modeling of cable-moored floating breakwaters connected with hi

bs

iHinge jointsGeneralized modesCable tensionsEffectivenessResponseFluidstructure interaction

generalized degrees of freedomare derivedhere in the general form. A rigorous parametric study is carriedout in order to investigate the effect of different configurations (number of hinge joints and number ofmooring lines) on the performance of the cable-moored array of floating breakwaters. Moreover, theperformance of the various configurations of cable-moored floating breakwaters connected by hingesexamined is comparedwith the performance of a single cable-moored floating breakwaterwith no hinges.It is found that the number of hinge joints and mooring lines have a direct effect on the performance ofthe cable-moored array of floating breakwaters.

2011 Elsevier Ltd. All rights reserved.

1. Introduction

The traditional type of breakwater is the bottom-founded struc-ture. The construction of this type of breakwater is not alwayseconomical, especially for deep water depths; furthermore, break-waters of this type are potentially associated with environmentalproblems, such as intense shore erosion, water quality problemsand aesthetic considerations. The aforementioned disadvantagesmotivated the search for an alternative type of breakwater, namelythe floating ones. The application of such kind of structures iscontinuously increasing, because of the fast and inexpensive con-struction as well as the possibility of mobility and reallocation. Thefloating breakwaters are usually pile-restrained or cable-moored.Reviews of the general design of floating breakwaters are pre-sented in [14]; furthermore, Isaacson [4] provides an overview ofwave effects on floating breakwaters. As far as the hydrodynamicanalysis of the floating body is concerned, 2D models have beendeveloped that describe the complete linear hydrodynamic prob-lem of the wavestructure interaction [414]. These 2D modelsuse four methods: (i) finite element method, (ii) boundary integralmethod, (iii) finite differences using Boussinesq type equations,(iv) volumeof fluid and (v) particlemethods. Analytical solutions ofthe hydrodynamic problemare available for simple geometries and

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

regularwaves [15]. Loukogeorgaki andAngelides [16] andDiaman-toulaki et al. [17] used a 3D hydrodynamicmodel to investigate theperformance of floating breakwaters. A 3D analysis for a V-shapedfloating breakwater was used by Briggs et al. [18], including hydro-elasticity.

The phenomenon of hydroelasticity has also been investigatedin various studies using (i) 2D linear theories [1922], (ii) 2Dnon-linear theories [23,24], (iii) 3D linear theories [2528] and(iv) 3D non-linear theories [29,30]. Bishop and Price [31] used freeundamped wet bendingmodes, while Gran [32] used orthogonalmodes of a uniform beam to express the vertical translations ofa slender ship. Newman [33] extended the linearized frequencydomain analysis of wave diffraction and radiation for a 3D bodyin a fixed mean position to a variety of deformable body motionsusing an expansion in arbitrary modal shape functions. Jensenand Pedersen [23] developed a non-linear quadratic strip theoryformulated in the frequency domain for predicting wave loads andship responses in moderate seas. Du [34] presented a completefrequency domain analysis for linear 3D hydroelastic responsesof floating structures moving in a seaway and Fu et al. [35]used 3D linear hydroelasticity theory to predict the responseof flexible interconnected structures. Finally, Wu et al. [29]used a 3D non-linear hydroelasticity theory for both frequencyand time domain analyses. Many researchers have dealt withthe application of hydroelasticity theories in the analysis ofVLFS [3640], since hydroelasticity is very important for this kindof structures. A comprehensive review of hydroelasticity theoriesEngineering Structures

Contents lists availa

Engineering

journal homepage: www.el

Modeling of cable-moored floating breakIoanna Diamantoulaki , Demos C. AngelidesDepartment of Civil Engineering, Aristotle University of Thessaloniki, University Campus, T

a r t i c l e i n f o

Article history:Received 29 June 2010Received in revised form30 November 2010Accepted 30 January 2011Available online 1 March 2011

Keywords:Cable-moored floating breakwaters

a b s t r a c t

In the present paper, the oveby hinges is investigated undperformance is defined hereand its effectiveness and (iibased on a 3D hydrodynamanalyses of the mooring linesvertical translations are consgeneralized modes. The stiffn0141-0296/$ see front matter 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2011.01.024

Downloaded from http://www.elearnica.ir33 (2011) 15361552

le at ScienceDirect

Structures

evier.com/locate/engstruct

waters connected with hinges

hessaloniki 54124, Greece

rall performance of a cable-moored array of floating breakwaters connecteder the action of monochromatic linear waves in the frequency domain. Theas: (i) demonstration of acceptable levels of both response of the array

) non-failure of the mooring lines. The numerical analysis of the array isic formulation of the floating body coupled with the static and dynamic. The motions of the array of floating breakwaters associated with the hingedered in the hydrodynamic analysiswith the implementation of appropriateess and damping coefficients caused by the mooring lines in both rigid and

eI. Diamantoulaki, D.C. Angelides / Engin

is presented in [41], and in each of the aforementioned studiesregarding hydroelasticity the floating body is assumed to be free-floating (unrestrained). The effect of hydroelasticity on flexiblefloating breakwaters has been studied by Williams et al. [10] forone compliant beam-like breakwater (idealized as 1D beam ofuniform flexural rigidity) using an appropriate Green function,and by Abul-Azm [42] using an eigenfunction approach; bothof the researches use 2D hydrodynamic analysis. Diamantoulakiet al. [43] and Manolis et al. [44] have also investigated theeffect of hydroelasticity on flexible floating breakwaters using 3Dhydrodynamic analysis. Furthermore, the effect of hydroelasticityphenomena on floating breakwaters connected by hinge jointshas been investigated numerically by Newman [33], Lee andNewman [45] and Diamantoulaki and Angelides [46,47]. In each ofthese hydroelastic studies the floating body is assumed to be eitherfree [33,45,47] or pile-restrained [46].

Besides free and pile-restrained floating bodies, cable-mooredones have also been investigated by several researchers, assumingrigid body conditions as mentioned in [48]. Chakrabarti and Cot-ter [49] proposed a time domain analysis, assuming rigid bodymo-tion, and generated the solution by a forward integration schemeso as the non-linear effects due to the mooring lines are taken intoconsideration. The performance of cable-moored floating break-waters was investigated experimentally by Martinelli et al. [50]and Johanning and Smith [51], theoretically and experimentally byBhat [52], Bhat and Isaacson [8], and theoretically byWilliams et al.[9,53]. Bhat [52] as well as Bhat and Isaacson [8] took into accountthe non-linear behavior of the mooring lines through an iterativecoupling procedure between the 2D hydrodynamic analysis of thefloating body and the analysis of themooring lines in terms of con-vergence of the steady drift forces. Williams et al. [9,53] modeledthe effect of mooring lines using appropriate modification of thehydrodynamic equations that refer to the 2D motion of a floatingbody.

