Theoretical and experimental predictions of the hydroelastic response of a very large floating structure in waves

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Applied Oceofi

Research

E L S E V I E R Applied Ocean Research 20 (1998) 135-144

Theoretical and experimental predictions of the hydroelastic response of a

very large floating structure in waves

Hiroshi Kagemoto*, Masataka Fujino, Motohiko Mural

Department of Environmental and Ocean Engineering, Graduate School of Engineering, Tlie University' of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan

Received 7 January 1998; accepted 11 February 1998

Abstract

A prediction method for the hydroelastic behavior of a very large box-shaped flexible stmcture in regular waves is proposed. The structure considered is representative of such structures as a floating international airport and thus the horizontal dimensions are expected to be as large as several kilometers in both length and width. In the analysis, the structure is divided into a number of substructures and the continuous deformation is approximated by the succession of a discrete displacement of each substructure. The displacement of each component is determined from the equation of motion of a uniform free-free plate representing the substmcture, while taking structural constraints into account as an additional restoring force. The hydrodynamic forces on the substructure are determined by enforcing the normal velocity of the flow to be equal to that of the corresponding body surface. Thus, the fluid motion and the body motion interact with each other, which is termed 'hydroelastic interaction', requiring the simultaneous solution of the structure and fluid problems. © 1998 Elsevier Science Ltd. A l l rights reserved.

Keywords: Very large floating structure; Hydroelastic interaction; Hydrodynamic interaction

1. Introduction

A very large floating structure (VLFS) which is several kilometers in length and width is now being considered in some countries, including Japan, as an alternative to such large, land-based facilities as airports. The reason for this is that, in these countries, the land space that can be used for human activities is quite limited. Even in a country of suffi-cient land area, a floating airport may be preferable over land-based airports near big cities, because various environ-mental issues such as jet noise can be relaxed. A floating internafional airport should be at least 5000 m long and 1000 m wide in order to accommodate commercial aircraft, whereas the total thickness of the stracture is expected to be approximately 10 m. The bending stiffness of the platform w i l l thus be much smaller than that of existing floating structures such as a ship, resulting in much lower natural frequencies of bending oscillations, which may be within the region of dominant wave frequencies. Hence, it is con-sidered that large elastic deformations may be induced by wave excitation. For the analysis of elastic responses of conventional floating stractures such as a large tanker, the

* Corresponding author. Tel.: +81 3 3812 2111; fax: +81 3 3815 8360; e-mail: kagemoto@yuiko.nasi.t.u-tokyo.ac.jp

hydrodynamic forces are first predicted while neglecting the elastic deformations, because they are usually far smaller than the displacements due to rigid-body motions. The elas-tic deformations are then analyzed by taking the hydrody-namic forces and the inertia forces due to rigid-body motions as external forces. However, i f elastic deformations are comparable to, or even larger than, the displacements due to rigid-body motions, which may be the case for the VLFS considered here, they have to be accounted for in the prediction of hydrodynamic forces, although the deforma-tions can be determined only after the hydrodynamic forces are known. The interaction between hydrodynamic forces and elastic deformations is a phenomenon encountered in the analysis of such VLFSs.

One method that is usually used for the hydroelastic ana-lysis of a floating body is a so-called mode anaana-lysis [1], in which the deformation of a corresponding stiucture is expressed as a superposition of a certain set of mode func-tions. Differential equations for each mode are developed while using the hydrodynamic forces as generalized added mass and damping coefficients. However, since, in obtain-ing the hydrodynamic forces, radiation problems due to each mode of motion of unit amplitude should be calculated, the computational burden increases rapidly as the corre-sponding stracture becomes large, requiring more modes

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136 H. Kagemoto et al./Applied Ocean Research 20 (1998) 135-144

Substructure

Fig. 1. Discretization of a stracture into substructures and the approximation of a continuous deforiuation with a succession of discrete displacements of each substructure.

