## 5 Numerical Example
### Cases of Time-Dependent Reliability Evaluation via Available Quantified Results  
Here, a 12-layer composite laminates plate \(\left[\theta/\theta/\theta-\theta/-\theta\right]_{sym}\) under one concentrated load in the geometric center with four clamped supported sides is studied. The length \(l\) and width \(b\) of the plate are \(l=b=120\)mm, with a thickness of \(t=0.3\)mm \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \begin{tabular}{c} **The Grey** \\ **Constant** \\ **Coefficient** \\ \end{tabular} & \begin{tabular}{c} **The Mean** \\ **Value** \\ \end{tabular} & \
\begin{tabular}{c} **The Lower** \\ **Bound** \\ \end{tabular} &
\begin{tabular}{c} **The Upper** \\ **Bound** \\ \end{tabular} \\ \hline \(E_{1}\) & \(c_{E_{1}}=2.8361\) & \(E_{1}^{c}=131.2819\) & \(E_{1}^{L}=117.7189\) & \(E_{1}^{U}=144.8448\) \\ \(E_{2}\) & \(c_{E_{2}}=2.7582\) & \(E_{2}^{c}=9.3050\) & \(E_{2}^{L}=8.1233\) & \(E_{2}^{U}=10.4867\) \\ \(\
u_{12}\) & \(c_{\
u_{12}}=2.6793\) & \(\
u_{12}^{U}=0.3263\) & \(\
u_{12}^{U}=0.2754\) & \(\
u_{12}^{U}=0.3771\) \\ \(G_{12}\) & \(c_{G_{12}}=2.5181\) & \(G_{12}^{c}=5.1456\) & \(G_{12}^{c}=4.5625\) & \(G_{12}^{U}=5.7287\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Quantification results for static uncertainty characteristics \(E_{1}\)\(E_{2}\), \(\
u_{12}\), and \(G_{12}\). Figure 5: Quantification results for dynamic uncertainty characteristics \(F(t)\) (Case (**a**)–(**d**)). for each lamina. The material of the lamina is transversely isotropic with a density of \(\rho=1.38\times 10^{3}\)kg/m\({}^{3}\). Table 3 shows the basic strength variables of this composite plate, in which the strength degeneration index is defined as \(\alpha(t)\) (\(t\in[0,\ T],\ T=20\) years). In common deterministic analysis, the key point of the first-layer failure judgment relies on solving the allowable load \(F_{allow}\), which can be confirmed by finite element analysis. In this problem, each lamina is divided into 2500 elements. Therefore, there are 30,000 elements for the whole plate (as shown in Figure 6), which may guarantee a high accuracy result. According to the Tsai-Wu strength criterion, \(F_{allow}\) can be obtained from  
\[F_{allow}=\min(F^{i},T) \tag{59}\] \[s.t.\ F_{1}(T)\cdot\sigma_{1}(F^{i},T)+F_{2}(T)\cdot\sigma_{2}(F ^{i},T)+F_{11}(T)\cdot\sigma_{1}^{2}(F^{i},T)+F_{22}(T)\cdot\sigma_{2}^{2}(F^ {i},T)\] \[+F_{66}(T)\cdot\tau_{12}^{2}(F^{i},T)+2F_{12}(T)\cdot\sigma_{1}(F ^{i},T)\cdot\sigma_{2}(F^{i},T)=1\]  
where \(F_{1}(T)=\alpha(T)\cdot\left(\frac{1}{X_{t}}-\frac{1}{X_{c}}\right)\), \(F_{2}(T)=\alpha(T)\cdot\left(\frac{1}{Y_{t}}-\frac{1}{Y_{c}}\right)\), \(F_{11}(T)=\frac{\alpha(T)}{X_{t}\cdot X_{c}}\), \(F_{22}(T)=\frac{\alpha(T)}{Y_{t}\cdot X_{c}}\), \(F_{66}(T)=\frac{\alpha(T)}{S^{2}}\), \(F_{12}(T)=-\frac{\alpha(T)}{2\sqrt{X_{t}\cdot X_{c}\cdot Y_{t}\cdot Y_{t}}}\), \(F^{i}\) means the failure load of the \(i\)th lamina, \(\sigma_{1}(F^{i},T)\), \(\sigma_{2}(F^{i},T)\), and \(\tau_{12}(F^{i},T)\) are the longitudinal normal stress, the transverse normal stress, and the shear stress under \(F^{i}\) and \(T\), respectively. Nevertheless, once the uncertainties, which exist in material properties, are considered (from the quantitative results in Table 2), it will be more complicated to solve Equation (59), and the allowable load \(F_{allow}\) will be converted into an interval process, rather than a specific value any more. In such circumstances, the proposed uncertainty propagation method cited in Section 3.1 can be directly adopted by a combination of the finite difference conceptions and the interval mathematics, and the fast boundary solutions with changingly angle \(\theta\) are given. Figure 7 summarizes some results of \(F_{allow}^{L}(t)\) and \(F_{allow}^{U}(t)\) under typical cases of \(\theta\).