Thin-shell structure

"[T]he early work of Lambot... was one of the first applications of , but [was] also... a form of . His patent on wire-reinforced boats that was issued in 1847... This was the birth of reinforced concrete, but subsequent development differed from Lambot’s concept. The technology of the period could not accommodate the time and effort needed to make mesh of thousands of wires. Instead, large rods were used to make what is now called conventional reinforced concrete, and the concept of ferrocement was almost forgotten for 100 years."

"In the twentieth century, lightly reinforced brick shells were inspired by timbrel vaulting, a common building method in the Mediterranean. ... upon emigrating to the United States from 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."

"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 ... 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..."

"[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."

"The method of geometric modeling of multi-shell roofs depends mostly on the surface's properties forming the shell; their curvature, as well as continuity between them. ...s play a specific role, due to their characteristics. Catalan surfaces are s... They are oblique ruled surfaces which can be divided into two groups... second order—hyperbolic paraboloid... [and] of more than second order—s, cylindroids... The difference between hyperbolic paraboloid, conoid, and cylindroid results from different path of movement of a surface's ruling during formation. In all cases of Catalan surfaces' creation... each ruling is parallel to the fixed plane (not containing the surface's directrices)."

"... 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..."

"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."

"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 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..."

"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..."

"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."

"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 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. 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."

"At the time of construction of the Wyss shell, three-dimensional 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... 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."

"At the beginning, the 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 team."

"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."

"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, , 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 lattice tower by... in 1896. In the 1920s, Anton Tedesko first introduced reinforced concrete thin shells in the United States. This expansion was pursued... by , 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."

"Nature does not apply the construction principle of a beam supported by two s. 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."

"The design of shells... implicates the design of internal stress fields of form dependent shapes... meeting the compatabilites of all boundary conditions..."

"The title of first pioneer of the HP hyperbolic paraboloid or hypar construction] in concrete in the 1930s belongs to Fernand Aimond for the projects that he constructed.., for the formulation of the theoretical structural membrane model, and for his influence both on [Giorgio] Baroni in Italy in the late 1930s and on Candela in Mexico in the 1950s."

"[[w:Membrane theory of shells|[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."

"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."

"The true Mathematical and Mechanical Form of all manner of es 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."

"The meandor compression spring... or corrugated spring... is a promising design variant for compression axle springs of composite material... The middle surface of the meander spring is the... "." ...[T]he middle surface ...is the ...[i.e.,] can be unrolled onto a flat plane without stretching or tearing it. ...[A] developable surface ...in three-dimensional space and ...complete ...is necessarily ruled. This property is of primary importance for manufacturability... The meander spring... is modeled for mechanical analysis as a thin shell."

"My invention shows a new product which helps to replace timber where it is endangered by wetness, as in wood flooring, water containers, plant pots, etc. The new substance consists of a metal net of wire or sticks which are connected or formed like a flexible woven mat. I give this net a form which looks in the best possible way, similar to the articles I want to create. Then I put in hydraulic cement or similar bitumen tar or mix, to fill up the joints."

"Cement Hall, Swiss National Exhibition, Zurich (1939)... was built by architect and engineer Hans Leuzinger and ... 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."

"Like Candela, Isler has concentrated his practice on thin s... 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."

"There is... [a] need to code cheaper and accessible programs in line with using sustainable methods to better the livelihood of mankind. To address this issue a theory is formulated based on the Euler-Bernoulli beam model. This model is applicable to thin elements which include plate and membrane elements. This paper illustrates a finite element theory to calculate the master stiffness of a curved plate. The master takes into account the stiffness, the geometry and the loading of the element. The of this is established from which the load which is unknown in the matrix is evaluated by the principle of bifurcation."

"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 -like, -like, or saddle-like. All three of these geometric shaped can be used as the basis for thin-shell structures."

"[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 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 ... can be neglected, originally made by [Noah] Burger, is introduced to derive a simplified set of nonlinear... differential equations."

"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... ... ... Felix Candela... ... ... 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... s... 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 ing preceded the works of Candela... looked to nature and natural forms—an approach today called biomimicry..."

"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."

"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, S, of a thin spherical shell is very nearly given by formula:S = 2 \pi rtwhen t represents the thickness, and r = the inner radius of the shell, and t is supposed to be very small compared with r. The whole force, F, to be resisted by the tenacity of section 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, orF = \pi r^2 pAssuming that every portion of section S is equally strained by F, and designating by 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: \pi r^2 p = 2 \pi rtk; hencep = \frac{2tk}{r} { I.}and the proper ratio of the thickness to the radius of a thin hollow sphere is given by the formula:\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, F, producing the first-named tension is represented by the formula:F = \pi r^2 p;and the sectional area, 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:S = 2 \pi rt.The value of p, when it becomes the bursting pressure, is found from the equation,r^2 \pi p = 2 \pi rtk; hence 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:F = 2rp,and the area of these sections byS = 2t.The bursting pressure is, therefore, found from the equation:2rp = 2tk; hence 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."

"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 l = the length, d = the diameter, and t the thickness of the shell, all expressed in the same unit of measure, and let p = the collapsing pressure in pounds per unit of area; thenp = 9,672,000 \frac{t^{2.19}}{l d}. { IV.}In case a tube is stiffened by T-iron rings or by flanges, 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 d the [following]... let a be the greater and b the less semi-axis of the ellipse; then we are to maked = \frac{2 a^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 S the greatest shearing stress, p the pressure in pounds per square inch, t the thickness of the tube in inches, 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 S—viz.:p = \frac{2tS}{L}. { VI.}The writer deduces then a general value 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 S becomesS = 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."

"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 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."

"[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 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 deformations produced by an applied moment are not more than half as great in a spherical dish shaped plate as in a plane plate..."

"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."