Method for nonstationary random vibration of inelastic structures

Probabilistic Engineering Mechanics 9 (1994) 115-123

Method for nonstationary random vibration of inelastic structures Y.K. Wen & D. Eliopoulos Department of Civil Engineering, University of Illinois, Urbana, Illinois 61801-2397, USA A method of response analysis of inelastic structures under nonstationary random excitation is presented. Emphasis is on realistic consideration of the time-varying intensity and frequency content of the excitation, detailed nonlinear response behavior of the structural frame, and computational efficiencyof the method such that it can be applied to response and safety evaluation of real structural systems. The method is verified by comparison of results with detailed time history analyses of inelastic structural frames and Monte Carlo simulation. Examples on application to reliability evaluation of steel buildings designed according to current code procedures are given. based on the random vibration method are verified by comparison with those from detailed numerical time history analyses of the inelastic structural frames using the well tested computer program for inelastic structures, DRAIN-2DX. 3

INTRODUCTION The method of random vibration has been used successfully in many engineering applications, for example in gust response analysis of structures under wind and offshore platforms under random wave excitations. Although research efforts have been going on in earnest for a long time on application to earthquake response and safety analysis, the results have not been as encouraging. The difficulties in implementing the method are primarily: (1) that earthquake ground motions are almost always nonstationary with time-varying intensity and frequency content; and (2) that structures under severe earthquake excitation become nonlinear and inelastic with restoring forces that depend on the time history of the response. As a result, recent efforts have been more on development of methods for more realistic modeling of the ground motion and the structural response behavior as well as computationally efficient solution procedures. A method based on a time-varying filtering technique for the random excitation, a differential equation model for the restoring force, and an equivalent linearization method for the response analysis has shown some promise in application to real complex structural systems. ]'2 The purpose of this paper is to demonstrate the viability of this method by performing a response and safety study of a class of steel buildings located in southern California. The nonstationary ground motion parameters depend on the characteristics of the seismic source, path, and site condition. The response statistics

MODELING OF THE GROUND MOTION Earthquake ground motions are known to be superposition of seismic waves of different propagation speeds, amplitudes and frequencies. They are generally highly nonstationary in intensity and spectral content. A nonstationary random process model has been recently proposed which allows easy identification of the model parameters from recorded accelerograms and can be used for both simulation and random vibration analysis.2 This model is used herein. The effort is concentrated on identification of the model parameters from characteristics of seismic source, path, and local site condition. The ground motion is obtained by passing a white noise through a filter with time-varying parameters as follows:

a(t) = I(t)([~p(t)]

(1)

in which I(t) is the intensity envelope function; ((4~) is a zero mean, unit variance stationary filtered white noise. ~b, however, varies with time and serves as a frequency modulation function. Therefore, the nonstationary behavior of the process is governed by I(t) and 0(t). The general frequency content of the process is controlled by the filter parameters. Two linear filters in cascade are used to generate random processes with a Clough-Penzien (C-P) spectral density. Such a spectral

Probabilistic Engineering Mechanics 0266-8920/94/$07.00 © 1994 Elsevier Science Limited. 115

116

Y.K. Wen, D. Eliopoulos

form has four parameters allowing modeling of the ground frequency and damping and approaches zero according to w4 as w goes to zero which is in agreement with physics of wave propagation. It has been shown that the instantaneous power spectral density function of ((t) has a general shape of that of the C - P spectrum and it varies with time depending on the first derivative of 0:

S¢¢(t, w) = 1/¢'(t) Scp[W/¢'(t)]

(2)

This approach by no means gives the most general nonstationary model. It allows, however, easy identification of the model parameters and the filter model is compatible with the time domain solution procedures for the response statistics developed for inelastic systems. To identify the model parameters from a given ground acceleration record, those of I(t) and ¢(t) are first estimated from the energy function and the zero crossing rate of the record, respectively. With these two functions the record can be reduced to a stationary process ~(¢) and from which the parameters of the C - P filter are estimated by conventional methods for stationary processes. Details can be found in Refs 2 and 4. This procedure can be used for sites where past records are available and are statistically representative of future earthquakes. For sites that these conditions do not apply, the potential future earthquakes that present a threat to the site may be classified as either characteristic earthquakes (CE) or noncharacteristic earthquakes (NE). The former are major events along the major fault with relatively better understood magnitude and recurrence time behavior, 5 therefore treated as a renewal process. The latter are local events so that their occurrence collectively can be treated as a Poisson process. 6 Besides occurrence time the major relevant parameters of a CE are magnitude M, epicentral distance to the site R, and intensity attenuation; whereas for a NE, the major relevant parameter is local modified Mercalli intensity (MMI) L The frequency modulation function for ground motions due to these events may be constructed from the consideration of the seismic wave propagation pattern for the region, or from recorded motion in stations close by. The intensity function will depend on the duration of the future event. Using the definition of significant duration tD as the time interval between 5 and 95% of the Arias intensity of the record. The dependence of the tD on M and R for a CE can be given by the results of regression analysis as: 7 log to = --0"14 + 0 " 2 M + 0"002 R + e

