November 1999

by

Ronald S. HarichandranProfessor and Chairperson

Department of Civil and Environmental EngineeringMichigan State University

East Lansing, MI 48824-1226

Phone: (517) 355-5107Fax: (517) 432-1827

E-Mail: harichan@egr.msu.eduWeb: http://www.msu.edu/~harichan

SPATIAL VARIATION OF EARTHQUAKEGROUND MOTION

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

Abstract

Observations from closely-spaced seismograph arrays since the late 1970’s have shown thatearthquake ground accelerograms measured at different locations within the dimensions of typicallarge engineered structures are significantly different. This has led to considerable research in thelast 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 soon. A modification of the popular response spectrum method has also been developed to includeSVEGM. An overview of SVEGM, modeling approaches, methods for computing structural re-sponses, and case studies are presented.

Table of Contents1. What is SVEGM and is it Important?.........................................................................12. Causes of SVEGM......................................................................................................23. Measuring SVEGM ....................................................................................................24. Analyzing Recorded Data...........................................................................................35. Observations and Synthesis ........................................................................................46. Probabilistic Modeling of SVEGM ............................................................................67. Analysis of Structural Response .................................................................................68. Theoretical Background on Stationary RVA..............................................................7

8.1 Direct Transfer Function Approach..........................................................88.2 Modal Decomposition Approach..............................................................8

9. Random Vibration Analysis using ANSYS..............................................................1010. Response Spectrum Method .....................................................................................1011. Response of Structures to SVEGM...........................................................................11

11.1 Structures on Rigid Mat Foundations .....................................................1111.2 Long-Span Bridges .................................................................................1211.3 Earth Dams..............................................................................................1511.4 Other Selected References Related to SVEGM......................................17

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

engi-rded

or larg-

uctures

e with

1. What is SVEGM and is it Important?

Earthquake accelerograms measured at different locations within the dimensions of anneered structure are typically different. This is SVEGM! Fig. 1 shows two accelerograms recoat stations separated by 200 m. In spite of the similarities, there are also some differences. Fer 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 fullycoherent.

Can differences in earthquake accelerograms over the dimensions of engineered strbe 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 thesthose 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

Acc

eler

atio

n(g

als)

Station C00

Station I06

Figure 1 Recorded accelerograms at stations 200 m apart

1

me path

pli-

ered

ns are

ns that

own

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 froan extended source, and (b) scattered by irregularities and inhomogeneities along thand at the site, causes a loss of coherency.

3. “Local” site effect: Differences in local soil conditions at each station may alter the amtude and frequency content of the bedrock motions differently.

An illustration of the scattering of seismic waves from an extended source through laystrata and a dense soil pocket is given in Fig. 2.

3. Measuring SVEGM

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

2. The seismographs must be synchronized.

3. For engineering purposes, the seismographs must be closely-spaced, with separatiospan 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 shin Fig. 3.

Site

Source

Path

Direct waves

Reflected wave

Figure 2 lllustration showing seismic wave propagation and scattering

2

elero-ate

t fre-

-hown in

4. Analyzing Recorded Data

Recorded signals are usually treated as random time series. If and are accgrams at locationsl and m, then using spectral estimation techniques, we estim(Harichandran 1991):

1. The real-valued, positive,auto spectral density functions(SDFs),Sl(f) andSm(f). Thesecharacterize the power at different frequencies. Fig. 4(a) shows a typical SDF.

2. The complex-valuedcross SDF, Slm(f).

3. The complex-valuedcoherency spectrum

(1)

|γlm(f)| describes the maximum correlation between the harmonics of and aquencyf. |γlm(f)|2 is called thecoherence. Fig. 4(b) shows a typical coherency spectrum.

4. The real-valued phase spectrum

(2)

This is the phase change required at frequencyf to achieve the correlation |γlm(f)|. i.e., It de-scribes the lead/lag of the harmonics of and at frequencyf. Fig. 4(c) shows atypical phase spectrum that is wrapped between−90° and 90°. By piecing together the segments of the wrapped phase spectrum, an unwrapped phase spectrum such as that sFig. 4(d) can be obtained.

