Ship Load Model on Large Amplitude Waves
Albert Zamarin
Faculty of Engineering, University of Rijeka
Department of Naval Architecture and Ocean Engineering
Vukovarska 58, 51000 Rijeka, CROATIA
ABSTRACT: Non-linearity arising from the ship motion at large amplitude waves can’t be neglected, especially in wave load, as in case of linear model. The wave load numerical model in large amplitude waves is
presented. Main source of non-linearity, focused on in this work, involves time dependent hydrodynamic coefficients and excitation due to changing of cross section drought because of ship motion and wave passing.
The load problem has been solved on harmonic waves in the frequency domain within vertical longitudinal
plane. Vertical shear force and vertical bending moment in head waves are analyzed. As the source for linear
and non-linear equation formulation, for motion, as well as for the wave load, modified Salvensen-TuckFaltinsen strip theory was used. The results are presented for S175 container ship and show non-linear effects
in motion and hull load. They are in the form of high order wave load harmonic transfer functions and longitudinal distribution of the wave load along the ship hull.
1 INTRODUCTION
During ship’s advance on sea waves, her structural
elements are under the influence of changeable
weather acts of wind and waves. Among all the
loads, the dynamic wave load is the most important
and has the essential role in ship hull structure design. Therefore, it is necessary to have structure load
response as accurate as possible in the basic design
phase. The available methods and computer programs can determine more accurate structure load
response when we have proper load definition. Previous to this calculation, ship motion equation must
be defined and solved. Basic characteristic of ship
motion problems is modeling the hydrodynamic influence.
Unequal distributions of the hydrodynamic pressures and inertial forces, during ship oscillating on
sea waves, do cause additional ship structural loads
1. Together with calm sea structural loads (hydrostatic pressures and gravity force), these loads on extreme waves determine the load level limit ship
structure is about to endure. This problem rationalization is, in modern engineering practice, alternative
to empiric approach to ship structure elements dimensioning and includes methods based on existing
problem’s real shape numerical model i.e. interaction between ship hull and sea waves. In extreme sea
conditions, on large amplitude waves, real model description must include the variation of the wetted
surface, influenced by hydrodynamic pressures that
resulted from wave load on ship hull. So, the relation
between wave forces and ship hull response is becoming a non-linear one with non-symmetrical response characteristics containing first order harmonic components as well as harmonics of higher and
lower orders 2. Incident wave stochastic nature
causes additional difficulties in analysis of the event
where final result is the extreme stress level and can
be interpreted only in the probability domain.
Existing “reserves” that we use today during primary ship strength structural elements dimensioning,
are the area in which we can further improve the
ship structure design towards higher safety and efficiency levels.
Non-accuracy in wave load determination arises
from the fact that hydrodynamic problems describing flow around ship advancing on waves are very
complex. Therefore, some simplifications are necessary and important. Furthermore, there are several
different hydrodynamic models that may be linear or
non-linear but can be defined in terms of frequency
or time domains.
In the frequency domain, perturbation analysis
with wave amplitude, as the small parameter, has
been applied to access the nonlinearities of hydrodynamic boundary value problem, aimed at assessing
the loads. The first-order solutions are linear ones.
The solution of the second-order theory keeps terms
proportional to the wave amplitude and to its square.
They enable the determination of mean and slowly
varying wave loads 3. The perturbation methods
