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


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

The reference result was obtained by analytical calculation using the Airy function method.


* Plane constraints:


    *
        * :math:`{\sigma }_{\mathrm{xx}}=(\mathrm{12.Py.}(x-L))/{\mathrm{2.H}}^{3}`

:math:`{\sigma }_{\mathrm{yy}}=0`

:math:`{\sigma }_{\mathrm{xy}}=\mathrm{6.P.}(({H}^{2}/4)-{y}^{2})/{\mathrm{2.H}}^{3}`


* Travel:


    *
        * :math:`u=\frac{\mathrm{12P}}{{\mathrm{EhH}}^{3}}[y(\frac{{x}^{2}}{2}-\mathrm{Lx})-(1+\frac{\nu }{2})\frac{{y}^{3}}{3}]+\mathrm{Ay}+B`

:math:`v=\frac{-\mathrm{12P}\nu }{{\mathrm{EhH}}^{3}}\frac{{y}^{2}}{2}(x-L)+\frac{\mathrm{12P}}{{\mathrm{EhH}}^{3}}[\frac{-{x}^{3}}{3}+\frac{{\mathrm{Lx}}^{2}}{2}+(1+\nu )\frac{{H}^{2}x}{4}]-\mathrm{Ax}+C`


* The constants :math:`A,B,C` depend on the boundary conditions on the movements:


    *
        * :math:`u(\mathrm{0,0})=v(\mathrm{0,0})=\frac{\partial v}{\partial x}(\mathrm{0,0})=0`

:math:`u(\mathrm{0,}-\frac{H}{2})=v(\mathrm{0,}-\frac{H}{2})=u(\mathrm{0,}\frac{H}{2})=v(\mathrm{0,}\frac{H}{2})=0`


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

Move from :math:`y` to point :math:`x=L;y=0`: :math:`v=0.3413\cdot {10}^{-3}` :math:`m`

Constraint according to :math:`x` at point: :math:`x=0;y=-H/2` :math:`{\sigma }_{\mathrm{xx}}=80.\cdot {10}^{6}` :math:`\mathrm{Pa}`


Uncertainties
------------

Analytical solution