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MICHIGAN STATE UNIVER ITY LIBRARIE

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This is to certify that the
dissertation entitled
Suspension Bridge Response to Spatially
Varying Ground Motion

presented by
Ahmad Radi Hawwari

has been accepted towards fulfillment
of the requirements for

Doctor of Philosophy degreein Civil Engineering

 

 

 

Major professor

Date May 19, 1992

 

MSU is an AffirmatiwAction/Equal Opportunity Institution 0.12771

 

 

 

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1 University

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MSU Is An Affirmative Action/Equal Opportunity institution
emails-n1

 

 

SUSPENSION BRIDGE RESPONSE TO SPATIALLY
VARYING GROUND MOTION

BY
Ahmad Radi Hawwari

A DISSERTATION

submitted to

Michigan State University
in partial fullfilment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Civil and Environmental Engineering

1992

6:7,:L 32: ‘/l'

ABSTRACT

SUSPENSION BRIDGE RESPONSE TO SPATIALLY
VARYING GROUND MOTION

By

Ahmad Radi Hawwari

The stochastic lateral responses of the Golden Gate suspension bridge, which has a
center span length of 4,200 feet and a side span length of 1,125 feet was investigated. A two
dimensional finite element model of the bridge was used. A space-time earthquake ground
motion model that accounts for both coherency decay and seismic wave propagation was
used to specify the support motions. The double-filter spectrum fitted to an artificial accel-
erogram similar to the El Centro earthquake was used.

Linear stationary random vibration analysis was used to compute the bridge respons-
es. Three models of excitations were considered at the supports: (1) correlated ground mo-
tion model accounting for both wave propagation and coherency decay; (2) identical
support motion; (3) delayed excitation caused by wave propagation. Transient response
analysis was also performed to determine whether the suspension bridge will attain its sta-
tionary response during typical durations of strong shaking (10 to 20 seconds). The effects
of shear deformation on the natural frequencies and mode shapes, and their corresponding

effect on the linear stationary random vibration responses was investigated. Inclusion of

shear deformation drastically lowers the frequencies of a group of modes, resulting in

smaller moment and shear responses, but slightly higher displacement responses.

Results indicate that the use of identical excitations significantly over-estimates the
responses at some locations and under-estimates the responses at others, the relative devi-
ation being more severe for the longer center span. The use of delayed excitations gives ac-
ceptable results for the side span, but shows greater deviations for the center span in which
the moment and shear are sometimes significantly under-estimated. The increase in the ap-
parent wave velocity causes progressively higher responses at some locations of the span
and progressively lower responses at others. Results of transient analyses indicate that for
common ground motion durations, the assumption of stationarity may grossly over-esti-
mate the side span responses. The transient displacement response of the center span can
overshoot the stationary response considerably, but the moment and shear responses grad-

ually approach their stationary values in about 40 seconds.

(To my parents
Sofia and Rodi Hawwori
for tfieir [072e, confidence, devotion,

and sacrifice

iv

Acknowledgments

I am gratefully indebted to my thesis advisor, Dr. Ronald Harichandran, for his invalu-
able guidance throughout the course of this research.

I would like to express my gratitude to all members in my Ph.D. guidance committee,
Dr. Robert Wen, Dr. Parvis Soroushian, Dr. William Sledd, and Dr. John Masterson.

I would also like to give special thanks to the loved ones in my family for their contin-

uous love, understanding, and encouragement.

TABLE OF CONTENTS

 

 

LIST OF TABLES _ .................. -- ---viii
LIST OF FIGURES - ______ -— —- -ix
LIST OF SYMBOLS -- - -- ----------- _ _ _ __ -- —— . _ -— xiv

 

1. General Introduction and Background..................................................1

 

 

1.1 Literature Review .......................................................................................... 1
1.2 Purpose and Scope ......................................................................................... 3
2. Free Lateral Vibration of Suspension Bridges- _- ______ 5
2.1 Description of the Bridge ............................................................................... 5
2.2 Basic Assumptions for Analysis .................................................................... 6
2.3 Derivation of the Equations of MOtion .......................................................... 9
2.4 Finite Element Formulation of Lateral Vibration ........................................ 13
3. Random Vibration Analysis - — -- —_ - - - 24
3.1 Modal Analysis ............................................................................................. 25
3.2 Random Vibration Theory ............................................................................ 26
3.2.1 Variance of Dynamic Displacements ........................................ 27
3.2.2 Variance of Pseudo-Static Displacements ................................. 30

3.2.3 Covariance between Pseudo-Static and
Dynamic Displacements ............................................................ 31
3.2.4 Variance of Dynamic Element End Forces ................................ 32
3.2.5 Variance of Static Element End Forces ..................................... 33

3.2.6 Covariance of Pseudo-Static and

Dynamic Element End Forces ................................................... 33
3.3 Transient Response ....................................................................................... 34
3.4 Ground Motion Model .................................................................................. 35
3.5 Computation Steps ........................................................................................ 39

vi

4. Numerical Results and Analysis" - ________ .... - _ ------43

 

 

 

4.1 Free Vibration Analysis ................................................................................ 43
4.2 Ground Motion Models ................................................................................ 49
4.3 Side Span ...................................................................................................... 49
4.3.1 Side Span Response Components ............................................... 49

4.3.2 Lateral Response of the Side Span ............................................. 50

4.3.3 Effect of Apparent Wave Velocity ............................................. 54

4.3.4 Modal Contributions ................................................................... 58

4.3.5 Transient Response ..................................................................... 64

4.4 Center Span ................................................................................................... 74
4.4.1 Center Span Response Components ........................................... 76

4.4.2 Lateral Response of the Center Span .......................................... 83

4.4.3 Effect of Apparent Wave Velocity ............................................. 86

4.4.4 Modal Contributions ................................................................... 88

4.4.5 Transient Response .................................................................... 97

4.5 Shear Deformation ...................................................................................... 110
4.5.1 Side Span .................................................................................. 110

4.5.2 Center Span ............................................................................... 114

5. Summary and Conclusions ....... _ _ _ __ __________ 125
5.1 Summary ..................................................................................................... 125
5.1.1 Finite Element Model ............................................................... 125

5.1.2 Response Components .............................................................. 125

5.1.3 Lateral Response ....................................................................... 126

5.1.4 Effect of Apparent Wave Velocity ........................................... 127

5.1.5 Modal Contributions ................................................................. 128

5.1.6 Transient Response ................................................................... 128

5.1.7 Effect of Shear Deformation ..................................................... 129
Bibliography ..... __ _____________ _ 130

vii

LIST OF TABLES

Table 2.1 : Structural properties related to the lateral vibration of the Golden Gate
Bridge. .......................................................................................................... 8
Table 3.1 : Ground Motion Model Parameter.37

Table 4.1 : Golden Gate Bridge side span natural frequencies and periods

of lateral vibration ....................................................................................... 44
Table 4.2 : Golden Gate Bridge center span natural frequencies and periods of lateral

vibration ...................................................................................................... 45
Table 4.3 : Relation between the number of modes and the number

of integrations ............................................................................................. 74
Table 4.4 : Relative contribution of response components to cable

displacement response at quarter and mid span. ........................................ 81
Table 4.5 : Relative contribution of response components to deck

displacement response at quarter and mid span ......................................... 82

Table 4.6 : Relative modal contributions to the dynamic deck displacement.94
Table 4.7 : Relative modal contributions to the dynamic deck moment.98

Table 4.8 : Relative modal conuibutions to the dynamic cable displacements. 101
Table 4.9 : Ratio of corresponding modal responses of the side span

from analyses including and excluding shear deformation ...................... 116
Table 4.10 : Ratios of corresponding modal responses of center span
including and excluding shear deformation. ............................................ 123

viii

Figure 2.1
Figure 2.2
Figure 2.6
Figure 2.7
Figure 2.8

Figure 3.1
Figure 3.2
Figure 4.1
Figure 4.2
Figure 4.3
Figure 4.4

Figure 4.5
Figure 4.6
Figure 4.7
Figure 4.8

Figure 4.9

Figure 4.10
Figure 4.11:

Figure 4.12:

LIST OF FIGURES

: Definition diagram of the Golden Gate Bridge ............................................ 7
: Latcrally vibrating suspension bridge ......................................................... 10
: Node numbering scheme for side span ....................................................... l7
: Node numbering scheme for center span .................................................... 18
: Consistent mass matrix when shear deformation

is included ................................................................................................... 22
: Coherency function at two separations ....................................................... 38
: Estimated and fitted autospectra for Type-B accelerogram ....................... 40
: First nine mode shapes of the bridge side span .......................................... 46
: First set of the center span mode shapes47
: Second set of the bridge center span mode shapes ..................................... 48
: Variation of normalized displacement variances of the

side span cables ........................................................................................... 51
: Variation of normalized displacement variances of the

side span deck ............................................................................................. 51
: Variation of normalized moment variances of the

side span deck ............................................................................................. 52
: Variation of normalized shear variances of the

side span deck ............................................................................................. 52
: Normalized displacement variances of the side span

cables due to three ground motion models ................................................. 53
: Normalized displacement variances of the side span

deck due to three ground motion models .................................................... 53
: Normalized moment variances of the side span

deck due to three ground motion models .................................................... 55

Normalized shear variances of the side span

deck due to three ground motion models .................................................... 55

Relative modal contributions to the dynamic deck shear

response- Node 15 ..................................................................................... 56

ix

Figure 4.13
Figure 4.14
Figure 4.15
Figure 4.16
Figure 4.17
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

Figure 4.30

Figure 4.31 :

Figure 4.32

fiymifi

: Effect of apparent wave velocity on the side span

cable displacement ...................................................................................... 57

: Effect of apparent wave velocity on the side span

deck displacement ....................................................................................... 57

: Effect of apparent wave velocity on the side span

deck moment ............................................................................................... 59

: Effect of apparent wave velocity on the side span

deck shear ............................................................................................... ....59

: Relative modal contributions to the dynamic deck displacement

response - Node 7 ....................................................................................... 60

: Relative modal contributions to the dynamic deck displacement

response - Node 15 ..................................................................................... 61

: Relative modal contributions to the dynamic deck moment

response - Node 7 ....................................................................................... 62

: Relative modal contributions to the dynamic deck moment

response - Node 15 ..................................................................................... 63

: Relative modal contributions to the dynamic deck shear

response - Node 7 ....................................................................................... 65

: Relative modal contributions to the dynamic cable displacement

response - Node 8 ....................................................................................... 66

: Relative modal contributions to the dynamic cable displacement

response - Node l6 ..................................................................................... 67

: Variation of normalized transient displacement variances of

the side span cables ..................................................................................... 68

: Variation of normalized transient displacement variances of

side span deck ............................................................................................. 68

: Variation of normalized transient moment variances of

the side span deck ....................................................................................... 7O

: Variation of normalized transient shear variances of

the side span deck ....................................................................................... 70

: Normalized transient displacement variances of the side span

cables due to three ground motion models at t=15 seconds ....................... 72

: Normalized transient displacement variances of the side span

deck due to three ground motion models at t=15 seconds .......................... 72

: Normalized transient moment variances of the side span

deck due to three ground motion models at t=15 seconds .......................... 73

Normalized u'ansient shear variances of the side span
deck due to three ground motion models at t=15 seconds .......................... 73

: Variation of normalized displacement variances of the

center span cables ....................................................................................... 75

: Variation of normalized displacement variances of the

center span deck ......................................................................................... 75

Figure 4.34
Figure 4.35
Figure 4.36
Figure 4.37
Figure 4.38
Figure 4.39
Figure 4.40
Figure 4.41
Figure 4.42
Figure 4.43
Figure 4.44
Figure 4.45
Figure 4.46
Figure 4.47
Figure 4.48
Figure 4.49
Figure 4.50
Figure 4.51
Figure 4.52
Figure 4.53

Figure 4.54

Figure 4.48 :

Variation of normalized moment variances of the

center span deck .......................................................................................... 77
: Variation of normalized shear variances of the

center span deck .......................................................................................... 77
: Variation of normalized displacement variances of the

center span cables (General ground motion model) ................................... 78
: Variation of normalized displacement variances of the

center span cables (Fully correlated ground motion model) ....................... 78
: Variation of normalized displacement variances of the

center span cables (Wave propagation ground motion model) ................... 79
: Variation of normalized displacement variances of the

center span deck (General ground motion model) ...................................... 79
: Variation of normalized displacement variances of the

center span deck (Fully correlated ground motion model) ......................... 80
: Variation of normalized displacement variances of the

center span deck (Wave propagation ground motion model) ..................... 80
: Normalized displacement variances of the center span

cables due to three ground motion models ................................................. 84
: Normalized displacement variances of the center span

deck due to three ground motion models .................................................... 84
: Normalized moment variances of the center span

deck due to three ground motion models .................................................... 85
: Normalized shear variances of the center span

deck due to three ground motion models .................................................... 85
: Effect of apparent wave velocity on the center span

cable displacement (General ground motion model) .................................. 87
: Effect of apparent wave velocity on the center span

cable displacement (wave propagation ground motion model) .................. 87
: Effect of apparent wave velocity on the center span

deck displacement (General ground motion model) ................................... 89
: Effect of apparent wave velocity on the center span

deck displacement (Wave propagation ground motion model) .................. 89
: Effect of apparent wave velocity on the center span .

deck moment (General ground motion model) ........................................... 90
: Effect of apparent wave velocity on the center span

deck moment (wave propagation ground motion model) ........................... 90
: Effect of apparent wave velocity on the center span

deck shear (General ground motion model) ............................................... 91
: Effect of apparent wave velocity on the center span

deck shear (wave propagation ground motion model) ................................ 91
: Relative modal contributions to the dynamic deck displacement

response - Node 41 ..................................................................................... 92

Ratio of modal responses for lateral displacement ..................................... 92

xi

Figure 4.55
Figure 4.56
Figure 4.57
Figure 4.58
Figure 4.59
Figure 4.60
Figure 4.61
Figure 4.62
Figure 4.63
Figure 4.64
Figure 4.65
Figure 4.66
Figure 4.67
Figure 4.68
Figure 4.69
Figure 4.70

Figure 4.71

Figure 4.72 :

Figure 4.73 :

Figure 4.74
Figure 4.75
Figure 4.76
Figure 4.77

: Relative modal contributions to the dynamic deck displacement

response - Node 83 ..................................................................................... 93
: Relative modal contributions to the dynamic deck moment
response - Node 41 ..................................................................................... 95
: Relative modal contributions to the dynamic deck moment
response - Node 83 ..................................................................................... 96
: Relative modal conuibutions to the dynamic cable displacement
response - Node 42 ..................................................................................... 99
: Relative modal contributions to the dynamic cable displacement
response - Node 84 ................................................................................... 100
: Variation of normalized transient displacement variances of
the center span cables .............................................................................. 102
: Variation of normalized transient displacement variances of
the center span deck ................................................................................. 102
: Integrand function for mode 1 of the
center span ................................................................................................ 105
: Integrand function for mode 1 of the
side span .................................................................................................... 105
: Variation of normalized transient moment variances of
the center span deck ................................................................................. 107
: Variation of normalized transient shear variances of
the center span deck ................................................................................. 107
: Normalized transient displacement variances of the center
span cables due to three ground motion models at F20 seconds ............. 108
: Normalized transient displacement variances of the center
span deck due to three ground motion models at t=20 seconds ................ 108
: Normalized transient moment variances of the center
span deck due to three ground motion models at t=20 seconds ................ 109
: Normalized transient shear variances of the center
span deck due to three ground motion models at t=20 seconds ................ 109
: Undamped natural frequencies of the side span
excluding shear deformation ..................................................................... 110
: Undamped natural frequencies of the side span ‘
including shear deformation .................................................................... 111
First nine mode shapes of the side span
including shear deformation .................................................................... 112
Normalized displacement variances of the side span cables .................... 113
: Normalized displacement variances of the side span deck ....................... 113
: Normalized moment variances of the side span deck ............................... 115
: Normalized shear variances of the side span deck ................................... 115

: Undamped natural frequencies of the Golden Gate bridge

center span excluding shear deformation ................................................. 117

xii

Figure 4.78 : Undamped natural frequencies of the Golden Gate bridge

center span including shear deformation .................................................. 117
Figure 4.79 : First set of mode shapes of the center

span including shear deformation ............................................................. 119
Figure 4.80 : Second set of mode shapes of the center

span including shear deformation ............................................................. 120
Figure 4.81 : Normalized displacement variances of the center span cables ................. 121
Figure 4.82 : Normalized displacement variances of the center span deck ................... 121
Figure 4.83 : Normalized moment variances of the center span deck ........................... 122
Figure 4.84 : Normalized shear variances of the center span deck ................................ 122

xiii

LIST OF SYMBOLS

[A] , AI, = matrix of static displacement due to unit re
strained displacements and its elements;

[C] = overall damping matrix;

Cov (us, ad.) = covariance between static and dynamic dis-
placements;

C pp. C pg. C”. C M = partitions of the damping matrix;

{D cg} i, {Dem},- = global and local member end -displacements
corresponding to the ith mode;

DOF = acronym for degree-of-freedom;

Esi = modulus of elasticity corresponding to the ith
deck;

f = linear frequency;
e(t) = temporal modulating function;

{f}, = element end forces corresponding to the ith
mode shape;
G ,- = generalizedmodal excitation;

g = gravitational acceleration;
Hw = horizontal component of cable tension;
h(x,-) = length of hanger corresponding to ith span;
H j(w), HJ(-w) = modal frequency response function for mode j

and its conjugate;
121-(t) = impulse function due to an impulse excitation
G j = 5 (t);
1,7, = nodal covariances contributing to the overall
dynamic response;
1,,- = area moment of inertia of the deck;

[K] = overall stiffness matrix;
KFF’ Km. K”. K RR = partitions of the overall stiffness matrix;

[Kse]e, [ng]e, [Kce]e, [chL = subelement matrices corresponding to elastic
stiffness of deck, gravitational stiffness of

xiv

deck, elastic stiffness of cables, and gravita-
tional stiffness of cables;

L = length of element;
[M] = overall consistent mass matrix;
MFF’ MFR’ MRF’ MRR =partiti0ns Ofmass matrix;

["13] e. [me] e = subelement consistent mass matrices corre-
sponding to deck and cables, respectively;

in}, = mass of two cables per unit length of the span,
and mass of the deck per unit length of the ith
span;

n = number of free DOF;

[P] = static end force matrix;

P .. = the i‘h element end force due to a unit displace-
ment along the jth restrained DOF;

r = number of restrained DOF;
Re[ ] = real part of argument;

Ru, (1) , Ru, (1) , Ru, (1) = autocorrelation functions of the ith element of
' ' i {up}, {us}:and {“4};
Ru “d (1:) = cross autocorrelation function of the ith element
"' " of {us} and the i‘h element of {ad} ;
final: (1:) = cross autocorrelation of the nodal force excita-

tions at I‘h and m‘h element of {u d} ;
SDF = acronym for degree(s)~of-freedom;
Sn, ((0) , Su (0)) , S“d (to) = SDF’s of the ith element of {uF} , {us} , and
i " r
' {“4};

S“ u (to) = cross SDF’s of the i‘h element of {us} and the
a. (I.

