svegm

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November 1999

by

Ronald S. Harichandran

Professor and Chairperson

Department of Civil and Environmental EngineeringMichigan State University

East Lansing, MI 48824-1226

Phone: (517) 355-5107

Fax: (517) 432-1827

E-Mail: [email protected]

Web: http://www.msu.edu/~harichan

SPATIAL VARIATION OF EARTHQUAKEGROUND MOTION

What is it, how do we model it, andwhat are its engineering implications?

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Abstract

Observations from closely-spaced seismograph arrays since the late 1970s have shown that

earthquake ground accelerograms measured at different locations within the dimensions of typical

large engineered structures are significantly different. This has led to considerable research in the

last decade on modeling spatially varying earthquake ground motion (SVEGM) and on determin-ing its effect on the seismic response of large structures such as bridges, pipelines, dams, and so

on. A modification of the popular response spectrum method has also been developed to include

SVEGM. An overview of SVEGM, modeling approaches, methods for computing structural re-

sponses, and case studies are presented.

Table of Contents

1. What is SVEGM and is it Important?.........................................................................1

2. Causes of SVEGM......................................................................................................2

3. Measuring SVEGM ....................................................................................................2

4. Analyzing Recorded Data...........................................................................................3

5. Observations and Synthesis ........................................................................................4

6. Probabilistic Modeling of SVEGM ............................................................................6

7. Analysis of Structural Response.................................................................................6

8. Theoretical Background on Stationary RVA..............................................................7

8.1 Direct Transfer Function Approach ..........................................................8

8.2 Modal Decomposition Approach..............................................................8

9. Random Vibration Analysis using ANSYS..............................................................10

10. Response Spectrum Method .....................................................................................10

11. Response of Structures to SVEGM...........................................................................1111.1 Structures on Rigid Mat Foundations .....................................................11

11.2 Long-Span Bridges .................................................................................12

11.3 Earth Dams..............................................................................................15

11.4 Other Selected References Related to SVEGM......................................17

12. References.................................................................................................................18

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1. What is SVEGM and is it Important?

Earthquake accelerograms measured at different locations within the dimensions of an engi-

neered structure are typically different. This is SVEGM! Fig. 1 shows two accelerograms recorded

at stations separated by 200 m. In spite of the similarities, there are also some differences. For larg-

er separations the differences become more noticeable.

Current engineering practice assumes:

1. Excitations at all support points are the same; or

2. Excitations are different by only a wave propagation time delay.

i.e., Excitations at all locations are assumed to be fully coherent.

Can differences in earthquake accelerograms over the dimensions of engineered structures

be neglected? Is current engineering practice reasonable/conservative?

To answer these questions we must:

1. Measure earthquake ground accelerations at closely-spaced locations.

2. Analyze and quantify the differences in observed accelerations.3. Build suitable models for use in structural analysis.

4. Compute structural responses using models that include SVEGM and compare these with

those obtained using models that neglect SVEGM.

5. Classify the effect of SVEGM on the response of different classes of structures.

0 2 4 6 8 10

Time (sec)

-20

0

20

-20

0

20

Acceleration(gals)

Station C00

Station I06

Figure 1 Recorded accelerograms at stations 200 m apart

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2. Causes of SVEGM

1. Wave passage effect: Seismic waves arrive at different times at different stations.

2. Incoherence effect: Differences in the manner of superposition of waves (a) arriving from

an extended source, and (b) scattered by irregularities and inhomogeneities along the path

and at the site, causes a loss of coherency.

3. Local site effect: Differences in local soil conditions at each station may alter the ampli-

tude and frequency content of the bedrock motions differently.

An illustration of the scattering of seismic waves from an extended source through layered

strata and a dense soil pocket is given in Fig. 2.

3. Measuring SVEGM

1. Seismograph arrays that can record ground motions simultaneously at several locations are

required.

2. The seismographs must be synchronized.

3. For engineering purposes, the seismographs must be closely-spaced, with separations thatspan the dimensions of most engineered structures.

Desirable features of arrays are that they should:

be useful to both engineers and seismologists

record all three components of ground motion

be located in highly seismic areas and in a variety of site conditions

The configuration of the SMART 1 seismograph array located in Lotung, Taiwan is shown

in Fig. 3.

