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Page 1: Moving resonance in nonlinear response to fully nonstationary stochastic ground motion

Probabilistic Engineering Mechanics $ (1993) 157-167

Moving resonance in nonlinear response to fully nonstationary stochastic ground motion

J.L. Beck Division of Engineering and Applied Science, California Institute of Technology, Pasadena, California 91125, USA

&

C. Papadimitriou Department of Civil Engineering, Texas A&M University, College Station, Texas 77843, USA

A parsimonious stochastic seismic ground-motion model is used to study the effect of ground-motion nonstationarities on the response of simple linear and softening nonlinear systems. This model captures with at most nine parameters the features of the ground motion which are important for computing dynamic response, including the amplitude and frequency-content nonstationarities of the earthquake. Simple approximate expressions for the mean-square response statistics are obtained and are used to demonstrate analytically the importance of modeling the temporal nonstationarity in the frequency content of the ground motion, not only as expected for the nonlinear system, but also for linear systems. For the nonlinear systems, the phenomenon of 'moving resonance' is demonstrated whereby the shortening of the system frequencies, due to stiffness softening with increasing amplitudes, tracks the shift of the dominant frequencies of the ground motion, leading to a large resonant build-up in response amplitudes.

1 INTRODUCTION

Measured earthquake motions are obviously non- stationary time series. The nonstationarity is mani- fested in two different ways. First, the intensity of the ground acceleration varies with time; after arrival of the first seismic waves, it builds up rapidly to a maximum value over several seconds and then decreases more slowly until it vanishes into the background noise. Second, the frequency content varies with time, with a tendency to shift to lower frequencies as time increases. These nonstationarities can be attributed partly to the different intensity, frequency content and arrival times of the P-wave, S-wave and surface-wave groups, and partly due to the finite rupture-time and finite fault area.

Past stochastic models of ground motion have usually included the amplitude nonstationarity but have often neglected the temporal change of the frequency content. This is partly because it was difficult to incorporate this change in simple ground-motion models and to identify it from earthquake records, and also partly because

Probabilistic Engineering Mechanics 0266-8920/93/$06.00 © 1993 Elsevier Science Publishers Ltd.

157

including nonstationary frequency content comphcates random-vibration analyses. It was also commonly believed that this frequency nonstationarity would only have a minor effect on linear structural response, although it was expected that it would be important for inelastic response of structures.

Recently, Papadimitriou & Beck 1 have developed a 'fully' nonstationary ground-motion model which includes both the time variation of the frequency content and the amplitude nonstationarity. This stochastic model has been shown to capture, with at most nine parameters, those features of earthquake ground acceleration time histories which have an important influence on the dynamic response of linear and nonlinear systems. In its equivalent discrete-time form, it can be efficiently identified from a 'target' accelerogram and it can be used for efficient numerical simulations. In its equivalent continuous-time form, it can be used in conjunction with a recently developed simplified approximate method 2 to calculate efficiently nonstationary linear and nonlinear response in analy- tical random-vibration studies.

In this paper, the new ground-motion model is briefly described and then it is used to study the effect of

Page 2: Moving resonance in nonlinear response to fully nonstationary stochastic ground motion

158 J.L. Beck, C. Papadimitriou

ground-motion nonstationarities on the response of simple linear and softening nonlinear systems. Simple approximate expressions for the mean-square response statistics are obtained and these are used to demonstrate the importance of modeling the temporal nonstationar- ity in the frequency content of the ground motion, not only as expected for the nonlinear system but also for linear systems. For the nonlinear system, the phenomenon of 'moving resonance' described next is demonstrated.

1.1 Moving resonance

Beck & Skinner, 3 in an assessment of the seismic performance of a 'stepping pier' concept for a railway viaduct, observed an apparent phenomenon they described as 'moving resonance' in computing the deterministic response of this lightly damped softening system to recorded ground motions. The system seemed to be resonantly excited in such a way that the decreasing equivalent linear frequency produced by a steadily increasing response amplitude was 'tracking' the tem- poral shift in the ground-motion spectrum to lower frequencies. This quasi-resonant 'lock-in' produced large response amplitudes and also produced additional sensitivity of the response to changes in the structural parameters, depending on whether this 'lock-in' was inhibited or enhanced by the changes. It proved difficult, however, to confirm directly that this postulated 'moving resonance' phenomenon was actually occurring.

Recently, Yeh & Wen 4 have also commented on this phenomenon for a softening hysteretic system, the Bouc-Wen model, subject to a stochastic earthquake excitation. In their study, they produced a fully nonstationary ground-motion model by using a fre- quency modulation, as well as the usual amplitude modulation, as introduced by Grigoriu et al. 5 They suggest that the much larger response of the hyster- etic system to their fully nonstationary ground- motion model, compared with the purely amplitude- nonstationary model of the same 'target' accelerogram, is due to the shortening of the system frequency coinciding with the shifting of the dominant frequencies of the ground excitation. We show later, however, that even linear systems, where this phenomenon cannot occur, can produce much larger responses for a fully nonstationary excitation compared with a purely amplitude-nonstationary model of the same accelero- gram. A more direct demonstration of the existence of 'moving resonance' is therefore desirable.

2 FULLY NONSTATIONARY STOCHASTIC GROUND-MOTION MODELS

Much of the early work on nonstationary stochastic modeling of seismic ground motions focused on the

more obvious amplitude nonstationarity, whereas the focus of this paper is on 'fully' nonstationary stochastic models where frequency nonstationarities are included as well. Examples of previous work in this latter area can be found in Refs 1 and 4-12. The typical approach is to model the ground acceleration time history as the output of a low-order time-varying linear system which is subject to a white-noise input, either in discrete time by using time-varying ARMA(2,1) or AR(2) models, or in continuous time by using a time-varying SDOF linear oscillator. The equivalence between these discrete and continuous representations has been described in, for example, Ref. 1.

