The seventeenth-century scientist Robert Hooke’s theory that a hanging chain will mirror the shape of an upstanding rigid arch is well-known to structural engineers. New research, however, shows that this widely held belief is incorrect because, despite their similarities, the hanging chain and the arch are incompatible mechanical systems.
One of the fundamental rules of construction has been turned on its head by an Aston University researcher. For centuries, a hanging chain has been used to demonstrate how masonry arches stand. The seventeenth-century scientist Robert Hooke’s theory that a hanging chain will mirror the shape of an upstanding rigid arch is well-known to structural engineers.
However, research from Aston University’s College of Engineering and Physical Sciences, shows that this common-held belief is incorrect because, regardless of the similarities, the hanging chain and the arch are two incompatible mechanical systems.
The analogy between inverted hanging chains and the optimal shape of masonry arches is a concept deeply rooted in our structural analysis practices. Curved structures have enabled masons, engineers, and architects to carry heavy loads and cover large spans with the use of low-tensile strength materials for centuries while creating the marvels of the world’s architectural heritage.
Dr Haris Alexakis
Dr Haris Alexakis used the transition in science from Newtonian to Lagrangian mechanics, which led to the development of modern physics and mathematics, to prove this with mathematical rigor.
He revisits the equilibrium of the hanging chain and the arch in his paper Vector Analysis and the stationary potential energy for Assessing the equilibrium of Curved Masonry Structures, explaining that the two systems operate in different spatial frameworks. As a result, the hanging chain requires only translational force to be in equilibrium, whereas the inverted arch requires both translational and rotational force. As a result, solutions are always unique.
Dr. Alexakis’ analysis revealed subtle inconsistencies in the way Hooke’s analogy has been interpreted and applied for the design and safety assessment of arches over the centuries, as well as its practical limitations.
He said: “The analogy between inverted hanging chains and the optimal shape of masonry arches is a concept deeply rooted in our structural analysis practices. Curved structures have enabled masons, engineers, and architects to carry heavy loads and cover large spans with the use of low-tensile strength materials for centuries while creating the marvels of the world’s architectural heritage.”
“Despite their long history, finding optimal structural forms and assessing the stability and safety of curved structures remains as relevant as ever.” This is because there is a growing interest in preserving heritage structures and reducing material use in construction by replacing steel and concrete with low-carbon natural materials.”
His paper, published in the journal Mathematics and Mechanics of Solids, proposes a new structural analysis method based on the principle of stationary potential energy that would be faster, more flexible, and aid in the calculation of more complex geometries. As a result, analysts will no longer need to consider the equilibrium of each individual block or describe the load path of thrust forces geometrically in order to obtain a rigorous solution.
“The analysis tools discussed in the paper will enable us to assess the condition and safety of heritage structures and build more sustainable curved structures, such as vaults and shells,” Dr. Alexakis added. Apart from their appealing aesthetics, the main advantage of these structures is that they can be made of low-cost, low-tensile-strength, and low-carbon natural materials, contributing to net zero construction.”
