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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
A ~ I
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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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-300
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MIL
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RST
MODE
FIG
UR
E8
Ab
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on
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- 20-
-
-
l-
I-
I I I I I I I I I I I I
RELATIVE DISPLACEMENT IN CMo~ 2 seconds
300
200[;:len"uwen;C 100uz
ZEl
;:: 0a:a::
~uua: -100~:::l..JElenaJa: -200
-300-6 -IJ -2 o 2 6
300
u 200 c-wen"-uwen"- 100 '-EU
Z
~;:: aa:a::w..JWUUex: -100 IwI-:::l..JElen00ex: -200 f-.
-300~ ~ ~ 0 2 l!
RELRTIVE OJSPLACEMENT IN eM2 ~ 4 seconds
6
u 200 I- u 200w wen en"- "-u uw
Iw
en en"- "- 100E 100 - EU U
Z Z~ ~
z ZEl EJ....
0 ~
0l- I ~yI-ex: ex:a:: a::w w
..J ..JW WU
Uu
u ua: -100 - a: -100w wl- I-:::l :::l..J ..Je:J EJen enllO aJex: -200 - 0: -200
300
-300 I I I I I I I I I I I I
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
;:: 0ex:ex:~wuucr: -100wI-=:J-'E:l(fJCD
cr: -200
-300-6 -Ii, -2 a 2 l1
RELRTIVE DISPLRCEMENT IN CM8 ~ 1 a seconds
6
300
u 200w(fJ......uw(fJ
"-t3 100z~
ZE:J-I- 0cr:ex:w-'wuuex: -100wI-=:J-'e::I(fJ
00a: -200
-300-6 -Ii, -2 a 2 1.,\
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
-300
300
u 200w(fJ
"-WI.!.J(fJ
"-t3 100z-Ze::I
;:: 0ex:a:l!.J....ll!.JUW
ex: -100l!.JI-=:J-'o(fJcoex: -200
-300-6 -Ii, -2 0 2 1.,\
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)
;;; 100uz,...,Z\D
;:: 0a:ex:I.IJ..JI.IJuua: -100I.IJI-~.J<D<.ncoa: -200
-300
RELRTIVE OISPLRCEMENT IN CM16~18seconds
300
u 200wtn.....Wwen"-B 100z......
£5;:: 0Oi:ex:w.JI.IJWWOi: -100I.IJI-::J-l<DencoCl: -200
-300-6 -ll -2 0 2 II
RELRTIVE OISPLRCEMENT IN CM18 ~ 20 seconds
6
300
u 200UJ<.n.....uUJ<.n
~ 100u
z
Z10
;:: 0a:ex:w..JI.IJUUa: -100I.IJI-~.Jto<.n00a: -200
-300
300
w 200I.IJ<.n.....Ww<.n
i: 100uz,...,
~;:: 0~W-lWUUa: -100l::!::J-l\D<.nCDCl: -200
-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
;:: 0ex:c:ew-IWuua: -100w....=>-'e:J(f)00a: -200
-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
u 200wen"-uwen"-B 100
Z\D
::: 0ex:c:ew-IWUU
ex: -100wl-=>-'\D(f)COex: -200
-300-6 -l,I -2 0 2 L!
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
0
o0
o
I N co !3
2
G0
L-~
28
~.
L
e0
0o
0G
oo
o0
oo
0
~2
~6
20
24
T~ME
iNS
EC
ON
DS
o
l1
~I
0.~--L_~
o
oo
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G
4o
0020
00i0
0,00
11IL
.
o
at:
r~
Zu
w~
~u.:
~(!)
~z
::)-
OeL
w~ ~ o
EQU~VALENT
DAMP~NG
FAC
TOR
FRO
MTH
EF~RST
MO
DE
HYSTERES~S
FIG
UR
E10
Eq
uiv
ale
nt
da
mp
ing
facto
rs
dete
rm
ined
fro
ma
na
lysi
so
fm
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:
aU
w"-'
wl-I-(f)
.::J
a::
'-JO
J:u
Oa
lJJw
(f)-li
fJ
oo
w'
a:w
:ii:
wU
cr: -3
00 I.LO
w>
>
I-U
........
