Vessel stability — metacentric height and free surface effect

What the rule requires

The vertical order of G and M decides stable or unstable

Three points do all the work. G, the center of gravity, is where all the vessel's weight is treated as concentrated, acting straight down at right angles to the waterline. B, the center of buoyancy, sits at the geometric center of the underwater hull, so it moves off the centerline as the hull heels and changes shape. M, the metacenter, is where two successive lines of buoyant force intersect as the vessel heels through a very small angle NAVEDTRA 14057 Ch. 12 — Analysis of Stability.

GM, metacentric height, is the distance from G to M, and its sign is the whole answer to whether the vessel is stable. With M above G, GM is positive, the moments that develop on inclination are righting moments, and the vessel is stable. With M below G, GM is negative, the moments that develop are upsetting moments, and the vessel is unstable NAVEDTRA 14057 Ch. 12 — To a Very.

GM also sizes the ride. A large GM produces large righting arms at small angles: the vessel is stiff and resists roll. A small GM produces small righting arms and a tender vessel that rolls slowly. Large GM and large righting arms are what you want for resistance to damage, though a naval GM value is a compromise, since a smaller GM buys an easier roll.

GZ and righting moment are not the same quantity

GZ, the righting arm, is the horizontal separation between the lines of gravity and buoyancy, measured in feet. At small angles of heel, GZ = GM sin θ, which is why GM serves as a measure of the initial righting arm. Righting moment is displacement times righting arm, in foot-tons NAVEDTRA 14057 Ch. 12 — Review Questions.

Ten thousand tons with a 2-foot righting arm at 40° of inclination gives 20,000 foot-tons of righting moment NAVEDTRA 14057 Ch. 12 — Center of Buoyancy Above Base (Kb). Plot GZ against successive angles of heel and you have the curve of static stability; static means the vessel need not be in motion for the curve to apply — stop her at any angle during a roll and the GZ from the curve still holds NAVEDTRA 14057 Ch. 12 — Stability Curves.

Weight moves G, and G moving is what you control

G is fixed for a given condition of loading and shifts only when weight is added, removed, or moved. Add weight and G moves toward it; remove weight and G moves away from it. Weight added above G decreases stability, and — the half candidates forget — removing weight from below G also decreases stability, because G rises either way Boat Crew Seamanship Manual (M16114.5C) Ch. 9 — Understanding Stability ¶1.

Icing is this mechanism at its worst: it increases displacement by adding weight above G, G rises, and swells, sharp turns, or quick speed changes can capsize a vessel carrying ice topside. Correct it by changing course or speed to reduce freezing spray and rolling, and by physically removing the ice Boat Crew Seamanship Manual (M16114.5C) Ch. 9 — Losing Stability ¶1.

Added weight also costs you freeboard. Freeboard is the distance from the waterline to the weather deck edge, calculated at the midship section, and freeboard plus draft always equals the depth of hull. Add weight and draft and displacement rise by the same amount while freeboard and reserve buoyancy fall NAVEDTRA 14057 Ch. 12 — Waterline Displacement. Increased draft reduces the righting arms throughout the entire range of stability, because B shifts closer to G NAVEDTRA 14057 Ch. 12 — Waterline.

You land 800 lb of dive gear on the deckhouse top instead of stowing it in the lazarette. What happens to GM?

GM decreases. G moves toward the added weight, so G rises; M has not moved, so the distance G-to-M shrinks. Weight added above the center of gravity decreases stability, and the higher the stow, the more you lose.

Free surface: the virtual rise in G

A tank pressed full behaves as a solid weight at its own location. The same tank slack does not. Liquid free to move as the vessel heels tends to stay level, and that surface is a free surface; liquid having a free surface is loose water, and liquid completely filling a tank or compartment is neither NAVEDTRA 14057 Ch. 12 — Effects of Loose Water. The penalty is a reduction in GM by way of a change in the center of gravity, separate from and independent of any effect of the liquid's weight, and it must be carried into stability calculations as a free surface correction.

Two properties of free surface decide exam questions, and both are counter-intuitive.

Location does not matter. A free surface of a given length and breadth causes the same GM reduction at any given angle of heel no matter where it is in the vessel — forward or aft, high or low, on the centerline or off it NAVEDTRA 14057 Ch. 12 — Angle of Heel in Degrees.
Depth does not matter either, unless the water is shallow enough or deep enough to cause pocketing — the free surface contacting the deck or overhead and thereby reducing the breadth of the free surface. Pocketing improves stability, and the benefit grows at larger angles of heel, but it is not credited when evaluating stability; leaving it out is a deliberate margin of safety.

Breadth is the variable that matters. The rise in G goes as b³l / 12(35W), where b is the athwartship breadth of the compartment, l its fore-and-aft length, and W the displacement — so the effect varies as the cube of breadth but only as the first power of length. Two consequences follow straight off that exponent: double the breadth of a free surface and its effect is multiplied by eight, and a single bulkhead cutting a compartment in half fore-and-aft quarters the free surface effect NAVEDTRA 14057 Ch. 12 — C Almost Full. Holding fuel oil tanks 95 percent full trades on the same geometry from the other end, since an almost full compartment pockets at very small angles of heel.

