Stochastic dynamics of hysteretic media

Structural Safety, 6 (1989) 259-269 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

259

S T O C H A S T I C D Y N A M I C S OF H Y S T E R E T I C M E D I A * F. Casciati Department of Structural Mechanics, University of Pavia, Via Abbiategrasso 211, 27100 Pavia (Italy)

Key words: numerical analysis; plastic analysis; random excitation; soil; structural dynamics.

ABSTRACT In classic plasticity theory the yielding condition is a discontinuity between the elastic and plastic phases. This discontinuity has obvious negative consequences in the mathematical features of the algorithm to be used for the solution of solid and structural mechanics problems. In the framework of endochronic theory smoothed plasticity models, without discontinuities, are available. In particular a three-dimensional tensorial smoothed idealization of the Prager's model is considered in this paper. Moreover multivariate smoothed constitutive laws for beam sections are provided Their use in non-linear stochastic dynamics is discussed

1. INTRODUCTION The non-linear analysis of a medium and its three-dimensional extension are considered. The medium is discretized into 3-D finite elements and a step-by-step integration of the resulting equilibrium (or motion) equations is conducted. The incoveniences of such a procedure are not of minor nature: (i) a time history of the external action has to be assigned, (ii) the computational effort required is still significant even with the current computational facilities; and (iii) the results generally do not provide the engineer with enough information for subsequent decision-making. The computational effort could be significantly reduced by using smoothed forms of the elasto-plastic constitutive law [1]. This would also open the way to the application of stochastic analysis techniques that incorporate the variability of the external actions and give a probabilistic description of the resulting response [2,3]. For structural systems, a further reduction of the computational effort could be obtained by using the appropriate force-displacement relationship at the member level, rather than working * Paper presented at an International Symposiumon Methods of Stochastic Mechanics and Applications, Urbana, IL, U.S.A., October 31-November 1, 1988. 0167-4730/89/$03.50

© 1989 Elsevier Science Publishers B.V.

260 directly with the stress-strain relationship. Thus, the analysis of the structural system is still three-dimensional, but the use of 3-D finite elements is avoided [4,5]. Unfortunately, no experimental results are yet available which may be used to support any model of the elasto-plastic behaviour under cyclic biaxial loads [6]. Section 2 of this paper considers the problem of smoothing the plasticity law. Stochastic equivalent linearization is summarized in Section 3. Section 4, finally, deals with the problem of developing a similar m e t h o d at a structural level. For appropriate numerical examples the reader is referred to Refs. [5,7].

2. S M O O T H E D PLASTICITY

Independently of endochronic theory, with which relations were successively stated [8], Bouc [1] proposed a functional form which provides a mathematical representation of the hysteresis in a system on condition that it does not depend on time (for this extension see Ref. [9]). It was generalized to the multivariate case in Refs. [10,11]. 2.1. Univariate case

Let ,I, denote a generalized force, U a generalized displacement and Z an auxiliary variable. The model proposed by Bouc [1] is written

ff'(t)=a2U(t) + ftlA exp[-fl~tldU/dv, dvl(dU/ds) ds =a2U(t)+Z(t)

fl>O,

A>O

(1)

In eqn. (1) Z(t), i.e. the integral over the interval (to, t), is also the solution of the differential equation

2 + fllOIZ=AO

(2)

Such a relation between U and Z is of hysteretic nature. Improved models for this relation were provided by Bouc [12] (eqn. (3B)), Wen [2] (eqn. (3W)) and the author [13] (eqn. (3C)), respectively, based on eqn. (2):

2+BIOIZ+YOIZI

=At)

2 +/31t)llzl"-lz+

t)lzl" =At)

z, +/31t)llzl"-

z + vt)lzl"-[/311t)lizl"'-1z +/31t)lzl "1] =At)

(3B)

<,,

(3W) (3C)