In this paper, the performance of a cable-moored array offloating breakwaters connected with hinge joints under the actionof linear monochromatic waves is investigated numerically inthe frequency domain. The performance is defined here as:(i) demonstration of acceptable levels of both response of the arrayand its effectiveness (in terms of the reduction of transmittedenergy behind it) and (ii) non-failure of themooring lines. It shouldbe mentioned that the objective of the present paper, comprisestwo facets. Firstly, the formulation in three dimensions of a cable-moored array of floating breakwaters connected with hinges ispresented. The array of floating breakwaters experiences motionsalong its length and is interacting with the wave field. The totalnumber of degrees of freedom needed to describe the array offloating breakwaters are the six conventional rigid body modes(surge, sway, heave, roll, pitch, yaw), plus the extra generalizedmodes, equal to the number of the hinge joints. The generalizedhinge modes are introduced to facilitate the effect of the verticaltranslations of the hinges. All the hinges permit each moduleto pitch, while affecting the response of the rest. The numericalanalysis of the array is based on a 3D hydrodynamic formulationof the floating body coupled with the static and dynamic analysisof mooring lines. The stiffness and damping coefficients caused bythemooring lines are derived here in a general form for all degreesof freedom, including the generalized ones. Secondly, a rigorousparametric study is carried out in order to investigate the effectof different configurations in terms of number of hinge joints andmooring lines on the performance of the cable-moored array offloating breakwaters. The performance of the various examinedconfigurations of cable-moored floating breakwaters connected

with hinges is also compared with the corresponding one of asingle cable-moored breakwater with no hinges.ering Structures 33 (2011) 15361552 1537

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

2. Definition of generalized hinge modes

A longitudinal array of floating breakwaters is considered tobe interacting with the wave field. The floating breakwaters(modules) are connectedwith each other by transverse hinge jointspermitting each module to pitch, while affecting the response ofthe rest. The array, as a whole, could undergo small oscillationsin the six degrees of freedom j (j = 1, . . . , 6), correspondingto surge, sway, heave, roll, pitch and yaw as defined in Fig. 1.The vertical displacements of the array due to hinge verticaltranslations lead to non-trivial hydroelastic effects that need tobe considered in the hydrodynamic analysis by implementingappropriate generalized hinge modes [37].

The generalized hingemodes fj(q) are expressed here accordingto the definition included in [46,47] using appropriate sets of thetent functions fj(q) given by the equation:

fj(q) = f (q qh) = 1 |q qh|, h = 1, . . . ,H (1)where q is the non-dimensional coordinate q = xH+1Lf , x is thelongitudinal coordinate of the array (Fig. 1), Lf is the total length ofthe array of floating breakwaters (Fig. 1) and qh is the q coordinatecorresponding to the h = j 6 hinge joint. The origin x = 0 orq = 0 is at the midpoint of the array. In this case, fj(q) is eithersymmetric or antisymmetric function about q = 0 and is describedby Eq. (2a)(2b) and Eqs. (3a)(3c) for even and odd total numberof hinge joints respectively:

f7(q) = f7(q)+ f8(q) (2a)fj(q) = fj+p(j1)1(q) f6+p(j1)(q),

j = 8, 10, . . . , 2[6+ H2

](2b)

fj(q) = fj+p(j)(q)+ f6+p(j)(q),j = 9, 11, . . . , 2

[6+ H2

] 1

(2c)

f7(q) = f 13+H2(q) (3a)

fj(q) = fj+p(j)(q) f6+p(j)(q),j = 8, 10, . . . , 2

[7+ H2

] 2

(3b)

fj(q) = fj+p(j1)1(q)+ f6+p(j1)(q), [ ]

j = 9, 11, . . . , 2 7+ H

2 1 (3c)

eof hinged floating breakwaters at the initial equilibrium positionassuming zero level of any external static loads, OXYZ , where thecenter of gravity of the floating array is located at the origin O(Figs. 1 and 3), (ii) the coordinate system of the moored array ofhinged floating breakwaters at the final equilibrium position, afterexperiencing static movements, OX Y Z , and finally (iii) the localcoordinate system corresponding to eachmooring line, oxy (Fig. 3).The quantities related to the geometry of eachmooring line are alsodepicted in Fig. 4.

The displacement vector X of the floating array, assuming to berigid, is given by:

XT = [Xo Yo Zo X Y Z Z1 Z2 . . . ZH ] (5)where Xo, Yo, Zo and X , Y , Z correspond to the translationsof the center of gravity of the floating array across the axesX, Y , Z and the rotations of the floating body around the aforesaidaxes respectively. As far as the terms Zh (h = 1, 2, . . . ,H) are

Fig. 3. Description of coordinate systems OXYZ , OX Y Z and oxz and relevantquantities.

The linear transformation described by Eq. (6) [48]:

Uf = TUf + X (6)relates the position vectors of the fairlead of each mooring line inOXYZ and OX Y Z coordinates systems, Uf and U f respectively.More specifically, Uf and U f , are given by:

UTf = [XP YP ZP ] (7)U Tf = [XPO YPO ZPO]. (8)T is the rotational-transformation matrix described in Box I:1538 I. Diamantoulaki, D.C. Angelides / Engin

Fig. 2. Generalized hinge modes fj(q) (j = 7, . . . , 10) for total number of hinge

where p(j) is given by:

p(j) = |7+ H j|2

. (4)

The generalized hinge modes (Eqs. (2a)(2c), and (3a)(3c)) for anarray 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 withhinges is subjected to static movements relatively to its initialequilibrium position, due to second order steady drift forces.Consequently, the mooring lines experience changes relatively totheir initial configuration and their initial level of static tensions aswell. It should be mentioned that the aforesaid changes determinethe stiffness of the mooring lines and thus, the final equilibriumposition of the moored array. In the present study, the stiffnessmatrix due to the presence of the mooring lines is derived foran array of hinged floating breakwaters in three dimensions,considering that the stiffness coefficients are strongly affected bythe static equilibrium position of the array; namely the stiffnesscoefficients are affected by the differential changes of both statictension and static angle at the fairlead of each mooring line, Tstand .

