# Flexural Buckling Failure of Rock Slopes: A Review of Influencing Factors and Analysis Methods
## 4 Analysis Methods for Flexural Buckling Failure
### Analytical Methods
Based on the multi-layer beam model, [PERSON] et al. [23] derived the analytical solutions for the critical buckling load and the critical buckling length of dip slopes using the energy equilibrium principle. Considering the effects of water pressure and earthquakes, [PERSON] et al. [48] proposed theoretical buckling curves for each rock layer based on special function theory and deeply discussed the buckling failure characteristics of stratified rock slopes. [PERSON] et al. [36] derived analytical solutions for the critical buckling length and critical buckling depth of stratified rock slopes using the energy equilibrium principle and introduced the generalized Hoek-Brown criterion to account for the impact of scale effect on buckling stability. Similarly, utilizing the energy equilibrium principle, [PERSON] et al. [32] \begin{table}
\begin{tabular}{c c c} \hline
**No.** & **Analytical Approaches** & **Literature Sources** \\ \hline
1 & \(l_{cr}^{3}=\frac{\pi^{2}Eh^{2}}{2.25(\gamma\sin\alpha-\gamma\cos\alpha\tan\varphi -\frac{1}{\gamma})}\) & [PERSON] [18] \\ \hline
2 & \(\frac{1}{l_{cr}^{2}}=\frac{1}{l_{cr}^{2}}+\frac{1}{l_{cr}^{2}}\); & \\  & \(\frac{1}{l_{cr}^{2}}=\frac{1}{l_{cr}^{2}}\); & \\
2 & \(\frac{1}{l_{cr}^{2}}=\sqrt{\frac{\pi^{2}Eh^{2}}{1.44(\gamma\hbar\sin\alpha- \cos\alpha\tan\varphi)-c}}l_{cr2}=\sqrt{\frac{\pi^{2}Eh^{2}}{5.88\gamma}}\) & [PERSON] et al. [49] \\ \hline
3 & \(l_{bcr}=\sqrt{C_{1}^{2}+2C_{1}I-C_{1}}\); & \\  & \(\frac{}{}\) & \(\frac{}{}\) & [PERSON] [69] \\ \hline
4 & \(\frac{}{}\) & \(\frac{}{}\) & \\
4 & \(\left[\gamma h\sin\alpha-\gamma h(\cos\alpha-\beta_{s}K_{s}\sin\alpha)\tan \varphi-c\right](l-l_{bcr})\); & \\  & \(\left[\gamma h\sin\alpha-\gamma h(\cos\alpha-\beta_{s}K_{s}\sin\alpha)\tan \varphi-c\right](l-l_{bcr})\); & \\  & \(\left[\tilde{\gamma}h\cos\alpha-\left[\gamma h\cos\alpha-\gamma h\beta_{s}K_{s} \sin\alpha-0.5\gamma_{br}(l_{br}-l_{bcr})^{2}\cos\alpha\right]\tan\varphi-c \right\}(l-l_{bcr})\); & [PERSON] et al. [53] \\ \hline \end{tabular}
\end{table} 
Table 2: Typical analytical approaches based on the single beam (column) model. provided a calculation formula for the critical buckling length of dip slopes, and they verified it in combination with numerical simulation and field data. Later, [PERSON] et al. [70] suggested that critical sliding displacement is more suitable for quantitatively analyzing the progressive evolution process of buckling landslides. They proposed a calculation method for determining critical sliding displacement and established corresponding criterion for assessing flexural buckling stability, taking into account both the tensile and compressive strengths of rock. Based on the mechanical analysis, [PERSON] et al. [40] established the buckling curve equation of multi-layered rock masses under the action of earthquake and top loading and presented the analytical formula of critical slope length for buckling failure using the energy method, as well as the method for determining the position of failure surface according to the maximum deflection point. Then, this approach was successfully applied to typical landslide cases. Typical methods for the stability evaluation of flexural buckling based on the multi-layer beam model are given in Table 3.  
(3) Elastic (plastic) slab model  
Considering the presence of joints, fissures, and fractures in dip rock slopes, the longitudinal and transverse lengths of the buckled rock layers are usually limited. Therefore, simplifying the flexural buckling of dip slopes as thin plate instability problems is more in line with the actual situation. This model regards the rock layer as a plate with a hinged bottom, peripheral roller supports, and finite length, and it can be analyzed according to the stability theory of elastic (plastic) compressed plates.