3. Modélisation A#
3.1. Caractéristiques de la modélisation#
Le modèle est composé de 10 éléments poutre droite d’Euler.
Section S1: section circulaire variable
à l’encastrement, |
\(\mathrm{R1}=0.1m\) (section pleine) |
à l’extrémité libre, |
\(\mathit{R2}=0.05m\) (section pleine) |
Section S2: section rectangulaire variable
à l’encastrement, |
\({H}_{\mathit{y1}}=0.05m\) |
\({H}_{\mathit{z1}}=0.10m\) |
à l’extrémité libre, |
\({H}_{\mathit{y2}}=0.05m\) |
\({H}_{\mathit{z2}}=0.05m\) |
Section S3 : section générale variable
à l’encastrement, |
\({A}_{1}={10}^{2}{m}^{4}\) |
\({I}_{\mathit{y1}}=8.3333{10}^{6}{m}^{4}\) |
à l’extrémité libre, |
\({A}_{2}=2.5{10}^{3}{m}^{2}\) |
\({I}_{\mathit{y2}}=5.20833{10}^{7}{m}^{4}\) |
3.2. Caractéristiques du maillage#
3 sections \(\times\) 10 éléments POU_D_E
3.3. Grandeurs testées et résultats#
Cas de charge |
Section |
Identification |
Référence |
Tolérance % |
1 |
\(\mathit{S1}\) |
\(u(l)\) |
3.1831E–08 |
1.00E-05 |
\(n(0)\) |
1.0000E+02 |
1.00E-05 |
||
\(n(l)\) |
1.0000E+02 |
1.00E-05 |
||
\({\sigma}_{xx}(0)\) |
3.1831E+03 |
1.00E-05 |
||
\({\sigma}_{xx}(l)\) |
1.2732E+04 |
1.00E-05 |
||
2 |
\(\mathit{S1}\) |
\(v(l)\) |
4.2441E–06 |
1.00E-05 |
\({\theta}_{z}(l)\) |
8.4882E–06 |
1.00E-05 |
||
\({v}_{y}(0)\) |
1.0000E+02 |
1.00E-05 |
||
\({v}_{y}(l)\) |
1.0000E+02 |
1.00E-05 |
||
\({\mathit{mf}}_{z}(0)\) |
1.0000E+02 |
1.00E-05 |
||
\({\mathit{mf}}_{z}(l)\) |
0.0000E+00 |
1.00E-05 |
||
\({\sigma}_{xx}(0)\) |
1.2732E+05 |
1.00E-05 |
||
\({\sigma}_{xx}(l)\) |
0.0000E+00 |
1.00E-05 |
||
3 |
\(\mathit{S1}\) |
\({\theta}_{x}(l)\) |
3.8621E–05 |
1.00E-05 |
\({m}_{x}(0)\) |
1.0000E+02 |
1.00E-05 |
||
\({m}_{x}(l)\) |
1.0000E+02 |
1.00E-05 |
||
4 |
\(\mathit{S1}\) |
\(w(l)\) |
–8.4882E–06 |
1.00E-05 |
\({\theta}_{y}(l)\) |
2.9708E–05 |
1.00E-05 |
||
\({v}_{z}(0)\) |
0.0000E+00 |
1.00E-05 |
||
\({v}_{z}(l)\) |
0.0000E+00 |
1.00E-05 |
||
\({\mathit{mf}}_{y}(0)\) |
1.0000E+02 |
1.00E-05 |
||
\({\mathit{mf}}_{y}(l)\) |
1.0000E+02 |
1.00E-05 |
||
\({\sigma}_{xx}(0)\) |
1.2732E+05 |
1.00E-05 |
||
\({\sigma}_{xx}(l)\) |
1.0185E+06 |
1.00E-05 |
||
5 |
\(\mathit{S1}\) |
\(u(l)\) |
1.2296E–08 |
1.00E-02 |
\(n(0)\) |
1.0000E+02 |
1.00E-05 |
||
\(n(l)\) |
0.0000E+00 |
1.00E-03 (absolu) |
||
\({\sigma}_{xx}(0)\) |
3.1831E+03 |
1.00E-05 |
||
\({\sigma}_{xx}(l)\) |
0.0000E+00 |
1.00E-03 (absolu) |
||
6 |
\(\mathit{S1}\) |
\(v(l)\) |
1.3486E–06 |
1.00E-02 |
\({\theta}_{z}(l)\) |
2.1220E–06 |
1.00E-02 (absolu) |
||
\({v}_{y}(0)\) |
1.0000E+02 |
1.00E-05 |
||
\({v}_{y}(l)\) |
0.0000E+00 |
1.00E-03 (absolu) |
||
\({\mathit{mf}}_{z}(0)\) |
5.0000E+01 |
1.00E-02 |
||
\({\mathit{mf}}_{z}(l)\) |
0.0000E+00 |
1.00E-03 (absolu) |
||
\({\sigma}_{xx}(0)\) |
6.3662E+04 |
