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


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

The reference solution is the one given in sheet TPLA05 /89 of the guide VPCS


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* Temperature according to :math:`z`: :math:`T(z)=\frac{-\varphi }{\lambda r}z(z-L)+{T}_{1}\mathrm{.}\frac{z}{L}`


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

The cylinder is assumed to be infinitely long (:math:`L\gg r`)

The minimum temperature is :math:`-4.17°C` in :math:`z=0.083m`


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

Temperature in :math:`z=0.,0.1,\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