Hanging Chain Form-Finding

Linking Excel API with ROBOT API using VB.NET to facilitate form finding of MULTIPLE hanging chains. With care, the engineering platform ROBOT can be used to facilitate the form-finding of multiple shapes. Inverting the form will theoretically give the optimum shape for a purely compressive action. By varying the catenary profile lengths in Excel we can arrive at an infinite range of possible shapes. There is no explicit mathematical solution for multiple hanging chains. Geometric non linear solutions are needed or physical models a la Gaudi.

Robot has a geometric non-linear solver. Typically for a linear analysis the stiffness matrix is not updated when the geometry changes. For typical rigid structures this assumption is fine. Obviously for chains, tensile structures, soap films this is not the case. Furthermore, we have to find the shape that is in a static equilibrium - we can’t simply dictate the shape of a soap film tensile structure for instance.

So basically robot updates the matrix as the geometry changes. It still uses a stiffness matrix approach hence can be very unstable and you have to "nudge" teh model along by not applying too much load at once. I basically start with a network of flat chains at the start of the analysis. Their connectivity, geometry of end points, final hung lengths and other attributes are defined in excel. Hence we can quickly vary the hung length to get a new profile. I then apply a nominal vertical load on each chain of the network to "nudge" it to start the analysis process of finding the stable equilibrium shape of all chains. We then invert teh chains and rotate to get the inverted cupola form.

Unfortunately a stiffness matrix approach is not the best or most stable solution for geometrically non-linear structures. The best approach is to use Dynamic Relaxation - see my images where I form-found the shape of a soap film from an initially flat shape.