The stresses and displacements are determined in an infinite circular cylindrical shell reinforced at its midlength by a single frame. Acting at the ends of a diameter of the ring are two self-equilibrating radial forces and moments. Solutions are obtained using the Donnell, the Love-Reissner, and the Fliigge equations. Numerical results indicate that the three solutions are in excellent agreement. Stress distributions and displacements in the shell are presented along three generators and along a circumference. They indicate high stresses in the neighborhood of the ring. The variation of the axial bending stress in the shell with the eccentricity between the shell median surface and the ring centerline is studied for the case of the radial forces. The results indicate that for an inner ring the axial bending stress is tensile at the inner surface of the shell, and for an outer ring this stress is compressive. Furthermore, for the case of the outer ring the maximum deflection of the shell does not occur at the ring.
IntroductionE NGINEERS designing submarines, missiles, aircraft, pressure vessels, etc. use reinforced shells so extensively that investigations in this area are of continuing interest. In this connection, the stress analysis problem of a circular cylindrical shell reinforced by rings, one or more of which is externally loaded, is important in design considerations. In this paper the authors solve the closely related problem of an infinitely long, circular cylindrical shell reinforced at its midlength by a single uniform ring. The Donnell, Fliigge, and Love-Reissner shell theories are used in the analysis.General analyses of the Donnell equations which are applicable to the present work have been given previously. 1 ' 2 In the present work solutions of the same form are obtained for the Fltigge equations 3 and the Love-Reissner equations. 4 Since all quantities must be periodic in the circumferential direction, it is convenient to use corresponding Fourier series expansions. Thus, the radial and circumferential ring-shell interaction loads are represented by such series with unknown amplitudes a n and b n . The homogeneous equations of equilibrium are used to determine the shell solutions with the line loadings appearing in the boundary conditions. This analysis yields the displacements as homogeneous functions of a n and b n . Similarly, the ring displacements are determined as functions of a n , b n , and the external loads.Once the analyses of the ring and shell problems have each been completed, it is necessary to impose the kinematical conditions at the line of contact that the radial and circumferential displacements of the ring and shell, respectively, are equal. These conditions yield an infinite number of pairs of simultaneous equations for the interaction load harmonics a n and b n . After these quantities have been computed, the calculations can be continued for the displacement and stresses in the ring and shell.Since the mean radius of the ring and shell are, in general, not the same, an ...
