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Analytical model for tensile armors lateral deflections and buckling in flexible pipes
Marine Structures 64 (2019) 211–228
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Analytical model for tensile armors lateral deflections and buckling in flexible pipes
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 Tensile armor wires Lateral buckling Perturbation
The present paper addresses the mechanical behavior of tensile armor wires in a flexible pipe subjected to bending and longitudinal loads. An analytical model is formulated to evaluate the equilibrium state of a single armor wire on a frictionless torus surface by solving a consistent system of six nonlinear differential equations through the application of a perturbation technique. Cases studies are performed where the armor wire's responses to both tensile and compressive loads are discussed and compared with the results obtained from a numerical program. Under the circumstance of axial compression, two potential lateral buckling modes, global and periodic, are uncovered and their corresponding buckling limits are identified by establishing eigenvalue problems. While the global buckling limits are found conservative compared with the test data available in the literature, good agreements are observed for the estimated buckling limits regarding the periodic buckling mode.
1. Introduction Flexible pipes are tubular composite structures widely used in the offshore oil and gas field developments. Fig. 1 shows the crosssection of a typical unbonded flexible pipe, which is basically composed of metallic layers (carcass, zeta layer, and tensile armors), polymer layers and high strength tapes. The tensile armors, which conventionally consist of an even number of layers of helically wound steel wires, are the key functional components to resist the gravity loads and cyclic bending induced by the ocean environment. In the last decades, the decrease in the available number of oil and gas reservoirs in shallow water has driven the offshore industry to move into deeper waters and harsher environments, posing technical challenges for the safe operations of flexible pipes. For deepwater flexible risers, significant dynamic tensile loading may be experienced, which raises the concerns of fatigue failure of tensile armor wires. Besides, significant longitudinal compressive loads may also be experienced during flexible pipes installation and operation at large water depths due to the hydrostatic end-cap effect [1], which may promote the tensile armor wires to buckle in either radial or lateral direction, especially when the armor annulus is flooded. While studies have shown that the armor wires buckling in the radial direction, also known as birdcage, to some extent can be prevented by the application of high strength tapes over the tensile armors [2], the risks of lateral buckling still remains [3–5]. Thus, to guarantee the safe operation of flexible pipes in deepwater applications, reliable models are required to predict the tensile armor wires mechanical behavior when combined bending and longitudinal loads, tensile or compressive, are experienced. The mechanics of armor wires in flexible pipes have been investigated by many authors throughout the past few decades. Ref. [6] ∗
Corresponding author. E-mail address: [email protected] (M.A. Vaz).
https://doi.org/10.1016/j.marstruc.2018.11.003 Received 16 August 2018; Accepted 5 November 2018 0951-8339/ © 2018 Elsevier Ltd. All rights reserved.
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Fig. 1. Typical unbonded flexible pipe.
presented a theoretical approach to evaluate the tensile armors radial and axial deformations of a straight flexible pipe on basis of the constant lay angle. Subsequently, assuming the armor wires seek an equilibrium path corresponding to the shortest lines on the torus surface, i.e., the geodesic curve, an analytical approximation of the armor wires deflections on an uniformly bent torus was derived by Ref. [7]. Nevertheless, in reality, the armor wires may not be able to reach the geodesic paths. The study of Ref. [8] demonstrated that, due to the frictional resistance, the armor wires slippages are primarily axial when the flexible pipes are subjected to tension and cyclic bending as proposed by Ref. [9]. Assuming the armor wires transverse deflections are negligible, also known as the loxodromic curve, finite element formulations were developed by Ref. [10] for the prediction of the armor wires slips and stresses. Also, assuming that the armor wires can slide only along their own axes, the slip initiation and progression of the armor wires subjected to bending and tension have been investigated by Ref. [11]. Nevertheless, the over-constrained transverse displacements may lead to overestimated stresses in the armor wires. Also, the pre-defined configurations regarding the armor wires equilibrium paths, geodesic or loxodromic, prevent these models from being applicable for the lateral buckling analysis. Without pre-assumed configurations, more general models were proposed by Refs. [12–17] to investigate the armor wires deflections in longitudinal loads associated with repeated bending, where the armor wires are free to slip into any configuration between loxodromic and geodesic depending on friction, thus are applicable for lateral buckling analysis. However, large computational times are needed. Besides, in an effort to identify the lateral buckling limit, Ref. [18] proposed an analytical prediction based on a predefined harmonic deflection mode without considering the frictional resistance and bending effect. Similarly, with the absence of frictional resistance and bending, an analytical model was developed by Ref. [19] for the prediction of the armor wire's deflections on a cylindrical surface without predefined kinematic configurations. Nevertheless, the lateral buckling limits predicted by both models lie on the conservative side since the frictional effect is disregarded. In addition, Ref. [20] developed an analytical model to access the stability limit of flexible pipes under axisymmetric loading on the basis of perturbation methods by establishing an eigenvalue problem. Moreover, Ref. [21], developed an analytical model for simulating the buckling and post-buckling behavior of armor wires based on a total strain energy approach, nevertheless, with no detail presented. Besides, some FE models were developed to explore the physics of this failure mode [1,22–25], however, all include simplifications, thus not representing the effects of cyclic bending and frictional resistance. More recently, an empirical model was proposed by Ref. [26] using symbolic regression to evaluate the lateral buckling limit of armor wire, again, without considering the bending and frictional effects. In the present study, the mechanical behavior of an armor wire in a flexible pipe will be investigated without predefined configurations such as geodesic or loxodromic curves. For the sake of simplicity, the radius of the armor wire is assumed constant, the twist of armor wire is assumed geometrically governed by the underlying torus surface, the frictional resistance is disregarded and the lateral contacts between adjacent armor wires are not considered. On this basis, an analytical model is formulated which is capable of capturing the armor wire's equilibrium state on a uniformly bent frictionless torus in both tension and compression. Through which, two lateral buckling modes, global and periodic, are revealed and the corresponding buckling limits are identified. This is followed by case studies comparing with experimental results available in the literature. 2. Differential equations system A single armor wire is modeled first as a curved beam with a rectangular cross-section helically wound over a cylinder with radius r bent to a torus of constant radius R = 1/ κ , see Fig. 2. The torus surface is parameterized by an arc length coordinate u along the 212
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Fig. 2. Wire geometry and coordinate system.
torus centerline and a circumferential angular coordinate θ. The armor wire's central line is represented by a directed curve parameterized by the arc length parameter s and the angle between the armor wire's tangent and its projection in the longitudinal direction is denoted by ϕ. Assuming that the armor wire's cross-sections remain plane and retain their orientation relative to the → → n , b ⎞⎟ are defined on the armor wire representing the centerline, a right-handed and orthonormal triad of local unit vectors ⎛⎜ t , → ⎝ ⎠ tangential, normal and transverse directions respectively. Using the equilibrium equations of a curved beam in the tangent plane and applying the concepts of curved beam differential geometry, the equilibrium state of an armor wire on a torus is defined by Ref. [12] through a consistent system of six ordinary nonlinear differential equations, which are summarized here:
cos ϕ du = ds 1 + κr cos θ
(1a)
sin ϕ dθ = ds r
(1b)
dϕ κ sin θ cos ϕ + κ g =− ds 1 + κr cos θ
(1c)
dPt = κn Pn − κ g Pb − pt ds
(1d)
dPb = κ g Pt − τPn − pb ds
(1e)
dMn = −κn Mt + τMb + Pb ds
(1f)
In which, κ g , κn and τ respectively denote the wire's geodesic curvature, normal curvature, and torsion, Mt , Mn and Mb respectively denote the sectional moment components in wire's tangential, normal and transverse directions, Pt , Pn , Pb are respectively the sectional forces in the armor wire's tangential, normal and transverse directions. pt and pb respectively represent the distributed external loads in the armor wire's tangential and transverse directions, which are taken as zero since the frictional effect is disregarded in the present study. pn is the squeezing force impeding the armor wires to move radially, which can be obtained from the force equilibrium equation in the normal direction of a curved beam on a torus surface as:
pn = −
dPn + κn Pt + τPb ds
(2)
Moreover, using the moment equilibrium equation in the binormal direction of a curved beam on a torus surface Pn can be given by:
Pn = −
dMb + κ g Mt − τMn ds
(3)
Considering the armor wire is not allowed to freely rotate due to the constraint induced by the neighboring layers, herein it is assumed that the armor wire's twist around its local tangent is solely governed by the underlying torus surface. In other words, the normal vector on the armor wire is considered to coincide with the surface normal. Hence, based on Kirchhoff's theory for thin rods 213
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and the Darboux frame, the armor wire's normal curvature and torsion on a torus surface can be derived respectively as:
κ cos θ cos2ϕ sin2ϕ − 1 + κr cos θ r
(4a)
κ cos θ 1 τ=⎛ − ⎞ sin ϕ cos ϕ r⎠ ⎝ 1 + κr cos θ
(4b)
κn = −
In the initial state, when no longitudinal loads nor bending are applied, the armor wire is assumed stress-free and laid as a helix over a cylinder with a constant lay angle ϕini . Substituting κ = 0 and ϕ = ϕini into Eqs. (1c) and (4a), (4b), the geometrical curvature, normal curvature, and torsion in the initial state, respectively denoted by κ gini , κnini and τ ini , can be obtained as:
κ gini = 0
(5a)
κnini = −
sin2ϕini r
(5b)
τ ini = −
sin 2ϕini 2r
(5c)
Besides, considering the armor wire's dimensions small relative to the minor and major torus radii, it is fair to assume that the sectional moment components are linearly proportional to the change of curvatures with respect to curvature components in the initial un-deformed state, thus:
Mt = GJ (τ − τ ini )
(6a)
Mn = EIn (κ g − κ gini )
(6b)
Mb = EIb (κn − κnini )
(6c)
where E and G are respectively the elastic and shear moduli of the armor wire, In and Ib are respectively the inertia moments of the armor wire's cross-section in normal and transverse directions. Considering the cross-section of armor wire as a narrow rectangular, the stiffness of torsion in the armor wire's tangential direction can be approximately evaluated by Ref. [27]:
1 64 b J = ab3 ⎛ − 5 ⎞ π a⎠ ⎝3
(7)
Note that the previous equations are derived as functions of the armor wire's arc length in the deformed state. Considering that the armor wire is linearly elastic and its axial strain ε is small, the aforementioned system of differential equations in terms of the unknown deformed arc length parameter s can be converted into a system in terms of the initial un-deformed arc length parameter s ini through Cauchy's definition of strain:
ds P =1+ε=1+ t ds ini EA
(8)
where A = a × b (width × thickness) is the cross-sectional area of the armor wire. Consequently, the system of differential equations, Eqs. (1a)–(1f), can be solved as a boundary value problem of sixth order, where u, θ, ϕ, Pt , Pb , Mn are taken as unknowns, and the remaining terms κ g , κn , τ, Mt , Mb , Pn , pn can be respectively expressed in terms of these six unknowns by Eqs. (6b) and (4a), (4b), (6a), (6c), (3), (2). Considering that the flexible pipe is twist balanced subjected to longitudinal tensile or compressive loads when no buckling is formed in the armor layers, the boundary conditions corresponding to the torsional fixed pipe ends are applied:
u (0) = 0
u (S ) = Lini (1 + ΔL¯ )
θ (0) = θini (0)
θ (S ) = θini (0) +
(9a)
sin ϕini ini S r
(9b)
ϕ (0) = ϕ (S ) = ϕini
(9c)
S ini
θini (0)
In which, and S are respectively the armor wire's total arc length in the initial and deformed states, corresponds to the armor wire's circumferential position at s = 0 , Lini = S ini cos ϕini is the pipe length in the initial state and ΔL¯ denotes the longitudinal strain which is defined by the longitudinal elongation divided by the original pipe length. 3. Analytical treatments Applying a perturbation technique, Ref. [19] have proposed an analytical solution to the aforementioned differential equations system, Eqs. (1a)–(1f), in an axisymmetric case, i.e., for κ = 0 . Nevertheless, the introduction of the pipe bending significantly raises the nonlinearity of the differential equations system, thus posing great challenges for the analytical treatment. Note that, in practical applications of flexible pipes, the bending radius is generally much larger than the flexible pipe's diameter, 214
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thus it is reasonable to assume the ratio of the minor torus radius to the major torus radius as a small quantity, i.e., |κr| ≪ 1. Besides, note that the armor wire's deflections are very small when no buckling takes place, the change of armor wire's lay angles with respect to the wire's lay angles in the initial state, denoted by Δϕ (s ) , and the change of armor wire's circumferential positions with respect to that in the initial state, denoted by Δθ (s ) , both can be deemed as small quantities. On this basis, this complex nonlinear differential equations system might be therefore solved analytically through the application of a multi-parameter perturbation technique. Hence, the armor wire's lay angles ϕ (s ) and circumferential positions θ (s ) are herein expressed by adding small variations to their values in the initial un-deformed state as:
ϕ (s ) = ϕini + Δϕ (s )
(10a)
θ (s ) = θini (s ini ) + Δθ (s )
(10b)
where the armor wire's circumferential positions in the initial state θini can be obtained by integrating Eq. (1b) as:
θini (s ini ) = θini (0) +
sin ϕini ini s r
(11)
Ignoring the second and higher order small terms of Δϕ (s ) , Δθ (s ) and κr , the following approximations can then be used:
1 = 1 − κr cos θ 1 + κr cos θ
(12a)
ϕini
(12b)
cos ϕ (s ) = cos ϕini − Δϕ (s )sin ϕini
(12c)
sin ϕ (s ) = sin
ϕini
θini
(12d)
cos θ (s ) = cos θini − Δθ (s )sin θini
(12e)
sin θ (s ) = sin
θini
+ Δϕ (s )cos
+ Δθ (s )cos
Applying the approximations given by Eqs. (12a)–(12e) and ignoring the products between Δϕ (s ) , Δθ (s ) and κr , the armor wire's geodesic curvature, normal curvature, and torsion can then be approximately rewritten from Eqs. (1c), (4a), (4b) respectively as:
1 ⎡ d Δϕ r + κr cos ϕinisin θini⎤ r ⎣ ds ⎦
κ g (s ) =
(13a)
1 κn (s ) = − [sin2ϕini + sin 2 ϕini Δϕ (s ) + κr cos2ϕini cos θini] r
τ (s ) =
(13b)
sin 2ϕini 1 ⎡ sin 2 ϕini − − cos 2 ϕini Δϕ (s ) + κr cos θini⎤ ⎥ ⎢ r⎣ 2 2 ⎦
(13c)
Substitution of Eqs. (13a)–(13c) and Eqs. (5a)–(5c) into Eqs. (6a)–(6c) yields: ini
⎡−cos 2 ϕini Δϕ (s ) + κr sin 2 ϕ cos θini⎤ ⎥ ⎢ 2 ⎦ ⎣
Mt (s ) =
GJ r
Mb (s ) =
EIb [−sin 2 ϕini Δϕ (s ) − κr cos2ϕini cos θini] r
(14b)
Mn (s ) =
EIn ⎡ d Δϕ r + κr cos ϕinisin θini⎤ r ⎣ ds ⎦
(14c)
(14a)
Subsequently, substituting Eqs. (13a), (13c) and (14a)–(14c) into Eq. (3) and retaining only the first order small terms containing Δϕ (s ) and κr , the armor wire's sectional force in the normal direction can be approximately evaluated by:
Pn (s ) = (EIn + 2EIb )
sin 2ϕini d Δϕ κ + (EIn − EIb)sin ϕini cos2ϕinisin θini 2r ds r
(15)
Likewise, substituting Eqs. (13b), (13c) and (14a)–(14c) into Eq. (1f), rearranging and retaining only the first order small terms containing Δϕ (s ) and κr , the sectional force in the armor wire's transverse direction can be approximately given by:
Pb (s ) = EIn
sin2ϕini cos 2 ϕini sin 2 ϕini sin22ϕini d 2Δϕ (EIn − EIbcos2ϕini − GJ sin2ϕini )cos θini Δϕ (s ) + GJ Δϕ (s ) + κ − EIb 2 2 2 2r 2r r ds
(16)
Thereafter, substituting Eqs. (13a), (13b) and (15), (16) into Eq. (1d), integrating in terms of the arc length and retaining only the first order small terms containing Δϕ (s ) and κr , the sectional force in the armor wire's tangential direction can be approximately evaluated by:
Pt (s ) = Pt (0) −
sin22ϕini EIn − EIb EIn + 2EIb 3 ini [cos θini (s ini ) − cos θini (0)] sin ϕ cos ϕini Δϕ (s ) + κr 4 r2 r2 215
(17)
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The distributed external load in the armor wire's normal direction pn can hence be approximately evaluated by substituting Eqs. (13b), (13c) and (15)–(17) into Eq. (2) as:
pn = −
sin2ϕini Pt (0) r
(18)
Alternatively, substituting Eqs. (13a), (13c) and (15), (17) into Eq. (1e), integrating in terms of the arc length and retaining only the first order small terms containing Δϕ (s ) and κr , the sectional force in the armor wire's transverse direction can also be approximately given by:
Pb (s ) = Pb (0) + ⎡Pt (0) + ⎣ − κr cot ϕini ⎡Pt (0) + ⎣
EIn + 2EIb sin22ϕini ⎤ Δϕ (s ) 4 r2 ⎦
EIn − EIb sin22ϕini ⎤ [cos 4 r2 ⎦
θini (s ini ) − cos θini (0)]
(19)
Combining Eqs. (16) and (19), a second-order differential equation is obtained which governs the equilibrium state of an armor wire on a frictionless torus surface:
d 2Δϕ P (0) − P1 P (0) κr κr Δϕ (s ) = b [Pt (0) − P3]cot ϕini cos θini (0) + [P2 − Pt (0)]cot ϕini cos θini − t + ds 2 EIn EIn EIn EIn
(20)
In which P1, P2 , P3 are constants corresponding to the armor wire's properties:
P1 = −
sin2ϕini (EIn cos2ϕini + 4EIbcos2ϕini − GJ cos 2ϕini ) r2
(21a)
P2 = −
sin2ϕini [EIn (1 + cos2ϕini ) − 2EIbcos2ϕini − GJ sin2ϕini] r2
(21b)
P3 =
sin22ϕini (EIb − EIn ) 4r 2
(21c)
Applying the boundary conditions given from Eq. (9c), the changes of the armor wire's lay angles on a curved frictionless torus surface can be obtained by solving Eq. (20) as:
( ) ⎤⎥ + κrγ cotϕ ( )⎥⎦ s
1 − cos 2χ S P (0) + κr (Pt (0) − P3)cot ϕini cos θini (0) ⎡ ⎢ Δϕ (s ) = − b ⎢− tan χ sin 2χ s Pt (0) − P1 S ⎣
( ) ⎫⎪ ( )⎬⎪
⎧ cos θini (s ini ) − cos θini (0)cos 2χ Ss ⎪ ini ini ⎨+⎡ cos θ (0)cot 2 χ ⎤ sin 2χ s S ⎪ ⎢−cos θini (S ini)sin−12χ ⎥ ⎦ ⎩ ⎣
⎭
(22)
where χ and γ are non-dimensional parameters corresponding to the force in the armor wire's tangent:
χ=
γ=
P1 − Pt (0) S EIn 2
(23a)
Pt (0) − P2 Pt (0) − P1 +
EIn sin2ϕini r2
(23b)
This solution is valid for all the three cases χ < 0 , χ = 0 and χ > 0 . Note that the changes of the armor wire's lay angles given by Eq. (22) depend on the sectional forces in both tangential and bi-normal directions at s = 0 , which could be related to each other through the boundary conditions in the circumferential direction. Substituting the approximation given by Eq. (12b) into Eq. (1b) and integrating in terms of the wire's arc length, the armor wire's circumferential positions can be approximately expressed by:
θ (s ) = θini (0) +
sin ϕini cos ϕini s+ r r
∫0
s
Δϕ (s ) ds
(24)
Thus, substituting Eq. (22) into Eq. (24) and applying the boundary conditions given by Eq. (9b), Pb (0) can be expressed in terms of Pt (0) as:
Pb (0) =
+
Pt (0) χ (Pt (0) − P1) tan EA χ − tan χ
ϕini − κr (Pt (0) − P3)cot ϕini cos θini (0)
⎧ 2χr ⎡ κr cot ϕini γ (Pt (0) − P1) ⎪ S sin ϕini ⎢sin ⎣ 2 χ − tan χ ⎨
⎫ θini (S ini ) − sin θini (0) ⎤⎪ ⎥ ⎦⎬ ⎪ −tanχ [cos θini (0) + cos θini (S ini )] ⎪ ⎩ ⎭
(25)
Subsequently, substituting Eq. (25) into Eq. (22), the changes of the armor wire's lay angles can be rewritten in the form solely depending on the tangent force. In order to reduce the size of the expression, herein let us consider that an armor wire begins at θini (0) = π /2 with an integer pitch number. Consequently, the changes of armor wire's lay angles can be expressed as the 216
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superposition of global and periodic components:
Δϕ (s ) = Δϕg (s ) + Δϕp (s )
(26)
with functional Δϕg (s ) and Δϕp (s ) being:
Δϕg (s ) = −
Pt (0) χ tan ϕini ⎡ s s 1 − cos ⎛2χ ⎞ − tan χ sin ⎛2χ ⎞ ⎤ EA χ − tan χ ⎣ ⎝ S⎠ ⎝ S ⎠⎦
(27a)
Δϕp (s ) = κrγ cot ϕini cos θini
(27b)
cosθini
In which Δϕg (s ) denotes the variation of the armor wire's lay angles in the global pattern, Δϕp (s ) contains thus representing the variations of armor wire's lay angles in the periodic form. Note that χ and γ are both functions of the tangential force Pt (0) . Thus, once Pt (0) and the bending curvature κ are given, the armor wire's deflections can be expressed in terms of the wire's lay angles through Eqs. (26), (27a) and (27b). Thereafter, with the armor wire's lay angles identified, the curvature components κ g , κn , τ can be evaluated by substituting the expression of Δϕ (s ) into Eqs. (13a)–(13c) respectively, the sectional moment components Mt , Mb , Mn can be obtained by substituting Δϕ (s ) into Eqs. (14a)–(14c) respectively, and the sectional forces Pt , Pb , Pn can be calculated by substituting Δϕ (s ) into Eqs. (15)–(17) respectively. Similarly, substituting Δϕ (s ) into Eq. (24), the changes of the armor wire's circumferential positions Δθ (s ) can also be expressed as the superposition of global and periodic components as:
Δθ (s ) = Δθg (s ) + Δθp (s )
(28)
with functionals Δθg (s ) and Δθp (s ) being:
Δθg (s ) =
( ) ( ( ) )
s s ⎧ ⎤⎫ ⎡ 2χ S − sin 2χ S ⎪ Pt (0)sin ϕiniS ⎪ s 1 1 ⎥ ⎢ − ⎨S EAr 2 χ − tan χ ⎢+ tan χ cos 2χ s − 1 ⎥ ⎬ ⎪ ⎪ S ⎥ ⎢ ⎦⎭ ⎣ ⎩
Δθp (s ) = κrγ cot2ϕini (sin θini − 1)
(29a) (29b)
Besides, substituting the expression of Δϕ (s ) into Eq. (13a), ignoring the higher order small terms and setting the geodesic curvature equal to zero, the lay angle of the geodesic curve on a torus can be approximately evaluated by:
Δϕgeo (s ) = κr cot ϕini cos θini
(30)
In addition, substituting the expression of Δϕ (s ) into Eq. (1a) and applying the boundary conditions given by Eq. (9a), the longitudinal strain ΔL¯ can be determined. Since the analytical expression of ΔL¯ is very complex, in order to simplify its expression, herein the armor wire is assumed very long, thus all the terms containing 1/ S can be deemed negligible. Consequently, retaining both first and second order small terms of Pt (0)/ EA and κr , the longitudinal strain can be approximately evaluated by: 2
ΔL¯ =
γ γ2 3χ χ 2 tan2χ ⎤ 1 1 Pt (0) P (0) tan ϕini ⎞ ⎡ (tan2χ + 2) ⎤ −⎛ t + + κ 2r 2 ⎡ + − 2 ini ini 2 2 ⎢ ⎥ 2 ⎠ ⎢ (χ − tan χ ) ⎥ 2sin ϕ 4sin2ϕini EA cos ϕ ⎝ EA ⎣ χ − tan χ ⎦ ⎣2 ⎦ ⎜
⎟
(31)
From which it can be seen that while the longitudinal strain primarily follows a linear relationship with the wire's tangential force, it is quadratically affected by the bending curvature. The present model is consistent with the model proposed by Ref. [19] under the circumstance κ = 0 . 4. Results and discussions 4.1. Wire response to bending and tension Firstly, a parametric study is performed to discuss the armor wire's response to bending and tension. Consider a single armor wire with cross-section 10 mm × 3 mm helically wound over a torus surface with minor and major radii respectively equal to 0.1 m and 10 m. The pitch length of armor wire is set to 1.25 m, and Young's modulus and Poisson's Ratio are 210 GPa and 0.3 respectively. An armor wire with four pitches length is modeled with ends fixed at the neutral plane, i.e., θini (0) = π /2. In order to calibrate the proposed analytical model, Eqs. (1a)–(1f) are solved numerically as a boundary value problem using a solver known as bvp4c in the MATLAB environment. To illustrate the transverse deflections of the armor wire subjected to bending and tension, the changes of armor wire's lay angles given by the analytical formulation Eq. (26) and the numerical approach are plotted in Fig. 3. Good correlations between the analytical and numerical results are demonstrated. It can be seen that the bending tends to drive the armor wire to deflect from the initial loxodromic curve towards the geodesic curve evaluated by Eq. (30), see the case with Pt (0) = 0 . Such process is further promoted when the armor wire is tensioned, see the case with Pt (0) = 5000N . To illustrate how the armor wire's transverse deflections are constituted, the global and periodic components of the changes of armor wire's lay angles given respectively by Eq. (27a) and Eq. (27b) for both cases are depicted in Fig. 4, demonstrating that the armor wire's transverse deflections are primarily dominated by the periodic component when the armor wire is subjected to bending 217
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Fig. 3. Changes of wire's lay angles in bending and tension.
Fig. 4. Global and periodic components of wire's deflections in bending and tension.
and tension. Such transverse migration tendency can also be revealed through the changes of armor wire's curvature components, see Figs. 5–7. From which it can be noted that, while the changes of the armor wire's normal curvature and torsion increase when the armor wire is tensioned, the geodesic curvature is observed approaching zero, which corresponds to the geodesic curve. Besides, despite the fact that good agreements are shown for most arc lengths, deviations are observed at both ends in Fig. 5, which might be due to the limited precision of numerical arithmetic and the linearization technique applied in the analytical model development. In addition, Fig. 8 shows the plots of Eq. (30) and the numerical results for a single armor wire with 10 pitches length subjected to tension and different bending curvatures κ = 0 , 1//20m−1, 1//15m−1, 1/10m−1, 1//5m−1. The results indicate that the armor wire tends to be elongated when subjected to pure bending due to the transverse deflections towards the geodesic curve, which is depicted by the term containing (κr )2 in Eq. (30). Besides, a small deviation between the analytical and numerical results is observed in the case with κ = 1//5m−1. This is because the present analytical model is developed on basis of the assumption |κr| ≪ 1. Thus, the present 218
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Fig. 5. Change of wire's geodesic curvature in bending and tension.
Fig. 6. Change of wire's normal curvature in bending and tension.
model is not precise for a flexible pipe subjected to large bending curvature. Nevertheless, for the cases with zero or small bending curvatures, e.g., κ = 0 , 1//20m−1, 1//15m−1, 1//10m−1, it can be concluded that the analytical predictions coincide well with the numerical results. It needs to be emphasized that the frictional resistance is not taken into consideration in the present context. The presence of friction may hinder the armor wire from reaching the equilibrium path predicted by the proposed model and lead the armor wire to be stabilized in the positions between the path corresponding to the loxodromic curve and the equilibrium path predicted by the present model. In other words, the equilibrium path given by the present model represents the maximum transverse deflections that 219
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Fig. 7. Change of wire's torsion in bending and tension.
Fig. 8. Equilibrium path for an armor wire in bending and tension.
might take place on the armor wire.
4.2. Wire response to bending and compression To investigate the armor wire's response to compression, a simple case study is carried out first considering a single armor wire on a straight frictionless cylindrical surface subjected to pure compressive loads. Under this circumstance, the present model degenerates 220
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Fig. 9. Changes of wire's lay angles in compression without bending.
into the model proposed by Ref. [19]. Using the same properties of the armor wire as previously described, the armor wire's transverse deflections in pure compression are revealed through the changes of the wire's lay angles and the changes of wire's circumferential positions in Fig. 9 and Fig. 10 respectively using Eq. (27a) and Eq. (29a). From Figs. 9 and 10 it can be found that the first half and second half of the armor wire have the tendency to slip in opposite directions when compressive loads are experienced. While the armor wire's transverse deflections are found to slowly progress when
Fig. 10. Changes of wire's circumferential positions in compression without bending. 221
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Fig. 11. Analytical solutions to the wire's lay angles variations in bending and compression.
the compressive load is relatively small, large transverse deflections are triggered suddenly when the compressive load approaches a critical value, see the case with Pt (0) = −1213N given by the analytical model and case with Pt (0) = −1216N given by the numerical program. This critical value corresponds to the lateral buckling limit, which can be evaluated by identifying the lowest compressive force causing a singularity in either Eq. (27a) or Eq. (29a), i.e., χ = tanχ , as proposed by Ref. [19]:
Pt , global = −8.18π 2cos2ϕini
sin2ϕini EIn − (EIn cos2ϕini + 4EIbcos2ϕini − GJ cos 2 ϕini ) ini 2 (L ) r2
(32)
It needs to be noted that, while good correlations between the analytical and numerical results are demonstrated for the moments before the lateral buckling forms, differences are observed when the buckling limit is approached, see Figs. 9 and 10. This is due to the large transverse deflections triggered by the lateral buckling which invalidate the small deflections assumption introduced in the analytical development. Thus, the present model is not applicable to the post-buckling analysis. Then, the armor wire's response to combined bending and compression will now be considered. Using Eq. (26) and Eq. (28), Fig. 11 and Fig. 12 respectively show the changes of armor wire's lay angles and circumferential positions on a frictionless torus surface with uniform bending curvature κ = 1/10m−1 subjected to axial compressive loads. Comparing with Figs. 9 and 10, it can be observed that the armor wire on a torus surface tends to buckle in the same manner at same buckling limit as on a straight cylindrical surface, which is promoted by the rapid growth of the global components described by Eq. (27a) and Eq. (29a). Nevertheless, different buckling configurations are obtained from the numerical program. Fig. 13 and Fig. 14 respectively show the numerical results of the changes of armor wire's lay angles and circumferential positions on a frictionless torus surface bent to the same curvature κ = 1/10m−1 subjected to axial compressive loads, demonstrating that the armor wire tends to buckle globally in one direction instead of the center-symmetric configuration predicted by the analytical model. This difference is probably due to the limitation of the applied perturbation technique that the secular terms might not be properly addressed. Fig. 15 shows the plots of Eq. (30) and the numerical results of end shortenings for an armor wire with 10 pitches length subjected to axial compressive loads and different bending curvatures κ = 0 , 1//20m−1, 1//15m−1, 1//10m−1. Despite the fact that the lateral buckling limits captured by the numerical model are found slightly lower than the limits predicted by the analytical formulation, since the buckling modes predicted by them are different, it can be observed that the lateral buckling limit is to a very small extent influenced by the bending curvature as proposed by Ref. [28]. Moreover, it needs to be noted that, since the frictional resistance is disregarded in both analytical and numerical models, the lateral buckling limits predicted by either Eq. (31) or the numerical program both lie on the conservative side compared with the experimental results [12,19]. Motivated by the observations that the armor wires are always found to buckle in periodic patterns in experimental tests instead of the global modes depicted in Figs. 11–14, herein let us assume that the growth of the global deflection components, given by Eq. (27a) and Eq. (29a), are restricted by the frictional resistance, only the periodic components, given by Eq. (27b) and Eq. (29b), are considered. Using Eq. (27b), Fig. 16 illustrates the periodic deflections components of the changes of an armor wire's lay angles subjected to a 222
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Fig. 12. Analytical solutions to the wire's circumferential positions variations in bending and compression.
