Dynamics of a taut mooring line accounting for the embedded anchor chains

Ocean Engineering 121 (2016) 403–413

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Dynamics of a taut mooring line accounting for the embedded anchor chains Lingzhi Xiong a,b, Jianmin Yang a,b,n, Wenhua Zhao c a

State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, China Collaborative Innovation Center for Advanced Ship and Deep-sea Exploration, Shanghai Jiao Tong University, Shanghai 200240, China c Faculty of Engineering, Computing and Mathematics, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia b

art ic l e i nf o

a b s t r a c t

Article history: Received 14 November 2015 Received in revised form 24 March 2016 Accepted 7 May 2016

Mooring line responses have been widely analyzed in the past three decades. Most of these studies have assumed that the anchor point is fixed at the seabed. However, with the increasingly employment of the taut mooring system, the suction anchors are more and more often used. To achieve an appropriate hold capacity, the anchor points are embedded several meters beneath the seabed, making part of the mooring lines embedded, whereby complex soil-chain interaction occurs. The soil-chain interaction has not been incorporated in previous mooring dynamic analyses. In this study, a dynamic method using the lumped mass model was used to consider the soil-chain interaction for the embedded anchor chain. Validation has been conducted against both analytical results and those from the commercial code. To clarify the influence of the embedded chain, case studies were conducted to compare the line's dynamic response at the anchor point, touchdown point and fairlead in three different configurations. Recommendations have been given on two frequently-used in simplifying the mooring dynamic calculation. & 2016 Published by Elsevier Ltd.

Keywords: Taut mooring line Dynamic analysis Soil-chain interaction Lumped mass method

1. Introduction Mooring system is of vital importance for offshore floating facilities, which is usually divided into three categories based on line profiles: the catenary mooring system, semi-taut mooring system and the taut mooring system. Most of the mooring systems under operation is catenary or semi-taut. However, there is an increasing application of taut mooring systems (TMS) in deep waters. To hold the large vertical pullout loads in TMS, embedded anchors with large uplift capacity are used, such as suction anchors, suction embedded plate anchors and vertically loaded anchors. The padeyes on these anchors are designed at an appropriate penetration depth beneath the mud line, to maximize holding capacity. In such a scenario, part of the mooring line will be embedded in the seabed and strongly interact with the soil, which is a very complex process and may cause failures of mooring system (Arslan et al., 2015; O'Loughlin et al., 2015). As such an example, big holes have been observed near all the 9 suction anchors of an FPSO (Floating Production Storage and Offloading unit) operating in West Africa due to trenching by the embedded anchor chain (Bhattacharjee n Corresponding author at: State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, China. E-mail address: [email protected] (J. Yang).

http://dx.doi.org/10.1016/j.oceaneng.2016.05.011 0029-8018/& 2016 Published by Elsevier Ltd.

et al., 2014), which is illustrated in Fig. 1. The trenching induced a decrease of the anchor's holding capacity and further affected the integrity of the mooring system (Arslan et al., 2015). There have been studies emphasizing on the chain-soil interactions. Vivatrat et al. (1982) proposed a method to predict the profile of the anchor chain, but no details on the computation procedure was provided. Yen and Tofani (1984) conducted model test to investigate the soil resistance to the movements of the stud link chain. The chain was dragged in the soil in three different ways: sliding, cutting and surface dragging. Dimensionless parameters had been derived from the tests and can be further used in the calculation of the soil resistance force. Degenkamp and Dutta (1989) employed an incremental-integration method to calculate the line tension at the anchor point and predicted the line profile inside the soil. The line was divided into two parts: the embedded chain and the suspended component. Each part was calculated separately. By comparing the numerical results with laboratory results, some parameters for the chain were recommended. Those parameters were further applied by Neubecker and Randolph (1995), whereby they compared the results obtained from a closed-formed expression with the laboratory results. In their study, the self-weight of the chain has been neglected in the expression. Based on the closed-form expression, Neubecker and O'Neill (2004) presented a simple but quick method to evaluate the chain slippage versus load. However, Martins and Lages (2014)

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L. Xiong et al. / Ocean Engineering 121 (2016) 403–413

model is adopted in this study. The present study carried out the static and fully-dynamic analysis for a mooring line. A general expression has been proposed to calculate both the line suspended in water and embedded in soil simultaneously. The soil resistance and chain elongation has been taken into account. Based on the proposed method, case studies are conducted to elucidate the influence of the embedded mooring chain on the static and dynamic analysis. Recommendations have been given on two frequently-used simplifications of mooring configuration, which can avoid calculating the embedded line.

Fig. 1. Illustration of trenching.

