1. health and fitness
2. disease
3. diabetes

SEISMIC ANALYSIS OF THE RC INTEGRAL BRIDGES USING PERFORMANCE-BASED
DESIGN APPROACH INCLUDING SOIL STRUCTURE INTERACTION
by
Kianosh Ashkani Zadeh
B.Sc.E. University of Isfahan, 1993
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF APPLIED SCIENCE
in
THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES
(Civil Engineering)
THE UNIVERSITY OF BRITISH COLUMBIA
(Vancouver)
October 2013
© Kianosh Ashkani Zadeh, 2013
Abstract
Bridges in high seismic risk zones are designed and built to withstand damage when
subjected to earthquakes. However, there have been cases of bridge collapse due to design flaws
around the world in the last few decades. To avoid failure and minimize seismic risk, collapse
issue should be appropriately addressed in the next generation bridge design codes. One of the
important subjects that needs to be addressed in bridge design codes is Soil-Structure Interaction
(SSI), especially when the supporting soil is soft.
In this research, SSI is incorporated within a performance-based engineering framework
to assess the behaviour of RC integral bridges. 3-D nonlinear models of three types of integral
bridges with different skew angles are built. For each bridge type, two archetype models are
constructed with and without considering the effect of SSI. CALTRANS spring and multipurpose dynamic Winkler models are employed to simulate the effect of soil in the SSI
simulation.
In this study, relative displacement and drift of the abutment backwall and pier columns
are considered as engineering demand parameters (EDPs). Spectral acceleration of ground
motions is chosen as the intensity measure (IM). Incremental dynamic analysis (IDA) is
employed to determine the engineering demand parameters and probability of collapse using a
set of 20 well-selected ground motions.
Current study shows that for the integral abutment bridges considering soil structure
interaction mostly demonstrate smaller relative displacement capacity/demand ratio. Therefore,
neglecting SSI can result in overestimating relative displacement capacity of the structural
components in this type of bridges. In addition, it is shown that SSI can cause an increase in
ductility of the pier columns while it can cause a decrease in the ductility of the abutments.
ii
Collapse Margin Ratio (CMR) is considered here as a primary parameter to characterize
the collapse safety of the structures. It is found that the probability of collapse of the SSI
archetype models is higher than probability of collapse of their corresponding non-SSI models.
Consequently, CMR value of the SSI archetype model is smaller than CMR value of its
corresponding non-SSI models.
iii
Preface
This thesis titled “Seismic analysis of the RC integral bridges using performance-based
design approach including soil structure interaction” presents the research performed by Kianosh
Ashkani Zadeh. This research was supervised by Dr. Ventura and co-supervised by Dr. Liam
Finn and Dr. Mahdi Taiebat at the University of British Columbia (UBC). The research was
carried out as a part of 'Soil Structure Interaction in Performance Based Design of Bridges'
project at UBC sponsored and funded by Natural Sciences and Engineering Research Council of
Canada (NSERC).
Analytical fragility fitting functions used in Chapter 4 are based on documentation and
tools provided in Baker Research Group webpage, as described in the following paper:
Baker, J. W. (2013). “Efficient analytical fragility function fitting using dynamic structural
analysis.”Earthquake Spectra, (in review).
The author of this thesis is responsible for reviewing the literature, developing models,
conducting analysis, data processing, and interpreting the results. The author of this thesis is
responsible for preparing the tables and figures. The manuscripts were drafted by the author of
this thesis and finalized in an iterative process and discussed during SSI meeting with the thesis
supervisor, Dr. Carlos Ventura and co-supervisors Dr. Liam Finn and Dr. Mahdi Taiebat.
iv
Table of Contents
Abstract .......................................................................................................................................... ii
Preface ........................................................................................................................................... iv
Table of Contents ...........................................................................................................................v
List of Tables ................................................................................................................................ xi
List of Figures ............................................................................................................................. xiii
List of Symbols ......................................................................................................................... xxiv
List of Abbreviations .............................................................................................................. xxvii
Glossary .................................................................................................................................... xxix
Acknowledgements .................................................................................................................. xxxi
Dedication ................................................................................................................................ xxxii
Chapter 1: Introduction ................................................................................................................1
1.1
Overview and Motivation for Study ............................................................................... 1
1.2
Objective and Scope ....................................................................................................... 3
1.3
Outline............................................................................................................................. 7
Chapter 2: Modeling of the Bridge Structures............................................................................9
2.1
Archetype Models ........................................................................................................... 9
2.2
Sources of Uncertainty.................................................................................................. 11
2.2.1 Modeling Uncertainty ............................................................................................... 12
2.2.1.1
Structural Nonlinearity...................................................................................... 12
2.2.1.2
P-Δ Effect .......................................................................................................... 12
2.2.1.3
Soil Structure Interaction (SSI) ......................................................................... 13
2.2.1.3.1 Soil Effect behind the Abutment Backwall ................................................. 13
v
2.2.1.3.2 Soil Effect around the Abutment and Pier Piles .......................................... 17
2.2.2 Ground Motion Uncertainty...................................................................................... 21
2.3
Structural Damping ....................................................................................................... 22
2.4
Additional Mass Assignment ........................................................................................ 22
2.5
Element Behavior - Bridge Type 1 (Model M1 & M2) ................................................ 22
2.5.1 Abutments - Bridge Type1 (Model M1 & M2) ........................................................ 24
2.5.1.1
Abutment Sectional Response - Bridge Type1 (Model M1 & M2).................. 25
2.5.2 Abutment Piles Bridge Type1 (Model M2) .............................................................. 25
2.5.3 Bridge Deck Slab and Pre-stress Precast Girders - Bridge Type 1 (Model M1 & M2)
27
2.5.3.1
Deck-girder Sectional Response - Bridge Type 1 (Model M1 & M2) ............. 29
2.5.4 Performance Criteria - Bridge Type 1 (Model M1 & M2) ....................................... 29
2.6
Element Behavior - Bridge Type 2 (Model M3 & M4) ................................................ 31
2.6.1 Abutments - Bridge Type2 (Model M3 & M4) ........................................................ 32
2.6.1.1
Abutment Sectional Response - Bridge Type2 (Model M3 & M4).................. 32
2.6.2 Bridge Deck Slabs and Pre-stress Precast Girders - Bridge Type 2 (Model M3 &
M4)
33
2.6.2.1
Deck-girder Sectional Response - Bridge Type 2 (Model M3 & M4) ............. 34
2.6.3 Pier Columns - Bridge Type 2 (Model M3 & M4) ................................................... 35
2.6.3.1
Pier Columns Sectional Response - Bridge Type 2 (Model M3 & M4) ........... 36
2.6.4 Bridge Pier Pilecap and Pile Head - Bridge Type 2 (Model M3 & M4) .................. 37
2.6.5 Abutment and Pier Piles - Bridge Type 2 (Model M4) ............................................ 38
2.6.6 Performance Criteria - Bridge Type 2 (Model M3 & M4) ....................................... 38
vi
2.7
Element Behavior - Bridge Type 3 (Model M5 & M6) ................................................ 39
2.7.1 Abutments - Bridge Type 3 (Model M5 & M6) ....................................................... 40
2.7.1.1
Abutment Sectional Response - Bridge Type 3 (Model M5 & M6)................. 41
2.7.2 Bridge Deck Slabs and Pre-stress Precast Girders - Bridge Type 3 (Model M5 &
M6)
42
2.7.2.1
Deck-girder Sectional Response - Bridge Type 3 (Model M5 & M6) ............. 43
2.7.3 Pier Columns - Bridge Type 3 (Model M5 & M6) ................................................... 44
2.7.3.1
Pier Columns Sectional Response - Bridge Type 3 (Model M5 & M6) ........... 44
2.7.4 Pier Pilecap and Pier Head - Bridge Type 3 (Model M5 & M6) .............................. 45
2.7.5 Steel Intermediate Diaphragm - Bridge Type 3 (Model M5 & M6)......................... 45
2.7.6 Performance Criteria - Bridge Type 3 (Model M5 & M6) ....................................... 46
Chapter 3: Analysis......................................................................................................................48
3.1
Eigen Value Analysis .................................................................................................... 48
3.1.1 Eigen Value Analysis - Bridge Type 1 ..................................................................... 48
3.1.1.1
Model M1.......................................................................................................... 48
3.1.1.2
Model M2.......................................................................................................... 50
3.1.2 Eigen Value Analysis - Bridge Type 2 ..................................................................... 51
3.1.2.1
Model M3.......................................................................................................... 51
3.1.2.2
Model M4.......................................................................................................... 52
3.1.3 Eigen Value Analysis - Bridge Type 3 ..................................................................... 53
3.1.3.1
Model M5 & M6 ............................................................................................... 53
3.2
Nonlinear Static Pushover Analysis.............................................................................. 54
3.3
Hazard Analysis ............................................................................................................ 59
vii
3.3.1 Vancouver Uniform Hazard Spectrum ..................................................................... 64
3.3.2 Input Ground Motions............................................................................................... 65
3.4
Incremental Dynamic Analysis (IDA) .......................................................................... 69
Chapter 4: IDA Results and Probability of Collapse of the Models .......................................71
4.1
Non-Simulated Collapse Mode IDA Results and Probability of Collapse - Model M1 &
M2
73
4.2
Non-Simulated Collapse Mode IDA Results and Probability of Collapse - Model M3 &
M4
79
4.2.1 Pier Columns Hysteretic Graphs - Bridge Type 2 (Model M3 & M4) ..................... 85
4.3
Non-simulated Collapse Mode IDA Results and Probability of Collapse - Model M5 &
M6
89
4.3.1 Pier Columns Hysteretic Graphs - Bridge Type 3 (Model M5& M6) ...................... 93
4.4
Relative Displacement Capacity/Demand Ratio (λ) and Period-Based Ductility (
) of
The Archetype Models .............................................................................................................. 96
4.5
Percentage of the Failure Mode and Comparison of the All Obtained Probability of
Collapses ................................................................................................................................. 100
4.5.1 Collapse Margin Ratio for the Model M1 & M2 .................................................... 102
4.5.2 Collapse Margin Ratio for the Model M3 & M4: ................................................... 104
4.5.3 Collapse Margin Ratio for the Model M5 & M6: ................................................... 105
4.5.4 Summary of the Obtained Collapse Margin Ratios: ............................................... 106
4.6
Proposed Design Procedures....................................................................................... 108
Chapter 5: Concluding Remarks ..............................................................................................111
5.1
Conclusion .................................................................................................................. 111
viii
5.2
Recommendations, Limitations and Future Works .................................................... 112
References ...................................................................................................................................114
Appendices ..................................................................................................................................117
Appendix A Significant Duration (5%-95% Arias Intensity) Acceleration Time History of the
Selected Ground Motions ....................................................................................................... 117
Appendix B Fragility Fitting Functions for Use with Incremental Dynamic Analysis Data.. 122
Appendix C 3_D Schematic View of the Obtained IDA Results ........................................... 123
C.1
3-D Schematic View of the Obtained IDA Results - Model M1 & M2 ................. 123
C.2
3-D Schematic View of the Obtained IDA Results - Model M3 & M4 ................. 124
C.3
3-D Schematic View of the Obtained IDA Results - Model M5 & M6 ................. 125
Appendix D ............................................................................................................................. 126
D.1
Pier Column Hysteretic Plot - Total Column Drift Ratio vs. Total Base Shear -
Model M3 & M4. ................................................................................................................ 126
D.2
Pier Column Hysteretic Plot - Column's Actual Drift Ratio vs. Total Base Shear -
Model M3 & M4. ................................................................................................................ 130
D.3
Pier Column Hysteretic Plot - Column's Actual Drift Ratio vs. Total Base Shear -
Model M5 & M6. ................................................................................................................ 134
Appendix E ............................................................................................................................. 138
E.1
Pier Column Hysteretic Plot - Column's Actual Drift Ratio vs. Base Moment -
Model M3 & M4. ................................................................................................................ 138
E.2
Pier Column Hysteretic Plot - Column's Actual Drift Ratio vs. Base Moment) -
Model M5 & M6. ................................................................................................................ 142
Appendix F Bridge Drawings ................................................................................................. 146
ix
F.1
Bridge Type 1 ......................................................................................................... 146
F.2
Bridge Type 2 ......................................................................................................... 150
F.3
Bridge Type 3 ......................................................................................................... 154
Appendix G Site Classification for Seismic Site Response - NBCC2010 (Table 4.1.8.4.A) . 159
Appendix H Deaggregation Charts - Soil Site Class C (
) ......................... 160
H.1
Period 0.284 sec and Amplitude 0.86g ................................................................... 160
H.2
Period 0.335 sec and Amplitude 0.798g ................................................................. 161
H.3
Period 0.374 sec and Amplitude 0.757g ................................................................. 162
x
List of Tables
Table 1-1Assessment framework for performance based earthquake engineering - Source: G.
Deierlein, 2004. ............................................................................................................................... 6
Table 2-1 Summary of the bridge types and archetype models are developed and used in this
study .............................................................................................................................................. 10
Table 2-2 Summary of calculated stiffness of the bridge abutments due to the embankment
passive pressure force resisting movement ................................................................................... 16
Table 2-3 Presents calculated parameters which obtained fitting a tri-linear curve to the derived
p-y curves ...................................................................................................................................... 20
Table 2-4 Calibrating parameters for the nonlinear material model- Model M1 & M2 ............... 24
Table 2-5 Calibrating parameters for the elastic material model used in the pile elements ......... 26
Table 2-6 Required calibrating parameters for the elastic material model- Model M1 & M2 ..... 29
Table 2-7 Shows defined performance criteria for the model M1 & M2 ..................................... 31
Table 2-8 Required calibrating parameters for the elastic material model - Model M3 & M4 .... 34
Table 2-9 Calibrating parameters for the nonlinear material model used to model pier columns Model M3 & M4 ........................................................................................................................... 36
Table 2-10 Calibrating parameters for the nonlinear material model used to model pier heads and
pilecaps- Model M3 & M4 ........................................................................................................... 38
Table 2-11 Shows defined performance criteria for the model M3 & M4 ................................... 39
Table 2-12 Dimensions of the steel T-section used as bracings in the intermediate diaphragm Model M5 & M6 ........................................................................................................................... 46
Table 2-13 Shows defined performance criteria for the model M5 & M6 ................................... 47
Table 3-1 Period and cumulative modal mass for the first 6 modes - Model M1 ........................ 49
xi
Table 3-2 Period and cumulative modal mass for the first 6 modes - ModelM2 ......................... 50
Table 3-3 Period and cumulative modal mass for the first 6 modes - Model M3 ........................ 51
Table 3-4 Period and cumulative modal mass for the first 6 modes - Model M4 ........................ 52
Table 3-5 Period and cumulative modal mass for the first 6 modes - Model M5 & M6 .............. 54
Table 3-6 Summary of the obtained ultimate capacities for the abutment and pier column of the
models ........................................................................................................................................... 58
Table 3-7 Summary of the performed Hazard analysis results ..................................................... 63
Table 3-8 Summary of the selected ground motion records ......................................................... 68
Table 3-9 Acceleration are applied to the restrained nodes in X & y directions .......................... 69
Table 3-10 Summary of the input/output frequencies of the performed IDAs ............................. 70
Table 4-1 Summary of the obtained failure modes performing IDA - Model M1 & M2............. 73
Table 4-2 Summary of the obtained failure modes performing IDA - Model M3 & M4............. 80
Table 4-3 Summary of the obtained failure modes performing IDA - Model M5 & M6............. 90
Table 4-4 Summary of the calculated relative displacement capacity/demand ratio (λ) and periodbased ductility (µT) using the obtained results from static pushover and incremental dynamic
analysis .......................................................................................................................................... 97
Table 4-5 Summary of the percentage of failure for the failure modes defined as performance
criteria in all the models .............................................................................................................. 100
Table 4-6 Summary of the calculated collapse margin ratios for median (50%), 10%, and 20%
and for all models........................................................................................................................ 106
Table G-1 Site classification for seismic site response based on top 30 meter soil average
properties - Source: NBCC 2010 (Table 4.1.8.4.A) ................................................................... 159
xii
List of Figures
Figure 1-1 Left photo: Shows collapse of a 630m segment of the elevated Hanshin Expressway
due to Kobe Earthquake Japan (1995) (Source: Wikipedia) .......................................................... 1
Figure 1-2 Interstate 35W bridge, which stretches between Minneapolis and St. Paul, was
suddenly collapsed during rush hour due to design flaw on Aug.1 2007, caused thirteen people to
die and 145 injuries. (Source: AP Photo/Pioneer Press, Brandi Jade Thomas, by permission). ... 2
Figure 1-3 PEER framing equation and example parameters for seismic shaking (Moehle, 2003)5
Figure 1-4 General framework of the study research ..................................................................... 7
Figure 2-1 3-D view of the developed bridge models .................................................................. 11
Figure 2-2 Soil separation due to the cyclic pressure on the embankment soil behind the bridge
abutment backwall - Source: Thevaneyan K. David and John P. Forth (2011). ........................... 14
Figure 2-3 Shows effective abutment width for skewed bridges, Source: SDC 1.7 (2013) ......... 15
Figure 2-4 Response curve of a link used to simulate soil effect behind the abutment backwall 16
Figure 2-5 Shows assumed soil sub-layer around the piles and location of the assigned
CALTRANS springs and SSI p-y links in the SSI models ........................................................... 17
Figure 2-6 The obtained parameters from Cone Penetration Test (CPT) for the chosen bore holes
from the soil the soil investigation report- Source: H5M, 2009. .................................................. 18
Figure 2-7 Left: Developed p-y curves based on the assumed sub-layers of soil surrounding piles
using API code. Right: Obtained p-y curve for sand layer around the piles using L-Pile Manual.
....................................................................................................................................................... 19
Figure 2-8 Shows backbone curve of the Winkler model used in the SSI p-y links - Source:
SeismoStruct Manual .................................................................................................................... 20
xiii
Figure 2-9 Illustrates implementation of inelasticity distribution along beam elements using fiber
approach in SeismoStruct software (Source: SeismoStruct Manual) ........................................... 23
Figure 2-10 Abutment discretized pattern and reinforcement arrangement - Model M1 & M2 .. 24
Figure 2-11 Shows abutment moment curvature diagram and M-V interaction obtained using
Reponse-2000 software - Model M1 & M2 .................................................................................. 25
Figure 2-12 Indicates piles local stiffness matrix-Source: SeismoStruct Manual ........................ 26
Figure 2-13 Deck-girder reinforcement arrangement and discretized pattern - Model M1 & M2 27
Figure 2-14 Hysteretic loop used to simulate the effect of the pre-stressing tendons in the models
....................................................................................................................................................... 28
Figure 2-15 Shows deck-girder moment curvature diagram and M-V interaction obtained using
Reponse-2000 software - Model M1 & M2 .................................................................................. 29
Figure 2-16 Shows a typical structural performance and associated damage states (A.Ghobarah,
2004) ............................................................................................................................................. 30
Figure 2-17 Abutment discretization and reinforcement arrangement - Model M3 & M4 .......... 32
Figure 2-18 Shows abutment moment curvature diagram and M-V interaction obtained using
Reponse-2000 software - Model M3 & M4 .................................................................................. 32
Figure 2-19 Abutment longitudinal strain (mm/m) - Model M3 & M4 ....................................... 33
Figure 2-20 Deck-girder reinforcement arrangement and discretized pattern - Model M3 & M4 33
Figure 2-21 Shows deck-girder moment curvature diagram and M-V interaction obtained using
Reponse-2000 software - Model M3 & M4 .................................................................................. 35
Figure 2-22 Deck-girder longitudinal strain (mm/m) - Model M3 & M4 .................................... 35
Figure 2-23 Pier column reinforcement arrangement and discretized pattern - Model M3 & M436
xiv
Figure 2-24 Shows pier column moment curvature diagram and M-V interaction obtained using
Reponse-2000 software - Model M3 & M4 .................................................................................. 37
Figure 2-25 Pier column longitudinal strain (mm/m) - Model M3 & M4 .................................... 37
Figure 2-26 Shows discritized pattern of the pier pilecaps and heads - Model M3 & M4 ........... 38
Figure 2-27 Abutment discretization and reinforcement arrangement - Model M5 & M6 .......... 40
Figure 2-28 Shows abutment moment curvature diagram and M-V interaction obtained using
Reponse-2000 software - Model M5 & M6 .................................................................................. 41
Figure 2-29 Abutment longitudinal strain (mm/m) - Model M5 & M6 ....................................... 41
Figure 2-30 Deck-girder reinforcement arrangement and discretized pattern - Model M5 & M6 42
Figure 2-31 Deck-girder moment curvature diagram and M-V interaction obtained using
Reponse-2000 software - Model M5 & M6 .................................................................................. 43
Figure 2-32 Deck-girder longitudinal strain (mm/m) - Model M5 & M6 .................................... 43
Figure 2-33 Pier column reinforcement arrangement and discretized pattern - Model M5 & M644
Figure 2-34 Pier column moment curvature diagram and M-V interaction obtained using
Reponse-2000 software - Model M5 & M6 .................................................................................. 44
Figure 2-35 Pier column longitudinal strain (mm/m) - Model M5 & M6 .................................... 45
Figure 2-36 Shows discritized pattern of the pier pilecaps and heads - Model M5 & M6 ........... 45
Figure 2-37 Shows intermediate diaphragm bracings in red color - Model M5 & M6 ................ 46
Figure 3-1 Mode shapes of the model M1for the first six modes ................................................. 49
Figure 3-2 Mode shapes of the model M2for the first six modes ................................................. 50
Figure 3-3 Mode shapes of the model M3 for the first six modes ................................................ 51
Figure 3-4 Mode shapes of the model M4 for the first six modes ................................................ 52
Figure 3-5 Mode shapes of the model M5 and M6 for the first six modes................................... 53
xv
Figure 3-6 Obtained abutment pushover curves for all archetype models when abutment
No.1(south abutment) is pushed along the deck direction ............................................................ 55
Figure 3-7 Obtained abutment pushover curves for all archetype models when abutment No.1
(south abutment) is pushed along and across the deck directions ................................................ 56
Figure 3-8 Obtained pier column pushover curves for all archetype models when the column
No.1 is pushed across and across-along the deck directions ........................................................ 57
Figure 3-9 Seismic history of earthquake magnitudes in Canada - Source: Natural Resources
Canada (NRC) ............................................................................................................................... 59
Figure 3-10 Demonstrates tectonic plates in southwestern Canada - Source: Natural Resources
Canada (NRC) ............................................................................................................................... 60
Figure 3-11 Illustrates the contributed seismic zones in H& R model-source: EZ-FRISK.......... 61
Figure 3-12 Magnitude-Distance deaggregation spectral response @ 5% damping - horizontal
component obtained from EZ-FRISK for the period 0.246sec ..................................................... 62
Figure 3-13 Shows mean hazard for spectral response at 5% damping - Source: EZ-FRISK
software ......................................................................................................................................... 63
Figure 3-14 Target and H & R model uniform hazard spectra (50%ile) for soil site class C Source OPEN File 4459 (Geological Survey of Canada, 2003) ................................................... 65
Figure 3-15 Shows ground motion spectra along with the target spectrum, and the range of
period of interest [T1-model M1 (0.246 sec) -T1-model M4, M5 & M6 (0.374 sec)] ................. 67
Figure 4-1 Shows relative displacements of a rocking element (abutment) ................................. 74
Figure 4-2 Shows obtained abutment rocking - Model M2 .......................................................... 75
Figure 4-3 Maximum relative displacement of the abutment along the bridge deck - Model M1 &
M2 ................................................................................................................................................. 76
xvi
Figure 4-4 Shows obtained probability of collapse for the archetype model M1 & M2 based on
the fragility fitting functions illustrated in Appendix B................................................................ 77
Figure 4-5 Compares obtained probability of collapse for the archetype model M1 & M2 ........ 78
Figure 4-6 Maximum relative displacement of the abutment along the bridge deck - Model M3 81
Figure 4-7 Maximum relative displacement of the pier column across (x-direction) and along(ydirection) of the bridge deck - Model M3 ..................................................................................... 82
Figure 4-8 Left plot shows the obtained abutment rocking and right plot shows abutment nonsimulated collapse mode IDA results along the bridge deck for the model M4 ........................... 83
Figure 4-9 Shows obtained pier column actual relative displacement across (x-direction) and
along (y-direction) of the bridge deck for the model M4 ............................................................. 84
Figure 4-10 Shows pier column total drift ratio (including column's rocking drift) across the
bridge deck verses total base shear obtained from performed IDA - Model M3 & M4. .............. 85
Figure 4-11 Shows pier column actual drift ratio across the bridge deck verses total base shear
obtained from performed IDA - Model M3 & M4. ...................................................................... 86
Figure 4-12 Shows pier column actual drift ratio across the bridge deck verses total moment
obtained from performed IDA - Model M3 & M4. ...................................................................... 87
Figure 4-13 Shows obtained probability of collapse for the archetype model M3 & M4 based on
the fragility fitting functions illustrated in Appendix B................................................................ 88
Figure 4-14 Compares obtained probability of collapse for the archetype model M3 & M4 ...... 89
Figure 4-15 Shows obtained actual relative displacement for abutment and pier column along
and across the bridge deck respectively - Model M5 ................................................................... 91
Figure 4-16 Shows obtained actual relative displacement for abutment and pier column along
and across the bridge deck respectively - Model M6 ................................................................... 92
xvii
Figure 4-17 Shows pier column actual drift ratio across the bridge deck verses total base shear
obtained from performed IDA - Model M5 & M6. ...................................................................... 93
Figure 4-18 Shows pier column actual drift ratio across the bridge deck verses total moment
obtained from performed IDA - Model M5 & M6. ...................................................................... 94
Figure 4-19 Shows obtained probability of collapse for the archetype model M5 & M6 based on
the fragility fitting functions illustrated in Appendix B................................................................ 95
Figure 4-20 Compares obtained probability of collapse for the archetype model M5 & M6 ...... 96
Figure 4-21 Shows effective yield displacement
- Source FEMA P695 ....................... 98
Figure 4-22 Compares obtained probability of collapse for all the archetype models ............... 101
Figure 4-23 illustrates median collapse intensity (SCT) and the intensity at MCE-level ground
motions (SaMT) for the model M1 & M2 .................................................................................. 103
Figure 4-24 Illustrates median collapse intensity (SCT) and the intensity at MCE-level ground
motions (SaMT) for the model M3 & M4. ................................................................................. 104
Figure 4-25 Illustrates median collapse intensity (SCT) and the intensity at MCE-level ground
motions (SaMT) for the model M5 & M6. ................................................................................. 105
Figure 4-26 Demonstrates the proposed design procedure when SSI effect needed to be
considered in a non-SSI model ................................................................................................... 108
Figure A-1 Significant duration of Chi Chi, Superstition, Loma Prieta, and Northridge
acceleration time histories with 5-95% arias intensity - Source: PEER Strong Motion Database
117
Figure A-2 Significant duration of Imperial Valley, Victoria- Mexico, Morgan Hill, and Duzce
acceleration time histories with 5-95% arias intensity - Source: PEER Strong Motion Database
..................................................................................................................................................... 118
xviii
Figure A-3 Significant duration of Cape Mendocino, Mammoth Lakes, N.Palm Springs, and
Tabas acceleration time histories with 5-95% arias intensity - Source: PEER Strong Motion
Database ...................................................................................................................................... 119
Figure A-4 Significant duration of San Fernando, Gazli, Managua, and Whittier Narrows
acceleration time histories with 5-95% arias intensity - Source: PEER Strong Motion Database
..................................................................................................................................................... 120
Figure A-5 Significant duration of Coalinga, Westmorland, Kobe, and Spitak acceleration time
histories with 5-95% arias intensity - Source: PEER Strong Motion Database ......................... 121
Figure C-1 Shows dispersion of the obtained IDA results for the archetype model M1 & M2
123
Figure C-2 Shows dispersion of the obtained IDA results for the archetype model M3 & M4 . 124
Figure C-3 Shows dispersion of the obtained IDA results for the archetype model M5 & M6 . 125
Figure D.1-1 Shows pier column total drift ratio across the bridge deck verses total base shear
obtained from performed IDA using the Imperial Valley, Victoria-Mexico, Morgan Hill, and
Duzce ground motions - Model M3 & M4. 126
Figure D.1-2 Shows pier column total drift ratio across the bridge deck verses total base shear
obtained from performed IDA using the Cape Mendocino, Mammoth Lakes, N.Palm Springs,
and Tabas ground motions - Model M3 & M4. .......................................................................... 127
Figure D.1-3 Shows pier column total drift ratio across the bridge deck verses total base shear
obtained from performed IDA using the San Fernando, Gazli, Managua, and Whittier Narrows
ground motions - Model M3 & M4. ........................................................................................... 128
xix
Figure D.1-4 Shows pier column total drift ratio across the bridge deck verses total base shear
obtained from performed IDA using the Coalinga, Westmorland, Kobe, and Spitak ground
motions - Model M3 & M4 ......................................................................................................... 129
Figure D.2-1 Shows pier column actual drift ratio across the bridge deck verses total base shear
obtained from performed IDA using the Imperial Valley, Victoria-Mexico, Morgan Hill, and
Duzce ground motions - Model M3 & M4. ................................................................................ 130
Figure D.2-2 Shows pier column actual drift ratio across the bridge deck verses total base shear
obtained from performed IDA using the Cape Mendocino, Mammoth Lakes, N.Palm Springs,
and Tabas ground motions - Model M3 & M4. .......................................................................... 131
Figure D.2-3 Shows pier column actual drift ratio across the bridge deck verses total base shear
obtained from performed IDA using the San Fernando, Gazli, Managua, and Whittier Narrows
ground motions - Model M3 & M4. ........................................................................................... 132
Figure D.2-4 Shows pier column actual drift ratio across the bridge deck verses total base shear
obtained from performed IDA using the Coalinga, Westmorland, Kobe, and Spitak ground
motions - Model M3 & M4 ......................................................................................................... 133
Figure D.3-1 Shows pier column actual drift ratio across the bridge deck verses total base shear
obtained from performed IDA using the Imperial Valley, Victoria-Mexico, Morgan Hill, and
Duzce ground motions - Model M5 & M6. ................................................................................ 134
Figure D.3-2 Shows pier column actual drift ratio across the bridge deck verses total base shear
obtained from performed IDA using the Cape Mendocino, Mammoth Lakes, N.Palm Springs,
and Tabas ground motions - Model M5 & M6. .......................................................................... 135
xx
Figure D.3-3 Shows pier column actual drift ratio across the bridge deck verses total base shear
obtained from performed IDA using the San Fernando, Gazli, Managua, and Whittier Narrows
ground motions - Model M5 & M6. ........................................................................................... 136
Figure D.3-4 Shows pier column actual drift ratio across the bridge deck verses total base shear
obtained from performed IDA using the Coalinga, Westmorland, Kobe, and Spitak ground
motions - Model M5 & M6. ........................................................................................................ 137
Figure E.1-1 Shows pier column actual drift ratio across the bridge deck verses total moment
obtained from performed IDA using the Imperial Valley, Victoria-Mexico, Morgan Hill, and
Duzce ground motions - Model M3 & M4. ................................................................................ 138
Figure E.1-2 Shows pier column actual drift ratio across the bridge deck verses total moment
obtained from performed IDA using the Cape Mendocino, Mammoth Lakes, N.Palm Springs,
and Tabas ground motions - Model M3 & M4. .......................................................................... 139
Figure E.1-3 Shows pier column actual drift ratio across the bridge deck verses total moment
obtained from performed IDA using the San Fernando, Gazli, Managua, and Whittier Narrows
ground motions - Model M3 & M4. ........................................................................................... 140
Figure E.1-4 Shows pier column actual drift ratio across the bridge deck verses total moment
obtained from performed IDA using the Coalinga, Westmorland, Kobe, and Spitak ground
motions - Model M3 & M4. ........................................................................................................ 141
Figure E.2-1 Shows pier column actual drift ratio across the bridge deck verses total moment
obtained from performed IDA using the Imperial Valley, Victoria-Mexico, Morgan Hill, and
Duzce ground motions - Model M5 & M6. ................................................................................ 142
xxi
Figure E.2-2 Shows pier column actual drift ratio across the bridge deck verses total moment
obtained from performed IDA using the Cape Mendocino, Mammoth Lakes, N.Palm Springs,
and Tabas ground motions - Model M5 & M6. .......................................................................... 143
Figure E.2-3 Shows pier column actual drift ratio across the bridge deck verses total moment
obtained from performed IDA using the San Fernando, Gazli, Managua, and Whittier Narrows
ground motions - Model M5 & M6. ........................................................................................... 144
Figure E.2-4 Shows pier column actual drift ratio across the bridge deck verses total moment
obtained from performed IDA using the Coalinga, Westmorland, Kobe, and Spitak ground
motions - Model M5 & M6. ........................................................................................................ 145
Figure F.1-1 Abutment layout plan - Br. Type 1 ........................................................................ 146
Figure F.1-2 Abutment plan and side view - Br. Type 1 ............................................................ 147
Figure F.1-3 Abutment sectional elevation and RC detail - Br. Type 1 ..................................... 148
Figure F.1-4 Deck-girders general arrangement and deck-girder and abutment connection detail Br. Type 1 ................................................................................................................................... 149
Figure F.2-1 Bridge sectional view and typical abutment and pier pile layouts - Br. Type 2 .... 150
Figure F.2-2 Abutment layout plan, front and side elevation views - Br. Type 2 ...................... 151
Figure F.2-3 Abutment RC detail and pier pilecap and columns general arrangement - Br. Type 2
..................................................................................................................................................... 152
Figure F.2-4 Pier pilecap and columns RC detail - Br. Type 2 .................................................. 153
Figure F.3-1 Bridge foundations and side elevation views - Br. Type 3 .................................... 154
Figure F.3-2 Abutment shear key, layout plan, and front and side elevation views and detail of
bridge intermediate diaphragms - Br. Type 3 ............................................................................. 155
Figure F.3-3 Pier pilecap and columns concrete outline detail - Br. Type 3 .............................. 156
xxii
Figure F.3-4 Pier pilecap and columns RC detail - Br. Type 3 .................................................. 157
Figure F.3-5 Abutment RC detail and end diaphragm detail - Br. Type 3 ................................. 158
Figure H.1-1 Mean Epsilon and Magnitude-Distance Deaggregation spectral responses @ 5%
damping for the period 0.284 sec and amplitude 0.86g and soil class C- Source: EZ-FRISK ... 160
Figure H.2-1 Mean Epsilon and Magnitude-Distance Deaggregation spectral responses @ 5%
damping for the period 0.335sec and amplitude 0.8g and soil class C- Source: EZ-FRISK ...... 161
Figure H.3-1 Mean Epsilon and Magnitude-Distance Deaggregation spectral responses @ 5%
damping for the period 0.374sec and amplitude 0.757g and soil class C- Source: EZ-FRISK .. 162
xxiii
List of Symbols
Element x-section
Modulus of elasticity
Fy
Soil yield strength
fc
Compressive strength
ft
Tensile strength
Modulus of rigidity
hdia*
Effective height if diaphragm is not designed for full soil pressure
hdia** Effective height if diaphragm is designed for full soil pressure
Arias intensity
&
Moment of inertia about local axis 2 and 3 respectively
Torsional constant
K0
Initial stiffness
Stiffness of abutment due to embankment's passive pressure force resisting movement
behind the abutment backwall
Length of a structural element (pile, abutment, etc.)
P-Δ
Geometrical non-linearity
Pu
Soil ultimate strength
The design spectral response acceleration, expressed as a ratio to gravitational
acceleration, for a period of
The 5% damped spectral response acceleration, expressed as a ratio to
gravitational acceleration, for a period of
xxiv
Period of a structure
1
Fundamental period of structure
Base shear of the structure
Time average 30 meter shear wave velocity
kc
Confinement factor
Su
Un-drained shear strength
qt
Resistance soil or tip resistance
fs
Sleeve friction reading
Rf
Friction ratio
SCT
Spectral acceleration correspond to median collapse
SCT10 % Spectral acceleration correspond to10% probability of collapse
SCT20 % Spectral acceleration correspond to20% probability of collapse
SaMT
Intensity at MCE-level ground motions correspond to fundamental period of a structure
α
Second segment coefficient of stiffness of Allotey and Elnagar nonlinear dynamic soilstructure interaction model
β
The stiffness ratio parameter in the SSI model which defines the stiffness of the 3rd
segment in proportion to K0
βn
Strength ratio parameter in the SSI model
σ
Stress or standard deviation
ϒ
Specific weight or Ramberg-Osgood parameter
η
Average ratio of the CMR@ median of SSI model to the correspond CMR@ median of
the non-SSI model
Maximum relative displacement of the structure at direction
xxv
Maximum relative displacement of the structure at direction
Effective yield relative displacement
ε
Strain in structural analysis and Number of standard deviations by which an observed
logarithmic spectral acceleration differs from the mean logarithmic spectral
acceleration in hazard analysis
εc
Strain at peak stress
λ
Relative displacement capacity/demand ratio
µ
Period-based ductility or mean value

