Linear elastic analysis of spatial thin-walled rod system with orthogonal structure | Mekhanika | kompozitsionnykh | materialov i konstruktsii
> Volume 25 > №1 / 2019 / Pages: 122-139

Linear elastic analysis of spatial thin-walled rod system with orthogonal structure


A linear elastic analysis of a regular spatial thin-walled rod system of orthogonal structure is presented. The system is formed from three mutually orthogonal families of straight homogeneous rods and thin rectangular plates of constant thickness located between them. According to the assumption, the rods work only for tension-compression and in the plates a state of uniform pure shear is realizable. For the elastic analysis of such systems, a strict discrete linear theory of elasticity constructed using the gluing method is proposed. In accordance with its procedure, the thin-walled rod system was divided into elements (rods, plates and nodes – intersections of elastic rod lines). The given external and unknown internal forces were applied to them and a linear analysis of the mechanical behavior of isolated elements was carried out taking into account the geometric conditions of their conjugation. The theory is formulated in terms of nodal displacements, the generalized strains (full rod elongations and shifts of the plates and their rod frames) and the generalized internal forces (initial rod forces and tangent force flows). All these variables are functions of integer parameters used for numbering system elements. The complete closed system of defining relations of the theory consists of geometrical and physical dependences, static relations and equations of compatibility of generalized deformations. Geometric relationships express generalized deformations through nodal displacements, and physical relationships represent a linear dependency between generalized internal forces and deformations. The role of static relations, establishing a connection between the given external and the unknown internal forces are the equilibrium equations of free nodes. With the help of all these dependences, two statements of discrete boundary value problems are given: one in nodal displacements and shifts of plates, another in generalized internal forces. The latter formulation is illustrated by the example of an arbitrarily loaded one-closed caisson of any finite length for which an exact analytical solution in Chebyshev polynomials is constructed.