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, Russia.

Quotes

A groundbreaking advance on the way to efficient structures in steel construction was the principle of using tetrahedra as a basic module instead of rectangular geometries. The inventor is considered to be the all-round genius Alexander Graham Bell, who became famous for the invention of the telephone and who built kites that were big enough to lift people into the air.
Another engineering genius was Vladimir Grgorevic Suchov, whose genius can definitely be compared to Gustav Eiffel. In 1919 he designed towers up to 350 m high on the principle of hyperbolic paraboloids. The lightness of his constructions is seldom achieved even today, in Moscow there is a television tower with a height of 160 m. His tent constructions with suspended steel grids can be seen as the forerunners of the Olympic roof in Munich or the new Center Pompidou Metz. In Vyksa, Suchov realized the first double-curved lattice shells on the floor plan of rectangular halls as early as the 19th century. Most of the structures still in existence are massively endangered by corrosion and destruction; current rescue operations are trying to preserve this legacy.
- Frank Kaltenbach, Wendepunkt im Bauen-Ausstellungsbesuch (2010) Detail, 05/04/2010.

Shukhov's water tower['s]... double curved surface... was generated by a mesh of straight members overlapping in contrary directions... supported by horizontal rings. While... constructed from steel... Shukhov's 1896 patent application... initially mentions straight wooden beams as a material option. ...[T]he ...application describes ...being able to resist extreme forces while using very little material. As a result, Shukhov's... design was used extensively throughout Russia in the first half of the twentieth century.
- Ramsey K. Leung, History and Theory of Gridshell Structure, Thinking While Doing: Explorations in Educational Design/Build (2019) pp. 47 ff.

The theory of minimal surfaces and surfaces of constant mean curvature is a vigorously developing branch of mathematics which has a broad range of applications in physics, chemistry and biology where it investigates, for example, soap films and bubbles, bimaterial interfaces or capillaries. Over the past decades, it was also boosted by the anticipated nanotechnology applications. Applications of minimal surfaces have been extensively explored in the sculpture (see e.g. the works by Robert Engman) and in architecture (tensile structures, the Shukhov radio tower ... works by Frei Otto etc.) The first mention of minimal surfaces goes back to Lagrange (1798), who considered the... variational problem...
- Tatiana Pavlyukevich, "Spinor representation of Bryant surfaces with catenoidal and smooth ends" (Oct, 2014) dissertation.

Construction history teaches us that excellent, huge structures like this one are invariably the result of a specific and singular solution, in which all aspects were reflected upon before or after early experiences in the building sequence. The author wants to compare the achievement of Segovia Aqueduct with more recent singular structures: the Oka electricity pylons (1929) by... Vladimir Gregoryevich Shukhov... and the roof of the 1972 Olympic Games in Munich, by a team directed by Gunter Behnish, Frei Otto, Fritz Leonhardt/Jörg Schlaic, and Jürgen Linkwitz...
- Jos Tomlow, "Observations on the design and building of the Roman Segovia Aqueduct" (2021) History of Construction Cultures ed. João Mascarenhas, Ana Paula Pires, Proceedings of the 7th International Congress on Construction History July 12-16, 2021, Lisbon, Portugal, 7ICCH 2021, Vol. 2, p. 231.

Hyperbolic Structures (1983)

: Shukhov's lattice towers - forerunners of modern lightweight construction by Matthias Beckh with Forward (June 2012) by Rainer Barthel

The structures of the great Russian engineer Vladimir Grigor'evič Šuchov (Shukhov) are among the world's most sophisticated and distinctive in the history of steel construction. These extremely stable structures, such as cable-stabilised arches, doubly curved grid-shells and above all lattice towers hold a great fascination... They result from the desire to achieve an engineering objective using as little material as possible. ...[T]hey are a testament to the extraordinary creativity and inventiveness of an extensively educated engineer ...
- Forward by Rainer Barthel (June 2012)

Many engineering structures of today were anticipated in Shukhov's works. Some of his others have no modern equivalent or have remained unmatched in their visual impact and... technical efficiency. Among these are... the lattice towers... Countless towers with this new type of construction and geometries defined by just a few parameters were built [by him]... The fine-lined tower structures served as water towers, lighthouses, power transmission masts and fire brigade watchtowers—with some of them still in use today
- Forward, Ibid.

