Modeling of cable-moored floating breakwaters connected with hinges

Engineering Structures 33 (2011) 1536–1552

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Modeling of cable-moored floating breakwaters connected with hinges Ioanna Diamantoulaki ∗ , Demos C. Angelides Department of Civil Engineering, Aristotle University of Thessaloniki, University Campus, Thessaloniki 54124, Greece

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Article history: Received 29 June 2010 Received in revised form 30 November 2010 Accepted 30 January 2011 Available online 1 March 2011 Keywords: Cable-moored floating breakwaters Hinge joints Generalized modes Cable tensions Effectiveness Response Fluid–structure interaction

abstract In the present paper, the overall performance of a cable-moored array of floating breakwaters connected by hinges is investigated under the action of monochromatic linear waves in the frequency domain. The performance is defined here as: (i) demonstration of acceptable levels of both response of the array and its effectiveness and (ii) non-failure of the mooring lines. The numerical analysis of the array is based on a 3D hydrodynamic formulation of the floating body coupled with the static and dynamic analyses of the mooring lines. The motions of the array of floating breakwaters associated with the hinge vertical translations are considered in the hydrodynamic analysis with the implementation of appropriate generalized modes. The stiffness and damping coefficients caused by the mooring lines in both rigid and generalized degrees of freedom are derived here in the general form. A rigorous parametric study is carried out in order to investigate the effect of different configurations (number of hinge joints and number of mooring lines) on the performance of the cable-moored array of floating breakwaters. Moreover, the performance of the various configurations of cable-moored floating breakwaters connected by hinges examined is compared with the performance of a single cable-moored floating breakwater with no hinges. It is found that the number of hinge joints and mooring lines have a direct effect on the performance of the cable-moored array of floating breakwaters. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction The traditional type of breakwater is the bottom-founded structure. The construction of this type of breakwater is not always economical, especially for deep water depths; furthermore, breakwaters of this type are potentially associated with environmental problems, such as intense shore erosion, water quality problems and aesthetic considerations. The aforementioned disadvantages motivated the search for an alternative type of breakwater, namely the floating ones. The application of such kind of structures is continuously increasing, because of the fast and inexpensive construction as well as the possibility of mobility and reallocation. The floating breakwaters are usually pile-restrained or cable-moored. Reviews of the general design of floating breakwaters are presented in [1–4]; furthermore, Isaacson [4] provides an overview of wave effects on floating breakwaters. As far as the hydrodynamic analysis of the floating body is concerned, 2D models have been developed that describe the complete linear hydrodynamic problem of the wave–structure interaction [4–14]. These 2D models use four methods: (i) finite element method, (ii) boundary integral method, (iii) finite differences using Boussinesq type equations, (iv) volume of fluid and (v) particle methods. Analytical solutions of the hydrodynamic problem are available for simple geometries and

∗

Corresponding author. Tel.: +30 2310 995702; fax: +30 2310 995740. E-mail address: [email protected] (I. Diamantoulaki).

0141-0296/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2011.01.024

regular waves [15]. Loukogeorgaki and Angelides [16] and Diamantoulaki et al. [17] used a 3D hydrodynamic model to investigate the performance of floating breakwaters. A 3D analysis for a V-shaped floating breakwater was used by Briggs et al. [18], including hydroelasticity. The phenomenon of hydroelasticity has also been investigated in various studies using (i) 2D linear theories [19–22], (ii) 2D non-linear theories [23,24], (iii) 3D linear theories [25–28] and (iv) 3D non-linear theories [29,30]. Bishop and Price [31] used free undamped ‘‘wet’’ bending modes, while Gran [32] used orthogonal modes of a uniform beam to express the vertical translations of a slender ship. Newman [33] extended the linearized frequency domain analysis of wave diffraction and radiation for a 3D body in a fixed mean position to a variety of deformable body motions using an expansion in arbitrary modal shape functions. Jensen and Pedersen [23] developed a non-linear quadratic strip theory formulated in the frequency domain for predicting wave loads and ship responses in moderate seas. Du [34] presented a complete frequency domain analysis for linear 3D hydroelastic responses of floating structures moving in a seaway and Fu et al. [35] used 3D linear hydroelasticity theory to predict the response of flexible interconnected structures. Finally, Wu et al. [29] used a 3D non-linear hydroelasticity theory for both frequency and time domain analyses. Many researchers have dealt with the application of hydroelasticity theories in the analysis of VLFS [36–40], since hydroelasticity is very important for this kind of structures. A comprehensive review of hydroelasticity theories

I. Diamantoulaki, D.C. Angelides / Engineering Structures 33 (2011) 1536–1552

is presented in [41], and in each of the aforementioned studies regarding hydroelasticity the floating body is assumed to be freefloating (unrestrained). The effect of hydroelasticity on flexible floating breakwaters has been studied by Williams et al. [10] for one compliant beam-like breakwater (idealized as 1D beam of uniform flexural rigidity) using an appropriate Green function, and by Abul-Azm [42] using an eigenfunction approach; both of the researches use 2D hydrodynamic analysis. Diamantoulaki et al. [43] and Manolis et al. [44] have also investigated the effect of hydroelasticity on flexible floating breakwaters using 3D hydrodynamic analysis. Furthermore, the effect of hydroelasticity phenomena on floating breakwaters connected by hinge joints has been investigated numerically by Newman [33], Lee and Newman [45] and Diamantoulaki and Angelides [46,47]. In each of these hydroelastic studies the floating body is assumed to be either free [33,45,47] or pile-restrained [46]. Besides free and pile-restrained floating bodies, cable-moored ones have also been investigated by several researchers, assuming rigid body conditions as mentioned in [48]. Chakrabarti and Cotter [49] proposed a time domain analysis, assuming rigid body motion, and generated the solution by a forward integration scheme so as the non-linear effects due to the mooring lines are taken into consideration. The performance of cable-moored floating breakwaters was investigated experimentally by Martinelli et al. [50] and Johanning and Smith [51], theoretically and experimentally by Bhat [52], Bhat and Isaacson [8], and theoretically by Williams et al. [9,53]. Bhat [52] as well as Bhat and Isaacson [8] took into account the non-linear behavior of the mooring lines through an iterative coupling procedure between the 2D hydrodynamic analysis of the floating body and the analysis of the mooring lines in terms of convergence of the steady drift forces. Williams et al. [9,53] modeled the effect of mooring lines using appropriate modification of the hydrodynamic equations that refer to the 2D motion of a floating body. In this paper, the performance of a cable-moored array of floating breakwaters connected with hinge joints under the action of linear monochromatic waves is investigated numerically in the frequency domain. The performance is defined here as: (i) demonstration of acceptable levels of both response of the array and its effectiveness (in terms of the reduction of transmitted energy behind it) and (ii) non-failure of the mooring lines. It should be mentioned that the objective of the present paper, comprises two facets. Firstly, the formulation in three dimensions of a cablemoored array of floating breakwaters connected with hinges is presented. The array of floating breakwaters experiences motions along its length and is interacting with the wave field. The total number of degrees of freedom needed to describe the array of floating breakwaters are the six conventional rigid body modes (surge, sway, heave, roll, pitch, yaw), plus the extra generalized modes, equal to the number of the hinge joints. The generalized hinge modes are introduced to facilitate the effect of the vertical translations of the hinges. All the hinges permit each module to pitch, while affecting the response of the rest. The numerical analysis of the array is based on a 3D hydrodynamic formulation of the floating body coupled with the static and dynamic analysis of mooring lines. The stiffness and damping coefficients caused by the mooring lines are derived here in a general form for all degrees of freedom, including the generalized ones. Secondly, a rigorous parametric study is carried out in order to investigate the effect of different configurations in terms of number of hinge joints and mooring lines on the performance of the cable-moored array of floating breakwaters. The performance of the various examined configurations of cable-moored floating breakwaters connected with hinges is also compared with the corresponding one of a single cable-moored breakwater with no hinges.

