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


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

The reference solution is the one given in sheet TPLA06 /89 of the guide VPCS


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* Temperature according to :math:`z`: :math:`T(z)={T}_{1}\frac{\mathrm{sinh}(\mathrm{az})}{\mathrm{sinh}(\mathrm{aL})}` or :math:`a=\sqrt{(\frac{\mathrm{2h}}{\lambda r})}`


* :math:`T(z=0)=0` :math:`T(z=L)={T}_{1}`.


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

Temperature in :math:`z=0.,0.1,0.2,\mathrm{...},0.8,0.9,1.0`


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

:math:`<1\text{\%}`

Approximate analytical solution (approximation: :math:`T=\mathrm{cte}`, for all :math:`r`)


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


1. Guide to the validation of structural calculation software packages. French Society of Mechanics, AFNOR 1990 ISBN 2-12-486611-7