Data - Marine Formulae #1

For propeller and shaft calculations, see:
                        www.pelaginox.com/data/d-driveline1.html

Where possible, alternative methods of calculation have been given. Some of the constants used, and the range of results deduced, are a matter of lively debate, in which I'm always willing to join; and I'm also on the lookout for more formulas and conversions.

Note: if using these formulas to compare one boat with another, one should apply comparisons only to similar boats of similar lengths. Application to boats of widely differing lengths and materials is an unreliable method of comparison. Also, our commonly-held interpretations of results really only holds good in the popular 26 - 45 foot length group; outside this group, interpretation is different.

Note that deductions are, to a great extent, a value judgement; two naval architects might hold opposing views, even though both design successful boats.

In the examples, where given, a heavily-built sailboat of the following dimensions (in feet and tons) has been used:
LOA 37', LWL 29', Bmax 11.5', Bwl 10', Draught 5.25', Ballast 4 t, Displacement 10 tons, body type = full.

The most commonly used ratios for sailboat comparison are often abbreviated in speech and informal writing to the D/L, SA/D, S/L and Ballast ratios.

Naming conventions: most standard measurements are normally stated in the form Avs, Lwl, Boa, and so on, where in print the suffix letters are (or should be) in subscript: small letters below the normal level, as for instance in H2O, the chemical symbol for water, where the '2' should be in subscript. The opposite is superscript as in 3² = 9. The examples given here are incorrect as they do not place the '2' below or above the normal font line.

Since that option is not available in most PC programs, to avoid confusion all-capitals are used here; with a result for instance of BOA for beam overall, and LWL. The alternative, e.g. Boa or Lwl, is undesirable due to confusion with other words, or misleading fonts.

Displacement should be stated here as Disp, to avoid confusion with draught as D, except where it obviously refers to the weight. In some places I will no doubt have forgotten all this and typed it wrongly.

D/LR (displacement / length ratio) = weight / wl cubed / .000001
or: weight in long tons over waterline in feet cubed over .five noughts 1
or: disp in tons / (.01 x LWL)cubed

e.g. 10 / (29x29x29 / .000001) = 410

An indicator of relative heaviness, therefore overall performance and downwind speed, contrasted with comfort and, to a certain extent, safety offshore. Lighter boats will sail faster in moderate conditions, and downwind in any sea state; heavier boats will be slower but more comfortable offshore, and will go to weather better in rough seas. They may be wetter as a consequence – but then they get there with less histrionics.

Approximate meaning of values:
<100           = ultra-light;
100 - 160    = very light;
160 - 250    = light;
250 - 320    = moderate;
320 - 380    = heavy;
>380           = extra heavy

Interpretation of these values is changing, as boats become lighter due to a combination of factors: better and lighter construction materials and methods; better design technology; boatbuilders striving to build lighter, thereby building cheaper/lighter/faster boats; and boats are no longer required to be strongly built except where specified as such, at extra cost. A future interpretation might therefore look like:
<70             = ultra-light;
70 - 120      = light;
120 - 180    = moderate;
180 - 250    = heavy;
250 - 320    = extra heavy;
320 - 380    = a half-tide rock;
>380           = you've sunk.

Most people currently seem to favour moderate displacement, with equal numbers supporting light and heavy displacement. What can't be argued is that lighter boats go faster, and that offshore you need weight for comfort. How you balance the two is why there are ten thousand boat designs. It's easier to design a 60-footer since there don't need to be so many compromises, but in the 26 - 45 foot range, it's tough to get the right blend – at an attractive price.

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BR (ballast ratio) = ballast x 100 / disp.
or: weight of ballast, keel, and tankage if in keel, by 100 over displacement.

e.g. 4 t x 100 / 10 t = 40%
Metric & Imperial come out the same.

Keep the units of measurement the same. Expressed in the UK as a percentage (e.g. 40%) or in the US as a decimal (e.g 0.4).

This is one of the main stability factors (and a static factor, not of course a dynamic stability factor) of a sailing monohull of up to 50 feet length; form stability is the next most important. Above this, big displacements, long waterline lengths, speed capability and sheer size cloud the issue. The full picture of stability is clearer if the LOA, RP, and CG are known; Roll Inertia would help, and AVS clinch it...

