Summary The objective of this contribution is to present some new recent developments regarding the evaluation of the ultimate bearing capacity of massive three‐dimensional reinforced concrete structures which cannot be modeled as 1D (beams) or 2D (plates) structural members. The approach is based upon the implementation of the lower bound static approach of yield design through a discretization of the three‐dimensional structure into tetrahedral finite elements, on the one hand, the formulation of the corresponding optimization problem in the context of semi‐definite programming techniques, on the other hand. Another key feature of the method lies in the treatment of the concrete‐embedded reinforcing bars not as individual elements, but by resorting to an extension of the yield design homogenization approach. The whole procedure is first validated on the rather simple illustrative problem of a uniformly loaded simply supported beam, then applied to the design of a bridge pier cap taken as an example of more complex and realistic structure.

This contribution is addressing the ultimate limit state design of massive three-dimensional reinforced concrete structures based on a finite-element implementation of yield design theory. The strength properties of plain concrete are modeled either by means of a tension cutoff Mohr Coulomb or a Rankine condition, while the contribution of the reinforcing bars is taken into account by means of a homogenization method. This homogenization method can either represent regions of uniformly distributed steel rebars smeared into the concrete domain, but it can also be extended to model single rebars diluted into a larger region, thereby simplifying mesh generation and mesh size requirements in this region. The present paper is mainly focused on the implementation of the upper bound kinematic approach formulated as a convex minimization problem. The retained strength condition for the plain concrete and homogenized reinforced regions are both amenable to a formulation involving positive semidefinite constraints. The resulting semidefinite programming problems can, therefore, be solved using state-of-the-art dedicated solvers. The whole computational procedure is applied to some illustrative examples, where the implementation of both static and kinematic methods produces a relatively accurate bracketing of the exact failure load for this kind of structures. K E Y W O R D S finite element, homogenization, reinforced concrete structures, semidefinite programming, tension cutoff Mohr-Coulomb strength condition, upper bound kinematic approach, yield design 1 INTRODUCTION Assessing the ultimate load bearing capacity of constructions involving massive three-dimensional reinforced concrete components requires a specific analysis, such as the widely acknowledged "strut-and-tie" model (see among many other Refs. 1-4), which can be related to the lower bound static approach of yield design. Indeed, such "D" (Disturbed or Discontinuity) regions, as opposed to "B" (Beam) regions, in the context of the strut-and-tie method cannot be modeled with enough accuracy using simple beam or plate/shell models. Such regions often require a fully 3D analysis, which is difficult to perform manually and therefore require some computational assistance. Typical examples are corbels, deep beams, pier or pile caps, footings, and so on. With a special attention to assessing the ultimate shear capacity of 2216

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