Final Report CIE619

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REPORT OF EXAMPLE 5 BRIDGE PLACED
ON A SITE IN TACOMA
Eight-Span Continuous Steel Girder Curved
Bridge
Group 6
Course: CIE 619
Structural Dynamics and Earthquake Engineering II
Report Prepared by:
Lemuria Pathfinders
Supratik Bose
Sathvika Meenakshisundaram
Sharath Chandra Ranganath
Sandhya Ravindran
Amy Ruby

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ACKNOWLEDGEMENTS
Lemuria Pathfinders would like to acknowledge that this seismic bridge design
has been adapted from design example 5 in the US Department of
Transportation Federal Highway Administrations Seismic Design of Bridges,
from October 1996. In addition, the original nine span viaduct steel girder bridge
was prepared by BERGER/ABAM Engineers Inc.
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TABLE OF CONTENTS
ACKNOWLEDGMENTS ................................................................................................. 2
LIST OF TABLES ................................................................................................................. 4
LIST OF FIGURES .................................... .......................................................... 5
CHAPTER
1. UNIFORM LOAD METHOD – ELASTIC ANALYSIS............................................. 12
1.1 General Description of Bridge ............................................................................... 12
1.1.1 Structural System................................................................................................. 13
1.1.2 Superstructure...................................................................................................... 14
1.1.3 Substructure ......................................................................................................... 15
1.1.4 Location of Bridge................................................................................................ 17
1.1.5 Site Conditions..................................................................................................... 18
1.2 Objectives.................................................................................................................. 18
1.3 Modeling Description ............................................................................................. 19
1.3.1 Superstructure...................................................................................................... 19
1.3.2 Substructure ......................................................................................................... 22
1.4 Initial Elastic Analysis............................................................................................. 27
1.4.1 Uniform Load Method........................................................................................ 27
1.4.2 Results and Discussions...................................................................................... 28
1.5 Summary and Conclusions.................................................................................... 37
2. MODAL ANALYSIS, DEVELOPMENT OF RESPONSE SPECTRA AND
SCALING OF GROUND MOTIONS........................................................................... 39
2.1 Introduction.............................................................................................................. 39
2.2 Eigen Value Analysis .............................................................................................. 39
2.2.1 Natural Periods and Mode Shapes of Structure.............................................. 42
2.2.2 Higher Modes associated with Vibration of Piers .......................................... 44
2.2.3 Comparison with Elastic Analysis Results in SAP 2000 ................................ 45
2.2.4 Analytical Calculations of Bridge Stiffness along local directions ............... 47
2.2.5 Analytical Calculations of Bridge Stiffness along global directions............. 52
2.3 Response Spectra..................................................................................................... 54
2.3.1 Seismic Design Spectra ....................................................................................... 55
2.3.2 Seismic Design Spectra of our Site .................................................................... 56

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2.3.3 Ground Motion Selection ................................................................................... 58
2.3.4 Development of Response Spectra and Scaling of Ground Motions ........... 58
2.4 Development of SDoF Model ................................................................................ 60
2.4.1 Modeling Assumptions....................................................................................... 61
2.4.2 Analysis Procedure.............................................................................................. 62
2.5 Summary and Conclusions.................................................................................... 64
3. UNIFORM LOAD, DYNAMIC MULTIMODE AND PUSHOVER ANALYSIS . 65
3.1 General Overview ................................................................................................... 65
3.2 Uniform Load Method............................................................................................ 65
3.2.1 Introduction.......................................................................................................... 65
3.2.4 Summary............................................................................................................... 72
3.3 Dynamic Multi-Mode Analysis............................................................................. 72
3.3.1 Introduction.......................................................................................................... 72
3.4 Push-Over Analysis................................................................................................. 81
3.4.1 Introduction.......................................................................................................... 81
3.4.2 Description of Model........................................................................................... 82
3.4.3 Plastic Hinge Model ............................................................................................ 84
3.4.4 Non-linear models for pushover analysis........................................................ 84
3.4.7 Comparison of stiffness with analytical results ............................................ 101
3.5 Summary and Conclusions.................................................................................. 101
4. TIME HISTORY ANALYSIS........................................................................................ 103
4.1 General Overview ................................................................................................. 103
4.2 Selected Ground Motions..................................................................................... 103
4.3 Linear Elastic Time History Analysis ................................................................. 104
4.3.1 Analysis Procedure............................................................................................ 104
4.3.2 Results 106
4.3.3 Summary............................................................................................................. 108
4.4 Non Linear Dynamic Time History Analysis.................................................... 109
4.4.1 Introduction........................................................................................................ 109
4.4.2 Code Specification ............................................................................................. 109
4.5 Non Linear SDoF Time History Analysis .......................................................... 109
4.5.2 Results 110
4.5.3 Summary............................................................................................................. 113
4.6 Non Linear MDoF Time History Analysis......................................................... 113

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4.6.1 Introduction........................................................................................................ 113
4.6.2 Description of Model......................................................................................... 113
4.6.4 Results and Discussions.................................................................................... 117
5. CAPACITY SPECTRUM AND FLOWCHARTS...................................................... 124
5.2 Capacity Spectrum Analysis................................................................................ 124
5.3 Flowcharts .............................................................................................................. 127
6. FINAL CONCLUSIONS................................................................................................ 133
6.2 Comparison from Various Analysis Procedure................................................ 133
6.3 Performance of Structure...................................................................................... 135
6.4 Scope of Future work............................................................................................ 138
6.5 Recommendations for Improvement of Performance...................................... 138
7. APPENDIX A – VALIDATION OF MODEL ........................................................ 13340
Validation of Elastic Analysis in SAP 2000 ................................................................. 139
Validation of spring stiffness in SAP 2000 .................................................................. 141
Validation of equivalent concrete rectangular section in SAP 2000 ........................ 143
Calibration of Eigen Value Analysis in SAP 2000...................................................... 147
Validation of USGS Ground Motion Information and Response Spectra .............. 150
Calibration of SDoF Model in NONLIN Program..................................................... 151
Calibration of the Program used for Response Spectrum Development................ 153
Validation of Pushover Analysis and Fiber PMM hinge in SAP 2000 .................... 160
Validation of Time History Analysis .......................................................................... 1647
8. APPENDIX B – TEAM MANAGEMENT PLAN.................................................. 13368
REFERENCES …....................................................................................................................178
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LIST OF TABLES
Table 1.1 Deflection, moment and shear force along the spans under gravity load 30
Table 1.2 Variation of axial forces in superstructure under gravity load ..................30
Table 1.3 Deflection, moment and shear force along the spans under transverse
load......................................................................................................................33
Table 1.4 Maximum resultant forces along piers under transverse load...................34
Table 1.5 Maximum resultant forces along piers under longitudinal load on deck 35
Table 2.1 Natural periods and cumulative mass participation of different modes..41
Table 2.2 Modal mass participation of first three modes.............................................42
Table 2.3 Comparison of periods of the modified bridge and the FHWA original
bridge..................................................................................................................44
Table 2.4 Calculation of period of bridge from uniform load method.......................47
Table 2.5 Deflected shape corresponding to 2nd mode (Transverse) ..........................48
Table 2.6 Calculation of overall transverse stiffness analytically................................48
Table 2.7 Calculation of overall transverse stiffness analytically................................51
Table 2.8 Comparison of stiffness of the piers obtained analytically and in SAP 2000
..............................................................................................................................52
Table 2.9 Calculation of overall stiffness analytically along global direction ...........53
Table 2.10 Stiffness and mass used in the development of the SDoF model...............54
Table 2.11 Response Spectra parameters obtained from USGS ....................................56
Table 2.12 Scaled ground motions selected from PEER Database................................60
Table 2.13 Results of time history analysis in NONLIN using elastic linear SDoF
models along local directions (longitudinal and transverse)......................63
Table 2.14 Results of time history analysis in NONLIN using elastic linear SDoF
models along global directions (X and Y)......................................................63
Table 3.1 Summary of uniform load method results obtained from SAP 2000 ........69
Table 3.2 Forces, moments and displacements - 100% EE_Trans + 40% EE_long....70
Table 3.3 Forces, moments and displacement - 40% EE_Trans + 100% EE_long .....70
Table 3.4 Forces, moments and displacements - 100% MCE_Trans + 40% MCE_long
..............................................................................................................................71
Table 3.5 Forces, moments and displacements - 40% MCE_Trans + 100% MCE_long
..............................................................................................................................71
Table 3.6 Forces and moments under dead load...........................................................77
Table 3.7 Forces and moments for EE - 100% EE_Long + 40% EE_Trans..................78
Table 3.8 Forces and moments for EE - 100% EE_Trans + 40% EE_Long..................78

