A general model to predict torsion and curvature increments of tensile armors in unbonded flexible pipes

Marine Structures 67 (2019) 102632

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A general model to predict torsion and curvature increments of tensile armors in unbonded flexible pipes

T

Leilei Donga, Zixin Qua, Qi Zhanga,∗, Yi Huanga,b, Gang Liua,b a b

School of Naval Architecture, Dalian University of Technology, Dalian, China State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian, China

A R T IC LE I N F O

ABS TRA CT

Keywords: Unbonded flexible pipe Tensile armor Local torsion and bending General formulation

An analytical model is proposed to investigate the local torsion and bending behavior of tensile armors in unbonded flexible pipes. The general expressions of torsion and curvature increments, including the effects of axial strain and twisting rotation of tendon cross-section, are firstly derived based on the generalized Frenet–Serret equations. Then the expressions corresponding to the strained helix assumption are obtained, which are further simplified by ignoring the twisting rotation. Afterwards, explicit expressions are given for axisymmetric case and tendon stick and slip states when bending is included. The developed model is finally verified with a finite element simulation. The results show that the model prediction of normal curvature increment is in good agreement with the numerical solution while the changes in torsion and transverse curvature cannot be obtained accurately through simple analytical formulas. Several earlier models are also discussed briefly and are demonstrated to be special cases of the present general model.

1. Introduction Unbonded flexible pipes have been widely used as the offshore industry advances into deeper waters due to their ability to withstand large bending curvatures without damage. Tensile armor layers are largely responsible for such great flexibility of the pipe, because the armoring tendons can slide against surrounding layers. So many authors put substantial effort into slip behavior modelling of flexible pipes [1–12], based on the assumption that the slip path of the tendon follows either the strained helix (also known as the loxodromic path) or a geodesic curve. Tendon slip releases part of the axial strain only, and the tendon still undergoes local torsion and bending as the pipe is loaded. These local deformations have a significant effect on the fatigue performance of dynamic applications of flexible pipes, such as a deep-water flexible riser. On one hand, local torsion and bending of individual tendons contribute to the elastic post-slip bending stiffness of the pipe to a considerable extent, sometimes even larger than the stiffness provided by the polymeric layers [10]. On the other hand, longitudinal stresses are induced by local bending about the tendon strong axis (the transverse curvature stress) and bending about the tendon weak axis (the normal curvature stress). Considering the slope parameter in the S–N curve, a small change in stress can have a great impact on the resulting fatigue life. Therefore, it is important to calculate the curvature increments as accurately as possible, especially for cases that the fatigue life is dominated by local bending stresses. The prediction of torsion and curvature increments of helical elements has been studied extensively for structures incorporating helical layers, such as flexible pipes, umbilicals, cables and wire ropes. Several analytical models are available in the literature [2,3,8,9,13–20]. However, the obtained expressions show significant differences, even for the simple axisymmetric case. Tang et al. ∗

Corresponding author. E-mail address: [email protected] (Q. Zhang).

https://doi.org/10.1016/j.marstruc.2019.102632 Received 30 January 2019; Received in revised form 4 April 2019; Accepted 16 May 2019 0951-8339/ © 2019 Elsevier Ltd. All rights reserved.

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[21] compared the analytical results of seven models with finite element (FE) solutions, and recommended using the spring theory based model. However, it is noticed that friction is not considered in the FE simulation which thus cannot represent adequately the physical characteristics. Recently Komperød [22,23] employed numerical techniques to calculate the changes in torsion and curvature components. The results disclosed some properties that the analytical approximations used for comparison failed to capture. In this paper, a general analytical model is presented and verified. The word “general” can be interpreted in two ways: (a) this model is generally applicable to both axisymmetric and bending cases; and (b) this model can be reduced to several earlier models through different simplifications.

2. Analytical model 2.1. Assumptions and definitions The main assumptions used throughout the analysis are: 1. The tendon is treated as a thin rod with its center-line a helix in the unloaded state. 2. The tendon has a uniform rectangular cross-section. The change in its thickness, i.e. the Poisson effect, is ignored since the tendon stiffness is relatively high. 3. The pipe is subjected to axisymmetric deformations and uniform bending which take place far from terminations so that the end restraint effect can be ignored. 4. The bend curvature experienced by longitudinal lines on the cylindrical surface is the same as that of the pipe center-line. The error induced by this assumption is very small (see Dong et al. [24]), as the pipe radius is typically much smaller than the bend radius. 5. The tendon slip follows the strained helix path. This assumption has been validated by recent experimental tests [25,26]. 6. All strains are small and within the elastic range of material behavior. The basic definitions are in line with those used by Dong et al. [24,27]. Fig. 1 shows the geometry for an infinitesimal helix element ds of lay angle α 0 wound on a cylindrical surface of length dsp subjected to tension, torsion and bending. Here s and sp are the arc lengths along the tendon and the pipe center-line, respectively, in the undeformed state. As the pipe is loaded, the lay angle of the armor wire must change slightly, with a small angle β deviated from the initial lay angle. According to Assumption 5, β is actually the angle between the original and strained helices, which can be determined as [24]:

β = εp sin α 0 cos α 0 +

u2 sin α 0 cos α 0 − Rτp cos2 α 0 R

(1)

where u2 and R are the radial displacement and the helical radius of the wire, respectively; τp is the pipe torsion and is defined as the change of the pipe twisting angle, ϕp , per unit length of the pipe center-line, i.e. τp = dϕp / dsp ; εp is the varying strain of longitudinal lines on the supporting surface and has two components:

εp = εTEN + εBEN = εTEN − κR cos θ

(2)

here, εTEN and εBEN are the strains caused by tension and bending, respectively; κ is the curvature imposed to the pipe; θ is the wire polar angle measured as shown in Fig. 2. It should be noted that no distinction is made in Eq. (2) between the radius of the tendon center-line and the external radius of the underlying layer considering the fact that the tendon is relatively thin. The supporting surface can also be taken as a virtual cylindrical surface that supports the tendon center-line as if the center-line is on this surface but can slide along the strained helix path. The supporting surface undergoes a displacement in the lateral direction along the x3 axis (see Fig. 2) due to the surface strain, which is the integral of β :

Fig. 1. Element of original and strained helix. 2

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Fig. 2. Principal axes orientation of layer and tendon cross-section.