Matthias Beckh analyzes these hyperbolic lattice towers... [and] demonstrates how... Shukhov was already parameterizing his structures as part of the design... Modern methods... provide... Beckh with the tools to place Shukhov's achievements in a historical context and validate their considerable contribution to the history of structural engineering. He also...demonstrate[s] their relevance to modern structures. ...Beckh was part of the first research project into Shukhov's structures ...The research project ...included in-depth studies of building history ...and detailed investigations into the way the structures were built. Later wind tunnel investigations were intended to provide ...knowledge about the load assumptions... also... relevant to modern structures.
- Forward, Ibid.

The object of this book is gain deeper knowledge on the architectural history of hyperbolic structures. The focus of the investigation is the first hyperbolic lattice towers ever built, the work of... Shukhov.

This form of construction, which had no predecessors... is notable for its strength and economy of materials. Added to this is the high visual impact...

Even today, Shukhov's load-bearing system can be found in one form or another in architecture...

This book presents the results of the first ever extensive analysis of the way these structures work.

Hyperboloid of one sheet	Conical surface between	Hyperboloid of two sheets

The ruled surface of a one-sheeted hyperboloid is resolved into three different mesh variants to create open lattices and their structural behaviour investigated.

Particular attention is paid to evaluation and analysis of Shukhov's tower calculations and the assumptions made for the structural model. His historical calculations are compared to the results of modern calculations.

Shukhov's design process is reconstructed and the development of the water towers... illustrated.

Until now, the form and geometry of the hyperbolic lattice structures have only been investigated to a limited extent.

A paper by Peter de Vries discussed the stiffness of simple hyperbolic lattice structures and highlighted a single connection between geometry and structural behaviour. The focus... is... on a simple form of hyperbolic lattice structure which always has the intermediate rings positioned at intersection points of the lattice members.

[T]he form and structure of the Sukhov-built towers were developed not on geometrical or constructional criteria alone, their designs specifically took into account structural engineering considerations.

The chapter "Geometry and form of hyperbolic lattice structures"... deals with the form and geometry... A precise description of the mathematical principles of a one-sheeted hyperboloid precedes an explanation of the parametrisation...

[T]he chapter "Structural analysis and calculation methods"... considers the principle means of transfer of vertical and horizontal loads and describes the interactions between geometry and structural behaviour. Then... an explanation of the theoretical principles of determining a lattice tower's ultimate load capacity.

The chapter "Design and analysis of Sukhov's towers"... is devoted to consideration and analysis of Sukhov's structural calculations... Sukhov's design process is reconstructed based on... calculations of five different water towers. From... tables stored in the Moscow city archives, a summary... is produced to chart the development of the towers over more than three decades...

An analysis of the design and construction of the NiGRES tower on the Oka... The initial ensemble of four electricity transmission masts represents the consummation of Shukhov's tower construction method. Only one... remains... today.

The final chapter "Towers in comparison"... contains an extensive table and drawings of 18 towers.

It is hoped that the examination of Sukhov's form of construction made public in this book will give an impetus to new applications in architecture.

Design of Reconfigurable doubly-curved canopy structure (2013)

F. Madden, K. Korkmaz, Y. Akgun, Structures and Architecture: New concepts, applications and challenges pp. 1040 ff.

[A] new reconfigurable doubly-curved structure has been developed for a canopy roof.

[A] series of kinetic structures which are capable of geometric transformations have been developed. ...[T]he most impressive ones are deployable bar structures with [a] single degree-of-freedom ...These structures ...may become stable and carry loads ...Therefore ...may offer viable solutions for architectural applications, especially for temporary buildings, emergency shelters, exhibition halls, outdoor recreation facilities or sporting fields ...[M]ost of them are composed of scissor like elements (SLEs) ...which is a complex structural system ...[P]resent solutions are insufficient to constitute real form flexibility, because they are limited to... forms such as singly-curved vaults and doubly-curved synclastic domes.

Due to... complexity... doubly-curved anticlastic surfaces such as hyperboloid and hyperbolic paraboloid (HP) have been rarely used for deployable structures. In fact, anticlastic structures can be easily constructed using simple straight bars rather than SLEs since their geometric forms can be generated by ruled surfaces...

[A]nticlastic structures are capable of resisting... various design loads through their curvatures and twists. Thus, their structural efficiency is more than ...others. ...[W]e have decided to use HP ...as a [deployable] canopy roof structure.

