# Thin-shell structure

John F. Kennedy International Airport's TWA Flight Center, by Eero Saarinen.
Membrane roof of Forest Opera in Sopot, Poland.

Thin-shell structures are also called plate and shell structures. They are lightweight constructions using shell structural elements. These elements, typically curved, are assembled to make large structures. Typical applications include aircraft fuselages, boat hulls, and the roofs of large buildings.

## Quotes

• Thin shells — Three-dimensional spatial structures made up of one or more curved slabs or folded plates whose thicknesses are small compared to their other dimensions. Thin shells are characterized by their three-dimensional load-carrying behavior, which is determined by the geometry of their forms, by the manner in which they are supported, and by the nature of the applied load.
• In the twentieth century, lightly reinforced brick shells were inspired by timbrel vaulting, a common building method in the Mediterranean. Rafael Guastavino... upon emigrating to the United States from Catalonia in 1881, introduced the method with great success. His son, Rafael Guastavino, Jr. ...appears to be the first to have introduced steel reinforcing to thin brick shells. ...[T]wo patents... [1910, 1913] documented this system, which is a precursor to the thin shells of reinforced concrete developed widely in the ensuing decades.
Heinz Isler's Highway service cover (1968) Solothurn, Switzerland.
• Heinz Isler has designed some of the most striking thin shells in reinforced concrete of the second half of the twentieth century. He creates thin shells by hanging small membranes in tension and creating smooth curving surfaces that are then inverted and scaled up to create large-scale structures in compression. ...Within the constraint of economy, he discovered new forms from purely structural considerations and demonstrated the unlimited possibilities for thin compression shells to be found in hanging models.
• Candela did not invent the concrete shell; nor is he the first to make use of the hyperbolic paraboloid... Other people... have contributed more to the theoretical analysis of shell structures. But nobody else can claim credit for such an exciting variety of shell structures... [H]e has concentrated his effort in one particular sphere: the construction of light concrete roofs.
• [Candela] is not just an engineer, or an architect, or a contractor and constructor, but all three... [W]hen he thinks out a new scheme, the method of construction and its economy is constantly in his mind. ...He prefers to obtain his economy by using his inventive skills as an engineer to reduce... material... [H]e recognizes the value of structural analysis... but he is also very conscious of its limitations. Especially is he skeptical about the value of the theory of elasticity as applied to concrete... of... calculations suggesting an accuracy which is purely fictitious... Designing... proceeds from a structural feeling acquired by experience and guided by rough calculations, a refinement of design... further analysis... and so on. ...[A] flair for making the right guess yields quicker and better results than a lot of mathematics... this is no reason for despising theoretical analysis... But one cannot design by theory...
• Ove Arup, Forward (1962) in Colin Faber, Candela, the Shell Builder (1963)
• [M]ost of Candela's structures are almost complete in themselves... the forms and proportions bear witness to his artistic sensibility. ...[B]alanced perfection ...makes a... structure into a work of art. ...[T]he whole must take precedence over any of its parts.
Eduardo Torroja's Algeciras market hall (1934).
• Heinz Isler... 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...
• David P. Billington, The Tower and the Bridge: The New Art of Structural Engineering (1985)
Pier Luigi Nervi's design for the Norfolk Scope Arena (Norfolk, VA) recalls closely his Little Sports Palace (Palazzetto dello Sport), Rome.
• 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... Nervi's 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.
• David P. Billington, The Tower and the Bridge: The New Art of Structural Engineering (1985)
• 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.
• David P. Billington, David P. Billington Jr., Power, Speed, and Form: Engineers and the Making of the Twentieth Century (2006)
Anton Tedesko's Hershey Arena
opening night (1937)
• The Hershey Arena... Tedesko designed a thin reinforced concrete, barrel-shell roof, three and one-half inches thick, supported across its width by eight arches. ...A roof posed a different problem than a bridge or dam. On a bridge, live load from traffic is significant, and a dam must resist the live load of water on its upstream face. On a long-span concrete roof, live load (mainly rain and snow) is a small fraction of the dead load of the structure itself. Tedesko realized that the supporting arches did not need to be of uniform depth. ...[he] designed the arches to be able to... support the entire roof load, including the thin shell and all of the live load. He also made calculations to show that the thin shell could carry its own weight and the live load without help from the arches, except near their lowest edges. It was thus a conservative design...
• David P. Billington, David P. Billington Jr., Power, Speed, and Form: Engineers and the Making of the Twentieth Century (2006)
• Shells were not being done in the United States at all, and I started seeing pictures of these buildings coming out of South America... Italy and Spain... they happened to be all Latin countries. ...I thought, "God, how do I do this?" you know, these three-dimensional curved structures... [T]he greatest of them all... was Nervi... I found out that I couldn't find out how to design... them. ...I realized after many years of striving that a lot of them really didn't know how to design them. They were just doing it intuitively, and that was not satisfactory to an engineer. I needed to know a rational way of doing it, and that sent me back to school and... to studying... [I]t took me years and years of very hard intellectual work to find ways to do this. ...[W]hen I finally did it, I was one of the few guys in the country that had really made that much effort, and so I became a pioneer...
• Richard R. Bradshaw, "Richard Bradshaw on the Construction of LAX (Modern Architecture in Los Angeles)" (Apr 1, 2013) 0:31-1:52, video from Getty Research Institute.
Félix Candela's Cosmic Ray Pavilion, University of Mexico City.
• In 1958, Felix Candela completed his most significant work, the Los Manantiales Restaurant shell, in Xochimilco, Mexico City. ...[He] was taking a risk... The form was original, unexplored, and impossible to analyze precisely. Candela’s career, however, habitually flew in the face of precise analysis. In his first acclaimed shell, the Cosmic Ray Pavilion of the University of Mexico City, he also designed an unprecedented form, using almost no calculation... Candela’s subsequent designs relied increasingly on structural understanding and practical experience. As a designer-contractor, he had the unique responsibility of building his own solutions. By closely observing his buildings, and using smaller projects to test new ideas, he developed an acute sense of concrete shell behavior.
• Noah Burger, David P. Billington, "Felix Candela, Elegance and Endurance: An Examinations of the Xochimilco Shell," Journal of the International Association for Shell and Spatial Structures: IASS (2006) Vol. 27 No. 3, Dec., no. 152, pp. 271-278.
• 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.
• For around 2000 years single and double curved shells structures, such as barrel and vault domes, 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... Hagia Sophia... Santa Maria del Fiore... and St. Peter's Basilica... have a span to thickness ratio of less than 50 to 1, which is relatively thicker than a... typical hen's egg. ...[T]he stone vaulting of... medieval Gothic cathedrals... demonstrate the mason's 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.
• John Chilton, Heinz Isler, Heinz Isler: The Engineer's Contribution to Contemporary Architecture (2000)
Heinz Isler's Wyss garden center (1962) Solothurn, Switzerland.
• At the time of construction of the Wyss shell, three-dimensional solid modelling computer software was not available and it would have been extremely difficult to convey, using only normal engineering drawings, the required form of the concrete at the feet of the shell... To overcome the problems... Heinz Isler proposed that, rather than making sketches, drawings, or even a model of the detail, they should resort to modelling it at full-scale on site.
• John Chilton, Heinz Isler, Heinz Isler: The Engineer's Contribution to Contemporary Architecture (2000)
Forest Opera open-air amphitheatre, Sopot, Poland
• At the beginning, the Forest Opera was not covered by any type of roof. During the reconstruction... in 1964... the roof covering... was created. ...reinforced by cotton threads and covered by rubber layers. The material underwent large rheological displacements. In... 1968... catastrophe occurred, caused by wind and high humidity. In the eighties... a polyester fabric, intended only [for] seasonal application, was used. ...age and... repeated disassembling of the membrane caused its gradual destruction. ...In 2007 ...complete rebuilding ...assumed the roof to be a permanent structure. ...[V]erification of ...internal forces was conducted in 2012 ...by Gdańsk University of Technology team.
• J. Chróścielewiski, M. Miśkiewicz, Ł. Pyrzowski, K. Wilde, "Assessment of tensile forces in Sopot Forest Opera membrane by in situ measurements and interative numerical strategy for inverse problem," Shell Structures: Theory and Applications (2014) Vol. 3
• The resistant virtues of the structure that we make depend on their form; it is through their form that they are stable and not because of an awkward accumulation of materials. There is nothing more noble and elegant from an intellectual viewpoint than this; resistance through form.
• Eladio Dieste, as quoted by Remo Pedreschi, The Engineer's Contribution to Contemporary Architecture: Eladio Dieste (2000) p. 21.
• The emergence of lightweight structures can be traced back to the second half of the nineteenth Century. This period witnessed the advent of new material technologies such as steel, reinforced concrete, resistant glass and, later, fabric membrane. Together with advances in analysis and design tools, engineers and architects have been challenged to build increasingly lighter structures. ...[A] pioneering structure was the hyperboloid lattice tower by... Vladimir Shukhov in 1896. In the 1920s, Anton Tedesko first introduced reinforced concrete thin shells in the United States. This expansion was pursued... by Félix Candela, Heinz Isler and André Paduart...
