The refined theory of deep rectangular beams for symmetrical deformation  

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 

出　　处：《Science China(Physics,Mechanics & Astronomy)》2009年第6期919-925,共7页中国科学：物理学、力学、天文学（英文版）

基　　金：Supported by the National Natural Science Foundation of China (Grant Nos.10702077,10672001,and 10602001);the Beijing Natural Science Foundation (Grant No.1083012);the Alexander von Humboldt Foundation in Germany

摘　　要：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.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 

分 类 号：O302[理学—力学]