Factors affecting GZ curve shape (MAR)

1

FACTORS AFFECTING THE SHAPE OF THE CURVE OF STATICAL STABILITY

1. Effect of a change of KG

Consider a ship heeled by an external force to some angle .

G Z

B1

The righting lever is GZ.

If G was lowered GZ would increase and the ship would become more stable.

If G was raised GZ would decrease and the ship would become less stable.

The increase or decrease in GZ can be easily calculated.

K

Factors affecting GZ curve shape (MAR)

2

If G raised to G1

GX = Loss of righting lever

In GG1X

Sine = OPP = GX HYP GG1

GX = GG1 Sine

G Z

G1 Z1

X

If G lowered to G2

GY = Increase in righting lever

In GG2Y

Sine = OPP = GY HYP GG2

GY = GG2 Sine

G2 Y Z2

If G is raised or lowered (KG increased or decreased) the effect on GZ at any angle of heel can be determined by:

LOSS/INCREASE IN GZ = GGv Sine

Factors affecting GZ curve shape (MAR)

3

Effect on curve of statical stabilityImagine a ship with an initial range of stability of 90°.

If G is raised (KG increased) the loss of GZ at any particular angle of heel can be found by:

LOSS OF GZ = GGv Sine

Since Sine 0° = 0.00 and Sine 90° = 1.00 it follows that:

LOSS OF GZ WILL INCREASE AS THE ANGLE OF HEEL INCREASES.

The effect on the curve of statical stability is shown. Remember that if KG is increased then initial GM will be reduced.

GZ

57.3°

= Loss of GZ = Loss of GZ at 90° heel= GGv Sine = GGv Sine 90° = GGv 1.00 = GGv

Factors affecting GZ curve shape (MAR)

4

SummaryIf KG is increased:

* GM is reduced;* All GZ values are reduced, particularly at larger angles

of heel;* Dynamical stability is reduced, particularly at larger

angles of heel;* Range of stability is reduced;* Angle of deck edge immersion (DEI) remains the same.

ALL ASPECTS OF STABILITY ARE WORSENED!

If G was lowered (KG reduced) then all values of GZ would be increased. Initial GM would be increased. All aspects of stability would be improved.

Factors affecting GZ curve shape (MAR)

5

G Z

B1

2. Effect of an athwartships shift of weight

Consider a ship initially upright with a weight ‘w’ on one side heeled by an external force to some angle .

The righting lever is GZ.

If the weight is moved athwartships to the other side the centre of gravity of the ship (G) will move horizontally off the centre-line in a direction parallel to the shift of the weight.

Consider the same ship heeled to the same angle with G now off the centre-line at GH.

K

w

Factors affecting GZ curve shape (MAR)

6

The righting lever has been reduced from GZ to GHZ1 as a result of the atwartships shift of weight.

The loss of GZ can be easily calculated.

w

GGH is the horizontal distance that G has moved off the centre-line.

G ZGH Z1

B1

Factors affecting GZ curve shape (MAR)

7

GX = Loss of righting lever as a result of moving the weight.

In GGHX

Cosine = ADJ = GX HYP GGH

GX = GGH Cosine

LOSS OF GZ = GGH Cosine

G X Z

GH Z1

Factors affecting GZ curve shape (MAR)

8

= Loss of GZ = Loss of GZ at 0° heel= GGH Cosine = GGH Cosine 0° = GGH 1.00 = GGH

Effect on curve of statical stabilityImagine a ship with an initial range of stability of 90°.

If G is caused to move off the centre-line to GH the loss of GZ at any particular angle of heel is found by:

LOSS OF GZ = GGH Cosine

Since Cosine 0° = 1.00 and Cosine 90° = 0.00 it follows that:

LOSS OF GZ WILL DECREASE AS THE ANGLE OF HEEL INCREASES i.e. THE GREATEST LOSS OF GZ WILL BE WHEN THE SHIP IS UPRIGHT (The ship will come to rest at an angle of list in still water conditions!).

The effect on the curve of statical stability is shown.

GZ

57.3° 90°

LIST

Factors affecting GZ curve shape (MAR)

9

SummaryIf G moves off the centre-line causing the ship to list:

* GM remains the same;* All GZ values are reduced, particularly at smaller

angles of heel;* Dynamical stability is reduced, particularly at smaller

angles of heel;* Range of stability is reduced (since the ship is initially

listed);* Angle of deck edge immersion (DEI) remains the same,

however, less dynamical stability is available before deck edge immersion is reached on the listed side.

NoteThe M.S. (Load Line) Regulations 1998 do not specify that the ship must be upright prior to sailing. It should now be evident that list has a detrimental effect on ship stability, particularly since less work by external forces will be required to heel the ship to a dangerous angle of heel on the listed side. The International Grain Code (IMO) recognises this fact and to minimise the adverse effects of grain shift it is a requirement that the Master ensures that the ship is upright prior to sailing. It should always be borne in mind that in extreme circumstances there may be a potential for cargo shift, which may further significantly reduce the stability of an already listed ship.

Factors affecting GZ curve shape (MAR)

10

3. Effect of a change in freeboard

Consider two ships of similar characteristics, at the same draughts with the same KG (and GM), but with one having a greater freeboard as shown.

At angles of heel up to the angle of deck edge immersion of the smaller freeboard ship the curve of statical stability will be the same.

At angles of heel beyond the angle of deck edge immersion of the smaller freeboard ship values of GZ will be greater for the larger freeboard ship because curve steepness continues to increase up to a larger angle of heel i.e. inflexion of the curve will occur at a larger angle of heel since the angle at which the deck edge becomes immersed is greater.

G G

K K

Factors affecting GZ curve shape (MAR)

11

Effect of the curve of statical stabilityThis is clearly seen below.

GZ

57.3°DEI DEI

SummaryIf freeboard is increased:

* GM remains the same;* GZ values are increased at angles of heel beyond DEI

of the smaller freeboard ship;* Dynamical stability is increased at angles of heel

beyond DEI of the smaller freeboard ship;* Range of stability is increased;* Angle of deck edge immersion (DEI) occurs at a larger

angle of heel.

If freeboard was to be decreased then GZ values would be reduced.