PNA

The Principles of
Naval Architecture Series
Intact Stability
Colin S. Moore
J. Randolph Paulling, Editor
Published by
The Society of Naval Architects
and Marine Engineers
601Pavonia Avenue
Jersey City, New Jersey 07306

Copyright O 2010 by The Society of Naval Architects and Marine Engineers.
The opinions or assertions of the authors herein are not to be construed as of cia1or
re ecting the views of SNAME or any government agency.
It is understood and agreed that nothing expressed herein is intended or shall be construed
to give any person, rm, or corporation any right, remedy, or claim against SNAME or any of its
of cers or member.
Library of Congress Cataloging-in-PublicationData
Moore, Colin S.
Intact stability 1Colin S. Moore. -- 1st ed.
p. cm. -- (Principles of naval architecture)
Includes bibliographical references and index.
ISBN 978-0-939773-74-9
I. Stability of ships. I. Title.
VM159.M59 2010
623.8'171--dc22
2009043464
ISBN 978-0-939773-74-9
Printed in the United States of America
First Printing, 2010

This page has been reformatted by Knovel to provide easier navigation.
INDEX
Index Terms Links
A
Adequate stability, U.S. Navy criteria 47 47
Advanced marine vehicles 76
ACVs 78 79
catamarans 77
hydrofoil craft 77 78
planning hull 77
SES 79
small waterplane area hull 77
stability criteria, hazards 76
trimarans, pentamarans 77
WIG crafts 79
Air cushion vehicle (ACV) 78 79
Antiroll tanks, free-surface effect, free liquids 37 37
B
Beam effect 20 20
Beam winds 6 6 46
Bilge fining 21 21
Bulk carriers carrying grain 50
Bulk dry cargo, free-surface effect, free liquids 37
C
Cargo effect on stability, changes in weight 40
Catamarans 77
Center of buoyancy 2
Center of floatation 13
Center of gravity (CG)
changes in weight effect 38
height, stability 4 5
weight and 1
see also Displacement, CG location determinations

Index Terms Links
Changes in weight
cargo effect on stability 40
CG impact 38
displacement impact 38
initial heel compensation 39
large trim changes 39
liquid and stores consumption 40
stability impact 38
submarines 39
Critical roll axis 27
Cross curves of stability, statical stability curves 19 19
Crowding of personnel, U.S. Navy stability criteria 48 48
D
Dangerous phenomena combinations 45
Depth effect, statical stability curves 20 20
Displacement, buoyancy interaction
floating body equilibrium 2 3
longitudinal equilibrium 5 5
overturning moments 3 5
righting moments 3 5
stable equilibrium, floating body 3 4
submerged floating body 3 4 6
watertight rectangular body 3
see also Draft, trim, heel, and displacement calculations
Displacement, CG location
CG margins 10 10
classification systems 9
detailed estimates 9
sample loading computer display 11
sample summaries 9
variation with ship loading 10 11
weight margins 10 10
Diving ballast 67
Diving trim 68
Double hulls, free-surface effect, free liquids 35 36

Index Terms Links
Draft readings, inclining experiment 61
Draft, trim, heel, and displacement calculations
center of flotation 53
displacement and CG from drafts 55 55
draft after change in loading 56
drag 58
heel computation 59
hog and sag 58
moment to trim 1 cm 54
navigational drafts 57 57
reference planes 59
tons per centimeter immersion 54
trim 53
weight, CG and 54
Drafts, trim, inclining experiment 60
Drag 58
Dynamic stability assessments 53
Dynamic stability, rolling effect 24 25
E
Equilibrium concepts 1
Equilibrium polygon, submerged equilibrium 69 70 71
F
Fishing vessels, stability criteria 49
Fluid shift for wall-sided tank, free-surface effect, free liquids 30 30
Form changes, statical stability curves 21 21
Free surface in tanks, inclining experiment 60
Free-surface effect, free liquids
antiroll tanks 37 37
approximate vs. exact calculations 34 34 35
bulk dry cargo 37
double hulls 35 36
fluid shift for wall-sided tank 30 30
large angles 31 32
longitudinal subdivision 35 35
metacentric height 30
moment of inertia 33 33

Index Terms Links
Free-surface effect, free liquids (Cont.)
moment of transference 31 32
numerical example 34 34
relative filling level 31 32
relative free-surface effect 31 32
righting arm effect 31 32
tank fill 36 36
top and bottom effects 32 32
trim 34
two liquids 36
wing ballast tank 33 33
Fuel ballast tanks, submerged equilibrium 66
Full load departure condition, metacentric height 15 16
G
GM, GZ curves, stability criteria 41 43 44
Gravitational stability 1
Grounded ships
dry dock stability 74 74
stranded stability 75
Grounding effect 7 7
H
Heel
computation 59
forces, inclining experiment 61
heeling moment, statical stability curves 25 26
initial heel compensation, changes in weight effect 39
see also Draft, trim, heel, and displacement calculations;
Upsetting forces, heeling moments
High-speed turning, U.S. Navy stability criteria 48
Hog and sag 58
Hydrofoil craft 77 78
I
IMO Resolution A.167 43
IMO Resolution A.749 43

Index Terms Links
Impulsive moment response 25 25
Inclining experiment
accuracy 64
basic principles 59
draft readings 61
drafts, trim 60
forces affecting heel 61
free surface in tanks 60
inclination measurement 61 62
inclining in air 63 65
inclining weights selection 61
induced rolling, sallying 65
inventory 62
list 60
personnel aboard 61
personnel movement 62
plot of tangents 62
preparation for inclining 60
report 62 63
schedule 60
swinging weights 61
transfer of liquids 61
water density 61
weight movements 62
Inclining in air 63 65
Inclining weights selection 61
Induced rolling, sallying 65
Initial heel compensation, changes in weight effect 39
International Convention for the Safety of Life at Sea (SOLAS) 15
International Convention on Load Lines (ICLL) 15
International Marine Organization (IMO)
MSC Circular 1228 44
Resolution A.167 43
Resolution A.749 43
stability criteria 42 51

Index Terms Links
J
Jack-up platform, statical stability curves 28 28 29
L
Lead, solid ballast, submerged equilibrium 66
Lead, variable tankage adjustment, submerged equilibrium 70 71
Lifting weights effect, U.S. Navy stability criteria 47
Lightship, submerged equilibrium 66
Liquid and stores consumption, changes in weight effect 40
List, inclining experiment 60
Load to submerge 66
Loading conditions, metacentric height 15 16
Longitudinal equilibrium 5 5
Longitudinal metacentric height 12 13
Longitudinal stability, upsetting forces, heeling moments 8 8
Longitudinal subdivision, free-surface effect, free liquids 35 35
M
Main ballast tanks, submerged equilibrium 66
Maximum righting moment, statical stability curves 22 23
Merchant ship stability criteria 42
Metacentric height (GM)
applications 14
arrival conditions 15
center of floatation 13
free-surface effect, free liquids 30
full load departure condition 15 16
inclining experiment 60
loading computer software 17
loading conditions 15 16
longitudinal metacenter 12 13
metacenter, submerged submarines 14
minimum operating conditions 15
moment to heel 1 degree 14
moment to trim 1 degree 14
Navy ships 16
partial load departure conditions 15

Index Terms Links
Metacentric height (GM) (Cont.)
passenger ships 16
period of roll 14
Ship Status for Proposed Weight Changes 17 17
stability curves 16
statical stability curves 16
submerged submarines 14
suitable conditions 16
transverse metacenter 11 11 12
trim effects 14
Midcolumn draft, statical stability curves 27 27 28
Minimum operating conditions, metacentric height 15
Mobile offshore drilling units (MODUs) 50
Moment diagram, submerged equilibrium 65
Moment of inertia, free-surface effect, free liquids 33 33
Moment of transference, free-surface effect, free liquids 31 32
Moment to heel 1 degree, transverse metacentric height 14
Moment to trim 1 degree 1cm, longitudinal metacentric height 14
MSC Guidance to Masters, Circular 1228 44
N
Neutral equilibrium 1 3
Normal fuel-oil tanks, submerged equilibrium 66
O
Offshore structures, non-ship-shape vessels
statical stability curves 27 27 28 29
Offside weight, upsetting forces, heeling moments 6 6
Overturning moments, weight and buoyancy interaction 3 5
P
Parametric rolling motion 45
Passenger ships, metacentric height 16
Period of roll, metacentric height 14
Personnel aboard, inclining experiment 61 62
Planning hull 77
Pontoon-based offshore structure, statical stability curves 27 27

Index Terms Links
Potential energy surface, jack-up platform, statical stability
curves 28 29
Preparation for inclining, inclining experiment 60
R
Reference planes, draft, trim, heel, and displacement
calculations 59
Relative filling level, free-surface effect, free liquids 31 32
Relative free-surface effect, free-surface effect, free liquids 31 32
Reserve buoyancy, submerged equilibrium 67
Residual water, submerged equilibrium 67
Righting arm (GZ), statical stability curves 17 18
Righting arm curve computation, statical stability curves 18
Righting arm effect, free-surface effect, free liquids 31 32
Righting moments
statical stability curves 26 26
weight and buoyancy interaction 3 5
Roll axis, critical, statical stability curves 27
Rolling effect, dynamic stability, statical stability curves 24 25
S
Sallying, see Induced rolling, sallying
Ship main body appendages, buoyancy contribution 19
Ship Status for Proposed Weight Changes 17 17
Slope of GZ curve at origin, statical stability curves 22 22 23
Small waterplane area hull 77
Stability criteria
bulk carriers carrying grain 50
dangerous phenomena combinations 45
dynamic stability assessments 53
fishing vessels 49
GM, GZ curves 41 43 44
hazards 76
IMO 42 51
merchant ships 42
MODUs 50
parametric rolling motion 45
surf-riding, broaching-to 45

Index Terms Links
Stability criteria (Cont.)
synchronous rolling motion 45
topside icing 48
towboats 50
wave crest at midship 45
Stability criteria, U.S. Navy
adequate stability 47 47
beam winds, rolling 46
crowding of personnel 48 48
general 45
heeling arms 47
high-speed turning 48
lifting weights effect 47
wind heeling moment 46 46
wind pressure vs. height 47
wind velocities 46 46
Stability evaluation standards 41
Stability in depth, submerged equilibrium 71
Stability, drafts, and list improvement
changes in form 73
load adjustment 73
loading instructions 73
permanent ballast 73
weight removal 73
Stable equilibrium 1
floating body 2 4
submerged floating body 3 4 6
Statical stability curves
beam effect 20 20
bilge fining effect 21 21
cross curves of stability 19 19
depth effect 20 20
dynamic stability, rolling effect 24 25
form changes 21 21
heeling moment 25 26
impulsive moment response 25 25
jack-up platform 28 28 29
maximum righting moment 22 23

