MarineStructures8 (1995)481-499
© 1995ElsevierScienceLimited
Printed in Great Britain. All rights reserved
0951-8339/95/$9.50
ELSEVIER
Analysis of Moorings of a Berthed Ship
R. Natarajan & C. Ganapathy
Ocean Engineering Centre, Indian Institute of Technology, Madras 600 036, India
(Received 19 July 1993)
ABSTRACT
Ships are moored to the shore structures with a system of multiple moorings. The determination of the forces in the cables is essential for the design
of moorings and for berthing structures. An optimum number and length of
cables can be arrived at for a given ship of known dimensions and environmental conditions. This paper is intended to derive the equations of static
equilibrium for the mooring lines when the ship is subjected to surface wind,
current and restricted waves. It also gives a methodology to assist the
designer to predict the mooring forces on the mooring cables, bollards, etc.
A computer p r o g r a m - S H I P M O O R - - w a s developed for the analysis of
the mooring system. Experiments were also conducted on a barge model,
moored in a 30 m × 2 m flume in which both wave and current can be
generated. The experimental results were compared with the results
obtained from the computer program and they were found to be in good
agreement. Using this computer program, a few case studies were carried
out and the results are presented.
Key words." berthed ship, environmental forces, m o o r i n g ropes, restoring
forces, static analysis.
NOTATION
A
Acx
Acy
Cross-sectional area of the mooring line
Projected area of the ship in the yz-plane below the
waterline
Projected area of the ship in the xz-plane below the
waterline
481
482
hw
Zwx
Awy
B
Bi
Ci
Cm
Cw
d
D
E
F~,Fy, F~
g
GM~.
GMr
H
ki
K
!
li
L
Zs
Mx, My, Mz
n
N
a
T
Ux
Vy
Vw
W
ot
R. Natarajan, C. Ganapathy
Waterplane area of the ship
Projected area of the ship in the yz-plane above the
waterline
Projected area of the ship in the xz-plane above the
waterline
Breadth of the ship
Coordinates of the ith bollard
Drag coefficient for current force
Coordinates of the ith chock in the x, y and z directions
Added mass coefficient of the ship
Drag coefficient for wind force
Draught of the ship
Depth of water
Young's modulus of the mooring line
Reactive force of the ith mooring line
Force components along the x, y and z axes
Acceleration due to gravity
Longitudinal metacentric height of the ship
Transverse metacentric height of the ship
Incident wave height
Stiffness of the ith mooring line along its own axis
Global stiffness matrix
Stiffness of the ith mooring line along its axes of the ship
Length of mooring line from the mooring bitt to the bollard
Reactive moment of the ith mooring line about the axes of
the ship
Global reactive moment acting through the centre of the
gravity of the ship
Length of the ship
Moments of the force about the x, y and z axes
Number of mooring lines
Number of fenders
Environmental forces
Tension of the mooring line
Translation of any point of the ship along the x-axis
Translation of any point of the ship along the y-axis
Translation of any point of the ship along the z-axis
Current velocity
Wind velocity
Displacement of ship
Direction of wind
Direction of current
Analysis of moorings of a berthed ship
Ox
Oz
#
Pa
Pw
483
Angular displacement of the ship due to heel
Angular displacement of the ship due to trim
Angular displacement of the ship due to yaw
Wave number
Direction of wave
Mass density of air
Mass density of water
1 INTRODUCTION
Mooring is the operation of securing a ship to a wharf or quay by means
of ropes or chains. A moored ship need not necessarily be truly stationary.
It may be free to rise and fall with the tide or the loading and unloading of
cargo or to oscillate in response to the action of the environmental forces.
In this respect a moored ship is restricted to a limited amount of movement and is restrained only to the extent necessary to keep that movement
within well-defined bounds.
The forces acting on a moored ship include those imposed by the
environment and those opposing forces that are applied to keep the ship
stationary in a desired position and altitude. The environmentally imposed
forces result from wind, current, tidal action and wave action. Because of
these environmental forces, a moored ship will undergo heel, trim and
immersion displacements. Due to these displacements, the ship will be
subjected to restoring forces because of changes in the distribution of
buoyancy. Hence, the selection and location of equipment for the fixed
mooring of a ship presents a major task in the preliminary design stage.
Moreover, the available literature indicates an overall behaviour of the
moored vessels at berth for a specific environmental condition. 1"6 In
addition, in these references, no parametric studies are available for
studying the relative influence of various parameters of berthed ships.
Hence, the present experimental and theoretical investigations will enable
the designer to evolve efficient mooring systems.
2 F O R M U L A T I O N OF THE M O O R I N G PROBLEM
The mooring arrangement is usually composed of bow lines, spring lines,
breast lines and stern lines, all of which may have different angular
orientations as shown in Fig. 1. The cumulative elastic behaviour of
moorings poses a complicated non-linear problem since the mooring lines
are relatively short and hang in the air as non-coplanar catenaries, while
484
R. Natarajan, C. Ganapathy
'L
V
L_----. . . . . . . . . . .
~nnr~
r
"
+"
"
"
"
"'"
"+ ++++"
Fig. I. Vessel moored to a pier and its excursion.
the ship in shallow water may be subjected to considerable change in
elevation from alteration of the tide level, variable cargo load and parasitic motions of surge, sway, heave, roll, pitch and yaw. The size and
number of lines will depend on the external forces of wind, current, tide
and loading or unloading of the cargo.
The wind forces on the exposed surfaces of the ship, particularly in light
draught conditions, cause large mooring loads. The ratio of the wind
velocities at two different heights above the surface varies as approximately the seventh root of the heights. 5 For design purposes, the prediction of wind loads is based on uniform wind speed measured at least 10 m
above the water surface. 3 A moored vessel in a current stream may
experience large current forces and yawing moments. In most cases, an
adequate mooring arrangement against wave action is usually not a
problem because the vessel is moored in a sheltered location. The forces
due to various loading and unloading operations of large vessels or the
forces due to tide transmitted to a mooring system can reach large values
acting vertically upon the vessel.
To analyse a mooring system as a six-degrees-of-freedom system, the
hydrostatic restoring forces must be considered. A moored vessel undergoing heel, trim, and immersion displacements will be subjected to restoring forces because of changes in the distribution of 0 buoyancy. Under
mooring conditions, the hydrostatic restoring forces can be assumed to
vary linearly with the vessel movement. 4
When the vessel rotates about its longitudinal x-axis through an angle
of 0x from its position of equilibrium, as shown in Figs 2 and 3, the
restoring heel moment will be
L rest ~ W ×
N
GMr x 0~
(1)
When the vessel rotates about its transverse y-axis through an angle, 0y,
from its position of equilibrium, the restoring trim moment resulting from
the angular deflection will be
t~ est=
m x a m L x Ov
(2)
485
Analysis of moorings of a berthed ship
~MWIND
f
,
t
Ill
1I
. L I H g
T
•
•
I
.
Fig. 2. External and buoyant restoring forces.
When the vessel is displaced downward along the vertical z-axis by a
distance Uz from its position of equilibrium, the immersion force becomes
Frest ~-
P
W X g × A., × Uz
(3)
Mooring lines which stretch under a load absorb mechanical energy,
which is equal to the work done by the external loads' on the moored
vessel. The actual length of a rope from mooring bitt to bollard is important. The elastic behaviour of mooring ropes under tension results in a
strongly non-linear relationship between stress and strain. The elastic
stiffness of the mooring line in its principal direction is given by
,4,
i
9z ~
~
Fig. 3. Mooring coordinates.