indicated that the chain weight is important in soft soils. An expression in Frenet-Serret frames was derived to predict the threedimensional (3D) inverse catenary in soil. It is regarded as an extending of the expression provided by DNV Recommended Practices (DNV, 2002). Wang et al. (2010) introduced a 3D quasi-static method to study anchor chain slippage during pretension stage. This method coupled the inverse catenary chain with the catenary mooring line during the analysis. Different pretension levels and pretension angles at the touchdown point were compared. It is concluded that both higher pretension loads and lower angles can reduce the in-service slippage of the chain in soil. More recently, O'Loughlin et al. (2015) pointed out that recent field observations have showed that the trenching phenomenon will significantly reduce the anchor system capacity, which challenges the integrity of mooring systems. It is suggested that a better modelling of the chain-seabed interaction should be incorporated. Although efforts have been made to clarify the chain-soil interaction, it is still not clear how the complex interaction process works (Wang et al., 2010). In order to obtain an accurate prediction mooring line response and to provide a better understanding of the dynamics of a mooring system, it is necessary to include the contribution from the embedded anchor chain. Plenty of numerical studies have been conducted for the mooring dynamic analysis. However, most of these studies focus on the suspended component in the water such as the method adopted in commercial codes Riflex, Cable3D and Orcaflex (Fylling et al., 2008; Ma and Webster, 1994; Orcina, 2009) . In these studies, anchors are assumed to be a fixed point located at the surface of the seabed (Kim et al., 2013, 2005; Tahar and Kim, 2003) and the seabed is modelled as a rigid plane or an elastic spring (Chai et al., 2002; Ma and Webster, 1994). This process has ignored the chainsoil interactions, which may not be safe for a semi-taut or taut mooring system as stated above. There are two dominating treatments of the mooring line in the dynamic analysis, i.e., the slender rod model or lumped mass model. The governing equation of the slender rod is solved by the finite element method (FEM) in many studies (Chai and Varyani, 2006; Chen et al., 2001; Garrett, 1982; Tahar et al., 2002). The slender rod model has shown its versatility in accurate spatial discretization and handling various boundary conditions. But the mathematic expression is complex, especially when dealing with large elongation problem (Chai and Varyani, 2006). In the lumped mass model, mooring line is spatial discretized into lumped mass nodes which are connected by massless springs. Discretization in this way simplifies the problem and provides numerical efficiency, which has been performed in many other studies (Chai et al., 2002; Huang, 1994; Leonard and Nath, 1981; Low and Langley, 2006; Orcina, 2009). Furthermore, each item in the lumped mass model has a clear physical meaning, which facilitates the inclusion of the soil-chain interactions. As a consequence, the lumped mass

2. Mathematical formulations The lumped mass method is widely used in mooring line analyses. It has been addressed in detail in previous studies (Chai et al., 2002; Huang, 1992; Orcina, 2009). In this study, the associated equations and notations are addressed in a form similar to Low and Langley (2006). The bending and torsion effects are considered to be negligible and thus are ignored in this study. In the lumped mass method, mooring lines are divided into several segments. The weight of each segment and the external force distributed on the segment are assumed to be lumped at two end nodes, which are connected by a massless spring. Fig. 2 illustrates the line model, whereby the nodes is arbitrarily numbered from 1 to 3, connecting the element j and k. The node coordinates is represented by the vector y1⃗ , y2⃗ and y3⃗ . Each node should reach to an equilibrium position during both static and dynamic status. The governing equation for each node is

My¨ =F,

(1)

where M represents the mass matrix and F refers to the external force at each time step. As shown in Fig. 3, the external force consists of the elastic tension from two adjacent elements Tj⃗ − 1 and D D I I Tj⃗ , wet weight ( W⃗ j − 1, W⃗ j ), hydrodynamic loads F j⃗ − 1, F j⃗ and F j⃗ − 1, F j⃗ ⃗ , j − 1 and Fsoil ⃗ , j ) for the embedded chains. The and soil resistance ( Fsoil

details in dealing with these external forces will be addressed in the following sections. 2.1. Elastic tension For an arbitrarily element, i.e., element j in Fig. 3, the element's elastic energy can be given by

V j=

2 1 EA y2⃗ − y1⃗ − L j , 2 Lj

(

)

(2)

where L j is the unstretched length, E the elasticity modulus, A the

Fig. 2. Illustration of the lumped mass method (Low and Langley, 2006).

L. Xiong et al. / Ocean Engineering 121 (2016) 403–413

1 M2j = ρπD2L j C An [I−τ ⃗T τ ⃗], 4

405

(7)

where I is the identity matrix. 2.4. Soil resistance The soil resistance Fsoil is composed of two perpendicular components: the normal resistance Q and the tangential resistance Fl , whose magnitude is given by (Degenkamp and Dutta, 1989) Fig. 3. External force on lumped node and definition of line tension.

cross-section area. As shown in Fig. 3, the tension T1,⃗ j and T2,⃗ j denote the tension in Element j ( Tj⃗ ) that acts on the node No. 1 and the node No. 2, respectively. Tension T1q, j represents the components of tension

T1,⃗ j Ti⃗, j=⎡⎣Tix, j Tiy, j Tiz, j ⎤⎦ , which is illustrated in the last illustration of

(

)

Fig. 3 and given by

∂Vj EA (y2q −y1q ) EA (y2q −y1q ) T1q, j= = − . ∂y1q Lj y2⃗ −y1⃗

(3)

2.2. Wet weight Wet weight should be used to consider the element's gravity and buoyancy. For those embedded elements, the buoyancy contains a component that is contributed by soil. For simplicity, this component has been ignored in the present study. Then, the wet weight is given by

(4)

2.3. Hydrodynamic loads For the mooring line elements that suspended in water, they are subjected to the hydrodynamic loads, which can be divided D I into two part: the drag force ( F j⃗ ) and the inertia force ( F j⃗ ). Morison's equation is used to compute the two components (Faltinsen, 1993): D

Fj⃗ =

I

Fj⃗ =

1 ⎡ ⎤ ρDL j ⎣⎢Cdn Vr⃗ − Vr⃗ ∙τ ⃗ τ ⃗ Vr⃗ − Vr⃗ ∙τ ⃗ τ ⃗ +Cdt (Vr⃗ ∙τ ⃗) τ ⃗ |(Vr⃗ ∙τ ⃗) τ ⃗ |⎦⎥, 2

( ( )) ( )

1 ρπD2L j 4

{ (1 + C )u⃗̇−C (u⃗∙̇ τ ⃗) τ }⃗ , n A

n A

(5)

⃗ j −y1, ⃗ j y2, ⃗ j| 2, j −y1,

Q = (En D) qdl,

(9)

where Eτ and En are the multipliers to provide the effective width in the tangential and normal direction, dl is the segment length. The average normal pressure q and friction f per unit area of chain in clay soil can be taken as

f =Su,

(10)

q=Nc Su,

(11)

(12)

Su = Su0−kz,

where Su0 is the undrained shear strength at the seabed surface, k is obtained by the in-situ measurement and z is the depth. When the anchor point is beneath the mudline, z should be negative.

