A method is developed for studying the dynamic deformation of structurally inhomogeneous beams consisting of homogeneous isotropic layers with different mechanical characteristics. The method is based on the virtual-displacement principle. The equation of motion is derived in vector and scalar forms for arbitrary loads, boundary conditions, and cross-sections with one and two axes of symmetry. The efficiency of the method is demonstrated by solving, as an example, the dynamic deformation problem for a hinged layered beam with a rectangular cross-section under harmonic loading. Mechanical effects are revealed, which describe the influence of the beam structure and the mechanical properties of beam components on the dynamic compliance in comparison with the relevant homogeneous beam with the same geometry Keywords: structurally inhomogeneous beam, analytical method, dynamic deformation, virtual-displacement principleIntroduction. Beams with an arbitrary cross-section that is symmetric about an axis or a pair of axes and with discretely distributed material are widely used in engineering [1,8]. For example, bimetallic beams are used as a basic element in instrument making, composite columns, floors, etc. Gradient changes in the mass and mechanical characteristics across the cross-section necessitate a search for generalized solutions to relevant problems of mechanics [7,11], which can be found using integral principles, the virtual-displacement principle being the most general among them.This paper studies the dynamic deformation of structurally inhomogeneous beams and proposes a method for analyzing dynamic processes based on the virtual-displacement principle. We will derive equations of motion for a cross-section of arbitrary shape with one or two axes of symmetry, boundary conditions, and types of loading and discuss a solution describing the dynamic deformation of a hinged layered beam of rectangular cross-section under harmonic loading.1. Mechanical Object of Study. Multilayered beams of dissimilar materials are widely used in developing new structural elements. Some layers may be reinforced with high-modulus wires. Beams may be of various cross-sections. Such mechanical systems are used in creating various types of thermostats, rotors for modern helicopters, shock absorbers, etc. Compound rods fall into a separate group. Their structure, engineering applications, and results of research are presented in the monograph [4]. Timoshenko's studies show that inhomogeneous rods with effective characteristics [6, etc.] can be used instead of plane or spatial truss structures in solving certain problems of mechanics.Out of structurally inhomogeneous beam systems, we choose to consider extended beams with the following structure: their cross-section is symmetric about some axes, the material is discretely distributed over the cross-section, and the areas occupied by the material have complex configuration and consist of materials with different mechanical characteristics. An example of such systems is cables with comp...

The dynamic deformation of glued beams with a thin adhesive interlayer under harmonic loading is studied. The virtual-work principle is used. The system of ordinary differential equations for the amplitudes in the approximations in coordinate functions is derived and solved. The proposed approach allows determining the stresses and strains at an arbitrary point of the beam, especially in the adhesive interlayer. Numerical results are presented Keywords: glued beam, adhesive interlayer, harmonic loading, stress-strain state, virtual-work principle Introduction. The mechanical system to be considered here is two beams glued together and subjected to harmonic loading. There is a thin hard adhesive interlayer (AI) between them. The beams and AI are elastic, homogeneous, isotropic and have different geometrical and mechanical characteristics.Methods of analysis of the static and dynamic deformation of layered materials are addressed in many studies, including [1-3, 5, 9].There are just a few publications [6-8] on glued structural members under loading (harmonic, cyclic) dynamic. Considerable difficulties are associated with the small thickness of the AI. However, the AI is important (in aircraft construction, shipbuilding, etc.) [4] because it holds the beams together, on the one hand, and, on the other hand, transfers stresses between the beams, promoting longitudinal and flexural vibrations of the glued beam (GB) as a whole. A literature review reveals that problems of deformation of adhesive joints were in most cases solved in static formulation.We will propose a model to describe the dynamic deformation of the GB and to analyze the stress-strain state (SSS) at an arbitrary point of the beams and AI. To solve the problem, we will follow an integral approach based on the virtual-work principle (VWP). All required formulas are derived using the linear theory of elasticity. The efficiency of this method is in that it allows considerable flexibility (incorporation of some complicating factors, selection of approximations, etc.).1. Coordinate Systems. Simplifying Assumptions. Consider a mechanical system consisting of two rectangular beams (1, 2) glued together and an adhesive interlayer (strip) between them (Fig. 1). The geometrical and mechanical parameters of the components are the following: b is the width; l is the length; 2h a is the height; E ( ) a , G ( ) a , n a ( ) for beams (a =1 2 , ); 2 3 h ,

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