2. Benchmark solution

2.1. Calculation methods used for reference solutions

In the local coordinate system of the beam, the equation relating to the displacement \({U}_{x}\) (without neglecting the elongation) is given by:

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

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

2.1.1. 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:

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

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

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

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

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

with \(r=\sqrt{{X}^{2}+{Y}^{2}+{Z}^{2}}\).

2.1.2. 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.

2.2. Benchmark results

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

2.3. Uncertainty about the solution

Analytical solution for A, B, C, and D models.

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