1. Introduction#
1.1. Global modelling#
Reinforced concrete is a heterogeneous material constituted of ribbed or plain steel rebar and concrete. The rebar have regular spacing on the two directions \(x\) and \(y\) in the plane. We consider moderate cycling or alternate loading conditions: the material is therefore considered as a continuum.
A global constitutive plate model devoted to structural element generally means that the constitutive law is written directly in terms of the relationship between the generalized stresses and generalized strains. The overall approach of the structures behavior modeling is particularly applicable to composite structures, such as reinforced concrete (see Figure), and this represents an alternative to the so-called local approaches or semi-global ones, which are finer but more expensive models (see [bib5] and [bib6]). In the local approach a thin model is used for each phase (steel, concrete) and their interactions (adhesion). In the semi-global approach one exploits the slenderness of the structure to simplify the description of the kinematics, it leading to « PMF » models (multi-beam fiber) or multi-layer shells.
The use of the theory of plates and thin shells can effectively describe the mechanical behaviour of reinforced concrete structures, which are usually slender; indeed we use this constitutive modelling in the context of Love-Kirchhoff’s kinematics, see [bib 16] and [bib 17].
The interest of the overall model lies in the fact that the structural finite element requires only a unique point of integration in the element thickness and also in the use of a homogenized behaviour. This advantage is even more important in the analysis of reinforced concrete, since it bypasses the localization problem encountered in the modelling of concrete without reinforcement. Obviously, a global model idealizes local phenomena of a coarse manner and requires more validation prior to its application to industrial situations. Finally, it is impossible as soon as we consider the non-linear phase behaviour back locally to provide field values, except strain fields.
This simplified modelling approach can be enhanced by an appropriate calibration of the overall parameters.
Figure 1.1-a : reinforced concrete slab .
1.2. Objectives of the DHRCconstitutive model#
The DHRCconstitutive model, named for « dissipative homogenised reinforced concrete », is able to idealize the stiffness degradation of a reinforced concrete plate, for a quite moderate load range, i.e. without reaching the collapse, as the GLRC_DMconstitutive model does, see [R7.01.32], We can refer to the papers [], We can refer to the papers [], We can refer to the papers [bib12, bib13], We can refer to the papers [] for extensive explanations; however this*Code_Aster* Reference document reuse the main parts of [ bib13]. Nevertheless, DHRC brings an important enhancement of the GLRC_DM model: (1) it is based on a full theoretical formulation from the local analysis, (2) it natively involves the membrane-bending coupling, (3) accounts for any type of rebar grids in the thickness (e.g. different upper and lower grids, or rebar with distinct sections on the two \(\mathrm{Ox}\) and \(\mathrm{Oy}\) directions), In that sense, DHRCconstitutive model is more representative than GLRC_DM.
The reference frame is constituted by the two directions \(x\) and \(y\) of steel rebar in the plane. The user can define these two local directions with the command, see [U4.42.01], with respect to the global mesh reference frame:
AFFE_CARA_ELEM (COQUE = _F (ANGL_REP = \((\alpha ,\beta )\)))
It is built by a periodic homogenization approach using the averaging method and it couples concrete damage and periodic debonding between steel rebar and surrounding concrete. It leads to a better modelling of the energy dissipation during loading cycles.
By construction, the DHRC model has comparable performance in terms of computational cost and numerical robustness with GLRC_DM model.
A restricted number of geometric and material characteristics are needed from which the whole set of model parameters are identified through an automatic numerical procedure performed on a Representative Volume Elements (RVE) of the RC plate.
As GLRC_DM model does, DHRC ones accounts for thermal strains, idealized in the plate thickness, see [R3.11.01].
1.3. Ratings#
A Cartesian orthonormal coordinate system is chosen so that covariant and contra-variant components are assimilated. Uppercase letters will refer generally to the macroscopic scale and lowercase ones to the microscopic scale. The simply observed tensorial product will be noted by a simple dot « \(\mathrm{.}\) » and the double one by two dots « \(:\) » or « \(\text{}\otimes \text{}\) ». - Tensor components will be given through subscripts relatively to the differential manifold used: Greek subscripts will set for integers ranging from 1 to 2 and Latin ones for integers ranging from 1 to 3.