Analytical study of a repair methodology for flexible pipes anti-birdcage tapes

Marine Structures 63 (2019) 289–303

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Analytical study of a repair methodology for flexible pipes antibirdcage tapes

T

Xiaotian Lia, Murilo Augusto Vaza,∗, Anderson Barata Custódiob a b

Ocean Engineering Program, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil Petrobras, CENPES, Rio de Janeiro, Brazil

A R T IC LE I N F O

ABS TRA CT

Keywords: Flexible pipes Repair technique Anti-Birdcage tape Birdcage Capstan effect

Flexible pipes may be exposed to high risk of birdcage failure mechanism if the anti-birdcage tapes are damaged. A feasible repair methodology has been proposed by winding multi-layers of high strength filaments with specific pre-tension over the damaged area and its neighboring regions. To investigate the mechanism of this repair technique, an analytical model is developed by simplifying the problem as superimposed tape and filaments over an expandable frictional cylindrical surface. Through which, the effects of the pivotal parameters regarding this repair technique, including the geometrical and material properties of the filaments, pre-tension in the filaments, number of filamentous layers, as well as the length of the repaired section surpassing the damaged region on both sides, on the effectiveness of this repair methodology are investigated extensively. In addition, the effectiveness of this repair methodology in the long term is evaluated considering the potential pre-tension reduction phenomenon of the filaments. A series of case studies are performed and some useful suggestions are presented in view of this repair methodology.

1. Introduction Flexible pipes are widely used in offshore oil and gas field development for transportation of liquid or gas, well control and monitoring. A typical unbounded flexible pipe, as depicted in Fig. 1, is composed of a number of layers with specific functions. The metallic layers provide structural stiffness and the polymeric layers offer fluid leak-proof capacity and reduce the friction between metallic layers. Among these layers, tensile armor layers, which conventionally consist an even number of layers of helically wound steel wires, are the principal layers to bear the tensile loads. Despite the fact that tensile loads are the principal loads carried by the flexible pipes during most of their service life, significant axial compressive loads may also be experienced. Under scenarios typically found during deep water installation or some phases of its service life, such as well shut down operations, the flexible pipe bore may be empty or with very small internal pressure. These conditions may generate significant longitudinal compressive loads on the flexible pipes due to the hydrostatic end-cap effect [1]. Such axial compressive loads, mainly carried by the tensile armors, promote radial dilatation of flexible pipes. If such radial dilatation is not properly impeded, a radial failure mechanism, characterized by large localized radial dislocation, will be formed as depicted in Fig. 2. This radial failure mode, also known as birdcage, has lately attracted extensive interest for its complicated mechanism [1–6,8–11]. Studies have shown that the birdcage failure mode to some extent can be prevented by the application of anti-birdcage tapes over

∗

Corresponding author. E-mail address: [email protected] (M.A. Vaz).

https://doi.org/10.1016/j.marstruc.2018.10.001 Received 20 March 2018; Received in revised form 14 August 2018; Accepted 4 October 2018 0951-8339/ © 2018 Elsevier Ltd. All rights reserved.

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Fig. 1. Typical unbonded flexible pipe.

Fig. 2. Birdcage reproduced in the laboratory.

the tensile armors [12]. The anti-birdcage tapes, commonly made of aramid fibers, are loosely wound over the tensile armors as depicted in Fig. 3. Nevertheless, during flexible pipe installation and operation, a considerable incidence of damages has been observed. One of the most common damages is the external sheath and anti-birdcage tapes rupture or abrasion which is generally caused by contact with other risers or platform structure, or abrasion from the pipe-soil interaction. Some repair techniques are proposed by industry to mitigate the progression of damages on the external sheath, including the application of adhesive tapes and polyurethane sleeves [13–15]. However, little attention has been devoted to investigating the repair technique for anti-birdcage tapes preventing the birdcage failure. The damage to external sheath results in wet armor annulus condition, therefore vanishing the radial constraint from the external water pressure. Worse still, the damage on anti-birdcage tapes

Fig. 3. Application of anti-birdcage tapes in flexible pipes. 290

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Fig. 4. Schematic diagram of damaged (up) and repaired (down) flexible pipes.