Most of this research has demonstrated the adequacy of such simple low-order ground-motion models by comparing features of simulated samples with the corresponding 'target' accelerogram used to estimate the parameters of the model. Some of the researchers have also studied the stochastic response of linear and inelastic systems to the fully nonstationary ground motion by using numerical simulation, but there is a scarcity of previous work involving analytical studies. An advantage of the approximate analytical method presented herein is that it gives better insight into the stochastic response of dynamic systems subject to fully nonstationary excitation.

In the parsimonious stochastic model developed by Papadimitriou & Beck, 1 the earthquake ground accel- eration a(t) at a site is represented by the displacement output of two cascaded SDOF linear oscillators described by the set of linear differential equations:

y + 2(g(t)~og(t))~ + o~(t)y =fg(t)e(t) /J + 2o~ctJ + ~o2a = y

(1)

(2)

where (g(t), ~og(t) and fs(t) are deterministic time- varying functions described below, and e(t) is a Gaussian white-noise process with E[e(t)] = 0 and E[e(t)e(s)] = 6 ( t - s ) . Equation (2) mimics Brune's source model. 13,14 The comer frequency 0Jc can be computed from source mechanism studies of the earthquake generating the ground motion. For poten- tially damaging earthquakes, it is usually less than 0.2 Hz, well below the dominant frequencies of several hertz which are typically found in strong-motion accelerograms. For this reason, eqn (2) gives a ground- motion model with the correct behavior of the spectral amplitudes at very low frequencies, without substan- tially affecting the spectral amplitudes produced by eqn (1) at frequencies greater than about 2we. For the purposes of this study, 0~c can be set to zero because the very low frequencies are not important in the dynamics of the systems investigated here. This would not be the case if inelastic systems were of interest, where the drift is influenced by the frequency content of the excitation at very low frequencies, is

Page 3: Moving resonance in nonlinear response to fully nonstationary stochastic ground motion

Moving resonance in response to nonstationary ground motion 159

In order to produce a desired time variation of the intensity Is(t ) = v/-E-[a2(t)] of the ground motion process, fs(t) in eqn (1) can be taken as: 2

A(t) = 2u8(/) a g(t)Ig(/) (3)

where as = ~s(t)ug(t). This relationship is valid for wide-band processes with slowly varying correlation structure, which is usually the case for earthquake motions.

Based on analyses of earthquake accelerograms I the intensity Is(t) is adequately modeled by a slight variation of the envelope function proposed by Saragoni & Hart: 6

Is(t;-0) =/max ~-# exp [/3(1 - ~-)] (4)

where T = (t + to)/(tmax + to). The parameters/max and tmax are the maximum intensity and the time of the maximum intensity of the ground acceleration, respec- tively. The variable to is the time of the first non-zero acceleration before the triggering time, introduced to provide flexibility in fitting the data, and it is not needed when predicting response with the model. The non- dimensional parameter /3 controls the duration of the accelerogram and is typically of order unity. The duration tdur can be defined as the time interval over which Is(t) is greater than, say, 50% of the maximum, /max-

The slowly varying functions Ws(t ) and as(t ) can be roughly interpreted as the dominant frequency and its associated bandwidth respectively, in the ground acceleration. They are adequately parametrized by: l

ug( t; -0) =Ur -F (Up - Ur) ( us - u---~zr ~ r (5) \Up - u~/

c~g(t; _0) = Up~p q- (Ur~ r - - U p ' p ) t / t d u r (6)

where Up, us and Wr may be interpreted approximately as the dominant frequencies of the P, S and surface waves present in the ground motion, respectively, and where ~p and (r measure the frequency ranges around Up and Ur respectively that strongly contribute to the earthquake process. Papadimitriou & Beck ] have also presented a Bayesian method for estimating the optimal model, i.e. for determining the most probable value of the parameter vector _0 : (/max, tmax, tdur, Up, Us, u r, (p, (r), which provides, in a statistical sense, the best fit to a 'target' accelerogram.

3 FORMULATION OF NONSTATIONARY PROBLEM OF INTEREST

The governing equation of motion of a SDOF nonlinear system subjected to the stochastic ground excitation a(t) is given by

~(t) + 2¢0~00:~(t) +u2R( t ) /Ko = a(t) (7)

where u0 and C0 are the initial (small-amplitude) system frequency and viscous damping ratio respectively. The restoring force R(t) is modeled by a nonlinear elastic softening restoring force

R(t) = 2/¢r(Koxy) arctan ['rcx/(2Xy)] (8)

where the nominal 'yield' displacement x x = Ra/Ko is similar to the elastic limit displacement of a yielding system, with R, and K0 being the ultimate strength and the initial stiffness of the system, respectively. Linear oscillators are included as the special case Xy ~ oo. The nonlinear model is chosen to gain analytical insight into the effects on nonlinear response of temporal nonsta- tionarity in the frequency content of the ground motion.

We wish to investigate the differences in the mean- square response of this system to two types of excitation which differ only in the way they model the frequency content of the ground motion. The first excitation, denoted by (TV), is fully nonstationary and so has time- varying frequency content, while the second excitation, denoted by (TI), has time-invariant frequency content throughout its duration. Both the (TV) and (TI) accelerations have identical mean-square time histories, and so have the same amplitude nonstationarity. More specifically, (TV) is our nine-parameter model described earlier but with wc set to zero, while (TI) involves the further simplification of setting Wp = w s = Wr : O3g, say, and ep = (r = (s' say. The parameters of the (TV) and (TI) models are estimated by applying our Bayesian method to a 'target' accelerogram.