are presented in 4. Bruzzone at al 5 developed a
second-order theory, which is an extension of STF
strip theory.
Ship wave load calculation in frequency domain
includes Response Amplitude Operator (RAO) determination. The response is achieved as response
spectra for given wave spectra. Design load defines
the level that is not going to be overcome most
probably during the ship exploitation. To define design load, sea surface is divided into stationary sea
conditions. By stationary condition, we presume
constant speed and ship course values and constant
wave kinematics values. These presumes are good
for the several hours period; and in that case we
have a short-term prediction of structure response.
Linear methods give, in general, good results for
light and moderate sea conditions but we have a
problem for heavy sea conditions when load is extremely non-linear compared to wave height 6.
Therefore, non-linear methods should be used for
the sea conditions whose effects can be described
clear and exact.
The possibility of predicting large amplitude nonlinear wave induced loads and responses of the
structure in heavy weather condition is very important in ship design as well in the strength assessment of ship in service, particularly in critical cases
such as for old and corroded ships. Basic loads for
these calculations are gained by usage of classification societies’ formulas or as a result of complex hydrodynamic model calculations that require significant mathematical and fluid mechanics support.
2 NON-LINEAR WAVE LOAD IN HEAD
WAVES
2.1 Introduction
The moving co-ordinate system 0(x,y,z) fixed with
respect to the mean position of the ship is defined,
with z in the vertical upward direction and xy plane
which coincide with the undisturbed free surface.
Considering a ship advancing in waves and oscillating as an unrestrained rigid body, the oscillatory motions will consist of three translations and three rotations. Following standard strip theory 7 through
dynamic load equation, six wave load components
are as presented on Figure 1. Although the theory,
which will be presented, can be applied for arbitrary
headings relative to waves, the present work is restricted to head waves, thus wave loads to be studied
are the vertical shear force and vertical bending
moment.
Ship structure wave loads considers forces and
moments in ship hull cross section (internal loads)
that form equilibrium with exciting forces and mo-
ments (external loads) and inertial forces and moments.
V3
V6
z
V2
y
V5
V4
V1
V2 – horizontal shear force
V3 – vertical shear force
V4 – torsion moment
V5 – vertical bending moment
V6 – horizontal bending moment
Figure 1. Convention for wave loads caused by oscillatory motions
In the case of large amplitude motions, it is necessary to define the relationship between the inertial
reference system and the ship fixed reference system. This relation is defined through a rotation matrix, which is dependent of the modified Euler’s angles. If the angular motions are small, the forces and
moments may be represented on the inertial reference system, and the equations of motion solved directly 8.
2.2 Dynamic load equation on harmonic wave
Total dynamic force and moment VjT(t) at hull cross
section can be represented as a difference between
inertial force and moment, and restoring, exciting
and hydrodynamic forces and moments:
V jT  e , t =I j  e , t   R j  e , t   E j  e , t   D j  e , t  (1)
where j=3, 5 represents wave load components in
longitudinal vertical plane; total vertical shear force
(V3T) and total vertical bending moment (V5T), which
are results of ship motion in large amplitude head
waves. On the other hand it is assumed that total
cross section forces and moments are given by the
sum of linear and perturbation parts:
V jT  e , t   V j  e   V jP t ,
j  3,5 .
(2)
Linear part is encountered frequency dependent,
while non-linear part is time dependent and both
parts of (2) are in function of position along the ship.
The total dynamic cross section force and moment
can be assumed as the N-th order trigonometric series:
Re
Im
N
V3T  
 V3n  iV3n  ni et  


