### What is the Difference Between a Parabola and a Catenary?

This is a fairly common question, and understandably so. They are both quite similar superficially, and can leave you wondering what – if any – differences there are. The confusion is perhaps caused in part by such statements as, “The shape of the main overhead supporting cables of a suspension bridge forms a parabola, while the shape of a simple suspension bridge is a catenary“. Huh? Let’s clarify the difference between a suspension bridge and a simple suspension bridge.

The Golden Gate Bridge and Akashi Kaikyo Bridge are both suspension bridges, while simple suspension bridges are those whose weight runs parallel to the supporting cables – no towers where supporting cables swoop down from. The crucial difference between the two lies in the fact that the former is not “free-hanging” while the latter is. The deck of a suspension bridge is essentially being supported by the two parabolic cables suspended between the two towers.

But these cables are only parabolic because they must support the weight of the bridge deck – which essentially pulls them out of shape. Were it allowed to hang freely, as it does prior to connection to the deck, it would be a catenary. But once the vertical cables connect the main cable to the bridge deck, it’s over for the catenary. A simple suspension bridge however, has no “interferences”, and simply hangs, letting gravity shape it into a catenary. (See image above.)

Structural Benefits of a Catenary

Simply put, inverted catenaries are the strongest possible arch shape under gravity. By taking the profile of a hanging chain and inverting it, you obtain a shape that is structurally ideal for an arch. Why? The shape of a catenary minimizes the potential energy throughout the chain as is evidenced by the way it naturally returns to a catenary after it is disturbed, thereby creating and maintaining a state of equilibrium.

The stability of a catenary is based off of this principle – where all internal forces are balanced. You can manipulate the chain-end placement, length of the chain, and sag, to obtain the desired span and height for your particular needs. And by simply transferring this profile to formwork, you have the initial tools needed to build a catenary arch.

However, there are cases in free-standing arches, such as the St. Louis Gateway Arch (the tallest stainless steel monument in the world), where the arch tapers towards the vertex. Such arches that don’t maintain a constant thickness throughout often don’t follow an exact catenary. Technically speaking, the shape of the St. Louis Gateway Arch follows what’s called a “weighted catenary“, creating a flatter point at the vertex – apparently chosen to make it look less steep.