1. Reference problem

1.1. Geometry

The concrete beam is straight and rectangular in cross section.

Its dimensions are: \(L\mathrm{\times }h\mathrm{\times }p\mathrm{=}10m\mathrm{\times }\mathrm{0,4}m\mathrm{\times }\mathrm{0,2}m\) (\(y\mathrm{=}h\mathrm{/}2\)).

The cable crosses the beam parallel to the middle fiber of the beam, halfway up. Its eccentricity with respect to the middle plane is \(e\mathrm{=}\mathrm{0,05}m\) (\(z\mathrm{=}e\)).

The cross-sectional area of the cable is \({S}_{a}\mathrm{=}\mathrm{1,5}{.10}^{\mathrm{-}4}{m}^{2}\).

1.2. Material properties

Concrete material constituting the beam:	Young’s Module \({E}_{b}\mathrm{=}{3.10}^{10}\mathit{Pa}\)
Steel material constituting the cable:	Young’s Module \({E}_{a}\mathrm{=}\mathrm{2,1}{.10}^{11}\mathit{Pa}\)

The Poisson’s ratio is taken to be equal to 0 for both materials. We therefore cancel the Poisson effects in the \(y\) and \(z\) directions. Travel only has components in plan \((x,z)\).

Since voltage losses in the cable are neglected, the various parameters used to estimate them are set to 0.

1.3. Boundary conditions and loads

Point \(A\) located at the bottom of the left edge of the beam, with coordinates \((\mathrm{0 };–h\mathrm{/}\mathrm{2 };0)\), is blocked in translation in all three directions and in rotation around the axis \(y\).

The blocking of the \(\mathit{DRY}\) degree of freedom of rotation implies a zero slope in the deformation of the mean fiber in \(x\mathrm{=}0\).

The left end of the cable, with coordinates \((\mathrm{0 };\mathrm{0 };e)\), is blocked in translation in all three directions.

A normal force of traction \(({F}_{\mathrm{0 }};\mathrm{0 };0)\) or \({F}_{0}\mathrm{=}{2.10}^{5}N\) is applied to the right end of the cable, with coordinates \((L;\mathrm{0 };e)\).