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Nonlinear System Identification of Reinforced Concrete Steel
Structure: Using Pseudodynamic Testing Data
Olga V. Pavlenko1 and Chin-Hsiung Loh, M.ASCE2

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COAbstract: The research results are presented of the joint project of the National Center for Research on Earthquake Engineering
共Taiwan兲 and Stanford University on the pseudodynamic testing of a three-story full-scale reinforced concrete steel moment frame.
Nonlinear stress–strain relations, corresponding to oscillations in the frame, were estimated for successive time intervals from a pseudodynamic test of strong ground motion excitation. Numerical models of the behavior of the frame under the pseudodynamic loading of fourlevels of input ground motion were constructed and used for estimating changes in the shear moduli in the frame during its loading andfor the nonlinear identification of the frame behavior. The contents of linear and nonlinear quadratic and cubic components in the responseof the frame were estimated. It was found that the frame possesses mostly odd-order nonlinearities, such as the third, fifth, seventh, etc.
order nonlinearities, whereas, even-order nonlinearities, such as, quadratic, the fourth, sixth, eighth, etc. order nonlinearities, are weak.
Temporal changes in the behavior of the frame were observed during the test, indicating a substantial decrease in the elastic shear moduli,
increase in the absorption of seismic waves and in the contents of the nonlinear part in the response of the frame.

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DOI: 10.1061/共ASCE兲0733-9399共2004兲130:7共1兲

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IntroductionM

CE Database subject headings: Pseudodynamic method; Data analysis; Numerical models; Full-scale tests; Building frames;
Ground motion.

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Visiting Research Fellow, National Center for Research on Earthquake Engineering, Taipei 106-17, Taiwan, Republic of China.
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Professor, Dept. of Civil Engineering, National Taiwan Univ., Taipei
106-17, Taiwan, Republic of China. E-mail: lohc0220@ccms.ntu.edu.tw
Note. Associate Editor: Roger G. Ghanem. Discussion open until December 1, 2004. Separate discussions must be submitted for individual
papers. To extend the closing date by one month, a written request mustbe filed with the ASCE Managing Editor. The manuscript for this paper
was submitted for review and possible publication on August 4, 2003;
approved on December 23, 2003. This paper is part of the Journal ofEngineering Mechanics, Vol. 130, No. 7, July 1, 2004. ©ASCE, ISSN
0733-9399/2004/7-1–12/$18.00.

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A joint project of the National Center for Research on Earthquake
Engineering 共NCREE兲 in Taiwan and Stanford Univ. on testing ofa three-story full-scale reinforced concrete steel 共RCS兲 composite
moment frame was carried out in October 2002. The test specimen, 12 m tall and 21 m long, was a three-story, three-bay‘‘RCS’’ moment frame, consisting of reinforced concrete columns
and composite steel beams, as shown in Fig. 1. The frame wasamong the largest specimens of its type ever tested with pseudodynamic loading. This three-story prototype structure was designed for areas of high seismicity following provisions for composite structures of the International Building Code 2000. During
the test, the frame was loaded pseudodynamically by using input
ground motions from the 1999 Chi-Chi and 1989 Loma Prieta
earthquakes. The tests were conducted according to the following
schedule:
1. October 11, 9:00 a.m.—the imposed motion was the 1999
Chi-Chi Earthquake, station TCU082, EW component,
scaled to represent the seismic intensity level of the 50%
probability of exceedance in 50 years (PGA⫽0.276g).
2. October 14, 9:00 a.m.—the imposed motion was the 1989
Loma Prieta Earthquake, station LP89g04, NS component,

