Reliability of linear structures with parameter uncertaintyunder non-stationary earthquake

Abhijit Chaudhuri a, Subrata Chakraborty b,*,1

a Department of Civil Engineering, Indian Institute of Science, Bangalore 560012, Indiab Department of Civil Engineering, Bengal Engineering and Science University, Howrah 711103, India

Received 10 April 2004; received in revised form 30 June 2005; accepted 1 July 2005Available online 16 September 2005

Abstract

The present work aims towards the development of a general framework of time varying unconditional reliability eval-uation of linear elastic multi degree of freedom structures with uncertain parameter subjected to the generalized earth-quake ground motion, a non-stationary process both in amplitude and frequency content. The formulation isdeveloped in double frequency spectrum to derive the generalized power spectral density function of the structuralresponses. The time varying reliability is evaluated using conditional crossing rate following the Vanmarcke�s modification.The perturbation based stochastic finite element method is utilized in deriving unconditional reliability. An idealized threedimensional dam structure subjected to El Centro (1940) earthquake is taken up to elucidate the proposed unconditionaltime varying reliability computation procedure based on the maximum top displacement and base shear criteria. Theresults are presented to compare the change in reliabilities of the uncertain system with that of deterministic systemand associated variance of the reliability due to parameter uncertainty.� 2005 Elsevier Ltd. All rights reserved.

Keywords: Generalized non-stationary earthquake; Double frequency spectrum; Time varying reliability; Parameter uncertainty; Pertu-rbation

1. Introduction

The earthquake phenomenon is associated with a large number of uncertainties, which are normallyviewed, in two different scales. Firstly, the uncertainties associated with the long time features (i.e., number,size, and relative location of earthquake) are taken care by the so-called seismic risk analysis at the structuralsite. The outcomes of the risk analysis are obtained as a set of probability estimates for the peak ground accel-eration in a typical design period. The second type of uncertainty, associated with the details of the groundmotion at the structure�s site during single episode of seismic action, is an important aspect in the structural

0167-4730/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.strusafe.2005.07.001

* Corresponding author. Fax: +91 33 2668 4564/2916.E-mail address: schak@civil.becs.ac.in (S. Chakraborty).

1 Presently, visiting the Department of Civil Engineering and Engineering Mechanics, University of Arizona, Tucson, USA in theBOYSCAST Fellowship programme, DST, Govt. of India.

Structural Safety 28 (2006) 231–246

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STRUCTURAL

SAFETY

design. A structure subjected to earthquake produces random response as it is a random phenomenon. And,the analysis of a deterministic structure subjected to earthquake loading is a random vibration problem. Thewell-established random vibration theory [1–4] can be readily applied to obtain the probabilistic description ofthe responses. Extensive contributions [5,6] are made on this field. However, most of the works mainly con-sider the structural parameters as deterministic. But uncertainties in structural parameters are bound to occur.The uncertainties result from numerous assumptions made with the geometry, boundary conditions, and con-stitutive behaviour of materials can have significant effect on the reliability. Hence, a more complete stochasticanalysis is felt essential to integrate the seismic risk analysis, stochastic dynamic response and design reliabilityestimation, so that the design values of the response parameters of interest for specified reliability and excita-tion can be obtained.

The stochastic finite element method (SFEM) [7–11] is developed to incorporate the uncertainties in struc-tural parameters. Various attempts are found in the literature to combine the random vibration theory withSFEM [12–16]. Interestingly, in most of the cases [17–19] the computations are restricted to statistics of powerspectral density and mean square displacement. The reliability analysis under uncertain structural parametersis tried with simple structural and dynamic load model [20–22] and the time variant reliability under parameteruncertainty is scare in the literature [23–26] particularly for realistic non-stationary earthquake model.

In present study an attempt has been made to develop a time varying reliability analysis for linear elasticdynamic structural system under generalized non-stationary earthquake incorporating uncertainty in thestructural system parameters. In doing so, a fully non-stationary earthquake model based on the family ofsigma oscillatory process as described in Conte and Peng [27] has been readily extended to derive the general-ized power spectral density function (PSDF) in double frequency spectrum. The time varying reliability is eval-uated using conditional crossing rate following the Vanmarcke�s modification based on the two states Markovprocess [28,29] rather than the commonly used unconditional crossing rate based on the Poisson�s assumption.The perturbation method is adopted to tackle the parameter uncertainties. Finally, a typical three dimensionaldam subjected to ground motion correspond to El Centro 1940 earthquake is taken up to elucidate the timedependent reliabilities computation. The unconditional reliability and its variance are evaluated with respectto maximum displacement and base shear as well.

2. The generalized non-stationary ground motion model

The effect of natural seismicity on the structural responses is markedly non-stationary due to the non-sta-tionary random excitation process and also for the transition nature of responses. Various non-stationarymodels of ground motion are proposed in the literature, i.e., Priestley evolutionary power spectrum, the energyspectra, double frequency spectrum, Weiner process, random pulse train model, sigma oscillatory process etc.Though most of the literatures mainly emphasize on the extreme random response computation consideringdifferent ground motion model, the probabilities that these extreme responses have ever been exceeded somecritical levels during the time interval are mainly tried with simplified assumptions primarily restricted to thestationary model or that of a evolutionary model [30]. But the evolutionary earthquake models normallyneglects the frequency non-stationary character. Moreover, it is well verged that the maximum responseand its likely duration can be determined approximately from the shape of the envelope function without car-rying out any non-stationary response analysis [31].