further reduces the radial constrain capacity, causing significant risk of formation of the birdcage in the damaged region. Thus, it is worthwhile devoting effort to develop a repair technique to prevent the impending formation of the birdcage. A feasible repair methodology to regain the radial constraint capacity entails developing an apparatus able to wind multiple layers of high strength filaments over the damaged section with ends immobilized on the flexible pipe's outer surface, see Fig. 4. By applying sufficient layers of filaments, the radial constraint capacity can be reestablished in the repaired section. However, besides the recovery of radial constraint capacity, concerns about the potential formation of the birdcages at the edges of the repaired section have arisen. When the repaired flexible pipe experiences axial compressive loads, the tensile armors tend to expand outwardly, which tightens the remaining portions of the damaged tape and tends to pull it outwardly from the repaired section. The slippage of a tape wound over a cylindrical surface is essentially inhibited by capstan effects, which have been widely applied in the tribological area [7,17–21]. If the frictional force between the tape and adjacent layers inside the repaired section is insufficient to impede the tape slippage, the tape will be huddled close to the edges of the repaired section and localized birdcages may be formed, as depicted in Fig. 5. Such localized birdcages, generally coexisting with plastic deflections in the tensile armors, should be precluded for the flexible pipes repaired by this technique. The mechanical behavior of the superimposed tape and filament over a fixed frictional cylindrical surface has been studied by Liu et al. [22]. However, their study is only applicable if the radial dilatation of the flexible pipe generated by axial compressive loads is not considered. This paper aims at investigating the mechanism of this repair technique for the damaged anti-birdcage tape in flexible pipes. Ignoring the effect of the outer polymeric sheath and considering the flexible pipe in a straight condition, typically encountered in flowlines, an analytical model is formulated by simplifying this problem as superimposed tape and filaments over an expandable frictional cylindrical surface. For a given filament that has fixed geometrical and material properties, analytical solution corresponding to the minimum number of filamentous layers demanded to reestablish the radial constraint capacity in the damaged region is obtained. Besides, the mechanical behavior of the damaged tape inside the repaired section under different loading conditions are discussed. To impede the tape sliding outwardly from the repaired section, the demanded length of the section covered by filaments surpassing the damaged region on both sides is assessed considering capstan effect. Moreover, the potential pre-tension reduction effect of the filaments on the effectiveness of this repair methodology in the long term is evaluated and some practical suggestions are presented. 2. Mechanical model for an undamaged tape over a cylindrical surface To obtain sufficient radial constraint capacity, a number of anti-birdcage tapes are usually superimposed, depending on the tape strength and the estimated compressive loads sustained by the flexible pipe. Besides, to ensure the tensile armors are entirely covered by the tapes when the flexible pipe is bent, typically the tapes are wound over the flexible pipe in straight condition with partial overlap at each turn. Intuitively, the overlapped areas may sustain higher radial restriction than the non-overlapped areas. To simplify the problem, in the present context only one tape is considered, which is assumed evenly wound over a straight cylindrical

Fig. 5. Sketch of localized birdcages formed at the edges of the repaired section. 291

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Fig. 6. Schematic diagram of a tape wound over a cylindrical surface.

surface with neither gap nor overlap at each turn. Fig. 6 shows the schematic diagram of a tape with width W1 wound over a cylindrical surface with diameter denoted by D . The underlying cylindrical surface is parameterized by the column coordinates, where u and θ respectively describe the longitudinal and circumferential positions of the points on the cylindrical surface. Besides, the tape's central line is parameterized by the arc length → → n , b ⎞⎟ is defined on the tape's central line representing the tangential, normal and parameter s , and a triad of local unit vectors ⎛⎜ t , → ⎝ ⎠ bi-normal directions respectively. Since the tape is wound with neither gap nor overlap at each turn, from geometrical relationship its lay angle with respect to the longitudinal axis α can be determined by:

W α = arccos ⎛ 1 ⎞ ⎝ πD ⎠

(1)

Ignoring the tape's thickness, i.e., its bending stiffness, the equilibrium state of the tape over a cylindrical surface can be defined through the system of equations:

κn Q1 + qn = 0

(2a)

dQ1 cos α − qt = 0 du

(2b)

where Q1 is the tensile load in the tape, qt and qn are respectively the distributed loads in the tangential and normal directions. κn is the normal curvature component of the tape, which can be calculated by:

κn = −

sin2 α D /2

(3)

When an intact flexible pipe is subjected to axial compressive loads in the straight condition, the flexible pipe tends to expand uniformly and the tape is uniformly tensioned. Thus, according to Eq. (2b), no tangential distributed load is generated, i.e., qt = 0 . Besides, substituting Eq. (3) into Eq. (2a) and considering the radial dilatation small relative to the cylinder diameter, the radial distributed load generated by the tensile load in the tape can be obtained as:

qn =

sin2 α Q1 D /2

(4)

Consequently, the corresponding contact pressure between the tape and the underlying cylindrical surface generated by the tensile load in the tape can be evaluated by:

σ1 =

qn 2 sin2 α Q1 = W1 DW1

(5)

Note that the tensile load in the tape is not allowed to exceed its breaking strength Q1,max , otherwise, the tape will be ruptured and the radial buckling will be triggered. Substituting Q1 = Q1,max into Eq. (5), the maximum contact pressure on the tape inner interface can be approximately determined by:

σ1,max =

2 sin2 α Q1,max DW1

(6)

3. Mechanical model for the filaments over a cylindrical surface The helically wound filaments exhibit similar mechanical behavior as the tape previously described. To regain sufficient radial constraint capacity in the damaged region, a number of filamentous layers, denoted by n , are applied in superimposed position. Within each filamentous layer, the filament is wound with negligible interval spacing between each turn. Since their cross-section 292

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dimensions are small relative to the cylinder diameter, the filaments are nearly perpendicular to the longitudinal axis. Comparable to Eq. (5), ignoring the filamentous thickness and using α = π/2, the contact pressure on the inner interface of filamentous layers generated by the tensile load in the filaments can be evaluated by:

σ2 =

2n Q2 DW2

(7)

In which W2 denotes the width of a single filament on the cylindrical surface, Q2 corresponds to the tensile load in the filaments. Let Q2,max denote the breaking strength of a single filament, the maximum contact pressure on their inner interface that the filamentous layers are capable to hold can be approximately determined by substituting Q2 = Q2,max into Eq. (7) as:

σ2,max =

2n Q2,max DW2

(8)

Despite the fact that the remaining portions of the damaged tape can still provide some radial constraint in the damaged section, for the sake of safety and model simplicity, this contribution is herein disregarded. In other words, it is assumed that the tape in the damaged section is totally ineffective. Thus, to regain the radial constraint capacity in the damaged section, the ultimate strength of the deployed filamentous layers needs to be at least the ultimate strength of the original intact tape. Consequently, combining Eqs. (6) and (8), the number of filamentous layers to regain the radial constraint capacity in the damaged section needs to be at least:

n≥⌈

Q1,max W2 sin2 α ⌉ Q2,max W1

(9)

where the symbol ⌈⌉ represents the top integral function. 4. Mechanical model for the superimposed tape and filaments over a cylinder Besides the recovery of radial constraint capacity in the damaged section, concerns about impending slippage of the remaining portions of the damaged tape have arisen. Based on the principles of capstan effects, the slippage of the damaged tape over a cylindrical surface is governed by the frictional restriction between the tape and the adjacent layers. Since the maximum frictional restriction on the tape is proportional to the length and contact pressures on the tape's interfaces with adjacent layers, to prevent the tape sliding outwardly from the repaired section, measures can be taken by either applying a specific pre-tension in the filaments to increase the contact pressures or extending an adequate length of the section covered by the filaments away from the damaged region on both sides. However, extending the section covered by filaments entails more filamentous material and time consumptions. Thus, to develop sufficient frictional restriction with lower costs, it is preferable to apply a large pre-tension in the filaments with a shorter repaired section. Let Q2, pre denote the deployed pre-tension in the filaments, comparable to Eq. (7), the contact pressure generated by the pretension in the filaments on their inner interface can be evaluated by:

σ2, pre =

2n Q2, pre DW2

(10)

It is conceivable that the pre-tension in filaments gives birth to a slight radial contraction of the flexible pipe in the repaired section. However, since the support from the inner layers, such as carcass and pressure armor, is sufficiently stiff, the radial contraction due to the pre-tension in the filaments is taken as negligible in the present context. When axial compressive loads are sustained, the tensile armors tend to expand outwardly such that it yields a contact force between the armor layer and the adjacent outer layer. It is herein assumed that the corresponding contact pressure on the outer interface of the armor layer generated by axial compressive loads is uniform, which is denoted by σ . Note that this expansion force is not allowed to exceed the ultimate strength of the original intact tape, i.e., σ < σ1,max . Obviously, if the tensile armors’ expansion force is insufficient to overcome the radial restriction generated by the pre-tension in the filaments, i.e., σ2, pre ≥ σ , the tensile armors will be compacted on the inner core in the repaired section, while radial dilatations are generated outside the repaired section. On the contrary, if σ2, pre < σ , radial dilatations will be generated inside of the repaired section as well. To explore the influence of the filamentous pre-tension on the effectiveness of this repair technique, the mechanical behavior of the remaining portions of the damaged tape in both conditions will be discussed in this section. 4.1. Condition where expansion is impeded in the repaired section (σ2, pre ≥ σ ) Consider a flexible pipe with damages on the tape overlapped by sufficiently long filamentous layers, and the expansion force of tensile armors generated by axial compressive loads is insufficient to overcome the restriction induced by the pre-tension in the filaments. Under this circumstance, radial dilatation is generated outside the repaired section and inhibited inside the repaired section, thus generating dilatation gradients close to the edges of the repaired section. Away from the edges of the repaired section, the pipe dilatation tends to stabilize as the effect of the filaments declines. Considering the radial dilatation small relative to the pipe dimension, in the section far away from the repaired section the tensile load in the tape can be approximately evaluated by substituting σ1 = σ into Eq. (5) as: 293

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Fig. 7. Simplified schematic of superimposed tape and filaments over a fixed cylinder.