For a fishing vessel the correction is mandatory and specified: the virtual rise in the vertical center of gravity due to liquids in tanks must be calculated for the tank or transverse pair of tanks having the greatest free surface effect for each type of consumable, plus a service tank correction, and for each partially filled tank or hold holding a non-consumable liquid or fish that can shift as the vessel heels, including loose water from fish processing. Tanks fitted with cross-connection piping are calculated as one common tank unless valves that will be kept closed are installed in the piping 46 CFR §28.540.

Afloat, the corrective actions are short and physical. Minimize the number of partially filled tanks and ballast with seawater as necessary. Keep gear from rolling across the deck, and stow cargo low and close to the centerline. Fish wells that are neither emptied nor kept full will hurt you badly .

The numbers, which you can look up

You are open book, so know which section holds each value rather than the value itself. Intact righting energy requires, in each condition of loading, an initial GM of at least 1.15 feet (0.35 m), a righting arm of at least 0.66 feet at a heel angle not less than 30°, the maximum righting arm at not less than 25°, and positive righting arms through 60° — reducible to 50° only where the hatch conditions of paragraph (b) are met 46 CFR §28.570.

For the simplified stability proof test, the vessel is tested with each fuel and water tank approximately three-quarters full and any sewage tank empty or full, and the heel limits run by hull type — half the freeboard immersed on a flush deck vessel, a quarter on an open boat — with one absolute cap: in no case may the angle of heel exceed 14 degrees 46 CFR §178.330.

Telling it apart

Heel, list, roll, trim — the criterion is the axis and whether it persists

Heel — transverse inclination that is temporary Boat Crew Seamanship Manual (M16114.5C) Ch. 9 — Understanding Stability ¶1.
List — transverse inclination that is permanent, caused by G lying off the centerline; the vessel heels until B comes into vertical alignment with G and stays there.
Rolling — the side-to-side motion itself, distinct from both.
Trim — longitudinal inclination, measured by the difference between forward and after draft, about an athwartship axis through the center of flotation. Greater draft aft is trimmed by the stern; greater draft forward is trimmed by the bow or by the head. Longitudinal stability resists a change in trim, and the standard measure of that resistance is the moment to change trim one inch (MTI) NAVEDTRA 14057 Ch. 12 — Longitudinal Stability.

The three effects of loose water — the criterion is what the water changes

Added weight — the weight of the flooding water alone, which increases displacement and draft.
Free surface — the reduction in GM from the shifting wedge of liquid, present whether or not the compartment is open to the sea.
Free communication — the unobstructed movement of seawater into and out of the hull, which reduces both GM and GZ, and does so in addition to the free surface and added weight effects. It requires an off-center compartment ruptured badly enough for the sea to flow freely as the vessel rolls; a compartment symmetrical about the centerline is not usually reduced by free communication, and any loss there comes from increased free surface instead NAVEDTRA 14057 Ch. 12 — Displacement in Tons.
Added buoyancy — not one of the three, and it is the standard distractor NAVEDTRA 14057 Ch. 12 — Review Answers.

Working a question

A 120-long-ton fishing vessel carries a fish hold 10 feet athwartships and 12 feet fore-and-aft. In the departure condition the hold is partly full of slack water from processing. Uncorrected GM is 1.60 feet. Does she meet the intact righting energy requirement, and what righting moment does she develop at 10° of heel?

Fix the governing section. Initial GM of at least 1.15 feet is required in each condition of loading 46 CFR §28.570. "Each condition" means this departure condition, not a favourable one.
Decide whether free surface enters. It does. A partially filled hold containing loose water associated with the processing of fish is expressly named for correction 46 CFR §28.540. Had the hold been pressed full, there would be no free surface and only its weight would count.
Compute the virtual rise in G. b³l / 12(35W) = (10 × 10 × 10 × 12) / (12 × 35 × 120) = 12,000 / 50,400 = 0.24 ft NAVEDTRA 14057 Ch. 12 — C Almost Full. Note what did not appear in that calculation: how deep the water is, and how high or low the hold sits.
Correct GM and test it. 1.60 − 0.24 = 1.36 ft, which clears 1.15 ft. She passes on initial GM, with 0.21 ft of margin.
Take the righting moment at a small angle. GZ = GM sin θ = 1.36 × 0.174 = 0.237 ft at 10°. Righting moment = W × GZ = 120 × 0.237 ≈ 28 foot-tons NAVEDTRA 14057 Ch. 12 — Review Questions.
Stop before 30°. The 0.66-foot arm required at not less than 30° comes off the corrected curve of static stability, not from GM sin θ, which holds only for small angles. Passing step 4 does not establish compliance with the righting arm and area criteria.
Recognise the second rule if she lifts. Put a trawl or a weight over the side and a different section may switch on, triggered only when the maximum heeling moment exceeds 0.67(W)(GM)(F/B) — with W and GM both including the lifted weight or gear force, F the freeboard to the lowest weather deck at amidships, and B the maximum beam 46 CFR §28.545. A suspended load is assumed to act at the boom tip unless its transverse movement is restricted.