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Fig. 1. Description of geometry and definition of basic quantities.

2. Definition of generalized hinge modes A longitudinal array of floating breakwaters is considered to be interacting with the wave field. The floating breakwaters (modules) are connected with each other by transverse hinge joints permitting each module to pitch, while affecting the response of the rest. The array, as a whole, could undergo small oscillations in the six degrees of freedom ξj (j = 1, . . . , 6), corresponding to surge, sway, heave, roll, pitch and yaw as defined in Fig. 1. The vertical displacements of the array due to hinge vertical translations lead to non-trivial hydroelastic effects that need to be considered in the hydrodynamic analysis by implementing appropriate generalized hinge modes [37]. The generalized hinge modes fj (q) are expressed here according to the definition included in [46,47] using appropriate sets of the tent functions fˆj (q) given by the equation: fˆj (q) = f (q − qh ) = 1 − |q − qh |,

h = 1, . . . , H

(1)

where q is the non-dimensional coordinate q = x HL+1 , x is the f

longitudinal coordinate of the array (Fig. 1), Lf is the total length of the array of floating breakwaters (Fig. 1) and qh is the q coordinate corresponding to the h = j − 6 hinge joint. The origin x = 0 or q = 0 is at the midpoint of the array. In this case, fj (q) is either symmetric or antisymmetric function about q = 0 and is described by Eq. (2a)–(2b) and Eqs. (3a)–(3c) for even and odd total number of hinge joints respectively: f7 (q) = fˆ7 (q) + fˆ8 (q)

(2a)

fj (q) = fˆj+p(j−1)−1 (q) − fˆ6+p(j−1) (q),



j = 8, 10, . . . , 2

[

6+H

] (2b)

2

fj (q) = fˆj+p(j) (q) + fˆ6+p(j) (q),



j = 9, 11, . . . , 2

[

6+H

]

 −1

2

(2c)

f7 (q) = fˆ13+H (q)

(3a)

2

fj (q) = fˆj+p(j) (q) − fˆ6+p(j) (q),



j = 8, 10, . . . , 2

[

7+H

]

 −2

2

(3b)

fj (q) = fˆj+p(j−1)−1 (q) + fˆ6+p(j−1) (q),



j = 9, 11, . . . , 2

[

7+H 2

]

 −1

(3c)

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Fig. 2. Generalized hinge modes fj (q) (j = 7, . . . , 10) for total number of hinge joints H = 1, . . . , 4 located on the plane z = 0 and parallel to the y-axis [47].

where p(j) is given by: p(j) =

|7 + H − j| 2

.

(4)

The generalized hinge modes (Eqs. (2a)–(2c), and (3a)–(3c)) for an array of up to five floating breakwaters, are depicted in Fig. 2. 3. Formulation of mooring lines’ stiffness coefficients 3.1. Definition of basic quantities The moored array of floating breakwaters connected with hinges is subjected to static movements relatively to its initial equilibrium position, due to second order steady drift forces. Consequently, the mooring lines experience changes relatively to their initial configuration and their initial level of static tensions as well. It should be mentioned that the aforesaid changes determine the stiffness of the mooring lines and thus, the final equilibrium position of the moored array. In the present study, the stiffness matrix due to the presence of the mooring lines is derived for an array of hinged floating breakwaters in three dimensions, considering that the stiffness coefficients are strongly affected by the static equilibrium position of the array; namely the stiffness coefficients are affected by the differential changes of both static tension and static angle at the fairlead of each mooring line, Tst and φ . More explicitly, the coordinate systems introduced for the present analysis are: (i) the coordinate system of the moored array of hinged floating breakwaters at the initial equilibrium position assuming zero level of any external static loads, OXYZ , where the center of gravity of the floating array is located at the origin O (Figs. 1 and 3), (ii) the coordinate system of the moored array of hinged floating breakwaters at the final equilibrium position, after experiencing static movements, O′ X ′ Y ′ Z ′ , and finally (iii) the local coordinate system corresponding to each mooring line, oxy (Fig. 3). The quantities related to the geometry of each mooring line are also depicted in Fig. 4. The displacement vector X of the floating array, assuming to be rigid, is given by: XT = [Xo

Yo

Zo

θX

θY

θZ

Z1

Z2

...

ZH ]

(5)

where Xo , Yo , Zo and θX , θY , θZ correspond to the translations of the center of gravity of the floating array across the axes X , Y , Z and the rotations of the floating body around the aforesaid axes respectively. As far as the terms Zh (h = 1, 2, . . . , H) are concerned, they denote the vertical translations of the array described by the fj (q) (j = 6 + h) hinge mode (Fig. 2).

Fig. 3. Description of coordinate systems OXYZ , O′ X ′ Y ′ Z ′ and oxz and relevant quantities.

The linear transformation described by Eq. (6) [48]: Uf = TU′f + X′

(6)

relates the position vectors of the fairlead of each mooring line in OXYZ and O′ X ′ Y ′ Z ′ coordinates systems, Uf and Uf′ respectively. More specifically, Uf and Uf′ , are given by: UTf = [XP U′f T = [XPO

YP

ZP ]

YPO

ZPO ].

(7) (8)

T is the rotational-transformation matrix described in Box I: X ′ denotes the translational displacement vector of the fairlead of each of the M total mooring lines restraining the hinged array,

I. Diamantoulaki, D.C. Angelides / Engineering Structures 33 (2011) 1536–1552

cos θY · cos θZ T = cos θY · sin θZ − sin θZ

sin θX · sin θY · cos θZ − cos θX · sin θZ sin θX · sin θY · sin θZ − cos θX · cos θZ sin θX · cos θY



cos θX · sin θY · cos θZ − sin θX · sin θZ cos θX · sin θY · sin θZ − sin θX · cos θZ cos θX · cos θY

1539

 (9)

Box I.

From Fig. 3 it follows: lX lY lZ

 

 ′  −lX · cos θ =  −l′X · sin θ  = −l′Z   −l · cos ω · cos θ = −l · cos ω · sin θ −l · sin ω XA − XP YA − YP ZA − ZP





(17)

where XA , YA and ZA are the constant coordinates of the anchor of each mooring line with respect to OXYZ coordinate system. 3.2. Derivation of the stiffness matrix of the mooring lines Fig. 4. Description of geometry of a mooring line and relevant quantities.

with regard to the coordinate system OXYZ , and is defined by:

 ′

X =

 Xo

Zo +

Yo

H −

 Zhm

(10)

h=1

where the term denotes the translational displacement of the fairlead of mooring line m due to vertical translations of the hinged array caused by the contribution of the hinge joint h. The term Zhm is defined here as: for m = 1, 2, . . . , M and h = 1, 2, . . . , H

(11)

where cm h is the coefficient that corresponds to each combination of mooring line, m, and hinge mode, fj (q), and relates the maximum (unit) vertical displacement at the position of hinge joints with the vertical displacement at the initial position of the fairlead of the m mooring line with respect to OXYZ coordinate system. Considering Eqs. (7), (8), Box I, Eqs. (10) and (11) it holds:

 XP YP ZP

 

XPO  = T YPO +  





ZPO

Xo Yo

Zo +

H −

  . m

(12)

Zh

h=1

Since the various displacements of the hinged array due to static loads result to modification of the initial configuration and level of static loads, with the assumption that all mooring lines are not only of identical geometry and material characteristics but also experience equal pretension level, it follows that the static quantities Tst and φst can be expressed as function of their total length, l: Tst = Tst (l)

(13)

φst = φst (l)

(14)

where l can be expressed as function of the projection of the length l of the mooring line in the OXYZ coordinate system, lX , lY , and lZ : l = l(lX , lY , lZ ).

