Flexible riser-bend stiffener top connection analytical model with I-tube

Marine Structures 71 (2020) 102707

Contents lists available at ScienceDirect

Marine Structures journal homepage: http://www.elsevier.com/locate/marstruc

Flexible riser-bend stiffener top connection analytical model with I-tube Yangye He, Murilo Augusto Vaz *, Marcelo Caire Ocean Engineering Department, Federal University of Rio de Janeiro, COPPE/UFRJ, P.O. Box 68508, Rio de Janeiro-RJ, 21941-972, Brazil

A R T I C L E I N F O

A B S T R A C T

Keywords: Flexible riser Top connection Bend stiffener Nonlinear elastic I-tube interface

The flexible riser top connection to the floating unit is a critical region considering extreme loading and fatigue lifetime assessment and is generally protected by a bend stiffener to limit the curvature in this region. The top connection usually interface the floating unit with two main configurations: i) end-fitting and bend stiffener directly connected to a riser balcony or ii) riser connected to the floating unit in the end of an I-tube, which reduces the end-fitting bending loading, and bend stiffener assembled to a bellmouth with a given inclination in relation to the Itube longitudinal axis. The traditional modeling approach considers the riser/bend stiffener system attached to the floating unit, representative of the first configuration. A more realistic modeling approach, capturing the complex interactions of flexible riser/bend stiffener with I-tube interface can be employed for preliminary assessment with less conservatism. In this work, a large deflection analytical beam model is developed for the riser top connection with I-tube considering the bellmouth transition region with a straight rigid surface followed by a curved section. The riser follows a nonlinear bending behavior described by a bilinear moment vs curvature function and the bend stiffener polyurethane material exhibits nonlinear elastic symmetric response rep­ resented by a power law function. It is assumed that there is no gap between the riser and the bend stiffener and the riser is fixed in the end-fitting position. The mathematical formulation of the statically indeterminate system results in three systems of coupled differential equations combined with the corresponding multipoint boundary conditions to be numerically solved by an iterative procedure. A case study is carried out with a 7” flexible riser protected by a bend stiffener connected to an inclined I-tube bellmouth. The system is subjected to extreme loading conditions and the influence of the sleeve shape and I-tube length on the riser curvature distri­ bution, including the end-fitting position, and contact forces between the riser/sleeve and riser/ bend stiffener sections are assessed.

1. Introduction Flexible risers are widely used in offshore oil fields, connecting floating production systems (FPSO) and subsea facilities. A flexible riser is generally made up of several different layers for specific field development requirements, which leads to low bending stiffness combined with high axial and torsional stiffness. The riser top connection is a critical area as it sustains the highest tensions and often the maximum curvatures in the riser system and is generally protected by a polymeric conical shape structure known as bend stiffener.

* Corresponding author. E-mail addresses: [email protected] (Y. He), [email protected] (M.A. Vaz). https://doi.org/10.1016/j.marstruc.2020.102707 Received 15 July 2019; Received in revised form 28 November 2019; Accepted 7 January 2020 Available online 21 January 2020 0951-8339/© 2020 Elsevier Ltd. All rights reserved.

Marine Structures 71 (2020) 102707

Y. He et al.

Fig. 1. Schematic of riser-bend stiffener system with I-tube interface. Adapted from Ref. [2].

The top connection usually presents two main floating unit interface configurations: i) end-fitting and bend stiffener directly connected to the riser balcony or ii) riser connected to the floating unit in the top end of an I-tube, reducing the end-fitting bending loading, and bend stiffener assembled to a bellmouth with a described inclination in relation to the I-tube longitudinal axis. Fig. 1 illustrates the I-tube configuration in a FPSO with riser balcony, which is similar to a FPSO with turret. The I-tube bellmouth system is usually designed to receive the bend stiffener with an end angle, as adopted by Petrobras for the FPSOs in Albacora Leste field [1], for example. A polymeric curved plate sleeve is assembled at the bellmouth in the bend stiffener top base to avoid wear between the riser and I-tube and riser overbending. The traditional modeling approach, representative of the first configuration, considers the riser/stiffener system attached to the floating unit, where both the riser end-fitting and bend stiffener base are considered to be fixed in the same point. A large deflection beam model is usually adopted to represent the system response and has been employed by a number of authors for design purposes [3–6] and to investigate the riser curvature at top connection under different modeling assumptions, such as, bend stiffener nonlinear elastic behavior [7], nonlinear viscoelastic behavior for time domain [8] and steady-state formulations [9] and bilinear riser bending moment vs curvature relationship [10]. For the I-tube configuration, the flexible riser outer sheath contact pressure and relative displacement with the metallic interfaces have been considered to be a cause of excessive wear [1,11,12]. As a consequence, a polymeric sleeve has been introduced in the stiffener design to reduce abrasion on the external sheath. Full scale dynamic riser tests have been performed [2] to evaluate the wear of both riser’s outer sheath and the polymeric insert followed by a design methodology investigation [13,14] based on combining contact pressure estimations with finite element models and wearing rates. For the global dynamic analyses of the top connection system with I-tube interface and riser tensile armour fatigue assessment, the pipe-in-pipe modeling approach implemented in Orcaflex [15] has been recently employed [16,17], highlighting that the traditional approach is considered to be conservative. It allows to take into account the separation between the riser, bend stiffener and bellmouth and to estimate the contact force distribution along length. The Orcaflex pipe-in-pipe approach, including riser bending hysteresis, has also been compared [18] with a finite element model developed in Abaqus [19], where the case study demonstrated that the contact stiffness between the riser and the sleeve has a significant influence in the curvature distribution. It is observed that a realistic approach to capture the interaction between the riser and its interfaces is essential for a more accurate integrity assessment in terms of external sheath wear and tensile armour lifetime, considering the contact force and curvature distribution estimations. In this work, an improvement to the traditional riser/bend stiffener top connection analytical modeling approach is presented. It includes the riser interaction with an I-tube bellmouth where the polymeric sleeve is considered rigid with an initial straight part followed by a curved section. A large deflection beam formulation is developed incorporating the riser nonlinear bending behavior with a bilinear moment vs curvature function and a nonlinear bend stiffener polyurethane response with a power law function. The mathematical formulation of the statically indeterminate system is presented in Section 2 and results in three systems of coupled differential equations combined with the corresponding multipoint boundary conditions to be numerically solved by an iterative procedure described in Section 3. A case study is carried out with a 7” flexible riser protected by a 1.8 m bend stiffener connected to an I-tube with 7∘ inclination, as detailed in Section 4. The system is subjected to extreme loading conditions and the sleeve shape and Itube length effects on the riser curvature distribution, including the end-fitting position, and the contact forces between the riser/ sleeve and riser/bend stiffener sections are investigated. 2. Mathematical formulation The mathematical formulation for the flexible riser/bend stiffener system with I-tube interface is presented in this section. The problem description, hypotheses and simplifications adopted are presented in Section 2.1. The geometrical relations and the equi­ librium of internal forces and moments of an infinitesimal element are presented in Sections 2.2 and 2.3, respectively. The bend stiffener polyurethane constitutive relation and its resulting nonlinear bending moment vs curvature relationship is presented in Section 2.4 together with the flexible riser bilinear behavior under bending. The resulting governing equations are separately described 2

