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Ocean Engng, Vol. 25, No. 1, pp. 27~-8, 1998 1997 Elsevier Science Ltd. All rights reserved

Printed in Great Britain 0029~018/97 $17.00 + 0.00

MOORING FORCES AND MOTION RESPONSES OF PONTOON-TYPE FLOATING BREAKWATERS

S. A. Sannasiraj, V. Sundar and R. Sundaravadivelu Ocean Engineering Centre, Indian Institute of Technology, Madras 600 036, India

(Received 1 May 1996; accepted in final form 12 June 1996)

Abstract- -The experimental and theoretical investigations on the behaviour of pontoon-type floating breakwaters are presented. A two-dimensional finite element model is adopted to study the behaviour of pontoon-type floating breakwaters in beam waves. The stiffness coefficients of the slack mooring lines are idealized as the linear stiffness coefficients, which can be derived from the basic catenary equations of the cable. The theoretical model is supported by an experimental programme conducted in a wave flume. The motion responses and mooring forces are measured for three different mooring configurations, and the results are reported and discussed in detail in this paper. The wave attenuation characteristics are presented for the configurations studied. 1997 Elsevier Science Ltd.

1. INTRODUCTION

Over the past two decades, interest in the study of the behaviour of floating breakwaters, FBWs, has increased owing to the requirement for the development of large number of small marinas and recreational harbours. The lower initial investment and the mobility of the structure of FBWs is attractive to the designer. FBWs are evaluated as a viable alterna- tive when the cost of a fixed structure exceeds the economic return to be gained at that location. Several configurations for the FBWs have been studied by many of the investi- gators. The pontoon-type FBW performs wave attenuation as well as being useful for boat docks, moorings or walkways.

Adee (1975) developed a two-dimensional, linear, theoretical model to predict the performance of catamaran type FBWs in deep water and compared the results with measurements in a model tank and from a prototype installation in the field. Adee and Martin (1974) observed that roll motion does not contribute significantly to the wave transmission performance. An analysis of mooring lines was done from static equations of cable equilibrium and included as a restoring force term in the equations of motion for the breakwater (Adee, 1977). They concluded that theory consistently under-predicts mooring forces. Sutko and Haden (1974) reported that a square cross-section gives slightly better wave reduction than a triangular, circular or trapezoidal section. Carver (1979) reported that uncrossing the anchor chains had a negligible effect and adding a vertical barrier-plate has little effect on wave-attenuation characteristics and, therefore, appears to be of questionable benefit in reducing transmitted energy. The transmission coefficient was found to be strongly dependent on relative breakwater length (the ratio of the characteristic dimensions of the breakwater to wave length) and weakly dependent on wave steepness.

Hales (1981) reviewed the five concepts--pontoons, sloping floats, scrap tires, cylinders, and tethered floats--which are the dominant FBW types. It was suggested that the designs

27

28 s.A. Sannasiraj et al.

should be kept as simple, durable and maintenance free as possible; avoiding highly complex structures that are difficult and expensive to design, construct and maintain. The reinforced-concrete pontoon-type structure with a foam core for floatation is reported as the most widely accepted FBW meeting the standard wave climate. Miller and Christensen (1984) predicted the dynamic response of FBWs using a frequency domain analysis and compared this with full-scale field measurements. The range of predicted response vari- ances is from 0.5 to 1.3 of those obtained from field measurements.

Yamamoto et al. (1980) solved the problems of wave transformation and motions of elastically moored floating objects by direct use of Green's identity formula, and validated their solutions with the experimental investigations. They found that if the mooring system is properly arranged, the wave attenuation by a small draft breakwater can be improved several times compared to the same FBW conventionally moored. Yamamoto (1981) conducted large-scale model tests in a large wave-tank on elastically moored floating breakwaters under the action of regular and random waves. The response of an elastically moored floating breakwater to random waves is reported as essentially the same as the response of the breakwater to periodic waves.

The scale effects apparently affect the end result (Veldee, 1983). Wave attenuation is reported to be strongly dependent on wave length and only slightly affected by incident wave steepness. The mooring forces at all relative breakwater lengths were shown to increase with increasing wave height. Isaacson and Bytes (1988) reported the development of a numerical model, based on linear diffraction theory, to investigate FBW motions, transmission coefficients and mooring forces, in obliquely incident waves. A field survey of FBWs in British Columbia, Canada is also reported.

Johansson (1989) showed how the wave protection can be significantly improved by using a rectangular breakwater with a horizontal protruding bottom-plate rather than a rectangular one. This is confirmed both by potential theory and by measurements. The response of a moored vessel to beam waves has been investigated by Isaacson and Wu (1995), based on an idealized representation of the nonlinear stiffness characteristics of the mooring system. Oliver et al. (1994) compiled various types of pontoon-type FBW installations in the field and reported the analysis process and analysed the cost comparison of FBWs.