The limit of lightness was achieved with tensile structures constructed of prestressed cable nets and fabric membranes; the strength coming from the anticlastic curvature of the geometric surface. ...Nowadays, lightweight structures should be designed... by including the multitude of design contraints. This will result in hybrid systems lying at the boundary of different typologies.
• Benoit Descamps, Computational Design of Lightweight Structures: Form Finding and Optimization (2014)
• Nature does not apply the construction principle of a beam supported by two columns. Forms developed by nature are following the rational attempt to achieve distinct functionalities with the smallest possible material - and energy consumption. An impressive example is the phenomena of egg shells... The shell principle is adopted by humans... in building construction, in order to achieve wide spanning and material saving 'slender' structures.
• Philipp Eisenbach, Processing of Slender Concrete Shells - Fabrication and Installation (2017)
• The design of shells... implicates the design of internal stress fields of form dependent shapes... meeting the compatabilites of all boundary conditions...
• Philipp Eisenbach, Processing of Slender Concrete Shells - Fabrication and Installation (2017)
• [M]embrane theory... is the theory of shells whose bending rigidity may be neglected. The spectacular simplification... makes it possible to examine a wide variety of shapes and support conditions. In particular, the stress problems of tanks and shell roofs... There is, of course a heavy penalty... [T]he inadequacies... can be discovered by a critical inspection of the... solutions, without any need for... solving the bending problem—a task which is often out of the reach of the practical engineer and even of the research worker. On the other hand, membrane theory is more than a first approximation... If a shell is so shaped and so supported that it can carry its load with a membrane stress system, it will be thin, light, and stiff and, therefore, the most desirable solution to a design problem. Membrane theory will guide the shell designer toward such structure.
• Wilhelm Flügge, Stresses in Shells (1973) 2nd edition, Preface. Originally published in German as Statik und Dynamik der Schalen (1937).
• Shell-like structures are familiar enough in nature but the use of such structures as containers, aircraft fuselages, submarine hulls and roofing structures is only of recent origin. That the inherent strength of shells... has not been utilised much in the past is probably due to the difficulty in obtaining suitable material... [S]hell structures in general are these days constructed of such varied materials as steel, light alloy, plastics, wood and reinforced concrete. ...[T]o simplify analysis it will be assumed that the material... is homogeneous, isotropic and perfectly elastic. ...[A]lthough reinforced concrete behaves in a reasonably elastic manner only in the lower stress ranges the majority of reinforced concrete shell roofs that are constructed in practice are designed as elastic structures.
• J. E. Gibson, Thin Shells: Computing and Theory (1980) p. 3.
• The true Mathematical and Mechanical Form of all manner of Arches for building with the true butment necessary to each of them, a Problem which no Architectonick Writer hath ever yet attempted, much less perform'd. ...Ut pendet continaum flexile, sic stabit contiguum rigidum, which is the Linea Catenaria.
• Cement Hall, Swiss National Exhibition, Zurich (1939)... was built by architect and engineer Hans Leuzinger and Robert Maillart... to demonstrate the potential of thin shells. ...The width of this parabolic shell is 50.32 ft... built by the Gunite method, [it] is only 2.36 in thick. ...A sculptural piece by Robert Maillart entititled "Endless Ribbon" [1935-1936]... is a thin shell built of reinforced concrete. This... can be classified within Constructivism or... constructive spatial art. [I]t represents... the direct correlation between the language of sculpture and that of modern thin-shell architecture that moves the sensitive engineer such as Maillart to express himself also as a sculptor. Maillart's structural shells as a whole testify to this similarity of language... He was the father of flared columns connecting with floor slabs to eliminate supporting beams and designed unique bridge forms. Maillart's name remains associated with innovative concrete forms that extend to the structural virtuosity of thin shells.
• Michele Melaragno, An Introduction to Shell Structures: The Art and Science of Vaulting (1991)
• Like Candela, Isler has concentrated his practice on thin concrete shells... Isler derives his forms not from analytical geometry (as were Candela's hypars) but directly from physical and funicular models - flexible membranes that assume the least energy, or minimal surface, for a specific boundary and force patterns. In the mid-1950s Isler invented two new form-making techniques, the first by using pneumatic models and the second by experimenting with hanging cloth models sprayed with water and put out to freeze in wintertime. Later, in 1965, he added a third technique that made shapes "by the flow method, by... the advancing velocity of a liquid inside a tube... At the wall, velocity is zero because of friction, whereas in the center there is maximum velocity... and forms a dome shape." While the frozen cloths conform to the funicular shape given by gravity, the other methods, pneumatic and flow, are hydraulic.
• Guy Nordenson, "Constellations," in Seven Structural Engineers: The Felix Candela Lectures (2008) ed., Guy Nordenson
• It was the great nineteenth century mathematician, Carl Gauss who proved mathematically that any curved surface, natural or man-made, can be characterized as only one of three different possible shapes: as cylinder-like, dome-like, or saddle-like. All three of these geometric shaped can be used as the basis for thin-shell structures.
• Stephen Ressler, "Amazing Thin Shells—Strength from Curvature", The Great Courses, Understanding the World's Greatest Sructures, The Teaching Company, LLC.
• [T]he basic assumption in the linear theory of shells that the displacements of the shell are considered to be small in comparison to the thickness is abandoned in the present nonlinear analysis of shells. A shell is called thin if the maximum value of the ratio h/R, where h is the thickness of the shell and R is the principal radius of curvature of the middle surface... is less than or equal to 1/20 ...beyond this range... the shell is regarded as thick. ...in a large number of practical applications the ratio... lies in the range between 1/50 and 1/1000, making the theory of thin shells of great practical importance.
In this chapter the nonlinear equations... in terms of orthoganal curvilinear coordinates are derived assuming the material... is isotropic, homogeneous, and elastic. An important simplification based on the assumption that second invariant of the median surface strains in the expression for the extensional strain energy... can be neglected, originally made by [Noah] Burger, is introduced to derive a simplified set of nonlinear... differential equations.
• Muthukrishnan Sathyamoorthy, Nonlinear Analysis of Structures (1997)
• Concrete being such a fluid and dynamic material... finds its identity once it is contained. ...A few... who used the forming materials at hand [were]... Antoni Gaudi... Robert Maillart... Pier Luigi Nervi... Felix Candela... Eladio Dieste... Heinz Isler... Miguel Fisac... Many of these early innovators pushed the computational envelope... Some, like Antoni Gaudi, looked to nature for inspiration. The question... Do we need to "reinvent forming" or just draw from nature, i.e., gravity—catenary action? as Gaudi did. Alan Chandler in fabric framework notes "...for Felix Candela and Christopher Alexander fabric acted as a permanent shutter (framework)..." Chandler speaks of the family of fabric construction that includes... Tensile structures... Pneumatic structures... Hydrostatic structures and... Shell structures derived from membrane form-finding.
When faced with extremely complicated and complex shapes Heinz Isler and Antoni Gaudi used fabric as a modeling tool. These visionaries recognized that hanging chains and fabrics, forming catenaries, are in pure tension and when inverted are in pure compression and very stable. Gaudi, whose Catalan vaulting preceded the works of Candela... looked to nature and natural forms—an approach today called biomimicry...
• R. Schmitz, "Is there a future for fabric-formed concrete structures?", Structures and Architecture: Beyond their Limits (2016) ed., Paulo J. da Sousa Cruz, pp. 1087-1088.
• Shells under compressive loading investigated under the assumption of perfect properties may be considered to be optimal structures. Their load carrying capacity is significantly larger compared to shells which show deviations in geometry, material behaviour, loading and boundary conditions. ...Unfortunately, comparatively little quantitative information exists about the initial imperfections in actual structures... One possibility to improve this situation is to perform systematic numerical simulations... Classical numerical concepts of the load carrying capacity of imperfect structures focus on the model of a perfect shell configuration and on the analytical estimation of unstable, postcritical equilibrium paths. This was first demonstrated by Koiter, whose postbuckling theory describes the nonlinear static load carrying behaviour of structures in the initial stages of buckling. ...[I]nitial unfavourable imperfections will lead to a reduction in load carrying capacity. This approach has certain restrictions as the results are evaluated by linearisation around the bifurcation point of the perfect shell.
For the numerical simulation of the load carrying behaviour of imperfect shells it is commonly assumed that the initial geometric imperfections have the shape of the lowest bifurcation mode of the respective shell. ...In the cases of high imperfection sensitive shells [with] multi-mode-buckling... the lowest bifurcation mode is not always the ”worst” imperfection shape.
Recently, a specific concept employing finite element procedures... directly evaluates the ”worst” imperfection shape and [is based upon] analysis of the imperfect shell space.
• W. Wunderlich, U. Albertin, "Analysis and Load Carrying Behaviour of Imperfection Sensitive Shells" (1998) Computational Mechanics: New Trends and Applications, ed., S. Idelsohn, E. Oñate and E. Dvorkin.

