Deck-Stepped Mast or Keel Stepped

Steel42,

"the horizontal loads (sail forces) would be dominant"

While the sail loads are significant, as you describe them, they are not dominant, because you have not considered the induced loads in the rig that are holding the mast in column. The mast is held up by rigging wires which, by the nature of how a boat is built, necessarily make acute angles to the axis of the mast. Rigging wires support the mast by their tension strength, which necessarily has two components to support the mast--one transverse to the mast which counteracts the sail load, and a very much higher vertical component that induces compression in the mast. This induced compression load is more critical than the sideways load caused by the sails as transfered through the sail track.

If you work through any given example, particularly on a monohull which necessarily has a narrow staying base resulting in very acute rigging wire angles, you'll see that the induced compression load has a lower factor of safety against buckling than does the bending stress caused by the sail load against the yield strength of the material. Mast compression and column bucklling, therefore, is the more critical condition, and that is why we designers design to critical buckling loads instead of bending loads.

Note that the more spreaders and wires that are in the rig design, the more compression builds up in the mast from top to bottom. Each spreader induces another transverse component at its juncture on the mast which adds to the transverse reaction that must support it. That, in turn, requires more tension in the shroud attached at that point, which in turn induces additional compression in the mast below that point. This is why the lower shrouds typically are larger in diameter than the upper shrouds--until stretch comes into play and must be considered in some cases--bigger wires stretch less than smaller wires.

You also have to consider column buckling in the transverse direction in the plane that the shrouds are oriented, as well as column buckling in the fore/aft direction in the plane that the stays are oriented. If necessary, to keep buckling from happening, a rig may require running backstays, check stays, and baby stays to make sure that the mast does not buckle fore/aft.

I hope that helps.

Eric

 

Eric, the calculation of masts is a very interesting topic and there is little written. There are books that give formulas to calculate the forces acting on masts and rigging and therefore the stresses to which they are subjected. However there are few texts that explain why, the theoretical part, of the equations. So I greatly appreciate your explanations. Thank you.
But a couple of questions arise: is it possible to speak of longitudinal or transverse buckling?. What exactly do you mean when talking about this phenomenon?. By reading what you say seems to be confused, for example, a longitudinal buckling with simple longitudinal flexural.
It is clear that the shrouds / spreaders transmit loads to the mast, but the vertical component of these forces is much smaller than the horizontal, is not it?.
Sorry if there is anything I misunderstood and I'm misinterpreting what you say.

 

Many thanks Eric, that clears that up. I see now why Euler is the way to go.

 

Tansl--I understand what you say about rig engineering and the fact that little is written on the subject. That is why I have proposed to IBEX this coming October that I present a seminar on rig engineering. They have not notified me that this idea is accepted yet, so I just have to wait and see.

At any rate, your first question is about longitudinal buckling. As you know, in the Euler column buckling equation, the length factor is the length between supports. In a rig in the transverse direction, this is basically the length between the spreaders, or the length between the deck partners and the spreaders, or the length between the upper-most spreaders and the top of the mast. So these are relatively short distances compared to the overall length of the mast.

In the longitudinal direction, the unsupported length is usually between the deck and the top of the headstay where it attaches to the mast. This is if the mast does not have a baby stay or running backstays. So the unsupported length is very much greater, practically the length of the mast. And since, in the Euler column buckling equation, length in the denominator is squared, it has a huge effect on column buckling--it greatly reduces the critical buckling load, which is not what you want to have happen. The only things under your control as an engineer in this regard is the modulus of elasticity of the mast material and the moment of inertia in the fore/aft direction. That is, without changing the rigging arrangement. This is why we sometimes elect to have masts made of carbon fiber (if we can get a modulus of elasticity greater than that of aluminum, which is possible) and why you might select a longer section (fore and aft dimension) rather than a shorter one because its moment of inertia fore/aft is greater. Otherwise, your only other option is to break up the length of the mast into shorter sections, such as adding baby stays and/or running backstays. By cutting the length of the unsupported panel in half, you get a 4x increase in column buckling strength--a very efficient way to increase rig strength.

As for the spreader loads, a spreader is loaded in nearly pure compression by virtue of the fact that it turns the tension load of the wire it supports to a different angle down to the deck. The compression load is the component of the rig tension load x cosine of the angle that the shroud makes to the spreader. And you have to consider the rig wire both above and below the spreader tip, both angles. So if the spreader bisects the rig wire angle equally above and below the spreader, the compression load is the tension load x 2 x cosine of the spreader angle. Otherwise, just take the tension of the rigging wire x cosine of each part's (above and below the spreader) angle and add them together. That gives the total compression in the spreader. Yes, the vertical component at the spreader tip is supposed to be nil--that's why you make sure the spreader angles to the rigging wire are equal above and below the spreader.

So this spreader is pushing sideways on the mast, and the only thing that is resisting this compression load is the next wire attachment directly beneath the spreader, i.e. the next diagonal down. This is in addition to the normal sailing load that this diagonal is already carrying. That's why the next diagonal down is probably going to be a bit bigger in diameter than the next diagonal above it. And so on down the rig--lower shrouds will be larger than upper shrouds.

I hope that helps.

Eric

 

Eric, your explanation perfectly clear forces acting on the mast and which items are the transmitter of forces. But all that, precisely, increases my confusion because, according to what you say and what I thought, the mast works primarily in pure bending (lets forget by the momento torsional that it can also experience). Pure compression efforts on the mast are small.
Euler's formula applies to long columns working in pure compression. In this formula we find the radius of gyration of the section and the length of the column, no other physical property of the column. The radius of gyration is a function of the "polar" second moment of area. Not longitudinal second moment appears.
Thanks for your time.