(15)

The quantities lX , lY , and lZ depend on the displacement vector of the center of gravity of the array of hinged breakwaters, X (Eq. (5)), which includes the vertical displacements due to motion of hinge joints: lX = lX (X),

lY = lY (X),

K=

M −

Km =

m=1

Zhm

Zhm = cm h · Zh

The definition of the stiffness matrix Kij for a system of M mooring lines is given by:

lZ = lZ (X).

(16)

M − m=1

−

∂ Fm ∂X

(18)

where K m denotes the stiffness matrix of m mooring line (m = 1, 2, . . . , M ), Fm is the vector of the reaction loads that the m mooring line exercises on the total floating array without hinges. Particularly, Fm is given by:

(Fm )T = [(fX )m (fY )m (fZ )m (MX )m (MY )m (MZ )m (fZ )m (fZ )m . . . (fZ )m ]

(19a)

where

(fX )m = T cos φ · cos θ (fY )m = T cos φ · sin θ (fZ )m = T sin φ (MX )m = (YP − Yo )m · (V )m − (ZP − Zo )m · (H · sin θ )m

(19b) (19c) (19d) (19e)

(MY )m = a{(ZP − Zo )m · (H · cos θ )m − [(XP − Xo) − XVR ]m · (V )m }

(19f)

(MZ )m = (XP − Xo )m · (H · sin θ )m − (YP − Yo )m · (H · cos θ )m

(19g)

with a = 0 when there is a hinge joint located in the middle of the floating array, while otherwise a = 1. XVR is given by: XVR = XP − Xho

(20)

where XP is the initial X coordinate of fairlead of a mooring line with respect to OXYZ , Xho is the x coordinate of the hinge joint located at the same module as the fairlead of the m mooring line and also is closer to the middle of the moored array with respect to OXYZ . Similarly to the matrix X , Fm also consists of 6 + H terms, equal to the total degrees of freedom describing the motions of the floating array. Comparing the reaction forces and moments calculated for an array of floating breakwaters connected with hinges derived here with the respective ones calculated for a single rigid floating breakwater [48], it can be seen that for the reaction loads (fX )m , (fY )m , (fZ )m , (MX )m and (MZ )m no differences of definitions are observed. Therefore, the stiffness coefficients of the hinged array, (Kijm )H , for i = 1, 2, 3, 4, 6 and j = 1, 2, . . . , 6 are equal to the

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I. Diamantoulaki, D.C. Angelides / Engineering Structures 33 (2011) 1536–1552

respective coefficients calculated for a single floating breakwater, Kijm , namely:

(Kijm )H = Kijm for i = 1, 2, 3, 4, 6 and j = 1, 2, . . . , 6.

(21a)

Extended definitions of the stiffness coefficients for the case of a rigid floating breakwater, Kijm are included in [48]. On the contrary, according to evaluations of the present investigation, the equation giving the moment (MY )m in the case of an array of hinged floating breakwaters appears modified with regard to the respective one for a single floating breakwater presented in [48]. Therefore, the stiffness coefficients of the array of hinged floating breakwaters due to (MY )m , (K5jm )H , (j = 1, . . . , 6) differ from the respective ones for a single floating breakwater, Kijm (see also Appendix, Eqs. (A.1.1)–(A.1.6)).

(K5jm )H = a · (K5jm + XVR · K3jm ) for j = 1, 2, . . . , 6.

(21b)

The stiffness coefficients related to the hinge modes are derived in the present study as follows (see also Appendix, Eqs. (A.2.1)– (A.2.6), (A.3.1)–(A.3.7)):

The longitudinal vertical translations of the array of floating breakwaters are considered in the hydrodynamic analysis by implementing appropriate generalized hinge modes [37]. The hinge modes implemented are equal to the number of hinge joints. The vertical translations of the hinged floating structure, expressed as the complex amplitude ξ tot ′ (q), is equal to the sum of an appropriate set of modes including the contribution of heave (j = 3), pitch (j = 5) and each of the H total generalized hinge modes introduced, fj (q) (j = 7, . . . , H + 6):

ξ tot (q) =

for h = 1, 2, . . . , H

( (

) =

K6m,6+h H K6m+h,j H

chm

·

m K63

(21d)

(21e) for h = 1, 2, . . . , H

(21f)

)

K3jm chm

=

6+H −

[−ω2 (Mij + Aij ) + iω(Bij + BEij ) + (cij + KijH )]ξj = Xi ,

j =1

j = 1, 2, . . . , 6 and h = 1, 2, . . . , H j = 7, . . . , 6 + h and h = 1, 2, . . . , H .

(21g)

4. Description of the numerical model The numerical analysis of the response of the moored array of floating breakwaters connected with hinges includes two components: (a) the 3D hydrodynamic analysis of the aforesaid array and (b) the static and dynamic analysis of the mooring lines. A brief description of these two components is given in the next two subsections. It should be emphasized that components are coupled together through the application of an appropriate iterative procedure in terms of the steady drift forces and the response of the floating body, as described in [48]. 4.1. Hydrodynamic analysis of the array of floating array connected with hinges The numerical investigation of the performance of the array of floating breakwaters connected by hinges includes a 3D hydrodynamic analysis of the array. The 3D hydrodynamic analysis is carried out in the frequency domain under the action of monochromatic waves and is based on a linear wave diffraction theory. A velocity potential Φ , satisfying the Laplace equation, since the fluid is considered inviscid and incompressible and the flow irrotational, describes the fluid motion. This velocity potential Φ is defined by the relationship:

Φ = ΦD + Φr = (Φo + Φ7 ) + Φr

(22)

where the terms ΦD Φr , Φo , Φ7 correspond to the diffraction, radiation, incident and scattered potentials respectively, and are defined in [54,55].