The present paper deals with a comprehensive study on the behaviour of pontoon-type floating breakwaters both theoretically and experimentally. The theoretical model is based on a two-dimensional finite element technique to evaluate the hydrodynamic coefficients and wave exciting forces on the floating structure. The stiffness of the slack mooring lines is modelled as linearized stiffness coefficients derived from basic catenary equations. The equations of motion are solved to evaluate responses and, hence, mooring forces. An experimental study has been carried out in a wave flume. The motion responses in three modes of motion, viz. sway, heave and roll, the mooring forces and the wave field in front and on the lee side of the structure were measured. The responses are presented in terms of response amplitude operator, RAO (response due to unit amplitude wave), and mooring forces are presented in a normalized form. The experimental predictions are compared with the theoretical model and the results are reported.

Mooring forces and motion responses of pontoon-type floating breakwaters 29

2. THEORETICAL INVESTIGATION

2.1. General formulation Fluid is assumed to be ideal, flow is considered as irrotational and application of linear

wave theory is valid. The body is assumed to be rigid. It is assumed that no flow of energy takes place through the bottom surface or the free surface. Energy is gained or lost by the system only through waves arriving or departing at infinity or due to the external forces acting on the body. The motions are assumed to be small, so that the body boundary conditions are satisfied very close to the equilibrium position of the body.

A Cartesian coordinate system is employed, with the origin at the mean free surface, Oz directed positive upwards and Ox directed positive in the direction of propagation of waves. The state of the fluid can be completely described by the velocity potential, alP(x, z, t) satisfying the Laplace equation.

Va~(x, z, t) = 0 (1)

The general configuration of an infinitely long floating structure interacting with a mono- chromatic linear wave of height, H, and wave angular frequency, to = 27r/T is shown in Fig. 1. It is generally convenient to separate the total velocity potential into incident potential, ~i, radiation potentials, ~;, j = 1, 2, 3 in three modes, viz. sway, heave and roll, and scattered potential, ~4. This is mathematically represented as

3

(I)(x, Z, t) = (I) I + E (I)J + (I)4 (2) j=l

where % = L ~bj in which, ~bj is the radiation potential per unit body velocity, L .

2.2. Diffraction problem For the present wave structure interaction problem, the incident velocity potential is

represented by

I 7 , i I r ' I

I I ' !

' [ , i

', i !

I f / / / / / / f f /77 / / f / f / f / " f / f~

Fig. 1. Definition sketch.

30 S.A. Sannasiraj et al.

*,(x, Z, t )= Re[ -iagw cshcoshk(z + d)kd ei(k'~ 6t) 1 (3)

where g is the gravitational constant, d is the water depth, a is the amplitude of wave and k is the wave number satisfying the dispersion relation,

oo 2 = gk tanh kd (4)

The linear wave diffraction problem is described by a sinusoidally varying diffracted velocity potential, ~D in time, given by

dPD(X, Z, t) = ~l(x, Z, t) + (I)4(x , z, t)

= Re[{c~l(x,z) + ~b4(x,z)}e i{o,] (5)

The boundary value problem for the diffracted potential can be defined by the governing Laplace equation and the boundary conditions as defined below:

724}4 = 0 in the fluid domain ~1 (6)

. . . . . . 4}4 -- 0 at the free surface, FF, z = 0 (7) bz g

a+4 = 0 at the sea bed, I't~, z = - d (8) Oz

04~4 Ox T- ikq~4 = 0 at the radiation boundary, F ..... r --~ + ~c (9)

O+4 a+, - on the body surface, I'o (10)

On On

where n is the unit outward normal from the fluid domain. The infinite boundary F~ is fixed at a finite distance, x = xR. The position of the radiation boundary relative to the characteristic dimension of the structure and water depth is described in detail by Bai (1977).

2.3. Radiation problem The wave radiation problem can also be described by a radiated potential varying sinus-

oidally in time given by:

j(x,z) = Re[ - i t .oX j f~ j (x ,z )e hot] (11)

The linear radiation boundary value problem is defined by the Laplace equation as a governing equation, and the boundary conditions are as given below:

~72(~j = 0 in ~ (12)

0+~ ,0 2 0z g ~bi--0 atFF, Z=0 (13)

3+, = 0z 0 at FB,z = - d (14)


Oxx ~ ik~b~ =0 atr~,x--- , +oo (15)

a6j 0n -n~ onFo, j= 1,2,3 (16)

where n~ and n2 are the x and z component of the unit inward normal to the body and

n3 = (x - Xc)n2 - (z - Zc)nl (17)

in which (Xc, zc) are the coordinates of the centre of rotation.

2.4. Hydrodynamic forces

The hydrodynamic pressure at any point in the fluid can be expressed as,

00 p(x, z, t) = - P 0 t = itop~ (18)

where p is mass density of fluid. The hydrodynamic forces can be determined by integrat- ing the pressure over the wetted body surface Fo.

F~j = f pnjdF (19)

F o

where j = 1, 2, 3 correspond to sway, heave and roll modes, respectively. The hydrodynamic forces, thus evaluated, can be separated into wave exciting forces

governed by the diffraction problem and the hydrodynamic restoring forces governed by the radiation problem.