To incorporate the fully non-stationary character of the ground motion, i.e., both amplitude and frequencynon-stationarity, it is described as a sigma-oscillatory process [27].The closed form expression of the autocor-relation function of response of a linear structure subjected to the ground motion modelled as a sigma oscil-latory process was derived in Peng and Conte [33]. The model views the earthquake ground motion assuperposition of the component process described by individual arrival time, frequency content, and timeintensity function. The ground acceleration is expressed as

€ugðtÞ ¼Xpk¼1

X kðtÞ ¼Xpk¼1

AkðtÞZkðtÞ: ð1Þ

Here, Ak(t) is the time modulating function of the kth component process defined as

232 A. Chaudhuri, S. Chakraborty / Structural Safety 28 (2006) 231–246

AkðtÞ ¼ akðt � fkÞbk expð�ckðt � fkÞHðt � fkÞÞ; ð2Þwhere ak and ck are the positive constants, bk is a positive integer and fk is the arrival time of the kth sub-pro-cess and H(t � fk)is the unit step function. Zk(t) is the kth Gaussian stationary process characterized by itsautocorrelation function

RZkZk ðsÞ ¼ RZkZk ðt1 � t2Þ ¼Z 1

�1/ZkZk

ðxÞeiðxt1�xt2Þ dx; ð3Þ

where /ZkZkðxÞ is the PSD function defined as

/ZkZkðxÞ ¼ mk

2p1

m2k þ ðxþ gkÞ2þ 1

m2k þ ðx� gkÞ2" #

: ð4Þ

Here, mk and gk are the two free parameter representing the frequency band width and predominant or centralfrequency of the sub-process Zk(t). The parameters are normally determined by the adaptively least squarefitting of the analytical evolutionary PSD of the model to that estimated from the target actual earthquake.The extended Thomson�s multiple window spectrum estimation method is normally applied. Note that if pis the number of component process then total 6p parameters need to be estimated and more the componentsincluded, the model will better fit the estimated data. Details of the method, its fruitful estimation ability etc.with associated references may be seen in Conte and Peng [27].

The autocorrelation function and the evolutionary PSD function of €ugðtÞ can be expressed as

R€ug€ugðt1; t2Þ ¼Xpk¼1

Akðt1ÞAkðt2ÞRZkZk ðt1 � t2Þ; ð5Þ

/€ug€ugðx; tÞ ¼Xpk¼1

j AkðtÞj2/ZkZkðxÞ: ð6Þ

The generalized non-stationary cross-spectral density matrix for ground motion can be represented in dou-ble frequency domain as a function of two frequencies by applying the double Fourier transform [32] to theassociated auto covariance function (details of which are provided in Appendix A1). The generalized PSDfunction of the ground motion is finally obtained as

S€ug€ugðx1;x2Þ ¼Xpk¼1

Z 1

�1Mkðx� x1Þ/ZkZk

ðxÞM�kðx� x2Þdx: ð7Þ

In above,

MkðxÞ ¼ 1

2p

Z 1

�1AkðtÞeixt dt ¼ akeixfk

ðck � ixÞðbkþ1Þ Cðbk þ 1Þ. ð8Þ

Here C(Æ) is the gamma function and * denotes complex conjugate.In subsequent numerical study, the six parameters needed for each component process are taken from

Conte and Peng [27] corresponding to the El Centro (1940) earthquake record. The parameters and associatedPSD function in double frequency domain are furnished in Table 1 and Fig. 1, respectively.

3. Stochastic dynamic response

The equation of motion for a multi degree of freedom (MDOF) system under ground acceleration can bereadily expressed as

½M �f€uðtÞg þ ½C�f _uðtÞg þ ½K�fuðtÞg ¼ �½M �fLg€ugðtÞ; ð9Þwhere [M], [C] and [K] are the global mass, damping and stiffness matrixes of the structure, respectively. {u(t)}is the vector of the nodal displacement relative to the base of the structure and €ugðtÞ is the ground acceleration.{L} is the influence coefficient vector which represents the pseudo-elastic response in all degrees of freedomdue to unit support motion.

A. Chaudhuri, S. Chakraborty / Structural Safety 28 (2006) 231–246 233

The ground motion that has only one horizontal component is considered in present study. The evolution-ary zero mean vector process can be expressed in Fourier–Stieltjes integral form [6] as

€ugðtÞ ¼Z 1

�1€Ugðx; tÞ expðixtÞdZðxÞ: ð10Þ

And for linear structural behaviour, the response u(t) can be also expressed as

fuðtÞg ¼Z 1

�1fUðx; tÞg expðixtÞdZðxÞ: ð11Þ

Table 1Parameters of non-stationary ground motion model correspond to El Centro (1940) Earthquake

k ak bk ck fk mk gk

1 37.2434 8 2.7283 �0.5918 1.4553 6.76032 104.0241 8 2.9549 �0.9857 2.4877 11.08573 31.9989 8 2.6272 1.7543 3.3024 7.36884 43.8375 9 3.1961 1.686 2.1968 13.59175 33.1958 9 3.1763 �0.0781 3.1241 14.38256 41.3111 9 3.1214 �7.096 6.7335 25.15327 4.2234 10 2.9904 �0.9464 2.6905 48.06178 19.9802 6 1.895 1.402 7.2086 37.61639 2.4884 10 2.6766 5.3123 6.1101 19.461210 24.1474 10 3.3493 8.8564 1.9862 9.0411 2.5916 2 0.224 3.2558 2.4201 9.338112 2.2733 3 0.5285 16.2065 1.5244 14.106713 24.2732 3 1.0361 17.5331 1.7141 24.044414 41.3111 9 3.1214 �7.096 6.7335 25.153215 1.3697 10 2.5936 21.683 1.9362 12.919816 15.4646 2 0.7044 27.2979 1.7897 12.020517 0.0174 10 1.8451 �2.4168 4.9373 98.62818 2.9646 10 3.1137 1.5751 1.9726 61.831619 0.0007 10 1.3686 2.5173 3.2479 43.9067520 0.8092 4 0.5969 6.4396 3.6749 26.336521 16.7115 2 0.7294 12.493 1.7075 37.1139

Fig. 1. Generalized non-stationary power spectral density correspond to El Centro (1940) Earthquake.

234 A. Chaudhuri, S. Chakraborty / Structural Safety 28 (2006) 231–246

For a stable system, i.e., damping ratio, f > 0, the response of the system may be assumed to die out soon afterthe excitation dies out and the dynamic equilibrium in the frequency domain can be obtained as

½�x2½M � þ ix½C� þ ½K��fUðx; tÞg ¼ �½M �fLg €Ugðx; tÞ; i:e:; ½DðxÞ�fUðx; tÞg ¼ �fF g €Ugðx; tÞ; ð12Þwhere the dynamic stiffness matrix is defined by

½DðxÞ� ¼ �x2½M � þ ix½C� þ ½K�: ð13ÞTo obtain the evolutionary PSD function of response, Eq. (12) is expressed in a simplified form as

below

½DðxÞ�fHuðxÞg ¼ fF g; i:e:; fHuðxÞg ¼ ½DðxÞ��1fF g; ð14Þwhere {U(x,t)} = {Hu(x)}Ug(x, t) and {F} = �[M]{L}. For a linear system with known complex frequencyresponse function, the evolutionary PSD function [/uu(x, t)] of the displacements at any d.o.f. can be readilywritten as

½/uuðx; tÞ� ¼ fHuðxÞg/€ug€ugðx; tÞfH �uðxÞgT; ð15Þ

where fH �uðxÞg is the complex conjugate of {Hu(x)}. The PSD function of the structural response under evo-

lutionary ground motion /€ug€ugðx; tÞ as presented above is obtained assuming that the amplitude of the forc-ing function is a slowly varying function of time such that it can be treated as being nearly independent oftime. Otherwise, computation of the non-stationary response involves accurate evaluation of {Hu(x)} at eachinstant of time. Thus, the computation for each frequency is equivalent to performing a time history anal-ysis. The response statistics is then computed using the pdf approximation of the non-stationary process un-der the assumptions that a non-stationary process behaves like a stationary process at each instant of time[31].