Marine Structures 71 (2020) 102707

Y. He et al.

Fig. 2. (a) Top connection system with I-tube interface beam model; (b) Infinitesimal element; (c) Bend stiffener cross-section.

in Section 2.5 for each beam model section, including: i) riser section from end-fitting up to the contact point in the sleeve; ii) sleeve region and iii) flexible riser/bend stiffener section. Section 2.6 summarizes the geometrical compatibility and the problem boundary conditions are presented in Section 2.7. 2.1. Problem description The top connection system, consisting of a flexible riser segment, bend stiffener and I-tube interface is represented by a large deflection beam model subjected to a tip tension F, angle φL between the riser tangent and the vertical axis, and angle α between the tension and the riser tangent, as schematically presented in Fig. 2 (a). The riser is fixed in the end-fitting position located at point O and starts contacting the rigid sleeve in point A. The sleeve consists of a curved section AB, with constant radius RAB , followed by a straight section BC. It is considered to be fixed at point C and rotated by an angle φ0 in relation to the Y axis of the Cartesian coordinates (X, Y). By assuming that there is no separation in the sleeve region, the riser exhibits a constant curvature value κAB in the curved section AB and in the straight section BC the riser is considered to have zero curvature until the bend stiffener starts at point C. The mathematical formulation is based on the following assumptions and simplifications, � � � � � � �

Euler-Bernoulli large deflection beam theory is employed; the beam is inextensible (neutral axis length is constant); the flexible riser has nonlinear bending behavior represented by a bilinear moment vs curvature function; the bend stiffener polyurethane material exhibits nonlinear elastic symmetric response adjusted by a power law function; self-weight, frictional forces and dynamic effects are disregarded; the gap between the riser and bend stiffener is disregarded; the sleeve in the I-tube transition area is considered to be rigid.

2.2. Geometrical relations An infinitesimal element ds of the beam system is schematically shown in Fig. 2(b) in the X and Y Cartesian coordinate system considering that there is no gap between the riser and bend stiffener. Applying trigonometrical relations to it and considering the curvature as the slope angle rate of change with respect to the distance ds along the neutral axis, leads to the following relations, dX ¼ cosφðsÞ ds

(1a)

3

Marine Structures 71 (2020) 102707

Y. He et al.

dY ¼ sinφðsÞ ds

(1b)

dφ ¼ κðsÞ ds

(1c)

where s is the arc-length along the riser top connection system (0 � s � L), L is the total length, XðsÞ and YðsÞ are the deflected riser/ bend stiffener coordinates, φðsÞ is the angle between the tangent to the beam axis and the X axis, and κðsÞ is the curvature. 2.3. Equilibrium of internal forces and moments A schematic of the internal forces and moments in a riser and bend stiffener infinitesimal element is also shown in Fig. 2(b). Eliminating multiplication of differential terms, the equilibrium of normal and tangential forces and bending moments yields, dVi ds

Ti

dφi ds

(2a)

ð 1Þi f ¼ 0

dTi dφ þ Vi i ¼ 0 ds ds

(2b)

dMi ds

(2c)

Vi ¼ 0

where the subscripts i ¼ 1 and 2 refer to the riser and bend stiffener, respectively, Vi ðsÞ, Ti ðsÞ and Mi ðsÞ are the shear forces, axial forces and bending moments, and fðsÞ is the contact force between the riser and bend stiffener, which is zero outside the bend stiffener and sleeve sections, assuming there is no flexible riser contact in the I-tube region from point O to A. Contact force is positive “þ” or negative “-” depending on its direction in the bending plane, as presented in Fig. 2(b). Algebraically manipulating the horizontal and vertical infinitesimal equilibrium Eqs. (2a) and (2b), they may also be described according to, Horizontal ​ ðY

Vertical ​ ðX

axisÞ :

axisÞ :

d ðTi cosφ þ Vi sinφÞ ds d ðTi sinφ ds

ð 1Þi f sinφ ¼ 0

(3a)

Vi cosφÞ þ ð 1Þi f cosφ ¼ 0

(3b)

2.4. Constitutive relations and pure bending formulation The flexible riser and bend stiffener bending moment vs curvature relationships are described in this section being further incorporated in the governing equations of the large deflection beam problem presented in Section 2.5. 2.4.1. Flexible riser The bending behavior of a flexible riser is governed by interlayer friction mechanisms leading to a hysteretic response when subjected to cyclic loading. For low values of curvature, the interlayer friction forces are able to prevent tensile armours slippage, which results in a high bending stiffness EIns value. Slippage starts gradually between layers after the curvature reaches a certain critical value, nonlinearly reducing the bending stiffness until the full slippage value EIfs is reached. This nonlinear behavior and stiffness transition are highly affected by the interlayer contact pressure resulting from axisymmetric loading (tension, internal and external pressure), but in the present formulation, for simplification purposes, the transition from the stick to the slip domain is simplified into a bilinear relationship, as follows, � jκðsÞj � jκcr j EIns κðsÞ; � M1 ðsÞ ¼ (4) EIfs κðsÞ þ EIns EIfs κcr ; jκðsÞj > jκcr j where κcr is the critical curvature at which the stiffness transition occurs, EIns is the no-slip bending stiffness, and EIfs is the full-slip bending stiffness. 2.4.2. Bend stiffener The polyurethane employed for bend stiffener manufacture presents a nonlinear mechanical behavior that is different under tension and compression and highly dependent on temperature and loading rate. In general, higher rates lead to a stiffer response and when this effect is disregarded, a hyperelastic modeling approach may be employed considering the material at a given constant temperature. Further simplification can be employed for a beam bending formulation, where a general nominal stress vs nominal strain function can be used to fit experimental data obtained from uniaxial tensile tests. In the present work, the polyurethane nonlinear elastic behavior is considered to be symmetric (same response for tension and compression) and adjusted by a two parameters power function defined by, 4