The wave exciting force, ~ due to the diffracted potential can be expressed as

F~j = i topI(~ I + ~4)njdF = Re~e - io,,] (20)

Fo

where fj is the complex force amplitude in the jth mode of motion. From the radiation potential, the hydrodynamic restoring forces, F~j can be evaluated as

F~j = Ii~oo2k6knidF = - tZjkJ~k -- h~k2k (21) , / FO

where /zjk is the added mass coefficient proportional to the body acceleration and hjk is the damping coefficient proportional to the body velocity. /xjk and hjk are evaluated from the real and imaginary parts of the complex radiation potential, respectively.

3. NUMERICAL MODEL

In order to solve the above diffraction-radiation boundary value problem, a finite element method is used. The infinite fluid domain can be made finite by incorporating either plane or higher order boundary-dampers at the radiation boundary, which is at a


finite distance, x = + XR, from the structure. For the present study, a plane damper is used to model the radiation boundary.

Bai (1977) formulated the finite element system of equations for the diffraction problem using a variational principle. For the present study, finite element formulations for a diffraction-radiation boundary value problem in a two-dimensional vertical plane have been done based on the Galerkin approximation (Sannasiraj et al., 1994). The fluid domain is divided into discrete elements with a total of M nodes. The velocity potential, q~j is approximated by a linear combination of interpolating functions, and may be represented as

dO 9 = ~'~NrCh~re ~,o, j= 1,2,3,4 (22) r 1

where Nr(x, z) are the shape functions and ne denotes the number of element nodes. Using the Galerkin approximation, qbj should satisfy

fNrv2qbj dl~ = (23) 0

Application of the divergence theorem and using Equations (6)-(10) and Equations (12)-(16), the finite element formulation of a diffraction-radiation problem leads to linear algebraic equations of the form

[K 1 + K 2 + K3]{~bj } = {Pj} (24)

where

K~.~ = ~ f (Vgr)(VN,.)m e

(25)

oJ2f Kr2~ = E -- g NrUs e

r~

(26)

Kr3s2 ~- ik I N,N~ dr ~ (27)

The load vector is given by

J P j r=~ Nr~nn dF~, e

j = 1, 2, 3,4 (28)

where j = 4 corresponds to the diffraction problem and j = 1, 2, 3 corresponds to the

where e denotes a finite element. The summation implies assembly of the element property matrices over the entire fluid domain 12. Furthermore, the contributions by the free surface and radiation conditions are as follows:


radiation problem in the three modes. The assemblage of the element matrices has been done in the usual manner (Zienkiewicz and Taylor, 1989) and the resulting simultaneous equations can be solved for ~bjs, s = 1, 2 ..... M taking advantage of the symmetric banded nature of the matrix.

4. LINEARIZED CABLE STIFFNESS

The stiffness components of the slack mooring lines are assumed to remain unaffected by the motion of the structure, which is induced by waves, and the stiffness values are computed at the mean position. The line tension acting on the buoy is represented as a linear function of displacement by using the integrated equations of deep sea mooring lines in static equilibrium (Ogawa, 1984). The present derivations are restricted to two dimensions following the procedure given by Jain (1980), and it is assumed that the current drag on the cable is negligible compared to the cable weight. It is assumed that the cable is perfectly flexible, inextensible and heavy. The basic equations for the cable equilibrium are used to derive the linearized stiffness coefficients in sway, heave and roll modes.

The definition sketch showing a floating two-dimensional structure moored by a cable of length, L to an anchor at B with the bottom angle of 00 is shown in Fig. 2. The cable is extended hypothetically beyond B for a length of l, to O' so that the angle, 0 at O' is zero. Two sets of rectangular coordinates (X, Z) and (~, 7/) are chosen in the plane of the cable through B and O', respectively. The coordinates of A and B with respect to the origin, O' are

ea= o sinh- ,(wLl w \To /

[{ ] =To 1+ -1 rlA W \To] J (29) ~B = To sinh- ' /wll

7 I ToJ (30)

-q Z

and

0 ' 1

L

J

, x Ep

I F - -1 ! I ! I i I ! j J

d

1 Fig. 2. Cable configuration.

34 S .A . Sannasiraj et al.

nB = W \To / } - 1

where To is the initial horizontal tension, w is unit weight of cable in water and L = L + 1. TA and TB are the initial cable tensions at A and B, respectively, expressed as

[ (wLtq"2 j TA=T 0 1 +\To]

[ ( ,lq 'p- TB=To 1 +\To/ J (32)

The linearized stiffness coefficients thus evaluated are

[ [TBL- TA/~[XA- (TBL- TAI)I- T(2)ZA] - . ro[W irA- K )lr; rat. /j rAr.J (33) w [LTB - ITA'] U

K~ = To ~ TA TB )K,2 (34)

W ( .TATB tIXA [LTB -- ITA\ ] M

[W(~,!- ~c) (LTB - ITa] ] K i~ = [ T , , \ ira ~ T. / + (x~. -- x0 K~ (36)

142(X c ~ X!! ( TAT ~ tIXa [LT B -ItA~]t M g~ = (z, - z~,)+ To \TA -- TB][To --~ TATB-)IJK,2 (37)

K~ = (38)

W(Z, 2 ~!2 (LTB -- (TA] W(X~ ---r-02 ( TA~ '],XA (LT~-- ITa]I l T,, \ TA-- TB ] + To ~Ta - Ts/[T, - \ TATB ]J K~

+ 2(& - xt)(z~ - z~)

where (x,, z,) is the point of attachment of the mooring cable with the floating structure with respect to the x-z coordinate system. Also, K~j = Kj~, i = 1, 2, 3; j = 1, 2, 3.

