Flexible Pipes

Abstract

An unbonded flexible pipe is one of the most important equipment in offshore engineering, transporting oil and gas between the floater on the sea and the well located on the seabed. Flexible pipes consist of several metallic helical-reinforced layers and internal and external polymer sheaths, and relative slip between the layers is allowed, so that the structure show high axial stiffness and radial stiffness associated with relatively low bending stiffness. During the operation and installation, the flexible pipe will be subjected to complex and coupled loads such as tension, internal pressure, external pressure, torsion and bending, which lead to multiple structural failures. This chapter will present the current theoretical models and research progress to effectively evaluate the response of such composite structure, providing reference ideas for the engineering design of the flexible pipes.

1. Introduction

Pipelines are the “lifeline” equipment for the production and development of offshore oil and gas resources and mineral resources. The traditional steel pipeline technology is mature, but the limitations in bendability, corrosion protection, installation and laying speed have significantly increased its comprehensive cost, so the concept of offshore flexible pipeline came into being. It can be bent into a small bending radius without reducing the mechanical bearing capacity, and can be installed and recycled by the reel method, so the comprehensive cost is relatively low；Meanwhile, in some medium water depth environments, using offshore flexible pipes as risers is the only solution. Therefore, since the 80s of last century, flexible pipelines have gradually been paid attention to and applied in the development of offshore oil and gas resources, and flexible pipelines account for more than 80% of the marine risers currently in service [1].

Each reinforcement layer is specifically designed to accommodate a specific load. Under the axisymmetric loads, such as tension, internal pressure, external pressure, and torsion, the bearing capability of the reinforced layer usually depends not only on the structural design of the layer itself, but also on the interaction with the adjacent layers [2], so that the pipe wall is formed by the armor layers. While, under the combined non-axisymmetric loads such as bending loads, the interlayered interaction becomes more complicated and makes the mechanical behavior of the reinforced components show the strong nonlinearity. These bring challenges into accurately evaluating the structural strength and fatigue performance of flexible pipes.

The objective of the current study is to comprehensively establish the latest theoretical models of structural responses of offshore flexible pipes under various typical loads, and analyze the coupling relationship between the structural deformation and interlayered interaction under different loads. Especially for the tension armor wires, the calculation method of nonlinear responses under combined tension and bending is developed, and the theoretical models are verified by one case. These can not only help readers clearly understand the bearing principles and the response rules of flexible pipelines, but also provide a comprehensive, convenient and effective tool for carrying out the quantitative design and evaluation of offshore flexible pipeline in practical engineering.

This chapter is organized as follows. Section 2 puts forward the design requirements of flexible pipelines from the perspective of engineering applications; Section 3 describes the mechanical principles and structural types of flexible pipes; Section 4 introduces theoretical models of pipeline structural responses for axisymmetric loads such as tension, internal pressure, external pressure and torsion; The mechanical responses of pipes under non-axisymmetric loads such as bending and complex loads are discussed in Section 5; The last section takes one 8 inches internal diameter pipeline in practical engineering as example to quantitatively describe the mechanical responses of flexible pipeline structure under various loads.

2. Offshore flexible pipe design requirements

1. On the basis of ensuring normal flow, the design of offshore flexible pipelines should first achieve better bendability, which means compliance with strenuous movement of the floater in high sea conditions, as well as rapid manufacturing, laying and recovery by the reel method. The flexible pipe with an inner diameter of 8 inches can be rolled on a drum with a radius of 2 meters for storage, and the speed of the reeling method is about 7 times that of conventional steel pipe installation.

2. Flexible pipelines need to ensure that the internal high temperature (up to 130°C), the complex fluid medium does not leak outward, and the external seawater does not erode inward, so as to form a complete pipe wall space and protect the internal metal components from corrosion.

3. In order to transport the medium from the seabed of thousands of meters to the sea surface, the pipeline needs to have a high resistance to the continuous action of internal pressure (the general working pressure of a 4-inch inner diameter pipe is 25Mpa).

4. As the water depth gradually increases, the pipeline body will be subjected to huge external pressure. In order to prevent local or global buckling collapse failure, the flexible pipe wall needs to be able to continuously resist the external pressure from seawater (about 30Mpa at a depth of 3000 meters).

5. During the installation and operation of flexible risers (as shown in Figures 1 and 2), the upper end of the pipeline is suspended on the floater on the sea surface [3]. Due to the self-weight, the flexible riser close to the floater will be subjected to huge axial tension (when the flexible riser of an 8-inch inner diameter is used in water depth of 1500 meters, the axial tension can generally reach more than 100 tons). Therefore, the pipeline should have the low bending stiffness associated with the high axial stiffness and radial stiffness.

6. During the installation using reeling method, flexible pipes are subjected to repeated axial torsion. And during normal operation, the pipeline itself will also twist due to the movement of the floater and dynamic environment. Therefore, the torsion resistance in the clockwise/counterclockwise direction of flexible pipes needs to be considered during the design process, and the torsional equilibrium is also required, i.e. the pipeline does not twist freely under other loads.

7. Owing to its self-weight coupled with the floater motion and dynamic environment, the flexible riser at the interface with the floater (generally within 50 m) will encounter relatively alternative curvatures as well as high axial loads and internal pressure in operation, and stress fatigue failure is more likely to occur on the tension armor layer in the pipeline [4]. Currently, the service life of Flexible pipes is designed to reach 25 years at least, so accurately assessing the fatigue resistance of pipelines is a constant need for oil companies and owners.

Figure 1.

Schematic diagram of the flexible pipe laying- J lay.

In summary, the flexible pipeline structure needs to have excellent bending performance and corrosion resistance, and at the same time, it must have good resistance in internal pressure, external pressure, tension, torsion and fatigue. That is, while giving full play to the advantages of flexibility, it can meet the safety and reliability of the structure.

3. Mechanical principle and structural type of offshore flexible pipeline

3.1 Mechanical principle

According to the design requirements of flexible pipes, it is no longer feasible to use homogeneous materials for the entire pipe wall. Considering the need to seal inside and outside the pipe, the innermost and outermost layers of the pipe are usually made of polymer materials of a certain thickness. In order to meet the bearing capacity needs in all directions at the same time, the metal reinforcement structure in the form of a spiral is required. Considering that the single-layer helical structure is difficult to balance and achieve the design resistances, it is necessary to increase the spiral reinforcement layer according to different bearing capacity requirements, and maintain an unbonded form between the layers to reduce the minimum bending radius of the pipeline. The following describes the strengthening principle of pipeline structure according to different load resistance requirements.

