2. Benchmark solution

The analytical reference solution is determined by the theory of beams. We consider a free-standing embedded beam. The geometric characteristics are those defined in paragraph [§2.1]. A normal compression force \((–F;\mathrm{0 };0)\) and a bending moment \((\mathrm{0 };{e}_{z}\mathrm{.}F;–{e}_{y}\mathrm{.}F)\) are applied to the free end.

The solution to this problem is as follows:

Stress tensor:

with

Eq 2-1

Movations: neglecting the Poisson effects we get

Eq 2-2

with boundary conditions

In the expressions above, \(F\) refers to the residual normal force in the cable after elastic shortening of the beam, which can be explained as a function of the initial tension \({F}_{0}\).

The axial deformation rate of concrete at the cable level is written

\({\varepsilon }_{\mathrm{xx}}^{\mathrm{béton}}=\frac{{\sigma }_{\mathrm{xx}}}{{E}_{b}}=-\frac{F}{{E}_{b}{a}^{2}}\left[1+\frac{12{e}_{y}^{2}}{{a}^{2}}+\frac{12{e}_{z}^{2}}{{a}^{2}}\right]\)

The residual normal force in the cable is deduced from the initial tension \({F}_{0}\) by the relationship

and \({\varepsilon }_{\mathrm{xx}}^{\mathrm{acier}}=\frac{F-{F}_{0}}{{E}_{a}{S}_{a}}\); from where:

either

eq 2-3

The numerical reference values are calculated using the formulas [éq 2-1], [éq 2-2], and [éq 2-3].