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Table 3.9 Forces and moments for MCE - 100% MCE_Long + 40% MCE_Trans .....79
Table 3.10 Forces and moments for MCE - 100% MCE_Trans + 40% MCE_Long .....79
Table 3.11 Displacement under Expected Earthquake...................................................80
Table 3.12 Displacement under Maximum Credible Earthquake.................................80
Table 3.13 Comparison of stiffness..................................................................................101
Table 4.1 Selected Ground Motions ..............................................................................104
Table 4.2 Mass and stiffness values...............................................................................104
Table 4.3 Scaled PGA (g) of respective GMs................................................................106
Table 4.4 Resultant forces and displacements in local directions .............................106
Table 4.5 Resultant forces and displacements in global directions ..........................107
Table 4.6 Resultant forces and displacements obtained in linear elastic time history
analysis in SAP 2000 program at EE.............................................................107
Table 4.7 Resultant forces and displacements obtained in linear elastic time history
analysis in SAP 2000 program at MCE.........................................................108
Table 4.8 Comparison of maximum values recorded for linear time history analysis
in SAP 2000 and NONLIN.............................................................................108
Table 4.11 Nomenclature used for defining the GMs...................................................116
Table 4.12 Maximum resultant forces and displacement recorded at piers during
expected earthquake.......................................................................................119
Table 4.13 Maximum resultant forces and displacement recorded at piers during
maximum credible earthquake .....................................................................121
Table 5.1 Comparison of results from various analysis procedure ..........................126
Table 5.2 Summary of Cc values at EE...........................................................................127
Table 5.3 Summary of operational performance level at MCE .................................127
Table 5.4 Summary of life safety performance level at MCE.....................................127
Table 6.1 Comparison of results from various analysis procedure ..........................134
Table 6.2 Calculation of R factor at EE and MCE........................................................136
Table 6.3 Performance evaluation of the structure .....................................................137
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LIST OF FIGURES
Figure 1.1 Plan View of 8-span continuous; [adapted from FHWA-SA-97-010 Figure
1a] ........................................................................................................................12
Figure 1.2 Elevation View of 8-span continuous; [adapted from FHWA-SA-97-010
Figure 1]..............................................................................................................13
Figure 1.4 Typical Cross Section; [adapted from FHWA-SA-97-010 Fig 1b] ..............14
Figure 1.5 Section at Seat-Type-Abutment; [adapted from FHWA-SA-97-010 Fig 1c]
..............................................................................................................................15
Figure 1.6 Intermediate Pier Elevations; [adapted from FHWA-SA-97-010 Fig 1c]...16
Figure 1.8 Sliding action of the bearings [adapted from FHWA-SA-97-010 Figure 2]
..............................................................................................................................17
Figure 1.9 Location of the bridge (Source: Google Maps)..............................................18
Figure 1.10 Subsurface Soil Conditions; [adapted from FHWA-SA-97-010 Fig A1]....18
Figure 1.11 Stick element bridge model in SAP 2000 .......................................................19
Figure 1.12 Rigid link element connecting the pier to the superstructure in SAP .......21
Figure 1.13 Relationship between actual pier and stick model of 3-D frame elements
[adapted from FHWA-SA-97-010]..................................................................22
Figure 1.14 Typical view of an intermediate pier in SAP 2000 .......................................23
Figure 1.15 Details of sliding bearings at piers [adapted from FHWA-SA-97-010
Figure 10]............................................................................................................24
Figure 1.16 Releases provided in SAP 2000 at top of pier to simulate bearing action.24
Figure 1.17 Typical plan view of the pile arrangements [adapted from FHWA-SA-97-
010] ......................................................................................................................25
Figure 1.18 Details of support for spring foundation model [FHWA-SA-97-010 Figure
11] ........................................................................................................................26
Figure 1.19 Details of foundation springs in SAP 2000....................................................26
Figure 1.20 Details of abutment supports [FHWA-SA-97-010 Figure 16] .....................27
Figure 1.21 Deflected shape of modeled bridge under gravity load..............................28
Figure 1.22 Bending moment diagram (major) of modeled bridge under gravity load
..............................................................................................................................29
Figure 1.23 Shear force diagram (major) of modeled bridge under gravity load ........29
Figure 1.24 Settlement of the foundation under pier-1 ....................................................31
Figure 1.25 Deflected shape of modeled bridge under transverse loading...................31
Figure 1.26 Bending moment diagram (major) of modeled bridge under transverse
load......................................................................................................................32
Figure 1.27 Shear force diagram (major) of modeled bridge under transverse load...32

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Figure 1.28 Bending moment diagram of modeled bridge under longitudinal load on
deck .....................................................................................................................35
Figure 1.29 Shear force diagram (major) of modeled bridge under longitudinal load
on deck................................................................................................................35
Figure 1.30 Deflected shape of modeled bridge under loangitudinal load...................36
Figure 1.31 Bending moment diagram of modeled bridge under longitudinal load on
piers.....................................................................................................................36
Figure 1.32 Shear force diagram of modeled bridge under longitudinal load on piers
..............................................................................................................................37
Figure 2.1 Mass source defined for modal analysis in SAP 2000..................................40
Figure 2.2 3D view of the mode shape corresponding to first mode (Longitudinal) 42
Figure 2.3 Plan view of the mode shape corresponding to first mode (Longitudinal)
..............................................................................................................................43
Figure 2.4 3D view of the mode shape corresponding to second mode (Transverse)43
Figure 2.5 Plan view of the mode shape corresponding to second mode (Transverse)
..............................................................................................................................43
Figure 2.6 3D view of the mode shape corresponding to third mode (Torsional).....43
Figure 2.7 Plan view of the mode shape corresponding to third mode (Torsional) ..44
Figure 2.8 Mode shape corresponding to vibration of pier (4th Mode)........................45
Figure 2.9 Mass Source considering only the weight of the superstructure ...............46
Figure 2.10 Load applied in local directions for stiffness calculations of 50ft and 70ft
piers.....................................................................................................................51
Figure 2.11 Displacement recorded in local directions at top of the piers ....................52
Figure 2.12 Construction of design response spectra using 2-point method
[MCEER/ATC 49].............................................................................................54
Figure 2.13 Response Spectra used in the design example..............................................55
Figure 2.14 Response Spectra obtained for our site from USGS website for MCE and
EE.........................................................................................................................57
Figure 2.15 Response Spectra in PEER Ground motion Database..................................58
Figure 2.16 Resultant ground motion spectra compared with target spectra in PEER59
Figure 2.17 Comparison of the mean response spectra of the selected GMs with the
target spectra and 85% of target spectra at EE..............................................59
Figure 2.18 Comparison of the mean response spectra of the selected GMs with the
target spectra and 85% of target spectra at MCE..........................................60
Figure 3.1 Distribution of load Po in transverse direction .............................................66
Figure 3.2 Distribution of load Po in longitudinal direction..........................................66
Figure 3.3 Maximum displacement recorded in transverse direction .........................67
Figure 3.4 Maximum displacement recorded in longitudinal direction......................67
Figure 3.5 Response Spectrum function for MCE...........................................................74

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Figure 3.6 Response Spectrum function for EE...............................................................74
Figure 3.7 Load case 100 MCE - Long + 40 MCE - Trans...............................................75
Figure 3.8 Load case 100 MCE - Trans + 40 MCE - Long...............................................75
Figure 3.9 Load case 100 EE - Long + 40 EE - Trans .......................................................76
Figure 3.10 Load case 100 EE - Trans + 40 EE - Long .......................................................76
Figure 3.11 Column cross-section at base...........................................................................83
Figure 3.12 Column reinforcement details.........................................................................83
Figure 3.13 Pushover Model of the 70 feet pier.................................................................84
Figure 3.14 Reinforcement detailing in section designer for the column top section..85
Figure 3.15 Reinforcement detailing in section designer for the column bottom section
..............................................................................................................................85
Figure 3.16 Fiber model of column top in section designer.............................................86
Figure 3.17 Fiber model of column base in section designer...........................................86
Figure 3.18 Bilinear Stress strain model of concrete.........................................................86
Figure 3.19 Non-linear material property of concrete......................................................87
Figure 3.20 Bilinear Stress strain model of rebar...............................................................87
Figure 3.21 Material Property input in SAP 2000..............................................................87
Figure 3.22 Plastic hinge definition in SAP 2000 program at pier bottom ....................88
Figure 3.23 Fiber hinge model in SAP 2000 .......................................................................88
Figure 3.24 Triangular loading pattern used in SAP 2000 program...............................89
Figure 3.25 Typical pushover load case in SAP 2000 program.......................................90
Figure 3.26 Cross section of the pier considered for Push over analysis......................91
Figure 3.27 Typical plastic hinge assignment at pier bottom in SAP 2000....................92
Figure 3.28 Typical plastic hinge assignment at the column neck in SAP 2000 ...........93
Figure 3.29 Typical deflected shape of the 70 feet pier in transverse direction............94
Figure 3.30 Force displacement relationship of the 70 feet pier in transverse direction
..............................................................................................................................94
Figure 3.31 Moment rotation plot of the plastic hinge at the bottom of the 70 feet pier
in transverse direction......................................................................................95
Figure 3.32 Typical deflected shape of the 70 feet pier in longitudinal direction ........96
Figure 3.33 Force displacement relationship of the 70 feet pier in longitudinal
direction..............................................................................................................96
in longitudinal direction ..................................................................................97
Figure 3.35 Force displacement relationship of the 50 feet pier in transverse direction
..............................................................................................................................97

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in transverse direction......................................................................................98
Figure 3.37 Force displacement relationship of the 50 feet pier in longitudinal
direction..............................................................................................................98
in longitudinal direction ..................................................................................99
Figure 3.39 Typical deflected shape of the pier in transverse direction ........................99
Figure 3.40 Force displacement relationship of the bridge in transverse direction ...100
Figure 3.41 Typical deflected shape of the pier in longitudinal direction...................100
Figure 3.42 Force displacement relationship of the bridge in longitudinal direction101
Figure 4.1 Inputs in NONLIN program for linear analysis along longitudinal
direction............................................................................................................105
Figure 4.2 Inputs in NONLIN program for nonlinear analysis along chord direction
............................................................................................................................110
Figure 4.3 Definition of a time history function in SAP 2000 program......................114
Figure 4.4 Typical Time History Load Case defined in SAP 2000..............................115
Figure 4.5 Type of direct integration procedure followed in SAP 2000.....................115
Figure 4.6 Mass and stiffness coefficients for damping ...............................................116
Figure 4.7 Definition of mass source for time history analysis...................................116
Figure 4.8 Time history load cases defined in SAP 2000..............................................117
Figure 4.9 Maximum displacement response recorded for pier 4 during North Ridge
GM.....................................................................................................................117
Figure 4.10 Base shear in global X direction with time recorded during North Ridge
GM.....................................................................................................................118
Figure 4.11 Hysteretic loop of the base shear observed during North Ridge GM.....118
Figure 4.12 Hysteretic loop of the plastic moment rotation observed at one of the
bottom hinges during North Ridge GM ......................................................119
Figure 5.1 Flowchart as applicable to our bridge..........................................................131
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CHAPTER 1
1. UNIFORM LOAD METHOD – ELASTIC ANALYSIS
UNIFORM LOAD METHOD – ELASTIC ANALYSIS
1.1 GENERAL DESCRIPTION OF BRIDGE
The bridge being evaluated here is an adapted version of a nine-span viaduct steel
girder bridge, totaling in 1488 feet, presented by a report via the FHWA. The afore-
mentioned bridge has varying span lengths on the left side of the bridge. In addition,
the bridge has expansion joints. The bridge being analyzed in this report is an eight-
span curved continuous bridge, having no expansion joints. The total length of this
bridge is 1384 feet. The eight-spans are a mirror image of the four spans to the right of
the original bridge. All of the properties of the original bridge are mirrored, such that
on each side there are four 173’ spans as shown below in Figure 1.1. The radius of this
curved bridge is 1300 feet. The superstructure consists of four steel plate girders and a
concrete composite cast-in-place deck. The substructure elements, abutments and piers
are all cast-in-place concrete and supported on steel H-piles. The plan and the elevation
views are shown in Figure 1.1 and Figure 1.2.
Figure 1.1 Plan View of 8-span continuous; [adapted from FHWA-SA-97-010 Figure 1a]

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Figure 1.2 Elevation View of 8-span continuous; [adapted from FHWA-SA-97-010 Figure 1]
1.1.1 Structural System
The structural system of the bridge can be classified into two broader sections:
superstructure and the substructure. The superstructure consists of the deck and the
steel girders while the substructure comprises the abutments and pier columns, pile
foundations and bearings to connect the piers to the girders. The load from the deck is
transferred to the girders which transfer the entire load to the foundation through
bearings thus acting as a rigid element.

