The Seismic Design Handbook || Earthquake Ground Motion and Response Spectra

Chapter 2

Earthquake Ground Motion and Response Spectra

Bijan Mohraz, Ph.D., P.E.Southern Methodist University, Dallas, Texas

Fahim Sadek, Ph.D.Mechanical Engineering Department, Southern Methodist University, Dallas, Texas. On assignment at the NationalInstitute ofStandards and Technology, Gaithersburg, Maryland

Key words: Earthquake Engineering, Earthquake Ground Motion, Earthquake Energy, Ground Motion Characteristics,Peak Ground Motion, Power Spectral Density, Response Modification Factors, Response Spectrum, SeismicMaps, Strong Motion Duration.

Abstract: This chapter surveys the state-of-the-art work in strong motion seismology and ground motioncharacterization. Methods of ground motion recording and correction are first presented, followed by adiscussion of ground motion characteristics including peak ground motion, duration of strong motion, andfrequency content. Factors that influence earthquake ground motion such as source distance, site geology,earthquake magnitude, source characteristics, and directivity are examined. The chapter presentsprobabilistic methods for evaluating seismic risk at a site and development of seismic maps used in codesand provisions. Earthquake response spectra and factors that influence their characteristics such as soilcondition, magnitude, distance, and source characteristics are also presented and discussed. Earthquakedesign spectra proposed by several investigators and those recommended by various codes and provisionsthrough the years to compute seismic base shears are described. The latter part of the chapter discussesinelastic earthquake spectra and response modification factors used in seismic codes to reduce the elasticdesign forces and account for energy absorbing capacity of structures due to inelastic action. Earthquakeenergy content and energy spectra are also briefly introduced. Finally, the chapter presents a brief discussionof artificially generated ground motion.

F. Naeim (ed.), The Seismic Design Handbook© Kluwer Academic Publishers 2001

2. Earthquake Ground Motion and Response Spectra 49

2.1 INTRODUCTION

Ground vibrations during an earthquake canseverely damage structures and equipmenthoused in them. The ground acceleration,velocity, and displacement (referred to asground motion) when transmitted through astructure, are in most cases amplified. Thisamplified motion can produce forces anddisplacements, which may exceed those thestructure can sustain. Many factors influenceground motion and its amplification; therefore,an understanding of how these factors influencethe response of structures and equipment isessential for a safe and economical design.

Earthquake ground motion is usuallymeasured by strong motion instruments, whichrecord the acceleration of the ground. Therecorded accelerograms, after they are correctedfor instrument errors and baseline (see nextsection), are integrated to obtain the velocityand displacement time-histories. The maximumvalues of ground motion (peak groundacceleration, peak ground velocity, and peakground displacement) are of particular interestin seismic analysis and design. Theseparameters, however, do not by themselvesdescribe the intensity of shaking that structuresor equipment experience. Other factors, such asearthquake magnitude, distance from the faultor epicenter, duration of strong shaking, soilcondition of the site, and the frequency contentof the motion also influence the response of astructure. Some of these effects such as theamplitude of the motion, frequency content, andlocal soil conditions are best representedthrough a response spectrum (2-1 to 2-4) whichdescribes the maximum response of a dampedsingle-degree-of-freedom (SDOF) oscillatorwith various frequencies or periods to groundmotion. The response spectra from a number ofrecords are often averaged and smoothed toobtain design spectra which specify the seismicdesign forces and displacements at a givenfrequency or period.

This chapter presents earthquake groundmotion and response spectra, and the influenceof earthquake parameters such as magnitude,duration of strong motion, soil condition,source distance, source characteristics, anddirectivity on ground motion and responsespectra. The evaluation of seismic risk at agiven site and development of seismic maps arealso discussed. Earthquake design spectraproposed by several investigators and thoserecommended by various agencies andorganizations are presented. The latter part ofthe chapter includes the inelastic earthquakespectra and response modification factors (Rfactors) that several seismic codes andprovisions recommend to account for theenergy absorbing capacity of structures due toinelastic action. Earthquake energy content andenergy spectra are also presented. Finally, thechapter presents a brief discussion of artificiallygenerated ground motion.

2.2 RECORDED GROUNDMOTION

Ground motion during an earthquake ismeasured by strong motion instruments, whichrecord the acceleration of the ground. Threeorthogonal components of ground acceleration,two in the horizontal direction and one in thevertical, are recorded by the instrument. Theinstruments may be located on free-field ormounted in structures. Typical strong motionaccelerograms recorded on free-field and theground floor of the Imperial County ServicesBuilding during the Imperial Valley earthquakeof October 15, 1979 (2-5) are shown in Figure 21.

Analog accelerographs, which recordground accelerations on photographic paper orfilm, were used in the past. The records werethen digitized manually, but later the processwas automated. These instruments weretriggered by the motion itself and some part ofthe initial motion was therefore lost, resulting inpennanent displacements at the end of therecord. Today, digital recording instruments

50

NORTH

Free field

IMPERIAL COUNTY SERVICES BUILDING

OCTOBER 15, 1879

,.... 24 8

Chapter 2

......28 8_GrOW'd_tloor__..".NlWN\;WJ.;\IV: vv""\,,,.,.,.....~-.,,...,..-.r..- ____

UPF," flelcl ,... •2 7 ;

-.. ·~·~~'~M,~\Y#.~""·,IfI~""---""",-,,,-------------

Ground floor

- ---..4N.""I"~~~_.,~looi'......~+~~------------------.. ,........,.'--.-- ....r-. -~...,......-

"""'.18;

EAST

Free field

-.24 8Grola'ld floor ~ ~---------ral#Jv"lJtJ C -~~~-------.....,.,...,.,.....-_--..--------

-.32;

TIME, IN SECONDS

Figure 2-1. Strong motion accelerograms recorded on free field and ground floor of the Imperial County Service Buildingduring the Imperial Valley earthquake of October 15, 1979. [After Rojahn and Mork (2-5).]

using force-balance accelero-meters are findingwider application because these instrumentsproduce the transducer output in a digital formthat can be automatically disseminated via dialup moderns or Internet lines. In addition tohaving low noise and greater ease in dataprocessing, digital instruments eliminate thedelays in recording the motion and the loss ofaccuracy due to digitizing traces on paper orfilm. They also permit recovery of the initialportion of the signal.

Accelerations recorded on accelerographsare usually corrected to remove the errors

associated with digitization (transverse play ofthe recording paper or film, warping of thepaper, enlargement of the trace, etc.) and toestablish the zero acceleration line before thevelocity and displacement are computed. Smallerrors in establishing the zero accelerationbaseline can result in appreciable errors in thecomputed velocity and displacement. Tominimize the errors, a correction is applied byassuming linear zero acceleration and velocitybaselines and then using a least square fit todetermine the parameters of the lines. For adetailed description of the procedure used in


digitizing and correcting accelerograms, oneshould refer to Trifunac (2-6), Hudson et al. (2-7),

and Hudson (2-8). Another procedure whichassumes a second degree polynomial for thezero acceleration baseline has also been used inthe past (2-9). The Trifunac-Hudson procedure,however, is more automated and has been usedextensively to correct accelerograms. Thecorrected accelerograms are then integrated toobtain the velocity and displacement timehistories.

For accelerograms obtained from digitalrecording instruments, the initial motion ispreserved, thereby simplifying the task ofdetermining the permanent displacement. Iwanet al. (2-10) proposed a method for processingdigitally recorded data by computing theaverage ordinates of the acceleration andvelocity over the final segment of the recordand setting them equal to zero. A constantacceleration correction is applied to the strongshaking portion of the record which may bedefined as the time segment between the firstand last occurrence of accelerations ofapproximately 50 cm/sec2

• A different constantacceleration correction is applied to theremaining final segment.

In the United States, the digitization,correction, and processing of accelerogramshave been carried out by the EarthquakeEngineering Research Laboratory of theCalifornia Institute of Technology in the pastand currently by the United States GeologicalSurvey (USGS) and other organizations such asthe California Strong Motion InstrumentationProgram (CSMIP) of the California Division ofMines and Geology (CDMG). A typicalcorrected accelerogram and the integratedvelocity and displacement for the SOOEcomponent of El Centro, the Imperial Valleyearthquake of May 18, 1940 are shown inFigure 2-2.

2.3 CHARACTERISTICS OFEARTHQUAKE GROUNDMOTION

The characteristics of ground motion thatare important in earthquake engineeringapplications are:1. Peak ground motion (peak ground

acceleration, peak ground velocity, and peakground displacement),

2. duration of strong motion, and3. frequency content.

Each of these parameters influences theresponse of a structure. Peak ground motionprimarily influences the vibration amplitudes.Duration of strong motion has a pronouncedeffect on the severity of shaking. A groundmotion with a moderate peak acceleration and along duration may cause more damage than aground motion with a larger acceleration and ashorter duration. Frequency content stronglyaffects the response characteristics of astructure. In a structure, ground motion isamplified the most when the frequency contentof the motion and the natural frequencies of thestructure are close to each other. Each of thesecharacteristics are briefly discussed below:

2.3.1 Peak ground motion

Table 2-1 gives the peak groundacceleration, velocity, displacement, earthquakemagnitude, epicentral distance, and sitedescription for typical records from a number ofseismic events from the western United States.Some of these records are frequently used inearthquake engineering applications. Peakground acceleration had been widely used toscale earthquake design spectra andacceleration time histories. Later studiesrecommended that in addition to peak groundacceleration, peak ground velocity anddisplacement should also be used for scalingpurposes. Relationships between ground motionparameters are discussed in Section 2.6.

52 Chapter 2

Table 2-1. Peak Ground Motion, Earthquake Magnitude, Epicentral Distance, and Site Description for Typical RecordedAccelerograms

Earthquake and location Mag. Eplcentral Compo Peak Peak Peak Site Descriptiondlatanee Ace. (g) Vel. Dlap.

(km) (In/sec) (In)Helena, 10/31/1935 6.0 6.3 SOOW 0.146 2.89 0.56 RockHelena, Montana Carroll College S90W 0.145 5.25 1.47

Vert 0.089 3.82 1.11Imperial Valley, 511811940 6.9 11.5 SOOE 0.348 13.17 4.28 Alluvium, several 1000 ItEI Centro site S90W 0.214 14.54 7.79

Vert 0.210 4.27 2.19Western Washington, 4/13/1949 7.1 16.9 N04W 0.165 8.43 3.38 Deep cohesionless soil,Olympia, Washington Highway Test N86E 0.280 6.73 4.09 420ftLab Vert 0.092 2.77 1.59Northwest California, 1017/1951 5.8 56.2 S44W O.t04 1.89 0.94 Deep cohesionless soil,Ferndale City Hall N46W 0.112 2.91 1.08 500ft

Vert 0.027 0.87 0.64Kern County, 7/21/1952 7.2 41.4 N21E 0.156 6.19 2.64 40 ft of alluvium overTaft Lincoln School Tunnel S69E 0.179 6.97 3.60 poorly cemented

Vert 0.105 2.63 1.98 sendstoneEureka, 12/21/1954 6.5 24.0 Nl1W 0.168 12.44 4.89 Deep cohesionless soil,Eureka Federal Building N79E 0.258 11.57 5.53 250 ft deep

Vert 0.083 3.23 1.83Eurkea, 12/21/1954 6.5 40.0 N44W 0.159 14.04 5.58 Deep cohesionless soil,Ferndale City Hall N46E 0.201 10.25 3.79 500 It deep

Vert 0.043 2.99 1.54San Francisco, 3/22/1957 5.3 11.5 Nl0E 0.083 1.94 0.89 RockSan Francisco Golden Gate Park S80E 0.105 1.82 0.33

Vert 0.038 0.48 0.27Hollister, 4/811961 5.7 22.2 SOIW 0.065 3.06 1.12 Unconsolidated alluviumHollister City Hall N89W 0.179 6.75 1.51 over parity consolidated

Vert 0.050 1.85 0.85 gravelParkfield, 6/27/1966 5.6 56.1 NOSW 0.355 9.12 2.09 AlluviumCholame Shandon, Calilornia Array N89E 0.434 10.02 2.80No.5 Vert 0.119 2.87 1.35Borrego Mountain, 4/811968 6.4 67.3 SOOW 0.355 9.12 2.09 AlluviumEI Centro site S90W 0.434 10.02 2.80

Vert 0.119 2.87 1.35San Fernando, 2/9/1971 6.4 21.1 NOOW 0.255 11.81 5.87 Alluvium8244 Orion Blvd., 1'1 Roor S90W 0.134 9.42 5.45

Vert 0.171 12.58 5.76San Fernando, 2/9/1971 6.4 29.5 N21E 0.315 6.76 1.66 SandstoneCastaic Old Ridge Route N69W 0.271 10.95 3.74

Vert 0.156 2.54 1.38San Fernando, 2/911971 6.4 7.2 SISW 1.170 44.58 14.83 Highly jointed dioritePacoima Dam S74W 1.075 22.73 4.26 gneiss

Vert 0.709 22.96 7.61San Fernando, 2/9/1971 6.4 32.5 SOOW 0.180 8.08 2.87 GraniticGriffith Park Observatory S90W 0.171 5.73 2.15

Vert 0.123 2.92 1.33Loma Prieta, 10117/1989 7.0 7.0 90 0.479 18.70 4.54 Landslide depositsCorralitos, Eureka Canyon Road 360 0.630 21.73 3.76

Vert 0.439 7.33 3.06Loma Prieta, 10117/1989 7.0 16.0 90 0.409 8.36 2.68 LimestoneSanta Cruz - UCSC/L1CK Lab 360 0.452 8.36 2.60

Vert 0.331 4.71 2.65Loma Prieta, 10117/1989 7.0 43.0 360 0.219 13.16 5.45 Bay sediments/alluviumSunnyvale - Colton Avenue 270 0.215 13.42 4.98

Vert 0.103 2.91 1.21Landers, 6/2811992 7.3 1.8 350 0.800 12.91 28.32 Stiff alluvium overlyingSCE Lucerne Valley Station 260 0.730 58.84 107.21 hard granitic rock

Vert 0.860 16.57 17.03Northridge, 1/17/1994 6.6 19.3 265 0.434 12.05 1.97 Highly jointed dioritePacoima Dam 175 0.415 17.59 1.83 gneiss

Vert 0.184 6.33 1.03Northridge, 1/17/1994 6.6 22.5 90 0.883 16.44 5.64 AlluviumSanta Monica - City Hall Ground 360 0.370 9.81 2.57

Vert 0.232 5.52 1.49Northridge, 1/1711994 6.6 15.8 90 0.604 30.29 5.99 AlluviumSylmar - County Hosp~al Parking 360 0.843 50.74 12.81Lot Vert 0.535 7.34 2.97

2. Earthquake Ground Motion and Response Spectra

Time, sec

i60

i60

i60

53

Page or Bolt

Figure 2-2. Corrected accelerogram and integrated velocity and displacement time-histories for the SOOE component of EICentro, the Imperial Valley Earthquake of May 18, 1940.

"~ "n", ~o'"d~ ~ J .. t _ ~. .JIo. ... I. • ~ "o lI~~••~.~t'- ",,,,,,"', -h

-75

75~o lIN~IoII.";\IIIJI1I---------------

-75

"~ """'OC 000 Brod,• J .. I _~".A. J •

o *w,,¥of\IIlIrno '"~t'''

-75

]:J!!!~~'b:_M":#:~'O~d Shch

o ~ M • ~

Time. sec.

i50

i60

Figure 2-3. Comparison of strong motion duration for the S69E component of the Taft, California earthquake of July 21,1982 using different procedures.

54

2.3.2 Duration of strong motion

Several investigators have proposedprocedures for computing the strong motionduration of an accelerogram. Page et al. (2-11)

and Bolt (2-12) proposed the "bracketed duration"which is the time interval between the first andthe last acceleration peaks greater than aspecified value (usually 0.05g). Trifunac andBrady (2-13)1 defined the duration of the strongmotion as the time interval in which asignificant contribution to the integral of thesquare of acceleration (fa2dt) referred to as theaccelerogram intensity takes place. Theyselected the time interval between the 5% andthe 95% contributions as the duration of strongmotion. A third procedure suggested byMcCann and Shah (2-15) is based on the averageenergy arrival rate. The duration is obtained byexamining the cumulative root mean squareacceleration (rms)2 of the accelerogram. Asearch is performed on the rate of change of thecumulative rms to determine the two cut- offtimes. The final cut-off time T2 is obtainedwhen the rate of change of the cumulative rmsacceleration becomes negative and remains sofor the remainder of the record. The initial timeTI is obtained in the same manner except thatthe search is performed starting from the "tailend" of the record.

Figure 2-3 shows a comparison among thestrong motion durations extracted from atypical record using different procedures. Table2-2 gives the initial time TI , the final Time T2,

the duration of strong motion !:1T, the rmsacceleration, and the percent contribution to (fa2dt) for several records. The comparisonsshow that these procedures result in differentdurations of strong motion. This is to beexpected since the procedures are based ondifferent criteria. It should be noted that sincethere is no standard definition of strong motionduration, the selection of a procedure forcomputing the duration for a certain study

I An earlier study by Husid et aI. (2-14) used a similardefinition for the duration of strong motion.

2 See Section 2.3.3 for defInition of rms

Chapter 2

depends on the purpose of the intendedapplication. For example, it seems reasonable touse McCann and Shah's definition, which isbased on rms acceleration when studying thestationary characteristics of earthquake recordsand in computing power spectral density. Onthe other hand, the bracketed duration proposedby Page et al. (2-11) and Bolt (2-12) may be moreappropriate for computing elastic and inelasticresponse and assessing damage to structures.

Based on the work of Trifunac and Brady (2

13), Trifunac and Westermo (2-16) developed afrequency dependent definition of durationwhere the duration is considered separately inseveral narrow frequency bands. They definethe duration as the sum of time intervals duringwhich the integral <f f 2(t)dt) -- where fit) is theground acceleration, velocity, or displacement - has the steepest slope and gains a significantportion (90%) of its final value. This definitionconsiders the duration being composed ofseveral separate segments with locationsspecified by the slopes of the integral. Theprocedure is to band-pass filter the signal fit)using two Ormsby filters in different frequencybands with specified central frequencies. Theduration for each frequency band is computedas the sum of several time intervals where thesmoothed integration function <ff 2(t)dt) has thesteepest slope. The study by Trifunac andWestermo (2-16) indicated that the duration ofstrong motion increases with the period ofmotion.

2.3.3 Frequency content

The frequency content of ground motion canbe examined by transforming the motion from atime domain to a frequency domain through aFourier transform. The Fourier amplitudespectrum and power spectral density, which arebased on this transformation, may be used tocharacterize the frequency content. They arebriefly discussed below:


Table 2-2. Comparison of Strong Motion Duration for Eight Earthquake RecordsRecord Compo Method* T, (sec) Tz(sec) .1T (sec) RMS fa2dt

(cm/sec2)EI Centro, SOOE A 0.00 53.74 53.74 46.01 1001940 B 0.88 26.74 25.86 65.16 97

C 1.68 26.10 24.42 64.75 90D 0.88 26.32 25.44 65.60 96

S90W A 0.00 53.46 53.46 38.85 100B 1.24 26.64 25.40 54.88 95C 1.66 26.20 24.54 54.39 90D 0.80 26.62 25.82 24.73 96

Taft, 1952 N21E A 0.00 54.34 54.34 25.03 100B 3.44 22.94 19.50 38.50 85C 3.70 34.24 30.54 31.70 90D 2.14 36.46 34.32 30.85 96

S69E A 0.00 54.38 54.38 26.10 100B 3.60 18.72 15.12 44.61 82C 3.66 32.52 28.86 33.96 90D 2.34 35.30 32.96 32.71 95

EICentro, SOOW A 0.00 90.28 90.28 19.48 1001934 B 1.92 14.78 12.86 46.89 83

C 2.82 23.92 21.10 38.27 90D 1.92 23.88 21.96 38.38 94

S90W A 0.00 90.22 90.22 20.76 100B 1.98 20.10 18.12 44.58 93C 2.86 23.14 20.28 41.57 90D 1.62 20.10 18.48 44.26 93

Olympia, N04W A 0.00 89.06 89.06 22.98 1001949 B 0.74 22.30 22.30 44.25 93

C 1.78 25.80 25.80 40.51 90D 0.08 22.94 22.94 43.73 93

N86E A 0.00 89.02 89.02 28.10 100B 1.00 21.04 21.04 56.00 94C 4.34 18.08 18.08 59.22 90d 0.28 21.52 21.52 55.48 94

• A: Entire RecordB: Page or Boll (2-11 or 2-12)c: Trifunac and Brady (2-13)D: McCann and Shah (2-15)

Where T is the duration of the accelerogram.Fourier ampLitude spectrum. The finite The Fourier amplitude spectrum FS(w) IS

Fourier transform F(w) of an accelerogram art) defined as the square root of the sum of theis obtained as squares of the real and imaginary parts of F(w).