However, it can be easily expressed as a function of two different frequency, i.e., x1 and x2 by the doubleFourier transform similar to the ground motion model. When the forcing function on the right-hand side ofEq. (9) is non-stationary, the cross correlation matrix of the responses can be expressed as

½Ruuðt1; t2Þ� ¼Z 1

�1

Z 1

�1fHuðx1ÞgS€ug€ugðx1;x2ÞfH �

uðx2ÞgTeiðx1t1�x2t2Þ dx1 dx2: ð16Þ

The PSD function [Su(x1,x2)] of any generalized response quantity for a linear system can be readily ob-tained as

½Suuðx1;x2Þ� ¼ fHuðx1ÞgS€ug€ugðx1;x2ÞfH �uðx2ÞgT: ð17Þ

Similar to the displacement, force developed at any section can be obtained. The FRF of the total base shearforce along the direction of ground motion is

HbsðxÞ ¼ fLgT½K�fHuðxÞg: ð18ÞAnd the PSD of the base shear force can be obtained similarly as in Eq. (17) using Eq. (18).

The PSD function for any generalized response quantity x(t), autocorrelation function and its derivativeprocess are detailed in Appendix A2. Finally, the time varying spectral moments which are useful in reliabilityevaluation can be obtained from the evolutionary PSD function and utilizing the relations in Eq. (A2.4).(Appendix A2) as

kjðtÞ ¼Z 1

�1j xjj/xxðx; tÞdx ¼

Z 1

�1

Z 1

�1j xjjSxxðx1;xÞ expðiðx1 � xÞtÞdx1 dx

¼Z 1

�1

Z 1

�1j xjjSxxðx;x2Þ expðiðx� x2ÞtÞdx2 dx; ð19Þ

where j is the order of the spectral moment. The integrations in the above equations are performed using fastFourier Transform. The RMS value of the response and its derivatives are obtained as

rxðtÞ ¼ffiffiffiffiffiffiffiffiffiffik0ðtÞ

pand r _xðtÞ ¼

ffiffiffiffiffiffiffiffiffiffik2ðtÞ

p: ð20Þ

A. Chaudhuri, S. Chakraborty / Structural Safety 28 (2006) 231–246 235

4. The time variant conditional reliability

The reliability is defined as the probability that the absolute value of the response process will not exceed aspecified threshold level from time t = 0 to any time t. Then, the time dependent reliability of the structurebased on the first passage failure criterion for double barrier problem can be expressed as [4]

Rcðx0; tÞ ¼ Rðx0; 0Þ exp �Z t

0

aðx0; sÞds� �

; ð21Þ

where x0 is the barrier level and a(x0, t) is the rate of crossing with positive slope of the level x(t) = x0 at time, t.Note that R(x0,0) is the reliability at time t = 0 and for the non-stationary case it can be assumed as unity.When the value of the barrier level is large and the response is a wide band process the up-crossings are inde-pendent. In this situation, the reliability analysis using the unconditional crossing rate is a good approxima-tion based on the Poisson assumption that the times between two up-crossings are independent. But for anarrow band process with slowly varying amplitude A(t) and comparatively lower barrier level, a single up-crossing by A(t) is associated with several (almost uniformly spaced, i.e., in clump) up-crossing by x(t). Thisis inconsistent with the Poisson assumption. In this situation reliability is evaluated by a conditional crossingrate rather than the unconditional crossing rate. The conditional up crossing rate is determined here using theVanmarcke�s modification based on the two states Markov process [28,29]. For non-stationary process thecrossing rate is obtained from the time dependent RMS values as

aðx0; tÞ ¼ 1

2pr _xðtÞrxðtÞ

1� exp � ffiffiffiffiffiffi2p

px0ðqðtÞÞ1:2

rxðtÞ

� �exp 1

2x0

rxðtÞ

� �2� �� 1

0BB@

1CCA; ð22Þ

where

qðtÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� k21ðtÞ

k0ðtÞk2ðtÞ

sð23Þ

is called the measure of narrowness. Finally the reliability is evaluated using Eq. (22).

5. Uncertainty in structural parameter

The analysis described so far assumes all the input structural design parameters as deterministic. If theparameters uncertainties are included in the analysis, the dynamic stiffness matrix [D(x)] as described byEq. (13) will involve the random variables di (i = 1,2, . . .,N) (obtained on discretization of the randomfield, N is the number of discretized element), which represents the spatial variation of any structuralparameters. Thus, {H(x)} will also involve the random variables. The random design parameter di canbe viewed as the superposition of the deterministic mean component ð�diÞ with a zero mean deviatoriccomponent (Ddi).