1. In order to withstand the internal pressure, the outside of the innermost polymer material needs to be spiral strengthened, and the helical winding angle should reach more than 85° (the angle to the axis of the pipeline). At the same time, in order to ensure the stability of the pipe body section during the dynamic operation, the adjacent spiral members usually adopt a lap structure, so the pressure armor layer is usually made of metal components with special-shaped sections spiral wound at a large angle, as shown in Figure 3. Depending on the amount of internal pressure, the cross-sectional shape, thickness and winding angle of the wire can be specially designed.

2. In order to withstand high external pressure, spiral reinforcement is usually required on the innermost side of the pipe. This layer is similar to the pressure-resistant armor layer, which needs to be wound at a large angle. Considering that the layer is directly washed by the internal transport fluid medium, and needs to have higher structural stability and lumen smoothness, the carcass layer usually adopts the form of interlocking structure. By changing the shape, thickness, angle and other parameters of the cross-section of the layer component, the bearing capabilities to different grades of external pressure can be realized.

3. The individual helical wire is similar to a spring. Under the axial tension, the bending moment and torsion exist on the cross section without the tensile force along the helical direction, so the spring resistance to the axial tension is very limited. And the larger the helical angle, the smaller the cross-sectional moment and the lower the resistance to axial tension. When the helical wire is supported by a cylindrical component inside, the tensile force along the helical direction will dominate under the axial tension [5] and as the helical angle reduces, the resistance to the axial tension improves significantly. Therefore, the tension armor layer is usually formed by many steel wires with rectangular cross sections winding at an angle between 30 degrees to 60 degrees on the internal pressure armor layer. According to different tension loads, the wire cross section size, the number of wire and the helical angle can be specially designed.

4. The demand on the helical components for the torsional resistance of pipeline is similar to that for the tension resistance, so the tension armor layer also provides torsional resistance. In order to achieve the torsional balance [6], the pipe is usually provided with two tension armor layers, and the inner and outer tension armor layers are spirally wound at opposite angles.

Figure 2.

Schematic diagram of flexible riser in operation.

It can be seen that in addition to the inner and outer polymer sheaths, the flexible pipe can simultaneously realize the overall flexibility and the resistance to loads in other directions by setting different types of metal spirals in the annulus.

3.2 Typical structure

According to the mechanical principle, the typical structure of the offshore flexible pipeline used in deep water is shown in Figure 4. From the inside to the outside, they are respectively:

• Carcass layer: interlocked section wire helically wound, mainly used to resist uniform external pressure from seawater, and avoid the erosion of the inner sheath from the medium. The helical angle between the direction of the wire and the axial direction of the pipeline is close to 90 degrees, and the material is usually stainless steel to avoid corrosion caused by the internal medium.

• Inner sheath: cylindrical polymer sheath formed by continuous extrusion, and mainly used to seal the internal medium while transferring internal pressure. Its common materials are: HDPE (high-density polyethylene), XLPE (cross-linked polyethylene), PA (nylon series), PVDF (polyvinylidene fluoride) and so on. Different materials have different mechanical properties, permissible temperature and permeability, and need to be selected according to the types of the internal medium. Among them, the permissible temperature of PVDF can reach 130 degrees Celsius, which is the upper limit of the current design temperature of flexible risers.

• Pressure armor layer: special-shaped section (such as “Z” type, “C” type and “T” type, etc.) steel wire helically wound, mainly used to resist the internal pressure from the inner sheath, and also play a certain role in resistance to the external pressure. The helical angle of this layer is close to 90°, and the material is usually pipeline steel or special carbon steel, which yield strength is above 800Mpa.

• Tension armor layer: rectangular section steel wire helically wound, mainly used to resist the axial tension and ensure the torsional balance of the pipeline itself. The helical angle of this layer is usually between 30 degrees and 60 degrees, and the material is usually pipeline steel or special carbon steel, which yield strength is also above 800Mpa.

• Anti-wear layer: the polymer tape with a very small friction coefficient helically wound, mainly used to reduce the possible friction and wear between the metal layers without bearing the loads.

• Outer sheath: cylindrical polymer sheath formed by continuous extrusion, and mainly used to seal the external sea water while transferring the external pressure. Its common materials are HDPE, PA, etc.

Figure 3.

Typical cross-section of the pressure armour layer.

In addition, according to the specific needs from users, layers can be added or subtracted on the basis of the above basic structure. The composite pipe wall is unbonded, and the whole is formed through the interaction between the layers, so it is of great significance to accurately understand the structural responses under different loads for designing and evaluating the safety of the pipeline.

4. Mechanical behavior of pipelines under symmetrical loads

According to the symmetry of the load on the pipe relative to the central axis, the loads including axial tension, internal pressure, external pressure and torsion are collectively referred to as the axisymmetric loads. Under this type of loads, the adjacent layers in the pipe wall contact and squeeze with each other.

4.1 Resistance to internal pressure

Flexible pipes are usually designed using the pressure armor layer to independently bear the internal pressure. The internal pressure is transferred through the inner sheath layer to the pressure armor layer. Considering that the cross-sectional area of the armor wire has a direct impact on its internal pressure resistance, the special-shaped section can be simplified to a rectangular section with the same thickness and material and certain voids in the design stage. Based on the assumption of line elasticity and small deformation, an analytical model is established for the general mechanical behavior of the spiral steel strip with rectangular section under internal pressure, in which the axial stress along the direction of the steel wire is concerned.

In case of the helical angle of pressure armor wire 90°, a closed plane ring is acted with radial pressure $P$ along circumference of cylinder core, as shown in Figure 5a.

Figure 4.

Schematic diagram of typical structure of flexible pipeline.

Referring to the most flexible pipe structures, the thickness of wires is much smaller than the diameter of the internal core. Then the classic Lame formula of the plane problem [7] can be used and equilibrium can be written as:

$$P=\frac{{\sigma }_0{h}_0}{R}$$E1

In which, $R$ means the helical radius of the pressure armor layer, ${\sigma }_0$ denotes the circumferential stress, that is the axial stress of the wire, $h_0$ is the equivalent thickness of the wire.