486
R. Natarajan, C. Ganapathy
Figure 3 presents a view of a ship with a set of stationary axes x, y and z
and coincident set of axes x', y' and z' fixed in the vessel origin; both sets
of axes through the centre of gravity of the vessel. The excursion of the
moored ship due to external forces is described by the movement of the
ship's axes x', y' and z' relative to the stationary x, y and z coordinate
system. The translation of the centre of gravity is as well as any point of
the ship is Ux, Uy and Uz in the x, y and z directions, respectively, and is
expressed by a vector
Uc =
u~
(5)
Uz
The rotational displacements of the ship are characterised by the angles of
rotation 0x, Oy and 0~ about the x, y and z axes and may be expressed as a
vector
Oc=
(Ox)
Oy
Oz
(6)
A layout of a typical mooring line i is shown in Fig. 3 where the origin of
the line coordinates is located at the mooring chock. The angles ~bx, q~y
and ~bz lying between the line axis and its coordinates define the direction
of the mooring line. The mooring chock restraining the line is displaced by
a distance.
ui=uc-GOc
(7)
where 6",.is a matrix representing the location of chock i
ci=
0
G
-~
0
Gj
-~
-C x
~
0
(8)
i
where C,=,Cy and C~ are point coordinates of the chock as shown in Fig. 3.
The stiffness of a mooring line oblique to the ship's axes may be
expressed as a matrix
I COS 0 x
Ki = ki | c o s Ox cos Oy
cos Oxc o s
0y
cos%
L cos Oxcos O~ COS 0y COS 0 z
cos
Oxc o s Ozl
cos0,cos
Oz
COS20z
where ki is the stiffness of an individual line along its axis.
J
i
(9)
Analysis of moorings of a berthed ship
487
The reactive force f,. of a mooring line has three components parallel to
the vessel axes x', y' and z' and is given in the form of a vector
f" = g i × U i
(10)
The reactive moment li of a mooring line i about the vessel coordinate axes
may be expressed as a vector
[i ~- Ci >(A"
(11)
If a mooring arrangement has n mooring lines connected between the
bollard and the vessel chocks at CI, C2 ... C,, the resultant reactive force
/~ acting through the centre of gravity of the vessel may be expressed in the
matrix form
~" = ~
(K, x Vc + Ai Oc)
(12)
i=1
where Ai = -K~ x C~ and a moment/S, about the centre of gravity of the
vessel is given by
/1
L= ~
( A ri × Uc + BiOc)
(13)
i=l
where B; = C; × A/and the superscript T signifies a transposed matrix.
Generally, a berthing facility consists of one or more fenders attached
to a pier. The fenders absorb the berthing forces and form a protection for
ship and berthing structure.
Let the coordinates of the point of the j t h fender which makes contact
with the ship be rxj, rrj and rzj- as shown in Fig. 3. The relationship
between the deflection of the fender (¢) and the reaction force in the
fender (Di) is given by
19] = f . ¢
(14)
where ]) is the stiffness of the j t h fender.
The deflection of the j t h fender amounts to
d1 = Uy + {rjx - Ux}Oz - {rjz - Uz} Ox
(15)
The corresponding fender reaction is found from eqn (5) and the resulting
forces and moments on the ship are
Njk = 0
k = 1,3,5
Nj2 = - n j
Nj4 = {ryz- Uz} Dj
Nj6 = { Vx - rjx } Dj
(16)
(17)
(18)
(19)
488
R. Natarajan, C. Ganapathy
3 SOLUTION OF M O O R I N G EQUATIONS
A set of static equilibrium equations is obtained by considering the equilibrium of a moored ship subjected to external force (/~ext) and elastic and
hydrostatic restoring forces (~'aast) and (Pr,st), respectively.
(20)
/~elast +/g'rest = /Text
The generalised force vector (/Z'ext) consists of forces and moments due to
wind, current, tide and loading or unloading cargo as shown in Fig. 2.
Hence
K x Ui = Q
(21)
where
O =
Fx wind + Fx current + Fx wave
Fy wind + Fy current + Fy wave
Fz tide + Fz load/unload + F~ wave
Mx wind + Mx current + Mx wave
My wind + My current + My wave
Mz wind + Mz current + Mz wave
(22)
Wind and current loads result mainly from velocity forces, which can be
calculated from the drag force equation using projected areas of the ship.
The total resultant wind force acting on the ship from any direction is
given by 2
Pa X C w (~) x V 2w (Awx x cos2o{ "-~ A wy x
Fwind (0{) = -~-
sin2a)
(23)
From the wind tunnel studies on a wide range of vessel types, it is recommended that for different wind directions, the wind force coefficient (Cw)
will vary from 0-6 to 1.4 in the lateral direction and from 0.4 to 1.2 in the
longitudinal direction.
