Explaining the five cases of elastic bending

At the moment, this is how I would explain the geometry of the 5 cases of elastic bending (see previous post):

Case #1 This one follows the elastica curve, which means the curvature varies with the sin of distance along the curve (explanation here). The curve equals half a cycle of Sin (180 degrees) which means the curvature will be zero at start point and endpoint.
[EDIT 2010-06-13] This case probably involves the Cornu spiral (clothoid), see here and here.

Case #2 This is probably* a part of a clothoid curve (Cornu spiral). Curvature is maximum at the clamped end and zero at the loose end. There is a linear change in curvature in between. (A loose end cannot store any bending energy and the curvature there must be zero).
*(This could possibly be simply half an elastica curve, but I find that less likely).

Case #3 This is a circle (cylinder). Curvature is constant along the curve.

Case #4 This is a helix. Curvature is constant along the curve and there is also a constant twist. This could also be called a cylindrical plank line, which means it has the shape of a thin (straight) strip that has been wrapped around a cylinder.

Case #5 This is a conic plank line, which means it has the shape of a thin (straight) strip that has been wrapped around a cone.

Five cases of elastic bending

I have so far identified five cases of elastic bending in a thin strip:

Case #1
Two loose ends pushed together (2d).

Case #2
One clamped end, the loose end is pushed (2d).

 Case #3
Both ends clamped together to form a loop (2d).

Case #4Both ends clamped to form a loop, but with a distance sideways between endpoints (3d).

Case #5Both ends clamped and twisted to form a loop (3d).

"Conic plank line"

I would like to introduce the term "Conic plank line". (See these images).
The meaning being what I have earlier described as "Cone strip" or "Wrapping a cone with a straight strip".


Loop comparison. Investigation completed?

I hesitated for some time before I compared the new Grasshopper-generated Cone Strip with the original digitized saw blade surface. It took some fine tuning of the "Cone Angle" parameter and some scaling+rotating to find the corresponding shape.
After having compared the two surfaces closely I have found that they are almost identical!
Please download the 3d-model (rhino .3dm-file) and have a look for yourself.

Well, this pretty much concludes my investigation I think! You are welcome to prove me wrong!
Thanks for your interest in this blog!


Single curved Cone Strip

This time it's single curved...

Download GHX-file.

More cone testing

Generating the cone plank line directly in Grasshopper (no ToyCar). The strips turn out slightly double curved. Why? Probably because the surface is a loft between lines that are perpendicular to the plank line curve (they shouldn't be, they should all point towards the tip of the cone and vary in length).


Comment: Volker Mueller on Plank Lines

A comment by Volker Mueller on the post "Wrapping cones" was so interesting that I'm making it a post of it's own. Thank you very much Volker!

Hello Marten,
A curve that might behave the way you explore with strips of paper, rubber, or metal could be what is sometimes called a plank line (GC.BSplineCurve.PlankLine).
The idea of a plank line is that it bends in one direction, twists in the second direction, and is stiff in the third, like a thin but relatively broad board of wood.
Julius Natterer did structures like that (http://ibois.epfl.ch/page12022.html; I believe also the Polydome at the Ecole Polytechnique Federale Lausanne) and Judit Leuppi did some research about plank lines published as a paper at the ACADIA 2000 Conference (Plank Lines of Ribbed Timber Shell Structures; available from http://cumincad.scix.net/data/works/att/f197.content.pdf).
Another example of such a gridshell seems to be at the Weald and Down Open Air Museum in Singleton, Sussex (images at http://www.flickr.com/search/?w=all&q=downland+gridshell&m=text).
Volker Mueller

ToyCar plug-in for Rhino by David Rutten

David Rutten at Robert McNeel was kind enough to revive his plug-in named "ToyCar" for me. The ToyCar runs along a surface and finds a plank line path on it. PLEASE NOTE that the plug-in is not yet finished!

ToyCar + Grasshopper helped me create this:

Thanks for your help David!


Wrapping cones

Some more testing with wrapping cones.

This is what the unrolled strip should look like. The yellow lines represent a fixed change of direction.


Collar on the cone

Today I discovered that the loop I have been trying so hard to understand may in fact be a part of a cone!
The loop sits like a collar on the cone.

The loop photographed from the cone focal point.

This is a bit of a breakthrough! It's interesting that such a simple paper model can be so useful. In CAD, how would you constrain a straight strip to follow the shape of a (developable) cone?