Poisson's ratio
w
Width of the backwall or diaphragm for seat and diaphragm abutments
ξ
Rayleigh damping ratio
Subscript
MT
Maximum Considered Earthquake (MCE) spectral acceleration at the period of the
system, T.
CT
Spectral acceleration of the collapse level ground motions
SSI
Soil structure interaction
Non-SSI Ignored soil structure interaction
n%
Value relates to n% probability of collapse
Superscript
ˆ
Estimated value
xxvi
List of Abbreviations
ACMR:
Adjusted Collapse Margin Ratio
ACMRmedian
Adjusted Collapse Margin Ratio correspond to median (50%)
ACMR10% or 20% Adjusted Collapse Margin Ratio correspond to 10% or 20% probability of
collapse
API:
American Petroleum Institute
CDF:
Cumulative Distribution Function
CMR:
Collapse Margin Ratio
CMRmedian
Collapse Margin Ratio correspond to median
CMR10% or 20% Collapse Margin Ratio corresponding to 10% or 20% probability of collapse
CPT:
Cone Penetration Test
CSD:
Caltrans Seismic Design Criteria
DB:
Displacement Base
DV:
Decision Variable
EDP:
Engineering Demand Parameter
FB:
Force Base
FEMA:
Federal Emergency Management Agency
PBEE:
Performance Based Earthquake Engineering
IM:
Intensity Measure
IDA:
Incremental Dynamic Analysis
MCE:
Maximum Considered Earthquake
:
NRC:
Multiple Degrees of Freedom
Natural Resources Canada
xxvii
NSC:
Non-Simulated Collapse Modes
NSERC
PEER:
Natural Sciences and Engineering Research Council of Canada
The Pacific Earthquake Engineering Research Center
:
Peak Ground Acceleration expressed as a ratio to gravitational acceleration
PSHA:
Probabilistic Seismic Hazard Analysis
RC:
Reinforced Concrete
:
Response Spectrum Method
RTH:
Response Time History Analysis
SBT:
Soil Behaviour/classification Type
SC:
Simulated Collapse Modes
:
Single Degree of Freedom
:
Seismic Force Resisting System(s)
SSF:
Spectral Shape Factor
SSI:
Soil Structure Interaction
:
Uniform Hazard Spectra
xxviii
Glossary
A measure of the strength of a ground motion that determines
the intensity of shaking by measuring the acceleration of
transient seismic waves It is defined as the time-integral of the
square of the ground acceleration:
Displacement Base
In the displacement base element formulation approach
Formulation
displacement shape functions are used corresponding for instance
to a linear variation of curvature along the element. Therefore, DB
formulation depends on the assumed sectional constitutive
behavior.
Epsilon (ε )
the number of standard deviations by which an observed
logarithmic spectral acceleration differs from the mean logarithmic
spectral acceleration of a ground-motion prediction (attenuation)
equation
Force Base Formulation
In force base element formulation approach a linear moment
variation is imposed. FB formulation is more accurate than DB as
it does not depend on the assumed sectional constitutive behavior.
The only approximation in this case is disserting number of the
controlling section.
xxix
Lanczos Algorithm
An algorithm that introduced by Hughes in 1987 for evaluation of
the structural natural frequencies and mode shapes.
Jacobi Algorithm
An alternative algorithm to evaluate the structural natural
frequencies and mode shapes using Ritz transformation.
Least Square Method
Minimizes the total square errors between the estimated
probability of collapse and the observed probability of collapse
over all of the Sa level
Maximum Likelihood Method Counts for non-constant variance
:
The part of the structural system that has been considered in the
design to provide the required resistance to the earthquake forces
xxx
Acknowledgements
I would like to express my deepest gratitude to my academic supervisor, Dr. C.Ventura,
for his outstanding guidance in research and generous advice in teaching and many other aspects
offered by him. I feel truly honoured to have the opportunity of being his student. The
completion of this thesis would have not been possible without his insightful comments and
valuable feedback.
My many special thanks to Dr. Liam Finn and Dr. Mahdi Taiebat, my co-supervisors, for
their amazing support/advising/guidance through all the years of my studies. I am indebted to
Mr. Don Kennedy, P.Eng, and Mr. Alfred Kao, P.Eng. for providing me with very helpful bridge
documents and drawings. This research has been funded by Canadian Seismic Research Network
(CSRN). Their generous support is graciously acknowledged. I would like to gratefully
recognize research team at Soil-Structure interaction in performance Based Design of Bridges for
their support, helpful discussions and joyful company.
I am utterly thankful to my wife, Maryam, and my son, Mohammad, for their continual
support, inspiration, and being there for me at hard times and sharing the happy moments
throughout the years of studying and living in Vancouver. Words cannot express my appreciation
to them. I am eternally grateful to my parents who always listened to me and shared a piece of
advice on the different challenges I faced. Their love and comfort was present every moment and
they taught me the path of success under God’s will, being patient and working hard to succeed
while serving others. This degree and all my achievements would not be possible without their
encouragements and continuous support.
xxxi
Dedication
To my family for their unconditional love and support
xxxii
Chapter 1: Introduction
1.1
Overview and Motivation for Study
Many bridges have been collapsed due to occurrence of devastating earthquakes and
design flaws. So far, this matter resulting in numerous fatalities and huge financial loss. Figure
1-1 shows catastrophic failure of two bridge structures due to Kobe and Loma Prieta earthquake:
Figure 1-1 Left photo: Shows collapse of a 630m segment of the elevated Hanshin Expressway due to Kobe
Earthquake Japan (1995) (Source: Wikipedia)
Right photo: Shows collapse of the Cypress Structure, the freeway approach to the Bay Bridge from Oakland
due to Loma Prieta Earthquake (1989) (Source: Wikipedia).
Based on results obtained from analytical study of the Hanshin Expressway, soil role in
the collapse of the structure had been doubled: Initially, soil modified the seismic waves so that
the frequency of the surface motion at the site became disadvantageous for the structure.
Moreover, the compliance of soil and foundation increased the period of the system and moved it
to a region of stronger response, i.e., increase of ductility demand on the piers exceeded 100% as
compared to piers fixed at the base (G. Mylonakis et al., 2000). The Cypress Structure was built
on loose soils that shook much more vigorously than surrounding regions on stronger ground.
1
On the other hand, a few bridge structures suddenly collapsed because of design flaw
without being hit by any sever Earthquake. One of these bridges that are collapsed due to design
flaw was Interstate 35W bridge. The picture portion of this bridge immediately after the
occurrence of collapse is shown in Figure 1-2.
Figure 1-2 Interstate 35W bridge, which stretches between Minneapolis and St. Paul, was suddenly collapsed
during rush hour due to design flaw on Aug.1 2007, caused thirteen people to die and 145 injuries. (Source:
AP Photo/Pioneer Press, Brandi Jade Thomas, by permission).
To minimize seismic risk and to avoid bridge catastrophic failure issue in future, collapse
issue should be sufficiently addressed in the next generation bridge design codes. One of the
important factors that need to be investigated and effectively assessed and incorporated in the
future code is soil structure interaction (SSI) of bridges, especially for event of strong earthquake
when supporting soil is soft as presence of the soft soils can contribute to unexpected seismic
demands to the bridge structures. This matter can happen in two ways:
2
 Modifying the frequency content of the seismic wave such that the bridge base motion
can increase the structural responses.
 Modifying dynamic parameters of the bridge structures - soil foundation (presence of the
surrounding soil shifting the entire soil-structure system to a region of sensitive
response).
In addition, soil structure interaction may cause basement motions to differ from those of
the free field. Furthermore, load distribution in members may be different when SSI is
considered. This variation of load in members can lead to differential settlements and as a result
crack propagation in the structural members of the bridge structures.
In this study, performance based approach including the soil structure interaction is considered in
investigation of the seismic performance of RC integral bridges.
1.2
Objective and Scope
Creating a system that can lead to a desired performance in a efficient way is the main
challenge not only in design of new bridges, but also in retrofitting the existing bridges. Many
researchers believe that using performance-based earthquake engineering, important
infrastructures facilities such as bridges can be designed with the better performance level,
higher safety, and lower life-cycle costs associated with seismic risk.
The main objective of this research is to assess nonlinear seismic response of three
different types of the integral abutment bridges and to obtain their probability of the collapse
3
through performance-based earthquake engineering methodologies performing incremental
dynamic analysis using SeismoStruct1 software (version 6.3.2).
As it is mentioned in Section 1.1, in this research methodology adapted to investigate seismic
performance of the RC integral bridges considering the soil structure interaction is relying on
performance based approach. Although structural damage can be viewed in terms of the
occurrence of 'excessive forces' or 'excessive deformations', performance-based approach is
chosen to view the bridge structural damage due to the following reasons:
1) The performance assessment procedure follows a logical progression of steps
such as: Seismic hazard characterization, simulation of structural response,
and damage and loss modeling and assessment.
2) The results of each procedure are fully understandable through four well
established output variables: Earthquake Intensity Measure (IM), Engineering
Demand Parameters (EDP), Damage Measures (DM) , and Decision Variables
(DV)
3) The performance-based approach has ability to enhance seismic risk decisionmaking through assessment and design methods that have a sound scientific
base
4) Performance-based methodology can be employed in providing a rigid
probabilistic framework for the next generation of seismic design codes and
criteria
1 SeismoStruct is an award-winning Finite Element software here is used as a tool for predicting displacement behavior and other structural
response of bridge models under static or dynamic loading.
4
The relationship between decision variable, engineering demand parameter, and damage
and intensity measure in performance based earthquake engineering (PBEE) is introduced by
Pacific Earthquake Engineering Research Center (PEER) and shown in Figure 1-3:
Figure 1-3 PEER framing equation and example parameters for seismic shaking (Moehle, 2003)
Assessment framework along with stereotype parameters for seismic shaking in
performance based earthquake engineering (PBEE) for each process is illustrated in Table 1-1.
5
Table 1-1Assessment framework for performance based earthquake engineering - Source: G. Deierlein, 2004.
Process
Seismic Hazard
Analysis
Site→IM
Output Variable
IM: Intensity Measure




Structural
Analysis
IM→EDP
Damage
Assessment
EDP→DM
EDP: Eng. Demand Parameter



Analysis
DM→DV
Peak & residual interstory drift
Floor acceleration
Component forces & deformations
DM: Damage Measure



Loss & Risk
Sa(T1)
PGA,PGV
Arias Intensity
Inelastic Spectra
Component damage and repair
states
Hazard (falling, egress, chemical
release, etc.
Collapse
DV: Decision Variable




Casualties
Closure issues (post EQ safety)
Direct $ loss
Repair duration
Disciplines
Key Parameters
Seismology;
Geotechnical
Engineering


Structural and
Geotechnical
Engineering



Foundation & structural
system properties
Model parameters
Gravity loads
Structural &
Construction
Engineering;
Architecture
loss modeling




Structural and components
HVAC & plumbing systems
Cladding & partition details
contents
Construction
Cost estimating;
Loss modeling;
Risk
Management