The initial horizontal tension, To and the hypothetical projected length, l can be evaluated from the cable configuration and the length of the cable, L using the basic cable catenary equations.

5. MOTION RESPONSE PROBLEM

The response of the structure in waves of amplitudes sinusoidally varying in time can be evaluated from the basic equations of motion. In the frequency domain, the equations of motion can be written as

_ "e [ - w2(M + Ix) - io)A + C + KM]fj - f j , j=1 ,2 ,3 (39)


where M,/x, C and A are the body mass, added mass, restoring force and damping matrices of the system. 6j is the complex amplitude of the motion response, Xj = 6je - i.,,. The response of the body can generally be expressed in terms of response amplitude operator, RAO, defined as the response of the structure in unit amplitude wave. The equations of motion can be solved using the Gauss elimination method for the displacement vector in the three modes of motion. On establishing the displacement vector, the mooring forces, F M can be evaluated from

F~ = [K,~.]{Xj}, j = 1, 2, 3. (40)

6. EXPERIMENTAL PROGRAMME

6.1. Testing facility

In order to investigate the behaviour of the pontoon-type FBWs in regular and random waves, a well-controlled experimental programme was carried out in a wave flume at the Ocean Engineering Centre, Indian Institute of Technology, Madras, India. The wave flume, 4 m wide and 90 m length, with a constant water depth of 2.35 m, is equipped with a hinged-type twin flap wave-maker. The wave absorbers are a rubble mound capable of absorbing effectively, in particular, the short period waves. The wave-maker is capable of generating regular and random long-crested waves of predefined spectral characteristics.

6.2. Model details

The experimental model is made of fibre reinforced plastic covered by a marine plywood board on its top. The model inner walls and the bottom plate are reinforced with the fibre coated wooden stiffeners. The required draft of the body is obtained by placing firmly additional weights inside the model. The details of model parameters are presented in Table 1.

Table 1. Particulars of the model

Particulars Symbol Unit Value

Length L rn 3.78 Breadth B m 0.40 Draft T m 0.10 Mass M kg 150.5 Mass center (x~, z,) m (0.0, 0.026) Transverse Metacentric GMr m 0.057 height Mass MI about transverse lxx kg m 2 188.0 axis, XX Mass MI about lrv kg m 2 5.33 longitudinal axis, YY Mass MI about vertical lzz kg m 2 189.8 axis, ZZ Roll natural frequency fr Hz 0.592 Heave natural frequency fz Hz 1.012


1

CONFIGURATION_C 1

/ CONFIGURATION_C2

F ig . 3. Mooring line configurations.

CONFIGURATION_C3

6.3. Model setup

The single pontoon-type FBW with three different types of mooring configurations are considered (Fig. 3), viz.: (i) mooring at water level (configuration Ct); (ii) mooring at base bottom (configuration, C2); and (iii) cross moored at base bottom level (configuration, C3). In all the cases the length of the mooring line is fixed at twice the water depth, and four mooring lines are used for each configuration. The horizontal projected length and the angle with the vertical at the top of the anchor, the point of attachment of the moorings with the model and at the bottom of the flume for the three configurations CI, C2 and C3 are given in Table 2. The unit weight of chain is 1.25 N m ~ in air.

6.4. Instrumentation

The instrumentation set-up is shown in Fig. 4. The motions in the three modes were measured using a gyroscope. A gyroscope amplifier was used to amplify the signals. The gyroscope was able to measure the accelerations in the X, Y and Z directions in the range

Tab le 2. S lack mooring line details

Particulars Configuration C I Configuration C2 Configuration C3

Horizontal projected length, / (m) Angle at anchor point, 0o Angle made with the body, 0 Point of attachment with the body (m) (x~,y~,zt ~)

(x?,y?,z?) (x?,y?,z~)

Point of attachment with the flume bottom (m)

-4 ,4 ~4 (x, ,y . . . . ) I I I

(xb ,yb ,zb)

2 2 .2 (Xb 7h ,,t,)

3 3 3 (xb,yb,zb)

,4 ,4 4 (Xh,yh,Zh)

4.05 4.11 4.11 20 20 20 39 36 .6 36 .6

(0 .15 , - 1 .0 , - 0 .1 ) ( 0 .15 , - 1 .0 , - (0 .2 , - 1 .0 ,0 .0) 0 .1 )