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1.1.2 Superstructure
The two main components of the superstructure to be designed and analyzed are the
deck and the girder. The deck is simply the surface of the bridge on which the vehicles
run. It’s generally made of concrete covered with another layer of asphalt concrete or
pavement to account for the wearing of the surface due to friction and damage from the
vehicle loads. In the present project, also the bridge is made of concrete. The deck is
supported on steel girders which effectively take the loads of the vehicles running on
the deck and the self-weight of the deck itself. In this case the bridge has ‘I’ shaped steel
sections for girders.
The geometric properties of the superstructure are as follows:
 The bridge consists of eight spans, with all the spans 173 feet long. The right four
spans are mirror image of the other four spans.
 The width and thickness of the deck is 42 ft and 9 inch throughout the length of
the bridge.
 The bridge slab is made of concrete of characteristic compressive strength 4 ksi
and supported by four steel girders.
 Chevron bracings are provided to connect the girders to the deck. The bracings
are used to transfer the lateral internal load of the superstructure to the bearing.
The cross-section of the superstructure is shown below in Figure 1.3.
Figure 1.3 Typical Cross Section; [adapted from FHWA-SA-97-010 Fig 1b]

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1.1.3 Substructure
The substructure of a bridge is mainly used to transfer the loads from the superstructure
to the soil through the foundation and is a combination of all the components that
support the superstructure. It mainly consists of abutments, piers, piles and bearings.
Abutments
Abutments are the part of the substructure which, in case of a multi-span bridge,
supports the ends near the approach slab. They are meant to resist and transfer loads
like the self-weight, lateral loads (wind loads) and the ones from the superstructure to
the foundation elements. The abutments are mainly provided in the design bridge to
accommodate the thermal movement of the superstructure which will also allow for a
tolerance of movement in the longitudinal direction, and restraint in the transverse
direction. A clearance of 4 in was provided at the end of the girder-abutment connection.
The typical cross-section of a seat-type abutment of the design bridge is presented in
Figure 1.4.
Figure 1.4 Section at Seat-Type-Abutment; [adapted from FHWA-SA-97-010 Fig 1c]

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Piers
When bridges are too long to be supported by abutments alone, that is, in case of multi-
span bridges the intermediate support is provided by piers which are built like walls
shaped like girders. Piers are supported by elements called piles. These are slender
columns that are generally placed in a group to support loads transferred from the piers
via a pier cap. They are designed in such a way that they support loads through bearing
at the tip, friction along the sides, adhesion to the soil or a combination of all these.
Figure 1.5 shows the elevation of the piers of the design bridge.
Figure 1.5 Intermediate Pier Elevations; [adapted from FHWA-SA-97-010 Fig 1c]
Bearings
The devices that transfer the loads and movements from the deck to the substructure
and the foundation are called bearings. These movements are accommodated by the
basic mechanisms of internal deformation (elastomeric), sliding (PTFE) or rolling.

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Conventional types of pinned bearings are assumed at the piers 2, 3, 5 and 6 to ensure
transfer of both longitudinal and transverse seismic forces to the substructure through
anchor bolts. For the piers 1, 4 and 7 bearings were provided to accommodate expected
displacements. Elastomeric bearing with provisions for sliding between the bearing and
girder under large displacements was used for this purpose. Polytetraflouralethylene
(PTFE) bearings were provided against the sliding surface (stainless steel). In addition,
no expansion joints are present in the modified bridge used in the present project.
Figure 1.6 Sliding action of the bearings [adapted from FHWA-SA-97-010 Figure 2]
Figure 1.6 shows the action of the bearings during longitudinal deflection. During
longitudinal loads only the pinned piers (Pier 2,3,5,6) participate and the piers with
elastomeric bearing will slide (Pier 1,4,7) without resisting any longitudinal forces.
However transverse shear will be transferred in all the bearings during transverse
loading.
It is also to be noted that the values and numbering systems in the figures taken from
the previous report done by the FHWA do not necessarily coincide with the numbering
system and calculated values for this configuration. The height of the middle five and
outer two piers will be 70’ and 50’, respectively and enclosed between two abutments,
one on either side.
1.1.4 Location of Bridge
The bridge is located at coordinates 47.2663 N and 122.395105 W, in Tacoma,
Washington. Figure 1.7 present the location of the bridge from google maps. Tacoma is
a mid-sized port city named after the nearby Mount Rainier, originally called Mount
Tahoma. Known as the ‘City of Destiny’ because it was chosen to be the western
terminal of the Northern Pacific Railroad in the late 19th century.

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The Tacoma fault, is an active east-west striking north dipping reverse fault with close
to 35 miles of identified surface rupture, capable of generating earthquakes of atleast
magnitude 7.
Figure 1.7 Location of the bridge (Source: Google Maps)
1.1.5 Site Conditions
Although the soil in Tacoma, Washington is generally gravelly loam, for purpose of
analysis, the soil conditions will be taken as the same as the conditions given in the
FHWA report. Therefore, the soil profile will be taken as Type I- “Stable deposits of
sands and gravels where the soil depth is less than 200 feet.” The soil properties are
summarized in the Figure 1.8.
Figure 1.8 Subsurface Soil Conditions; [adapted from FHWA-SA-97-010 Fig A1]
1.2 OBJECTIVES
The bridge analyzed here is the fifth of seismic design examples developed using
AASHTO for the FHWA. The bridge was relocated in Tacoma nearby Mount Rainer
from Pacific Northwest to evaluate the seismic performance of the bridge. The analysis

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presented in the present project was done in accordance with the provisions of MCEER-
ATC/49 document and AASHTO 2009 LRFD Seismic Design Guide Specifications.
The primary objective was to evaluate the bridge response using various analysis
procedures given in the codes and compare the results obtained from them and critical
assessments were made from the results. The elastic analysis approach based on
uniform load method is carried out in the present chapter and the results are presented.
1.3 MODELING DESCRIPTION
The bridge model was developed in a commonly used structural analysis program SAP
2000 v. 16.0.1 [CSI, 2009].
Figure 1.9 shows the stick model used to simulate its behaviour in SAP 2000 program in
which single line frame elements were used for both superstructure and intermediate
piers. The nodes and the work line elements were located at the center of gravity of the
superstructure, which is 8 feet above the top of the piers. Dimensions of the bridges are
presented earlier in the report.
Figure 1.9 Stick element bridge model in SAP 2000
1.3.1 Superstructure
Some the basic modeling assumptions are listed as follows:
 Only bridges which subtend an angle of more than 30 degrees are required to be
analyzed as a curved structure, else they are allowed be analyzed as a straight

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one. In our case, the bridge has a span of 1384 ft (173*8) and a radius of curvature
of 1300 ft, thus subtending an angle of theta= 1384/1300 = 1.065 radians = 60.097
degrees> 30 degrees. Therefore, the superstructure of the bridge was analyzed
using the actual curved geometry.
 The bridge superstructure considered in this project has 8 spans over which a
uniformly distributed load (dead load) of 9.3 kips/feet was acting. The
calculations for the dead load are similar to the design example 5 and presented
as follows.
Weight of the superstructure is calculated as following:
concrete 0.15 kip/feet3 Unit weight of concrete
Deck 42’ X 9 “ Width and thickness of bridge deck
wslab 5 kip/feet Weight of concrete deck and girder pads
wsteel 1.9 kip/feet
Weight of steel plate girders and cross
frames
wmisc 2.4 kip/feet
Weight of barriers, stay-in-place metal
forms and future overlay
wsuper = wslab + wsteel + wmisc
wsuper 9.3 kip/feet Weight per length of the superstructure
 The superstructure is a composite structure comprising of I shaped steel girders
and a concrete deck. To simulate this model in SAP 2000, we consider an
equivalent concrete cross-section which has the same Area and Moment of
Inertia as that of the composite cross section. The modifiers used to model the
superstructure is calculated in the succeeding sections
 While analyzing ,the additional loads due to traffic barriers, wearing surface
overlay and stay-in-place metal forms are included and taken to be 2.4 kips per
lineal foot of superstructure.
 To account for the height of the bearings and the levelling pedestal, the centroid
of the superstructure is taken at a height of 8 feet above the top of the pier. The
girders are modeled as rigid link element in SAP 2000 program which was done

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by providing end length offset to the elements with rigid zone factor 1 indicating
full rigidity (Figure 1.10).
 To compute the bending stiffness full composite action between deck and girder
was assumed. The slipping at higher levels of loadings were neglected.
 The torsional properties are simulated considering that only the deck was
effective in providing torsional stiffness.
 Strength of concrete was taken to be 4000 psi, while steel was assumed to be
A615Gr60. Uncracked section properties were used to determine area and
moments of inertia assuming full composite action between deck and girders.
Figure 1.10 Rigid link element connecting the pier to the superstructure in SAP
Mass and Stiffness property of superstructure
In the design example, the spans are divided into four parts and the masses are lumped
in the nodes based on tributary area consideration. However, in SAP 2000 program, the
superstructure is modelled as frame elements with each span divided into eight stations.
Also, the gravity load calculated as 9.3 kips/feet (same as the design example 5 as the
cross-section of the superstructure remains same) was applied as uniformly distributed
throughout the spans. So, masses were not needed to be lumped at the nodes in SAP
2000 model.
Calculation of modifiers used in SAP 2000 to model the superstructure
For analysis, the deck and girder are considered to be a composite concrete structure
which has the same Area and the moment of inertia as that of the composite beam. Also
the torsional constant of the deck alone was used to model the superstructure.
Rigid Link