In case of the helical angle of steel wire not 90°, that is, the helical wire is pressured in the radial direction illustrated in Figure 5b. Let $f$ be the axial force along the wire and $\sigma$ represents axial stress, the component force in hoop direction from the axial force can be expressed as $f_0=f\times sin\alpha$. And the mapping sectional area in hoop direction of the wire is $a=A/sin\alpha$. Then the stress of the wire in hoop direction is:

$$\sigma =\frac{f}{A}=\frac{f_0/sin\alpha }{asin\alpha }=\frac{f_0}{a{sin}^2\alpha }=\frac{{\sigma }_0}{{sin}^2\alpha }$$E2

Substituting Eq. (2) into (1), the relationship between radial pressure from single armor layer and axial strain along the wire can be obtained as [8]:

$$P=\frac{h_0\sigma {sin}^2\alpha }{R}$$E3

It can be seen that as the internal pressure increases, the tensile stress along the direction of the wire axis increases. Considering the thickness, helical angle and radial radius of the wire unchanged, the stress along the wire direction and internal pressure basically show a linear relationship.

Figure 5.

Helical steel wire with radial pressure- (a) helical angle is 90°, (b) helical angle is not 90°.

4.2 Resistance to external pressure

Flexible pipes are usually designed using the innermost carcass layer to independently bear the external pressure. Although the sea water pressure generally acts on the outer sheath, the outer sheath leaks in extreme cases and the seawater passes through the annulus and directly acts on the outside of the inner sheath. In the design phase, the interlocked carcass layer can be equivalent to a homogeneous cylinder with a certain thickness. Then the problem can be simplified to analyze the buckling collapse of the cylinder under external pressure. This section establishes an analytical model of the general mechanical behavior of a homogeneous ring (cylinder) under external pressure based on the theory of elastic stability.

Timoshenko and Gere [9] first gave the deflection of a thin bar with a circular cross-section as shown in Eq. (4), for the problem of the elastic buckling of a ring or tube. It was assumed that radial displacement is small and the displacement in the tangential direction can be ignored.

$$\frac{d^2\omega }{d{\theta }^2}+\omega =-\frac{1}{\mathit{EI}}q{R}^3\left(\omega +{\omega }_{1}cos2\theta \right)$$E4

where $\omega$ denotes radial deformation of a thin bar, $\theta$ is the angle in the hoop direction, $M$ is the bending moment loading on the bar, $\mathit{EI}$ is the bending stiffness of the bar cross section, $R$ is the mean radius of curvature, ${\omega }_1$ is the maximum initial radial deviation, and $q$ means a uniform external pressure on the plane ring.

Considering the continuity condition on points A, B, C and D on the plane ring in Figure 6, the analytical solution of Eq. (4) is:

Figure 6.

Schematic diagram of a plane ring with an initial imperfection (ovality).

$$\mathrm{\omega }=\frac{{\mathrm{\omega }}_1\mathrm{q}}{{\mathrm{q}}_{\mathrm{cr}}-\mathrm{q}}cos2\mathrm{\theta }$$E5

Eq. (5) gives the analytical expression for the elastic buckling of a plane ring under uniform external pressure, where the critical pressure $q_{\mathit{cr}}$ of the ring is given by:

$${\mathrm{q}}_{\mathrm{cr}}=\frac{3\mathrm{EI}}{{\mathrm{R}}^3}$$E6

In the case of a rectangular ring section and the plane strain condition, the inertia moment $I$ per unit length can be written as $t^3/12$ and $E$ is converted to $E/\left(1-{\nu }^2\right)$. Then the critical pressure value under the plane strain conditions can be expressed as follows:

$${\mathrm{q}}_{\mathrm{cr}}=\frac{\mathrm{E}}{4\left(1-{\mathrm{\nu }}^2\right)}{\left(\frac{\mathrm{t}}{\mathrm{R}}\right)}^3$$E7

where $t$ represents the equivalent thickness of the carcass layer resulted from equivalent methods. It can be seen that the critical pressure and the equivalent thickness of the carcass layer basically show a cubic relationship. It should be noted that in the design stage, it is critical to use conservative and effective equivalent methods to determine thickness. Current equivalent methods include area equivalence [10], bending stiffness equivalence per unit length [11], bending stiffness equivalence per unit area and strain energy equivalence methods [12]. Through the experimental verification, the equivalent thickness obtained by the strain energy equivalent method can keep the critical pressure in a conservative state, which is convenient for engineering applications.

4.3 Resistance to axial tension

The double tension armor layers of flexible pipes are designed mainly to resist the axial tension, meanwhile the structure of internal core is designed to provide a support in the radial direction. To capture structural character, a simplified model that a column is wound by helical steel wires is established to analyze the tensile stiffness as illustrated in Figure 7.

Figure 7.

Mechanical model for tensile stiffness of flexible pipeline.

To simplify the deducing of the tensile stiffness, the following assumptions are made in the model: (i) Helical wires in any layer are equally spaced around the circumference of the flexible pipe. (ii) Only the axial deformation of the wires is considered during tension and the bend stiffness and torsional stiffness of wires are neglected. (iii) The internal core is modeled as a cylinder which has radial deformation under pressure from armor wires. (iv) All elements meet the assumption of small deformation and all possible frictions are neglected. (v) The helical wires and cylinder core element are homogeneous, isotropic and linearly elastic.

The analysis will be carried out on the plane for convenience by unfolding the cylinder with a helical pitch length as shown in Figure 8a. Considering axial tension $F$ on the pipe model and the axial strain $\epsilon$, we analyze the two situations below [13].

1. If only the axial deformation of the pipeline is considered, $\mathrm{\Delta }l$ is defined as axial deformation just as shown in Figure 8b. According to the geometrical relationship, the elongation along wire axial direction can be expressed as $\mathrm{\Delta }l\cdot cos\alpha$, in which $\alpha$ means the angle between umbilical axial direction and wire axial direction. Considering the length of the helix wire before deformation $l_0=l/cos\alpha$, we can obtain the strain in the wire axial directions as follows:

   $$\epsilon =\frac{\mathrm{\Delta }l\cdot cos\alpha }{l_0}=\frac{\mathrm{\Delta }l}{l}{cos}^2\alpha$$E8

2. If we only consider the radial deformation of pipeline and let $\mathrm{\Delta }R$ be radial elongation as shown in Figure 8c, the elongation along wire axial direction is $\mathrm{\Delta }R\cdot sin\alpha$. If we define the length of the helix wire before deformation $R_0=R/sin\alpha$, where $R$ means radius of the helix wires, the strain of wire can be expressed as eq. (9).

$$\epsilon =\frac{\mathrm{\Delta }R\cdot sin\alpha }{R_0}=\frac{\mathrm{\Delta }R}{R}{sin}^2\alpha$$E9

Figure 8.