Equation (3W), for a 2 = 0 (i.e. in the absence of post-yielding hardening), provides a relationship , I , - U which tends to be elasto perfectly plastic as n tends to ~ [2]. Note that the yielding value of q is obtained for Z = 0 as

qlty= [A/(/3 ÷ y)] 1/n

(4)

In this paper attention is focused on n = 1 (and hence n I = 0), i.e. on a s m o o t h e d model of the elasto-plastic behaviour. It is easy to show [14] that the energy dissipated in every hysteresis cycle is positive for /3 + y > 0 and y - / 3 ~< 0 (/3 being positive by assumption). This ensures the

261

positiveness of the energy dissipated over the cycle, i.e. satisfies Drucker's stability postulate of the classic plasticity theory. The so-called complementarity rule ~(xI/) • 0p = 0

(5)

is however violated. Here ~(*) = (,I,- %) = 0

(6)

is the yielding surface and Up is the plastic part of the total displacement

Up= U - Ue

(7)

Ue being the elastic displacement. In the unloading stage, eqn. (5) is satisfied only when "~= ft. In this case the unloading occurs along a straight line of slope .4, and hence, Up = 0. For fl > "~, Up is greater than 0 even when Z decreases (unloading). During the loading stage, eqn. (5) is never satisfied since ~'p is greater than 0 for any fl and ~,, ensuring the positiveness of the dissipated energy. Nevertheless, since Up increases as Z increases, one can regard eqn. (3B) as a smoothed elasto-plastic constitutive law. Its main drawback is that the hysteresis loops during an unload-reload cycle for which * does not change sign do not close. As a consequence, the work done on the material during this cycle is negative and, hence, Drucker's stability postulate is violated [15]. Of course, the entity of this violation can be reduced by increasing n. High values of n make obviously cumbersome a step-by-step integration of the non-linear equilibrium/motion equations; also they reduce the accuracy achievable by using stochastic equivalent linearization techniques [16], Therefore, the author tried to reduce the violation of Drucker's postulate without increasing the exponent n. This is obtained by adding (see eqn. (3C)) a second hysteretic item 12 to the Bouc's constitutive law equation contributed by a single hysteretic device I r This additional item is counterclockwise, as fll > 0 appears with negative sign in eqn. (3C), and is effective only during the loading (~q = ill). During the loading phases, therefore, it gives rise to "negative" inelastic displacements Up with associated negative dissipated energy linearly proportional to U 2 for nl = 0. The energy dissipated by /1 is initially unable to compensate the negative dissipation of 12, but the rate of dissipating energy is higher for/1, since its equation is characterized by an exponent n higher than nl in 12. Therefore, in the initial stage of loading the e n e r g y [fttoZVdt] is that of an elastic system of stiffness (A + 2fll ) rather than A and, in case of immediate unloading, the dissipated energy turns out to be negative. With reference to Fig. 1 it was pointed out [14] that the requirement to satisfy Drucker's stability postulate (AE 1 >/AE 2) in the unload-reload cycle leads to violation in the load-unload cycle. In fact it has been recently concluded by Iwan [17]: "when endochronic models are adopted, local violations of the Drucker's stability postulate cannot be avoided". Therefore, if these smoothed forms of plasticity law are accepted because of their computational advantages, the objective should be the reduction of the violations of Drucker's postulate either by increasing n in eqn. (3W) or, better, by adopting appropriate parameters in the linear form (n = 1, n 1 = 0) of eqn. (3C). The experience of Ref. [13] suggests fl = - 2 y (i.e. Upm > 0 in Fig. 1) in order to reduce load-unload cycle violations and fll = 2 in order to reduce small-amplitude cycle violations: the global result makes AE 1 - AE 2 a small negative quantity but not a positive one as Drucker's postulate would require. About the need of substituting eqn. (3B) with eqn. (3C), it is worth noting that comparisons were made for systems dynamically loaded by a

262

q~=Z

AE2 A

/

C

///

D

U

Fig. 1. Drucker's stability postulate is violated whenever AE 1 - AE 2 < 0.