Q1′ =

DW1 σ 2 sin2 α

(11)

Considering the mechanical complexity of the transition zones close the edges of the repaired section, the tensile load given by Eq. (11) is herein approximately assumed as the tensile load in the tape at the edges of the repaired section. Since the tape in the damaged section is assumed ineffective as previously described, this problem may be simplified as superimposed tape and filaments over a fixed frictional cylindrical surface with one tape end relaxed and the other tape end tensioned as depicted in Fig. 7. In which L0 and L respectively denote the length of the damaged section and the length of the section covered by the filaments surpassing both sides of the damaged section. Due to the symmetry, the right side is herein chosen for the study with reference position located at the broken end of the tape. From Eq. (2b) it can be seen that the existence of the frictional restriction drives the tensile load in the tape decrease from the edges of the repaired section towards the damaged region. If the repaired section is sufficiently long, the tensile load in the tape will decrease to zero after a certain length and remain zero in the remaining portion. The section that encompasses tension variation in the tape is herein named recovery section and the length of which is named recovery length. Let X1 denote the length of the recovery section close to the edges of the repaired section, the tensile load in the tape inside the repaired section can be approximately expressed as:

(0 ≤ u ≤ L − X1) ⎧Q1 = 0 ′ ⎨ 0 < Q ≤ Q 1 1 (L − X1 < u ≤ L) ⎩

(12)

In the recovery section, the frictional force on the tape in the tangential direction can be approximately calculated on basis of the Coulomb friction model as:

qt = (σin μin + σex μex ) W1

(13)

where μin and μex respectively represent the Coulomb friction coefficients on the inner and outer interfaces of the tape. σin and σex respectively denote the contact pressures on the corresponding interfaces inside the recovery section, which can be approximately evaluated by:

⎧ σin (u) = σ2, pre + σ1 (u) σ = σ2, pre ⎨ ⎩ ex

(14)

Combining Eqs. (2b), (5) and (10), (13), (14) and using Eq. (11) as the boundary condition at u = L , the tensile load in the tape inside the recovery section can be approximately expressed in a non-dimensional form as:

ξnQ¯ 2, pre ⎛ ξnQ¯ 2, pre ⎛ μex ⎞ μex ⎞ ⎤ 2μ sin2 α (u¯ − L¯ ) in cos α Q¯ 1 (u¯) = ⎡ − ⎜1 + ⎟ 2 α ⎢σ¯ + sin2 α ⎜1 + μ ⎟ ⎥ e sin μin ⎠ in ⎠ ⎦ ⎝ ⎝ ⎣ In which those non-dimensional parameters are defined as follows:

σ¯ = σ / σ1,max

ξ = (Q2,max W1)/(Q1,max W2) u¯ = u/ D

L¯ = L/ D

Q¯ 1 = Q1/ Q1,max

X¯1 = X1 / D

Q¯ 2, pre = Q2, pre / Q2,max

Consequently, Eq. (9) can be rewritten as: 294

(L¯ − X¯1 < u¯ ≤ L¯ ) (15)

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n≥⌈

sin2 α ⌉ ξ

(16)

In addition, the condition that no expansion generated in the repaired section σ2, pre ≥ σ can be rewritten as:

sin2 α Q¯ 2, pre ≥ σ¯ ξn

(17)

Applying the boundary condition Q¯ 1 = 0 at u¯ = L¯ − X¯1 in Eq. (15), the non-dimensional length of the recovery section close to the edges of the repaired section can be approximately determined by:

X¯1 =

cos α σ¯ sin2 α 1 ⎞ ln ⎛⎜1 + ⎟ 2μin sin2 α ⎝ ξnQ¯ 2, pre 1 + μex / μin ⎠

(18)

4.2. Condition where the repaired section expands (σ2, pre < σ ) Now consider the condition that the pre-tension in filaments is insufficient to impede the radial dilatation of tensile armors inside the repaired section, i.e., σ2, pre < σ . Unlike the previously discussed situation, the entire tape in the repaired section is tensioned due to the radial dilatation under this circumstance. Intuitively, the tensile load in the tape increases departing from the broken end due to the frictional effect, thus generating another recovery section close to the damaged region. If the repaired section is sufficiently long, the tensile load in the tape will approach a stable value after a certain length from the damaged region where no relative slide tendency is experienced between the tape and the adjacent layers. Thereafter, the tensile load in the tape continues to vary in the recovery section close to the edges of repaired section as previously described. The section between those two recovery sections where the tensile load in the tape remains constant is herein named stabilized section, see Fig. 9. On the contrary, if the repaired section is insufficiently long to cover these two recovery sections, the tape will be pulled outwardly from the repaired section. Considering the repaired section sufficiently long, and let the length of the recovery section close to the damaged region be denoted by X2 and the constant tensile load in the tape inside the stabilized section be represented by Q″1, the tensile load in the tape inside the repaired section can be approximately expressed as:

⎧ 0 ≤ Q1 < Q″1 (0 ≤ u < X2 ) Q1 = Q″1 (X2 ≤ u ≤ L − X1) ⎨ ⎩Q″1 < Q1 ≤ Q′1 (L − X1 < u ≤ L)

(19)

4.2.1. Mechanical analysis in the stabilized section Since the tensile load in the tape remains constant in the stabilized section, the generated radial constraint is uniform, thus uniform radial dilatation will be observed in this section. Let ΔD denote the radial dilatation in the stabilized section, which is considered small relative to the pipe dimension. Since there is no relative slide between the tape and the adjacent layers in the stabilized section, from the geometrical relationship shown in Fig. 8, the tape tensile strain inside this section can be expressed in terms of the radial dilatation by:

Δε =

πΔD sin α ΔD = sin2 α πD /sin α D

(20)

Assuming the tape is linear elastic with axial stiffness set to κ1, the generated tensile load in the tape inside the stabilized section can be approximately given by:

Fig. 8. Geometrical relationship between radial dilatation and tensile strain in the tape. 295

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Fig. 9. Simplified schematic of superimposed tape and filaments over an expanded cylinder.