(3)

in which e is the uncertainty term following a normal distribution with a zero mean and a standard deviation of 0.135. For a NE, the relationship is: log tD = 1'96-- 0"123I+ e (4) in which e has a zero mean and a standard deviation of 0"2.

The parameters of the C-P spectrum are determined from the ground acceleration Fourier amplitude spectrum as an empirical function of the source, path, and site parameters, s This relationship has been established based on regression analyses of large number of earthquake records in southern California. For a CE, the Fourier amplitude spectrum FS(T) is given by: log [FS(T)] = M + Att(A, M, T) + bl(T)M

+ b2(T)s + bs(T) + b6(T)M 2 +

(5) in which Att is the frequency dependent attenuation; A is a measure of the site-to-source distance; s is a site characteristic parameter; and b are regression coefficients. The uncertainty term is again a normal variate N (0, 0.205). The major contribution to the uncertainty is from the attenuation law. There has been a large body of literature dealing with the uncertainty in attenuation. The value used in this study is based on the result of a survey of recent literature and consideration of excluding variability due to site-to-site variation in attenuation. Note that it corresponds to a coefficient of variation of 50% in the FS(T). For a NE, the corresponding relationship is:

log[FS(T)] = b l ( T ) I + b 2 ( T ) s + b 4 ( T ) + e

(6)

Details can be found in Refs 7 and 8. It can be shown that FS 2 is proportional to the spectral density function; therefore, the CP spectrum parameters can be determined from the above relationships. A nonlinear optimization method based on the Gauss method has been developed for the evaluation of the model parameters according to the procedures outlined in the foregoing.7

MODELING OF THE STRUCTURAL SYSTEM The class of structures under study herein is low-tomedium rise moment resisting steel frames designed according to the 1988 Uniform Building Code (UBS) 9 requirements. To facilitate the random vibration analysis, the large number of degrees of freedom need to be reduced. For this particular class of structures this is achieved by the strong-column-weak-beam model (SCWB). Following the SCWB design philosophy, the model localizes the inelastic behavior at the base and at the floor levels it is assumed that all columns remain elastic and all yielding occurs at the beams. A linear beam-column element at each story and a rotational inelastic spring at the base and each floor level can be used for this purpose as shown in Fig. 1. Masses are lumped at the floor levels and the rotational inertia is neglected. A 5% damping ratio is assumed for the first two modes. The moment of inertia of the equivalent

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beam-column element at a story is assumed to be equal to the sum of the moments of inertia of the columns of that story. The restoring moment of the rotational spring of the ith story is given by: M i = t~iGiOi+ (1 - oq)GiYi

(7)

in which 0/is the joint rotation at the ith level. Gi is the elastic stiffness coefficient and ~i is the post-to-preyielding stiffness ratio of the ith story rotational inelastic spring. The hysteresis of the inelastic spring is modeled by the smooth differential equation model 1 which has been shown to reproduce the structural inelastic behavior well and compatible with the time domain equivalent linearization solution procedure. The hysteretic component of the joint at the ith story Y/is given by:

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The parameters of this restoring force model is identified by matching the SCWB model with the original

(1) The linear stiffness coefficients of the rotational springs Gi are determined by requiring that the first modes of vibration of the model and the structure are the same. (2) The inelastic system parameters are determined by requiring that the response of the model and structure in a quasi-static test into the inelastic range are the same. It is assumed that the real structural response behavior can be described by the response analysis using the well known and well tested finite element program DRAIN2DX. The system identification is done by a nonlinear programming procedure based on the Gauss method. 7 The SCWB model can capture the stiffness coupling between adjacent floors and therefore should reproduce well the response behavior of the structural frame. This is verified by time history analyses of the resulting SCWB system for different acceleration records and comparisons with those of the original structures by the program DRAIN-2DX. Figure 2 shows the comparison of interstory drift time histories of a five-story, three-bay frame under the excitation of the Imperial Valley earthquake of 1979, one of the most severe ground motions ever recorded. The design of the frame according to 1988 UBC is also shown in the figure. The foregoing ground motion model is then used to generate 20 time histories of the Imperial Valley

Y.K. Wen, D. Eliopoulos

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earthquake and the comparison of the root-meansquare (RMS) interstory drifts as a function of time based on these 20 samples is shown in Fig. 3. It is seen from these comparisons that the response levels are high and well into the inelastic range and the SCWB model gives very good results.

