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.




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 :math:`X`, 2 in the direction :math:`Y` for the lower and upper parts). The reinforcement rate for each reinforcement sheet is :math:`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 :math:`\pm 0.04m` *:* it is therefore assumed here that the steels in :math:`X` and in :math:`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.




Material properties
------------------------

**Concrete (model** ENDO_ISOT_BETON **)**:

Young's module: :math:`{E}_{b}=32308.0\mathrm{MPa}`

Poisson's ratio: :math:`{\nu }_{b}=0.20`

Single pull damage threshold :math:`{\text{SYT}}_{\text{EIB}}`: :math:`3.4\mathrm{MPa}`

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

**Steel**:

Young's module: :math:`{E}_{a}=200000.0\mathrm{MPa}`

Linearity limit :math:`{\sigma }_{e}^{\text{acier}}`: :math:`570.0\mathrm{MPa}`

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



.. _RefNumPara__30712112: