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MICHIGAN STATE UNIV

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This is to certify that the
dissertation entitled ‘
STOCHASTIC RESPONSE OF DECK ARCH BRIDGES T0 CORRELATED
' SUPPORT EXCITATIONS
presented by

BASHEER NMAIR SWEIDAN

has been accepted towards fulfillment
of the requirements for

Eh . D _ degree in _C_i3Lil_and_Environmental
Engineering

WM

Major professor

Date May 8, 1990

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c:\clm\datedue.un&n.'

STOCHASTIC RESPONSE OF DECK ARCH BRIDGES

TO CORREJAIED SUPPORT EXCITATIONS

By

Basheer Nmair Sweidan

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Civil and Environmental Engineering

1990

 

Sto
formed on m
California
Virginia.

A 5
both cohere
SUpport mc
ments was \
tYPES of e
lated SUpp-
and (3) cu
COherency
model Par
Sidered.

Tl
Variation
Plane res
ments of 1
Velocity
bridges.
on the l;

an°ther_

IA

kl!

 

ABSTRACT

STOCHASTIC RESPONSE OF DECK ARCH BRIDGES
TO CORRELATED SUPPORT EXCITATIONS

By

Basheer Nmair Sweidan

Stochastic analysis to correlated support excitations was per—
formed on models of the 700 foot Cold Spring Canyon Bridge (CSCB) in
California, and the 1700 foot New River Gorge Bridge (NRGB) in West
Virginia.

A space-time earthquake ground motion model that accounts for
both coherency decay and seismic wave propagation is used to specify the
support motion. A random vibration approach combined with finite ele-
ments was used to develop expressions for the structural response. Three
types of excitations were considered at the supports: (1) fully corre-
lated support motion; (2) delayed excitation caused by wave propogation;
and (3) correlated excitation accounting for both wave propagation and
coherency decay. For each type of support motion, two seperate sets of
model parameters representing stiff and soft site conditions were con—
sidered.

The results of the study indicate that the effect of the spatial
variation of ground motion is very significant, especially on the in-
plane responses of axial forces, bending moments and vertical displace-
ments of the bridge. The ground motion parameters and seismic wave
velocity is found to substantially influence the responses of the two
bridges. The influence of different correlation models of ground motion
on the lateral responses was irregular and differs from one member to
another. The lateral displacements were not as greatly influenced as
the vertical displacements by the type of correlation of support excita—

tion. The response to the wave propogation effect and the more general

 

case (inclu
the most, a

Non
an earthqua

the respons

 

case (including wave propogation and coherency decay) were within 20% at

the most, and in general within 7% to 10% from each other.
Nonstationary response was also examined. It was found that for

an earthquake having a duration of strong shaking of 5 seconds or more,

the responses were close to those for stationary exctiation.

 

To my wife Carol .
and my children Eric and Gina

for their love and encouragement

 

Wi

substanti

Without tr

this study

I
Environm
DI. Robe:

serving 0]

ACKNOWLEDGEMENTS

With deep gratitude and thanks I would like to acknowledge the
substantial contribution made to this study by Dr. Ronald Harichandran.
Without the assistance, cooperation and the help of Dr. Harichandran
this study would not have been possible.

I would also like to thank the Department of Civil and
Environmental Engineering. Finally, I Would also like to thank
Dr. Robert K. Wen, Dr. Parvis Soroushian and Dr. Norman Hills for

serving on the dissertation committee and for their helpful comments.

 

LIST OF TAB'

LIST OF FIG
LIST OF SYM
1. GENERAL
1.1

1.2

1.3

2. ARCH Bi
2.:

2.1

2.

2.

TABLE OF CONTENTS

 

LIST OF TABLES

LIST OF FIGURES

LIST OF SYMBOLS

1. GENERAL INTRODUCTION
1.1 LITERATURE REVIEW
1.2 OBJECTIVE AND SCOPE

1.3 ORGANIZATION .

2. ARCH BRIDGES AND BRIDGE MODELING
2.1 ARCH BRIDGES
2.1.1 CLASSIFICATION OF ARCH BRIDGES
2.2 DECK ARCH BRIDGES USED IN THE STUDY
2.2.1 COLD SPRING CANYON BRIDGE
2.2.2 THE NEW RIVER GORGE BRIDGE
2.3 BEAM ELEMENT, STIFFNESS MATRIX, LOCAL AND
GLOBAL COORDINATES
2.3.1 BEAM ELEMENT WITH WARPING AND
SHEAR DEFORMATION .
2.3.2 STIFFNESS MATRIX
2.3.3 LOCAL AND GLOBAL COORDINATE AXES.
2.4 MODELING OF THE TWO BRIDGES
2.4.1 VERSIONS OF THE ONE-PLANE MODEL .
2.4.2 ARCH AND DECK EQUIVALENT BEAM
STIFFNESSES
2.4.3 MODELING OF THE DECK OF THE
CSCB BRIDGE

2.4.4 MAIN SPAN COLUMN AND CABLE MODELING

ii

Vi

XV

11

14

14

16

18

18

20

29

31

 

 

I»

3.

RANDOM

.5

2.4.5

2

2.

2.

.4.

4.

4.

6

7

8

CSCB APPROACH SPAN MODELING
CSCB MASS DISTRIBUTION
FINAL CSCB MODEL

FINAL NRGB MODEL

LINSTRUCT PROGRAM

2

.5.

1

FURTHER REDUCTION IN THE

DEGREES-OF-FREEDOM

3. RANDOM VIBRATION ANALYSIS

3.1 FINITE ELEMENT FORMULATION OF

.2

.3

EQUATION OF MOTION .

MODAL ANALYSIS AND NORMAL COORDINATES

RANDOM VIBRATION ANALYSIS

3.3.1 THE VARIANCE OF DYNAMIC

3.

3.

7

DISPLACEMENT

VARIANCES OF PSEUDO-STATIC
DISPLACEMENT

COVARIANCE OF PSEUDO-STATIC

AND DYNAMIC DISPLACEMENT

THE VARIANCE OF DYNAMIC END FORCES
THE VARIANCE OF STATIC END FORCES
THE COVARIANCE OF PSEUDO-STATIC
AND DYNAMIC FORCES

SUMMARY .

3.4 THE INPUT MOTION .

3.

3.

5

6

COMPUTATIONAL PROCEDURE

NONSTATIONARY RESPONSE

45

45

47

47

56

59

65

73

75

77

77

79

81

 

 

II»

 

4. ANALYST.

4.1

4.

ANALYSIS RESULTS

4.1 MODAL ANALYSIS

4.

MODELS

l.

1

CSCB MODE SHAPES AND
NATURAL PERIODS
NRGB MODE SHAPES AND
NATURAL PERIODS

OF GROUND MOTION

ANALYSIS RESULTS

4.

3.

1

RESPONSE COMPONENTS OF

CSCB AND NRGB

IN-PLANE RESPONSE OF THE CSCB
IN-PLANE RESPONSE OF THE NRGB
OUT-OF-PLANE RESPONSE OF THE
CSCB AND NRGB .

THE EFFECT OF GROUND MOTION
PARAMETERS

THE CSCB RESPONSE DUE TO

TWO SUPPORT MOTION VERSUS
FOUR SUPPORT MOTION .

THE EFFECT OF INCREASING THE

BRIDGE STIFFNESS ON THE RESPONSE

COMPONENTS

118

139

154

156

THE EFFECT OF WAVE VELOCITY ON THE

RESPONSE OF THE TWO BRIDGES
NONSTATIONARY RESPONSE OF CSCB

AND NRGB

COMPARISON BETWEEN DETERMINISTIC AND

RANDOM VIBRATION STUDIES

iv

163

170

178

 

5.

SUMMARY

REFERENCE

5. SUMMARY AND CONCLUSIONS

5.

5.

REFERENCES

1

.3

SUMMARY

5

.1.

1

MODELS OF THE CSCB AND NRGB

CSCB IN-PLANE RESPONSE

NRGB IN-PLANE RESPONSE

NRGB AND CSCB OUT-OF-PLANE RESPONSE
RELATIVE CONTRIBUTIONS OF RESPONSE

COMPONENTS

THE EFFECT OF SEISMIC WAVE VELOCITY .

NONSTATIONARY RESPONSE OF CSCB

AND NRGB

CONCLUSIONS

RECOMMENDATIONS

183

183

184

185

186

187

187

187

188

190

191

 

 

 

 

Table
Table
Table
Table
Table
Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

2-1

2-2

2-3

2-4

3—1

3-2

4-3

4-4

4-5

4-6

L13T OF TABLES

Element Stiffnesses for CSCB
Lumped Masses for CSCB .
Lumped Masses for NRGB .
Element Stiffnesses for NRGB

Model Parameters for p(v,f)

Parameters for Double-Filter Autospectra .

Relative Contributions of the
Component of CSCB Members End
General Case of Ground Motion
Relative Contributions of the
Component of CSCB Members End
General Case of Ground Motion

Relative Contributions of the

Dynamic

Forces to the

H

Static

Forces to the

1

Covariance of

Static and Dynamic Component of CSCB Members

End Forces to the General Case of

Ground Motion 1

Relative Contributions of the
Component of CSCB Response to
Ground Motion - Spectra 2
Relative Contributions of the
Component of CSCB Response to
Ground Motion - Spectra 2

Relative Contributions of the

Dynamic

General Case of

Static

General Case of

Covariance of

Static-Dynamic Component of CSCB Response to

Ground Motion - Spectra 2

vi

38

39

39

42

83

83

99

100

101

102

103

“.5" _ .. I

 

 

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

4-15

4-2

4-2

3‘
l
5‘

J_\
' A

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

4-8

4-9

4-10

4—12

4-13

4-14

4—15

4—16

4-17

4-18

Normalized In-plane CSCB Responses -
Ground Motion 1
Normalized In-plane CSCB Responses —
Ground Motion 2

Normalized In-plane NRGB Responses

Ground Motion 1

Normalized In-plane NRGB Responses

Ground Motion 2
Normalized Out-of—plane CSCB Responses -
Ground Motion 1
Normalized Out-of-plane CSCB Responses -
Ground Motion 2

Normalized Out-of—plane NRGB Responses

Ground Motion 1

Normalized Out-of—plane NRGB Responses
Ground Motion 2

In-plane CSCB Responses to the General Case:
Ratio of Ground Motion 1 to Ground Motion 2
Responses

In-plane CSCB Responses to the Wave
Propagation Case: Ratio of Ground Motion 1
to Ground Motion 2 Responses

In-plane CSCB Responses to the Fully
Correlated Case: Ratio of Ground Motion 1 to
Ground Motion 2 Responses

Out-of—plane CSCB Responses to the General
Case: Ratio of Ground Motion 1 to

Ground Motion 2 Responses

105

106

115

116

129

130

131

132

140

141

142

143

 

 

 

Table

Table

Table

Table

Table

Table

Table

4_\
.'_

4_\
:

.;_\

Table

Table

Table

Table

Table

Table

Table

4—29

4-30

4-31

4-32

4-34

4-35

Normalized In-plane CSCB Responses to
Support Motion - Ground Motion 2
Normalized Out-of—plane CSCB Respones
Four Support Motion - Ground Motion 1
Normalized Out-of-plane CSCB Respones
Four Support Motion - Ground Motion 2
Ratio of In-plane CSCB Responses with
Support Motion in the General Case to
Responses with Two Support Motion -
Ground Motion 1

Ratio of In-plane CSCB Responses with
Support Motion in the General Case to
Responses with Two Support Motion -

Ground Motion 2

Four

Four

Ratios of CSCB Response at One and Five

Seconds to the Stationary Response

Ratios of NRGB Response at One and Five

Seconds to the Stationary Response

ix

158

159

160

161

162

179

179

 

 

Figure 2-1

Figure 2-2

Figure 2-3

Figure 2-4

Figure 2-5

Figure 2.1

Figure 2.

Figure 2-

Figure 2.

Figure 2-

Figure 2.

FiSure 2.

Figure 2

Figure 2-1

Figure 2-2

Figure 2-3

Figure 2-4

Figure 2—5

Figure 2-6

Figure 2—7

Figure 2-8

Figure 2-9

Figure 2-10

Figure 2-11

Figure 2—12

Figure 2—13

 

LIST OF FIGURES

Typical Deck Arch Bridge

(Excerpted from Dusseau (1985)) . . . . . . . 8
CSCB Elevation View

(Excerpted from Dusseau (1985)) . . . . . . . 9
CSCB Typical Cross-section

(Excerpted from Dusseau (1985)) . . . . . . . 10
NRGB Elevation View

(Excerpted from Dusseau (1985)) . . . . . . . 12
NRGB Typical Cross-section

(Excerpted from Dusseau (1985)) . . . . . . . 13
LINSTRUCT Straight Beam Element End Displacements
(Excerpted from Dusseau (1985)) . . . . . . . 15
Local Coordinate Axes

(Excerpted from Dusseau (1985)) . . . . . . . 19
Cantilevered Segment End Fixity

(Excerpted from Dusseau (1985)) . . . . . . . 22
CSCB Cantilevered Segment End Loads

(Excerpted from Dusseau (1985)) . . . . . . . 23
NRGB Cantilevered Segment End Loads

(Excerpted from Dusseau (1985)) . . . . . . . 24
CSCB Lateral Cables

(Excerpted from Dusseau (1985)) . . . . . . . 33
CSCB Cable Models

(Excerpted from Dusseau (1985)) . . . . . . . 34
CSCB Tower Elevation View

(Excerpted from Dusseau (1985)) . . . . . . . 36

 

Figure 2-11

Figure 2-1

Figure 2-1

Figure 2-1

Figure 3-1

Figure 3-1

Figure 3-

Figire 3-

Flgure 3.

Figure 4.

Figure 4.

FigUre 1..

Figure 4.

Figure 1.

FigUre 4

Figure 2—14

Figure 2—15

Figure 2-16

Figure 2-17

Figure 3-1

Figure 3-2

Figure 3-3

Figire 3-4

Figure 3-5

Figure 4-1

Figure 4-2

Figure 4-3

Figure 4-4

Figure 4-5

Figure 4-6

 

CSCB One-plane Model

(Excerpted from Dusseau (1985))

CSCB Node and Element Numbers
(Excerpted from Dusseau (1985))

NRGB One-plane Model

(Excerpted from Dusseau (1985))

NRGB Node and Element Numbers
(Excerpted from Dusseau (1985))
Finite Element Model of CSCB
(Excerpted from Dusseau (1985))
Finite Element Model of NRGB
(Excerpted from Dusseau (1985))

The Coherency Function p(v,f)
Spectra of Ground Acceleration Using
Estimated Parameters

Spectra of Ground Displacement Using
Estimated Parameters

CSCB In-plane Modes

(Excerpted from Dusseau (1985))

CSCB Out-of—plane Modes

(Excerpted from Dusseau (1985))
NRGB In-plane Modes

(Excerpted from Dusseau (1985))
NRGB Out-of—plane Modes

(Excerpted from Dusseau (1985))
Normalized Variances of CSCB Vertical
Displacements for Ground Motion 1
Normalized Variances of CSCB Vertical

Displacements for Ground Motion 2

xi

4O

41

43

44

49

84

85

85

91

92

93

94

110

111

 

 

Figure 4-7

Figure 4-8

Figure 4-9

Figure 4-1

Figure 4-1

Figure 4.‘

Figure 4.

Figure 4-

Figure 4-

Figure 4

FigUre 4

Figure 4-7

Figure 4-8

Figure 4-9

Figure 4-10

Figure 4-11

Figure 4-12

Figure 4—13

Figure 4-14

Figure 4-15

Figure 4-16

Figure 4-17

 

CSCB Normalized Variances of Vertical
Displacements of Ground Motion 1 with Respect
to the General Case

CSCB Normalized Variances of Vertical
Displacements of Ground Motion 2 with Respect
to the General Case

CSCB Ratio of Variances of Ground Motion 1
to Ground Motion 2 for the Three Cases

of Ground Motion

Deck Longitudinal Force Transfers Mechanism
(Excerpted from Dusseau (1985))

Normalized Variances of NRGB Vertical
Displacements for Ground Motion 1
Normalized Variances of NRGB Vertical
Displacements for Ground Motion 2

NRGB Normalized Variances of Vertical
Displacements of Ground Motion 1 with Respect
to the General Case

NRGB Normalized Variances of Vertical
Displacements of Ground Motion 2 with Respect
to the General Case

NRGB Ratio of Variances of Ground Motion 1
to Ground Motion 2 for the Three Cases

of Ground Motion

Normalized Variances of CSCB Lateral
Displacements for Ground Motion 1
Normalized Variances of CSCB Lateral

Displacements for Ground Motion 2

112

112

113

119

120

121

122

122

123

124

125

 

 

Figure 4-1

Figure 4-1

Figure 4-2

Figure 4-1

Figure 4-

Figure 4-

Figure A.

Figure A

Figure 4

Figure 4

FiSllre l.

F1Sure z

Figure 4-18

Figure 4-19

Figure 4—20

Figure 4-21

Figure 4-22

Figure 4-23

Figure 4-24

Figure 4-25

Figure 4-26

Figure 4-27

Figure 4-28

Figure 4-29

 

CSCB Normalized Variances of Lateral
Displacements of Ground Motion 1 with Respect
to the General Case

CSCB Normalized Variances of Lateral
Displacements of Ground Motion 2 with Respect
to the General Case

CSCB Ratios of Lateral Displacements of
Ground Motion 1 to Ground Motion 2

Normalized Variances of NRGB Lateral
Displacements for Ground Motion 1

Normalized Variances of NRGB Lateral
Displacements for Ground Motion 2

NRGB Normalized Variances of Lateral
Displacements of Ground Motion 1 with Respect
to the General Case

NRGB Normalized Variances of Lateral
Displacements of Ground Motion 2 with Respect
to the General Case

NRGB Ratios of Lateral Displacements of
Ground Motion 1 to Ground Motion 2

Two Support Motion vs. Four Support Motion .
The Relative Contributions of Axial Force in
Longitudinal Bracing #1

The Relative Contributions of Response
Components to Bending Moment in

Deck Member #3

The Relative Contributions of Response
Components to Axial Force in

Deck Member #6

xiii

126

127

128

134

135

136

137

138

155

164

165

166

 

.Ib

Figure 4-3l

Figure 4-3

Figure 4-3

Figure 4-I

Figure 4-

Flgure A.

Figure 4-

Figure 4-

Figure 4.

 

Figure 4-30 The Relative Contributions of Response

Components to Axial Forces in

Arch Member #20 . . . . . . . . . . . . . . . 167
Figure 4-31 The Relative Contributions of Response

Components to Axial Force in

Arch Member #24 . . . . . . . . . . . . . . . 168
Figure 4-32 Responses of the Axial Forces and Bending

Moments in CSCB Arch Members No. 20 and 26 . . 171
Figure 4-33 Responses of the Axial Forces and Bending

Moments in CSCB Deck Members No. 2 and 7 . . . 172

 

Figure 4-34 Responses of the Axial Forces in NRGB Arch
Members No. 34, 40 and 47 . . . . . . . . . . 173
Figure 4-35 Responses of the Bending Moments in NRGB Arch
Members No. 34, 40 and 47 . . . . . . . . . . 174

Figure 4-36 Responses of the Bending Moments in NRGB Deck

 

Members No. 2, 7 and 12 . . . . . . . . . . . 175
Figure 4-37 Responses of the Axial Forces in CSCB

Longitudinal Bracing . . . . . . . . . . . . . 176
Figure 4-38 Responses of the Axial Forces in NRGB

Longitudinal Bracing . . . . . . . . . . . . . 177

 

 

The follov

A

LIST OF SYMBOLS

The following symbols are used in this dissertation:

A

Cov(us.,ud.)
1 1

Cov(si,fi)

(D

H

H

cross section area, or a parameter of ground motion
model;
matrix related to pseudo-static support displacements;

entries of matrix [A];
shear area in x and y directions;

parameter of ground motion model;
damping matrix;

partitions of matrix [C];

covariance of pseudo-static and dynamic displacements;

covariance of pseudo-static and dynamic element end

forces;
vector of nodal displacements in the global coordinate
system;

translation due to a moment Mz;
vector of displacements in member coordinates;
translation due to a force p;

modulas of elasticity;
linear frequency (Hz);
vector of element end forces;

. .th
element end force corresponding to the 1 d.o.f. and

the jth eigenvector;

XV

 

 

ex’ ey

ii

 

shear modulas;

dimensionless shear constant with respect to x and y

axes;

generalized modal excitation;

modal impulse response function;

modal frequency response function;

moment of inertia of cross section with respect to x

and y axes;
stiffness matrix of structure;

partitions of matrix [K];

element length;

"effective" length with respect to x and y axes;
"effective" length with respect to warping;

mass matrix;

partitions of matrix [M];
force in the direction of x and y axes;
rotation due to a force and a moment;

autocorrelation function of the dynamic component of

free displacement;

autocorrelation function of the free displacement;

autocorrelation function of the pseudo-static

component of free displacement;

 

 

cross correlation function of pseudo-static and

dynamic displacements;

cross correlation function of support displacements;

cross correlation function of support displacement and

acceleration;

cross correlation function of support acceleration;
intensity parameter;
spectral density function of dynamic displacement;

spectral density function of pseudo-static

displacement;

spectral density function of free displacement;

spectral density function of pseudo-static and dynamic

displacements;

cross spectral density function of ground

accelerations;

cross spectral density function of ground displacement

and acceleration;

cross spectral density function of ground

displacement;
time;

dynamic nodal displacement of free node;

xvii

iamg'=/

fl

pseudo-

 

static nodal displacement of free node;

free nodal displacement;

vector

vector

vector

vector

vector

of absolute displacements;

of absolute free displacements;

of restrained displacements;

of pseudo-static displacement;

of dynamic displacement;

generalized modal displacement;

empirical parameters describing spectral density

function;

modal participation factor;

separation distance;

coherency of ground acceleration for frequency f and

station seperation u;

variance of dynamic displacement;

variance of psuedo-static displacement;

variance of free dynamic displacement;

variance of dynamic end force;

variance of psuedo-static end force;

variance of end force;

xviii

and

H

 

variance of member shear forces in the direction

of local x-axis;

variance of the dynamic component of member shear

forces in the direction of local x-axis;
variance of the static component of member shear
forces in the direction of local x—axis;

covariance of the static-dynamic component of member

shear forces in the direction of local x-axis;
variance of shear forces in the direction of local y-
axis;

variance of axial forces;

variance of the static component of member shear
forces in the direction of local z-axis;

variance of the dynamic component of member axial
forces;

covariance of the static dynamic component of member
shear forces in the direction of local z-axis;

variance of bending moment about local x-axis;

variance of bending moment about local y-axis;

variance of static component of member bending moments

about local y-axis

 

be.

SUbs\cripl

F. FF
FR, RF

RiRR

 

 

aim)
aim)
ai<*>.a22<*>

oimou) ,ai(ST)

0:07)
(I)

wg, wf
(A).

J

[‘1']

{j

Subscripts

F, FF
FR, RF
R, RR

ll

covariance of the static-dynamic component of the
bending mome about y-axis

variance of dynamic component of member axial forces;

l'h

variance 0 torsional moment;

H)

variance 0 warping moment;

H1

variance 0 any response due to fully correlated case;

H)

variance 0 any response due to general case;

H)

variance 0 any response due to ground motion 1 and 2;

variance of any response due to nonstationary and

stationary excitations;
variance of any response due to

wave propogation case;

circular frequency (rad/sec);

empirical parameters describing spectral density

function;

modal circular frequency;

eigenvectors;

modal damping ratio;

quantity corresponding to free displacement;
quantity corresponding to free and restrained
displacement or vice versa;

quantity corresponding to restrained displacement;

XX

 

 

 

Superscrir

 

 

W

. = first partial derivative with respect to time;
= second partial derivative with respect to time;
" = second partial derivative with respect 7;

T = matrix transpose.

 

very impot
lifelines
T
structuri
or under
to the sp
and desig
T
0f grounr
strong g
Variatic
accounte
motion a
Obtained
determil
travell

Variatio

seiSmic
(DUSSeau
Califorr
t0 dete
the two

presente

CHAPTER 1

GENERAL INTRODUCTION

1.1 LITERATURE REVIEW

Lifeline systems such as bridges and pipelines are part of the
very important infrastructures serving society. The ability of these
lifelines to function after an earthquake is of great importance.

The difference between lifeline systems and conventional
structures is that typical lifelines extends for large distances above
or under the ground surface. Because of this, they are very sensitive
to the spatial variation of earthquake ground motion, and their analysis
and design should take this into consideration.

The traditional method of analyzing structures under the effect
of ground motion is to perform time history analysis based on recorded
strong ground motion. For a long structure with multiple supports, the
variation of seismic ground motion due to travelling waves can be
accounted for by considering the recorded time history as the input
motion at one support, with the input motion at the other supports being
obtained by considering a delay in the arrival of the shear waves. This
deterministic approach, however, is only capable of representing
travelling waves, which is only one feature of realistic space-time
variation of ground motion.

A comprehensive deterministic study on the effect of unequal
seismic support motion was conducted on two steel deck arch bridges
(Dusseau and Wen, 1985). In that study the Cold Spring Canyon Bridge in
California and the New River Gorge Bridge in West Virginia were studied
to determine the effect of unequal support motion on the responses of
the two bridges. An outline of the conclusions of this study will be

presented in Chapter 4.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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7

In recent years, studies have been conducted on the response of
lifeline structures using stochastic models of ground motion. The
response of a suspension bridge subject to multiple support excitations
by means of random vibration theory was conducted by Abdel-Ghaffar
(1982). He found that the response values associated with correlated
multiple-support excitations are significantly different from those
obtained through the uncorrelated case.

In a study on the response of a burried pipline to random ground
motion, Hindy and Novak (1980) concluded that the lack of correlation of
seismic excitation could produce excessive stresses in the pipe. These
stresses depend on the degree of correlation of the excitation and its
frequency content.

The effects of spatially varying stochastic model of ground
motion on the responses of pipelines and bridges of various span lengths
in the longitudinal, lateral and vertical directions were also studied
by Harichandran and wang (1988), and also by Zerva and Wen (1988).
Based on these, the following conclusions can be made:

1. The assumption of perfectly correlated earthquake ground
motion is not always safe in the seismic evaluation of
pipelines.

2. The effect of differential ground motion is not significant
for typical single-span, simply supported bridges, and
assumming identical support excitations will lead to
conservative stress response estimates. The assumption of
perfectly correlated support motion is therefore a valid
approximation for spans up to 200 m.

3. The spatial variation of earthquake ground motion is

important for the analysis of indeterminate structures, and

 

 

 

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neglecting this can result in significant error in stress
estimation.
All the above mentioned studies indicate the importance of
considering the effect of spatial variations of earthquake ground motion

on the response of long structures.

1.2 OBJECTIVE AND SCOPE

In this study the models of two deck arch bridges, the Cold
Spring Canyon Bridge and the New River Gorge Bridge are examined under
the effect of partially correlated multiple support excitation. The
formulation of the equations of motions is developed using finite
element and random vibrations methods. The responses of the bridges are
obtained for different types of ground motion inputs, including those
commonly used in current practice. The responses of the bridges are
analyzed and compared to determine the worst type of ground motion that

induces the highest responses.

1.3 ORGANIZATION

 

A brief summary of the contents of each chapter is presented
here.

Chapter 2 covers in some detail the description of the two
bridges. The method used by Dusseau (1985) to obtain the one-plane
model is diSCussed in some detail, since this same model is used in this
work. Also, the chapter contains discussions about the local and global
coordinate system, transformations and stiffness matrices. A
description of the computer program used in the analysis is provided.

Chapter 3 presents the development of the equations of motions
using finite element and random vibration methods. Detailed

deriviations of the response components is presented. The model used

 

 

 

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discussed.

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A

for the cross spectral density function of the ground acceleration is
discussed.

In Chapter 4, results of fairly extensive analysis of the two
bridges are presented. The responses of the two bridges to different
types of ground motions are discussed in detail. The effects of
structural stiffness and shear wave velocity are studied. The
assumption that the ground motion constitute a stationary random field
is checked. A comparison between this study and the deterministic study
conducted by Dusseau and Wen (1985) is presented.

In Chapter 5, major conclusions of the results from this study
are summarized. the avenues in which future research in this area may

proceed are also discussed.

 

 

 

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CHAPTER 2

ARCH BRIDGES AND BRIDGE MODELING

This chapter contains a summary of description of the Cold
Spring Canyon Bridge (CSCB) and the New River Gorge Bridge (NRGB)
(Dusseau and Wen, 1985). Also, this chapter contains the procedure of
modeling the (CSCB) and (NRGB) into in-plane and out-of—plane models
(Dusseau and Wen, 1985), so that the reader of this dissertation will
have the full information to understand the models that are used to
analyze the two bridges.

The first section of this chapter discusses the basic features
of arch structures and their classification.

The second section describes the two deck arch bridges used in
this study, and includes the geometric parameters of the bridges and
their main structural components.

The third section discusses the type of finite element used in
the analysis, their stiffness matrices, and the local and global
coordinate systems.

The fourth section discusses the modeling of the two bridges as

two-dimensional planar structures.

2.1 ARCH BRIDGES

Arch Bridges made of steel and reinforced concrete have been
used in transportation networks throughout the world. With the use of
structural steel it is possible to economically construct long span arch
bridges ranging from a minimum of about 190 ft. to a maximum of about
1700 ft. With present high-strength steels and under favorable soil
conditions, spans of the order of 2000 ft. are feasible for economical

arch construction.

 

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6

Cross sections of the arch are designed for axial thrust,
bending moment and shear forces, with magnitudes depending on the
location of the pressure line (funicular polygon of applied loads). If
the pressure line coincides with the axis of the arch (as in uniformly
loaded parabolic arches), all cross sections are subject to compression
with no moment or shear. If the pressure line falls within the kern of
the section, there will exist thrust, bending moment and shear, but no
tension on the cross section. Finally, if the shape of the structure

differs from the pressure line, moment may become dominant.

2.1.1 CLASSIFICATION OF ARCH BRIDGES

Arch Bridges are classified as trussed or solid ribbed. If the
horizontal thrust is taken by structural ties between the reaction
points then the arch is referred to as a tied arch.

Arches are also classified according to the degrees of static
indeterminancy. A fixed arch in which rotation is prevented at the ends
is a three degrees indeterminate structure. A one-hinged arch is a two
degrees indeterminate structure. A two-hinged, and three-hinged arch is
a one and zero degrees indeterminate structure, respectively.

In addition arch bridges are classified as "deck construction"
when the arches are entirely below the deck, "through arch", when the
arch is entirely above the deck and the tie is at deck level, "half
through arch" when the deck is at some intermediate elevation between
springing and crown.

In this study we are concerned with the deck arch bridges,
because the mass of the bridge is concentrated in the deck at a high
elevation from the springing. This makes the deck arch bridges
extremely vulnerable to seismic ground motion relative to the other

types of arch bridges. Figure 2-1 shows a typical deck arch bridge.

 

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7
2.2 DECK ARCH BRIDGES USED IN THE STUDY

In this study two bridges were chosen: The Cold Springs Canyon
Bridge (CSCB) in California and the New River Gorge Bridge (NRGB), in

West Virgina, the worlds longest steel deck arch Bridge.

2.2.1 COLD SPRING CANYON BRIDGE

The Bridge is located about 13.5 miles North of Santa Barbara,
California. It is a two lane solid-ribbed steel deck arch bridge. All
major structural steel members are made of A373 steel which has a
minimum yield strength of 33 ksi. Figure 2-2 shows an elevation view of
the CSCB, while Figure 2-3 depicts a typical cross section. As shown in
Figure 2—2, the bridge consists of 19 panels with two of 46.5 feet
length, 13 of 63.64 feet length and four of 74.385 feet length, yielding
an over all length of 1217.8 feet. The two hinge arch consists of two
rectangular steel box girders spaced 26 feet apart and hinged at their
abutments. The arch has 11 panels of 63.64 feet length, yielding a
total arch span of 700 feet.

The configuration of the arch is based on a seventh degree
polynomial with the southern hinges being 46.48 feet above the northern
hinges and with the rise at the highest point of the arch being 144.5
feet above the northern hinges, seventh degree polynomial was used to
minimize dead load moments in the arch. This configuration also makes
the main span column heights symmetric about the center of the arch span
despite the overall deck slope of 6.64%.

The arch ribs are connected laterally by a system of crossframes
with one crossframe at each panel point and three crossframes spaced

equally between panel points. The ribs are also connected laterally by

 

 

 

 

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Figure 2-3: CSCB Typical Cross-Section. (Excerpted from Dusseau (1985)

 

 

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11
top and bottom lateral bracing which, along with crossframes and the
arch ribs, creates a box shaped cross-section with the arch ribs acting
as sides.

The columns located at panel points 2 to 5, 7 through 16, 18 and
19 are steel box sections with hinge connections at the top and bottom.
The towers at panel points 6 and 17 consist of steel box section columns
that are rigidly fastened at their bases and are connected laterally by
two steel box girders, intermediate struts, composite steel box girder,
and concrete slab at the top.

The deck consists of a 7 inch two-way reinforced concrete slab
which acts compositely with four longitudinal plate girder stringers,
the latter being supported by plate girder floorbeams. The deck is
divided into three continuous segments by hinged tower connections at
panel points 6 and 17, which provide a release for in plane bending
moment and warping bimoment at these points.

Between panel points 11 and 12, a system of cable x-bracing is
provided between the deck and the arch in the longitudinal direction,

and a system of cable v-bracing in the lateral direction.

2.2.2 THE NEW RIVER GORGE BRIDGE

The New River Gorge Bridge is a four lane, box truss, steel deck
arch carrying U.S. 19 over New River Gorge and Route 82 in West
Virginia. The principal material used in the bridge is ASTM A588 grade
A steel with a minimum yield stress of 50 ksi.

Figure 2-4 shows an elevation View of NRGB, while Figure 2-5 is
a typical cross-section. As shown in the figures, both the deck and the
arch in the NRGB are essentially box trusses consisting of four box
girder chords connected by lateral and vertical truss members. Each

panel in the deck is divided into 6 subpanels, while the arch panels are

 

 

 

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bent Y
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L 72- J

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Figure 2-5: NRGB Typical Cross-section. (Excerpted from Dusseau (1985)

divided i
panels in
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14

divided into 1% or 3 subpanels. The deck in the NRGB consists of four
panels in the north approach span, each of length 143.5 feet, five
panels in the south approach span, each of length 126.5 feet, and 14
panels in the main span, each of length 129.75 feet, yielding a total
bridge span of 3030.5 feet. The two hinged arch consists of 12 center
panels, each of length 129.75 feet, and two end panels, each of length
71.5 feet, yielding a total arch span of 1700 feet.

The configuration of the arch is based on a symmetric five
centered series of circular arcs which results in a maximum arch height
of 370 feet above the hinges. The deck and arch are connected at each
panel point in the main span by bents consisting of two box section
columns joined laterally by diagonal truss elements. Similar bents
connect the deck to concrete pedestals at each panel point in the north
and south approach spans. The approach span deck segments are isolated
from the main span deck segments by expansion joints at the top of bents
5 and 19. At these points, the bottom chords of the approach span deck
are pinned to the top of the bents while the bottom chords of the main
span deck are attached to the bents by rollers. Thus the expansion
joints provide deck axial force, bending moment and warping bimoment

releases at these points.

2.3 BEAM ELEMENT STIFFNESS MATRIX LOCAL AND GLOBAL COORDINATES

 

2.3.1 BEAM ELEMENT WITH WARPING AND SHEAR DEFORMATION
The Beam Element used in performing the random vibration
analysis of the two bridges includes the warping deformation (W.F. Chen

and T. Atsuta 1977). Each node has seven degrees of freedom:

Translations Ux

U , and Uz’ rotations 0x, 93x and O 2 and warping

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1977) in<
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matrix ar

K(3,3) =

MW) 4

KW) s
K(7,11)

K014)

K(1,1) .

 

l6
displacemern: 9w. The shear deformation is also included in the
stiffness matrix formulation. Figure 2—6 illustrates the beam element
and the local coordinate system. Also, a standard space truss element
with three translational degrees of freedom at each node is used to

model the truss members.

2.3.2 STIFFNESS MATRTX

The stiffness matrix used in this study (W.F. Chen and T. Atsuta
1977) includes the warping deformation, with some modification to
account for shear deformation. The non-zero entries of the stiffness
matrix are listed below.

E

 

 

K(3,3) = K(10,10) = if (2.1)
K(3,10) = K(10,3) = -%A (2 2)
12EI GK
K(4,4) = K(ll,ll) = 3W + %% “I; (2 3)
L
K(4,ll) = K(ll,4) = -K(4,4) (2.4)
4EIW 4
K(7,7) = K(14,14) = L + 56 GKtL (2.5)
6EIW 3
K(4,7) = K(7,4) = K(4,14) = K(l4,4) = ——E— + 35 GKt (2.6)
L
K(7,ll) = K(ll,7) = K(ll,l4) = K(l4,ll) = -K(4,7) (2.7)
2E1w GKtL
K(7,l4) = K(l4,7) = —L—_ ~ ‘ga— (2.8)
12EI
K(l,l) = K(8,8) = 3 (2.9)
L (1+2.o cl)
(2.10)

 

K(l,5) =

K(8,5) =

K(6,131

Where 1

K(l,8) =

K(1,5) =

K(8,5) =

K(5,5) =

K(5,l2) = K(l2,5) =

K(2,2) =

K(2,9) =

K(2,6) =

K(9,6) =

K(6,6) =

K(6,13) =

where I
w

Kt

l7

-12EI

L3 (1+2.o G1)

K(8,l)

K(5,l) = K(l,12) = K(12,l) =

K(5,8) = K(8,l2) = K(l2,8) =

4E1 (1+G1/2.O)

K(12,12) = L (1+2.o G1)

2EI (1+cl/2.0)
L (1+2 0 cl)
12EI
X

K(9,9) = X

L3 (1.0+2.o c2)

12EI
xx

L3 (1 o+2.o c2)

K(9,2)

K(6,2) = K(2,l3) = K(13,2) =

K(6,9) = K(9,13) = K(l3,9) =

6E1
______JCL_____

L2 (1+2.o G1)

-6EI
___.__11_____

L2 (1+2.o G1)

-6EI
XX

L2 (1 0+2.o c2)
6E1
XX
L2 (1.0+2 c2)

aerxx (1.0+c2/2.0)

K(l3,13) =

2EIXX (1.0-G2)

K(13’6) = L (1.0+2.o 02)

- warping moment of inertia.

- torsional constant.

L (l.0+2.0 G2)

.11)

.12)

.13)

.14).

.15)

.16)

.17)

.18)

.19)

.20)

.21)

.22)

 

 

 

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18
G1 and G2 dimensionless shear constants with respect to y and x
coordinates.
If the warping moment of inertia is equal to zero then the
stiffness matrix will be the same as for the standard space frame

elements.

2.3.3 LOCAL AND GLOBAL COORDINATE AXES

Figure 2-7 illustrates the local and global coordinate system
used in the analysis. The transformation matrix formulation is based on
the K node method (W.Weaver Jr., and J.M. Gere, 1965). This method was
the coordinates of a third point that lies in one of the principal
planes of the member but is not on the member axis itself. In this
study, the third point must lie in the x-z plane of the elements local
coordinate system, and it cannot lie on the centroidel z axes of the

element.

2.4 MODELING OF THE TWO BRIDGES

As mentioned earlier the Cold Spring Canyon Bridge in California
and the New River Gorge Bridge in West Virginia were studied. It was
very important that the models of both bridges have the fewest number of
degrees of freedom. It would have been very difficult from a
computational time point of view to study the probabilistic response of
the two bridges without having a model of the bridges with reduced
degrees of freedom. The so-called "one-plane—model" is used to
represent each bridge, and the structural parameters derived by Wen and
Dusseau (1985) for the two bridges were used. For completeness, the
next few sections give a summary of the method used by Wen and Dusseau

(1985) to model the two bridges, so that the reader will have a clear

 

 

 

 

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local coordinates

    
 

global
coordinates

Figure 2-7: Local Coordinate Axes. (Excerpted from Dusseau (1985))

 

 

 

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20

idea of the "one-plane-model" approach which was used in this study.More

detail is given in Dusseau (1985).

2.4.1 VERSIONS OF THE ONE—PLANE MODEL

The first version of the one-plane model is the "in-plane” model
and was used in the analysis of the two bridges in the X-Y plane for the
horizontal ground motion. Each node has two translational degrees of
freedom in the Global X and Y directions and one rotation about the
global Z axes.

The second version of the one-plane model is the "out-of—plane"
model and was used in the analysis of the two bridges in the lateral
direction. Each node has four degrees-of—freedom (d.o.f.): one
translational d.o.f. in the direction of the Z axis; two rotational

d.o.f.’s about the X and Y axes and one warping d.o.f.

2.4.2 ARCH AND DECK EQUIVALENT BEAM STIFFNESSES

To use the one~plane models for the NRGB and CSCB, it was
necessary to determine the equivalent beam stiffnesses for the arch and
deck components of both bridges. The stiffnesses of the deck for the
CSCB were based on the composite action of the concrete slab and steel
girders. To derive the equivalent beam stiffnesses for the arch of the
CSCB and both the arch and deck of the NRGB, special "cantilevered"
segments of the arch and deck components were analyzed by Dusseau
(1985).

Each cantilevered segment of arch or deck was one panel in
length and included all of the structural components in the actual
bridge. The member stiffnesses and lengths used in these cantilevered
segments were determined as follows: for the deck in the NRGB, three

cantilevered segments were selected to derive the equivalent bean

 

 

 

 

stiffness
the deck;
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21

stiffnesses that will represent the properties of all existing panels in
the deck; for the arch in the NRGB, the cantilevered segment near the
abutment was used to derive the equivalent beam properties of the
strongest section of the arch, and a segment at the crown of the arch
was used to represent the weakest section in the arch; for the CSCB
arch, average member stiffnesses and lengths in the end panels, in the
quarter point panels and in the crown were used to develop three
cantilevered segments, respectively.

Each cantilevered segment was "fixed" at one end and loads were

applied at the other end. For the CSCB arch which consists of two box
girders connected by lateral members, "fixing" one end of each
cantilevered segment meant preventing all translations and global z axis
rotations of the box girders at one end as shown in Figure 2-8a.
For the NRGB deck and arch, which each consist of four box girder chords
connected by lateral and vertical truss members, "fixing" one end of
each cantilevered segment meant preventing all translations of these box
girder chords at one end as shown in Figure 2-8b.

With one end "fixed", a series of equivalent beam loads were
applied to the free end of each cantilevered segment and the resulting
displacements were then used to determine equivalent beam stiffnesses.
For the CSCB arch assembly, which contains two box girders, forces and
moments equivalent to the desired beam end loads were applied at the
free ends of the box girder ribs, as depicted in Figures 2-8a and 2-9.
For the NRGB arch and deck assemblies, each of which has four box
girders chords, equivalent beam end loads were derived by applying point
loads to the free ends of the box girder chords as shown in Figures 2-8b
and 2-10.

After fixing the cantilevered section at one end and performing

the analysis due to different loading conditions at the other end, the

 

 

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b) CSCB Cantilevered Segment

‘4

Figure 2-8: Cantilevered Segment End Fixity.

(Excerpted from Dusseau (1985))

 

 

23

"bracing"

 

 

 

 

 

 

 

 

? t

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

b)Mz
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/ I
/ 4,
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Figure 2-9: CSCB Cantilevered Segment End Loads

(Excerpted from Dusseau (1985))

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

Figure 2-10: NRGB Cantilevered Segment End Loads

(Excerpted from Dusseau (1985))

 

 

 

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aspeci

calcula

where E

moment

the NR

and 2.

torsio

Where

cantp

the f1

regul

25

fixed and loaded ends were reversed so that two sets of end
displacements were obtained. For the in-plane model the two sets of end
displacements were the same. For the out-of—plane model the two sets of
end displacements were different, and the equivalent straight beam
stiffnesses were based on the larger set of end displacements for both
bridges.

To determine the axial area of the equivalent beam element, an
axial force P was applied to the free end of each cantilevered segment
(see Figures 2—9a and 2-10a). Knowing the axial displacement due to
aspecified axial force, the area of an equivalent beam element is

calculated using the formula
A = —— (2.23)
where P is the applied force 6 is the calculated axial displacement.
Similarly, to determine the torsional constant Kt’ a torsional
moment Mx was applied to the free end of each cantilevered segment of
the NRGB and CSCB arches, where warping was ignored (see Figures 2-9b

and 2-10b). The resulting rotation Oz wasrmed to calculate the

torsional constant by the formula
M L
= ———z (2 24)
t Gez '

where G is the shear modulus.

In order to determine the out-of—plane shear area Ay for each
cantilevered segment, an equivalent beam shear force Py was applied to

the free end of each cantilevered segment (see Figures 2-9c and 2-lOc)

resulting in a translation Dp and a rotation RP. Similarly, an

 

 

 

 

 

 

 

equivaler

resulting

types of

The two

the syr
the sam
be so];
element

0f thr

Called

exPress

SubSti.

Finau

26

equivalent bending moment M2 was applied (see Figures 2-9d and 2-10d)
resulting in a translation Dm and a rotation Rm'

The two pair of flexibility equations that govern these two
types of loading are

P L3 P L

_ .X___ _l_
Dp — 3E1 + GA (2.25)
xx y

R = y— (2.26)

D = __ (2.27)

R =_ (2.28)

The two unknowns in the last four equations are IXX and Ay. Because of

the symmetry of the flexibility equations, (2.26) and (2.27) will yield
the same results, thus only equations (2.25), (2.26) and (2.28) need to
be solved in order to achieve equivalence. Letting the length of the
element be a variable equations (2.25), (2.26) and (2.28) yield a system

of three equations with three unknowns Ix , A and Lex’ where LeX is

X y

called the "effective" beam length with respect to out-of—plane motion.
Dividing equation (2.26) by (2.28) yields the following

expression for L .
(ex

2R MX
Lex = P R (2.29)
y m
Substituting (2.29) into (2.28) yields the expression for Ixx'
I =2R MZ/ (EP R2) (2.30)
xx p x y m

Finally, equation (2.25) yields the following expression for Ay.

 

 

 

 

 

I , Sh!
Y)’

obtained

shown ir

equiva‘.
determi1

each ea

in an a
M was
w

andaw

types (

27

3E1
xx

P L3
A = P L / [a [n - —X——§§— ] ] (2.31)
Y Y ex P
Similar expressions for moment of inertia with respect to y axes

Iyy’ shear area AK and the in- plane “effective" beam length Ley were

obtained using similar procedures. The applied free end forces are
shown in Figures 2-9e, 2-9f, 2-lOe and 2-10f.

For the deck in the NRGB, where warping was not ignored, the
equivalent beam warping constant Iw and torsion constant Kt were
determined by applying an equivalent beam torque MZ to the free end of
each cantilevered segment (see Figures 2—9b and 2-10b), which resulted
in an axial rotation Rt and a warping displacement Wt. Then, a bimoment
Mw was applied (see Figures 2-9g and 2-lOg), resulting in a rotation Rw
and a warping displacement Mw'

The two pairs of stiffness equations that governs these two

types of loadings are

 

 

12EI GK 6E1
_ [ w 36 c] T _ [ w 3 (2_32)

+
L3 30 L

 

___r _é
L + 30 GKtIJ wt (2.33)
W + —3 GK w (2 34)
2 30 t w '

 

EIW
L + % GKtL] Mw (2.35)

 

 

 

 

 

in these

symmetry
the same
need to l
letting

(2.33) a

where 1.1

torsion,

followir

Lew = 4r

where A
B

C
K=-
tG<
for al
bridge

28

In these four equations, the two unknowns are KC and Iw. Because of

symmetry of stiffness equations, equations (2.33) and (2.34) will yield
the same results. Therefore, only equations (2.32), (2.33) and (2.35)
need to be solved to achieve equivalence. As in the case of bending, by
letting the length of the element L be a variable in equations (2.32),

(2.33) and (2.35), three equations in the three unknowns Kt’ Iw and Lew,

where Lew is the “effective" beam length with respect to warping and
torsion, were obtained.
Solving equations (2.32), (2.33) and (2.35) results in the

following expressions for Lew, KC and Iw'

= /—_— .
Lew (_B + (132-4 AC) / 2A (2 36)
2
where A = (-Rm wt Mz) + (wIn Rt Mz) + (MW wt) (2.37)
B = ant MW Rt (2.38)
c = -15 MW R2 (2.39)
t

30MzLew (-3Rt+2WtLew)
t G(36Rt-3WtLew)(—3Rt+2WtLew)+(6Rt—3WtLew)(3Rt—4WtLew)

 

K (2.40)

2 2
(GKtLew) (3Rt-4WtLew)

Iw = (60E) (-3Rt+2WtLew) (2.41)

 

By deriving the beam equivalent stiffness and effective length
for all segments of the arch and deck, the equivalence between the real

bridge structure and the model is assured.

 

 

 

were cl
stiffnes
by usin
and weal
panels

panelsi
and Lev

were di

average
Thus,

five pa
corresp

element

calcul
the c<
Strlny

(111055 .

modere
fOur

Cross.
and t

SiHCe

29

As mentioned earlier, two cantilevered segments of the NRGB arch
were chosen to represent the strongest and weakest sections. The
stiffness of intermediate segments were calculated at segments midpoints
by using linear interpolation between the stiffnesses of the strongest
and weakest sections. The straight beam elements representing the end
panels in the NRGB arch were 1% subpanels in length, while the other
panels were 3 subpanels. For this case the equivalent lengths Lex, Ley
and Lew, which were calculated for the midpoints of the end elements,
were divided by two.

For the CSCB arch, the three cantilevered segments were based on
average member sizes and lengths over five of the eleven arch panels.
Thus, the stiffnesses for the straight beam elements representing these
five panels were taken to be the same as the values calculated for the
corresponding cantilevered segments. For the intermediate straight beam

elements, the stiffnesses were determined by linear interpolation.

2.4.3 MODELING OF THE DECK OF THE CSCB BRIDGE

Equivalent straight beam stiffnesses for the CSCB deck were
calculated based on the composite action of the four floor stringers and
the concrete roadway deck. Because there are four average floor
stringer cross-sections in the CSCB deck and hence four average deck
cross-sections, four sets of deck stiffnesses were calculated.

Since the roadway slab can be expected to crack under relatively
moderate loads, the first step in calculating the stiffnesses of the
four CSCB deck cross-sections was an estimation of the portion of the
cross-sectional area of the roadway slab that would be in compression
and thus contributing to overall deck stiffness at any given time.
Since the portion of the slab area in compression could be anything from

0 to 100%, a compromise value of 50% was chosen. This assumption

 

 

 

coupled '
"effecti
by a fac
four £1:
to calcu

torsion

beans,

with the
stringe
convers:
divide'
effecti
center
calcule
channe
a recta
inerti
the st:
of On.
dimens

warpir

COnsta

Semi-r
Warpij
connec
my i.

joint:

30

coupled with a modular ratio of steel to concrete of 10, led to an
"effective" modular ratio of 20. Thus the area of concrete was reduced
by a factor of 20 and then used in conjunction with the areas of the
four floor stringers, the slab reinforcing steel and the deck laterals
to calculate the axial area, shear areas, moments of inertia and the
torsion constant for each deck cross-section.

In determining the warping constants for the equivalent deck
beams, each deck cross-section was first converted to a channel section
with the concrete roadway slab acting as the channel web and the outside
stringers acting as the channel flanges. The first step in this
conversion was to reduce the area of concrete by a factor of 20 and then
divide by 28 feet (the distance between the outside stringers) to get an
effective channel web thickness. Next, the average distance from the
center of the roadway slab to the bottoms of the floor stringers was
calculated and used as a channel flange width. Then, an effective
channel flange thickness was calculated by determining the thickness of
a rectangular flange moment of inertia equal to 1.33 times the moment of
inertia of one floor stringer. The factor of 1.33 was based on 100% of
the stiffness of one exterior floor stringer plus 33% of the stiffness
of one interior floor stringer. Finally, with all of the channel
dimensions determined, the values were substituted into the general
warping constant formula for a channel section and thus the warping
constants for the four deck cross-sections were determined.

The deck expansion connection at panel point 1 was modeled as
semi-rigid with global X axis translation, Y and Z axis rotations, and
Warping displacements of the deck, allowed at this point. The bearing
connection at panel point 20 was also modeled as semi-rigid but with
only Y and Z axis rotations and warping displacements allowed. The deck

joints at panel points 6 and 17 were modeled such that no Z axis moments

 

or warp:

decks an

main sp
truss e2
the col
and bot

to be t‘

panel p
the CSC
one la
dimens:
result

beams

L is t

12 in

in the

simil;

and r7
deck

bimom

31

or warping bimoments could be transferred between the approach span

decks and the main span deck.

2.4.4 MAIN SPAN COLUMN AND CABLE MODELING

Except for the columns at panel points 11 and 12, each pair of
main span columns at a given panel point were represented by a single
truss element in the CSCB models. Truss elements were chosen because
the columns in the CSCB (excluding the towers) are hinged at both top
and bottom. The cross-sectional area of these truss elements was taken
to be twice the cross-sectional area of one column.

The systems of columns and transverse cables (Figure 2-11) at
panel points 11 and 12 were each represented by a single beam element in
the CSCB models. A pair of truss elements representing one column and
one lateral cable as illustrated in Figure 2-12 were analyzed in two

dimensions. Loads Py and P2 were applied in turn at the free joint
resulting in translations Dy and Dz. The axial areas of the equivalent
beams at panel points 11 and 12 were determined using A=2PyL/(EDy) where

L is the distance between the arch and deck nodes at panel points 11 and
12 in the one plane models of the CSCB.

The moment of inertia about the global X axis, the shear area in
in the global 2 direction and the effective length were determined in

similar way as discussed in section 2.4.2.

2.4.5 CSCB APPROACH SPAN MODELING

 

The approach spans in the CSCB were represented by translation
and rotation springs. These springs were located at the centroid of the
deck at panel points 6 and 17. Because of the Z axis moment and the

bimoment releases in the deck at panel points 6 and 17. Because of the

 

 

Z axis m
and 17, 1

these po

parallel
inertia
the rep
represei

springs

translz
17, the
with t
element
centrt
approa
elemen
deck a'
in the
the di
the s
disple

3X15 p

32

Z axis moment and the bimoment releases in the deck at panel points 6
and 17, no Z axis rotation springs or warping springs were needed at
these points to represent the approach spans.

The beam elements used to represent the approach span decks were
parallel with the global X axis and were of equal length. The moment of
inertia with respect to Y axis and the Z axis shear area were taken as
the represented segment properties. The other section properties were
represented as part of the stiffnesses of the translation and rotation
springs at panel points 6 and 17 as described below.

In order to derive the stiffnesses of the X and Y axis
translation springs and X axis rotation springs at panel points 6 and
17, the north and south approach spans were analyzed in their entirety
with the tower columns, tower struts and deck represented as beam
elements and the remaining columns represented as truss elements. The
centroid of the continuous beam that represented the deck in each
approach span was fastened to the tops of the columns using rigid
elements. Three loads were then applied in turn at the centroid of the
deck at panel point 6 in the south approach span and at panel point 17

in the north approach span. The first load applied was a force FX in
the direction of the global X axis, which resulted in a displacement Dx’
the second load was a Y direction force Fy which resulted in a
displacement Dy and the third load was a moment Mx about the global X
axis which resulted in a rotation ¢x'

The stiffnesses of the X and Y axis translation springs

representing each approach span were determined by Sx=Fx/Dx and

syaFy/Dy’ respectively. The stiffness of the X axis rotation spring was

calculated using RX=Mx/¢X. The X axis translation and rotation springs

 

 

 

dec

colt

arcl
ril

Figur

33

lateral cables

deck"Z>

  
   

column

é?— column

arch

arch rib
rib ’Z) 5

Figure 2-11: CSCB Lateral Cables. (Excerpted from Dusseau (1985))

 

18.1

 

“gin

 

34

free 50111”

lateral cable

   
   

 

 

   
 

 

8. .
column
Z 10.667' J
K fl]
a) Lateral Cable Yodel
w free joint
Y 14 . longitudinal
l d cable
’1}? column

\ L 63.635- J

X; lé-f panel point 11
panel point 12 —/L"l
b) Longitudinal Cable Model 1
longitudinal .
cable free Joint -—-\ I

5.6'

column l
L 63.635' J Y
'\ —71
L; Panel point 11 A
panel point 12
c) Longitudinal Cable Model 2 x

Figure 2-12: CSCB Cable Models. (Excerpted from Dusseau (1985))

 

 

 

 

were re]

using A
be 10 f

(L=lO f

y axis

(Figure
global
result:
the z
determ
the y

While

the fl

elemer

2‘54jL
Per f
dead

for t
aPPro
nodeE
17 f0

and 1

 

35
were represented by beam elements. The axial areas were calculated

L
using A="E—)—(, where L is the length of the beam element and was taken to

be 10 feet. The torsion constants were determined using Kt=(L Rx)/G

(L=lO feet).
To derive the stiffnesses of the Z axis translation springs and
y axis rotation springs at panel points 6 and 17, only the towers

(Figure 2-l3) were analyzed. First, a force F2 in the direction of
global Z axis then a moment My about the y axis were applied separately
resulting in a displacement D2 and a rotation qSy, respectively. Then

the z axis translation springs and the Y axis rotation springs were

determined using S =F /D and R =M /4S . The torsional constant about
2 z z y y y
the y axis of the beam element was determined using KT=(L Ry)/G, (L=lO),

while the axial area of the Z direction beam element was calculated by

the formula A=(lO Sz)/E' All other stiffness elements for the beam

elements representing the CSCB approach span were taken to be zero.