to be used in the dynamic response analysis. In order to overcome this difficulty, we propose a new method in which a structure is divided into small substructures, as shown in Fig. 1. Each substructure is treated as i f it were an independent rigid floating body, and the continuous deformation is approximated by a succession of discrete vertical displacements of each substructure. The effect of the structural constraints associated with the relative displacement between the adjacent substructure is taken into account as additional restoring forces to the rigid-body motions of each substructure. A similar idea has already been proposed by Kagemoto and Yue [2] for the hydroelastic analysis of a VLFS supported on a large number of legs. In the present study, we extend their idea to a structure of box-shaped geometry. The hydrodynamic forces on the coiTesponding substructure that appear in the equations of motion are determined by enforcing the normal velocity of the flow to be equal to that of the corresponding body surface, which is a function of the motion of the cor-responding substructure. Thus, the motions of the substmc-ture and the hydrodynamic forces are functions of each other and are hence coupled. In the present analysis, this hydroelastic interaction is accounted for in one computation by solving the equations of motion of the substructures and the equations for the hydrodynamic analysis simulta-neously. As will be shown later in this paper, although the present approach gives almost identical results with those given by conventional modal analysis, the present method seems more promising, as the corresponding structure becomes large because a further simplification can be incor-porated by exploiting the fact that the structure is very large.

2. Theory

As stated previously, we are dealing with hydroelastic behavior in waves of a box-shaped VLFS and we assume that the usual small-amplitude linear potenüal theory can be applied. As shown in Fig. 1, a space-fixed coordinate system (x,y,z) is used, in which z is vertically upwards and x-y lies on the undisturbed free surface. The structure is divided into A''s substructures and its continuous deformation is approxi-mated by a succession of discrete displacements of each

substmcture. Each substructure is assumed to act as i f it were an independent rigid body, while taking the stmctural constraints into account as additional restoring forces to the rigid-body motions. Since the slope of each substructure is expected to be very smaU as the corresponding stmcture becomes large, it is neglected in the present study. The equation of the vertical motion of each substructure is then written as follows:

M,- •.F. + F^j + Fkj ( / • = ! , 2, (1) where M,- and Zj represent respeétively the mass and the vertical displacement of the jth substructure, and F^j is the hydrodynamic force, F^j the hydrostatic force and f y the restoring force due to structural restraints.

Here, the steady responses in a regular wave train of radial frequency co are considered, and everything is assumed to be oscillating with a frequency co, e.g. Zj = 9?{Z/x,y)e"'"'}, where For this reason, the time-dependent term e~'"' will be omitted hereafter unless otherwise stated.

The hydrostatic force F^j is expressed as:

(2) where p and g represent the density of water and the accel-eration due to gravity respectively, and A„,y is the cross-sectional area of the j t h substmcture at an undisturbed free surface. For an isotropic plate element, Fj^j is approximately expressed as follows:

-D d'Z d'Z _{' „ 7} (3)

at {Xj,yj)

where D [ = Eh^/12(l - p,^)] is the bending rigidity of the stmcture approximated as a plate of thickness /), p. is Poisson's ratio and E Young's modulus, and Z{x,y) repre-sents the continuous vertical displacement of the structure. The derivatives are evaluated at the center of the jth sub-structure {xj,yj). Since it may be difficult to evaluate numeri-cally the fourth derivatives in Eq. (3), we express Z by the superposition of a set of certain functions:

(4) h i = O h = 0

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H. Kagemoto et al./Applied Ocean Research 20 (1998) 135-144 137 For w„„ and w,y„ the modal functions of free oscillations of a

free-free beam can be used: / s \nX . X,„A' >Vv W = cos - y - + cosh cos X„, - cosh X„ sin X„, - sinh X„ sm \ i X • sinh X,„A- (5) Ky

w„y(}>) = cos + cosh - cos X„ — cosh X„

sin X,, - sinh X„ X ( s i n ^ + s i n h ^

^3'

where X,„ and X„ satisfy

cosh X,,,-cos X,„ = 1 for m > 2 ₍₆₎

cosh X„-cos X„ = 1 for n > 2

h i order to account for rigid-body motions (heave roll and pitch), Wa, — 1, Wu = x and WQy= 1, Wxy=y should be included. Then, for example, d^Zldx^ can be evaluated ana-lytically as:

T u J "

X X

"mil " • ^ H i = 0 ; i = 0

d w,„,{x)

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in Eq. (1) is written in terms of the velocity potential 0y, which expresses the flow field around the;'th substructure as follows:

!7 . — _ _{io:p(f>jn^ dS}

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where is the z component of the outward normal unit vector at the corresponding surface and the integral is car-ried out over the surface of the j t h substructure under the undismrbed free surface.

The velocity potentials 4>j{j ^\,2, N,) can be calcu-lated by the hydrodynamic interaction theory developed by Kagemoto and Yue [3] as follows.

Since the wave which is incident on the jth substructure is the summation ofthe original regular broadcrested incident wave, the reflected wave due to the other stationary sub-structures and the radiated wave due to the motions of the other substructures, the total velocity potential (/>] that repre-sents the wave incident on the 7th substructure is written as follows: a7 + A's

I

A7T:, --1 (9)

where the superscript T represents a transpose and bold face (e.g. a^) indicates that the corresponding variable is a vector or a matrix, aj^j in Eq. (9) represents the original regular incident wave on the j t h substructure expressed as a

i - t h

substructure

i-th substructure

' X

Fig. 2. Coordinate systems.

superposition of component waves:

a^^î = _ 'gâcoshô(z + /i)^,.„.„, ^^^f^^y

ÜJ cosh kgh (10)

_ cosh ko [zj + h)

cosh k^h

I

Here, {x,y,z) is the space-fixed coordinate system and

{ f j f i p Z j ) is the cylindrical coordinate system fixed to the

;'th substmcture (see Fig. 2). / „ is the nth order Bessel func-tion of the first kind, and are the amphtude and the incident angle of the incident wave, and k^ and h are the wavenumber and the water depth, ej is defined as follows:

e,- = kQXj cos 6-^+ ko Yj sin (11) where, as shown in Fig. 2, ( X j j j ) is the x,y coordinate of the

center of the ;th substmcture in the (x,y,z) coordinate sys-tem. The summation in Eq. (10) is trancated at a certain number and is a vector consisting of

ig^,CO&hko(Zj+h),„^„fg^_

w cosh kgh

and is a vector consisting of Jnikorj) [or, i f we account for evanescent waves together with progressing waves, \pj con-sists of In{k,„rj) {nth order modified Bessel function of the first kind), as well as J„(/:o'7), where k,„ represents the wavenum-ber of the evanescent waves]. S f l 1 (,.^^.)A?'T,y;/'j in Eq. (9) represents the waves at the j'th substiaicture due to the reflec-tions by the other substi'ucmres, where A; is the amphtude of each component wave. Ty is a matrix which converts the flow field expressed in terms of a coordinate system fixed to the ith substmcture to that expressed in terms of a coordinate system fixed to the Jth substmcture. T,;^- is known in advance from the geometric positions of the two substmctures / and j . The third term in Eq. (9), E f i j (,.^.jZ,R/'T^-t/'], represents waves radiated due to the motion of the ;th substructure, in which R^T/yi/'j expresses the radiated waves due to unit amphmde motion of the ith substmcture. Since R,- represents the radia-tion characteristics of the ;'th substmcture, it can be deter-mined in advance by solving the radiation problem of the

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1 3 8 H. Kagemoto et al./Applied Ocean Research 20 (1998) 135-144

ith substructure in the absence of other substmctures. The

total velocity potential for the flow field at the jth substructure can be written as the sum of the total incident wave [Eq. (9)], the reflected wave and the radiated wave:

C /

N, \ A's

I V i=Hi*j) / i=l(i*J)

T . ; . s

+ ZjR}xl. (12)

where 4^] is a vector composed of H„(kofj) (nth order Hankel function of the first kind), K„{k,„rj) (nth order modified Bessel function of the second Idnd), and is related to \pj as i/'- = Tij^j-in Eq. (12) is a matrix that relates the wave incident at the j t h substmcture and the scattered flow field around the j t h substmcture, which, like R^, can be deter-mined in advance by solving a diffraction problem of the Jth substructure in the absence of other substructures.

Substituting 4>j into Eq. (8), is expressed in terms of

A,-(,1=1,2, ...,N,).

Finally, the equations of vertical motion of each substmc-ture [Eq. (1)] are rewritten as follows:

+ X Z,RJTA^] +

i=l{i*j) I

+ X Z^RjTABjrj+ZjRjfj n,dS-pgA,,j-Zj

D Ö^Z (13)

at (xj,yj)

Since A^- represents the reflected wave of the waves incident to the j t h substmcture, the following equation should be

satisfied:

A's \ A's

X T j A , + X Z,-TjR,. (14)

From Eqs. (13) and (14), the motions of ah the substruc-tures, Z y ( / = 1,2, ...,N,), as weh as A ^ ( j = 1,2, . . . . A ^ s ) , are then determined simultaneously. Hydrodynamic forces are determined by substituting Aj in Eqs. (8) and (12). (Although there is some ambiguity i n applying the interac-tion theory at the side faces of adjacent substmctures between which there exists no water, it has been shown numerically that the interaction theory works well even for such a case [4].)

3. Experiment

3.1. The model anci. the experimental set-up

In order to obtain experimental data for comparison with the results of the numerical calculations based on the theory described so far, experiments were caiTied out in a wave tank with an acryhc flexible model, as shown in Fig. 3. Table 1 shows the principal details of the model, which consists of 100 (20 X 5) buoyant rectangular solids. Each of the solids is attached to the upper flexible deck. There is a nanow gap (2 mm) between the adjacent solids so that the forces acting on each solid are not transfened to the adjacent solids, but only to the upper plate. Because of the gap, the solids do not restrict the deformation of the plate and thus do not contribute to the bending rigidity of the total stmcture. (Since the gap is so narrow, it has little effect on the hydro-dynamic characteristics of the total stiucture, which is sup-posed to be continuous.) The upper plate is a thin plate (thickness: 5 mm) of uniform rigidity. Since the stracture we have in mind has very large horizontal dimensions (typically 5000 m in length) while the thickness is around 10 m, i f we constract an experimental model 2 m long, the thiclmess should be as small as 4 mm in order to satisfy the

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H. Kagemoto et al. /Applied Ocean Research 20 (1998) 135-144 139

Table 1

The principal details of the model used f o r the experiment (L and B refer to the cross-section at the wateriine)

position-4 positiorr-3 PQ5ition-2 position-1

Length (L) 2.000 m

Width (B) 0.500 m

Thickness of upper deck 0.005 m

Draft 0.050 m

E (Young's modulus) 2.9 X l O ' N / m ^

geometrical similarity law. It can be easily imagined that it is quite difficult to use such a thin model in waves because, for example, waves would tend to wash over the model easily. The basic idea of the model is that the upper plate takes care of the elastic deformations, while the rectangular solids under the deck take care of the hydrodynamic and hydrostatic (buoyancy) forces and transfer these fluid forces to the deck. In this way, since the plate is located at a sufficient distance from the water surface, such difficulties as deck wetness that may be encountered in using a simple thin plate model are avoided. The model was moored by a weak spring in order to prevent drifting in waves, as shown in Fig. 4. Experiments were conducted in a sinusoidal monochromatic wave train with periods of 0.5-2.0 s. The wave amplitude was maintained around 10 mm. The quan-tities measured in the experiments are local displacements and local bending strains of the upper deck. The local dis-placements were measured by an optical system, in which the instantaneous positions of light-emitting diodes (LEDs) are detected by a CCD camera. From this, the time histories of the local vertical, horizontal and rotational displacements are identified. Four sites were selected for the measure-ments, as indicated in Fig. 5, which also shows that the bending strains were measured by strain gauges at five locations.

