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

Calculation method used for the reference solution
--------------------------------------------------------

Analytical solution:

For an extendable (elastic) cable, subject to its own weight, the displacement is equal to:

:math:`x(s)=a\mathit{Argsh}\left(\frac{s}{a}\right)+\frac{\rho g}{E}as`

:math:`z(x)=a\sqrt{1+\frac{{s}^{2}}{{a}^{2}}}+\frac{\rho g}{E}\frac{{s}^{2}}{2}-a\sqrt{1+\frac{{l}_{0}^{2}}{{a}^{2}}}-\frac{\rho g}{E}\frac{{l}_{0}^{2}}{2}`

:math:`a` solution of equation :math:`L=a\mathrm{Argsh}(\frac{{l}_{0}}{a})+\frac{\rho g}{E}a{l}_{0}=f(a)`

With :math:`s` curvilinear abscissa, :math:`s\in [-{l}_{o},{l}_{o}]`. Here we are interested in the arrow in the center (point :math:`C`):

:math:`z(C)=a-a\sqrt{1+\frac{{l}_{0}^{2}}{{a}^{2}}}-\frac{\rho g}{E}\frac{{l}_{0}^{2}}{2}`

:math:`a` solution of equation :math:`L=a\mathrm{Argsh}(\frac{{l}_{0}}{a})+\frac{\rho g}{E}a{l}_{0}=f(a)`

The only difficulty in calculating this solution is solving equation :math:`L=f(a)`. This resolution was done numerically (Fortran program using the zero search routine from Aster ZEROFO).

**Note:**

*In the case of thermal expansion, the solution is the same as before, considering that the initial length* :math:`2{l}_{0}` *is equal to its initial length* :math:`\mathrm{2L}` *increased by the linear expansion:* :math:`{l}_{0}=L(1+\alpha T)`


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

Move to :math:`Z` to point :math:`C`


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

Semianalytic solution: the numerical resolution of equation :math:`L=f(a)` gives a value to within :math:`{10}^{-3}`.


Bibliographical references
---------------------------

[:ref:`1 <1>`] C. CONEIM "On the approximation of the equations of the statics of aerial cables in the presence of electromagnetic force fields". Thesis and note HI/3640-02 (February 1981)