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

Abstract

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

Elastoplastic and limit analysis of reinforced concrete with an equilibrium-based finite element formulation

“…Special attention will be paid to both cases of elastoplastic and limit analysis. For the latter, comparisons will be made against the kinematic upper-bound limit analysis approach of 3D reinforced concrete structure based on the work of [VABDB20] and which has been extended to account for 1D rebars instead of 3D smeared regions as considered in the original paper. In addition to providing an error estimation with respect to the proposed lowerbound static approach, kinematic results also offer an additional understanding of the underlying phenomena when inspecting the corresponding collapse mechanisms or plastic dissipation regions.…”

“…However, it appears that the two specimens with a particularly high tensile reinforcement ratio, precisely T 7 and T 11 , failed because of the concrete crushing limitation. A safeguard regarding this crushing limitation would be to analyse the compressive strains resulting from the kinematic elastoplastic analysis (whom theory is not described in this article but can partly be found in [VABDB20]).…”

Strength-based material layout optimization of solid reinforced concrete

“…For example, Poulsen and Damkilde [18] who treated concrete wall structures with reinforcement modeled as embedded bar and beam elements, and Jensen et al [19] who treated slab bridges including limitation on shear forces. Several studies of solid reinforced concrete structures have also been made, with examples of pier caps [20,21], and tension connection with overlapping U-bars [22].…”