stationary stochastic external excitation [5] and no significant difference was detected in terms of dissipated energy. However, this is a general result of non-linear r a n d o m vibration (i.e. for stationary excitation the dissipated energy does not depend on the form of the system constitutive law) and, hence, comparisons under non-stationary external actions are presently in progress. In the previous models, the hardening can be included by putting

qff = KHU + Z

(8a)

instead of ,t, = Z and, for instance for eqn. (3W), 2 + fllOllZ] " - I Z + y O l z l " = ( K -

KH)(/

(85)

with K denoting the small-cycle stiffness (initial stiffness; it coincides with A in eqn. (3W) when a 2 = 0), and K H = a 2 = a K (0 < a < 1) the large-cycle stiffness (post-yielding stiffness). The ratio a is called the hardening ratio. Note that eqn. (4) in this case becomes

Xtty = KHXIty/K + [( K - K H ) / ( 3 + It)] 1/n = K( K - g H ) l / n - 1 / / ( ~ + ]/)l/n

(9)

Alternatively one can write ,t, = KHU + ( K - K . ) Z a

(8a')

Za + fla[Za[n-l[(f[Za +,YaO[Za[n= (]

(8b')

,I,, =

+ VaJ,1/,

(9')

In the report [9] and in the successive paper [3,13,16,18,4,5] eqn. (9') is written as ~y = ( K -

KH)/(fla + "Ya),1/,

(9")

It corresponds to a different definition of the yielding level, aiming at defining the amplitude of the loop of the hysteretic item rather than the actual yielding value of plasticity theory. The difference between the values provided by eqns. (9') and (9"), however, is meaningless for low values of the ratio a = K H / K . These low values characterize the constitutive laws eqns. (8a') and

263 (8b') of potential plastic hinges in Refs. [3-5]. By contrast for large values of a, eqn. (9') should be preferred when comparisons with results of elasto-plastic analyses are pursued. 2.2. Multivariate c a s e

According to the classic plasticity theory (Prager's model), the plastic strain tensor c~(t) in any point of an elasto-plastic medium should be computed by integrating along the appropriate path F the plastic strain rate tensor i~ [7] cP(t) =

dr

(10)

The strain rate tensor is a function of the plastic potential, which, in turn, is a function of the stress tensor 04 and of the internal variables ~ j depending on the previous o-e history:

~ = (O0(%j, 7bj)/Oo, j)~

(11)

where the function to be derived (i.e. the plastic potential) coincides with the yielding condition for materials with associate flow rule. The term X is the corresponding plastic multiplier. Equation (11) must be completed by the coherency rule

X ~ O, 4,k = o,

~i)(Oij,tl,j) < 0

(12)

k(oep/o%)~i~ >1o

and by the kinematic hardening rule

~ij =C~P"

(13)

A classical form of O(oij, ~ j ) is the Von Mises model

~(o,,, ,,,)=lls.- ~.11- sM~0

(14)

where Sty denotes the deviatoric part of the stress tensor and

IIs,,- ,,,11 = [(st,- ,,,)(s,,- ,,,)],j2

(15)

In eqn. (14) S M is the yielding constant. Equation (15) leads to

O*/Ooij = ( Stj - W,,)/ll Sij- w,,ll

(16)

o f course, if # denotes the elastic shear modulus,

S,, -= 2t t { eij - }~( Sij -

,7.)/11x.- n.ll}

(17)

where etj is the deviatoric part of the strain tensor. In order to simplifiy the notation, from now on any tensor component Yij is denoted by y, the tensorial product of y by x is marked (y : x) and the position

Y= S - ~ is introduced. In Refs. [10,11] it is noted that:

(18)

264

(i) by the third constraint of eqn. (12) the plastic multiplier is:

X=0

for [ ~ < 0 +=0,

:k>0

forO=0,

(a+/oo).6<0 (0q~/0o): 6>~0

(19a) (19b)

(ii) the consistency condition (i.e. ~ = 0, which is implicit in the third of eqn. (12)) states that during plastic flow (00/ao): 6 + (a0/an):/7 = 0

