## 4 Engineering Applications
### Finite Element Simulation
#### 4.3.3 Slope Stability Analysis  
There are three main criteria for discriminating slope instability with the strength discounting method, among which the slope displacement inflection point has a clear physical meaning and can be combined with slope displacement monitoring in practical engineering. The strength discounting method in ABAQUS is not applied directly but rather is realized by adding field variable exhaustion to the material properties in the model so that the material properties follow the field variable changes. The linear increase is allowed until the model calculation does not converge, and the factor of safety of the slope is calculated according to the instability assessment criteria [43], as given by the following:  
\[c_{m}=c/F_{r} \tag{3}\]  
\[\varphi_{m}=\arctan(\tan\varphi/F_{r}) \tag{4}\]  
where \(F\) is the factor of safety, \(F_{r}\) is the discount factor and is the discounted strength index obtained by dividing the cohesive force and the angle of internal friction simultaneously by an ever-changing discount factor \(F_{r}\).  
Field variables were added to the materials in the model to discount their shear strength, and the calculations were rerun to obtain a factor of safety for the slope. From the previous analysis, it is obtained that the maximum displacement area is found at the foot of the excavation surface (i.e., the main distribution area of plastic strain); therefore, this is taken as the horizontal displacement characteristic point. The horizontal displacement at the foot of the slope is plotted versus the change in the reduction factor, and the reduction factor at the time of the inflection point mutation is the safety factor of the slope. The relationship curve between the displacement and discount factor was obtained after finite element calculations, and the safety coefficients of the plain soil slope and improved soil slope under rainfall conditions were found to be 1.06 and 1.42, respectively; the safety coefficient of the improved soil increased by 25.35% compared with that of plain soil. The factor of safety for the vegetative soil is close to 1, indicating that the vegetative soil slopes are about to reach a state of limiting equilibrium after rainfall, whereas the improved soil slopes remain in a stable state after rainfall. The addition of fly ash to granite residual soils is recommended to improve their slope stability.  
[PERSON] et al. [24] also studied the use of fly ash to improve granite residual soils. In their study, they showed that the shear strength of granite residual soil was improved and the permeability coefficient was reduced after a sufficient age of maintenance (e.g., 14 days), and the optimum dosage of fly ash was also 15%. They concluded that fly ash can effectively improve the stability of granite residual soils with respect to water, i.e., with the addition of water, the strength of granite residual soils improved by fly ash will be higher than that of plain soils. The numerical simulation results in this chapter show that granite residual soil modified by 15% fly ash is more stable than plain soil after rainwater erosion. This coincides with the results of [PERSON] et al. Therefore, it can be concluded that fly ash can improve the stability of granite residual soil slopes under rainfall conditions, and 15% fly ash is the optimum dosage.