## 5 Numerical Example
### Discussions on the Results  
Synthesizing the computational results of the laminate composite example, the following points can be inherited:  
1. From the aspect of uncertainty quantification, the quantitative results derived from the [PERSON] systematic theory (with respect to either the static variables \(E_{1}\), \(E_{2}\), \(\
u_{12}\), \(G_{12}\), or the dynamic process \(f(t)\)) may better envelop the initial data, as is embodied in  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(\theta\) & \(R_{s}(t)\) & \(R_{s}(t\leq 14)\) & \(R_{s}(t=15)\) & \(R_{s}(t=16)\) & \(R_{s}(t=17)\) & \(R_{s}(t=18)\) & \(R_{s}(t=19)\) & \(R_{s}(T)\) \\ \hline \(\theta=15^{\circ}\) & 1 & 0.9997 & 0.9704 & 0.9223 & 0.8998 & 0.8394 & 0.8185 \\ \(\theta=30^{\circ}\) & 1 & 1 & 1 & 0.9993 & 0.9913 & 0.9871 & 0.9771 \\ \(\theta=45^{\circ}\) & 1 & 1 & 1 & 0.9910 & 0.9757 & 0.9526 & 0.9338 \\ \(\theta=60^{\circ}\) & 1 & 0.9918 & 0.9855 & 0.9528 & 0.9398 & 0.9039 & 0.8370 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Non-probabilistic time-dependent reliability results versus different values of ply angel \(\theta\).  
Figure 10: The reliability results comparison of the composite laminates plate between the proposed method and the Monte Carlo method.  
Figure 9: Non-probabilistic time-dependent reliability results versus different values of ply angel \(\theta\).  
Table 2 and Figure 5. Moreover, as an improvement for the traditional Grey model, the way of determination of the Grey constant coefficients is a more precise treatment for UQ analysis under small sample limitations. Indeed, it will, to a certain degree, avoid undesirable interval extension effects by using the relationship of the topological locations among all the sample points/curves.
2. From the point of uncertainty propagation, the developed methodology combined with the state-space transformation and the second-order Taylor expansion may effectively calculate \(F^{L}_{allow}(t)\) and \(F^{U}_{allow}(t)\), although the factors of material dispersion (as summarized in (1)), the material degeneration \(\alpha(t)\), and the changing ply patterns (\(\theta=15^{\circ}\), \(30^{\circ}\), \(45^{\circ}\), \(60^{\circ}\)) are all taken into account. Furthermore, it should also be indicated that, compared with metal structures, the uncertainty influences in composite structures are much more obvious, and the mechanical performances under different ply angels \(\theta\) vary even more (if \(\theta=45^{\circ}\to F^{L}_{allow}(T)=1651.85\ N\), but if \(\theta=60^{\circ}\to F^{L}_{allow}(T)=1161.12\ N\)).
3. From the prospect of safety estimation, the time-dependent reliability obtained by the present analytical method matches the one derived from the Monte Carlo method, as we expected (the accuracy verification). However, there are two aspects that should be stressed: For one thing, the results on the basis of former non-probabilistic method are a little more conservative because of fewer assumptions made about the quantified uncertainties (based on the engineering practice). Furthermore, the accuracy of the reliability results from the Monte Carlo method greatly depends on the amount of samples (1,000,000 samples in the example), which means a computation-intensive and time-consuming situation has to be faced (the superiority of efficiency).