Any response variable that depends on the system parameters is uncertain can be expanded in the Taylorseries about the mean value of the uncertain system parameter (with the assumptions that the random vari-ation is small). Let H({Dd})is any such variable. Now the Taylor series expansion about mean is

HðfDdgÞ ¼ �HþXNk¼1

HIkDdk þ 1

2

XNk¼1

XNl¼1

HIIklDdkDdl þ � � � ; ð24Þ

where

�H ¼ Hð0Þ;HIk ¼

oHðfDdgÞodk

����fDdg¼0

and HIIkl ¼

o2fHðfDdgÞgodkodl

����fDdg¼0

: ð25Þ

The mean value of the dynamic stiffness matrix and its derivatives with respect to the uncertain parameters atthe mean system parameters can be derived from Eq. (13) as

236 A. Chaudhuri, S. Chakraborty / Structural Safety 28 (2006) 231–246

½�DðxÞ� ¼ ½Dðx; 0Þ� ¼ �x2½ �M � þ ix½�C� þ ½�K�; ð26aÞ½DI

kðxÞ� ¼ �x2½M Ik� þ ix½CI

k� þ ½KIk�; ð26bÞ

½DIIklðxÞ� ¼ �x2½M II

kl� þ ix½CIIkl� þ ½KII

kl�: ð26cÞIf the Young�s modulus is taken as the random parameter, the first order derivative of the mass matrix van-ishes and the second order derivatives of the mass, damping and stiffness matrix become zero, i.e.,½M I

k� ¼ 0 and ½M IIkl� ¼ ½CII

kl� ¼ ½KIIkl� ¼ 0. The derivatives of the force vector also vanish in this case. With the

strength of Eqs. (24) and (25), the mean value and the derivatives of {H(x)} can be obtained from Eq. (14) [34]

f �HuðxÞg ¼ ½�DðxÞ��1fF ðxÞg; ð27aÞfH I

u;kðxÞg ¼ ½�DðxÞ��1ð�½DIkðxÞ�f �HuðxÞgÞ; ð27bÞ

fH IIu;klðxÞg ¼ ½�DðxÞ��1ð�½DI

lðxÞ�fH Iu;kðxÞg � ½DI

kðxÞ�fH Iu;lðxÞgÞ: ð27cÞ

For base shear, the expression of FRF at mean and the derivative with respect to the uncertain parameters canbe obtained as

�HbsðxÞ ¼ fLgT½�K�f �HuðxÞg; ð28aÞH I

bs;kðxÞ ¼ fLgT ½K�Ikf �HuðxÞg þ ½�K�fH Iu;kðxÞg

� �; ð28bÞ

H IIbs;klðxÞ ¼ fLgT ½KI

k�fH Iu;lðxÞg þ ½KI

l�fH Iu;kðxÞg þ ½�K�fH II

u;klðxÞg� �

: ð28cÞSimilarly, from the expression of (A2.1) the mean value of the PSD function and its derivatives can be ob-tained as

�Sxxðx1;x2Þ ¼ �Hxðx1ÞS€ug€ugðx1;x2Þ �H �xðx2Þ; ð29aÞ

SIxx;kðx1;x2Þ ¼ S€ug€ugðx1;x2Þ H I

x;kðx1Þ �H �xðx2Þ þ �Hxðx1ÞH �I

x;kðx2Þh i

; ð29bÞ

SIIxx;klðx1;x2Þ ¼ S€ug€ugðx1;x2Þ H II

x;klðx1Þ �H �xðx2Þ þ H I

x;kðx1ÞH �Ix;lðx2Þ þ H I

x;lðx1ÞH �Ix;kðx2Þ þ �Hxðx1ÞH �II

x;klðx2Þh i

:

ð29cÞApplication of the perturbation with respect to the random parameter to the spectral moments and asso-

ciated RMS values as described in Eqs. (19) and (20), one can derive the following first and second orderterms,

�kðtÞ ¼Z 1

�1

Z 1

�1j xjj�Sðx1;xÞ expðiðx1 � xÞtÞdx1 dx; ð30aÞ

kIj;kðtÞ ¼Z 1

�1

Z 1

�1j xjjSI

xx;kðx1;xÞ expðiðx1 � xÞtÞdx1 dx;

rIx;kðtÞ ¼

1

2�rxðtÞ kI0;kðtÞ and rI

_x;kðtÞ ¼1

2�r _xðtÞ kI2;kðtÞ

ð30bÞ

and

kIIj;klðtÞ ¼Z 1

�1

Z 1

�1j xjjSII

xx;klðx1;xÞ expðiðx1 � xÞtÞdx1 dx;

rIIx;klðtÞ ¼

1

2�rxðtÞ kII0;klðtÞ �

1

�rxðtÞ rIx;kðtÞrI

x;lðtÞ; rII_x;klðtÞ ¼

1

2�r _xðtÞ kII2;klðtÞ �

1

�r _xðtÞ rI_x;kðtÞrI

_x;lðtÞ.ð30cÞ

6. Unconditional reliability evaluation

The total probability concept is one of the most important theorems in evaluating the overall reliability ofthe structure under parameter uncertainties [22,35]. Knowing the conditional second order information of re-

A. Chaudhuri, S. Chakraborty / Structural Safety 28 (2006) 231–246 237

sponse quantities, the unconditional reliability can be obtained from the joint pdf of the parameter vector,f({Dd}), i.e.,

Rucðx0; tÞ ¼Z1;...;N

� � �Z

Rcðx0; t j fDdgÞf ðfDdgÞdDd1; . . . ; dDdN . ð31Þ

The time-varying reliability under random excitation becomes random in nature due to the uncertaintyassociated with the system parameters. Thus, Ru(x0, t) defined in above equation is the expected value ofthe reliability. In Eq. (31), Rc(x0, tj{Dd}) is the conditional reliability to the given set of parameters ({Dd}).And it can be expanded in Taylor series about the mean value as

Rcðx0; t j fDdgÞ ¼ �Rcðx0; tÞ þXNk¼1

RIc;kðx0; tÞjfDdg¼0Ddk þ 1

2

XNl¼1

XNk¼1

RIIc;klðx0; tÞjfDdg¼0DdkDdl þ � � � ; ð32Þ

where

�Rcðx0; tÞ ¼ Rcðt; x0 j fDdg ¼ 0Þ ¼ Rðx0; 0Þ exp �Z t

0

�aðx0; sÞds� �

; ð33aÞ

RIc;kðx0; tÞ ¼ �

Z t

0

aIkðx0; sÞds� �

�Rcðx0; tÞ; ð33bÞ

RIIc;klðx0; tÞ ¼

Z t

0

aIkðx0; sÞds� � Z t

0

aIkðx0; sÞds� �

�Rcðx0; tÞ �Z t

0

aIIklðx0; sÞds� �

�Rcðx0; tÞ. ð33cÞ

The details of the derivatives of a(x0, t) are shown in Appendix B. Substituting Eqs. (32) in (31), Ruc(x0, t) canbe expressed in terms of various order of statistical moments of the system parameters. Finally, the uncondi-tional reliability (by second order approximation in the mean) can be evaluated using Eqs. (22) in (31),

Rucðx0; tÞ ¼ �Rcðx0; tÞ þ 1

2

XNl¼1

XNk¼1

RIIc;klðx0; tÞCkl. ð34Þ

The variance of the reliability (the first order approximation) is obtained as

r2Rðx0; tÞ ¼

XNl¼1

XNk¼1

RIc;kðx0; tÞRI

c;lðx0; tÞCkl; ð35Þ

where Ckl is the covariance matrix for the deviatoric part of the random design parameter. The related discret-ization of the random filed is performed through local averaging method and detailed in Appendix C.

