Catenary Arch

Anyone know how to make catenary arches? Or maybe parabolic? I'm doing a bridge design and I'm not sure which I need. The arch is in compression and has suspension cables attached to the deck of the bridge beneath. I know the catenary is ideal when not loaded and it has no shear (like St Louis arch), but I'm hanging weight from it, so I'm not sure. Well theory aside, all I'm really looking for is to make them in GH. I know the function component can do COSH and SINH, but I figured I'd ask if someone had done this, before I try to remember trig.

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Permalink Reply by Luis Fraguada on November 20, 2009 at 9:24am

Wolfram Mathworld Catenary

I am pretty sure I've seen Giulio messing with something like this...

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Permalink Reply by Chris Wilkins on November 20, 2009 at 9:59am

Thanks.

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Permalink Reply by Manuel on November 20, 2009 at 12:52pm

giulio piacentino solved for non symmetrical points:
http://www.grasshopper3d.com/xn/detail/2985220:Comment:25775

I couldn't solve the catenary equation so I made myself a definition to solve parabolic curves from three points if you are interest (this one was before giulio solved).

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Permalink Reply by Roberto on November 30, 2009 at 12:52pm

Hi.
I have been interested in this problem for a while.
Apart from a mathematical solver, there is room for a numerical one, based in strings and an iterative process that finds the relaxed state of the catenary. This approach, though being less precise than the mathematical one, can be used to solve undetermined configurations just by applying the concept to more general cases. All the script is based on cloth relaxation procedures, which are widely described on the net.

In the definition I dared to include Giulio Piacentino's catenary2d solver in order to check the distortion between the mathematical solution and mine. Thank you Giulio, very inspiring work!

Besides, there is a romantic way of solving a catenary arc by applying the principles of graphic statics. In my second attachment you will find a definition that calculates the funicular polygon of a uniform distribution of forces between two points (even asymmetric). In this case, target length is not an input of the definition, but could be calculated numerically if the whole definition was scripted.

Please, take this definitions as they are, modify and share. Further development of the numerical solver shall include mesh relaxation and better iterative algorithms.

Hope it helps.

References:
http://freespace.virgin.net/hugo.elias/models/m_string.htm
http://www.cs.ucl.ac.uk/research/vr/Projects/3DCentre/cloth_simulat...
http://www.designexplorer.net/newscreens/cadenarytool/cadenarytool....

Be happy
R.

Attachments:

catenary3d_num_explicit_solver.ghx, 214 KB
catenary3d_num_explicit_solver.3dm, 27 KB
catenary3d_funicular_polygon.ghx, 352 KB

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Permalink Reply by Chris Wilkins on November 30, 2009 at 2:09pm

Tonight, on Grasshopper Confessions... The Fake Catenary!

After fiddling with Giulio's code for some bit of time, I ended up scaling a 3-point arc in one dimension to create a fake catenary. Sometimes it is counterproductive to be too realistic with school projects, especially when nobody can even tell that you even used a real catenary. But I applaud Roberto's further explorations!

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Permalink Reply by Ivan Kiryakov on March 1, 2012 at 5:58pm

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Permalink Reply by Roberto on November 30, 2009 at 4:29pm

Hi Karthik.
To be honest, this was the first thing I did based on the principles of graphic statics. It is something we learnt in the school of architecture and never used again until now. I have never seen anything of 3d graphic statics but if you come up with a interesting problem I am sure we will have fun.

This is the reference I used.
http://books.google.es/books?id=Fn-1j6bOphwC&dq=funicular+polyg...

Seems interesing to me that there is also a simple solution for the problem with three constrained points...
I will take a look at the paper you mention, seems amazing!