Occupancy
Time of Earthquake
Post Earthquake recovery
resources

Fault location & type
Location & length of
rupture (M_R)
Site & soil condition
As discussed earlier, this research seeks to determine seismic performance of the RC
integral bridges through performance design approach including soil-structure interaction.
However, this research focused damage assessment and loss and risk analysis are not considered.
To assess the structural damage, 3-D nonlinear models of three types of integral bridges
with different skew angles are constructed. For each bridge type, two archetype models are
simulated with or without considering the soil structure interaction. In this study, relative
displacement and drift of the abutment backwalls and pier columns are considered as engineering
demand parameters (EDPs). In addition, spectral acceleration of ground motions is chosen as
6
intensity measure (IM). Incremental dynamic analysis (IDA), however, is used to determine the
engineering demand parameters.
To determine probability of collapse, IDA is performed for each archetype model using a
set of 20 well selected and un-scaled ground motions. Probability of collapse and collapse
margin ratio (10%, 20%, or median) for each bridge prototype are calculated using the results
obtained through performing IDA. Finally, a design procedure is proposed when the SSI effects
need to be considered assuming that the collapse margin ratio (10%, 20%, or median) for each
bridge prototype were determined and provided in the future bridge design codes.
A general framework for the methodology adapted in this research is illustrated in Figure 1-4.
Main Objective
Upstream
Robust
Bridge Model
Type 1
IDA
Probability of
Collapse of The
Archetype Models
Type 3
Type 2
Non-SSI & SSI
Models (M1 &M2)
Non-SSI & SSI
Models (M3 &M4)
Downstream
Non-SSI & SSI
Models (M5 &M6)
Proposed Design
Procedure
Figure 1-4 General framework of the study research
1.3
Outline
This report is developed in five chapters. In chapter two, modeling of the bridge
structures is described in detail. Eigen value, static pushover, and incremental dynamic analysis
are discussed in chapter three. In the following chapter, IDA results and probability of collapse
7
curve for each archetype model are provided. In this chapter, collapse margin ratio (CMR) is
considered as damage indicator for each model for different probabilities (110%, 20%, and
50%). The calculated CMR values are compared to each other. Lastly, summary of the work
done in this study are presented in chapter five and it is followed by a proposed design
procedure, limitations and problem for future work.
8
Chapter 2: Modeling of the Bridge Structures
2.1
Archetype Models
To determine and compare the probability of collapse of the integral abutment bridges a
total three different types of bridges is considered that here named: type one, type two, and type
three. For each bridge type, two archetype models are defined: One with consideration of the soil
structure interaction, and another archetype model without including SSI effect. Therefore, total
six multi degree of freedom (MDOF) analytical models are developed: Model M1, M2, M3, M4,
M5, & M6.
Model M1 is a simplified model of a single span integral bridge with 30 degree skew
angle. Model M2 is a SSI version of the model M1 considering the effect of the soil around the
abutment piles assuming a soil sub-layer arrangement along the piles that simulated with SSI p-y
links corresponding to each layer of soil based on the nonlinear dynamic SSI soil model which
developed and introduced by Allotey and El Naggar2 in 2008.
Model M3 is a simplified MDOF model of a three span integral bridge with 15 degree
skew angle. Model M4 is SSI version of the model M3 considering the effect of the soil behind
the abutment backwall using CALTRANS springs and soil around the abutment and pier piles
assuming the same soil sub-layer arrangement and the SSI p-y links along the piles which is used
in the model M2.
Finally, Model M5 is simplified model of a two bonds semi-integral three span bridge
with 6 degree skew angle. Here, it is called semi integral as vertical end edge of the pre-stressed
girders in this bridge are connected to the abutment, but bottom of girders are supported by
2 Research associate dean, Geotechnical research director, and professor at department of Civil an Environmental Engineering University of
Western Ontario, London - ON.
9
elastomeric rubber pad bearings to allow minor rotation and displacement of the girders.
However, here for simplicity bearings are not modeled but effect of it was considered in the
assigned damping ratio during performing analysis. Model M6 is SSI version of the model M5
considering just the effect of the soil behind the abutment backwall using CALTRANS springs
as abutment and piers are rested on deep pilecap foundation without having piles. Detail of the
above numeric bridge models is summarized in Table 2-1 and constructed prototype models are
shown in Figure 2-1.
Table 2-1 Summary of the bridge types and archetype models are developed and used in this study
Bridge
Type
Span Length
(m)
Skewness
Angle
(°)
Integral abutment with pilecap
foundation and pre-stressed
precast girder
38
Integral abutment bridge with
pile foundation and total 9 Nos.
pre-stressed precast girder
Archetype Model
Structural Feature
M1
M2
Horizontal
Radius
(m)
ρLong
(%)
ρTrans
(%)
SSI Feature
30°
1-4
0.4-1.0
N/A
38
30°
1-4
0.4-1.0
CALTRANS springs and
Allotey –El Naggar
nonlinear SSI p-y links
17-29-19
15°
1-4
0.4-1.0
N/A
1-4
0.4-1.0
CALTRANS springs and
Allotey –El Naggar
nonlinear SSI p-y links
1-4
0.4-1.0
N/A
Type 1
M3
Integral abutment bridge with
pier & abutment pile
foundations and total 4 Nos.
pre-stressed precast girders
1009.95
Type 2
M4
M5
Type 3
M6
Integral abutment bridge with
pier & abutment pile
foundations and total 4 Nos.
pre-stressed precast girders
Two bonds semi-integral bridge
with common seat abutment,
pier pilecap deep foundation,
intermediate diaphragms, and
total 8 Nos. pre-stressed precast
girders
Two bonds semi-integral bridge
with common seat abutment,
pier pilecap deep foundation,
intermediate diaphragms, and
total 8 Nos. pre-stressed precast
girders
17-29-19
15°
27.5-37.5-27.5
6°
1009.95
CALTRANS springs
27.5-37.5-27.5
6°
1-4
0.4-1.0
10
Figure 2-1 3-D view of the developed bridge models
2.2
Sources of Uncertainty
In this study, two major types of uncertainties exist: Modeling uncertainty and ground
motion uncertainty. Modeling uncertainty, however, consists of three uncertainties: uncertainty
due to the structural nonlinearities which inherited in the material properties of the structural
components, geometrical non-linearity (P-Δ effect), and soil structure interaction effect (SSI).
11
The modeling and ground motion uncertainties are considered and discussed in the upcoming
sub-sections.
2.2.1
Modeling Uncertainty
Generally modeling uncertainty exists due to structural uncertainty, P-Δ effect, and soil
structure interaction. The first one is considered in modeling structural components which have
important role in the response of the structures, e.g., modeling structural components in Shear
Force Resisting System (SFRS) using fiber approach illustrated in Chapter 2. The latter is
considered by selecting nonlinear geometry option during Response History Analysis (RHA).
Finally, soil structure interaction is considered by assigning a series of spring and links to the
abutment backwall and abutment and pier piles which are discussed in detail in Section 2.2.1.3.
2.2.1.1
Structural Nonlinearity
To minimise structural nonlinearity, elements which have important role in the overall
structural response of the bridge structures are defined in the models using force-based
formulation. However, some structural components such as shear key and wing walls are not
modeled for simplicity.
Methodology requires detailed modeling of nonlinear behaviour of archetypes. However,
using appropriate limits on the controlling response parameter which illustrated in the element's
sectional response sub-sections, collapse failure modes that cannot be explicitly modeled are
evaluated and imposed to the models.
2.2.1.2
P-Δ Effect
Large displacements/rotations and large independent deformations relative to the frame
element's chord due to cumulative gravity forces is known as P-Δ effect. Considering P-Δ effect
in analysis is very important as small deformations relative to the element's chord may result in
12
larger progressive lateral displacements. P-Δ effects can be very significant for bridge structures
with long and relatively flexible pier columns as flexibility of these columns is usually resulting
in having a plastic flexural mechanism and as a result progressive larger lateral displacements.
This increase in displacements can result in a loss of stability of the structure. In SeismoStruct,
this effect can be captured through the employment of a developed total co-rotational
formulation based on an exact description of the kinematic transformations associated with large
displacements and three-dimensional rotations of the beam-column members. The implemented
total co-rotational formulation results in correct definition of the element's independent
deformations and forces and correct definition of geometrical non-linearity on the stiffness
matrix (Correia and Virtuoso, 2006).
2.2.1.3
Soil Structure Interaction (SSI)
To consider soil structure interaction, effect of soil interaction behind the abutment
backwall (model M2, M4 and M6) and around the abutment and pier piles (model M2 and M4)
are considered using the CALTRANS springs and SSI p-y links provided in the software based
on a dynamic Winkler model for nonlinear soil structure interaction respectively.
2.2.1.3.1
Soil Effect behind the Abutment Backwall
Bridge abutments attract large seismic forces, especially, in longitudinal direction. Soilabutment interaction can have a big effect on overall bridge response. However, as soil has
limited ability to take tension, separation may occur at embankment behind the abutment
backwall due to cyclic loading (gapping effect). This gap causes large compression stresses to
develop in front of the structure and tensile stresses behind the structures. The gapping effect is
demonstrated and shown in Figure 2-2.
13
Expansion of superstructure
Contraction of superstructure
Embankment
Backfilling
Separation due to the cyclic pressure
on the soil over a time of period
Superstructure in
static position
Embankment
Soils experiences
cyclic loading
Figure 2-2 Soil separation due to the cyclic pressure on the embankment soil behind the bridge abutment
backwall - Source: Thevaneyan K. David and John P. Forth (2011).
Previous studies demonstrate that if the bridge is analyzed using a procedure that
acknowledges backfill stiffness reduction, displacements at the piers are greater by 25%-75%,
depending on soil properties (Thevaneyan K. David and John P. Forth, 2011).
In this study, however, to account for the backfilling passive pressure force resisting
movement at the bridge abutments CALTRANS springs are assigned to the abutment backwalls
in the model M2, M4 & M6. Stiffness of the abutment due to passive pressure of backfilling
behind it is obtained based on section 7.8.1-2 SDC 2013 as per eq. 2.1 and summarized in Table
2-2.
h
h
K abut  Ki  w  (
)or Ki  w  (
)
1.7m
5.5 ft
(eq. 2.1)
14
where, Ki is the initial stiffness of the embankment fill material behind the abutments, and based
on section 7.81-1 SDC 2013 can be calculated as per equation below:
(Ki 
28.7kN / mm
50kips / in
) or
)
m
ft
(eq. 2.2)
in which,
w is the projected width of the backwall or diaphragm for seat and diaphragm abutments
respectively
hdia* is effective height if diaphragm is not designed for full soil pressure
hdia** is effective height if diaphragm is designed for full soil pressure
In this study, effective height is chosen considering the abutment diaphragms are
designed for full soil pressure. The above parameters are shown in Figure 2-3 for clarity.
Figure 2-3 Shows effective abutment width for skewed bridges, Source: SDC 1.7 (2013)
15
Table 2-2 Summary of calculated stiffness of the bridge abutments due to the embankment passive pressure
force resisting movement
Models
Ki
W
h
Kabut
(kN/mm/m)
(m)
(m)
(kN/mm)
M2
28.7
19.2
4.33
1404
M4
28.7
13.69
5.1
1178.54
M6
28.7
34.88
4
2355.43
An idealized link with linear behavior and an initial stiffness equal to 156 kN/mm, 294.63
kN/mm, and 294.43 kN/mm are assigned to the abutment backwall at end of intersection of each
pre-stressed precast girder for the model M2, M4 and M6 respectively. Response curve of the
implemented link is shown in Figure 2-4.
Figure 2-4 Response curve of a link used to simulate soil effect behind the abutment backwall
16
2.2.1.3.2
Soil Effect around the Abutment and Pier Piles
To account for soil effect around the abutments' and piers' piles, following soil sub-layer
arrangement are considered around the piles based on an assumed soil sub-layer. These
simplified sub-soil layer arrangement is shown in Figure 2-5.
Figure 2-5 Shows assumed soil sub-layer around the piles and location of the assigned CALTRANS springs
and SSI p-y links in the SSI models
The above soil sub-layer arrangement is considered based on geotechnical
recommendation report which was provided by H5M Company in the design stage of one of
newly constructed bridges as a part of PORT MANN Highway 1 Project in Vancouver. The
obtained parameters from Cone Penetration Test (CPT) for a chosen bore hole are shown in
Figure 2-6.
17
Figure 2-6 The obtained parameters from Cone Penetration Test (CPT) for the chosen bore holes from the
soil the soil investigation report- Source: H5M, 2009.
in which,
Su is un-drained shear strength in kPa,
qt represents of resistance soil or tip resistance in bar,
fs is sleeve friction reading in bar
Rf is friction ratio and calculated as per eq. 2.3:
(eq. 2.3)
Vs is shear wave velocity in m/s, and
SBT is a parameter that represents soil behaviour/classification type introduced by
Robertson et al in 1986.
18
According to the above assumption, a p-y curve for each layer is derived based on the
API code and L-Pile as they are shown in Figure 2-7.
p-y curves
180
Stiff Clay - D=1m
Soft Clay - D=3m
Soft Clay - D=5m
Soft Clay - D=7.5m
160
140
p-y curves - Sand Layer (D=10.85m)
9000
8000
7000
100
6000
80
p(kN/m)
p(kN/m)
120
60
5000
4000
3000
40
2000
20
1000
0
0
0.05
0.1
y(m)
0.15
0.2
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
y(m)
Figure 2-7 Left: Developed p-y curves based on the assumed sub-layers of soil surrounding piles using API
code. Right: Obtained p-y curve for sand layer around the piles using L-Pile Manual.
For simplicity, soil layers around the abutment and pier piles are considered identical for the
model M2 and M4. For considering the soil effect around the piles in the model M2 & M4, a
multi-purpose dynamic Winkler model for nonlinear soil structure interaction analysis is
employed which developed by Nii Allotey and M.Hesham El Naggar. The backbone curve in
this model is an adaptation of existing multi-linear approaches. In addition, features such as:
loading and unloading rules, modeling of radiation damping and cyclic degradation, and slack
zone for the analysis of shallow and deep foundation is considered in this model (N.Allotey & El
Naggar, 2008). As the model shows in good agreement with the experimental results, it is
incorporated in many software including SeismStruct. Schematic views of the loading and
unloading curves of this model are shown in Figure 2-8.
19
Figure 2-8 Shows backbone curve of the Winkler model used in the SSI p-y links - Source: SeismoStruct
Manual
Fitting a tri-linear curve to each obtained p-y curve, the following parameters are
obtained and defined in the models for pile links in each subdivided layer. Values of these
parameters are presented in Table 2-3.
Table 2-3 Presents calculated parameters which obtained fitting a tri-linear curve to the derived p-y curves
Thickness
Depth
K0
Pu
Fy
Layer
(m)
(m)
(kN/m)
α
β
(kN)
(kN)
βn=Pu/Fy
Fc/Fy
Stiff Clay
2
1
11490.59
0.0969
0.0404
117.21
98.56
1.189
0.595
Soft Clay
2
3
15651.47
0.1313
0.051
289.31
208.31
1.389
0.458
Soft Clay
2
5
15814.1
0.1313
0.051
292.32
210.47
1.389
0.458
Soft Clay
3
7.5
20756.07
0.1313
0.051
383.67
276.24
1.389
0.458
Silty Sand
3.7
10.85
685185.18
0.3484
0.0599
27624.23
25900
1.067
0.714
20
in which,
K0 is initial stiffness
α is the second segment coefficient of stiffness of the nonlinear dynamic soil structure
interaction model illustrated in Figure 2-8.
β is the stiffness ratio parameter in the SSI model which defines the stiffness of the third
segment in proportion to K0
Pu is soil ultimate strength
Fy is soil yield strength
βn is strength ratio parameter in the SSI model
2.2.2
Ground Motion Uncertainty
To minimize the ground motion uncertainly, a set of 20 well selected ground motions is
used in performing nonlinear response history analysis. The ground motions are selected in a
manner that their ε(T1) is close to mean ε(T1) obtained during hazard analysis and listed in Table
3-7. These un-scaled ground motions are directly obtained from PEER strong motion database.
However, to save the time, the significant duration of motion with more than %95 of arias
intensity is used for each of them in performing incremental dynamic analysis (see Appendix A
). This estimation, however, expected to have a minimal effect on the obtained engineering
demand parameters. This matter was investigated and verified by running a series of nonlinear
response history analysis with different ground motions using both entire motion record and the
above significant duration of them. Comparing the obtained results, it was found that this
estimation had a minimal effect on the obtained engineering demand parameters for various
motions as mainly peak acceleration of the motions were not truncated in the adapted significant
duration.
21
2.3
Structural Damping
A Rayleigh damping ratio (ξ) equal to 1%, 2%, and 2.5% is assigned for the bridge type
1 (model M1 & M2), bridge type 2 (model M3 & M4), and bridge type 3 (model M5 & M6) as
bridge global damping in analysis respectively. No additional damping at the element level is
defined in the models.
2.4
Additional Mass Assignment
To count for additional bridge deck concrete slab and barriers, a distributed mass equal to
0.458 tone/m, 0.425 tone/m, and 0.425 tone/m is assigned along each pre-stressed girder for
bridge type 1 (model M1 & M2), bridge type 2 (model M3 & M4), and bridge type 3 (model M5
& M6) respectively. In addition, an additional lumped mass of 0.925 tone/m is assigned to the
abutments bridge type 3 (model M5 & M6) to count for abutment median bulk head.
2.5
Element Behavior - Bridge Type 1 (Model M1 & M2)
For deck-girder and abutment section, Force Based (FB) inelastic frame element
(inlfrmFB) is adapted as this type of element doesn't depend on the assumed sectional
constitutive behavior. In SeismoStruct for FB elements, Gauss-Lobatto quadrature is employed.
Generally a minimum number of three Gauss-Lobatto integration sections are required to avoid
under-integration; however, such option will not generally simulate the spread of inelasticity in
an acceptable way. To simulate spread of inelasticity in deck-girder and abutments, total 5 and
10 numbers of integration points are considered respectively as fallow:
 5 integration sections: [-1 -0.655 0.0 0.655 1] x L/2
 10 integration sections: [-1 -0.920 -0.739 -0.478 -0.165 0.165 0.478 0.739 0.920 1]
x L/2
in which, L represents length of an element.
22
To present the cross-section behavior fiber approach is used in the SeismoStruct. In this
approach each fiber is associated with a uniaxial stress-strain relationship. As a result, the
sectional stress-strain state of beam-column elements can be obtained through the integration of
the nonlinear uniaxial stress-strain response of the individual fibers. As material nonlinearity is
implicitly defined by the material constitutive models, there is no need to introduce any element
hysteretic response or previous moment curvature analysis.
In the model M1 & M2, deck-girder and abutment section are subdivided to 150 and 200
fibers respectively. Integration sections and fibers in each section are shown schematically in
Figure 2-9.
Figure 2-9 Illustrates implementation of inelasticity distribution along beam elements using fiber approach in
SeismoStruct software (Source: SeismoStruct Manual)
23
2.5.1
Abutments - Bridge Type1 (Model M1 & M2)
Reinforced concrete rectangular section is used to model abutment with section height
and width 22.17m and 1.3 m respectively. A uniaxial nonlinear constant confinement model
(Con_ma) proposed by Madas in 1993 that follows the constitutive relationship proposed by
Mander et al. [1988] and the cyclic rules proposed by Martinez-Rueda and Elnashai [1997] is
employed as material model for the abutments in the model. In this material model, the
confinement effects provided by the lateral transverse reinforcement are incorporated through the
rules proposed by Mander et al. [1988] whereby confining pressure is assumed constant
throughout the entire stress-strain range. Five model calibrating parameters are defined to fully
describe the mechanical characteristics of the material are shown in Table 2-4.
Table 2-4 Calibrating parameters for the nonlinear material model- Model M1 & M2
Parameter
Value
Compressive strength - fc
45 (MPa)
Tensile strength - ft
2 (MPa)
Strain at peak stress - εc
0.002 (mm/mm)
Confinement factor - kc
1.2
Specific weight - ϒ
24 (kN/m^3)
Discretization and reinforcement arrangement of the abutment are shown in Figure 2-10.
Figure 2-10 Abutment discretized pattern and reinforcement arrangement - Model M1 & M2
24
2.5.1.1
Abutment Sectional Response - Bridge Type1 (Model M1 & M2)
Abutment sectional responses are obtained using Response-2000 software. Obtained
abutment Moment-Curvature and Moment-Shear interaction graphs are shown in Figure 2-11.
AASHTO-99 M-V Interaction
Moment-Curvature
42000.0
12000.0
Shear Force (kN)
Moment (kNm)
35000.0
28000.0
21000.0
14000.0
9000.0
6000.0
Nu = -4600 kN Mu = 31226 kNm
dv = 936 mm bv = 22170 mm
3000.0
7000.0
Phi = 1.00
sx = 111 mm
Less than minimum reinforcement
0.0
0.0
9.0
18.0
27.0
36.0
45.0
54.0
0.0
0.0
Curvature (rad/km)
7000.0
14000.0
21000.0
28000.0
35000.0
42000.0
Moment (kNm)
Figure 2-11 Shows abutment moment curvature diagram and M-V interaction obtained using Reponse-2000
software - Model M1 & M2
2.5.2
Abutment Piles Bridge Type1 (Model M2)
Reinforced concrete circular section is used to model 12.7 m long abutment piles with
section diameter 0.406 m. To consider the effect of casing and piles’ vertical and transversal
reinforcements, a simplified uniaxial elastic material model with symmetric behaviour in tension
and compression is chosen. Two model calibrating parameters required to describe the
mechanical characteristics of the material are defined in Table 2-5.
25
Table 2-5 Calibrating parameters for the elastic material model used in the pile elements
Material Parameters
Value
Modulus of elasticity - Es
200000 (MPa)
Specific weight - ϒ
25 kN/m3
In this element class, elastic mechanical properties of the piles:
are
automatically calculated by the program and local stiffness matrix of the pile can be defined as
per matrix shown in Figure 2-12.
n3 (lies in 1-3 plane)
Z
(3)
(1)
(2)
n2
n1
Y
X
4EI2
1/L
0
2EI2
0
4EI3
0
0
0
2EI3
0
0
0
2EI2
0
0
0
0 2EI3
4EI2 0
0
0
0
0
0
0
0
4EI3
0
0
EA
0
0
0
0
GJ
Figure 2-12 Indicates piles local stiffness matrix-Source: SeismoStruct Manual
26
in which;
is the length of the pile,
is the pile x-section,
is modulus of elasticity,
&
are moment of inertia about local axis 2 and 3 respectively,
s the modulus of rigidity, obtained as

