
on the stresses and strains (J(u), A(u), G(u), and as a result the shear stiffness klu) is also
nonlinear.
The flexibility matrix for joint j including shear deformations has the form
[
det(k) l F = 1 k22 + k -k12 det (k) s
-k21 kll
(EQ 3.36)
Note that when the shear stiffness ks is infinite, (no shear deformations) the origi-
nal flextural model is recovered.
60
Inverting the flexibility matrix will give the stiffness matrix which includes shear
deformations contribution.
where
and
that
~kll k12 ] - 1 1 k = det (F) det (k) k k + det (k)
21 22 k s
(EQ 3.37)
(EQ 3.38)
(EQ 3.39)
Using equilibrium the stiffness terms k13 ,. k23 ,. k14 ,. k24 can be derived such
- - - - - - - - --k13 = -kll ,. k14 = kllL - k21 ,. k32 = -k21 ,. k42 = -k22 + k12L (EQ3.40)
In the same manner (using equilibrium), the stiffness matrix for the other side of
the beam can be derived. The element stiffness matrix is which includes bending and shear
is defined as:
where
[
kll k12 ] k k +det(kll)
21 22 k s
(EQ 3.41)
[
k33 k34 ] k k det (k22)
43 44 + k s
61
(EQ 3.42)
In the limit when ks =0 (shear failure):
[: k d~'(kll)] [~31 ~3J 22 + k k41 k42
k = II
[~31 ~3J k41 k42 [: k" + d~' i~2)]
It should be noted that all the elements of k matrix can change in time in nonlinear
structure. If the element reaches yielding in flexure or in shear, the failure mechanism is
controlled by the lowest capacity of the two mechanisms. When the limit state of either of
the two mechanisms is reached, total failure of the element occurs.
As mentioned earlier this formulation was adopted since both the flexure and the
shear stiffness can be highly nonlinear. It is beneficial to derive separately the flextural
stiffness matrix and the shear stiffness for increased modularity in handling and allowing
the separation of two complicated procedures.
In order to illustrate the shear or moment dominant failure mechanism, the simple
frame in Fig. 3.9 is analyzed. First the frame is assumed to have only flexural stiffness.
Then both flexural and shear are considered in two cases: (i) The flextural capacity is
assumed to be smaller than the shear capacity. (ii) The shear capacity is assumed to be
62
F '" rigid
1= 10m
" .""
FIGURE 3.9. Example Frame, for llIustration of Combined Shear and Flexure Model
smaller than flexural capacity. The constitutive relations of shear and flexture for all the
cases are shown in Fig. 3.10 .
The influence of shear on the capacity of the structure is demonstrated in Fig. 3.11
The first case corresponds to pure bending model. For this case (no shear at
Fig. 3.11 ) yielding in bending can be observed. In the case where shear deformations are
considered but the shear capacity is high, again flextural yielding occurs, with increased
deformations because of the contribution of shear deformations. When the shear capacity
is low, the element is not able to develop it's full flextural capacity and fails in shear. If the
shear model is assumed to be brittle (which is the case in many short RIC bridge piers)
then catastrophic failure may occur.
63
2.00
1.50
~ z ~ C 1.00 ~
E 0 :;
0.0 I:-~--'_~....J...~_-'-~---:! 0.00 0.05 0.10 0.15 0:20
Curvature (11m]
Moment curvature
0.4 0.4
0.3 - 0.3
Z ~
~ 0.2
V 0.2 oS
iii ~ rn
0.1 - 0.1
0.0 0:0 0.5 1.0 1':5
0.0 0:0 0.5 1.0 1:5
Shear disp.[m] Shear disp [m]
Low shear capacity High shear capacity
FIGURE 3.10. Constitutive Relations in Shear and Bending of the Frame's Columns
64
0.8
0.7
0.6
0.5 Z .!!.
iB 0.4 '" 0 ~ 0
III 0.3
0.2
0.1
Olsp[m)
F1GURE 3.11. Influence of Shear on Frame's Capacity
3.2.3 Shear and Bending Wall Panel Model (for modeling Abutment)
The abutments of a bridge can be modeled by using a four nodded shear wall ele-
ments. This element's nonlinear behavior has similar characteristics to the hysteretic beam
described in the previous section which includes flexture and shear. The wall is modeled
by using the a shear/flexure beam. The element has four nodes of which two of them are
coupled to the other two. The shear wall element, shown in Fig. 3.12 (a), can be physically
defined by a flexural-shear beam connected between nodes 2 and 4 Fig. 3.12 (b) with rigid
arm to the center of the wall. The actual properties are related to the theoretical beam
located at the center of the shear wall. The stiffness properties are transformed to the phys-
65
ical beam by using rigid arms, as described in section 3.2.1.1 . Joints 1 and 2 are coupled
using the coupling procedure described in section 3.4 to ensure rigid rotation of line 1-2
such that
U Lu I 1- 2
U 3_U 3 I - 2
(EQ3.43)
(EQ 3.44)
(EQ 3.45)
The current formulation of the shear wall is a straight forward extension of the
basic hysteretic beam element into wall element by using existing tools such as rigid arms
and coupled motions without any additional special formulation.
3.2.4 Supports and Base Isolation Models
Base isolation is a strategy to reduce the seismic hazard for a bridge. An isolation
system combines two elements. The first one is, to provide a soft medium between the
bridge and the ground and thus isolate the bridge from the ground. The other element is to
provide energy dissipating capability between the ground and the bridge. The two ele-
ments can be used together, in which case the energy dissipating capability is used to
reduce displacements, or as it is the case in Menshin design isolation system, where it is
used to dissipate energy only, and shifting the period of the bridge is not an objective.
Two models are introduced to model the behavior of such devices. The first one is
based on a model developed by Bouc(l971) and enhanced by Wen(1976), which represent
elastic - plastic constitutive relations with smooth transition. The other model, is based on
triaxial interaction to model the behavior of an isolation system which includes the infiu-
66
y
Ushear Uflexure u/t 3 "\ V I
3 ++---,1- rigid "\ v2
rhP~""1 ~--•
3
theoreticaJ beam I
¥ I I I I I
I I
I
I rigid arm
--
4
~ d
x
(a)
• I I I I I I
theoretical beam I
~: I
beam for analysis
2
4
rigid arm
(b)
FIGVRE 3.12. Shear Wall Definition (a) Shear Wall (h) Shear Wall Modeling for Analysis
ence of the vertical load on the lateral force, and the dependency of the lateral forces on
velocity.
3.2.4.1 Boue-Wen's Model for Seismic Isolators
In order to represent the behavior of an isolator element as shown in Fig. 3.13(a),
the smooth bilinear model shown in Fig. 3.13(b) is used.
67
Bouc Wen's equation requires solution of a first order differential equation that can
solved by enhanced Runga Kuta method (Nagarajaiah et al. 1989). In order to obtain a
more accurate and faster solution, and to better understand the model, an analytical solu-
tion of the basic differential equation is derived.
F ,
~ Kinitial
F Kyield ~ ~ ~Y .' L __
~ . - - ..
F U by .. (a) (b)
F ~
F~
(c)
FIGURE 3.13. Elastomeric Isolator. a) side view, b) force displacements relations, c) top view of deformed isolator
The basic equation for an isolator with elastic perfectly plastic behavior is (Nagar-
ajaiah et aI. 1989):
F = ZF y
where Z is defined by the differential equation:
(EQ 3.46)
68
. 0 . 0 Z = A- -12111 (ysgn UZ + ~) -Uy Uy
(EQ 3.47)
Z is a nondimensional parameter which basically represents the "nondimensional reactive
force", and its value assumed to be limited -1 ::; Z::; 1 , U is the total displacement, A ~
and y are nondimensional parameters which control the unloading slope of the hysteretic
loops, Uy is the displacement at yield.
To obtain strain hardening EQ. 3.46 is modified
(EQ 3.48)
or
(EQ 3.49)
where F is the restoring force, 0: is the ratio of initial elastic stiffness to the post yielding
stiffness, Fy is the yield force.
EQ. 3.47 can be written as
t = g {A-IZl 11 ((ysgn(UZ) +~)} y
(EQ3.50)
Rearranging EQ. 3.50 we obtain:
dZ d(U/U) . - = d y {A -12111 (ysgn (UZ) +~)} dt t
(EQ 3.51)
and by multiplying both sides by dt yields
69
dZ U dU = y
A -IZl11 (y sgn (UZ) + ~) (EQ3.52)
This equation is time independent and can be solved directly. Integrating both
sides:
U=UJ dZ y A-IZl11(y sgn(UZ) +~)
(EQ 3.53)
A close form solution is possible only for small number of.., values, .., = 1 or
.., = 2; the one chosen for this model is.., = 2.
New variables a and b are defined as:
A = a2 (EQ 3.54)
b = Jlysgn (UZ) + ~I (EQ3.55)
Several different cases should be distinguished in the solution of this equation:
For y>~:
y sgn (UZ) + ~ = b2 sgn (UZ) (EQ3.56)
Introducing EQ. 3.54 . and EQ. 3.56 into the integrand of EQ. 3.53 will give
U=UJ dZ =UJ dZ Y A-IZl2y sgn(UZ) +~ y a2 -sgn(UZ) b2z2
(EQ 3.57)
The solution of EQ. 3.57 can be derived for the following cases:
(A) sgn (UZ) > 0
J dZ U = Uy 2 2z2 a -b
(EQ 3.58)
70
and after integration
u = u _1_10gla+b~ +c Y2ab a-bZI
(EQ3.59)
This solution is divided into two cases:
(i) Case when b2Z2 < a2
1 bZ U = U -atanh-+c
Yab a (EQ3.60)
and
Z = -tanh -+c a (Uab ) b Uy
(EQ 3.61)
(ii) Case when b2z2 > a2 the solution is
U bZ U = .2acoth- + c
ab a (EQ 3.62)
and
Z = -coth -+c a (Uab ) b Uy
(EQ 3.63)
(B) sgnUZ<O
Uy bZ U = -atan-+c
ab a (EQ3.64)
Z = ~tan( Uab + c) b Uy
(EQ3.65)
The case y > ~ and b2Z2> 1 leads to Z larger then 1 which it invalid, so that the
only valid solution for case (B) is for b2Z2<1.
71
The instantaneous stiffness of the isolator can be derived using the solutions for Z
(EQ. 3.61 . EQ. 3.63 and EQ. 3.65 ). The solution for Z can be used to define directly the
instantaneous stiffness k as a function of deformations by differentiating EQ. 3.49
dF Fy dZ k = - = a-+ (l-a)F-
dU Uy YdU (EQ3.66)
The term :~ in EQ. 3.66 is obtained as follows:
(i) case sgn UZ = I and b2z2 < a2
( dZ) _ d (a Uab) _ a2 2Uab dU - dU btanhU - U sech U
1 Y Y Y (EQ 3.67)
(ii) case sgn UZ = I and b2z2 > a2
( dZ) = .!!...(~ctanh Uab) = _ a2
csech dU 2 dU b Uy Uy (EQ3.68)
(iii) case sgn UZ = -I
( dZ) d (a Uab) a2
2 (Uab) dU 3 = dU btan Uy = uy
sec Uy (EQ3.69)
The values of the instantaneous stiffness k are used in the analysis to update the
stiffness of the structure as the analysis progresses.
The parameters required to define this model are: the yield force F yo the yield dis-
placements Uyo the ratio between Kinitial' and Kyield a (see Fig. 3.13(b)), and the parame-
ters A ~, and 'Y, which control the shape of the unloading branch. In order to limit the
extreme value of Z = ± 1.0 the ratio alb in EQ. 3.61 , EQ. 3.63 and EQ. 3.65 must be
72
equal to 1.0. One of the ways to achieve it, is by fulfilling the conditions A= 1 and
"( + ~ = 1. When ~ = "( the unloading branch of the hysteresis loop is a straight line
with slope (stiffness) equal to the slope of the loading branch (initial stiffness kinitial=F/
uy), when ~ is larger than ,,(, the unloading stiffness is higher than Kinitial and the unload
ing curve is convex (as shown in Fig. 3.13(b)). When "( is larger than ~, the initial
unloading stiffness is smaller then Kinitial and the unloading curve is concave.
3.2.4.2 Triaxial Isolator
Triaxial interaction model was developed to model the behavior of sliding isolators
subjected to triaxial loading. Modeling of sliding isolators with biaxial interaction was
treated previously using pseudo force technique in which equivalent forces are added into
the right hand side of the global equilibrium equation, while there are no stiffness or
damping tenns related to these D.O.F's (Nagarajaiah et al. 1989). Iterative procedures are
required to solve this problem. The pseudo force method is used because it allows to keep
the initial stiffness of the structure, and update only the right hand side of the differential
equation of motion, while using iterations to find equilibrium. It allows the use of linear
solution procedures such as mode superposition to solve nonlinear systems (Bathe et al.
1981). However, there are two significant deficiencies to this procedure: (i) In the presence
of nonlinearities in several locations in the structure, many or all the modes of vibration
will have to be considered to achieve accurate modeling. (Bathe et al. 1981) (ii) When
direct integration method is used, in the presence of large number of nonlinear elements,
the search for eqUilibrium is multidirectional. While convergence in one direction might
73
F ~ I p I
t ~ I'd' ~rf P 51 mgmte ace
I I I I
(a)
Jlmax
Jl
smooth function
bilinear
F
(b)
~ ____ L-______________ ~O
Vlimit
(c)
FIGURE 3.14. Triaxial Sliding Isolator, a) side view, b) sliding force· displacements relation. c) dependancy of coeficlent of friction on velocity
be reached, divergence might occur in other directions, which result in infinite oscillation
around the equilibrium point.
Another approach for the modeling of hysteretic behavior of isolation systems is
by using first order differential equation to represent the hysteretic behavior of the isola-
74
tion systems (Tsopelas et al. 1994). This method is very accurate and gives very good
results for relatively small structural systems, and when no other types of nonlinear ele
ments are present. In the presence of elements with other types of nonlinear rules, the sec
ond order differential equation of motion can be solved using a numerical method
(Newmark 1959). For this kind of procedure a better way is to represent the behavior of
nonlinear elements by their instantaneous stiffness and damping. This type of modeling
provides standard and unified procedure for the different types of nonlinear elements.
The sliding element model developed here has the capability to represent triaxial
behavior of a sliding isolator which includes displacements in the global X and Z direction
and the influence of the variation of vertical force on the sliding force. The friction sliding
isolators have two stages, slip and stick. In the stick stage the element has elastic stiffness,
Kinitia/' After the transition to sliding (slip) stage, at the first sliding (slip) stage, the coeffi
cient of friction is function of velocity (Fig. 3.14 (c). When the velocity exceeds the veloc
ity limit, the coefficient of friction is constant and equal to I1max' The data required for
this model is: P - the initial normal static force on the sliding surface without the contribu
tion of dynamic forces, Kinitiar the sticking stiffness (no sliding), Ksecondary the stiffness of
the recentering spring during the slip mode, VUmit - velocity limit above which the friction
coefficient becomes constant and equal to I1max' and, I1min' I1max' I1static - the minimum,
maximum and static coefficients of friction, as described in Fig. 3.14 - (b).
The lateral force in the isolator, F(t), is represented by incremental formulation as
a combination of three components (i) a linear rise; (ii) a viscous component; and (iii) a
"Coulomb" friction component as defined by the following rules:
75
~ ~ ~ F(t) = F(t-Ilt) +AF(t) (EQ 3.70)
where
(EQ 3.71)
where A indicates the increment from time t - Ilt to time t, N is the variable normal force
)
to the sliding surface, and i is the sliding direction vector
(EQ 3.72)
The parameters I coefficients in EQ. 3.71 take the following values:
(a) Forj= 1 when IF(t) I ~!1b_awayN(t)
(EQ 3.73)
where kois the initial stiffness and !1b_ awayN (t) is the breakaway friction coefficient.
(b) For j=2, after sliding occurred, when
!1minN (t) ~ IF (t) I ~ !1maxN (t) and Iv (t) I ~ Vlirnit (EQ 3.74)
(EQ3.75)
where the equivalent damping coefficient is:
(EQ3.76)
Note that !1max and !1min are the maximum and the minimum friction coefficient
respectively, and UUmit is a constant depending on friction properties which can be
obtained from the inversion of the exponent a in the model by Mokha et al. (1990). This
76
coefficient (in velocity units) can be taken as 2-4 in.ls. (Tsopelas et al. 1994). It should be
noted that N(t) is time variable, therefore, Ceq is also time variable.
(c) For j = 3 the maximum sliding force achieved ClFtl = llmaxN (t) ) and
Iv (t) I> Vlimit (EQ 3.77)
(EQ 3.78)
It should be noted that coefficient Sj is an indicator for biaxial effects in the sliding
mode. When the isolator's force, IF(t)I, drops below llminN (t) the system is transferred to
the first stage U= 1).
Using the above formulation, the model can be integrated into regular numerical
formulations without loss of generality. The vectorial formulation indicated above for the
directional movement in the plane of the isolator can be conveniently transformed back to
a cartesian formulation for simplicity of usage.
Test results obtained from an experiment performed on teflon sliding bearings,
subjected to biaxial motion, with high and low velocity (Mokha et aI. 1993), were used to
verify the capacity of the analytical model to reproduce experimental results.
The properties of the isolator used for the analysis are:
p= 95.98kN, llmax = 0.13, llmin = 0.04, vlimit = 100~. sec
The input displacements are defined by the function (see Fig. 3.15):
77
u = umaxsincot x x
u = umax sin2rot y y
The frequency co for the case with high velocity was taken as 2.22 rad/sec, and for
the case of low velocity 0.5 rad/sec.
The analytical modeling was able to predict well (Fig. 3.16) the experimental
results (Fig. 3.17). It should be noted though that the input motion used for the experi-
ment did not follow exactly the input motion as described by the functions above because
of the influence of the testing equipment. This caused some discrepancy between the ana-
Iytical and experimental results in the behavior in the initial stage (breakaway stage) in
which the actual velocity from the experiment is lower than the velocity obtained theoreti-
cally. The general shape of the experimental results is also distorted, because of the influ-
ence of the testing equipment.
78
50
30
- 10 E E ........ >-
::> -10
-30
-30 -10 10 30 50 Ux(mm)
FIGURE 3.1S. Graph of Theoretical Input Motion of 8 Shaped Test
79
16
12
8 /' " ./ " / I""--.. rngl 'l. .,.., '\ ..-
..-. 4 Z ...It: 0 -X
/. "" Ie wvelo ity "- ~'\ y "\ "- A
LL. -4 \. "- ./ /
-8 \. I--- .......... / "- /' "'- ./
-12
-16 -50 -30 -10 10 30 50
Ux [mm]
16
12 hi~h elocit
8 " ........
L , ..-. 4 Z
:..,....- wn -.... '/ ,
...It: 0 ->. LL.
-4 ;-.... .....
-8 J , 1/ -12 I'-.. ,/
-16 -50 -30 -10 10 30 50
Uy [mm]
FIGURE 3.16. Analytical Loops of Frictional Force and Displacements in 8 Shaped Motion Test
80
- .05 ·4 Test
3
2 ,..... 1 VI
Co :i 0 ........ x -1
lL.
-2
-3
-2
4
3
2
,..... 1 VI Co :i 0 ........
(,05
Test
·
· V'
~ ·
No.
-1
-0~25
No. 3~ 6\
>--1 lL. ·
-2 ~. t--3 \.
-4 - 2 -1
,
Ux (m) 0.0,25
•
Ux (inch)
Uy (m) ?5 0.0,..
1
I
w
0 i Uy (inch)
0.05·
15
10
5
3 0
-5
-10
. -15
0.0 5
15 l-
10
5
• o
-5
-10
-15
,..... Z ~ ........ x
lL.
FIGURE 3.17. Recorded Loops or Frictional Force and Displacements In S·Sbaped Mollon Tests
81
3.2.5 Three Dimensional Spring. Connection Element Model
An important basic element is the 3D spring, which can have linear or nonlinear
properties in every direction. This element can be viewed as shown in Fig. 3.18 as six
separate spring subelements corresponding to the six global D.O.F's.
y
z
joint 2 ky
+
joint 1
x
kx
+
kz
+ rotational stiffness lax, kry, krz
FIGURE 3.18. Spring Element Representation
82
The stiffness matrix k of this element has the form:
kx 0 0 0 0 0 -k x 0 0 0 0 0
ky 0 0 0 0 0 -k y 0 0 0 0
kz 0 0 0 0 0 -k z 0 0 0
krx 0 0 0 0 o -kr x 0 0
kry 0 0 0 0 0 -kr y 0
krz 0 0 0 0 0 -kr k= z (EQ 3.79)
kx 0 0 0 0 0
ky 0 0 0 0
SYM. kz 0 0 0
krx 0 0
kry 0
krz
The elements of the matrix k can be linear or nonlinear(k (u (t)) ). In the case
when interaction between the springs exists it can be added directly to the elements stiff-
ness matrix in the proper locations (EQ. 3.79 is revised and the proper stiffness terms are
replacing the 0 values).
It should be noted that there is certain similarity between the end springs formula-
tion presented earlier, and this element. The differences between the two cases are: (i) The
end spring can be connected only to a single element, while the spring element can be con-
nected to a joint in which several elements are connected. (ii) for postprocesing purpose,
in the locations where the displacements must be monitored it is easier to use the connec-
tion spring element and directly obtain displacements since there is an actual joint at that
location. This is especially important when those springs are connected to the ground,
since the relative displacements between the bridge and the ground are of interest. (iii)
83
when mass or force is connected to the joint between the. "stick" elements and the springs
(mass of the ground), the end spring formulation cannot be used since it assumes that there
are no forces at this location.
