2. Benchmark solution#
2.1. Calculation method used for the reference solution#
Analytical solution [bib1]: movements in \(B\) in the coordinate system \((\mathit{Oxyz})\) linked to the beam.
Single pull: \({u}_{x}=\frac{{F}_{x}L}{ES}\)
Simple bending: \({u}_{y}=\frac{{F}_{y}{L}^{3}}{3E{I}_{z}}\) \({t}_{z}=\frac{{L}^{2}{F}_{y}}{2E{I}_{z}}\)
Simple bending: \({u}_{z}=\frac{{F}_{z}{L}^{3}}{3E{I}_{y}}\) \({\theta }_{y}=\frac{-{L}^{2}{F}_{z}}{2E{I}_{y}}\)
Twist: \({\theta }_{x}=\frac{{M}_{x}L}{G{J}_{x}}\)
Pure flex: \({u}_{z}=\frac{-{M}_{y}{L}^{2}}{2E{I}_{y}}\) \({\theta }_{y}=\frac{{M}_{y}L}{E{I}_{y}}\)
Pure flex: \({u}_{y}=\frac{{M}_{z}{L}^{2}}{2E{I}_{z}}\) \({\theta }_{z}=\frac{{M}_{z}L}{E{I}_{z}}\)
Pressure: \({u}_{r}=\frac{P{a}^{2}r}{E({b}^{2}-{a}^{2})}\left[(1-\nu )+(1+\nu )\frac{{b}^{2}}{{r}^{2}}\right]\) calculated in \(r=\frac{a+b}{2}\)
actually \({u}_{r}\in \left[7.12E\text{-}06,7.78E\text{-}06\right]\) for \(r\in \left[b,a\right]\)
Here, the values are obtained with:
\(S=1.809557E\text{-}03{m}^{2}\) \({I}_{y}={I}_{z}=1.18707E\text{-}06{m}^{4}\) \({J}_{x}=2.37414E\text{-}06{m}^{4}\) \(L=5m\)
For generalized beam deformations, using the law of behavior, we obtain:
Single pull: \({\epsilon }_{x}=\frac{{F}_{x}}{ES}\)
Simple bending: \({\gamma }_{\mathrm{xy}}=\frac{{F}_{y}}{GS}\) \({\kappa }_{z}=\frac{{F}_{y}(L-x)}{E{I}_{z}}\)
Simple bending: \({\gamma }_{\mathrm{xz}}=\frac{{F}_{z}}{GS}\) \({\kappa }_{y}=\frac{{F}_{z}(L-x)}{E{I}_{y}}\)
Twist: \({\kappa }_{x}=\frac{{M}_{x}}{G{J}_{x}}\)
Pure flex: \({\kappa }_{y}=\frac{{M}_{y}}{E{I}_{y}}\)
Pure flex: \({\kappa }_{z}=\frac{{M}_{z}}{E{I}_{z}}\)
Gravity loading and line loading:
If \(p\) refers to the distributed load, the moment at the origin is equal to: \(M(o)=\frac{p{L}^{2}}{2}\) and the next displacement is set to \(z\) at the end B is equal to: \({u}_{z}(B)=\frac{p{L}^{4}}{8EI}\).
The thermal expansion load leads to an axial displacement (in the local direction \(x\)):
\({U}_{x}(B)=L(\alpha T)\)
The free expansion deformations of the pipe surface are simply, as a local coordinate system:
\({\epsilon }_{\mathrm{xx}}={\epsilon }_{\mathrm{yy}}=\alpha T\)
Finally, to validate the calculation of the mass matrix, a modal analysis of the first 12 natural modes (with embedding in \(O\)) must give, for the bending modes:
\({f}_{i}={(\frac{{\lambda }_{i}}{L})}^{2}\sqrt{\frac{EI}{\rho S}}\)
Mode |
\({\lambda }_{i}\) |
Frequency |
1 |
1.87510407 |
2.9030234 |
2 |
4.69409113 |
18.192937 |
3 |
7.85475744 |
50.9407506 |
4 |
10.9955407 |
99.8235399 |
5 |
14.1371684 |
165.015464 |
6 |
17.2787596 |
246.504532 |
7 |
20.4203522 |
344.291453 |
8 |
23.5619449 |
458.376195 |
9 |
26.7035376 |
588.758758 |
10 |
29.8451302 |
735.43914 |
11 |
32.9867229 |
898.417343 |
12 |
36.1283155 |
1077.69337 |
2.2. Benchmark results#
Movement to point \(B\), efforts, stresses and deformations in the vicinity of point \(O\).
Generalized deformities.
Natural frequencies
2.3. Uncertainty about the solution#
Analytical solution.
2.4. Bibliographical references#
Validation manual, test SSLL102 Embedded beam subjected to unit efforts [V3.01.102]