An amateur's contribution to the design of Telford's Menai Suspension Bridge: a commentary on Gilbert (1826) ‘On the mathematical theory of suspension bridges’

Abstract

Davies Gilbert's work on the catenary is notable on two counts. First, it influenced Thomas Telford in formulating his final design for the Menai Strait suspension bridge (1826); and second, it established for the first time the form of the ‘catenary of equal strength’. The classical catenary is a uniform flexible chain or cable hanging freely under gravity between supports. The ‘catenary of equal strength’ is the form of a cable whose cross-sectional area is made proportional to the tension at each point, so that the tensile stress is uniform throughout. In this paper I provide a sketch of the lives and achievements of Gilbert and Telford, and of their interaction over the Menai Bridge. There follows a commentary on Gilbert's 1826 paper, and on his two related publications; and a brief sketch of the earlier history of the catenary. I then describe the development of the suspension bridge up to the present time. Finally, I discuss relations between mathematical analysts and practical engineers. This commentary was written to celebrate the 350th anniversary of the journal Philosophical Transactions of the Royal Society.

Theoretically optimal forms for very long-span bridges under gravity loading

“…The fact that a high span-to-dip ratio in a cable-supported bridge can lead to a high volume of material being required has been known for many years (indeed this prompted the seminal work of Davies Gilbert, who wanted to demonstrate to Thomas Telford that the initial shallow suspension bridge design for the Menai Straits was structurally inefficient [ 24 ]). However, a low span-to-dip ratio also requires the use of tall pylons, which can be problematic to construct.…”

“…In fact, when self-weight is taken into account it has been known for almost two centuries that each (non-vertical) element in an optimal structure must take the form of a catenary of equal strength [23,24]. This is an element which is free of bending and has a cross section which varies along its length, thus ensuring no excess material is present, a requirement in a rigorously optimal solution (notwithstanding that uniform cross sections are normally preferred in practice, for practical reasons).…”

Constant stress arches and their design space

“…The first contribution in this area was due to Gilbert [6] who described the curve as a 'catenary of equal strength'. His derivation, given in Newton's fluxional calculus notation, and characterized by many divergences from contemporary terminology, was made accessible by Calladine [7]. A more concise form of the 'catenary of equal strength' equation, given in Cartesian coordinates, was later derived by Routh [8].…”

Constant Stress Arches

“…The first contribution in this area was due to by Gilbert [5] who described the curve as a 'catenary of equal strength'. His derivation, given in Newton's fluxional calculus notation, and characterised by many divergences from contemporary terminology, was made accessible by Calladine [6]. A more concise form of the 'catenary of equal strength' equation, given in Cartesian co-ordinates, was later derived by Routh [7].…”