Mathematical modelling of structural behaviour during earthquakes

REVIEW PAPER Mathematical modelling of structural behaviour during earthquakes Linda J. Branstetter* and Garrett D. Jeong

School of Civil Engineering. Purdue University, West Lafayette, IN 47907, USA James T. P. Yao

Department of Civil Engineering, Texas A&M University, College Station, TX 77843, USA with Appendices by Y. K. Wen

Department of Civil Engineering, University of Illinois, Urbana, IL 61801, USA and Y. K. Lin

Center for Applied Stochastic Research, Florida Atlantic University, Boca Raton, FL 33431, USA The dynamical behaviour of structures can be modelled and simulated satisfactorily in the linear and slightly nonlinear ranges using finite elements and other analytical models in conj unction with modern computers. However, the highly nonlinear behaviour of structures remains to be studied further. The nonlinear response of complex structures which have been damaged by strongmotion earthquakes lies in this area. It is extremely difficult to collect sufficient field or laboratory data for the construction of accurate mathematical models of such structures for practical applications. This is partly due to the fact that the failure behaviour of a given structure is highly load-history dependent. In the absence of reliable analytical tools, system identification techniques may occasionally be applied to verify and/or update the mathematical model of a structure. The development of mathematical models useful for analysis of structural behaviour to earthquake excitation is an area too large to be covered in detail by a single review. In this paper, an attempt is made to critically review available mathematical models in the random earthquake response analysis of structures. By random, we refer to the excitation, to the properties of the structure itself, or to both. We are especially interested in those models dealing with slightly to highly nonlinear behaviour. Some recent work which utilizes these analytical techniques is highlighted. Also included is a review of system identification techniques as applied to structures. The concept of damage accumulation is mentioned, and a damage index is discussed which may be used in conjunction with an analytical model. A selective bibliography is also given for a more complete introduction into the areas discussed herein. Key Words: damage, earthquakes, hysteresis, mathematical models, nonlinear behaviour, random vibration, structural dynamics, system identification, uncertain systems. 1. I N T R O D U C T I O N The objective of this paper is to summarize and discuss the state-of-the-art of several subject areas related to the Editorial Note: This review paper consists of a main text coauthored by L. J. Branstetter, G. D. Jeong and J. T. P. Yao and two appendices. The appendices represent two shorter review articles: one on equivalent linearization methods written by Y. K. Wen (Appendix I) and the other on the effect of vertical ground motion on structural response authored by Y . K . Lin (Appendix II). The main text and appendices are coordinated in contents but written independently by these authors and therefore carry their own equation and reference numbers. * O n educational leave from Sandia National Laboratories, Albuquerque, N M 87185. Supported in part by National Science Foundation through Grant No. CEE 8412569.

mathematical modelling of structural behaviour during strong earthquakes. This includes an overview of available mathematical modelling techniques, a brief description of some hysteretic response models which have been used for earthquake analysis, and a look at some work done on systems which have parametric uncertainties. The role of system identification and its potential application to structural dynamics and reliability studies is also explored, and an explicit model for damage is reviewed. The review presented herein is not intended to be all inclusive. To construct mathematical models, it is necessary to simplify and idealize the structural system and its environment. Using field data and other relevant information, a preliminary design is made and the

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Mathematical modelling of structural behaviour during earthquakes: L. J. Branstetter et al. idealized mathematical model is analysed for expected or specified loading conditions. Based on these analytical studies, the design may be revised and re-analysed in an iterative manner until all design criteria are satisfied. The completed design is then constructed accordingly. Because (a) it is difficult to predict future loading conditions, and (b) material properties are random in nature, stochastic processes have been used to represent these quantities for the estimation of failure probabilities. However, the behaviour of the as-built structure is usually different from that as predicted with the original mathematical model, because the real-world structure is an extremely complex system. Even with the use of finite element methods and modern computers, it is essential to idealize at least some of the features of real structures in the mathematical modelling process. Moreover, the damage path and failure behaviour of most large structures remain unknown because few experimental studies of full-scale structures have been conducted to date. It becomes even more critical to develop sound mathematical techniques for analysis when there is a lack of observable data. Researchers have worked for many years on various aspects of the random earthquake response problem. A complete analysis of the structural response to strong earthquakes must include nonlinear effects caused by accumulation of damage in the structure. A system of coupled, nonlinear stochastic differential equations is usually used to represent such effects in real systems. During the past thirty years the field of random vibrations has grown tremendously. Especially, the stochastic analysis of linear multi-degreeof-freedom systems has been well developed. However, there exist only a few general techniques for attacking the nonlinear problem. In addition, most of the available techniques are based on restrictive assumptions about the nature of the random load process or the extent of system nonlinearity. The problem of highly nonlinear systems having random parameters as well as random loading has been left largely unaddressed because of its mathematical complexity and the difficulties in modelling the dynamic behaviour of damaged structures. System identification techniques are useful for the improvement of existing models. In recent years, nondestructive dynamic tests have been conducted on many structures for the estimation of their as-built dynamic properties. Such test data may then be used to obtain 'improved' or 'more realistic' equations of motion. These equations of motion are applicable within the range of the test amplitude, which is usually small and within the linear behaviour of the given structure. Therefore, results of such analyses should not be applied where destructive or damaging loading conditions are considered. Nevertheless, these mathematical representations can be useful for comparison purposes.

2. OVERVIEW OF MATHEMATICAL MODELLING TECHNIQUES IN PROBABILISTIC NONLINEAR STRUCTURAL DYNAMICS Several levels of complexity may be considered in discussing probabilistic models for nonlinear structural dynamics. The first level of complexity involves a single-

degree-of-freedom deterministic oscillator model with a slightly nonlinear spring, subjected to white-noise excitation. Starting with such a model, available solutions would not be applicable as the excitation became nonstationary or as the spring became highly nonlinear. At a higher level of complexity, the spring stiffness of the oscillator could be represented with a random variable. At an even higher level of complexity, the spring could undergo inelastic deformations during high-amplitude oscillations, making the stiffness a random process correlated to both the random load and random response processes. Finally, the system could be expanded to a multi-degree-of-freedom model of a structure which accumulates damage during an earthquake. In that case, the base excitation becomes a correlated random process vector, while the structural stiffness random process matrix would have an associated fourth-order correlation tensor. Regardless of the level of complexity, a nonlinear stochastic analysis has one inherent mathematical difficulty, i.e., the problem of closure. In the case of linear systems, the response is determined from a finite set of stochastic differential equations. A number of response moments equal to the number of equations is uniquely determined. For nonlinear systems the number of unknown moments always exceeds the number of equations. In order to obtain solutions, researchers have either neglected higher order moments or assumed some model (either Gaussian or non-Gaussian) which relates the higher order moments to the lower ones. These assumptions are referred to as closure techniques or truncation schemes by various authors. Ibrahim 1 discussed several of these. For a further introduction to the concept of closure, refer to Beaman and Hedrick 2, Crandall 3, and Dash and Iyengar4. Another basic ingredient in any analysis involving earthquakes is the form of the ground-motion excitation. Various models have been used. A white-noise approximation has been used for the period of most severe oscillation at points removed from the earthquake source. Early investigations in this area include those by Housner 5, Rosenblueth and Bustamante6, BycroftT, and Thomson 8. Ruiz and Penzien 9 proposed an approximate power spectral density function for Gaussian white-noise of a specified intensity. The excitation is often assumed to be white-noise in numerical solutions of nonlinear systems. For example, this is done when the FokkerPlanck equation is used with a Markov process assumption. To better approximate the spectral power distributions observed in real earthquakes, Kanai 1° and Tajimi 11 suggested a filter transfer function for stationary filtered white-noise excitation. Nonstationary filtered white-noise was used by multiplying the stationary filtered process by a specified time-varying function. For example, this was done by Amin and Aug 12 and Jennings, Housner, and Tsai 13, who used an envelope function with several segments, and Shinozuka and Sato ~4, who used a continuous envelope function. Nonlinear stochastic models have been summarized by Lin et al. ~5, Benaroya ~6, and Roberts 17'~s in a comprehensive manner with extensive bibliographies. An account of the mathematical background of concern in the solution of the differential equations involved in nonlinear stochastic modelling has been given by

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Mathematical modelling of structural behaviour during earthquakes: L. J. Branstetter et al. Caughey ~9. Following Benaroya, available nonlinear stochastic models may be categorized into the following six major types: (1) Markov methods based on the Fokker-Planck equation, (2) equivalent linearization methods, (3) perturbation methods, (4) functional series representations, (5) decomposition method of Adomian, and (6) simulation methods. Highly nonlinear behaviour of stochastic structures has been left largely unexplored because many of the existing solution methods are not applicable in this regime. For example, the Fokker-Planck equation approach is not valid if the excitations are non-Gaussian. Equivalent linearization techniques cannot be applied to systems with non-Gaussian response statistics. Perturbation theory is applicable only to systems having small nonlinearities. One very promising method which has yet to be fully explored in the literature is the decomposition method of Adomian 2°'zl , and Adomian and Malakian 22. This method involves fewer assumptions than other methods. Benaroya 23 has shown that Adomian's restriction that system parameters and the excitation are uncorrelated may be removed. Beyond this and the other solution methods mentioned above, Monte Carlo type simulations have been extensively used in practice. In the following, various solution techniques are briefly reviewed. 2.1 Markov methods based on the Fokker-Planck equation A large body of literature exists describing this method and its many variations for different types of systems. For a mathematical description, the references cited in the following discussion are excellent. Roberts 17 explores the assumptions behind the method and its applicability to various problems, including cases in which exact solutions may be obtained. Basically, if the elements of the excitation process are broad-band in character, then a white-noise approximation may be used. This introduces certain difficulties in interpreting the equation of motion, which have often been resolved by treating it as an It6 equation 24. When this is done, the system displacements and velocities are Markovian 25, and the transition probability densities are governed by a diffusion equation. This diffusion equation is called the Fokker-Planck-Kolmogorov (FPK) equation. It is often referred to as the forward Kolmogorov equation. Roberts cites some historical reviews of this method, including a survey by Fuller 26. The method has been used, under various restrictions, to study nonlinearities associated with various types of oscillators. This has been done by many researchers, including Crandal127, Caughey 2s, Lyon 29'3°, and Smith 31. Chonan 32 approximated a continuous system as a single-degree-of-freedom system with nonlinear stiffness. Most nonlinear system solutions of the F P K equation involve iterative schemes (e.g., see Mayfield33). Series expansion techniqes have often been employed as well. Two eigenfunction expansions have been suggested by Stratonovitch 34 and Atkinson 35. Wen 36 and Bhandari and Sherrer 37 have used the Galerkin method, a special case of the method of weighted residuals where the trial and weighting functions are identical. Other methods used for solution of the F P K equation include finite difference, stochastic averaging, series expansions, and the method of moments. Each of these techniques, along with those

