# Numerical Investigations of Pile Group Foundations under Different Pile Length Conditions
## 7 Plastic Zone and Factor of Safety  
As this model adopts the Mohr-Coulomb constitutive model, it adheres to the shear-tension damage criterion. In this context, the term \"shear\" represents the shear failure element, and \"tension\" represents the tension failure element. The addition of \"n\" signifies that the element is currently on the yielding surface during the ongoing calculation time step. Conversely, \"p\" denotes the past state, indicating that the element was previously on the yielding surface but has since exited and entered the elastic range. When examining plastic zones in practical engineering problems, it is crucial to focus on elements in the \"now\" plastic state, which actively undergo plastic flow. Only elements in the \"now\" state influence the instability of the model. Therefore, understanding the elements that are currently yielding is essential for analyzing plastic zones in practical engineering scenarios. The penetration of the plastic zone does not necessarily indicate destruction. Instead, it signifies the extent to which the plastic development of the rock layer has occurred, leading to overall failure and the appearance of multiple plastic zones. The slope's destructive process involves the gradual development of plastic zones, ultimately culminating in overall failure. Therefore, the penetration of the plastic zone is often used as a criterion for assessing slope instability.  
From Figure 8, it can be observed that only the uppermost layer of gravel soil exhibits a continuous series of elements in the \"now\" plastic state. However, under the influence of the rear pile row, this continuity is disrupted. This indicates that the slope in this soil layer is in a stable state due to the reinforcing effect of the group piles.  
The Factor of Safety (FOS) is another indicator used to assess slope stability, FLAC3D can use the strength reduction method to obtain the factor of safety. In the strength reduction method, the factor of safety for slope stability is defined as the extent to which the shear strength of rock or soil is reduced when the slope is just at the critical failure state. Specifically, the safety factor is defined as the ratio of the actual shear strength of the rock or soil to the reduced shear strength at the critical failure state. Equation (1) can be used to express this.  
\[\begin{array}{l}c_{F}=\frac{c}{F_{trial}}\\ \phi_{F}=\tan^{-1}(\frac{\tan\phi}{F_{trial}})\end{array} \tag{1}\]  
The above formula is used to adjust the strength indicators \(c\) and \(\phi\) (in this formula, \(c_{F}\) is the adhesive force after reduction, \(\phi_{F}\) is the friction Angle after reduction, and F\({}_{trial}\)  
Figure 8: The plastic zone and factor of safety in FLAC3D.  
is the reduction coefficient). Then, numerical analysis is performed on the slope stability. By continuously increasing the reduction factor, the analysis is repeated until reaching the critical failure state. The reduction factor obtained at this point is the factor of safety.  
The output results file from FLAC3D provides the FOS of the model. According to the graph, the FOS is calculated as 2.4. Based on the relevant literature, a slope's FOS should be greater than 1, indicating stability. Thus, the model's calculation results suggest a stable condition. Additionally, the simulation also recorded the FOS under uniformly distributed axial loads ranging from 100 kN to 450 kN, with detailed data available in Table 3.  
From the table, it is clear that as the load increases, the Factor of Safety (FOS) decreases. When the load reaches 450 kN, the FOS is recorded as 1.06. This indicates that the slope's stability is at a critical point. Consequently, if the uniformly distributed axial load exceeds 450 kN, the foundation will become unstable due to the excessively large load.