One of the first questions people ask when they discover I mess around with boat designs is: "How do you know it will float?"
Well, making it float is just Archimedes' principle of buoyancy, which we all know about from elementary school: A floating boat displaces water equal to its own weight, and the water pushes upward on the boat with a force equal to its weight. What people usually mean when they ask "How do you know it will float" is really "How do you know it will float upright?"
That's a little bit more complicated, but it's something every skipper and potential boat buyer should understand, at least conceptually. (Warning: High school mathematics is necessary for today's article.)
A yacht at an angle of heel
Let's consider a boat at rest, sitting level in calm water. The boat's mass is centred on a point G, the centre of gravity, and we can think of the force of gravity as acting straight down through this point. The centroid of the boat's underwater volume is called B, the centre of buoyancy. The force of buoyancy is directed straight up through this point.
We now heel the boat over by an angle "phi". Point G doesn't move, but point B does: by heeling the boat, we've lifted her windward side out of the water and immersed her leeward side. The centre of buoyancy, B, therefore shifts to leeward.
The force of buoyancy, acting upward through B, is now offset from the force of gravity, acting downward through G. The perpendicular distance between these two forces, which by convention we call GZ, can be thought of as the length of the lever that the buoyancy force is using to try to bring the boat upright. GZ is the "righting arm".
If we draw a line straight upward from B, it will intersect the ship's centreline at a point called M, known as the "metacentre". (Strictly speaking, the term "metacentre" applies only when phi is very tiny, but a pseudo-metacentre exists at any given angle of heel.) The metacentric height is a useful quantity to know when calculating changes in trim and heel.
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We can easily draw a few conclusions simply by looking at the geometry:
- The boat will be harder to heel, i.e. more stable, if GZ is increased.
- Lowering the centre of gravity, G, will increase GZ.
- Moving the heeled centre of buoyancy to leeward will increase GZ.
- If GZ changes direction- i.e. if Z is to the left of G- the lever arm will help to capsize the boat instead of righting it.
Stability Curves: GZ at all angles of heel
To prepare a stability curve, the designer must find GZ for each angle of heel. To do this, she must compute the location of B at each angle of heel, and determine the height of G above the base of the keel (the distance KG).
In the early 20th century, finding B at each angle of heel was an extremely tedious process involving a lot of trial-and-error, a lot of calculus, and days or weeks of an engineer's time. Today, this can be computerized, and takes only a few seconds once the hull is modelled in a CAD program. Finding KG, though, is still a tedious process: it can either be measured by moving weights around on an existing boat and measuring the resulting angle of heel, or it can be calculated by tallying up every piece of structure, ballast, equipment and cargo on the boat.
Once that math is done, the designer can plot GZ (or righting moment, i.e. displacement times GZ) over all possible angles of heel. This produces the familar stability curve:
All yacht skippers should be at least somewhat familiar with their own boat's stability curve, and it's a useful thing to study when buying a boat. To read the curve, we look at the following features:
- The slope of the curve at low angles of heel tells us whether the boat is tender (shallow slope) or stiff (steep slope).
- The righting moment at 15 to 30 degrees of heel tells us about the boat's sail-carrying power. A large righting moment indicates a boat that can fly a lot of sail; a boat with a lower righting moment will need her sails reefed down earlier.
- The maximum righting arm (or righting moment), and the heel angle at that point, tells us where the boat will be fighting her hardest to get back upright. If this is at a low angle of heel, we have a boat with high initial stability- she'll feel very stable under normal conditions, but a bit touchy at her limits, and relies on her skipper's skill to avoid knock-downs. If the maximum righting arm occurs at a very large angle of heel, the designer chose to emphasize ultimate stability- she'll be hard to capsize, but will heel more than you might expect in normal sailing.
- The angle of vanishing stability is the point where the boat says "Meh, I'm done" and stops trying to right herself. Looking at the diagram above, this means that Z is now at the same point as G. A larger AVS indicates a boat that's harder to capsize.
- The region of positive stability is the region in which the boat will try to right herself. The integral of the righting moment curve (i.e. the area of the green region) is an indicator of how much energy is needed to capsize her.
- In the region of negative stability, the boat will give up and roll on her back, her keel pointing skyward. The integral of this region (i.e. the blue area) tells us how much energy it'll take to right her from a capsize; if this area is relatively small, the waves that helped capsize her might have enough energy to bring her back upright.
Try it on a real boat
How does this apply to some real boats? Let's consider a 10 metre, 8 tonne double-ender yacht of fairly typical layout and proportions. The parent hull looks something like this:
Keeping her draught (1.5 m), displacement (8 tonnes), length (10 m), freeboard, deckhouse shape, etc. the same, we'll adjust the shape of the midship section to yield four boats that are directly comparable in all respects except beam and section shape. Hull A is a deep "plank on edge" style, hulls B and C are moderate cruising yacht shapes, and the wide, shallow-bilged hull D resembles an old sandbagger- or a modern racing sloop.
Now, assuming that G lies on the waterline (so KG = 1.5 m), we can compute the righting arm GZ as a function of the heel angle. If we multiply the righting arm GZ by the displacement, we get the righting moment.
Some immediate observations from this graph:
- The narrow hull "A" has relatively little sail-carrying power at low angles of heel, but will self-right from any capsize. Her good "ultimate stability" comes from using ballast to get G as low as possible.
- The wide hull "D" can fly a lot more sail, but if she goes over, she ain't coming back up. She gets her high "initial stability" from her wide beam, which moves the heeled centre of buoyancy farther to leeward.