(24)

where Mij is the mass matrix; Aij is the added mass matrix; Bij is the radiation damping matrix, BEij , is the damping matrix due to external causes, cij the stiffness matrix caused by buoyancy and gravity forces, KijH is the stiffness matrix due to the mooring lines and. Xi represents the exciting forces and moments corresponding to the i degree of freedom. Specifically, BEij is given by the following equation: E (D)

m · K33

(23)

where ξj is the unknown complex amplitude of each mode. The response of the floating breakwater, considering H hinge modes, is calculated by solving the following (6 + H ) × (6 + H ) system of equations [33,47]:

BEij = Bij



j = 3, 5, 7, 8, 9, . . . , 6 + H

i = 1, 2, . . . , H + 6 (21c)

m (K4m,6+h )H = chm · (K43 − Tst · cos φ · sin θ )

for h = 1, 2, . . . , H m m (K5m,6+h )H = a · chm · (K53 + XVR · K13 + Tst · cos φ · sin θ )

ξj · fj (q),

j

(Kim,6+h )H = chm · Ki3m for i = 1, 2 and 3 and h = 1, 2, . . . , H

−

+ BEij(V )

E (D)

where Bij

(25)

is the damping matrix due to the drag damping of E (V )

is the viscous damping matrix. The the mooring lines and Bij latter matrix is calculated according to the empirical relationship described in [52], after its appropriate modification according to Diamantoulaki et al. [17] in order to accommodate the occurrence of negative added mass: E (V )

Bij

 = 2ζ |(Mij + Aij )| · Cij

(26)

where ζ is the damping ratio. Calculation of the response constitutes a boundary value problem. The solution of this problem is based on the 3D panel method, where Green’s theorem is applied, considering appropriate boundary conditions [55,54]. The response of the floating body is described by the Response Amplitude Operator (RAOj ), which is given by the following equation: RAOj =

|ξj | A

,

j = 1, . . . , 6 + H

(27)

where A is the incident wave amplitude. Similarly, the total vertical displacement across the length of the floating array is defined as follows:

δ=

|ξtot (q) | A

,

−

H +1 2

≤q≤

H +1 2

.

(28)

The effectiveness of the floating breakwater is expressed by the ratio of the wave elevation behind the breakwater (shadow zone) to the incident wave amplitude: Kb (x, y) =

η(x, y) A

,

(y > 0)

where η(x, y) represents the wave elevation at (x, y).

(29)

I. Diamantoulaki, D.C. Angelides / Engineering Structures 33 (2011) 1536–1552

Another expression of the effectiveness behind the floating breakwater is introduced in this study: Kbav =

∑

Kb (x, y)

ZC = ZCR + ZCI · i = ξ3 · cos(a3 ) − ξ5 · XP · cos(a5 ) + ξ4 · YP · cos(a4 )

+

(30)

s where s is the number of the field points in the rectangular area behind the floating breakwater (−Lf /2 ≤ x ≤ Lf /2) and B/2 ≤ y ≤ Lf , for Lf and B denoting the total length and width of the floating array within which the term Kb (x, y) is computed. Both the expressions of effectiveness given by Eqs. (29) and (30) are used here. 4.2. Static and dynamic analyses of mooring lines Mooring lines are used for anchoring the floating breakwater against the action of waves, currents and wind. A static and a dynamic analysis is necessary to be carried out in order to calculate the total loads exercised on the mooring lines, as well as the stiffness and drag damping coefficients imposed on the floating body by the mooring lines. 4.2.1. Static analysis of mooring lines The static analysis aims at the calculation of: (a) the initial static configuration and the static tensions Tst of the mooring lines, (b) the new equilibrium position of the floating array-mooring lines system due to the action of the steady drift forces and the corresponding Tst of the mooring lines and (c) the stiffness matrix KijH that is applied on the breakwater by the mooring lines at the new equilibrium position. The calculations of (a) and (b) items mentioned above are based on the equations that are reported in [56,57].

E (D)

all the non-diagonal terms are assumed zero (Bij = 0 for i ̸= j). At first, the complex horizontal and vertical motion amplitudes, xd and zd , respectively, are calculated at the fairlead of each mooring line. The amplitudes xd and zd , are in the x and z direction respectively of the oxz local coordinate system (Figs. 3 and 4). Furthermore, the amplitudes xd and zd are attributed to the sinusoidal motions of the floating array (RAOj = 1, 2, . . . , 6 + H ) including the vertical displacements due to the hinge joints that need to be considered in the dynamic analysis of the mooring lines as well. The complex motions XC , YC and ZC observed at the fairlead of each mooring line in the X , Y and Z directions of the global coordinate system OXYZ (Fig. 3) respectively are given by: XC = XCR + XCI · i = ξ1 cos(a1 ) − ξ6 · YP cos(a6 ) + ξ5 · ZP cos(a5 ) + [ξ1 · sin(a1 ) − ξ6 · YP sin(a6 ) + ξ5 · ZP sin(a5 )]i

(31a)

YC = YCR + YCI · i = ξ2 · cos(a2 ) + ξ6 · XP · cos(a6 ) − ξ4 · ZP · cos(a4 ) + [ξ2 · sin(a2 ) + ξ6 · XP · sin(a6 ) − ξ4 · ZP · sin(a4 )] · i

(31b)

H − [chm · ξ6+h · cos(a6+h )] h=1 

ξ3 · sin(a3 ) − ξ5 · XP · sin(a5 ) + ξ4 · YP · sin(a4 )  H −  m  + ch · ξ6+h · sin(a6+h ) ·i (31c)

+

h=1

where aj is the phase of the amplitudes ξj (Fig. 1). Obviously, the contribution of the hinge motions is considered only in the calculation of the vertical motion ZC . It should be noticed that Eqs. (31a)–(31c) reduce to the corresponding ones for rigid floating body presented in [48] when the effect of the hinges is eliminated. If the motion amplitudes XC , YC and ZC are analyzed in the oxz plane (Fig. 3), it holds: QX = QXR + i · QXI = [XCR · cos(θf ) + YCR · sin(θf )] + [XCI · cos(θf ) + YCI · sin(θf )] · i

(32a)

QZ = ZCR + ZCI · i

(32b)

where θf represents the angle of each mooring line on X –Y at the new static equilibrium position. From Eqs. (32a) and (32b), it is derived that the motions of the fairlead of each mooring line, xd and zd , are equal to: xd = zd =

4.2.2. Dynamic analysis of mooring lines The dynamic analysis of the mooring lines includes the calculation of: (a) the dynamic tensions of them Tdyn at the fairlead at the new equilibrium position and (b) the drag damping matrix E (D) Bij . Extended description of the calculation of the dynamic tensions E (D) are included in [56,57]. As far as the damping coefficient, Bij , is concerned, its calculation is based on linearizing the hydrodynamic drag force using an equivalent linearization technique [57,48]. This technique is extended here so as the vertical displacements, due to the hinge joints, be considered. E (D) The diagonal coefficients Bij (i = j) are considered here since

1541



2 QXR + QXI2

(33a)

2 QZR + QZI2 .

(33b)



The terminal impedances Sxx , Sxz , Szx and Szz are considered as functions of the static and dynamic tension and angle at the fairlead of each mooring line and are given by [57]:

[

Sxx Szx

Sxz Szz

] =

[ ] [ ] xd F · x zd

Fz

(34)

where Fx and Fz denote the excitation forces in x and z directions of the oxz coordinate system respectively. Next, the complex reaction forces and moments are derived after analyzing the terminal impedances of Eq. (35) in Box II in the OXYZ system (Fig. 3), in a similar manner as Loukogeorgaki and Angelides [48] with proper modifications to include the effect of the hinges, and are given in Box II: Apparently, according to Eq. (35) the reaction force corresponding to the vertical displacements due to hinge motions is equal to the reaction force in the vertical direction of the Z axis of the OXYZ coordinate system (Fig. 3). R R I Szdi can also been given by Eq. (35) after substituting Sxx , Szx , Sxx and I R R I I Szx with Sxz , Szz , Sxz and Szz respectively. Then, the amplitudes of the reaction loads, Si , and the corresponding phases, βi , can be easily computed according to:



[(Sxd(i) )R + (Szd(i) )R ]2 + [(Sxd(i) )I + (Szd(i) )I ]2 [ ] (Sxd(i) )I + (Szd(i) )I βi = tan−1 (Sxd(i) )R + (Szd(i) )R Si =