### Steam Boilers: Their Design, Construction, and Management (1880)

by William Henry Shock, source.
• Resistance of Spherical Shells to an Internal Fluid Pressure.—An elastic fluid contained in a closed vessel presses each unit of area of the surrounding walls with equal force. The resistance offered by the walls depends on their superficial area, their form, their thickness, and the coefficient of resistance of the material.
• The hollow sphere encloses the largest space in proportion to the superficial area of its shell, and all vessels that are not spherical, exposed to an internal fluid pressure, experience distortion on account of their tendency to assume the spherical form. A hollow sphere, having a shell of uniform thickness composed of a homogeneous material, experiences the same tension at all sections of metal formed by diametrical planes.
• The area of a [circular] diametrical section, ${\displaystyle S}$, of a thin spherical shell is very nearly given by formula:
${\displaystyle S=2\pi rt}$
when ${\displaystyle t}$ represents the thickness, and
${\displaystyle r}$ = the inner radius of the shell,
and ${\displaystyle t}$ is supposed to be very small compared with ${\displaystyle r}$.
The whole force, ${\displaystyle F}$, to be resisted by the tenacity of section ${\displaystyle S}$ is equal to the excess of the internal fluid pressure per unit of area over the external pressure, into the area of the plane passing through this section, or
${\displaystyle F=\pi r^{2}p}$
Assuming that every portion of section ${\displaystyle S}$ is equally strained by ${\displaystyle F}$, and designating by ${\displaystyle k}$ the coefficient of the ultimate tenacity of the material of which the shell is composed, the bursting pressure will be found from the equation: ${\displaystyle \pi r^{2}p=2\pi rtk}$; hence
${\displaystyle p={\frac {2tk}{r}}}$ { I.}
and the proper ratio of the thickness to the radius of a thin hollow sphere is given by the formula:
${\displaystyle {\frac {t}{r}}={\frac {p}{2k}}}$...
• Resistance of Cylindrical Shells to an Internal Fluid Pressure.—The tension produced in a cylindrical shell by an internal fluid pressure may be considered as being of two different kinds—viz., first, a tension acting in a longitudinal direction, tending to pull the ends of the cylinder apart; and, secondly, a tension acting in a diametrical direction, tending to split the cylinder from end to end.
• The force, ${\displaystyle F}$, producing the first-named tension is represented by the formula:
${\displaystyle F=\pi r^{2}p}$;
and the sectional area, ${\displaystyle S}$, of a thin shell resisting this force may be represented with sufficient accuracy, as in the case of thin spherical shells, by the formula:
${\displaystyle S=2\pi rt}$.
The value of p, when it becomes the bursting pressure, is found from the equation,
${\displaystyle r^{2}\pi p=2\pi rtk}$;
hence ${\displaystyle p={\frac {2tk}{r}}}$ { II.}
the same as that of a spherical shell of equal radius and thickness.
• To find the value of p which would split the cylinder [of unity length] from end to end... The force tending to rupture such a ring at the sections formed by any diametrical plane is given by formula:
${\displaystyle F=2rp}$,
and the area of these sections by
${\displaystyle S=2t}$.
The bursting pressure is, therefore, found from the equation:
${\displaystyle 2rp=2tk}$;
hence ${\displaystyle p={\frac {tk}{R}}}$...
only half as great as... equation { II.}
• Resistance of Cylindrical Shells to an External Fluid Pressure.—Thin hollow cylinders exposed to an external fluid pressure never give way by direct crushing, but by collapsing; it may be assumed that, other things equal, the resistance of tubes to collapsing is greater as their form is more truly cylindrical and their shell more perfectly homogeneous.
William Fairbairn's
Tube Testing Apparatus
Philosophical Transactions,1858
• Fairbairn has deduced the following formula from experiments made mostly on very thin cylindrical tubes of various lengths and diameters—viz., for wrought-iron cylindrical tubes let
${\displaystyle l}$ = the length,
${\displaystyle d}$ = the diameter, and
${\displaystyle t}$ the thickness of the shell, all expressed in the same unit of measure, and let
${\displaystyle p}$ = the collapsing pressure in pounds per unit of area; then
${\displaystyle p=9,672,000{\frac {t^{2.19}}{ld}}}$. { IV.}
In case a tube is stiffened by T-iron rings or by flanges, ${\displaystyle l}$ represents the distance between two such adjacent rings or flanges.
• Fairbairn finds that the collapsing pressure of... an elliptic form of cross-section is found approximately by substituting... for ${\displaystyle d}$ the [following]... let ${\displaystyle a}$ be the greater and ${\displaystyle b}$ the less semi-axis of the ellipse; then we are to make
${\displaystyle d={\frac {2a^{2}}{b}}}$. { V.}
• In an article in the Annales d'u Génie civil, March, 1879, on the “Resistance of Tubes subjected to an External Pressure,” by Théodore Belpaire, an attempt has been made to deduce a new formula for the collapsing strength of tubes. ...The writer ... considers the case of a tube with ends rigidly fixed, and supposes that under an external pressure it changes its form in such a manner that its generatrix becomes the arc of a circle, the centre of which lies on a perpendicular erected in the centre of the generatrix; and, neglecting, the elastic forces due to flexure or elongation of the fibres—which are very small as long as the curvature is slight—he investigates the shearing stresses; these attain their greatest value at the fixed ends.
Calling ${\displaystyle S}$ the greatest shearing stress,
${\displaystyle p}$ the pressure in pounds per square inch,
${\displaystyle t}$ the thickness of the tube in inches,
${\displaystyle L}$ the length of the tube in inches,
he deduces the following approximate formula for the external pressure which a given tube can bear with a degree of safety depending on the value attributed to ${\displaystyle S}$—viz.:
${\displaystyle p={\frac {2tS}{L}}}$. { VI.}
The writer deduces then a general value ${\displaystyle S}$ from two experiments made by Fairbairn with elliptical tubes, because the uncertain and variable elements of strength due to the cylindrical form and to homogeneity of the material do not enter here. When the factor of safety in the foregoing equation is to be four, the value of ${\displaystyle S}$ becomes
${\displaystyle S=428,394{\frac {t}{D}}-7,111,550({\frac {t}{D}})^{2}}$;
...With reference to those cases where the factor of safety exceeded four greatly, the writer claims that the high pressures necessary to produce collapse indicate merely the great increase of strength derived in the particular instances from the uncertain element of circular form.

