Use of internal and external variables and extremum principle in limit equilibrium formulations with application to bearing capacity and slope stability problems

Limit equilibrium methods, satisfying both force and moment equilibrium can be formulated using assumptions on the internal variables or the external variables. Even though most stability methods are based on force and moment equilibrium, as well as the Mohr–Coulomb yield criterion, there are great differences between the results of the different formulations due to variations in the assumptions. The authors believe that the use of the interslice force function f(x), the thrust line or the base normal forces should provide an equivalent concept at the ultimate/failure state. In the present study, the authors have used the well-known bearing capacity solutions to determine f(x), the thrust line and the base normal forces for a “horizontal slope”. The equivalence between the different formulations under the ultimate condition is demonstrated. It is shown that it is not important which forces are used in the stability formulation, external boundary forces or internal forces, if only that the ultimate state is considered. It is also demonstrated in the present paper that the maximum extremum from the limit equilibrium analysis is equivalent to the slip line solution using a classical bearing capacity problem.