Diagrid Structures (2014)

: Systems, Connections, Details by Terri Meyer Boake

In 1829... Nikolai Ivanovich Lobachevsky published a disproof of Euclid's fifth or parallel postulate using the case of a doubly curved surface, thereby establishing non-Euclidean geometry.

According to Elizabeth C. English, there was a direct connection between the mathematical discoveries of Lobachevsky and the structural explorations of Vladimir Shukhov

The "idea of the diagrid" and the first... diagrid structure have been credited to... Shukhov. The design evolved as an efficient and easily constructed tower for carrying a large gravity load... a water tower. The "Shukhov Tower"... 1896, relies on the use of a diagonal lattice of steel angles, constrained laterally... by steel rings. ...The tower is hollow, requiring little resistance to wind loads ...

Shukhov is credited with the construction of some 200 towers using this method ...

Although the slender steel sections, when formed into the hyperbolic paraboloid shape, did undergo some bending, a diagrid shape emerges, using a combination of straight segments that are joined at their nodal points of intersection.

The Shukhov towers tended to use much longer steel sections and overlap them at their crosspoints, rather than using the crosspoints as "nodes" in the fashion of later geodesic or spaceframe structures.

[T]he diagrid form could support both the gravity loads and the lateral loads without requiring additional means. In comparison, the new skyscraper types that were under development at this time used a steel frame to support the gravity loads... while the central core provided the stiffness required to resist wind loads. Where additional resistance was required, it was usually the core that carried the bracing.

Soviet Architecture (2014)

: The Search for New Solutions in the 1920s and 1930s by Selim O. Khan-Magomedov, Tr. Alexander Lieven, ed. Catherine Cooke

The School of Engineering and Construction which came into being in Russia after the reforms of the 1860s, and in connection with large-scale railway construction towards the end of the nineteenth century, soon acquired a worldwide reputation. It produced distinguished theorists and practitioners who included Belelyubsky, Loleit, Proskuryakov, Shukhov and Yasinsky. The engineering work incorporated in the most diverse kinds of buildings in Russia during the last quarter-century before the Revolution show that this country not only kept pace with the more developed parts of Europe and the USA, but outstripped them in certain fields, in terms of engineering design, as well as the use of modern construction techniques and new materials. Thus Shukhov produced many original designs unparalleled in work done abroad. At the Nizhny-Novgorod Exhibition in 1896... several original designs by Shukhov appeared, the most important of which were latticed: suspended latticed roofs for exhibition halls with a circular, elliptical and rectangular ground plan, latticed roof vaults and hyperbolic latticed towers. This category of structure also included the dual curvature roof designed by Shukhov in 1897-98 to cover the workshop at the Vykhsusnk factory.

Forms of spanning which have come into use in recent years were foreshadowed in various forms of latticed structures designed by Shukhov before the end of the last century—guyed structures, the spanning of dual curves by structures of standardized rods, the provision of curved 'hyperbolic' profiles by means of a straight component, using straight rods in Shukhov's case.

Russian engineers made many new and stimulating contributions to the design and construction of multiple-span lattice metal bridges, a great number of which were needed in railway work because of the multitude of large rivers in Russia.
Ferro-concrete was first used in buildings there at the beginning of the twentieth century. Russian engineers... made their own contribution to the development of structural techniques in this area.

The creative principles of innovative trends such as Rationalism and Constructivism were well understood by engineers and constructors bent on applying the latest technological achievements. Great engineers such as Shukhov, [Artur] Loleit and [Hermann] Krasin, were happy to work with innovative architects. They jointly explored new ways of developing architecture, and engineers became active members of Asnova and Osna. ...In the twenties... Soviet architecture set tasks for the building industry which prompted the employment of new materials and structural elements, and raised building standards. Engineering technology in the building industry achieved great successes during this period, such as the metal structures by Shukhov and Krasin, Loleit's work in the field of ferro-concrete, the elaboration of modern timber work by Karlson and the production of new hyperbolic paraboloid roofing by Markarova. The very latest structures and building materials, as well as modern production methods, were applied in the construction of engineering and industrial buildings such as Shukhov's radio tower in Moscow in 1922 and Krasin's viaduct at the Shatura power station in 1925.

