Benchmark solution
=====================


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: :math:`{f}_{i}\mathrm{=}\frac{{\lambda }_{i}^{2}}{2\pi {R}^{2}}\sqrt{\frac{E{I}_{z}}{\rho A}}` :math:`i\mathrm{=}\mathrm{1,2}`;


* transverse flexure: :math:`{f}_{i}\mathrm{=}\frac{{\mu }_{i}^{2}}{2\pi {R}^{2}}\sqrt{\frac{G{I}_{p}}{\rho A}}` :math:`i=\mathrm{1,2}`;

The values :math:`{\lambda }_{i}^{2}` and :math:`{\mu }_{i}^{2}` are taken from an abacus.

This formulation can only be used for very slender pipes:


* Slender right parts greater than :math:`\frac{l}{{d}_{e}}>20`


* Thin elbow such as :math:`\alpha R>100\sqrt{\frac{{I}_{z}}{A}}` with :math:`\alpha`, center angle in radians. It is not necessary to use an elbow flexibility coefficient here.


Benchmark results
----------------------


* First four natural frequencies,


* Four first natural modes (2 transverse modes, 2 modes in the plane).


.. csv-table::

    "* Frequency (transverse mode 1)", ":math:`17.9\mathrm{Hz}`"
    "* Frequency (mode in plan 1)", ":math:`24.8\mathrm{Hz}`"
    "* Frequency (transverse mode 2)", ":math:`25.3\mathrm{Hz}`"
    "* Frequency (mode in plan 1)", ":math:`27.0\mathrm{Hz}`"


Table 2.2-1


Bibliographical references
---------------------------


1. VPCS: Guide to the validation of structural calculation software packages: "test SDLL14", SFM, AFNOR technique.


2. R.D. Blevins, formulas for natural frequency and mode shape, New York, Van Nostrand, 1979, p. 215.