1 INTRODUCTION

In the presently binding rules of classification societies the requirements referring to the dynamic load components (wave bending moments and shear forces, external sea pressures, internal dynamic pressures, ship motion and accelerations) are given in the form of formulae. The formulae represent the amplitudes of the loads that can be exceeded with an assumed probability (normally 10-8). The load combination factors, intended to be an approximation of the phase shifts between particular wave loads used to analyze the strength of ship structure, are given as tabulated values and are calculated by application of the equivalent design wave approach (IACS CSR, 2010).

Shigemi and Zhu 2003 and Zhu and Shigemi 2003 developed methods for the practical estimation of the design loads based on the following definitions:

• design sea state is the sea state that generates response value equivalent to the long-term prediction of the considered ship response to waves (e.g. stresses in structure member),

• design regular wave is the regular wave that generates response values equivalent to the response values generated in design irregular wave, and

• design load is the load generated by design regular wave used in designing ship hull structure.

The values of stresses estimated with the use of the proposed design loads are claimed to be equivalent to the long-term predictions of stresses for typical load cases.

In the design regular wave approach, the dominant load is determined for such wave-heading angle, wave period and height, at which the response is at its maximum. The dominant load is determined for each load case. Then the load combination factors, representing the relationship between the response to the dominant load and response to the secondary load, are determined (Shin et al, 2004). This regular wave approach has been widely used in local scantling and finite element analysis of ship structure.

However, it is questionable whether:

• the simple formula approximating the amplitudes of wave load components – with the assumed probability of exceeding thereof,

• the phase shifts between loads in form of factors, and

• the small number of load cases assumed in the simplified methods

can approximate the complex process of sea waving acting on the moving ship in waves.

The exact phase shifts between particular wave loads used to analyze the strength of the ship structure are naturally taken into account in the simulation (in time domain) of wave loads acting on the ship and the response (stresses) of the ship to the waves. However, due to their complexity the simulations of stresses are not used in design practice.

The wave-ship system is normally assumed to be a linear one in naval architecture analysis. This allows computing transfer functions of ship response to waves, including the transfer functions of wave loads on ship. The complex functions are normally

Testing the impact of simplifications adopted in models describing wave loads on ship

P. A. Wroniszewski & J.A. Jankowski Polish Register of Shipping, Gdańsk, Poland

ABSTRACT: The wave-ship system in naval architecture is generally assumed to be a linear one. This allows for the computation of a transfer function of ship response to waves including wave loads on ship. The transfer function and the wave spectral density function enable the development of the stochastic process of wave loads on ship, corresponding to the irregular wave generating that loads. If the mathematical models of wave load transfer functions were to be used in the designing process, they would require simple mathematical forms. The paper presents the attempt of developing a simplified models of wave loads on ships.

used to determine the wave loads on ship as they include information on the amplitudes of: wave components, wave loads, and the responses of the ship to the wave loads; as well as information on phase shifts between them. Therefore, having the real and imaginary load components for different wave frequencies and having the wave spectral density function, the wave loads on ship in the time domain can be developed. Such an approach can be used in the strength analysis in designing process, provided the functions determining the wave loads have a simple mathematical form.

The paper presents a study on various simplifications in simulating wave loads on ships for design purposes. The method used is based on the linear approach in the determination of wave loads in the first step, and then on simulation of the wave loads in time domain, in the sense of discrete Fourier transform. The results of this method are compared with the simulation of vessel motions in waves based on numerical solutions of non-linear equations of motion (Jankowski, 2006).