‘ ith element of { “4} ;

(0)) = cross SDF of the nodal force excitations at 1‘h

12 ii ,,
m R and mm elements of {up} ;

S a (to) = point auto SDF of the ground acceleration;

Sir ta (0)) = cross SDF between the acceleration of two loca—
‘ ’ tions A and B;

Tc (t) , TS (t ) ) = kinetic energies caused by the lateral vibrational
displacements wc and w,, respectively;

{ii} , {u} , {u} = vectors of nodal accelerations, velocities, and
displacements; '

{up} , {u R} = free and restrained DOF vectors;

XV

{“3} , {u d} = pseudo-static and dynamic components of
{up};
ujm = bridge element nodal displacement;

V6 (1‘) , VS (1) = potential energy of laterally vibrating cable and
deck, respectively;

V“ (t) , V (t) 38 = elastic potential and gravitational potential ener-
gy of deck;

V = apparent wave propagation velocity in the direc-
tion AB;
w = displacement components of cable and deck
along z axes, respectively;

w . = dead weight of the two cables per unit length
and deck per unit length of the it1 span, respec-
tively;

Yj = generalized modal displacement;
z(t) = stationary random process;
{Pi} = vector of participation factor for mode j;
6 (t) = Dirac delta function;
e = machine precision;
11 = performance index;

6. = angle of rotation of the deck with respect to the
vertical plane passing through the deflected
position of the cable (at section x9;

~

vc, vs = upward displacements of the cables and deck;
v = separation between locations A and B;
221 (i) , 52 (if) = normalized coordinates;
i]. = ratio of critical damping;
p (v, f) = coherency function;

oi , oi , o: = variance of the ith element of {up} , {14,} , and
’r ‘r ‘r {u } .
d a
a} = variance of the ith element end force;
I
02 = variance of the ith static element end force;

3:
t = time delay in seconds;
(pi = angle of rotation of the cable plane (at section
xi);

4; = shear deformation parameter.

xvi

[‘I’] = matrix of mode shapes;

{\V j} = mode shape corresponding to natural frequency

(0].,

vi} = elements of mode shape matrix;

(08, 5,8, (of, §f = parameters of the Clough-Penzien SDF;

subscripts
F, FF = quantity corresponding to free displacement;
FR, RF = quantity corresponding to free and restrained.
displacement or vice versa;
R, R = quantity corresponding to restrained displace-
ment;
e = quantity corresponding to element;
superscripts

= first partial derivative with respect to time;
= second partial derivative with respect to time;
* = complex conjugate;

T = matrix transpose;

xvii

1. General Introduction and Background

1 .1 Literature Review

Lifeline structures, such as pipelines, bridges and communication transmission sys-
tems, are important infrastructures of cities and urban communities. The functional reliabil-
ity of these lifelines after an earthquake, is therefore essential to the safety and health of

society.

Lifelines differ from conventional ‘point’ structures in that they extend for long dis-
tances along or close to the ground surface, and tend to have long periods of vibration ( e.g.,
long-span suspension bridges). If the base dimensions of the structure are small relative to
the vibration wavelength in the soil, the assumption that the wavelengths of earthquake
ground waves are long compared to the structural dimensions is acceptable. For example,
if the velocity of the wave propagation is 6,000 ft/sec, a sinusoidal wave of 3.0 Hz frequen-
cy will have a wave length of 2,000 ft, and a building with a base dimension of 100 ft will
be subjected to essentially the same motions over its entire length. 0n the other hand, a
long-span suspension bridge, which might have a length of several thousand feet, obviously

would be subjected to drastically different motions at its foundations.

In classical deterministic analysis, a recorded time history at one point is used as the
input motion, and the differential motion between two points is estimated by considering a
delay in the arrival of the seismic wave between the points. This deterministic approach is
capable of realistically describing the response of conventional structures subjected to
earthquakes, but is restrictive for long-span suspension bridges because it neglects the loss

of coherence between support excitations.

In the stochastic approach, the spatial variation of seismic ground motion is modelled

as a random process with a given power spectral density, and the spatial variation is de-

scribed by a correlation function and a phase shift. Recorded earthquake data from seismo—

graph arrays are used to estimate the power spectral density and the correlation function.

In recent years, the earthquake response of suspension bridges has been studied using
a frequency-domain random vibration approach to take into account not only the differenc-
es in ground motion inputs, but also the correlation among the various input motions ( Ru-
bin and Abdel-Ghaffar 1983, Abdel-Ghaffar and Suigfellow 1984, Abdel-Ghaffar and
Rubin 1983 and 1982). It was found that the transmission time can have a significant effect

on the response.

In a study on pipeline response to spatially variable ground motions, Zerva, et. a1.
( 1988) concluded that the differential ground motion is of major importance. Whereas per-
fectly correlated support motion will yield zero differential displacements and forces be-
tween the pipe systems, partially correlated support motion can give high differential

displacements.

In a study of the response of one— and two-span beams to spatially varying seismic
excitation Harichandran and Wang (1988), concluded that it is important to consider the
spatial variation of earthquake ground motion in the analysis of structures, especially for

long statically indeterminate ones. They found that:

1. Both wave propagation and spatial correlation effects can be significant, but for
cases where the apparent wave propagation velocity is large compared to the struc-

tural length the latter efi‘ect is more important.

2. For indeterminate structures, the pseudo-static stress is very significant especially
for stiff structures, and neglecting this can result in a significant error.

3. Fully correlated support motions do not excite anti-symmetric modes which are ex-
cited by general support motions, and therefore in a few cases the former can result

in a lower dynamic displacement than the latter.

The effects of spatially varying ground motion on the response of deck arch bridges

was studied by Sweidan (1990), and the following conclusions were made:

1. The most important component of the response of arch bridges is the dynamic re-

sponse (as opposed to the pseudo-static response).

2. The most important effect of the differential support excitation is the substantial

increase in arch axial forces and bending moments.

3. The seismic wave velocity has a very important effect on the response of long

SIIUCIUICS.

4. The arch bridges studied by Sweidan attained stationary response during a strong

ground motion of five seconds or more.

In a conclusion, all of the studies mentioned above indicate the importance of the ef-

fect of spatial variation of earthquake ground motion on the response of long structures.

1 .2 Purpose and Scope

This research is concerned with the effects of spatially varying ground motion on the
lateral response of an actual long span suspension bridge. The study was conducted on the
Golden Gate Bridge in California with a 4,200 feet center span and 1,125 feet side spans.
Two dimensional finite element model which accounts for the cable’s uplift developed by
Abdel-Ghaffar (1976) is used, with the corrections made by Castellani and Felloti (1986).
A ground motion model proposed by Harichandran and Vanmarcke (1986) is used, where
the model accounts for the correlation between the accelerations at two different points in
the form of a coherency function. The effects of spatial variation in the excitation was stud-
ied in detail using linear stationary random vibration analysis. The transient response of the
suspension bridge, was also performed to determine whether the suspension bridge will
reach its stationary response during typical durations of su'ong shaking (10 to 20 seconds).

The effects of shear deformation on the natural frequencies and mode shapes, and their cor-

responding effect on the linear stationary random vibration analysis of the bridge also irr-

vestigated.

The bridge modeling and theoretical formulation is presented in Chapters 2 and 3.
Chapter 2 is concerned with the finite element modeling to formulate the equations of mo-
tion and its finite element solution. Chapter 3 presents the derivation of the response com-
ponents using linear random vibration theory and, the approach used to study the transient
response. The ground motion model is also discussed in Chapter 3. The results from the
analyses are presented in Chapter 4. A rigorous analysis of the response components and
the response due to different ground motion models is investigated. The effects of apparent
wave velocities and the relative modal conuibutions of the responses are investigated and
discussed in details. Results from the transient response of the suspension bridge are pre-
sented. The effect of shear defamation on the response is discussed and presented. Finally,
Chapter 5 summarize the main conclusions and contributions of this research and suggests

possible direction for future research.

2. Free Lateral Vibration of Suspension Bridges

Analysis of suspension bridges subjected to lateral dynamic loads were developed by
Moisseif, et. a1. (1933), Silverman (1957), Selberg (1958), Hirai, et. al. (1960), Konishi,et.
al. (1965), and Ito (1966), before the discovery of digital computers; and by Abdel-Ghaffar

(1978), and Sigbjonsson, et. a1. (1981), in recent times.

The contribution of the first group of researchers is mainly, confined to solving, in an
approximate way, the system of equations governing the dynamic equilibrium, and to find-
ing a closed form solution for the first natural frequency of vibration. Ito, however, dis-
agrees with his precursors by including, as a restoring force for the cable, the effect of the
cable’s uplift which accompanies the lateral displacement. This effect is also the cause of
disagreement between the analysis of Abdel-Ghaffar, who includes it, and that of Sigb-
jomsson and Hjorth-Hansen, who neglect it. The importance of this effect on the period of
the first natural mode is limited to a few percent for short span bridges, for which the pre-
dominant restoring action is the pendulum effect exerted by the suspended deck (which is
the same in both approaches). Greater influence is expected for long bridges (say with a
span of 2,000 ft or more).

In this work, the model developed by Abdel-Ghaffar is adopted in formulating of the
equations of motion. The incorrect sign in the expression of the strain energy of the cables
discussed by Castellani and Felotti (1986), is followed through carefully through the for-

mulation of the equations of motion and the finite element modeling.

In this study, the Golden Gate Bridge is used as a typical example.

I 2.1 Description of the Bridge

The Golden Gate Bridge which lies across the entrance to San Francisco Bay and

joins the northern and southern peninsulas was built in 1937. The main span is 4,200 feet,

the largest ever constructed at the time. Each of the side spans is 1,125 feet long and is sus-
pended from the main cables. The width of the roadway is 90 feet, and provides six traffic
lanes and two sidewalks. The roadway initially consisted of a slab, a floor system, two stiff-
ening trusses, and a lateral bracing system. The lateral bracing was in the plane of the top
chords of the stiffening trusses (Strauss 1937). Since 1937 the bridge was subjected to sev-
eral strong wind storms. After the storm of December 1“, 1951 a decision was made to stiff-
en the lateral bracing system. Lateral bracing in the plane of the bottom chords of the
stiffening trusses were added. The addition of the bottom laterals made a closed box of the
floor system for resisting torsion, greatly increasing the torsional rigidity of the roadway
(Paine 1970).The general layout and the principal dimensions of the bridge are given in
Figure 2.1, and the lateral structural properties of the Golden Gate Bridge are summerized
in Table 2.1. For more description and the complete details of the structural components of

the bridge see Strauss (1937) and Paine (1970).

2.2 Basic Assumptions for Analysis

1. The vibration amplitude around the equilibrium position is small, so that nonlinear
terms in the differential equation of equilibrium can be neglected. As a conse-

quence, lateral displacements are uncoupled from torsional motion.
2. The bending stiffness of the cables is neglected.
3. It is assumed that the hangers are pin-ended struts, and inextensible.

4. In modeling the bridge as a 2-D structure the two main cables are assumed to move

in tandem as if they were connected by horizontal struts.
5. The ends of the cables are taken to be fixed.

6. As a corollary to the assumption in step 1, the increment of the horizontal compo-
nent of cable tension H(t), due to lateral vibration is small in comparison with the

initial dead-load horizontal component of cable tension Hw.

‘1
VI

702 ft

 

I;
I‘

 

7 I]
1,125 ft _.- 4,200 ft __ 1425 ft

7"

i _'
1'.

Figure 2.1 : Definition diagram of the Golden Gate Bridge

Table 2.1 : Structural properties related to the lateral vibration of the Golden Gate

 

 

 

 

Bndge.
Parameter Center Span Side Span
Span length 1.2-4,200 ft L1=L2= 1,125 ft
Spanwidth b=90ft b=90ft
Deck weight 87,2 = 16.02 km W,(1,3,=16.42 k/ft
Modulus Esz = 29,000 ksi 5,1” =29,000 ksi
Pmperties Moment of inertia 1,, = 7,639.58 re 1,1,3 =7,639.58 a“
Shear deformation
parameter 266.3 118.36
Modulus EC: 29,000 ksi
. _ . 2
Cable Cross sectional area Ac - 831.9 m
Length LE = 7698 ft
. Honz. component of H” = 53 4 67 kips
Properties tensron

Weight W, = 6.68 k/ft

 

 

 

 

 

7. All suesses in the bridge remain within the elastic limit and therefore obey Hooke’s

law.

8. The initial curvature of the stiffening structure is considered small in comparison

with the cable curvature and is therefore neglected.

9. The Golden Gate bridge was studied by Baron, et. al. (1976). The first nine trans-
verse modes were obtained by means of the 3-dimensional model. The period of vi-
bration of the first mode was found to be 20.23 seconds, no transverse modes of the
towers were obtained in the calculations of the structure as a whole. It is observed
that the towers act as rigid frames in the transverse direction and their periods of
vibration are considerably smaller than those of the deck and cables. Therefore, the

tower-piers are assumed to move as rigid bodies under ground motion excitation.

2.3 Derivation of the Equations of Motion

The derivation of the governing differential equations of lateral vibration of the cable
and deck system is carried out in a general form by using Hamilton’s variational principle.
The resulting equations are linearized and reduced to a standard form through use of the
previously stated simplifying assumptions. The exhaustive derivation was done by Abdel-
Ghaffar (1976 and 197 8). Applying the sign correction mentioned by Castellani and Felotti
(1986), a summary of the corrected energy equations and the final form of equations of mo-

tion are briefly described.

The coordinate systems and vibrational displacements are described diagrammatical-
ly in Figure 2.2. By considering Figure 2.2, the upward displacements vc and vs of the ca-
bles and deck, respectively, may be expressed as

vc (xvi) = -yc (xi) [1 —cos<pi] ,i=1,2,3 (2.1)
and

vs (xi, t) = -yc (xi ) (1 - cosoi) -- h (xi) (1 — cosOi) , i=1,2,3 (2.2)

 

v '1“. s-------s-s§.s.~cs-§.‘

WWbh~i§bhiitbh~hhhlhbbbbh’

OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

 

3) Lateral (bending) deformation

 

 

 

 

 

 

 

4 :
Ye (I)
bl
w. v
M")
Bridge Deck
Siféjé? —*u
vI Tower
a“
b) Coordinate system

Figure 2.2 : Laterally vibrating suspension bridge

10

in which (pi is the angle of rotation of the cable plane (at section xi ) with respect to the
vertical plane passing through the tower top; and 9‘. is the angle of rotation of the deck with

respect to the vertical plane passing through the deflected position of the cable at section xe-

Since we and we are very small quantities compared with ye and h, we may write

 

W3 2 3
Va 3-2—ye ( ° )
2 2
—Wc (W: - we)
Vs ~ 2)"; - T (2.4)

where we , we , ye , h are described in Figure 2.2.

The potential energy of the laterally vibrating cable, Ve (t) , is comprised of two
parts: the strain energy Vee (t) and the gravitational potential energy Veg (t) . Thus, the to-

tal potential energy of the cable is expressed as
Ve(t) = Vee(t) +Vee (t) (2.5)
The linear strain energy expression of the two cables can be formed as
3 11‘ awe 2 1 i704..- 1’ W3
were) arm—Jim
where Hw = horizontal component of cable tension.
We = dead weight of the two cables per unit length of the span.
We, = dead weight of deck per unit length of the ith span.

The expression for the gravitational potential energy Veg (t) of the two cables due to
the upward deflection v e which occurs together with their lateral movement we can be writ—

ten as

11

_ w (x,.t)
Vee(r) = -2jw (7—:c(x) )dx, (2.7)

i=10

The potential energy of the laterally vibrating deck Ve (I) also consists of two parts:
(1) the elastic potential energy (i.e, the main energy) Vee (t) due to the effects of bending
moments, shear forces and normal forces; and (2) the gravitational potential energy Vee (2‘)

due to upward movement.
The energy stored in the deck due to bending can be written as
It
van) = .212 [55,13,673 (w (xi, r) ) ) dx (2.8)
i— - 10 8xi
where E ee- = modulus of elasticity of the deck in the ith span.
J si = area moment of inertia of the deck about its vertical axis yi in the ith span.

The gravitational energy Veg (t) of the deck due to upward displacement vs is

(2.9)

 

3 l.- 2 2
I _ We (xi: I) (W 5' (xi, 0 —W‘xir t )
V38 (t) - "5.2194 yexi + hxi )dxi

The kinetic energies caused by the lateral vibrational displacements we and we of the

two cables and of the deck, respectively, are expressed as
(awe (x, t)) 2

3 i,
Te(t)=-2-21Ime( )dx, (2.10)
=10

and

12

3 I, 2
Te(t) = 5 ij,,(T—) dx, (2.11)
i=10

in which He = we/ g is the mass of the two cables per unit length of the span,
E“. = 173/ g is the mass of the deck per unit length of the ith span; and g is the acceleration
due to gravity.