Index Terms Links
Statical stability curves (Cont.)
midcolumn draft 27 27 28
offshore structures, non-ship-shape vessels 27 27 28 29
pontoon-based offshore structure 27 27
potential energy surface, jack-up platform 28 29
righting arm curve 17 18
righting arm curve computation 18
righting moment 26 26
roll access, critical 27
rolling effect, dynamic stability 24 25
ship main body appendages, buoyancy contribution 19
significance 22 22 23 24
slope of curve at origin 22 22 23
static stability curves 18 18
transverse righting arms 17 17 18
tumble-home flare effect 21 21
typical stability curves, different ships 24
waves effect 21 22
work and energy determination 24 25
Submarines
stability criteria 50 51
submerged, metacentric height 14
Submerged equilibrium 8
definition 66
diving ballast 67
diving trim 68
equilibrium conditions 69
equilibrium polygon 69 70 71
fuel ballast tanks 66
lead, solid ballast 66
lead, variable tankage adjustment 70 71
lightship 66
load to submerge 66
main ballast tanks 66
maximum condition, surfaced 67
moment diagram 65

Index Terms Links
Submerged equilibrium (Cont.)
normal condition, surfaced 67
normal fuel-oil tanks 66
reserve buoyancy 67
residual water 67
stability in depth 71
submerged displacement 66
variable ballast 67
variable load 67
water seal, fuel ballast tanks 67
weight items 66 66 67
weight items relationship 66 67 67
Submerged floating body, weight and buoyancy interaction 3 4 6
Surf-riding, broaching-to 45
Surface effect ships (SES) 79
Suspended cargo or weight, effect on stability 38
Swinging weights, inclining experiment 61
Synchronous rolling motion 45
T
Tank fill level, free-surface effect, free liquids 36 36
Top and bottom effects, free-surface effect, free liquids 32 32
Topside icing, stability criteria 48
Towboats, stability criteria 50
Transfer of liquids, inclining experiment 61
Transverse metacenter, metacentric height 11 11 12
Transverse righting arms, statical stability curves 17 17 18
Transverse stability, upsetting forces, heeling moments 8 8
Trim
free-surface effect, free liquids 34
Trim dive 67
basic principles 72
calculations, report 72
conducting 72
Trimarans, pentamarans 77

Index Terms Links
Tumble-home and flare effects, statical stability curves 21 21
Turn effect, upsetting forces, heeling moments 7 7
Two liquids, free-surface effect, free liquids 36
U
Unstable equilibrium 1
Upsetting forces, heeling moments
beam wind 6 6
grounding effect 7 7
longitudinal stability 8 8
offside weight 6 6
transverse stability 8 8
turn effect 7 7
weight lifting over the side 6 6
V
Variable ballast, submerged equilibrium 67
Variable load, submerged equilibrium 67
W
Water density, inclining experiment 61
Water seal, fuel ballast tanks, submerged equilibrium 67
Watertight rectangular body, stability 3
Waves effect, statical stability curves 21 22
Waves effects, dynamic 45
Weight estimate 9
Weight items, submerged equilibrium 66 67 67
Weight lifting over the side, upsetting forces, heeling moments 6 6
Weight movements, inclining experiment 62
Wind heeling moment, U.S. Navy stability criteria 46 46
Wind pressure vs. height, U.S. Navy stability criteria 47
Wind velocities, U.S. Navy stability criteria 46 46
Wing ballast tank, free-surface effect, free liquids 33 33
Wing-in-ground (WIG) crafts 79
Work and energy determination, statical stability curves 24 25

Nomenclature
A
&I
AP
B
B
BI
BL-
BM
-
BML
b
C
CL
CB
CG
G c p
D
D
DWT
E
e
F
F
F P
FW
G
G,
GM
-
GML
-
GZ
9
9
H
h
I
IL
IT
i ,
i,.
K
-
K B
KG
KM-
KMI,
k
L
L
stands for area, generally
area of waterplane
after perpendicular
maximum molded breadth
center of buoyancy
etc., changed positions of center of buoyancy
molded baseline
transverse metacentric radius, or height of M
above B
longitudinal metacentric radius, or height of ML
above B
width of a compartment or tank
constant or coefficient
centerline; a vertical plane through centerline
block coefficient,VILBT
center of gravity
waterplane area coefficient,&/LB
molded depth
diameter, generally
deadweight
energy, generally
base of Naperian logarithms, 2.7183
force, generally
center of flotation (center of area of
waterplane)
forward perpendicular
fresh water
center of gravity of ship's mass
etc., changed positions of the center of gravity
transverse metacentric height, height of M
above G
longitudinal metacentric height, height of MI,
above G
righting arm; horizontal distance from G to Z
acceleration due to gravity
center of gravity of a component
head
depth of water or submergence
moment of inertia, generally
longitudinalmoment of inertia of waterplane
transverse moment of inertia of waterplane
longitudinalmoment of inertia of free surface in
a compartment or tank
transverse moment of inertia of free surface in
a compartment or tank
any point in a horizontal plane through the
baseline
height of B above the baseline
height of G above the baseline
height of M above the baseline
height of MLabove the baseline
radius of gyration
length, generally
length of ship
LBP
LPP
LOA
LwL
Lw
LCB
LCF
LCG
LWL
I
M
M
ML
MT
MTcm
MTI
m
m
rnL
0
ox
OY
OZ
P
P
P
Q
R
S
SW
T
T
Tw
TCB
TCG
TPcrn
TPI
t
t
v
Vk
v c
VCB
VCG
vcg
W
WL
WL1
v
v
W
length between perpendiculars
length between perpendiculars
length overall
length on designed load waterline
length of a wave, from crest to crest
longitudinalposition of center of buoyancy
longitudinalposition of center of flotation
longitudinalposition of center of gravity
load, or design,waterline
length of a compartment of tank
moment, generally
transverse metacenter
longitudinal metacenter
trimming moment
moment to trim 1em
moment to trim 1inch
mass, generally (W/g or w/g)
transverse metacenter of liquid in a tank or
compartment
longitudinal metacenter of liquid in a tank or
compartment
origin of coordinates
longitudinal axis of coordinates
transverse axis of coordinates
vertical axis of coordinates
(upward) force of keel blocks
pressure (force per unit area) in a fluid
probability,generally
fore and aft distance on a waterline
radius, generally
wetted surface of hull
salt water
draft
period, generally
period of a wave
transverse position of center of buoyancy
transverse position of center of gravity
tons per em immersion
tons per inch immersion
thickness, generally
time, generally
linear velocity in general, speed of the ship
speed of ship, knots
speed of a surface wave (celerity)
vertical position of center of buoyancy
vertical position of center of gravity
vertical position of g
weight of ship equal to the displacement (pgV)
of a ship floating in equilibrium
any waterline parallel to baseline
etc., changed position of WL
volume of an individualitem
linear velocity
weight of an individualitem

xvi NOMENCLATURE
x distance from origin along X-axis
Y distance from origin along Y-axis
x distance from origin along Z-axis
Z a point vertically over B, opposite G
A,, displacement mass = pV
A displacement force (buoyancy) = pgV
6 specificvolume, or indicating a small change
0 angle of pitch or of trim (about OY-axis)
P permeability
P density; mass per unit volume
4) angle of heel or roll (about OX-axis)
$ angle of yaw (about OZ-axis)
V displacement volume
cc) circular frequency,2r/T, radians

Preface
Intact Stability
During the twenty years that have elapsed since publication of the previous edition of this book, there have been
remarkable advances in the art, science and practice of the design and construction of ships and other floating
structures.In that edition,the increasing use of high speed computers was recognized and computational methods
were incorporated or acknowledged in the individual chapters rather than being presented in a separate chapter.
Today, the electronic computer is one of the most important tools in any engineering environment and the laptop
computer has taken the place of the ubiquitous slide rule of an earlier generation of engineers.
Advanced concepts and methods that were only being developed or introduced then are a part of common
engineering practice today. These include finite element analysis, computational fluid dynamics, random process
methods, numerical modeling of the hull form and components, with some or all of these merged into integrated
design and manufacturing systems. Collectively,these give the naval architect unprecedented power and flexibility
to explore innovation in concept and design of marine systems. In order to fully utilize these tools, the modern
naval architect must possess a sound knowledge of mathematics and the other fundamental sciences that form a
basic part of a modern engineering education.
In 1997,planning for the new edition of Principles of Naval Architecture was initiated by the SNAME publica-
tions manager who convened a meeting of a number of interested individuals including the editors of PNA and
the new edition of Ship Design and Construction on which work had already begun. At this meeting it was agreed
that PNA would present the basis for the modern practice of naval architecture and the focus would be principles
in preference to applications. The book should contain appropriate reference material but it was not a handbook
with extensive numerical tables and graphs. Neither was it to be an elementary or advanced textbook although it
was expected to be used as regular reading material in advanced undergraduate and elementary graduate courses. It
would contain the background and principles necessary to understand and to use intelligentlythe modern analytical,
numerical, experimental and computational tools availableto the naval architect and also the fundamentals needed
for the development of new tools. In essence, it would contain the material necessary to develop the understanding,
insight, intuition, experience andjudgment needed for the successful practice of the profession. Followingthis initial
meeting, a PNA Control Committee, consisting of individuals having the expertise deemed necessary to oversee and
guide the writing of the new edition of PNA, was appointed. This committee, after participating in the selection of
authors for the various chapters, has continued to contribute by critically reviewing the various component parts as
they are written.
In an effort of this magnitude, involvingcontributions from numerous widely separated authors, progress has not
been uniform and it became obvious before the halfway mark that some chapters would be completedbefore others.
In order to make the material availableto the profession in a timely manner it was decidedto publish each major sub-
division as a separate volume in the "Principlesof Naval Architecture Series"rather than treating each as a separate
chapter of a singlebook.
Although the United States committed in 1975to adopt SI units as the primary system of measurement the transi-
tion is not yet complete.In shipbuildingas well as other fields,we still find usage of three systems of units: English or
foot-pound-seconds,SIor meter-newton-seconds,and the meter-kilogram(force)-secondsystem commonin engineer-
ingwork on the European continent and most of the non-Englishspeaking worldprior to the adoption of the SIsystem.
In the present work, we have tried to adhere to SI units as the primary system but other units may be found particu-
larly in illustrations taken from other, older publications. The symbols and notation follow,in general, the standards
developed by the International Towing Tank Conference.
Severalchanges fromprevious editions of PNA may be attributed directly to the widespread use of electronic com-
putation for most of the standard and nonstandard naval architectural computations. Utilizing this capability,many
computations previously accomplishedby approximate mathematical, graphical or mechanical methods are now car-
ried out faster and more accurately by digital computer.Many of these computations are carried out within more com-
prehensive software systems that gather input from a common database and supplyresults, often in real time, to the
enduser orto other elements of the system.Thusthe hydrostatic and stability computations may be contained in a hull
form design and development program system, intact stability is often contained in a cargo loading analysis system,
damaged stability and other flooding effects are among the capabilities of salvageand damage control systems.

