Estimating strength and stiffness of a carbon-wrapped 2-piece wooden mast

Look up composite beams.
The areas of the materials are scaled in width proportional to the Young's modulus of the other material.
Mechanics eBook: Composite Beams https://www.ecourses.ou.edu/cgi-bin/ebook.cgi?topic=me&chap_sec=06.1&page=theory

My advice is don't go there! Design for wood or carbon fiber only.

When I want to make longer spar I use scarfs:


AlanX

 

Thanks for that link. Your suggestion to scale the widths actually makes a lot of sense to me. This might simplify calculations by a lot. I suppose you say "don't go there" in the sense that it is very complicated to figure out the math... I mean, if I were to take an already working wooden mast, and then add carbon sleeve to it, it's not like it would be come weaker, right?

As for scarfs - yes, I use them too when I need to join pieces of wood, but what you have there is a glued joint, and not a removable one like I need... I think I saw somewhere a two-piece mast that had scarph cuts on both mating pieces of the mast, and collar rings that hold the two together on both ends of the scarph. But it seems to me that such a joint would not be isotropic in it's properties (strength would depend on how the mast is rotated in relation to the bending force). That wouldn't matter for a glued scarph, but it would for a non-glued one, because if the force is perpendicular to the scarph, then the mating surfaces can simply slide out sideways, with nothing to hold them. I could be wrong, though.

 

Okay, I banged some numbers into an old spreadsheet for your 5.5 m long, 70 mm diameter mast and 96 MPa UCS (~62 Mpa WS).
I agree the mast design and sail area (8.6 m^2) is about right. I use a wind pressure of about 102 kPa (~43 km/hr wind speed).
I calculated the Section Modulus (Z) and then the bending moment for your hollow mast: 0.000029 m^3 and 1.83 kNm
I then redesigned the mast for fibreglass (152 MPa) and calculate a 4 mm thickness (15 layers of 4 oz/yd^2 cloth).

Estimating the WS of Carbon Fiber is tricky but it has to be limited to 2000 MPA or it will debond from the epoxy.
Further, the estimated fill ratio is 30% (as this is what I use for glass fiber).
This equates to 600 MPa yield strength.
Then I used a 65% working stress factor.
So I ended up with working stress of 390 Mpa (no idea if this is right!).
With this factor I calculate a 1.3 mm thickness.

By the way, Z=Pi*(Do^4-Di^4)/32/D0 and BM=Z*WS

I have not checked these numbers and I do not actually know the right numbers to use for Carbon Fiber so user beware.

Regards AlanX

 

Last night I was thinking about my magic numbers so I did some research this morning to re-derive them.

First the 2000 MPa strain limit for carbon fiber (C/F) source:

• Figure 13.9 "Principles of Yacht Design" by Larsson and Eliasson.

Next the fill factor (which is v/v):

• David Gerr (The Nature of Boats) says (p331) that the weight of glass in the hull laminate is about 35% w/w.
• Give density of Epoxy is 2.09 t/m^3, E-Glass 2.58 t/m^3 and C/F is 1.6 t/m^3:
  • For E-Glass that equates to 30.4% v/v

Next the Oz/sq.yd per mm:

• David Gerr (The Nature of Boats) says (table on p330) that the weight of glass in is 255 Oz/sq.yd per inch.
• Or 25 Oz/sq.yd. per mm for E-Glass
• This equates to 15.5 Oz/sq.yd. per mm (=25*1.6/2.58) for C/F

Therefore the estimate yield/ultimate strength for C/F is:

• 608 Mpa (= 2000 MPa x 30.4% v/v)

And the working stress:

• 395 MPa (= 608 MPa x 65%)

• Figure 13.10 "Principles of Yacht Design" by Larsson and Eliasson, suggests:
  • C/F tensile strength of 640 MPa
  • C/F flexural strength of 980 MPa
  • C/F compressive strength of 490 MPa
• But according to Larsson and Eliasson, these are for manufacturer's "pre-preg laminates", which have about a 60% w/w fill ratios!
• Not much on the Internet but I did find:
  • Mechanical Properties of Carbon Fibre Composite Materials http://www.performance-composites.com/carbonfibre/mechanicalproperties_2.asp
  • Which suggest similar numbers for standard C/F fabric of 600 MPa UTS and 570 MPa UCS.

So my estimates look good at this point.

Regards AlanX

 

Alan, the information you posted here is a gold mine of knowledge. Thank you so much for taking the time to look up and write all this. This is exactly what I was looking for. All this information is more than enough now for me to make some educated estimations of the strength of my hypothetical mast and determine how much wood can I save by using CF. Again, thanks a lot. Hope others find their way into this topic as well to see this info.