The kinetic energies caused by vertical movements, v e and vs, of the cables and the

deck, respectively, are given by

1 l‘ W2 2
~ — - _ a C
and
"' 1 -— a We (Ws-wc
Ts(t) = —-2—.;l£me ('8';('2—y:+ TJJ dxe. (2.13)

Hamilton’s principle is used to derive the linearized equations of motion (Abdel-

Ghaffar 1976). The equations of motion are found to be

" azwe 211 32w“ " (———w‘—W‘) 0 ° 1,23 (214)
m — "' _ “W ' = 91: 9 '
Catz Wax? SI h

for the cable; and

dzwe 32( 32w w —w

a .— +_— E J .—‘)+w .(——‘———°) = 0,i=l,2,3 (2.15)
2 :1 st 2 at
“8:2 6x,- e— h

for the deck.

2.4 Finite Element Formulation of Lateral Vibration

The method of analysis based on the finite element technique takes into account the

characteristics of both the cable and deck. The cable is idealized by a set of string elements,

13

while the deck is idealized by a set of beam elements. The two types of elements, connected
by rigid hangers, form the bridge element. The stiffness and inertia properties for each set
of elements are derived and assembled to obtain the gross assemblage characteristics. The
description of the bridge element and the finite element discretization of the bridge side
span and center span are described diagrammatically in Figure 2.3 to Figure 2.5. The node
numbering scheme used for the bridge side and center spans is illustrated in Figure 2.6, and
Figure 2.7.

The interpolation functions associated with the two degrees of freedom of the nodal
point in the deck subelement are taken to be cubic Hermitian polynomials. The lateral vi-
bration of the suspended-structure can now be expressed in terms of the bridge element

nodal displacements u}. (t) , j = 1,2,3,4,5, and 6, as

Wse<§v§23> = [§¥(3-2§1>-L§¥§20§§ <3-2§2)-L§1§§0] {W}. (246)
01'
Wse(§p§2;t) = Us(§p§2)}:{u(t)}¢ (2.17)

where e is a subscript indicating element, L is the length of element, and Q, £2 are the nor-

malized coordinates defined as

(2.18)

(“It-<1

5,10?) -- (1—§)md§,(i) =

The interpolation function associated with the one degree of freedom of the cable
nodal point is taken to be linear. Thus, the cable lateral displacement can be expressed in

terms of the six nodal displacements of the bridge element, as

wee (5.1, €23) = [o 0610 0 big {a (t) }, (2.19)

14

Node 2

1 /\
Iu3 . Cable Node4

u6

 

 

 

 

 

x 9/
“1 Node 1
l‘ L -l
T‘ '1

Figure 2.3 : Finite element and nodal displacements

/€ Tl"

 

500ft.

@

 

 

 

 

   

1,125 ft.

 

TV
JL

Figure 2.4 : Finite element discretization of the side span

15

5% .888 05 no :83?me 80836 BEE " mfi 2:»:—

7r
JL

d oouJe
ova ova mvfl mu— m m N

e E rrlnuIL ii.“
EN new new Nam ofi a c m _
t ® 9 i

S:

IF

4

 

 

 

 

 

d 8m

 

 

SN o—

E\ /e

 

 

 

 

emu

 

 

 

 

l6

 

500ft.

 

 

 

 

 

 

 

 

1,125 fr.

it
L

Figure 2.6 : Node numbering scheme for side span

a 8n

[4

 

 

2w.

 

5% .250 to. 689.8 9.2863: 802

 

 

e£\

 

 

dooflw

1 mm _

 

 

 

 

 

 

" ha 959“.

 

 

 

 

 

18

01'

ch(§1’§2;t) = {fc (S1: S2) } f {u (I) }e

(2.20)

The application of equations (2.17) and (2.20) in equation (2.8) yields the elastic stiff-

ness mauix of the deck:

[k,,] .=

 

[k]=Esereo 000 001
“e L3 —126L0126L0

L
l 8.21.. if".}.{f".}fdi

' 12 -6L 0 —12 —6L 6
—6L 4L2 0 6L 21.2 0

 

 

—6L 2L2 0 6L 4!.2 o
_o 0 o o ooJ

(2.21)

(2.22)

If shear defamation is included in the deck, the elastic stiffness mauix becomes

(Przemieniecki 1968)

ES¢JS€

[kse]¢= 3
L (l-l-(p)

I— —

12 -6L 0-12 —6L 0
—6L (4+¢)L20 6 (2-¢)L20
o o o 0 0 o
—12 6 o 12 6L 0

—6L (2-¢)L20 6L (4+¢)L20

 

 

L0 000 00_

where 4) is the shear deformation parameter.

(2.23)

The use of equations (2.17) and (2.20) in equation (2.9) yields the gravitational stiff-

ness mauix of the deck:

19

[k] _ E
38 e " 420he

' 156 22L
22L 4L2

—147
-21L
-147 -21L 140 (1 + —‘

e

54 13L
—13L -3L2

—63
14L

 

e

h he
) -63 14L 70(1+—)
y ye

he he
-63 -14L 70 (1+_) -147 -21L140(1+—)

-63
- 14L

54 —13L
13L —3L2

156 —22L
—22L 4L2

—l47
-21L

 

ed

where Wee is the weight of the deck subelement per unit length.

(2.24)

Similarly, using equations (2.17) and (2.20) in the equation (2.5) yields the elastic

stiffness mauix of cables:

[keel = w

oococo

OOHCO
oooooo
oooocc
HooLoo

N
:
'ocoooo‘
I
p—a

 

 

b00000
_ _ 000000
_(Fu+wa)002001

we 000000
000000
001002

 

(2.25)

The gravitational stiffness matrix of cables is formed by applying equations (2.17)

and (2.20) in equation (2.7):

w..L
6ye

 

[we =

000000
000000
002001
000000
000000

 

'901002

(2.26)

The use of equations (2.17) and (2.20) in equation (2.11) yields the consistent mass

matrix of the deck:

20

 

"156 -22L0 54 13L 0
UL —22L 4L2 0-13L —3L20I
m, 0 0 0 0 0 0
=___ 227
[Me 420 54 -13L0 156 22L 0 ( )

13L -3L2 0 22L 4L2 0
0 0 0 0 0 Q

 

 

b

If shear deformation in the deck is accounted for, then the following consistent mass

matrix (Przernieniecki 1968) is obtained:
[me] e = [mel] (2.28)
where [mel] is given by Figures 2.8.

Finally, using equations (2.17) and (2.20) in equation (2.10) yields the mass matrix of
the cable:

b00000
000000

000000
000000
001002

3

 

e

 

 

The various matrices corresponding to the overall structure are assembled from the

element matrices in the standard way. The following su'ucture matrices are assembled:
l. The elastic stiffness mauix of the deck [K s E] from the element matrices [keel e.
2. The gravitational stiffness matrix of the deck [K so] from the element matrices
[k,gl .-

3. The elastic stiffness matrix of the cables [K C E] from the element matrices [ke e] e.

21

I

 

Buses a
coszLoBo Beam :23 5.8:. «was 2.22200 .u ad 2:96.

C o o o o o
o aimw+wfie+nlwwv iwwlemw+flmv o~iWw+mm+olmw .. 4Am+mm+oM~W
oeim+mw+elm ~w+m+m eeim+mm+mw4el w+mw+ew 13:

o o o o o o loeumlnzg
OEWW+WM+O+LTQAW+W+ON~H§IO Nimww+W+mw3 AW+WTA$~ u

 

22

4. The gravitational stiffness mauix of the cables [K CG] from the element matrices
[ks-g] e'
5. The mass matrix of the deck [M s] from the element matrices [me] e.

6. The mass matrix of the cables [Me] from element matrices [me] e.

The free vibration equations of the bridge are

[M] {ii} + [K] {14} = {0} (2.37)
in which
[M] = [M5] + [MC] (2.38)
and
[K] = [K35] + [K35] + [Keg] + [Keg] (2.39)

23

3. Random Vibration Analysis

The damped equations of motion of the bridge can be written as

[M] {11} + [C] {u} + [K] {it} = {0} (3.1)

where [M] is the overall consistent mass matrix of equation (2.39), [C] is the damping
matrix, [K] is the overall stiffness matrix of equation (2.40), and {ii} , {u} , and {u}

are vectors of nodal accelerations, velocities, and displacements. Equation (3.1) represents
the equations of motion for all nodal displacements, regardless of whether they are free or

restrained.

Equation (3.1), can be rearranged and partitioned as follow:

MFF MFR {fir} + CFF CFR {'11:} + KFF Kim ”H = [{0}] (3.2)
MRF MRR {an} CRF CRR {an} KRF KRR {“R} {0}
The subscript F refers to free nodal displacements, while the subscript R denotes restrained

nodal displacements.

The free nodal displacement vector {u F} can be decomposed into pseudo-static and

dynamic parts, {ue} and {ad} ,respectively:
{up} = {u,} + {ad} (33)

The pseudo-static displacements are obtained from the support displacements. The

static equilibrium equations, with no external loading are:
[Kpp] {up} 'I' [Kirk] {uR} = {0} (3-4)

{u e} are the free displacements from the above equation due to prescribed displacements

{u R} and is therefore given by

{“3} = ‘IKppl -1 [Kirk] {14R} (35)

24

Equation (3.5) represents the instantaneous free displacements of the structure due to sup-
port movement {u R} at time t Substituting equation (3.5) into equation (3.2), and assum-
ing stiffness proportional damping (for which [C] =0: [K] ) yields (Harichandran and
Wang 1988)

(tn-,1 {12,} + 1%] {12,} + [Km in,» ~ UM”) 1K,,1 '4 urn] — mph) {11,} (3.6)

in which the term ([CFF] [KFF] [K FR] - [Ce-RD {rig} is dropped. Equation(3.6) is
also approximately true for any light damping.

3.1 Modal Analysis

The free vibration equations of motion are
[Map] {”4} + [Kpp] {ud} = {0} (3.7)

For free vibrations of the undamped structure, we seek solutions of equation (3.7) in the

form

{12,} = [‘1'] them" (3.8)

in which [‘1‘] = [{W1} {W2} - - - {Wing is the matrix of mode shapes, and
{Y} are a set of generalized coordinates. Substituting equation (3.8) inequation (3.7)

yields the generalized eigenvalue problem

([Kppl - [diag(w2)l [MppD [‘P] = [0] (3.9)

The solution of these equations yield the natural frequencies of vibration 0) 1., and the mode

shapes, {Wj} , of the structure. Substituting
{ud} = [‘1’] {Y} (3.10)

into equation (3.6), premultiplying by [‘1'] T and assuming that mode shapes are orthogo-

nal to the damping matrix (classical damping). results in the uncoupled modal equtions

25

.. . 2 _ ,_
Yj+2§jijj+r0j Yj — GJ’J’1’2’3’ .......... ,n (3.11)

where

{ -}T[[M 11K 1-11K l-[M 11
07= V’ FF 7,. F” F" {in} = {I‘ertuR} (3.12)
J

 

[ [MN] urn.) -1 [Km] - [Mn] 1 T {V1}

”1

Mi = {Vj}T[MFF] {‘71} (3.14)

 

{rj} = (3.13)

In practice it is common to assume modal damping ratios £1. rather than to assemble the ma-
trix [C] in equation (3.1). It is convenient to collect the excitations Gj(t) into a vector

{G (t) } and the modal participation factors {1}} into a mauix

[F] = [{I‘l} {r2} — — — {133] (3.15)
in which case equation (3.12) may be written as
{G} = {Wm} (3.16)

The modal participation factor matrix [I‘] is of size r x n, where r is the number of re-

mained degrees-of-freedom and n is the number of mode shapes considered in the analysis.

3.2 Random Vibration Theory

The autocorrelation function of the im free displacement is defined as

Repit) = E{“F,(T)“F,('+T)} (3.17)

Using equation (3.3) in equation (3.17), we obtain the following expression

1%,, (r) = Rum +11% (0 +11... ., (0 +12... (0 (3.18)

26

where Ru, , Ru , and Red“

’3 i ‘5

are the autocorrelations of the dynanric displacement compo-
nent, the static displacement component, and the cross correlation between the dynamic

and static component, respectively.
For stationary response

Rum (z) = Rude (—r) (3.19)

The Fourier Transform of equation (3.18) yields the spectral density function of the i'h free

displacement

Salem) = See; ((0) +S“4.-“'.-(w) +Su.,uee(c°) +S“’.-(w) (3.20)
For stationary response

5,], (m) = fwd“) (3.21)

t ‘r
where the asterisk denotes the complex conjugate.

The variance of the i‘h free displacement can be obtained by integrating equation

(3.20)

0'2 = j 5., ((0) d0) + is“, ((0) d0) + 2Re[ j Snead (m) din]

“F3
(3.22)
035.. = Oiae+ 0313+ ZCOV (usi, udr)

where Re[ ] denotes the real part of the argument, oi and oil are the variances of the

pseudo-static and dynamic im displacement, and Cov (u 3:” ud.) is the covariance between

the static and dynamic displacements.

3.2.1 Variance of Dynamic Displacements

Applying the definition of the autocorrelation function in equation (3.17) to the i‘h dy-

namic displacement, and using equation (3.10) we obtain

27

Rel” = 2 2v,,-v,-,,E{Y,-(t)¥,(t+r)} (3.23)
‘ j=lk=l

where the index n is equal to the number of mode shapes considered in the analysis.

The equation of motion expressed by equation (3.10) can be solved using Duhamel’s

integral as

13(2) = j Gj(t-0)hj(0)d6 (3.24)

where hj(t) is the impulse response function for mode j and h,(t) is the response function due

to an impulse excitation 01- = 8 (r) , where 8 (t) is the Dirac delta function.

Substituting equation (3.24) into equation (3.23), yields

Rum = Z X ‘I’y‘I’trE{161("91)hj(°r)dor j Ge(t+1:—02) h,(ee)dee} (3.25)

j-rr-r

Equation (3.25) shows that the impulse response does not depend on the time lag 1:, thus it

can be written as

Rum = 2 2 We’ll!“ j j 11401))",(02)12{Gj(r—el)o,(:+'c-02)melee2 (3.26)
I 1.1k31 —ee-oe

substituting equation (3.12) into equation (3.26) yields

keen) = X 2 2 Z eyeitye,‘r,jt'mt [ j Rina. (1—024-01)hj(01)he(02)d61d92 (3.27)
‘ j=1kxli=lnsl _.._.. '

The Fourier Transform of the above equation yields the spectral density function of

the i'h dynamic displacement :

28

5..._(¢°) = 2:1: 2 2 2 Z wearer”. j j j Rani. (1-02+91)hj(01)hk(02)e_imd01d02dt (3.28)
' jslksllslm-l _.._.._.. "

The impulse function hi (0) is related to the frequency response function H j (00)
through

hj(9) = 217. [H (co)e“°°de (3.29)

using equation (3.29) and a change of variables to (t — 02 + 01) in equation (3.29) yields

n

5.59) = 2 2 Z 2 9.1! .r .- ..H,-< w>H.<w>S... ._<w> (3.30)

j=lk=ll=lm=l

Integrating equation (3.30) yields the variance of the free dynamic response “(1.0) :

 

03‘. = 2 2112 21% Wik I‘lj rmkiHj(- m)Hk(w)SiiR ,3... ((0)610)
' j: lk=1 I: m:
,. n (3.31)
= z Z‘I’ijw wik Ijk
j: lk=l
where H j (0)) may be obtained directly from equation (3.11), and has the form
H-(tu) = 1 (3 32)
’ (09-002) +21§.to.to '
J J J
and
1].. = ’21 21131.1)". j H]. (-m) H. (0)) 5...... (.5) d0) (3.33)
= m: _..

are the nodal covariances contributing to the overall dynamic response.

29

3.2.2 Variance of Pseudo-Static Displacements

The static displacements of the free nodes due to static support motion is expressed

by equation (3.5), and can be rewritten in a compact form as
{u,} = [A] {MR} (334)
Where [A] = "' [KFF] -1 [KFR]

[A] represents an n x r matrix, where each column in it represents the static displace-
ments of the free nodes due to a unit displacement of the corresponding support, while all

other support displacements are zero.

Applying the autocorrelation function to the ith pseudo-static displacement and using

the summation expansion of equation (3.33) yields
R... (1):; 2 A ,Ae-mR “my. (12) (3.35)
‘ i=1m=1

and the spectral density function of the pseudo-static displacement is obtained through the

Fourier transform of equation (3.34)
s... ((1)): 2 2 AuAe-mS “M.“ ((1)) (3.36)
I l- — 1m: 1
Equation (3.35) can be expressed in terms of the spectral density function of acceleration as

Sn..(‘”)= 2 Z A-IA Heme Steal”) (3.37)

l=lm=l

The variance of the ith pseudo-static displacement is then

= 2 2 Apt..." j ésemehmmm (3.38)

l=1m=l

30

3.2.3 Covariance between Pseudo-Static and Dynamic Displacements

The cross correlation between static and dynamic displacements is expressed in equa-

tion (3.18), using this expression and equations (3.10) and (3.34), we obtain

“4") = Z ZAuweeEwR (:)Y,.(:+c)} (3.39)

l=lk=l

Using Duhamel’s integral (equation (3.24)), the relation between the impulse response
function hj (0) and the frequency response function of equation (3.29), and applying a
change of variable to equation (3.38), the spectral density function between the static and

dynamic displacement can be obtained as (Harichandran and Wang 1988)

f

5.9.400) = 2 Z 2 Auwiel‘meruoHe . (m) (3.40)
i =1 RI Run

l=lk=lm

Differentiating the cross correlation function R “.1“. (1:) twice, yields

R u .u._(‘t) = R -- (t) (3.41)

allulrl

Differentiating the Fourier Transform relation between the autocorrelation function and the

spectral density function, we obtain

(1:): j -(°25....._ (0)) e‘mdm (3.42)

m

Rnuniukn

From equation (3.40) and equation (3.41), it follows that

1”
=TnIR" w “a: x (3.43)

“kt“an(

OL

31

=._ 2
Sana... ((0) (.0 Sumac,” ((0) (344)
Using the fact that
1
Sukluln (m) = EDTISiikfin. (C0) (3'45)

and substituting in equation (3.43) yields

= .._l_s.. .. (0)) (3.46)

“mun- (1)2 "mun.