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listed above, are discussed and referenced thoroughly by Benaroya 16 and Roberts 17. Besides the white-noise excitation assumption the F P K method requires a very large computational effort for systems of second or higher order. Most of the theoretical methods developed for either exact of inexact F P K solutions have been applied to single-degree-of-freedom systems. Therefore, the applicability of this general method to the solution of complex nonlinear structural systems subjected to earthquake excitation has not yet been well established. 2.2 Equivalent linearization methods* Krylov and Bogoliubov 3s introduced this approximate solution technique as the describing function method in control theory for nonlinear systems having deterministic excitation. It was independently extended to the case of random excitation by several researchers including Booton 39 and Caughey 4°. In this method, the stochastic equation governing a nonlinear system is replaced by an 'equivalent' linearized version. This introduces a random error between the true nonlinear and the linearized systems. This error is minimized, usually in a meansquare sense, by setting to zero the partial derivatives of the expected value of the squared error with respect to coefficients appearing in the linearized equation. These partial derivatives define a set of equations which are then solved simultaneously for the required coefficients. In this way, the 'equivalent' linearized equation differs from the nonlinear one by the least possible error in the mean square sense. The inherent danger in this method lies in the loss of certain possibly important nonlinear effects in the linearized equation. Consider a single-degree-of-freedom stochastic system governed by the following nonlinear differential equation: 5~+ h(x, 2c)= G( t)

(1)

where G(t) is the system forcing function, and the function h contains nonlinear terms related to the system damping and stiffness. Assume that the nonlinear terms may be approximately written as the sum of two linear components, one pertaining to damping behaviour and the other to stiffness: 5~+ sic + fix = G(t)

(2)

The error E introduced by the linearization is simply the difference between the nonlinear h function and the two linear components:

~=h(x,X)-~-/~x

(3)

This error is a random process. If the mean-square error is minimized, we have the equations E{e z} = 0

(4)

E{~2}--0

(5)

and

* See Appendix I for further comments.

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Mathematical modelling of structural behaviour durin9 earthquakes: L. J. Branstetter et al. Substituting the error equation into the partial derivatives, it is possible to solve for the coefficients ct and fl which minimize 5. Many researchers have used this solution technique, which has one major advantage: the ease of extension into the analysis of multi-degree-of-freedom systems. Adaptations of the method may be used for systems having either stationary or nonstationary excitation. In theory, it may be argued that only systems having small nonlinearities may be studied. However, it has been shown that mean square errors remain relatively small even in the case of large nonlinearities is. The field of control engineering has produced a large amount of literature in the area 4L42. The equivalent linearization method has been reviewed by Spanos 43 and Roberts ~8 The method has been applied to the field of earthquake engineering by several authors, including Goto and Iemura 44, Kobori, Minai, and Suzuki 45, Crandall, Lee, and Williams 46, and Crandall and Lee 47. It is not always easy to categorize a subject into well defined segments. Such is the case for the procedures known as 'stochastic averaging', which will be mentioned in this section because, like the equivalent linearization procedure, they also have their origins from theories for deterministic excitations as developed by Krylov and Bogoliubov. Basically, these procedures use as their mathematical basis a theorem by Khasminskii 48. Rapid fluctuations in random excitation lead to complex equations which may be replaced by a simpler set when the excitation is replaced by some approximation of its 'averaged' effect. This averaged effect is multipled by a correlated response. Crandall and Zhu 49 discuss three such procedures. Stochastic averaging has been used in the study of nonlinear systems having parametric uncertainties. When used for nonlinear hysteresis, Iwan and Lutes 5° showed that stochastic averaging, as opposed to less specialized linearization techniques, gives inaccurate results for systems with large nonlinearities.

2.3 Perturbation methods This method may be applied to any system having small nonlinearities, where the linear equation which remains after all nonlinear terms have been dropped is solvable. It may be applied either to continuous systems as done, for example, by Lyon 51, or to discrete systems with either single or multiple degrees-of-freedom. The basic idea as explained by Benaroya 16 is summarized here. Other introductory summaries are given by Lin 52, Roberts la, and others. Perturbation theory has been used for deterministic vibration analysis for many years s3, and was generalized to the case of stochastic excitation by Crandal154. Recent work using this approach has been done by Huang 55, Szopa 56 (for a linear system), and Spanos 57. Consider the equation of motion for a single-degree-offreedom system having small nonlinearities q. It may be written as: 5~+ ~;Yc+ ¢OoX + et1(x, Yc)= F(t)

For example, if t/(x, ~)= x~, then the resulting linear equations for terms of order zero, one, and two in 5 become 5~o + 7:~o + COoXo = F(t)

(8)

J~l "~Y'~I "~ (DoX1 =

(9)

--Xo'~o

"~2 "-1-~)¢2 -Jr-(DoX 2 -~- - -

(X1XO -~-X1XO)

(10)

One could write as many such equations as desired for higher orders of e. However, the mathematical problems associated with finding solutions including higher than first order terms are usually too complicated to be of practical use. Therefore, normally only terms of the order e are included. The solution of equation (8) above may be written using a convolution as:

Xo(t ) =

h ( t - z ) F ( z ) dz

(1 i)

oo

where h(- ) is the impulse response function for the linear system. Knowing Xo(t ), equation (9) may then be solved using the following convolution:

x 1(t)=

h(t-z)(-Xo(Z)&o(Z)) dr

(12)

In this way, as many terms as desired in the series expansion for x(t) may be evaluated, which when summed yield the complete solution. If x(t) is stochastic, then to find the mean response one takes the expected value of each convolution, and then sums these expected values in the series expansion. The criteria of convergence of the series expansion may be viewed in several different ways, but each depends on the magnitude of the perturbation parameter e. These criteria have not been rigorously established, so that the appropriateness of the perturbation approach can only be judged when the results are compared to those obtained with methods which do not depend on smallness of the nonlinear terms, such as Fokker-Planck. It may be stated, however, that perturbation series are often asymptotic, such that higher order perturbations may improve the approximation for small e at the expense of worsening it for large e49.

2.4 Functional series representations Roberts la summarized this method in the following manner. The response of a linear system may be expressed in the form of a convolution as

(6)

where 7, COo, and 5 are constants, and e,~l. If the perturbations are of the order 5 or smaller, then the response may be written as a power series in 5:

x(t) = Xo(t ) + 5x 1(t) + e2X2(t) + . . .

at terms which may be substituted for x, ~, and 5~in the equation of motion. Equating terms of the same order in e, one arrives at a series of linear equations for x 0, x~, x z,

(7)

By taking derivatives of the series expansion, one arrives

x(t) = f ~

hl (q)F(t - t 1) dt~

(13)

where F(t) is the input and hx(tl) is the system impulse response function. It has been shown that the convolution may be generalized to what is called a Volterra series expression when the system is nonlinear:

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Mathematical modelling of structural behaviour during earthquakes: L. J. Branstetter et al. x(t)= ~=,

~. . . .

,~

As summarized by Benaroya, consider the following linear system where the stiffness k(t) is a function of time only, and is therefore not dependent on the load

h,(tl,t 2. . . . . t,)

×F(t-ta)... F(t-t.)dt 1 ...dt.

(14)

m2(t) + c2(t) + k(t)x(t) = F(t)

The function h, may be regarded as the nth degree impulse response function. This generalization has been done by several researchers, including Wiener 58, Barrett 59, Bedrosian and Rice 6°, and Dalzell 6~ . Bedrosian and Rice considered a system having harmonic input of the form F(t)= ~ ei'''k k

where m and c are constants and k(t) is a random process. Define the linear operator L as d2 d L = m dtz +C-d[+k(t)

.

.

.

Then the equation of motion in operator form is:

X

L[x(t)] = F(t)

k(t) = E[k(t)] + K(t)

OC

e i['°;q +'''+U~"t"]do91

. do),,

(16)

h,(q . . . . t,,)

... -cc

x e-

so i[v)l,, +...

+ o.t.]

dt 1 . . .dt,,

(23)

R=K(t)

(24)

Substituting L = D + R equation of motion,

(25)

Now let x be written as the sum of a series as

(18)

2.5 Decomposition method of Adomian This operator method has yet to be fully explored in the literature, yet it shows great promise as a technique for the solution of nonlinear problems without the necessity of using any of the assumptions used in the FokkerPlanck, equivalent linearization, perturbation, or functional series methods. Benaroya 23 presented perhaps the first explanation of the method as it may be applied to problems in structural dynamics. The method, proposed by Adomian 2°'21 provides a sound mathematical approach to the solution of systems having random coefficients as well as random parameters, and is equally useful for linear and nonlinear systems. Benaroya has proposed a generalization to the method which allows the solution of systems in which the input and the coefficients are correlated. With this generalization, the method may in theory be applied to virtually any structural dynamics problem.

into the operator form of the

x=D-1F-D-1Rx

x(t)= ~

(26)

xi(t)

i-O

Indications are that the functional series approach yields accurate results only when the system nonlinearities are small, and so in that regard resembles the perturbation method. The true relationship between these two methods is not yet well understood.

