page_title: Hyperboloid structure
section_title: Work of Shukhov
hierarchical_section_title: Hyperboloid structure / Work of Shukhov
context_page_description: 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 off the ground, but hyperboloid geometry is also often used for decorative effect as well as structural economy. The first hyperboloid structures were built by Russian engineer Vladimir Shukhov. The world's first hyperboloid tower is located in Polibino, Dankovsky District, Lipetsk Oblast, Russia.
context_section_description: 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 gravity load at the top – the problem of the water tower. His solution was inspired by observing the action of a woven basket holding up 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 twenty years, he designed and built close to two hundred 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.
  By 1918 Shukhov had developed this concept into the design of a nine-section stacked hyperboloid radio transmission tower for Moscow. Shukhov designed a 350m tower, which would have surpassed the Eiffel Tower in height by 50m, while using less than a quarter of the amount of material. His design, as well as the full set of supporting calculations analyzing the hyperbolic geometry and sizing the network of members, was completed by February 1919; however, the 2200 tons of steel required to build the tower to 350m were not available. In July 1919, Lenin decreed that the tower should be built to a height of 150m, and the necessary steel was to be made available from the army's supplies.  Construction of the smaller tower with six stacked hyperboloids began within a few months, and Shukhov Tower was completed by March 1922.
caption: The gridshell of Shukhov Tower in Moscow.
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