Marine Structures 71 (2020) 102707

Y. He et al.

Fig. 3. Schematic of the riser-bend stiffener top connection model: (a) Configuration of the riser extended section in the I-tube area; (b) Contact force of the riser with the sleeve; (c) Configuration of riser/bend stiffener section.

�

(5)

σðεÞ ¼ signðεÞEq �εjq

where σ and ε are the nominal stress and strain respectively, and Eq and q are the material parameters. The function sign ðεÞ is defined as ε=jεj to determine positive or negative strain under tension or compression, respectively. Tensile tests of typical bend stiffener polyurethane samples have been previously performed in a servo hydraulic testing machine at room temperature with three different stretch rates (5, 50 and 500 mm=min), as presented by Ref. [8]. In the present work, the stress vs strain response obtained for the 50 mm=min loading rate is selected for the case study and adjusted up to 15% by Eq. (5), leading to the following material parameters: Eq ¼ 20:19MPa and q ¼ 0:4738 [20]. Considering Euler-Bernoulli beam bending theory for the riser/bend stiffener system, the strain ε at a distance η from the neutral axis can be expressed by εðη; sÞ ¼ ηκðsÞ. As the bend stiffener material is assumed to be nonlinear elastic symmetric in tension and compression, the neutral axis coincides with the cross section centroid during bending. Consequently, the equilibrium of bending moment for the bend stiffener cross-section area yields, Z � �q � � M2 ðsÞ ¼ σ BS ηdA ¼ signðκðsÞÞEq �κðsÞ� IBS ðsÞ (6) ABS

with the geometrical function IBS ðsÞ given by, �qþ3 � �qþ3 � Z π �� Z 2� � �qþ1 � De ðsÞ Di �sinθ�qþ1 cos2 θdθ IBS ðsÞ ¼ �η� dA ¼ 2 π 2 2 2

(7)

ABS

where, as shown in Fig. 2(c), the subscript BS refers to the bend stiffener cross-section in the Cartesian coordinates (ξ, η), De ðsÞ and Di are the external and internal diameters, respectively, dA is an infinitesimal element of area, and θ is the angle of infinitesimal element position related to the ξ axis. 2.5. Governing equations The top connection system governing equations may be separately obtained for different sections and combined with appropriate multipoint boundary conditions. The three sections adopted are schematically presented in Fig. 3 and defined as follows, 1. Section I (0 � s � s1 ): flexible riser from fixed boundary condition at point O up to the contact point A in the sleeve, with an arclength given by s1 ; 2. Section II (s1 � s � s3 ): flexible riser contact region with the sleeve, from point A to C. The distances s2 and s3 define the arc-lengths from point O to B and C, respectively. The segment AB consists of a curved section with constant radius RAB and arc-length given by sAB ¼ s2 s1 . The straight section BC has an arc-length defined by LBC ¼ s3 s2 . The sleeve contact constraint enforces a reaction force NA at point A and a reaction moment MB at point B to the flexible riser, as shown in Fig. 3(a); 5

Marine Structures 71 (2020) 102707

Y. He et al.

3. Section III (s3 � s � L): flexible riser/bend stiffener section CD with an arc-length of LCD ¼ L s3 . The arc-length s4 is defined from the point O to the bend stiffener tip end. The bend stiffener has a total length of LBS . The loading (F, φL , α) is applied to the end of the flexible riser segment at s ¼ L. 2.5.1. Section I (0 � s � s1 ) - flexible riser extended section Isolating the first section, where there is no contact force, the tension and shear forces at point A can be respectively defined as, TA ¼ T1 ðs1 Þ and NA ¼ V1 ðs1 Þ, with a neutral axis slope given by φðs1 Þ ¼ φA . Integrating Eqs. (3a) and (3b) considering these boundary constants, leads to, NA cosφA

(8a)

T1 ðsÞcosðφðsÞÞ þ V1 ðsÞsinðφðsÞÞ ¼ TA cosφA þ NA sinφA

(8b)

T1 ðsÞsinðφðsÞÞ

V1 ðsÞcosðφðsÞÞ ¼ TA sinφA

Further manipulating Eqs. (8a) and (8b) yields the following shear force equation, V1 ðsÞ ¼ TA sinðφðsÞ

φA Þ þ NA cosðφðsÞ

(9)

φA Þ

Introducing Eq. (9) into Eq. (2c) with the riser bending moment formulation (4) yields, 8 1 > > ðTA sinðφðsÞ φA Þ þ NA cosðφðsÞ φA Þ Þ; jκðsÞj � jκcr j > dκðsÞ < EIns ¼ > ds 1 > > ðTA sinðφðsÞ φA Þ þ NA cosðφðsÞ φA Þ Þ; jκðsÞj > jκcr j : EIfs

(10)