Figure 3 The SMART 1 seismograph array

ul t( ) um t( )

γ lm f( )Slm f( )

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

ul t( ) um t( )

φlm f( ) tan 1–Im Slm f( )[ ]Re Slm f( )[ ]---------------------------=

ul t( ) um t( )

3

Exten-986,

struc-

a at lowes atig. 6

on of

f

single

-

ase

5. Observations and Synthesis

The various spectra are estimated for numerous (or all available) accelerogram pairs.sive analysis of data from SMART 1 indicate that (Harichandran and Vanmarcke 1Harichandran 1991):

1. The auto SDFs of accelerograms at locations within the dimensions of engineeredtures are similar. i.e., Local site effect can often be neglected.

2. Typically, coherency becomes smaller as the distance between stationsl andm increase.

3. Typically, coherency decreases with increasing frequencyf.

4. The decay of |γlm(f)| is not overly direction sensitive.

5. The gross lead/lag between signals (estimated using linear trends in the phase spectrfrequencies) display some deterministic features. Fig. 5 shows contours of the timwhich the seismic waves arrive at each station, relative to the center station C00. Fshows a plot of the relative arrival times vs. the absolute separation along the directipropagation used to estimate the gross apparent propagation velocity.

The observations suggest the following simplifications:

1. The auto SDF at any location can be given bya point SDF S(f) estimated as the average oall the auto SDFs.

2. The absolute coherency decay between all pairs of stations can be described by afunction |γ(ν, f)|, whereν = separation betweenl andm.

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

(3)

whered = V.ν/|V|2 = gross propagation time delay, andV = gross apparent propagation velocity vector.

f

S(f)

f

|γ(f)|

0

1

φ(f)

f0

90o

-90o f

φ(f)

0

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

(a) (b)

(d)(c)

φ ν f,( ) 2πd–=

4

abso-

To aid visualization of |γ(ν, f)|, the pair-wise coherenciesγlm(f) can be smoothed using:

(4)

whereνi = scalar separation between stationsl andm, w(x) = exp(–x2/2) = a smoothing window,and∆ν = a smoothing parameter. Figs 7 and 8 show the variation of the smoothed estimatedlute coherency for the radial components of two events recorded by the SMART 1 array.

Figure 5 Contours of relative arrival times forSMART 1 Event 20

Figure 6 Plot of estimated gross propagation velocityfor SMART 1 Event 20

γ v f,( )

γ νi f,( ) wν νi–

∆ν-------------

i 1=

n

∑

wν νi–

∆ν-------------

i 1=

n

∑------------------------------------------------------=

Figure 7 Smoothed estimated coherency for radialcomponent of SMART 1 Event 20

Figure 8 Smoothed estimated coherency for radialcomponent of SMART 1 Event 24

5

it is alts. In-987a).

d as a

d

q. 7ground

e

ndfunc-

1993).vent

of

A word of caution is in order regarding spectral estimation, which is as much an art asscience. Various parameters and smoothing options must be used to obtain meaningful resucorrect application of the techniques can produce poor or erroneous results (Harichandran 1

6. Probabilistic Modeling of SVEGM

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

(5)

Note that the cross SDF for accelerograms atl andm is

(6)

whereνlm = separation between stationsl andm.

A popular functional form forS(f) is the filtered Kanai-Tajimi spectrum (Clough anPenzien 1993)

(7)

The parametersξg, fg, ξf, andff control the shape of the spectrum.S0 is an intensity factor.ξg andfgmay be interpreted as the “soil damping” and “soil frequency.” The first term on the RHS in Eis the Kanai-Tajimi spectrum, and the second term is a modifier that makes the mean squaredisplacement finite. Fig. 9 shows typical estimated and fitted auto SDFs.

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

(8)

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

Novak (1980), Loh (1985), Luco and Wong (1986) and others have used single exponentialtions, which may all be written in the form

(9)

Other more complex models have been proposed by Hao et al. (1989) and Abrahamson (Fig. 10 shows the double exponential coherency function fitted to the radial component of E20 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 termscross SDFs.