(0 .2 ,1 .0 ,0 .0 ) (0 .15 ,1 .0 , - 0 .1 ) ( 0 .15 ,1 .0 , - 0 .1 ) ( 0 .2 , - 1 .0 ,0 .0) ( - 0 .15 , - 1.0, - (0 .15 , - 1.0, - 0 .1)

0 .1) ( - ( I .2 ,1 .0 ,0 .0) ( - 0 .15 ,1 .0 , 0 .1) (0 .15 ,1 .0 , - 0 .1 ) (4 .25 , 1.0, - (4 .26 , - 1.0, - (3 .96 , - 1.0, -

2 .35) 2 .35) 2 .35)

(4 .25 ,1 .0 , - 2 .35) (4 .26 ,1 .0 , - 2 .35) (3 .96 ,1 .0 , - 2 .35) ( - 4 .25 , - 1.0, - ( - 4 .26 , - 1.0, - ( - 3 .96 , - 1.0, -

2 .35) 2 .35) 2 .35) I - 4 .25 ,1 .0 , ( - 4 .26 ,1 .0 , ( - 3 .96 ,1 .0 , -

2 .35) 2 .35) 2 .35)


I /D H-':-- COMPUTER ]

POR I I DATA I

AQUIS IT ION

COMPUTER I

D/A I

1. GYROSCOPE 2. WAVE PROBES

' ~ I 3. GYROSCOPE AMPLIFIER 4. RING TYPE LOAD CELLS

WAVE I CFA CARRIER FREQUENCY AMPLIFIER

f / / / / / /

Om

222 4%

i l / / / ~ " ~ / ' / / / / / / / / / / / J i / / , n ,~ / / / J

30 m 40 m 90 m

Fig. 4. Test setup.

of + 19.62 m 2 s - i with an accuracy of + 0.1%, and the rotation along the X and Y axes with an accuracy of 0.5% in the range of + 60 . Three resistance-type wave gauges were placed in a line on the sea side of the model to measure the wave time history. The signals from the three wave gauges were later processed to obtain the incident and the reflected waves from the composite wave field. Similarly, a wave gauge was mounted on the leeward side of model to trace the transmitted wave time series. The position of the wave gauges relative to the position of the model and the wave maker is also shown in Fig. 4.

The mooring forces exerted in the slack mooring lines were measured using ring-type load cells. The ring-type load cells were fabricated using stainless steel rings, diameter 30 mm with thickness 2 mm, and made waterproof. The load cells were connected at the top between the mooring line and the barge. The strain gauges were fixed in such a manner that a full Wheatstone bridge was achieved, such that the errors in measurement could be minimized. The load cells were calibrated and found to be linear up to 120 N, and the calibration constant was 0.012 N m-~V -~.

A dedicated personal computer, PC, was used for the generation of waves. The data collection was accomplished by a PC that was equipped with the necessary A/D cards and a special purpose programme.

6.5. Input parameters In the case of regular wave tests, the wave height was varied from 0.03 to 0.08 m. The

frequency range selected for experimental runs for the tests was 0.3-1.5 Hz in steps of

38 s.A. Sannasiraj et al.

0.1 Hz. For random waves, the tests were carried out by generating time series following the Bretschneider spectrum with peak periods, fp = 0.72 and 0.96 Hz.

6.6. Data processing

The response was measured for 30 s with a sampling time interval of 0.05 s for regular waves. The arrival time of wave at the model depends on the wave frequency; and, hence, each time series of waves and motion responses have been analysed separately. The incident wave and motion responses were calculated by averaging the crest to trough height of two steady response cycles just after the transient response.

The response amplitude operators in sway, heave and roll and mooring force are defined as follows:

Sway RAO -- sway amplitude/wave amplitude; Heave RAO = heave amplitude/wave amplitude; Roll RAO -- roll amplitude/wave amplitude; Mooring force RAO = force amplitude/wave amplitude.

The wave and response records were collected for 60 s for random wave tests with a sampling interval of 0.05 s. The effective time series processed was 52 s after removing the transient response to obtain the spectral characteristics. A typical measured time-series of wave elevation, motion responses in three modes and sea-side mooring force are shown in Fig. 5 for the test run with fp = 072 Hz and Hs = 0.045 m. The time series is subjected to the fast Fourier analysis to obtain the response spectra; and, hence, spectral characteristics such as significant value and peak period are evaluated.

7. RESULTS AND DISCUSSION

7.1. Response amplitude operator

The response amplitude operator, RAO, of the motion responses in regular waves in the three modes of motion viz. sway, heave and roll is reported as a function of normalized frequency, ~o2B/2g.

A comparison of the measured and theoretical prediction of RAO of sway response for the different configurations of the mooring line shown in Fig. 6 reveals a good agreement. The theoretical prediction of RAO (sway) for configurations C2 and C3 is found to be almost the same for the entire normalized frequency range, except at the roll resonant frequency. The RAO at lower frequencies is found to reduce drastically for frequencies less than resonant frequency beyond which the reduction is gradual. The RAO for configuration C1 is found to be significantly less for normalized frequencies lower than 0.2 owing to the greater resistance being offered by such moorings, thus restraining the dominant motions expected at lower frequencies. At frequencies beyond 0.2, the RAO is found to yield an identical response for all the configurations leading to the conclusion that the configuration C1 can be quite effective at lower frequencies. Beyond the roll resonant frequency, the type of mooring configurations does not have any effect on the RAO.

