Rotating Free Standing Mast Design

Hi all,
Over the last week or so I have been studying the design of free standing, rotating masts. I am not a NA and I am not studying to be a NA. Nor am I an engineer. I just like intellectual challenges and I like to understand how things work and why. This is just for my own personal knowledge and enjoyment. I have been fascinated by cat ketches and free standing masts since Hoyt first came out with his Freedom 40. The actual first cat ketch that caught my interest I believe was designed by a prof at MIT and did quite well in the SORC back in the early 70's but it didn’t have free standing masts as I recall.
I have pretty much figured out how your basic free standing mast is designed and the calculations. By basic I mean a mast that is inserted in a tube and allowed to rotate. The mast in a sabot dinghy is an example. When you go up in boat size, say 30' and larger, this doesn’t seem like it would work very well to me. Probably too much friction and not enough distance between the bearings to spread out the loads. Using a round stub mast that a separate sail carrying mast sets on and rotates around seems like a better idea. The design of the sub mast seems pretty straight forward. It’s a cantilever beam that has to be designed to handle the load at the partners where it comes through the deck. My understanding is that the calculation for this is Bending Stress = Righting Moment x Radius / Moment of Inertia. You are looking for a tube size and thickness that gives you a bending stress that is less than the material that the stub mast is made out of plus a safety factor of 3.
But what about the sail carrying mast? Does it also have to be designed to the same strength as the stub mast, or more, or less. Does the same formula apply or is a different formula used. What about an elliptical mast verses a round mast. Do you need to take into consideration the force that the sail applies to the mast. Skene’s gives a force of 1.5 lbs per square foot of sail area. Is this in the ball park? What about the load on the bearings and how would you calculate that?
I don’t know if some numbers would help with this discussion. But I am going to throw some out just in case. I’ll use numbers from my Gulfstar 37. My best estimate is that the righting moment is 23,100 lbs. The other numbers are LOA 37', LWL 34', Beam 11'10", Draft 4'9", Ballast 8000 lbs. Total designed sail area with full main and a 150% genoa is 820 sq. ft. My actual sail area is quite a bit less than this. If my boat was rigged as a cat ketch it would have 2 equal size elliptical sails of 400 sq. ft. each. Luff 39', Foot 13'. The luff and foot dimensions are the same as my current mainsail. The total length of the sail carrying mast would be 44' and the stub mast would go 10' into the sail carrying mast. By the way, I am not looking to convert my boat to a cat ketch. These are just numbers that are available to me and will make my understanding of these concepts easier. If someone would like to propose and use different numbers, be my guest.
All of the above assumes that we are talking about a round or elliptical tube with no internal supports for the sail carrying mast. I have read the articles on Eric Sponberg’s website and a lot of his posts here and also those of Rob Denny. Some of the masts that they talk about have a transverse member that adds to the structural strength. I could not find anything on the web that discusses this situation. So my second area of question is how would incorporating a transverse member affect the mast and how would you calculate that. Would using a transverse member allow a smaller section and or thinner tube to be used?
Finally, does the tube that the stub mast slips into affect anything and if so how and how would this figure into the calculations. My guess is that it is just along for the ride and merely holds the bearings in place.
Thanks for any help you can provide.

 

This may be more low-tech than what you're looking for. But the old working sharpies used free-standing stick masts, with brass rings for bearings. To quote feeblecrew.com (which is almost certainly quoting someone else, possibly Chapelle):

The mast revolved [while] tacking in order to prevent binding of the sprit under the tension of the heel tackle. The tenon at the foot of the mast was round, and to the shoulder of the tenon a brass ring was nailed or screwed. Another brass ring was fastened around the mast step. These rings acted as bearings on which the mast could revolve.

http://feeblecrew.com/new_haven_sharpie/new-haven-sharpie_rigging.140.html

They also lined the mast partners with greased rawhide.

 

Troy,
Interesting site but not what I am looking for. I am looking for the calculations to design a free standing mast that sits on and rotates around a stub mast. My assumption is that it is more complex than designing a simple round rotating mast. But what do I know. That's why I am asking the question.
Chuck

 

It seems to me you might be trying to over-engineer the problem. But I'm a rank amateur without much experience in boats bigger than bathtubs, and I certainly wouldn't advise you to take my answer to heart.

You're asking an interesting question, and I'll be checking back to see what the pro's have to say.

 

After more research I think that I have found the answer to the effect of adding a transverse member. My research indicated that you would add the Moment of Inertia of the transverse member to the Moment of Inertia of the mast and use this total in the calculations. Is this correct? I am I missing something here?

