Skip to main content
Log in

General analytic solution of dynamic response of beams with nonhomogeneity and variable cross-section

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

In this paper, a new method, the step-reduction method, is proposed to investigate the dynamic response of the Bernoulli-Euler beams with arbitrary nonhomogeneity and arbitrary variable cross-section under arbitrary loads. Both free vibration and forced vibration of such beams are studied. The new method requires to discretize the space domain into a number of elements. Each element can be treated as a homogeneous one with uniform thickness. Therefore, the general analytical solution of homogeneous beams with uniform cross-section can be used in each element. Then, the general analytic solution of the whole beam in terms of initial parameters can be obtained by satisfying the physical and geometric continuity conditions at the adjacent elements. In the case of free vibration, the frequency equation in analytic form can be obtained, and in the case of forced vibration, a final solution in analytical form can also be obtained which is involved in solving a set of simultaneous algebraic equations with only two unknowns which are independent of the numbers of elements divided. The present analysis can also be extended to the study of the vibration of such beams with viscous and hysteretic damping and other kinds of beams and other structural elements with arbitrary nonhomogeneity and arbitrary variable thickness.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Yeh, K. Y., General solutions on certain problems of elasticity with nonhomogeneity and variable thickness, Part IV: Bending, buckling, and free vibration of nonhomogeneous variable thickness beams,Journal of Lanzhou University, Special Number of Mechanics 1 (1979), 133–157. (in Chinese)

    Google Scholar 

  2. Chu, C. H. and W. D. Pilkey, Transient analysis of structural members by the CSDT Riccati transfer matrix method,Computers and Structures,10 (1979), 599–611.

    Article  MATH  Google Scholar 

  3. Finlayson, B. A.,The Methods of Weighted Residuals and Variational Principles, Academic Press, New York (1972).

    Google Scholar 

  4. Mikhlin, S. G.,Variational Methods in Mathematical Physics, Macmillan, New York (1964).

    MATH  Google Scholar 

  5. Laura, P. A. A., B. Valerga De Greco, J. C. Vtjes and R. Carnicer, Numerical experiments on free and forced vibrations of beams of nonuniform cross-section,Journal of Sound and Vibration,120 (1988), 587–596.

    Article  Google Scholar 

  6. Bathe, K. J.,Finite Element Procedures in Engineering Analysis, Prentice-Hall, Engliwood Cliffs, New Jersey (1982).

    MATH  Google Scholar 

  7. Ghali, A. and A. M. Neville,Structural Analysis Intext Educational Publishers, Pennsylvania (1972).

    MATH  Google Scholar 

  8. Beskos, D. E., Boundary element methods in dynamic analysis,Applied Mechanics Reviews,40, (1987), 1–23.

    Article  Google Scholar 

  9. Just, D. J., Plane frameworks of tapered box and I-section,Journal of the Structural Division, Proceedings of ASCE,103 (1977), 71–86.

    Google Scholar 

  10. Ovunk, B. A., Dynamics of frameworks by continuous mass methods,Computers and Structures,5 (1974), 1061–1089.

    Article  Google Scholar 

  11. Beskos, D. E., Dynamics and stability of plane trusses with gusset plates,Computers and Structures,10 (1979), 785–795.

    Article  MATH  Google Scholar 

  12. Beskos, D. E. and G. V. Narayanan, Dynamic response of frameworks by numerical Laplace transform,Computer Methods in Applied Mechanics and Engineering,37 (1983), 289–307.

    Article  MATH  Google Scholar 

  13. Wang, H. C., Generalized hypergeometric function solutions on transverse vibrations of a class of nonuniform beams,ASME Journal of Applied Mechanics,34 (1968), 702–708.

    Google Scholar 

  14. Taleb, N. J. and E. W. Suppiger, Vibration of stepped beams,Journal of Aerospace Science,28 (1961), 295–298.

    MATH  MathSciNet  Google Scholar 

  15. Lindberg, G. M., Vibration of nonuniform beams,The Aeronautical Quarterly,14 (1963); 387–395.

    Google Scholar 

  16. Yeh, K. Y., et al., Elastic and plastic analysis of high-speed rotating disks with nonhomogeneity and variable thickness under nonhomogeneous steady temperature field, presented at the 15th ICTAM, Toronto (1980).

  17. Yeh, K. Y. and P. Lieu, Steady heat conduction of a disk with nonhomogeneity and variable thickness,Appl. Math. and Mech. (English Ed.),5, 5 (1984), 1587–1593.

    MATH  Google Scholar 

  18. Yeh, K. Y. and P. Lieu, Equi-strength design of nonhomogeneous, variable thickness high-speed rotating disk under a steady temperature field,Appl. Math. and Mech. (English Ed.),7, 9 (1986), 825–834.

    Article  MATH  Google Scholar 

  19. Yeh, K. Y., Recent Investigations of structural optimization by analytic methods,Structural Optimization, edited by G. I. N. Rozvany & B. L. Karihaloo, Kluwer Academic Publishers (1988), 379–386.

  20. Yeh, K. Y. and H. R. Yu, Discussion of problems existing in the calculation of reinforced concrete structures (I), (II),Journal of Lanzhou University, Special Number of Mechanics,19 (1983), 1–36. (in Chinese)

    Google Scholar 

  21. Yeh, K. Y. and C. Y. Hsu, Circular shallow spherical shells with central circular hole under simultaneous actions of arbitrary unsteady temperature field and arbitrary dynamic loads,Appl. Math. and Mech. (English Ed.),1, 3 (1980), 187–209.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kai-yuan, Y., Xiao-hua, T. & Zhen-yi, J. General analytic solution of dynamic response of beams with nonhomogeneity and variable cross-section. Appl Math Mech 13, 779–791 (1992). https://doi.org/10.1007/BF02481798

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02481798

Key words

Navigation