Y (s ) =

∫0

s

βds

(3)

where s is measured from the fixed end. As the tendon is assumed to slide only along its axial direction, i.e. no transverse slip occurs, Y also represents the lateral displacement of the tendon. 2.2. Change in torsion and curvature components 2.2.1. Moving reference frames The expressions of torsion and curvature increments, considering the effects of axial strain and tendon twisting rotation, were derived independently by Chen [28] and Dong et al. [29] based on the generalized Frenet–Serret equations. However, the reference frames which are important to the derivation, were not defined clearly. This issue was noted and addressed by Dong et al. [24], but still one of the moving frames was established incorrectly. The armor wire is naturally curved in the unstressed state. The tangent at a point on the center-line and the principal axes of the cross-section issue from this point form a triad of orthogonal axes of x10 , x 20 , x 30 , with the x10 axis oriented along the tangent. In the deformed state, a local orthonormal coordinate system with base vectors t , n and b is attached to each point on the strained centerline. As shown in Fig. 3, t is directed along the tangent of the center-line, while n is oriented along the inward normal of the supporting surface, and b is determined by the right-hand rule. The system formed by these vectors is a Darboux–Ribaucour frame if the tendon center-line is considered on the virtual supporting surface. However, Dong et al. [24] misdefined it as a Frenet–Serrete frame, i.e. directed the vector n to the center-line normal rather than the inward surface normal. The base vectors t , n and b define the center-line only, and do not hold any information about the orientation of the tendon cross-section. Therefore, a new triad is introduced at the same point, defined by the coordinate axes x i and associated base vectors Gi (i = 1, 2, 3). The orientation is such that G1 is coincident with t , while G2 points towards the supporting surface but is fixed to the strong principal axis of the tendon crosssection, and G3 is directed to complete the right-hand frame. The coordinate axes x1, x2 , x3 constructed as above for any point on the strained center-line represent the “principal torsion–flexure axes” of the tendon at the point [30]. From the definition of the two frames it is seen that the Gi triad coincides with the {t, n, b} triad only if the direction of the strong principal axis is parallel to the surface normal. However, this is generally not the case due to the twisting rotation of the cross-section. The angle φ is thus introduced

Fig. 3. Definition of coordinate systems in deformed state. 3

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to define the orientation of Gi system relative to the {t, n, b} system, as shown in Fig. 3. 2.2.2. General expressions The derivation of torsion and curvature increments presented in Ref. [29] is summarized briefly here for completeness. The relationship between the two frames shown in Fig. 3 is:

0 0 ⎤ t ⎧ G1 ⎫ ⎡ 1 ⎧ ⎫ G2 = ⎢ 0 cos φ sin φ ⎥ n ⎨ ⎬ ⎢ ⎨b ⎬ ⎥ ⎩ G3 ⎭ ⎣ 0 − sin φ cos φ ⎦ ⎩ ⎭

(4)

As the base vectors are mutually orthogonal, the following equation holds for the Gi system [31]:

G1 0 κ3 − κ2 ⎤ ⎧ G1 ⎫ d ⎧ ⎫ ⎡ κ1 ⎥ G2 G2 = ⎢− κ3 0 ds ⎨ ⎬ ⎢ ⎥⎨ ⎬ ⎩ G3 ⎭ ⎣ κ2 − κ1 0 ⎦ ⎩ G3 ⎭

(5)

κi = κi0 + Δκi (i = 1, 2, 3)

(6)

where

κi0

here, and Δκi are the initial values and increments of torsion and curvature components of the tendon, respectively. According to the generalized Frenet–Serret formula, an equation similar to Eq. (5) can be written for the {t, n, b} frame, which is:

κb − κn ⎤ t 0 d ⎧t ⎫ ⎡ ⎧ ⎫ n = ⎢− κb 0 κt ⎥ n ⎨ ⎬ ds¯ ⎨ b ⎬ ⎢ ⎥ − κ κ 0 ⎩ ⎭ ⎣ n t ⎦⎩b ⎭

(7)

where s¯ is the arc length along the tendon center-line in the deformed state; κt is the geometric torsion of the deformed tendon centerline corresponding to the {t, n, b} frame; κn and κb are the associated curvature components. Writing the last equality of Eq. (7) and considering Eq. (4) results in:

db = κn t − κt n = κn G1 − κt (G2 cos φ − G3 sin φ) ds‾

(8)

Considering Eq. (4) again and that the axial strain in the tendon is ε1 =

d s¯ − d s , ds

the left-hand side of Eq. (8) can be rewritten as:

db db d ≈ (1 − ε1) = (1 − ε1) (G2 sin φ + G3 cos φ) ds¯ ds ds

(9)

Substituting Eq. (6) into the latter two equalities of Eq. (5) then into Eq. (9) and ignoring the smaller terms gives:

