A solution of vibration and stability problem of an axially loaded cylindrical shell with a four lobed cross section of variable thickness

Based on the thin-shell theory and using the computational transfer matrix approach and the Romberg integration method, the free vibration and elastic buckling behaviors of a cylindrical shell with a four lobed cross section of circumferential variable thickness subjected to axially compressive loads is presented. Modal displacements of the shell can be described by trigonometric functions and Fourier s approach is used to separate the variables. The governing equations of the shell are formulated in terms of eight first-order differential equations in the circumferential coordinate, and by using the transfer matrix of the shell, these equations are written in a matrix differential equation. The transfer matrix is derived from the differential equations of the cylindrical shells by introducing the trigonometric functions in the longitudinal direction and applying a numerical integration in the circumferential direction. The natural frequencies and critical loads beside the mode shapes are calculated numerically in terms of the transfer matrix elements for the symmetric and antisymmetric vibration-modes. The influences of the thickness variation of cross-section and radius variation at lobed corners of the shell on the natural frequencies, mode shapes and critical loads are illustrated.