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An Analytical Approach for the Green's Functions of Biharmonic Problems with Circular and Annular Domains

Published online by Cambridge University Press:  05 May 2011

J. T. Chen ,
H. Z. Liao  and
W. M. Lee
J. T. Chen*
Affiliation:
Department of Harbor and River Engineering, Department of Mechanical and Mechatronic Engineering, National Taiwan Ocean University, Keelung, Taiwan 20224, R.O.C.
H. Z. Liao*
Affiliation:
Department of Harbor and River Engineering, Department of Mechanical and Mechatronic Engineering, National Taiwan Ocean University, Keelung, Taiwan 20224, R.O.C.
W. M. Lee*
Affiliation:
Department of Mechanical Engineering, China Institute of Technology, Taipei, Taiwan 11581, R.O.C.
*
*Life-time Distinguished Professor
**Master student
***Associate Professor
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Abstract

In this paper, an analytical approach for deriving the Green's function of circular and annular plate was presented. Null-field integral equations were employed to solve the plate problems while kernel functions were expanded to degenerate kernels. The unknown boundary data of the displacement, slope, normal moment and effective shear force were expressed in terms of Fourier series. It was noticed that all the improper integrals were avoided when the degenerate kernels were used. After determining the unknown Fourier coefficients, the displacement, slope, normal moment and effective shear force of the plate could be obtained by using the boundary integral equations. The present approach was seen as an “analytical” approach for a series solution. Finally, several analytical solutions were obtained. To see the validity of the present method, FEM solutions using ABAQUS were compared well with our analytical solutions. The displacement, radial moment and shear variations of radial and angular positions were presented.

Keywords

Green's functionNull-field integral equationDegenerate kernelFourier series.
Type
Articles
Information
Journal of Mechanics , Volume 25 , Issue 1 , March 2009 , pp. 59 - 74
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2009

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References

1.Boley, B. A., “A Method for the Construction of Green's Functions,” The Quarterly Journal of Mechanics and Applied Mathematics, 14, pp. 249–257 (1956).Google Scholar
2.Wang, X. and Sudak, L. J., “Antiplane Time-Harmonic Green's Functions for a Circular Inhomogeneity with an Imperfect Interface,” Mechanics Research Communications, 34, pp. 352–358 (2007).CrossRefGoogle Scholar
3.Timoshenko, S. and Woinowsky-Krieger, S., Theory of Plates and Shells, McGraw-Hill, New York (1959).Google Scholar
4.Melnikov, Y. A. and Melnikov, M. Y., “Green's Function for Mixed Boundary Value Problems in Regions of Irregular Shape,” Electronic Journal of Boundary Elements, 4, pp. 82–104 (2006).Google Scholar
5.Melnikov, Y. A. and Sheremet, V. D., “Some New Results on the Bending of a Circular Plate Subject to a Transverse Point Force,” Mathematics and Mechanics of Solids, 6, pp. 29–45 (2001).CrossRefGoogle Scholar
6.Melnikov, Y. A., “Influence Functions of a Point Source for Perforated Compound Plates with Facial Convection,” Journal of Engineering Mathematics, 49, pp. 253–270 (2004).CrossRefGoogle Scholar
7.Sharafutdinov, G. Z., “Stress and Concentrated Forces in Thin Annular Plates,” Journal of Applied Mathematics and Mechanics, 68, pp. 39–51 (2004).CrossRefGoogle Scholar
8.Adewale, A. O., “Isotropic Clamped-Free Thin Annular Circular Plate Subjected to a Concentrated Load,” Journal of Applied Mechanics, 73, pp. 658–663 (2006).CrossRefGoogle Scholar
9.Chen, J. T., Wu, C. S. and Chen, K. H., “A Study of Free Terms for Plate Problems in the Dual Boundary Integral Equations,” Engineering Analysis with Boundary Element, 29, pp. 435–446 (2005).CrossRefGoogle Scholar
10.Chen, J. T., Wu, C. S., Chen, K. H. and Lee, Y. T., “Degenerate Scale for the Analysis of Circular Thin Plate Using the Boundary Integral Equation Method and Boundary Element Methods,” Computational Mechanics, 38, pp. 33–49 (2006).CrossRefGoogle Scholar
11.Chen, J. T., Hsiao, C. C. and Leu, S. Y., “A New Method For Stokes' Flow with Circular Boundaries Using Degenerate Kernel and Fourier Series,” International Journal for Numerical Methods in Engineering, 74, pp. 1955–1987 (2008).CrossRefGoogle Scholar
12.Szilard, R., Theory and Analysis of Plates Classical and Numerical Methods, Englewood Cliffs, New Jersey (1974).Google Scholar
13.Melnikov, Y. A., “Influence Functions of a Point Force for Kirchhoff Plates with Rigid Inclusions,” Journal of Mechanics, 20, pp. 249–256 (2004).CrossRefGoogle Scholar
14.Liao, H. Z., “Analytical Solutions for the Green's Functions of Laplace and Biharmonic Problems with Circular Boundaries,” MS Thesis, National Taiwan Ocean University, Taiwan (2007),Google Scholar
15.Wu, A. C., “Null-Field Approach for Multiple Circular Inclusion Problems in Anti-Plane Piezoelectricity,” MS Thesis, National Taiwan Ocean University, Taiwan (2006).Google Scholar
16.Chen, J. T. and Chen, P. Y., “A Semi-Analytical Approach for Stress Concentration of Cantilever Beams with Holes Under Bending,” Journal of Mechanics, 20, pp. 211–221 (2007).CrossRefGoogle Scholar