The chosen accelerogram is the north-south compo- nent of record C04816 which is shown in Fig. 1. The time variation of the intensity, Is(t), is the same for both the (TV) and (TI) models, and it is plotted in Fig. 2 [/max = 71 crn/s 2, /3 = 1-34, tmax = 7"4s and to = 0.66s in eqn (4)]. The frequency content of the (TI) model is chosen to fit the segment of the accelerogram with the stronger intensity, i.e. from 2 to 12s. This gives ~3g = 3-32Hz and (g = 0.258. The time-varying frequency content of the (TV) model is described by the parameters Wp=7"43Hz, ws=3"12Hz, ~Vr= I ' l l Hz, ~p = 0"096 and fir = 0"655. The spectral content of the two ground-motion models is illustrated graphically in Fig. 3 using the normalized EPSD 17'18 (evolutionary power spectral density). The normalized EPSD of the

( " 400

0 2o0

t,. (]) o

0 -2oo

! !

C048, COMP NOOW

T i m e ( s )

Fig. 1. Component of Orion Blvd accelerogram (1971 San Fernando earthquake).

Page 4: Moving resonance in nonlinear response to fully nonstationary stochastic ground motion

160 J.L. Beck, C. Papadimitriou

60

40

0 10 20 30

TIME (s)

Fig. 2. Time variation of the intensity function Is(t) of the (TI) and (TV) models for the accelerogram in Fig. 1.

ground acceleration is defined by:

SN(W, t) = S(w, t)/I2(t) (9)

The EPSD of the (TV) ground motion model is given by:

4 ag (t) w~(t) I2(t) (10) S(w, t) = (w 2 _ w2g(t)) 2 + 4~g2(t)w2

under the assumption that the ground motion is broadband and has a slowly varying correlation structure. 2 The EPSD for the (TI) model is also given by eqn (10) but with wg(t) and as(t ) replaced by the constants &g and &g = ~g&s respectively. It is clear that the predominant frequencies of the recorded ground motion shift to lower frequencies with increasing time, as reflected by the EPSD for the (TV) model. This is often observed in earthquake ground motions.

o

a t t i m e s 1 , 4 , 7 , 1 0 , 1 6 , 2 2 s e c o n d s

11 , , , ,

I 0.51_. • • - ~..., . . . . . .

F - - ' _ . - - , ~ . , , ~', % .,.¢, ,., . ' , - , , _ , , . > % , , . ,

I-" - - . _ ' ~ . ~ : " ' , " ~ \ ""-

• >. ) . . ' . . - . - . . 0.05 a t :::".': o.oz 2 - _ - - 2 " "" " "

. . . . . . . . . . . . 22 seconds [ 0 .005 I

I I I I I 0 2 4 6 8 i 0

FREQUENCY (FIz)

Fig. 3. The normalized EPSD function computed for the (TV) (dashed curves) and (TI) (solid curve) models for the

aeeelerogram in Fig. 1.

4 S I M P L E A P P R O X I M A T E M E T H O D FOR NONSTATIONARY MEAN-SQUARE RESPONSE

The equivalent linearization method 19'2° is used to obtain the second-moment response statistics. The nonlinear equation (7) is replaced by the equivalent linear one:

Jr(t) + 2((t)w(t)Sc(t) + w2(t)x(t) = a(t) (11)

where w(t) and ((t) are the equivalent linear frequency and damping factor given by:

(0w0 (12) w(t)wo = [x/~7 exp (9 ,2) erfc (7)1 ]/2 ((t) = w(t)

with 7 = v / ' 2 X y / [ V r ~ ] and qll(t) = E [x2(t)]. 21 The mean-square statistics of the response are obtained by solving the nonlinear moment equations corresponding to the system of equations (11), (1) and (2).

An approximate method is next developed which simplifies the formulation for the mean-square response and provides valuable insight into the response characteristics of SDOF oscillators subjected to fully nonstationary excitations. For mathematical conveni- ence, eqn (11) is rewritten in state-space form as:

5c(t) = A(t)x_(t) + g(t) (13)

where / 0 1 \

\/c( t) ] ,. -w2(t) -2(oWo / (14) (o)

g ( t ) = a(t)

Let Q(t) = E[x(t)x_r(t)] denote the mean-square matrix of the response vector _x. Assuming zero initial conditions for x_(t), Q(t) is obtained by the solution of the Liapunov differential matrix equation: 22

Q(t) = A(t) Q(t) + Q(t)At(t) + L(t) + LV(t) (15)

Q(0) = 0

where

0 L( t )= J~ ~ ( t , t - 7 - ) ( ~ r.(t,t-•-) ) dT (16)

in which rg(t, t-7-) = E[a( t )a( t - ~-)], the auto-covari- ance function of the excitation process. The matrix • (t, -r) in eqn (16) is the principal matrix solution ofeqn (13), obtained from the solution of the system:

4}(t, r) = A(t)ep(t, r) (17)

~,(r , ~-) = I

where 4} denotes differentiation of • with respect to the independent variable t.

It has been shown in the Appendix that the mean- square displacement response qll (t) = E[x2(t)] satisfies

Page 5: Moving resonance in nonlinear response to fully nonstationary stochastic ground motion

Moving resonance in response to nonstationary ground motion 161

the first-order differential equation:

qu(t) + 2[ffoWo + 8(t)]qll(t) = 2r(t) (18)

ql (0) = 0

with the excitation r(t) satisfying the second-order differential equation:

~(t) + 4[(0w0 - ½6(t)]k(t) + 4w2(t)r(t) = g(t) (19)

r(0) = 0 k(0) = 0

In these equations, 8(t) can be expressed approximately in terms of only the damped natural frequency tad(t) = (ta2(t) __ (2ta2)1/2:

