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Concept
Parabolic arch
Summary
A parabolic arch is an arch in the shape of a parabola. In structures, their curve represents an efficient method of load, and so can be found in bridges and in architecture in a variety of forms.
While a parabolic arch may resemble a catenary arch, a parabola is a quadratic function while a catenary is the hyperbolic cosine, cosh(x), a sum of two exponential functions. One parabola is f(x) = x2 + 3x − 1, and hyperbolic cosine is cosh(x) = ex + e−x/2. The curves are unrelated.
Unlike a catenary arch, the parabolic arch employs the principle that when weight is uniformly applied above, the internal compression (see line of thrust) resulting from that weight will follow a parabolic curve. Of all arch types, the parabolic arch produces the most thrust at the base. Also, it can span the widest area. It is commonly used in bridge design, where long spans are needed.
When an arch carries a uniformly distributed vertical load, the correct shape is a parabola. When an arch carries only its own weight, the best shape is a catenary.
File:Parabola graphed against a catenary upside down view.png|Parabola (red) graphed against a catenary (blue), view to simulate an arch.
File:Parabola graphed against a catenary upside-down, zoomed out.png|Parabola (red) graphed against a catenary (blue), view to simulate an arch. Zoomed out.
Natural arch
A hen's egg can be fairly well described as two different paraboloids connected by part of an ellipse.
Self-supporting catenary arches appeared occasionally in ancient architecture, for examples in the main arch of the partially ruined Sassanian palace Taq Kasra (now in Iraq), the largest single-span vault of unreinforced brickwork in the world, and the beehive huts of southwestern Ireland. In the modern period, parabolic arches were first used extensively from the 1880s by the Catalan architect Antoni Gaudí, deriving them from catenary arched shapes, constructed of brick or stone, and culminating in the catenary based design of the famous Sagrada Familia.
Official source
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