### A Treatise on the Mathematical Theory of Elasticity (1906)

by Augustus Edward Hough Love, source; originally published in volumes 1 (1892) and 2 (1893).
• The problem of curved plates or shells was first attacked from the point of view of the general equations of Elasticity by H. Aron. He expressed the geometry of the middle-surface by means of two parameters after the manner of Gauss, and he adapted to the problem the method which Clebsch had used for plates. He arrived at an expression for the potential energy of the strained shell which is of the same form as that obtained by Kirchhoff for plates, but the quantities that define the curvature of the middle-surface were replaced by the differences of their values in the strained and unstrained states.
• E. Mathieu adapted to the problem [of curved plates or shells ] the method which Poisson had used for plates. He observed that the modes of vibration possible to a shell do not fall into classes characterized respectively by normal and tangential displacements, and he adopted equations of motion that could be deduced from Aron's formula for the potential energy by retaining the terms that depend on the stretching of the middle-surface only.
• Lord Rayleigh... concluded from physical reasoning that the middle-surface of a vibrating shell remains unstretched, and determined the character of the displacement of a point of the middle-surface in accordance with this condition. The direct application of the Kirchhoff-Gehring method led to a formula for the potential energy of the same form as Aron's and to equations of motion and boundary conditions which were difficult to reconcile with Lord Rayleigh's theory. Later investigations have shown that the extensional strain which was thus proved to be a necessary concomitant of the vibrations may be practically confined to a narrow region near the edge of the shell, but that, in this region, it may be so adjusted as to secure the satisfaction of the boundary conditions while the greater part of the shell vibrates according to Lord Rayleigh's type.

### Concrete-steel Construction (1919)

Part I—Buildings, 2nd edition, by Henry Turner Eddy, ‎Claude Allen Porter Turner, A Treatise upon the Elementary Principles of Design and Execution of Reinforced Concrete Work in Buildings. Also see 1st (1914) edition
• [P]rinciples as developed by Kelvin and by Love show that it is impossible to bend a nearly flat dish shaped shell about one horizontal axis without at the same time bending it in the opposite direction about a second horizontal axis at right angles to the first.
• [P]icture... a circular piece of a plate which has an approximately spherical curvature at its center point O. Pass a plane XY tangent to the surface at O and let OZ be normal to it at O. Then if a bending moment be applied about OX it will not only make the curvature of the plate greater in the plane ZOY but at the same time it will make its curvature less in the plane ZOX to an equal amount as is evident by experiment on a shell of any elastic material, and as is proven in Gauss' theorem of the curvature of thin shells. Since the force applied to produce the given bending moment must produce both these equal changes of curvature simultaneously by producing elongations and compressions in twice as much material as in a plane plate of equal cross section, each of them is only half as great as would be produced in a plane plate in a single direction by this same moment. Hence it. appears that the extensometer deformations produced by an applied moment are not more than half as great in a spherical dish shaped plate as in a plane plate...
Planes of principal curvature, normal vector & tangent plane.
• The curvature at any point P of a curved surface is most readily measured by finding the radii of curvature of two curved plane sections of the surface made by a pair of planes drawn normal to the surface at P and at right angles to each other. Normal planes are those perpendicular to the surface at P, and they intersect each other in the normal to the surface at P. If R is the radius of curvature of any plane section at the point P, then 1/R is defined as its curvature at P. At every point of a convex surface there must, except in case when the curvature of all of the sections is the same, be some one of the normal sections in which the curvature is the greatest, and also another section in which the curvature is least. These are called principal planes and principal curvatures. According to Euler's Theorem these principal sections lie in normal planes which are at right angles to each other, and further, the sum of the curvatures of any pair of rectangular normal sections whatever, at a given point P is constant, so that in rotating a pair of normal planes that remain perpendicular to each other about the normal the increment of the curvature of either normal section is equal to the decrement of the other, and the sum of the two normal curvatures is equal to that of the principal curvatures.
• In his great treatise on the Mathematical Theory of Elasticity Love, following the original investigations of Gauss, demonstrates... that when a piece of a thin elastic shell or plate that has a spherical curvature of 1/R is deformed by a small bending without stretching, then in the case of initial spherical curvature one principal curvature of the deformed surface will exceed the initial curvature 1/R by the same amount as the other will be less than l/R. That this is the fact in this case seems evident without following the abstruse analysis of Love... because if x and y be lines drawn tangent to two rectangular normal sections at P, and the spherical surface be bent slightly about x so as to alter the curvature of the section in the normal plane at right angles to x by alternately increasing and decreasing it, it is evident from symmetry that the curvatures of the section in the normal plane at right angles to y will undergo at the same time alterations of curvature which are equal and opposite to those in the first normal plane, altho this equality does not in general hold true in shells that are not spherical in shape. Nevertheless, the same statement evidently holds true for surfaces of revolution about the normal at P as an axis.
• [C]onsider the bending moments that must be applied to produce such a deformation. It is evident from the fundamental equation ${\displaystyle EI/R=M}$ of the ordinary theory of bending that
${\displaystyle EI\;d({\frac {1}{R}})=dM}$
is the equation which expresses the relation between the increment (or decrement) of curvature ${\displaystyle d(1/R)}$ and the increment (or decrement) ${\displaystyle dM}$ of the applied moment which produces this change of curvature about x (or y) in which the value of ${\displaystyle I}$ may be calculated in case of only slight curvature just as in a flat plate.
• It is evident that in order to produce this kind of deformation it is sufficient to apply one positive and one negative moment increment, each of the same magnitude ${\displaystyle dM}$, simultaneously about each of the rectangular axes x and y, respectively. These moments produce elongations and shortenings in the exterior and interior fibers of the plate or shell independently in two directions at right angles to each other in the same manner as in a beam but none whatever are produced in the neutral surface, so that the resistance that is offered to this bending arises from the resistance which the fibers offer to such elongation and shortening. The principal curvatures of the surface after bending will be ${\displaystyle 1/R+d(1/R)}$ and ${\displaystyle 1/R-d(1/R)}$ respectively.

