Yield design‐based numerical analysis of three‐dimensional reinforced concrete structures

Abstract

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.

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

“…An alternative approach for the finite element modelling of 1D steel inclusions in 3D concrete volumes has been recently proposed by Vincent et al (2017). Considering one individual 1D-inclusion embedded in a 3D-concrete block, a cylindrical volume of concrete with the inclusion placed along its axis is defined, as shown in Figure 5(a).…”

“…The advantage of such a modelling procedure, is that the characteristic size of the homogenized zone (namely s) is significantly larger than the inclusion's diameter, thus allowing for example a much easier finite element discretization of the reinforced concrete structure as a 3D-3D composite, since a refinement of the mesh around the inclusion is no more required for obtaining accurate and reliable predictions. Of course, the choice of s being arbitrary, it will be necessary to make sure that the results of the computations performed on the basis of this model, remain rather insensitive to the value of s, which has been checked in (Vincent et al, 2017).…”

“…It is to be noted that there are as many stress tensors attached to any geometrical node of the mesh as there are tetrahedral elements sharing this node as an apex. It can be shown (Vincent et al, 2017) that the finite element implementation of the lower bound static approach of yield design finally reduces to the following convex optimization problem:…”

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

“…• Under such conditions, it is shown that the implementation of the upper bound kinematic approach with such discretized velocity fields amounts to solving a convex minimization problem, which can be treated by means of SDP techniques, 10 in the same way as for the lower bound static approach to the same kind of problem. 6 The whole computational procedure is first illustrated and validated on the simple example of a uniformly loaded deep beam, where the calculated upper bound estimates are favorably compared to the previously derived lower bound estimates. 6 It is then applied to evaluating the ultimate bearing capacity of a reinforced concrete bridge pier cap subjected to concentrated vertical loads, on the one hand, a circular reinforced concrete foundation slab on the other hand, where in the latter case, a semianalytical design method has already been proposed.…”

“…6 The whole computational procedure is first illustrated and validated on the simple example of a uniformly loaded deep beam, where the calculated upper bound estimates are favorably compared to the previously derived lower bound estimates. 6 It is then applied to evaluating the ultimate bearing capacity of a reinforced concrete bridge pier cap subjected to concentrated vertical loads, on the one hand, a circular reinforced concrete foundation slab on the other hand, where in the latter case, a semianalytical design method has already been proposed. 11…”