More explicitly, the coordinate systems introduced for thepresent analysis are: (i) the coordinate system of themoored arrayconcerned, they denote the vertical translations of the arraydescribed by the fj(q) (j = 6+ h) hinge mode (Fig. 2).ering Structures 33 (2011) 15361552

joints H = 1, . . . , 4 located on the plane z = 0 and parallel to the y-axis [47].X denotes the translational displacement vector of the fairleadof each of the M total mooring lines restraining the hinged array,

ecc

foredng)m,rel 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 ofthe center of gravity of the array of hinged breakwaters, X (Eq. (5)),which includes the vertical displacements due to motion of hingejoints:

equal to the total degrees of freedom describing themotions of tfloating array.

Comparing the reaction forces and moments calculatedan array of floating breakwaters connected with hinges derivhere with the respective ones calculated for a single rigid floatibreakwater [48], it can be seen that for the reaction loads (fX(fY )m, (fZ )m, (MX )m and (MZ )m no differences of definitions aI. Diamantoulaki, D.C. Angelides / Engin

T =cos Y cos Z sin X sin Y cos Z cos X sin Zcos Y sin Z sin X sin Y sin Z cos X cos Z sin Z sin X cos Y

Box I.

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 Yo

Zo +

Hh=1

Zmh

(10)

where the term Zmh denotes the translational displacement of thefairlead ofmooring linem due to vertical translations of the hingedarray caused by the contribution of the hinge joint h. The term Zmhis defined here as:

Zmh = cmh Zh form = 1, 2, . . . ,M and h = 1, 2, . . . ,H (11)where cmh is the coefficient that corresponds to each combination ofmooring line, m, and hinge mode, fj(q), and relates the maximum(unit) vertical displacement at the position of hinge joints with thevertical displacement at the initial position of the fairlead of the mmooring line with respect to OXYZ coordinate system.

Considering Eqs. (7), (8), Box I, Eqs. (10) and (11) it holds:

XPYPZP

= T

XPOYPOZPO

+

XoYo

Zo +H

h=1Zmh

. (12)Since the various displacements of the hinged array due to staticloads result to modification of the initial configuration and levelof static loads, with the assumption that all mooring lines arenot only of identical geometry and material characteristics butalso experience equal pretension level, it follows that the staticquantities Tst and st can be expressed as function of their totallength, l:

Tst = Tst(l) (13)st = st(l) (14)where l can be expressed as function of the projection of the lengthlX = lX (X), lY = lY (X), lZ = lZ (X). (16)ering Structures 33 (2011) 15361552 1539

os X sin Y cos Z sin X sin Zos X sin Y sin Z sin X cos Z

cos X cos Y

(9)

From Fig. 3 it follows:lXlYlZ

=XA XPYA YPZA ZP

=lX cos lX sin lZ

=l cos cos l cos sin

l sin

(17)

where XA, YA and ZA are the constant coordinates of the anchor ofeach mooring line with respect to OXYZ coordinate system.

3.2. Derivation of the stiffness matrix of the mooring lines

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

K =M

m=1Km =

Mm=1

FmX

(18)

where Km denotes the stiffness matrix of m mooring line (m =1, 2, . . . ,M), Fm is the vector of the reaction loads that the mmooring 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 cos (19b)(fY )m = T cos sin (19c)(fZ )m = T sin (19d)(MX )m = (YP Yo)m (V )m (ZP Zo)m (H sin )m (19e)(MY )m = a{(ZP Zo)m (H cos )m

[(XP Xo) XVR]m (V )m} (19f)(MZ )m = (XP Xo)m (H sin )m

(YP Yo)m (H cos )m (19g)with a = 0 when there is a hinge joint located in the middle of thefloating 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 linewith respect to OXYZ , Xho is the x coordinate of the hinge jointlocated at the same module as the fairlead of the m mooring lineand also is closer to the middle of the moored array with respectto OXYZ .

Similarly to the matrix X , Fm also consists of 6 + H terms,heobserved. Therefore, the stiffness coefficients of the hinged array,(Kmij )H , for i = 1, 2, 3, 4, 6 and j = 1, 2, . . . , 6 are equal to the

e1540 I. Diamantoulaki, D.C. Angelides / Engin

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

(Kmij )H = Kmij for i = 1, 2, 3, 4, 6 and j = 1, 2, . . . , 6. (21a)Extended definitions of the stiffness coefficients for the case of arigid floating breakwater, Kmij are included in [48].

On the contrary, according to evaluations of the presentinvestigation, the equation giving the moment (MY )m in the caseof an array of hinged floating breakwaters appears modified withregard to the respective one for a single floating breakwaterpresented in [48]. Therefore, the stiffness coefficients of the arrayof hinged floating breakwaters due to (MY )m, (Km5j )H , (j = 1, . . . , 6)differ from the respective ones for a single floating breakwater, Kmij(see also Appendix, Eqs. (A.1.1)(A.1.6)).

(Km5j )H = a (Km5j + XVR Km3j ) for j = 1, 2, . . . , 6. (21b)The stiffness coefficients related to the hinge modes are derivedin the present study as follows (see also Appendix, Eqs. (A.2.1)(A.2.6), (A.3.1)(A.3.7)):

(Kmi,6+h)H = cmh Kmi3for i = 1, 2 and 3 and h = 1, 2, . . . ,H (21c)

(Km4,6+h)H = cmh (Km43 Tst cos sin )for h = 1, 2, . . . ,H (21d)

(Km5,6+h)H = a cmh (Km53 + XVR Km13 + Tst cos sin )for h = 1, 2, . . . ,H (21e)

(Km6,6+h)H = cmh Km63 for h = 1, 2, . . . ,H (21f)(Km6+h,j)H

=Km3j j = 1, 2, . . . , 6 and h = 1, 2, . . . ,Hcmh Km33 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 arrayof floating breakwaters connected with hinges includes twocomponents: (a) the 3D hydrodynamic analysis of the aforesaidarray and (b) the static and dynamic analysis of the mooringlines. A brief description of these two components is given in thenext two subsections. It should be emphasized that componentsare coupled together through the application of an appropriateiterative procedure in terms of the steady drift forces and theresponse of the floating body, as described in [48].

4.1. Hydrodynamic analysis of the array of floating array connectedwith hinges

The numerical investigation of the performance of the arrayof floating breakwaters connected by hinges includes a 3Dhydrodynamic analysis of the array. The 3D hydrodynamic analysisis carried out in the frequency domain under the action ofmonochromatic waves and is based on a linear wave diffractiontheory.