1.00E-05 |
||
\({\sigma}_{xy}(0)\) |
3.1831E+03 |
1.00E-05 |
Cas de charge |
Section |
Identification |
Référence |
Ecart % |
1 |
\(\mathit{S2}\) |
\(u(l)\) |
1.3862E–07 |
1.00E-05 |
\(n(0)\) |
1.0000E+02 |
1.00E-05 |
||
\(n(l)\) |
1.0000E+02 |
1.00E-05 |
||
\({\sigma}_{xx}(0)\) |
2.0000E+04 |
1.00E-05 |
||
\({\sigma}_{xx}(l)\) |
4.0000E+04 |
1.00E-05 |
||
2 |
\(\mathit{S2}\) |
\(v(l)\) |
1.8969E–04 |
2.30E-02 |
\({\theta}_{z}(l)\) |
3.0238E–04 |
2.70E-02 |
||
\({v}_{y}(0)\) |
1.0000E+02 |
1.00E-05 |
||
\({v}_{y}(l)\) |
1.0000E+02 |
1.00E-05 |
||
\({\mathit{mf}}_{z}(0)\) |
1.0000E+02 |
1.00E-05 |
||
\({\mathit{mf}}_{z}(l)\) |
0.0000E+00 |
1.00E-03 (absolu) |
||
\({\sigma}_{xx}(0)\) |
2.4000E+06 |
1.00E-05 |
||
\({\sigma}_{xx}(l)\) |
0.0000E+00 |
1.00E-03 (absolu) |
||
\({\sigma}_{xy}(0)\) |
2.0000E+04 |
1.00E-05 |
||
\({\sigma}_{xy}(l)\) |
4.0000E+04 |
1.00E-05 |
||
3 |
\(\mathit{S2}\) |
\({\theta}_{x}(l)\) |
8.3506E–04 |
5.70E-02 |
\({m}_{x}(0)\) |
1.0000E+02 |
1.00E-05 |
||
\({m}_{x}(l)\) |
1.0000E+02 |
1.00E-05 |
||
\({\sigma}_{xy}(0)\) |
1.5600E+06 |
1.00E-05 |
||
\({\sigma}_{xy}(l)\) |
4.0371E+06 |
5.00E-02 |
||
\({\sigma}_{xz}(0)\) |
1.5600E+06 |
1.00E-05 |
||
\({\sigma}_{xz}(l)\) |
4.0371E+06 |
5.00E-02 |
||
4 |
\(\mathrm{S2}\) |
\(w(l)\) |
–1.2000E–04 |
3.00E-03 |
\({\theta}_{y}(l)\) |
3.600E–04 |
4.00E-03 |
||
\({v}_{z}(0)\) |
0.0000E+00 |
1.00E-03 (absolu) |
||
\({v}_{z}(l)\) |
0.0000E+00 |
1.00E-03 (absolu) |
||
\({\mathit{mf}}_{y}(0)\) |
1.0000E+02 |
1.00E-05 |
||
\({\mathit{mf}}_{y}(l)\) |
1.0000E+02 |
1.00E-05 |
||
\({\sigma}_{xx}(0)\) |
1.2000E+06 |
1.00E-05 |
||
\({\sigma}_{xx}(l)\) |
4.8000E+06 |
1.00E-05 |
||
5 |
\(\mathit{S2}\) |
\(u(l)\) |
6.1370E–08 |
1.00E-05 |
\(n(0)\) |
1.0000E+02 |
1.00E-05 |
||
\(n(l)\) |
0.0000E+00 |
1.00E-03 (absolu) |
||
\({\sigma}_{xx}(0)\) |
2.0000E+04 |
1.00E-05 |
||
\({\sigma}_{xx}(l/2)\) |
1.3333E+04 |
1.00E-05 |
||
\({\sigma}_{xx}(l)\) |
0.0000E+00 |
1.00E-03 (absolu) |
||
6 |
\(\mathrm{S2}\) |
\(v(l)\) |
6.8626E–05 |
2.00E-02 |
\({\theta}_{z}(l)\) |
9.4847E–05 |
2.40E-02 |
||
\({v}_{y}(0)\) |
1.0000E+02 |
1.00E-05 |
||
\({v}_{y}(l)\) |
0.0000E+00 |
1.00E-03 (absolu) |
||
\({\mathrm{mf}}_{z}(0)\) |
5.0000E+01 |
1.00E-02 |
||
\({\mathrm{mf}}_{z}(l)\) |
0.0000E+00 |
1.00E-03 (absolu) |
||
\({\sigma}_{xx}(0)\) |
1.2000E+06 |
1.00E-05 |
||
\({\sigma}_{xx}(l)\) |
0.0000E+00 |
1.00E-03 (absolu) |
||
7 |
\(\mathrm{S3}\) |
\(w(l)\) |
–3.8259E–05 |
1.00E-02 |
\({\theta}_{y}(l)\) |
5.7388E–05 |
1.00E-02 |
||
\({v}_{z}(0)\) |
–4.4633E+02 |
1.00E-03 |
||
\({\mathrm{mf}}_{y}(0)\) |
1.7535E+02 |
1.00E-02 |
3.4. Remarques#
La modélisation étant faite en poutres d’Euler, les coefficients de cisaillement sont \(\mathit{ky}=\mathit{kz}=1\) .