Playing around with the bending stress formula gave me results that were backward from what I expected. As I increased the size of the mast the MoI became larger and the bending stress decreased. This just didn't make sense to me. I guess I don't really understand what is meant by "bending stress".

Further research gave another formula which I believe is more useful and gives the maximum force that a given mast can withstand. That formula is F=3.1415^2*EI/L^2. E is Young's modulus, I is the MoI and L is the length of the mast. My best guess is that L is in inches because E and I are in inches. This formula makes more sense to me because as MoI increases the maximum force increases. Using this formula I should be looking for a force "F" that is 3 times the righting moment. Am I correct in my analysis?

Still trying to figure out the stub mast and sail carrying mast relationship.
Thanks for any help you guys can give me.
Chuck

 

I am an engineer, but I did not quite follow all of your musings, but I think you might be a little confused. The force used is the max righting moment of the hull, the assumption is that if the wind is strong enough to blow the boat over by the sail, the max load on it would be the max righting moment the hull can generate (plus the safety factor). If you have a fully cantilevered mast, whether rotating or not, you will transmit that full righting moment through the mast (and stub), rather than distributing it through shrouds and other rigging, to the deck/hull. the max moment on the mast occurs at the base of the mast if the mast is fully cantilevered off the stub.

As the mast section gets larger (larger MOI), the stress on the material is smaller for the same given bending moment. A larger dia tube will be stronger in bending than a smaller dia tube, even if the same weight (presuming you are not approaching the buckling limit of the material). I do not understand why this does not make sense. It makes perfect sense to me.

 

Think of the rotating mast as a keel stepped mast with the lower bearing the mast step and the upper bearing the deck collar. All the calculations for a free standing mast that does not rotate apply. The mast does not know if the support is external step and deck collar or internal bearings.

It would be more practical IMO to put the mast in an external bearing rather than on a stub mast. If the internal stub mast extends even 6 feet from the deck, things like halyards become a new challenge. With an external bearing the halyards etc can use mast mounted hardware and not limit the rotation of the rig.

R

 

Thanks to Petros and Rhough for your replies.

Petros, I have a clear understanding that the maximum force that can be applied to a mast is the righting moment and the mast has to be strong enough to resist this force plus a safety factor. I think that I am finally starting to understand what is meant by "Bending Stress." Not 100% yet but I am getting there. What I haven't figured out is how you use the Bending Stress to know that you have a mast that would be strong enough to resist the force of the righting moment plus a safety factor. My best guess at this point is that the Bending Stress multiplied by the safety factor should be equal to or less than the righting moment. Or does it have to be less than the strength of the material that the mast is made out of plus a safety factor?

The other formula that I found which I understand calculates the maximum force that a given mast can handle before it fails was real easy for me to understand. You are looking for a result that gives you a number that is equal to or greater than the righting moment multiplied by the safety factor.

Rhough, My intuition suggested to me that the you would design the sail carrying mast to the same load as the stub mast with the lower bearing acting the same as the heal at the mast step and the upper bearing acting the same as the mast collar at the deck. But I didn't know if that was correct. Thanks for the input.

From reading Eric's posts and website, my understanding of the purpose of using a stub mast is to spread out the distance between the bearings which decreases the load on the bearings. If you use external bearings this would limit the distance between the bearings because the top bearing would have to be below all the hardware on the mast. The heal of the rotating mast would also have to go through the deck which I believe would be impossible to seal and the bearings might fail rather quickly from salt water getting into the bearings.

By the way, I liked your post about your Sabre in the thread about suitable boats for use in British Columbia and Alaska. Very informative. I wish other posters would give out that kind of information.

Thanks again.
Chuck

 

I see your confusion, let see if I can clear this up. Lets start with basics. First of all you can just ingore the term "bending stress", and just think of it as "stress" on the material. Each material has an allowable stress limit, it has to do with the properties of the material, not the shape or configuration of the structure. There are some materials, like wood or unidirectional composite layups, that have different allowable stress limits in bending than in tension, shear or compression, so for those materials they would refer to the allowable "bending stress" of a material, but what they mean is the allowable stress level when in bending (which might be different for other loading conditions). Most metals have the same bending stress limits as compression or tension stress, so it is just refered to as the stress limit. IF your different choices of masts are all aluminum, they would all have the same allowable bending stress. The total strength of the mast however, or total allow bending moment, would than be dependent on the cross sectional shape.

The term "stress" is a very precise engineering term (that was later applied to emotions but is unrelated to the now common use of this term). In engineering, "stress" means a force applied over an area, as in "pounds per square inch", i.e. 1000 PSI means for each square inch of area it has 1000 pounds of force on it. So do not confuse strength of the mast with allowable bending stress, they are indirectly related and dependent on other factors. A wood mast can be stronger than a metal or composite, even though wood has a much lower stress limit than either metal or composite.