3.2.6 Modeling of Damping in Bridges
The previously described structural elements are contributing to the global stiff-
ness matrix k. In this section global modeling of structural damping, and modeling of indi-
vidual damping elements is described.
3.2.6.1 Global Damping
The global damping matrix is used to describe the inherent damping in the struc-
tural system. Since it is difficult to quantify the damping in specific locations, it is consid-
ered a property of the structure as a whole. Usually it is defined as a function of the
frequency of the structure. A common way to model damping is by using Rayleigh pro-
portional damping (Clough &Penzien 1993)
(EQ3.80)
Where M is the global mass matrix, k is the global stiffness matrix, u and ~ are
determined so that specific damping ratios are obtained for two selected frequencies, say
those of modes 1 and 2 co 1 and co2 :
1; - uM+ ~k 1 - 2Mco
1
1; - uM+ ~k 2 - 2Mco
2
(EQ3.81)
In the case of inelastic structure, the values of k and co are changing with time.
Several options are available to update the damping matrix (e.g. Fajfar et. el 1994): 1)
84
Keep the initial damping matrix constant for the whole time of analysis 2) Update the
damping matrix such that C = a.M + ~k/ (t) , where k/ is the tangential instantaneous
stiffness. 3) Update the damping matrix considering both the change of stiffness and the
change of frequency, in time
(EQ 3.82)
In the case were elements with individual damping properties such as viscous
dampers are considered, they contribute to the damping matrix in the same manner as the
individual stiffness elements are contributing to the global stiffness matrix. The individ-
ual damping elements has damping matrix of the size l2x12 with 6 D.O.F's at each side of
the element.
3.2.6.2 Linear Damping Element
The basic damping element as mentioned earlier has a l2x12 element damping
matrix. The element represents damping in all degrees of freedom as shown in Fig. 3.19.
joint 2
+
joint 1
~z + rotational damping
crx, cry, crz
FIGURE 3.19. Damper Element Representation
85
And the element damping matrix c has the form
Cx 0 0 0 0 0 -c 0 x 0 0 0 0
cy 0 0 0 0 0 -c y 0 0 0 0
Cz 0 0 0 0 0 -c z 0 0 0
crx 0 0 0 0 0 -crx 0 0
cry 0 0 0 0 0 -cr y 0
crz 0 0 0 0 0 -crz c = (EQ 3.83) Cx 0 0 0 0 0
cy 0 0 0 0
SYM. CZ 0 0 0
crx 0 0
cry 0
crz
3.2.6.3 Nonlinear Damping
The behavior of each of the subelements in the damper element can be nonlinear as
well as linear. The c value can vary with time such that c = c (Ii (t) , u (t) ). In this case
the global damping matrix of the all structure needs to be updated in time.
3.2.6.4 Damping Devices
An example of the use of nonlinear damping is certain viscous dampers (Reinhom
et. al. 1995). The force in a fluid damper is a result of flow through orifices leading to a
pressure differential across the piston head. Most of the practical devices are built using
differential shaped orifices in which the pressure differential is depending on a fractional
power of velocity:
(EQ 3.84)
86
------ -_ .. _-
where sgn (Ii) indicated the sign of velocity Ii and IX is a power between 0.5 to
2.0. A lower power is used for high velocity shocks.
This model can be linearized instantaneously by differentiating EQ. 3.84
(EQ 3.85)
which gives the equivalent instantaneous damping coefficient
It should be noted that the values of the instantaneous damping in EQ. 3.85 are:
dF dF e(t) -. ~ 00 for IX < 1.0 and e(t) -d. ~ 0 for IX;:: 1.0
duu .... o uu .... o
3.2.7 Nonlinear, Yielding in Tension, Stiffening in Compression Element
Fig. 3.20 shows a typical expansion joint detail with a restrainer and a gap in
extension of the size of tgap and gap in compression of the size egap. It may be assumed
that during travel form -egap to +tgap no force is transmitted between the deck and the
abutment. When the compression gap closes, the joint exhibits stiffening characteristics
which can be approximated as bilinear stiffening elastic behavior. This is depicted in Fig.
3.21. In tension, the behavior is more complex: (a) the behavior is hysteretic with soften-
ing characteristics, as approximately depicted in Fig. 3.21 and (b) the tension gap
increases with each inelastic extension from tgap to tgap' and so on, as depicted in Fig.
3.21.
87
Fig. 3.21 depicts an approximate force-displacement relation for an expansion
joint. A one directional element has been developed with these characteristics to represent
the behavior of the tributary section of an expansion joint. The behavior of the whole
expansion joint can be defined by a combination of several elements of this kind.
soft material
Abutment
Beam
Restrainer
Bearing
FIGURE 3.20. Typical Expansion Joint
F
FIGURE 3.21. Force Displacements Relations of Expansion Joint
88
3.3 Modeling of Bridge Subsystems
In the previous section the mathematical modeling of the basic structural elements
required for performing analysis of a bridge was presented. In the current section the mod
eling of the bridge components described in section 3.1.1 , using the basic structural ele
ments is described.
3.3.1 Modeling of Foundation and Soil System
The foundation system is the medium that connects the bridge to the ground. In
general, the bridge and the ground form a single structural system, and therefore it should
be analyzed as a single unit. Realistic limitations does not allow integral modeling and
analysis. Therefore each one of the systems is modeled and analyzed separately with sub
sequent interaction defined as soil - structure interaction. The methods used to analyze the
behavior of the ground by itself are: finite elements, boundary elements and several other
approximation methods. The use of the finite elements method for analyzing large volume
like the one of the soil, requires descritization of large portion of the soil to avoid creating
refraction of the propagating waves from the artificial boundaries of the model.
The use of boundary elements is much more appropriate for analyzing infinite
diamines, such as the ground medium Benerjee (1994). The boundary elements method
however has also significant drawbacks in approaching the problem: the problem of 3D
structural dynamics cannot be solved, since the required Green function to solve this kind
of problem is not readily available. Furthermore, this method can be viewed as a linear
superposition method, and therefore it cannot be used straightforward for nonlinear analy
sis, as required for inelastic dynamic analysis of bridges
89
3.3.1.1 Elastic Linear Models for Soil Structure Interaction
The simplest models used for analyzing soil structure interaction are the once
using equivalent linear springs and dampers to model surface or embedded foundations.
According to this method the soil behavior is represented by a spring and a dashpot. Their
properties are depending on the size of the foundation interface and the soil properties.
However simple, these model was found to be very suitable to represent the behavior of
the foundation interface as compare to more complicated models (Cofer et al. 1994).
3.3.1.2 Frequency Dependant Modeling
A more advanced method for analyzing soil structure interaction with dynamic
effects, considers the soil medium as a combination of equivalent springs and dampers in
which their main properties are function of frequency. The soil is modeled therefore, as a
combination of spring and damper elements, described earlier. These elements however
can be function of frequency such that k = k ( 0) and c = c ( 0) , where 0) is the instan
taneous frequency of the nonlinear system. If the frequencies of the bridge with changing
stiffness are calculated at each time step of analysis, then the stiffness and the damping
properties of soil-foundation medium can be updated at each time step too. A more effi
cient computation scheme however provides such an adjustments only when substantial
changes occur in the stiffness properties.
3.3.2 Abutment Modeling
The abutment is usually made of three walls as shown in Fig. 3.22, two wing walls
and one central load bearing wall, on which the deck is supported. The soil is supported
90
laterally by all the walls and apply passive pressure on them. A major roll is played by the
abutment in supporting lateral loads which are applied to the bridge during seismic events.
bearing wall
soil spring
FIGURE 3.22. Typical Arrangement of Bridge Abutment. The Soil Stiffness Modeled By Springs.
The abutment can be modeled in two ways: the first one is more rigorous, in which
the soil is represented by several spring and damper elements, and the walls are modeled
by several "wall" elements. The other way is to lamp the all abutment system into few
springs and dampers, similarly to the way the foundations system is modeled.
3.3.3 Bents modeling
Since the deck of a bridge is more critical and expensive to repair in seismic
design, permanent damage due to seismic loading is allowed only in piers and bents.
91
"".j1..-The bents are the parts of the bridge which ls..the most susceptible to damage dur-
ing earthquakes. The bents and piers are designed so that they will experience damage but
will not collapse when subjected to seismic loading. Therefore the behavior of the bent
can be modeled by one or more of elements of the type of hysteretic beam I column. In
short piers and walls, the failure mechanism is dominated by shear mechanism. Therefore
the combined flexture shear model (see flextural-shear model in section 3.2.2.2) should be
used for modeling of these elements. In frame type bents, the top beam is also modeled as
an inelastic concrete element since it is also a part of the energy dissipation mechanism.
3.3.4 Deck modeling
The deck is usually made of multiple beams supporting a plate or of box girders.
The deck can be modeled using very fine finite elements. From past experience however it
appears that this approach is very expansive computationaly, especially for nonlinear anal-
ysis. Approximation methods including condensed macro-modeling can be suitable for
this task even if some uncertainties are involved in deriving the correct properties. There-
fore approximation methods are preferred for modeling of the deck. Two approximation
methods that can be used for this purpose are suggested.
According to the first modeling method the deck is represented by a single beam in
the centroid ofthe deck. Thus the structural properties of the deck are collapsed to it's cen-
terline. For this purpose the basic elastic beam-column element can be used. Since in
many cases the ratio between the length and width of the deck is smaller than 4, shear
deformations in the horizontal plane should be considered. The deck is usually supported
by bearings at its ends. Since the deck can be assumed to be rigid at its ends around the
vertical axis, the connection between the deck which is modeled as an elastic beam and
92
the bearings can be done using suitable rigid body transformations. Those rigid body
transformations can be performed using the coupled motions method (explained in the
next section). An example of transformation using coupled motions is presented in
Fig. 3.23 ,where the top joints of the bearings Goints I to 4) are coupled to the single joint
at the end of the deck (joint 5) using the coupling equations:
Joints 2 and 3 are coupled to joint 5 in the same way.
The other method of modeling the deck behavior for the case where irregular set
ting of the deck exist is shown as an example of the use of the coupled motions in section
3.4.
93
bearings
deck edge
(a) Deck Section Supported on Bearings
2
joints on the ~ __ ... 3 "\
top of bearings \ c; rx z 4 deck end
joint deck modeled by stick elements
(b) Modeling of Deck and Bearings
_y-_ kJdzl . dS-1 : , , -,-1- • - • .-=-:--:-::--:-:-_C":'"_ .... _ --...--:...--:...:-~- • - - - - - - - •
d • a.." 9 5-4 ,,----
---'-- ~'"" :dx4
(c) Displaced Joint
FIGURE 3.23. Modeling of Connection Between Deck and Bearings
94
3.3.5 Expansion Joints and Special Connections
bearings
FIGURE 3.24. Typical Expansion Joints in Bridge
A typical expansion joint on a bent is shown in Fig. 3.24. The left deck section is
supported directly by the bent, the right section, is supported by the bent through bearings.
The two sections of the deck can be connected together by restrainers (see Fig. 3.25), that
prevent the right section of the deck from dropping from it's support, while moving in the
longitudinal direction
9S
-H-~----~ r--~------------~
restrainer
bearing
(a) Expansion Joint Cross Section
F tension in restrainer
u
compression in a concrete
(b) Force Displacements Relationships
FIGURE 3.25. Expansion Joint Modeling
96
A possible model of an expansion joint is shown in Fig. 3.25. In principle the joint
should be modeled using several elements. Since the deck edge can be assumed to move
as a rigid body around the vertical axis, all the nodes along the joint can be represented by
two nodes only. The bearings and restrainers, can be transformed from their actualloca-
tion to the two nodes at the centerline of the joint using rigid body transformation, by
defining this elements with rigid arms.
3.4 Solution Procedures for Coupled motions
The efficiency of the solution procedures can be substantially improved if degrees
of freedom representing motions / movements of rigid bodies in the structures are coupled
by rigid'links. Such cases are customary in bridges with decks which can be considered
rigid in-plane. The rigid links provide large stiffness terms into the structure global stiff-
ness matrix as compare to the other more flexible elements of the structure, leading to
severely ill-conditioned matrices. Moreover, many degrees of freedom can be eliminated
using appropriate geometrical relations. The solution procedure suggested herein is auto-
matically processing and reducing the number of active degrees of freedom and the stiff-
ness matrices for effective processing. This is done using an optimal scheme as outlined
below (Bathe 1982).
The conditions on the relative motions are specified using the variational formula-
tion of the structural system, by introducing Lagrange multipliers, which include the con-
strains provided by the coupled motions:
(EQ3.86)
97
Where II is the energy, U is the displacements vector, K is the stiffness matrix, R
is the load vector, and A is. the Lagrange multiplier; c and c' are coefficients of the con-
straint equations provided by the rigid links (see example).
Taking the variation of the expression in Eq. 3.86 and equating it to zero leads to:
(EQ3.87)
Since the terms associated with 0 are arbitrary, will get a set of linear equations
KU-R = 0 (EQ 3.88)
cU +c' = 0 (EQ3.89)
Introducing EQ. 3.89 into EQ. 3.88 and taking c' as zero the number of equations
that needs to solves will be reduced from the initial number of Degrees Of Freedom
(D.O.F) to reduced number, equal to the original number of D.O.F's minus the number of
coupled equations. The terms in the structure global stiffness matrix will change such that
where Ci is an index on all the D.O.F's coupled to the i'th dof, Cj is an index on all
the D.O.F's coupled to the j'th dof. While the coupled (slaved) D.O.F's are eliminated
from the stiffness matrix. The terms in the global coupled load vector are defined with
respect to the original load vector as:
Rcoupled - R - ~ c R i - i "'-' iLL (EQ 3.91)
L= Ci
98
Example:
rigid deck modeling
A typical case rigid deck is used to explain the procedure of coupling.
The deck section in Fig. 3.26 is rigid in plane. The behavior of this deck section
can be represented by three D.O.F's only, i.e. displacements in two directions and rotation.
The procedure for performing the transformation using coupled motion is presented
below.
(a) Define the constrained conditions (coupling):
J4 Jl
FIGURE 3.26. Rigid Floor Displacements
Since the rotation of the whole deck is identical for the all joints, the coupling read:
(EQ 3.92)
(EQ 3.93)
99
(EQ 3.94)
The first tenn in each equation is the coupled tenn, and the second is the master
term. For example in Eq. 3.92 the master D.O.F is 3 and the coupled D.O.F is 6. The cou
pled equation related to Eq. 3.92 can be written as:
(EQ 3.95)
from which the coefficient c3-6 =-1, (see Eq. 3.90 Eq. 3.91 ). Rewriting the next two cou
pling equations gives c3-9 =-1 and c3_12 =-1
Defining dXij as the distance in the x direction between joint i and j and similarly
dYij as the same distance in the y direction, the additional relations between the coupled
displacements are defined as:
U4 = UI + dY2_I U3
Us = U2 +dx2 _ IU3
U7 = UI + dY3_I U3
U8 = U2 + dX3 _ 1 U3
UIO = UI + dY4_1 U3
Ull = U2 +dY4_I U3
(EQ 3.96)
(EQ 3.97)
(EQ 3.98)
(EQ 3.99)
(EQ 3.100)
(EQ 3.101)
In the same manner by rewriting the equations into the form of Eq. 3.95 , the coef
ficients for the other variables are derived as:
Cl_4 =-1, c3_4 =-dY2_I, c2_5 =-1, c3_5 =-dY2_1 etc.
lOO
Flexible Deck modeling
In order to capture the nonlinear (hysteretic) behavior of the deck, it is assumed
that the deck is composed of several parallel hysteretic beams, assuming that the deforma-
tions in the vertical and horizontal direction are related through shear interaction.
A deformed section of such a bridge is shown in Fig. 3.27 . Assuming that the deck
cross section remains plan and vertical to the beam axis (Bernoulli assumption) the defor-
mations in the horizontal X direction of all the beams (bj)can be coupled to a single beam
bi using the following equations:
Ubi = u bj+6 B .. x x y IJ (EQ 3.102)
where the index bi indicate that the deformations of the i'th beam are coupled to
the displacements of the j'th beam and Bij is the distance between beam i and beam j.
Assuming that the in-plane stiffness of the deck is infinite the following coupling equation
can be written:
Ubi = ubj z z (EQ 3.103)
Using the same approach for the vertical direction, the coupling between the
degrees of freedom in the global Y direction is:
Ubi = u bj +6 B .. y y X I)
(EQ3.104)
Where u;i are the deformations in the vertical direction of the i'th beam.
Since the height of the deck is substantial, the distance between the theoretical
beam center line and the actual support should be considered when modeling the connec-
tion between the deck and bearings, which are placed on the top beam of a bent. To con-
101
sider the offset, the beams can be modeled with offset connections (rigid arms).
Furthermore, in order to consider flexibility in the support connections, beam end springs
can be added.
r-I
z
x
1 •
!.:..................................... b:.~ __ ~~
FIGURE 3.27. Deck Section Modeled by Several Beams
102
3.5 Evaluation of Bridges via Approximated Methods
The evaluation of a bridge implies determining the internal strain resultant in the
bridge elements as well as the relative deformations in the elements and of the entire
bridge in case of load disturbance. In the case of earthquakes when the system responds
inelastically, the internal force distribution influences the deformations and accelerations
implicitly depending on the loading path. It is therefore impossible to determine accu
rately the force (strain resultants) and deformation distribution without a time history anal
ysis for a specific earthquake.
However, results of approximated analyses (Reinhorn et a1.1996, Valles 1996)
show that the maximum response has the tendency to fit more simplified, independently
obtained functions. One representing the envelope of force-deformations obtained by
monotonic loading and the second by determining the expected maximum response for a
specific or a group of ground motions:
(a) Monotonic structure's capacity will be defined as the above mentioned force
deformation envelope independently determined.
(b) Seismic demand will be defined as the maximum expected response for a
ground motion or a group of ground motions for a structural model similar to
the one describing the structural system for evaluation. This is defined using
the seismic response spectrum.
A plot of the structural capacity, [see Fig. 3.28 (a)] indicating possible loading
paths, can be combined with a composite spectrum (Reinhorn et. al 1996) [see Fig. 3.28
(b)] to define the response at the intersection of the curves [see Fig. 3.28 (c)].
103
F/w
. •
... elastic
inelastic
(a) capacity d
Salg
Sd (b )demand spectrum
elastic response
Salg=F/w
(c) response
FIGURE 3.28. Evaluation of Inelastic Response Using Composite Spectrum
Sd=d
It should be noted that an inelastic system can be defined as independent force-
deformation description versus the spectral seismic demand.
The current section attempts to show various techniques to determine the response
by determining the capacity and the seismic demand separately, as well as it explains more
rigorous inelastic analysis to verify this evaluation approach.
It should be noted that a good evaluation can be obtained if the monotonic "capac-
ity curve" is determined for increasing inertial forces best simulating those that might
develop in a single large cycle during an earthquake. At the same time the calculation can
be obtained by proper evaluation of equivalent damping influences or actual inelastic
response in the preparation of the demand spectrum.
104
3.5.1 Methods for Determining Seismic Demand
The seismic demand can be defined in terms of the expected forces and deforma
tions which will develop in the structural system due to seismic event. This can be done by
determining the response specific for a single degrees of freedom (s.d.o.f) or using pre
defined code averaged data for such spectra. However, the force-deformation demands in
m.d.oJ's require farther evaluation as will be outlined below. Several methods currently
used are outlined for sake of completness.
3.5.1.1 Modal (Spectral) Analysis by Seismic Codes
Current seismic codes including AASHTO suggest the use of modal analysis pro
cedure, which transfers the structural problem from the physical space domain to the
domain defined by the eigenvectors of the homogeneous equation of motion. In the fol
lowing a review of the method is presented along with the presentation of efficient compu
tational method to determine the dynamic properties.
The homogeneous dynamic equation of structural system is
mii+ku = 0 (EQ 3.105)
By using the customary transformation u can be transformed to it's eigenvector's
domain such that
u = :E<PPi (EQ 3.106)
and EQ. 3.105 can be written for each mode i using its eigen vector <Pi as
(EQ 3.107)
The eigenvectors of this equations are derived as the roots of the polynomial
105
(EQ 3.108)
The eigenvectors can be derived iteratively from the equation
(EQ 3.109)
In order to solve this problem efficiently it is suggested to use the Lanczos method.
A matrix X comprised of the m orthonormal vectors (xj>x2,x3 ... xn) is constructed
(Bathe 1982). This matrix satisfies the condition:
(EQ 3.110)
where Tn is a tridiagonal matrix of the size of n. The basic problem (EQ. 3.109 ) can be
rewritten as
(EQ 3.111)
The transformation
$ = X$ (EQ 3.112)
is introduced into EQ. 3.111 and premultiplying by XTwill yield
- 1 -T$ =-$
n co2 (EQ 3.113)
The eigenValues of Tn can be found using one of the iterative techniques. The
eigenvalues of Tn are the reciprocal of the eigenValues of the original problem and the
eigenvectors can be derived using EQ. 3.112
106
The big advantage of using this method is that finding the roots of the polynomial
of tridiagonal system which is significantly reduced is much easier than working on much
larger system. This method is using iterations to check convergence.