6 ( t ) - ¢ba(t) (20) 2tad(t)

and the forcing term is given by:

g(t) = 2L22(t) + 4(0ta0Ll2(t) + Ll2(t) (21)

where L12 and L22 are given by eqn (16). The above formulation is approximate since appro-

priate slowly varying conditions for ta(t) were utilized to neglect small-order terms. For a time-invariant matrix A, however, there are no approximations involved. In that case, Papadimitriou & Beck 23 have shown that if the forcing term g(t) is slowly varying relative to the time scale given by the period 21r/ta0, the dynamics of the first-order differential equation (18) provide all the essential characteristics of the response. The dynamics of the second-order differential equation (19) have only a secondary effect on the response, producing a small oscillatory ripple superposed on the solution of the first- order differential equation. Similarly, for slowly varying ta(t) relative to the time scale 2~r/ta(t), the dynamics of eqn (19) continue to be insignificant. Therefore, the dominant solution of eqn (19) can be approximated by:

r(t) = g(t) (22) 4ta2(t)

Substituting this solution into eqn (18), a good approximation for the mean-square displacement of the response can be obtained by solving the simpler first- order nonlinear differential equation:

g(t) (23) qll(t) -F 2[(0ta0 + 6(t)]qll(t) : 2ta2(t )

To evaluate the forcing term g(t) in eqn (23), the integrals in eqn (16) have to be evaluated, which in turn requires the knowledge of the principal matrix solution • (t, t - r) and the auto-correlation rg(t, t - r) of the excitation. For constant matrix A ( t ) = A, a simple closed-form solution for the principal matrix exists in terms of exponential and trigonometric functions. For time-variant A(t), a closed-form solution for the principal matrix cannot be obtained in general. In particular, when the matrix A(t) = A(Q(t)), so that it depends on the mean-square matrix of the response, eqns

(17) and (23) are no longer independent and they have to be solved simultaneously. In what follows, a formulation is developed which avoids finding the exact solution of eqn (17); it imposes general conditions on system and excitation characteristics for analytically approximating the integrals by simpler algebraic expressions.

Since I2(t) is the variance of the excitation process a(t) at time t, its auto-covariance function can be expressed as:

rg(t, t - ' r ) = I2(t)r('r;_Og(t)) with r(0;_0g(t)) -- 1

(24)

where r(r; _0g(t)) is the auto-correlation function of a(t) at time t and _0g(t)= (wg(t), ag(t)) is the parameter vector of the excitation model accounting for the time variation of its frequency content. Assuming that the auto-correlation functions decays sufficiently fast with increasing time lag r, we can write:

I r(r; _0g(/)) I < Roe -r/re(t) (25)

where Rc = O(1), and %(t) is the characteristic length measuring the time interval of significant correlation of the stochastic process a(t) at time t. Equation (25) is valid for the (TI) and (TV) ground-motion models [see eqn (34), later]. The more broadband the process is, the less the time re(t). For models of most earthquake ground motions, the auto-correlation function at time t reduces from its peak value to a negligibly small amount in a relatively short time lag, so %(t) is relatively small compared with the time scales of the slowly-varying parameters in _0g(t).

The slowly varying assumption on ta(t) implies that A (t) can be considered to remain constant in the interval [t - %(t), t], provided re(t) is small compared with the time scale over which ta(t) changes significantly. In this case, the principal matrix ff~(t, t - r) can be approxi- mated by ~(r ; t) which is obtained for r E [0, re(t)] by solving the time-invariant differential equation d ~ ( r ; t ) / d r = A ( t ) ~ ( r ; t ) with the fixed value of A(t). The integrals Li2(t) from eqn (16) can therefore be approximated by

Li2(t) = I2(t) J o xI//2(T; t)r(r; _0g(t)) dr i = 1 , 2

(26)

where, because of eqn (25), the integrand decays exponentially fast. The e r r o r ei(t ) - - - / ~ i 2 ( t ) - L i 2 ( t ) becomes small for t sufficiently large and for sufficiently slowly varying w(t). The stochastic excitation process has to be broadband in order to ensure that the error is negligible after a small time t. Usually, over this initial time segment, the error is significant and the approxi- mate expression (26) does not apply. However, the modulation Ig(t) has small values at these times, so that the error made by using the approximation (26) does not affect the accuracy of the response at later times.

Page 6: Moving resonance in nonlinear response to fully nonstationary stochastic ground motion

162 J.L. Beck, C. Papadimitriou

Substituting eqn (26) into eqn (21), and using the form of the approximate principal matrix ~('r; t), the forcing term g(t) finally takes the form:

g(t) = lg2(t)Rg(a)(t), (0w0; _0g(t)) {1 + eR(t)} (27)

where

Rg(w(t), ~0w0; _0g(t)) = 2I¢(t) + 2((0w0)/s(t ) (28)

It(t) - I¢(w(t), (0oJ0; 0_g(t))

I o exp [-(0O~o r] r(r; O_g(t)) cos [wd(t ) 7-] dT- (29)

1 /s(t) --/s(a)(t), (0~0; _0g(t)) -- Wd(t )

X JO exp [--(0w07-] r(7-; 0_.g(t)) sin [wd(t ) 7-] d7-

(30)

21g(t) Is(t) ls(/) ,R(w(t), (0~o0; ~( t ) ) - is(1 ) Rg(t-----] ~- Rg(t-"--~ (31)

For specific functions r(7-;_0g(t)) of % such as polynomials, exponentials, trigonometric functions or a combination of these, the integration in eqns (29) and (30) can be carried out analytically without requiring the exact time variation of the excitation model parameters _0s(t ). For slowly-varying Is(t) and _0g(t), usually the case in most applications, the functions Is(w(t), (oW0; _0g(t)) and Is(w(t),~oWo;Og(t)) are expected to be slowly varying functions of time as well. In such cases, the term eR(t) can be neglected in eqn (27) since it involves derivatives of slowly varying functions.

In the special case of a linear oscillator subject to ground motion with a stationary frequency content, w(t) and _0g(t) are time-invariant, and the variance of the oscillator can be approximately obtained by its variance when subjected to white-noise excitation with power spectral density Rg(w0, ~0oJ0; _0g) which is modulated by the function Is(t). A similar argument has been used in the past 24 to approximate the response of lightly damped oscillators, with the quantity Rg(w0, (0~0; 0g) being replaced by its limit SF(wO) as (0 ~ 0, i.e. the value of the constant power spectral density of the filter computed at the oscillator frequency w0.