(κ 0 − κ 0 ε1 + Δκ2)cos φ ⎞ dφ ⎞ db 0 ⎛ 0 (G2 cos φ − G3 sin φ) = ⎜⎛ 2 0 2 0 ⎟ G 1 − (κ1 − κ1 ε1 + Δκ1) − ds ⎠ ds¯ ⎝ ⎝− (κ3 − κ3 ε1 + Δκ3)sin φ ⎠

(10)

Comparing the coefficients of the same terms on the right-hand side of Eq. (8) and Eq. (10) yields:

κt = −

dφ + (κ10 − κ10 ε1 + Δκ1) ds

(11a)

κn = (κ 20 − κ 20 ε1 + Δκ2)cos φ − (κ30 − κ30 ε1 + Δκ3)sin φ

(11b)

κb can be obtained using a similar procedure as: κb = (κ 20 − κ 20 ε1 + Δκ2)sin φ + (κ30 − κ30 ε1 + Δκ3)cos φ

(11c)

Noticing the fact that

κ10 = κt0, κ 20 = κn0, κ30 = κb0

(12)

in the initial state and solving Δκi from Eq. (11a-c) leads to:

Δκ1 =

dφ + κt − κt0 (1 − ε1) ds

(13a)

Δκ2 = κn cos φ + κb sin φ − κn0 (1 − ε1) Δκ3 = −κn sin φ + κb cos φ −

κb0 (1

(13b)

− ε1)

(13c)

Eq. (13a-c) hold for both axisymmetric and bending cases, because no loading conditions are involved in the derivation. It is shown that the torsion and curvature increments of the tendon depend not only on the geometric torsion and curvature of its center-line but also on the twisting rotation of its cross-section relative to the center-line. The definition of curvature is clear, but the term torsion is used in an inconsistent way in literature, sometimes used interchangeably with the term twist. To illustrate this point, 4

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the axial strain effect is excluded provisionally, then Eq. (11a) becomes:

κt = −

dφ + κ1 ds

(14a)

or can be rearranged as:

κ1 =

dφ + κt ds

(14b)

Love [30] found an expression similar to Eq. (14b), i.e. τ = df / ds1 + 1/ Σ , and termed τ twist and 1/ Σ tortuosity to avoid confusion. The counterparts of τ and 1/ Σ in Love's expression are κ1 and κt in Eq. (14b), respectively. Following Love Podvratnik [32] also called 1/ Σ tortuosity, but referred to τ as torsion. The other component of torsion, dψ/ ds , was called internal twist, i.e. τ = dψ/ ds + 1/ Σ , where ψ is the third Euler angle (see Refs. [30,32] for details). Brokaw [33] gave an expression similar to Eq. (14a), which is κt = dθ / ds + κ1 using the present notation for his variables τ and κ z , where θ is the angle between the curvature vector and G2 . κt and κ1 were termed torsion and twist, respectively, and the relation between them was described as “twist is torsion, but not all torsion is twist”. Other terms used for κt in literature include: (geometric) torsion [34] and mathematical torsion [35], and for κ1: (physical) twist [34], physical torsion [35] and twist density [36]. In this paper, κt is referred to as the geometric torsion, as it is a geometric property of the tendon center-line; κ1 is called the (physical) torsion, because it is a physical measure of the rotation of the body coordinate system Gi ; φ is termed the twisting angle which describes the material twisting of the tendon about its center-line; dφ / ds is referred to as the (internal) twist which represents the rate of change of the twisting angle φ . 2.2.3. Explicit expressions The twisting angle φ is very small in practical cases, so it is reasonable to fix the strong principal axis of the tendon cross-section to the surface normal, which means that the vectors G2 and n are now coincident, i.e. φ = 0 . This restraint has been shown to be fulfilled when the tendon is exposed to reasonably high stresses, and the fact that one layer is restrained by the next layer should make this assumption reasonable even if compression occurs [15]. Following this assumption, Eq. (13a-c) are reduced to:

Δκ1 = κt − κt0 (1 − ε1) = Δκt + κt0 ε1

(15a)

Δκ2 = κn − κn0 (1 − ε1) = Δκn + κn0 ε1

(15b)

Δκ3 = κb − κb0 (1 − ε1) = Δκb + κb0 ε1

(15c)

The axial strain ε1 is also very small compared with unity, and therefore is not taken into account in most analytical models. Here it is retained in the present model, as it will be shown in the following that whether the axial strain effect is included or not is a key factor to explain the differences among several sets of expressions. Now consider the geometric torsion and curvature components in the {t, n, b} frame, involved in Eq. (15a-c), before and after deformations. Their initial values are given by the well-known formulas:

κt0 =

sin α 0 cos α 0 sin2 α 0 , κn0 = 0, κb0 = R R

(16)

In the deformed state, the lay angle and radius of the tendon center-line change to α 0 + β and R − u2 , respectively, which induces the variations in its geometric torsion and curvature. Besides, the global curvature imposed to the pipe makes another contribution to the variations, which can be obtained by examining the degree of alignment between the global curvature vector and the base vectors of the {t, n, b} frame. Therefore, the total changes in geometric torsion and curvature components caused by these two effects are:

Δκt =

sin(α 0 + β )cos(α 0 + β ) sin α 0 cos α 0 − + κ sin(α 0 + β )cos(α 0 + β )cos θ R − u2 R

Δκn = −

Δκb =

dβ − κ cos(α 0 + β )sin θ ds

sin2 (α

0 + β) − R − u2

sin2 α R

0

(17a) (17b)

− κ cos2 (α 0 + β )cos θ

(17c)