The model of the structural system of the bridge is assembled from individual
matrices of elements to get the system matrices k, c, m, and the load vector F.
The eigenvectors and eigenvalues of the bridge are calculated using the aforemen
tioned procedure. It is assumed that the bridge can be modeled by a limited number of
modes. The site response spectrum is used to derive the equivalent forces on the bridge
using the expected acceleration profile to derive the distribution of the inertial forces on
the bridge:
(EQ 3.114)
where Fj is the force at thej D.O.F for mode i, Mj the mass at thej D.O.F, <Pij is the
ordinate at the j D.O.F of the i'th mode, and rSj the response spectrum value related to the
j eigenvalue (r represents the modal contribution from modal participation factor).
Using this force profile an equivalent static approach can be used to determine the
internal forces in the elements and deformations. The deformations can be also determined
directly from displacements spectra if available. If only few of the mode shapes need to be
considered, the results from the static analysis for the various modes can be combined
together using SRSS rule.
Fi = JU 5
ui = Jru5
(EQ 3.115)
(EQ 3.116)
107
where Fj is the end force in element j F ij is the j element end force due to the response in
the i'th mode. The indices for the displacements u are defined similarly to the once for the
forces.
It should be noted that this method can be used also for inelastic response evalua-
tion if proper spectra are used (Reinhorn et al. 1996)
3.5.1.2 Static Equivalent Procedure
The simplest way to evaluate the demand from a bridge, is by representing the
dynamic behavior of the bridge (explained in the previous section) by equivalent static
one. This approach is widely used by the design codes (UBC 94, AASHTO, etc.).
This procedure is based on solving the static equilibrium equations of the bridge in
which the forces f are determined however using simplified empirical formulation to dis-
tribute the total force. The distribution of the forces is obtained according to the force fi
relative to mass Mi, and to the expected total force I.F j as defined in design codes:
(EQ 3.117)
solution of the static linear and nonlinear equilibrium equation
The complex system of a bridge can be analyzed also incrementally to determine
it's nonlinear response. The equilibrium equation of the structural system is
ku = f (EQ 3.118)
The matrix k can be separated into four submatrices related to the free and con-
strained D. O.F' s
108
(EQ3.1J9)
In the case where Uc equals to 0, the problem can be viewed as force input prob-
lem. The free deformations ufcan be derived by solving the equation
(EQ3.120)
In the case of nonlinear analysis, this procedure is performed incrementally, and
the instantaneous equilibrium equation is
(EQ 3.121)
where kif is the tangential stiffness matrix, l1uf the incremental displacements of the free
D.O.F.'s, and Aft the incremental load.
The reactions on the restrained D.O.F's can be calculated from the equation
(EQ 3.122)
The same procedure can be used to apply displacements input in which case the
equilibrium equation has the form
(EQ 3.123)
3.5.2 Time History Analysis
Time history analysis provides the most accurate information about the response of
bridges to seismic loading. If such loading is well known, structural parameters well
defined and the model is complete. Two procedures can be used to perform time history
analysis (Clough and Penzin 1993) i.e. (i) mode superposition and (ii) direct integration.
109
The mode superposition procedure even though much more efficient numerically can be
used only for linear analysis. Differential support excitation cannot be considered ade
quately by this method. The direct integration procedure is the one that is recommended in
the current approach due to its versatility to handle elastic and inelastic structures.
3.5.2.1 Bridges Under Seismic Excitation ·Uniform and Differential Excitations
Bridges span over large distances, and therefore the ground deformations, acceler
ations and velocities at one support might be significantly different than at other supports,
in magnitude and phase due to several reasons: (i) When the bridge is located over a fault
the excitation on one side of the fault differs significantly from the other side. (ii) Different
soil types amplify differently the magnitude of traveling shear waves, and cause excitation
magnitude difference between different foundations. (iii) The difference in arrival time of
the shear waves to largely distanced footings causes phase difference between the ground
excitations.
All this affects the response of the bridge both by changing the dynamic excitation
of the bridge, and by applying quasistatic relative deformations which result in quasistatic
forces.
The solution procedures for both, uniform support excitation, and differential sup
port excitation are presented in the following two sections.
3.5.2.1.1 Uniform I Rigid Base Motion Analysis
The governing equation for nonlinear time history analysis can be expressed in
incremental form, as:
uo
Mdu + Cdu + Kdu = dF for ti ::; t::; ti +!:J.t (EQ 3.124)
in which the nonlinear properties C (u, u) and K (u, u) are assumed constant during the
time step interval. The excitation which is a uniform ground motion, in which the acceler-
ation in all the supports is the same:
dF = -Mdu g
where dUg is the uniform ground acceleration increment.
(EQ3.125)
This equation is solved using an implicit scheme based on the Newmark ~ method
(Bathe 1982).
The acceleration at the time t +!!.t is
du - utdt - dt2 (112 - a) ut ut +dt = ---=---d-:t2:-a------.:
The velocity at the same time is
.. [ .. dU-utdt-dt2 (1I2-ex)Ut] ut + dt = ut + dt (I - Ii) ut + /) 2
dt ex
and the velocity and acceleration increments are:
du = (1-/))Ut+~[~~-u,- (1I2-ex)U ldt]
du ut UI (112 - ex) .. du = -=.:,:.:;,---- -u
(dt) 2ex adt ex I
(EQ3.126)
(EQ 3.127)
(EQ 3.128)
(EQ 3.129)
Introducing EQ. 3.128 and EQ. 3.129 into EQ. 3.124 ,will give the numerical
incremental equation of motion.
111
- -kdu = F (EQ 3.130)
where
- 1 ° k=M +C-+K (dt) 2ex. ex.dt
(EQ3.131)
= + u - + u - + u + - - - ex. u + -u F- FM[,,1 .IJC[(O-I)" 0(1 )"O'J t2ex. t dtex. dt ex.dt 2 ex.
3.5.2.1.2 Differential Ground Motion Analysis
Differential ground motion cannot be applied through acceleration histories. Such
motions can be expressed in terms of displacements and velocity input.
The global equation of motion in matrix form, for the acceleration input is:
(EQ 3.132)
in which mf efand kf are the super-structure mass, damping and stiffness matrices respec-
tively. Those matrices are related only to the free degrees of freedom. The relative dis-
placements between the structure and the ground are denoted as u.
The equation can be written explicitly with respect to the total displacements ut
such that:
CEQ 3.133)
112
where the sub matrices with the c index corresponds to the restrained degrees of freedom
or the degrees of freedom where ground displacements are applied. The first block of
equations in Eq. 3.133 represents the behavior of the superstructure and can be written as:
(EQ 3.134)
where the acceleration, velocity and displacements marked f are the total displacements
of the structure, while the displacements and velocity marked g are those related to the
ground. The solution of these equations is similar to the solution for the uniform ground
motion with the proper adjustments for the forcing vector, i.e. the right hand side of EQ.
3.134.
3.5.2.1.3 Verification procedure
In order to verify the analysis procedure with displacements excitation, a compari
son between deformations resulting from analysis with acceleration input motion and
analysis with displacements input motion is performed. The verification model shown in
Fig. 3.29 is analyzed using displacement input shown in Fig. 3.30 -(b) and acceleration
input shown in Fig. 3.30 -(a). The displacements results extracted from the base displace
ment analysis are the total displacements ufin Fig. 3.30 (b) which includes the ground dis
placements. The results from the acceleration input analysis are the relative deformations
(u in Fig. 3.30 (a)). To compare the relative displacements u for both cases, the ground dis
placements are subtracted from the total displacements to get the relative displacements
between the base and the structure (Fig. 3.30 - b). A comparison of the results is shown in
Fig. 3.30 . It is apparent that the relative displacements are the same. However it should be
noted that the acceleration input does not require any restriction on the initial conditions of
113
\.
the ground motion. However with displacements input the initial conditions should be
properly adjusted so that the first and the second derivative will equal to zero.
-it t+ , , , I I I I I I I I
I I I I I I I I
I I I
(a) ~ Ug
(b)
FIGURE 3.29. Verification Frame Model
114
(a) displacement input
Input displacement 20 r----====--'-----,
11O C ~ 0
~ !f·10 '6
(b) acceleration input
Input acceleration ~r---~~=~---_,
I I g 1"500
·201r-------,~,------_,\, • ·'00c0'n. -------,-:-....,.-----,,\,0 time (sec] time [sec]
20r-___ ~ro~m~l~d~is~l~ac~e~m~en~t ___ _,
I'0 w ~ 0
~ !f·10 is
·20 ir------:---:--:----__,\, time [sec]
20r-__ ;rr~e~lm~lv~e~d~lso~lla~c~em~en~t ___ -,
I'o f c ~ 0t\! ;-10 I'6
·201r--.L.---":~_...lL. __ .,\, time [sec]
20r-__ ;rr~el~at~lv~e~d~IISSO~lla~~~m~en~t ___ -,
I 10 ~
.201r--.L.---:--":~-...lL.---,b time [sec]
FIGURE 3.30. Frame Response to Displacement and Acceleration Input.
\
115
3.5.3 Numerical Solution for Bridge Systems
The solution of the system equations
kx =/ (EQ3.135)
is obtained by solving the reduced system:
- -kx = / (EQ 3.136)
- -where k is the reduced stiffness matrix considering the coupled or slaved motions, and /
is the reduced force vector.
3.5.3.1 Identification of Special Issues in Solution Procedures
(1) Usually the stiffness matrix k is a banded matrix, for efficient solution of the
system of equations it is stored as a smaller rectangular matrix, with dimensions dictated
by the band-width and the length of the diagonal. This type of storage is not efficient in
many cases. In order to improve this storage, most of the commonly used solution proce-
dures perform reordering of elements to achieve smaller band width. This issue might
especially be significant when coupled motions are used, because the band width might
increase in the presence of coupled motions, since the D.O.F's that didn't have interaction
terms initially, will be interacting after the modification related to coupled motions.
The global solution procedure used for solving EQ. 3.136 is usually the Gaus elim-
ination or frontal solver. These solvers usually have problems of round off error when the
stuffiness matrix has terms with drastically different values. This might happen when the
structure has stiffness discontinueties, in which soft and stiff elements are present in the
same structural bridge system.
116
In order to reduce the difficulties mentioned above, a sparse matrix storage com-
bined with a solution using the Conjugate Gradient Iterative Solver are used. In the fol-
lowing sections detailed description of solution procedures is presented.
3.5.3.2 Sparse Matrix Formulations of Structures Equations
Iterative solvers are based on vector matrix multiplication. This can be done very
efficiently by using sparse format storage of the stiffness matrix. The sparse format used
here, is based on a package developed at Yale University (Kincaid et al. 1990).
A symmetric matrix K can be represented by three vectors A, JA and IA. All the
nonzero elements are stored in the vector A. JAG) contains the column number at which
the term AU) is located in the K matrix. The IA vector denotes the elements serial number
on the diagonal, and used to define the number of nonzero elements at each row. JA has
the size of number of nonzero elements, and IA has the size of number of rows + 1.
An example of the use of Yale sparse format is shown below
The matrix K is
and the equivalent vectors are:
11. O. O. 14. 15.
O. 22. O. O. O. O. O. 33. O. O. 14. O. O. 44. O.
15. O. O. O. 55.
A = [11. 14. 15.22.33.44.45.55.]
JA =[1 4523455]
(EQ 3.137)
117
IA=[1 4 5 6 8 9)
3.5.3.3 Conjugate Gradient Iterative Solver (Kincaid et al. 1990)
An iterative solver was chosen for two major reasons: (i) The accuracy of the solu-
tion is not influenced by large differences between the values of the terms in the stiffness
matrix. (ii) The solution from one step can be used as a first guess for the solution at the
next step which is useful in incremental static analysis, and in step by step dynamic analy-
sis where there are no significant changes expected in the incremental solution between
two consecutive steps. A detailed description of the method is presented below.
CG is the most popular iterative method for solving large systems of linear equa
tions. It is effective on systems of the form
Ax = b (EQ 3.138)
where A is a known symmetric, positive definite matrix. Iterative methods are suit
able for use with sparse matrices because they usually use matrix vector multiplications
operations.
The basic quadratic form is a scalar, it is quadratic function of a vector
(EQ 3.139)
where A is a matrix, x, b are vectors, and c is a scalar.
For positive definite matrices the surface defined by f(x) has the shape of parabola
for the case where the size of vector x is 2.
The gradient off(x), i.e./(x) is:
(EQ 3.140)
and if A is symmetric
118
f (X) = Ax-b (EQ 3.141)
Setting the gradient to be zero will retrieve EQ. 3.138 . If A is positive-definite and
symmetric, this solution is the minimum off.
Method of Steepest Descent
In this method we start with an initial guess Xo and take a series of steps to values
of x: x j>x2'" until we are close enough to the solution x.
The direction of the step is chosen to be the direction of the gradient to the function
fwhich is also the residual
(EQ 3.142)
where ri is the residual, the residual can be also defined as
(EQ 3.143)
where e i is the error term.
Moving from one point to another in the search for the minimum point will be
done in this manner.
(EQ 3.144)
After finding the direction of the step ri' to find a line search is performed along
the direction of ri' such that a minimize f where
dx. 1 d(x.+ar.) - 0 - fT ( ) -' +- - fT ( ) , , - - xi+l da - xi+l da
(EQ 3.145)
Using the last term and EQ. 3.143 and EQ. 3.144 the value of a is derived as
rTr. a=-'-'
rTAri
The error reduction between consecutive steps is calculated as
(EQ 3.146)
119
(EQ 3.147)
where 0)2 is defined as
(EQ 3.148)
Al ~2 and k = r- ;~ = ~
2 ':>1
where Al ' A2 are the eigen vector of matrix A and ~I ; ~2 are the length's of the projec
tion of the error term on the eigenvectors.
It shows that the smaller 0) is, the faster is the convergence. The term k relates to
the ratio of the eigen values and the term ~ relates to the ratio of properties related to the
initial guess. The drawback of steepest descent is that steps are taken in the same direction
several times, and the number of iterations might be large.
The Method of Conjugate Directions
This method is based on the idea of choosing the search directions so they will be
A orthogonal, in this way the search in each direction will be done only once and after
moving n times in each direction, the solution is found.
The length of each search vector is found from
(EQ 3.149)
To get the A orthogonal vectors, the conjugate Gram-Schmid's process is used, in
this process the vectors d; are calculated
and
;-1
d; = u j + L ~jkdk k=o
(EQ 3.150)
120
-uTAd. ~ _ , J
ij - dTAd. J J
(EQ 3.151)
The drawback of the method of cojugate directions is that to derive each new vec
tor, all the old vectors should be used and kept, this requires large number of operations
O(n3).
To improve the performance of conjugate directions, instead of using arbitrary set
of independent vectors uj the set of vectors used is the set of the residuals rj
, using this
set of vectors simplifies the solution considerably because of the special properties of rj
,
and the final procedure is
where
and
rTr. a. = -'-'
, dTAd. , ,
~i+ 1 =
and the initial searching vector is
The convergence of CG is function of the condition number of A, k
"'min "'max are the minimum and maximum eigenValue of the matrix A,
(EQ 3.152)
(EQ 3.153)
(EQ 3.154)
(EQ 3.155)
(EQ 3.156)
(EQ 3.157)
(EQ 3.158)
121
and the rate of convergence of CO (or the decrease of the error norm) is
(EQ 3.159)
Preconditioning
In order to decrease the condition number k, a preconditioning technique is used.
The system of equations is premultiplied by a symmetric, positive definite matrix M 1
(EQ3.160)
and the full solution algorithm is
(EQ 3.161)
(EQ 3.162)
r!M-I r . ex. = I I
, d!Ad. , , (EQ 3.163)
(EQ 3.164)
(EQ 3.165)
(EQ 3.166)
(EQ 3.167)
A well known preconditioner is the diagonal or Jacobi preconditioner in this case
the matrix M is taken to be the diagonal of matrix A.
122
3.6 Methods of Analysis to Evaluate Bridge Capacity
Capacity is a limit state which can be defined as: (i) The maximum force or defor
mation that a system or subsystem can experience before it reaches collapse. (ii) A system
failure is reached when it cannot play it's designated roll. A failure of a system occurs
when one or few of it's subsystems fails.
Each element in the system by it self can fail in different ways: (i) Section failure,
such as failure in flexure (ii) Zone failure, such as failure in shear. (iii) A failure of mecha
nism combined from flexure and shear.
The reliability of a system can be derived in two basic ways. (I) By first deriving
the reliability of each individual member and then combining them into reliability of the
global system. For example by using "Fault Tree" approach, in which the probabilities of
each component of the system are arranged in a system "Fault Tree". And the critical path
in the tree defines the probable failure path of the system. (2) By using analysis procedure
which provides directly the probability of failure of the whole structure.
Similarly the capacity of the system, can be assessed by deriving first the capacity
of the basic elements directly from the constitutive relations of the individual element and
expending it to the global system, or by assessing the capacity of the whole system
directly.
In the following sections, a brief review of the commonly used techniques to eval
uate ultimate state or capacity of a section are presented, followed by presentation of pro
cedures to evaluate the structural capacity of the whole bridge system, namely by
monotonic push-over or dynamic analysis procedures in which the capacity of the struc-
123
"---------------------------- -----
ture as a whole is derived by applying simple loading to the bridge and detecting condi-
tions which are defined as failure.
3.6.1 Direct Section Evaluation
The derivation of the moment curvature relations for a section in flexure is well
known. Two methods are used usually for this purpose: (i) the approximate one based on
stress block and, (ii) the more rigorous one: fiber analysis. Both methods are based on the
following assumptions: plane sections remain plane, so the concrete strain distribution can
be defined just by two variables (e.g., the strain at the top face and the strain at the bottom
face) or by the strain at the centroid, Ecen and the curvature $. The curvature is equal to
the change in slope per unit length along the member, and is also equal to the strain gradi-
ent over the depth of the member.
Thus the concrete strain at any level y is given by
(EQ 3.168)
The strain in a reinforcing bar at any level y is equal to the strain in the surrounding
concrete; hence
(EQ 3.169)
The following equilibrium conditions must be also satisfied.
JfcdAc + JfsdAs = N (EQ 3.170)
Ac As
JfcydAc+ JfsydAs = -M (EQ 3.171)
Ac As
124
In which! c,! s are stresses in concrete and steel, Ac. A s are the areas cross sections
of concrete and steel. M, N the global moment and axial force.
3.6.1.1 Mechanical Properties Modeling
The mechanical properties or the moment curvature relations can be evaluated
directly by assuming that the stresses in the concrete are concentrated in a rectangular
block with dimensions related to the parabolic distribution of the stress.
The moment curvature relations are derived by iterating on the eqUilibrium equa
tions, for a given axial force and moment using the stress strain relationships for steel and
concrete.
3.6.1.2 Fiber Analysis
Instead of assuming distribution of the stresses over the height of the section, the
section is divided into number of layers where the concrete and the steel micro mechanics
are used. The integral of the stresses of the layers over the sections height should satisfy
the equilibrium conditions presented earlier. Iterative procedure is used to find the stresses
at the layers, and the related curvature. The ultimate state is defined as a condition where,
no equilibrium can be achieved for increase in moment.
3.6.2 Incremental Monotonic Static Analysis
In this section the concept of evaluation of structural components, or structural
system capacity to sustain forces and deformations before reaching a point of failure is
presented.
125
The capacity of a single element is usually defined as the extreme value on the
constitutive relation curve of the element, when the failure mechanism is expected to be
flexure, the failure happens in a single section. However when shear failure occurs it can
be defined as failure of a region.
The commonly recommended way in the design codes to evaluate the strength
demand of a single element is by reducing the forces derived using elastic analysis for
seismic loading Felastic by a reduction factor R which relates to the assumed capacity of
this element type to dissipate energy by experiencing inelastic deformations.
F _ Felastic design - R (EQ3.172)
Using this procedure it is assumed that the inelastic behavior increases the seismic
capacity of the elements. However the inelastic behavior doesn't influence the distribution
of the forces in the bridge due to inelastic behavior which might transfer significantly
larger force to certain elements. In order to give more emphasis to the plastic behavior
(Seismic Retrofit Manual for Highway Bridges - Buckle and Friedland 1995), a mono-
tonic incremental static lateral analysis or dynamic analysis with a predefined dynamic
load, defined also as "Push-Over" analysis is used to evaluate the capacity of the whole
structural system, or major parts of it such as frame bents. In such a case a single demand
to capacity ratio, which represents the reliability of the whole system can be defined. The
derivation of single system demand to capacity ratio, is desirable for accurate road sys-
terns evaluation.
In order to derive the capacity of a structural system: (i) the geometry, (ii) the con-
stitutive law ofthe elements and (iii) the loading distribution should be known. Assuming
126
that the geometry can be well defined by geometric modeling and the constitutive law can
be also well defined, the main difficulty is to define an appropriate distribution of the total
force F. The distribution of the total force F between the elements of a bridge should be
representative of the distribution of the forces during the seismic event.
If the spatial force distribution in the bridges is defined as:
f(x, y, z) = FS (x, y, z) (EQ 3.173)
in which F is and intensity function and S (x, y, z) a nondimensional distribution.
