Hyperboloid structure

Hyperboloid structures are architectural structures designed using a hyperboloid in one sheet. Often these are tall structures, such as towers, where the hyperboloid geometry's structural strength is used to support an object high above the ground. Hyperboloid geometry is often used for decorative effect as well as structural economy. The first hyperboloid structures were built by Russian engineer Vladimir Shukhov (1853–1939), including the Shukhov Tower in Polibino, Dankovsky District, Lipetsk Oblast, Russia.

Hyperbolic structures have a negative Gaussian curvature, meaning they curve inward rather than curving outward or being straight. As doubly ruled surfaces, they can be made with a lattice of straight beams, hence are easier to build than curved surfaces that do not have a ruling and must instead be built with curved beams.

Hyperboloid structures are superior in stability against outside forces compared with "straight" buildings, but have shapes often creating large amounts of unusable volume (low space efficiency). Hence they are more commonly used in purpose-driven structures, such as water towers (to support a large mass), cooling towers, and aesthetic features.

A hyperbolic structure is beneficial for cooling towers. At the bottom, the widening of the tower provides a large area for installation of fill to promote thin film evaporative cooling of the circulated water. As the water first evaporates and rises, the narrowing effect helps accelerate the laminar flow, and then as it widens out, contact between the heated air and atmospheric air supports turbulent mixing.^{[citation needed]}

In the 1880s, Shukhov began to work on the problem of the design of roof systems to use a minimum of materials, time and labor. His calculations were most likely derived from mathematician Pafnuty Chebyshev's work on the theory of best approximations of functions. Shukhov's mathematical explorations of efficient roof structures led to his invention of a new system that was innovative both structurally and spatially. By applying his analytical skills to the doubly curved surfaces Nikolai Lobachevsky named "hyperbolic", Shukhov derived a family of equations that led to new structural and constructional systems, known as hyperboloids of revolution and hyperbolic paraboloids.

The steel gridshells of the exhibition pavilions of the 1896 All-Russian Industrial and Handicrafts Exposition in Nizhny Novgorod were the first publicly prominent examples of Shukhov's new system. Two pavilions of this type were built for the Nizhni Novgorod exposition, one oval in plan and one circular. The roofs of these pavilions were doubly curved gridshells formed entirely of a lattice of straight angle-iron and flat iron bars. Shukhov himself called them azhurnaia bashnia ("lace tower", i.e., lattice tower). The patent of this system, for which Shukhov applied in 1895, was awarded in 1899.

Shukhov also turned his attention to the development of an efficient and easily constructed structural system (gridshell) for a tower carrying a large load at the top – the problem of the water tower. His solution was inspired by observing the action of a woven basket supporting a heavy weight. Again, it took the form of a doubly curved surface constructed of a light network of straight iron bars and angle iron. Over the next 20 years, he designed and built nearly 200 of these towers, no two exactly alike, most with heights in the range of 12m to 68m.

At least as early as 1911, Shukhov began experimenting with the concept of forming a tower out of stacked sections of hyperboloids. Stacking the sections permitted the form of the tower to taper more at the top, with a less pronounced "waist" between the shape-defining rings at bottom and top. Increasing the number of sections would increase the tapering of the overall form, to the point that it began to resemble a cone.