Thus,

F(m) = J;a(t)e-iOXdt, i=H (2-1)

56

FS(W) =

[T J2 [T J2 (2-2)

fo a(t)sinoxdt + fo a(t)cosOXdt

Chapter 2

closely related. The equation of motion of thesystem can be written as

(2-3)

x(t) = f;-a(r)cosw/I(t-r)dr (2-5)

The relative velocity x(t) follows directlyfrom Equation 2-4 as

In which x and x are the relativedisplacement and acceleration, and COn is thenatural frequency of the system. UsingDuhamel's integral, the steady-state responsecan be obtained as

Since art) has units of acceleration, FS(m)has units of velocity. The Fourier amplitudespectrum is of interest to seismologists incharacterizing ground motion. Figure 2-4 showsa typical Fourier amplitude spectrum for theSOOE component of EI Centro, the ImperialValley earthquake of May 18, 1940. The figureindicates that most of the energy in theaccelerogram is in the frequency range of 0.1 to10 Hz, and that the largest amplitude is at afrequency of approximately 1.5 Hz.

It can be shown that subjecting anundamped single-degree-of-freedom (SDOF)system to a base acceleration a(t), the velocityresponse of the system and the Fourieramplitude spectrum of the acceleration are

1 itx(t)=- -a(r)sinwn(t-r)drW 0

n

(2-4)

1.1Cl>

~ 80.S

E::J~ 601.1Cl>Q..III

~::J- 40'--a..EI)

\..Cl>'t 20::J

~

00 5 70 15 20 25

Frequency, cps.

Figure 2-4. Fourier amplitude spectrum for the SOOE component of EI Centro, the Imperial Valley earthquake of May 18,1940.


Equation 2-5 can be expanded as occurs at td then

x(t) = -[J:a(r)cosmnrdrJcosmnt

[J:a(r)SinmnrdrJsinmnt(2-6)

PSV(m) = roSD(m) =

[J:d a(r)sinmrdrr+[J:d a(r) cos mrdrr(2-8)

Denoting the maximum relative velocity(spectral velocity) of a system with frequency 0>

by SV(OJ) and assuming that it occurs at time tv,one can write

SV(m) =

[J:v a(r)sinmrdrr+ [J:v a(r)cosmrdrr(2-7)

The pseudo-velocity PSV(OJ) defined as theproduct of the natural frequency 0> and themaximum relative displacement or the spectraldisplacement SD( OJ) is close to the maximumrelative velocity (see Section 2.7). If SD(OJ)

Comparison of Equations 2-2 and 2-7 showsthat for zero damping, the maximum relativevelocity and the Fourier amplitude spectrum areequal when tv=T. A similar comparisonbetween Equations 2-2 and 2-8 reveals that thepseudo-velocity and the Fourier amplitudespectrum are equal if td=T. Figure 2-5 shows acomparison between FS(OJ) and SV(OJ) for zerodamping for the SOOE component of El Centro,the Imperial Valley earthquake of May 18,1940. The figure indicates the close relationshipbetween the two functions. It should be notedthat, in general, the ordinates of the Fourieramplitude spectrum are less than those of theundamped pseudo-velocity spectrum.

'00

sv

[$. .

252015105

o.n-----L---=---;~~.::::....~...:::~==:;::::~=~o

Frequency, cps.

Figure 2-5. Comparison of Fourier amplitude spectrum and velocity spectrum for an undamped single-degree-of-freedomsystem for the SOOE component of El Centro. the Imperial Valley earthquake of May 18, 1940.

58

Power spectral density. The inverse Fouriertransform of F(w) is

Chapter 2

densities of an ensemble of N accelerograms(see for example reference 2-17). Therefore,

1 fllJO .aCt) = - F(m)e

1fadm

1r 0(2-9)

1 NG(m) =-IGi (01)

N i=l(2-15)

where Wo is the maximum frequency detected inthe data (referred to as Nyquist frequency).Equations 2-1 and 2-9 are called Fouriertransform pairs. As mentioned previously, theintensity of an accelerogram is defined as

I = J:a2(t)dt (2-10)

where G;{w) is the power spectral density ofthe ith record. The power spectral density isfrequently presented as the product of anormalized power spectral density G<n>(w)

(area =1.0) and a mean square acceleration as

(2-16)

Based on Parseval's theorem, the intensity Ican also be expressed in the frequency domainas

The intensity per unit of time or thetemporal mean square acceleration ","'" can beobtained by dividing Equation 2-10 or 2-11 bythe duration T. Therefore,

Figure 2-6 shows a typical example of anormalized power spectral density computedfor an ensemble of 161 accelerograms recordedon alluvium. Studies (2-17, 2-18) have shown that

strong motion segment of accelerogramsconstitutes a locally stationary random processand that the power spectral density can bepresented as a time-dependent function G(t,w)in the form:

1 fllJo1 1

2I = - F(m) dm

1r 0(2-11)

G(t,m) = tj/2 S(t)G<n> (01) (2-17)

The temporal power spectral density isdefined as

",2=

1 rT1 rllJo

I 12T Jo a

2(t)dt = Til' Jo F(m) dm

(2-12)

(2-13)

where Set) is a slowly varying time-scale factorwhich accounts for the local variation of themean square acceleration with time.

Power spectral density is useful not only as ameasure of the frequency content of groundmotion but also in estimating its statisticalproperties. Among such properties are the rmsacceleration 'fI, the central frequency We, andthe shape factor {j defined as

Combining Equations 2-12 and 2-13, themean square acceleration can be obtained as

(2-18)

(2-14)(2-19)

In practice, a representative power spectraldensity of ground motion is computed byaveraging across the temporal power spectral

(2-20)


o.~

...I

Wa..~

:i- 0.3'-IIIc:~

g(JQ)

~ 0.2I...Q)

~0a..~Q)

~ 0.1

E~

0.00 2 4 S I) 7 8 9 10

Frequency, cps

Figure 2-6. Normalized power spectral density of an ensemble of 161 horizontal components of accelerograms recorded onalluvium. [After Elghadamsi et al. (2-18).]

where A,. is the r-th spectral moment defined as

Smooth power spectral density of theground acceleration has been commonlypresented in the form proposed by Kanai (2-19)

and Tajimi (2-20) as a filtered white noise groundexcitation of spectral density Go in the form

The Kanai-Tajimi parameters ~g. 109, and Gorepresent ground damping, ground frequency,and ground shaking intensity. These parametersare computed by equating the nns acceleration,the central frequency, and the shape factor,

1+4~i (m / mg )2~~= 2 ~

[1- (m / mg )2

] + (2~gm/ mg )2

(2-22)

Equations 2-18 to 2-20, of the smooth and theraw (unsmooth) power spectral densities (2-18. 2-

21). Table 2-3 gives the values of ~g> 109, and Gofor the normalized power spectral densities ondifferent soil conditions. Also shown are thecentral frequency 10c and the shape factor o.Using the Kanai-Tajimi parameters in Table 23, normalized power spectral densities forhorizontal and vertical motion on various soilconditions were computed and are presented inFigures 2-7 and 2-8. The figures indicate that asthe site becomes stiffer, the predominantfrequency increases and the power spectraldensities spread over a wider frequency range.This observation underscores the influence ofsite conditions on the frequency content ofseismic excitations. The figures also show thatthe power spectral densities for horizontalmotion have a sharper peak and span over anarrower frequency region than thecorresponding ones for vertical motion.

(2-21 )A.,. = J~mrG(m)dm

60 Chapter 2

Table 2-3. Central Frequency, Shape Factor, Ground Frequency, Ground Damping, and Ground Intensity for Different SoilConditions. [After Elghadamsi et al. (2·18)]

Site Category No. of Recocds Central

Frequency Ie(Hz)

Shape Factor 0 Ground

Frequency Ig(Hz)

Ground

Damping~

Ground

Intensity Go

(11Hz)

Horizontal

Alluvium

Alluvium on rockRock

161

60

26

4.10

4.58

5.41

0.65

0.59

0.59

2.92

3.64

4.30

0.34

0.30

0.34

0.1020.Q78

0.070

VerticalAlluvium

Alluvium on rockRock

78

29

13

6.27

6.68

7.53

0.63

0.62

0.55

4.17

4.63

6.18

0.46

0.46

0.46

0.080

0.072

0.053

0.35

10

0.05

Alluvium

'\ Alluviu!'l/R~a.oc~ __

-'\",, ,\ "

, "'--. ' ........j.--.----....---r"----r--.--...;.~:::~.~-~-~-=-~=~0.00a 5 6 t

rrequency, CPS

civi 0,20a:"~ 0.15

'5E 0.10~

... 0.30I~

Ie8. 0.25

Figure 2-7. Normalized power spectral densities for horizontal motion. [After Elghadamsi et al. (2-18).]

Clough and Penzien (2·22) modified the KanaiTajimi power spectral density by introducinganother filter to account for the numericaldifficulties expected in the neighborhood ofw=O. The cause of these difficulties stems fromdividing Equation 2-22 by rJ and 0/,respectively, to obtain the power spectraldensity functions for ground velocity anddisplacement. The singularities close to w=Ocan be removed by passing the process throughanother filter that attenuates the very lowfrequency components. The modified powerspectral density takes the form

G(CO) =

1+4~: {COl COg Y--[I--~-CO-Ico-g---iyf+{~gCOI COg YX (2-23)

__~~colcoit (G )11- (co ICOI)2 ]+ 4~12 (CO ICOlr 0

Where WI and ;1 are the frequency anddamping parameters of the filter.


0.35

.... 0.30I .......

VI0..

~ 0.25

civi 0.20a:".~ 0.15

"0E5 0.10Z

0.05

Alluvium

~I.Rock

Rock- - --

--......

10984 5 6

Frequency, CPS

O.OO-j----,---.----,----,--...,.----r--,---,---r--..,o

Figure 2-8. Normalized power spectral densities for vertical motion. [After Elghadamsi et aI. (2-18).)

Lai (2-21) presented empirical relationships forestimating ground frequency wg and centralfrequency We for a given epicentral distance Rin kilometers or local magnitude M L' Theserelationships are

Using these relationships and theacceleration attenuation equations (see Section2.4.1), Lai proposed a procedure for estimatinga smooth power spectral density for a givenstrong motion duration and ground damping.

Once the power spectral density of groundmotion at a site is established, random vibrationmethods may be used to formulate probabilisticprocedures for computing the response ofstructures. In addition, the power spectraldensity of ground motion may be used for otherapplications such as generating artificialaccelerograms as discussed in Section 2.12.

{J)g = 27 -O.09R 10::;; R ::;; 60 (2-24)

(2-25)

(2-26)

2.4 FACTORS INFLUENCINGGROUND MOTION

Earthquake ground motion is influencedby a number of factors. The most importantfactors are: 1) earthquake magnitude, 2)distance from the source of energy release(epicentral distance or distance from thecausative fault), 3) local soil conditions, 4)variation in geology and propagation velocityalong the travel path, and 5) earthquake sourceconditions and mechanism (fault type, slip rate,stress conditions, stress drop, etc.). Pastearthquake records have been used to studysome of these influences. While the effect ofsome of these parameters such as local soilconditions and distance from the source ofenergy release are fairly well understood anddocumented, the influence of source mechanismis under investigation and the variation ofgeology along the travel path is complex anddifficult to quantify. It should be noted thatseveral of these influences are interrelated;consequently, it is difficult to discuss themindividually without incorporating the others.Some of the influences are discussed below:

62 Chapter 2

100

N.UWCII,~1!

~

=II:~III

Bcxcr

\'\

5

1.0

l••• It"'." \. D5.0 t. 6.0'.0 •• '.01.0 te 1.0,~••t.· ch.l"f

'\, ,

... ",.,,,:11:,,...t; ftlI8!".kIll4.6 ",.,,,.kvtll.l' ••,n,t....II .....,,,.tw••

lIIUI

r-----~"---------------------.... IOOO"-

"-,,,"-

~ --..... ' .... IIUN _\ STANDARD'~""'.L' D[,IATION "5- . :-...~ '\

. ....... .. , ' ,,\...... . . '\" ..~, '\ ',/"[U -1 STAIDUD, .... , DHUlIONS

~ \ S .., " ," .SQUAllS , J&~t' '~ "

" f't,/'~~f"-': \ret, '5'\5 • ,, Ii ''\ ', ,., 5 ,,,,. II' ,. \J

~ , ,_s '\ \ ', "'Ii' ,5 ,,, ,5 5 ,5 " 15 5 ~" , ,5 f1i~ , ", ,,,

-- .... - '" .' ~, ...'," • 5 \............... , 5 "i ~ 5" '\' ~' \

'. ~ S ,'~~5~' '~'5' ""1 ", ,., \

" 5 ,,' ~ ,., \~ Ss "'" l5Spt' ,ft 5, \

5"-. 5 , \ ' \

'-'\ Ii , 'Ii U ".1666' • \11[." .1 STANDUO J \ 1 \', ~ ClOUD-'Ull

-.............. DnlATlON V ~ 1 \, \ [NVU.,[..... ' '\ " , \1 \" '\ ~'\ \

............ , "\...... ,~ . \~ ", 5 61i'" 6 6 ~1Ii, ~ , \ ' \

IIUI -, STAiDUD I ,).. 5',' 66 , ' "OnlATlOU '-/', S' 1 \'. \, " \' \

,~ \' \, s S "'\ 15 \' \

, S5 __ Ii ~ \',5 S ,6 \

S'\r 5 ~ 5 '6 "5' 5 ~, '\" .\" .," .\\S 16 \1 "'.' , \5~ 6 \ ,,, , ,

',\ , ,\ \

~ \, ' 1, \\ ,\ \

\ \\ " \

\ \, ,\ \\ \\ \, ,

1--1--'--1-1....1-1-1='-- ....L.J.--I.--LI...JI.......~b:-_--L-'-- ..........-L.l......................J...J.,,~\!'=-~o ~

HYPOCENTRAL

Figure 2-9. Peak ground acceleration plotted as a function of fault distance obtained from worldwide set of 515 strongmotion records without normalization of magnitude. [After Donovan (2-23).]


2.4.1 Distance

The variation of ground motion withdistance to the source of energy release hasbeen studied by many investigators. In thesestudies, peak ground motion, usually peakground acceleration, is plotted as a function ofdistance. Smooth curves based on a regressionanalysis are fitted to the data and the curve orits equation is used to predict the expectedground motion as a function of distance. Theserelationships, referred to as motion attenuation,are sometimes plotted independently ofearthquake magnitude. This was the case in theearlier studies because of the lack of sufficientnumber of earthquake records. With theavailability of a large number of records,particularly since the 1971 San Fernandoearthquake, the database for attenuation studiesincreased and a number of investigators reexamined their earlier studies, modified theirproposed relationships for estimating peakaccelerations, and included earthquakemagnitude as a parameter. Donovan (2-23)

compiled a database of more than 500 recordedaccelerations from seismic events in the UnitedStates, Japan, and elsewhere and later increasedit to more than 650 (2-24). The plot of peakground acceleration versus fault distance fordifferent earthquake magnitudes from hisdatabase is shown in Figure 2-9. Even thoughthere is a considerable scatter in the data, thefigure indicates that peak acceleration decreasesas the distance from the source of energyrelease increases. Shown in the figure are theleast square fit between acceleration anddistance and the curves corresponding to meanplus- and mean minus- one and two standarddeviations. Also presented in the figure is theenvelope curve (dotted) proposed by Cloud andPerez (2-25).

Other investigators have also proposedattenuation relationships for peak groundacceleration, which are similar to Figure 2-9. Asummary of some of the relationships, compiledby Donovan (2-23) and updated by the authors, isshown in Table 2-4. A comparison of variousrelationships (2-24) for an earthquake magnitude

of 6.5 with the data from the 1971 SanFernando earthquake is shown in Figure 2-10.This figure is significant because it shows thecomparison of various attenuation relationshipswith data from a single earthquake. While thefigure shows the differences in variousattenuation relationships, it indicates that theyall follow a similar trend.

D

1000

IUil:Kll

o Il:l)(,~l,l"'l "rn:• 'inll "rn

10 100DISTA\JCE TO [~[RGY C[~HR (KMS)

Figure 2-10. Comparison of attenuation relations with datafrom the San Fernando earthquake of February 9, 1971.

[After Donovan (2-24).]

Housner (2-38), Donovan (2-23), and Seed andIdriss (2-39) have reported that at fartherdistances from the fault or the source of energyrelease (far-field), earthquake magnitudeinfluences the attenuation, whereas at distancesclose to the fault (near-field), the attenuation isaffected by smaller but not larger earthquakemagnitudes. This can be observed from theearthquake data in Figure 2-9.

64

Table 2-4. TyPical Attenuation Relationships

Chapter 2

Data Source

I. San Fernando earthquakeFebruary 9. 1971

2. California earthquake

3. California and Japanese earthquakes

4. Cloud (1963)

5. Cloud (1963)

6. U.S.c. and G.S.

7. 303lnslrUmental Values

8. Western U.S. records

9. U.S.• Japan

10. Western U.S. records. USSR. and Iran

II. Western U.S. records and worldwide

12. Western U.S. records and worldwide

13. Western U.S. records

14. Italian records

15. Western U.S. and worldwide (soil sites)

16. Western U.S. and worldwide (rock sites)

17. Worldwide earthquakes

18. Western North American earthquakes

Relationship·

1.83log PGA =190/ R

2PGA =Yo /(1 + (R'/ h) )

where logy. =-(b + 3) + 0.81M - 0.027~ and b is a site factor

0.0051 (0.61M-ploaR.0.167-1.83IRjPGA=--IO,p;

where P= 1.66 + 3.(J)JR and TG is the fundamental period of the site

1.6oIM I.IM 2PGA =0.0069.. /(1.1.. + R )

0.8M 2PGA =1.254.. /( R + 25)

log PGA = (6.5 - 210g(R'+80» /981

0.67M 1.6PGA = 1.325.. /( R + 25)

0.8M 2PGA = 0.0193<' /(R + 400)

0.58M 1.52PGA = 1.35.. /(R + 25)

0.732MIn PGA = -3.99 + 1.28M -1.751n(R = 0.147.. I

M is the surface wave magnitude for Mgreater than or equal to 6. or it isthe local magnitude for M less than 6.

log PGA = -1.02 + 0.249M _ log ~R2

+ 7.32

_ 0.0025s.JR2 + 7.32

log PGA =0.49 + 0.23(M - 6) - log~ - 0.002~In PGA = Ina(M) - (l(M)ln(R + 20)

M is the surface wave magnitude for M greater than or equal to 6. or it isthe local magnitude for smaller M.R is the closest distance to source for M greater than 6 and hypocentraldistance for M smaller than 6.~M) and (J(M) are magnitude-dependent coefficients.

In PGA =-1.562 + 0.306M - log ~R2

+ 5.82

+ 0.169SS is 1.0 for soft sites or 0.0 for rock.

For M less than 6.5.0.418M

In PGA =-2.611 + I.1M - 1.751n( R + 0.822..

For M greater than or equal to 6.5.0.629M

In PGA = -2.611 + I.lM - 1.751n[R + 0.316e

For M less than 6.5,O.406M

In PGA = -1.406 + I.1M - 2.051n(R + 1.353<'

For M greater than or equal to 6.5,0.537M

In PGA = -1.406 + I.1M - 2.051n( R + 0.579.. I

In PGA =-3.512 + O.904M -1.328In~ R2 + (O.149..0.6oI7M 12

+ (1.125 - O.ll21n R - 0.0957M IF + [0.440 - 0.1711n RIS"

+ [0.405 - 0.222 In RIS Itr

F=0 for strike-slip and normal fault earthquakes and I for reverse,reverse-oblique, and thrust fault earthquakes.S" = I for soft rock and 0 for hard rock and alluviumS", = I for hard rock and 0 for soft rock and alluvium

~ , 2 VsIn PGA =b + 0.527(M - 6.0) - 0.778 In R· + (5.570) - 0.371 In--

1396where b =-0.313 for strike-slip earthquakes= -0.117 for reverse-slip earthquakes= -0.242 if mechanism is not specified

V, is the average shear wave velocity of the soil in (mlsec) over the upper30 meters]he equatiOll can be used for magnitudes of 5.5 to 7.5 and for distancesnot greater than 80 km

Reference

Donovan(2-23)Blume(2-26)

Kanai(2-27)

Milne and Davenport(2-28)

Esteva(2-29)

Cloud and Perez(2-25)

Donovan(2-23)

Donovan(2-23)

Donovan(2-23)

Campbell(2-30)

Joyner and Boore(2-31)

Joyner and Boore(2-32)

Idriss(2-33)

Sabella and Pugliese(2-34)

Sadigh et al.(2-35)

Sadigh et al.(2-35)

Campbell and Bozorgnia(2-36)

Booreet al.(2-37)

• Peak ground acceleration PGA in g, source distance R in km, source distance R' in miles, local depth h in miles, and earthquake magnitude M. Refer tothe relevant references for exact definitions of source distance and earthquake magnitude.