RANDOM VIBRATION ANALYSIS

The above methods of modeling of ground motions and structural systems allow one to solve for the response statics by the method of random vibration using time domain equivalent linearization. The theoretical background and fundamentals of this method are available in the literature and therefore are not repeated here. Suffice it to say that the above formulation facilitates the linearization procedure in that one can obtain the coefficients of the equivalent linear systems as function of the response statistics in closed form. l The linearized

equations of the restoring forces can be written as: Y = [Ceq]0 + [Keq]Y

(9)

where [C~q] and [Keq] are diagonal matrices with elements which are functions of the response statistics. The equation of motion under ground excitation in terms of the interstory drift vector u and rotation vector 0 is: ii = - a u - [ K l ( u + b u ) - [ A l ( O + b O )

-elag

(10)

= - O / b - 1/b[F] -I [E] (u + 6u) - 1/b[F]-I[D]Y

(11)

in which a and b are Rayleigh damping coefficients; and matrices [A], [D], [E], IF] and [K] are functions of story mass, height, Eli, Gi and ~i. ag is the ground acceleration obtained from: ag = 2~f~f.,~f/q~t(t) + ¢o~xf + 2(gwgJCg/¢'(t) + w2xg (12)

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(15)

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(16)

in which the only nonzero term in [B] is: Bkk = 27r[dp'(t)]312(t)So

(17)

and k = 4n + 4 where n is number of stories of the structural frame. Equation (16) can be solved numerically. An explicit predictor-corrector method or an implicit backward differentiation may be used; the latter is more suitable if the set of covariance matrix differential equations is stiff. 7 Note that matrix [G] contains the linearization coefficients which are updated continuously during the solution. Note also that modeling of the ground motion by filters and the restoring force by differential equations makes the foregoing solution procedure possible.

NUMERICAL EXAMPLES

The response of the five-story, three-bay moment resisting space frame mentioned in the foregoing is considered. Two sites are considered, one located at Santa Monica Boulevard, Los Angeles, 60 km from the Mojave Segment of the southern San Andreas Fault; and one at the Imperial Valley, 5 km from the Imperial

Y.K. Wen, D. Eliopoulos

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Fault. At the Los Angeles site, because of the large distance to the closest major fault segment and the relatively long recurrence time for characteristic earthquakes at this segment, local events (NE) present a large risk. At the Imperial Valley site, however, the seismic risk is dominated by the CE because of the short recurrence time and close proximity of the site to the fault. Therefore, examples on response due to NE at the Los Angeles site and due to CE at the Imperial Valley site are given. Figure 4 shows the ground motion model functions and parameters for a NE with an intensity I 100 g--.

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large acceleration pulses which are known to cause severe structural response. The RMS interstory drifts and joint rotations as functions of time are shown and compared with simulation results again based on a sample size of 40 in Figs 9 and 10. The analysis slightly overestimates the responses at the upper stories. The accuracy of the analytical solutions is generally satisfactory.

Valley site, the ground motion functions and parameters identified from the E1 Centro differential array record of the 1979 Imperial Valley earthquake are shown in Fig. 7. A time history of the ground acceleration generated by the model is shown in Fig. 8. Note that although the magnitude of CE at this fault segment has been estimated at 6.5, because of the close distance to the site the acceleration level is much higher. Notice also the

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Y.K. Wen, D. Eliopoulos

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APPLICATION TO SAFETY EVALUATION

The above method of analysis has been applied to reliability evaluation of steel buildings designed according to UBC. The random vibration analysis of the structure provides the response statistics from which the probabilities of limit states can be evaluated. These probabilities, however, are conditional on the occurrence of the earthquake and a set of ground motion parameters. These parameters, for example attenuation and duration, are known to have large variabilities and may be correlated or functionally dependent. They often play a dominant role in the evaluation of the overall risk. A literature survey has been carried out on these

parameters and their uncertainties,7 the results of which were used to model these parameters as random variables. A fast integration technique based on the first-order reliability analysis l° was then used to include these uncertainties in the evaluation of limit state probabilities given the occurrence of the earthquake. These probabilities were then combined with the earthquake occurrence probability to arrive at the risk of limit state within a given time window. At the Los Angeles site, although both types of earthquakes contribute to the overall risk, the CE contribute much less primarily because of the large distance from the site to the closest fault segment. Tables 1 and 2 show the conditional probabilities of limit states given the

