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

For each modeling, the objective is to find the correct initial prestress tension of the cable after applying CALC_PRECONT but also to determine the maximum displacement of the plate at node :math:`D` along the :math:`z` axis after applying the pressure.


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

To determine the maximum displacement, the theory of beams working under bending is used. For a beam embedded at one end and free at the other under distributed loading, the maximum displacement at the free end, called an arrow, is given by:

:math:`f\mathrm{=}\mathrm{-}\frac{{\mathit{qL}}^{4}}{\mathrm{8EI}}`

:math:`q`: the force distributed in :math:`N/m`.

:math:`L`: the length of the beam in :math:`m`.

:math:`E`: the Young's modulus of the plate, therefore of concrete in :math:`\mathit{Pa}`.

:math:`I=\frac{{t}^{3}l}{12}`: the quadratic moment of the beam with respect to the y axis in :math:`{m}^{4}`.

*For pressing force:*

Thanks to its eccentricity, the cable contributes to the rigidity of the model:

:math:`{(\mathit{EI})}_{\mathit{eq}}\mathrm{=}{E}_{b}\frac{{t}^{3}l}{12}+{E}_{a}{a}_{x}l\mathrm{\times }{e}_{z}^{2}`

:math:`{a}_{x}=\frac{A}{\mathit{dx}}`: is the reinforcement rate.

:math:`{(\mathit{EI})}_{\mathit{eq}}=13.50{\mathit{MN.m}}^{2}`

The arrow under the pressure load is calculated as follows:

:math:`{f}_{p}=\frac{-{P}_{0}l\times {L}^{4}}{8{(\mathit{EI})}_{\mathit{eq}}}`

So:

:math:`{f}_{p}=-\mathrm{0.118552m}`

*For the tensioning of the pretension cable:*

The tension cable then applies a compression force :math:`-{F}_{0}` and a bending moment :math:`-{e}_{z}{F}_{0}` to the free end of the plate.

With the principle of superposition, the expression for normal stress is:

:math:`{\sigma }_{x}=\frac{-{F}_{0}}{\mathit{tl}}(1+\frac{12{e}_{z}z}{{t}^{2}})`

If we neglect the effects of the Poisson's ratio the displacement field is given by:

:math:`\{\begin{array}{c}u(x,y,z)=\frac{-{F}_{0}}{{E}_{b}\mathit{tl}}(1+\frac{12{e}_{z}z}{{t}^{2}})x\\ v(x,y,z)=0\\ w(x,y,z)=\frac{{F}_{0}}{{E}_{b}\mathit{tl}}(\frac{6{e}_{z}}{{t}^{2}}{x}^{2})\end{array}`

The displacements are given at the free end of the beam at node :math:`D`, i.e. in :math:`(\mathrm{4,}0.5,0)`:

:math:`\{\begin{array}{c}u(x,y,z)=-\mathrm{0.375mm}\\ v(x,y,z)=0\\ w(x,y,z)=\mathrm{16.875mm}\end{array}`

By superposition, the theoretical maximum displacement of the prestressed plate under the pressure force is:

:math:`{f}_{\mathit{tot}}={f}_{p}+w`

:math:`{f}_{\mathit{tot}}=-\mathrm{0.101677m}`


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

Analytical solution.