The Mohr-Coulomb yield criterion takes on the simplest form in the Mohr stress space, which has thus been most extensively applied in limit analysis and limit equilibrium methods because of its accuracy. However, the Mohr-Coulomb yield surface in the stress space is non-smooth, causing huge troubles to the constitutive integration in the deformation based finite element plasticity analysis. In addressing strength problems, meanwhile, solvers based on the load controlled method (LCM) are hard to bring the finite element model to the limit equilibrium state. Aiming at these issues, the solution schemes are proposed as follows. First, an algorithm named GSPC is designed for the constitutive integration for plasticity with non-smooth yield surfaces. GSPC is always convergent for arbitrary large strain increments, with far more excellent numerical properties than the return-mapping methods available. A solver based on the displacement controlled method (DCM) is developed fitted for the finite element plasticity analysis. The DCM solver is able to bring easily the finite element model into the limit equilibrium state, with no convergence issue, and far more efficient and robust than any LCM solvers. At the same time, a formula is derived for the computation of partial derivatives of principal stresses with respect to stress components. At last, combined with the strength reduction method, the secant method for the factor of safety of slopes is developed, and the location and depth of tension cracks at the slope top are proposed.