# Random Field-Based Time-Dependent Reliability Analyses of a PSC Box-Grider Bridge
## 3 Structural Reliability Considering Random Field  
A critical issue of structural reliability analysis is to accommodate the calculation accuracy and efficiency. For large and complex structures with very small failure probabilities, the direct Monte-Carlo sampling (MCS) method requires a large number of random samples to ensure estimation accuracy, which leads to low calculation efficiency and is difficult to apply in practice. To minimize the required number of samples, an IS method [31] is used to analyze the structural reliability. The basic idea of this method is to change the sampling center, which would be established at the most likely failure point of the structure (i.e., \"checking point\") to improve the chance of the sample points falling into the failure region. The other independent random variables of the structure are assumed as \(R_{j}\) (\(j\) = 1, 2,..., \(m\)) represented in the vector \([R]\) = \([R_{1}\), \(R_{2}\),..., \(R_{m}]\). The total random variables considered in the reliability analysis constitute the vector \([X]\) = \([H]\) + \([R]\), including the random variables discretized from random fields. The failure probability based on the IS method can be written in the following form:  
\[P_{f}=\int_{-\infty}^{+\infty}\frac{I[G(X)]f(X)}{\rho(X)}\rho(X)\mathrm{d}X=E \bigg{\{}\frac{I[G(X)]f(X)}{\rho(X)}\bigg{\}}, \tag{12}\]  
where \(G(X)\) = performance function of the structure;\(f\) (\(X\)) = joint probability density function; and \(\rho(X)\) = IS density function. The specific expressions of these three functions are shown below. \(I\) [\(G(X)\)] denotes the indicator function, where \(I\) [\(G(X)\)] = 1 if [\(G(X)\)] \(\leq\) 0 and \(I\) [\(G(X)\)] = 0 if [\(G(X)\)] > 0. Applying the probability distribution of \(\rho(X)\), the \(N\) sets of samples for the random variables \(X_{k}\) (\(k\) = 1, 2,..., \(N\)) are generated, and then the failure probability of the structure can be estimated as follows:  
\[\hat{P}_{f}=\frac{1}{N}\sum_{k=1}^{N}\bigg{\{}\frac{I[G(X_{k})]f(X_{k})}{\rho( X_{k})}\bigg{\}}. \tag{13}\]  
For the IS method, \(\rho(X)\) plays a crucial role in the calculation accuracy and efficiency of structural reliability, and its sampling center is established at the checking point. When the performance function of a structure is described as an explicit function of random variables, the checking point is obtained by solving a constrained optimization problem. However, the performance function is usually implicit in the probabilistic FE method. Moreover, when the reliability analysis takes into account the influence of the random field, the number of random variables will be generally very large after the discretization of the random fields. As a result, the structural performance function will hardly be expressed as the explicit function of random variables. The approximate checking point can be found via a search process based on a certain number of Monte-Carlo samplings (MCSs) and cyclic finite element analysis, as presented in Figure 4.  
As shown in Figure 4, the proposed reliability analysis method considering the influence of the random field mainly includes three steps: Discretization of the random field, determination of the checking point, and calculation of the failure probability. First, the random field of a bridge structure should be discretized into a set of correlated random variables by the LAM (represented as \([H]\) = \([H_{1}\), \(H_{2}\),..., \(H_{m}]^{\mathrm{T}}\)), and then the covariance matrix \([C_{H}]\) of \([H]\) can be  
following conditions are satisfied: \(G\left(\underline{H}^{*},\underline{R}^{*}\right)\approx 0\), which guarantees that the checking point is located on the failure boundary of the structure; and the increment in \(H_{i}^{*}\) and \(R_{i}^{*}\) between two iterations is less than the predefined errors, \(\varepsilon_{i}\) and \(\tau_{j}\), respectively. The final \(\underline{H}^{*}\) (or \(\underline{Y}^{*}\)) and \(\underline{R}^{*}\) can be used as the approximate checking point.