Angle of heel when turning (MAR)

1

ANGLE OF HEEL WHEN TURNINGConsider a ship turning to starboard.

The sequence of events is as follows:

1. Rudder put over to starboard.

2. The athwartships component of thrust (F) acts on the face of the rudder at P, P being the centre of pressure which coincides with the geometric centre of the rudder face.

SHIP’S FORWARD MOTION

Thrust acting normal torudder face at P

PAthwartships componentof thrust

F

Angle of heel when turning (MAR)

2

3. An equal and opposite reaction (F1) resists the athwartships motion at the centre of lateral resistance (CLR).

(The CLR is at the centroid of the ship’s longitudinal area below the waterline.)

F1

OUTWARD INWARD

FP

CLR Q

4. An inward heeling couple is set up for which the heeling moment is:

F PQ

F1 Q

P F

Angle of heel when turning (MAR)

3

5. When the ship achieves a steady rate of turn the inward heel is overcome by the effect of centrifugal force acting outwards through the ship’s centre of gravity (G).

Centrifugal Force = WV2 tonnes gR

where: W is ship displacement;V is ship speed in metres per second;g = 9.81 m/sec2;R is the radius of the turning circle.

The centrifugal force is opposed by an equal and opposite centripetal force which acts through the

CLR.

The CLR is assumed to be at the same height above the keel as the centre of buoyancy (B).

Consider the diagram.

Angle of heel when turning (MAR)

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WV2

gR

OUTWARD INWARD

CLRB

G WV2

gRCentripetal force

Centrifugal force

The original inward heeling moment is overcome by this outward heeling couple which develops in the steady turn state.

In the turn the ship will settle at an angle of steady heel when the outward heeling moment balances the normal righting moment (RM = GZ Displacement).

At small angles of heel: GZ = GM Sine

Angle of heel when turning (MAR)

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Z G

B1 B

d

HEEL

OUTWARD INWARD

B

GZ

M

HEEL

WV2

gR CLR

WV2

gR

B1

d

B and B1 are assumed to be at thesame depth.

Cos = ADJ = d HYP BG

Therefore: d = BG Cos

(At small angles of heel)

Angle of heel when turning (MAR)

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At the small angle of heel shown:

RIGHTING MOMENT = HEELING MOMENT

Therefore: W GZ = WV2 d gR

If: d = BG Cos and: GZ = GM Sin

then: W GM Sin = WV2 BG Cos gR

Transposing gives:

gR W GM Sin = WV2 BG Cos

Divide both sides by Cos :

gR W GM Sin = WV2 BGCos

Gives:gR W GM Tan = WV2 BG

Thus: Tan = WV2 BGgR W GM

Finally: Tan = V2 BGg R GM

Angle of heel when turning (MAR)

7

Note

In practice the outward angle of heel will be slightly less than that given by the formula because of the small inward heeling moment set by the athwartships component of thrust on the rudder.

However, if the rudder is returned quickly to the amidships position, the outward angle of heel due to turning will instantaneously increase. If the rudder is suddenly reversed i.e. put hard-a-port on a starboard turning circle, an even more serious outward angle of heel would arise (albeit temporarily) which could cause excessive heeling and in extreme situations cargo shift or even capsizing!