A Survey on Hyperbolic Cooling Towers (2014)

E. Asadzadeh & M. Alam, World Academy of Science, Engineering and Technology, International Journal of Civil, Architectural, Structural and Construction Engineering Vol:8 No:10, 2014. A source.

[H]yperbolic cooling towers are associated with nuclear and thermal power plants... they are also used... in some large chemical and other industrial plants. [T]hey are high rise reinforced concrete structures in the form of doubly curved thin walled shells of complex geometry...

The in-plane membrane actions primarily resist the applied forces and bending plays the secondary role in these special structures.

The first hyperboloid shaped cooling tower was introduced by the Dutch engineers Frederik van Iterson and Gerard Kuypers and built in 1918 near Heerlen having 35 meter height.

The hyperbolic geometry has advantage of a negative Gaussian curvature which makes it superior in stability against external pressures...

The widened bottom of the tower accommodates large installation of fill to facilitate the evaporative cooling of the thin film of circulated water. Narrowing effect of the tower accelerates the laminar flow of evaporation and diverging top promotes turbulent mixing which increases the contact between hot inside air and cooler outside air.

The... structure is made of high-strength [cement,] Reinforced... Concrete... in the form of hyperbolic thin shell standing on diagonal, meridional, or vertical supporting columns and radial supports. The shell is sufficiently stiffened by upper and lower edge members.

[T]o achieve sufficient resistance against instability, large cooling tower shells may be stiffened by additional internal or external rings which may also be used as repair or rehabilitation...

The hyperbolic form of thin-walled towers provides optimum conditions for good aerodynamics, strength, and stability.

[T]he first cooling tower shell [to be] analyzed by means of a shell bending theory [was in 1967].

[T]he most preferred method of the modeling and analysis of NDCT [natural draft cooling towers] is [the] Finite Element Method (FEM).

Determination [utilizing nonlinear static analysis] of the ultimate strength of the cooling towers... subjected to the severe quasistatic wind loads is one of the [research] objectives... for... structural engineers. Various nonlinear factors, such as the material nonlinearities in the concrete and reinforcing steel, tensile cracking, the bond effects between concrete and steel in the cracked concrete which is known as the tension stiffening, the large displacement effects, and so on; need to be taken into account...

Mahmoud and Gupta... concluded that the failure of the Port Gibson Tower [in] Mississippi was caused by the circumferential buckling in the vicinity of the throat rather than the yielding of the reinforcement, which contradicts... previous researchers.
- Ref: B.E.H. Mahmoud, and A.K. Gupta, "Inelastic large displacement behavior and buckling of cooling tower," Journal of Structural Engineering, ASCE, vo. 121, no. 6, pp.981–985, 1995.

Deformation response and ultimate strength of RC shell structures are governed predominantly by material response of concrete and reinforcing steel, tensile cracking of concrete, [and] bond between concrete and steel.... Softening response of concrete due to quasi-brittle cracking in tension also... influences the nonlinear response by inducing loss of strength and stiffness... Due to all [of] these, analysis requires attention for realistic modeling of the layer of shell concrete confined between the reinforcement layers. ...[O]ne of the most challenging areas... is the modeling techniques using the layered elements.

The NDCT [natural draft cooling tower] shell structures are submitted to environmental loads such as wind, earthquake and thermal gradients that are stochastic in nature. The dead loads, settlement, and construction load are... common... and various accidental loads, e.g., explosion [are]... experience[d]... in their lifetime.

[I]n 1967, closed-form expressions [were] derived by Gould and Lee... for determining... stresses in the shell and corresponding deformations under static seismic design load.
- Ref: Gould, P.L. Lee, S.L., Hyperbolic cooling towers under seismic design load, ASCE Journal of Structural Engineering, 1967, 93(3), 87–109.

Abu-Sitta and Davenport... investigated the effects of dynamic earthquake loading. ...[I]nduced dynamic stresses were related to equivalent membrane stresses from static loads, resulting in... a simplified earthquake analysis procedure.
- Ref: S.H. Abu-Sitta, and A.G. Davenport, "Earthquake design for cooling towers," ASCE Journal of Structural Engineering, vol. 96, no. 9, pp. 1889–1902, 1970.