4.1 Special case of lightly-damped oscillators

An oscillator is considered to be lightly damped here if the damping ratio ((t) is sufficiently small that the term exp[-((t)w(t)7-] ~ 1 over the interval [0, re(t)], i.e. ((t)w(t)%(t) << 1, where "rc(t ) is the correlation time in eqn (25). In addition, it is assumed that the excitation process is broadband and that _0g(t) and Is(t) are slowly varying functions so that they remain approximately constant over the duration %(t) of significant correla- tion of the excitation process at time t. Therefore, from

eqn (29), we get the approximation:

_0g(/)) ~ 2 I o r(7-; _0g(t)) cos [w(t)r] dr 2Ie(w(t), (o~0;

= SN(a); 0_g(t)) (32)

where SN(W; _0g(t)) is the normalized EPSD function of the excitation process defined in eqn (9).

As ( approaches zero, the O(() terms can be neglected in eqn (28), so the term g(t) takes the simple form:

g(t) = S(w(t), t) (33)

showing that it depends only on the form of the EPSD function of the excitation process. This forcing term includes the case of oscillators with slowly varying angular frequency and damping ratio, and therefore it is a generalization of the result obtained by Spanos 25 for time-invariant oscillators. Equation (27) is the proper replacement of eqn (33) when the conditions of lightly damped systems that apply to eqn (33) are violated.

4.2 Application to ground-motion model

Under the assumption that the ground-motion process a(t) is broadband, i.e. its correlation time rc(t) is relatively small, then the time variation of t~g(t), wg(t) and fg(t) in eqn (1) over the time interval [ t - "re(t), t] can be ignored without significant loss of accuracy. In this case, the statistical properties of a(t) in the neighborhood of t can be obtained by an equivalent stationary process generated by eqn (1) with the values of the corresponding constant coefficients being the instantaneous values ag(/), Wg(/) and fg(t). This results in the auto-covariance function: 2

rg(t, s) = E[a(t)a(s)] = I2(t) exp { -ag( t ) ( t - s)}

cos {Wg, d( t)( t -- s) + ~bg(t)} (34) × COS {~bg(/)}

where Is(t) is given by eqn (4) and ~bg(t) is given by:

as(l) tan [~bg(/)] - Wg, d(t) (35)

and

Wg, d(t) = ~/w2(t)--a2(t) (36)

Evaluation of the integrals in eqns (29) and (30) can be carried out analytically, resulting in

is(1 ) _ A(t) C(t) (37a)

Ie(t) - - B(t) C(t) (37b)

A(t) = [¢0w 0 + OLg(t)] 2 + wg(t) - W2,d(/)

+ 2 [~0w0 + ag(t)]wg, d(t) tan [~g(t)] (37c)

Page 7: Moving resonance in nonlinear response to fully nonstationary stochastic ground motion

Moving resonance in response to nonstationary ground motion 163

B(t) = [~otao + o~g(t)] {[ffoWo + c~s(t)] 2 + oj2(t)

+ a;g,d(t)} + WS, d(t ) tan [q~s(t)] {[¢oWo

+ as(t)] 2 -- o;2(t) + a;2,d(t)} (37d)

C(t) = [w2(t) - a;g(t)] 2 + 4o~(t)wg(t)[¢oWo

+ c~g(t)][w(t)~s(t ) + a;g(t)~(t)] (37e)

~d(t) = ~/~2(t) -- ~02 w02 (37f)

By substituting these equations in eqn (28), an analytical expression for g(t) in eqn (27) can be obtained, where the small term ~ ( t ) can be neglected•

The (TV) version of the ground-motion model described earlier is used to demonstrate the accuracy of the approximations developed above for the mean- square response. The exact variance response qn(t) is obtained by rewriting the coupled second-order differ- ential equations (1) and (11) as a four-dimensional first- order vector differential equation subjected to the appropriate amplitude-modulated white noise• The corresponding Liapunov matrix equation for the covariance response results in a ten-dimensional system

0.6

Z O 0 . 4

~0.2

of first-order coupled nonlinear differential equations. These can be numerically integrated to give a numerical solution which provides the basis for assessing the approximate results. It should be noted, however, that using the derived approximate expressions, the comput- ing time is reduced by two orders of magnitude.

Define the STD response as the time history of for the equivalent linear oscillator [eqns (11)

and (12)] of the nonlinear oscillator given by eqns (7) and (8). Comparisons between the exact (solid curves) and the approximate (dashed curves) STD response are shown in Figs 4(a) and 4(c) for two nonlinear oscillators with initial frequency ~0 = 5 Hz and w0 = 1 Hz respec- tively. The initial damping ratio and 'yield' displacement for both oscillators are ~0 -- 0.05 and xy = I/3 respec- tively. The time-varying frequency w(t) is computed by the expression (12) and it is shown in Figs 4(b) and 4(d) for the two oscillators, along with the temporal variation of the frequency ws(t ) of the (TV) ground- motion model• The differences between the exact and the approximate solutions observed in Figs 4(a) and 4(b) for the oscillator with initial frequency w0 = 5 Hz are of the order of a few per cent. . Discrepancies of similar order were also observed for

other variations of a)(t) and c~(t), provided that the

10

. . . . l . . . .

(a)

20

TIME (s)

F z 3 ¢/) Z 0 O~

20

10

0 0 0 3 0 0

. . . . i ' , . , o ' i . . . .

..oo.'* • •

i0 20 30

TIME (s )

~ 6 N

z 4

2

"\

i . . . .