(20)

which ensures Y: S = Y: 4/

(21)

By eqn. (17), eqn. (21) provides after re-arrangement 2/*11Y]I }~= 2/*(Y: O) - (Y: S) = 2/*(Y: O) - (Y: 4/) and, in view of eqn. (13), during plastic flow one can write = 2/*(Y: 0 ) / [ ( 2 / * + C)II Yfl]

(22)

With the help of two Heaviside functions:

H,(x) = 1 H 2(x) = 1

for x > S M for x > 0

(23)

eqns. (11), (19) and (22) can be summarized in a single expression ge = 2t, [( Y: 0)/((2/* + C) II Y II)]( Y/II Y II) H i ( II Y II)//2 (0O/0o : 6)

(24)

With the substitution of eqn. (24) into eqn. (17), the latter equation becomes, (remembering that = ~+ cC) Iy = 2/*[(~ - [(Y" ~)/11 Y II](Y/II Y II)HI( II Y II)H2(Ock/Oo" d)

(25)

Smoothing the plasticity laws means now to smooth the Heaviside function /41(- ). For this purpose Bouc proposed:

H 1( x ) = x"/S~t

(26)

Writing the other Heaviside function in the form

H2(Oq~/ao:6) = [1 + s i g n ( 0 + / 0 o : 6 ) ] / 2 = [ ( r : ~ ) + IY: OI]/[Z(Y: 0)]

(27)

one finds the three-dimensional Bouc's model, for/3 = t~/S~ and A = 2/*:

~>+ BllYfl°-2(Y • o)Y + BIIYIIn-21Y • OIY=AO

(28)

Its generalization is

?+BiiYii.-=ly:

~]y+ yllyll. 2(y. O)Y=AO

(29)

for which it is easy to prove (see eqn. (4)) IfYII <~[A/(B+ y)] a/" f o r / 3 + y > 0

(30)

265

In Ref. [7] a constitutive equation alternative to eqn. (29) is proposed. It is founded on a piece-wise linearization of the yielding surface and the Koiter's hardening rule [4,5]. Its discussion is p o s t p o n e d to Section 4.

3. STOCHASTIC EQUIVALENT LINEARIZATION Consider an elasto-plastic medium; its behaviour can be regarded as the sum of the elastic response to the loads W and the one due to the inelastic strain Cp at the Gauss points of the finite elements by which the c o n t i n u u m is discretized. Since the response is always elastic, the equation of motion can be written: mii + ckil + k u - k,cp - W = 0

(31)

where the matrices of mass, m, and stiffness, k and k,, are assembled over the structure. The d a m p i n g matrix is assumed to be proportional to k by the coefficient c. The matrix Ap groups the vectors Cp in the different Gauss points. Each c o m p o n e n t of the matrix k, is evaluated by the elastic analysis of the system subject to a distorsion in the single Gauss point: a single c o m p o n e n t of the deformation tensor is imposed equal to 1. The stress at the Gauss points is O -----Eli/-t- E2A P

(32)

where E 1 and E 2 are static matrices to be estimated by elastic analysis: EIj k is the stress in the k t h Gauss point due to a value 1 of the j t h displacement; E2k~jt) is the stress in the k t h Gauss point due to a value 1 of the j t h strain c o m p o n e n t in t h e / t h Gauss point. The constitutive law in the k th Gauss point is then % = e k + e8

(33a)

Ck P = e k -- e~,

(33b)

t~k = S k + %8

(33c)

e~ = S k / 2 ~

(33d)

Os= (3A + 2t~),s

(33e)

S~ = (1 - a ) g k Z i , + a K k e k

(34a)

Z~k = Zk(O ~, Z )