### Equilibrium of Shell Structures (1977)

by Jaques Heyman, Clarendon Press, Oxford.
• [S]uppose a uniform thin-walled hemisphere... is subjected only to its own weight, and is supported round its [base] by forces which produce compressive stresses σ. If the shell has radius ${\displaystyle a}$ and thickness ${\displaystyle t}$, and the material has unit weight ρ, then [the force applied to the base is equal to the weight of the structure, where ${\displaystyle 2\pi a}$ is the circumference and ${\displaystyle 2\pi a^{2}}$ is the surface area of the hemisphere]
${\displaystyle \sigma (2\pi at)=\rho (2\pi a^{2}t)}$
or [dividing by ${\displaystyle 2\pi at}$]${\displaystyle \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad }$
${\displaystyle \sigma =\rho a}$
Thus the compressive stress necessary to support the dome has a magnitude independent of the thickness of the dome. ...Now the expression ${\displaystyle \rho a}$ ...is typical of the order of magnitude of stresses in more general shapes of shell... [I]t may be expected that externally applied loads will at most equal, and will usually be less than the self-weight loads of shells of reasonable size. ...Thus stresses resulting from snow or wind may be expected to be of the same order as those resulting from dead load.
• Very thin shells may be in danger of buckling locally. The problem hardly arises for civil engineering applications over moderate spans, but may be very important if the spans are very large. Local buckling of a thin shell will occur at a typical stress
${\displaystyle \sigma _{cr}=kE{\frac {t}{R}}}$
where ${\displaystyle E}$ is Young's modulus [and ${\displaystyle R}$ is the minimum radius of curvature of the shell]. The value of the constant ${\displaystyle k}$ varies from author to author, but a reasonable value is about 0.25. Thus for a concrete shell for which ${\displaystyle E}$ is about 20 000 N/mm2, and for which ${\displaystyle {\frac {t}{R}}}$ is as small as 1/1000, the critical stress is determined as say 5 N/mm2.

### Theory of Shell Structures (1983)

Arseniusz Romanowicz & Piotr Szymaniak's Warszawa Ochota train station, Warsaw, Poland.
Heinz Isler's Sports hall ("CIS") in Solothurn, Switzerland.
• [S]ome will claim that they give the best service to engineers by concentrating mainly on the form and structure of the governing equations... for once the foundations have been laid properly (they say), the solution of all problems becomes merely a mathematical or computational exercise... and indeed unless the foundations have been laid properly (they say), any resulting solutions are of questionable validity. Another group will argue... they can serve engineers best by providing a set or 'suite' of computer programmes, which are designed to solve a range of relevant problems... and... such programmes renders obsolete... the theory of shell structures... I have rejected both... the important thing is for engineers to understand how shell structures behave...
• Any load which is applied to the shell is sustained in general partly by the stretching surface and partly be the bending surface; and the balance in load-sharing is regulated mainly by the 'interface pressure' between the two surfaces, which varies from point to point over the surface. This interface pressure becomes a prime variable of the problem. In terms of classical structural mechanics it is a variable 'redundancy'.
• Preface
• Gauss (1828) pointed out that there are two distinct but complimentary ways of thinking about the curvature of surfaces. ...this point ...is absolutely fundamental to a clear understanding of the subject. Gauss's dual view of curvature fits precisely the 'two surface' description, and it provides succinctly the geometrical conditions which are necessary if the deformation of the two surfaces is to match. The key variable in this connection turns out to be a scalar quantity; and paradoxically... conventional treatment... in terms of general curvalinear coordinates is... too elaborate to reveal this crucial... quantity.
• Preface
• The development of the plastic theory of structures from the 1930s onwards was healthy not only because many real materials are still structurally useful when they have passed beyond the elastic range, but also because the new theory was able to shift the emphasis... to more profound questions about the kind of information which the engineer requires in order to design satisfactory structures. ...plastic theory can contribute to the understanding of the response of shells to localized loading.
• Preface
• The essential ingredients of a shell structure are continuity and curvature. ...an ancient masonry dome or vault is not obviously continuous... it may be composed of separate... sub-units or voussoirs not necessarily cemented... But in general... are held in a state of compression throughout... thus in compressive contact... [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, as required.
These structures have a quite different mode of action from skeletal structures... [which are] only capable of transmitting forces along their discrete structural members.
• Introduction
• [T]he basic model of a shell which we shall use... takes as its first step the replacement of a shell by a surface... [M]uch of the detail of the stress distribution will be suppressed precisely in the step of shrinking the three-dimensional physical shell into a zero-thickness surface. ...[T]he 'surface' theory... is extremely simple in comparison with other theories... [T]he regions in which the theory is inadequate are all highly localized, and there are very many practical problems in which... the local details are either unimportant or else can be treated more or less in isolation. All if this is closely analogous to the classical methods for analyzing beam and frame structures.
• Introduction

### Fundamentals of the Analysis & Design of Shell Structures (1987)

Vasant S. Kelkar, Robert T. Sewell
• The analysis and design of shell structures is a of interest in... the design of large-span roofs, liquid storage facilities, silos... pressure vessels, including nuclear reactor containment vessels and pipes... structural design of aircrafts, rockets and aerospace vehicles. All... require the analysis and design of shells... [T]he derivation of equations for plates or shells is only an extension of... bars or beams, based on equilibrium, kinematics, and Hook's law. ...Using the understanding of shell behavior... including the approximate methods and... tables for quick solutions... a reader may be able to judge the computer results, before designing a shell structure. ...Theory of circular rings, concepts of stress resultants and middle surface are... introduced... Circular plate theory also forms an intermediate step for a gradual introduction or transition to shell analysis from the analysis of bars and beams.
• Preface.
• [A] shell element will have, in general, 10 unknown internal stress reactants... [B]y making suitable assumptions, we try to obtain simpler a solution... for practical purposes. We first assume that the shell is thin. Such shells are... very flexible for resisting bending moments and shearing forces. ...We would always prefer the shell to resist any loading by development of in-plane forces... we assume that the moments Mx, My, Mx,y and My,x are zero... Then, by taking moments about the x- and y-axes... we conclude that Qy and Qx must also be zero, and by taking moments about the z-axis, we get Nxy = Nyx. Thus there remain only three unknown internal stress resultants, Nx, Ny, and Nxy, to support a given loading. Also... three equations of equilibrium
${\displaystyle \sum _{}F_{x}=0\quad \sum _{}F_{y}=0\quad \sum _{}F_{z}=0}$
...determine the three unknowns. Such simplified theory... is called membrane theory, as opposed to the more general and complex bending theory...
• [W]e can consider the general equation for the deflection [${\displaystyle w}$] of a shell as
${\displaystyle {\mathcal {L}}[w(x,y)]=f[p(x,y)]}$
where ${\displaystyle {\mathcal {L}}}$ is a differential operator and ${\displaystyle f[p]}$ is some function of the given loading ${\displaystyle p}$. The general solution... will be
${\displaystyle w=w_{h}+w_{p}}$
where ${\displaystyle w_{p}}$ is the particular solution... that satisfies equilibrium and compatibility at all internal points... but not necessarily satisfying the boundary conditions. ...${\displaystyle w_{h}}$ is the solution of the homogeneous equation with ${\displaystyle p=0}$ (...only edge loads can be present). ...[F]or solving practical problems, we can [find] ${\displaystyle w_{p}}$...by assuming moments and shears to be zero... the "membrane solution." ...similar to obtaining (fairly correctly) the forces in Truss members by assuming moments and shears in the members as zero (i.e., assuming the joints are perfect pins)... For obtaining ${\displaystyle w_{h}}$... we must... use the exact differential equation... the "bending theory" solutions... Fortunately, for most types of shells, they die out quickly as we move away from the boundaries. ...[O]ur general procedure ...obtain membrane forces under a given loading... then superimpose... the bending theory solutions for edge loads. ...[T]he membrane and edge-load solutions together satisfy the boundary conditions; i.e., the edge loads are obtained by solving equations of compatibility at the boundaries.

