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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

The approach to the problem of a variable thickness plate used in this paper is different from the usual approach in that this paper starts with general stress-strain relations and a generalized form of the position vector as used by Green and Zerna in "Theoretical Elasticity". They use R̅=L[ r̅ (θ₁,θ₂)+ λθ₃a̅₃(θ₁,θ₂)] where θ₁,θ₂, and θ₃ are curvilinear coordinates with θ₁ and θ₂ being the coordinates of the middle surface and λ=t/L being a constant for a plate of constant thickness t. This paper takes λ = λ(θ₁,θ₂) as a function of θ₁ and θ₂ so that the variable thickness may be taken into account. General tensor notation is used so as to work independent of coordinate systems. Making simplifying assumptions only when necessary, the equations of equilibrium and stress-strain relations are derived in terms of tensors connected with the middle surface as was done by Green and Zerna for a constant thickness plate. The additional terms obtained in these equations due to the variation in λ help us to evaluate the effects of the varying thickness. Expressions for stress are developed and they include the effects of transverse shear deformation and normal stress as well as the variation in thickness. These expressions are very much like those used by Essenburg and Naghdi in a paper presented at the Third U.S. National Congress of Applied Mechanics, June, 1958. However, they assumed the form for the stresses while the present paper arrived at their assumed forms with some additional terms after starting with general stress-strain relations. Using the notation of Green and Zerna, a set of nine equations involving the nine unknowns, m αβ, w, nαβ, and vα is derived and under appropriate boundary conditions, this set will yield a solution to the problem which will be better than the classical solution. Two problems are solved and numerical results are obtained and compared with the classical solutions. One of the problems involves a rectangular plate clamped on one edge with a uniform shear load on the other. The other problem involves a circular ring plate clamped on the outer edge with a uniform shear load on the inner edge. A much better correlation for the deflection of the middle surface is obtained for the rectangular than for the circular ring plate. The deflection at the inner edge of the ring plate obtained by the theory of this paper is over twice that obtained in the classical solution of the same problem. In the previously mentioned set of nine fundamental equations, we have the stress resultants nαβ and the deflections vα. With appropriate boundary conditions, these equations could lead to a study of in-plane forces and buckling of variable thickness plates, a field in which not much progress has been made. This paper does not include any numerical work in this direction. It is felt, however, that one of the principal contributions of this paper to the literature is that the set of nine fundamental equations includes the stress resultants in nαβ thus enabling us to study the effect of in-plane forces as well as that of transverse shear deformation, normal stress, and surface tractions. Ph. D.