One design note on step 3: run a single bulkhead down the centerline of that hold and the two 5-foot halves together produce about 0.06 ft of rise instead of 0.24 ft. Cube of breadth, quartered effect.

Same vessel, same slack hold, but she is loaded 8 inches deeper. Is the free surface loss smaller?

No. Free surface effect at a given angle does not depend on the depth of the loose water, and increasing draft does not lessen it. Worse, increased draft reduces the righting arms throughout the whole range of stability, so the deeper condition is the poorer one.

Where candidates lose the point

Topping a tank off, and marking it down as a loss because weight went aboard. The pressed-full tank has no free surface; its contents may be treated as a solid acting at their own location. It is the slack tank that costs you GM. Held at 95 percent, pocketing works in your favour at small angles.
"Only a few inches, and it's low in the bilge anyway." Neither figure enters the calculation. Breadth does, cubed — and pocketing, the single exception, is refused credit on purpose.
Added buoyancy. Water inside the hull brings added weight, free surface, and, where the sea flows freely in and out, free communication. Three effects. The fourth option on the answer sheet is bait.
Carrying GM sin θ out to 30° or 40° and then claiming the 0.66-foot criterion is met. That relation is stated for small angles. Past them, read GZ off the curve of static stability, which climbs to roughly 40° before the rate of increase levels off, then falls to zero at a very large angle of heel.
Reciting the 15 foot-degree area requirement the moment a boom or a trawl appears in the stem. The lifting-gear criteria bind only once the maximum heeling moment exceeds 0.67(W)(GM)(F/B). Compute the trigger before you quote the standard.
Pumping out a low tank on the low side to take a list off her. Weight removed from below G raises G and stability goes down with it. Where the hull is in free communication with the sea, the actions are to patch the opening, put weight on the high side, and take weight off above G on the damaged side.
Treating the SST freeboard immersion limits as the whole test. The 14-degree ceiling sits on top of them; a cockpit vessel or a flush-deck sailing vessel can satisfy the immersion limit and still fail on angle.

Check yourself

You put two crew and a portable dewatering pump aboard a partly flooded skiff. What are you weighing, and when may you take her in tow?

Every pound you send across is added weight to a vessel whose righting moment is already in question, and the added weight of assisting personnel or equipment can take the last of it and capsize her. A flooded vessel may look stable when it is not, so compare her motion in the sea with your own boat's. Where she is visibly unstable — listing, trimmed by the bow or stern, or downflooding — do not make fast to her or tow her at all; tow only after the loss of stability has been corrected, keep your own equipment aboard your vessel where you can, and adjust course and speed to hold the rolling down rather than turning sharply or at speed.

You are dewatering a partly flooded 42-foot vessel and the water in her engine space stands 4 inches deep across the full beam. Your mate says there is too little water to matter. Correct him.

Depth is not the measure. The free surface effect of a given area of loose water at a given angle of heel does not depend on its depth, unless the layer is shallow enough for pocketing to reduce the breadth of the free surface — and pocketing is never credited when evaluating stability. Water lying across the full beam has maximum breadth, and the effect varies as the cube of breadth.

Your vessel displaces 90 long tons and her corrected GM is 1.5 feet. What righting moment does she develop at 8° of heel?

GZ = GM sin θ = 1.5 × sin 8° = 1.5 × 0.139 = 0.209 ft. Righting moment = W × GZ = 90 × 0.209 ≈ 18.8 foot-tons. The small-angle relation is valid here; it would not be at 30°.

Your fuel tanks are cross-connected and the crossover valve is left open at sea. How must the free surface be calculated, and why does that matter?

Tanks fitted with cross-connection piping must be calculated as one common tank unless valves that will be kept closed to prevent transfer of liquid as the vessel heels are installed in the piping. It matters because the combined tank has a far greater athwartship breadth, and free surface effect varies as the cube of breadth — closing the crossover restores two narrow surfaces in place of one wide one.

A 60-foot vessel takes green water on deck repeatedly in a following sea. Name what that does to her stability and the actions available to you.

Water on deck increases displacement, which increases draft and decreases stability and trim; it contributes free surface effect; and it amplifies her rolling motion, which may result in capsizing. Decrease trim and increase freeboard, change course or speed or both, and ensure the drain openings are unobstructed.

She takes the bottom on a falling tide and begins to lean, though nothing was moved and nothing flooded. Account for it.

The volume of water beneath her is shrinking, and with it the buoyant force that water was supplying. At the same time the upward force acting at the point of grounding increases, and the unsupported part of the hull falls to one side. A vessel can capsize aground for exactly that reason, with her loading unchanged.