Buckling Analysis of Hydrostatically Loaded Conical Shells by the Collocation Method

Abstract

A collocation method is proposed as an alternative to the conventional methods of solving buckling problems. The method is applied to hitherto unsolved buckling problems. Those are elastically supported conical shells of given geometry subjected to hydrostatic pressure. Five possible elastic supports are assumed and the variations of the critical load due to change of the supports' stiffnesses are determined. The suitability of the collocation method for buckling problems analysis is displayed in its ability to satisfy complicated boundary conditions and range differential equations with continuous derivatives. COMMON method for solving differential equations of a shell is the well-known Galerkin procedure. This method constructs a well-conditioned stability matrix, secures good con- vergence and enables to cross check at some intermediate stages of the computations. Similar in nature are the Rayleigh-Ritz and the least squares methods. The use of the Galerkin pro- cedure is usually subject to the condition that the assumed displacement functions fulfill the geometric and the natural boundary conditions. However, several hundreds of integrals are involved in the stage of preparation for computer processing. Another problem arises when the critical load (eigenvalue) appears at the boundary conditions. Other methods, such as the known finite difference and the finite element methods, have their own shortcomings regarding convergence and fulfillment of complicated boundary conditions. A considerable amount of work is now devoted in the application of these methods to more complicated buckling cases. The Collocation Method overcomes the difficulties en- countered in the satisfaction of the boundary conditions and in the preparatory work. However, two major problems inherent in the method, namely the determination of collocation points and the ill behavior of the stability matrix, prevented its wide application. These difficulties are treated and overcome in the present work. For more details and references on the various numerical methods see Ref. 1. The effect of the boundary conditions on the buckling behavior has been a subject of many investigations following the initial ones of Ohira2 for circular cylindrical shells and Baruch et al. 3 for conical shells. (See also Refs. 1 and 3.) Very little has been done in analyzing shells with elastic boundary conditions. The importance of solutions which satisfy elastic boundary conditions lies in two major areas: a) in tests, where the actual boundary conditions never behave exactly as the analytically defined boundary conditions. Solutions with elastic boundaries may enable to obtain a better correlation between analysis and tests, b) In the field of shell design, one may find many instances in which several shells are connected together whereby the edge of one shell becomes the elastic boundary for the other shell. The practical cases always include shells' boundaries which should be defined as elastic. A purely analytical solution for the buckling of a semifmite cylindrical shell, axially compressed, with three types of uncoupled elastic supports (kNx, kNxtj>, /cMv), has been presented by Kobayashi. 4 Although this work deals with a specialized case of the cylindrical shell and does not include the cases of radial elastic constraints, it points out characteristic phenomena in the buckling behavior of the cylindrical shell with elastic supports. Another work by Cohen 5 presents a solution for a stiffened conical shell under hydrostatic pressure with elastic rings at its edges. This type of support, although a practical one, does not represent the pure case of uncoupled elastic supports, and, this restricts one to foresee the influence of each type of displacement on the buckling load. Nevertheless, the results drawn in the work of Cohen5 are important and check with the results of the present work. Other works which deal with the special case of buckling of various lengths of cylinders with both edges free (this boundary condition is assigned in this work as SW1) were published by Nachbar and Hoff6 and Hoff and Soong.7 Their results are similar in nature to those obtained here for conical shells.