C ν f,( ) S f( ) γ ν f,( ) eiφ ν f,( )

=

Slm f( ) C νlm f,( )=

S f( )1 4ξg

2 f f g⁄( )2+

1 f f g⁄( )2–[ ]2 4ξg2 f f g⁄( )2+

-------------------------------------------------------------------------f f f⁄( )4

1 f f f⁄( )2–[ ]2 4ξ f2 f f f⁄( )2+

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

γ ν f,( ) Aexp2ν

αθ f( )-------------- 1 A– αA+( )– 1 A–( )exp

2νθ f( )---------- 1 A– αA+( )–+=

γ ν f,( ) exp λ fν( )µ–[ ]=

6

m-

tions.mod-sim-

rac-

ita-

thod

ained

Disadvantages:Not commonly used in practice. Including non-stationary effects is cubersome. Non-linear analysis is very difficult.

2. Time history analysis.This requires deterministic acceleration time-histories to be used as input ground moThe time-histories can be obtained from: (a) those measured at a suitable array; (b) byeling the seismic source and propagation of waves in an elastic medium; or (c) throughulation 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 ptice 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 exction.

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 memay be written in the partitioned form:

(10)

where

• M , C andK are mass, damping and stiffness matrices associated with the unrestrDOF u

Figure 9 Estimated and fitted auto SDFs for radialcomponents of SMART 1 Event 20

Figure 10 Fitted coherency function for radialcomponent of SMART 1 Event 20

M M C

M CT M R

u

uR C CC

CCT CR

u

uR K K C

K CT K R

u

uR

+ +0

R

=

7

ained

e

ck-usedd. Al-mustnumer-

fficient

struc-ficient-

g forc-

-ti-

s, and

• MR, CR andKR are mass, damping and stiffness matrices associated with the restrDOF uR

• MC, CC andKC are coupling mass, damping and stiffness matrices

• R is the reaction force vector

The structure is excited byr stationary support accelerations, ,l = 1, 2, …,r. The sup-port excitations are characterized by their cross SDF,Slm(ω), wherel andm denote the indices onthe support accelerations.

8.1 Direct Transfer Function Approach

If the transfer (frequency response) function relating thelth harmonic excitation to a givendisplacement, strain or stress responsez is denoted by for support excitations, then thmean-square response is given by (Newland 1984)

(11)

in which is the complex conjugate of . The RVA capability of many software paages (e.g., I-DEAS Master Series 1.3) is limited to this. Versions of ANSYS prior to 5.0 alsothis approach, and were even more restrictive because only a single excitation was allowethough simple to implement, the main shortcoming of this method is that transfer functionsbe obtained for each excitation and each response through harmonic analyses, and separateical integrations must be performed for each physical response. Hence, the method is not eif 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 entireture or component requires the computation of numerous r.m.s. responses, which is most efly 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 dampines, and are

(13)

in which A = −K−1KC = matrix of influence coefficients whoseith column constitutes the unrestrained nodal displacements due to a unit value of theith restrained support displacement. Substuting 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 forceare identically zero for stiffness proportional damping.

uRlt( )

H l ω( )

σz2 H l

* ω( )Hm ω( )Slm ω( )m 1=

r

∑l 1=

r

∑

ωd

0

∞

∫=

H l* ω( ) H l ω( )

u ud us+=

us K 1– K CuR– AuR= =

Mu d Cud Ku d+ + MA M C+( )– uR CA CC+( )– uR MA M C–( ) uR≈=

8

he eqs.

tic

to 21ad-sultsation

The undamped free vibration modes of the restrained structure are used to uncouple tof motion into

(15)

where

(16)

is a modal participation vector,φφj = mode shape vector,ωj = modal circular frequency (in radi-ans), andξj = modal damping ratio.

