PB 295661

CAl~FORNIA ~NST~TUTE OF TECHNOLOGY

EARTHQUAKE ENGINEERING RESEARCH lABORATORYCenter for Research on the Prevention of Natural Disasters

HYS1flE~IE¥~C ~tES~ONSE Of

A N~NIE=§iO~lf ~1E~NfORCED

CONC~lEl1!E BU~l[Dl~NG DUR~NG

THE SAN f[E~NAN[)O [EA~1THQUAKE

by

H. lemura and P. C. Jennings

EERL 73-07

A Report on Research Conducted under Grantsfrom the National Science Foundation and

the Earthquake Research Affiliates Programat the California Institute of Technology

Pasadena, CaliforniaOctober/ 1973 REPRODllCED BV

NATIONAl TECHNICAILINFORMATIOI'i !!;EI~Vi~~

u. S. DEPARTMENT OF c: il \ IERCESPRI~GFJELD, V~

CALIFORNIA INSTITUTE OF TECHNOLOGY

EARTHQUAKE ENGINEERING RESEARCH LABORATORY

Center for Research on the Prevention of Natural Disasters

HYSTERETIC RESPONSE OF A NINE-STORY

REINFORCED CONCRETE BUILDING

DURING THE SAN FERNANDO EAR THQUAKE

by

H. Iemura and P. C. Jennings

Any opinions, findings, conclusionsor recommendations expressed in thispublication are those of the author(s)and do not necessarily reflect the viewsof the National Science Foundation.

Report No. EERL 73-07

A Report on Research Conducted under Grants from theNational Science Foundation and

the Earthquake Research Affiliates Program.at the California Institute of Technology

Pasadena. California

October. 19 7 3

HYSTERETIC RESPONSE OF A NINE-STORY

REINFORCED CONCRETE BUILDING

DURING THE SAN FERNANDO EARTHQUAKE

by

* **H, Iemura and P, C, Jennings

ABSTRACT

The Millikan Library on the campus of the California Institute of

Technology was strongly shaken during the San Fernando earthquake of

February 9. 1971, The building was not dam.aged structurally, but the

observed E- W response of the building showed a fundam.ental period of

about LOsee, significantly longer than the 0, 66 sec observed in pre-

earthquake vibration tests, In this study, the response of the fundaITlental

m.ode was treated as that of a single-degree-of-freedom. hysteretic structure,

and four siITlple m.odels, two stationary and two with changing properties,

were exam.ined to see if they could describe the observed response, It was

found that an equivalent linear m.odel and a bilinear hysteretic ITlodel both

could m.atch the response, provided their properties were changed during

the earthquake, (Four changes were used), A linear m.odel with constant

properties and a stationary, bilinear hysteretic ITlodel did not give nearly as

good agreem.ent as the nonstationary models, The results indicated, in

general, a degrading of the stiffness and energy dissipation capacity of the

building, but it could not be determined whether the change s were sudden

or graduaL

-,--.-Graduate Student in Mechanical Engineering, California Institute ofTechnology,

...1,.. ..l~

-''''-Professor of Applied Mechanics, California Institute of Technology,

-1­

INTRODUCTION

It is the intent of ITlost ITlodern approaches to earthquake- resistant

design to produce a structure capable of responding to ITloderate shaking

without dam.age, and capable of resisting the unlikely event of very strong

shaking without seriously endangering the occupants. In the second case,

however, structural daITlage and large deflections are permis sible provided

collapse is not iITlITlinento To achieve this goal, it is necessary to under­

stand the way buildings and other structures respond to deflections beyond

the elastic UITlit, and ITluch analytical and experiITlental work has been

directed in recent years toward developing the required knowledge. In this

effort, the development of analytical models for hysteretic behavior has been

guided alITlost exclusively by static tests of structural elements and as­

seITlblages because it is not yet pos sible to excite full- scale structures

significantly into the yielding range, and because the response of structures

that have been heavily damaged under the action of strong earthquake ITlotion

has not yet been recorded. Thus, the desired full- scale, dynaITlic, confirm.a­

Hon of the approaches to the analysis of earthquake response of yielding

structures have not yet been obtained.

In many studies of nonlinear response to earthquake ITlotions or other

dynamic forces, the yielding properties of structures have been modelled

by the well-known elasto-plastic or bilinear force-deflection relations.

References 1 and 2 are among the earliest works, and Reference 3 is one

of the several studies presented at the Fifth World Conference on Earthquake

Engineering which used these relations. In addition to these sim.ple yielding

relations, trilinear (4) and smoother but more cOITlplex ITlodels of yielding

- 2-

behavior (5) have also been used in studies of earthquake response. Som.e

of the most recent work in this area includes the developm.ent of models for

the deteriorating hysteresis evidenced by structures that are weakened by

excur sions beyond the ela stic lim.it (6 ~ 7).

The occurrence of an earthquake can be viewed as a full- scale experi­

m.ent and it is possible to learn m.uch about the properties of structures from.

exam.ination of their response to strong shaking (8), The largest collection

of data of this type is from. the recent San Fernando earthquake (9) in which

responses of about 50 instrum.ented buildings in the Los Angeles area were

obtained, None of the instrum.ented buildings was heavily dam.aged~ but

som.e did show evidence of nonlinear behavior in the form. of lengthening of

periods of the lower m.odes of vibration over those found from low-level

vibration tests. A particular exam.ple of this occurred in the E- W response

of the Millikan Library on the cam.pus of the California Institute of Technology.

The earthquake m.otion was measured at the basement and at the roof by two

RFT- 250 accelerographs which recorded the N-S, E-Wand vertical com.ponents

of the earthquake m.otion and building response, During the earthquake, the

E=W motion at the roof reached a peak value of 340, 5 cm./sec 2 (35%g)~ and

clearly showed the fundam.ental period to be about one second, which is 50%

greater than the value of 0,66 sees determ.ined from. forced vibration tests

performed before the earthquake (10, ll)(the N-S response showed about a

20% reduction in the fundam.ental natural frequency. Visual exam.ination of

the E- W accelerogram. and Fourier analysis of the record (12, 13) suggested

that the library responded to the earthquake m.otion as a hysteretic structure

to a degree that might make it a useful object of study.

The only observed effects in the building after the earthquake that m.ight

bear on the E- W response were sm.all cracks at som.e floors in the interior plaster

-3-

at the points of supports of the precast window wall panels. The exterior of

the panels can be seen on the south face of the building in Figure 1. Because

the building suffered no observable structural damage and because only the

earthquake response at the top floor was "lvailable. it was not considered

justified to make a detailed. nonlinear model of the structure of the library.

It was decided instead to treat the response of the library in its fundamental

mode as a single-degree-of-freedom hysteretic structure. The intent of

this approach was to learn in a general way about the response of the building

during the earthquake. and also to develop techniques of analysis that may be

useful when the response of damaged structures is obtained in the future.

In particular. we were interested in finding out if the response of the library

in its fundamental mode could be satisfactorily described by one of the simpler

models for hysteretic behavior.

As described in this report, several methods were tried in the attempt

to find simple descriptions for the nonlinear response of the building during

the earthquake. First, the measured motion of the fundamental mode of the

structure was examined to see if the hysteretic relation could be determined

from the measured earthquake response. It was found that, with some care.

the dynamic force- deflection relation of the fundamental mode could be re­

covered. Another portion of this analysis was concerned with the nonstationary

characteristics of the response in terms of the parameters of equivalent funda­

mental frequency and equivalent viscous damping. The changes of these

variables during the response guided the selection of nonlinear models of

the structure. The second major portion of the study was devoted to a com­

parison of the recorded response of the fundamental mode with that predicted

by analyses using various hysteretic models. The models included a

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- 5-

stationary linear model with damping and frequency characteristics chosen

to ITlatch the recorded response. a stationary bilinear ITlodel. and two non-

stationary ITlodels. The nonstationary ITlodels were of two types. an equiva-

lent linear ITlodel with daITlping and fundaITlental frequency that changed at

selected tiITles during the earthquake. and a nonstationary D bilinear hysteretic

ITlodel whose properties also were changed during the response. The study

closes with a discussion of the accuracy of the various ITlethods proposed,

and conclusions about their application to the response of Millikan Library.

