Deformation of a Step Composite Beam in a Temperature Field

Abstract

Consideration is given to thermal force deformation of an elastic three-layer beam with a stepwise thickness of one supporting layer. To describe the kinematics of a rod that is asymmetric across the bundle, the hypotheses of a broken normal are adopted. A system of equilibrium equations is derived and its general analytical solution in displacements is obtained. A numerical parametric analysis of the stressed-strained state of a metal-polymer threelayer beam is carried out.Introduction. Since composite structural elements are widely used in industry, the problem of their strength calculation for various external effects and internal confi gurations is topical. Static and dynamic deformations of three-layer (sandwich) composite structural elements with a smooth confi guration were studied in [1]. The theory of homogeneous thinwalled structures of stepwise varying thickness was developed in [2]. In [3,4], consideration was given to the deformation of a three-layer rod with an irregular boundary under isothermal loading. Studies [5,6] were devoted to the dynamic deformation of three-layer building structures.Problem Formulation. Consideration is given to an asymmetric-across-the thickness elastic three-layer beam with a stiff fi ller (Fig. 1), i.e., with regard for the work of shear stresses. The system of coordinates x, y, and z is tied to the fi ller midplane. The bundle kinematics is described using the hypothesis of a "broken" normal: for thin supporting layers 1 and 2, the Bernoulli hypothesis is valid, in the stiff fi ller 2, which is incompressible across the thickness, the normal remains rectilinear and does not change its length but turns by a certain angle ψ(x). Presumably, deformations are small, but the external layer 1 of the beam is acted upon by a distributed force load q(x) and p(x) and the temperature fi eld T k in the rod is known. In accordance with the adopted assumptions, the defl ections will be the same at all points of the rod cross section. There are stiff diaphragms on the rod faces, which hinder the relative shear of the layers. The layer thickness h k is defi ned as