Site

Source

Path

Direct waves

Reflected wave

Figure 2 lllustration showing seismic wave propagation and scattering

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4. Analyzing Recorded Data

Recorded signals are usually treated as random time series. If and are accelero-

grams at locations l and m, then using spectral estimation techniques, we estimate

(Harichandran 1991):

1. The real-valued, positive, auto spectral density functions (SDFs), Sl(f) and S

m(f). These

characterize the power at different frequencies. Fig. 4(a) shows a typical SDF.

2. The complex-valued cross SDF, Slm(f).3. The complex-valued coherency spectrum

(1)

|lm

(f)| describes the maximum correlation between the harmonics of and at fre-

quency f. |lm

(f)|2 is called the coherence. Fig. 4(b) shows a typical coherency spectrum.

4. The real-valued phase spectrum

(2)

This is the phase change required at frequency fto achieve the correlation |lm

(f)|. i.e., It de-

scribes the lead/lag of the harmonics of and at frequency f. Fig. 4(c) shows a

typical phase spectrum that is wrapped between 90 and 90. By piecing together the seg-ments of the wrapped phase spectrum, an unwrapped phase spectrum such as that shown in

Fig. 4(d) can be obtained.

Figure 3 The SMART 1 seismograph array

ul t( ) um t( )

lm f( )Slm f( )

Sl f( )Sm f( )------------------------------=

ul t( ) um t( )

lm f( ) tan 1Im Slm f( )[ ]

Re Slm f( )[ ]---------------------------=

ul t( ) um t( )

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5. Observations and Synthesis

The various spectra are estimated for numerous (or all available) accelerogram pairs. Exten-

sive analysis of data from SMART 1 indicate that (Harichandran and Vanmarcke 1986,

Harichandran 1991):

1. The auto SDFs of accelerograms at locations within the dimensions of engineered struc-

tures are similar. i.e., Local site effect can often be neglected.

2. Typically, coherency becomes smaller as the distance between stations l and m increase.

3. Typically, coherency decreases with increasing frequency f.4. The decay of |

lm(f)| is not overly direction sensitive.

5. The gross lead/lag between signals (estimated using linear trends in the phase spectra at low

frequencies) display some deterministic features. Fig. 5 shows contours of the times at

which the seismic waves arrive at each station, relative to the center station C00. Fig. 6

shows a plot of the relative arrival times vs. the absolute separation along the direction of

propagation used to estimate the gross apparent propagation velocity.

The observations suggest the following simplifications:

1. The auto SDF at any location can be given by a point SDF S(f) estimated as the average of

all the auto SDFs.

2. The absolute coherency decay between all pairs of stations can be described by a singlefunction |(, f)|, where = separation between l and m.

3. The phase spectra can be (grossly) simplified as

(3)

where d= V./|V|2 = gross propagation time delay, and V= gross apparent propagation ve-locity vector.

f

S(f)

f

|(f)|

0

1

(f)

f0

90o

-90o f

(f)

0

Figure 4 Typical shapes of: (a) Auto SDF; (b) absolute coherency; (c) wrapped phase; (d) unwrapped phase

(a) (b)

(d)(c)

f,( ) 2d=

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To aid visualization of |(, f)|, the pair-wise coherencies lm

(f) can be smoothed using:

(4)

where i= scalar separation between stations l and m, w(x) = exp(x2/2) = a smoothing window,

and = a smoothing parameter. Figs 7 and 8 show the variation of the smoothed estimated abso-lute coherency for the radial components of two events recorded by the SMART 1 array.

Figure 5 Contours of relative arrival times for

SMART 1 Event 20

Figure 6 Plot of estimated gross propagation velocity

for SMART 1 Event 20

v f,( )

i f,( ) w i

-------------

i 1=

n

w i

-------------

i 1=

n

------------------------------------------------------=

Figure 7 Smoothed estimated coherency for radial

component of SMART 1 Event 20

Figure 8 Smoothed estimated coherency for radial


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A word of caution is in order regarding spectral estimation, which is as much an art as it is a

science. Various parameters and smoothing options must be used to obtain meaningful results. In-

correct application of the techniques can produce poor or erroneous results (Harichandran 1987a).

