Bird's Mouth Spars revisited By Gaétan Jetté

Introduction

A few months ago, an interesting article on bird's mouth spars was published right here, at Duckworks. The author, David Farless, presented a series of equations to accurately calculate the size of such a construction. This has inspired me to study this subject in greater depth. In this article, we will explore some of the limits and potentials of bird's mouth spars. You will also be provided with tools to let your computer do the size calculations for you. But first, a quick reminder...

What is a bird's mouth spar?

A bird's mouth spar is a way to make a hollow mast or spar from wood for use on a sailboat. Figure 1 below shows what the cross section looks like. It typically uses 8 identical pieces of wood or staves glued together. The outside is then trimmed to produce a round spar. The main goal is to save weight.

Figure 1- Bird's Mouth Spar : Definitions O.D. is the outside diameter I.D. is the largest inside diameter R1 is the shortest inner radius R2 is the largest inner radius or half I.D. N is the number of sides (eight in this case) (alpha) is the angle between adjacent staves or 360° / N L is the width of material H is the thickness of material Important Ratios K is the thickness to width ratio of staves or (H / L) M is the "conversion factor" or (O.D. / L) A is the inside to outside diameter ratio or (I.D. / O.D.)

The definitions used here are pretty much the same that were used in the previous article. I have only added a few names for the ratios that will be of interest here. The model shown in Figure 1 is of "classic" proportions, that is, a stave's thickness ratio of 0.5 in an 8-sided configuration. Although I have never seen other configurations, it is possible to build a bird's mouth spar with a different number of sides. Figure 2 shows some of the possibilities.

Figure 2- The world of Bird's Mouth Spars

Okay, you might think that the math for size calculations would be far too complex for such configurations, right? Well, no! The equations presented by David Farless in the previously mentioned article had a hidden potential. With just a minor modification, these equations can handle any possible configuration! From now on, we will refer to these as the Farless equations. (Always give credit when credit is due!) The revised equations can be found in the Appendix section.

The trick for building a spar with more or less than 8 sides is in the V-notch angles used on the staves. Figure 3 shows them in a little bit more detail than Figure 2. Table 1 gives you the angles used and a few other points to take in consideration.

Figure 3- Stave shape

Angle 1, shown in Figure 4 below, is located on the inside of the spar. Its value is always the same as "" (alpha), shown in Figure 1. Angle 2 is on the outside and is equal to 90 degrees minus the first angle.

Figure 4- Stave Angles

Table 1 also lists minimum and maximum limits for the thickness ratio of the staves (K). The minimum theoretical limit is reached when I.D. is equal to O.D. Figure 5 below shows an example of K going too far in that direction. In practice, you obviously need to use a higher ratio to have some thickness left! Otherwise your spar will fall apart when you try to round the outside. That's where the practical range comes into play. The minimum practical limits shown in Table 1 corresponds to the maximum weight saving possible for a given strength. (More on that later.)

Figure 5- Going too far: K is too low, I.D. is larger than O.D.!

The maximum theoretical limit is reached when I.D. equals zero (See Figure 6B). You can actually exceed that limit: the inside will open up again (Figure 6C). However, the formulas for calculating size can produce some weird results when you go in that zone. Better stay below that maximum limit. Here again, a practical upper limit for K has been shown. This maximum practical limit was chosen at the point where the weight saving falls to 5 percent compared to a solid spar of equal strength. (More on that later too.)

Figure 6- Going too far, high values of K A- "Practical Maximum" B- "Theoretical Maximum", I.D. = 0 C- Gone too far, formula results unpredictable

You may have noticed that both the 5 and 20-sided configuration have the comment "not recommended". Here is why. For these, one of the angles used is very sharp. This means that the corresponding side of the V-notch becomes very thin (See Figure 3). This may be too fragile, at least during assembly. But this may very well be a weak spot under stress, even when glue has cured. This potential problem would only get worse for any larger number of sides and sets the limit of what you can do. This is one reason why 8-sided spars are far more common. For an 8-sided spar, the angles are equal and each side is of equal strength. And the glue area of the V-notch is also the largest at 45 degrees. So it is probably not a good idea to go very far from that 45-degree value. It may be wise to use a number of sides between 6 and 12. The smallest angle never falls below 30 degrees with these.

Configurations using 7, 11, 13, 14, 17 and 19 sides are not discussed here: the angles required with these are just too odd.

Now that we have looked at some of the limits of bird's mouth spars, let's look at ways to simplify size calculations.

Size Calculations

For those of you who have read the previous article written by David Farless, you already know that accurate size calculations are not really simple. That article also mentions a "rule-of-thumb" solution originating from a WoodenBoat article (I must admit that I have never read that article). The outside diameter was defined as 2.5 times the stave width. However that solution is not entirely accurate. In fact, both the width and thickness of the staves have an effect on the resulting diameter.

Wouldn't it be nice to have a solution that offers the simplicity of the WoodenBoat rule-of-thumb and the accuracy of the Farless equations at the same time? Better still, wouldn't it be great to have a computer program that does the calculations for you? Well, that is exactly what we'll do!

First, our new rule-of-thumb. If you look back at Figure 1, among the definitions, there is a ratio called "M". It is defined as O.D. divided by L. That is exactly the same thing as the "2.5x" from the WoodenBoat article. We "only" need to figure out what is the real value for "M". That ratio is defined in the Appendix if you are really "math curious". Table 2 below, however, will save you time. It gives you pre-calculated "M" values for 6 to 12-sided spars. The cells are highlighted in pink when K falls outside the "Practical Range" defined in Table 1. You should avoid these. No values are shown when K falls outside the theoretical limits of Table 1.

Choose the stave thickness ratio and the number of sides that you want to use, look up the corresponding value of "M" in the table, and you have everything you need to do the size calculations. Let's do an example.

You can download a text version of Table 2 and print it on a single page. (You may have to adjust the page margins and the font size to achieve that). With this table at your disposal, you can do the calculations by hand if you have to.

Let's now look at the easiest method: your PC doing the calculations for you!

Bird's Mouth Spar Size Calculators

Instructions

Choose the Calculator that suits your needs. Enter the values of your choice on the left, then click on the "Calculate!" button. Results will appear on the right. An error message will appear in the Status box if your inputs don't work out. You can clear all entries by clicking on the Reload or Refresh button of your Web browser. Or you can just enter new inputs and click on "Calculate!" again. Clicking on the pictogram in the middle will lead you back to Figure 1, in case you need a reminder for all those definitions. Dimensions must all be in the same units. Ex.: Inputs in inches give results in inches.

Hints to avoid error messages (in the Status box)

Input values must be numbers higher than zero (no negatives, no letters).
The number of sides must be 5 or more.
I.D. must be smaller than O.D. (obviously).
H or K must not be too small or too large (less obvious).
H must be smaller than the radius or half the diameter.