Asadzadeh et al... studied the structural response... under static wind and pseudo static seismic forces. ...[T]wo types of... column supports at the base of the towers had been considered... The structural response... under wind and earthquake was... completely different for the towers supported on different columns.
- Ref: E. Asadzadeh, A. Rajan, M.S. Kulkarni, and S. Asadzadeh, "Finite Element Analysis for Structural Response of RCC Cooling Tower Shell Considering Alternative Supporting Systems," International Journal of Civil Engineering and Technology (IJCIET), vol. 3, no. 1, pp. 82-98, 2012.

Asadzadeh et al... studied the effects of the inclination angle of the supporting columns on the dynamic response of the cooling towers. ...[S]tiffness of the structure increases with increase in inclination angle of the supporting columns resulting in decrease of the period therefore altering the resistance... against the earthquake loading. ...[T]he hyperbolic structure... can be optimized by finding the optimum inclination angle of the supporting columns.
- Ref: E. Asadzadeh, M. Alam, S. Asadzadeh, "Dynamic response of layered hyperbolic cooling tower considering the effects of support inclinations," Structural Engineering and Mechanics, vol. 50, no. 6, pp. 797-816, 2014.

Wind is the prime lateral load and its combination with self weight of the tower shell can cause the buckling instability leading to catastrophic failure.

After the sudden collapse of three immense cooling towers at Ferry-Bridge Power Station in England in 1965, experimental and theoretical investigations had been done in... the stability of hyperbolic shells to study the parameters increasing the wind resistance and buckling safety...

The wind-induced response... is the key factor to improve safety and to reduce tower crack[ing]...

Form... found that the construction of the stiffening rings significantly influenced the buckling safety factor (BSF) and structural buckling stability... He showed that adding 2, 3, or 4 stiffening rings to the cooling tower increase the BSF of the R.C. shell by a factor of 1.65, 2.32, and 2.80, respectively.
- Ref: J. Form, "The ring-stiffened shell of the ISAR II nuclear power plant natural-draught cooling tower," Engineering Structures, vol. 8, no. 3, p. 199-207, 1986.

Sabouri-Ghomi et al... investigated... buckling stability of R.C. cooling towers supported on X-shaped columns by considering... parameters such as the number, dimension, and location of stiffening rings... The stiffening rings behaved flexibly or rigidly, depending on their dimensions. The number of flexible stiffening rings required to maximize the buckling safety factor was found to be higher than the number of rigid stiffening rings...
Ref: S. Sabouri-Ghomi, M.H.K. Kharrazi, and P. Javidan, "Effect of stiffening rings on buckling stability of R.C. hyperbolic cooling towers," Thin Walled Structures, vol. 44, no. 2, pp. 152-158, 2006.

Zhang et al... studied the effect of the stiffening ring... to improve the dynamic properties and the wind resistance... They... concluded that for the latitude and meridian directions, the stiffness in the latitude direction contributes more... and... the location where the maximum modal displacement of the unstiffened HCT appears will not always be the most effective place for an additional stiffening ring unless the mode shapes of the unstiffened and stiffened HCTs are similar.
- Ref: J. F. Zhang, H. Chen, Y. G. Ge, L. Zhao, and S. H. Ke, "Effects of stiffening rings on the dynamic properties of hyperboloidal cooling towers," Wind and Structures, vol. 49, no. 5, pp. 619-629, 2014.

Consideration of the soil–structure interaction... is extremely important when the soil or the foundation medium is not very firm. ...[U]nder ...dynamic loadings, the structure interacts with the surrounding soil imposing soil deformations. These... change the response of the structure.

Noorzaei et al... analyzed the cooling tower–foundation–soil system under vertical and lateral load generated due to self-weight and wind loads. In this study, the unsymmetrical wind pressure distribution in terms of Fourier series [were] given...
- Ref: J. Noorzaei, A. Naghshineh, M.R. Abdul Kadir, W.A. Thanoon, and M.S. Jaafar, "Nonlinear interative analysis of cooling tower–foundation–soil interaction under unsymmetrical wind load," Thin-Walled Structures, vol. 44, pp. 997-1005, 2006.

The collapse of three natural draft cooling towers at the Ferry bridge power station in 1965... [was] due to the inadequate design for the wind forces...

In recent years... numerical simulation... has been applied to describe the collapse of structures, e.g., the collapse of cooling towers under blasting demolition... and the collapse of the World Trade Center...

Due to the complexity of the building procedure, uncertainties in the material properties as well as differences between the theoretical and the real geometry... reliability analysis... seems... indispensible...

