Kangaroo Curvature Graphs

I used Kangaroo to generate these bending curves and plotted their Curvature Graphs above.


Comparing 5 curves

This diagram shows five different versions of a bending curve.
1. Elastica curve, curvature varies with Sine (0-180°)
2. Clothoid curve, linear increase / linear decrease of curvature
3. The original traced saw blade (it was difficult to meassure the curvature of the scanned line)
4. Curve from Daniel Pikers Kangaroo (see previous post)
5. A curve with curvature made up of two different Sine curves.
The last curve is based on a diagram by Maarten Kuijvenhoven, see image below from his thesis. The curve is a combination of two sine waves, the first one with aplitude 1 (0-180°) and the other one with amplitude -0.1 (0-540°). The amplitude relation was something I had to experiment with, but the one used above seemed to work quite well. In the illustration below, both sine waves have amplitude 1.

Image: Maarten Kuijvenhoven


Work by Maarten Kuijvenhoven and Matthijs Toussaint

I received an interesting email from Maarten Kuijvenhoven, structural engineer at DHV in The Netherlands. He studied bending geometry in his thesis work at TU Delft (February 2009) 
http://homepage.tudelft.nl/p3r3s/IASSpaperKuivenhovenHoogenboom.pdf (reworked into a paper)
"About three years ago I wrote my thesis at TU Delft about timber grid shells and also tried to answer what geometry an elastically bent beam will have. The problem was that standard engineering formulas for deformation of beams exist, but are only valid as long as deformations remain small. Therefore I had to work it out in a more elaborate way using the concept of minimal potential energy."
Image: Maarten Kuijvenhoven

One of Maartens colleagues, Matthijs Toussaint, wrote his thesis on Timber grid shells as well (May 2007):
including some very nice tests with physical models:

Image: Matthijs Toussaint