Catenary and Parabola

Catenary(悬垂线）

Hooke discovered that the catenary is the ideal curve for an arch of uniform density and thickness which supports only its own weight. When the centerline of an arch is made to follow the curve of an up-side-down (i.e. inverted) catenary, the arch endures almost pure compression, in which no significant bending moment occurs inside the material. If the arch is made of individual elements (e.g., stones) whose contacting surfaces are perpendicularto the curve of the arch, no significant shear forces are present at these contacting surfaces. (Shear stress is still present inside each stone, as it resists the compressive force along the shear sliding plane.) The thrust (including the weight) of the arch at its two ends is tangent to its centerline.

However the conditions for a catenary to be the ideal arch are almost never fulfilled: arches usually support more than their own weight, and on the rare occasions when they are freestanding they are sometimes not of uniform thickness. The ideal shape for an arch supporting a large weight is more like a parabola than a catenary. As a result, there are very few arches that have been deliberately built as catenaries, though there are quite a few incorrect claims that various arches are catenaries.

Parabola（抛物线）

In nature, approximations of parabolae and paraboloids are found in many diverse situations. The best-known instance of the parabola in the history of physics is the trajectory of a particle or body in motion under the influence of a uniform gravitational fieldwithout air resistance.

Simple Suspension Bridge:

Free-hanging chains follow the catenary curve, but suspension bridge chains or cables do not hang freely since they support the weight of the bridge. In most cases the weight of the cable is negligible compared with the weight being supported. When the force exerted is uniform with respect to the length of the chain, as in a simple suspension bridge, the result is a catenary.When the force exerted is uniform with respect to horizontal distance, as in a suspension bridge, the result is a parabola.

When suspension bridges are constructed, the suspension cables initially sag as the catenary curve, before being tied to the deck below, and then gradually assume a parabolic curve as additional connecting cables are tied to connect the main suspension cables with the bridge deck below.

The Gateway Arch, Earo Saarinen, St. Louis

is sometimes said to be an (inverted) catenary, but this is incorrect. It is close to a more general curve called a flattened catenary, with equation y=Acosh(Bx). (A catenary would haveAB=1.) A catenary is the ideal shape for a freestanding arch of constant thickness, but the gateway arch is not of constant thickness as it gets narrower near the top. According the U.S.National Historic Landmark nomination for the arch, it is a "weighted catenary" instead. Its shape corresponds to the shape that a weighted chain, having lighter links in the middle, would form.

Parabola: http://en.wikipedia.org/wiki/Parabola

Catenary:http://en.wikipedia.org/wiki/Catenary