2.4.6 CSCB MASS DISTRIBUTION

 

The lumped nodal masses were based on the average dead weights
per foot (Merritt, F.S., 1972) of the bridge components. These average
dead load weights per foot are 3930 pounds, 5335 pounds and 210 pounds
for the arch, deck and columns, respectively. Portions of the CSCB
approach span deck, tower and column masses were lumped at the deck
nodes at panel point 6 for the south approach span, and at panel point
17 for the north approach span. For the north span, half of the deck

and tower masses and one-fourth of the column masses were lumped at

 

 

 

Dec]

'Ibwer

Figurl

 

36

\

5 1 I '
Deck Slab j 1 stringers
pr Strut

 

 

>,-'Ibwer Column

 

 

Tower Column 29

Intermediate
Struts

 

 

top of
concrete

 

 

 

 

 

 

Figure 2-13 CSCB Tower Elevation View. (Excerpted from Dusseau (1985))

 

panel p<
half of

lumped a

equival'
ll. Tl
longitu
shown

connect
finite
specif;

Table

the N}
nunbe
repre
to ti
relea

Span
Figu
the

indi

Spec

37
panel point 17 in the x and y directions. For the south approach span,
half of deck and tower masses and one—fourth of the column masses were

lumped at panel point 6 (Figure 2-2).

2.4.7 FINAL CSCB MODEL

The one-plane model of the CSCB is shown on Figure 2-14. The
equivalent beam elements of the arch and the deck are numbered from 1 to
11. The two truss elements representing the cables which transfer
longitudinal loads from the deck to the arch are labelled l and 2. Also
shown are the two deck moment releases resulting from the hinge
connection at the towers. Figure 2—15 shows the numbering used in the
finite element model. The stiffness properties of all the elements are
specified in Table 2-l. The listings for the lumped masses are shown in

Table 2-2.

2.4.8 FINAL NRGB MODEL

The same procedures were used to derive the one-plane model of
the NRGB. The equivalent beam elements of the deck and the arch are
numbered from 1 to 14 (see Figure 2-16). The two truss elements
representing the cables which transfer longitudinal loads from the deck
to the arch are labelled l and 2. The deck axial force and moment
releases resulting from the expansion joints at the ends of the main
span are also shown.

The numbering used in the finite element model is shown in
Figure 2-l7. The masses lumped at each node of the model are based on
the total weights of the arch, main span deck, approach span decks and
individual bents.

The lumped masses and the element stiffness parameters are

Specified in Tables 2—3 and 2-4.

 

 

p"?
o‘

\H.

Truss I

Column

Deck
Membe1

38

 

 

Table 2-1 Element Stiffness Parameters for CSCB
ASB AGXS AGYS I I KTS IWS
xx vv
Truss Elem 1 0.014 - - - —
0.014 - - — -
Column 3 0.667 - — - -
4 0.667 - - - -
5 0.667 - - - -
6 0.667 - - - -
7 0.667 - - - -
8 0.667 - - - -
9 0.667 - - - -
10
Deck 1 2.227 O 451 1.589 225.820 8.320 0.126 720.685
Member 2 2.201 0.451 1.551 225.820 7.631 0.126 702.852
3 2.167 0.451 1.528 219.204 7.435 0.126 683.990
4 2.167 0.451 1.528 219.204 7.435 0.126 683.990
5 2.167 0.451 1.528 219.204 7.435 0.126 683.990
6 2.167 0.451 1.528 219.204 7.435 0.126 683.992
7 2.167 0.451 1.528 219.204 7.435 0.126 683.992
8 2.167 0.451 1.528 219.204 7.435 0.126 683.992
9 2.167 0.451 1.528 219.204 7.435 0.126 683.992
10 2.201 0.451 1.551 222.521 7.631 0.126 702.852
11 2.227 0.451 1.589 225.821 8.320 0.126 720.685
12 0.667 0.000 0.021 274.624 0.000 1.604 0.000
13 0.667 0.000 0.021 274.624 0.000 1.604 0.000
14 0.009 0.000 1.540 220.602 0.000 3434.668 0.000
15 0.250 0.000 0.000 0.000 0.000 1.329 0.000
16 0.002 0.000 0.000 0.000 0.000 0.000 0.000
17 2.174 0.000 1.548 221.584 0.000 2017.002 0.000
18 0.250 0.000 0.000 0.000 0.000 1.329 0.000
19 0.017 0.000 0.000 0.000 0.000 0.000 0.000
20 4.752 2.810 0.382 997.029 57.039 33.047 0.000
21 5.549 2.811 0.342 1165.350 74.760 37.615 0.000
22 6.347 2.813 0.302 1333.680 92.481 42.183 0.000
23 5.960 2.818 0.282 1184.220 83.812 39.954 0.000
24 5.573 2.812 0.261 1034.750 75.143 37.725 0.000
25 5.187 2.812 0.241 885.285 66.474 35.496 0.000
26 5.573 2.812 0.261 1034.750 75.143 37.725 0.000
27 5.960 2.813 0.282 1184.220 83.812 39.954 0.000
28 6.347 2.813 0.302 1333.680 92.481 42.183 0.000
29 5.549 2.811 0.342 1165.350 74.760 37.615 0.000
30 4.752 2.810 0.382 997.029 57.039 33.047 0.000

 

 

 

 

 

 

 

 

 

 

 

 

Table 2-2 ,Lumped Masses for CSCB

 

 

 

 

 

 

 

 

 

Node Mass Node Mass

8 62.429 23 10.619
12 11.977 24 7.587
13 8.358 25 10.643
14 10.886 26 8.474
15 8.507 27 10.730
16 10.731 28 8.874
17 8.201 29 10.886
18 10.643 30 8.764
19 7.439 31 11.977
20 10.619 35 25.166
21 6.876

22 6.919

Table 2-3 Lumped Masses for NRGB
Node Mass Node Mass

9 62.8167 10 70.5313
33 62.816 12 66.235
11 65.953 14 63.421
31 65.953 16 61.067
13 57.021 18 59.399
29 57.021 20 58.576
15 49.833 22 56.902
27 49.933 24 58.576
17 44.499 26 59.399
25 44.499 28 61.067
19 40.675 30 63.421
23 40.675 32 66.235
21 36.703 34 70.531
8 79.040 36 77.513

 

 

 

 

 

 

 

 

 

 

 

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42

Table 2-4 Element Stiffness Parameters for NRGB

 

 

ASB AGXS AGYS IXX IW KTS IWS
Truss Members
1.956 - - - - _ _
2 1.956 - - - - - _
Beam Members

1 0.996 0.304 0.118 1304.52 80.654 12.804 82338
2 0.996 0.304 0.118 1304.52 80.654 12.804 82338
3 0.996 0.304 0.118 1304.52 80.654 12.804 82338
4 0.996 0.304 0.118 1304.52 80.654 12.804 82338
5 0.996 0.304 0.118 1304 52 80.654 12.804 82338
6 0.996 0.304 0.118 1304.52 80.654 12.804 82338
7 0.996 0.304 0.118 1304.52 80.654 12.804 82338
8 0.996 0.304 0.118 1304.52 80.654 12.804 82338
9 0.996 0.304 0.118 1304.52 80.654 12.804 82338
10 0.996 0.304 0.118 1304.52 80 654 12.804 82338
11 0.996 0.304 0.118 1304 52 80.654 12.804 82338
12 0.996 0.304 0.118 1304.52 80.654 12.804 82338
13 0.996 0.304 0.118 1304.52 80.654 12.804 82338
14 0.996 0.304 0.118 1304.52 80.654 12.804 82338
15 3.529 0.000 0.550 3459.72 0.000 0.000 0
16 3.437 0.000 0.587 3438.65 0.000 0.000 0
17 3.395 0.000 0.803 3201.43 0.000 0.000 0
18 4.061 0.000 0.792 3115.42 0.000 0.000 0
19 4.914 0.000 1.531 5805.94 0.000 0 000 0
20 5.922 0.000 0.182 2655.29 0.000 0.000 0
21 3.042 0.000 0.769 6020.25 0.000 3960.960 0
22 5.922 0.000 0.182 2655.29 0.000 0.000 0
23 4.914 0.000 1.531 5805.94 0.000 0.000 0
24 4.061 0.000 0.792 3115.42 0.000 0.000 0
25 3.395 0.000 0.803 3201.43 0.000 0.000 0
26 3.437 0.000 0.587 3438.65 0.000 0.000 0
27 3.529 0.000 0.550 3459.75 0.000 0.000 0
28 0.000 0.000 0.000 0.00 0.000 347.475 0
29 0.085 0.000 0.000 0.00 0.000 0.000 0
30 0.003 0.000 0.000 0.00 0.000 0.000 0
31 0.000 0.000 0.000 0.00 0.000 352.731 0
32 0.089 0.000 0.000 0.00 0.000 0.000 0
33 0.003 0.000 0.000 0.00 0.000 0.000 0
34 14.428 0.484 1.622 24946.80 9774.56 1355.360 0
35 13.836 0.482 1.537 23921.40 8823.50 1277.170 0
36 13.077 0.480 1.428 22608.40 7605.71 1177.040 0
37 12.366 0.478 1.327 21377.80 6464.36 1083.190 0
38 11.689 0.475 1.230 20206.30 5377.82 993.854 0
39. 11.036 0.473 1.136 19076.60 4329.99 907.700 0
40 10.399 0.472 1.045 17973.20 3306.59 823.553 0
41 10.399 0.471 1.045 17973.20 3306.59 823.553 0
42 11.036 0.473 1.136 19076.60 4329.99 907 700 0
43 11.689 0.475 1.230 20206.30 5377.82 993.854 0
44 12.366 0.478 1.327 31277.80 6464.36 1083.190 0
45 13.077 0.480 1.428 22608.40 7605.71 1177.040 0
46 13.836 0.482 1.537 23921.40 8823.50 1277.170 0
47 14.428 0.484 1.622 24946.80 9774.56 1355.360 0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

43

    

          

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2.5 LINSTRUC PROGRAM

The LINSTRUC Program was developed by Ralph A. Dusseau and
Robert K. Wen (1985) to perform the analysis of arch bridges subject to
unequal seismic support motions using time-history analysis. The
program has the following special features:
a. It uses a straight beam element with both shear and warping
deformation with "effective" member lengths. These effective
lengths were used to achieve equivalence between the stiffnesses of
beam elements in the model and in the real structure.
It has a "master" and "slave" node feature, which allows the user to
declare a displacement at a given node (slave) to be equal to the "
corresponding displacement at another (master) node. For example,
if node 21 is slave to node 26 with respect to X and Y translation
and Z rotation, then these displacements will be the same for the
two nodes, and a rotation at node 26 will not cause corresponding X
and Y translations at node 21. Thus slave nodes cannot be used to
create rigid links in the program.
It has a static condensation procedure that allows the user to

remove some nodal degrees-of-freedom before dynamic analysis.

2.5.1 FURTHER REDUCTION IN THE DEGREES-OF-FREEDOM
In the CSCB, the deck and arch are connected by two columns at
each panel point. Assuming that the axial deformation of these columns
are nominal and that the deck and arch cross-sections do not deform,
then the deck, the arch and the two columns at each panel point must
maintain a parallelogram configuration under all loads. Since the
columns in the CSCB are truss members which allow no shear transfer
between the deck and the arch, the pair of column at each panel point

were represented in the model by a single truss element. In order to

 

 

maintai
necessai

the sam

models,
panel 1

both th

 

46
maintain the parallelogram configuration described above, it was
necessary to require the arch and deck nodes at each panel point to have
the same longitudinal X-axis rotation.

To further reduce the number of degrees-of-freedom in the bridge

models, the additional requirement that the arch and deck nodes at each

panel point have the same vertical Y-axis translation was imposed for

both the NRGB and CSCB models.

 

 

y

 

elemen
motions

are

where

e“tr:

free

equa

CHAPTER 3

RANDOM VIBRATION ANALYSIS

3.1 FINITE ELEMENT FORMULATION OF EQUATIONS OF MOTION
The model of the two bridges are discretized into finite
elements as shown in Figure 3-1 and Figure 3-2. The equations of

motions of a multiple degree—of—freedom system subject to support motion

are
[M] {X} + [C] {in + [k] {X} = (0) (3.1)
where [M] = lumped or consistant mass matrix
[C] = damping matrix
[k] = stiffness matrix

{X}, {R}, (X) are the absolute displacements, velocity

and acceleration vectors
By partitioning the matrices [M], [C] and [k] such that the

entries of these matrices correspond to the partioning of {X} into the

free displacements (XF), and restrained displacements {XR}, the

equations of motion may be expressed as
mm] [MM] ”80 [ORR] [CRF] {RR}
[MFR] [MW] {in + [cm] [and DEF)

[km] [kRF] {KR} {0)
+ bkm] [kFFl]]{XFl={01 <32)

47

 

 

 

where (X

pseudo

Then,

follo

the

The

Strl

48

where {XF}, {R }, {RF} are the absolute displacement, velocity and

F

acceleration vectors of free nodes

{XR}, {RR}, {XR} are the absolute displacement, velocity and

acceleration vectors of restrained (support) nodes due

to ground motion

{ }
Thus {X} ={XR} (3.3)

(XFl

The free nodal displacement vectors {XF) can be decomposed into

. d
pseudo—static {Xi} and dynamic (XF) components

(3.4)

Then, the total absolute displacement vector may be written in the

following form

{X} = {new} = {{XR} d} (3.5)
{x } s
F {XF) + (XFl

The pseudo-static displacements {XE} are the displacements of
the free structural nodes due to static support displacements (KR).

These are obtained from the static equilibrium equations of the

structure with no applied external loads, i.e.,

 

 

 

 

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Taking

and so

hmst

The

51

0
0 } (3.6)

Taking the second system of equations (3.6), gives

[kFR] (XR) + [kFF] {Xi} = 0 (3.7)

and solving for {X3}, yields

{x3} = '[kpel [kFR] (XR) (3.8)

Substituting equation (3.5) into equation (3.2) yields

[[MRR] [MRFJ] { {KR} } + [[CRR] [CRFl]

s d
[MFR] [MFF] {XE} + {XF} [CPR] [OFF]
{RR} [kRR] [kRF] {XR} d
C U S
(X?) + {x2} + [kFR] [kFFl {XFT + {XFl

= {0) (3.9)
{0}

The second system of equation in (3.9) is

 

Separat

(3.8) g

 

SECOT‘M

of 11

motio

exp;
und

EXP

52

+ [kFR] {XR} + [kFF] {XE} + [kFF] {x3} = {0) (3.10)

Separating the dynamic response from static response and using equation

 

(3.8) gives
[MFF] (X?) + [CH] (kg) + [km] (x3)
= [- MFR + [MFF] [kFFJ' [kFRll {xR)
+ [— [CFR] + [OFF] [kFFJ'1 [kFRJ] (iR) (3.11)
For stiffness proportional damping, for which [c1 = a [k], the

second term of the right hand side is equal to zero; and for other forms
of light damping it may be neglected. Thus, the final equations of

motions for the free displacements become

-d -d d
[MFF] (xF) + [cFF] (XE) + [kFF] {XF}

 

-1 .
= [[MFF] [kFF] [kFR] — [MFR]] {XR} (3.12)

3.2 MODAL ANALYSIS AND NORMAL COORDINATES

 

Using normal coordinates, the dynamic displacements can be
eXpanded in terms of the undamped free vibration mode shapes; thus for
undamped harmonic free vibration the dynamic displacements may be

expressed as

 

in whi<

{Y} ar

free w

Subst

eigen

The 1

int(

53

d .
F} = [n] {Y} e1”t (3.13)

in which [T] = [{ul} ($2) H.. (¢n}] is the matrix of mode shapes and

(Y) are the normal coordinates.

The mode shape vectors {pi} are obtained by solving the undamped

free vibration equations of motion.

"d d _
[MFFl {XF} + [kFFl {XF} — {0} (3.14)

Substituting equation (3.13) into equation (3.14) yields the generalized

eigenvalue problems.
[[kFF] - [diag <w2>1 [MFFll [w] = {01 (3.15}

The solution of these equations yields the undamped natural frequencies

wj and mode shapes, {uj}, of the structure. Substituting

{x3} = [r] {Y} (3.16)
into equations (3.12) yields

[M [g] {Y} + {CFF} {w} {i} + [KFF] [i] {Y}

FF]

-1 " 3.17

 

 

 

Multipl

of an

yield:

Where

54

Multiplying equation (3.17) through by the transpose of the modal vector

[gb]T and making use of the orthogonality conditions
{u}T[M]{u}—0
j FF k _

{{pj}T [cFF] {wk} = 0 for J 7‘ K (3.18)

T

yields the uncoupled model equations

M. Y. + CY. + k.Y. = F. (3.19)
J J J J J J J

where the scalor quantities

= T 3.20
Mj {uj} [MFF] {ij} ( >
c. = {114T {C 1 {17.} {3.21}
J J FF J
k. = {WT {k 1 {11.} {3.22}
J J FF J

-1 e

F, = {11f [[MFF] [kFF] [km] - [MFRJ] {xR} (3.23}

 

are the

Dividir

where

 

Inp

rati

mat

55
are the generalized mass, damping, stiffness and excitation force.

Dividing equation (3-19) by Mj gives

.. . 2

Y. 2 . . Y. . Y. = c. _
J+ {JoJJ+oJJ J (324)

where
l_ T -1 . _ T "
cj = Mj {uj} [[MFF][kFF] [kFR] - [MFR]](XR) - {rj} (KR) (3.25)
El

2 (joj = Mj (3.26)
2 k.

w- = ‘l (3.27)
J Mj

T
- [MFRll {wj} (3.28)

In practice it is common to assume typical values for the modal damping

ratios g. rather than to assemble the physical damping matrix [C]. It
J
is convenient to collect the modal participation factors {Fj) into a

matrix

— 3.29)
[F] — [{F1} {F2} .... {Pn}] (

then, the right hand side of equation (3.24) may be written as

(3.30)

 

 

 

r is th

mode ST

and

Then,

the

56
The modal participation factor matrix [F] is of size r x n where
r is the number of restrained degrees—of—freedom and n is the number of

mode shapes considered in the analysis.

3.3 RANDOM VIBRATION ANALYSIS

For notational convenience, let

d
XF “ ud
S
XF _ us
and
XF = “F

u = u + u (3-31)

.th
u u and u are functions of time t. For the 1 degree of freedom,
5

F’ d

the autocorretation function of the displacement is defined as

R (r) = E [u (t) u (t+r>] (3.32)
u F. F.
F. 1 1
1
where T is time delay. Substituting equation (3 31) into equation

(3.32) yields

 

 

 

where

57

RuFi(T) = E [[udi(t) + usi(t)] [udi(t + r) + udi(t + T)y

= E [ud.(t) ud.(t + 1)] + E [ud (t) us (t + 7)]
l l l l

+ E [uS (t) ud (t + 1)] + E [us (t) us. (t + 7)]

1 1 1 1
= Ru (7)+Ru u (r)+Ru u (r)+Ru (r) (3.33)
d. d. s. s. d. s.
1 1 1 1 1 1
where
Ru (T) is the autocorrelation of the dynamic component of the
i
displacements
Ru (T) is the autocorrelation of the static component of
Si
displacements
R (7) and R (r) are the cross correlation between
u u u u
s. d. d. s.
1 1 1 1

static and dynamic components.

For a stationary response

R (r) = R (-r) (3.34)
u U. U
d. s. s. .
1 1 1 1
The fourier transform of equation (3 33) yields the SPeCCral density 0f

the free displacements

 

 

 

 

For st;

Whe

+ S ud.(w) + Su u (w) (3.35)

s d (o) = s: (-w) (3.36)

where the asterisk denotes the complex conjugate. The variance of the

.th .
1 free displacement can be obtained by integrating equation (3.35)

00 00
a F = I Sud (w) dw + I S u (w) d(w)
i m 1 i -w 51 Si
so
+ 2 Re ] I S u (w) dw]
s.
—m 1 1
= 02 + 2 2
u au + cov (us , ud.) (3.37)
d. s. 1 1
1 1
Where Re [ ] denotes the real part of the argument
0 , 03 = variances of the pseudo-static and dynamic
s d.
1 1,

displacements for the 1th degree of freedom.

 

 

 

Usin

Std

cov (uS ,u.d ) = covariance between the static and dynamic
i i

displacements for the 1th degree of freedom.

3.3.1 THE VARTANCE OF DYNAMIC DISPTACFMFNTS

The autocorrelation function of the dynamic displacement for the

ith degree of freedom is defined as

R (T) = E [ud (t) ud (t + 1)] (3.38)
ud 1 i

i
Using the normal coordinates

) = [W] {Y}

or

= .. Y. (3.39)
u ¢1J J

Substituting equation (3.39) into (3~38) gives

n n
Rud (T) = E ]E: uij Yj(t) E: uik Yk(t + 7)]

i j=1 k=1

n
3.40
2 11,3. eik E {196) Yk<t + 7}] < >
k:

 

 

Where t

the ana

equatit

where

respo

where
excit

Subst

The

equ

60
Where the index n is equal to the number of mode shapes considered in

the analysis.

The equation of motion for the jth mode corresponding to

equation (3.24) can be solved using Duhamel’s integral

w

. = . - . 3.41
YJ(t) ] GJ(t a) hJ(€) d6 ( )

~CD

where hj(0) is the impulse response function for mode j. h (g) is the
response Yj in equation (3.24) due to an impulse excitation Gj=6(t),

where 6(t) is the Dirac delta function. The response for a general
excitation is given by the superposition integral in equation (3.41).

Substituting equation (3.41) into equation (3.40), yields

1'1 n 00 co
Rud (r) = E: E: uijuik E ] ] Gj(t - 91) hj(€1) dal ] Gk(t + r - 92)
i j—l k=1 -8 -e

hk (62) d 02] (3.42)

The impulse response function does not depend on time lag 7, thus

equation (3.42) becomes

 

 

 

Refer

and

Ther

 

61

u

l

1 -oo .00

R (r) =
d. .

MS
TT\/l”

1

L4
||

Gk(t + 1 - 62)] d 91 dfiz (3.43)

Referring to equation (3.30), we can write the following
T
- = " — 3.44
cj (c 91) {rj} {xR(e 91)} < >

T
Gk (t + r - 92) ={1‘k} {KR (t + ¢ - {92)} (3.45)
Then, from equation (3.43)

E [Gj(t - 91) ck(c + r — 92)]

r r
= E ]E: Ffij XR£(C ' 61) E: ka XRm(t + T ' 62)
m=1

i=1

TT\’1”

H

if\’l”

Prj ka E [XR£(t ‘ 91) XRm(t + T ' 92)]
1

r
, _ (3.46)
E: Fflj ka Rxae XRm(T 02 + 01)

1 m=1

Tf\/1“

 

Putting

 

The n
degre

autoc

Subs

Wit

62

Putting equations (3.43) and (3.46) together gives

n
Rudi (r)= Z

=1

TM”

r r coco

22% [I 46
j ¢ikr 2j ka

1£= m= -ooRx-R£Rm

1 1

La.

(3.47)

6 l) hj(€l) hk(62) d 61 d 62

The spectral density function of the dynamic displacement for the 1th

degree of freedom is obtained through the fourier transform of the

autocorrelation function as follows

m
l - iw‘r
= —— 3.48
Sudi(w) 2n -1 Rud (1) e d7 ( )

Substituting equation (3.47) into equation (3.48) gives

00

r 1 {4...

=1

r (0
E: I’bijwik 1“lljrmk ]
1113 on

=1

TM”
TM”

1

H
L_J.

-i 7
(1 — 0 + 61) hj(61) hk (92) e w dfil d02 dr (3.49)

2

with the change of variables

Equation (3.49) can be written in the form

 

1
S w = ‘—
d_< ) 2n ¢ij¢ik Ffljrmk

fiTF\/l”
Tf\’15
TF\/l”
if\/l”

1

' . -iw(7 + 9 - 6 )
dfil d02 d7

00

r r
iwfi
E: E: wij¢ik rfijrmk ] ] hj(61) e 1 dgl]

n
‘ 1 £=1 m=1 -w

=1

TT\/l”

L4

CO co

-iw€ l_ -iw7
x ] J hk(62) e 2 c102:|]:21r I RXRfime(7) e d7] (3.50)

-(D

The impulse response function hj(0) and the frequency response function

Hj(w) are related through

dw (3.51)

II
N
a
‘5
E
A
8
V
(D
H-
8

hj(€)

Hj(w) = ] hj(€) e_' dd (3.60)

Using equations (3.51), (3.60) and (3 48) we can write

 

 

 

Then

Thi

deI

 

64

03

] hj(61) oiw91 dol = Hj(-w) (3.61)
] hk(62) e'iwg2 dflz = Hk(w) (3.62)

—l - (7) e'iwl d7 = s" - (w) (3-63)

2“ -£ Rinexam an XRm

Then, equation (3.50) takes the form

F\/l”

r
sud (w) = E: ¢ij ¢ik Frj ka Hj('w) Hk(w)
i j =1

ELM”

7:

1M5

H
H

2

m

XRZXRm

This equation represents the transfer relation between power spectral
density function of the stationary random excitation {XR} and the

response ud (t).
i

The variance of the free dynamic displacement response udi(t)

with zero mean is

(3.65)

 

 

Substi

The

eque

noti

w
U)

mot

65

Substituting equation (3.64) into (3.65) gives

m

Ibijll’ ik Pfij ka JHj("")Hk(“’)

Q

C M
Q.
was

7:.

none

lM”

2 m

H
La.

Sn - (w) dw (3.66)

xazxam

The modal frequency response function Hj(w) may be obtained from

equation (3.51), or more directly from the decoupled equations of

motions, equation (3.24), and has the form

1
2

2 (3.67)
(wj - w ) + 2i§jij

Hj (w) =

3.3.2 VARIANCES OF PSEUDO-STATIC DISPLACEMENT

 

The static displacements of the free nodes due to static support

motion is determined by equation (3.8)
S -1

s .
using the same notation for X: as in paragraph 3.3 (XF=Us)’ and letting

[A]= -[K [K for notational convenience, gives

FF]- FR]

_ (3.68)
{Us} — [A] {XR}

 

[A] is
freedo
of-fr
the f
displ

words

toa

The

deg:

 

66
[A] is an nxr matrix, where n is equal to the number of degrees-of-
freedom for the free nodes, and r is the restrained (support) degrees-
of—freedonL Each column in [A] represents the static displacements of
the free nodes due to a unit value of the corresponding support

displacement, while all other support displacement are zero. In other

words, the rth column represents the displacements of the free nodes due

th

to a unit static displacement of the r ground degree of freedom only.

Equation (3.68) can be written in scalar form as

1‘

“Si: E: A12XR2 (3'69)

£=1

. . . .th
The autocorrelation function of pseudo-static displacements for the 1

degree of freedom previously defined in equation (3.33) is

Ru (1) = E [uS (t) us (t+1)]
s. 1 i
1
Using equation (3.69) gives
r r
Si i=1 m=1

-3.

£=1 m

iflAim E [XR£(t) XRm(t+T)]

.th/J”

 

 

The

obtai

 

rel;

67

(3.70)

A. A. (1')
i2 1m RXR2XRm

2c

LM”
M”

E!
II
H

The spectral density function of the pseudo-static displacement is

obtained through the fourier transform of equation (3.70)

r r
S ( ) = E: E: A. A. S (w) (3.71)
us. w =1 lfl 1m XR2XRm

m=1

The spectral density function of support displacements SXRZX (w) is

related to the acceleration spectra through

3 (o) = —% s" - (o) (3.72)
2 Rm

Thus equation (3.71) can be written as

r r

1
su (o) = E: E: Ai, Aim 57 s- - (o) (3.73)
Si 2=1 m=1 XR2XRm

. . .th d f
The variance of the pseudo-static displacement of the 1 agree 0

freedom is

r r °°
2 _1 - - d (3.74)
Cu = E: E: Ai2Aim I w4 S X (w) w
51 2=1 m=1 -w XR2 Rm

 

 

disple

From

From

Sub

68

3.3.3 COVARTANCE OF PSEUDO-STATIC AND DYNAMIC DISPTACFMF‘NT

Using equations 3.33 the cross correlation of static and dynamic

displacements is

u u
s. d
1 1

R (T) = E [uS (t) ud-(t+1)]
. 1 1

From equation (3.39)

udi(t+1) = Ipinj(t+r)

From equation (3.69)

usi(t) = Aifi SR£(C)

Substituting equations (3.39) and (3.69) into (3.75) yields

1‘ n
Ru u (T) = E 2 A12 X'R,2<t) Z IflikYk<t+T>
i 1 i=1 k=1

r n

= Z Au ¢ik E [xR2(c) Yk(c+r)1
[=1 k=l

Substituting equation (3.41) into 3.76 yields

(3.75)

(3.76)

 

 

Using

C01

 

69

r n Go
Rug ud (r) = E: E: A12 wikE [XR£(t) I Gk(t+r-€)hk(€)d€] (3.77)
‘ 2:1 k=l _m

1 i

Using equation (3.45) gives

r 1'1 00 r
Ru “d (r) = E: E: AU2 ¢ik E[XR£(t) f E: ka XRm(t+T-€)hk(9)d6
5' i 2=1 k=1 1

1 —oo m:

CI.)

