Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-28T10:46:12.885Z Has data issue: false hasContentIssue false
  • English
  • Français

A Semi-Analytical Approach for Stress Concentration of Cantilever Beams with Holes Under Bending

Published online by Cambridge University Press:  05 May 2011

J.-T. Chen  and
P.-Y. Chen
J.-T. Chen*
Affiliation:
Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung, Taiwan 20224, R.O.C.
P.-Y. Chen*
Affiliation:
Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung, Taiwan 20224, R.O.C.
*
*Life-Time Distinguished Professor
**Master student
Get access

Abstract

In the paper, the degenerate kernels and Fourier series expansions are adopted in the null-field integral equation to solve bending problems of a circular beam with circular holes. The main gain of using degenerate kernels in integral equations is free of calculating the principal values for singular integrals. An adaptive observer system is addressed to fully employ the property of degenerate kernels for circular boundaries in the polar coordinate. After moving the null-field point to the boundary and matching the boundary conditions, a linear algebraic system is obtained without boundary discretization. The unknown coefficients in the algebraic system can be easily determined. The present method is treated as a “semi-analytical” solution since error only attributes to the truncation of Fourier series. Stress concentration is also our concern. Finally, several examples, including two holes and four holes, are given to demonstrate the validity of the proposed method. The results are compared with those of Naghdi and Bird and Steele. Also, the position where the maximum concentration factor occurs is examined. The present formulation can be extended to handle beam problems with arbitrary number and various positions of circular holes by using the developed general-purpose program.

Keywords

Null-field integral equationDegenerate kernelFourier seriesCircular holesCantilever beamStress concentration.
Type
Articles
Information
Journal of Mechanics , Volume 23 , Issue 3 , September 2007 , pp. 211 - 222
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Hwu, C. B., “Stroh Formalism and its Extensions to Coupled Inplane-bending Problems,” The Chinese Journal of Mechanics — Series A, 19, pp. 41–53 (2003).Google Scholar
2.Shiah, Y. C., Fang, J., Wei, C. Y. and Liang, Y. C., “In-plane Bending Fracture of a Large Beam Containing a Circular-arc Crack,” The Chinese Journal of Mechanics— Series A 18, pp. 145–151 (2002).Google Scholar
3.Huang, C. S., “Analysis of Stress Singularities at Bi-material Corners in Reddy's Theory of Plate Bending,” Journal of Mechanics, 22, pp. 67–75 (2006).CrossRefGoogle Scholar
4.Sokolnikoff, I. S., Mathematical Theory of Elasticity, McGraw-Hill, New York (1956).Google Scholar
5.Timoshenko, S. P. and Goodier, J. N., Theory of Elasticity, McGraw-Hill, New York (1970).Google Scholar
6.Chou, S. I., “Stress Field around Holes in Antiplane Shear Using Complex Variable Boundary Element Method,” ASMEJ. Applied Mechanics, 64, pp. 432–435 (1997).CrossRefGoogle Scholar
7.Ang, W. T. and Kang, I., “A Complex Variable Boundary Element Method for Elliptic Partial Differential Equations in a Multiply-connected Region,” Int. J. Computer Mathematics, 75, pp. 515–525 (2000).CrossRefGoogle Scholar
8.Hromadka, T. V. and Lai, C., The Complex Variable Boundary Element Method in Engineering Analysis, Springer-Verlag, New York (1986).Google Scholar
9.Honein, E., Honein, T. and Herrmann, G., “Further Aspects of the Elastic Field for Two Circular Inclusions in Antiplane Elastostatics,” ASME J. Applied Mechanics, 59, pp. 774–779 (1992).CrossRefGoogle Scholar
10.Chen, J. T., Shen, W. C. and Wu, A. C., “Null-field Integral Equations for Stress Field around Circular Holes under Anti-plane Shear,” Engineering Analysis with Boundary Elements, 30, pp. 205–217 (2006).CrossRefGoogle Scholar
11.Naghdi, A. K., “Bending of a Perforated Circular Cylindrical Cantilever,” Int. J. Solids and Structures, 28, pp. 739–749 (1991).CrossRefGoogle Scholar
12.Bird, M. D. and Steele, C. R., “A Solution Procedure for Laplace's Equation on Multiply Connected Circular Domains,” ASME J. Applied Mechanics, 59, pp. 398–404 (1992).CrossRefGoogle Scholar
13.Chen, J. T. and Hong, H.-K., “Review of Dual Boundary Element Methods with Emphasis on Hypersingular Integrals and Divergent Series,” Applied Mechanics Reviews, ASME, 52, pp. 17–33 (1999).CrossRefGoogle Scholar
14.Chen, J. T. and Chiu, Y. P., “On the Pseudo-differential Operators in the Dual Boundary Integral Equations Using Degenerate Kernels and Circulants,” Engineering Analysis with Boundary Elements, 26, pp. 41–53 (2002).CrossRefGoogle Scholar
15.Chen, J. T., Lin, J. H., Kuo, S. R. and Chiu, Y. P., “Analytical Study and Numerical Experiments for De-generate Scale Problems in Boundary Element Method Using Degenerate Kernels and Circulants,” Engineering Analysis with Boundary Elements, 25, pp. 819–828 (2001).CrossRefGoogle Scholar
16.Chen, J. T., Lee, C. F., Chen, I. L. and Lin, J. H., “An Alternative Method for Degenerate Scale Problem in Boundary Element Methods for the Two-dimensional Laplace Equation,” Engineering Analysis with Boundary Elements, 26, pp. 559–569 (2002).CrossRefGoogle Scholar
17.Chen, J. T., Kuo, S. R. and Lin, J. H., “Analytical Study and Numerical Experiments for Degenerate Scale Prob-lems in the Boundary Element Method for Two-Dimensional Elasticity,” Int. J. Numerical Methods in Engineering, 54, pp. 1669–1681 (2002).CrossRefGoogle Scholar
18.Bird, M. D. and Steele, C. R., “A Solution Procedure for Laplace's Equation on Multiply Connected Circular Domains,” Asme J. Applied Mechanics, 59, pp. 398–404 (1992).CrossRefGoogle Scholar