where  is the Poisson's
ratio, and
is torsional constant.
Drawings of the bridge type 1 including abutment and piles layout arrangements are provided in
Appendix F.1.
2.5.3
Bridge Deck Slab and Pre-stress Precast Girders - Bridge Type 1 (Model M1 &
M2)
Bridge deck slab and pre-stressed precast girders are modeled together. Reinforced
concrete rectangular section is used to model them. Then total nine number of deck-girder
elements are stitched together to model the bridge deck and precast girders.
Section of the deck-girder and its discretized model are shown in Figure 2-13.
Figure 2-13 Deck-girder reinforcement arrangement and discretized pattern - Model M1 & M2
27
In SeismoStruct software pre-stressed tendons cannot be assigned directly; however, prestressing effect can be considered using a non linear link (nlin_el) which its hysteresis loop is a
simplified version of the Ramberg Osgood model. In this hysteretic loop no hysteretic dissipation
is allowed (the same curve is employed for loading and unloading). This hysteretic loop is shown
in Figure 2-14.
Figure 2-14 Hysteretic loop used to simulate the effect of the pre-stressing tendons in the models
Four parameters are defined in order to fully characterize this response curve:
 Yield strength ( Fy) calculated 5208kN for 20 numbers 270 grade 15.24 mm dia. tendons
 Yield displacement (Dy~ Du)
 Ramberg-Osgood parameter (ϒ) is considered 5.5
 Convergence limit for Newtown-Raphson procedure (Β) is considered 0.001
In this model, a simplified uniaxial elastic material model with symmetric behavior in tension
and compression is assigned as material of integrated Deck-girders.
To fully describe the mechanical characteristics of the material, two calibrating parameters are
defined in the model. These parameters are summarized in Table 2-6.
28
Table 2-6 Required calibrating parameters for the elastic material model- Model M1 & M2
2.5.3.1
Material Parameters
Value
Modulus of elasticity - Es
31522 (MPa)
Specific weight - ϒ
24.5 (kN/m3)
Deck-girder Sectional Response - Bridge Type 1 (Model M1 & M2)
Deck-girder sectional responses are also obtained using Response 2000 software.
Deck-girder Moment-Curvature and Moment-Shear interaction graphs are shown in Figure 2-15.
AASHTO-99 M-V Interaction
Deck-Girders' Moment-Curvature Diagram
1800.0
10000.0
1500.0
Shear Force (kN)
Moment (kNm)
8000.0
6000.0
4000.0
1200.0
900.0
600.0
Mu = 8459 kNm
dv = 1618 mm bv = 127 mm
2000.0
300.0
0.0
0.0
5.0
10.0
15.0
20.0
25.0
30.0
0.0
0.0
Phi = 1.00
Code Fe = 0.100
2000.0
Curvature (rad/km)
4000.0
6000.0
8000.0
Moment (kNm)
Figure 2-15 Shows deck-girder moment curvature diagram and M-V interaction obtained using Reponse2000 software - Model M1 & M2
2.5.4
Performance Criteria - Bridge Type 1 (Model M1 & M2)
Performance-based seismic design requires structures are design based on a damage state
that loss of life or life threatening injury is prevented; however, it could sustain extensive
structural and non-structural damage and be out of service for an extended period of time. To
estimate potential collapse level, analysis was performed in a design level earthquake (2% in 50
year), and ultimate capacity of structural components is considered as collapse damage state.
Different damage states for a typical reinforced concrete member are indicated in Figure 2-16.
29
Elastic
Inelastic
Collapse
Behaviour
Repairable Irreparable
Severe
Extreme
Damage
Lateral Load
Vision 2000
IO Op
LS
CP
NC
Ultimate Capacity
Yield of steel
Reinforcement
Concrete cracking
IO: Immediate Occupancy
OP: Operational
CP: Collapse Prevention
NC: Near Collapse
Drift
Figure 2-16 Shows a typical structural performance and associated damage states (A.Ghobarah, 2004)
To obtain the ultimate capacity of the important structural components in the models
Rsponse-2000 software is employed. Using the obtained sectional response for the abutment and
deck-girder presented in section 2.5.1.1 & 2.5.3.1, following target values are considered as
performance criteria and introduced to model as the collapse prevention criteria. The above
criteria with their correspond value are listed in Table 2-7.
30
Table 2-7 Shows defined performance criteria for the model M1 & M2
Collapse Prevention Criterion
Element/core
Targeting Value
Action
CP1
Abutment backwall curvature (rad/m)
+/-0.10692
Stop
CP2
Girders curvature (rad/m)
+/-.01537
Stop
CP3
Girders concrete strain (mm/mm)
-0.0045
Stop
CP4
Abutment backwall concrete strain (mm/mm)
-0.0015422
Stop
CP5
Girders steel rupture strain (mm/mm)
+0.062
Stop
CP6
Abutment steel rupture strain (mm/mm)
+0.04877
Stop
CP7
Abut. backwall shear force (kN)
+/-14969.3
Stop
CP8
Girders shear force approaching Abutments (kN)
+/-14969.3
Stop
CP9
Girders shear force (kN)
+/-1820.09
Stop
When a response exceeds the above specified targeting values, analysis is forced to stop.
2.6
Element Behavior - Bridge Type 2 (Model M3 & M4)
Force based inelastic frame element (inlfrmFB) is use for deck-girders, abutments, pier
columns, pier pilescap, and pier head as this element class doesn't depend on the assumed
sectional constitutive behavior. Total five numbers of integration points are considered to
simulate spread of inelasticity in deck-girders, abutments, and pier columns, pier pilecap and pier
heads as fallow:
 5 integration sections: [-1 -0.655 0.0 0.655 1] x L/2
in which, L represents length of an element.
In the model M3 & M4, all integration sections of the above mentioned elements are subdivided
to 150 fibers.
31
2.6.1
Abutments - Bridge Type2 (Model M3 & M4)
Reinforced concrete rectangular section is used to model abutment with section height
and width 14.171 m and 0.8 m respectively. Similar to the Model M1 & M2, Con_ma material is
employed for the abutments. Similarly, the same calibrating parameters which previously
introduced in Table 2-4 section 2.5.1 for the model M1 & M2 are used for these models.
Abutment discretization and reinforcement arrangement are shown in Figure 2-17.
Figure 2-17 Abutment discretization and reinforcement arrangement - Model M3 & M4
2.6.1.1
Abutment Sectional Response - Bridge Type2 (Model M3 & M4)
Abutment sectional responses are obtained using Response-2000 software. Obtained
Abutment Moment-Curvature and Moment-Shear interaction graphs are shown in Figure 2-18.
Moment-Curvature
AASHTO-99 M-V Interaction
12000.0
6000.0
5000.0
Shear Force (kN)
Moment (kNm)
10000.0
8000.0
6000.0
4000.0
2000.0
4000.0
3000.0
Nu = -1100 kN Mu = 9113 kNm
2000.0
1000.0
dv = 649 mm bv = 14171 mm
Phi = 1.00
sx = 217 mm
Less than minimum reinforcement
0.0
0.0
20.0
40.0
60.0
Curvature (rad/km)
80.0
100.0
0.0
0.0
2000.0
4000.0
6000.0
8000.0
10000.0
12000.0
Moment (kNm)
Figure 2-18 Shows abutment moment curvature diagram and M-V interaction obtained using Reponse-2000
software - Model M3 & M4
Abutment's principal tensile and compressive strains (mm/m) are shown in Figure 2-19.
32
Longitudinal Strain
top
-4.83
51.38
bot
Figure 2-19 Abutment longitudinal strain (mm/m) - Model M3 & M4
2.6.2
Bridge Deck Slabs and Pre-stress Precast Girders - Bridge Type 2 (Model M3 &
M4)
Similar to the model M1 & M2, bridge deck slab and pre-stressed precast girders are
modeled together. Reinforced concrete rectangular section is used to model them. Then total four
number of deck-girder elements are stitched together to model the bridge deck and precast
girders. Section of the deck-girder and its discretized model are shown in Figure 2-20.
Figure 2-20 Deck-girder reinforcement arrangement and discretized pattern - Model M3 & M4
Likewise model M1 & M2, in these models the nlin_el non linear link is used to simulate
effect of pre-stressing tendons. Four parameters are defined in order to fully characterize the
response curve:
33
 Yield strength ( Fy) calculated 4408.8 kN for 24 numbers 270 grade 12.7 mm dia.
tendons
 Yield displacement (Dy~ Du)
 Ramberg-Osgood parameter (ϒ) is considered 5.5
 Convergence limit for Newtown-Raphson procedure (Β) is considered 0.001
In these models, similar to the model M1 & M2 a simplified uniaxial elastic material model with
symmetric behavior in tension and compression is assigned as material of integrated Deckgirders. Two model calibrating parameters are defined. These parameters are summarized in
Table 2-8.
Table 2-8 Required calibrating parameters for the elastic material model - Model M3 & M4
Material Parameters
2.6.2.1
Value
Modulus of elasticity - Es
33667 (MPa)
Specific weight - ϒ
24.5 (kN/m3)
Deck-girder Sectional Response - Bridge Type 2 (Model M3 & M4)
Deck-girder sectional responses are also obtained using Response 2000 software. Deck-
girder Moment-Curvature and Moment-Shear interaction graphs are shown in Figure 2-21.
34
AASHTO-99 M-V Interaction
Moment-Curvature
8000.0
1200.0
Shear Force (kN)
Moment (kNm)
6000.0
4000.0
2000.0
900.0
600.0
Mu = 5520 kNm
300.0
0.0
0.0
20.0
40.0
60.0
80.0
100.0
0.0
0.0
dv = 1310 mm bv = 127 mm
Phi = 1.00
Code Fe = 0.100
800.0
Curvature (rad/km)
1600.0
2400.0
3200.0
4000.0
4800.0
Moment (kNm)
Figure 2-21 Shows deck-girder moment curvature diagram and M-V interaction obtained using Reponse2000 software - Model M3 & M4
Deck-girder's principal tensile and compressive strains (mm/m) are shown in Figure 2-22.
Longitudinal Strain
top
-1.66
29.87
bot
Figure 2-22 Deck-girder longitudinal strain (mm/m) - Model M3 & M4
2.6.3
Pier Columns - Bridge Type 2 (Model M3 & M4)
Reinforced concrete circular section is used to model 12 m pier column with section
diameter 1.5m. In these models, Con_ma material model is employed for the Pier columns. Five
model calibrating parameters are defined to fully describe the mechanical characteristics of the
material are shown in Table 2-9.
35
Table 2-9 Calibrating parameters for the nonlinear material model used to model pier columns - Model M3 &
M4
Parameter
Value
Compressive strength - fc
50 (MPa)
Tensile strength - ft
0 (MPa)
Strain at peak stress - εc
0.002 (mm/mm)
Confinement factor - kc
1.2
Specific weight - ϒ
24 (kN/m^3)
Pier Columns reinforcement arrangement and section digitization is shown in Figure 2-23.
Figure 2-23 Pier column reinforcement arrangement and discretized pattern - Model M3 & M4
Layout plan of the pier columns and pilecaps are shown in Appendix F.2.
2.6.3.1
Pier Columns Sectional Response - Bridge Type 2 (Model M3 & M4)
Pier columns sectional responses are also obtained using Response 2000 software.
Deck-girder Moment-Curvature and Moment-Shear interaction graphs are shown in Figure 2-24.
36
Moment-Curvature
AASHTO-99 M-V Interaction
3000.0
6000.0
2500.0
Shear Force (kN)
Moment (kNm)
5000.0
4000.0
3000.0
2000.0
2000.0
1500.0
1000.0
Nu = -2425 kN Mu = 5392 kNm
500.0
1000.0
dv = 1080 mm bv = 1500 mm
Phi = 1.00
0.0
0.0
6.0
12.0
18.0
24.0
30.0
36.0
0.0
0.0
1000.0
2000.0
3000.0
4000.0
5000.0
6000.0
Moment (kNm)
Curvature (rad/km)
Figure 2-24 Shows pier column moment curvature diagram and M-V interaction obtained using Reponse2000 software - Model M3 & M4
Pier column's principal tensile and compressive strains (mm/m) are shown in Figure 2-25.
Longitudinal Strain
top
-3.65
15.42
bot
Figure 2-25 Pier column longitudinal strain (mm/m) - Model M3 & M4
2.6.4
Bridge Pier Pilecap and Pile Head - Bridge Type 2 (Model M3 & M4)
Reinforced concrete rectangular section is used to model pier pilecap and pier head with
section height and width 2.5 m X 10 m and 1.6 m X 13.202 m respectively.
In these models, Con_ma material model is used as a material model for the Pier columns. Five
model calibrating parameters are defined to fully describe the mechanical characteristics of the
material are shown in Table 2-10.
37
Table 2-10 Calibrating parameters for the nonlinear material model used to model pier heads and pilecapsModel M3 & M4
Element
Parameter
Pier Head
Pier Pilecap
Compressive strength - fc
50 (MPa)
45 (MPa)
Tensile strength - ft
0 (MPa)
0 (MPa)
Strain at peak stress - εc
0.002 (mm/mm)
0.002 (mm/mm)
Confinement factor - kc
1.2
1.2
24 (kN/m^3)
24 (kN/m^3)
Specific weight - ϒ
Digitization pattern of the pier pilecaps and heads are shown in Figure 2-26.
Figure 2-26 Shows discritized pattern of the pier pilecaps and heads - Model M3 & M4
Reinforcement detail for the pier pilecaps and heads of the model M3 &M4 are provided in
Appendix F.2.
2.6.5
Abutment and Pier Piles - Bridge Type 2 (Model M4)
Abutment and pier piles are chosen identical to the abutment piles for the model M2
described in section 2.5.2. Layout plans of the abutment and pier piles can be found in Appendix
F.2.
2.6.6
Performance Criteria - Bridge Type 2 (Model M3 & M4)
Using the obtained sectional response for the abutments, deck-girders, and pier columns
presented in section 2.6.1.1 , 2.6.2.1, and 2.6.3.1 limit state values for these models are obtained.
38
These values are listed in Table 2-11as collapse prevention criteria and introduced to the model
as performance criteria.
Table 2-11 Shows defined performance criteria for the model M3 & M4
Collapse Prevention Criterion
Element/core
Targeting Value
Action
CP1
Abutment backwall curvature (rad/m)
+/-.1281
Stop
CP2
Girders curvature (rad/m)
+/-.015
Stop
CP3
Girders concrete strain (mm/mm)
-0.0017
Stop
CP4
Abutment backwall concrete strain (mm/mm)
-0.006
stop
CP5
Girders steel rupture strain (mm/mm)
+0.03
Stop
CP6
Abutment steel rupture strain (mm/mm)
+0.09
stop
CP7
Abutment backwall shear force (kN)
+/-7416.6
Stop
CP8
Girders shear force approaching Abutments (kN)
< +/-7416.6
Stop
CP9
Girders shear force (kN)
+/-1443.02
Stop
CP10
Pier curvature (rad/m)
+/-.0442
Stop
CP11
Pier concrete strain (mm/mm)
-0.0036
Stop
CP12
Pier steel rupture strain (mm/mm)
+0.0873
Stop
CP13
Pier shear force (kN)
+/-449.32
Stop
Incremental dynamic analysis is stopped when a response exceeds the above specified targeting
values.
2.7
Element Behavior - Bridge Type 3 (Model M5 & M6)
Similar to the model M3& M4, for deck-girders, abutments, pier columns, pier pilescap,
and pier head section force based inelastic frame element (inlfrmFB) is adapted as this type of
element doesn't depend on the assumed sectional constitutive behavior. Total five integration
points are considered to simulate spread of inelasticity in deck-girders, and pier columns, pier
39
pilecap and pier heads; however, total 10 number of integration is considered for abutments as
fallow respectively:
 5 integration sections: [-1 -0.655 0.0 0.655 1] x L/2
 10 integration sections: [-1 -0.920 -0.739 -0.478 -0.165 0.165 0.478 0.739 0.920 1]
x L/2
in which, L represents length of an element.
In these models, each integration section of deck-girders, pier columns, pier pilescaps, and pier
heads is subdivided to 150 fibers. However, abutment integration sections are subdivided to 300
fibers to obtain more accurate results.
2.7.1
Abutments - Bridge Type 3 (Model M5 & M6)
Reinforced concrete rectangular section is used to model abutment with section height
and width 1.4 m and 35.07 m respectively.
Similar to the previous models, Con_ma material model is employed for the abutments.
Similarly, the same calibrating parameters which previously introduced in section 2.5.1for model
M1 & M2 are used for these models. Mechanical characteristics of the material are summarized
in the following table:
Abutment discretized model and reinforcement arrangement are shown in Figure 2-27.
Figure 2-27 Abutment discretization and reinforcement arrangement - Model M5 & M6
40
2.7.1.1
Abutment Sectional Response - Bridge Type 3 (Model M5 & M6)
Abutment sectional responses are obtained using Response-2000 software. Obtained
Abutment Moment-Curvature and Moment-Shear interaction graphs are shown in Figure 2-28.
AASHTO-99 M-V Interaction
10000.0
Shear Force (kN)
8000.0
6000.0
4000.0 Nu = -2200 kN Mu = 14839 kNm
dv = 659 mm bv = 35070 mm
Phi = 1.00
2000.0 sx = 220 mm
Less than minimum reinforcement
0.0
0.0
4000.0
8000.0
12000.0
16000.0
20000.0
Moment (kNm)
Figure 2-28 Shows abutment moment curvature diagram and M-V interaction obtained using Reponse-2000
software - Model M5 & M6
Abutment's principal tensile and compressive strains (mm/m) are shown in Figure 2-29.
Longitudinal Strain
top
-3.31
52.89
bot
Figure 2-29 Abutment longitudinal strain (mm/m) - Model M5 & M6
41
2.7.2
Bridge Deck Slabs and Pre-stress Precast Girders - Bridge Type 3 (Model M5 &
M6)
Similar to the previous models, bridge deck slab and pre-stressed precast girders are
modeled integrally and a reinforced concrete rectangular section is used to model them. In each
bridge bond, four elements of the deck-girder are stitched together to model the bridge deck and
precast girders. Reinforcing detail and discretized model of the deck-girder are shown in Figure
2-30.
Figure 2-30 Deck-girder reinforcement arrangement and discretized pattern - Model M5 & M6
Likewise previous models, in these models the nlin_el non linear link is used to simulate
effect of pre-stressing tendons. Four parameters are defined in order to fully characterize the
response curve:
 Yield strength (Fy) calculated 4776.2 kN for 26 numbers 270 grade 12.7 mm dia.
tendons
 Yield displacement (Dy~ Du)
 Ramberg-Osgood parameter (ϒ) is considered 5.5
 Convergence limit for Newtown-Raphson procedure (Β) is considered 0.001
42
In these models, similar to the previous models a simplified uniaxial elastic material model with
symmetric behavior in tension and compression is assigned as material of integrated Deckgirders. Two model calibrating parameters required for the material model are defined identical
to those parameters in the model M3 & M4.
2.7.2.1
Deck-girder Sectional Response - Bridge Type 3 (Model M5 & M6)
Deck-girder sectional responses are also obtained using Response 2000 software.
Deck-girder Moment-Curvature and Moment-Shear interaction graphs are shown in Figure 2-31.
Figure 2-31 Deck-girder moment curvature diagram and M-V interaction obtained using Reponse-2000
software - Model M5 & M6
Deck-girder's principal tensile and compressive strains (mm/m) are shown in Figure 2-32.
Longitudinal Strain
top
-1.73
29.32
bot
Figure 2-32 Deck-girder longitudinal strain (mm/m) - Model M5 & M6
43
2.7.3
Pier Columns - Bridge Type 3 (Model M5 & M6)
Reinforced concrete circular section is used to model 14m long pier column with section
diameter 1.2m. In These models, material model and its calibrating parameters are identical to
those are employed in the model M3 & M4. Pier Columns reinforcement arrangement and
digitization model are shown in Figure 2-33.
Figure 2-33 Pier column reinforcement arrangement and discretized pattern - Model M5 & M6
2.7.3.1
Pier Columns Sectional Response - Bridge Type 3 (Model M5 & M6)
Likewise previous members, pier columns sectional responses are obtained using
Response 2000 software. Deck-girder Moment-Curvature and Moment-Shear interaction graphs
are shown in Figure 2-34.
AASHTO-99 M-V Interaction
Moment-Curvature
2400.0
3600.0
2000.0
Shear Force (kN)
Moment (kNm)
3000.0
2400.0
1800.0
1200.0
1600.0
1200.0
800.0
Nu = -2425 kN Mu = 3380 kNm
400.0
600.0
dv = 864 mm bv = 1200 mm
Phi = 1.00
0.0
0.0
5.0
10.0
15.0
Curvature (rad/km)
20.0
25.0
30.0
0.0
0.0
600.0
1200.0
1800.0
2400.0
3000.0
3600.0
Moment (kNm)
Figure 2-34 Pier column moment curvature diagram and M-V interaction obtained using Reponse-2000
software - Model M5 & M6
44
Pier column's principal tensile and compressive strains (mm/m) are shown in Figure 2-35.
Longitudinal Strain
top
-3.40
12.10
bot
Figure 2-35 Pier column longitudinal strain (mm/m) - Model M5 & M6
2.7.4
Pier Pilecap and Pier Head - Bridge Type 3 (Model M5 & M6)
Reinforced concrete rectangular section is used to model pier pilecap and pier head with
section height and width 5 m X 12 m and 1.4 m X 11.614 m respectively.
In these models, the same material model and its calibrating parameters are used are similar to
those model and parameters are used for the pier pilecap in the model M3 & M4. However,
model material is used for the deck-girder is employed for the pier head with the same
calibrating parameters. Pier pilecap and pier head reinforcement arrangement and section
digitization are shown in Figure 2-36.
Figure 2-36 Shows discritized pattern of the pier pilecaps and heads - Model M5 & M6
2.7.5
Steel Intermediate Diaphragm - Bridge Type 3 (Model M5 & M6)
A symmetric steel T section is used to cross-brace the girders at outer bays in the middle
of the intermediate span as intermediate diaphragm in each bond as it is shown in red color in
Figure 2-37.
45
Figure 2-37 Shows intermediate diaphragm bracings in red color - Model M5 & M6
Dimensions of the used T-section in these models are listed in Table 2-12.
Table 2-12 Dimensions of the steel T-section used as bracings in the intermediate diaphragm - Model M5 &
M6
Location
Dimension (m)
Bottom flange width
0.2
Bottom flange thickness
.01
Top flange width
0.2
Top flange thickness
0.015
Web height
0.3
Web thickness
0.015
Drawings of the bridge type 3 are presented in Appendix F.3.
2.7.6
Performance Criteria - Bridge Type 3 (Model M5 & M6)
Based on obtained sectional response for the abutments, deck-girders, and pier columns
presented in section 2.7.1.1, 2.7.2.1, and 2.7.3.1, following target values introduced to model are
46
considered as performance criteria. The above criteria with their corresponding value are listed in
Table 2-13.
Table 2-13 Shows defined performance criteria for the model M5 & M6
Collapse Prevention Criterion
Element/core
Targeting Value
Action
CP1
Abutment backwall curvature (rad/m)
+/-.1247
Stop
CP2
Girders curvature (rad/m)
+/-0.015
Stop
CP3
Girders concrete strain (mm/mm)
-0.0017
Stop
CP4
Abutment backwall concrete strain (mm/mm)
-0.005
stop
CP5
Girders steel rupture strain (mm/mm)
+0.03
Stop
CP6
Abutment steel rupture strain (mm/mm)
+0.09
stop
CP7
Abutment backwall shear force (kN)
+/-10609.5
Stop
CP8
Girders shear force approaching Abutments (kN)
+/-10609.5
Stop
CP9
Girders shear force (kN)
+/-1492.72
Stop
CP10
Pier curvature (rad/m)
+/-.056
Stop
CP11
Pier concrete strain (mm/mm)
-0.003
Stop
CP12
Pier steel rupture strain (mm/mm)
+0.09
Stop
CP13
Pier shear force (kN)
+/-241.43
Stop
47
Chapter 3: Analysis
3.1
Eigen Value Analysis
Elastic frame elements are employed in the creation of the structural model in Eigen
value analysis as this analysis is a purely elastic type of structural analysis. Therefore, in this
type of analysis material properties are taken as constant throughout the entire computation
procedure.
In SeismoStrurct, efficient Lanczos algorithm introduced by Hughes in 1987 and Jacobi
algorithms using Ritz transformation can be used for evaluation of the structural natural
frequencies and mode shapes. However, in this study Lanczos algorithm is used to the following
obtain bridge periods and mode shapes.
3.1.1
3.1.1.1
Eigen Value Analysis - Bridge Type 1
Model M1
Mode shapes of the model M1 are shown in Figure 3-1.
48
Figure 3-1 Mode shapes of the model M1for the first six modes
The effective modal of a mode is the fraction of the total static mass (static inertia for
rotation modes) that can be attributed to that mode. Period and cumulative effective modal
masses for the first six modes of the model M1are presented in Table 3-1.
Table 3-1 Period and cumulative modal mass for the first 6 modes - Model M1
Mode
Period (sec)
Ux (tone)
Uy (tone)
Uz (tone)
Rx (tone)
Ry (tone)
Rz (tone)
1
0.246
0.00
0.00
426.16
0.13
0.25
165.04
2
0.151
72.13
178.21
426.16
1711.82
73.49
165.04
3
0.085
72.14
178.21
430.47
1712.50
74.49
5479.81
4
0.078
368.51
197.50
430.47
10607.49
319.72
5481.00
5
0.071
400.98
259.92
430.47
46987.32
343.69
5481.77
6
0.046
401.18
260.09
430.47
51202.42
12503.29
5481.85
49
3.1.1.2
Model M2
Mode shapes of the model M2 are shown in Figure 3-2.
Figure 3-2 Mode shapes of the model M2for the first six modes
Periods and corresponding cumulative effective modal masses for model M2 are presented in
Table 3-2.
Table 3-2 Period and cumulative modal mass for the first 6 modes - ModelM2
Mode
Period (sec)
Ux (tone)
Uy (tone)
1
Uz (tone)
Rx (tone)
Ry (tone)
Rz (tone)
0.284
430.20
1060.86
4.97
237.42
40.03
54.53
2
0.251
1547.98
1501.59
5.82
246.16
57.15
317.73
3
0.240
1548.94
1528.35
440.48
270.47
57.15
38344.37
4
0.199
1549.87
1528.86
486.91
274.72
57.38
441870.47
5
0.112
1559.86
1559.59
513.00
628.01
185.30
441924.50
6
0.081
1560.78
1562.32
532.19
