# Forecasting Landslides via Three-Dimensional Discrete Element Modeling: Helong Landslide Case Study
## 3 Methodology  
The stability analysis of a slope is based on LEM. LEM is a classical and widely used method for the calculation of the FOS. LEM is used to statically analyze the slope stability. Because the slip surface is generated by shear damage, the FOS is defined as the ratio of the shear strength to the slide force on the slip surface in the LEM, as shown in Equation (1):  
\[\text{FOS}=\frac{\int(c+\sigma\tan\varphi)dA}{\int\tau dA}, \tag{1}\]  
where \(c\) is the cohesion of material, \(\varphi\) is the internal angle of friction of material, \(\sigma\) is the normal compressive stress on slip surface, and \(\tau\) is the shear strength per unit area on the slip surface. The calculation of forces on the slip surface is based on discrete soil slices (Figure 6). In the LEM, many supposed slip surfaces of the slope are calculated to acquire the FOS, and the most critical slip surface corresponds to the minimum FOS value.  
For post-failure analysis, a DEM model was used. DEMs are built based on [PERSON]'s second law and the force-displacement law, shrinking the study scale into a single particle and using the interactions between particles to reflect macroscopic mass behavior. The method includes two kinds of basic elements, a ball and a wall, which are considered as considers the rigid body. Each element is permitted to overlap with adjacent elements to trigger a normal and a tangential force. The inter-particle contacts are usually represented by elastic springs and viscous dampers (Figure 7a). The vibration motion of the particle contact is decomposed into normal and tangential directions. The equation of normal vibrational movement on contact points is expressed in Equation (2), whereas the tangential vibrational movement on contact points is expressed as tangential sliding and rolling movement (Equations (3) and (4)):  
\[\frac{m_{1,2}d^{2}u_{n}}{dt^{2}}+\frac{c_{n}\text{d}u_{n}}{dt}+K_{n}u_{n}=F_{n}, \tag{2}\]  
\[\frac{m_{1,2}d^{2}u_{s}}{dt^{2}}+\frac{c_{s}\text{d}u_{s}}{dt}+K_{s}u_{s}=F_{s}, \tag{3}\]  
\[\frac{I_{1,2}d^{2}\phi}{dt^{2}}+\left(\frac{c_{s}\text{d}u_{s}}{dt}+K_{s}u_{s} \right)\mathfrak{s}=M, \tag{4}\]  
Figure 6: Slice discretization and slice force in a sliding mass.  
where \(m_{1,2}\) is the equivalent mass of particle \(P_{1}\) and \(P_{2}\); \(I_{1,2}\) is the equivalent moment of inertia of the particle; \(S\) is the rotation radius; \(u_{n}\) and \(u_{s}\) are the normal and tangential relative displacements of the particles, respectively; \(\theta\) is the rotation angle of the particle; \(F_{n}\) and \(F_{s}\) are the normal and tangential components of the external force on the particle, respectively; \(M\) is the external torque received by the particle; \(K_{n}\) and \(K_{s}\) are the normal and tangential elastic coefficients in the contact model, respectively; and \(c_{n}\) and \(c_{s}\) are the normal and tangential damping coefficients in the contact model, respectively. For contact points, the magnitude of the force is determined by the magnitude of the overlap. The displacement and velocity are calculated using the original location and force, which are updated according to [PERSON]'s second law of movement in every time step (Figure 7b). The macroscopic physical-mechanical behavior of material is reflected by the microscopic granular interaction using this method.