7. Numerical study

A concrete gravity dam has been taken up to elucidate the proposed time varying reliability evaluationalgorithm. The finite element discretization of the structure is shown in Fig. 2. Isoparametric 8-noded brickelement with three degrees of freedom at each node has been used to discretize the dam into 36 elements hav-ing 80 nodes. The mass density is taken as 2400 kg/m3. The Poisson�s ratio is taken as 0.2. The proportionaldamping ratio (Rayleigh type) is taken as 0.05. The modulus of elasticity has been considered as the randomdesign parameters having mean value of 2.0 · 1010 N/m2. The correlation length in all three directions (Eq.(C1)) is assumed to be same. The ground motion correspond to the El Centro (1940) earthquake as detailedin Section 2 is considered to act along Y-direction.

The reliability is computed with respect to the displacement along Y-direction at node P (as show in Fig. 2)and the maximum base shear along Y-direction as well. The time varying reliability of the deterministic system(i.e., fixing the value of the random modulus of elasticity of all elements at their mean values) is compared inFig. 3 with that of stochastic system (for 10% COV of modulus of elasticity). It may be seen that there is def-inite change in the reliability of the uncertain system with that of deterministic one for various barrier level. Ingeneral, the uncertainty reduces the reliability of the system. It is also found that the uncertainty effect is lesswhen the time varying reliability is evaluated based on the base shear as failure criteria due to the fact that the

238 A. Chaudhuri, S. Chakraborty / Structural Safety 28 (2006) 231–246

base shear is directly related to the ground acceleration and the mass rather than the stiffness of the structure.Fig. 4 depicts that the reliability (after a fixed time of 10 s) decreases with increasing coefficient of variation ofYoung�s modulus (COVE) as expected. The variation of reliability with COVE follows a parabolic profile. Thesecond order terms in Eq. (34) confirms such behaviour.

Fig. 2. The idealized dam model.

Fig. 3. Comparison between the reliability of a deterministic and stochastic system with respect to (a) displacement at P and (b) shearforce at base (10% COV, k = 60 m).

Fig. 4. Reliability of the stochastic system with respect to (a) displacement at P and (b) shear force at base with increasing COV of therandom parameter (after 10 s, k = 60 m).

A. Chaudhuri, S. Chakraborty / Structural Safety 28 (2006) 231–246 239

The change of time varying reliability of structure with respect to the displacement at point P and the baseshear force along Y-axis for different correlation lengths are shown in Figs. 5 and 6, respectively. It is seen thatfor a comparatively smaller values of the correlation length representing relatively high rate of fluctuation ofthe random field in the space, the change in the reliability is small due to the fact that the uncertainty con-tained in the excitation process dominates in such case. Moreover, there are compensation effects betweenthe finite elements, and the effect of system parameter uncertainty is very small on the time varying reliability.But the random variability of the structural parameters have significant contribution for larger k. The varianceof the time dependent reliability of the uncertain system (shown in Figs. 7 and 8) increases with the increasing

Fig. 5. The change of reliability of the structure with respect to the displacement at P along X-axis for various correlation length ofrandom field and barrier level (x0). (a) x0 = 3max(rx(t)) and (b) x0 = 4max(rx(t)).

Fig. 6. The change of reliability of the structure with respect to the shear force at base along X-axis for various correlation length ofrandom field and barrier level (x0). (a) x0 = 3max(rx(t)) and (b) x0 = 4max(rx(t)).

Fig. 7. The variance of reliability of the structure with respect to the displacement at P along X-axis for various correlation length ofrandom field and barrier level (x0). (a) x0 = 3max(rx(t)), and (b) x0 = 4max(rx(t)).

240 A. Chaudhuri, S. Chakraborty / Structural Safety 28 (2006) 231–246

value of k. The off-diagonal elements of the covariance matrix increases and become closer to the variance ofthe random field as k increases. It results more change in the reliability and its variance through Eqs. (34) and(35) which is obvious from Eqs. (C2) and (C3).

The variance of the time dependent reliability of the uncertain system is shown in Figs. 7 and 8 with increas-ing correlation length k. Here also it is observed that the variance of reliability is less in the case of base shear.It may be observed that the variance of the reliability based on the base shear have peak approximately at time4 s, also the reliability curve changes slope at that time (Fig. 8). Fig. 9 shows the COV of reliability with chang-ing barrier level for different correlation length with 10% COV of the random parameter. It may be observedthat, in general the variance of the reliability increases with increase in COVE (imply more uncertainty), butthe rate of change is dependent on barrier level as well.

8. Conclusions

A time variant reliability analysis algorithm is developed for a linear elastic multi degree of freedom systemwith uncertain parameters subjected to generalize non-stationary earthquake excitation. The fully non-station-ary random vibration analysis under generalized earthquake motion model and system parameter uncertain-ties are integrated in time varying total reliability evaluation and its possible variance. Though the uncertaintycontained in the excitation process dominates, the random variability of the structural parameters contributessignificantly on the reliability of the structure. However, the change of reliability of uncertain system with thatof deterministic one depends on the barrier level. The effect of uncertainty is seen to be less for higher barrierlevel as the reliability for deterministic system becomes almost unity with increase in the barrier level. Itpossibly indicates that the structure remains almost safe under the uncertainties associated with the load as

Fig. 8. The variance of reliability of the structure with respect to shear force at the base along X-axis for various correlation length ofrandom field and barrier level (x0). (a) x0 = 3max(rx(t)), and (b) x0 = 4max(rx(t)).

Fig. 9. COV of reliability for various barrier level and correlation length with respect to (a) displacement at node P and (b) base shearforce (after 10 s, 10% COV of random parameter).