Interpretation:
>50%         = a leadmine;
46 - 50%    = very heavily ballasted, likely to be slow, probably safer offshore.
42 - 46%    = heavily ballasted;
38 - 42%    = well ballasted;
34 - 38%    = moderate;
30 - 34%    = lightly ballasted;
<30%         = inshore fair-weather use only.

The picture is confused nowadays by the use of water ballast, sliding ingots and canting keels, which all provide much more stability than their weight alone would suggest.

As stated, it is by far the most important as a static factor, especially in this size range. A simplistic explanation of static and dynamic factors is that static applies to stationary structures, and dynamic applies to moving ones. The most important factor in the dynamic stability of a boat is speed – fast boats have inherently more dynamic stability (until they stop, of course).

A good demonstration of this is a bicycle rider. At rest, the rider has no static or dynamic stability, and will soon fall over. At speed, the dynamic stability is considerable, and even a substantial pressure applied briefly at the side may not knock the rider over – compared with a featherweight applied while stationary. If, though, the rider was a circus performer on a cycle without tyres crossing a highwire, with a weight suspended below the cycle, then the cycle starts out with some static stability.

Another way to understand this is to look at an alternative type of structure such as rope. Because this distinction is so important when applied to rope, it is the single most important method of classifying rope; no other method of classifying rope is nearly so important as the distinction between static and dynamic ropes.

Spectra rope is a good example of a static rope; it has very little stretch. Nylon is the best example of a dynamic rope: it has the best stretch of all. You can immediately see that a sail is best attached by Spectra, and nylon would be useless for a halyard (the sail would perform as if attached with elastic, and would hardly work at all); and a climber is best attached by nylon, Spectra being disastrous, and virtually the equivalent of chain (a fall would cause injury due to the sudden arrest, links would be subject to unacceptable stress, and rock / ice anchors might be torn out).

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S/LR (speed / length ratio) = C x root WL
or: C by the square root of the waterline length.

C = The constant to be employed.
e.g. 1.45 x root 29 = 1.45 x 5.38 = 7.8 knots

Gives the displacement hull speed, i.e. the maximum speed of a conventional hull driven by an engine of reasonable size, or by sails. This equals the max speed on a reach for a monohull sailboat, though very light boats could exceed this. More sail area on a conventional boat won't improve matters, as you will simply fall over. The speed drops slightly to windward, and also downwind under white sails only; nylon sails will improve that, and a light boat can fly downwind with wave assistance. In theory, almost any hull can be made to exceed its hull speed by fitting a large-enough engine; hardly desirable or even possible in practice.

Heavy boats with a high D/L value will have trouble reaching these speeds; light boats with low D/Ls may exceed them.

Note that a Constant of 1.34 is often wrongly applied.

The Constant varies for waterline length as follows:

      15 - 20' W/L   = 1.34;
      20 - 30'           = 1.4;
      30 - 40'           = 1.45;
      40 - 50'           = 1.5

Ensure that the method of calculation you use, and the Constant employed, results in a boat with a 45-foot waterline achieving 10 knots (or thereabouts – you can debate the second and third decimal places in the Constant, and, who knows,1.49 might be more accurate); any method of calculation which gives a slower speed than this is wrong. Using C as 1.34 gives just under 9 knots, for instance. That is not to say that a given 45' w/l boat will achieve 10 knots; that is another matter entirely.

If you run the full calcs for a very small or a large vessel you will see that the constant needs to be larger than 1.34 for both tiny boats and large boats. Run it through for a model yacht, for instance, and you'll find it should be slower than a snail; which is not possible. Much the same applies at the other end of the scale.
This indicates that a graph of the constant versus boat length will therefore be a shallow U-shape, probably more of a dish shape. Presumably this dip in the value of C is related to the physical characteristics of water – its specific gravity (or 'weight') and its viscosity; if it were lighter and thinner, or heavier and more dense, the value of C would change. I don't know the answer to this and similar problems because I'm an engineer not a rocket scientist. I deal with simple real-world problems, and find simple but obvious solutions; which nevertheless don't seem so obvious to others.

The popular constant of 1.34 was apparently introduced by a designer working on heavy dayboats and small cruisers of just under 20 feet in length. He needed a quick guide to their likely hull speed, and was fed up using the long-winded calculations previously employed. The same thing happened with astronav, spearheaded by Mary Pera. The old spherical-trig method went out the window, and the Air Tables came in.