Q″1 = κ1 Δε = κ1 sin2 α

ΔD D

(21)

Consequently, the corresponding contact pressure on the inner interface of the tape in the stabilized section generated by the tape tensile load can be obtained by substituting Eq. (21) into Eq. (5) as:

σ1 =

2κ1 sin4 α ΔD D 2W1

(22)

Similarly, assuming the filaments are linear elastic with axial stiffness equal to κ2 , the tensile load in the filaments in the stabilized section can be approximately expressed in terms of the radial dilatation as:

Q2 = Q2, pre + κ2

ΔD D

(23)

The corresponding contact pressure on the inner interface of the filamentous layers generated by the tensile load in the filaments can be evaluated by substituting Eq. (23) into Eq. (7) as:

σ2 =

2n ΔD ⎞ ⎛Q2, pre + κ2 DW2 ⎝ D ⎠

(24)

Since the radial constraint provided by the external polymeric layer is deemed negligible, the following equilibrium equation should hold: (25)

σ = σ1 + σ2 Combining Eqs. (22), (24) and (25), the radial dilatation in the stabilized section can be approximately evaluated by:

nQ2, pre ⎞ ΔD σD W1 W2 =⎛ − D W2 ⎠ nκ2 W1 + κ1 W2 sin4 α ⎝ 2 ⎜

⎟

(26)

Substituting Eq. (26) into Eq. (21), the constant tensile load in the tape inside the stabilized section can be approximately determined as:

nQ2, pre ⎞ σD κ1 W2 W1 sin2 α − Q″1 = ⎛ 4 2 W nκ 2 ⎠ 2 W1 + κ1 W2 sin α ⎝ ⎜

⎟

(27)

Besides, substituting Eq. (26) into Eqs. (23) and (24), the tensile load in the filaments in the stabilized section and the corresponding contact pressure on their inner interface can be obtained respectively as:

nQ2, pre ⎞ σD κ2 W1 W2 Q2 = Q2, pre + ⎛ − 2 W nκ W + κ1 W2 sin4 α 2 2 1 ⎝ ⎠ ⎜

σ2 =

2nQ2, pre

⎟

2nQ2, pre ⎞ nκ2 W1 + ⎛σ − DW2 ⎠ nκ2 W1 + κ1 W2 sin4 α ⎝ ⎜

DW2

(28)

⎟

(29)

4.2.2. Mechanical analysis in the recovery sections It is conceivable that when the repaired section expands outwardly, non-uniform radial dilatation will be generated in the damaged section and the recovery sections. For the sake of simplicity, herein the radial dilatation in the entire repaired section is assumed uniform and equal to the radial dilatation evaluated by Eq. (26). Consequently, this problem is simplified as superimposed tape and filaments over a uniformly expanded frictional cylindrical surface with one tape end relaxed and the other tape end tensioned as depicted in Fig. 9. Hence, the contact pressures on the inner and outer interfaces of the tape inside the recovery sections in the expanded condition 296

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can be approximately evaluated by:

⎧ σin (u) = σ2 + σ1 (u) σ = σ2 ⎨ ⎩ ex

(30)

Combining Eqs. (2b), (5) and (13), (29), (30) and using Eq. (11) as the boundary condition at u = L , the tensile load in the tape inside the recovery section close to the edges of the repaired section in the expanded condition can be approximately determined in a non-dimensional form as:

μex ⎞ μex ⎞ ⎤ 2μ sin2 α (u¯ − L¯ ) ξnQ¯ 2 ⎛ ξnQ¯ 2 ⎛ ⎡ 1 − (L¯ − X¯1 < u¯ ≤ L¯ ) Q¯ 1 (u¯) = ⎢σ¯ + ⎜1 + ⎟ e in cos α ⎜1 + ⎟ 2 α 2 α ⎥ sin sin μin ⎠ μ in ⎠ ⎦ ⎝ ⎝ ⎣

(31)

where Q¯ 2 is a non-dimensional parameter given by:

Q2 σ¯ κ¯ = Q¯ 2, pre + ⎛⎜ sin2 α − nQ¯ 2, pre ⎞⎟ Q2,max ξ nκ + ¯ sin4 α ⎝ ⎠

Q¯ 2 =

(32)

In which κ¯ = (κ2 W1)/(κ1 W2) is a non-dimensional parameter corresponding to the stiffness ratio between a single filament and the tape per unit width. ′ Thereafter, using the boundary condition Q¯ 1 = Q¯ ′ 1 at u¯ = L¯ − X¯1, from Eq. (31) the length of the recovery section close to the edges of the repaired section in the expanded condition can be determined as: ξnQ¯

