Analytical solution and semi-analytical solution for anisotropic functionally graded beam subject to arbitrary loading  

作　　者：HUANG DeJin1,2,DING HaoJiang1 & CHEN WeiQiu3 1 Key Laboratory of Soft Soils and Geoenvironmental Engineering,Ministry of Education,Zhejiang University,Hangzhou 310027,China 2 Faculty of Engineering,Ningbo University,Ningbo 315211,China 3 Department of Engineering Mechanics,Zhejiang University,Hangzhou 310027,China 

出　　处：《Science China(Physics,Mechanics & Astronomy)》2009年第8期1244-1256,共13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. 10472102, 10432030, and 10725210)

摘　　要：Analytical and semi-analytical solutions are presented for anisotropic functionally graded beams subject to an arbitrary load,which can be expanded in terms of sinusoidal series.For plane stress problems,the stress function is assumed to consist of two parts,one being a product of a trigonometric function of the longitudinal coordinate(x) and an undetermined function of the thickness coordinate(y),and the other a linear polynomial of x with unknown coefficients depending on y.The governing equations satisfied by these y-dependent functions are derived.The expressions for stresses,resultant forces and displacements are then deduced,with integral constants determinable from the boundary conditions.While the analytical solution is derived for the beam with material coefficients varying exponentially or in a power law along the thickness,the semi-analytical solution is sought by making use of the sub-layer approximation for the beam with an arbitrary variation of material parameters along the thickness.The present analysis is applicable to beams with various boundary conditions at the two ends.Three numerical examples are presented for validation of the theory and illustration of the effects of certain parameters.Analytical and semi-analytical solutions are presented for anisotropic functionally graded beams subject to an arbitrary load, which can be expanded in terms of sinusoidal series. For plane stress problems, the stress function is assumed to consist of two parts, one being a product of a trigonometric function of the longitudinal coordinate (x) and an undetermined function of the thickness coordinate (y), and the other a linear polynomial of x with unknown coefficients depending on y. The governing equations satisfied by these y-dependent functions are derived. The expressions for stresses, resultant forces and displacements are then deduced, with integral constants determinable from the boundary conditions. While the analytical solution is derived for the beam with material coefficients varying exponentially or in a power law along the thickness, the semi-analytical solution is sought by making use of the sub-layer approximation for the beam with an arbitrary variation of material parameters along the thickness. The present analysis is applicable to beams with various boundary conditions at the two ends. Three numerical examples are presented for validation of the theory and illustration of the effects of certain parameters.

关 键 词：ANISOTROPIC material functionally GRADED ARBITRARY loading stress function analytical SOLUTION SEMI-ANALYTICAL SOLUTION 

分 类 号：O344[理学—固体力学]