MILLIKAN LIBRAR Y AND THE RECORDEDEAR THQUAKE RESPONSE

Brief Description of Millikan Library

The Millikan Library at the California Institute of Technology is a

nine- story. reinforced concrete building constructed in 1966 (11). The

lateral load re sistance in the N - S direction is provided by reinforced concrete

shear walls and the resistance in the E- W direction is provided by a central

elevator and stairwell core also of reinforced concrete. In addition. the

structure possesses a reinforced concrete fraITle. The shear walls cOITlprise

the east and west faces of the building, whereas the north and south faces

consist of precast concrete window-wall panels which are attached three per

floor between reinforced concrete coluITlns. It was deterITlined froITl forced

vibration tests of the structure during construction that these precast window-

wall panels added appreciable stiffness to the structure for motions in the

E- W direction. An exterior view of the building is shown in Figure 1 which

also includes sketches of the foundation. More detailed inforITlation about

the structure can be found in References 10 and 1 L

-6-

Results of Vibration Tests Before the Earthquake

During the final stages of construction, the library was subjected to

an extensive series of dynamic tests by J. Kuroiwa and one of the authors

(10,11). In these tests it was found that the fundamental period in the E- W

direction was o. 66 sees. This value increased roughly 3% over the ampli­

tude range of testing. The mode shape corresponding to this fundamental

frequency was found from measurements taken at every other floor of

the structure. In the vibration test the damping in the fundamental E- W

mode varied between O. 7 and 1. 5 percent of critical, increasing with the

amplitude of response. Measurements of the foundation motions and mo­

tions on the nearby surface of the ground showed that the building responded

nearly as if it were fixed at the foundation; rocking contributed less than one

percent to the total roof motion of the structure, and foundation transla­

tion les s than about two percent.

Within a few days after the earthquake an ambient vibration test was

performed on the structure during which the fundamental E- W period was ob­

served to be 0.80 sees (12,14). Hence, there appeared to be a permanent

change in the fundamental period of small vibrations in the E- W direction.

It has been found that since the post- earthquake test, the structure has

partially recovered and it exhibited a fundamental period of 0.73 sees in

the E-W direction in December, 1972 (12).

Accelerograms Recorded During the Earthquake

Two accelerograms, one at the basement and one at the roof, were

obtained at the Millikan Library during the San Fernando earthquake. The

accelerograms and the calculated velocities and displacements are shown

-7-

in Figures 2 and 3 (15,16). In these figures, 80 secs of m.otion is shown,

but not all of this m.otion is im.portant to the present study. The first

40 secs of the accelerogram. at the basem.ent m.ay be separated for discus­

sion into two parts. The first part of the accelerogram. (from. 0 to about

15 secs) has a high acceleration level with a relatively high predom.inant

frequency, whereas the second part of the record (from. 15 to 40 sees)

shows a relatively low acceleration level, and a lower predom.inant frequency.

Com.paring Figures 2 and 3, it seem.s that the different character of the

response in Figure 3 in the early and latter parts of the record m.ay be due

to the different types of excitation that arrived during these two portions of

tim.e (17); there appears to be a larger fraction of surface waves in the latter

portion of the basem.ent accelerogram., The first part (0-15 secs) of the

accelerogram. shown in Figure 3 consists of a m.ixture of the first and second

m.odes of response. The period of the second m.ode in the E- W direction is

approxim.ately 0, 17 sec. During the second portion of the response (15-

40 secs) the m.otion consists alm.ost exclusively of the fundam.ental m.ode.

Com.paring the levels of m.easured acceleration at the base and at the roof, it

appears that the ground m.otion was such as to excite the structure in a quasi­

resonant fashion during the latter part of the record,

The displacem.ent record in Figure 3 consists of a short-period

(about L 0 sec) portion and fluctuations at longer periods. Since the accelera­

tion at the roof records the absolute m.otion of the structure, it is considered

that the displacem.ent record shows a com.bination of the m.otion of the

structure with respect to the base, which is the m.otion of shorter period,

superim.posed upon a longer-period m.otion which represents the displace­

m.ent of the foundation seen in Figure 2.

There are two m.ajor characteristics of the m.otion which are most

apparent from. exaTIlination of the records of earthquake response. First,

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the fundamental period of the E- W vibration during the strong motion is

about 50% longer than that measured at small amplitudes during vibration

tests; it is clear from Figure 3 that the period of the E- W fundamental mode

during the earthquake is near one second, Second, the records show that

the library responded primarily in its fundamental mode in this direction.

Although there is some vibration of the second mode apparent in the first

part of the response, it is generally small with respect to the response of

the fundamental mode, From these observations it was thought possible to

consider the library to be a single- degree- of- freedom hysteretic structure

responding to the earthquake, filtering or disregarding components of higher

modes of response.

ANALYSIS OF RECORDED MOTIONS

Calculation of Relative Velocity and Displacement

The calculation of relative velocity and displacement is required

to determine the hysteretic character of the restoring force acting on the

structure as a function of amplitude of response. Considering the library as

a single-degree-of-freedom oscillator, the acceleration, velocity and displace­

ment shown in Figure 3 may be considered as the absolute response of the

oscillator, whereas those in Figure 2 may be considered as the base motion.

Therefore, if the two recorded accelerograms have an accurate time cor­

respondence, the relative velocity and displacement of the oscillator can be

obtained by subtracting the calculated ground velocity and displacement from

the calculated values of velocity and displacernent obtained from the record

measured on the roof.

Fortunately, the two accelerograms were recording a common time

signal and were, in fact, a part of a more extensive network (18) which

included accelerographs at the Jet Propulsion Laboratory, Millikan Library

and the Caltech Seismological Laboratory.

- 11-

When the calculated values of velocity and displacement were sub-

tracted from each other to obtain the relative motion, it was found in

preliminary analyses that the relative displacement included long fluctua-

tions with a period of about 11 sees. It was subsequently pointed out by

T. C. Hanks that these were due to a processing error in the digitizing of

some accelerograms, which has since been corrected. To eliminate this

II-second period motion from the records analyzed in this study, a low-pass filter

proposed by Jennings, Housner and Tsai (9) was employed. The subtracted

and corrected relative acceleration, velocity and displacement are plotted in

Figure 4. Comparing Figures 3 and 4 (which are at differen.t time scales) it

is seen that the changes in the acceleration are slight, but the smoothing

process of the integration, the subtraction of the long-period ground displace­

ment, and the elimination of the digitizing error, have led to comparatively

smooth curves for relative velocity and displacement. Some possible con­

tributions of the second mode to the acceleration and relative velocity can be

seen, but the relative displacement is essentially only that of the fundamental

mode. It should be noted in Figure 4 that the motions beyond 40 sees have no

longer been included in the analysis.

Analysis of Natural Frequencies and Amplitudes

From the relative displacement shown in Figure 4 and replotted in

Figure 5, it is easy to see that the period of the motion is much longer than

the period of vibration exhibited at small amplitudes. To investigate the

nature of the nonlinearity of the restoring force, the period and corresponding

amplitude of each whole cycle of displacement were measured from Figure 5

and plotted in Figure 6. The number of each point in Figure 6 corresponds

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to the number of the cycle as indicated in Figure 5, The results obtained

from the vibration tests before and after the earthquake are also plotted in

Figure 6, The points in Figure 6 show considerable scatter, which is

expected in a measure this crude, and it is hard to find clear relations in

the figure, However, the points numbered 1 to 11 and the re suIts of the tests

before the earthquake suggest an approximately linear relation between the

amplitude and period of vibration, with the larger amplitudes corresponding

to the longer-period motion. The remaining points, numbers 12 to 29, are

scattered about one second over a fairly wide range, but above the approximately

linear band shown by points 1 to 11.

