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The Delightful Catenary Curve

When teaching how tension and compression relate to geometrical structures such as bridges, arches, and domes, I show a picture of the Notre Dame Cathedral in Paris (Figure 1A), completed in the 14th century. I point out the elaborate buttresses that keep the walls from pushing outward while supporting its weight. Architects of the day had not yet learned how to hold up a very large, massive building without external propping. This was accomplished in the 17th century in the construction of St. Paul's Cathedral in London (Figure 1B). Why, I ask, is St. Paul's Cathedral free of such buttresses? Aha, inside its famous dome is an inner "secret dome" that provides structural support. To understand this, let's first investigate the roles of tension and compression in structures. [FIGURE 1 OMITTED] Tension I stretch a length of rope taut, explaining that the stretching force we call tension acts in a direction parallel to the direction of the rope. When I let the rope sag between my hands, tension vectors within the sagging rope continue to align with the rope. The curved shape of the sag is determined by this alignment of tension vectors. Likewise for a sagging chain or sagging cable. A rope, chain, or cable supported at its ends and hanging only by its own weight takes the shape of a special curve called a catenary. I sketch a sagging chain on the board and show that tension vectors between links of the chain are everywhere parallel to the curve with no components of tension perpendicular to the curve (Figure 2). The chain ends can be held at different distances apart, making the curve deep or shallow. As long as the chain supports only its own weight, it's a catenary. [FIGURE 2 OMITTED] If a sagging chain or cable supports weight that is distributed uniformly in a horizontal direction, as is approximately true in a suspension bridge, then the shape of the curve is a parabola, the same curve followed by a tossed ball. The curved cables of a suspension bridge or suspended roadway are approximately parabolas. Only if the cable supports only its own weight--such as sagging clotheslines, power lines, and strands of spider webs--is the shape a catenary. Compression--and the inverted catenary (an arch) Of particular interest is an inverted catenary, where internal forces are of compression rather than tension. When a free-standing arch takes the shape of an inverted catenary, the weight of the arch is supported by compression forces pressing along the arch's curve. There are then no compressive forces perpendicular to the curve. My grandson Manuel delightfully shows two catenaries in Figure 3, one of a suspended chain and in the background the Gateway Arch in St. Louis, Missouri. [FIGURE 3 OMITTED] I sketch the Gateway Arch, showing that compression vectors between adjacent slabs that make up the arch are everywhere parallel to the curve (Figure 4). I tell students that they could make a stable mini-arch out of slippery blocks of ice if the shape of the arch is a catenary! But if the shape were any other, such as a semicircle, blocks of ice would squeeze free, and the arch would collapse. Where strength is important, modern arches are usually catenaries. The three-dimensional catenary: a dome I ask students to imagine rotating an arch through a complete circle. …