...w
l-U
ifJO
a:a'
.-J-J
2b
ww
Ll
oc>
-1,40 10
l- Z
Ww
>:i
i:.....
.w
I-U
a:a
::ii
:O.-J-'u
wo.
...o
c(f) -. a
J W o n
-10
o5
1015
20T
IME
IN5E
CGNO
S25
II
I30
35~O
RESP~NSE
OFST
RTIO
NRRY
EQUI
VRLE
NTLI
NER
RM
ODEL
FIG
UR
E11
Ab
solu
teaccele
rati
on
,re
lati
ve
velo
cit
yan
dre
lati
ve
dis
pla
ceIn
en
to
fth
esta
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
l!O35
3025
1520
TIM
EIN
SECC
lN05
10
300
z DU
l.l.
.I'-
"wI-
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f'l
:::l
a:'
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lWW
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...J(
f"j
ro
W'
o:u~
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a:: -3
00 lJ,O
W2>
-:>
I-U
,...,
.....,
I.!J
I-U
iflO
IX
E)'
-1..
..J:
?r:
ww
Uo
c>
-!.LO 10
II-
ZW
W:>
z:
......
wI-U
dr:O
o:C
Cu
-1..
..J
wQ
...
occ
n .......
0
-10
05
RESP
ONSE
OFST
RTIO
NRRY
BILI
NER
RM
ODEL
FIG
UR
E1
3
Ab
solu
teaccele
rati
on
,re
lati
ve
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cit
yan
dre
lati
ve
dis
pla
ceIT
len
to
fth
esta
tio
nary
,b
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ear
hy
ste
reti
cIT
lod
el.
-35-
300
u 200UJrn"-UUJ(J1
"-15 100:z....ZEl
I- 0a:ex:UJ-.JUJUU
a: -100UJI-::J-.JElrn<Xla: -200
-300
f-
f-
f-
'-
I I I I I I I I I I I I
300
u 200UJrn"-uUJen"-i5 100z....ZEl....I- 0cr:a:UJ-.JUJUU
cr: -100UJI-::J-.JE:lrntna: -200
-300
I--
I--
IIi
I--
f-
I I I I I I I I I I I I
300
200[l'U1
~;;C 100u;il
;J<;J
;! 0
~Uu:2 -100I::J-.JE:l(J1tnex: -200
-300
-6 -1,\ -2 0 2 1,\
RELAT!VE OlSPLRCEMENT IN eMo~ 2 seconds
~ ~ ~ 0 2 1,\
RELATIVE OISPLRCEMENT IN CM4 ~ 6 seconds
6
6
300
u 200UJ(J1
"-uUJen"-:lC 100uz
ZEl....I- 0a:a:UJ-.JUJUU
a: -100UJI-::J-.JE:l(J1CD
a: -200
-300
-6 -1,\ -2 0 2 1,\
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
a: -100UJI-~.-JE:lenCDex: -200
-300-6 -4 -2 o 2 4 6
-36-
300
u 200UJen......uUJU1......x: 100uz
ZD
;::: aa:0::UJ--1WUU
a: -100UJI-~.-JDenCDa: -200
-300-6 -l!, -2 a 2 4
RELRTIVE DISPLRCEMENT IN CMlO~12 seconds
5
300
u 200UJen......uwU1......i5 100z~
zo;::: 0 r-----------,oL-I-r---------Ja:ex:UJ--1UJUU
a: -100UJI~.-JDenCDa: -200
-300
300
u 200wen......uwen......i5 100z:
zo~
I- 0ex:0::w--1UJUU
a: -100UJI-~--1E:lenCDex: -200
-300~ ~ ~ 0 2 4
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,