Deformation of steel wire under axial tension.

However, the actual deformation is the synthetization of the axial deformation from tension and the radial deformation from the contraction of cylinder core caused by the pressure of the outer tension armor layers as shown in Figure 8d. Add the eq. (8) and (9), the synthetization strain can be written as:

$$\epsilon =\frac{\mathrm{\Delta }l}{l}{cos}^2\alpha -\frac{\mathrm{\Delta }R}{R}{sin}^2\alpha$$E10

From eq. (10), the axial tension of all the armor wires can be expressed as:

$$\begin{array}{l}F=\sum _{i=1}^m\mathit{EA}{\epsilon }_{i}cos{\alpha }_i\\ =\frac{\mathrm{\Delta }l}{l}\left(\sum _{i=1}^m\mathit{EA}{cos}^3{\alpha }_i\right)-\mathrm{\Delta }R\left(\sum _{i=1}^m\frac{\mathit{EA}{sin}^2{\alpha }_{i}cos{\alpha }_i}{R_i}\right)\\ ={\mathrm{\Theta }}_1\frac{\mathrm{\Delta }l}{l}-{\mathrm{\Theta }}_2\mathrm{\Delta }R\end{array}$$E11

where $E$ is Young modulus of steel wire, $A$ is cross-sectional area of the wire, $m$ identifies the amount of armor layers and $i$ means layer number, $n_i$ is the wire number in $i$th layer, ${\mathrm{\Theta }}_1$ and ${\mathrm{\Theta }}_2$ are the representative symbols of the corresponding analytical algebraic formulas.

From eq. (3), the relationship between radial pressure from single armor layer and axial strain along the wire can be extended as [14]:

$$P=\frac{\mathit{\text{nEAε}}{sin}^2\alpha }{2\pi R^{2}cos\alpha }$$E12

There are usually two or more armor layers for common flexible pipes. Then the radial pressure on cylinder core should be added by the pressure from all the armor layers. By substituting (10) into (12), the equilibrium equation about $P$ and $\epsilon$ is further written as:

$$\begin{array}{l}P=\sum _{i=1}^m\frac{n_i\mathit{EA}{\epsilon }_i{sin}^2{\alpha }_i}{2\pi {R_i}^{2}cos{\alpha }_i}\\ =\frac{\mathrm{\Delta }l}{l}\left(\sum _{i=1}^m\frac{n_i\mathit{EA}{sin}^2{\alpha }_{i}cos{\alpha }_i}{2\pi {R_i}^2}\right)-\mathrm{\Delta }R\left(\sum _{i=1}^m\frac{n_i\mathit{EA}{sin}^4{\alpha }_i}{2\pi {R_i}^{3}cos{\alpha }_i}\right)\\ ={\mathrm{\Psi }}_1\frac{\mathrm{\Delta }l}{l}-{\mathrm{\Psi }}_2\mathrm{\Delta }R\end{array}$$E13

where ${\mathrm{\Psi }}_1$ and ${\mathrm{\Psi }}_2$ are the representative symbols of the corresponding analytical algebraic formulas. When the cylinder core is pressed by armor steel wires, it comes to the radial contraction. In order to describe the mechanical phenomenon, the radial stiffness $\mathrm{\Omega }$ of the core is introduced and defined as:

$$\mathrm{\Omega }=P/\mathrm{\Delta }R$$E14

Eqs. (11), (13) and (14) together form a close equation set (15), which leads to the equilibrium of the steel wires with tensions considering the compressible deformation of the internal core.

$$\left\{\begin{array}{c}F={\mathrm{\Theta }}_1\frac{\mathrm{\Delta }l}{l}+{\mathrm{\Theta }}_2\mathrm{\Delta }R\\ P={\mathrm{\Psi }}_1\frac{\mathrm{\Delta }l}{l}+{\mathrm{\Psi }}_2\mathrm{\Delta }R\\ \mathrm{\Omega }=\frac{P}{\mathrm{\Delta }R}\end{array}\right.$$E15

By eliminating $P$ and $\mathrm{\Delta }R$, the relation of the tension and axial strain can be obtained as:

$$F=\left({\mathrm{\Theta }}_1+\frac{{\mathrm{\Theta }}_2{\mathrm{\Psi }}_1}{\mathrm{\Omega }-{\mathrm{\Psi }}_2}\right)\frac{\mathrm{\Delta }l}{l}$$E16

The relationship between the axial tension and the generated interlayer pressure can be analytically expressed as:

$$F=\frac{{\mathrm{\Theta }}_1\mathrm{\Omega }+{\mathrm{\Theta }}_2{\mathrm{\Psi }}_1-{\mathrm{\Theta }}_1{\mathrm{\Psi }}_2}{{\mathrm{\Psi }}_1\mathrm{\Omega }}P$$E17

If the radial contraction of the internal cylindrical core is not considered, that is, the radial stiffness $\mathrm{\Omega }$ tends to infinity, Eq. (17) can be further simplified to obtain the explicit expression of axial tension and radial pressure:

$$P=\frac{{\Psi }_1}{{\Theta }_1}F=\frac{n{\mathit{tan}}^2\alpha }{2\pi R^2}F$$E18

It can be found that as the axial tension increases, both the tensile stress along the direction of the wire and the radial pressure on the internal cylindrical core increases, and the interlayer pressure and axial tension basically show a linear relationship. It is precisely because the radial stiffness provided by the pressure armor layer is relatively large, the axial tensile stiffness of the pipeline is sufficient, and the helical armor wire can better withstand the tension. As the radial stiffness of the internal cylindrical components gradually decreases, the deformation of the helical wire will no longer be a small geometric deformation, and each wire will tend to have the independent spring deformation, therefore the tension resistance will be greatly reduced.

