7. Modeling E

Modeling E is carried out with finite elements of pipes on the axis of the tube.

7.1. Characteristics of the mesh

There is a « SEG3 » mesh along the axis of the \(Oz\) tube.

7.2. Characteristics of modeling

The finite element is « TUYAU_3M », with 3 shell modes on the circumference (Fourier). In this modeling, cf. [r3.08.06], the kinematics represent axial and circumferential deformations, distortion deformations in the tangent plane and transverse distortion in the plane of the section, but not the radial deformation or the transverse distortion orthogonal to the plane of the section. The behavioral relationship produces axial and circumferential stresses, shear stresses in the tangential plane, and transverse stresses in the plane of the section. We choose the option « MODI_METRIQUE = OUI » in order to express the gradients in the thickness with the local metric (and not the average metric as in ordinary shell elements). There are 16 sectors on the circumference, each with 3 sub-points for calculating stress fields and internal variables. There are 5 layers in the thickness, each with 3 sub-points for calculating stress fields and internal variables. So that’s \(4 \times 2 + 3 = 11\) subpoints in the thickness and \(15 \times 2 + 3 = 33\) subpoints on the circumference, so that’s \(363\) subpoints in total.

7.3. Loading conditions

The lower edge is affected by a total blocking of all 6 degrees of freedom of the beam; the upper edge is affected by a total blocking of the 5 degrees of freedom of the beam other than that of translation in the axial direction \(Oz\). The degrees of freedom of the « out-of-plane » Fourier modes are also blocked on the lower and upper edges: UO2, UO3, VO2, VO3, WO2, WO3.

A kinematics identical to that of a very long tube is thus simulated.

The values of the pressures applied are corrected to respect the exact metric of the internal and external walls, so that they are identical to those applied in modeling A. The keyword factor used in « AFFE_CHAR_MECA » is « FORCE_TUYAU =_F (PRES =) ». As this modeling only accepts pressures exerted on the internal surface of the pipe, the application of external pressure is simulated by negative internal pressure, so that its effect on the normal circumferential force is identical: \(p_i^{éq}=-p_e \times {R_e}/{R_i}\).

7.4. Characteristics of nonlinear static modeling

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 fixed: math: 10^ {-6} `. The precision required to meet the plane stress condition is: « RESI_CPLAN_RELA =1e-06 ». Newton’s method uses the option « MATRICE =” TANGENTE “`. To calculate the limit pressure, the load is controlled by controlling the load by the radial displacement WO of the node at the lower edge. (To draw the curves, a calculation performed with a maximum time step of \(0.1\) is used.)

7.5. Post-treatment

The post-processing of the radial displacement is done at any time in a node on plane \(z=0.\). The post-processing of the axial displacement is done at any time in a node on the \(z=1.\) plane. Generalized forces (beam theory) are produced at the nodes of the elements: N, VY, VZ, MT, MFY, MFZ. The post-processing of constraints and internal variables is done on an integration sub-point, close to plane \(z=0.\).

7.5.1. Route 1

Fig. 7.1 Radial displacement

Fig. 7.2 Axial displacement

Fig. 7.3 Circumferential stresses at time 1

Fig. 7.4 Circumferential stresses at time 4

Fig. 7.5 Plastic deformation at instant 1

Fig. 7.6 Plastic deformation at instant 4

7.5.2. Route 2

Fig. 7.7 Radial displacement

Fig. 7.8 Circumferential stresses at time 3

Fig. 7.9 Circumferential stresses at time 5

Fig. 7.10 Plastic deformation at instant 3

Fig. 7.11 Plastic deformation at instant 5

7.6. Tested sizes and results

The quantities tested are confronted with a non-regression criterion and also with their reference values produced by modeling A, see Modeling A.

7.6.1. Route 1

• Radial displacement on the mean surface (tube \(h/R = 1/20\))

WO displacement on average surface	Tolerance (%)
Instant t = 0.5	1
Instant t = 1.	8
Instant t = 2.	10
Instant t = 3.	10
Instant t = 4.	35
Instant t = 5.	30

• Radial displacement on the mean surface (tube \(h/R = 1/10\))

WO displacement on average surface	Tolerance (%)
Instant t = 0.5	2
Instant t = 1.	15
Instant t = 2.	16
Instant t = 3.	20
Instant t = 4.	100
Instant t = 5.	75

• Radial displacement on the mean surface (tube \(h/R = 1/5\))

WO displacement on average surface	Tolerance (%)
Instant t = 0.5	3
Instant t = 1.	40
Instant t = 2.	40
Instant t = 3.	50
Instant t = 4.	4000
Instant t = 5.	500

• Uniform axial displacement \(h/R = 1/20\) tube

DZ displacement on average surface	Tolerance (%)
Instant t = 0.5	3
Instant t = 1.	4
Instant t = 2.	4
Instant t = 3.	3
Instant t = 4.	30
Instant t = 5.	35

• Uniform axial displacement \(h/R = 1/10\) tube

DZ displacement on average surface	Tolerance (%)
Instant t = 0.5	6
Instant t = 1.	10
Instant t = 2.	10
Instant t = 3.	10
Instant t = 4.	85
Instant t = 5.	70