scaled to represent the seismic intensity level of the 10%
probability of exceedance in 50 years (PGA⫽0.426g).
3. October 15, 9:00 a.m.—the imposed motion was the 1999
Chi-Chi Earthquake, station TCU082, EW component,
scaled to represent the seismic intensity level of the 2%
probability of exceedance in 50 years (PGA⫽0.622g).
4. October 16, 9:00 a.m.—the imposed motion was the 1989
Loma Prieta Earthquake, station LP89g04, NS component,
scaled to represent the seismic intensity level of the 10%
probability of exceedance in 50 years (PGA⫽0.426g).
The input acceleration waveform is shown in Fig. 2. Displacements and force 共collected from the load cell of actuator兲 wererecorded at the floor level of the three stories of the frame. Fig. 3
shows the recorded displacements at each floor level from thefour different tests.
As a whole, the test had the following primary objectives: toprovide data for evaluation and validation the design provisions
for composite moment frames 共strong-column weak-beam criterion, composite action of concrete slab and steel beams, integrityof the precast column and composite beam–column connections,and overall frame response兲; to provide information for validating
simulation models and computer codes for nonlinear simulationand performance assessment; and to support the use of innovative
composite moment frames as alternatives to conventional steel
and concrete systems for regions of high seismicity. Based on theassumption of time-varying system, the modal natural frequencies
were estimated 共Loh et al. 2003兲. Because of severe intensity of
excitation on the frame structure it is believed that nonlinear responses of the frame structure can be observed from the experimental data. To observe the dynamic characteristics of the structure nonlinear system identification must be used.
The successful development of identification procedures for anonlinear system depends upon the model which is used to represent the system under investigation. Experimental analysis ofnonlinear systems may be accomplished by using time domain
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Fig. 1. Dimensions of three-story reinforced concrete steel frame from side view and top viewM1980兲. Unfortunately, functional series models require excessive
kernel values. However, by expanding the system output in termsof past input and output using a nonlinear autoregressive moving
average model with exogenous inputs model, a very concise representation for a wide class of nonlinear systems can be obtained
共Korenberg et al. 1987; Hunter et al. 1990; Loh and Duh 1996兲.

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models. Efforts have been devoted to establish the models of anonlinear hysteretic system and various techniques are developedby many researchers to identify the model parameters. Experimental analysis of nonlinear systems may be accomplished byusing time domain models. Traditionally the functional series descriptions of Volterra and Wiener have been used 共Schetzen

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Fig. 2. Two original ground acceleration time histories from Chi-Chi earthquake and Loma Prieta earthquake for input acceleration in pseudodynamic test
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Fig. 3. Recorded displacement at each floor level from four different tests

8⌬z i⫺1i⫽2,3, . . .
(1b)in which subscripts i and i⫺1/2 refer to levels z i 共of the ith
accelerometer兲 and (z i⫺1 ⫹z i )/2 关halfway between accelerometers
I and (i⫺1)] respectively, ␶ i (t)⫽␶(z i ;t), ␶ 1 ⫽␶(0;t)⫽0 at thestress-free surface; ␳ i⫺1 ⫽mass density of the z i⫺1 to z i layer;ü i ⫽ü(z i ,t)⫽absolute acceleration at level z i ; and ⌬z i ⫽spacing
interval between accelerometers. The corresponding second-orderaccurate shear strains may be expressed as

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⌬z i⫺1⌬z i
1⫹ 共 u⫺u i⫺1 兲
共 u i⫹1 ⫺u i 兲⌬z i⫺1 ⫹⌬z i⌬z i⌬z i⫺1for i⫽2,3, . . .
␥ i⫺1/2共 t 兲 ⫽u i ⫺u i⫺1⌬z i⫺1

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for i⫽2,3, . . .

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(2a)
(2b)

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(1a)

3ü i⫺1⬘ ⫹ü i]85

A simple identification procedure, which was proposed and used
to study the downhole earthquake response, was adapted in thisstudy to identify the nonlinear response of the frame structure.
Using the shear beam model to describe the lateral response,
shear stresses at levels z i and (z i⫺1 ⫹z i )/2 may be evaluated asfollows:i⫽2,3, . . .

␶ i⫺1/2共 t 兲 ⫽␶ i⫺1 共 t 兲 ⫹␳ i⫺1

␥ i共 t 兲 ⫽Shear Stress–Strain Response Analysisü i⫺1 ⫹ü i
␶ i 共 t 兲 ⫽␶ i⫺1 共 t 兲 ⫹␳ i⫺1⌬z i⫺1
2

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Based on the assumption of a time-varying system the response of
the test data was analyzed and the time-varying modal frequency
was investigated 共Loh et al. 2003兲.
In this study the results of the test were used for validating thepossibility of applying the method for estimation of nonlinear
stress–strain relations 共Pavlenko and Irikura 2003兲, for studying
the response of the frame. Another goal of the present work was
the nonlinear identification of the behavior of the frame, and estimation of the contents of linear and nonlinear components in the
response of the frame in conditions of the applied dynamic loading.