A. Chaudhuri, S. Chakraborty / Structural Safety 28 (2006) 231–246 241

well as structural properties. The change in reliability due to the parameter uncertainty is a function of thecoefficient of variation as well as the correlation length of the random field. In general, the increase uncertaintyin the structural parameter produces larger variance of the reliability hence reduced the safety margin of thesystem. The reliability evaluated based on the base shear force criteria is less affected by the system uncertaintycompare to that of displacement criteria. In present study the barrier level of the base shear force related to thepermissible shear force (or shear stress) is considered as deterministic. But most of the standard codes providean empirical relation between the Young�s modulus and the permissible shear stress. In such case, the barrierlevel for base shear will also be stochastic in nature. The effect of system uncertainty may be more when uncer-tainty in barrier level is taken into account. The numerical study presented here for parameter uncertaintyrestricted to Young�s modulus only. However, the methodology can be readily extended to analyze the effectof uncertainty associated with the density and damping ratio using the derivative of the mass and dampingmatrices, respectively.

Acknowledgment

The financial support received in OYS Scheme No. SR/FTP/ET-40/2000 dated 3.10.2000 from the Depart-ment of Science and Technology, Government of India in connection with this work is gratefullyacknowledged.

Appendix A1

The generalized non-stationary cross-spectral density matrix for the ground motion can be represented indouble frequency domain as a function of two frequencies by the double Fourier transform [32] to the asso-ciated auto covariance function as defined by Eq. (5), i.e.,

S€ug€ugðx1;x2Þ ¼ 1

4p2

Z 1

�1

Z 1

�1R€ug€ugðt1; t2Þ expð�iðx1t1 � x2t2ÞÞdt1 dt. ðA1:1Þ

The use of (A1.1) becomes

S€ug€ugðx1;x2Þ ¼ 1

4p2

Z 1

�1

Z 1

�1

Xpk¼1

Akðt1ÞRZkZk ðt1 � t2ÞAkðt2Þ !

expð�iðx1t1 � x2t2ÞÞdt1 dt2. ðA1:2Þ

With the strength of (A1.2) further yields,

S€ug€ugðx1;x2Þ ¼ 1

4p2

Xpk¼1

Z 1

�1

Z 1

�1Akðt1Þ

Z 1

�1/ZkZk

ðxÞeiðxt1�xt2Þ dx

� �Akðt2Þ expð�iðx1t1

� x2t2ÞÞdt1 dt2. ðA1:3ÞAnd simplifying above, the generalized PSD function of the ground motion is finally obtained as

S€ug€ugðx1;x2Þ ¼Xpk¼1

Z 1

�1

1

2p

Z 1

�1Akðt1Þ expðiðx� x1Þt1Þdt1

� �/ZkZk

ðxÞ

� 1

2p

Z 1

�1Akðt2Þ expð�iðx� x2Þt2Þdt2

� �dx. ðA1:4Þ

Appendix A2

In general, if x(t) is any generalized response quantity, the PSD function and autocorrelation function ofx(t) can be defined by

Sxxðx1;x2Þ ¼ Hxðx1ÞS€ug€ugðx1;x2ÞH �xðx2Þ; ðA2:1Þ

Rxxðt1; t2Þ ¼Z 1

�1

Z 1

�1Sxxðx1;x2Þ expðiðx1t1 � x2t2ÞÞdx1 dx2; ðA2:2Þ

242 A. Chaudhuri, S. Chakraborty / Structural Safety 28 (2006) 231–246

where Hx(x1) and Hx(x2) are the well known transfer functions at two different frequencies. Again the auto-correlation function can also be expressed (based on the evolutionary PSD function of response x(t)) as

Rxxðt1; t2Þ ¼Z 1

�1/xxðx; t1Þ expð�ixðt2 � t1ÞÞdx. ðA2:3Þ

From above, a relation between the evolutionary PSD function (/xx(t,x)) and the PSD function in doublefrequency domain (Sxx(x1,x2)) can be derived as

/xxðx; tÞ ¼Z 1

�1Sxxðx1;xÞ expðiðx1 � xÞtÞdx1 ¼

Z 1

�1Sxxðx;x2Þ expðiðx� x2ÞtÞdx2; ðA2:4Þ

where /xx(x, t) represents the evolutionary PSD function of the non-stationary process x(t). The autocorre-lation functions for the derivative process of the generalized response quantity cab be easily derived as

Rx _xðt1; t2Þ ¼Z 1

�1

Z 1

�1ð�ix2Sxxðx1;x2ÞÞ expðiðx1t1 � x2t2ÞÞdx1 dx2

¼Z 1

�1ð�ix/xxðx; t1ÞÞ expð�ixðt2 � t1ÞÞdx; ðA2:5Þ

R _x _xðt1; t2Þ ¼Z 1

�1

Z 1

�1ðx1x2Sxxðx1;x2ÞÞ expðiðx1t1 � x2t2ÞÞdx1 dx2

¼Z 1

�1ðx2/xxðx; t1ÞÞ expð�ixðt2 � t1ÞÞdx. ðA2:6Þ

Appendix B

The conditional expected crossing rate as described in Eq. (22) can be written as

aðx0; tÞ ¼ aðtÞbðtÞcðtÞ ; ðB1Þ

where

aðtÞ ¼ 1

2pr _xðtÞrxðtÞ ; bðtÞ ¼ 1� exp �

ffiffiffiffiffiffi2p

p x0q1:2ðtÞrxðtÞ

� �; cðtÞ ¼ exp

1

2

x0rxðtÞ� �2

!� 1. ðB2Þ

The first derivative with respect to the uncertain parameter can be obtained directly as

aIkðtÞ ¼aIkðtÞ�aðtÞ þ bIkðtÞ

�bðtÞ � cIkðtÞ�cðtÞ

� ��aðtÞ; ðB3Þ

where

aIkðtÞ ¼ �aðtÞ rI_x;kðtÞ�r _xðtÞ � rI

x;kðtÞ�rxðtÞ

� �;

bIkðtÞ ¼ffiffiffiffiffiffi2p

px0

1:2ð�qðtÞÞ0:2qIkðtÞ�rxðtÞ � ð�qðtÞÞ1:2rI

x;kðtÞð�rxðtÞÞ2

!ð1� �bðtÞÞ; cIkðtÞ ¼ � x20

ð�rðxtÞÞ2

rIx;kðtÞð1þ �cðtÞÞ. ðB4Þ

The bar indicates the value of the variables at mean system parameters. The second derivative can also be ob-tained directly as

aIIklðtÞ ¼aIIklðtÞ�aðtÞ � aIkðtÞaIlðtÞ

ð�aðtÞÞ2 þ bIIklðtÞ�bðtÞ � bIkðtÞbIlðtÞð�bðtÞÞ2 � cIIklðtÞ�cðtÞ � cIkðtÞcIlðtÞ

ð�cðtÞÞ2 !