819.19
225.32
443458.80
50
3.1.2
3.1.2.1
Eigen Value Analysis - Bridge Type 2
Model M3
Mode shapes of the model M3 are shown in Figure 3-3.
Figure 3-3 Mode shapes of the model M3 for the first six modes
Period and cumulative effective modal masses for model M3 are presented in Table 3-3.
Table 3-3 Period and cumulative modal mass for the first 6 modes - Model M3
Mode
Period (sec)
Ux (tone)
Uy (tone)
Uz (tone)
Rx (tone)
Ry (tone)
Rz (tone)
1
0.335
31.44
272.28
0.10
29001.41
286.91
2.15
2
0.240
853.15
273.39
0.10
29383.18
6586.26
203.98
3
0.233
861.33
273.90
0.78
29650.11
6653.75
564.93
4
0.201
861.48
273.94
162.21
29743.57
6653.97
572.14
5
0.119
865.58
1311.38
162.63
33186.20
6710.01
994.13
6
0.112
865.84
1315.99
162.65
33197.66
6710.89
94720.24
51
3.1.2.2
Model M4
Mode shapes of the model M4 are shown in Figure 3-4.
Figure 3-4 Mode shapes of the model M4 for the first six modes
Period and cumulative effective modal masses for model M4 are presented in Table 3-4.
Table 3-4 Period and cumulative modal mass for the first 6 modes - Model M4
Mode
Period (sec)
Ux (tone)
Uy (tone)
1
Uz (tone)
0.374
109.41
805.84
0.06
2
0.304
1511.26
856.97
3
0.296
1519.23
4
0.255
5
6
Rx (tone)
Ry (tone)
Rz (tone)
30635.75
920.60
34.43
0.08
32248.52
9962.26
34.45
872.29
0.62
32689.50
10008.76
9732.68
1520.09
1628.55
0.70
32856.96
10008.76
9893.57
0.209
1520.21
1628.63
93.11
32986.36
10009.21
348973.93
0.205
1520.28
1628.63
155.64
33034.62
10009.23
890950.94
52
3.1.3
3.1.3.1
Eigen Value Analysis - Bridge Type 3
Model M5 & M6
Mode shapes of the model M5 & M6 are identical as archetype model M6 is the SSI
version of the Model M5 which the only SSI feature considered in this model is CALTRANS
springs to simulate the effect of the soil embankment behind the abutment backwall. Mode
shapes of these models are shown in Figure 3-5.
Figure 3-5 Mode shapes of the model M5 and M6 for the first six modes
Due to the reason mentioned in the above, model M5 & M6 have the identical period and
cumulative effective modal mass. Periods and cumulative effective modal masses for these
models are presented in Table 3-5.
53
Table 3-5 Period and cumulative modal mass for the first 6 modes - Model M5 & M6
Mode
Period (sec)
Uz (tone)
Rx (tone)
Ry (tone)
1
0.374
1.44
0.00
174.26
224.79
174.45
8.66
2
0.373
3.81
0.00
174.30
469.45
18720.24
9.35
3
0.350
1910.89
0.01
174.47
504.13
82508.92
2622.79
4
0.332
1910.89
0.01
174.47
504.31
82523.86
5202.91
5
0.229
1911.50
24.01
174.47
241648.75
82534.91
5277.63
6
0.225
1911.50
24.01
174.48
241649.77
82534.92
7382.27
3.2
Ux (tone)
Uy (tone)
Rz (tone)
Nonlinear Static Pushover Analysis
To determine ultimate capacity of the structures and quantify period-based ductility
illustrated in Section 4.4, a series of static pushover analysis are carried out. To do this, forcebased pushover analysis with load control phase is performed for each 3D archetype models, i.e.,
abutment and pier column of the structures are pushed along and across the bridge deck
respectively.
In addition to the above load strategy, pushover load are applied in two directions
simultaneously (along and across the bridge deck) using the same incremental step. The later
applied load strategy is also conducted in static pushover analysis to compare the obtained
capacity results in this way with the obtained demand from the incremental analysis provided in
the next chapter as ground motion accelerations in IDA are applied in two directions to the
restrained nodes in each model.
The static pushover analysis is terminated by program automatically when the collapse
occurs (until a structural responses reaches one of the collapse criteria defined in the
performance criteria of the models). Obtained static pushover curves for the models when
54
abutment and pier column No.1 (the outer pier column of the fist span in bridge type 2 &3) are
pushed in X, Y and X-Y directions are shown in Figure 3-6, Figure 3-7, and Figure 3-8.
Figure 3-6 Obtained abutment pushover curves for all archetype models when abutment No.1(south
abutment) is pushed along the deck direction
55
Figure 3-7 Obtained abutment pushover curves for all archetype models when abutment No.1 (south
abutment) is pushed along and across the deck directions
56
Figure 3-8 Obtained pier column pushover curves for all archetype models when the column No.1 is pushed
across and across-along the deck directions
The last points shown on the Figure 3-6, Figure 3-7, and Figure 3-8 are not representative
of analysis result. There is only used to represent the total base shear coresponding to the
collapse level. The points previous the last points represent non-simulated collapse points that
analysis is forced to stop. Obtained ultimate capacities for the abutment and pier column of the
models are summarized in Table 3-6.
57
Table 3-6 Summary of the obtained ultimate capacities for the abutment and pier column of the models
Element
Abutment No.1
Pushing Direction
Capacity
Model
Y
Δy
(mm)
Pier Column No.1
X&Y
Base Shear
(kN)
X
Δy
Base Shear
Δx
(mm)
(kN)
(mm)
X&Y
Base Shear
(kN)
Δx
(mm)
Base Shear
(kN)
M1
9.3
26884.2
0.84
13499.3
-
-
-
-
M2
8.5
31000
3.0
21500
-
-
-
-
M3
40.3
14658.3
10.5
6558.4
22.5
5635.5
15.5
5053.5
M4
45.3
14700
8.5
6900
34.1
6187.5
25.8
5625
M5
2.9
15258.2
2.9
15329
82.7
5475
58.6
5197.1
M6
2.8
15287.4
2.2
22630.7
78.5
5888.4
77.6
5432.4
in which, direction X & Y are across and along the bridge deck respectively.
58
3.3
Hazard Analysis
The Pacific Coast is the most earthquake-prone region of Canada. In the offshore region
to the west of Vancouver Island, more than 100 earthquakes of magnitude 5 or greater have
occurred during the past 70 years. High concentration of earthquakes in the west coast is due to
the following reasons: presence of active faults or breaks in the earth’s crust and tectonic plate
movement. In this area, the tectonic plates can slide past one another, collide, or diverge.
Likewise other part of the world that have these three plate movements together, there is
significant earthquake activity in the west coast region of Canada. Concentration and magnitude
of the Earthquakes that occurred in Canada are shown in Figure 3-9.
Figure 3-9 Seismic history of earthquake magnitudes in Canada - Source: Natural Resources Canada (NRC)
Earthquakes in southwestern British Columbia occur in three distinct source regions:
59
1) Relatively close to the surface in the North American Plate (continental crust)
2) Deeper in the sub-ducting Juan de Fuca Plate (oceanic crust)
3) Along the boundary between the North American Plate and the sub-ducting Juan de Fuca
Plate (locked zone)
Tectonic plates in south west of Canada are shown in Figure 3-10.
Figure 3-10 Demonstrates tectonic plates in southwestern Canada - Source: Natural Resources Canada
(NRC)
In this study probabilistic seismic hazard is adapted from the Open File 4459 (2005) for
city of Vancouver. However, a probabilistic seismic hazard analysis (PSHA) and deaggregation
is also performed for the period of 0.246sec, 0.284sec, 0.335sec, and 0.374sec using EZ-FRISK
software to obtain mean magnitude of the ground motions required for performing the response
time history analysis. For the hazard analysis, damping ratio is considered 5% and soil site class
is classified as class C based on Table 4.1.8.4.A. NBCC 2010 (see Appendix G ) assuming that
time average of the shear wave velocity for the top 30 m of the sub-soil layer of the site is 450
m/s. Two different models are considered in the analysis: H model and R model. The H model is
used to consider relatively small source zones drawn around historical seismicity clusters, and
the R model is used to consider larger regional zones in the analysis. H West and R West seismic
60
sources are considered to represent the tectonic region that include all the potentially active
seismic sources that can contribute to the earthquake ground motions within 1000 km of city of
Vancouver. Cascadian subduction zone is taken into account as a seismic source as it represent
continental crest and it is relatively close to the surface in the North American Plate. Contributed
seismic zones in H and R West model are presented in Figure 3-11.
Attenuation equations used in the analysis were: Boore-Atkinson (NGA-2008),
Campbell-Bozorgnia (NGA-2008), Zhao, et. al (2006, USGS 2008), Abrahamson-Silva (NGA2008), and Chiou-Youngs (NGA-2008).
Figure 3-11 Illustrates the contributed seismic zones in H& R model-source: EZ-FRISK
61
Deaggregation is performed for the period of 0.246 sec, 0.284sec, 0.335sec, and 0.374sec
at 5% damping. Obtained Magnitude-Distance deaggregation from PSHA for city of Vancouver
for the period of 0.246sec is shown in Figure 3-12.
Figure 3-12 Magnitude-Distance deaggregation spectral response @ 5% damping - horizontal component
obtained from EZ-FRISK for the period 0.246sec
Obtained epsilon deaggregation spectral response for the period of 0.246 sec at 5%
damping is presented in Figure 3-13.
62
Figure 3-13 Shows mean hazard for spectral response at 5% damping - Source: EZ-FRISK software
Overall, obtained Earthquake hazard analysis results for city of Vancouver are
summarized in Table 3-7.
Table 3-7 Summary of the performed Hazard analysis results
Period (sec)
Parameter
0.246
0.284
0.335
0.374
Amplitude(g)
0.9
0.86
0.798
0.757
Mean magnitude (M)
6.58
6.62
6.67
6.7
Mean Distance (km)
21.3
23.7
25.5
25.2
Mean Epsilon
1.25
1.27
1.25
1.21
Mean Hazard
0.000261633
0.00027283
0.000289151
0.000288606
Other deaggregation charts corresponding to the period 0,284sec, 0.335sec, and 0.374 sec are
provided in Appendix H .
63
3.3.1
Vancouver Uniform Hazard Spectrum
So far intensive studies have been made by experts for better understanding of seismic
hazard and the relationships between structural damage and ground-motion intensities. Based on
these investigations, adaption of 2% in 50 years (return period of 2,475 years) tends to be more
closely related with the probability of structural collapse. Subsequently, this adoption is
considered as the design level of hazard in many codes such as: 1997 NEHRP guidelines, IBC
2000 (Leyendecker et al., 2000), and 2005 edition of the Canadian seismic code (Heidebrecht,
2003). Based on the above matter, 50 percentile H & R model probabilistic seismic hazard
estimates for the city of Vancouver with the hazard level of 2% probabilities in 50 year (return
period of 2475) and when horizontal spectral acceleration 5% damped, is taken from the Open
File 4459 (Geological Survey of Canada, 2003). However, average of them is considered as
target uniform hazard spectrum. All the hazard spectra are shown in Figure 3-14.
64
Probabilistic Seismic Hazard Estimates - Vancouver (2%/50 year)
1
H Model (50%ile)
R Model (50%ile)
Average H & R Model (50%ile)
0.9
0.8
0.7
Sa(g)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Period (sec)
Figure 3-14 Target and H & R model uniform hazard spectra (50%ile) for soil site class C - Source OPEN
File 4459 (Geological Survey of Canada, 2003)
3.3.2
Input Ground Motions
To account for the uncertainties of the structural response due various range of motion
intensities, a set of 20 numbers of un-scaled far field ground motion records are selected from
PEER strong ground motions database. In this study, un-scaled ground motions are preferred to
use due to two main reasons: First, scaling ground motions to match the target spectrum over a
range may have an impact to the structural responses such as displacement. Second, here the
strategy is to increase the acceleration intensity of ground motions in performing IDA until
collapse is occurred.
To avoid adjusting the collapse fragility curve to account for the spectral shape effect
explained in introduction, these motions were selected in a manner that mostly they have positive
65
epsilon (ε) and their ε(T1) value is close to ε(T1) value of the target spectrum. This approach is
used to account for the ground motion spectra shape characteristics to ensure an unbiased
estimate of the collapse probabilities. Here, epsilon (ε) is defined as number of standard
deviations by which an observed logarithmic spectral acceleration differs from the mean
logarithmic spectral acceleration of a ground-motion prediction values at the fundamental period
of the structures (J.W. Baker3, 2005). Spectra of the selected motions, target spectrum, and
interested period range are shown in Figure 3-15.
3 Associate Professor at department of Civil and Environmental Engineering Stanford university.
66
Target and Ground Motion Hazard Spectra
3
UHS_Vancouver(2%/50)
Cape Mendocino
Chi Chi
Coalinga
2.5
Duzce
Imperial Valley
Kobe
2
LomaPrieta
Mammoth Lakes
Sa (g)
Morgan Hill
Northridge
1.5
N.Palm Springs
San Fernando
Superstition
1
Tabas
Victoria-Mexico
Westmorland
Whittier
0.5
Spitak
Managua
Gazli
0
Fundamental Period - Model M1
0
0.5
1
1.5
2
2.5
3
Period (sec)
3.5
4
Fundamental Period - Model M6
Figure 3-15 Shows ground motion spectra along with the target spectrum, and the range of period of interest
[T1-model M1 (0.246 sec) -T1-model M4, M5 & M6 (0.374 sec)]
67
Detail information of these ground motions are presented in Table 3-8.
Table 3-8 Summary of the selected ground motion records
No.
Event
PEER NGA No.
Year
Station
Magnitude
Mechanism
1
Chi-Chi- Taiwan
1503
1999
TCU065
7.62
Reverse-Oblique
2
Superstition Hills-02
727
1987
Superstition Mtn Camera
6.54
Strike-Slip
3
Loma Prieta
779
1989
LGPC
6.93
Reverse-Oblique
4
Northridge-01
1084
1994
Sylmar - Converter Staion
6.69
Reverse
5
Imperial Valley-06
180
1979
El Centro Array #5
6.53
Strike-Slip
6
Victoria- Mexico
265
1980
Cerro Prieto
6.33
Strike-Slip
7
Morgan Hill
451
1984
Coyote Lake Dam (SW Abut)
6.19
Strike-Slip
8
Duzce- Turkey
1617
1999
Lamont 375
7.14
Strike-Slip
9
Cape Mendocino
828
1992
Petrolia
7.01
Reverse
10
Mammoth Lakes-01
232
1980
Mammoth Lakes H. S.
6.06
Normal-Oblique
11
N. Palm Springs
529
1986
North Palm Springs
6.06
Reverse-Oblique
12
Tabas- Iran
139
1978
Dayhook
7.35
Reverse
13
San Fernando
77
1971
Pacoima Dam (upper left abut)
6.61
Reverse
14
Gazli- USSR
126
1976
Karakyr
6.8
Unkown
15
Managua- Nicaragua-01
95
1972
Managua- ESSO
6.24
Strike-Slip
16
Whittier Narrows-01
668
1987
Norwalk - Imp Hwy- S Grnd
5.99
Reverse-Oblique
17
Coalinga-05
412
1983
Pleasant Valley P.P. - yard
5.77
Reverse
18
Westmorland
319
1981
Westmorland Fire Staion
5.9
Strike-Slip
19
Kobe- Japan
1116
1995
Shin-Osaka
6.9
Strike-Slip
20
Spitak- Armenia
730
1988
Gukasian
6.77
Reverse-Oblique
To minimize the analysis time, a significant duration of the above un-scaled ground
motions which at least fulfill 95% of arias intensity is considered in performing incremental
analysis. As it is discussed, this simplification did not result in missing the peak response
68
comparing to the case that entire duration of a ground motion record is used in analysis.
Acceleration time history for the significant duration of these ground motions is shown in
Appendix A .
3.4
Incremental Dynamic Analysis (IDA)
In incremental dynamic analysis (IDA), structure subjected to a succession of transient
loads (in this study acceleration time-histories) of increasing intensity until the structure reaches
a collapse point. As Vamvatsikos and Cornell (2002) suggested, the median collapse intensity
can be obtained through the concept of IDA. Consequently, here IDA is used to collect collapse
data and to define a collapse fragility curve for each archetype model through a cumulative
distribution function (CDF) which relates the ground motion intensity to the probability of
collapse. In the performed IDA, both starting scaling factor and scaling factor step are considered
0.25. In addition, acceleration time histories are applied in both X & Y directions (along and
across the bridge decks) to the nodes that were restrained in all directions and specified in Table
3-9.
Table 3-9 Acceleration are applied to the restrained nodes in X & y directions
Archetype Model
Nodes
M1
End node both abutment elements
M2
Abutment Piles' toe
M3
End node abutments and pier pilecaps
M4
Abutments and piers piles' toe
M5
End node of the abutment and pier pilecap elements
M6
End node of the abutment and pier pilecap elements
69
Detail of the input ground motions frequencies and output frequencies of the performed IDAs are
summarized in Table 3-10.
Table 3-10 Summary of the input/output frequencies of the performed IDAs
Archetype Model
No
Ground Motion
M1
GM Time Step (Sec)
M2
M3
M4
M5
M6
IDA Output Frequency
1
Chi Chi - Taiwan
0.005
1
8
6
6
6
6
2
Superstition
0.01
1
1
3
3
6
6
3
Loma Prieta
0.005
1
1
3
3
6
6
4
Northridge
0.005
1
1
3
3
6
6
5
Imperial Valley
0.005
1
1
3
3
6
6
6
Victoria-Mexico
0.01
1
1
3
3
6
6
7
Morgan Hill
0.005
1
1
3
3
6
6
8
Duzce
0.01
1
4
3
6
6
6
9
Cape Mendocino
0.02
1
1
3
6
6
6
10
Mammoth Lakes
0.005
4
4
3
3
6
6
11
N.Palm Springs
0.005
1
1
3
3
6
6
12
Tabas
0.02
1
1
3
6
6
6
13
San Fernando
0.01
1
1
6
6
6
6
14
Gazli-USSR
0.005
1
1
6
6
6
6
15
Managua
0.01
1
1
6
6
6
6
16
Whittier Narrows
0.005
4
4
6
6
6
6
17
Coalinga
0.005
1
1
3
3
6
6
18
Westmorland
0.005
1
1
6
6
6
6
19
Kobe
0.01
4
4
6
6
6
6
20
Spitak
0.01
4
4
6
6
6
6
70
Chapter 4: IDA Results and Probability of Collapse of the Models
Collapse of bridges can lead to a large number fatalities and threatening injuries
depending on importance level of the structures. Due to this matter, methodology of this study
rely on quantifying probability of collapse of the structure equivalent to 'collapse prevention'
rather than 'life safety' defined in FEMA 273/356 as performance levels. Therefore, the main
focus here is on partial and global instability of the seismic-force resisting system, and nonstructural systems and their potential life-threatening failure are not addressed.
In this study, non-simulated collapse modes are indirectly evaluated using alternative limit
state checks on structural response quantities measured in the analyses (performance criteria).
Performance criteria are employed when it was not directly possible to simulate deterioration
modes contributing to collapse behavior of the seismic-force resisting system. Therefore, here
identified potential deterioration and collapse mechanisms are addressed both through the
explicit simulation of failure modes through nonlinear analyses and evaluation of “nonsimulated” failure modes using performance criteria illustrated and defined in the model
description section.
However, in Non-simulated collapse modes it is generally considered that initial
occurrence of the failure mode will lead to collapse of entire of the structure. As it is indicated in
the FEMA P695 (June 2009), non-simulated limit state checks may result in lower estimates of
the median collapse comparing to the case that all the local failure modes directly simulated.
Simulating all the local failure modes directly is ideal, but not practically possible. Therefore, a
combination of simulating direct failure mode of the important components and non-simulated
limit state checks is an optimal method for evaluating the effects of deterioration and collapse
mechanisms, and it is used to capture local failure modes in the models.
71
Performing incremental dynamic analysis and considering abutment and pier column
relative displacement as engineering demand parameters and spectral acceleration as an intensity
measure, probability of collapse fragility curve for each prototype is provided. In providing the
collapse fragility curves a function fits a lognormal CDF to observed probability of collapse data
using optimization on the likelihood function for the data and least square method as illustrated
in the Appendix B (J. W. Baker, 2013). These provided graphs are presented in the following
sections. In addition, to obtain actual relative displacement for the abutment and pier column in
the model M2 & M4, rocking of these elements is determined and graph of them are presented in
the relevant sub-sections.
72
4.1
Non-Simulated Collapse Mode IDA Results and Probability of Collapse - Model M1 & M2
Obtained failure mode results at collapse level from the performed incremental dynamic analysis are summarized for the
model M1 & M2 in
Table 4-1.
Table 4-1 Summary of the obtained failure modes performing IDA - Model M1 & M2
Archetype Model
M1
Chi Chi - Taiwan
Sa
(g)
0.841
Superstition
1.448
Loma Prieta
1.208
Northridge
1.335
Imperial Valley
M2
2.103
Girder shear force
Sa
(g)
0.824
2.5
3.621
Abutment shear force
1.241
2.25
2.717
Abutment shear force
1.419
1.75
2.336
Girder shear force
1.144
3
3.431
Victoria-Mexico
1.092
3.25
Morgan Hill
1.924
Duzce
Ground Motion
Collapsed
Factor
2.5
Sa-Collapse
Failure Mode
Collapsed
Factor
1.75
Sa-Collapse
Failure Mode
1.443
Girder shear force
1
1.241
Abutment shear force
1
1.419
Girder shear force
1.523
0.75
1.142
Girder shear force
Abutment shear force
1.022
1.25
1.278
Girder shear force
3.549
Abutment shear force
0.708
1.5
1.062
Abutment shear force
2.5
4.809
Abutment shear force
2.388
0.75
1.791
Girder shear force
1.008
3.5
3.527
Abutment shear force
1.321
1.75
2.311
Abutment shear force
Cape Mendocino
0.778
3
2.334
Abutment shear force
0.861
2
1.723
Abutment shear force
Mammoth Lakes
0.611
5.5
3.361
Abutment shear force
0.814
2.5
2.034
Girder shear force
N.Palm Springs
1.131
3
3.394
Abutment shear force
1.332
1
1.332
Girder shear force
Tabas
0.736
4.5
3.312
Abutment shear force
0.557
1.5
0.836
Abutment shear force
San Fernando
1.937
1.5
2.906
Abutment shear force
2.266
0.75
1.700
Abutment shear force
Gazli-USSR
1.061
2.75
2.918
Abutment shear force
1.200
1.25
1.500
Girder shear force
Managua
1.116
4.5
5.022
Abutment shear force
1.140
1
1.140
Abutment shear force
Whittier Narrows
0.370
5.75
2.126
Girder shear force
0.265
3.25
0.862
Girder shear force
Coalinga
1.187
2.75
3.264
Abutment shear force
1.224
1.25
1.530
Girder shear force
Westmorland
1.010
3.5
3.535
Abutment shear force
0.691
1.25
0.864
Girder shear force
Kobe
0.426
7
2.985
Abutment shear force
0.420
3.5
1.469
Abutment shear force
Spitak
0.353
9
3.181
Abutment shear force
0.408
6.75
2.754
Girder shear force
73
In SeismoStruct, actual relative displacement of an element (d2-2) that bottom node of it is
not completely restrained cannot be obtained directly due to rocking of the element. At this case,
to find actual relative displacement of an element (d2-2), relative displacement of the element due
to rocking (d2-1) should be subtracted from the total relative displacement of the element (d2).
This matter is shown in Figure 4-1.
Figure 4-1 Shows relative displacements of a rocking element (abutment)
Accordingly, to obtain abutment relevant displacement, abutment rocking along the deck
is determined for the model M2 and deducted from the total relative displacement to obtain
actual relative displacement of the abutment. Abutment rocking in the model M2 is shown in
Figure 4-2.
74
IDA -Model M2
3
Chi Chi
Superstition
Loma Prieta
Northridge
Imperial Valley
Victoria_Mexico
Morgan Hill
Duzce
Cape Mendocino
Mammoth Lakes
N.Palm Springs
Tabas
San Fernando
Whittier Narrows
Coalinga
Westmorland
Kobe
Gazli
Managua
Spitak
Whittier Narrows
2.75
2.5
Sa (T1=0.284 sec) [g]
2.25
2
1.75
1.5
1.25
1
0.75
0.5
0.25
0
0
0.001
0.002
0.003
Abutment Rocking (rad)
Figure 4-2 Shows obtained abutment rocking - Model M2
The last points shown on the figure are not representative of analysis result. There is only used to
represent the Sa value of the collapse. The points prior to the last points represent non-simulated
collapse points that analysis is forced to stop.
75
Non-simulated collapse mode IDA results for abutment of the model M1 and M2 are presented in Figure 4-3
IDA- Model M2
IDA-Model M1
Chi Chi
Superstition
Loma Prieta
Northridge
Imperial Valley
Victoria_Mexico
Morgan Hill
Duzce
Cape Mendocino
Mammoth Lakes
N.Palm Springs
Tabas
San Fernando
Whittier Narrows
Coalinga
Westmorland
Kobe
Gazli
Managua
Spitak
5
4.5
Sa (T1=0.246 sec) [g]
4
3.5
3
2.5
2
1.5
1
0.5
0
0
0.001
0.002
Maximum Abutment Relative Displacement (m)
3
2.75
2.5
2.25
Sa (T1=0.284 sec) [g]
5.5
2
1.75
1.5
1.25
1
0.75
0.5
0.25
0
0
0.002
0.004
Chi Chi
Superstition
Loma Prieta
Northridge
Imperial Valley
Victoria_Mexico
Morgan Hill
Duzce
Cape Mendocino
Mammoth Lakes
N.Palm Springs
Tabas
San Fernando
Whittier Narrows
Coalinga
Westmorland
Kobe
Gazli
