Converting wooden mast specs to aluminum

I am completing a cold molded Bill Garden catboat, 12’6” x 6’. He spec’d wooden spars. I wish to use aluminum in the interests of getting her sailing! (As Garden said “ Hop to it before they have to carry you aboard!” I’ve been at this project entirely too long already.)
The mast is 16’6”. If built of solid or hollow wood, he calls for 3” or 3.5” diameter respectively. He has detailed calculations on the plans. I’m thinking to use a 16’6” section of 3’” OD 6061 T6, with an 1/8th wall thickness aluminum tubing. How do I determine if this is enough, too much, or just right? This mast has a forestay only, no side stay.
thanks in advance for your thoughts.

 

Is the actual formula on the previous page? Curious as to the constant etc

 

Barry, Garden does have drawings and calcs on the previous page that support the numbers in this formula. I could post the page as a pdf if necessary although I was hoping we could assume his calcs are good and just plug in the aluminum numbers. Having said that, I must admit I am NOT a math guy. I can’t even get this formula to work out as he wrote it…
Looking up aluminum tubing on the net shows numbers vastly greater than wood. I’m hoping someone can tell me how to calculate and interpret those numbers in the real world of sailing. Grateful for all assistance!

 

Here are the supporting descriptions and numbers from Garden. Perhaps they will shed light on the calculations. It seems like the original formula is all apples and oranges…

 

The wooden mast is tapered which can't be compared directly to a constant section of aluminum. You can calculate the breaking strength of a wooden mast and then compare that to an aluminum section. However, the bending characteristics will be very different. A solid wooden mast can be made in an hour or so with a power planer. That's less than the time to do the calculations.

 

Giving this some thought it is not difficult to come up with an aluminum cross section that will carry the load that the existing solid 3 inch spruce mast would carry. I am assuming that
the exising 3 inch diameter is designed to allow for whatever factor of safety that you are comfortable

Not sure where you are on terminology therefore:
Load is a force-------------------units--pounds force
Stress is the force per area------unit--pounds force per square inch= sigma = symbol S
Moment is the multiplication of the equivalent force acting at some distance from the bottom of the mast to the spot where the variable loads acting on the mast can be replaced
by a single force. ------------------units-- inch x pounds (pound inches) = symbol M
c is the distance from the center to the outer edge of the cross section---unit inches
I is the moment of inertia of the cross section=units--inches to the fourth power = symbol I

The moment in a cantilevered beam will produce the maximum allowable stress of 6700 psi for spruce at the bottom of the mast. ( might not not precisely correct)
The calculations will be for a 3 inch solid spruce mast

Sigma =Mc/I

Sigma max = 6700 inch pounds
c - 1.5 inches diameter divided by 2
I for a 3 inch cross section is 4 inches to the fourth.

The Maximum moment that this mast can carry is
6700 = M x 1.5 /4 M then equals the moment that the forces on the sail will produce until the allowable stress is reached which is about 18,000 inch pounds

Taking this Moment and inputting into an aluminum hollow cross section

More definitions
Yield stress = the maximum stress that 6061T6 before it permanently stretches, ie will not rebound elastically to its original shape. 40,000 psi
Allowable stress= probably open to push back here but often a factor is introduced to deal with other metal failure issues. This is often taken as .8 of yield. So this number is 32,000 psi

So back to Sigma =Mc/I

Sigma max is 32,000, M is the same moment that the original mast was designed and that is 18,000 inch pounds. You want to see if a 3 inch x 1/8 wall mast will carry the same
as the 3 inch solid wood

32,000 = 18,000 x 1.5 / I I = 18,000 x 1.5 / 32000 = .844 inches to the 4th This is the moment of inertia that you would need for a hollow aluminum mast.

A 3 inch 6061T6 tube with 1/8 inch wall has an I of 1.2 prox. So the mast will carry the load.

Other factors to consider
1) you need to ensure that the mounting will not cause a stress concentration then failure
2) The modulus of elasticity of aluminum is about 9 times that of white spruce so it is stiffer

THE BIGGEST ASSUMPTION IS THAT THE 3 INCH SOLID WOOD MAST MEETS ALL OF THE LOADING THAT THE MAST MAY BE EXPOSED TO AND THAT A SUITABLE FACTOR OF SAFETY HAS BEEN INCLUDED.

 

That assumes the mast will not buckle at the partners.

 

Not sure what you mean buckle at the partners.

The mast would exhibit the max stress at the base of the mast. The loads and hence stresses would diminish moving from the base to the top of the mast. The aluminum is 8 or 9 times stiffer
than wood so that instability induced by deflection would be less than the wood mast.

Why would the aluminum mast be more prone to buckling? Just curious

 

The mast is fixed at the step and then goes through the partners. It is not a cantilever but a beam fixed at two points. The section at the partners is where they usually break. On a thin walled tube it will probably fail from buckling. Doubling that section or inserting a wooden plug would help.

 

Barry, I thank you for the calculations and results. My thought to use a 3” x 0.125” aluminum tube was basically a wild-*** guess on my part but it seems to bear out in your calcs. I bought that section in a 20’ stick. It’s something of an unnwieldy beast and I’m now thinking that the only way I’ll be able to step it is to create a tabernacle which I had hoped to avoid…. Could you, using your calcs, suggest the lightest section I might use, in both diameter and wall thickness? To your assumption that the wooden mast meets all the loading requirements plus a safety factor, Garden calls for a 3 1/4” solid wood mast..
Gonzo’s point about failure at the partners is well-taken. The mast comes through the deck at 24” above the step. That’s the most heavily loaded point on the mast and sailing dead downwind would exert the greatest load from the 120 sq ft sail.
I had a conversation with a local rigger of some renown here in Santa Cruz this morning. He suggested the following approach to the problem:
Take the distance from the deck at the partners to the center of effort of the sail. That’s 87”. Multiply that times the load on the sail at various wind speeds. I don’t know that number yet. Haven’t found the tables or a formula. The result is the pounds per sq inch on the mast/beam. Then figure the mast/beam diameter and wall thickness required to deal with those psi. He also thought an inner or outer reinforcing tube at the partners might be wise. I can tell you that I will not be sailing this little cat in more than 20 knots. And I have two sets of reef points specified for the sail…
Thank you for your help so far.

 

Your rigger has not mentioned that the main sheet will be taking some of the sail load which reduces the load on the mast.

I would not redo any calculations to suggest to you to use anything with a thinner wall. Nor would I say that your profile will work. Only suggested that the 3 x 1/8 wall will support the bending moment and keep the stresses below the alloys allowable stress.

The calculations made many assumptions mainly based on the fact that your first comment that a solid would mast with a 3 inch diameter would do the job. Certainly the 3 1/4 inch diameter
were in the calculations and but took 3 inches as this had been mentioned first.

While Gonzo said that the beam is a simply supported it is not past the spot where the mast leaves the deck of the boat.
The mast is cantilevered from that point up with the load reducing due to the shape of the sail.

The comment by Gonzo to reinforce the potential buckling area is a good idea.

I would check Facebook Market Place as I have seen many used masts for sale for pocket change. Including stays, the vertical track, the top hook and pulley.