3. Solution of the equations The solution consists of two parts: (1) static analysis is used to obtain the equilibrium position of the whole line, which will then serve as a basis for the following calculations; (2) dynamic analysis, which is used to predict the line's motion and dynamic tension. 3.1. Static analysis The static problem can be expressed by the external force equilibrium at each node. Considering the acceleration y¨ is zero in static problem, the Eq. (1) can be rewritten into Eq. (13). D

I

⃗ =0, F ⃗=T ⃗+W⃗ +F ⃗ +F ⃗ +Fsoil (6)

where ρ denotes the water density, D the line's nominal diameter, Vr⃗ the relative velocity between the fluid particle and the line's segment, u⃗ ̇ the acceleration of water particle. The tangential unit vector is determined by τ ⃗= |y⃗

(8)

where Su is the undrained shear strength and Nc is the bearing capacity factor for plane strain failure. The undrained shear strength of soil is given by Eq. (12) and the possible changes of mud line due to trenching caused by the embedded anchor chain is not included in the present study.

where q represents the directions x, y or z.

⎧ → → ⎪ Ww , Line weight in water W = ⎨→ . → ⎪ ⎩Ws = Ww, Line weight in soil

Fl=(Eτ D) fdl,

. The dot between two vectors

refers to the product of two vectors. The operational symbol ‘| |’ refers to the norm of vector. In terms of the fluid particle velocity, it is determined at the axis of the structure based on the current speed and the incident wave, assuming the fluid is undisturbed by the structure. More details are available at Ran et al. (1999) The normal drag coefficient Cdn , tangential drag coefficient Cdt and normal added mass coefficient C An should be empirically determined. Meanwhile, the added mass matrix is derived from

(13)

where T ⃗ has been derived in Eq. (3). For those elements suspended D I ⃗ =0; in water, F ⃗ andF ⃗ can be expressed by Eqs. (5), (6) and Fsoil D I ⃗ can be derived while for those embedded in soil, F ⃗ , F ⃗ =0 and Fsoil from Eqs. (8) and (9). The Newton-Raphson iteration method (Ran, 2000) is used to solve the nonlinear Eq. (13). The Jacobian matrix needed for the iteration can be derived from the following derivation.

∂F ⃗ F ⃗ y⃗m +1 =F ⃗ (y⃗m )+ Δy⃗ , ∂y

(

)

⎛ ∂F ⃗ ⎞−1 Δy⃗ = − ⎜⎜ ⎟⎟ F ⃗ (y⃗m )= − J −1 ∙F ⃗ (y⃗m ), ⎝ ∂y⃗ ⎠

(14)

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L. Xiong et al. / Ocean Engineering 121 (2016) 403–413

y⃗m +1=y⃗m +Δy⃗ , where y⃗ is the displacement vector, m is the iteration step and ∂F ⃗

∂T ⃗

J = ∂y⃗ = ∂y⃗ +

⃗ ∂Fsoil =KTe+K S . ∂y⃗

The stiffness matrix KTe and K S are con-

tributed by the elastic tension and soil resistance, respectively. Once these two stiffness matrixes are obtained, the static problem will be solved. The derivation of matrix KTe is presented in Sections 3.1.1 and 3.1.2 provides the details to obtain the matrix K S . 3.1.1. Stiffness matrix K Te The matrix KTe can be derived from the elastic tension, which has been given by Eq. (3). Then, the stiffness matrix ( K1Te ) for Node 1 in element j is denoted by

K1Teqr =

⎛ EA ∂ 2Vj EA ⎞ EA (y2q −y1q )(y2r −y1r ) =δqr ⎜⎜ − ⎟⎟− , 3 ∂y1q ∂y1r Lj ⎠ y2⃗ − y1⃗ ⎝ y2⃗ −y1⃗

(15)

Fig. 4. Distribution of the soil resistance (The first subscript refers to the node number and the second one stands for the element number.).

mass method. For example, as shown in Fig. 4, the force F ⃗ acting ⃗ =F2,1 ⃗ = 1F⃗ on element 1 will be distributed to Node 1 and 2, with F1,1 2

⃗ denotes the node force acting on Node 2 in element 1. , where F2,1 The 6 × 6 soil stiffness matrix K S1 which is contributed by Fl will be derived by

where δqr is given by

⎧1,q=r δqr = ⎨ . ⎩ 0,q≠r

S1 K1:3,1 =

(16)

⎯⎯⎯⎯→ ∂ F1,1 = 0.5* ⎡⎣A + 0.5*B (y13 + y23 ) ⎤⎦ ∂y11 ⎯→ ⎯ ⎯→ ⎯ 2 ⎯→ ⎯ ⎯→ ⎯ ⎯→ ⎯ − y2 − y1 e1 − ( y2 − y1 ) (y11 − y21 )

Thus, the stiffness matrix for element j can be assembled by

⎯→ ⎯ ⎯→ ⎯ 3 y2 − y1

K1Te :