(36) (37)

where in both Eqs. (36) and (37) the index i varies from 1 to 6 + H. Finally, after the reaction loads that are in phase with the velocity, ξ˙i (with i = 1, 2, . . . , 6 + H), have been computed, the drag damping coefficients for the m mooring line can be given by:

(BEij(D) )m =

|(Si )m · cos((βi )m − aj − π /2)| . ω · ξj

(38)

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 (Sxx )R cos(θf ) + (Sxx )I cos(θf ) · i   (Sxx )R sin(θf ) + (Sxx )I sin(θf ) · i     (Szx )R + (Szx )I · i   R R I I   [(Sxx ) sin(θf ) · ZP − (Szx ) · YP ] + [(Sxx ) sin(θf ) · ZP − (Szx ) · YP ] · i   R R I I a · [(Szx ) sin(θf ) · (XP − XVR ) − (Sxx ) cos(θf )ZP ] + a · [(Szx ) sin(θf ) · (XP − XVR ) − (Sxx ) cos(θf )ZP ]i Sxd =   R R I I   [(Sxx ) sin(θf ) · XP − (Sxx ) cos(θf ) · YP ] + [(Sxx ) sin(θf ) · XP − (Sxx ) cos(θf ) · YP ] · i   R I   (Szx ) + (Szx ) · i   R I   (Szx ) + (Szx ) · i   ... R I (Szx ) + (Szx ) · i 

(35)

Box II.

Considering Eq. (38), the drag damping coefficients for the system consisting of all mooring lines is: E (D)

Bij

=

M −

(BEij(D) )m ,

i = j = 1, 2, . . . , H + 6.

(39)

m=1

Table 1 Characteristics of mooring lines. Diameter Total initial length Submerged weight (Elasticity modulus) × (Area) Breaking tension T break

33 (mm) 30 (m) 191.25 (N/m) 342,119,440 (N) 400,000 (N)

5. Results and discussion The effect of different configurations (in terms of number of hinge joints and mooring lines) of an array of cable-moored floating breakwaters on its performance is studied through a rigorous parametric study. The hydrodynamic analysis is performed using the 3D radiation/diffraction code WAMIT [54]. A cable-moored array of floating breakwaters (Fig. 1) with dimensions (Lf = 20.00 m, B = 4.00 m, Hf = 2.00 m, dr = 0.77 m) is placed in water depth Dw = 10 m and is freely interacting with the wave field. Various configurations of arrays that consist of multiple floating breakwaters connected by hinges are examined by setting the total number of hinge joints H equal to 0, 1, 2 and 3. Each value of the parameter H correspond to a different configuration of arrays Ck, where k = 0, . . . , 3. It is obvious that C 0 is equivalent to a single floating breakwater with no hinges. Moreover, two different configurations of mooring systems, A and B, are considered. All the mooring lines of these two mooring systems are located at water depth equal to 10 m and are identical (Table 1). Configuration A is symmetric and consists of four identical mooring lines, where each one forms an angle of 45° with respect to the x axis on the x–y plane. As far as configuration B, it is also symmetric and consists of six mooring lines; where four of them form an angle of 45° with respect to the x axis on the x–y plane and the rest of them are perpendicular to the x axis on the x–y plane. Five cases, C 0A, C 1A, C 2A, C 3A and C 2B corresponding to Fig. 5(a)–(e) respectively (Table 2), are examined here in order to investigate the effect of the number of: (a) hinge joints and (b) mooring lines on the performance of cable-moored floating arrays of breakwaters connected with hinges. Twenty-one (21) or more wave frequencies are examined, where the non-dimensional wave length ratio B/L varies from 0.1 to 1.5, for each case examined assuming normal incident wave conditions. 5.1. Response The variation of δ (Eq. (28)) versus x, across the length of the array of floating breakwaters, is plotted in Fig. 6(a)–(c) for three representative wave frequencies B/L = 0.3, 0.6 and 1.1 respectively; where L is the corresponding wave length. According to this figure, a drastic reduction is observed for the values of δ when B/L = 0.6 (Fig. 6(b)) and B/L = 1.1 (Fig. 6(c)) in comparison to the respective values of δ when B/L = 0.3 (Fig. 6(a)). This is attributed to the significantly lower amplitude levels of the exciting loads, X3 and/or X7 , observed for all configurations for B/L = 0.6 and 1.1 (variation of the exciting loads is representatively

Table 2 Characteristics of the configurations CkA or CkB (k = 0, . . . , 3). Configuration

Number of hinge joints H

Number of floating modules FB (FB = H + 1)

Length of floating module Ls (Ls = Lf /FB m)

C 0A C 1A C 2A or C 2B C 3A

0 1 2 3

1 2 3 4

20.00 10.00 6.67 5.00

shown here for configuration C 1A in Fig. 7) compared to the respective ones computed for B/L = 0.3. Moreover, Fig. 6(a)–(c) depicts the visible effect of the total number of hinge joints, H, introduced upon the vertical translations of the array. In more detail, increase of H leads to either increase or reduction of δ depending on the value of x for all configurations CkA (k = 0, 1, 2 and 3). The variation of δ across the length of the array can be: (a) significant for B/L = 0.3 (Fig. 6(a) and (b)) mediocre for B/L = 0.6 (Fig. 6(b)) and (c) minimal for B/L = 1.1. Fig. 8(a) and (b) presents the effect of the variation of hinge joints upon the vertical translations of the CkA (k = 0, 1, 2 and 3) arrays as a function of the parameter B/L, at the bow, x = −Lf /2, (Fig. 8(a)) and in the middle, x = 0 m, (Fig. 8(b)) of the various arrays. Similarly to Fig. 6(a)–(c), Fig. 8(a) and (b) demonstrates the significant effect of the total number of the hinge joints on the response of the array for B/L ≤ 0.9. Moreover, considering Fig. 8(a) and (b) it is obvious that the increase of the total number of hinge joints can result to increase or decrease of δ values depending on both the combination of: (i) the B/L, and (ii) the position along the longitudinal axis of the array. This fact is attributed to variation of: (i) phase difference of modes contributing to vertical motions for different B/L values (relevant results are representatively shown for configuration C 2A in Fig. 10) and (ii) generalized hinge modes fj (q) for j ≥ 7 along the length of the floating array (Fig. 2). Furthermore, significant decrease of δ is observed for B/L ≥ 0.9, attributed to the drastic decrease of exciting loads contributing to vertical translations, X3 and/or X7 (X2 and X4 do not contribute to vertical translations), for B/L ≥ 0.9 (Fig. 7). Fig. 9(a) and (b) demonstrates the variation of δ versus B/L at the bow and in the middle of C 2A and C 2B, which are different only in terms of the number of restraining mooring lines. Obviously, an increasing number of restraining mooring lines hardly has a

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Fig. 5. Description of the configurations (a) C 0A, (b) C 1A, (c) C 2A, (d) C 2B and (e) C 3A (_ _ _ denotes the position of hinge joints and . . . . . . denotes position of mooring lines).

a

b

c

Fig. 6. δ versus x corresponding to (a) B/L = 0.3, (b) B/L = 0.6 and (c) B/L = 1.1.