X¯1 =

In which

Q¯ ″1 =

( (

μ

) ⎤⎥ ) ⎥⎥⎦

⎡ σ¯ + 2 2 1 + ex μin sin α cos α ln ⎢ 2μin sin2 α ⎢ Q¯ ′ ′ + ξnQ¯ 2 1 + μex ⎢ 1 sin2 α μin ⎣ ′ Q¯ ′ 1

(33)

is a non-dimensional parameter defined by:

ξnQ¯ 2, pre ⎞ sin4 α Q″1 = ⎜⎛σ¯ − ⎟ sin2 α ⎠ nκ¯ + sin4 α Q1,max ⎝

(34)

On the other hand, combining Eqs. (2b), (5) and (13), (29), (30) and using the boundary condition Q1 = 0 at u = 0 , the tensile load in the tape inside the recovery section close to the damaged region in the expanded condition can be approximately determined in a non-dimensional form as:

μex ⎞ 2μ sin2 α u¯ ξnQ¯ 2 ⎛ ⎛ − 1⎞ (0 ≤ u¯ ≤ X¯2 ) Q¯ 1 (u¯) = ⎜1 + ⎟ e in cos α sin2 α ⎝ μin ⎠ ⎝ ⎠

(35)

In which X¯2 = X2 / D is a non-dimensional parameter corresponding to the length of the recovery section close to the damaged ′ region. Using the boundary condition Q¯ 1 = Q¯ 1′ at u¯ = X¯2 , from Eq. (35) X¯2 can be identified as:

X¯2 =

′ cos α Q¯ ′ sin2 α 1 ⎛ ⎞ ln ⎜1 + 1 2 2μin sin α Q¯ 2 nξ 1 + μex / μin ⎟ ⎝ ⎠

(36)

Thus, once the filamentous pre-tension is insufficient to impede the radial dilatation in the repaired section, to prevent the tape sliding outwardly from the repaired section, the repaired section surpassing both sides of the damaged region needs to be at least larger than the sum of these two recovery sections, i.e.:

L¯ > X¯1 + X¯2 =

cos α σ¯ sin2 α 1 ⎞⎟ ln ⎛⎜1 + 2 2μin sin α ⎝ ξnQ¯ 2 1 + μex / μin ⎠

(37)

It needs to be noted that the pre-tension in the filaments may decrease in the long term due to temperature variation, material viscoelasticity and tribological effects. Even though a large pre-tension is applied in the filaments in the initial state, which properly impedes the radial dilatation in the repaired section, its long-term effects may decrease such that the dilatation may still be triggered in the repaired section. Thus, the expanded condition described in section 4.2 may be more commonly encountered in engineering application. In the following section, the effect of the filamentous pre-tension on the mechanical behavior of the tape will be investigated through a series of case studies and the potential pre-tension reduction effect of the filaments on the recovery lengths will be discussed. 5. Results and discussion 5.1. Condition where expansion is impeded in the repaired section (σ2, pre ≥ σ ) Consider a tape wound over a cylindrical surface with lay angle equal to 80°, the frictional coefficients on the inner and outer interfaces of the tape with the adjacent layers are constant and equal to 0.1. Assume the ultimate strengths per unit width of the tape 297

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Fig. 10. Tape tensile load variations with different σ¯ in the unexpanded condition.

and the filament are identical, i.e., ξ = 1. According to Eq. (16), the ultimate constraint capacity can be regained in the damaged region by deploying only one layer of the filament. To investigate the mechanical behavior of the tape inside the repaired section when the dilatation is impeded, herein five layers of filaments with the pre-tension parameter Q¯ 2, pre = 0.2 are applied over the damaged section surpassing L¯ = 0.4 on both sides. From Eq. (17) it can be seen that no dilatation is generated in the repaired section for 0 ≤ σ¯ < 1 under this circumstance. According to Eq. (15), Fig. 10 shows the tensile loads variations in the tape inside the repaired section for σ¯ = 0.2, 0.4, 0.6, 0.8. It can be clearly seen that in the unexpanded condition the recovery length is proportional to the tensile armors’ expansion force. For instance, in those four cases, the increase of σ¯ leads the recovery length respectively to reach X¯1 = 0.08, 0.16, 0.23, 0.29. Besides, Fig. 11 shows the recovery length given in Eq. (18) as a function of σ¯ for different tape lay angles α = 70°, 75°, 80°, 85°, indicating that for a certain σ¯ larger recovery section is generated for the tape with a smaller lay angle. Thus, once the pre-tension in the filaments is determined, a longer repaired section is demanded for a damaged tape with a smaller lay angle. According to the model previously discussed, the number of the filamentous layers and the pre-tension in the filaments play pivotal roles in the recovery length. Fig. 12 shows plots of the tensile load in the tape inside the recovery section given in Eq. (15) as a function of u¯ for four cases with n × Q¯ 2, pre = 1, 2, 3, 4 when σ¯ = 0.8, evidencing that increasing either the number of the filamentous layers or the pre-tension in the filaments can significantly increase the tensile load reduction rate, thus effectively reducing the recovery length. As increasing the number of filamentous layers entails more material and time consumptions, to reduce the recovery length, it is hence preferable to deploy fewer filamentous layers with larger pre-tension. Fig. 13 shows plots of the recovery length given in Eq. (18) in terms of Q¯ 2, pre for different values of σ¯ = 0.2, 0.4, 0.6, 0.8 when five layers of filaments are deployed. Whence, it can be found that the reduction rate of the recovery length decreases as the pre-tension increases and the recovery length gently reduces when the

Fig. 11. Effect of the tape lay angle on the recovery length in the unexpanded condition. 298

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Fig. 12. Tape tensile load variations with different n × Q¯ 2, pre in the unexpanded condition.