6. Probabilistic Modeling of SVEGM

Based on the observations, if local site effects are neglected SVEGM can be modeled as ahomogeneous random field with cross SDF

(5)

Note that the cross SDF for accelerograms at l and m is

(6)

where lm

= separation between stations l and m.

A popular functional form for S(f) is the filtered Kanai-Tajimi spectrum (Clough and

Penzien 1993)

(7)

The parameters g, f

g,

f, and f

fcontrol the shape of the spectrum. S

0is an intensity factor.

gand f

g

may be interpreted as the soil damping and soil frequency. The first term on the RHS in Eq. 7

is the Kanai-Tajimi spectrum, and the second term is a modifier that makes the mean square ground

displacement finite. Fig. 9shows typical estimated and fitted auto SDFs.

Different empirical forms have been suggested for |(, f)|. Harichandran and Vanmarcke(1986) suggested the sum of two exponentials:

(8)

in which (f) = k[1 + (f/f0)b]1/2 is the frequency-dependent spatial scale of fluctuation. Hindy and

Novak (1980), Loh (1985), Luco and Wong (1986) and others have used single exponential func-

tions, which may all be written in the form

(9)

Other more complex models have been proposed by Hao et al. (1989) and Abrahamson (1993).

Fig. 10 shows the double exponential coherency function fitted to the radial component of Event

20 recorded by the SMART 1 array (c.f. Fig. 7).

7. Analysis of Structural Response

Three techniques are available for analyzing structural response due to SVEGM:

1. Random vibration analysis (RVA).

Advantages: Consistent with probabilistic modeling. Input is specified directly in terms of

cross SDFs.

C f,( ) S f( ) f,( ) ei f,( )=

Slm f( ) C lm f,( )=

S f( )1 4g2 f fg( )2+

1 f fg( )2[ ]2 4g2 f fg( )2+

-------------------------------------------------------------------------f ff( )4

1 f ff( )2[ ]2 4f2 f ff( )2+

-------------------------------------------------------------------------- S0=

f,( ) Aexp 2 f( )-------------- 1 A A+( ) 1 A( )exp 2

f( )---------- 1 A A+( )+=

f,( ) exp f( )[ ]=

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Disadvantages: Not commonly used in practice. Including non-stationary effects is cum-

bersome. Non-linear analysis is very difficult.

2. Time history analysis.

This requires deterministic acceleration time-histories to be used as input ground motions.

The time-histories can be obtained from: (a) those measured at a suitable array; (b) by mod-

eling the seismic source and propagation of waves in an elastic medium; or (c) through sim-ulation based on the probabilistic SVEGM model.

Advantages: Can include non-stationary excitation and non-linear behavior.

Disadvantages: Results are specific to the selected excitation time histories. Used in prac-

tice only for important structures.

3. Response spectrum method.

This should include the effect of SVEGM.

Advantages: Commonly used in practice. Inherently includes non-stationarity of excita-

tion.

Disadvantages: Cannot include non-linear behavior. Is approximate.

8. Theoretical Background on Stationary RVA

The dynamic equations of motion of a structure discretized using the finite element method

may be written in the partitioned form:

(10)

where

M, C and K are mass, damping and stiffness matrices associated with the unrestrained

DOF u

Figure 9 Estimated and fitted auto SDFs for radial

components of SMART 1 Event 20

Figure 10 Fitted coherency function for radial


M MC

MCT MR

u

uR C CC

CCT CR

u

uR K KC

KCT KR

u

uR

+ +0

R

=

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MR

, CR

and KR

are mass, damping and stiffness matrices associated with the restrained

DOF uR

MC, C

Cand K

Care coupling mass, damping and stiffness matrices

R is the reaction force vector

The structure is excited by rstationary support accelerations, , l = 1, 2, , r. The sup-

port excitations are characterized by their cross SDF, Slm

(), where l and m denote the indices onthe support accelerations.

8.1 Direct Transfer Function Approach

If the transfer (frequency response) function relating the lth harmonic excitation to a given

displacement, strain or stress response z is denoted by for support excitations, then the

mean-square response is given by (Newland 1984)

(11)

in which is the complex conjugate of . The RVA capability of many software pack-

ages (e.g., I-DEAS Master Series 1.3) is limited to this. Versions of ANSYS prior to 5.0 also used

this approach, and were even more restrictive because only a single excitation was allowed. Al-

though simple to implement, the main shortcoming of this method is that transfer functions must

be obtained for each excitation and each response through harmonic analyses, and separate numer-

ical integrations must be performed for each physical response. Hence, the method is not efficient

if a very large number of responses are required.

8.2 Modal Decomposition Approach

The graphical display of contours of root-mean square (r.m.s.) responses over an entire struc-

ture or component requires the computation of numerous r.m.s. responses, which is most efficient-ly performed using modal decomposition (Harichandran and Wang 1990b).

The displacements are decomposed into dynamic and pseudo-static components:

(12)

The pseudo-static displacements are obtained from Eq. 10 by neglecting inertia and damping forc-

es, and are

(13)

in which A = K1KC

= matrix of influence coefficients whose ith column constitutes the unre-

strained nodal displacements due to a unit value of the ith restrained support displacement. Substi-tuting Eqs. 12 and 13 into the equation of motion yields

(14)

The damping forces neglected on the RHS of Eq. 14 are small compared to the inertia forces, and

are identically zero for stiffness proportional damping.

uRl

t( )

H l ( )

z2 H l* ( )H m ( )Slm ( )

m 1=

r

l 1=

r

d0

=

H l* ( ) H l ( )

u ud us+=

us K1 KCuR AuR= =

Mud Cud Kud+ + MA MC+( ) uR CA CC+( ) uR MA MC( )uR=

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The undamped free vibration modes of the restrained structure are used to uncouple the eqs.

of motion into

(15)

where

(16)

is a modal participation vector, j

= mode shape vector, j

= modal circular frequency (in radi-

ans), and j

= modal damping ratio.

Any displacement, strain or stress response z is separated into dynamic and pseudo-static

parts:

(17)

The variance (or mean-square since the mean is zero)) of the response is

(18)

where

(19)

(20)

(21)

In Eqs. 19 to 21

n = no. of modes, r= no. of support DOF;

j

= response z from the jth mode (for the ith displacement response j

ij);

Bl= response z due to a unit displacement of support DOF l (for the ith displacement re-

sponse Bl A

il);

lj = lth element of the participation vector j; H

j() = (

j2 2 + 2i

jj)1 = jth modal frequency response function; and

Slm

() = cross SDF of accelerations along DOF l and m.

The efficiency of the modal analysis method lies in the fact that the integrals in Eqs. 19 to 21

are independentof the response quantity z, and need to be computed and stored only once. In ad-

dition, for some forms of excitation cross SDFs commonly used in practice, closed-form results

can be used to compute the integrals in a fraction of the time required for numerical integration

Y j 2jjYj j2Yj+ + jTuR=

jMA MC+( )Tj

jTMj------------------------------------------=

z zd zs+=

z2 zd2 zs

2 2Cov zs zd,( )+ +=

zd2 jk ljmkHj* ( )Hk ( )Slm ( )

m 1=

r

l 1=

r

d

k 1=

n

j 1=

n

=

zs BlBm1

4------Slm ( ) d

0

m 1=

r

l 1=

r

=

Cov zs zd,( ) jBl1

2------ mjH j ( )Slm ( )

m 1=

r

d0

l 1=

r

j 1=

n

=

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(Harichandran 1992). Previously computed integrals can then be re-used to efficiently compute

mean-square values for a very large number of responses (Harichandran 1993).

The mean peak response may be obtained through

(22)

in which pz = peak factor for given duration s of stationary response (Der Kiureghian 1980).

9. Random Vibration Analysis using ANSYS

The author has collaborated with ANSYS, Inc., to implement extensive random vibration

analysis capability in the popular ANSYS finite element package. Version 5.0 of ANSYS is capa-

ble of performing random vibration analysis under SVEGM.

Coherency functions supported are:

Fully coherent excitations: |lm

(f)| = 1

Uncorrelated excitations: |lm

(f)| = 0

Linear variation:

Exponential variation:

10. Response Spectrum Method

Der Kiureghian and Neuenhofer (1992) have developed the most comprehensive response

spectrum method, and this is based on random vibration theory. The method accounts for cross cor-

relations between support motions and between different modes of vibration (the latter being im-portant for closely-spaced modes).