Substituting equation (3.45) into equation (3.39) and using equation (3.22) yields

Cov(us,udi) = 2 2 2 41.17..ka _I-MH*((°)S ”(audio (3.47)
l=lk=lm=l

The covariance has the property
Cov (net, “3.) = Cov (“3,2 “4.) (3.48)

3.2.4 Variance of Dynamic Element End Forces
From the solution of the eigenvalue problem we obtain mode shapes in the global co-
ordinate system. For a general beam element with d.o.f. (j, k) at the left end, and (l, m) at

the right end, the member end—displacements corresponding to the ith mode are

‘I’jj

{Dee}. = W“ (3.49)
I V1:

34’...-

 

 

If the beam element is inclined, then the member end-displacements in local coordinates
can be obtained by

{Dem}: = [T] {Deg}. (3.50)

32

in which [T] is rotational transformation matrix.

The element end forces corresponding to the i‘h mode shape can be computed as
where [Ice] is the element stiffness mauix in local coordinates.

Performing this sequence of operations for all the eigenvectors, the resulting element
end forces can be collected into a matrix [F] in which the F U element is the i‘h element end
force corresponding to the jth mode shape. Using the same procedure described in Section

3.2.1, it can be shown that the variance of the i‘h dynamic element end force is expressed as

“12 ElFijFikrljrka”1(w03)Hk((D)S“((1))do)
= m: (3.52)
pa thk

"M; "M;

where I j,‘ is defined in equation (3.33).

3.2.5 Variance of Static Element End Forces

Similar to the dynamic element end forces, making use of equation (3.33), we can as-
semble a static element end force matrix [P] in which Pij is the ith element end force due
to a unit displacement along the j‘h restrained degree of freedom. Using the same formula-
tion described in Section 3.2.2, it can be shown .that the variance of the i‘h static element

end force is expressed as
=2 2 Pup... j— (.45 e 4.. (0))d0) (3.53)
l=lm=l

3.2.6 Covariance of Pseudo-Static and Dynamic Element End Forces

Following the procedure described in Section 3.2.3, the covariance between the pseu-

do-static and dynamic element end forces can be shown to be

33

f n r

Cov(si,fi) = 2 2 Z Pulsar” j (-£§)Hk(m)sfimuh(m)dm (3.54)

l=lk=lm=l —oo

3.3 Transient Response

The theory summarized in Sections 3.1 to 3.3 is valid for stationary seismic excita-
tion. However, earthquake acceleration amplitudes are characterized by a finite build-up
time, a period of uniform intensity and a period of decay. It follows that responses of qui-
escent systems to such excitation are non-stationary. For a single degree-of-freedom sys-
tem with undamped circular natural frequency (on and damping ratio é, the rate at which
the response grows to the stationary state depends on the value of go)" and the duration of
strong shaking. However, for a multi-degree-of-freedom system, the rate at which the total
response grows depends on film]. for each mode, and on how much the lower modes con-
tribute to the overall response. If the lower modes with small éjwj do not contribute signif-
icantly, then the total response may reach stationarity rather quickly. In this study of
suspension bridge response, the first few modes have extremely low frequencies m j and
therefore may not reach the stationary state within the earthquake duration. Therefore, it is
of interest to study the transient response of suspension bridges, subjected to non-stationary

seismic excitation.

In most earthquake engineering applications it is reasonable to represent the non-sta-
tionary excitation by an envelope-modulated stationary random process that may be ex-
pressed as a product of a stationary random process with a deterministic envelope

modulating function as:
am = e(t)z(r) (3.55)
where z (t) is a stationary random process, and e (t) is a temporal modulating function.

The generalized displacement response for the fh mode may be expressed as:

t
Y]. (t) = Ihj(t-'c)e (1)2 (t) dt (3.56)
0

where h,(t) is the impulse response function of the 1“ mode. In frequency domain analysis,

it is convenient to define a “time—dependent frequency response function” as
t o
H]. (0), z) = th (t— r) e (r) e'mdt (3.56)
o

The response variance at a given time t is evaluated by substituting the function
H j (03, t) in place of the normal frequency response function H]. (to) in the expressions ob-
tained for the stationary response. However, it is very difficult to express ”1((0’ t) in
closed form for arbitrary e(r), and closed form expression have been derived only for a few

functional forms of e(t).

The purpose of this work is to study the effect of correlated support excitations, and
not really to determine the absolute response variances. Thus the exact form used for e(t)
is not very crucial, and the use of a Heaviside modulating function is sufficient to assess the

effect of transient responses.

For the Heaviside modulation, Lin (1963) derived the expression for H j (to, r) as

 

- . .co.+ im)
[11(0), t) = H} (0)) [I _ e gjmlte-rmr[cosmjdt + (g) (3)-d sinmjdt):| (3.57)

where (01.4 = w}. 1 - if, and H j (to) is shown in equation (3.32).

3.4 Ground Motion Model

A mathematical model for the acceleration cross spectrum of ground acceleration

33.131. (0)) , is needed for the random vibration analysis of structures.

The ground motion model proposed by Harichandran and Vanmarcke (1986), is used

in this study. The model considers the spatial as well as the temporal variation of earth-

35

quake 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 be-

tween the acceleration of two locations A and B is expressed as:

-icov

 

 

Sign, = Sa(m)p (v, %)e V (3.58)
where
”2" (l-A+a.A) fl(l—A+0A)
p(v,f) = Aeaom + (1-A)e°m (3.59)
-1
em = k[1+(f£)b:| 2 (3.60)
and

v = separation between locations A and B.
f = linear frequency.
V = apparent wave propagation velocity in the direction AB.
and S a (to) = point auto spectral density function of the ground acceleration.

A, a, k, f0, and b are model parameters where typical values are shown in Table 3.1
(Harichandran 1991). The function p (VJ) is known as the coherency function and equa-

tion (3.59) is one of the more suitable forms based on the analysis of events recorded by the

SMART-1 array. In general the absolute value of coherency decreases with increasing fre-
quency and increasing separation, as shown in Figure 3.1 for separations corresponding to

the side and center span lengths of the Golden Gate bridge.

36

Table 3.1 : Ground Motion Model Parameter.

 

 

 

Model Parameter Ground Motion
038 15.0
[38 0.55
Coherency
function (of 3'0
B, 0.6
So 0.1387
A 0.636
a 0.0186
Double-filter
autospectrum k 31200
f, (Hz) 1.51
b 2.95

 

 

 

 

 

The functional form suggested by Clough and Penzien (1975) for the auto spectral

density function is used in this study. This function is expressed as
s, ((1)) = [H1 (or) flu, (co) [250 (3.61)

where [H1 (0)) |2 is the Kanai-Tajimi spectrum

37

 

 

 

 

 

 

1.0 rrrrr fi ''''' l‘ I 1 IIIIII l ' I Y I ft 1

———u=342.9m ‘

.9 - — — u=12eo.tem 1

.J
>~
0
C
O
L
O
J:
O
U
V
O
O
2
O
>
vi
.0
<

.1- ..

’ '1

o J L A J L l a l L l 1 l J 1 a 1 A 14

 

 

 

o 2 4 6 81012141618202224262830
Frequency (Hz)

Figure 3.1 : Coherency function at two separations

 

 

(1+4 2((1)/(0)2)

[H1(m)|2= (05232 8 m 2 (3.62)

1—(—) J +452(_)

( “’1 f (”I

and
2 _ ((1)/(of)4

|Hz(w)| _ 2 2 2 (3.63)

(1-(9) ] +4B}(-‘i’)

“’f “’r

in which the parameters mg, Bs’ (of, and Bf, control the shape of the spectra, and S o is an
intensity parameter. These parameters can be estimated by fitting the function expressed in
equation (3.61) to observed acceleration spectra. The auto spectral density function of the

ground displacement is

38

su (0)) = 5175:; (0)) (3.64)

For studies requiring the spectrum of ground displacement, the Kanai-Tajimi spectrum be-
comes undefined as co —> 0 and the double-filter spectrum given by equation (3.61) over-

comes this problem.

The double-filter spectrum was fitted to the artificial accelerogram of Type-B (Jen-
nings, Housner, and Tsai, 1968). The Type-B accelerogram, is one of four types generated
to model accelerograrns corresponding to different earthquake magnitudes. Each of the ar-
tificial accelerogram is a section of a random process with a prescribed power spectral den-
sity, multiplied by an envelope function chosen to model the changing intensity at the
beginning and end of real accelerograms. The Type-B motion has a duration of 50 seconds
and is intended to model the shaking close to the fault in a magnitude 7 earthquake, similar
to the El Centro earthquake of 1940 and the Taft, California, earthquake of 1952. The shape
and intensity parameters of equations (3.59) to (3.61) were evaluated using a least squares
fit to the spectrum estimated from an accelerogram. The band width of the smoothing win-
dow used for the spectral estimation was 0.5 Hz. The normalized autospectrum and the fit-
ted model are'plotted in Figure 3.2.

3.5 Computation Steps
A computer program was written to perform the analysis. The main segments of this

program are summarized in the following steps:

1. The overall stiffness and consistent mass matrices were assembled from the element
matrices using equations (2.38) and (2.39). These matrices are of order 48 x 48 for
the side span and of 255 x 255 for the center span.

2. The partitioning of the overall mass and stiffness matrices were performed to obtain
the [Km] , [Km] , [MFF] , and [MFR] , where the subscript F refers to free nod-

a1 displacements, while the subscript R denotes restrained displacements. The free

39

10°

 

 

 

 

 

 

 

 

~ ————£eflmohd
; — — ooueu: FILTER
E r-
: h
.5 1
310":/ \ -
m P
o I \
u.-
3 r-
< »I
'8 ' .A
'5 -2 I \ ‘
B 10 :- \ 1
E :f \ ‘ 1
6 : \-
z I
10-3 1 1 1 1 . 1 . 1 1 1 1 1 1 I 1
o 2 4 6 8 to 12 14 16

Frequency (Hz)

Figure 3.2 : Estimated and fitted autospectra for Type-B accelerogram

free mauices were of order 44 x 44 for the side span and of order 251 x 251 for the
center span. The [K FF] and [M FF] matrices were banded and their half band
width was equal to 6. Both [M FF] and [K FF] were positive definite symmetric
matrices.

. The mauix [A] was established from the static displacements of the structure due
to unit restrained displacements using equation (3.34). This matrix was of order 44

x 2 for the side span and of order 251 x 2 for the center span.

. The generalized eigenvalue problem given by equation (3.9) was solved using the
IMSL (1987 ) subroutine DGVCSP. This routine is designed to compute all of the

40

eigenvalues and eigenvectors of the real symmetric generalized eigenvalue prob-
lem, with symmetric positive definite [M FF] . In this routine the Cholesky factor-
ization [MW] = [R]T[R] , with [R] a triangular matrix, is used to convert

equation (3.9) into the standard eigenvalue problem

([R] '1 [Kpp] 1R1") [R] [‘1'] = [diagmzn ([R1 [‘13) (3.65)

Theeigenvalues, [diag(m2)] andeigenvectors {iii} of [R] "T [KN] [R]"1 are
then computed. Equation (3.65) has the same eigenvalues as the original problem;
and the eigenvectors of the original problem are found using
{V’s} = [R] -1 {\Tri} . The eigenvectors are normalized such that a modified oo-

norm of each eigenvector is one.

The Cholosky factorization is computed by IMSL routine DLFTDS. The eigenval-
ues and eigenvectors of the real symmetric matrix [R] ’T [K FF] [R] ‘1 are com-
putesd as follow: first, accumulating orthogonal similarity transformations are used
to reduce the matrix to an equivalent symmetric tridiagonal matrix; second, the im-
plicit QL algorithm is used to compute the eigenvalues and eigenvectors of this trid-

iagonal matrix.

The performance index for the generalized real symmetric eigensystem of equation
(3.9) is computed using IMSL routine DGPISP. In this routine a performance index
n , is defined to be

H [Kpp] {Wj} '03}? [Mpp] {WI} "1
n = max
15j5"8("[KFF]“1+|m12|”[MFF]||1)||{Wj}"1 (3.66)

 

where e is the machine precision.

While the exact value of 11 is highly machine dependent, the performance of
DEVCSF is considered excellent if 11 < 1 (which is the case for the side span

41

T} = 0.56), good if 1 S n S 100 (which is the case of the center span 1] = 13.09),
and poor if n > 100.

. Compute the participation factors matrix [F] using equation (3.13).

. Calculate the upper triangular part of the integration matrix of equation (3.33) and

. + 1 .
store the values in a one dimensional array of order 293—)— r2, where n rs the

number of modes considered and r is the number of nodal excitation points.

. Calculate the integrals I 61353.,“ (to) do) and store the values in a two dimen-

sional r x r array.

. Calculate the integrals I izH,‘ (to) 533.13. (to) doc and store the values in a three
(0 I

dimensional array of order n x r x r.

. Calculate the dynamic and static displacement variances, and the covariances be-

tween dynamic and static responses using equations (3.31), (3.38), and (3.47), re-

spectively.

10. Calculate the element end forces corresponding to the ith mode shape using equation

(3.51), and collect the resulting element end forces into a matrix [F] in which the

Fij element is the im element end force corresponding to the j‘h mode shape.

11. Calculate the force responses (dynamic, static, and cross covariances) using equa-

tions (3.52), (3.53), and (3.54) respectively.

42

4. Numerical Results and Analysis

4.1 Free Vibration Analysis

The study of the Golden Gate Bridge was divided into two parts, the side span and the
center span of the bridge. The side span of 1,125 feet length was subdivided into 15 ele-
ments, each of length 75 feet. The center span of 4,200 feet was subdivided into 84 ele-
ments each of length 50 feet (see Figure 2.3 and 2.4). All mode shapes and frequencies were
extracted for the side and center spans using the IMSL library (see step 5 of section 3.5). A
summarized description of the first eighteen frequencies and mode shapes of the side and
center spans are presented in Table 4.1 and 4.2, respectively. The first nine mode shapes of
the side span are shown in Figure 4.1, while the first eighteen mode shapes of the center
span are shown in Figures 4.2 and 4.3.

It can be seen from Figure 4.1 that in the first two modes there is a coupled motion
between the cables and the deck, the cables and the deck are moving in phase for the first
mode, while in the second mode they are moving 180° out of phase. The first and second
mode represent one fourth wave lateral motion of the cables and the deck, respectively. The
third mode is a half-wave cable mode with hardly any participation from the deck, while
the fifth mode is a half-wave deck mode with hardly any participation from the cables. The
fourth mode represents a full-wave lateral motion of the cables. The smoothness of the
higher mode shapes is degraded due to the coarseness of the discretization. For smoother

shapes at higher frequencies one should consider increasing the number of elements per
span length.

The first eighteen mode shapes of the center span are shown in Figures 4.2 and 4.3.
Many of the modes reflect coupled lateral motion between the cables and the deck. Modes

1, 2, 5, and 7 are examples of in phase lateral motion between cables and deck, while modes

43

Table 4.1 : Golden Gate Bridge side span natural frequencies and periods

of lateral vibration.

 

 

 

 

 

 

 

Mode # Frequency Period Primary Type of
(Hz) (sec) contributer mode

1 0.3133 3.1918 deck & cables symmetric
2 0.3435 2.9112 deck & cables symmetric
3 0.6561 1.5242 cables anti-symm.
4 0.9841 1.0162 cables symmetric
5 1.2450 0.8032 deck anti-symm.
6 1.3235 0.7556 cables anti-symm.
7 1.6769 0.5963 cables symmetric
8 2.0484 0.4882 cables anti-symm.
9 2.4404 0.4098 cables symmetric
10 2.7965 0.3576 deck symmetric
1 1 2.8546 0.3503 cables anti-symm.
12 3.2890 0.3040 cables symmetric
13 3.7368 0.2676 cables anti-symm.
14 4.1837 0.2390 cables symmetric
15 4.6034 0.2172 cables anti-symm.
16 4.9564 0.2018 cables symmetric
17 4.971 1 0.2012 deck anti-symm.
18 5.1953 0.1925 cables anti-symm.

 

 

Table 4.2 : Golden Gate Bridge center span natural frequencies and peri-

ods of lateral vibration.

 

 

 

 

 

 

 

Mode # Frequency Period Primary Type of
(Hz) (sec) contributer mode
1 0.0480 20.8514 deck & cables symmetric
2 0.1079 9.2708 deck & cables anti-symm.
3 0.1985 5.0387 deck & cables symmetric
4 0.2189 4.5676 cables anti-symm.
5 0.2238 4.4681 deck & cables symmetric
6 0.3249 3.0775 cables symmetric
7 0.3537 2.8273 . deck & cables anti-symm.
8 0.3910 2.5577 deck & cables anti-symm.
9 0.4548 2.1986 cables symmetric
10 0.5357 1.8667 cables anti-symm.
1 1 0.5732 1.7447 deck symmetric
12 0.6205 1.61 15 cables symmetric
13 0.7026 1.4233 cables anti-symm.
14 0.7877 1.2695 cables symmetric
15 0.8184 1.2220 deck anti-symm.
16 0.8732 1.1452 cables anti-symm.
17 0.9590 1.0427 cables symmetric
18 1.0455 0.9564 cables anti-symm.

 

45

 

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48

3, 6, and 8 are examples of 180° out of phase between cables and deck. The first mode rep-
resents one fourth wave lateral motion of the cables and deck with a period of 20.85 sec-
onds, while the second represents one half wave lateral motion of the cables and deck with
a period of 9.27 seconds. A full-wave lateral motion is seen in the fifth mode. Relatively

smooth mode shapes are achieved for high frequencies as a result of the fine discretization.