Fig. 13. Effect of the pre-tension in the filaments on the recovery length in the unexpanded condition.

pre-tension is larger than a certain value. For instance, increasing the pre-tension parameter Q¯ 2, pre from 0.2 to 0.5 leads the recovery lengths to reduce approximately 60% in all those four cases. However, increasing the pre-tension parameter Q¯ 2, pre from 0.2 to 0.8 gives birth to approximately 70% reduction of the recovery length for those four cases, only 10% in addition, demonstrating that the effect of the pre-tension on reducing the recovery length declines as the pre-tension increases. 5.2. Condition where the repaired section expands (σ2, pre < σ ) Now consider the situation that the pre-tension in the filaments is insufficient to prevent the tensile armors expansion in the repaired section. In this case study, two filamentous layers with the pre-tension parameter Q¯ 2, pre = 0.2 are applied over the damaged region surpassing L¯ = 1.0 on both sides, and the lay angle of the tape is set to 80°. The ultimate strengths and the stiffness of the filament and tape per unit width are both assumed identical, i.e., ξ = κ¯ = 1. According to Eqs. (15), (31) and (35), Fig. 14 shows the tensile load variations in the tape inside the repaired section for σ¯ = 0.2, 0.4, 0.6, 0.8, indicating that while the tensile armors expansion is not capable to overcome the radial constraint induced by the pre-tension in filaments for the cases σ¯ = 0.2, 0.4 , radial dilatations are generated in the repaired section when σ¯ = 0.6, 0.8. As previously discussed, once the repaired section expands, a recovery section close to the damaged region will be generated, as depicted in the cases σ¯ = 0.6, 0.8 in Fig. 14. It can be noted that the slope of the tensile load variation in the tape inside both the recovery sections increases with rising σ¯ . This is due to the increase of frictional restriction induced by the rising contact pressures on the interfaces of the tape with neighboring layers. According to Eqs. (18), (33) and (36), Fig. 15 shows plots of the recovery lengths in terms of σ¯ for different tape lay angles α = 70°, 75°, 80°, 85°, where the solid and dotted lines respectively represent X¯1 and X¯2 . The sum of the recovery lengths X¯1 and X¯2 299

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Fig. 14. Tape tensile load variations with different σ¯ in both unexpanded and expanded conditions.

Fig. 15. Recovery lengths variations in terms of σ¯ with different tape lay angles.

Fig. 16. Sum of the recovery lengths in terms of σ¯ with different tape lay angles. 300

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Fig. 17. Pre-tension reduction effect on the recovery lengths.

for above four cases are shown in Fig. 16. It is interesting to note that while the recovery length X¯1 increases with σ¯ when the radial dilatation is impeded in the repaired section, it decreases after the repaired section expands. This is due to the diminution of tension gradient in the recovery section close to the edges of the repaired section after the expansion, see Fig. 14. However, despite the recovery length X¯1 decreases with σ¯ when the repaired section expands, recovery length X¯2 is generated, which in conjunction results in the monotonical increase of the total recovery length in terms of σ¯ as depicted in Fig. 16. Moreover, from Fig. 16 it can also be noted that the increase rate of the sum of the recovery lengths reduces suddenly after the repaired section expands for all the four cases. 5.3. Pre-tension reduction effect of the filaments on the recovery lengths Note that the pre-tension in the filaments may decrease in the long term which may affect the effectiveness of this repair technique. To illustrate the potential pre-tension reduction effect on the repaired section, case studies are carried out by considering two layers of filaments with the pre-tension parameter Q¯ 2, pre decreases from the initial value 0.8 to 0.0. According to Eqs. (18), (33) and (36), Fig. 17 shows the variations of the recovery lengths X¯1 and X¯2 with decreasing pre-tension in the filaments for different tape lay angles α = 70°, 75°, 80°, 85°, and the sums of these two recovery lengths are depicted in Fig. 18. From Figs. 17 and 18 it is found that despite the fact that the recovery length X¯1 decreases as the pre-tension decreases when the repaired section expands, the sum of the recovery lengths increases monotonically with decreasing pre-tension in both expanded and unexpanded conditions. For instance, in those four cases, the sums of recovery lengths increase respectively 164%, 158%, 154% and 152% when the pre-tension parameter Q¯ 2, pre reduces from 0.8 to 0.0. Thus, it is essential to consider the pre-tension reduction effect of

Fig. 18. Pre-tension reduction effect on the sum of the recovery lengths. 301

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Fig. 19. Effect of the filamentous stiffness on the recovery lengths.

