# Dynamic Optimization of Buckling Problems for Panel Structures with Stiffening Characteristics
## 4 Results and Discussion
### Further Analysis of Buckling and Discussion  
Considering the nonlinear phenomenon that occurs after optimization, where the first-order buckling eigenvalue increases and the high-order eigenvalue decreases, the nonlinear buckling analysis of the panel with ribs and frames will be conducted below [27].  
Figure 23 shows the process of nonlinear buckling analysis. Firstly, perform linear buckling analysis on the panel with ribs and frames and calculate the first-order buckling mode. Then multiply the 1 st-order buckling mode eigenvector by a coefficient to yield the initial displacement vector applied to the panel with ribs and frames, which is regarded as an imperfection. Finally, analyze the stability of the panel with ribs and frames using nonlinear methods. In the nonlinear analysis section, unlike linear static analysis, the load is loading from zero to the set load size [28]. Some basic parameters are shown in Table 10. The measurement point is shown in Figure 24 and the displacement-load curve calculated from nonlinear buckling analysis is shown in Figure 25.  
Figure 21: Buckling factor of the optimized results with 8 components: (**a**) 1 st-order mode; (**b**) 2 nd-order mode; (**c**) 3 rd-order mode.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Nodes** & **Initial** & **4 Components** & **8 Components** & **16 Components** \\ \hline
1 & 71.8846 & 85.0776 & 85.9576 & 85.1749 \\
2 & 116.488 & 89.8292 & 90.7172 & 91.4845 \\
3 & 128.889 & 107.185 & 103.803 & 107.386 \\ \hline \hline \end{tabular}
\end{table}
Table 9: Buckling factors of the optimized panel with ribs and frames.  
Figure 22: Buckling modes of the optimized results with 16 components: (**a**) first-order mode; (**b**) second-order mode; (**c**) third-order mode.  
Figure 23: The process of nonlinear buckling analysis.  
Figure 24: Measurement point.  
Figure 25: Displacement-load curve.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline Maximum load (N) & 150 \\ Duration (s) & 10 \\ Initial incremental step (s) & 0.01 \\ Minimum incremental step (s) & \(1\times 10^{-6}\) \\ Maximum incremental step (s) & 0.05 \\ \hline \hline \end{tabular}
\end{table}
Table 10: Parameters of nonlinear buckling analysis.  
From Figure 25, it can be seen that the displacement of the panel with ribs and frames initially increases approximately linearly as the load gradually increases. Until the load increases to a certain value, the rate of displacement change increases. And it is considered that the structure has buckled. For different cases, the buckling resistance calculated through nonlinear buckling analysis is close to the buckling factor which is calculated through linear buckling analysis. In a comparison of the displacement-load curves of the panel with ribs and frames before and after optimization, it can be found that the displacement inflection point of the optimized panel with ribs and frames has been improved. This means that the buckling resistance of the structure is improved. The displacement of the measuring point has also decreased compared to the result before optimization.