Steady current forces may be evaluated from the following equation: 2
Fcurrent (]~) = ~
× Cc (~) × V,2 (Acx × cos2fl -k- Acy × sin2B)
(24)
Depending upon the hull form, appendages, and roughness of the vessel,
the current force coefficient (Co) ranges from 0.5 to 1-5 in the lateral
direction and from 0.1 to 1.0 in the longitudinal direction, based on the
direction of the current.
Analysis of moorings of a berthed ship
489
The total static wave load on the vessel can be computed from Ref. 2:
~2 (sinhxD - sinhx(D - d))
Fwave(#)=pwxgxCmxHx9
2xY
-
coshxd
(25)
in which
Do = (Ls - B) sin # + B
Usually, the added mass coefficient (Cm) varies from 0.13 to 0.25 in the
longitudinal direction and from 1.45 to 1.75 in the lateral direction,
according to the size of the vessels and angle of incident waves.
The displacements and rotations of the ship due to the above environmental forces are given by
Ui =
Uz
(26)
Ox
Oy
Oz
All the elements of the stiffness matrix K are defined in Table 1, i.e.
"all
a12
. . . . . .
a16
a21
a22
. . . . . .
a26
K=
(27)
a61
a62
......
a66
With the use of a computer, the equilibrium equations can be solved
for the unknown vessel translation Uc and rotation 0c from which the
elongations and forces on the mooring lines may be determined. A
computer p r o g r a m - - S H I P M O O R - - w a s developed for the analysis
and design of mooring systems of ships. The flow chart is shown in
Fig. 4.
4 M O D E L EXPERIMENTS A N D C O M P U T E D RESULTS
The computer p r o g r a m - - S H I P M O O R - - w a s validated by the model
experiments. The mooring experiments were conducted on a barge model
with the following particulars:
490
R. Natarajan, C. Ganapathy
Length o f model
Breadth
Depth
Draught
Displacement
Block coefficient
4-8750
0.7500
0-2175
0.1373
451.5000
0.8994
m
m
m
m
kg
TABLE 1
Elements of the Stiffness Matrix
all
=
Kxx
a12
~
a13
=
Kxy
Kxx
a14 = K x ~ G - K~yC,
a15 = Xx.C~ - K=C~
a~6 = K ~ C x - K ~ C ~
a21
~
a22
=
a23
~
a24
=
Kxy
Kyy
Kyz
KyzCy -- KyyCz
a25 = K~yC~ - Ky~C~
az6 = KyyCx - KxyCy
a31
=
a32
=
grz
Kyz
a33
: Kzz + Pwg A.,
a3. = K = C y - ryzC~
a35
a36
a4~
a.:
=
=
=
=
Kx~C~
K~Cx
Kx~C~
r~c~
-
K~Cx
r~c~,
KxyC~
K~y C~
043
a~
a45
046
asi
KzzCz 2 - KyzCz
+ K=Cf - 2K~C~C~ + WGMr
= K.~C~C~ + K ~ C ~ C ~ - K = C x C ~ - K ~ y C f l
= K~yCyC~ + KyzC~C, - K,,C~C~ - KxzCy 2
= K~C~ - K.~C,
= K.C,
a52 = K.yC~ - ~ z C z
a53 = K=C~ - K=C~
a54 = K=Cs, Cz - KxyC~ z + Ky~C~C, - K = C x C y
a55
KxxC~ 2 + K ~ C ~ 2 - 2K=C~C~ + W G M L
a56 = KxyCxC~ + Kx~CxCy - K x x G C ~ - K y z G 2
a61 = K ~ y C . - K . ~ C y
a62 = KyvC~ - KxyCy
a63 =
K,~C~ - K ~ G
a64 = K x y C y f z q- KyzCxCy - KyyCxCz - KxzC'y 2
a6s = KxyCxCz + KxzC~Cy - KxxCyCz - KyzCx 2
a66
:
gxxfy 2 + gyyfx 2 - 2gxyCxfy
Analysis of moorings of a berthed ship
Read A, GML, GMT , Aw ,n, ~w,g , '[Q}, N
1
For i = 1, n ; j =I,N
Read Cxi, Cy i , Czi
Bxi, Byi ,Bzi.rxj.ryj.rzj
i
Generate[K] of Mooring
I_
line with respect to ship's axes F
t
I
I
I Solve fo~{U}i~rom {Qr = ["] {U} i J
i
i
t
~
J Calculate '[f}i from {f}i=[K]i{U}i I
Yes
_~ Eliminate ith
-I mooring line
o
JCotculate{=} i
from{l~[C]i{,}iJ
L
I Calculate Global reactive forces
and moments about C.G.of the ship
Fig. 4. Flow chart for SHIP MOOR computer program.
