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


Calculation method used for the reference solution
--------------------------------------------------------

The solution is analytical.


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Thermal loading is equivalent to a loading defined by a uniform distribution of moments at the edges as shown in the figure.

The value of these moments per unit length is equal to:

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.

Either:

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. This leads to an even distribution of :math:`M` in the plate.


Benchmark results
----------------------

So we have :math:`M=2380.95238N`; the plate being rotated by an angle :math:`\theta \mathrm{=}53°.1301`, we have components whose absolute value is: :math:`M\mathrm{\times }\mathrm{cos}\theta \mathrm{=}1428.5715N` and :math:`M\mathrm{\times }\mathrm{sin}\theta \mathrm{=}1904.76184N`

The reactions are defined by a distribution of moments equal to the previous one in absolute value and of opposite sign.

Meshes are squares whose length is equal to :math:`0.05m`, so the moments in each node must be equal to :math:`{M}_{1}\mathrm{=}M\mathrm{\times }\mathrm{cos}\theta \mathrm{\times }0.05\mathrm{=}71.42857\mathit{N.m}`

and :math:`{M}_{2}\mathrm{=}M\mathrm{\times }\mathrm{sin}\theta \mathrm{\times }0.05\mathrm{=}95.2381\mathit{N.m}`

either

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Uncertainty about the solution
---------------------------

Uncertainty is zero.


Bibliographical references
---------------------------


1. TIMOSHENKO: Theory of plates and shells chapter 2, article 14.