Benchmark solution
=====================


Calculation method
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The total deformations in steel and in concrete are:

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

Let :math:`\epsilon` be the deformation of the middle plane of the plate and :math:`\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:

 :math:`{\epsilon }_{a}^{t}\mathrm{=}\epsilon \mathrm{-}e\chi` and :math:`{\epsilon }_{b}^{t}\mathrm{=}\epsilon \mathrm{-}y\chi`

The normal force imposed on the plate is zero:

 :math:`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:

 :math:`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:

 :math:`\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:

 :math:`\epsilon \mathrm{=}\frac{\alpha \Delta T}{A}` and :math:`\chi \mathrm{=}\frac{\mathrm{-}\alpha \Delta T}{B}`

with:

 :math:`A\mathrm{=}1+\frac{{E}_{b}{S}_{b}}{{E}_{a}{S}_{a}}+\frac{{S}_{b}{e}^{2}}{{I}_{b}}` and :math:`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: :math:`\Delta L\mathrm{=}\epsilon L`


* the rotation of the plate: :math:`{R}_{y}\mathrm{=}\chi L`


* the arrow at the end of the plate: :math:`f\mathrm{=}\mathrm{-}\chi \frac{{L}^{2}}{2}` or in the middle of the plate :math:`f\mathrm{=}\mathrm{-}\chi \frac{{L}^{2}}{8}`


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


* the normal force in concrete: :math:`{N}_{b}\mathrm{=}{E}_{b}{S}_{b}\epsilon`


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


Uncertainty about the solution
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Exact solution.