   ## Basic Buoyancy Calculation

QUESTION: Is there an easy way for me to calculate how much buoyancy the hulls of my new design will have?

### Basic Buoyancy Calculation—an old, but easy way

So, you sketched out your dreamboat, but now you want to know IF it will float - and to what waterline !
If you've drawn your hull shape with boat or ship-based software, there's an excellent chance that you already have all the basic displacement information that you need. No need to read further, if so. But if you've done this 'the good old way', by hand, with pencil and paper, then you'll need some other approach to find out whether you've enough volume below the waterline to support your estimated weight. For 200+ years, shipyards the world over have used a very close approximation for volume, based on something called Simpson's Rule. It basically uses the sectional area of each station, equally spaced over the boat length, and multiplies them by that spacing to estimate the volume—though there are a few tricks to making this work. We'll get back to how Simpson's Rule does this, but first, we need to figure out the sectional area of each station, up to whatever waterline you wish to calculate. This waterline will typically be the Design Waterline for the main hull of a trimaran, but could be the total immersion waterline for an ama! We used to do this using a relatively ancient device called a 'planimeter' that has little wheels that freely roll around while you run a pointer over the line and there's perhaps still the odd one available on eBay—but here's an easy way to do this that is about as accurate IF done with a little care.

For an ama, one way is to divide the centerline of each section into an equal number of parts (say 6) and then measure the waterline ½ breadths. From this information, we could even use Simpson's Rule, provided the spacing of the waterlines was equal all the way up. But for a small boat like we are considering here, these areas are pretty small and would need drawing out large scale to measure the waterline widths accurately.

So let's look at another approach that works well for do-it-yourselfers. This requires that you draw out your sections on graph paper that has squares drawn on it. Typically, one will use a paper with say 1" squares divided into 10 in both directions, though metric paper can achieve the same result as long as the paper is divided the same, both vertical and horizontal. (You can also use the cheaper squared-paper used by school kids, but then you'll need to draw out your sections even larger for enough accuracy). So after having decided on a suitable scale for your drawing, mark each one off on your graph paper, and here's how a section will look. Now mark off all the squares that are totally inside your station line and divide them into logical blocks for easy counting. Then, running down the curved edge of your section, 'trade-off' what is 'in or out' of the line, so that you come close to approximating the equivalent number of FULL squares.  Little point in trying to be more accurate than ½ a square—you'd be surprised how accurate this method is, as you are only making your approximation on a very small part of the total. Jot down the totals and then add them at the side. Check the totals a couple of times as it's easy with so many figures, to add them in twice—note the corrections on this sheet ;-)  Do this for each equally-spaced station and then you'll be ready to use these figures to calculate the hull volume—this time using Simpson's Rule. Remember that you ONLY need to measure the section area up to the waterline that you are considering. It's essential of course to do one of these checks at the design waterline, but it's also good to do one say 50 or 100 mm further up, as this will help to determine how many lbs or kgs it will take to sink the boat say 1 inch or 1 cm. Also, if you take moments about amidships of these section areas, (by multiplying their distance from amidship) you'll then be able to figure out where is the LCB (Longitudinal Center of Buoyancy).  Your center of gravity of all the weights, including crew, will need to be vertically over the calculated LCB if the boat is to trim EXACTLY to your designed line. (This is seldom achieved on a small boat unless the crew are well placed—but the results will show what we learn in practice, that quite often the crew needs to be more forward to keep the bow down, until your speed develops enough hydrodynamic lift to carry more weight aft). Initial changes of trim will take place pivoting about the LCF (longl center of flotation), as that's the center of your current water-plane.

I'll attach a sample spreadsheet at the end of this article that will guide you for entering this information. Keep in mind, that you've probably only calculated the area of HALF the full station, so somewhere towards the end, you'll need to multiply by 2, to cover for the other half.

OK—now on to the most famous Simpson's Rule (yep…he created more than one).

