Numerical upper bounds to the ultimate load bearing capacity of three‐dimensional reinforced concrete structures

This contribution is addressing the ultimate limit state design of massive three-dimensional reinforced concrete structures based on the finite element implementation of both the upper and lower bound methods of yield design. The strength properties of plain concrete are modelled by means of a tension cutoff Mohr Coulomb condition, while the contribution of the reinforcing bars is taken into account by means of an extended homogenization method. Following a previous article [9], more specifically dedicated to the finite element implementation of the lower bound static approach, the present paper is focused on the upper bound kinematic approach. Similarly to what has been previously done for the lower bound static approach, the reinforced structure is discretized into ten-nodded tetrahedral finite elements, with a quadratic variation of the velocity fields inside each element and velocity jumps across the triangular facets separating any two adjacent elements. This discretization of the velocity fields used in the kinematic approach leads to the formulation of a convex minimization problem which can be solved by resorting to Semi-Definite Programming (SDP) optimization techniques. The whole computational procedure is applied to some illustrative examples, where the implementation of both the static and kinematic methods produces a relatively accurate bracketing of the exact failure load for this kind of structures.