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


Calculation method used for the reference solution
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The vibration equation for a prestressed beam is:

:math:`{\mathrm{EI}}_{z}\frac{{\partial }^{4}y}{\partial {x}^{4}}+P\frac{{\partial }^{2}y}{\partial {x}^{2}}=-\rho S\frac{{\partial }^{2}y}{\partial {x}^{2}}`

tensile prestress if :math:`P>0`, compression if :math:`P<0`, and leads to the natural frequencies of bending (Euler-Bernoulli hypothesis)

:math:`{f}_{i}\mathrm{=}\frac{{i}^{2}\pi }{2{L}^{2}}{(1+\frac{{\mathit{PL}}^{2}}{{\mathit{EI}}_{z}{i}^{2}{\pi }^{2}})}^{1\mathrm{/}2}{(\frac{{\mathit{EI}}_{z}}{\rho S})}^{1\mathrm{/}2}`, :math:`i=\mathrm{1,2}\mathrm{,3},\mathrm{...}`


Benchmark results
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5 first natural frequencies.


Uncertainty about the solution
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Analytical solution (hypothesis of Euler-Bernouilli beams).


Bibliographical references
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1. Robert D. BLEVINS Formulas for natural frequency and mode shape - 1979 p.144 (corrected formula 8.20).