Mathematical curves and Hooke’s chain theory
line within the structure. As stated by Robert Hooke, when a structure is subjected to tension (such as a chain), the shape placed upside down would be the path of the compressive force (path of the thrust line). And this can be proven mathematically using a similar method to that of section 2.1. 10
Knowing the path of the thrust line enables more optimal structures to be designed, for example, the arch. For the arch which needs to hold its own weight, its optimal shape would be that of a catenary, as that supports its own weight in tension making such a shape, which means that the thrust line when held in compression will have such shape (but upside down). Material will only be needed in an amount to form such a shape and have sufficient cross-sectional area and compressive strength not to be crushed at the bottom (assuming the absence of live load such as wind or earthquake).
Uses of previously mentioned models
3.1 Antonio Gaudi (1852 - 1926)
Gaudi is one of the most famous architects in Spain and is known for his bizarre and unique style of constructions. Gaudi was quite interested in the application of Robert Hooke’s chain theory, as expressed in his works. For example, in Casa Milá, the top roof has a catenary shape (as shown in figure 8), because the roof is only supporting its own weight in this case. This design allows for a relatively thin and stable rooftop built using concrete.
Figure 8 Graph of y = −coshx plotted using Desmos and layered on Casa Milá’s drawing set.
In his work, the cathedral of Sagrada Familia, the use of Hooke’s chain is more obvious, as shown in figure 9. As the model is turned over, the configurations of the arches and beams are revealed. The sandbags are used to represent the compressive forces experienced by the structural elements, while the form of the strings takes the most effective form by which the arches and beams can save materials. The string model will be carefully photographed in various angles upside down. As the string model only acts in tension, according to Hooke’s chain theory, the cathedral of la Sagrada Familia is also supported with mostly only compression. The structures are constructed according to the
Figure 9 String model for Sagrada Familia
10 See https://www.researchgate.net/publication/335137167_THEORY_AND_HISTORY_OF_THE_THRUST_LINE_FOR_M ASONRY_ARCHES_A_BRIEF_ACCOUNT_ARCH_%2704.
288
Made with FlippingBook - Online catalogs