3.2. Discussion on similarity law

From the viewpoint of the similarity between the model and a projected real structure (e.g. 5000 m long, 1000 m wide, 10 m thick and 1 m draft), the following problems exist in the present experiment.

1. In order to satisfy the geometrical similarity of the wetted portion of the structure, which is necessary for the similarity of hydrodynamic forces, since the scale ratio of the horizontal dimension is —1:2500, the draft

40cffl ^ -lOcm 40cm 40CÏÏI (a)posilinu of L E D s

position-4 positioii-3 position-2 position-t

' 2 0 ™ 40crii 4 0 ™ 40CII1 40CII1

{b)positiuiiof strain gauges

^^^^

Fig. 5. The locations of LEDs and strain gauges, (a) Positions of LEDs. (b)

should be 0.0004 m. However, the actual draft of the model is 0.05 m.

2. The wave period used in the experiments (0.5-2.0 s) corresponds to 25-100 s for the projected real stiucture. 3. The amplitude of the waves used in the experiments

(10 mm) coiTcsponds to 25 m for the real structure. 4. The similarity in elasticity is satisfied i f EIIL^ is the same

provided the geometrical similarity is satisfied, where L = length of the structure and £•ƒ is the bending rigidity of the flexible deck. EIIÜ of the projected real stracture is expected to be around 10"^ Wnr', while that of the pre-sent model is about 5 X 10"^ N/m"'.

Since the period of the waves that are observed i n near-shore areas, where the stractures that are dealt with in the present study are to be located, is 3-10 s, the waves used i n the experiments are too long. The amplitude of the waves is also too large. In order to overcome these problems, we either have to malce the model much larger or produce much shorter wave and much smaller wave-heights. How-ever, the first option is difficult to achieve, because a model of 2 m length is already quite large, and the second option is also difficult because of the characteristics of the wave-maker. The bending rigidity of the model is also too large from the viewpoint of the similarity in elasticity. Although this problem w i l l also be averted i f we could make a model much larger or i f we could use a much thinner deck, it is quite difficult in practice. Overall, although the present model enables us to use a more flexible model than a simple box shape, it is still quite difficult to stricfly satisfy the geometrical similarity of a model and waves, as well as

incident wave

508mn

Fig. 4. The experimental set-up.

Table 2

The major details of the structure used in the analysis (Figs 6 and 7)

Length (L) Width (B) Draft

Thickness of the structure

E (Young's modulus) Water depth 10.00 m 0.500 m 0.008 m 0.038 m 9.8 X l O ' N / m ^ 1.100m

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140 H. Kagemoto et al./Applied Ocean Research 20 (1998) 135-144

the similarity in elasticity. These problems are the subject of a future study.

4. Results and discussion

4.1. Effect ofthe number of substructiu-es and the number of modes

In order to examine the effect of the number of sub-divisions and the number of modes, parametric calculations were caiTied out based on the method described in Section 2. The major details of the structure used in this parametric study are shown in Table 2. Fig. 6(a) and (b) show the longitudinal distributions of the vertical displacement amplitudes in head waves (L/X = 20, where X = wave-length) along the side-edge of the structure and along the

0 . 1 ' ! ' ! ' ! 1 I 1 1 1 1 1 1 1 1 1 1 1 \ — r 6 0 x 1 5 ; i ; — 4 8 x 1 2 - i 1 f-\ f-\ i" s i x 8 , i V r i' . i - 0 . 5 - 0 . 4 - 0 . 3 - 0 . 2 - 0 . 1 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 x / L