22
For this we consider the composite section to be a square and thus calculate its width as
follows:
Area of the composite section = b2 = 60 ft2
Calculation was done by equating MIX of the transformed section to that of the actual
section
Moment of Inertia about horizontal axis= 518 ft4
= b^4/12
= b = 8.879 ft ~ 8.8 ft
Therefore, the Area modifier = 60 / 8.8792 = 0.76
The moment about the Y axis is given to be 9003 ft4.
The modifier used for Moment of Inertia along vertical axis = 9003/518 = 17.37.
1.3.2 Substructure
Piers
In both transverse and longitudinal directions the pier base was assumed to be fixed
against rotation at the pile cap to account for expected lack of foundation flexibility.
Gross moment of inertia was used for the modeling of pier sections. These assumptions
provide a conservative estimate of the foundation stiffness and hence can be used for
simplification of model in SAP 2000.
Figure 1.11 Relationship between actual pier and stick model of 3-D frame elements
[adapted from FHWA-SA-97-010]

23
The intermediate piers are modeled as 3D frame elements that represent the represent
the individual columns. The relationship between the stick element and the actual pier
cross section is presented in Figure 1.11. Three elements were used to model the pier in
SAP 2000 to take into account the varying cross-section by interpolating between the
member end notes. All the properties are based on uncracked sectional details.
Foundation stiffness were attached to the bottommost nodes of the piers (2XX) by means
of spring supports. The intermediate pier modeled in SAP 2000 program is shown in
Figure 1.12.
Figure 1.12 Typical view of an intermediate pier in SAP 2000
Connection of piers to superstructure
In the actual bridge, the internal forces are transferred from the superstructure to the
piers through the bearings. In the SAP 2000 program, the forces are transferred through
a single point where the superstructure and the pier intersects, node 6XX in Figure 1.11.
At the pinned piers, node 6XX transfers shears in all directions from the superstructure,
but is released in moment along longitudinal direction. To account for this, the M3
moment is released at the top of the piers in SAP 2000 program (Figure 1.14). The other
sliding piers with elastomeric bearing are free to move longitudinally and hence only
transverse shear were transferred. So, in addition to M3, V2 are also released at the top
of those piers.

24
Figure 1.13 Details of sliding bearings at piers [adapted from FHWA-SA-97-010 Figure 10]
Translational and rotational releases were provided at the top of the piers with sliding
bearings to allow unrestrained longitudinal motion. The releases were made in local
coordinate system in SAP 2000 program to ensure its tangential orientation with respect
to the point of curvature at the center of the pier.
Figure 1.14 Releases provided in SAP 2000 at top of pier to simulate bearing action
Foundation Stiffness’s
Generally, soil contribution under a pile cap is not included because it is assumed that
soil will settle away from the cap. The piers are assumed to be located in flood plain of
a large river. The scour and loss of contact of soil around and beneath the pile cap, only

25
the stiffness of the pile group will be considered and the resulting forces at the
foundation level will only be applied to pile group to determine design loads to the pile.
Flexibility of pile cap is also neglected. To compute linear springs, elastic subgrade
approach is used as described in the seismic design, FHWA.
Since the relative stiffness of the foundation to the stiffness of the pier column is very
large, the resulting force for design of the pier and foundations will not vary
significantly, generally less than 5 percent. Generally, any reasonable development of
spring stiffness will produce acceptable results.
Considering he pile group, as shown in Figure 1.15, the foundation stiffness is calculated
in FHWA-SA-97-010. As the soil conditions are similar to the design example 5, the
spring stiffness obtained for foundation can be directly used in SAP 2000 model. Figure
1.16 and Figure 1.17 shows the modeling of foundation stiffness. The values of the
spring constants used in SAP 2000 program are as follows:
k11 2.66 × 104 Kip/ft
k22 7.847 × 105 Kip/ft
k33 1.70× 104 Kip/ft
k44 7.96 × 107 Kip-ft/rad
k55 4.785 × 106 Kip-ft/rad
K66 9.628 × 107 Kip-ft/rad
Figure 1.15 Typical plan view of the pile arrangements [adapted from FHWA-SA-97-010]

26
Figure 1.16 Details of support for spring foundation model [FHWA-SA-97-010 Figure 11]
Figure 1.17 Details of foundation springs in SAP 2000
Abutments
The abutments were modeled as simple nodes with a combination of full restraints
(vertical translation and superstructure torsional rotation) and an equivalent spring
stiffness along transverse direction as shown in Figure 1.18. The calculation of the spring
stiffness was based on the pile stiffness of the intermediate piles and it was similar to
the one calculated in design example 5. Spring stiffness of 4663.64 kips/feet was
provided in transverse direction to model the abutments in SAP 2000 program. The
restraints and the springs are all provided relative to the local coordinate geometry.

27
Figure 1.18 Details of abutment supports [FHWA-SA-97-010 Figure 16]
1.4 INITIAL ELASTIC ANALYSIS
1.4.1 Uniform Load Method
The objective of the uniform load method is to estimate the displacement demand for
the simplistic model of the superstructure done in SAP 2000 program. In this analysis
procedure, the structure was subjected to gravity load (9.3 kips/feet) only considering
the weight of the superstructure and an arbitrary distributed load (40 kip/feet) applied
both longitudinally and transversely, separately, to study the behaviour of bridge
subjected to longitudinal and transverse forces.
The following basic assumptions were made during elastic analysis in SAP 2000
 The superstructure was subjected to uniformly distributed load of 40 kips/feet
to ensure high workable displacement.
 Linear elastic analysis was done, no plastic hinges were assumed to be formed
throughout the analysis.
 Lateral load along transverse direction was subjected only on the superstructure
while the lateral load along longitudinal direction was subjected both on the
superstructure and piers separately.

28
1.4.2 Results and Discussions
The bridge modeled in SAP 2000 program was subjected to both gravity load and lateral
loads and elastic analysis was performed. The results obtained from the analysis in
terms of deflected shapes, bending moment and shear forces are discussed in this
section. As seen from the Figure 1.19 to Figure 1.30, the bridge behave symmetrically
under the gravity load which further validate the model produced in SAP 2000 to
simulate the bridge behaviour.
Gravity Load
The deflected shape, bending moment and shear force diagrams under gravity load of
9.3 kips/feet are presented in Figure 1.19 to Figure 1.21. The deflection observed was
more along the end spans compared to the intermediate spans as expected. Maximum
displacement of 0.25 feet was observed under the gravity loads at the end spans. The
bending moment and shear force diagrams obtained for the bridge model are similar to
that obtained for a multi-span continuous beam, which was expected. Also, it was
observed that there was no deflection at the nodes of the superstructure, as rigid
elements were considered to model the girders thereby allowing zero displacement.
Table 1.1 shows the deflection, bending moment and shear force in the spans under
gravity load. As the bridge is symmetric in geometry only the first four spans were
considered for critical assessment of the bridge. The maximum values were also
obtained and presented in the Tables so that the critical sections can be identified.
Figure 1.19 Deflected shape of modeled bridge under gravity load

29
Figure 1.20 Bending moment diagram (major) of modeled bridge under gravity load
Figure 1.21 Shear force diagram (major) of modeled bridge under gravity load

30
Table 1.1 Deflection, moment and shear force along the spans under gravity load
Span Location
Deflection
(feet)
Bending
Moment
(Kips-feet)
Shear Force
(Kips)
GravityLoading
Span-1
Left 0 -2504.1 -694.4
Middle 0.25 22225.2 161.2
Right 0 -32125.2 1020.25
Maximum 0.25 32125.2 1020.25
Span-2
Left 0 -30325.3 -863.2
Middle 0.06 9555.0 -54.1
Right 0 -21050.7 756.0
Maximum 0.06 -30325.3 -863.2
Span-3
Left 0 -21213.5 -778.3
Middle 0.08 11290.3 12.3
Right 0 -21817.0 785.5
Maximum 0.08 -21817.0 785.5
Span-4
Left 0 -21799.2 -774.3
Middle 0.07 10381.2 -11.7
Right 0 -20944.8 763.9
Maximum 0.07 -21799.2 -774.3
It can be seen from the Table 1.1, that maximum deflection for all the spans were
observed at the middle with the value maximum for end span. Negative moments were
observed at all the supports, while positive bending moment were observed at the
middle, indicating double curvature bending of the spans. Also, it was observed that
for all the spans the bending moments and shear forces are maximum at the same
sections, mostly along the girder supports. Maximum shear force and moment was
observed at the right end of the first span.
Table 1.2 Variation of axial forces in superstructure under gravity load
Spans Span-1 Span-2 Span-3 Span-4
Axial Force (Kips) 12.7 24.9 17.1 17.9
Resultant Torsion (kips-feet) -4321.1 700.2 -342.2 261.1

31
Figure 1.22 Settlement of the foundation under pier-1
The variation of the axial forces under gravity load was not much, however it can be
seen from Table 1.2, that the superstructure was subjected to some amount of torsion
under gravity loading. Figure 1.22 shows the settlement of the foundation at the pier-1.
There was slight settlement observed in the foundations of the order of 0.0010 feet, as
they were not modelled as fixed supports. The restraints were provided in form of
spring constants as described earlier. Similar observations were also made with the
other foundation supports.
Transverse Load
A transverse load of 40 kips/feet was applied along the superstructure throughout the
entire length of the bridge. The deflected shape, bending moment and shear force
diagrams under transverse load are presented in Figure 1.23 to Figure 1.25. As it can be
seen from the deflected shape, the entire superstructure moves like a rigid body in the
direction of the force. Maximum deflection of 1.36 feet was observed at the center of the
bridge as expected.
Figure 1.23 Deflected shape of modeled bridge under transverse loading

32
Figure 1.24 Bending moment diagram (major) of modeled bridge under transverse load
Figure 1.25 Shear force diagram (major) of modeled bridge under transverse load

33
Table 1.3 Deflection, moment and shear force along the spans under transverse load
Span Location
Absolute
Deflection
(inch)
Bending
Moment
(Kips-feet)
Shear
Force
(Kips)
TransverseLoading
Span-1
Left 0 0 -2055.9
Middle 0.87 71364.8 565.0
Right 0.81 -106905.9 3401.6
Maximum 0.87 -106905.9 3401.6
Span-2
Left 0.86 -106904.5 -2668.5
Middle 0.89 3298.9 133.1
Right 0.92 -137042.6 3140.7
Maximum 0.92 -137042.6 -2668.5
Span-3
Left 0.96 -137047.9 -3518.9
Middle 1.11 34346.2 -543.1
Right 1.22 -45548.6 2504.9
Maximum 1.22 -135047.9 -3518.9
Span-4
Left 1.24 -45555.2 -3076.3
Middle 1.35 81644.6 0
Right 1.36 -43980.1 3081.3
Maximum 1.36 81644.6 3081.3
As it can be observed from Table 1.3, the deflection of the superstructure was observed
to be more or less similar throughout the length of the beam with the maximum value
being observed at the end of span 4, which is actually the center point of the bridge. The
maximum bending moment and shear force was observed at same sections with one
exception in span 4. Again change in sign of bending moment and shear force was
observed indicating double curvature bending. In almost all the cases the maximum
resultant forces were recorded at the supports as also observed under gravity load. So
the sections near the girder support are critical sections and needs tension reinforcement
at the top as negative bending moment (hogging) was observed both during transverse
and gravity loading.

