Non-watertight longitudinal subdivision (longitudinal bulkheads or wash bulkheads that are not watertight) of tanks serves to:

• Reduce the free-surface effect: by dividing a wide tank into narrower compartments, the second moment of area of the free surface (i = L B^3/12) is greatly reduced, so the free-surface loss of GM is reduced.
• Reduce the sloshing of the liquid: the wash bulkheads damp the movement of the liquid in a partially full tank, reducing the dynamic loads on the tank structure and the effect on stability.
• Provide structural support: the longitudinal bulkheads act as girders that stiffen the tank and the hull, and support the deck and bottom.
• Reduce the free-surface effect during ballasting and cargo operations.

The subdivision is non-watertight so that the liquid can flow between the compartments (for filling/emptying and for equalising), while still reducing the free-surface and sloshing effects.

A box-barge 30 m long and 9 m beam floats at a draught of 3 m. The centre of gravity lies on the centreline and KG is 3.50 m. A mass of 10 t, already on board, is moved 6 m across the ship.

(i) Estimate the angle of heel using GZ = sin(theta)(GM + 1/2 BM tan^2 theta).

Displacement = L x B x d x rho = 30 x 9 x 3 x 1.025 = 830.25 t.

KB = d/2 = 1.5 m. BM = B^2/(12 d) = 81/(12 x 3) = 2.25 m. KM = 1.5 + 2.25 = 3.75 m. GM = KM - KG = 3.75 - 3.50 = 0.25 m.

Transverse shift of G: GG' = w x d/Delta = 10 x 6/830.25 = 0.0723 m.

At equilibrium, GG' = GZ = sin(theta)(GM + 0.5 BM tan^2 theta).

0.0723 = sin(theta)(0.25 + 0.5 x 2.25 tan^2 theta) = sin(theta)(0.25 + 1.125 tan^2 theta).

Solving iteratively: try theta = 13 deg. sin(13) = 0.225, tan(13) = 0.231, tan^2 = 0.0533. GZ = 0.225(0.25 + 1.125 x 0.0533) = 0.225(0.25 + 0.06) = 0.225 x 0.31 = 0.0698. Close to 0.0723.

Try theta = 13.3 deg: sin = 0.230, tan = 0.236, tan^2 = 0.0557. GZ = 0.230(0.25 + 0.0627) = 0.230 x 0.3127 = 0.0719. Very close.

So theta = 13.3 deg.

(ii) Compare with the metacentric formula.

Metacentric formula: tan(theta) = GG'/GM = 0.0723/0.25 = 0.2892, theta = atan(0.2892) = 16.1 deg.

Answer: the wall-sided formula gives about 13.3 deg, while the metacentric (small-angle) formula gives about 16.1 deg. The difference is because the metacentric formula assumes small angles and a constant GM, while the wall-sided formula accounts for the change in BM with angle.