Thin-shell structure

"The 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 s. 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 EI/R = M of the ordinary theory of thatEI\;d(\frac{1}{R}) = dMis the equation which expresses the relation between the increment (or decrement) of curvature d(1/R) and the increment (or decrement) dM of the applied moment which produces this change of curvature about x (or y) in which the value of 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 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 1/R + d(1/R) and 1/R - d(1/R) respectively."

"[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 a and thickness t, and the material has unit weight ρ, then [the force applied to the base is equal to the weight of the structure, where 2\pi a is the circumference and 2\pi a^2 is the surface area of the hemisphere]\sigma(2\pi at) = \rho(2\pi a^2t) or [dividing by 2\pi at] \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \sigma = \rho aThus the compressive stress necessary to support the dome has a magnitude independent of the thickness of the dome. ...Now the expression \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 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\sigma_{cr} = kE \frac{t}{R}where E is [and R is the minimum radius of curvature of the shell]. The value of the constant k varies from author to author, but a reasonable value is about 0.25. Thus for a concrete shell for which E is about 20 000 N/mm2, and for which \frac{t}{R} is as small as 1/1000, the critical stress is determined as say 5 N/mm2."

"[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 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 it is a variable 'redundancy'."

"Gauss (1828) pointed out that there are two distinct but complimentary ways of thinking about the 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 is... too elaborate to reveal this crucial... quantity."

"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."

"The essential ingredients of a shell structure are continuity and '. ...an ancient masonry or vault is not obviously continuous... it may be composed of separate... sub-units or s 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."

"[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."

"The analysis and design of shell structures is 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, , and . ...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 also forms an intermediate step for a gradual introduction or transition to shell analysis from the analysis of bars and beams."

"[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\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 [w] of a shell as\mathcal{L}[w(x,y)] = f[p(x,y)]where \mathcal{L} is a and f[p] is some function of the given loading p. The general solution... will bew = w_h + w_pwhere w_p is the particular solution... that satisfies equilibrium and compatibility at all internal points... but not necessarily satisfying the boundary conditions. ...w_h is the solution of the homogeneous equation with p = 0 (...only edge loads can be present). ...[F]or solving practical problems, we can [find] 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 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."

"A revival of interest in curvilinear structures is under way... es, 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 s 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 and made a thin shell possible."

"The of a shell can be of the same sign throughout... In such a case the surface is called synclastic. s 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."

"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 s 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."

"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 . 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 s, 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."

"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 , 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 s 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 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 s, 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..."

"Shells can be singly curved (e.g., cylinders and cones) or doubly curved (e.g., sphere or hyperbolic paraboloid)."

"[A] hyperboloid of one sheet... is often used for s, because it can be formed of straight lines... Another doubly curved shape formed of straight lines is the ... for which a straight line travels along another straight line at one end and a curve at the other end."

"At any point on a surface, there are two principal radii of curvature that uniquely define the surface. Of all the curves on the surface that can be drawn through the point, the two principal radii of curvature will be the maximum and minimum that can exist at the point. The maximum radius of curvature for a is infinity, while [its] minimum is the radius of the circle..."

"[Take] two pieces from the outside and inside of a , respectively. In the former, both radii of curvature lie on the same side of the surface, and [so] the curvature is considered positive, while in the latter they are on opposite sides of the surface, and [therefore] the curvature is negative."

"All shells have either positive (bowl-shape) or negative (saddle-shape) curvature. ...Positive curvature shells are subject to , as the entire shell is subject to compression forces. In contrast, material failure is more common in negative curvature shells with brittle materials such as concrete."

"The structural analysis of shells has had a long and difficult history."

"Shells were developed and reached their peak popularity just before the ready availability of computers and the FE method."

"The analysis of shells requires that the three conditions of equilibrium, compatibility, and constitutive laws be satisfied simultaneously. The latter are the stress-strain properties for the materials.... Most shells are designed with isotropic materials."

"With the development of advanced composites, their orthotropic and anisotropic behavior must be considered. However, composites have not been used for architectural shells to date."

"Roof structures are seldom designed for dynamic loads. Earthquake and wind loads may be treated as equivalent static loads."

"The above three conditions result in three partial differential equations, two of the 2nd order and one of the 4th order, for the most general case with two different radii of curvature and with combined bending and membrane actions. An early representation of these equations for cylindrical shells may be found in Donnell (1933). Bradshaw... extended those equations to the general case of double curvature, which can describe any 1D member (beam), 2D member (plate), or 3D singly or doubly curved member (shell)."

"If the 4th order -related terms are left out, the equations will represent only the membrane action, which is usually sufficient for part of the shell away from the s because flexural resistance of thin shells contributes little in this region."

"Equal radii of curvature result in equations for a . Equations of a cylindrical shell are derived when one radius of curvature is set to infinity; when both are set to infinity, it will result in bending of a flat plate. Finally, the ordinary differential equation for bending of a beam is derived when plate width is set to unity."

"Stress analysis of complete shells, such as s, is much simpler than for architectural roofs because of the boundary conditions."

"When the shell is a portion of the sphere, it tends to spread outward at the discontinuous edge. To counteract this a ring is added, but the ring and the shell distort by different amounts, which results in stresses in the shell. These incompatible strains must be reconciled analytically, which is not too difficult a task for simple spherical shells. However, when the shell has isolated supports and few (if any) planes of symmetry, it is a severe problem..."

"The resulting disturbance at the edge may be thought of as causing stress redistributions to flow across the entire shell with diminished effect as they move away from the edge."

"In many cases... the stresses resulting from the discontinuous edges will dominate the design."

"Ribs are frequently added at the edges, though visually disruptive. One of the graceful aspects of Candela’s shells is their lack of ribs. It is also possible to design the shell with the rib integrated within the shell itself."

"There is a remarkable property of shells supported vertically at their edges. [Take] a spherical dome supported on a wall. A tension tie is required around the perimeter at the intersection of the dome and the wall. This tie will be funicular, i.e., it will only carry axial tension forces. This principle has been known since antiquity for circular domes and ties. However... the tie will be funicular for any shape of either the plan or elevation (Csonka 1962) provided that the shell has positive curvature and continuous vertical support. The support may be a continuous wall or stiff beams between adequately spaced columns. ...The thrusts are taken by shear forces through the width of the shell, and only tension forces exist in the tie."

"Beles and Soare... have reported [on] buckling failure of shells. Unlike shells of positive curvature that are subject to buckling, in shells of negative curvature, such as hyperbolic paraboloids, buckling is prevented through the tension curvature in the other direction."

"Virtually all studies on shell buckling have focused on cylindrical, conical, and spherical shells made of metals, and usually on full 360° models rather than the much more complex architectural shells. ...Applicability of these tests to large-scale concrete shells... is questionable."

"Initial imperfections in shells can result in their at loads far below their theoretical capacity. Once a shell buckles, its collapse tends to be complete, contrary to plates, which have high post-buckling capacity."

"When is placed on a , the weight distorts the balloon. This means the shell will not be exactly the initial shape of the balloon. This is not important for a small span shell. For a long span shell, however, this deviation from the spherical shape could be serious as shells are sensitive to buckling due to the initial roughness effect."

"Shells were seldom the most economical way of covering a large space, especially when compared to lightweight tension membranes. ...[F]ormwork has always been a major cost factor."