Eq. (10) and the geometrical relations (1a) - (1c) form a system of four first-order nonlinear differential equations for the co­ ordinates XðsÞ and YðsÞ, angle φðsÞ and curvature κðsÞ in the riser extended section. 2.5.2. Section II (s1 � s � s3 ) - flexible riser/sleeve contact region The riser deflection in the sleeve contact region is assumed to follow the curved sleeve geometrical shape in the segment AB followed by the straight segment BC, yielding the following angle relations, � dφðsÞ=ds ¼ κAB ; s1 � s � s2 (11) φðsÞ ¼ φ0 ; s2 � s � s3 Introducing the constant curvature κAB into the flexible riser bending moment vs curvature relation given by Eq. (4) and considering the flat sleeve region BC leads to the following relation, 8� jκAB j � jκcr j < EIns κAB ; � ; s � s � s2 M1 ðsÞ ¼ (12) EIfs κAB þ EIns EIfs κcr ; jκAB j >jκcr j 1 : 0; s2 � s � s3 which from Eq. (2c) results in a zero shear force V1 distribution in this region and employing Eq. (2b) a constant tension T1 ðs1 Þ ¼ T1 ðs3 Þ ¼ TA is obtained. It should be noted that in the transition from the curved to the flat region in the sleeve, the bending moment distribution drops from a constant value caused by the concentrated reaction moment MB ¼ M1 ðs2 Þ to zero. The contact force between the riser and the sleeve can be calculated employing Eq. (2a) considering the constant curvature κAB in section AB and the zero curvature in section BC, as shown in Fig. 3(b), leading to, � TA κAB ; s1 � s < s2 f¼ (13) 0; s2 � s � s3 2.5.3. Section III (s3 � s � L) - flexible riser/bend stiffener For the riser/bend stiffener section (segment CD) in Fig. 3(c), summing up Eqs. (2a)–(2c) for the riser i ¼ 1 and bend stiffener i ¼ 2, yields, dV ds

dφ ¼0 ds

(14a)

dT dφ þV ¼0 ds ds

(14b)

dM ds

(14c)

T

V¼0

where M ¼ M1 þ M2 , T ¼ T1 þ T2 and V ¼ V1 þ V2 are the total bending moment, tension and shear force of riser/bend stiffener section respectively. Considering the tip loading condition defined by the force F and angles α and φL and integrating Eqs. (3a) and (3b), leads to, 6

Marine Structures 71 (2020) 102707

Y. He et al.

VðsÞcosðφðsÞÞ ¼ FsinðφL þ αÞ

(15a)

TðsÞcosðφðsÞÞ þ VðsÞsinðφðsÞÞ ¼ FcosðφL þ αÞ

(15b)

TðsÞsinðφðsÞÞ

that can be solved to find the total shear force relation given by, VðsÞ ¼

FsinðφL þ α

(16)

φðsÞÞ

Introducing Eq. (16) into Eq. (14c) with the bending moment formulations (4) and (6) of the riser and bend stiffener, respectively, yields, � 8 � �q dI ðsÞ � � � 1 � � BS > > ; �κðsÞ� � jκcr j þ α φðsÞ Þ þ signðκðsÞ ÞE Fsinðφ �κðsÞ � � � q > L > EI þ E q�κðsÞ�q 1 I ðsÞ ds < ns q BS dκðsÞ ¼ (17) � � �q dI ðsÞ � � > ds � 1 > � � BS > > ; �κðsÞ� > jκcr j FsinðφL þ α φðsÞ Þ þ signðκðsÞ ÞEq �κðsÞ� � �q 1 : ds EIfs þ Eq q�κðsÞ� IBS ðsÞ Eq. (17) and the geometrical Eqs. (1a)–(1c) form the system of four first-order nonlinear differential equations in the riser/bend stiffener section. Generally, a geometrical discontinuity exists at the bend stiffener tip end due to the tip thickness, which is taken into account by ensuring the total moment continuity in the riser/bend stiffener section. 2.6. Geometrical compatibility The arc-length sAB of the curved sleeve section may be written as a function of the initial contact angle φA and the sleeve inclination φ0 , as follows, sAB ¼ RAB ðφ0

(18)

φA Þ

and arc-lengths of riser/sleeve and riser/bend stiffener sections can be described by summations of riser extended arc-length s1 , sAB , LBC and LBS , s2 ¼ s1 þ sAB

(19a)

s3 ¼ s1 þ sAB þ LBC

(19b)

s4 ¼ s1 þ sAB þ LBC þ LBS

(19c)

and the length of riser/bend stiffener section is presented by, LCD ¼ L

s1

sAB

(20)

LBC

as schematically presented in Fig. 3. As the sleeve is fixed in the point C with given coordinates, the coordinates of initial contact point A in the sleeve are geometrically related to RAB , φA and φ0 according to, XA ¼ X C

LBC cosφ0

RAB ðsinφ0

sinφA Þ

(21a)

YA ¼ YC

LBC sinφ0 þ RAB ðcosφ0

cosφA Þ

(21b)

and the coordinates of point B yield, XB ¼ X C

(22a)

LBC cosφ0

(22b) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The contact angle φA and the arc-length s1 are under the following system geometrical restrictions: 0 � φA � φ0 and X2A þ Y 2A � YB ¼ YC

LBC sinφ0

s1 � XA þ YA .

2.7. Multipoint boundary conditions The riser-bend stiffener top connection beam system is under multipoint boundary conditions, which consists of a built-in point O in the end-fitting position, fixed constraints due to the sleeve curved section AB and straight section BC, a bend stiffener inclination in the connection point C with the sleeve and a flexible riser tip end rotation at point D. The boundary conditions are defined, for each point, as follows, 1. Point O - fixed boundary condition in the end-fitting position (s ¼ 0): 7

Marine Structures 71 (2020) 102707

Y. He et al.

Table 1 Top connection system geometrical and loading parameters. Input data

ðXB ; YB Þ, ðXC ; YC Þ - sleeve coordinates φ0 - sleeve rotation angle

RAB - sleeve radius (κAB ¼ 1=RAB ) L, LBS and LBC - total riser, bend stiffener and sleeve flat segment lengths

F, φL and α - loading conditions Unknowns

s1 - arc-length of segment OA

φA - initial contact angle NA - reaction force

Fig. 4. Flowchart of numerical solution procedure for flexible riser-bend stiffener top connection system.