34
Table 1.4 Maximum resultant forces along piers under transverse load
Piers
Shear(kips) Moment (kips-feet)
Displacement
(feet) Axial force
(kips)
Long Trans Trans Long Long Trans
Left Abut 285.3 3115.9 0.0 1095.3 0.60 0.65 51.8
Pier 1 0.0 7098.1 405837.3 0.0 0.49 0.97 797.6
Pier 2 1393.0 7502.9 490020.7 80793.5 0.35 1.11 1165.0
Pier 3 361.3 6218.5 509082.6 28182.6 0.19 1.39 782.3
Pier 4 0.0 6696.7 547189.6 0.0 0.00 1.50 253.6
Pier 5 35.0 6309.6 514862.2 2724.7 0.19 1.39 754.4
Pier 6 952.3 7518.7 491541.5 55232.9 0.35 1.11 1142.9
Pier 7 0.0 7105.7 406414.8 0.0 0.49 0.97 797.6
Right Abut 285.0 3115.4 0.0 1097.1 0.61 0.65 51.8
Table 1.4 presents the maximum resultant forces in the piers under transverse load both
in its weak and strong direction. The resultant forces were observed to be more in its
strong direction compared to weak direction, as the load was applied along transverse
direction. High negative moment was observed along the piers in strong direction with
the pier-4 having maximum value. The bending moment in the piers are much higher
than the superstructure as evident from Figure 1.24 and Table 1.4. The deflection of the
pier along the direction of loading increases from the ends to the center with a maximum
displacement of 1.00 feet at the center pier. However, along weak direction the
deflection of the pier is not varying much.
Longitudinal Load on Superstructure
Longitudinal load of 40 kips/feet was applied to the superstructure of the bridge to
investigate its behaviour under longitudinal forces. Thus it can be seen from the Figure
1.26 and Figure 1.27 that the siding piers (pier 1, 4 and 7) don’t participate in the
longitudinal direction which is in accordance with the assumption made in Figure 2 of
FHWA design example. In order to take into account the sliding action of those piers
only transverse shear was transferred and hence no shear and bending moment was
observed under longitudinal loading in the corresponding piers.

35
Figure 1.26 Bending moment diagram of modeled bridge under longitudinal load on deck
Figure 1.27 Shear force diagram (major) of modeled bridge under longitudinal load on deck
Table 1.5 Maximum resultant forces along piers under longitudinal load on deck
Piers
Shear(kips) Moment (kips-feet)
Displacement
(feet) Axial force
(kips)
Long Trans Trans Long Long Trans
Left Abut 46.8 165.7 0.0 102.6 5.75 0.04 2.8
Pier 1 0.0 786.5 40110.1 0.0 5.72 0.12 280.0
Pier 2 19407.5 403.5 24143.2 1125633.3 5.65 0.07 84.3
Pier 3 8243.4 192.2 3540.7 642988.9 5.66 0.03 470.8
Pier 4 0.0 877.4 38889.6 0.0 5.68 0.00 1341.6
Pier 5 8192.2 2545.3 159735.9 638990.5 5.66 0.03 1917.7
Pier 6 19181.3 501.2 3814.0 1112512.6 5.65 0.07 1291.7
Pier 7 0.0 907.3 42467.7 0.0 5.72 0.12 573.3
Right Abut 82.3 231.2 0.0 102.6 5.75 0.04 3.8

36
Longitudinal Load on Piers
Longitudinal load of 40 kips/feet was applied to the piers to investigate the behaviour
bridge under longitudinal forces. The load was applied in SAP 2000 program in global
X direction along the piers, therefore, it was not applied in purely longitudinal direction
due to curved geometry. Hence, some transverse displacement was also evident from
the Figure 1.28. As it can be seen from the Figure 1.29 and Figure 1.30, the bending
moments and shear forces were maximum at the pier bottom, where the foundation
stiffness’s were provided. Also, the resultant forces (V and M) was more in the piers
compared to the superstructure. This was mainly because, the deflection of the
superstructure was much less compared to that of the piers.
Figure 1.28 Deflected shape of modeled bridge under loangitudinal load
Figure 1.29 Bending moment diagram of modeled bridge under longitudinal load on piers

37
Figure 1.30 Shear force diagram of modeled bridge under longitudinal load on piers
1.5 SUMMARY AND CONCLUSIONS
A uniform load method of analysis was used to get response of a simplified model of
the bridge in SAP 2000 program. The general description of the bridge and assumptions
made in the model are discussed in details and the results obtained from the analysis
are presented. The bridge used in the project is symmetric in geometry and hence
symmetry is also observed in the resultant forces. It can be observed that all the bending
moment and shear force diagrams are symmetric in nature. The behaviour of the bridge
under gravity and lateral loads can be summarized as follows:
 The superstructure of the bridge almost behave as a rigid body under transverse
loading with partial restrain at both abutment and at pier location.
 Maximum deflection was observed at the end spans under gravity loading, however
the deflection was maximum at the center of the bridge under transverse loading.
 The bending moment diagrams indicated that the superstructure was under double
curvature bending both under gravity and transverse loads.
 Maximum shear forces and bending moments were observed at the girder supports
for both gravity and transverse loading.
 The maximum displacement of the superstructure observed in transverse direction
for 40 kips/feet of uniformly distributed load was 1.36 feet, while the maximum
deflection was observed to be 0.25 feet for gravity loading.

38
 The maximum deflection of the substructure (pier 4) was 1.00 feet under transverse
direction along the direction of loading.
 The variation in axial force in the superstructure was not much due to gravity load
along the length of the bridge. However, torsional moments were present in the
superstructure under the action of gravity loads.
 It was also observed that the foundation nodes have undergone some settlement, as
springs were used for modeling.
-o-o-o-

39
CHAPTER 2
2. MODAL ANALYSIS, DEVELOPMENT OF RESPONSE SPECTRA AND SCALING OF GROUND MOTIONS
MODAL ANALYSIS, DEVELOPMENT OF
RESPONSE SPECTRA AND SCALING OF
GROUND MOTIONS
2.1 INTRODUCTION
In the previous chapter, the general description of the bridge was presented and its
behaviour under generic lateral load, both transverse and longitudinal, was
investigated. Therefore, the two principle directions were considered for analysis. In
this chapter multimode analysis of the bridge is carried out in SAP 2000 program
considering all the modes which contribute significantly to the overall behaviour of the
structure. The response spectra for our site (Tacoma) has been obtained for both design
earthquake (DE) and maximum credible earthquake (MCE). A suite of ground motions
is selected for time history analysis and scaled by comparing their corresponding
response spectra to the design spectra for our site. Further, a simplified single degree of
freedom (SDoF) model of the bridge was developed in NONLIN software to examine
its behaviour and compared with the response obtained in SAP 2000 program.
2.2 EIGEN VALUE ANALYSIS
The model developed in SAP 2000 program for the analysis using uniform load method
(described in previous chapter) is also used for the multimode method of analysis and
therefore, the same modeling assumptions are valid. The load considered for the modal
analysis in SAP 2000 program is the total dead load of the superstructure coming from
the element self-weight. The live load and other miscellaneous loads are neglected in

40
modal analysis to avoid complications. The load of the structure is defined in SAP 2000
by defining mass source as shown in Error! Reference source not found..
Figure 2.1 Mass source defined for modal analysis in SAP 2000
The maximum number of modes were initially set to 12 in SAP 2000 program, as it was
expected that the modal participation factor of the first 12 modes will be greater than
90%. However, as the analysis was carried out, it was observed that about 83 Eigen
values were needed to capture 100% mass participation in both translation and rotation
along all the three directions. However, modal participation factor of 90% was observed
in the 27th mode for the principle directions (X and Y). Therefore, the results obtained
from the first 30 modes are shown in Table 2.1 to also demonstrate the contribution of
the higher modes on the structure. The natural periods and the corresponding mode
shapes are presented in the succeeding sections. It can be seen from Table 2.1, that the
cumulative modal mass participation had reached 90% first in rotation along vertical
axis at 13th mode, while for translational motion it is reached only after 20th and 27th
modes for transverse and longitudinal directions, respectively.

41
Table 2.1 Natural periods and cumulative mass participation of different modes
Mode
Period
(s)
Cumulative Modal Mass Participation
SumUX SumUY SumUZ SumRX SumRY SumRZ
1.00 1.54 0.57 0.00 0.00 0.00 0.00 0.06
2.00 0.88 0.57 0.59 0.00 0.02 0.00 0.06
3.00 0.75 0.62 0.59 0.00 0.02 0.00 0.60
4.00 0.75 0.67 0.59 0.00 0.02 0.00 0.60
5.00 0.71 0.67 0.59 0.00 0.02 0.02 0.60
6.00 0.69 0.67 0.87 0.00 0.03 0.02 0.60
7.00 0.68 0.67 0.87 0.01 0.09 0.02 0.60
8.00 0.62 0.70 0.87 0.01 0.09 0.02 0.89
9.00 0.62 0.70 0.87 0.01 0.09 0.07 0.89
10.00 0.54 0.70 0.87 0.03 0.18 0.07 0.89
11.00 0.52 0.70 0.88 0.03 0.18 0.07 0.89
12.00 0.48 0.70 0.88 0.03 0.18 0.15 0.89
13.00 0.45 0.76 0.88 0.03 0.18 0.15 0.90
14.00 0.45 0.78 0.89 0.03 0.18 0.15 0.91
15.00 0.43 0.78 0.89 0.03 0.18 0.15 0.93
16.00 0.43 0.78 0.89 0.07 0.35 0.15 0.93
17.00 0.39 0.78 0.89 0.07 0.35 0.47 0.93
18.00 0.37 0.78 0.89 0.42 0.38 0.47 0.93
19.00 0.36 0.78 0.89 0.42 0.38 0.47 0.93
20.00 0.31 0.78 0.91 0.42 0.42 0.47 0.93
21.00 0.30 0.78 0.91 0.42 0.42 0.47 0.93
22.00 0.30 0.78 0.91 0.42 0.43 0.47 0.93
23.00 0.27 0.78 0.95 0.42 0.52 0.47 0.93
24.00 0.27 0.86 0.95 0.42 0.52 0.47 0.93
25.00 0.27 0.86 0.96 0.42 0.56 0.47 0.93
26.00 0.27 0.86 0.96 0.42 0.56 0.47 0.94
27.00 0.26 0.90 0.96 0.42 0.56 0.47 0.94
28.00 0.25 0.96 0.96 0.42 0.56 0.47 0.95
29.00 0.24 0.96 0.97 0.42 0.56 0.47 0.95
30.00 0.24 0.96 0.97 0.42 0.57 0.47 0.95