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Fig. 7. Xj (j = 2, 3, 4 and 7) versus B/L corresponding to C 1A.

minimal effect on δ which is observed mainly for 0.3 ≤ B/L ≤ 0.45. This is attributed to the combined effect of: (i) different values of phase difference of ξ3 and ξ7 modes calculated when 0.3 ≤ B/L ≤ 0.45 (Fig. 10) and (ii) significant values of amplitudes of ξ3 and ξ7 modes (which contribute to vertical motions) for 0.3 ≤ B/L ≤ 0.45 (Fig. 12(b) and (d)). Fig. 11(a)–(d) shows the variation of sway (RAO2 ), heave (RAO3 ), roll (RAO4 ) for CkA (k = 0, 1, 2 and 3) versus B/L and generalized hinge modes (RAOj for j ≥ 7, 9) for CkA (with k values indicated in Fig. 11(d)); while Fig. 12(a)–(d) shows the

a

variation of RAO2 , RAO3 , RAO4 and RAO7 versus B/L for C 2A and C 2B configurations. It is observed that all configurations exhibit a peak value of RAO4 when B/L = 0.34 (Figs. 11(c) and 12(c)). Besides, the intense decrease of RAO2 for B/L = 0.34 (Figs. 11(a) and 12(a)) is associated with the intense increase of RAO4 values (Figs. 11(c) and 12(c)). This behavior is attributed to the strong coupling between sway and roll modes due to presence of mooring lines. In more detail, the aforesaid coupling leads to an increased effect of sway behavior on roll behavior and vice versa at the wave frequencies that peak values are exhibited. C 2A demonstrates significantly higher RAO3 for B/L ≤ 0.3 compared to the rest configurations plotted in Fig. 11(b). It can be also been shown that increase of hinge modes leads to increase or decrease of all modal amplitudes depicted depending on the B/L value (Fig. 11(a)–(d)). It should also be mentioned that RAO3 and RAO9 of C 3A exhibit similar patterns of variation. This happens because for C 3A, X9 = 0 (Fig. 13), and thus, motion of ξ9 mode occurs due to radiation effect activated by motion of ξ3 mode. Moreover, considering Fig. 12 (a)–(c), it can be shown that the introduction of two supplementary mooring lines in the middle of the array consisting of three floating breakwaters mainly affects RAO2 , RAO4 and RAO7 . In particular, increase of mooring lines leads to decrease of RAO2 and RAO7 , especially for B/L ≈ 0.35, whereas it causes increase of RAO4 . Finally, all modal responses exhibit noticeable decrease for B/L ≥ 0.9 (Figs. 11(a)–(d) and 12(a)–(d)), which is in complete accordance with the decrease of exciting loads computed for this frequency range (Fig. 7).

b

Fig. 8. δ versus B/L for CkA (k = 0, 1, 2 and 3) configurations (a) at the bow and (b) in the middle of the array.

a

b

Fig. 9. δ versus B/L for C 2A and C 2B configurations (a) at the bow and (b) in the middle of the array.

I. Diamantoulaki, D.C. Angelides / Engineering Structures 33 (2011) 1536–1552

Fig. 10. Phase difference of ξ3 and ξ7 versus B/L for C 2A and C 2B configurations.

1545

joints is associated with snapping phenomena (Tst < Tdyn ) which are observed for 0.3 ≤ B/L ≤ 0.4; namely the region of high RAO4 values (Figs. 11(c) and 12(c)). The peak RAO4 values observed for the aforesaid B/L values (Fig. 11(c)) lead to peak values of Tdyn and consequently to peak values Ttot . Moreover, it can be seen that the higher the RAO4 peak value is, the higher is the Tdyn or Ttot level for bow or stern mooring lines (this statement is not valid for the mooring lines located in the middle of C2B since pitch motion has no effect in the middle of the array). Besides, there is no chance for the total tensions to exceed breaking tension, since the maximum ratio of the breaking tension to the total tension observed is equal to 0.089 (Ttot /Tbreak = 0.089). The lower Ttot levels are exhibited by C 0A configuration; thus it can be claimed that hinge joints lead to increase of Ttot (Fig. 14(c)). Besides, increase of the number of mooring line leads to higher Ttot levels. As regards C2B, the most heavily loaded mooring lines are the ones located in the middle of the array (Fig. 15(a)) since they correspond to larger area of effect.

5.2. Static and dynamic tensions of mooring lines

5.3. Effectiveness

Fig. 14 shows the variation of the static (Tst ), the dynamic (Tdyn ) and total tensions (Ttot = Tst + Tdyn ) tensions versus B/L at the top (fairlead) of the mooring lines. These forces are exercised either on the front mooring lines (y = −2 m, Fig. 1) or back mooring lines (y = 2 m, Fig. 1). Obviously, the front mooring lines are the most heavily loaded compared to the back mooring lines (Fig. 14(a) and (b)), due to the action of incident waves in the normal direction. According to Figs. 14(a), (b) and 15(a), (b) the presence of hinge

The effectiveness of the floating array discussed in this subsection is expressed in terms of Kb (Eq. (29)) and Kbav (Eq. (30)). The wave elevation, in all cases, is calculated at the field points with coordinates ranging within x = −Lf /2, . . . , Lf /2 m and y = B/2, . . . , Lf m in the rear of the floating array according to the definition of the body coordinate system depicted in Fig. 1. For each configuration CkA (k = 0, 1, 2 and 3), C 2A and C 2B examined, the effectiveness is calculated considering the complete problem

a

b

c

d

Fig. 11. Modal responses RAOj (j = 2, 3, 4, 7 and 9) versus B/L corresponding to CkA (k = 0, 1, 2 and 3) configurations.

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b

a

c

d

Fig. 12. Modal responses RAOj (j = 2, 3, 4 and 7) versus B/L corresponding to C 2A and C 2B configurations.

Fig. 13. Exciting loads Xj (j = 2, 3, 4 and 7) versus B/L corresponding to C 2A and C 2B configurations.

(i.e. considering both diffraction and radiation). The effect of diffracted waves on the effectiveness is also discussed separately. Fig. 16 depicts the variation of the non-dimensional efficiency parameter Kb with y (x = 0 m) for three representative wave frequencies of the low (0.1 ≤ B/L ≤ 0.4), middle (0.4 < B/L < 0.9) and high (0.9 ≤ B/L ≤ 1.5) wave frequency ranges corresponding to B/L values equal to 0.3, 0.6 and 1.1. Obviously, the efficiency parameter Kb corresponding to the diffracted waves, namely Kbd , is the same for all configurations examined (Figs. 16 and 17) due to the identical geometry of all configurations examined. Thus, any

differences observed among Kb patterns are exclusively attributed to the effect of radiation waves caused by the motion of the floating breakwaters. Regarding B/L = 0.3 (low wave frequency range), it is shown that the variation of all Kb patterns is quite different from the variation of the pattern corresponding to Kbd (Figs. 16(a) and 17(a)). This statement proves the intense effect of radiation waves on the effectiveness for all configurations examined in the aforementioned figures (Figs. 6, 11 and 12). The configurations performing the highest effectiveness among all CkA (k = 0, 1, 2 and 3) configurations, are C 0A and C 3A; the former configuration for 2.5 m ≤ y ≤ 7.5 m and the latter 7.5 m ≤ y ≤ 20 m. On the contrary, the effectiveness performed by C 1A is inadequate, since for y ≥ 8 m it holds Kb ≥ 1.0 as shown in Fig. 16(a). The reason that C 1A exhibits inadequate effectiveness is attributed to the fact that C 1A exhibits the highest δ (Fig. 6(a)), RAO3 (Fig. 11(b)) and RAO4 (Fig. 11(c)) values. As far as for B/L = 0.6 (middle wave frequency range), all CkA (k = 0, 1, 2 and 3) configurations exhibit acceptable levels of performance. In more detail, C 3A appears to be the most effective configuration for 2.5 m ≤ y ≤ 4.5 m and 7.0 m ≤ y ≤ 20.0 m whereas C 0A or C 1A are the most effective configurations for 4.5 m ≤ y ≤ 7.0 m. A clear resemblance among Kb patterns of C 0A and C 1A is observed, since low and very similar levels of response have been calculated for B/L = 0.6 (Fig. 6(b)). Finally, for B/L = 1.1 (high wave frequency range), the variation of Kb is of similar trends with the variation of Kbd (Fig. 16(c)), given the reduced contribution of radiation effect on the effectiveness due to extremely low response levels (Figs. 6(c) and 11). The Kb patterns of all CkA (k = 0, 1, 2 and 3) are almost