4.4 Resistance to torsion

Flexible pipes are primarily designed to resist torque in the clockwise/counterclockwise direction through the double tension armor layers [15]. The basic theory and assumptions are the same as those in the previous section, and this section focuses on the relationship among the torque, the radial pressure and the axial stress along the wire helical. It can be seen from the eq. (10) that the circumferential component of the axial force along the wire direction resists the overall torque of the pipeline, so the relationship between the torque on the single tension armor layer of the pipeline and the deformation along the wire direction can be expressed as:

$$Q=\mathit{\text{nREAε}}sin\alpha$$E19

Combined with eq. (12), the analytical relationship between interlayered pressure and torque without consideration of the radial contraction can be written as:

$$P=\frac{\mathit{\text{tanα}}}{2\pi R^3}Q$$E20

It can be noticed that as the torque increases, both the tensile stress along the direction of the wire and the radial pressure on the internal cylindrical core increases, and the interlayer pressure and torque also show a linear relationship. Considering both the axial and radial deformation, the derivation process is similar to the previous section, resulting in a more complete expression for eq. (20). It is worth noted that due to the different helical directions of tension armor layers, the direction of each layer’s resistance to torque is different. The neighboring tension armor layers will tend to squeeze or separate with each other due to the different torsion direction.

5. Mechanical behavior of pipelines under complicated loads

When the pipeline is subjected to non-axisymmetric loads, such as bending, the relative sliding between neighboring layers occurs due to the unbonded condition, so that the pipeline has excellent flexibility. When the pipeline is subjected to both non-axisymmetric loads and axisymmetric loads at the same time, the tangential friction is generated between the layers due to radial pressure and relative slippage. Especially for tension armor wire, the stress state becomes very complicated as the pipeline curvature changing, which will firstly cause the fatigue failure.

5.1 Bending performance

When the flexible pipe is only subject to bending, the interlayered interaction is weak, and both the deformation and relative slippage exist for each layer. Since the minimum bending radius of the inner and outer sheath and the interlocked armor layer can be obtained by the classical material mechanics, this section focuses on the mechanical behavior of tension armor wire bending under weak interlayered interaction conditions, which provides a theoretical basis to evaluate the bending performance of pipelines.

One helical wire with a rectangular cross-section wrapped around a cylindrical shell with radius $r$ is considered to represent the initial condition of the tension armor. When the model is bent (the radius of curvature is $\rho =1/\kappa$), a curve is created on the toroid, as illustrated in Figure 9a. The angular coordinate $\theta$ located along the torus radius and the arc length coordinate $u$ located along the torus centerline are chosen as parameters of the torus surface. The space vector $\mathbf{R}$on the surface can be expressed in $\left(\theta ,u\right)$-coordinates instead of in Cartesian coordinates as [16]:

Figure 9.

Wire geometry on a toroid under the curvature- (a) overall view, (b) cross-section A-A view.

$$\mathrm{R}\left(\mathrm{u},\mathrm{\theta }\right)=\left[\begin{array}{l}\left(\frac{1}{\mathrm{\kappa }}+\mathrm{r}·\text{cosθ}\right)cos\left(\mathrm{\kappa u}\right)-\frac{1}{\mathrm{\kappa }}\\ \left(\frac{1}{\mathrm{\kappa }}+\mathrm{r}·\text{cosθ}\right)sin\left(\mathrm{\kappa u}\right)\\ \mathrm{r}·\text{sinθ}\end{array}\right]$$E21

Correspondingly, $\left(x_2,x_3\right)$-rectangular coordinates can be built in the cutting plane perpendicular to the centerline of the wire (see Figure 9b), where ${\mathit{x}}_{\mathbf{2}}$ is perpendicular to the local unit wire normal $\mathbf{n}$ and is located along the rectangular width, and ${\mathit{x}}_{\mathbf{3}}$ points to the local unit wire binormal $\mathbf{b}$ and is located along the thickness.

When the cylindrical shell is bent with a specified curvature, the initial equilibrium state of the helical wire is broken, and the wire is forced to slip on the torus surface. Correspondingly, the changed space vector of the wire can be expressed as ${\mathbf{R}}^{\prime }\left({\theta }^{\prime },u^{\prime }\right)$. Based on differential geometry theory, the variation of curvature components of the bending wire can be defined as:

$$\left\{\begin{array}{l}∆{\mathrm{\kappa }}_{\mathrm{n}}={\mathrm{\kappa }}_{\mathrm{n}}-{{\mathrm{\kappa }}_{\mathrm{n}}}^0\\ ∆{\mathrm{\kappa }}_{\mathrm{b}}={\mathrm{\kappa }}_{\mathrm{b}}-{{\mathrm{\kappa }}_{\mathrm{b}}}^0\\ ∆\mathrm{\tau }=\mathrm{\tau }-{\mathrm{\tau }}^0\end{array}\right.$$E22

In which, the stress on the wire cross section caused by the normal curvature $∆{\mathrm{\kappa }}_{\mathrm{n}}$ and the binormal curvature $∆{\mathrm{\kappa }}_{\mathrm{b}}$ is the positive stress and along the wire direction; the stress on the wire cross section caused by the torsional curvature $∆\mathrm{\tau }$ is the shear stress and perpendicular to the wire direction. During the wire bending, the fatigue failure usually occurs firstly at the corner point of the wire rectangular section due to high local stress, which needs to be concerned. Since the shear stress at the corner point is zero, the stress is in the uniaxial stress state and can be expressed as:

$${\mathrm{\sigma }}^{\mathrm{y}}={\mathrm{\sigma }}_{\mathrm{n}}+{\mathrm{\sigma }}_{\mathrm{b}}=\left(∆{\mathrm{\kappa }}_{\mathrm{n}}\times {\mathrm{x}}_{2\mathrm{y}}\right)·\mathrm{E}+\left(∆{\mathrm{\kappa }}_{\mathrm{b}}\times {\mathrm{x}}_{3\mathrm{y}}\right)·\mathrm{E}$$E23

where $\mathrm{y}$ represents the corner point number.