(34b)

where 8 is a vector of Kronecker indexes and A is the Lame' elastic constant. Strain, e, deviatoric strain, e, and isotropic strain, es, were already defined in the previous section, as well as the stress contributions. When W is a vector of stochastic processes W ' , eqn. (31) can be solved by stochastic equivalent linearization techniques [19,20,3,7]. At the k th Gauss point, the endochronic constitutive law (34b), expressed in terms of the auxiliary variable Z, is linearized and usual matrix algebra [3,7] leads eventually to d + ad = w

(35)

with d T = {u T, ~iT, eT}. Note that some elements of the matrix f~ in eqn. (35) d e p e n d on t h e linearization coefficients which are yet to be determined. The covariance matrix of d is the solution of the matrix equation d Y ' d / d t + flied + ~"d ~'~T = T

(36)

266 The r.h.s. T = E[w d T] + E[d wT] can be easily computed when the system is excited by a single white noise (e.g. the ground acceleration due to a seismic excitation). For the problem considered the task of the analyst is therefore twofold: (1) to express the structural matrices which form ~ by finite element discretization and successive algebra; (2) to solve Eq. (36) by numerical integration, updating at each step the linearization coefficients. Significant improvements in the solution procedure [21-23] are in progress: they are making possible the adoption of this approach also in the analysis of large hysteretic structural systems. Note, in fact, that the size of the system of first-order differential equations (35) depends on: (i) the number of degrees of freedom of the dynamical system multiplied by a factor two (two first-order differential equation for each equation of motion, which is a second-order differential equation); and (ii) the number of Gauss points multiplied by a factor six. Moreover, each equation of this second set is characterized by several linearization coefficients to be determined step by step.

4. STRUCTURAL CONSTITUTIVE LAW

As usual in structural engineering, after the analysis problem is formulated in the context of solid mechanics, a simplification can be pursued at a structural level. With reference to frames, this can be done by stating the constitutive law either for the single beam section [3-5,21] or the whole storey [24]. Both these constitutive laws are of the multivariate type due to the multi-dimensional nature of the frame and the bidirectional nature of the external excitation. In Ref. [24], the constitutive law was put in the form

= e

Bli, xl Izxlz =i,y-

-

1 ' 11zylZy- vayz

-

1 ,11zylZx Izxlz

-

, zxzy

(37)

- Vi, xZxZ

where u x and Uy are the two horizontal displacements of the storey., They are related with the corresponding forces by the auxiliary variables zx and Zy. The hardening is not included. Equation (37) is quite similar to eqn. (29) with n = 2. However not all the terms of the product (Y: e) are considered in it. This also occurs in its extension to the continuous case of Ref. [25], where some mechanical requirements, however, are not satisfied (the relationship is not stated for the deviatoric tensors; the yielding curve is not convex etc.). In Ref. [8] it is emphasized that different results are reached by using eq. (37) or the constitutive relationship proposed in Refs. [3-5]. The yielding surface is here introduced in a piece-wise linear form. For the hth (h = 1. . . . . nh) boundary line, forming an angle q'h with the first axis x, the deformation Xh is introduced. It is normal to the yielding surface and its time derivative plays the role of a plastic multiplier in classic plasticity theory: f sin Oh

/ Uyxl ,,cos

with implicit sum over h. Its static counterpart is: q~ ) ~h = Q~hTtl qy

(39)

267 The hardening characteristics can be included by writing [4,5] q~h = (1 - ot)khz~, h + akh~ h

•

im_l

(40)

In this way the normality rule is satisfied, the Koiter's hardening rule is introduced and any form of yielding surface can be considered. It is proved in the following that eqn. (41) for m = 1 can be derived from eqn. (29) with a value of the exponent n equal to 1. The different results achievable by eqn. (37) (n = 2) and eqn. (41) (n = 1) are therefore theoretically explained. Writing eqn. (29) for n = 1 and making use of eqn. (22)--which, of course, holds when the von Mises yielding criterion is c o n s i d e r e d - - t h e result is

+ Yfl [(2# + C)/2/*]

IX I+ v [(2/* +

C)/2/*] ~Y = A0

(42)

or by dividing both members by A = 2/, 2~ +/3[(2/* + C)/2/*] Z I ~, I +V[(2/* + C)/2/*] ~ Z = ~