134

d2 d D = m ~ 2 +c ~ + E[k(t) ] (17)

H 1 is the usual linear frequency function. Once H, is found, h, is evaluated from the Fourier transform. Then the response of the Volterra expressions has been completely determined. The Volterra expression may also be used to evaluate response statistics. Dalzell 6a gave a detailed example of this method for a nonlinear system having the equation of motion:

A12+A35c3+B12+B323-{-C1x+C3x3=F(t)

(22)

Similarly, the operator L may be separated into a determinstic part D and random part R ( L = D + R ) , where

and H,(~o 1. . . . . ~o,)=

(21)

The stiffness may be written as the sum of deterministic and random parts. The deterministic part is the mean value of k(t), E[k(t)], and the random part, denoted by K(t), is the random fluctuation about the mean:

H,,(coI . . . . . co,) sf3

(20)

(15)

1

Using this input and substituting into the Volterra expression, then into the equation of motion, it is possible to obtain the nth degree frequency response function H, in terms of system parameters. Note that H . and h, are related by the following Fourier transforms:

h,(q . . . . . t,) .

(19)

so that

x=D-1F-D-1R(xo+xl

+...)

(27)

In practice, this series expression for the solution will be truncated to a finite number of terms. Some error is introduced at this point, but the solution converges to the exact solution as the number of terms goes to infinity. The series expression for x(t) may be regrouped as follows:

Xo(t ) = D- 1F

Xl(t)= - D - 1 R x o = _ D - 1 R D - 1 F x2(t ) = - D - 1Rx 1 = D - 1RD - 1RD - 1F

(28)

Thus, x(t) as the sum of these terms becomes

Probabilistic Engineering Mechanics, 1988, Vol. 3, No. 3

x(t)= ~ ( - 1)i(D-tR)iD-~F i=O

(29)

Mathematical modelling of structural behaviour during earthquakes: L. J. Branstetter et al. If the inverse D-1 of the deterministic operator D exists and has a corresponding Green's function g(t,z), then the above solution may be written in integral form as

x(t)=fl g(t,r)F(z)d*-fl g(t,~)K(r)x(r)dr + c 1~b1(t) + e2~b2(t)

(30)

where the c~i(t) solve the homogeneous equation L[x] = 0 and the ci are determined by initial conditions. Statistics of the response may now be computed using this last expression. No assumptions have been made on the nature of the forcing function to obtain this expression, although independence of the forcing function and the coefficients was required. As mentioned earlier, Benaroya has applied the technique in such a way as to remove this restriction. Shinozuka and Dasgupta 62 used an approach similar to the decomposition method of Adomian to develop a stochastic finite element solution technique along with a consistent Monte Carlo method. Application of their technique requires the inversion of the deterministic operator D only once and successive improvements to the solution are made by means of repeated application of the stationary operator. Furthermore, evaluation of the partial derivatives of the operator is not necessary, unlike conventional perturbation methods used for stochastic finite element solutions, making the technique an efficient and highly competitive alternative.

2.6 Simulation methods Most simulation methods are known by the name of Monte Carlo. An introduction to this technique is provided by many authors, including Elishakoff63 and Ang and Tang 64. In this approach, sample functions of the excitation process are generated, and sample functions of the response are then computed using the excitation samples. Each individual computation is deterministic. However when a large number of calculations are performed, these results may be used to infer statistical information about the response. Automatic random number generators and filters are often used to alter the statistics of the random input to a desired state. Alternatively, the summation of sine waves with random coefficients may be used. Numerous references pertaining to this technique with many of its variations are given by Roberts is. One disadvantage of this approach is that particularly for nonlinear systems, the computational effort and expense involved in performing a large enough number of computations to allow making reliable statements about the response statistics may be prohibitive. According to Crandall and Z h u '.9, the statistical uncertainty in the response statistics decreases in proportion to n-1/2 while the cost increases essentially in proportion to n. To gain one additional significant figure in a result requires a hundredfold increase in cost. Nevertheless, the advent of modern electronic computers and efficient computational methods has enabled the use of the Monte Carlo method in solving many practical problems. 3. REVIEW OF HYSTERESIS MODELS WITH SOME APPLICATIONS Several surveys of current literature on the response of systems having hysteretic response behaviour have been

published 43'46'65'66. An excellent review is given by Spencer 67. Hysteretic restoring forces are difficult to compute because they are history dependent. Models of these forces have ranged in complexity from bilinear, to piecewise linear in several segments, to fully nonlinear polynomial approximations. Jennings 68, for example, used a polynomial approximation for the skeletal forcedisplacement curve, first proposed by Ramberg and Osgood 69, to study the response of yielding structures to generated ground motion. For the most part, explicit solution of system equations where hysteresis is included has not been possible, although a variety of approximate solution techniques have been used with varying degrees of success. B o u c 7° suggested a versatile hysteresis model which was later revised by Wen 71 and then used by Wen and others. Baber and Wen72'73 modelled zero mean response statistics for multiple degree-of-freedom shear-beam structures, and then for plane frames TM using the modified Bouc model. The plane frames were idealized as having discrete hysteretic hinges of zero length, with lumped nodal masses. An equivalent linearization technique was used. With the avoidance of stochastic averaging, Baber and Wen obtained results which compared well with Monte Carlo simulation over a wide range of parameters. Equivalent linearization has been the most widely used technique to date for highly nonlinear multiple degree-offreedom hysteretic systems. Other researchers using this approach include Atalik and Utku 75, Baber 76, Iwan and Mason 7v, Kazakov 7s, Nabavi-Noori and Baber 79, Spanos, Noori, Choi, and Davoodi sl, and Chang, Mochio, and Samaras s2. Other approaches to hysteretic behaviour have been used, including series expansions s3, and numerical solutions of the FPK diffusion equations s*. Of the above references pertaining to hysteretic degrading systems, by far the majority consider the zeromean response problem. However, the non-zero mean response problem often arises in real applications. If a system has nonsymmetric nonlinearities, for example, it will exhibit nonzero mean response. Seismic ground motions are another example. Although nominally of zero mean, such excitation may produce nonzero mean response. Systems having nonsymmetric stiffness characteristics have been addressed by Spanos 8°'s5, Baber and Wen 72, Baber 76, and Nabavi-Noori and Baber 79. Recently, Baber 86 presented a direct stiffness frame model with hysteresis which uses discrete hinge elements to incorporate nonzero mean coupling effects between gravity and earthquake responses. Noori, Choi, and Davoodi 81 developed a model of a smooth hysteresis element in series with a hardening spring element. Besides addressing the zero mean response problem, they applied their model to the response of a system under nonzero mean excitation. The bibliography given at the end of their paper references an extensive body of work on hysteretic models.

4. APPLICATIONS FOR SYSTEMS HAVING PARAMETRIC UNCERTAINTIES Most available numerical techniques for nonlinear systems address only the load as being random in nature, and do not consider the possibility of random system parameters such as stiffness and damping coefficients.

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Mathematical modelling of structural behaviour during earthquakes: L. J. Branstetter et al. When parametric uncertainties are discussed, studies which would have been classified as pertaining to linear systems enter the realm of nonlinear analysis, and thus are included in this discussion. Frisch 87 provided insight as to why this is true as he discussed linear random equations (i.e. differential equations which would be classified as linear before considering the added complication of parametric uncertainties). To quote Frisch: 'The principle difficulty of the problem stems from the fact that in spite of its apparent linearity, it is nonlinear to stochastic quantities, because the solution of a linear equation depends nonlinearly upon the coefficients. This gives rise to very serious difficulties, which are of a mathematical nature, and are shared in c o m m o n . . , by all linear stochastic equations . . . . Typical among these difficulties are the divergence of perturbation expansions, the appearance of secular terms, and the fact that even (generally) the lowest order moments of the solution depend on the complete set of all the moments of the coefficients.' Many papers concerning parameter uncertainty have appeared in the literature. Early work by Tikhinov 88, Kozin 89, and Bogdanoff and Chenea 9° was concerned with the character of the response itself. Later work focused more on the modal characteristics of parametrically uncertain systems. Soong and Bogdanoff91 characterized modal statistics for structures with random stiffnesses. They established that the lower modes of a structure are much less sensitive to perturbations in mass or stiffness than the higher modes. Boyce92, Collins and Thomson 93, and Hart 94 worked in the same area. In his 1973 paper, Hart performed a static analysis in which two levels of randomness were accounted for in development of an overall stiffness matrix. The geometric stiffness matrix depended upon imprecisely known member axial forces found in a static response problem. This matrix was then added to a random elastic stiffness matrix. An eigenvalue analysis was then performed, and statistical moments of system natural frequencies and mode shapes were found. Other analyses of complex parametrically uncertain systems have been performed. Researchers in this area include Collins, Kennedy, and Hart 95, Hart and Collins 96, Hasselman and Hart 97'98, and Schiff and Bogdanoff99. Static displacements and member-end forces have been computed for linear systems with uncertain stiffness 100,101, as well as linear elastic buckling loads96, 102. Prasthofer 1o3 used a perturbation approach to develop general integral solutions for arbitrary forcing functions acting on either single or multiple degree-of-freedom systems with uncertain stiffnesses. His goal was to look at the effect of input uncertainty on both modal and actual response uncertainty. He found actual solutions for certain systems having impulsive input, although his solutions are theoretically valid for any type of forcing function. His method does involve some arbitrariness in finding required correlation coefficients. Some researchers have written computer programs in a finite element framework to establish response moments for uncertainly parametered systems on a step-by-step basis. For example, Bennett 104 and Chang 105 considered

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linear and nonlinear structures, respectively. Branstetter and Paez 1°6 also developed a computer code for the linear analysis of multiple degree-of-freedom structures, avoiding some of the approximating assumptions of Bennett and Chang. By assuming all random load, response, and stiffness quantities to be normally distributed, and assuming stiffness to be independent of both load and response random variables, Branstetter and Paez developed and evaluated first and second order response moment equations. A necessary step in extending this technique to nonlinear structures will be allowing correlation between stiffness and load and between stiffness and response. An excellent review of the state-of-the-art in the analysis of parametrically uncertain systems is provided in a recent article by Ibrahim a°v. Anyone interested in pursuing research in this area would be well advised to begin there. 5. SYSTEM I D E N T I F I C A T I O N IN S T R U C T U R A L DYNAMICS It is now possible to simulate structural response to extreme forces such as strong earthquakes or windstorms with the use of digital or hybrid computers, and to evaluate the serviceability and safety conditions of the modelled structures. However, there exists the paradox that (1) the applicability of 'realistic' models of the structure are limited to the small-amplitude response range, (2) catastrophic loading conditions are likely to cause the structures to behave beyond the assumed linear or 'near-linear' responses, and (3) severe loadings may cause serious damage to the structure and thus change the structural behaviour appreciably. It is important that the extent of damage in structures can be assessed following each major earthquake or at regular intervals for the evaluation of the effects of aging and deterioration. On the basis of such a damage assessment, appropriate decisions can be made as to whether a particular structure can and should be repaired to salvage its residual value. System identification is a process for constructing a mathematical model of a physical system when both the input to the system and the corresponding output are known. For most civil engineering applications, the input is usually a forcing function and the output is the displacement, velocity, or acceleration response of the structure to these forces. The mathematical model obtained from the identification process should produce a response that in some sense matches closely the system output, when it is subjected to the same input. System identification techniques have been widely used in many branches of science and engineering for the estimation of various characteristics of a physical system (e.g., Eykhoffl°s and Sage and Melsal°9). Their applications in civil engineering have been studied with increasing interest during the last two decades. The primary objective of applying system identification in structural engineering is to obtain an improved mathematical model which can better represent the characteristics of the real-world structure. In the available literature, a set of differential equations describing either a lumped-mass or simple continuous model in the time domain, or a transfer function (black box model or lumped-mass model) in the freuqncy domain are often used to represent the structural behaviour. A set of

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Mathematical modelling of structural behaviour during earthquakes: L. J. Branstetter et al. parameters is to be estimated from the measured response of the real structure to a known disturbance. The application of system identification techniques to solve structural engineering problems has been called structural identification by several investigators 110-114Because of their simplicitly the linear lumpedparameter models are widely used in structural identification. More complex models such as the linear continuous-parameter models and nonlinear-parameter models are used when the lumped-parameter model cannot be used to provide an adequate representation of the structural behaviour. For lumped systems or continuous systems with lumping approximations, the disturbance must also be represented in a discrete form. On the other hand, the disturbances can be either discrete or continuously distributed in a continuous system. Various least-squares estimation methods (including repeated and generalized least squares), the instrumental variables method, and maximum likelihood estimation have been used to identify linear structural models. The least-squares estimation minimizes the summation of square errors between the predicted response and the measured structural response. In the generalized leastsquares method, the criterion function for evaluating the 'goodness of fit' is the summation of square generalized errors which is defined to include the additive noise covariance matrix. Repeated application of the least squares method can be used to modify the usual least squares procedure by increasing the order of the mathematical model in an iterative process until the desired accuracy is obtained. The instrumental variables method is applicable to the problem of bias with noise-polluted response. The method involves an iterative process in the calculation of revised estimates and an instrumental variables matrix function. The maximum likelihood method is widely used for parameter estimation in statistics. It determines the parameter estimate by minimizing a criterion function through an iterative procedure. The method appears to have the advantage of providing the best estimation for a wide range of contamination intensity in the external excitation and the structural response 115. Estimation methods are generally applied to time-domain analysis, and usually involve complicated iterative procedures. Nevertheless, they can be used to treat nonlinear models for which the modal expansion and transfer functions in the frequency domain are not defined. In contrast with the linear models, relatively little work seems to have been done for nonlinear models 116. It is in part due to the mathematical difficulties in considering the nonlinear terms. Some common techniques in dealing with linear systems, such as the modal expansion and transfer function, are not appropriate in the nonlinear case. Nevertheless, it is possible to apply the modal expansion analysis to obtain approximate solutions for slightly nonlinear problems. Current developments in structural identification have mostly dealt with structural parameters with limited range of application or parameter~ for highly simplified structural behaviours. For example, in the evaluation of vibratory parameters of structures, the models are often limited to the smallamplitude response range and time-invariant structural behaviour. However, catastrophic loading conditions such as strong earthquakes and windstorms are likely to cause the structure to behave beyond the assumed linear

range of responses. Using the theory of invariant imbedding, a best a priori estimate can be obtained by minimizing an error function 117. The method is applicable for some general boundary conditions. Invariant imbedding is a special case of dynamic programming filtering. Instead of going through the Euler-Lagrange equations to determine the best estimate that minimizes the error function, dynamic programming may be applied directly. In such cases, the decomposition of the error function can lead to a system of partial differential equations. The optimal least squares filter satisfies the governing differential equation which describes the structural model and minimizes the quadratic error function. The error function is defined in terms of observed error vectors (weighting matrices) and the best a priori estimate of the parameters 11s,119. The Kalman filter has been used to obtain optimum sequential linear estimation and has been extended filter to deal with nonlinearities. It provides a good approximation for high sampling rates as demonstrated in simulation studies of parameter estimation 1°9. The maximum likelihood method has been applied to both linear and nonlinear systems. It can be used to treat both the measurement noise and the process noise, and may also be used to estimate the covariances of the noises 111. It is also suggested that the extended Kalman filter may be introduced in the calculation of the likelihood function. An input-output relationship of multiple integral form is assumed to represent the model 11°'121. The kernel functions which represent model parameters can be estimated by a cross-correlation technique. In theory, the relationship can be written in the Laplace domain and thus the kernels are identified in terms of the Laplace parameter. Their values in the time domain are then obtained by the usual inversion techniques. An integral form of the fomulation of the excitation-response relationship has also been used when the transfer function is used for the linear model. The integral formulation may be extended to include nonlinear kernels. However, the computational effort involved for the second or higher order models is much greater than for first order systems. Hart and Yao 11° presented a review of identification theories and their applications in structural dynamics as of 1976. They included identification problems which require a prior structural model with or without a quantification of experimental and modelling errors. The review also contained a brief description of the algorithms and sample data. Ibanez 122 presented a comprehensive review of various techniques for the improvement of mathematical models in structural dynamics. Liu and Yao 123 discussed the concept of structural identification in the context of system identification and unique characteristics in its structural engineering applications. Basically, structural engineers are interested in identifying the damage and reliability functions, respectively D(t) and L(t), in addition to the equation of motion. From another viewpoint, the updated equation of motion using test data and system identification can be a tool for the estimation of damage and reliability of existing structures. System identification tests are always conducted at extremely low-level vibrations. Therefore, they can be performed as many times as it is needed without causing any apparent damage to the structure. In most cases, only

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Mathematical modelling of structural behaviour during earthquakes: L. J. Branstetter et al. earthquake records are available for the purpose of damage analysis. In general, the complete record of an earthquake can be separated into the following three portions with different characteristics: (a) a stronglyexcited portion at the beginning of an earthquake showing contributions from higher modes, (b) a much larger amplitude portion with nonlinear behaviour, and (c) a very low-level vibration portion at the end of an earthquake. One approach used by Chen 124 is to deal with the identification of structural characteristics only in portion (b) by dividing this portion into several segments in order to study and compare the changes among these characteristics. This method has been applied to analyse response records of two buildings collected during the 1971 San Fernando Valley earthquake 12~. Results of Chen's method agreed with those of the modal minimization method by Beck 116 Toussi 113 developed a nonparametric identification method for a multi-storey building frame, where the relative acceleration )~(t) and the applied force f(t) were available as recorded data. In between local maximum and minimum values, polynomial functions were used to describe the relationships between (a) the damping force fp(t) and the total velocity 2(0 and (b) the spring force fk(t) and the total displacement x(t) 117.128. The response data of two test structures 129'13o were used to evaluate the effectiveness and applicability of the proposed method. Toussi then applied his hysteresis identification method to estimate the inter-storey load-deflection relationships of the frame. According to Toussi's model, as the intensity of the earthquake excitation increased, more nonlinearity in the structural behaviour appeared. Another interesting observed feature was the 'soft-to-stiff' type of behaviour that the structure experienced under different levels of load. Finally, the identified loaddeflection curves became rather wide area-wise which indicated energy dissipation. Iwan and Cifuentes~31 developed a physically motivated model for the gross behaviour of degrading hysteretic structures subjected to damaging earthquake motions by accurately modelling the restoring force behaviour of the structure. The restoring force was modelled by a series of linear and elasto-plastic elements. Stiffness degradation was achieved by elasto-plastic elements which would 'break' when their displacement exceeded a certain limiting value. The parameters of the model elements were determined from experimental data such that the difference between the gross restoring force of the real structure and the model was minimized. Using an indentified model developed using this approach, the response of a structure to different excitations may be examined. 6. A P A R T I C U L A R M O D E L F O R D A M A G E

ACCUMULATION In 1983, Stephens and Yao 13z reviewed available information concerning (a) real-world structures in actual earthquakes, (b) full-size structural systems in controlled tests, and (c) small-size to full-size models of structural systems in laboratory experiments. On the basis of a low-cycle failure criterion 133, Stephens ~34 developed the following damage function:

,=i L \ A ' $ . I ) J,

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where the fatigue damage exponent, a, is given by the expression 1 - ( b ' r ) for which b is the deformation ratio coefficient and r is the relative deformation ratio. The relative deformation ratio is defined as the ratio of the negative change in plastic deformation in cycle i, A6pc, to the positive change in plastic deformation in cycle i, A6pt; and A6pl is the positive change in plastic deformation in a one-cycle test to failure conducted at the relative deformation ratio of cycle i. The value of the cumulative damage, D, is postulated to range from 0 to 1.0 with a value of 0 corresponding to no damage (safe), and a value of 1.0 to failure (critically damaged). To develop such a damage function for structural systems, it was assumed that deformation under reversed cyclic load in one direction produced 'indirect damaging response' (e.g., 'less-than-ultimate' compression strains) at some locations and 'direct damaging response' (e.g., tension strains) at other locations in the structure. The response condition at each location reversed when the direction of deformation reversed. Effects of 'indirect damaging response' were considered through the cycleshape-dependent parameters r and A6pl. Damage was accumulated independently in each deformation direction for both forms of the expressions. Total damage was conservatively estimated as the larger value of the damage indices as calculated in both directions. Several acceleration records of measured structural response to earthquake ground motions were processed and analysed to estimate the force-deformation response of the structure. Building structures studied included laboratory test models 129,135 and the Imperial County Services Building 136. Information from the forcedeformation response was then substituted into Stephens' damage function ot obtain quantitative measures of the damage condition of the structures TM. In addition, the damage of the structures was compared to the damage estimated by an index proposed by Park 13v The damage indices obtained for these structures were also correlated with independently formulated descriptive damage assessments of the type safe, lightly damaged, damaged, and critically damaged 138. Based on these correlations, the damaged functions produced reasonable and potentially useful measures of the damage conditions of the structures. However, the available data were considered to be insufficient to reliably determine the specific correlation between damage index and damage state x34 7. S U M M A R Y O F RESEARCH N E E D S