• SUPERSTIl IONNOUNTA I N PARACNUTE

e TEST FACILITY •

.10

• NO. 11.NO.12

·NO.I)

EXPLANATION

• SMAT-I GROUND STATION• DSA-I GROUND STATION• eRA-1 IUILDING,. CRA-' IAI DCE• DCA-lOO SPECIAL ARRAY

HOLTVILLE

C>

65

Figure 2-11. Strong motion stations in the Imperial Valley, California. [After Porcella and Matthiesen (2-40); reproducedfrom (2-39).]

FAUlT DISTANCE nun)

IMPERIAL VALLEY EARTHQUAKEOCTOBER IS. 1979

Figure 2-12. Observed and predicted mean horizontalpeak accelerations for the Imperial Valley earthquake ofOctober 15, 1979 plotted as a function of distance from

the fault. The solid curve represents the medianpredictions based on the observed values and the dashed

curves represent the standard error bounds for theregression. [After Campbell (2-30).]

The majority of attenuation relationships forpredicting peak ground motion are presented interms of earthquake magnitude. Prior to theImperial Valley earthquake of 1979, the vastmajority of available accelerograms wererecorded at distances of greater thanapproximately 10 or 15 km from the source ofenergy release. An array of accelerometersplaced on both sides of the Imperial Fault (2-40)

prior to this earthquake (See Figure 2-11)provided excellent acceleration data for smalldistances from the fault. The attenuationrelationship from this array presented byCampbell (2-30) is shown in Figure 2-12. Thefigure indicates the flat slope of the accelerationattenuation curve for distances close to thesource, a phenomenon which is not observed in

66

50 PERCENTILE

III I.0F--- _

zo~

a:ocUJ

u1 0.1UUa:...Ja:~

zo~ 0.01ocoJ:

M=50

7.57065605.5

8~ PERCENT IL.E

Chapter 2

7570656055

M:50

I0.001 L....J...L.J...LU.l---l-....l..-L....-JL.J.U.l.::IO-...I....-..L-J...u..u..uIOO,..,.---l-....l..-LJ L....J.-L..........U:----L---I.....L..J-J....lJl...LII..:-O-.L-l......J....L.J...L.l....L.l:IOO=--.....l...-....l..-LJ

DISTANCE • KM DISTANCE • KH

Figure 2-13. Predicted values of peak horizonlal acceleration for 50 and 84 percentile as functions of distance and momentmagnitude. [After Joyner and Boore (2-31).]

Figure 2-14. Comparison of attenuation curves for theeastern and western U.S. earthquakes. (Reproduced from

2-39.)

en

.; 0.7~...<:: 0.6""'"...'" 0.5uu<::...

O.~<::...z0N 0.3;;;:0:::w: 0.2<::'"a.

0.1

01 2 5 10 20 SO 100

CLOSEST HORIZONTAL DISTANCE FROM ZONEOF ENERGY RELEASE, km

the attenuation curves for far-field data. Similarobservations can also be made from theattenuation curves (Figure 2-13) proposed byJoyner and Boore (2-31). The majority ofattenuation studies and the relationshipspresented in Table 2-4 are primarily from thedata in the western United States. Severalseismologists believe that ground accelerationattenuates more slowly in the eastern UnitedStates and eastern Canada, i.e. earthquakes ineastern North America are felt at much greaterdistances from the epicenter than westernearthquakes of similar magnitude. Acomparison of the attenuation curves for thewestern and eastern United States earthquakesrecommended by Nuttli and Herrmann (2-41) isshown in Figure 2-14. Another comparison foreastern North America prepared by Milne andDavenport (2-28) is presented in Figure 2-15.Both these figures reflect the slower attenuationof earthquake motions in the eastern UnitedStates and eastern Canada. According toDonovan (2-23), a similar phenomenon also existsfor Japanese earthquakes. Due to the lack of


sufficient earthquake data in the eastern UnitedStates and Canada, theoretical models whichinclude earthquake source and wavepropagation in the surrounding medium areused to study the effect of distance and otherparameters on ground motion. The reader isreferred to references (2-42 to 2-44) for the detailedprocedure.

67

In(PGA) = -3.5 12+ 0.904Mw

-1.3281n ~R; + [0.14gexp(0.647M w )]2

+[1.125-0.1121n Rs -0.0957Mw]F

+ [0.440-0.1711n Rs]Ssr

+[0.405-0.2221n Rs]Shr +e(2-27)

where PGA is the mean of the two horizontalcomponents of peak ground acceleration (g),Mw is the moment magnitude, Rs i~ the closestdistance to the seismogenic rupture on the fault(km), F = 0 for strike-slip and normal faultearthquakes and = I for reverse, reverseoblique, and thrust fault earthquakes, Ssr = 1 forsoft rock and = 0 for hard rock and alluvium, Shr= I for hard rock and = 0 for soft rock andalluvium, and e is the random error term witha zero mean and a standard deviation equal to- In(PGA) which is represented by

"'00,.

,.

{

0.55

O'ln(PGA) = 0.173-0.l40ln(PGA)

0.39

PGA < 0.068

0.068 S; PGA S; 0.21

PGA > 0.21

(2-28)

3 Seismogenic rupture zone was determined from thelocation of surface fault rupture, the spatial distributionof aftershocks, earthquake modelling studies, regionalcrustal velocity profl.1es, and geodetic and geologicdata.

with a standard error of estimate 0.021.More recently, Boore et aI. (2-37) used

approximately 270 records to estimate the peakground acceleration in terms of I) the closesthorizontal distance Rjb (km) from the recordingstation to a point on the earth surface that liesdirectly above the rupture, 2) the momentmagnitude Mw, 3) the average shear wavevelocity of the soil Vs (m1sec) over the upper 30meters, and 4) the fault mechanism such that:

Figure 2-15. Intensity versus distance for eastern andwestern Canada. [After Milne and Davenport (2-28).]

In addition to source distance and earthquakemagnitude, recent attenuation relationshipsinclude the effect of source characteristics (faultmechanism) and soil conditions. As anexample, Campbell and Bozorgnia (2-36) used645 accelerograms from 47 worldwideearthquakes of magnitude 4.7 and greater,recorded between 1957 and 1993, to developattenuation relationship for peak horizontalground acceleration. The data was limited todistances of 60 km or less to minimize theinfluence of regional differences in crustalattenuation and to avoid the complexpropagation effects at farther distancesobserved during the 1989 Lorna Prieta andother earthquakes. The peak groundacceleration was estimated using a generalizednonlinear regression analysis and given by

In(PGA) =b+ 0.527(M W - 6.0)

~ 2 2 Vs- 0.778ln R"b + (5.570) - 0.371ln-J 1396

(2-29)

68 Chapter 2

Where b is a parameter that depends on thefault mechanism. They recommended

Figure 2-16. Peak ground acceleration versus distance forsoil sites for earthquake magnitudes of 6.5 and 7.5. [After

Boore et aI. (2-37).)

{

- 0.313 for the strike - slip earthquakes

b = -0.1l7 for the reverse - slip earthquakes (2-30)- 0.242 if the fault mechanism is not specified

Equation 2-29 is used for earthquakemagnitudes of 5.5 to 7.5 and distances less than80 lan. Although Equations 2-27 and 2-29 usedifferent definitions for the source distance, theequations indicate the decaying pattern of thepeak ground acceleration with distance. Figure2-16 shows the variation of the peak groundacceleration with distance computed fromEquation 2-29 for earthquakes of magnitude 6.5and 7.5 with an unspecified fault mechanismand for soils with a shear wave velocity of 310m1sec. Also shown in the figure is theattenuation relationship proposed by Joyner andBoore (2-32) (Equation 12 Table 2-4).

The variation of peak ground velocity withdistance from the source of energy release(velocity attenuation) has also been studied byseveral investigators such as Page et al. (2-11),

Boore et al. (2-45.2-46), Joyner and Boore (2-31), andSeed and Idriss (2-39). Velocity attenuationcurves have similar shapes and follow similartrends as the acceleration attenuation. Typicalvelocity attenuation curves proposed by Joynerand Boore are shown in Figure 2-17.Comparisons between Figures 2-13 and 2-17indicate that velocity attenuates somewhatfaster than acceleration.

The variation of peak ground displacementwith fault distance or the distance from the

0.1

0.2

1001

10 202

Mechanism: unspecified

- Boore 81 at (l994a): V", =310 mls_••• Joyner and lloo<e (19821; soil

fIf

••••• I

f 002I--,,---~-,,,,,,,,,,,,, --,-~-,-~~.-+! 0.01

2 10 20 100

Distance (km)

11 +I_--'-~_~~.L...-_'-~~~-'-'-

§c:0e

0.1 ~Q)1)uu« j

,:,r. Jell

~Q)ll.

0.02]0.01

50 PERCENTILE

.'00.0,ru

>-::g 10.0..JUJ>

rtI-Z0N • .0cr0J: - rock

-- soil

0.1 10

DISTANCE • K'"

e" PERCENTILE

10

DISTANCE • K'"

Figure 2-17. Predicted values of peak horizontal velocity for 50 and 84 percentile as functions of distance. momentmagnitude. and soil condition. [After Joyner and Boore (2-31).)

10


Qor---,----,-----,--,---.-------.------,,--,-----,-----,--,---.--.......

"..801---+---"c---f---+--t---+---+--I---+---+--+--t---+---l

'\ '\ I70t----t---t---.....,-+--t----j------;---t__-+---j---+--t---+---l

" LEGE~O

~ I'~ A M07.1III 60 t----;->oo,,---t----t-,>o,---+----t--+---+----+-__+_ oM, 6.7-7.1 -+----+--1~ "', '\ 0 M, 6.506.6~I " , SOIL SITE , M, 5.1'6.0u ~. I . 1M, 5.3 ·5.6U SO t----+---t-'->.r---+--+--"<-"--t--+---+----+--j-------+----+------'----j

~ ',', I~ ", " ~'.t5

4 0 I----+---"..,..J----+--->O",---+----+--=--..I::,------J'----+---+----+--'-----'-~

~' ,," "" "~, I--t-~,--_ 1 I

~ 30f---i~~:-+""--~,,---+---"oc:"'-1-,~-,9-'+-,--+-I--+-----+I----=::::.......,.-~_--l+----+-I----l

1'---8.. 0 "'-.,r--.- -"-, ~I~~__ I -"I'201--+--="""'==--_+....;:"",~"o_'''7--t---+-r----=~......-:=_'"'""-_+--t__-_;_--i

r--.- ~~,."~'.LLL J I -I,-- ~I !I -----. I Q J ---- 1--

~- "". '0 : -I-~ I~.,~5S r-a...-! .'f~ I \I~65 ,i -----_ I

"""'---.~ '50 " '1 ..., I i I'II 0; I, ~ In ~l I! II I,

o0'---~1.L0..:.....::..----J2-0--:3="=0-r-4,=,"0--S="=0r-=----:6"':-O--'---=-70~----=8:':-0---1..---='9 0:::----<-:17:00=---:-:"110~r-1:-:-27"0 ..LJ

EPtCENTRAL DISTANCE, KM

Figure 2-18. Duration versus epicentral distance and magnitude for soil. [After Change and Krinitzsky (2-47).]

69

source of energy release (displacementattenuation) can also be plotted. Boore et aI. (2

45.2-46) have presented displacement attenuationsfor different ranges of earthquake magnitude.Only a few studies have addressed displacementattenuations probably because of their limiteduse and the uncertainties in computingdisplacements accurately.

Distance also influences the duration ofstrong motion. Correlations of the duration ofstrong motion with epicentral distance havebeen studied by Page et al. (2-11), Trifunac andBrady (2.13), Chang and Krinitzsky (2-47), and

others. Page et aI., using the bracketed duration,conclude that for a given magnitude, theduration decreases with an increase in distancefrom the source. Chang and Krinitzsky, alsousing the bracketed duration, presented thecurves shown in Figures 2-18 and 2-19 forestimating durations for soil and rock as afunction of distance. These figures show that

for a given magnitude, the duration of strongmotion in soil is greater (approximately twotimes) than that in rock.

Using the 90% contribution of theacceleration intensity (/ a2 dt) as a measure ofduration, Trifunac and Brady (2-13) concludedthat the average duration in soil isapproximately 10-12 sec longer than that inrock. They also observed that the durationincreases by approximately 1.0 - 1.5 sec forevery 10 km increase in source distance.Although there seems to be a contradictionbetween their finding and those of Page et aI.and Chang-Krinitzsky, the contradiction stemsfrom using two different definitions. Thebracketed duration is based on an absoluteacceleration level (0.05g). At longer epicentraldistances, the acceleration peaks are smallerand a shorter duration is to be expected. Theacceleration intensity definition of duration isbased on the relative measure of the percentile

70 Chapter 2

60

70

,....,a>~ 50oIIU

~ 40'-

uWIII• 30

zQ~a:: 20:>o

10

i I,I iI

I

I!!

i i!

h·~1 I ROCK SiTEI

, I'". ~~.,,~ I', ;

I

I :1,,1 "'-.. !

~~! i"-"" """k~"-~ "" I ...~tI~ --l '-.. ..... -.. '-..... ............ ~,

I~ --' .'--i?- .....~'- ~""

I'~~~ 1'----. ...................-- - r---.... ::,-...::::::::::- -I-r-.::-- ""'- ~- '--10 20 30 40 50 60 ,0

EPICENTRAL DISTANCE, KM80 90 100 110

Figure 2-19. Duration versus epicentral distance and magnitude for rock. [After Chang and Krinitzsky (2-47).]

contribution to the acceleration intensity.Conceivably, a more intense shaking within ashorter time may result in a shorter durationthan a much less intense shaking over a longertime. According to Housner (2-38), at distancesaway from the fault, the duration of strongshaking may be longer but the shaking will beless intense than those closer to the fault.

Recently, Novikova and Trifunac (2-48) usedthe frequency dependent definition of durationdeveloped by Trifunac and Westermo (2-16) tostudy the effect of several parameters on theduration of strong motion. They employed aregression analysis on a database of 984horizontal and 486 vertical accelerograms from106 seismic events. Their study indicated anincrease in duration by 2 sec for each 10 km ofepicentral distance for low frequencies (near 0.2Hz). At high frequencies (15 to 20 Hz), theincrease in duration drops to 0.5 sec per each 10kIn.

Near-Source Effects. Recent studies haveindicated that near-source ground motions

contain large displacement pulses (grounddisplacements which are attained rapidly with asharp peak velocity). These motions are theresult of stress waves moving in the samedirection as the fault rupture, thereby producinga long-duration pulse. Conse-quently, nearsource earthquakes can be destructive tostructures with long periods. Hall et al. (2-49)

have presented data of peak groundaccelerations, velocities, and displacementsfrom 30 records obtained within 5 kIn of therupture surface. The ground accelerationsvaried from 0.31g to 2.0g while the groundvelocities ranged from 0.31 to 1.77 rnIsec. Thepeak ground displacements were as large as2.55 m. Figures 2-20 and 2-21 offer twoexamples of near-source earthquake groundmotions. The first was recorded at the LADWPRinaldi Receiving Station during the Northridgeearthquake of January 17, 1994. The distancefrom the recording station to the surfaceprojection of the rupture was less than 1.0 kIn.The figure shows a uni-directional ground


"orthridgeLADWP Rinaldi Receivin~ StationJan 17,1994

Acceleration (cm/s/s) "AX =-829.88

I~8 18

i

28

Velocity (CII/S)

Displacement (cm)

38 48Tillte (second)

"AX = 188.48

rIA)( = 92.48

Figure 2-20. Ground acceleration, velocity, and displacement time-histories recorded at the LADWP Rinaldi ReceivingStation during the Northridge earthquake of January 17, 1994.

displacement that resembles a smooth stepfunction and a velocity pulse that resembles afinite delta function. The second example,shown in Figure 2-21, was recorded at the SeELucerne Valley Station during the Landersearthquake of June 28, 1992. The distance fromthe recording station to the surface projection ofthe rupture was approximately 1.8 km. Apositive and negative velocity pulse thatresembles a single long-period harmonicmotion is reflected in the figure. Near-sourceground displacements similar to that shown inFigure 2-21 have also been observed with azero permanent displacement. The two figuresclearly show the near-source grounddisplacements caused by sharp velocity pulses.For further details, the reader is referred to thework of Heaton and Hartzell (2-50) andSomerville and Graves (2-51).

2.4.2 Site geology

Soil conditions influence ground motion andits attenuation. Several investigators such asBoore et al. (2-45 and 2-46) and Seed and Idriss (2-39)

have presented attenuation curves for soil androck. According to Boore et aI., peak horizontalacceleration is not appreciably affected by soilcondition (peak horizontal acceleration is nearlythe same for both soil and rock). Seed andIdriss compare acceleration attenuation for rockfrom earthquakes with magnitudes ofapproximately 6,6 with acceleration attenuationfor alluvium from the 1979 Imperial Valleyearthquake (magnitude 6.8). Their comparisonshown in Figure 2-22 indicates that at a givendistance from the source of energy release, peakaccelerations on rock are somewhat greater thanthose on alluvium. Studies from otherearthquakes indicate that this is generally thecase for

72 Chapter 2

Acceleration (cm/s/s) "AX =-585.88

I ='==

LandersSCE Lucerne Valley StationJun 28,1992

Velocity (cm/s) "AX = 115.38

8 18I

28

Displacement (cm)i i

38 48Tillie (second)

"AX =179.88

Figure 2-21. Ground acceleration, velocity, and displacement time-histories recorded at the SeE Lucerne Valley Stationduring the Landers earthquake of June 28, 1992.

0.7

'" 0.6 ~MEAN FOR ROCK SITES,.z Ms " 6.6 (From Fig. III~t- 0.5~

'"....'" 0.4uu<t....<t 0.3t-z0N

MEAN FOR IMPERIAL VALLEY"""'";;: 0.20:l: 1979 EARTHQUAKE. M = 6.8

"" (From Fig. 15) s<t0.1'"0-

01 2 5 10 20 50

CLOSEST HORIZONTAL OISTANCE FROM ZONEOF ENERGY RELEASE, km

Figure 2-22. Comparison of attenuation curves for rocksites and the Imperial Valley earthquake of 1979. [After

Seed and Idriss (2-39).]

acceleration levels greater than approximatelyO.1g. At levels smaller than this value,accelerations on deep alluvium are slightlygreater than those on rock. The effect of soilcondition on peak acceleration is illustrated by

Seed and Idriss in Figure 2-23. According tothis figure, the difference in acceleration onrock and on stiff soil is not that significant.Even though in specific cases, particularly softsoils, soil condition can affect peakaccelerations, Seed and Idriss conclude that theinfluence of soil condition can generally beneglected when using acceleration attenuationcurves. In a more recent study, Idriss (2-52\ usingthe data from the 1985 Mexico City and the1989 Lorna Prieta earthquakes, modified thecurve for soft soil sites as shown in Figure 2-24.In these two earthquakes, soft soils exhibitedpeak ground accelerations of almost 1.5 to 4times those of rock for the acceleration range of0.05g to O.1g. For rock accelerations larger thanapproximately O.lg, the acceleration ratiobetween soft soils and rock tends to decrease toabout 1.0 for rock accelerations of O.3g to 0.4g.The figure indicates that large rockaccelerations are amplified through soft soils toa lesser degree and may even be slightly deamplified.


0.6 r--,....---r--"'T"""--r----r---.,..-,...... The effect of site geology on peak groundacceleration can be seen in Equation 2-27proposed by Campbell and Bozorgnia (2-36). Theratios of peak ground acceleration on soft rockand on hard rock to that on alluvium (defined assite amplification factors) were computed fromEquation 2-27 and are shown in Figure 2-25.The figure indicates that rock sites have higheraccelerations at shorter distances and loweraccelerations at longer distances as compared toalluvium sites, with ground accelerations onsoft rock consistently higher than those on hardrock.