These results suggest that the library may have behaved like one

hysteretic structure up until about 15 sec s, and then changed to a different

hysteretic structure. This is also consistent with the observed loss of struc­

tural stiffnes s indicated by the post- earthquake vibration test, To investigate

this suggestion further, the measured periods of vibration are plotted on the

time axis in Figure 7, which also includes, as lines, the results from the vibra­

tion tests before and after the earthquake. It is seen from this figure, which

also shows considerable scatter, that the natural period tends to increase

gradually until about 14 sees, Points number 12 and 13 show unusually long

periods but these points may be subject to more error than others as the

amplitude of response is quite small (Figure 5), After these points, most

of the values fluctuate around one second. Shnilar trends were obtained by

F, E. Udwadia and M. D. T rifunac from their analysis of the accelerograms

using Fourier transform techniques (12, 13). The results from their work

confirm that the fundamental period of vibration in the E- W direction increased

about 50% during the first part of the strong shaking, and remained at about

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one second for the rest of the first 40 sees of response. even though the

amplitude of the response decreased. Their analysis also showed unusual

behavior at about t = 15 sees and, in the period of weak response from 40 to

80 sees, a tendency for the period to shorten from LOsee to values in the

range of 0.8-0.9 sees.

Experimental Hysteretic Diagrams

The analysis of the previous section identifies the nonlinearity of the

restoring force as the reason for the lengthening fundamental period ob­

served during the earthquake. In this section, the time-dependence of the

hysteretic behavior of the library is studied by plotting the measured values

of acceleration against the calculated values of relative displacement.

Consider the equation of motion of a single-degree- of- freedom system

excited by an earthquake:

F(x, x) = -MG + ;;] (1)

in which F(x, x) represents the nonlinear restoring force due to relative

velocity x and displacement x; M is the mass and ;; is the ground accelera­

tion. Equation 1 shows that the total restoring force divided by the mass is

the negative of the absolute acceleration. Using this relation, a preliminary

version of the hysteretic response of the library was obtained by plotting the

relative displacement shown in Figure 4 vs. the absolute acceleration shown

in Figure 3. This trajectory, plotted every 0.02 sees, gave a reasonable

estimate of the first-mode hysteresis of the library during the second portion

of the response, because the first mode of the vibration predominates at this

time. However, the first portion of the response (0 to 15 sees) showed marked

fluctuations along the trajectory of the supposed first-mode hysteresis. This

- 18-

fluctuation was thought to be the result of the non-negligible contributions

of the second mode of vibration of the library, which is discernible in this

part of the acceleration records.

Because of the assumptions of the study it was considered appropriate

to eliminate the effect of the second mode of vibration as well as any higher

modes that may have participated in the response. If this could be done, the

desired trajectory between the absolute acceleration and the relative displace­

ment of the fundamental mode would be obtained. The simple low-pass filter

(19) mentioned above was again used to eliminate these higher mode responses

from the absolute acceleration record and then the relative velocity and displace­

ment were calculated. These curves are shown in Figure 8, which can be

compared with the absolute acceleration in Figure 3 and the relative velocity

and displacement in Figure 4. It can be seen from this comparison that the

response of the higher modes has been greatly diminished, but not completely

eliminated, especially in the region from about 5 to 8 sees. Using the results

shown in Figure 8, the trajectory between the first mode absolute acceleration,

which is porportional to the restoring force by Equation I, and the relative

displacement was plotted every 0.02 sees and is given in Figure 9, In plotting

these trajectories it was found that there was a small phase error of about

,04 to ,06 sees between the absolute acceleration and the relative displacement,

This phase error significantly affected the shape of the trajectories and, unless

corrected, some of the trajectories indicated negative hysteretic damping. By

close examination of the digitized data, the amount of this phase errOr was

found to differ over the first 12 seconds of the response when compa red to the

part after 12 sees. The source of this small phase error could not be identified,

but it is small enough that it is a possibility that it is a phase difference in the

300

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- 20-

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RELATIVE DISPLACEMENT IN CMo~ 2 seconds

300

200[;:len"­uwen;C 100uz

ZEl

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RELRTIVE OJSPLACEMENT IN eM2 ~ 4 seconds

6

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300

-300-6 -l! -2 0 2 lj

RELATIVE DISPLACEMENT IN CM4~6 seconds

6 -6 -1,\ -2 0 2 1,\

RELATIVE DISPLACEMENT IN CM6 ~ 8 seconds

6

FIRST MODE HYSTERESIS OF MILLIKAN LIBRARY FRGM RECGRO

FIGURE 9aHysteretic response of single-degree-of-freedom. oscillatormodelling fundam.ental mode. as determined from recorded motions,

- 21-

300

u 200w(fJ......uw(fJ

~ 100uz~

ZEJ

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cr: -200

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RELRTIVE DISPLRCEMENT IN CM8 ~ 1 a seconds

6

300

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RELATIVE DISPLRCEMENT IN CMIO~12 seconds

6

300

u 200w(fJ......uw(fJ

"-t3 100

ZE:J

;:: 0a:ccw-'wuua: -100wI-=:J-'E:J(fJcoex: -200

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300

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RELATIVE OISPLACEMENT IN CMIZ ~ 14 seconds

6 -6 -Ii, -2 0 2 Ii,

RELATIVE DISPLACEMENT IN eM14~16 seconds

6

FIRST MGDE HYSTERESIS GF MILLIKAN LIBRARY FRGM RECGRDFIGURE 9b

Hysteretic response of single-degree-of-freedom. oscillatorm.odelling fundam.ental m.ode. as determ.ined from. recordedm.otions. (Cont' d. )

-22-

300

u 200I.IJ<.n.....uI.IJIf)

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-300

RELRTIVE OISPLRCEMENT IN CM16~18seconds

300

u 200wtn.....Wwen"-B 100z......

£5;:: 0Oi:ex:w.JI.IJWWOi: -100I.IJI-::J-l<DencoCl: -200

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RELRTIVE OISPLRCEMENT IN CM18 ~ 20 seconds

6

300

u 200UJ<.n.....uUJ<.n

~ 100u

z

Z10

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i: 100uz,...,

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-300~ 4 ~ 0 2 ij

RELATIVE DISPLRCEMENT IN CM20 ~ 22 seconds

6 -6 -ll -2 0 2 ij

RELATIVE DISPLACEMENT IN CM22 ~ 24 seconds

6

FIRST M~DE HYSTERESIS GF MILLIKAN LIBRARY FRGM REC~RD

FIGURE 9cHysteretic response of single-degree-of-freedom oscillatormodelling fundamental mode, as determined from recordedmotions, (Cont'd,)

- 23-

300

u 200wen"-uwen"-tJ 100z......ZE:l

;:: 0ex:c:ew-IUJUU

ex: -100UJ....:::J-IE:len0Clex: -200

-300

300

u 200wen"-Uwen"-tJ 100

ZE:l

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-300-6 -\,\ -2 0 2 L!

RELATIVE DISPLACEMENT IN CM24 ~ 26 seconds

6 -6 -l! -2 0 2 l!