In this paper, an attempt has been made to review the state-of-the-art in mathematical modelling of structural behaviour during earthquakes. Owing to the massive amount of literature which pertain to this area, the treatment was necessarily brief. Only those mathematical techniques which have been shown useful for the random earthquake response analysis of structures, particularly in the nonlinear regime, were discussed. Numerical work incorporating some of these solution techniques for hysteretic systems, or for systems having parametric uncertainties, was presented. Because of space limitations, the important question of nonlinear constitutive modelling was omitted in this discussion, with the exception of a brief introduction to hysteretic models. System identification techniques which are applicable to earthquake structural analysis were

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M a t h e m a t i c a l modelling o f structural behaviour during earthquakes: L. J. B r a n s t e t t e r et al.

introduced, along with one example of a d a m a g e index t h r o u g h which c o m p u t e d response quantities m a y be used to estimate damage. There is a need for further research on the r a n d o m a n d n o n l i n e a r response of structures. It is difficult to o b t a i n precise representations of (a) the physical structure a n d its properties a n d (b) the expected future loading. F o r structures which a c c u m u l a t e d a m a g e d u r i n g strongm o t i o n earthquakes, the response range m a y fall outside limits of applicability of available linear or slightly n o n l i n e a r solution techniques. M e t h o d s for h a n d l i n g the moderate-to-highly n o n l i n e a r system are necessary. Also, the n o n l i n e a r response b e h a v i o u r of such structures is critically d e p e n d e n t on the imprecisely defined l o a d i n g history. Reliable prediction of future e a r t h q u a k e loads is complicated by the lack of sufficient response data, either for actual structures or l a b o r a t o r y models. M u c h work r e m a i n s to be d o n e in estimating the a c c u m u l a t e d d a m a g e a n d the reliability of actual structural systems. Existing a n d deteriorating structures are extremely complex systems for which the d a m a g e state is difficult to evaluate. D a m a g e assessment is by n a t u r e a subjective process which does n o t lend itself to precise numerical results. T h e application of expert systems to this area has been suggested in c o n j u n c t i o n with system identification techniques in which m a n y sources of data, calculation, a n d other i n f o r m a t i o n c o n c e r n i n g the structure may be considered 139. Practical i m p l e m e n t a t i o n of such expert systems remains to be a challenging task. This review is n o t i n t e n d e d to be a n exhaustive c o m p i l a t i o n of all i m p o r t a n t work d o n e to date in this subject area. However, it is hoped that this p a p e r m a y serve as a valuable reference for those investigators who are interested in a wide range of m e t h o d s a n d approaches for c o n s t r u c t i n g m a t h e m a t i c a l models of structural b e h a v i o u r d u r i n g strong earthquakes.

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Conference on Earthquake Engineering, 1965, I1, 783 796 Ramberg, W. and Osgood, W. R. Description of Stress-Strain Curves by Three Parameters, NACA Technical Note, TN-902, 1943 Bouc, R. Forced Vibration of Mechanical Systems with Hysteresis, Abstract, Proceedings of the Fourth Conference on Non-linear Oscillation, Prague, Czechoslavakia, 1967 Wen, Y. K. Method for Random Vibration of Hysteretic Systems, J. Engr. Mech. Div., ASCE, 1976, 102(EM2), 249-263 Baber, T. T. and Wen, Y. K. Stochastic Equivalent Linearization for Hysteretic, Degrading Multistorey Structures, Civil Engineering Studies, Structural Research Series No. 471, University of Illinois, Urbana, IL, 1979 Baber, T. T. and Wen, Y. K. Random Vibration of Hysteretic Degrading Systems, J. Engr. Mech. Div., ASCE, 1981, 107(EM6), 1069-1089 Baber, T. T. and Wen, Y. K. Stochastic Response of Multistorey Yielding Frames, Earthquake Engineering and Structural Dynamics, 1982, 10(3), 403416 Atalik, T. S. and Utku, S. Stochastic Linearization of Multidegree of Freedom Nonlinear Systems, Earthquake Engineering and Structural Dynamics, 1976, 4(4), 411420 Baber, T. T. Nonzero Mean Random Vibration of Hysteretic Systems, J. Engr. Mech. Div., ASCE, 1984, ll0(EM7), 1036 1049 Iwan, W. D. and Mason, A. B., Jr. Equivalent Linearization for Systems Subjected to Non-stationary Random Excitation, Intl. J. Non-Linear Mech., 1980, 15, 7 1 4 2 Kazakov, I. E. Statistical Analysis of Systems of MultiDimensional Non-Linearities, Automation and Remote Control, 1965, 26, 458464 Nabavi-Noori, M. and Baber, T. T. Random Vibration of Degrading Systems with General Hysteresis, Rept. No. UVA/ 526378/CE85/104, Department of Civil Engineering, University of Virginia, 1984 Spanos, P.-T. D. Stochastic Linearization Method for Dynamic Systems with Asymmetric Nonlinearities, Rept. EMRL No. 1126, Engr. Mech. Res. Lab., University of Texas at Austin, TX, 1978 Noori, M., Choi, J.-D. and Davoodi, H. Zero and Nonzero Mean Random Vibration Analysis of a New General Hysteresis Model, Prob. Engr. Mech., 1986, 1(4), 192-201 Chang, T.-P., Mochio, T. and Samaras, E. Seismic Response Analysis of Nonlinear Structures, Prob. Enqr. Mech., 1986, i(3), 157-166 Jahedi, A. and Ahmadi, G. Application of Wiener-Hermite Expansion to Nonstationary Random Vibration of a Duffing Oscillator, J. Appl. Mech., ASME, 1983, 50(2), 436442 Bergman, L. A. and Spencer, B. F., Jr. Solution of the First Passage Problem for Simple Linear and Nonlinear Oscillators by the Finite Element Method, T. and A.M. Rept. No. 461 (UILU-ENG-83-6007), University of Illinois, Urbana, IL, 1983 Spanos, P.-T. D. Formulation of Stochastic Linearization for Symmetric or Asymmetric MDOF Nonlinear Systems, J. Appl. Mech., ASME, 1980, 47(I), 209-211 Baber, T. T. Nonzero Mean Random Vibration of Hysteretic Frames, Computers and Structures, 1986, 23(2), 265-277 Frisch, V. Wave Propagation in Random Media. In Probabilistic Methods in Applied Mathematics, (Ed. A.T. Bharucha-Reid), Academic Press, NY, 1968, Vol. 1 Tikhinov, V. I. Fluctuation Action in the Simplest Parametric Systems, Automat. Remote Control, 1958, 19, 705-711 Kozin, F. On the Probability Densities of the Output of Some Random Systems, J. Appl. Mech., 1961, 28(2), 161-165 Bogdanoff, J. L. and Chenea, P. F. Dynamics of Some Disordered Linear Systems, Intl. J. Mech. Sci., 1961,3, 157 169 Soong, T. T. and Bogdanoff, J. L. On the Natural Frequencies of a Disordered Linear Chain of N Degrees of Freedom, Intl. J. Mech. Sci., 1963, 5, 237-265 Boyce, W. E. Random Eigenvalue Problems. In Probabilistic Methods in Applied Mathematics, (Ed. A.T. Bharucha-Reid), Academic Press, 1968, 1, 1-73 Collins, J. D. and Thomson, W. T. The Eigenvalue Problem for Structural Systems with Statistical Properties, AIAA Journal, 1969, 7, 642~i48 Hart, G. C. Eigenvalue Uncertainty in Stressed Structures, J. Engr. Mech. Div., ASCE, 1973, 99(EM3), 481494 Collins, J. D., Kennedy, B. and Hart, G. C. Bending Vibrational Data Accuracy Study, Tech. Rept. No. 70-1066, J.H. Wiggins, Co., 1970 Hart, G. C. and Collins, J. D. The Treatment of Randomness in

