Engineering Resistance Theory of Heterogeneous Rods made of Composite Materials
Authors: Gorbachev V.I. | Published: 06.12.2016 |
Published in issue: #6(69)/2016 | |
DOI: 10.18698/1812-3368-2016-6-56-72 | |
Category: Mechanics | Chapter: Dynamics, Strength of Machines, Instruments, and Equipment | |
Keywords: heterogeneous rod, problem of the heterogeneous body elasticity theory, theory of zero-order approXimation, effective rigidity |
To construct the engineering resistance theory of heterogeneous rods, we used an integral formula which presents the displacement of the body points in the initial problem of the heterogeneous body elasticity theory by means of the points displacement in a similar problem, but for a homogeneous elastic body (an accompanying task). The integral formula implies an equivalent notion of displacements series in a heterogeneous rod. The displacements are compared to the derivatives in the accompanying homogeneous rod. We approximately defined the points displacement of the accompanying rod by classical strength of materials methods through the three components of the points displacement vector relative to its axis. As a result, we presented the displacement vector components of any point of the heterogeneous rod in the form of series of derivatives displacement of the longitudinal axis of a homogeneous rod. According to the displacement, we found the series for stresses in the heterogeneous rod. Furthermore, by longitudinal stress we determined the internal force factors in the heterogeneous rod cross section - longitudinal force and two bending moments, presented in series of derivatives of the three components of the rod axis displacement vector. Then, from Zhuravsky equations we derived a system of three ordinary differential equations of infinite order with respect to the three components of the longitudinal axis displacement vector. This paper studies the so-called theory of zero-order approximation, which takes into account only the rod axis longitudinal deformation and curvature (kinematic factors) to express internal force factors. The coefficients within the kinematic factors are the effective rigidity of the rod - longitudinal rigidity, four bending rigidities and four rigidities of mutual influence, which are calculated after solving the supporting planar and antiplanar problems in cross-section of the heterogeneous rod.
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