Based on the existing experimental results and theoretical models, the bending behavior of the helical wire considering the weak interlayered interaction is closer to that of a spring [4]. Therefore, according to the classical principle of elasticity, the curvature variation in the three directions of the helical wire can be obtained as eq. (24) when the pipeline has a certain curvature $\kappa$.

$$\left\{\begin{array}{l}∆{\mathrm{\kappa }}_{\mathrm{n}}=\frac{\text{Bcosθ}}{{\mathrm{EI}}_{\mathrm{n}}}\mathrm{\kappa }\\ ∆{\mathrm{\kappa }}_{\mathrm{b}}=-\frac{\text{Bsinθcosα}}{{\mathrm{EI}}_{\mathrm{b}}}\mathrm{\kappa }\\ ∆\mathrm{\tau }=\frac{\text{Bsinθsinα}}{\mathrm{GJ}}\mathrm{\kappa }\end{array}\right.$$E24

Use of Eq. (24) in Eq. (23) leads to the stress on a corner point of the helical wire:

$${\mathrm{\sigma }}^{\mathrm{y}}=\frac{\text{Bcosθ}}{{\mathrm{EI}}_{\mathrm{n}}}\mathrm{\kappa E}·{\mathrm{x}}_{2\mathrm{y}}-\frac{\text{Bsinθcosα}}{{\mathrm{EI}}_{\mathrm{b}}}\mathrm{\kappa E}·{\mathrm{x}}_{3\mathrm{y}}$$E25

in which $B$ represents the bending stiffness of the spring and is expressed as $2\mathit{\text{cosα}}/\left(\frac{1}{{\mathit{EI}}_n}+\frac{{cos}^2\alpha }{{\mathit{EI}}_b}+\frac{{sin}^2\alpha }{\mathit{GJ}}\right)$, ${\mathit{EI}}_n$ and ${\mathit{EI}}_b$ represent the bending stiffness of the rectangular cross-section with respect to the $x_3,x_2$ axis, respectively, and $\mathit{GJ}$ indicates the torsion stiffness of wire section.

It can be seen that under the condition of weak interlayered interaction, the bending behavior of flexible pipes can be evaluated layer by layer, and then the minimum bending radius (MBR) can be comprehensively obtained. In general, the MBR of the pipeline is first determined by the stress at corner points of tension armor wires.

5.2 Structural response under complicated loads

When the pipeline is subjected to complicated multiaxial loads, the friction will be generated between the layers, which brings challenges to the evaluation of the overall mechanical responses of the pipeline. This section focuses on the mechanical behavior of tension armor wires under complicated loads, that is, considering the large interlayered pressure caused by the axial tension, internal and external pressure or torsion, and as the curvature increases from zero, the interlayered friction will force the steel wire on the surface of internal cylindrical core to go through three stages [17] of no-slip, stick–slip and full-slip, as shown in Figure 10. The theoretical model is based on the common assumptions: i) The lay angle of the wire remains constant during helical wire tension and bending, which yields a loxodromic curve on a torus surface; ii) The size of wire cross section is very small compared to the helical radius; iii) Small deformations occur in the linear elastic range of the material; iv) The helical wires are evenly distributed along the circumference and there is no mutual influence between neighboring wires.

Figure 10.

Schematic diagram of nonlinear bending behavior of offshore flexible pipes.

5.2.1 No-slip stage (O-A)

When the curvature is very small, the helical wire remains firmly attached to the bent cylinder without any relative movement due to the interlayered static friction. Let $\mathrm{R}$ be the helical radius of the wire, $\mathrm{\alpha }$ is the helical angle, $\mathrm{\theta }$ represents the phase angle in the circumferential direction, which varies from zero to $\mathrm{\pi }/2$. Considering the pipeline bent to a small curvature and the helical angle of the wire unchanged, the spatial three-dimensional coordinates $\mathrm{\kappa }\left({\mathrm{x}}^{\prime },{\mathrm{y}}^{\prime },{\mathrm{z}}^{\prime }\right)$ of the deformed helical wire can be described parametrically as eq. (26) based on the differential geometry [18].

$$\left\{\begin{array}{l}{\mathrm{x}}^{\text{'}}=\text{Rcosθ}\\ {\mathrm{y}}^{\text{'}}=sin\left(\frac{\mathrm{R}}{\text{tanα}}\mathrm{\theta \kappa }\right)\frac{1}{\mathrm{\kappa }}-\text{Rsinθsin}\left(\frac{\mathrm{R}}{\text{tanα}}\mathrm{\theta \kappa }\right)\\ {\mathrm{z}}^{\prime }=\left(1-cos\left(\frac{\mathrm{R}}{\text{tanα}}\mathrm{\theta \kappa }\right)\right)\frac{1}{\mathrm{\kappa }}+\text{Rsinθ}\left(\frac{\mathrm{R}}{\text{tanα}}\mathrm{\theta \kappa }\right)\end{array}\right.$$E26

The corresponding axial strain along the bent helical strip is given by:

$$\mathrm{\epsilon }\left(\mathrm{\varphi }\right)=\frac{{\mathrm{ds}}^{\prime }}{\mathrm{ds}}-1=\sqrt{\frac{\mathrm{d}{{\mathrm{x}}^{\prime }}^2+\mathrm{d}{{\mathrm{y}}^{\prime }}^2+\mathrm{d}{{\mathrm{z}}^{\prime }}^2}{\mathrm{d}{\mathrm{x}}^2+\mathrm{d}{\mathrm{y}}^2+\mathrm{d}{\mathrm{z}}^2}}-1$$E27

where $\mathrm{s}$ represents the arch length along the helical wire. By using eq. (26) in eq. (27), and assuming small deflections, a linear expression for the axial strain can be obtained by eliminating second and higher order terms. The linearized axial strain is given by:

$$\mathrm{\epsilon }\left(\mathrm{\varphi }\right)=-\mathrm{R}{cos}^2\text{αsinθκ}$$E28

It can be seen that the maximum stress of the wire in the no-slip stage located at the intrados and extrados, and can be expressed as:

$${\mathrm{\sigma }}_{\mathrm{f}-\mathrm{ns}}^{\max }=\mathrm{RE}{cos}^2\mathrm{\alpha \kappa }$$E29

