Understanding monohull sailboat stability curves

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.

Sailboat's midship section, heeled, with key points K, B, G, M, B(phi), Z marked.

(Can't see the images? Click here for now, then go update your web browser.)

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:

 

Righting arm (GZ) curve showing regions of positive and negative stability, maximum and zero stability points, and typical sailing range.

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:

Perspective view of a 10 metre, 8 tonne generic double-ender sailboat hull with a medium fin keel.

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.

Midship sections for four related hulls, A through D, ranging from narrow to very beamy.

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.

Righting moments for hulls A through D if the centre of gravity is on the waterline in all four ships.

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.

Righting moments for hulls A through D with centre of gravity adjusted to true values (lower in deep narrow boat, higher in shallow wide boat).

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.

Energy needed to capsize and to right each of the four hulls. The energy to capsize is similar (5500 to 6000) for all four; only the widest hulls require energy (~2000) to right 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.

Lessons Learned

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).

Confounding Factors

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.

Further Reading

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.

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Comments

Other confounding factors

One major confounding factor which most English-speaking designers still seem to routinely dismiss, or overlook, is to do with the nature of knockdown lever moments in a 'survival storm' situation:

You specifically state you're not taking waves into account, so this is addressed at those who do, in the conventional way -- generally led by the insights of academics and researchers tracing their conceptual methodology back to the likes of Marchaj.

The lever moments I'm thinking of arise from the vertical offset between:
Where the wave force vector acts, and
Where the hull resistance vector is located.

It has long been contended by the school of expedition yacht designers, going back to around the days of Damien II, from France in the 70s, that the greatest risk ... and arguably the only one worth worrying about for such vessels ... was due to the tripping moment caused by the vertical offset between the centre of effort of a true breaking wave, and the centre of resistance of the hull AND UNDERWATER APPENDAGES

When a large ocean wave breaks entirely forwards, the part which was formerly the crest avalanches down the front of the wave. Admittedly this behaviour is VERY rare offshore - where almost all 'breakers' actually spill most of the water down the back, but it's these events which present a real survival threat, and which define the limits to a vessel's capability.

Unlike the water particles in the body of the wave, which are circulating in the well known way of text book diagrams, and effectively not going anywhere over time, this "former crest" water has escaped from the wave system and is travelling rapidly under the influence of gravity down a steep ramp whose geometry (as opposed to constituent particles), in the case of a Southern Ocean wave of truly heroic proportions, might itself be advancing as fast as 30 to 40 knots.

So we have an aerated but still rather massive entity tumbling down above this already very fast moving ramp, hitting the topsides and cabin coamings, in the worst case, perpendicularly.

The contention of the French school was that, in this situation, while a high freeboard is clearly undesirable, the absolute last thing you want, which trumps everything else, is deep appendages providing lots of lateral grip, situated down in green water. This would provide a lever arm converting the sideways impulse (which is at a height not very far from the centre of mass, and hence not inherently an insuperable problem) into a very dangerous overturning moment.

The insight was based on simple empirical observations, such as of a flat wooden plank, or a surfboard with no appendages, floating side on to breaking waves at a surf beach. Despite having no ballast whatsoever, and a zero GZ in the plank case, this will sideslip down those waves and stay happily the same way up, in conditions where (say) a windsurf board with a deep centreboard (whether ballasted or not) will be tumbled repeatedly.

They reasoned that the thing to avoid at all costs, for a well found expedition yacht, was a knockdown with an angular acceleration sufficient to snap the rig.

This turned everything on its head with regard to the conventions of stability calculations: the relative positions of the centre of mass and the centre of buoyancy become largely irrelevant: the former should if anything ideally be high, so the vector from the striking crest passes through or near it, (to minimise the inertial overturning moment) while the latter is almost irrelevant because on the face of such a steep wave, the hull is in virtual freefall, and the hull is largely disengaged from green water. Aerated water offers little buoyancy.

This is so divorced from statics (which are arguably most useful for calculating how to prevent ships capsizing at a dock) that it is a shame to see so much reliance on static measures persisting to this day, in educating sailors, defining ultimate seaworthiness, and framing regulations and recommendations.

Be that as it may: this insight led to a completely different school of storm management by the adventurous people who sailed off to places like the subAntarctic and Antarctic in the new generation of beamy, generally low-freeboard # hulls, equipped with swing (or even dagger) ballasted keels capable of retracting - in many cases - right within the canoe body.

# ideally, no cabin trunk - which on the face of it is bad for self-righting...

In survival conditions, these sailors began retracting these keels, even though on the face of static stability calcs, this is entirely wrong. And (AFAIK*) not one of these yachts has yet been lost in the deep south, despite them making up the majority of the fleet, and I'm not even aware of a single 180deg knockdown to such a vessel configured in this way.

There have been, and continue to be, numerous knockdowns and dismastings of fixed-keel yachts designed to the other, older paradigm.

*(The first two losses of private expedition yachts in Antarctic waters both occurred within the last two years, and neither was a vessel of this type)

So even if these sailors are not right, they're clearly not VERY wrong.

