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


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

The reference solution is the one given in sheet SSLV07 /89 of the guide VPCS, which presents the calculation method as follows:

Travel:

:math:`\begin{array}{c}u\mathrm{=}\mathrm{-}\frac{\nu \rho gxz}{E}\\ v\mathrm{=}\mathrm{-}\frac{\nu \rho gyz}{E}\\ w\mathrm{=}\frac{\rho g{z}^{2}}{2E}+\frac{\nu \rho g}{2E}({x}^{2}+{y}^{2})\mathrm{-}\frac{\rho g{L}^{2}}{2E}\end{array}`

Constraints:

:math:`\begin{array}{c}{\sigma }_{\mathit{zz}}\mathrm{=}\rho gz\\ {\sigma }_{\mathit{xx}}\mathrm{=}{\sigma }_{\mathit{yy}}\mathrm{=}{\sigma }_{\mathit{xy}}\mathrm{=}{\sigma }_{\mathit{yz}}\mathrm{=}{\sigma }_{\mathit{zx}}\mathrm{=}0\end{array}`


.. image:: images/10001E60000011FC000013A345A9480F67F3F1C4.svg
    :width: 231
    :height: 253


.. _RefImage_10001E60000011FC000013A345A9480F67F3F1C4.svg:


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

Move points :math:`B`, :math:`C`, :math:`D`, and :math:`E`.

Constraints :math:`{\sigma }_{\mathrm{zz}}` in :math:`A` and :math:`E`.


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

Analytical solution.


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


1. S.P. TIMOSHENKO. Theory of elasticity. Paris. Polytechnic bookstore. Ch.Béranger, p.279 to 282 (1961).