A velocity potential , satisfying the Laplace equation, sincethe fluid is considered inviscid and incompressible and the flowirrotational, describes the fluid motion. This velocity potential isdefined by the relationship:

= D + r = (o + 7)+ r (22)where the terms D r , o, 7 correspond to the diffraction,

radiation, incident and scattered potentials respectively, and aredefined in [54,55].ering Structures 33 (2011) 15361552

The longitudinal vertical translations of the array of floatingbreakwaters are considered in the hydrodynamic analysis byimplementing appropriate generalized hinge modes [37]. Thehingemodes implemented are equal to the number of hinge joints.

The vertical translations of the hinged floating structure,expressed as the complex amplitude tot (q), is equal to the sumof an appropriate set of modes including the contribution of heave(j = 3), pitch (j = 5) and each of the H total generalized hingemodes introduced, fj(q) (j = 7, . . . ,H + 6): tot(q) =

j

j fj(q), j = 3, 5, 7, 8, 9, . . . , 6+ H (23)

where j 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]:

6+Hj=1[2(Mij + Aij)+ i(Bij + BEij)+ (cij + KHij )]j = Xi,

i = 1, 2, . . . ,H + 6 (24)where Mij is the mass matrix; Aij is the added mass matrix; Bij isthe radiation damping matrix, BEij , is the damping matrix due toexternal causes, cij the stiffness matrix caused by buoyancy andgravity forces, KHij is the stiffness matrix due to the mooring linesand. Xi represents the exciting forces and moments correspondingto the i degree of freedom. Specifically, BEij is given by the followingequation:

BEij = BE(D)ij + BE(V )ij (25)where BE(D)ij is the damping matrix due to the drag damping ofthe mooring lines and BE(V )ij is the viscous damping matrix. Thelatter matrix is calculated according to the empirical relationshipdescribed in [52], after its appropriate modification according toDiamantoulaki et al. [17] in order to accommodate the occurrenceof negative added mass:

BE(V )ij = 2|(Mij + Aij)| Cij (26)

where is the damping ratio.Calculation of the response constitutes a boundary value

problem. The solution of this problem is based on the 3Dpanel method, where Greens theorem is applied, consideringappropriate boundary conditions [55,54].

The response of the floating body is described by the ResponseAmplitude Operator (RAOj), which is given by the followingequation:

RAOj = |j|A , j = 1, . . . , 6+ H (27)where A is the incident wave amplitude.

Similarly, the total vertical displacement across the length ofthe floating array is defined as follows:

= |tot(q)|A

, H + 12

q H + 12

. (28)

The effectiveness of the floating breakwater is expressed by theratio of the wave elevation behind the breakwater (shadow zone)to the incident wave amplitude:

Kb(x, y) = (x, y) , (y > 0) (29)Awhere (x, y) represents the wave elevation at (x, y).


Another expression of the effectiveness behind the floatingbreakwater is introduced in this study:

K avb =

Kb(x, y)s

(30)

where s is the number of the field points in the rectangular areabehind the floating breakwater (Lf /2 x Lf /2) and B/2 y Lf , for Lf and B denoting the total length and width of thefloating 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 breakwateragainst the action of waves, currents and wind. A static and adynamic analysis is necessary to be carried out in order to calculatethe total loads exercised on the mooring lines, as well as thestiffness and drag damping coefficients imposed on the floatingbody by the mooring lines.

4.2.1. Static analysis of mooring linesThe static analysis aims at the calculation of: (a) the initial

static configuration and the static tensions Tst of themooring lines,(b) the new equilibrium position of the floating array-mooringlines system due to the action of the steady drift forces and thecorresponding Tst of the mooring lines and (c) the stiffness matrixKHij that is applied on the breakwater by the mooring lines at thenew equilibrium position. The calculations of (a) and (b) itemsmentioned above are based on the equations that are reportedin [56,57].

4.2.2. Dynamic analysis of mooring linesThe dynamic analysis of the mooring lines includes the

calculation of: (a) the dynamic tensions of them Tdyn at the fairleadat the new equilibrium position and (b) the drag damping matrixBE(D)ij .

Extended description of the calculation of the dynamic tensionsare included in [56,57]. As far as the damping coefficient, BE(D)ij , isconcerned, its calculation is based on linearizing the hydrodynamicdrag force using an equivalent linearization technique [57,48]. Thistechnique is extended here so as the vertical displacements, due tothe hinge joints, be considered.

The diagonal coefficients BE(D)ij (i = j) are considered here sinceall the non-diagonal terms are assumed zero (BE(D)ij = 0 for i = j).At first, the complex horizontal and vertical motion amplitudes,xd and zd, respectively, are calculated at the fairlead of eachmooring line. The amplitudes xd and zd, are in the x and zdirection respectively of the oxz local coordinate system (Figs. 3and 4). Furthermore, the amplitudes xd and zd are attributed to thesinusoidal motions of the floating array (RAOj = 1, 2, . . . , 6 + H)including the vertical displacements due to the hinge joints thatneed to be considered in the dynamic analysis of the mooring linesas well.

The complex motions XC , YC and ZC observed at the fairleadof each mooring line in the X , Y and Z directions of the globalcoordinate system OXYZ (Fig. 3) respectively are given by:

XC = XCR + XCI i= 1 cos(a1) 6 YP cos(a6)+ 5 ZP cos(a5)

+ [1 sin(a1) 6 YP sin(a6)+ 5 ZP sin(a5)]i (31a)YC = YCR + YCI i

= 2 cos(a2)+ 6 XP cos(a6) 4 ZP cos(a4)

+ [2 sin(a2)+ 6 XP sin(a6) 4 ZP sin(a4)] i (31b)ering Structures 33 (2011) 15361552 1541

ZC = ZCR + ZCI i= 3 cos(a3) 5 XP cos(a5)+ 4 YP cos(a4)

+H

h=1[cmh 6+h cos(a6+h)]

+3 sin(a3) 5 XP sin(a5)+ 4 YP sin(a4)

+H

h=1

cmh 6+h sin(a6+h)

i (31c)where aj is the phase of the amplitudes j (Fig. 1).

Obviously, the contribution of the hinge motions is consideredonly in the calculation of the vertical motion ZC .

It should be noticed that Eqs. (31a)(31c) reduce to thecorresponding ones for rigid floating body presented in [48] whenthe effect of the hinges is eliminated.

