3. Modeling A#
Modeling A is the reference calculation, based on elastoplastic modeling (isotropic work hardening model with bi-linear tension curve) in axisymmetric massive finite elements. The calculated quantities depend only on the radial direction. The three tubes (with their three different thickness/radius ratios) are calculated together.
3.1. Characteristics of the mesh#
The quadratic mesh is very refined: 100 mesh QUAD8 in thickness, but one mesh along the axis of the tube. There are 503 knots.
3.2. Characteristics of nonlinear static modeling and loading conditions#
Solid finite elements are axisymmetric. The length of the modelled tubes is \(10 \mathit{mm}\). The lower face is affected by total blocking in the axial direction; the upper face is affected by a uniform connection in the axial direction in order to simulate kinematics identical to that of a very long tube. The time step chosen is \(0.5\), fixed for all non-linear calculation steps; however, a cut is authorized in case of convergence failure, the criterion « RESI_GLOB_RELA » being set to \(10^{-6}\). Newton’s method uses the option « MATRICE =” TANGENTE “``. (To draw the curves, we use a calculation carried out with a maximum time step of \(0.1\).)
The pressures are respectively applied to the inner and outer edges. To calculate the limit pressure (path 2), load control is used by controlling the load by the radial movement of a node in the external wall.
3.3. Tested sizes and results#
For information, the calculated values are indicated. These values are used as reference values for other models. The tests correspond to non-regression tests, except for tangential stresses, where we compare to the average value indicated in ce tableau (The average value is not necessarily the value calculated at the level of the middle sheet, but it gives an order of magnitude).
3.3.1. Route 1#
Radial displacement on the mean surface (tube \(h/R = 1/20\))
DX displacement on average surface |
value (m) |
Instant t = 0.5 |
0.011902 |
Instant t = 1. |
0.176855 |
Instant t = 2. |
0.176007 |
Instant t = 3. |
0.150506 |
Instant t = 4. |
0.105688 |
Instant t = 5. |
0.130341 |
Radial displacement on the mean surface (tube \(h/R = 1/10\))
DX displacement on average surface |
value (m) |
Instant t = 0.5 |
0.011668 |
Instant t = 1. |
0.165026 |
Instant t = 2. |
0.163329 |
Instant t = 3. |
0.136599 |
Instant t = 4. |
0.071341 |
Instant t = 5. |
0.096374 |
Radial displacement on the mean surface (tube \(h/R = 1/5\))
DX displacement on average surface |
value (m) |
Instant t = 0.5 |
0.011158 |
Instant t = 1. |
0.141588 |
Instant t = 2. |
0.138094 |
Instant t = 3. |
0.108989 |
Instant t = 4. |
0.003604 |
Instant t = 5. |
0.029314 |
Uniform axial displacement \(h/R = 1/20\) tube
DY displacement on average surface |
value (m) |
Instant t = 0.5 |
-0.00034568 |
Instant t = 1. |
-0.00805735 |
Instant t = 2. |
-0.00798468 |
Instant t = 3. |
-0.00714781 |
Instant t = 4. |
-0.00542527 |
Instant t = 5. |
-0.00618936 |
Uniform axial displacement \(h/R = 1/10\) tube
DY displacement on average surface |
value (m) |
Instant t = 0.5 |
-0.00032818 |
Instant t = 1. |
-0.00720990 |
Instant t = 2. |
-0.00706445 |
Instant t = 3. |
-0.00611717 |
Instant t = 4. |
-0.00332040 |
Instant t = 5. |
-0.00412222 |
Uniform axial displacement \(h/R = 1/5\) tube
DY displacement on average surface |
value (m) |
Instant t = 0.5 |
-0.00029454 |
Instant t = 1. |
-0.00564851 |
Instant t = 2. |
-0.00535760 |
Instant t = 3. |
-0.00418669 |
Instant t = 4. |
-0.00009860 |
Instant t = 5. |
-0.00010608 |
Circumferential surface stress (mean tube \(h/R = 1/20\)) interpolated at the nodes of the mesh
Component SIZZ on an average area |
value (MPa) |