42
2.2.1 Natural Periods and Mode Shapes of Structure
The first three natural periods coming from the modal analysis are 1.54 s, 0.88 s and
0.75 s, respectively. First mode is primarily associated with translation in longitudinal
direction coupled with some rotation about vertical axis while the second mode is
associated with translation in transverse direction coupled with some rotation about
longitudinal axis of the bridge. It can be verified from the values of modal mass
participation presented in Table 2.2 for the first 2 modes. It must be noted that the
rotation present in the first two mode shapes are much less compared to the
translational components, and hence the period associated with the first and the second
modes can be considered as the period of the bridge for translational motion along
longitudinal and transverse directions, respectively. The third mode is predominantly
rotation about vertical axis.
Table 2.2 Modal mass participation of first three modes
Mode Period (s)
Cumulative Modal Mass Participation
Translational Rotational
UX UY UZ RX RY RZ
1.00 1.54 0.57 0.00 0.00 0.00 0.00 0.06
2.00 0.88 0.00 0.59 0.00 0.02 0.00 0.00
3.00 0.75 0.05 0.00 0.00 0.00 0.00 0.54
Figure 2.2 to Figure 2.7 show the mode shapes corresponding to first three natural
modes of vibration as obtained from modal analysis in SAP 2000 program.
Figure 2.2 3D view of the mode shape corresponding to first mode (Longitudinal)

43
Figure 2.3 Plan view of the mode shape corresponding to first mode (Longitudinal)
Figure 2.4 3D view of the mode shape corresponding to second mode (Transverse)
Figure 2.5 Plan view of the mode shape corresponding to second mode (Transverse)
Figure 2.6 3D view of the mode shape corresponding to third mode (Torsional)

44
Figure 2.7 Plan view of the mode shape corresponding to third mode (Torsional)
For comparison of the multi-mode analysis results obtained in SAP 2000 program with
the periods obtained in FHWA Design Example 5, the results are presented in Table 2.3.
It can be seen from the Table that the results obtained from SAP 2000 program are in
close agreement with the results obtained in the design example. The longitudinal
periods of unit-1 and unit-2 of the original bridge in the design example are 1.52 s and
1.20 s respectively. Since the modified bridge analyzed in this project is eight span
bridge similar to the unit-2 of the original bridge, therefore its longitudinal period
obtained from modal analysis in SAP 2000 matches closely with that obtained for Unit-2
of the design example. In addition, the period associated with translational motion in
transverse direction is also similar in both SAP 2000 and the design example. However,
the small difference is due to presence of expansion joints in the original bridge.
Therefore, the similarity in time periods of the bridge in principle directions obtained
from SAP 2000 with the periods of the original bridge presented in FHWA design
example further validates our model.
Table 2.3 Comparison of periods of the modified bridge and the FHWA original bridge
SAP 2000
Analytical Calculation in
Design Example
Multimode analysis in
Design Example
Mode Period Mode Period Mode Period
1 Longitudinal 1.54
Longitudinal
Unit 2 1.55 1 Unit-2 Long 1.52
2 Transverse 0.88 Unit 1 1.26 2 Unit-1 Long 1.20
3 Torsion 0.75 Transverse 0.43 3 Transverse 0.80
2.2.2 Higher Modes associated with Vibration of Piers
Piers are rigid compared to the bearings provided at the top of the piers, as a result of
which, the initial modes of vibration are mostly dominated by the vibrations of the
bearings, particularly at the top of the piers 1,4 and 7, which allows sliding. The first

45
mode associated only with vibration of pier is the fourth mode with period of 0.75 s,
with vibration of pier 4 along longitudinal direction, as presented in Figure 2.8. The next
modes that are dominated by vibration of piers have natural period less than 0.4 s. Thus
it can be concluded that the vibration of pier was negligible in the first few modes and
hence the contribution of piers to the inertia forces can be neglected for those modes.
Therefore, for the simplified SDoF model that is developed to consider the vibration of
the bridge along its principal directions, it is safe to neglect the inertia of the piers and
only the weight of the superstructure is considered.
However, for better results it is recommended that the weight of the substructure should
also be considered and a comparative study is carried out in the later section. It can be
found that the period obtained by considering the weight of the superstructure and the
piers are in better agreement with the SAP 2000 results and actual period of the structure
obtained analytically.
Figure 2.8 Mode shape corresponding to vibration of pier (4th Mode)
2.2.3 Comparison with Elastic Analysis Results in SAP 2000
As stated in Chapter 1, the primary objective of the uniform load method is to estimate
the displacement demands of the superstructure under generic lateral loads. A
transverse and longitudinal lateral load of 40 kips/feet were applied along the
superstructure. Based on the following equation, the lateral stiffness of the bridge can
be estimated for longitudinal and transverse vibrations.
max
Lat
wL
K
v

where, w = 40 kips/feet, L = total length of the superstructure along which the uniformly
distributed load is acting and vmax is the maximum displacement recorded in SAP 2000
program along longitudinal and transverse directions. So, once the lateral stiffness is
obtained, the periods can be calculated based on the following equation.
2m
Lat
W
T
K g


46
The period of the bridge obtained from the above method is presented in Table 2.4. As,
it can be seen, the periods obtained from SAP 2000 was higher (almost 25%) than those
calculated based on displacement recorded during uniform load method. This was
probably because, the weight used to calculate the periods was the weight of the
superstructure alone, which is 9.3 kips/feet. Therefore, the modal analysis is repeated
in SAP 2000 by using the mass source as 9.3 kips/feet (Figure 2.9) and it was observed
that the periods exactly matches with those calculated based on uniform load method,
which further validates our model in SAP 2000.
Figure 2.9 Mass Source considering only the weight of the superstructure
The time period was also calculated considering both the weight of the superstructure
and piers. The weight of the 50 feet and the 70 feet piers are 690 and 880 kips,
respectively as reported in the design example. It can be seen that the periods along both
longitudinal and transverse direction are in good agreement with the values obtained
from SAP 2000 program considering the element weights. Therefore, the lumped mass
is considered as 18461.2 kips considering both the weight of the superstructure and the
piers half the height above the pile cap.
The stiffness of the bridge obtained from this simplified procedure is presented in Table
2.4. It can be seen later that the longitudinal stiffness calculated using fixed base is in
close agreement with the analytical calculations, but the transverse stiffness is much
lesser compared to the analytical solution. The possible reason is stated in the
succeeding section and a more rigorous calculation of mass and stiffness is presented
which is to be further used for the development of the SDoF model in NONLIN.

47
Table 2.4 Calculation of period of bridge from uniform load method
Notations
Considering only weight
of superstructure
Considering the weight of
superstructure and piers
Longitudinal Transverse Longitudinal Transverse
UniformLoad
Method
vmax (feet) 5.75 1.52 5.75 1.52
wL(kips) 55360 55360 55360 55360
KLat (kips/feet) 9629.5 36514.74 9629.5 36514.74
W (kips) 12871.2 12871.2 18271.2 18271.2
Tm (s) 1.27 0.66 1.52 0.79
SAP
2000
T (Element mass) 1.54 0.88 1.54 0.88
T (9.3 kips/feet) 1.27 0.66 1.27 0.66
2.2.4 Analytical Calculations of Bridge Stiffness along local directions
The stiffness of the bridge along the longitudinal and transverse directions are
calculated analytically and compared with the results obtained from the simplified
procedure presented in the preceding section. Two procedures were used for analytical
calculation of bridge stiffness and designated as method 1 and 2 in this report.
Method 1: The piers are assumed to be fixed at the base and the springs attached to the
foundation is neglected. The objective of such assumption is to check if this simplified
model can efficiently predict the stiffness of the bridge.
Method 2: The foundation springs are considered at the pier base and the stiffness of
the individual piers are calculated in local directions. It must be noted that this will
capture the bridge behaviour with more efficiency, however the calculations will be
more complex.
Method 1: Piers assumed fixed at base
Longitudinal
It can be seen from the mode shape corresponding to first mode, the entire
superstructure moves like a rigid body along longitudinal direction. The piers that will
contribute in the longitudinal direction are the pinned piers as the sliding piers are
taking only transverse shear. So the stiffness of the piers in longitudinal direction can
be calculated by considering the pinned piers in parallel. The values of the pier stiffness
are taken directly from the calculations presented in FHWA design example.

