2. Benchmark solution#
2.1. Calculation method used for the reference solution#
The reference solution is obtained analytically at a given node of the calculation. Here we are only interested in the validation of post-treatment in fatigue; the values of the stresses and displacements at the node in question are those resulting from the calculation with Code_Aster.
2.2. Benchmark results#
We are particularly interested in:
to the maximum allowable amplitude coefficient associated with node \(\mathit{N111}\) of mesh \(294\), which is the most statically loaded node;
at the maximum vibration amplitude admissible for node \(\mathit{N194}\) (at the end of the fin).
Two cases are considered: a solicitation to the first natural mode only; a solicitation to the first two natural modes only. In the latter case, it is assumed that the weight of the second natural mode is equal to half the weight of the first natural mode.
The table below corresponds to the first calculation. The static constraint \({\sigma }_{\mathrm{stat}}\) (von Mises signed) at the node in question is \(\mathrm{307,71}\mathit{MPa}\); the modal constraints of the first two modes are \(\mathrm{9,80}\mathit{MPa}\) and \(-\mathrm{31,15}\mathit{MPa}\) respectively.
Modal constraints correspond to standardized constraints. The objective is, knowing the static stress, to calculate the maximum amplitude of variation in the dynamic stress (sum of the modal constraints taken into account) allowing unlimited endurance of the structure. The coefficient \(\alpha\) corresponds to this maximum amplitude. It is calculated either using the Goodman line or using the Gerber parabola:
\({\alpha }_{\mathit{Goodman}}\mathrm{=}{S}_{l}(1\mathrm{-}\frac{{\sigma }_{\mathit{stat}}}{{S}_{u}})\mathrm{/}{\sigma }_{\mathit{dyn}}\) and \({\alpha }_{\mathit{Gerber}}\mathrm{=}{S}_{l}(1\mathrm{-}{(\frac{{\sigma }_{\mathit{stat}}}{{S}_{u}})}^{2})\mathrm{/}{\sigma }_{\mathit{dyn}}\)
In these two formulas, \({S}_{l}\) represents the endurance limit. \({S}_{l}\mathrm{=}500\) in this test is associated with the amplitude of the stress alternating to \(1.E6\) cycles. \({S}_{u}\) is the maximum stress of the material. \({S}_{u}\mathrm{=}1000\) in this test.
Case |
Static Constraint \({\sigma }_{\mathit{stat}}\) |
Dynamic Constraint \({\sigma }_{\mathit{dyn}}\) |
|
|
|
Mode 1 |
\(\mathrm{307,71}\mathit{MPa}\) |
|
46.18 |
35.32 |
|
Mode 1 + 0.5 Mode 2 |
|
|
17.83 |
13.64 |
To pass from the coefficient \(\alpha\) to the admissible vibration amplitude at a given point \(\mathrm{\partial }\tilde{u}\) (corresponding for example to the position of a sensor), an additional operation is required, see the documentation [U4.83.02].
We note \({\tilde{u}}_{\mathit{mod}}^{i}\) the displacement at the point of interest associated with the \(i\) mode; the admissible vibration amplitude at this point is then:
\(\mathrm{\partial }\tilde{u}\mathrm{=}\mathit{min}(\alpha )\mathrm{\sum }_{i\mathrm{=}1}^{N}{\beta }_{i}{\tilde{u}}_{\mathit{mod}}^{i}\)
where \({({\beta }_{i})}_{1\le i\le N}\) the relative weights of the various eigenmodes considered.
This operation is explained below for node \(\mathit{N194}\) (at the end of the fin). Note that the values of \({\beta }_{i},{\tilde{u}}_{\mathit{mod}}^{i}\) are obtained with Code_Aster.
Note that the most statically loaded node does not correspond to the most penalizing node: the coefficient \(\alpha\) calculated above is not the minimum value over the entire mesh (see below for the minimum value for the Gerber correction).
Case |
\(\mathit{DX}\) (\(\mathrm{mm}\)) |
\(\mathrm{DY}\) (\(\mathrm{mm}\)) |
\(\mathrm{DZ}\) (\(\mathrm{mm}\)) |
\({\alpha }_{\mathit{Gerber}}^{\mathit{min}}\) |
(\(\mathrm{mm}\)) |
\({\mathrm{DY}}^{\mathrm{max}}\) (\(\mathrm{mm}\)) |
\({\mathrm{DZ}}^{\mathrm{max}}\) (\(\mathrm{mm}\)) |
\({D}^{\mathrm{max}}\) (\(\mathrm{mm}\)) |
||||
Mode 1 |
0.38 |
1 |
-0.05 |
-0.05 |
31.36 |
11.91 |
31.36 |
1.56 |
1.56 |
33.58 |
||
Mode 2 |
0.24 |
0.27 |
0.27 |
0.92 |
/ |
/ |
/ |
/ |
/ |
|||
Mode 1 + 0.5Mode 2 |
0.49 |
0.49 |
1.14 |
1.14 |
0.51 |
8.73 |
3.28 |
7.56 |
3.56 |
3.40 |
3.40 |
9.92 |
2.3. Uncertainty about the solution#
None, the solution is analytical.