(b)

L) Z

. . . . i . . . . . . . . i

6 "\

\

4 \ \

%

2

, ,

0 10

,",. . , . . , .7 - ' - 0 10 2 0 3 0 2 0 3 0

TIME ( s ) TIME ( s )

Fig. 4. Comparison between the exact (solid curves) and approximate (dashed curves) response of an equivalent linear oscillator (xy = 1/3, ¢0 = 0.05). STD response for initial frequencies: (a) w0 = 5 Hz, (c) w0 = 1 Hz; equivalent linear frequency ~(t) for initial

frequencies: (b) w0 = 5 Hz, (d) w0 = 1 Hz.

Page 8: Moving resonance in nonlinear response to fully nonstationary stochastic ground motion

164 J.L. Beck, C. Papadimitriou

conditions validating the approximations were satisfied. For example, for the oscillator with initial frequency w0 = 1 Hz, shown in Figs 4(c) and 4(d), the discrepan- cies between the exact and the approximate mean-square displacements are of the order of 20% or higher. In this case, the excitation parameters Wg(/) and/g(t) shown in Figs 4(d) and 2 respectively, vary significantly over the equivalent period of the oscillator, which is approxi- mately 10s over most of the response (see Fig. 4(d)). Therefore, the larger discrepancies between the exact and the approximate solutions are due to the violation of the slowly varying conditions in this case.

In summary, the simple approximate method based on eqns (23) and (27) is accurate under broadband excitation with slowly varying amplitude and frequency content. It provides valuable analytical insight into the mean-square response characteristics and their depen- dence on the nonstationary amplitude (via Ig(t)) and frequency content (via Rg(t)). This is exploited in the following sections.

5 EFFECT OF NONSTATIONARY FREQUENCY CONTENT ON LINEAR OSCILLATORS

For linear oscillators and for the (TI) ground motion, the spectral part Rg(w0, (0a;0; _0g) in eqn (27) is constant and the shape of the forcing term in eqn (23) is controlled only by the amplitude nonstationarity Ig(t) of the ground motion. However, for the time-varying frequency content in the (TV) model, the spectral part Rg(a; 0, ~0a~0; _0g(t)) in eqn (27) varies with time, altering the shape of the forcing term in eqn (23). For lightly damped oscillators, Rg approaches EPSD, and so its temporal variations is depicted in Fig. 3. The character- istics of the response to the (TV) excitation are therefore expected to be different from those for the (TI) excitation, even for linear oscillators. In this section, we use the simple approximate method to investigate this difference in response.

The maximum of the STD response (i.e. max ~ ) , the corresponding time when the maxi- mum occurs and the duration of the STD response are compared in Fig. 5 for the (TI) and (TV) excitations and

for a linear oscillator with fundamental frequencies ranging from 1 to 8 Hz. The duration is defined as the difference between the two times when the STD of the response upcrosses and downcrosses 50% of its maximum value. All numerical results correspond to 5% initial damping ratio, i.e. ~0 = 0.05.

For the (TI) excitation, the times of the maximum STD and the STD duration, which control the shape of the nonstationary STD response, do not vary signifi- cantly over the linear oscillator frequencies examined. This is because the time variation of the forcing term for eqn (23), which is given in eqn (27), depends only on Ig(t) in this case. The values of the oscillator parameters w0 and (0 control only the overall amplitude of the forcing term. Therefore, the resulting shape for the STD response is controlled mainly by the product ~0~0. However, as (0w 0 increases, the shape becomes independent of this product, since qll(t) eventually becomes proportional to g(t) (see eqn (23)).

For the (TV) excitation, however, the shape of the forcing term given in eqn (27) is sensitive to the individual values of the oscillator parameters w0 and ~0, and so, therefore, is the resulting shape of the STD response. The time of the maximum STD responses as well as the duration of the responses corresponding to the (TI) and (TV) excitations can differ by a factor of 2 or more. The actual STD time histories for ~0 = 1.5 Hz, 3.3Hz and 6Hz are shown in Figs 6(a), 6(b) and 6(c). These figures illustrate that even for linear systems, the (TV) excitation can produce substantially different, and larger, motions than the (TI) excitation, despite the fact that both ground-motion models are identified from the same 'target' accelerogram. This demonstrates directly the importance of modeling the frequency nonstation- arity of seismic ground motion even for the dynamic response of linear systems.

6 EFFECT OF NONSTATIONARY FREQUENCY CONTENT ON THE NONLINEAR OSCILLATOR: MOVING RESONANCE

For the nonlinear oscillator, it may happen that the decrease of the equivalent linear frequency of the

31' 0 "~"

0 2 4 6 8 FREQUENCY (Nz)

x

1C

I I !

2 I I I 0 2 4 6

FREQUENCY (Hz)

18

10

6 2 4 6 FREQUENCY (Hz)

Fig. 5. Nonstationary linear response characteristics for the (TV) (solid curves) and the (TI) (dashed-dotted curves) excitations for various oscillator frequencies w 0 (Xy ~ oo, (0 = 0.05).

Page 9: Moving resonance in nonlinear response to fully nonstationary stochastic ground motion

Moving resonance in response to nonstationary ground motion 165

2.0

1.5 5 9 Z o

1.0 e~

590.5

0 0

. " - (a)

/ .I

o

y. , ,Wo=,IT5,Hz , , . . . . I 0 20 30

TIME (s)

0 .6

59 Z o ~0 .4

590.2

0 , , , , I . . . . I

0 tO 20

' • " ~ I . . . . I . . . .

/ " / '~" ' , , . , , (b)

// \ , , // \ , . " " %

i i i J

30

TIME (s)

0 .15

5 9

Z 0 .10 o 5 9

0,05 59

0 0 i0 20 30

TIME (s)

Fig. 6. STD response for linear oscillators (Xy --'-> OO, (0 = 0"05) for the (TV) (solid curves) and (TI) (dashed-dotted curves) excitations for various oscillator frequencies w0: (a) w0 =

1.5 Hz, (b) w0 = 3.3 Hz, (c) w0 = 6.0Hz.

softening system tracks the decrease of the predominant frequency of the ground motion, i.e. w(t) ,~ wg(t) over some time interval, giving rise to the 'moving resonance' effect. In this case, eqns (37) show that Rs(t ) in eqn (28) is controlled by Ic(t), and that this term is large. Therefore, g(t) in eqn (27) is large, resulting in significant amplification of the mean-square response.