48
K50 3509 kips/feet
K70 1413 kips/feet
Klong = 2(K50 + K70) 9844 kips/feet
Transverse
In the transverse direction all the piers and the abutments participate, but it can be seen
from the corresponding mode shape that the superstructure does not move like a rigid
body. The maximum transverse displacement was observed at the center (Pier 4) of the
bridge. The mode shape corresponding to the 1st mode is used to calculate the
participation of each piers to the overall stiffness of the bridge along transverse
direction. Table 2.5 shows the deflection recorded at each piers in transverse direction
normalized with the maximum deflection observed at the center pier for mode shape
corresponding to 2nd mode.
Table 2.5 Deflected shape corresponding to 2nd mode (Transverse)
A P1 P2 P3 P4 P5 P6 P7 B
0.0024 0.0046 0.0216 0.0494 0.0628 0.0494 0.0216 0.0046 0.0024
0.038 0.073 0.344 0.787 1 0.787 0.344 0.073 0.038
So the stiffness of the bridge along the transverse direction is calculated by considering
the stiffness of the individual elements (piers and abutments) to be proportional to the
normalized displacement and calculated as follows:
Table 2.6 Calculation of overall transverse stiffness analytically
K50 Trans = 35928 kips/feet K70 Trans =14474 kips/feet
A P1 P2 P3 P4 P5 P6 P7 B
Factor 0.038 0.073 0.344 0.787 1 0.787 0.344 0.073 0.038
K 176.1 2622.7 12359.2 11391.0 14474 11391.0 12359.2 2622.7 176.1
Overall Transverse Stiffness 67572 kips/feet
The longitudinal stiffness calculated based on the displacement recorded in the uniform
load method in SAP 2000 program as shown in Section 2.2.3 is 9629.5 kips/feet, which
is in good agreement with the value calculated analytically (9844 kips/feet). In this
simplified procedure, the foundations were considered to be fixed at the base of the pile

49
cap while calculating the stiffness of the individual piers. So, it can be concluded that
assuming the foundations to be fixed at the base of the pile cap gives a close
approximation of the longitudinal foundation spring stiffness’s used in the modal
analysis. However, the transverse stiffness calculated analytically is much higher than
value reported in Section 2.2.3. This is mainly because, it is not correct to calculate the
pier stiffness in transverse direction considering it to be fixed at the base.
Method 2: Calculation of local stiffness: Considering foundation springs at pier bottom
Longitudinal Direction (Mode 1)
As mentioned in the previous chapter, the piers 1, 4 and 7 are sliding in nature and
hence does not contribute to the longitudinal stiffness of the bridge. The stiffness of the
individual piers are first calculated both analytically and in SAP 2000 program by
applying an unit load and then the total stiffness is obtained by considering the piers in
parallel. The stiffness of the individual piers are obtained analytically as follows by
considering the translational spring, the rotational spring and the stiffness of the
columns in series.
7
4
4
503
4
70
7.96 10 /
2.67 10
3
3509 / ( 56.5 , 408 )
1414 / ( 76.5 , 408 )
rot
tran
pier pier
pier
k kip feet rad
k kip feet
EI
k k kip feet h feet I feet
h
k kip feet h feet I feet


  
  
   
  
eff pier
rot tran pier
eff pier
rot tran pier
L
k L feet
k k k
kip feet
L
k L feet
k k k
kip feet
1
2
50
50
1
2
70
70
1 1
, 56.5 6.5 63
2983 /
1 1
, 76.5 6.5 83
1259 /

 


 

 
      
  

 
      
  

50 702 2
8484 /
long eff pier eff pierk k k
kip feet
    


50
 
sup
sup
50 70
9.3 1384 12871.2
Weight from half of the participating piers
2 2 2 690 2 880 3140
16011.2
long er sub
er
sub
long
W W W
W kips
W
W W kips
W kips
 
  

      

(Mode 1) 2 1.52
long
long
long
W
T s
gK
 
Thus the period of the bridge obtained analytically in longitudinal direction (Mode 1) is
in good agreement with that obtained from SAP 2000 program.
Transverse Direction (Mode 2)
As mentioned in the previous chapter, all the piers contribute to the transverse stiffness
of the bridge. The stiffness of the individual piers are first calculated both analytically
and in SAP 2000 program by applying an unit load and then the total stiffness is
obtained by considering the piers in parallel along with the abutment stiffness. The
stiffness of the individual piers are obtained analytically as follows by considering the
translational spring, the rotational spring and the stiffness of the columns in series.
7
4
4
503
70
9.63 10 /
1.71 10
3
35928 / ( 56.5 , 4166 )
14474 / ( 76.5 , 4166
rot
tran
pier pier
pier
k kip feet rad
k kip feet
EI
k k kip feet h feet I feet
h
k kip feet h feet I feet


  
  
   
   4
)
eff pier
rot tran pier
eff pier
rot tran pier
L
k L feet
k k k
kip feet
L
k L feet
k k k
kip feet
1
2
50
50
1
2
70
70
1 1
, 56.5 6.5 63
7841 /
1 1
, 76.5 6.5 83
5022 /

 


 

 
      
  

 
      
  


51
Table 2.7 Calculation of overall transverse stiffness analytically
A P1 P2 P3 P4 P5 P6 P7 B
Factor 1 0.073 0.344 0.787 1 0.787 0.344 0.073 1
K 4664 572.4 2697.3 3952.3 5022 3952.3 2697.3 472.4 4664
Overall Transverse Stiffness 28793 kips/feet
28793 / calculated based on the participation
factor in above Table
transk kip feet
 
sup
sup 9.3 1384 12871.2
Weight from half of the piers calculated based on participation of each piers
2840.6
15711.8
trans er sub
er
sub
trans
W W W
W kips
W
kips
W kips
 
  



(Mode 2) 2 0.82trans
trans
trans
W
T s
gK
 
Thus the period of the bridge obtained analytically in transverse direction (Mode 2) is
in good agreement with that obtained from SAP 2000 program.
A load of 40 kips, was applied in the longitudinal and transverse directions at the top of
the piers in SAP 2000 (Figure 2.10) and the maximum deflection was recorded for the 50
feet and 70 feet piers (Figure 2.11) based on which the effective stiffness of the individual
piers were obtained.
Figure 2.10 Load applied in local directions for stiffness calculations of 50ft and 70ft piers

52
Figure 2.11 Displacement recorded in local directions at top of the piers
The stiffness obtained based on this displacement was compared with those obtained
analytically and presented in Table 2.8. A good agreement was observed between the
results which further validate the analytical procedure for stiffness calculation.
Table 2.8 Comparison of stiffness of the piers obtained analytically and in SAP 2000
Direction
50 feet pier 70 feet pier
SAP 2000 Analytical SAP 2000 Analytical
Longitudinal 3478 2983 1476 1259
Transverse 8163 7841 5128 5022
The simplified procedure presented in Section 2.2.3 and simplified method 1 of section
2.2.4 gives a quick and good approximation of the actual periods of the structure in
longitudinal and transverse directions. However, for this project the stiffness obtained
according to method 2 in this section along the longitudinal and transverse directions
are used for development of the SDoF model in NONLIN program.
2.2.5 Analytical Calculations of Bridge Stiffness along global directions
However, it must be taken into account, that for better understanding of the behaviour
of the bridge during seismic activity, its response must also be investigated in the two
principal directions (X and Y) orthogonal to each other and hence the stiffness was also
calculated for global X and Y directions.

53
The stiffness obtained in the local directions were transferred to the global directions, as
shown in the Table 2.9, based on the angle of the respective piers with the global axes.
2
2
2
2
cos and sin
y long
x trans
y long trans
x long trans
k kc cs
c s
k kcs s
k c k csk
k csk s k
 
    
      
    
 
 
Table 2.9 Calculation of overall stiffness analytically along global direction
Piers
Angle
(rad)
klong
(kips/feet)
ktrans
(kips/feet)
Ky
(kips/feet)
Kx
(kips/feet)
Abut A 0.53 0 4664 2034 1192
P1 0.41 2983 7841 5375 2336
P2 0.28 2983 7841 4838 1391
P3 0.13 1259 5022 1883 246
P4 0.00 1259 5022 1259 0
P5 0.13 1259 5022 1883 246
P6 0.28 2983 7841 4838 1391
P7 0.41 2983 7841 5375 2336
Abut B 0.53 0 4664 2034 1192
Total Global Stiffness 29521 10331
A uniformly distributed load was applied in the superstructure in SAP 2000 model
along global X and Y directions and a maximum displacement of 0.1343 and 0.0465 feet
was recorded, respectively. The stiffness, thus obtained was 10305 and 29763 kips/feet
along X and Y directions, respectively, and hence are in good agreement with the values
obtained analytically.
Therefore, the final values of mass and stiffness along the principal directions (both local
and global) that were used for development of the SDoF model is shown in Table 2.10.

54
Table 2.10 Stiffness and mass used in the development of the SDoF model
Direction Mass (kips) Stiffness (kips/feet)
Longitudinal (Local X) 16011.2 8484
Transverse (Local Y) 15711.8 28793
Chord (Global X) 18271.2 10331
Radial (Global Y) 18271.2 29521
2.3 RESPONSE SPECTRA
The ATC 49 report suggests to consider two level of earthquakes for analysis and design
of the structures. The design expected earthquake (EE) is considered to be the one
associated with 50% probability of exceedance in 75 years, while maximum credible
earthquake (MCE) corresponds to 3% probability of exceedance in 75 years (Table 3.2-1
of ATC-MCEER 49). The construction of the design response response spectra using the
two point method and the definition of the parameters as presented in the ATC 49 is
shown in the Figure 2.12.
Figure 2.12 Construction of design response spectra using 2-point method [MCEER/ATC 49]
The input Response spectra graph (as specified in FHWA 1996 design example 5) that
gives information about the effect of earthquake for the given bridge is shown below in
Figure 2.13. However, the site of the modified bridge being analyzed in the present
project differs from the one given in design example and hence the response spectra is
developed in USGS website as presented in the succeeding sections.

55
Figure 2.13 Response Spectra used in the design example
2.3.1 Seismic Design Spectra
Since the time histories with respect to ground acceleration vary for each earthquake,
the resulting response spectrum will also be different. Hence when a structure is
designed for earthquake, the design spectra is generated based on average values of the
previous earthquakes. In order to provide loading for the model, a design response
spectrum was created following the specifications in Article 3.6.2 of MCEER/ATC 49
for both transverse and longitudinal directions.
The parameters of seismic design spectra were obtained from the USGS website based
on the following assumptions:
 The expected life span of the bridge is considered to be 75 years
 Presence of any active fault in the nearby region is not considered
 The bridge is expected to overcome the EE level ground motion with minimal
damage and the MCE level ground motion without collapse.
 The soil profile is considered to be same as that reported in the design example
which is site class C.

