# Slope Failure of Noncohesive Media Modelled with the Combined Finite-Discrete Element Method
## 3 Example I: Rectangular Noncohesive Soil Heap under Gravity  
In this example, an initially rectangular soil area (4 m \(\times\) 2 m) subject to gravity is modelled with the FDEM (see Figure 3). It should be noted that in the current 2D FDEM program, only constant strain triangular (CST) elements are available, and thus entities are all meshed with triangular elements.  
There are 7234 elements in total, with 7142 elements for rectangular soil heap and 92 elements for support. Characteristic element size for soil heap is 0.048 m. The support is made of steel. Soil and steel material parameters are given in Table 1 from reference [5]. Friction coefficient between soil-soil and soil-steel are defined as 0.6. The support is fixed and gravity (9.8 m/s\({}^{2}\)) is applied to the soil. Time step is 0.1 \(\upmu\)s. Since the FDEM program used is nonparallel and the CPU frequency is not high, it costs 298.6 hours to bring the simulation to 2.5 s on the previously mentioned computer. However, the parallel version of the program with GPU acceleration is under testing and will come out soon.  
It is understood that since there is no support for the right side of the noncohesive soil, it will collapse naturally under gravity. Figure 4a shows the failure process of the soil at representative time points from the FDEM modelling. Corresponding results simulated by SPH [5] are given opposite in Figure 4b for comparison. It can be observed from Figure 4 that good agreement is reached. Large deformation and failure of soil have been well simulated out within the FDEM framework. Soil moves to the right and a repose angle \(\psi\) has been formed in the FDEM simulation. With the time elapsing, the angle of repose gets smaller. A summary on the repose angle is tabulated in Table 2.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Soil** & **Steel** \\ \hline Young’s modulus (MPa) & 1.8 & 200,000 \\ Poisson’s ratio & 0.3 & 0.3 \\ Density (kg/m\({}^{3}\)) & 1850.0 & 7800.0 \\ Friction angle (\({}^{\circ}\)) & 25 & - \\ \hline \hline \end{tabular}
\end{table}
Table 1: Material properties of soil and steel.  
Figure 3: Mesh configuration of the noncohesive soil in the FDEM modelling (unit: m).  
Figure 2: Two adjacent discrete elements in noncohesive media: (**a**) initial; (**b**) separated.  
Besides the repose angle, failure processes produced by both the FDEM and the SPH agree with each other well. Since no cohesion is considered in the FDEM modelling, soil acts as granular material. At the early stage of the failure (t = 0.8 s), a steeper slope is formed with smaller horizontal distance \(L\). Later, soil moves rightwards gradually and eventually rests after t = 1.8 s. A graph on the evolution of \(L\) versus time is plotted in Figure 5. Similar results were also obtained using the SPH, and the horizontal distance (\(L\)) that soil collapses to the right is close to that in the FDEM simulation. Through comparison, results of the FDEM modelling on noncohesive soil failure are highly consistent with those from SPH modelling. In spite of the consistency on the results from the two methods, the solid-mechanics-based FDEM has advantages over the SPH method which is based on hydrodynamics in simulating the behaviour of solid materials, like slope instability and dynamic sliding motion in this study. Moreover, engineering applications of SPH method are limited due to its own limitations, such as difficulty in setting boundary conditions and high computational cost.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Time (s)**} & \multicolumn{2}{c}{**Repose Angle \(\boldsymbol{\psi}\) (\({}^{\circ}\))**} \\ \cline{2-3}  & **FDEM** & **SPH** \\ \hline
0.8 & 34 & 39 \\
1.1 & 24 & 23 \\
1.4 & 21 & 20 \\
2.5 & 20 & 17 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison on the repose angle at different time points.  
Figure 4: Snapshots of failure process of noncohesive soil at different instants of time: (**a**) FDEM; (**b**) SPH (after [5]).