 

Years ago Eric Sponberg shared info on how to design freestanding carbon fiber masts. Do a search. All calculations are based on the righting moment of the boat. It takes very little carbon fiber to achieve the necessary strength in a mast as small as yours. The problem then becomes resisting buckling of the mast. By the time you add enough wall stiffness to resist buckling the carbon fiber is just along for the ride. A two piece aluminum tube mast like what is used on a lasar is probably a better option then trying to combine wood, fiberglass and carbon fiber. Probably lighter too.

 

Free standing mast design requires you to calculate the load on the mast every 12" (30cm) along the length of the mast. You calculate the load by dividing the sail into 12" sections and multiplying the area of each section by the force of the wind. The righting moment of the boat is the maximum amount of force that resists the heeling moment from the sail. If the mast at any point along its length is not strong enough to carry the load from the sail against the load from the righting moment the mast will break at that point. To determine if the joint is strong enough you need to calculate the load on the mast at this point.

Once you have done this you will see that a straight section mast that does not taper is way over the needed strength the further up the mast you go. Your biggest weight saving will come from tapering the mast. Not from adding a strip of carbon fiber to the mast.

I also think that your proposed joint is more complicated that it needs to be. Once you have done the calculations you may find that a simple plug that fits the interior shape of the master provides sufficient strength to keep the mast from breaking at the joint

 

While it may seem complex, it can be simplified. The math has been there for a long time.

Free standing mast is really a cantelevered load model with diminishing load towards the end. The max load starts at a distance from the base and diminishes towards the end. With a triangular sail. the load is not exactly zero as wind force is stronger the higher it is from sea level. This approximates a tapered wing and the load approaches Perry's equation.

Eric Sponberg has written an excellent article about free standing mast and the proportions of the winding angle is discussed. That is the hoop (90 degrees) for compression, the unidirectional (0 degrees) for bending stress, and the helical (+45/-45 degrees for torsion. This is consistent with the data in the filament winding method bearing similar loads.

M. Hollman has written also an equation for a cantelevered tapered load for a wing spar (mast) and the full article deals with rotation (torque) of the wing. Filling in the blanks will show you the mast needs equation for COMBINED LOADS (see inset)

Understanding the basic will show that you are on the right track. Wood is mainly longitudinal (0 degrees) in fiber alignment. It is also good in perpendicular compression (hoop strength or 90 degree) so that leaves you with torsion (+45/-45) to deal with. Wrap carbon fiber around it and that takes care of torsion.

The only thing to watch out for is that the load is not exactly linear which accounts for the mast being thickest where the greatest load are and tapers off when the load diminishes.

Maybe one day given the initiative I will complete the equation and write the code.

 

That procedure has been illustrated by M Hollmann in his book Composite Aircraft Design (vol 1 I think). He subdivided the spar into section to give the thickness of each layer required for a given spar thickness and material property. The only difference is that his load starts from mid span. He also had a modified bending moment to approximate Perry's equation for wing load.

 

I did something similar many years in a spreadsheet.
I divided my mast (5500 mm) into 100 sections (i.e. 55 mm).
You can add loads, section dimensions etc. at each division.
It calcs stress and deflection. The maths is not hard if you have studied statics at uni.
Looking at that particular design I dimensioned the mast using Skene's mast design rules.

I went on to do a yard for a lateen sail. Which was a bit more complicated.
I realised that the section against the mast would break first so I added some reinforcement in that area.
Thus my weird looking yard.

Happy to upload the spreadsheet if anyone is interest.
They are not complicated but they are not user friendly.
Only of interest really if you want to build your own spreedsheets and want to look at a worked example, and understand the maths.

AlanX

 

I have skimmed through this thread and despite there being a large amount of great information, there isn't what it takes to produce a successful mast in one try. In fact, I am pretty sure that this all greatly underestimates what it takes to produce a carbon fiber mast like the one on a Fin that is has proper deflection properties (much stiffer laterally than longitudinally, nonlinear longitudinally).
To start, consider the way materials share stress and strain -proportional to the inverse of their modulus. This is why you were warned early that the wood in your sandwich was 'doing nothing'. But there is more, the stress rise created by the dissimilar materials can be a source of failure. To make a high-performance, high strain mast you need to get all the way down to the buckling of the carbon fibers and the strain in the epoxy matrix. There is software that does this HyperSizer Express | HyperSizer.

About the Wood/Carbon masts Eric S. commented on some time ago -those were wing masts of large section and very low deflection. That is not what you are trying to do.

The 45 alternating layup is best for minimizing interlayer failure. Forgiveness for skills you haven't proven and things you can't calculate is more important than using the ultimate material properties. Note that those expensive, professional production masts didn't have to do this.
Early windsurf masts went through a phase of mixed glass and carbon fibers -less stiff but didn't fail.

The whole gust response feature is an art. How do you even know what you are going for? There is a thread about trying to do this combined analysis, good stuff, nobody (qualified) even commenting. Log/attempt for open source CFD / FEM framework for mast/sails simulation | Boat Design Net

Some alternative ideas.
-Do a wingmast instead -no deflection, no problem. What you build is what you get -predictable.

-If you want to do the gust response, build an I beam (low deflection) in carbon for the lateral stiffness, then sandwich that in wood fore and aft for the more flexible and tunable gust response. This is my own idea, I have never seen done successfully -be warned.

-Do like the Clarks did on the UFO. They added carbon fiber to the base and rigging to the top of a production CF mast (windsurfer?). If you dig in to the calculations, that rig is pretty clever, and because it is based on an existing rig the gust response at the top is already developed/known.

 

Mast loads are all driven by righting moment. The mast needs only be strong enough to tip the boat over. How much it bends before it hits the water is something else. The sailboard mast you are using is plenty strong, it just isn't stiff enough. More specifically it isn't stiff enough in the bottom 1/3. The easiest way to improve this is to find a larger and sleeve the current mast into it. Or you can laminate additional carbon fiber ..
Off axis fiber is good to stop the compression side from crimping--which is the predominate mode of failure for freestanding masts. It is a safe bet to match the unidirectional fiber weight with off axis fiber weight.
If you are wrapping a wood core, you can rely on the core to mostly stabilize the carbon.
I believe it is easy to over think this. As kids we bent and added and then bent and planed until we got the bend just right. This is a little boat and this is more in the range of tuning. The improvement you seek are smaller than the resolution of simple engineering and should best be approached empirically.
You can pretty easily calculate the righting moment and then the maximum sail force. You can fixture the mast and apply that load to the mast at the center of effort and measure the deflection.
Standard 300 g/m^2 unidirectional comes in 300mm wide strips. This would go around a 50mm diameter mast about 2x. You would then slide a braided sleeve over the top and then spiral wrap the laminate with vinyl tape or peel ply to consolidate the laminate. You can then do the same test and measure what the difference is. Add more until you have something that behaves the nicely with your luff curve.
As an aside, I have burned the resin out of some sailboard masts that were old as having 40-60% carbon fiber and found them to be 100% fiberglass. So incrementally adding I and E is a pretty easy way to drastically improve things. It is easy to overshoot the mark. I have done it more than once.
SHC
Thanks to Skyak for the shout out. On a one off, I wouldn't bother with the whole UFO approach. We needed to make the mast taller as well as stiffer, and we needed to be able to produce many of them. in this case it is a one off and I recommend the tuners approach.
SHC

 

Laukejas_ I was working on something similar, carbon fiber driveshafts and the math used is identical to the topic.

I fed the data you posted and the wooden mast does not seem so good. I get a lot of shear failure and displacement angle failure. And that is for the basic torsion formula only and not for combined load. My wood property data maybe at fault.

Can you give me the correct dimension of the mast and sail as shown? Include the material (wood) you used.

 

So this is a calculation of wood only? It makes sense to me that it would show failure because the dimensions are more appropriate for a CF mast and it includes wind force but doesn't have limits for boat righting force. You could limit the forces based on boat righting, but I think your calculation is right that the section is too small to work in wood.
Cross grain strength varies quite a bit for different woods, does this address that?

Could I get a copy of the spreadsheet in post #23? I want to try and add the effects of preload for luff curvature.

 

Wood or CF, the math remains the same. The torsion is defined by the moment of arm (centroid of sail) and the polar moment of inertia which is the radius of the mast. That p value diminish as you go towards center. In this case, radius is 35 mm (diameter 70 mm). Even as a core the wood is failing.

I will be happy to give you the spreadsheet in post 23 but it is only for thickness layup of a cantelever load with diminishing load towards tip (and it is for a C or box beam, not round). The inset is more appropriate but very difficult for unidirectional carbon fiber. Mast and carbon fiber are calculated with the least strain method due to low strain of CF and wood. The full math calculates the angle of twist (angle of displacement or strain) with diminishing load. I will post that as soon as I get the proper data.