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


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


Analytical model of the plate
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

We consider that the response of the plate corresponds to the response of a console beam (because :math:`\nu \mathrm{=}0`). In this case, the movement of the plate is given by the movements of each plate section, in the direction :math:`x(u)`, :math:`z(w)` and their rotation with respect to the axis :math:`y({\theta }_{y})`.

:math:`{U}_{x}(x,z)\mathrm{=}u(x)+z{\theta }_{y}(x)` :math:`{U}_{z}(x,z)\mathrm{=}w(x)`

At all points, the deformations are:

:math:`{\varepsilon }_{xx}(x,z)\mathrm{=}u\text{'}+z{\theta }_{y}\text{'}` and :math:`2{\varepsilon }_{xz}(x,z)\mathrm{=}{\theta }_{y}+w\text{'}\mathrm{=}{\gamma }_{z}`

     The beam is embedded in :math:`x\mathrm{=}0` and subjected to a vertical force :math:`F` in :math:`x\mathrm{=}L`.

The resolution (principle of virtual work and the balance of forces) gives the expression of generalized efforts in the case of an elastic linear response: :math:`N` is the axial force, :math:`{M}_{y}` the moment along the :math:`y` axis and :math:`T` the shear force.

:math:`N(x)\mathrm{=}EAu\text{'}(x)` and :math:`>`

:math:`{M}_{y}(x)\mathrm{=}EI{\theta }_{y}\text{'}(x)` :math:`>`

:math:`T(x)\mathrm{=}GkA{\gamma }_{z}` :math:`>`

     where parameter :math:`k` is the shear correction factor

with the boundary conditions, we deduce the movements :math:`u,w\mathit{et}{\theta }_{y}`:

:math:`>`, :math:`>`, :math:`{\theta }_{y}(x)\mathrm{=}\frac{F}{EI}\frac{x(x\mathrm{-}2L)}{2}`

The deformations and stresses are:

:math:`{\varepsilon }_{xx}\mathrm{=}z\frac{\mathrm{-}F}{EI}(L\mathrm{-}x)` and :math:`{\sigma }_{xx}\mathrm{=}z\frac{\mathrm{-}F}{I}(L\mathrm{-}x)`

:math:`2{\varepsilon }_{xz}\mathrm{=}\frac{F}{GkA}\mathrm{=}{\gamma }_{z}` and :math:`{\sigma }_{\mathit{xz}}\mathrm{=}\frac{\mathrm{-}F}{A}6({z}^{2}\mathrm{/}{h}^{2}\mathrm{-}1\mathrm{/}4)`

where :math:`h` is the height of the plate. The distribution of shear stresses is parabolic in the thickness of the plate to respect free surface conditions.

(:math:`{\sigma }_{\mathit{xz}}(z\mathrm{=}h\mathrm{/}2)\mathrm{=}{\sigma }_{\mathit{xz}}(z\mathrm{=}\mathrm{-}h\mathrm{/}2)\mathrm{=}0`).

The vertical response when moving point :math:`x\mathrm{=}L` is:

:math:`{U}_{z}(L)\mathrm{=}\frac{F}{3EI}{L}^{3}+\frac{FL}{GkA}`


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

The reference solution is analytical. So there is no uncertainty.