1. Reference problem#

1.1. Geometry#

Consider a reinforced concrete cube with side \(L\mathrm{=}1m\).

_images/10000200000001CC0000015FEE937DCCD077ECF4.png

The reinforcement sheets belong to the plane defined by the four \(\mathit{N05}\mathrm{-}\mathit{N02}\mathrm{-}\mathit{N03}\mathrm{-}\mathit{N08}\) nodes.

Two reinforcement layers are defined: one in the local direction \(X\) and one in the local direction \(Y\).

The reinforcement rate for each reinforcing sheet is \(0.1{m}^{2}\mathrm{/}\mathit{ml}\) (Section per linear meter).

1.2. Characteristics of the models#

The concrete mesh is modelled with a hexa8 element.

The models considered for reinforcing plies are:

  • modeling A (§3): use of grid_membranewith tria3 support meshes

  • modeling B (§4): use of grid_membrane with quad4 support meshes

  • C modeling (§5): use of grille_excenter with tria3 support meshes

1.3. Material properties#

The concrete material is isotropic elastic whose properties are:

  • \({E}_{b}=20000\mathrm{MPa}\)

  • \(\nu =0.2\)

The law of behavior of reinforcements follows an elastoplastic model whose properties are:

  • \({E}_{a}=200000\mathrm{MPa}\)

  • \(\nu =0\)

  • \({E}_{\mathrm{ecr}}^{\mathrm{acier}}=20000\mathrm{MPa}\)

  • \({\sigma }_{e}^{\mathrm{acier}}=200000\mathrm{MPa}\)

The grid_isot_line model for isotropic work hardening plasticity is used in stat_non_line.

1.4. Boundary conditions and loads#

_images/10000200000001A5000001425BD1DA55CB9AE19C.png

Boundary conditions:

  • Embedding in \(\mathit{A1}\)

  • \(\mathit{DX}\mathrm{=}0\) on the \(\mathit{A1}\mathrm{-}\mathit{A4}\) ridge

  • \(\mathit{DY}\mathrm{=}0\) on the \(\mathit{A1}\mathrm{-}\mathit{A2}\) ridge

  • \(\mathrm{DZ}=0\) on the bottom surface of the cube (N01-N02-N03-N04)

Charging by mandatory trips:

  • \(\mathrm{DX}=1\) on the \(\mathit{A2}\mathrm{-}\mathit{A3}\) ridge

  • \(\mathit{DY}\mathrm{=}1t\) on the \(\mathit{A3}\mathrm{-}\mathit{A4}\) ridge

where \(t\) is the pseudotime parameter.

1.5. Initial conditions#

Initially, travel and constraints were zero everywhere.