We denote the displacement response spectrum by D(, ), and the maximum ground dis-placement by u

g,max. Note that u

g,max= D(0, ). Neglecting local site effects (i.e., assuming S

l() =

Sm()), the mean peak response is approximated by

(23)

The only new parameters are the correlation coefficients , and which depend

on the response spectrum, coherency function and propagation time delay, and must be obtained

by numerical integration of the following expressions:

E max z t( )[ ] pzz=

lm1

max

lm

lm f( ) exp flm( )

[ ]=

E max z t( )[ ] jkljmkjklmd D j j,( )D k k,( )m 1=

r

l 1=

r

k 1=

n

j 1=

n

BlBmlms ug max,2

m 1=

r

l 1=

r

2 Bljmjjlmsd ug max, D j j,( ) ]1 2m 1=

r

l 1=

r

j 1=

n

+ +

jklmd lms jlmsd

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(24)

(25)

(26)

(27)

(28)

The cross SDF Slm

() is obtained from the SVEGM model

(29)

with S() being the auto SDF that is equivalentto the design response spectrum D(, ), |lm

()|is an appropriate coherency function for the site, and dis the wave propagation delay from l to m.

To first-order

, 0 (30)

in which ps() = peak factor for oscillator with frequency and response duration s

(Der Kiureghian 1980), and is the modal damping ratio assumed to be the same for all modes.Fortunately p

s() is not too sensitive to the parameters s and and may be assumed to be constant

if necessary.

11. Response of Structures to SVEGM

11.1 Structures on Rigid Mat Foundations

A simplfied model of a structure on a non-embedded rigid mat foundation is shown in

Fig. 11. Harichandran (1987b) investigated the effect of SVEGM on the response of the oscillator.The foundation averages the ground accelerations along its bottom surface to produce

an effective acceleration at the base of the structure of

(31)

jklmd1

jk----------- Hj

* ( )Hk ( )Slm ( )d

=

lms 1

ug2--------

1

4------S

lm ( )d

=

jlmsd1

ugj-------------

1

2------Hj ( )Slm ( ) d

=

j2 Hj ( ) 2S ( )d

=

ug21

4------ S ( )d

=

Slm ( ) S ( ) lm ( ) e id=

S ( ) 23

-------------D ,( )

ps ( )------------------

2

ug x y t, ,( )

ueff t( )1

12------------ ug x y t, ,( )dxdy

0

2

0

1

=

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We define the base reduction factorwhich describes the effect of base averaging on the oscillator

response as

(32)

The BRF for square foundations of area 2,500 m2

and 5,000 m2

are shown in Fig. 12 for ap-parent propagation velocities of 1,000 m/s and 3,600 m/s along the x direction. The following con-

clusions emerged from this study:

1. Base averaging always reduces the effective excitation, with high frequencies being filtered

out more severely than low frequencies.

2. Response of stiff structures is reduced more than the response of flexible structures.

3. The response reduction is more sensitive to the aspect ratio of rectangular foundations for

smaller apparent wave propagation velocities.

11.2 Long-Span Bridges

Harichandran, Hawwari and Sweidan (1996) investigated the effect of SVEGM on: (a) the

lateral response of the Golden Gate suspension bridge (GGB) in California with a 4,200 ft center

span and 1,125 ft side spans; and (b) the longitudinal and lateral responses of the 1,700 ft New Riv-

er Gorge arch bridge (NRGB) in West Virginia, and the 700 ft Cold Spring Canyon arch bridge

(CSCB) in California.

Two-dimensional finite element models were used for all the bridges. For the suspension

bridge, the model developed by Abdel-Ghaffar and Rubin (1983) was used, with the corrections

made by Castellani and Felloti (1986). For the arch bridges, the models developed by Dusseau and

Wen (1989) were used.

Linear stationary random vibration analysis was performed. Total mean-square displacement

and force responses were computed from the dynamic and static variances, and the (possibly neg-

ative) covariances between the dynamic and static responses. Due to the flexibility of the bridges,

the dynamic variances were the most dominant, typically contributing between 80% and 110% to

the arch bridge member forces, and about 100% to the suspension bridge member forces.