in which u i ⫽u(z i ,t)⫽absolute displacement 关evaluated throughdouble integration of the recorded acceleration history ü(z i ,t)].
In this study, u i (i⫽1 – 3) and ü i (i⫽1 – 3) represent the disJOURNAL OF ENGINEERING MECHANICS © ASCE / JULY 2004 / 3

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frame under the applied dynamic loading. A rather good agreement between the simulated and observed accelerograms indicates that the model is a rather good approximation to the real
behavior of the frame. The stress–strain relations obtained show
temporal changes in the frame behavior, as is seen in Fig. 5共a兲.
Noticeable changes in the slopes of the hysteretic curves withtime can also be observed, as shown in Fig. 5共a兲. On the third
stage of the test, the areas within the curves substantially increase,indicating increase in the absorption of seismic waves 关Fig. 5共a兲兴.
The obtained stress–strain relations also show the difference inthe responses of the frame to the same input signals, applied to
the frame on the second and fourth stages of the test, with a 2 day
delay, which is obviously due to changes in the stiffness of thetested frame, as a result of the applied loading.
The estimated stress–strain relations were used to calculate
their slopes, averaged over the three floors of the frame and over
the oscillations within each time interval; their values are shownin Fig. 5共b兲. Changes in the slopes of the stress–strain relations
indicate changes in the shear moduli of the frame. According to
the estimates, the decrease in the shear moduli was observed during the first 共⬃50% of the initial values兲, second 共⬃15% of theinitial values兲, and fourth 共⬃15% of the initial values兲 stages of
the test, whereas at the third stage they remained approximately at
the same level. As far as these conclusions and the following
nonlinear identification of the frame behavior under the dynamic
loading are based on the constructed model of the behavior of theframe, it would be useful to validate our conclusions by analyzing
records, obtained in the test. An agreement of estimates made forthe model with the experimental data would testify to the validityof the model and of our conclusions.

placement and force recorded from the actuator attached at eachfloor of the RCS frame. Simultaneous recording of ␶ and ␥ gives
us a representation of the behavior of the frame.

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Methods of Data Processing

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To carry out the nonlinear identification of the frame responses,
knowledge of the nonlinear stress–strain relations in the frame,
corresponding to the applied loading in various time intervals, isnecessary. To estimate stresses and strains, occurring in the frame
during the process of its loading, the modified method of theestimation of stress–strain relations in successive time intervals
was used. The procedure of estimation of stress–strain relations
in soils in strong ground motion is described in detail in the paper
by Pavlenko and Irikura 共2003兲. To estimate stresses and strainsin this system studied 共frame structure兲, in this study we calculate
acceleration time histories at points of locations of the recording
devices, at the three floors of the frame. For calculations, sets ofparametric stress–strain relations of a certain type were generated
共250 stress–strain relations兲, and item-by-item examination wasapplied to identify the relations showing the best-fit approximation between the observed and simulated records.
For calculations, we used the modified algorithm by Joyner
and Chen 共1975兲. To describe the behavior of the frame under theapplied dynamic loading, we selected stress–strain relations of
the most common type, similar to those used to describe the behavior of water-saturated soils. This choice was based on therecords obtained in the test. In the domain of small strains, thesestress–strain curves are close to linear ones, then they decline tothe strain axis with increasing strain 共i.e., stress and stiffness degradation兲 and, if strains increase more, the curves decline to thestress axis. This type of stress–strain relationship, if defined parametrically, represents the most common case because it can account for all the features in the behavior of a frame in conditionsof loading. The records obtained in the test indicate that this typeof stress–strain relation is also observed in the behavior of theframe under dynamic loading.
To account for temporal changes in the behavior of the frame,
the records were divided by time intervals of 2.8 s duration, andsuccessive time intervals were analyzed. The value of 2.8 s was
selected to satisfy, on one hand, the condition of the invariabilityof stress–strain relations within each time interval 共the intervals
should be short enough兲; on the other hand, the condition of arather long duration of analyzed intervals of records for obtaining
reliable estimates.

Based on the proposed method the impulse characteristics of theframe h 1 (␶) was estimated, obtained in the tests 共Fig. 6兲. Eachcharacteristic reflects the behavior of the frame at a certain stageof the test 共first, second, third, or fourth兲. Oscillating character of
impulse characteristics h 1 (␶) at the first and second stages of thetest 共Fig. 6, 1 and 2兲 indicates multiple reflections of seismic
waves in the frame at these stages. At the third stage of the test,
amplitudes of oscillations substantially decrease, and absorption
increases 共Fig. 6, 3兲, which is in good agreement with the results
obtained for the model.
The maxima of h 1 (␶), corresponding to times of propagationof seismic waves from the bottom of the frame to the third floor,
shift to larger time delays during the test 共Fig. 6兲, was also examined. The propagation times increased from ⬃0.14 s at the firststage of the test to ⬃0.18 –0.2 s at the second and third stages andto ⬃0.24 –0.26 s at the fourth stage of the test. This confirms ourconclusions, obtained for the model, concerning the decrease of
the shear moduli in the frame at the first and fourth stages of thetest, if we take into account that in calculating h 1 (␶), the main
contribution comes from intervals of higher intensity, i.e., themiddle and beginning parts of the records.

Nonlinear Identification

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Since stress–strain relations determine the transformations of incident seismic waves into the system response, nonlinearity ofstress–strain relations leads to the appearance of nonlinear components in the response, because it causes distortions in the
shapes and spectra of propagating seismic waves. If an input signal is the Gaussian white noise, the relation between an input andEM

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The results of the simulations are given in Figs. 4共a–d兲. Thesefigures show the observed and simulated accelerograms for thefour stages of the test. The ‘‘observed’’ accelerations were calculated based on the recorded force at the three floors of the frame.
A fairly good agreement between the observed and simulated
records testifies to the validity of the representation obtained.
Fig. 5共a兲 represents the stress–strain relations obtained for thefour stages of the test 共denoted as 1, 2, 3, and 4兲, which characterize the behavior of the frame under the applied dynamic loading in successive time intervals. According to our estimates, strain
values did not exceed the limits of about ⫾0.04, and stresses lay
within the interval of about ⫾8 kPa 关Fig. 5共a兲兴. The stress–strain
relations obtained represent the model of the behavior of the

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Identification of Impulse Response Function

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Fig. 4. Simulated and observed acceleration time histories of response of full-scale composite frame recorded at three stories; dashed lines
indicate time intervals selected for nonlinear identification of behavior of frame; the imposed ground motion is also shown; 共c兲 for case of October
15 test and 共d兲 for case of October 16 test

兺 G m关 h m共 ␶ 1 ,...,␶ m 兲 ;x 共 t ⬘ 兲 ,t ⬘ ⭐t 兴 ,

(3)

m⫽0

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h 2 共 ␶ 1 ,␶ 2 兲 x 共 t⫺␶ 1 兲 x 共 t⫺␶ 2 兲 d␶ 1 d␶ 2⬁

0

G 1 关 h 1 ;x 共 t 兲兴 ⫽

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⬁

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h 1 共 ␶ 兲 x 共 t⫺␶ 兲 d␶

冕冕 冕
⬁

0

h 3 共 ␶ 1 ,␶ 2 ,␶ 3 兲 x 共 t⫺␶ 1 兲 x 共 t⫺␶ 2 兲(4)⫻x 共 t⫺␶ 3 兲 d␶ 1 d␶ 2 d␶ 3 ⫺3 P

(5)

⫻

冕冕
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⬁

(6)

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G 0 关 h 0 ;x 共 t 兲兴 ⫽h 0

G 3 关 h 3 ;x 共 t 兲兴 ⫽h 2 共 ␶ 1 ,␶ 1 兲 d␶ 1

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where y(t)⫽output; G m ⫽orthogonal functionals if x(t) is theGaussian white noise with a zero mean; 兵 h m (␶ 1 ,...,␶ m ) 其⫽sequence of the Wiener kernels; and ␶ 1 ,...,␶ m ⫽time delays.
Each kernel is a symmetric function. The first four Wiener functionals are represented in the form

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y 共 t 兲⫽G 2 关 h 2 ;x 共 t 兲兴 ⫽01

an output of a nonlinear system can be represented as the Wiener
series 共Marmarelis and Marmarelis 1978兲

h 3 共 ␶ 1 ,␶ 2 ,␶ 2 兲 x 共 t⫺␶ 1 兲 d␶ 1 d␶ 2

(7)