Fig. 13. Numerical solutions to the wire's lay angles variations in bending and compression.
constant bending curvature κ = 1//10m−1 and axial compressive loads. It can be seen that the periodic deflections approach the loxodromic curve when the axial compressive load approaches P2 = −1728N as indicated by Eq. (27b). Hereafter, the periodic deflections of the armor wire tend to grow on another side of the loxodromic curve when larger compressive loads are applied. When the axial compressive load approaches a critical value, abrupt growth will be generated resulting in a periodic buckling mode. Such critical load corresponds to the singular point in either Eq. (27b) and Eq. (29b), which can be identified by setting the denominator of γ equal to zero as: 223
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Fig. 14. Numerical solutions to the wire's circumferential positions variations in bending and compression.
Fig. 15. Equilibrium path for an armor wire in bending and compression.
Pt , periodic = P1 −
sin2ϕini EIn 2 ini sin ϕ = − [EIn (1 + cos2ϕini ) + 4EIbcos2ϕini − GJ cos 2 ϕini] 2 r2 r
(33)
Eq. (33) coincides with the formula independently developed by Ref. [18] when the buckling length is set to half pitch of wire. Comparing Eq. (32) and Eq. (33) it can be seen that, for a long flexible pipe, the lateral buckling limit corresponding to the periodic buckling mode is larger than that corresponding to the global buckling mode. Now the lateral buckling limit for a flexible pipe will be considered. Assuming that the armor wires in each armor layer deflect in 224
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Fig. 16. Periodic components of wire's lay angles in bending and compression.
a similar manner and the longitudinal compressive loads carried by each armor layer are approximately identical, the critical longitudinal compressive load that triggers armor wires lateral buckling in global and periodic modes can hence be approximately evaluated by multiplying the corresponding critical loads predicted by Eq. (32) and Eq. (33) for a single armor wire by the total number of wires in the flexible pipe n respectively as:
Pglobal = ncosϕiniPt , global
(34a)
Pperiodic = ncosϕiniPt , periodic
(34b)
A series of experiments concerning the lateral buckling behavior of flexible pipes were conducted in laboratory environment by Ref. [12] applying monotonic compressive loads combined with numerous bending cycles, which will be used to validate the proposed analytical models. The properties of the armor layers in the tested flexible pipes are summarized in Table 1. The test conditions and results are presented in Table 2. Using the properties of the inner armor wire, the lateral buckling limits for the three tested pipes in both global and periodic buckling patterns are calculated respectively through Eq. (34a) and Eq. (34b). The comparisons between the experimental results and the analytical predictions for both global and periodic buckling modes, denoted respectively by the blue and red dotted lines, are depicted in Fig. 17. From Fig. 17 it can be seen that in the cases 4, 7, 8, 9, 10 the applied compressive loads are larger than the predicted buckling limits corresponding to the global buckling modes, however, no buckling failures are observed. That corresponds to state that the Table 1 Flexible risers properties [12]].
Inner armor
Outer armor
Steel properties
Parameter
6″ riser
8″ riser
14″ jumper
Outer diameter (mm) Lay angle (deg) Wire size (mm) Number of wires Outer diameter (mm) Lay angle (deg) Wire size (mm) Number of wires Young's Modulus (GPa) Yield stress (MPa) Poisson's ratio
201 26.2 3×10 52 209 −26.2 3×10 54 210 650 0.3 3.96
276 30 5×12.5 54 289 −30.3 5×12.5 56 210 1350 0.3 3.39
442 31.5 4×15 70 452 −31 4×15 72 210 1350 0.3 3.34
Pitch number of inner armor
225
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Table 2 Tests conditions and results [12]]. Pipe ID
Case number
Applied compression (kN)
Results
Pglobal (kN)
Pperiodic (kN)
6″ riser
1 2 3 4 5 6 7 8 9 10 11 12
265 80 210 160 265 700 300 400 277 269 411 950
Failure No failure Failure No failure Failure Failure No failure No failure No failure No failure Failure Failure
113
197
286
463
173
307
8″ riser
14″ jumper
Longitudinal compressive force (kN)
1000
Experimental results Failure 800 No Failure
600
Pglobal Pperiodic
14'' jumper
8'' riser
6'' riser
400
200
0
1
2
3
4
5
6
7
8
9
10
11
12
Case number Fig. 17. Analytical buckling loads versus tests results.
buckling limits corresponding to the global buckling modes are conservative compared to the experimental results. On the contrary, all the cases subjected to the compressive loads larger than predicted buckling limits corresponding to the periodic buckling modes, including cases 1, 3, 5, 6, 11, 12, are found failed due to lateral buckling. In the other cases with compressive loads smaller than Pperiodic , such as cases 2, 4, 7, 8, 9, 10, no lateral buckling failure were observed after numerous bending cycles. In other words, the test results of all the cases agree well with the buckling limits corresponding to the periodic buckling modes. Thus, Eq. (34b) can be taken as an estimate for the flexibe pipe armor wires lateral buckling limit, which should be further carefully validated by additional experimental data. 5. Conclusions In this work, an analytical model is formulated for the prediction of the armor wire's equilibrium state in the flexible pipe assuming no friction. This model is based on the thin curved beam theory where a six order differential equations system was formulated to describe the equilibrium state of a slender beam helically wound over a torus surface. Assuming that the bending curvature of the torus is much smaller compared with the pipe diameter and the armor wire's transverse deflection is very small before the lateral buckling takes place, an analytical solution to this differential equations system is approximately obtained through the application of a multiparameter perturbation technique. Consequently, the equilibrium path of the armor wire is identified, which is found to be described by superposed global and periodic components. 226
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Under the circumstance of axial tension, cases studies were performed which indicate that the armor wire's transverse deflections are primarily dominated by the periodic components, and the armor wire tends to migrate towards the geodesic curve when large tensile loads are applied. The analytical results were compared with the results obtained from a numerical program. Excellent correlations were demonstrated for the cases with small bending curvature, however, slight differences are observed for the cases with large bending curvature since it violates model assumption. Besides, it needs to be emphasized that, in reality, the transverse deflections of armor wires are impeded by the frictional force between the armor wires and the adjacent layers. Thus, the equilibrium path predicted by the present analytical model represents the maximum transverse deflections that might take place on the armor wire. In a physical flexible pipe structure, the equilibrium path of the armor wire should locate between the path estimated by the present analytical model and the path corresponding to the loxodromic curve. Under the circumstance of axial compression, different lateral buckling configurations were identified through the analytical and numerical approaches when bending is introduced, which might be due to the limitations of the applied perturbation technique. Despite the difference between the buckling shapes, the lateral buckling limits predicted by the analytical and numerical models are found similar, which, however, both lie on the conservative side compared with the experimental results available in the literature since the frictional effect is disregarded. It is interesting to note that, assuming the global deflection components are eliminated by the frictional resistance, a periodic buckling mode is uncovered and the corresponding buckling limits show excellent agreements with existing test data. According to the proposed formula for the armor wires lateral buckling limit corresponding to the periodic shape, to improve the design against the lateral buckling failure, measures may be taken by modifying the cross-sections and lay angles of armor wires. Nevertheless, how the armor wires buckling shapes are affected by the frictional resistance and whether the armor wires are guaranteed to buckle in such periodic shape when exposed to cyclic loads still remain in question which requires further investigations. Finally, it needs to be emphasized that the proposed analytical model is not applicable to the post-buckling study since the potential transverse contacts between neighboring armor wires are not considered, the pipe twist is restricted, the armor wire's deflection is assumed small, and the elastoplastic behaviors of the armor wires are not modeled. In addition, it needs to be noted that the armor wires may have some space to rotate along their own axis due to the gap formed between the armor wires and the pipe core when the axial compressive loads are experienced. The armor wire's twist, ignored in the present analytical model, may facilitate the armor wire transverse deflections thus lower the lateral buckling limit, which needs to be carefully investigated in future studies. Acknowledgements The authors acknowledge the support from Petrobras, the National Council for Scientific and Technological Development (CNPq) (No 302380/2013-2) and the Brazilian Coordination for the Improvement of Higher Education Personnel (CAPES) (No 1987/2014) 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. Halkidiki, Greece: American Society of Mechanical Engineers; 2005. p. 821–9. OMAE2005-67404. [2] API R P 17B. Recommended practice for flexible pipe. latest ed. Washington: American Petroleum Institute; 2014. [3] Bectarte F, Coutarel A. Instability of tensile armour layers of flexible pipes under external pressure. Proceedings of the 23rd international conference on offshore mechanics and arctic engineering, vol. 3. Vancouver, British Columbia, Canada: American Society of Mechanical Engineers; 2004. p. 155–61. OMAE2004-51352. [4] Braga MP, Kaleff P. Flexible pipe sensitivity to birdcaging and armor wire lateral buckling. Proceedings of the 23rd international conference on offshore mechanics and arctic engineering, vol. 1. Vancouver, British Columbia, Canada: American Society of Mechanical Engineers; 2004. p. 139–46. OMAE2004-51090. [5] Secher P, Bectarte F, Felix-Henry A. Lateral buckling of armor wires in flexible pipes: reaching 3000 m water depth. Proceedings of the 30th international conference on ocean, offshore and arctic engineering, vol. 4. Rotterdam, The Netherlands: American Society of Mechanical Engineers; 2011. p. 421–8. OMAE2011-49447. [6] Feret JJ, Bournazel CL. Calculation of stresses and slip in structural layers of unbonded flexible pipes. J Offshore Mech Arctic Eng 1987;109(3):263–9. [7] Out JMM, Von Morgen BJ. Slippage of helical reinforcing on a bent cylinder. Eng Struct 1997;19(6):507–15. [8] Sævik S. A finite element model for predicting stresses and slip in flexible pipe armouring tendons. Comput Struct 1993;46(2):219–30. [9] Witz JA, Tan Z. On the axial-torsional structural behaviour of flexible pipes, umbilicals and marine cables. Mar Struct 1992;5(2–3):205–27. [10] Sævik S. Theoretical and experimental studies of stresses in flexible pipes. Comput Struct 2011;89(23–24):2273–91. [11] Kraincanic I, Kebadze E. Slip initiation and progression in helical armouring layers of unbonded flexible pipes and its effect on pipe bending behaviour. J Strain Anal Eng Des 2001;36(3):265–75. [12] Østergaard NH. On lateral buckling of armouring wires in flexible pipes (Ph.D. thesis). Department of Mechanical and Manufacturing Engineering, Aalborg University; 2012. [13] Caleyron F, Leroy J-M, Guiton M, Duchêne P, Estrier P, Vraniskoski K, Damiens A. Stresses in tensile armour layers of unbounded flexible risers loaded with external pressure: application to lateral buckling mode. Proceedings of the 36th international conference on ocean, offshore and arctic engineering Trondheim, Norway: American Society of Mechanical Engineers; 2017. V05AT04A006, OMAE2017-61133. [14] Sævik S, Ji G. Differential equation for evaluating transverse buckling behavior of tensile armour wires. Proceedings of the 33rd international conference on ocean, offshore and arctic engineering, 6B. San Francisco, California, USA: American Society of Mechanical Engineers; 2014. V06BT04A021, OMAE2014-24158. [15] Sævik S, Li H. Shear interaction and transverse buckling of tensile armours in flexible pipes. Proceedings of the 32nd international conference on ocean, offshore and arctic engineering, 4A. Nantes, France: American Society of Mechanical Engineers; 2013. V04AT04A013, OMAE2013-10130. [16] Zhou C, Sævik S, Ye N, Ji G. Effect of lay angle of anti-buckling tape on lateral buckling behavior of tensile armors. Proceedings of the 34th international conference on ocean, offshore and arctic engineering, 5A. St. Johns, Newfoundland, Canada: American Society of Mechanical Engineers; 2015. V05AT04A040, OMAE2015-41527. [17] Zhou Y, Vaz MA. A quasi-linear method for frictional model in helical layers of bent flexible risers. Mar Struct 2017;51:152–73. [18] Sævik S, Thorsen MJ. An analytical treatment of buckling and instability of tensile armors in flexible pipes. J Offshore Mech Arctic Eng 2017;139(4):041701. [19] Li X, Vaz MA, Custódio AB. Analytical prediction for lateral buckling of tensile wires in flexible pipes. Mar Struct 2018;61:268–81.
