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A review of the state of the art for the membrane analysis of hyperbolic parabolic (HP) shells is presented. Membrane solutions using simple statics are given for HP shells having square, rectangular, or parallelogram shapes in plan and subjected only to uniform vertical loading over the horizontal projection of the shell. Statics of shell-edge beam systems are discussed for saddle shells, inverted umbrellas, gable shells, and shells on two supports. Membrane stresses for general loadings on parallelogram-shaped HP shells obtained using a differential equation approach are described. Stress transformation formulas are given that can be used to find principal membrane stresses in the shell and boundary stresses to be transmitted to the edge members. Membrane stresses in HP shells having an arbitrary quadrilateral shape are discussed and procedures for determining them are discussed.

Construction of hyperbolic paraboloidal concrete shell roof structures is discussed, including the phases for selection, economy, forms, reinforcing, concrete materials, placing, and curing. Umbrella shells have the greatest potential for economy for large commercial or industrial construction. Gabled shells and saddle shells are suited to long span monumental buildings. Shells require high standards of concrete construction and the close cooperation of the architect, the engineer, and the contractor.

Umbrella and gable configurations are two of the most popular hyperbolic parabolic geometries. Early design of those shells was accomplished through the use of membrane theory. This determinant theory predicted overall behavior of these shells as a double cantilever beam for the inverted umbrella and as a simple beam for the gable shell. As long as the dimensions of these shells were small enough, this theory was adequate. However, with increased spans and with flatter applications, the theory proves to be inadequate. Bending solutions achieved by a finite element analysis prove to be necessary. Results from such analysis demonstrate the influence of several design parameters. The major role played by the dead weight of the edge and ridge beams is demonstrated.

Shell structures mobilize geometry to activate both the membrane and the flexural internal force systems to efficiently support any distributed loads applied to those structures. Based primarily on their efficiency, these geometric structural forms are employed in a number of industrial applications such as pressure vessels, containment structures, etc., where the principal function of the structure is to contain or sustain a particular loading environment. This selection is especially true when substantial loads such as high internal pressures are involved. The shell structures selected for these applications are simple forms, most frequently combinations of various shells of revolution such as cylinders, cones, spheres, and (for the pressure vessels) the torospherical or ellipsoidal shapes, because of their ability to respond to most loads by a membrane state over the major area of the shell. Edge effects are confined to a narrow zone near the edge or around a zone of discontinuity that may be present as a result of geometry changes. In addition, shell forms contain a number of functional, economic and aesthetic virtues that make them logical choices for applications to building structures. For repetitive structures and for those needing long, column-free spans, reinforced concrete shell roofs have often been chosen. Furthermore, these structures provide a clean inner surface, often in a pleasant geometric shape. They have good fire-resistance qualities. By proper orientation or shape selection, glass areas can be placed so that natural lighting can be directed onto all or nearly all the covered ground plan.

The state of the art for the analysis and design of hyperbolic parabolic (HP) shells is reviewed. The necessary expressions for determining the internal membrane stresses in the shell and the internal axial forces, shears, and bending moments in the groin arch due to surface dead and live loads on the shell are given. A special-purpose computer program GROINV, for the membrane analysis of HP groined vaults with a minimum of input data, is briefly described. The program is available from the author. Numerical results for several groined vaults with varying amounts of curved shell overhang are presented and their structural behavior with respect to concrete and reinforcing steel requirements is compared. Recent developments in the nonlinear finite element analysis of reinforced concrete shells that account for membrane and bending actions are reviewed, and results obtained for HP groined vaults loaded to ultimate failure using computer programs developed at Berkeley are briefly described with respect to their design implications.