This subject and related issues are never discussed, probably for the following very good reason: there are many yacht designers who haven't figured out that C @ 1.34 is incorrect. Naval architects who know better, certainly aren't going to tell the opposition. Why tell a competitor? You'd think that the sheer number of people reporting better than 'hull' speed for every kind of boat, with calibrated instruments, would eventually make them wake up...

My reasons for providing this information

First of all I am fed up with arguing about this and having to state all these facts repeatedly – it's easier to write them down once and then just point people to the data. It is also easier to justify a statement if all the facts are presented lucidly; after a couple of pints or more there is a risk that clarity is receding rapidly.

As an engineer I abhor myth, mystery, and crooked thinking – which, let's be honest, is almost a definition of the state of mind of us sailors. It takes one to know one. I would love to be able to help drag UK sailors, kicking and screaming, into the 21st Century; though it will probably take another 80 years to accomplish that...

Our activity is founded on the perception that old is best, and the old way is the best way. Once something has achieved acceptance among our fraternity, it is there for good – be it true or false. Look how many people, for instance, will tell you that ropes are never called such aboard ship. Rope should apparently be renamed as line, sheets, and so on, as soon as it comes aboard – but this is utterly, totally wrong and has certainly never been the case on ships (a square rigger couldn't even be sailed without the foot ropes on the yards; and the bell ropes on the pumps, or even the man ropes on the gangway). It may of course apply in the strictest sense to Solent yachts; but that's another story...

It mainly seems to be sailing instructors who spout this type of garbage, as well; so if they don't know what they are talking about, what hope has the poor old ordinary yottie?

I do hope you take my point: that what you are told, what you read, and what you believe to be true can, in the sailing world, be a long way from the truth.

In the case of the Hull Speed Constants given in the above section, these are the facts. I know a (real) Naval Architect who does all the calculations for three famous Yacht Designers who you will certainly have heard of, since they are household names in the sailing world. At least, there are the three I know of – but there may be more he has not mentioned. This NA works in the field of steel ships and mega yachts, and is in demand worldwide for his technical ability. The designers draw the nice boats – he does the maths. That's the way it is; and if you think that Famous Yacht Designers are all capable of understanding or carrying out the complex calculations sometimes necessary, then you will probably also believe in the Tooth Fairy and the honesty of advertising.

Naval Architects do the maths; Yacht Designers do the style and the marketing. There are very, very few Designers like Bergstrom or Lavranos, capable of understanding the science behind their work. Many Designers are ex-boatbuilders, who know exactly what it takes to build a good, reliable, stylish, efficient top-quality sail or power boat. They design boats, get them built, and the best are then slightly modified to provide the next model. This is modified again, to provide the next design after. The results are often excellent boats – since this is the standard method of progression in boat design.

But don't start ascribing them qualifications in marine science or naval engineering, as this just isn't their field. They employ a professional NA to do the maths and tell them what the figures mean. He will be based in the shipbuilding field, where they can afford him, and where budgets are expressed in multi-millions and answers have to be right: look what happened to the poor men who designed the P&O Canberra, with its incurable propshaft vibration; and the Titanic, with its half-height bulkheads (Damaged Stability calcs sure took on a new significance after that). One committed suicide after the maiden voyage; one went down with the boat, and both live on in shipbuilding lore as The Men Who Got It Wrong In A Big Way.

And if a Yacht Designer tells you that a Hull Speed Constant of 1.34 can be applied across the board, just smile brightly and turn the conversation to golf...

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L/B (length to beam ratio) = length / beam
LOA and Bmax is used.
e.g. 37 / 11.5 = 3.22

An indicator of the overall fineness of the hull. A more useful value for functional purposes might be LWL / BWL; however, BWL is rarely available.

Note that you must evaluate L/B ratios entirely differently for different hull lengths, which applies to this formula more than any other. These could be divided into groups such as 20 - 28 feet, 28 - 40 feet, 40 - 55 feet, and so on. The interpretation below, of values for boats of around 28 - 40 feet, has no relevance at all to those outside of this group. For instance, in a 60-footer, a figure of 4 would merely be moderate. The opposite applies at the lower end of the 20-foot range.