The distribution of the mass in building structures can be considered in the vertical direc
tion only, assuming that the floor diaphragms are stiff in there plan. Using this assumption
the distribution of forces in a building can collapse to:
f(x, y, z) = FS (y) (EQ 3.174)
However, in bridges with significant spatial distribution of masses the relations in
EQ. 3.173 may not be simplified as easy. In bridges of significant length where the number
of spans is more then two, the modes of vibration are much more complex and so is the
bridge response to earthquakes. To get better evaluation of the bridge response, several
types of "Push-Over analyses" are suggested. Two of them, i.e. the incremental force load
ing and incremental displacements loading, were previously defined in various versions
for analysis of buildings (Valles et. el. 1996). To be able to represent fue accurately the
influence of the changing structural properties of the bridge due to inelastic behavior adap
tive (modal) loading is suggested. The other two options are dynamic versions of the static
"Push-Over procedure", in which the expected dynamic loading and the adaptable distri
bution of the load are addressed.
127
Both the previously developed push over procedures (arbitrary force loading) and
the newly developed procedures (displacement quasistatic, modal adaptable and the quasi
dynamic procedures: pulse and ramp) are described hereafter
3.6.2.1 Arbitrary Force Loading
The simplest and most commonly used method for "Push-Over analysis" is per-
formed by applying incrementally lateral forces to the bridge until failure criteria is
reached. The incremental static analysis with force input presented earlier is used. Several
criteria (Valles et. el 1995) are used as a definition of failure: 1) first component reached
failure, 2) gravity load carrying element reached failure, 3) mechanism is formed, 4) ser-
viceability displacement criteria is reached.
As mentioned earlier the distribution of the forces along the bridge is an important
issue and two types of distribution which are similar to the once which are used in build-
ings (Valles et. al. 1995) are suggested: 1) uniform load distribution 2) load distribution
equivalent to one or more modes of vibration.
F- w«p(u) - 'Y1.:w«p(u)
in which «p (w) is spatial function of u(u;x> uy uz).
(EQ 3.175)
There are several drawbacks in using this procedure: 1) if the modes of vibration
represent the displacement response of the bridge, the acceleration response will be pro-
portional to the modes only under harmonic loading,
u (x, y, z) = (j)2u (x, y, z) (EQ 3.176)
In the general case the force distribution is the fourth derivative of displacements:
128
d ( 1 d2u) [(x) = dx2 Eldx2 (EQ 3.177)
In the case of complex modes of vibration of the bridge. The deformation u and the
external force [are distributed as two different function in space. 2) usually it is consid-
ered that only few first modes of vibration are enough to represent the actual displacement
response (Clough and Penzien 1993) however it does not apply for acceleration response.
3.6.2.2 Arbitrary Displacement Input
Because of the limitations mentioned in the previous section (the forces in the
bridge cannot be accurately described by the modes of vibration) analysis based on the
previously described incremental displacement input can be performed. The analysis pro-
cedure for this case is similar to the one used in the previous section for arbitrary force
input. However instead of applying forces on the bridge, a certain displacements pattern is
applied incrementally. A possible assumption is the first mode of vibration.
3.6.2.3 Adaptable (Modal) Loading
In the previous two sections, the distribution of the load is not a function of the
damage the structure sustain due to inelastic behavior. To provide more compatibility
between the load distribution and the change in the structure stiffness, adaptable modal
loading push over procedure is developed (Reinhorn et al. 1996). This procedure applies
both to the force input and displacement input procedures. In this procedure at each time
increment the mode shapes are calculated using the instantaneous stiffness of the structure
which changes due to plastic behavior, the incremental force at location i along the bridge
/:;.Fi is defined as:
129
(EQ 3.178)
in which <l>i,j (u) is the ordinate of the j'th mode vector at location "i";, rj (u) is the
modal participation factor for the j'th mode; Ftolal is the total force, or total displacements)
and F11d is the quantity level at location "i" in the previous loading step. With the load
distribution defined in EQ. 3.178 a nonlinear incremental analysis can be performed.
3.6.2.4 Acceleration Ramp Load
In the previous section the basic assumption is that the effect of a random dynamic
loading can be represented by static equivalent loads and the effect of the dynamic behav-
ior is achieved by distributing the load proportionally to the mode shapes.
A better approximation for the behavior of the bridge under dynamic loading can
be achieved by sUbjecting the bridge, to predefined dynamic loading, in which the
dynamic characteristics of the bridge response are inherently accounted for.
One type of such analysis is the dynamic loading using a base "Ramp Accelera-
tion" history (or Ramp Acceleration Loading) (see Fig. 3.31 )
130
Ground Acceleration
Application rate
Time
FlGURE 3.31. Ramp Acceleration Loading
In this type of push-over procedure the acceleration is increased monotonically in
time as shown in Fig. 3.31 until the "failure" condition is reached. The rate in which the
acceleration is increased influence the results. When the acceleration is increased slowly,
the behavior is similar to applying static loading. Significant dynamic effects appear when
the rate of application is high. A range of application rates should be used, to estimate reli
ably the bridge capacity to resist seismic - dynamic effect.
3.6.2.5 Acceleration Pulse Input
In many of the recent strong earthquakes the major damage to bridges was attrib
uted to a single strong short impulse. This pulse has the effect of one strong "Push" on the
structure. A way of simulating the expected response of a bridge near failure is to apply a
single pulse with high magnitude of acceleration to the structure. The shape of this pulse is
not significant, if it is short enough relative to the structures natural period (Clough and
Penzin 1993).
The basic equation of motion for this system is:
131
mx+kx = -mx g (EQ 3.179)
Integrating the equation on both sides will give the impulse-momentum relation-
ship as
I,
mllV = f [p (t) - kv (t) 1 dt
o (EQ 3.180)
Since the integral of kv is of the order of r2 and the integral of p(t) is of the order of
t, it can be shown to be approximately
'I mllV;: fp (t) dt (EQ 3.181)
o
Therefore an initial velocity analysis can be performed.
Since the level of energy that will cause the structure to fail is not known appriori,
an iterative procedure is used. The initial energy required to bring "failure condition" is
estimated for the first iteration, and then increased, until failure is observed.
3.6.3 Comparing Demand to Capacity
In order to evaluate the reliability of the system, the demand and capacity of the
system's components, or of ihe system as"Whole, should be compared. In the following A
sections. different procedures of evaluation are presented. An overview of those methods
and their advantages and disadvantages was presented in section 1.0
3.6.3.1 Demand/Capacity comparison in Inelastic structures (Damage Indexing)
During nonlinear time history analysis, the component's capacity and demand can
be compared instantaneously, at each time instant during the analysis, since this compari-
132
son represents the amount of damage sustained by the element it is defined as "damage
index" (DI) (Reinhorn et. el. (1996»
(EQ 3.182)
where U d (t) is the maximum displacement attained in case of shear or the maximum cur-
vature in case of bending, uy is the yield displacement, Uu is the ultimate displacement,
Eh (t) is the hysteretic energy dissipated until time t, and Qy is the yielding force
(moment). The numerator
(EQ 3.183)
is the displacement (or curvature) demand while the denominator
(EQ 3.184)
is the displacement capacity. It should be noted that the capacity is reduced as the hyster-
etic energy Eh (t) increases during the inelastic excursion (see EQ. 3.184 ). Therefore the
damage index DI is a function of time, and it is not possible to derive it by simply compar-
ing the maximum demand to the capacity, following the analysis.
The final damage index DIf is the largest value attained during the analysis
DIf = max (DI (t» (EQ 3.185)
133
3.6.3.2 Evaluation of Bridges in ModeratelLow Seismicity Zones· Procedure
In order to evaluate bridges with low risk, costly and complex procedures are not
required. Simplified equivalent elastic procedures are enough to provide adequate evalua-
tion, especially since other load cases might control the design. The procedures described
hereafter are described in the AASHTO code, and brief summery of them is presented for
sack of completness.
3.6.3.2.1 Static Equivalent
The static equivalent procedure is very simple and easy to implement. The
response of the bridge is assumed to be similar to the response of a single mass, which is
the attributed weight of the deck and the upper part of the bents, on a single spring, in the
transverse direction. In the longitudinal direction the deck is supported by the bents later
iJ-ally, so the response can be approximated by single mass of the deck and upper part of the
A
column, supported on several springs.
In the transverse direction, the bridge can be divided into separate frames, where
each frame represents the bent with the tributary section of the deck. By doing that, the
interaction between the different parts of the deck, is ignored.
The lateral force on the frames is derived, using code formula, considering the
mass the stiffness, and the PGA (Peak Ground Acceleration). This procedure is very sim-
pie although not very accurate, and can be used, as any other static loading case. The
resulting element's forces are compared to the nominal strength of the element increased
by ductility factor which represents it's ability to dissipate energy. The deformations of the
134
bridge are compared to design requirements in the code, which represent stability and ser-
viceability requirements.
3.6.3.2.2 Modal Spectral Analysis
Modal spectral analysis is a more rigorous procedure than the static one. In this
procedure the dynamic behavior of the structure is considered, and therefore the engineer
using this method should have some knowledge in structural dynamic.
The bridge modeled as framed structure, represented by it's stiffness and mass.
Modal analysis that was presented in the previous sections is performed to derive the
mode shapes, and the modal participation factors.
As it was described earlier using the mode shapes and the response spectrum the
loading for each mode can be derived. The structural analysis performed on the bridge
provides the forces and deformatio~ required for the evaluation. A
The comparison of the forces demand and capacity, is similar to the one performed
for the previously presented static procedure. The element forces are compared to there
nominal strength, and the displacements are compared to the once in the code.
3.6.3.3 Evaluation of Bridges in Severe Seismicity Zones
In severe seismicity zones, the response of the bridge to earthquake will be usually
the lateral loading case that dominates the design of several components of the bridge.
Usually performing design with the methods presented for the low/moderate seismicity
will result in unnecessary excessive design forces.
135
The appropriate methods to be used in severe seismicity zones are suppose to con
sider more rigorously both the dynamic behavior of the bridge and the resulting design
forces, and the element's nonlinear capacity under cyclic dynamic loading. The methods
presented in the following sections address this issues. The first procedure uses the spec
tral method presented in the previous section to derive the demand. The capacity however
is addressed more rigorously using "push over" techniques. The second procedure, evalu
ates directly and instantaneously both demand and capacity, and "damage index" is used
to get accurate evaluation of reliability.
3.6.3.3.1 Spectral Demand vs. "Push Over" Capacity
Comparing the demand derived from the spectral procedure, and the capacity from
the "push over" procedure, enables direct calculation of the reliability of the bridge. The
required steps for this procedure are presented below.
• The nonlinear force displacement curve (Fig. 3.32) is derived using one of the proce
dures presented earlier. The choice of the procedure should be done according to the
nature of the expected earthquake. For example if an earthquake is characterized by a
single pulse, such as Northridge, which excite many modes, the appropriate procedure
would be "dynamic pulse". While "adaptive modal" procedure is not appropriate, since
only few modes are used to represent the load. On the other end when the expected
earthquake has vibratory nature "adaptive modal" or "ramp loading" would be appro
priate for the same reasons. In the longitudinal direction, there is single displacement
that need to be considered. In the transverse direction however, the displacements
might be different in different locations along the bridge, and equivalent displacements
should be considered for the force displacements curve.
136
The limit state that defines failure, is defined according to the performances expected
from the bridge. If the bridge is expected to remain serviceable following an earth-
quake, the service limit state should be used. While if the bridge is not essential the lim-
iting state is collapse (Fig. 3.32).
Total force
service limit coUapse limit state state
I
I F. IF. I I
I
u.
Equivalent dlsp.
FIGURE 3.32. Force Displacements Relations for ''Push Over" Analysis
• The displacement ductility factor is derived as
u. 11. = "ii"
y (EQ 3.186)
. • Using the displacements ductility factor 11., the reduction factor Rtis derived using
equivalent displacements approach for long period bridge, (Priestley et al. 1992)
for T> 1.5sec Rf = 11. (EQ 3.187)
137
and equivalent energy approach for short period structures
. T forT<1.5sec Rf = 1+0.67(l1u-l)1'""<l1u
o (EQ 3.188)
Where T is the dominant period, or equivalent period, and To is the period at which
response spectrum has maximum.
• The inelastic capacity of the bridge Fu is converted into elastic capacity Fe' such that
(EQ 3.189)
• The total force demand F demand is derived using the regular modal spectral procedure
described earlier
• The total reliability of the system R can be defined directly as:
R = 1- Fe
Fdemand (EQ 3.190)
The total reliability of the bridge system is a very important parameter, and can be
cv used to evaluate the reliability of bridge system as lifeline following an earthquake.
A
Although this procedure compromise the accuracy by using procedure which only
approximate the dynamic response, and the capacity evaluation procedure is not address-
ing the issue of capacity reduction due to cyclic loading, the advantages of this procedure
are obvious.
3.6.3.3.2 Time History Analysis with Multiple Earthquakes
In all the previously described procedures the capacity and the demand are derived
separately, both are only approximations of the actual response, and there is no direct
interaction between them.
138
U sing time history analysis provides a much more accurate tool, for the evaluation
of bridge reliability. The dynamic properties of the bridge such as the mass and the stiff-
ness, are directly introduced into the analysis procedure, as well as the nonlinear elemen~
properties which include the variation of force and stiffness as function of displacements,
as well as their ultimate capacities.
Using the directly defined properties and the variation of the properties during the
analysis, the interaction between demand and capacity is naturally included.
The damage to the nonlinear elements is evaluated instantaneously using the pro-
cedure described previously, and the final damage to the element is taken as the maximum
of the damage values calculated at every time interval during the duration of the analysis.
Although the analytical procedure by itself is accurate, the earthquake that the
bridge will experience in the future is unknown. Therefore the analysis should be per-
formed several times, using set of several earthquakes that can reliably represent the site
specific response spectrum. Those earthquakes might be previous earthquakes scaled to
the design response spectrum level, or artificially generated earthquakes.
The damage index or the reliability derived by the analysis apply only to individ-
ual elements or sections of elements. The desirable reliability index however is the one of
0-the system a~ whole. To achieve this target, engineering judgment should be used as of the
contribution of each element to the reliability of the whole system with the different per-
formance criteria. An example of a method to combine the reliability of a single element
into the reliability of a global system is the "fault tree" method (Melchers 1987).
139
4.0 CASE STUDIES FOR BRIDGE EVALUATIONS
4.1 Evaluation of Retrofit of Existing Bridges
4.1.1 Case Study #1: Analytical Evaluation of Small Scale Bridge System
Results of an experimental study performed by Tsopelas et. al. (1994) are used to
verify the capacity of the computational models to evaluate the response of structural sys-
terns which include "base isolation systems" placed at arbitrary locations in the structure
(not necessary at the base).
A shaking table test was performed on a simple scaled bridge model (1:4) (a sche-
matic of the bridge on the shaking table is shown in Fig. 4.1 ). The bridge consisted of a
deck of 140kN weight, supported by four flexible columns. On top of the columns were
bM.I,,~ 1\\ ~~ four Friction Pendulum System (FPS) bargains (Zayas et. al. 1987). The bridge was sub-
jected to ground motion defined by the Japanese seismic design code as "level 2" and
"ground condition I". The simulated motion is shown in (Fig. 4.2).
Fig. 4.3 shows the geometry of the FPS bearings. Each of these bearings carried a
load of 35kN. The frictional characteristics of the bearings were experimentally deter-
mined as shown in Fig. 4.4.
The bridge was modeled as a single frame as shown in Fig. 4.5. The deck was
modeled using a very stiff elastic beam element, and the weight of the deck was concen-
trated at joints 5 and 6 (deck ends). The isolators were modeled using a continuous kine-
matic model, known as Wen's model (see Sect. 3.2.4). It should be noted that the model
used for the bearings is not velocity dependant although the bearings friction characteris-
tics are velocity dependent. The dominant coefficient of friction as observed from the hys-
140
teretic loops obtained from the shaking table test experimental results was found to be 9%.
This value is related to the behavior of the bearing in low velocity (Fig. 4.4). The
required parameters for this model, Le.: the initial stiffness kinirial, the secondary stiffness,
kyie/d, the yielding force Fy and the post yielding ratio n are derived using the geometrical
properties of the bearing, the friction coefficient of the bearing and the supported weight.
The yielding force is Fy = WIl = 35 x 0.09 = 3;'..J5kN. The other parameters, the yield
ing displacements uy was determined from the component testing hysteretic loop as
0.255mm. Using Fy and uy the initial stiffness is kinirial = 13059kN/m. The post yield stiff
ness of the isolation system is calculated from the radius of surface R and the weight W:
kyield = (j)2m = (~)41t2 :.~ = 62.4 kN/m which gives post yield ratio nof 0.006. The
simulated ground motion is applied to the frame in the longitudinal direction. The results
of the analysis are shown in Fig. 4.6 and Fig. 4.7 .
The stiffness of the columns was determined from their tabulated structural prop
erties. Because of flexibility of the shaking table the supports to the bridge cannot be con
sidered fixed. Therefore the degree of fixity was determined from the experimental results,
using the free vibration characteristics of the bridge in very low amplitudes when the bear
ings are not sliding.
One of the parameters required for the analysis procedure is the global structural
equivalent damping. Evaluating this parameter has significant influence on the results
obtained from the analysis. As mentioned in the previous section, the global structural
damping, can be represented as stiffness proportional, mass proportional, or combination
141
1- -_.
I
I of the two (Rayliegh) damping. In order to observe the influence of global structural
damping and types of damping on the analytical solution, several analyses with different
levels of damping were performed. A comparison between the analytical and experimental
results for different levels of damping, is presented in Fig. 4.6 and Fig. 4.7 .
A good agreement is observed between the analytical and experimental displace-
ments of the deck (Fig. 4.6) for most types of damping. However In the case where there
is significant amount of stiffness proportional damping a mismatch can be observed in the
maximum peak.
The displacements at the top of the piers (point 3, 4) (Fig. 4.7), are related
directly to the moments and shear forces at the piers. Those displacements are more sensi-
tive to the global damping representation. In most cases the analytical results overestimate
the experimental, the best results are achieved with 3.4% mass proportional damping, and
a small contribution from stiffness proportional damping. This happens probably because
the high modes of the piers are damped, and therefore it is more difficult for sliding to
occur
The change in damping representation influence the results because the of the
change in the stiffness matrix when the bearing are softening. Tsopelas et al (1994)
applied damping. terms only to the degrees of freedom which has only linear elements con
nected to then and got very good results.
The presented comparison shows that using careful representation of global damp-
ing, the behavior of isolated bridges with flexible piers can be adequately evaluated using
the analytical model.
142
II AISC r--------;====--....;,, ___ ....,...;- WI4.VO
0.!5 ..
~ ~ ~ . . - - I AISC 1'S ..... /11
I 3 .....
'.0... I 4.10...
FIGURE 4.1 Bridge Experiment Set Up
.•..
0,4 .--.,.--___ -.-___ -_ ...... ___ -_--,
0,2
~ c
'~ 0,0
.£
~ ·0,2
~,4 L---!..--_-'-__ ~ __ ...,.. _____ ..J,
o 10 15
, Time [sec,)
FIGURE 4.llnpul Ground MOllon (Japanese code, levell, ground condition 1)
143
units:mm
'" ...
IUII------ sc"-1-'----.l!'!L....,
'-__ IQISIHG "'.,' "'4" ,,,.,,,.,,
FIGURE 4.3 Construdlon of Frldlon Pendulum System Bearing
144
0.15 -r-----------.... C.."O .... M:=P""O""S=IT='E":":M,...A=TE="R::':'IA..-:L"""N,..,..O....,.3,...., z Q t o.12 ·
V IE IS ~ 0.09
- o
fmax-0.120, PRESSURE 17.2 MPa (2.5 ksl) •• •
• • •
~ fmax-0.062, PRESSURE 275.8 MPa (40 ksl) w 0.06.!.-------e----......:--..;.....-----_--.:. ___ ....:....:--j
8 ~ 0.03 o IDENTIFICATION TEST
uj • SEISMIC TEST
O.oo-t----I---.,·'----I-___ I-__ -+ __ --l o 100 200 300 400
VELOCITY (mm/sec) 500 600
FIGURE 4.4 Experimental Derivation of The Sliding Bearings Frictional Characteristic (Coefficient of Friction as Function of Velocity)
145
5
3 ~ Isolator
1
:;::;
Deck,L = 6.0 m
Column, L = l.4m
FIGURE 4.5 Base Isolated Bridge Model
146
-Eltpcriment
1.7% mass prop. damp. -Analysis
0.01 0 r---'--"T"'""':"""::"'""T"-=-~-'
0.005 ~
~~ ]: ~ ~O.ooo
i <>0.005
.0.0100.L;;0-"-~5.':.0:;---'--;-;1O!c.0;--"--;-}15.0
3.4% mass prop. damp. 0.010 ,---r--,--...:.,...:-,.....:.--.---,
0.005
]: -e O.OOO
i &0.005
1.7% mass 0.010
0.005
0.005
0.000
-0.005
+ small stiff prop. damp.
damp, + small stiff prop. damp
.0.0100.'::0-''--5;'.0;;--'--;-;1O~.0;--''-''7!15.0 -0.0 100.'::0-''-7.:;---'--;-;!O-'-~
1.7% stif. prop. damp. 0.010 r--'-":""":":':'::':"::';::'::":"::.:.r.:......---,
]:
• 0.005
~ 0.000
i -0.005
r_~I~.7~%~~~~d~am~p~i~ng~ __ 0.010
0.005
0.000
-0.005
.0.011O~'::---'--;';;----'~7t-;;--"-"7!15.0 .0.0100'.'::0;--''--5''.0;;--'--;-;~-'-~~ Time (sec.) Time (sec.)
