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The refined theory of deep rectangular beams for symmetrical deformation

Science China(Physics,Mechanics & Astronomy) 2009年第6期52卷 919-925页

作者： GAO Yang1 & WANG MinZhong2 1 College of Science, China Agricultural University, Beijing 100083, China 2 State Key Laboratory for Turbulence and Complex Systems and Department of Mechanics and Aerospace Engineering, Peking University, Beijing 100871, China College of Science China Agricultural University Beijing China State Key Laboratory for Turbulence and Complex Systems and Department of Mechanics and Aerospace Engineering Peking University Beijing China

Based on elasticity theory, various one-dimensional equations for symmetrical deformation have been deduced systematically and directly from the two-dimensional theory of deep rectangular beams by using the papkovich-neuber solution and the Lur'e method without ad hoc assumptions, and they construct the refined theory of beams for symmetrical deformation. It is shown that the displacements and stresses of the beam can be represented by the transverse normal strain and displacement of the mid-plane. In the case of homogeneous boundary conditions, the exact solutions for the beam are derived, and the exact equations consist of two governing differential equations: the second-order equation and the transcendental equation. In the case of non-homogeneous boundary conditions, the approximate governing differential equations and solutions for the beam under normal loadings only and shear loadings only are derived directly from the refined beam theory, respectively, and the correctness of the stress assumptions in classic extension or compression problems is revised. Meanwhile, as an example, explicit expressions of analytical solutions are obtained for beams subjected to an exponentially distributed load along the length of beams.

关键词： deep rectangular beams the refined theory symmetrical deformation the papkovich-neuber solution the Lur’e method

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