• Uniform axial displacement \(h/R = 1/5\) tube

DZ displacement on average surface	Tolerance (%)
Instant t = 0.5	12
Instant t = 1.	30
Instant t = 2.	30
Instant t = 3.	30
Instant t = 4.	600
Instant t = 5.	5300

• Average circumferential surface stress tube \(h/R = 1/20\)

Component SITT on an average area	Tolerance (%)
Instant t = 0.5	0.1
Instant t = 1.	0.2
Instant t = 2.	0.2
Instant t = 3.	0.3
Instant t = 4.	0.2
Instant t = 5.	0.2 (ABSOLU)

• Average circumferential surface stress tube \(h/R = 1/10\)

Component SITT on an average area	Tolerance (%)
Instant t = 0.5	1
Instant t = 1.	1
Instant t = 2.	1
Instant t = 3.	1
Instant t = 4.	1
Instant t = 5.	0.6 (ABSOLU)

• Average circumferential surface stress tube \(h/R = 1/5\)

Component SITT on an average area	Tolerance (%)
Instant t = 0.5	0.5
Instant t = 1.	0.5
Instant t = 2.	0.5
Instant t = 3.	1
Instant t = 4.	0.5
Instant t = 5.	3 (ABSOLU)

• Cumulative plastic deformation on the average surface (tube \(h/R = 1/20\))

Component V1 on average area	Tolerance (%)
Instant t = 0.5
Instant t = 1.	10
Instant t = 2.	10
Instant t = 3.	10
Instant t = 4.	8
Instant t = 5.	8

• Cumulative plastic deformation on the average surface (tube \(h/R = 1/10\))

Component V1 on average area	Tolerance (%)
Instant t = 0.5
Instant t = 1.	18
Instant t = 2.	18
Instant t = 3.	18
Instant t = 4.	15
Instant t = 5.	15

• Cumulative plastic deformation on the average surface (tube \(h/R = 1/5\))

Component V1 on average area	Tolerance (%)
Instant t = 0.5
Instant t = 1.	45
Instant t = 2.	45
Instant t = 3.	45
Instant t = 4.	25
Instant t = 5.	25

7.6.2. Path 2 monotonous internal pressure to ruin

• Radial displacement on the mean surface (tube \(h/R = 1/20\))

Coefficient factor of the applied pressure	Tolerance (%)
Instant t = 0.5	1
Instant t = 1.	1
Instant t = 2.	3
Instant t = 3.	3
Instant t = 4.	3
Instant t = 5.	3

• Radial displacement on the mean surface (tube \(h/R = 1/10\))

Coefficient factor of the applied pressure	Tolerance (%)
Instant t = 0.5	2
Instant t = 1.	2
Instant t = 2.	5
Instant t = 3.	3
Instant t = 4.	3
Instant t = 5.	3

• Radial displacement on the mean surface (tube \(h/R = 1/5\))

Coefficient factor of the applied pressure	Tolerance (%)
Instant t = 0.5	3
Instant t = 1.	3
Instant t = 2.	9
Instant t = 3.	6
Instant t = 4.	6
Instant t = 5.	6

• Average circumferential surface stress tube \(h/R = 1/20\). In particular: at the node of the upper edge of the tube, sector 1, layer 1.

Component SITT on an average area	Tolerance (%)
Instant t = 3.	5
Instant t = 5.	3

• Average circumferential surface stress tube \(h/R = 1/10\). In particular: at the node of the upper edge of the tube, sector 1, layer 1.

Component SITT on an average area	Tolerance (%)
Instant t = 3.	6
Instant t = 5.	6

• Average circumferential surface stress tube \(h/R = 1/5\). In particular: at the node of the upper edge of the tube, sector 1, layer 1.

Component SITT on an average area	Tolerance (%)
Instant t = 3.	15
Instant t = 5.	15

• Cumulative plastic deformation on the average surface of tube \(h/R = 1/20\). In particular: at the node of the upper edge of the tube, sector 1, layer 1.

Component V1 on average area	Tolerance (%)
Instant t = 3.	6
Instant t = 5.	3

• Cumulative plastic deformation on the average surface of tube \(h/R = 1/10\). In particular: at the node of the upper edge of the tube, sector 1, layer 1.

Component V1 on average area	Tolerance (%)
Instant t = 3.	10
Instant t = 5.	6

• Cumulative plastic deformation on the average surface of tube \(h/R = 1/5\). In particular: at the node of the upper edge of the tube, sector 1, layer 1.

Component V1 on average area	Tolerance (%)
Instant t = 3.	20
Instant t = 5.	15

• Generalized efforts (beam theory) at the nodes of the elements: N, VY, VZ, MT, MFY, MFZ. In particular: at the node at the bottom edge of tube \(h/R = 1/20\).

Instant component t = 2.5	worthiness
N	-2.37
VY
VZ
MT
MFY
MFZ

• Nodal forces (degrees of freedom of the beams and the 3 shell modes) at the nodes of the elements. In particular: at the nodes of tube \(h/R = 1/20\).

Instant component t = 2.5	worthiness
DZ top edge
WO top edge	172.89
DZ middle
WO middle	691.58