So when you apply a bending moment to a mast, the stress on the material of the mast, is totally dependent on the shape of the cross section. If for example it was a solid rod, the surface of the rod would have the very high stress (and fail at a much lower bending moment) as compared to the same amount of metal shaped as a large diameter thin walled tube. The stress on the material, in bending is determined by the equation:

Fb = M/S (in psi)

Fb is "bending stress" (psi), M is bending moment (in-lbs), and where S = I/c (in3)

I= area moment of inertia (in4) and c= (in) max distance from the centroid of the cross sectional area to the outer most fiber or surface.

In a symmetrical shape (like a tube) "c" is simply half the diameter. It is important to know this when analyzing irregular shapes, like a tear drop shaped mast when not loaded across an axis of symmetry.

This can also be written as:

Fb = [M x c]/I

As you can see for any given bending moment, the larger the I, the lower the stress. So that is why a large thin walled tube will have less stress on the material than a solid rod of the same cross sectional area for the same moment. Or will tolerate a much larger bending moment before it fails. It has different I (larger), that is why a thin walled tube is stronger for the same amount of weight.

So you have keep your terms correct. Allowable stress is the max loading the material can take before failure. The moment is the bending force on the mast. And the strength of any given mast (or the maximum bending moment it can carry) is a matter of the both material, and the shape of the cross sectional area of the mast.

I hope this helps.

 

Thanks Petros. That clears up things. So let me try to relate the theory to some numbers. The maximum stress that a mast has to withstand is the force applied by the righting moment. Righting moments are usually given in pounds per foot whereas the allowable stress for a given material is given in pounds per square inch (psi) and everything in the Fb formula is in inches. So to keep the numbers the same you need convert the righting moment to pounds per inch by multiplying by 12. Thus a righting moment of 20,000 foot pounds is the same 240,000 inch pounds. A round mast 4” in diameter with a .25” wall thickness would have an I of 5.2 where I = pi/64(Do^4-Di^4) or 3.1415/64(4^4-3.5^4). Using Fb = Mc/I you get an Fb of 92,308=240,000*2/5.2. Fb is pounds per square inch. The maximum allowable stress of aluminum is 35,000 psi. If I understand all this correctly in this example the mast would fail because the Fb is more than double the maximum allowable stress of aluminum. If the mast diameter was increased to 8” keeping everything else the same you would have an I of 45.75 and the Fb would be 20,986 psi. In this case the mast would not fail because the Fb is less than the maximum allowable stress of aluminum. Do I have all this right so far?

Taking this to the next step how do you determine the safety factor? My intuition tells me that you would divide the maximum allowable stress of the material by Fb to get the safety factor. In the case of the 8” mast above, the safety factor would be 1.67 (35,000/20986). Is this correct?

If the mast is an ellipse, you have two I’s, Ix & Iy. Ix is always smaller than Iy because Ix is the I for the minor axis and Iy is the I for the major axis. Because Ix is always smaller my intuition tells me that this is the I that you would use in the Fb formula. Is this right?

Again thanks for your help.
Chuck

 

Sounds like you got that all that correct. One minor point, bending moment and/or righting moment is in ft-lbs, not lb per ft.

You determined the safety factor correctly, and the amount of safety factor is usually set by an industry standard if you are going for a rating. Sometimes for special designs or racing, you determine your own safety factors (what you are willing to risk). Typically for cruising industry will use any where from 3 to 5 safety factor, sometimes even more. It is set based on the predictability of the loads, and the reliability of the material. Metals are more predictable than wood in terms of max allowable stress, so the more reliable material will have a lower safety factor. You have to go by those factors set by industry, based on long observations, or by your own good judgment AFTER you have many years experiance with different materials and conditions.

Some loads are difficult to predict reliably, so the pile on the safety factor.

Yes, for an elliptical mast section, you have to use the I in the axis of load direction. You can load a 2x4 wall stud in either direction in bending, but when loaded on edge it is will hold about 6 times the weight than when loaded flat-wise. The max allowable stress is the same either way.

One other minor point is the max allowable loads on any given material can vary depending on the application. Wood for example allows double the stress for impact loads vs. permanent or continual loading, and 25 percent over for transient loading. And sometimes you will see ultimate strength, and yield strength for metals. Yield is where it first takes a permanent bend (does not spring back all the way) some designs require you to work to yield (like a bridge, you do not want it bending when you use it), other designs you can use ultimate, like when I design fall arrest anchors for safety lines, bending the anchor is okay as long as it does not fail. You will also see working load or fatigue limit, some materials, like aluminum and magnesium alloys, have high yield strength but do not have good fatigue life, so for repeating loads you have to design to a much lower working stress level.

