Mathematical curves and Hooke’s chain theory
2.2 Quadratic parabola
Other erroneous solutions previous to that of catenary have wrongly assumed the string to be of negligible mass and that there are weights with even spaces in between them hanging off the string, or that the weightless string has a uniformly distributed load across its x length. The reason why such proofs have resulted in a parabola curve will be deduced below:
Figure 4 Free body diagram of a section of the string
The string is in static equilibrium, which means:
∑𝑀 =0
Balance the moments at the end point of the section:
𝑤𝑥 2 2
𝑦 =
(1)
Rearrange:
𝑤𝑥 2 2𝑇 0
𝑦 =
(2) 4
𝑤 2𝑇 0 is a constant, y is a quadratic function of x, which means that the string takes the shape
As we can see, as
of a parabola. This modelling method, although it was inappropriate for that of a free-hanging string under its own weight, is quite an accurate model for a string that is under uniformly distributed loads which are far greater than weight of the string. The parabolic model can be used for the modelling of the main cables of suspension bridges of small intervals.
Importance of Robert Hooke’s findings
Architects have known since ancient times that the uses of arches can effectively save materials, simplify the structure, and make an aesthetic and elegant construct, as the arches are elements capable of holding massive loads by conducting only compressive force across the element. The properties of which allows materials which have low compressive strengths, such as masonry and concrete, to be used for their construction without the need to use logs or steel, as these materials might be inaccessible in ancient times; the compressive force is also capable of holding the structure together without the need for mortar mixture as attachment. 5
4 See https://www.youtube.com/watch?v=FcaVxy5YnTo. 5 See Structures, or why things don't fall down 188-190.
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