A12 ¢ik Pmk I E[XR£(t) XRm(t+¢-6)] hk(0)d€

-<D

ALMS
M”

kn

it")

7‘.)

E
H
H

LM”
rm“

.. _ . 8
A12 ¢ik rmk I kaz XRm(T 9>hk(9)d€ (3 7 )

h

LM”

5"

'Consider f(r) = I RXR §Rm(T-€)hk(9)d0 (3.79)
2

The fourier transform of equation (3.79) is

00 co

1 " —iwr (3.80)
= —- -9 h 0)e dfldT
foo) 2" J JRXMXRmu > k(
w —w
Let 7-6 = T '

3.81
Then d7 = dr ’ ( )

Substituting equations (3.81) into equation (3.80) gives

 

 

 

 

Now,

equa

The

 

7O

f(w) = I h (6)d0 —l I " (T,)e-iw(r’+6)d ,
_m k 2" _w RXRfime T
= Jhk(6>e‘i“’9do 2‘; IRXRBHLRmU’) e-inIdr’
= H (w) s " (w) (3_81)
k XRBXRm

Now, taking the fourier transform of equation (3.78) and considering

equation (3.81) we have

r n r
sus ud (w) = E: E: E: Ai£¢ikrmk Hk(w) SXRfime(w) (3.82)
1 1 i=1 k=l m=1
The covariance of pseudo-static and dynamic displacements is
r n r m
cov(us ,ud )= E: E: Ai£¢ikrmk I Hk(w) SXRg (m) (3.83)
l l 2=1 k=1 m=1 -m

In this equation, we have to express SXR " (w)dw through the spectrum
3 Rm

density function of the ground acceleration. Starting with

RXRngm(T) = E[XR£(t) me(t+r)] (3.84)

Differentiating twice with respect to 1 gives

 

 

 

 

 

 

 

diffe

diff

Con

71

H

(r) = E [ (c) - (t+r)] (3.85)
RXRfime XR£ XRm

differentiating equation (3.84) once with respect to T gives

I

(r) = E [ (t) ’ <t+r>1
RxRflme XRz XRm

= E [xMu-r) Khan (3.86)

differentiating equation (3.86) again with respect to 7 yields

RXRfime(T) = -E [XMu-r) XRm<c)]

-E [XR£(t) XRm(t+r)] (3.87)

Comparing equations (3.85) and (3.87) shows that

RXR XRmm = E [xmm iRm(t+v)1=-E [XM<c>XRm(c+r)1 (3.88)
2

or

(T) = (T) = -R. . (T) (3.89)
Ram... Rem 31“me

Taking the fourier transform of equation (3.89) given

_5 (w) (3.90)

SXRfiéRm(w) = XRfime

 

 

 

 

The a‘

follo)

The

Dif

72

The autocorrelation function R. ' (r) can be expressed in the

xRfime

following form.

R. . (r) = I s. . (w) ein dw (3.91)
XszRm ‘” XR£XRm
Then
RXR k (r) = I -s. . (w)ei“’ dw (3 92)
2 Rm -m XRflme

The auto correlation function of support displacement is expressed as

(w) ein dw (3.93)

RXRRXRm(T) = I_@ SXR2XRm

Differentiating equation (3.93) twice with respect to 1 yields

CK)

.. _ _w2
RXRfime(T) _ J_w SXR2XRm

(w) ein dw (3.94)

Using equations (3.88), (3.91) and 3.94) show that

S (w) (3.95)

2
88...”) = °’ 5812me

Also, from equation (3.72)

 

 

 

Then

Now

Sub

k»:

 

73

1
S (w) = -; s- - (w)
XRXR a.)
1 Rlme

Then, from the above two equations, it shows that

s ((3)337 s.. .. (to) (3.96)

XRJZme 822%

Now using, equation (3.90) yields

5 " (w)=—1—s.. .. (ca) (3.97)

XszRm “’2 xR 12me

Substituting equation (3.97) into equation (3.83) gives

(0

r n r
2 X Z X ‘1
”u u = AUZ wik 1‘ka Hk(w) [(0—2] s.. .. (to) do.) (3.98)
S' ‘1' =1 .00 XRJZXRm

I"
l"
M
II
[—1
W
H
H
5

Using the same procedure, it can be shown that the covariance of

dynamic and pseudo‘static displacement is

a)

r
E -1
A. $. P J H (w)[-3] S" u (w)dw (3.99)
12 1k mk k w
R XRm

m=1 -w 3

1M”

r
cov (udi'usi) = E:
i=1

k
3.3.4 THE VARTANCF OF DYNAMIC END FORCES

The eigenvectors represent the displaced shapes of the nodal
points for each mode in the global coordinate system. Each node had a

number of displacement vectors equal to the number of degrees of

 

 

 

 

freedc
extra
displ

to th

wher

syst

The

Wh¢

 

74

freedom. In order to calculate the element end forces, we need to
extract the values of the eigenvectors corresponding to the nodal
displacements at the two ends of the members and transform these values

to the local coordinate system through

} (3.100)

where {Dm} is the vector of nodal displacement in the local coordinate

system

[T] is the transformation matrix

{D } is the vector of nodal displacement in the global

G

coordinate systems.
The element end forces are then given by

_ (3.101)
(f) — [K]{Dm)

where {f} is the vector of element end forces [K] is the element of
stiffness matrix.

This sequence of operations is performed for all the
eigenvectors, or for the subset of them that are to be used in the
analysis. Finally, the variances of each end force may obtained as

described below.

. .th
Let F.. be the end force of the element corresponding to the 1
1

.th . .
degree of freedom of a given element and the J e1genvector. Fij is

 

 

calcu

dynam

Usf

V3]

'92

75

calculated by equation (1.105). The autocorrelation function of the 1th

dynamic end force is defined as

n T1
Rf f, (r) = E [ E: Fij Yj(t) E: Fik Yk (t+r)]
l l j=l k=1

F.. F

1] ik E [Yj(t) Yk (t+1)]

1M”

(.1.
7;.

LM” 1M5

7“

1M8

Fij Fik RYij (r) (3.102)

(_1.

Using the same procedure as in section 3.3.1, it can be shown that the

variance of this end force is

a)

F

f\/1”

Hj(-w)Hk(w) S" u dw (3.103)

F. F .F
1‘ 1k 2 mk I
J J -00 ZXRIH

n r
”Mi Z Z
1 . =1

n
3-1 =1 £

7‘.
II
'—I

In

3.3.5 THE VARTANCE OF STATIC END FORCES
To determine the static nodal displacements in the global

coordinates, we use equation (3.69)

(us) = [A]{XR)

We then determine the element end forces due to the movement of each

support in the local coordinate system in the same manner as explained

earlier.

 

 

the i

freed

statf

V2

 

 

 

 

 

76

Let Sij be the static end force of the element corresponding to

the ith degree of freedom of a given element and the jth degree of

freedom of ground motion.

. . .th
Then, the autocorrelation function of the 1 component of the

static end forces is defined as

r

r
Rs.s.(7) = E [ E: Sifi XR£(t) E: Sim XRm (t+T)]
m=1

£=1

r
E S” Sim E [XR}2(C) XRm(t+r)]
m:

r r
_ . 04
‘ E: E: 512 Sim RXszRm (T) (3 l )
=1

Using the same procedures as in section 3.3.2, it can be shown that the

variance of the ith pseudo—static end force of a given element is

CO

_1_
Sifisim I wa

m XR£XRm

r
02 = E: (w) dw (3.105)
Si =1

3

1M”

m

 

 

force

Usi

 

COV

3.

 

 

 

 

77
3.3.6 THE COVARTANCF OF PSEUDO-STATIC AND DYNAMIC FORCES
The cross correlation function of pseudo-static and dynamic

forces is defined as

1' 1’1

Rsif.(T) = E [ E: Si2XR2 (t)

FikYk(t+1) ]
£=1 k=1

1'1
E: SifiFik E [XR£(t) Yk (t+r) ] (3.106)
=1

Pa

||
1M“

7;.

Using the same procedure as in section 3.3.3, it can be shown that the
covariance between the pseudo-static and dynamic end forces is expressed

as:

r n r co 1
L“ 1 n d 3.107
cov (Si’fi) = E: E: Si£Fikak I Hk(w)[w2 ]S (w) w ( )
£=1 k=l m=1 -m 2
3.3.7 SUMMARY
. .th
The variance of the displacement or rotation of the 1 degree

of freedom in the global coordinate system is expressed in the following

form

a = a + 02 + 2 COV (Us ,Ud )
L1 U i i

where

 

 

 

elen

whe

78
1pi j¢ikF flj Pmk J Hj(-w) Hk(w)SiR iRm(w) dw
-m 2

AM“
M”

E
H
H

a)

r

.l
E: AifiAim I w4
=1

3" -
-w XR£XRm

(w) dw

r n
cov (us',ud‘)=E: E:
i 1 =1

2 k=1 m

1M”

-1
Ai P H (w) S" u (w)dw
Ipik mk I k [wZ ]
- m X‘RJZX'Rm

. .th . .
The variance of the 1 component of end forces for each finite

element is expressed in the following form

2
a = a + a + 2 cov (5., f.)
. . s. 1 1
i 1 1
where

(I)

r r
E: E: F. ijFikFZijk i Hj(-w)Hk(w) Si n (w) dw<
=1 m=1 szRm

F\/1“

n
02 =2

f.
1 . 1

J= 2

r
=E: $12Sim

m= =1

k

II
H

(w) dw

é‘-—>8
8
'—I
V
xix.
Ex.

Pg

q
H.
H
HF\/1H

. ;i
Fiksifirmk J Hk (—w) [ 2 (w) dw

-w w ] 8%R2XRm

F\/1“

n
cov (si’fi)= E:

k=1 2

LM“

5
ll
H

All the above mention notations were previously identified.

 

 

 

 

L“.

we m1

grou

and

eque

Har
wel
has

an

79
3.4 THE INPUT MOTION
In order to solve this problem using the probabilistic approach
we must have a mathematical model for the acceteration spectrum of the

ground motion S~ - (w). For stationary excitation, the displacement

XszRm

and velocity spectrum are related to the acceleration spectrum by
equation (3.72) and (3.95), respectively.

The ground motion model used in this study is that proposed by
Harichandran and Vanmarcke (1986). The model considered the spatial as
well as the temporal variation of earthquake ground motion, and was
based on the analysis of recordings made by the SMART-1 seiSmograph
array in Lotung, Taiwan. In this model the cross spectral density

function between the acceleration of two locations A and B is expressed

as
s- .. - s- (w) p (v, f -2—;’) 6 '1—31 (3.108)
xAxB x
where
- -2
p(u,f) - A exp[a§%%)(l-A+aA)]+(l-A)exp[;?%) (1-A+aA)] (3.109)
f b -8 (3 110)
0(f) - k [1 + (ES) ] .

A. a, k, f0 and b are model parameters where typical values are shown in
Table 3-1 (Harichandran 1988). The function p(V,f) wiflithese

parameters are plotted in Figure 3-3(a) and (b).

 

 

 

 

 

 

 

 

 

accele

whel

and

ir

80

S (w)- istiw auto spectral density function of ground
X

acceleration.
u - seperation between locations A and B
f — Linear frequency
v - apparent wave propogation velocity in the direction AB.
The functional form suggested by Claugh and Penzien (1974) is used for

S~(w):

x
s (w) = [Hl(w)|2 |H2(w)l2 30 (3.111)
x

where [Hl(w)|2 is the Kanai—Tajimi spectrum function and has the form

 

 

2 2
{1+4/8 [w/w ] )
2 g g
”g g ”g
and
4
(w/w )
2 f
IH2(w)| = 1 _9 2]2 + 452 [ _fl ]2 (3.113)
[ _ “f f “f

in which cog, pg, 0) f p fand S are model parameters that can be

estimated by fitting the above function to observed acceleration
Spectra. Two acceleration spectra with the parameters given in
Table 3-2 were used in this study. These spectra are plotted in
Figure 3-4 . Ground motion 1 has a wide excitation frequency range and
is characteristic of motion recorded on rock, while ground motion 2 has
a narrow excitation frequency range is charactertistic of motion on

soil. For the values of S given in Table 3—2, the variances of ground
0

acceleration (area under the spectrum in equation (3.111) are 2n gal

 

 

for a

is re

are'
Note
tota

subs

81

for all the events. The variances of ground displacements Sx(w) which

is related to the acceleration spectrum. S (w), through
X

l
S (w) = —— S» (w)
X wa X

7 7

are l.O9(lO)- and A.09(10)_ m2 for spectra 1 and 2, respectively.
Note that although the normalized acceleration spectra have the same

total power for both events, the corresponding displacement spectra have

substantially different total power as depicted in Figure 3-5.

3-5 COMPUTATIONAL PROCEDURES.

The LINSTRUC computer program (Dusseau l985) was modified so
that it can be used on the VAX/VMS. The program was used to obtain the
overall stiffness and mass matrices. The computational procedures that
was used is summarized here.

1. Compute the eigenvalues and the eigenvectors using the
generalized Jacobi method (Bathe, K.J. 1982).
2. Calculate matrix [A] using equation (3.8). In the case of CSCB,

[A] is a 58 x 2 matrix. The entry Aij is the free nodal

displacement at mode i due to a unit support displacement at j.
3. Calculate the entries of the modal participation factor matrix
[P] using equation (3.28).

«J

4. Calculate the integrals J Hj(-w)Hk(w) S~ " (w)dw and store the
X112

C0
results in a 34 x 34 x 2 x 2 array. (The integrations were done

using the IMSL subroutine "DCADRE").

 

 

 

 

 

 

 

 

82

(1)
Calculate the integrals I 7]: S“ ~- (0)) doc and store the
-0. ‘0 XR

12me

results in 4 X 4 or 2 x 2 arrays depending on the number of
moving supports.

a:

2

-<=0 (0 XR£XRm

the results in a 34 x 2 x 2 array.

Calculate the integrals J-Hk (-w) [5;] Sn -_ (w) dw and store

Calculate the variances of dynamic and static displacements, and
the covariance between the static and dynamic displacement, using
equations (3.66), (3.74) and (3.99).

Calculate the variances of members end forces using the following

sequence of operations:

a. For nodes i and j of each element extract the corresponding
static and dynamic displacements from matrix [A] and from the
eigenvectors. For each element the number of displacement
sets of the static components is equal to the number of moving
supports. For the dynamic component the number is equal to
the mode shapes considered in the analysis.

b. Transform the end displacements to the local coordinate, then
determine the members end forces and store them in an array.
For the static component the array is of r x q where r is the
number of moving supports, and q is the number of degrees
of freedom for nodes i and j, in this case 14. For the
dynamic component the array is of order n x q, where n is the

number of mode shapes.

 

 

83
c. To calculate the variance of the dynamic, static and the
covariance of static-dynamic components use equations (3.103),
(3.105) and (3.107), respectively.

d. To calculate the total response use equation (3.37).

Table 3—1: Model Parameters for p(v,f)

 

 

 

Model Parameter Ground Motion 1 Ground Motion 2
A 0.626 0.355
Double a 0.022 0.086
Exponen— k 19700 23100
tial 5

0.5
fo(HZ) 2.02

b 3.47 2.35

 

 

 

 

 

 

Table 3-2: Model Parameters for Autospectra

 

 

 

 

 

Model Parameter Ground Motion 1 Ground Motion 2
wg(rad/s) 20.22 5.05
Double fig 0.53 0.62
Filter wf(rad/S) 5.45 6.41
fif 0.46 0.27
So(gal.32) 0.0957 0.3068
i___________________________________

 

 

 

 

 

 

 

84

 

1.0
0.9;; \
0.393,
0.7: 1;
o.e= '
0.5:
0.4.:
0.3:

 

Abs. Value of Coherency

(12-
0.1-

4

n "I

-V'I‘I'IrT’I'T'l*r'I‘fi
0 100 200 300 400 500 600 700 800 900 1000

 

 

 

Separation (m)

 

 

 

 

 

 

 

Abs. Value of Coherency

 

' 1 r T‘r l r l ' I ' I ' l ' I ' 1_r——1
0 100 200 300 400 500 600 700 800 900 1000
Separation (m)

Figure 3-3: The Coherency Function p(u,f). (a) Ground Motion 1;

(b) Ground Motion 2.

 

 

Eahdovnmmnvagdu nvamuflEhoz

 

 

 

85

 

 

 

10.09905'l'l'l'l'l'l'l'l'l'l'l'l'l
2 GROUND uono~ 1 —
g : _3
5 ' 1
La
‘5 1.0000: f
0 i 1‘
% \
O I
"’ I
53 0.1000; \
'o 5’ \
O I \
.53. \
a ' \
8 0.01001 \\
s 5 ‘\
z 7 r =
1 ‘ \
Q0019 - r7 Ir] ‘I '1 'I '1 'l' 1' l I l l -
Y Y 1' rI—fi
0123456789101112131415

Frequency (Hz)

Figure 3-4: Spectra of Ground Acceleration Using Estimated Parameters.

 

lfln‘)
~v— lvllI'YVIIVfiIII'lr'l‘lIVIfiTYI—rY—V’Y—I’YIY

POINT SPECTRUM 1

 

 

lE-OS

P ——— POINT SPECTRUM 2

 

 

 

 

AUTOSPECTRA 0F GROUND DISPLACEMENT

VlIllI‘f‘j’r'jfiI‘lrlivl'IlI’vl

0 1 2 3 4 5 . 6 I 7
FREQUENCY (Hz)

o;l_._.LllllllJ__Ll.l|lllll_L_.l LuqunL.|_I unld..1..14uufll.4.uuul

 

Figure 3-5: Spectra of Ground Displacement Using Estimated Parameters.

 

 

w
ox
E

\'

excit
from
excit

syst

of]
he
hig

1110):

 

 

 

 

 

86
3.6 NONSTATIONARY RESPONSE

The theory developed until now is valid only for stationary
excitation. Earthquake acceleration amplitudes, however, initially grow
from zero, then have a steady phase and eventually decay. The
excitation is therefore nonstationary. For a single degree-of—freedom

system with undamped circular natural frequency on and damping ratio fl,

the response may not attain its stationary state for very small values

of fiwn or small durations of strong shaking. For a multi-degree—of—

freedom system, each modal response grows at a different rate, with
higher modes having large natural frequencies attaining stationarity
more quickly. The rate at which the total response grows depends on how
much the lower modes contribute to the overall response. If the lower
modes do not contribute significantly, then the total response may
attain stationarity rather quickly. For the arch bridges considered in
this study, the first few modes have long periods (low frequencies) and
therefore may not reach stationary conditions for short earthquakes. It
is therefore of interest to compute the transient response of the
bridges due to nonstationary excitation.

In most earthquake engineering applications it is reasonable to
account for nonstationarity in the amplitude of the ground accelera-
tions, while the frequency content may be assumed not to change with

time. For these cases the ground accelerations may be written as

8(t) = A(t)Z(t) (3.114)

in which Z(t) is a stationary process, and A(t) is a temporal modulating
function. Various forms have been suggested for A(t) based on the

fitting of modulating functions to measured accelerograms. The response

 

 

 

 

 

of t]

whe

fre

fre

87
of the generalized displacement for the jth mode may then be expressed

as

t ..
Y(t) = J hj(t - T)A(T)X(T)dT (3.115)
0

where hj(t) is the impulse response function of the jth mode. For

frequency domain analysis, it is convenient to define a "time-dependent

frequency response function“, as

t .
H.(w,t) = I h.(t - T)A(T)eledT (3.116)
J O J

To evaluate the response variance at a given time t, the function

Hj(w,t) can be substituted for the normal frequency response function
Hj(w) in the expressions for the stationary response. The main
difficulty in this is that while Hj(w) has a closed form expression,
Hj(w,t) cannot easily be expressed in closed form for any A(t).
However, a closed form expression can be derived for Hj(w,t) if A(t) is

the unit Heaviside function

This corresponds to an excitations that suddenly starts at time t=0 with
stationary intensity, and is not the same as stationary excitation.
Gasparini and DebChaudhury (1980) considered structural response

to two modulating functions A(t) using a time domain approach. The

 

 

 

”flute—V

 

firs

fina

A51

dif

sta

 

88

first modulating function grew linearly from zero, then was steady and
finally decayed linearly to zero, while the second began suddenly (like
the Heaviside function) but decayed linearly to zero after some time.
As would be intuitively expected, the initial growth of the response
differed for the two modulations, but they gradually approached the same
stationary value and began to decay as soon as the excitations started
to decay.

The present study is concerned more with comparing the relative
differences in the responses due to different models of multiple support
excitation, and not so much with finding absolute response variances.
Thus the exact form used for A(t) is not very crucial, and the use of a
Heaviside modulating function is sufficient to assess the effect of
transient modal responses.

For the Heaviside modulation, the closed form expression for

Hj(w,t) is (Lin 1963)

Hj(w,t) = Hj(w) 1 - exp(-wjfijt)exp(-iwt)[cos wjdt

(w.fi.+iw)
+ —J-J— sin (.0. t (3.117)
wjd Jd

The transient responses in this study are computed by replacing
H(w) in the results derived in Section 3.3, with the expression for

H(w,t) given in equation (3.117).

 

 

 

 

 

prc

ext

[3"

 

CHAPTER 4

ANALYSIS RESULTS

4.1 MODAL ANALYSIS

The Generalized Jacobi method was used to solve the eigenvalue
problem for both bridges. All mode shapes and frequencies were
extracted and considered in obtaining the response values of the two

bridges for the in—plane and out-of—plane models.

4.1.1 CSCB MODE SHAPES AND NATURAL PERIODS

The first four modes for the in—plane model of the CSCB are
shown in Figure 4-1. The first mode has a natural period of 2.32
seconds and is a full wave vertical motion of the deck and the arch.
The second mode is a 1% wave of vertical motion for the deck and the
arch with a natural period of 1.19 seconds. The third mode has a
natural period of 0.65 seconds and is characterized by large two full
waves of vertical motion of the deck and the arch. Finally, the fourth
mode has a natural period of 0.63 seconds and is characterized by large
two full waves of vertical motion for the arch and the deck and a
moderately large longitudinal translation of the deck.

Figure 4-2 illustrates the first four modes for the out-of—plane
model of the CSCB. The first mode has a natural period of 2.67 seconds
and is a half wave lateral motion of the deck and the arch. The second
mode has a natural period of 1.54 seconds and is characterized by a
large full wave lateral motion of the deck with a small lateral half
wave motion of the arch. The third mode has a natural period of 1.01
seconds and is characterized by a large 1% wave lateral motion of the
deck accompanied by a small half wave lateral motion of the arch.

Finally, mode four has a natural period of 0.69 seconds and is

89

 

 

 

 

char

acc01

dep
sec

Mod

he

d

 

90
characterized by a large half wave lateral motion of the arch

accompanied by a moderately large two wave lateral motion of the deck.

W

The first four modes for the in-plane model of the NRGB are
depicted in Figure 4-3. The first mode has a natural period of 4.18
seconds and is a full wave vertical motion of the deck and the arch.
Mode two is a 1”: wave vertical motion of the deck and the arch with a
natural period of 2.00 seconds. The third mode has a natural period of
1.43 seconds and represents a two wave vertical motion of the deck and
the arch. Finally, mode four is characterized by large horizontal
motions of the deck toward the center of the bridge. This latter mode
has a natural period of 1.21 seconds and also exhibits a small vertical
deck and arch motion in the form of 111 waves.

Figure 4-4 illustrates the first four modes for the out-of—plane
model of the NRGB. The first mode has a natural period of 6.78 seconds
and is a half wave lateral motion of the deck and the arch. Mode two
has a natural period of 3.48 seconds and is a full wave lateral motion
of the deck accompanied by a small full wave lateral motion of the arch.
The third mode has a natural period of 2.40 seconds and is characterized
by a large 1’»: wave lateral motion of the deck and a small 1;: wave
lateral motion of the arch. Finally, mode four has a natural period of
1.89 seconds and is characterized by a large two full wave lateral
motion of the deck accompanied by a small full wave lateral motion of

the arch.

 

 

 

Mode 1

 

 

Mode 2

X

Mode 3

Mode 4

><

Figure 4-1: CSCB In-Plane Modes (Excerpted from Dusseau (1985))

 

 

92

 

 

 

 

 

 

 

 

 

 

 

T = 2.675
X
2 %s L f . *
Mode 1
T = 1.545
X
Z i V
Made 2
V Wis
X
z ._. ‘ ‘
Mode 3
' WV ' ' T = 0.695

Pbde 4

Figure 4-2: CSCB Out-Of-Plane Modes (Excerpted from Dusseau (1985))

 

93

Made 1

 

 

 

 

X Mode 3
T = 1.213
Y rIIlIr’a,a4g5Fin—‘tpfl-=:=Hh~““\~\“
x Mode 4

Figure 4-3: NRGB ln-Plane Modes (Excerpted from Dusseau (1985))

 

 

 

 

 

 

 

v . v v . r v r "Vi v v v -
VT: 3.1485

 

 

 

 

 

X
Z
Fade 2
I v A 7 v 1 Av; : ¢ ; I v v, 1
T = 2.405
X
Z _____ _ . Li A #_____
r V r—‘
Mode 3

 

Mode 4

Figure 4-4: NRGB Out-Of-Plane Modes (Excerpted from Dusseau (1985))

 

 

 

 

Ll

 

CORE

Cas‘

 

95

4.2 MODELS OF GROUND MOTION

The three specialized coherency models of ground motion

considered in this study are the following:

Case 1:

Case 2:

Case 3:

Fully correlated ground motion. In this case we assumed that
the supports of the bridges are moving identically. This
corresponds to the current practice of designing bridges for
earthquake ground motion. For this case the term

p(v,f=§$)e-le/Vin equation (3.108) is equal to one.

Wave propogation case. In this case we assumed that there is
no loss of coherency between the support excitations, but there
is a time delay corresponding to the time required for the
seismic waves to travel from one support to another. For this
case the term p(v,f) in equation (3.108) is equal to one.

The general case of ground motion. In this case, the wave
propogation factor as well as the frequency-dependent spatial
correlation function p(y,f) were considered in the ground
motion model.

In each case two different sets of ground motion parameters
were used to study the responses of the two bridges (see Tables
3-1 and 3-2). Note that although the variances of ground
acceleration corresponding to the two sets of parameters were
normalized to be the same, the variances of ground

displacements are different.

4.3 ANALYSIS RESULTS

As mentioned earlier, each bridge has two models; the in-plane

model and the out-of-plane model. For the in-plane model the ground

motion acceleration was applied in the global X-axis direction. For

 

 

 

 

 

 

 

96
each element, the variances of bending moments, shear and axial forces
were obtained. Also, the variances of nodal rotations and displacements
in the global coordinate system were determined.

For the out-of—plane model, the ground motion was applied in the
global Z-axis direction. The corresponding variances of member end
forces, nodal displacements and rotations were determined.

For the CSCB Bridge models only, the ground acceleration was
additionally applied to the support of the approaching span to study the
effect of that on the response of the CSCB.

Throughout this chapter the following symbols are used:

a: - The variance of member shear forces in the direction of
z
local x-axis.
2 . .
0F - The variance of aXial forces.
2
2 . . . .
0F - The variance of shear forces in the direction of local
y
y-axis.
2 . . .
UM - The variance of bending moment about local X-aXlS.
x
2 . .
GM - The variance of torSional moment.
2
2 . . .
GM - The variance of bending moment about local y—aXis.
y
2 . .
GM — The variance of warping moment.
w

2
0*(W), 03(H), a:(G) - The variances of any member end forces resulting

from the wave propogation case (Case 2), fully correlated case (Case 1)

and general case (Case 3) of ground motion, respectively.

 

 

 

 

 

5F:

 

res

com

 

97
4.3.1 RESPONSE COMPONENTS OF CSCB AND NRGB

As discussed earlier in chapter 3, the variance of the total
response consists of three components; the variance of the dynamic
component, the variance of the static component and the covariance
between the static and dynamic components. The relative contributions
of response components were found by dividing the variance of a specific
response by the variance of the total response. In CSCB Bridge, for the
first set of ground motion parameters (ground motion 1), the relative
contributions of dynamic and static components are shown in Tables 4-1
and 4-2. For the second set of parameters (ground motion 2) the same
relative contributions are shown in Tables 4-4 and 4-5. The relative
contributions of the covariance between the dynamic and static
components for both ground motions are shown in tables 4—3 and 4-6,
respectively.