8

Marine Structures 71 (2020) 102707

Y. He et al.

Xð0Þ ¼ 0;

Yð0Þ ¼ 0

and

(23a-c)

φð0Þ ¼ 0

2. Point A - flexible riser initial contact point with the sleeve (s ¼ s1 ): Xðs1 Þ ¼ XA ;

Yðs1 Þ ¼ YA ;

φðs1 Þ ¼ φA

and

dφðs1 Þ ¼ κAB ds

(24a-d)

where φA is the riser initial contact angle with the sleeve and ðXA ; YA Þ are the contact point coordinates. The sleeve normal reaction force NA is applied to the riser at this point. 3. Point B - end of sleeve curved section (s ¼ s2 ): Xðs2 Þ ¼ XB ;

Yðs2 Þ ¼ YB

and

(25a-c)

φðs2 Þ ¼ φ0

where φ0 is the sleeve inclination and ðXB ; YB Þ are the coordinates of point B. The sleeve reaction moment MB is applied to the riser at this point. 4. Point C - end of sleeve flat section and beginning of bend stiffener (s ¼ s3 ): Xðs3 Þ ¼ XC ;

Yðs3 Þ ¼ YC

and

(26a-c)

φðs3 Þ ¼ φ0

where ðXC ; YC Þ are the coordinates of point C and φ0 is the bend stiffener base inclination relative to the X axis, which is the same as the sleeve inclination. 5. Point D - flexible riser tip (s ¼ L): (27)

φðLÞ ¼ φL where φL is the rotation angle in the riser tip end. 3. Numerical solution procedure

The system of sections I, II and III coupled governing differential equations, subjected to boundary conditions (23)–(27) and geometrical compatibility Eqs. (18)–(22) forms a multipoint boundary value problem (BVP) of the riser/bend stiffener top connection system with I-tube. Table 1 summarizes the input and unknown geometrical and loading parameters. The initial contact angle φA and arc-length s1 of segment OA are geometrical unknown parameters. The riser initial contact point coordinates ðXA ; YA Þ with the sleeve are unknown parameters that can be calculated with Eq. (21) once the angle φA and arc-length s1 are iteratively found. The sleeve reaction force NA at contact point A is another unknown to be solved. The Mathematica package [21] is employed for the iterative numerical solution following the flowchart presented in Fig. 4 and detailed as follows: i. Start the loop (m ¼ 0) with initial values given by: angle φ0A ¼ φ0 , arc-length s01 ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X2B þ Y 2B and tension T0A ¼ F;

ii. Calculate the riser/bend stiffener section length Lm CD with Eq. (20); iii. Solve section III BVP with governing equations given by Eqs. (17), (1a)–(1c) and boundary conditions (26a-c) and (27) with the shooting method, that converts the BVP problem into an equivalent initial value problem (IVP). As there is a curvature discontinuity at the bend stiffener tip, a modified shooting method is employed to capture this effect, as follows: a) Guess an initial curvature κðs3 Þ, and integrate two differential Eqs. (17) and (1c) with κðs3 Þ and φðs3 Þ ¼ φ0 until the bend stiffener tip, s4 ¼ s3 þ LBS . Therefore, curvature κðs4 Þ and angle φðs4 Þ at the left side of bend stiffener tip are obtained. Ensure the moment continuity at this point, i.e., total moment at the left side of bend stiffener tip (composed of riser and bend stiffener), Mðs4 Þ ¼ M1 ðs4 Þ þ M2 ðs4 Þ, is equal to the total moment at the right side of the tip (only riser). With this condition, the riser curvature discontinuity κðs4 Þþ is calculated at the right side of bend stiffener tip, 8 1 > > jMðs4 Þj � jMcr j Mðs4 Þ; > < EI ns κðs4 Þþ ¼ (28) � � > 1 > > Mðs4 Þ EIns EIfs κcr ; jMðs4 Þj > jMcr j : EIfs where Mcr ¼ EIns κcr is the critical bending moment in the nonlinear riser bending curvature and moment relationship. With angle φðs4 Þ and “jumped” curvature κðs4 Þþ , integrate two differential Eqs. (17) and (1c) from the bend stiffener tip s ¼ s4 to the riser end s ¼ L, and compare the calculated angle φðLÞ with the specified boundary condition φL ; b) Update the initial curvature with the bisection method and restart the process until the calculated angle φðLÞ matches the specified angle φL with a given convergence criteria φðLÞφ 9

φL L

� 10 5 ;

Marine Structures 71 (2020) 102707

Y. He et al.

Fig. 5. Schematic drawing of top connection system with I-tube interface.

Fig. 6. Geometry of the riser/bend stiffener section.

c) After this BVP solution is found, calculate T1 ðsÞ employing Eqs. (2c) and (2b) to update Tmþ1 ¼ T1 ðs3 Þ; A m iv. Check stop criteria with ðTmþ1 Tm A Þ=T A after the first m loop. If satisfied go to (v.). If not, set n ¼ 1 and proceed with Section I A BVP loop as follows: (a) Guess angle φnA , arc-length sn1 , reaction force NnA and set tension TA ¼ T mþ1 ; A (b) Solve section I BVP with governing equations defined by (10), (1a) - (1c) and boundary conditions (23a-c) and (24c). The Mathematica FindRoot function is employed to search for a numerical solution to the simultaneous equations until the roots are found to a specified accuracy; n m n (c) Update m ¼ m þ 1 and set φm A ¼ φA and s1 ¼ s1 . Go to (ii.); v. Calculate the moments Mi ðsÞ, shear forces Vi ðsÞ, axial forces Ti ðsÞ and contact force fi ðsÞ with Eqs. (2a)–(2c); vi. Calculate Section II with Eqs. (11)–(13).