e

 T    Re
Im 
 
V5  
 n 1 V5n  iV5n 


(3)
where N = 1 reach to linear solution, which is linear
load model calculation result.
V3  V3Re  iV3Im  i et
e
    Re
Im 
V5  V5  iV5 
(4)
In case of N = 2,3 the second and third order vertical shear force and vertical bending moment, as
perturbation part of total force and moment are:
Re
Im
3
V3P  
 V3n  iV3n  niet  

e
 .
 P     Re
Im 
V5  
 
 n 2 V5n  iV5n 


(5)
2.3 Cross section draught variation
The hull wetted surface S in (11) is changing with
time, which makes the problem strongly non-linear.
Instead of solving the problem with the non-linear
boundary condition, variable cross section hydrodynamic forces coefficients are introduced. In addition,
assuming all of the present non-linear wave load
phenomena, the strongest influence on the accuracy
has the variation of the wetted surface S(t). The variation of cross section drought around mean position,
or instantaneous cross section drought zv, in the vertical longitudinal plane, is caused because of ship position changing during oscillation and because the
free surface elevation, i.e. wave profile changing
along the ship.
(6)
where free surface elevation is given by the equation:
 x, t   a w cos kx  i sin kxe iet ,
L
 e , x   b33V  e , x, t 
b33  e , x, t   b33
L
x   c33V x, t 
c33 x, t   c33
f 3  e , x, t  
(8)
 e , x    e , x, t 
h3  e , x, t   h3L  e , x   h3V  e , x, t  ,
f 3L
f 3V
with
V
 e , x, t    33  e , x z v x, t 
a33
All wave load components, Ij, Rj, Ej, Dj, are complex function consisting also of linear part and second and third order harmonics. These components
can be determined after the non-linear motion calculation is performed. The reason is, as in case of linear procedure, that wave load components are ship
motion dependent. So, the high order motion component will be directly included in terms for high order forces and moments of the same order. Besides,
second order motion component will influenced the
third order load component, because of combination
with added mass and damping coefficients, due to
cross section draught variation during the ship motion in waves.
z v ( x, t )   3 (t )  x 5 (t )   ( x, t ) ,
L
 e , x   a33V  e , x, t 
a33  e , x, t   a33
V
 e , x, t    33  e , x,z v x, t 
b33
V
x, t   g  33 x  z v x, t 
c33
f 3V  e , x, t     3  e , x  z v x, t 
h3V  e , x, t     3  e , x  z v x, t  ,
where 33, 33, 33, 3 and 3 are the coefficients of
dependence on cross section immersion, which is
calculated by the linear regression method. Coefficients 33 i 33 represent the gradient of linear variation of sectional added mass and damping coefficients for an unit immersion of the cross section,
related to the mean draught. Linear dependence coefficient 33 represents breadth variation of the cross
section multiplied with g, where  is mass density
of water and g is gravitational acceleration, represent
cross section restoring coefficient variation. Coefficients 3 and 3 represent the change of the FroudeKrylov and diffraction two-dimensional force coefficients.
The fluid motion isn’t rotational, and hydrodynamic problem may be formulated in terms of potential flow theory, thus the fluid velocity vector may
be represented by the gradient of a total velocity potential, which is separated into two parts assuming a
slender hull at slow forward speed. The first one is a
steady contribution due to forward motion of the
ship, and the other one is unsteady part associated
with the incident wave system and the unsteady
body motion:
(x, y, z; t) = [-Ux +  S (x, y, z)] +  T (x, y, z) eit
(9)
Complex amplitude of the unsteady velocity potential is linearly decomposed:
6
(7)
and 3 and x5 being cross section variation due to
heave and pitch motion.
The instantaneous section added mass, damping
and restoring coefficients, and two-dimensional exciting force for the cross section, oscillating with the
encounter frequency e, are expressed as the sum of
encounter frequency e dependent part, and time dependent part, due to draught variation:
T =  I + D +  j  j
(10)
j=1
where the incident and diffraction potentials I, D
are assumed to be proportional to the incident wave
amplitude (t), and the radiation potential j, j=3,5 is
proportional to the heave and pitch motion amplitude. Substitution of the potential equation (9) and
(10) into the Bernoulli’s equation results in hydrodynamic pressure. Integration of the oscillatory pressure terms over the wetted surface of the hull S, results in total hydrodynamic force and moment:
H j  
 
1
 pn ds      t  2 
2
j
S
S

 gz n j ds (11)

and by linearising the pressure and applying basics
of the strip-theory 7 and equation (10), total hydrodynamic force and moment is:
(12)
H j= F j +G j
where Fj is exciting force and moment:
F j  
 n
 

 e  U  I   D dS
x 

j i
S
(13)
G j  
 n
 

 e  U   k  k dS 
x  k 1

j i
S

A   η    B   η    C η    ΔF   e 
ΔA   η    iω ΔA    ΔB   η    C   η   ;
ωe
jj
ωe ,t
jj
and Gj is radiation force and moment due to j-mode
of body motion.
6
Using (16), (17) and (18) and applying (6) to (8)
in (15) results in coupled non-linear equation of
heave and pitch motions. Linear part of that equation
defines linear solution (17) of motion equation. The
remaining part of the same equation, after neglecting
the small second order values in multiplication,
gives the coupled linear equations for the perturbation heave and pitch displacement vector (18), and is
expressed in matrix form:
6
T
jk
k
(14)
k 1
2.4 Motion equations
For the ship oscillating in head waves and advancing
with the constant speed, two coupled equations for
heave and pitch motions, that include total derivative
of the momentum are expressed as:
M + A33  e ,t 3 d M + A33  e ,t  3+B33  e ,t  3+C 33 t  3+
ωe ,t
jP
ωe
jj
ωe
jL
ωe ,t
jP
ωe ,t
jj
e
ω e ,t
jP
jj
ωe ,t
jj
ωe
jL
I 5 x, t    m x   x  3T  5T d
where
 3 L   3 L Re  i 3 L Im  i e t
e
    Re
Im 
 5 L   5 L  i 5 L 
Re
Im
N
 3P  
  