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Fig. 5. 共a兲 Obtained stress–strain relations for different time windows from four tests referred to in Fig. 4. Stresses are given in kPa and strainsare given in strains. Dashed lines indicate time intervals selected for nonlinear identification of behavior of frame 共1, 2, 3, and 4 correspond tofour successive stages of test兲; and 共b兲 decrease in slopes of obtained stress–strain relations with respect to time, reflecting reduction of shear
moduli of frame during four tests

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As is seen from formulas 共3兲–共7兲, the contents of linear andnonlinear components in the response of a nonlinear system depend on the intensity of the testing signal. Therefore, an adequate
choice of the intensity of the testing Gaussian white noise is important for obtaining faithful results. Evidently, we should choosesuch intensity of the testing Gaussian white noise that the contents of the nonlinear components in the response of a system
studied to the Gaussian white noise were similar to that in its
response to the dynamic loading applied in the test. Since thenonlinearity of the system response is determined by the nonlinearity of its stress–strain relations, this study chose Gaussian
white noise as the testing input signal so that average stresses andstrains occurring in the frame during the propagation of theGaussian white noise, coincide with stresses and strains occurringin the frame under the dynamic loading in the test.
Based on the study of nonlinear acoustics 共for example, Zarembo and Krasil’nikov 1966兲, monochromatic signals, passed
through a nonlinear filter, acquire higher harmonics of their mainfrequencies. To emphasize these phenomena, a Gaussian white
noise was selected as an input, and then the nonlinear quadratic,
cubic, and higher-order components of the frame response were
calculated by estimating the Wiener functionals by the method ofcross-correlation functions. The intensities of the quadratic, cubic,
and higher-order components of the response can be expressed inpercent of the whole intensity of the response.

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Fig. 6. Identified impulse characteristics based on records obtained
from four different tests

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Nonlinear Identification w.r.t. Different Time Windows
Figs. 7, 8, 9, and 10 represent the results of the nonlinear identification of the response of the frame at the four stages of the test,
respectively. Four time intervals are analyzed at each stage of thetest, which are marked in Figs. 4共a–d兲 and Fig. 5共a兲. The Gaussian white noise was used as an input signal, and in each timeinterval we chose the intensity of the Gaussian white noise so thatstresses and strains that occurred in the frame were similar to
those that occurred in the frame in conditions of the dynamic
loading applied in the test. In the upper parts of Figs. 7–10, theidentified first-order impulse characteristics of the model of theframe behavior h 1 (␶) and the diagonal values of its impulse characteristics of the second and third orders h 2 (␶ 1 ,␶ 2 ) andh 3 (␶ 1 ,␶ 2 ,␶ 3 ) are shown. The oscillating character of h 1 (␶) indicates an important role of reflections in the propagation of seismic
waves in the frame. A decrease in the amplitudes of oscillations of
the first-order impulse characteristics h 1 (␶) is observed, mostclearly appearing at the fourth stage of the test. This indicates anincrease in the absorption of seismic waves in the frame andagrees well with that observed in the test data. Maxima of thefirst-order impulse characteristics h 1 (␶) shift to larger time delays, and the periods of oscillations of h 1 (␶) increase, which isdue to decrease in the elastic shear moduli of the frame.
Diagonal values of the second- and third-order impulse characteristics h 2 (␶ 1 ,␶ 2 ) and h 3 (␶ 1 ,␶ 2 ,␶ 3 ) are scaled in the samemanner for comparison. It is concluded that coefficients at thequadratic terms in expressions describing the respective stress–
strain relations, are usually substantially smaller than coefficientsat the cubic terms. Multiple reflections of seismic waves in theframe appear in the oscillating shapes of the second- and thirdorder impulse characteristics h 2 (␶ 1 ,␶ 2 ), and h 3 (␶ 1 ,␶ 2 ,␶ 3 ). The
variability of the amplitudes of the diagonal values of the secondand third-order impulse characteristics, which does not show any
definite tendency, is due to the fact that these amplitudes depend
not only on the shapes of the stress–strain relations, but also on