4. Case study A case study is carried out with a 7” flexible riser with a total arc-length of L1 þ 4m, surrounded by a fixed sleeve inside the I-tube end and protected by a 1.8 m bend stiffener, as presented in Fig. 5. An I-tube turning point with 7∘ inclination occurs at L1 , followed by a 1:2m segment length up to the bend stiffener base. The sleeve consists of a curved section with a constant bending radius RAB and a straight section of 0:5m. Two loading conditions on the riser-bend stiffener top connection system are considered, corresponding to static and dynamic extreme values, respectively: (F ¼ 450kN, φL ¼ 10∘ ) and (F ¼ 1200kN, φL ¼ 15∘ ), both with a zero angle α. Parametric assessments of sleeve curved radius RAB and I-tube length L1 are respectively presented in sections 4.2 and 4.3 to investigate their influence on the curvature distribution response. The following geometrical and material parameters are generally considered in the case study: 1. The riser bilinear bending behavior, presented in Eq. (4), is defined by the following parameters: bending stiffness EIfs ¼ 40kNm2 , EIns ¼ 2800kNm2 , and critical curvatures κcr ¼ 0:00075m 1 and 0:002m 1 estimated according to the equations developed in Ref. [22] for effective tensions respectively given by 450 kN and 1200 kN. The flexible riser safely sustains a minimum bending radius (MBR) of 2m or higher, equivalent to a maximum curvature of 0.5m 1 .

10

Marine Structures 71 (2020) 102707

Y. He et al.

Fig. 7. Top connection response for loading condition (450kN, 10∘ ): (a) Configuration; (b) Angle; (c) Curvature; (d) Bending moment; (e) Shear force and (f) Tension distributions.

11

Marine Structures 71 (2020) 102707

Y. He et al.

Fig. 8. Top connection response for loading condition (1200kN, 15∘ ): (a) Configuration; (b) Angle; (c) Curvature; (d) Bending moment; (e) Shear force and (f) Tension distributions.

2. The nonlinear elastic model presented in Eq. (5) is employed for the bend stiffener polyurethane behavior with the following material parameters: Eq ¼ 20:19MPa and q ¼ 0:4738. The bend stiffener geometrical parameters are shown in Fig. 6, consisting of a 1:65m conical shape followed by a 0:15m thin cylindrical tip. The riser/bend stiffener section length from bend stiffener top to riser tip is described by LCD with an initial straight length LICD ¼ 2:8m and the coordinates of point C are given by, 12

Marine Structures 71 (2020) 102707

Y. He et al.

Fig. 9. Contact force distribution along riser/sleeve and riser/bend stiffener sections. Table 2 Sleeve radius influence in the top connection response (450kN, 10∘ ). Sleeve A

φ(∘) A

7 6.70

s1 ðmÞ κ0 ðm

1

Þ

NA ðkNÞ ΔLCD ðmmÞ

RAB ðmÞof sleeve B 3

4

5

6

7

8

9

10

6.14 6.65

4.70 6.54

3.80 6.42

3.18 6.30

2.72 6.17

2.35 6.05

2.04 5.92

1.77 5.78

0.027

0.026

0.025

0.022

0.019

0.016

0.012

0.007

�0

5.16

44.8

33.6

26.8

22.4

19.2

16.8

14.9

13.4

3.77

fAB ðkN =mÞ

3.76

3.62

3.42

3.19

2.95

2.70

2.43

2.15

150.0

112.5

90.0

75.0

64.3

56.2

50.0

45.0

XC ¼ L1 þ 1:2cosφ0

(29a)

YC ¼ 1:2sinφ0

(29b)

4.1. Large deflection assessment of top connection with I-tube interface The riser-bend stiffener top connection response with I-tube interface is initially assessed considering an I-tube length of L1 ¼ 6m and a sleeve curved section with a radius of RAB ¼ 8m. The resulting configuration, angle, curvature, bending moment, shear force and axial force distribution along the X axis are presented in Figs. 7 and 8, for the two loading conditions: (450kN, 10∘ ) and (1200kN, 15∘ ). The dotted lines define the coordinates of “I-tube”, “Sleeve” and “Bend stiffener” sections. Figs. 7a, 7b, 8a and 8b show that the riser slightly deflects along the I-tube length up to the initial contact point A with the sleeve, keeps contact with the sleeve following its shape, and continues the large deflection attached to the bend stiffener, as no gap is considered in the mathematical formulation. From point A to B a constant moment is observed, as defined by Eq. (12), followed by a section with zero bending moment from B to C (Figs. 7d and 8d). Along the riser/sleeve contact length, a concentrated reaction force can be observed in the initial contact point A followed by a zero shear force distribution up to the point C (Figs. 7e and 8e). After that point, the shear force and bending moment distribution are presented separately for the riser and the bend stiffener, being mostly supported by the bend stiffener. As it is expected for practical applications and shown in Figs. 7d, 7f, 8d and 8f, the bending moment and axial forces are respectively absorbed by the bend stiffener and riser. The curvature distribution is presented in Figs. 7c and 8c for both loading conditions. It can be observed that for the higher loading (1200kN, 15∘ ), the curvature in the bend stiffener root, 0:19m 1 , is higher than the sleeve curvature given by κAB ¼ 0:125m 1 , while for the loading case (450kN, 10∘ ), the maximum riser curvature occurs in the sleeve region. A curvature discontinuity is observed for both cases in the bend stiffener tip. The curvature in the endfitting position is given by 0:012m 1 and 0:017m 1 for the static (450kN, 10∘ ) and dynamic (1200kN, 15∘ ) extreme loading, respec­ tively. The contact force distribution is shown in Fig. 9 for both loading conditions employed in the case study, presenting a constant value after the initial contact point A and dropping to zero in the straight sleeve section, according to Eq. (13). It can be observed that the highest loading condition results in larger forces for both the sleeve and bend stiffener region and also a longer contact length in the curved sleeve section.

13

Marine Structures 71 (2020) 102707

Y. He et al.

Table 3 Sleeve radius influence in the top connection response (1200kN, 15∘ ). Sleeve A

φ(∘) A

RAB ðmÞ of sleeve B

7 6.69

s1 ðmÞ

2

3

4

5

6

7

8

9

10

5.79 6.65

4.02 6.54

3.11 6.42

2.53 6.31

2.12 6.19

1.81 6.06

1.56 5.94

1.34 5.81

1.13 5.67

Þ

0.052

0.052

0.049

0.044

0.038

0.032

0.025

0.017

0.004

�0

NA ðkNÞ

135.1

109.7

73.1

54.9

43.9

36.7

31.5

27.6

24.6

22.2

4.12

4.10

3.93

3.71

3.48

3.24

2.98

2.71

2.44

2.16

599.6

399.8

299.8

239.9

199.9

171.3

149.9

133.3

119.9

κ0 ðm

1

ΔLCD ðmmÞ fAB ðkN =mÞ

Fig. 10. Initial contact angle and end-fitting curvature versus curved sleeve radius (I-tube length L1 ¼ 6m).

