Ultimate limit state design of three-dimensional reinforced concrete structures: A numerical approach

Vincent, Hugues; Arquier, Mathieu; Bleyer, Jérémy; Belletti, P. de Buhan

doi:10.1201/9781315182964-67

Cited by 1 publication

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“…It can be proved (see Refs. 14, 20, or 21 for more details) that the macroscopic strength condition at any point of the homogenized layer of concrete reinforced in directions x and y is given by:

F^{r c} false(\underset{̲}{\underset{̲}{σ}} false) \leq 0 \Leftrightarrow \{\begin{matrix} left \underset{̲}{\underset{̲}{σ}} = {\underset{̲}{\underset{̲}{σ}}}^{c} + \sum_{α} σ_{α}^{r} {\underset{̲}{e}}_{α} \otimes {\underset{̲}{e}}_{α}, α = x, y \\ left normalwith F^{c} false({\underset{̲}{\underset{̲}{σ}}}^{c} false) \leq 0 and - k σ_{α 0} \leq σ_{α}^{r} \leq + σ_{α 0} \end{matrix}

with

σ_{α 0} = η_{α} f_{y}^{s},

where, as in Equation (7),

η_{α}, α = x, y

is the reinforcement volume fraction in the direction α and

f_{y}^{s}

the uniaxial strength of the constituent steel material. The corresponding support functions are:

\begin{matrix} true \\ left π^{r c} (\underset{̲}{\underset{̲}{d}}) = = π^{c} (\underset{̲}{\underset{̲}{d}}) + \sum_{α = x, y} σ_{0 α} sup \{- k d_{α α}, d_{α α}\} \\ left π^{r c} (\underset{̲}{n}; \underset{̲}{V}) \end{matrix}

…”

Section: Load Bearing Capacity Of a Reinforced Concrete Footingmentioning

confidence: 99%

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

Vincent

Arquier²,

Bleyer

et al. 2020

Num Anal Meth Geomechanics

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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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