Is a cycloid arch the most stable of arches?

Question

I figured I could find this information pretty readily on the internet, but AFAICT it's not there, at least not in any obvious form. So, I have a vague recollection of years ago being taught that the shape of an arch that could be made simply by stacking blocks next to each other was a cycloid. And other arch would need mortar to make it hold together, but the cycloid would naturally retain its shape. (This is obviously, even if true, an idealization.) Alternatively, if helium balloons were spaced at equal lengths along an anchored string and allowed to rise, they pull the string into an approximate cycloid, if I am remembering correctly.

Is this true? I have tried to use this idea to derive the differential equation of a cycloid, but I am getting the wrong equation.

Answer 1

I see quite a few articles asserting that the catenary is the ideal form for an arch. The wikipedia article on Catenary arch states:

For an arch of uniform density and thickness, supporting only its own weight, the catenary is the ideal curve.

This page on the catenary, from Making Math Visible website, explains that

... the force of gravity on each small segment now results in a compression force which is aimed as a tangent to the curve ... Before the mathematical properties of catenaries were fully understood, stone masons in medieval times knew this principle and designed arches by inverting the curves they measured from hanging chains.

And this paper describes the analogy between an arch and a hanging cable (which takes the form of a catenary):

Another way to understand the behaviour of masonry arches was proposed by Robert Hooke: “As hangs the flexible line, so but inverted will stand the rigid arch” [Hooke 1675] (fig. 8). The equilibrium of cables and arches is the same problem, and this was Hooke’s genial analysis. Another English mathematician, David Gregory, completed Hooke’s assertion: “None but the catenaria is the figure of a true legitimate arch, or fornix. And when an arch of any other figure is supported, it is because in its thickness some catenaria is included”.

As for the helium balloon connection, this MAA Found Math image depicts a chain of helium-filled balloons tied at each end, forming a catenary.