FIGURE 4.6 Displacements at the Top of the Pier (point 3 and 4)
147
- Experiment 1.7% mass pro. damp. - Analysis
0.10 ,-----,,..---,-.....;.--i---.---,
g 0.05
~ ! 0.00
is -0.05
1. 71"'%_m_as..,s..:p_ro_._da,.m--,-p._+_s~m_a1_I_S_til"'f . .:;.p_ro.:;.p..,. d;.:am~p__, 0.10
0.05
0.00
-0.05
-0.100. "'0---'--c51;;.0---'--;-;101-;.0;----'--dI5.0 -0·100."'0---'--c5.1;;0---'--;-;101-;.0;----'--;-;I15.0
3.4% mass prop. damp. 0.10 ,--...-::..:..:.:r=::.:.!.;.:::::-==r:--.----, 3.4% mass prop. damp. + small slif. prop. damp.
0.10 ,--'--';-....,----..--'0'----;----,
g 0.05 0.05
I 0.00 0.00
is -0.05 -0.05
-0. %.1,;0,-----'---5,-l.0;:-~~-1,.,0!-;.0,---~--;-;!15.0 -0·100.1,;0---'---'5.-J:.0;:-~~--;-;:1O!-;.0;----'---;-;!15.0
1.7% stiff prop damp 0.10 ....---~-,_-..:.-"---,-:-~-__,
1.7% Riley damping 0.10 r--..--.,.--..:.----'--,-:-~--,
g 0.05 0.05
t: 0.00
-0.05
-0.10 '---"--..1----'--:-'-::--'---' -0.10 '::--"-,----f;:----'--~;__~---;; M ~ 1M 1~ M ~ 1M 1~
Time (sec) Time (sec)
FIGURE 4.7 Displacements of deck (point 5 and 6)
148
4.1.2 Case Study #2: East Aurora Bridge - Field Test
A full scale field test performed by Wendichansky (1996) on the southbound high
way bridge crossing Cazanovia Creek and laboratory component experiments (Mander et.
al (1996) were used to evaluate the capability of the analytical platform to adequately pre
dict the transient dynamic response behavior of bridges with different types of bearings
during the experiment. The bridge is located in East Aurora, New York, on Route 400. It is
a three spans continuous structure, supported on two abutments at its ends, and two piers
in the interior (Fig. 4.8). The south bound bridge deck is supported on seven girders,
which are connected laterally by stringers (Fig. 4.9, Fig. 4.10 and Fig. 4.11 ). For the
original condition of the bridge, the girders were supported on steel bearings. Three of the
supports had expansion bearings, and one pier support provides the longitudinal fixity.
The bridge was retrofitted and the steel bearings were replaced by lead rubber bearings at
the abutments, and laminated rubber isolation bearings at the piers. Several snap back tests
were performed on the bridge with both types of bearings. The bridge was pulled by two
jacks connected to the bridge at points on the deck above the bents and then quick released
to observe its free vibration response. The history of the force applied to bridge measured
by a load cell is presented in Fig. 4.12.
The bridge was analyzed using the analytical platform for both types of bearing
settings. The structural model used for the analysis is presented in Fig. 4.13. The deck
and the bents are modeled as elastic beam with shear deformations modeling capability
because of the low length to width ratio of this elements. The soil at the bent's foundation
level and at the abutments is modeled as elastic foundation element. The participating
mass of the soil is introduced into the corresponding joints.
149
Although each deck section between the bents is modeled as a single beam ele
ment, and the deck and the bent are connected only at one point, there is no loss of accu
racy (see Fig. 4.11 ), since the rigid body motions at this connection are represented by
rigid arms (see section 3.2.1.1). The finite stiffness of the stringers in the lateral direction
can be modeled by using beam end springs (see section 3.2.1.3). The linear properties of
the bridge were taken from Wendichansky (1996).
Both types of bearings were modeled as Wen's model type isolator element (see
section 3.4.2.1). The required properties of the bearings were retrieved form laboratory
experimental results (Mander et. al.1996).
The displacements measured during the experiment for the steel bearings (Fig.
4.15 ) and the elastomeric bearings (Fig. 4.16) were compared to the results from the
analysis. In both cases a very reasonable agreement may be observed.
It should be noted that although the bridge response is primarily governed by first
mode behavior, due to the angle of the applied load and the skew inherent in the bridge, all
modes are excited by the snap test. It is therefore essential to use 3D nonlinear modeling
to capture all of the behavior attributes. A 2D nonlinear program can be used to achieve
reasonable results, in an approximate fashion, but not without great difficulty as demon
strated by Wendichansky (1996)
ISO
H --" N.'. Lo." " 12150
~::::7/:::: __ ::~a~::~ __ ~a:3:30~:j~:-~::i ~ I ;' 21213 110''--: . I IfpPMllh ,"I
IZZOO
11"0
'N40
• 2f213
FIGURE 4.81Op VIew or the Bridges
151
'" N
'" -
-_ ... _-------------
f.l.711~
.. 7391 ' . I 7391
h· .2210 ~
! I I
rr"" ~L,' ,)., '--rr .., rr~ ; , rr'..,
~'0450r-- ~ on on '" - '" t f -
686
lIlI'!l
, "
0 N on -
_ 1753 __ •
FIGURE 4.9 'Jyplc:a\ Bent
152
".
13869 StIffeners 178x 16
2210 16WF'36 at S. Abut. ~ 14C33.9 at N. Abut. - • r I-
i i i i ! I ' , i
~ JoL ;~
Ii ... .. 11 L
-~
ABUTMENT
FI~URE 4.10 Typical Abutment
153
r: .. 147U
.. ~ l 7;l91 -80475 + I
I I
I ,II. ~ ,II.
~ &~ , \ ,II. JI JI. "'-'L.271.93m
12850
-~
lL:" £L.284.21 m £L.264.S2 "\ I I I 1520 1220U J
I SOUTH IOUND "
NORTH BOUND
FIGURE 4.11 Side VIew or-the Bents
".
154
.".
100.0
~ 50.0 If
0.0
".
-50.0 '--"--..L..--'---'-"""--'--,--,---,-~ 0.0 2.0 4.0 6.0 8.0 10.0
Time [Sec.]
FIGURE 4.12 Input Force History, Measured During the ThsL
155
FIGURE 4.13 Analytical Model or the Bridge, Used ror Analysis
1$6
--~-~-------c----------------------------
bearing element
soil mass
soil springs (longltudmal and lateral)
stiff support
(a) Modeling of connection between bridge deck and abutment without rigid body transformatIOn
deck modeled as shear and ~arn
bearing
rigid arm soil mass (rotational
~;~~~~~;~~~~~~~~~~~~~~tr:anslational)
soil spring (longitudinal transverse and rotational) stiff support
(b) Modeling of conq,,\,tion between bridge deck and abutment using rigid body transformatIOn (ngld arms)
FlGURE 4.14 Modeling of Bridge Deck Connection to Abutment (a) Modeling witb Stiff Elements (b) Modeling witb Rigid Body Transformation
157
E .§.
~
10
- an$fis - ex ment
5
~ l'i 0 'Q
] ·5
·10 Time (Sec]
FIGURE 4.15 Comparison of Analytical and Experimental Results for Bridge on Steel Bearing at Abutment.
~r-----~----~----~-----r-----r-----r----~----~
30
20 / experimental
10
·100."'0----'-----:-1.'::-0---...... ---,2::\.0,---........ ---;3:\;.0:---........ --......,.J4.0
Time (Sec]
FIGURE 4.16 Comparison of Analytical and Experimental Results for Bridge on Rubber Bearing at Abutment.
158
4.1.3 Case Study #3: Bridge in Los Angeles· Retrofit Solution
A bridge model based on a prototype bridge in Los Angeles was evaluated to dem
onstrate the 3D capacity of the computational model to: (i) evaluate a bridge subjected to
differential ground motion and (ii) to evaluate the efficiency of a retrofit scheme using
base isolation systems.
Using a simplified evaluation technique, the bridge was found to have deficient .
strength capacity to resist seismic loading.
(a) Original bridge Structure and Modeling
The bridge (see Fig. 4.17 ) has seven spans, ranging between 30m to 39m in
length. The deck is supported on eight supports. Five of them (Bl-B5) are bents of the
type shown in Fig. 4.17 -(b). Support Al is a very stiff abutment. Supports A2 and A3
are also very stiff but have some flexibility due to their flexible foundations. Abutment
AI, piers Bl, B2 and support A2 are skewed at 48°,54°,61° and 67° respectively.
The deck's vertical load is supported by concrete arches through ribs (see Fig.
4.17 -b, Fig. 4.18 -a). The horizontal forces from the deck are transferred through a wall
1.2m thick, at the support centerline (Fig. 4.17 -b). The arches are supported by a shear
wall type bent (Fig. 4.17 -b). The bents are supported on spread footings.
The bridge is modeled as a space frame system (see Fig. 4.19). The combined
section of the deck, the ribs and the arches is modeled using four equal longitudinal seg
ments of beam type elements in each span. The segments have different section properties,
relating to the arch depth. Since the ratio of length to width of the deck is low (1: 1.4 -
159
1 :2.0) shear deformations are considered. The bents are modeled as beam colunms, with
the adequate rigid body transformations, to account for stiff connection between the deck
and the bent. The foundations system is modeled as elastic springs, using information pro
vided by a geothech consulting firm. Support A 1 is considered as fixed support. Supports
A2 and A3 are modeled as stiff elements with flexible foundations, using the given soil
properties.
The soil conditions are varying along the bridge, from very stiff soil under bent B 1
and supports A I, A3 to very soft soil under the other supports.
(b) Retrofit solution and modeling
Because of the low seismic capacity of the bents and foundations, a retrofit solu
tion using PTFE bearings system is considered. According to this solution the bents (B 1-
B5) are cut at a level of 2m below the arches spring line. Eight isolators with elastomeric
restoring springs are inserted in the space created by the cut, elevating the deck on the iso
lation system. It is assumed that by isolating the deck, the force transferred to the bents
will be reduced, by transferring some of the horizontal forces to supports Al-A3 while
also reducing the seismic demand, by shifting the bridge period and increasing the dissipa
tion of energy at the isolators (damping).
The group of isolators was modeled as single isolator element for bents B3-B5.
Several isolation elements were used however to model the isolation system of bents B 1
and B2, to be able to capture the behavior related to the skewed bents. The model of the
original bridge is presented in Fig. 4.19 and the model of the isolated bridge is presented
in Fig. 4.20.
160
(c) Spatial Ground Excitation
Since the soil conditions under the supports vary significantly, so does the ground
motion. The ground motion (Fig. 4.21) under the supports on the soft soil (Bents B2-B5
and abutment A2) is amplified and it is much larger then the ground motion (Fig. 4.22)
under the supports (abutments Al,A3, bent Bl) on stiff soil.
(d) Analysis of Bridge
In order to evaluate the seismic demand of the bridge, before and after retrofit, and
the influence of the variable ground motion, four different analysis cases were performed:
original bridge with variable and uniform ground motion and the same for the isolated
bridge.
Bridge Response for Seismic Excitation
(a) Uniform motion
It should be emphasized, that the bridge is extremely stiff in both X (longitudinal)
and Z (transversal) directions. The period of the original bridge is about 0.2 sec (Fig. 4.23
and Fig. 4.28). The period is increased to about 0.4 sec for the isolated case (the bridge is
only isolated at bents BI-B5 and at A2 in X direction). The responses in X and Z direc
tions are interacting trough the skew of bents B 1, B2. which leads to similar natural peri
ods in both directions.
In X direction the isolated bridge is vibrating mostly between supports Al and A2
while floating (sliding) over the other supports. In Z direction the bridge is bending like a
beam on three supports A I-A3.
161
In X direction there is no significant reduction in shear forces in bents I and 5 due
to isolation solution (Fig. 4.27) for the case of uniform motion. The displacements, how
ever, are significantly larger in the isolated bridge due to change of structural system from
a beam on 8 supports to beam on 2 supports (see Fig. 4.23).
In Z direction, the influence of isolation is more notable especially in the bents
which are straight (not skewed). This translates into reduction of 50% in shear forces in
bent B5 (straight), and a reduction of only 20% in bent BI (skewed) (see Fig. 4.32 - uni
form motion).
(b) differential motion
The most significant influence obtained using base isolation for this bridge, can be
observed in the case when differential motion is applied to the bridge. Supports AI, BI
and A3 experience smaller displacements than the other supports, which result in differen
tial motion between the supports. This differential motion introduce large stresses in the
bridge components. The bents of the original bridge subjected to differential motion have
significant deformation (relative movement between their top and bottom) in X direction
(Fig. 4.24), this results in significantly larger shear forces as compared to the "uniform"
ground motion case (Fig. 4.26). The isolated bridge however experiences much smaller
shear forces in X direction (Fig. 4.27) due to the "floating behavior" mentioned earlier.
The deformations of the piers are reduced at the expense of the relative deformations of
the isolators.
The effect of isolation on the bridge response when subjected to differential
motion is very significant in Z direction. While the increase of shear forces between the
162
uniform and differential ground motion for the original bridge is of the order of ten (Fig.
4.31 ) the increase for the isolated case is of the order of 2.5 (Fig. 4.30). And the signifi
cant differences between the shear forces of the isolated and original bridge are shown in
Fig. 4.32.
Remarks on Case Study #3
The case study presented herein demonstrates the ability of the new computer plat
form to model a three dimensional problem, using condensed macro elements, including
triaxial isolation models (see also section 3.2.4), and soil springs (see also section 3.3.1).
The alternative models, i.e. planer models could not capture the influence of skewness of
supports, or general purpose three dimensional models such as ANSYS require extremely
large resourcese to complete one analysis.
The case study presented also the ability to determine influence of differential
ground motion.
The analytical results show that each of the modeling details, i.e. isolators, soils,
skewness, differential motion, has indeed substantial influence on the force and deforma
tion response of the bridge.
163
AI BI B2 A2 B3 B4 85
30 30 30 36 ,
39 ,
39 ,
36 1 "'
"
f ~, '- ---J: !L --/j \~- '} ~rJ ~lo-7 ~-- ---
isolator location
(a) Bridge Top View
ribs
arch
'.hoI~'--_ bent
...--- foundation
3.0
6.0
(b) Section A-A (bent side view)
FIGURE 4.17 Bridge Top View and Typical Bent Cross Section
A3 , "'
ci -~
8
164
rib /
arch
deck
21m
(a) Section B-B (deck cross section)
21.0
inserted isolator
(b) Section C-C (bent cross section)
FIGURE 4.18 Deck and Bent Cross Section
165
".
J • •
• 1 ,
• , ! ,
•
FlGURE 4.19 Structural Model or Original Bridge· Before RetroDt .
• •
166
• , !
• o , •
FIGURE 4.20 Structural Model or Isolated Bridge
•
167
x direction 20.0
10.0
0.0
·10.0
·20.0 0 O. 5.0 10.0 15.0 20.0
z direction 40.0
30.0 .-. § 20.0 ~
P. 10.0 '" ..... 0
0.0
.10.00 .0 5.0 10.0 15.0 20.0
vertical
10.0
0.0 \-."""'"'
·1 0.00.':-0-'--'---'--'--:5"'0,.......-"---'---'--:1-:!-0'O::-'---'--'--'--1:-:5'::0-'--'---'--'--::-:!20 0 . Time Lsec.]· .
FIGURE 4.21 Ground Displacements on Stiff Soil
168
x direction 40.0
30.0 20.0
10.0
0.0 -10.0
-20.0 -30.0 -40.0
-50.00
.0 5.0 10.0 15.0 20.0
z direction 60.0 50.0
40.0
'8 30.0 () 20.0 '-'
~ 10.0 ..... 0.0 Q
-10.0
-20.0
-30.00
.0 5.0 10.0 15.0 20.0
vertical 20.0
10.0
0.0
-10.0
-20.00
.0 5.0 T' 10[0 ] 15.0 20.0
lme sec.
FIGURE 4.22 Ground Displacements on Soft Soils
169
A2 B5 0.10 0.10
0.05 0.05
0.00 0.00
-0.05 -0.05
~O.IO 0 2 4 6 8 10
-0.100 2 4 6 8 10
B2 B4 0.10 0.10
0.05 0.05
§ .. 0.00 0.00 ~
l5
-0.05 -0.05
-0.10 ·0.10 0 2 4 6 8 10 0 2 4 6 8 10
Bl B3 0.10 0.10
0.05 0.05
0.00 0.00
-0.05 -0.05
·0.10 -0.10 0 2 4 6 8 10 0 2 4 6 8 10
Time [Sec] -- isolated -- not isolated
FIGURE 4.23 Relative Deformations in X Direction, Between Ground and Deck Disp., at Piers Centerline for Uniform Ground Motion
170
I •
A2 B5 0.20 ,---r-'---'.-,--r-,-,..--.--.--, 0.20
O.lS O.IS
OW OW
O.OS O.OS
0.00 1-_"""''''''''.... 0.00 L..J"'="'" .().05 -0.05
-O.W -CliO
-O.lS -O.lS
.(l.20 .(l.20
-0.2S -0.2S
.(l.30 OL-L-2L....L...-..J4L-,-.....J6--,--.1S-.&--iW .(l.30 OL-L-2L....L...-..J4L-,-.....J6--'-.....JS--'-....IW
B2 B4 0.20 0.20 ,--.---,-,.--,--.--.---,--..--.--
0.15 0.15
O.W O.W
O.OS O.OS
0.00 0.00 i-.J""""''''''''"" -O.OS .(l.OS
-0.10 .(l.lO
-O.lS .(l.IS
-0.20 -0.20
-0.25 -0.25
-0.30 0!--'--:'-2--'--4!--'--:'-6-,-..JSL.J...-:'W -0.30 0!--'---!-2--'--4!--'----!-6--'-...,S~.-'-..,JW
Bl B3 0.20 0.20 '--'---,.-..-.,---.-,--,--,--.---,
0.15 0.15
0.10 0.10
O.OS O.OS
0.00 0.00 1---""""""'", -O.OS .(l.OS
-0.10 -0.10
-O.lS .(loiS
-0.20 .(l.20
.(l.25 -0.2S
.(l.30 OL-L-2L....L...-..J4C--'--:'-6--'--!-S-.L--i
10 .(l.30 0~--'---'2L....L...-..J4c--'--:'-6--'--:'-S--'--:'10
Time [Sec] -- isolated -- not isolated
FIGURE 4_24 Relative Deformations in X Direction, Between Ground and Deck Disp., at Piers Centerline for Variable Ground Motion
171
BI B5
above isolator 3000
2000
1000 1000
o !-11m
-1000
-2000 -2000
under Isolator under isolator 3000 r-'--'--'--,--.--,---r--,-..---, 3000 ,--r---r-'-,-.--,--,--,---r-,
1000 1000
-1000 -1000
-2000
-3000 !--'---!---'--~-'-+-'--!,---L-,J -3000 J,---,--!--,---.J!--'-~-'---!,---'--!. 2 8 10 0 2 4 6 8 10
bottom 3000
2000
1000 1000
o
-1000 -1000
-2000 -2000
-3000 !--'---J2L....&.....~-"-+--'--!-8-"--!JO -30000!--'--!2:--'--!-4 -'-.J
6L....&.....--!-S-"--,l10
dme[sec] -- differentia] motion -- urufonn motion
FIGURE 4.25 Shear Forces in The Pier's X Direction, for Piers Bl and B5, for Isolated Bridge, at Different Heights Along the Pier
172
A2 85 7000 7000 6000 6000 5000 5000 4000 4000 3000 3000 2000 2000 1000 1000
0 0 -1000 -1000 -2000 -2000 -3000 -3000 -4000 -4000 -5000 -5000 -6000 -6000 -7000
4 -7000
10 0 2 6
82 84 7000 7000 6000 6000 5000 5000 4000 4000 3000 3000
C 2000 2000 ~ 1000 1000 ~ 0 0 .E -1000 -1000
~ -2000 -2000 -3000 -3000 -4000 -4000 -5000 -5000 -6000 -6000 -7000
0 2 4 6 8 -7000
10 0 2 4 6 8 10
81 83 7000 7000 6000 6000 5000 5000 4000 4000 3000 3000 2000 2000 1000 1000
0 0 -1000 -1000 -2000 -2000 -3000 -3000 -4000 -4000 -5000 -5000 -6000 -6000 -7000 -7000
6 10 time (sec]
-- differential motion --_. uniform motion
FIGURE 4.26 Shear Forces in X Direction, for Nonisolated Bridge, at Base of piers
173
uniform motion differential motion B5 B5
4000 4000
3000 3000
2000 2000
1000 1000
0 0
-1000 -1000
-2000 -2000
-3000 -3000
-4000 -4000
'0 -5000 -5000 g 0 2 4 6 8 10 0 2 4 6 8 10 ~ .2
~ BI BI 4000 4000
3000 3000
2000 2000
1000 1000
0 0
-1000 -1000
-2000 -2000
-3000 -3000
-4000 -4000
-5000 -5000 0 2 4 6 8 10 0 2 4 6 8 10
orne [sec] - isolated - not isolated
FIGURE 4.27 Comparison of Shear Forces in X Direction, for Piers Bl and BS
174
A2 B5 0.10 I I I 0.10
0.05 f- - 0.05
0.00 .k 0.00 "I
·0.05 f- - ·0.05
·0.10 0 I I I I
10 ..().10 0 2 4 6 8 2 4 6 8 10
B2 B4 0.10 0.10
0.05 0.05
:§: Q. 0.00 0.00 .:!l Q
·0.05 .0.05
.0.100 2 4 6 8 10 .0.10 0 2 4 6 8 10
Bl B3 0.10 0.10
0.05 0.05
0.00 0.00
·0.05 -0.05
.0.100 2 4 6 8
Time [Sec) 10 .0.10 0 2 4 6 8 10
-- isolated -- not isolated
FIGURE 4.28 Relative Deformations in Z Direction, Between Ground and Deck Disp., at Piers Centerline for Uniform Ground Motion
175
A2 B5 0.10 0.10
0.05 O.OS
0.00 0.00
-0.05 -0.05
-0.100 2 4 6 8 10 -0.10 0 2 4 6 8 10
B2 B4 0.10 0.10
0.05 0.05
:§:
~ 0.00 0.00
-0.05 -0.05
-0.10 0 2 4 6 8 10 -0.10 0 2 4 6 8 10
Bl B3 0.10 0.10
0.05 0.05
0.00 0.00
-0.05 -0.05
-0.100 2 4 6 8
Time [Sec] 10 -0.10 0 2 4 6 8 10
-- isolated -- not isolated
FIGURE 4.29 Relative Deformations in Z Direction, Between Ground and Deck Disp., at Piers Centerline for Variable Ground Motion
176
bl b6
above isolator above isolator 6000 r-~~--r-~~~--_~~--' 6000 ""--"'-"'--'--,--.---r--r--,-,.--,
5000
4000
3000
2000
1000
o I-~''-,J\..,-/\,~ ,J
-1000
5000
4000
3000
2000
1000
o -1000
-2000 -2000
-3000 OL--I---1-.J--L-,--..L-L_.l.--I---1 -3000 OL--L..--12--'--4L.-L..--1
6-.J--L
S -,--.J
IO-
under isolator uruler isolator 6000 r-~~~~~~~~~--~ 6000 r-~~~~r-~~~~r-~~
5000 5000
4000 4000
3000 3000
2000 2000
1000 1000
o 0
-1000 -1000
-2000 -2000
-3000 L-'--L--'~.L--'--L----'_L--'-..J_ -3000 L..I..--L---,-__ L--,--...L---,--'L--,-.J o 2 4 6 8 10
bottom bottom 6000 6000 '--~-'-""""T-.-r--~-,-~---,r--~-,
5000 5000
4000 4000
3000 3000
2000 2000
1000 1000
o 0
-1000 -1000
-2000 -2000
-3000 OL--I---1--L:.-L-,--..L-L-.l.--1---1-3OOO0 !--,--.J2l.-.J--L4-,--.J6l.--I--!-S--L-..JIO
timelsec)
-- differential motion .. _-_. uniform motion
FIGURE 4.30 Shear Forces in The Pier's Z Direction, for Piers Bl and B5, for Isolated Bridge, at Different Points Along the Pier
177
------------------------ ----------
A2 BS 20000 20000
15000 IS000
10000 10000
SOOO SOOO
0 0 ~,.