Approx. values:
4         = a pencil;
3.75    = slim;
3.25    = average, enough volume for cruising purposes;
3         = slightly beamier than average, and plenty of volume;
2.9      = any modern AWB (average white boat)
2.75    = fatboy;
2.5      = a caravan.

Again, perceived values are changing, and boats becoming beamier, as more accomodation and better downwind performance is sought, at the expense of upwind performance (the engine can be used) and ultimate stability (no longer seen as a vital consideration for the majority of boats). In any case, the EU RCD says a 105-degree Vanishing Angle (Avs) is OK for Category 'A' Ocean, and that's a new boat ex-factory, which will certainly see a good ten degrees off that figure a couple of years down the line – so why worry?

Therefore, a future interpretation might read:
3.5      = surely that wouldn't float?
3.25    = anorexic;
3         = on the slim side;
2.5      = normal, perfect;
2.25    = superb accomodation;
2         = palatial internal accomodation, but seems to require a lot of engine power to shift it.

In the final analysis, these figures don't indicate either a 'good' or a 'bad' hull shape; though they give pointers as to performance, i.e. as measured by speed overall. In a large racing boat, a slim hull (and enough ballast, together with a sufficiently-deep keel) indicates good performance to windward; but the old Norwegian Colin Archer sailing lifeboats were not slim by any definition. They did however fulfill their designed purpose admirably, which was to rescue fishermen, and their craft if possible, in appalling conditions: they could beat off a lee shore in a gale, while towing several disabled fishing boats. Speed was unimportant, but they needed muscle. They also have amongst the highest CFs measured (Ted Brewer's Comfort Factor), reportedly 60 for some models – and that's high.

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Waterline Beam, or BWL

This is a useful number for various calculations, but hard to come by, even on your own boat. An accepted average figure is 0.82 BOA – i.e. 82% of beam overall, the standard beam dimension.

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Half-Entry Angle

This is half of the included angle presented by the immersed bow at the waterline. It gives an excellent guide to the relative fineness of the foreship, and is easier to obtain than the foreship CP or Prismatic Coefficient.

Range: from about 10 degrees for long slim racers, 12° for long and super-skinny racer/ cruisers, to 25° or higher for the everyday cruiser.

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CB (Block Coefficient) = disp / ( LWL x BWL x HD x 64)

where:
HD = hull draught, without foils

A measure of how 'blunt' the boat is, and therefore how much power is needed to push it through the water.

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WSA (wetted surface area) = LWL (BWL + D) x C
or, waterline length, by waterline beam plus draught, by C

e.g. {29 x (10 + 5.25)}.65 = 287.46 sq ft
or {8.86 x (3.06 + 1.6)}.65 = 26.83 sq m

The Constant is x .5 for fin keel racing yachts, .65 for full-bodied or bilge-keel yachts, and .75 for motor boats. Interpolate for a constant for your boat.

This is an indicator of light / medium air performance; also useful for antifouling paint calculations.

For painting the immersed hull, divide the metric final figure by the paint coverage in litres per sq m, usually 8 - 10, for the amount of paint per coat. Thus:
26.83 / 9 = 2.98; or about 3 litres per coat.

In my experience, this can prove a little optimistic; I add .1 to the Constant for my boat, and use the minimum coverage figure (maybe I'm heavy on the brush or the paint manufacturers are fibbing – all of them). For the boat in the example, 3.5 litres per coat is nearer the mark.

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Topside area = (LOA + BOA) x 2FBa
or: length + beam, by twice the freeboard.
where:
FBa = average freeboard

This is the area of the topsides, for paint calcs. The metric value is used, since this is how paint is sold.

e.g. for a Centaur, which is a monohull sailboat of approx. 26 feet x 9 x 2.5 freeboard, to get the total # of tins required for 2 coats, the paint being sold in .75 litre tins, then convert to metric and:
(8m + 2.75m) x 2 x .75m = 16.1 sq m.

Divide the resulting topside area by the paint coverage per sq m, to get an answer in litres required per coat; e. g. 16.1 / 12 = 1.34 litres.

Multiply by the number of coats, to get the total litres needed;
e.g. 1.34 x 2 coats = 2.68 litres

Divide by the amount per tin (as it is unlikely to be sold in 1-litre tins);
2.68 / .75 = 3.57 tins, e.g. 4 tins will be needed for 2 coats. Undercoat and topcoat would usually be required.

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