Balance Lug Design Ideas.

The choice of symmetrical and asymmetrical stave layout is spurious for most, but I've found it's a lot easier to round a symmetrical arrangement, than the asymmetric you've shown. The symmetrical layout also places the corner of the stave closer to the inside of the mast wall, so there's more "meat" to lever against, with twisting loads. Lastly, I've had masts of both arrangements under similar loads, where the asymmetric arrange gave up before the symmetric, so I'll assume my assumptions about strength are justified, if only by limited empirical data.

This type of spar really needs to be full stayed or some serious calculation effort needs to be performed, to get uniform loading in use as free standing or partly stayed. I don't have a lot of experience with lugs, admitting a preference for other types of them. Why discount a rectangular section, when in this application, you'll gain in the fore and aft dimension and be able to fall back of the lateral shrouding to a degree, so the section would be smaller, easier to build and likely as or more advantageous then the tear drop of those general proportions.

 

I don't want to be critical because we are mostly throwing around ideas here. There appears to be a considerable amount of staying complexity in your last post. The fact that it is all tabernacled is great from the stepping stand point and the fact that there is minimal adjustment on stepping. I have visions of all of the drag from all of the stays associated with it.

I'm not sure I follow this comment. The drawing was crude and I was relying on reference to earlier drawings. I'm still depending on the "head strap" at the top of the mast to anchor the shrouds to the mast. Were you referencing any particular configuration or maybe all three that were presented?

 

I'll buy into your experience factor when it comes to sym v. asym.

I've not discounted the rectangular mast section. Exploring the possibility of a free air mast on the BL though warrants a bit of investigation into more aerodynamic shapes IMHO. With the traditional rig, I can see "sail effect" coming into play on the bad tack and actually assisting with airflow around the sail and mast. On the good tack, is it possible that the mast is causing more drag than on the bad tack because it is in free air, but the better sail shape overcomes the added drag? I certainly agree that it is hard to beat a rectangular mast for strength and simplicity.

 

Load Pathing

I've drawn up load pathing for possible staying on one side only. The plan is to have a mast dimensioned longitudinally such that no staying is required there. As drawn the green pathing is utilized while on a leeward tack (sail is to leeward) and the red pathing is utilized on a windward tack (sail is to windward). There is only one free shroud attachment when rigging. Windage is comparable to having a single shroud on either side, but may actually be more advantageous as there is some mid-span staying of the mast.

As a side note: Sorry to all for going the PDF route with my drawings. I've found that it is the best the best way to go for maintaining drawing detail.

 

LP.

I like your new drawing much better.

I think the spreaders might have to be a bit longer, particularly the lowest one.

You also need a shroud upwind of the yard.

This is to keep the mast from sagging off to leeward.

Counting on the halyard/down haul tension to do that, probably will not work. This is because of stretch.

You will be asking the halyard/down haul system to do double duty.

First it has to hold the sail taught, which in itself is a lot. Then it has to keep the mast top from sagging off to leeward (when the sail is upwind of the mast). This can only multiply the loads, increasing the compression on the mast.

Even with a shroud upwind of the sail, there is not enough angled tension to keep the middle of the mast from bending to leeward.

Once that starts, a compression load becomes a bending load, which creates more leverage for further bending, and so on.

The cure for that is running additional diagonal bracing from the bottom of each mast/spreader 'panel', at the mast, to the spreader tips. This would create the 'X' pattern shown in my last sketch.

The only difference would be that the compression strut, shown in my drawing, would be replaced with a nearly vertical shroud.

This, itself, will work, providing the loads are all lined up with the bracing.

The reality is this will almost never be the case. The sail will be pulling forward as well as sideways.

You said you hoped to account for that by making the mast chord long enough so it wouldn't need fore and aft bracing.

I am skeptical that you can accomplish this.

