Hyperboloid structure

"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 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."

"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."

"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."

"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."

"Wind is the prime lateral load and its combination with self weight of the tower shell can cause the 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 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."

"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."

"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 [were] given..."

"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."

"We present a new mechanical model of interatomic bonds, which can be used to describe the elastic properties of the carbon allotropes, such as , diamond, , and s. The interatomic bond is modeled by a hyperboloid–shape 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 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 s. These methods can be considered as the bridges between the s 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 , 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 (1896)... Being widely demanded in architecture and engineering, such models still haven’t found a wide use in micro- and . It appears that the analogy drawn from the macro level will allow to describe correctly the properties of carbon materials at the micro level."

"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 ’..."

"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 ..."

"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..."

"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."

"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..."

"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..."

"[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..."

"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 . The rings between the different segments offer additional reinforcement to create an equilibrium between minimal material consumption, structural efficiency and geometry."

"As hyperboloid structures are double curved, that is simultaneously curved in opposite directions, they are very resistant to . 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 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."

"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 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 ..."

"... 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."