Sayan Gupta1*, Pieter van Gelder1 and Mahesh Pandey2

1 Department of Civil Engineering, Technical University of Delft, The Netherlands
2Department of Civil Engineering, University of Waterloo, Canada
* Corresponding Author: gupta. sayan@gmail. com


Failures in randomly vibrating structures are defined to occur if the response exceeds permissible thresholds, within specified time durations. Estimating the failure probability involves characterizing the probability of the exceedance of the structure response, which are modeled as random processes. An elegant approach for addressing this problem lies in expressing the failure probability in terms of the probability distribution function (PDF) of the extreme values associated with the response.

In structural series systems, failure of any of the individual components signals system failure. The system reliability is thus expressible in terms of the joint probability of exceedance of the component response processes. Often, the loads acting on the various components of a system have common source, and hence, the component responses, and in turn, their extremes, are mutually dependent. This emphasizes the need to characterize the joint PDF of these extreme values for estimating the system reliability.

A common approach in characterizing the extreme value distributions for random processes, is to study the associated first passage failures, based on the assumption that level crossings can be modeled as Poisson counting processes. The parameter of the counting process is related to the mean outcrossing rate, which in turn, can be estimated if the joint probability density function (pdf) of the process and its time derivative, is available (Rice, 1956). For Gaussian random pro­cesses, this is readily available and closed form expressions for the extreme value distributions have been developed (Lin, 1967; Nigam, 1983). This knowledge is, however, seldom available for non­Gaussian processes. A literature review on the various approximations developed for the extreme value distributions for scalar and vector non-Gaussian processes is available (Manohar and Gupta,

2005) . Outcrossing rates of vector random processes have been studied in the context of problems in load combinations and in structural reliability. The focus of many of these problems have been in de­termining the probability of exceedance of the sum of the component processes, and the outcrossing event has been formulated as a scalar process outcrossing. Some of these results have been used in the geometrical approach (Leira, 1994, 2003) in the studies on development of multivariate extreme value distributions for vector Gaussian/non-Gaussian random processes. Multivariate extreme value distributions associated with a vector of Gaussian random processes have been developed (Gupta and Manohar, 2005), based on the principle that multi-point random processes can be used to model the level crossing statistics associated with the vector Gaussian processes. Similar principles have been applied in developing approximations for the multivariate extreme value distributions associated with a vector of non-Gaussian processes, obtained as nonlinear transformations of vector Gaussian processes (Gupta and van Gelder, 2005).

Here, we extend the above formulation to illustrate its usefulness in estimating the reliability of a randomly vibrating structural system, in series configuration. The response of the structural components have been modeled as a vector of mutually correlated log-normal loads and approxim­ations have been developed for the joint extreme value distribution for the response of the structural components. This is of particular importance in the context of risk analysis of nuclear plants, where the dynamic loads arising from various load effects, are modeled as log-normal random processes.


M. Pandey et al. (eds), Advances in Engineering Structures, Mechanics & Construction, 747-759.

© 2006 Springer. Printed in the Netherlands.

Problem Statement

We consider a linear structural system consisting of m components in series configuration. We assume that the structural system is excited by a и-dimensional vector of mutually correlated, sta­tionary, log-normal loads [Yk(t)}nk=1. The structure response of the jth component is given by

Zj(t) = gj [71(f),Yk(t)] = gj [X1(t), …, Xk(t)l (1)

where Yj(t) = eXj(t (j = 1,… ,k) and gj [eX1(t),…, eXk(t)] = gj [X1(t),…, Xk(t)]. Here, {Xj (t)}k=1 constitutes a vector of mutually correlated Gaussian random processes, gj [■] is a determ­inistic nonlinear function which relates the random processes Xj(t) to the component response Zj(t) and t is time. It is clear that Zj(t) is a non-Gaussian process whose probabilistic characteristics are difficult to estimate. A component failure is defined to occur when Zj (t) exceeds specified threshold levels and is given by

Pfj = 1 – P[Zj(t) < aj; Vt є (0, T)], (2)

where aj denotes the threshold level, T is the duration of interest and P [■] is the probability measure. Equation (2) can be recast into the following time invariant format

Pfj = 1 – P[Zmj < aj] = 1 – PZmj (aj), (3)

where Zmj = max0<t<T Zj(t) is a random variable denoting the extreme value of Zj(t) in [0, T] and PZm. (■) is the corresponding PDF. We assume that aj, (j = 1,… ,m) to be high and the spectral bandwidth ratio (Vanmarcke, 1972) of the processes Zj(t), (j = 1,.. .,m) to be such that the outcrossings of Zj(t) can be modeled as a Poisson point process. This leads to the following expression for PZm (aj):

PZm.(aj) = exp[-v+(aj)T ]. (4)

Here, v+ (aj) is the mean outcrossing rate of Zj(t) across level aj. An estimate of v+ (aj) can be determined from the well known expression (Rice, 1956)

p TO

v+ (aj) = J zPZjZj (aj, Z; t, № (5)

where pZjZj(z, z; t, t) is the joint pdf of the process Zj(t) and its instantaneous time derivative Z j(t), at time t. A crucial step in this formulation lies in determining the joint pdf pZjZj(z, z).

For a structural system comprising of m components, the structure is deemed to have failed if any of the constituent m components fail. Thus, the system failure, denoted by Pfs, is expressed as

Pfs = 1 – P [nm=1{Zmj. < aj}] = 1 – PZm1 …Zmm(a1, …, am). (6)

Here, PZmi…Zmm (■) is the m-dimensional joint PDF for the vector of extreme value random variables {Zmj }m=j. Assuming that the respective thresholds, aj, corresponding to each component process Zj(t), are sufficiently high for the respective outcrossings to be rare, the level crossings, denoted by {^j(aj’)}m=1, can be modeled as Poisson random variables. Since the different components have common source of excitations, {Zj (t)}m=1, and, in turn, {^j(aj’)}m=1, are mutually correlated. Consequently, {Zmj }m=1 are also expected to be mutually dependent. This implies the need for devel­oping approximations for the joint multivariate PDF for the level crossings. Based on recent studies (Gupta and Manohar, 2005; Gupta and van Gelder, 2005) we construct the multivariate PDF for the
extreme values of the vector of non-Gaussian random processes [Zj(t)}’J=1. We first illustrate the proposed method for the case when m = 2 and then extend it to the more general multi-dimensional situation.

Bivariate Vector

We first consider the case when m = 2 and Z1(t) and Z2(t) constitute a vector of mutually dependent non-Gaussian random variables, given by

Z1(t) = g[X1(t),…,Xn(t)], (7)

Z2(t) = h[X1(t),…,Xn(t)], (8)

where g[-] and h[-] are deterministic nonlinear functions. Let Ni(«i) and N2(a2) be the number of level crossings for Z1(t) and Z2(t), across thresholds a1 and a2, in time duration [0, T]. For high thresholds, N1(a1) and N2(a2) can be modeled as mutually dependent Poisson random variables. Introducing the transformations,

N1(a1) = U1 + U3,

N2(0,2) = U2 + U3, (9)

where [Uj}3=1 are mutually independent Poisson random variables with parameters [Lj}3=1, it can be shown that N1(a1) and N2(a2) are Poisson random variables with parameters (L1 + L3) and (L2 + L3) respectively and covariance equal to L3. This construct for multivariate Poisson random variables has been discussed in the literature (Johnson and Kotz, 1969). The parameters [Lj }3=1 are, as of yet, unknowns.

Taking expectation on both sides of Equation (9), it can be shown that

"10 1′



0 1 1




■ . (10)

0 0 1



Here, Cov[N1 (a1), N2(a2)] = (N1 (a1)N2(a2)) – (N1(a1)) – (N2(a2)) and if Zj(t) are stationary random processes,



T f nЖ о Ж Л

(T – Iхl)J J Z1Z2PZ1Z2Z1Z2(a1,a2,Z1,Z2; r)dz1dz^ dr, (12)

where t = t2 – t1. Details of the derivation for Equation (12) is available (Gupta and Manohar, 2005). Thus, a solution for [Lj}3=1 can be obtained from Equation (10). Furthermore, it has been shown (Gupta and Manohar, 2005) that the joint PDF for the extreme values are related to [Lj}3=1 through the relation


A crucial step in this formulation, however, lies in evaluating the expressions {N1(a1)), {N2(a2)) and {Ni(ai)N2(a2)), for which, a knowledge of the pdfs pZiZ1 (■), pZiZl(■) and pZ1z2ZjZ2(■), is essential. This, however, is seldom available, especially when Zj(t) are non-Gaussian. In the following section, a methodology has been presented for developing approximate models for these pdfs.