Managua
Spitak
Whittier Narrows
Maximum Abutment Relative Displacement (m)
Figure 4-3 Maximum relative displacement of the abutment along the bridge deck - Model M1 & M2
A 3-D Schematic view of the obtained IDA results which shows dispersion of the results for the model M1 & M2 is provided in
Appendix C.1.
76
Based on the above obtained Non-Simulated Collapse Mode IDA results and using fragility function fittings which elaborated
in Appendix B and a MATLAB code in which developed by Dr. J.W Baker for this purpose in 2013, fragility curves for the
probability of collapse of the model M1 & M2 are plotted. These curves are shown in Figure 4-4.
Fitted Fragility Curves - Model M2
1
0.9
0.9
0.8
0.8
0.7
0.7
Probability of collapse
Probability of collapse
Fitted Fragility Curves - Model M1
1
0.6
0.5
0.4
0.3
0.2
0.6
0.5
0.4
0.3
0.2
Observed Fractions of Collapse
Using MLE Method
Using Leaset Square errors
Observed Fractions of Collapse
0.1
0
0.1
Using MLE Method
Using Least Square Method
0
1
2
3
4
IM= Sa(T1=0.246 sec) [g]
5
6
0
0
0.5
1
1.5
2
2.5
3
3.5
4
IM= Sa(T1=0.284 sec)[g]
Figure 4-4 Shows obtained probability of collapse for the archetype model M1 & M2 based on the fragility fitting functions illustrated in Appendix B.
77
In MLE method (Maximum likelihood Method), a lognormal CDF fits to observed
probability of collapse data using optimization on the likelihood function for the data while in
the least square method sum of the squared errors are minimized.
To compare the obtained probability of collapse for the archetype model M1 & M2, these curves
are plotted together and presented in Figure 4-5.
Probability of Collapse - Comparison
1
0.9
Probability of Collapse
0.8
0.7
0.6
0.5
0.4
0.3
Model M1
Model M2
0.2
0.1
0
0
1
2
3
4
5
6
7
Sa(T1)[g]
Figure 4-5 Compares obtained probability of collapse for the archetype model M1 & M2
78
As can be seen in Figure 4-5, probability of collapse of the model M2 has a significant shift to
the left and has a smaller spectral acceleration for a specific probability of collapse. This results
an increase in probability of collapse.
4.2
Non-Simulated Collapse Mode IDA Results and Probability of Collapse - Model M3
& M4
Obtained IDA results for the model M3 & M4 are summarized in Table 4-2. As it can be
seen in this table, some of the failure modes in the model M4 comparing for their corresponding
IDA in the model M3., i.e.; abutment shear force failure is changed to abutment confined
concrete strain and steel rupture or abutment section failure mode or vice versa for the performed
IDA with Chi Chi, Loma Prieta, Imperial Valley, Mammoth Lakes, Managua, and Spitak ground
motions.
79
Table 4-2 Summary of the obtained failure modes performing IDA - Model M3 & M4
M3
M4
Archetype Model
Ground Motion
Sa (g)
Collapsed
Factor
Chi Chi - Taiwan
0.804
2
SaCollapse
(g)
1.609
0.75
Sa
Collapse
(g)
0.546
Abutment conf. conc. and steel strain
Superstition
Loma Prieta
0.904
1.215
1.25
1.5
1.116
1.053
0.75
1
0.837
1.053
Abut section curvature
Abutment conf. conc. and steel strain
Northridge
Imperial Valley
1.078
1.186
Abutment shear force
Abutment shear force
1.257
1.001
1
1.75
1.257
1.752
Abutment shear force
Abutment conf. conc. and steel strain
Victoria-Mexico
1.697
Abutment shear force
0.924
0.75
0.693
Unable to apply the next step load
2
3.75
Abutment shear force
1.748
1.25
2.185
Unable to apply the next step load
1.322
1.25
1.652
Abutment conf. conc. and steel strain
0.76
1
0.76
Abutment section curvature
Cape Mendocino
0.97
0.75
0.728
Unable to apply the next step load
0.955
0.75
0.716
Unable to apply the next step load
Mammoth Lakes
0.62
4
2.481
Abutment shear force
0.594
2
1.189
Abut section curvature
N.Palm Springs
Tabas
1.225
0.637
2.5
1.25
3.062
0.796
Abutment shear force
Unable to apply the next step load
1.015
0.672
1.75
1.5
1.776
1.008
Abutment shear force
Abutment conf. conc. and steel strain
San Fernando
1.729
1.25
2.161
Unable to apply the next step load
1.845
0.75
1.384
Abutment shear force
Gazli-USSR
1.234
1.75
2.16
Abutment shear force
1.298
1.5
1.948
Abutment shear force
Managua
1.168
1
1.168
Abutment conf. conc. and steel strain
1.147
0.75
0.86
Abut section curvature
Whittier Narrows
0.279
3.5
0.978
Abutment shear force
0.339
3
1.017
Abutment shear force
Coalinga
1.023
2
2.045
Abutment shear force
0.992
1.5
1.488
Abutment shear force
Westmorland
Kobe
0.713
0.534
2.25
2.5
1.605
1.336
Abutment shear force
Unable to apply the next step load
0.715
0.497
1.75
1.75
1.251
0.87
Unable to apply the next step load
Unable to apply the next step load
Spitak
0.501
2.5
1.253
Abutment conf. conc. and steel strain
0.447
2.5
1.118
Abut section curvature
Sa
(g)
Collapsed
Factor
Abutment shear force
0.729
1.129
1.822
Unable to apply the next step load
Abutment shear force
1.25
3
1.347
3.559
0.849
2
Morgan Hill
1.875
Duzce
Failure Mode
Failure Mode
80
Non-simulated collapse mode IDA results for the abutment and pier column of the model
M3 are presented in Figure 4-6 and Figure 4-7 respectively.
IDA-Model M3
4
Chi Chi
3.75
Superstition
3.5
Loma Prieta
3.25
Northridge
Imperial Valley
Sa (T1=0.335 sec) [g]
3
Victoria_Mexico
2.75
Morgan Hill
2.5
Duzce
Cape Mendocino
2.25
Mammoth Lakes
2
N.Palm Springs
1.75
Tabas
1.5
San Fernando
Whittier Narrows
1.25
Coalinga
1
Westmorland
0.75
Kobe
0.5
Gazli
Managua
0.25
Spitak
0
0
0.001
0.002
0.003
0.004
0.005
Maximum Abutment Relative Displacement (m)
Figure 4-6 Maximum relative displacement of the abutment along the bridge deck - Model M3
81
IDA-Model M3
IDA-Model M3
3.5
3.25
3
2.75
2.5
2.25
2
1.75
1.5
1.25
1
0.75
0.5
0.25
0
0
0.01
0.02
0.03
0.04
0.05
Maximum Pier Column Relative Displacement- X direction (m)
Superstition
3.75
Loma Prieta
3.5
Northridge
3.25
Imperial Valley
3
Sa (T1=0.335 sec) [g]
Chi Chi
Superstition
Loma Prieta
Northridge
Imperial Valley
Victoria_Mexico
Morgan Hill
Duzce
Cape Mendocino
Mammoth Lakes
N.Palm Springs
Tabas
San Fernando
Whittier Narrows
Coalinga
Westmorland
Kobe
Gazli
Managua
Spitak
3.75
Sa (T1=0.335 sec) [g]
Chi Chi
4
4
Victoria_Mexico
2.75
Morgan Hill
Duzce
2.5
Cape Mendocino
2.25
Mammoth Lakes
2
N.Palm Springs
1.75
Tabas
1.5
San Fernando
1.25
Whittier Narrows
Coalinga
1
Westmorland
0.75
Kobe
0.5
Gazli
0.25
Managua
0
0
0.005
0.01
Spitak
Maximum Pier Column Relative Displacement- Y direction (m)
Figure 4-7 Maximum relative displacement of the pier column across (x-direction) and along(y-direction) of the bridge deck - Model M3
82
Due to the reason illustrated in Section 4.1, abutment rocking for the model M4 is determined prior to obtaining the actual
relative displacement of the abutment. Obtained abutment rocking and abutment non-simulated collapse mode IDA results for the
model M4 are presented in Figure 4-8.
IDA-Model M4
Chi Chi
Superstition
Loma Prieta
Northridge
Imperial Valley
Victoria_Mexico
Morgan Hill
Duzce
Cape Mendocino
Mammoth Lakes
N.Palm Springs
Tabas
San Fernando
Whittier Narrows
Coalinga
Westmorland
Kobe
Gazli
Managua
Spitak
2.25
2
Sa (T1=0.374 sec) [g]
1.75
1.5
1.25
1
0.75
0.5
0.25
0
0
0.0005
0.001
0.0015
Maximum Abutment Rocking (rad)
2.5
Chi Chi
Superstition
Loma Prieta
Northridge
Imperial Valley
Victoria_Mexico
Morgan Hill
Duzce
Cape Mendocino
Mammoth Lakes
N.Palm Springs
Tabas
San Fernando
Whittier Narrows
Coalinga
Westmorland
Kobe
Gazli
Managua
Spitak
2.25
2
1.75
Sa (T1=0.374 sec) [g]
2.5
IDA-Model M4
1.5
1.25
1
0.75
0.5
0.25
0
0
0.005
0.01
Maximum Abutment Relative Displacement (m)
Figure 4-8 Left plot shows the obtained abutment rocking and right plot shows abutment non-simulated collapse mode IDA results along the bridge
deck for the model M4
83
Similarly, to obtain pier column actual relevant displacement, pier rocking in both directions is considered for the model M4.
Pier column's non-simulated collapse mode IDA results for the model M4 are presented in Figure 4-9.
IDA-Model M4
IDA-Model M4
2.5
2.25
2
Sa (T1=0.374 sec) [g]
1.75
1.5
1.25
1
0.75
0.5
0.25
0
0
0.005
0.01
0.015
0.02
Maximum Pier Column Relative Displacement-X direction(m)
2.5
Chi Chi
Superstition
Loma Prieta
Northridge
Imperial Valley
Victoria_Mexico
Morgan Hill
Duzce
Cape Mendocino
Mammoth Lakes
N.Palm Springs
Tabas
San Fernando
Whittier Narrows
Coalinga
Westmorland
Kobe
Gazli
Managua
Spitak
2.25
2
1.75
Sa (T1=0.374 sec) [g]
Chi Chi
Superstition
Loma Prieta
Northridge
Imperial Valley
Victoria_Mexico
Morgan Hill
Duzce
Cape Mendocino
Mammoth Lakes
N.Palm Springs
Tabas
San Fernando
Whittier Narrows
Coalinga
Westmorland
Kobe
Gazli
Managua
Spitak
1.5
1.25
1
0.75
0.5
0.25
0
0
0.005
0.01
Maximum Pier Column Relative Displacement-Y direction(m)
Figure 4-9 Shows obtained pier column actual relative displacement across (x-direction) and along (y-direction) of the bridge deck for the model M4
A 3-D schematic view of the obtained IDA results for the model M3 & M4 is provided in the Appendix C.2
84
4.2.1
Pier Columns Hysteretic Graphs - Bridge Type 2 (Model M3 & M4)
To compare pier column drift ratio of the model M3 & M4, hysteretic graph of the pier
column No.1 for the both models are plotted at collapse level. The following figures show pier
column total drift ratio (including column's rocking drift) of these models verses total base shear
for Chi Chi, Superstition, Loma Prieta, and Northridge ground motions:
1.5
Model M3
Model M4
1
0.5
0
-0.5
-1
-1.5
-2
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
Pier Column Drift Ratio Across The Bridge Deck (%)
Total Base Shear Along The Bridge Deck (kN)
Pier Column No.1 Hysteretic Plot At Collapse Level - Loma Prieta
4
2
1.5
x 10
Model M3
Model M4
1
0.5
0
-0.5
-1
-1.5
-2
-0.4
-0.2
0
0.2
0.4
Pier Column Drift Ratio Across The Bridge Deck (%)
Pier Column No.1 Hysteretic Plot At Collapse Level - Superstion
Total Base Shear - Along The Bridge Deck (kN)
4
x 10
4
1
x 10
Model M3
Model M4
0.5
0
-0.5
-1
-1.5
-0.2 -0.15 -0.1
-0.05
0
0.05
0.1
0.15
Pier Column Drift Ratio Across The Bridge Deck (%)
Pier Column No.1 Hysteretic Plot At Collapse Level - Northridge
Total Base Shear Along The Bridge Deck (kN)
Total Base Shear Along The Bridge Deck (kN)
Pier Column No.1 Hysteretic Plot At Collapse Level - Chi Chi
4
1.5
1
x 10
Model M3
Model M4
0.5
0
-0.5
-1
-1.5
-2
-0.4
-0.2
0
0.2
0.4
Pier Column Drift Ratio Across The Bridge Deck (%)
Figure 4-10 Shows pier column total drift ratio (including column's rocking drift) across the bridge deck
verses total base shear obtained from performed IDA - Model M3 & M4.
More hysteretic graph of the pier column total drift at collapse level for the model M3 and M4
are provided in Appendix D.1.
However, drift due to rocking separated from the total pier column's drift to obtain the
actual drift of the column. The results are shown in Figure 4-11.
85
1.5
Model M3
Model M4
1
0.5
0
-0.5
-1
-1.5
-2
-0.4
-0.2
0
0.2
0.4
Pier Column Drift Ratio Across The Bridge Deck (%)
Total Base Shear Along The Bridge Deck (kN)
Pier Column No.1 Hysteretic Plot At Collapse Level - Loma Prieta
4
3
2
x 10
Model M3
Model M4
1
0
-1
-2
-3
-0.4
-0.2
0
0.2
0.4
Pier Column Drift Ratio Across The Bridge Deck (%)
Pier Column No.1 Hysteretic Plot At Collapse Level - Superstition
Total Base Shear - Along The Bridge Deck (kN)
4
x 10
4
1.5
x 10
Model M3
Model M4
1
0.5
0
-0.5
-1
-1.5
-0.15
-0.1
-0.05
0
0.05
0.1
Pier Column Drift Ratio Across The Bridge Deck (%)
Pier Column No.1 Hysteretic Plot At Collapse Level - Northridge
Total Base Shear Along The Bridge Deck (kN)
Total Base Shear Along The Bridge Deck (kN)
Pier Column No.1 Hysteretic Plot At Collapse Level - Chi Chi
4
3
2
x 10
Model M3
Model M4
1
0
-1
-2
-3
-4
-0.4
-0.2
0
0.2
0.4
Pier Column Drift Ratio Across The Bridge Deck (%)
Figure 4-11 Shows pier column actual drift ratio across the bridge deck verses total base shear obtained from
performed IDA - Model M3 & M4.
Figure 4-12 indicates pier column drift ratio of these models versus total base moment for the
same ground motions:
86
Pier Column No.1 Hysteretic Plot At Collapse Level - Chi Chi
Pier Column No.1 Hysteretic Plot At Collapse Level - Superstition
1500
2000
Total Moment - About Y axis (kN.m)
Total Moment - About Y axis (kN.m)
2000
Model M3
Model M4
1000
500
0
-500
-1000
-1500
-2000
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
Pier Column Drift Ratio - X Direction (%)
1000
500
0
-500
-1000
-1500
-0.3
-0.2
-0.1
0
0.1
Pier Column Drift Ratio - X Direction (%)
0.2
500
0
-500
-1000
-0.15
-0.1
-0.05
0
0.05
0.1
Pier Column Drift Ratio - X Direction (%)
0.15
Pier Column No.1 Hysteretic Plot At Collapse Level - Northridge
2000
Total Moment - About Y axis (kN.m)
Total Moment - About Y axis (kN.m)
Model M3
Model M4
Model M3
Model M4
1000
-1500
-0.2
0.3
Pier Column No.1 Hysteretic Plot At Collapse Level - Loma Prieta
2000
1500
1500
1500
Model M3
Model M4
1000
500
0
-500
-1000
-1500
-2000
-0.3
-0.2
-0.1
0
0.1
Pier Column Drift Ratio - X Direction (%)
0.2
Figure 4-12 Shows pier column actual drift ratio across the bridge deck verses total moment obtained from
performed IDA - Model M3 & M4.
As it can be seen from the Figure 4-11 and Figure 4-12, there is a significant reduction of
the pier column drift ratio for the model M4 which SSI features are considered in it.
Pier column hysteretic graphs for the other ground motions also show significant reduction in the
column's drift ratio for the SSI models. They can be found in the Appendices D.2 & E.1.
87
Based on the above obtained Non-Simulated Collapse Mode IDA results and using the above mentioned MATLAB code,
fragility curve for the probability of collapse model M3 & M4 are obtained. These curves are shown in Figure 4-13.
Fitted Fragility Curves - Model M3
Fitted Fragility Curves - Model M4
1
1
0.9
0.9
0.8
0.7
Probability of collapse
Probability of collapse
0.8
0.6
0.5
0.4
0.3
0.2
0.6
0.5
0.4
0.3
0.2
Observed Fractions of Collapse
Using MLE Method
Using Least Square Method
0.1
0
0.7
0
0.5
1
1.5
2
2.5
3
3.5
IM= Sa(T1=0.335 sec) [g]
4
4.5
5
Observed Fractions of Collapse
Using MLE Method
Using Least Square Method
0.1
5.5
0
0
0.5
1
1.5
2
2.5
3
3.5
4
IM= SA(T1=0.374 sec) [g]
Figure 4-13 Shows obtained probability of collapse for the archetype model M3 & M4 based on the fragility fitting functions illustrated in Appendix B.
88
The obtained probability of collapse for the archetype model M1 & M2 are plotted
together for the purpose of comparison and presented in Figure 4-14.
Probability of Collapse - Comparison
1
0.9
Probability of Collapse
0.8
0.7
0.6
0.5
0.4
0.3
Model M3
Model M4
0.2
0.1
0
0
1
2
3
4
5
6
7
Sa(T1)[g]
Figure 4-14 Compares obtained probability of collapse for the archetype model M3 & M4
4.3
Non-simulated Collapse Mode IDA Results and Probability of Collapse - Model M5
& M6
Obtained IDA results are summarized for the model M5 & M6 in Table 4-3.
89
Table 4-3 Summary of the obtained failure modes performing IDA - Model M5 & M6
M5
M6
Archetype Model
Sa
(g)
Collapsed
Factor
Sa-Collapse
(g)
Sa
(g)
Collapsed
Factor
Sa-Collapse
(g)
Chi Chi - Taiwan
Superstition
Loma Prieta
Northridge
Imperial Valley
Victoria-Mexico
Morgan Hill
Duzce
0.729
1.25
0.911
Abutment shear force
0.729
1.25
0.911
Abutment shear force
1.116
3.25
1.053
0.75
3.627
Abutment shear force
1.116
3.25
3.627
Abutment shear force
0.790
Abutment shear force
1.053
1
1.053
Abutment shear force
1.257
1
1.257
Girder conf. Concrete strain
1.257
1
1.257
Girder conf. Concrete strain
1.001
2.75
2.753
Abutment shear force
1.001
2.75
2.753
Abutment shear force
0.924
1.75
1.617
Abutment shear force
0.924
1.75
1.617
Abutment shear force
1.748
0.760
1.75
3
3.059
2.279
1.748
0.760
1.75
3
3.059
2.279
Girder conf. Concrete strain
Abutment Conf. Concrete and Steel Strain
Cape Mendocino
0.955
1.25
1.193
0.955
1.25
1.193
Unable to apply the next step load
Mammoth Lakes
0.594
4.5
2.674
Girder conf. Concrete strain
Abutment section curvature
Unable to apply the next step
load
Abutment Conf. Concrete and
Steel Strain
0.594
1.5
0.891
Abutment Conf. Concrete and Steel Strain
N.Palm Springs
1.015
1.75
1.776
1.015
1.75
1.776
Girder conf. Concrete strain
Tabas
0.672
1.25
0.840
0.672
1
0.672
Unable to apply the next step load
San Fernando
Gazli-USSR
Managua
Whittier Narrows
Coalinga
Westmorland
Kobe
Spitak
1.845
1.25
2.306
Girder conf. Concrete strain
Unable to apply the next step
load
Abutment shear force
1.845
1.25
2.306
Abutment shear force
1.298
1.25
1.623
Abutment shear force
1.298
1.25
1.623
Abutment shear force
1.147
3.5
4.013
Abutment shear force
1.147
3.5
4.013
Abutment shear force
0.339
3.75
1.271
Pier column shear force
0.339
3.75
1.271
Pier column shear force
0.992
3.5
3.473
Girder conf. Concrete strain
0.992
3.5
3.473
Girder conf. Concrete strain
0.715
1.75
1.251
Abutment shear force
0.715
1.75
1.251
Abutment shear force
0.497
3.25
1.616
Abutment shear force
0.497
2.5
1.243
Abutment shear force
0.447
3.25
1.453
Abutment shear force
0.447
5.25
2.347
Abutment shear force
Ground Motion
Failure Mode
Failure Mode
90
Non-simulated collapse mode IDA results for abutment and pier column of the model M5 are presented in Figure 4-15.
IDA-Model M5
4.25
4
3.75
3.5
3.25
Sa (T1=0.374 sec) [g]
3
2.75
2.5
2.25
2
1.75
1.5
1.25
1
0.75
0.5
0.25
0
0
0.001
0.002
0.003
Maximum Abutment Relative Displacement (m)
Chi Chi
Superstition
Loma Prieta
Northridge
Imperial Valley
Victoria_Mexico
Morgan Hill
Duzce
Mammoth Lakes
N.Palm Springs
Tabas
San Fernando
Whittier Narrows
Coalinga
Westmorland
Kobe
Gazli
Managua
Spitak
Cape Mendocino
Sa (T1=0.374 sec) [g]
IDA-Model M5
4.5
4.25
4
3.75
3.5
3.25
3
2.75
2.5
2.25
2
1.75
1.5
1.25
1
0.75
0.5
0.25
0
Chi Chi
Superstition
Loma Prieta
Northridge
Imperial Valley
Victoria_Mexico
Morgan Hill
Duzce
Cape Mendocino
Mammoth Lakes
N.Palm Springs
Tabas
San Fernando
Whittier Narrows
Coalinga
Westmorland
Kobe
Gazli
Managua
Spitak
0
0.05
0.1
0.15
Maximum Pier Column Relative Displacement (m)
Figure 4-15 Shows obtained actual relative displacement for abutment and pier column along and across the bridge deck respectively - Model M5
91
Non-simulated collapse mode IDA results for abutment and pier column of the model M6 are presented in Figure 4-16.
IDA-Model M6
IDA-Model M6
4.25
4
Superstition
3.75
Loma Prieta
3.5
Northridge
3
Imperial Valley
3.25
Victoria_Mexico
N.Palm Springs
2
Tabas
1.75
San Fernando
1.5
Whittier Narrows
Coalinga
1.25
Westmorland
1
Kobe
0.75
Gazli
0.5
Managua
0.25
Spitak
0
0
0.001
0.002
0.003
Northridge
Victoria_Mexico
Mammoth Lakes
2.25
Loma Prieta
3.75
3.5
Duzce
2.5
Superstition
4
Imperial Valley
Morgan Hill
2.75
Chi Chi
4.25
Cape Mendocino
Maximum Abutment Relative Displacement (m)
Sa (T1=0.374 sec) [g]
3.25
Sa (T1=0.374 sec) [g]
4.5
Chi Chi
3
Morgan Hill
Duzce
2.75
2.5
Cape Mendocino
2.25
Mammoth Lakes
N.Palm Springs
2
Tabas
1.75
San Fernando
1.5
Whittier Narrows
1.25
Coalinga
1
Westmorland
0.75
Kobe
0.5
Gazli
Managua
0.25
Spitak
0
0
0.05
0.1
0.15
Maximum Pier Column Relative Displacement(m)
Figure 4-16 Shows obtained actual relative displacement for abutment and pier column along and across the bridge deck respectively - Model M6
A 3-D schematic view of the obtained IDA results for the model M5 & M6 showing dispersion of the results are provided in Appendix
C.3
92
4.3.1
Pier Columns Hysteretic Graphs - Bridge Type 3 (Model M5& M6)
To compare pier column drift ratio of the model M5 & M6, hysteretic graph of the pier
column of the both models are plotted at collapse level. Figure 4-17 shows pier column drift ratio
of these models verses total base shear and moment for Chi Chi, Superstition, Loma Prieta, and
Northridge ground motions:
3
Model M5
Model M6
2
1
0
-1
-2
-0.4
-0.2
0
0.2
0.4
0.6
Pier Column Drift Ratio Across The Bridge Deck (%)
Total Base Shear Along The Bridge Deck (kN)
Pier Column No.1 Hysteretic Plot At Collapse Level - Loma Prieta
4
3
2
x 10
Model M5
Model M6
1
0
-1
-2
-3
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Pier Column Drift Ratio Across The Bridge Deck (%)
Pier Column No.1 Hysteretic Plot At Collapse Level - Superstion
Total Base Shear - Along The Bridge Deck (kN)
4
x 10
4
4
x 10
2
0
-2
-4
-6
Model M5
Model M6
-8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Pier Column Drift Ratio Across The Bridge Deck (%)
Pier Column No.1 Hysteretic Plot At Collapse Level - Northridge
Total Base Shear Along The Bridge Deck (kN)
Total Base Shear Along The Bridge Deck (kN)
Pier Column No.1 Hysteretic Plot At Collapse Level - Chi Chi
4
3
x 10
2
1
0
-1
-2
-3
-4
-1
Model M5
Model M6
-0.5
0
0.5
1
Pier Column Drift Ratio Across The Bridge Deck (%)
Figure 4-17 Shows pier column actual drift ratio across the bridge deck verses total base shear obtained from
performed IDA - Model M5 & M6.
93
Pier column drift ratio of these models versus total base moment for the same ground motions is
shown in Figure 4-18.
Pier Column No.1 Hysteretic Plot At Collapse Level - Chi Chi
Pier Column No.1 Hysteretic Plot At Collapse Level - Superstition
1000
1500
Total Moment - About Y axis (kN.m)
Total Moment - About Y axis (kN.m)
1500
Model M5
Model M6
500
0
-500
-1000
-1500
-0.6
-0.4
-0.2
0
0.2
Pier Column Drift Ratio - X Direction (%)
Model M5
Model M6
500
0
-500
-1000
-1500
-1
0
-500
-1000
0
Pier Column Drift Ratio - X Direction (%)
0.5
Pier Column No.1 Hysteretic Plot At Collapse Level - Northridge
2000
Total Moment - About Y axis (kN.m)
Total Moment - About Y axis (kN.m)
1500
Model M5
Model M6
500
-1500
-0.5
0.4
Pier Column No.1 Hysteretic Plot At Collapse Level - Loma Prieta
1000
1000
-0.5
0
0.5
Pier Column Drift Ratio - X Direction (%)
1
1500
Model M5
Model M6
1000
500
0
-500
-1000
-1500
-1
-0.5
0
0.5
Pier Column Drift Ratio - X Direction (%)
1
Figure 4-18 Shows pier column actual drift ratio across the bridge deck verses total moment obtained from
performed IDA - Model M5 & M6.
As it can be seen from the above figures, pier column drift ratio for the both models are
almost identical as the only SSI feature is considered in these models was CALTRANS springs
to simulate the effect of the soil behind the abutment backwalls.
Pier column hysteretic graphs for the rest of the ground motions can be found in the Appendices
D.3 and E.2 respectively.
94
Based on the above obtained non-simulated collapse mode IDA results and using the above mentioned MATLAB code,
fragility curve for the probability of collapse model M5 & M6 are obtained. These curves are shown in Figure 4-19.
Fitted Fragility Curves - Model M6
1
0.9
0.9
0.8
0.8
0.7
0.7
Probability of collapse
Probability of collapse
Fitted Fragility Curves - Model M5
1
0.6
0.5
0.4
0.3
0.2
0.5
0.4
0.3
0.2
Observed Fractions of Collapse
Using MLE Method
Using Least Square Method
0.1
0
0.6
0
1
2
3
4
IM= Sa(T1=0.374 sec) [g]
5
6
Observed Fractions of Collapse
Using MLE Method
Using Least Square Method
0.1
0
0
1
2
3
4
5
6
IM=Sa(T1=0.374 sec) [g]
Figure 4-19 Shows obtained probability of collapse for the archetype model M5 & M6 based on the fragility fitting functions illustrated in Appendix B.
95
The obtained probability of collapse for the archetype model M5 & M6 are plotted
together for purpose of comparison and presented in Figure 4-20.
Probability of Collapse - Comparison
1
0.9
Probability of Collapse
0.8
0.7
0.6
0.5
0.4
0.3
Model M5
Model M6
0.2
0.1
0
0
1
2
3
4
5
6
7
Sa(T1)[g]
Figure 4-20 Compares obtained probability of collapse for the archetype model M5 & M6
4.4
Relative Displacement Capacity/Demand Ratio (λ) and Period-Based Ductility (
) of
The Archetype Models
Based on the presented static pushover and IDA results, relative displacement
capacity/demand ratio (λ) and period-based ductility
of the models are calculated and
summarized in Table 4-4.
96
Table 4-4 Summary of the calculated relative displacement capacity/demand ratio (λ) and period-based
ductility (µT) using the obtained results from static pushover and incremental dynamic analysis
Abutment
Pier Column
Element
Model
(mm)
(mm)
(mm)
(mm)
(mm)
(mm)
M1
0.84
1.3
0.84
1
0.65
-
-
-
-
-
M2
3.0
2.0
3.0
1
1.5
-
-
-
-
-
M3
10.5
3.5
3.3
3.2
3
22.5
15.5
8.8
2.6
1.45
M4
8.5
6.6
3.5
2.4
1.3
34.1
25.8
11
3.1
1.32
M5
2.9
2.2
2.9
1
1.32
58.6
116
37
1.6
0.5
M6
2.2
2.2
2.2
1
1
77.6
116
38
2.04
0.67
in which,
x & y directions are across and along the bridge deck respectively,
and
are ultimate relative displacement in y direction of the
abutment obtained from the static pushover and incremental dynamic analysis when
the elements is pushed in x & y directions respectively,
and