With an un stayed mast, the mast is expected to sag off to leeward a bit. And that includes bending forward a bit. The mast is designed to take these loads and to be a bit flexible as well. They can do this, because, except for halyard tension, bending loads are the only loads imposed. I've seen pictures of such rigs under sail with noticeable bend in the mast.

With stiff staying, especially when you are relying on fine angles for bracing, you can't afford any noticeable bending at all.

Once the bending starts, some of the bracing angles get finer. The finer they get, the more tension they impose on the wire for the same load. The more tension imposed, the more the wire stretches, creating finer angles, imposing even more tension for the same load.

I hope you can see how this could result in catastrophic failure.

Even the system I sketched would be subject to this sort of thing. That is why I mentioned twist.

The cure (for my system) would be to run a pair of shrouds from the top of both the mast top and the compression strut, running them to the lee of the mast, and giving them some fore and aft drift.

The fore and aft angles would be quite fine, but hopefully sufficient to keep the twist from starting.

As you mentioned earlier, about my sketch, there is a LOT of wire bracing.

This is the price for lighter weight.

In the early days of aviation, the engines had terrible HP to weight ratios. The '03 Wright Flyer had a 12 HP engine that weighed around 240 lbs.

To compensate, their flying machine had to have a lot of wing area AND it had to be quite light.

The only way to do this was with lots of wire bracing, so just about every component of the machine was in either tension or compression.

This sort of engineering was necessary up until the very end of WWI, when engines with much better HP/weight ratios became available.

Sailboats evolved in the opposite direction, as aerodynamic theory mandated ever taller masts, producing rigs of ever higher aspect ratios.
(remember. A mast twice as tall, has to weigh half as much, as the mast it replaces, to have the same vertical weight moment)

So, just as wire bracing was starting to disappear from airplanes, it was starting to multiply on sailboats.

The once ubiquitous masthead rig took that trend to its ultimate limit, with a fixed back stay, a fore stay, a pair of lower shrouds on each side, with fore and aft drift, and a single pair of upper shrouds, held off from the mast with a pair of spreaders. A total of eight standing rigging wires.

 

I'm going to disagree about needing a windward shroud on the a windward tack (sail to windward).

You should like the newest iteration, though. I've extended and raised the lower spreader. I also extended the upper spreader to bisect the lower. This balances the shroud angles and puts them all above 10 degrees. I've also added a lower cross shroud. This change increase the lower panel size, but otherwise improves the whole system. I think that you'll see the red load path will carry those (windward) loads. The lower strut now transmits lateral head loads for both tacks.

The temptation now is to extend the upper strut to pull the sail further from the mast. It will allow for less sail setting tension if conditions warrant and still keep the sail clear of the mast.

My original intend was to have the downhaul/halyard combination perform your double-duty. I think that the bigger problem here is that the mast is left vulnerable to failure loads when the sail is struck and it loses a vital means of support. Admittedly the geometry of the arrangement is not best suited for the role either.

 

LP.

I got a little curious about my so called ‘Gallows Rig’ idea, so I decided to do a scaled sketch (see attachment). It is drawn to proportion, including the likely mast widths and thickness‘. They may be a bit understated but they shouldn’t be too far off.

The gussets shown at the top of the masts are meant to keep the two roughly parallel to each other when they are being raised and lowered. Each mast is meant to set in a tabernacle, which will allow raising and lowering it, then locking it in place. No shrouds or other standing rigging is intended.

The sail is proportioned so, when it is reefed to the point where the Yard and Boom ends touch, there will be roughly one third the original Sail Area still up. With a guestimated starting S/D of 15.0, that will leave only a S/D of about 5.0, once the last reef is tucked in. Hopefully that would still be enough to make progress upwind.

As you can see, you don’t get much Sail Area for the amount of spars, but the thing should be quite simple, rugged and durable.

The problems are: the windage of two masts, the weight and windage of the spreader up top, not to mention that of the four gussets.

Another possible problem is sailing on a reach. Most likely the belly of the sail will touch the leeward mast then. It is also likely to disrupt the airflow over the lee side of the sail, then as well, causing considerable loss of lift.