The experimental results and the theoretical predictions of the heave RAO shown in Fig. 7 indicate that the theoretical results are insensitive to the mooring line configuration. This is because the slack mooring line stiffness is negligible compared to hydrostatic stiffness of the floating structures. However, the measurements indicate that the heave


0.05

o .oo

-0.05

0.05

t ! ! t !

0.00

-0.05

0.05

I I I I

g

-0.05

l 0 .00

o.oo

-10 .~

1.5

I I I

! i t i t

z 0.0

-1.5 0 10 20 30 40 50

Time, t (s)

Fig. 5. Typical time series of measured time series of wave and responses.


E

ca

0 .<

0.0

- - - Theory -C 1

-C2

. . . . . . . . -C3

Expt . -C1

O O o O O -C2

z~ zx z~ A ZX -C3

~,5. zx r,

,'~ m 0

0 0 0

I I I

0.5 1.0 1.5 2.0

~2B/2g

Fig. 6. Variat ion of Sway RAO tor different moor ing conf igurat ions.

response is more for configuration CI near the heave natural frequency. The variation of the theoretical roll RAO along with the experimental results for the three configurations are plotted in Fig. 8. It is seen that the type of mooring configuration significantly affects the resonant magnitude of roll motion, while the shift in the resonant frequency is insig- nificant. There is a significant deviation in the theoretical roll RAO in comparison with the experimental predictions near the normalized frequency of 0.28, which is the roll natural frequency. The theoretical roll RAO at resonance is 32, 30 and 72 rad m ' for the configurations C1, C2 and C3, respectively, indicating a significant increase for configuration C3.

A comparison of the measured and theoretical RAO of the force on the seaside mooring line, FM for the three mooring configurations is shown in Fig. 9. For configuration C3, the experimental values are found to be closer to the theoretical results. For the other two configurations, deviations are observed at frequencies closer to the resonant frequency. The responses in the three modes of motion dictate the mooring forces near the correspond- ing natural frequency of that mode. There are three distinctive peaks observed in the RAO of FM at ~o2B/2g---*O, 0.28 and 0.82 which correspond to the natural frequency in sway, roll and heave modes. The measured mooring forces for configuration C3 were observed to be more than the predicted values, while for configuration C 1 they were less.

The variation of transmission coefficient with normalized frequency for the three moor-

Mooring forces and motion responses of pontoon-type floating breakwaters

3

41

0 <

2 -

Theory-C1

-C2

. . . . . . . . . C3

Expt . -C l

OOOOO -C2

A z~ A A A -C3

9o~" O o

O

0.0 0.5 1.0 1.5 2.0

~2B/2g

Fig. 7. Variation of Heave RAO for different mooring configurations.

ing configurations shown in Fig. 10 reveals that the Kt for all the configurations generally decreases with an increase in the frequency of the incident wave. The theoretical prediction of Kt exhibits the same value for all the mooring configurations. At low frequencies the type of mooring configuration does not influence the variation of Kt. For o~B/2g>0.2, the range of variation in measured Kt could be up to a maximum of 40%. The floating breakwaters are normally designed with an efficiency of about 50% in attenuating the incident wave energy. With this in mind, it was clearly seen that a single floating box-type of barrier considered in the present study was found effective when o~B/2g>0.9 (B/A>0.29, where h is the wave length). Here again, among the three configurations, C3 yields a higher Kt compared to the other two configurations. The configuration C2 is observed to be efficient in attenuating the incident wave energy and in addition the forces on the mooring lines for this is found to be less.

7.2. Response spectra The random wave test results for configuration C2 are only discussed here, since it is

found to be optimum based on regular wave test results. The random wave tests were carried out by generating time series of wave elevation following Bretschneider spectrum with two different peak frequencies of 0.72 and 0.96 Hz with a significant wave height of 0.045 m.


@ <

80

70

60

50

40

30

20

tO

I I I I I

00000

Theory-C1 -C2

-C3

Expt.-C 1

-C2

-C3

- - I - - I - - ~

0.0 0.5 1.0 i .5

(o2B/2g

Fig. 8. Variation of Roll RAO for different mooring configurations.

2.0

The trend in the variation of both the measured and the theoretical response spectra 0rp = 0.96 Hz, H~ = 0.045 m) in the sway and heave mode is found to be similar with incident wave spectra and reasonable comparison is obtained except with the deviations around the peak frequency as can be seen in Fig. 11. The deviation in roll spectra [Fig. 11 (c)] near the roll natural frequency is more compared to that of sway and heave.