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[20] Custódio AB. Modelo Analítico para Avaliação de Instabilidade nas Armaduras de Dutos Flexíveis (Ph.D. thesis). Department of Ocean Engineering, Federal University of Rio de Janeiro; 2005. [21] Tan Z, Loper C, Sheldrake T, Karabelas G. Behavior of tensile wires in unbonded flexible pipe under compression and design optimization for prevention. Proceedings of the 25th international conference on offshore mechanics and arctic engineering, vol. 4. Hamburg, Germany: American Society of Mechanical Engineers; 2006. p. 43–50. OMAE2006-92050. [22] Caleyron F, Guiton M, Leroy J-M, Perdrizet T, Charliac D, Estrier P, Paumier L. A multi-purpose finite element model for flexible risers studies. Proceedings of the 33rd international conference on ocean, offshore and arctic engineering, 6A. San Francisco, California, USA: American Society of Mechanical Engineers; 2014. V06AT04A015, OMAE2014-23250. [23] Li X. Application of finite element method for axisymmetric analysis of flexible pipe armor wires instability (Master's thesis). Department of Ocean Engineering, Federal University of Rio de Janeiro; 2014. [24] Vaz MA, Rizzo NAS. A finite element model for flexible pipe armor wire instability. Mar Struct 2011;24(3):275–91. [25] Yang X, Sævik S, Sun L. Numerical analysis of buckling failure in flexible pipe tensile armor wires. Ocean Eng 2015;108:594–605. [26] Paiva LF, Vaz MA. An empirical model for flexible pipe armor wire lateral buckling failure load. Appl Ocean Res 2017;66:46–54. [27] Timoshenko S, Timoshenko S, Goodier JN. Theory of elasticity. McGraw-Hill book Company; 1951. [28] Østergaard NH, Lyckegaard A, Andreasen JH. Imperfection analysis of flexible pipe armor wires in compression and bending. Appl Ocean Res 2012;38:40–7.
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Marine Structures journal homepage: www.elsevier.com/locate/marstruc
Analytical model for tensile armors lateral deflections and buckling in flexible pipes
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 Tensile armor wires Lateral buckling Perturbation
The present paper addresses the mechanical behavior of tensile armor wires in a flexible pipe subjected to bending and longitudinal loads. An analytical model is formulated to evaluate the equilibrium state of a single armor wire on a frictionless torus surface by solving a consistent system of six nonlinear differential equations through the application of a perturbation technique. Cases studies are performed where the armor wire's responses to both tensile and compressive loads are discussed and compared with the results obtained from a numerical program. Under the circumstance of axial compression, two potential lateral buckling modes, global and periodic, are uncovered and their corresponding buckling limits are identified by establishing eigenvalue problems. While the global buckling limits are found conservative compared with the test data available in the literature, good agreements are observed for the estimated buckling limits regarding the periodic buckling mode.
1. Introduction Flexible pipes are tubular composite structures widely used in the offshore oil and gas field developments. Fig. 1 shows the crosssection of a typical unbonded flexible pipe, which is basically composed of metallic layers (carcass, zeta layer, and tensile armors), polymer layers and high strength tapes. The tensile armors, which conventionally consist of an even number of layers of helically wound steel wires, are the key functional components to resist the gravity loads and cyclic bending induced by the ocean environment. In the last decades, the decrease in the available number of oil and gas reservoirs in shallow water has driven the offshore industry to move into deeper waters and harsher environments, posing technical challenges for the safe operations of flexible pipes. For deepwater flexible risers, significant dynamic tensile loading may be experienced, which raises the concerns of fatigue failure of tensile armor wires. Besides, significant longitudinal compressive loads may also be experienced during flexible pipes installation and operation at large water depths due to the hydrostatic end-cap effect [1], which may promote the tensile armor wires to buckle in either radial or lateral direction, especially when the armor annulus is flooded. While studies have shown that the armor wires buckling in the radial direction, also known as birdcage, to some extent can be prevented by the application of high strength tapes over the tensile armors [2], the risks of lateral buckling still remains [3–5]. Thus, to guarantee the safe operation of flexible pipes in deepwater applications, reliable models are required to predict the tensile armor wires mechanical behavior when combined bending and longitudinal loads, tensile or compressive, are experienced. The mechanics of armor wires in flexible pipes have been investigated by many authors throughout the past few decades. Ref. [6] ∗
Corresponding author. E-mail address: [email protected] (M.A. Vaz).
https://doi.org/10.1016/j.marstruc.2018.11.003 Received 16 August 2018; Accepted 5 November 2018 0951-8339/ © 2018 Elsevier Ltd. All rights reserved.
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Fig. 1. Typical unbonded flexible pipe.
presented a theoretical approach to evaluate the tensile armors radial and axial deformations of a straight flexible pipe on basis of the constant lay angle. Subsequently, assuming the armor wires seek an equilibrium path corresponding to the shortest lines on the torus surface, i.e., the geodesic curve, an analytical approximation of the armor wires deflections on an uniformly bent torus was derived by Ref. [7]. Nevertheless, in reality, the armor wires may not be able to reach the geodesic paths. The study of Ref. [8] demonstrated that, due to the frictional resistance, the armor wires slippages are primarily axial when the flexible pipes are subjected to tension and cyclic bending as proposed by Ref. [9]. Assuming the armor wires transverse deflections are negligible, also known as the loxodromic curve, finite element formulations were developed by Ref. [10] for the prediction of the armor wires slips and stresses. Also, assuming that the armor wires can slide only along their own axes, the slip initiation and progression of the armor wires subjected to bending and tension have been investigated by Ref. [11]. Nevertheless, the over-constrained transverse displacements may lead to overestimated stresses in the armor wires. Also, the pre-defined configurations regarding the armor wires equilibrium paths, geodesic or loxodromic, prevent these models from being applicable for the lateral buckling analysis. Without pre-assumed configurations, more general models were proposed by Refs. [12–17] to investigate the armor wires deflections in longitudinal loads associated with repeated bending, where the armor wires are free to slip into any configuration between loxodromic and geodesic depending on friction, thus are applicable for lateral buckling analysis. However, large computational times are needed. Besides, in an effort to identify the lateral buckling limit, Ref. [18] proposed an analytical prediction based on a predefined harmonic deflection mode without considering the frictional resistance and bending effect. Similarly, with the absence of frictional resistance and bending, an analytical model was developed by Ref. [19] for the prediction of the armor wire's deflections on a cylindrical surface without predefined kinematic configurations. Nevertheless, the lateral buckling limits predicted by both models lie on the conservative side since the frictional effect is disregarded. In addition, Ref. [20] developed an analytical model to access the stability limit of flexible pipes under axisymmetric loading on the basis of perturbation methods by establishing an eigenvalue problem. Moreover, Ref. [21], developed an analytical model for simulating the buckling and post-buckling behavior of armor wires based on a total strain energy approach, nevertheless, with no detail presented. Besides, some FE models were developed to explore the physics of this failure mode [1,22–25], however, all include simplifications, thus not representing the effects of cyclic bending and frictional resistance. More recently, an empirical model was proposed by Ref. [26] using symbolic regression to evaluate the lateral buckling limit of armor wire, again, without considering the bending and frictional effects. In the present study, the mechanical behavior of an armor wire in a flexible pipe will be investigated without predefined configurations such as geodesic or loxodromic curves. For the sake of simplicity, the radius of the armor wire is assumed constant, the twist of armor wire is assumed geometrically governed by the underlying torus surface, the frictional resistance is disregarded and the lateral contacts between adjacent armor wires are not considered. On this basis, an analytical model is formulated which is capable of capturing the armor wire's equilibrium state on a uniformly bent frictionless torus in both tension and compression. Through which, two lateral buckling modes, global and periodic, are revealed and the corresponding buckling limits are identified. This is followed by case studies comparing with experimental results available in the literature. 2. Differential equations system A single armor wire is modeled first as a curved beam with a rectangular cross-section helically wound over a cylinder with radius r bent to a torus of constant radius R = 1/ κ , see Fig. 2. The torus surface is parameterized by an arc length coordinate u along the 212
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Fig. 2. Wire geometry and coordinate system.