1

2

x

y

Figure 11 Idealized oscillator on a rigid mat

foundation

Figure 12 BRFs for two different foundation sizes and

two different apparent propagation velocities

BRFmax. response of oscillator accounting for SVEGM

max. response of oscillator for identical excitation---------------------------------------------------------------------------------------------------------------------------=

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Suspension Bridge Lateral Responses

Figs. 13 and 14 show schematics of the finite element models used for the side and main

spans of the Golden Gate Bridge, and the node numbering scheme. The main towers are extremely

stiff compared to the suspended structure, and were assumed to be rigid. As a result, the excitations

at the cable and deck supports at each end were assumed to be identical, and the side and main

spans were analyzed independently. The hangers were considered to be inextensible and the deck

was modeled using beam elements. All elements were 50 ft long in the longitudinal direction. Con-sistent mass matrices were used in the analyses.

Table 1 shows force response ratios for the two bridges due to longitudinal and lateral exci-

tations, respectively. The ratios were computed by dividing each mean-square response due to

identical and delayed excitation by the corresponding mean-square response due to general exci-

tation. The maximum and minimum ratios that were obtained from all deck and cable members,

respectively, are shown in the tables, and the node numbers at which the extreme ratios occur are

shown within parentheses. The increase in cable tension due to the excitations was small, and

hence for the cable the response ratios are given only for displacements. The ratios indicate how

close responses computed using the more common excitation types are when compared to respons-

es due to the general spatially varying ground motion model, and indicate the following trends:

1. The use of identical excitations significantly over-estimates the responses at some locationsand under-estimates the responses at others, the relative deviations being more severe for

the longer main span. The shear near the mid-spans is drastically under-estimated because

anti-symmetric modes are not excited by identical excitations.

2. The use of delayed excitations gives acceptable results for the side span, but shows greater

deviations for the main span in which the deck moment and shear are sometimes signifi-

cantly under-estimated. This indicates that the loss of correlation is important for long span

suspension bridges.

1 3 5

24

6

161 163 165

162164

166

81 83 85

82 84 86

Figure 13 Model of GGB side span Figure 14 Model of GGB main span

1 3 5

2 46

23 25 27

2426

28

13 15 17

1416

18

TABLE 1 LATERAL RESPONSE RATIOS FOR GGB SIDE AND MAIN SPANS

Ratio

Range Side Span Main Span

Deck Nodes Cable Deck Nodes Cable

Moment Shear Displ. Displ. Moment Shear Displ. Displ.

Max. 1.23 (15) 1.41 (5) 1.17 (13) 1.18 (14) 2.24 (83) 1.54 (93) 1.54 (83) 1.84 (84)

Min. 0.80 (23) 0.00 (13) 1.04 (27) 0.98 (28) 0.37 (69) 0.04 (81) 0.84 (23) 0.78 (100)

Max. 1.09 (17) 1.03 (27) 1.11 (15) 1.11 (16) 1.16 (129) 1.16 (117) 1.22 (89) 1.21 (96)

Min. 0.96 (27) 0.85 (19) 1.05 (1) 1.01 (2) 0.72 (115) 0.78 (131) 0.97 (153) 0.99 (6)

IdenticalGeneral---------------------

DelayedGeneral--------------------

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Arch Bridge Responses

Figs. 15 and 16 show schematics of the finite element models used for the two arch bridges

and member numbering along the decks and arches. Beam elements with additional warping d.o.f.

were used to model the decks and end towers. Equivalent properties for each element were derived

from the complex arrangement of members in the actual bridges. The columns connecting the

decks to the arches and the diagonal bracing were modeled as struts, with the former assumed to

be inextensible. Bracing elements in the CSCB were cables. Lumped mass matrices were used in

the analyses. At each end, identical excitations were used at the deck, tower and arch supports.

Tables 2 and 3 show force response ratios for the two bridges due to longitudinal and lateralexcitations, respectively. The ratios were computed by dividing each mean-square response due to

identical and delayed excitation by the corresponding mean-square response due to general exci-

tation. The maximum and minimum ratios that were obtained from all deck, arch and bracing mem-

bers, respectively, are shown in the tables, and the member numbers at which the extreme ratios

occur are shown within parentheses. The response ratios for the two bridges indicate the following

trends:

1. For the longitudinal response, the use of identical excitations severely under-estimates the

axial force in all arch members, and the moment and shear in some deck and arch members;

and over-estimates the axial force in the deck and bracing.

2. For the longitudinal response, the use of delayed excitations yields acceptable results for

most members. However, for the NRGB, the shear in some arch members and the axialforce in the bracing is significantly under-estimated.

