Mathematics

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

"[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."

"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."

"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."

"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 , 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."

"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."

"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."

"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."

"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..."

"[H]yperbolic s 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 structures in the form of doubly curved thin walled shells of complex geometry..."

"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."

"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..."

"[T]he most preferred method of the modeling and analysis of NDCT [natural draft cooling towers] is [the] (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..."

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

"[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..."

"Mahmoud and Gupta... concluded that the failure of the Port Gibson Tower [in] Mississippi was caused by the circumferential in the vicinity of the throat rather than the yielding of the reinforcement, which contradicts... previous researchers."

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

"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."

"The hyperbolic geometry has advantage of a negative 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 of evaporation and diverging top promotes turbulent mixing which increases the contact between hot inside air and cooler outside air."

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

"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."

"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."

"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."

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

"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..."

"In 1829... Nikolai Ivanovich Lobachevsky published a disproof of Euclid's fifth or 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"

"[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."

"[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 s."

"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 s..."

"The "idea of the " 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 ..."

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

"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..."

"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."

"[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."

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

"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."

"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."

"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..."

"[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 considerations."

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

"... 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."

"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 ..."

"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."

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

"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."

"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."

"[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..."