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 [:ref:`§2.1 <§2.1>`]. A normal compression force :math:`(–F;\mathrm{0 };0)` and a bending moment :math:`(\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:**


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with

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.. _RefEquation 2-1:

                         Eq 2-1

**Movations**: neglecting the Poisson effects we get


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.. _RefEquation 2-2:

                                         Eq 2-2

with boundary conditions

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.. _RefImage_Object_8.svg:

In the expressions above, :math:`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 :math:`{F}_{0}`.

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

:math:`{\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 :math:`{F}_{0}` by the relationship

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


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either

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.. _RefImage_Object_13.svg:


.. _RefEquation 2-3:

                         eq 2-3

The numerical reference values are calculated using the formulas [:ref:`éq 2-1 <éq 2-1>`], [:ref:`éq 2-2 <éq 2-2>`], and [:ref:`éq 2-3 <éq 2-3>`].