# Dynamic Optimization of Buckling Problems for Panel Structures with Stiffening Characteristics
## 2 Panel Structure with Stiffening Characteristics
### Buckling  
Structural buckling refers to the sudden change in the original deformation mode of a structure subjected to compressive loads during the loading process, resulting in a transition to another equilibrium state. Buckling can significantly reduce the load-bearing capacity, which makes the structure lose its load-bearing capacity before reaching the strength limit of its material.  
The eigenvalue buckling analysis method is also known as a kind of linear buckling analysis method. The eigenvalue buckling analysis method considers the stress stiffening effect of the structure. As the compressive loads gradually increase, the ability of the structure to resist lateral loads gradually decreases. If the compressive loads increase to a certain extent, the stress stiffness will exceed the structural stiffness, so it will lead to the buckling behavior of the structure.  
Similarly to in dynamic characteristic analysis, based on the principle of minimum potential energy, the governing equation of elastic stability analysis can be written as [23].  
\[(\mathbf{K}+\mathbf{K}_{\sigma})\delta=\mathbf{P}. \tag{10}\]  
Here, \(\mathbf{P}\) is the external force vector. \(\mathbf{K}_{\sigma}\) is the geometric stiffness matrix, which is determined by stress and displacement, as shown in Equation (11).  
\[\mathbf{K}_{\sigma}=\int_{\Omega}\mathbf{B}^{T}\sigma\mathbf{B}\mathrm{d}\Omega \tag{11}\]  
Figure 6: Natural modes of the panel with ribs and frames: (**a**) first-order mode; (**b**) second-order mode; (**c**) third-order mode; (**d**) fourth-order mode; (**e**) fifth-order mode; (**f**) sixth-order mode.  
[PERSON]'s stability problem refers to a situation where any disturbance to the critical displacement of a system can cause it to lose stability when its initial stress state is in a critical state. The initial stress is \(p\sigma\), while the geometric stiffness matrix is \(p\mathbf{K}_{\sigma}\); then, Equation (10) can be written as  
\[(\mathbf{K}+p\mathbf{K}_{\sigma})\delta=\mathbf{P} \tag{12}\]  
Assuming that a critical state is reached at this point, there exists a perturbed displacement state \(\delta+\Delta\delta\) that causes the system to remain in equilibrium even under constant external forces:  
\[(\mathbf{K}+p\mathbf{K}_{\sigma})(\delta+\Delta\delta)=\mathbf{P} \tag{13}\]  
Then subtracting Equations (12) and (13) can yield a generalized characteristic equation similar to dynamic characteristic analysis.  
\[(\mathbf{K}+p\mathbf{K}_{\sigma})\Delta\delta=0 \tag{14}\]  
The condition for Equation (14) to have non-zero solutions is that the coefficient determinant is equal to zero. According to this condition, it can be transformed into a standard eigenvalue problem.  
\[\left(-\mathbf{K}_{\sigma}{}^{-1}\mathbf{K}\right)\delta=p\delta \tag{15}\]  
Here, the generalized eigenvalue is \(p\), presenting the buckling factor in buckling analysis; the eigenvector is \(\delta\), corresponding to the buckling mode in linear buckling analysis.  
The eigenvalue after solving Equation (15) is also called the buckling factor. When the load is multiplied by the buckling factor, the structure will experience buckling failure.  
Tables 5 and 6 show the buckling factors while Figures 7 and 8 show the buckling modes of the present panels.  
With the previous analysis, it is seen that stiffening characteristics can also increase the buckling eigenvalue and improve the buckling resistance of the structure significantly.