This review is a complete collection of the studies done for cooling towers and... [gives] updated and sufficient materials for the researches in this field.

A hyperboloid structure as a mechanical model of the carbon bond (2016)

I. E. Berinskii & A. M. Krivtsov, International Journal of Solids and Structures (1 October 2016) Vol. 96, pp. 145-152. A source.

We present a new mechanical model of interatomic bonds, which can be used to describe the elastic properties of the carbon allotropes, such as graphite, diamond, fullerene, and carbon nanotubes. The interatomic bond is modeled by a hyperboloid–shape truss structure.

[T]here are... approaches that are closer to the field of the classical mechanics... so–called structural or discrete-continuous methods... The most straightforward example... is a covalent bond modeled by the solid deformable rod... [T]he interatomic bonds are modeled as a deformable body or a construction. ...[T]hese approaches... can be implemented in standard computing packages based on the finite–element, boundary–element, or finite difference methods. These methods can be considered as the bridges between the parameters of atomistic and continual models of the material.

In this paper, the carbon bond model is built on the symmetry properties of the hyperboloid. These properties allow it to achieve a high ratio of the lateral and longitudinal stiffness, therefore the hyperboloid shapes are widely used in the engineering to create lightweight constructions consisting of straight beams that are known for being able to carry a large load while achieving a low use of raw materials.

[T]he first hyperboloid tower was built by Russian engineer V.G. Shukhov in Nizhny Novgorod (1896)... Being widely demanded in architecture and engineering, such models still haven’t found a wide use in micro- and nanomechanics. It appears that the analogy drawn from the macro level will allow to describe correctly the properties of carbon materials at the micro level.

Rethinking Complexity (2016)

: Vladimir Shukhov’s Steel Lattice Structures by Elizaveta Edemskaya, Asterios Agkathidis, Journal of the International Association for Shell and Spatial Structures Vol. 57, No. 3 (2016) DOI:10.20898/j.iass.2016.189.806

Alongside Shukhov’s drawings and sketchpads... a typescript produced by Shukhov’s former employee Grigory Kovelman... put together an extensive overview of Shukhov’s inventions and projects, both as a biographer and as a specialist who had worked with Shukhov.... Elena Shukhova... grand-daughter... presents extensive biographical details in Vladimir Grigorevich Shukhov. The First Engineer in Russia... Art of Construction... is a valuable collection of articles about Shukhov and his various inventions, written by... specialists. It includes... examples of his calculations. Shukhov’s own book Rafters... discusses the mathematical investigations which led him to conceptualise the spatial lattice structure, describing it as 'an optimisation process'.

In 2010, the journal Detail published an analysis of Shukhov’s constructions, calling his approach to design 'an early example of parametric design’...
- Ref: R. Graefe, M. Schuller, R.Barthel, "Wendepunkt im Bauen-Ausstellungsbesuch" (Nov, 2010) Detail.

Vaulted gridshell constructions... were formed with thin metal arches turned away from the frontal position at a particular angle. They thus worked as one continuous resilient truss. ...Each arch was made with rigid metal strips of equal length... during the assembling process, each piece was bent equally. ...It was the first time in the world’s building practice that double-curved spatial vaults were created with single type rod elements ...
- Ref: S. O. Khan-Magomedov, Vladimir Shukhov (2010)

Shukhov’s lattice-suspended and vaulted structures represented a carrying surface, which could be shaped in any form. ...The density of the grid made it possible to attach it to the shell without additional structures. ...[T]he grids were two to three times lighter than roofs with conventional frames...
- Ref: G. M. Kovelman, The great Russian engineer Vladimir Grigorevich Shukhov (1853-1939) (1958) Ch. 2. Metal structures of industrial buildings and civil buildings. Archive of the Russian Academy of Science, p. F.1508/Op.2/37.

The final and most unusual of the gridshell structures presented at the Exhibition was the 32-metre-tall lattice hyperboloid water tower. Everything was amazing in that first Shukhov tower—everything in it was some structural and geometric puzzle: straight rods and the external silhouette double curvature, the openwork lightness below and the solid heaviness above.
- Ref: S. O. Khan-Magomedov, op. cit.

The water tower was a unique structure of its time... According to Cooper, the idea... came directly from an imaginary hyperboloid geometry, invented by... Lobachevski in 1829...
- Ref: Elizabeth Cooper, Arkhitektura I mnimosti: the origins of Soviet Avant Garde Rationalist architecture in the Russian mystical, philosophical and mathematical intellectual tradition (2000) University of Pennsylvania Ph.D. dissertation.