4.2 Ground Motion Models

The following three specialized ground motion models were considered in studying
the response of the side and center spans of the Golden Gate bridge to seismic support ex-
citation:

1. The most general form which includes both the wave propagation effect as well as
correlation effects, as expressed in equation (3.58).

2. Fully correlated ground motion (which is commonly used in practice) for which
S i“. (to) = Sii (to) (i.e., p (v, (0) = 1 and V —> oo in equation (3.58)).

3. Wave propagation without coherency loss for which p (v, (u) = 1 and V is finite

in equation (3.58).

4.3 Side Span

The response variances are normalized by dividing by the maximum total response
along the span.
4.3.1 Side Span Response Components

As discussed earlier in Chapter 3, the variance of the total response comprises of three
components: the variance of the dynamic response, the variance of the pseudo-static re-
sponse, and the covariance between the pseudo-static and dynamic responses. It is instruc-

tive to deduce which component contributes most to the total response.

The first natural frequency of the side span is 0.31 Hz which is indicative of a flexible

structure. The contribution of the three components to the total lateral displacement re-

49

sponse along the cables and deck are presented in Figures 4.4 and 4.5, respectively for the
general ground motion model. All response quantities are normalized by dividing by the
maximum total response along the span. It is clear from these figures that the dynamic com—
ponent dominates the total response. Examining the components of the total lateral dis-
placement response at node 7 (which is 300 feet from the left support) reveals that the
dynamic component contributes 100.04%, the static component contributes 4.1%, and the
covariance contributes -4.2%. In a similar manner for node 15 (which is 600 feet from the
left support), the dynamic component contributes 101.12%, the static component contrib-

utes 2.2%, and the covariance contributes -3.3%.

The moment and shear responses are dominated completely by the dynamic compo-
nent and the effect of pseudo-static and covariance terms are negligible (see Figures 4.6 and

4.7).

4.3.2 Lateral Response of the Side Span

A comparison of the side span responses due to the three ground motion models are
presented in Figures 4.8 to 4.11. The responses in each figure are normalized by dividing
by the maximum response along the span due to the general ground motion model. Figures
4.8 and 4.9 represent the total lateral displacement response of the side span cables and
deck, respectively, and show that the response due to fully correlated ground motion is the
highest, while that due to the general ground motion model is the lowest. The responses of
node 8 (on the cable) due to fully correlated and propagating excitations are essentially the
same, and are 6.5% larger than the response due to general excitation. At node, 16 the re—
sponse due to fully correlated and propagating excitations are 17.12% and 10.91% larger
than that due to general excitation. The responses of node 7 (on the deck) due to fully cor-
related and propagating excitations are 13.67% and 10.41% larger than that due to general
excitation. At node 15, the responses due to fully correlated and propagating excitaions are

16.87% and 11.21% larger than that due to general excitation.

50

 

1 . 4 v 1 v I r j I | t T I I v I v I Y r I I 1 I 1 I
r ——-— Dynamic response I
— — Static response
— - —— - Cross response
- — — - Total response «

 

1.2

l
l

 

 

 

I '

Normalized Displacement

 

 

 

 

_.2 1 1 1 l 1 l l l A L L l

2 4 6 8 10 12 14 16 18 20 22 24 26 28
Node number

 

Figure 4.4 : Variation of normalized displacement variances of the
side span cables

 

1 e 6 fit I I I 1 l 1 T t I v 1 I I I l v I 1 I v I I I v I 1
Dynomlc response 4

l' ——
1 .4 L — — Static response _,
— - — - Cross response

 

 

 

 

 

 

E 1 2 :- - -— — - Total response
8 1.0- 4
O . .
'6.
.2 -B- -
D — I
‘D .6» I
Q)
.5 ' r
‘6 .4- -1
s - d
O _2. ..
Z _ .,
0- I
. -l
_ 2 l 1 IL 1 l 1 I l 1 l 14 1 l s l l l 1

 

 

 

O 2 4 6 8 10 12 14 16 18 20 22 24 26 28
Node number

Figure 4.5 : Variation of normalized displacement variances of the
side span deck

51

Normalized Moment

Normalized Shear

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Node number

Figure 4.7 : Variation of normalized shear variances of the
' side span deck

52

1 . 4 v r I T v f I l v I v T 1 r v I v I ' I ' I 7 l T I
—— Dynamic response .
1 2 _ — — Static response
' -— - — - Cross response '"
— -— - — Total response .
1 .0 L 1
.8 ~ .
e 6 '- 1
. 4 F -
.2 - .
0 - _ .
_ . 2 1 1 A 14 1 1 1 1 1 L 1 a L 1 1 J 1 1 1 1 1 1 1 1 1 L
O 2 4 6 8 10 12 14 16 18 20 22 24 2 28
Node number
Figure 4.6 : Variation of normalized moment variances of the
side span deck
1 . 4 . , v , t 1 . I . r v 1 . , v . . 1 r , a 1 v , 1 1 .
-— Dynamic response 1
1 2 _ ---- —— Static response
' — - — - Cross response ‘
_ — - — - Total response
1 . O - 4
r- 1
. 8 - -
. 6 b -
a 4 " 1
O 2 I. d
L “i
0 » ...._ .
_ o 2 1 1 1 41 1 1 1 l 1 l 1 l 1 1 1 l 1_ l 1 l J l 1 l 1 l 1
0 2 4 6 8 10 12 14 l 6 1 8 20 22 24 26 28

 

1 . 4 I IiI I I I T I T I I I I If I r I I I I I I f I I
General Case

-—— — Fully Correlated

1 . 2 *- — - — - Wave Propagation

 

L

 

 

 

Normalized Displacement

 

 

l

 

 

O 1 l 1 1 1 1 1 l 1 l 1 l 4 l 1 l J l 1

2 4 6 8 10 12 14 16 18 20 22 24 26 28
Node number

Figure 4.8 : Normalized displacement variances of the side span
cables due to three ground motion models

 

 

 

 

 

 

1.4 W I T I I I I I I I r I I I v I r I r I I I I I v I
————Genera|Case .
—— —— Fully Correlated
1. 2 +- — - — - Wave Propagation u
.0—
c _ .
E
0 1.0- -
o 'l
.9.
Q .. .I
.2 .8
D .1
'8 .6. .
«'1‘ .
'5
E e4.” -‘
L
o ..
Z
1
o 1 l 1 l 1 l 1 l 1 l 1 l 1 4 1 l 1 _L 1 l 1 I

 

 

 

o 2 4 6 8101214161820222 2628
Node number

Figure 4.9 : Normalized displacement variances of the side span
deck due to three ground motion models

53

The deck bending moment response due to the three ground motion models are pre-
sented in Figure 4.10. The maximum moment response for all three ground motion models
occurs at the midspan. While the moments due to fully correlated excitation are greater than
those due to general excitation near the midspan, the trend is reversed near the quarter
spans. The moment response at node 7 due to fully correlated excitation is 17.4% lower

than that due to general excitation.

The lateral shear responses of the side span deck due to the three ground motion mod-
els are presented in Figure 4.11. The figure indicates that the shear response due to fully
correlated excitation underestimate the response due to general excitation by about 20% at
the supports; overestimates the response by about 40% at the quarter span locations; and
gives zero shear at the midspan. The rather unexpected behavior at midspan where the shear
response drops to zero for fully correlated excitation can be explained by examining the
modal contributions. Figure 4.12 shows the relative contributions of the dynamic modal co-
variances F 11F ”(II-k to the total dynamic variance (see equations 3.33 and 3.52) for the first
23 modes, due to general excitation. Since the modal covariances are symmetric, i.e.,
F er “I it = F i If 11'1ij the off diagonal elements that are shown, are twice the value of the
corresponding covariance (i.e., the if“ value shown is the relative contribution of
2F ijF‘. kl it to the overall dynamic variance). Figure 4.12 indicates that mode 5 contributes
about 95% of the total dynamic shear response, while mode 17 contributes 4% to give a
total contribution of 99%. The fully correlated excitation does not excite either mode 5 or

mode 17 since they are anti-symmetric modes (see Table 4.1).

4.3.3 Effect of Apparent Wave Velocity

By choosing different apparent wave velocities in equation (3.58), a study was con-
ducted to examine its effect on the side span response due to the general excitation. Figures
4.13 and 4.14 represent the effect of the apparent wave velocity on the displacement re-

sponse of the cables and deck. It is found that increasing the lateral wave velocity from

54

Normalized Moment

Normalized Shear

1.

1.

1.

‘d—b-‘N
NONLODON$GQO

‘

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

General Case .
-— —- Fully Correlated
2 - / \ — - -— - Wave Propagation -
0P 1
o l 1 1 1 l 41 l 1 l 4 l 1 l 1 l 1 1 1 J 1 l 1 L 1 L J
O 2 4 6 8 1O 12 14 16 18 20 22 24 26 28
Node number
Figure 4.10 : Normalized moment variances of the side span
deck due to three ground motion models
‘ ———— General Case "
r —— — Fully Correlated ‘
- -— - — - Wave Propagation ‘
i- q
- 1
Z J
L- .—
O 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Node number

Flgure 4.11 : Normalized shear variances of the side span
deck due to three ground motion models

55

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56

Normalized Displacement

Normalized Displacement

020 ' l ' I ' l V I

.15

.10: ""‘“"="

 

020. ' I ' T ' I ' T ‘ I r I V l ' I V T

 

.70 1 1 1 1 1 1
9

 

d
.1
.1
q

I I l I I I
v=1700 m/sec.
—— —— v=3000 m/sec.
— - — - v=6000 m/sec.

 

VIITT'I

 

 

 

 

111111111111111111111111111111111111111

 

 

 

75;-
.70 1 l 1 1 L 1 1 1 11 1 1 1 L l 1 1 1 l 1
10 11 12 13 14 15 16 17 18 19 20

Node number

Figure 4.13 : Effect of apparent wave velocity on the side span
cable displacement

 

 

v=l700 m/sec.
—— —— v=3000 m/sec.
—- - —— - v=6000 m/sec.

--——— v=ee

 

 

 

 

  

1111111111111111111111111111111111111

 

1 1 l 1 1

1O 11 12 13 14 15 16 17 18

 

 

.- /
d 1L11111 u 1

Node number

Figure 4.14 : Effect of apparent wave velocity on the side span
deck displacement

57

1,700 m/sec to co increases the cable displacement response by about 5.0% at midspan and

the deck displacement response by about 4.0% at midspan.

The effect of the apparent wave velocity on the moment response of the deck is pre-
sented in Figure 4.15. It can be seen that the moment response at midspan increases as the
apparent wave velocity is increased, while in the moment response near the quarter span

(node 9) decreases as the apparent wave velocity is increased.

The effect of apparent wave velocity on the shear response of the deck is presented in
Figure 4.16. The shear response at midspan (node 15) and near supports (nodes 3 and 27)
decreases as the apparent wave velocity increases, while the shear response near the quarter

span (node 7) increases as the velocity increases.

4.3.4 Modal Contributions

The dynamic response variances are composed of individual modal response varianc-
es and covariances between pairs of modal responses as indicated in equations (3.31) and
(3.33). The relative modal contributions at nodes 7 and 15 in the deck located at (300 feet
and 600 feet from left support, respectively), are presented in Figures 4.17 to 4.20 for the
displacement and moment responses. Figures 4.17 and 4.18 reveal that modes 1, 2, and 5
contribute most to the lateral displacement. Mode l contributes 99.1% and 101.4% to the
dynamic response at nodes 7 and 15, respectively. Only a very few of the off diagonal mod-
al covariances have a noticeable contribution at these nodes. Figures 4.19 and 4.20 show
that modes 1, 2, 5, 10, and 19 contribute most to the total dynamic moment response at
nodes 7 and 15 with their total contribution being 102.4% for node 7 and 101% for node
15. It is clear that the contribution of mode 1 to the moment response of node 7 is less than
that of node 15. The modal covariances participate more to the moment response than for
the displacement response. A comparison between the displacement response and the mo-
ment force response indicates that the number of modes required to compute the moment

response is higher than that required for the displacement response.

58

Normalized Moment

Normalized Moment

.20

.15

.85

.80

 

 

l l l l l 1 1 1 l

l I 1 r l I

 

v=1700 m/uc.
— —- v=3000 rn/sec.
— - — - v=6000 m/ue.

 

 

 

 

4 l 1 l J L 1—

[11114 LlllLllIllllllllLLlJlllJJ

111/114

 

 

O 2 4 6 8 10 12 14 16

9 10 11 12 13 14 15

Node number

Figure 4.15 : Effect of apparent wave velocity on the side span

deck moment

16 17 18

 

 

I ' I ' T ' I 7 T

 

—— v=1700 m/uc.
— — =3000 m/ue.

———— v=¢

 

-— - — - v=6000 m/sec. "

 

 

 

l l A l

 

 

Node number

. Figure 4.16 : Effect of apparent wave velocity on the side span

deck shear

59

18 20 22 24 26

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63

Figure 4.21 shows the modal variances and covariances for the deck shear response
at node 7. The modal variances and covariances for the cable displacement at nodes 8 and
16 are presented in Figures 4.22 and 4.23. Figure 4.21 indicates that a relatively large num-
ber of modes contribute to the dynamic shear response (modes 1, 2, 5, 10, 17, 19, 20, 21,
22, and 23). The higher frequency modes contribute about 10.0% at node 7. A greater num-
ber of modal covariances contribute to the shear responses than to the displacement and
moment responses. Figure 4.12, presented earlier shows the contribution of the modal co-
variances to the shear response at node 15, and indicates that the first mode has zero con-

tribution and 95% of the contribution is from mode 5.

Figures 4.22 and 4.23 indicates that mode 1 contributes about 54.5% and 53% to the
lateral dynamic displacement at nodes 8 and 16, respectively. The relative contributions in
the figures are shown only to the third decimal figure and smaller contributions are simply
shown as 0.000. This is a little misleading since the sum of all the non-zero values is 70.8%
in Figure 4.22 and 66.1% in Figure 4.22, and a very large number of small contribution
from variances and covariances shown as 0.000 in the figures must additively contribute
the remaining 29.2% and 33.9% at nodes 8 and 16, respectively. This indicates the need to

consider a very large number of modes in the overall analysis.

4.3.5 Transient Response

It is of interest to determine whether the side span will reach its stationary response
during typical durations of strong shaking (10 to 20 seconds). The variances of the cable
displacement, deck displacement, deck moment, and deck shear are evaluated at times of
5, 15, 30, and 40 seconds and compared with the stationary responses for the general
ground motion model in Figures 4.24 to 4.27. All variances have been normalized by di-

viding by the maximum stationary variance for the corresponding response.

Figure 4.24 shows the lateral displacement response of the side span cables. At node
8, 47.4%, 79.2%, 91.7%, and 96.9% of the general stationary response is achieved at times

n 802 - 3:039 52m 208 2:856 on. 2 2259280 .805 o>=2om " Rd 2:9“.

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67

Normalized Displacement

Normalized Displacement

1.

_s
N
I

d

1.

1.

1.

 

O

 

I

I

r 1 r v r v I ' I

 

 

Stationary Response

——-- tBSsec.
—-—- t=1Ssec.
———— t=30eee.
-------- t=40sec.

 

 

 

r I 4 I r I r I

 

 

Figure 4.24 : Variation of normalized transient displacement variances of

r r I r
12 14

16

18 20 22 24 26

Node number

the side span cables

 

 

T V I V I T 1 0 I f I

 

.Statlonary Response
— — t = 5 see.
— - — - i I: 15 sec.
- — - — t = 30 sec.

 

........ i = 40 see.

 

 

 

I J I r I

I

 

 

Flgure 4.25 : Variation of normalized transient displacement variances of

12 14

16 18 20 2 24 26

Node number

side span deck

28

of 5, 15, 30, and 40 seconds, respectively, while for node 16 the corresponding numbers are
46.2%, 77.9%, 90.8%, and 96.5%.

Figure 4.25 illustrates the lateral displacement of the deck. At node 7, 30.4%, 71.3%,
90.9%, and 95% of the general stationary response is achieved at times of 5, 15, 30, and 40
seconds, respectively, while for node 15 the corresponding numbers are 29.1%, 70.6%,
90.9%, and 95%. A comparison of the results between cable and deck responses reveal that
the rate at which the responses grow is sensitive to the percentage contribution of the lower
modes to the overall responses. While the contribution of mode 1 to the cable response at
nodes 8 and 16 is 54.5% and 52.8%, it is 99.1% and 101.4%, respectively for deck nodes 7
and 15. The lower modes with longer periods take longer to attain stationarity, and there-

fore, total responses dominated by the lower modes take longer to attain stationarity.

Figure 4.26 shows the moment response of the side span deck. Results for node 7
show that 45.3%, 79.9%, 93.4%, and 96.2% of the stationary response is achieved at times
of 5, 15, 30, and 40 seconds, respectively, while for node 15 the corresponding numbers are
32.3%, 71.7%, 90.8%, and 94.8%.

Finally Figure 4.27 illustrates the shear response of the side span deck. Mode 1 con-
tributes 79.1% and 0.0% of the overall dynamic response at nodes 7 and 15 respectively
(see Figures 4.12 and 4.21). The earlier statement reading the effect of the percentage con-
tribution of the lower modes on the rate at which the responses grow is exemplified by the
shear responses. At node 7, 45.1%, 78.4%, 94.2%, and 97.4% of the stationary response is
achieved at times of 5, 15, 30, and 40 seconds, respectively, while for node 15 the corre-
sponding numbers are 79.3%, 98.7%, 99.6%, and 99.6%. The results presented here indi-
cate that for common ground motion durations, the assumption of stationarity may grossly

over estimate the side span responses, and the transient nature of the responses should be

taken into account.