The effect of the peak frequency on the measured response spectra in three modes along with the measured incident wave spectra is shown in Fig. 12. It is seen that the sway and heave response spectra follow the incident wave spectra, while, the roll response spectra exhibit a single peak at roll resonance frequency of 0.592 Hz irrespective of the peak frequency of the wave spectra. This is due to the low GM x adopted for the model. The significant values of the measured and predicted responses in the three modes of motion are tabulated in Table 3. The significant responses of the sway, heave and roll motions were observed to decrease with the increase in the peak frequency of wave spectra. The decrease in roll motion is found to be related to the increase in wave peak frequency. This is because large floating structures are sluggish compared to the high frequency wave components.


50

40

30

0 20

1o

0.0

(a)

Configuration CI

i i i

0.5 1.0 1.5

~2B/2g

2.0

80

60

z

40

0 <

2O

0.0

(b)

Configuration C2

0

o

o o

o

i i i

0.5 1.0 1.5 2.0

~2B/2g

125

100

75

u,

O 5O <

25

Configuration C3 f

I

A A

A t~

r I i

0.0 0.5 1.0 1.5 2.0

to2B/2g

Fig. 9. Variation of mooring force on seaward side mooring line.

7.3. Mooring force spectra The comparison of the measured and theoretical spectra of the force exerted on the

seaward side and the leeward side mooring lines of the model is shown in Fig. 13 (a) and (b), respectively. A significant deviation is observed at frequencies less than about 0.25 Hz and at frequencies near the peak frequency of incident wave spectra. Such large deviations have been reported by Adee (1977). A spike is seen for the theory at a frequency corre- sponding to the roll natural frequency, which is quite dominant and nearly twice the dominant peak for the leeward side mooring line. The agreement between the experiment and

44 S, A. Sannasiraj et al.

1.4

1.2

1.0

O

0.8 - '1

0.6

0 .4 -

0.2--

0.0

__ . .~

Theory -C 1

-C2

. . . . . . . . . C3

Zx "~. ,~ Expt . -C l

aa a ~ ooooo -C2

,, ~ ,', ~, za a ~ -C3

9

. . . . . . . . . . _< .~- __ _

A

o ~ o 0

l

I I I 1 ] 0.0 0.4 0.8 1.2 1.6 2.0

to2B/2g

Fig. 10. Variation of transmission coefficient for different mooring configurations.

theory for the leeward side mooring was observed to be good over the entire frequency range tested. The linear theory adopted to model the present problem does not consider the low frequency drift motion of the floating structure that dictates the dominant forces observed at the low frequency range.

8. CONCLUSIONS

Detailed theoretical and experimental investigations on the pontoon-type floating breakwaters have been studied. The hydrodynamic coefficients and wave exciting forces are evaluated using a two-dimensional finite element technique using an eight-noded isopara- metric element. The stiffness components for the slack mooring lines are linearized from the basic cable catenary equations. The pontoon-type floating breakwaters with three different mooring configurations, viz. (i) mooring at water level; (ii) mooring at base bottom; and (iii) cross moored at base bottom level, were subjected to the action of both regular and random waves.

The salient conclusions drawn from the present study are detailed below.

1. The linear theoretical model predicts the motion responses in the three modes of motion viz. sway, heave and roll satisfactorily. The comparison between the theoretical and experimental measurements shows good agreement except at the roll resonance frequency.


0.0002

r .~

O.O00l I

0.0000 0.0

(a ) , , , , , Expt. - - Theory

f = 0.96 Hz P

o ~ o o

o

o o

0.5 1.0 1.5

f (Hz)

o

e41

0.0006

0.0004

0.0002 r ,q

0.0000 2.0 0.0

oo*o*Expt . - - Theory

o**OO o

o oO o i i i

0.5 1.0 1.5

f (Hz)

O

~0

20

15

1o

5

o 0

0.0

. . . . . Expt. (c ) - - Theory

c

0.5

o

1.0 1.5 2.0

f (Hz)

Fig. 11. Comparison of measured and predicted response spectra in a random wave field.

2. The cross moored FBW exhibits a high resonant roll response near the roll natural frequency than the other two configurations.

3. The roll response spectrum shows single dominant peak at the roll resonance frequency irrespective of the peak frequency of the wave spectra. As the wave peak frequency increases, the roll response significantly reduces.

4. The transmission coefficient is not significantly affected by the mooring configurations studied. However, the tests showed a higher transmission coefficients for the FBW with cross moorings.

5. The mooring forces are significantly affected by the mooring line configuration. The moorings at water level and at the bottom of FBW yield significantly lesser mooring forces than that with crossed moorings. This is a factor which significantly alters the design concepts.

(b )

2.0

0.0003

0.0002

g 0.0001

0.0000 r 0.0

J

0.0006

0.0004

I

0.0002

f (Hz)

(a)

__ fp = 0.72 Hz ........ fp = 0.96 Hz

T" i 1

0.5 1.0 1.5 2.0 0.0000

0.0

%" H I

0.0006

0.0004

0.0002

0.5 1.0 1.5


f (Hz)

(c)

(b)

/'/'i

i

0.5 1.0 1.5

i ol

60

40

20 o~ ,O

0 ~"

0.0

/ - - ,

0.0ooo ~ " / "~ ' 0.0 2.0 0.5 1.0 1.5

f (Hz) f (Hz)

Fig. 12. Variation of response spectra with different wave spectra.

