2. Benchmark solution#
2.1. Calculation method used for the reference solution#
The Rayleigh method applied to slender straight beam elements and to a thin curved beam element makes it possible to determine parameters such as:
bending in the plane: \({f}_{i}\mathrm{=}\frac{{\lambda }_{i}^{2}}{2\pi {R}^{2}}\sqrt{\frac{E{I}_{z}}{\rho A}}\) \(i\mathrm{=}\mathrm{1,2}\);
transverse flexure: \({f}_{i}\mathrm{=}\frac{{\mu }_{i}^{2}}{2\pi {R}^{2}}\sqrt{\frac{G{I}_{p}}{\rho A}}\) \(i=\mathrm{1,2}\);
The values \({\lambda }_{i}^{2}\) and \({\mu }_{i}^{2}\) are taken from an abacus.
This formulation can only be used for very slender pipes:
Slender right parts greater than \(\frac{l}{{d}_{e}}>20\)
Thin elbow such as \(\alpha R>100\sqrt{\frac{{I}_{z}}{A}}\) with \(\alpha\), center angle in radians. It is not necessary to use an elbow flexibility coefficient here.
2.2. Benchmark results#
First four natural frequencies,
Four first natural modes (2 transverse modes, 2 modes in the plane).
|
\(17.9\mathrm{Hz}\) |
|
\(24.8\mathrm{Hz}\) |
|
\(25.3\mathrm{Hz}\) |
|
\(27.0\mathrm{Hz}\) |
Table 2.2-1
2.3. Bibliographical references#
VPCS: Guide to the validation of structural calculation software packages: « test SDLL14 », SFM, AFNOR technique.
R.D. Blevins, formulas for natural frequency and mode shape, New York, Van Nostrand, 1979, p. 215.