2. Benchmark solution#
2.1. Calculation method used for the reference solution#
The reference formula is the one given in sheet SDLS02 /89 of the guide VPCS, which presents the calculation method as follows:
The formulation of M.V. BARTON, for a side plate a, led to:
\({f}_{i}=\frac{1}{2\pi {a}^{2}}{\lambda }_{i}^{2}\sqrt{\frac{E{t}^{2}}{12\rho (1-{\nu }^{2})}}\) \(i=\mathrm{1,2},\mathrm{...}\)
where: \({\lambda }_{i}^{2}=g(\alpha )\)
with, for a Poisson’s ratio \(\nu =0.3\) and \(\alpha =30°\):
\(\alpha =30°\) |
|
\({\lambda }_{1}^{2}\) |
3.961 |
\({\lambda }_{2}^{2}\) |
10.19 |
M.V.Barton mentions the sensitivity of the result to the order of the mode and to angle \(\alpha\).
This reference solution applies to thin plates such as: \(t/a<0.1\).
The \({\lambda }_{i}\) coefficients were established with limited and insufficient development.
2.2. Benchmark results#
The first two eigenmodes given by:
M.V.Barton’s formula,
the average of 5 software packages for calculating by the finite element method.
2.3. Uncertainty about the solution#
Semi-analytical solution \(\text{< 2\%}\).
2.4. Bibliographical references#
M.V. BARTON, Vibrations of rectangular and skew cantilever plates. Journal of Applied Mechanics, vol.18, p.129-134 (1951).