Fig. 11. Sleeve geometry influence in the curvature distribution (I-tube length L1 ¼ 6m).

4.2. Sleeve geometry influence The sleeve geometry influence in the riser-bend stiffener top connection response with I-tube interface subjected to loading con­ ditions (450kN, 10∘ ) and (1200kN, 15∘ ) is assessed considering an I-tube length of L1 ¼ 6m with the following sleeve geometrical parameters: i) Sleeve A - straight section of 0:5m without curved section; ii) Sleeve B - straight section of 0:5m þ curved section with sleeve radius varying from RAB ¼ 1; 2; …; 9; 10m. The initial contact angle φA , the arc-length s1 , the reaction force NA , the tension TA 14

Marine Structures 71 (2020) 102707

Y. He et al.

Table 4 I-tube length influence in the top connection response (450kN, 10∘ ). L1 ðmÞ

φ(∘) A s1 ðmÞ

2

3

4

5

6

7

8

2.91 2.13

2.60 3.08

2.47 4.06

2.39 5.05

2.35 6.05

2.31 7.04

2.29 8.04

Þ

0.046

0.028

0.019

0.015

0.012

0.009

0.007

NA ðkNÞ

16.76

16.77

16.78

16.78

16.78

16.79

16.79

2.55

2.63

2.66

2.68

2.70

2.70

2.71

κ0 ðm

1

ΔLCD ðmmÞ

Table 5 I-tube length influence in the top connection response (1200kN, 15∘ ). L1 ðmÞ

φ(∘) A s1 ðmÞ

2

3

4

5

6

7

8

2.16 2.02

1.84 2.98

1.69 3.96

1.61 4.94

1.56 5.94

1.52 6.93

1.50 7.93

Þ

0.081

0.049

0.034

0.024

0.017

0.011

0.004

NA ðkNÞ

27.39

27.46

27.52

27.57

27.59

27.61

27.61

ΔLCD ðmmÞ

2.58

2.66

2.68

2.71

2.71

2.72

2.73

κ0 ðm

1

Fig. 12. I-tube length influence in the curvature distribution (Curved sleeve radius RAB ¼ 8m).

and the riser end-fitting curvature κ0 are calculated by the numerical solution procedure in Fig. 4 with the given loading conditions. Once the global problem is solved, the riser/bend stiffener section length variation ΔLCD (defined by Eq. (20)) in relation to the initial length LICD ¼ 2:8m and the contact force fAB (defined by Eq. (13)) along section AB can be obtained. Tables 2 and 3 present the calculated results φA , s1 , κ0 , NA , ΔLCD and fAB of the top connection system for loading conditions (450kN, 10∘ ) and (1200kN, 15∘ ) respectively. It can be observed that φA , s1 , NA , ΔLCD and fAB decrease as the sleeve radius increases. The tension TA presents no variation with the sleeve radius and is given by 449:9kN and 1199:2kN for both load cases, respectively. The smallest radius results in the highest contact force between the riser and the sleeve. Fig. 10 shows the initial contact angle φA and the end-fitting curvature κ0 versus the sleeve radius RAB for both load cases. For the end-fitting curvature, higher sleeve radius lead to smaller values of κ0 for both loading conditions, which may be an important consideration for preliminary design purposes. In terms of the contact angle φA , it can be observed that as the sleeve radius decreases below a certain limit value, 1:6m for (1200kN, 15∘ ) and 2:6m for (450kN, 10∘ ), the riser does not interact with the curved sleeve section and enters into direct contact with the straight section, that presents an inclination angle of 7∘ . Below this sleeve radius limit in sleeve B , the riser curvature presents the highest values: 0:38m 1 for (450kN, 10∘ ) and 0:65m 1 for (1200kN, 15∘ ), observed in Fig. 11, as calculated for the straight sleeve configuration A . The figure also presents the curvature distribution for other sleeve geometries (RAB ¼ 4; 7 and 10m) where, as expected, higher sleeve radius lead to lower curvature values in the curved section followed by zero curvature in the 0:5m straight section. It is important to highlight that the 15

Marine Structures 71 (2020) 102707

Y. He et al.

Fig. 13. Initial curvature versus I-tube length (Curved sleeve radius RAB ¼ 8m).

curvature distribution in the bend stiffener region is not affected by the sleeve geometry. 4.3. I-tube length influence To investigate the I-tube length influence in the top connection system response, a sleeve geometry with a curved section radius given by RAB ¼ 8m and a 0:5m straight section is considered combined with the following values of I-tube lengths: L1 ¼ 2; 3; 4;…;8m. The same loading conditions employed in the previous section are used here. The initial contact angle φA , the arc-length s1 , the reaction force NA , the riser end-fitting curvature κ0 and the riser/bend stiffener section length variation ΔLCD in relation to the initial length LICD are presented in Tables 4 and 5. The length variation presents a small influence in the reaction force NA and a slight increase in the relative sliding ΔLCD is observed as the I-tube length increases. The tension TA presents no variation with the I-tube length and is given by 449:9kN and 1199:2kN for both load cases, respectively. As there is no variation in the curvature and tension in the sleeve region, the contact force fAB is constant and given by 56kN=m and 150kN=m for each load case, respectively. The flexible riser curvature distribution in the I-tube and sleeve region is presented in Fig. 12 for both loading conditions. A decrease in the length between the end-fitting and the riser initial contact point A with the sleeve leads to a larger curvature variation in the region, which is highlighted in Fig. 13, demonstrating that the end-fitting curvature κ0 decreases as the I-tube length increases. 5. Conclusions A large deflection beam formulation is developed for riser-bend stiffener top connection considering the riser interaction with an Itube. The end-fitting is assumed clamped in the I-tube top and the riser interacts with a straight rigid surface followed by a curved section in the bellmouth region. The riser and bend stiffener are assumed to have the same deflection as no gap is considered. The formulation incorporates the riser nonlinear bending behavior and nonlinear elastic bend stiffener polyurethane and allows the calculation of the contact forces between the riser/sleeve and riser/bend stiffener sections. The mathematical formulation of the statically indeterminate system results in three systems of coupled differential equations combined with corresponding multipoint boundary conditions. An iterative numerical procedure has been presented and employed to solve the boundary value problem with the Mathematica package. A case study is carried out with a 7” flexible riser protected by a 1.8 m bend stiffener connected to an I-tube with 7∘ inclination and subjected to extreme loading conditions. A parametric assessment is performed to evaluate the influence of the sleeve shape and I-tube length on the riser curvature distribution. It has been observed that the end-fitting curvature is affected by both parameters, where, i) as the sleeve radius RAB increases, the initial curvature k0 decreases and, ii) as the I-tube length L1 is increased, the initial curvature decreases. It has also been observed that the sleeve radius not only controls the initial contact angle and curvature distribution in the contact region, but also that, below a given radius, the riser does not interact with the curved section but directly contact the straight sleeve region, which leads to a peak in the riser curvature. The contact force between the riser and the sleeve is highly affected by its radius but is not influenced by the I-tube length. The proposed model allows a more realistic assessment of the flexible riser top connection response, contributing to a better understanding of the complex riser interaction with the bend stiffener and I-tube, which can also be employed for preliminary design. Acknowledgments The authors acknowledge the support of CAPES (Coordination for the Improvement of Higher Education Personnel) and CNPq (National Council of Scientific and Technological Development). 16

