2. Benchmark solution

2.1. Calculation method

The deformation is estimated from the relative elongation of the beam.

The elongation of a beam of length \(L\) due to a longitudinal force \(F\) is written as:

\(\Delta L=\frac{FL}{ES}\)

In our case, we apply a force per unit area \({F}_{B}\) to the free end of the beam, so the relative elongation of the beam is in the following form:

\(\frac{\Delta L}{L}={\epsilon }_{\mathrm{xx}}=\frac{{F}_{B}}{E}\)

For this test case, the results from a static calculation, a harmonic calculation, a transitory calculation and a modal calculation are calculated.

For the static case, we get:

\({\varepsilon }_{\mathrm{xx}}=\frac{{F}_{B}}{E}\) and \({\varepsilon }_{\mathrm{yy}}={\varepsilon }_{\mathrm{zz}}=-\nu {\varepsilon }_{\mathrm{xx}}\)

For dynamic cases, the system is governed by the following equation:

\(M\frac{{\partial }^{2}u}{\partial {t}^{2}}+Ku={F}_{\mathrm{ext}}\)

For a girder in tension - compression, if we consider a model that contains only one element, the matrices of mass \(M\) and stiffness \(K\) take the following form:

\(M=\frac{\rho SL}{6}\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right]\) \(K=\frac{\mathrm{ES}}{L}\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]\) with: \(u=\left[\begin{array}{c}{u}_{1}\\ {u}_{2}\end{array}\right]\)

\({u}_{1}\) and \({u}_{2}\) are the movements of the element’s nodes.

In harmonic pulse response \(\omega\), the movements of the nodes of the element are governed by the following relationship:

\(\frac{\mathrm{ES}}{L}\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]\left[\begin{array}{c}{u}_{1}\\ {u}_{2}\end{array}\right]-{\omega }^{2}\frac{\rho SL}{6}\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right]\left[\begin{array}{c}{u}_{1}\\ {u}_{2}\end{array}\right]=\left[\begin{array}{c}{F}_{1}\\ {F}_{2}\end{array}\right]\)

By exploiting the second line of this relationship, and applying the boundary conditions (\({u}_{1}=0\) and \({F}_{2}=F={F}_{B}S\)), we obtain:

\({u}_{2}=\frac{{F}_{B}}{\frac{E}{L}-{\omega }^{2}\frac{\rho L}{3}}=\Delta L\)

The deformation at the free end of the beam is written as:

\({\varepsilon }_{\mathrm{xx}}=\frac{\Delta L}{L}=\frac{{F}_{B}}{E-{\omega }^{2}\frac{\rho {L}^{2}}{3}}\) and \({\varepsilon }_{\mathrm{yy}}={\varepsilon }_{\mathrm{zz}}=-\nu {\varepsilon }_{\mathrm{xx}}\)

The reference solution for the transitory solution can be obtained in the same way. If we apply a longitudinal force \(F(t)=Ft={F}_{B}St\) with an initial condition of a beam in balance (zero initial displacement and zero initial speed), we obtain:

\({u}_{2}(t)=\frac{3{F}_{B}}{\rho L{\omega }_{0}^{2}}\left[t-\frac{\mathrm{sin}({\omega }_{0}t)}{{\omega }_{0}}\right]=\Delta L\) with: \({{\omega }_{0}}^{2}=\frac{3E}{\rho {L}^{2}}\)

And the deformation at the free end of the beam is written as:

\({\varepsilon }_{\mathrm{xx}}=\frac{\Delta L}{L}=\frac{3{F}_{B}}{\rho {L}^{2}{\omega }_{0}^{2}}\left[t-\frac{\mathrm{sin}({\omega }_{0}t)}{{\omega }_{0}}\right]\) and \({\varepsilon }_{\mathrm{yy}}={\varepsilon }_{\mathrm{zz}}=-\nu {\varepsilon }_{\mathrm{xx}}\)

In the case of modal calculation, a non-regression test is carried out on the deformations calculated at the middle point of the beam.

We also simulate a 90-degree rotation in order to check the change of frame of reference in OBSERVATION.

2.2. Reference quantities and results

We test the value of the mean deformation on the surfaces: \(\mathrm{surf3}\), \(\mathrm{surf4}\), \(\mathrm{surf5}\) and \(\mathrm{surf6}\). The results obtained are then projected onto the « measurement » model which includes only the nodes \(\mathrm{P3}\), \(\mathrm{P4}\), \(\mathrm{P5}\) and \(\mathrm{P6}\) associated with the surfaces \(\mathrm{surf3}\), \(\mathrm{surf4}\), \(\mathrm{surf5}\) and \(\mathrm{surf6}\).

For the validation of the static solution, we chose: \({F}_{B}=1000.N/{m}^{2}\)

For the validation of the harmonic solution, we chose: \({F}_{B}=1000.(1+\mathrm{2j})N/{m}^{2}\) and: \(\omega =2\pi 200\mathrm{rd}{s}^{-1}\)

For the validation of the transitory solution, we chose: \({F}_{B}=1000.tN/{m}^{2}\) and the solution is tested at the moment \(t=1s\)

The values of the fields obtained by mixed observation are also tested.

2.3. Uncertainties about the solution

Analytical solution for the static case, the harmonic case and the transitory case.

A non-regression solution is proposed for the case of modal deformation.

2.4. Bibliographical reference

[R3.08.01] « Exact » elements of beams (straight and curved).