2. Benchmark solutions#
2.1. Calculation method used for reference solutions#
2.1.1. Distributed traction-compression loading#
A straight beam of length \(L\) working only under traction-compression is subjected to a loading distribution that is constant along \(x\) but varies sinusoidally as a function of time. It is embedded at both ends.
\(\mathrm{\{}\begin{array}{c}\rho S\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{t}^{2}}\mathrm{-}\mathit{ES}\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^{2}}\mathrm{=}f(t)\\ u(0)\mathrm{=}\mathrm{0,}u(L)\mathrm{=}0\end{array}\)
To solve, we apply the Fourier transform in time to the equation:
\(\frac{{\mathrm{\partial }}^{2}\stackrel{ˆ}{u}}{\mathrm{\partial }{x}^{2}}\mathrm{=}\mathrm{-}\frac{\rho }{E}4{\pi }^{2}{\omega }^{2}\stackrel{ˆ}{u}+\frac{1}{\mathit{ES}}\stackrel{ˆ}{f}(\omega )\)
\(\stackrel{ˆ}{u}\) |
: |
Fourier transform of \(u\), |
\(\stackrel{ˆ}{f}\) |
: |
Fourier transform of \(f\). |
So, for \(f(t)\mathrm{=}F\mathrm{cos}(2\pi {\omega }_{0}t)\), we have:
with: \({a}^{2}\mathrm{=}\frac{E}{\rho }\).
Using the law of behavior gives us the tensile force compression:
\(N(x,t)=\frac{aF}{2\pi {\omega }_{0}}\left\{\left[1-\mathrm{cos}\left(\frac{2\pi {\omega }_{0}}{a}L\right)\right]\frac{\mathrm{cos}\left(\frac{2\pi {\omega }_{0}}{a}x\right)}{\mathrm{sin}\left(\frac{2\pi {\omega }_{0}}{a}L\right)}-\mathrm{sin}\left(\frac{2\pi {\omega }_{0}}{a}x\right)\right\}\mathrm{cos}\left(2\pi {\omega }_{0}t\right)\)
2.1.2. Punctual loads#
A console beam of length \(L\) working only in tensile compression (or torsional) is subjected to a sinusoidal force in time, (or a moment) applied to its free end.
2.1.2.1. Traction#
The resolution technique is equivalent to that in paragraph [§2.1.1.1].
For \(f(t)\mathrm{=}F\mathrm{cos}(2\pi {\omega }_{0}t)\), we have:
and
2.1.2.2. Torsion#
2.2. Benchmark results#
Domestic efforts (\(N\) and \(\mathrm{MT}\))
2.3. Uncertainty about the solution#
Analytical solution.
2.4. Bibliographical references#
Report no. 2314/A from the Aerotechnical Institute « Proposal and implementation of new test cases lacking in the validation of beams ASTER »