Any displacement, strain or stress responsez is separated into dynamic and pseudo-staparts:

(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;

• ψj = responsez from thejth mode (for theith displacement responseψj ≡ φij);

• Bl = responsez due to a unit displacement of support DOFl (for the ith displacement re-sponseBl ≡ Ail);

• Γlj = lth element of the participation vectorΓj;Γ• Hj(ω) = (ωj

2 − ω2 + 2iωjξjω)−1 = jth modal frequency response function; and

• Slm(ω) = cross SDF of accelerations along DOFl andm.

The efficiency of the modal analysis method lies in the fact that the integrals in Eqs. 19areindependentof the response quantityz, and need to be computed and stored only once. Indition, for some forms of excitation cross SDFs commonly used in practice, closed-form recan be used to compute the integrals in a fraction of the time required for numerical integr

Y j 2ξ jω j Y j ω j2Yj+ + ΓΓ j

T uR=

ΓΓ j

MA M C+( )T φφ j

φφ jTM φ jφ

------------------------------------------–=

z zd zs+=

σz2 σzd

2 σzs

2 2Cov zs zd,( )+ +=

σzd

2 ψ jψk Γlj ΓmkH j* ω( )Hk ω( )Slm ω( )

m 1=

r

∑l 1=

r

∑

ωd

∞–

∞

∫k 1=

n

∑j 1=

n

∑=

σzsBlBm

1ω4------Slm ω( ) ωd

0

∞

∫m 1=

r

∑l 1=

r

∑=

Cov zs zd,( ) ψ jBl1

ω2------ ΓmjH j ω( )Slm ω( )

m 1=

r

∑ ωd

0

∞

∫l 1=

r

∑j 1=

n

∑–=

9

pute

tionapa-

ponsess cor-g im-

-

endbtained

(Harichandran 1992). Previously computed integrals can then be re-used to efficiently commean-square values for a very large number of responses (Harichandran 1993).

The mean peak response may be obtained through

(22)

in whichpz = peak factor for given durations of stationary response (Der Kiureghian 1980).

9. Random Vibration Analysis using ANSYS

The author has collaborated with ANSYS, Inc., to implement extensive random vibraanalysis capability in the popular ANSYS finite element package. Version 5.0 of ANSYS is cble of performing random vibration analysis under SVEGM.

Coherency functions supported are:

• Fully coherent excitations: |γlm(f)| = 1

• Uncorrelated excitations:|γlm(f)| = 0

• Linear variation:

• Exponential variation:

10. Response Spectrum Method

Der Kiureghian and Neuenhofer (1992) have developed the most comprehensive resspectrum method, and this is based on random vibration theory. The method accounts for crorelations between support motions and between different modes of vibration (the latter beinportant for closely-spaced modes).

We denote the displacement response spectrum byD(ω, ξ), and the maximum ground displacement byug,max. Note thatug,max= D(0, ξ). Neglecting local site effects (i.e., assumingSl(ω) =Sm(ω)), the mean peak response is approximated by

(23)

The only new parameters are the correlation coefficients , and which depon the response spectrum, coherency function and propagation time delay, and must be oby numerical integration of the following expressions:

E maxz t( )[ ] pzσz=

γlm

1

νmaxνlm

γ lm f( ) exp λ f νlm( )µ–[ ]=

E maxz t( )[ ] ψ jψkΓlj Γmkρ jklmd D ω j ξ j,( )D ωk ξk,( )

m 1=

r

∑l 1=

r

∑k 1=

n

∑j 1=

n

∑≈

BlBmρlms ug max,

2

m 1=

r

∑l 1=

r

∑ 2 Blψ jΓmjρ jlmsd ug max, D ω j ξ j,( ) ]1 2⁄

m 1=

r

∑l 1=

r

∑j 1=

n

∑+ +

ρ jklmd ρlms ρ jlmsd

10

des.nt

n inllator.oduce

(24)

(25)

(26)

(27)

(28)

The cross SDFSlm(ω) is obtained from the SVEGM model

(29)

with S(ω) being the auto SDF that isequivalentto the design response spectrumD(ω, ξ), |γlm(ω)|is an appropriate coherency function for the site, andd is the wave propagation delay froml to m.To first-order

