Tensile cracks are often observed near the crest of slopes in both model tests and actual landslides. Therefore, tensile failure should be considered together with shear failure in slope stability analysis. In our previous study (Gao et al., 2026), a lower-bound rigid-plastic finite element method was developed to evaluate slope stability while considering both shear and tensile failure. In that method, the tensile failure condition was expressed by limiting the maximum principal stress to the tensile strength, and Gershgorin’s circle theorem was used to approximate this condition by linear inequalities. The previous results showed that the proposed linear constraint can represent tensile failure during slope failure and capture the change in failure mechanism caused by tensile strength. However, since the Gershgorin-based formulation is a conservative approximation, its influence on the calculated stability number should be evaluated. Therefore, the present study focuses on the error evaluation of the Gershgorin-based tensile failure constraint.
For this purpose, the stability numbers obtained from the Gershgorin-based constraint are compared with those obtained from a quasi-exact plane strain constraint. The same lower-bound formulation is used in both cases, and only the tensile failure condition is changed. The comparison is carried out for different slope inclinations, internal friction angles, and tensile strengths.
The results show that the two conditions give almost identical stability numbers under the examined plane strain cases, with a maximum relative difference of about 0.23%. This indicates that the influence of the Gershgorin-based approximation on the calculated stability number is negligible. Therefore, the proposed constraint is useful for incorporating tensile failure into rigid-plastic finite element analysis while maintaining a simple linear formulation. Furthermore, the velocity field obtained from the dual problem indicates the failure mechanism considering tensile failure, as shown in Fig. 1.

slope stability,tensile failure,numerical analysis,finite element method