Table 1. Probability of 1.5% drift being exceeded for the time window 1991-2041 (u~ = 0.015 h)

Story

~3ch

eF(Umax > ufl Och)

Z3,c

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PF(Umax > Uf)

1 2 3 4 5

3"76 2'87 2'82 3"13 3"33

0"00009 0'00205 0'00240 0"00087 0"00043

2"23 2"28 2.34 2-42 2"32

0"0129 0"0113 0"0097 0'0078 0"0102

0'0723 0"0647 0"0558 0"0446 0-0577

123

Nonstationary random vibration of inelastic structures Table 2. Probability of 1% drift level being exceeded for the time window 1991-2041 (uf = 0.010h) Story 1 2 3 4 5

3ch

PF(Umax > uf]Och)

3nc

PF(Umax > uf[ One)

eF (Um~ ___uf)

2.80 1.83 1.83 2.11 2.33

0.0026 0.0336 0.0336 0.0174 0.0099

2.08 2.00 2.00 2.07 1.99

0.0188 0.0228 0.0228 0.0192 0.0233

0.1048 0.1408 0.1408 0.1146 0.1318

occurrence of CE or N E and the overall risk for a time window of the next 50 years. Details of the analysis can be found in Ref. 7. The risks of 1.5% drift being exceeded are approximately 5%, which corresponds to about 10 -3 per year, if the time dependency of the occurrence probability of the CE is neglected.

CONCLUSIONS A major obstacle in using the random vibration method in earthquake engineering has been the inability to account fully for the nonstationarity in the excitation and response, and the nonlinear, inelastic response behavior of the structure. Presented herein is a method based on recent results on modeling of restoring force, structural frame, and ground motion. It has been demonstrated that the method can take the nonlinearity and nonstationarity into consideration satisfactorily. Examples are given on application to response and safety evaluation of steel buildings designed according to a current code.

ACKNOWLEDGMENT

This study is part of an effort on reliability evaluation of steel buildings under seismic loads supported by the National Science Foundation under Grants N S F CES88-22690 and BCS-91-06390~ The support is gratefully acknowledged. Suggestions and contributions from D. Foutch and C. Yu are much appreciated.

REFERENCES

1. Wen, Y.K. Methods of random vibration for inelastic structures. AppL Mechanics Rev., Amer. Soc. Mechan. Engrs, 1989, 42(2), 39-52. 2. Yeh, C.H. & Wen, Y.K. Modeling of nonstationary ground motion and analysis of inelastic structural response. J. Structural Safety, 1990, 8, 281-98. 3. Kannaan, A.E. & Powell, G.H. Drain-2D; A general purpose computer program for inelastic dynamic analysis of plane structures. Report UBC/EERC-73/06. Earthquake Engineering Research Center, Berkeley, CA, Apr. 1973. 4. Yeh, C.H. & Wen, Y.K. Modeling of nonstationary earthquake ground motion and biaxial and torsional response of inelastic structures. Civil Engineering Studies, Structural Research Series No. 546, University of Illinois, Urbana, IL, Aug. 1989. 5. US Department of Interior, US Geological Survey 1988. Probabilities of large earthquake occurring in California on the San Andreas Fault. Open File report 88-398. 6. Cornell, C.A. & Winterstein, S.R. Temporal and magnitude dependence in earthquake recurrence models. Bull. Seismological Soc. Amer., 78, 1522-37. 7. Eliopoulos, D. & Wen, Y.K. Method of seismic reliability evaluation for moment resisting frames. Civil Engineering Studies, Structural Research Series No. 562, University of Illinois, Urbana, IL, Sept. 1991. 8. Trifunac, M.D. & Lee, V.W. Empirical models for scaling Fourier amplitude spectra of strong earthquake accelerations in terms of magnitude, source to station distance, site intensity and recording site conditions. Soil Dynamics & Earthquake Engng, Vol. 8, No. 3, 1989. 9. Uniform Building Code, 1988 Edition. International Conference of Building Officials, Whittier, CA, 1988. 10. Wen, Y.K. & Chen, H.-C. On fast integration for time variant structural reliability. Probabilistic Engng Mechan., 1987, 2, 156-62.