 3Pn  i 3Pn 
 ni et 


e

  

Re
Im 




 5P  

 n  2   5 Pn  i 5 Pn 


Inertial force and moment due to ship motion are obtained by integration of the cross sectional inertial
force and moment over the length of the ship forward of the cross section being considered:
 

L
I 3 x, t   m x 3T  5T d
x

L


x
where mx is sectional mass per unit length of the ship
and the total sectional heave 3T t  and pitch 5T t 
acceleration is obtained by double derivation of
equation (18) with respect to time:
C 2i e t
C 3i e t
3T  e , t    e2 3CL e i et  4 e2 32
e
 9 e2 33
e
C 2i e t
C 3i e t
5T  e , t    e2 5CL e i et  4 e2 52
e
 9 e2 53
e
C
I 32
x, t   4 m
x




C
C
 e2  32
 52
d
x
L
(18)
(21)
By separating the part of inertial force and moment connected with e2it and the part connected
with e3it, the second and third order inertial force
and moment is obtained as follows:
L
(17)
ωe
jL
 Tj  e , t    jL  e e iet   j 2  e , t e 2iet   j 3  e , t e 3iet
(20)
where Ajj, Bjj, Cjj are added mass, damping and restoring coefficients, and Fj is exciting force, for the
whole ship, and are time and encountered frequency
dependent that are results of draught variation during
ship motion. These coefficients are obtained by integration over the ship length following standard strip
theory 7 and by using (6) to (8).
The total heave and pitch displacement is assumed to be composed of linear part and perturbation part:
 5T   5 L   5 P
t
jj
(19)
Ajj, Bjj, are time dependent added mass and
damping matrix containing correction coefficients.
Fj is time dependant excitation force vector containing correction coefficients, too. The details of
the elements of these matrices are given in 8 and
9. System equation solution of (19) results in total
heave and pitch displacement process:
2.5 High order forces and moments
(16)
iω e t
j  3,5
dt
d
A35  e ,t 5 A35  e ,t  5 B35  e ,t  5+C 35 t  5 F3 t ei et
dt
d
A53  e ,t 3+ A53  e ,t  3 B53  e ,t  3+C 53 t  3+
dt
(15)
I 5 + A55  e ,t 5 d I 5 + A55  e ,t  5 B55  e ,t  5+
dt
C 55  e ,t  5  F5 t ei et
 3T   3 L   3 P
ωe ,t
j
C
x, t   9 m x  e2  33C  53C d
I 33

x
L




C
x, t   4 m x   x  e2  32C  52C d
I 52

C
E 32
C
x, t   9 m x   x  e2  33C  53C d
I 53

Hydrostatic restoring force R3 and moment R5 are
given by:


R3 x, t    g bvx ,t  3T  5T d

x

L

R5 x, t   g bvx ,t   x   3T  5T d

x
where 3 and 5 are total heave and pitch displacement complex function (20) and bv(x,t) is sectional breadth which depends of cross section position x and cross section draught variation zv (6):
T
T
bv x, t   Bx    33 x  z v x, t 
(22)
Following the same procedure as in case of inertial force and moment hydrostatic restoring force
end moment of the second and third order are:
L

C
x, t    g  B x  32C   52C   3x z vC  3CL   5CL d 
R32
 x

 



 






L

C
x, t    g  B x  33C   53C   3x z vC  32C   52C d 
R33
 x

 



Exciting forces and moments over the portion of
the ship forward of the cross section x can be obtained directly from 7:
 
 





N 3 e iky sin  e kz dl
 iN
3

(25)
Therefore, the third order exciting force and moment are as follows:
L

C
x, t      3C   3C z vC2  d  U  3C z vC2  x 
E 33
i e


x

L
 


C
x, t        x   3C   3C  U  3C  zvC2 d 
E53
i e 


x 

 
 





and using instantaneous cross section added mass
and damping (8), with the variation of cross section
drought around mean position (6) and total displacement formulation (20), following the same procedure
of separating high order components, hydrodynamic
force and moment of the second and the third order
are given by:
 a xZ  b xZ   xZ   xZ d 
a xZ  b xZ   xZ   xZ 
  a x Z  b x Z   x Z   x Z d 
a xZ  b xZ   xZ   xZ 
   x a x W b x W  x W   x W d
L
*C
32

