# Flexural Buckling Failure of Rock Slopes: A Review of Influencing Factors and Analysis Methods
## 4 Analysis Methods for Flexural Buckling Failure
### Numerical Simulations  
Currently, numerical research on the flexural buckling of dip rock slopes can be mainly classified into the stability analysis and the run-out analysis. The stability analysis generally accounts for the small deformation of slope, such as the formation of sliding surface and the local bending of rock layers. In terms of run-out analysis, more attention is paid to large deformation and complex motions, such as the fracturing of rock layers and the run-out and accumulation of landslides. Accordingly, there are various numerical methods that are suitable for analyzing the different phases of flexural buckling of dip rock slopes. Mostly, the finite difference method (FDM) and the finite element method (FEM) are capable of modelling the instability process of the flexural buckling of dip slopes. For instance, combining the FDM with the Cosserat continuum model, [PERSON] and [PERSON] [74] deeply investigated the buckling deformation process and failure mechanism of a stratified rock slope, and they found that the explicit joint model would overestimate the critical buckling coefficient of the rock masses with fewer layers. Subsequently, [PERSON] et al. [24] introduced the large deformation Cosserat continuum model into the AFENA code and numerically recreated the entire process of buckling instability of a stratified rock slope. [PERSON] and [PERSON] [26], and [PERSON] and [PERSON] [28] used the finite element software Phase2 to simulate the buckling deformation process of phyllite rock layers, focusing on the influences of the shear and normal stiffness of discontinuities, cohesion, and in situ stress on the buckling failure process of a stratified slope. Recently, [PERSON] et al. [75] regarded the \begin{table}
\begin{tabular}{c c c} \hline
**No.** & **Analytical Approaches** & **Literature Sources** \\ \hline \(1\) & \(\frac{\pi^{2}D}{b^{4}}l_{bcr}^{4}+(H-\frac{\gamma h}{2}\sin\alpha)l_{bcr}^{3}+( \frac{2\pi^{2}D\sqrt{\Psi_{\rm t}}}{b}-IH)l_{bcr}^{2}+\pi^{2}D\psi_{\rm t}=0\) & [PERSON] et al. [68] \\  & _where_\(H=\gamma h(\sin\alpha-\cos\alpha\tan\varphi)-c,D=\frac{EI^{3}}{2(1-\mu^{2})}; \psi_{\rm t}=E_{\rm t}/E\) & \\ \hline \(2\) & \(+\bigg{\{}\frac{[\mu(1+\psi_{\rm t})+2\sqrt{\psi_{\rm t}}(1-\mu)]}{b^{2}}\pi^{2} K(t)-HI\bigg{\}}l_{bcr}^{2}=0\) [PERSON] et al. [41] \\  & _where_\(K(t)=\sum\limits_{i=1}^{n}\frac{E(t)h_{i}^{3}}{12(1-\mu^{2})},H=\gamma h(\sin \alpha-\cos\alpha\tan\varphi)-c,\psi_{\rm t}=E(t)/E\) \\ \hline \(3\) & \((\frac{\pi^{2}D}{b^{4}}-\frac{\gamma_{\rm e}\sin\alpha\tan\varphi}{2})l_{bcr}^{ 4}+(Q+\gamma_{\rm e}w\sin\alpha\tan\varphi-\frac{\gamma h(\sin\alpha+\beta_{ \rm e}K_{\rm e}\cos\alpha)}{2})l_{bcr}^{3}\) \\  & \(+\Big{\{}\frac{\pi^{2}D}{b^{2}}[2\mu\psi_{\rm t}+2(1-\mu)\sqrt{\psi_{\rm t}}]- IQ-\frac{\gamma_{\rm e}\mu^{2}\sin\alpha\tan\varphi}{2}\Big{\}}l_{bcr}^{2}+\pi^{2}D \psi_{\rm t}=0\) & [PERSON] et al. [72] \\  & _where_\(D=\frac{Eh^{3}}{12(1-\mu^{2})},Q=\gamma h(\sin\alpha-\cos\alpha\tan\varphi)+ \beta_{\rm s}K_{\rm s}\gamma h(\cos\alpha+\sin\alpha\tan\varphi),\) \\  & \(\psi_{t}=E_{t}/E\) \\ \hline \end{tabular}
\end{table} 
Table 4: Typical analytical approaches based on the elastic (plastic) slab model. inherent anisotropic characteristics of the rock mass as a discrete crack network, and they implemented numerical analysis of the buckling deformation of the stratified rock mass using the FEM. In terms of numerical modelling of the entire evolution of flexural buckling failure, the Universal Distinct Element Code (UDEC), Particle Flow Code (PFC), Discontinuous Deformation Analysis (DDA), and Numerical Manifold Method (NMM) are more suitable. For example, by using the UDEC program, the buckling instability and failure process of the slope at the Westfield open-pit coal mine was detailed analyzed [76,77]. [PERSON] et al. [78] investigated the effect of long-term weathering on buckling deformation and found that changes in the in situ stress state of a slope and the tensile strength of rock significantly affect the flexural buckling stability of the slope.