, ω ≥ 0 (30)

in which ps(ω) = peak factor for oscillator with frequencyω and response durations(Der Kiureghian 1980), andξ is the modal damping ratio assumed to be the same for all moFortunatelyps(ω) is not too sensitive to the parameterssandω and may be assumed to be constaif 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 showFig. 11. Harichandran (1987b) investigated the effect of SVEGM on the response of the osciThe foundation averages the ground accelerations along its bottom surface to pran effective acceleration at the base of the structure of

(31)

ρ jklmd 1

σ jσk----------- H j

* ω( )Hk ω( )Slm ω( )dω∞–

∞

∫=

ρlms 1

σug

2-------- 1

ω4------Slm ω( )dω

∞–

∞

∫=

ρ jlmsd 1

σugσ j

------------- 1ω2------H j ω( )Slm ω( ) ωd

∞–

∞

∫–=

σ j2 H j ω( ) 2S ω( )dω

∞–

∞

∫=

σug

2 1ω4------ S ω( )dω

∞–

∞

∫=

Slm ω( ) S ω( ) γ lm ω( ) e iωd–=

S ω( )2ξω3

π------------- D ω ξ,( )

ps ω( )------------------

2≈

ug x y t, ,( )

ueff t( )1

∆1∆2------------ ug x y t, ,( )dxdy

0

∆2

∫0

∆1

∫=

11

tor

ltered

.

ns for

) theenterw Riv-idge

nsionctionsau and

emently neg-idges,0% to

We define thebase reduction factorwhich describes the effect of base averaging on the oscillaresponse 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 thex direction. The following con-clusions emerged from this study:

1. Base averaging always reduces the effective excitation, with high frequencies being fiout 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 foundatiosmaller apparent wave propagation velocities.

11.2 Long-Span Bridges

Harichandran, Hawwari and Sweidan (1996) investigated the effect of SVEGM on: (alateral response of the Golden Gate suspension bridge (GGB) in California with a 4,200 ft cspan and 1,125 ft side spans; and (b) the longitudinal and lateral responses of the 1,700 ft Neer Gorge arch bridge (NRGB) in West Virginia, and the 700 ft Cold Spring Canyon arch br(CSCB) in California.

Two-dimensional finite element models were used for all the bridges. For the suspebridge, the model developed by Abdel-Ghaffar and Rubin (1983) was used, with the corremade by Castellani and Felloti (1986). For the arch bridges, the models developed by DusseWen (1989) were used.

Linear stationary random vibration analysis was performed. Total mean-square displacand force responses were computed from the dynamic and static variances, and the (possibative) covariances between the dynamic and static responses. Due to the flexibility of the brthe dynamic variances were the most dominant, typically contributing between 80% and 11the arch bridge member forces, and about 100% to the suspension bridge member forces.

∆1

∆2

x

y

Figure 11 Idealized oscillator on a rigid matfoundation

Figure 12 BRFs for two different foundation sizes andtwo different apparent propagation velocities

BRF max. response of oscillator accounting for SVEGMmax. response of oscillator for identical excitation---------------------------------------------------------------------------------------------------------------------------=

12

maintremelyitationsnd mainhe deckCon-

exci-due toal exci-bers,cur arell, andte howespons-ds:

ationsere forecause

greaterignifi-span

4)0)6))

Suspension Bridge Lateral Responses

Figs. 13 and 14 show schematics of the finite element models used for the side andspans of the Golden Gate Bridge, and the node numbering scheme. The main towers are exstiff compared to the suspended structure, and were assumed to be rigid. As a result, the excat the cable and deck supports at each end were assumed to be identical, and the side aspans were analyzed independently. The hangers were considered to be inextensible and twas modeled using beam elements. All elements were 50 ft long in the longitudinal direction.sistent mass matrices were used in the analyses.

