2. 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 \(D\) along the \(z\) axis after applying the pressure.

2.1. 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:

\(f\mathrm{=}\mathrm{-}\frac{{\mathit{qL}}^{4}}{\mathrm{8EI}}\)

\(q\): the force distributed in \(N/m\).

\(L\): the length of the beam in \(m\).

\(E\): the Young’s modulus of the plate, therefore of concrete in \(\mathit{Pa}\).

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

For pressing force:

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

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

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

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

The arrow under the pressure load is calculated as follows:

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

So:

\({f}_{p}=-\mathrm{0.118552m}\)

For the tensioning of the pretension cable:

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

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

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

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

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

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

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

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

\({f}_{\mathit{tot}}={f}_{p}+w\)

\({f}_{\mathit{tot}}=-\mathrm{0.101677m}\)

2.2. Uncertainty about the solution

Analytical solution.