### An Introduction to Shell Structures: The Art and Science of Vaulting (1991)

by Michele Melaragno
Lattice thin-shell roof of the British Museum's Great Court, by Buro Happold, Norman Foster.
• A revival of interest in curvilinear structures is under way... Arches, vaults, and thin-shelled structures must be re-discovered. ...Why is there a revival in shell structures, and where might it lead? Cost factors, materials availability, labor supply, housing crises, solutions to domestic and Third World problems all play a part... With today's almost unlimited computer technology and the knowledge that can be gained from understanding domes and vaults built both in the past and present, it is hoped that this work on the practical aspects of designing curvilinear forms will contribute to further exploration and encourage the application of thin shells...
• A thin shell is a special kind of vault whose geometry may include many shapes. ...a three-dimensional form made thicker than a membrane, so that it can not only resist tension as membranes do, but also compression. On the other hand, a thin shell is made thinner than a slab, which makes it unable to resist bending, as a slab does. In short, thin shells are structures thicker than membranes, but thinner than slabs.
Thin shells are made possible by the use of materials that work well under tension and compression. Masonry has no tensile strength... Only the availability of reinforced concrete and ferrocement made a thin shell possible.
• The curvature of a shell can be of the same sign throughout... In such a case the surface is called synclastic. Domes are synclastic surfaces... The curvature of a shell can also be of a different sign... both concave and convex... which is known as anticlastic. An example... is the hyperbolic paraboloid.

### "Concrete Shell Structures Practice and Commentary" (1992)

American Concrete Institute, ACI 334.1R-92, Reported by ACI Committee 334, Anton Tedesko, Chairman; members included David P. Billington, Felix Candela, Wilhelm Flugge, and Mario G. Salvadori; (Reapproved 2002).
• While size and support conditions have an important bearing on the degree of accuracy needed in the analysis, the distribution of load has a less important effect on stresses. This is due to the fact that bending moments in the shell are more closely related to the boundary conditions than to the load. Hence, it is usually unnecessary to analyze a thin shell for partial live loads even though the supporting members must be analyzed for such partial loads. For this reason, snow load on thin shells may be assumed either uniformly distributed on the horizontal projection or uniformly distributed over the surface of the shell. On the other hand, local bending moments due to large concentrated loads on the shell must be considered.
• Part II, Ch. 2-Analysis of Shell
• Shells of double curvature both the synclastic... and anticlastic... are inherently better suited to resist loads by direct forces than are shells of single curvature. The reason for this is obvious from the fact that this type of shell possesses arch action along both curvatures. But in order that surfaces curved in two directions behave as a shell, it is important that proper support or edge members be provided.
The direct stresses throughout the major portion of the shell are usually of little significance except as they relate to buckling. A careful evaluation should be made of the bending moments produced in the vicinity of the edge members by the interaction of the edge member and the shell. For moderate size shells, this effect usually is confined within a few feet of the vicinity of the edge member. ...
An exception to this are some anticlastic shells, like the hyperbolic paraboloid, wherein bending can prevail throughout a greater portion of the shell. To a limited extent, this also occurs in domes, when the supports do not provide a reaction tangent to the shell surface. In these cases, the bending moments may extend a significant distance into the shell.
• Part II, Ch. 2, 2.3-Thin shells of double curvature

### "Contributions of André Paduart to the Art of Thin Concrete Shell Vaulting" (2003)

Bernard Espion, Pierre Halleux, Jacques I. Schiffmann, Proceedings of the First International Congress on Construction History, Madrid, 20th-24th January 2003 (2003) ed., S. Huerta, Madrid: I. Juan de Herrera, SEdHC, ETSAM, A. E. Benvenuto, COAM, F. Dragados.
• The great era of thin concrete shells... was an attempt to cover large spans with the most widely used construction material of the Twentieth Century and yielded structures that are now regarded as architectural masterpieces. The design of thin concrete shells also fostered theoretical developments in structural analysis, in the mathematical theory of shells and in the theory of finite elements.
• In Belgium, the key figure in the design, construction and popularisation of concrete thin shells was certainly André Paduart (1914-1985). ...Paduart was also and particularly an active member of the International Association for Shells Structures (IASS) founded by E. Torroja in Madrid in 1959. ...[He] organized in Brussels in 1961 one of the very first symposia of this association... Shortly after, he published [1961] in French a remarkable small book covering essential theory, design and construction of thin concrete shells [Introduction au calcul et a l' exécution des voiles minces en béton armé]... translated in English [Introduction Shell Roof Analysis] in 1966...
• [A] significant breakthrough was achieved with... two celebrated huge airship hangars built by Freyssinet at Orly in the early 1920s... [whereby] the principle of the corrugated form for the concrete shell was introduced to obtain the necessary stiffness...
• Cylindrical barrel vaults have probably been the most used form of concrete shells. The reconstruction after the devastations of the Second World War required forms of building which offered economy of material. This gave an enormous boost to the use of shell roofing... since materials... were in short supply... [N]early 50000 square meters of warehouses at the docks of Antwerp harbour [were built] between 1947 and 1950...by [André] Paduart and [C.] Wets... [H]angars were built 1950-1952 at... airfields... one arch of a hangar under construction at Chièvres collapsed... a [short] time after decentering. ...[M]easurements made in the early 1990s [indicated that] several of the [arches] at Chièvres ...were significantly deformed.
• For the 1958 Brussels international exhibition, Paduart and architect J. Van Dooselaere received an official commission... to design a structural symbol testifying of the "victory of [Belgian] civil engineering over nature"... The final structure... the "Civil Engineering Arrow" [Pavilian of Civil Engineering], was a spectacular thin wall... cantilever beam... a bold impression of equilibrium and "tour de force". [They] received the 1962 Construction Practice Award for their "Arrow". ...dismantled in 1970.
• Paduart was working at the edge between academia and engineering practice. ...[H]is production during thirty years... is eclectic, with barrel vaults, corrugated shells, hypar shells and folded plates. He could teach... mathematical theory of shells at the university, but used... very simple methods derived from the Strength of Materials to design his own shells. This did not deter him from conceiving bold structures, at the limits of the utilization of the materials and construction techniques of his time, but he looked always forward with anxiety to the decentering of the shells...

