Hollow unstayed birdsmouth mast

[Gary Dierking]

If I know the proper end diameters for a tapered solid unstayed mast, does anyone know the formula for how much increased diameter is needed for a hollow spar?

[Jack Heinlen]

Seat of the pants, none. A spar, hollow or solid, carries the majority of its lateral stress in its skin. Taking out the center does virtually nothing to that strength, though you will lose a modicum of compressive strength, which can be an issue. Not much in an unstayed rig. Build it to design scantlings.

[ 08-31-2005, 06:48 PM: Message edited by: Jack Heinlen ]

[George Roberts]

Gary Dierking ---

5-10%

[Frank Wentzel]

The information you want is in "Skene's". I'll post it when I get home tonight. If memory serves it is between 10 and 15 percent.

/// Frank ///

[Keith Wilson]

An unstayed mast is essentially a cantilevered beam with the load distributed along its length; exactly how the load is distributed is hard to figure out, but depends on the sail – anyway, for this case it doesn’t matter. Here’s a page with the formulas. (http://www.engineersedge.com/beam_bending/beam_bending8.htm) This is for a beam of constant cross-section, but if the two masts you're comparing have essentially the same taper, that cancels out.

Deflection, when you are comparing two masts of essentially identical shape, is proportional to the moment of inertia, which for a round tubes (or solid rounds) in bending is proportional to (D^4-d^4). D is the OD, d is the ID. Stress is proportional to the section modulus, which (again, when comparing two similar masts) is proportional to (D^4-d^4)/D

So if we want a round hollow mast with OD=D2 and ID=d2, that has the same section modulus (comparable strength) as a solid mast of known diameter D1, you have (D1^4)/D1=(D2^4-d2^4)/D2. For example, if you have a 3” solid mast, and you want a comparable hollow mast with a ½” wall, 3.25” OD would work. Because the section modulus is proportional to the cube of the diameter, you don’t have to make it much bigger. A hollow mast of comparable strength will be somewhat stiffer than the solid mast of comparable strength.

Here’s a simple calculator (http://www.engineersedge.com/calculators/section_square_case_12.htm) for the moment of inertia and section modulus of round tubes; you can put in zero for the ID of a solid mast.

This assumes that you don’t make the wall very thin; the failure mode is different in that case.

If you play around with it, you’ll find that the larger diameter and thinner the wall, the lighter the weight for a given strength. For example, a 4-1/2” diameter mast with a 3/16” wall has the same section modulus as a 3” solid, but 35% of the weight. That’s way too thin a wall in wood, I think; I expect it would fail by splitting along the grain unless perhaps you could find a way to make a hollow plywood tube.

Anyway, for reasonable wall thicknesses (ID is 65-70% of OD), 10% additional diameter is fine, 20% is conservative. Thinner walls need more diameter. Hollow masts are much stronger for their weight.

A disclaimer - I design industrial machines for a living, not sailboat masts. If mmd or John Hardiman or somebody who does this all the time comes along and points out where I goofed, believe them, not me. If you are going to build a boat based on this and sail it around Cape Horn, get a professional to do the calculations.

[ 09-01-2005, 10:51 AM: Message edited by: Keith Wilson ]

[Dave Hadfield]

I dunno.... I sailed in my dinghy one day last July and got caught out in winds of 25-30. It has a leg-o-mutton sprit rig (fortunately with a set of reef points). Unstayed.

The mast had to bend like hell to survive, and did. I was amazed at the curve, but it did tend to feather the gusts. It's made from 2 full-length pieces of construction-grade spruce, glued together. Lots of little knots (cut from a 2 x 12). It's 16ft long.

I think "bendiness" may be better served with a solid stick. After all, whoever made a hollow archery bow?

In a mast of this size, which hardly weighs anything anyway (2 1/4" sq at the bottom, 7/8" square at the top) making it hollow wouldn't save much.

[Jack Heinlen]

There was an article in WB five years back about the ins and outs of the birdsmouth method. That would be worth looking up.

It's interesting, to a point, reading the engineering details of stressed beams.

