# Plank curvature equations

Discussion in 'Boat Design' started by erik818, Dec 7, 2011.

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### erik818Senior Member

When building a plank on frames boat, what will the curvature of the planks be? I've tried to google to find equations or a model for the bending of planks over support points (like frames) but have not succeded. Can anyone help me with a model or the equations?

Erik

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### viking northVINLAND

The question indicates you are new at the game,To basically answer your question --planks are cut to width as required by their alloted space on a perticular frame, steamed in a steam box which allows ease of bending along the curvature of the hulls frames and fastened. --one of my favourite reference books on the subject of "planking her down" as we say is "Modern Boatbuilding" by Edwin Monk first published 1939 re published 1973 by Charles Scribner's Sons-- ISBN-0-684-15257-6This is an excellent book on all aspects of "hands on boatbuilding" and will be a great addition to your growing library.---Geo.

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### erik818Senior Member

Maybe I have to clarify my question. I'm asking about the behavior of planks, not desired hull forms. When designing a boat you design a nice curved hull. From the hull model the offsets are calculated for the stations where there will be frames. When building the boat, you first manufacture the frames and then fasten planks on the frames. The planks don't go the straightest way between the frames resulting in an angled shape, but hopefully bends nicely to produce the curved hull the designer intended.

I have no problems visualising how the planks will bend. My question is how the bending is described with mathematics.

Erik

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### jehardimanSenior Member

You spile planks, even if you model them. If you want to make a CNC milled plank, it is far easier to take the points off after construction of the prototype than to try to make it match beforehand. Unlike a CNC cut steel panel you can't butter up the joints with welding.

As for the equations, it is a simple superposition of mechanics of materal concepts (Castigliano's method and moment distribution via slopes) which will form a system of equations similiar to a n-dimensional non-uniform rational basis spline (NURBS) where you have n-2 frames with a stem and transom landing. If you use a NURBS based surface model you can get the apparent net shape of the plank (and this works fairly wells for thin sheet methods like stitch&glue or thin steel) but for more conventional carvel or lapstrake construction you will need to configure the lands, laps, and gains by hand to account for the plank edge treatment and thickness effects (also true with thick steel construction).

(Edit: X-post with Viking North, above)

Last edited: Dec 7, 2011
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### baeckmoHydrodynamics

Each plank is (or should be...) a developable surface. As such it is handled by most hull design programs. Take a look at the algorithms in "Hullform" for instance.

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### DCockeySenior Member

That assumes the planks will not be steamed and twisted.

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The amount a plank will bend depends on the stiffness of the material.....2" fir....not so much........

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### PARYacht Designer/Builder

It will if it's long enough, but yep, edge set is a common though preferably avoided condition in carvel builds, though not necessarily so in other build methods.

You could "line off" the hull mathematically if desired, fairly easily, but this will only provide the seam lines, which from experience with lapstrake designs, is difficult to do on a screen and get them to look right in real life.

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### viking northVINLAND

Sorry about that" new in the game" Eric818 -Here I am cutting and plaining by hand, steaming/ bending / fastening by hell (thats where I'm going as a result) and there you are manupilating the input of computers which in turn will control robots that will replace me. One generation later I would have had a good chance in going to Heaven While the math associated with computer software is light years above me I can visualize your concept. Looking at a plank layed flat on the floor you are attempting to develop a program that will cut that plank on the proper left/right curve as well as it's varying width over it's length to fit it's location on the frames. In addition as pointed out you will also have to provide software input to compensate for twist. Funny the more things change the more they stay the same, by the time you get that perfected like me you'll be headed for that firey place down below and the generation after you will rise into fluffy clouds-- Just kiddin. Getting back to your task, it might be easier to begin with designing the frames with multiple chines and developing planks to fit each chine. From that it should be easier to work the formulae into the unlimited chines of a rounded hull. Anyhow what do I know, all my software is still three dimensional --Good Luck and have Fun ---Geo

A yacht is not determined by the vessel but by the care and love of her owner---

Last edited: Dec 7, 2011
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### PARYacht Designer/Builder

Milling a plank on a 5 axis machine might get you to heaven George. Just think, you insert a thumb drive, punch a few key strokes with your only good working finger and watch it carve a perfect plank. No edge set, perfect rolling bevels, fastener holes pre-drilled for both pilot and clearance, caulking seams of equal dimension it's full length. Of course, you'd have to have a whole tree, milled into wood chips, per plank, because of the twist and typical shapes employed on a yacht, but hey, if they can afford CNC planking, by God what's a 24" diameter tree per plank.

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### baeckmoHydrodynamics

Naah, David. A twisted and bent plank is still in general a single-curvature surface (in fact that is the point in using planks instead of fullwidth panels). In order to create double-curvature, the plank would have to be steamed and stretched over a D-C plug. The criterion for a S-C surface is that you may find straight "rulings" or generatrix along the full length of the panel. They may be differing in angle (creating a conical part of the plank surface), but there may never be rulings crossing each other within the boundaries of the panel.

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### PARYacht Designer/Builder

In theory this is partly correct, but in reality edge set is going to be there, which introduces some compound. Some building methods have quite a bit, though I assumed we were talking about carvel construction, in which case edge set is avoided by most skilled builders, though there still is some, especially in the ends of full bellied designs. Then there's the practical matter, of finding stock of sufficient size, to accommodate the vertical to nearly horizontal then back to vertical set, a typical garboard would have, without killing a whole tree worth of material, in the milling process. Grain run out would be a nightmare in this situation too.

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### DCockeySenior Member

Take a straight plank. Steam it and twist it. No longer a developable surface. (Term you used in your previous post).

Necessary but not sufficient for a developable surface. Another requirement for a developable surface is that the normals to the surface along any ruling line must be parallel, which doesn't happen with a twisted surface. A conical surface meets this requirement, a helix does not even though a helix can be described by a set of ruling lines.

Developable surface vs "single curvature" and twisted surface is important because a flat surface can be rolled into a developable surface without any in-plane shear distortion. If there is twist then there will also be in-plane shear distortion when forcing a flat surface into the twisted surface. The narrower the plank the less critical the twist is in terms of stresses, but it still prevents the use of a developable surface algorithm to accurately develop the unrolled plank shape.

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### viking northVINLAND

One tree per plank-- the Norse did better than that, essentially doing the same by hand using an axe. They actually introduced some twist in those long oak planks. Then again they used a trainable brain that learned by repitition and compensated in real time when chopping those chips. The human brain , never totally happy with the wood choice, the cutting, steaming , bending, fitting and fastening. However once that f-en garboard is in place, it throws another hunk of wood in the stove, sits itself down with a cold beer pats itself on the back as the smartest piece of grey matter on earth. May they struggle until eternity to make a computer capable of doing that.

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### erik818Senior Member

I want to thank everybody for the responses. Although it will take me awhile to digest and follow up the information in jehardimans answer, I believe it will lead to the answer to my question.

Viking north, I don't hold any illusions that knowing the mathematics behind plank bending will in any way beat practical experience with wood when it comes to building a boat. I just wanted to know. Useless knowledge might turn out to be useful some day.

Thanks,
Erik

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