If the motion amplitudes XC , YC and ZC are analyzed in the oxzplane (Fig. 3), it holds:

QX = QXR + i QXI = [XCR cos(f )+ YCR sin(f )]+ [XCI cos(f )+ YCI sin(f )] i (32a)

QZ = ZCR + ZCI i (32b)where f represents the angle of each mooring line on XY at thenew static equilibrium position. From Eqs. (32a) and (32b), it isderived that the motions of the fairlead of each mooring line, xdand zd, are equal to:

xd =Q 2XR + Q 2XI (33a)

zd =Q 2ZR + Q 2ZI . (33b)

The terminal impedances Sxx, Sxz , Szx and Szz are considered asfunctions of the static anddynamic tension and angle at the fairleadof each mooring line and are given by [57]:[Sxx SxzSzx Szz

]=[xdzd

][FxFz

](34)

where Fx and Fz denote the excitation forces in x and z directionsof the oxz coordinate system respectively.

Next, the complex reaction forces and moments are derivedafter analyzing the terminal impedances of Eq. (35) in Box II inthe OXYZ system (Fig. 3), in a similar manner as Loukogeorgakiand Angelides [48] with proper modifications to include the effectof the hinges, and are given in Box II: Apparently, according toEq. (35) the reaction force corresponding to the vertical displace-ments due to hingemotions is equal to the reaction force in the ver-tical direction of the Z axis of the OXYZ coordinate system (Fig. 3).Szdi can also been given by Eq. (35) after substituting SRxx, S

Rzx, S

Ixx and

S Izx with SRxz , S

Rzz , S

Ixz and S

Izz respectively.

Then, the amplitudes of the reaction loads, Si, and thecorresponding phases, i, can be easily computed according to:

Si =[(Sxd(i))R + (Szd(i))R]2 + [(Sxd(i))I + (Szd(i))I ]2 (36)

i = tan1[(Sxd(i))I + (Szd(i))I(Sxd(i))R + (Szd(i))R

](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, i (with i = 1, 2, . . . , 6 + H), have been computed, thedrag damping coefficients for the m mooring line can be given by:(BE(D)ij )m = |(Si)m cos((i)m aj /2)|

j . (38)

e)

)

)

x

x

)

)

)

geonheof:fornes2)..9,toto

helywith hinges. Twenty-one (21) ormorewave frequencies are exam-ined,where the non-dimensionalwave length ratio B/L varies from0.1 to 1.5, for each case examined assuming normal incident waveconditions.

5.1. Response

The variation of (Eq. (28)) versus x, across the length of thearray of floating breakwaters, is plotted in Fig. 6(a)(c) for threerepresentative wave frequencies B/L = 0.3, 0.6 and 1.1 respec-tively; where L is the corresponding wave length. According to thisfigure, a drastic reduction is observed for the values of whenB/L = 0.6 (Fig. 6(b)) and B/L = 1.1 (Fig. 6(c)) in comparisonto the respective values of when B/L = 0.3 (Fig. 6(a)). This isattributed to the significantly lower amplitude levels of the excit-

response of the array for B/L 0.9.Moreover, considering Fig. 8and (b) it is obvious that the increase of the total number of hinjoints can result to increase or decrease of values dependingboth the combination of: (i) the B/L, and (ii) the position along tlongitudinal axis of the array. This fact is attributed to variation(i) phase difference of modes contributing to vertical motionsdifferent B/L values (relevant results are representatively showfor configuration C2A in Fig. 10) and (ii) generalized hinge modfj(q) for j 7 along the length of the floating array (Fig.Furthermore, significant decrease of is observed for B/L 0attributed to the drastic decrease of exciting loads contributingvertical translations, X3 and/or X7 (X2 and X4 do not contributevertical translations), for B/L 0.9 (Fig. 7).

Fig. 9(a) and (b) demonstrates the variation of versusB/L at tbow and in the middle of C2A and C2B, which are different on1542 I. Diamantoulaki, D.C. Angelides / Engin

Sxd =

(Sxx)R cos(f )+ (Sxx(Sxx)R sin(f )+ (Sxx

(Szx)R + (Szx[(Sxx)R sin(f ) ZP (Szx)R YP ] + [(Sx

a [(Szx)R sin(f ) (XP XVR) (Sxx)R cos(f )ZP ] + a [(Sxx)R sin(f ) XP (Sxx)R cos(f ) YP ] + [(Sx

(Szx)R + (Szx(Szx)R + (Szx

. . .

(Szx)R + (SzxBox II.

Considering Eq. (38), the drag damping coefficients for the systemconsisting of all mooring lines is:

BE(D)ij =M

m=1(BE(D)ij )

m, i = j = 1, 2, . . . ,H + 6. (39)

5. Results and discussion

The effect of different configurations (in terms of numberof hinge joints and mooring lines) of an array of cable-mooredfloating breakwaters on its performance is studied througha rigorous parametric study. The hydrodynamic analysis isperformed using the 3D radiation/diffraction code WAMIT [54].

A cable-moored array of floating breakwaters (Fig. 1) with di-mensions (Lf = 20.00m, B = 4.00m,Hf = 2.00m, dr = 0.77m)is placed in water depth Dw = 10 m and is freely interacting withthewave field. Various configurations of arrays that consist ofmul-tiple floating breakwaters connected by hinges are examined bysetting the total number of hinge jointsH equal to 0, 1, 2 and 3. Eachvalue of the parameterH correspond to a different configuration ofarrays Ck, where k = 0, . . . , 3. It is obvious that C0 is equivalentto a single floating breakwater with no hinges. Moreover, two dif-ferent configurations of mooring systems, A and B, are considered.All the mooring lines of these two mooring systems are located atwater depth equal to 10 m and are identical (Table 1). Configura-tion A is symmetric and consists of four identical mooring lines,where each one forms an angle of 45with respect to the x axis onthe xy plane. As far as configuration B, it is also symmetric andconsists of six mooring lines; where four of them form an angle of45with respect to the x axis on the xy plane and the rest of themare perpendicular to the x axis on the xy plane. Five cases, C0A,C1A, C2A, C3A and C2B corresponding to Fig. 5(a)(e) respectively(Table 2), are examined here in order to investigate the effect ofthe number of: (a) hinge joints and (b)mooring lines on the perfor-mance of cable-moored floating arrays of breakwaters connecteding loads, X3 and/or X7, observed for all configurations for B/L =0.6 and 1.1 (variation of the exciting loads is representativelyering Structures 33 (2011) 15361552

I cos(f ) iI sin(f ) iI i)I sin(f ) ZP (Szx)I YP ] i[(Szx)I sin(f ) (XP XVR) (Sxx)I cos(f )ZP ]i)I sin(f ) XP (Sxx)I cos(f ) YP ] iI iI iI i

(35)

Table 1Characteristics of mooring lines.

