TRANSVERSE STABILITY, LARGE ANGLES
When the angle of heel becomes larger a number of the simplifying assumptions made above no longer apply:
• The ship will, in general, trim as well as heel. This means that the CB moves fore and aft as well as transversely. In what follows the change in trim is ignored but it should be remembered that in talking of positions of B or M it is strictly their projections on to a transverse plane that is meant.
• The deck edge and the turn of bilge will become immersed or exposed.
• The heeled waterline will no longer intersect the upright waterline on the centreline of the ship.
• M will no longer remain in one position and it is no longer of direct relevance to the calculation. By definition, M remains the point at which successive lines of buoyancy intersect as the ship heels.
The weight and buoyancy are still equal and, referring to picture, the heeling moment is now expressed as:
W.GZ, where Z is the foot of the perpendicular from G on to the vertical through the new position.
This is a very simple expression but the calculation of GZ for a one-off ship condition for a range of heel angles is not easy. The approach used is to calculate instead a value SZ where Z is the foot of the perpendicular to the line action of the buoyancy force from S, an arbitrary point on the ship’s centreline. One great advantage of using SZ is that it is independent of the weight distribution within the ship. It is dependent only on the ship geometry.
CROSS CURVES OF STABILITY Rather than heel the ship at constant displacement it is easier to calculate the displacement and CB position for a range of draughts and angles of heel. In practice the calculations are made simpler by using a suitable computer program or by using a special instrument that records areas and moments of area. The end result is a series of curves showing how SZ varies with displacement at a number of heel angles. Such curves are known as cross curves of stability. Once derived for the ship by the designer they remain accurate as long as the ship’s hull form is not changed. The curves are provided to the master to assist in stability calculations.
Now, for any given loading of the ship for which the vertical centre of gravity has been assessed, the variation of GZ with heel angle can be quickly obtained. For the known displacement the values of SZ can be read from the cross curves. Then:
GZ = SZ – SG Sin ϕ, where ϕ is the angle of heel and G is above S.