69

Normalized Moment

Normalized Shear

1.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Stationary Response ‘
— — t = 5 sec.
1.2: —-—-t=15uc. 4
— - — - t = 30 see.
-------- t=40mu. 1
1 .0 r- '-
.8- -
l. a
_ 6 l- .-
.4- a
.2- -
o r L r I L I r I r I 1 J r I r I r I r I r I r l r I
O 2 6 8 10 12 14 16 18 2 22 24 26 28
Node number
Figure 4.26 : Variation of normalized transient moment variances of
the side span deck
1 0 4 j I I I I I I I I I T I I I I I I I I I
L Statlonary Response ‘
— — t = 5 sec.
1.2- —-—-t=158ec. -
— - — - t = 30 see.
-------- t=40eun ‘
1.0- a
.8- _
.6» -
.4- ‘
.2- a
o J I r J r IJr l r I I I n J r I J

 

 

 

O 2 4 6 8 10 12 14

Node number

16 18 20 22 24 26

Figure 4.27 : Variation of normalized transient shear variances of
the side span deck

70

The transient response of the side span was computed for the three ground motion
models at time t=15 seconds. The responses of this study are normalized by dividing by the
maximum response along the span due to the general ground motion model at time t=15
seconds. The results are presented in Figures 4.28 to 4.31 for displacement of the side span
cable and for the displacement, moment, and shear of the side span deck, respectively.
These figures are very similar to Figures 4.8 to 4.1 1, which are for stationary response. This
indicates that the general conclusions drawn based on comparisons between the stationary

response due to the three types of excitation are also valid for the transient response.

71

 

I .4 I I I I I I I T I I I f I I I I I I I u I I I I
General Case

—— — Fully Correlated

Wave Propagation

 

 

 

.0
N
l
I

I

 

d
0

Normalized Displacement

 

 

 

I I I

I I I I I I
2 4 6 8 to 12 14 16 18 20 22 24 26 28
Node number

J I I I L 4 I I I 4 I I I L

 

Figure 4.28 : Normalized transient displacement variances of the side span
cables due to three ground motion models at t=15 seconds

 

 

 

 

 

 

1'4Il'I'I‘I'ITIfTIIIIIIIfirfiII
L ————6eneralCase
—- — Fully Correlated
1.2 r- __ — - — - Wave Propagation «
E / \
g P
l. _
o 1.0
0 . l
2
O. _ _
.22 .8
O
.3 .6: d
.t‘
'6
E e4~ -'
L
O i-
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.2»- a
r 1
0 I I I L I I I I I I L IL L I I I I J

 

 

 

o 2 4 6 810121415182022242628
Node number

Figure 4.29 : Normalized transientdisplacement variances of the side span
deck due to three ground motion models at t=15 seconds

72

Normalized Moment

Normalized Shear

1

 

.4 I I I I I I I I I I I I I I I I j T I I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Node number

L General Case
— — Fully Correlated
1 . 2 - / \ — - — - Wave Propagation -
. 4- __-_._->\-~
1 .0 - -
. 8 - -
. 6 .- _
. 4 —- —
. 2 p a
o L J I L I L I I I I I I I I 4 ILI I I LI I I I I I I I
0 2 4 6 8 1 0 12 14 1 6 18 20 22 24 26 28
Node number
Figure 4.30 : Normalized transient moment variances of the side span
deck due to three ground motion models at t=15 seconds
2 O o fir I I I I I I I j I I I I I j I I I I I
-—-——--—- General Case ‘
1 . 8 - — -— Fully Correlated “
- — - — - Wave Propagation -*
1 . 6 P .
1 . 4 r j
' '1
1 . 2 r
1 . 0 1
. 8 r
h J
. 6 ..
. 4 r
h '1
. 2 ‘
o .1
>- 4
_ . 2 L I I I I I I I I I I L I I I I I I J I I I I I I I I
0 2 4 6 8 10 12 14 1 6 1 8 2 22 24 26 28

Figure 4.31 : Normalized transient shear variances of the side span
deck due to three ground motion models at t=15 seconds

73

4.4 Center Span

It is very costly to consider all 251 modes in estimating the response of the Golden
Gate bridge center span. For example, in equation (3.33) the required number of integra-

tions is (n2 + n) r2/2. Table 4.3 presents the number of integrations that are required as a

Table 4.3 : Relation between the number of modes and the number
of integrations.

 

 

No. of modes Required .No. of

considered Integrations
10 220
20 840
40 3,280
60 7,320
80 12,960
104 21,840
180 65,160
251 126,504

 

 

 

 

function of the number of modes used in the analysis. Therefore, it is important to deter-
mine the minimum number of modes required to acurately evaluate the response of the
bridge center span. Several runs were performed with varying number of modes and it was
found that using 104 modes gave at least 99% accuracy for all responses. Results for dis-
placement and force responses using 20, 40, 60 and 104 modes are presented in Figures

4.28 to 4.31. Figure 4.32 indicates that if 40 modes are considered, the lateral displacement

74

 

 

 

 

 

 

 

 

 

 

L 104 modes ‘
1-1’ —— —— 20 modes ‘
' —-—- 40mdes I
'E 1.0r -——- BOmodes a
o > «
E .9 - -«
8 . .
2 .8 - a
O. t ‘
e9 '7 t- -
o . .
'U 06 '- -‘
0 I- .
05 .5 " -I
T: . .l
E .4 r- -
L
o b d
z .3 — ~
.2 - \-
i- -I
.1 I L I I I J I I IL I I I J
0 12 24 36 48 60 72 84 96 108 120 132 144 156 168

Node number

Figure 4.32 : Variation of normalized displacement variances of the
center span cables

 

I I

 

L I L L

. 1 . 1 . .

104 modes
— — 20 modes
— - — - 4O modes
60 modes -

 

I L4

 

 

 

 

ILILIILLLII

I I I I I

 

 

1.2
1.1-
31.0-
0 r
E e9-
3
o .8-
0.
e2! '7b
0 .
1, .6~
O ..
._._"—‘ .s~
O ,,
E .u
L.
O .
Z .3-
.2-
.1
0

12

24

36 48

60 72 a4 96
Node number

 

108 120 132 144 156 168

Figure 4.33 : Variation of normalized displacement variances of the
center span deck

75

response of the cables will be represented with high accuracy, while the consideration of
60 modes will give almost an identical response to that of 104 modes. Figure 4.33 shows
that the response of the center span deck lateral displacement estimated using 20 modes will

give identical results to that of 104 modes.

Figure 4.34 and 4.31 show that using 60 modes represent the force responses accu-
rately. It is clear that accm'ate computation of the force responses requires larger number of

modes. Based on these results, 60 modes were considered in this study.

4.4.1 Center Span Response Components

The components of the total response variances are presented in Figures 4.32 to 4.37.
The first three figures show the center span cable response for the three ground motion
models, while the remaining figures show the displacement response of the center span
deck for the three ground motion models. In Figures 4.32 to 4.37 the dynamic displacement
response dominates the total response, but on the other hand the pseudo-static and covari-

ance components contribute significantly more than for the side span.

Table 4.4 and 4.5 present the relative contribution of response components to the ca-
ble and deck displacement response at quarter span (node 42) and midspan (node 84). The
results show the domination of the dynamic response and the significant relative contribu-
tion of the covariance components. The dynamic response contributes about 110% or more
to the total response for all excitation models, and the covariance response contributes

about 20% or more.

The results indicate that the correlation effect is not significant on the response, and
that wave propagation alone yields good accuracy. In terms of the response components,
neglecting the static variance and the covariance would over-estimate the lateral cable re-

sponse by about 10% and the deck response by about 24%.

The force response components of the deck (i.e., moment and shear) are dominated

totally by the dynamic component and the effect of the static component and cross covari-

76

Normalized Moment

Normalized Shear

1.

1.

‘.

 

 

I I I I L I

0
0 12 24 36 48 60 72

I I I I T

I

I

 

 

— — 20 modes

--—-- GOmodes

104 modes .

40modes -

 

 

LL I I I L

(,r’

\
//
5

 

l L

84 96 108 120 132 144 156 168
Node number

Figure 4.34 : Variation of normalized moment variances of the
center span deck

 

 

 

I

I

 

 

-— —- 20 modes

——-- 50modes

104 modes ‘

40 modes 4

 

 

 

 

 

 

o I
O 12 24 36 48 60 72
Node number

84 96 108 120 132 144 156 168

Figure 4.35 : Variation of normalized shear variances of the
center span deck

77

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1.4

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Dynamic response
— — Static response
— - — - Cross response
— - — — Total response

 

 

 

 

 

I I I I J I I

 

 

60 72 8 96

Node number

108 120 132 144 156

168

Figure 4.36 : Variation of normalized displacement variances of the

center span cables (General ground motion model)

 

rfrT‘IfiTrrI I

U I r t T r V

 

I I I I I I I I I I 1 r r r I

 

Dynamic response
— — Static response
— - — - Cross response
- — -— - Total response

 

 

 

  
 

 

 

\- - -—-—-—-_--—-- - -/

I I I I I I I I

I
IIII

 

 

12 24 36 48 60 72 84 9 108 120 132 144 156

Node number

168

Figure 4.37 : Variation of normalized displacement variances of the

center span cables (Fully correlated ground motion model)

78

Normalized Displacement

Normalized Displacement

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2 e O I I I I I I I I r g I r I I Y I
Dynamic response ‘
I ’8 ' — — Static response ‘
1 6 _ — - — - Cross response I
' _ — - — - Total response :1
1 .4 ~ -
1 .2 -
1 .0 e
.8 d
. 6 ~
.4 ..
.2 ~
0 .1
- . 2
- . ‘ I L I I J I L I L I I I I
0 12 24 36 48 60 72 84 96 108 120 132 144 156 168
Node number
Figure 4.38 : Variation of normalized displacement variances of the
center span cables (Wave propagation ground motion model)
1 . 6 . , . . . r . , . , 5 r r , . .
Dynamic response -
1 . 4 r — — Static response -
r — - — - Cross response ..
- — — — Total response _,
1 .2 ~
1 C
0 ~ .
I- \ ‘ ‘ / 1
-.2~ \. /..——--"""'“‘\- / _
P \ . .— - """— - \ " ~_. - / ‘

 

 

 

4 I
o 12 2 :56 4 60 72 a4 96 108 120 132 144 156 168
Node number

Figure 4.39 : Variation of normalized displacement variances of the
center span deck (General ground motion model)

79

Normalized Displacement

Normalized Displacement

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 8 1 Dynamic response :
' _ —— -—- Static response .
1 . 6 __ — - —— - Cross response .
- — — — Total response .
1.4 -
1.2 I
1 .o 3
.8 J
.6 l
.4 L
.2 I
o .;
- 2 .. \ ~ '/ ..
' i \. ./ 1
_,4+ \\w___ _.—‘*""—--_.--‘— “--__-_.— ’/t -
_ . 6 p I I I I I I I I I I I I I T
D 12 24 36 48 60 7 84 96 108 120 132 144 156 168
Node number
Figure 4.40 : Variation of normalized displacement variances of the
center span deck (Fully correlated ground motion model)
2 o 0 j I I f I T 1 I I I ' I
1 8 '_ Dynamic response 1
. l- — — Static response q
1 . 6 r- --— - -— - Cross response -
. - - - — Total response T
1.4- -
1.2 d
1.0 -
.8 n
O 6 q
.4 l
.2 3
° i
-.2 ‘
-.4 q
_ . 6 I I I J I I I I I I I I

 

 

 

O 12 24 36

48

60 72 84 96
Node number

108 120 132 144 156 168

Figure 4.41 : Variation of normalized displacement variances of the
center span deck (Wave propagation ground motion model)

80

 

 

 

 

 

 

 

 

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82

ance between pseudo-static and dynamic components is neglegible. The pseudo—static and
cross covariance terms are essentially zero and the total response will essentially be the dy-

namic component similar to those in Figures 4.6 and 4.7.

4.4.2 Lateral Response of the Center Span

A comparison of the center span responses due to the three ground motion models are
I presented in Figures 4.42 to 4.45. The responses in each figure are normalized by dividing
by the maximum response along the span due to the general ground motion model. Figures
4.42 and 4.43 represent the total lateral displacement response of the center span cables and

deck, respectively.

Examining the total cable displacement response in Figure 4.42 at quarter span (node
42) reveals that the use of identical excitations overestimates the response by 18.5%, while
the use of delayed excitation overestimates the response by 13% when compared to the re-
sponse due to the general ground motion model. Similarly at mid-span (node 84), the use
of identical excitation overestimates the response significantly (84.1%), while the use of
delayed excitation overestimates the response slightly by 5.8% when compared to the re-
sponse due to the general ground motion model. It is clear that the overestimation at mid-
span when identical support excitation is used is very high. The response due to the delayed
excitation also overestimates the general response near the midspan by as much as 20.6%

(e.g. at node 96).

The lateral deck displacement shown in Figure 4.43 is overestimated by 53.6% and
21.9% at mid-span (node 83) for the fully correlated and wave propagation ground motion

models, respectively, when compared to the general ground motion model.

The moment response of the center span deck presented in Figure 4.44 indicates that
the use of fully correlated and wave propagation ground motion models overestimate or un-
der estimate the response due to general excitation depending on the location. The moment

response at node 83 is overestimated by 124.44% and underestimated by 11.52% for the

83

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 e 6 I I I l T l I 1 v I r T I I V T
'- —-— General Case
— — Fully Correlated
1 '4 b —— - — — Wave Propagation
4—
S
E 1 .
O
0
g 1 .
o.
.2!
c:
'o
O
.L‘
'6
E
L
o
z
o I I I L I I I I I I I I J
o 12 24 36 48 60 72 8 96 108 120 132 144 156 168
Node number
Figure 4.42 : Normalized displacement variances of the center span
cables due to three ground motion models
1 e 6 V I ' l V T Fr 1 V T V T ' I T I
\ —— General Case .
— — Fully Correlated
1 ' 4 ” / — - — - Wave Propagation ‘
4...
c / \ -
° 1
E e
O
0
g 1 .
a.
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o
'o
0
£1
'6
E
I.
o
z

 

 

 

0
0 12 24 36 4 60 72 8 96 108120132144156168
Node number

Figure 4.43 : Normalized displacement variances of the center span
deck due to three ground motion models

Normalized Moment

Normalized Shear

““

o‘nbsmmvmoodnu

O-‘NM#UIOO\IOIOO-‘NOI

 

V I r' TT'

 

I V I I I

 

 

 

 

A _ __ 333731-321..- .

l \ — - — - Wave Propagation 3

\ , / \ [I \\ ./‘\ ,'\. /
\\\ ./ \‘/F~ / "x' \ l/ -

I I I I I_ I I

ILJIIII

 

I I I I I

 

O

12 24 36 48 60 72 84 96
Node number

108 120 132 144 156 168

Figure 4.44 : Normalized moment variances of the center span
deck due to three ground motion models

 

I

W

Tjj

'YTTIrr'

 

 

1 I I I I I

 

 

_ — Fully Correlated
__ - — - Wave Propagation _,

General Case

 

 

 

i
ii
\i

ll“
.l’

I I I I I I

 

.-/' fl

J I L I I

 

 

O

12 24 3 48 60 72 84 96
Node number

108 120 132 144 156 168

Figure 4.45 : Normalized shear variances of the center span
deck due to three ground motion models

85

fully correlated and wave propagation ground motion models, respectively, when com-
pared to that of the general ground motion model response. The identical support excitation
may result in a serious underestimation of the moment response, for example at node 71 the
moment is underestimated by 61.2% when compared to that of. the general ground motion
model. The large underestimation in the response is due to the fact that some of the anti-
symmetric modes which are excited by the general response are not excited by the fully cor-

related ground motion model.

The shear response of the deck is presented in Figure 4.45. At node 83, the response
due to fully correlated excitation underestimates the general response by 95.71% and the
wave propagation case overestimates it by 7.1%. The reason for the 95.71% underestima-
tion is again due to the fact that the fully correlated excitation does not excite many of the
anti-symmetric modes. The results indicate that contribution of general correlated excita-
tion is important for estimating the force response to avoid serious underestimation or over-

estimation of the force response at some locations along the bridge.

4.4.3 Effect of Apparent Wave Velocity

By choosing different apparent wave velocities in equation (3.58), a study similar to
that done for the side span was conducted to examine its effect on the center span response

due to the general and wave propagation excitation models.

Figures 4.46 and 4.47 show the effect of the apparent wave velocity on the displace-
ment due to the general and wave propagationground motion models, respectively. The in-
crease in the velocity causes progressively higher displacements near nodes 83 and 130,
and progressively lower displacements near node 100. The wave propagation ground mo-
tion shows a similar behavior to that of the general ground motion model but with higher
increases and decreases in the response. As is expected, the response due to wave propaga-

tion excitation approaches that due to fully correlated excitation as V —-> oo.

86

 

 

 

 

 

 

 

 

 

 

 

 

 

1 0‘ I V I I I I I I I I I I I 1 I I
” v=1700 m/sec. "
1 '3 b -— —— “33000 m/sec. 1
1.2 - -—-—- v=6000 m/sec. q
E - -' - - - Vg. a
g I .1 l- , u-l
. / ' \ .
O 1 .0 " r- l' / \\' \ "
U .- I ' \ ‘ .1
2 e9 " fl \ I/A "‘
.8- . 8 L I O \\ '/ :
CD I \\. \ .
g ‘7 I i
5 .6 . 1
.5 .5 l J
:5. - i
0 e4 "’ "1
z .3 — ~
- 1
.2- -
1 P I J I I I I I L I I I I I d
O 12 24 36 48 60 7 84 96 108 120 132 144 156 168
Node number
Figure 4.46 : Effect of apparent wave velocity on the center span
cable displacement (General ground motion model)
2.0 . - . . . . . . . . . , a , . T
v:- (Fully Correlated) ‘
1 .8 - --- — =17DD m/sec. 1
i — - — - vSSDDD m/sec. 1
"E 1 6 — — — - v=6000 m/sec. q
. i .
6.... .
8 .
-' 1.2- ~ ._ _
o.
.g _ \ I, ' /"'\ / /. -\\‘ \ .
010- .// \ // <\\{/’—~~\_\; J
g ‘ \\ I 0 \ . ' II
N 8 - ./ - '1
= l- .l
o -
E .6 . J
O _ 4
:Z .4
r 4
.2 -
O f I I I I I I I I I I I I I

 

 

 

O 12 24 36 48 60 72 8 96 108120132144156168
Node number

Figure 4.47 : Effect of apparent wave velocity on the center span
cable displacement (wave propagation ground motion model)

87

Figure 4.48 and 4.49 show the effect of the apparent wave velocity on the lateral dis-
placement of the center span deck due to the general and wave propagation ground motion
models, respectively. The behavior of the response is similar to that of the cable, and for
the wave propagation ground motion model the response approaches that of the fully cor-

related resPonse as the velocity goes to co.