2.0

(d)

2.0

Table 3. Comparison of measured and predicted significant responses for the mooring configuration C2

Run no. Sway (m) Heave (m) Roll (deg) Measured Theoretical Measured Theoretical Measured Theoretical

1 O~p = 0.72 Hz, 0.041 0.035 0.047 0.046 11.24 17.5 H, = 0.045 m) 2 (fp = 0.96 Hz, 0.028 0.026 0.045 0.044 3.85 4.25 H~ = 0.045 m)


I

0.50

0.40

0.30

0.20

0.10

. . . . . Expt. (a)

- - Theory

0.00 0.0 2.0

:o o

o o

o

o o

o ~ o

% :

- ! [ I

0.5 l.O 1.5

f (Hz)

13

I

0.50

0.40

0.30 -

0.20 -

0.10 -

. . . . . Expt. (b) - - Theory

0.00 0.0 0.5 2.0

i i

1.0 1.5

f (Hz)

Fig. 13. Comparison of measured and predicted mooring force spectra in a random wave field.


6. Mooring forces are dominant at lower frequencies. 7, Based on this investigation on responses, transmission characteristics and mooring

forces, crossed mooring for FBW is not efficient. REFERENCES

Adee, B. H. (1975) Analysis of floating breakwater performance. In Proceedings eft' Symposium on Modeling Techniques, ASCE, pp. 1585-1602.

Adee, B. H. (1977) Analysis of floating breakwater mooring torces. Ocean Engineering Mechanics, Winter Annual Meeting, ASME, Houston, Texas, pp. 77-92.

Adee, B. H. and Martin, W. (1974) Theoretical analysis of floating Breakwater performance. 1974 Floating Breakwaters Conference Papers, University of Rhode Island, Marine Technical Report Series no. 24, pp. 21-39.

Bai, K. J. (1977) A localized finite element method tot steady, three-dimensional free surface flow problems. In Proceedings on Second International Conference on Numerical Ship Hydrodynamics, ed. J. V. Wehausen and N. Salvesen. University of California, Berkeley.

Carver, R. D. (1979) Floating breakwater wave-attenuation tests .]or East bay marina Olympia harbour, Wash- ington. Technical report HL-79-13, US Army Engineer Waterways Experiment Station, Vicksburg, Miss.

Hales, L. Z. (1981) Floating Breakwaters--State of the art. CERC, Technical report 81-1. Isaacson, M. and Byres, R. (1988) Floating breakwater response to wave action. In Proceedings of the 21st

Coastal Engineering Conference, Vol. 3, pp. 2189-2200. Isaacson, M. and Wu, S. R. (1995) A numerical study of moored vessel response to beam waves. In Proceedings

of the Fifth International Offshore and Polar Engineering Conference, The Netherlands, Vol. 3, pp. 499-506. Jain, R. K. (1980) A simple method of calculating the equivalent stiffnesses in mooring cables. Applied Ocean

Research 2, 139-142. Johansson, M. (1989) Barrier-type breakwaters--Transmission, Reflection and Forces. Ph.D. dissertation, School

of Civil Engineering, Chalmers University of Technology, Sweden. Miller, R. W. and Christensen, D. R. (1984) Rigid body motion of a floating breakwater. In Proceedings of

Coastal Engineering Conference, Chapter 179, pp. 2663-2679. Ogawa, Y. (1984) Fundamental analysis of deep sea mooring line in static equilibrium. Applied Ocean Research

6, 140-147. Oliver, J.G. et al. (1994) Floating Breakwaters~A practical guide.fi)r design and construction. Report no. 13,

Permanent International Association of Navigation Congresses, Belgium. Sannasiraj, S. A., Sundaravadivelu, R. and Sundar, V. (1994) Dynamics of floating bodies subjected to regular

waves. In Indian National Conjerence on Harbour and Ocean Engineering, Pune, India, Vol. 2, pp. I 11 -I20. Sutko, A. A. and Haden, E. L. (1974) The effect of surge, heave and pitch on the performance of a floating

breakwater, In 1974 Fhmting Breakwaters Conference Papers, University of Rhode Island, Marine Technical Report Series no. 24, pp. 21-39.

Veldee, M. A. (1983) Floating breakwater wave attenuation and mooring force performance. M.Sc. thesis, Uni- versity of Washington.

Yamamoto, T., Yoshida, A. and Ijima, T. (1980) Dynamics of elastically moored floating objects. Applied Ocean Research 2, 85-92.

Yamamoto, T. (1981) Moored floating breakwater response to regular and irregular waves. Applied Ocean Research 3, 114-123.

Zienkiewicz, O. C. and Taylor, R. L. (1989) The Finite Element Method, Vol. 1. McGraw Hill, London.

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