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New journal paper addresses a 300-year old question

A new paper by Professor Wanda Lewis addresses a question which has remained unresolved since the 17th century. The paper, entitled “Mathematical Model of a Moment-less Arch” is published in the highly prestigious Proceedings of the Royal Society A; a rare achievement for a single-author paper.

The article addresses the question of optimal form of arch bridges, the question dating back to the famous academic Robert Hooke, who demonstrated to the Royal Society in 1670 that the optimal shape of an arch (arch developing minimal stresses under loading) is that resembling a line/shape of an upside down chain (catenary form). The only other such form proposed by the classical theory is that of an inverted parabola. However, each of these shapes corresponds to a particular load condition.

The paper fills the gap in the classical theory, as it provides a general mathematical solution to the optimal arch question. It reflects the author’s interest in form-finding as a methodology for designing structures shaped by forces that are allowed to develop in them- a process observed in highly optimised objects in nature.

Professor Lewis has been with the Warwick School of Engineering for 30 years, where she combines her passion for architecture with her wealth of knowledge in structural engineering and mathematics. Here she explains the reasoning and background behind this research:

“The work presented in this paper arises from my interest in minimal structural forms and their relationship with principles of shape formation observed in natural objects, such as minimum potential energy of surface tension in soap films, for example. Soap films are being used as analogues in the design of tension fabric membranes/roofing forms, and this produces ‘stable minimal surface’ configurations, i.e. surfaces of constant surface tension. Known to be optimal for the case of a permanent load, i.e., pre-stress, these surfaces are found /calculated using a variety of numerical form-finding techniques, which, in principle, use loads to shape the overall geometry of such structures." More information can be found in this article:

"Having studied fabric structures for over 25 years, my attention turned to optimal ‘rigid’ structures and the development of form-finding techniques that can produce them. The paper submitted here represents continuation of this work, and proposes a mathematical model that amounts to an analytical form-finding method for shaping moment-less arches. By eliminating bending moments, the moment-less arch is shaped by the chosen span/rise ratio, and thrust action only - in response to the permanent loads, i.e., uniformly distributed load per unit span, and arch weight density. To the best of my knowledge, there is no research paper available addressing the question of uniformly distributed loads and arch self-weight density, together with the span/rise ratios, as parameters responsible for shaping the moment-less arch.”

For the website of the Proceedings of the Royal Society A visit:

 Arched Bridge