# Size-Dependent Mechanical Behaviors of Defective FGM Nanobeam Subjected to Random Loading
## 6 Conclusions  
Owing to superior mechanical properties, nanostructures have become a hot research area. Further, some of the nanostructures are made of FGMs, due to the superior material properties. Currently, most of the previous work has been performed on the deterministic framework. However, the properties of nanostructures are always uncertain, due to the material defect of the nanostructures. Anyway, the external load applied on nanostructures is always uncertain because of the randomness. Due to multi-source uncertainties existing, the uncertain propagation in nanostructures is necessary to analyze. In this paper, the nanobeam made of FGM is investigated, considering the size-dependent effect, based on non-local elastic theory. The SFPM is established to analyze the bending behavior of the nanobeam-based Timoshenko theory. In comparison with FEM, SFPM has higher accuracy, taking analytical solution as a benchmark. Considering the multi-source uncertainties, the material properties are quantified as interval parameters, due to the material defect, and loads are taken as random parameters. To deal with the problems with two types uncertainties coexisting, a hybrid uncertain analysis model was established, and the PCE-DW was proposed to predict the response of nanobeam in the \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\(\mu\)} & \multicolumn{3}{c}{**Lower Bound**} & \multicolumn{3}{c}{**Upper Bound**} \\ \cline{2-7}  & **PCE–DW (m)** & **MCS (m)** & **Error** & **PCE–DW (m)** & **MCS (m)** & **Error** \\ \hline
0 & \(8.46\times 10^{-9}\) & \(8.38\times 10^{-9}\) & \(0.924\%\) & \(9.35\times 10^{-9}\) & \(9.43\times 10^{-9}\) & \(-0.851\%\) \\
1 & \(9.20\times 10^{-9}\) & \(9.11\times 10^{-9}\) & \(0.921\%\) & \(1.02\times 10^{-9}\) & \(1.02\times 10^{-9}\) & \(-0.399\%\) \\
2 & \(9.94\times 10^{-9}\) & \(9.85\times 10^{-9}\) & \(0.838\%\) & \(1.10\times 10^{-9}\) & \(1.12\times 10^{-9}\) & \(-1.609\%\) \\
3 & \(1.07\times 10^{-8}\) & \(1.06\times 10^{-8}\) & \(0.412\%\) & \(1.18\times 10^{-9}\) & \(1.20\times 10^{-8}\) & \(-1.900\%\) \\
4 & \(1.14\times 10^{-8}\) & \(1.13\times 10^{-8}\) & \(0.622\%\) & \(1.26\times 10^{-8}\) & \(1.28\times 10^{-8}\) & \(-1.275\%\) \\
5 & \(1.21\times 10^{-8}\) & \(1.21\times 10^{-8}\) & \(0.687\%\) & \(1.34\times 10^{-8}\) & \(1.34\times 10^{-8}\) & \(0.470\%\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The bounds of the expectation calculated by PCE–DW and MCS. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\(\mu\)} & \multicolumn{3}{c}{**Lower Bound**} & \multicolumn{3}{c}{**Upper Bound**} \\ \cline{2-7}  & **PCE–DW (m)** & **MCS (m)** & **Error** & **PCE–DW (m)** & **MCS (m)** & **Error** \\ \hline
0 & \(7.31\times 10^{-10}\) & \(7.19\times 10^{-10}\) & \(1.57\%\) & \(8.08\times 10^{-10}\) & \(8.88\times 10^{-10}\) & \(-9.09\%\) \\
1 & \(8.03\times 10^{-10}\) & \(7.49\times 10^{-10}\) & \(7.28\%\) & \(8.88\times 10^{-10}\) & \(9.63\times 10^{-10}\) & \(-7.84\%\) \\
2 & \(8.76\times 10^{-10}\) & \(7.90\times 10^{-10}\) & \(10.95\%\) & \(9.68\times 10^{-10}\) & \(1.08\times 10^{-9}\) & \(-10.20\%\) \\
3 & \(9.49\times 10^{-10}\) & \(8.27\times 10^{-10}\) & \(14.74\%\) & \(1.05\times 10^{-9}\) & \(1.17\times 10^{-9}\) & \(-10.31\%\) \\
4 & \(1.02\times 10^{-9}\) & \(9.51\times 10^{-10}\) & \(7.44\%\) & \(1.13\times 10^{-9}\) & \(1.26\times 10^{-9}\) & \(-10.69\%\) \\
5 & \(1.10\times 10^{-9}\) & \(1.04\times 10^{-9}\) & \(5.12\%\) & \(1.21\times 10^{-9}\) & \(1.32\times 10^{-9}\) & \(-8.34\%\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: The bounds of the variance calculated by PCE–DW and MCS. hybrid uncertain system. Numerical examples ultimately illustrate the effectiveness of the model and solution techniques, compared to MCS. The results furtherly verify the efficiency and accuracy. Locking behavior is a numerical problem that often occurs in the process of finite element analysis. This paper focuses on modeling and solving the uncertainty of the material defects and load dispersion of nanobeams.