[A] shell element will have, in general, 10 unknown internal stress re — Thin-shell structure

"For around 2000 years single and double curved shells structures, such as barrel and vault s, have been used to cover large spans in buildings. Until the twentieth century these were generally constructed either from masonry or some form of unreinforced concrete, materials strong in compression but relatively weak in tension. Well known examples such as the Pantheon... ... Santa Maria del Fiore... and St. Peters Basilica... have a span to thickness ratio of less than 50 to 1, which is relatively thicker than a... typical hens egg. ...[T]he stone vaulting of... medieval Gothic cathedrals... demonstrate the masons art in the construction of... complex masonry shells. With the advent of reinforced concrete... strong in both compression and tension, it became possible... to construct thin shells with much higher span to thickness ratios... commonly... in the region of 500 to 1."

"Torroja was a specialist in stress analysis... and he wrote a... book on the mathematical theory of elasticity. This... led him to see a connection at Algeciras between the stresses in the shell and the reinforcement... but not to express those stresses in... visually evident ribs. We contrast... Nervis Little Sports Palace... whereas Nervi sees shells as ribbed, Torroja sees them as ribless... since domes tend to spread, Nervi designed ribbed buttresses... whereas Torroja avoids buttresses by connecting vertical supporting columns with a... polygonal ring of horizontal ties... prestressed to counteract dead load and to lift the shell slightly off its scaffold... probably the first application of prestressing to a doubly curved shell. In the Nervi dome... the buttresses are supported below ground on a ring which carries the horizontal thrust and... transmits the vertical weight to the ground. ...[These] choices related to the [respective] local traditions in Italy and Spain."

"... all through school had a reputation of working alone and of doing his work in an unusual way. ...in 1950 he graduated with a degree in civil engineering. For his final-year design project, he chose to study thin shells... Following graduation... he helped [Pierre] Lardy with teaching, and also worked on the many cases of structural failure [both at his alma mater, the Federal Technical Institute]... When Isler left his position... he considered... [a] career as a painter, but challenged by shell design problems... while doing free-lance engineering work.. in late 1954, he designed a pneumatic form, thin shell factory for the Trösch Company. It was the first work in which he set the form completely on his own. In 1955, at an international congress in Amsterdam, he presented publicly for the first time his new designs..."

"The idea of form over mass also developed in Europe in the pioneering work of Dyckerhoff and Widmann... in Weisbaden, Germany. Working in reinforced concrete, the firm experimented with new ways to cover large spaces in the 1920s. The firm built domes and cylindrical "barrel" shells to serve as large roofs of extraordinary thinness. The possibilities... fascinated an Austrian civil engineering student, Anton Tedesko... who joined the firm in 1930."

"The essential ingredients of a shell structure... are continuity and curvature. ...[S]hells are structurally continuous in the sense that they can transmit forces in a number of different directions in the surface of the shell... These have a quite different mode of action from skeletal structures... only capable of transmitting forces along discrete structural members. ...There seems to be a principle that closed surfaces are rigid. This principle is used in many areas of engineering construction. ...[A]lthough the ideas of closed and open shells... are fairly clear, it is difficult to quanitify intermediate cases into which... the majority of actual shell structures fall. ...There is a theorem, due to Cauchy, which states that a convex polyhedron is rigid. ...[N]on-convexity may produce deformability. ...While rigidity and strength are in many cases desirable attributes of shell structures, there are some important difficulties which can occur... [involving] unavoidable rigidity. ...[A] second broad principle... may be stated thus: efficient structures may fail catastrophically. Here I use the term efficient to describe the consequences of employing the first principle. By designing a structure as... closed... we may be able to use thinner sheet material, and hence produce an economical, or efficient, design."

"[I]n 1967, closed-form expressions [were] derived by Gould and Lee... for determining the resultant stresses in the shell and corresponding deformations under static seismic design load."