UDC 539.3 Methods for calculating the buckling loads of shells with geometrical imperfections have been developed by many authors [2, 13-15]. A detailed survey of this activity and a relevant bibliography can be found in several review articles and books [2, 3, 7, 8]. In the present article we are concerned primarily with methods pioneered by Koiter [14, 15]. A "special theory" is proposed in [14] for determining the maximum loads or bifurcation loads of axially compressed, cylindrical shells in the presence of initial axisymmetric bending deflections. Koiter has also developed a general theory of buckling and initial postbuckling behavior [15] for investigating the influence of nonaxisynunetric initial imperfections on buckling loads. The theory is based on an asymptotic analysis of the nonlinear equilibrium equations in the vicinity of bifurcation loading of an ideal shell. The general theory is applicable both in the case of an isolated minimum eigenvalue ~ of the linearized problem and in the case when the load ~'c corresponds to a set of buckling (loss of stability) mode shapes. Byskov and Hutchinson [11] have generalized Koiter's asymptotic method to a spectrum of very closely spaced eigenvalnes X. The method of [11] has been used by the present authors and by Manevich [5] to investigate the interaction of local and general buckling shapes of stringer-stiffened cylindrical shells. When shells of composite materials are optimized, a situation also arises where the eigenvalues of the linearized problem coincide or are close together. A procedure for calculating the interaction between buckling shapes of composite sbells is discussed in [6]. A distinguishing feature of the Byskov--Hutchinson method [I I] is the fact that it can be used not only to find the maximum loads for imperfect shells, but also to determine the bifurcation loads solely in the presence of certain imperfections corresponding to shapes whose interaction is taken into account. In particular, the branching off of a nonaxisymmetric solution is possible under axisymmetric initial bending. It is instructive to compare the results obtained in this case with the buckling loads calculated by Koiter's special theory. The latter, in contrast with the asymptotic procedure in [11], can be regarded as exact. The approximate nature of the the Byskov--Hutchinson procedure [11] and its error limits have also been investigated in [12]. In addition to comparing the remits of the calculations with Koiter's special theory [14] and the Byskov--Hutchinson procedure [11], we propose to analyze certain details of the interaction of axisymmetric and nonaxisymmetric imperfections and the influence of the mechanical properties of composite materials on the sensitivity of shells to imperfections. 1. We use a previously described [1] modification of Timoshenko's nonlinear theory. In the same paper, a procedure is given for calculating the buckling of composite cylindrical shells with axisymmetric initial bending deflections according to Koiter's special theory [14]. The application of Byskov--Hutchinson asymptotic analysis [11] to the basic nonlinear equations makes it possible to obtain a system of nonlinear algebraic equations in the amplitudes ~i of the buckling modes with the interaction between such modes taken into account: