Thin cylindrical shells under surface torques

Abstract

Introduction I a previous paper the bending problem of an infinitely long cylinder loaded with concentrated equal and opposite forces, acting at the ends of a vertical diameter, was discussed. The loading function in this case was presented by a Fourier integral in the longitudinal direction and by a Fourier series in circumferential direction. The integral representation has the advantage that the boundary conditions are automatically taken care of, and no subsequent determination of Fourier coefficients is necessary. The Fourier coefficients and the undetermined function in the Fourier integral are determined simply from the loading condition. The radial displacement was then obtained from the solution of the eighth-order differential equation. In the present paper, the shearing stress-resultant of an infinitely long cylindrical shell subjected to two equal and opposite torques acting about the radial axis on the surface of the shell, as shown in Fig. la, has been analyzed. The solution of this problem was achieved by replacing the torque with two equal and opposite forces acting about the same axis at an infinitely small distance apart. The moment produced by the pair of forces is of the same magnitude and sign as the torque. In the case of a plate, the shearing stresses produced by two equal and opposite forces acting perpendicularly to either axis are identical. However, in the present problem, the phenomenon is quite different because of the effect of the curvature of the shell on the displacements. For this reason, in the present investigation, a combination of two pairs of forces with moments are introduced to replace the applied torque. The forces are so arranged that one pair of forces, equal and opposite in magnitude, act at an infinitely small distance apart in the direction of x axis and the other pair of forces under the same condition in the direction of s axis. The loading distribution functions under consideration may be presented by a combination of a Fourier series and a Fourier integral along the circumference and the generatrix, respectively.