ИСТИНА

Flat periodic frames are considered consisting of thin weightless elastic rods rigidly interconnected by nondeformable massive nodes. The problem of deforming such frameworks contains two small parameters. These are ratios of the thickness of the rod to its length and the length to the size of the frame. This fact makes it difficult to solve this problem numerically. The method for determining the averaged characteristics of the framework is proposed. It consists of the following steps: - the formulas known from the strength of materials are used, which relate the displacements and angles of rotation of the rod ends on the one hand and the forces and moments that occur in the rods on the other hand; - equilibrium conditions of an arbitrary node are written; - three continuous functions are introduced that coincide at the intersection points of the rods with displacements and angles of rotation of the nodes; - it is assumed that the functions introduced satisfy the equations that coincide at the nodal points with the equilibrium equations of the nodes; - the functions included in the system of equations are expanded into a Taylor series up to the second order terms with respect to a small parameter (the ratio of the thickness of the rod to the length); - the equations obtained describe the equilibrium of the averaged medium, the elastic moduli of which are the coefficients of the equations. The method is applied to several types of frameworks: with square, rectangular, triangular and rhombic periodicity cells.