-SOOO -5000
-10000 -10000
-15000 -IS000
-20000 0 2 4 6 8
-20000 10 0 2 4 6 8 10
B2 B4 20000 20000
IS000 IS000
10000 10000
I SOOO sooo
8 .E 0 0
~ -SOOO -SOOO
-10000 -10000
-15000 -IS000
-20000 0 2 4 6 8
-20000 10 0 2 4 6 8 10
B1 B3 20000 20000
IS000 IS000
10000 10000
SOOO SOOO
0 0
-SOOO -SOOO
-10000 -10000
-IS000 -IS000
-20000 0 2 4 6 8
-20000 10 0 2 4 6 8 10
time [sec] -- differentia1 motion -- umfonn motion
FIGURE 4.31 Shear Forces in The Pier's Z Direction, for Nonisolated Bridge, at Their Base
178
uniform motion differential motion
BS BS 4000 20000
3000 1S000
2000 10000
1000
0
·1000
·2000
·3000
~ -4000 0 2 4 6 8 0 2 4 6 8 10
~ .2
~ BI BI
4000 20000
3000 1S000
2000 10000
1000 sooo -1\'\
0 0
·1000 ·SOOO \ '
·2000 V ·3000
·4000 0 2 4 6 8 2 4 6 8 10
time (sec] - isolated - Dot isolated
FIGURE 4.32 Shear Forces in The Pier's Z Direction, for Piers Bl and BS
179
4.1.4 Case Study #4: Bridge over Los Angeles River ·Retrofit Solution with Base Isolation and Restrained Expansion Joints.
A large size bridge modeled from a prototype bridge in Los Angeles with deficient
seismic capacity, is analyzed to demonstrate the capacity of the computational model to
evaluate the efficiency of retrofit of a bridge using base isolation, when the deck has mul·
tiple expansion joints restrained, or unrestrained. The bridge suffered minor damage dur-
ing past earthquakes, but using simplified methods, it was found that the seismic capacity
of the bridge is smaller then the predicted demand in future events. The original bridge
was analyzed and a selected retrofit scheme based on introduction of seismic isolation sys-
tem similar to case study #3 is evaluated.
Several techniques of retrofit were initially considered. Among them the conven-
tional techniques of strengthening the piers with steel jackets. It was found that this tech-
nique cannot provide a good retrofit solution to the problem, since this type of retrofit
increases the bearing demand on the foundations due to the additional weight, and
increase the seismic forces to a level of which the foundations will not be able to resist.
For this type of retrofit the foundations will have to be retrofitted also, at large cost, proba-
bly prohibitive. On the other hand, retrofit using base isolation, doesn't change the capac-
ity, but is expected to reduce the demand, and therefore increase the reliability of the
bridge.
The bridge, shown in Fig. 4.33-a is a mUltiple concrete arches bridge with 12
spans of an average opening of 24.0m (between 17.5m and 35.0m) supported by bents of
changing height, of 8.0m to 19.0m. The bridge supports 4 traffic lanes, and has width of
about 20m, and height of O.5m. It is supported by 5 parallel arch girders in each span. The
180
original bridge has thermal expansion joints, 20 rnrn in size, in span B3-B4 near bent B3
(shown in Fig. 4.33-b), and similar joints near bents B6 and B9. The two abutments are
very rigid, and can be considered as rigid supports. The foundations of the bents, sup
ported by soft soil, are considered as flexible. The passive pressure on the piers below
grade is considered as horizontal flexible support.
For evaluation, the bridge is modeled as a space frame (see Fig. 4.34). In the cur
rent analysis the bridge is assumed to be elastic, and the nonlinear behavior is concen
trated, in the base isolation system and in the expansion joints only. The deck sections are
modeled as a single beam elements. Since the ratio of length to width is small, the shear
deformations of the beam should be considered. The deck elements are connected to the
bents, using rigid body transformations (coupled motions - see section 3.2.1). The expan
sion joints are modeled using several "gap" elements (see section 3.3.5), which are trans
formed from the bridge center line by using rigid body transformations (rigid arms see
section 3.2.1).
Two retrofit schemes are considered for the bridge:
1) The bridge is fully isolated as shown in Fig. 4.35-(a) friction (teflon and stain
less steel) devices are inserted in bents 3, 6 and 9 just under the bents crown, and
above ground level in the rest of the bents. At the abutments, a horizontal and vertical
cut is created and the isolation system is inserted between the bridge deck and the
abutment. Springs in the form of elastomeric springs are suggested to be inserted near
the sliding supports (see Fig. 4.35 (c». Those springs are not transferring any vertical
load because they are softer then the isolation system. The natural period of the iso-
181
lated bridge was chosen to be T = 2.5 sec. The ratio between the initial vertical force
P initial on the isolator and the restoring spring stiffness kisolated is in this case:
k = 4rt2P
initiai isolated r g (EQ4.1)
The friction coefficient was assumed to change, from 5% at 0 velocity to 10% at veloc-
ity of 0.1 mlsec and above. The isolation system is modeled using the triaxial isolation
element (see section 3.2.4).
2) The retrofit according to the second alternative, has a similar arrangement to the
first alternative, except that the connection between the deck and the abutments
remains rigid (not isolated) and can transfer forces without relative m~vement. This
type of solution is considered because of the significant cost involved in isolating the
abutment
The two retrofit schemes are compared to each other and to the original unretrofit-
ted bridge, and the importance of various modeling components is evaluated .
.!. Behavior of bridge under seismic loading
Since the deck of the bridge is sloped from the elevation of 8.3 m at the west (left)
abutment to 19.2 m at the east (right) abutment (Fig. 4.33-a), the bridge is supported by
columns with changing height. The longer columns are much more flexible then the
shorter columns. Under seismic excitations, in the original bridge, there is transfer of force ";1'
through the deck from the long columns to the short ones. The transverse shape of the
deformed bridge at time 2.0 seconds is shown in Fig. 4.1 (a), shows that bents b7 b8 and
b9 experience the largest displacements while bent b I hardly moves. At the same time the
182
displacements of the retrofitted bridge according to alternative 1 (see Fig. 4.1(b» are
almost uniform along the bridge, showing that the deck is floating over the isolators. As a
consequence the forces of the isolated bridge, at the tall part of the bridge at bents b9 and
b 1 0 are up to three times smaller then the forces in the original bridge, as shown in Fig.
4.39.
The deformations of a typical frame can be observed for the original bridge in Fig.
4.37(a) and for the bridge retrofitted according to alternative I in Fig. 4.37(b). The defor
mations in the original bridge resulting from the bending of the piers, while in the isolated
case, the movement is mostly due to relative displacements in the isolation system and the
piers bending is minimal.
For the retrofit with alternative II, the behavior of the structure is predominantly
similar to the behavior of the isolated bridge retrofitted according to alternative I.
Although the displacements at the two ends are much smaller, as shown in Fig. 4.40.
Therefore the forces in the bents are close to the forces in alternative I of the isolated
bridge, except for the end bents, which transfer some of their force, directly to the abut
ments (Fig. 4.39).
In both retrofit alternatives, there was significant reduction in shear force demand
especially in the long columns, and as a result reduction in bending moments, which
increase significantly the reliability of the bridge.
Remarks on Case Study #4
183
The modeling <!f the bridge included transversely distributed isolators and flexibil
ity of transversely distributed piers with soil springs influence. This modeling could be
achieved alternatively either with large general purpose computer codes i.e ANSYS,
ADINA, etc.) inefficient in carrying the computations, or with much simplified coeds like
(DRAIN 2DX, IDARC-2D etc.) but with restrictive assumptions for isolators behavior
and spacial behavior of the overall structure. The solution was obtained in presence of
highly nonlinear expansion joints models (gap. elements) with stiffness changing from
zero to extremely high values. The solutions converged for long time records as shown in
the time histories displayed.
184
... ,
[
'"
.... ,
r '"
A +19.2
. -I.! "" 18,0 , .... "., l!1.~ '" 19.5 n.' '" '" ".,
a) Bridge Side View (As Built)
ppJoint PI'JoInr
811' joint 'tt' B
B
, .... 18.0 '" "., '" '" 19.5 n.' '" '" ".,
b) Bridge Side View With Expansion Joint (After first retrofit) Ifo· ___ :2om
deck surface "I'D o.s:L --I j!:m
• • .j,!' ~
SecB-B Section through joint
T
elastomeric or sliding bearings
SecC-C
Sec A-A Cross Section
FIGURE 4.33 Original Bridge Before and After First Retrofit
.... ,
....
185
.....
"
FIGURE 4.34 Analytical Model of the Bridge
186
A +19.2 ... , 1 1
lI- I , .... , ... '" ,,~ 18.0 ,,~ ,,~ '" '" 19.5 21.0 ,,~
'" 35.0
GGGGG8G8GB8 a) Base Isolated Bridge With Expansion Joints
~.:~.:,~< , ',-,'
g g g Sec A-A
+ .,., ... , r .... ,
"'" '" , .. 18.0 "" ,,~ '" ,,~ ,,~ 17.0 ,,~ '" ".0
b) Base Isolated Bridge Without Expansion Joints
bridge deck
bridge pier
c) Retrofit Isolation System Detail
FIGURE 4_35 Base Isolated Bridge with and without Expansion Joints
187
~. · .... ~·t .... ·4· ...... J .......... ·t'; ....... }, .. · .... t:z .... ·~\ ...... i= ... 1 ........ t= .. ~--==t:::=".-I ., -- -~-~-.- -""- --1 " \ \ ,.. , ,-- . -.. \ - ," -
\1 \ \1
. ,
(a)
(b)
FIGURE 4.36 Displaced Shape or (a)Orlglnal Bridge, (b) Isolated Bridge
188
~--~------~----------"
I I
I I
•
I
(a)
(b)
,
FIGURE 4.37 Displaced Shape or Typical Frame In the (a) Original Bridge, (b) Isolated Bridge
189
-- no gap -- not isolated
b2 bll 1000 1000
800 800
600 600
400 400
200 200
0 0
-200 -200
-400 -400
-600 -600
-800 -800
-1000 -1000 0 2 4 6 8 10 0 2 4 6 8 10
bl·n blO 100 1000
80 800
60 600
40 400
C 20 200 g 0
" .s 0 0
" -20 -200 0
-Ii -40 -400
-60 -600
-80 -800
-100 -1000 0 2 4 6 8 10 0 2 4 6 8 10
bloc b9 100 1000
80 800
60 600
40 400
20 200
0 0
-20 -200
-40 -400
-60 -600
-80 -800
-100 -1000 0 2 4 6 8 10 0 2 4 6 8 10
time {sec)
FIGURE 4.38 Comparison of Shear Force in Non Isolated Bridge, and Bridge Without Gap
190
-- isolated .... _-- not isolated
b2 bll 1000 1000
SOO SOO
600 600
400 400
200 200
0 0
-200 -200
-400 -400
-600 -600
-800 -800
-1000 -1000 0 2 4 6 S 10 0 2 4 6 8 10
bl·n blO 100 1000
SO SOO
60 600
40 400
" 20 200 g ~ 0 0
O!
1 -20 -200
-40 -400
-60 -600
-SO -soo
-100 -1000 0 2 4 6 S 10 0 2 4 6 S 10
bloc b9 100 1000
SO SOO
60 600
40 400
20 200
0 0
-20 -200
-40 -400
-60 -600
-SO -800
-100 -1000 0 2 4 6 8 10 0 2 4 6 8 10
time [sec)
FIGURE 4.39 Comparison of Shear Forces in Original and Retrofitted (AIt.1) Bridge
191
-- isolated -- nogap
b2 bll 1000 1000
800 800
600 600
400 400
200 200
0 0
-200 -200
-400 -400
-600 -600
-800 -800
-1000 -1000 0 2 4 6 8 10 0 2 4 6 8 10
bI-D blO 100 1000
80 800
60 600
40 400
'0 20 200 g ~ 0 0
oS ~ -20 -200 • 'Ii
-40 -400
-60 -600
-80 -800
-100 -1000 0 2 4 6 8 10 0 2 4 6 8 10
bloc b9 100 1000
80 800
60 600
40 400
20 200
0
-20 -200
-40 -400
-60 -600
-80 -800
-100 -1000 0 2 4 6 8 10 0 2 4 6 8 10
time (sec)
FIGURE 4.40 Comparison of Shear Force In Isolated Bridge, and Bridge Withont Gap
192
- isolated b5 blO -nogap
0.10 0.10
0.05 0.05
0.00 0.00
-0.05 -0.05
·0.10 ·0.10
-0.15 0 2 4 6 8 10..0,15 0 2 4 6 8 10
b4 b9 0.10 0.10
0.05 0.05
0.00 0.00
-0.05 -0.05
-0.10 -0.10
.0,150 2 4 6 8
10-0,150 2 4 6 8 10
b3 b8 0.10 0.10
0.05 0.05
S 0.00 0.00 ~
~ ._ -0.05 -0.05 Q
-0.10 -0.10
·0.15 0 2 4 6 8
-0.15 10 0 2 4 6 8 10
b2 b7 0.10 0.10
0.05 0.05
0.00 0.00
-0.05 -0.05
-0.10 -0.10
-0.15 0 2 4 6 8
-0.15 10 0 2 4 6 8 10
bI b6 0.10 0.10
0.05 0.05
0.00 0.00
-0.05 -0.05
-0.10 -0.10
-0.150 2 4 6 8 10-0.15 0 2 4 6 8 10
Time [Sec)
FIGURE 4.41 Comparison of Deck Displacements in Isolated Bridge, and Bridge Without Gap
193
- isolated b5 blO - Dot isolated
0.10 0.10
0.05 0.05
0.00 0.00
-0.05 -0.05
-0.10 -0.10
-0.150 2 4 6 8 10 -0.15 0 2 4 6 8 10
b4 b9 0.10 0.\0
0.05 0.05
0.00 0.00
-0.05 -0.05
-0.10 -0.10
-0.15 0 2 4 6 8 10-0.15 0 2 4 6 8 10
b3 b8 0.10 0.10
0.05 0.05
E 0.00 0.00 0. ~ -0.05 -0.05 is
-0.10 -0.10
~ -0.150 2 4 6 8 JO ·0.15 0 2 4 6 8 10
b2 b7 0.\0 0.10
0.05 0.05
0.00 0.00
-0.05 -0.05
-0.10 -0.10
-0.150 2 4 6 8 10-0.15 0 2 4 6 8 10
bI b6 0.10 0.10
0.05 0.05
0.00 0.00
-0.05 -0.05
-0.10 -0.10
-0.150 2 4 6 8 10-0.15 0 2 4 6 8 10
Time [Sec]
FIGURE 4.42 Comparison of Deck Displacements In Isolated Bridge, and Not Isolated Bridge
194
-nogap b5 blO - not isolated
0.10 0.10
0.05 0.05
0.00 0.00
-0.05 .0.05
-0.10 -0.10
-0.15 0 2 4 6 8 10 -0.15 0 2 4 6 8 10
b4 b9 0.10 0.10
0.05 0.05
0.00 0.00
-0.05 -0.05
-0.10 -0.10
-0.150 2 4 6 8 10 -0.15 0 2 4 6 8 10
b3 b8 0.10 0.10
0.05 0.05
E 0.00 0.00
0. -0.05 -0.05 '" is ·0.10 -0.10
-O.IS 0 2 4 6 8 10-0.15 0 2 4 6 8 10
b2 b7 0.10 , 0.10
0.05 0-"
0.05
0.00 ~ _L:I L'- 0.00 V
-0.05 0-"
-0.05
-0.10 E- , -0.10
-0.15 0 1 ,
10 -0.15 0 2 4 6 8 2 4 6 8 10
bi b6 0.10 0.10 , I I I
0.05 ~ -:: 0.05
0.00 0.00
-0.05 ~ -: -0.05
-0.10 ~ -: -0.10
..0.15 0 I 1 1 1
10 -0.15 0 2 4 6 8 2 4 6 8 10
Time [Sec]
FIGURE 4.43 Comparison of Deck Displacements In Non Isolated Bridge, and Bridge without Gap
195
4.1.5 Case Study 5: Miyagawa Bridge
An existing bridge in Japan is used to demonstrate the difference between elastic
and inelastic behavior of base isolated bridges. The bridge was used as a pilot project for
base isolated bridges in Japan. Those bridges are called in Japan Menshin bridges. The
Menshin design is aimed toward increasing the energy dissipation capacity of the bridge
system rather then shifting the period of the bridge.
Bridl:e Description
The Miyagawa bridge is a three span continuous bridge, the total length of the
bridge is 105.5m, the width is lO.5m and it was erected across the Keta river as part of
National Highway No. 326. A view of the bridge is shown in Fig. 4.44 and a geometric
description is given in Fig. 4.45 (Buckle 1992). Lead rubber isolators were used to isolate
the deck from the abutment and piers in the longitudinal direction only; the bearings are
locked in the transverse direction. A geometrical description of the lead rubber bearings is .
shown in Fig. 4.46 for the abutment bearings and in Fig. 4.47 for the pier bearings. The
total weight of the deck is 1320 metric tons
The cross section and reinforcement of the bridge piers is shown in Fig. 4.48. As
shown in Fig. 4.45 the cross section of the pier is changing along it's height and so is the
reinforcement as shown in Fig. 4.48. The force displacement curves used for the bearings
as derived from field test are shown in Fig. 4.49.
The structural model of the bridge used in the analysis is shown in Fig. 4.50. The
supporting pier was modeled by several sections in order to take into account the changing
stiffness and the changing mass along its height. The pier foundations are represented by
196
two infinitely stiff elements (elements 1,2,3 and 4) and the weight of the foundations of
300 tf is applied in between them (joints 5 and 6 in Fig. 4.50). The inelastic pier is split
into several elements (elements 5 to 11 and 6 to 12) for the following two reasons: (i)The
geometric section of the pier and the hysteretic properties are changing over the height of
the pier (ii) It enables the application of the mass of the column in intermediate locations
along the height of the pier. The isolator elements are located above the piers, and above
the abutment. The deck weight is concentrated at the joints between the deck and the piers
and between the deck and the abutment. The deck is assumed to act as an elastic beam.
