A finite element procedure for rigorous numerical enclosures on the limit load in the analysis of multibody structures

A rigorous finite element numerical procedure is proposed for the computation of guaranteed lower and upper bounds for the limit load of failure in a system of linear-elastic blocks in mutual non-penetrative contact with given friction. First the static and kinematic principles are formulated as continuous optimization problems and existence of a solution to the corresponding limit load problems at the infinite dimensional level is established. Two numerical approaches are devised, one for each limit load problem, to obtain actual numerical bounds on the unique critical load. The first approach uses the static limit load problem involving stresses in conjunction with a non-standard conforming finite element method to obtain a linear program from which one can derive a lower (safe) bound for the limit load and an expression for the corresponding stress field. The second approach uses the kinematic limit load problem to obtain a linear optimization problem from which one can determine an upper (unsafe) bound for the limit load and an expression for the failure mode. Together, these procedures give rise to rigorous numerical enclosures on the limit load.