Recent studies on the influence of sitegeology on ground motion use the averageshear wave velocity to identify the soilcategory. Boore et al. (2·37) used the averageshear wave velocity for the upper 30 meters ofthe soil layer to characterize the soil conditionin the attenuation relationship in Equation 2-29.The equation indicates that for the samedistance, magnitude, and fault mechanism, asthe soil becomes stiffer (i.e. a higher shear wavevelocity), the peak ground acceleration becomessmaller. The recent UBC code and NEHRPrecommended provisions use shear wavevelocities to identify the different soil profileswith a shear wave velocity of 1500 m1sec orgreater defining hard rock and a shear wavevelocity of 180 m1sec or smaller defining softsoil (Section 2.9).

There is a general agreement among variousinvestigators that the soil condition has apronounced influence on velocities anddisplacements. According to Boore et al. (2-45 and

2-46), Joyner and Boore (2-31), and Seed and Idriss(2-39); larger peak horizontal velocities are to beexpected for soil than rock. A statistical studyof earthquake ground motion and responsespectra by Mohraz (2-53) indicated that theaverage velocity to acceleration ratio forrecords on alluvium is greater than thecorresponding ratio for rock.

Using the frequency dependent definition ofduration proposed by Trifunac and Westermo (2

16\ Novikova and Trifunac (2-48) determined thatfor the same epicentral distance and earthquakemagnitude, the strong motion duration for

0.1 O.:! O..l O~ O.

eeel ration on Rock ilcs (g)

.-~IUc:~ 0._c;; 0.1OJ"0:Jo«

Figure 2-24. Variation of peak accelerations on soft soilcompared to rock for the 1985 Mexico City and the 1989

Lorna Prieta earthquakes. [After Idriss (2-52).]

Figure 2-25. Variation of site amplification factors (ratioof peak ground acceleration on rock to that on alluvium)with distance. [After Campbell and Bozorgnia (2-36).]

0.1 02 0.3 0.4 0.5 0.6 0.7

Puk Horizontal Acceleration on Rock - g

~ 0.6r--------------~JI

~ ;;; 0..>

Figure 2-23. Relationship between peak accelerations onrock and soil. [After Seed and Idriss (2-39).]

1.3 , ,£ 1.2

, , ,() ,0

1.1 "'u.. "' "'c: "'.2 1.0,

"'-0 "'() "'~ 0.9 "' "'a. "' "'E 0.8 "' "'<: - - - - Soft Rock~

- Hard Rock

iii 0.7

0.61 10 100

Distance to Seismogenic Rupture (km)

~ 0.5co'; 0.4

! 0.3-;co.~ 02~ .

:%:

~.

:. 0.1

74 Chapter 2

Table 2-5. Maximum Ground Accelerations and Durationsof Strong Phase of Shaking [after Housner (2-54)]

Figure 2-26. Relationship between magnitude andduration of strong phase of shaking. [After Donovan (2

23).]

Magnitude Maximum DurationAcceleration (%g) (sec)

5.0 9 25.5 15 66.0 22 126.5 29 187.0 37 247.5 45 308.0 50 348.5 50 37

(2-47) give approximate upper-bound for durationfor soil and rock (Table 2-7). Their values forsoil are close to those presented by Page et al.,and the ones for rock are consistent with thosegiven by Housner and by Donovan. The studyby Novikova and Trifunac (2-48) which uses thefrequency dependent definition of durationpresents a quadratic expression for the durationin terms of earthquake magnitude. Their studyindicates that the duration of strong motiondoes not depend on the earthquake magnitude atfrequencies less than 0.25 Hz. For higherfrequencies, the duration increasesexponentially with magnitude.

6 7RICHTER MAGNITUDE - M

• ,/

o .4.11(M-5)~V

V;/• •

;/. •:.~ • •. :

~•• • •, •• ••• ••••

••• • ••o

4

u.o

'" 10o....~::>o

~

V>«

'"Q.

'" 20

~....v>

50

'"'"'"V> 30u.o

V>o

'"ou~

V>o

~ 40

records on a sedimentary site is longer than thaton a rock site by approximately 4 to 6 sec forfrequencies of 0.63 Hz and by about 1 sec forfrequencies of 2.5 Hz. The records onintermediate sites, furthermore, exhibited ashorter duration than those on sediments. Theyindicated that for frequencies of 0.63 to 21 Hz,the influence of the soil condition on theduration is noticeable.

2.4.3 Magnitude

Different earthquake magnitudes have beendefined, the more common being the Richtermagnitude (local magnitude) M L' the surface

wave magnitude Ms, and the moment

magnitude Mw (see Chapter 1). As expected,

at a given distance from the source of energyrelease, large earthquake magnitudes result inlarge peak ground accelerations, velocities, anddisplacements. Because of the lack of adequatedata for earthquake magnitudes greater than 7.5,the influence of the magnitude on peak groundmotion and duration is generally determinedthrough extrapolation of data from earthquakemagnitudes smaller than 7.5. Attenuationrelationships are also presented as a function ofmagnitude for a given source distance asindicated in Equations 2-27 and 2-29. Bothequations show that for a given distance, soilcondition, and fault mechanism, the larger theearthquake magnitude, the larger is the peakground acceleration. Figure 2-16, plotted usingEquation 2-29, confirms this observation.

The influence of earthquake magnitude onthe duration of strong motion has been studiedby several investigators. Housner (2-38 and 2-54)

presents values for maximum acceleration andduration of strong phase of shaking in thevicinity of a fault for different earthquakemagnitudes (Table 2-5). Donovan (2-23) presentsthe linear relationship in Figure 2-26 forestimating duration in terms of magnitude. Hisestimates compare closely with those presentedby Housner in Table 2-5. Using the bracketedduration (0.05g), Page et al. (2-11) give estimatesof duration for various earthquake magnitudesnear a fault (Table 2-6). Chang and Krinitzsky

75

Soil812162332456286

Rock468II16223143

Magnitude5.05.56.06.57.07.58.08.5


Table 2-6. Duration of Strong Motion Near Fault [after earthquakes. Joyner and Boore (2-59) believe thisPage et aI. (2-II)) ratio should be 1.25.

Magnitude Duration (sec) Recent attenuation relationships include the5.5 10 effects of fault mechanism on ground motion as6.5 17 indicated in Equations 2-27 and 2-29. Equation7.0 ;~ 2-27 by Campbell and BOlorgnia (2-36) indicates~:~ 60 that reverse, reverse-oblique, and thrust fault8.5 90 earthquakes result in larger ground

accelerations than strike-slip and normal faultearthquakes. Figure 2-27, computed fromEquation 2-27, shows the variation of peakground acceleration with distance forearthquakes with different magnitudes and faultmechanisms on alluvium. Similar observationscan also be made from Equation 2-29 by Booreet a1. (2-37) where reverse earthquakes result inhigher accelerations than strike-slipearthquakes.

Table 2-7. Strong Motion Duration for DifferentEarthquake Magnitudes [after Chang and Krinitzsky (247))

2.4.5 Directivity

10 100Distance to Seismagenic Rupture (km)

Figure 2-27. Peak ground acceleration versus distance fordifferent magnitudes and fault mechanisms. [After

Campbell and BOlorgnia (2-36).]

Directivity relates to the azimuthal variationof the angle between the direction of rupturepropagation (or radiated seismic energy) andsource-to-site vector, and its effect onearthquake ground motion. Large groundaccelerations and velocities can be associatedwith small angles since a significant portion ofthe seismic energy is channeled in the directionof rupture propagation. Consequently, when alarge urban area is located within the smallangle, it will experience severe damage.

- Strike Slip- - - - Reverse

:§c.2'0v 0.1Qjuu<:~

0.,a.

0.01

2.4.4 Source characteristics

Factors such as fault mechanism, depth, andrepeat time have been suggested by severalinvestigators as being important in determiningground motion amplitudes because of theirrelation to the stress state at the source or tostress changes associated with the earthquake.Based on the state of stress in the vicinity of thefault, many investigators believe that largeground motions are associated with reverse andthrust faults whereas smaller ground motionsare related to normal and strike-slip faults.

The above observations agree with the studyby McGarr (2-55, 2-56) who concluded that groundacceleration from reverse faults should begreater than those from normal faults, withstrike-slip faults having intermediateaccelerations. McGarr also believes that groundmotions increase with fault depth. Kanamoriand Allen (2-57) presented data showing thathigher ground motions are associated withfaults with longer repeat times since theyexperience large average stress drops. Usingempirical equations, Campbell (2-58) found thatpeak ground acceleration and velocity inreverse-slip earthquakes are larger by about 1.4to 1.6 times than those in strike-slip

76 Chapter 2

used to estimate the recurrence intervals offuture earthquakes with certain magnitudeand/or intensity. These models together with theappropriate attenuation relationships arecommonly utilized to estimate ground motionparameters such as peak acceleration andvelocity corresponding to a specifiedprobability and return period. Among theearthquake recurrence models mostly used inpractice is the Gutenberg-Richter relationship (2

63. 2-64) known as the Richter law of magnitudewhich states that there exists an approximatelinear relationship between the logarithm of theaverage number of annual earthquakes andearthquake magnitude in the form

where N(m) is the average number ofearthquakes per annum with a magnitudegreater than or equal to m, and A and Bareconstants determined from a regression analysisof data from the seismological and geologicalstudies of the region over a period of time. TheGutenberg-Richter relationship is highlysensitive to magnitude intervals and the fittingprocedure used in the regression analysis (2-65. 2

66.2-33). Figure 2-28 shows a typical plot of theGutenberg-Richter relationship presented bySchwartz and Coppersmith (2-66) for the southcentral segment of the San Andreas Fault. Therelationship was obtained from historical andinstrumental data in the period 1900-1980 for a40-kilometer wide strip centered on the fault.The box shown in the figure representsrecurrence intervals based on geological datafor earthquakes of magnitudes 7.5-8.0 (2-67). It isapparent from the figure that the extrapolatedportion of the Gutenberg-Richter equation(dashed line) underestimates the frequency ofoccurrence of earthquakes with largemagnitudes, and therefore, the model requiresmodification of the B-value in Equation 2-31for magnitudes greater than approximately 6.0(2-33)

According to Faccioli (2-60l, in the Northridgeearthquake of January 17, 1994; the rupturepropagated in the direction opposite fromdowntown Los Angeles and San FernandoValley, causing moderate damage. In theHyogoken-Nanbu (Kobe) earthquake of January17, 1995, the rupture was directed toward thedensely populated City of Kobe resulting insignificant damage. The stations that lie in thedirection of the earthquake rupture propagationwill record shorter strong motion durations thanthose located opposite to the direction ofpropagation (2-61).

Boatwright and Boore (2-62), believe thatdirectivity can significantly affect strongground motion by a factor of up to 10 forground accelerations. Joyner and Boore (2-59)

indicate, however, that it is not clear how toincorporate directivity into methods forpredicting ground motion in future earthquakessince the angle between the direction of rupturepropagation and the source-to-recording-sitevector is not known a priori. Moreover, forsites close to the source of a large magnitudeearthquake, where a reliable estimate of groundmotion is important, the angle changes duringthe rupture propagation. Most ground motionprediction studies do not explicitly include avariable representing directivity.

2.5 EVALUATION OF SEISMICRISK AT A SITE

Evaluating seismic risk is based oninformation from three sources: 1) the recordedground motion, 2) the history of seismic eventsin the vicinity of the site, and 3) the geologicaldata and fault activities of the region. For mostregions of the world this information,particularly from the first source, is limited andmay not be sufficient to predict the size andrecurrence intervals of future earthquakes.Nevertheless, the earthquake engineeringcommunity has relied on this limitedinformation to establish some acceptable levelsof risk.

The seismic risk analysis usually begins bydeveloping mathematical models, which are

log N(m) =A- Bm (2-31)


where f(Ri , a) is a function which can beobtained from the attenuation relationships.Substituting Equation 2-33 into Equation 232, one obtains

where Ai and Bi are known constants for

the ith sub-source.3. Assuming that the design ground motion is

specified in terms of the peak groundacceleration a and the epicentral distancefrom the ith sub-source to the site is Ri , themagnitude ma,i of an earthquake initiated atthis sub-source maybe estimated from

(2-33)

(2-34)

~.\

\\ \ 0

\\\\

\

o 1900-1932• 1932-1980e:. 1857 ittershocks0.001

~....>

8-E

".8E:0Z...~..:;E:0

U

Figure 2-28. Cumulative frequency-magnitude plot. Thebox in the figure represents range of recurrence based on

geological data for earthquake magnitudes of 7.S-8. [AfterSchwartz and Coppersmith (2-66); reproduced from Idriss

(2-33).]

Cornell (2-68) introduced a simplified methodfor evaluating seismic risk. The methodincorporates the influence of all potentialsources of earthquakes. His procedure asdescribed by Vanmarcke (2-69) can besummarized as follows:1. The potential sources of seismic activity are

identified and divided into smaller subsources (point sources).

2. The average number of earthquakes perannum N;{m) of magnitudes greater than orequal to m from the ith sub-source isdetermined from the Gutenberg-Richterrelationship (Equation 2-31) as

Magnitude. m

log N;(m) = Ai - Bim

(2-35)Na = LNi(ma,i)all

Assuming the seismic events are independent(no overlapping), the total number ofearthquakes per annum Na which may resultin a peak ground acceleration greater than orequal to a is obtained from the contributionof each sub-source as

4. The mean return period Ta in years isobtained as

1Ta =N (2-36)

a

In the above expression, Na can be alsointerpreted as the average annual probability Aathat the peak ground acceleration exceeds acertain acceleration a. In a typical designsituation, the engineer is interested in theprobability that such a peak exceeds a duringthe life of structure h. This probability can beestimated using the Poisson distribution as

P = 1- e-'\h (2-37)

9

(2-32)

87654J

78

Another distribution based on a Bayesianprocedure (2-70) was proposed by Donovan (2-23).

The distribution is more conservative than thePoisson distribution, and therefore moreappropriate when additional uncertainties such

Chapter 2

as those associated with the long return periodsof large magnitude earthquakes areencountered. It should be noted that otherground motion parameters in lieu ofacceleration such as spectral ordinates may be

• •

37 0 ~LRICHTER

SYMBOL MAGNITUDE

• 4

• 5• 6

• 7• 8

•

•

I

--~-• I

••

II

121<'

Figure 2-29. Instrumental or estimated epicentrallocations within 100 kilometers of San Francisco. [After Donovan (223).]


used for evaluating seismic risk. Otherprocedures for seismic risk analysis based onmore sophisticated models have also beenproposed (see for example Der Kiureghian andAng,2-71).

The evaluation of seismic risk at a site isdemonstrated by Donovan (2-23) who used as anexample the downtown area of San Francisco.The epicentral data and earthquake magnitudeshe considered in the evaluation were obtainedover a period of 163 years and are depicted inFigure 2-29. The data is associated with threemajor faults, the San Andreas, Hayward, andCalaveras. Using attenuation relationships forcompetent soil and rock, Donovan computedthe return periods for different peakaccelerations (see Table 2-8). He thencomputed the probability of exceeding variouspeak ground accelerations during a fifty-yearlife of the structure which is shown in Figure 230. Plots such as those in Figure 2-30 may beused to estimate the peak acceleration forvarious probabilities. For example, if thestructure is to be designed to resist a moderateearthquake with a probability of 0.6 and asevere earthquake with a probability of between0.1 and 0.2 of occurring at least once during thelife of the structure, the peak accelerationsusing Figure 2-30(b) for rock, are 0.15g andO.4g, respectively.

79

map, which shows contours of peakacceleration on rock having a 90% probabilityof not being exceeded in 50 ~ears. The AppliedTechnology Council ATC (2- 4) used this map todevelop similar maps for effective peakacceleration (Figure 2-32) and effective peakvelocity-related acceleration (Figure 2-33). Theeffective peak acceleration Aa and the

effective peak velocity-related acceleration Av

are defined by the Applied Technology Council(2-74) based on a study by McGuire (2-75). They

are obtained by dividing the spectralaccelerations between periods of 0.1 to 0.5 secand the spectral

\,0 r=::---------------,

0.8

0.6

0.1

1.0 2.0 J.O 4.0 5.0NUHBEP OF OCCURR{NC£.S • CUMULATIVE

(a) Based on return periods using relationship 9 in Table 2-4

\,0 ~---=::==__---------,

Table 2-8. Return Periods for Peak: Ground Accelerationin the San Francisco Bay Area [after Donovan (2-23)]

Peak: Return Period (years)Acceleration Soil Rock

QM 4 8QW 20 300.15 50 60QW 100 1000.25 250 2000.30 450 3000.40 2000 700

0.6

0.'

0.1

1.0 2.0 3.0 4.0rWI-1HfR OF OCCURRfNC£S • CUI'll'UT IV£

S.O

2.5.1 Development of seismic maps(b) Based on return periods using attenuation equation for rock

Using the seismic risk principles of Cornell(2~), Algermissen and Perkins (2-TI, 2-73)

developed isoseismal maps for peak groundaccelerations and velocities. Figure 2-31 is a

Figure 2-30. Estimated probabilities for a fifty yearproject life. [After Donovan (2-23).]

00 o

ii' '-:

It'

..IC

l' I I~I

...'".L

r,I

'.I I

~I

......J

Jl',

i..

I~ ...

-1,...

I'. ...

"d~;+--:'

'II'

.n: 1....

.,,

,,.

.,

.---"\

~.'

~r

" "

~

.,

vS

<;""

"CII

,•••

•,0

'...

·'.1

."

.,...

,'1

-4'1

..,."

"

.. .----.----

-l

Pr~limino')'

MO

pot

Hor

ilon

tal

Acc

ele,

ohon

(E.p

r.ss

edA

tP

lftt

n'

Of

Gro

vi.y

)In

Roe

ll.W

ir,..

90

P,r

C'"

'P

rob

abil

ity

01N

ollle

inV

Eac

..d

.dIn

~O

Y••r

.

'"_

...lo

cu

l".t

_•."

'-'

''''

10

""Ct

'"C

..."

""..

....

,...

","•

.•••

•,..

'-"

ttl

''''

''II

I(til

t••

,..,

'I1

0,•.

(t"

,or

,.IU

•."

,'~

.'111

cO

fSc

"""'

'''a

,.,•

..,1

11

r-

'n""

Ill'

-----1

"'-.

r'

II'

U'I

_

II'..

It

,..I

I..

.II

'1

\..

e'n'

Fig

ure

2-31

,Se

ism

icri

skm

apde

velo

ped

byA

lger

mis

sen

and

Perk

ins.

(Rep

rodu

ced

from

2-74

,)Q {; ~ .... N

Fig

ure

2-32

.C

onto

urm

apfo

ref

fect

ive

peak

acce

lera

tion

(AT

C,

2-74

).

!'J ~ ~ ~ ~ C':l 2l ::: s.. ~ ..... o· ~ ~ s.. ::tl

(\:)

~ c ~ '"(\:) ~ (\:)

~ ~ 00 ......

82 Chapter 2

ALASKA

PUERTO RICO

CONTOUR MAP FOR EFFECTIVE PEAK ACCELERATION

Figure 2-32. (continued)


84 Chapter 2

ALASKA

HAWAIIPUERTO RICO

CONTOUR MAP FOR EFFECTIVE PEAK VELOCITY-RELATEDACCELERATION COEFFICIENT

Figure 2-33. (continued)


velocity at a period of approximately 1.0 sec bya constant amplification factor (2.5 for a 5%damped spectrum). It should be noted that theeffective peak acceleration will generally besmaller than the peak acceleration while theeffective peak velocity-related acceleration isgenerally greater than the peak velocity (2-75).

The Aa and Av maps developed from theATC study are in many ways similar to theAlgermissen-Perkins map. The most significantdifference is in the area of highest seismicity inCalifornia. Within such areas, the AlgermissenPerkins map has contours of 0.6g whereas theATC maps have no values greater than OAg.This discrepancy is due to the differencebetween peak acceleration and effective peakacceleration and also to the decision by theparticipants in the ATC study to limit thedesign value to OAg based on scientificknowledge and engineering judgment. TheATC maps were also provided with the contourlines shifted to coincide with the countyboundaries.

