This research presents an analytical investigation of the buckling of eccentrically stiffened orthotropic cylinders and includes the influence of prebuckling deformations. Nonlinear equilibrium equations and boundary conditions are derived by using energy principles. The stiffened cylinder consists of a cylindrical shell made of a homogeneous orthotropic material with eccentric stiffeners on its surface. The rings, or circumferential stiffeners, are considered to be located discretely on circumferential lines along the length of the cylinder and the stringers, or longitudinal stiffeners, are considered to be closely spaced so that their properties can be averaged (smeared out) over the stringer spacing. The stiffeners are considered to be beam elements, to be equally spaced, and to have the stiffener twisting accounted for in an approximate manner. Non-linear Donnell type strain-displacement relations for the shell and the stiffeners are defined and the strain energy of the stiffener-cylinder system is formulated. The governing nonlinear equilibrium equations and boundary conditions are then obtained by the principle of minimum potential energy and the fundamental lemma of calculus of variations. The discrete ring terms are included in the nonlinear equilibrium equations by use of a Dirac delta function. By a perturbation of the nonlinear equilibrium equations and boundary conditions, a set of nonlinear prebuckling equations and boundary conditions and linear buckling equations and boundary conditions are obtained which govern the prebuckling deformations and stresses and buckling of a stiffened orthotropic cylinder with discrete rings. Solutions of the prebuckling and buckling equations are obtained for classical simple support boundary conditions and for loadings of axial compression, lateral pressure, and combinations of axial compression and external or internal pressure. The solutions are obtained by the method of finite differences in which the governing equations and boundary conditions are changed to a system of second order differential equations which are then written in terms of finite differences at stations along the length of the cylinder. The difference equations are formulated in terms of a matrix equation which is solved by a modified G~ussian elimination technique. Solutions of the prebuckling and buckling equations for the case where the rings are considered to be smeared out are presented for comparison with the discrete case. A Galerkin solution of the buckling equations for discrete rings assuming classical prebuckling deformations is also presented in the Appendix. Computed results for two types of contemporary stiffened cylinders are presented in order to study and illustrate the importance of prebuckling deformations, discrete rings, and eccentrically applied compressive loads. The results show that the predicted buckling loads for stiffened cylinders may be substantially affected by using an analysis which takes into account prebuckling deformations. / Doctor of Philosophy