For lightly damped oscillators, an alternative point of view is as follows. Recall that in this case g(t) is the

EPSD, and the EPSD temporal variation is shown in Fig. 3. During 'moving resonance', the oscillator frequency a;(t) in eqn (33) is close to wg(t), which corresponds approximately to the peak in the EPSD at that time. Therefore, once again, we conclude that g(t) in eqn (27) is large.

This amplification effect is less likely to occur for the (TI) excitation, since the softening at resonance will cause a decrease in the equivalent linear frequency, which will move the oscillator out of resonance with the ground motion. These effects are investigated in this section, again using the simple approximate method for the mean-square response.

The full time history of the STD displacement response for the (TI) and (TV) excitations (denoted by NTI and NTV) are compared in Figs 7(a) and 7(c), corresponding to two different nonlinear oscillators with initial frequencies w0 = 6 Hz and ~0 = 3 Hz respectively. The time variation of the corresponding equivalent linear frequencies w(qll (t)) computed from eqn (12) are plotted in Figs 7(b) and 7(d). For comparison purposes, each figure also includes the time variation of the dominant frequency ~g(t) of each ground-motion model. Considerable amplification of the response is expected to occur in the 'moving resonance' situation, i.e. when the dominant frequency of the ground motion approximately coincides with the equivalent oscillator frequency w(t) over some time interval.

In Fig. 7(a), the maximum STD response correspond- ing to the (TV) excitation is larger than the maximum response corresponding to the (TI) excitation by a factor as high as three. For comparison, the response to the (TI) and (TV) excitations of a linear oscillator with the small-amplitude parameters of the nonlinear oscillator is also plotted in Figs 7(a) and 7(b), labeled by LTI and LTV respectively. The linear STD responses which correspond to the (TI) and (TV) excitation differ only by a factor of two. For the (TI) excitation, the nonlinear system is behaving almost linearly since the equivalent frequency (Fig. 7(b)) does not vary significantly. The much larger STD response for the nonlinear oscillator compared with the linear oscillator is attributed to the moving-resonance effect occurring for the (TV) excita- tion. This is apparent from Fig. 7(b) from approxi- mately the first to the seventh second of the excitation, where the equivalent linear frequency tracks the changing predominant ground-motion frequency.

In Figs 7(c) and 7(d), the initial oscillator frequency was chosen to be close to the dominant frequencies of the strong S waves of the ground motion. For the (TI) excitation, the equivalent linear system (11) is never in resonance with the ground motion. Similarly, for the (TV) excitation, over the first 10-15s of the highest ground intensity, the equivalent linear system is not in resonance with the ground motion. However, at later times when the weaker surface waves of the ground motion are arriving, the equivalent linear system

Page 10: Moving resonance in nonlinear response to fully nonstationary stochastic ground motion

166 J.L. Beck, C. Papadimitriou

0,3 r ~

z o m~0.2

~0.1

1.5 ' • I . . . . I . . . .

-131!I (a) . . ../2

• • 2,,,b ° , ~ , ' ~

10 20

CO

Z 21.0

= a

~o.5

0 0 0 30 0

' ' ' ' I ' ' I ' ' ' ' '

(c)

. - - 1 . / /

' f , ~ , , , i . . . . I i j i ' ~ " l

\ - , , .

10 20

TIME (s) TIME (s)

30

. . . . . , . . . . , ( ' b i '

4 r., "" " 6 . . . . ".,, . . . . . . . . . . . . _ / . . . . . . . . .

z . . ./5 0 I0 20 30

,~ 6

~ 4 D

2

• ' ' ' • I . . . . I . . . .

°

(d) 'F" 5

6 " . °

1 "" ' . . . . / . 2 . . --"-'~

10 20 30

TIME (s) TIME (s)

Fig. 7. Nonstationary nonlinear response characteristics for the (TV) (solid curves) and (TI) (dashed-dotted curves) excitations (xy = 1/3, Go = 0.05). STD response for initial frequencies: (a) w0 = 6 Hz, (c) w0 = 3 Hz; equivalent linear frequency w(t) for initial frequencies: (b) w0 = 6 Hz, (d) w0 = 3 Hz. [wg(t) for the (TV) and (TI) models are also shown in (b) and (d)]. 1, NTV; 2, NTI; 3,

LTV; 4, LTI; 5, (TV) wg(t); 6, (TI) wg.

resonates with the ground motion from approximately 15-22s, causing an amplification of the response. Therefore, in this case, the maximum STD response is controlled primarily by surface waves rather than the S waves. Modeling the ground motion by the (TI) excitation where the S waves control the response results in an underestimation of the importance of the weaker-intensity surface waves.

7 CONCLUSION

frequency content towards lower frequencies, it is important to capture this behavior in ground-motion models used for dynamic response studies. In particular, this temporal nonstationarity can have a substantial effect on the response of nonlinear systems of softening type, especially when the shortening of the system frequencies due to the stiffness softening tracks the shift of the dominant frequencies of the ground motion, leading to 'moving resonance', which can produce a greatly amplified response•

A fully nonstationary ground-motion model was used along with an approximate random-vibration formula- tion to analyse the effect of the temporal nonstationarity in the frequency content of the ground motion on the response of simple linear and nonlinear systems. It was shown that the characteristics of both linear and nonlinear response can strongly depend on the time variation of the frequency content of the excitation, so time-invariant frequency content models are inappropri- ate to model ground motions with time-varying frequency content. Since many recorded earthquake accelerograms appear to show such a temporal shift in

R E F E R E N C E S

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2. Papadimitriou, C. Stochastic characterization of strong ground motion and applications to structural response. Ph.D Thesis, California Institute of Technology, Report No, EERL 90-03, 1990.