2 THE LINEAR EQUATIONS OF SHIP MOTION IN REGULAR WAVES

The linear equations of ship motion in regular waves have the following form (e.g. Jankowski, 2006):

( )[ ] [ ] ,2ADWAEE YYNiCM +=++− ξωω (1)

where the matrices in the equation are equal to:

;6,...,1,,

,6,...,1, remainingfor ,

3,2,1,, 3,33,3

==

==

=+=+= ++++

jinN

jimM

imJMmmM

ijij

ijij

iiiiiiiii

the restoring forces – elements of matrix C , are defined by the formulae:

( ) ( ),,

,,

13552344

1533533

SVSV

Sw

JMgcJMgc

gIccgSc

+=+=

−===

ρρ

ρρ (2)

where Aξ is the vector of ship motion amplitudes, m is the mass of the ship, 3,...,1, =jJ j , are its moments of inertia, wS is the area of ship water plane, 1SI is the longitudinal static moment of water plane, 1SJ and 2SJ are the longitudinal and transverse moments of inertia of the water plane, respectively, and 3VM is the static moment of underwater ship volume in relation to the water plane, whereas, the added masses ijm , 6,...,1, =ji , and damping coefficients ijn , 6,...,1, =ji , are defined by the following integrals:

6,...,1,,

,

10

10

0

0

=

∂∂+=

∂∂−−=

∫

∫

kjdsnx

un

dsnx

um

j

RRkI

RkESjk

j

IRkR

RkESE

jk

ϕϕωρ

ϕϕωωρ

(3)

where the encounter frequency is defined as follows:

−= βωωω cos1 0

g

uE ,

0u is the forward speed of the ship, ω is the wave frequency, 0S is underwater part of a ship surface, β is the angle between the wave vector k and the 1x axis (wave-heading angle), and

( )0,sin,cos ββ kk=k .

The exciting forces occurring in equations (1) are represented by the Froude-Krylov forces, determined by the following integrals:

,)sin(

,)cos(

3

0

3

0

dsnxegyY

dsnxegyY

jkx

SAIWj

IWj

jkx

SARWj

RWj

•=≈

•−=≈

∫

∫k

k

ζρ

ζρ (4)

where ,6,...,1=j gk 2ω= , ( )0,sin,cos ββ kk=k , ( )0,, 21 xxx = ; and diffraction forces:

,

,

10

10

0

0

dsnx

uyY

dsnx

uyY

j

IDR

DESIDj

IDj

j

RDI

DESRDj

RDj

∂∂−=≈

∂∂+−=≈

∫

∫

ϕϕωρ

ϕϕωρ (5)

where .6,...,1=j The radiation Rϕ and diffraction Dϕ potentials

are determined by the following boundary-value problem (Haskind 1973):

• Laplace equation:

( ) ),,,(,\,0 3213 xxxxVRxx =∈=∆ −ϕ (6)

where V is a closed domain occupied by the ship, and by the following boundary conditions:

• on the free surface:

( ) ( ) ( ) ( )

{ } ,\0:

,01

2

3

321

2

01

VxxSx

x

x

x

x

kx

xix

F ==∈

=∂

∂+∂

∂+∂

∂+− ϕϕϕτνϕ

where gE2ων = , guE 0ωτ = and 2

00 ugk =

• on the wetted surface of the ship: − for the diffraction potential:

( ) ( ),, 0Sx

n

x

n

x WD ∈∂

∂−=∂

∂ ϕϕ

where ( )xWϕ is the potential of the incident wave, and 0S is the wetted surface of the ship;

− for the radiation potential:

( )

( )0,6,...,1,0

,

Sxin

x

nn

x

IRi

i

RRi

∈==∂

∂

=∂

∂

ϕ

ϕ

where in are the components of the normal vectors to 0S : 23324 nxnxn −= , 31135 nxnxn −= ,

12216 nxnxn −= ;

• and appropriate conditions at infinity.

The formulae determining the internal forces in a given cross section 1x of the ship hull structure can be derived from equations (1), which yields (Jankowski, 2006):

( ) ( ) ( ) ( )[ ]

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ),

,

,

,6,...,2

,

121161

131151

12141

6

11111

xFxxFxM

xFxxFxM

xdFxFxM

i

xnixmxYxF

H

V

s

jAjijijii

+=+=+=

=

−−= ∑=

ξ

(7)

where 6,...,1, =jAjξ , are the components of ship motion computed from (1); ( )1xYi are the components of exciting forces vector DW YYY += ;

Ik

Rkk iFFF += , 6,...,2=k , ( )13,2 xF are

respectively horizontal and vertical shear forces; ( )1xM H , ( )1xMV and ( )1xM s are respectively

horizontal, vertical bending moments and torsional moment; d is the centre of torsion in the considered cross section 1x .

