A reduction in the size of the righting arm usually

To understand this stability curve, it is necessary to

consider the following facts:

means a decrease in stability. When the reduction in GZ

is caused by increased displacement, however, the total

1. The ship's center of gravity does NOT change

effect on stability is more difficult to evaluate. Since the

position as the angle of heel is changed.

RIGHTING MOMENT is equal to W times GZ, it will

2. The ship's center of buoyancy is always at the

be increased by the gain in W at the same time that it is

geometric center of the ship's underwater hull.

decreased by the reduction in GZ. The gain in the

righting moment, caused by the gain in W, does not

3. The shape of the ship's underwater hull changes

necessarily compensate for the reduction in GZ.

as the angle of heel changes.

In summary, there are several ways in which an

If these three facts are considered collectively, you

increase in displacement affects the stability of a ship.

will see that the position of G remains constant as the

Although these effects occur at the same time, it is best

ship heels through various angles, but the position of B

to consider them separately. The effects of increased

changes according to the angle of inclination. When

displacement are the following:

the position of B has changed so that B and G are not in

the same vertical line, a righting arm GZ must exist.

1. RIGHTING ARMS (GZ) are decreased as a

The length of this righting arm depends upon the angle

result of increased draft.

at which the ship is inclined (fig. 12-25). GZ increases

2. R I G H T I N G M O M E N T S ( f o o t - t o n s ) a r e

as the angle of heel increases, up to a certain point. At

decreased as a result of decreased GZ.

about an angle of 40°, the rate of increase of GZ begins

to level off. The value of GZ diminishes and finally

3. RIGHTING MOMENTS are increased as a

reaches zero at a very large angle of heel.

result of the increased displacement (W).

The position of the center of buoyancy at any given

FORCE OF BUOYANCY AT

18 FOOT DRAFT AND 20o HEEL

angle of inclination depends upon the draft. As the

o

20

26 FOOT WATERLINE AT 20o ANGLE OF HEEL

draft increases, the center of buoyancy moves closer to

Z

26

o

20

Z18

18 FOOT WATERLINE AT 20o ANGLE OF HEEL

G

the center of gravity, thereby reducing the length of the

B

26 F

26

OOT

WAT

righting arms. To determine this effect, the design

B 18

ERL

18 F

INE

OO

AT E

TW

VEN

ATE

activity inclines a line drawing of the ship's lines at a

RLI

KEE

NE

L

AT E

VEN

FORCE OF BUOYANCY AT

given angle, and then lays off a series of waterlines on

KEE

26 FOOT DRAFT AND 20oHEEL

L

it. These waterlines are chosen at evenly spaced drafts

DCf1225

throughout the probable range of displacements. For

each waterline the value of the righting arm is

calculated, using an ASSUMED center of gravity,

rather than the TRUE center of gravity. A series of such

c a l c u l a t i o n s i s m a d e f o r va r i o u s a n g l e s o f

heel--usually 10°, 20°, 30°, 40°, 50°, 60°, 70°, 80°,

and 90°--and the results are plotted on a grid to form a

series of curves known as the CROSS CURVES OF

A change in displacement will result in a change of

STABILITY.

draft and freeboard; and B will shift to the geometric

center of the new underwater body. At any angle of

Figure 12-26 is an example of a set of cross curves.

Note that, as draft and displacement increase, the

inclination, a change in draft causes B to shift both

curves all slope downward, indicating increasingly

horizontally and vertically with respect to the keel. The

smaller righting arms.

horizontal shift in B changes the distance between B

and G, and thereby changes the length of the righting

The cross curves are used in the preparation of

arm, GZ. Thus, when draft is increased, the righting

stability curves. To take a stability curve from the cross

arms are reduced throughout the entire range of

curves, draw a vertical line (such as line MN in

stability. Figure 12-25 shows how the righting arm is

fig. 12-26) on the cross curve sheet at the displacement

reduced when the draft is increased from 18 feet to 26

that corresponds to the mean draft of the ship. At the

feet when the ship is inclined at an angle of 20°.

intersection of this vertical line with each cross curve,

12-13