The problem of plane bending a direct composite beam of arbitrary cross-section and the prerequisites for its approximate analytical solution

The approach to the reduction of the spatial problem of plane bending a composite discrete-inhomogeneous beam of arbitrary cross-section to the approximate two-dimensional bending problem of the equivalent multilayer beam has been discussed here. The result is represented in the form of relations for determination the characteristics of the equivalent multilayer structure by physical and mechanical materials characteristics of the original beam’s phases and the system of static, geometric and physical relations of the corresponding two-dimensional problem. The obtained equations are similar to the plane problem equations of the elasticity theory, but instead of stresses, they contain internal efforts consolidated to the main plane of the beam. The equations of the approximate two-dimensional problem were used to solve the problem of static bending a composite console of arbitrary structure with a load on the free end, taking into account the uniform change of the temperature field. The given system of equations and relations is the starting point for the construction of non-classical deformation models and solving a wide range of problems concerning the deformation of a direct composite beams.


Introduction
Beams of discrete-inhomogeneous, composite structure are increasingly used in various fields of mechanical engineering and construction. The research of mechanics deformation of such elements contributes to the development of methods for their design, which is a necessary condition for effective practical implementation of composites. However, the construction of theoretical models of composite beams is complicated by anisotropy of mechanical properties of their material, heterogeneity of the structure, significant susceptibility to transverse displacements, uneven thermal expansion of composite components, even with a uniform temperature field, etc.
For most traditional structural materials, the model of a linear-elastic body is acceptable when solving problems of mechanics. In [1,2] it was experimentally established that carbon plastics, even at elevated temperatures, shows an almost linear relationship between stresses and strains before failure. Despite the significant physical nonlinearity of the reinforced concrete matrix material [3], for such composites, with some caveats, it is also permissible to assume the linear-elastic work of the components. It has become topical to develop deformation models of composite elements based on the linear theory of elasticity.
The exact solutions of spatial deformation problems of composite beams have been obtained only for simple shapes and cross-sectional structures (circle, annular multilayer structure) [4][5][6][7] and in cases of loading (tension-compression, pure bend).
A lot of works  have suggested exact solutions of plane problems of elasticity theory concerning composite beams, directly or indirectly. Such solutions preferably allow one of the types of external loads to be taken into account. There are also works devoted to bending by force and moment at the end [8][9][10][11][12], uniformly distributed normal load on the longitudinal faces [8,[13][14][15][16][17][18][19][20][21], linearly distributed load [22][23][24][25], load distributed by the law of sinusoids [26,27]. Several works consider more complex types and combinations of loads. In works [28][29][30][31], the load is considered as the sum of  [32,33,35] it is studied as trigonometric series, and in [34] it is discussed as their combinations. Most of these works are devoted to composite homogeneous or continuously inhomogeneous beams with different types of elastic materials symmetry. Only in some works [9,10,12,17,21,25,35] multilayer beams are presented. Plane solutions for composite beams are relatively simple and give precise distributed components of the stress-strain state. They can be directly used only for the calculation of beams with a rectangular cross-section. Their structure is formed by a package of flat layers perpendicular to the force plane. However, in practice, composite beams have a more complex shape and cross-sections structure.
Non-classical models of bending a composite beams, for example [36,37,38,39], are also constructed as approximate plane problem solutions of the elasticity theory. Іn some works, while constructing such models, there is a possibility to take into account the arbitrary structure of the crosssections. In particular, the construction of an iterative model in [38] is carried out by introducing static and kinematic hypotheses at the level of the spatial problem. Next, integrating by the width of the section, the three-dimensional equations of the elasticity theory are reduced to generalized twodimensional ones, but without taking into account the susceptibility of beam materials to transverse compression. The combination of such an approach with methods for constructing exact solutions of plane problems of the elasticity theory for multilayer beams is quite promising. This will allow to obtain practical ratios for determining stress-strain state when bending composite beams with an arbitrary cross-sectional structure.