Grigory Kovelman writes that Shukhov told him he had been thinking about the properties of hyperboloid structures for a long time, that he had studied hyperboloid forms at the Technical School, and that apparently the moment of enlightenment came about when he saw an up-ended wicker wastepaper basket with a focus on top of his desk. According to Shukhov, this was when he understood clearly how a hyperboloid structure with its curved surface [was] generated by straight rods...
- Ref: G. M. Kovelman, op. cit.

[T]he structure of the lattice tower was a spatial system, where the load was equally spread along the surface. ...Aiming to optimise the design process, soon after building the tower Shukhov presented the standardised elements of the tower structure in a table format... with the aid of which it became possible to design a new water tower according to a client’s requirements in twenty-five minutes...
- Ref: G. M. Kovelman, op. cit.

The Radio Tower in Moscow is a gridshell which aims for structural efficiency. Its minimal surface and open lattice structure help in reducing the wind load, one of the main challenges in high-rise building design. ...Shukhov's design logic focuses on structural stiffness. The rings between the different segments offer additional reinforcement to create an equilibrium between minimal material consumption, structural efficiency and geometry.

Design of Hyperboloid Structures (2017)

Sebahattin Bektas, Faculty of Engineering, Ondokuz Mayis University, Turkey, MOJ Civil Engineering 2017;3(6):414-420 (December 27, 2017) DOI:10.15406/mojce.2017.03.00089 Alt source.

As hyperboloid structures are double curved, that is simultaneously curved in opposite directions, they are very resistant to buckling. This means that you can get away with far less material than you would otherwise need...

Single curved surfaces, for example cylinders, have strengths but also weaknesses. Double curved surfaces... are curved in two directions and thus avoid... weak directions.

[T]his is the magical part... despite the surface being curved in two directions, it is made entirely of straight lines. Apart from the cost savings of avoiding curved beams or shuttering, they are far more resistant to buckling because the individual elements are straight...

This is an interesting paradox: you get the best local buckling resistance because the beams are straight and the best overall buckling resistance because the surface is double curved.

The hyperboloid is the design standard for all nuclear cooling towers and some coal-fired power plants.

When designing... cooling towers, engineers are faced with two problems:
I. The structure must be able to withstand high winds and
II. They should be built with as little material as possible... The hyperbolic form solves both...

For a given diameter and height of a tower and a given strength, this shape requires less material than any other form. ...Hyperboloidal towers can be built from reinforced concrete or as a steel lattice, and is the most economical such structure for a given diameter and height.

The presented orthogonal fitting algorithm can be applied easily for ellipsoid, and sphere also other surface[s] such as paraboloid.

Degradation Theory of Long Term Operated Materials and Structures (2021)

by Grzegorz Lesiuk pp. 95 ff.

In 1899... V. Shukhov... patented the principle of constructing hyperboloid gridshell structures based on a hyperboloid of revolution. ...A one-sheet hyperboloid is a connected surface having negative Gaussian curvature at each point. Through any point of this surface, two intersecting [straight] lines can be drawn, completely belonging to it. Thus the... surface can be formed by the set of straight lines. It the steel beams would be placed along these lines the shape... could be retained under... external loading... [I]n Shukhov's towers horizontal rims [rings] were used... located at different levels... The stability... was ensured by numerous riveted connections... During the installation... the straight angles of the metal profile was somewhat deformed... [at] the... intersecting elements in order to ensure maximum contact... by rivets.

The main hazard for high-rise structures is the loads caused by wind gusts. The grid shell design... minimize[d] their influence. Open-work design... ensured... sufficient strength, high stability and low metal consumption. ...[C]onsumption of metal per unit height... was three times less than... the Eiffel Tower...

Kobe Port Tower... Ještěd in Czechia, Guangzhou in China... and Khan Shatir Shopping-Entertaining Center in Kazakhstan, Aspire Tower in Qatar... are vivid examples of using... the hyperboloid principle... [in] modern buildings.

191 of the Shukhov's Towers (from known near 200...) have been irretrievably lost during... the 20th century... [Some] towers were destroyed because their continued use for water supply has become impractical. To use them for another purpose... large investments were necessary.

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Hyperboloid structure