The inelastic properties of the piers segments shown in Table 4.1 . Those values
were calculated from the section geometry, reinforcement, and the applied axial force.
TABLE 4.1. Hysteretic Pier Nonlinear Properties
member Crack Crack Yield Yield Ultimate Ultimate section # Moment Curvature Moment Curvature Moment Curvature
(Fig. 4.50) MeCtm) cI> (11m) My Ctm) cI> (11m) MuCtm) cI> (11m)
5 819.75 0.9389E-4 1509.50 0.1I29E-2 1632.42 0.3005E-l
7 859.89 0.8832E-4 1568.16 0.II09E-2 1687.06 0.3207E-l
9 886.02 0.8279E-4 1126.48 0.1057E-2 1217.59 0.3939E-1
II 1001.87 0.7714E-4 1135.92 0.1029E-2 1225.71 0.4402E-l
The ground was modeled by elastic rotational and translational spring.
Two input ground motions were used for the analysis, as it is required in the Japa-
nese design code. According to the Japanese code, it is required that the bridge remains in
the elastic range for a moderate earthquake described as level 1 design earthquake, and
survive a strong earthquake designated as the level 2 design earthquake (see Fig. 4.2)
197
without collapse. The ground conditions of the site is stiff soil which are under the cate-
gory of ground condition 1.
Two types of analysis were performed: (i) elastic analysis was performed using the
initial stiffness of the piers as presented in Table 4.1 . (ii) nonlinear analysis were the stiff-
ness of the piers changing according to the properties in Table 4.1 .
Since in the current study, the interest is concentrated in analysis of structures sub-
jected to severe earthquakes, the presented results are for the level 2 earthquake only. The
anal sis results for th I v
the structure is subjected to large displacements at about 2.0 sec from the beginning of the
excitation, and the base of the pier undergoes large inelastic deformation in which the cur-- -~
vature is about 116 of the ultimate curvature, so that the structure can survive this excita-
tion. The displacements of the deck are larger then the design displacements which control
the size of the required gap in the abutment, the relative displacements of the pier from the
analysis where about 230mm compare to 130mm which are the displacements calculated
from the linearized design procedure.
In order to compare inelastic analysis, where the strength of the element is limited
to elastic analysis, where the strength of the element is infinite, both type of analysis were
performed on the isolated bridge. As mentioned earlier, the properties presented earlier
and marked on the bridge model, were used. The results of those analysis are presented in
Fig. 4.51 and Fig. 4.52.
198
The displacements of the isolated deck are hardly influenced, by the limited capac-
ity analysis, but the displacements at the top of the bent under the isolator are much
smaller in the elastic case.
The moments at the bottom ofthe pier are kept low in the case of the inelastic anal-
ysis while for the case of the elastic analysis they are 50% higher.
-..;:~?~ 1 ~.' ""', ' , "
FIGURE 4.44 Completed Mlyagawa Bridge
199
.................. .. .... ............ ., ,
, ... ~ .,
,
t , .. " •
~~:'~. ! - " f ~ ,
! .... , .-.... . ... .... r ...... ~ , .
L4l • Side View
--
At suppon AI Center of span
• .... '" 1100 - ...
AUI\" • •• ~_IJ, I. J.,... ... !OD 10 "" '00!l
~ ."""'.~!..I I _If_ . C.-.c.,n ... , .... , .. ". "''C ....... I .,..
bI! -~ • -
r.,l, ", . '.~ "" I~k~
8 V\ .I !Vi : , " . ".:! .. '
-,~, • • ,.oo_tUO UO
Supersaucrure
' . . II
! II
I~
l ,
~
;:
, ~ .'
: '""f.. : . Jll " ' ~ .. ' "'~: "'r ....... .-:(:.'. ' ... . .~. . .-.. ................. ... .. --l i":r .".. . Lf-1
L _
, • - ---too .Il00 toO
e ...... ' J .towc··l r-"-'" .-"" \!wh.
I ,
1---
~\ --- --
i h , ...... ~ .. ~ .... (' ... , ....... l\ •. __ .1 ..... YI ,. ....... e· .. " .... .000 ~x- .. --·-
j !
I I I '10 ."
Bridge Pier
FIGURE 4,45 Bridie Geometry
200
·AI
i' • '1$ ~ •
FIGURE 4.46 Lead Rubber Burlngs for BridIe Abutment
IS 41+SUSlII
1-22
;'1
A-A
J: ..... :=J ~;..~-·.fA-+:143 :[, I, 1000
ISO I,XlO ~..:S30=--rIO IXI-M 20 111-.....lii:O::"'o -lll
• • • • .~ 4· ~+. . + ..
• 4- 70
10
FIGURE 4.47 Lead Rubber Burlnp for BridIe Pier
201
At Mid·height
2@300 -600
At Bottom
3100
10@300=SOOO 20@150=3000
--irt----- ,-------.----- ---H-
3989.5 20@150=3000
3100
4833 4000
D22
FIGURE 4.48 Cross Section or Plen
.... ""'" 70 • • '! r! ,
~ ~F - U curve'uled for dellgn .. 50 ·····f·····r .. ···;·····,····· ····:·····f·· .0; : ..... .0 ·····'·····1··· .. ·;·····,,· .. ··· .... ":...... ..:- ... : .....
30 ·····i·····i·····i·····i····· .... -!o.. ··i····· - " I . . .~ 20 ·····'·····1·· .. • ..... j..... .. .. .. I ..... .. 10 ••••• i··· •. i· .. ·.i..... . .. -} ......... i. ·.·i.·· .. o O. .::
- • •• I • -10 ..... : .... : .......... .:..... . ... , ..... ; ..... : .... . - -20 ••••• ! ..... t ••• :.. • .0' •••• l .... ~.:. ..... :. ..... : .... . • • I • • •
30 I • t.. .. - ...•. !.. . tOO. ··1····· ····7·····;·····:·····!····· II -.0 ··· .. ·i·· ::' ·l·····:····.. ····T·····:·····r·····!····· o -so ·····1· s·····,···· .. ,,····· .... .,. ..... , ..... , .....•..... H -&0 .•.•• , ....... ,.................. • .•.. ;. ............ ; ...•• i .... . - ...... .. . :; -JYSO-I20-eo -~ -30 0 lIO ~ eo 120 ISO :c Horizontal dhpllcelDent (IDIII) ,
FIGURE 4.49 Load Displacements CU"es
202
., ...
." ..
t
+13.0
Y
+9.0 .
Y:-
+4.0 . -y:--
+2.0
Y: +1.0 .
Y: +0.0
Y
IS
3
lumped mass 20
M '= co Y'
13
= 1126t~
I y = 1126t~
y = 1570t~
14
8
5 6
r ' 8 , . 4
• • • ' ••• J • •• ; •• 6
6 k, = 2.77xlO tmlTad
18
16
w=114t
. . ,..,326l' ....•.. ,
5 kT = 2.21xlO tim
FIGURE 4.50 Structural Model of the Bridge
DECK
. , . 4
203
2000
1000
E 0
6 " c
j -1000
-2000
-3000
2000
1000
E 6 " 0 c
~ Q
::;
-1000
-2000
moment at bottom of foundation
I
~~ ~ rf"
~ o 5
~k ~h\ rr
~ o 5
~ '11 ~
it ,
10 Time (Se<;.)
II f-
IJ I tI A
IJV
15
-- inelastic elastl.c
moment at bottom of pier
K~ V
-
10 Time (Sec.)
IAt./\ pv
15
FIGURE 4.51 Moment at Bottom of Pier and Foundation
20
20
204
!
I !
! " c
~ ~ s-is
0.05
-0.05
.().l5
·0.25
0.03
om
0.00
·0.01
·0.02
Pier displacement above isolator
I - M ~ ~
fJ
o 5
rJ\~ ~ V
to Time (Sec.)
fI
~ I" " v~-0v
15 20
elastic --"'-'-- -inerastic
Pier displacements, under isolator
~A ~
l!. Id ~I~ , 11'1111 IJ 1/1 ~ f\/\
~ ,
11'1
l' I W ~ I M 1"
HI ·0.03 0 5 10
lime (Sec.)
15 20
FIGURE 4.52 Displacements at Pier Centerline, Under and Above Isolator
205
4.1.6 Case Study 6: Evaluation of Bridge Capacity Using Different Push· Over Procedures and Dynamic Analysis
The concept of using push-over procedures for the entire bridge assembly instead
of using this procedure on each frame separately where the interaction between the frames
is neglected to evaluate the capacity of a bridge was presented in section 3.6.2. In this sec-
tion a bridge that was subjected to the Northridge earthquake and experienced collapse
was evaluated using various push-over procedures: (1) uniform incremental static load. (2)
modal adaptable incremental load. (3) acceleration ramp input (4) pulse input. The
response to the various push-over evaluation procedures is compared to the response of
the bridge to measured earthquake record with impulsive nature.
The southbound SR14/1-5 separation and overhead structure is located in Los
Ang(!lesCounty approximately 24 miles to the northwest-of downtown-Los-Angeles.-A
general plan and elevation of the bridge is shown in Fig. 4.53 , a typical cross section of
the box girder deck is shown in Fig. 4.54 and the geometry and reinforcement of the piers
is shown in Fig. 4.55 .
The bridge was modeled using combination of elastic and nonlinear hysteretic
concrete beam column elements. The prestressed deck and the top part of the piers was
modeled as elastic beams. The bottom part of the piers were inelastic behavior is expected
was modeled as concrete hyste,etic beam column element. The piers are supported on
shafts. In order to consider the flexibility of the shafts, the height of the piers is increased
for the analysis the increased height is shown in Table 4.2 . The analytical model of the
bridge is shown in Fig. 4.56 . The heights of the piers and the elastic properties of all the
elements are presented in Table 4.2 . The moment - curvature relations and the shear
206
capacities of the nonlinear elements are presented in Table 4.3 . Since the current analysis
focus on ~he behavior of the bridge in the transverse direction, the properties related to the
behavior of the bridge in this direction are presented only. The stiff part at the top of the
piers is modeled using rigid arm transformations.
TABLE 4.2. Geometry and Structural Elastic Properties of the Bridge Elements
Height (for Moment orinertla
Element analysis) Area gross longitudinal transversal torsional
# m m2 m4 m4 m4
pier 2 . 13.4 4.2 0.49 4.23 1.74
pier 3 13.7 4.2 0.49 4.23 1.74
pier 4 17.5 4.2 0.49 4.23 1.74
pier 5 18.0 4.2 0.49 4.60 1.74
pier 6 29.2 3.7 0.85 5.20 2.37 -
-
pier 1 41.2 4.0 1.35 5.20 3.70
pier 8 30.9 4.0 1.35 5.20 3.70
pier 9 32.6 4.0 1.35 5.20 3.70
pier 10 17.6 4.2 0.49 4.23 3.70
deck nla 8.8 5.85 188.56 16.30
TABLE 4.3. Transversal Moment Curvature Relations and Shear Capacity of Piers
Cracking Yielding Ultimate Shear
pier moment corvo moment curv. moment curv. capacity
# kN·m radlm kN·m radlm kN·m radlm kN
2 22600 0.21 68740 1.29 75000 3.99 6925
3 23100 0.21 69740 1.31 75000 3.95 7065
4 20435 0.21 54570 1.18 70000 4.65 6345
5 20435 0.21 54570 1.18 70000 4.65 6365
6 26160 0.20 78250 1.39 85000 3.89 7347
7 24900 0.18 64260 1.25 70000 5.34 7041
8 19130 0.15 56360 1.23 70000 5.27 5770
9 22960 0.18 61670 1.27 70000 4.76 6608
10 22660 0.21 57320 1.16 70000 4.46 7885
207
•
/' 1582'.()·
< Ii II~ II (I II if» Pier 3 Pier4 PierS~-...(~ P~r8 ~erlO
Pier 6 PI.r 7 Pier 9
c "
Abulmenll Pier 2 Abutmenlll
ELEVATION ,. 53'..Q· II
T ~
d I. : I
~ P4 P5 P6 P7
PLAN P10 AI All
FIGURE 4,53 SR1411·5 Separation and Overhead (Southbound) • General Plan and Elevation
208
• I PRESTRESSED BOX GIRDER SECTION REINPORCED BOX GIRDER SECTION
~
,;
r--~ ~ 0.305 ~
col TYP. ~
Ill' ~
,;
- --1.118 --- - -
0 p5 1.118 ..2.,892 2.743 .
0.152
• TYPICAL !ECTIONS
I • I • I
I • I 0.203 • TYP. I
: I
-
13.810 --
• I
-
1.372 ~ 1.372 2.743 •
18.154
DIMENSIONS IN 1m)
~ ~
,;
~ ]r-~
~ ~
CII' ., -- ~ - --,;
1-1.118
O. 5
2.743 1.118
0.102
FIGURE 4.54 SR1411·5 Separation and Overhead (Southbound) • Typical Deck Section
209
".
." .. , ..
tt· ,
---. .:~---'!""-4!":::'::.J
FIGURE 4.55 SR1411.5 Separation and Overhead (Southbound) .Pler Sections and Relnrorcement.
210
FIGURE 4.56 SR141I·5 Separation and Overhead (Southhound) ·Structural Model for Analysis
211
Evaluation procedures
The bridge was subjected to several push-over procedures: (1) Modal adaptable:
static force distributed relative to the changing mode shapes due to change in the stiffness
of the elements when experiencing inelastic behavior. (2) Uniform pseudo acceleration:
the static incremental load is distributed relative to the mass (the distribution doesn't
change during the analysis). (3) Ramp loading (0.25 mlsec3, 0.5 mlsec3, 5.0 mlsec3): lin
early increasing acceleration (Fig. 4.60 -b, Fig. 4.61 -b, Fig. 4.62 -b), is applied in differ
ent rates to the bridge until failure is reached. (4)Pulse loading (0.1 sec, 0.05 sec): a short
duration acceleration pulse (Fig. 4.63 -b, Fig. 4.64 -b) with two different durations is
applied to the bridge and its failure is monitored.
An impulsive ground motion, measured during the Northridge earthquake, at the
Santa Monica City Hall, increased to 150% was also used to analyze the bridge for earth
quake loading.
Results
The limit state of the bridge was defined by two conditions: (i) flexure failure,
where one of the piers, reached its ultimate flexture capacity (see Table 4.3 ). (ii) shear
failure, where one of the piers reached its shear capacity (see Table 4.3 ). The loading his
tory, the shape of the bridge at time of failure, and the history of shear forces in piers 2 to
7 is presented in Fig. 4.58 to Fig. 4.65 for the various analysis cases. The distribution of
the shear forces at failure is presented in Table 4.4 and graphically in Fig. 4.57 . The dis
tribution of moments at failure is presented in Table 4.5 . Indications on the performances
of the bridge subjected to the various loading procedures are presented in Table 4.6 .
212
----~- -
Results evaluation
Significant deformations were observed at the time of failure when the bridge was
analyzed for the first five push-over procedures (modal, uniform, three ramp loading pro
cedures). The failure mechanism for all these procedures was found to be shear domi
nated. However the moments at the time of failure in al these cases were very close to the
ultimate moment capacity. The failure occurred in all cases in short piers located in prox
imity to long piers (pier 3: 13.7m/pier4: 17.5m;pier5: 18m/pier 6, 7: 29m, 41m) due to
transfer of shear forces from the flexible support to the stiffer support (Fig. 4.57 ).
The external inertia forces on the bridge F can be related to its global acceleration
iilO/al such that:
F = wiilolal (EQ4.2)
where w is the weight. The total acceleration can be divided into two parts: the ground
acceleration ii g and the structure relative acceleration to its base ii as
(EQ4.3)
The forces developed in the bridge in the case of modal adaptable analysis are related
directly to relative acceleration ii only. Higher forces are distributed to the softer parts of
the bridge and very low forces are applied to the stiffer parts.
The distribution of the forces in the cases of uniform and ramp loading is related to
the distribution of the ground acceleration ii g' The distribution in all this cases is similar.
However, the force in pier 2 are higher by 30% in the case of steep ramp loading (5.0 mI
213
sec3) then the other cases because of significant dynamic amplification for stiff elements.
The total force in the bridge F can be defined as:
F = ku+mii (EQ4.4)
The part of the force that control the failure is ku, therefore higher ground acceler
ation is required in the dynamic cases (Table 4.6 ) then in the static and low acceleration
cases.
The case of pulse loading provides very different behavior then the previous cases,
the failure is dominated by shear at the stiff piers, where most of the shear force is concen
trated (Fig. 4.57 ) which results in failure of pier 2. The maximum deformations are very
low (0.058m, 0.046m) and all the piers have almost the same displacements, because at
this stage only the inertia effect influence the response, while the effect of the piers stiff
ness was not developed yet. Very high ground acceleration is required in a very short
period of time to bring the bridge failure. The moments in the piers are relatively low at
the time of failure.
A very impulsive ground motion was chosen as input to perform nonlinear time
history analysis. This kind of ground motion is characteristic to near fault seismic events.
The ground motion was increased to 150% to be able to bring the bridge to failure during
the first large pulse of the earthquake. The response of the bridge at failure is characterized
by a higher mode (see Fig. 4.65 -a). The larger shear forces are concentrated in piers 2 and
3. relatively low maximum deformations (0.07m) are reached at failure. The failure is
reached before the ground acceleration reached its maximum, therefore the ground accel-
214
eration at failure (O.35g) is lower than the maximum ground acceleration of the earth
quake.
Conclusion
It is evident that two different types of behavior were observed: The first one is
dominated by first mode response, which is typical to vibratory earthquake. The second is
response to a uniform acceleration loading, which is typical to impulsive earthquake. This
kind of behavior can be represented relatively well by the pulse loading in which the mag
nitude of deformations and the concentration of shear force was close to the one observed
in the pulse push-over procedure. The shape of the bridge at the time of failure however
was different. It should be noted that the only procedure that was able to predict the actual
failure of pier 2 during the northridge earthquake is the pulse push-over procedure.
215
TABLE 4.4. Comparison of Shear Force Distrihution Between the Piers at the TIme of Fallure of Critical Pier In Shear
pier 2 pier 3 pier 4 pier 5 pier 6 pier 7 Total
kN kN kN kN kN kN kN
Modal adaptable -21.9 2530 3480 6365 3680 1410 17470 ;%"ftolll1:::' .•... : ':'-.', '.,'<c- :,,,,~,'(-" .
Unifonn pseudo 3720 acceleration
::o/~(,ftotilI.;i;: ... , " C" ,' . .'Jc't''''';'':\'':;
Ramp 0.25 mlsec3 6365 ":/",,,"
;'l®ftotal •.
Ramp 0.5 mlsec3
%bftotal
Ramp 5.0 mlsec3
%oftotal
Pulse 0.1 sec
%of total
2160
%oftotal
Ground motion 4830 14242 (THA)
%oftotal
Piers Height [mJ 10.5 10.8 14.7 15.2 26.3 38.4
Tributary length 55 57 47 48 54 43 502 for gravity load [mJ
%oftotal .18 19 15
216
40
30 I- -20 I- - Modal 10 -40 r 30 -20 - Uniform 10 -
,......., 40 -'" ... 30
.9 ..... 20
0
~ 10
........ 40
---
I
Ramp 0.25
I: 30 0 I- -..... ... 20 ..6 ..... 10
~ 40 .....
I- -I- -
I
Ramp 0.5
"0 ~ 30 -~ 20 .s
10
~ ~ 40
I- -I- -
I
Ramp 5.0
..c: '" 30 I- -
20 - Pulse 0.1 10 · 40 I
30 · 20 - Pulse 0.05 10 -40
30 l- · 20 I- 50% (-) THA 10 l- · 0
pI.r2 plcr3 P1cr4
FIGURE 4.57 Distribution of Shear Forces in Piers 2-7 for Different Analysis Procedures
217
TABLE 4.5. Moment at Bridge Piers due to Different Push·over Procedures at Shear FaUure
pier 2 pier 3 pier 4 pier 5 pier 6 pier 7
kN·m kN·m kN·m kN·m kN·m kN·m
Modal adaptabie ·1980 40100 60000- 67600-- 84900-- 60700
Unifonn load 38400 73100'- 59300' 65800' 80900-- 46500
Ramp 0.25 mlse2 32800 68300- 56700' 62400' 78600' 50200
Ramp 0.5 mlsec3 49700 74400" 61400' 69900"" 81200"" 62200
Ramp 5.0 mlsee3 53800 74000"" 61700" 69500"" 82200"" 65300'
Pulse 0.1 sec 64600 71100" 43700 45200 27500 51000
Pulse 0.05 53500 48300 32300 32300 15600 39100
THA 43300 63500 38800 3820 ·20000 18000
Yielding moment 68740 69740 54570 54570 78250 '64260
Ultimate moment 75000 75000 70000 70000 85000 70000
• Yielding moment is exceeded."" within 5% of ultimate
TABLE 4.6. Comparison of Performances of The SR141I5 Bridge Subjected to Different Push· Over Procedures
Critical Maximum Type of FaUure FaUedPier Acceleration Deformation
G m
Modal adaptable flexure/shear p5 0.21 0.590
Unifonn pseudo aeeel· flexure/shear p3 0.29 0.550 eration
Ramp 0.25 mlsee2 shear p5 0.34 0.470
Ramp 0.5 mlsee2 flexure/shear p5 0.38 0.530
Ramp 5.0mlsee2 flexure/shear p5 0.54 0.610
Pulse 0.1 sec shear p2 1.50 0.058
Pulse 0.05 see shear p2 3.00 0.046
THA shear p3 0.35 0.070
218
I !
