## 2 Materials and Methods
### Geotechnical Model  
The implementation of lower and upper bound plasticity theorems using finite elements was described by [PERSON] [40,41]. The limit analysis theorems are employed to determine the maximum and minimum stress that the soil could achieve (according to the stress boundary surface), ensuring than their actual value is within its theoretical range. The theory of plasticity assumes the soil to be a perfectly plastic material. Implementing the lower bound theorem in finite element establishes that the estimation of the lower bound of the actual load limit is given by any statically admissible stress field, where stress discontinuities are only permitted at the interface of the triangular elements of the model. In the software used in this investigation, first-order triangular elements with a linear variation of the stresses between the nodes and a Gauss point are used [42]. Meanwhile, implementing the upper bound theorem in finite elements establishes that the upper bound of the actual load limit can be deduced by comparing the dissipated power of any kinematically admissible velocity field and the power dissipated by external loading, where the displacements between elements are continuous. In this case, triangular elements with six nodes, linear interpolation of stresses, quadratic interpolation of displacements, and three Gauss points are used [42]. Figure 1 shows a diagram of the two types of elements used in this study. The shear strength reduction method (SSRM) [43] was employed to obtain finite elements' safety factor values. It represents the same safety factor concept that is typically used in conventional calculations of slopes using analytical or limit equilibrium methods, applying a reduction factor to the strength parameters. The shear strength reduction method is an algorithm used to complement the analysis with finite elements. To perform any finite element analysis, the first step is to discretize the area of study with elements that replace the original geometry in computations. As shown in Figure 1, each of these elements is composed of different types of nodes. Conventional finite element analysis calculates the node displacements, and based on them, the stresses and deformations produced in the elements that constitute the model are determined. To use the finite element method, a constitutive model that allows deformation to be related to stress is necessary. The most typical constitutive model in soil mechanics is the perfect elastoplastic model with the Mohr-Coulomb failure criterion, which is used in this investigation. The shear strength reduction method introduces an equal reduction factor to the strength parameters and searches for the maximum value it can have for the model to remain stable. This parameter's maximum value is the analyzed slope's strength reduction factor (SRF). In its most typical form, the Mohr-Coulomb criterion uses a linear envelope, such that the parameters that define it, called strength parameters, are cohesion (\(c\)) and friction angle (\(\phi\)). The maximum shear stress (\(\tau\)) as a function of the effective normal stress (\(\sigma\)') can be obtained using the linear Mohr-Coulomb failure envelope, which is defined using Equation (1). \[\tau=c+\sigma^{\prime}.\tan(\phi) \tag{1}\]  
Slope stability is usually analyzed by introducing a material safety factor (SF), that is, the value by which the value of the strength parameters can be reduced until a strict equilibrium is reached. To this end, the available shear stress is calculated using Equation (2). \[\tau_{s}=\frac{\tau}{SF}=\frac{c+\sigma^{\prime}.\tan(\phi)}{SF}=\frac{c}{SF}+ \sigma^{\prime}.\frac{\tan(\phi)}{SF} \tag{2}\]  
When the finite element method is used, the instability situation is reached through the iterative reduction of strength parameters via a strength reduction factor (SRF), which divides the soil strength parameters, resulting in a new value of the strength parameters used in the calculation. This process is carried out until it induces a slope collapse.