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where ␶⫽time delay; and P⫽intensity of the white noise not depending on frequency.
Using the Gaussian white noise as an input signal provides
essential advantages in describing nonlinear systems: orthogonality of terms in the Wiener series with respect to the Gaussian
white noise allows application of the effective methods of estimating kernels, based on determination of cross-correlation functions; from the other side, the Gaussian white noise as an input
signal allows the most effective testing of the system, because it
contains components of all frequencies and amplitudes.
By analogy with an ordinary impulse characteristic h(t), thekernel series 兵 h m 其 can be treated as a generalized composed impulse characteristic of a system. The first-order kernel determines
the linear part of the system response, whereas higher-order kernels describe interactions between the values of the input signal in
the past with respect to their influence on the response at present
共Marmarelis and Marmarelis 1978兲. If a system is linear,h 2 (␶ 1 ,␶ 2 )⫽h 3 (␶ 1 ,␶ 2 ,␶ 3 )⫽...⫽0, and knowledge of the firstterm of the Wiener series, i.e., knowledge of the first-order kernelh 1 (␶), is sufficient to describe the system. To describe a nonlinear
system, several terms of the Wiener series are necessary. Thus, toestimate the contents of the linear and nonlinear components in
the response of the frame, we calculated the propagation of theGaussian white noise in the frame by using previously obtained
stress–strain relations. Kernels h 0 , h 1 (␶), h 2 (␶ 1 ,␶ 2 ), andh 3 (␶ 1 ,␶ 2 ,␶ 3 ) were estimated by the method of cross-correlation
functions, described in Marmarelis and Marmarelis 共1978兲.
Dash lines in Figs. 4共a–d兲 and Fig. 5共a兲 mark time intervals
selected for the nonlinear identification of the behavior of theframe. Nonlinear identification allows us to distinguish a linear
part of the system response and nonlinear corrections, which are
due to various types of nonlinearity, and estimate their contributions to the response of the system; it allows determination of thetypes and quantitative characteristics of the system nonlinearity
共Marmarelis and Marmarelis 1978兲.

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Fig. 7. 共a兲 First-order, second-order, and third-order impulse response functions identified from four different time windows 共shown in Fig. 4兲
first test data; and 共b兲 Gaussian white noise was used as input to identified system 关shown in 共a兲兴 and results of nonlinear identification of different
time window

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response predicted by the linear model with the nonlinear quadratic and cubic corrections. Discussions are made based on these
nonlinear identifications.
1. The contents of linear and nonlinear components in theframe response are estimated in percent of the intensities of
the response 共Figs. 7–10兲. Table 1 shows estimates of thenonlinear components averaged over the stages of the test.
The last two columns represent the ratios of the nonlinear
quadratic and cubic components and the residual parts of theframe response, which are the differences between the responses of the frame and the responses predicted by linear
models with the nonlinear quadratic and cubic components
共rows 8 in Figs. 7–10兲.
2. As is seen from Table 1, the whole nonlinear component andthe nonlinear quadratic and cubic components of the frame
response increase in successive loadings of the frame tested.

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the average stresses and strains excited in the frame, which varyduring its loading, according to changes in the intensities of input
signals.
In Figs. 7–10, the results of the nonlinear identification of the
behavior of the frame under the dynamic loading are presented.
‘‘1’’ indicates the Gaussian white noise, approximating the imposed motion in the test, which was used as an input signal; ‘‘2’’is the calculated response of the frame to the Gaussian white
noise; ‘‘3’’ is the response predicted by a linear model; ‘‘4’’ is thedifference between the response of the frame and the response
predicted by the linear model; ‘‘5’’ is the nonlinear correction due
to quadratic nonlinearity of the frame; ‘‘6’’ is the difference between the response of the frame and the response predicted by thelinear model with the nonlinear quadratic correction; ‘‘7’’ is thenonlinear correction due to cubic nonlinearity of the frame; and‘‘8’’ is the difference between the response of the frame and thePROOF COPY [EM/2003/023585] 012407QEM

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4.