L
33
2
1
L
33
L
33
2
2
L
3
33
L
4
33
2
1U
L
33
2
2U
L
3U
L
33
3
2
33
2
3
3
2U
33
2
3U
33
L
4U
33
 x
L

*C
G33
L
33
3
1
2
4
33
x
L
33
3
1U
L
33
L
33
2
1
2
4U
33
 x
L
*C
G52
L
33
2
2
33
L
3
L
4
33
x
(23)
C
h3L   e ik cos 

C
C
z v2  x, t   z vC2  e 2iet   32
 x 52
e 2iet
x
by using (8) in linear Froude-Krylov and diffraction
sectional force component correction
f 3L  g e ik cos 

For the purpose of evaluating the third order exciting force and moment it is necessary to introduce
second order cross section draught variation. Instead
of the linear heave and pitch motion in (6), second
order displacement components 32C and 52C are introduced, together with second order wave elevation
neglected.
G


C
x, t      3C   3C z vC2  d  U  3C z vC2  x 
E 33
i e


x

L
 


C
x, t        x   3C   3C  U  3C  zvC2 d 
E53
i e 


x 

L

 
C C
3 zv x
t 33   e2 33  i e b33 ,
L

C
x, t    g  B x  32C   52C   3x z vC  3CL   5CL d 
R32
 x

 

Complex amplitude of the hydrodynamic heave
force and hydrodynamic pitch moment can be obtained directly from (14) by applying section hydrodynamic force:
 L x C

C
C
C
R x, t    g  B  33   53
  3x z vC  32
  52
d 
 x

C
33


x
L
U
i e


x

L
 


C
x, t        x   3C   3C  U  3C  z vC d 
E 52
i e



x 

x
L
L
x, t      3C   3C z vC d 
 N 2 sin   e iky sin  e kz 3 dl (24)
C
By separating the part of exciting force and moment connected with e2it and the part connected
with e3it, the second order exciting force and moment is obtained as follows:


L
L
x W1U2 b33L x W22U  33 x W3LU   33 x W4LU d
 a 33
x
L


*C
L
x W13 b33L x W23  33 x W32   33 x W42 d
G53
   x  a 33

x

L

L
x W1U3 b33L x W23U  33 x W32U   33 x W42U d
 a 33
x
(26)
Coefficients Z and W in (26) are listed in Appendix, and details are given in 9.
3 NUMERICAL EXAMPLE
The non-linear load model based on presented theory is implemented through software GIOP 9. The
result of the numerical example is given for the container ship S175, shown in Figure 2 and 3. The ship
is advancing with constant speed v=20 kn in head
waves, =180. The wavelength to ship length ratio
range in calculation is /Lpp= 0,5 – 2,2. The wavelength to wave height ratio is /aw= 80. The characteristics of the ship are given in Table1.
0
over the range of the harmonic wave and are results
of linear load model calculation. Figure 5 and 6
show transfer functions of second and third order
vertical bending moment over the same range of
harmonic wave, which are large amplitude wave
load model results.
WAVE LOADS TRANSFER FUNCTIONS
PROJECT:S175
heading:180 deg, speed:20,1 kn
20
CROSS SECTION
19
18
4
5
6
7
9
10
8
1
3 2
WL
17
Figure 4. Transfer function of the linear vertical bending
moment (VBM=V5) for a series of cross sections
16
15
14
13
12
WAVE LOADS TRANSFER FUNCTIONS (2)
11
PROJECT:S175
heading:180 deg, speed:20,1 kn
BL
Figure 2. Body plan of container ship S175 hull
mx, tm-1
S175
Figure 3. Distribution of structural mass of
CONTAINER
S175 hull
200
SHIP
150
CROSS SECTION
100
Figure 5. Transfer function of second harmonic vertical bending moment (VBM2)
50
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
WAVE LOADS TRANSFER FUNCTIONS (3)
PROJECT:S175
heading:180 deg, speed:20,1 kn
Figure 3. Distribution of structural mass of S175 hull
Table 1. Principal characteristics of the S175 ship
Length between perpendiculars
Lpp
175,0
Breadth moulded
B
25,4
Depth moulded
H
17,4
Draught, design
T
9,5
Displacement