3• Beck, J.L. & Skinner, R.I., The seismic response of a reinforced-concrete bridge pier designed to step. Earth- quake Engng Struct. Dyn., 1974, 2, 343-58.

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Moving resonance in response to nonstationary ground motion 167

4. Yeh, C.H. & Wen, Y.K. Model of nonstationary earth- quake ground motion and applications. Proc. 4th Nat. Conf. Earthquake Engineering, EERI, E1 Cerrito, Califor- nia, 1990, Vol. I, pp. 505-14.

5. Grigoriou, M., Ruiz, S.E. & Rosenblueth, E. The Mexico earthquake of September 19, 1985 - - nonstationary models of seismic ground acceleration. Earthquake Spectra, 1988, 4(3), 551-68.

6. Saragoni, G.R. & Hart, G.C. Simulation of artificial earthquakes. Earthquake Engng. Struct. Dyn., 1974, 2, 249-67.

7. Kozin, F. Estimation and modeling of nonstationary time series. Proc. Syrup. Applications of Computer Methods in Engineering, ed. L.C. Wellford. University of Southern California, Los Angeles, 1977, Vol. I, pp. 603-12.

8. Jurkevics, A. & Ulrych; T.J. Representing and simulating strong ground motion. Bull. Seism. Soc. Am., 1978, 68(3), 781-801.

9. Gerseh, W. & Kitagawa, G. Time varying AR coefficient model for modeling and simulating earthquake ground motion. Earthquake Engng & Struct. Dyn., 1985, 13, 243-54.

10. Safak, E. Optimal-adaptive filters for modeling spectral shape, site amplification, and source scaling. Soil Dyn. & Earthquake Engng., 1989, 8(2), 75-95.

11. Deodatis, G. & Shinozuka, M. Auto-regressive model for nonstationary stochastic processes. J. Engng. Mech. ASCE, 1988, 114(11), 1995-2012.

12. Conte, J.P., Pister, K.S. & Mahin, S.A. Variability of structural response parameters within an earthquake. Proc. 4th Nat. Conf. Earthquake Engng., EERI, El Cerrito, California, 1990, Vol. I, pp. 791-800.

13. Brune, J.N., Tectonic stress and the spectra of seismic shear waves from earthquakes. J. Geophys. Res., 1970, 75, 4997-5009.

14. Brune, J.N. Corrections. J. Geophys. Res., 1971, 76, 5002. 15. Iwan, W.D. & Paparizos, L.G. The stochastic response of

strongly yielding systems. Prob. Engng Mech., 1988, 3(2), 75-82.

16. Caltech, Strong motion earthquake accelerograms, Vol. II, Part C, Report No. EERL 72-51, Earthquake Engineering Research Laboratory, California Institute of Technology, Pasadena, California, 1973.

17. Priestley, M.B. Evolutionary spectra and nonstationary processes. J. Roy. Statist. Soc., 1965, 27, 204-28.

18. Priestley, M.B. Power spectral analysis of nonstationary random processes. J. Sound and Vibration, 1967, 6, 86-97.

19. Caughey, T.K. Equivalent linearization techniques. J. Acoust. Soc. Am., 1963, 35(11), 1706-11.

20. Roberts, J.B. & Spanos, P.D. Random Vibration and Statistical Linearization. J. Wiley & Sons, New York, 1990.

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23. Papadimitriou, C. & Beck, J.L. Approximate analysis of nonstationary random vibrations of MDOF systems. Computational Stochastic Mechanics, ed. P.D. Spanos & C.A. Brebbia. Elsevier Science, New York, 1991, pp. 371- 82.

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APPENDIX COMPLETE SET OF EQUATIONS FOR THE MEAN-SQUARE DISPLACEMENT OF THE NONLINEAR RESPONSE

Eliminating q12(t) = E[x(t)Jc(t)] and q22(t) = E[x2(t)] from the set of equations (15), the following third-order differential equation in terms of qll(t) is obtained.

q(3) 1, (t) + 6(owoq~)(t) + [4w2(t) + 8¢2 wg]q~l)(t)

+ [8(oWow2(t) + 4w(t)&(t)]qll(t)

= 4L22(t) + 8~owoL12(t) + 2L12(t) (A.1)

Guided by the formulation developed for a linear oscillator, 23 the third-order differential equation can be split into a first-order differential equation

qll(t) + 2s*(t)qll(t) = 2r(t) (A.2)

qH(0) = 0

with the excitation r(t) satisfying the second-order differential equation

F(t) + [2w*(t ) ]2r( t ) = g( t ) (A .3 )

r (0) = 0 = 0

where

e*(t) = (oWo - ½6(t) (A.4)

s*(t) = (oWo + 6(t) (A.5)

[2w*(t)] 2 = [2w(t)] 2 - e(owo6(t) - 6(t) + 462(t) (A.6)

g(t) = 2L22(t) + 4(0w0 L12(t) + Ll2(t) (A.7)

and 6(t) satisfies the nonlinear differential equation

6 - 6 6 6 + 4 ( w 2 - ( z w 2 ) 6 + 4 6 3 =2w& (A.8)

Let w(t) be a slowly varying function so that

w(n)(t) w - - ~ <- ,~(t)w(t) (A.9)

where 0 < e,:(t) << 1 and w(n)(t) is the nth derivative of w(t) with respect to time t. The dominant solution for 6(t) becomes

0J~) __ ~)d (A. 10) 6(t) = 2(c02 2 2 -- ~0 W0 ) 2Wd

and it can be used to simplify the expressions (A.2) to (A.7) in the form shown in eqns (18) and (19) respectively.