Mathematical curves and Hooke’s chain theory
model with the optimal shape, after calculations are performed to account for the crushing load of materials and the life loads to determine which materials to use and how much to use. 11
4.2
Heinz Isler (1926-2009)
Isler was a structural engineer who is famous for thin-shelled structures made out of concrete. Before designing the ones made out of concrete, Isler used to create structures by hanging clothes sprayed with water in his backyard and freezing them. These structures, by definition of Hooke’s chain theory, if flipped upside down perfectly, the frozen clothes should be in the most optimal form to support themselves in compression. 12 For modelling the thin concrete structures, a hanging model is used. Normally a small piece of fabric or plastic net is hanged off of a few corners, the number of corners corresponds to the number of points of contact the desired structure has with the ground. Then, similar to the method of Gaudi, the models are photographed and analysed, and the thin-shelled structures are constructed. The piece of material is subjected to tension mainly when hanged and only under its own weight, after being turned upside down. It means that the constructed structure is subjected mainly to compression, which is ideal for concrete, and the structure holds its own weight with the minimum material.
4.3
Mike Schlaich
Schlaich is a specialist in lightweight structures, similar to Isler. His most famous, award-winning work, the stainless-steel footbridge in Ditzingen, is designed and modelled applying Hooke’s chain theory. In an interview, Schlaich demonstrated the simplified process by which the bridge was modelled, where a plastic net with square holes is used, which covers a series of masses. A reverse example is shown in figure 10. 13
Figure 10 Demonstration done during the interview (a reversed demonstration for convenience)
The oranges, in this case, make the plastic net subject to tension. When turning the structure upside down, it would make the most optimal shape for compression. In the actual modelling process, the plastic net hangs by six points and the block of mass representing the mass of the walkway over the bridge is placed on top of the net. Then turning the model upside down produces the optimal structure for Figure 11 Schlaich’s bridge
11 See https://mathstat.slu.edu/escher/index.php/The_Geometry_of_Antoni_Gaudi#:~:text=Gaudi%20used%20catenary
%20arches%20in,suspended%20chains%20is%20on%20view. 12 See https://www.youtube.com/watch?v=8c8SEemAXO0. 13 See https://www.youtube.com/watch?v=VeahtDy7n8I.
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