1. Reference problem#

1.1. Geometry#

We consider a reinforced concrete plate, comprising two perpendicular reinforcement layers within it. This plate is held at one end, and a force is applied to the section of the steel bars at the other end.

_images/10000200000002A00000028EA3702F92EEFF1011.png

Figure 1: Extraction of a reinforcing sheet from a reinforced concrete plate.

To limit the calculation cost, only one section of a plate is modelled, at the edges of which periodicity conditions are imposed. The dimensions of the plate and the frames are shown in Figure.

_images/100002000000039D00000280E8FBA71A807406B6.png

Figure 2: Dimensions of the modelled plate section. (a) General view; (b) longitudinal view; (c) section of the modelled section.

The two reinforcement layers are represented by a membrane model linked to the volume of concrete by an interface law. This makes it possible to greatly limit the calculation cost (see Figure).

_images/Cadre2.gif

Figure 3: Representation of reinforcement sheets by a membrane

1.2. Material properties#

Concrete has an isotropic homogeneous elastic behavior, characterized by the Young’s modulus and the Poisson’s ratio shown below:

\(\mathrm{\{}\begin{array}{c}{E}_{B}\mathrm{=}30\text{GPa}\\ {\nu }_{B}\mathrm{=}0.22\end{array}\)

The Young’s modulus of steel is \({E}_{A}\mathrm{=}200\text{GPa}\), the spacing between the steel bars is \(e\mathrm{=}20\text{cm}\), and the bar diameter is \(d\mathrm{=}2\text{cm}\).

The behavior of the steel-concrete bond is of type CZM_LAB_MIX, with the following parameters:

Size

Value

\({\sigma }_{C}\)

\(10\text{MPa}\)

\({\delta }_{C}\)

\(1\text{mm}\)

\(\alpha\)

\(0.7\)

\(\beta\)

\(1\)

1.3. Boundary conditions and loads#

The limit conditions applied to the plate are indicated below, corresponding to the limit conditions at the end of the plate, to the periodicity conditions and to the force exerted on the reinforcements:

\(\mathrm{\{}\begin{array}{c}{u}_{Y}\mathrm{=}0\text{sur}\text{A\_FOND et B\_FOND}\\ {u}_{Z}\mathrm{=}0\text{sur}\text{A\_FOND}\\ {u}_{X}\mathrm{=}0\text{sur}\text{A\_GAUC et B\_GAUC}\\ {u}_{X}\mathrm{=}0\text{sur}\text{A\_DROI et B\_DROI}\\ {F}_{Y}\mathrm{=}\mathrm{-}\frac{9810.T}{e}\text{sur}\text{BOUT}\end{array}\)