2. Benchmark solution

2.1. Bending plate

We are trying to calculate the resultant \({F}_{Z}\) of the forces applied to a reinforced concrete plate (2 reinforcement layers) of dimension \({L}_{1}\times {L}_{2}\times e\) (\({L}_{1}\) is the dimension in the main direction of the reinforcements), embedded on one edge and which is subjected to a flexural displacement on the opposite edge (\({U}_{z}\)).

Force is written as:

with \({K}_{z}\) the stiffness according to \(z\) given by:

with \({(\mathrm{EI})}_{\mathrm{tot}}\) equal to

where

with \({E}_{\mathrm{armat}}\) the Young’s modulus of steel, \(s\) the section of the reinforcements per linear meter and \({e}_{\mathrm{exc}}\) the eccentricity of the reinforcing sheets with respect to the mean sheet

where \({E}_{\mathrm{béton}}\) is the Young’s modulus of concrete.

Knowing the vertical displacement imposed and using the previous formulas, it is possible to go back to the analytical value of the force.

2.2. Effect of gravity

We are now interested in a reinforced concrete plate embedded at both ends and subjected to the effect of gravity.

We want to calculate the resultant of the associated vertical forces \({F}_{z}\)

where \({F}_{z,\mathrm{beton}}\) and \({F}_{z,\mathrm{armat}}\) are gravity effects related to concrete and reinforcements respectively.

with \(g\) the acceleration of gravity

with \({\rho }_{\mathrm{armat}}\) the density of steel reinforcements, and \(s\) the cross section per linear meter.

By combining the preceding equations, it becomes possible to determine the value of the vertical force linked to gravity and to deduce the vertical resultant of the support reactions.

2.3. Pre-Deformations

We are trying to calculate the following average displacement \(\mathrm{Ux}\) of the free edge of a reinforced concrete plate embedded at the other edge. Pre-deformation \({\varepsilon }_{\mathrm{xx}}\) is applied to the reinforcements.

Considering the homogeneous and equal deformation on the reinforcement sheets and in the concrete, we simply write:

with \({L}_{x}\) the dimension of the plate in the \(x\) direction (equal to \({L}_{1}\) in this case)

It is thus possible to determine the desired displacement value.