2. Benchmark solution#
2.1. Static calculation: embedded beam#
The theory of beams working under bending provides a reference value for maximum displacement. For a beam embedded at one end and free at the other under distributed loading, the maximum displacement at the free end, called an arrow, is given by:
\(f\mathrm{=}\frac{\mathrm{-}{\mathit{qL}}^{4}}{{\mathrm{8EI}}_{\mathit{eq}}}\)
\(q\): the force distributed in \(N/m\).
\(L\): the length of the beam in \(m\).
\({\mathit{EI}}_{\mathit{eq}}\): in the mono-material case it is the product of the Young’s modulus and the quadratic moment.
In the multi-material case, and with the configuration described above, \({\mathit{EI}}_{\mathit{eq}}\) is calculated as follows:
\({\mathit{EI}}_{\mathit{eq}}\mathrm{=}{\mathrm{\int }}_{S}E(s){z(s)}^{2}\mathit{ds}\)
In Code_Aster this calculation is approximated by a sum on the fibers:
\({\mathit{EI}}_{\mathit{eq}}\mathrm{=}{\mathrm{\sum }}_{i\mathrm{=}1}^{{\mathit{nb}}_{\mathit{fibres}}}{E}_{i}{{z}_{i}}^{2}{A}_{i}\)
where \({z}_{i}\) is the \(Z\) coordinate of the center of the fiber \(i\) and where \({A}_{i}\) is its surface.
2.2. Modal calculation: simple supported beam#
For a beam with simple support, as defined in paragraph 1.3.1, the theory of beams provides reference values for the natural frequencies of the modal calculation.
The first natural frequency is: \({f}_{1}\mathrm{=}\frac{\pi }{2{L}^{2}}\sqrt{\frac{{\mathit{EI}}_{\mathit{eq}}}{m}}\).
where \(L\) is the length of the beam and \(m\) is the linear mass of the beam.
2.3. Uncertainties about the solution#
None.