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99

I00 101 102 103 104 105 106 107 108 109 110 111 112

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Finite Element Modeling, SAE Shock and Vibration Symposium, Los Angeles, CA, 1970 Hasselman, T. K. and Hart, G. C. Modal Analysis of Random Structural Systems, J. Engr. Mech. Div., ASCE, 1972, 98(EM3), 561-579 Hasselman, T. K. and Hart, G. C. Solution of the Structural Dynamics Eigenproblem by Modal Synthesis: Sensitivity to Coordinate Selection and Parameter Variation, UCLA Tech. Rept., UCLA-ENG-7239, University of California at Los Angeles, Los Angeles, CA, 1972 Schiff, A. J. and Bogdanoff, J. L. An Estimator for the Standard Deviation of a Natural Frequency - Part I and Part II, Paper No. 71-WA/APM-7 and 8, ASME, Winter Annual Meeting, Washington DC, November 28-December 2, 1971 Schaefer, R. D. Randomness of Lengths in Finite Element Structural Analysis, Masters Thesis, University of California at Los Angeles, Los Angeles, CA, 1971 Shigenura, T. M. Static Response of Random Propertied Structures, Masters Thesis, University of California at Los Angeles, Los Angeles, CA, 1971 Collins, J. D. The Eigenvalue Problem for Systems with Statistical Properties, PhD Thesis, University of California at Los Angeles, Los Angeles, CA, 1967 Prasthofer, P. H. Dynamic Response of Structures with Statistical Uncertainties in Their Stiffnesses, SLL-73-0261, Sandia National Laboratories, Livermore, CA, 1973 Bennett, R. Reliability of Protective Structures, PhD Thesis, School of Civil Engineering, University of New Mexico, Albuquerque, NM, 1985 Chang, F. Probabilistic Dynamics of Elasto-Plastic Structures, PhD Thesis, School of Civil Engineering, University of New Mexico, Albuquerque, NM, 1985 Branstetter, L. J. and Paez, T. L. Dynamic Response of Random Parametered Structures with Random Excitation, SAND851175, Sandia National Laboratories, Albuquerque, NM, 1986 Ibrahim, R. A. Structural Dynamics with Parameter Uncertainties, Appl. Mech. Rev., 1987, 40(3), 309-328 Eykhoff, P. System Identification-Parameter and State Estimation, John Wiley and Sons, 1974 Sage, A. P. and Melsa, J. L. System Identification, Academic Press, 1971 Hart, G. C. and Yao, J. T. P. System Identification in Structural Dynamics, J. Engr. Mech. Div., ASCE, 1977, 103(EM6), 10891104 Rodeman, R. Estimation of Structural Dynamic Model Parameters, PhD Thesis, School of Civil Engineering, Purdue University, W. Lafayette, IN, 1974 Ting, E. C., Chen, S. J. H. and Yao, J. T. P. System Identification, Damage Assessment, and Reliability Evaluation of Structures, Tech. Rept. No. CE-STR-79-1, School of Civil Engineering, Purdue University, W. Lafayette, IN, 1978 Toussi, S. System Identification Methods for the Evaluation of Structural Damage, PhD Thesis, School of Civil Engineering, Purdue University, W. Lafayette, IN, 1982 Natke, H. G. and Yao, J. T. P. Research Topics in Structural Identification, Proceedings of the Third Conference on Dynamic Response of Structures, ASCE, Los Angeles, CA, 1986, 542-550 Gersch, W. Parameter Identification: Stochastic Process Techniques, Shock and Vibration Digest, 1975 Natke, H. G. and Yao, J. T. P. System Identification Approach in Structural Damage Evaluation, Tech. Rept. No. CE-STR-8621, School of Civil Engineering, Purdue University, W. Lafayette, IN, 1986 Distefano, N. amd Pena-Pardo, B. System Identification of Frames Under Seismic Loads, ASCE National Structural Engineering Meeting, New Orleans, LA, 1975 Distefano, N. Some Numerical Aspects on the Identification of a Class of Nonlinear Viscoelastic Materials, ZAMM, 1972, 52,389 Distefano, N. and Todeschini, R. Modeling Identification and Prediction of a Class of Nonlinear Viscoelastic Materials, Int. J. Solids Struct. 1974, 9(I), 805-818; 1974, 9(II), 1431-1438 Marmarelis, P.-Z. and Udwadia, F. E. The Identification of Building Structural Systems - II. The Nonlinear Case, Bull. Seismol. Soc. Am., 1976, 66(1), 153-171 Udwadia, F. E. and Marmarelis, P.-Z. The Identification of Building Structural Systems - I. The Linear Case, Bull. Seismol. Soc. Am., 1976, 66(1), 125-151 Ibanez, P. Review of Analytical and Experimental Techniques for Improving Structural Dynamic Models, Bulletin 249, Welding Research Council, New York, June 1979

123 124 125

126 127 128

129

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Liu, S. C. and Yao, J. T. P. Structural Identification Concept, J. Struc. Div., ASCE, 1978, 104(ST12), 1845-1858 Chen, S. J. H. System Identification and Damage Assessment of Existing Structures, PhD Thesis, School of Civil Engineering, Purdue University, W. Lafayette, IL, 1980 Foutch, D. A., Housner, G. W. and Jennings, P. C. Dynamic Responses of Six Multistorey Buildings During the San Fernando Earthquake, Rept. No. EERL 75-02, California Institute of Technology, Pasadena, CA, 1975 Beck, J. L. Determining Models of Structures from Earthquake Records, Rept. No. EERL 78-01, California Institute of Technology, Pasadena, CA, 1978 Masri, S. F. and Caughey, T. K. A Nonparametric Identification Technique for Non-Linear Dynamic Problems, J. Appl. Mech., ASME, 1979, 46(2), 433~,47 Masri, S. F. and Anderson, J. C. Identification/Modeling Studies of Non-Linear Multidegree Systems, Vol. 3, Analytical and Experimental Studies of Non-Linear System Modeling, Progress Report AT (49-24-0262), US Nuclear Regulatory Commission, 1980 Healey, T. J. and Sozen, M. T. Experimental Study of the Dynamic Response of a Ten-Story Reinforced Concrete Frame with a Tall First Story, Civil Engineering Studies, Structural Research Series No. 450, University of Illinois, Urbana, IL, 1978 Cecen, H. Response of Ten-Story Reinforced Concerete Frames to Simulated Earthquakes, PhD Thesis, School of Civil Engineering, University of Illinois, Urbana, IL, 1979 Iwan, W. D. and Cifuentes, A. O. A Model for System Identification of Degrading Structures, Earthquake Engineering and Structural Dynamics, 1986, 14(6), 877-890 Stephens, J. E. and Yao, J. T. P. Survey of Available Structural Response Data for Damage Assessment, Tech. Rept. No. CESTR-83-23, School of Civil Engineering, Purdue University, W. Lafayette, IN, 1983 Yao. J. T. P. and Munse, W. H. Low-Cycle Fatigue Behavior of Mild Steel, Special Technical Publication No. 338, ASTM, 1962, 5-24 Stephens, J. E. Structural Damage Assessment Using Response Measurements, PhD Thesis, School of Civil Engineering, Purdue University, W. Lafayette, IN, 1985 Okamoto, S. et al. A Progress Report on the Full-Scale Seismic Experiment of a Seven Story Reinforced Concrete Building Part of the US Japan Cooperative Program, Building Research Institute, Ministry of Construction, Japan, 1982 McJunkin, R. D. and Ragsdale, J. T. Compilation of StrongMotion Records and Preliminary Data from the Imperial Valley Earthquake of 15 October 1979, Prelim. Rept. No. 26, Office of Strong-Motion Studies, California Division of Mines and Geology, Sacramento, CA, 1980 Park, Y. J. Seismic Damage Analysis and Damage-Limiting Design of R.C. Buildings, Report No. UILU-ENG-84-2007, Structural Research Series No. 516, University of Illinois, Urbana, IL, 1984 Toussi, S. and Yao, J. T. P. Assessment of Structural Damage Using the Theory of Evidence, Structural Safety, 1982, 1(2), 107121 Yao, J. T. P. Safety and Reliability of Existing Structures, Pitman Advanced Publishing Program, Boston, 1985

APPENDIX

I

EQUIVALENT

LINEARIZATION

METHODS

Buildings and structures generally become nonlinear and g o i n t o inelastic r a n g e b e f o r e d a m a g e occurs. E x a c t s o l u t i o n of r e s p o n s e statistics in t h e n o n l i n e a r r a n g e u n d e r r a n d o m e x c i t a t i o n is in g e n e r a l difficult. A m o n g the a p p r o x i m a t e m e t h o d s , t h o s e b a s e d o n an e q u i v a l e n t l i n e a r i z a t i o n t e c h n i q u e h a v e p r o v e d to b e m o s t useful in t e r m s of a p p l i c a t i o n s to p r a c t i c a l m u l t i - D . O . F , systems. T h e p u r p o s e o f this p a p e r is to give a b r i e f r e v i e w of t h e theoretical background of t h e m e t h o d and its a p p l i c a t i o n s in e a r t h q u a k e e n g i n e e r i n g , e s p e c i a l l y to inelastic a n d h y s t e r e t i c s y s t e m s , i.e., r e s p o n s e a n d d a m a g e p r e d i c t i o n . F o r m o r e c o m p r e h e n s i v e r e v i e w of the m e t h o d o l o g y a n d a p p l i c a t i o n s , r e a d e r s a r e referred to S p a n o s 1.

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141

Mathematical modelling of structural behaviour during earthquakes." L. J. Branstetter et al. THEORY

equations may be set up for the mean responses.

The equivalent linearization method was first developed for random vibration analysis by Booton 2 and Caughey 3 and has since been extended for more complex systems with varied nonlinear restoring forces including hysteretic systems. The theoretical basis of this method is the replacement of the nonlinear system by an equivalent linear one such that the means square difference between the two systems (i.e., equations of motion) is minimized. Recent work by Iwan et al. 4'5 have laid further theoretical foundation for this method for stationary as well as nonstationary analysis. The linearized system coefficients so obtained are in general functions of the response variable covariance matrix and other statistics involving the restoring force. They may be in an implicit form for general random excitation which could hamper the solution procedure, especially for multi-D.O.F, systems. Explicit solutions of the coefficient may be obtained if the excitation is a zeromean Gaussian process 6'7. The equation of motion of a general multi-D.O.F, nonlinear system may be given by g(~,~,x) = f(t)

(1)

in which g = a vector representing the total internal force acting on the system, being single valued and odd; and f = zero-mean Gaussian vector process. Equation (1) is replaced by a linear system with the following equations of motion M/t + Cx + Kx = f(t)

(2)

in which the elements in the mass, damping and stiffness matrices are determined by the requirements that the mean square error E[~re] is minimized, where = g -

M/~

-

Cx

-

Kx

(3)

The necessary conditions for the minimum are that the partial derivatives of E[,%] with respect to the undetermined coefficient vanish. To find the expectation, however, the joint density of the response variables is required, which is yet unknown and in general nonGaussian. If as an further approximation, the joint density function is assumed to be Gausian, because of the Gaussian excitation one obtains the following solutions

in which the coefficients can be expressed explicitly in terms of the response statistics for most nonlinear systems without difficulty. Since M, C and K are functions of the response statistics, an iterative solution procedure is generally required. In this regard, the solution to corresponding linear system may be used to start the iteration. Alternatively, a set of ordinary differential equations of the response covariance matrix may be obtained, which reduces to a matrix equation and can be solved iteratively for stationary solution and integrated numerically for nonstationary solution. The extension of the method for system under nonzero-mean excitation has been developed 8 where a separate set of differential

