A numerical-analytic approach is used to study the stress-strain state of orthotropic toroidal shells of variable thickness. The problem is solved with the use of a nonclassical Timoshenko-type shell theory based on the model of a rectilinear element. The system of partial differential equations is reduced to a one-dimensional problem by applying the method of spline approximation in one coordinate direction. The boundary-value problem for the system of ordinary differential equations is solved by the stable numerical method of discrete orthogonalization. We also present the data on the distribution of the fields of displacements and stresses depending on the curvature of the axis of the shell and the parameter of variability of its thickness.With toughening of the requirements to contemporary structures, the shells of complex shapes with variable parameters produced of composite materials are more and more extensively applied as structural elements [5,9,13,14,18,19]. Among these shells, one especially can mention toroidal shells of variable stiffness made of orthotropic materials [2,11]. They are used in various branches of the contemporary engineering, including machine building, rocket and spacecraft engineering, etc. Note that shells of variable thickness are now extensively used parallel with the shells of constant thickness. The investigations of the characteristics of strength of these shells encounter significant difficulties of the mathematical and computational character caused by the complexity of the system of partial differential equations and the corresponding boundary conditions. Thus, it is necessary to perform calculations of the stress-strain state of shells of this kind on the basis of an improved of Timoshenko-type shell theory [4,8,15].In the present work, for the investigation of the influence of variable thickness of a toroidal shell and the curvature of its axis on its stress-strain state, we apply an approach consisting of two stages:(1) application of the method of spline approximation along a generatrix to reduce the analyzed twodimensional problem to a one-dimensional problem [1,6,7,9,10];(2) solution of the obtained one-dimensional boundary-value problem by the stable numerical method of discrete orthogonalization [3,5,9,16,17].

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