The latter is further reduced if the initial imperfections are considered symmetrically distributed on the cross-section plane. The symmetry along the longitudinal direction of the pipe can be taken into account in order to further reduce the number of operations, so that it is useful to consider half of the ring, as shown in Figure 2.5.
The chosen geometry is as shown in Figure 2.3, the imported profile validates the real one since only two of the actual pitches are involved. In fact, a full corrugate cross-section and two adjacent halves are considered, as shown in Figure 2.4. The chosen dimensions are listed in Table 2.1 relatively to the cross-section geometry. Material properties and other parameters needed to the computation are listed in Table 2.2.
The inertia matrix for the abovementioned cross-section dimension is computed, showing the following results:
Figure 2.3 Parameterized carcass profile.
Figure 2.4 Full carcass cross section imported.
Table 2.1 Interlocked carcass cross-section parameters.
| L1 [mm] | 8. 00 | R1 [mm] | 1.00 |
| L2 [mm] | 3. 00 | R2 [mm] | 1.00 |
| L3 [mm] | 9. 00 | R3 [mm] | 3.00 |
| L4 [mm] | 4. 50 | Rtip [mm] | 0.50 |
| L5 [mm] | 10. 00 | f1 [°] | 60 |
| L6 [mm] | 3. 00 | f2 [°] | 45 |
| L7 [mm] | 2. 00 | f3 [°] | 90 |
Table 2.2 Interlocked carcass parameters.
| n | 1 |
| K | 1 |
| E [MPa] | 200,000 |
| sp [MPa] | 600 |
| n | 0.3 |
| Lp [mm] | 16.00 |
| Rinn [mm] | 76.20 |
| t [mm] | 6.40 |
(2.15)
The initial imperfections are given using a helical path that can simulate the initial displacements along both x and y directions using sweep command for two different initial radii that account for ovality equal to L = 0.04. The cross-section is imported in the xz plane and follows half of the elliptical path. The greater diameter is considered along the y direction, instead the other lays on the x direction, as it is shown in Figure 2.5.
Figure 2.5 Carcass model geometry.
Due to the complex shape, proper of the geometry of the carcass and many possibilities of contact, “General contact” is chosen to simulate the interactions among the three parts, until the buckling collapse is reached. For this unbonded condition, “Frictionless” tangential behavior and “Hard contact” normal behavior with “Allow separation after contact” are chosen. The latter is defined by (p-h)model, which relates p: contact pressure among surfaces and h: overclosure between contact surfaces respectively. When h < 0, it means no contact pressure, while for any positive contact h is set equal to zero [6].
Also, the plastic behavior of the material is taken into account to verify if the collapse happens in elastic or plastic field and if the hypothesis of considering only elastic behavior in the theoretical model makes sense. As it is shown in Figure 2.6, the material properties for the carcass layer consider a linear elastic behavior, following the Hooke’s law during the first stage then again, a linear behavior due to the plastic tangent modulus for simulating high strain against low stress increments in the plastic field, which accounts for an isotropic hardening law [7].
The external pressure is considered as constant along the width, in z direction and applied directly on the external surface. The kinematic is governed by the boundary conditions that must mainly avoid rigid body displacements. The possibility of assigning symmetric boundary condition with respect to the xz plane is exploited in order to allow displacement in x direction (U1) at the bases of the ring, which also fix the pitch as constant during the simulation. For simulating the occurrence of the local bucking of the ring, the only allowed displacements in the middle of the ring are in y direction (U2), as depicted in Figure 2.7.
Figure 2.6 Steel strain-stress relationship.
Figure 2.7 Load and boundary conditions.
A dynamic implicit analysis is employed because of the necessity of capture the variation of the stiffness changes step by step in case of non-linearity