3. For the lateral response, the use of identical excitations over-estimates the forces in some

members and under-estimates them in others. The moments in some arch members are

quite significantly under-estimated.

1

2 3 4 5 6 7 8 9 10 11 12 1 3 14

34

35

3637

38 3940 41 42 43

4445

46

47

1

2 3 4 5 6 7 8 9 10 11

20

21

2223 24

4344

45

46

47

25

Figure 15 Model of NRG bridge Figure 16 Model of CSC bridge

TABLE 2 LONGITUDINAL FORCE RESPONSE RATIOS FOR ARCH BRIDGES

R

atio

Range

BridgeDeck Members Arch Members Bracing

Moment Shear Axial Moment Shear Axial Axial

Max. NRGB 0.87 (5) 1.03 (5) 2.03 (6) 1.19 (40) 2.69 (39) 0.15 (34) 2.02

CSCB 0.70 (7) 1.61 (6) 1.30 (10) 0.72 (26) 1.60 (25) 0.06 (30) 0.95

Min. NRGB 0.27 (1) 0.27 (1) 1.66 (1) 0.10 (39) 0.11 (38) 0.06 (40) 1.82

CSCB 0.07 (5) 0.03 (5) 1.23 (1) 0.08 (24) 0.03 (24) 0.01 (25)

Max. NRGB 1.22 (4) 1.22 (3) 1.09 (1) 1.18 (34) 1.22 (38) 1.20 (40) 0.91

CSCB 1.10 (8) 1.14 (10) 0.98 (1) 1.10 (27) 1.14 (29) 1.16 (all) 0.97

Min. NRGB 1.07 (5) 1.09 (5) 0.85 (6) 0.93 (40) 0.68 (39) 1.19 (34) 0.83

CSCB 0.94 (2) 0.92 (6) 0.97 (10) 0.93 (28) 0.92 (25) 1.16 (all)

Identical

General

---------------------

Delayed

General

--------------------

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4. For the lateral response, the use of delayed excitations over-estimates the forces in some

members and under-estimates them in others. The level and number of members for which

the forces are under-estimated is greater for the shorter and stiffer CSC bridge.

General Conclusions

The use of identical excitations is in general unacceptable for these long-span bridges. The

use of delayed excitations is acceptable for the longitudinal response of short arch bridges, and the

lateral response of short suspension bridge side spans; however, it is unacceptable for the longitu-

dinal response of long arch bridges, the lateral response of short and long arch bridges, and the lat-

eral response of long suspension bridge main spans.

11.3 Earth Dams

Chen and Harichandran (1996) studied the effects of SVEGM on the Santa Felicia earth dam

located in Southern California. Fig. 17 shows the cross section of the dam. A 3-D finite elementmodel of the dam was used for the analysis. The contours in Figs. 18 and 19 show the variation of

the mean plus three standard deviation values of the maximum shear stress (max

) along the base

and mid-length cross section due to general SVEGM, identical and delayed (wave propagation

only) excitations. While the magnitude and distribution ofmax

is very similar for identical and de-

TABLE 3 LATERAL FORCE RESPONSE RATIOS FOR ARCH BRIDGES

Ratio

Range

BridgeDeck Members Arch Members

Moment Shear Torsion W. Mom. Moment Shear Torsion

Max. NRGB 1.40 (8) 1.70 (13) 1.70 (7) 1.27 (6) 1.31 (45) 1.07 (45) 2.37 (40)

CSCB 1.64 (4) 1.56 (5) 1.46 (4) 1.25 (6) 1.10 (23) 1.37 (23) 1.57 (25)

Min. NRGB 0.88 (3) 0.74 (11) 0.59 (9) 0.69 (11) 0.38 (40) 0.63 (40) 0.89 (36)

CSCB 0.75 (10) 0.70 (6) 0.71 (2) 0.81 (7) 0.40 (25) 0.13 (25) 0.83 (28)

Max. NRGB 1.14 (7) 1.12 (11) 1.05 (7) 1.13 (7) 1.12 (40) 1.09 (44) 1.05 (36)

CSCB 2.67 (6) 1.11 (10) 1.30 (2) 1.21 (9) 1.72 (29) 1.30 (26) 1.22 (20)

