# Postbuckling and Free Vibration of Multilayer Imperfect Nanobeams under a Pre-Stress Load
## 3 Static Analysis
### Postbuckling of Imperfect, Multilayer Nanobeams under Pre-Stress Loading  
In this section, we present the postbuckling response of clamped-clamped nanobeams with imperfection. The beam is supposed to be under compressive pre-stress load due to manufacturing residual stresses. If this pre-stress load is beyond the buckling load, the beam buckles while its ends are kept fixed. As a result, a tensile force is created at the midplane due to bending. The net compressive force at the beam ends does not remain constant as the compressive pre-stress is reduced by the amount of the midplane stretching. First, we investigate the significance of the small size on the postbuckling for an initially straight column. Figure 3 shows the buckling amplitude as the pre-stress load is increased. Because the beam is perfect, the buckling point is called bifurcation or Euler buckling, where two symmetric stable equilibrium positions develop and grow gradually with the load. The initial straight position becomes unstable after buckling. The figure shows that as the size parameter becomes significant, i.e., size gets smaller and smaller, the buckling load reduces and more buckling amplitude is obtained at the same load. This is due to the softening effect of the nonlocal effect as highlighted before. As pointed out earlier, as the beam exhibits imperfection, the onset of buckling occurs at a limit point rather than a bifurcation point. Figures 4-7 show the postbuckling response of a beam with \(\alpha=1\), initial imperfection, \(a=1\), and size parameter, \(\overline{\mu}=0,\ 0.02,\ 0.04,\ 0.06\), respectiv \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 & 71.8484 & 42.4339 & 30.2078 & 23.3849 & 18.9855 & 15.8917 \\
2 & 78.9568 & 42.4957 & 27.0549 & 18.3317 & 12.6551 & 8.6339 \\
3 & 72.6622 & 30.2908 & 12.1536 & 1.8365 & \(-\)4.9113 & \(-\)9.7103 \\
4 & 54.5237 & 6.8768 & \(-\)13.6671 & \(-\)25.4069 & \(-\)33.1108 & \(-\)38.6041 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Non-dimensional buckling load for a variety of initial curvatures and size parameter for \(a=2\). Figure 2: Variation of the buckling load with the imperfection amplitude for \(\alpha=1\). solid lines represent stable positions and dashed lines represent unstable positions. The load-deflection curves show the limit point where buckling occurs. It can be noticed that for the case in hand and for all size parameters the buckling load is greater than the Euler buckling load. As the beam size becomes smaller, the buckling load decreases and the buckling amplitude increases as a result of the nonlocal effect. In each figure, we plot the load-deflection curves for the perfect beam having the same size for the sake of comparison. To investigate the significance of the lamination, we present the postbuckling response for (\(\alpha=3.434\)) for the same imperfection amplitude (\(a=1\)), as shown in Figure 8. As can be noted, the buckling load is higher than the Euler buckling as long as the imperfection amplitude is \(a=1\). The nonlocal effect of the size reduces the buckling load. The postbuckling response for beams with \(\alpha=3.434\) shows a smaller buckling amplitude compared with the case where \(\alpha=1\), shown in Figure 9. In order to see cases where the buckling load becomes less than the Euler buckling load, we use an imperfection amplitude of \(a=4\). Figure 9 shows the load-deflection curves for beams with \(\alpha=1\), an imperfection amplitude of \(a=4\), and a nonlocal parameter of \(\overline{\mu}=0,\ 0.02,\ 0.04,\ 0.06\), respectively. For the case where \(\overline{\mu}=0.06\), the buckling amplitude is less than the Euler buckling load, as can be noticed from Figure 9d.