The refined theory of deep rectangular beams for symmetrical deformation |
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Authors: | Yang Gao and MinZhong Wang |
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Institution: | (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 |
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Abstract: | 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.
Supported by the National Natural Science Foundation of China (Grant Nos. 10702077, 10672001, and 10602001), the Beijing Natural
Science Foundation (Grant No. 1083012), and the Alexander von Humboldt Foundation in Germany |
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Keywords: | deep rectangular beams the refined theory symmetrical deformation the Papkovich-Neuber solution the Lur’ e method |
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