force resistant structure of flexible pipe, which has an important influence on the mechanical properties of pipe. For pressure armor, it is necessary to simplify the section. In order to keep the radial continuity, the same thickness of the original section is used. In addition, the axial stiffness of the helix is mainly considered, so keep the area same as the original section. The exact formula for axial strain of a helical element was derived by Knapp [8]. In order to obtain a linear equilibrium equation of helical tendons, the exact formula is simplified as
(3.5)
The radial strain is derived as:
(3.6)
Therefore, the internal work can be expressed as these two parts above adding together:
(3.7)
Similarly, the stiffness matrix of helix layers can be derived:
(3.8)
3.2.3 The Stiffness Matrix of Pipe as a Whole Helix Layers
Subjected to the axisymmetric load, all layers at all cross-sections present the same twist per unit of pipe length and the same elongation see Figure 3.3. It will be assumed that all layers of the pipe are numbered consecutively from 1 (the innermost layer) to N (the outermost layer), where N is the total number of layers of the pipe. Combine the stiffness matrix together, Eqs. (3.9) to (3.12) are derived:
(3.10)
(3.11)
Figure 3.3 Helix mathematical parameters definition.
Still, lacking of equations for solving all unknown variables, the consistent in the radial direction is added to solve the problem:
(3.13)
where j denotes the layer number. For some applications, the calculated contact pressure between layers may be found negative. Treating such a case requires a change of unknowns. The contact pressure is known and equal to zero and the gap between the two layers becomes the new unknown.
3.2.4 Blasting Failure Criterion
In order to use the above method to predict the burst pressure of flexible pipe, it is necessary to introduce the failure criterion into the program. API17J [9] stipulates that the bearing capacity of the structure can be determined by the yield strength of the material or 0. 9 times of the ultimate tensile strength. In order to simulate the properties of steel used in practice, the Ramberg-Osgood model expressed in Eq. (3.14) is used to describe the non-linear stress-strain relationship of the armor layers. For this model, the pressure armor’s ultimate strength is 630 MPa.
Where σy = 600MPa, E = 207GPa, n = 13.
Specifically, the total load is decomposed into a limited increment for loading. Attention should be paid to the step size of the incremental step is small enough to ensure the convergence of the results. In this chapter, Mises stress is used as failure criterion. When Mises stress reaches ultimate strength during loading, it is considered that the pressure armor failures.
3.3 FEA Modeling Description
In this chapter, ABAQUS model is used to verify the correction of theoretical model. The finite element model consists of a Z-type self-locking pressure armor layer, two layers of tension armor layer with opposite winding angles and other polymer layers. Specific size information is shown below in Table 3.1, Table 3.2.
As the structure of the model is quite complex and there are a lot of contact between layers, implicit static analysis is not only time-consuming but also brings a lot of convergence problems. Therefore, this chapter uses dynamic explicit method to carry out the analysis. Due to the non-bonding between the layers of flexible pipe, contact and slip might occur between the layers during the loading process and self-contact will occur in the pressure armor layer, which makes the contact situation more complex. Therefore, the “All with itself” algorithm is used to simulate contact in the ABAQUS model. This algorithm can automatically identify contact pairs and can consider the possible layer separation. The contact normal direction is set by hard contact, and the tangential direction is set by non-friction.
Table 3.1 Geometric and material parameters of FEM.
| Layer name | Parameters |
|---|---|
| 1. Pressure sheath | Internal radius: 76 mmThickness: 6 mmYoung modulus: 1,040 MpaPoisson ratio: 0.45Thickness: 10 mmNumber of wires: 1 |
| 2. Pressure armor | Profile: shown in Figure 3.4 Winding angle: +88.8° Young modulus: 207,000 Mpa Poisson ratio: 0.3 Thickness: 1.5 mm |
| 3. Anti-wear tape | Young modulus: 301 MpaPoisson ratio: 0.45Thickness: 5 mmNumber of wires: 46Profile: 5 mm × 11 mm |
| 4. Tensile armor | Winding angle: +54.7°Young modulus: 207,000 MpaPoisson ratio: 0.3Thickness: 1.5 mm |
| 5. Anti-wear tape | Young modulus: 301 MpaPoisson ratio: 0.45Thickness: 5 mmNumber of wires: 47Profile: 5 mm × 11 mm |
| 6. Tensile armor |
Winding angle: −54.7°Young
|
