A new computational method based on the minimum lithostatic deviation (MLD) principle to analyse slope stability in the frame of the 2-D limit-equilibrium theory

摘要

The stability of a slope is studied by applying the principle of the minimum lithostatic deviation (MLD) to the limit-equilibrium method, that was introduced in a previous paper (Tinti and Manucci, 2006; hereafter quoted as TM2006). The principle states that the factor of safety F of a slope is the value that minimises the lithostatic deviation, that is defined as the ratio of the average inter-slice force to the average weight of the slice. In this paper we continue the work of TM2006 and propose a new computational method to solve the problem. The basic equations of equilibrium for a 2-D vertical cross section of the mass are deduced and then discretised, which results in cutting the cross section into vertical slices. The unknowns of the problem are functions (or vectors in the discrete system) associated with the internal forces acting on the slice, namely the horizontal force E and the vertical force X, with the internal torque A and with the pressure on the bottom surface of the slide P. All traditional limit-equilibrium methods make very constraining assumptions on the shape of X with the goal to find only one solution. In the light of the MLD, the strategy is wrong since it can be said that they find only one point in the searching space, which could provide a bad approximation to the MLD. The computational method we propose in the paper transforms the problem into a set of linear algebraic equations, that are in the form of a block matrix acting on a block vector, a form that is quite suitable to introduce constraints on the shape of X, but also alternatively on the shape of E or on the shape of X. We test the new formulation by applying it to the same cases treated in TM2006 where X was expanded in a three-term sine series. Further, we make different assumptions by taking a three-term cosine expansion corrected by the local weight for X, or for E or for A, and find the corresponding MLDs. In the illustrative applications given in this paper, we find that the safety factors associated with the MLD resulting from our computations may differ by some percent from the ones computed with the traditional limit-equilibrium methods.