# Size-Dependent Mechanical Behaviors of Defective FGM Nanobeam Subjected to Random Loading
## 3 Deterministic Static Analysis of FGM Timoshenko Nanobeam
### Material Parametric Study of the Bending Response  
To verify the pressed method, SFPM is a good alternative choice to FEM, in terms of accuracy, and a numerical example of FGM Timoshenko nanobeam subject to uniform load is investigated. An FGM Timoshenko nanobeam with rectangle section is examined, where \(b/h=1\) and \(L/h=10\). Then, the displacement results are compared with results of previous published results, which are defined in dimensionless, i.e.,  
\[w_{\text{max}}=100\delta_{\text{max}}\frac{EI}{q_{0}L^{4}} \tag{31}\]  
The model describing the FGM nanobeam can be degenerated into that to describe the nanobeam made up of single material, i.e., \(n=0\). The parameters of this case are as follows, [PERSON]'s modulus \(E=30\times 10^{6}\) Pa, [PERSON]'s ratio \(v=0.3\), and \(G\!=\!11.538\,\times\,10^{6}\) Pa. Actually, \(\delta_{\text{max}}\) in (31) is the max deflection of the nanobeam, which is related to the [PERSON]'s modulus \(E\). The deflection \(\delta_{\text{max}}\) is inversely proportional to [PERSON]'s modulus \(E\). Thus, \(w_{\text{max}}\) is independent on [PERSON]'s modulus \(E\). To illustrate the effectiveness and accuracy, a simple supported nanobeam, made up of a single material, is investigated, and the results are listed in Table 2. It is observed that the dimensionless deflection at mid-span of the nanobeam, obtained by SFPM and FEM, have a good agreement with those calculated by analytical method. Furthermore, it can be seen that SFPM had higher accuracy, compared with FEM. Further, an FGM Timoshenko nanobeam is investigated by SFPM and FEM. The dimensionless deflection of the FGM Timoshenko nanobeam obtained by SFPM and FEM are listed in Table 3. As the results show, the deflection calculated by the two methods have a good agreement. The results of SFPM list in Table 3 are calculated according to the following formula,  
\[\widetilde{w}_{\text{max}}=100\delta_{\text{max}}\frac{E_{\text{m}}I}{q_{0}L^ {4}} \tag{32}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{\(\mu\)} & **Analytical [44]** & \multicolumn{2}{c}{**FEM [17]**} & \multicolumn{2}{c}{**SFPM**} \\ \cline{2-6}  & **Value** & **Value** & **Error** & **Value** & **Error** \\ \hline
0 & 1.3483 & 1.3264 & \(-\)1.62\% & 1.3346 & \(-\)1.02\% \\
1 & 1.4937 & 1.4514 & \(-\)2.83\% & 1.4596 & \(-\)2.28\% \\
2 & 1.6391 & 1.5764 & \(-\)3.83\% & 1.5846 & \(-\)3.33\% \\
3 & 1.7845 & 1.7014 & \(-\)4.66\% & 1.7096 & \(-\)4.20\% \\
4 & 1.9299 & 1.8264 & \(-\)5.36\% & 1.8346 & \(-\)4.94\% \\
5 & 2.0754 & 1.9514 & \(-\)5.97\% & 1.9596 & \(-\)5.58\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of the results of SFPM with those of analytical method and FEM. To further study the effect of the material property on the bending response of the nanobeam, the Young's and shear moduli are taken as uncertain parameters to discuss the effect they have on the dimensionless deflection. Figure 5 demonstrates the variation of the deflection at the midspan point under the varying Young's and shear moduli in the range \([p-\gamma p,\ p+\gamma p]\), where \(p=(E,G)\), and \(\gamma\) is a concept to describe the variation magnitude of material properties. As Figures 5 and 6 shown, some traits can be summarized as follows. 1. The dimensionless deflection of the nanobeam decreases with the increase of the Young's and shear moduli. The reduction of the dimensionless deflection is because the beam becomes stiffer, as contributed by the Young's and shear moduli. It means that the dimensionless deflection of the nanobeam with uncertain material properties is indeterministic. Thus, it is necessary to analyze the response of the nanobeam with defects. 2. Comparatively, the shear modulus has less impact on dimensionless deflection than Young's modulus. Dimensionless deflection is mainly determined by Young's modulus. Thus, the impact of shear modulus on the static response can be ignored during the uncertainty analysis in problems with too many uncertain parameters.