x PREFACE
In this new editionof PNA,theprinciples of intact stabilityin calm water are developedstartingfrominitial stability
at small angles of heel then proceeding to large angles. Various effects on the stability are discussed such as changes
in hull geometry, changes in weight distribution, suspended weights,partial support due to grounding or drydocking,
and free liquid surfaces in tanks or other internal spaces.The concept of dynamic stability is introduced startingfrom
the ship's response to an impulsive heeling moment.The effects of waves on resistance to capsize are discussed not-
ing that, in some cases,the wave effect may result in diminished stability and dangerous dynamic effects.
Stabilityrules and criteria such as those of the International Maritime Organization,the US Coast Guard,and other
regulatory bodies aswell asthe US Navy arepresented with discussion of their physicalbases and underlyingassump-
tions. The section includes a brief discussion of evolving dynamic and probabilistic stability criteria. Especial atten-
tion is given to the background and bases of the rules in order that the naval architect may more clearly understand
their scope,limitations and reliabilityin insuring vessel safety.
There are sections on the special stability problems of craft that differ in geometry or function from traditional
seagoing ships including multihulls, submarines and oil drilling and production platforms. The final section treats
the stability of high performance craft such as SWATH, planing boats, hydrofoils and others where dynamic as well
as static effects associated with the vessel's speed and manner of operation must be considered in order to insure
adequate stability.
J. RANDOLPHPAULLING
Editor

1
Elementary Principles
1.1 Gravitational Stability. A vessel must provide
adequate buoyancy to support itself and its contents or
working loads. It is equally important that the buoyancy
be provided in a way that will allow the vessel to float
in the proper attitude, or trim, and remain upright. This
involvesthe problems of gravitational stability and trim.
These issues will be discussed in detail in this chapter,
primarily with reference to static conditions in calm
water. Consideration will also be given to criteria for
judging the adequacy of a ship's stability subject to both
internal loading and external hazards.
It is important to recognize, however, that a ship or
offshore structure in its natural sea environment is sub-
ject to dynamic forces caused primarily by waves, wind,
and, to a lesser extent, the vessel's own propulsion sys-
tem and control surfaces. The specific response of the
vessel to waves is typically treated separately as a ship
motions analysis. Nevertheless, it is possible and advis-
able to consider some dynamic effects while dealing
with stability in idealized calm water, static conditions.
This enables the designer to evaluate the survivability
of the vessel at sea without performing direct motions
analyses and facilitates the development of stability
criteria. Evaluation of stability in this way will be ad-
dressed in Section 7.
Another external hazard affecting a ship's stability is
that of damage to the hull by collision, grounding, or
other accident that results in flooding of the hull. The
stabilityand trim of the damaged shipwill be considered
in Subdivision and Damage Stability (Tagg, 2010).
Finally, it is important to note that a floating struc-
ture may be inclined in any direction. Any inclination
may be considered as made up of an inclination in the
athwartship plane and an inclination in the longitudi-
nal plane. In ship calculations, the athwartship inclina-
tion, called heel or list, and the longitudinal inclination,
called trim, are usually dealt with separately. For float-
ing platforms and other structures that have length to
beam ratios of nearly 1.0,an off axis inclination is also
often critical, since the vessel is not clearly dominated
by either a heel or trim direction. This volume deals pri-
marily with athwartship or transverse stability and lon-
gitudinal stability of conventional ship-like bodies hav-
ing length dimensions considerably greater than their
width and depth dimensions. The stability problems of
bodies of unusual proportions, including off-axis stabil-
ity, are covered in Sections 4 and 7.
1.2 Concepts of Equilibrium. In general, a rigid body
is considered to be in a state of static equilibrium when
the resultants of all forces and moments acting on the
body are zero. In dealing with static floating body sta-
bility, we are interested in that state of equilibrium as-
sociated with the floating body upright and at rest in a
still liquid.In this ease, the resultant of all gravity forces
(weights) acting downward and the resultant of the
buoyancy forces acting upward on the body are of equal
magnitude and are applied in the same vertical line.
1.2.1 Stable Equilibrium. If a floating body, ini-
tially at equilibrium, is disturbed by an external mo-
ment, there will be a change in its angular attitude. If
upon removal of the external moment, the body tends to
return to its original position, it is said to have been in
stable equilibrium and to have positive stability.
1.2.2 Neutral Equilibrium. If, on the other hand,
a floating body that assumes a displaced inclination be-
cause of an external moment remains in that displaced
position when the external moment is removed, the
body is said to have been in neutral equilibrium and has
neutral stability. Afloating cylindrical homogeneous log
would be in neutral equilibrium in heel.
1.2.3 UnstableEquilibrium. If, for a floating body
displaced from its original angular attitude, the dis-
placement continues to increase in the same direction
after the moment is removed, it is said to have been in
unstable equilibrium and was initially unstable. Note
that there may be a situation in which the body is stable
with respect to "small"displacements and unstable with
respect to larger displacements from the equilibrium
position. This is a very common situation for a ship, and
we will consider cases of stability at small angles of heel
(initial stability) and at large angles separately.
1.3 Weight and Center of Gravity. This chapter deals
with the forces and moments acting on a ship afloat
in calm water. The forces consist primarily of grav-
ity forces (weights) and buoyancy forces. Therefore,
equations are usually developed using displacement,
A, weight, W, and component weights, w. In the "Eng-
lish" system, displacement, weights, and buoyant forces
are thus expressed in the familiar units of long tons (or
lb.). When using the International System of Units (SI),
the displacement or buoyancy force is still expressed
as A=pgV,but this is units of newtons which, for most
ships, will be an inconveniently large number. In order
to deal with numbers of more reasonable size, we may
express displacement in kilonewtons or meganewtons.
A non-SI force unit, the "metric ton force," or "tonnef,"
is defined as the force exerted by gravity on a mass of
1000 KG. If the weight or displacement is expressed in
tonnef, its numerical value is approximatelythe same as
the value in long tons, the unit traditionally used for ex-
pressing weights and displacement in ship work. Since
the shipping and shipbuilding industries have a long
history of using long tons and are familiar with the nu-
merical values of weights and forces in these units, the
tonnef (often written asjust tonne) has been and is still
commonly used for expressing weight and buoyancy.

2 INTACT STAB1llTY
With this convention, righting and heeling moments are
then expressed in units of metric ton-meters, t-m.
The total weight, or displacement, of a ship can be
determined from the draft marks and curves of form,
as discussed in Geometry of Ships (Letcher, 2009). The
position of the center of gravity (CG) may be either cal-
culated or determined experimentally. Both methods
are used when dealing with ships. The weight and CG of
a ship that has not yet been launched can be established
only by a weight estimate, which is a summation of the
estimated weights and moments of all the various items
that make up the ship. In principle, all of the compo-
nent parts that make up the ship could be weighed and
recorded during the construction process to arrive at
a finished weight and CG, but this is seldom done ex-
cept for a few special craft in which the weight and CG
are extremely critical. Weight estimating is discussed in
Section 2.
After the ship is afloat, the weight and CG can be ac-
curately established by an inclining experiment, as de-
scribed in detail in Section 9.
To calculate the position of the CG of any object, it
is assumed to be divided into a number of individual
components or particles, the weight and CG of each be-
ing known. The moment of each particle is calculated
by multiplying its weight by its distance from a refer-
ence plane, the weights and moments of all the particles
added, and the total moment divided by the total weight
of all particles, W. The result is the distance of the CG
from the reference plane. The location of the CG is com-
pletely determined when its distance from each of three
planes has been established. In ship calculations, the
three reference planes generally used are a horizontal
plane through the baseline for the vertical location of
the center of gravity (VCG), a vertical transverse plane
either through amidships or through the forward per-
pendicular for the longitudinal location (LCG), and a
vertical plane through the centerline for the transverse
position (TCG). (The TCG is usually very nearly in the
centerline plane and is often assumed to be in that
plane.)
1.4 Displacement and Center of Buoyancy. In Sec-
tion 1, it has been shown that the force of buoyancy is
equal to the weight of the displaced liquid and that the
resultant of this force acts vertically upward through a
point called the center of buoyancy, which is the CG of
the displaced liquid (centroid of the immersed volume).
Application of these principles to a ship, submarine, or
other floating structure makes it possible to evaluate
the effect of the hydrostatic pressure acting on the hull
and appendages by determining the volume of the ship
below the waterline and the centroid of this volume.
The submerged volume, when multiplied by the specific
weight of the water in which the ship floats is the weight
of displaced liquid and is called the displacement, de-
noted by the Greek symbol A.
1.5 Interaction of Weight and Buoyancy. The attitude
of a floating object is determined by the interaction of
the forces of weight and buoyancy.If no other forces are
acting, it will settle to such a waterline that the force of
buoyancy equals the weight, and it will rotate until two
conditions are satisfied:
1.The centers of buoyancy B and gravity Gare in the
same vertical line, as in Fig. l(a), and
2. Any slight clockwise rotation from this position,
as from WL to WILlin Fig. l(b), will cause the center
of buoyancy to move to the right, and the equal forces
of weight and buoyancy to generate a couple tending to
move the object back to float on WL (this is the condi-
tion of stable equilibrium).
For every object,with one exception as noted later, at
least one position must exist for which these conditions
are satisfied, since otherwise the object would continue
to rotate indefinitely. There may be several such posi-
tions of equilibrium. The CG may be either above or be-
low the center of buoyancy, but for stable equilibrium,
the shift of the center of buoyancy that results from a
small rotation must be such that a positive couple (in a
direction opposing the rotation) results.
An exception to the second condition exists when the
object is a body of revolution with its CG exactly on the
Fig. 1 Stable equilibrium of floating body

