7. Summary of results#
Maximum differences (in%) between AXIS_FOURIER models and 3D modeling, observed at points \(E,F,B\) (in plane \(\theta =0°\)), on the combined load cases.
Location |
Variance AXIS_FOURIER /3D In (%) |
|
Travel \(U\): = \(u\) in 3D = \({u}_{r}\) in AXI |
|
1.5 |
Constraints \({\sigma }_{\mathrm{zz}}\) |
|
2.8 |
Constraints \({\sigma }_{\mathrm{xx}}(\mathrm{3D})\) = \({\sigma }_{\mathrm{rr}}(\mathrm{AXI})\) |
|
—14.1 |
Constraints \({\sigma }_{\mathrm{yy}}(\mathrm{3D})\) = \({\sigma }_{\theta \theta }(\mathrm{AXI})\) |
|
14.6 |
The results between the 3D models on the one hand and AXIS_FOURIER on the other hand, are consistent with respect to the displacements (difference of 1.5%) and the flexural stress \({\sigma }_{\mathrm{zz}}\) (difference of 2.8%).
At embedding, the \({\sigma }_{\mathrm{xx}}={\sigma }_{\mathrm{yy}}=0\) relationship results in:
\({\sigma }_{\mathrm{xx}}={\sigma }_{\mathrm{yy}}=\frac{\nu }{1-\nu }{\sigma }_{\mathrm{zz}}\)
The embedding relationship is well verified at point \(B\), in 3D modeling.
In addition, in point \(B\), we also have:
\({\sigma }_{\mathrm{xx}}={\sigma }_{\mathrm{rr}}\)
\({\sigma }_{\mathrm{yy}}={\sigma }_{\theta \theta }\)
In modeling AXIS_FOURIER, the difference between the two constraints is around 25%.
A second calculation on model AXIS_FOURIER was carried out with a finer mesh: 4 elements in the thickness instead of 2, denser mesh in the vicinity of the \(\mathrm{AB}\) embedment (total 800 TRIA6).
The difference observed in the constraints \({\sigma }_{\mathrm{rr}}\) and \({\sigma }_{\theta \theta }\) at point \(\mathrm{AB}\) remains: \({\sigma }_{\mathrm{rr}}=1.51\times {10}^{6}\), \({\sigma }_{\theta \theta }=2.08\times 106\) (combined load case).
The embedding relationship \({\sigma }_{\mathrm{xx}}={\sigma }_{\mathrm{yy}}\) is therefore much better verified on the 3D model, with a thick mesh that is nevertheless coarse.