This document provides definitions and explanations of key concepts related to a ship's transverse stability. It discusses heel and list, stability reference points like the metacenter, center of buoyancy, and center of gravity. It introduces the stability triangle and explains positive, neutral, and negative stability. Key terms are defined, such as displacement, draft, and the laws of buoyancy. Graphs demonstrate concepts like the righting arm curve and how stability changes with angle of heel. The roles of factors like GM, GZ, and the angle of loll in capsizing are also summarized.

1. Ship’s Transverse Stability

2. DEFINITIONS
1. Heel. A ship is said to be heeled when she is inclined by an
external force. For example, when the ship is inclined by
the action of the waves or wind.
2. List. A ship is said to be listed when she is inclined by forces
within the ship. For example, when the ship is inclined by
shifting a weight transversely within the ship. This is a fixed
angle of heel.

3. Introduction to Stability
CL
M
G
B
K
BM
KM
DISPLACEMENT
TONS
W
EIGHT
TON S
W
EIGHT
B
G

4. STABILITY
The tendency of a ship to rotate one way or the other (to right
itself or overturn)
INITIAL STABILITY
The stability of a ship in the range from 0° to 7°/10°
OVERALL STABILITY
A general measure of a ship's ability to resist capsizing
in a given condition of loading.
DYNAMIC STABILITY
The work done in heeling a ship to a given angle of
heel.

5. LAWS OF BUOYANCY
 A floating object has the property of
buoyancy
 A floating body displaces a volume of water
equal in weight to the weight of the body.

6. LAWS OF BUOYANCY
• A floating object has the property of buoyancy
• A floating body displaces a volume of water
equal in weight to the weight of the body.
• A body immersed (or floating) in water will be
buoyed up by a force equal to the weight of the
water displaced.

7. DISPLACEMENT
• The weight of the volume of water that the ship
hull is displacing
• Units of displacement is metric ton / ton.

8. DISPLACEMENT
00
G
DISPLACEMENT

9. DISPLACEMENT
04
G
B
DISPLACEMENT

10. DISPLACEMENT
09
G
B
DISPLACEMENT

11. DISPLACEMENT
16
G
B
DISPLACEMENT

12. DISPLACEMENT
20
G
B
DISPLACEMENT

13.  Weight :
- Gravitational force which direction towards the
centre of the earth.
Units : tons, pounds, etc
• Moment:
The tendency of a force to produce rotation
about an axis
Moment = F x d
a
d
F

14. 2. Stability Reference Points
 Metacentre
 Gravity
 Buoyancy
 Keel

15. STABILITY REFERENCE POINTS
CL
M
G
B
K
etacenter
ravity
uoyancy
eel

16. STABILITY REFERENCE POINTS
CL
otherM
ooseG
eatsB
idsK

17. B
WATERLINE
RESERVE BUOYANCY
B
THE CENTER OF BUOYANCY
B1

18. WATERLINE
B
RESERVE BUOYANCY
RESERVE BUOYANCY, FREEBOARD, DRAFT
AND DEPTH OF HULL
DRAFT
FREEBOARD
DEPTH

19. CENTER OF BUOYANCY
B
WLWL
B
WL
B
WL
B
WL
B

20. CENTER OF BUOYANCY
B
BB
B
BB
B
B
B

21. CENTER OF GRAVITY
• Point at which all weights could be concentrated.
• Center of gravity of a system of weights is found by taking moments
about an assumed center of gravity, moments are summed and divided
by the total weight of the system.