The 1985, 1988, 1991 and 1994 NationalEarthquake Hazard Reduction Program(NEHRP) Recommended Provisions forSeismic Regulations for New Buildings (2-76 to 2

79) include the ATC Aa and Av maps whichcorrespond to a 10% probability of the groundmotion being exceeded in 50 years (a returnperiod of 475 years). The 1991 NEHRPprovisions (2-78) also introduced preliminaryspectral response acceleration maps developedby the United States Geological Survey (USGS)for a 10% probability of being exceeded in 50years and a 10% probability of being exceededin 250 years (a return period of 2,375 years).These maps, which include elastic spectralresponse accelerations corresponding to 0.3 and1.0 sec periods, were introduced to present newand relevant data for estimating spectralresponse accelerations and reflect the variabilityin the attenuation of spectral acceleration and infault rupture length (2-78).

The 1997 NEHRP recommended provisions (2-80) provide seismic maps for thespectral response accelerations at the shortperiod range (approximately 0.2 sec) and at a

period of 1.0 sec. The maps correspond to themaximum considered earthquake, defined as themaximum level of earthquake ground shakingthat is considered reasonable for design ofstructures. In most regions of the United States,the maximum considered earthquake is definedwith a uniform probability of exceeding 2% in50 years (a return period of approximately 2500years). It should be noted that the use of themaximum considered earthquake was adoptedto provide a uniform protection against collapseat the design ground motion. While theconventional approach in earlier editions of theprovisions provided for a uniform probabilitythat the design ground motion will not beexceeded, it did not provide for a uniformprobability of failure for structures designed forthat ground motion. The design ground motionin the 1997 NEHRP provisions is based on alower bound estimate of the margin againstcollapse which was judged, based onexperience, to be 1.5. Consequently, the designearthquake ground motion was selected at aground shaking level that is 1/1.5 or 2/3 of themaximum considered earthquake groundmotion given by the maps.

The 1997 NEHRP Guidelines for theSeismic Rehabilitation of Buildings (2-81),

known as FEMA-273, introduce the concept ofperformance-based design. For this concept, therehabilitation objectives are statements of thedesired building performance level (collapseprevention, life safety, immediate occupancy,and operational) when the building is subjectedto a specified level of ground motion.Therefore, multiple levels of ground shakingneed to be defined by the designer. FEMA-273provides two sets of maps; each set includes thespectral response accelerations at short periods(0.2 sec) and at long periods (1.0 sec). One setcorresponds to a 10% probability of exceedancein 50 years, known as Basic Safety Earthquake1 (BSE-l), and the other set corresponds to a2% probability of exceedance in 50 years,known as Basic Safety Earthquake 2 (BSE-2),which is similar to the Maximum ConsideredEarthquake of the 1997 NEHRP provisions (2

80). FEMA-273 also presents a method for

86

adjusting the mapped spectral accelerations forother probabilities of exceedance in 50 yearsusing the spectral accelerations at 2% and 10%probabilities.

The Aa and Av maps, developed during theATC study, were also used, after somemodifications, in the development of a singleseismic map for the 1985, 1988, 1991, 1994,and 1997 editions of the Uniform BuildingCode (2-82 to 2-86). The UBC map shows contoursfor five seismic zones designated as 1, 2A, 2B,3, and 4. Each seismic zone is assigned a zonefactor Z, which is related to the effective peakacceleration. The Z factors for the five zonesare 0.075, 0.15, 0.20, 0.30, and 0.40 for zones1, 2A, 2B, 3, and 4; respectively. The onlychange in the UBC seismic map occurred in the1994 edition (2-85) reflecting new knowledgeregarding the seismicity of the PacificNorthwest of the United States.

2.6 ESTIMATING GROUNDMOTION

In the late sixties and early seventies, theseverity of the ground motion was generallyspecified in terms of peak horizontal groundacceleration. Most attenuation relationshipswere developed for estimating the expectedpeak horizontal acceleration at the site.Although structural response and to someextent damage potential to structures can berelated to peak ground acceleration, the use ofthe peak acceleration for design has beenquestioned by several investigators on thepremise that structural response and damagemay relate more appropriately to effective peakacceleration Aa and effective peak velocityrelated acceleration Av. Early Studies byMohraz et al. (2-9), Mohraz (2-53), Newmark andHall (2-87) and Newmark et al. (2-88)

recommended using ground velocity anddisplacement, in addition to groundacceleration, in defining spectral shapes andordinates.

Prior to the 1971 San Fernando earthquakewhere only a limited number of records wasavailable, Newmark and Hall (2-89. 2-90)

Chapter 2

recommended that a maximum horizontalground velocity of 48 inlsec and a maximumhorizontal ground displacement of 36 in. beused for a unit (l.Og) maximum horizontalacceleration. Newmark also recommended thatthe maximum vertical ground motion be takenas 2/3 of the corresponding values for thehorizontal motion.

With the availability of a large number ofrecorded earthquake ground motion,particularly during the 1971 San Fernandoearthquake, several statistical studies (2-9.2-91.2-53)

were carried out to determine the average peakground velocity and displacement for a givenacceleration. These studies recommended tworatios: peak velocity to peak acceleration viaand peak acceleration-displacement product tothe square of the peak velocity ad/v2 be used inestimating ground velocities and displacements.Certain response spectrum characteristics suchas the sharpness or flatness of the spectra can berelated to the ad/v2 ratio as discussed later.According to Newmark and Rosenbleuth (2-92),

for most earthquakes of practical interest, ad/v2

ranges from approximately 5 to 15. Forharmonic oscillations, ad/v2 is one and forsteady-state square acceleration waves, the ratiois one half.

A statistical study of via and ad/v2 ratios wascarried out by Mohraz (2-53) who used a total of162 components of 54 records from 16earthquakes. A summary of the via and ad/v2

ratios for records on alluvium, on rock, and onalluvium layers underlain by rock are given inTable 2-9. It is noted that via ratios for rock aresubstantially lower than those for alluvium withthe via ratios for the two intermediatecategories falling between alluvium and rock.Table 2-9 also shows that the via ratios for thevertical components are close to those for thehorizontal components with the larger of thetwo peak accelerations. The 50 percentile viaratios for the larger of the two peakaccelerations from Table 2-9 (24 (inlsec)/g forrock and 48 (inlsec)/g for alluvium) and thosegiven by Seed and Idriss (2-39) (22 (inlsec)/g forrock and 43 (inlsec)/g for alluvium) are in closeagreement. The ad/v2 ratios in Table 2-9


Table 2-9. Summary of Ground Motion Relationships [after Mohraz (2-53)]Soil Category Group· via (in/sec)/g adII

L 24 5.3Rock S 27 5.2

V 28 6.1

<30 ft of L 30 4.5alluvium S 39 4.2

underlain by rock V 33 6.8

30-200 ft of L 30 5.1alluvium S 36 3.8

underlain by rock V 30 7.6

L 48 3.9Alluvium S 57 3.5

V 48 4.6

• L: Horizontal components with the larger of the two peak accelerations

S: Horizontal components with the smaller of the two peak accelerations

V: Vertical components

d/a (in/g)81012

111719

121318

232927

87

a".Tfkal(alwrizOllkJJJL

0.48

0.47

0.40

0.42

indicate that, in general, the ratios for alluviumare smaller than those for rock and those foralluvium layers underlain by rock. The d/aratios are also presented in Table 2-9. Thevalues indicate that for a given acceleration, thedisplacements for alluvium are 2 to 3 timesthose for rock. The table also includes the ratioof the vertical acceleration to the larger of thetwo peak horizontal accelerations where it isapparent that the ratios are generally close toeach other indicating that soil condition doesnot influence the ratios. The ratio of the verticalto horizontal acceleration of 2/3 whichNewmark recommended is too conservative,but its use was justified to account for thevariations greater than the median and theuncertainties in the ground motion in thevertical direction (2-91).

Statistical studies of via and ad/v2 ratios forthe Lorna Prieta earthquake of October 17, 1989were carried out by Mohraz and Tiv (2-93). Theyused approximately the same number ofhorizontal components of the records on rockand alluvium that Mohraz (2-53) used in hisearlier study. Their study indicated a mean viaratio of 51 and 49 (in/sec)/g and a mean ad/v2

ratio of 2.8 and 2.6 for rock and alluvium,respectively. The differences in via and ad/v2

ratios from the Lorna Prieta and previousearthquakes indicate that each earthquake isdifferent and that site condition, magnitude,epicentral distance, and duration influence thecharacteristics of the recorded ground motion.

2.7 EARTHQUAKE RESPONSESPECTRA

Response spectrum is an important tool inthe seismic analysis and design of structuresand equipment. Unlike the power spectraldensity which presents information about inputenergy and frequency content of groundmotion, the response spectrum presents themaximum response of a structure to a givenearthquake ground motion. The response

d b B· ~I ~~ dspectrum introduce y lot . anHousner (2-3) describes the maximum responseof a damped single-degree-of-freedom (SDOF)oscillator at different frequencies or periods.The detailed procedure for computing andplotting the response spectrum is discussed inChapter 3 of this handbook and in a number ofpublications (see for example 2-54, 2-22, 2-94,2-95). It was customary to plot the responsespectrum on a tripartite paper (four-waylogarithmic paper) so that at a given frequency

88 Chapter 2

50

10

....../ ......

fe. •.... . .

SV

PSV

~ ....... ,...,..".t•••••.,.,

,'''\',

,,'"

e .......,2510I

Frequency, cps.0.1

0.5 +-......--.-".-----,---,.-.---.--.,--,...-..,...,----r---,---,-....,....-...,-..--,-,----,--,0.05

Figure 2-34. Comparison of pseudo-velocity and maximum relative velocity for 5% damping for the SOOE component of EICentro. the Imperial Valley earthquake of May 18.1940.

or period, the maximum relative displacementSD, the pseudo-velocity PSV, and the pseudoacceleration PSA can all be read from the plotsimultaneously. The parameters PSV and PSAwhich are expressed in terms of SD and thecircular natural frequency (J) as PSV = WSDand PSA = oJSD have certain characteristicsthat are of practical interest (2-87). The pseudovelocity PSV is close to the maximum relativevelocity SV at high frequencies (frequenciesgreater than 5 Hz), approximately equal forintermediate frequencies (frequencies between0.5 Hz and 5 Hz) but different for lowfrequencies (frequencies smaller than 0.5 Hz) asshown in Figure 2-34. In a recent study bySadek et at. (2-96\ based on a statistical analysisof 40 damped SDOF structures with periodrange of 0.1 to 4.0 sec subjected to 72accelerograms, it was found that the maximumrelative velocity SV is equal to the pseudovelocity PSV for periods in the neighborhood of

0.5 sec (frequency of 2 Hz). For periods shorterthan 0.5 sec, SV is smaller than PSV while forperiods longer than 0.5 sec, SV is larger andincreases as the period and damping ratioincrease. A regression analysis was used toestablish the following relationship between themaximum velocity and pseudo-velocityresponses:

(2-38)

where av = 1.095 +0.647{3 -0.382{32,

bv =0.193+0.838{3 -0.621{32, T is the

natural period, and {3 is the damping ratio. The

relationship between SV and PSV is presentedin Figure 2-35.


-~.O2 - - - ~.O5-~.IO - C1- ~.15

-~.20 - C1- ~.30

-~.40 - 06- ~.50

-R=O.60

2.5 ,..----~------------_=_____a

2 ~

0:;";:. 1.5C/)Q........;:.C/)

CIIIQJ

~

0.5 ",

43.532.521.50.5

o+-'-""--'--'-+-'...........-'-+.....................,t-'-'...........+-'-..........'-+-'...........-'-+..........""--'--t-'-'~

o

Period (s)

Figure 2-35. Mean ratio of maximum relative velocity to pseudo-velocity for SDOF structures with different dampingratios. [After Sadek et at. (2-96).]

10

~1..._.....PSA

tJl

1:'.~e 0.1

~\J\J

"l:

0.01

25/0/

Frequency, cps.0.1

0.00' +-,....,r-r-.-r---.,--.,---.----r--.--,-,.,-,-------r-----r-.,-...-....--,,""T"T---...--,0.05

Figure 2-36. Comparison of pseudo-acceleration and maximum absolute acceleration for 5% damping for the SOOEcomponent of EI Centro, the Imperial Valley earthquake of May 18, 1940.

'0

10

10

Chapter 2

,

• 6Period, sec

• 6Period, sec

• 6Period, sec

ot---r---,------,--,-----,o

i J

E~ 2

.!2

.~0,

1.5 2 2.5 3 3.5 4

Period (5)

0.5 .j..o...........+o-.........+o-.........+-'-".........+-'-".........+-'-".........+-'-".........+-'-"-.........j

o 0.5

Figure 2-37. Mean ratio of maximum absoluteacceleration to pseudo-acceleration for SDOF structureswith different damping ratios. [After Sadek et aI. (2-96).]

90

3.5

3

i 2.5cr

~ll..'- 2

~CIII41

1.5~

Figure 2-39. Acceleration, velocity, and displacementamplifications plotted as a function of period for 5%damping for the SOOE component of El Centro, the

Imperial Valley earthquake of May 18, 1940.

For zero damping, the pseudo-accelerationPSA is equal to the maximum absoluteacceleration SA, but for dampings other thanzero, the two are slightly different. For theinherent damping levels encountered in mostengineering applications, however, the two canbe considered approximately equal (see Figure2-36). When a structure is equipped withsupplemental dampers to provide large dampingratios, the difference between PSA and SAbecomes significant, especially for structureswith long periods. Using the results of astatistical analysis of 72 earthquake records,Sadek et al. (2-96) described the relationshipbetween PSA and SA as:

c:,g 2

!!~II 1"l

2 • 6 , I 10

Frequency, cps

..c:",2

~ '"~ ~l~ 1"l

00 2 • 6 10

Frequency, cps

C J

•E• 2I>.!2.~0 1

00 • 6 10

Frequency, cps

Figure 2-38. Acceleration, velocity. and displacementamplifications plotted as a function of frequency for 5%

damping for the SOOE component of El Centro, theimperial Valley earthquake of May 18. 1940.

SA--=l+a T b

•PSA a

(2-39)


where Q a = 2.436/3J·895 and

ba = 0.628 +0.205/3 . The relationship

between SA and PSA is presented in Figure 237.

Arithmetic and semi-logarithmic plots havealso been used to represent response spectra.Building codes have presented design spectra intenns of acceleration amplification as a functionof period on an arithmetic scale. Typicalacceleration, velocity, and displacementamplifications for the SOOE component of EICentro, the Imperial Valley earthquake of May18, 1940 are shown in Figures 2-38 and 2-39 the fonner plotted as a function of frequencyand the latter as a function of period.

To show how ground motion is amplified indifferent regions of the spectrum, the peakground displacement, velocity, and acceleration

91

for the SOOE component of EI Centro areplotted together on the response spectra, Figure2-40. Several observations can be made fromthis figure. At small frequencies or longperiods, the maximum relative displacement islarge, whereas the pseudo-acceleration is small.At large frequencies or short periods, therelative displacement is extremely small,whereas the pseudo-acceleration is relativelylarge. At intennediate frequencies or periods,the pseudo-velocity is substantially larger thanthose at either end of the spectrum.Consequently, three regions are usuallyidentified in a response spectrum: the lowfrequency or displacement region, theintennediate frequency or velocity region, andthe high frequency or acceleration region. Ineach region, the corresponding ground motionis amplified the most. Figure 2-40 also shows

50

2

.5.01 .02 .05 .1 .2 .5 1 2

Frequency, cps5 10 20 50 100

Figure 2-40. Response spectra for 2, 5, and 10% damping for the SOOE component of EI Centro, the Imperial Valleyearthquake of May 18,1940, together with the peak ground motions.

92

that at small frequencies (0.05 Hz or less), thespectral displacement approaches the peakground displacement indicating that for veryflexible systems, the maximum displacement isequal to that of the ground. At large frequencies(25-30 Hz), the pseudo-acceleration approachesthe peak ground acceleration, indicating that forrigid systems, the absolute acceleration of themass is the same as the ground. As indicated inFigure 2-40, the response spectra for a givenearthquake record is quite irregular and has anumber of peaks and valleys. The irregularitiesare sharp for small damping ratios, and becomesmoother as damping increases. As discussedpreviously, the ratio of ad// influences theshape of the spectrum. A small ad/v ratioresults in a pointed or sharp spectrum while alarge ad/v ratio results in a flat spectrum in thevelocity region. Response spectra may shifttoward high or low frequency regions accordingto the frequency content of the ground motion.

While response spectra for a specifiedearthquake record may be used to obtain theresponse of a structure to an earthquake groundmotion with similar characteristics, they cannotbe used for design because the response of thesame structure to another earthquake record willundoubtedly be different. Nevertheless, therecorded ground motion and computed responsespectra of past earthquakes exhibit certainsimilarities. For example, studies have shownthat the response spectra from accelerogramsrecorded on similar soil conditions reflectsimilarities in shape and amplifications. For thisreason, response spectra from records withcommon characteristics are averaged and thensmoothed before they are used in design.

2.8 FACTORS INFLUENCINGRESPONSE SPECTRA

Earthquake parameters such as soilcondition, epicentral distance, magnitude,duration, and source characteristics influencethe shape and amplitudes of response spectra.While the effects of some parameters may bestudied independently, the influences of severalfactors are interrelated and cannot be discussed

Chapter 2

individually. Some of these influences arediscussed below:

2.8.1 Site geology

Prior to the San Fernando earthquake of1971, accelerograms were limited in numberand therefore not sufficient to determine theinfluence of different parameters on responsespectra. Consequently, most design spectrawere based on records on alluvium but they didnot refer to any specific soil condition. Studiesby Hayashi et al. (2-97) and Kuribayashi et al. (2-

98) on the effects of soil conditions on Japaneseearthquakes had shown that soil conditionssignificantly affect the spectral shapes. Otherstudies by Mohraz et al. (2-9) and Hall et al. (2-91)

also referred to the influence of soil conditionon spectral shapes.

The 1971 San Fernando earthquakeprovided a large database to study the influenceof many earthquake parameters including soilcondition on earthquake ground motion andresponse spectra. In 1976, two independentstudies, one by Seed, Ugas, and Lysmer (2-99),

and the other by Mohraz (2-53) considered theinfluence of soil condition on response spectra.The study by Seed et al. used 104 horizontalcomponents of earthquake records from 23earthquakes. The records were divided into fourcategories: rock, stiff soils less than about 150ft deep, deep cohesionless soil with depthsgreater than 250 ft, and soft to medium clay andsand. The response spectra for 5% damping'were normalized to the peak groundacceleration of the records and averaged atvarious periods. The average and the mean plusone standard deviation (84.1 percentile) spectrafor the four categories from their study ispresented in Figures 2-41 and 2-42. Theordinates in these plots represent theacceleration amplifications. Also shown inFigure 2-42 is the Nuclear RegulatoryCommission (NRC) design spectrum proposed

4 they limited their study to 5% damping, although theconclusions can easily be extended to other dampingcoefficients.


by Newmark et al. (2-88.2-100), see Section 2.9. It

is seen that soil condition affects the spectra toa significant degree. The figures show that forperiods greater than approximately 0.4 to 0.5sec, the normalized spectral ordinates(amplifications) for rock are substantially lowerthan those for soft to medium clay and for deepcohesionless soil. This indicates that using thespectra from the latter two groups mayoverestimate the design amplifications for rock.

two studies are somewhat different. Normalizedresponse spectra corresponding to the meanplus one standard deviation (84.1 percentile) forthe four soil categories from the Mohraz studyare given in Figure 2-45. The plot indicates thatfor short periods (high frequencies) the spectralordinates for alluvium are lower than the others,whereas, for intermediate and long periods theyare higher.

TOIOI number of recordS ol'lOlysed Q4

RoCk • 28 records

/ Soh 10 medIum cia, ond 50no· IS 'ecords

/r Dee. """''''''''' """ t>Z;O ",. 30 """0'$1,11 SOli cond,hatlS 1<150111- Jlltcords

°o!:----o*,---:',O""'--""'-','=-,--2='=0---='2'-=-----='30Period· seconds

0~..,.L-----L.--~==:=:==;::===:3o O~ to l~ 20 2~ )0

Period - seconO'

Figure 2-41. Average acceleration spectra for differentsoil conditions. [After Seed et aI. (2-99).]

Figure 2-42. Mean plus one standard deviationacceleration spectra for different soil conditions. [After

seed et al. (2-99).]

5r----.----.r---...,------,.---,----,.---,

32.51.5 2

PERIOD, SEC.

- ALLUVIUM

_.- LESS THAN 30 FT. ALLUVIUMON ROCK

_..- 30 - 200 FT. ALLUVIUMON ROCK

--- ROCK

.5

t-, \I', \1\ iJ \ .