(42')

Linearize now the yielding condition by a hyperplane with normal defined by the direction of ~, i.e. the h t h b o u n d a r y line of a piecewise linear yielding surface is activated. R e m e m b e r that: (i) in eqn. (38) u plays the role of e in the previous section; and (ii) in eqn. (39) q plays the role of S. Equation (34) multiplied by QT (i.e. the transposed vector of the direction cosines) is then written in the k t h Gauss point:

Q'~q = Q~(1 - a ) K Z + aQff Ku

(43)

where the index k is d r o p p e d for simplifying the notation and K is the stiffness diagonal matrix of eqn. (34a). Equation (43) can also be put in the form

Q~q = (1 - a)Q~KQhz~, h + aQ~KQhX h

(44)

and equivalently q~h = (1 - a)khz h + otkh)~h

(44')

which coincides with eqn. (40). For this purpose, the following positions were introduced:

k h = Q'~KQh

(45a)

as shown in Ref. [5], and

Z = QhZ~h

(45b)

Equation (44) makes implicitly use of the Koiter hardening rule: the stress in the hardening branch increases proportionally to the plastic deformation associated with the activated yielding hyperplane. Therefore the sum over h, which is implicit in eqn. (38), does not apply in this case. The same remark coupled with eqn. Eq. (45b) permits one to write eqn. (42') as 2~,h +/3 [(2/* + C)/2/*]l~hlz,h + 3'[(2/* +

c)/2/*]J, hz ,h= Xh

This is again eqn. (41), for m = 1, when /3h =/3[(2/* + C)/2/*]

Vh = V[(2/* + C)/2/*]

The only difference is the absolute value of z,h in the last term of the 1.h.s.. This absolute value is

268 introduced in order to include two symmetric branches of the yielding surface in a single relation. Equation (41) for n > 1 is obtained by appropriate modifications in eqn. (29). The comparison of eqn. (41) with eqn. (29) gives rise to the following remarks: (i) Equation (29) depends on the particular shape (von Mises) of the yielding criterion; eqn. (41) is independent of it. (ii) Equation (29) holds at a tensorial level; in a single point it provides the relation between the stress and strain tensors. By contrast, eqn. (41) was originally proposed for a beam section under bi-axial bending moments. However, eqn. (41) also holds at a tensorial level, as well as it can express the relation between storey displacement and force vectors. (iii) The hardening rule which characterizes eqn. (29) is the classical Prager's rule. The one of eqn. (41) is presently the Koiter's hardening rule: some improvements may be required to remove this limitation in view of more accurate practical applications.

5. CONCLUSIONS At a single Gauss point of a hysteretic medium, a smoothed form of the relationship between the stress and strain tensors is obtained as an extension of the univariate Bouc's endochronic model. Explicit use of von Mises's yielding criterion is made and Prager's hardening rule is adopted. An alternative model was originally proposed for biaxial bending moments in Refs [4,5], It makes use of a piece-wise linear yielding condition. It can also be used to describe the relationship between the stress and strain tensors. The paper shows that this alternative smoothed form of the stress-strain relationship is a special case of the previous Bouc's multivariate model. However, it leads to a much more flexible constitutive law. In fact, it can be applied to modeling of bi-axial bending moments in beam sections, stress and strain tensors in Gauss points and, also, displacement and force vectors in frame structures. Moreover, it is quite independent of the shape of the yielding condition provided it is given in a piecewise linear form.

ACKNOWLEDGEMENT This study has been supported by grants from the Italian Ministry of Public Education (M.P.I.) and from the National Research Council (C.N.R.)