### Analysis of Thin Concrete Shells Revisited (2008)

: Opportunities due to Innovations in Materials and Analysis Methods, Bart Peerdeman, Master's Thesis, Delft University of Technology
• The construction of thin concrete shells ended abruptly at the end of the 1970s, mainly caused by the high costs... However, uncertainties in the structural behaviour of shells did not help either. Contemporary progress in finite element software discards these uncertainties as it allows the engineer to closely approach the actual behaviour of thin concrete shells by performing geometrically and physically nonlinear finite element analyses. ...The combination of advanced finite element analyses and ultra high performance fibre reinforced concrete may lead to shells with even greater spans and thinner thicknesses than achieved so far.
• Preface
Zeiss Planetarium Dome, Jena, under construction (1924)
• Shell structures have been constructed since ancient times. The Pantheon in Rome and the Hagia Sophia in Istanbul are well-known examples. After the Roman times the traditions of domes continued up to the 17th century. Since then they seemed forgotten. Stimulated by the newly developed reinforced concrete and the demand to cover long-spans economically and column free the shell made a comeback in the early 20th century. Franz Dischinger and Ulrich Finsterwalder designed in 1925 the first thin concrete shell of the modern era, the Zeiss planetarium in Germany. The modern era of shell construction is recognised by the trend towards greater spans and thinner shells. Guided by well-known engineers as Pier Luigi Nervi, Eduardo Torroja, Anton Tedesko, Nicolas Esquillan, Felix Candela and Heinz Isler a blooming period of widespread shell construction took place between 1950 and 1970. Shell construction suddenly vanished at the end of the 1970s, mainly caused by the high costs [relative] to other structural systems. Moreover, inflexible usability and uncertainties in the structural behaviour of shells and difficulty of proper analysis methods did not help[,] neither did the stylistic identification with the 1950s and 1960s. Today the great era of thin shells is over, however, nowadays natural free-form shapes and blobs attract more and more attention. In addition, recent developments in concrete technology have led to ultra high performance fibre reinforced concrete with revolutionary performance in tension and compression. Eventually this may lead to a revival of the thin concrete shell.
• Summary
Zeiss Planetarium Dome, Jena, the oldest existing planetarium in the world (1926)
• In case of a failure, the shell may fail due to large deformations (buckling) or due to material nonlinearity (cracking and crushing) or by a combination of both (so-called inelastic or plastic buckling). ... Opposite to columns and plates, shells experience a sudden decrease in load carrying capacity after the bifurcation point (which can be obtained by a simple linear buckling analysis). ...compound buckling ...refers to several buckling modes associated with the same critical load. In the postbuckling range the modes... start to interact resulting in a significantly reduced load carrying capacity. As discovered by Koiter... geometrical imperfections in the shell cause the bifurcation point never to be reached and lead to... buckling at a considerably lower load. The size of the imperfections determines the limit load at which the shell fails. In case of plastic buckling, the fall-back is further intensified by material nonlinearity.
• Summary
• [T]wo primary research questions can be formulated: [1] What is for a shell of hemispherical geometry, with given material properties, given support conditions, and subjected to a given load, the knock-down factor which indicates the difference between the linear critical buckling load and the actual critical buckling load taking into account imperfections and geometrical and physical nonlinearities? ...[2] Can high strength fibre reinforced concrete add to the trend towards greater spans and thinner shells with possibilities for even more slender structures? To obtain an answer to the research questions a series of analyses (linear elastic, stability, geometrically nonlinear and geometrically and phy sically nonlinear) is performed on a given hemispherical shell: the Zeiss planetarium shell.
• Summary

### A Thin Shell Trolley Barn for Seattle's Water Front (2010)

Michael W. Weller, thesis proposal, Master of Architecture, University of Washington, Dept. of Architecture.
• In the early twentieth century reinforced concrete was a new building technology. Its novelty inspired experimentation, both from architects, such as Le Corbusier, and from engineers, who dreamed up different applications for the new ferroconcrete. One application for reinforced concrete that developed rapidly was its use in thin shells. These shells spanned great distances or stretched out in dramatic cantilevers, their thinness seemingly impossible for the distance they extended. This technology quickly grew ever more common, especially in long-span utilitarian settings, where thin shell concrete was able to cover large areas economically.
Max Berg’s Centennial Hall in Breslau (now Wrocław), Poland
Eduardo Torroja's Algeciras market hall (1934).
• The German engineering firm of Dyckerhoff & Widmann['s]... design of the concrete dome for Max Berg’s Century [or Centennial] Hall in Breslau... in 1913... became the first modern building whose clear span exceeded Rome’s Pantheon. Other notable structures of this early phase are the elegant works of Eduardo Torroja in Spain, including the Algecira market hall (1934), and Freyssinet’s economical segmented system for an aircraft hangar at Orly (1921).
• By the 1970s the use of thin shell concrete had all but disappeared... This change was due to a combination of [economic] factors... This disappearance was also caused by the design challenges [of] thin shell concrete. Shell structures, because of their thinness, must be shaped to conform to the forces present in the structure. Until recently this shell form-finding could only be done... with specialized computer expertise, or through... physical model testing and measurement. ...In recent years ...simple computational models ...have been adopted ...to explore rapid form-finding in ...early design phase.
• Forces present in the structure, shape thin shell concrete. Areas of uniform load present smooth, catenary curvatures, while areas of concentrated force express themselves as sharp bends or spikes in the surface form.
Heinz Isler’s structure for the former Kilcher company in Recherswil, Switzerland.
• By their very nature all funicular structures, including thin shell, use... less material... By designing only for pure tension or compression these structures experience very little bending force. These pure forces require less material to resist...
• Dieste’s hypar masonry roofs are inexpensive, utilitarian statements about space enclosure...
• [T]he second wave of shell building (1940-1960s)... focused almost exclusively on ruled surfaces... [to] include hyperboloids and hyperbolic paraboloids. ...[T]hey were definable through mathematical formulas, which allowed the designer to understand the forces... Ruled surfaces are... more constructible, because they can be created out of linear elements, such as... boards and pipes...
• The most prominent designer to eschew ruled surfaces is the Swiss engineer Heinz Isler. In 1954... Isler hit on the idea that a “bubble” (in this case a pillow) takes the optimal shape for its edge boundaries. Isler began to construct models by inflating surfaces or by hanging and then hardening them.