,~ "1;\ '.',' \ \.\\

\ .\ \\" 1\,,\, ......

"-'>-' '.'----':::~~ .....

"..:~:::::.~2=..::~~=.:..~==~_:.:::O'--_--'-_---l'--_....l.-_--'-__........._--'---l

o

Figure 2-43. Average horizontal accelerationamplifications for 2% damping for different soil

categories. [After Mohraz (2-53).]

z24:c!:!...:; 3

i~ 2

~.....':1u~

The study by Mohraz (2-53) considered a totalof 162 components of earthquake recordsdivided into four soil categories: alluvium,rock, less than 30 ft of alluvium underlain byrock, and 30 - 200 ft of alluvium underlain byrock. Figure 2-43 presents the averageacceleration amplifications (ratio of spectralordinates to peak ground acceleration) for 2%damping for the horizontal components with thelarger of the two peak ground accelerations.Consistent with the study by Seed et al. (2-99),

the figure shows that soil condition influencesthe spectral shapes to a significant degree. Theacceleration amplification for alluvium extendsover a larger frequency region than theamplifications for the other three soilcategories. A comparison of accelerationamplifications for 5% damping from the Seedand Mohraz studies is shown in Figure 2-44.The figure indicates a remarkably closeagreement even though the records used in the

94 Chapter 2

Figure 2-44. Comparison of the average horizontalacceleration amplifications for 5% damping for rock.

[After Mohraz (2-53).]

greater than 6 to formulate a relationship forpseudo-velocity in terms of various earthquakeparameters. The response spectra for 5%damping were computed for four sitecategories; rock, soft rock or stiff soil, mediumstiff soil, and soft soil classified as soil class Athrough D, respectively. A regression analysiswas performed for periods in the range of 0.1 to4.0 sec. Their proposed equation for thepseudo-velocity (PSV) in cm/sec is given as

.5

_ MOHRAlI2-63)

-_. SEED. UGAS, LYSMER 12"')

1.5 2 2.5PERIOD, SEC.

3

In(PSV) = a +bM s

+d In[R+cI exp(c2Ms)]+eF(2-40)

(2-41)

In(PSA) = bl +b2(M W -6)

+b3(M w _6)2 +b5In~""'R-~b-+-h-2

V+bv In-s

VA

where Mwand Rjb are the moment magnitudeand distance (see section 2.4.1), respectively.The parameter b] is related to the fault type andis listed for different periods for strike-slip andreverse-slip earthquakes, and the case where the

w.here Ms is the surface wave magnitude, R isthe closest distance from the site to the faultrupture in km, and F is the fault type parameterwhich equals 1 for reverse-slip and 0 for strikeslip earthquakes. The parameters a, b, Cj, Cz, dand e are given in tabular form for differentperiods and soil categories (2-101). Parameters b,C], and Cz are greater than zero whereas d is lessthan zero for all periods and different soilconditions. Figure 2-46 presents the spectralshapes for the four soil categories at a distanceof 10 km from the source for a strike-slipearthquake of magnitude 7. The figure indicateshigher spectral values for softer soils.

A similar study was carried out by Boore etal. (2-37) using the average shear wave velocityVs (m1sec) in the upper 30 m of the surface toclassify the soil condition. In their study, thepseudo-acceleration response PSA in g is givenby

5 10 20.5 1 2

PERIOD, SEC.

2 .......--'---L...---'-_........._--'-_.........--l.1 .2

500

200

100

vIII<II 50~

!~.

20!:u0...III 10>

5

Figure 2-45. Mean plus one standard deviation responsespectra for 2% damping for different soil categories.

normalized to LOg horizontal ground acceleration. [AfterMohraz (2-53).]

/'.. rJ \".. .r

.".' . .I\J,IN;~>'~··-ll.(1". v,·· ~,j \ '-, ...-......... '.:J~ ,A"...._' " ,,\ \

."i' ..' ........_, /"'~ \ '\ \~! v \\~

I . .' \ \

f " -r·.~.' , '. \'... \ ..

\ ..- ALLUVIUM ',,_

_.- LESS THAN 30 FT. ALLUVIUMON ROCK

. --- 30- 200 FT. ALLUVIUMON ROCK

--- ROCK

Recent studies indicate that the spectralshape not only depends on the three peakground motions, but also on other parameterssuch as earthquake magnitude, source-to-sitedistance, soil condition, and sourcecharacteristics. Similar to ground motionattenuation relationships (Section 2.4), severalinvestigators have used statistical analysis ofthe spectra at different periods to developequations for computing the spectral ordinatesin terms of those parameters. For example,Crouse and McGuire (2-101) used 238 horizontalaccelerograms from 16 earthquakes between1933 and 1992 with surface wave magnitudes


.......... 0.9 bD

....... o.a :.

s:: -o 0.7~e0.6Q)

Q) 0.5oo< 0.4

~ 0.3~

o~ 0.2

en0.1

0.0 1-+--------------------

95

Figure 2- 46. Response spectra for 5% damping for different soil conditions for a magnitude 7 strike-slip earthquake. [AfterCrouse and McGuire (2-101).]

fault mechanism is not specified. Factors b2, b3,

bj , b", VA, and h for different periods are alsopresented in tabular form (2-37). The parametersb2, VA, and h are always positive whereas b3, bj ,

and bv are always negative. Consistent with thestudy by Crouse and McGuire (2-101), Equation2-41 indicates that, for the same distance,magnitude, and fault mechanism, as the soilbecomes stiffer (a higher shear wave velocity),the pseudo-acceleration becomes smaller sincebv is always negative.

2.8.2 Magnitude

In the past, the influence of earthquakemagnitude on response spectra was generallytaken into consideration when specifying thepeak ground acceleration at a site.Consequently, the spectral shapes andamplifications in Figures 2-41 and 2-42 wereobtained independent of earthquake magnitude.Earthquake magnitude does, however, influencespectral amplifications to a certain degree. Astudy by Mohraz (2-102) on the influence ofearthquake magnitude on response

amplifications for alluvium shows largeracceleration amplifications for records withmagnitudes between 6 and 7 than those withmagnitudes between 5 and 6 (see Figure 2-47).While the study used a limited number ofrecords and no specific recommendation wasmade, the figure indicates that earthquakemagnitude can influence spectral shapes andmay need to be considered when developingdesign spectra for a specific site.

Equations 2-40 and 2-41 in the previoussection include the influence of earthquakemagnitude on the pseudo-velocity and pseudoacceleration, respectively. The equationsindicate that spectral ordinates increase with anincrease in earthquake magnitude. Figure 2-48presents the spectral ordinates computed usingEquation 2-41 by Boore et al. (2-37) for soil witha Vs = 310m/sec at a zero source distance forearthquakes with magnitudes 6.5 and 7.5 and anunspecified fault mechanism. The figureindicates that the effect of magnitude is morepronounced at longer periods and it also showsa comparison with the spectra computed froman earlier study by Joyner and Boore (2-32).

96 Chapter 2

Period (sec)

oL-_...1-_--L_~L_. _ _L__ ___'__ ____JL_.J

o ~ ~ NPERIOD, SEC.

acceleration divided by the peak groundacceleration) for the records on rock and onalluvium for the three groups are shown inFigure 2-49. The plots indicate that for sites onrock, the amplifications for the near-field aresubstantially smaller than those for mid- or farfield for periods longer than 0.5 sec. For shorterperiods, however, the amplifications for thenear-field are larger. The effect of distance isless pronounced for records on alluvium.

Equation 2-40 proposed by Crouse andMcGuire (2-101) shows that the spectral ordinatesdecay with the logarithm of the distance(parameter d in the equation is always negative)for a given soil, earthquake magnitude, andsource characteristics. A similar trend is alsoobserved from Equation 2-41 by Boore et al. (2

37). Figure 2-50 shows the pseudo-velocityresponse computed using Equation 2-41 forsites on soil for a magnitude of 7.5 at varioussource distances for strike-slip and reverse-slipfault mechanisms. The figure indicates that thespectral ordinates decrease with distance. Sincethe spectral shapes are nearly parallel to eachother for the distance range of 10 to 80 km, itmay be concluded that distance does notsignificantly affect the spectral shape butinfluences the spectral ordinates throughattenuation of ground acceleration.

10'

2

..................'"

_.- BETWEEN 5&6

- BETWEEN 6&7

- Boore et aI. (1994.): v,. =310 mls••• JoY'l"r and Boore (19l12): soil

0.2

Mechanism: unspecified

D=O.OIem

zo;: 4c~~ 32:czo 2

~'"...;;: 1u~

~.Q. 102

Z''g~Q)>!10' j~ j'" ia. I

10° .)..1~'-rl_.-r-~~~~......----,-~-.--+-10°0.1

Figure 2-47. Effect of earthquake magnitude on spectralshapes. [After Mohraz (2-102).]

2.8.4 Source characteristics

Figure 2-48. Pseudo-velocity spectra for 5% damping onsoil and earthquake magnitudes 6.5 and 7.5 at a zero

distance. [After Boore et al. (2-37).]

2.8.3 Distance

Recent studies have considered the effect ofdistance on the shape and amplitudes of theearthquake spectra. Using the data from theLorna Prieta earthquake of October 17, 1989;Mohraz (2-103) divided the records into threegroups: near-field (distance less than 20 km),mid-field (distance between 20 to 50 km) andfar-field (distance greater than 50 km). Theaverage acceleration amplification (pseudo-

Fault mechanism may influence the spectralordinates. Using Equation 2-40, Crouse andMcGuire (2.J01), computed the ratios of thespectral ordinates for a reverse-slip fault toordinates for strike-slip fault for two soilcategories: soft rock or stiff soil (site class B)and medium stiff soil (site class C). The ratios,plotted in Figure 2-51, show that the spectralordinates for reverse-slip faults are greater thanthe ordinates for strike- slip faults for shortperiods but not for long periods. Crouse andMcGuire concluded, however, that it is difficultto attach any significance on the influence offault mechanism on the spectral shape. Similartrends and conclusion can also be depicted from


5% DampingRoc" Heo,-Fi.ld

,!!!!-I(_IJj1:.!l~__

8!'f~..'..O!:!i!'1.........

5% Damping

.....oL- ---'- ....... --'

o

(a) (b)

Figure 2-49. Average acceleration amplification for 5% damping for different distances from the 1989 Lorna Prietaearthquake for sites on (a) rock and (b) alluvIUm. [After Mohraz (2-103).]

..._... Site Class B-- Site Class C

o "'f'iiijiillj'IJijlllljilllj-rr-n-rrrrr

:.:i 1.4..0:

<:o

~ 1.2:E

.:-iii,~ 1.0

U;, , ~-....-f 0.6!1 _....- ... ,

~ 0.6, I"" I"" I,!"",!", [,!",!"",! I, [ ,[ t ,~0.0 0.5 1.0 I.~ 2.0 2.5 3.0 3.~ ....0

Oscillator Period [sec]

Figure 2-51. Ratio ofreverse-slip to strike-slip spectralordinates for soft rock or stiff soil referenced as site classB and medium stiff soil referenced as site class C. [After

Crouse and McGuire (2-101).]

o .10km

D=Okm

2

••••• D=Bokm

....... O.20km

•••••• D.40km

Period (sec)

0.2

.'.', .

, ••• Mechanism: strike slip- Mechenism: reverse slip

0.1

SoilM=7.5

QlE 102.!:!.~'00

Q)>Q)

,~

iCQ) 10'ex:0"0:::lQ)VI0..

Figure 2-50. Pseudo-velocity spectra for 5% damping onsoil and for earthquake magnitude 7.5 at different

distances. [After Boore et al. (2-37).]2.8.5 Duration

. I (2-37) h thFIgure 2-50 by Boore et a. were ereverse-slip faults result in a larger response forshort periods and the strike-slip faults result in alarger response for long periods. The differencebetween the response from the two faultmechanisms, however, is not that significant.

While earthquake response spectra providethe best quantitative description of the intensityand frequency content of ground motion, theydo not provide information on the duration ofstrong shaking -- a parameter that manyresearchers and practitioners consider to beimportant in evaluating the damaging effects ofan earthquake. The influence of the duration of

98 Chapter 2

Table 2-10. Relative Values of Spectrum AmplificationFactors (after Newmark and Hall, 2-90)

The first earthquake design spectrum wasdeveloped by Housner (2-109. 2-110). His designspectra shown in Figure 2-52 are based on the

of different sets of records for different regionsof the spectrum.

The difference between response spectra anddesign spectra should be kept in mind. Aresponse spectrum is a plot of the maximumresponse of a damped SDOF oscillator withdifferent frequencies or periods to a specificground motion, whereas a smooth or a designspectrum is a specification of seismic designforce or displacement of a structure having acertain frequency or period of vibration anddamping (2-101).

Since the peak ground acceleration,velocity, and displacement for variousearthquake records differ, the computedresponse cannot be averaged on an absolutebasis. Various procedures are used to normalizeresponse spectra before averaging is carried out.Among these procedures, two have been mostcommonly used: 1) normalization according tospectrum intensity (2-108) where the areas underthe spectra between two given frequencies orperiods are set equal to each other, and 2)normalization to peak ground motion where thespectral ordinates are divided by peak groundacceleration, velocity, or displacement for thecorresponding region of the spectrum.Normalization to other parameters such aseffective peak acceleration and effective peakvelocity-related acceleration has also beensuggested and used in development of designspectra for seismic codes.

Amplification Factor forDisplacement Velocity Acceleration

6.45.85.24.32.61.91.51.2

4.03.63.22.81.91.51.3l.l

2.52.22.01.81.41.2l.l1.0

Percent ofCritical

Dampingo

0.512571020

strong motion on spectral shapes has beenstudied by Peng et al. (2-104) who used a randomvibration approach to estimate site-dependentprobabilistic response spectra. Their studyshows that long durations of strong motionincrease the response in the low andintermediate frequency regions. This isconsistent with the fact that accelerograms withlong durations have a greater probability ofcontaining long-period wave components whichcan result in a large response in the long periodor low frequency region of the spectrum.

2.9 EARTHQUAKE DESIGNSPECTRA

Because the detailed characteristics of futureearthquakes are not known, the majority ofearthquake design spectra are obtained byaveraging a set of response spectra from recordswith similar characteristics such as soilcondition, epicentral distance, magnitude,source mechanism, etc. For practicalapplications, design spectra are presented assmooth curves or straight lines. Smoothing iscarried out to eliminate the peaks and valleys inthe response spectra that are not desirable fordesign because of the difficulties encountered indetermining the exact frequencies and modeshapes of structures during severe earthquakeswhen the structural behavior is most likelynonlinear. It should be noted that in some cases,determining the shape of the design spectra fora particular site is complicated and cautionshould be used in arriving at a representative setof records. For example, long periodcomponents of strong motion have apronounced effect on the response of flexiblestructures. Recent strong motion data indicatesthat long period components are influenced byfactors such as distance, source type, rupturepropagation, travel path, and local soilconditions (2-50. 2-105. 2-106). In addition, the

direction and spread of rupture propagation canaffect motion in the near-field. For thesereasons, the selection of an appropriate set ofrecords in arriving at representative designspectra is important and may require selection


characteristics of the two horizontalcomponents of four earthquake ground motionsrecorded at EI Centro, California in 1934 and1940, Olympia, Washington in 1949, and Taft,California in 1952. The plots are normalized to20% acceleration (0.2g) at zero period (groundacceleration). For any other acceleration, theplots or the information read from them aresimply scaled up or down by multiplying themby the ratio of the desired acceleration to 0.2g.

In the late sixties, Newmark and Hall (2-89, 2-

90) recommended straight lines be used torepresent earthquake design spectra. Theysuggested that three amplifications(acceleration, velocity, and displacement)which are constant in the high, intermediate,and low frequency regions of the spectrum(Table 2-10) together with peak groundacceleration, velocity, and displacement of1.0g, 48 in/sec, and 36 in. be used to constructdesign spectra. Their recommended groundmotions and the amplifications were based on

80

100

60

Period of vibration, T(sec)

100 ,Y/ vA '" ',',IVA', /80 ., /X/X / A X , / / X / ,I, X , / ,< / /' ,/

ov YV A" x, /\/ -< / ".

0\0 ~",0

Figure 2-52. Design spectra scaled to 20% ground acceleration. [After Housner (2-110).]

100

.>.-

Frequency, cps

Figure 2-53. Design spectra normalized to l.Og. [After Newmark and Hall (2-90).]

Chapter 2

the characteristics of several earthquake recordswithout considering soil condition. The spectralordinates which are obtained by multiplying thethree ground motions by the correspondingamplifications are plotted on a tripartite (fourway logarithmic) paper as shown in Figure 253. The spectral displacement, spectral velocity,and spectral acceleration are plotted parallel tomaximum ground displacement, groundvelocity, and ground acceleration, respectively.The frequencies at the intersections of spectraldisplacement and velocity, and spectral velocityand acceleration define the three amplifiedregions of the spectrum. At a frequency of

approximately 6 Hz, the spectral acceleration istapered down to the maximum groundacceleration. It is assumed that the spectralacceleration for 2% damping intersects themaximum ground acceleration at a frequency of30 Hz. The tapered spectral acceleration linesfor other dampings are parallel to the one for2%. The normalized design spectra in Figure 253 can be used for design by scaling theordinates to the desired acceleration.

In the early seventies with increased activityin the design and construction of nuclear powerplants in the United States, the Atomic EnergyCommission AEC (later renamed the Nuclear


.:

0.2 0.5 2 5Frequency. in cycle~ / ~ec

10 20 so 100

Figure 2-54. NRC horizontal design spectra scaled to LOg ground acceleration. A, B, C, and D are control frequenciescorresponding to 33, 9, 2.5, and 0.25 HZ, respectively.

Regulatory Commission) funded two studies one by John A. Blume and Associates (2-111) andthe other by N. M. Newmark ConsultingEngineering Services (2-9) to developrecommendations for horizontal and verticaldesign spectra for nuclear power plants. Thesestudies which used a statistical analysis of anumber of recorded earthquake ground motionsand computed response spectra were the basisfor the Nuclear Regulatory Commission (NRC)Regulatory Guide 1.60 (2-88. 2-H)(». The studiesrecommended that the mean plus one standarddeviation (84.1 percentile) response be used for

the design of nuclear power plants andequipment. The NRC design spectra areconstructed using a set of amplificationscorresponding to four control frequencies(Figure 2-54). The spectra are normalized to1.0g horizontal ground acceleration. While theNRC spectra were developed for design ofnuclear power plants, they were also used todevelop and compare design spectra for otherapplications.

In 1978, the Applied Technology CouncilATC (2-74) recommended a smooth version ofthe normalized spectral shapes proposed by

102 Chapter 2

Seed et al. (2-99) be used in developingearthquake design spectra for buildings. Thespectral shapes in Figures 2-41 and 2-42 weresmoothed using four control periods (2-39). Inaddition, the four soil categories were reducedto three: rock and stiff soils (soil type 1), deepcohesionless or stiff clay soils (soil type 2), andsoft to medium clays and sands (soil type 3).The ATC spectra which was adopted by theSeismology Committee of the StructuralEngineers Association of California, SEAOC (2

112) is presented in Figure 2-55. A comparisonof the spectral shapes from the study by Mohraz(2-53) and those proposed by ATC is shown inFigure 2-56. The 1985, 1988, 1991, and 1994editions of the Unifonn Building Code (2-82102

85) use the spectral shapes for the three soilconditions recommended by ATC. The designspectra for a given site is computed bymultiplying the spectral shapes in Figure 2-55

by the seismic zone factor Z (or the effectivepeak acceleration) obtained from the seismicmaps.

".--.,...----,.--.,...---,--,.----.

/

SOFT TO MEDIUM CLAYS ANDSANDS (SOIL TYPE 3)

DEEP COHESIONLESS OR STIFFn--y-.... _, CLAY SOILS (SOIL TYPE 2)

ROCK AND STIFF SOilS.... (SOIL TYPE 1)

...." ' .... ........ ...... .......-... -

Figure 2-56. Normalized spectral curves recommended foruse in building codes. (Reproduced from 2-39).

4-r-------------------------------,SEAOC---- ---

Soil type 1

Soil type 2

Soil type 3

RockLess than 30' alluviumlrock

30'·200' alluvium/rockAlluvium

......... : ~.~, ~ ~'':~~~ ~ '.~"'"........ .......

.. - -............. .. ...•.................