Materials, hypotheses and research methods
Consider a direct composite beam with a constant structure and cross-sectional dimensions relative to length -see figure 1. The beam consists of m discrete continuous single-connected or multiconnected phases 1 2 3 , , , , , km P P P P P , which include the matrix, reinforcement, adhesive layers and other elements of the composite, made of appropriate different materials ( figure 1, a). The phases of the beam are rigidly connected on common surfaces, there is no relative displacement and separation. The beam has a longitudinal plane of symmetry, relative to which its cross-sections are symmetrical in shape and structure.
The beam is under the action of external loads distributed on its longitudinal faces with a width of Despite this conditionality, we assume that the load system is balanced. The material of the composite beam's phases can be homogeneous or continuously inhomogeneous (functionally gradient) in the plane of the cross section. It can also be isotropic or orthotropic with planes of elastic symmetry parallel to the coordinate planes of the coordinate system xyz . Physicomechanical characteristics of the materials of the beam phases are known: For the whole beam, the characteristics of the inhomogeneous material will be piecewise continuous functions with discontinuities of the first kind at the boundaries of the phases, which can be conditionally represented in the following form: For the analytical description of the characteristic functions (2), the Heaviside function can be used. For example, for an arbitrary phase of a composite beam, which is limited in cross-section on the sides by continuous curves   For a beam with an arbitrary cross-sectional structure, it is always possible to divide it in height by k z horizontal lines into a number of generalized layers. Within these layers, the characteristic functions of the phases can be represented as (2). This gives a generalized approach to the analytical representation of the mechanical characteristics functions of an inhomogeneous beam (3). Due to the accepted condition of absolutely rigid contact of phases, the material of the considered beam is continuous. Accordingly, the classical equilibrium equations and geometric relations of the elasticity theory are valid for such an element: where ,, VVV x y z FFF -projections of bulk forces per unit mass of beam material;   -function of density distribution of beam materials; t -time. The physical dependences of Hooke's law also remain valid within individual phases. However, for all inhomogeneous beams, they must take into account the change in mechanical characteristics in its cross-section. This can be achieved by replacing the elastic constants in the classical equations with the distribution functions of physical and mechanical characteristics: where T  -change in temperature field inside the beam.
Given the received load (figure 1) static boundary conditions on the longitudinal faces and ends of the beam will be written as For a symmetrical section, the side curves of its contour are distinguished only by a sign: where   Given the absence of loads on the side surfaces, the static boundary conditions for them, taking into account (10), will be written as Equations (4),(5) and (6) constitute a complete system of equations of the elasticity theory spatial problem for the considered composite beam. Their analytical solution in general in accordance with the boundary conditions (7), (8), (11) encounters significant mathematical difficulties. However, by introducing some physical assumptions about the properties of phase materials, the dimension of such a problem can be reduced, which simplifies its research and analytical solution.

Reduction of the spatial problem of plane bending a composite discrete-inhomogeneous beam of arbitrary cross-section to the approximate two-dimensional problem
We consider the beam narrow enough to neglect the deformation of the composite's phases along the axis Oy . This is equivalent to the assumption of absolute stiffness of the materials of all composite beam's phases in the specified direction: , , , , 0, , , .
We also assume a uniform distribution of the temperature field along the axis Oy : Substituting (12) for the physical dependences (6), we have: Taking into account (14), the geometric relations (5) have the form: Solving the third, fifth and sixth relations (15), taking into account the assumption of plane deformation of the beam, we obtain: , .
Solving (14) with respect to stresses and integrating the width of the section (within , taking into account (13) and (16) we have: In equation (18) it is taken into account that for a composite beam with isotropic and orthotropic phases: EE x zx z xz       , as a result we have SS zx xz    . Solving (17) with respect to deformations and comparing with the corresponding expressions (14) we obtain the summary relations of Hooke's law for a composite beam: , .
and ratios for consolidated mechanical characteristics: ,.
We equate the right-hand sides of the corresponding relations (14), (20) and solve the obtained equations with respect to stresses ,, x z xz    . As a result, we obtain the dependences for the transition from the consolidated stresses ,, x z xz    to distributed across the width of the cross-section: Integrate the equilibrium equation (4) In the case where the load on the lateral cylindrical surfaces , , , That is, the loads on the lateral cylindrical surfaces will directly enter the system of consolidated static equations and will affect the stress distribution along the cross-section height.