the filaments when calculating the minimum repair length exceeding the damaged region on both sides. Since the sum of the recovery lengths increases monotonically with decreasing pre-tension, to ensure the operational security in the long term, the repaired section should be sufficiently long to cover the recovery sections even in the extreme condition that the pre-tension reduces to zero, i.e., the filaments are loosely wound over the pipe. Under this circumstance, any longitudinal compressive load will promptly generate radial dilatation in the repaired section. Using Q¯ 2, pre = 0 , Eq. (37) reduces to:

L¯ > X¯1 + X¯2 =

cos α sin4 α ⎞ 1 ⎤ ln ⎡1 + ⎛1 + 2 ⎢ 2μin sin α ⎣ nκ¯ ⎠ 1 + μex / μin ⎥ ⎝ ⎦ ⎜

⎟

(38)

Considering μex = μin , it can be noted that the demanded minimum length covered by the filaments surpassing the damaged region on both sides is inversely proportional to the frictional coefficient. According to Eq. (38), for a certain damaged flexible pipe, once the number of filamentous layers is determined, the sum of recovery lengths is principally governed by the non-dimensional parameter κ¯ , which corresponds to the stiffness of filaments. To illustrate the effect of the stiffness of filaments on the repaired section when the filaments are loosely wound over the pipe, the variations of the recovery lengths X¯1 and X¯2 in terms of κ¯ for different tape lay angles α = 70°, 75°, 80°, 85° are shown in Fig. 19 using Eqs. (33) and (36), and the sum of these two recovery lengths are depicted in Fig. 20. Whence, it is found that while the recovery length X¯1 increases with κ¯ , the recovery length X¯2 decreases with κ¯ , which in conjunction results in the monotonical reduction of the sum of recovery lengths with κ¯ . This corresponds to state that, to reduce the necessary length of the section covered by filaments considering the pre-tension reduction effect of the filaments, it is recommended to deploy the filaments with relatively larger stiffness. For instance, when κ¯ = 0.2 , the sums of the recovery lengths X¯1 + X¯2 are respectively 1.75, 1.32, 0.88, 0.44, however, when κ¯ = 0.6, the sums of the recovery lengths become 1.16, 0.86, 0.57, 0.28 and the decrease rate for these four cases is approximately 35%. Nevertheless, it should be noted that the effect of the stiffness of

Fig. 20. Effect of the filamentous stiffness on the sum of the recovery lengths. 302

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the filaments on reducing the necessary length of the section covered by the filaments declines when the filamentous stiffness is larger than a certain value. For instance, when κ¯ increases from 0.2 to 1.0, the reduction rate of the sums of the recovery lengths for those four cases is less than 45%, which are only less than 10% in addition compared with the cases when κ¯ increases from 0.2 to 0.6. 6. Conclusions In this work, a repair methodology for the damaged anti-birdcage tape in flexible pipes is envisaged by winding multiple layers of pre-tensioned filaments over the damaged area and its neighboring regions. An analytical model is formulated to investigate the mechanism of this repair methodology. The results indicate that to fully reestablish the radial constraint capacity in the damaged region an adequate number of filamentous layers is demanded. Besides, to impede the tape sliding outwardly from the repaired section when axial compressive loads are sustained, the repaired section surpassing the damaged region on both sides needs to be sufficiently long, the minimum necessary length of which principally depends on the pre-tension in the filaments. Moreover, the pretension reduction effect of the filaments on the repaired section is discussed through case studies, indicating that it has an essential impact on the recovery lengths and should thus be carefully considered when calculating the demanded length of the section covered by filaments. Based on the discussion above, some practical recommendations are presented as following: (1) It is better to apply the filaments with high pre-tension, even though its long-term effectiveness may decrease since the pretension may reduce due to temperature variation, material viscoelasticity and tribological effects; (2) As the expanded condition may be commonly encountered due to the pre-tension reduction effect of the filaments, to ensure the structural integrity in the long term, it is recommended to adopt the filaments with large stiffness; (3) To impede the tape sliding outwardly from the repaired section, the length of the repaired section surpassing the damaged region on both sides should be sufficiently long to cover the recovery sections even in the extreme condition that the maximum axial compressive load is sustained and the pre-tension in the filaments decreases to zero. It needs to be emphasized that the presented analytical solution is based on the assumption that the flexible pipe is wound by only one anti-birdcage tape with no overlap. However, in some situations, 2 or 4 superimposed high strength tapes are applied with partial overlap at each turn. In future studies, the analytical model will be updated to take multiple anti-birdcage tapes into account. In addition, the experimental verification is also planned to be performed nearly. Acknowledgments The authors acknowledge the support from Petrobras, the National Council for Scientific and Technological Development (CNPq) and the Brazilian Coordination for the Improvement of Higher Education Personnel (CAPES) for this work. References [1] Brack M, Troina LMB, Sousa JRM. Flexible riser resistance against combined axial compression, bending, and torsion in ultra-deep water depths. Proceedings of the 24th international conference on offshore mechanics and arctic engineering, vol. 1. 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