Figures 4.50 and 4.51 show the effect of the apparent wave velocity on the moment
response of the center span deck due to general and wave propagation ground motion mod-
els. For the general ground motion model the maximum moment response occurs at nodes
13 and 155. For the general ground motion model the maximum moment response occurs
at node 13. For the wave propagation ground motion model the maximum moment re-
sponse occurs at node 83. Figures 4.52 and 4.53 represent the effect of apparent wave ve-

locity on the shear response for the general and wave propagation ground motion models.

4.4.4 Modal Contributions

The relative contributions of the dynamic modal covariances to the total dynamic
variances were computed for 23 modes at nodes 41 and 83 for the deck and nodes 42 and
84 for the cables due to the general ground motion model. The modal covariance mauix is
symmetric as explained in section 4.3.2, and the off diagonal elements that are shown are

twice the value of the corresponding covariances.

Figures 4.54 and 4.55 show the relative modal contributions to the total dynamic dis-
placement response at quarter and mid-spans (nodes 41 and 83, respectively). The main
conclusions are highlighted in Table 4.6. Significant conuibutions are obtained from three
anti-symmetric modes at quarter-span, and there modes are not excited by identical ground
excitations. The diagonal terms contribute most of the total dynamic response. Significant
contributions are obtained from the off-diagonal terms at quarter-span, and neglecting them

will result in an error of about 5%

Figures 4.56 and 4.57. show the relative modal contributions to the total dynamic mo-

88

NOI‘

‘ d d ‘
.
‘

Normalized Displacement
to '6- 3. in 'a- l: in in 'o

O
‘

Normalized Displacement

 

I I

IjII

I] Y I fi'jffi' 1 I I

 

I

r I I 1 T I I I I I I T I I

 

 

v=ee

v=1700 m/sec.
v=3000 m/sec.
v=6DOD m/sec.

 

 

   

I I I I I I I I I I

IIIIIIII

IIILIJI

 

 

O

12

72 84 96
Node number

24 36 48 60

108 120 132 144 156 168

Figure 4.48 : Effect of apparent wave velocity on the center span

deck displacement (General ground motion model)

 

 

T I I ' I j I ' I I I 1 I

 

v=- (Fully Correlated)
v=17DD m/sec.
"3000 m/sec.
v=6000 m/sec.

 

 

 

I I I I I I I I L I I I

 

 

 

12

24 36 48 60 72 84 96 108120132144

Node number

156 168

Figure 4.49 : Effect of apparent wave velocity on the center span
deck displacement (Wave propagation ground motion model)

89

Normalized Moment

Normalized Moment

 

I-4 ' I - I - I I I I I v I I I . I

 

 

 

 

l- —— v=i700 m/sec.
— — v-ISODD m/eec.
1.2- —-—-v=6000m/sec.-
— _ — — vg.
1 .0 i- -
t \
8 - ’

. /‘
. \ , rm .
.5. ./'/ <5 I// A}, \\ .

.2~ .

d

 

 

I I I I I I I I I I l I

o I
O 12 2 36 4 60 72 8 96 108 120 132 144 156 168
Node number

 

Figure 4.50 : Effect of apparent wave velocity on the center span
deck moment (General ground motion model)

 

 

 

 

 

 

. 6 - , . f - , fl , . r , , , t r l
v:- (Fully Correlated)
4 _ —- — v=17DO m/sec. ..
' -— - — - "3000 m/sec.
r — - - - va=6000 m/sec. A
.2 - .4
i
0 ‘- p‘ 1
\ . 'q 1
8r \\ / /" //\ //\(A‘\ A/\ "
\
5 ’ \é '/// / / ‘\..' ‘ \ - \\\‘\ J
J\ / \ / \\ I,\I / 4
4 - / \J'\v "
. .J
. 2 .-
i
o I I I I I I I I L I I

 

 

o 12 24 36 4 60 7 84 96 108120132144156168
Node number

Figure 4.51 : Effect of apparent wave velocity on the center span
deck moment (wave propagation ground motion model)

Normalized Shear

Normalized Shear

1.

1.

_s
O

0
0

0

 

 

 

 

 

 

 

 

 

v=t700 m/sec. ‘
— — v-IJOOD m/sec.
>- — - — - v=6000 m/sec. e
L -- — — — v=eo
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- ' =44“ 'H‘rr .
- ix! 7‘
I \ .-
P - I
I J I I I L I I I J I I I
0 12 24 36 48 60 72 8 96 108 120 132 144 156 168

Node number

Figure 4.52 : Effect of apparent wave velocity on the center span
deck shear (General ground motion model)

 

 

 

I f 1 I T

 

 

v:- (Fuily Correlated)
v=1700 m/sec.
v83000 m/sec.

 

 

 

 

 

Node number

— - - — v=6000 m/sec.

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Figure 4.53 : Effect of apparent wave velocity on the center span
deck shear (wave propagation ground motion madel)

91

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ment response at quarter and mid-spans, the results are summarized in Table 4.7. A larger
number of diagonal terms contribute to the moment response at quarter span. The anti-sym-
metric modes are large contributors to the moment response at quarter-span (about 53.1%
of the total response). The total contribution from all terms with individual relative conui-
butions less than 0.0005 add to 16% of the total dynamic response at quarter-span, while it
is 46% at mid-span. These results show the importance of considering a large number of

modes and the various cross-covariance terms associated with them.

Figures 4.58 and 4.59 show the relative modal contributions to the total dynamic ca-
ble displacement responses at quarter and mid-spans and Table 4.8 summarizes this infor-
mation. A large number of diagonal and off—diagonal terms are needed to estimate the total
dynamic response at quarter-span. The individual off-diagonal terms have small relative
contributions, whith 76 of them contributing about 5.4% at quarter-span. The anti-symmet-
ric modes are important in estimating the displacement response, with contributions of

about 24.4%.

4.4.5 Transient Response

The results discussed in Section 4.3.5 for the side span revealed that the percentage
contribution of the first mode of the side span influenced the rate at which the displacement
responses grow to attain stationary. It is also of interest to determine the time required for
the center span to attain its stationary response especially since its first few modes have fre-
quencies far lower than that of the first mode of the side span. The transient variances of
the cable displacement, deck displacement, deck moment, and deck shear were computed
for various durations of stationary excitation; and compared with the stationary responses
for the general ground motion model. All variances were normalized by dividing by the

maximum stationary variance.

Figures 4.60 and 4.61 show the lateral displacement response of the center span ca-

bles and deck, respectively. The behavior is opposite to that observed for the side span and

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O 12 24 36 48 60 72 84 96 108120132144156168

Figure 4.60 : Variation of normalized transient displacement variances of

Node number

the center span cables

 

 

 

 

 

 

 

 

 

0122

Figure 4.61 : Variation of normalized transient displacement variances of

36 48 60 72 8 96 108 120 132 144 156 168

Node number

the center span deck

102

 

 

to what would intuitively be expected. Instead of the response gradually growing to the sta-
tionary response, there is a strong overshoot of the response for short durations, and then it
gradually approaches the stationary variance from above. In order to explain this behavior
a thorough study was conducted on the integrand in equation (3.31). This function can be

written as
H]. (-w, r) 11,, (co, t) Sana... ((0) (4.1)

where H j ((0, t) is expressed by equation (3.57). For the diagonal element j, equation (4.1)

can be written as
2
|Hj (a), t)| Sana... (0)) (4.2)

Caughy and Stumpf (1961) studied the transient response of a single-degree-of-freedom
system under ideal white noise excitation using equation (4.2) . The response was evaluated
for different natural frequencies and different damping factors of the single d.o.f. system.
As expected the response variance approaches the stationary value as time increases, and
the larger damping values result in lower stationary values and allow the response to be-
come stationary in a shorter time. The transient variance did not overshoot the stationary
variance. This, however, is not always the case with the transient response. Barnoski and
Maurer (1969) studied the response of a single-degree-of-freedom system, with excitation

having the input autocorrelation function
R F (r) = Roe'“"' cosm (4.3)

where a = decay coefficient of noise correlation function
(u = frequency of noise correlation function

and found that the mean-square response does indeed overshoot the stationary value.
Whether or not an overshoot is obtained depends on the shape of the excitation spectrum

and on the frequency and damping of the oscillator. An overshoot is obtained only when

103

the frequency of the oscillator is very low and the excitation spectrum varies sharply at low

frequencies.

The integrand function of equation (4.2) is studied for two different modes (mode 1
of the center span and mode 1 of the side span) at two different times (5 and 10 seconds).
A plot of the integrand function is presented in Figure 4.62 and 4.63. Figure 4.62 indicates
that the area under the integrand function for t = 5 seconds is substantialy larger than that
for t = 10 seconds, which would result in the first mode response at 5 seconds being much
larger than that at 10 seconds. In Figure 4.63, however, the area under the integrand func-
tion at t = 5 seconds is smaller than that at t = 10 seconds, which results in the first mode
response at 5 seconds being smaller than that at 10 seconds. These results indicate that the
integration of equation (4.2) is dominated by the behavior of the cross spectral density func-
tion around the natural frequencies (0].. The first natural frequency of the center span is very
low compared to most structures and, because of this the nature of the fitted autospectra at
low frequencies su'ongly affects the estimated displacement response of the low frequency
modes.

Both the estimated autospectrum and the fit of the autospectrum shown in Figure 3.2
are not accurate at very low frequencies. In estimating the spectrum from the accelerogram,
the band width of the smoothing window used to achieve stability was 0.5 Hz. This ad-
versely affects the resolution in the estimated spectrum, and the estimates are expected to
be biased, especially at very low frequencies. It is not possible to get good resolution unless
the recorded accelerogram has a very long duration of strong motion, or several similar ac-
celerograms are recorded. In view of the inaccuracy in the excitation spectrum at very low
frequencies, and in view of the fact that the nature of the excitation autospectrum at very
low frequencies strongly affects the transient response of the low frequency modes, the
overshoot in the transient displacement response should be considered as qualitative infor-

mation, and the magnitude of the overshoot should not be considered as accurate.

104

Integrand for mode I

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t: 5. sec
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nlannnlnlnnlnrln

35 4O 45

Frequency (rad/sec)

Figure 4.62 : Integrand function for mode 1 of the

center span

 

50

 

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.10
.09
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4:484:01112l14 16.
Frequency (rad/sec)

l #1 I

 

 

 

 

18 20 22 24

l I I A l a I

Figure 4.63 : Integrand function for mode 1 of the

side span

105

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l

 

26

The modal contributions to the center spanforce response presented in Section 4.4.4
reveal that the lower modes hardly contribute. Since only the first few modes display the
overshoot behavior, and these do not contribute significantly to the force responses, the
transient force responses do not overshoot their stationary values. This behavior is seen in
Figures 4.64 and 4.65. Figure 4.64 show the transient moment response of the center span
deck at various times. Results of node 41 indicates that 76.6%, 98.3%, and 99.1% of the
stationary response is achieved at times of 5, 40, and 60 seconds, while for node 83 the cor-
responding numbers are 62.7%, 97.9%, and 99.2%. 4

The results presented here indicate that for usual ground motion durations, the as-
sumption of stationarity may grossly overestimate the center span force responses, and sig-
nificantly under-estimate the displacement response. Therefore transient response must be
considered when analyzing the center span. However, to accurately estimate the transient
displacement response, the excitation autospectrum should be specified carefully at very

low frequencies based on more abundant data.

The transient responses of the center span using the three ground motion models were
computed at t=20 seconds. The responses, normalized by dividing by the maximum re-
sponse along the span due to the general ground motion model at t=20 seconds, are present-
ed in Figures 4.66 to 4.69 for the cable and deck displacements , moment and shear of the
center span. The normalized transient force responses (moment and shear) of the center
span deck presented in Figures 4.68 and 4.69 reveal almost an identical behavior to the nor-
malized stationary responses presented previously in Figures 4.44 and 4.45. This means
that the conclusions based on comparing the force responses along the center span due to
the three stationary ground motion models are also valid for the transient responses. How-
ever, the normalized transient displacement response of the center span cable and deck
(Figures 4.66 and 4.67) are significantly different from the normalized stationary responses
in Figures 4.42 and 4.43. This is due to the strong participation from the first few low fre-

106

Normalized Moment

Normalized Shear

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48 60 72 8 96 108 120 132 144 156 168

Node number

Figure 4.64 : Variation of normalized transient moment variances of

the center span deck

 

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12 24 36 48 60 72 84 96 108

Node number

120 132 144 156 168

Flgure 4.65 : Variation of normalized transient shear variances of

the center span deck

107

 

2.0 I I I I Y I j I f r ' I I' T I r

 

I

1

r f —— General Case
1 .8 - \ —— — Fully Correlated
. / — - — - Wave Propagatlon

 

 

 

1

Normalized Displacement

 

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o L
O 12 24 36 4 60 72 84 96 108120132144156168
Node number

 

Figure 4.66 : Normalized transient displacement variances of the center
span cables due to three ground motion models at t=20 seconds

 

 

 

 

 

2'0 T 1 I T 1 r I r I v —[ 1— T
’ ————-GeneralCaee -
1.8» /\ —— Fully Correlated -
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o
0 12 2 36 48 60 72 84 96 108120132144156168
Node number

Figure 4.67 : Normalized transient displacement variances of the center
span deck due to three ground motion models at t=20 seconds

108

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Normalized Shear

 

I 1 r r r v F v j ' fi— ' a r I

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l
O 12 24 36 48 60 72 84 96 108120132144156168
Node number

 

Flgure 4.68 : Normalized transient moment variances of the center
span deck due to three ground motion models at t=20 seconds

 

I I I fl I I I I I I I I 1 T I
—— General Case
— — Fully Correlated

,_ —- - —— - Wave Propagation

 

 

 

 

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l

 

 

 

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0 12 24 36 48 60 72 84 96 108120132144156168
Node number

 

O-‘NU‘UO‘JUQO-‘NU

Figure 4.69 : Normalized transient shear variances of the center
span deck due to three ground motion models at t=20 seconds

109

quency modes, whose relative contributions are quite different for transient and stationary

responses.

4.5 Shear Deformation

It is of interest to study the effect of including shear deformation on the seismic re—
sponse of the suspension bridge. Shear deformation was included in calculating the re-
sponse of the side and center spans by substituting equations (2.23) and (2.28) in place of
equations (2.22) and (2.27), respectively, with the remainder of the analysis being the same.

4.5.1 Side Span

The eigenvalue problem in equation (3.9) was assembled and solved for the side span
natural frequencies and mode shapes. A softening behavior of the side span occured with a
drop in the value of the first natural frequency to 0.27 Hz compared to 0.31 Hz when shear
deformation was excluded. The spectrum of the undamped natural frequencies with and
without the effect of shear deformation are presented in Figures 4.62 and 4.63. Figure 4.71

0 25 50 75 100 125 150 175 200 225 250 275 300 325 350
Spectrum of Undamped Natural Frequencies (Hz)

Figure 4.70 : Undamped natural frequencies of the side span
excluding shear deformation

indicates that many of the modes are closely spaced and span a small range of frequencies

when the shear deformation is included. Comparing Figure 4.71 with Figure 4.70, shows

110

l l l

lTllllllTlIllllllllllllllflllllllIllllllllllllllllllllllllTllllllllqll

O 25 50 75 100 125 150 175 200 225 250 275 300 325 350

 

Spectrum of Undamped Nuturol Frequencies (Hz)

Figure 4.71 : Undamped natural frequencies of the side span
including shear deformation

that modes with frequencies between 11.2 Hz and 63 Hz are shifted to between 4.6 Hz and
7.6 Hz.

The first nine mode shapes of the side span are presented in Figure 4.72. A compari-
son of the mode shapes for the two cases (including and excluding shear deformation) in
Figures 4.1 and 4.64 reveal that the shapes of the first mode are similar, with the lateral dis-

placement of the cables being smaller when shear deformation is included. The same

behavior is seen for the second mode but the lateral displacement of the deck is now smaller
when shear deformation is included. The mode order is also switched, with modes 4,5, 6,
and 7 (for the analysis excluding shear deformation) becoming modes 5, 4, 7, and 8 (for the

analysis including shear deformation).

The total lateral displacement response of the side span cables and deck are presented .
in Figures 4.73 and 4.74, respectively, for the two analyses including and excluding shear
deformation using the general ground motion model. Figure 4.7 3 indicates that the cable
displacement response when shear deformation included is consistently lower than that
when shear deformation is excluded. Figure 4.74 shows a similar behavior for the deck dis-

placement response.

111

 

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Normalized Displacement

 

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Shear defamation Excluded
— — Shear deforrnatlon Included

 

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d

 

 

 

o I r I r I r I I r I r I J I r I

2 4 6 8 10 12 14 16 18 20 22 24 26‘28
Node number

I

 

Figure 4.73 : Normalized displacement variances of the side span cables

 

1 e 2 I I I I I I I I I I I I I I I I 1' I I I I I I I I I
Shear deformatlon Excluded
— — Shear deformatlon Included

 

 

 

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0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

o I r J I I I I I r I I I I I

 

Node number

Flgure 4.74 : Normalized displacement variances of the side span deck

113

Figures 4.75 and 4.76 show the side span deck moment and shear responses for the
analyses excluding and including shear deformation using the general ground motion mod-
el. Figure 4.75 shows a severe drop in the deck moment response (53% at mid-span) when
shear deformation is included. Figure 4.76 indicates a similar behavior for the shear re-

sponse, with the maximum drop of 60% occuring at nodes close to the supports.