(a)distribution along the side-edge

centerline of the structure respectively. In the figures, results obtained by three different methods of subdivision are com-pared. In the legends, '60 X 15' indicates that the structure is divided into 60 substmctures in the longitudinal direction and 15 substructures in the transverse direction. The number of modes M + I and A' -|- 1 used for the expression of the displacement [Eq. (4)] is the same as the number of sub-structures used in the longitudinal direction and in the trans-verse direction respectively. (For example, the '60 X 15' computation was conducted with M = 59 and N = 14.) As expected, the result converges as the number of substmc-tures increases. It is our experience that at least two sub-stmctures are needed for one wavelength, which means that at least 40 substmctures are required in the longitudinal direction for the calculations shown in Fig. 6(a) and (b). In fact, the computation conducted with 48 substmctures

0 . 1 I ' ! ' I 1 1 I 1 1 1 1 1 1 I 1 I 1 ; — r 4 8 x 1 2 i i ; — 2 1 x 7 - ; 1 f-i \ i" l i x 3 , 1 . 1 7 i ' > - < i ^ ' ^ > * ' T ™ T ' ^ - 0 . 5 - 0 . 4 - 0 . 3 - 0 . 2 - D . 1 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 x / L

(a)distribution along the side-edge

Fig. 6. The longitudinal distributions of the vertical displacement ampli-tudes in head waves (L/X = 20). Effect of the number of substructures, (a) Distribution along the side-edge, (b) Distribution along the centeriine.

Fig. 7. The longitudinal distributions of the vertical displacement ampli-tudes in head waves (Ll\ = 20). Effect of the number of modes, (a) Dis-tribution along the side-edge, (b) DisDis-tribution along the centerline.

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H. Kagemoto et al. /Applied Ocean Researcli 20 (1998) 135-144 141

in the longitudinal direction converged sufficiently, while that obtained with 32 substmctures has not yet converged. As for the terms M and A'^ needed in Eq. (4), we can afford to cairy out the analysis with a relatively small number of terms, because the relevant elastic modes that appear i n the responses are rather limited, even for the VLFS consid-ered in this study. Fig. 7(a) and (b) show the longitudinal distributions of the vertical displacement amplitudes in head waves ( L / \ = 20) along the side-edge of the structure and along the centerline of the stmcture respectively, which were obtained by discredzing the structure into 48 X 12 substructures, while varying the number of modes M + 1 and N + \ used in the expression of the vertical displace-ment [Eq. (4)]. In the figures, '21 X 7' indicates that the number of modes used in Eq. (4) is M - M = 21 and A' - M = 7. It can be observed that, with only M -|- 1 = 11 and V -f-1 = 3, the results have already converged.

4.2. Comparisons with conventional methods

For hydroelastic analyses of a ship-like floating stmcture, a so-called mode analysis is often used [1]. In this method, the hydrodynamic forces due to each mode of oscillations of unit amplitude are calculated in advance of the analysis. After obtaining the forces, simultaneous equations for each mode of motion are solved by using the forces obtained as the generalized added mass and damping coefficients. The present method should give identical results with those obtained by this conventional method. In order to confirm this, calculations were canied out for an example floating stiucture, the main details of which are shown i n Table 3. Fig. 8 compares the longitudinal distributions of the vertical displacement amplitudes along the centerline of the stracture obtained using the present method and those obtained using conventional mode analysis in head waves; the agreement between the results obtained using the two different methods is close.

4.3. Comparison with experimental results

After the convergence test that was described in Section 4.1 and the comparisons with results obtained by a mode analysis described in Section 4.2, the present method is compared with the experiment explained in Section 3. Fig. 9 and Fig. 10 compare the vertical displacements and the bending strains in head waves respectively. [The phase shown in Fig. 9(b) is the phase difference from that of position 1.] The numbers in the figures indicate that the vertical displacements and the bending strains were

Table 3

The main details of the structure used in the analysis (Fig. 8)