Marine Structures 71 (2020) 102707

Y. He et al.

References [1] Longo C, Neto S, Paula M, Lopes F, Godinho C. Albacora leste field - subsea production system development. In: Offshore technology conference; houston, Texas; 2006. Offshore Technology Conference, OTC-18044-MS. [2] Clevelario J, Sheldrake T, Pires F, Falcao G, Souza I, Aquino F. Flexible riser outer sheath full scale wearing simulation and evaluation. In: Offshore technology conference; houston, Texas; 2009. Offshore Technology Conference, OTC-20099-MS. [3] Boef W, Out J. Analysis of a flexible riser top connection with bend restrictor. In: Houston, Texas: offshore technology conference; 1990. p. 131–42. OTC-6436MS. [4] Tanaka R, Silveira L, Novaes J, Barros E, Martins C. Bending stiffener design through structural optimization. In: Proceedings of the 28th international Conference on offshore Mechanics and arctic engineering, vol. 3. New York: American Society of Mechanical Engineers; 2009. p. 411–8. OMAE2009-79505. [5] Sodahl N, Ottesen T. Bend stiffener design for umbilicals. In: Proceedings of the 30th international Conference on offshore Mechanics and arctic engineering, vol. 4. New York: American Society of Mechanical Engineers; 2011. p. 449–60. OMAE2011-49461. [6] Drobyshevski Y. Investigation into non-linear bending of elastic bars with application to design of bend stiffeners. Mar Struct 2013;31:102–30. [7] Vaz M, Lemos C, Caire M. A non-linear analysis formulation for bend stiffeners. J Ship Res 2007;51(3):250–8. [8] Caire M, Vaz M, Costa M. Bend stiffener nonlinear viscoelastic time domain formulation. Mar Struct 2016;49:206–23. [9] Caire M, Vaz M. A nonlinear viscoelastic bend stiffener steady-state formulation. Appl Ocean Res 2017;66:32–45. [10] Ruan W, Bai Y, Yuan S. Dynamic analysis of unbonded flexible pipe with bend stiffener constraint and bending hysteretic behavior. Ocean Eng 2017;130: 583–96. [11] API 17L1. Specification for flexible pipe ancillary equipment. latest ed. Washington: American Petroleum Institute; 2014. [12] API 17L2. Recommended practice for flexible pipe ancillary components. latest ed. Washington: American Petroleum Institute; 2014. [13] Nogueira V, Pires F, Aquino F, Clevelario J, Sheldrake T. Numerical analysis assessment of the contact pressure between the flexible pipe outer sheath and the Itube interface equipment for wearing research program. In: Proceedings of the 30th international Conference on offshore Mechanics and arctic engineering, vol. 4. New York: American Society of Mechanical Engineers; 2011. p. 247–53. OMAE2011-49267. [14] Pires F, Nogueira V, Aquino F, Clevelario J, Sheldrake T, Takey T. Development of a test methodology and a design tool to predict the wear rate of flexible risers outer sheaths installed in I-tubes. In: Proceedings of the 31th international Conference on offshore Mechanics and arctic engineering, vol. 3. New York: American Society of Mechanical Engineers; 2012. p. 733–9. OMAE2012-83874. [15] Orcina. OrcaFlex help file and user manual. UK. 2019. https://www.orcina.com/. [16] Hou Y, Yuan J, Zhang Y, Tan Z, Sheldrake T. Direct hang-off model to evaluate fatigue damage at riser hang-off. In: Proceedings of the 32th international Conference on offshore Mechanics and arctic engineering, 4B. New York: American Society of Mechanical Engineers; 2013. OMAE2013-10847. [17] Elosta H, Gavouyere T, Garnier P. Flexible risers lifetime extension: riser in-service monitoring and advanced analysis techniques. In: Proceedings of the 36th international Conference on offshore Mechanics and arctic engineering, 5A. New York: American Society of Mechanical Engineers; 2017. OMAE2017-62700. [18] He Y, Lu H, Vaz M, Caire M. Flexible riser top connection analysis with I-tube interface and bending hysteresis effect. In: Proceedings of the 38th international Conference on offshore Mechanics and arctic engineering, vol. 4. New York: American Society of Mechanical Engineers; 2019. OMAE2019-95826. [19] Hibbit D, Karlsson B, Sorensen P. ABAQUS theory manual; 6.14 ed. Providence, RI.: ABAQUS; 2014. [20] He Y, Caire M, Vaz M. An inverse problem methodology for multiple parameter estimation in bend stiffeners. Appl Ocean Res 2019;83:37–47. [21] Wolfram Research. Wolfram Mathematica; 10.0 ed. Champaign, Illinois: Wolfram; 2015. [22] 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;36:265–75.

17