5.2.2 Stick–slip stage (A-B)

In the stick–slip stage, the axial force of the helical wire and the interlayered friction between are balanced with each other. However, as the curvature increases, the axial force on the wire section increases, and when the maximum static friction is exceeded, the wire begins to slide relative to internal components. Thus, the slipping condition can be written as [19]:

$$\frac{\mathrm{dN}}{\mathrm{ds}}>f$$E30

in which, $\mathrm{N}$ is the internal force along the wire direction, $f$ means the friction force on the unit length of the wire, $s$ and denotes the length along the wire axis. According to the eq. (28) and taking into account the geometric relationship $\theta =s·\text{sinα}/R$, the critical curvature (Point A in Figure 10) corresponding to the wire starting to slip can be derived as:

$${\mathrm{\kappa }}_0=\frac{\mathrm{f}}{-\mathrm{EA}{cos}^2\text{αsinαcos}\theta }$$E31

It can be found that when $\theta$ equals to $\mathit{k\pi }\left(k=0,1,2,\dots \right)$, the critical curvature is minimized. That implies the helical wire at the neutral axis first beginning to slip as the curvature increasing. Assuming that the contact pressure on the inner and outer surfaces of the wire layer are separately ${\mathrm{q}}_{\mathrm{i}}$ and ${\mathrm{q}}_{\mathrm{o}}$, and the corresponding static friction coefficients are ${\mathrm{\mu }}_{\mathrm{i}}$ and ${\mathrm{\mu }}_{\mathrm{o}}$, the critical curvature of the wire can be further expressed as:

$${\mathrm{\kappa }}_0=\frac{{\mathrm{q}}_{\mathrm{i}}{\mathrm{\mu }}_{\mathrm{i}}+{\mathrm{q}}_{\mathrm{o}}{\mathrm{\mu }}_{\mathrm{o}}}{-\mathrm{EA}{cos}^2\text{αsinα}}$$E32

When the curvature is greater than the critical curvature, the slipping area of the wire rapidly extends from the neutral axis to both ends, during which one part of the wire is slipping while the other part still in the stick stage. According to the equilibrium relationship of the axial force and interlayered friction in the stick area, the initial slipping curvature at the phase angle $\theta$ of the wire can be obtained:

$${\mathrm{\kappa }}_{\mathrm{f}}=\left(\frac{\mathrm{f}}{-\mathrm{EA}{cos}^2\text{αsinα}}\right)\frac{\theta }{sin\theta }={\mathrm{\kappa }}_0\frac{\theta }{sin\theta }$$E33

If $\theta =\mathrm{\pi }/2$ in the above equation, the curvature when all parts of the wire enter the slipping state (Point B in Figure 10) can be described as:

$${\mathrm{\kappa }}_{\mathrm{f}}=\frac{\mathrm{\pi }}{2}{\mathrm{\kappa }}_0$$E34

From point A to point B, the wire completes the transition from the no-slip state to the full-slip state. As the axial stress of the wire is gradually released, the overall bending stiffness of the layer decreases.

5.2.3 Full-slip stage (B-)

Assuming that the friction along the wire direction dominates, the friction on the inner and outer surfaces of the rectangular wire produces the axial stress evenly distributed by the wire section. According to the eq. (33), the axial stress generated by the total friction ${\mathrm{F}}_0$ of the slipping wire can be written as:

$${\mathrm{\sigma }}_{\mathrm{f}-\mathrm{fs}}=\frac{{\mathrm{F}}_0}{\mathrm{A}}=\left[\frac{\mathrm{R}\theta }{\text{Asinα}}\right]\mathrm{f}$$E35

where $\mathrm{A}$ is the cross-sectional area of the wire. It can be seen that the friction varies with the phase angle, but the stress at the four corner points of the wire at the same phase angle is the same. Therefore, the maximum axial stress of the wire generated by the total friction in the full-slip state is further obtained:

$${\mathrm{\sigma }}_{\mathrm{f}-\mathrm{fs}}^{\max }=\frac{\mathrm{R\pi }}{2\text{Asinα}}\left({\mathrm{q}}_{\mathrm{i}}{\mathrm{\mu }}_{\mathrm{i}}+{\mathrm{q}}_{\mathrm{o}}{\mathrm{\mu }}_{\mathrm{o}}\right)$$E36

In summary, the theoretical model for calculating the maximum axial stress of the helical wire at each stage during the curvature increasing is established considering the interlayered friction. For conservative consideration in practical engineering, the above three stages are usually instead of two stages: one is the no-slip stage where the curvature varies from zero to ${\mathrm{\kappa }}_{\mathrm{f}}.$, and the other is the full-slip section where the curvature is greater than ${\mathrm{\kappa }}_{\mathrm{f}}$

Additionally, due to the complicated loads, it is also necessary to consider the axial stress ${\sigma }_T$ (as shown in Eq. (11)) of the wire caused by the tension and the axial stress ${\sigma }_{\kappa }$ (as shown in Eq. (25)) caused by the helical “spring” itself during bending in the full-slip stage. Then the nonlinear stress of the helical wire under complicated loads with the consideration of the interlayered interaction can be expressed as Eq. (37) and shown as Figure 11.

Figure 11.

Schematic diagram of nonlinear stress at the corner point of the wire cross-section.

$$\mathrm{\sigma }=\left\{\begin{array}{l}{\mathrm{\sigma }}_{\mathrm{T}}+{\mathrm{\sigma }}_{\mathrm{f}-\mathrm{ns}},\mathrm{\kappa }\le {\kappa }_{\mathrm{f}}\\ {\mathrm{\sigma }}_{\mathrm{T}}+{\mathrm{\sigma }}_{\mathrm{\kappa }}+{\mathrm{\sigma }}_{\mathrm{f}-\mathrm{fs}},\mathrm{\kappa }>{\kappa }_{\mathrm{f}}\end{array}\right.$$E37

In order to perform the fatigue evaluation [20], it is necessary to calculate the maximum alternating stress amplitude ${\mathrm{K}}_{\mathrm{c}}\left(\mathrm{\kappa }\right)$ of offshore flexible risers as the curvature changes. Removing the average stress term ${\mathrm{\sigma }}_{\mathrm{T}}$ and taking into account the maximum stress caused by the curvature and interlayered friction, the maximum alternating stress can be described as:

$$∆{\sigma }^{\mathit{max}}=\left\{\begin{array}{l}{\mathrm{\sigma }}_{\mathrm{f}-\mathrm{ns}}^{\max },\mathrm{\kappa }\le {\kappa }_{\mathrm{f}}\\ {\mathrm{\sigma }}_{\mathrm{\kappa }}^{\max }+{\mathrm{\sigma }}_{\mathrm{f}-\mathrm{fs}}^{\max },\mathrm{\kappa }>{\kappa }_{\mathrm{f}}\end{array}\right.$$E38

It can be seen that the alternating stress exhibits a nonlinear relation with the curvature. Therefore, $K_c\left(\kappa \right)$ is not a constant value and cannot be directly used in the existing commercial software to calculate the fatigue life. In this case, it is necessary to carry out the secondary development in order to input the alternating stress with curvature, and then the fatigue life calculation can be performed by the damage accumulation using the Miner formula.

