# Random Field-Based Time-Dependent Reliability Analyses of a PSC Box-Grider Bridge
## 4 Verification of Numerical Examples
### Example Three  
As shown in Figure 9, an elastic rectangular plate is subjected to a distributed load (q = 1 kN/m\({}^{2}\)) on one side, whose side length and plate thickness are 10 and 1 m, respectively, and Poisson's ratio is 0.2. The elastic modulus, \(E\), of the plate is modeled as a 2D uniform random field expressed as \(E=\overline{E}(1+k\lambda_{e}(x,y))\), where \(\overline{E}=1\times 10^{4}\) kN/m\({}^{2}\) and \(k=0.1\). The correlation function is EXP, and the correlation length is described as \(L=d\sqrt{\pi}\), where \(d=1,2,5\). The limit state function of deflection is defined as follows.  
\[G=1.1\times\overline{\delta}_{y}-\delta_{y}, \tag{19}\]  
where \(\overline{\delta}_{y}\) denotes the displacement in the Y direction at the A point when the mean values of the random variables are taken; and \(\delta_{y}\) denotes the displacement in the Y direction at this point.  
Figure 8: Reliability of the continuous beam vs. different correlation models: (**a**) Deflection reliability; (**b**) bending moment reliability.  
The purpose of this example is to investigate the effect of the correlation length on structural reliability. The mesh discretization of the random field of the elastic modulus is the same as that of the FE model of the rectangular plate. The proposed method is adopted to calculate the reliability index of displacement at point A under a distributed load, and the results are compared with those obtained from the PSFE method and the MC method, as shown in Figure 10[34].  
It can be seen from Figure 10 that the reliability index estimated by the proposed method is in agreement with the results obtained by the PSFE method and the MC method (10,000 samples) [34], which shows the validity and correctness of this method. The computation time by using the MC method, PSFE method, and proposed method is 12,208, 428, and 289 ms, respectively, and it can be seen that the proposed method has the highest computational efficiency. It can also be found that the correlation length has a significant effect on the reliability of the plate, and the reliability index decreases rapidly with increasing correlation length. This is because the correlation between two points in the distribution space of the random field increases with an increasing correlation length. In other words, the local variation of the random field will cause a wider range of variations. When the correlation length tends to be infinite, the random field will degenerate into a random variable. For example, the local variation in the elastic modulus will cause random fluctuations in the whole structure, which reduces the structural reliability. It is the main reason that when the structural parameters are considered as random variables, the calculated failure probability is greater than that obtained by considering the spatial randomness of the parameters.  
Figure 10: Reliability of the rectangular plate vs. the correlation length: (**a**) \(3\times 3\) (discrete elements); (**b**) \(5\times 5\) (discrete elements).  
Figure 9: Rectangular plate subjected to a distributed load.