# Postbuckling and Free Vibration of Multilayer Imperfect Nanobeams under a Pre-Stress Load
## 3 Static Analysis
### Buckling Load of Imperfect, Multilayer Nanobeams  
In this section, we present the lowest buckling load for imperfect multilayer nanobeams. Namely three parameters are considered: The imperfection amplitude, the small-size parameter, and the lamination coefficient.  
Before the numerical results are presented, we shed light on the range of the nanobeam length considered in this study. The nondimensional size parameter is varied from 0 to 0.1, which varies the beam length up to 6 nm, according to \(e_{0}a=2\) nm. The effects of the initial curvature and size parameter on the buckling load according to Equation (25) are given in Table 1 for \(\alpha=1\). The first row of Table 1 is identical to the results reported by [PERSON] [70] and [PERSON] [42] for nonlocal initially straight beams, which validates the current model. The table shows that as the size parameter increases, the buckling load decreases. On the other hand, the existence of the initial curvature may enhance the buckling load. For instance, the buckling load at \(\overline{\mu}=0.1\) is one-fifth of the classical buckling load (\(\overline{\mu}=0.0\)) for a perfect beam. This means that the buckling load significantly decreases as the nonlocal parameter is increased for a given initial curvature. It is also noticed that the imperfection tends to increase the buckling load up to a certain limit then it inverses its effect. The buckling load increases from 39.48 to 78.86 as the initial curvature increases from 0 to 3 before the buckling loads drops with the increase of initial curvature. At a certain beam curvature, the buckling load may change its sense (i.e., from compression to tension), which is consistent with the shell structure theories [71]. The loads at which the buckling load changes sense are dubbed 'null-loads'. The buckling load for \(\alpha=2\) for different values of the imperfection amplitude and the size parameter is shown in Table 2. First, one can notice the significance of the layup on the buckling load as the two parameters are varied. The beam shows the same trend as the case discussed before.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{1}{c}{**Nondimensional**} & \multicolumn{5}{c}{**Nondimensional Size Parameter (\(\overline{\mu}=\mu/\text{L}^{2}\))**} \\ \cline{2-7}
**Imperfection Amplitude (\(a\))** & **0** & **0.02** & **0.04** & **0.06** & **0.08** & **0.1** \\ \hline
0 & 39.4784 & 22.0603 & 15.3068 & 11.7192 & 9.4939 & 7.9789 \\
1 & 66.6198 & 39.6802 & 28.5831 & 22.4276 & 18.4767 & 15.7086 \\
2 & 76.6099 & 44.0773 & 30.4287 & 22.7649 & 17.8004 & 14.2962 \\
3 & 78.8606 & 41.6370 & 25.8483 & 16.9195 & 11.1047 & 6.9832 \\
4 & 74.6095 & 33.1988 & 15.4999 & 5.4418 & \(-\)1.1318 & \(-\)5.8044 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Nondimensional buckling load for a variety of initial curvatures and size parameter for \(a=1\).  
Now, we turn our attention to the significance of the initial imperfection on the buckling load, according to Equation (26). Figure 2 shows the variation of the buckling load with the beam's imperfection amplitude as the nondimensional small-size parameter that is varied. Typically, the buckling load increases as the imperfection amplitude increases up to a threshold where the buckling load starts to decrease as the imperfection amplitude increases. It can also be noted that beyond the null-load, the buckling load switches from being positive (compressive) to being negative (tensile). The significance of the size parameter is evident from the figure where a considerable reduction in the buckling load is noticed.