At first inspection the image above is not terribly interesting. It’s a picture of some helium balloons attached to a piece of string, which is tied to an advertising board at either end. Helium is of course lighter than air and therefore the balloons have risen until they are restrained by the string. The resulting shape is recognisable as an arch.
On closer inspection the image is possibly more interesting than you might think and that’s why I took the photograph. To understand why it's interesting you would need to know something about arches.
Everyone is familiar with the form of an arch and have likely seen many examples. Perhaps the most obvious examples would be arched bridges, which were traditionally built of masonry. More modern examples are built of iron and steel.
From antiquity arch bridges had been designed by trial and error with rules of proportion being gradually developed from the classical period onwards. This worked reasonably well, however engineers were never quite sure how far an arch design could be pushed before it became unstable. A reliable design theory was required, which did not rely on having to build lots of examples to see what happened. In fact how to determine the minimum thickness of an arch remained a major challenge until the mid nineteenth century.
Notwithstanding the desire to avoid trial and error it was an important challenge for several other reasons. Thin arches were generally viewed as being more elegant than thick ones. Thin arches are also cheaper because they are constructed of fewer raw materials and because they are lighter. Lighter arches could have smaller abutments and the timber supports (centring), used to support the arch while it was being constructed, could be less substantial.
One of the first people to make progress on the structural behaviour of arches was none other than Robert Hooke; one of the foremost thinkers of his day and a contemporary of Issac Newton.
Hooke of course new what is intuitive to almost everyone. Arches are compression structures that convey loads to the ground by squashing together the masonry blocks from which they are constructed. He would also have known that, much like building a house of cards, the two sides of an arch will tend to spread, and will eventually collapse, unless retrained by abutments.
Hooke's genius was to figure out the most efficient shape for an arch to be. He expressed his insight as “Ut contiuum flexile, sic stabit contiguum rigidum inversum”. [It was the 17th century so everything smart was written in Latin]. A rough English translation would be “as hangs the flexible line, so inverted will stand the rigid arch”.
In other words hook had realised that the best form for an arch with a given span and a given rise is that formed by inverting the shape of a chain suspended between the arch supports.
Self evidently the hanging chain is a tension structure, which modern engineers would call a catenary. Its shape conveys visually the idealised load-path. We recognise catenary cables in the form of a suspension bridge or as shown below in a rudimentary barrier.
Hooke also realised that the compressive thrust in an arch is the inverse of the tension in a hanging chain, which would allow restraints to be designed more efficiently.
Building on Hooke’s insight later engineers showed mathematically that providing an arch’s load path is contained within its thickness the arch would remain stable.
So why does this make my balloon photograph interesting; why did I take that picture? The reason, if you are not ahead of me already, is simple. As we have said already arches are solid structures that convey compression forces while catenaries are flexible structures that convey tension. Both their shapes and the forces they carry are opposite.
The balloon structure I photographed is interesting, because it is a tension structure that has taken the form of an arch. In actual fact it is an inverted catenary and not an arch at all. It has effectively inverted Hooke’s inversion.
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