Benchmark solutions
======================

Calculation method used for reference solutions: analytical
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On section :math:`(A,B)(0<\theta <\frac{\pi }{2})`, we have:

:math:`N=\frac{F}{2}\mathrm{cos}\theta`, :math:`\mathrm{Vy}=-\frac{F}{2}\mathrm{sin}\theta`, :math:`{M}_{z}=-\frac{\mathrm{FR}}{2}(1-\mathrm{cos}\theta )+\Gamma`.

On section :math:`(B,C)(\frac{\pi }{2}<\theta <\pi )`, we have:

:math:`N=-\frac{F}{2}\mathrm{cos}\theta`, :math:`\mathrm{Vy}=\frac{F}{2}\mathrm{sin}\theta`, :math:`{M}_{z}=-\frac{\mathrm{FR}}{2}(1+\mathrm{cos}\theta )+\Gamma`.

By using the law of behavior relating :math:`M` to the rotation of the normal and taking into account that the latter is zero in :math:`A` and in :math:`B`, we have:

:math:`{\int }_{0}^{\frac{\pi }{2}}Md\theta =0`,

from where: :math:`\Gamma ={M}_{A}={M}_{C}=\frac{\pi -2}{2\pi }FR`


Benchmark results
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Domestic efforts for :math:`\theta \mathrm{=}0°` and :math:`90°`.


Uncertainty about the solution
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Analytical solution.


Bibliographical references
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1. Report no. 2314/A from the Aerotechnical Institute "Proposal and implementation of new test cases lacking the validation of *Aster* beams"