In Eq. (17b), the first term is introduced to account for the fact that the tendon would experience transverse curvature if it follows a line other than the geodesic path. Using trigonometric transformation formulas and assuming that β is much smaller than α 0 , Eqs. (17a)–(17c) are simplified to:

Δκt = −

cos 2α 0 sin α 0 cos α 0 u2 β+ + κ sin α 0 cos α 0 cos θ R R R

(18a)

Δκn = −β′ − κ cos α 0 sin θ Δκb = −

(18b)

sin 2α 0 sin2 α 0 u2 β+ − κ cos2 α 0 cos θ R R R

(18c) 5

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where the prime denotes the derivative with respect to the arc length s . Substituting Eq. (16) and (18a-c) into Eq. (15a-c) and considering Eq. (3) results in:

Δκ1 = −

cos 2α 0 sin α 0 cos α 0 u ⎛fASE ε1 + 2 ⎞ + κ sin α 0 cos α 0 cos θ Y′ + R R R⎠ ⎝

(19b)

Δκ2 = −Y ′′ − κ cos α 0 sin θ Δκ3 = −

(19a)

sin 2α 0 sin2 α 0 u ⎛f ε1 + 2 ⎞ − κ cos2 α 0 cos θ Y′ + R R ⎝ ASE R⎠

(19c)

where fASE is the flag variable indicating whether the axial strain effect is introduced:

1 if ASE is considered fASE = ⎧ ⎨ ⎩ 0 if ASE is not included

(20)

It is seen from Eqs. (19a) and (19c) that the change in torsion and normal curvature is caused by two effects. The first is due to the displacement of the tendon in axial, radial and lateral directions, reflected by the terms relating to ε1, u2 and Y ′, respectively. Because of the strained helix assumption, the lateral displacement is governed by the straining of the supporting surface, which produces a component of strain induced displacement in the lateral direction. The axial displacement is controlled by the surface straining only if the tendon is in stick state, as the strain induced displacement also has a component in the axial direction. The other effect is due to the fact that the applied curvature can be decomposed into the relevant components given by the last terms in Eqs. (19a) and (19c). Eq. (19b) shows that only the lateral displacement contributes to the transverse curvature in addition to the global curvature. The axial strain in the tendon, ε1, involved in Eqs. (19a)–(19c), consists of two components: (21)

ε1 = εa + εb where εa is caused by axisymmetric loads and is given by Refs. [15,17]:

εa = εTEN cos2 α 0 −

u2 sin2 α 0 + Rτp sin α 0 cos α 0 R

(22)

and εb is induced by bending loads, whose magnitude depends on the tendon state. Before gross slip initiates in the tendon, following the usually-made assumption that plane sections remain plane, εb is expressed as:

εb = ε fstk = −κR cos2 α 0 cos θ

(23)

On the other hand, if the tendon is in full-slip state, its axial strain is totally controlled by the available friction, which is determined as [8]:

εb = ε fslp =

f0 R π ⎛θ − ⎞ EA sin α 0 ⎝ 2⎠

(24)

where E is Young's modulus of the tendon material; A is the cross-sectional area of the tendon; f0 is the friction force per unit length and is presented as [11]:

f0 = μi Ta κ30 + (μi + μo ) po w

(25)

here, μi and μo are the friction coefficients on the internal and external surfaces of the tendon, respectively; Ta = EAεa , is the axial tension induced by axisymmetric loads; po is the contact pressure on the external surface; w is the tendon width. Dong et al. [11] proposed rigorous formulations to describe the gross slip initiation and progression in the tendon. The total tension caused by both axisymmetric and bending loads, instead of Ta only, is used to determine the friction force, and the continuity of the axial tension is considered in their model. The obtained axial tension gives the axial strain in slip state as follows:

εb = ε fslp =

T0 μi sin α0 θ Tp − Ta e + EA EA

(26)

where

Tp = −

R (μ + μo ) po w μi sin2 α 0 i

(27)

and

T0 =

μi π sin α 0 (Ta − Tp) e μi π sin α0 − 1

(28)

It is worth pointing out that Eqs. (24) and (26) are valid for the interval [0 π], and should be adapted slightly to use for the tendon part within the interval [π 2π]. The tendon strain in partial-slip state can be obtained in a straightforward manner from the continuity condition of the axial tension (see Sævik [8] and Dong et al. [11] for further details). Eqs. (19a)–(19c) are derived directly from Eqs. (15a)–(15c) which are obtained from Eqs. (13a)–(13c) by ignoring the twisting rotation, and are therefore applicable to both axisymmetric and bending cases. Substituting Eqs. (3) and (21) into Eqs. (19a)–(19c) 6

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and letting κ = 0 and εb = 0 gives the torsion and curvature increments due to axisymmetric loads as:

Δκ1 = f1t εTEN + f1u

u2 + f1τ τp R

(29a) (29b)

Δκ2 = 0

u2 + f 3τ τp R

(29c)

sin α 0 cos α 0 (fASE cos2 α 0 − cos 2α 0) R

(30a)

Δκ3 = f3t εTEN + f 3u where

f1t =

f1u = (2 − fASE )

sin3 α 0 cos α 0 R

(30b)

f1τ = cos2 α 0 (fASE sin2 α 0 + cos 2α 0 ) f3t = (fASE − 2) f 3u = −

f 3τ

sin2 α

0

R

(30c)

sin2 α 0 cos2 α 0 R

(30d)

(fASE sin2 α 0 + cos 2α 0)

= sin α 0 cos α 0 (fASE

sin2 α

0

+2

cos2 α

(30e) 0)

(30f)