However, an increase in the nonlinear quadratic componentof the response of the frame during its loading is observed
共Figs. 7–10 and Table 1兲, which is obviously due to changes
in the shapes of the stress–strain relations: the stress–strain
relations gradually become more sloping and stiffness degradation is observed. This process resembles liquefaction insoils, where similar phenomena are observed 共Pavlenko and
Irikura 2002兲. Increase in the nonlinear components in theframe response is obviously related to the decrease in theelastic moduli and strength of the frame tested during its
loadings. In this process, stress–strain relations change theirshapes and slopes, and the domain of the linear elastic behavior of the frame decreases and shifts to smaller strains. In
this case, the nonlinear components of the response obviously increase.
Inaccuracies in the estimates are analyzed, and correction
methods are given in the paper by Marmarelis and Marmarelis 共1978兲. The main causes of the inaccuracies are: the finite
duration and the boundedness of the spectral band of the

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The residual part of the frame response is due to the nonlinear components of higher orders, such as, the fourth, fifth,
sixth, etc., as well as to inaccuracies in the estimates. Thispart of the response also shows the tendency of increasing
with time. Zero-order components of the responses h 0 arevery small, and their values are not given.
Intensities of the nonlinear cubic components of the response
are substantially higher than that for the nonlinear quadraticcomponents 共Figs. 7–10 and Table 1兲. This means that thetested frame possesses strong cubic and weak quadratic nonlinearity. Nonlinear systems defined by hysteretic stress–
strain relations, similar to those shown in Fig. 5共a兲, possess
not only cubic nonlinearity, but also, other odd-order nonlinearities, such as, the fifth, seventh, ninth, 11th, etc. ordernonlinearities, whereas, the role of even-order nonlinearities
is usually small 共Pavlenko 2001兲. Therefore, it is concluded
that the tested frame possesses mostly odd-order nonlinearities, such as the third, fifth, seventh, ninth, etc. order nonlinearities.

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Fig. 8. 共a兲 First-order, second-order, and third-order impulse response functions identified from four different time windows 共shown in Fig. 4兲 ofsecond test data; and 共b兲 Gaussian white noise was used as input to identified system 关shown in 共a兲兴 and results of nonlinear identification ofdifferent time window

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Fig. 9. 共a兲 First-order, second-order, and third-order impulse response functions identified from four different time windows 共shown in Fig. 4兲 ofthird test data; and 共b兲 Gaussian white noise was used as input to identified system 关shown in 共a兲兴 and results of nonlinear identification ofdifferent time window

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and calculation errors usually do not exceed 1–2%, if we use
testing Gaussian white noise of a long duration, such as,
about 500,000 points, or 3 h. Multiple reflections increase
errors in the estimates of linear and nonlinear components up
to 3– 4%, because in this case, exact knowledge of damping
is very important, however, corrections should be similar for
all the stages of the test, and they could not influence thetendency of increasing the nonlinear components in the response of the frame.

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Gaussian noise, truncation of tails of normal distribution ofnoise amplitudes, and inaccuracies in calculations of impulse
characteristics, increasing with increase in the order of thecharacteristic 共Marmarelis and Marmarelis 1978兲. Additional
factors causing inaccuracies are evident, such as deviationsin the constructed model of the frame behavior from the real
behavior of the frame, following inaccurate estimation ofstress–strain relations and their working intervals; discrepancies in the contents of nonlinear components in the response of the frame in the test and in the response of its
model tested by the Gaussian white noise; and inaccuracies
in the profiling data. However, an accurate estimation of thelinear and the whole nonlinear component is most important,because further analysis is based on these estimates. Preliminary calculations show us that the accuracy of the estimatesin the nonlinear identification in this case can be rather high,In this study it is concluded that the method of estimation of
nonlinear stress–strain relations, which was previously developed
for soils, can be applied for studying the response of engineering

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Response Averaged over Stages of Test

10.2
11.1
13.6
13.3

0.17
0.25
0.32
0.27

6.2
6.6
7.1
8.1

0.027
0.029
0.036
0.046

8.7
8.9
11.9
10.6

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Ratio ofnonlinear
Whole
Nonlinear Nonlinear quadratic tononlinearquadratic
cubic
nonlinearStage of component component component
cubic
Residual
the test
共%兲
共%兲
共%兲
components
共%兲

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Table 1. Estimates of Contents of Nonlinear Components of Frame

estimate of the contents of linear and nonlinear components in the
response can be obtained in the case of a proper choice of parameters of testing signals.
Estimated nonlinear stress–strain relations in the three-story
full-scale composite moment frame loaded by scaled motions of
the 1999 Chi-Chi Earthquake and the 1989 Loma Prieta Earthquake in the joint tests of the National Center for Research in
Earthquake Engineering in Taipei and Stanford Univ. show a substantial decrease in the elastic shear moduli in the frame by⬃70% of their initial values during the tests. At the same time,absorption of seismic waves in the frame increased, and the contents of nonlinear components in the frame response increased
from about 8.5–11% to about 15.5–19.5%, which was due to theincrease of odd- and even-order components in the response. It
was found that the frame possesses mostly odd-order nonlinearities, such as the third, fifth, seventh, ninth, etc. order nonlinearities, whereas, the even-order nonlinearities are weak. However,part of the even-order nonlinearities increased during the tests,
which is due to changes in shapes of the stress–strain relations.
Similar phenomena are observed in liquefaction of soils.

