Abstract
Based on the refined theory for narrow rectangular deep beams, two different displacement boundary conditions of the fixed end of a cantilever beam are used to study the deformation of the beam. One is the conventional simplified displacement boundary condition, and the other is a new boundary condition determined by the least squares method. Three load cases are investigated, which are a transverse shear force at the free end of the beam, a uniformly distributed load at the top surface, and a linearly distributed load at the top surface, respectively. Solutions are given for both the refined theory and the Timoshenko beam theory and are compared with the known solutions from the elastic theory and results by the finite element method. It is shown that the solutions of the refined theory coincide with those of the elastic theory; the solutions from the Timoshenko theory by using the two different displacement boundary conditions are the same; the refined theory by using the new boundary condition provides better results than using the conventional boundary condition and also better than those of the Timoshenko beam theory.
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References
Dym C.L., Shames I.H.: Solid Mechanics: A Variational Approach. McGraw-Hill, New York (1973)
Timoshenko S.P.: On the correction for shear of the differential equation for transverse vibration of prismatic bars. Philos. Maga. Ser. 6 41(245), 744–746 (1921)
Levinson M.: A new rectangular beam theory. J. Sound Vib. 74(1), 81–87 (1981)
Lur’e A.I.: Three-Dimensional Problems of the Theory of Elasticity. Interscience, New York (1964)
Gao Y., Wang M.Z.: A refined theory for the rectangle deep beam based on the general solution in elasticity. Sci. China Ser. G Phys. Mech. Astron. 36(3), 286–297 (2006)
Gao Y., Wang M.Z.: A refined beam theory based on the refined plate theory. Acta Mech. 177(1–4), 191–197 (2005)
Gao Y., Xu S.P., Zhao B.S.: Boundary conditions for elastic beam bending. Comptes Rendus Mecanique 335, 1–6 (2007)
Timoshenko S.P., Goodier J.C.: Theory of Elasticity. McGraw-Hill, New York (1970)
Timoshenko S.P., Gere J.M.: Mechanics of Materials. Van Nostrand Company, New York (1972)
Hang, W.B.: Problems on the shear deflection of a planer elastic cantilever. Mech. Eng. 19(2):61–62 (1997) (in Chinese)
Wang, M.Z.: About “Problems on the shear deflection of a plane elastic cantilever”. Mech. Eng. 26(6), 66–68 (2004) (in Chinese)
Cowper G.R.: The shear coefficients in Timoshenko’s beam theory. ASME J. Appl. Mech. 33, 335–340 (1966)
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Yang, LZ., Shang, LG., Gao, Y. et al. Further study on the refined theory of rectangle deep beams. Acta Mech 224, 1999–2007 (2013). https://doi.org/10.1007/s00707-013-0850-1
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DOI: https://doi.org/10.1007/s00707-013-0850-1