Now you'll already find several articles explaining this on the Internet—but if you look them over (even the 3 links that I'll include here), you might find them looking more complicated than they really need be. So here's my attempt at assisting you to use this method, without too much mental agony.

### Calculation of Volume of Body by classic Simpson's Rule

(As used for 200+ years in basic naval architecture—Simpson was an 18th century Brit)

First, check these out—see Composite Simpson's Rule, 2nd formula…
http://en.wikipedia.org/wiki/Simpson's_rule
An easier to follow version is found here
http://www.intmath.com/integration/6-simpsons-rule.php
But if you want to get totally rattled … just watch this YouTube video!
(That's what can happen when you ask a mathematician to explain something simple ;-)

So now let me take you through this my way…

This rule is based on a version of an older trapezoidal rule that's then compounded.
So - STEP ONE
It can be proven (you don't need to know how ;-) that if you take almost any shape (without significant reverse curves) that sits on a straight baseline, that its area can be closely approximated by: 1. Dividing it into two with a line at center. ie: by adding line C with h = h (h is then known as 'the common interval')
2. Now , just multiply the value of ordinate 'a' by ONE, ordinate 'c' by FOUR and ordinate 'b' by ONE and add the results
3. You then multiply your result by 'h/3' to find the Area!

(Why '3' you may ask?  I'll explain later.)

So if you want to find the area of something larger, you can compound this simple 1‑4‑1 rule by imagining adjacent areas with the same pattern repeated.
See here… Now this can be expanded as required—to any EVEN number of divisions—8 are shown here.

So , to calculate the area of a figure with 8 ordinates (or stations), we multiply the length of each ordinate by the multiplier shown—namely, 1st one by 1, the 2nd by 4 the 3rd by 2 etc etc , to always finish with the 'last one by 1'.

If you now add the results of these together, you end up with what we call a 'Function of Areas' [F of A].
(this is because it's 'a function' of the area but not 'the actual' area…not yet.)

To get there, we have to introduce the horizontal spacing of each ordinate, 'h'.
This we divide by '3' as noted in the first example. Why '3'?
Well, that's the AVERAGE of the multipliers! In the above example, add all the multipliers together and you'll find it gives 24. Now being as you have 8 divisions, the average multiplier will be 24/8, or 3!  Regardless of how many even-number of ordinates you have, it will ALWAYS be 3!

So, we have our first FORMULA for Area:

 Area = [F of A] × h 3

But now…how about the VOLUME ?

Well, IF the ordinate heights on your plotted shape now represents the AREA instead of some line length or ½ breadth, then the result of your new 'area' calculation, will indeed BE the volume!

So, looking at this sketch below, the vertical ordinates can EITHER be 'measured LINE lengths' (like ½ breadths of a waterplane), in which case you'll get the AREA of the shaded shape. OR, if you plot the SECTION areas of each station (taken from a sectional view called a 'Body Plan'), then the Area of the shaded shape will in fact allow you to compute the VOLUME of that shape—again, after multiplying (what is now a 'Function of Volume') by h/3. You may also need to multiply by 2 for both sides. So you just need to establish a scale for the ordinates: such as 1 cm = 100 cm or something that gives you a workable diagram. As little as 4 subdivisions of your shape give a rough result, but typically, we use 10–12 for a boat hull—even 20 for a ship. With 10 or more, the end result is surprisingly accurate and totally adequate for all calculations—having been used for centuries for even major ship design work—right up to the 1960's when computers came in.

Fifty years later, there were still a few individuals that use Simpson's Rule and it's something useful that one can even do on a scrap of paper. Typically you just enter some columns on a spread sheet and let the computer do the simple mathematics. Also, if you take moments of the ordinate values from a specific point (like Station 0 or the amid-ship point), you can also calculate the center of the area to find the longitudinal center of flotation (LCF) or the longitudinal center of buoyancy (LCB) if working from the curve of volume.

A worksheet for making such a hand calculation for volume, might look like this. Hope this helps the do-it-yourselfers out there — Have fun, but check yourself 2 or 3 times!

mike…jan 2012

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