CHAPTER 9
INTACT STABILITY CRITERIA
Stability is one of the most important safety features of ships, and in particular of small ships
which tend to suffer from insufficient stability which could lead to capsizing the vessel and loss
of the crew. It is, therefore, essential to design a ship with adequate stability and to maintain it in
all conditions of loading during its operation.
9.1. History
Development of the concept of metacentric height apparently originated with Bouguer in 1746.
Derivation and calculation procedures for the righting lever curves was published by Atwood in
1796. Development of quasi dynamic stability and the concept of the energy balance method was
advanced by Moseley in 1850.
Several proposals for th use of a GM based stability criteria were offered in the late 1800s, and
proposals for criteria based on righting energy have existed since the early 1900s.
The major historical work on the stability of ships was by rahola in 1939. Rahola’s work
involved a detailed analysis of Baltic ship capsizings and included a proposal for a GZ based
criteria.
Wind heel GM requirements have been applied in the US since the 1940s and became a US
requirement for cargo ships in 1952.
Based on recommendations from the 1960 International Conference on the Safety of Life at Sea
(SOLAS 60), the IMCO sub-committee on Subdivision and Stability was formed in 1962. The
first international stability criterion, Resolution A.167, largely based on Rahola’s GZ criteria,
was adopted by the IMO in 1968 for ships under 100m.
The IMO assembly adopted Resolution A.562 in 1985. This resolution is an energy balance
criterion, but also includes a wind heel recommendation, and is to be used as a supplement to
A.167.
9.2. Types of Intact Stability Criteria
The types of stability criteria can be divided into the following groups






GM or initial stability
Wind heel
GZ or quasi dynamic stability
Energy balance
Wave adjusted stability
Dynamic motion stability
9.2.1. Criteria based on GM or Initial Stability
GM is based on the geometrical relationship the centre of gravity, G, and the centre of
buoyancy, B, of a floating body. The initial metacentric height or metacentre, M, is calculated
based on the centre of buoyancy, KB, and the moment of inertia of the ship’s waterplane, BM.
Then GM is the vertical distance between the centre of gravity and the metacentre and is linearly
1
related to the righting arm by the sine of the heel angle. Since the sine of a small angle is
approximately equal to the angle and since the metacentre is relatively stationary for small
angles of heel, GM can also be expressed as the initial slope of the righting arm curve. As such,
it provides a realistic approximation of the resistance of a vessel to small angles of heel in the
manner of a linear spring constant, M H  GM , where MH is the heeling moment,  is the
equilibrium heel angle, and  is the displacement.
GM is the most basic stability criterion and one of the earliest methods used to quantify a ship’s
stability. Specification of a minimum value of initial GM and freeboard sufficed as the sole
numerical stability criterion in use until 1940s. Many national and international regulations have
a minimum GM as one component of their requirements since a value of GM is easy to calculate,
its meaning is relatively simple to understand, and the effect of its magnitude on the heel
response is readily predictable. However, since the position of the metacentre varies as the ship
heels, GM is only a valid measure within small angles of the initial unheeled position. In most
cases, GM tells surprisingly little about the character of the vessel at other than very small angles
of heel.
GZ
Cargo vessel
Tug
Heel angle
Figure 9.1.
The use of GM based criterion on two different types of ships, shown in Figure 9.1, serves well
to illustrate the problem in choosing criteria to represent a type of rather than a form of vessel.
The much greater initial GM of the tug is misleading. The high value of initial GM does not
reflect the tug’s greatly shortened range of positive stability compared with a conventioanl cargo
vessel, or its low value of maximum righting arm and low value of the angle at which that
maximum occurs.
The principal strength of GM based criteria lies in their simplicity and their accuarcy in
predicting equilibrium heel angles when very specific heeling moment descriptions can be
provided. For this reason, GM has been widely used in the past in criteria that establish a
2
maximum heel angle under the influence of such forces as those due to lateral wind loading,
towline pull, lifting heavy weights over the side, passenger group movements, steady high-speed
turning, and others.
Despite these shortcomings, GM is still used as a useful stability criteria for special cases.
Typical examples of specific GM based stability criteria are give below
Passenger vessels : For passenger vessels, the US Coast Guard require the satisfaction of the
following minimum GM value
GM R 
Nb
24 tan 
where
GMR : required minimum metacentric height at any particular draught (ft)
N
: number of passenger ships
b
: distance from vessel’s centreline to geometrical centre of passenger deck area on one
side of centreline (ft)