Table 1 shows force response ratios for the two bridges due to longitudinal and lateraltations, respectively. The ratios were computed by dividing each mean-square responseidentical and delayed excitation by the corresponding mean-square response due to genertation. The maximum and minimum ratios that were obtained from all deck and cable memrespectively, are shown in the tables, and the node numbers at which the extreme ratios ocshown within parentheses. The increase in cable tension due to the excitations was smahence for the cable the response ratios are given only for displacements. The ratios indicaclose responses computed using the more common excitation types are when compared to res due to the general spatially varying ground motion model, and indicate the following tren

1. The use of identical excitations significantly over-estimates the responses at some locand under-estimates the responses at others, the relative deviations being more sevthe longer main span. The shear near the mid-spans is drastically under-estimated banti-symmetric modes are not excited by identical excitations.

2. The use of delayed excitations gives acceptable results for the side span, but showsdeviations for the main span in which the deck moment and shear are sometimes scantly under-estimated. This indicates that the loss of correlation is important for longsuspension 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 4 6

23 25 27

2426

28

13 15 17

14 1618

TABLE 1 LATERAL RESPONSE RATIOS FOR GGB SIDE AND MAIN SPANS

Rat

io

Ran

ge

Side Span Main SpanDeck 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 (8Min. 0.80 (23) 0.00 (13) 1.04 (27) 0.98 (28) 0.37 (69) 0.04 (81) 0.84 (23) 0.78 (10Max. 1.09 (17) 1.03 (27) 1.11 (15) 1.11 (16) 1.16 (129) 1.16 (117) 1.22 (89) 1.21 (9Min. 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--------------------

13

ridgesg d.o.f.derivedg themed to

used inports.

ateraldue to

al exci-em-

e ratioslowing

es thembers;

lts foraxial

someers are

Arch Bridge Responses

Figs. 15 and 16 show schematics of the finite element models used for the two arch band member numbering along the decks and arches. Beam elements with additional warpinwere used to model the decks and end towers. Equivalent properties for each element werefrom the complex arrangement of members in the actual bridges. The columns connectindecks to the arches and the diagonal bracing were modeled as struts, with the former assube inextensible. Bracing elements in the CSCB were cables. Lumped mass matrices werethe analyses. At each end, identical excitations were used at the deck, tower and arch sup

Tables 2 and 3 show force response ratios for the two bridges due to longitudinal and lexcitations, respectively. The ratios were computed by dividing each mean-square responseidentical and delayed excitation by the corresponding mean-square response due to genertation. The maximum and minimum ratios that were obtained from all deck, arch and bracing mbers, respectively, are shown in the tables, and the member numbers at which the extremoccur are shown within parentheses. The response ratios for the two bridges indicate the foltrends:

1. For the longitudinal response, the use of identical excitations severely under-estimataxial force in all arch members, and the moment and shear in some deck and arch meand over-estimates the axial force in the deck and bracing.

2. For the longitudinal response, the use of delayed excitations yields acceptable resumost members. However, for the NRGB, the shear in some arch members and theforce in the bracing is significantly under-estimated.

3. For the lateral response, the use of identical excitations over-estimates the forces inmembers and under-estimates them in others. The moments in some arch membquite significantly under-estimated.

1

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

34

3536

3738 39 40 41 42 43

4445

46

47

1

2 3 4 5 6 7 8 9 10 11

20

21

2223 24 43 44

4546

47

25

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

TABLE 2 LONGITUDINAL FORCE RESPONSE RATIOS FOR ARCH BRIDGES

Rat

io

Ran

ge

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) 2.02CSCB 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) 1.82CSCB 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) 0.91CSCB 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) 0.83CSCB 0.94 (2) 0.92 (6) 0.97 (10) 0.93 (28) 0.92 (25) 1.16 (all) –

Iden

tical

Gen

eral

--------

--------

-----D

elay

edG

ener

al----

--------

--------

14

somewhich

s. Thend the

ongitu-the lat-

h dammenttion of

gation

4. For the lateral response, the use of delayed excitations over-estimates the forces inmembers and under-estimates them in others. The level and number of members forthe 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 bridgeuse of delayed excitations is acceptable for the longitudinal response of short arch bridges, alateral response of short suspension bridge side spans; however, it is unacceptable for the ldinal response of long arch bridges, the lateral response of short and long arch bridges, anderal response of long suspension bridge main spans.