Diameter 33 (mm)Total initial length 30 (m)Submerged weight 191.25 (N/m)(Elasticity modulus) (Area) 342,119,440 (N)Breaking tension T break 400,000 (N)

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

hinge joints HNumber offloatingmodules FB(FB = H + 1)

Length of floatingmoduleLs (Ls = Lf /FBm)

C0A 0 1 20.00C1A 1 2 10.00C2A or C2B 2 3 6.67C3A 3 4 5.00

shown here for configuration C1A in Fig. 7) compared to therespective ones computed for B/L = 0.3.

Moreover, Fig. 6(a)(c) depicts the visible effect of thetotal number of hinge joints, H , introduced upon the verticaltranslations of the array. In more detail, increase of H leads toeither increase or reduction of depending on the value of x for allconfigurations CkA (k = 0, 1, 2 and 3). The variation of across thelength 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 forB/L = 1.1.

Fig. 8(a) and (b) presents the effect of the variation of hingejoints 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 variousarrays. Similarly to Fig. 6(a)(c), Fig. 8(a) and (b) demonstrates thesignificant effect of the total number of the hinge joints on the

(a)in terms of the number of restraining mooring lines. Obviously,an increasing number of restraining mooring lines hardly has a

Fig. 5. Description of the configurations (a) C0A, (b) C1A, (c) C2A, (d) C2B and (e) C3A (_ _ _ denotes the position of hinge joints and . . . . . . denotes position of mooring lines).

a b

cI. Diamantoulaki, D.C. Angelides / Engineering Structures 33 (2011) 15361552 1543Fig. 6. versus x corresponding to (a) B/L = 0.3, (b) B/L = 0.6 and (c) B/L = 1.1.

Fig. 8. versus B/L for CkA (k = 0, 1, 2 and 3) configur

aFig. 9. versus B/L for C2A and C2B configurationations (a) at the bow and (b) in the middle of the array.

bminimal effect on which is observed mainly for 0.3 B/L 0.45. This is attributed to the combined effect of: (i) different valuesof phase difference of 3 and 7 modes calculated when 0.3 B/L 0.45 (Fig. 10) and (ii) significant values of amplitudes of3 and 7 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 andgeneralized hinge modes (RAOj for j 7, 9) for CkA (with kvalues indicated in Fig. 11(d)); while Fig. 12(a)(d) shows the

(Fig. 13), and thus, motion of 9 mode occurs due to radiationeffect activated by motion of 3 mode. Moreover, consideringFig. 12 (a)(c), it can be shown that the introduction of twosupplementary mooring lines in the middle of the array consistingof three floating breakwaters mainly affects RAO2, RAO4 and RAO7.In particular, increase of mooring lines leads to decrease of RAO2and RAO7, especially for B/L 0.35, whereas it causes increaseof RAO4. Finally, all modal responses exhibit noticeable decreasefor B/L 0.9 (Figs. 11(a)(d) and 12(a)(d)), which is in completeaccordance with the decrease of exciting loads computed for thisfrequency range (Fig. 7).

a b1544 I. Diamantoulaki, D.C. Angelides / Engineering Structures 33 (2011) 15361552

Fig. 7. Xj (j = 2, 3, 4 and 7) versus B/L corresponding to C1A.

variation of RAO2, RAO3, RAO4 and RAO7 versus B/L for C2A andC2B configurations. It is observed that all configurations exhibita 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 strongcoupling between sway and roll modes due to presence ofmooringlines. In more detail, the aforesaid coupling leads to an increasedeffect of sway behavior on roll behavior and vice versa at the wavefrequencies that peak values are exhibited. C2A demonstratessignificantly higher RAO3 for B/L 0.3 compared to the restconfigurations plotted in Fig. 11(b). It can be also been shown thatincrease of hinge modes leads to increase or decrease of all modalamplitudes depicted depending on the B/L value (Fig. 11(a)(d)).It should also be mentioned that RAO3 and RAO9 of C3A exhibitsimilar patterns of variation. This happens because for C3A, X9 = 0s (a) at the bow and (b) in the middle of the array.

cFig. 11. Modal responses RAOj (j = 2, 3, 4, 7 and 9) versus BdI. Diamantoulaki, D.C. Angelides / Engineering Structures 33 (2011) 15361552 1545

Fig. 10. Phase difference of 3 and 7 versus B/L for C2A and C2B configurations.

5.2. Static and dynamic tensions of mooring lines

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 themooring lines. These forces are exercised either onthe front mooring lines (y = 2 m, Fig. 1) or back mooring lines(y = 2 m, Fig. 1). Obviously, the front mooring lines are the mostheavily 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

joints is associated with snapping phenomena (Tst < Tdyn) whichare observed for 0.3 B/L 0.4; namely the region of high RAO4values (Figs. 11(c) and 12(c)). The peak RAO4 values observed forthe aforesaid B/L values (Fig. 11(c)) lead to peak values of Tdyn andconsequently to peak values Ttot . Moreover, it can be seen that thehigher the RAO4 peak value is, the higher is the Tdyn or Ttot levelfor bow or stern mooring lines (this statement is not valid for themooring lines located in the middle of C2B since pitch motion hasno effect in the middle of the array). Besides, there is no chance forthe total tensions to exceed breaking tension, since the maximumratio of the breaking tension to the total tension observed is equalto 0.089 (Ttot/Tbreak = 0.089). The lower Ttot levels are exhibitedby C0A configuration; thus it can be claimed that hinge joints leadto increase of Ttot (Fig. 14(c)). Besides, increase of the number ofmooring line leads to higher Ttot levels. As regards C2B, the mostheavily loaded mooring lines are the ones located in the middleof the array (Fig. 15(a)) since they correspond to larger area ofeffect.

5.3. Effectiveness

The effectiveness of the floating array discussed in thissubsection is expressed in terms of Kb (Eq. (29)) and K avb (Eq. (30)).The wave elevation, in all cases, is calculated at the field pointswith coordinates ranging within x = Lf /2, . . . , Lf /2 m and y =B/2, . . . , Lf m in the rear of the floating array according to thedefinition of the body coordinate systemdepicted in Fig. 1. For eachconfiguration CkA (k = 0, 1, 2 and 3), C2A and C2B examined,the effectiveness is calculated considering the complete problem

a b/L corresponding to CkA (k = 0, 1, 2 and 3) configurations.

eFig. 12. Modal responses RAOj (j = 2, 3, 4 and 7) versus B/L corresponding to C2A and C2B configurations.