⎡ K Te − K Te ⎤ 1 1 ⎥ K Te= ⎢ . ⎢⎣− K1Te K1Te ⎥⎦

(17)

τ⃗=

y2⃗ −y1⃗ y2⃗ −y1⃗

,

⎛ y −y y −y ⎞ n⃗= ⎜⎜ 13 23 , 0 , 21 11 ⎟⎟, y2⃗ −y1⃗ ⎠ ⎝ y2⃗ −y1⃗

2

− y2⃗ − y1⃗ e2⃗ −(y2⃗ −y1⃗ )(y12 −y22 )

⎛ y −y y −y ⎞ Q⃗ =Qn⃗=⎡⎣A2 +0. 5B2 (y13 +y23 ) ⎤⎦ ⎜⎜ 13 23 ,0, 21 11 ⎟⎟, y2⃗ −y1⃗ ⎠ ⎝ y2⃗ −y1⃗

(20)

(21)

where

A1= − Eτ DSu0 dl, B1=Eτ Dkdl, A2 =En DNc Su0 dl, B2= − En DNc kdl,

(22)

The soil resistance force distributed on each segment is assumed to be lumped at the two end nodes equally in the lumped

3

,

y ⃗ −y ⃗ ∂F1,1⃗ =0. 25*B 2 1 +0. 5* ⎡⎣A+0. 5*B (y13 +y23 ) ⎤⎦ ∂y13 y2⃗ −y1⃗

S1 K1:3,3 =

2 − y2⃗ − y1⃗ e3⃗ −(y2⃗ −y1⃗ )(y13 −y23 )

y2⃗ − y1⃗ S1 = K1:3,4

∂F1,1⃗ S1 = − K1:3,1 , ∂y21

S1 = K1:3,5

∂F1,1⃗ S1 = − K1:3,2 , ∂y22

S1 K1:3,6 =

3

,

y ⃗ −y ⃗ ∂F1,1⃗ =0. 25*B 2 1 +0. 5* ⎡⎣A+0. 5*B (y13 +y23 ) ⎤⎦ ∂y23 y2⃗ −y1⃗ 2

y2⃗ − y1⃗ e3⃗ −(y2⃗ −y1⃗ )(y23 −y13 ) y2⃗ − y1⃗

Substitute Eqs. (10, 12) and (18, 19) into Eqs. (8) and (9), the soil resistance can be expressed as follows:

y ⃗ −y ⃗ Fl⃗ = − Flτ ⃗=⎡⎣ (A1+0. 5B1(y13 +y23 ) ⎤⎦ 2 1 , y2⃗ −y1⃗

y2⃗ − y1⃗

(18)

(19)

(23)

∂F1,1⃗ =0. 5* ⎡⎣A+0. 5*B (y13 +y23 ) ⎤⎦ ∂y12

S1 = K1:3,2

The element stiffness matrix can be further assembled to the global stiffness matrix. 3.1.2. Stiffness matrix K S In the static analysis, the tangential force is assumed to be along the segment and point to the padeye, while the normal force points upwards in the vertical plane and perpendicular to the segment. The tangential and normal unit vector is defined in Eqs. (18) and (19). As the nodes are numbered increasingly from the padeye to the fairlead, the tangential unit vector always deviates from the padeye. It should be noted that Eq. (19) is only valid for the mooring line in two-dimensional (2D) plane. And thus, the employment of the present method is restrict in 2D cases, although other equations (except Eqs. (24) and (30)) are valid for three-dimensional case. It is expected that the Eqs. (19), (24) and (30) should be defined in three-dimensional (3D) space, if someone wants to extend this study to solve 3D problems.

,

3

,

where e1⃗ ,e2⃗ ,e3⃗ represents the unit vector in x, y and z direction respectively. The 6 × 6 soil stiffness matrix K S2 which is contributed by Q will be derived by

⎯⎯⎯⎯→ ∂ Q1,1 = 0.5* ⎡⎣A2 + 0.5B2 (y13 + y23 ) ⎤⎦ ∂y11 ⎛ ⎯→ ⎯ ⎯→ ⎯ 2 − y2 − y1 + (y11 − y21)2 ⎜ (y23 − y13)(y11 − y21) , 0, ⎜ ⎯→ ⎯ ⎯→ ⎯ 3 ⎯→ ⎯ ⎯→ ⎯ 3 ⎜ y2 − y1 y2 − y1 ⎝ ⎯⎯⎯⎯⎯→ ∂ Q1,1 S2 = K1:3,2 =0. 5* ⎡⎣A2 +0. 5B2 (y13 +y23 ) ⎤⎦ ∂y12 ⎛ ⎞ ⎜ (y23−y13)(y12−y22 ) ,0, (y21−y11)(y22−y12 ) ⎟, 3 3 ⎜ ⎟ y2⃗ − y1⃗ y2⃗ − y1⃗ ⎝ ⎠ S2 = K1:3,1

⎞ ⎟ ⎟ ⎟ ⎠

L. Xiong et al. / Ocean Engineering 121 (2016) 403–413

⎛ ⎯⎯⎯⎯→ y21 − y11 ∂ Q1,1 ⎜ y13 − y23 S2 K1:3,3 = = 0.25*B2 ⎜ ⎯→ ⎯ , 0, ⎯→ ⎯ ⎯→ ⎯ ⎜ ⎯y − ⎯→ ∂y13 y1 y2 − y1 ⎝ 2

(

ε= − J1−1 f (∆y), +1 m ∆ytm+∆ t =∆yt +∆ t +ε ,

where , J1≈

(24)