Min. NRGB 0.92 (11) 0.78 (9) 0.69 (10) 0.84 (10) 0.78 (45) 0.89 (45) 0.84 (43)

CSCB 0.62 (8) 0.70 (6) 0.73 (5) 0.88 (8) 0.88 (23) 0.79 (27) 0.75 (26)

Identical

Gene

ral

---------------------

Delayed

General

--------------------

ImperviousCore

PerviousShell

PerviousShell

Existing Stream GravelsExisting Stream Gravels

Bedrock

Upstream Downstream

y

x

0.33:1 4:1

2.25:1 2:1

3:1

Figure 17 Cross section of the Santa Felicia earth dam

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FRAME OF REF: GLOBAL

DISPLACEMENT - Z MIN: 507.72 MAX: 38806.00

507.72

8000.00

15500.00

23000.00

30500.00

38806.001.86

1.49

1.12

0.76

0.39

0.02

(a)

:


180.11

2223.89

4267.67

6311.44

8355.22

10399.000.50

0.40

0.30

0.20

0.11

0.01

(b)

:


204.75

2223.89

4267.67

6311.44

8355.22

10669.00

0.51

0.40

0.30

0.20

0.11

0.01

(c)

Figure 18 +3 contours ofmax

(MPa) at thebase for: (a) general, (b) identical, and (c) delayed excitations

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layed excitations, it is markedly different within the stiff gravel streambed for general SVEGM.

General conclusions that emerged were that:

1. SVEGM significantly increases the maximum shear stress in the stiff gravel streambed,

mostly due to the incoherence.

2. The wave passage effect is not as significant as coherency loss for the SVEGM model con-

sidered.

3. For displacement and maximum shear strain responses, and for maximum shear stresses

within the core, the use of identical ground motion yields slightly conservative results and

is acceptable.

4. A preliminary reliability analysis indicates that a larger variety of sliding failures may be

possible under SVEGM than under identical excitation.

11.4 Other Selected References Related to SVEGM

The modeling of SVEGM, techniques for analyzing structural response excited by it, its ef-

fect on various types of structures, and its simulation, have been studied by several investigators

and selected references are provided here:

Analysis and modeling of ground motion(Harichandran 1987a, Loh and Yeh 1988, Zerva

and Shinozuka 1991, Spudich 1994, Boissieres and Vanmarcke 1995a, Boissieres and

(a)

(b)

(c)

0.50

0.40

0.30

0.20

0.11

0.01

Figure 19 +3 contours ofmax

(MPa) on the mid-length cross section for: (a) general, (b) identical, and

(c) delayed excitations

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Vanmarcke 1995b, Chiu et al. 1995, Der Kiureghian 1996a, Der Kiureghian 1996b,

Nakamura 1996)

Techniques for analyzing structural response (DebChaudhury and Gazis 1988, Yamamura

and Tanaka 1990, Heredia-Zavoni and Vanmarcke 1994, Heredia-Zavoni et al. 1996)

Response spectrum techniques (Berrah and Kausel 1993, Zembaty and Krenk 1994)

Response of beam-like structures (Harichandran and Wang 1988, Zerva et al. 1988, Dattaand Mashaly 1990, Harichandran and Wang 1990b, Harichandran and Wang 1990a,

Zerva 1990, Zerva 1991)

Response of bridges (Abdel-Ghaffar and Rubin 1982, Wilson and Jennings 1985,

Zerva 1988, Loh and Lee 1990, Nazmy and Abdel-Ghaffar 1992, Hao 1993, Hao 1994,

Nazmy and Konidaris 1994)

Response of buildings (Hahn and Liu 1994, Hao and Duan 1995, Herdia-Zavoni and

Barranco 1996, Hao 1997)

Response of dams (Haroun and Abdel-Hafiz 1987, Novak and Suen 1987, Zhang and

Chopra 1991)

Response of foundations; soil structure interaction (Luco and Wong 1986,Harichandran 1987b, Luco and Mita 1987, Veletsos and Prasad 1989)

Response of transmission lines (Ghobarah et al. 1996)

Simulation of SVEGM (Shinozuka and Jan 1972, Wittig and Sinha 1975,

Abrahamson 1992, Zerva 1992, Ramadan and Novak 1993, Vanmarcke et al. 1993, Ra-

madan and Novak 1994)

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