Fig. 13. Exciting loads Xj (j = 2, 3, 4 and 7) versus B/L corresponding to C2A andC2B configurations.

(i.e. considering both diffraction and radiation). The effect ofdiffracted waves on the effectiveness is also discussed separately.

Fig. 16 depicts the variation of the non-dimensional efficiencyparameter Kb with y (x = 0 m) for three representative wave fre-quencies 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 correspond-ing to B/L values equal to 0.3, 0.6 and 1.1. Obviously, the efficiencyparameter Kb corresponding to the diffracted waves, namely Kbd, is

differences observed among Kb patterns are exclusively attributedto the effect of radiationwaves caused by themotion of the floatingbreakwaters.

Regarding B/L = 0.3 (low wave frequency range), it is shownthat the variation of all Kb patterns is quite different from thevariation of the pattern corresponding to Kbd (Figs. 16(a) and17(a)). This statement proves the intense effect of radiationwaves on the effectiveness for all configurations examined in theaforementioned figures (Figs. 6, 11 and 12). The configurationsperforming the highest effectiveness among all CkA (k = 0, 1, 2and 3) configurations, are C0A and C3A; the former configurationfor 2.5 m y 7.5 m and the latter 7.5 m y 20 m. On thecontrary, the effectiveness performed by C1A is inadequate, sincefor y 8m it holdsKb 1.0 as shown in Fig. 16(a). The reason thatC1A exhibits inadequate effectiveness is attributed to the fact thatC1A exhibits the highest (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 per-formance. Inmore detail, C3A appears to be themost effective con-figuration for 2.5m y 4.5mand7.0m y 20.0mwhereasC0A or C1A are the most effective configurations for 4.5 m y 7.0 m. A clear resemblance among Kb patterns of C0A and C1A isobserved, since low and very similar levels of response have beencalculated for B/L = 0.6 (Fig. 6(b)).

Finally, for B/L = 1.1 (high wave frequency range), thevariation of Kb is of similar trends with the variation of Kbd(Fig. 16(c)), given the reduced contribution of radiation effect on1546 I. Diamantoulaki, D.C. Angelides / Engin

a

cthe same for all configurations examined (Figs. 16 and 17) due tothe identical geometry of all configurations examined. Thus, anyering Structures 33 (2011) 15361552

b

dthe 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

cFig. 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 bI. Diamantoulaki, D.C. Angelides / Engineering Structures 33 (2011) 15361552 1547

a bFig. 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 C2A and C2Bconfigurations.

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

c

b1548 I. Diamantoulaki, D.C. Angelides / Engineering Structures 33 (2011) 15361552

a

c

bFig. 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 C2A and C2B configurations.

Fig. 19. Kb contours for (a) C2A and (b) C2B considering

aFig. 20. Kb,av versus B/L corresponding to configuraB/L = 0.3 (the dotted lines indicate the mooring lines).

bb

Fig. 18. Variation of Kb in the rear of the array of floating breakwaters for (a) C2Aand (b) C2B considering B/L = 0.3.

perform acceptable level of effectiveness (Kb < 1.0) and each ofthem can be the most effective configuration depending on theB/L value (i.e. C3A and C2A appear to be the most effectiveconfigurations for 0.1 B/L 0.17 and 0.17 B/L 0.32respectively). As far as the middle and high wave frequencyranges are concerned, all CkA configurations exhibit significantimprovement of the effectiveness. In more detail, C0A and C3Aare the most effective configurations for 0.4 B/L 0.5 and0.5 B/L 0.9 respectively, whereas for 0.9 B/L 1.5 allconfigurations exhibit almost identical level of effectiveness dueto reduced response levels (Figs. 6(c), 8(a), (b) and 11(a)(d)). Theeffect of the number of hinge joints is apparent only in the lowand middle wave frequency ranges due to either high or mediocreresponse 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 C2A and

a bI. Diamantoulaki, D.C. Angelides / Engineering Structures 33 (2011) 15361552 1549

a

identical as are also the low response levels. All configurations ofFig. 16(c) exhibit adequate effectiveness irrespective of the hingejoints introduced.

According to Fig. 17(a)(c), an increasing number of mooringlines has a direct effect on the effectiveness only for B/L = 0.3 (lowwave frequency range, Figs. 17(a), 18 and 19), where the effect ofradiation caused by heave motion is significant (Figs. 9(a), (b) and12(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 plottedfor CkA (k = 0, 1, 2, 3) configurations. Regarding the low wavefrequency range, C1A configuration demonstrates unacceptableeffectiveness, since for 0.1 B/L 0.3Kb 1.

The rest of CkA configurations, namely C0A, C2A and C3Ations (a) CKA for k = 0, 1, 2, 3 and (b) C2A and C2B.


C2B configurations. According to this figure, it can be seen thatthe effect of the number of mooring lines on the effectiveness isnoticeable only in the lowwave frequency range, where increasingthe number of mooring lines can have a positive or negativeinfluence on the effectiveness, depending on the B/L value. Theaforesaid statement is also apparent considering Figs. 18 and 19.

6. Conclusions

In the present investigation, the overall performance of a cable-moored array of floating breakwaters connected by hinges isinvestigated under the action of monochromatic linear waves inthe frequency domain. The numerical analysis of the array is basedon a 3D hydrodynamic formulation of the floating body coupledwith the static and dynamic analysis of the mooring lines.

The motions of the array of floating breakwaters due to thehinge vertical translations are considered in the hydrodynamicanalysis with the implementation of appropriate generalizedmodes. The stiffness and damping coefficients caused by themooring lines in both rigid and generalized degrees of freedom arederived here in general form. A rigorous parametric study is carriedout in order to investigate the effect of different configurations,namely number of hinge joints andnumber ofmooring lines, on theperformance of the cable-moored array of floating breakwaters.The main conclusions generated by this research are:

1. A strong dependence of the arrays response on the numberof hinge joints is exhibited, mainly in the case of low andmiddle wave frequency ranges. Increasing the number of hingejoints can either increase or decrease response level dependingon the combination of both the wave frequency parameterB/L and the position along the longitudinal axis of the array.Vertical translations exhibit high variation in the low wavefrequency range, and this variation becomes smoother for anincreasingwave frequency. All modes exhibit a drastic decreaseof response in the high wave frequency range.