 

Petros,
thanks for all your help.
Chuck

 

I'm still missing something here, I guess. Why would you want to slip rotating masts over stubs, when it's so much simpler just to have external bearing surfaces at the mast steps and partners? What's the advantage? It seems to me you're over-engineering, and forgetting the KISS factor.

 

Troy,
Eric Sponberg can give you the real correct answer. From reviewing Eric's website, his most recent designs all use a stub mast. I doubt that his clients would pay for the added cost and complications of a stub mast if there weren't some real benefits. To my understanding there are several benefits of using the stub mast. But my understanding of the benefits of using a sub mast may be incorrect.

By using a stub mast the deck can be sealed so no water goes below deck and no water gets into the bearings. The bearings are protected inside the mast. There is no way that I can think of to make a bearing at the partners water tight without adding fiction against the mast. I also believe that the sub mast becomes a structural part of the boat helping to distribute the load. The major benefit as I understand it is that you can spread the bearings further apart which diminishes the load on the bearings. Less load on the bearings makes it easier for the mast to rotate.

Maybe Eric wil chime in here and educate all of us as to the benefits of using a stub mast.

Chuckl

 

Hi Guys,

I have been away for a few days helping my wife promote her new novel. Just found this thread.

First, most of the engineering questions I think have been answered. To summarize, the only load on the mast is that caused by the righting moment of the boat. You do not necessarily add in any other sailing loads, although sometimes you might want to consider local stresses caused by the attachment of a headsail, as in a sloop rig, and/or running backstays.

You match the stress calculated in the mast with the strength of the material, which you determine separately. Unidirectional carbon fiber laminates typically have a strength of about 95,000 to 100,000 psi. However, there will be off-axis fiber in the mix (as much as 40% of the total perhaps) and this has the overall effect of reducing the strength of the total laminate down to about 65,000 to 70,000 psi. So you know the stress caused by the bending moment, and the strength of the material. But you want to make sure the mast does not break at that load, so you need a safety factor. This gives me an allowable stress. I use FofS = 3.0, typically, so my 70,000 psi strength (and this is compression strength because masts and poles always fail on the compression side) becomes 70K/3 = 23,333 psi.

The reason I use FofS = 3.0 is because of the failure mechanisms in composites. Generally, there are three failure mechanisms. The first is intersticial fracturing of the resin between fibers and it happens at very low loads. The second, at slightly higherloads, is the failure of the resin bond to the fibers. Both of these modes of failure do not really detract from the overall strength of the mast. The third mode of failure is the actual breaking of fibers, and this does indeed reduce overall strength of the laminate. This third mode typically happens at about 50% of the laminate's strength. So you never want to get that high, and that would imply a FofS = 2.0. I don't want to go that far, so I use FofS = 3.0. To go higher would make the mast heavier and more expensive--they are heavy and expensive enough as it is, so 3.0 is a good number and it has lasted me well for years (touch wood!).

My mast designs, which most typically are not round in section but elliptical, almost always have a shear web (Saint Barbara's does not). This effectively holds the sides of the mast in place in relation to themselves--that is, they carry the shear stress in the mast. There is always shear stress present in any body undergoing bending. You can think of it this way: As the mast bends, one side is in compression and the other side is in tension. If unrestrained, those sides would physically move in relation to each other. But we don't want that to happen, so if we put a shear web in the design, it prevents movement; it carries the shear load, which is very high betwen the partners and the step. It contributes only a very little bit to the overall bending strength of the mast because it is located on the neutral axis and through the center of the mast.

For elliptically shaped masts, yes, the transverse width was lower moments of inertia and section moduli than the fore/aft length, so the width is the controlling direction, that is the direction of assumed load. In reality, the mast actually twists somewhat, and you really end up experiencing the polar moment of inertia most of the time. But you can't count on that--the mast will from time to time be loaded directly transversely.

The reason I like stub masts with the bearings above the deck is because bearings are difficult to seal at the deck level. Seawater gets through them and the inside of the boat gets very damp and wet. By using the concept of the stub mast, the deck joint can be sealed against all water ingress. The halyards and reefing lines can exit the stub mast just above deck level, and the stub mast can be heavily reinforced to allow for the cutting of these running rigging holes. It all works very nicely.

The main wing also has to be designed to carry the full bending load because that load is maximum right at deck level. So I usually design and engineer the wing mast and stub mast together, allowing space for the bearings. The bearings do have to be very low friction, and they have to be made out of metal with generously sized metal rollers because the bearing stresses on the rolling elements are quite high.

I hope that clears some things up.

Eric