Don't sail around the horn on it but the engineering numbers on a hollow unstayed spar are going to be near identical to a solid spar. If it had stays the pressures would change to largely compressive, but unstayed it's like a wand in the wind, blown sideways according to the press of sail. And, I think, the hollow is going to react the same way as a solid, with just a bit more flex.

How a solid wood v a built spar, both without rigging, reacts is not likely found in Skenes.

If it makes you more comfortable, beef it up by ten percent, but I don't think it's necessary.

[kc8pql]

Practial Junk Rig by Hasler and McLeod has a lot of information on unstayed hollow wooden spars for larger boats. If the sail area is 150 sq.ft. or more, it may be worth a look. Unstayed masts are huge compared to stayed masts for the same sail area. A 7' dia. stayed mast goes to 11" dia. unstayed, given the same sail area and depth between the partners and step.

[Frank Wentzel]

Data taken from Skene's Elements of Yacht Design, Third edition, copyright 1925. He compares a 4 inch diameter hollow spruce mast (length is not specified) with a 1/2 inch wall to three variations

4" Hollow --- Weight 11.5lb --- Strength 8,500lb
3 5/8" Solid --- Weight 21.6lb --- Strength 8,500lb
2 5/8" Solid --- Weight 11.5lb --- Strength 2,250lb
4" Solid --- Weight 26.2lb --- Strength 12,700lb

Therefore:
-A 10.35% increase in diameter over a solid mast yields equal strength with a weight savings of 53%.
-A solid mast of equal weight is only 26.5% as strong as a hollow mast.
-A solid mast of the same diameter as a hollow mast is 50% stronger than a hollow mast, but 128% heavier.

/// Frank ///

[ 09-02-2005, 08:04 AM: Message edited by: Frank Wentzel ]

[ishmael]

Corrected again, Frank.

With a small spar on a small boat, people ought relax a little. Yeah, build the spar hollow to guessed scantlings. Or build it solid to published scantlings. When it fails report back. Even the stoutest small-boat spar fails on occasion.

Interesting discussion, carry on, by all means.

[George Roberts]

"A 10.35% increase in diameter over a solid mast yields equal strength with a weight savings of 53%."

I think that should be 47%.

[Boatmik]

http://members.ozemail.com.au/~storerm/rowbutton.jpg

Howdy,

There are a couple of downloadable utilities that the wonderful Mr Carlson has made available to assist in working out the relative strengths of birdsmouth masts.

Be aware that the utility with the more complex interface sometimes does not provide quite enough meat for the finished diameter - so check the graphics carefully and do a little bit of math or countercheck using the other program. The shortfall is only very small so is probably not important in strength terms

We have done several mast designs this way - so below is the method for producing a tapered hollow mast - and lucky it is easy for you to work out because of Mr Carlson's generosity - we had to work it out for ourselves!

You can calculate the strength/stiffness of a solid mast by manipulating the stave thicknesses to fill the interior of the mast.

Then the solid mast and the birdsmouth mast have to be designed to the same bending stress.

Be aware that there are significant crushing loads at deck level which have to be dealt with for a freestanding mast - perhaps make that section solid, or build up glass on the outside or deal with it by some other method of your choice.

Similar with loads from the boom - they need to be considered.

As far as taper you can take the original taper schedule of the solid mast and work out the dimensions of the stave mast to match the solid mast at different heights.

Then fair the stave widths at the different heights along the length of the birdsmouth staves to provide a smooth taper.

It is a few hours of careful work, but it is not too difficult.

Best Regards,
Michael
web page (http://www.ozemail.com.au/~storerm)

[Boatmik]

Originally posted by Boatmik:

You can calculate the strength/stiffness of a solid mast by manipulating the stave thicknesses to fill the interior of the mast.

Hi,

Just realised this was a little ambiguous.

Use the Carlson program to calculate the stiffness/strength of a solid mast by inputting the diameter and then setting the stave thickness to fill the interior of the mast. The program then outputs the physical properties of the solid mast.

Then you start again and manipulate the diameter and wall thickness of the hollow mast to match the physical properties of the solid one.

MIK
web page (http://www.ozemail.com.au/~storerm)