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


Calculation method used for the reference solution
--------------------------------------------------------

The reference solution is obtained analytically for a Timoshenko beam, taking into account the shear force deformation and the rotary inertia of the sections. Theoretical aspects are developed in the reference given in :ref:`2.4 <RefNumPara__3063_280392679>`.

Define the following dimensionless quantities:

:math:`{\Omega }_{n}=\frac{\rho A{L}^{4}}{\mathit{EI}}{\omega }_{n}^{2}` eigenvalues

:math:`j=\frac{I}{A{L}^{2}}` rotary inertia

:math:`g=\frac{\mathit{EI}}{k\text{'}AG{L}^{2}}` shear coefficient

The natural frequencies of the first modes of flexure are given by the following expression:

:math:`{\Omega }_{n}=\frac{(g+j){\lambda }_{n}^{2}+1-\sqrt{{(g-j)}^{2}{\lambda }_{n}^{4}+2(g+j){\lambda }_{n}^{2}+1}}{2gj}`

with

:math:`{\lambda }_{n}=n\pi`, :math:`n=\mathrm{1,}\mathrm{2,}\mathrm{3,}\mathrm{...}`

The frequencies of the expansion modes are given by:

:math:`{f}_{n}=(\mathrm{2n}-1)\frac{1}{4L}\sqrt{\frac{E}{\rho }}`, :math:`n=\mathrm{1,}\mathrm{2,}\mathrm{3,}\mathrm{...}`


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


.. csv-table::

    "Fashion", "Shape", "Frequency (:math:`\mathit{Hz}`)"
    "1", "bending", ":math:`115.7`"
    "2", "bending", ":math:`442.2`"
    "3", "extension", ":math:`648.6`"
    "4", "bending", ":math:`931.6`"
    "5", "bending", ":math:`1534.0`"



Uncertainty about the solution
---------------------------


* Analytical solution.


.. _RefNumPara__3063_280392679:


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


* ROBERT G., Analytical solutions in structural dynamics, Samtech Report No. 121, Liège, 1996.