491
492
R. Natarajan, C. Ganapathy
The model was constructed of wood coated with wax of 3-5 m m thickness. The bottom and side shells were stiffened by wooden frames. Nylon
ropes of 5 m m in diameter were used as mooring ropes. The model was
ballasted to the required displacement with concrete blocks. The complete
mooring arrangement is shown in Fig. 5.
The model was tested in a wave-current flume which has an overall
length of 30 m and a cross-section of 2 m × 1.5 m. The far end of the
flume is provided with a rubble stone beach to absorb incident waves
and to reduce reflection. At the other end, the wave generator is installed. The wave generator has a plunger-type wave maker driven by an
electric motor system. The maximum stroke and speed of the plunger are
46 cm and 0.56 rev/s, respectively. The wave height can be controlled by
adjusting the stroke of the plunger. An electrical control device is capable
of producing regular waves of different frequencies. Suitable pumping
devices are also provided to develop water current of velocity of 20 cm/s,
in the flume. Maximum wave length of 5 m waves can be produced in
the flume.
Fabricated steel frame structures were used for simulating harbour
basin conditions, for handling the model, fixing transducers, and for
connecting the equipments. A block diagram of instrumentation used
for recording data is shown in Fig. 6. This consists mainly of the
following:
(1) Wave p r o b e - - 2 Nos
(2) Wave m o n i t o r s - - 2 Nos
(3) Carrier frequency amplifier six c h a n n e l s - - 2 Nos
" O " - FORCE TRANSOUCER
,.
-woooE.
FE.OER
~ ) 6 ( ] ~ - " O O m , . G U.ES
'
~-BERTHING STRUCTURE
Fig. 5. Barge model.
Analysis of moorings of a berthed ship
,it
1
I ~
"
.
I........
'~l.~_.JcoNv[~rtl
493
I
I
.
Fig. 6. Instrumentation of the model.
(4) Apple II computer
(5) Printer
Two. resistance type wave probes were used for wave monitoring. They
were electrically connected to two wave monitors to amplify the signals.
To measure three motions of the model in the longitudinal vertical
plane, three linear variable differential transformers (LVDTs) were
connected through frictionless pulley-twine arrangements. Two LVDTs
to measure vertical movements of the model were installed at the bow
and stern. A third one was at the bow to measure the surge motion of
the model. Four force transducers were attached to the four mooring
lines of the barge model. All the LVDTs and force transducers were
connected electrically to six-channel carrier frequency amplifiers. Output
from amplifiers and wave monitors were fed to an Apple II computer
through an analogue digital converter. A 16-channel data acquisition
program package was used to collect the data. The gains of the amplifier for each channel and the sampling intervals were adjusted such that
the recorded data accommodate a sufficient number of waves and such
that the magnitudes of recorded responses had sufficient resolution.
All the measurements were recorded for the following different environrnental conditions:
(a) Waves of different frequencies for a wave height of 7.5 cm
(b) Current alone
(c) Combinations of (a) and (b)
The coordinates of the mooring arrangement of the model are shown in
Table 2. The experimental results were compared with the results obtained
494
R. Natarajan, C. Ganapathy
from the computer program and found to be in satisfactory agreement as
shown in the Table 3.
