1. Reference problem

1.1. Geometry

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Figure 1.1-a : geometry of the reinforced concrete square plate

Length: \(L\mathrm{=}1.0m\); Plate Thickness: \(e=0.1m\)

Steel diameter: \(0.01m\). Coating the lower and upper reinforcing plies of the steels along the \(z\) axis: \(0.01m\).

The modeling is based on a semi-global modeling in a multilayer plate. Concrete and reinforcement are modelled separately. For each sheet of reinforcements, we consider a layer that behaves only in the longitudinal direction of the reinforcements. So we will have 4 layers for the frames.

On the same mesh, 5 models representing the reinforced concrete plate are defined: 1 model DKT for the concrete and 4 models GRILLE for the reinforcements (2 in the direction \(X\), 2 in the direction \(Y\) for the lower and upper parts). The reinforcement rate for each reinforcement sheet is \(8.0\times {10}^{-4}{m}^{2}/m\).

The position of the reinforcements (lower or upper) is defined by the keyword EXCENTREMENT under the keyword factor GRILLE in the operator AFFE_CARA_ELEM, which is equal to \(\pm 0.04m\) : it is therefore assumed here that the steels in \(X\) and in \(Y\) are in the same position, which constitutes the usual approximation for multilayer models.

1.2. Material properties

Concrete cracking is modelled by the BETON_REGLE_PR law of behavior, while steel is modelled by the GRILLE_ISOT_LINE law.

Concrete (template BETON_REGLE_PR):

Young’s module: \({E}_{b}=32308.0\mathrm{MPa}\)

Poisson’s ratio: \({\nu }_{b}=0.20\)

Single pull damage threshold \({\sigma }_{y}^{t}\): \(3.4\mathrm{MPa}\)

Softening slope: \({E}_{T}\mathrm{=}\mathrm{-}0.2{E}_{b}\)

Simple compression damage threshold \({\sigma }_{y}^{c}\): \(32.308\mathit{MPa}\)

Deformation at compression threshold \({\varepsilon }_{c}\): \(0.002\)

Exponent of the law of work hardening in compression \(n\): \(2\)

Steel:

Young’s module: \({E}_{a}=200000.0\mathrm{MPa}\)

Linearity limit \({\sigma }_{e}^{\text{acier}}\): \(570.0\mathrm{MPa}\)

Post-elastic slope \({E}_{\text{écrouis}}^{\text{acier}}\): \(\text{}=0.0015{E}_{a}=300\mathrm{MPa}\).

1.3. Boundary conditions and loads

Different models*A to H* are considered for different types of characteristic loads and different plate behaviors. In all cases, the loads are movements (or rotations) imposed on the edges of the plate, differently for each model.

The models considered are:

modeling A (§ 3 ): traction - compression - pure traction;

modeling B (§ 4 ): pure alternating bending;

C modeling (§ 5 ): coupling of traction - compression and flexure;

D modeling (§ 6 ): pure shear and in-plane distortion;

E modeling (§ 7 ): traction — pure compression, high stresses;

F modeling (§ 8 ): pure alternating flexure, high stresses;

G modeling (§ 9 ): traction/compression and flexure coupling, high stresses;

H modeling (§ 10 ): thermal loading.

1.4. Initial conditions

Initially, the movements and the constraints are identically zero.