Bilging a centreline compartment (one that spans the full width, with the centreline running through it) that is located away from amidships causes:

• A loss of buoyancy in that compartment, so the ship sinks (increases draught) and, because the compartment is away from amidships, the loss of buoyancy is not symmetrically distributed about the centre of flotation, producing a change of trim (the ship trims towards the damaged end).
• The loss of buoyancy also reduces the waterplane area and hence the BM and GM, so the stability is reduced.
• Because the compartment is on the centreline, there is no list (the loss is symmetric about the centreline), but there is a change of trim and a reduction of GM.
• The free-surface effect of the flooded water further reduces the effective GM.

The ship must be able to survive the flooding with adequate residual stability and freeboard (damage stability criteria).

Ship of 5,000 t displacement has a double-bottom tank 12 m long. The half-breadths of the top of the tank are 5, 4 and 2 m respectively. The tank has a watertight centreline division. Calculate the free-surface effect if the tank is partially full of fresh water on one side only.

Because of the centreline division, the free surface exists on one side only, and its breadth is the half-breadth (5, 4, 2 m) at the three stations over the 12 m length (spacing 6 m).

Second moment of area of the free surface about its longitudinal axis:

i = (1/12) x integral of (breadth^3) dx = (1/12) x (h/3)[b0^3 + b2^3 + 4 b1^3]

= (1/12) x (6/3)[5^3 + 2^3 + 4 x 4^3] = (1/12) x 2[125 + 8 + 4 x 64] = (1/12) x 2[133 + 256] = (1/12) x 2 x 389 = (1/12) x 778 = 64.83 m4.

Free-surface effect = rho x i/Delta = 1.000 x 64.83/5000 = 0.01297 m.

Answer: the free-surface effect is about 0.013 m (13 mm) reduction in GM.