By varying environmental load conditions, the computer program SHIP
MOOR was further used to investigate characteristics of the mooring lines
and the behaviour of a ship with the following particulars:
TABLE 2
Coordinates of Mooring Arrangement of the Model
Line no
1
2
3
4
Chock
Bollard
C~i (m)
Cvi (m)
C~i (m)
xi (m)
yi (m)
2.4
1.0
-1.0
-2.4
0
0.375
0.375
0
0.117
0.117
0.117
0-117
3.15
-0.75
1.05
-3.45
0.42
0.10
0.10
0-42
Zi
(m)
0.0168
0.0168
0.0168
0.0168
TABLE 3
Experimental Results
Mooring particulars
Mooring line number a
®
1
®
H
I
®
H
I
®
H
I
H
Condition (a) - - Wave alone, direction head sea, wave height 7.5 cm, period 2 s
Mooring line
10.62 il.89
1 4 . 2 5 13.68
9.66 1 0 . 5 9 10.47 11.52
force (N)
Line tension (N)
12-26 1 2 - 9 1 1 4 . 7 5 1 4 - 1 6 1 0 - 0 0 1 1 . 2 1 1 4 . 8 0 15.10
Chock excursion in
0.052 0.049
0.058 0.056
0.056 0.059
0.042 0.046
x - direction (m)
Condition (b) - - Water current alone, Velocity 0.2 m/s
Mooring line
1.81
1.75
3-04
3-16
force (N)
Line tension (N)
2.10
2.02
3.16
3.28
Chock excursion in
0.038 0.045
0.042 0.028
x-direction (m)
Condition ( c ) - - Combined wave and water current
Mooring line
8.89 1 0 . 6 7 1 1 . 5 0 12.08
force (N)
Line tension (N)
10.26 1 1 . 8 5 1 1 . 9 1 13.45
Chock excursion in
0-047 0-042
0.052 0.050
x - direction (m)
2-54
2.41
1-66
1.16
2.63
0.026
2.49
0-032
2.32
0.034
2.66
0-031
7.07
6-79
9.47
10.03
9-80 1 1 . 2 7
0-058 0-038
"I, Results of model experiments; II, Results of computer program.
1 0 . 2 0 12.01
0-038 0.039
Analysis of moorings of a berthed ship
Length of the ship
Displacement
Longitudinal metacentric height
Transverse metacentric height
122.0
13413.0
133.8
0.8
495
m
tonnes
m
m
The coordinates of the mooring arrangement are given in Table 4. The
results of these case studies are illustrated in Figs 7-9.
5 DISCUSSION
F r o m the analysis of results, it is observed that the tension of the
mooring ropes increased non-linearly with the increased intensity of
current forces for a constant wind velocity, for the combined wind and
current forces acting on the ship in bow, quartering and beam directions as shown in Fig. 7. However, the magnitudes of the tension of
the :ropes varied without following any specific pattern. In the case of
ship motion, the displacements of the ship in surge and sway directions
varied non-linearly with stiffness of the mooring ropes as in Fig. 8.
The excursions were reduced with higher stiffness for all the load
conditions. The forces on the mooring lines and fender reactions
depended on the direction of wind and current acting on the berthed
ship. When the direction of wind and current, i.e. ~ = fl was 90 ° with
respect to axis of the ship, all the mooring lines were in tension and
TABLE 4
Coordinates of the Mooring Arrangement of the Ship
(a)/Vlooring Lines
Line no
Cxi (m)
Cyi (m)
Czi (m)
B.ri (m)
Byi (m)
B.~i (m)
1
2
3
4
5
6
-65.1
-54.9
-36-6
36.6
54.9
62.0
0-0
5.3
8.4
8.4
5.3
0.0
3.1
3.1
3.1
3-4
3.1
5-6
-75.0
-53.4
38.4
-38.4
53-4
72.0
9.4
9.4
9.4
9-4
9.4
9.4
1.2
1.2
1-2
1.2
1.2
1.2
(b) Fenders
~n~r
r.~j(m)
ry.j(m)
r:j(m)
A
B
-30.0
30-0
9.0
9.0
0,5
0-5
R. Natarajan, C.
496
Ganapathy
fender reactions were zero. When 0~ = fl -- 0, the tension in the mooring
lines and fender reactions varied non-linearly from 0 to 367 kN and
270-692 kN, respectively. If a = fl = 45 °, the reaction in one of the
fenders, i.e. fender B became zero and the tension in stern and aft
breast mooring ropes reduced considerably. All the fender reactions
and tension of the mooring ropes are presented in Table 5.