22. G
G1
KGo
KG1
THE CENTER OF GRAVITY
G
G1
KGo
KG1

23. MOVEMENTS IN THE CENTER OF GRAVITY
G moves towards a weight addition
 G moves away from a weight removal
G moves in the direction of a weight shift

24. MOVEMENTS IN THE CENTER OF GRAVITY
•G MOVES TOWARDS A WEIGHT ADDITION
G
G1
KGo
KG1

25. G
KGo
G1
KG1

26. MOVEMENTS IN THE CENTER OF GRAVITY
 G moves away from a weight removal
G
G1
KGo
KG1

27. G
G
G
G
G
G
G1
KG1
KGo
G

28. MOVEMENTS IN THE CENTER OF GRAVITY
 G moves in the direction of a weight shift
G G

29. G2
G

30. THE METACENTER
CL B
B20
B45
M
M20
M45
M70
B70
METACENTER
M
B B1 B2

31. METACENTER
BBBBBBBBB

32. METACENTER
B SHIFTS
M

33. MOVEMENTS OF THE
METACENTER
The metacenter will change positions in the vertical
plane when the ship's displacement changes.
the metacenter moves law these two rules:
1. When B moves up M moves down.
2. When B moves down M moves up.

34. M
G
B
M
G
B
G
M
B
M1
B1
G
M
B
M1
B1
G
M
B
M1
B1
G
M
B
M1
B1
MOVEMENT OF THE METACENTRE

35. CL
B
M
0o
-7/10o
MOVEMENT OF THE METACENTRE

36. CL
B B20
M
M20
MOVEMENT OF
THE METACENTRE

37. C
L M
M20
M45
B
B20 B45
MOVEMENT OF
THE METACENTRE

38. CL
B
B20
B45
M
M20
M45
M70
B70
MOVEMENT OF
THE METACENTRE

39. CL
M20M45
M70
M90
B
B20
B45
B70
B90
M
MOVEMENT OF
THE METACENTRE

40. MOVEMENTS OF THE METACENTER
The metacenter will change positions in the vertical
plane when the ship's displacement changes
The metacenter moves law these two rules:
1. when B moves up M moves down.
2. when B moves down M moves up.

41. G
B
G
M
B
M1
B1
MOVEMENT OF THE METACENTRE
G
WHEN BMOVES UPMMOVES DOWN.

42. GM
KG
CLK
M
G
B
BM
KM
LINEAR MEASUREMENTS IN STABILITY
KB

43. RECAPITULATION
1. The centre of gravity of a body `G' is the point through which the
force of gravity is considered to act vertically downwards with a
force equal to the weight of the body. KG is VCG of the ship.
4. KM = KB + BM Also KM = KG + GM
3. To float at rest in still water, a vessel must displace her own weight of
water, and the centre of gravity must be in the same vertical line as
the centre of buoyancy.
2. The centre of buoyancy `B' is the point through which the force of
buoyancy is considered to act vertically upwards with a force equal
to the weight of water displaced. It is the centre of gravity of the
underwater volume. KB is VCB of the ship.

44. 3. Stability Triangle
M
G Z

45. M
G Z
CL
K
B
G
M
THE STABILITY TRIANGLE
CL
K
B
G
M

46. CL
K
B
G
MM
CL
G
B
K
B1
CL
M
G
B
K
B1
CL
M
G
B
K
B1
CL
K
B
G
M
B1
Z
THE STABILITY TRIANGLE

47. M
G Z
Sin θ = opp / hyp
Where :
opposite = GZ
hypotenuse = GM
Sin θ = GZ / GM
GZ = GM x Sin θ
Growth of GZ α GM

48. CL
K
B
G
M
G1

49. G
M
Z
G1 Z1
 As GM decreases righting arm
also decreases
Growth of GZ α GM

50. INITIAL STABILITY
G
B
M
0 - 7°CL

51. M
ZG
B B1
CL
OVERALL STABILITY
RM = GZ x Wf

52. G
B1
M
Z
G
B1
M
B
G
B1
M
B
THE THREE CONDITIONS
OF STABILITY
POSITIVE
NEUTRAL
NEGATIVE

53. CL
K
B
G
M
POSITIVE STABILITY

54. CL
K
B
G
M
B1
Z
POSITIVE STABILITY

55. CL
K
B
GM
NEUTRAL STABILITY

56. CL
K
B B1
NEUTRAL STABILITY
GM

57. CL
K
B
G
M
NEGATIVE STABILITY

58. CL
K
B
G
M
B1
NEGATIVE STABILITY

59. EQUILIBRIUM
Stable equilibrium
 A ship is said to be in stable equilibrium if, when inclined, she tends
to return to the initial position. For this to occur the centre of
gravity must be below the metacentre, that is, the ship must have
positive initial metacentric height.
 The lever GZ is referred to as the righting lever and is the
perpendicular distance between the centre of gravity and the vertical
through the centre of buoyancy. At a small angle of heel (less than
150
). GZ = GM sin θ and
Moment of Statical Stability = W x GM sin θ
 If moments are taken about G there is a moment to return the ship to
the upright. This moment is referred to as the Moment of Statical
Stability and is equal to the product of the force 'W' and the length
of the lever GZ. i.e. Moment of Statical Stability = W x GZ
(tonnes-metres).

60.  When a ship which is inclined to a small angle tends to heel over still
further, she is said to be in unstable equilibrium. For this to occur
the ship must have a negative GM. Note how G is above M. Figure a
shows a ship in unstable equilibrium which has been inclined to a
small angle. The moment of statical stability, WGZ, is clearly a
capsizing moment which will tend to heel the ship still further.
UNSTABLE EQUILIBRIUM
Note. A ship having a very small negative initial metacentric height GM need
not necessarily capsize. This point will be examined and explained later. This
situation produces an angle of loll.

61. NEUTRAL EQUILIBRIUM
 When G coincides with M as shown in Figure a, the ship is said to be
in neutral equilibrium, and if inclined to a small angle she will tend
to remain at that angle of heel until another external force is applied.
The ship has zero GM. Note that KG = KM.
 Moment of Statical Stability = W x GZ, but in this case GZ = 0;
Moment of Statical Stability = 0 see Figure b. Therefore there is no
moment to bring the ship back to the upright or to heel her over still
further. The ship will move vertically up and down in the water at the
fixed angle of heel until further external or internal forces are applied.

62. CORRECTING UNSTABLE
AND NEUTRAL EQUILIBRIUM
When a ship in unstable or neutral equilibrium is to be made
stable, the effective centre of gravity of the ship should be
lowered. To do this one or more of the following methods may
be employed:
1. weights already in the ship may be lowered,
2. weights may be loaded below the centre of gravity of the ship,
3. weights may be discharged from positions above the centre of
gravity,
or4. free surfaces within the ship may be removed

63. STIFF AND TENDER SHIPS
o The time period of a ship is the time taken by the ship to roll from
one side to the other and back again to the initial position.
o When a ship has a comparatively large GM, it will thus require
larger moments to incline the ship. When inclined she will tend to
return more quickly to the initial position. The result is that the ship
will have a comparatively short time period, and will roll quickly -
and perhaps violently - from side to side. A ship in this condition is
said to be `stiff', and such a condition is not desirable. The time period
could be as low as 8 seconds. The effective centre of gravity of the
ship should be raised within that ship.
o When the GM is comparatively small, The ship will thus be much
easier to incline and will not tend to return so quickly to the initial
position. The time period will be comparatively long and a ship, for
example 30 to 35 seconds, in this condition is said to be `tender'. As
before, this condition is not desirable and steps should be taken to
increase the GM by lowering the effective centre of gravity of the
ship.

64. The officer responsible for loading a ship should aim at a
happy medium between these two conditions whereby the
ship is neither too stiff nor too tender. A time period of 20 to
25 seconds would generally be acceptable for those on board
a ship at sea.
WHAT SHOULD THE OFFICER DO?

65. NEGATIVE GM AND ANGLE OF LOLL
It has been shown previously that a ship having a negative initial
metacentric height will be unstable when inclined to a small angle.
As the angle of heel increases, the centre of buoyancy will move out still
further to the low side.

66. If the centre of buoyancy moves out to a position vertically under G,
the capsizing moment will have disappeared as shown in Figure b.
The angle of heel at which this occurs is called the angle of loll. It
will be noticed that at the angle of loll, the GZ is zero. G remains on
the centre line.
If the ship is heeled beyond the angle of loll from θ1 to θ2, the centre
of buoyancy will move out still further to the low side and there will
be a moment to return her to the angle of loll as shown in Figure c.

67. From this it can be seen that the ship will oscillate about the angle of loll
instead of about the vertical. If the centre of buoyancy does not move out
far enough to get vertically under G, the ship will capsize.
The angle of loll will be to port or starboard and back to port depending
on external forces such as wind and waves. There is always the danger
that G will rise above M and create a situation of unstable
equilibrium. This will cause capsizing of the ship.

68. GM is crucial to ship stability. The table below shows typical
working values for GM for several ship-types all at fully-loaded
drafts.
The GM value
At drafts below the fully-loaded draft, due to KM tending to be larger
in value it will be found that corresponding GM values will be higher
than those listed in the table above. For all conditions of loading
the Tp stipulate that the GM must never be less than 0.15 m.

69. Body Axis

70. Curve of Statical Stability

71. RIGHTINGARMS(FT)
ANGLE OF HEEL (DEGREES)
9060300 10 20 40 50 70 80
WL
20°
G
B
Z
W
L
40°
G
B
Z
WL
60°
G
B
Z
GZ = 1.4 FT GZ = 2.0 FT GZ = 1 FT
RIGHTING ARM CURVE

72. RIGHTINGARMS(FT)
ANGLE OF HEEL (DEGREES)
9060300 10 20 40 50 70 80
WL
WL
20°
G
B
Z
W
L
40°
G
B
Z
60°
G
B
Z
GZ = 1.4 FT GZ = 2.0 FT GZ = 1 FT
MAXIMUM RIGHTING
ARM
ANGLE OF
MAXIMUM
RIGHTING
ARM
DANGER
ANGLE
MAXIMUM
RANGE OF
STABILITY

73. DRAFT DIAGRAM AND FUNCTIONS OF FORM
17
16
15
14
13
12
11
800
4000
3500
3000
2550
750
700
650
600
550
AFTER
DRAFT
MARKS
(FT)
MOMENT TO
ALTER TRIM
ONE INCH
(FOOT-TONS) DISPLACEMENT
(TONS)
22.2
22.3
22.4
22.5
22.6
22.7
22.8
TRANSERSE
METACENTER
ABOVE BOTTOM
OF KEEL
(FT)
28
29
30
31
32
33
5
4
3
2
1
1
2
3
4
5
11
12
13
14
15
16
17
TONS
PER
INCH
(TONS/IN)
LONGITUDINAL
CENTER OF
BUOYANCY
(FEET)
FORWARD
DRAFT
MARKS
(FT)
CURVE OF CENTER OF FLOTATION
30 20 10
Length Between Draft Marks 397' 0"
DRAFT FWD = 14 FT 6 IN
DRAFT AFT = 16 FT 3 IN
Wo
Wo = 3850 TONS
KM =
TPI =
LCB =
LCF =MT1" =
MT1"
778 FT-TONS/IN
KM
22.28 FTTPI
32.7 TONS/IN
LCB
3.5 FT AFT
LCF
24 FT AFT

74. FFG 7
CROSS CURVES OF STABILITY
CENTER OF GRAVITY ASSUMED
19.0 FT ABOVE THE BASELINE
DISPLACEMENT (TONS)
RIGHTINGARMS(FT)
3000 3500 4000 4500
40
30
20
15
10
60
5545
50
3.0
2.5
2.0
1.5
1.0
0.5
10o
=
15o
=
20o
=
30o
=
40o
=
45o
=
50o
=
55o
=
60o
=
.55 FT
.85 FT
1.1 FT
1.73 FT
2.35 FT
2.55 FT
2.6 FT
2.5 FT
2.3 FT

75. 0
1
2
3
4
5
STATICAL STABILITY CURVE PLOTTING SHEET
RIGHTINGARMS(FT)
10o
=
15o
=
20o
=
30o
=
40o
=
45o
=
50o
=
55o
=
60o
=
.55 FT
.85 FT
1.1 FT
1.73 FT
2.35 FT
2.55 FT
2.6 FT
2.5 FT
2.3 FT
X
X
X
X
X
X
X X X
X
10 20 30 40 50 57.3
60 70 80 90
ANGLE OF INCLINATION - DEGREES