142

APPLICATIONS The kind of nonlinear systems that are amenable to the method of equivalent linearization are really quite general. In most applications, it is assumed that the restoring force vector g are algebraic functions of the displacement and velocity. Nonlinearities associated with the inertial force as well as self-oscillatory systems have been considered 7'9. The restriction that the function has to be an anti-symmetric (odd) function of its argument has also been relaxed 1o. A special class of nonlinear systems which is of particlar interest in study of performance of structure under earthquake loads is that of inealstic systems. However, a major difficulty in treating inelastic system is the hereditary behaviour of the restoring force, i.e., it depends on the response time history. Invoking a slowly varying parameters (Krylov-Bogoliubov, K-B) assumption, Caughey 11 obtain the linearized system coefficient in integral form. However, the K-B assumption is tantamount to that of the system being narrow banded which is only true for systems with small to moderate inelastic response 12. Also, it prohibits drifts in the system and may cause serious underestimates in the r.m.s, response. A recent method of modelling the hysteretic system by a nonlinear differential equation proves to be versatile and efficient in reproducing structural hysteretic behaviour. It is capable of modelling system deterioration and biaxial interaction and is compatible with the equivalent linearization method that system coefficients can be obtained in close form without resort to the K-B assumption 13'14. As this method of modelling and solution procedure has been successfully applied to response and damage prediction of a variety of structural systems under seismic excitation. Some details of this method is given in the following. M O D E L L I N G , RESPONSE AND D A M A G E P R E D I C T I O N O F INELASTIC HYSTERETIC SYSTEMS Consider a single-D.O.F. (degree-of-freedom) system. Details of this model and extension to multi-D.O.F. systems can be found in Refs 12 and 13. The restoring force is given by

Q(x, ~, t) = 9(x, 5:) + h(x)

t5)

in which g = a nonhysteretic component, an algebraic function of the instantaneous x and ~. h = a hysteretic component, a function of the time history of x. As an example, the restoring force of a nearly elasto-plastic system may be modelled by

Q (x, t) = ~kx + (1 - ~)kz

(6)

in which k = t h e pre-yielding stiffness; a = ratio of postyielding stiffness to pre-yielding stiffness; and ( 1 - ~)kz= the hysteretic part of the restoring force in which z is described by the following nonlinear differential equation

Probabilistic Engineering Mechanics, 1988, Vol. 3, No. 3

t

~= ~7[A~-v{~l~llzl"-'z-v~lzl")]

(7)

with the force parameters A, r, 7, t/, v and n governing the

Mathematical modellin9 of structural behaviour durin9 earthquakes: L. J. Branstetter et al. amplitude, shape of the hysteretic loop, and the smoothness of transition from elastic to inelastic ranges. Proper choices of parameters give various softening as well as hardening systems. Degradation of the restoring forced can be included by prescribing the parameters to be functions of the severity of the response, such as amplitude of the response and total hysteretic energy dissipation. The hysteretic energy dissipation which is a measure of the cumulative effect of severe response and repeated oscillations, is given by,

eT(t) = (1 - ~)k

z(z))~(z)dz

(8)

The above model has been extended to include the pinching of the hysteresis loops as well 15 For two-dimensional structures under biaxial excitations, the interaction of the restoring forces in the two directions may significantly alter the response behaviour. The above restoring force has been extended to include such interaction by requiring that the hysteretic components in the two directions zx and zr satisfy the following coupled differential equations (16) zx = A f i x - Blaxzxlz

- 7axZ

-

la,z, lzx - va::

zr = atiy - fllfiyzylzy - yurzy - fllUxZ~lZy-?UxZxZy •

2

•

•

(9)

in which ux and uy are, respectively, the displacement in the x and y direction. Comparison of this model with analytical 17 and experimental is studies of biaxial system indicate that it successfully reproduces all the essential features of a biaxial restoring force, with or without degradation. It is pointed out here that although equations (7) and (9) are for restoring force and displacement and phenomenological in nature, they correspond closely to the rate-type constitutive equations in plasticity therefore do have a sound theoretical basis. This also explains why such model works so well. It has been applied to continuous systems in a finite element context by Mochio and Shinozuka 19. This method of modelling is especially suited for equivalent linearization method. For example, for a S.D.O.F. system with a restoring force given by equations (6) and (7) ( n = r / = v = 1) under zero-mean, Gaussian excitation, application of equations (3) and (4) gives the auxiliary equation.

~+cl~+c2z=O

(10)

in which,

ca =

~- ~E(~z)+fla z] - A

~/ ~ k

(11)

a~

c2= ~--[ya~+fl E(~z) ] ~L az J

(12)

a~=~, az= E x i t , and E[ ]=expected value. Note that the coefficients are given in closed form and no additional approximation such as the K-B assumption is required. The resulting system is a 3rd-order linear oscillator. It can be shown that unlike a second order system, such a third order system allows 'drift' in the

response, therefore preserving one of the most important properties of an inelastic system. It can be shown 13,14 that under a zero-mean excitation modelled by a filtered Gaussian shot noise the one-time response variable (displacement, velocity and the hysteretic part of the restoring force) covariance matrix IS] of the linearized system satisfies the matrix ordinary differential equation

d[S] dt

F[G][S]+[S][G]'=[B]

(13)

in which [G] is the matrix of structural system (including linearization coefficients) and excitation (filter) parameters, t indicates transpose. [B] is a matrix of the expected values of the product of the response vector and the shot noise excitation. The stationary solution for nondeteriorating system is obtained by solving equation (13) with d[S]/dt=O. For nonstationary solution (including degrading systems) a computer routine is needed to solve equation (13). General purpose routines available in ordinary differential equation routine package had been found satisfactory for this purpose. The power spectral density matrix of the response variables (in the case of stationary response) can be obtained through an eigen analysis of the matrix [G]. Note that this can be performed only after the linearized system is determined, since [G] contains linearization coefficients that depend on response statistics. A response quantity that is particularly useful for predicting potential structural damage is the total hysteretic energy dissipation gr(t) of equation (8). The mean value of eT(t ) is t

E[er(t)]=(1-a)k

i E[z(z)~(z)] dz

do

(14)

A more general measure of the cumulative damage may be given by

E[Dr(t)]=c

f0

E[z(z)m~(z)]dz

/ m + 2 \ 2,.+2 /~r

=cF --

--

(15)

where m allows one to model the effect of response (or stress) amplitude as in fatigue damage. As E(zY¢) is part of the solution of [S], E[er(t)] and E[Dr(t)] can be easily evaluated. Evaluation of the variance of er(t) and Dr(t) requires the covariance matrix between two time instants 2°. The applications of the hysteretic energy based damage measure in combination with large deflection to damage prediction of reinforced concerete, masonry buildings and soil deposit have been successfully demonstrated by Park et al. 21, Kwok 22, and Pires et al. 23. The method of analysis has been extended to system under nonzero-mean excitation and for systems with pinched hysteresis 14,24. ACCURACY OF THE METHOD Comparisons of the results based on equivalent linearization with exact solution (e.g., from Fokker Planck equation) and Monte Carlo simulations indicate that the accuracy of this method is generally very good.

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143

Mathematical modelling o f structural behaviour during earthquakes: L. J. Branstetter et al. An error in the range of 0 ~ 2 0 percent is representative. F u r t h e r m o r e , unlike most a p p r o x i m a t e methods, the accuracy of this m e t h o d is quite i n d e p e n d e n t of the severity of the n o n l i n e a r i t y ; be it of the geometric or material source. However, experience indicates that the errors tend to be on the u n c o n s e r v a t i v e side, i.e., the statistical m o m e n t s of the response are usually underestimated. It must also be noted that since the linearization is based on m i n i m i z a t i o n of the m e a n square errors, the prediction of the response statistics other than the m o m e n t s (covariance matrix), such as extreme value a n d auto correlation function m a y not be as reliable. It appears that the accuracy of the m e t h o d can be improved, if instead of assuming a G a u s s i a n d i s t r i b u t i o n for the response variable, one uses a G r a m - C h a r l i e r type expansion to include higher order term ( n o n - G a u s s i a n ) effect 2s~27. However, the ensuing p r o b l e m is that the equations to be solved multiply at a fast rate that serious c o m p u t a t i o n a l difficulty may arise for systems with even a small n u m b e r of degrees of freedom 25 a n d direct M o n t e Carlo simulations may prove to be more economical.

11 12 13 14 15 16

17

18

SUMMARY OF RESEARCH NEEDS The m e t h o d of equivalent linearization has u n d e r g o n e signficant changes in the recent past, i.e., more rigorous theoretical f o u n d a t i o n has been developed a n d there have been more applications to complex n o n l i n e a r systems, especially inelastic systems. The current state of this m e t h o d seems healthy a n d promising. Additional works are needed to further refine this powerful m e t h o d a n d define its limits such that it can be properly used in evaluation of even a wider class of engineering systems u n d e r severe hazards such as e a r t h q u a k e motions. In particular, further developments for nonlinear c o n t i n u o u s systems a n d systems u n d e r vector excitation (e.g., bi-axial, m u l t i - c o m p o n e n t e a r t h q u a k e excitation) are needed. Also, there is a need for refinement of the m e t h o d such that accurate extreme response statistics can be o b t a i n e d without excessive c o m p u t a t i o n a l effort.