torus centerline and a circumferential angular coordinate θ. The armor wire's central line is represented by a directed curve parameterized by the arc length parameter s and the angle between the armor wire's tangent and its projection in the longitudinal direction is denoted by ϕ. Assuming that the armor wire's cross-sections remain plane and retain their orientation relative to the → → n , b ⎞⎟ are defined on the armor wire representing the centerline, a right-handed and orthonormal triad of local unit vectors ⎛⎜ t , → ⎝ ⎠ tangential, normal and transverse directions respectively. Using the equilibrium equations of a curved beam in the tangent plane and applying the concepts of curved beam differential geometry, the equilibrium state of an armor wire on a torus is defined by Ref. [12] through a consistent system of six ordinary nonlinear differential equations, which are summarized here:
cos ϕ du = ds 1 + κr cos θ
(1a)
sin ϕ dθ = ds r
(1b)
dϕ κ sin θ cos ϕ + κ g =− ds 1 + κr cos θ
(1c)
dPt = κn Pn − κ g Pb − pt ds
(1d)
dPb = κ g Pt − τPn − pb ds
(1e)
dMn = −κn Mt + τMb + Pb ds
(1f)
In which, κ g , κn and τ respectively denote the wire's geodesic curvature, normal curvature, and torsion, Mt , Mn and Mb respectively denote the sectional moment components in wire's tangential, normal and transverse directions, Pt , Pn , Pb are respectively the sectional forces in the armor wire's tangential, normal and transverse directions. pt and pb respectively represent the distributed external loads in the armor wire's tangential and transverse directions, which are taken as zero since the frictional effect is disregarded in the present study. pn is the squeezing force impeding the armor wires to move radially, which can be obtained from the force equilibrium equation in the normal direction of a curved beam on a torus surface as:
pn = −
dPn + κn Pt + τPb ds
(2)
Moreover, using the moment equilibrium equation in the binormal direction of a curved beam on a torus surface Pn can be given by:
Pn = −
dMb + κ g Mt − τMn ds
(3)
Considering the armor wire is not allowed to freely rotate due to the constraint induced by the neighboring layers, herein it is assumed that the armor wire's twist around its local tangent is solely governed by the underlying torus surface. In other words, the normal vector on the armor wire is considered to coincide with the surface normal. Hence, based on Kirchhoff's theory for thin rods 213
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and the Darboux frame, the armor wire's normal curvature and torsion on a torus surface can be derived respectively as:
κ cos θ cos2ϕ sin2ϕ − 1 + κr cos θ r
(4a)
κ cos θ 1 τ=⎛ − ⎞ sin ϕ cos ϕ r⎠ ⎝ 1 + κr cos θ
(4b)
κn = −
In the initial state, when no longitudinal loads nor bending are applied, the armor wire is assumed stress-free and laid as a helix over a cylinder with a constant lay angle ϕini . Substituting κ = 0 and ϕ = ϕini into Eqs. (1c) and (4a), (4b), the geometrical curvature, normal curvature, and torsion in the initial state, respectively denoted by κ gini , κnini and τ ini , can be obtained as:
κ gini = 0
(5a)
κnini = −
sin2ϕini r
(5b)
τ ini = −
sin 2ϕini 2r
(5c)
Besides, considering the armor wire's dimensions small relative to the minor and major torus radii, it is fair to assume that the sectional moment components are linearly proportional to the change of curvatures with respect to curvature components in the initial un-deformed state, thus:
Mt = GJ (τ − τ ini )
(6a)
Mn = EIn (κ g − κ gini )
(6b)
Mb = EIb (κn − κnini )
(6c)
where E and G are respectively the elastic and shear moduli of the armor wire, In and Ib are respectively the inertia moments of the armor wire's cross-section in normal and transverse directions. Considering the cross-section of armor wire as a narrow rectangular, the stiffness of torsion in the armor wire's tangential direction can be approximately evaluated by Ref. [27]:
1 64 b J = ab3 ⎛ − 5 ⎞ π a⎠ ⎝3
(7)
Note that the previous equations are derived as functions of the armor wire's arc length in the deformed state. Considering that the armor wire is linearly elastic and its axial strain ε is small, the aforementioned system of differential equations in terms of the unknown deformed arc length parameter s can be converted into a system in terms of the initial un-deformed arc length parameter s ini through Cauchy's definition of strain:
ds P =1+ε=1+ t ds ini EA
(8)
where A = a × b (width × thickness) is the cross-sectional area of the armor wire. Consequently, the system of differential equations, Eqs. (1a)–(1f), can be solved as a boundary value problem of sixth order, where u, θ, ϕ, Pt , Pb , Mn are taken as unknowns, and the remaining terms κ g , κn , τ, Mt , Mb , Pn , pn can be respectively expressed in terms of these six unknowns by Eqs. (6b) and (4a), (4b), (6a), (6c), (3), (2). Considering that the flexible pipe is twist balanced subjected to longitudinal tensile or compressive loads when no buckling is formed in the armor layers, the boundary conditions corresponding to the torsional fixed pipe ends are applied:
u (0) = 0
u (S ) = Lini (1 + ΔL¯ )
θ (0) = θini (0)
θ (S ) = θini (0) +
(9a)
sin ϕini ini S r
(9b)
ϕ (0) = ϕ (S ) = ϕini
(9c)
S ini
θini (0)
In which, and S are respectively the armor wire's total arc length in the initial and deformed states, corresponds to the armor wire's circumferential position at s = 0 , Lini = S ini cos ϕini is the pipe length in the initial state and ΔL¯ denotes the longitudinal strain which is defined by the longitudinal elongation divided by the original pipe length. 3. Analytical treatments Applying a perturbation technique, Ref. [19] have proposed an analytical solution to the aforementioned differential equations system, Eqs. (1a)–(1f), in an axisymmetric case, i.e., for κ = 0 . Nevertheless, the introduction of the pipe bending significantly raises the nonlinearity of the differential equations system, thus posing great challenges for the analytical treatment. Note that, in practical applications of flexible pipes, the bending radius is generally much larger than the flexible pipe's diameter, 214
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thus it is reasonable to assume the ratio of the minor torus radius to the major torus radius as a small quantity, i.e., |κr| ≪ 1. Besides, note that the armor wire's deflections are very small when no buckling takes place, the change of armor wire's lay angles with respect to the wire's lay angles in the initial state, denoted by Δϕ (s ) , and the change of armor wire's circumferential positions with respect to that in the initial state, denoted by Δθ (s ) , both can be deemed as small quantities. On this basis, this complex nonlinear differential equations system might be therefore solved analytically through the application of a multi-parameter perturbation technique. Hence, the armor wire's lay angles ϕ (s ) and circumferential positions θ (s ) are herein expressed by adding small variations to their values in the initial un-deformed state as:
ϕ (s ) = ϕini + Δϕ (s )
(10a)
θ (s ) = θini (s ini ) + Δθ (s )
(10b)
where the armor wire's circumferential positions in the initial state θini can be obtained by integrating Eq. (1b) as:
θini (s ini ) = θini (0) +
sin ϕini ini s r
(11)
Ignoring the second and higher order small terms of Δϕ (s ) , Δθ (s ) and κr , the following approximations can then be used:
1 = 1 − κr cos θ 1 + κr cos θ
(12a)
ϕini
(12b)
cos ϕ (s ) = cos ϕini − Δϕ (s )sin ϕini
(12c)
sin ϕ (s ) = sin
ϕini
θini
(12d)
cos θ (s ) = cos θini − Δθ (s )sin θini
(12e)
sin θ (s ) = sin
θini
+ Δϕ (s )cos
+ Δθ (s )cos
Applying the approximations given by Eqs. (12a)–(12e) and ignoring the products between Δϕ (s ) , Δθ (s ) and κr , the armor wire's geodesic curvature, normal curvature, and torsion can then be approximately rewritten from Eqs. (1c), (4a), (4b) respectively as:
1 ⎡ d Δϕ r + κr cos ϕinisin θini⎤ r ⎣ ds ⎦
κ g (s ) =
(13a)
1 κn (s ) = − [sin2ϕini + sin 2 ϕini Δϕ (s ) + κr cos2ϕini cos θini] r
τ (s ) =
(13b)
sin 2ϕini 1 ⎡ sin 2 ϕini − − cos 2 ϕini Δϕ (s ) + κr cos θini⎤ ⎥ ⎢ r⎣ 2 2 ⎦
(13c)
Substitution of Eqs. (13a)–(13c) and Eqs. (5a)–(5c) into Eqs. (6a)–(6c) yields: ini
⎡−cos 2 ϕini Δϕ (s ) + κr sin 2 ϕ cos θini⎤ ⎥ ⎢ 2 ⎦ ⎣
Mt (s ) =
GJ r
Mb (s ) =
EIb [−sin 2 ϕini Δϕ (s ) − κr cos2ϕini cos θini] r
(14b)
Mn (s ) =
EIn ⎡ d Δϕ r + κr cos ϕinisin θini⎤ r ⎣ ds ⎦
(14c)
(14a)
Subsequently, substituting Eqs. (13a), (13c) and (14a)–(14c) into Eq. (3) and retaining only the first order small terms containing Δϕ (s ) and κr , the armor wire's sectional force in the normal direction can be approximately evaluated by:
Pn (s ) = (EIn + 2EIb )
sin 2ϕini d Δϕ κ + (EIn − EIb)sin ϕini cos2ϕinisin θini 2r ds r
(15)
Likewise, substituting Eqs. (13b), (13c) and (14a)–(14c) into Eq. (1f), rearranging and retaining only the first order small terms containing Δϕ (s ) and κr , the sectional force in the armor wire's transverse direction can be approximately given by:
Pb (s ) = EIn
sin2ϕini cos 2 ϕini sin 2 ϕini sin22ϕini d 2Δϕ (EIn − EIbcos2ϕini − GJ sin2ϕini )cos θini Δϕ (s ) + GJ Δϕ (s ) + κ − EIb 2 2 2 2r 2r r ds
(16)
Thereafter, substituting Eqs. (13a), (13b) and (15), (16) into Eq. (1d), integrating in terms of the arc length and retaining only the first order small terms containing Δϕ (s ) and κr , the sectional force in the armor wire's tangential direction can be approximately evaluated by:
Pt (s ) = Pt (0) −
sin22ϕini EIn − EIb EIn + 2EIb 3 ini [cos θini (s ini ) − cos θini (0)] sin ϕ cos ϕini Δϕ (s ) + κr 4 r2 r2 215
(17)
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The distributed external load in the armor wire's normal direction pn can hence be approximately evaluated by substituting Eqs. (13b), (13c) and (15)–(17) into Eq. (2) as:
pn = −
sin2ϕini Pt (0) r
(18)
Alternatively, substituting Eqs. (13a), (13c) and (15), (17) into Eq. (1e), integrating in terms of the arc length and retaining only the first order small terms containing Δϕ (s ) and κr , the sectional force in the armor wire's transverse direction can also be approximately given by:
