Mathematical curves and Hooke’s chain theory
Figure 5 Forces in the arch of a bridge 6
Before such discoveries, the arches in history, whether of an arched bridge, or of a dome or that of on top of a doorway, had the shape of a semicircle (named semi-circular arches), because it was easier for the architects of the time to trace or design such shapes by using a compass. However, through analysis of the thrust line, the semi-circular design is not the most optimal of all.
Figure 6 Thrust line in a minimally thin semi-circular arch 7
The line of thrust is defined as a theoretical line that represents the path of the resultants of the compressive forces. By architectural convention, for the structure to not collapse, the thrust line should be kept within the structure itself. It is also important to recognize that if the thrust line exceeds over one-third of the structural element, the sections which are subjected to such conditions would have some parts subjected to tension. 8
Figure 7 Stress/strain analysis of a 2D element (below is the force distribution across the width) 9
When stone or concrete elements are subjected to tension, they are highly likely to crack. Consequently, the crack would spread due to weathering conditions and thus cause structural failure. When observing Figure 5, we can see that most of the materials are wasted as they do not contribute to containing the thrust
6 See ibid. 189, figure 14. 7 See THEORY AND HISTORY OF THE THRUST LINE FOR MASONRY ARCHES. A BRIEF ACCOUNT (ARCH '04). 8 See Structures, or why things don't fall down 178-181. 9 See ibid. 177, figure 4.
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