### Form Finding, Force and Function: Mass-Spring Simulation for a Thin Shell Concrete Trolley Barn (2011)

Michael W. Weller, thesis proposal, Master of Architecture, University of Washington, Dept. of Architecture.
• I have selected the use of hanging models to simulate compressive forces. This is one of the longest-used form finding techniques. It has its roots in physical models, but in recent years it has also given rise to a range of digital tools that are fairly accessible to an uninitiated designer... Hanging models... can be used to simulate... funicular structures. ...derived from the Latin word for “rope”... a structure takes its shape in response to the magnitude and location of forces acting upon it. For example, a rope suspended from two level points will form a “V” when a single point load is added at midpoint, but will form a catenary when under an evenly-distributed load. While a suspended rope is a purely tensile system, if inverted and made rigid that same form converts into a system that is in pure compression. This was first postulated (and wonderfully expressed) by the English scientist Robert Hooke... The value of a structure that is purely in compression is that it experiences no bending due to structural loads. With no bending present materials can be used very efficiently, allowing for the use of extremely thin elements... materials that are strong in compression... as tiles or masonry, can... be employed.
• History shows that not all thin shell concrete buildings are funicular—other families... chosen for pragmatic reasons such as their constructability, or because they were geometrically simple enough [that] through calculation... bending... was [found to be] within... tolerances... for the material.
Antoni Gaudí's Houses in Park Güell, Barcelona, Spain
• [T]he infill fields of a Gothic cathedral’s rib vaulting behave as masonry shells, and by the 1860s masonry vaults of great beauty were developed by Rafael Guastavino in both Spain and the United States. Gustavino’s work and graphic design methods... influence[d] Antonio Gaudí... whose work has been shaped by an interest in force-derived architectural form. While early engineers such as Guastavino pushed unreinforced masonry’s limits in thin shell construction, it was the introduction of steel reinforcement, initially in concrete, that sparked the Twentieth century’s interest in thin shell construction. Early [twentieth century] shell-builders were often engineers, and were engaged to design economical long-span structures... as aircraft hangars, train sheds and factories.
• [A] list of factors combined to make thin shell concrete a less desirable solution for long spans by the 1970s. ...material efficiency could no longer off set the labor premiums demanded by complex formworks ...shapes were more difficult to create when an insulating layer was required ...A third challenge posed to compressive structures ...was the rapid development of tensile structures. Frei Otto’s experiments with tensile membranes were mature enough in 1972 that they were used on Munich’s Olympic Stadium.

### Structural Analysis of Thin Concrete Shells (June, 2015)

Hanibal Muruts Ghebreselasie, Yuting Situ, Master Thesis, NTNU Open, Trondheim, Norwegian University of Science and Technology, source
• Concrete shell structures, often referred to as ’thin shells’ are suitable structural elements for building spacious infrastructures. ...Loads acting on the surface of shell structures are mainly carried by the so called membrane action. This is a general state of stress [which] consists of the in-plane normal and shear stress resultants only. In comparison, other structural forms such as beams and plates carry loads acting on their surfaces by bending action, which can be said [to be] structurally less efficient. Usually the in-plane stresses in shells are low such that with a relatively small thickness it is possible to span over large distances.
• Compared to structural elements such as beams, slab and walls, the structural behaviour of shells in not easy to predict. Hence evaluating the accuracy of the results obtained from FEA of shell structures is a challenging task. Having the knowledge and understanding of the analytical solution method can provide the basis for this verification and... give ...much needed insight into the structural behaviour of shells. ...for some of the most commonly constructed concrete shell structures, a complete analytical solution procedure is available. The two types of concrete shell structures considered in this paper are axisymmetric shells and cylindrical shell roofs. ...[A]xissymetrical shells include structures such as containment buildings, tanks and silos. ...[C]ylindrical shell roofs are often preferred structural elements for large span concrete roof structures.
• A shell can be defined as a body that is bounded by two surfaces parallel to its middle surface, and is deformed in any arbitrary manner. This is true for shells of a constant thickness... considered in this study. ...One particular way of classifying shell surfaces is according to their Gaussian curvature.
${\displaystyle \kappa _{g}=\kappa _{1}\cdot \kappa _{2}={\frac {1}{r_{1}}}\cdot {\frac {1}{r_{2}}}}$
Another way of describing shell surfaces is according to how the surfaces are generated. Using this method, in 1980 Heinz Isler classified shell surfaces into Geometric, Structural and Sculptural surfaces. Geometric shells are well defined mathematically and... easily calculated analytically. These type of shells were quite significant in the development of shell structures [when] computer[s]... were not available. ...geometrically shell surfaces can be classified as cylindrical... spherical... conical... paraboloidal... etc.
• [C]losed surfaces are more rigid than open surfaces. ...Therefore to achieve ...rigidity the openings [are] compensated.
• [U]sing concrete shells as roofing provides the possibility of constructing spacious columnless buildings... reinforced concrete has enhanced this possibility.
• Concrete shells can be built by the assembly of several cast units, or cast in one piece (monolithic). Monolithic concrete shells are structurally stronger...
• [D]evelopment of the theory employs Hooke's law (elastic material), equilibrium and compatibility. Hooke’s law relates strains with stresses, equilibrium relates stress resultants with external loading and compatibility relates strains with deformation/displacements. These three sets of equations together with appropriate boundary conditions make up the mathematical aspect of the problem.
• [T]he ratio [of] radii of curvature to thickness of the shell, ${\displaystyle {\frac {R}{t}}}$... greater than 20 can be characterized as thin shells... an egg shell has a ratio of around 55...
• [W]e will mainly be dealing with uniform shells. The shells are uniform in the sense that the material properties do not vary through the thickness. Reinforced concrete (RC) is... regarded as sufficiently uniform... [since] the difference in Young's modulus between steel and concrete is not large...

### Richard Bradshaw Oral History (May 9, 2016)

: Part 1 of 2, video of an interview of structural engineer, Richard R. Bradshaw, founder of Richard R. Bradshaw, Inc., Van Nuys, California, from Getty Research Institute, source.
• The greatest of them all, in my opinion was Nervi. ...First of all, he did a tremendous volume of work. Now Torroja, from Spain, was very good, but the bulk of his work was quite small. ...Candela came along and... was very experimental and very creative, but he stuck almost entirely with the hyperbolic paraboloid form of the shells. ...[H]e did... a lot of great buildings, but they were... limited in that they were all this one type... Nervi stuck with all kinds of different stuff: precast... and Nervi had a laboratory in Milan... where he made experimental models. ...Torroja had a lab too and... finally became... just a professor... [so he] didn't do a great deal of volume. ...Nervi kept on doing these things his whole life... working with models, and working with whatever theory he could learn... [T]he great Lamella roof hangar he built before the war, that the Germans finally blew up when they had to retreat from Italy, was a magnificent... and a huge structure... [T]he other two... Torroja and Candela, were excellent engineers. Candela became a friend of mine later on... but they never had the tremendous variety and the huge buildings that Nervi did... Nervi's the guy that I really admire... most...
• 7:59-10:06. Note: Nervi built lamella grid roof pattern hangars in Orvieto (1935) entirely out of reinforced concrete, and a second set in Orbetello and Torre del Lago (1939) using a lighter roof, precast ribs, and modular construction.
• The first shell I did was in Honalulu, and there I worked with an architect, that I had worked with at the Navy yard, named Pete Wimberly... Wimberly's architectural firm went on to be one of the biggest in the world. ...Pete and I were good friends, and when I went to Honolulu I'd stay in his house, and we'd start barnstorming at night. ...A lot of our stuff was just crazy talk, but a lot of it was freeing ourselves of inhibitions that later on we remembered... Pete had a very good intuitive sense for structure. He could... guess a structure that would make sense. ...[T]he first shell I did was a bowling alley near Pearl Harbor... a single curvature shell, cylindrical shapes... a span of about 93 feet... I could not find any way that... other shells were engineered. If you tried... you'd always get some kind of double-talk... [T]he world didn't have translations of everything then, like it has now. If a guy did something behind the Iron Curtain it stayed... [T]he way I finally did it, I figured out that it must behave an awful lot like a... simple beam, even though it was a... curved structure, and so largely I figured it that way and it worked fine, and it's still up, and it was quite thin, it was 2 1/2 inches thick. In that day that was pretty daring...