# Flexural Buckling Failure of Rock Slopes: A Review of Influencing Factors and Analysis Methods
## 4 Analysis Methods for Flexural Buckling Failure
### Analytical Methods  
At present, analytical methods for evaluating flexural buckling stability of dip slopes mainly adopt three theoretical mechanical schemes, namely, the single beam (column) model, the multi-layer beam model, and the elastic (plastic) slab model, as illustrated in Figure 6. By simplifying the flexural buckling failure of dip slopes under various loads into the instability problems of the previous mechanical models, the deflection equations of the column, beam, or slab models are assumed according to the boundary constraint conditions, and then the explicit or implicit solutions to the critical slope length or critical load are determined based on methods such as the Euler buckling theory, the energy balance principle, the catastrophe theory, or the force balance conditions. Subsequently, the critical slope length or critical load can be adopted to evaluate the flexural buckling stability of dip rock slopes, that is, when the longitudinal load at the slope crest is greater than the critical buckling load (\(P_{cr}\)) or the actual slope length is greater than the critical slope length (\(I_{cr}\)), the slope will undergo buckling. (1) Single beam (column) model  
The instability and evolution process of the flexural buckling of dip slopes may involve a single rock layer or multiple rock layers. The single beam (column) model simplifies the single buckled rock layer or the pack of buckled rock layers as a single beam (column) with assumed end conditions, and the beam (column) stability theory is capable of analyzing the flexural buckling of the dip slope. Adopting the model of column with pinned ends, [PERSON] [18] derived the critical slope length against flexural buckling based on the Euler  
Figure 6: Simplified mechanical model. (**a**) Single beam (column) model (modified from [PERSON] et al. [53]). (**b**) Multi-layer beam model (modified from [PERSON] et al. [36]). (**c**) Elastic (plastic) slab model (modified from [PERSON] et al. [68]). buckling theory. Then, by solving the eigenvalue of the basic differential equation of beam deflection under the combined action of transverse load, axial concentrated load, and distributed load, [PERSON] and [PERSON] [19] proposed the analytical solution to critical buckling load considering the self-weight of rock layer and the hydrostatic pressure. Based on the cusp catastrophe theory, [PERSON] et al. [42] determined the necessary and sufficient conditions for the flexural buckling failure of stratified rock slopes. By considering the effects of earthquakes and hydrostatic pressure along the basal sliding surface, [PERSON] et al. [47] developed an analytical approach in the frame of energy equilibrium theory and analyzed the typical buckling failure of the Tangiiashan landslide. In order to investigate the flexural buckling failure of stratified rock slopes under top loading, [PERSON] et al. [49] presented an explicit analytical approach by combining [PERSON]'s method and the superposition principle. Under equilibrium conditions, [PERSON] [69] evaluated the flexural buckling stability of stratified rock slopes using the tangent-modulus theory and classified the buckling failure modes into three types: brittle, elastic, and inelastic failure. Recently, accounting for the longitudinal and transverse bending of layered rock mass under the combined action of the self-weight of rock layer, the hydrostatic pressure, and the seismic load, [PERSON] et al. [53] presented the conditional equation for the length of the buckling segment based on the elastic column stability theory and energy principles. Representative analytical approaches for the stability evaluation of flexural buckling based on the single beam (column) model are shown in Table 2.  
(2) Multi-layer beam model  
This model regards the pack of buckled rock layers as a multi-layer beam with fixed ends, assuming that multiple rock layers bend in the same way under the combined action of self-weight and external loads, and no relative slip occurs in the buckled rock layers.