24764
Block coefficient
CB
0,57
LCG (aft of midship)
LCG
2,8
Transverse metacentric height
GM
0,98
Vertical center of displacement
KB
5,3
Vertical center of gravity
zg
9,55
Longitudinal radius of gyration
kyy
42,8
Transverse radius of gyration
kxx
8,3
Water plane area
Awp
3147
Midship section coefficient
CMS
0,97
m
m
m
m
t
m
m
m
m
m
m
m2
Figure 4 show transfer functions of the vertical
bending moment for a series of hull cross sections
CROSS SECTION
Figure 6. Transfer function of the third harmonic vertical bending moment (VBM3)
Amplitude of the vertical bending moment of the
linear load model and amplitudes of second and
third order from non-linear model are then used for
vertical bending moment process simulation, Fig. 7.
Three different wavelengths at midship are chosen
and comparison of linear and large amplitude wave
load model is illustrated.
WAVE LOADS TRANSFER FUNCTIONS
/Lpp=1,2
linear
nonlinear sagg (+)
nonlinear hogg (-)
time
linear
nonlinear
Figure 9. Transfer function of the linear and non-linear (sagging +, hogging -) vertical shear force at 3/4Lpp
/Lpp=0,9
4 CONCLUSION
time
linear
nonlinear
/Lpp=0,6
time
linear
nonlinear
Figure 7. Simulation of linear and non-linear vertical bending
moment at midship for a different ratio /Lpp
Then, maximum value of bending moment is calculated over certain time interval and these values
are used for results in form of transfer functions,
Figures 8 and 9.
WAVE LOADS TRANSFER FUNCTIONS
linear
nonlinear hogg (-)
nonlinear sagg (+)
Figure 8. Transfer function of the linear and non-linear (sagging +, hogging - ) vertical bending moment at midship
Solution of proposed non-linear load model based on
modified STF method clearly shows non-linear effects during motion and for loaded structure of container ship taken into consideration.
Non-linear load simulation shows that included
non-linearity is mostly expressed for the waves of
length equal to the ship length, i.e. in the range
/Lpp=1,0 to 1,3. While decreasing wavelength, the
non-linearity influence is also decreased. So, for the
ratio /Lpp=0,5, differences are almost neglectable.
Similar happens when wavelength is increasing up
to ratio /Lpp=2,0. The late can be seen through the
amplitudes of non-linear transfer function of vertical
shear force and bending moment in frequency domain. Amplitudes of transfer load function of second
and third order get the greatest values in the area
where /Lpp=1,0. Generally speaking, it can be concluded that the differences between linear and nonlinear load process are much greater than the differences in motion process. Throughout the presentation of transfer load functions amplitude distribution
Figures 8 and 9, or non-linear vertical shear force
amplitude and non-linear longitudinal bending moment distribution, Figure 10, it can be concluded that
the bending moments resulted from the sag are much
higher that the one from the hog over the frequency
range and along the ship length. Also, the non-linear
vertical shear forces are higher for the sag than for
the hog. The presented results are compared with
ones of a similar procedure 10, Figure 11. Additional analysis of total force and cross section moment components indicates the dominant role of restoring force and moment throughout complete
frequency range. The exciting force and moment become more relevant at lower frequencies, and are
getting equal to the restoring forces and moments.
At higher frequencies, inertial and hydrodynamic
forces and moment are becoming more relevant. By
analysis of second and third order force and moment
components, trend of changing, with reference to
frequency noticed is similar to one of linear components. Additionally, throughout the frequency range
the influence of third order inertial force and moment turning into the second order ones is emphasized. More emphasized is hydrodynamic force and
moment of second order within higher frequencies.
Exciting force and moments of third order are,
throughout the frequency range, much lower than
the other components of third order and can be neglected.
WAVE LOADS DISTRIBUTION
cross section (AP=0; FP=20)
linear
nonlinear sagg (+)
nonlinear hogg (-)
WAVE LOADS DISTRIBUTION
REFERENCES
[1] JENSEN, J. J.: “Load and Global Response of Ship”, Elsevier Science Ltd., Oxford, UK, 2001.
[2] JENSEN, J. J., PEDERSEN, P. T.: “Wave-induced bending
moment in ship – a quadratic theory”, Proceedings RINA
Supplementary Papers, 121, pp. 151-165, 1979.
[3] FALTINSEN, O. M.: “Wave and current induced motions
of floating production systems”, Applied Ocean Research,
Vol. 15 pp. 351-370, 1994.
[4] OHKSU, M.: “Hydrodynamics of Ships in Waves”; Computational Mechanics Publications 1996.
[5] BRUZZONE, D., PITTALUGA, A., PODENZANA, B.:
“Feasibility of second-order strip-theory for longitudinal
strength of ships”, Proceedings 6th International Symposium on Practical design of Ship and Mobile Units, Seul,
p1.530-1.540., 1995.
[6] SCLAVOUNOS, P. D.: ”Computation of wave ship interaction”,
Advanced
in
Marine
HydrodynamicsComputational Mechanics Publications, pp. 172-231.
[7] SALVESEN, N., TUCK, E. O., FALTINSEN, O.: “Ship
Motion and Sea Loads”, SNAME Trans., vol. 78, 1970.
[8] PRPIĆ-ORŠIĆ, J.: “STF Ship Motion Theory Modification
for Non-linear Response Characteristics”, Ph.D. Thesis,
Faculty of Engineering, University of Rijeka, 1998.
[9] ZAMARIN, A.: “Numerical Modeling of Ship Extreme
Loads on Large Amplitude Waves”, Ph.D. Thesis, Faculty
of Engineering, University of Rijeka, 2002.
[10] WANG, Z. H.: “Hydroelastic Analysis of High-Speed
Ships”, PhD Thesis, Dept. of Naval Architecture and Offshore Eng., Technical University of Denmark, 2000.
APPENDIX - COEFFICIENTS IN EQUATIONS
OF HYDRODYNAMIC FORCE AND MOMENT
Z 1j   e2 3Cj  iU e 5Cj   e2 5Cj , j  2,3
cross section (AP=0; FP=20)
linear
Z 2j  i e  3Cj  U 5Cj  i e  5Cj , j  2,3
nonlinear sagg (+)
nonlinear hogg (-)

