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

[Gregg Senne]

Gregg, have a look at post #38. Choosing the model you have presented is quite a simplification of what might be considered appropriate for addressing a real mast condition.

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.

 

[gonzo]

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

 

[sharpii2]

I've been arguing with a sailor friend of mine for a while now, as we are trying to find a reliable method of sizing a sailboat mast. For wooden masts, Norman L. Skene provides the following formula:

Mast diameter = ∛(16*P*L*SF/15700), where P is wind pressure (sail area in sq.ft. multiplied by a coefficient of 1.15 - 1.5, depending on the size of the boat), L is mast length in inches, and SF is safety factor. 15700 is π multiplied by fiber stress of spruce, which is 5000.

Obviously, this formula only works for solid, round masts made from spruce, because that's what Skene had in mind. It isn't too difficult to size a different type of mast - for example, a box section, or a hollow one. All you have to do is run the Skene's formula, get a diameter for a round solid mast, determine the Moment of Inertia of it's profile, and then run your numbers for a different profile type mast you want to use until it's Moment of Inertia matches. All should be good, because the material properties remain the same.

Example:
Say my boat has 60ft of sail area, and a 100" long mast. Since it is a small boat, the wind pressure coefficient is 1.15. Skene's formula says such a mast should be 2.2" in diameter. Moment of Inertia of a round solid profile is around 1.15 in^4. But I want to build a box section, hollow mast. I fiddle around with numbers, and determine that a square 2" diameter section with 0.4" (13/32") walls would also have a 1.15 in^4 Moment of Inertia, so it should be just as stiff and strong as the round solid 2.2" mast.

So far, so good. I can even determine how much this mast would bend in a breeze. We used the 1.15 coefficient, so it's 69 lb of pressure on that 60 ft^2 sail. Center of effort is, say, 50" above mast partner, and the luff is attached, so using the Cantilever Beam deflection formula for concentrated load P at any point (it's not really concentrated, but let's do concentrated for simplicity), we get:
θ = Pa^2 / 2EI, where P is is the force, a is center of effort above mast partner, E is Elastic Modulus of spruce (9 GPa), and I is Moment of Inertia. Once you plug in the numbers, the result is 6 3/4" of deflection at the mast top. Sounds about right.

Now, the problem. What if I decide I want to make an aluminum mast instead, using a simple aluminum tube commonly found in hardware stores? Obviously, aluminum is much stiffer, yet much heavier. Running all the numbers again, I could easily calculate the diameter for an aluminum mast which would have the same deflection as the wooden one when subjected to the same wind pressure. I guess the stiffness would match. But I doubt the ultimate strength would match too.

I am an amateur in engineering field, so this is where my knowledge ends. I know that each material has it's Yield Point, past which it will deform permanently, and break soon after. No sailor wants to see that happen to his mast. Obviously, different materials have different amount of elasticity. A brick won't bend as much as a piece of rubber before breaking. Wood and aluminum probably also have a very different Yield Points. For example, I can calculate that my 100" mast will bend 6 3/4" in specific load conditions. But will it recover from such bend? Or will it stay deformed? How much can it bend before it breaks? 7"? 10"? 15"? These are the questions I cannot find an answer to. My sailor friend says that the deflection which would still be okay for a wooden mast might deform an aluminum one, or possibly vice versa. We do not know how to find out how much of a deflection would force the mast past it's Yield Point.

So, for those still reading, I'd like to ask for some help - can you teach me the way to find out how much can a mast bend before it is permanently deformed or broken? How do you calculate that critical amount of deflection?

Thank you in advance. And sorry for the long post, but I wanted to show what I already know to save you the trouble of explaining the basic part of beam deflection

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.