Engineering

"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."

"The strength of latex concrete is [negatively] affected by excess water in the mix. For that reason it is best to work with dry sand. ...[I]f you add the latex to wet sand, you will be watering down what latex you have, and not getting the advantageous physical properties ...Use dry sand. ...Try for a ratio of polymer solids to cement by weight of 0.12 to 0.15, and a water to cement ratio by weight of 0.35 to 0.38."

"Latex liquids are a mixture of polymer latex solids suspended in water... typically at a mix of about 50 percent solids to 50 percent water by weight. This material is then used by manufacturers of construction liquids such as latex paints and cement additives by adding additional water, coloring, and special chemicals such as antifoaming agents. ...[T]rade named materials ...all have properties in common. They improve mechanical strength and adhesion properties, and impart the ability of the concrete to air cure. They remain stable in Portland cement, and resist the penetration of water, hence provide ...good freeze-thaw resistance, and low moisture penetration. They can readily bond with themselves. Latex paints are typically... 20 percent solids by weight... [cement additives] are... about 47 percent solids."

"The latex concrete is typically made by mixing Portland cement with dry sand to a 1 to 3 ratio by weight, then adding latex liquid until you get a broomable mix."

"An LC HP shell is an HP shell made of latex concrete. Latex concrete is fiberglass fabric stretched over a large frame... and the resulting fabric surface is then saturated with first a slurry of latex and Portland cement (a liquid paste), then when that dries, a ¼ to ⅜ inch thick layer (or layers) of latex concrete made of latex liquid, Portland cement, and dry sand is broomed on."

"Thin shell latex concrete shell roofs are one answer to low cost construction of immediate shelter of displaced people groups. liquid and are generally available in... regions where poverty and the need for housing abound."

"This construction is a means of fighting poverty. The work can unite war torn communities, and communities disoriented by natural disaster."

"This is a Manual on low cost, permanent shelter for needy families built using unskilled labor."

"When designed properly, [shells] are among the most beautiful and efficient of architectural structures. They will present both problems to be solved and opportunities to create for those who take the time to understand them."

"There are... new materials such as fibercrete concrete and fiber reinforced polymer (FRP) composites that may be used in shells."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

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

"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."

"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..."

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

"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."

"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."

"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)."

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

"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."

"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."

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

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

"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."

"[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."

"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..."

"[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."

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

"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..."

"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."

"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."

"[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..."

"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..."

"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."

"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."

"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."

"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."

"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."

"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..."

"[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] 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..."

"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."

"[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 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."