The reduction in the lateral displacement response of the side span deck when shear
deformation is included is primarily due to the drop in the first natural frequency which
contributes the most (see Figures 4.17 and 4.18). The excitation spectrum has lower power
at the reduced first modal frequency, hence giving rise to a smaller response. This behavior
is illustrated in Table 4.9 which shows the ratios of corresponding modal displacement re-
sponses for the analyses excluding and including shear deformation. The first mode re-

sponse is reduced by about 9% when shear deformation is included.

The moment response of the side span deck has dominant contributions from modes
1, 2, and 5 for nodes 7 and 8 (see Figures 4.19 and 4.20). The ratios of corresponding modal
moment responses shown in Table 4.9 reveal that the moment response of mode 1 when
shear deformation is included is 44% and 33% of that when shear deformation is excluded
at quarter and mid-span, respectively. Mode 5 is switched to mode 4 when shear deforma-
tion is included, and its response is 24% and 21% of that when shear deformation is exclud-
ed. The large reductions in the responses of modes 1 and 5 explain the severe dr0p in

moment response when shear is included.

4.5.2 Center Span

As for the side span, the eigenvalue problem of equation (3.9) was assembled and
solved for the center span natural frequencies and mode shapes. The first natural frequency
remains the same as that when shear deformation is excluded. Results indicate that the first
120 modes span a frequency range between 0 Hz to 7.25 Hz compared to a range of 0 Hz

to 30.95 Hz when shear deformation is not included. Furthermore, the first 166 modes span

114

Normalized Moment

Normalized Shear

dd

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Node number

1 e 2 I I I I l I I I I I l I I I I I I I I I j I I I
' Shear deformation Excluded '
1 '1 _' — -— Shear deformation Included “
1 . O - -
.9 - l
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o .- I r I I I r l r I r I I I r I j I r I r I r I r I r
0 2 4 5 8 1 0 12 1 4 1 6 18 20 22 24 26 28
Node number
Figure 4.75 : Normalized moment variances of the side span deck
. 2 I I I I I I I I I I I I I I I I I I I I I I I I I
’ Shear deformation Excluded "
'1 : —- -——- Shear deformation Included '
. 0 - _
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Figure 4.76 : Normalized shear variances of the side span deck

115

 

 

 

 

 

 

 

 

 

 

 

 

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a frequency range from 0 Hz to 11.82 Hz, and mode 167 jump to a frequency of 150.78 Hz.
Therefore, modes in the frequency range between 12 Hz and 150 Hz are completely lost
when shear deformation is included, and their corresponding contribution on the response
vanishes. The spectrum of the undamped natural frequencies (excluding and including
shear deformation) are presented in Figures 4.77 and 4.78, respectively.

 

0 25 50 75 100 125 150 175 200 225 250

Spectrum of Undamped Natural Frequencies (Hz)

Figure 4.77 : Undamped natural frequencies of the Golden Gate bn'dge
center span excluding shear deformation

 

 

llllflllllllllllllllllllllllllll lllllljlllllll

O 25 50 75 100 125 150 175 200 225 250

Spectrum of Undamped Natural Frequencies (Hz)

Figure 4.78 : Undamped natural frequencies of the Golden Gate bridge
center span including shear deformation

The change in the natural frequencies when shear deformation is included is dramatic

and somewhat unexpected. This behavior may be partly due to the use of a simple beam

117

model to represent a stiffening u'uss which has a complex arrangement of members. Seis-
mic response studies using 3-D models that have been conducted (Baron, et. al. 1976), have
all used beam elements to simplify stiffening trusses and therefore no literature appears to

be available to confirm the behavior seen here when shear deformation is included.

The first eighteen mode shapes of the center span are shown in two sets in Figures
4.79 and 4.80. The first five modes when shear deformation is included are similar to those
when shear deformation is excluded (see Figures 4.79 and 4.2). A switching of the mode

rank starts at mode 6, which switches to mode 7 when shear deformation is included.

The total lateral displacement response of the center span cables and deck are present-
ed in Figures 4.81 and 4.82, respectively, for the analyses of including and excluding shear
deformation using a general ground motion model. Figure 4.81 indicates that the displace-
ment response when shear deformation included is 15% to 17% lower than that when shear
deformation is excluded. Figure 4.82 on the other hand indicates that when shear deforma-
tion is included the lateral displacement response of the deck increases at most locations,

while decreasing slightly near mid-span.

Figures 4.83 and 4.84 show the center span deck moment and shear responses for the
analyses excluding and including shear deformation using a general ground motion model.
Figure 4.83 show a severe drop in the deck moment response, when shear deformation is

included. Figure 4.84 indicates similar drop for the shear response.

The increase in the lateral displacement response of the center span deck can be ex-
plained by studying the contributions of the modes contributing most to the response. Fig-
ure 4.54 indicates that modes 1, 2, 3, and 4 contribute 53%, 26.6%, 3.5%, and 0.6% of the
total dynamic response when shear deformation is excluded. Table 4.10 shows the ratios of
modal variances when shear deformation is included to corresponding modal variances
when shear deformation is excluded . The modal response ratios reveal that the displace-

ment response of modes 1 and 2 are not affected significantly when shear deformation is

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121

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Figure 4.83 : Normalized moment variances of the center span deck
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123

included. The displacement response of mode 3, however, is is increased significantly

when shear deformation is included.

The moment response of the center span deck at node 41 is mainly contributed to by
mode 15 when shear deformation is excluded. The relative conuibution of this mode is
found to be 49.6% (see Figure 4.56). This mode becomes mode 12 when shear deformation
is included and its response is reduced to 21.7%. Figure 4.57 indicates that the relative mod-
al contribution of the diagonal terms of the first 23 modes contribute only 51.5% of the total
dynamic response, and the relative modal contribution of the cross terms sum to only 2.5%.
The remaining contributions to the moment response are therefore from higher modes.
When shear defamation is included, the frequencies of most of these higher modes shift to
low values (see Figure 4.78), and as a result, the transformed modes no longer contribute

significantly to the moment response.

The results of this section indicate that a dramatic change in the dynamic properties

and response of the center span occm's when shear deformation is included in the analysis.

124

 

 

5. Summary and Conclusions

5.1 Summary

This research was conducted to study the effects of spatially varying ground motion
on the lateral response of the Golden Gate suspension bridge. The bridge has a center span
of length 4,200 feet and side spans of length 1,125 feet. The ground motion model proposed
by Harichandran and Vanmarcke (1986), which accounts for the propagation and correla-

tion between the accelerations at two different points, was used.

Three ground motion models were used in the study. The first was the general ground
motion model, which included b0th the travelling wave effect as well as the correlation ef-
fects between the acceleration at two different points characterized by a coherency func-
tion. The second model was for fully correlated ground motion in which all supports move

identically. The third model included only wave propagation and neglected coherency loss.

5.1.1 Finite Element Model

The method adopted for analysis was a 2—D finite element based technique, which
takes into account the characteristics of both the cable and deck. In this model the cable is
idealized by a set of string elements, while the deck is idealized by a set of beam elements.
The two types of elements, connected by rigid hangers, form the bridge element This tech-
nique was applied to the spans of the Golden Gate bridge to assemble the overall mass and

stiffness matrices.

5.1 .2 Response Components
The first mode of the 'side span has a relatively low natural frequency (0.31 Hz). This

,however, did not strongly influence the components that comprise the total response. The
most important component of the response was found to be the dynamic one and it conuib-

utes about 100% to the total response. In the displacement response the maximum static

125

 

 

contribution was found to be 4% and the covariance contribution to be around -3%. The
variances of the force responses are completely dominated by the dynamic component and

contributions from the static and covariance components are negligible.

The first natural frequency of the center span is extremely low (0.048 Hz). The dy-
namic component dominates the cable displacement response and it is found to be about
110%, the static component contributes about 10%, and the covariance component contrib-
utes about -20%. A similar trend is found for the deck displacemens, but the dynamic com-
ponent contributes about 124%, the static component contributes about 18%, and the
covariance component contributes about -42%. These results indicate that the pseudo-static
and covariance components contribute significantly to the total response and neglecting

these terms would result in about 10% to 24% over-estimate in the displacement responses.

5.1.3 Lateral Response

Response variances due to the more common types of excitation consisting of identi-
cal or delayed support motions were computed and compared to responses due to the gen-
eral spatially varying ground motion model. For the side span, the use of identical support

excitations

0 over-estimates the moment response by as much as 23% (at node 15) and under-es-

timates it by as much as 20% (at node 23); and

- over-estimates the shear response by as much as 41% (at node 5) and under-esti-

mates it drastically at mid-span.
The use of delayed excitations gives acceptable results for the side span response, with

a a maximum over-estimation of about 11% for the displacement response of the

deck and cable, respectively, (at nodes 15 and 16); and

o a maximum under-estimation of the shear response by 15% (at node 19).

For the center span, the use of identical support excitations

126

- over-estimates the moment response by as much as 124% (at mid-span) and under-

estimates it by as much as 63% (at node 69);
o over-estimates the shear response by as much as 54% (at node 93) and drastically
under-estimates it near mid-span; and

0 over-estimates the displacement response by as much as 84% (at node 84) and un-

der-estimates it by as much as 22% (at node 100).

The use of delayed excitations for the center span yields significantly different responses
compared to those due to the general ground motion model, with the moment and shear re-

sponses being under-estimated by as much as 28% (at node 115).

 

5.1.4 Effect of Apparent Wave Veloclty

For the general ground motion model the effect of increasing the apparent wave ve-
locity from 1,700 m/sec to co was examined. For the side span, the increase in the velocity

produced
0 at most a 5% and 4% increase in the displacement response variance at mid-span of
the cable and deck, respectively;

0 at most an 8% increase in the moment response at midspan and an 18% decrease

near the quarter span; and

0 at most a 25% increase in the shear response at quarter-span and a 70% decrease at

midspan.

 

For the center span the increase in the velocity produced

0 at most an increase of 35.7% in the cable displacement response (at node 84) and

an increase of 13% in the deck displacement response at mid-span;
- at most an increase of 58.3% in the deck moment response at mid-span; and

o a decrease in the deck shear response by as much as -37% at mid-span.

127

5.1.5 Modal Contributions

The relative modal contributions to the dynamic responses were examined in detail to
assist in understanding the contribution of modal covariances and to understand which
modes are important for particuler responses. The study was performed for both the side

and center spans at quarter and mid-span locations.

For the side span the dynamic deck displacement response is contributed to mainly
by the first 5 modes and their corresponding covariances, while for the cable the first 8
modes are significant. The dynamic moment and shear responses require a larger number
of modes, with the moment requiring about 19 modes and the shear requiring more than 20

modes.

The center span deck displacement response has a greater number of participant
modes, with mode 15 being a strong contributer to the response at quarter—span. The force
responses have contributions from a large number of modes, but the contributions from the

first few modes is very small.

5.1 .6 Translent Response

Transient response analyses show that the side span displacement response variances
attain about 96% of their stationary values after 40 seconds of stationary excitation. A sim-
ilar behavior was found for the moment response. The rate at which the responses grow is
greatly dependent on the percentage contribution of the lower modes. The results for the
side span indicate that transient response should be considered if the responses are not to

be grossly over-estimated.

Transient response analyses of the center span show that due to the first natural fre-
quency being extremely low, the displacement responses which have a su'ong contribution
from the first mode greatly exceed their stationary values. The level of exceedance, how-
ever, is strongly dependent on the nature of the excitation spectrum at very low frequencies, .

and extreme care should be used in specifying the excitation spectrum if the results are to

128

be considered seriously. The transient force responses, however, do not overshoot their sta-
tionary values, since they do not have strong contributions from the low frequency modes.
The time taken for the transient responses to settle to their stationary values is much larger
for the center span, and it is clear that transient effects must be considered when analyzing
the center span for realistic durations of strong earthquake excitation.
5.1 .7 Effect of Shear Deformation

For the general ground motion model the effect of including shear deformation was
examined. For the side span, the inclusion of shear deformation produced

- a softening behavior of the side span occured with a drop of about 13% in the value

of the first natural frequency;

0 a shift in the band of frequency containing nine modes from 11.2 - 63 Hz to 4.6 -
7.6 Hz;

0 a shifting in the mode order , with modes 4, 5, 6, and 7 (excluding shear deforma-
tion) becoming modes 5, 4, 7, and 8 (including shear deformation);

- consistently lower displacement responses of the side span cable and deck;
- a severe drop in the deck moment response (53% at mid-span) and shear response
(60% at nodes close to support).

For the center span the inclusion of shear deformation produced

- a shift in the band of frequency containing 120 modes from 0 - 33 Hz to 0 - 7.25 Hz;
0 a switching in mode order;
. a lower cable displacement response of at most 17%;

0 an increase in the lateral displacement response of the deck at most locations, with

a slight decrease near mid-span; and

o a severe drop in the deck moment and shear response.

129

Bibliography

Abdel-Ghaffar, A. M. and Stringfellow, R. G. (1984). “Response of suspension bridges to travelling earth-
quake excitations : Part lI-Lateral response,” Soil Dynamics and Earthquake Engineering, 3(2).

Abdel-Ghaffar, A. M. and Rubin, L. I. (1983). “Lateral earthquake response of suspension bridges,” Journal
of Structural Engineering, ASCE, 109(3).

Abdel-Ghaffar, A. M. and Rubin, L. I. (1982). “Suspension bridge response to multiple support excitations,”
Journal of Engineering Mechanics Division. ASCE,108(EM2).

Abdel-Ghaffar, A. M. (1978). “Free lateral vibration of suspension bridges,” Journal of Structural Mechan-
ics, ASCE, 104(3),503. ‘

Abdel-Ghaffar, A. M. (1976). “Dynamic analysis of suspension bridge structures,” EERL 76-01, California
Institute of Technology, Pasadena, California.

Barnoski, R. L., and Maurer, J. R. (1969). “Mean-square response of simple mechanical systems to nonsta-
tionary random excitation,” Journal of Applied Mechanics, June, 221-227.

Baron, R, Arikan, M., and Hamati, R. E. (1976). “The effects of seismic disturbances on the Golden Gate
Bridge,” EERC Report No. 76-31 , Earthquake Engineering Research Center of California, Berkeley.

Castellani, A. and Felotti, P. (1986). “Lateral vibration of suspension bridges,” J. Struct. Eng., 112(9), 2169-
2173.

Caughy, T. K., and Stumpf, H. J. (1961). “'I‘ransient response of a dynamic system under random excitation,”
Journal of Applied Mechanics, December, 563-566.

Clough, R. W., and Penzien, J. (1975). Dynamics of strucuras, McGraw-I-Iill Book Company, New York.

Harichandran, R. S. (1991). “Estimating the spatial variation of earthquake ground motion from dense array
recordings,” Structural Safety, 10, 219-233.

Harichandran, R. S. and Wang, W. (1988). “Response of one- and two-span beams to spatially varying seis-
mic excitation,” Report No. MS U-ENGR-88-002, College of Engineering, Michigan State University,
East Lansing, Michigan.

Harichandran, R. S., and Vanmarcke, E. (1986). “Stochastic variation of earthquake ground motion in space
and time,” Journal of Engineering Mechanics, ASCE, 112(2), 154-174.

Hirai, A., etal. ( 1960). “Lateral stability of suspension bridge subjected to foundation-motion,” Proc. , Second
World Conference on Earthquake Engineering, Japan, Vol. II, 931-945.

IMSL (1987). IMSL User's Manual , International Mathematical and Statistical Libraries. Inc., Vol. 1, Hous-
ton, Texas.

Ito, M. (1966). “Lateral rigidity of suspension bridge,” Symposium on suspension, Lisbon, paper No. 7, 7.1-
7.8.

Jennings, P. C., Housner, G. W., and Tsai, N. C. (1968). “Simulated earthquake motions,” Earthquake Engi-
neering Research Laboratory, California Institute of Technology, Pasadena, California.

Konishi, I. and Ymada, Y. “Earthquake resistant design of long span suspension bridges,” Proc., Third World
Conference on Earthquake Engineering. New Zealand, Vol. II, pp. IV/K/l2.

Moisseif, L. S. and Leinhard, F. (1933). “Suspension bridges under the action of lateral forces,” Trans.
ASCE, 98, 1080-1109.

130

Paine, C. D. (1970). “Supplement to the final report of the chief Engineer-Golden Gate Bridge,” Golden Gate
Bridge, Highway and Transportation District.

Przemieniecki, J. S. (1968). Theory of matrix structural analysis, Dover Publications, Inc. New York.

Rubin, L. I., Abdel-Gluffar, A. M., and Stringfellow, R. G. (1983). “Earthquake response of long-span sus-
pension bridge,” Report No. 83-SM-13. Department of Civil Engineering, Princeton, NJ.

Selberg, A. (1958)., Discussion of “The lateral rigidity of suspension bridges,” by Silverman, K., Proc.,
ASCE, No. EMl, 1520-29-1520-31.

Sigbjonsson, R. and Hjorth-Hansen, E. (1981). “Along-wind Response of Suspension bridge with special ref-
erence to stiffening by horizontal cables,” Eng. Struct.. 3, 27-37.

Silveman, I. K. (1957). “The lateral rigidity of suspension bridges,” Proc., ASCE, 83(3), 1292-1-1292-17.

Strauss, J. B. (1937). “The Golden Gate Bridge,” Report to the Board of Directors of the Golden Gate Bridge
and Highway District. California.

Sweidan, B. N. (1990). “Stochastic response of Deck Arch Bridges to correlated support excitations,” Ph.D.
Thesis, Department of Civil and Environmental Engineering, Michigan State University, East Lansing,
Michigan.

Zerva, A., Ang, A. HS, and Wen, Y. K. (1988). “Lifeline response to spatially variable ground motions,”
Earthquake Engineering and Structural Dynamics, 16,361-379.

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