⎞ ⎟ ⎟⎟ ⎠

+ 0.5* ⎡⎣A2 + 0.5B2 (y13 + y23 ) ⎤⎦ ⎛ ⎯→ ⎞ ⎯ ⎯→ ⎯ 2 2 ⎜ − y2 − y1 + (y13 − y23) (y23 − y13)(y21 − y11) ⎟ , 0, ⎜ ⎟ ⎯→ ⎯ ⎯→ ⎯ 3 ⎯→ ⎯ ⎯→ ⎯ 3 ⎜ ⎟ y2 − y1 y2 − y1 ⎝ ⎠ S1 S1 S2 S2 ⃗ =Q 2,1 ⃗ . The It is noted that K4:6 =K1:3 =K1:3 , K4:6 due toF1,1⃗ =F2,1⃗ , Q 1,1 S soil stiffness matrix K for element 1 is given by

K S =K S1+K S2,

(29)

+1 m +1 ytm+∆ t =yt +∆yt +∆ t ,

⎛ ⎯→ ⎞ ⎯ ⎯→ ⎯ 2 2 ⎜ y2 − y1 − (y13 − y23) (y23 − y13)(y11 − y21) ⎟ , 0, ⎜ ⎟ ⎯→ ⎯ ⎯→ ⎯ 3 ⎯→ ⎯ ⎯→ ⎯ 3 ⎜ ⎟ y2 − y1 y2 − y1 ⎝ ⎠ ⎯⎯⎯⎯→ ∂ Q1,1 S2 S2 = , K1:3,4 = − K1:3,1 ∂y21 ⎯⎯⎯⎯⎯→ ∂ Q1,1 S2 S2 = , K1:3,5 = − K1:3,2 ∂y22 ⎛ ⎯⎯⎯⎯→ y11 − y21 ∂ Q1,1 ⎜ y23 − y13 = − 0.25*B2 ⎜ ⎯→ ⎯ , 0, ⎯→ ⎯ ⎯→ ⎯ ⎜ ⎯y − ⎯→ ∂y23 y y2 − y1 1 ⎝ 2

)

+1 f ∆ytm+Δ =f ∆ytm+Δt +J1 ε, t

⎞ ⎟ ⎟⎟ ⎠

+ 0.5* ⎡⎣A2 + 0.5B2 (y13 + y23 ) ⎤⎦

S2 K1:3,6 =

) (

407

(25)

3.2. Dynamic analysis

∂F (yt ) 1 1 − M (t )= − K Te (yt ) + K S (yt ) − M (t ). ∂Δy β ∆t2 β ∆t2

As the Jacobian matrix is used to provide convergence of the algorithm, it does not need to be its exact value (Low and Langley, 2006). There are three components in the Jacobian matrix. The first component KTe denotes the contribution from the elastic tension. It can be obtained by the method which has been illustrated in Section 3.1.1. The second component K S is made up of two parts: K S1 and K S2. In the dynamic analysis, K S1 has the same expression as in static analysis. But K S2 is different due to the different direction assumption of the normal component in soil resistance. It will be solved later in this section. The third component is related to the mass matrix, which is easy to obtain. In taut mooring configuration, the mooring line experiences high tension all the time, which can lead to the assumption that the embedded chain is cutting the soil during the dynamic process. And therefore, the normal soil resistance and tangential component will be opposite to the segment normal velocity ( Vn⃗ ) and tangential velocity ( Vτ⃗ ), and given by

⎧ ⎪− Qn⃗ , V ⃗ ∙n⃗ >0 n , Q⃗ = ⎨ ⎪ ⎩ Qn⃗ , Vn⃗ ∙n⃗≤0

(30)

⎧ ⃗ ⎪− F τ , l ⃗ Vτ ∙τ ⃗≥0 . Fl⃗ = ⎨ ⎪ ⎩ Flτ ,⃗ Vτ⃗ ∙τ ⃗<0

(31)

In the static analysis above, the line's profile can been obtained, which will serve as the initial position of the dynamic simulation. The governing Eq. (1) can be rewritten to obtain the dynamic response of the mooring line in time domain, which is given by:

The 6 × 6 soil stiffness matrix K S2 in dynamic analysis can still be derived by Eqs. (23) and (24).

M (t ) y¨ (t )=F (t ),

4. Validation of the numerical model

(26)

where t represents time, M the mass matrix and F the external force. The mass matrix M should include both the node mass and the added mass matrix.The external resultant force F (t ) can be obtained by Eq. (13). The numerical solution of Eq. (26) is based on an incremental procedure using dynamic time integration scheme according to the Newmark- β method (Newmark, 1959). NewtonRaphson iteration is used to assure force equilibrium at every time step. The Newmark- β method is expressed by

y¨t +∆ t =

⎛ 1 ⎞ 1 1 ∆y − ẏ −⎜ −1⎟ y¨ , β ∆t2 β∆t t ⎝ 2β ⎠ t

The validation of the developed model has been carried out in both static and dynamic analyses. At each analysis stage, the embedded chain component and the suspended component are validated. 4.1. Validation of the static analysis results 4.1.1. Embedded Chain Neubecker and Randolph (1995) presented the analytical expression to predict the profile of the embedded mooring line. The soil bearing resistance is assumed to increase linearly with depth.