01

structures subjected to strong ground motion excitation. Application of this method allows us to obtain information about the
behavior of the construction studied and its elastic moduli in successive time intervals. Estimated stress–strain relations can beused for the nonlinear identification of the response of the construction studied in strong ground motion, and a rather accurate

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Fig. 10. 共a兲 First-order, second-order, and third-order impulse response functions identified from four different time windows 共shown in Fig. 4兲
of fourth test data; and 共b兲 Gaussian white noise was used as input to identified system 关shown in 共a兲兴 and results of nonlinear identification ofdifferent time window

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Joyner, W. B., and Chen, T. F. 共1975兲. ‘‘Calculation of nonlinear groundresponse in earthquakes.’’ Bull. Seismol. Soc. Am., 65, 1315–1336.
Korenberg, M., Billings, S. A., Liu, Y. P., and McIlroy, P. J. 共1987兲.
‘‘Orthogonal parameter estimation algorithm for nonlinear stochastic
systems.’’ Int. J. Control, 48, 193–210.
Loh, C. H., and Duh, J. Y. 共1996兲. ‘‘Analysis of nonlinear system using
NARMA model.’’ J. Struct. Eng./Earthquake Eng., 537, I-35.
Loh, C. H., Wu, T. Z., Tseng, C. C., and Kao, C. Y. 共2003兲. ‘‘Experimental study of identification of RCS structure: Using pseudo-dynamic
test data.’’ Proc., Int. Conf. on Structural Health Monitoring and Intelligent Infrastructure, Z. Wu and M. Abe, eds., Vol. 1, A. A.
Balkema, Tokyo, 437– 445.
Marmarelis, P. Z., and Marmarelis, V. Z. 共1978兲. Analysis of physiologicalsystems, the white-noise approach, Plenum, New York.
Pavlenko, O. V. 共2001兲. ‘‘Nonlinear seismic effects in soils: Numericalsimulation and study.’’ Bull. Seismol. Soc. Am., 91, 381–396.
Pavlenko, O. V., and Irikura, K. 共2002兲. ‘‘Changes in shear moduli ofliquefied and nonliquefied soils during the 1995 Kobe earthquake andits aftershocks at PI, SGK, and TKS vertical array sites.’’ Bull. Seismol. Soc. Am., 92共5兲, 1952–1969.
Pavlenko, O. V., and Irikura, K. 共2003兲. ‘‘Estimation of nonlinear timedependent soil behavior in strong ground motion based on verticalarray data.’’ Pure Appl. Geophys., in press.
Schetzen, M. 共1980兲. The Voltera and Weiner series of nonlinear systems,
Wiley, New York.
Zarembo, L. K., and Krasil’nikov, V. A. 共1966兲. Introduction to nonlinear
acoustics, Nauka, Moscow.

Acknowledgments

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This research was supported by the National Science Council
共Taiwan兲 under Grant No. NSC91-2211-E-002-079. Pseudodynamic test data of the RCS frame provided by Professor K. C.
Tsai 共NCREE兲 are greatly appreciated.

Notation

References

PY
COThe following symbols are used in this paper:G m 关 • 兴 ⫽ orthogonal functions;h 1 (␶ 1 ) ⫽ first order impulse response function;h 2 (␶ 1 ,␶ 2 ) ⫽ second order impulse response function;h 3 (␶ 1 ,␶ 2 ,␶ 3 ) ⫽ third order impulse response function;
P ⫽ intensity of white noise;ü i (t) ⫽ recorded acceleration at level z i ;␥ i ⫽ shear strain at level z i ;␳ i ⫽ mass density; and␶ i ⫽ shear stress at level z i .

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Hunter, N. F. 共1990兲. ‘‘Analysis of nonlinear systems using ARMA models.’’ Proc., 8th Int. Modal Analysis Conf., 341–347.

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