�aðtÞ þ aIkðtÞaIlðtÞ�aðtÞ ; ðB5Þ

A. Chaudhuri, S. Chakraborty / Structural Safety 28 (2006) 231–246 243

where

aIIklðtÞ�aðtÞ � aIkðtÞaIlðtÞð�aðtÞÞ2 ¼ rII

_x;klðtÞ�r _xðtÞ � rI_x;kðtÞrI

_x;lðtÞð�r _xðtÞÞ2

� rIIx;klðtÞ�rxðtÞ � rI

x;kðtÞrIx;lðtÞ

ð�rxðtÞÞ2ðB6aÞ

bIIklðtÞbðtÞ � bIkðtÞbIlðtÞð�bðtÞÞ2 ¼ 1:2

ffiffiffiffiffiffi2p

px0ð�qðtÞÞ0:2 0:2qIkðtÞqIlðtÞ þ �qðtÞqIIklðtÞ

�qðtÞ�rxðtÞ�

� ðrIx;kðtÞqIlðtÞ þ qIkðtÞrI

x;lðtÞÞð�rxðtÞÞ2

� �qðtÞðrIIx;klðtÞ�rxðtÞ � 2rI

x;kðtÞrIx;lðtÞÞ

1:2ð�rxðtÞÞ3!ð1� �bðtÞÞ

�bðtÞ

� 2px20ð�qðtÞÞ0:41:2qIkðtÞ�rxðtÞ � �qðtÞrI

x;kðtÞð�rxðtÞÞ2

!1:2qIlðtÞrxðtÞ � �qðtÞrI

x;lðtÞð�rxðtÞÞ2

!1� �bðtÞð�bðtÞÞ2 ðB6bÞ

and

cIIklðtÞ�cðtÞ � cIkðtÞcIlðtÞð�cðtÞÞ2 ¼ � x20ðrI

x;klðtÞ�rxðtÞ � 3rIx;kðtÞrI

x;lðtÞÞð�rxðtÞÞ4

1þ �cðtÞ�cðtÞ � x40r

Ix;kðtÞrI

x;lðtÞð�rxðtÞÞ6

1þ �cðtÞð�cðtÞÞ2 . ðB6cÞ

The measure of narrowness in Eq. (23) is also random. Its mean value and derivatives can be obtained fromthe various moments of PSD function as below

�qðtÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� ð�k1ðtÞÞ2

�k0ðtÞ�k2ðtÞ

s; ðB7aÞ

qIkðtÞ ¼1� ð�qðtÞÞ2

2�qðtÞkI0;kðtÞ�k0ðtÞ

þ kI2;kðtÞ�k2ðtÞ

� 2kI1;kðtÞ�k1ðtÞ

!; ðB7bÞ

qIIklðtÞ ¼1� �q2ðtÞ2�qðtÞ

kII0;klðtÞ�k0ðtÞ � kI0;kðtÞkI0;lðtÞð�k0ðtÞÞ2

þ kII2;klðtÞ�k2ðtÞ � kI2;kðtÞkI2;lðtÞð�k2ðtÞÞ2

� 2kII1;klðtÞ�k1ðtÞ � kI1;kðtÞkI1;lðtÞ

ð�k1ðtÞÞ2 !"

� 1þ �q2ðtÞ�qðtÞð1� �q2ðtÞÞq

IkðtÞqIlðtÞ

#.

ðB7cÞAppendix C

The Young�s modulus is considered as spatially varying random variable and decomposed into a man anddeviatoric random component (with a zero mean). The random component is assumed to follow the Gaussiandistribution. The Young�s modulus in space can be expressed as

Eðx; y; zÞ ¼ Eðx; y; zÞ þ DEðx; y; zÞ ðC1ÞHere, random component DE(x,y,z) is assumed to follow a normalized covariance function defined by,

qðrx; ry ; rzÞ ¼ exp � rxkx

� �2

þ ryky

� �2

þ rzkz

� �2( )" #

; ðC2Þ

where kx, ky and kz are the correlation parameters parallel to x, y and z axis, respectively, rx, ry and rz are thecorresponding distances. The correlation lengths along three directions are taken same, i.e., statistically iso-tropic random field is considered. The mean value of the modulus of elasticity is taken as 2.0 · 1011 N/m2.The covariance matrix of the uncertain parameter needed for the evaluation of the mean and variance ofthe reliability through Eqs. (34) and (35), respectively, is obtained through the local averaging method. Thedetails may be seen in Vanmarcke [36], Bhattacharyya and Chakraborty [37]. Two arbitrary finite elementsof a structure are shown in Fig. C1 to illustrate the computation of covariance of the Young�s modulus bythe local averaging method. Here, ri = {xi,yi,zi} and rj = {xj,yj,zj}are the two vectors correspond to the centrepoints in the elements. Let, (xik,yik,zik) is the coordinate of kth node, Nik = Nk(ni,gi,1i) is the kth shape func-

244 A. Chaudhuri, S. Chakraborty / Structural Safety 28 (2006) 231–246

tion of ith element. Then, the coordinates of points (xi,yi,zi) and (xj,yj,zj) for ith and jth element can be writ-ten as

xi ¼X8k¼1

Nikxik; yi ¼X8k¼1

Nikyik and zi ¼X8k¼1

Nikzik;

xj ¼X8l¼1

Njlxjl; yj ¼X8l¼1

Njlyjl and zj ¼X8l¼1

Njlzjl.

The shape function for 8-noded isoparametric 3-D element (as shown in Fig. C2) is given as

Ni ¼ ð1þ ninÞð1þ gigÞð1þ fifÞ8

.

The covariance between two elements by local averaging method can be written as

Cij ¼ 1

ViVj

ZVi

ZVj

qðri; rjÞdri drj

¼Z Z Z Z Z Z

exp �X8k¼1

X8l¼1

ðNikNilAkl � 2NikNjlBkl þ NjkNjlCklÞ !

dni dgi dfi dnj dgj dfj; ðC3Þ

where

Akl ¼ xikxilk2x

þ yikyilk2y

þ zikzilk2z

!; Bkl ¼ xikxjl

k2xþ yikyjl

k2yþ zikzjl

k2z

!and Ckl ¼ xjkxjl

k2xþ yjkyjl

k2yþ zjkzjl

k2z

!.

