How does one determine the yield point of a wooden or aluminum mast?

My project is a 15' gaff sloop. The mast will be a solid round section of Douglas fir with no taper for the lower 11'. There is no standing rig, so the mast is for practical purposes a cantilevered beam. Still, figuring out the loading is a bit tricky. Some of it is distributed, some of it is point loaded. The deflection at the point of support will be the sum of the loads, I assume. The model offered is just an illustration of a simple case for the purposes of getting some idea of the magnitude of the forces involved. Unless someone offers a free FEA download, I'm limited to the various formulae I can find. The question remaining is, how close is close enough? If my mast weighs an extra two or three pounds I'm good with that.

 

Try this, they are the classic solutions
Beams - Fixed at One End and Supported at the Other - Continuous and Point Loads https://www.engineeringtoolbox.com/beams-fixed-end-d_560.html

 

I think you need to consider everything that is happening to the mast, not just the bending loads.

There are compression loads as well, especially at the partners.

The Lee side of the mast gets squished against the Lee partner. The windward side gets stretched as it tries to hold the mast straight.

Your task is to figure out which side will fail first.

As you probably already know, there are four kinds of loads in structural engineering:

Compression,
Tension,
Bending, and
Shear.

But the most important ones are the first two, especially now that we are working with a uniform material--aluminum.

The problem with an aluminum tube is that the bending resistance is uniform the entire length of the tube, but the bending loads are concentrated at the partner, and diminish up and down from there.

What this means is that the top of the mast is going to tend to stay straight, moving the bending stress further down, so it will fail in bending sooner than a more tapered mast.

All this means is that the wall thickness of your mast is going to have to be greater than if it were either tapered or locally reenforced.

What you need are:

1.) the yield strength of the particular alloy you are using (there may be two or more. Pick the lowest one. If the yield strength is least in compression, that's the side your mast is going to fail in).

2.) The maximum righting moment of your boat, converted to inch pounds or cm kilograms, and

3.) the Sectional Modulus at the partner (or the whole mast).

The Sectional Modulus can be found by measuring the distance between the center of the mast section, refered to as the "neutral axis", and the outside of this section, called the "extreme fiber", and dividing this number into the into the inertia of this section.

This Sectional Modulus is then compared to (the bending load, divided by the yield syrength).

If it is greater, your mast should be strong enough.

But this is in a theoretically perfect world. You should have some sort of safety factor. In the aircraft world, such is typically 1.10.

But, since sailboats tend to suffer more severe shock loads, I'd go with at least 1.20.

This safety factor is multiplied by the original bending load, divided by the yield strength, then compared to the sectional modulus.