INTACT STAB1llTY
I
Fig. 2 Neutral equilibrium of
axis of revolution, as illustrated in Fig. 2. When such an
object is rotated to any angle, no moment is produced,
since the center of buoyancy is always directly below
the CG. It will remain at any angle at which it is placed
(this is a condition of neutral equilibrium).
A submerged object whose weight equals its buoy-
ancy that is not in contact with the seafloor or other ob-
jects can come to rest in only one position. It will rotate
until the CG is directly below the center of buoyancy. If
its CG coincides with its center of buoyancy, as in the
case of a homogeneous object, it would remain in any
position in which it is placed since in this case it is in
neutral equilibrium.
The difference in the action of floating and sub-
merged objects is explained by the fact that the center
of buoyancy of the submerged object is fixed relative to
the body, while the center of buoyancy of a floating ob-
ject will generally shift when the object is rotated as a
result of the change in shape of the immersed part of
the body.
As an example, consider a watertight body having a
rectangular section with dimensions and CG as illus-
trated in Fig. 3. Assume that it will float with half its
volume submerged, as in Fig. 4. It can come to rest in
either of two positions, (a) or (c), 180 degrees apart. In
either of these positions, the centers of buoyancy and
gravity are in the same vertical line. Also, as the body
is inclined from (a) to (b) or from (c) to (d), a moment
is developed which tends to rotate the body back to its
original position, and the same situation would exist if
it were inclined in the opposite direction.
1- 20 cm -4Fig. 3 Example of stability of watertight rectangular body.
floating body.
If the 20-emdimension were reduced with the CG still
on the centerline and 2.5 em below the top, a situation
would be reached where the center of buoyancy would
no longer move far enough to be to the right of the CG as
the body is inclined from (a) to (b). Then the body could
come to rest only in position (c).
As an illustration of a body in the submerged condi-
tion, assume that the weight of the body shown in Fig.
3 is increased so that the body is submerged, as in Fig.
5. In positions (a) and (c), the centers of buoyancy and
gravity are in the same vertical line. An inclination from
(a) in either direction would produce a moment tending
to rotate the body away from position (a), as illustrated
in Fig. 5(b).An inclinationfrom (c) would produce a mo-
ment tending to restore the body to position (c). There-
fore, the body can come to rest only in position (c).
Aship or submarine is designed to float in the upright
position. This fact permits the definition of two classes
of hydrostatic moments, illustrated in Fig. 6, as follows:
Righting moments: A righting moment exists at any
angle of inclination where the forces of weight and buoy-
ancy act to move the ship toward the upright position.
Overturning moments: An overturning moment
exists at any angle of inclination where the forces of
weight and buoyancy act to move the ship away from
the upright position.
The center of buoyancy of a ship or a surfaced sub-
marine moves with respect to the ship, as the ship is
inclined, in a manner that depends upon the shape of
the ship in the vicinity of the waterline. The center of
buoyancy of a submerged submarine, on the contrary,
does not move with respect to the ship,regardless of the
inclination or the shape of the hull, since it is station-
ary at the CG of the entire submerged volume. This con-
stitutes an important difference between floating and
submerged ships. The moment acting on a surface ship
can change from a righting moment to an overturning
moment, or vice versa, as the ship is inclined, but this
cannot occur on a submerged submarine unless there is
a shift of the ship's CG.
It can be seen from Fig. 6 that lowering of the CG
along the ship's centerline increases stability. When a
righting moment exists, lowering the CG along the cen-

INTACT STAB1llTY
A
(c) (d)
Fig. 4 Alternate conditions of stable equilibrium for floating body.
terline increases the separation of the forces of weight
and buoyancy and increases the righting moment. When
an overturning moment exists,sufficient lowering of the
CG along the centerline would change the moment to
a righting moment, changing the stability of the initial
upright equilibrium from unstable to stable.
In problems involvinglongitudinal stability of undam-
aged surface ships, we are concerned primarily with de-
Fig. 5 Singe condition of stable equilibrium for submerged floating body

INTACT STABlllTY
SURFACE
- SHIP -
POSITIVE
STABlLlTY
NEGATIVE
STABlLlTY
(a) RIGHTING MOMENT WHEN HEELED (b) OVERTURNING MOMENT WHEN HEELED
POSITIVE
STABlLlTY
SUBMERGED
SUBMARINE
NEGATIVE
STABlLlTY
(c) RIGHTING MOMENT WHEN HEELED (d) OVERTURNING MOMENT WHEN HEELED
Fig. 6 Effect of height of CG on stability.
termining the ship's draft and trim under the influence
of various upsetting moments, rather than evaluatingthe
possibility of the ship capsizingin the longitudinal direc-
tion. If the longitudinal centers of gravity and buoyancy
are not in the samevertical line,the ship will change trim
as discussed in Section 8 and will come to rest as illus-
trated in Fig. 7, with the centers of gravity and buoyancy
in the same vertical line. A small longitudinal inclination
will cause the center of buoyancy to move so far in a fore
and aft direction that the moment of weight and buoy-
ancy would be many times greater than that produced by
the same inclination in the transverse direction. The lon-
gitudinal shift in buoyancy creates such a large longitudi-
nal righting moment that longitudinal stability is usually
very great compared to transverse stability.
Thus, if the ship's CG were to rise along the center-
line, the ship would capsize transversely long before
there would be any danger of capsizing longitudinally.
However,a surface ship could, theoretically, be made to
founder by a downward external force applied toward
one end, at a point near the centerline, and at a height
near or below the center of buoyancy without capsizing.
It is unlikely,however,that an intact ship would encoun-
ter a force of the required magnitude.
Surface ships can, and do, founder after extensive
flooding as a result of damage at one end. The loss of
buoyancy at the damaged end causes the center of buoy-
ancy to move so far toward the opposite end of the ship
that subsequent submergence of the damaged end is not
Fig. 7 Longitudinal equilibrium.

6 INTACT STAB1llTY
adequate to move the center of buoyancy back to a posi-
tion in line with the CG, and the ship founders, or cap-
sizes longitudinally. The behavior of a partially flooded
ship is discussed in Tagg (2010).
In the case of a submerged submarine, the center of
buoyancy does not move as the submarine is inclined in
a fore-and-aft direction. Therefore, capsizing of an in-
tact submerged submarine in the longitudinal direction
is possible and would require very nearly the same mo-
ment as would be required to capsize it transversely. If
the CG of a submerged submarine were to rise to a posi-
tion above the center of buoyancy, the direction, longi-
tudinal or transverse, in which it would capsize would
depend upon the movement of liquids or loose objects
within the ship. The foregoing discussion of submerged
submarines does not take into account the stabilizing
effect of the bow and stern planes which have an impor-
tant effect on longitudinal stability while the ship is un-
derway with the planes producing hydrodynamic lift.
1.6 Upsetting Force. The magnitude of the upsetting
forces, or heeling moments, that may act on a ship deter-
mines the magnitude of moment that must be generated
by the forces of weight and buoyancy in order to prevent
capsizing or excessive heel.
External upsetting forces affecting transverse stabil-
ity may be caused by:
Beam winds, with or without rolling.
Lifting of heavy weights over the side.
High-speed turns.
Grounding.
Strain on mooring lines.
Towline pull of tugs.
Internal upsetting forces include:
Shifting of on-board weights athwartship.
Entrapped water on deck.
Section 7 discusses evaluation of stability with re-
gard to the upsetting forces listed above. The discussion
below is general in nature and illustrates the stability
principles involved when a ship is subjected to upsetting
forces.
When a ship is exposed to a beam wind, the wind
pressure acts on the portion of the ship above the water-
line, and the resistance of the water to the ship's lateral
motion exerts a force on the opposite side below the wa-
terline. The situation is illustrated in Fig. 8.Equilibrium
with respect to angle of heel will be reached when:
The ship is moving to leeward with a speed such that
the water resistance equals the wind pressure, and
The ship has heeled to an angle such that the moment
produced by the forces of weight and buoyancy equals
the moment developed by the wind pressure and the wa-
ter pressure.
As the ship heels from the vertical, the wind pres-
sure, water pressure, and their vertical separation re-
main substantially constant. The ship's weight is con-
PRESSURE /
CL
Fig. 8 Effect of a beam wind
stant and acts at a fixed point. The force of buoyancy
also is constant, but the point at which it acts varies
with the angle of heel. Equilibrium will be reached when
sufficient horizontal separation of the centers of grav-
ity and buoyancy has been produced to cause a balance
between heeling and righting moments.
When a weight is lifted over the side, as illustrated
in Fig. 9, the force exerted by the weight acts through
the outboard end of the boom, regardless of the angle
of heel or the height to which the load has been lifted.
Therefore, the weight of the sidelift may be considered
to be added to the ship at the end of the boom. If the
ship's CG is initially on the ship's centerline, as at G in
Fig. 9,the CG of the combined weight of the ship and the
sidelift will be located along the line GA and will move
to a final position, GI, when the load has been lifted
clear of the pier. Point GIwill be off the ship's centerline
and somewhat higher than G. The ship will heel until the
Fig. 9 Lifting a weight over the side

INTACT STAB1llTY 7
CL
Fig. 10 Effect of offside weight.
center of buoyancy has moved off the ship's centerline
to a position directly below point GI.
Movement of weights already aboard the ship, such
as passengers, liquids, or cargo,will cause the ship's CG
to move. If a weight is moved fromA to B in Fig. 10,the
ship's CG will move from G to GI in a direction parallel
to the direction of movement of the shifted weight. The
ship will heel until the center of buoyancy is directly be-
low point GI.
When a ship is executing a turn, the dynamic loads
from the control surfaces and external pressure accel-
erate the ship towards the center of the turn. In a static
evaluation, the resulting inertial force can be treated as
a centrifugal force acting horizontally through the ship's
CENTRIFUGAL FORCE ? G I
'o B,
CG. This force is balanced by a horizontal water pres-
sure on the side of the ship, as illustrated in Fig. ll(a).
Except for the point of application of the heeling force,
the situation is similar to that in which the ship is acted
upon by a beam wind, and the ship will heel until the
moment of the ship's weight and buoyancy equals that
of the centrifugal force and water pressure.
If a ship runs aground in such a manner that contact
with the seafloor occurs over a small area (point con-
tact), the seabottom offerslittle restraint to heeling,as il-
lustrated in Fig. ll(b), and the reaction between ship and
seafloor of the bottom may produce a heeling moment.
As the ship grounds, part of the energy due to its forward
motion may be absorbed in lifting the ship, in which case
a reaction,R, between the bottom and the ship would de-
velop. This reaction may be increased later as the tide
ebbs. Under these conditions, the force of buoyancy
would be lessthan the weight of the ship because the ship
would be supported by the combination of buoyancy and
the reaction at the point of contact. The ship would heel
until the moment of buoyancy about the point of contact
became equal to the moment of the ship's weight about
the same point, when (W - R) x a equals W x 6.
There are numerous other situations in which ex-
ternal forces can produce heel. A moored ship may be
heeled by the combination of strain on the mooring
lines and pressure produced by wind or current. Tow-
line strain may produce heeling moments in either the
towed or towing ship. In each ease, equilibrium would
be reached when the center of buoyancy has moved
to a point where heeling and righting moments are
balanced.
In any of the foregoing examples, it is quite possible
that equilibrium would not be reached before the ship
(a) EFFECT OF A TURN (b) EFFECT OF GROUNDING
Fig. 11 Effect of a turn and grounding.