I. Diamantoulaki, D.C. Angelides / Engineering Structures 33 (2011) 1536–1552

1547

b

a

c

Fig. 14. Variation of: (a) Tst and Tdyn for front lines, (b) Tst and Tdyn for back lines and (c) Ttot as function of B/L at the top of the front mooring lines corresponding to CkA (k = 0, 1, 2 and 3) configurations.

a

b

Fig. 15. Variation of: (a) Tst and Tdyn for front mooring lines and (b) maximum Ttot as function of B/L at the top of the front mooring lines corresponding to C 2A and C 2B configurations.

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a

b

c

Fig. 16. Kb and Kbd versus y (x = 0 m) for (a) B/L = 0.3, (b) B/L = 0.6 and (c) B/L = 1.1 corresponding to CkA (k = 0, 1, 2 and 3) configurations.

a

b

c

Fig. 17. Kb and Kbd versus y (x = 0 m) for (a) B/L = 0.3, (b) B/L = 0.6 and (c) B/L = 1.1 corresponding to C 2A and C 2B configurations.

I. Diamantoulaki, D.C. Angelides / Engineering Structures 33 (2011) 1536–1552

a

b

Fig. 18. Variation of Kb in the rear of the array of floating breakwaters for (a) C 2A and (b) C 2B considering B/L = 0.3.

identical as are also the low response levels. All configurations of Fig. 16(c) exhibit adequate effectiveness irrespective of the hinge joints introduced. According to Fig. 17(a)–(c), an increasing number of mooring lines has a direct effect on the effectiveness only for B/L = 0.3 (low wave frequency range, Figs. 17(a), 18 and 19), where the effect of radiation caused by heave motion is significant (Figs. 9(a), (b) and 12(b)) as opposed to B/L = 0.6 (middle wave frequency range, Fig. 17(b)) or B/L = 1.1 (high wave frequency range, Fig. 17(c)). In Fig. 20(a) the variation of Kb,av versus B/L is plotted for CkA (k = 0, 1, 2, 3) configurations. Regarding the low wave frequency range, C 1A configuration demonstrates unacceptable effectiveness, since for 0.1 ≤ B/L ≤ 0.3Kb ≈ 1. The rest of CkA configurations, namely C 0A, C 2A and C 3A perform acceptable level of effectiveness (Kb < 1.0) and each of them can be the most effective configuration depending on the B/L value (i.e. C 3A and C 2A appear to be the most effective configurations for 0.1 ≤ B/L ≤ 0.17 and 0.17 ≤ B/L ≤ 0.32 respectively). As far as the middle and high wave frequency ranges are concerned, all CkA configurations exhibit significant improvement of the effectiveness. In more detail, C 0A and C 3A are the most effective configurations for 0.4 ≤ B/L ≤ 0.5 and 0.5 ≤ B/L ≤ 0.9 respectively, whereas for 0.9 ≤ B/L ≤ 1.5 all configurations exhibit almost identical level of effectiveness due to reduced response levels (Figs. 6(c), 8(a), (b) and 11(a)–(d)). The effect of the number of hinge joints is apparent only in the low and middle wave frequency ranges due to either high or mediocre response levels (Figs. 6(a), (b), 8(a), (b) and 11(a)–(d)). Finally, Fig. 20(b) depicts the variation of K,b,av versus B/L for C 2A and

a

b

Fig. 19. Kb contours for (a) C 2A and (b) C 2B considering B/L = 0.3 (the dotted lines indicate the mooring lines).

a

1549

b

Fig. 20. Kb,av versus B/L corresponding to configurations (a) CKA for k = 0, 1, 2, 3 and (b) C 2A and C 2B.

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I. Diamantoulaki, D.C. Angelides / Engineering Structures 33 (2011) 1536–1552

C 2B configurations. According to this figure, it can be seen that the effect of the number of mooring lines on the effectiveness is noticeable only in the low wave frequency range, where increasing the number of mooring lines can have a positive or negative influence on the effectiveness, depending on the B/L value. The aforesaid statement is also apparent considering Figs. 18 and 19. 6. Conclusions In the present investigation, the overall performance of a cablemoored array of floating breakwaters connected by hinges is investigated under the action of monochromatic linear waves in the frequency domain. The numerical analysis of the array is based on a 3D hydrodynamic formulation of the floating body coupled with the static and dynamic analysis of the mooring lines. The motions of the array of floating breakwaters due to the hinge vertical translations are considered in the hydrodynamic analysis with the implementation of appropriate generalized modes. The stiffness and damping coefficients caused by the mooring lines in both rigid and generalized degrees of freedom are derived here in general form. A rigorous parametric study is carried out in order to investigate the effect of different configurations, namely number of hinge joints and number of mooring lines, on the performance of the cable-moored array of floating breakwaters. The main conclusions generated by this research are: 1. A strong dependence of the array’s response on the number of hinge joints is exhibited, mainly in the case of low and middle wave frequency ranges. Increasing the number of hinge joints can either increase or decrease response level depending on the combination of both the wave frequency parameter B/L and the position along the longitudinal axis of the array. Vertical translations exhibit high variation in the low wave frequency range, and this variation becomes smoother for an increasing wave frequency. All modes exhibit a drastic decrease of response in the high wave frequency range. 2. An increasing number of restraining mooring lines has a small effect on the vertical translations and the modal responses confined in the low wave frequency range. 3. The presence of hinge joints is associated with snapping phenomena and higher level of dynamic and, consequently, total tensions in the mooring lines. These phenomena are confined around some B/L values in the low wave frequency range. Proper choice of dimension B can overcome this problem. 4. Variations of the effectiveness among all configurations examined are attributed to the radiation effect, since the diffracted waves are not affected by neither the number of hinge joints nor the number of mooring lines. Radiation effect is more intense in the low and middle wave frequency range, where higher response levels have been computed. 5. The number of hinge joints strongly affects the array effectiveness in both the low and middle wave frequency ranges. The number of mooring lines affects the array effectiveness in the low wave frequency range. Moreover, an increasing number of hinge joints or mooring lines can have a positive or negative influence on the effectiveness, depending on the wave frequency ratio B/L. 6. Based on all the above, for a given range of dominant wave frequencies the proper combination of dimension B, number of hinge joints and number of mooring lines has to be determined. Acknowledgements The authors would like to thank the reviewers for their valuable comments and suggestions.