Pb (s ) = Pb (0) + ⎡Pt (0) + ⎣ − κr cot ϕini ⎡Pt (0) + ⎣
EIn + 2EIb sin22ϕini ⎤ Δϕ (s ) 4 r2 ⎦
EIn − EIb sin22ϕini ⎤ [cos 4 r2 ⎦
θini (s ini ) − cos θini (0)]
(19)
Combining Eqs. (16) and (19), a second-order differential equation is obtained which governs the equilibrium state of an armor wire on a frictionless torus surface:
d 2Δϕ P (0) − P1 P (0) κr κr Δϕ (s ) = b [Pt (0) − P3]cot ϕini cos θini (0) + [P2 − Pt (0)]cot ϕini cos θini − t + ds 2 EIn EIn EIn EIn
(20)
In which P1, P2 , P3 are constants corresponding to the armor wire's properties:
P1 = −
sin2ϕini (EIn cos2ϕini + 4EIbcos2ϕini − GJ cos 2ϕini ) r2
(21a)
P2 = −
sin2ϕini [EIn (1 + cos2ϕini ) − 2EIbcos2ϕini − GJ sin2ϕini] r2
(21b)
P3 =
sin22ϕini (EIb − EIn ) 4r 2
(21c)
Applying the boundary conditions given from Eq. (9c), the changes of the armor wire's lay angles on a curved frictionless torus surface can be obtained by solving Eq. (20) as:
( ) ⎤⎥ + κrγ cotϕ ( )⎥⎦ s
1 − cos 2χ S P (0) + κr (Pt (0) − P3)cot ϕini cos θini (0) ⎡ ⎢ Δϕ (s ) = − b ⎢− tan χ sin 2χ s Pt (0) − P1 S ⎣
( ) ⎫⎪ ( )⎬⎪
⎧ cos θini (s ini ) − cos θini (0)cos 2χ Ss ⎪ ini ini ⎨+⎡ cos θ (0)cot 2 χ ⎤ sin 2χ s S ⎪ ⎢−cos θini (S ini)sin−12χ ⎥ ⎦ ⎩ ⎣
⎭
(22)
where χ and γ are non-dimensional parameters corresponding to the force in the armor wire's tangent:
χ=
γ=
P1 − Pt (0) S EIn 2
(23a)
Pt (0) − P2 Pt (0) − P1 +
EIn sin2ϕini r2
(23b)
This solution is valid for all the three cases χ < 0 , χ = 0 and χ > 0 . Note that the changes of the armor wire's lay angles given by Eq. (22) depend on the sectional forces in both tangential and bi-normal directions at s = 0 , which could be related to each other through the boundary conditions in the circumferential direction. Substituting the approximation given by Eq. (12b) into Eq. (1b) and integrating in terms of the wire's arc length, the armor wire's circumferential positions can be approximately expressed by:
θ (s ) = θini (0) +
sin ϕini cos ϕini s+ r r
∫0
s
Δϕ (s ) ds
(24)
Thus, substituting Eq. (22) into Eq. (24) and applying the boundary conditions given by Eq. (9b), Pb (0) can be expressed in terms of Pt (0) as:
Pb (0) =
+
Pt (0) χ (Pt (0) − P1) tan EA χ − tan χ
ϕini − κr (Pt (0) − P3)cot ϕini cos θini (0)
⎧ 2χr ⎡ κr cot ϕini γ (Pt (0) − P1) ⎪ S sin ϕini ⎢sin ⎣ 2 χ − tan χ ⎨
⎫ θini (S ini ) − sin θini (0) ⎤⎪ ⎥ ⎦⎬ ⎪ −tanχ [cos θini (0) + cos θini (S ini )] ⎪ ⎩ ⎭
(25)
Subsequently, substituting Eq. (25) into Eq. (22), the changes of the armor wire's lay angles can be rewritten in the form solely depending on the tangent force. In order to reduce the size of the expression, herein let us consider that an armor wire begins at θini (0) = π /2 with an integer pitch number. Consequently, the changes of armor wire's lay angles can be expressed as the 216
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superposition of global and periodic components:
Δϕ (s ) = Δϕg (s ) + Δϕp (s )
(26)
with functional Δϕg (s ) and Δϕp (s ) being:
Δϕg (s ) = −
Pt (0) χ tan ϕini ⎡ s s 1 − cos ⎛2χ ⎞ − tan χ sin ⎛2χ ⎞ ⎤ EA χ − tan χ ⎣ ⎝ S⎠ ⎝ S ⎠⎦
(27a)
Δϕp (s ) = κrγ cot ϕini cos θini
(27b)
cosθini
In which Δϕg (s ) denotes the variation of the armor wire's lay angles in the global pattern, Δϕp (s ) contains thus representing the variations of armor wire's lay angles in the periodic form. Note that χ and γ are both functions of the tangential force Pt (0) . Thus, once Pt (0) and the bending curvature κ are given, the armor wire's deflections can be expressed in terms of the wire's lay angles through Eqs. (26), (27a) and (27b). Thereafter, with the armor wire's lay angles identified, the curvature components κ g , κn , τ can be evaluated by substituting the expression of Δϕ (s ) into Eqs. (13a)–(13c) respectively, the sectional moment components Mt , Mb , Mn can be obtained by substituting Δϕ (s ) into Eqs. (14a)–(14c) respectively, and the sectional forces Pt , Pb , Pn can be calculated by substituting Δϕ (s ) into Eqs. (15)–(17) respectively. Similarly, substituting Δϕ (s ) into Eq. (24), the changes of the armor wire's circumferential positions Δθ (s ) can also be expressed as the superposition of global and periodic components as:
Δθ (s ) = Δθg (s ) + Δθp (s )
(28)
with functionals Δθg (s ) and Δθp (s ) being:
Δθg (s ) =
( ) ( ( ) )
s s ⎧ ⎤⎫ ⎡ 2χ S − sin 2χ S ⎪ Pt (0)sin ϕiniS ⎪ s 1 1 ⎥ ⎢ − ⎨S EAr 2 χ − tan χ ⎢+ tan χ cos 2χ s − 1 ⎥ ⎬ ⎪ ⎪ S ⎥ ⎢ ⎦⎭ ⎣ ⎩
Δθp (s ) = κrγ cot2ϕini (sin θini − 1)
(29a) (29b)
Besides, substituting the expression of Δϕ (s ) into Eq. (13a), ignoring the higher order small terms and setting the geodesic curvature equal to zero, the lay angle of the geodesic curve on a torus can be approximately evaluated by:
Δϕgeo (s ) = κr cot ϕini cos θini
(30)
In addition, substituting the expression of Δϕ (s ) into Eq. (1a) and applying the boundary conditions given by Eq. (9a), the longitudinal strain ΔL¯ can be determined. Since the analytical expression of ΔL¯ is very complex, in order to simplify its expression, herein the armor wire is assumed very long, thus all the terms containing 1/ S can be deemed negligible. Consequently, retaining both first and second order small terms of Pt (0)/ EA and κr , the longitudinal strain can be approximately evaluated by: 2
ΔL¯ =
γ γ2 3χ χ 2 tan2χ ⎤ 1 1 Pt (0) P (0) tan ϕini ⎞ ⎡ (tan2χ + 2) ⎤ −⎛ t + + κ 2r 2 ⎡ + − 2 ini ini 2 2 ⎢ ⎥ 2 ⎠ ⎢ (χ − tan χ ) ⎥ 2sin ϕ 4sin2ϕini EA cos ϕ ⎝ EA ⎣ χ − tan χ ⎦ ⎣2 ⎦ ⎜
⎟
(31)
From which it can be seen that while the longitudinal strain primarily follows a linear relationship with the wire's tangential force, it is quadratically affected by the bending curvature. The present model is consistent with the model proposed by Ref. [19] under the circumstance κ = 0 . 4. Results and discussions 4.1. Wire response to bending and tension Firstly, a parametric study is performed to discuss the armor wire's response to bending and tension. Consider a single armor wire with cross-section 10 mm × 3 mm helically wound over a torus surface with minor and major radii respectively equal to 0.1 m and 10 m. The pitch length of armor wire is set to 1.25 m, and Young's modulus and Poisson's Ratio are 210 GPa and 0.3 respectively. An armor wire with four pitches length is modeled with ends fixed at the neutral plane, i.e., θini (0) = π /2. In order to calibrate the proposed analytical model, Eqs. (1a)–(1f) are solved numerically as a boundary value problem using a solver known as bvp4c in the MATLAB environment. To illustrate the transverse deflections of the armor wire subjected to bending and tension, the changes of armor wire's lay angles given by the analytical formulation Eq. (26) and the numerical approach are plotted in Fig. 3. Good correlations between the analytical and numerical results are demonstrated. It can be seen that the bending tends to drive the armor wire to deflect from the initial loxodromic curve towards the geodesic curve evaluated by Eq. (30), see the case with Pt (0) = 0 . Such process is further promoted when the armor wire is tensioned, see the case with Pt (0) = 5000N . To illustrate how the armor wire's transverse deflections are constituted, the global and periodic components of the changes of armor wire's lay angles given respectively by Eq. (27a) and Eq. (27b) for both cases are depicted in Fig. 4, demonstrating that the armor wire's transverse deflections are primarily dominated by the periodic component when the armor wire is subjected to bending 217
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Fig. 3. Changes of wire's lay angles in bending and tension.
Fig. 4. Global and periodic components of wire's deflections in bending and tension.
and tension. Such transverse migration tendency can also be revealed through the changes of armor wire's curvature components, see Figs. 5–7. From which it can be noted that, while the changes of the armor wire's normal curvature and torsion increase when the armor wire is tensioned, the geodesic curvature is observed approaching zero, which corresponds to the geodesic curve. Besides, despite the fact that good agreements are shown for most arc lengths, deviations are observed at both ends in Fig. 5, which might be due to the limited precision of numerical arithmetic and the linearization technique applied in the analytical model development. In addition, Fig. 8 shows the plots of Eq. (30) and the numerical results for a single armor wire with 10 pitches length subjected to tension and different bending curvatures κ = 0 , 1//20m−1, 1//15m−1, 1/10m−1, 1//5m−1. The results indicate that the armor wire tends to be elongated when subjected to pure bending due to the transverse deflections towards the geodesic curve, which is depicted by the term containing (κr )2 in Eq. (30). Besides, a small deviation between the analytical and numerical results is observed in the case with κ = 1//5m−1. This is because the present analytical model is developed on basis of the assumption |κr| ≪ 1. Thus, the present 218
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Fig. 5. Change of wire's geodesic curvature in bending and tension.
Fig. 6. Change of wire's normal curvature in bending and tension.
model is not precise for a flexible pipe subjected to large bending curvature. Nevertheless, for the cases with zero or small bending curvatures, e.g., κ = 0 , 1//20m−1, 1//15m−1, 1//10m−1, it can be concluded that the analytical predictions coincide well with the numerical results. It needs to be emphasized that the frictional resistance is not taken into consideration in the present context. The presence of friction may hinder the armor wire from reaching the equilibrium path predicted by the proposed model and lead the armor wire to be stabilized in the positions between the path corresponding to the loxodromic curve and the equilibrium path predicted by the present model. In other words, the equilibrium path given by the present model represents the maximum transverse deflections that 219
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Fig. 7. Change of wire's torsion in bending and tension.
Fig. 8. Equilibrium path for an armor wire in bending and tension.
might take place on the armor wire.