56
2.3.2 Seismic Design Spectra of our Site
Using USGS Website
The spectral acceleration values for 0.2 second and 1 second time periods for the location
(Tacoma, WA) were obtained from the USGS website. The soil condition in the region
was assumed to be dense and hence classified as ‘site class C’ as mentioned earlier in
the report. Based on 2013 ASCE 41 Design Code reference document, the earthquake
hazard level was custom designed for the analysis. The percentage probability of
ground motion exceedance in 50 years was taken as 2% for MCE (Maximum Credible
Earthquake) and 37% for EE (Expected Earthquake). The calculation for probability of
exceedance of EE are shown below. The values obtained from USGS are presented
below. The 2 response spectra are shown in Figure 2.14.
For EE, the probability of exceedance in 75 years is 50%.
3
(9.24 03 50)
1 , where for T = 75 yrs, p = 50%
9.24 10
For T = 50 yrs, 1 37%
T
E
e p
p e




  
 
 
  
Table 2.11 Response Spectra parameters obtained from USGS
Hazard Latitude Longitude Ss S1 Site Class Fa Fv
MCE 47.24879 -122.442 1.298 0.527 C 1.0 1.3
EE 47.24879 -122.442 0.330 0.115 C 1.2 1.685

57
MCE EE
Figure 2.14 Response Spectra obtained for our site from USGS website for MCE and EE
Using PEER Ground Motion Database
The values of S1 and Sd obtained from USGS were input into PEER Ground Motion
Database to obtain the scaled design spectra. The target spectrum was generated based
on ASCE Spectrum. The long period transition period (TL) was taken as 6 second as
obtained from USGS website. The values for the site coefficients Fa (Short period range)
and Fv (long period range) were obtained from MCEER/ATC-49(Part 1) Table 3.4.2.3-1
and 2. The Sds and Sd1 values were calculated as follows.
MCE – 2% in 50 years
Fa = 1
Fv = 1.3
Sds = Ss *Fa= 1.298 * 1= 1.298
Sd1 = S1*Fv= 0.527 * 1.3= 0.685
EE – 37% in 50 years
Fa = 1.2 (Interpolated)
Fv = 1.685 (interpolated)
Sds = Ss *Fa= 0.330 * 1.2 = 0.396 g
Sd1 = S1*Fv= 0.115 * 1.685 = 0.193 g

58
MCE EE
Figure 2.15 Response Spectra in PEER Ground motion Database
2.3.3 Ground Motion Selection
Ground motion selection is one of the most important factors for performing time
history analysis and should not be affected by performance characteristics of the
structure. In the present project the ground motions were selected and scaled in PEER
ground motion database.
2.3.4 Development of Response Spectra and Scaling of Ground Motions
The selected ground motions are scaled w.r.t. MCE and EE level spectra according to
MCEER ATC 49 such that
 The mean response spectra never lies below 15% of the design spectra for any
period and,
 The average ratio of the mean spectra and the target spectra shall not be less than
unity over the period range of significance.
So these two considerations were made while scaling the ground motions to the target
spectrum in PEER database.
After generating the target spectrum with the above values in PEER Ground Motion
Database, a magnitude range of 6.5-8.5 was selected to generate a list of ground motions
in that range. The period of interest was given between 0.2 and 1.8 s which more or less
capture the first 30 modes of the structure to ensure better match between the response
spectra of the selected GMs and the target spectra. 3 locations were selected from the
list of records each for MCE and EE hazard level to scale the target spectrum using the

59
average of their spectral acceleration curves. The curves of the selected locations were
chosen so that the scaling factor would be 2.5 or less. In addition, the ground motions
already present in the NONLIN database was selected, so that scaling will be easy for
the time history analysis in NONLIN program, the result of which is presented in the
succeeding sections. The resultant ground motion average spectra is compared with the
target spectra in Figure 2.16 to Figure 2.18. The list of ground motions and their
corresponding scale factors are presented in Table 2.12. It can be seen from the Figures
that the mean response spectra of the selected ground motions after scaling never falls
below 0.85 times the target spectra and also the ratio of the mean spectra to the target
spectra is approximately 1.02 for both MCE and EE. Thus, both the considerations of
ATC 49 are duly met.
MCE EE
Figure 2.16 Resultant ground motion spectra compared with target spectra in PEER
Figure 2.17 Comparison of the mean response spectra of the selected GMs with the target
spectra and 85% of target spectra at EE

60
Figure 2.18 Comparison of the mean response spectra of the selected GMs with the target
spectra and 85% of target spectra at MCE
Table 2.12 Scaled ground motions selected from PEER Database
No Ground Motion NGA# Scale M Year Station
MCE
1 Cape Mendocino 828 1.0 7.01 1992 Petrolia
2 North Ridge 960 1.0 6.69 1994
Canyon Country
-W Lost Cany
3 Loma-Prieto 753 1.0 6.93 1989 Corralitos
EE
1 North Ridge 1048 0.3788 6.69 1994
North Ridge 17645
Saticoy St
2 Imperial Valley 181 0.4671 6.53 1979 El-Centro #6
3 Kobe Japan 1116 0.7236 6.90 1995 Shin-Osake
2.4 DEVELOPMENT OF SDOF MODEL
A simple elastic SDoF analysis of the bridge was performed using the program
NONLIN, in which a lumped mass model was developed with the entire mass of the
superstructure and the piers lumped at the node. The piers of the bridge were modeled
as a single column with effective stiffness values in longitudinal and transverse
directions (local) and global X (chord) and Y (radial) directions. The primary objective
to carry out this simplified analysis was to get a preliminary idea about the response of
the bridge along two principal directions, when subjected to different level of ground
motions scaled with the design spectra of our site. The maximum resultant forces of the
piers in both strong and weak directions can also be obtained from this simplified
analysis.

61
2.4.1 Modeling Assumptions
Elastic analysis of the bridge was performed using NONLIN software with a simple
SDoF model in which representative mass and stiffness was assigned to evaluate its
performance in two orthogonal directions. The scaled ground motions were used for
analysis and the maximum resultant forces are reported. The modeling assumptions are
presented as follows:
Mass
As presented earlier in this Chapter, close to 80 modes of vibration are necessary to
entirely capture the overall response of the structure. The following assumptions were
made to consider the mass of structure in local directions (longitudinal and transverse)
and in the two orthogonal directions (chord and radial).
 Local Directions: The mass lumped at the node of the SDoF model is the weight
of the superstructure and part of the weight of the participating piers based on
their participation in the respective directions. The masses considered in the
SDoF model in the longitudinal and transverse directions as shown in Section
2.2.4 is 16011.2 and 15711.8 kips, respectively.
 Global Directions: The mass lumped at the node of the SDoF model is the weight
of the superstructure and half the weight of the piers, which is 18271.2 kips,
active in both the principal directions along global X and Y.
Stiffness
It is very difficult to characterize the bridge response with a single value of stiffness and
therefore the method 2 presented in Section 2.2.4 and Section 2.2.5 was used to calculate
the bridge lateral stiffness along local ( longitudinal and transverse) and global (chord
and radial) directions, respectively. Therefore, the key assumptions related to stiffness
of the bridge are as follows:
 The stiffness of the bridge was calculated along the local and global directions
of bridge both analytically and based on the displacement recorded by applying
unit load in SAP 2000 program.
 Local Directions: The values of the equivalent bridge stiffness in the transverse
and longitudinal directions are 28793 kips/feet and 8484 kips/feet, respectively.

62
 Global Directions: The values of the equivalent bridge stiffness in the chord (X)
and radial (Y) directions are 10331 kips/feet and 29521 kips/feet, respectively.
Damping
The value of damping used for SDoF analysis was 5% of critical damping, which is
typical for concrete bridges.
2.4.2 Analysis Procedure
A series of time history analysis was performed in NONLIN program using the
simplistic SDoF model of the concerned bridge as described in the preceding sections.
The ground motions selected are scaled according to two seismic hazard levels as
described in Section 2.3.2. The elastic dynamic analysis was performed for the principal
directions, longitudinal and transverse direction (local X and Y), chord and radial
direction (global X and Y) which are corresponding to the weak and strong directions
of the piers respectively. In all there were twelve ground motions (6 for EE and 6 for
MCE) and so, in total, 48 time history analysis were run, 12 along each of the four
directions as described above.
2.4.3 Results and Discussions
The results of the time history analysis considering simplified elastic linear SDoF model
are presented in Table 2.13. Expected earthquakes has lesser demand on the structure
and hence impose smaller displacement on the piers as compared to the maximum
considered earthquake which impose a demand about 3 times of that of EE as far as
displacements and shear forces are concerned in both transverse and longitudinal
direction. This comes from the difference in response spectra itself. The spectral
acceleration of MCE at short period was 1.298g which is 3.3 times the spectral
acceleration at same period for EE and hence the difference in demands between MCE
and EE is justified.

63
Table 2.13 Results of time history analysis in NONLIN using elastic linear SDoF models
along local directions (longitudinal and transverse)
Hazard
Level
Ground Motion
Longitudinal Transverse Maximum
Shear
(kips)
Disp.
(feet)
Shear
(kips)
Disp.
(feet)
Shear
(kips)
Disp
(feet)
MCE
NGA 753-FN 2108.23 .2485 6866.14 .2385
Long
7496.66 0.8464NGA 753-FP 5494.80 .6477 21294.58 .7396
NGA 828-FN 3475.63 .4097 19715.00 .6847
NGA 828-FP 7181.12 .8464 20814.34 .7229
Trans
21294.5 0.7396NGA 960-FN 7496.66 .6738 8520.73 .2959
NGA 960-FP 3135.31 .3696 6486.10 .2253
EE
NGA 181-FN 2349.51 .2769 6424.55 .2231
Long
3669.45 0.4325NGA 181-FP 2807.07 .3309 3862.65 .1342
NGA 1048-FP 2373.31 .2797 2698.36 .0397
NGA 1048-FP 2132.93 .2514 4507.14 .1565
Trans
7075.49 0.2457NGA 1116-FP 1748.23 .2061 4438.99 .1542
NGA 1116-FP 3669.45 .4325 7075.49 .2457
Table 2.14 Results of time history analysis in NONLIN using elastic linear SDoF models
along global directions (X and Y)
Hazard
Level
Ground Motion Global X Global Y Maximum
Shear
(kips)
Disp.
(feet)
Shear
(kips)
Disp.
(feet)
Shear
(kips)
Disp
(feet)
MCE
NGA 753-FN 2757.22 .2669 7356.05 .2492
X
8426.75 .8157
NGA 753-FP 6905.20 .6684 20409.00 .6913
NGA 828-FN 4165.44 .4032 18875.97 .6394
NGA 828-FP 8426.75 .8157 20743.91 .7027
Y
20743.9
1
.7027
NGA 960-FN 6136.84 .5940 9195.69 .3115
NGA 960-FP 3801.43 .3680 7144.12 .2420
EE
NGA 181-FN 2950.20 .2856 6803.53 .2305
X
4051.63 .3922
NGA 181-FP 3160.45 .3059 4206.73 .1425
NGA 1048-FP 3048.14 .2950 3168.14 .1073
NGA 1048-FP 3105.05 .3006 5007.08 .1696
Y
6803.53 .2305
NGA 1116-FP 2296.96 .2223 5644.35 .1912
NGA 1116-FP 4051.63 .3922 6282.22 .2128

Final Report CIE619

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Final Report CIE619