Appendix. Definitions of mooring lines’ stiffness coefficients for an array of floating breakwaters connected with hinges A.1. Stiffness coefficients (K5jm )H for j = 1, 2, . . . , 6

∂ MY ∂ =− {a · [(ZP − Zo ) · fX ∂ Xo ∂ Xo − ([XP − Xo − X VR ) · fZ ]}  ∂ fX a · (ZP − Zo ) · − ∂ Xo ] ∂ fZ − (XP − Xo − XVR ) · − ∂ Xo m m a · [(ZP − Zo ) · K11 − (XP − Xo − XVR ) · K31 ] m m m a · [(ZP − Zo ) · K11 − (XP − Xo ) · K31 + XVR · K31 ] m m a · (K51 + XVR · K31 ) (A.1.1)

m (K51 )H = −

=

= = =

∂ ∂ MY =− {a · [(ZP − Zo ) · fX ∂ Yo ∂ Yo − ([XP − Xo − X VR ) · fZ ]}  ∂ fX a · (ZP − Zo ) · − ∂ Yo ] ∂ fZ − (XP − Xo − XVR ) · − ∂ Yo m m a · [(ZP − Zo ) · K12 − (XP − Xo − XVR ) · K32 ] m m m a · [(ZP − Zo ) · K12 − (XP − Xo ) · K32 + XVR · K32 ] m m a · (K52 + XVR · K32 ) (A.1.2)

m (K52 )H = −

=

= = =

∂ ∂ MY =− {a · [(ZP − Zo ) · fX ∂ Zo ∂ Zo − ([XP − Xo − X VR ) · fZ ]}  ∂ fX a · (ZP − Zo ) · − ∂ ZO ] ∂ fZ − (XP − Xo − XVR ) · − ∂ ZO m m − (XP − Xo − XVR ) · K33 ] a · [(ZP − Zo ) · K13 m m m a · [(ZP − Zo ) · K13 − (XP − Xo ) · K33 + XVR · K33 ] m m a · (K53 + XVR · K33 ) (A.1.3)

m (K53 )H = −

=

= = =

∂ ∂ MY =− {a · [(ZP − Zo ) · fX ∂θX ∂θX − ([XP − Xo − XVR ) · fZ ]} ∂ ∂ fX a · −f X · (ZP − Zo ) − (ZP − Zo ) · ∂θX ∂θX ] ∂ ∂ fZ + fZ · (XP − Xo − XVR ) + (XP − Xo − XVR ) · ∂θX [ ∂θX   ∂ ZP m a · −f X · − 0 − (ZP − Zo ) · K14  ∂θX  ] ∂ XP m + fZ · − 0 − 0 + (XP − Xo − XVR ) · K34 ∂θX [  ∂ lZ m a · −f X · − − (ZP − Zo ) · K14 ∂θ X   ] ∂ lX m + fZ · − + (XP − Xo − XVR ) · K34 ∂θX m m a · (K54 + XVR · K34 ) (A.1.4) ∂ MY ∂ − =− {a · [(ZP − Zo ) · fX ∂θY ∂θY − ([XP − Xo − XVR ) · fZ ]} ∂ ∂ fX a · −f X · (ZP − Zo ) − (ZP − Zo ) · ∂θY ∂θX ∂ + fZ · (XP − Xo − XVR ) ∂θY

m (K54 )H = −

=

=

=

= m (K55 )H =

=

I. Diamantoulaki, D.C. Angelides / Engineering Structures 33 (2011) 1536–1552

] ∂ fZ + (XP − Xo − XVR ) · [  ∂θY ∂ ZP m = a · −f X · − 0 − (ZP − Zo ) · K15 ∂θY   ] ∂ XP m + fZ · − 0 − 0 + (XP − Xo − XVR ) · K35 [ ∂θY  ∂ lZ m = a · −fX · − − (ZP − Zo ) · K15 ∂θ Y ]  ∂ lX m + (XP − Xo − XVR ) · K35 + fZ · − ∂θY m m = a · (K55 + XVR · K35 ) (A.1.5) ∂ ∂ MY =− {a · [(ZP − Zo ) · fX ∂θZ ∂θZ − ([XP − Xo − XVR ) · fZ ]} ∂ ∂ fX a · −fX · (ZP − Zo ) − (ZP − Zo ) · ∂θZ ∂θZ ] ∂ ∂ fZ + fZ · (XP − Xo − XVR ) + (XP − Xo − XVR ) · ∂θY [ ∂θZ   ∂ ZP m a · −f X · − 0 − (ZP − Zo ) · K16   ∂θZ ] ∂ XP m − 0 − 0 + (XP − Xo − XVR ) · K36 + fZ · ∂θ  [ Z ∂ lZ m − (ZP − Zo ) · K16 a · −fX · − ∂θ Z   ] ∂ lX m + (XP − Xo − XVR ) · K36 + fZ · − ∂θZ m m a · (K56 + XVR · K36 ). (A.1.6)

m (K56 )H = −

=

=

=

=

A.2. Stiffness coefficients (Kim ,6+h )H for i = 1, 2, . . . , 6

∂ ∂ fX =− (−Tst · cos φ · cos θ ) ∂ Zh ∂ Zh m m = ch · K13

(A.2.1)

∂ ∂ fY =− (−Tst · cos φ · sin θ ) ∂ Zh ∂ Zh m m = ch · K23

(A.2.2)

∂ fZ ∂ =− (−Tst · sin φ) ∂ Zh ∂ Zh m m = ch · K33

(A.2.3)

(K1m,6+h )H = −

(K2m,6+h )H = −

(K3m,6+h )H = −

∂ MX ∂ [(YP − Yo ) · fZ − (ZP − Zo ) · fY ] =− ∂ Zh ∂ Zh m m = (YP − Yo ) · K37 − (ZP − Zo ) · K27 − Tst · cos φ · sin θ · chm m = chm · (K43 − Tst · cos φ · sin θ ) (A.2.4)

(K4m,6+h )H = −

∂ MY ∂ Zh ∂ = − [a · (ZP − Zo ) · fX − a(XP − Xo − XVR ) · fZ ] ∂ Zh m = −a · Tst · cos φ · cos θ · chm + a · (ZP − Zo ) · (K17 )H m − a · (XP − Xo − XVR ) · (K37 )H m m = a · chm · (K53 + XVR · K13 + Tst · cos φ · cos θ ) (A.2.5)

(K5m,6+h )H = −

∂ MZ ∂ =− [(XP − Xo ) · fY − (YP − Yo ) · fX ] ∂ Zh ∂ Zh m m = (XP − Xo ) · (K27 )H − (YP − Yo ) · (K17 )H m = chm · K63 . (A.2.6)

(K6m,6+h )H = −

1551

A.3. Stiffness coefficients (K6m+h,j )H for j = 1, 2, . . . , 6 + H

∂ fZ m m = (K31 )H = K31 ∂ Xo ∂ fZ m m (K6m+h,2 )H = − = (K32 )H = K32 ∂ Yo ∂ fZ m m = (K33 )H = K33 (K6m+h,3 )H = − ∂ Zo ∂ fZ m m = (K34 )H = K34 (K6m+h,4 )H = − ∂θX ∂ fZ m m (K6m+h,5 )H = − = (K35 )H = K35 ∂θY ∂ fZ m m = (K36 )H = K36 (K6m+h,6 )H = − ∂θZ ∂ fZ m = (K3m,6+h )H = chm · K33 . (K6m+h,6+h )H = − ∂ Zh (K6m+h,1 )H = −

(A.3.1) (A.3.2) (A.3.3) (A.3.4) (A.3.5) (A.3.6) (A.3.7)

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