Deformation of glued beams under harmonic loading

Abstract

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 ,