Thermoelasticity of a plane regular truss with orthogonal structure

Linear thermoelasticity is studied of a plane regular truss formed by four families of straight homogeneous rods. Rods of two of them are mutually orthogonal and form rectangular cells that are repeated in two perpendicular directions. The other two families combine differently oriented diagonal rods of the cells. All the rods are only in tension-compression and their elastic lines belong to the same plane. Adjacent rods are rigidly connected together in the truss nodes, i.e. the points of intersection of the elastic lines of mutually orthogonal rods. The regularity of the truss implies the invariance of the thermoelastic and geometrical parameters of the rods within the same family. External actions on the truss belong to its plane and generally include the external nodal forces, the linear axial forces of the rods and their non-uniform heating. The rigorous linear thermoelasticity of the truss is constructed by the gluing method. According to this method, the truss was split into rods and nodes, i.e. the elements of the truss. The given external actions were applied to the isolated elements as well as the forces of interaction with their neighbors. Then the analytical study of thermoelastic rods with geometric conditions of the conjugation of the adjacent elements and the analysis of the equilibrium of the nodes were carried out. The theory was formulated in terms of nodal displacements and total elongations and internal initial forces of the rods. All these variables are the functions of two integer arguments used for numbering of the elements of the truss. The complete closed system of equations of the truss thermoelasticity is represented by geometric and physical relationships, the equilibrium equations of the nodes and the compatibility equations of the total elongations of the rods. Alternative formulations of the discrete boundary-value problems are presented with their help. An application of the theory is illustrated by the exact analytical solution of the problem of thermoelastic deformation of the truss without internal nodes.