J2.521.5

Period, sec

o-t-----,,-------r-------r-------r-------r-----jo 0.5

Figure 2-55. Comparison of spectral shapes for 5% damping proposed by Mohraz with those recommended by SEAOC.


0.3 1.0

Building Period. T (sec)

Figure 2-57. Two-factor approach for constructing sitedependent design spectra recommended by the 1994

NEHRP recommended provisions.

(2-42)

F oA 02.5

Aao 2.5.--,. : "-:....... A" Soil: ........... -..... T. -

: ----·Rock

..(J)

c'.2iiiQ)Qj(,)(,)

<~ti AaQlQ.

(J)

In 1992, a workshop on site response duringearthquakes was held by the National Center forEarthquake Engineering Research (NCEER),the Structural Engineers Association ofCalifornia (SEAOC), and the Building SeismicSafety Council (BSSC). The workshop (2-113)

recommended that the spectral amplifications atdifferent periods should depend not only on thesoil condition but also on the intensity ofshaking due to soil nonlinearities.Consequently, a two-factor approach wassuggested for constructing the design spectra inorder to account for the dependence of thespectral shape on the shaking intensity. Thetwo-factor approach was introduced in the 1994NEHRP provisions (2-79), see Figure 2-57. Theapproach uses new seismic coefficients Ca andCv in terms of the effective peak acceleration Aaand the effective peak. velocity-relatedacceleration Av such that

where Fa and Fv are site amplificationcoefficients that vary according to soilcondition and shaking intensity (seismic lone).The provisions included tables for computingcoefficients Fa and Fvas well as Ca and Cv. Sixsoil categories, designated as A through F, wereintroduced in the provisions. The first five arebased primarily on the average shear wave

The 1985, 1988, and 1991 NEHRPrecommended provisions (2-76 to 2-78) present

design spectra using the effective peakacceleration Aa and the effective peak velocityrelated acceleration Av• These two factors whichare obtained from seismic maps are used todefine the constant acceleration and velocitysegments of the design spectrum, respectively.Since Aa and Av for the vast majority of the sitesin the United States are the same, the computedspectra are similar to the UBC spectra. Whilethe 1985 NEHRP provisions included the threesoil categories defined by ATC (2-74), the 1988NEHRP provisions (2-77) and the 1988 UniformBuilding Code (2-83) included a fourth soilcategory S4 based on the experience from theMexico City earthquake of September 19, 1985where most of the underlying soil is very sofe.Aexible structures (periods in the neighborhoodof 2 sec) in that earthquake experienced largeacceleration amplifications which resulted insevere and widespread damage. Consequently,it was recommended to compute the spectralshape in the velocity region from that of rockusing an amplification of 2.

A new procedure for constructing designspectra and computing the base shears wasrecommended in the 1991 NEHRP provisions(2-78) by obtaining the spectral accelerationordinates at periods of 0.3 and 1.0 sec from thespectral maps (see Section 2.5). The ordinate at0.3 sec is used for the constant accelerationlone whereas the ordinate at 1.0 sec is dividedby the period T for the velocity lone. Thespectral ordinates from the maps are modifiedaccording to the soil category of the site. Themaps in the 1991 NEHRP provisions wereprovided for the soil category S2 (deepcohesionless or stiff clay soils). The provisionsrecommended that the spectral ordinatescorresponding to the 1.0 sec period be reducedby a factor of 0.8 for soil type S/ and amplifiedby factors of 1.3 and 1.7 for soil types S3 and S4,respectively.

5 the shaking was most intense within a region underlainby an ancient dry lake bed composed of soft claydeposits.

104 Chapter 2

amplifications in terms of the average shearwave velocity Vs in the upper 30 meters of thesoil profile as:

Where Va is the average shear wave velocityfor a referenced soil profile (VA = 1050 mlsecfor firm to hard rock). Parameters ma and mv

represent the influence of the ground motionintensity on amplification (see Figure 2-58),Substitution for Va results in

The coefficients Fa and Fv recommended byBorcherdt were the basis for those presented inthe 1994 NEHRP provisions by computing thecoefficients for each site category bysubstituting the appropriate value for Vs'

Borcherdt also provided values for thecoefficients Fa and Fv for constructing designspectra in association with the spectralaccelerations at periods of 0.3 and 1.0 sec.Since seismic maps for spectral accelerationsare for deep cohesionless or stiff clay soils, thecoefficients are presented with reference to softto firm rocks and stiff clays. For this case,Equation 2-43 can be used to compute thecoefficients Fa and Fvusing a Vo= 450 mlsec.

After the Northridge earthquake of January17, 1994, Borcherdt (2-117) computed coefficientsFa and Fv for accelerograms recorded ondifferent soils in the Los Angeles area. Theresults indicate that the coefficients are in goodagreement with those suggested in his earlierstudy (2-114) and also those included in the 1994NEHRP provisions (2-79) for small shakingintensities. For large intensities, however, thecoefficients computed from the Northridge dataare greater than those recommended previously.

(2-43)

(2-44)Fa = (1050/VJm

•

Fv =(1050/VJm

v

6 in addition to the shear wave velocity, other parameterssuch as average standard penetration, undrained shearstrength, and plasticity index are used in theclassification.

velocity6 Vs (mlsec) in the upper 30 meters ofthe soil profile and the sixth is based on a sitespecific evaluation. The categories include: (A)hard rock (Vs > 1500), (B) rock (760 < Vs :S1500), (C) very dense soil and soft rock (360 <Vs:S 760), (D) stiff soil profile (l80 < Vs:S 360),(E) soft soil profile (Vs :S 180), and (F) soilsrequiring site-specific evaluations such asliquefiable and collapsible soils, sensitive clays,peats and highly organic clays, very highplasticity clays, and very thick soft/mediumstiff clays.

The site coefficients Fa and Fv are basedprimarily on the work of Borcherdt (2-114) whoused the strong motion data from the LornaPrieta earthquake of October 17, 1989 tocompute average amplification factorsnormalized to firm to hard rock (NEHRP siteclass B) for short-periods (0.1-0.5 sec),intermediate-periods (0.5-1.5 sec), mid-periods(0.4-2.0 sec), and long-periods (1.5-5.0 sec).Data for ground accelerations of approximatelyO.lg were used in an empirical procedure tofind amplifications Fa and Fv. Amplificationfactors for ground accelerations greater thanO.1g (0.2g, O.3g, and O.4g) were computed byextrapolation of amplification estimates at O.lgsince few strong motion records were availablefor ground motions greater than O.1g for softsoil. The extrapolations were based on resultsfrom laboratory experiments and numericalmodeling. The amplifications were in goodagreement with those computed by Seed et al.(2-115) based on a numerical modeling of the datafrom the Lorna Prieta records and those byDobry et al. (2-116) based on a parametric studyof several hundred soil profiles.

The amplifications Fa and Fv correspondingto short- and mid- periods with respect to firmto hard rock for different shaking intensities areshown in Figure 2-58. The figure indicates thatsite amplifications decrease with an increase inshear wave velocity and an increase in groundaccelerations. Borcherdt also presented the site


I I I I I I I/ Fa = (vsc." Iv)"'· = ( J050m/s Iv)"'·. ,

sC·lY / --/-0·/8; _=0.35

Softsoils /-"-1=0.28; _ .0.25

- • - • 1=0.38; _=0.10/ S~.fll •.••• '1-0.48; _ = .Q.05

~Stiffclays and SC·1l - Fa (O.lg)lor Sil~Ckw InletWls

IFa (or Sill CkwaSandy sails Gra_eUy soils and •

!- ...... """- Softroclr.s ISC.lb I!- •• -

-"-4~- Firm to Hard rocks-'-'4 '-' .- :"T':- .'::~-..• -. .... ...

(8)

5

o100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400

Mean Shear-Wave Velocity to 30 m(100 fO (v. mls)

SC·IY I I I I I I I

~Soft soils V F. =(vsc." Iv)"" =(J050m/s Iv)"',·

~V SC.11I

1-0.18; mv" 0.6's---1=0.2g; mv=O.60

Stiffclays and -"-1=0.38; mv =0.53Sandy sails.. "-

~.. ····1..0.48; mv =0.45. "- SC·1l -NO.lgUor Sil~Ckw IlIletWls. ,

Gravelly sails and...... • F. for Sill Clanu' ..... :-..... I Soft roclr.s...• ..~.f.~ ISC·lb I......

'" '":~::'... Firm 10 Hard rocks

(b)

s

o100 200 300 400 500 600 100 800 900 1000 1100 1200 1300 1400

Mean Shear-Wave Velocity to 30 m(100 fl) (v. mls)

Figure 2-58. Variation of short-period Fa and long-period Fv amplification factors normalized to firm to hard rock withmean shear wave velocity. [After Borcherdt (2-114).]

The 1997 Unifonn Building Code (2-86) useda method similar to that in the 1994 NEHRPprovisions to construct the design spectrum.The design spectrum, Figure 2-59, is defined intenns of the seismic coefficients Co and C.These coefficients are presented for the fiveUBC seismic zones for different soil categories,which are the same as those used in the 1994NEHRP provISIons. The only differencebetween the design spectra in the 1997 UBCcode and the 1994 NEHRP provisions is thatthe fonner includes the near-source factors.

These factors were introduced to amplify thespectral ordinates for sites close to a seismicsource in the zone with the highest seismicity(zone 4). The near-source factors depend on thedistance to the closest active fault and thesource type (maximum magnitude, rate ofseismic activity, and slip rate).

Design spectra presented in the 1997NEHRP recommended provisions (2-80) can beconstructed from the maps of spectral responseaccelerations at short periods Ss (defined as 0.2sec) and at 1.0 sec period 51 corresponding to

106 Chapter 2

Figure 2-59. Design spectrum recommended by the 1997Uniform Building Code (2-86).

The 1997 NEHRP Guidelines for theSeismic Rehabilitation of Buildings, FEMA273 (2.81), uses a procedure similar to that of the1997 NEHRP Provisions (2-80) to establish the5% damped design spectra. In addition,FEMA-273 uses damping modification factorsin the short- and long-period ranges to reducethe spectral ordinates for damping ratios largerthan 5% due to the use of supplementaldamping devices in the structure.

2.10 INELASTIC RESPONSESPECTRA

Structures subjected to severe earthquakeground motion experience deformations beyondthe elastic range. To a large extent, the inelasticdeformations depend on the intensity ofexcitation and load-deformation characteristicsof the structure and often result in stiffnessdeterioration. Because of the cycliccharacteristics of ground motion, structuresexperience successive loadings and unloadingsand the force-displacement or resistancedeformation relationship follows a sequence ofloops known as hysteresis loops. The loopsreflect a measure of a structure's capacity todissipate energy. The shape and orientation ofthe hysteresis loops depend primarily on thestructural stiffness and yield displacement.Factors such as structural material, structuralsystem, and connection configuration influencethe hysteretic behavior. Consequently, arrivingat an appropriate mathematical model todescribe the inelastic behavior of structuresduring earthquakes is a difficult task.

A simple model which has extensively beenused to approximate the inelastic behavior ofstructural systems and components is thebilinear model shown in Figure 2-60. In thismodel, unloadings and subsequent loadings areassumed to be parallel to the original loadingcurve. Strain hardening takes place afteryielding initiates. Elastic-plastic (elastoplastic)model is a special case of the bilinear modelwhere the strain hardening slope is equal to zero(a = 0). Other hysteretic models such asstiffness and strength degrading have also beensuggested. The elastic-plastic model results in amore conservative response than other models.Because of its simplicity, it was widely used inthe development of inelastic response spectra.

Response spectra modified to account forthe inelastic behavior, commonly referred to asthe inelastic spectra, have been proposed byseveral investigators. The use of the inelasticspectra in analysis and design, however, hasbeen limited to structures that can be modeled

CONTROL PERIODS1; cCvI2.5C.To ·0.27;

PERIOD (SECONDS)

2.!C.

C.

the maximum considered earthquake (seeSection 2.5). Since the maps are provided forrock (site class B), the spectral accelerations forother soil categories are adjusted by multiplyingthe spectral accelerations for rock by the sitecoefficients Fa and Fv in the short and the midto long period ranges, respectively. Similar tothe 1994 provisions, Fa and Fv depend on thesoil category and the shaking intensity and aregiven in tabular form based on the study byBorcherdt (2-114). To construct the spectra for thedesign earthquake, the adjusted spectralordinates at the maximum consideredearthquake are multiplied by 2/3 (see Section2.5).


Force

Umu Displacement

Figure 2-60. Bilinear force-displacement relationship.

107

as a single-degree-of-freedom. Procedures forutilizing inelastic spectra in the analysis anddesign of multi-degree-of-freedom systemshave not yet been developed to the extent thatcan be implemented in design. Similar to elasticspectra, inelastic spectra were usually plottedon tripartite paper for a given damping andductility' or yield deformation. When thespectra are plotted for various ductilities,computations are repeated for several yielddeformations using an iterative procedure toachieve the target ductility. Depending on theparameter plotted, different names have beenused to identify the spectrum (Riddell and

50.0

20.0

10.0

5.0

i 2.0

! 1.0

0.5

0.2

0.1

0.050.01 0.02 0.05 0.1 0.2 0.5 1.0 2.0 5.0 10.0 20.0 50.0 100.0

Figure 2-61. Inelastic yield spectra for the S90W component of El Centro, the Imperial Valley earthquake of May 18, 1940.Elastic-plastic systems with 5% damping. [After Riddell and Newmark (2-118).]

, ratio of maximum deformation to yield deformation

108

¥•.......6

IeaE~

3

Frequency (cps)

Chapter 2

Figure 2-62. Total deformation spectra for the S90W component of EI Centro, the Imperial Valley earthquake of May 18,1940. Elastic-plastic systems with 5% damping. [After Riddell and Newmark (2-118).]

Newmark, 2-118). In the inelastic yieldspectrum (lYS), the yield displacement isplotted on the displacement axis; in the inelasticacceleration spectrum (lAS), the maximumforce per unit mass is plotted on theacceleration axis; and in the inelastic totaldisplacement spectrum (ITDS), the absolutemaximum total displacement is plotted on thedisplacement axis. For elastic-plastic behavior,the inelastic yield spectrum and the inelasticacceleration spectrum are identical. Examplesof inelastic spectra for a 5% damped elasticplastic system for the S90W component of ElCentro, the Imperial Valley earthquake of May18, 1940 are shown in Figures 2-61 and 2-62.The figures indicate that for inelastic yield andacceleration spectra, the curves for variousductilities fall below the elastic curve (ductilityof one), whereas for the inelastic totaldeformation spectra, they primarily fall above

the elastic, particularly in the accelerationregion. It should be noted that increasing theductility ratio smoothes the spectra andminimizes the sharp peaks and valleys that arepresent in the plots.

A different presentation of inelastic spectrawas proposed by Elghadamsi and Mohraz (2-119).

The spectrum, referred to as the yielddisplacement spectrum (YDS), is plottedsimilar to the inelastic total deformationspectrum except that it is plotted for a givenyield displacement instead of a given ductility.The ductility is obtained as the ratio of themaximum displacement to the yielddisplacement for which the spectrum is plotted.Their procedure offers an efficientcomputational technique, particularly whenstatistical studies are used to obtain inelasticdesign spectra.


~---II:"'-~A"

A''\ /DISPLACEMENT

"-"","-", A"

'"..,",

Figure 2-63. Construction of inelastic acceleration andinelastic total displacement spectra from the elastic

spectrum. [After Newmark and Hall (2-89).]

Before the Riddell-Newmark study ofinelastic response, the most common procedurefor estimating inelastic earthquake designspectra was the one proposed by Newmark (2-120.

2-121) and Newmark and Hall (2-89). Based on

results similar to those in Figures 2-61 and 262, and studies by Housner (2-122) and Blume (2

123 to 2.125), Newmark (2-121) observed that: I) at

low frequencies, an elastic and an inelasticsystem have the same total displacement, 2) atintermediate frequencies, both systems absorbthe same total energy, and 3) at highfrequencies, they have the same force. Theseobservations resulted in the recommendation byNewmark for constructing inelastic spectrafrom the elastic by dividing the ordinates of theelastic spectrum by two coefficients in terms ofductility J.l. Figure 2-63 shows the constructionof the inelastic spectrum from the elastic. Thesolid lines DVAAo represent the elasticresponse spectrum. The solid circles at theintersections of the lines correspond tofrequencies which remain constant in obtainingthe inelastic spectrum. The lines D'V'A'Aorepresent the inelastic acceleration spectrumwhereas the lines DVA"Ao" show the totaldisplacement spectrum. D' and V' are obtainedby dividing D and V by J.L A' is obtained by

dividing A by ~(2Jl-I) (to insure that the

same energy is absorbed by the elastic and the

inelastic systems). A" and Ao are obtained bymultiplying A' and Ao by J.l.

The Riddell-Newmark study (2-118) also

considered bilinear and stiffness degradingmodels and concluded that using the elasticplastic spectrum for inelastic analysis isgenerally on the conservative side.

2.10.1 De-amplification factors

When inelastic deformations are permittedin design, the elastic forces can be reduced ifadequate ductility is provided. Riddell andNewmark (2-118) presented a set of coefficientsreferred to as "de-amplification factors" bywhich the ordinates of the elastic designspectrum are multiplied to obtain the inelasticyield spectrum. Lai and Biggs (2-126), usingartificial accelerograms with variable durationsof strong motion, presented a set of coefficientsreferred to as "inelastic acceleration responseratios" by which the ordinates of the elasticspectrum are divided to give the inelastic yieldspectrum. Since these two approaches are theinverse of one another, the reciprocal of theLai-Biggs coefficients represent deamplification factors. De-amplification factorscan also be obtained from the Newmark-Hall (2

89) and from the Elghadamsi-Mohraz (2-119)

procedures for estimating inelastic spectra.Comparisons of the de-amplification factorsfrom the four procedures are shown in Figure 264 for a 5% damping ratio and ductilities of 2and 5. The figure indicates that the RiddellNewmark de-amplification factors are ingeneral the smallest (largest reduction in theelastic force) compared to the other three. BothRiddell-Newmark and Newmark-Hall deamplification ratios remain constant overcertain frequency segments, whereas those fromLai-Biggs and Elghadamsi-Mohraz followparallel patterns. While the de-amplificationratios are affected by ductility, they arepractically not influenced by damping. Sincethe elastic spectral ordinates decreasesignificantly with an increase in damping, thedecrease in inelastic spectral ordinates with

110 Chapter 2

0.8 ...

06 ; [ LLU.U. :: :.:

t.~..L.j.

: :": : : ~ :;:: ::

..

DUCnUTY=5 ::1

ELGHAOAMSI-l.OHRAZ • : : : :

!:!1-!!G2§ _ _ _.;. L .!!!.2!ill.!..:~1S. _ •••!!'~!'~ll.K:"!.A.L!- ••••• : • : •:

0.4

0.8 .

0.2

DUCTIUTY=2 hELGHAOAMSI-l.OHRAZ ..~

"!:!1-!!G2§ _ _ _#RIDDELL-NEWMARK :: : : :' /!f :

I0' , __TI~ti;:- 0.4 '-- ' , . , ,•.,•..,•....... r-- ... i. ..L ~.. i .

~ ji

o .:....: ....L:...0.2

10o+---;.-;---;--;-i-i-i-;+-----,i---T-i-...-i-T1-;f---i-j 0 -t---;.--;--;-;.+;-;-;f----i--+-i-1H+.-;+--H

0.1 10 25 0.1

FREQUENCY, CPS FREQUENCY, CPS

Figure 2-64. Comparison of de-amplification factors for 5% damping. [After Elghadamsi and Mohraz (2-119).]

damping stems primarily from the elasticspectral ordinates.

Elghadamsi and Mohraz (2-119) also presentedde-amplification factors for alluvium and rock.Typical de-amplification factors for alluviumand rock for 5% damping is presented in Figure2-65. According to the figure, de-amplificationsare not significantly affected by the soilcondition.

The influence of the duration of strongmotion on the inelastic behavior of structureshas also been studied. In a non-deterministicstudy of nonlinear structures, Penzien and Liu(2-127) concluded that structures with elasticplastic and stiffness degrading behavior aremore sensitive to the duration of strong motionthan elastic structures. Using a randomvibration approach and the extreme valuetheory, Peng et al. (2-128) incorporated theduration of strong motion in estimating themaximum response of structures with elasticplastic behavior. The effect of duration ofstrong motion on de-amplification factors fromPeng's study is shown in Figure 2-66 whichindicates that for a longer duration of strongmotion, one should use a larger de-

amplification (smaller reduction in elasticforce). It should be noted that Lai and Biggs (2

126) conclude that inelastic response spectra arenot significantly affected by strong motionduration. They emphasize, however, that thisconclusion is valid only when ground motionwith varying strong motion durations arecompatible with the same prescribed elasticresponse spectrum.

