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

Calculation method used for reference solutions
----------------------------------------------------------

The beam in figure [:ref:`Figure 1.1.1-a <Figure 1.1.1-a>`] verifies the equilibrium equations (plane problem).

:math:`{V}_{y}=\frac{\mathrm{dN}}{d\theta }`, :math:`N+\frac{{\mathrm{dV}}_{y}}{d\theta }=-pR`, :math:`\frac{\mathrm{dM}}{d\theta }+{\mathrm{RV}}_{y}=0`

(:math:`p`: normal constant distributed load at any point on the beam).

:math:`N(\theta )` *,* :math:`{V}_{y}(\theta )` *,* :math:`{M}_{z}(\theta )` refer to the forces (normal, sharp and bending moment) at a point of the arch expressed in the local coordinate system.

Their integration with the boundary conditions:

:math:`{V}_{y}(0)=0`, :math:`{M}_{z}(0)=0`

give:

:math:`{V}_{y}(\theta )=0`, :math:`M(\theta )=0`, :math:`N(\theta )=-\mathrm{pR}`


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

Domestic efforts for :math:`\theta \mathrm{=}0°,6°,42°` and :math:`60°`.


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

Analytical solution.


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

[:ref:`1 <1>`] Report no. 2314/A of the Aerotechnical Institute "Proposal and implementation of new test cases missing the validation of ASTER beams"