Euler Factors for Mast Design

Hi Guys,

I'd appreciate some help on the correct Euler factors to use for the different panels of a mast. Published Euler factors are 4 for both ends fixed, 2 for one end fixed and one end rounded, and 1 for both ends rounded.

The rig I am designing is keel stepped, with 2 spreaders. So, for transverse inertia I'll be calculating for 3 panels. The base of the lowest panel, at the deck, is sort of fixed, not absolutely though. The top of the lowest panel is some kind of indeterminate condition, not fixed and not rounded either.

The middle panel is the same indeterminate condition at both top and bottom.

The top panel is the indeterminate at bottom and is rounded (free to pivot) at the top.

This seems like a really basic question but I've gone thru a lot of literature on the subject and have not found this addressed. It's a very basic point and the factors used have a large effect on the end result. So I'd appreciate insight into how to select/derive the correct factors for the various mast panels.

Regards, Paul

 

I'm not a structures guy, but I believe the correct answer is to treat the ends of the panels as pinned (rounded in your terminology). The spreaders keep the station from moving sideways, so the panel from one spreader to the next behaves much like it was pinned at both ends.

The reason is because the section at the spreader can rotate as the panels buckle in opposite directions (making the mast S-shaped). So there's very little moment carried across the division between the two panels at the point of buckling. In order to be effectively fixed, there would have to be substantial bending moment at that station, but that doesn't happen because of the flexibility of the adjacent panel.

If nothing else, pinned ends are the conservative assumption...

 

Why are you calculating the moments required for each panel? Do you plan to build the tube to match the output? Is the tube aluminum, carbon, or ?

 

Most CF masts are tailored to match the required moments of each panel and aluminum masts can be tapered by cut and weld in the top panel and full length if they are two piece extrusions (fore and aft taper). A lot of race boats did this when they used aluminum masts.

 

Thanks for the replies.

Tom, what you say makes sense. And if it's the final word I'll go with it. But I'm still curious whether a formula exists for determining factors for Euler for various conditions of support. Euler's formulas are purely theoretically derived. To achieve the results Euler calculates for a pin ended (rounded) column, for example, requires a frictionless support. I agree that considering the spreader points as being pin ended is conservative, and that sure beats being optimistic for this stuff, but if there's more to know about the subject I'd like to learn it.

Paul, I'm not building a carbon or otherwise custom tapered mast. Calculating each panel lets me optimize spreader location and gives me some insight into how much additional loading the different panels can stand, relevant for different loading condtions when reefed, when runners are set or not, etc.

I'm going into this rig design in detail because it is unconventional. Being less able to apply the common rules of thumb that have evolved for conventional rigs, I need to do my best to engineer this as accurately as possible.

Thanks, Paul

 

You should be optimizing spreader location by other means.

If you want to do the calcs there is a suggested method in the book Principles of Yacht Design.


I often wonder why someone would not use conventional methods when building a spar. What is your reason?

 

If you check around I think you will find this statement to be false.

I have never seen a "race boat" mast with a full length taper, even when using the 2 pc style of construction. Can you tell me what companies did this sort of work?

 

I am a structural engineer and I think you are going about this all wrong. There is no easy way to calculate loads on mast installation as you describe. The buckling equations are approximations at best, for a failure mode that is not very predictable. And it is not so much a buckling issue, but the mast, spreaders, shrouds, boom, step, etc. all act like an indeterminate truss system. Also the factored loads are variable for various conditions. Any assumptions you make are likely to be wrong during some condition of operation, especially in heavy weather conditions.

It is possible to design this way, but that assumes you can accurately determine what all the possible load combinations can be in various conditions of sail/operation.

There are several methods of mast/rig design given in Principles of Yacht Design that are formulas and prescriptions based on many years of observed behavior of mast configurations. It is the most reliable way to design a rig. So do not reinvent the design process, but use what has be proven to work for many decades.

Good luck.

 

Out of interest What do you think is the quickest way of doing this? There'd be a bit of a design spiral in the locations of spreaders when there's more than one. Each one changes the positioning criteria in turn.