The advantages are: a clean leading edge to the sail, at least while sailing up wind, the ability to have the masts completely clear of the cabin, and a wider choice of sail airfoil shapes, as there is no ‘bad tack’.

And, of course, a very convenient way to deal with mutineers

 

Sharps,

Have you considered a forward rake on your masts? This would complement the aft rake of the lug and offer more clearance when running with the sail out.

I think that the biggest problem a see with twin/parallel masts is that there is no real way to get them to support each other, other than tied together at the head. Individually, their I values can never approach the I of a single mast, using their combined dimensions. Intuitively, I feel this, but to confirm it, I've dug out H. C. Hibbeler (Statics and Mechanics of Materials) and have been reacquainting myself with the material contained within.

My sources for mast design are quite limited (Skene's and E&L) and only one offers any type of guidance on unstayed masts. Interestingly, the masts in my current design are very similar in height, but one carries 40% more sail than the other and sizing according to the Skene reference would not be very different between the two. I find that modern references on mast sizing tend to only cover a limited spectrum of rigs and when stepping outside of the norm, information gets sketchy really fast.

I've been threatening to dig out my engineering texts for a while, now and I'm finding the information very useful and inspiring as I have to take my design investigation to a more basic level of design. Visually, a mast looks like a mast, but a stayed mast and an unstayed mast are two different beasts entirely. On the more basic level, an unstayed mast functions more as a beam than as a column as it visually suggests as the major loads are lateral in nature while the stayed mast functions almost entirely as a column with a whole different set of load parameters involved.

In pursuing the idea of a laterally stayed and longitudinally unstayed mast, there is no reference information for general application. My basic thought to dimension longitudinally with unstayed dimensions and laterally with stayed dimension is most likely fundamentally flawed. The stayed lateral dimension will reduce the I value in the longitudinal direction. The moment of inertia will need to be accounted for in both directions in order for the mast to perform properly. So....the studies continue.

 

I wish you the best of luck in your studies. I understand that structural engineering is at least as much of an empirical study as it is a theoretical one. Pure theory, as was made much more possible with the advent of personal computers, can and has lead to structural catastrophes.

 

Initial calculations regarding my cantilever/offset design.



These calculations are with regard to a free standing mast. The added moment when calculated at the deck equals about a 25% increase. A 0.5" increase in the diameter of a round, hollow mast covers this increased moment at the deck. A hollow square mast is much more efficient in this regard and I have included both in my spreadsheet. The couple created with the cantilever(not a true cantilever) would be handled with a head strap about spreaders and down a distance, back to the mast. Mast size could be reduced/tapered significantly above this point. An actual cantilever design could be used, but I've not investigated the mechanics of it, yet.

I think that engineering texts offer more than just theory. Many of the formulas that are found in boat design texts are straight out of engineering texts. Or, they combine several elements or formulas into a single, application specific formula for the boat designer. This simplifies it's use, but may also limit it's application to a specific set of parameters.

From Skene, there is a formula for calculating the diameter of a solid spruce mast.

Diam. (Inches)= (16PL*Safety Factor/15700)^.33

Where:
P=wind pressure=sail area*(1.15 - 1.5) depending on boat size.
L=Length of the Mast
15,7000=pi*fiber stress of spruce(5,000PSI)

It's a nice safe simple formula, but lurking beneath it's polished exterior are some serious engineering formulae. Take the PL combo. That code for M, or moment. A cubed root has some serious undertones, too. Let's not for forget the pi they snuck in on you and then slyly told you about afterwards.

You know know the Flexure Formula for stress within a structural member. Sigma=M*c/I where M=Moment, c=distance from neutral axis and I=Moment of Inertia for the section. If we start plugging in different representatives for this formula, it will transform into the formula from Skene's book.