The tendon may stick to or slide against the underlying surface when bending load is included. Substituting Eqs. (3) and (21) into Eqs. (19a)–(19c) again, and considering different axial strains in the tendon for different states, the following expressions can be obtained. In stick state:

Δκ1 = f1t εTEN + f1u

u2 + f1τ τp + f1stk κ cos θ R

(31a)

Δκ2 = f2stk κ sin θ

Δκ3 = f3t εTEN + f 3u

(31b)

u2 + f 3τ τp + f3stk κ cos θ R

(31c)

where

f1stk = (2 − fASE )sin α 0 cos3 α 0

(32a)

f2stk = −(1 + sin2 α 0)cos α 0

(32b)

f3stk = −cos2 α 0 (fASE sin2 α 0 + cos 2α 0)

(32c)

In full-slip state:

Δκ1 = f1t εTEN + f1u

u2 + f1τ τp + f1slp κ cos θ + fASE κt0 ε fslp R

(33a)

Δκ2 = f2slp κ sin θ

Δκ3 = f3t εTEN + f 3u

(33b)

u2 + f 3τ τp + f3slp κ cos θ + fASE κb0 ε fslp R

(33c)

where

f1slp = 2 sin α 0 cos3 α 0

(34a)

f2slp = −(1 + sin2 α 0)cos α 0 = f2stk

(34b)

f3slp = −cos2 α 0 cos 2α 0

(34c)

It is noted that the transverse curvature is unaffected by the axial strain effect, which can be explained simply through Eq. (19b). 2.3. Special cases The increments of torsion and curvature components for special cases derived from the present model would yield the expressions obtained by earlier models when appropriate value is assigned to the flag variable fASE or appropriate terms are disregarded. 7

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2.3.1. Axisymmetric case Model 1: Sævik [8,15], Sævik and Li [20]. Substituting fASE = 1 into Eqs. (30a)–(30f) and then back into Eqs. (29a) and (29c) results in:

Δκ1 =

sin3 α 0 cos α 0 sin3 α 0 cos α 0 u2 εTEN + + cos4 α 0 τp R R R

Δκ3 = −

sin2 α

0

cos2 α

0

R

εTEN −

sin2 α

0

cos2 α

R

0

(35a)

u2 + (2 sin α 0 cos3 α 0 + sin3 α 0 cos α 0) τp R

(35b)

Model 2: Kebadze [17], Pesce et al. [19], Skeie et al. [9]. Substituting fASE = 0 into Eq. (30a-f) and then back into Eqs. (29a) and (29c) leads to:

Δκ1 = −

sin 4α 0 2 sin3 α 0 cos α 0 u2 εTEN + + cos2 α 0 cos 2α 0 τp 4R R R

(36a)

Δκ3 = −

2 sin2 α 0 cos2 α 0 sin2 α 0 cos 2α 0 u2 εTEN − + 2 sin α 0 cos3 α 0 τp R R R

(36b)

It can be seen clearly that the differences between these two sets of expressions are due to that the axial strain effect is included in Model 1 but not in Model 2. For Model 2, the increments of torsion and normal curvature are simply due to the changes in lay angle and radius of the tendon center-line (see Eqs. (15a), (15c), (17a) and (17c)). 2.3.2. Uniform bending In order to determine the torsion and curvature increments caused entirely by bending, no axisymmetric loads are considered, in which case the related terms in Eqs. (31a), (31c), (33a) and (33c) vanish. The geodesic based models will be not discussed here following the strained helix assumption. Model 1: LeClair and Costello [13]. Substituting Y ′ = Y ′′ = 0 into Eq. (19a-c) and excluding the axial strain effect yields:

Δκ1 = κ sin α 0 cos α 0 cos θ

(37a)

Δκ2 = −κ cos α 0 sin θ

(37b)

Δκ3 = −κ cos2 α 0 cos θ

(37c)

These expressions are purely geometric, as they simply represent the projections of the global curvature on the axes associated with the {t, n, b} frame. Also by projecting the applied curvature vector on the three principal axes, Witz and Tan [2], however, obtained different expressions, with cos α 0 omitted in Eq. (37a-c). These authors might have ignored the fact that the wire at an angle to the pipe axis cannot experience as much curvature as the longitudinal line parallel to the pipe center-line does. Model 2: Sathikh [14], Huang and Vinogradov [16]. Substituting Y ′ = 0 into Eqs. (19a) and (19c) and leaving out the axial strain effect, retaining Y ′′ but dropping the other term related to global curvature projection in Eq. (19b) gives:

Δκ1 = κ sin α 0 cos α 0 cos θ

(38a)

Δκ2 = −κ sin2 α 0 cos α 0 sin θ

(38b)

Δκ3 = −κ cos2 α 0 cos θ

(38c)

It is revealed by Eq. (19b) that the transverse curvature is caused by the combined effects of global curvature projection and strain induced displacement in the lateral direction. However, the above two models each cover one aspect only, with Model 1 including the contribution of curvature projection and Model 2 considering that of lateral displacement. Model 3: Sævik [3,8,15], Sathikh et al. [18], Skeie et al. [9]. Substituting fASE = 1 into Eqs. (32a) and (32c) then substituting Eq. (32a-c) respectively into Eq. (31a-c) leads to:

Δκ1 = κ sin α 0 cos3 α 0 cos θ

(39a)

Δκ2 = −κ (1 + sin2 α 0)cos α 0 sin θ

(39b)

Δκ3 = −κ cos4 α 0 cos θ

(39c)

It is shown that, strictly speaking, Eq. (39a-c) can be used only for the tendon in stick state, as what these expressions give are the changes in torsion and curvature of a strained helix glued totally to the supporting surface. Model 4: Pesce et al. [19], Sævik and Li [20]. Substituting Eq. (34a-c) into Eq. (33a-c), respectively, and substituting fASE = 0 into Eqs. (33a) and (33c) yields:

Δκ1 = 2κ sin α 0 cos3 α 0 cos θ

(40a)

Δκ2 = −κ (1 + sin2 α 0)cos α 0 sin θ

(40b) 8

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Table 1 Model parameters for verification. Parameters

Values and Units

Lay angle (α 0 ) Helical radius (R ) Width of tendon (w ) Thickness of tendon (t ) Young's Modulus (E ) Poisson's Ratio (ν ) Contact pressure ( po )

38 deg 0.254 m 12 mm 4 mm 210 GPa 0.30 2.55 MPa 180 m

Bending radius ( ρ = 1/ κ )

Δκ3 = −κ cos2 α 0 cos 2α 0 cos θ

(40c)

It can be seen that Eq. (40a-c) are valid only for the full-slip tendons and the axial strain effect is not taken into account. In practical cases, while lateral slip of the tendon will be restricted by friction, axial slip is unavoidable due to its significant axial stiffness, which will modify the increments of torsion and normal curvature [20]. It is clear now the reason axial slip changes torsion and curvature of the tendon is that slip alters the axial strain in the tendon. The transverse curvature is irrelevant to the axial strain, as noted earlier. 3. FE verification It has been demonstrated that the present model can be reduced to Eq. (35a-b) for axisymmetric case and to Eq. (40a-c) for uniform bending. These expressions have been verified by Sævik and Li [20] using FE simulations performed by the computer programs BFlex and UFlex3D for axisymmetric case and BFlex for bending. It is also known that the axial strain effect is considered in Eq. (35a-b) but not in Eqs. (40a) and (40c). Therefore, only the simplified version of Eqs. (33a) and (33c), i.e. omitting the terms due to axisymmetric loads, needs to be further verified. A 16-inch oil export jumper, used in Ref. [37], is selected for this purpose. The tendon characteristics and loading conditions are shown in Table 1. 3.1. Descriptions of the FE model The FE simulation is performed using the software package ANSYS Mechanical APDL v17.0 [38]. The method of creating the FE model is similar to that used in Ref. [27]. The BEAM189 element is selected to model the tendon, and the SHELL181 element is

Fig. 4. FE mesh of tendon and supporting layer. 9

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Fig. 5. Radial constraint imposed using MPC184 rigid beam element.

employed to mesh the cylindrical tube which represents the supporting layer. The model covers a total length of four tendon pitches, using in total 400 beam elements and 24000 quadrilateral-shaped shell elements. The FE mesh is shown in Fig. 4. The contact interface is created between the tendon and the supporting layer to simulate their interaction. The tendon is designated as the contact surface and the layer as the target surface. The contact pair, comprised of the contact and target surfaces, is built by the 3-D line-to-surface contact element, CONTA177, together with the 3-D target element, TARGE170. The normal contact stiffness, defined by the real constant FKN, is the most important parameter in a contact analysis. A suitable tradeoff is necessary between accuracy and convergence behavior. In this verification, FKN is set to 0.003 based on a sensibility analysis, and the resulting contact stiffness is 2.10 × 109 kN/m3. The augmented Lagrangian algorithm is selected as it usually leads to better conditioning of the global stiffness matrix and is less sensitive to the magnitude of the contact stiffness. An underlying assumption of the analytical models is that the helix radius remains unchanged during bending. Therefore, the radial constraint is imposed to satisfy this assumption, and also to prevent the tube from ovalizing. A series of 401 reference nodes is arranged equally spaced along the pipe axis, and the MPC184 rigid beam element is used to connect the shell nodes on the circumference to the corresponding reference node on the axis, as shown in Fig. 5. The middle reference node is fully constrained to avoid rigid body motion. The end nodes of the beam are fixed against all degrees of freedom (DOFs) relative to the shell. According to Ref. [37], the fixed condition has no influence on the axial stress in the tendon close to the end when it starts at the intrados or extrados points. Considering that the axial strain affects the torsion and normal curvature, the tendon is set to start at the extrados point (θ = π ) to minimize the end effect. Two load steps are specified in the analysis. The external pressure is applied in the first one, and prescribed translational and rotational DOFs are defined at all reference nodes in the second to ensure constant curvature along the pipe axis. The solution is converged after 14 cumulative iterations. The CPU and wall time spent on the solution process, i.e. excluding the pre- and post-processing time, is around 130 s and 80 s, respectively, on a standard desktop computer (Intel Core i7-8700 CPU @ 3.20 GHz).

3.2. Solutions and comparisons The comparison of the torsion increment is shown in Fig. 6. Eq. (26) is employed in the analytical calculation and the use of Eq. (24) gives very close result which is not given in the figure though. The most distinct observation in Fig. 6 is the large difference in amplitude between the numerical solution and the analytical prediction, with the former giving an amplitude approximately 3.5 times as large as the latter. This result is in accordance with Komperød's work [23]. In his study, the author compared the numerically calculated torsion increment with those obtained from two analytical methods presented by Skeie et al. [9] and Kebadze [17], respectively, i.e. Eq. (39a) and the modified version of Eq. (37a) following Witz and Tan's model [2]. The amplitude of the numerical result was found to be roughly twice as large as those the analytical models obtain. The discrepancy was attributed to the oversimplification, s = X1 /cos α 0 , in the parameterization of the helical element path, which is strictly valid only for a straight cable 10

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Fig. 6. Torsion increment of tendon.