Tolerance (%) |
Instant t = 0.5 |
195 |
0.1 |
Instant t = 1. |
390 |
0.1 |
Instant t = 2. |
370 |
0.1 |
Instant t = 3. |
-60 |
0.5 |
Instant t = 4. |
-410 |
0.1 |
Instant t = 5. |
0.0 |
0.2 (ABSOLU) |
Circumferential surface stress (mean tube \(h/R = 1/10\)) interpolated at the nodes of the mesh
Component SIZZ on an average area |
value (MPa) |
Tolerance (%) |
Instant t = 0.5 |
190 |
0.5 |
Instant t = 1. |
380 |
0.1 |
Instant t = 2. |
340 |
0.1 |
Instant t = 3. |
-120 |
0.6 |
Instant t = 4. |
-420 |
0.1 |
Instant t = 5. |
0.0 |
0.8 (ABSOLU) |
Circumferential surface stress (mean tube \(h/R = 1/5\)) interpolated at the nodes of the mesh
Component SIZZ on an average area |
value (MPa) |
Tolerance (%) |
Instant t = 0.5 |
180 |
0.6 |
Instant t = 1. |
360 |
0.5 |
Instant t = 2. |
280 |
0.5 |
Instant t = 3. |
-240 |
1.2 |
Instant t = 4. |
-440 |
0.5 |
Instant t = 5. |
0.0 |
3 (ABSOLU) |
Cumulative plastic deformation on the mean surface (tube \(h/R = 1/20\)) interpolated at the nodes of the mesh
Component V1 on average area |
worthiness |
Instant t = 0.5 |
0.0 |
Instant t = 1. |
0.015307 |
Instant t = 2. |
0.015307 |
Instant t = 3. |
0.015307 |
Instant t = 4. |
0.017576 |
Instant t = 5. |
0.017576 |
Cumulative plastic deformation on the mean surface (tube \(h/R = 1/10\)) interpolated at the nodes of the mesh
Component V1 on average area |
worthiness |
Instant t = 0.5 |
0.0 |
Instant t = 1. |
0.014177 |
Instant t = 2. |
0.014177 |
Instant t = 3. |
0.014177 |
Instant t = 4. |
0.018703 |
Instant t = 5. |
0.018703 |
Cumulative plastic deformation on the mean surface (tube \(h/R = 1/5\)) interpolated at the nodes of the mesh
Component V1 on average area |
worthiness |
Instant t = 0.5 |
0.0 |
Instant t = 1. |
0.011940 |
Instant t = 2. |
0.011940 |
Instant t = 3. |
0.011940 |
Instant t = 4. |
0.020925 |
Instant t = 5. |
0.020925 |
3.3.2. Route 2#
Pressure reached for tube \(h/R = 1/20\)
Coefficient factor of the applied pressure |
worthiness |
Instant t = 0.5 |
4,201 |
Instant t = 1. |
8,402 |
Instant t = 2. |
16,594 |
Instant t = 3. |
16,718 |
Instant t = 4. |
16,722 |
Instant t = 5. |
16,726 |
Pressure reached for tube \(h/R = 1/10\)
Coefficient factor of the applied pressure |
worthiness |
Instant t = 0.5 |
4,285 |
Instant t = 1. |
8,570 |
Instant t = 2. |
16,689 |
Instant t = 3. |
16,924 |
Instant t = 4. |
16,928 |
Instant t = 5. |
16,932 |
Radial displacement on the mean surface (tube \(h/R = 1/5\))
Coefficient factor of the applied pressure |
worthiness |
Instant t = 0.5 |
4,480 |
Instant t = 1. |
8,961 |
Instant t = 2. |
16,879 |
Instant t = 3. |
17,334 |
Instant t = 4. |
17,339 |
Instant t = 5. |
17,343 |
Average circumferential stress on the surface of the \(h/R = 1/20\) tube
Component SIZZ in on average area |
value (MPa) |
Instant t = 3. |
322,26 |
Instant t = 5. |
321.94 |
Average circumferential stress on the surface of the \(h/R = 1/10\) tube
Component SIZZ in on average area |
value (MPa) |
Instant t = 3. |
312.97 |
Instant t = 5. |
312.19 |
Average circumferential stress on the surface of the \(h/R = 1/5\) tube
Component SIZZ in on average area |
value (MPa) |
Instant t = 3. |
290.33 |
Instant t = 5. |
288,79 |
Cumulative plastic deformation on the average surface of the \(h/R = 1/20\) tube
Component V1 on average area |
worthiness |
Instant t = 3. |
1,129E-3 |
Instant t = 5. |
3,210E-3 |
Cumulative plastic deformation on the average surface of the \(h/R = 1/10\) tube
Component V1 on average area |
worthiness |
Instant t = 3. |
1,281E-3 |
Instant t = 5. |
3,457E-3 |
Cumulative plastic deformation on the average surface of the \(h/R = 1/5\) tube
Component V1 on average area |
worthiness |
Instant t = 3. |
1,664E-3 |
Instant t = 5. |
4,073E-3 |