 

A response to Petros - Thanks for your reply. I have Principles of Yacht Design. As you describe, they have design formula for a variety of rigs. You pick a rig layout and use the formulas that they assign to it.

I'm not designing a standard rig, and Principles of Yacht Design does not address what I'm building. I am certainly able to use much of traditional rig design approach for some aspects of the rig, but other parts of it just require engineering.

Because, as you say, a rig is an indeterminate system with large variations in forces applied to it, the traditional approach is to take a measure of easily measured forces (like RM) and then apply factors to that to design the rest of the rig. A different rig requires different factors, which benefit from a better understanding of he principles they're based on. Hence my question about Euler factors.

Regards,

Paul

 

Paul,

that sound like something I would do too. Than I suggest you just do a free body diagram of each component and the worst loading conditions for each, and see what you come up with. Watch out of load combination, and also take into consideration the transmitted moment in "nodes" where the mast is continous (like at the deck, spreaders, etc.). Some of your nodes will be rotation fixed, some will be rotation free and translation free. If you think about it long enough you should find the right combination of conditions. It might help to make a little flexible modle out of string and sticks to get an idea of how each node behaves when it is loaded.

I would use a safety factor of at least 1.5, go for 2.0 if does not add too much weight.

Good luck.

 

Thanks. "If you think about it long enough.....". Isn't that the truth. It's an interesting trip putting together the pieces and understanding how they work and interact. It's certainly an education.

Cheers,

Paul

 

I have designed several rigs -for bermuda sloop and schooner and for gaff schooner. 2 of them were problematic -I was forced to use REALLY THIN sections. Much more thin than I would have liked. However, all the design calculations were made using methods similar to that in "Principles...".

It is hard for me to imagine stayed rig with 2 spreaders, so designed, that tried and tested calculation/design methods do not apply...

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Factors for rig dimensioning (like Principles of Yacht Design, or those of Classification Societies) include:
*all uncertainties of end conditions for Euler factor selection, mentioned in #1 post
*supporting effect of mainsail on the mast -it act to some degree, as "cushion" against transverse oscillations between rigging points (effect is not large, but I know about cases, when yachts lose their mast, when going closehauled with genoa only)
*Hard to quantify interaction effects,mentioned by Petros.
*safety factors, adequate for material/manufacturing quality/aging/etc. .

None of them could be reliably evaluated by calculation of whatever kind.

If there is a requirement for minimum weight of aluminum extrusion, the obvious approach is to select longitudinal stiffness for full height of fore triangle, than divide this height by spreaders in such a way, that maximum required transverse moment of inertia is about same for each panel (from deck up, each panel will be somewhat longer).

All that remain is to source an extrusion with just required cross section properties and minimum weight for them.

 

In response to Perm Stress, how do you see the following factors fitting into Principles of Yacht Design's selection of layouts:
1. Double spreaders, swept 30 degrees, no standing backstay, the rig designed to stand without runners, ie shrouds are taking both transverse and fore/aft loads.
2. A quadralateral main, which gives a very different distribution of sail pressure on the rig than a standard marconi.
3. A design which requires the rig to withstand occassional runner errors, allowing the rig to withstand a knockdown without assistance from the runners, and which uses runners to reduce the rig stress to a level giving a long, fatigue resistant life. In other words, a design to 2 different factors of safety, one with runners and one without.

If you're aware of an off the shelf approach to this I'd like to hear about it. The best published information I've found is Lloyds Germanische "Design of Large Yacht Rigs" which gives a detailed, sensible approach to calculating rig stresses, gives recommended safety factors for different components, recommends max forestay sag for different class yachts, etc.

Regards,

Paul

 

To Paul post #14:

From the three problems you have mentioned, only square top main is not readily solved by off the shelf methods.
As for points 1 and 2, it simply mean designing a rig to stand without any runners at all and guesstimate some smaller safety factor. Then enjoy runners as a source of increased peace of mind and reduced forestay sag.
In a few days I could provide more details, if you are interested (quite busy at work this week).

Regards