Those tables show that the dynamic component of the response is
the dominant one and represents more than 96% of the total response for
most elements, except the bracing members. For the general correlation
case of ground motion 1, the relative contributions of the dynamic and
static variances and the covariance of the axial forces in longitudinal

bracing are 78%, 8% and 12%, respectively. For ground motion 2, the

same relative contributions listed in the same order are 65%, 11% and-

23%. This indicates the relative sensitivity of the bracing members to
statically applied support displacements, since they transfer the forces
between the deck and the arch.

The variance of the static component represents the bridge
response due to a pseudo-static application of differential support
motion. In this case the inertia forces of the bridge mass do not

contribute to the increase of the bridge response. The variance of the

 

 

 

 

Table 4-1 Relative Contributions of the Dynamic Components of CSCB

98

Members End Forces to the General Case of Ground Motion 1.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

NODE I NODE J
ElemEntS UZFXEG) azedG) aszéG) azdeG) aZdeG) aZMvdG)
(3&0) azFéG) azmgc) (flags) azFéG) UZM‘SG)
Bracing
Elements
79.8 79.8
2 77.7 77.7
Deck
Elements
1 99.3 100.3 78.9 99.3 100.3 99.3
2 97.4 100.3 99.3 97.4 100.3 92.2
3 98.6 100.3 92.2 98.6 100.3 96.5
4 100.1 100.2 96.5 100.1 100.2 105.5
5 98.6 100.2 105.5 98.6 100.2 102.4
6 95.6 100.8 102.4 95.6 100.8 101.8
7 101.7 101.3 101.8 101.7 101.3 99.0
8 102.0 101.4 99.0 102.0 101.4 100.9
9 100.1 101.4 100 9 100.1 101.4 101.2
10 98.5 101.4 101.2 98.5 101.4 101.3
11 101.3 101.4 101.3 101.3 101.4 91.5
Arch
Elements
20 96.5 100.3 78.4 96.5 100.3 96.5
21 99.0 100.3 96.5 99.0 100.3 93.1
22 100.6 100.3 93.1 100.6 100.3 95.9
23 98.7 100.3 95.9 98.7 100.3 105.8
24 98.6 100.3 105.8 98.6 100.3 102.6
25 96.7 100.0 102.6 96.7 100.0 101.7
26 101.5 99.8 101.7 101.5 99.8 99.0
27 101.7 99.7 99.0 101.7 99.8 100.5
28 100.2 99.8 100.5 100.2 99.8 101.7
29 100.4 99.8 101.7 100.4 99.8 100.8
30 100.8 99.8 100.8 100.8 99.8 91.2

 

 

 

 

 

E?
Z

\

 

 

99

Table 4-2 Relative Contributions of the Static Components of CSCB
Members End Forces to the General Case of Ground Motion 1.

 

 

 

 

 

 

 

 

 

 

 

NODE I NODE J
Elements azFxéc) 02Fzéc) aszéc) azFxéc) azegG) UzMyéc)
“zFim azréc) ”2M$G) ”21146) ”ZFEG) ”Zr/him
Bracing
Elements
1 7.90 7.90
2 9.50 9.50
Deck
Elements
1 0.05 0.01 11.25 0.05 0.01 0.05
2 1.80 0.01 0.05 1.80 0.01 2.13
3 0.13 0.01 2.13 0.13 0.01 1.84
4 0.01 0.01 1.84 0.00 0.01 4.35
5 0.25 0.01 4.35 0.25 0.01 0.32
6 0.57 0.07 0.32 0.57 0.07 1.00
7 0.43 0.22 1.00 0.43 0.22 0.23
8 0.39 0.20 0.23 0.39 0.20 0.05
9 0.00 0.18 0.05 0.00 0.18 0.05
10 0.00 0.18 0.05 0.00 0.18 0.05
11 0.20 0.17 0.05 0.20 0.17 0.36
Arch
Elements
20 0.79 0.01 11.87 0.79 0.01 0.79
21 0.35 0.01 0.79 0.35 0.01 1.73
22 0.04 0.01 1.73 0.04 0.01 2.20
23 0.08 0.01 2.20 0.08 0.01 4.19
24 0.28 0.01 4.19 0.28 0.01 0.37
25 0.35 0.00 0.37 0.35 0.00 0.29
26 0.34 0.01 0.93 0.34 0.01 0.29
27 0.23 0.01 0.29 0.23 0.01 0.02
28 0.05 0.01 0.02 0.05 0.01 0.11
29 0.02 0.00 0.11 0.02 0.00 0.05
30 0.05 0.00 0.05 0.05 0.00 1.60

 

 

 

 

 

 

 

 

 

 

100

Table 4-3 Relative Contributions of the Static-Dynamic Components of
CSCB Members End Forces to the General Case of Ground Motion

1.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

NODE I NODE J
Elements 2 2 2 2 2 2
a Fxéc) a Fzéc) a MyéG) a Fxéc) a FzéG) a MVéG)
”214(6) ”ZFEG) ”Zr/his) ”216(6) ”ZFEG) ”21656)
Bracing
Elements
12.28 12.28
2 12.77 12.77
Deck
Elements
1 0.68 0.35 9.84 0.68 0.35 0.68
2 0.78 0.30 0.68 0.78 0.30 5.67
3 1.22 0.27 5.67 1.22 0.27 1.70
4 0.15 0.25 1.70 0.15 0.25 9.84
5 1.12 0.24 9.84 1.12 0.24 2.69
6 3.78 0.85 2.69 3.78 0.85 2.76
7 2.10 1.55 2.76 2.10 1.55 0.80
8 2.36 1.55 0.80 2.36 1.55 0.93
9 0.08 1.55 0.93 0.08 1.55 1.28
10 1.28 1.56 1.28 1.28 1.56 1.66
11 1.62 1.60 1.66 1.66 1.60 7.03
Arch
Elements
20 2.75 0.28 9.77 2.75 0.28 2.75
21 0.69 0.29 2.75 0.69 0.29 5.14
22 0.59 0.29 5.14 0.59 0.29 1.86
23 1.25 0.30 1.86 1.25 0.30 10.03
24 1.11 0.29 10.03 1.11 0.29 2.94
25 2.97 0.04 2.94 2.97 0.04 2.61
26 1.82 0.22 2.61 1.82 0.22 0.72
27 1.94 0.22 0.72 1.94 0.22 0.53
28 0.27 0.22 0.53 0.27 0.22 1.80
29 0.40 0.22 1.80 0.40 0.22 0.81
30 0.81 0.22 0.81 0.81 0.22 7.17

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 4-4 Relative Contributions of the Dynamic Components of CSCB
Members End Forces to the General Case of Ground Motion 2.
NODE I NODE J
Elements 2 2 2 2 2 2
a deG) a deG) a MvdG) a dec) a deG) a Mde)
2
‘7 161G) ”ZFEG) “2169C” ”21:59 02mm) 021636)
Bracing
Elements
66.9 66.9
2 64.1 64.1
Deck
Elements
1 98.6 101.2 72.5 98.6 101.2 98.6
2 83.7 101.0 98.6 83.7 101.0 90.8
3 97.3 100.9 90.8 97.3 100.9 87.0
4 100.2 100.8 87.0 100.2 100.8 112.3
5 95.3 100.7 112.2 95.3 100.7 103.8
6 92.5 102.5 103.8 92.5 102.5 104.3
7 105.7 104.3 104.3 105.7 104.3 98.3
8 103.2 104.1 98.3 103.2 104.1 102.5
9 100.2 104.0 102.5 100.2 104.0 101.4
10 92.0 103.8 101.4 92.0 103.8 102.9
11 102.9 103.8 102.9 102.9 103.8 89.4
Arch
Elements
20 94.3 101.6 72.7 94.3 101.6 94.3
21 92.5 101.7 94.3 92.5 101.7 92.0
22 101.4 101.7 92.0 101.5 101.7 85.8
23 98.0 101.8 85.8 98.0 101.8 112.2
24 94.8 101.7 112.2 94.8 101.7 104.1
25 94.3 100.2 104.1 94.3 100.2 104.1
26 105.5 98.6 104.1 105.5 98.6 98.6
27 102 8 98.6 98.6 102.8 98.6 101.4
28 100.9 98.6 101.4 100.9 98.6 101.9
29 102.2 98.7 102.0 102.2 98.7 101.3
30 101.3 98.6 101.3 101.3 98.6 89.5

 

 

 

102

 

Table 4-5 Relative Contributions of the Static Components of CSCB
Members End Forces to the General Case of Ground Motion 2.

 

 

 

 

 

 

 

 

NODE I NODE J
Elements ”zrxéc) azeéc) ”2Myéc) UZFXéG) ”zeéc) ”széc)
02F£G) azFéc) ”2M$G) 02F§c) azFéG) ”2M$G)
Bracing
Elements
10.41 10.41

2 12.32 12.32

Deck

Elements
1 0.06 0.03 12.02 0.06 0.03 0.06
2 10 30 0.02 0 06 10 30 0.02 1 80
3 0.18 0.02 1 80 0.18 0.02 4 80
4 O 00 0.01 4.79 0.00 0.01 5 61
5 0.53 0.01 5.61 0.53 0.01 0 32
6 0 77 0.13 0.32 0 77 0.13 1.07
7 0.85 0.40 1.07 0 85 0.40 0.45
8 0.37 0 36 0.45 0.37 0.36 0 10
9 0.00 0.32 0.10 0.00 0.32 0.04
10 0.87 0.29 0.04 0.87 0.29 0.46
11 0.46 0.27 0.46 0.46 0 27 1.48

Arch

Elements
20 0.87 0.05 12.20 0.87 0.05 0.87
21 1.85 0.06 0.87 1.85 0.06 1.41
22 0.07 0.07 1.41 0.07 0.07 5.51
23 0.08 0.07 5 51 0.08 0.07 5 20
24 0.65 0.07 5 20 0.65 0.07 O 36
25 0.47 0.00 0.36 0 47 0.00 1 01
26 0.76 0.03 1.01 0.76 0.03 0.51
27 0.23 0.03 0.51 0.23 0.03 0.03
28 0.12 0.03 0.03 0.12 0.03 0.09
29 0.09 0.03 0.09 0.09 0.03 0.06
30 0.06 0.03 0.06 0.06 0.03 1.48

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 4-6 Relative Contributions of the Static-Dynamic Components of
CSCB Members End Forces to the General Case of Ground Motion

103

 

 

 

 

 

 

 

 

2.
NODE I NODE J
Elements OZFXSG) OzeéG) UZMVSG) azFxéG) UzeéG) azMyéG)
UZBEG) UZFEG) ‘72ng” ”zFic) 021%” ”21416)
Bracing
Elements
22.73 22.73
2 23.61 23.61
Deck
Elements
1 1.30 1.22 15.46 1.30 1.22 1.30
2 5.96 1.03 1.30 5.96 1.03 7.45
3 2.49 0.89 7.45 2.49 0.89 8.21
4 0.21 0.81 8.21 0.21 0.81 17.86
5 4.18 0.74 17.86 4.18 0.74 4.15
6 6.72 2.60 4.15 6.72 2.60 5.32
7 6.54 4.69 5.32 6.54 4.69 1.30
8 3.59 4.46 1.30 3.59 4.46 2.59
9 0.21 4.28 2.59 0.21 4.28 1.48
10 7.19 4.14 1.48 7.19 4.14 3.40
11 3.40 4.04 3.40 3.40 4.04 9.12
Arch
Elements
20 4.79 1.67 15.08 4.79 1.67 4.79
21 5.60 1.74 4.79 5.60 1.74 6.61
22 1.52 1.77 6.61 1.52 1.77 8.65
23 1.95 1.82 8.65 1.95 1.82 17.42
24 4.56 1.81 17.42 4.56 1.81 4.51
25 5.25 0.23 4.51 5.25 0.23 5.10
26 6.25 1.35 5.10 6.25 1.35 0.86
27 3.07 1.37 0.86 3.07 1.37 1.46
28 1.03 1.36 1.46 1.03 1.36 2.05
29 2.30 1.32 2.05 2.30 1.32 1.39
30 1.39 1.33 1.39 1.39 1.33 8.99

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

104
dynamic components represent the structural response to dynamically
applied differential support motion where the inertia forces come into
play. The inertia forces will mainly be generated in the vertical
direction as well as in the horizontal direction resulting in a much

higher response of the bridge members
The difference in the relative contribution of the bracing member
responses due to different ground motion parameters can be justified as
follows. Although the variance of ground motion acceleration for the
two sets of parameters was the same, the variance of ground displacement
was higher in ground motion 2 than in ground motion 1. This difference
in ground displacements causes the observed increase in the variance of
the static response and in the covariance between static and dynamic

responses.

4.3.2 IN-PLANE RESPONSE OF THE CSCB

 

As mentioned earlier, three cases of ground motion correlation were
used to study the responses of the bridges. Each case has two different
sets of parameters (ground motions 1 and 2). Thus the CSCB was analyzed
six times for the in-plane response.

A comparison of the bridge responses due to the three correlation
cases was carried by dividing the variance of the members responses due
to fully corrtelated and wave propogation cases by the corresponding
variances of responses of the general case. For this part of the study,
the goal was to establish the correlation model of ground motion that
will generate the highest structural response of the CSCB.

Tables 4-7 and 4—8 show the results of this part of the study. The
entries in the tables that are shown as "-", indicate that the compared
values are zero. For the deck members we notice that the axial forces

were the highest in the fully correlated case where the movement of the

 

 

 

 

 

105

 

 

 

 

 

 

 

 

Table 4-7 Normalized In-plane CSCB Responses - Ground Motion 1
Elements Shear force Fx Axial force Fz Moment My
0F (w) of. (H) 0F (w) 012:, (H) a; (w) a; (H)
x X z z Y J
a; (G) 012;. (G) of, (G) a; (G) 05, (G) 051 (G)
x x z z y y
Deck
Elements
1 1.01 0.14 0.98 1.23 1.01 0.14
2 1.14 0.08 0.98 1.24 0.94 0.60
3 1.04 0.21 0.98 1.24 1.09 0.09
4 0.98 0.11 0.98 1.26 1.00 0.63
5 1.09 0.03 0.97 1.26 1.00 0.07
6 0.92 1.61 0.97 1.28 0.96 0.10
7 1.09 0.02 0.97 1.29 0.98 0.70
8 0.94 0.16 0.97 1.29 1.10 0.08
9 1.04 0.24 0.97 1.30 0.95 0.12
10 1.14 0.08 0.97 1.30 1.00 0.17
Bracing
Elements
0.97 0.95
Arch
Elements
20 1.00 0.14 1.16 0.015 1.00 0.14
21 1.14 0.06 1.16 0.012 0.94 0.10
22 1.07 0.22 1.16 0.01 1.09 0.09
23 1.00 0.08 1.16 0.01 0.99 0.63
24 1.10 0.03 1.16 0.01 1.00 0.08
25 0.92 1.60 1.16 0.008 0.93 0.11
26 1.10 0.03 1.16 0.01 0.96 0.72
27 0.97 0.11 1.16 0.01 1.10 0.08
28 1.06 0.24 1.16 0.01 0.93 0.12
29 1.14 0.06 1.16 0.03 0.99 0.16
30 0.99 0.16 1.16 0.06 - ~

 

 

 

 

 

 

 

 

 

 

 

 

 

 

106

Table 4-8 Normalized In-plane CSCB Responses - Ground Motion 2

 

 

 

 

 

 

 

 

 

 

Elements Shear force Fx Axial force Fz Moment My
012:. (w) 012:. (H) or; (w) of, (H) a; (w) a; (H)
x x z z y
of. (G) 012:. (G) of. (G) a: (G) a; (G) a; (G)
x x z z y y
Deck
Elements
1 0.96 0.1 0.98 1 26 0.95 0.1
2 1.06 0 28 0.98 1 26 0.94 0.05
3 0 97 0.17 0 98 1 26 0.97 0.19
4 0 95 0.05 0.98 1 26 0.97 0.54
5 0 97 O 04 0.98 1.27 0.96 0.04
6 0.97 1.35 0 98 1 29 0.42 0.07
7 0.98 0.03 0.98 1 32 0.91 0.93
8 0.93 0.07 0 98 1 31 1.00 0.12
9 0.93 0.24 O 98 1 31 0.95 0.05
10 1.05 0.23 0 98 1 31 0.93 0.13
Bracing
Elements 0.99 0.76
0 98 0 74
Arch
Elements
20 0.93 0.09 1 10 0.03 0.95 0.09
21 l 04 0.16 1 10 0.03 0.94 0.05
22 0.97 0.23 1 10 0.023 0.97 0.19
23 0.95 0.04 1 10 0.02 0.97 0.52
24 0.97 0.05 1 10 0.015 0.96 0.05
25 0 96 1 39 1.10 0.01 0.92 0.07
26 0.98 0.04 1.10 0.01 0.91 0.87
27 0 94 0.05 1 10 0.01 1.00 0.13
28 0 94 O 32 l 10 0.01 0.94 0.05
29 1 04 0 13 1 10 0.01 0.94 0.10
30 0 94 0 10 1 10 0.01 — -

 

 

 

 

 

 

 

 

 

107

supports are identical. The increase in the variances of the axial
forces with respect to the general case ranged from 23% to 30% for
ground motion 1, and 26% to 31% for ground motion 2. The second worst
case was the general correlation case where the deck experienced a 2% to
3% increase in axial force responses with respect to the wave
propogation case for both ground motions.

The variances of shear forces and bending moments developed in the deck
members were the highest in the wave propogation case where the deck
members experienced a 1% to 15% increase over the general case. For the
fully correlated case, the response of shear forces and bending moments
ranged from 8% to 25% of the response in the general case for most of
deck member, with a few members having a 63% to 70% increase of that
response.

For the arch members the variances of the axial forces were the
highest in the wave propogation case, where the variances of the axial
forces were 16% and 10% higher than the variances of the general case
responses for ground motions l and 2, respectively. In the fully
correlated case, the variances of axial forces ranged from 11% to 23% of
the response in the wave propogation case. For ground motion 1, the
variances of shear forces and bending moments were higher in the wave
porpogation case by 6% to 15% in comparison with the general case, but
for ground motion 2, the general case responses were higher by a maximum
of 6% compared to the wave propogation case. In the fully correlated
case, the variances of the shear forces and bending moments ranged from
6% to 87% of the corresponding variances in the general case of ground
motion.

These results indicate that for the deck members, the worst case of
ground motion for axial force response was the fully correlated one.

For the arch members the worst case is the wave propogation case. The

 

 

108

bending moments developed in the deck members in the fully correlated
case are very small relative to the other two cases. The same occured
in the arch members. Consequently, we can deduce that the fully
correlated case generated the highest axial force responses in the deck.
Intuitively this is expected since the supports are moving identically
resulting in inertia forces in the horizontal direction that have the
same phase. In the wave propogation case, there is a phase shift in
support motion resulting in smaller axial forces and larger bending
moments and shear forces. The general case of ground motion will have a
phase shift and loss of coherency between the support motions, resulting
in responses similar to the wave propogation case.

For the arch members it is clear that the wave propogation case of
ground motion is the worst, resulting in higher responses, especially in
the arch axial forces. This shows the sensitivity of long arch
structures to dynamically applied differential support motion or dynamic
pinching. The dynamically applied support motion will generate inertia
forces in the horizontal and vertical directions. These inertia forces
coupled with the sensitivity of arch structures to differential support
motion will generate the highest response in the arch members.

The response of the axial bracing members in the longitudinal
direction was the highest in the general correlation case. The
difference in response between the wave propogation and general case was
3% at most. The response in the fully correlated case was 6% and 25%
less than the response in the general case for ground motions 1 and 2,
respectively. The reason for that difference in response is that the
variance of ground motion displacement is higher for ground motion 2.

The higher the support displacement, the larger the longitudianl force

that has to be transferred from the deck to the arch.

 

 

109

The vertical displacements of CSCB resulting from the two ground
motions, and the comparison between the displacements from different
ground motion correlations are shown on Figures 4-5 through 4-9.
Figures 4-5 and 4-6 show the normalized vertical displacements of ground
motion 1 and 2, respectively. The normalization was done by dividing
the variances of vertical displacements for each case of ground motion
correlation by the maximum value of displacement. Figure 4-5 shows that
in the general case of ground motion the maximum displacement occurs at
the middle of the bridge. Meanwhile for the wave propogation and fully
correlated ground motions the maximum displacement occurs at the one
third points. In the fully correlated ground motion the variance of
the vertical displacements at midspan is very small. In ground motion
2, as shown on Figure 4-6, the maximum vertical displacements in the
general and wave propogation occur at midspan point of the bridge. The
response to fully correlated ground motion 2 is similar to the one for
ground motion 1. Figures 4-7 and 4-8 show the comparisons between the
variances of vertical displacements due to the three cases of ground
motion correlation. The mormalization was done by dividing the
variances of displacements resulting from the wave propogation and the
fully correlated cases of ground motion by the corresponding
displacements from the general case. Both figures indicate that the
vertical displacement from the wave propogation and the general case are
very close to each other and are much higher than the resulting
displacements from the fully correlated case of ground motion. Figure
4-9 shows the comparison of the vertical displacements of ground motions
1 and 2. The comparison was done by dividing the variance of the
displacement from ground motion 1 by the corresponding variance of
ground motion 2 for all ground motion correlations. The figure shows

that the ratios in the wave propogation and the general cases are very

 

 

 

 

 

 

 

 

 

 

 

 

GEL) 1'? I I I I I I I I I I I I 4
L1J2 1.0 —
D‘UJ 0_g ..
2'5 0.7 —
55 8‘2 -
0:3 024 ~
2U, 0.2 j
a 0.1
00 M I I I I E I j I fl
1 2 3 4 5 6 7 8 9 10 11 12 13 14
BRIDGE PANEL POINTS
(0)
1.2
CPU.) 1.1“ I I I I I I T r I I f I _
LUZ 1.0~ _
CSUJ C)8~
._J§ ' ‘
<(UJ Of7- _
av 12- -
O; 024— _
2m 0.2- —
E5 0.1% «
O'OJ I I I I I I fl I r
‘ 2 3 4 5 6 7 8 9 10 11 12 13 14
BRIDGE PANEL POINTS
(b)
1.2
OK.) 11 I I I I I I I I I I I V -
I32 1.0 _
—'-*-’ 0.8 a
“'5 07 4
(LU -
:E() 0.6 _
055 0.5 _
0,l 0.4 -
Zicn 0.2 1
5 0.1
00 M I I I I I I I j 7 7
1 2 3 4 5 6 7 8 9 10 11 12 13 14

BRIDGE PANEI(_ )POINTS
c

Figure 4—5: Normalized Variances of CSCB Vertical Displacements for
Ground Motion 1. (a) General Case; (0) Wave Propogation

Case; (c) Fully Correlated Case.

 

 

111

 

 

 

 

 

 

 

 

 

 

1.2
DEL.) 11 I I I I I I I I I I I I
LLIz 1.0 3
gm 0.8 —
<(EE 8:7 -
05:0 0'?) 7
0; 014 —
2(1) 0.2 “‘

—- 0.1 _

D

0-0 M 1 1 1 I Iil 1 1 r T
1 2 3 4 5 6 7 8 9 1O 11 12 13 14
BRIDGE PANEL POINTS
(0)

U) 12 1 r 1 1 a r 1 n 1 I 1 f
0,. 1.1- —
Lt—Iz 1.0~ 4
1:111: 08- _
45 0'74 4
<(uJ -
av 82: :
OS 04— _
2% 022— —

a 0.1— -1

0‘0 f I I | I I F I I I
1 2 3 4 5 6 7 8 9 10 11 12 13 14

BRIDGE PANEL POINTS
(b)
1.2

0‘92 11- I I I I I I I I I I I I .4
32 1.0— I
-L'J 0.8- n
3‘5 0.7— ~
5'6 32- -
0; 0:4- -
Zm 0.2- ~
5 0.1 -«

0'0 I I I I I I I I I E
1 2 3 4 5 6 7 8 9 10 11 12 13 14

 

BRIDGE PANEI(. POINTS
C

Figure 4-6: Normalized Variances of CSCB Vertical Displacements for

Ground Motion 2. (a) General Case;

Case; (c) Fully Correlated Case.

 

(b) Wave Propagation

NORMALIZED VERTICAL
DISPLACEMENTS

Figure 4-7:

NORMALIZED VERTICAL
DISPLACEMENTS

Figure 4~8z

 

112

 

 

 

 

 

 

 

 

   
 
  

  

 

 

 

1.5 1 1 , 1 T y 1 1 1 1 a 1
1.44 0'? (W)/0;' (G) -
1.24 ——01 (ID/0? (G) —
1.1-
0.9-
0.8—
0.6—
0.54
0.3%
0.24 I .m-a. v—vx /
0-0 1 ”741/1 1 1 f>T=;5CT 1 1 I\Jb 1
2 3 4 5 6 7 8 91011121314
BRIDGE PANEL POINTS
CSCB Normalized Variances of Vertical Displacements of
Ground Motion 1 With Respect to the General Case.
1-5 T 1 1 1 1 1 t 1 1 I . I
1.4- 0: (W)/<72 (G) —
1,2- ——‘7: (I‘D/0': (G) '1
1.1- 1
0.9—
0.84 I
0.6-
O.5~
033-
0.24 /~\\
00 f Evé I I I ’I/ j I I J/I I
*234567891011121314
BRIDGE PANEL POINTS
CSCB Normalized Variances of Vertical Displacements of

Ground Motion 2 With Respect to the General Case.

 

 

113

 

5-0 111

 

 

T I I I I I I N
_, 45. ___a: (GI/a: (C) .
2319 4.0- ———_a: (WW: <w> -
E5 3.54 — —— —ot (HI/o: (H) —
2 _ _
:8 3.0
10:5 2.5— -
N 20_ _
:4 '
gig 1.5 ” ~
g 1.0] k N- ‘ —" J 1
Z 0.534
0.0 1 1
'2734667801‘01'1121314

 

BRIDGE PANEL POINTS

Figure 4-9: CSCB Ratios of Variances of Ground Motion 1 to Ground

Motion 2 for the Three Cases of Ground Motion.

close to each other. It also shows that the displacements are higher in
ground motion 1 and that the ratio is changing along the span of the
bridge. Meanwhile, the ratio in the fully correlated ground motion is

almost constant along the span of the bridge.

4,3,3 IN-PLANE RESPONSE OF NRGB
The comparison of the bridge responses to the three correlation

cases of the two ground motions are summarized in Tables 4-9 and 4-10.

For the deck members, the variances of axial forces corresponding to the

1

 

 

114
fully correlated case were 66% to 100% higher than the other two cases.
The variances of shear forces and bending moments were the highest in
the wave propogation case. The normalized variances of bending moments,
shear and axial forces were higher than the corresponding values for the
CSCB responses. The lowest values of shear forces and bending moments
occured in the fully correlated case.

The responses of the arch members indicate that the highest
variances of bending moments, shear and axial forces occured in the wave
propogation case. The second highest responses were in the general
case, and the lowest were in the fully correlated case. The variances
of axial forces in the wave propogation case were about 20% and 800%
higher than the responses in the general and fully correlated cases of
ground motion, respectively.

The most evident difference in the responses of the two bridges is
the response of the longitudinal bracing. For the CSCB, the variances
of the axial forces were close to each other for ground motion 1, but,
for ground motion 2, the variances of axial forces corresponding to the
fully correlated case were about 25% less than the responses to the
other two correlation cases of ground motion. For the NRGB, the
variances of axial forces in the longitudinal bracing were the highest
in the fully correlated case. The variances were 82% to 100% higher
than those in the general and wave propogation cases of ground motion 1.
For ground motion 2, the variances of axial forces due to fully
correlated excitation were 44% to 78% higher than the variances due to
the general and wave propogation excitations.