8 INTACT STAB1llTY
capsized. It is also possible that equilibrium would not
be reached until the angle of heel became so large that
water would be shipped through topside openings, and
that the weight of this water, running to the low side of
the ship,would contribute to capsizing which otherwise
would not have occurred.
Upsetting forces act to incline a ship in the longitudi-
nal as well as the transverse direction. Since a surface
ship is much stiffer, however, in the longitudinal direc-
tion, many forces, such as wind pressure or towline
strain,would not have any significant effect in inclining
the ship longitudinally. Shifting of weights aboard in a
longitudinal direction can cause large changes in the
attitude of the ship because the weights can be moved
much farther than in the transverse direction. When
very heavy lifts are to be attempted, as in salvage work,
they are usually made over the bow or stern rather
than over the side, and large longitudinal inclinations
may be involved in these operations. Stranding at the
bow or stern can produce substantial changes in trim.
In each ease, the principles are the same as previously
discussed for transverse inclinations. When a weight is
shifted longitudinally or lifted over the bow or stern,the
CG of the ship will move, and the ship will trim until the
center of buoyancy is directly below the new position of
the CG. If a ship is grounded at the bow or stern, it will
assume an attitude such that the moments of weight and
buoyancy about the point of contact are equal.
In the case of a submerged submarine, the center of
buoyancy is fixed, and a given upsetting moment pro-
duces very nearly the same inclination in the longitudi-
nal direction as it does in the transverse direction (Fig.
12). The only difference, which is trivial, is because of
the effect of liquids aboard which may move to a differ-
ent extent in the two directions. A submerged subma-
rine, however, is comparatively free from large upset-
ting forces. Shifting of the CG as the result of weight
changes is carefully avoided. For example, when a tor-
pedo is fired, its weight is immediately replaced by an
equal weight of water at the same location.
1.7 Submerged Equilibrium. Before a submarine
is submerged, considerable effort has been expended,
both in design and operation, to ensure that:
The weight of the submarine, with its loads and bal-
last, will be very nearly equal to the weight of the water
it will displace when submerged.
The CG of these weights will be very nearly in the
same longitudinal position as the center of buoyancy of
the submerged submarine.
The CG of these weights will be lower than the center
of buoyancy of the submerged submarine.
These precautions produce favorable conditions that
are described, respectively, as neutral buoyancy, zero
trim, and positive stability. A submarine on the surface,
with weights adjusted so that the first two conditions
will be satisfied upon filling the main ballast tanks, is
said to be in diving trim.
I i ~ d = ~ ~ ~ ~ II
INITIAL POSITION
ZERO HEEL, UPSETTING NEW
NEUTRAL BUOYANCY MOMENT EQUILIBRIUM
a) TRANSVERSE STABILITY
INITIAL ZERO TRIM POSITION - NEUTRAL BUOYANCY
I-d---I
TRIMMING MOMENT - wd= WXG,
NEW EQUILIBRIUM POSITION
b) LONGITUDINAL STABILITY
Fig. 12 Effect of weight shift on the transverse and longitudinal stability of
a submerged submarine.
The effect of this situation is that the submarine, in-
sofar as transverse and longitudinal stability are con-
cerned, acts in the same manner as a pendulum. This
imaginary pendulum is supported at the center of buoy-
ancy, has a length equal to the separation of the ten-
ters of buoyancy and gravity, and a weight equal to the
weight of the submarine.

INTACT STAB1llTY 9
It is not practical to achieve an exact balance of It is also not necessary, since minor deviations can be
weight and buoyancy or to bring the CG precisely to the counteracted by the effect of the bow and stern planes
same longitudinal position as the center of buoyancy. when underway submerged.
Determining Vessel Weights and Center of Gravity
2.1 Weight and Location of Center of Gravity. It is im-
portant that the weight and the location of the CG be
estimated at an early stage in the design of a ship. The
weight and height of the CG are major factors in deter-
mining the adequacy of the ship's stability. The weight
and longitudinal position of the CG determine the drafts
at which the ship will float. The distance of the CG from
the ship's centerline plane determines whether the ship
will have an unacceptable list. It will be clear that this
calculation of weight and CG, although laborious and
tedious, is one of the most important steps in the suc-
cessful design of ships.
During the early stages of design, the weight and the
height of CG for the ship in light condition are estimated
by comparison with ships of similar type or from coef-
ficients derived from existing ships. At later stages of
design, detailed estimates of weights and CGs are re-
quired. It is often necessary to modify ship dimensions
or the distribution of weights to achieve the desired op-
timum combination of a ship's drafts,trim, and stability,
as well as to meet other design requirements such as
motions in waves and powering. Sample lightship, full
load, and ballast load conditions are shown in Table 1.
2.2 Detailed Estimates of Weights and Position of Cen-
ter of Gravity. The reader is referred to Chapter 12,
by W. Boze, of Ship Design and Construction (Lamb,
2003) for a detailed discussion of the methodology of
weight estimating for each design stage, starting with
concept design and ending with detail design.
Ordinarily in design, the horizontal plane of refer-
ence is taken through the molded baseline of the ship,
described in Letcher (2009). The height of the CG above
this base is referred to as KG and its position as VCG.
Sometimes, after a ship's completion, the reference
plane is taken through the bottom of the keel, which,
depending on the definition of the molded surface, may
be a few centimeters below the molded surface.
The plane of reference for the longitudinal position
of the CG may be the transverse plane at the midship
section, which is midway between the forward and af-
ter perpendiculars. In this case, the LCG is measured
forward or abaft the midship section. This practice in-
volves the possibility of inadvertently applyingthe mea-
surements aft instead of forward, or vice versa, and a
more desirable plane of reference is one through the af-
ter or forward perpendicular.
The plane of reference for the transverse position of
the CG is the vertical centerline plane of the ship, the
transverse position of the CG being measured to port or
starboard of this plane.
In weight estimates, it is essential that an orderly
and systematic classification of weights be followed.
Two such classifications are in general use in this
country: Classification of Merchant Ship Weights by
the U.S. Maritime Administration (MARAD, 1995),and
Expanded Ship Work Breakdown Structure (ESWBS)
by the U.S. Navy (NAVSEA, 1985). The MARAD system
uses three broad classifications of hull (steel, outfit, and
machinery) each further subdivided into 10 subgroups.
The ESWBS uses nine major classifications reflecting
the mission requirements of military vessels. Further
recommendations on weight control techniques can be
found in the Recommended Practice No. 12 produced
by the International Society of Allied Weight Engineers
Table 1 Sample summaries of loading condition weights and centers.
Post-PanamaxContainership Aframax Tanker 132,000 m3 LNG (Membrane Type) Handymax Bulk Carrier
Carrier
Mass* Displace- VCG' LCG** Mass Displace- VCG LCG Mass Displace- VCG LCG Mass Displace- VCG LCG
ment' ment ment ment
Lightship 24,510 240,223 61% -7% 19,004 186,258 53% -5% 28,017 274,595 75% -5% 7289 71,439 73% -7%
Full Load 76,318 747,993 71% -3% 129,032 1,264,643 57% 3% 99,899 979,110 73% 0% 35,453 347,475 60% 2%
Ballast 49,275 482,944 45% -4% 62,070 608,348 39% 1% 75,561 740,573 64% 1% 25,944 254,277 56% 2%
LBP (m) 262 239 275 160
Depth 24.3 21.0 20.1 13.6
(m)
*In tonnes.
'In Kilo-Newtons.
:Percent of depth.
**Percent of LBP fonvard (+) or aft (-)of midships.

INTACT STAB1llTY 9
It is not practical to achieve an exact balance of It is also not necessary, since minor deviations can be
weight and buoyancy or to bring the CG precisely to the counteracted by the effect of the bow and stern planes
same longitudinal position as the center of buoyancy. when underway submerged.
Determining Vessel Weights and Center of Gravity
2.1 Weight and Location of Center of Gravity. It is im-
portant that the weight and the location of the CG be
estimated at an early stage in the design of a ship. The
weight and height of the CG are major factors in deter-
mining the adequacy of the ship's stability. The weight
and longitudinal position of the CG determine the drafts
at which the ship will float. The distance of the CG from
the ship's centerline plane determines whether the ship
will have an unacceptable list. It will be clear that this
calculation of weight and CG, although laborious and
tedious, is one of the most important steps in the suc-
cessful design of ships.
During the early stages of design, the weight and the
height of CG for the ship in light condition are estimated
by comparison with ships of similar type or from coef-
ficients derived from existing ships. At later stages of
design, detailed estimates of weights and CGs are re-
quired. It is often necessary to modify ship dimensions
or the distribution of weights to achieve the desired op-
timum combination of a ship's drafts,trim, and stability,
as well as to meet other design requirements such as
motions in waves and powering. Sample lightship, full
load, and ballast load conditions are shown in Table 1.
2.2 Detailed Estimates of Weights and Position of Cen-
ter of Gravity. The reader is referred to Chapter 12,
by W. Boze, of Ship Design and Construction (Lamb,
2003) for a detailed discussion of the methodology of
weight estimating for each design stage, starting with
concept design and ending with detail design.
Ordinarily in design, the horizontal plane of refer-
ence is taken through the molded baseline of the ship,
described in Letcher (2009). The height of the CG above
this base is referred to as KG and its position as VCG.
Sometimes, after a ship's completion, the reference
plane is taken through the bottom of the keel, which,
depending on the definition of the molded surface, may
be a few centimeters below the molded surface.
The plane of reference for the longitudinal position
of the CG may be the transverse plane at the midship
section, which is midway between the forward and af-
ter perpendiculars. In this case, the LCG is measured
forward or abaft the midship section. This practice in-
volvesthe possibility of inadvertently applyingthe mea-
surements aft instead of forward, or vice versa, and a
more desirable plane of reference is one through the af-
ter or forward perpendicular.
The plane of reference for the transverse position of
the CG is the vertical centerline plane of the ship, the
transverse position of the CG being measured to port or
starboard of this plane.
In weight estimates, it is essential that an orderly
and systematic classification of weights be followed.
Two such classifications are in general use in this
country: Classification of Merchant Ship Weights by
the U.S. Maritime Administration (MARAD, 1995),and
Expanded Ship Work Breakdown Structure (ESWBS)
by the U.S. Navy (NAVSEA, 1985). The MARAD system
uses three broad classifications of hull (steel, outfit, and
machinery) each further subdivided into 10 subgroups.
The ESWBS uses nine major classifications reflecting
the mission requirements of military vessels. Further
recommendations on weight control techniques can be
found in the Recommended Practice No. 12 produced
by the International Society of Allied Weight Engineers
Table 1 Sample summaries of loading condition weights and centers.
Post-PanamaxContainership Aframax Tanker 132,000 m3 LNG (Membrane Type) Handymax Bulk Carrier
Carrier
Mass* Displace- VCG' LCG** Mass Displace- VCG LCG Mass Displace- VCG LCG Mass Displace- VCG LCG
ment' ment ment ment
Lightship 24,510 240,223 61% -7% 19,004 186,258 53% -5% 28,017 274,595 75% -5% 7289 71,439 73% -7%
Full Load 76,318 747,993 71% -3% 129,032 1,264,643 57% 3% 99,899 979,110 73% 0% 35,453 347,475 60% 2%
Ballast 49,275 482,944 45% -4% 62,070 608,348 39% 1% 75,561 740,573 64% 1% 25,944 254,277 56% 2%
LBP (m) 262 239 275 160
Depth 24.3 21.0 20.1 13.6
(m)
*In tonnes.
'In Kilo-Newtons.
:Percent of depth.
**Percent of LBP fonvard (+) or aft (-)of midships.

INTACT STAB1llTY
(ISAWE, 1997). Some design offices may use systems
differing in detail from either of these, but the general
classification will be similar.
2.3 Weight and Center of Gravity Margins. The
weight estimate will of necessity contain many approxi-
mations and, it may be presumed, some errors. The er-
rors will generally be errors of omission. The steel as
received from the mills is usually heavier, within the
mill tolerance, than the ordered nominal weight. It is
impossible, in the design stages,to calculate in accurate
detail the weight of many groups such as piping,wiring,
auxiliary machinery, and many others.
For these and similar reasons, it is essentialthat mar-
gins for error be included in the weight estimate. The
amount of these margins is derived from the experience
of the estimator and varies with the accuracy and ex-
tent of the available information.
Table 2 is a composite of the usual practice of sev-
eral design offices. In each instance, the smaller values
apply to conventional ships that do not involve unusual
features and for which there is a reliable basis for the
estimate. If the estimate is reviewed by several inde-
pendent interested agencies, there is less chance of
substantial error and smaller margins are in order. The
Table 2 Weight margins.
Margin of Weight (in percent of lightship weight)
Cargo ships 1.5-2.5
Tankers 1.5-2.5
Cargo-passenger ships 2.0-3.0
Large passenger ships 2.5-3.5
Smallnaval vessels 6.0-7.0
Large navalvessels 3.5-7.0
Margin in VCG Meters
Cargo ships 0.15-0.23
Tankers 0.15
Cargo-passenger ships 0.15-0.23
Large passenger ships 0.23-0.30
Smallnaval vessels 0.15-0.23
Large navalvessels 0.15-0.23
larger values apply to vessels with unusual features or
in which there is considerable uncertainty as to the ulti-
mate development of the design.
The amount of margin will also depend on the seri-
ousness of misestimating weight or CG. For example,
until the advent of the double bottom for tankers, there
was no real need for any margin at all in the VCG of a
conventional tanker because such ships generally have
considerably more stability than is needed. On the other
hand, if there were a substantial penalty in the contract
for overweight or for a high VCG,a correspondinglysub-
stantial margin in the estimate would be indicated.
The above margins apply to estimates made in the
contract-designstage, where the calculations are based
primarily on a midship section, arrangement drawings,
and the specifications. In a final, detailed finished-
weight calculation, made mostly from working draw-
ings, a much smaller margin, of 1%or 2%, or even, if
extremely detailed information is available, no margin
at all may be appropriate.
Margins assigned to U.S. military ships (NAVSEA,
2001) are called acquisition margins and include Pre-
liminary and Contact Design Margins, Detail Design
and Build Margins, Contract Modifications Margin, and
Government Furnished Material Margin. The U.S. Navy
also includes Service Life Allowances that range from
5%to 10%for weight and 0.5to 2.5 ft (0.15 to 0.75 m) for
the VCG to allow for future modifications and additions
to the ship.
For more detailed information on margins and allow-
ances, the reader is referred to Chapter 12 in Ship De-
sign and Construction (Lamb, 2003).
2.4 Variation in Displacement and Position of Center of
Gravity With Loadingof Ship. Thetotal weight (displace-
ment) and position of the CG of any ship in service will
depend greatly on the amount and location of the dead-
weight items discussed in Letcher (2009): cargo, fuel,
fresh water, stores, etc. Hence, the position of the CG is
determined for various operating conditions of the ship,
the conditions dependingupon the class of ship (see Sec-
tion 3.8). These are usually calculated using an onboard
loading computer that has capabilities for tracking cargo
weight,ship stability,and strength (Fig. 13).

INTACT STAB1llTY
Fig. 13 Sample loading computer display.
Metacentric Height
3.1 The Transverse Metacenter and Transverse Meta-
centric Height. Consider a symmetric ship heeled to
a very small angle, 64, shown, with the angle exagger-
ated, in Fig. 14. The center of buoyancy has moved off
the ship's centerline as the result of the inclination,
and the lines along which the resultants of weight and
buoyancy act are separated by a distance,m,the right-
ing arm. In the limit 64 + 0, a vertical line through the
Fig. 14 Metacenter and righting arm
center of buoyancy will intersect the original vertical
through the center of buoyancy,which is normally in the
ships centerline plane at a point M, called the transverse
metacenter. The location of this point will vary with the
ship's displacement and trim, but, for any given drafts,it
will always be in the same place.
Unless there is an abrupt change in the shape of the
ship in the vicinity of the waterline, point M will remain
practically stationary with respect to the ship as the
ship is inclined to small angles, up to about 7 degrees.
As can be seen from Fig. 14,if the locations of G and
M are known, the righting arm for small angles of heel
can be calculated readily, with sufficient accuracy for
all practical purposes, by the formula
- -
GZ = GM sin 64 (1)
The distancem i s therefore important as an index of
transverse stability at small angles of heel, and is called
the transverse metacentric height. Since mis consid-
ered positive when the moment of weight and buoyancy
tends to rotate the ship toward the upright position, ?%
is positive when M is above G, and negative when M is
below G.
Metacentric Height (rmis often used as an index of
stability when preparation of stability curves forlarge an-

INTACT STAB1llTY
Fig. 15 Locating the transverse metacenter.
gles (Section4) has not been made. Its use is based on the
assumption that adequate in conjunction with ade-
quate freeboard, will assure that adequate righting mo-
ments will exist at both small and large angles of heel.
3.2 Location of the Transverse Metacenter. When a
symmetric ship is inclined to a small angle, as in Fig. 15,
the new waterline will intersect the originalwaterline at
the ship's centerline plane if the ship is wall-sided in the
vicinity of the waterline because the volumes of the two
wedges between the two waterlines will then be equal,
and there will be no change in displacement. If v is the
volume of each wedge, V the volume of displacement,
and the CGs of the wedges are at gl and g,, the ship's
center of buoyancy will move:
In a direction parallel to a line connecting g, and g,.
A distance,Dl,equal to (v .glg2)lV.
As the angle of heel approaches zero, the line 9291,
and therefore m,becomes perpendicular to the ship's
centerline. Also, any variation from wall-sidedness be-
comes negligible,and we may say
If y is the half-breadth of the waterline at any point of
the ship's length at a distance x from one end, and if
the ship's length is designated as L, the area of a sec-
tion through the wedges is i(y)(y tan 64) and its cen-
troid is at a distance of 2 x y from the centroid of the
corresponding section on the other side v x 'g1g2=
I,
S+(y)(ytan 64)(2 x y)dx or -
o tan64 3 ,
The right hand side of this expression, 4 y"dx, is
recognized as the moment of inertia of an area bounded
by a curve and a straight line with the straight line as the
axis. If we consider the straight line to be the ship's cen-
terline, then the moment of inertia of the entire water-
plane about the ship's centerline (both sides) designated
z Las IT,is 1, = -Sy%d?:= 9'and, therefore, when
3 0 tan 64
This theorem was derived by the French hydrographer
Pierre Bouger while on an expedition to Peru to mea-
sure a degree of the meridian near the equator. It ap-
peared in his Trait6 du Navire, published in Paris in
1746.It can be shown that BM is equal to the radius of
curvature of the locus of B as 64 -0.
The height of the transverse metacenter above the
keel, usually called is just the sum of m,or IT&
and m,the height of the center of buoyancy above the
keel. The height of the center of gravity above the keel,
is found from the weight estimate or inclining ex-
periment. Then,
- - -
GM = KM- KG
3.3 The Longitudinal Metacenter and Longitudinal
Metacentric Height. The longitudinal metacenter is
similar to the transverse metacenter except that it in-
volves longitudinal inclinations. Since ships are usually
not symmetrical forward and aft,the center of buoyancy
at various even keel waterlines does not always lie in a
fixed transverse plane but may move forward and aft
with changes in draft. For a given even keel waterline,
the longitudinal metacenter is defined as the intersec-
tion of a vertical line through the center of buoyancy in
the even keel attitude with a vertical line through the
new position of the center of buoyancy after the ship
has been inclined longitudinally through a small angle.
The longitudinal metacenter, like the transverse
metacenter, is substantially fixed with respect to the
ship for moderate angles of inclination if there is no
abrupt change in the shape of the ship in the vicinity
of the waterline, and its distance above the ship's CG,
or the longitudinal metacentric height, is an index of
the ship's resistance to changes in trim. For a normal
surface ship, the longitudinal metacenter is always far
above the CG, and the longitudinal metacentric height
is always positive.
3.4 Locationof the LongitudinalMetacenter. Locating
the longitudinal metacenter is similar to, but somewhat
more complicated than, locating the transverse meta-
center. Since the hull form is usually not symmetrical
in the fore-and-aft direction, the immersed wedge and
the emerged wedge usually do not have the same shape.
To maintain the same displacement,however,they must
have the same volume. Fig. 16shows a ship inclined lon-
gitudinally from an even keel waterline WL, through a
small angle, 84, to waterline WILl.Using the intersec-
tion of these two waterlines, point F,as the reference for
fore and aft distances, and letting:
L = length of waterplane
& = distance from Fto the forward end of waterplane
y = breadth of waterline WL at any distance x fromF

INTACT STAB1llTY
Fig. 16 Longitudinal metacenter
the volume of the forward wedge is
and the volume of the after wedge is
Equating the volumes
Q L-Q
t a n ~ ~ J , x y d x = t a n ~ O J , x z j d x
0 0
These expressions are, respectively, the moment of
the area of the waterplane forward of F and the mo-
ment of the area aft of F, both moments being about a
transverse line through point F. Since these moments
are equal and opposite, the moment of the entire wa-
terplane about a transverse axis through F is zero,
and therefore F lies on the transverse axis through
the centroid of the waterplane, called the center of
flotation.
In Fig. 16,AB is a transverse vertical plane through
the initial position of the center of buoyancy, B, when
the ship was floating on the even keel waterline, WL.
With longitudinal inclination, B will move parallel to
gig,, or as the inclination approaches zero, perpendicu-
lar to planeAB, to a point B,. The height of the metacen-
ter above B will be
The distance of g,, the centroid of the after wedge,
from Fis equal to the moment of the after wedge about
Fdivided by the volume of the wedge, and a similar for-
mula applies to the forward wedge. If the moments of
the after and forward wedges are designated as ml and
m,, respectively, then the distance
The moments of the volumes are obtained by inte-
grating, forward and aft, the product of the section area
at a distance x from Fand the distance x, or
rn,=SOQ(y)(stanSO)(s)ds = tan 68 f? yds
L-Q
rn, = tan SO J,,s"ds
The integrals in the expressions for ml and m2 are
recognized as giving the moment of inertia of an area
about the axis corresponding to x = 0, a transverse axis
through F,the centroid of the waterplane. Therefore, the
sum of the two integrals is the longitudinal moment of
inertia, I,, of the entire waterplane, about a transverse
axis through its centroid. Then
m, + m, = v -g,g,= 11,tan SO
or
v .QQ,IL=-
tan 68
In the limit when S+ + 0
where 11,is the moment of inertia of the entire water-
plane about a transverse axis through its centroid, or
center of flotation.

14 INTACT STAB1llTY
The height of the longitudinal metacenter above the
keel is given by an expression similar to equation (3)
by replacing the transverse metacentric radius by the
longitudinal metacentric radius.
- - -
KML = KBL +K B
and
GML= KML - KG
3.5 Metacenter for Submerged Submarines. When a
submarine is submerged,as noted in Section 1,the center
of buoyancy is stationarywith respect to the ship at any in-
clination. It follows that the vertical through the center of
buoyancy in the upright position will intersect the vertical
through the center of buoyancy in any inclined position at
the centerof buoyancy,andthe centerof buoyancyis,there-
fore,both the transverse and longitudinalmetacenter.
To look at the situation from a different viewpoint,
the =of a surfaced submarineis equal to m p l u s
or plus IIV. As the ship submerges, the waterplane
disappears, and the value of I,and hence is reduced
to zero. The value of becomes plus zero, and B
and M coincide.
The metacentric height of a submerged submarine is
usually denoted rather than
3.6 Effects of Trim on the Metacenter. The discussion
and formulas for and all assumed that the
waterline at each station was the same; namely, no trim
existed. In cases where substantial trim exists, values for--
BM,KM, andm w i l l be substantially different fromthose
calculated for the zero trim situation. It is important to
calculate metacentric values for trim for many ship types,
and tables forvarious trims are often included in trim and
stabilitybooks. The use of computers makes these tables
lessuseful asthe effectsof trim are includeddirectlyinthe
computation of the righting arm by maintaining longitu-
dinal moment equilibrium; thus,mis computed directly
when needed. Section 4.4 includes the effects of trim in
computing cross curves. Letcher (2009))in describingthe
calculationof also discusses the effects of trim.
3.7 Applications of Metacentric Height
3.7.1 Moment to Heel 1 Degree. A convenient and
frequentlyused concept is the moment to heel 1 degree.
This is the moment of the weight buoyancy couple, or
WWwhen the ship is heeled to 1degree, and is equiva-
lent to the moment of external forces required to pro-
duce a 1-degreeheel. For a small angle,the righting arm
is given by msin 4 and, after this is substituted for
we have:
Moment to heel 1degree = A?%? sin(1deg) (5)
Withinthe range of inclinations where themetacenter
is stationary, the change in the angle of heel produced
by a given external moment can be found by dividingthe
moment by the moment to heel 1degree.
3.7.2 Moment to Trim 1 Degree. The same theory
and formula apply to inclinations in the longitudinal di-
rection, and we may say:
Moment to trim 1degree = A m L sin(1 deg) (6)
where mLis the longitudinal metacentric height. We
are more interested, however, in the changes in draft
produced by a longitudinal moment than in the angle of
trim. The expression is converted to moment to trim 1
cm by substituting 1cm divided by the length of the ship
in centimeters for sin 1deg. The formula becomes, with
metric ton units,
AGM I,
MTcm=-t - m
l0OL
where L is ship length in meters. As a practical matter,
mLis usually so large compared to m t h a t only a negli-
gible error would be introduced if mI,were substituted
for GMI,.Then II,/V may be substituted f o r m , where
IL is the moment of inertia of the waterplane about a
transverse axis through its centroid, and A = pV, where p
is density. Then, moment to trim 1cm:
For fresh water, p = 1.0;for salt water, p = 1.025 (t/m3).
Since the value of this function is independent of the
position of G but depends only on the size and shape of
the waterplane, it is usually calculated together with the
displacement and other curves before the location of G
is known. Although approximate, this expression may
be used for calculations involving moderate trim with
satisfactory accuracy for ships of normal proportions.
3.7.3 Period of Roll. The period of roll in still wa-
ter, if not influenced by damping effects, is given by:
Period =
constant><k - CxB
E -JZ
where k isthe radius of gyration of the ship's mass about
a fore and aft axis through its CG.
The factor "constant x k" is often replaced by C x B,
where C is a constant obtained from observed data for
different types of ships.
This formula may be used to estimate the period of
roll when data for ships of the same type are available, if
it is assumed that the radius of gyration is the same per-
centage of the ship's beam in each case. For example, if
a ship with a beam of 15.24m and a mof 1.22m has a
period of roll of 10.5seconds,then
If another ship of the same type has a beam of 13.72m
and a of 1.52 m, the estimated period of roll would
be:

INTACT STAB1llTY
Thevariation of the value of Cfor shipsof different types
is not large; a reasonably close estimate can be made if
0.80is used for surface types and 0.67is used for subma-
rines. In almost all cases, values of C for conventional,
homogeneously loaded surface ships are between 0.72
and 0.91. This formula is useful also for estimating rn
when the period of roll has been observed.
A snappy, short period roll may be interpreted as in-
dicating that a ship has moderate to high stability, while
a sluggish, slow roll (long period) may be interpreted
as an indication of lesser stability, or that other factors
such as free surface or liquids in systems may be in-
fluencing the roll period. However, the external rolling
forces due to waves and wind and the effects of forward
speed through the water tend to distort the relationship
of T =- CB . Hence, caution must be exercised in cal-
E
culatingm v a l u e s from periods of roll observed at sea,
particularly for small and/or high-speed craft.
The case of the ore carrier is an interesting illustra-
tion of the effect of weight distribution on the radius of
gyration, and therefore on the value of C. The weight of
the ore, which is several times that of the lightship, is
concentrated fairly close to the CG, both vertically and
transversely. When the ship is in ballast, the ballast wa-
ter is carried in wing tanks at a considerable distance
outboard of the CG, and the radius of gyration is greater
than that for the loaded condition. This can result in
a variation in the value of C from 0.69 for a particular
ship in the loaded condition to 0.94 when the ship is in
ballast. For most ships, however, there is only a minor
change in the radius of gyration with the usual changes
in loading.
If no other information is available, the metacentric
height, in conjunction with freeboard, is a reasonably
good measure of a ship's initial stability, although it
must be used with judgment and caution. On ships with
ample freeboard, the moment required to heel the ship
to 20 degrees may be larger than 20 times the moment
to heel 1degree, but on ships with but little freeboard it
may be considerably less. Littleeffort may be required to
capsize a ship with large mbut with small freeboard.
When the metacentric height is zero or negative, certain
types of ship would capsize,while other types might de-
velop fairly large righting moments at the larger angles
of heel. The metacentric height may be used, however,
as an approximate index of stability for an undamaged
ship with reasonable confidence if the ship can be com-
pared to another with similar lines and freeboard for
which the stability characteristics are known.
3.8 Conditions of Loading. A ship's stability, and
hence may vary considerably during the course of
a voyage or from one voyage to the next, and it is nec-
essary during its design to determine which probable
condition of loading is the least favorable and will there-
fore govern the required stability. (The general effect of
variations in cargo and liquid load during a ship's op-
eration is further discussed in Section 6). It is custom-
ary to study, for each design, a number of loaded condi-
tions with various quantities, locations, and densities of
cargo and with various liquid loadings. When a ship is
completed, the builder usually provides such informa-
tion for the guidance of the operator in the form of a
trim and stability booklet. Typical booklets contain a
general arrangement of the ship, curves of form, capaci-
ties and centers, and calculations of and trim for a
number of representative conditionsand blank formsfor
calculating new conditions. The information contained
in such a booklet is required for all general cargo ships,
tankers, and passenger ships by international conven-
tions, including both the International Convention on
Load Lines and International Convention for the Safety
of Life at Sea (SOLAS) (IMO, 2006). Similar information
is furnished for naval ships and mobile offshore drill-
ing units where it is often referred to as the operating
manual. An onboard loading computer is allowed as a
supplement to the trim and stability booklet, but cannot
replace it. Type approval requirements for loading com-
puter software or systems vary internationally from
none to explicit version approval.
The range of loading conditions that a ship might ex-
perience varies with its type and the service in which it
is engaged. Typical conditions usually included in the
ship's trim and stability booklet are:
Full load departure condition,with full allowance of
cargo and variable loads. All the ship's spaces are filled
to normal capacity with load items intended to be car-
ried in these spaces, which usually implies minimum
density homogeneous cargoes, whether general, dry
bulk, liquid,or containerized. Atypical example is given
in Table 3.
Naval combatant ships do not carry cargo in the
usual sense. Instead, cargo equivalent variable load on
such ships would be ammunition or fuel for onboard air-
craft.
Additional conditions may be included for other
heavier cargo densities, involving partially filled or
empty holds or tanks. For ships that carry deck cargoes
such as container ships and timber carriers, conditions
with cargo on deck should be included, since they may
be critical for stability. Some ships may have minimum
draft requirements, which may include immersion of
propulsors or minimum draft forward to limit slamming
in heavy seas.
Partial load departure conditions, such as half car-
go or no cargo. When no cargo is carried, solid or liquid
ballast may be required, located so as to provide suffi-
cient draft and satisfactory trim and stability.
Arrival or minimum operating conditions. These
describethe ship after an extended period at sea and are
usually the lowest stability conditions consistent with
the liquid loading instructions (see Section 6.8).Certain
cargo ships might be engaged in point-to-point service,
while others might make many stops before returning

16 INTACT STAB1llTY
Table 3 Typical full load departure condition-post-Panamax con-
tainership.
Item Weight VCG (m) LCG TCG FSMom
(KN) (m-MS) (m-CL) (m-KN)
Lightship
Constant
Low Sulphur
Fuel Oil
Diesel Oil
Lube Oil
Fresh Water
SW Ballast
Misc.
In Hold
On Deck
Misc.Weights
Displacement 716,703 16.409 6.625A 0.157P 179,406
Stability Calculation Trim Calculation
KMt 16.409 m LCF draft 11.967 m
VCG 16.409 m LCB (even 6.025A m-MS
keel)
GMt (Solid) 3.05 m LCF 11.414A m-MS
FSc 0.25 m MT 1cm 11,470 m-KNIcm
GMt 2.799 m Trim 0.375 m-A
(Corrected)
GMt 0.50 m List 3P degrees
(Required)
GMt (Margin) 2.299 m Propeller 145 %
immersion
to home port. The amount of cargo and consumables
would vary, depending on the service. Conditions for
naval ships would reflect the most adverse distribution
of ammunition, along with reduced amounts of other
consumables.
In all of the above conditions of loading, it is neces-
sary to make appropriate allowances for the effects on
stability of the free surface of liquids in tanks, as ex-
plained in Section 5.
U.S. Coast Guard (USCG) stability requirements are
given in the Code of Federal Regulations (2006).
3.9 Suitable Metacentric Height. The stability of a
ship design, as evidenced approximatelyby its metacen-
tric height ( m ) , should meet at least the following re-
quirements in all conditions of loading anticipated:
It should be large enough in passenger ships to pre-
vent capsizing or an excessive list in case of flooding a
portion of the ship as a result of an accident. The effect
of flooding is described in Tagg (2010).
It should be large enough to prevent listingto unpleas-
ant or dangerous angles in case all passengers crowd to
one side. This may require considerable ?%?in light dis-
placement ships, such as excursion steamers carrying
large numbers of passengers.
It should be large enough to minimize the possibil-
ity of a serious list under pressure from strong beam
winds.
For passenger ships, the first bullet point is often the
controlling consideration.The International Convention
requirements for stability after damage, or other crite-
ria for sufficient stability, may result in a metacentric
height that is larger than that desirable from the stand-
point of rolling at sea. Since the period of roll in still
water varies inversely as the squareroot of the metacen-
tric height, larger metacentric heights produce shorter
periods of roll, resulting in greater acceleration forces
which can become objectionable. The period of roll may
also be a factor in determining the amplitude of roll,
since the amplitude tends to increase as the period of
roll approaches the period of encounter of the waves.
Of these two conflicting considerations, that of safety
outweighs the possibility of uncomfortable rolling, and
adequate stability for safety after damage must be pro-
vided for passenger ships and is desirable for cargo
ships. However, the metacentric height should not be
permitted to exceed that required for adequate stability
by more than a reasonable margin.
Numerous international and national maritime orga-
nizations have established stability criteria which cover
to some degree almost all types of ships, be they com-
mercial or military. These are discussed further in Sec-
tion 7.
Since the required stability will vary with displace-
ment, it is convenient to express the required stability as
a curve of required Wplotted against displacement or
draft. Actual values for various loading conditions
including corrections for free surface of liquids in tanks
(Section 5) are compared to the required m.Condi-
tions of loading that are unsatisfactory must avoided
by issuing loading instructions that will prevent a ship
from loading to an unsatisfactory stability condition.
Required ?%? curves must be used with caution since
analysis of the righting arm curve, which defines the
stability at large angles, is the only rigorous method of
evaluating adequacy of stability. The righting arm curve
takes into account freeboard, range of stability, and the
other features discussed in Section 4. Hence, stability
criteria are usually based on righting arm curves,rather
than on malone. Further recent positions taken by na-
tional authorities are increasingly requiring the direct
evaluation of stability for the specific loading condition
rather than a single criterion for a specific draft (see
Section 7).
Navy shipsmust meet allthe stability requirements of
commercial ships, including the ability to operate safely
in severe weather. In addition, they must have the ca-
pability of withstanding considerable underwater hull
damage as a result of weapons effects. For these rea-

INTACT STAB1llTY 17
sons, navy ships may have larger initial ?%?than similar
sized commercial ships.
An alternative approach is to make use of the "allow-
able KG" curve, derived from the righting arm curves,
which has the advantage that no stability calculations
are necessary tojudge the suitability of a loading condi-
tion. Thus, it is more amenable to implementation as a
criterion in load-planning software that does not have
access to the hull geometry information.
While loading computer software can rapidly evalu-
ate a potential loading condition against stability crite-
ria, a useful tabulation (NAVSEA,1975),can be prepared
for ships to permit a quick judgment as to whether a
proposed weight change will generally be acceptable
or unacceptable with regard to the limits on draft and
stability. The most useful part of this is the gauge on
sensitivity of the ship stability to weight changes. This
tabulation is titled Ship Status for Proposed Weight
Changes and takes on the followingformat:
1Ship 1Status 1Allowable KG for Governing
Loading Conditions
Status 1 means that the ship has adequate weight and
stability margins with respect to these limits. Thus, a
reasonable weight change at any height is generally ac-
ceptable.
Status 2 means that a ship is very close to both the
limiting drafts and the stability (E)limits. Thus, any
weight increase or rise in the CG is unacceptable.
Status 3 means that a ship is very close to the stabil-
ity limit but has adequate weight margin. If a weight ad-
dition is above the allowable m v a l u e and would thus
cause a rise in the ship's CG,the addition of solid ballast
low in the ship may be a reasonable form of compensa-
tion.
Status 4 means that adequate stability margin exists
but that the ship is operating at departure very close to
its limiting drafts. Tankers and beach landing ships usu-
ally fall into this category. A weight addition is at the
expense of cargo deadweight, or else may adversely af-
fect the ability of a landing ship to land at a designated
beach site.
To reduce any necessary compromise between the
requirements of a large amount of initial stability to
withstand underwater hull damage and the desire to
reduce ??%fto obtain more comfortable rolling char-
acteristics, many large ships have antirolling tanks
or fin stabilizers which operate to reduce roll ampli-
tude. Antiroll tanks operate on the principle of active
or passive shifting of liquids from side to side out of
phase with the ship's rolling. The liquids may cause a
free surface effect problem (discussed in Section 5)
which must be taken into account when evaluating a
ship's stability.
Curves of Stability
4.1 Righting Arm. To determine the moment of
weight and buoyancy tending to restore the ship to the A
upright position at large angles of heel, it is necessary to
know the transverse distance between the weight vec-
tor and the buoyancy vector. This distance is called the
righting arm and is usually referred to as
An illustration of the ship drawn with waterlines in-
clined at angles +,and +2 is shown in Fig. 18.The figure
shows the initial upright center of buoyancy B, and new 1
centers of buoyancy B, and B,, corresponding to +,and
+2, respectively. The corresponding righting arms are
then GZ, and GZ,, computed for reference point 0.The
reference point 0would normally be the CG.Sometimes,
+ l aI] cc' B1
righting arms are calculated for an assumed location of
the reference point 0 (usually taken at the keel), and in A ,
this case they are often referred to as righting arms for D
a poleheight of zero. Fig. 17Transverse righting arms.

INTACT STAB1llTY
5
Effect of Free Liquids
5.1 Free-Surface Effect. The motion of the liquid in
a tank that is partially full reduces a ship's stability be-
cause, as the ship is inclined, the CG of the liquid shifts
toward the low side. This causes the ship's CG to move
toward the low side, reducing the righting arm. Ships
with bulwarks that trap water on deck experience simi-
lar reductions in the righting arm.
Prior to the widespread use of computers, direct
evaluation of the free-surface effect on a normal ship
required an excessive amount of calculation. In many
cases, the tanks are not rectangular, and when formed
by the shell and a longitudinal bulkhead, they are not
symmetrical. With computer models of ships easily de-
veloped, direct calculation of the free-surface effect is
practical through computation of the fluid CG in the
heeled (or trimmed) condition. However, approximate
methods are useful when assessing the impact of free
surfaces prior to performing computer-based calcula-
tions or when they are not available as is the case early
in the design process. Furthermore, while loading com-
puter software can easily handle direct calculation of
the shift in CG, some regulations explicitly require du-
plication of approximate methods.
Consider a tank containing liquid in ship that is
heeled to a small angle 4,as shown in Fig. 40.
As a result of the heel, some of the liquid will flow
from the high side to the low side of the tank and a heel-
ing moment will exist equal to the weight of the shifted
liquid, w, multiplied by the lateral distance between the
original and final position of the CG of w. If the tank
is approximately wall-sided, the weight of liquid shifted
in an incremental length of tank is
the total weight, w, is given by the following expres-
sion.
where b(x) is the width of the tank free surface at longi-
tudinal position x.
The moment arm of this couple, b,, equals the lateral
separation of the centroids of the two triangular wedge-
shaped volumes representing the original and final loca-
Fig. 40 Fluid shift for a wall-sided tank
tions of w. In order for the triangular wedges to be of
equal volume, the initial and final free-surface planes
must intersect in a longitudinal line that passes through
the centroid of the free-surfacearea. The heeling couple
is then given by
Here, b(x) is the breadth of the tank free surface at loca-
tion x along the length; i ,is the moment of inertia of the
free-surface area of the tank about a longitudinal axis
through the centroid of that area.
This formulation points out that the mathematical
processes applying to the motion of the CG of the liquid
are similar to those applying to the motion of the center
of buoyancy of a ship. At small angles of inclination, the
liquid in each tank has a metacenter located at a dis-
tance equal to i,lv above its CG in the upright position,
where i, is the moment of inertia of the surface of the
liquid about an axis through its centroid and parallel
to the centerline, and v is the volume of the liquid. It is
common practice to develop a tank table that shows the
moment of inertia for varying quantities of liquid.
In evaluating the effect of free surface in a ship's
tanks, the usual practice is to assume the most unfavor-
able disposition of liquids likely to occur. If a tank is
empty or completely full, there is no effect. The maxi-
mum effect occurs when a tank is about half full. There-
fore, it is customary to assume that the largest tank in
each of the systems, or the largest pair of tanks if they
are in pairs, is half full. This assumption is made even
when a full-load condition is being studied, since a free
surface will developshortly after the ship leaves port. In
the fuel-oil system, the settling tanks also are assumed
to be half full. Fuel-oil tanks which are nominally full
of fuel are considered to be either about 95%fullin naval
practice or about 98%full in the ease of a merchant ship,
to allow for expansion and off-gassing of the oil, and
will therefore have some free-surfaceeffect. Aballasted
fuel-oil tank or a nominally full water tank should be
completely full and have no effect. If "split plant" op-
eration is practiced, which involves dividing the system
into two or more independent sections to enhance reli-
ability in the event of damage, the largest tank or pair of
tanks in each section is assumed to be half full.
5.2 Evaluation of Effect of Free Surface on Metacentric
Height. For ships on which the effect of free liquid is

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