••:1-.,-----,....;..;..1-,-~_~_..;...;+-~fr.qu_ney. t:p.

Figure 2-65. De-amplification factors for alluvium androck for 5% damping. [After Elghadamsi and Mohraz (2

119).]


c: 0.6

~i .,t-"""""",,~~q ...........•... , ............•... ':/' ············1

~0.1

Frequency. cps,

111

that use multiple lines of framing in eachprincipal direction of the building.The ductility factor RfJ. is defined as the ratio

of the elastic to the inelastic displacement for asystem with an elastic fundamental period Tand specified ductility J.l such that

(2-46)

Figure 2-66. Effect of strong motion duration on deamplification factors for systems with 2% damping. [After

page et al. (l-128).]

2.10.2 Response modification factors

Current seismic codes recommend forcereduction factors and displacementamplification factors to be used in design toaccount for the energy absorption capacity ofstructures through inelastic action. The forcereduction factors (referred to as R-factors) areused to reduce the forces computed from theelastic design spectra. A recent study by theApplied Technology Council (2-129) proposes thefollowing expression for computing the Rfactors:

(2-45)

Where Ve is the base shear computed fromthe elastic response (elastic design spectrum),and V is the design base shear for the inelasticresponse. The response modification factor R isthe product of the following terms:1. the period-dependent strength factor Rs

which accounts for the reserve strength ofthe structure in excess of the designstrength,

2. the period-dependent ductility factor RfJ.which accounts for the ductile capacity ofthe structure in the inelastic range, and

3. the redundancy factor RR which accounts forthe reliability of seismic framing systems

where uy is the yield displacement. Stateddifferently, RfJ. is the ratio of the maximuminelastic force to the yield force required tolimit the maximum inelastic response to adisplacement ductility j.l, or the inverse of thede-amplification factors presented in Section2.10.1.

The relationship between displacementductility and ductility factor has been thesubject of several studies in recent years. Earlierstudies by Newmark and Hall (2-87, 2-89) provided

expressions for estimating the ductility factorRfJ. for elastic-plastic systems irrespective of thesoil condition. The expressions are

RfJ. (T ~ 0.03sec,Jl) = 1.0

RfJ. (0.12sec ~ T ~ O.5sec,Jl) = ~2Jl-l (2-47)

RfJ. (T ~ 1.0sec,Jl) = Jl

A linear interpolation may be used toestimate RfJ. for the intermediate periods. Theexpressions are plotted in Figure 2-64 forductility ratios of 2 and 5.

Using a statistical study of 15 groundmotion records from earthquakes withmagnitudes 5.7 to 7.7, Krawinkler and Nassar(2-130.2-131) developed relationships for estimatingRfJ. for rock or stiff soils for 5% damping. Theirproposed relationship is

(2-48)

where

112

TO bc= +-

I+To T(2-49)

Chapter 2

shows that the differences between theserelationships are relatively small and may beignored for engineering purposes.

and a and b are parameters that depend on thestrain hardening ratio a. They recommend a =1.00, 1.01, and 0.80 and b = 0.42, 0.37, and0.29 for strain hardening ratios of 0% (elastoplastic system), 2%, and 10%, respectively.

Miranda and Bertero (2-132) using 124accelerograms recorded on different soilconditions, developed equations for estimatingRJl for rock, alluvium, and soft soil for 5%damping. Their equation is given by

8~~~==~/ :",,'" -~ -- - .... - - _ _ J.l = 6.- =:-:..--._._.-

::'.- --.=.,=.."':., -:-_~---. =:~

11=2_ -~---.r - - -__Nassar & KrawinlderMiranda - RockMiranda: - Aluvium

oL-_~~_.-..._-,----,--~--,----,

o 0.5 1.5 2 2.5 3 3.5 4Period (seconds)

(2-50)Figure 2-67. Variation ofthe ductility factor with

period for ductility ratios of 2, 4, and 6. [Reproduced from

ATC-19 (2-129).]

where

1 2<I>(T. J.I.) =1+ --exp[-1.5(ln T -0.6) ]

TOO - J.I.) 2T

for rock sites

2 2<I>(T,J.I.)=I+ --exp[-2(lnT-0.2)]

T(12 - J.I.) 5T

for alluvium sites

Tg 3Tg T 2<I>(T,J.I.) =1+---exp[-3(ln--0.25) ]

3T 4T Tg

for soft soil sites

(2.51)

and Tg is the predominant period of the groundmotion defined as the period at which therelative velocity of a linear system with 5%damping is maximum throughout the entireperiod range. A comparison of the NassarKrawinkler and Miranda-Bertero relationshipsfor rock and alluvium for ductility ratios of 2, 4,and 6 is presented in Figure 2-67. The figure

2.11 ENERGY CONTENT ANDSPECTRA

While the linear and nonlinear responsespectra, presented in previous sections, havebeen used for decades to compute designdisplacements and accelerations as well as baseshears, they do not include the influences ofstrong motion duration, number of responsecycles and yield excursions, stiffness andstrength degradation, or damage potential tostructures. There is a need to re-examine thecurrent analysis and design procedures;especially with the use of innovative protectivesystems such as seismic isolation and passiveenergy dissipation devices. In particular, theconcept of energy-based design is appealingwhere the focus is not so much on the lateralresistance of the structure but rather on the needto dissipate and/or reflect seismic energyimparted to the structure. In addition, energyapproach is suitable for implementation withinthe framework of performance-based designsince the premise behind the energy concept isthat earthquake damage is related to thestructure's ability to dissipate energy.

Housner (2-122) was the first to recommendenergy approach for earthquake resistantdesign. He pointed out that ground motion


transmits energy into the structure; some of thisenergy is dissipated through damping andnonlinear behavior and the remainder stored inthe structure in the fonn of kinetic and elasticstrain energy. Housner approximated the inputenergy as one-half of the product of the massand the square of the pseudo-velocity,

lj2m(PSV)2. His study provided the impetus

for later developments of energy concepts inearthquake engineering.

For a nonlinear SDOF system with pre-yieldfrequency and damping ratio of m and j:J,respectively; subjected to ground accelerationart) the equation of motion is given by:

x+ 2{3mX + Fs [x( t )] = -a( t ) (2-52)

Es = Recoverable elastic strain energyF 2 (2-57)

S=2m 2

EH = Dissipative plastic strain energy

x F 2 t F 2 (2 58)=fFdx-_s =fF idt--S -

S 2m2 S 2m2o 0

The energy tenns in the above equations aregiven in energy per unit mass. Through theremainder of this section, the tenn "energy"refers to the energy per unit mass.

12

~ ~ ~ ~ ~ ~ ~ ~ ~ ss ~

TIME ISECONDS)

where Fs [x( t)] is the nonlinear restoring

force per unit mass. Integrating Equation (2-52)over the entire relative displacement history,results in the following energy balanceequation:

(2-53)

where

~ 8..i! •i 4W...

ENERGYINPUT

DAIoPINGENERGY

, 101 HZIJ· S'lI.,. • 3

TOTAlDISSIPATEDENERGY

EI = Input energy =

ED = Dissipative damping energy

x t (2-56)= 2{3mfXdx = 2{3mf i 2dt

o 0

Figure 2-68. Energy time histories for a low and a highfrequency. elastic-plastic structure subjected to EI Centro

ground motion. [After Zahrah and Hall (2-133).]

TOTAlDlSSl'ATEOENEllGY

DAMPIlIGENERGY

" ~ ~ so ~ 4DTIME (SECONDS)

~

,. 5.0"1"$'lI./0'$

,

(2-54)

(2-55)

EK = Kinetic energy =x .2

fXdx=~o 2

x tJa(t)dx = Ja(t)Xdto 0

Figure 2-68 presents the energy responsecomputed by Zahrah and Hall (2-133) as a

114 Chapter 2

S 10 ZO

(a)

(b)

~ OQI OJ 01 OS , 1

FREOUENCY. ICPSI

sco

200!f\ ~~

Ir-:IF" \, ,I

zoI

10I ~iS

,,'2 \\I ---,. • 51

\,\as

Q2

..~IOlIO

~lJ \500

~t\i200 An I'co

,1'1 \so 1/ •20

., \, -"2" \

!I

--- •• !!..---"10'"2

,O!

Q2QOI OQIO' 01 os' 1 SID zo

FREOUrNCY. ICl'SI

Figure 2-69. Input energy spectra for (a) linear systemswith 2, 5, and 10% damping using EI Centro ground

motion and (b) elstic-plastic systems with 2% dampingand ductility ratios of 2 and 5 using Taft ground motion.

[After zahrah and Hall (2-133).]

According to Uan~ and Bertero (2-134), theenergy equations in (2- 3) through (2-58) should beconsidered as "relative energy equations" sincethe integrations are performed for equations ofmotion using the relative displacements. Forthis system of equations, the relative inputenergy is defined as the work done by the staticequivalent lateral force on a fixed-base system.Uang and Bertero introduced the "absoluteenergy equations" by integrating the equation ofmotion using the absolute displacements. For

function of time for two elastic-plastic SDOFstructures; a low frequency (0.1 Hz) and a highfrequency (5 Hz) structure; both with a 5%damping and a ductility of 3.0 subjected to the1940 El-Centro ground motion. In these plots,the difference between the input energy and thedissipated energy (sum of damping andhysteretic) represents the stored energy (sum ofstrain and kinetic). The stored energy becomesvanishingly small at the end of motion and theenergy dissipated in the structure becomesalmost equal to the energy imparted to it. Thelarger peaks and troughs in the energy responseof a low-frequency structure as compared to ahigh-frequency structure indicate that for lowfrequency structures, a larger portion of theenergy imparted to the structure is stored in theform of strain and kinetic energies.

Zahrah and Hall (2-133) introduced an energyspectrum as a plot of the numerical value of theinput energy EI at the end of motion as afunction of period or frequency for differentdamping and ductility ratios. Examples of suchspectra are shown in Figure 2-69 for linearstructures with different damping ratios usingthe El-Centro record and for nonlinearstructures with 2% damping and ductility ratiosof 2 and 5 using Taft ground motion. Zahrahand Hall indicated that for linear structuresunder the same ground motion, input energyspectra are generally similar in shape toresponse spectra and that the quantitylj2m(PSV)2 for an undamped structure is agood estimate of the amount of input energyimparted to the structure. For dampedstructures, however, this quantityunderestimates the input energy. They alsoindicated that the energy spectral shapes fornonlinear systems are similar to those of linearsystems and that the amount of energy input isnearly the same for a linear and a nonlinearstructure (with moderate ductility) with thesame frequency.


Ei,

E"Ei+E,

~'I'" R~

.A .... ,,- A.

Ej - rE"

£.+E,

~ 'I'" E~

Energy (1cN-m) T • 5.0 3eC

0.06

0.04

0.02

0.00 2 • 6 8 10 12

Ti.IIle (sec)

(a)

Energy (kN-m) T - 5.0 sec0.4

0.3Ei

0.2

0.1 Ei+E,E..

E~

0000 2 • 6 8 10 12

(b)TiDe (sec)

12

12

10

10

T • 0.2 sec

T • 0.2 sec

4 6 8Time (sec)

4 6 8Time (sec)

2

2

0.1

0.2

0.0o

0.0o

0.4

0.2

Energy (kN-m,

0.6

Energy (kN-m)

0.6

Figure 2-70. (a) Absolute and (b) relative energy time histories for elastic-plastic systems with 5% damping and ductilityratios of 5 subjected to the 1986 San Salvador earthquake. [After Uang and Bertero (2-134).]

Figure 2-70 shows energy time-histories fora short and a long-period elastic-plastic

the absolute energy terms; ED' ES' and EHare the same as their relative counterparts while

the absolute input energy is given as Jxtdxg

and the absolute kinetic energy is given as

xl /2; where Xt and xg are the absolute and

ground displacement; respectively. Theabsolute input energy represents the work doneby the total base shear on the foundationdisplacement. The difference between theabsolute and relative, input and kinetic energiesis given by:

(2-60)

structure using the relative and absolute energyterms. In addition, Uang and Bertero (2-134)

converted the input energy to an equivalentvelocity such that

where EI can be the relative or absoluteinput energy per unit mass. Figure 2-71presents the relative and absolute input energyequivalent velocity spectra along with the peakground velocity for three earthquake records.As the plots indicate, the relative and absoluteinput energies are very close for the mid-rangeperiods (in the vicinity of predominant periodsof ground motion). For longer and shorterperiods, however, the difference betweenrelative and absolute energies is significant.The figure also shows that the absolute andrelative equivalent velocities converge to the

(2-59)x2

E Eg ••

K,abs - K,rel = 2"" +xxg

116 Chapter 2

50

~ t-.- ;.• .. .','

\

~~

..i

10.5

Vi orVi.-) san Salvador5

0.10.05

Mexico City

.:.-

Vi or Vi (1Il/Scc)

5.-----r--r--.~___r-r___,

h

0.1 k---t-t'-!-+--+-+--i

0.05F--+.'+ --4-+--+-+---1

50 O. OJ.L..0..A.l.uLl<OIllL.......l ....O.u~.lIlI..5·L.-I.oIol.LII5·......• .......50 O. OJ. 01 0.1 0.5 5

Period (sec) Period (see)

El Centro

L....

/ """.- ~ .- _._. ~ --.. ~

1

0.5

Vi or Vi (Dt/Icc)

5

0.10.05

0.01 50.01 0.1 0.5

Period (sec)

__ Absolute equivalent velocity _ _ _ _ _ Relative equivalent velocity Peak ground velocity

Figure 2-71. Absolute and relative input energy equivalent velocity spectra for elastic-plastic systems with 5% damping andductility ratio of 5 using three earthquake records. [After Uang and Bertero (2-134).]

peak ground velocity at very short and verylong periods, respectively. Subsequently, Uangand Bertero concluded that the absolute inputenergy can be used as a damage index for shortperiod structures, while the relative in~ut

energy is more suitable for long-penodstructures. Their study also showed, usingenergy spectra, that the input energy isinsensitive to the ductility ratio. Finally, Uangand Bertero (2-134) believed that for linear

structures, Housner's use of lj2m(PSV)2 to

estimate input energy reflects the maximumelastic energy stored in the structure withoutconsideration of damping energy.

It should be noted that at the time of thiswriting, the energy concept outlined in thissection does not provide the basis for seismicdesign, despite the body of knowledge that hasbeen developed. Further research is required toreliably estimate both the energy demand andenergy capacity of structures in order toimplement energy approaches in seismic designprocedures.

2.12 ARTIFICIALLYGENERATED GROUNDMOTION

One major drawback in using the responsespectrum method in analysis and design ofstructures lies in the limitation of the method toprovide temporal information on structuralresponse and behavior. Such information issometimes necessary in arriving at asatisfactory design. For example, the responsespectrum procedure can be used to estimate themaximum response in each mode of vibration,and procedures such as square root of the sumof the squares can be used to combine themodal responses. When the natural frequenciesare close to each other, however, the square rootof the sum of the squares can result ininaccurate estimate of the response. In suchcases, the complete quadrature combinationSCQC, or a time-history analysis may be used. Ifinelastic deformation is permitted in design, theinelastic spectra and the de-amplificationfactors presented in the previous sections

8 An improved procedure for computing modal responsesreferred to as complete quadrature combination CQCwas proposed by Der Kiureghian (see Chapter 3).


cannot be used to compute the response ofstructures modeled as multi-degree-of-freedom,and one therefore relies on a time-historyanalysis for computing the inelastic response. Inmany cases, structures house equipment aresensitive to floor vibrations during anearthquake. It is sometimes necessary todevelop floor response spectra from the timehistory response of the floor. In addition, whendesigning critical or major structures such aspower plants, dams, and high-rise buildings, thefinal design is usually based on a completetime-history analysis. The problem which oftenarises is what representative accelerogramshould be used. Artificially generatedaccelerograms which represent earthquakecharacteristics such as a given magnitude,epicentral distance, and soil condition of thesite have been used for this purpose as well asin research. For example, Penzien and Liu (2-127)

used artificial accelerograms to investigate thestatistical characteristics of inelastic systemsand Lai and Biggs (2-126) used them to obtaininelastic acceleration and displacementresponse ratios.

Random models have been used to simulateearthquake ground motion and generateartificial accelerograms. Both stationary andnonstationary random processes have beensuggested (see for example 2-135 to 2-138).Other studies have proposed site-dependentpower spectral density from recorded groundmotion, which can be utilized in generatingartificial accelerograms. One of the firstattempts in generating artificial accelerogramswas by Housner and Jennings (2-135) whomodeled ground motion as a stationaryGaussian random process with a power spectraldensity from undamped velocity spectra ofrecorded accelerograms. They developed aprocedure for generating a random function thathas the same properties of strong earthquakeground motion and used it to generate eightartificial accelerograms of 30 sec durationwhich exhibit the same statistical properties ofreal ground motion.

The detailed description of the proceduresfor generating artificial accelerograms is

beyond the scope of this chapter. It may,however, be useful to briefly mention the basicelements, which are generally needed togenerate an artificial accelerogram. In mostcases, these elements consist of a powerspectral density or a zero-damped responsespectrum, a random phase angle generator, andan envelope function. The simulated motion isthen obtained as a finite sum of severalharmonic excitations. Usually an iterativeprocedure is needed to check the consistency ofthe artificial motion by examining its frequencycontent through its response spectrum or itspower spectral density. A typical artificialaccelerogram and integrated velocity anddisplacement generated from the Kanai-Tajimi(2-19, 2-20) power spectral density for alluviumusing the peak acceleration and the duration ofstrong motion of the SOOE component of EICentro, the Imperial Valley earthquake of May18, 1940 is shown in Figure 2-72.

2.13 SUMMARY ANDCONCLUSION

The state-of-the-art in strong motionseismology and ground motion characterizationhas advanced significantly in the past threedecades. One can now estimate, with reasonableaccuracy, the design ground motion andspectral shapes at a given location. Earthquakemagnitude, source distance, site geology, faultcharacteristics, duration of strong motion, etc.influence ground motion and spectral shapes.While building codes and seismic provisionsaccount for some of these influences such assite geology, magnitude, and distance, otherssuch as fault characteristics, travel path, andduration require further studies before they canbe implemented.

Response spectrum is used extensively inseismic design of structures. Recent codesrecommend acceleration amplifications in termsof seismic coefficients, which account for sitegeology, shaking intensity, and distance forconstructing design spectra and computing thedesign lateral forces.

118 Chapter 2

500c:.2 ..- 0e (l)

~ ~ a(l) E0 00«

-5000 5 /0 /5 20 25 30 35 40 45 50

50

>.. 0- (l)'-0 ~ 0.2~ E

0

-500 5 /0 /5 20 25 30 35 40 45 50

- 20c:(l)

E(l) E 00

.2 0

0...!(?Q

-200 5 /0 /5 20 25 30 35 40 45 50

Time, sec.

Figure 2-72. Acceleration - time history and integrated velocity and displacement generated from the Kanai-Tajimi powerspectral density for alluvium using the peak ground acceleration and the duration of the SOOE component of EI Centro, the

Imperial Valley earthquake of May 18, 1940.

In moderate and strong earthquakes,structures can experience nonlinear behaviorand dissipate a portion of the seismic energythrough inelastic action. To account for theenergy absorption capacity of the structure,seismic codes allow the use of responsemodification factors, referred to as R-factors, toreduce the elastic design forces and amplify theelastic displacements (drifts). Although theapplication of inelastic spectra is limited tostructures which can be modeled as singledegree-of-freedom, inelastic spectra can be usedto estimate the ductility demands which areneeded to compute response modification or Rfactors.

In special cases such as design of critical oressential structures, a time-history analysis maybe warranted. Determination of a representative

set of accelerograms which reflects theearthquake characteristics expected at the site isimportant. Artificially generated ground motionmay be used to determine representativeaccelerograms.

In most cases, particularly for critical andessential structures, the advice of geologists,seismologists, geotechnical engineers, andstructural engineers should be obtained beforeground motion and spectral shape estimates arefinalized for design.

ACKNOWLEDGMENT

The authors wish to thank Dr. Fawzi E.Elghadamsi who co-authored this chapter in thefirst edition of the handbook. His contributions,some of which are reflected in this edition, aregratefully acknowledged.


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The Seismic Design Handbook || Earthquake Ground Motion and Response Spectra