1. Introduction#
1.1. Global behavior models#
The global models represent the evolutions of the material within the structure studied — beam and plate — on the basis of a relationship between the generalized quantities of deformations (extension, curvature, distortion) and the generalized forces (membrane forces, flexure, shear forces). These models are « calibrated » beforehand using a fixed local analysis (for example using the results of a boundary analysis of the sections), according to the characteristics of the concrete and steel materials constituting the plate and the distribution of these in the section, cf. []. The schematic diagram is as follows:
This local analysis must take into account the various couplings: for example, the evolution in flexure is dependent on the value of the normal force applied. The non-linear balance of the structure is treated at the global level on the generalized forces, via the plate kinematics in question.
Figure 1.1-a : Reinforced concrete slab.
As the local analysis is only implemented in pre-treatment (as part of a monotonic load analysis), there is no immediate way to return during the calculation to the local stress analysis based on generalized internal forces. In fact, the dissipative nature of irreversible laws of behavior makes it necessary to store the evolution of internal variables during cycles at any moment if one wants to calculate the constraints at a particular point. One could consider launching the three-dimensional law of behavior according to [éq 1.1-1] in parallel and integrating it into the thickness to return to global behavior, but the cost and complexity of such an approach seem to be an obstacle. It should be noted that this could be a way of estimating the error made by a global law of behavior. However, this approach is not yet adopted in*Code_Aster*.
This type of model can be usefully validated by comparison with a direct analysis carried out with a local model.
1.2. Objectives of law GLRC_DAMAGE#
The initial formulation of the global model of reinforced concrete with plate GLRC_DAMAGE, established by Koechlin in 2002, can be found in [bib1], [bib2] and [bib3].
This model was first developed for dynamic applications with impact damage to reinforced concrete structures. The elastoplastic response of the model is essential for this type of application. In fact, the dissipation of energy by plasticizing steels is important. Taking into account damage by cracking in concrete makes it possible to make the first phases of non-linear behavior more precise. In the context of seismic applications, we can expect a reverse situation: damage and the response after cracking are essential, while it is rare to go so far as to mobilize the widespread plasticization of steels. However, it seems advantageous to have the same model for dealing with these two families of applications.
The formulation of the model is established within the framework of the thermodynamics of irreversible processes. It combines plasticity with work hardening, in particular provided by steels, and the damage caused by concrete cracking during the bending of the plate. The plastic behavior is constructed on the basis of the flexural limit analysis of a reinforced concrete plate. It is described using the generalized standard materials framework. Based on experimental results, cf. [bib1], linear kinematic work hardening was chosen to treat cyclic behavior. The damage is introduced to represent the loss of elastic stiffness that occurs due to the cracking of concrete before the plasticization of the steels. The damage threshold is assumed to be constant. This behavior is assumed to be independent of stress rates (the dissipations are instantaneous).