RELATIVE DISPLACEMENT IN CM26~28 seconds

6

300

u 200wen"-uwen~ 100uz......ZE:l

;:: aex:c:ew-IWUUex: -100~:::J-I\0enCD

ex: -200

-300

300

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RELATIVE DISPLACEMENT IN CM28~30 seconds

6 -6 -l! -2 0 2 1,\

RELATIVE DISPLACEMENT IN CM30 ~ 32 seconds

6

FIRST MODE HYSTERESIS OF MILLIKAN LIBRARY FROM RECORD

FIGURE 9d

Hysteretic response of single- degree- of-freedom. oscillatorm.odelling fundam.ental m.ode, as determ.ined from. recordedmotions. (Cont'd,)

- 24-

in the digitization, which cannot be expected to be much more accurate than

about, 04 to , 06 sees, Other possibilities include Instrument malfunction

around t = 12 sees or a small error in phase that might have been introduced

because of the application of the filters to the record,

To adjust for the phase error, the time history of the relative dis­

placement was shifted to match the peak value of the absolute acceleration

during the two parts of the response, This was done because the maximum

restoring force should occur at the same time as the maximum relative dis­

placement for the small values of viscous damping associated with the library.

At the beginning of the response, from 0 to 4 sees as shown in Figure 9a,

the hysteretic properties of the library are not clear because of the small

amplitudes, As is well known, the tangent of the trajectory is equal to the

square of the fundamental natural frequency for a structure that responds

essentially in the linear range. The slope of the trajectory from 0 to 4 sees

appears to be close to that of a linear structure with a natural period of O. 66

sees (for which the tangent value is 90/ sec 2), indicating that the library was

vibrating at the beginning of the earthquake with the fundamental period found

during the pre-earthquake vibration tests.

The slope of the trajectory is still steep from 4 to 6 sees as shown in

Figure 9a. However, the plots show more hysteresis due to the high response

levels. There is a large loop on the minus side of the trajectory and after­

wards there is a sharp drop in the restoring force, perhaps indicating a

sudden change in some structural elements du.e to the strong vibration. From

6 to 8 sees the slopes of the hysteresis loops have become less and the areas

of the hysteresis loops have become larger. There are also some short­

period fluctuations along the supposed first-mode hysteresis loops which are

- 25-

thought to be the results of the incom.plete filtering of the absolute accelero­

gram. as discussed above. It is also possible that these fluctuations represent

sm.all errors in the calculation. which appears to be a sensitive one. Figure 9b

shows the response from. 8 to 10 sees, and it is seen that there are still som.e

fluctuations and sudden changes in the restoring force but. in general. the

loops are becom.ing sm.oother. The slopes of the hysteresis loops are clearly

less than during the early part of the earthquake. and the area of the hysteresis

loops is still large. As seen in the sam.e figure. the slope of the hysteresis

loops from. 10 to 12 sees are alm.ost as soft as the stiffness of the linear

structure with a natural period of one sec (a tangent value of 39/ sec 2). There

is a suggestion. however. that the areas of the hysteresis loops during the

period from. 10 to 12 sees are less than those for 6 to 8. or 8 to 10 sees.

There are still fluctuations in the trajectory which m.ay be associated

with the second m.ode of response. From. 12 to 14 sees the areas of the hystere­

sis loops have clearly becom.e sm.aller. suggesting that the energy dissipation

capacity of the library at this am.plitude has decreased because of the previous

vibrations. The hysteresis loops are also noticeably sm.oother. presum.ably

due to the predom.inance of the fundam.ental m.ode of vibration. The rem.aining

portion of the response. from. 14 to 32 sees (figures 9b, c, and d) shows that

the library continues to exhibit a softer restoring force with a relatively

sm.aller energy dissipation capacity. when com.pared to the earlier response.

This is true even though the response level is decreasing. Com.paring the

responses between 4 to 6 sees and between 28 to 30 sees, which have about

the sam.e absolute acceleration level. it is seen that there has been a degrada­

tion of the stiffness of the structure. It is also seen. from. com.paring the two

figures for the periods from. 6 to 8 sees. and from. 24 to 26 sees. that the

energy absorbing capacity of the library has changed during the earthquake

response.

-26-

The overall indication gained from. Figure 9 is that the library lost

not only some of its stiffness, but also SOITle energy dissipation capacity

due to the large am.plitude response during the Jirst part of the earthquake.

This nonstationary characteristic of the hysteretic behavior of the library

agrees, in principle, with those of siITlple theoretical ITlodels of deteriorating

structures, although the details of the hysteretic behavior are SOITlewhat

different from. the theoretical ITlodels so far sugge sted.

It was thought desirable to estiITlate m.ore precisely the loss of

stiffness and energy absorbing capacity evidenced during the response.

The stiffness of the library during each full cycle of relatively large aITlpli-

tude has already been estim.ated and is shown in Figure 7. To ITlake a sim.ilar

study of the nonstationary behavior of the energy absorbing capacity, the

hysteresis loops were used to estilnate an equivalent viscous damping factor

for each full cycle of response.

In this study the equivalent viscous dam.ping factor h was definedeq

by equating the energy dis sipated by hysteresis to that dis sipated by viscous

daITlping.I'

6 2h w tl dlJ,J eq eq( 2)

in which F(/.L, il) is the restoring for c e; 11-, /J- are the relative displace-

ITlent and velocity respectively, and W is the equivalent natural frequencyeq

m.easured from. that portion of the response. To evaluate the right-hand

side of Eq. 2, it was assum.ed that over a cycle the aITlplitude of the response

was a slowly varying sine wave, i. e .

.j..L (t) = -w IJ, (t) sin [w t + $(t) ]

eq 0 eq( 3)

- 27-

in which ~(t) is a slowly varying phase angle.

From. Eqs. 2 and 3 the equivalent viscous dam.ping factor h iseq

obtained as

h ==eq1

2TT W 2 fJ 2eq 0

( 4)

in which fJ is the m.easured am.plitude of the relative displacem.ent ando

~ F(fJ,. [J,) is evaluated from. the hysteresis loops given in Figure 9.v

The equivalent viscous dam.ping factors calculated this way are plotted

for each full cycle in Figure 10. From. the nature of the assum.ptions

involved and the inherent errors, it is not expected that this would be a

precise calculation. Figure 10 indicates that the library showed about 8 to

10% of critical dam.ping from. about 4 to 10 sees at which tim.e the am.plitude

of the response reaches a m.axim.um. value. After 10 sees the energy

absorbing capacity shows a reduction, which is consistent with the suggestion

that a relatively sudden change in the energy absorbing capacity took place

at about the time of maxim.um. response. As pointed out above, the only

observable earthquake effects on the structure were small cracking in

the plaster in the vicinity of the m.ounts of the precast window wall panels.

It is one possibility that the working loose of these m.ountings was the cause

of the observed behavior.

ANALYTICAL MODELS OF FIRST-MODE RESPONSE

A Stationary, Equivalent Linear Model

Before attempting to model the response by nonlinear hysteretic

behavior, a simple linear model was tried, both to establish a base for

further com.parisons and to investigate the capabilities of this sim.plest

possible approach. In order to m.odel the first mode of the library as a

00

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::)-

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EQU~VALENT

DAMP~NG

FAC

TOR

FRO

MTH

EF~RST

MO

DE

HYSTERES~S

FIG

UR

E10

Eq

uiv

ale

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ea

sured

resp

on

se.

- 29-

simple oscillator it was necessary to calculate the participation factor of

the fundamental modeo This was done using the experimentally determined

values obtained in the vibration te sts 0 As indicated in the Appendix, the

input acceleration level to the equivalent linear oscillator was adjusted by

the participation factor of the fundamental mode and by the weighting factor

for the response of the rooL

The period of the equivalent linear rnodel was taken as 1 0 0 sec in

agreernent with the second part of the response shown in Figure 7 0 The

equivalent darnping factor was chosen to be 5% of critical darnping, a repre­

sentative value taken from Figure 10. It might be noted that approxirnately

2% of this 5% can be associated with viscous-like dam.ping measured in the

pre-earthquake vibration testso

Using this simple linear model, the absolute acceleration, relative

velocity and relative displacement were calculated and plotted in Figure 11 0

During the early part of the response from 0 to 15 sees the calculated

response in Figure 11 does not coincide with the first mode response shown

in Figure 8 0 The difference is particularly noticeable around 10 sees,

where the calculated response is decreasing, whereas the measured

response is growing and showing its maximum valueo The reason for this

discrepancy is that in the beginning of the vibration the library has a funda­

mental period of about 0 0 66 sees, whereas the simple linear model has a

period of one second throughout the responseo In addition, the assumed

dis sipation value of 5% is les s than aet-ually shown by the library during

that early portion of the responseo The coincidence of the calculated

response in Figure 11 and the first mode response in Figure 8 is much better

300

2:

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fth

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tio

nary

lin

ear

Ino

deL

= 31-

during the second part of the response. from 15 to 40 sees. In particular, the

phase difference is very smalL The simple linear model works well for this

portion of the response, which is consistent with the smooth hysteresis loops

shown in Figure 9, and the generally constant value of energy dissipation as

indicated by Figure 10.