Q =(En D) dlNc Su=(En DdlNc k ) z=k′z,

⎛ ⎛α ⎞ α α⎞ yṫ +∆ t = ∆y +⎜1 − ⎟ yṫ −⎜ −1⎟ ∆ty¨t , ⎝ ⎝ 2β ⎠ β∆t β⎠

(27)

yt +∆ t =yt +∆y, where α and β are the parameters to adjust the accuracy and stability of the algorithm. Substituting Eq. (27) into Eq. (26) yields

(

)

f ∆ytm+Δt =F (yt +∆ t )−M (t +Δt ) y¨t +∆ t

⎛ ⎛ 1 ⎞ ⎞ 1 1 yṫ −⎜ −1⎟ y¨t ⎟, =F (yt +∆ t )−M ⎜t +Δt )( ∆y − 2 β ∆t β∆t ⎝ 2β ⎠ ⎠ ⎝

(28)

By using the Newton-Raphson iteration, ∆y can be solved from Eq. (28). From step t to t + Δt , the incremental of the nodes’ position Δyt +∆ t should be iteratively solved.

(32)

where the gradient of the soil bearing resistance is denoted by k′. Then, the profile of the embedded chain can be expressed by the T non-dimensional tension at the anchor point T *= DQa , the chain's inclination angle at seabed surface θ0 and the non-dimensional x z coordinate x*= H , z*= H , where H is the depth of the attachment. It should be noted that the self-weight of the chain was ignored. The non-dimensional expression of the profile is given in Eq. (33).

⎡ ⎤ T *θ 02 ⎢ 1+ +1 ⎥ 2 2 x*=ln ⎢ ⎥, T* ⎢ z*+ T *θ 02 + (z*)2 ⎥ ⎣ ⎦ 2

(33)

Table 1 lists the case matrix for the static analysis of the embedded chain. The inclination angle at seabed surface θ0 ranges from 1° to 30 ° and the non-dimensional tension at the anchor

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Table 1 Validation matrix for static analysis of the embedded chain.

Attachment tension T * Angle θ0 /deg

Case 1

Case 2

Case 3

Case 4

Case 5

Case 6

2 1

2 10

2 30

20 1

20 10

20 30

point is 2 and 20, representing a low pretension and high pretension configuration respectively. Numerical simulations are conducted based on the developed code in this study and 20 elements are employed in validating the embedded part. The obtained results are compared with the analytical results based on Eq. (33). The result comparison is illustrated in Figs. 5 and 6. Good agreement has been achieved. It is indicated that the present method can provide an accurate prediction of the embedded line's profile. 4.1.2. Suspended mooring component To validate the present method in predicting the line's profile in water, a mooring line is selected from a typical taut mooring system. The properties of the mooring line are tabulated in Table 2 and the water depth is 300 m. Numerical studies are conducted to obtain the static configuration of the suspended mooring component in the developed code. The horizontal distance between the fairlead and the anchor point varies from 190 m to 210 m, with a step of 10 m. Line's profile and tension are compared with the results based on commercial code Orcaflex. As shown in Fig. 7, excellent agreement has been achieved. Table 3 presents the line tension at the fairlead, including the total tension and the component in X and Z direction, Fx and Fz . The errors of the total tension are all within 0.3%, which indicates that the present method can predict not only the line's profile, but also has a good estimation of the line tension.

Fig. 6. Comparison of the embedded chain profile for T *=20 . Table 2 Mooring line properties. Parameter

Length (m) Diameter (mm)

Ground chain 100 Mid-section 218 Fairlead chain 50

147 208 147

Wet weight (kg/m)

EA (MN) Drag Coefficient Cdn

367 7.8 367

1220 348 1220

2.4 1.8 2.4

4.2. Validation of the dynamic analysis Dynamic analysis has been conducted in this section with the same configuration of the mooring line as shown in Table 2. Forced sinusoidal oscillation is imposed to the top of the mooring line in X direction, to excite the mooring line actions. Sinusoidal inputs with different oscillation frequencies are used as the excitation at the fairlead, which represent a low-frequency motion and wavefrequency motion, respectively. The configuration of the model set-up is illustrated in Fig. 8. The low-frequency motion has an amplitude of 12 m and a period of 100 s, while the wave-frequency motion has an amplitude of 6 m and a period of 10 s. Figs. 9 and 10 present the dynamic tension series at both the top and anchor point of the mooring line, under the low-frequency and wave-frequency excitation, respectively. In numerical

Fig. 7. Comparison of the line's profile in water (Fairlead at x ¼ 190 m, 200 m and 210 m, z ¼310 m). Table 3 Line tension at the fairlead. Fairlead position

Fx (kN)

Fz (kN)

Total tension (kN)

X¼ 190 m

252.1 251.2 885.2 885.7 4831.5 4824.1

661.8 659.9 1602.3 1603.7 7354.7 7346.2

708.2 706.1 1830.6 1832.0 8799.7 8788.5

X¼ 200 m X¼ 210 m

Present Orcaflex Present Orcaflex Present Orcaflex

Error 0.29%  0.08% 0.13%

simulations, it will last a certain period to achieve a stable response results, i.e. around 12 s in this study. As shown in Figs. 9 and 10, the unstable stage at the beginning of the numerical simulation has been truncated, showing the stable responses only. Excellent agreement has been obtained when comparing the results in the present study with those calculated by Orcaflex.

5. Case study

Fig. 5. Comparison of the embedded chain profile for T *=2.

Aiming to clarify the influence of the embedded chain on the static and dynamic results, case study is conducted in this section.

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Fig. 8. Illustration of the model set-up.