6. A case study

6.1 Description of the pipeline structure

In order to quantitatively illustrate the relation between interlayer action and the overall performance of offshore flexible pipelines, an 8-inch inner diameter flexible riser actually applied in a water depth of 1500 meters is taken in this section as an example, and the mechanical properties of pipelines are given in Table 1.

Layer	Thickness (mm)	Inside diameter (mm)	Outside diameter (mm)	Material
Carcass	8	200	216	AISI 316 L
Internal sheath	10	218	234	XLPE
Anti-wear 1	1	234	236	PA-11
Pressure armor	6	236	248	special carbon steel
Anti-wear 2	1	248	250	PA-11
Tension armor 1	4	250	258	special carbon steel
Anti-wear 3	1	258	260	PA-11
Tension armor 2	4	260	268	special carbon steel
Anti-wear 4	1	268	270	PA-11
External sheath 1	5	270	280	HDPE
External sheath 2	5	280	290	HDPE

Table 1.

Material and dimensional parameters of flexible pipelines.

The pipeline structure is designed to have two tension armor layers, of which material elastic modulus is 206Gpa, the Poisson’s ratio is 0.3 and the yield strength is 800Mpa. The rectangular cross-sectional dimensions of the steel wire in the two tension armor layers are the same (4 mm $\times$ 12 mm) and the helical angles are separately positive and negative 30 degrees. According to the configuration design, functional requirements and hydrodynamic analysis of the flexible riser, the design loads can be determined as the internal pressure of 20 Mpa, the external pressure of 15 Mpa, the tension of 170 tons, the MBR of 2.5 meters and the fatigue life of 25 years.

6.2 Analysis and discussion of structural responses

Using the theoretical models for calculating the interlayered interaction and structural responses proposed in this chapter, this section quantitively gives the interlayered pressure and the material utilization of the pipeline under axisymmetric and non-axisymmetric design loads, and the results are shown as in Table 2.

Design load		Value	Interlayered pressure (MPa)	Stress (MPa)	Material utilization	Remarks
Axisymmetric	Internal pressure	20Mpa	19.67	403	50.4%	Only the pressure armor layer resisting
	External pressure	15Mpa	15.25	—	66.3%	Only the carcass layer resisting
	Tension	170 t	5.38Mpa (Inner surface)	371	46.4%	Distributed according to the stiffness of layers
			2.64Mpa (Outer surface)	372	46.5%
Non-axisymmetric	MBR	2.5 m	Weak interlayered interaction	665	83.2%	In storage and installation
						The tenson armor layer dominate

Table 2.

Interlayered pressure and material utilization of the pipeline under different design loads.

It can be clearly seen that the pipeline structure meets the design requirements, and still have a large safety margin under various independent loads. Those illustrate that the theoretical models for the structural responses proposed in this chapter could provide effective tools to design and evaluate the flexible pipelines in actual engineering. For the tension armor wires, achieving the tensile resistance of 170 tons requires the internal cylindrical component to provide a radial supporting pressure of nearly 6Mpa. Therefore, the internal pressure will be partially compensated by the pressure generated by axial tension, so that the material utilization of the tension armor wire is further reduced during actual operation of the pipeline.

Since the fatigue failure occurs first on the helical tension armor wire of offshore flexible riser under alternative loads, the nonlinear stress at the corner points of the wire with the curvature is calculated by using Eq. (38) under combining the axial tension and bending curvature, as shown by the red dot line in Figure 12. The figure also shows the linear stress (blue dot line) of the wire due to friction only in the no-slip state and the linear stress (black line) of the slipping wire without consideration of the interlayered friction. It can be clearly seen that the nonlinear wire stress considering interlayer friction is between the two linear stress distribution curves, so the corresponding distributions of the fatigue life should also have the similar tendency.

Figure 12.

Calculation results of nonlinear stress on the helical wire.

It should be noted that the interlayered contact and friction cause the stress of helical wires obvious nonlinear, so their accurate prediction is of great significance for the calculation of fatigue life. However, for the current analyzing theory of fatigue stress, there is a lack of accurate description of the wire behavior in the stick–slip stage [21], which has a decisive effect on the fatigue life when the riser is subjected to large axisymmetric loads and small bending curvature. In addition, after the wire enters the full-slip stage, it is worth further exploring whether the bending stress of the wire under axial tension can be simulated through the spring theory [22]. Moreover, despite the anti-wear layer, long-term repeated interlayered friction will still cause wear [23], which impact on the stress variation of the wire and the fatigue life cannot be ignored. All of those discussion items bring big challenges for evaluating the fatigue life of flexible pipes, which will promote the further development of the theoretical and experimental methods.

7. Conclusion

This chapter systematically introduces the mechanical principles of offshore flexible pipes, and gives the theoretical methods for evaluating the structural responses under axisymmetric loads, bending and combined complicated loads. It can be seen from a case study that although different strengthening layers in the pipeline are separately designed to resist against different loads, the overall resistance of the pipeline strongly depends on the interaction between the layers due to the use of metal helical armor wires, and then the unbonded pipe wall forms a whole. The contact pressure dominates the interlayered interaction under the axisymmetric loads such as axial tension, internal pressure, external pressure and torsion. And while the non-axisymmetric bending load is introduced, interlayered friction and relative slippage occur, resulting in the obvious stick–slip nonlinear behavior of pipeline components, which has an important impact on the subsequent fatigue life analysis.

Acknowledgments

This work was supported by the Program of NSFC (grant number 5170924); the Key Technologies Research and Development Program (grant number 2021YFC2801600); and the High-tech Ship Scientific Research Program (grant number CY05N20). Their support is gratefully appreciated.