From these results it can be said that this simple linear model of the

structure gives good agreement only for the portion of the response between

15 and 40 sees, the portion of the response over which structural parameters

do not change significantly. The simple linear model does give a reasonably

good estimate of the maximum response of the structure, and may therefore

be useful from the point of view of design. From the point of view of research,

however, it would seem that much better agreement could be obtained using

a more detailed model of the hysteretic behavior.

Stationary, Bilinear Model

A stationary, bilinear hysteretic model was adopted in this section

to represent the nonlinear hysteretic characteristics of the restoring force

of the structure. Considering the shape of the hysteresis loops given in

Figure 9, and the trends in the equivalent linear parameters shown in

Figures 7 and 10, the bilinear model selected was chosen to have a small

yield displacement with respect to the maximum response, and a relatively

steep second slope. The yield level of this model was chosen to fit the

observed behavior and does not indicate yielding in the structural frame

of the Jibrary. A hysteretic model with these parameters will, for large

deflections, show a small Cl,mount of energy dissipation and an equivalent

natural frequency which is almost the same as that indicated by the second

slope of the hysteretic diagram. The first slope of the bilinear hysteretic

- 32-

model was chosen to give a natural period of 0, 66 sees, whereas the second

slope was chosen to correspond to a linear restoring force for a structure

with a natural period of 1.0 sees, The transition point between these two

slopes was set at 0,25 em. which is only slightly larger than the maximum

displacement during the vibration experiments, A typical hysteresis loop

for a structure of this type is shown in Figure 12,

The calculated response of this bilinear hysteretic structure sub­

jected to the recorded base acceleration is shown in Figure 13. which gives

the absolute acceleration. relative velocity and relative displacement of

the oscillator. The calculated hysteretic behavior comparable to Figure 9

is plotted in Figure 14, The response values plotted in Figure 13 show

very poor agreelTIent with those from the first-mode response. Figure 8.

except for the phase in the period from 12 to 17 sees, Comparing the hy­

steretic response frolTI 4 to 6 sees, as shown in Figures 9a and 14a, it is seen

that the bilinear model gives a stiffness of the restoring force that is too low.

Also it is seen frolTI Figures 9b. c and 14b. c that the bilinear relation shows

too lTIuch hysteretic damping frolTI 8 to 24 sees, This is consistent with the

calculated response being slTIaller than the lTIeasured response during this

interval. These comparisons indicate that a satisfactory description of the

response by a stationary hysteretic model is unlikely and that better agree­

lTIent could be attained using a nonstationary mode,

Nonstationary. Equivalent Linear Model

It was seen previously that stationary lTIodels of the fundalTIental 1TI0de

gave only lilTIited agreelTIent with response measured during the earthquake.

In this section. a nonstationary, equivalent linear model. which changes its

structural parameters at selected points during the response. was tried to

see if the agreement could be ilTIproved, This was done both to check the

-33-

l~ .0em

NU00iflJ

"Eu

=LO

STAT~ONARY 8~L~ NEAR MODELFIGURE 12

Typical hysteresis loop for the stationary, bilinear hysteretic m.odel.

I W ,.j::>. I

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-35-

300

u 200UJrn"-UUJ(J1

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a: -100UJI-::J-.JElrn<Xla: -200

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f-

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300

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cr: -100UJI-::J-.JE:lrntna: -200

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-300

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RELAT!VE OlSPLRCEMENT IN eMo~ 2 seconds

~ ~ ~ 0 2 1,\

RELATIVE OISPLRCEMENT IN CM4 ~ 6 seconds

6

6

300

u 200UJ(J1

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a: -200

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RELATIVE DISPLACEMENT IN eM2 ~4 seconds

-6 -1,\ -2 0 2 lj

RELRTIVE OISPLRCEMENT IN CM6 ~ 8 seconds

6

6

HYSTERETIC RESPONSE OF STRTIONRRY BILINERR MGOEL

FIGURE 14a

Hysteretic response of stationary 9 bilinear hysteretic modelof fundamental mode,

RELRTIVE DISPLRCEMENT IN eM

8~ 10 seconds

300

u 200UJen......uwen"-i5 100

zo;::: aa:0::UJ--1UJUU

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-36-

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RELRTIVE DISPLRCEMENT IN CMlO~12 seconds

5

300

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RELRTIVE OISPLRCEMENT IN CM12~14seconds

5 -6 -1,\ -2 0 2 1,\

RELRTIVE DISPLRCEMENT IN eM14~16 seconds

5

HYSTERETIC RESP~NSE ~F STRTI~NRRY BILINERR MDDELFIGURE 14b

Hysteretic response of stationary. bilinear hysteretic modelof fundamental mode, (Cont1d,)

-37-

300

u 200l.LI01......UW01......G 100z

ZID

;:: aex:a:l.LI-1l.LIUUex: -100l.LIf-::J-1ID01aJex: -200

-300

300

u 200l.LI01......Uwlfl......25 100z

ZID~

f- 0ex:a:w-1WUU

a: -100l.LIf-::J-1E:l01aJex: -200

-300

300

u 200l.LI(f)......uwlfl

;;: 100uz~

ZID~

f- aa:a:~l.LIUU

cr: -100l.LIf-::J-1ID01ma: -200

-300

-6 -4 -2 0 2 l.j

RELRTIVE DISPLRCEMENT IN CM16~18seconds

~ ~ ~ 0 2 l.j

RELRTIVE DISPLRCEMENT IN CM20 ~22 seconds

6

6

300

u 200l.LI01......UW01......25 100z....~;:: Da:a:w...JWUWa: -100l.LII-:=J...JE:l

~ex: -200

-300

-6 -4 -2 0 2 l.j

RELRTIVE DISPLRCEMENT IN CM18 ~20 seconds

-6 -1,\ -2 0 2 lJ

RELRTIVE DISPLRCEMENT IN CM22 ~24 seconds

6

6

HYSTERETIC RESPGNSE GF STATIONARY BILINEAR MGDELFIGURE 14c

Hysteretic response of stationary, bilinear hysteretic modelof fundamental inodeo (ContI do)

-

-

111#

-

-

I I I I I I I I I I I I

RELRTIVE DISPLACEMENT IN CM26 ~28 seconds

300

u 200I.!Jen"-uUJ(f)

;;: 100uz.....z~

;:: 00:oc:W-IWUU0: -100wf-::l-Ie::lenlD0: -200

-300-6 -1,\ -2 0 2 lJ

RELRTIVE DISPLRCEMENT IN eM24 ~26 seconds

6

-38-

300

200f:jen"­uw(f)

;;: 100uz.....ZEl

;:: 00:ceW

u:luua: -100wf-::l-IElenCDex: -200

-300-6 -1,\ -2 o 2 6

300

u 200!J.J(f)

"-UI.!Jen;;: 100u:z.....~;:: a0:ceL!.J-IWUWa: -100L!.Jf-::l..J10enCDex: -200

-300

-

-

11

J

-

-

I I I I I I I I I I I I

300

u 200l.!Jen"-ul.!J(f)

"-5 100z.....ZEl

;:: 0ex:oc:W

u:luua: -100wf-::l..JElenCOex: -200

-300

,......

~

Ifj

-

-

I I I I I I I I I I I I

-6 -1,\ -2 0 2 !J

RELRTIVE DISPLRCEMENT IN CM28 ~ 30 seconds

6 -6 -1,\ -2 0 2 l!

RELRTIVE DISPLRCEMENT IN eM30.~32 seconds

6

HYSTERETIC RESPONSE OF STRTIGNRRY BILINERR MGDELFIGURE 14d

Hysteretic response of stationary, bilinear hysteretic modelof fundamental mode, (Cont' d. )

-39-

accuracy of the equivalent linear parameters given in Figures 7 and 10

and also to investigate the nonstationary characteristics of the response.

The time-dependent equivalent natural frequency wand damping factoreq

h for the nonstationary model were selected from examination of the data,eq

and are shown in Figure 15.

During the computation of the response, the changes of stiffness were

implemented at times when the relative displacement was zero so as not to

cause any permanent deformation. The calculated response of this oscil-

lator, shown in Figure 16, agrees quite well with the response of the first

ITlode shown in Figure 8. Thus, very good agreement with the observed

behavior can be obtained by considering the structure to behave like a linear

oscillator whose natural frequency and damping factor change during the

course of the earthquake response. The good agreement suggests also that

the analysis presented above can give sufficiently accurate nonstationary

equivalent linear parameters. It is seen from Figure 15 that the stiffness of

the equivalent linear system degrades to a constant value, whereas the equiva-

lent damping factor of the system first increases and then decreases to a

value somewhat lower than the peak response, but higher than the initial

value o

Nonstationary, Bilinear Model

In this section, a nonstationary, deteriorating model of bilinear

hysteresis is proposed to describe the response of the fundamental ITlode.

The model chosen consists of four different bilinear hysteretic relations

all having the saITle second slope. The time-dependent characteristics of

the stiffness and energy dissipation capacity of the library are represented

by changing the stiffness of the first slope and the yielding displacement.

)=0

uLa

z w~6

O.8~

rw

w0

....J

oc:3

006

GO

o0

o0

0o

00

<JLLL

'"

o0

>cr

0

~~

w00

400

2:3

w::

J00

2~ <1C

000

Z

e=

'==

'IJ

!i

)2--

Jj

0E

_=

=,

Ii

I~

!~

48

12!6

20

24

28

32

TiM

EIN

SE

CO

ND

S

NO

NS

TA

TIO

NA

RY

EQ

UIV

AL

EN

TL

INE

AR

PA

RA

ME

TE

RS

G--

ME

AS

UR

ED

FR

OM

FIR

ST

MO

DE

RE

SP

ON

SE

DE: o

00

20

FF

ZU

W<

J[...

JLL

crc<

!00

0>~

JC00

I-z

::J-

OQ

..W~ <

I o0

00

0 a

o

o0

(30

00

oo

oo

0

oe

0o

o0

00

I ,.f>­ a I

NO

NS

TA

TiO

NA

RY

EQ

UiV

ALE

NT

LiN

EA

RP

AR

AM

ETE

RS

FIG

UR

E1

5

Eq

uiv

ale

nt

dam

pin

gfa

cto

rsan

dn

atu

ral

freq

uen

cie

sfo

rn

on

stati

on

ary

peq

uiv

ale

nt

lin

ear

mo

deL

300

I >1'>-

.

......

!

j__

LL

LJ

H-H

-H-+

+-H

-+-H

-H-l

--H

++

-t+

++

--H

-'H

--+

+-r

Hrl

*-f

-JH

'-\+

-\\-

1R

c-t

J-3

00 l,IO

z ~U

W"-

LU~

!-

en:=

la::

'­...

.JO

CU

OlU

WO

(f)

-J

(J")

OO

W"­

cru

z::

uU

cr.

w)­

;>

!-u

.-.

<-J

ill

~utno

CK

:iC

l'-

,.

~~E

I

WW

UO

C>

-1.;\

0 10

I-

:2:

!JJW

;>d

l:_

IJ..

I~uz:O

cr:C

I:u

-J-J

I.L

lCL

ocU

1e-J

iC1

-10

oI_

___

un

__

L__

II

IL

II

510

1520

2S30

3540

TIM

EIN

5ECG

NOS

RESP

ONSE

OFNO

NSTR

TION

RRY

EQU

IVRL

ENTL

INER

RM

ODEL

FIG

UR

E1

6

Ab

solu

teaccele

rati

on

,re

lati

ve

velo

cit

yan

dre

lati

ve

accele

rati

on

of

no

nsta

tio

nary

lin

ear

mo

deL

-42-

Guided by the results of previous analyses, the yielding displacement for the

bilinear model was taken as large as 1.0 cm during the initial portion of the

response; and for the latter portion of the response, a smaller value of 0.07 cm

was used. The four different bilinear relations employed to model the non­

stationary characteristics of the structure are consistent with the equivalent

linear parameters shown in Figure 15, and hysteresis loops for these relations

are shown in Figure 17. The loss of stiffness with time and the decreasing

capacity for dissipating energy are apparent from Figure 17.

During the computations of response the transition from one hysteretic

model to another was controlled to avoid jumps in the restoring force. The

calculated values of absolute acceleration, relative velocity and relative

displacement for the nonstationary bilinear hysteretic model are plotted in

Figure 18. Figure 19 shows the calculated hysteretic behavior of the model,

and is to be compared with Figure 9. Comparing the responses in Figure 18

with those of the first mode in Figure 8, it is seen that the two results agree

very well except for a few peaks around 8 sees. Comparing the hysteretic

diagrams in Figures 9 and 19, the calculated hysteretic behavior produced

by the nonstationary bilinear model seems to represent the deteriorating

characteristics of the restoring force of the structure fairly well. The

agreement might be improved by the introduction of another, fifth model,

or by changing the properties of the four used, but the main features of the

hysteretic characteristics seem to be represented reasonably well by the

nonstationary model used in the analysis.

CONCLUSIONS

Of the four simple m.odels used to describe the E- W response of

the fundamental mode of the library during the San Fernando earthquake,

the best agreement was achieved by the use of the two nonstationary

150

Nu@IOO~ 91Eu

O~6 SECONDS

100N

U@

~E 50u

1000.25

~tOri-I (39/~ec2)

9~16 SECONDS

.,..43-

150

i2! r---- "N 1~ 100 1 I«n

" IE IU I

IiI

~an-I <:39/se(2 )

6""'9 SECONDS

100iN

U@

~E 50u

2.0 3.0em

~tOri~! (39/sec2)

16;v40 SECONDS

NONSTATIONARY BILINEAR MODELFIGURE 17

Typical hysteresis loops for the nonstationary. bilinearhysteretic model.

I ~ ~ I

!I

II

I

II

II

II

II

o5

1015

2025

3035

40TI

ME

INSE

COND

S

300

z aU

w.....

..w

f-f-(f)

::Ja::

'--,

a::::

ue::

>w

wO

(/)--,(f)

COW

'-cru

::E

UU

a:: -3

00 40

w>-

>f-U

o-.

,...

...,

Wf-u

(f)O

erE

)'-

-,--

,::E

ww

U0

:>

-LW 10

I- Zw

W>

::E

......

,I.A

.JI-U

::E

cra::

-O

-,-

-,u

wQ

...

o:(

f) ......