4.2. Wire response to bending and compression To investigate the armor wire's response to compression, a simple case study is carried out first considering a single armor wire on a straight frictionless cylindrical surface subjected to pure compressive loads. Under this circumstance, the present model degenerates 220
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Fig. 9. Changes of wire's lay angles in compression without bending.
into the model proposed by Ref. [19]. Using the same properties of the armor wire as previously described, the armor wire's transverse deflections in pure compression are revealed through the changes of the wire's lay angles and the changes of wire's circumferential positions in Fig. 9 and Fig. 10 respectively using Eq. (27a) and Eq. (29a). From Figs. 9 and 10 it can be found that the first half and second half of the armor wire have the tendency to slip in opposite directions when compressive loads are experienced. While the armor wire's transverse deflections are found to slowly progress when
Fig. 10. Changes of wire's circumferential positions in compression without bending. 221
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Fig. 11. Analytical solutions to the wire's lay angles variations in bending and compression.
the compressive load is relatively small, large transverse deflections are triggered suddenly when the compressive load approaches a critical value, see the case with Pt (0) = −1213N given by the analytical model and case with Pt (0) = −1216N given by the numerical program. This critical value corresponds to the lateral buckling limit, which can be evaluated by identifying the lowest compressive force causing a singularity in either Eq. (27a) or Eq. (29a), i.e., χ = tanχ , as proposed by Ref. [19]:
Pt , global = −8.18π 2cos2ϕini
sin2ϕini EIn − (EIn cos2ϕini + 4EIbcos2ϕini − GJ cos 2 ϕini ) ini 2 (L ) r2
(32)
It needs to be noted that, while good correlations between the analytical and numerical results are demonstrated for the moments before the lateral buckling forms, differences are observed when the buckling limit is approached, see Figs. 9 and 10. This is due to the large transverse deflections triggered by the lateral buckling which invalidate the small deflections assumption introduced in the analytical development. Thus, the present model is not applicable to the post-buckling analysis. Then, the armor wire's response to combined bending and compression will now be considered. Using Eq. (26) and Eq. (28), Fig. 11 and Fig. 12 respectively show the changes of armor wire's lay angles and circumferential positions on a frictionless torus surface with uniform bending curvature κ = 1/10m−1 subjected to axial compressive loads. Comparing with Figs. 9 and 10, it can be observed that the armor wire on a torus surface tends to buckle in the same manner at same buckling limit as on a straight cylindrical surface, which is promoted by the rapid growth of the global components described by Eq. (27a) and Eq. (29a). Nevertheless, different buckling configurations are obtained from the numerical program. Fig. 13 and Fig. 14 respectively show the numerical results of the changes of armor wire's lay angles and circumferential positions on a frictionless torus surface bent to the same curvature κ = 1/10m−1 subjected to axial compressive loads, demonstrating that the armor wire tends to buckle globally in one direction instead of the center-symmetric configuration predicted by the analytical model. This difference is probably due to the limitation of the applied perturbation technique that the secular terms might not be properly addressed. Fig. 15 shows the plots of Eq. (30) and the numerical results of end shortenings for an armor wire with 10 pitches length subjected to axial compressive loads and different bending curvatures κ = 0 , 1//20m−1, 1//15m−1, 1//10m−1. Despite the fact that the lateral buckling limits captured by the numerical model are found slightly lower than the limits predicted by the analytical formulation, since the buckling modes predicted by them are different, it can be observed that the lateral buckling limit is to a very small extent influenced by the bending curvature as proposed by Ref. [28]. Moreover, it needs to be noted that, since the frictional resistance is disregarded in both analytical and numerical models, the lateral buckling limits predicted by either Eq. (31) or the numerical program both lie on the conservative side compared with the experimental results [12,19]. Motivated by the observations that the armor wires are always found to buckle in periodic patterns in experimental tests instead of the global modes depicted in Figs. 11–14, herein let us assume that the growth of the global deflection components, given by Eq. (27a) and Eq. (29a), are restricted by the frictional resistance, only the periodic components, given by Eq. (27b) and Eq. (29b), are considered. Using Eq. (27b), Fig. 16 illustrates the periodic deflections components of the changes of an armor wire's lay angles subjected to a 222
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Fig. 12. Analytical solutions to the wire's circumferential positions variations in bending and compression.
Fig. 13. Numerical solutions to the wire's lay angles variations in bending and compression.
constant bending curvature κ = 1//10m−1 and axial compressive loads. It can be seen that the periodic deflections approach the loxodromic curve when the axial compressive load approaches P2 = −1728N as indicated by Eq. (27b). Hereafter, the periodic deflections of the armor wire tend to grow on another side of the loxodromic curve when larger compressive loads are applied. When the axial compressive load approaches a critical value, abrupt growth will be generated resulting in a periodic buckling mode. Such critical load corresponds to the singular point in either Eq. (27b) and Eq. (29b), which can be identified by setting the denominator of γ equal to zero as: 223
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Fig. 14. Numerical solutions to the wire's circumferential positions variations in bending and compression.
Fig. 15. Equilibrium path for an armor wire in bending and compression.
Pt , periodic = P1 −
sin2ϕini EIn 2 ini sin ϕ = − [EIn (1 + cos2ϕini ) + 4EIbcos2ϕini − GJ cos 2 ϕini] 2 r2 r
(33)
Eq. (33) coincides with the formula independently developed by Ref. [18] when the buckling length is set to half pitch of wire. Comparing Eq. (32) and Eq. (33) it can be seen that, for a long flexible pipe, the lateral buckling limit corresponding to the periodic buckling mode is larger than that corresponding to the global buckling mode. Now the lateral buckling limit for a flexible pipe will be considered. Assuming that the armor wires in each armor layer deflect in 224
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Fig. 16. Periodic components of wire's lay angles in bending and compression.
a similar manner and the longitudinal compressive loads carried by each armor layer are approximately identical, the critical longitudinal compressive load that triggers armor wires lateral buckling in global and periodic modes can hence be approximately evaluated by multiplying the corresponding critical loads predicted by Eq. (32) and Eq. (33) for a single armor wire by the total number of wires in the flexible pipe n respectively as:
Pglobal = ncosϕiniPt , global
(34a)
Pperiodic = ncosϕiniPt , periodic
(34b)
A series of experiments concerning the lateral buckling behavior of flexible pipes were conducted in laboratory environment by Ref. [12] applying monotonic compressive loads combined with numerous bending cycles, which will be used to validate the proposed analytical models. The properties of the armor layers in the tested flexible pipes are summarized in Table 1. The test conditions and results are presented in Table 2. Using the properties of the inner armor wire, the lateral buckling limits for the three tested pipes in both global and periodic buckling patterns are calculated respectively through Eq. (34a) and Eq. (34b). The comparisons between the experimental results and the analytical predictions for both global and periodic buckling modes, denoted respectively by the blue and red dotted lines, are depicted in Fig. 17. From Fig. 17 it can be seen that in the cases 4, 7, 8, 9, 10 the applied compressive loads are larger than the predicted buckling limits corresponding to the global buckling modes, however, no buckling failures are observed. That corresponds to state that the Table 1 Flexible risers properties [12]].
Inner armor
Outer armor
Steel properties
Parameter
6″ riser
8″ riser
14″ jumper
Outer diameter (mm) Lay angle (deg) Wire size (mm) Number of wires Outer diameter (mm) Lay angle (deg) Wire size (mm) Number of wires Young's Modulus (GPa) Yield stress (MPa) Poisson's ratio
201 26.2 3×10 52 209 −26.2 3×10 54 210 650 0.3 3.96
276 30 5×12.5 54 289 −30.3 5×12.5 56 210 1350 0.3 3.39
442 31.5 4×15 70 452 −31 4×15 72 210 1350 0.3 3.34
Pitch number of inner armor
225
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Table 2 Tests conditions and results [12]]. Pipe ID
Case number
Applied compression (kN)
Results
Pglobal (kN)
Pperiodic (kN)
6″ riser
1 2 3 4 5 6 7 8 9 10 11 12
265 80 210 160 265 700 300 400 277 269 411 950
Failure No failure Failure No failure Failure Failure No failure No failure No failure No failure Failure Failure
113
197
286
463
173
307
8″ riser
14″ jumper
Longitudinal compressive force (kN)
1000
Experimental results Failure 800 No Failure
600
Pglobal Pperiodic
14'' jumper
8'' riser
6'' riser
400
200
0
1
2
3
4
5
6
7
8
9
10
11
12
Case number Fig. 17. Analytical buckling loads versus tests results.
buckling limits corresponding to the global buckling modes are conservative compared to the experimental results. On the contrary, all the cases subjected to the compressive loads larger than predicted buckling limits corresponding to the periodic buckling modes, including cases 1, 3, 5, 6, 11, 12, are found failed due to lateral buckling. In the other cases with compressive loads smaller than Pperiodic , such as cases 2, 4, 7, 8, 9, 10, no lateral buckling failure were observed after numerous bending cycles. In other words, the test results of all the cases agree well with the buckling limits corresponding to the periodic buckling modes. Thus, Eq. (34b) can be taken as an estimate for the flexibe pipe armor wires lateral buckling limit, which should be further carefully validated by additional experimental data. 5. Conclusions In this work, an analytical model is formulated for the prediction of the armor wire's equilibrium state in the flexible pipe assuming no friction. This model is based on the thin curved beam theory where a six order differential equations system was formulated to describe the equilibrium state of a slender beam helically wound over a torus surface. Assuming that the bending curvature of the torus is much smaller compared with the pipe diameter and the armor wire's transverse deflection is very small before the lateral buckling takes place, an analytical solution to this differential equations system is approximately obtained through the application of a multiparameter perturbation technique. Consequently, the equilibrium path of the armor wire is identified, which is found to be described by superposed global and periodic components. 226
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Under the circumstance of axial tension, cases studies were performed which indicate that the armor wire's transverse deflections are primarily dominated by the periodic components, and the armor wire tends to migrate towards the geodesic curve when large tensile loads are applied. The analytical results were compared with the results obtained from a numerical program. Excellent correlations were demonstrated for the cases with small bending curvature, however, slight differences are observed for the cases with large bending curvature since it violates model assumption. Besides, it needs to be emphasized that, in reality, the transverse deflections of armor wires are impeded by the frictional force between the armor wires and the adjacent layers. Thus, the equilibrium path predicted by the present analytical model represents the maximum transverse deflections that might take place on the armor wire. In a physical flexible pipe structure, the equilibrium path of the armor wire should locate between the path estimated by the present analytical model and the path corresponding to the loxodromic curve. Under the circumstance of axial compression, different lateral buckling configurations were identified through the analytical and numerical approaches when bending is introduced, which might be due to the limitations of the applied perturbation technique. Despite the difference between the buckling shapes, the lateral buckling limits predicted by the analytical and numerical models are found similar, which, however, both lie on the conservative side compared with the experimental results available in the literature since the frictional effect is disregarded. It is interesting to note that, assuming the global deflection components are eliminated by the frictional resistance, a periodic buckling mode is uncovered and the corresponding buckling limits show excellent agreements with existing test data. According to the proposed formula for the armor wires lateral buckling limit corresponding to the periodic shape, to improve the design against the lateral buckling failure, measures may be taken by modifying the cross-sections and lay angles of armor wires. Nevertheless, how the armor wires buckling shapes are affected by the frictional resistance and whether the armor wires are guaranteed to buckle in such periodic shape when exposed to cyclic loads still remain in question which requires further investigations. Finally, it needs to be emphasized that the proposed analytical model is not applicable to the post-buckling study since the potential transverse contacts between neighboring armor wires are not considered, the pipe twist is restricted, the armor wire's deflection is assumed small, and the elastoplastic behaviors of the armor wires are not modeled. In addition, it needs to be noted that the armor wires may have some space to rotate along their own axis due to the gap formed between the armor wires and the pipe core when the axial compressive loads are experienced. The armor wire's twist, ignored in the present analytical model, may facilitate the armor wire transverse deflections thus lower the lateral buckling limit, which needs to be carefully investigated in future studies. 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