Re: Other confounding factors

Matthew's picture

Hi Andrew,

You are quite correct that when you are facing breaking waves, static stability analysis is not going to show the whole picture. Being caught in large breakers is certainly one of the highest-risk situations a yacht can face.

The "let it slide sideways" approach can have considerable merit in such a situation, if the boat is designed with this in mind. On a monohull sailing vessel, this calls for a retractable keel and a canoe body with relatively little lateral resistance of its own. If you do this, of course, you also have to ensure that the vessel won't trip over the leeward gunwale when she's surfing sideways with the keel retracted. There are plenty of good, seaworthy vessels out there with such a configuration.

The price you pay for doing it that way is that it's harder to right the boat if she does capsize. Frankly, though, I would rather not capsize in a non-self-righting boat than be upside-down in one that will eventually get herself back up. There are tens of thousands of catamaran sailors out there who would seem to agree.

This is not to say that static stability traits are not important: they certainly are. Given two vessels of generally similar configuration, the stability curves will tell you quite a lot about what kind of behaviour can be expected from each.

Static stability curves are certainly not the whole picture. There are several important dynamic aspects- the lateral resistance effects and the roll moment of inertia, among other features- that can have a huge effect in extreme situations. I'll discuss these in more detail in future posts.

I am thinking about. Buying a

I am thinking about. Buying a 38 foot guimond lobster boat. I am thinking Of widening the stern to 10 feet from 8 ft 8 in. Also I want to add some fiberglass to the keel to make her a little deeper maybe 36 in from present 32 inches. Should I make the new hull water line 90 degrees? Will this be better than a round traditional edge? Should I add bilge keel fins for more stability?

Modifying a design

Matthew's picture

Hi Mark,

The kind of modifications you're describing are fairly extensive. You would be wise to arrange a meeting with a naval architect, or with a builder who has extensive experience with that type of boat. With the boat's drawings and a good description of what performance characteristics you want, the professional will be able to assess what modifications (if any) would be appropriate- or if you'd be better off choosing a different design from the start.

Twin keels

Hi Mike,

Static stability is determined by the hull shape and by the distribution of mass, i.e. the centre of gravity. Two identical hulls, one with a single fin and one with twin keels, will have approximately the same stability curve if they have the same centre of gravity. The twin keel configuration is usually chosen to allow shallower draught, though, so the centre of gravity will often be higher than for a single-fin boat.

There is a significant performance sacrifice with this configuration. A higher centre of gravity reduces the sail-carrying ability, the lower aspect ratio foils are not as efficient to windward, and the extra wetted surface increases drag. The flip side is that you can safely dry out at low tide in places where most monohulls would never be able to go.

Ultimately, though, the keel configuration is a fundamental part of a design, and there's no real answer to "How does a second keel affect stability". It's the performance of the entire boat that matters, and unless you have two boats that are identical except for keel configuration, it doesn't make much sense to separate out this one aspect of the design. The class's performance record and the experiences of skippers who have sailed that class in bad weather are better ways to assess the relative seaworthiness of an existing design.

Stability Curves for Hunter 34

Hi Matt,

I'm french and it's not that easy for me to understand all of this but here is my question:

Do you know who I can contact to know the stability curves of my sailboat. It's a Hunter Sloop 34' 1985

I asked directly at Marlow-Hunter, they said they don't have this information.

Someone told me that Hunter Manufacturer has it and that I can have it for some dollars but it seems that this is not the case.

Can you help me?

Danielle

Tracking down data for old boats

Matthew's picture

Danielle, if I'm not mistaken, that Hunter would be one of Cortland Steck's designs. There's a chance that he might have the data you're looking for.

Stability curves are incredibly tedious to calculate without a computer, though, so many- if not most- boats designed prior to the advent of modern 3D CAD never had one calculated at all. It's possible to build a computer model of an existing boat and calculate the required data, but for most practical purposes you can find the important information through an inclining experiment. This essentially consists of moving known weights around the boat and measuring how she heels in various load conditions, and it's one of the more common ways of measuring stability data for an existing vessel in commercial service where all of these details must, by law, be properly measured and documented.

Re: Righting a Capsized Vanguard Nomad 17

Matthew's picture

Gerardo,
A 625 pound boat with a beam of 8 feet is not going to be an easy thing to right.
You might find Sailing World's article on the boat interesting. They were advised by the manufacturer's rep that the boat can't be righted by one person in the way that you'd right something small like a Laser. But if you flood the tank (through the spinnaker well) on one side, you'll be able to roll her far enough to pull her back up like a dinghy, and then drain the tank again.
I agree that you would NOT want to test this in November!

37 Foot Sailboat

I am from the Maldives in the Indian Ocean. I am building a fiberglass sailing yacht using local boat builders. Its 37 feet and 11 feet with a long keel of 3 foot deep. And will use concrete in the keel. They will be putting 9 fiberglass mats. Interior and the bulkheads will be done using marine plywood. The hull is going to look more like a Fisher 37. And the cabins like a Nauticat. I am intending to use ketch style two masts. I was surfing the internet and am trying to understand what are the issues that I need to take into consideration. Your explanations is very helpful. I am just wondering whether you will comfortable if I communicate on this topic.
Thanking you.

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