 

Tansl--the mast is being both bent and loaded in compression, and the compression load prevails.

For Euler's column buckling, one equation is:

P = critical buckling load = C x (A x PI^2 x E)/(L/r)^2

Where:

C = fixity coefficient, = 1.0 for pinned-pinned connections
A = cross-sectional area of the section
E = Young's Modulus (modulus of elasticity)
L = unsupported length of column
r = radius of gyration
and L/r = slenderness ratio

We know that r = (I/A)^0.5 where I = moment of inertia of the section.

Or, r^2 = I/A

so if you substitute r^2 into the equation, you get

P = C x (PI^2 x E x I)/L^2

Which, to me anyway, is simpler because it deals with all the parameters you normally have to deal with, and it takes A and r out of the equation.

For transverse section buckling, use the smaller I of the mast section. For longitudinal sectin buckling, use the larger I of the mast section.

This, by the way, is why Ix and Iy of mast sections are always published, and you never see the cross-sectional area published.

Eric

 

I think my confusion is in the value of second moment of area to be considered.
I thought that in Euler's formula we must take a value "I" of polar moment of the section, being the value of "I" :

I = Ix + Iy

 

Tansl--Because of the nature of most mast sections, in which Ix and Iy are quite different, and because the mast is constrained against rotation (pretty much, anyway), and because the planes of the loads (athwartships and fore/aft) are fairly discrete, and finally because the staying arrangements are quite different between the fore/aft and athwartship planes, then we really must analyze each load plane differently, using the corresponding moment of inertia.

Eric

 

Clearly, I have a misperception because I am convinced that buckling caused by compression of the mast, can not be studied with two different moments of inertia. It is NOT correct.
But, if you like. Eric, let's close this issue here. Thanks for your efforts in teaching me.

 

Why not? Calculate both, and the result with the lower buckling load will represent the real-world buckling mode.

 

As just marginally mentioned in my previous post, there are several important limiting factors for the use of Euler's formula:

1. It ignores the eccentricity of the compression load respect to line connecting the locus of centroids of the mast sections. This issue becomes important when the mast has an initial curvature due to the rig tuning process. As a result, the mast can no longer be considered a nearly-straight beam (infinitesimal curvature) - which is the basic assumption of the Euler's theory. Hence any load applied at the mast head will act as a combination of a pure compression force and a bending moment, both varying along the height. This eccentricity-induced bending moment can significantly lower the value of the maximum buckling stress. In order to account for this fact, one can make use of the Secant formula, the Perry-Robertson formula or the Rankine-Gordon formula (which I am not going to write here, since they can be easily found in the Internet).
2. The Euler's formula is valid for beams with slenderness ratio higher than approximately 120. Some masts might get close to or even fall below this value, in which case the Euler's method will give a false sense of safety. In those cases it might indicate that the yield stress shall be reached well before the buckling limit, when the opposite might be true - especially when the previously-mentioned eccentric loads are present.
3. It doesn't take into account the stress risers, like holes for halyards or various fasteners attached to the mast. A thinly designed mast can be brought to failure by a couple of additional wrongly placed holes in the mast structure - and it can happen under loads which are well-below those indicated by the Euler calculations.

All this is to say - take care when using Euler formula, and rather opt for more inclusive methods, which include (at least) the load eccentricity and the initial curvature of the mast centreline.
Cheers

 

I agree. That is a common practice in the hand-calculations of columns loaded by compression forces.

 

Very clarifier and instructive. As regards, thanks to three, Eric, steel42 and daiquiri.

 

you only get pure bending on a mast if it is cantilevered without any stays, this I think is what is confusing people. In a stayed mast, if you attach the sail only at the top of the mast where you might have all the stays also attach, you will only get compression since the stays take the lateral loads to the deck near the gunwale.

Do not think of the mast as a cantilevered column, but more like truss. The real complication is that a conventional stayed mast has very complex load combinations, it is structurally like a truss, the stays can only take tension loads, and that leaves only the mast to take compression. But in addition, you have point loads introduced from where the boom is attached, and a continuous tapering lateral load along the track where the sail is attached to the back of the mast (which also introduces some torsion to the main mast). If your forstay and side stays do not run all the way to the top than you have bending loads above the attache point of the stays, and if you have multiple side and fore-stays you also introduce additional loading complications. that is why most rig design is done by perspective tables from class rules rather than by analysis. These were developed over lots of observations on what generally works well.

In addition, if you are designing for deep water cruising you have to add considerable overstrength to prevent demasting in case of a knock down, and to account for some wear over the long term. If you capsize in a race, you are done racing, for those out in the middle of the pacific, losing your mast can be much more life threatening than just losing a race.

Most economical masts are constant section extrusions, and the design tables work well to make a durable and functional mast. If you are trying to design the lightest and fastest racing boat, than a much more complex analysis has to be done, and you will design local reinforcements for the point loads, and a small safety factor, to optimize weight, strength, and flexibility. It would have to be hand laid up with exotic composites, so cost savings is not the highest driving factor.

Even so, the analysis is only as good as your assumptions, so if you want the best performance, you will be occasionally breaking the masts where your design assumption was perhaps too optimistic, and not finishing the race. but it seems with all racing, if you are not right at that raged edge of performance, you will not win. So you either win, or DNF with broken equipment.

Euler equations can be altered to consider eccentric loading, and a full stress analysis can be done by super positioning of the various point and bending loads on the mast. But this is only valid if your load and design assumptions are valid. Most times nature does not always follow your assumptions and behaves in ways not always anticipated.

For recreational sailing, just use what works, a lot less risky and troublesome.

 

But in the fore-aft direction the influence of the stays is significantly lower. And that is also the plane in which the mast curvature lays after the rig has been tuned - hence, bending moment. For the rest, I agree.