Finite pure bending of circular cylindrical tubes

Abstract

1. Introduction. The present paper is concerned with the determination of stresses, deformations and stiffness of originally straight circular tubes in pure bending. The non-linear problem of determining the stiffness of such a tube as a function of the applied moment and the determination of a critical moment for which flattening instability-occurs has originally been discussed by Brazier [1], An alternate more precise formulation of the problem of flattening instability of circular cross-section tubes is contained in a recent paper by one of the present authors [2], as a special case of results for pure bending of general cylindrical tubes. In this same paper approximate solutions of the non-linear differential equations of the problem were obtained as expansions in powers of a dimensionless parameter a. It was found that the first terms of these expansions give the results of linear theory and that consideration of two terms gave the results of Brazier [1], It was further found that consideration of three terms lead to results which differed from Brazier's to the order of ten per cent. •Since the calculation of additional terms in the a-series becomes progressively more complicated, an alternate determination of the results is of interest. The present paper presents such an alternate determination, involving the iterative solution of a system of two simultaneous non-linear integral equations. In addition to this, the previous three-term a-series are extended by the calculation of fourth terms. Our calculations lead to the noteworthy conclusion that Brazier's results for flattening instability are quite close to the results of precise calculations based on the equations given in [2], in the sense that consideration of three and even four terms in the a-series lead to results which are further from the correct results (in the critical a-range) than the results based on only two terms in the a-series. In addition to these conclusions for the problem of the flattening instability, we obtain in what follows quantitative results for the non-linear behavior of stresses and deformations in the tube. We find, in particular, that when the applied bending moment is of the order of the critical moment, the order of magnitude of the secondary circum-ferential wall bending stresses-associated with the flattening of the cross-section-is the same as the order of magnitude of the primary longitudinal direct fiber stresses in