ZVX BIX dIL EIlpgL' 20.0 5.0 3.5 X 10" 3.4 X 10" 1.5 1.4 1.3 1,2 1.1 1 0.9 0.0 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 h 0.3 0.2 0.1 0 mode analysis present m e t h o d ( a ) i / A : .4 .5 x/L 1.5 1.4 1.3 • t > . I 1 mode analysis — . present m e t h o d • 1.2 • 1.1 1 0.9 -0.8 0.7 0.6 0.5 j. 0.4 j . 0.3 0.2 — ^ ^ ^ ^ ^ 0.1 -0 1 1 1 I 1 1 1 1 -.4 ,3 ,2 -.1 0 .1 .2 ( b ) i / A = 3.21 A .5 xlL mode analysis present m e t h o d {c)L/\ = 1.16 .4 .5 x/L Fig. 8. Comparisons of the vertical displacement amplitudes obtained by the present method and those obtained using a mode analysis. Solid line: obtained using a mode analysis; dotted line: obtained using the present method.

measured by the LEDs and the strain gauges of the corre-sponding number shown in Fig. 5. The results obtained by the present method agree well with the experimental data. Although a ratio X/L (wavelength/stracture length) - 1 . 3 cor-responds to the natural frequency of the heave motion as a rigid body, a high peak value, which is typical for a rigid

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142 H. Kagemoto et al./Applied Ocean Research 20 (1998) 135-144 2.0 1.5 1.0 0.5 1.0 2.0 3.0 X/l (a) a m p l i t u d e Cal Exp O ® © ® O ® e _e

1 ₈

" p i MO

9il\

\ =

4 / f

\ \ \ \

^S/o

1 : i j 4.0 5.0 1.0 2.0 3.0 4.0 5.0 X/l ( b ) p h a s e

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H. Kagemoto et al./Applied Ocean Research 20 (1998) 135-144 143 1 / m 0 . 0 3 0 . 0 2 0. 01 ' ! ' ! •! 0 ® ; — 6 © . A d ' - i . - - - - o ® 1 a ® ! ® CÉ.^,:è..o\ j ' ^ ' v H , . 0 1.0 2 . 0 6 Lïi _{3 . 0} X / l 4 . 0 5 . 0

Fig. 10. The frequency respon.se characteristics of the amplitudes of local bending strains in head waves.

body, is not observed due to the elastic nature of the struc-ture. Fig. 11 shows the phase difference between the local vertical displacements and the ambient incident wave. For a long wavelength, X/L > 1.0, the amplitudes of the local vertical displacements at all of the four sites are almost the same as that of the incident wave, as shown in Fig. 9(a), while the phase of the local displacement is the same as that of the ambient incident wave, as shown in Fig. 11. From these facts, we know that at X/L > 1.0 the structure is deforming in the same shape as that of the incident wave as the wave progresses along the structure. On the other hand, in shorter incident waves, the phase of the local vertical displacement differs from that of the ambient incident wave, as can be observed in Fig. 11, which manifests the effects of the appearance of natural modes of elastic responses. .71 3 . 1 4 2 1.67 - 1 . 5 7 — ' ! ' ! ' ! ' ! © ; l i e ®

I I i ®

o i i i I Ö 1.0 2 . 0 3 . 0 X/L 4 . 0 5 . 0

Fig. 11. The phase difference between the local vertical displacements and the ambient incident wave.

S. Further approximations f o r the analysis of a very large floating structure

I f a structure is very large compared to the ambient wave-length, it may be justified to assume that the flow field around the structure (except around the perimeters of the stiucture) is almost the same as that of a stiucture of infinite length/width. This idea has aheady been used by Kagemoto and Yue [2] and Kagemoto et al. [5] for diffraction analyses and radiation analyses of a VLFS. This may be extended to the analyses of the elastic responses of a VLFS. For exam-ple, i f we consider elastic responses of a very long floadng structure in beam waves, we may be able to assume that the longitudinal distribution of the deformation is uniform in amplitude as well as in phase, except at both ends of the stmcture. I f this is the case, we need not compute the dis-placements of all the substructures, but only need to com-pute the displacement of one of the substmctures. In this

