## 5 Numerical Examples
### Uncertainty Quantification for Elastic Parameters of T300/QY8911 Laminates  
In this section, we will consider the uncertainty quantification of composite laminates at the component scale. In general, as a suitable material for the component design, the desired composite laminate is created by stacking plies in specific directions. The  
Figure 4: Convex polyhedral models of macro elastic mechanical parameters for composites. integration of ply or lamina properties to yield laminate properties is called the laminate theory. Here, the laminate variables such as ply orientations and stacking sequences can be designed to produce a composite laminate with desired material properties. In this section, the elastic mechanical parameter, namely the longitudinal tension modulus \(E_{x}\) of composite laminated plates, whose material is T300/QY8911, will be quantified by virtue of the proposed non-probabilistic quantification method, and the [PERSON]'s model is adopted. The stacking sequence of composite laminates is \(\left[90^{\circ}/0^{\circ}/-45^{\circ}/45^{\circ}/90^{\circ}/0^{\circ}\right]_{S}\). The corresponding uncertain input key parameters at the micro scale are listed in Table 4. Uncertain information can consist of the nominal values, lower bounds and upper bounds of uncertain parameters. One can take use of the non-probabilistic [PERSON]'s model to obtain macro elastic mechanical properties of the composite lamina. Furthermore, the off-axis engineering elastic constants of the laminate can be calculated by the normal-axis engineering constants as:  
\[\frac{1}{E_{x}}=\frac{m^{4}}{E_{1}}+\frac{n^{4}}{E_{2}}+m^{2}n^{2}(\frac{1}{G_ {12}}-\frac{2v_{12}}{E_{1}}) \tag{28}\]  
where \(m=\cos\theta\) and \(n=sin\theta\). \(\theta\) is the corresponding axis-off angle. In order to carry out the uncertainty quantification by virtue of experimental results, the static mechanical performance experiments of laminates are implemented for comparison, with the sketch map of the test specimens plotted in Figure 5. The measured 39 experimental points are listed in Table 5, which can be calculated by:  
\[E_{x}=\frac{\Delta P}{th\Delta\varepsilon} \tag{29}\]  
where \(\Delta P\) is the load increment of the load-strain curve and \(\Delta\varepsilon\) is the corresponding strain increment. \(t\) is the width of the test specimen and \(h\) is the thickness. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variables** & **Nominal Values** & **Lower Bounds** & **Upper Bounds** \\ \hline \(E_{11}^{f}(\text{GPa})\) & 221.00 & 190.06 & 251.94 \\ \(E_{22}^{f}(\text{GPa})\) & 13.80 & 11.868 & 15.732 \\ \(v_{12}^{f}\) & 0.20 & 0.172 & 0.228 \\ \(G_{12}^{f}(\text{GPa})\) & 9.00 & 7.74 & 10.26 \\ \(E_{m}(\text{GPa})\) & 3.60 & 3.096 & 4.104 \\ \(v_{m}\) & 0.30 & 0.258 & 0.342 \\ \(V_{f}\) & 0.65 & 0.60 & 0.70 \\ \(V_{V}\) & 0.005 & 0 & 0.01 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Uncertain information at the micro scale for T300/QY8911 laminates. Figure 5: The sketch map of test specimens for T300/QY8911 composite laminates (mm). Based on these experimental data points, the statistical methods can be utilized to characterize the longitudinal tension modulus \(E_{x}\) as normal variables, and the observed significance level of the Anderson-Darling statistics test is calculated to ensure that the conclusion that the experimental data are submitted to normal distribution is true. Here, the Anderson-Darling test statistic is:  
\[AD=\sum_{i=1}^{n}\frac{1-2i}{n}\{\ln[F_{0}(x_{i})]+\ln[1-F_{0}(x_{n+1-i})]\}-n \tag{30}\]  
in which \(F_{0}\) is the standard normal accumulated distribution function. The corresponding observation significance level is:  
\[\text{OSL}=1/\{1+\exp[-0.48+0.78\ln(\text{AD}^{*})+4.58\text{AD}^{*}]\} \tag{31}\]  
where:  
\[\text{AD}^{*}=(1+\frac{4}{n}-\frac{25}{n^{2}})\text{AD} \tag{32}\]  
Through calculations, the observation significance level for longitudinal tension modulus \(E_{x}\) is \(\text{OSL}=0.735>0.05\). As a consequence, characterizing \(E_{x}\) as a normal random variable is reasonable.