are ultimate relative displacement in x direction of the pier
column obtained from the static pushover and incremental dynamic analysis when
the elements is pushed in x & y directions respectively, and
is period-based ductility for an archetype model and obtained as follow:
For abutments:
(eq. 4.1)
97
For pier columns:
(eq. 4.2)
where,
is effective yield relative displacement in x and y direction for pier
columns and abutments respectively and it is defined as per Figure 4-21:
Figure 4-21 Shows effective yield displacement
- Source FEMA P695
is the calculated relative displacement capacity/demand ratio for the specified model and
element and obtained as follow:
For abutments:
(eq. 4.3)
For pier columns:
(eq. 4.4)
As it can be seen in Table 4-4, it seems that capacity/demand ratio of the SSI models is mainly
less than the capacity/demand ratio obtained for their corresponding non-SSI models.
98
It can be concluded that considering soil structure interaction mostly resulted in having
smaller relative displacement capacity/demand ratio. As a result, neglecting SSI features in the
numeric models can result in overestimating relative displacement capacity of the system and
structural components in the non-SSI models.
However, abutment relative displacement capacity/demand ratio of the model M2 (1.5) is
very larger than capacity/demand ratio obtained for its corresponding non-SSI model M1 (0.65).
In this particular case, relative displacement capacity/demand ratio of the SSI model is more than
the non SSI model for the single span integral bridge (bridge type 1). Therefore, it can be
interpreted that soil structure interaction at this case is in the favor side and resulted in having
larger capacity/demand ratio for the bridge abutments.
In addition, period-based ductility of the pier columns in SSI model is significantly
increased comparing to their corresponding non-SSI model. For instance, period-based ductility
of the pier columns in archetype model M4 and M6 is increased about 120% and 127%
respectively comparing to their corresponding non-SSI model (M3 & M6). However, periodbased ductility of the abutment in archetype model M4 shows a 25% decrease comparing to its
corresponding non-SSI model (M3). In general, it can be concluded that period-based ductility of
the pier columns increases by 125% while period-based ductility for abutments decreases by
25% in SSI models. In other words, soil structure interaction causes ductility demand increases
for the pier columns up to 125% and decreases for the abutments maximum by 25%.
99
4.5
Percentage of the Failure Mode and Comparison of the All Obtained Probability of
Collapses
Based on the presented IDA results, percentage of the failure modes are calculated and
summarized in Table 4-5.
Table 4-5 Summary of the percentage of failure for the failure modes defined as performance criteria in all
the models
Percentage of Failure
Archetype Model
Failure Mode
M1
M2
M3
M4
M5
M6
Abutment shear force
85%
40%
60%
30%
55%
55%
Abutment conf. concrete or steel strain
0
0
15%
20%
5%
10%
Abutment section curvature
0
0
0
25%
5%
0
15%
60%
0
0
0
0
Girder conf. concrete or steel strain
0
0
0
0
20%
20%
Girder section curvature
0
0
0
0
0
0
Pier shear force
-
-
0
0
5%
5%
Pier conf. concrete or steel strain
-
-
0
0
0
0
Pier section curvature
-
-
0
0
0
0
Unable to apply the next step load
0
0
25%
25%
10%
10%
Girder shear force
As can be seen in Table 4-5, percentage of abutment shear force failure decreases in SSI
model that include piles and effect of soil surrounded piles are considered in their models (model
M2 & M4). In regards girder, only model M2 among the SSI models has an increase in shear
force failure. In addition, based on the percentage provided in the above table majority of failure
100
modes are forced base and the pier column shear force failure is occurred in the model M5 and
M6 that have slender pier columns comparing to the model M3 & M4.
In general, it seems that support elements (abutments) having less force failure in the SSI
model while non-supported elements (girders) showing more force failure comparing to their
corresponding non-SSI models. The only girder confines concrete strain and steel rupture failure
is observed for the model M5 & M6 (20%) which have a massive gravity abutment support.
All the obtained probability of collapse of the archetype models are graphed to together for
comparison and presented in Figure 4-22.
Probability of Collapse - Comparison All The Models
1
0.9
Probability of Collapse
0.8
0.7
0.6
0.5
Model M1
Model M2
Model M3
Model M4
Model M5
Model M6
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
6
7
Sa(T1)[g]
Figure 4-22 Compares obtained probability of collapse for all the archetype models
101
Based on the obtained collapse fragility curves shown in Figure 4-22, probability of
collapse for the SSI model has an increase for a specific spectral acceleration comparing to its
corresponding non-SSI model.
Collapse Margin ratio (CMR) is a primary parameter used to characterize the collapse
safety of the structures. CMR is defined as ratio between the median collapse intensity (SCT )
and the intensity at MCE-level ground motions (SaMT) where MCE intensity itself is obtained
from the response spectrum of MCE ground motions at the fundamental period, T1 (FEMA
P695, June 2009).
^
CMR  S CT / SaMT (T1)
(eq. 4.5)
Assuming that shear wave velocity of the top 30 m of the site sub-layer soil is larger than
360 m/s and smaller than 750 m/s which is equivalent to soil site class C based on NBCC 2010
(see Appendix G ), collapse margin ratio for 10%, 20%, and 50% (median) are calculated and
presented in the following sub-sections.
4.5.1
Collapse Margin Ratio for the Model M1 & M2
Collapse margin ratio for the model M1 & M2 are obtained as they are shown in Figure
4-23 and the following calculation:
102
Uniform Hazard Spectra (UHS) 2%/50 year-Vancouver
Fragility Curve - Model M1
1
SaMT
1
50%ile Average H & R Model
0.9
0.9
0.8
0.7
0.7
Probability of Collapse
0.8
Sa(g)
0.6
T1=0.246 sec
0.5
0.4
0.3
0.6
0.5
0.4
0.3
0.2
0.2
0.1
0.1
0
0
0
0.5
1
1.5
2
2.5
Period (sec)
3
3.5
4
0
1
2
4
5
6
7
Fragility Curve - Model M2
1
0.9
0.9
50%ile Average H & R Model
SaMT
0.8
0.7
0.7
Probability of Collapse
0.8
0.6
0.5
T1=0.284 sec
Sa(g)
SCT
Sa(g)
Uniform Hazard Spectra (UHS) 2%/50 year-Vancouver
1
0.4
0.3
0.2
0.6
0.5
0.4
0.3
0.2
0.1
0.1
0
3
0
0
0.5
1
1.5
2
2.5
Period (sec)
3
3.5
4
0
0.5
1
1.5
SCT
2
2.5
Sa(g)
3
3.5
4
4.5
Figure 4-23 illustrates median collapse intensity (SCT) and the intensity at MCE-level ground motions
(SaMT) for the model M1 & M2
^
CMR  S CT / SaMT (T 1)
^
CMR _ M 1  S CT / SaMT (T1  0.246 sec)  3.218 / 0.8975  3.58
^
CMR _ M 2  S CT / SaMT (T 1  0.284 sec)  1.669 / 0.855  1.952
103
4.5.2
Collapse Margin Ratio for the Model M3 & M4:
Collapse margin ratio for the model M3 & M4 are obtained as they are shown in Figure
4-24 and the following calculation:
Uniform Hazard Spectra (UHS) 2%/50 year-Vancouver
Fragility Curve - Model M3
1
1
50%ile Average H & R Model
0.9
0.8
0.8
0.7
0.7
Probability of Collapse
SaMT
0.9
0.5
T1=0.335 sec
Sa(g)
0.6
0.4
0.3
0.2
0.5
0.4
0.3
0.2
0.1
0.1
0
0.6
0
0
0.5
1
1.5
2
2.5
Period (sec)
3
3.5
4
0
1
2
Uniform Hazard Spectra (UHS) 2%/50 year-Vancouver
5
6
1
50%ile Average H & R Model
0.9
0.9
0.8
0.8
SaMT
0.7
Probability of Collapse
0.7
0.6
0.5
T1=0.374 sec
Sa(g)
4
Fragility Curve - Model M4
1
0.4
0.3
0.2
0.6
0.5
0.4
0.3
0.2
0.1
0.1
0
3
Sa(g)
SCT
0
0
0.5
1
1.5
2
2.5
Period (sec)
3
3.5
4
0
0.5
1
1.5
SCT
2
2.5
Sa(g)
3
3.5
4
4.5
Figure 4-24 Illustrates median collapse intensity (SCT) and the intensity at MCE-level ground motions (SaMT)
for the model M3 & M4.
104
^
CMR  S CT / SaMT (T 1)
^
CMR _ M 3  S CT / SaMT (T 1  0.335sec)  1.896 / 0.798  2.376
^
CMR _ M 4  S CT / SaMT (T 1  0.374sec)  1.390 / 0.757  1.836
4.5.3
Collapse Margin Ratio for the Model M5 & M6:
Similarly, collapse margin ratio for the model M5 & M6 are obtained based on the
following calculation and shown in Figure 4-25:
Fragility Curve - Model M5
Uniform Hazard Spectra (UHS) 2%/50 year-Vancouver
1
1
50%ile Average H & R Model
0.9
0.9
0.8
0.8
SaMT
0.7
Probability of Collapse
0.7
0.5
T1=0.374 sec
Sa(g)
0.6
0.4
0.3
0.2
0.5
0.4
0.3
0.2
0.1
0.1
0
0.6
0
0
0.5
1
1.5
2
2.5
Period (sec)
3
3.5
4
0
1
2
Uniform Hazard Spectra (UHS) 2%/50 year-Vancouver
5
6
7
5
6
7
Fragility Curve - Model M6
50%ile Average H & R Model
0.9
0.9
0.8
0.8
0.7
0.7
Probability of Collapse
SaMT
0.6
0.5
T1=0.374 sec
Sa(g)
4
Sa(g)
1
1
0.4
0.3
0.2
0.6
0.5
0.4
0.3
0.2
0.1
0
3
SCT
0.1
0
0.5
1
1.5
2
2.5
Period (sec)
3
3.5
4
0
0
1
2
SCT
3
4
Sa(g)
Figure 4-25 Illustrates median collapse intensity (SCT) and the intensity at MCE-level ground motions
(SaMT) for the model M5 & M6.
105
^
CMR  S CT / SaMT (T 1)
^
CMR _ M 5  S CT / SaMT (T1  0.374sec)  2.075 / 0.757  2.741
^
CMR _ M 6  S CT / SaMT (T 1  0.374sec)  1.986 / 0.757  2.623
4.5.4
Summary of the Obtained Collapse Margin Ratios:
Likewise, CMR 10% & CMR20% obtained for all archetype models. Calculated collapse
margin ratios for 10%, 20%, and median and for all archetype models are summarized in Table
4-6.
Table 4-6 Summary of the calculated collapse margin ratios for median (50%), 10%, and 20% and for all
models
Model
T1 (sec)
SCT 10%
SCT 20%
SCT @ Median
SMT @ T1
[g]
[g]
[g]
[g]
CMR10% =
SCT10%/
SMT@ T1
CMR20%
= SCT20%
/ SMT@ T1
CMR@
Median =
SCT@
Median /
SMT@ T1
M1
0.246
2.594
2.864
3.461
0.902
2.876
3.175
3.837
M2
0.284
1.187
1.334
1.669
0.855
1.388
1.560
1.952
M3
0.335
1.125
1.344
1.896
0.798
1.410
1.684
2.376
M4
0.374
0.942
1.077
1.390
0.757
1.244
1.423
1.836
M5
0.374
1.209
1.448
2.075
0.757
1.597
1.913
2.741
M6
0.374
1.108
1.350
1.986
0.757
1.464
1.783
2.623
0.51
0.77
0.96
In the last column of Table 4-6, average of the obtained CMR@ median of SSI and non-SSI
model (η) is presented. The application of this ratio is explained in Section Error! Reference
source not found.where a design procedure is proposed. In addition, averaging of the entire
obtained η ratio for the archetype models,
η
ratio for the integral bridges is obtained 0.75. In
general, as it is shown in Table 4-6, obtained CMR value of the SSI archetype model is
106
consistently smaller than the CMR value which obtained for the corresponding non-SSI
archetype model. In contrast, probability of collapse of a SSI archetype model is greater than
probability of collapse for the same model when the SSI features are not considered for a specific
spectral acceleration. In general, it can be concluded that soil structure interaction causes an
increase in probability of collapse and a decrease in CMR for a specified intensity measure (Sa
(T1)).
107
4.6
Proposed Design Procedures
Assuming that collapse margin ratio (10%, 20%, or 50% (median)) and η ratio for each
bridge prototype are pre-determined and provided in the future bridge design code, a design
procedure when the analysis to be carried out using a non-SSI model when the SSI effects is
required to be considered is suggested in Figure 4-26.
Start
Develop non-SSI model of a structure
implementing required performance criteria
Perform IDA using
sufficient nos. of suitable ground motions
Calculate the CMR of the non-SSI model (CMRnon - SSI)
Estimate CMR of non-SSI model
CMRSSI,est=η X CMRnon-SSI
CMR SSI,est
≥
CMRSSI
No
Enhance the design until obtaining CMRSSI,est ≥ CMRSSI
considering the most frequently failure mode in the analysis
Yes
Model is reliable
Figure 4-26 Demonstrates the proposed design procedure when SSI effect needed to be considered in a nonSSI model
108
Based on the proposed procedure, designers need to run IDA for a non-SSI archetype
with sufficient number of the well selected ground motions. It is ideal if the collapse simulated
for all the seismic resisting structural components in the non-SSI model to minimise
uncertainties. However, creating fully simulated collapse model is almost impossible. As a result,
designers are advised to obtain appropriate performance criteria through investigating sectional
response of the main structural components and determining and implementing of the required
limit states in the analytical models.
Running the IDA, median collapse margin ratio for the non-SSI models can be obtained
as it is illustrated in the previous sections. After obtaining the CMRnon-SSI, designers should
apply scale factor, η, specified in the code for the specific required bridge archetype to obtain an
estimate collapse margin ratio (CMRSSI-est) and compare this value to the corresponding value of
collapse margin ratio provided in the code for the SSI version of the same archetype model
(CMRSSI). If the obtained estimated CMR value is larger than CMRSSI value in the code for that
type of bridge, the designed model is reliable. Otherwise, considering the most failure modes in
the non-SSI model and enhancing the relevant structural component(s), designers need to re-run
IDA to obtain a new CMR value and repeat the procedure until obtaining CMRSSI-est value
greater than the corresponding CMRssi value specified in the code. For instance, based on Table
4-6, CMRnon-SSI, η, and CMRSSI-est value for the integral bridge are obtained 2.985, 0.75, and
2.238 respectively. In this way, CMRSSI value for the integral bridge in the code is obtained
2.238. If the obtained CMRSSI-est value through performing IDA using a non-SSI model of an
integral bridge is larger than this value in the code, the analytical model is considered reliable.
Otherwise, the non-SSI model should be modified based on the observed failure modes by
109
enhancing the failure structural components and the above procedure need to be repeated until
obtaining CMRSSI-est value greater than CMRssi value specified in the code.
Performing IDA with appropriate number of ground motions can be time consuming;
however, defining the appropriate performance criteria for the important structural components
in the model such as shear force resisting systems, designers are able to find out the most
collapse prone failure modes and by modifying the relevant structural components designers can
reach the target CMR in the code with less effort and minimum number of iteration.
As mentioned, considering soil in the analytical models can be very complex and cumbersome,
especially if there is possibility of liquefaction in the sub-layer soil. In addition, performing IDA
is time consuming process. However, the above approach can be a useful and efficient way in
modeling of the important structures and it allows designers to implement the soil structure effect
in the non-SSI model of this type of structures.
110
Chapter 5: Concluding Remarks
5.1
Conclusion
The main conclusion of this thesis has indicated that including the soil structure
interaction effect in the numerical model leads to having smaller relative displacement and drift
ratio in structural components. This result is based on the obtained results presented in Section
4.2.1 and 4.3.1 and Appendix E
In addition to the above matter, period-based ductility (
) of the pier columns in the SSI
model is significantly increased compared to their corresponding non-SSI model, as seen in
Table 4-4. For instance, period-based ductility of the pier columns in archetype model M4 and
M6 is increased about 120% and 127% respectively comparing to their non-SSI corresponding
model (M3 & M6). While, period-based ductility of the abutment in archetype model M4 shows
a 25% decrease comparing to its corresponding non-SSI model (model M3). In general, for
integral bridges it can be concluded that period-based ductility of the pier column increases by
125% in the SSI model while it decreases for the abutment by 25%. In other words, when soil
structure interaction is considered in the numerical model, abutments function up to 25 % less
ductile instead pier columns function up to 125 % more ductile.
On the other hand, for some cases calculated relative displacement capacity/demand ratio
( ) for the abutments and pier columns for the SSI analytical models is smaller than the same
value obtained for their corresponding non-SSI model, as shown in Table 4-4. This matter can
result in overestimating displacement capacity of the main structural components in the non-SSI
models.
111
As discussed in Chapter 4 and shown in Figure 4-22, probability of collapse (regardless
of the hazard level) increases for the archetype models that include soil structure interaction
features. As shown in Table 4-6, this matter results in obtaining a smaller CMR value for the SSI
models comparing to their corresponding non-SSI models.
In conclusion, based on the obtained relative displacement capacity/demand ratio ( ),
period-based ductility (
) for both the abutment and pier structural components, and the
fragility collapse curves and the collapse margin ratio (CMR) for the studied models, it can be
concluded that neglecting the SSI effects on a integral abutment bridge numerical model will be
less conservative.
5.2
Recommendations, Limitations and Future Works
At present, soil structure interaction is not sufficiently addressed in most bridge design
codes. On the other hand, consideration of the SSI features in the analytical models is extremely
difficult for most design engineers due to its complexity. Therefore, it is appropriate if
researchers in any bridge code review committee investigate SSI effect for a wide range of
bridge prototypes and implement their findings in to the future versions of the bridge code as
reference for designers.
The collapse fragilities, CMR values, and other data presented in this study (e.g. periodbased ductility) are based on the assumptions and simplification made for the archetype models.
Consideration of all the complexities in the models is not practically possible. However, extra
care was taken in making assumptions and simplification of the models. As discussed, a great
effort is made during modeling of the prototype bridges to reflect the behavior of the real
structures and to obtain an accurate response of the structures from nonlinear response analysis.
112
Due to the time limitation, only three type of the various integral abutment bridge
prototypes and a total six archetype models for the soil site class C are studied. However, for the
purpose of comparison, different type of bridges and different soil site classes need to be
investigated to understand the impact of considering soil structure in the numerical model.
In this study, CALTRANS springs and the nonlinear links developed by Allotey and El
Naggar (2008), implemented in SeismoStruct, are employed to consider effects of soil behind the
abutment backwall and around the piles, respectively. Nevertheless, to obtain more accurate
results, it is more appropriate if continuum model of the soil is considered using more
sophisticated FEA software (e.g OpenSees and Abaqus).
In this research, median collapse margin ratio is considered as a collapse indicator
parameter. When non-simulated limit state for one component is exceeded, it is assumed that this
triggers collapse for the entire structure. However, collapse may not occur due to the
redistribution of the loads. This matter, considering non-simulated collapse modes, leads to a
larger probability of collapse and as a result a decrease in the CMR value. To ensure that the
collapse assessment process represents the behavior of the structural system, limit state checks
should be carefully determined and collapse should be mainly simulated in the numerical model.
In addition, as recommended by FEMA P695, it is suggested that an adjusted collapse
margin ratio (ACMR) shall be defined through calculating a spectral shape factor (SSF) for each
archetype model to account for spectral shape effect.
113
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 Seismosoft Ltd. (2012). SeismoStruct User manual for Version 6. Pavia-Italy: Seismosoft
Ltd.
 Seismosoft Ltd. (2012). SeismoStruct Verification Report for Version 6. Pavia-Italy:
Seismosoft Ltd.
 Shamsabadi, A. (2007). THREE-DIMENSIONAL NONLINEAR SEISMIC
SOILABUTMENT-FOUNDATION-STRUCTURE INTERACTION ANALYSIS OF
SKEWED BRIDGES. PhD dissertation. Los Angeles, California, U.S.A: FACULTY OF
THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA.
 Thevaneyan K. David and John P. Forth. (2011). Modelling of Soil Structure Interaction
of Integral Abutment Bridges. World Academy of Science, Engineering and Technology,
pp. 769-774.
116
Appendices
Appendix A Significant Duration (5%-95% Arias Intensity) Acceleration Time History of
the Selected Ground Motions
Chi Chi Tiwan -Significant Duration (5-95% Arias Intensity)
Superstition-Significant Duration (5-95% Arias Intensity)
0.6
0.6
Superstition
0.4
0.4
0.2
0.2
Acceleration (g)
Acceleration (g)
Chi Chi Tiwan
0
-0.2
-0.4
-0.6
-0.8
0
-0.2
-0.4
-0.6
0
5
10
15
20
Time (sec)
25
-0.8
30
0
5
10
Loma Prieta-Significant Duration (5-95% Arias Intensity)
Northridge-Significant Duration (5-95% Arias Intensity)
1
1
Loma Prieta
Northridge
0.5
Acceleration (g)
Acceleration (g)
0.5
0
-0.5
-1
15
Time (sec)
0
-0.5
0
2
4
6
8
Time (sec)
10
12
-1
0
2
4
Time (sec)
6
8
Figure A-1 Significant duration of Chi Chi, Superstition, Loma Prieta, and Northridge acceleration time
histories with 5-95% arias intensity - Source: PEER Strong Motion Database
117
Imperial Valley- Significant Duration (5-95% Arias Intensity)
Victoria-Mexico- Significant Duration (5-95% Arias Intensity)
0.4
0.6
Imperial Valley
Victoria-Mexico
0.4
Acceleration (g)
Acceleration (g)
0.2
0
-0.2
0.2
0
-0.2
-0.4
-0.4
-0.6
-0.6
0
2
4
6
Time (sec)
8
-0.8
10
Morgan Hill- Significant Duration (5-95% Arias Intensity)
0
2
4
6
Time (sec)
8
10
Duzce- Significant Duration (5-95% Arias Intensity)
0.8
0.6
Morgan Hill
Duzce
0.6
Acceleration (g)
Acceleration (g)
0.4
0.4
0.2
0
-0.2
0.2
0
-0.2
-0.4
-0.6
0
1
2
3
Time (sec)
4
5
-0.4
0
5
10
15
Time (sec)
Figure A-2 Significant duration of Imperial Valley, Victoria- Mexico, Morgan Hill, and Duzce acceleration
time histories with 5-95% arias intensity - Source: PEER Strong Motion Database
118
Cape Mendocino- Significant Duration (5-95% Arias Intensity)
Mammoth Lakes- Significant Duration (5-95% Arias Intensity)
0.6
0.3
Mammoth Lakes
0.4
0.2
0.2
0.1
Acceleration (g)
Acceleration (g)
Cape Mendocino
0
-0.2
-0.4
-0.6
-0.8
0
-0.1
-0.2
-0.3
0
5
10
Time (sec)
15
-0.4
20
N.Palm Springs- Significant Duration (5-95% Arias Intensity)
0.6
0
2
4
6
Time (sec)
8
10
Tabas- Significant Duration (5-95% Arias Intensity)
0.4
N.Palm Springs
Tabas
0.3
Acceleration (g)
Acceleration (g)
0.4
0.2
0
0.2
0.1
0
-0.1
-0.2
-0.2
-0.4
0
2
4
Time (sec)
6
-0.3
0
5
10
15
Time (sec)
Figure A-3 Significant duration of Cape Mendocino, Mammoth Lakes, N.Palm Springs, and Tabas
acceleration time histories with 5-95% arias intensity - Source: PEER Strong Motion Database
119
San Fernando- Significant Duration (5-95% Arias Intensity)
Gazli-USSR- Significant Duration (5-95% Arias Intensity)
1
Gazli-USSR
1
San Fernando
Acceleration (g)
Acceleration (g)
0.5
0
-0.5
-1
-1.5
0
4
6
8
Time (sec)
Managua- Significant Duration (5-95% Arias Intensity)
0.2
0
-0.2
-0.4
-0.5
0
2
4
Time (sec)
6
8
Whittier Narrows- Significant Duration (5-95% Arias Intensity)
0.2
Whittier Narrows
0.1
Acceleration (g)
Acceleration (g)
Managua
0
-1
2
0.4
0.5
0
-0.1
-0.2
0
5
10
Time (sec)
15
-0.3
0
2
4
6
Time (sec)
8
10
Figure A-4 Significant duration of San Fernando, Gazli, Managua, and Whittier Narrows acceleration time
histories with 5-95% arias intensity - Source: PEER Strong Motion Database
120
Coalinga- Significant Duration (5-95% Arias Intensity)
Westmorland- Significant Duration (5-95% Arias Intensity)
1
0.6
Coalinga
Westmorland
Acceleration (g)
Acceleration (g)
0.4
0.5
0
0.2
0
-0.2
-0.5
0
1
2
3
Time (sec)
4
-0.4
5
0
2
4
Time (sec)
6
Kobe- Significant Duration (5-95% Arias Intensity)
Spitak- Significant Duration (5-95% Arias Intensity)
0.4
0.2
Kobe
Spitak
0.2
Acceleration (g)
Acceleration (g)
8
0
-0.2
-0.4
0
5
10
15
0.1
0
-0.1
-0.2
0
Time (sec)
5
10
15
Time (sec)
Figure A-5 Significant duration of Coalinga, Westmorland, Kobe, and Spitak acceleration time histories with
5-95% arias intensity - Source: PEER Strong Motion Database
121
Appendix B Fragility Fitting Functions for Use with Incremental Dynamic Analysis Data
of collapse at Sa level Sa i
total number of record
^