: displacement, long tons

: angle of heel to deck edge or 14 degrees, whicever is less
Towing Forces : Towing related stability hazards have commonly been divided into two
categories; those relating to the tendency of a towing vessel to overturn itself under the influence
of the heeling couple created by the opposing towline pull and propeller forces, and those
relating to the tendency for the tow to veer off and create an unexpectedly large transverse
component of force on the tug with a large upsetting moment resulting. Typically, the tug
induced tripping situation, or self-tripping, has been considered to be the controlling one, based
on two reasons. First, the tow-induced mechanism is extremely difficult to quantify since it
involves a large number of unknown parameters such as towing speed, relative headings of the
tug and tow, towline length, relative sizes of the tug and tow, and the position of the towing bitt.
Secondly, the current trend toward increased power in towing vessels reduce the response time in
the operator’s margin for error and increases the available heeling force of propeller thrust
dramatically, without significantly altering the typical vessel’s ability to resist the resulting
upsetting moment. As a consequence, the self-tripping case has been more thoroughly examined,
and a fundamental analysis has generated some simple relationships.
The basic form of classical tug-induced tripping relationships is exemplified by the Argyriadis
formula:
GM R 
SHP  h
f
100
B
(ft )
where
SHP

f
B
h
: shaft horsepower per shaft
: displacement, long tons
: minimum freeboard along the length
: molded beam
: vertical distance from propeller shaft centreline at rudder to towing bitt
3
In this relationship it is assumed that the towline is directly athwartships and that the rudder is
put hard over with full power applied. The height h is used to represent the distance between the
centre of underwater resistance and the towing bit, and the maximum amount of heel is related to
the function of freeboard divided by beam, f/B. In order to account for multiple screws, the effect
of propeller diameter, and the effect of rudder area, Murphy modified the basic form as follows:
NSHP  D 
GM R 
f
76
B
where
N
D
S
2/3
Sh
(ft )
: number of screws
: propeller diameter
: fraction of propeller circle cylinder intercepted by rudder turned to 45 deg
The Murphy criterion assumes that with constant power an increase in the propeller diameter
will result in an increase in thrust and therefore an increase in the heeling moment, provided the
effect of the rudder area is assumed to remain constant. The US Coast Guard used the Murphy
criterion until 1971, when the factor of 76 in the denominator was changed to 38, resulting in a
twofold increase in the required GM.
The only tow-induced tripping criterion based on GM that was included in a survey of towing
criteria by Miller is a standard proposed for use in Norway which was experimentally derived by
towing a model with a block coefficient of 0.5 sideways at a speed of 4 knots. The standard
requires
GM R 
h1 
5f
d
2 ( m)
where
h1
d
f
: height of towing bitt above waterline
: draught
: minimum freeboard along length
In general, the major criticism of all the foregoing towing vessel criteria is that they are valid
only at angles of heel less than about 7 degrees. As long as these relationships are used to limit
the angle of heel to less than that at which downflooding might occur through hull openings, they
may be satisfactory. But when these expressions attempt to predict the response of a vessel near
the heel angle which will cause capsizing, their use is both arbitrary and inaccurate. A more
sound approach for larger angles of heel is based on balancing the heeling moment induced by
either the tug or tow with the available righting moment below a critical angle, such as either the
angle of downflooding or the threshold of capsize. This criterion is described in the following
sections.
Fishing Vessels : The IMO simplified criterion for fishing vessels under 30 m in length is
expressed as follows
4
2

f
B

f 

GM R  1.7388  2B 0.075  0.37  0.82   0.014  0.032 

B
D
L 
 B

where
B
f
D

L
(ft )
: beam at waterline
: minimum freeborad along length
: minimum depth
: length of superstructure
: waterline length
9.2.2. Criteria based on Wind Heel
Wind heel criteria are based on the principle of a heeling moment created by a pressure on the
lateral profile of a ship coupled with a drag force on the underwater hull. This heeling moment is
evaluated against the righting moment (either based on GM or the righting lever) to limit heel to
a specified angle.
An example of a wind heel criterion is the USCG wind heel criterion. Under this method, the
minimum required GMR is given by
GM R 
PAh
(feet )
 tan 
where
2
 L

P : is in the form x   BP  (tons/ft2) and represents pressure on the projected lateral surface
 14200 
of the vessel due to a steady beam winf. The value of x varies depending on the area of
vessel opeartion and has units of tons/ft2.
x = 0.005 for oceans, coastwise service, and for the Great lakes in winter
x=0.0033 for partially protected waters such as lakes, bays, sounds and for the Great lakes in
summer
x=0.0025 for protected waters such as rivers and harbours
LBP : Length between perpendiculars (ft)
A : Projected lateral area of vessel above waterline (ft2)
 : Displacement, long tons
h : Vertical distance from centroid of A to half-draught point (ft)
 : Angle to ½ freeboard or 14 deg (0.24 rad), whichever is less
The heeling moment being resisted is the couple created by the force due to beam wind pressure
distributed over the exposed lateral area of the vessel and the resisting drag force of the
submerged hull, assumed to be acting at the half-draught point. The limitation of maximum heel
as one-half the freeboard is intended to allow for wind gusts, wave action, and rudder forces. The
use of angles near the upper limit of 14 deg (0.24 rad) in this relationship may lead to
questionable results, however, since the nonlinearity of righting arms with heel angle is well
established by this point.
5
9.2.3. Criteria based on GZ or Quasi Dynamic Stability
The International Conference on Safety of Life at Sea, 1960, recognizing the importance of the
stability of ships, recommended that IMCO should undertake studies on intact stability of
passenger ships, cargo ships and fishing vessels, with a view to formulating such international
standards as may appear necessary.
In pursuance of the above recommendation, the Inter-Governmental Maritime Consultative
Organization established in 1960 the Sub-Committee on Sub-division and Stability which was
charged, among others, with the task of studying the intact stability of passenger ships and cargo
ships.
As a result of the Sub-Committee’s comprehensive studies on existing national requirements, on
results of analyses of intact stability casualty records and on stability calculations of ships which
have operated successfully, a Recommendation on Intact Stability for Passenger and Cargo Ships
under 100 metres in length was drawn up. This Recommendation was approved by the Maritime
Safety Committee in March 1968 and adopted by the Assembly of the Organization at its fourth
extraordinary session in November 1968.
The following criteria are recommended for passenger and cargo ships under 100 m in length:




The initial metacentric height GM0 should not be less than 0. 15m.
The area under the righting lever curve (GZ curve) should not be less than 0.055 metreradians up to  = 30° angle of heel and not less than 0.09 metre-radians up to  = 40° or the
angle of flooding f if this angle is less than 40°. Additionally, the area under the righting
lever curve (GZ curve) between the angles of heel of 30° and 40° or between 30° and f , if
this angle is less than 40°, should not be less than 0.003 metre-radians.
The righting lever GZ should be at least 0.20m. at an angle of heel equal to or greater than
30°
The maximum righting arm should occur at an angle of heel preferably exceeding 30° but not
less than 25°
GM
GZ
GZmax
A2
A1
0
15
30
40
57.3
Meyil açısı
6
The following additional criteria are recommended for passenger ships:


The angle of heel on account of crowding of passengers to one side should not exceed 10°.
The angle of heel on account of turning should not exceed 10° when calculated using the
following formula:
M R  0.0
V2 
d
 KG  
L 
2
where:
MR
V
L

d
KG
: heeling moment in metre-tons
: service speed in m./sec.,
: length of ship at waterline in m.,
: displacement in metric tons,
: mean draught in m. ,
: height of centre of gravity above keel in m.
The criteria fix minimum values, but no maximum values are recommended. It is advisable to
avoid excessive values, since these might lead to acceleration forces which could be prejudicial
to the ship, its complement, its equipment and to the safe carriage of the cargo.
For ships loaded with timber deck cargoes and provided that the cargo extends longitudinally
between superstructures transversely for the full beam of ship after due allowance for a rounded
gunwale not exceeding 4 per cent of the breadth of the ship and/or securing the supporting
uprights and which remains securely fixed at large angle of heel, an Administration may apply
the following criteria in substitution for criteria given above:



the area under the righting lever (GZ curve) should not be less than 0.08 metre-radians up to
 = 40°or the angle of flooding if this angle is less than 40. ‘’ is an angle of heel at which
openings in the hull, superstructures or deckhouses which cannot be closed weathertight
immerse. In applying this criterion, small openings through which progressive flooding
cannot take place need not be considered as open.
The maximum value of the righting lever (GZ) should be at least 0.25 m.
at all times during a voyage the metacentric height GM0 should be positive after correction
for the free surface effects of liquid in tanks, and, where appropriate, the absorption of water
by the deck cargo and/or ice accretion on the exposed surfaces. Additionally, in the
departure condition the metacentric height should be not less than 0.10 m.
9.2.4. Criteria based on Energy Balance
The concept of the energy balance method is that the restoring energy or area must be equal to or
greater than the capsizing energy. Integration of the righting arm curve is interpreted to represent
righting or heeling energy. An example of an energy balance criterion is the IMO resolution
A.562. This criterion is recommended for all ships over 75 m and was approved by IMO in 1985.
The vessel is assumed to heel to a static heel angle,  0 , under the action of a steady wind heeling
lever, Lw1. Resonant rolling of the vessel is assumed with an amplitude 1 about the equilibrium
position  0 . A gust wind heeling lever Lw2 is then applied. If the righting energy b exceeds the
capsizing energy a, the vessel meets the criterion.
7
The criterion also recommends that under the action of the steady wind heeling lever Lw1, the
angle of heel shall not exceed 16 degrees or 80 percent of the level of deck edge immersion,
whichever is less:
PAZ

 1.5L w1
L w1 
( m)
Lw2
( m)
where
P
A
Z

1
2
= 0.0514 (t/m2)
: projected lateral area of portion of ship and cargo above waterline (m2)
: vertical arm from centre of A to centre of underwater lateral area (m)
: displacement (t)
: roll angle
: angle of downflooding or 50 deg or c, whichever is less. c is the angle of second
intercept between wind heeling lever Lw2 and the GZ curve.
GZ
b
Lw2
Lw1
a
Meyil açısı
2
o
o
1
8
Exercise . Check whether the IMO stability criteria are satisfied or not for a ship with following
particulars:
LOA
LBP
B
D

KG
KB
BM
: 65.225 m
: 59.200 m
: 10.000 m
: 4.950 m
: 1529 t
: 3.691 m
: 1.905 m
: 2.400 m
0.
Heel (deg)
KZ (m)
10
0.736
20
1.442
30
2.100
40
2.663
1.
Heel (deg)
KZ (m)
10
0.746
20
1.452
30
2.110
40
2.673
2.
Heel (deg)
KZ (m)
10
0.756
20
1.462
30
2.120
40
2.683
3.
Heel (deg)
KZ (m)
10
0.766
20
1.472
30
2.130
40
2.693
4.
Heel (deg)
KZ (m)
10
0.776
20
1.482
30
2.140
40
2.713
5.
Heel (deg)
KZ (m)
10
0.786
20
1.492
30
2.150
40
2.723
6.
Heel (deg)
KZ (m)
10
0.796
20
1.502
30
2.160
40
2.733
7.
Heel (deg)
KZ (m)
10
0.806
20
1.512
30
2.170
40
2.743
8.
Heel (deg)
KZ (m)
10
0.816
20
1.522
30
2.180
40
2.753
9.
Heel (deg)
KZ (m)
10
0.826
20
1.532
30
2.190
40
2.763
9