0

-10

RESP

ONSE

OFNO

NSTR

TION

RRY

BIL

INER

RM

ODEL

FIG

UR

E1

8

Ab

solu

teaccele

rati

on

,re

lati

ve

velo

cit

yan

dre

lati

ve

dis

pla

cern

.en

to

fth

en

on

stati

on

ary

,b

ilin

ear

hy

ste

reti

crn

.od

el.

f-

-

,

i-

i-

I I I I I I I 1 1 I I I

i-

l-

I

f-

f-

I I I I I I I I I I I I

300

u 200IJJen"Uwen~ 100uz...,Zin

~ 0a:a::IJJ-IIJJUU

a: -100IJJI-:::>-Ic:JUlCD0: -200

-300~ 4 ~ a 2 ~

RELRTIVE DISPLRCEMENT IN eM

O~2 seconds

6

-45-

300

u 200wen"UWUl

~ 100(J

z...,ZEl

~ 0a:ex:w-IWUU

0: -100wI-:::>-'c:Jen00a: -200

-300-6 -1,\ -2 0 2 II

RELATIVE DISPLACEMENT IN CM

2 ~4 seconds

6

300

u 200wen

"(JwUl

~ 100(J

z<-<

ZEl

;:: aCl:cr:IJJ-IWU(J

0: -100w1-:::>-IElU"lCDa: -200

-300

300

u 200wUl

"UWU"l

"13 100z<-<

Zin

;:: a0:cr:W-'WWua: -100w1-:::>-'<Ji:lUl00a: -200

-300-6 -1..\ -2 0 2 l.I

RELRTIVE DISPLACEMENT IN CM4~6 seconds

6 -6 -1,\ -2 0 2 \1

RELATIVE DISPLRCEMENT IN eM6 ~8 seconds

6

HYSTERETIC RESPGNSE GF NGN5TRTIGNARY BILINERR MGDEL

FIGURE 19a

Hysteretic response of nonstationary. bilinearhysteretic model of fundamental modeo

300

u 200UJ(fl

"-uUJ(fl

;;: 100u

""-.2:lD

;:: 0~UJ-.lWUUCI: -100wI-::J-llD(flCD0:: -200

-300

300

u 200w(fl

"-uW(fJ

;;: ! 00u2:....,

5;:: 00::a:UJ-.lUJUUIX -100UJI-::J-l<!:l(flC!JCl: -200

-300

-6 -l! -2 0 2 l,\

RELATIVE DISPLRCEMENT IN eM8 ~ 1 0 seconds

~ ~ ~ a 2 l!

RELATIVE DISPLRCEMENT IN CM12 ~ 14 seconds

6

6

-46-

300

u 200wen"-uw(fl

;;: ! 00u

""-.2:El

;:: 0Cl:ex::liJ-.lUJUU

Cl: -100UJI-::::>-lEl(fJ00a: -200

300

u 200UJen"-uW!Il

;r 100u

z-.ZI!:l

;::: 0CK:a:wu::luuIX -100UJI-::J-le:J(flWCK: -200

-300

-6 -ll -2 0 2 II

RELATIVE DISPLACEMENT IN CM1 0 ~ 12 seconds

-6 -l! -2 0 2 II

RELRTIVE DISPLACEMENT IN CM14 ~ 16 seconds

6

6

HYSTERETIC RESPGNSE Gf NGNSTATIONARY BILINEAR MGDElFIGURE 19b

Hysteretic response of nonstationary. bilinearhysteretic model of fundamental mode. (Contlcl.)

300

u 200wen'"uwen~ 100uz...,z(0

::: 0~W-JWUWa: -100UJI-::J-llI:)en!Dex: -200

-300-6 -Il -2 a 2 I.!

RELATIVE DISPLRCEMENT IN CM16 ~ 18 seconds

6

-47-

300

u 200UJcn"-uUJcn~ 100uz,...,z(0

:= 0~l!J..Jl!JUU

cx: -100l!JI-::J..J\0cnCQcx: -200

-300-6 -!! -2 0 2 l.\

RELRTIVE DISPLRCEMENT IN eM18 ~ 20 seconds

6

300

200~cn'"u.....cn'"B 100z....5:= aex:a:w..JWUUex: -100wI-::J..J(0cncoex: -200

-300

300

u 200UJen

'"u!.!Jen~ 100u2,...,2lO

r-:: 0~W-lWUU

a: -100wI-::J...JlI:)<nCQ

ex: -200

-300-6 -1,\ -2 0 2 l.\

RELATIVE DISPLACEMENT IN CM20 ~22 seconds

6 -6 -I!, -2 0 2 \J,

RELATIVE DISPLRCEMENT IN CM22 ~ 24 seconds

B

HYSTERETIC RESPGNSE GF NGNSTRTIGNARY BILINEAR MGDEL

FIGURE 19cHysteretic response of nonstationary. bilinearhysteretic model of fundamental modeo (ContI do)

300

u 200I.!JU1"-UWU1

:;;: 100uz

ZID

;:: 0cr:Cl:I.!J...JI.!JUU

a: -100I.!JI-=:J...JIDU100a: -200

-300

300

u 200wU1"-UW(IJ

;;: 100uz....,z(!:J

~ 0cr:Cl:W...JWUU

cr: -100wl-=:J...J(0(IJotl0;: -200

-300

-6 -IJ -2 0 2 Il

RELATIVE DISPLACEMENT IN eM24 ~ 26 seconds

-6 -IJ -2 0 2 l,\

RELATIVE DISPLACEMENT IN CM28 ~ 30 seconds

6

6

-48-

300

u 200wen"uw(IJ

"'-5 100

-300

300

u 200ILlif)

"uw~5 1002-~-l- 0err:!Ii:w...JIJJUU

0.: -100IJJI-::J...JG:!if)00cr -200

-300

-6 -l! -2 0 2 l1

RELATIVE DISPLRCEMENT 1M CM26 ~ 28 seconds

-6 -l! -2 0 2 IJ

RELRTIVE DISPLRCEMENT IN CM30 .~ 32 seconds

6

6

HYSTERETIC RESPGNSE GF NGNSTATIGNARY BILINEAR MGDELFIGURE 19d

Hysteretic response of nonstationary 9 bilinearhysteretic model of fundamental mode (Cont' d, )

- 49-

oscillators, The two stationary models, an equivalent linear model and a

bilinear hysteretic model, also with constant properties, were not capable of

duplicating the earthquake response nearly so well as the nonstationary

oscillators, The simpler, stationary models did give maximum responses

close to that observed in the earthquake, however, so that their use would have

uroduced valid information in an analysis intended for design,

The two nonstationary models that gave good agreement were an

equivalent linear model with properties that were changed at four times during

the earthquake, and a bilinear hysteretic model that also changed properties

four times during the response, Equally good agreement was obtained with

either model, and it is concluded that any of the more common hysteretic

rrlOdels giving the general trend of equivalent natural frequency and equivalent

damping factor shown in Figure 15 probably could be made to give good agree­

ment between observed and calculated responses, In doing any such analyses,

however, it does appear necessary to change the properties of the model during

the course of the response; it seems doubtful that any of the simple, non­

degrading hysteretic models could be capable of giving the degree of agreement

shown by the nonstationary models,

The results of the analysis, and study of the observed E- W response

of the library, clearly indicate a significant decrease in the stiffnes sand

energy dissipation capability of the building during the course of the earth­

quake response, This is perhaps most easily seen in Figure 1 So It is not

possible to relate the changes, with confidence, to any observed damage to

the building, nor is it possible to ascertain whether the changes were sudden

or graduaL It seenlS quite possible, however, that the observed behavior is

at least partly a consequence of the behavior of the precast concrete panels

-50 -

that contain the windows, and it seems to the authors that relatively rapid

or sudden changes in properties are more likely to have occurred than

gradual ones,

The simultaneous measurement of the ninth floor and basement

motions allowed the calculation of the relative response which, in this case,

could be used to construct an experim.ental estimate of the hysteretic response

of an oscillator modelling the fundamental mode of the structure, To this

extent it was possible to study the actual hysteretic behavior of the library

and thereby to judge the type of hysteresis that best described the response.

