Yong Bai - Deepwater Flexible Risers and Pipelines

(4.1)

(4.2)

for which, Ieq is the equivalent moment of inertia per unit length, A ′ is the cross- sectional area of the pressure armor according to API 17B [17], n is the number of tendons per each layer, Lp is the pitch length, and E is the Young’s modulus of the material. Ieq is computed as in Ref. [1]:

(4.3)

where k is depending on both lay angle and moment of inertia of the cross section, and is the smallest moment of inertia, which can be computed, referring to Figure 4.3, as follows:

(4.4)

Then, as discussed by Yue et al . [15], if the orthotropic cylinder is loaded by radial confining pressure PC induced by the adjacent tensile armor layers, it can be simplified into a planar ring under the same pressure. The equivalent Young’s modulus Eeq and thickness heq for pressure armor can be used in the calculation, where the mean radius for the equivalent cylinder is kept the same as its actual one, as can be seen in Figure 4.4.

Figure 4.3 Pressure armor profile-principal outline.

Figure 4.4 Contact pressure and equivalent radii.

Keeping the main radius as reference, it is possible to figure out the radial stiffness of the pressure armor, defined as done by Lu [18]:

(4.5)

where, DRC is the radial displacement of the external surface of the cylinder due to PC.

Radial stiffness, according to the elastic theory for a thin-walled tube, as shown in [18], can be expressed as follows:

(4.6)

where ν is the Poisson ratio of the material.

Due to the helical shape of the tensile armor wires, once they are subjected to tensile load, they exhibit elongation strain along the wire axis εi (with i = 1, 2 stands for the inner and outer armor layers, respectively). It can be expressed as a combination of DL (displacements along the longitudinal direction) and DRW(displacements in the radial direction), as shown in Figure 4.5, where s and s ′ represent the undeformed and deformed length of the wire, respectively. The mathematical model adopted here was quoted from Knapp [4]. Ignoring the rotation term, the expression can be written as

(4.7)

where α is the winding angle of the wires, Rm,i is the mean radius, and L is length of the pipe.

Figure 4.5 Contraction and elongation for a representative pitch length of tensile wire.

The tensile force along the axial direction of the helix can be divided into two components: the hoop direction and the axial directions of the pipe. The hoop stresses per each wire can be expressed as

(4.8)

where Es is the secant Young’s modulus of the constituent material. It should be pointed out that, in the incremental process, Es changes in every step in order to take the plasticity of the material into account. As Es used in the current step is from the previous one, whose value is actually larger, the total tensile force obtained might be greater than its real situation. However, if the increments are small enough, this error would be controlled in the tolerable range.

The hoop stress of the tensile armor layer results in confining or extruding pressure to its adjacent layer. Due to the gaps between the wires in the same layer, the filling factor βi is introduced, which exhibits the relationship between the area filled by wires and gap. The equilibrium state of the tensile armor’s cylinder can be seen in Figure 4.6, and the contact pressures can be derived as

(4.9)

where h is the thickness of the wire.

Figure 4.6 Radial loading condition of tensile armor layer.

Taking the inner tensile armor layer as example, by substituting (4.7)into (4.8)and then into (4.9), and considering the effects of the confined pressure produced by the outer tensile armor layer, one can get the contact pressure PC as

(4.10)

(4.11)

(4.12)

In the hypothesis of no separation between layers, the radial displacement can be considered the same for each layer: DRC = DRW.

By solving Eqs. (4.5)and (4.10)simultaneously, the two unknowns, i.e., radial deformation DRW and contact pressure PC of the problem can be obtained. Once these two outputs are known, it is possible to compute the strain for each wire using Eq. (4.7). The total tensile strength of the pipe F can be obtained by summing up the tensile resistance from the force of each wire, as well as the contributions of the internal and external HDPE layers: