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


Calculation method
------------------

The reference solution for calculating the stresses in the beam is given in [:ref:`1 <1>`], [:ref:`2 <2>`].


Reference quantities and results
-----------------------------------

Constraints :math:`{\sigma }_{\mathit{xx}}`, :math:`{\sigma }_{\mathit{yy}}` and :math:`{\sigma }_{\mathit{xy}}` along the :math:`Y` axis.

:math:`{\sigma }_{\mathit{xx}}(Y)\mathrm{=}\mathrm{-}\mathrm{10Y}+10`

:math:`{\sigma }_{\mathit{yy}}(Y)\mathrm{=}0.`

:math:`{\sigma }_{\mathit{xy}}(Y)\mathrm{=}0.`


.. csv-table::

    ":math:`Y(m)` "," :math:`{\sigma }_{\mathit{xx}}` "," :math:`{\sigma }_{\mathit{yy}}` "," :math:`{\sigma }_{\mathit{xy}}`"
    ":math:`0.0` "," :math:`10.0\mathit{Pa}` "," :math:`0.0\mathit{Pa}` "," :math:`0.0\mathit{Pa}`"
    ":math:`0.5` "," :math:`5.0\mathit{Pa}` "," :math:`0.0\mathit{Pa}` "," :math:`0.0\mathit{Pa}`"
    ":math:`1.0` "," :math:`0.0\mathit{Pa}` "," :math:`0.0\mathit{Pa}` "," :math:`0.0\mathit{Pa}`"
    ":math:`1.5` "," :math:`\mathrm{-}5.0\mathit{Pa}` "," :math:`0.0\mathit{Pa}` "," :math:`0.0\mathit{Pa}`"
    ":math:`2.0` "," :math:`\mathrm{-}10.0\mathit{Pa}` "," :math:`0.0\mathit{Pa}` "," :math:`0.0\mathit{Pa}`"



Uncertainties about the solution
----------------------------

Analytical Solution


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


1. M.H. SADR - LAHIDJANI: "Modeling and analysis of thin elastic plates and shells subjected to temperature fields", Doctoral thesis UTC, 1984.


2. J. PITER, HARTEL H. "Improved stress evaluation under thermal load for simple finite element", I.J.N.M.E., Vol. 15, 1507-1515, 1980.