Arguably the dependence of bending moment on shape is intuitive, but the geometry of developable surfaces is not.

Just like in Maxwell's theory of hills and dales: the location of topographic peaks, saddles etc. is "obvious" but the constraints on where you get saddles and how many, etc. are not. (http://en.wikipedia.org/wiki/Morse_theory or http://www.maths.ed.ac.uk/~aar/surgery/hilldale.pdf)

Or the similar territory of the Euler characteristic. You could know polyhedra very well from a physical, practical point of view and never notice it. (http://en.wikipedia.org/wiki/Euler_characteristic#Polyhedra)

Maybe?

Anyway, I am commenting because I'd be interested to hear your reaction to the problem stated here: https://www.youtube.com/watch?v=36gOx3dguWs#t=17m35s