# Flexural Buckling Failure of Rock Slopes: A Review of Influencing Factors and Analysis Methods
## 4 Analysis Methods for Flexural Buckling Failure
### Analytical Methods
In combination with the elastic plate model and the energy equilibrium principle, [PERSON] and [PERSON] [71] systematically investigated the effect of the self-weight component of the rock layer in the direction parallel to the strata, interlayer friction, and cohesion on the buckling stability of stratified rock slopes. Considering that the rock mass will exhibit plastic deformation when the compressive \begin{table}
\begin{tabular}{c c c} \hline \hline
**No.** & **Analytical Approaches** & **Literature Sources** \\ \hline
1 & \(P_{\sigma}=\frac{\pi^{2}\pi Eth^{3}}{6i^{2}}-\frac{2Q}{\sqrt{4}}\) & [PERSON] et al. [23] \\  & \(6(\gamma nH\sin\alpha-\gamma nH\cos\alpha\tan\varphi-c)l_{\sigma}^{3}+3 Ql_{ \sigma}^{2}-\pi^{2}nEh^{3}=0\) & [PERSON] et al. [23] \\ \hline
2 & \(\min\,D_{\sigma}=\sum\limits_{i=1}^{n}\,h_{i}\,l_{\sigma\tau}\) & \\  & s.t. \(3[(\sin\alpha-\cos\alpha\tan\varphi_{n})\sum\limits_{i=1}^{n}\,\gamma_{i}h_{i} -c_{\tau}]l_{\sigma\tau}^{3}+6\pi\sum\limits_{i=1}^{n}\,h_{i}l_{b\sigma}^{2}-2 \pi^{2}\sum\limits_{i=1}^{n}\,E_{i}h_{i}^{3}=0\) & [PERSON] et al. [36] \\  & \(r\sum\limits_{i=1}^{n}\,h_{i}+l_{b\sigma}\sum\limits_{i=1}^{n}\,\gamma_{i}h_{i} \sin\alpha-l_{b\sigma}(\sum\limits_{i=1}^{n}\,\gamma_{i}h_{i}\cos\alpha\tan \varphi_{n}+c_{n})>0\) \\ \hline
3 & \(l_{b\sigma}^{3}+k_{1}l_{b\sigma}^{2}-k_{2}=0\), \(U_{\sigma}=l_{b\sigma}+U_{1}-\frac{(l_{b\sigma}+U_{1})\tan\alpha}{\sqrt{1+n \alpha^{2}}\,\alpha}\) & \\  & _where_\(k_{1}=\frac{2(l\pi\eta)\sin\alpha-\gamma\eta h\cos\alpha\tan\varphi-c)}{-\eta \gamma\eta\sin\alpha+2\pi\eta\cos\alpha\tan\varphi+2c}\), & [PERSON] et al. [70] \\  & \(k_{2}=\frac{8 Er^{2}h^{3}}{(nE\alpha r)^{2}}[\sigma_{1}+0.5\gamma l_{b\sigma} \sin\alpha+(l-l_{b\sigma})(\gamma\sin\alpha-\gamma\cos\alpha\tan\varphi-\frac{ c}{nd})]^{2}\) & [PERSON] et al. [70] \\  & \(U_{1}=\frac{l_{b\sigma}^{3}}{(nE\alpha r)^{2}}[\sigma_{1}+0.5\gamma l_{b\sigma} \sin\alpha+(l-l_{b\sigma})(\gamma\sin\alpha-\gamma\cos\alpha\tan\varphi-\frac{ c}{nd})]^{2}\) & \\ \hline
4 & \(\int_{0}^{l_{\sigma}}\!\!\left[EI(-\frac{Ry-M}{EI})^{2}-\frac{1}{2}R(y^{\prime} )^{2}+Ny+My^{\prime}\right]dx=0\), \\  & _where_\(R=\left[\gamma h\sin\alpha+\gamma hK_{s}\cos\alpha-\frac{N}{x}\tan\varphi\right]x\), & [PERSON] et al. [40] \\  & \(M=\frac{h}{2}\left[(2n-1)\frac{N}{x}\tan\varphi+2c\right]x+\frac{1}{2}Nx\), \\  & \(N=(\gamma h\cos\alpha+\frac{qh}{l}\cos\alpha-\gamma hK_{s}\sin\alpha)x\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Typical analytical approaches based on the multi-layer beam model. stress exceeds its elastic limit, [PERSON] et al. [68] regarded the rock layer as an elastic-plastic plate in the analysis of flexural buckling. Recently, in response to the buckling deformation caused by the rock mass deterioration in the hydro-fluctuation belt of the Wuxia Gorge, [PERSON] et al. [41] proposed a formula for the critical length of buckling segment considering the dynamic deterioration effect of the rock mass based on the elastic-plastic plate model. According to the energy equilibrium of the elastic-plastic plate model, [PERSON] et al. [72] and [PERSON] et al. [73] presented a criterion to judge whether the dip slope is sliding or structurally unstable under the combined effects of self-weight, hydrostatic pressure, and earthquakes, and they subsequently applied this method to analyze three practical buckling cases. Some typical methods for the stability evaluation of flexural buckling developed on the elastic (plastic) slab model are shown in Table 4.