The behaviour of the ship and the characteristics of the mooring ropes
were, however, different in the case of combined wave and current forces
as presented in Fig. 9. The tension of the stern rope and forward spring
line decreased with the increasing current velocity to a certain value and
thereafter the tension increased. Also, the stiffness of the mooring ropes
SO0
500
Legend:
I
(~
Legend :
E :l,SxlO4kNIm2
Vw = 45 Knots
400
4O0
Moorn
i~g J
E
:1.5 x l 0 4 # N I m 2
o
E
®
o 30O
300
2OO
20O
"3
o~
1oo
lo0
0
0
0.0
0.50
1.00
Current
(a)
1.50
2.00
2.~
3.00
3.50
a =p
=
0.0
I
0.50
I
1.00
Current
velocity (Knots)
(b)
O*
I
1.SO
I
2,00
I
2.S0
I
3.00
3.50
velocity (Knots)
~ = p = 45 °
000
Legend:
vw : 45 Knots
E : 1.5xlO&kNI m 2
Mooring lines :(~) - ( ~
600
o
400
._~
~,
20C
0.0
I
I
0 . S O 1.00
I
1.50
Current
(c)
I
2.00
I
2.50
I
3.00
3~50
velocity ( K n o t s )
~=~=90
°
Fig. 7. Variation of tension of mooring ropes with current velocity at constant wind velocity. (a) ct = # = 0°; (b) ct = / ~ = 45°; (c) 0t = / ~ = 90 °.
Analysis of moorings of a berthed ship
497
TABLE 5
Tension of Mooring Lines and Fender Forces of the Berthed Ship under Steady Wind and
Current Forces
Line tension
(kN)
(Degrees) (Degrees) (Knots) (Knots)
0
0
45
45
45
45
913,
90
45
1.0
2-0
3.0
1.0
2.0
3.0
1.0
2.0
3-0
1
2
3
4
Fender force
(kN)
5
6
A
B
1 0 0 - 0 56.7 216.7 160.0 86.7 166.7 270.3 324.3
50-0 33.3 283.3 73.3 85-0 210.0 378-4 464.9
13.3 20.0 366.7
0 160-0 270.0 540.5 691-9
53.3 143-3 140.0 166.7 300.0 160.0 540.5 0
26.7 93-3 183.3 226.7 373-3 206.7 810.0
0
16.7 66.7 250.0 300.0 473.3 273-3 1264.9 0
150.0 236.8 115.0 60.0 273.3 165.0
0
0
196.0 298.5 150.0 90.0 352.7 225.0
0
0
255.0 384.2 220.0 150.0 478-9 320.0
0
0
and movements of the ship differed completely from that of the combined
wind and current forces. It may be due to the wave and current forces,
which acted orthogonally with increasing influence of wave forces acting
on the ship.
6 CONCLUSIONS
This method of determining the design requirements for mooring of
ships; restrained by elastic mooring lines and subjected to wind, current
and wave forces may aid the designer to make the proper selection and
location of the mooring equipments and mooring fittings on board
ship and as well as at berth. This method is suitable for all types of
mooring lines, particularly synthetic ones with highly elastic characteristics.. The same type of quasi-static analysis can also be extended to
offshore floating platforms with spread moorings. In these cases, the
catenary effects of mooring lines will also be considered, by suitably
discretising the mooring lines, while generating the stiffness of the
mooring lines.
But the only limitation to using this method, is that the floating
struc, tures should be moored in the sheltered location, where the influence of the waves are not severely felt. In the case of significant wave
load]ngs, the dynamic analysis should be carried out to solve the
mooring problems.