(κ = 0 ). This study reveals that omitting the twisting rotation in the analytical model is another reason for the difference. The torsion increment obtained from Sævik and Li's model [20], i.e. Eq. (40a), is also shown in Fig. 6 for comparison. The discrepancy between the predictions of this model and the present model seems not that great due to the presence of the numerical solution. In fact, the maximum relative error in amplitude between these two models, occurring at the extrados point, reaches about 30.166%. Fig. 7 shows the comparison of the change in normal curvature. With the axial strain effect considered, the analytical result of the present model agrees well with the ANSYS output. Sævik and Li's model, however, underestimates the amplitude to a significant extent. Another observation is that, unlike Sævik and Li's model, the present model produces the result curve which differs somewhat from a harmonic shape. The normal curvature increment at the extrados point is slightly larger than that at the intrados point, as the bending strain increases the normal curvature at the tensile side but decreases it at the compressive side (see Eq. (33c)). It is expected from Eqs. (33a) and (33c) that the effect of axial strain on torsion and normal curvature would become smaller as the global curvature increases. In the verification performed by Sævik and Li [20], a pre-stress of 300 MPa was prescribed in the tendon prior to global bending and the curvature was increased from zero to 0.10 m-1, so the axial strain effect was not disclosed. The torsion increment comparison shown in Fig. 6 implies that the twisting rotation can also have a large effect on the change in transverse curvature. Although Eq. (40b) has been verified by BFlex simulations in Ref. [20], it is necessary to verify this expression using a general FE software such as ANSYS, as BFlex is a tailor-made program and some kinematic restraints were introduced in the element development [15]. Fig. 8 shows the comparison of the transverse curvature increment. Compared with the FE solution, the present analytical model, also representing the stick model (Model 3) and the full-slip model (Model 4) in Section 2.3.2, is seen to overestimate the change in transverse curvature significantly. This is primarily due to the strained helix assumption and the simplification related to the twisting rotation. Assuming the tendon follows the strained helix path means no slip occurs in the lateral direction. However, a small amount of transverse slip is allowed in ANSYS, which eliminates the transverse curvature to some degree. sin2 α

The effect of the twisting rotation is even more pronounced. An additional term, R 0 φ , would emerge in Eq. (19b) if the twisting rotation is included. For the case at hand, a twisting angle as small as 0.154 deg would eliminate the difference in amplitude between

Fig. 7. Normal curvature increment of tendon. 11

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Fig. 8. Transverse curvature increment of tendon.

the analytical and numerical results. It is seen from Figs. 7–8 that the amplitude of the transverse curvature increment is greater than that of the change in normal curvature, even using the numerical results for comparison. Actually, from Eq. (32b-c) and (34b-c) it can be concluded that the ratios of f2stk / f3stk and f2slp / f3slp in absolute value are always larger than unity. For a typical lay angle of 35 deg, f2stk / f3stk and f2slp / f3slp are about 2.418 and 4.744, respectively. The amplitude ratio in full-slip state would be smaller than f2slp / f3slp if the axial strain effect is taken into account, but it is believed that this ratio would be still greater than unity in practical applications. Further considering the fact that the width of the tendon is always larger than its thickness, the local bending stress induced by transverse curvature is thought to be relatively more important than that caused by normal curvature change. The shear stress due to tendon torsion is of little concern from the point of view of fatigue analysis, but should be included in the von Mises stress calculation.

4. Case study It has been clarified in Section 2.3.1 that the axial strain effect is responsible for the difference between the two axisymmetric models. Figs. 9–10 show the ratios of coefficients given by Eq. (30a-f) without axial strain effect to their corresponding values including this effect with respect to lay angle. For example, f1t ratio shown in Fig. 9 refers to the ratio f1t (fASE = 0)/ f1t (fASE = 1) . It is desired that these coefficient ratios are as close to unity as possible, but instead they deviate significantly from unity for most of the practical lay angle range, which highlights the necessity of introducing the axial strain effect. Considering that the armoring tendons are slender and thin in flexible pipes, the changes in torsion and normal curvature and associated stresses from axisymmetric loads are small and normally ignored. However, these stresses should be taken into account for umbilicals or new flexible pipe designs where the width and thickness of helical elements become significant [20]. It is recommended in these applications that the model including the axial strain effect should be used. The comparisons of the change in torsion and curvature components obtained from different bending models are shown in

Fig. 9. Coefficient ratios against lay angle for torsion increment. 12

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Fig. 10. Coefficient ratios against lay angle for normal curvature increment.

Fig. 11. Bending model comparison for torsion increment.

Fig. 12. Bending model comparison for transverse curvature increment.

Figs. 11–13 where the tendon properties and loading conditions listed in Table 1 are used. In each figure, the total solution of the present model is broken into its two components, i.e. the displacement induced contribution and the global curvature projection, which may also represent other model(s), as indicated in the legend. It is seen that the curvature projection gives rise to the greatest 13

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Fig. 13. Bending model comparison for normal curvature increment.

