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Abstract: The plane stress problem of beams is a typical one in elasticity theory. In this paper a new set of boundary conditions for the fixed end is proposed to improve the accuracy of the plane elasticity solution for beams with fixed end(s). Plane elasticity solutions are then derived for the cantilever beam, propped cantilever beam, and fixed-fixed beam. The new set of boundary conditions is constructed by combining two conventional ones with a parameter. The parameters for different kinds of beams are determined by minimizing the square sum of the longitudinal displacements through the thickness of the fixed end. Comparison with the results obtained by the finite element method (FEM) shows the efficiency of the new type of boundary conditions. When the beam is a deep one, it is found that different boundary conditions yield different errors, and the elasticity solution obtained by the new boundary conditions best approaches the FEM results.

This is a quite interesting and complete work on the seemingly old but important problem in elasticity. The paper suggests a new mathematical form to express the fixed boundary of a beam, which combines the two existing ones in Timoshenko and Goodier by introducing a parameter which is determined on a reasonable ground. Numerical comparison with FEM shows that the new form enables more accurate results.

含固支端梁的弹性力学解

[1]Ahmed, S.R., Idris, A.B.M., Uddin, M.W., 1996. Numerical solution of both ends fixed deep beams. Computers & Structures, 61(1):21-29.

[2]Ahmed, S.R., Khan, M.R., Islam, K.M.S., et al., 1998. Investigation of stresses at the fixed end of deep cantilever beams. Computers and Structures, 69(3):329-338.

[3]Bhimaraddi, A., 1988. Generalized analysis of shear deformable rings and curved beams. International Journal of Solids and Structures, 24(4):363-373.

[4]Cowper, G.R., 1966. The shear coefficient in Timoshenko’s beam theory. Journal of Applied Mechanics, 33(2):335-340.

[5]Dai, Y., Ji, X., 2008. A plane stress solution of deep beam with fixed ends under uniform loading. Journal of Tongji University (Natural Science), 36(7):890-893 (in Chinese).

[6]Ding, H.J., Huang, D.J., Wang, H.M., 2005. Analytical solution for fixed-end beam subjected to uniform load. Journal of Zhejiang University-SCIENCE A, 6(8):779-783.

[7]Ding, H.J., Huang, D.J., Wang, H.M., 2006. Analytical solution for fixed-fixed anisotropic beam subjected to uniform load. Applied Mathematics and Mechanics, 27(10):1305-1310.

[8]Ding, H.J., Huang, D.J., Chen, W.Q., 2007. Elasticity solutions for plane anisotropic functionally graded beams. International Journal of Solids and Structures, 44(1):176-196.

[9]Gao, Y., Wang, M.Z., 2006. The refined theory of deep rectangular beams based on general solutions of elasticity. Science in China: Series G Physics, Mechanics & Astronomy, 49(3):291-303.

[10]Ghugal, Y.M., Sharma, R., 2011. A refined shear deformation theory for flexure of thick beams. Latin American Journal of Solids and Structures, 8(2):183-195.

[11]Heyliger, P.R., 2013. When beam theories fail. Journal of Mechanics of Materials and Structures, 8(1):15-35.

[12]Heyliger, P.R., Reddy, J.N., 1988. A higher order beam finite element for bending and vibration problems. Journal of Sound and Vibration, 126(2):309-326.

[13]Huang, D.J., Ding, H.J., Chen, W.Q., 2010. Static analysis of anisotropic functionally graded magneto-electro-elastic beams subjected to arbitrary loading. European Journal of Mechanics-A/Solids, 29(3):356-369.

[14]Jiang, A.M., Ding, H.J., 2005. The analytical solutions for orthotropic cantilever beams (I): subjected to surface forces. Journal of Zhejiang University-SCIENCE A, 6(2):126-131.

[15]Kant, T., Gupta, A., 1988. A finite element model for a higher-order shear-deformable beam theory. Journal of Sound and Vibration, 125(2):193-202.

[16]Lekhnitskii, S.G., 1968. Anisotropic Plate. Gordon and Breach, New York, USA.

[17]Levinson, M., 1981. A new rectangular beam theory. Journal of Sound and Vibration, 74(1):81-87.

[18]Nie, G.J., Zhong, Z., Chen, S., 2013. Analytical solution for a functionally graded beam with arbitrary graded material properties. Composites Part B: Engineering, 44(1):274-282.

[19]Timoshenko, S.P., 1921. On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philosophical Magazine Series 6, 41(245):744-746.

[20]Timoshenko, S.P., 1922. On the transverse vibration of bars of uniform cross-section. Philosophical Magazine Series 6, 43(253):125-131.

[21]Timoshenko, S.P., Goodier, J.N., 1970. Theory of Elasticity, 3rd Edition. McGraw-Hill, New York, USA.

[22]Timoshenko, S.P., Gere, J.M., 1972. Mechanics of Materials. Van Nostrand Reinhold, New York, USA.

[23]Wang, M.Q., Liu, Y.H., 2010. Analytical solution for bi-material beam with graded intermediate layer. Composite Structures, 92(10):2358-2368.

[24]Zhao, L., Chen, W.Q., Lü, C.F., 2012. New assessment on the Saint-Venant solutions for functionally graded beams. Mechanics Research Communications, 43:1-6.

[25]Zhong, Z., Yu, T., 2007. Analytical solution of a cantilever functionally graded beam. Composites Science and Technology, 67(3-4):481-488.