R. Natarajan, C. Ganapathy
498
ACKNOWLEDGEMENTS
The authors express their sincere thanks to the Institute authorities for the
facilities offered for the preparation of this paper.
xlO 3
xtO 3
to
10
Le
z
o
a
Le
:
:::,SKnots
O--O Vc : 0 . 5
b - - ~ VC = 1,0
VC = 1,5
e - - e Vc : 2 . 0
Vc : 2-5
6
[
:
0
Knot
K not
Knot
K not
K not
O-,-O VC = 0.5
Vc : 1 • 0
Vc = 1 • 5
e~e Vc=2.0
Vc = 2 . 5
6
Knot
Knot
K not
Knot
Knot
g
E
4
o
2
I
I
I
2
I
3
Surge
I
4
I
5
t
I
0
0.0
i
0.50
a : p
:
[b)
O*
x 1o 3
xlO 3
10
10
~--O V¢ =O.S
~ V c
=1.O
D-.-Q VC = 1 . 5
H
VC:2.O
~
Vc:2,S
¢
=
I
2.00
2.50
fl = &5 t
L
Knots
A
I
I. 50
Surge motion of ship ( m )
motion of ship ( m l
(0)
1.00
:
not
Knot
Knot
Knot
Knot
Knot
0~0
Vc = 0.5
VC = 1-0
C)~(3 Vc : 1 . 5
VC : 2. O
4k~& V ¢ : 2 . 5
Knot
Knot
Knot
Knot
Knot
4
0
0-0
I
0-50
I
1-OO
I
1.50
Sway
motion at s h i p ( m )
(c)
=
=
I
2-00
I
2-50
P = 45"
I
3.00
i
3.50
i
2
Sway
(d)
I
4
6
motion of ship ( m )
cr =
/9
= 90"
Fig. 8. Variation of ship motions with stiffness of mooring rope under wind and current.
( a ) ~t = fl = 0 ° ; ( b ) e = fl = 4 5 ° ; ( c ) ~t = fl = 4 5 ° ; ( d ) ct = fl = 9 0 ° .
Analysis of moorings of a berthedship
500
z
~
x103
10
/*
L.,:o:2.
4o0 .
E : 1.5xlO4kNIm2
Mooring lines= I -S /
499
t.:.;,2: -
/
300
g
~
8
~
6
O--O Vc : O. S K not
g
E
g 2oo
S
4
o
100
0
2
0
0.0 O.S0
0.0
1.O0 t.SO 2.00 Z .SO 3,00 3.50
Current velocity (Knots)
(el
B = 0"
I
O. 50
I
t
I
1.00
1.50
2.00
Surge motion of 1;hip (m)
(b)
/z--90*
/?=0"
I
2.50
3.00
p:90*
xlO 3
10
1 I ~
Legend:
H:2m
O~OVc : 0.5 Knot
~,-LsYc : 1.0 Knot
I~\\
I ~ ' ~ v c : 2 s KnOt
8
6
o
O,.-,gtYC : 2,0 Knot
I
2
I
I
I
~
6
8
Swoy motion of ship (m)
Ic)
Fig. 9.
f)=0*
10
/~ = 90"
Variation of tension mooring ropes and ship motions under wave and current.
(a) fl = 0°,/~ = 90°; (b) fl = 0°,/~ = 90°; (c) fl = 0°, # = 90°.
REFERENCES
1. Chernjawski, M., Mooring of surface vessels to piers. Marine Technology, 17
(1980) 1-7.
2. Gaythwaite, J. W., Design of Marine Facilities for Berthing, Mooring and
Repair of Vessels. Van Nostrand Reinhold, New York, 1990.
3. Hunley, W. H. & Lemley, N. W., Ship manoeuvering, navigation and motion
control. In Ship Design and Construction. (Ed. R. Taggart) SNAME, NY, 1980.
4. Lewis, E. V., The motion of ships in waves. Principles of Naval Architecture.
(Ed. John P. Comstock) SNAME, NY, 1967.
5. Saunders, H., Hydrodynamics in Ship Design. SNAME, NY, 1957.
6. Wilson, B. W., Elastic characteristics of moorings. In Topics in Ocean Engineering (Vol. 1) (Ed. Charles I. Brestschneider) Gulf Publishing Company,
Houston, TX, USA, 1969.