19

20 21 22 23

24 25

REFERENCES 1 2 3 4 5 6 7 8 9

Spanos, P. D. Stochastic linearization in structural dynamics, Applied Mechanics Review, January 1981, 34(2) Booton, R. C. The analysis of nonlinear central systems with random inputs, IRE Trans. Circuit Theory, 1, 23-24 Caughey,T. K. Equivalent linearization technique, Journal of Acoustical Society of America, 35(11), 1706-1711 Iwan, W. D. A generalization of the concept of equivalent linearization, International Journal of Nonlinear Mechanics, 1973, 8, 279-287 Spanos, P. D. and Iwan, W. D. On the existenceand uniqueness of solution generated by equivalent linearization, International Journal of Nonlinear Mechanics, 1978, 13, 71-78 Atalik, T. S. and Utku, S. Stochastic linearization of multidegree-of-freedom nonlinear systems, Earthquake Engineering and Structural Dynamics, 1976, 4, 411-420 Kazakov, I. E. Statistical analysis of systems with multidimensional non-linearities, Automation and Remote Control, 1965, 26, 458~,64 Spanos, P. T. D. Formulation of stochastic linearization for symmetric or asymmetric M.D.O.F. nonlinear systems, Journal of Applied Mechanics, Trans. of ASME, 1980, 47,209-211 Caughey,T. K. Responseof Van Der Pol's oscillator to random excitation, Journal of Applied Mechanics, Trans.of ASM E, 1959,

81,345-348

26 27

Journal of Waterwaysand Harbours Division, Proc. ASCE, 1967, 93, 126-156 Caughey,T. K. Random excitation of a system with bilinear hysteresis, Journal of Applied Mechanics, Trans. of ASCE, 1960, 27, 649~52 lwan, W. D. and Lutes, L. O. Responseof the bilinear hysteretic system to stationary random excitation, Journal of the Acoustical Society of America, 1968, 43, 545 552 Baber,T. T. and Wen, Y. K. Random vibration of hysteretic degrading systems, Journal of Engineering Mechanics, ASCE, December 1981, 107(EM6), 1069-1087 Wen, Y. K. Equivalent linearization for hysteretic systemsunder random excitation, Journal of Applied Mechanics, Trans. of ASME, 1980, 47, 150-154 Baber,T. T. and Noori, M. N. Random vibration of degrading, pinching systems, Journal of Engineering Mechanics, ASCE, August 1985 Park, Y. J., Wen, Y. K. and Ang, A. H.-S. Random vibration of hysteretic system under bi-directional ground motions, Earthquake Engineering and Structural Dynamic, 1986, 14,548557 Chen, P. F.-S. and Powell, G. H. Generalized plastic hinge concepts for 3D beam-column elements, Report No. UCB/EERC 82-20, Earthquake Engineering Research Center, University of Califomia, Berkeley, CA, November 1982 Otani, S. Cheung, V. W.-T. and Lai, S. Reinforced concrete columns subjected to biaxial lateral load reversals, Proc., 6th World Conf. Earthquake Engineering, New Delhi, India, 1977, 525 532 Mochio, T. and Shinozuka, M. Stochastic equivalent linearization in a finite element-based reliability analysis, Proc., 4th International Conference on Structural Safety and Reliability, Kobe, Japan, May 1985 Wen, Y. K. Stochastic response and damage analysis of inelastic structures, International Journal of Probabilistic Engineering Mechanics, 1985, 1(1) Park, Y. J., Ang, A. H.-S. and Wen, Y. Seismicdamage analysis of reinforced concrete buildings, Journal of Structural Engineering, Proc. ASCE, April 1985, 111(4) Kwok,Y. H. Seismicdamage analysis and design of unreinforced masonry buildings,PhD Thesis,University of Illinoisat UrbanaChampaign, Urbana, Illinois, March 1987 Pires,J. E. A., Wen, Y. K. and Ang, A. H.-S. Stochastic analysis of liquefaction under earthquake loading, Civil Engineering Studies, Structural Research Series No. 504, University of Illinois at Urbana-Champaign, Urbana, Illinois, April 1983 Baber,T. T. Nonzero mean random vibration of hysteretic system, Journal of Engineering Mechanics, ASCE, July 1984, 110(7), 1036-1049 Beaman, J. J. and Hedrick, J. K. Improved statistical linearization for analaysis and control of nonlinear stochastic systems, Part I: an extended statistical linearization technique, Journal of Dynamic Systems, Measurement and Control, Trans. ASME, March 1981, 103, 15-27 Crandall, S. H. Non-Gaussian closure for random vibration of nonlinear oscillators, International Journal of Nonlinear Mechanics, 1980, 15, 303-313 Suzuki,Y. and Minai, R. Seismicreliabilityanalysis of hysteretic structures based on stoachastic differentialequations, Proc., 4th International Conference on Structural Safety and Reliability, Kobe, Japan, May 1985, II, 177

APPENDIX II EFFECTS O F V E R T I C A L G R O U N D M O T I O N W h e n c o l u m n - t y p e m e m b e r s are included in a structural system, the analysis of structural response to earthquake excitation becomes m u c h more complicated. This can be explained by n o t i n g that the stiffness property of a c o l u m n depends o n its axial load which, in the case of earthquake, may be separated into a static c o m p o n e n t a n d a d y n a m i c c o m p o n e n t . The latter arises from the vertical g r o u n d acceleration. T h u s , the properties of the structural system is n o longer time-invariant. Even when the entire system remains linear, the elementary spectral analysis a n d c o n v o l u t i o n integral are n o longer

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Mathematical modelling o f structural behaviour during earthquakes: L. J. Branstetter et al.

applicable. The earliest investigations into the effects of vertical ground motion on a single column were conducted by Wirsching and Yao 1, Iyengar and Shinozuka 2, and Gfirpinar and Yao 3. In Ref. 1, use was made of the theoretical results of Ariaratnam 4 for asymptotic stochastic stability of a hinged column under the excitation of a vertical stationary random loading. Ref. 2 was a Monte Carlo approach where earthquake sample functions were generated on the computer and numerical integrations were carried out to obtain the corresponding sample structural response functions in the time domain. In Ref. 3, spectral analysis was performed to obtain the structural response spectra at successive short time intervals. On the other hand, a deterministic analysis was given by Cheng and Oster 5 in which highly irregular vertical ground motion was substituted by a sinusoidal motion. A theoretically more satisfactory formulation was first proposed by Lin and Shih 6. Lumping the mass at the top of a column, it can be shown that the motion of the column is governed by the following equation6: mg ~i'+2(°~°°d;+092( 1 Per

m'" 6

~,-V)=-t~

(1)

where 6 = generalized displacement of the column relative to the ground, e~o = natural frequency of the column if the axial load is ignored, ( o = d a m p i n g ratio, Pc,=static Euler buckling load, m = t h e concentrated mass, 9 = gravitational acceleration, i?= vertical ground acceleration, and U = horizontal ground acceleration. Although tY and 17are both random excitations from the earthquake, they play very different roles in equation (1). The vertical ground acceleration f?, appearing in the coefficient of the unknown 6, is a parametric excitation which can affect the stability of the system, although stability is not a primary concern here since earthquake excitations are limited in duration. On the other hand, the horizontal ground acceleration U, appearing as an inhomogeneous term on the fight hand side of the equation, is the more familiar external excitation encountered in usual random vibration problems. Unfortunately, no general method of solution is known for a dynamic system under both parametric and external random excitations at the present time. However, if the correlation times of both types of excitations are much shorter than the relaxation time ( ~ ( o 1) of the system, then 6 and ~;can be approximated by the components of a Markov vector, and the mathematical tools of Markov processes are applicable. In Ref. 6, the horizontal and vertical ground accelerations t~ and 1? are approximated as white-noises modulated by deterministic time-envelopes. It was found that the vertical ground motion can amplify the power of the horizontal ground motion, and this effect is greater the closer the static component of the axial load m9 is to

the buckling load Pc,. F o r mg < 1/2 Per, the effect of the vertical ground motion is generally negligible. Extension of the above analysis to a multi-storey building was given in Ref. 7. The calculated results showed that for a 6-storey building the presence of vertical ground acceleration could cause an increase in certain response variables in the order of 10%. The percentage is expected to be larger for a taller building. In the design of earthquake endurable structures, deformation to inelastic range must be considered. The effects of vertical ground motion on the response of a hysteretic column was considered in Ref. 8, using a material model due to H a t a and Shibata 9. The numerical results showed the same general trend as that of the linear model but to a much greater degree. The vertical ground acceleration can increase substantially the peak response level due to the horizontal ground acceleration alone. However, the time-variation of the response level is found to be highly individualized; i.e., one hysteretic system can behave very differently from another, particularly the tendency for a permanent deformation to remain after the termination of an excitation. Analyses similar to those of Refs 6 and 7 have also been reported by Ahmadi and his associates 1°'11

REFERENCES 1 2

3 4 5 6

7 8 9 10 11

Wirsching,P. H. and Yao, J. T. P. Random Behavior of Columns, Journal of Engineering Mechanics Division, ASCE, June 1971, 97(EM3), Proc. Paper 8163, 605-618 Iyengar, R. N. and Shinozuka, M. Effect of Self-Weight and Vertical Acceleration on the Behavior of Tall Structures During Earthquake, Earthquake Engineerin9 and Structural Dynamic, 1972, 1 G/irpinar, A. and Yao, J. T. P. Design of Columns for Seismic Loads, Journal of Structural Division, ASCE, September 1973, 99(ST9), Proc. Paper 9978, 1875-1889 Ariaratnam, S. T. Dynamic Stability of a Column under Random Loading, Dynamic Stability of Structures, (Ed. G. Herrman), Pergamon Press, Inc., New York, NY, 1967 Cheng,F. Y. and Oster, K. B. Ultimate Instability of Earthquake Structures, Journal of Structural Division, ASCE, May 1976, 102(ST5), Proc. Paper 12117, 961-972 Lin, Y. K. and Shih, Teng-Yuan Column Response to Horizontal-Vertical Earthquakes, Journal of the EnoineerinO Mechanics Division, ASCE, December 1980, 106(EM6), Proc. Paper 15896, 1099-1109 Lin, Y. K. and Shih, T. Y. Vertical Seismic Load Effect on Building Response, Journal of the Enoineerin# Mechanics Division, ASCE, April 1982, 106(EM2), 331-343 Shih, T. Y and Lin, Y. K. Vertical Seismic Load Effect on Hysteretic Columns, Journal of the Engineerin9 Mechanics Division, ASCE, 1982, 106(EM2), 242-254 Hata, S. and Shibata, H. On the Statistical Linearization Technique for Memory Type Nonlinear Characteristics, Control Enoineering, 1956, 11(10), (in Japanese) Orabi, I. I. and Ahmadi, G. Horizontal-Vertical Response Spectra for El Centro 1940 Earthquake, Report No. MIE-125, Clarkson University, October 1985 Abdel-Rahman,S. Z. and Ahmadi, G. Stability of Frames Subjected to a White Noise Base Excitation, Report No. MIE124, Clarkson University, October 1985

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