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


Calculation methods used for reference solutions
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In the local coordinate system of the beam, the equation relating to the displacement :math:`{U}_{x}` (without neglecting the elongation) is given by:

:math:`\frac{\partial }{\partial x}\left(ES(x)\frac{\partial {U}_{x}}{\partial x}\right)+\rho S\left(x\right){\omega }^{2}\left(x+{U}_{x}\right)=0\text{.}\text{ }`

The boundary conditions associated with this equation are written as: :math:`\begin{array}{}{U}_{x}(0)=0\\ \frac{\partial {U}_{x}}{\partial x}(L)={\sigma }_{\mathrm{xx}}(L)=0\end{array}`.


Case of a constant section — Analytical solution
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

In the case of a constant cross section (models A, B, C, and D), the above equation can be simplified as follows:

:math:`\frac{{\partial }^{2}{U}_{x}}{\partial {x}^{2}}+\frac{\rho }{E}{\omega }^{2}(x+{U}_{x})=0`.

We're posing :math:`\alpha =\sqrt{\frac{\rho {\omega }^{2}}{E}}`. By integrating the previous differential equation, we obtain, in the frame of reference of the beam:

:math:`{U}_{x}(x)=\frac{\mathrm{sin}(\alpha x)}{\alpha \mathrm{cos}(\alpha L)}-x` :math:`{U}_{y}={U}_{z}=0`

In the global coordinate system, the displacement at any point of the beam is therefore written:

:math:`\begin{array}{}{U}_{x}(X,Y,Z)=\frac{1}{\sqrt{3}}(\frac{\mathrm{sin}(\alpha r)}{\alpha \mathrm{cos}(\alpha L)}-r)\\ {U}_{y}(X,Y,Z)=\frac{1}{\sqrt{3}}(\frac{\mathrm{sin}(\alpha r)}{\alpha \mathrm{cos}(\alpha L)}-r)\\ {U}_{z}(X,Y,Z)=\frac{1}{\sqrt{3}}(\frac{\mathrm{sin}(\alpha r)}{\alpha \mathrm{cos}(\alpha L)}-r)\end{array}`,

with :math:`r=\sqrt{{X}^{2}+{Y}^{2}+{Z}^{2}}`.


Case of a variable section — Numerical solution
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

In the case of the variable section beam in modeling E, the reference solution in question comes from an Aster calculation carried out with a 3D model, with 19268 quadratic elements (TETRA10) and 29122 nodes.


Benchmark results
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Values of the three movements in the center of the section farthest from the axis of rotation.


Uncertainty about the solution
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Analytical solution for A, B, C, and D models.

Numerical solution (3D Aster calculation) for E modeling.