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

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

In a local coordinate system, :math:`x` along the :math:`\mathit{OA}` axis of the beam, the bending moment, at the coordinate plane :math:`x`, has the expression:

:math:`{M}_{{F}_{y}}(x)=p{\int }_{x}^{L}\left[v(\xi )-v(x)\right]d\xi`.

Arrow :math:`v(x)` therefore satisfies the equation:

:math:`E{I}_{z}=\frac{{d}^{2}v}{{\mathrm{dx}}^{2}}=p{\int }_{x}^{L}\left[v(\xi )-v(x)\right]d\xi =-p\left[{\int }_{x}^{L}v(\xi )d\xi +(L-x)v(x)\right]`

By differentiating the two members, we get the differential equation:

:math:`\frac{{d}^{3}v}{{\mathrm{dx}}^{3}}+\frac{p}{E{I}_{z}}(L-x)\frac{\mathrm{dv}}{\mathrm{dx}}=0`

The function :math:`v\text{'}(x)=\frac{\mathrm{dv}}{\mathrm{dx}}` satisfies the linear and homogeneous differential equation of the second order:

:math:`\frac{{d}^{\mathrm{2v}}\text{'}}{{\mathrm{dx}}^{2}}+\frac{p}{E{I}_{z}}(L-x)v\text{'}=0`,

that can be solved using Bessel functions. We then find the value of the critical linear weight equal to:

:math:`{p}_{c}=\mathrm{7,837}\frac{E{I}_{z}}{{L}^{3}}`.

The analytical solution gives numerically:

:math:`{p}_{c}=\mathrm{7,837}2{10}^{11}\cdot \frac{{10}^{-8}}{12}=\mathrm{1,3061667}{10}^{3}`.


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

The critical value of the multiplier :math:`\lambda`: :math:`{\lambda }_{c}=\frac{{P}_{c}}{\rho Sg}`


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 lacking in the validation of *Aster* beams"