0.12 0.10 i j , 0.06 \ 0. 04 I 1 1 1 1 1 1 1 1 1 1 I ; ; ° W i l l i a p p r o x i m a t i o n • • • - - ^ ^ ^ ^ ^ A ^ . . ^ p - ^ • • . i . i , i . i . - 0 . 5 - 0 . 4 - 0 . 3 - 0 . 2 - 0 . 1 0 0 . 1 0 . 2 • 0.3 0 . 4 0.5 x / L ( a ) a l ö n g t h e s i d e - e d g e 0.10 0.06 \ 0.02 0 T — ' — I — ' — I — ' — I — ' — I— I —I — ' — r 1 i 0 l ï i t l i a p p r o x i m a t i o n I : . Witiiout.iapproxiraat i o n . - 0 . 5 - 0 . 4 - 0 . 3 - 0 . 2 - 0 . 1 0 0 . 1 0 . 2 0.3 0 . 4 0.5 x / L

(b)along the centerline

Fig. 12. The longitudinal distributions of the vertical displacement of a very long structure in beam waves (L/B = 25, L/X = 25). (a) Along the side-edge, (b) Along the centerline.

(10)

144

H. Kagemoto et al. /Applied Ocean Research 20 (1998) 135-144

Table 4

The main details o f the stmcture used in the analysis (Fig. 12)

6. Conclusions

A new method for the analysis of hydroelastic responses of a box-shaped floating structure has been proposed. The validity of the method was confirmed through comparisons with a conventional mode analysis, as well as with experi-mental data. Although, as shown in this paper, the present method gives almost identical results with those obtained by a conventional mode analysis, the new method should be more advantageous as the structure becomes larger, because a further approximation can be incorporated by exploiting the fact that the structure is very large.

References

[11 Bishop RED, Price W G . Hydroelasticity of ships. Cambridge: Cam-bridge University Press, 1979.

[2] Kagemoto H , Yue DKP. Hydroelastic analyses of a structure sup-ported on a large number of floating legs. In: Faltinsen O et al., editors. Hydroelasticity in marine technology. Trondheim: Balkema 1994¬ 417-431.

[3] Kagemoto H , Yue DKP. Hydrodynamic interaction among multiple three dimensional floadng bodies; an exact algebraic method. J Fluid Mech. 1986;166:189-209.

[4] Yoshida K, Suzuki H , Oka N , l i j i m a K , Shimura T, Arima T. Hydro-dynamic interaction effects on wave exciting force in large scale floating structures. J Soc Naval Arch Japan 1993;174:243-251. [5] Kagemoto H , Fujino M , Zhu T. On the estimation method of

hydro-dynamic forces acting on a very large floating structure. Applied Ocean Research 1997;19:49~60.

U\ 25.0

Bl\ 1.0

dIL 2.0 X 10"^

EIlpgL^ 1.1 X 10^'

way, even i f the number of the substructures is large, the number of unknowns is reduced to one and thus the compu-tational burden will be significantly reduced. Fig. 12(a) and (b) compare the longitudinal distributions of the vertical displacements of a very long structure {LIB = 25) in beam waves (L/X = 25) computed with and without the approxi-mation described in fliis section. The main details of the structure used in this analysis are shown in Table 4. Fig. 12(a) shows the distribution along the side-edge and Fig. 12(b) the distribution along the centerline. In the cal-culation, the stiucture is divided into 75 substiucmres in the longitudinal dhrection. In the analysis with the present approximation, among the 75 substructiires, all but the 12 substructures located at each end of the structure are assumed to move with the same amplitude and with the same phase. The results obtained with the present approxi-mation agree quite well with diose obtained without using the approximation. Similar approximations may be applied for the analyses in oblique waves and in head waves. Since this kind of approximation is difficult to incorporate into a conventional mode analysis, the suitability of the present approach will be greater as the stiucture becomes larger.