Elements ijm and ijn of the matrices occurring in (7) are equal to:

( ),

,2

ijEij

ijijEij

nn

cmm

ωω

=

+−=

( )1xYi , ( )1xmij and ( )1xnij are computed for the part of the ship separated by the cross section 1x , for example:

( ) ( ) ,1

01 10dsn

xuxY j

RDI

DExSR

Dj

∂∂+−= ∫ϕϕωρ

where ( )10 xS denotes the wetted part of the hull up to the cross section 1x .

3 FURTHER SIMPLIFICATIONS

The loads taken into account in the ship structure designing process occur in heavy seas in which the forward ship speed 0u is normally reduced, therefore, this speed is usually neglected in the

boundary-value problem and preserved only in the equations (1) and formulae (3), (5) and (10), in the encounter frequency Eω . Such a simplified boundary-value problem can be solved numerically, enabling determination of diffraction forces, added masses and damping coefficients. However, this model is still too complex to be applied in ship design practice due to the prerequisite of solving boundary value problem (6) and further simplification need to be applied.

Some ideas have been given by (Haskind, 1973). He proposed, basing on the strict solution of the radiation problem for the ellipsoid by (Kotchin et al, 1963), the following form of the radiation potential:

,,

,,

,,

21663155

3244333

222111

xxCxxC

xxCxC

xCxC

RR

RR

RR

−=−=

−=−=

−=−=

ϕϕ

ϕϕ

ϕϕ

(8)

where coefficients iC , ,6,...,1=i can be determined for certain classes of ships. Functions (8) are real functions. Basing on the symmetry of added masses matrix, the coefficients satisfy the following relations (Jankowski, 2006):

( )

( )

( )

,

1

,1

1

1

1

,,

332

33

224

0

335

33

05

33

151

2635

GV

GV

S

GV

GV

GV

S

xx

MGC

Vxx

ICC

R

xxC

xx

RC

Vxx

ICC

CCCC

−−=

=

−+−=

−≈

−+

=

=

−+

=

=−=

(9)

where R is large metacentric radius, MG is the metacentric height, 3Vx is the third component of displacement centre, and 3Gx is the third component of the mass centre.

Based on the approximations, the added masses can be computed according to (3). However, computations of damping coefficients according to (3) are impossible as Rjϕ in form of (8) are real. Again, (Haskind, 1973) developed the following formula determining these coefficients:

( ) ( ) ,6,...,1,,,,Re4

== ∫−

jidHHn jiE

ij θθνθνπνωρ

π

π

(10)

where

( ) ( )[ ] ,,0

322113∫

++−∂

∂= ⋅+

S

RjRjxix

j dsnninn

eH νννϕϕ

θν νν

( ) ( )0,sin,cos0,, 21 θθνννν == , gE

2ων = .

Function ( )θν ,jH , j = 1, 2,…,6, has the form of the so called Kotchin function.

To complete the simplified model, the diffraction force should be determined by a function and not by the numerical solution of the boundary-value problem (6). On the basis of the Green’s identity and formula (5) the following form of the diffraction force function can be derived (Jankowski, 2006):

( )[ ] ,0

22113∫ +−=S

wRjDj dsnknkiknY ϕϕρ (11)

By combining (4) and (11) the following function, determining the wave exciting function on ships, having the Kotchin form, is obtained:

( )[ ]{ }∫ +−−−=

=+=⋅+

0

322113

S

Rjjxikkx

A

DW

dsnknkiknneg

YYY

ϕζρ (12)

Radiation potential in formulae (10) and (12) is assumed to be a complex function.

4 COMPUTATIONAL MODELS AND RESULTS OF COMPUTATIONS

Methods determining the wave loads on ships for design purposes should be as simple as possible. Therefore, an attempt was made to simplify the general linear method based on wave sources (boundary value model (6)) and in effect to simplify the computation of the loads.