The differences in longitudinal bracing responses for the CSCB and
NRGB were caused by the differences in their deck longitudinal force
transfer mechanism. The CSCB deck has one expansion joint at the south

abutment and a pin connection at the north abutment as shown in Figure

 

115

Table 4-9 Normalized In-plane NRGB Responses - Ground Motion 1

 

 

 

 

 

 

 

 

 

 

 

 

 

Elements Shear force Fx Axial force Fz Moment My
012:, (w) a; (H) a: (w) 0F (H) of, (w) a; (H)
x z Z J L.
of. (G) of. (G) of. (G) a? (G) ,2, (G) a; (a)
x x z z y y
Bracing
Elements
1 0.91 1.82
2 0.83 2.02
Deck
Elements
1 1.19 0.27 1.09 1.66 1.19 0.27
2 1.14 0.54 1.09 1.66 1.17 0.35
3 1.22 0.37 1.09 1.66 1.16 0.73
4 1.18 0.86 0.85 2.03 1.22 0.21
5 1.09 1.03 0.85 2.03 1.07 0.87
6 1.19 0.39 0.85 2.03 1.14 0.66
Arch
Elements
34 1.18 0.29 1.19 0.15 1.18 0.29
35 1.10 0.63 1.19 0.13 1.13 0.45
36 1.21 0.15 1.19 0.11 1.12 0.38
37 1.14 0.48 1.20 0.08 1.04 0.92
38 1.22 0.11 1.20 0.07 1.07 0.95
39 0.68 2.69 1.20 0.06 1.20 0.1
40 0.86 2.18 1.20 0.06 0.93 1.19

 

 

 

 

 

116

Table 4-10 Normalized In-plane NRGB Responses - Ground Motion 2

 

 

 

 

 

 

 

 

 

 

 

 

 

Elements Shear force Fx Axial force Fz Moment My
a: (w) of. (H) of. (w) of, (H) E, (w) of, (H)
x x z z y y
of. (G) of. (G) oi (G) a? (G) ,2, (G) of, (G)
x x z z y y
Bracing
Elements
0.99 1.43

2 0.92 1.78

Deck

Elements
1 1.11 0.08 0.98 1.80 1.11 0.08
2 1.09 0.33 0.98 1.80 1.11 0.21
3 1.12 0.09 0.98 1.80 1.09 0.32
4 1.11 0.33 0.92 1.98 1.12 0.05
5 1.09 0.33 0.92 1.98 1.08 0.46
6 1.12 0.10 0.92 1.98 1.10 0.38

Arch

Elements
34 1.11 0.16 1.12 0.02 1.11 0.16
35 1.06 0.47 1.12 0.02 1.09 0.35
36 1.12 0.07 1.12 0.02 1.07 0.28
37 1.09 0.37 1.12 0.02 1.12 0.04
38 1.09 0.31 1.12 0.02 1.06 0.65
39 1.13 0.03 1.12 0.02 1.06 0.75
40 0.78 3.20 1.12 0.02 1.12 0.12
41 0.92 2.28 1.12 0.02 1.01 0.93

 

 

 

 

 

 

 

117
4-10. The deck longitudinal force will be mainly transferred to the
support through the north abutment, and a little portion will be
transferred to the arch through the bracing.

In the response of the NRGB arch, the normalized variances of shear
forces and bending moments were relatively higher than for the CSCB arch
responses.

This can be related to the rise to span ratio. For the NRGB the
rise to span ratio is 0.22. For the CSCB the ratio is 0.21 for the
south hinge and 0.14 for the north hinge. Thus, the NRGB will respond
more in bending than the CSCB bridge.

The vertical displacements of NRGB resulting from the two ground
motions and the comparison between the different correlations of support
excitations are shown as Figures 4-11 through 4-15. Figures 4-11 and
4-12 show the normalized vertical displacements for ground motion 1 and
2, respectively. The normalization was done by dividing the variances
of vertical displacements for each case of ground motion correlation by
the maximum value of displacment. Figure 4-11 and 4-12 show that the
maximum vertical displacments of the general and wave propogation cases
occur at the one third span points. The minimum vertical displacements
occurs in the middle of bridge span. For the fully correlated ground
motion the maximum vertical displacement occurs at points close to the
middle of the bridge, where the vertical displacement are very small.

Figures 4-13 and 4-14 show the comparison between the variances of
Vertical displacements due to the three cases of ground motion
correlation. The normalization was done by dividing the wave
propogation and fully correlated cases of ground motion by the
corresponding displacements from the general case. Both figures
indicate that the vertical displacements resulting from the wave

propogation case were slightly higher than the displacements from the

 

 

 

118

general case for the two ground motions. The figures also show that the
displacements form the fully correlated case of ground motion is much
smaller comparing with the other two ground motion correlations.

Figure 4-15 shows the comparison of the vertical displacements of
ground motions 1 and 2. The comparison was done by dividing the
variances of the displacements from ground motion 1 by the corresponding
variance of the displacement of ground motion 2 for all ground motion
correlations. The figure shows that the ratio of the wave propogation
and the general case are very close. It also shows that the
displacements are higher in ground motion 2 by about 200% comparing with
ground motion 1. For the fully correlated ground motion the difference
between the two ground motions is not as high as in the other two cases.
This difference in response between ground motions l and 2 is related to

the frequency content of the ground motion.

4.3.4 OUT OF PLANE RESPONSE OF THE CSCB AND NRGB

Tables 4-11 and 4-12 show comparisons of the CSCB responses to the
three correlation cases of ground motions l and 2, respectively. The
study of the responses Show that the bridge members responded
differently to different ground motions. For some members the highest
response was in the wave propogation case, for others it was either
in the general or fully correlated case. Thus, it was difficult to
predict which is the worst case for the out-of—plane response of the
arch members or a group of members.

Tables 4—12 and 4-14 show comparisons of the NRGB responses for the
three correlation cases of ground motions l and 2, respectively. The
conclusions that can be drawn from the NRGB responses are the same as

the ones for the CSCB responses.

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(f) 1.2 1 1 1 r 1 1 1 1 1 Ifi 1 1 1
O}— 1.1- —1
£51 1.0“ ..

0. - _
:15 07— _
go 0.6- _
0:3 O.5~ —
OD. 0.4- —
ZU) 0.2-1 .1
O
0-0 r 1 1 1 1 1 1 1 1 1 r 1 1
‘ 2 3 4 5 6 7 8 91011121314151617
BRIDGE PANEL POINTS
(0)

(D 1-2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0}. 1,14 s
I'LiIJLIZJ 5.0- —
33 0:7: :
§0 06— —
0:5 0.5"1 -
of; 0.4— —
20) 0.2- _

a 0.1 -I

0-0 r [if t 1 T 1 1 WI 1 1 1 1
12 3 4567891011121314151617
BRIDGE PANEL POINTS
(b)

U) 1-2 ' 1 1 1 1 1 1 r I I 1 1 fr 1 1
O)— 1.1 -(
SE! 1.0 —
33 8:? 1
EU 0.6 _
QCL 0.4 _
2U) 0.2 ..

a 0.1 _

0.0

 

12 3 4 5 6 7 8 91011121314151617
BRIDGE PANEI(_ POINTS
0

Figure 4-11: Normalized Variances of NRGB Vertical Displacements for
Ground Motion 1. (a) General Case; (b) Wave Propogation

Case; (c) Fully Correlated Case.

 

 

NORMALIZED
DISPLACEMENTS

NORMALIZED

DISPLACEMENTS

NORMALIZED
DISPLACEMENTS

Figure

 

 

 

 

 

 

 

 

 

 

 

 

1:?“ I 1 1 1 I I I I I I 1 1 1 1 -
1.0— _
(18— _
Of7— _
0.6— -
0.5— _
O.4~ _
0.2—
0.14 I
013 ’1 I 1 I I I I I I 1 I I I I
1 2 4 5 6 7 8 9 H311121314151617
BRIDGE PANEL POINTS
(0)
1:3_ I 1 I 1 I I I I 1 I I I I I 1 g
1.0- _
(18- _
0.74 _
0.6— _
0.5- _
0.4“ -1
0.2-4 —<
0.1 3
C40 1‘1 I I I I 1 1 I 1 1 I 1 1 I T
I 2 4 5 6 7 8 9 IO 11 12 13 14 15 16 17
BRIDGE PANEL POINTS
(b)
1:?_ I I I 1 I I 1 I 1 1 I 1 I I I 4
1.0- -
0.8- -
0.7— -
0.6— -
0.5— a
0.44 a
0.2— -
0.1 ‘
CLO "T 1 I I I I I T I I I I 1 I rt
12 3 4 5 6 7 8 91011121314151617
BRIDGE PANEL POINTS
- (C)
4-12: Normalized Variances of NRGB Vertical Displacements for

Ground Motion 2. (a) General Case; (b) Wave Propogation

Case; (c) Fully Correlated Case.

122

 

 

 

 

 

15 I I I I I I I I I I I I I I I
21 1.4- 0? (W)/0? (C).——Uf (HI/<7? (C) —
31%? 1.2-
2:
(ILL, 1.1-
“J:
:8 0.9—
35 0.8—1
2% 0.6“
55 0.5“
O 0.3“
Z 0.2—
0'0 I I I I I I I I I I I I I I I
4 2 3 4 5 6 7 8 91011121314151617

 

BRIDGE PANEL POINTS

Figure 4-13: NRGB Normalized Variances of Vertical Displacements of

Ground Motion 1 With Respect to the General Case.

 

 

 

 

 

 

1‘5111111111111111
.1 1.4- 0: (W)/02 (G).———<72 (ID/02 (G) —
<{
1798 1.2~
05E 1.14
L”:
:8 0.9—
mi 0.8‘
N 06-
3% '
<— -
EC) 0-5 /\ /\
g 0.3— .n / \ /\
Z 02— / \ / I / \
. ‘J \j \
O'OIIIIIIITIIIIIII
‘ 2 3 4 5 6 7 8 91011121314151617
BRIDGE PANEL POINTS
Figure 4-14: NRGB Normalized Variances of Vertical Displacements of

Ground Motion 2 With Respect to the General Case.

123

 

5'0 T I I I

 

 

 

I I I I Ifi I I I I I
a) 4.54 ——-—— UT (CI/0: (G) a
(3g) 4.04 ————— Of (W)/0: (W) _
E3 3.5— — — —01 ((0/0: (H) —
2 _ 2
:5 3.0
LL15 2.5“ _(
E0. 20- _
_, .
gg 1.54 \ A s
n: / A \_
o 1'07 /’ ‘\ ./' \/
2 0.5- —
0'0 I I I I fI I I I I I I I I I
‘ 234567891011121314151617

BRIDGE PANEL POINTS

Figure 4-15: NRGB Ratios of Variances of Ground Motion 1 to Ground

Motion 2 for the Three Cases of Ground Motion.

The lateral displacements of CSCB resulting from the two ground
motions and the comparison between the displacement from different
correlations of ground motion are shown on Figures 4-16 through 4—20.
Figures 4-16 and 4-17 show the normalized lateral displacements of
thearch and the deck for ground motion 1 and 2, respectively. The
normalization was done by dividing the variances of lateral
diSplacements for each case of ground motion correlation by the maximum
value of displacement. Figures 4-16 and 4-17 indicate that the maximum

lateral displacements in CSCB occur at points close to the midspan point

124

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(I) 1 1
LLJ Deck
'EE 0.6“ ‘
2'12 0,0! m I 1 I 1 1 Y I v 1 I 1
28 2 3 4 5 6 7 8 9 1011121314
g3 1.1 v I I I I l 1 v
2% 06“ Arch
0 0.0 . . . . . . a . . . . .
1 2 3 4 5 6 7 8 9 10 11 12 13 14
BRIDGE PANEL POINTS
(0)
m 1.1 1 v
OI— D
L.1J _ eck
EEI 0.6
J: 0.0‘ j T I I I I I I I | I I
gm 2 3 4 5 6 7 8 9 10 11 12 13 14
050 I 1
(23% 0'6 ‘ I | ' . I I I I [Arch
(9 I 7‘ A
O 0.0 Y I l l I l l l I l l V
‘ 2 3 4 5 5 7 8 9 10 11 12 13 14
BRIDGE PANEL POINTS
(b)
1.1 . . . . 1 1 .
OK) Deck
LUZ 0.6‘ '1
N01
:12 0.0 I l I I l I I I l l I I
<L1J 2 3 4 5 6 7 8 9 10 11 12 I3 14
5% 1 1
O . r . . . . .A .
Q _, rch _
21) 0.6
D 0.0 Y . . . . . . . . . . .
‘ 2 3 4 5 6 7 8 9 10 II 12 I3 14
BRIDGE PANEL POINTS
(c)
Figure 4-16: Normalized Variances of CSCB Lateral Displacements for

Ground Motion 1. (a) Fully Correlated Case; (b) General

Case; (c) Wave Propogation Case.

1.1

125

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

U) . .
OF- Deck
LIJ .
BIL—T3 0.6 4
42 O-O‘ ' l I I I I I I T I
gm 2 3 4 5 6 7 8 91011121314
01% 1 1
O y I f ' 'Arch
Q 4 .
Z!) 0.61 A
O 0.0 v . . v . . 4 . . . . 4
1 2 3 4 5 6 7 8 9 10 11 12 I3 14
BRIDGE PANEL POINTS
(0)
1.1 T . .
OE.) Deck
LUZ 0.6‘ -
EL” 00
«IE ‘234567891011121314
50 1 1
O; . . I I I I I I Y . 'Arch
Z(i') 0.6'1 A 7
O 0.0 v . 1 . . . . T . . . v
1 2 3 4 5 6 7 8 9 10 11 12 13 14
BRIDGE PANEL POINTS
(b)
1.1 . . .
BE 0 Deck
(:15 '6‘.
-‘J 010 I V I I I I I I I V I I
<5 ‘ 2 3 4 5 6 7 8 9 IO 11 12 13 14
02:0
CZ); 1.1 r T 7 'Arch
(L) 0.6‘ ‘
O 0.0 v , . . . . . . . . Y
‘ 2 3 4 5 6 7 8 9 IO 11 12 13 14
BRIDGE PANEL POINTS
(C)
Figure 4-17: Normalized Variances of CSCB Lateral Displacements for

Ground Motion 2. (a) Fully Correlated Case; (b) General

Case. (c) Wave Propogation Case.

126

 

“’ I I I I I I T I I I I I
1.4— 0? (W)/0§ (C).——<7? (ID/0? (G) —
1.2— _
1.1— /“‘ “e r
0.9— {N/ ,__, ”‘ / -
0.8- / _
0.6-1 —1
0.5—I
0.3-
0.2—
0-0 1 1 1 1 l 1 1 1 1 1 1 1

‘ 2 3 4 5 6 7 8 9 1o 11 12 13 14

BRIDGE PANEL POINTS

 

NORMALIZED LATERAL
DISPLAC EM ENTS

Deck

 

 

 

 

1‘5 I I I I I I I I I I I I
1.4— 0? (W)/02 (G). ——_U? (I‘D/0? (C) —
1.2— -

 

1.1q °-~ q
09— \~ / \ / \ s
0.8-1 —1
0.6- _
0.5— ..
0.3— .1
0.2-
0.0 T

‘5313é'7éé1'o1'11‘21314

BRIDGE PANEL POINTS

NORMALIZED LATERAL
DISPLACEMENTS

Arch

 

 

 

Figure 4-18: CSCB Normalized Variances of Lateral Displacements of

Ground Motion 1 With Respect to the General Case.

127

 

 

 

 

 

 

 

 

 

 

15 I I I I I I I I I I I I
2’ 1.4— 02 (WI/02 (C).——02 (FD/U: (G) —
Eff 1.2- f. \ s
133 1.1- /\ _
2
OLIJ 0.9-1 / \l \ M. \—
KJJE’ 0.84 f v _
283 0'5“ / I
‘55 0.5—
% 0.3—
2 0.24
Deck
0-0‘ 1 1 f 1 1 1 1 1 1 1 1 1
2 3 4 6 6 7 8 9 1O 11 12 13 14
BRIDGE PANEL POINTS
1.5 1 1 1 1 1 l 1 1 1 1 1
_’ 1_4_ a; (W)/og (o),____o: (HI/0: (G) -
<1
mm 1.2- _
F— r-~.
L132 1 1— \ -
5L” '
of. 0.9— Ae// J —
uJQ) O.8- / ‘
N31
:10. 0-6~ ‘
$3 0.5— -
C) 0.3-1 _
2 0.2— a
A h
0.0 1 1 1 1 1 1rc1 1 1 1 1 T
‘ 2 3 4 5 6 7 8 9 1O 11 12 13 14
BRIDGE PANEL POINTS
Figure 4-19; CSCB Normalized Variances of Lateral Displacements of

Ground Motion 2 With Respect to the General Case.

50

128

 

45-
40-
354
30—
25—
2.0-4
L5—
LOfi
05—

NORMALIZED LATERAL
DISPLACEMENTS

 

 

T I I I 1 1 1 1 1 1 1 1
”I (CD/02 (G) 3
———— 0? (W)/o: (W) 3
— —— ——0": (H)/a; (H) q
Deck _
A j —" :

1 1

 

00
1

30

D47 1 1 1 1 1 1 1 1
2 3 4 5 6 7 8 9 10 11 12 13 14
BRIDGE PANEL POINTS

 

4.5—
4.0—
35-
3.0-«
2.5—1
201
1.5‘
L0—
0.5—

NORMALIZED LATERAL
DISPLACEMENTS

a: 161/0:15 _
———— a: (We: (w) .-
— —— —01 <H)/cr: (H) _

Arch —

 

 

00

 

Figure 4-20:

I I I

1 1 1 1 1 1 1 1
2 3 4 5 6 7 8 9 1O 11 12 13 14
BRIDGE PANEL POINTS

CSCB Ratios of Lateral Displacements of Ground Motion 1

to Ground Motion 2.

 

129

Table 4-11 Normalized Out-of-plane CSCB Responses - Ground Motion 1

 

 

 

 

 

 

 

 

Element TV Mx Mz MW
2 2 2 2 2 2 2 2
0 (W) 0 (H) U (W) 0 (H) 0 (W) U (H) U (W) 0 (H)
F F M M M M M M
V V X x z z w w
2 2 2 2 2 2 2 2
0 (G) 0 (G) U (G) U (G) U (G) 0 (G) 0 (G) U (G)
F F M M M M M M
y y x x z z w w
Deck
Elements
1 O 81 1 52 1.07 1.02 0 80 l 31 - -
2 0 71 1 51 1.33 0.88 l 30 O 71 1.15 0.98
3 0 89 1 20 1.01 0.95 1 10 1.25 l 10 1.00
4 1.08 1.19 1.12 1.64 O 79 1.46 1.13 0.89
5 0 75 1.56 0 99 1.21 0 73 l 32 1.05 1.17
6 1 04 O 70 2.67 0.43 O 86 1.09 O 98 1.25
7 0 90 0.92 1.06 0.84 0.89 1.08 1 17 0.81
8 0 72 1.24 O 62 1.23 1.18 O 79 O 88 0.87
9 0.95 0.91 0 71 0.80 1.22 O 77 1 21 0.86
10 1.11 0.82 1.38 0.75 1.03 0.97 0.98 0.92
11 l 10 0.89 1.02 0.95 0 82 1.06 1.12 0.89
20 1 24 0.91 1.09 O 99 1.22 O 92
21 l 24 0 90 1.60 0 68 1.10 0 99
22 1.10 O 97 1.10 1 00 O 99 1 O7
23 1.07 1 37 0 88 1 10 1.14 0 91
24 1.09 0 88 0 92 1 05 1.11 1 06
25 l 21 0.13 1.41 0.40 0.97 1 57
26 1 30 0.85 0.91 0.98 0.75 1 39
27 0 79 l 23 0.91 l 04 O 82 0.86
28 0 84 O 81 1.03 0 87 1.18 0.83
29 l 28 0 81 1.72 0 56 0 88 0 99
30 1 ll 0 80 1.05 1 00 1.09 O 91

 

 

 

 

 

 

 

 

 

 

 

130

Table 4-12 Normalized Out-of-plane CSCB Responses - Ground Motion 2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Elements I‘v Mx Mz Mw
2 2 2 2 2 2 2 2
U (W) 0 (H) 0' (W) U (H) 0 (W) 0 (H) U (W) 0 (H)
F F M M M M M M
V V X x z z w w
2 2 2 2 2 2 2 2
a (G) a (G) a (G) a (G) a (G) a (G) a (G) 0 (G)
F F M M M M M M
y y X X z z w w
Deck
Elements
1 0 88 1.72 1.02 1.06 0.98 1 55 - -
2 0.92 2.07 1.01 0.67 0 98 0 55 1.02 0.95
3 1.01 1.37 1.01 1.18 O 88 0 83 1.01 0.93
4 0 99 0.92 0.97 1.04 0 85 1 72 1 01 0.85
5 0 88 1.53 0.95 1.12 O 95 1 80 0.98 1.03
6 1.04 0 74 0.94 0.95 1.03 1 35 O 93 1.15
7 0.99 l 24 1.01 0.85 1.02 1 30 1.00 0.74
8 0.99 1.63 0.99 1.32 l 01 0 63 1.01 0.95
9 1.03 1.15 1.02 0.95 0 98 0.60 1.02 0.91
10 1.03 0 74 1.03 0.63 0 94 1.05 1.02 0.89
11 0.99 0 75 1.03 1.01 O 96 1.46 l 04 0.90
Arch
Elements
20 1 02 0 75 1.02 0.92 1.02 O 81
21 1.02 0 77 1.02 0 43 1.02 1 01
22 1.02 O 98 1.02 1 06 l 01 l 14
23 1.02 l 17 1.01 1 34 1 01 O 86
24 1 01 0 88 1 01 1 20 1 01 O 94
25 l 02 0 22 O 96 O 35 1 01 1.35
26 1 02 0 84 1 02 l 21 1.01 1.19
27 l 01 1 04 1.02 1.22 1.02 0 97
28 1.03 0.93 1.04 0.92 1 02 0.88
29 1.03 0.75 1.04 O 38 1.02 1.06
30 1.02 0 76 1.02 0 85 1.03 0 94

 

 

 

 

 

 

-‘fl

131

 

Table 4-13 Normalized Out-of-plane NRGB Responses - Ground Motion 1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Elements l‘v Mx Mz MW
2 2 2 2 2 2 2 2
a (W) a (H) a (W) a (H) a (W) a (H) a (W) a (H)
F F M M M M M M
V Y x x z z w w
2 2 2 2 2 2 2 2
a (G) a (G) a (G) a (G) a (G) a (G) 0 (G) a (G)
F F M M M M M M
y y x x z z w w
Deck
Elements
1 1 01 0 97 1.01 0 92 1.01 0.97 1 03 0.87
2 0 91 1 89 1.03 0.92 0.97 1.27 1.00 1 04
3 1 00 0 91 1.05 0.88 1.00 1.08 1.03 0 84
4 l 00 0.98 0.99 0 92 1.00 0.69 1 02 0 97
5 O 99 1 19 1.01 0 94 0.99 0 61 0.99 0 86
6 0 99 1 04 0.99 1 14 0.96 l 58 1.05 1 27
7 1 01 1.33 1 14 1.35 1.05 1.70 1 13 1 70
8 0 94 1.75 1 11 1.40 1.02 1.33 1 04 1 21
9 0 78 1.44 0 93 1.22 1.02 0 59 0 93 O 96
10 1.04 1.05 0 94 1.02 0.69 1 23 0.84 1 22
11 1.12 0 74 0 92 1.02 0.94 1 19 1.13 0.69
12 0 89 1.13 0 98 1 01 1.04 1.01 0 91 1.15
13 0 98 1 70 O 99 0.97 0.99 1.25 0 99 0.96
Arch
Elements
34 1.05 0.86 1.00 1.00 1.00 0.96
35 1 05 0.88 1 03 1 00 1.05 0 87
36 1 02 0.90 1 04 0 88 1.05 0 89
37 1.05 0.83 0 99 0.93 1.04 0.95
38 1 06 0.78 1 03 1.16 1.01 1.06
39 1 05 0.88 1 02 1 03 0.97 1 65
40 0 92 0.63 1 12 0 38 0.85 2 37
41 1 01 0.65 1 04 0 47 0.99 1 48
42 0 95 1.04 0 92 1.20 0.89 l 23
43 l 02 0.79 1.09 0.97 0.84 1.24
44 1 09 0.81 0.95 0 99 0.93 1.09
45 0 89 1.07 0 78 1 31 0.95 1.01
46 0 93 1.04 0.99 1.03 0.95 1 02
47 0 95 1.00 1 01 0.98 0.93 1 05 J

 

 

 

 

 

132

Table 4-14 Normalized Out-of-plane NRGB Responses - Ground Motion 2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Elements I‘V Mx Mz Mw
2 2 2 2 2 2 2 2
a (W) a (H) a (W) a (H) a (W) a (H) a (W) a (H)
F F M M M M M M
V v x X z z w w
2 2 2 2 2 2 2 2
a (G) a (G) a (G) a (G) a (G) a (G) a (G) a (G)
F F M M M M M M
y y x x z z w w
Deck
Elements
1 1.01 0.98 1.01 0 89 1.01 0.98 1.02 0.83
2 0.86 2.40 1.02 0 89 0.95 1.40 0.97 1 31
3 0 99 1.12 1.04 0 89 1.01 1 05 1.05 0 53
4 1.01 0.82 0.97 0 95 0.98 0 99 0.99 1 21
5 1 00 1.12 1.03 0 93 1.01 O 63 1.03 0.72
6 0 97 1.27 1.00 1 23 0.95 1 77 1.02 1.26
7 0 99 1 51 1.06 1 15 0.96 1.98 0.94 2 61
8 0 82 2 70 1.05 1 17 0.95 1.49 1.05 0 90
9 0 82 l 96 0.93 1 44 1.00 0.65 0.96 0 93
10 1 04 0 86 0.98 0 99 0.74 2 21 0.83 1 85
11 1 05 O 63 0.91 1 32 1024 1 04 1.09 0 38
12 0.88 1 56 0.94 l 20 1.00 1 01 0.87 1 66
13 0.95 l 84 0.99 0 95 0.96 1 36 1.00 0 92
14 0.98 1 02 1.00 0 93
Arch
Elements
34 1.04 0 86 1 00 0 98 1.00 1 00
35 1.04 0 88 1 03 0 94 1.03 0 88
36 1.01 1 04 1.02 1 01 1.03 0.93
37 1.05 0 66 0.97 1.11 1.02 1.03
38 1.06 0 64 1 02 1.03 0.99 1 29
39 1.04 0 85 1 03 0 97 0.91 2 12
40 1.05 0 14 1 O8 0 23 0.72 3 64
41 1.05 0 22 1.06 0.31 0.89 1 91
42 0.97 1 11 0.97 1.19 0.83 1.80
43 1.04 0 65 1 06 0 72 0.88 1.47
44 1.04 0.69 0 91 1 30 0.93 1.23
45 0.89 1.42 0 84 1 71 0.96 1.09
46 0.93 1.19 0.98 1.02 0.96 1 14
47 0.95 1 11 0.99 0.99 J 0.93 1.19

 

 

 

 

133

for all ground motion cases. The reason behind that is due to the fact
that CSCB is not symmetric.

Figures 4-18 and 4—19 show the comparison of the lateral
displacements of ground motion 1 and 2 for the arch and the deck. The
comparison was done in the same way as for the in-plane response of
CSCB.The figures indicate that the lateral displacements were not
greatly influenced by the different ground motion cases. The lateral
displacement for the three cases were within 10-15% from each other for
most of bridge panel points.

Figure 4-20 shows the comparison of the lateral displacements of
ground motion 1 and 2. The comparison was done in the same way as for
the in-plane response. The figure indicates that the lateral
displacements were larger in ground motion 1 by about 20%.

The lateral displacement response of NRGB is shown in Figures 4-21
Through 4—25. Figure 4—21 shows that the maximum displacement in the
fully correlated case of ground motion 1 occured at midspan. In the
other two cases, the maximum displacements in the deck occured at points
closed to the supports. In the arch the maximum displacements occured
at midspan. The maximum lateral displacements in ground motion 2
occured at one quarter points in the three cases of ground motion in
both the arch and the deck as shown in Figure 4-22. Figures 4-23 and
4-24 show that the maximum lateral displacements for the arch and the
deck occured in the fully correlated case of ground motion. Figure 4—25

shows that the lateral displacements are about 1.5 times higher in

ground motion 1 than in ground motion 2.

 

 

 

 

0.6~

Deck

 

 

0.0‘

2 3 4 5 6 7 6 5 1b1'11'21'31r41'61'617

 

1.1
0.6‘
0.0

NORMALIZED

m
’—
2
Lu
2
DJ
‘5
CL
(Q
a

 

Arch

 

 

2734 6 6 ? 693101'11'21'3141'61'617
BRIDGE PANEL<P)O|NTS
o

 

0.64
0,0

 

I 1' I v T 1 I Ifi

Deck ~

 

 

2 3 4 6 6 7 6 931'01'11'21'31'41'51'617

 

NORMALIZED
DISPLACEMENTS

0.61

I I I I l l ‘1 I I7
Arch

.1
‘

 

 

0.0

 

2 345761 891101T112131711'51Y617
BRIDGE PANEng’OINTS

 

0.61

Deck 1

 

 

0.0

 

2 3 4 5 6 7 8 é1'01'11'21'31'41'51'617

 

NORMALIZED
DISPLACEMENTS

0.61
0.0

 

I T fi' I I T I I I fl T I l I
Arch

M

 

 

1

Figure 4-21:

Ground Motion 1. (a) Fully Correlated Case;

Y

2 3 4 6 6 1 6 61011 1213141551617
BRIDGE PANEL( P)O|NTS
c

Normalized Variances of NRGB Lateral DiSplacements for

Case; (c) Wave Propogation Case.