The foregoing theory has been implemented on the mooring line tabulated in Table 2 to provide a comprehensive case study on the influence of the embedded chain. The undrained shear strength of soil is determined by Su= − 1.45∙z kPa (Andersen et al., 2005), which is presented in Fig. 11. In terms of the parameters Eτ andEn , the present study uses the recommendations from Degenkamp and Dutta (1989) ( Eτ =8.0, En=2.5). To simplify the simulation, the Nc is chosen to be 7.6 at any depth. The segment length for the embedded chain is 2 m and that for the suspended part ranges from 2 m to 20 m. Then, a total of 36 segments are employed in this case and approximate seven of them are buried in the soil. Fig. 10. Comparison of the line tension in dynamic analysis: amplitude ¼ 10 m, period¼ 10 s; (a) top tension, (b) anchor tension.

5.1. Influence of the embedded chain on the static analysis

Fig. 9. Comparison of the line tension in dynamic analysis: amplitude ¼ 12 m, period¼ 100 s; (a) top tension, (b) anchor tension.

Four different configurations of the mooring line are considered, whereby Case A is the normal configuration with the anchor point located beneath the mud line and part of the chain embedded in the soil, Case B is a simplification of Case A with the anchor point shifted to the mud line, Case C is another simplification of Case A by truncating the line at touchdown point and Case D ignores the soil-chain interaction. As shown in Fig. 12, the anchor point in Case A is located at point (0,0,  10 m), which means the attachment point is 10 m below the mud line. The anchor point in Case B is located at the seabed surface, (0,0,0). The line length in A and B keeps the same. The anchor point in Case C is located at the initial touchdown point as shown in Case A, while the line length is identical to suspended component of the mooring line in water in Case A. In Case D, the line length and anchor point is identical to that in Case A but the soil between −10 m ≤ z ≤ 0 m has been removed, which means the seabed is modelled at z = − 10 m . In all configurations, the initial position of the fairlead is located at the point (200, 0, 300). The Case A and C has the same suspended line profile in water and the profile in Case D is close to that in Case A, while the line profile of Case B shows a significant difference with the others. The line tension at the fairlead and the anchor point in three cases is calculated and compared in Table 4. It is noted that the line tension in Case A and C is almost the same, which is slightly lower than that in Case D but is significantly higher than that in Case B. The decrease of the tension from Case A to Case B at the fairlead and touchdown point (TDP) is 63.7% and 82.7% respectively. (The touchdown point in Case B and C coincide with the anchor point.)

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Fig. 11. Undrained shear strength, Su .

Fig. 12. Four different mooring line configuration. Table 4 Line tensions in different configuration. Config.

A B C D

Fairlead tension

Fig. 13. Stiffness curve in different configuration: (a) tension at fairlead, (b) tension at anchor point, (c) reduced tension from fairlead to anchor. Touchdown point tension

Fx (kN)

Fz (kN)

Total (kN)

Fx (kN)

Fz (kN)

Total (kN)

850.2 228.8 850.4 872.1

1544.9 598.2 1545.2 1584.9

1763.4 640.4 1763.8 1809.0

852.1 228.8 850.4 872.1

1034.4 41.4 1033.3 1028.1

1340.2 232.5 1338.2 1348.2

Fig. 13 presents the line's stiffness curves in different configurations, which shows the variations of the line tension against the position of the fairlead. Fig. 13(a) is the tension at fairlead. It is shown that when the distance between the anchor point and the fairlead is relatively small, all the three configurations provides similar tension. However, with the increase of the distance, the curve in Case B separates from the other curves and has a much smaller increment, indicating that the line stiffness is relatively small in Case B. As the distance grows further, the curves in Case A, C and D show a significant uprising after 200 m and keeps almost identical. This is because for a taut mooring line, the restoring force is mainly provided by the line's stretch and the soil force

acting on the line in Case A is minimal. The same tendency has been observed in the tension at the anchor point, as shown in Fig.13(b). Fig. 13(c) presents the difference between the tension at fairlead and anchor. It is an indication of the load loss from the fairlead to the anchor. The load loss is obvious larger in Case A than that in the other cases, while the other cases have a similar load loss. It is believed that the load loss is mainly caused by the weight of the suspended mooring segments. In Cases B, C and D, the horizontal component of the tension at fairlead keeps identical to that at anchor, but the vertical component is different. Considering the fact that the whole suspended weight is imposed on the fairlead, the similar suspended weight in Case B, C and D leads to a similar load reduction. It is more complicate in Case A, because of the effect of the embedded mooring chain. The tangential soil resistance on the embedded mooring chain is opposite to the pretension direction and induces a positive effect on the load reduction. In contrast, the normal soil resistance will decrease the loss. But the resultant soil effect on the present case has increased the load loss.

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It can be inferred that when dealing with mooring system including embedded line, simplification as in the Case B, which ignores the embedded depth and specifies the anchor point at the seabed surface directly, will introduce significant discrepancy of the stiffness. As to Case C and D, the line tension agrees well with Case A. 5.2. Influence of the embedded chain on the dynamic results Dynamic analyses are conducted for the four configurations stated above. Two forced sinusoidal oscillations in X direction are imposed to the top of the line. The excitation amplitude and period combination is selected as: (1) 12 m and 100 s; (2) 6 m and 10 s, to represent the low-frequency and wave-frequency motion respectively. The dynamic tension at the fairlead and anchor point are monitored and compared. Particularly, the position variation and the corresponding tension at the touchdown point in Case A are provided. The TDP is defined as the point on mooring line, which has a z-coordinate of zero during the dynamic analysis. Fig. 14 presents the dynamic response under the excitation amplitude of 12 m and period of 100 s. As shown in Fig. 14(a), the top tension in all cases is in the same phase with the fairlead motion. But the dynamic tension at the top of the mooring line in Case B is remarkably lower than that in other three cases, which implies the simplification of ignoring the embedded depth fails to predict the dynamic line tension. The line tension in Case A, C and D is almost identical to each other. The maximum tension at fairlead in the three cases is also identical in practice (Case A: 10,262.8 kN, Case C: 10,386.7 kN, Case D: 10,238.9 kN). Similar phenomenon is observed when analyze the anchor tension, which is shown in Fig. 14(b). Fig. 14(c) illustrates the response at the TDP, the X position (curve X.TDP in Fig. 14(c)) is under cyclic motion which is anti-phase to the motion at fairlead. The amplitude of TDP's motion is 5.65 m, which is relatively large when compared to the amplitude at fairlead (12 m). When the fairlead is superimposed with the wave-frequency motion (amplitude ¼6 m, period ¼ 10 s), as shown in Fig. 15, the tension response will be quite different. The top tension and anchor tension estimated by Configuration B still shows poor agreement with that in Case A. In Case A, C and D, the maximum top tension and anchor tension occurs at the farthest fairlead position. However, the minimum tension at fairlead and padeye appears around π ahead of the fairlead motion. The reason for the 8