There's a problem, though: We've assumed an identical centre of gravity for all four boats. That's not realistic. The deep, narrow hull will have her engine and tanks low in the bilge; the wide hull must mount these heavy components higher up. Let's reduce hull A's KG measurement to 1.35 m, and increase hull D's KG measurement to 1.65 m, a more realistic value. We'll scale KG for the other two accordingly.
The overall conclusions don't change much, but we now have some realistic numbers to play with.
- Hull A, the narrow one, will have a hard time flying much sail. She'll heel way over in a breeze. But she can't get stuck upside down.
- Hull B, a moderately slender cruising shape, also can't get stuck upside down- her AVS is 170 degrees. Her extra beam causes the centre of buoyancy to move farther to leeward when she heels, so she has more initial / form stability than hull A and can carry more sail.
- Hull C, which is typical of modern cruising yachts, has over twice the sail-carrying power of the slender hull A. She'll heel less, and since her midship section is much larger, she'll have more space for accommodations. The penalty is an AVS of 130 degrees. That's high enough that she can't be knocked down by wind alone, but wind plus a breaking wave- such as in a broach situation- could leave the boat upside down until a sufficiently large wave comes along.
- Hull D, the broad-beamed flyer, can hoist more than three times the sail of hull A at the same angle of heel. She'll be quite a sight on the race course with all that canvas flying. Her maximum righting moment, though, is only 37% more than hull A's, which leaves less of a margin for error- hull D is more likely to get caught with too much sail up, and will reach zero stability at a lower angle of heel. If she does go over, she has considerable negative stability, making it unlikely that she'll get back upright.
Work to capsize
If you're one of that slim percentage who paid attention in high school physics, you're probably looking at those curves and thinking: "Force (or moment) as a function of distance (or angle).... hey, if you integrate that, you get the work done!
And so you do, with the caveat that we're using a static approximation to a dynamic situation. The results are valid for comparison, but the actual numbers may not mean very much.
Let's do that for each of our hulls. We'll integrate the righting moment curve as a function of heel angle, up to the angle of vanishing stability, to get the work done to capsize the boat. We'll also integrate from the AVS to 180 degrees to get the work done to right the boat from a capsize.
Our four boats require roughly the same work to capsize! Changing the shape of the midsection affected the shape of the stability curve- a wider boat had more initial stability and less ultimate stability. In this case, though, our vessels are all about the same size and require about the same amount of work to capsize.
Righting from a capsize is another matter. The narrow, deep hulls A and B will self-right without any outside influence- a nice confidence-booster if you're heading into the open ocean, although the reduced sail-carrying power and limited interior space of these vessels will probably be more important to most skippers.
The moderate cruising hull, C, needs a bit of help to self-right, but any combination of wind and waves that can do 95 kN.m.rad of work on the boat is likely to produce a wave that can do 10 kN.m.rad of work on that same boat.
Our broad-beamed racer, hull D, is not so fortunate. Righting her from a capsize takes one-third the work that capsizing her in the first place did, and her acres of canvas were probably a major factor in the initial capsize- they're now underwater, damping her roll motion instead of catching the wind. The odds are that this boat will stay upside-down until someone comes along with a tugboat or crane.
What's the take-home message from all this?
If you're buying a new boat: Look at her stability curve, and compare it to other boats.
- Good: Large region of positive stability, small region of negative stability, high angle of vanishing stability, steep slope at low heel angles.
- Iffy: Shallow slope at low heel angles (makes it hard to fly lots of sail, excessive heeling when underway).
- Risky: Low angle of vanishing stability, large region of negative stability.
If you already have a boat:
- Use the boat's stability curve to help avoid, and plan for, emergencies.
- If you know her point of maximum stability, you can be sure to reef the sails well before that point.
- If you know her AVS and the shape of the curve in that region, then when a broach or knockdown happens, you already know how hard she'll fight to come back upright.
- If you know how much area is covered by the negative stability region of the curve, you'll have some idea of whether she'll come back from a capsize on her own or else have to wait for help.
- Know the location of the designer's intended centre of gravity (point G).
- Determine if anything you've changed- a dinghy added on the deck, perhaps- has moved the centre of gravity.
- If G has moved, adjust your mental model of the stability curve accordingly: just shift the curve up or down by (change in height KG) * sin(heel angle).
What we've discussed here is just about how to read the stability curve- it's not a complete picture.
There are many other factors that must be considered to get a complete understanding of a boat's stability. Among them:
- Dynamic effects. Everything discussed so far is for the static case, and is good for comparison purposes. But in practice, boats move.
- Waves. Stability curves are calculated for flat water, ignoring the effect of waves.
- Differences in rigging. Weight aloft has a much larger effect on the boat than weight down low- particularly where the roll moment of inertia, an important property for dynamic stability, is concerned.
- Keel shape. Keels tend to damp rolling motion; this behaviour is quite different with a long keel than with a fin keel, or with a fin keel underway versus a fin keel at rest.
- Downflooding. Everything we've discussed here assumes that the boat is watertight in any position. If she takes on water when rolled, everything changes.
- Cockpits. Our demonstration boat doesn't have a cockpit. A large cockpit could hold several tonnes of water- and with a free surface, no less. That means that G will move all over the place, usually in the wrong direction.
Steve Dashew's article "Evaluating Stability and Capsize Risks For Yachts", and others on his site, discuss stability-related risks as they relate to cruising yachts.
Technically-minded readers should refer to a naval architecture textbook, of which my present favourite is Larsson & Eliasson "Principles of Yacht Design" (McGraw-Hill).
Don't even think about buying a cruising yacht without first reading John Harries' extensive series of articles on boat and gear selection.