11.3 Earth Dams

Chen and Harichandran (1996) studied the effects of SVEGM on the Santa Felicia eartlocated in Southern California. Fig. 17 shows the cross section of the dam. A 3-D finite elemodel of the dam was used for the analysis. The contours in Figs. 18 and 19 show the variathe mean plus three standard deviation values of the maximum shear stress (τmax) along the baseand mid-length cross section due to general SVEGM, identical and delayed (wave propaonly) excitations. While the magnitude and distribution ofτmaxis very similar for identical and de-

TABLE 3 LATERAL FORCE RESPONSE RATIOS FOR ARCH BRIDGES

Rat

io

Ran

ge

BridgeDeck Members Arch Members

Moment Shear Torsion W. Mom. Moment Shear TorsionM

ax. 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)

Iden

tical

Gen

eral

--------

--------

-----D

elay

edG

ener

al----

--------

--------

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

15

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)

FRAME OF REF: GLOBAL

DISPLACEMENT - Z MIN: 180.11 MAX: 10399.00

180.11

2223.89

4267.67

6311.44

8355.22

10399.000.50

0.40

0.30

0.20

0.11

0.01

(b)

FRAME OF REF: GLOBAL

DISPLACEMENT - Z MIN: 204.75 MAX: 10669.00

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 µ+3σ contours ofτmax(MPa) at the base for: (a) general, (b) identical, and (c) delayed excitations

16

M.

bed,

l con-

tressesand

be

s ef-igators

ervas and

layed excitations, it is markedly different within the stiff gravel streambed for general SVEGGeneral conclusions that emerged were that:

1. SVEGM significantly increases the maximum shear stress in the stiff gravel streammostly due to the incoherence.

2. The wave passage effect is not as significant as coherency loss for the SVEGM modesidered.

3. For displacement and maximum shear strain responses, and for maximum shear swithin the core, the use of identical ground motion yields slightly conservative resultsis acceptable.

4. A preliminary reliability analysis indicates that a larger variety of sliding failures maypossible 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, itfect on various types of structures, and its simulation, have been studied by several investand selected references are provided here:

• Analysis and modeling of ground motion(Harichandran 1987a, Loh and Yeh 1988, Zand Shinozuka 1991, Spudich 1994, Boissieres and Vanmarcke 1995a, Boissiere

(a)

(b)

(c)

00.50

0.40

0.30

0.20

0.11

0.01

Figure 19 µ+3σ contours ofτmax(MPa) on the mid-length cross section for: (a) general, (b) identical, and(c) delayed excitations

17

96b,

amura)

)

Datta990a,

985,994,

and

g and

86,

75,, Ra-

ics-

ion and

metric

een

Vanmarcke 1995b, Chiu et al. 1995, Der Kiureghian 1996a, Der Kiureghian 19Nakamura 1996)

• Techniques for analyzing structural response (DebChaudhury and Gazis 1988, Yamand 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,and Mashaly 1990, Harichandran and Wang 1990b, Harichandran and Wang 1Zerva 1990, Zerva 1991)

• Response of bridges (Abdel-Ghaffar and Rubin 1982, Wilson and Jennings 1Zerva 1988, Loh and Lee 1990, Nazmy and Abdel-Ghaffar 1992, Hao 1993, Hao 1Nazmy and Konidaris 1994)

• Response of buildings (Hahn and Liu 1994, Hao and Duan 1995, Herdia-ZavoniBarranco 1996, Hao 1997)

• Response of dams (Haroun and Abdel-Hafiz 1987, Novak and Suen 1987, ZhanChopra 1991)

• Response of foundations; soil structure interaction (Luco and Wong 19Harichandran 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 19Abrahamson 1992, Zerva 1992, Ramadan and Novak 1993, Vanmarcke et al. 1993madan and Novak 1994)

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21