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269 6 G. Ceradini, Un legame costitutivo elasto-plastico pluridimensionale per materiali con degradazione, in: Sandro Dei Poll, A Festschrift for the 70th Birthday, Politecnico di Milano, 1985, pp. 195-207 (in Italian). 7 F. Casciati and L. Faravelli, Stochastic equivalent linearization for dynamic analysis of continuous structures, Proc. ASME/SES Meeting on Computational Probabilistic Methods, Berkeley, 1988, pp. 205-210. 8 F. Casciati and L. Faravelli,, Endochronic theory and nonlinear stochastic dynamics of 3D-frame, ASCE Spec. Conf. on Probabilistic Methods in Civil Engineering, Blacksburg, VA, 1988, pp. 400-403. 9 T.T. Baber and Y.K. Wen, Stochastic equivalent linearization for hysteretic degrading multistory structures, UILU-ENG-80-2001, SRS 471, Univ. of Illinois, Urbana, IL, 1980. 10 M.A. Karray, Etude de l'Efficacit6 d'un Sys'tme d'Isolation a la Base avec Ammortissement par Plasticit6, Ph.D. Thesis, Univ. d'Aix-Marseille II, 1987 (in French). 11 M.A. Karray and R. Bouc, Etude Dynamique d'un Syst~me d'Isolation Anti-Seismique, Annales de l'Ecole National d'Ingenieur Tunis (ENIT), (3)1 (1989) 43-60. 12 R. Bouc, Forced vibrations of a mechanical system with hysteresis, Proc. 4th Conf. on Non-linear Oscillations, Prague, Czechoslovakia, 1967. 13 F. Casciati, Non-linear stochastic dynamics of large structural system by equivalent linearization, Proc. ICASP5 (Int. Conf. on Application of Statistics and Probability in Soil and Structural Engineering), Vancouver, 1987, pp. 1165-1172. 14 F. Casciati, Smoothed plasticity laws and elasto-plastic analyses, in: G. Ceradini, A Festschrift for the 70th Birthday, Rome, Italy, 1988, pp. 188-203. 15 Z.P. Bazant, R.J. Krizek and C.-L. Shieh, Hysteretic endochronic theory for sand, J. Eng. Mech., ASCE, 109 (1983) 1073-1095. 16 F. Casciati, L. Faravelli and M.P. Singh, Non-linear structural response and modeling uncertainty on system parameters and seismic excitation, Proc. 8th ECEE, Lisbon, 1986, 6.3, pp. 41-48. 17 W.D. Iwan, Private communication, 1988. 18 F. Casciati and L. Faravelli, Non-linear stochastic dynamics by equivalent linearization, in F. Casciati and L. Faravelli (Eds.), Methods of Stochastic Structural Mechanics, SEAG, Pavia, Italy, 1986, pp. 571-586 19 P. Spanos, Stochastic linearization in structural dynamics, AppI. Mech. Rev., 34 (1981) 1-11 20 F. Casciati and L. Faravelli, Equivalent linearization in non linear random vibration problems, Proc. Int. Conf. on Vibration Problems in Eng., Xian, Cina, 1986, pp. 986-991. 21 T.T. Baber, Modal analysis for random vibration of hysteretic frames, Earthquake Eng. Struct. Dyn., 14 (1986) 841-859. 22 F. Casciati, L. Faravelli and M.P. Singh, Stochastic equivalent linearization algorithms and their applicability to hysteretic systems, Meccanica, 23 (2) (1988) 107-112. 23 M.P. Singh, M. Maldonado, R. Heller and L. Faravelli, Modal analysis of nonlinear hysteretic structures for seismic motions, in F. Ziegler, and G. SchuSller (Eds.), Non Linear Structural Dynamics in Engineering Systems, Springer-Verlag, Berlin, 1988, pp. 443-454. 24 Y.J. Park, Y.K. Wen and A.H.-S. Ang, Random vibration of hysteretic systems under bi-dimensional ground motion, Earthquake Eng. Struct. Dyn., 14 (1986) 543-557. 25 Y.J. Park and A.H.-S. Ang. Seismic damage analysis of r / c nuclear structures, Proc. 9th SMiRT Conference, Lausanne, Switzerland, 1987, Vol. M, pp. 229-236.