Two-Dimensional Complete Rational Analysis of Functionally Graded Beams Within the Symplectic Framework

ZHAO Li; CHEN Wei-qiu; Lü Chao-feng

doi:10.3879/j.issn.1000-0887.2012.10.001

Volume 33 Issue 10

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ZHAO Li, CHEN Wei-qiu, Lü Chao-feng. Two-Dimensional Complete Rational Analysis of Functionally Graded Beams Within the Symplectic Framework[J]. Applied Mathematics and Mechanics, 2012, 33(10): 1143-1155. doi: 10.3879/j.issn.1000-0887.2012.10.001

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Two-Dimensional Complete Rational Analysis of Functionally Graded Beams Within the Symplectic Framework

doi: 10.3879/j.issn.1000-0887.2012.10.001

1.
¹Department of Civil Engineering, Ningbo University of Technology, Ningbo, Zhejiang 315016, P.R.China;²Department of Engineering Mechanics, Zhejiang University, Hangzhou 310027, P.R.China;³Department of Civil Engineering, Zhejiang University, Hangzhou 310058, P.R.China

Received Date: 2012-01-18
Rev Recd Date: 2012-03-21
Publish Date: 2012-10-15

Abstract

Abstract

Exact solutions for generally supported functionally graded plane beams were given within the framework of symplectic elasticity. The Young’s modulus was assumed to vary exponentially along the longitudinal direction while Poisson’s ratio remained constant. The state equation with a shiftHamiltonian operator matrix had been established in our previous work, but limited to the SaintVenant solution. Here it was presented that a complete rational analysis of the displacement and stress distributions in the beam by exploring the eigensolutions which were usually covered up by the SaintVenant principle. These solutions played a significant role on local behavior of materials that was usually ignored by the conventional elasticity methods but may be crucial to the failure of the materials/structures. The analysis made full use of symplectic orthogonality of the eigensolutions. Two illustrative examples were presented to compare the displacement and stress results with those for homogenous materials, and to demonstrate the effect of material inhomogeneity.
- functionally graded materials,
- exact solutions,
- expansion of eigenfunctions,
- symplectic elasticity

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References(27)

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