2. An increasing number of restraining mooring lines has a smalleffect on the vertical translations and the modal responsesconfined in the low wave frequency range.

3. The presence of hinge joints is associated with snappingphenomena and higher level of dynamic and, consequently,total tensions in the mooring lines. These phenomena areconfined around some B/L values in the low wave frequencyrange. Proper choice of dimension B can overcome this problem.

4. Variations of the effectiveness among all configurations exam-ined are attributed to the radiation effect, since the diffractedwaves are not affected by neither the number of hinge joints northe number of mooring lines. Radiation effect is more intensein the low and middle wave frequency range, where higher re-sponse levels have been computed.

5. The number of hinge joints strongly affects the array effective-ness in both the low and middle wave frequency ranges. Thenumber of mooring lines affects the array effectiveness in thelow wave frequency range. Moreover, an increasing number ofhinge joints or mooring lines can have a positive or negative in-fluence on the effectiveness, depending on the wave frequencyratio B/L.

6. Based on all the above, for a given range of dominant wavefrequencies the proper combination of dimension B, number ofhinge joints and number of mooring lines has to be determined.

AcknowledgementsThe authorswould like to thank the reviewers for their valuablecomments and suggestions.ering Structures 33 (2011) 15361552

Appendix. Definitions of mooring lines stiffness coefficientsfor an array of floating breakwaters connected with hinges

A.1. Stiffness coefficients (Km5j )H for j = 1, 2, . . . , 6

(Km51)H = MYXo

= Xo

{a [(ZP Zo) fX (XP Xo XVR) fZ ]}

= a [(ZP Zo)

fXXo

(XP Xo XVR)

fZXo

]= a [(ZP Zo) Km11 (XP Xo XVR) Km31]= a [(ZP Zo) Km11 (XP Xo) Km31 + XVR Km31]= a (Km51 + XVR Km31) (A.1.1)

(Km52)H = MYYo

= Yo


= a [(ZP Zo)

fXYo

(XP Xo XVR)

fZYo


(Km53)H = MYZo

= Zo


= a [(ZP Zo)

fXZO

(XP Xo XVR)

fZZO


(Km54)H = MYX

= X


= a [fX

X(ZP Zo) (ZP Zo) fX

X

+ fZ X

(XP Xo XVR)+ (XP Xo XVR) fZX

]= a

[fX

ZPX

0 (ZP Zo) Km14

+ fZ XPX

0 0+ (XP Xo XVR) Km34

]= a

[fX

lZX

(ZP Zo) Km14

+ fZ lXX

+ (XP Xo XVR) Km34

]= a (Km54 + XVR Km34) (A.1.4)

(Km55)H = MYY

= Y


= a [fX

Y(ZP Zo) (ZP Zo) fX

X+ fZ Y

(XP Xo XVR)


+ (XP Xo XVR) fZY

]= a

[fX

ZPY

0 (ZP Zo) Km15

+ fZ XPY


]= a

[fX

lZY

(ZP Zo) Km15

+ fZ lXY

+ (XP Xo XVR) Km35

]= a (Km55 + XVR Km35) (A.1.5)

(Km56)H = MYZ

= Z


= a [fX

Z(ZP Zo) (ZP Zo) fX

Z

+ fZ Z

(XP Xo XVR)+ (XP Xo XVR) fZY

]= a

[fX

ZPZ

0 (ZP Zo) Km16

+ fZ XPZ


]= a

[fX

lZZ

(ZP Zo) Km16

+ fZ lXZ

+ (XP Xo XVR) Km36

]= a (Km56 + XVR Km36). (A.1.6)

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

(Km1,6+h)H = fXZh

= Zh

(Tst cos cos )= cmh Km13 (A.2.1)

(Km2,6+h)H = fYZh

= Zh

(Tst cos sin )= cmh Km23 (A.2.2)

(Km3,6+h)H = fZZh

= Zh

(Tst sin)= cmh Km33 (A.2.3)

(Km4,6+h)H = MXZh

= Zh

[(YP Yo) fZ (ZP Zo) fY ]= (YP Yo) Km37 (ZP Zo) Km27 Tst cos sin cmh= cmh (Km43 Tst cos sin ) (A.2.4)

(Km5,6+h)H = MYZh

= Zh

[a (ZP Zo) fX a(XP Xo XVR) fZ ]= a Tst cos cos cmh + a (ZP Zo) (Km17)H a (XP Xo XVR) (Km37)H= a cmh (Km53 + XVR Km13+ Tst cos cos ) (A.2.5)

(Km6,6+h)H = MZZh

= Zh

[(XP Xo) fY (YP Yo) fX ]m m= (XP Xo) (K27)H (YP Yo) (K17)H= cmh Km63. (A.2.6)ering Structures 33 (2011) 15361552 1551

A.3. Stiffness coefficients (Km6+h,j)H for j = 1, 2, . . . , 6+ H

(Km6+h,1)H = fZXo

= (Km31)H = Km31 (A.3.1)

(Km6+h,2)H = fZYo

= (Km32)H = Km32 (A.3.2)

(Km6+h,3)H = fZZo

= (Km33)H = Km33 (A.3.3)

(Km6+h,4)H = fZX

= (Km34)H = Km34 (A.3.4)

(Km6+h,5)H = fZY

= (Km35)H = Km35 (A.3.5)

(Km6+h,6)H = fZZ

= (Km36)H = Km36 (A.3.6)

(Km6+h,6+h)H = fZZh

= (Km3,6+h)H = cmh Km33. (A.3.7)

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Modeling of cable-moored floating breakwaters connected with hingesIntroductionDefinition of generalized hinge modesFormulation of mooring lines' stiffness coefficientsDefinition of basic quantitiesDerivation of the stiffness matrix of the mooring lines

Description of the numerical modelHydrodynamic analysis of the array of floating array connected with hingesStatic and dynamic analyses of mooring linesStatic analysis of mooring linesDynamic analysis of mooring lines

Results and discussionResponseStatic and dynamic tensions of mooring linesEffectiveness

ConclusionsAcknowledgementsDefinitions of mooring lines' stiffness coefficients for an array of floating breakwaters connected with hingesStiffness coefficients (K5jm)H for j = 1, 2, ..., 6 Stiffness coefficients (Ki, 6+ hm)H for i = 1, 2, ..., 6 Stiffness coefficients (K6+ h, jm)H for j = 1, 2, ..., 6 + H

References

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