 

(b) General

 

 

   

 

 

 

135

 

0.6 ~

0 O Deck
' 1 2 3 47> 6 i 661'01'11'21'31'41'51'617

 

 

 

 

1.1 I I I I I I I r n 1 I I

NORMALIZED
DISPLACEMENTS

'Aréh
0.6 .
0.0 Y.......f4..44.
12 3 4 5 6 7 8 91011121314151617
BRIDGE PANEL )POINTS
0

 

 

 

 

1.1 1 v . 1 . . . .fir . . $4
0.61
0.0‘ r T I I I‘Ifi' r I I r 1 I
12 3 4 5 6 7 8 91011121314151617

r

Deck

 

 

I.I I I I I I ‘r I I I I I ‘I W

0.6-

NORMALIZED
DISPLACEMENTS

Arch

 

 

 

1 2 3 4 6 67 6 61'01'11'2173141'61’617
BRIDGE PANEng’OINTS

 

O-PJ Deck ‘
0.0 I fiI I I I I I I T I I 7 l I
12 3 4 5 6 7 8 91011121314151617

1.1 I I I I I WTj‘rfi I

 

 

' T Arch '
0.64 -
0.0

NORMALIZED
DISPLACEMENTS

 

 

 

1237476 676 6 1'01r11'21'31'41'51'617
BRIDGE PANEL )POINTS
c

Figure 4-22: Normalized Variances of NRGB Lateral Displacements for
Ground Motion 2. (a) Fully Correlated Case; (b) General

Case. (c) Wave Propagation Case.

136

 

2.0 1 1 1 1 1 1 I I I I I I I I I
1.84 01 (We: (G).——Ui (HI/a: (o) 1
1.6— r
1.4- _
1.2- / \ / \ _
"0' / \ \ /‘\ .
0.8— .J v _
0.5-I

0.4-

0.2—
Deck

00 I I I l I I I I I I I I I II
2 3 4 5 6 7 8 91011121314151617

BRIDGE PANEL POINTS

 

NORMALIZED LATERAL
DIS PLACEMENTS

 

 

 

 

2‘0 I I I I I l 1 I I I I I I I I

1.8— 0? (w)/a: (c),___a: (111/07 ((3) -
1.61 _
1.4-
1.2-
1.04
C184
O.6~
0.4—
o.2~
0-0 T I 1 1 1 1 1 1 I 1 1 1 1 1 T

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

BRIDGE PANEL POINTS

 

NORMALIZED LATERAL
DISPLACEMENTS

 

Arch

 

 

Figure 4-23: NRGB Normalized Variances of Lateral Displacements of

Ground Motion 1 With Respect to the General Case.

 

 

   

 

137

 

2.0
1.8
1.6-
1.4-1
1.2—
1.0—
(18-
(16~
(14-
02—

NORMALIZED LATERAL
DISPLACEMENTS

I

 

J; 216/6; (1;). __o: (HI/o: (G) —

I I I

/\
/\ r -

Deck

 

 

(10

 

I

1 1 1 1 I 1 I 1 1 1 1 T
4 5 6 '7 8 9 1011 1213 P1151617
BRIDGE PANEL POINTS

 

2.0
1.8—
1.6-
1.4-
1.2—
1.0—
0.8-
0.6—
0.4—
o.2~
0.04

 

NORMALIZED LATERAL
DISPLACEMENTS

 

I

I

 

I 1 I I I I ”I I I I I I 1

0: (I‘D/<7: (C). ——-02 (ID/0: (C) _
Cd —1

/ \ P, -

\v
Lax _
Arch
I I

 

 

I T TF1 I l I I l f
4 5 6 7 8 91011121314151617
BRIDGE PANEL POINTS

Figure 4-24: NRGB Normalized Variances of Lateral Displacements of

Ground Motion 2 With Respect to the General Case.

 

 

 

NORMALIZED LATERAL
DISPLACEMENTS

NORMALIZED LATERAL
DISPLACEMENTS

Figure 4-25:

138

 

 

 

 

 

 

 

 

 

5‘0 I I I I I I I I I I I I I I
4.5 0? (CD/02 (C) -
4.o~ ————— at (W)/0: (W) —
3.54 —— — -—<71 (*0/02 (H) -
3.0— —
2.5- Deck "
2.0m -
1.5— “N —
1.0— fl, . Jr)“ ‘ _
0.54 ‘ A
0'0 I I I I I I I I I I I I I Ifi

‘ 2 3 4 5 6 7 8 91011121314151617

BRIDGE PANEL POINTS

5‘0 I I I I I I E I I I I I I
4.5— 0? (C)/02 (G) 4
4.0— ———_a: (WI/o: (w) -
3.5- —— — -—0’ (FD/0: (H) —
3.0“ q
2'57 Arch
2.0— _
1.5— “'7. _
1.0- . A
0.6- a
0'0 I I T I I r I I I I fiT I I I

1 2 5 6 7 8 91011121314151617

 

BRIDGE PANEL POINTS

NRGB Ratios of Lateral Displacements of Ground Motion 1

to Ground Motion 2.

139

4.3.5 THE EFFECT OF GROUND MOTION PARAMETERS

As mentioned earlier, two sets of characteristic ground motions
were used to analyze the responses of the CSCB and NRGB. The parameters
characterizing these ground motions are shown in Tables 3—1 and 3-2.
For both models the variances of ground acceleration (area under the
spectrum) are equal, but the variance of ground displacement in ground
motion 1 is larger than the variance in ground motion 2.

The comparison between the CSCB and NRGB out-of—plane and in-plane
responses for the three correlation cases of ground motions 1 and 2 are
shown in Tables 4-15 through Table 4-26. Tables 4-15 to 4-17 show the
comparisons of the in-plane responses of CSCB to the three correlation
cases of support motion in ground motions l and 2. The tables show that
the CSCB responses were much higher in ground motion 1 in the three
cases of support motion correlation. The axial forces in the arch were
more than 6 times higher than the response in ground motion 2. The
axial forces in the deck were about 65% higher in ground motion 1. The
variances of shear forces and bending moments in most members were
higher in ground motion 1. The increase in response ranged from 16% to
400%. For some members the shear forces and bending moments Were higher
in ground motion 2.

The results of the CSCB response to the out—of—plane ground motion
are shown in Tables 4-18 to 4-20. The responses of the arch members
to three correlation cases are higher in ground motion 2. The members
experienced an increase in end force response ranging from 5% to 87%.
The response of the deck members was not uniform. For some members
ground motion 1 generated the highest response, while for others it was

ground motion 2.

 

140

 

 

 

 

 

Table 4-15 In-plane CSCB Responses to the General Case:
Ratio of Ground Motion 1 to Ground Motion 2 Responses
Elements Shear Force Fx Axial Force Fz Bending moment Mv
0%. (cl>/a§ (<22) of. (op/of. (CZ) .734 (69/0?4 (G2)
x X z z y y
Bracing
Elements
1.14
2 1.12
Deck
Elements
1 1.04 1.76 1.04
2 4.96 1.73 0.73
3 1.22 1.70 2.25
4 0.82 1.66 1.11
5 1.85 1.62 0.85
6 1.16 1.61 0.93
7 1.73 1.61 1.71
8 0.81 1.58 1.69
9 1.51 1.55 0.69
10 3.82 1.53 1.12
11 1.12 1.51 -
Arch
Elements
20 0.95 6.05 0.95
21 4.52 6.10 0.70
22 1.55 6.16 2.16
23 0.87 6.20 1.07
24 2.03 6.23 0.85
25 1.17 6.10 0.94
26 1.92 5.94 1.55
27 0.87 5.94 1.68
28 1.95 5.93 0.67
29 3.61 5.90 0.98
30 0.98 5.86 —

 

 

 

 

 

 

 

 

 

 

 

141

 

 

 

 

 

Table 4-16 In-plane CSCB Responses to the Wave Propogation Case:
Ratio of Ground Motion 1 to Ground Motion 2 Responses
Elements Shear Force Fx Axial Force Fz Bending moment Mv
of. (1111/03 (112) of. (119/of. (112) of4 (119/of4 (112)
x x z z y y
Bracing
Elements
1.12
2 1.10
Deck
Elements
1 1.11 1.75 1.11
2 5.36 1.72 0.73
3 1.32 1.68 2.54
4 0.85 1.64 1.14
5 2.08 1.60 0.89
6 1.11 1.60 0.96
7 1.94 1.60 1.84
8 0.82 1.56 1.86
9 1.67 1.53 0.69
10 4.14 1.51 1.20
11 1.20 1.49 -
Arch
Elements
20 1.01 6.35 1.01
21 4.98 6.41 0.70
22 1.71 6.46 2.43
23 0.92 6.51 1.09
24 2.29 6.54 0.89
25 1.12 6.41 0.97
26 2.15 6.25 1.64
27 0.90 6.26 1.85
28 2.20 6.24 0.66
29 3.95 6.22 1.03
30 1.03 6.19 —

 

 

 

 

 

 

 

142

Table 4-17 ln-plane CSCB Responses to the Fully Correlated Case:
Ratio of Ground Motion 1 to Ground Motion 2 Responses

 

 

 

 

 

Elements Shear Force Fx Axial Force Fz Bendina moment Mv
of. (Hp/cf. (H2) 0% (Hp/0% (H2) of; (Hp/of; (H2)
x x z z y y
Bracing
Elements
1.42
2 1.40
Deck
Elements
1 1.55 1 72 1.55
2 1.40 1 70 1.39
3 1 52 1 67 1.11
4 1 76 1 64 1.30
5 1 35 1 61 1.38
6 1.37 1.60 1.36
7 1.40 1.58 1.28
8 1 7o 1 55 1.10
9 1 50 1 53 1.43
10 1 34 1 52 1.51
11 1 51 1 50 -
Arch
Elements
20 1.54 2 83 1.54
21 1.65 2 64 1.41
22 1.47 2 31 1.10
23 1.83 1 93 1.30
24 1.37 1 57 1-35
25 1.35 1 53 1'29
26 1.38 4 61 -
27 1.79 6 89 1.09
28 1.46 6 95 1.44
29 1.61 6 83 1.52
30 1.52 6 53 -

 

 

 

 

 

 

 

 

 

 

 

 

 

 

143

Table 4-18 Out-of-plane CSCB Responses to the General Case:
Ratio of Ground Motion 1 to Ground Motion 2 Responses

 

 

 

 

 

 

Elements FV Mx Mz Mw
2 2 2 2
a (G) U (G) 0 (G) 0 (G)
F 1 M 1 M 1 M l
V X z w
2 2 2 2
a (G ) a (G ) a (G ) a (G )
F 2 M 2 M 2 M 2
y x z w
Deck
Elements
1 1 35 1.26 1 00 1.18
2 1 29 0.97 0.79 1.12
3 O 89 2.08 1 21 1.11
4 O 65 1.81 1 10 1.22
S 1.01 1.07 0 73 0.98
6 1 43 0.40 0 79 0.94
7 1.06 1.09 1 01 1.45
8 O 78 2.64 O 95 1.13
9 1.16 2.29 0 71 1.13
10 1 27 1.00 0 81 1.20
11 1 04 1.25 1 20 -
Arch
Elements
20 1.03 1.06 1.03
21 1.05 0.79 1 20
22 1 55 1.20 1 20
23 1.59 1.23 1.06
24 1 17 1.20 l 33
25 1.71 0.85 1 66
26 1.07 1.23 1 86
27 1 69 1.18 1.66
28 1.85 1.23 1.08
29 1.08 0.79 1 20
30 1 14 0.96 1 21

 

 

 

 

 

 

 

 

 

144

Table 4—19 IOut-of-plane CSCB Responses to the Wave Propogation Case:
Ratio of Ground Motion 1 to Ground Motion 2 Responses

 

 

 

 

Elements FY MX Mz Mw
2 2 2 2
a (W ) 0 (W ) 0' (W ) U (W )
F 1 M 1 M 1 M 1
. .2 X Z w
2 2 2 2
a (W ) a (w ) a (w ) a (w )
F 2 M 2 M 2 M 2
y x z w
Deck
Elements
1 1.26 1.32 O 83 -
2 1 00 1.28 0.67 1.32
3 O 71 2.08 0.98 1.22
4 O 71 2.10 1.13 1.25
5 O 86 1.12 0.85 1.32
6 1.43 1.11 O 62 1.03
7 0.97 1.15 0.69 1.10
8 0.57 1.66 1.17 1.26
9 1.07 1.61 1.19 1.34
10 1.37 1.34 0.79 1.09
11 1.16 1.24 O 69 1.30
Arch
Elements
20 1.26 1.14 1 24
21 1.28 1.24 1.29
22 1.67 1.27 1.18
23 1.64 1.07 1.21
24 1.27 1.10 1.46
25 1.87 1.24 l 59
26 1.36 1.10 1 38
27 1 32 1.05 1.34
28 1 51 1.22 1.25
29 1.35 1.31 1.04
30 1.23 1.00 1.27

 

 

 

 

 

 

 

145

Table 4-20 Out:of-p1ane CSCB Responses to the Fully Correlated Case:
Ratio of Ground Motion 1 to Ground Motion 2 Responses

 

 

 

 

 

 

Elements Fy MX Mz Mw
0% (H1) ”51 (H1) ”121 (H1) UM (H1)
Y X z w
2
0F (H2) “121 (H2) ”12 (H2) ”12 (H2)
y X Z w
Deck
Elements
1 1.19 1.21 0.85 -
2 0 94 1.27 0.65 1 22
3 O 78 1 68 1.19 1.20
4 O 84 2 87 1.03 1.16
5 1.03 1.15 O 81 1 39
6 1 35 1.15 0.60 1.07
7 O 79 1.09 0.66 1.03
8 0.59 2.46 1.26 1 34
9 0.92 1.93 1 21 1.08
10 1.40 1.18 O 66 1.16
11 1 24 1.18 O 59 1.19
Arch
Elements
20 1.26 1.14 1.18
21 1 24 1.26 1.18
22 1.54 1.13 1.13
23 1.85 1.02 1.13
24 1.18 1.06 1 50
25 1 07 0.99 l 93
26 1 10 1.00 2.17
27 l 99 1.00 1.46
28 1.60 1.18 1.03
29 1.17 1.16 1.13
30 1.21 1.14 1.19

 

 

 

 

 

 

 

 

146

Table 4-21 In-plane NRGB Responses to the General Case:
Ratio of Ground Motion 1 to Ground Motion 2 Responses

 

 

 

 

 

Elements Shear Force Fx Axial Force Fz Bending moment My
of. (cg/of. <92) oi: (cg/4%w (62> afi (cg/of1 <02)
x x z z y y
Bracing
Elements
0.91
2 1.02
Deck
Elements
1 0.49 1.61 0.49
2 o 62 1.61 0.49
3 o 50 1.61 0.69
4 0 80 0.75 0.46
5 0 74 0.75 o 57
6 o 51 0.75 o 53
7 o 63 0.48 o 46
8 o 63 0.48 o 53
9 0.51 0.80 0 58
10 0 77 0.80 0.46
Arch
Elements
34 0.45 0.46 0.45
35 0.67 0.45 0.52
36 0.43 0.45 0.56
37 0.52 0.45 0.43
38 0.51 0.45 0.51
39 0.45 0.45 0-59
40 0.89 0.45 3.:2
41 0.78 0.45 .
42 0.45 0-45 0-51

 

 

 

 

 

 

147

Table 4-22 In-plane NRGB Responses to the Wave Propogation Case:
Ratio of Ground Motion 1 to Ground Motion 2 Responses

 

Elements Shear Force Fx Axial Force Fz Bending moment My

 

of. (119/of. (112) oi (wp/afi 012) of1 (wp/af4 (W2)
x x z z y y

 

 

 

Bracing
Elements
0.84
2 0.92
Deck
Elements
1 0.53 1.79 0.93
2 0.65 1.79 0.52
3 0.55 1.79 0.74
4 0.84 0.70 0.50
5 0.75 0.70 0.57
6 0.54 0.70 0.56
7 0.62 0.52 0.50
8 0.62 0.52 0.53
9 0.54 0.75 0.59
10 0.77 0.75 0.50
Arch
Elements
34 0.48 0.49 0-48
35 0.69 0.49 0 53
36 0.46 0.48 0-59
37 0 54 0.48 0.47
38 0 53 0.48 0 50
39 0.48 0.48 0-59
40 0.77 0.48 0.49
41 0.73 0-48 0 54
42 0.48 0.48 0 51

 

 

 

 

 

 

 

 

148

Table 4-23 In-plane NRGB Responses to the Fully Corrleated Case:
Ratio of Ground Motion 1 to Ground Motion 2 Responses

 

 

 

 

 

Elements Shear Force FX Axial Force Fz Bendinv moment My
0% (Hp/0% (H2) 0:? (Hp/oi (H2) afi (Hp/of4 (H2)
x x z z y y
Bracing
Elements
1.15
2 1.15
Deck
Elements
1 1.61 1.49 1.61
2 1.00 1.49 0.82
3 1.89 1.49 1.53
4 2.07 0.77 1.83
5 2.28 0.77 1.09
6 1.87 0.77 0.92
Arch
Elements
34 0.81 2.48 0 31
35 0.89 2.28 0 65
36 0.93 2.00 0 76
37 0 93 2.00 0 76
38 0 78 1.50 0 72
39 1 57 1.32 0 74
40 O 74 1.23 '

 

 

 

 

 

 

 

149

Table 4-24 Out-of-plane NRGB Responses to the General Case:
Ratio of Ground Motion 1 to Ground Motion 2 Responses

 

 

 

 

 

 

 

 

 

 

 

Elements Fy MX Mz MW
2 2 2 2
0(6) 0(9) 0(0) 0(8)
F l M 1 M l M 1
Y X z w
2 2 2 2
0(0) 0(C) 0(G) 0(6)
F 2 M 2 M 2 M 2
y X Z w
Deck
Elements
1 0.66 0.85 0.66 0.83
2 1.01 0.87 0.86 0.99
3 0.65 0.89 0.92 1.22
4 0.81 1.42 0.96 1.44
5 0.98 2.30 0.78 1.83
6 O 83 1.92 1.03 2.70
7 1 32 2.42 1.18 3.61
8 l 79 2.31 1.05 2.13
9 0.94 2.08 0.81 2.02
10 0.85 2.37 1.27 1.77
11 0.78 1.45 0.84 1.12
12 0 75 1.10 0.80 1.11
13 O 84 0.84 0.79 0.81
14 O 72 0.82 - -
Arch
Elements
34 0.82 0.78 1.19
35 0.74 0.75 0.66
36 1.76 0.70 0.61
37 1.51 0.86 0.70
38 0.87 1.19 0.93
39 1.34 0.75 1.19
40 0.97 0.75 1.45
41 1.13 0.75 1.21
42 1.47 0.78 1.12
43 0.86 1.06 0.77
44 1.56 0.92 0.65
45 1.99 0.83 0.69
46 0.84 0.79 1.30
47 0.91 0.78 0.89

 

 

 

150

Table 4-25 'Out-of-plane NRGB Responses to the Wave Propogation Case:
Ratio of Ground Motion 1 to Ground Motion 2 Responses

 

 

 

 

Elements Fv Mx Mz MW
2 2 2 2
0 (W) 0 (W) U (W) a (W)
F 1 M 1 M l M 1
47—_ X Z W
2 2 2 2
a (W) a (W) a (W) a (W)
F 2 M 2 M 2 M 2
y x z w
Deck
Elements
1 0.66 0.86 O 66 0 84
2 1.07 0 88 O 88 1.02
3 0.67 0.89 0 91 1.19
4 0.81 1.45 O 98 1.47
5 0.97 2.27 0 76 1.75
6 0.85 1.89 1 04 2 78
7 1.46 2 62 1.29 3.62
8 2.08 2 43 1.13 2.12
9 0.90 2.09 0.83 1.95
10 0.85 2.27 1.17 1.79
11 O 84 1.46 0.77 1.15
12 O 76 l 14 0.83 1.16
13 O 87 O 84 0 82 O 80
Arch
Elements
34 O 83 0.78 1.19
35 O 75 0.75 0.68
36 1 78 O 72 0.62
37 1.51 0 87 0.72
38 0.87 1 21 0.95
39 1.36 0 74 1.26
40 O 89 0 76 1.71
41 1.08 0.73 1.35
42 1.45 0.75 1.21
43 0.85 1.08 0.73
44 1.65 0.96 0.65
45 1.99 0.77 0.69
46 O 84 0 79 1.30
47 0.91 O 80 0.89

 

 

 

 

 

 

 

 

 

151

Table 4-26 Out-of-plane NRGB Responses to the Fully Correlated Case:
Ratio of Ground Motion 1 to Ground Motion 2 Responses

 

 

 

 

Elements Fv Mx Mz Mw
012:. (H1) 0; (H1) 0; (H1) 051 (H1)
2 X Z W
6127 (H2) 0131 (H2) 0; (H2) 0M (H2)
y x z w
Deck
Elements
1 0.65 0.89 0.65 O 88
2 O 80 0.90 0 78 O 79
3 0.53 0.88 O 94 1.94
4 0 97 1.16 1 54 1.15
5 1 04 2.34 0 77 2.19
6 0.69 1.78 O 92 2.72
7 1.17 2.83 1.01 1 97
8 1.16 2.75 0.93 2.88
9 0.69 1.78 0.73 2.07
Arch
Elements
1.16
0.66
0.58
0.64
0.77
0.92
0.95

 

 

 

 

 

 

 

 

 

 

152

Table 4-27: CSCB and NRGB In-Plane and Out-Of-Plane Frequencies of The

First Twelve Modes

CSCB in-plane CSCB out-of-plane
mgradgsec) fiiflgl mgradgsecz f Hz
2.71 0.43 2.30 0.36
5.27 0.84 3.74 0.59
9.73 1.55 5.54 0.88
9.93 1.58 8.87 1.41
14.6 2.33 9.31 1.48
15.3 2.44 10.03 1.60
21.8 3.47 15.72 2.50
28.9 4.60 19.64 3.13
29.9 4.76 23.80 3.78
36.8 5.80 24.54 3.91
44.5 7.08 31.62 5.03
50.9 8.10 34.41 5.47

NRGB in—plane NRGB out-of-plane
1.50 0.24 0.92 0.15
3.14 0.50 1.80 0.29
4.38 0.70 2.62 0.42
5.21 0.82 3.32 0.53
5.57 0.88 3.98 0.63
5.97 0.95 4.74 0.75
7.30 1.16 5.69 0.90
8.37 1.33 6.84 1.09
10.11 1.61 7.12 1.13
10.95 1.74 7.43 1.18
12.38 1.97 7.62 1.21
12.69 2.02 8.67 1.38

 

 

 

153

The comparison of the NRGB In-plane responses to the three
correlation cases of ground motions l and 2 are shown in Tables 4-21 to
4-23. The tables clearly show that the highest responses of the bridge
occurs due to the general and wave propogation cases of support
excitations in ground motion 2. For the fully correlated case, the
highest response occured in ground motion 1.

The out-of-plane response of the NRGB to the three correlation
cases with ground motions l and 2 are shown in Tables 4-24 to 4-26. The
responses of the arch members show that the highest responses occured
due to ground motion 2. For the deck members the response in general
was the highest in ground motion 1 with some members for which the
highest response occured due to ground motion 2.

The in-plane response of the NRGB and CSCB for ground motions l and
2 are determined by their frequency content. The natural frequencies of
the first twelve modes of the CSCB and NRGB in-plane and out—of—plane
models are shown in Table 4—27. If we compare the first 12 natural
frequencies of the in-plane model of the CSCB with the normalized
autospectrum of ground acceleration (Figures 3-3), it can be seen that
the area enclosed between the autospectrum and the frequency range from

O to 40 Hz is much larger in ground motion 1 than in ground motion 2.

Consequentely, ground motion 1 will contribute to a higher in-plane
response of the CSCB.
For the NRGB, the first 12 natural frequencies ranging from 0.24 to

2.12 Hz. Those frequencies are very close to each other. Also, one can

see that the corresponding values of the normalized autospectrum are
higher for ground motion 2 than for ground motion 1 (Figure 3-3). Thus,

a higher response of the NRGB will result due to ground motion 2. The

 

 

 

154
same conclusions can be made for the out-of—plane response of the CSCB
and NRGB.

The in-plane displacement response of the CSCB and NRGB (Figures
4-5, Through 4-9, and Figures 4-11 through 4-15) indicate that larger
displacements resulted from ground motion 2. As mentioned in Chapter 3
; the variance of ground displacement is higher in ground motion 2.
Thus, the responses of element end forces is governed by the frequency
content of the ground motion, and the displacement response is governed
by the displacement of the ground motion.

The out—of—plane displacement responses are shown in Figures
4—16,through 4-25. The figures show that the displacements resulting
from ground motion 2 were higher at the ends of both bridges. The
reason for this behavior could be due to the fact that the lateral
stiffnesses of the bridges are not as high as the in—plane stiffnesses.
Thus, the ground motion displacements will be effective at both ends of
the two bridges, but at the middle of the bridges the frequency content

of the ground motion will play a bigger role.

4.3.6 THE CSCB RESPONSE DUE TO TWO SUPPORT MOTION VERSUS FOUR
SUPPORT MOTION

In this section, additional support motions were considered at the
north and south supports of the approaching span of the CSCB as shown on
Figure 4-26. This was done to study the effect of the additional
support motion on the CSCB in-plane and out-of—plane responses.
Tables 4-28 and 4-29 show the normalized values of in-plane responses of
the CSCB due to the three correlation cases of ground motions 1 and 2,
respectively, when four support excitations are considered. Comparing

Tables 4-7 and 4-8 with 4-28 and 4-29. We can see that the general

 

 

 

 

155

"A. "A.”
X1 X4
X2 X3

a) Ground Motion Applied to Four Supports

 

 

 

 

/Rigid B100k\

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

7" 7' r ,Vrrrr r7
W’- W.-
Xl ' X2

b) Ground Motion Applied to Two Supports

Figure 4-26: Two support Motion Vs. Four Support Motion

 

 

 

156
behavior did not change, except that the normalized values of axial
forces resulting from fully correlated ground motion were higher when
the ground motion was applied to four supports.

Tables 4-30 and 4—31 show the normalized values of the out-of—plane
response of the CSCB due to the three correlation cases for ground
motions 1 and 2, respectively. The general behavior is not different
from the case with two support motion. In some members the normalized
values were higher for the 4 support excitation.

Tables 4~32 and 4-33 compare the responses of the CSCB when two and
four support motions are considered for the general correlation case of
ground motions 1 and 2, respectively. The tables show that except for
the increase in the deck axial forces by 7% and 9% and a decrease in
bracing axial forces by 10% and 18%, ground motions 1 and 2,
respectively. The responses for two and four support motions are within

2-3% from each other.

4.3.7 THE EFFECT OF INCREASING THE BRIDGE STIFFNESS ON THE
RESPONSE COMPONENTS

As mentioned earlier, the total variance of the responses consists
of three components; the variance of the dynamic component, the variance
of the static component and the covariance of static and dynamic
components. In Tables 4-1 through 4-6, it can be seen that at a given
stiffness of a structure, the dynamic component are dominant, and the
contribution of static and static-dynamic components is minor.