(a)
8000
6000
./
)~ o
·2000
; ! / ) /'
p7
/ /'
/ I'
~ -V~ V --I ---- /"
Step number [#]
(c)
Load step [#]
(b)
/pS
L po
/.,... ..... p4
.,,/ I p3
.... '-p7
....
p2
FIGURE 4.58 Response of the Bridge to Adaptable Proportional to Instantaneous Mode Shape Distributed Load (relative to mass). (a) Displaced Shape at Failure, (b) Loading History, (c) Shear
History for Piers 2-7
219
force
liiiiiiiil ".' .... /
p2 p3
(a)
40000
~3~ j ~~ ~ FI~
Load step 1#1
(b)
8~ ~---'-----r----T-----~---T----~----~---'
2 " ~
<E t;; " .c '"
rooo ~---+----+----+----4---~----~~~~~~
4000
10 20 Loading step [#]
(c)
30
p4
p7
p
40
FIGURE 4.59 Response of the Bridge to Non Adaptable Uniform Distributed Load (relative to mass). (a) Displaced Shape at FaUure, (b) Loading History, (c) Shear History for Piers 2-7
220
~y2.
p6 p7 p4 P~.<····· ".
p3 ,
~
Time [sec]
(b)
(a)
8000 r----,.----,-----,,..-----r---,------,
p3
p5 rooo~---~----~------4------4----~~~---4
" ~~r_--_+---~--~~-~_+----+_-~~ ,£
R
2000 i-----+-T--:;,f----:::;;o"'3 ..... ""--+---+-----I p7
Time [sec.]
(c)
FIGURE 4.60 RespoDSe oftbe Bridge to Ramp Loading. Acceleration Rate: o.lSmlsecl (8) Displaced Shape at FaDure, (b) Loading HIstory, (c) Shear HIstory for Piers 2·7
221
4r---__ ----~--------_,
p6 p7
p4 }.S/ i' .. ~~ ... ,:: ... _~3~ ... _---,_-,--,----,--,-_',-' '~
o~--~--~----~--~ o 5 10 Time [sec)
(b)
(a)
8000.0
~ p3
pS
j V 6000.0
~ t?'
~ V ~ 1/""" ..... '/'" p6
~
ib v k? ~ ./'
V/~ ~ .-10 p7
2000.0
~~ --0.0 Time (c)
FIGURE 4.61 Response of the Bridge to Ramp Loading. Acceleration Rate: 0.5 mfsec 3 (a) Displaced Shape at Failure, (b) Loading History, (c) Shear History for Piers 2-7
222
8000
6000
2000
o 0.0
(a)
b
/ VJ-"'"
/ ~ //J V/
VI/ V,. ~Z ~
o~~~~----~--~~ 0.0 0.5 1.0 1.5
Time [sec]
(b)
I' p3 7ps
V/ , nO
V/ l7
V p4
/ p6
1/
~ p7
io-"'"
0.5 Time [sec]
(c)
1.0 1.5
FIGURE 4.62 Response of the Bridge to Ramp Loading. Acceleration Rate: 5m1se~. (a) Displaced Shape at Failure, (h) Loading History, (c) Shear History for Piers 2·7
223
p2 p3 p4 p5 p6 p7 " ....
/)
(a)
8000
6000
2000
%.)0 0.05
/
15 .... _ ........ __ ....... __ ...-_......,
14 13 12 II
_ 10 N 9 i 8 - 7 ~ 6 '" 5 4
3 2 I O~~~~--~--~~~~ 0.00 0.05 0.10 0.15 0.20
Time [Sec.]
(b)
/ p2
7 _0
/ 1/
/ V v: "'-
p4 n<
~ p7
F-'" .......
"- p6
TIme [sec.] v. IV
(c)
F1GURE 4.63 Response of the Bridge to Pulse Loading 0.1 sec. (a) Displaced Shape at Failure, (b) Loading History, (c) Shear History for Piers 2·7
224
~ T
24
p2 p3 p4 p5 p6 p7 22
.. 20f. : 18
~ " 1: .s ~
'" " ..c
'"
...... ,." .. ". ,
(a)
8000
6000
4000
2000
~ 16 !14 ] 12 <10
8 6 4 2
8~~"~,,,~~u~ .. ,"~~,,~,,,~~~,,.0 Time (Sec.)
(b)
pI
p2
p5
p4 p3
p6 O~~~~~~--~--~~~~~~~~~~ 0.00 0.Q2 0.04 0.06 0.08 0.10
time [sec.]
(c)
FIGURE 4.64 Response of the Bridge to Pulse Loading 0.05 sec. (a) Displaced Shape at FaHure, (b) Loading History, (c) Shear History for Piers 2·7
225
10000
8000
6000
4000
~ 2000 t;J
" ..c
'" 0
-2000
-4000
-6000
\ .. p5
..
(a)
i .'
\,: ... ;' / p7 p6
rfI. Ia.
1.2
0.9
0.6
0.3
.... " -0.6
-0.9
-1.2
" I !f' "
~ \j v
7 [sec.l Time
(c)
time sec.]
(b)
_.
. I!M
I r\ I II p4
~ ,. p7
~ I!~
\I1fJI\ p6
\
FIGURE 4.65 Response of the Bridge to Dynamic Impnlsive Loading. (a) Displaced Shape at Failure, (b) Northridge Earthquake Record From Santa Monica City HaIl (input motion), (c) Shear History
for Piers 2-7
226
5.0 DISCUSSIONS AND CONCLUDING REMARKS
• The current work, laid up the ground work for the analysis of bridges were the nonlin-
ear behavior of elements (concrete or steel), and protective systems is considered. Sev-
eral types of analysis, such as time history, monotonic pushover and modal were
I introduc~ to extract information about the safety of new design or retrofit design of
bridges. Since every one of the analysis methods has strong and weak points, a good
evaluation procedure should be combined of the different analysis types presented. The
time history analysis can be used as indicator to the global behavior of the bridge dur-
ing severe earthquake, and approximate procedures such as pushover analysis, and
elastic analysis for preliminary design.
• Several case studies were presented in the current work, which were used to demon-
strate the capacities of an analysis program IDARe-BRIDGE developed in this work,
and to verify the analytical procedures, used in the program.
The special end connections and the coupled motions were used in almost all the
case studies and proved to provide very efficient tools to model complicated connec-
tions between the deck and the substructure. This contribution is especially apparent in
the modeling of the East Aurora bridge in which very simple and efficient modeling of
the connection was used as compare to the very sophisticated modeling used for the
same problem by Wendechansy(l996).
The modeling of base isolation systems placed in arbitrary locations in the bridge
was verified with experimental results. It was observed however, that care should be
taken when using stiffness proportional damping in structural systems which include
base isolation elements.
227
The modeling of the triaxial behavior which includes dependency of the friction
coefficient on velocity of sliding isolator, was modeled by the new triaxial isolator ele
ment and verified with experimental results.
The expansion joint element was developed to model efficiently the behavior of
thermal expansion joints in bridges.
The nonlinear damping concept is introduced and used for the modeling of the tri
axial element. The modeling of this element is verified using biaxial experiment as
mentioned earlier.
The use of displacements / velocity input, which allow direct representation of dif
ferential ground motion was introduced. The use of this method was verified by com
paring the response of a simple structural system subjected both to equivalent
displacement and acceleration input.
All the models mentioned above which include: special connections, different
types of base isolation, expansion joints, differential ground motion and nonlinear
damping were used in analysis of large scale real bridges. Parametric studies were per
formed to evaluate their contribution to the overall response of the bridge.
• A first step in the direction of developing evaluation methods based on time history
nonlinear procedures which include damage indexing combined with push-over analy
sis was made. A bridge in Los Angeles county was subjected to several types of push
over procedures as well as real ground motion exciton. The influence of the type of the
earthquake excitation on the distribution of forces in the bridge was studied using sev
eral push-over procedures and compared to the response of the bridge to actual impul
sive ground motion.
228
--- ------------------------------------------------------------------,
The modal adaptable push over procedure seems to be deficient in modeling the
bridge response. The results from the uniform force (relative to mass distribution) and
the slower ramp loading are similar in nature, and seems to provide reasonable repre
sentation of the response of the bridge to vibratory earthquake. The ramp push over
procedure, with faster application rate introduce dynamic effect to the distribution of
the forces along the bridge. Larger part of the seismic load is transferred to stiffer and
shorter piers. Because the dynamic effect results in additional inertia forces, larger
acceleration is required to bring the bridge to failure than the static and slower proce
dures.
Impulsive push-over procedures as well as impulsive ground motion excite the
bridge in different manner than the procedures related to vibratory earthquakes. Large
transfer of forces to the stiffer and shorter piers can be observed while experiencing
very small deformations.
The actual failure mechanism of the bridge during the Northridge earthquake was
identified when the bridge was subjected to pulse push-over procedure.
• The program provides basic approach based on the idea that the modeling of a struc
tural system should be done as much as possible considering physical models there fore
the nonlinear behavior of elements will be mostly modeled by nonlinear stiffness and
damping properties, and not by defining the nonlinear forces to the forcing vector. This
type of element is the main prototype element in the program IDARC-BRIDGE. Other
programs (ANSYS, DRAIN-2DX) are using only the stiffness properties and the force
vector to model nonlinear behavior.
229
• A close form solution of the governing differential equation is used to represent the pla
nar behavior of base isolation systems. This solution gives insight and understanding
on how the system works and explain why a three-dimensional additional element was
necessary to be developed.
• Several techniques which were developed previously such as: rigid end connections,
flexible connections, and free connection (Weaver and Gear 1990) and coupled motions
(Bathe (1982» were modified to be more versatile such that they can be used each one
separately and altogether to enable easier modeling of complicated bridge connections.
and avoid computational ill-conditions.
• A new triaxial base isolation element that can be incorporated in a general computer
program was developed. This element allows for a more realistic modeling of sliding
base isolation system and can be easily extended to model other types of isolation sys
tems, such as high damping rubber and lead rubber bearings.
• A modern iterative solver was incorporated in the program to provide faster and more
efficient solution of the equilibrium equations. A solver for large eigenvalue problems
was developed for use in the dynamic and monotonic analysis.
• The procedure developed by Priestley et. al. (1992) for the use of Lateral Strength
Method, was expanded. The original method provides solution to a simple planar
frames. The current work extends the idea to model the whole bridge system as an inte
gral unit. Several patterns of loading were developed to enhance the efficiency of this
method.
230
5.1 Further Research Suggestions
In order to develop further the use of the evaluation procedure, additional verifica
tion studies should be performed. The results of the different analysis procedures should
be compared with experimental results, with measured response of bridges which experi
enced real earthquakes and with analysis results from other programs, which were verified
with experimental results.
The concept of bridge design using the combined evaluation method should be
evaluated by using case studies on existing bridges, and comparing the design with the
existing design procedures used currently in practice.
The current evaluation methodology can be used to develop simplified design pro
cedures, by calibrating the elastic design methods.
Additional parametric studies should be performed to evaluate the influence of
base isolation systems on the behavior of bridge structures, especially the size of the
device and the influence of adding damping devices, with variety of locations in the
bridge.
The influence of gap elements on the seismic behavior of the bridge should be
studied and the gap element should be further verified in the program IDARe-BRIDGE.
The efficiency of the analysis program should be evaluated and if possible the
numerical procedures should be upgraded, to accelerate the speed of the analysis, using
currently developing methods such as explicit integration etc.
231
The influence of biaxial interaction for concrete elements and protective systems
elements on bridges need to be studied farther.
A very important issue that should be considered is the influence of the limitations
on input properties of the elements, both linear and nonlinear, on the response of the
bridge system. The procedure of evaluation of this part, should be done using initially,
bridges with a small number of elements. Then extend it to more complex systems, and try
to find out if there are limiting values that can be used for design. The verifications and the
definition of limitations should be developed to add confidence to the use of nonlinear
time history as a primary evaluation tool. Although it is a gloryless large afford, this step
should be taken to depart from the elastic equivalent design procedure to a more rigorous
design method. The tools developed in this work can play ar roll in such an afford.
Additional comparison between the different pushover procedures should be per
formed and the usefulness of this procedure should be evaluated.
Additional structural elements can be incorporated into the program to enlarge the
elements library of the program, and provide more flexibility and accuracy in modeling
bridge structures. The program can be used as a tool, for system identification, and for
evaluation of different active control schemes.
•
232
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A. APPENDIX A
A. 1 IDARClBridge -Inelastic !!amage Analysis of Reinforced ,Construction = Platform for Bridges
The major parts of the program are presented in Fig. A.I. A detailed description of
these parts follows.
L-_ID_A_R_C_-_B_R_ID_G_E_-li I Input
I Preprocessing
I Processing
FlGURE A.I Program General Structure
The description of the major functions of the input routines is provided in Table
A.I
TABLE A.I Input Routines Description
Function
initiate
define analysis
Routines names
gecanalysis_type
gecanalysis_options
Description
calculate the size of the arrays of: nodes, coordinates, properties
define if the analysis is 2d or 3d, and what type of analysis to perfonn (dynamic, modal, push-over etc.)
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TABLE A.1 Input Routines Description
Function
define nodes and restrains
define elements properties
define elements
define elements end conditions
loadings
push-over conditions
dynamic and modal analysis parameters
Routines names
gecnode300rdinates
geCboundary 30nditions
geccoupled_motions
gecelemencproperties
geCbiJ..gap
get_e_b
geCfoundation
geCh_b
geCh_b_s
geUso_sli
geUsolator_1
geClin_damper
gecmember_connectivity
assign_elements_property
assign_elements_types
get_member_boundarY30 nditions
gecmember_springs
geuigid_arms
get.,joinUoad
get.,joinc weight
gecmembecuniformJoad
gecnumber_oUoad_steps
get_usecpushover
get_romp_pushover
gecnumber_oCmodes
geUotal_analysis_duratio n
gecanalysis_time_step
get_damping coefficients
geCdirection of excitation
gecexcitation_file_name
gecexication..groups
geCpealcground_accelera tio
geUype_oCexciation
read_number_oCmodes
Description
Read joints coordinates, the boundary conditions of the joints and the coupled motions
The main routine - gecelemencproperties calls the other routine to retrieve the required properties of the elements.
This group of routines defines the elements. Including the connectivity of the elements to the nodes, the property number of the element, and its type.
The element matrices are created first assuming they are fixed at the ends. Releases, end springs, and rigid arms as described in chapter 3 are applied later for easier modeling
Some of the parameters defining the right hand side of the equation of equilibrium are defined in this group of routines. The joint load (for incremental static analysis) and the number of steps to which the load is divided to.
Joint weights are used to define the mass for dynamic analysis
Those conditions are used to indicate the stopping conditions for the adaptable pushover procedures, and the pseudo dynamic procedures
The parameters required for: modal analysis (number of modes, displacements velocity (excitation groups). and some general parameters like global damping and integration time are read
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The general description of the routines included in the preprocessing portion is
provided in Table A.2
TABLE A.2 Preprocessing Routines
Function
mapping routines
mass matrix
beam length
preparation routines
Routines names
biJ--l!ap_list
dampers_list
eJoad_list
elements_to--l!lobal
finaCorder
numbcoCdampers
slUso_list
pre_coupled
pre_release
pre_spring
Description
Some of the arrays require mapping to other arrays to prevent the need for unnecessary search and reduce the number of operations
Since the mass matrix is not changing during the analysis and it is assumed to be lumped, it is defined as a vector, prior to the analysis
The length of each element is calculated from the node coordinates, where the rigid zones are subtracted. This calculation is the base for the calculation of the rotation matrix.
These routines are used to prepare arrays required for the processing phase
The general description of the structure of the processing routines is presented in
Fig. A.2. A detailed description of each one of the blocks in the processing section is pre-
sented in the following tables.
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Processing update elements
apply end conditions
assemble global matrices
assemble load vector
solve
process results
FIGURE A.2 Processing Routines Arrangements
TABLE A.3 Update Elements Properties
Function Routines names
elements stiffness h_bjorce3alcualtion
(force calculation) rigid_arm_force
elements stiffness create3_b3d_stiffness
(general elements) createjound_stiffness
update_bi'-.gap_stiff
update_iso_l_stiff
Description
The first step in updating the elements properties. The end forces in the element are calculated using the stiffness from the previous step. and the calculated deformations. Since the forces are calculated at the theoretical joints. they have to be transformed to the actual joints through the rigid zones
Corresponding to the end forces the stiffness in the nonlinear elements is updated. The stiffness in the linear elements is created only at the first step and kept the same for the next steps.
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TABLE A.3 Update Elements Properties
Function
elements stiffness
(hysteretic beam)
elements damping
element general
Routines names
h_b_stiffness_main
ddnsth
damaged_length
local_stiffness
h_b_add_non_shear
calculate_damage
update_elements_dampin g
create_elements_damping
update_elements-!leneral
update_iso_sli
TABLE A.4 End Conditions Description
Function
apply end conditions
coordinates transformation
Routines names
end springs
releases
stiffnessJigid_arms
local to global
Description
The stiffness of hysteretic beam is updated in several stages: (1) the flexibility at the beam ends is calculated from the end moments (ddnsth). (2) from the moment diagram, the damaged length is calculated (damaged_length). Using both the flexibility and the damaged length, by integrating the flexibility over the beam length the flexibility matrix is derived, and inverting it yields the stiffness matrix (locaLstiffness). The nonlinear shear influence is added (h_b_add_non_shear) and the damage index is calculated.
The nonlinear damping of elements is controlled by the routine: update3lements_damping. The linear damping element is created.
Element of the type "general" are elements that contributes to three global matrices: (1) stiffness matrix (2)damping matrix (3) load vector. An example of this type of element is the isolator slider element
Description
The special end conditions (rigid zones, releases, end springs) are applied on the fixed ends stiffness matrix of the element.
The stiffness matrix of the elements is transformed from its local coordinate system to the global one.
TABLE A.S Global Matrices Building Description
Function
assemble stiffness
effective stiffness
Routines names
assemble
accel_ velocity
effective_stiffness
Description
The global stiffness matrix is assembled from the local once.
Using the acceleration and velocity from the previous time step the effective stiffness matrix is calculated from the mass and stiffness matrices (Newmark Beta method) only for dynamic analysis.
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TABLE A.S Global Matrices Building Description
Function Routines names
add damping to the effectivcstifCdamp effective stiffness matrix
I
coupling conditions coupled
TABLE A.6 Load Vector Description
Function Routines names
load (acceleration) forcinll-acceleration
load (disp-velocity) forcinll-disp_velocity
load (dynamic forcinll-dynamicjorce force)
load quasistatic) forcing_quasistatic
load (ramp push- ramp_pushover over)
load (modal-push- pushover_modal over)
load (pushover- pushovecuser user)
elements loads forcinll-elementsJoads
effective load effective_load_vector
damping effect forcinll-damping
coupling effect coupledJoad_vector
final load finaUoad_ vector
Description
The effect of stiffness proportional damping and individual damping elements is added to the global stiffness matrix.
The coupled motions conditions are applied to the effective stiffness matrix. As well as the restrains (global boundary conditions)
Description
Incremental ground acceleration is used to assemble the force vector, by mUltiplying the ground acceleration by the mass
The force vector is created by multiplying the ground displacements and velocity, by the interaction matrix between the superstructure and the ground
History of dynamic load is directly read.
Static load is applied incrementally
This load case is similar to the incremental acceleration loading case. However the acceleration is not read from a file but increases linearly until limiting condition is reached
The increase in the total force on the bridge is distributed proportional to the modes of vibration
The load distribution on the bridge is defined by the user using the joint load input as the guideline for the force distribution
The contribution of the individual elements force is added to the global load_ vector
The effective load vector is calculated considering the acceleration and velocity from the previous time step. (Newmark Beta method)
The effect of stiffness proportional damping, and individual elements damping is added to the effective load vector
The effect of the coupling ofD.O.F's is added to the effective load vector
The degrees of freedom related to coupled degrees of freedom, and restrained once are eliminated
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TABLE A.7 Solution Description
Function
solution
Routines names
gecarray _a_size
sparse_fonnat
solution
src2c
TABLE A.S Results Processing Description
Function
displacements results
force results
Routines names
disp_coupled_to~lobal
disp_out
h_bjorce_calculation
ejorce_out
TABLE A.9 Modal Analysis Description
Function
modal analysis
Routines names
modal_main
sparsejonnaLmodal
sparse_mass_modal
lanz
write_mode_shapes
description
The effective (dynamic analysis) stiffness matrix is transferred to sparse fonnat to use JeG solver.
. First the number of non-zero elements in the stiffness matrix is found. The sparse stiffness matrix is created (vectors: a, II, IJ). The solution driver (solution) is used to call the solver (src2c)
description
The first routine calculates the displacements of the couples D.O.F's, so that full solution of all the D.O.F's will be available. The second routine calculates the total displacements of the nodes out of the incremental once. At required time steps snap shots of the defonned bridge are printed.
The forces of selected elements are calculated and printed in each time step. Moreover the maximum force of larger number of elements is calculated.
description
The routine "modaCmain" is the driver for the modal analysis, where memory is allocated. The lanz modal solver requires that the stiffness and masS matrix will be in a fonn of sparse fonnat.
after the analysis the eigen-values are written to the screen, and the mode shapes to a file.
254
I