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The paper outlines a method of analyzing layered anisotropic shells of revolution for stability using complex Fourier series. This simplifies the derivation of the basic equations compared with complete trigonometric Fourier series. Anisotropic shells in the form of a torus segment are analyzed for stability. This method allows optimizing the structure of the material and the geometry of the shell Keywords: stability, torus-like shells of revolution, Fourier series, complex numbers, anisotropy of material, external pressure, axial compression, torsionIntroduction. The stability of anisotropic cylindrical shells under loading of various types was analyzed in [8][9][10][11][12][13]. Stability problems were solved using the Donnell-Mushtari-Vlasov theory (DMV). The basic feature of the solutions is that the unknown functions are represented by series of a complete (in the range of the circumferential coordinate j p Î ( , ) 0 2 ) system of trigonometric functions. Such series are used to describe spiral buckling modes of anisotropic shells induced by coupled tensile, shear, bending, and torsional strains. In the case of isotropic and orthotropic shells, the bending and torsion problems are uncoupled. Therefore, anisotropy of shells complicates the analysis: the order of systems of ordinary differential equations is doubled after reducing the dimension of problems. This is also true of other problems for anisotropic shells [3].To approximate the deflection (W) and force (F) functions, we will use more compact notation of complex trigonometric series. We will analyze shallow anisotropic shells of revolution for stability. The equilibrium equations of such shells differ from those for cylindrical shells only by terms multiplied by 1 1 / R , where R 1 is the meridian radius [2, 5, 6]. Solving these equations, we obtain new mechanical results though the method is very similar to that outlined in [8][9][10][11]. However, the chief result of the present study is the conformation that complex series can be used to reduce the number of governing equations.1. Problem Formulation. The shallow shells of revolution under consideration are generated by revolving a rather shallow arc around the axis of revolution z [1], the chord OO1 of the meridian is parallel to the axis of revolution (Fig. 1). The coordinate system x, y on the mid-surface of the shell is such that the x-axis is aligned with the meridian of this surface and the y-axis with the parallel circle. For surfaces of revolution, meridians and parallels are the principal lines of curvature, and

A solution is obtained to describe the stability and initial postbuckling behavior of cylindrical shells made of composites with one plane of symmetry. The solution is based on the Donnell-Mushtari-Vlasov nonlinear theory of anisotropic shells and Koiter's theory of buckling and postbuckling behavior. Calculated results are presented for boron plastic shells with reinforcement of different types under external pressure. It is shown that the conventional model of a composite as an orthotropic material is erroneous for many types of reinforcement Keywords: stability, initial postbuckling behavior, composite material, cylindrical shell, different types of reinforcement, external pressureIntroduction. Most studies on the stability of shells made of composite materials modeled a composite as an orthotropic material [2,3,5]. This model is justified for the most popular shell making techniques; however, there are cases where it was not proved to be applicable [4]. One example is a fiber-reinforced composite layer that, being orthotropic within its own layers, deforms as an anisotropic body with one symmetry plane when the reinforcement directions are not aligned with the coordinate axes of the shell. Analytic and numerical methods of stability analysis of shells with a similar kind of anisotropy were developed in [6][7][8][15][16][17][18][19]. Note that all results on the initial postcritical behavior of composite shells apply only to orthotropic materials [3].In what follows, we will study the relationship of the initial postcritical behavior and sensitivity of shells to geometrical imperfections and deviations of composites from orthotropy. To this end, we will use the above-mentioned technique, which is based on the Donnell-Mushtari-Vlasov (DMV) theory of anisotropic shells [1] and Budiansky's version [12] of Koiter's asymptotic theory of buckling and postbuckling behavior.1. DMV Equations. Consider shells of constant thickness t consisting of N layers, also of constant thickness t k . Each layer is an anisotropic body with one symmetry plane coinciding with the mid-surface. The elastic relations for laminated anisotropic shells can be written in the following form accurate to an extent sufficient for the engineering theory of shells:

The paper presents an analytical solution describing the stability and postbuckling behavior of a cylindrical shell made of an anisotropic material with one plane of symmetry and subjected to torques at the ends. The solution is found using Koiter's buckling theory and the Donnell-Mushtari-Vlasov theory of anisotropic shells. The force and deflection functions are approximated by trigonometric series that satisfy hinged boundary conditions. The system of algebraic equations to which the problem is reduced at the main stage of solution is analyzed. Specific results on stability and sensitivity to imperfections of boron-plastic shells consisting of layers with different reinforcement directions are obtained Keywords: cylindrical shells, stability, postbuckling behavior, anisotropic materials, one plane of elastic symmetry, torsion, Koiter's buckling theory, Donnell-Mushtari-Vlasov shell theory Introduction.Recently, there has been much attention focused on the stability and postbuckling behavior of composite cylindrical shells subject to torsion [6,[11][12][13][14][15][16][17][18][19][20]. Traditional design models are improved, profound experimental research is conducted, and attempts are made to refuse formulations based on previous experience [4,6,11,12]. A very important issue for cylindrical shells made of fibrous composites is the applicability of an orthotropic model in the case of many types of layups [15,16,19]. Anisotropic shells, unlike isotropic and orthotropic ones, under torsion have two critical loads depending on the torque direction. Since this direction is constant in some structures, the load-bearing capacity of the shell can significantly be increased by choosing an appropriate layup. If the torsion direction is unknown, then it is necessary to be guided by the lower of the two critical loads and to tend to choose the number and orientation of layers so as to minimize the difference among them.The purpose of the present study is to develop a method to analyze the stability and initial postbuckling behavior of anisotropic shells subject to end torques. The solution will be represented in the form of trigonometric series as in similar problems for isotropic shells [1][2][3]. The method is based on the Donnell-Mushtari-Vlasov (DMV) theory of anisotropic shells [5,7] and Budiansky modification of Koiter's asymptotic theory of buckling [1,2,8,10].1. DMV Equations for Anisotropic Shells. Consider shells of constant thickness t consisting of N layers of thickness t k each. Each layer is an anisotropic body with one symmetry plane coinciding with its midsurface. The elastic relations for layered anisotropic shells can be written with accuracy sufficient for the engineering theory of shells [1]: