Free Surface Effect on Ships: Formula and Worked Example

Why does a slack tank reduce stability?

When a ship heels slowly, the liquid surface remains horizontal while the tank tilts. Liquid redistributes towards the low side, shifting the liquid’s centre of gravity and therefore the ship’s G.

“Virtual rise” is an equivalent calculation. The liquid physically moves, but we model its effect on initial stability by raising G vertically by the free surface correction. This makes the calculation easier without treating that virtual point as the ship’s actual G.

Reference: AMSA NSCV C6A, definitions and Figure 4.

Free surface effect formula

For an initial transverse stability calculation, find the free surface moment for each relevant tank, add the moments and divide by the ship’s displacement.

The rectangular formula assumes a freely communicating, rectangular surface at small angles of heel. For a real ship, use the tank data and correction method in its approved stability information.

Check the table heading. If the given FSM already includes liquid density, do not multiply it by density again. Where a table uses a reference density, follow its instructions for adjusting to the actual liquid.

What if more than one tank is slack?

Add the moments before dividing by displacement. If a second tank contributes 500 t·m in a condition with the same 10,000 t displacement:

FSC = (1,000 + 500) ÷ 10,000 = 0.15 m

If uncorrected GM is still 0.80 m, corrected GM becomes 0.65 m. Include all moments required by the question or the approved onboard calculation method.

Why does tank breadth matter so much?

The rectangular formula contains b³. With length and density unchanged, doubling the breadth makes the tank’s FSM eight times as large.

Now divide the example tank into two equal, isolated compartments using a watertight longitudinal bulkhead. Each compartment is 12 m long and 5 m broad.

• Each compartment: i = (12 × 5³) ÷ 12 = 125 m⁴.
• Both together: total i = 2 × 125 = 250 m⁴.
• Total FSM for fresh water = 250 t·m.
• At the same displacement, FSC = 250 ÷ 10,000 = 0.025 m.

The combined correction is one quarter of the original. For n equal, isolated longitudinal compartments with the same total breadth and length, the combined FSM becomes 1/n² of the undivided value.

This assumes the compartments cannot freely exchange liquid. A perforated wash plate or an open cross-connection does not automatically qualify for this subdivision calculation.

Common exam mistakes

• Adding FSC to GM. Free surface reduces effective GM, so subtract it.
• Using sea-water density for every tank. Use the density of the tank’s actual liquid.
• Using displacement volume instead of tonnes. Keep units consistent with the formula above.
• Cubing the length. For transverse stability, the rectangular formula cubes the breadth across the ship.
• Correcting twice. If GM already includes free surface, do not subtract FSC again. Similarly, do not both raise KG by FSC and subtract FSC from the resulting GM.
• Assuming half-full is always the worst level. In a vertical-sided rectangular tank, the initial FSM stays the same while the free-surface dimensions stay the same. In shaped tanks, those dimensions can change with filling level.

Short answers for revision

Does a completely full tank have free surface effect?

A completely filled tank with no free liquid surface has no free surface effect. For a tank described as “full” in practice, follow the approved assumptions and tank tables; do not assume a small ullage can always be ignored.

Does a low double-bottom tank still cause it?

Yes, if it is slack. The stabilising effect of placing liquid weight low down and the reduction in GM from free surface are separate effects.

How can free surface effect be reduced on board?

Use the approved cargo or ballast plan to limit unnecessary slack tanks and check intermediate transfer conditions. Where the plan permits, completely full or empty tanks avoid a free liquid surface. Use the vessel’s approved stability information and loading instrument to assess the resulting condition.

For the underlying distinction between liquid weight and free surface, see MCA Fishing Vessel Stability Guidance, “Tanks”. The basic physical principle applies across vessel types.