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.)
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:
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:
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:
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.
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.
If you already have a boat:
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:
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.