2. Benchmark solution#

2.1. Calculation method used for the reference solution#

The vibration equation for a prestressed beam is:

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

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

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

2.2. Benchmark results#

5 first natural frequencies.

2.3. Uncertainty about the solution#

Analytical solution (hypothesis of Euler-Bernouilli beams).

2.4. Bibliographical references#

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