The following models have been used to perform the computations: Model 1. The model represents the direct approach –

based on the ship motions and loads in waves – determined by motion equation (1), restoring forces (2), added masses and damping coefficients (3), Froude-Krylov forces (4), and diffraction forces (5). In this approach the radiation and diffraction potentials occurring in formulae (3) and (5) are obtained by solving the boundary-value problem (6), using 3D, zero speed wave source (Green’s function).

Model 2. This model is based on equations as in model 1, with damping coefficients and the exciting forces computed on the basis of (10) and (12) and on radiation potential

Rjϕ , ,6,...,2,1=j obtained from the boundary-value problem (6). This approach is not a simplification of model 1 as solving the boundary value model (6) is required. Model 2 is only used to verify formulae (10) and (12).

Model 3. This model is based on equations as in model 1 with damping coefficients and the exciting forces computed using (10) and (12) and radiation potential Rjϕ in simplified form (8).

Different ships were accounted for to investigate the affect of their shape and approximation of the radiation potential Rjϕ on the results of computations. The results of computations in the case of zero speed are presented in Fig. 1, 2, 3. Figure 4 presents the results of computation with forward speed of smu /53.2= compared with experimental data.

The computations of ship motions and loads were performed for four following ships:

• large container of block coefficient equal to 0.7, length 281 m:

• bulk carrier of panamax size of block coefficient equal to 0.8, length 225 m:

• ship of series 60, block coefficient equal to 0.7, length 123 m:

• cuboid, block coefficient equal to 1.0, length 123 m:

Fig. 1. Response Amplitude Operator of bulk carrier heave for

°= 180β , determined by different models

Fig. 2. Response Amplitude Operator of bulk carrier pitch for

°= 180β , determined by different models

Fig. 3. Response Amplitude Operator of bulk carrier vertical bending moment at midship cross section, for °= 180β , determined by different models

Fig. 4. Response Amplitude Operator of vertical bending moments for °= 180β and smu /53.2= at midship cross section of different ships, obtained from the direct approach (model 1); experiment is presented in (Drummen, 2009)

The study shows that:

• the simplified model 3 can be used to compute the transfer function of ship motions in waves – Fig. 1 and 2

• model 3 cannot be used to determine internal forces (e.g. bending moments) as formulae (10) and (12) are based on integral identities applied to the whole ship and not to its part, as required in determination of internal forces in ship cross section 1x (equation (7)) – Fig.3

• the internal forces (computed with the use of model 1) are very sensitive to the shape of the underwater part of the hull – Fig. 4.

5 SIMULATION OF WAVE LOADS ON SHIP

The simulation of vessel motions in waves is based on numerical solutions of non-linear equations of motion. The non-linear model used is presented in (Jankowski, 2006).

This method assumes that: • Froude-Krylov forces are obtained by integrating

the pressure caused by irregular waves, undisturbed by the presence of the ship, over the actual wetted ship surface;

• the diffraction forces are determined as a superposition of diffraction forces caused by the harmonic components of the irregular wave;

• the radiation forces are determined by added masses for infinite frequency and by the so-called memory functions given in the form of convolution. The non-linear equations of motion are solved

numerically.

The simulation based on the linear solution is carried out according to the formula:

( ) ( ) ( )[ ]

( ) ( ) ( )[ ] ,sincos

,sincos,

1

11

i

n

iii

Ijiii

Rjij

i

n

iii

Ijiii

Rjij

attt

atFtFtxF

∑

∑

=

=

+−+=

+−+=

εωξεωξξ

εωεω(13)

where ( )txF j ,1 , ,5,3=j are internal forces in the cross section 1x of the ship structure in the time instant t , R

jiF and IjiF , ,5,3=j are respectively real

and imaginary part of the internal forces in the cross section 1x of the ship structure as a response to regular wave of frequency iω and unit amplitude, ia is the regular wave amplitude obtained from the wave spectrum and iε is the random phase shift between the regular wave components ni ,...,2,1= . The vertical bending moment in cross section 1x is computed according to formula (7).