part of the torsion increment while the displacement induced torsion is so small that it can be ignored for this particular case. For the transverse curvature, the curvature projection still plays the major role, but the tendon displacement also makes a good contribution. In fact, the ratio between these two components is 1/sin2 α 0 (see Eqs. (37b) and (38b)), implying that both of them should be retained. As for the normal curvature, the two components are opposite in sign and are both greater than the total solution in absolute value, yielding finally a total solution in phase with the curvature projection. This means that keeping the curvature projection only, as Model 1 and Model 2 do, will always overestimate the normal curvature increment considerably, as shown in Fig. 13. It is noted in Figs. 11 and 13 that the result curve of the present model lies between those of the stick and full-slip models. This can be explained easily by Eqs. (33a) and (33c). The full-slip model assumes free axial motion [20], i.e. ε fslp = 0 , while ε fstk is used in the stick model. Actually, the axial strain in full-slip state does exist but is smaller than ε fstk due to tendon slip, i.e. 0 < ε fslp < ε fstk . The present model is equivalent to the stick model in stick state and tends to “converge” to the full-slip model as the applied curvature increases. This implies that the full-slip model will not introduce significant errors for large curvatures. However, the critical curvature beyond which the full-slip model can produce satisfactory results is case dependent and is dominated by friction. For the current pure bending case, the maximum relative error related to the change in normal curvature falls below 5% (4.8%) until the global curvature is increased to 0.08 1/m. The associated maximum error of torsion increment is approximately 1.636%. 5. Discussion It is visualized that the tendon would experience no transverse curvature, i.e. f2stk and f2slp , given by Eqs. (32b) and (34b), respectively, should be zero if its lay angle is 0 or π/2 . However, this physical requirement is satisfied only for α 0 = π/2 but not for α 0 = 0 . It is mainly due to the fact that the process of linearization involved in the derivation cannot produce good approximations around α 0 = 0 . Fig. 14 shows the relative error caused by linearizing the function sin2 (α 0 + β ) when simplifying Eq. (17c) to Eq. (18c), which illustrates clearly the problem to be anticipated in the vicinity of α 0 = 0 . In practical applications, the lay angle used is far from

Fig. 14. Relative error caused by linearization (β = −0.5 deg). 14

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0, and therefore this problem is of little importance. The bending induced axial strain of the tendon in stick state (Eq. (23)) is obtained from the simple beam theory assuming that plane sections remain plane before gross slip begins. However, this might be not the case if the tendon is supported by a thick plastic layer in which shear deformation occurs. If so, the shear strain between the tendon and the underlying layer is governed by the shear deformations and the shear modulus of the plastic material, and can be determined using the shear interaction model [11,12,20]. If the shear deformation effect is considered, Eq. (32a-c) become:

f cos2 α 0 − cos 2α 0 ⎞ f1stk = sin α 0 cos α 0 ⎛⎜1 − ASE ⎟ fk ⎠ ⎝ f2stk = −⎜⎛1 + ⎝

(41a)

sin2 α 0 ⎞ ⎟ cos α 0 fk ⎠

(41b)

(2 − fASE )sin2 α 0 ⎞ f3stk = −cos2 α 0 ⎜⎛1 − ⎟ fk ⎠ ⎝

(41c)

and Eq. (34a-c) change to:

f1slp = sin α 0 cos α 0 ⎛⎜1 + ⎝ f2slp = −⎜⎛1 + ⎝

cos 2α 0 ⎞ ⎟ fk ⎠

(42a)

sin2 α 0 ⎞ stk ⎟ cos α 0 = f 2 fk ⎠

f3slp = −cos2 α 0 ⎜⎛1 − ⎝

(42b)

2 sin2 α 0 ⎞ ⎟ fk ⎠

(42c)

where fk is the parameter characterizing the shear interaction behavior and approaches unity gradually as the shear stiffness increases towards infinite (see Refs. [11,12,20] for details). In previous shear models, the effect of shear deformations was focused on the pre-slip behavior of the tendon in terms of critical curvature at which gross slip initiates but was not considered in the determination of the torsion and curvature increments [11,12,20]. However, Eqs. (41a) and (42c) show that shear deformations in the plastic layer also affect the tendon's local deformations, as they affect both the axial strain and lateral displacement of the tendon. Although the bend curvature imposed to the longitudinal lines on the supporting surface is assumed to be the same as that of the pipe center-line (Assumption 4), the present model can be further refined by removing this assumption. This is achieved by replacing the global curvature, κ , involved in the last terms of Eq. (19a-c) with the linearized local applied curvature, κ (1 + κR cos θ) . 6. Concluding remarks This paper presented an analytical model to predict the torsion and curvature increments of tensile armors in unbonded flexible pipes under tensile, torsional and bending loads. The general expressions were derived on the basis of the generalized Frenet–Serret equations, and were further simplified by ignoring the tendon twisting rotation, which finally resulted in the explicit expressions for different cases. The model avoids getting too much use of differential geometry involved in the derivation, and the physical interpretation is clear. The FE verification showed that the model prediction of normal curvature increment was quite satisfactory while the changes in torsion and transverse curvature were significantly underestimated and overestimated, respectively. The torsion discrepancy is caused by some oversimplification which is strictly valid only for straight pipes and the exclusion of twisting rotation in the analytical model. The latter is also a main reason for the transverse curvature difference between the analytical and numerical results. The differences in the obtained expressions among several earlier models were clarified. For certain sets of expressions, the differences lie in that some models consider the axial strain effect but other models do not. The applicability of these models was also discussed. The reason why the expression of transverse curvature violates the physical requirement for α 0 = 0 was explained, which is that the linearization of trigonometric functions cannot provide good approximations around this extreme lay angle. The proposed model was also refined by adding the effect of shear deformations in the underlying plastic layer. Another possible refinement would be to use the local applied curvature instead of the global curvature for projection. Funding This work was supported by the National Key R&D Program of China [grant number 2018YFC0310502]. References [1] Feret JJ, Bournazel CL. Calculation of stresses and slip in structural layers of unbonded flexible pipes. J Offshore Mech Arct Eng 1987;109(3):263–9. [2] Witz JA, Tan Z. On the flexural structural behaviour of flexible pipes, umbilicals and marine cables. Mar Struct 1992;5(2–3):229–49.

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