Uniqueness of Finite Element Limit Analysis solutions based on weak form lower and bound methods

Finite Element Limit Analysis (FELA) is increasingly used for calculating the ultimate bearing capacity of structures made of ductile materials. Within FELA for reinforced concrete structures the elements have been based on rigorous lower bound or boundary mixed formulations. The lower bound element formulation may be overly constrained for certain meshes and the dual displacement interpretation may contain spurious modes, moreover the boundary mixed element formulation has an internal equilibrium node with no associated displacement field. Here a consistent and general weak formulation based on virtual work is presented specifically for both the lower and the upper bound problem, and it is shown that they are each others dual ensuring uniqueness of the optimal solution. As a consequence there is no difference between the solutions based on the weak upper and lower bound methods. Here a plane element is presented, with a linear stress variation and a quadratic displacement field, optionally including a concentrated bar element with a linear variation of the normal force. These elements are applied in a verification example and two reinforced concrete examples where they show very good results for both load level, stress distribution and collapse mechanism even for coarse meshes.