In this part of the study, the influence of the structural
stiffness is examined. For this purpose, the stiffness of the CSCB
members were increased, and the relative contribution of each component

was monitored. The stiffness was increased by multiplying each member

 

 

 

 

157

Table 4-28 Normalized In-plane CSCB Responses to Four Support Motion-
Ground Motion 1

 

 

 

 

 

 

 

Elements Shear force Fx Axial force Fz Moment My
2 2 2 2 2
(W) U (H) 0 (W) 0 (H) (W) a (H)
F F F F M M
x x z z y y
2 2 2 2 2
(G) 0 (G) 0 (G) 0 (G) (G) 0 (G)
F F F M M
x x z z y y
Deck
Elements
1 1.01 0.15 0.94 1 34 1.01 0.15
2 l 14 0.08 0.94 1 34 0.94 O 10
3 l 04 O 21 0.94 1 35 1.09 O 09
4 0.98 0.11 0.94 1 35 0.98 0.67
5 1 09 0.03 0.94 1 36 1 00 0.07
6 0.87 1 80 0.94 1 37 0.95 0 10
7 1.09 0.02 0.94 1 39 0.96 0.71
8 0 94 0.16 0.94 1 39 1 10 0.08
9 1.03 0.24 0.94 1 39 0.94 0 12
10 1.14 0.08 0.94 1.39 1.00 O 17
11 l 00 0.17 0.94 1.39
Arch
Elements
20 1.00 0.14 1.16 0.01 1 00 0.14
21 1.47 0.06 1.16 0.01 0 93 0.10
22 1.07 0.22 1.16 0 00 l 08 0.09
23 0.99 0.09 1.16 0 00 0 98 0.67
24 1.10 0.03 1.16 0 00 1 00 0.08
25 0 86 1.80 1.16 0.00 0 95 0.10
26 1 10 0.03 1.16 0.00 O 94 0.74
27 0.97 0.11 1.16 0.00 1 10 0.08
28 1.06 0.25 1.16 0 00 0.93 0.12
29 1.14 0.06 1.16 0 00 0.99 0.16
30 0.99 0.16 1.16 O 01 - -

 

 

 

 

 

 

 

 

 

 

 

Table 4-29 Normalized In-plane CSCB Responses to Four Support Motion

Ground Motion 2

158

 

 

 

 

 

 

 

Elements Shear force Fx Axial force Fz Moment My
2 2 2 2 2 2
(W) 0 (H) 0 (W) 0 (H) (W) a (H)
F F F F M M
x x z z y y
2 2 2 2 2 2
(G) U (G) a (G) a (G) (G) U (G)
F F F F M M
x x z z y
Deck
Elements
1 0.45 0.10 0.96 1 36 0.95 0.10
2 1.05 0.29 0.96 l 34 0.94 0.05
3 0.96 0.17 0.96 1.37 0.96 0.18
4 0.95 0.05 0.96 1.37 0.97 0.58
5 0 97 0 04 0.95 1 36 0.96 0.04
6 0 94 1.48 0 96 1.41 0.91 0.06
7 0 97 0.02 0 96 1.45 0.89 0.95
8 0.92 0.07 0 96 1 44 1.00 0.12
9 0.93 0.24 0 96 l 44 0.94 0.05
10 1.06 0.23 0 96 1.44 0.93 0.13
11 0.93 0.13 0 96 1.43
Arch
Elements
20 0.95 0.09 1.10 0.03 0.95 0.09
21 1.03 0.16 1.10 0.02 0.94 0.05
22 0.97 0 24 1.10 0.02 0.96 0.19
23 0 95 0.04 1.10 0.01 0.97 0.56
24 0 97 0.05 1 10 0.01 0.96 0.05
25 0.94 1 51 1 10 0.00 0.91 0.07
26 0.98 0.04 1 10 0.00 0.88 0.88
27 0.93 0.05 1.10 0.00 1.00 0.13
28 0.93 0.33 1 10 0.00 0.94 0.05
29 1.05 0.13 l 10 0.00 0.93 0.10
30 0 93 O 10 l 10 0.00 - -

 

 

 

 

 

 

 

 

 

 

 

159

Table 4-30 Normalized Out-of-plane CSCB Responses to Four Support
Motion-Ground Motion 1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Elements Fy Mx Mz Mw
2 2 2 2 2 2 2 2
0 (W) 0 (H) 0 (W) <7 (H) U (W) 0 (H) U (W) 0 (H)
F F M M M M M M
__1___ __X___ X x z z w w
2 2 2 2 2 2 2 2
0 (G) a (G) 0 (G) U (G) a (G) a (G) U (G) U (G)
F F M M M M M M
y y x x z z w w
Deck
Elements
1 0.86 1.60 1 04 1.02 O 96 1 59 — -
2 0 90 1.98 1 01 0.70 O 91 O 51 1 O4 0 92
3 O 97 1 72 1.03 0.96 O 82 0 91 1 02 O 94
4 0 92 1 13 O 98 1.42 O 83 1.51 1.01 O 84
5 O 80 1 70 0 93 1.17 0 92 1 83 0.96 1 13
6 1 04 O 78 O 96 0.14 1 O4 1 58 0 91 1 16
7 0 99 1 O7 1 01 0.83 1.04 1 45 1 02 O 71
8 0.90 1 81 0 89 1.72 1.05 O 72 O 98 1.00
9 1.06 1 14 0 96 1.05 1 00 0.63 1 05 0.78
10 1.07 0 82 1.04 0.59 O 88 0.83 0.99 O 95
ll 1 01 O 79 1.07 1.00 0 87 l 38 1.07 O 86
Arch
Elements
20 1.01 0 77 1.05 0.95 1.03 O 82
21 1.02. O 78 1.03 0.46 1.05 O 97
22 1 02 0.93 1.06 0.98 l 04 1 11
23 1.04 1.28 1.03 1.33 1.00 0 85
24 1.02 O 84 1.00 1.22 0.99 1 01
25 0.98 0 12 0.92 0.28 1 03 1 64
26 1.06 0.71 1.03 1.17 0 96 1 74
27 0 96 1.46 1.05 1.23 O 96 1 04
28 0.99 0 96 1.06 0.90 1 O4 0 77
29 1.05 O 69 1.06 0.35 l 01 1 14
30 1.01 O 75 1.02 0.98 1 07 0.91

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

160

Table 4-31 Normalized Out-of—plane Responses to Four Support Motion
-Ground Motion 2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Elements Fy Mx Mz Mw
2 2 2 2 2 2 2 2
U (W) U (H) 0 (W) U (H) 0 (W) 0 (H) 0 (W) 0 (H)
F F M M M M M M
__X___ __1___ X X z z w w
2 2 2 2 2 2 2 2
0 (G) 0' (G) 0 (G) a (G) 0 (G) 0 (G) a (G) 0 (G)
F F M M M M M M
y y X X Z Z w w
Deck
Elements
1 O 90 1.70 1 01 1.10 0 97 1 64 — —
2 O 92 2.26 l 00 0.72 0 99 0 58 1 01 0.99
3 1 00 1.65 1 01 1.22 0 92 O 80 1 01 0.95
4 1 00 1.06 0.97 1.05 O 88 1 74 1 00 0.91
5 0 9O 1 61 0.96 1.16 0 94 2 11 0.97 1.11
6 1 02 O 85 0.96 0.14 1.02 1 71 0 94 1.15
7 O 99 1.42 1.00 0.87 1.01 1.57 l 00 0.75
8 0.96 2.13 0 98 1.38 1.03 O 65 1.00 1.00
9 1.03 1.38 1 01 0.95 O 99 0 60 1.01 0.95
10 1.05 0.78 1 02 0.66 0 93 1 15 1.01 0.91
11 1 OO 0 75 1.03 1.02 0.93 1 87 1.03 0.92
Arch
Elements
20 1 01 O 80 1.01 0.91 1.01 0 86
21 1 01 0.83 1 01 0.46 1 01 1 05
22 1 00 1.05 l 02 1.08 1 01 1 14
23 1 01 1.16 1 00 1 36 1 00 O 92
24 1 00 O 92 0.99 1 27 O 99 1 O4
25 1 00 0 23 0.95 O 36 1 00 1 43
26 1 02 0 86 1.00 1 27 1.01 1 21
27 1.01 1 04 1.01 1.24 1.00 1 03
28 1.02 0 96 1.03 0.92 1.01 O 92
29 1.02 O 78 1.03 0 39 1.02 1.08
30 1 01 0 79 1 03 0.39 1.02 1.08

161

Table 4—32: Ratio of In-plane CSCB Responses with Four Support Motion

in the General Case to the Responses with Two Supports
Motion-Ground Motion 1

 

 

 

 

 

 

Elements
2 2
a (G)(4 supports) a (G)(4 Supports) a (G)(4 Supports)
F F M
X Z L
2
a (G)(2 supports) a (G)(2 Supports) a (G)(2 Supports)
F F M
x z y
Bracing
Elements
0.89
2 0.87
Deck
Elements
1 1 02 1.09 1 02
2 1.01 1.08 1.00
3 1.03 1.08 1.00
4 1.02 1.07 1 O6
5 1.01 1.07 1 O3
6 l 13 1.07 O 98
7 0.99 1.07 1 02
8 0.99 1.07 0.99
9 1 01 1.07 0.99
10 1.00 1.07 1.01
11 1.01 1.07
Arch
Elements
20 1.01 0.99 1.01
21 1.01 0.99 1.00
22 1.03 0.99 0.99
23 1.02 0.99 1.06
24 1 01 0.99 1.03
25 l 12 0.99 O 98
26 0.99 1.00 1 02
27 0.99 1.00 0.99
28 1.02 1.00 0.99
29 1.00 1.00 1.00
30 1.00 1.00 -

 

 

 

 

 

162

Table 4-33: Ratio of In-plane CSCB Responses with Four Support Motion
in the General Case to the Responses with Two Support
Motion-Ground Motion 2

 

 

 

 

 

Elements
2 2
a (G)(4 supports) a (G)(4 Supports) a (G)(4 Supports)
F F M
x z Y
2 2
a (G)(2 supports) a (G)(2 Supports) a (G)(2 Supports)
F F M
x z y
Bracing
Elements
0.83
2 0.80
Deck
Elements
1 1.01 1.08 1.01
2 1.01 1.08 1.00
3 1.03 1.08 1.00
4 1.02 1.08 1.07
5 1.01 1.08 1.03
6 1 09 1.09 0.98
7 0.99 1.10 1.02
8 0.99 1.09 1.00
9 1.01 1.09 0.99
10 1.00 1.09 1.00
11 1.01 1 09 -
Arch
Elements
20 1 01 0.99 1.01
21 1.01 0.99 1.00
22 1.04 0.99 0.99
23 1.01 0.99 1.07
24 1.01 0.99 1.03
25 1.09 0.99 0.98
26 0.99 0.99 1.02
27 0.99 0.99 1.00
28 1.02 0.99 0.99
29 1.00 0.99 1.00
30 1.00 0.99 0.99

 

 

 

 

 

 

163

stiffness by the same factor. Five members were studied and the results
are shown in Figures 4-27 through 4-31. All figures indicate that at
the initial bridge stiffness, the main response contribution came from
the dynamic component and the other two components were very small. By
increasing the stiffness, the variance of the static component starts
increasing and at one point it becomes equal to the dynamic component.
Beyond that point by increasing the stiffness further, the static
component becomes the major contributor to the response and the dynamic
component keeps decreasing until it's contribution becomes very small.
The relative contribution of the covariance of static and dynamic
components is different from one member to another. What was noticed is
that if the member response is sensitive to static displacement, the
covariance component is higher. Also, it was noticed that the static
component of axial forces increases faster than the other components
such as bending moments.

The figures indicate that the stiffness can be increased by a large
factor before the static response becomes significant. This implies
that for bridges of this type, the static response is not expected to be
significant. Only bracing members responses have a noticeable static

component for a moderate increase in stiffness.

4.3.8 THE EFFECT OF WAVE VELOCITY ON THE RESPONSE OF THE TWO BRIDGES
In literature the central frequency of ground motion is calculated

using one of the following formulas

Iw S(w) dw
01 = (4.1)
Zn I S(w) dw

 

 

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___ ______________ (4.2)

425(6) dw 0.5
01 =21
" J S(w) dw

WIIere 53(w) is the spectral density function of ground motion

acceleration and has the form

I41 W

M10». 212+ 4:041 [WP/61214 WWII

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Using the ground motion 1 the values of 01 calculated by equations

(4.1) and (4.2), are found to be 5.71 Hz and 10.77 Hz’ respectively.

For ground motion 2 those values are 2.25 Hz and 5.61 Hz. By choosing

different wave velocity in equation (3.108) and investigating the
responses of some deck and arch members, a study on the effect of
seismic wave velocity was conducted.

Figure 4-32 shows the responses of axial forces and bending moments
in CSCB arch members No. 20 and 26 as a function of wave velocity. The
figure indicates that the bending moments and axial forces are much
higher at low velocity and have a periodic behavior. Both responses
tend to approach a unique level with increasing wave velocity.

Figure 4-33 shows the responses of axial forces and bending moments
in CSCB deck members No. 2 and 7. The figure shows that the axial force

in deck members increases with the increase of wave velocity, but the

 

 

 

 

 

 

 

170

bending moments decreases, and both responses are approaching a unique
level.

Figures 4-34 and 4—35 show the axial forces and bending moments in
NRGB arch members No. 34, 40 and 47 as a function of wave velocity.
Both responses in all the three members show that they are decreasing
with increasing the wave velocity and tend to approach a unique level.

The same conclusion can be drawn from Figure 4-36, where the
bending moment responses is depicted for NRGB deck members No. 2, 7 and
12.

Figures 4—37 and 4—38 show the axil force responses in longitudinal
bracing members in CSCB and NRGB, respectively.

Figure 4-37 shows that the wave velocity has no effect on the axial
force responses in CSCB, while it procuces a slight increase in NRGB

response.

4.3.9 NONSTATIONARY RESPONSE OF THE CSCB AND NRGB

 

To determine whether the CSCB and NRGB will reach their stationary
response during strong ground motions of normal duration (10—15 sec),
the variances of responses in both bridges were calculated at one and
five seconds after the beginning of strong ground motion. The ground
acceleration was assumed to start at time zero with its full stationary
intensity (i.e., a Heaviside temporal modulating function was assumed).
The results were compared against the stationary responses of the two
bridges. The calculations were performed by replacing the frequency
response function (equation 3.67) by the "time—dependent frequency
response function" (equation 3.117).

Tables lI-34 and 4-35 show the response ratios of the CSCB and NRGB

at one and five seconds relative to the stationary response. The tables

 

 

 

 

 

 

 

171

 

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178

indicate that the responses of the CSCB and NRGB are very close to
their stationary values after about five seconds of strong shaking.

These results indicate that considering only the stationary
response is reasonable for both bridges. The reason for this is that
the first few modes, which have long periods and would therefore be
expected to have slowly growing responses, do not contribute signifi-
cantly to the overall responses. If we compare the in-plane frequencies
of the CSCB and NRGB (Table 4—27) with the normalized acceleration
spectra (Figure 3-4), it can be seen that the first two modes for the
CSCB and the first five modes for the NRGB have frequencies lower than

the dominant excitation frequencies of the ground acceleration.

4.4 COMPARISON BETWEEN DETERMINISTIC AND RANDOM VIBRATION STUDIES

The relative response between fully correlated and wave propagation
excitations Were obtained by Dusseau (1985) using deterministic
analysis, and in this study using random vibration analysis. These are
compared here.

It should be noted that there were some differences between the
input excitations used in the two studies. Dusseau used scaled versions
of the Caltech B1 and B2 simulated accelerogram, while in this study
normalized ground acceleration l and 2 with the parameters given in
Tables 3-1 and 3-2 were used. The frequency content of ground motion 1
is similar to that of the B1 record, and hence the comparisons are only
made for these excitations.

Although the acceleration spectrum of the Bl record is similar to
that of ground motion 1, there may be significant differences in the
ground displacements. The filter used to attenuate low frequencies of

the ground acceleration spectrum in the random vibration study

 

 

 

   

179

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 4-34: Ratios of the CSCB Responses at One and Five Seconds to the
Stationary Responses.
2 2 2
a (NON) a (NON) a (NON)
x z 1
Elements
2
0 (ST) 0 (ST) 0 (ST)
F M
x z v
t=lsec t=Ssec t=lsec t=Ssec t=lsec t=Ssec
Bracing
0.66 1.0
2 0.66 1.0
Deck
Elements
0.29 0.99 0.63 0.99 0.63 0.95
6 0.52 0.96 0.60 0.99 0.30 0.96
Arch
Elements
20 0.47 0.96 0.78 0.99 0.47 0.96
25 0.51 0.95 0.78 0.99 0.30 0.95
Table 4-35: Ratios of the NRGB Responses at One and Five Seconds to the
Stationary Responses.
0 (NON) a (NON) a (NON)
M
x Z 2
Elements
2
0 (ST) 0 (ST) 0 (ST)
F
x 2 v
t=lsec t=Ssec t=lsec t=Ssec t=lsec t=Ssec
Bracing
0.52 0.92
2 0.60 0.94
Deck
Elements
0.27 0.95 0.48 0.94 0.52 0.97
10 0.33 0.92 0.18 0.99 0.26 0.93
Arch
Elements
34 0.24 0.94 0.57 0.96 0.24 0.94
40 0.48 0.88 0.55 0.95 0.12 0.93

 

 

 

 

 

 

 

 

 

 

 

 

 

180

(IH2(w)|2) in equation (3.111) has a significant effect on the dis-

placement spectrum. Therefore similar acceleration spectra do not
necessarily have similar displacement spectra. The ground displacement
affects the pseudo-static responses but not the dynamic responses. In
the bridges studied, the contribution of the static response to the
total response is small (about 10 to 12% at most in the bracing elements
and much smaller in the other elements), and therefore any differences
in the ground displacements between the deterministic and random
vibration studies is not expected to contribute significantly to the
total responses (except perhaps to the bracing elements).

The apparent wave velocity along the length of the bridge also
differed slightly for the results compared here: Dusseau used 1706 m/s
and 853 m/s while this study used 1000 m/s. The effect of increasing
velocity from 1000 m/s to 1706 m/s in the random vibration analysis was
found to give similar tendencies, and therefore this difference in the
velocities in not very important for the purpose of the comparisons.

To compare the results of both studies, we can only look at the
response of the NRGB and CSCB under Bl-Bl, Bl-Bl', and BI-Bl” loading
conditions of the deterministic study, where Bl-Bl represents fully
correlated excitations, and Bl-Bl' and Bl-Bl" represent wave propogation
cases with seismic wave velocities of 1706 m/s and 853 m/s, respective-
ly. The results from these cases are compared to the responses of fully
correlated wave propogation cases in this study. The results are
discussed for bracing, deck and arch members, and also the vertical
displacements.

l. Bracing: In the deterministic study (see Tables 4-9 through 4-12

in Dusseau (1985)) the axial forces in the CSCB bracing were

 

 

181

105.2, 97.3 and 121.8 Tons for loadings Bl-Bl, Bl-Bl', and B1-
31“, respectively. Decreasing the wave velocity from 1706 m/s to
853 m/s, the axial forces increased from 97.3 Ton to 121.8 Ton.
These results are compared with the ground motion 1 results of
this study. From Table 4-7 (of this study) it can be seen that
the difference in the bracing response to the fully correlated
and wave propogation cases is negligible. The influence of wave
velocity on the CSCB bracing response was minimal as shown in
Figure 4-37. Since the static response contributes about 10% to
the bracing response, the difference in the ground displacements
between the two studies (as pointed out earlier) may be the
reason for the observed differences.

. Deck members: In the deterministic study, the stress in the deck
element near the center were 15.18, 13.53 and 17.04 ksi (see
Tables 4-9 through 4—12 in Dusseau (1985)). In this study, the
combined normal stresses due to axial forces and bending moments
were not computed. However, based on the results for varying
velocity, (see Figure 4—33 in this study) the axial force
decreases slightly with decreasing wave velocity, but the bending
moment increases with decreasing velocity. Figure 4-33 shows
that when the velocity reduces from w to 1706 m/s, the bending
moment does not change much but the axial force decreases. This
corresponds to the decrease in stress from 15.18 to 13.53 ksi in
the deterministic study. However, as the velocity decreases
further to 853 m/s, the axial force reduces a little more while
the bending moment increases sharply, which would give a higher
combined stresses.

. Arch members: The stresses in the arch elements at the abutment

(see Tables 4-9 through 4-12 in Dusseau (1985)) were 9.78, 15.40

 

 

 

182
and 19.06 ksi for loadings Bl-Bl, Bl-Bl', and Bl-Bl", respective-
ly. These results have the same behavior as in this study, but
the ratios of responses between wave propogation and fully
correlated excitation is larger in this study.

4. Vertical displacements also have the same tendencies in the two
studies, but again the ratios between wave propogation and fully
correlated excitation are larger in this study.

5. The main difference between the CSCB and NRGB responses is in the
response of the bracing elements. The results in both studies
confirm that the bracing axial force response is much higher in
the fully correlated case of ground motion.

Some of the differences between the two studies may be due to the
differences in the input ground motion acceleration and displacement
models. The fact that the deterministic analysis was only one sample of
the excitation process, which may produce lower or higher responses than

the average, is also expected to contribute to these differences.

 

 

CHAPTER 5

SUMMARY AND CONCLUSIONS

W

The research conducted here was to study the effect of spatial
variation of earthquake induced ground motion on the response of (The
New River Gorge Bridge (NRGB), and Cold Springs Canyon Bridge (CSCB)).

The equations of motion were developed using the finite element
technique and the random vibration approach.

In order to perform this study, a suitable ground motion model
is required. In this study a space-time ground motion model proposed by
Harichandran and Vanmarcke (1986) is used. In this model the ground
acceleration are assumed to constitute a homogeneous random field. The
point spectral density function of the ground acceleration, is therefore
assumed to be the same at all spatial locations. The correlation
between the accelerations at two different points is characterized by
the coherency function p(y,f) and the phase due to the time delay caused
by wave propagation is accounted for by an exponential function
exp(—iwu/v).

In this study three cases of ground motion models were used.
Case 1 is the fully correlated ground motion in which all the support
points are moving identically. This assumption is not realistic for
long span bridges, but it is the current practice of designing. Case 2,
is the wave propogation model where only the wave travelling effect is
considered with no coherency loss. Case 3, is the general case of
ground motion model where the travelling wave effect as well as the
correlation between the acceleration at two different points which is

characterized by a coherency function are considered.

183

 

 

 

184

Each case of ground motion has two different sets of fitting
parameters. Those cases of ground motion were applied in the horizontal
direction and the responses of the two bridges were determined and

analyzed.

5.1.1 MODELS OF THE CSCB AND NRGB

The models used in this study are the ones obtained by Dusseau
and Wen (1985) who studied the same two bridges using the deterministic
approach to analyze the response of the bridges under unequal seismic
support motion. Using those models made the task of analyzing the two
bridges much easier because, they transformed the two bridges from a
structure with hundreds of d.o.f. to a structure with less than 100

d.o.f.

5.1.2 CSCB IN-PLANE RESPONSE

The CSCB in-plane responses were determined for the three cases
of ground motion. The deck axial forces were the highest in the fully
correlated case, meanwhile the bending moments and shear forces were
minimum. The deck bending moment was maximum in general and wave
propogation case. The most striking difference is the response of the
bridge's arch to the general and wave propogation cases comparing with
the response to the fully correlated case. The difference in the
response between the wave propogation and the general case was about 7%
in bending moments and shear forces, and about 16% in axial forces where
the wave propogation response was higher than the general case response.
The arch axial forces response in the wave propogation case was 4 to 6
times higher than in the fully correlated case, meanwhile the bending

moment and shear forces were about 2 to 4 times higher.

 

 

 

185

The longitudinal bracing response in ground motion 1 was very
close in all three cases of ground motion. In ground motion 2, the
highest bracing response was in the general case followed by the wave
propogation case and fully correlated case, respectively. The
difference between the wave propogation and the general case was very
small and both were about 25% higher than the fully correlated case
response. The vertical displacement in both the general and wave
propogation cases were close to each other and both were about 3 to 10
times higher than the response in the fully correlated case.

The responses of ground motion 1 comparing to ground motion 2
were much higher, and this can be explained by the frequency content of

ground motion.

5.1.3 NRGB IN-PLANE RESPONSE

The deck axial forces were the highest in the fully correlated
case of ground motion, where the axial forces were about 66% to 100%
higher than the axial response in the general and wave propogation
cases. The deck bending moments responses were the highest in the wave
propogation case, where they were about 19% higher than the general case
response, and about 73% to 400% higher than the fully correlated case
response.

The arch responses were the highest in the wave propogation
case, where the axial forces response was about 2.7 to 4.4 times higher
than the response in the fully correlated case. The responses in the
wave propogation and the general case of ground motion were within 20%
of each other in ground motion 1, and within 12% in ground motion 2.

The response of the longitudinal bracing was the highest in the
fully correlated case of ground motion. In comparison with the general

case of ground motion the bracing axial forces in the fully correlated

 

 

186
case were about 80% to 100% higher in ground motion 1, and 44% to 78%
higher in ground motion 2. The bracing axial forces in wave propogation
and general cases of ground motion were within 10% to 17% in ground
motion 1 ,and within 8% in ground motion 2.

The responses of ground motion 1 is about 50% less than the
responses in ground motion 2. This can be explained by the frequency
content of ground motions l and 2.

The vertical displacement was the highest in the wave
propogation case at 1/3 points of the span. The displacements in the
wave propogation and general cases were very close and both were in
about 10 times higher than the vertical displacements in the fully

correlated case of ground motion.

5.1.4 NRGB AND CSCB OUT—OF-PLANE RESPONSE

The study of the out—of-plane responses of both bridges shows
that the members responded differently to different ground motion
correlation. For some members the highest response was in the wave
propogation case, for others it was the general or fully correlated
case. Thus it was difficult to predict what is the worst case of ground
motion.

For NRGB the maximum lateral displacement occured in the fully
correlated case. The displacements in the wave and general cases were
close the each other. The maximum lateral displacements occured at the
ends of NRGB, and that was more evident in ground motion 2 where the
variance of ground displacement is higher.

The lateral displacements of CSCB show that the three cases of

ground motion had similar effect on the out—of-plane response.

 

 

 

 

 

187
5.1.5 RELATIVE CONTRIBUTION OF RESPONSE COMPONENTS

The variance of total response consists of three different
components; the variance of dynamic component, the variance of static
component and the covariance of pseudo-static and dynamic components.
It was found that the dominant component is the dynamic one. The two
other components could be ignored when the stiffnesses of the structure
is normal.

By increasing the stiffness of the two bridges the static
component starts increasing to a point where it becomes equal to the
dynamic component. By increasing the stiffness more, the static
component continues its increase and the dynamic component continues its

decrease until it becomes negligible.

5.1.6 THE EFFECT OF SEISMIC WAVE VELOCITY

By choosing different wave velocities and calculating the
responses of the two bridges, it was noticed that the responses of the
two bridges were decreasing with the increase of wave velocity except,
the axial forces in the deck of CSCB where they were increasing. Also,
it was noticed that the wave velocity did not affect the response in the
longitudinal bracing of CSCB, while increasing the velocity caused
slight increase in the NRGB longitudinal bracing response. The decrease
in response with increasing velocity can be related to the fact that the

wave propogation effect decreases with the increase of velocity.

5.1.7 NONSTATIONARY RESPONSE OF CSCB AND NRGB

It was found that the responses of both bridges reach the
stationary state response in about five seconds. Consequently, the
assumption that the ground motion constitute a stationary random field

is a valid one when the duration of strong ground motion is more than 5

 

 

 

 

 

 

 

188

seconds. The reason that both bridges reach the stationary response in

that short period of time is due to the fact that the first modes do not

contribute

significantly to the response of both bridges.

5 . 2 CONCLUS IONS

This study of the response of NRGB and CSCB to differential

support excitation illustrates that it is very important to consider the

spatial variation of earthquake ground motion in the analysis of such

S tructures .

l.

The following conclusions are made based on this study.

The most important component of the structural response of
both bridges is the variance of the dynamic component. The
variance of the static and the covariance of static and
dynamic components could be ignored when the structures is
not stiff. For stiff structures the variance of static
componenets and the covariance of static and dynamic
component must be considered in the analysis. This
conclusion is true for a structure with the same end fixity
conditions. For different conditions a study must be
performed.

The most important effects of the differential support
exictation is the substantial increase in arch axial forces
and bending moments. The increase of the arch axial forces
is much higher than the increase of bending moments.
Comparing with the fully correlated case of ground motion,
the wave propagation case increases the axial forces in the
arch by 6 to 10 times, meanwhile the increase of bending
moments was 2 to 4 times. The responses due to wave

propagation and general cases of ground motion were close to

 

 

 

 

 

 

 

189

each.cnfl1er in most cases with a maximum difference of 20% in
some cases.

The vertical displacements in the two bridges were high and
close to each other in the wave propagation and the general
case of ground motion“ In the fully correlated case of
ground motion the vertical displacements were very small
comparing with the other two cases.

The lateral displacements of the two bridges were the highest_
in the fully correlated case of ground motion. But in all
three cases of ground motion the difference iJIILateral
displacements was not far away from each other.

The response of the two bridges is very dependent on the
parameters of ground motion which eventually influence the
frequency content of ground motion models. For example the
axiaJ_:Eorces in CSCB increased by 6 times when changing from
ground motion 1 to ground motion 2, but in the NRGB the axial
forces decreases by about 50% . Of course, these results are
also related to the structural frequencies.

The seismic wave velocity has a very important effect on the
response of long structures. .At low*wave velocity the
responses of the two bridges were high and periodic, but with
increasing wave velocity the responses decreased and
approached a unique value.

Both bridges will reach the stationary values of response

during a strong ground motion of five seconds or more.

 

 

 

 

190
warms

The scope for future research in this area is quite wide. Some

of the main points that need to be addressed are as follows:

1. Other steel deck arch bridges, and other types of long span
bridges need to be studied, so that we can generalize the
findings, and study the different parameters that affect the
differential excitation on the structural response.

2. Perform non-linear random vibration analysis on steel deck
arch bridges to better and more in depth understand the
structural behavior.

3. The numerical results of the random vibration analysis is as
good as the model of ground motion. So, more studies should
be directed towards the modeling of ground motion on
different site conditions and different ground motion

intensities.

 

 

H

10.

ll.

12.

13.

191

REFERENCES

. Abdel-Ghaffar, A. M. and Rubin, L.I. (1982). "Suspension Bridge

Response to Multiple Support Excitations", Journal of Engineering
Mechanics, ASCE, 108(EM2).

Bathe, K. J., (1982). "Finite Element Procedures in Engineering
Analysis", Prentice-Hall, Inc., Englewood Cliffs, New Jersey.

Chen, W. F. and Atsuta, T. (1977). "Theory of Beam-Columns, Volume
2: Space Behavior and Design", McGraw-Hill, New York.

Claugh, R. W. and Penzien, J. (1975). "Dynamics of Structures",
McGraw~Hill Book Company, New York.

Cook, D, C., (1974). "Concepts and Applications of Finite Element
Analysis", John Wiley & Sons, Inc., New York, New York.

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