out-of-phase phenomenon is that the line tension is related to not only the fairlead position, but also the direction and the magnitude of the segment's acceleration and velocity, due to the inertia force ( F I ), drag force ( F D ) and the soil resistance ( Fsoil ) on the mooring line. The acceleration and velocity are always in different phase with the fairlead position. In the wave-frequency case, the dynamic effects are significant. The external forces reach to their own peak at different phase. Thus, the summation of these force exhibits a complex pattern and peaks at different phase. In the low-frequency case, the dynamic effects are limited. The elastic force, which is influenced mainly by the fairlead position, dominates in the external force. Therefore, the tension has the same phase with the fairlead position, as shown in Fig. 14. The line tension in Case B is notably smaller than the other two cases, as shown in Fig. 15. Unlike the case in the low-frequency situation, the maximum top tension in Case A is 5604.3 kN, which is 4.4% and 2.8% lower than that in Case C and Case D, respectively. In terms of the average tension at fairlead, Case C and D is 4.6% and 3.9% lower than that in Case A. For the anchor tension, the maximum in Case C and D is 5.4% and 3.0% higher than that in Case A. The average in Case C and D is 3.1% and 3.7% lower than that in Case A. Therefore, truncating the mooring line at touchdown point

Fig. 14. Dynamic line response when oscillation amplitude of 12 m and period of 100 s: (a) tension at top, (b) tension at anchor point and (c) tension at TDP and TDP's motion response.

or ignoring the soil-chain interaction will overestimate the maximum line tension. Comparison between Case A and D shows the influence of the soil. It is noted that the load on the fairlead and anchor without considering the soil is bigger than that when considering the soil effect. This implies that when trench is developed, anchors will not only experience a possible loss of capacity, but also will be exposed to a heavier load than before. Thus, the integrity of the anchor may be threatened. In terms of the X position of the TDP, it varies with the fairlead motion, ranging from 6.7 m to 15.4 m, as shown by the curve X. TDP in Fig. 15(c). The motion is no longer sinusoidal but still nearly anti-phase to the fairlead's motion. This means that when the embedded mooring chain is accounted for, the mooring line's boundary condition at the seabed is no longer a fixed point but a point with obvious cyclic motion.

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been conducted against a commercial code and an analytical expression. Good agreement has been achieved, which proves the accuracy and reliability of the proposed model. To clarify the influence of the embedded mooring chain, the static and dynamic response of three simplified mooring line configurations are compared with the original configuration. The shift of pad-eye leads to underestimation of mooring loads at fairlead and anchor in both static and dynamic analysis, and thus not recommended for the mooring analysis. The simplification of truncating the mooring line at the mud line may induce a certain discrepancy, which is acceptable for those analysis that are not sensitive to mooring loads, i.e. the preliminary mooring design. It should be noted that the touchdown point should be predicted accurately before the truncation. For the simplification in which the soil-chain interaction is ignored, although it will overestimate the maximum line tension, the discrepancy is relatively small. In low-frequency, the top tension keeps an identical phase with the fairlead motion. But in wave-frequency case, strong dynamic effects occurs in the top and anchor tension. The minimum tension occurs around π ahead of the fairlead motion. 8

Although the embedded chain segment accounts for a small part of the total line, it is observed to have a certain influence on the static and dynamic performance of the mooring line. The soil resistance on the embedded chain will increase the load loss from fairlead to anchor, reducing the effective load on the anchor. In addition, the cases without embedded parts have a boundary condition of fixed point at seabed, but the mooring line has a cyclical motion at the seabed in the case with the embedded parts. This may further cause the trenching problem. These differences will further influence the tension acting on the anchor and the fairlead, which is related to the safety of the mooring system.

Acknowledgement This work was financially supported by National Natural Science Foundation of China (Project no. 51239007). The third author is supported by the Shell EMI offshore engineering initiative at the University of Western Australia. These sources are greatly appreciated.

References

Fig. 15. Dynamic line response when oscillation amplitude ¼6 m, period¼ 10 s: (a) tension at top, (b) tension at anchor point and (c) tension at TDP and TDP's motion response.

The significant motion at TDP will definitely contribute to the trenching phenomenon near the anchor, which reduces the anchor capacity and may cause economy loss. On the other hand, the motion introduces uncertainty in predicting the mooring line tension at the fairlead and pad-eye. This further makes it difficult to estimate both the anchor performance and the reliable response of the floating structures when coupled with mooring system.

6. Conclusion The present study proposed a numerical method to predict the dynamic responses of mooring lines, which can account for the embedded anchor chains. The simple and efficient lumped mass method was applied to model the mooring line. Validation has

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