The method used in this report appear s to be a. promising one for studying

earthquake response of hysteretic structures, even though some difficulties

exist in obtaining hysteretic trajectories from the response, To study the

hysteretic behavior in ITlore detail, in particular to determine where in the

structure the hysteresis ITlight be concentrated, would require more instru­

mentation than is present in the library. It is concluded that one instrument

per floor, all with a cornmon timing signal, would be the minimum required

to give the information needed.

ACKNOWLEDGMENT

The authors wish to acknowledge the financial support of the Rotary

Foundation which made it possible for the senior author to study at the

California Institute of Technology as a visiting student. The author swish

also to thank Professors D. E. Hudson and M. D. Trifunac for their discus­

sions concerning the data processing and recording of the accelerograms

used in this analysis,

This research was supported in part by the National Science Foundation

and the Earthquake Research Affiliates of the California Institute of Technology.

The financial support of these organizations is gratefully acknowledged.

L

2.

3.

4.

5,

6,

7

8.

9.

10.

lL

REFERENCES

Tanabashi, R" I'Studies on the Nonlinear Vibrations of StucturesSubjected to Destructive Earthquakes, II Proceedings of the First WorldConference on Earthquake Engineering, Berkeley, CaliL, 1956.

Berg, G, V., and D, A. Da Deppo, "Dynamic Analysis of Elasto­Plastic Structures, If Journal of the Engineering Mechanics Division,ASCE, Vol. 86, No. EM2, April, 1960, pp. 35- 58.

Goto, H., and H. Iemura, llEarthquake Response of Single-Degree­of- Freedom Hysteretic Structures, " Preprint No o 266, Proceedingsof the Fifth World Conference on Earthquake Engineering, Rome, 19730

Umemura, H., H. Aoyama and H. Takizawa, "Analysis of the Behaviorof Reinforced Concrete Structures During Strong Earthquakes Basedon Empirical Estimation of Inelastic Restoring Force Characteristicsof Members, II Preprint No. 275, Proceedings of the Fifth WorldConference on Earthquake Engineering, Rome, 19730

Jennings, P, C" l'Earthquake Response of a Yielding Structure,"Journal of the Engineering Mechanics Division, ASCE, VoL 91,EM4, August,1965, pp,41-68.

Clough, R, W" and S. B, Johnston, "Effect of Stiffness Degradationon Earthquake Ductility Requirements, II Proceedings of the JapanEarthquake Engineering Symposium, 1966, October, 1966, pp. 227­232.

Iwan, W. D., "A Model for the Dynamic Analysis of DeterioratingStructures, " Preprint No, 222, Proceedings of the Fifth WorldConference on Earthquake Engineering, Rome, 1973.

Hudson, D. E., "Dynamic Properties of Full-Scale StructuresDetermined from Natural Excitations," Dynarnic Response ofStructures, Edited by Herrman,G. and N. Perrone, Pergarnon Press,1 972, pp. 159- 177.

Jennings, P, C" Ed., "Engineering Features of the San FernandoEarthquake, II Earthquake Engineering Research Laboratory ReportNo, EERL 71- 02, California Institute of Technology, June, 197 L

Jennings, P. C. and H, Kuroiwa, IIVibration and Soil-StructureInteraction Tests of a Nine-Story Reinforced Concrete Building, IIBulletin of the Seismological Society of America, Vol. 58, No.3,June, 1968, pp,891-916,

Kuroiwa, H., "Vibration Test of a Multistory Building, II EarthquakeEngineering Research Laboratory, California Institute of Technology,June, 1967,

.- 52-

12 0 Udwadia, F oEo, and Mo Do Trifunac, IITim.e and Am.plitude DependentResponse of Structures, II October, 1973 (in m.anuscript),

13 0 Udwadia, F oEo, and Mo D, Trifunac, IIAm.bient Vibration Tests ofFull-Scale Structures, II Proceedings of the FifthWorld Conferenceon Earthquake Engineering, ROIne, 19730

14. McLam.ore, V oRo, "post Earthquake Vibration Measurem.ents,Millikan Library, II 1970 (unpublished report).

15. IIStrong-Motion Earthquake Accelerogram.s, Uncorrected Data,VoL I, Part G, II Earthquake Engineering Research LaboratoryReport EERL 72- 20, California Institute of Technology, June, 1972.

160 IIStrong-Motion Earthquake Accelerogram.s, Corrected Accelerogram.sand Integrated Ground Velocity and Displacem.ent Curves, VoL II;Part G, II Earthquake Engineering Research Laboratory ReportEERL 73- 52, California Institute of Technology, 1973 (in m.anuscript),

17 0 Trifunac, M oDo, "Stress Estim.ates for the San Fernando, California,Earthquake of February 9, 1971: Main Event and Thirteen Aftershocks, IIBulletin of the Seism.ological Society of Am.erica, Vol. 62, Noo 3,ppo 721-750, June, 19720

18 0 Keightley, W o00' IIA Strong-Motion Accelerograph Array withTelephone Line Interconnections, It Earthquake Engineering ResearchLaboratory Report EERL 70-05, California Institute of Technology,Septem.ber, 19700

190 Jennings, P. Co, Go W. Housner and No Co Tsai, IISim.ulated EarthquakeMotions, II Earthquake Engineering Research Laboratory, CaliforniaInstitute of Technology, April, 1968.

APPENDIX

This appendix describes the calculation of the participation factor of

the fundamental mode and the weighting factor for the first mode for response

as measured on the roof, These factors are required to scale the measured

response on the roof to the response of the single-degree-of-freedom oscil-

lator that models the fundamental mode of the building.

The equation of motion for earthquake response of an n degree-of-

freedom system such as the library can be written as

M{x} + cUd + K[x} ::: -MfI };;(t) (All

in which M. C and K are the n x n mass. damping and stiffness matrices.

respectively, The vector [x}. ([x} T::: [Xl' x 2• "" x } 1 denotes the relativen

displacements. [l} symbolizes the vector {I} T::: fl. 1•.. ,. I}. and z(t)

is the acceleration of the base of the structure.

The matrix of mode shapes ep is defined by

(A2l

in which the colunm vectors are the individual mode shapes, 1. e" [~.} T :::1

Letting

(A3)

substituting into Eq, AI. and multiplying by q, T gives

Under the assumption that the damping matrix C can be diagonalized by the

same transform.ation ep which diagonalizes M and K, the individual equa-

Hon in matrix equation A4 will be uncoupled. A typical equation will have

- 54-

the forITl

f+1

o

;. +1

(A5)

The coefficient of ;;(t) is the participation factor, f h .th da., 0 r tel ITlO e,1

and in particular

:::

[~l}TM[l }

[~l}TM[~l}(A6)

As sUITling that the solutions to equation AS are known, the response of

each of the n ITlasses can be found by use of equation A3. Ifanindexofl inaITlode

corresponds to the roof, the roof displaceITlent is

(A7)

The ITlodal ordinate ~ 11 is herein called the weighting factor.

FroITl the test data (10 ), the first ITlode of the nine- story library was

found as

[~l}T::: [1,00,0.87,0.74,0.62,0.51,0.40,0.28,0.20,0.11, 0.04} (A8)

where values for interITlediate floors have been interpolated froITl the ITleasure-

ITlents, The ITlass ITlatrix is given in the saITle reference as

M::: 2600 kipsg

1,0

0.75

0.75

0.75

o

0.75

o

0.75

0.75

0.94

0.88

(A9\

-55 -

Evaluating equation A6 it is found that

al = 1 0 44

and from equation AS

~ll = 1,00

(AIO)

(All)

The product of these two factors, 1044, is the desired ratio, Leo,

the response of an oscillator subjected to the recorded base acceleration

should be multiplied by 1044 before comparison with the fundamental mode

response, as measured on the rooL For the calculations in this report,

the rounded value of lo 4 has been used,