Unified finite element limit analysis for solid reinforced concrete structures

Abstract

The application of Limit Analysis is effective in the pursuit of the collapse load of a structure. So far, whether it is an analytical model or a numerical model, the choice has been to apply either the lower or the upper bound theorem. Here, a unified approach for solid Finite Element models has eliminated this distinction and there is only one shared optimal solution. The unified solution is based on the two theorems by applying coinciding constitutive points for the fulfilment of the yield criteria together with weak forms of the equilibrium and the compatibility demands in the lower and upper bound formulation, respectively. This approach applies to any 3D stress state which may be formulated as a Semidefinite Program. Models based on either the lower or the upper bound theorem often give an indistinct result for the dual solution, interpreted as the displacements or the stresses, respectively. The present unified mixed solution renders accurate results for both stresses, displacements and load levels which in general are more accurate than the corresponding strict lower and upper bound solutions. A tetrahedral solid element with a linear stress variation and a quadratic displacement interpolation is presented along with a compatible embedded bar element. The effectiveness of this implementation is shown in three examples with regards to stresses, displacements, plastic work and the load capacity of reinforced concrete structures.