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The problems of synthesis of laminated bodies is a promising line of investigation in the field of structural optimization. These problems have been studied in a number of papers [1][2][3][4][5] concerning the questions of design of heat-shielding panels, multilayer wave filters, and elastic laminated bodies. The composition of a structure and its geometric dimensions are chosen as control parameters in the problems of synthesis of laminated structures. A control parameter that characterizes a laminated-body structure is a stepwise function with a discrete range of values. Therefore, in deducing the necessary optimum conditions and in constructing a numerical algorithm, one should use the methods of the optimum-control theory. The structure and sizes of the laminated structure are determined in the process of optimization, although the number, sizes, and materials of the layers are not known beforehand.In the present paper, we consider the problem of synthesis of a multilayer curvilinear beam of minimal weight, which is bent under a uniformly distributed load, from a finite set of elastic homogeneous orthotropic and isotropic materials under given constraints on the beam strength and thickness. The necessary optimum conditions are obtained, a computational algorithm is built, and an example of calculation of the optimum beam is given.1. Formulation of the Problem. Let a set W consist of k homogeneous orthotropic and isotropic materials. It is required to synthesize a laminated curvilinear beam of minimal weight from the given set.Let rl and r2 be the radii of the inner and outer surfaces of the curvilinear beam (see Fig. 1) which is hinge-supported at the ends and loaded by an external pressure q [6]. We shall use the common center of the circumferences that constrain the beam as the origin of coordinates and the axis of symmetry as the polar r axis. The support reactions form equal angles/~ with the axis of symmetry. Let us denote the angle between the end cross sections of the beam by 2 ~o. By symmetry of the problem, we can consider half of the beam. In the case of a plane stress state, the stress-strain state of the multilayer curvilinear beam is described in the polar coordinate system (r, 0) by the boundary-value problem including the equations of equilibrium

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