# Numerical Example

For illustrating the proposed formulation, we consider a simple structural system consisting of two components. The component responses, Z1(t) and Z2(t), are assumed to be lognormal random processes. We assume that Z1(t) and Z2(t) arise from a common source of load effects, and are
expressed as

Ziit) = exp[Xi(f) + X2(t) + X3(t)],

Z2(t) = exp[Xi(t) – X2(t) — X3(t)], (35)

and hence, are mutually correlated. For the sake of simplicity, we assume that {Xj (t)}3=1 are mutu­ally independent, zero-mean, stationary, Gaussian random processes, with auto-correlation function given by

Rjj(T) = Sj exp[-в’Т2], j = 1, 2, 3, (36)

where Sj and fij are constants. The prescribed safety levels are assumed to be deterministic and constant over time. The time duration considered is T = 10 s.

First, we develop the marginal extreme value distributions for Z1(t) and Z2(t). Following Equa­tion (21), we get

(Nj{aj)) = T ( ( ої, ЧІІ^)^^рХ2(х2)рх3(хз)йх2йх3, (37)

J-<xJ-<x aZj|x) aj

where, for zj = aj,

X1 = ln[aj ]— X2 — X3, (38)

Pj |x = aj({X 1> + (X 2> + (X 3» = 0, (39)

aiix = aj {aX 1 + aX2+aX 3} (40)

and {pXj (x; )}3=1 are Gaussian pdf with mean zero and standard deviation aj. It can be shown that Equation (37) can be further simplified to the form

/ pZ і |x 1

{Nj(af)) = TV*., Ф^ — (pxM’))- (41)

j Va^ j ix aj

The analytical predictions for the failure probability for Z1(t) and Z2(t), given by Equation (41), for various levels of a1 and a2 are compared with those obtained from full scale Monte Carlo simulations in Figures 2 and 3. respectively. The accuracy of the analytical predictions are observed to be acceptable.

Next, we construct the joint extreme value distribution for Z1(t) and Z2(t). For this example, Equation (29) can be written as

where, for Z1 = a1 and Z2 = a2,

 Fig. 2. Probability of exceedance, Pf, for Z(t) across threshold level a.

 Fig. 3. Probability of exceedance, Pf2, for Z2(t) across threshold level a^.

and F(t) = S0(t){si(t) + S2(t) + S3(t) + S4(t)}. Here,

S0(t) = 0.25 (жє22)-3/2,

ґ л 3/2

si(t) = nci2c22 ,

s2(r) = 2{c2C\ – с2С22)^ЛС22/{СиС22 – cf2), s3{r) = 2cj2(c22 – c2l2/cu)^Jnc22/(cuc22 – c2),

54(r) = 2c22cl2^{2cl2/[f^^ c2)(2 + 2cj2/(сцс22 – с2Щ + tan-1[ci2A/cnc22 – c22],

g1 = g2 = g3 = ai and hi = a2, A2 = h3 = – a2, and pjj(r) =

-d2Rjj(t)/dt2, (j = 1, 2, 3). The inner integral in Equation (42), given by

can be evaluated numerically. This leads to the following simplified form for Equation (42):

The joint extreme value distribution function for Z1(t) and Z2(t) are computed analytically and is shown in Figure 4.

The marginal distribution for the exceedance probability, for various threshold levels of a1, have been shown in Figure 5. In the same figure, the corresponding conditional exceedance probability, when conditioned on various threshold levels a2, have also been shown. The significant levels of difference in the probability levels indicate the importance of the correlations that exist between the extreme values of Z1(t) and Z2(t). The results obtained from Monte Carlo simulations are also shown in the same figure and are observed to have close resemblance with the analytical predictions. The corresponding results for Z2(t) are shown in Figure 6.

The analytical predictions are compared with those obtained from Monte Carlo simulations carried out on an ensemble of 5000 samples of time histories for Z1(t) and Z2(t). These results have been shown in Figure 6.