2. Benchmark solution

2.1. Calculation method used for the reference solution

The reference solution is the one given in sheet SDLL10 /89 of the guide VPCS, which presents the calculation method as follows:

Pulse \({\omega }_{i}\) is given by the roots of the equation:

with:

The translation components \(\nu\) of the \({F}_{i}(x)\) mode are then:

\({\phi }_{i}(x)\mathrm{=}{e}^{\alpha x}\left[\mathrm{cos}(\mathit{rx})\mathrm{-}\mathit{ch}(\mathit{rx})+\frac{\mathrm{cos}(\mathit{rL})\mathrm{-}\mathit{ch}(\mathit{sL})}{r\mathit{sh}(\mathit{sL})\mathrm{-}s\mathrm{sin}(\mathit{rL})}(s\mathrm{sin}(\mathit{rx})\mathrm{-}r\mathit{sh}(\mathit{sx}))\right]\)

2.2. Benchmark results

4 first natural frequencies and natural modes standardized to 1 for the largest translation component.

2.3. Uncertainty about the solution

Analytical solution.

2.4. Bibliographical references

• Dynamic Analysis Working Group. Structural Analysis Software Validation Committee. French Society of Mechanics (1988).