 ln Sai   ln Sa^ 

P(C | Sai )observed  1   
^


^



ln
Sa


P(C | Sai )observed 
#
(eq.B-1)
Least Square Method: Minimizing the total square errors between the estimated probability of
collapse and the observed probability of collapse over all of the Sa level:
^

 ln Sa

^
,
 ln Sa  min  ,  i P(C | Sai )observed  P(C | Sai ) pred 2



 ln Sai  μ ln Sa  
 min  ,  i  P(C | Sai )observed  1   
 
 ln Sa



2
(eq.B-2)
Maximum Likelihood Method: To account for non-constant variance, this method can be applied
(Rice, 1955):
^

 ln Sa

,

^

ln Sa


N ni 
ni


 ln Sai  μ ln Sa    ln Sai  μ ln Sa 


 max  ,  i  1   
   


 ln Sa
 ln Sa


 



(eq.B-3)
in which ni is the number of collapse observed at Sa level Sai, and N is the total number of
records analyzed at level Sai
122
Appendix C 3_D Schematic View of the Obtained IDA Results
C.1
3-D Schematic View of the Obtained IDA Results - Model M1 & M2
IDA Results Along The Bridge Deck - Model M1 & M2
4
x 10
3
2
1.5
1
0.5
6
5
4
Abut
-3
x 10
men
t Rel
1.5
ative
Disp
lacem
2
1
1
ent (
m)
0.5
0
(T
1
3
2
)[g
]
0
2.5
Sa
Total Base Shear (kN)
2.5
M1-Chi Chi
M1-Superstition
M1-Loma Prieta
M1-Northridge
M1-Imperial Valley
M1-Victoria-Mexico
M1-Morgan Hill
M1-Duzce
M1-Cape Mendocino
M1-Mammoth Lakes
M1-N.Palm Springs
M1-Tabas
M1-San Fernando
M1-Gazli
M1-Managua
M1-Whittier Narrows
M1-Coalinga
M1-Westmorland
M1-Kobe
M1-Spitak
M2-Chi Chi
M2-Superstition
M2-Loma Prieta
M2-Northridge
M2-Imperial Valley
M2-Victoria-Mexico
M2-Morgan Hill
M2-Duzce
M2-Cape Mendocino
M2-Mammoth Lakes
M2-N.Palm Springs
M2-Tabas
M2-San Fernando
M2-Gazli
M2-Managua
M2-Whittier Narrows
M2-Coalinga
M2-Westmorland
M2-Kobe
M2-Spitak
Figure C-1 Shows dispersion of the obtained IDA results for the archetype model M1 & M2
123
C.2
3-D Schematic View of the Obtained IDA Results - Model M3 & M4
IDA Results Along The Bridge Deck - Model M3 & M4
4
x 10
3
Total Base Shear (kN)
2.5
2
1.5
1
0.5
4
3
0
10
2
8
-3
x 10
Abut
6
men
t Rel
ativ
4
e Dis
1
place
2
0
m en
t ( m)
-2
0
S
]
)[g
1
T
a(
M3-Chi Chi
M3-Superstition
M3-Loma Prieta
M3-Northridge
M3-Imperial Valley
M3-Victoria-Mexico
M3-Morgan Hill
M3-Duzce
M3-Cape Mendocino
M3-Mammoth Lakes
M3-N.Palm Springs
M3-Tabas
M3-San Fernando
M3-Gazli
M3-Managua
M3-Whittier Narrows
M3-Coalinga
M3-Westmorland
M3-Kobe
M3-Spitak
M4-Chi Chi
M4-Superstition
M4-Loma Prieta
M4-Northridge
M4-Imperial Valley
M4-Victoria-Mexico
M4-Morgan Hill
M4-Duzce
M4-Cape Mendocino
M4-Mammoth Lakes
M4-N.Palm Springs
M4-Tabas
M4-San Fernando
M4-Gazli
M4-Managua
M4-Whittier Narrows
M4-Coalinga
M4-Westmorland
M4-Kobe
M4-Spitak
Figure C-2 Shows dispersion of the obtained IDA results for the archetype model M3 & M4
124
C.3
3-D Schematic View of the Obtained IDA Results - Model M5 & M6
IDA Results Along The Bridge Deck - Model M5 & M6
4
x 10
12
8
6
4
2
5
4
0
3
2.5
Abut
-3
x 10
2
2
men
1.5
t Rel
ative
1
Disp
lacem
ent ( 0.5
m)
1
0
0
(T
1
)[g
]
3
Sa
Total Base Shear (kN)
10
M5-Chi Chi
M5-Superstition
M5-Loma Prieta
M5-Northridge
M5-Imperial Valley
M5-Victoria-Mexico
M5-Morgan Hill
M5-Duzce
M5-Cape Mendocino
M5-Mammoth Lakes
M5-N.Palm Springs
M5-Tabas
M5-San Fernando
M5-Gazli
M5-Managua
M5-Whittier Narrows
M5-Coalinga
M5-Westmorland
M5-Kobe
M5-Spitak
M6-Chi Chi
M6-Superstition
M6-Loma Prieta
M6-Northridge
M6-Imperial Valley
M6-Victoria-Mexico
M6-Morgan Hill
M6-Duzce
M6-Cape Mendocino
M6-Mammoth Lakes
M6-N.Palm Springs
M6-Tabas
M6-San Fernando
M6-Gazli
M6-Managua
M6-Whittier Narrows
M6-Coalinga
M6-Westmorland
M6-Kobe
M6-Spitak
Figure C-3 Shows dispersion of the obtained IDA results for the archetype model M5 & M6
125
Appendix D
D.1
Pier Column Hysteretic Plot - Total Column Drift Ratio vs. Total Base Shear -
Model M3 & M4.
Pier Column No.1 Hysteretic Plot At Collapse Level - Imperial Valley
Pier Column No.1 Hysteretic Plot At Collapse Level - Victoria-Mexico
2
4
Model M3
Model M4
1.5
1
0.5
0
-0.5
-1
-1.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
Pier Column Drift Ratio Across The Bridge Deck (%)
Total Base Shear - Along The Bridge Deck (kN)
Total Base Shear Along The Bridge Deck (kN)
4
x 10
Pier Column No.1 Hysteretic Plot At Collapse Level - Morgan Hill
1
0
-0.5
-1
-1.5
-2
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
Pier Column Drift Ratio Across The Bridge Deck (%)
Pier Column No.1 Hysteretic Plot At Collapse Level - Duzce
4
x 10
Model M3
Model M4
1
0.5
0
-0.5
-1
-1.5
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
Pier Column Drift Ratio Across The Bridge Deck (%)
Total Base Shear Along The Bridge Deck (kN)
Total Base Shear Along The Bridge Deck (kN)
1.5
Model M3
Model M4
0.5
4
2
x 10
1
x 10
Model M3
Model M4
0.5
0
-0.5
-1
-0.15 -0.1
-0.05
0
0.05
0.1
0.15
0.2
Pier Column Drift Ratio Across The Bridge Deck (%)
Figure D.1-1 Shows pier column total drift ratio across the bridge deck verses total base shear obtained from
performed IDA using the Imperial Valley, Victoria-Mexico, Morgan Hill, and Duzce ground motions - Model
M3 & M4.
126
4
1
x 10
Model M3
Model M4
0.5
0
-0.5
-1
-1.5
-0.1
-0.05
0
0.05
0.1
Pier Column Drift Ratio Across The Bridge Deck (%)
Total Base Shear - Along The Bridge Deck (kN)
Total Base Shear Along The Bridge Deck (kN)
Pier Column No.1 Hysteretic Plot At Collapse Level - Cape Mendocino Pier Column No.1 Hysteretic Plot At Collapse Level - Mammoth Lakes
4
2
x 10
1.5
1
0.5
0
-0.5
Model M3
Model M4
-1
-0.15 -0.1 -0.05
0
0.05
0.1
0.15
0.2
Pier Column Drift Ratio Across The Bridge Deck (%)
x 10
Model M3
Model M4
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
-0.2
-0.1
0
0.1
0.2
Pier Column Drift Ratio Across The Bridge Deck (%)
Pier Column No.1 Hysteretic Plot At Collapse Level - Tabas
Total Base Shear Along The Bridge Deck (kN)
Total Base Shear Along The Bridge Deck (kN)
Pier Column No.1 Hysteretic Plot At Collapse Level - N.Palm Springs
4
1.5
4
1.5
1
x 10
Model M3
Model M4
0.5
0
-0.5
-1
-0.2 -0.15 -0.1 -0.05
0
0.05
0.1
0.15
Pier Column Drift Ratio Across The Bridge Deck (%)
Figure D.1-2 Shows pier column total drift ratio across the bridge deck verses total base shear obtained from
performed IDA using the Cape Mendocino, Mammoth Lakes, N.Palm Springs, and Tabas ground motions Model M3 & M4.
127
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-0.3
Model M3
Model M4
-0.2
-0.1
0
0.1
0.2
0.3
0.4
Pier Column Drift Ratio Across The Bridge Deck (%)
Total Base Shear Along The Bridge Deck (kN)
Pier Column No.1 Hysteretic Plot At Collapse Level - Managua
8000
6000
4000
2000
0
-2000
-4000
-6000
-8000
-0.2
Model M3
Model M4
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
Pier Column Drift Ratio Across The Bridge Deck (%)
Pier Column No.1 Hysteretic Plot At Collapse Level - Gazli
Total Base Shear - Along The Bridge Deck (kN)
4
x 10
4
2
x 10
1.5
1
0.5
0
-0.5
Model M3
Model M4
-1
-1.5
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
Pier Column Drift Ratio Across The Bridge Deck (%)
Pier Column No.1 Hysteretic Plot At Collapse Level - Whittier Narrows
Total Base Shear Along The Bridge Deck (kN)
Total Base Shear Along The Bridge Deck (kN)
Pier Column No.1 Hysteretic Plot At Collapse Level - San Fernando
4
1.5
x 10
1
0.5
0
-0.5
-1
-1.5
-2
-0.4
Model M3
Model M4
-0.2
0
0.2
0.4
Pier Column Drift Ratio Across The Bridge Deck (%)
Figure D.1-3 Shows pier column total drift ratio across the bridge deck verses total base shear obtained from
performed IDA using the San Fernando, Gazli, Managua, and Whittier Narrows ground motions - Model M3
& M4.
128
4
2
x 10
Model M3
Model M4
1.5
1
0.5
0
-0.5
-1
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
Pier Column Drift Ratio Across The Bridge Deck (%)
Pier Column No.1 Hysteretic Plot At Collapse Level - Westmorland
Total Base Shear - Along The Bridge Deck (kN)
Total Base Shear Along The Bridge Deck (kN)
Pier Column No.1 Hysteretic Plot At Collapse Level - Coalinga
4
1
x 10
Model M3
Model M4
0.5
0
-0.5
-1
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
Pier Column Drift Ratio Across The Bridge Deck (%)
Total Base Shear Along The Bridge Deck (kN)
Total Base Shear Along The Bridge Deck (kN)
Pier Column No.1 Hysteretic Plot At Collapse Level - Kobe
4
2
1.5
x 10
Model M3
Model M4
1
0.5
0
-0.5
-1
-1.5
-0.5
0
0.5
Pier Column Drift Ratio Across The Bridge Deck (%)
Pier Column No.1 Hysteretic Plot At Collapse Level - Spitak
4
x 10
1
Model M3
Model M4
0.5
0
-0.5
-1
-1.5
-0.3
-0.2
-0.1
0
0.1
0.2
Pier Column Drift Ratio Across The Bridge Deck (%)
Figure D.1-4 Shows pier column total drift ratio across the bridge deck verses total base shear obtained from
performed IDA using the Coalinga, Westmorland, Kobe, and Spitak ground motions - Model M3 & M4
129
D.2
Pier Column Hysteretic Plot - Column's Actual Drift Ratio vs. Total Base Shear -
Model M3 & M4.
4
3
x 10
Model M3
Model M4
2
1
0
-1
-2
-3
-0.25 -0.2
-0.15 -0.1
-0.05
0
0.05
0.1
Pier Column Drift Ratio Across The Bridge Deck (%)
Pier Column No.1 Hysteretic Plot At Collapse Level - Victoria-Mexico
Total Base Shear - Along The Bridge Deck (kN)
Total Base Shear Along The Bridge Deck (kN)
Pier Column No.1 Hysteretic Plot At Collapse Level - Imperial Valley
4
3
x 10
2
1
0
-1
-2
Model M3
Model M4
-3
-0.2
-0.15 -0.1
-0.05
0
0.05
0.1
0.15
Pier Column Drift Ratio Across The Bridge Deck (%)
Total Base Shear Along The Bridge Deck (kN)
Total Base Shear Along The Bridge Deck (kN)
Pier Column No.1 Hysteretic Plot At Collapse Level - Morgan Hill
4
1
x 10
0.5
0
-0.5
-1
-1.5
Model M3
Model M4
-2
-0.2
-0.15 -0.1
-0.05
0
0.05
0.1
0.15
Pier Column Drift Ratio Across The Bridge Deck (%)
Pier Column No.1 Hysteretic Plot At Collapse Level - Duzce
4
x 10
1.5
1
0.5
0
-0.5
-1
-1.5
Model M3
Model M4
-2
-0.1
-0.05
0
0.05
0.1
0.15
Pier Column Drift Ratio Across The Bridge Deck (%)
Figure D.2-1 Shows pier column actual drift ratio across the bridge deck verses total base shear obtained
from performed IDA using the Imperial Valley, Victoria-Mexico, Morgan Hill, and Duzce ground motions Model M3 & M4.
130
2
1.5
Model M3
Model M4
1
0.5
0
-0.5
-1
-1.5
-0.1
-0.05
0
0.05
0.1
Pier Column Drift Ratio Across The Bridge Deck (%)
Total Base Shear Along The Bridge Deck (kN)
Pier Column No.1 Hysteretic Plot At Collapse Level - N.Palm Springs
4
x 10
4
3
Model M3
Model M4
2
1
0
-1
-2
-0.15
-0.1
-0.05
0
0.05
Pier Column Drift Ratio Across The Bridge Deck (%)
Pier Column No.1 Hysteretic Plot At Collapse Level - Mammoth Lakes
Total Base Shear - Along The Bridge Deck (kN)
4
x 10
Total Base Shear Along The Bridge Deck (kN)
Total Base Shear Along The Bridge Deck (kN)
Pier Column No.1 Hysteretic Plot At Collapse Level - Cape Mendocino
4
1.5
x 10
1
0.5
0
-0.5
-1
-1.5
Model M3
Model M4
-2
-2.5
-0.02
-0.01
0
0.01
0.02
0.03
Pier Column Drift Ratio Across The Bridge Deck (%)
Pier Column No.1 Hysteretic Plot At Collapse Level - Tabas
4
x 10
2.5
2
1.5
Model M3
Model M4
1
0.5
0
-0.5
-1
-1.5
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
Pier Column Drift Ratio Across The Bridge Deck (%)
Figure D.2-2 Shows pier column actual drift ratio across the bridge deck verses total base shear obtained
from performed IDA using the Cape Mendocino, Mammoth Lakes, N.Palm Springs, and Tabas ground
motions - Model M3 & M4.
131
3
Model M3
Model M4
2
1
0
-1
-2
-3
-0.1
-0.05
0
0.05
0.1
Pier Column Drift Ratio Across The Bridge Deck (%)
Total Base Shear Along The Bridge Deck (kN)
Pier Column No.1 Hysteretic Plot At Collapse Level - Managua
4
1.5
x 10
1
Model M3
Model M4
0.5
0
-0.5
-1
-1.5
-0.1
-0.05
0
0.05
0.1
Pier Column Drift Ratio Across The Bridge Deck (%)
Pier Column No.1 Hysteretic Plot At Collapse Level - Gazli
Total Base Shear - Along The Bridge Deck (kN)
4
x 10
4
3
2
x 10
Model M3
Model M4
1
0
-1
-2
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
Pier Column Drift Ratio Across The Bridge Deck (%)
Pier Column No.1 Hysteretic Plot At Collapse Level - Whittier Narrows
4
x 10
3
Model M3
2
Model M4
Total Base Shear Along The Bridge Deck (kN)
Total Base Shear Along The Bridge Deck (kN)
Pier Column No.1 Hysteretic Plot At Collapse Level - San Fernando
1
0
-1
-2
-3
-4
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
Pier Column Drift Ratio Across The Bridge Deck (%)
Figure D.2-3 Shows pier column actual drift ratio across the bridge deck verses total base shear obtained
from performed IDA using the San Fernando, Gazli, Managua, and Whittier Narrows ground motions Model M3 & M4.
132
4
4
x 10
Model M3
Model M4
3
2
1
0
-1
-2
-0.2 -0.15 -0.1 -0.05
0
0.05
0.1
0.15
Pier Column Drift Ratio Across The Bridge Deck (%)
Pier Column No.1 Hysteretic Plot At Collapse Level - Westmorland
4
x 10
3
Model M3
2
Model M4
Total Base Shear - Along The Bridge Deck (kN)
Total Base Shear Along The Bridge Deck (kN)
Pier Column No.1 Hysteretic Plot At Collapse Level - Coalinga
1.5
1
Model M3
Model M4
0.5
0
-0.5
-1
-1.5
-0.1
-0.05
0
0.05
0.1
0.15
Pier Column Drift Ratio Across The Bridge Deck (%)
Total Base Shear Along The Bridge Deck (kN)
Total Base Shear Along The Bridge Deck (kN)
4
0
-1
-2
-3
-0.2 -0.15 -0.1 -0.05
0
0.05
0.1
0.15
Pier Column Drift Ratio Across The Bridge Deck (%)
Pier Column No.1 Hysteretic Plot At Collapse Level - Spitak
Pier Column No.1 Hysteretic Plot At Collapse Level - Kobe
x 10
1
4
2
1.5
x 10
Model M3
Model M4
1
0.5
0
-0.5
-1
-1.5
-2
-0.1
-0.05
0
0.05
0.1
Pier Column Drift Ratio Across The Bridge Deck (%)
Figure D.2-4 Shows pier column actual drift ratio across the bridge deck verses total base shear obtained
from performed IDA using the Coalinga, Westmorland, Kobe, and Spitak ground motions - Model M3 & M4
133
D.3
Pier Column Hysteretic Plot - Column's Actual Drift Ratio vs. Total Base Shear -
Model M5 & M6.
4
4
x 10
3
2
1
0
-1
-2
Model M5
Model M6
-3
-4
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Pier Column Drift Ratio Across The Bridge Deck (%)
Pier Column No.1 Hysteretic Plot At Collapse Level - Victoria-Mexico
Total Base Shear - Along The Bridge Deck (kN)
Total Base Shear Along The Bridge Deck (kN)
Pier Column No.1 Hysteretic Plot At Collapse Level - Imperial Valley
4
4
x 10
3
2
1
0
-1
-2
Model M5
Model M6
-3
-0.5
0
0.5
1
Pier Column Drift Ratio Across The Bridge Deck (%)
Total Base Shear Along The Bridge Deck (kN)
Total Base Shear Along The Bridge Deck (kN)
Pier Column No.1 Hysteretic Plot At Collapse Level - Morgan Hill
4
2
1
x 10
Model M5
Model M6
0
-1
-2
-3
-4
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
Pier Column Drift Ratio Across The Bridge Deck (%)
Pier Column No.1 Hysteretic Plot At Collapse Level - Duzce
4
x 10
3
2
1
0
-1
-2
-3
Model M5
Model M6
-4
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
Pier Column Drift Ratio Across The Bridge Deck (%)
Figure D.3-1 Shows pier column actual drift ratio across the bridge deck verses total base shear obtained
from performed IDA using the Imperial Valley, Victoria-Mexico, Morgan Hill, and Duzce ground motions Model M5 & M6.
134
Pier Column No.1 Hysteretic Plot At Collapse Level - Mammoth Lakes
4
3
Total Base Shear - Along The Bridge Deck (kN)
Total Base Shear Along The Bridge Deck (kN)
Pier Column No.1 Hysteretic Plot At Collapse Level - Cape Mendocino
x 10
Model M5
Model M6
2
1
0
-1
-2
-0.4
-0.2
0
0.2
0.4
Pier Column Drift Ratio Across The Bridge Deck (%)
4
4
x 10
3
2
1
0
-1
-2
-3
Model M5
Model M6
-4
-1
-0.5
0
0.5
1
Pier Column Drift Ratio Across The Bridge Deck (%)
Total Base Shear Along The Bridge Deck (kN)
Total Base Shear Along The Bridge Deck (kN)
Pier Column No.1 Hysteretic Plot At Collapse Level - N.Palm Springs
4
4
x 10
3
2
1
0
-1
-2
Model M5
Model M6
-3
-4
-0.4
-0.2
0
0.2
0.4
Pier Column Drift Ratio Across The Bridge Deck (%)
Pier Column No.1 Hysteretic Plot At Collapse Level - Tabas
4
1.5
1
x 10
Model M5
Model M6
0.5
0
-0.5
-1
-1.5
-0.15
-0.1
-0.05
0
0.05
0.1
Pier Column Drift Ratio Across The Bridge Deck (%)
Figure D.3-2 Shows pier column actual drift ratio across the bridge deck verses total base shear obtained
from performed IDA using the Cape Mendocino, Mammoth Lakes, N.Palm Springs, and Tabas ground
motions - Model M5 & M6.
135
4
3
2
1
0
-1
-2
Model M5
Model M6
-3
-4
-0.4
-0.2
0
0.2
0.4
0.6
Pier Column Drift Ratio Across The Bridge Deck (%)
Total Base Shear Along The Bridge Deck (kN)
Pier Column No.1 Hysteretic Plot At Collapse Level - Managua
4
4
x 10
3
Model M5
Model M6
2
1
0
-1
-2
-3
-4
-0.4
-0.2
0
0.2
0.4
Pier Column Drift Ratio Across The Bridge Deck (%)
Pier Column No.1 Hysteretic Plot At Collapse Level - Gazli
Total Base Shear - Along The Bridge Deck (kN)
4
x 10
4
3
x 10
2
1
0
-1
-2
Model M5
Model M6
-3
-0.5
0
0.5
Pier Column Drift Ratio Across The Bridge Deck (%)
Pier Column No.1 Hysteretic Plot At Collapse Level - Whittier Narrows
Total Base Shear Along The Bridge Deck (kN)
Total Base Shear Along The Bridge Deck (kN)
Pier Column No.1 Hysteretic Plot At Collapse Level - San Fernando
4
3
2
x 10
Model M5
Model M6
1
0
-1
-2
-3
-4
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Pier Column Drift Ratio Across The Bridge Deck (%)
Figure D.3-3 Shows pier column actual drift ratio across the bridge deck verses total base shear obtained
from performed IDA using the San Fernando, Gazli, Managua, and Whittier Narrows ground motions Model M5 & M6.
136
6
4
2
0
-2
Model M5
Model M6
-4
-1
-0.5
0
0.5
1
Pier Column Drift Ratio Across The Bridge Deck (%)
Total Base Shear Along The Bridge Deck (kN)
Pier Column No.1 Hysteretic Plot At Collapse Level - Kobe
4
4
x 10
3
2
1
0
-1
-2
-3
Model M5
Model M6
-4
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Pier Column Drift Ratio Across The Bridge Deck (%)
Pier Column No.1 Hysteretic Plot At Collapse Level - Westmorland
Total Base Shear - Along The Bridge Deck (kN)
4
x 10
4
3
x 10
2
1
0
-1
-2
Model M5
Model M6
-3
-1
-0.5
0
0.5
1
Pier Column Drift Ratio Across The Bridge Deck (%)
Pier Column No.1 Hysteretic Plot At Collapse Level - Spitak
Total Base Shear Along The Bridge Deck (kN)
Total Base Shear Along The Bridge Deck (kN)
Pier Column No.1 Hysteretic Plot At Collapse Level - Coalinga
4
3
x 10
2
1
0
-1
-2
-3
-4
Model M5
Model M6
-5
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
Pier Column Drift Ratio Across The Bridge Deck (%)
Figure D.3-4 Shows pier column actual drift ratio across the bridge deck verses total base shear obtained
from performed IDA using the Coalinga, Westmorland, Kobe, and Spitak ground motions - Model M5 & M6.
137
Appendix E
E.1
Pier Column Hysteretic Plot - Column's Actual Drift Ratio vs. Base Moment -
Model M3 & M4.
Pier Column No.1 Hysteretic Plot At Collapse Level - Imperial Valley
Pier Column No.1 Hysteretic Plot At Collapse Level - Victoria-Mexico
1500
2000
Total Moment - About Y axis (kN.m)
Total Moment - About Y axis (kN.m)
2000
Model M3
Model M4
1000
500
0
-500
-1000
-1500
-2000
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
Pier Column Drift Ratio - X Direction (%)
Pier Column No.1 Hysteretic Plot At Collapse Level - Morgan Hill
0
-500
Model M3
Model M4
-1000
-0.05
0
0.05
0.1
0.15
0.2
Pier Column Drift Ratio - X Direction (%)
0.25
Pier Column No.1 Hysteretic Plot At Collapse Level - Duzce
Total Moment - About Y axis (kN.m)
Total Moment - About Y axis (kN.m)
500
1500
Model M3
Model M4
1000
500
0
-500
-1000
-1500
-2000
-0.2
1000
-1500
-0.1
0.1
2000
1500
1500
-0.15
-0.1
-0.05
0
0.05
0.1
Pier Column Drift Ratio - X Direction (%)
0.15
1000
Model M3
Model M4
500
0
-500
-1000
-1500
-0.15
-0.1
-0.05
0
0.05
Pier Column Drift Ratio - X Direction (%)
0.1
Figure E.1-1 Shows pier column actual drift ratio across the bridge deck verses total moment obtained from
performed IDA using the Imperial Valley, Victoria-Mexico, Morgan Hill, and Duzce ground motions - Model
M3 & M4.
138
Pier Column No.1 Hysteretic Plot At Collapse Level - Cape Mendocino
2000
Total Moment - About Y axis (kN.m)
Total Moment - About Y axis (kN.m)
1500
Model M3
Model M4
1000
500
0
-500
-1000
-1500
-0.06
-0.04
-0.02
0
0.02
0.04
Pier Column Drift Ratio - X Direction (%)
Model M3
Model M4
1000
500
0
-500
-1000
-0.15 -0.1 -0.05
0
0.05
0.1
Pier Column Drift Ratio - X Direction (%)
0.15
Pier Column No.1 Hysteretic Plot At Collapse Level - Tabas
1500
Total Moment - About Y axis (kN.m)
1500
Total Moment - About Y axis (kN.m)
1500
-1500
-0.2
0.06
Pier Column No.1 Hysteretic Plot At Collapse Level - N.Palm Springs
Model M3
Model M4
1000
500
0
-500
-1000
-1500
-0.08
Pier Column No.1 Hysteretic Plot At Collapse Level - Mammoth Lakes
-0.06 -0.04 -0.02
0
0.02 0.04
Pier Column Drift Ratio - X Direction (%)
0.06
1000
Model M3
Model M4
500
0
-500
-1000
-1500
-0.06
-0.04
-0.02
0
0.02
Pier Column Drift Ratio - X Direction (%)
0.04
Figure E.1-2 Shows pier column actual drift ratio across the bridge deck verses total moment obtained from
performed IDA using the Cape Mendocino, Mammoth Lakes, N.Palm Springs, and Tabas ground motions Model M3 & M4.
139
Pier Column No.1 Hysteretic Plot At Collapse Level - San Fernando
Pier Column No.1 Hysteretic Plot At Collapse Level - Gazli
1500
2000
Total Moment - About Y axis (kN.m)
Total Moment - About Y axis (kN.m)
2000
Model M3
Model M4
1000
500
0
-500
-1000
-1500
-0.1
-0.08 -0.06 -0.04 -0.02
0
0.02
Pier Column Drift Ratio - X Direction (%)
Pier Column No.1 Hysteretic Plot At Collapse Level - Managua
0
-500
-1000
-1500
-0.2
-0.1
0
0.1
Pier Column Drift Ratio - X Direction (%)
0.2
Pier Column No.1 Hysteretic Plot At Collapse Level - Whittier Narrows
Total Moment - About Y axis (kN.m)
Total Moment - About Y axis (kN.m)
500
2000
Model M3
Model M4
500
0
-500
-1000
-1500
-0.1
Model M3
Model M4
1000
-2000
-0.3
0.04
1500
1000
1500
-0.05
0
0.05
Pier Column Drift Ratio - X Direction (%)
0.1
1500
1000
500
0
-500
-1000
-1500
-0.2
Model M3
Model M4
-0.1
0
0.1
0.2
Pier Column Drift Ratio - X Direction (%)
0.3
Figure E.1-3 Shows pier column actual drift ratio across the bridge deck verses total moment obtained from
performed IDA using the San Fernando, Gazli, Managua, and Whittier Narrows ground motions - Model M3
& M4.
140
Pier Column No.1 Hysteretic Plot At Collapse Level - Coalinga
Pier Column No.1 Hysteretic Plot At Collapse Level - Westmorland
1000
2000
Total Moment - About Y axis (kN.m)
Total Moment - About Y axis (kN.m)
1500
Model M3
Model M4
500
0
-500
-1000
-1500
-2000
-0.15
-0.1
-0.05
0
0.05
Pier Column Drift Ratio - X Direction (%)
0
-500
-1000
-0.15 -0.1 -0.05
0
0.05
0.1
Pier Column Drift Ratio - X Direction (%)
0.15
2000
Total Moment - About Y axis (kN.m)
Total Moment - About Y axis (kN.m)
500
Pier Column No.1 Hysteretic Plot At Collapse Level - Spitak
Pier Column No.1 Hysteretic Plot At Collapse Level - Kobe
Model M3
Model M4
1000
500
0
-500
-1000
-1500
-0.15
Model M3
Model M4
1000
-1500
-0.2
0.1
2000
1500
1500
-0.1
-0.05
0
0.05
Pier Column Drift Ratio - X Direction (%)
0.1
1500
Model M3
Model M4
1000
500
0
-500
-1000
-1500
-0.1
-0.05
0
0.05
Pier Column Drift Ratio - X Direction (%)
0.1
Figure E.1-4 Shows pier column actual drift ratio across the bridge deck verses total moment obtained from
performed IDA using the Coalinga, Westmorland, Kobe, and Spitak ground motions - Model M3 & M4.
141
E.2
Pier Column Hysteretic Plot - Column's Actual Drift Ratio vs. Base Moment) -
Model M5 & M6.
Pier Column No.1 Hysteretic Plot At Collapse Level - Imperial Valley
Pier Column No.1 Hysteretic Plot At Collapse Level - Victoria-Mexico
1500
Total Moment - About Y axis (kN.m)
Total Moment - About Y axis (kN.m)
2000
1500
1000
500
0
-500
Model M5
Model M6
-1000
-1500
-1
-0.5
0
0.5
Pier Column Drift Ratio - X Direction (%)
Model M5
Model M6
-500
-0.1
0
0.1
0.2
0.3
0.4
Pier Column Drift Ratio - X Direction (%)
0.5
1500
Total Moment - About Y axis (kN.m)
Total Moment - About Y axis (kN.m)
0
Pier Column No.1 Hysteretic Plot At Collapse Level - Duzce
Pier Column No.1 Hysteretic Plot At Collapse Level - Morgan Hill
Model M5
Model M6
500
0
-500
-1000
-1500
-1
500
-1000
-0.2
1
1500
1000
1000
-0.8
-0.6
-0.4
-0.2
0
0.2
Pier Column Drift Ratio - X Direction (%)
0.4
1000
Model M5
Model M6
500
0
-500
-1000
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
Pier Column Drift Ratio - X Direction (%)
0.3
Figure E.2-1 Shows pier column actual drift ratio across the bridge deck verses total moment obtained from
performed IDA using the Imperial Valley, Victoria-Mexico, Morgan Hill, and Duzce ground motions - Model
M5 & M6.
142
Pier Column No.1 Hysteretic Plot At Collapse Level - Cape Mendocino
Pier Column No.1 Hysteretic Plot At Collapse Level - Mammoth Lakes
1500
1000
Total Moment - About Y axis (kN.m)
Total Moment - About Y axis (kN.m)
1500
Model M5
Model M6
500
0
-500
-1000
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
Pier Column Drift Ratio - X Direction (%)
-500
-1000
-0.3
-0.2
-0.1
0
0.1
0.2
Pier Column Drift Ratio - X Direction (%)
0.3
1000
Total Moment - About Y axis (kN.m)
Total Moment - About Y axis (kN.m)
0
Pier Column No.1 Hysteretic Plot At Collapse Level - Tabas
2000
Model M5
Model M6
1000
500
0
-500
-1000
-1500
-0.6
Model M5
Model M6
500
-1500
-0.4
0.3
Pier Column No.1 Hysteretic Plot At Collapse Level - N.Palm Springs
1500
1000
-0.4
-0.2
0
0.2
0.4
0.6
Pier Column Drift Ratio - X Direction (%)
0.8
Model M5
Model M6
500
0
-500
-1000
-0.1
-0.05
0
0.05
Pier Column Drift Ratio - X Direction (%)
0.1
Figure E.2-2 Shows pier column actual drift ratio across the bridge deck verses total moment obtained from
performed IDA using the Cape Mendocino, Mammoth Lakes, N.Palm Springs, and Tabas ground motions Model M5 & M6.
143
1500
1000
Model M5
Model M6
500
0
-500
-1000
-1500
-0.6
-0.4
-0.2
0
0.2
Pier Column Drift Ratio - X Direction (%)
Pier Column No.1 Hysteretic Plot At Collapse Level - Gazli
Total Moment - About Y axis (kN.m)
Total Moment - About Y axis (kN.m)
Pier Column No.1 Hysteretic Plot At Collapse Level - San Fernando
0.4
Total Moment - About Y axis (kN.m)
Total Moment - About Y axis (kN.m)
1000
Model M5
Model M6
500
0
-500
-1000
-1500
-0.4
-0.2
0
0.2
Pier Column Drift Ratio - X Direction (%)
0.4
1000
Model M5
Model M6
500
0
-500
-1000
-1500
-0.5
0
Pier Column Drift Ratio - X Direction (%)
0.5
Pier Column No.1 Hysteretic Plot At Collapse Level - Whittier Narrows
Pier Column No.1 Hysteretic Plot At Collapse Level - Managua
1500
1500
2000
1500
Model M5
Model M6
1000
500
0
-500
-1000
-1500
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
Pier Column Drift Ratio - X Direction (%)
0.6
Figure E.2-3 Shows pier column actual drift ratio across the bridge deck verses total moment obtained from
performed IDA using the San Fernando, Gazli, Managua, and Whittier Narrows ground motions - Model M5
& M6.
144
Pier Column No.1 Hysteretic Plot At Collapse Level - Coalinga
Pier Column No.1 Hysteretic Plot At Collapse Level - Westmorland
1500
2000
Total Moment - About Y axis (kN.m)
Total Moment - About Y axis (kN.m)
2000
Model M5
Model M6
1000
500
0
-500
-1000
-1500
-1
-0.5
0
0.5
Pier Column Drift Ratio - X Direction (%)
0
-500
-1000
-0.5
0
0.5
Pier Column Drift Ratio - X Direction (%)
1
1500
Total Moment - About Y axis (kN.m)
Total Moment - About Y axis (kN.m)
500
Pier Column No.1 Hysteretic Plot At Collapse Level - Spitak
1500
Model M5
Model M6
500
0
-500
-1000
-1500
-0.5
Model M5
Model M6
1000
-1500
-1
1
Pier Column No.1 Hysteretic Plot At Collapse Level - Kobe
1000
1500
0
Pier Column Drift Ratio - X Direction (%)
0.5
1000
Model M5
Model M6
500
0
-500
-1000
-1500
-0.4
-0.2
0
0.2
Pier Column Drift Ratio - X Direction (%)
0.4
Figure E.2-4 Shows pier column actual drift ratio across the bridge deck verses total moment obtained from
performed IDA using the Coalinga, Westmorland, Kobe, and Spitak ground motions - Model M5 & M6.
145
Appendix F Bridge Drawings
F.1
Bridge Type 1
Figure F.1-1 Abutment layout plan - Br. Type 1
146
Figure F.1-2 Abutment plan and side view - Br. Type 1
147
Figure F.1-3 Abutment sectional elevation and RC detail - Br. Type 1
148
Figure F.1-4 Deck-girders general arrangement and deck-girder and abutment connection detail - Br. Type 1
149
F.2
Bridge Type 2
Figure F.2-1 Bridge sectional view and typical abutment and pier pile layouts - Br. Type 2
150
Figure F.2-2 Abutment layout plan, front and side elevation views - Br. Type 2
151
Figure F.2-3 Abutment RC detail and pier pilecap and columns general arrangement - Br. Type 2
152
Figure F.2-4 Pier pilecap and columns RC detail - Br. Type 2
153
F.3
Bridge Type 3
Figure F.3-1 Bridge foundations and side elevation views - Br. Type 3
154
Figure F.3-2 Abutment shear key, layout plan, and front and side elevation views and detail of bridge
intermediate diaphragms - Br. Type 3
155
Figure F.3-3 Pier pilecap and columns concrete outline detail - Br. Type 3
156
Figure F.3-4 Pier pilecap and columns RC detail - Br. Type 3
157
Figure F.3-5 Abutment RC detail and end diaphragm detail - Br. Type 3
158
Appendix G Site Classification for Seismic Site Response - NBCC2010 (Table 4.1.8.4.A)
Table G-1 Site classification for seismic site response based on top 30 meter soil average properties - Source:
NBCC 2010 (Table 4.1.8.4.A)
Other soils include:
a) Liquefiable soils, quick and highly sensitive clays, collapsible weakly cemented soils, and
other soils susceptible to failure or collapse under seismic loading.
b) Peat and/or highly organic clays greater than 3 m in thickness.
c) Highly plastic clays (PI > 75) with thickness greater than 8 m.
d) Soft to medium stiff clays with thickness greater than 30 m.
159
Appendix H Deaggregation Charts - Soil Site Class C (
H.1
)
Period 0.284 sec and Amplitude 0.86g
Figure H.1-1 Mean Epsilon and Magnitude-Distance Deaggregation spectral responses @ 5% damping for
the period 0.284 sec and amplitude 0.86g and soil class C- Source: EZ-FRISK
160
H.2
Period 0.335 sec and Amplitude 0.798g
Figure H.2-1 Mean Epsilon and Magnitude-Distance Deaggregation spectral responses @ 5% damping for
the period 0.335sec and amplitude 0.8g and soil class C- Source: EZ-FRISK
161
H.3
Period 0.374 sec and Amplitude 0.757g
Figure H.3-1 Mean Epsilon and Magnitude-Distance Deaggregation spectral responses @ 5% damping for
the period 0.374sec and amplitude 0.757g and soil class C- Source: EZ-FRISK
162