The comparison of the computation results using non-linear and linear models is presented in Fig. 5.

Fig. 5. Panamax bulk carrier heave3ξ , pitch 5ξ and vertical bending moment VM as a response to irregular wave, determined by:

significant wave height 15=sH m, average zero-up-crossing period 120 =T s, °= 180β ; ζ is the wave elevation

6 CONCLUSIONS

The requirements in binding rules of classification societies referring to the dynamic loads are given in the form of:

• formulae, determining amplitudes of the loads that can be exceeded with an assumed probability, and

• tabulated values of load combination factors, intended to be an approximation of the phase shifts between particular wave loads used to analyze the strength of ship structure.

However, it is questionable whether such a simplified method can approximate the complex process of sea waving acting on the ship moving in waves.

The exact phase shifts between particular wave loads used to analyze the strength of the ship structure are naturally taken into account in the simulation (in time domain) of wave loads on ship and the response of the ship to the waves (ship motions in waves, wave loads on ships, stresses in the structure). However, due to complexity the simulation of loads and stresses in ship structure (ship response to irregular wave) is not used in design practice. Therefore, a simplified method should be developed.

The paper presents the attempt of developing such a simplified method, starting from the linear model based on the boundary-value problem (6) and then by adopting the following simplifications: • in the first step, ship forward speed 0u is

neglected due to the fact that in high sea states, normally used as the design wave cases, ship speed 0u is significantly reduced, simplifying the problem;

• in the second step, the attempt is made to approximate radiation potential Rjϕ , 6,...,1=j , by formulae (8) and to compute the damping coefficients and the exciting forces using (10) and (12); this approach does not trigger the necessity to solve the boundary-value problem (6).

It turns out that:

1. Wave loads are very sensitive to the shape of the underwater part of the ship (see Fig. 5). Therefore, the models used to compute these loads (especially internal forces in the ship structure) should be based on precise description of the ship geometry.

2. Simplifications of the radiation potential Rjϕ , 6,...,1=j , can be applied only to the ship

motions in waves (see Fig. 1 and 2). The proposed simplifications – model 3, can not be applied to determine internal forces in ship cross section 1x as formulae (10) and (12) are

based on integral identities applied to whole ship and not to its part, as required in equation (7). Therefore, new ideas are necessary to develop the transfer functions of internal forces and then the simulation method for wave loads on ship.

The simulations with the use of formulae (13) and that based on the transfer function computed with the use of model 1, presented in Fig. 5, show that this approach is comparable to the non-linear method and can be applied in the designing processes, provided simpler than model 1 methods are developed for determining transfer function.

Therefore, necessary new ideas will be further developed to determine the transfer functions of internal forces of wave loads on ship with the use of model 3 in order to simulate them according to formulae (13).

REFERENCES

Drummen I., Wu MK., Moan T., 2009, Marine Structures 22, Experimental and numerical study of containership responses in severe head seas, pp.172-193.

IACS, 2010, Common Structural Rules for Bulk Carriers, International Association of Classification Societies.

Haskind M., D., 1973, Gosudarstvennoe Izdatelstvo fiziko – matematitcheskoi literatury, Moscow, Hydrodynamic theory of vessel motion in waves (in Russian).

Jankowski J.,A., 2006, Technical Report No. 52, Polish Register of Shipping, Ship facing waves, (in Polish).

Kotchin N., E., 1963, Gosudarstvennoe Izdatelstvo fiziko – matematitcheskoi literatury, Moscow, Theoretical hydrodynamics (in Russian).

Shigemi T., Zhu T., 2003, Marine Structures 16, Practical Methods of the design loads for primary structural members of tankers, pp 275-321.

Shin Y. S., Kim B., Fyfe A. J., 2004, OMAE, Load Combination for Fatigue Analyses of Ship Structural.

Zhu T., Shigemi T., 2003, Marine Structures 16, Practical Estimation Method of the Design Loads for Primary Structural Members of Bulk Carriers, pp. 489 – 515.