2. Benchmark solution

The reference solution is obtained by a semi-global modeling in a multi-layer plate , where the mesh and the loading are the same as for the corresponding models with the GLRC_DM law.

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.

In addition, several analytical results with the GLRC_DM model have been established.

2.1. Patterns

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.

Concrete cracking is modelled by behavior law ENDO_ISOT_BETON, while it is assumed that steel always remains in the elastic domain.

2.2. Material properties

Concrete (model ENDO_ISOT_BETON ):

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

Poisson’s ratio: \({\nu }_{b}=0.20\)

Single pull damage threshold \({\text{SYT}}_{\text{EIB}}\): \(3.4\mathrm{MPa}\)

Softening slope: \(-0.2{E}_{b}\) (\({\gamma }_{\text{EIB}}=5\)).

Steel:

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

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

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