 i 

,
Z 3j  z vC  e2 3Cj  iU e 5Cj   e2 5Cj , j  L,2
WAVE LOADS DISTRIBUTION
Z 4j  z vC
Z
j
1U
e
 iU e 
C
3j
C
3j
Z 2jU  U 3Cj  i

 U 5Cj  i e  5Cj
 iU 
2
U2
e
C
5j
 iU e x
C
5j
j  L, 2
, j  2,3
 5Cj  Ux  5Cj , j  2,3

Z 3jU  z vC  iU e  3Cj  iU 2 5Cj  iU e x 5Cj , j  L,2
cross section (AP=0; FP=20)
linear
nonlinear sagg (+)
nonlinear hogg (-)
Figure 10. Distribution of the linear and non-linear
(sagging +, hogging -) vertical bending moment along the ship for a
three different ratio /Lpp
2

 , j  L, 2


W2j  i e  3Cj  i e  5Cj , j  2,3

i 

,
0.05
W3 j  z vC   e2 3Cj   e2 5Cj , j  L,2
0.04
W4  z
j
0.03
V5 / (gBLp p aw)

U2 C
Z 4jU  z vC   U 3Cj  i
 5 j  Ux  5Cj
e

W1 j   e2 3Cj   e2 5Cj , j  2,3
j
1U
W
0.02
0.01
C
v
e
 iU e 
C
3j
C
3j
W2Uj  U 3Cj  i
0.00

-0.01
-0.02
-0.03
0
2
4
6
8
10
12
cross section
14
GIOP (lin.)
Experiment
Wang (lin.)
16GIOP (nonlin.)
18
Wang (nonlin.)
 i e 
 iU 
2
U2
e
C
5j
C
5j
j  L, 2
 iU e x 5Cj , j  2,3
 5Cj  Ux  5Cj , j  2,3

W3Uj  z vC iU e  3Cj  iU 2 5Cj  iU e x 5Cj , j  L,2
20
Figure 11. Amplitude comparison of the linear and non-linear
(sagging+, hogging-) vertical bending moment distribution
along the ship for /Lpp=1,2; Fn=0,25

U2 C
W4Uj  z vC U 3Cj  i
 5 j  Ux  5Cj

e



z vC  3CL  e   x5CL  e    C x

 , j  L, 2


,