Fig. C1. Two elements of a structure.

Fig. C2. A 3-D isoparametric element.

A. Chaudhuri, S. Chakraborty / Structural Safety 28 (2006) 231–246 245

References

[1] Bolotin VV. Statistical methods in structural mechanics. Holden-day, Inc.; 1961.[2] Lin YK. Probabilistic theory of structural dynamics. New York: McGraw-Hill; 1967.[3] Nigam NC. Introduction to random vibrations. The MIT Press; 1983.[4] Lutes LD, Sarkani S. Stochastic analysis of structural and mechanical vibration. Prentice Hall Inc; 1997.[5] Nigam NC, Narayanan S. Application of random vibration. Narosa Publishing House; 1994.[6] Lin YK, Cai GO. Probabilistic structural dynamics: advanced theory and application. McGraw-Hill; 1995.[7] Vanmarcke EH, Shinozuka M, Nakagiri S, Schueller GI, Grigoriu M. Random fields and stochastic finite elements. Struct Saf

1986;3(3):143–66.[8] Benaroya H, Rehak M. Finite element methods in probabilistic structural analysis: a selective review. Appl Mech Rev

1987;41(5):201–13.[9] Spanos PD, Ghanem RG. Stochastic finite element expansion for random media. ASCE J Engng Mech 1989;115(5):1035–53.[10] Kleiber M, Hien TD. The stochastic finite element method. John Wiley and Sons Ltd; 1992.[11] Schueller GI. A state-of-the–art report on computational stochastic mechanics. Prob Engng Mech 1997;12(4):197–321.[12] Kotulski Z, Sobczyk K. Effects of parameter uncertainty on the response of vibratory systems to random excitation. J Sound Vibr

1986;6, 119(1):159–71.[13] Igusa T, Kiureghian AD. Response of uncertain system to stochastic excitation. ASCE J Engng Mech 1988;114:812–32.[14] Chang T-P, Chang H-C. Perturbation method for probabilistic dynamic finite element analysis of a rectangular plate. Mech Struct

Mach 1997;25(4):397–415.[15] Manohar CS, Ibrahim RA. Progress in structural dynamics with stochastic parameter variations: 1987–1998. Appl Mech Rev

1999;52(5):177–97.[16] Muscolino G, Ricciardi G, Impollonia N. Improved dynamic analysis of structures with mechanical uncertainties under deterministic

input. Prob Engng Mech 2000;15:199–212.[17] RahamanMS, Hwang CW. A SFE analysis for the seismic response of soil sites. Stochastic structural dynamics, vol. 2. New Practical

Publishers; 1990. p. 156–86.[18] Ambros L, Katafygiotis S, Papadimitriou C. Dynamic response variability of structures with uncertain properties. Earthquake Engng

Struct Dynam 1996;25:775–93.[19] Chakraborty S, Dey SS. Stochastic finite element simulation of uncertain structures subjected to earthquake loading. Shock Vibr

2000;5:309–20.[20] Pradlwarter HJ, Schueller GI. Analysis of containment structures of NPP�s. In: Schuelelr GI, editor. Structural dynamics, recent

advances. Springer-Verlag; 1990. p. 467–75.[21] Wall FJ, Bucher CG. Exceedance rate including system uncertainties. In: Schueller GI, editor. Structural dynamics, recent

advances. Springer-Verlag; 1990. p. 467–75.[22] Zhang Y, Web B, Liu Q. First passage of uncertain single degree-of-freedom nonlinear oscillations. Comput Meth Appl Mech Engng

1998;165:223–31.[23] Balendra T, Quek ST, Teo YP. Time variant reliability of linear oscillator accounting for uncertainty in structures and input model

parameters. Prob Engng Mech 1991;6(1):10–7.[24] Yang CY, Cheng AH-D. Integrated earthquake reliability design of concrete arch dam-reservoir systems. Computational stochastic

mechanics. New York: Elsevier Science; 1993. p. 661–73.[25] Zhato YG, Ono T, Idota H. Response uncertainty and time-varaint reliability analysis for hysteretic MDF structures. Earthquake

Engng Struct Dynam 1999;28:1187–213.[26] Paolo Venini, Caludi Mariani. Reliability as a measure of active control effectiveness. Comput Struct 1999;73:465–73.[27] Conte JP, Peng BF. Fully nonstationary analytical earthquake ground-motion model. ASCE J Engng Mech 1997;123:15–24.[28] Corotis RB, Vanmarcke EH, Cornell CA. First passage of nonstationary random process. ASCE J Engng Mech

1972;98(EM2):401–15.[29] Vanmarcke EH. On distribution of the first-passage time for normal stationary random processes. J Appl Mech, ASME

1975;42:215–20.[30] Michaelov G, Sarkani S, Lutes LD. Spectral characteristics of nonstationary random processes – a critical review. Struct Saf

1999;21:223–44.[31] Gupta ID, Trifunac MD. Investigation of nonstationarity in stochastic seismic response of structures. Report No. CE 96-01,

University of Southern California, Department of Civil Engineering; 1996.[32] Holman RE, Hart GC. Nonstationary response of structural systems. ASCE J Engng Mech 1974;100(EM2):415–31.[33] Peng BF, Conte JP. Closed from solutions for the response of linear systems to fully nonstationary earthquake excitation. ASCE J

Engng Mech 1998;124:684–94.[34] Chaudhuri A, Chakraborty S. Sensitivity evaluation in seismic reliability analysis of structures. Comput Meth Appl Mech Engng

2004;93:59–68.[35] Wen YK, Ellingwood BR, Veneziane D, Bracci J. Uncertainty modelling in earthquake engineering, MAE Centre Project FD-2,

Report; 2003.[36] Vamnarck EH. Random field analysis and synthesis. The MIT Press; 1983.[37] Bhattacharya B, Chakraborty S. Sensitivity statistics of 3-D structures under parametric uncertainty. ASCE J Engg Mech

2002;127(9):909–14.

246 A. Chaudhuri, S. Chakraborty / Structural Safety 28 (2006) 231–246