Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-18T22:07:32.032Z Has data issue: false hasContentIssue false
  • English
  • Français

Vibration and Buckling Analyses of Beams by the Modified Differential Quadrature Method

Published online by Cambridge University Press:  05 May 2011

Y.-T. Chou  and
S.-T. Choi
Y.-T. Chou*
Affiliation:
Institute of Aeronautics and Astronautics, National Cheng Kung University, Tainan, Taiwan 70101, R.O.C.
S.-T. Choi*
Affiliation:
Institute of Aeronautics and Astronautics, National Cheng Kung University, Tainan, Taiwan 70101, R.O.C.
*
*Graduate Student
**Associate Professor
Get access

Abstract

In this paper the modified differential quadrature method (MDQM) is proposed for static and vibration analyses of beams. Modified weighting matrices are developed and a new formulation process is presented for incorporating boundary conditions such that the numerical error induced by using the δ-method in the original DQM is reduced. The present method is applied to various beam problems, such as static deflections of Euler beams, buckling loads of columns, and free vibrations of Timoshenko beams. Numerical results of the present method are shown to have excellent accuracy when compared to exact values and are more accurate than those obtained by the original DQM. The accuracy and efficiency of the present method have been demonstrated.

Keywords

BeamModified differential quadrature methodNatural frequencyBuckling
Type
Articles
Information
Journal of Mechanics , Volume 16 , Issue 4 , December 2000 , pp. 189 - 195
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2000

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

REFERENCES

1Bellman, R. E. and Casti, J., “Differential Quadrature and Long-Term Integration,” J. Math. Anal. Appl., 34, pp. 235–238 (1971).CrossRefGoogle Scholar
2Bellman, R. E., Kashef, B. G. and Casti, J., “Differential Quadrature: A Technique for The Rapid Solution of Nonlinear Partial Differential Equations,” J. Comput. Phys., 10, pp. 40–52 (1972).CrossRefGoogle Scholar
3Bert, C. W., Jang, S. K. and Striz, A. G., “Two New Approximate Methods for Analyzing Free Vibration of Structural Components,” AIAA J., 26, pp. 612–618 (1988).CrossRefGoogle Scholar
4Jang, S. K., Bert, C. W. and Striz, A. G., “Application of Differential Quadrature to Static Analysis of Structural Components,” Int. J. Numer. Meth. Engng., 28, pp. 561–577 (1989).CrossRefGoogle Scholar
5Sherbourne, A. N. and Pandey, M. D., “Differential Quadrature Method in the Buckling Analysis of Beams and Composite Plates,” Comput. Struct., 40, pp. 903–913 (1991).CrossRefGoogle Scholar
6Shu, C. and Richards, B. E., “Application of Generalized Differential Quadrature to Solve Two-Dimensional Incompressible Navier-Stokes Equations,” Int. J. Numer. Meth. Fluids, 15, pp. 791–798 (1992).CrossRefGoogle Scholar
7Gutierrez, R. H. and Laura, P. A. A., “Solution of the Helmholtz Equation in a Parallelogrammic Domain with Mixed Boundary Conditions Using the Differential Quadrature Method,” J. Sound Vibr., 178, pp. 269–271 (1994).CrossRefGoogle Scholar
8Bert, C. W. and Malik, M., “Differential Quadrature Method in Computational Mechanics: A Review,” Appl. Mech. Rev., 49, pp. 1–28 (1996).CrossRefGoogle Scholar
9Moradi, S. and Taheri, F., “Differential Quadrature Approach for Delamination Buckling Analysis of Composites with Shear Deformation,” AIAA J., 36, pp. 1869–1873 (1998).CrossRefGoogle Scholar
10Teo, T. M. and Liew, K. M., “A Differential Quadrature Procedure for Three-Dimensional Buckling Analysis of Rectangular Plates,” Int. J. Solids Struct., 36, pp. 1149–1168 (1999).CrossRefGoogle Scholar
11Wu, C. -P. and Hung, Y. -C., “Asymptotic Theory of Laminated Circular Conical Shells,” Int. J. Engng. Sci., 37, pp. 977–1005 (1999).CrossRefGoogle Scholar
12Choi, S. T. and Chou, Y. T., “Structural Analysis by the Differential Quadrature Method Using the Modified Matrices,” Proc. ASME Int. Comput. in Engng Conf., Atlanta, Georgia, USA (1998).Google Scholar
13Wang, X. and Bert, C. W., “A New Approach in Applying Differential Quadrature to Static and Free Vibrational Analyses of Beams and Plates,” J. Sound Vibr., 162, pp. 566–572 (1993).CrossRefGoogle Scholar
14Bert, C. W., Wang, X. and Striz, A. G., “Static and Free Vibrational Analysis of Beams and Plates by Differential Quadrature Method,” Acta Mech., 102, pp. 11–24 (1994).CrossRefGoogle Scholar
15de Boor, C., A Practical Guide to Splines, Springer-Verlag, New York (1987).Google Scholar
16Meirovitch, L., Principles and Techniques of Vibration, Prentice-Hall, New Jersey (1997).Google Scholar
17Gere, J. M. and Timoshenko, S. P., Mechanics of Materials, Wadsworth Inc, CA (1984).Google Scholar
18Timoshenko, S. P. and Gere, J. M., Theory of Elastic Stability, McGraw-Hill, New York (1961).Google Scholar
19Cowper, G. R., “The Shear Coefficient in Timoshenko's Beam Theory,” J Appl. Mech., 33, pp. 335–340 (1966).CrossRefGoogle Scholar
20Carr, J. B., “The Effect of Shear Flexibility and Rotatory Inertia on the Natural Frequencies of Uniform Beams,” Aero. Quart., 21, pp. 79–90 (1970).CrossRefGoogle Scholar