LINEAR THEORY OF A PLANE ORTHOGONAL LATTICE

The linear discrete-continuous analysis of a plane lattice is presented. The lattice consists of two mutually orthogonal sets of rectilinear parallel rods which are spatially deformable. The elastic axes of these rods lie in a common plane. The rods of each set are identical and homogeneous. One of the principal axes of the cross section of each rod lies in the plane of the lattice. The rods are rigidly attached to each other at the intersections of their elastic lines, i.e., at the nodes of the lattice. With these assumptions, the problem of the spatial deformation of the lattice splits into two independent problems, which are mathematically similar. One problem is connected with the deformation of the lattice in its plane, whereas the other problem with the lateral bending of the lattice. Using the sewing method [1, 2], we reduce each problem to the corresponding rigorous discrete theory. In these theories, the complete closed system of constitutive partial difference equations is expressed in terms of the generalized nodal displacements, total strains, and initial internal forces in the rods. This system consists of the geometrical and physical relations, equations of equilibrium of the nodes, and equations of compatibility of the total strains of the rods. Within the framework of the theories constructed, two alternative statements of the problems are presented-in terms of the generalized nodal displacements and in terms of the initial internal forces in the rods. The latter statement is illustrated by examples. The theories presented are discrete analogs of the plane stress state theory and the theory of bending of plates and follow from the moment theory of elasticity.