Introduction
The foundation of good statics is always stability. Particularly for difficult soil conditions, a detailed soil failure analysis may be necessary. In addition to classic slip circle or lamella methods, these can also be conducted by reducing the shear strength parameters.
A convenient option is to automatically determine the minimum strength required to achieve equilibrium. This involves a successive reduction of cohesion and internal friction angle until a stable numerical state can no longer be found. The result is a global safety factor. Further information is available in the Geotechnical Analysis manual under the following links:
- Online Manuals RFEM 6 | Geotechnical Analysis | Theory | Stability Analysis Using Reduction of Strength Parameters (ϕ-c-Reduction)
- Online Manuals RFEM 6 | Geotechnical Analysis | Input Data | Stability (Slope/Ground Failure)
Model Description
In this article, this method is carried out on a simple homogeneous slope according to Sysala et al. [1] and addresses the associated challenges.
The model consists of a slope with a 45° inclination. For the soil material, an elasticity modulus of 40 MPa, a Poisson's ratio of 0.3, and a unit weight of 20 kN/m³ are assumed. The plastic failure model uses the Mohr-Coulomb criterion, specifically with an unregulated failure surface. The strength parameters are: a cohesion of 6 kPa, an internal friction angle of 45°, and a dilation angle, which is varied between 45, 15, and 0°. Since numerical problems can arise with a dilation angle of 0°, a minimal value of 0.01° was instead set. Additionally, a value of 1° was applied to examine the behavior at these small angles more closely. The dimensions are shown in the following image with a link to the model.
Mesh Convergence
A point that should not be neglected is the study of the mesh influence on the obtained safety factor. This shows, due to its dependency on the local failure (plasticity) of the soil, a not fundamentally negligible relationship to the mesh size. Further information on mesh convergence can be found in the following article.
The meshing in this example was done with a coarse mesh in the outer area and a mesh refined by a factor of 5 in the area of the expected slip wedge. This was calculated using one element over the thickness. This means that a terrain section corresponding to the coarser element length was simulated. The following image shows the dependence of the safety factor on the length of the outer FE mesh elements, for all investigated dilation angles and below only for 15°.
As expected, the safety factor decreases with increasing mesh fineness, regardless of the chosen dilation angle. Sufficient mesh fineness was found in this example with an FE element size of 0.5 m for the outer, coarser mesh. This corresponds to the expected condition, as the bearing capacity strongly depends on the localization of the failure. Since the slip surface, unlike classic methods, is not defined but results during the calculation, its location depends on the mesh. This affects both the location of the first plastic zone and the position of the slip surface itself. A coarser mesh thus results in "smeared" shear bands, while increasingly finer meshes produce more distinctly defined shear bands. The obtained safety factor is therefore more realistic. This is well illustrated in the following image, which compares the deformations and plastic equivalent strains for a very coarse mesh (lFE=4.00 m) on the left to a sufficiently finely meshed simulation (lFE=0.50 m, right). Besides the deformation in the upper part, the plastic equivalent strains are even more indicative. Here, one can clearly see the more sharply defined shear band with mesh refinement.
Literature Comparison
As mentioned at the beginning, the model is based on the publication by Sysala et al. [1]. The safety factors obtained in this study and using RFEM (with a mesh size of 0.25 m) are shown in the following diagram according to the applied dilation angles. The second diagram shows the related relative deviations. As can be seen here, these are in an acceptable range. The greatest deviation was determined at a dilation angle of 45° with 9.3%. These deviations can be explained by different simulation software and meshing approaches.