M=PL (a moment)
I=pi(R^4)/4 (Moment of inertia for a round member)
c=r=D/2
Sigma=5000(for spruce)

Sigma = 5000 = M*c/I = P*L*r/(pi(r^4)/4) This cleans up to :

5000 = 4*P*L/(pi*r^3) , Multiplying through r^3/5000 gives us:

r^3 = 4*P*L/(pi*5000) , taking the cubed root and multiplying through with 2 leaves us with:

2r = D = 2(4PL/15700)^.33 = (32PL/15700)^.33

I was stuck with this last form of the equation for the longest time.

D = (32PL/15700)^.33

It finally dawned on me that this equation is applying the wind force to the top of the mast and that the L value early on should have been L/2, so replacing L with L/2, we finally get:

D = (16PL/15700)^.33 or Diam (inches) = (16PL*Safety Factor/15700)^.33

The nice part about knowing the engineering behind the details is that I can apply design factors to a larger selection of materials and shapes rather than the limited view that the original formula provides. The bigger concern is taking the known engineering elements and applying them properly and applying the proper safety factors.

 

I see you have done your homework.

I've done a little of mine.

I have found my gallows rig would probably weigh from 1/3 to 1/2 more than a more conventional set up.

I came up with these figures by a simple calculation.

For a pair of beams to have the same inertia as a single beam you need the cube root of 0.5 which is roughly 0.79. This would mean if my original single mast was 6.0 inches in depth, each of the two new masts would have to be 4.74 inches in depth.

To calculate the area of the section, since I'm assuming the depth will equal width, will be 0.79 * 0.79 * 36 or 22.47 SI. 22.47 SI * 2 = 44.94 SI total.

44.94/36 = 1.25 times the original weight of the original mast and that's just for the mast. The heavy spreader on top plus the gussets are going to add to that.

That's the best I can do.

I can compensate by increasing the level of technology.

For instance, I can make the new masts hollow when the old mast was solid, I can use more expensive, lighter, and stronger materials to make the masts, I can add some wire bracing to the top of the gallows, thus making the effective side to side bending length and the effective column length of the masts shorter, or I could do all of the above.

I might even end up with a mast system which is lighter than the original, and, if I was in the ad business, I could crow about how much lighter my new rig is. Never mind the extra windage, not to mention the extra costs.

I never learned the engineering version of the Greek alphabet, so I really can't understand your spread sheet. I use AY^2 tables and have a spreed sheet for doing them.

I hope you have taken account the added bending moment due to the offset sail pulling down on the offset halyard. This should be considerably greater than having the halyard just a few inches from the mast. My gallows rig deals with that quite effectively by turning almost all the halyard loads into compression loads, either on the masts or the spreader.

 

Sharps,

Wow, 36 sq in of mast is a lot of mast. Your boat must be bigger than I envisioned it. The nice aspect of going from free standing to a quasi-stayed gallows rig is that I believe that there are large load reductions in the rig.

Freestanding loads are essentially concentrated at the deck via load induced moments. Your gallows idea transforms the rig from a beam(with bending moments) to a column(s) with mostly compression loads. If you could get "gusseting" from stays or shrouds without the cross-section area of the gusset, you would be pushing past a major obstacle.

There are "K" factors involved in column theory, and thus mast theory, that modify mast panels effective length. The most effective reduction in apparent length is a fixed end as in a keel stepped mast. A pinned end is next on the effective list. The least effective is a free end. A free end column always has to have a fixed end. No deck stepped free standing masts allowed. This type of column has K=2 for a factor. Two pinned ends has K=1 for a factor. A pinned end with a fixed end has K=0.7 for a factor and finally two fixed ends has K=0.5 for a factor. The critical force(load) that a column can support is inversely proportional to the square of the length of the column so the K factor has a big effect on reducing apparent panel length and the strength or section required for the column or mast.

I would concentrate on design features that do the most to reduce apparent mast length. Stepping through the deck would be a first choice, especially if the gallows were cabin width and you had some hull depth to work with. If not, a large for-n-aft foot print at the deck would work also with some athwartship bracing to fix lateral position. The upper portions in the for-n-aft direction will have to be considered free without out adding stays, but longitudinal depth can be tolerated. With proper bracing, the upper lateral ends could be defined as fixed, or at least pinned, and so lateral dimensioning could take major advantage of K factor reductions.

Longitudinally, I can't see anything, but a 2 for a K factor without stays. Laterally, I think you could get to 0.7 for a k factor and maybe even 0.5. A close look at section requirements for the masts and and taper and thickness adjustments would go a long way towards reducing weight aloft.

I like your hoist design. It gives an automatic 2-1 advantage. Using the anticipated hoist dimensions and halyard positioning, I think you could utilize some 7x19 cross bracing for lateral stabilization of the rig that closely matches halyard position. I don't think it would count for reducing lateral panel length, but I think it would allow you to use a more opportune K factor. If the upper cross-member was to penetrate the masts, or vice versa, structural fairing fillets could be made to add rigidity to the joint. A parting agent could be used in the build and the joint could be disassemble if necessary. Just some thoughts. I like the option of placing the mast anywhere and not having it interfere with cabin accommodations.

 

Maybe as much as the top third of the mast could be cross braced as you have suggested, without the wires getting in the way of the yard.

 

Mast Offsets

I ran some numbers on an arbitrary sail.

It's a 100 SF on a 200 in. mast pair. I estimated halyard tension to be around 600 pounds with a tensioned downhaul. This may be excessive, but it's what I get with my wind factor set to 1.5 (equals 1.5 lbs per SF of sail) so it may be set a little high. With your halyard design, I estimate about 2.5 times downhaul tension is the force transmitted through the mast. I over estimated this by a bit and used a 3 times factor. I am estimating 1800 lbs of downward force in the mast system and worst cased it through a single mast.

I'm getting side-tracked here a little, but I wanted to leave a numbers trails. I've attached a set of mast offsets that fit these criteria. The mast tapers for-n-aft over it's whole length but it also tapers in width starting at maximum thickness at mid-span. It's able to do this for several reasons. If we were only dealing with compression forces, a one inch cross-section would be adequate. Rigidity is the bigger focus here and the sections required for rigidity will most often satisfy any compressional needs.

Since the longitudinal direction is fixed only at the base of the mast, the top is free and panel length is a function of distance from the mast head. The higher on the mast you get, the shorter your panel length is. Thus, the taper in the for-n-aft direction as height is gains on the mast.

Assuming we have fixed joints on both ends. the panel length is dependent on how far we are from the ends of the mast(head and foot). Mid-span is furthest from the ends and panel length equals mast length at this point. If we evaluate the spar 1/8 span away from mid point in both directions, we have essentially shortened the panel by 1/4 at these locations. All other forces being the same, the panel length at these points is equal to full span minus 1/4 span or 200" - 50" = 150" panel length. So as we progress from mid-span to the spar ends, apparent spar length gets shorter and shorter and approaches zero as we approach the ends.

My spreadsheet reflects this though I did try to keep some realistic mast dimensions. Theoretically, mast width could go to zero at the deck to be matched by an infinitely long for-n-aft section. Two aspects of your design that would need investigation would be the shear forces imposed by the lateral sail loads of the sail and the section modulus for the mast to carry those loads. Since the upper corners of the rig aren't truly "fixed", there would be tendency for the rig to flex to leeward if the mast sections aren't rigid enough. I'm still sorting out the use of section modulus for myself. Mostly, it has to do with the rigidity of a structural member.

I don't think that you see a dual taper on most masts as compression increases with proximity to the deck with a regular stayed mast. These are more of a free spar rather than a constrained mast.

All of my figures have a safety factor of three or more based on the numbers I gave earlier. Using RM30 for your vessel would probably reduce the loads considerable based on what you've told m about your design.

 

LP.

I've put some comments in your text.

I've attached hand sketch drawings of 'Lola', along with the split BL rig I settled on.