## 5 Numerical Example
### Cases of Time-Invariant and Time Variant Uncertainty Quantification Analysis  
As the necessary input conditions for the subsequent reliability assessment of the composite laminates, the achievement of the quantitative study with limited sample data is of great importance. Here, a total of two cases, namely, the static one under 16 groups of actual experimental samples corresponding to the engineering constants \(E_{1}\), \(E_{2}\), \(\
u_{12}\), and \(G_{12}\) of T300/QY8911 (as listed in Table 1, which is referred by [26]) and the dynamic one under six groups of virtual simulation histories subjected to external excitation \(F(t)\) (as shown in Figure 4, which is modified by [37; 38]) are analyzed by utilizing the improved Grey system theory as above discussed. For ease of understanding, we take the longitudinal elastic modulus \(E_{1}\) as an example, and its specific quantification procedure is summarized below. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
u_{12}\) & \(G_{12}\)**(GPa)** & \(E_{1}\)**(GPa)** & **No.** & \(E_{1}\)**(GPa)** & \(E_{2}\)**(GPa)** & \(\
u_{12}\) & \(G_{12}\)**(GPa)** \\ \hline
1 & 129.20 & 9.34 & 0.28 & 5.23 & 9 & 132.19 & 9.07 & 0.30 & 4.85 \\
2 & 131.59 & 9.53 & 0.33 & 4.97 & 10 & 132.00 & 9.73 & 0.35 & 5.00 \\
3 & 130.63 & 9.08 & 0.33 & 5.16 & 11 & 130.39 & 9.21 & 0.34 & 5.34 \\
4 & 132.01 & 9.34 & 0.33 & 5.15 & 12 & 128.28 & 8.67 & 0.33 & 4.98 \\
5 & 131.04 & 8.94 & 0.34 & 5.15 & 13 & 135.30 & 9.18 & 0.32 & 5.13 \\
6 & 120.61 & 9.04 & 0.33 & 4.81 & 14 & 137.33 & 9.28 & 0.33 & 5.25 \\
7 & 127.69 & 8.99 & 0.32 & 5.11 & 15 & 141.69 & 10.73 & 0.31 & 5.47 \\
8 & 133.65 & 9.36 & 0.35 & 5.08 & 16 & 126.91 & 9.39 & 0.33 & 5.45 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Experimental data of the engineering constants for T300/QY8911. Figure 4: \(Cont.\)  
In accordance with Equations (10) and (11), the experimental data points of \(E_{1}\) can be listed as  
\[ \begin{array}{rcl}E_{1}^{(1)}&=&\left\{E_{1}^{(1)}(1),\ E_{1}^{(1)}(2),\ldots, \ E_{1}^{(1)}(16)\right\}\\ &=&\left\{120.61,\ 126.91,\ 127.69,\ 128.28,\ 129.20,\ 130.39,\ 130.63,\ 131.04, \right.\\ &&\left.131.59,\ 132.00,\ 132.01,\ 132.19,\ 133.65,\ 135.30,\ 137.33,\ 141.69\right\} \\ \end{array}  \tag{52}\]  
After the accumulation, the above sequence will be expressed as \(E_{1}^{(2)}\), i.e.,  
\[ \begin{array}{rcl}E_{1}^{(2)}&=&\left\{120.61,\ 247.52,\ 375.21,\ 503.49,\ 632.69,\ 763.08,\ 893.71,\ 1024.75, \right.\\ &&\left.1156.34,\ 1288.34,\ 1420.35,\ 1552.54,\ 1686.19,\ 1821.49,\ 1958.82,\ 2100.51\right\} \\ \end{array}  \tag{53}\]  
Substituting Equation (53) into Equation (13) yields  
\[ \begin{array}{rcl}\Delta_{E_{1}}^{\max}=\max(\Delta_{E_{1}}(l))=\max\!\left( \frac{E_{1}^{(2)}(16)}{16}\cdot l-E_{1}^{(1)}(l)\right)=25.505\end{array}  \tag{54}\]  
In terms of Equations (15)-(17) cited in Section 2.3, we further obtain  
\[ \begin{array}{rcl}\Sigma_{E_{1}}&=&\left[\sigma_{E_{1}}(1),\ \sigma_{E_{1}}(2),\ldots,\ \sigma_{E_{1}}(15)\right]^{T}\\ &=&\left[4.4548,\ 3.8821,\ 3.5529,\ 3.4179,\ 3.4378,\ 3.3984,\ 3.3640, \right.\\ &&\left.3.3556,\ 3.3536,\ 3.3225,\ 3.2893,\ 3.3647,\ 3.5616,\ 3.9061,\ 4.6844\right]^{T}\end{array}  \tag{55}\]  
and  
\[ \begin{array}{rcl}\mathbf{D}_{E_{1}}&=&\left[d_{E_{1}}(1),\ d_{E_{1}}(2),\ldots,\ d_{E_{1}}(15)\right]^{T}\\ &=&\left[1.5750,\ 1.4867,\ 1.3156,\ 1.1856,\ 1.1400,\ 1.1180,\ 1.1339, \right.\\ &&\left.1.1599,\ 1.1846,\ 1.1819,\ 1.1835,\ 1.2188,\ 1.2744,\ 1.3632,\ 1.5941\right]^{T}\end{array}  \tag{56}\]  
After a second accumulation and substitution of Equations (55) and (56) together with (18), the optimal estimation \(c_{E_{1}}^{\varepsilon}\) reads as  
\[ \begin{array}{rcl}c_{E_{1}}\to c_{E_{1}}^{\varepsilon}=\left(\left( \mathbf{D}_{E_{1}}^{(\mathbf{2})}\right)^{T}\cdot\mathbf{D$ }_{E_{1}}^{(\mathbf{2})}\right)^{-1}\cdot\left(\mbox{\boldmath$D}_{E_{1}}^ {(\mathbf{2})}\right)^{T}\cdot\Sigma_{E_{1}}^{(\mathbf{2})}=2.8361\end{array}