Linear elastic analysis of a plane orthogonal rod system with different models of deformation of elements | Mekhanika | kompozitsionnykh | materialov i konstruktsii
> Volume 26 > №2 / 2020 / Pages: 157-173

### Linear elastic analysis of a plane orthogonal rod system with different models of deformation of elements

#### Abstract:

A linear theory of a plane regular elastic system formed of horizontal and vertical rods and ascending and descending inclined rods, rigidly connected to each other at the intersections of elastic lines of the rods of the first two families, is presented. The peculiarity of the system under study is the combination of different models of rod deformation. According to the assumption, all rods work for tension-compression, and horizontal rods are also endowed with the ability to perceive transverse bending loads. When constructing the theory, the gluing method is applied. By analyzing the behavior of isolated elements and the geometric conditions of their coupling, it is shown that the deformation of the system is described by nodal displacements and rotations, complete deformations and the initial internal force factors of the rods. All these dependent variables turned out to be functions of integer parameters used for numbering the elements of the system, and are related to each other by geometric and physical relations of the theory. The rest of its defining dependencies are derived from the Lagrange and Castigliano variational principles, which are based on the discrete analog of the calculus of variations, in which, unlike its classical version, functionals are formed by sums and depend on functions of discrete arguments. Static equations are identified from the Lagrange principle and the formulation of the discrete boundary value problem in nodal displacements and rotations is given. The general solution of static equations is presented up to three functions of integer parameters called force functions. In the constructed theory, they play the same role as the stress functions in the mechanics of elastic bodies. Using force functions, the compatibility equations for the complete deformations of the rods are derived from the Castigliano principle and the formulation of the discrete boundary value problem in the initial force factors and in the force functions is given.

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