2. Benchmark solution

2.1. Calculation method

The total deformations in steel and in concrete are:

\({\epsilon }_{a}^{t}\mathrm{=}{\epsilon }_{a}^{m}+{\epsilon }_{a}^{\mathit{th}}\) and \({\epsilon }_{b}^{t}\mathrm{=}{\epsilon }_{b}^{m}\) with \({\epsilon }_{a}^{\mathit{th}}\mathrm{=}\alpha \Delta T\)

Let \(\epsilon\) be the deformation of the middle plane of the plate and \(\chi\) the curvature of the plate, the two unknowns to be found. By respecting the kinematics (the sections remain flat), the steel being perfectly bonded to the concrete, we have:

\({\epsilon }_{a}^{t}\mathrm{=}\epsilon \mathrm{-}e\chi\) and \({\epsilon }_{b}^{t}\mathrm{=}\epsilon \mathrm{-}y\chi\)

The normal force imposed on the plate is zero:

\(N\mathrm{=}{N}_{a}+{N}_{b}\mathrm{=}{E}_{a}{S}_{a}{\epsilon }_{a}^{m}+{E}_{b}\mathrm{\int }{\epsilon }_{b}^{m}\text{}\mathrm{=}{E}_{a}{S}_{a}(\epsilon \mathrm{-}e\chi \mathrm{-}{\epsilon }_{a}^{\mathit{th}})+{E}_{b}{S}_{b}\epsilon \mathrm{=}0\)

Likewise, the bending moment imposed on the plate is zero:

\(M={M}_{a}+{M}_{b}=e{E}_{a}{S}_{a}{\epsilon }_{a}^{m}+{E}_{b}\int y{\epsilon }_{b}^{m}\text{}={E}_{a}{S}_{a}(e\epsilon -{e}^{2}\chi -e{\epsilon }_{a}^{\mathrm{th}})+{E}_{b}{I}_{b}\chi =0\)

We thus obtain two equations to determine the two unknowns:

\(\begin{array}{c}({E}_{a}{S}_{a}+{E}_{b}{S}_{b})\epsilon \mathrm{-}{E}_{a}{S}_{a}e\chi \mathrm{=}{E}_{a}{S}_{a}{\epsilon }_{a}^{\mathit{th}}\\ {E}_{a}{S}_{a}e\epsilon +({E}_{b}{I}_{b}\mathrm{-}{E}_{a}{S}_{a}{e}^{2})\chi \mathrm{=}{E}_{a}{S}_{a}e{\epsilon }_{a}^{\mathit{th}}\end{array}\)

We get:

\(\epsilon \mathrm{=}\frac{\alpha \Delta T}{A}\) and \(\chi \mathrm{=}\frac{\mathrm{-}\alpha \Delta T}{B}\)

with:

\(A\mathrm{=}1+\frac{{E}_{b}{S}_{b}}{{E}_{a}{S}_{a}}+\frac{{S}_{b}{e}^{2}}{{I}_{b}}\) and \(B\mathrm{=}e+(\frac{1}{{E}_{a}{S}_{a}}+\frac{1}{{E}_{b}{S}_{b}})+\frac{{E}_{b}{I}_{b}}{e}\)

From these values we can calculate:

• the elongation of the plate: \(\Delta L\mathrm{=}\epsilon L\)

• the rotation of the plate: \({R}_{y}\mathrm{=}\chi L\)

• the arrow at the end of the plate: \(f\mathrm{=}\mathrm{-}\chi \frac{{L}^{2}}{2}\) or in the middle of the plate \(f\mathrm{=}\mathrm{-}\chi \frac{{L}^{2}}{8}\)

• the normal stress in steel: \({\sigma }_{a}\mathrm{=}{E}_{a}(\epsilon \mathrm{-}e\chi \mathrm{-}{\epsilon }_{a}^{\mathit{th}})\)

• the normal force in concrete: \({N}_{b}\mathrm{=}{E}_{b}{S}_{b}\epsilon\)

2.2. Reference quantities and results

We calculate the elongation and the deflection of the plate (displacement \({U}_{x}\) and \({U}_{z}\) of a node of the edge \(C\) of the plate), the rotation (\({R}_{y}\) constant over the length), the normal force in the reinforcements, the normal force in the reinforcements, the normal force in the concrete \({N}_{b}\).

2.3. Uncertainty about the solution

Exact solution.