Siegfried Gass, Frei Otto & Wolfgang Weidlich

Architect Lorenz Lachauer (who is currenty a research assistent at the ETH Zürich) sent me this scan from a book he found in the library. The book is called "Experimente: Physikalische Analogmodelle Im Architektonischen Entwerfen" by Siegfried Gass, Frei Otto and Wolfgang Weidlich, published by the Institut für leichte Flächentragwerke, Universität Stuttgart in 1990. He says it's a really amazing book and I believe him! Thanks for letting us know, Lorenz!

Email from Oliver David Krieg

I was contacted by Oliver David Krieg, a student at the University of Stuttgart, Faculty for Architecture and Urban Planning. He is currently working on a project called "Reciprocities" led by prof. Achim Menges at the Institute for Computational Design (ICD) .

They are investigating the properties of wood very thoroughly and of course bending geometry is one of them.

Rhino and scripting are among their tools and the goal is to develop a material system with interconnected elements that can react (passively) to surrounding influences.

It will be very interesting to see how this will develop. Thanks for getting in touch Oliver!

(Photos by Oliver David Krieg)

SurfaceTurtle and CurvatureTurtle by Lorenz Lachauer

Lorenz Lachauer has a great blog called eat-a-bug. It has some very interesting topics. I used his Grasshopper file SurfaceTurtle and Rhino plugin TwistedBeams to generate these plank lines from an ellipsoid:

(The SurfaceTurtle works just like the ToyCar plugin by David Rutten).



I also used his Grasshopper file CurvatureTurtle to generate some elastica curves:


Thanks a lot Lorenz!

3 useful cones

These 3 cones unroll to 270, 180 and 90 degrees, which can be useful if you want to line up an orthogonal pattern. For example rectangular plywood sheets.

Self intersecting developable surfaces

Some sketch models:

The idea is to create one continous, developable, self intersecting surface that can create a useful space (a room) and perhaps extend to become a whole building. The surface is allowed to overlap itself when unrolled.

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.

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

Case #5
Both 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!

Mårten

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).

Regards,

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?

Twister

I discovered it was possible to replace a number with a list of numbers as an input to a VB.Net-script i Grasshopper and produced this rather weird model:

(Rendered in Maxwell)

Rubber band

I found this rubber band on the table looking just like this. The shape of the bending and twisting is similar to that of the saw blade. (Christmas tree in the background...)

Analysis of helix angle in loop surface

Grasshopper helped me plot the difference in angles between curve direction and curvature direction along the loop surface. Seems to start around 30° and reach 0° at surface mid point. It flips over to 180° and then decreases another 30° to 150° during the second half. Interesting! This would indicate a linear decrease/increase of the twist amount (measured as the helix angle). I think this pretty much solved the puzzle...

More helix testing

Some more helix testing in Grasshopper (no scripting yet as you may notice from the messy layout). The question is now: How to make a smooth transition between helixes with different angle?

Developable helix surface

The measurement for bending is obviously curvature, but how to measure the amount of twisting? The image above shows how the Curve Direction differs from the Curvature Direction in a developable helix surface. Perhaps this "Helix Angle" can be a useful measurent?

More elastica curves...

Dr. Christopher D. Rahn at the Pennsylvania State University has developed a mine-hunting vehicle with whiskers (!) that uses the "elastica equations" to identify objects. Read more here.
Below is an image of experimentally predicted whisker shapes:

Elastica curves!

I went back to read "The Curve of Least Energy" by B.K.P. Horn more carefully. He writes: "Unfortunately, the Cornu spiral is not optimal either...".
In this web page, prof. Albert K. Harris explains the Elastica curve: "If you compress a long thin metal rod, when it eventually kinks its shape will approximate one of the elastica. This is said to optimize the spatial distribution of bending stress..."
And finally, in "Non-linear Beam Analysis" Japaneese aircraft structures engineer Toshimi Taki uses Elastica curves to generate something very interesting:

This is the first time I've seen anything similar to my own diagram of bending curves. I'm really excited!

Maybe the Cornu Spiral isn't the answer to 2d-bending after all?

Bending Structure

I used mirrored segments of the cornu spiral to create these arches.
You can download the grasshopper file here:
http://omkrets.se/grasshopper/MN-002-bending-structure.zip

This script generates the cornu spiral.

Cornu spiral in Grasshopper

A test with construting a cornu spiral in Grasshopper.
Script written in VB.NET. Download .ghx-file below:

http://omkrets.se/grasshopper/MN-001-cornu-spiral.zip

Grasshopper

I have used the Grasshopper plug-in for Rhino developed by David Rutten to do some more testing with creating the Cornu Spiral. First manually...


And later by using the Script function.

Generative components

With GenerativeComponents (GC) I tried to mimic bending in 3d.

This shows an attempt to generate cornu spirals in GC. Not quite right - but beautiful!

Here I used scripting in GC to get a twisting strip.

Bending in 3D

So far, most examples have been 2d-curves, but what happens when you bend in 3 dimensions.


I made a 3d loop with the saw blade and scanned it with a Microscribe 3d digitizer. Download 3dm-file here.

Cornu spirals in MicroStation

The CAD-program MicroStation has a command for clothoids (cornu spirals) that I used to draw this figure.


When I combined these curves with the ones I had traced by hand, the match was very good!

The traced curves from the saw blade were double (mirrored) clothoid segments.

Finding the Cornu Spiral

I involved a friend (Ola J.) who helped me searching. He found an article by B.K.P. Horn that was called “The Curve of Least Energy”, which contained some pages about a spiral called the “Cornu Spiral” or “Clothoid”.

This spiral is very useful when designing roads, rails and roller-coasters. Segments of the spiral are used as transitions between for example straight and curved parts and give the smoothest possible changes in curvature.

Arranging curves

I digitized these curves and used a CAD-program to arrange them in different ways in order to find some kind of clue to the geometry. The curves could roughly be represented as NURBS-curves with two control points in the application Rhinoceros (Rhino).

Arranged in this way, the endpoints seemed to lie along two circles with the diameter 4/5 of the curves length. The displacement between the circles was 1/4 of the diameter.

I measured angles and distances in order to understand, but could not see any connections.

Tracing a saw blade

A saw blade, 90 centimetres (3 feet) long, was traced in different positions.


The result was 35 curves with equal length but different distance between the endpoints.

First steps

In order to answer these questions I started tracing strips of plywood. If the endpoints touched each other, there was almost a 90 degree angle and a drop-like shape.

Here, a strip of paper was bent and traced several times. The anchor points are fixed, but the curve length vary.

I scanned the curves and investigated them in a CAD-program. They were obviously not arcs.

Trying to understand the geometry of bending

When you bend a thin strip of an elastic material you get a beautifully shaped curve. What geometry does this curve follow? Can the curve be calculated if you know the length of the material and the position of the end points? Is it possible to calculate more complex situations with several forces in different directions? Can you make similar calculations in 3d? Can this geometry be useful in design/production?