Classical Bending Theory Research Articles

Effect of Bending on Vibrations of Spherical Shells

This paper is concerned with the vibration analysis of spherical shells closed at one pole and open at the other by means of the linear classical bending theory of shells. Frequency equations are derived in terms of Legendre functions with complex indices, and for axisymmetric vibration the natural frequencies and mode shapes are deduced for opening angles ranging from a shallow to a closed spherical shell. It is found that for all opening angles the frequency spectrum consists of two coupled infinite sets of modes that can be labeled as bending (or flexural) and membrane modes. This distinction is made on the basis of the comparison of the strain energies due to bending and stretching of each mode. It is also found that the membrane modes are practically independent of thickness whereas the bending modes vary with thickness. When the opening angle is large and the shell may be considered shallow, only then are the modes distinguished by predominantly large transverse or longitudinal displacements, which were investigated recently for shallow spherical shells by Kalnins [in Proc. 4th U. S. Congr. Appl. Mech. (1963)]. Previous analyses of axisymmetric vibration of spherical shells by Lamb, Love, and more recently by Baker [_J. Acoust. Soc. Am. 33, 1749 (1961)] with the use of membrane theory have shown that one of the two infinite sets of modes is spaced within a finite interval of the frequency spectrum. It is shown that this set of modes is a degenerate case of bending modes, and, if deduced by means of membrane theory, it is applicable only when the thickness of the shell is zero. When the bending theory is employed, then the frequency interval for this set of modes extends to infinity for every value of thickness that is greater than zero. [This work has been supported by a grant from the National Science Foundation.]

On Vibrations of Elastic Spherical Shells

This investigation is concerned with axisymmetric as well as asymmetric vibrations of thin elastic spherical shells. First, with the limitation to torsionless axisymmetric motion, the basic equations for spherical shells of the classical bending theory of Love’s first approximation are reduced to a system of two coupled differential equations in normal displacement of the middle surface and a stress function; this system of equations is applied to free vibrations of a hemispherical shell with a free edge and numerical results are obtained for the lowest natural frequency as a function of the thickness of the shell. The remainder of the paper is, in the main, devoted to a study of asymmetric vibrations of a hemispherical shell with a free edge according to the extensional theory. Numerical results for natural frequencies (of the four lowest circumferential wave numbers) and mode shapes are given and the results are compared with the prediction of Rayleigh’s inextensional theory.

The Additional Deflection of a Cantilever Due to the Elasticity of the Support

A beam is cantilevered from a semi-infinite solid with neutral axis normal to its bounding plane (two-dimensional problem). Deformation of the supporting solid allows the beam to rotate about the built-in end, producing a deflection in addition to that caused by bending and shear in the beam itself. Results of tests designed to measure this additional deflection are presented and compared with various theoretical values. Calculated deflections, based on classical bending and shear-deflection theory, are also compared with measured deflections. An experimentally verified expression for the additional deflection of a cantilever due to the elasticity of the support is given.