# Numerical Modeling of 3D Slopes with Weak Zones by Random Field and Finite Elements
###### Abstract  
This work investigates an analysis method for the stability of a three-dimensional (3D) slope with weak zones considering spatial variability on the basis of two-phase random media and the finite element method. By controlling the volume fractions of rock and weak zones, two-phase random media are incorporated into the 3D slope model to simulate the random distribution of rock and weak zones. Then, a rotation of a Gaussian random field is performed to account for the inclination of the weak zones. The validity of the proposed model for use in the analysis of the stability of 3D slopes with weak zones was verified by comparing it to existing results and analytical solutions. The failure mechanism of the slope is considered by examining the plastic failure zone at incipient slope failure. The safety factor is sensitive to the inclination of the weak zones, but it is predictable. Parametric studies on the inclination of the layer of weak zones demonstrate that when the rotation angle of the weak zones is approximately parallel to the slope inclination, the slope is prone to slippage along the weak zones, resulting in a significant reduction in the safety factor. The findings of this research can serve as the foundation for further research on the stability of slopes with weak zones. slope stability; two-phase random media; slope with weak zones; finite element analysis; random field 2021 18 October 2021 20  
given site and the spatial distribution of the mechanical characteristics of the slope should have similar characteristics [17]. Therefore, random fields can be applied to simulate the non-homogeneity of the spatial distribution of weak zones. Various investigations have been carried out on the effect of spatial randomness in rock and weak zones using the random field method [18; 19; 20; 21; 22; 23; 24; 25; 26]. However, most of these were limited to two-dimensional (2D) models. [PERSON] et al. [18] used representative slip surfaces together with Monte-Carlo simulation to conduct slope stability analysis. [PERSON] and [PERSON] [20] developed a multiple response surface approach to perform slope reliability analysis. [PERSON] and [PERSON] [23] explored the influence of the spatial variability of the strength parameter of weak zones on the critical slip surface of slopes using a 2D model. By comparison, research on three-dimensional (3D) slope stability remains less well documented [27; 28]. Generally, compared with 2D models, 3D models have more degrees of freedom, and more computational effort is needed. Despite possessing the advantages of simplicity and computational efficiency, 2D models obviously deviate from reality. Furthermore, 2D analysis is likely to obtain conservative results compared with 3D analysis when performing deterministic slope stability analysis, but it cannot be ensured that the results of 2D analysis will be more conservative than 3D analysis when the spatial variability of weak zones parameters is considered [29; 30]. These findings further illustrate that it is necessary to undertake 3D slope stability analysis in random weak zones and rock. When investigating slopes with weak zones, the two-phase random media characterization method can be applied to soil-rock mixtures, since the overall material properties of the slope are dependent on the spatial distribution and volume fraction of the rock and weak zones. [PERSON] et al. [31] simulated two-phase random media through the nonlinear translation of the Gaussian field, and combined the finite element method to characterize the spatial variability of rock and soil materials; [PERSON] et al. [32] proposed a nonlinear transformation based on the Gaussian random field to realize the simulation of composite materials using two-phase random media. This was achieved by Fourier transform, improving the calculation efficiency. This method has high applicability for large-scale soil-rock mixture models and the characterizations of large numbers of model samples; [PERSON] et al.