Analiza stateczności promieniowo ściskanych powłok walcowych metodą uogólnionych szeregów potęgowych
Stability analysis of radially compressed cylindrical shells, by means of the method of generalized power series
Formulae given in the literature express usually the critical pressure for a shell in function of not only the geometrical and material characteristics but also the number of circumferential and axial semiwaves, m and n, that are formed during buckling. For actual computation, it is necessary to select these parameters in such a manner that the critical pressure is minimum. It is recommended to do this by numerical or graphical methods.
In the present paper is proposed a method for a sufficiently accurate analytic determination of m and n. It is discussed in detail by means of an example of radially compressed circular cylindrical shells. A few formulae expressing the critical pressure are confronted, the equation (2.13) being adopted for analysis. This equation differs insignificantly from the formula of K. Girkmann (2.3). It is denoted, for simplicity m2 = U, n2 = V. The dimensionless quantities s and λ are determined by Eqs. (2.1) and (2.2).
The derivative of the critical pressure with respect to V is positive, therefore the least possible value of V should be assumed for computation. It is determined by Eq. (3.1). Setting equal to zero the derivative with respect to U, we obtain a fifth-order algebraic equation (3.2), which is solved by means of expansion in series of a small parameter for which 1/ λ is assumed. Formulae are obtained expressing the required values U and s in the form of generalized power series (3.6) and (3.7). However, the quantity U cannot be less than 4. If U < 4 is obtained from the equation (3.6), we must assume U = 4, which leads to the formula (3.11) (long shells).
In Sec. 4, by applying operations on generalized power series [22], equations are derived for the necessary wall-thickness for a prescribed critical pressure. In Sec. 5 distinction is made between shells of great and medium length in the case in which the critical pressure is sought-for in function of the dimensions of the shell and also for the purpose of shell design for a prescribed critical pressure. Also the limitations are discussed imposed by the assumption of elastic buckling and a thin-walled shell.
Figs. 1 and 2 summarize the results. A numerical example illustrates the convergence of the series and the approximation errors. These errors are, as a rule, less than one percent.
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