Finite Deflections and Buckling of Slightly Curved Beams and Shallow Spherical Shells under Lateral Loads. Part I. Buckling of Low Arches or Curved Beams of Small Curvature. Part II. A Nonlinear Theory of Bending and Buckling of Thin Elastic Shallow Spherical Shells

Abstract

This research consists in the experimental and theoretical investigation of the finite deflection and buckling of two similar structures; the low arch or slightly curved beam and the shallow spherical dome, both subjected to lateral loads. These structures are of interest because the large interaction between bending and axial forces causes their load-deflection behavior to become nonlinear at very low values of the deflection. Due to the wide difference in the methods of solution of these two problems they are separated into two parts, each having its own abstract. [Part I] When a low arch (a thin curved beam of small curvature) is subjected to a lateral loading acting toward the center of curvature, the axial thrust induced by the bending of the arch may cause the arch to buckle so that the curvature becomes suddenly reversed. The critical lateral loading depends on the dimensions and rigidity of the arch, the elasticity of the the end fixation, the type of load distribution, and the initial curvature of the arch. A general solution of the problem is given in this paper, using the classical buckling which is based on the stability with respect to infinitesimal displacements about the equilibrium positions. For a sinusoidal arch under sinusoidal loading, the critical load can be expressed exactly as a simple function of the beam dimension parameters. For other arch shapes and load distributions, approximate values of the critical load can be obtained by summing a few terms of a rapidly converging Fourier series. The effect of initial end thrust and axial and lateral elastic support are discussed. The buckling load based on the of Karman and Tsien is also calculated. The results for both the classical and the criteria are compared with experiments made on a series of centrally loaded, pin-ended arches. For larger values of a dimensionless parameter [gamma subscript l], which is proportional to the ratio of the arch rise to the arch thickness, the experimental critical buckling loads agreed quite well with the classical criterion, but, for smaller values of [gamma subscript l], the experimental critical loads were appreciably below those calculated from the classical criterion, although they were always above those obtained from the criterion. [Part II] The shallow spherical dome subjected to lateral pressure is a structure for which the deformation departs appreciably from the linear theory at very small values of the deflection amplitude. It is also. one for which the buckling process is characterized by a rapid decrease in the equilibrium load once the buckling load has been surpassed. For structures having this type of buckling characteristic, the question arises as to whether the proper buckling to apply is the classical criterion, which considers equilibrium with respect to infinitesimal displacements or the finite displacement energy criterion proposed by Tsien. In this paper the problem of the finite displacement and buckling of a shallow spherical dome is investigated both theoretically and experimentally. In the theoretical approach the nonlinear equations are converted into a sequence of linear equations by expanding all of the variables in powers of the center deflection and then equating the coefficients of equal powers. The basic parameter for the shallow dome is proportional to the ratio of the central height of the dome, h, to its thickness, t. For small values of this ratio the expansions converge rapidly and enough terms are computed to determine the buckling load. For higher values of h/t, convergence deteriorates rapidly and the buckling, load is not computed. However, even for these higher values of h/t the deflection shapes are determined for deflection amplitudes below the amplitude at which buckling occurs. These deflection shapes are characterized by their rapid change as h/t increases and by the fact that, over most of the range of h/t studied, the maximum deflection does not occur at the center of the dome. The experimental program was carried out on a series of clamped-edge, eight inch base diameter shells, subjected to uniform pressure. The deflection shapes and the buckling loads agreed quite well with the values computed theoretically. It was also found that there was no significant difference between the buckling loads observed using air pressure and those observed using oil pressure. Thus it is concluded that for the shallow domes studied the classical buckling holds rather than the energy criterion proposed by Tsien.