## 2 Linear Eigenvalue Analysis for Buckling  
Within the analysis of equations systems and matrix, use of eigenvalues and eigenvectors is a well-common practice. Linear tensile buckling can be defined as an eigenvalue problem. In this context, the eigenvectors correspond to the various buckling modes, while the eigenvalues indicate the loads that lead to buckling [29]. \[([K]-\lambda_{i}[S])\{\Psi\}_{i}=\{0\} \tag{1}\]  
As the first step in the buckling analysis, the given equation is composed of:  
* [\(K\) ] is the stiffness matrix. * [\(S\) ] is the stress stiffness matrix. * \(\lambda_{i}\) is the corresponding eigenvalue, which is utilized to scale the applied loads. The formula below outlines the response of the plate, whether buckled or intact, to uniaxial compression [29]. \[\frac{\partial^{4}w}{\partial x^{4}}+\frac{2\partial^{4}w}{\partial x^{2}\partial y ^{2}}+\frac{\partial^{4}w}{\partial y^{4}}=\frac{12\big{(}1-\
u^{2}\big{)}}{Et^ {3}}\bigg{(}-N_{x}\frac{\partial^{2}w}{\partial x^{2}}\bigg{)} \tag{2}\]  
Main components of ecuation are:  
* \(\
u\), [PERSON]'s ratio. * \(N_{x}\), in-plane force. * \(t\), thickness of the plate. * \(E\), [PERSON]'s modulus. * \(W\), the vertical displacement in the \(z\)-axis of a point positioned on the \((x,y)\) plane. For a plate that is supported on the four sides, the displacement can be approximated as [29]:  
\[w=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}A_{mn}\sin\frac{m\pi x}{a}\sin\frac{n \pi y}{b} \tag{3}\]  
When the \(x\)-axis is parallel to the plate's longer edge and the \(y\)-axis is perpendicular to it, \(m\) and \(n\) stand for the number of half-waves along the \(x\) and \(y\) axes, respectively. In the meantime, \(a\) and \(b\) show the dimensions of the plate in these directions, respectively. A nontrivial solution is obtained in accordance with the boundary requirements [29]:  
\[A_{mn}\bigg{[}\pi^{4}\bigg{(}\frac{m^{2}}{a^{2}}+\frac{n^{2}}{b^{2}}\bigg{)}^{ 2}-\frac{N_{x}}{D}\frac{m^{2}\pi^{2}}{a^{2}}\bigg{]}=0 \tag{4}\]  
where \(N_{x}=\frac{k\pi^{2}D}{b^{2}}\), \(D=\frac{Et^{3}}{12(1-\
u^{2})}\) is the stiffness of the plate, \(k=\big{(}\frac{m}{\pi}+\frac{\pi}{m}\big{)}^{2}\) is the element responsible for buckling, and \(\alpha=\frac{a}{b}\) symbolizes the aspect relationship. A well-known formula provides the critical buckling stress under elastic circumstances in the case of uniaxial compression [25]. \[\sigma_{buck}=\frac{k\pi^{2}E}{12(1-\
u^{2})}\bigg{(}\frac{t}{b}\bigg{)}^{2} \tag{5}\]  
The Johnson-Ostenfeld formula [30], which takes into account both elastic conditions and a plasticity adjustment for buckling strength, is used to determine the critical buckling strength. \[\sigma_{cr}=\left\{ \begin{array}{l}\sigma_{buck}\\ \sigma_{y}\Big{(}1-\frac{\sigma_{y}}{\omega_{buck}}\Big{)}\end{array} \right. \text{for}\;\sigma_{cr}=\left\{ \begin{array}{l}\sigma_{buck}\\ \sigma_{y}\Big{(}1-\frac{\sigma_{y}}{\omega_{buck}}\Big{)}\end{array} \right. \tag{6}\]  
Here, \(\sigma_{y}\) symbolizes the yield stress, \(\sigma_{cr}\) shows the critical buckling stress, and \(p_{r}\) is a measure of influence on plasticity phenomena, typically falling between 0.5 and 0.6. Buckling may arise from in-plane compressive stress brought on by shear pressures. In pure shear situations, this in-plane compressive stress acts at a \(45^{\circ}\) angle to the shear axis [31]. \[\tau_{cr}=\frac{k\pi^{2}E}{12(1-\
u^{2})}\bigg{(}\frac{t}{b}\bigg{)}^{2}\text {with}\left\{ \begin{array}{l}k=5.34+\frac{4}{a^{2}}\\ k=\frac{5.34}{a^{2}}+4\end{array} \right.\text{for}\;\;\; \begin{array}{l} \alpha>1\\ \alpha\leq 1\end{array} \right\} \tag{7}\]  
To clarify the elastic buckling behavior of a simply supported plate under various applied stress components, the following analytical technique is provided [30].