Tuesday, May 11, 2010

Arch designs: Corbels and Catenaries

We’re taking another small detour from description of applications for the masonry system I’ve developed to talk about some fundamental aspects of masonry. Yesterday we looked at thermal mass, today we’ll take a look at types of arches; another important aspect to this masonry system.

Very early in the development of masonry in the ancient world, masons developed what is known as a corbel arch. A corbel arch is not a “true” arch because it uses blocks that are cantilevered, and do not transfer the load directly down the arch. Corbelled arches are found in ancient Irish, India, Mayan, Greek, and Cambodian architecture.

Here’s a good description of corbelled arches from Wikipedia:
“A corbel arch (or corbeled / corbelled arch) is an arch-like construction method which uses the architectural technique of corbeling to span a space or void in a structure, such as an entranceway in a wall or as the span of a bridge. A corbel vault uses this technique to support the superstructure of a building's roof.

A corbel arch is constructed by offsetting successive courses of stone at the springline of the walls so that they project towards the archway's center from each supporting side, until the courses meet at the apex of the archway (often capped with flat stones). For a corbeled vault covering the technique is extended in three dimensions along the lengths of two opposing walls.

Although an improvement in load-bearing efficiency over the post and lintel design, corbeled arches are not entirely self-supporting structures, and it is sometimes termed a "false arch" for this reason. Unlike "true" arches, not all of the structure's tensile stresses caused by the weight of the superstructure are transformed into compressive stresses. Corbel arches and vaults require significantly thickened walls and an abutment of other stone or fill to counteract the effects of gravity, which otherwise would tend to collapse each side of the archway inwards.”

A much better arch design is a catenary arch. “Catena” is Latin for “chain.” If a chain is allowed to hang with some slack, and if the links of the chain are welded together and the chain is flipped upside down, the resulting curve is a catenary curve.
Here’s a good discussion of catenary from Wolfram Mathworld:

“In 1669, Jungius disproved Galileo's claim that the curve of a chain hanging under gravity would be a parabola (MacTutor Archive). The curve is also called the alysoid and chainette. The equation was obtained by Leibniz, Huygens, and Johann Bernoulli in 1691 in response to a challenge by Jakob Bernoulli.

Huygens was the first to use the term catenary in a letter to Leibniz in 1690, and David Gregory wrote a treatise on the catenary in 1690 (MacTutor Archive). If you roll a parabola along a straight line, its focus traces out a catenary. As proved by Euler in 1744, the catenary is also the curve which, when rotated, gives the surface of minimum surface area (the catenoid) for the given bounding circle.”

Catenary arches are the strongest possible arch under gravity. Catenary arches are commonly used in gas-fired kilns. They are found across many cultures and civilizations around the world. An interesting example is Musgum architecture found in Cameroon, Africa. These beautiful structures seem naturally derived, and are a fine example of using this optimal arch design to build houses.

If a slack chain is allowed to settle into a shape approximating a spherical curve, it gets pretty close. Thus a spherical dome is not too far from a catenary arch, and provides some indication that a spherical section is a strong arrangement.

In an environment with little or no gravity, the advantages of a catenary arch disappear, and a spherical structure stands alone as the strongest and best design for a shelled structure.

This opens the door to extraterrestrial applications, which we’ll be looking at tomorrow. Sounds like science fiction, but it’s really not.

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