2. Benchmark solution

2.1. Calculation method used for the reference solution

2.1.1. Flexion problem

The reference solution to the flexure problem is the one given in sheet SDLS03 /89 of guide VPCS, which presents the calculation method as follows.

The formulation of M.V. BARTON for a rectangular plate, placed on its four sides, leads to:

\({f}_{\mathit{ij}}=\frac{\pi }{2}\left[{\left[\frac{i}{a}\right]}^{2}+{\left[\frac{j}{b}\right]}^{2}\right]\sqrt{\frac{E{t}^{2}}{12\rho (1-{\nu }^{2})}}\)

with:

\(i\) = half-wavelength number according to \(y\) (dimension \(a\)),

\(j\) = number of half-wave lengths according to \(x\) (dimension \(b\)).

2.1.2. Membrane problem

For the search for the first vibration frequency, the problem treated with the membrane is equivalent to the following one-dimensional problem:

where:

\(k\) is the stiffness of the springs,

\(m\) is the mass of the plate.

So the frequency sought is: \(f=\frac{1}{2\pi }\sqrt{\frac{\mathrm{4k}}{m}}\)

2.2. Benchmark results

For the flexure problem, the first six vibration frequencies are calculated and for the membrane calculation, only the first frequency is calculated.

2.3. Uncertainty about the solution

Since the solutions are analytical, there is no uncertainty.

2.4. Bibliographical references

1. M.V. BARTON « Vibrations of rectangular and skew cantilever plates » - Journal of Applied Mechanics, vol.18, p.129-134 (1951).