Accurate Representation of External Force in Time History Analysis
Publication: Journal of Engineering Mechanics
Volume 132, Issue 1
Abstract
It is usually thought that the integration time step should be small enough to represent properly the variation of the dynamic loading with respect to time. However, there is no evaluation criterion that can be used to determine whether the external force is accurately represented. In this paper, criteria for accurate representation of external force are proposed based on analytical results. It is found that amplitude distortion both in the transient response and the steady-state response for each time step is closely related to the step discretization error of external force. In fact, for a negligible period distortion, an amplitude distortion will be less than 5% if the relative step discretization error is constrained to be less than 5% at each time step for the Newmark explicit method, Fox–Goodwin method, and linear acceleration method while for the constant average acceleration method it must be less than 2.5%. This criterion leads to the need of using of eight or more integration time steps to accurately represent a complete cycle of a harmonic loading for the Newmark explicit method, Fox–Goodwin method, and linear acceleration method while for the constant average acceleration method 12 or more integration time steps are required.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgment
The writer gratefully acknowledges financial support for this study provided by the National Science Council, Taiwan, Republic of China, under Grant No. NSCTNSC-91-2218-E-027-010.
References
Bathe, K. J. (1996). Finite element procedures, Prentice Hall, Upper Saddle River, N.J.
Bathe, K. J., and Wilson, E. L. (1973). “Stability and accuracy analysis of direct integration Methods.” Earthquake Eng. Struct. Dyn., 1, 283–291.
Belytschko, T., and Hughes, T. J. R. (1983). Computational methods for transient analysis, Elsevier Science Publishers B.V., North-Holland, Amsterdam, The Netherlands.
Burden, R. L., and Faires, J. D. (1989). Numerical analysis, PWS-KENT, Boston.
Chang, S. Y. (1996). “A series of energy conserving algorithms for structural dynamics.” J. Chin. Inst. Eng., 19(2), 219–230.
Chang, S. Y. (1997). “Improved numerical dissipation for explicit methods in pseudodynamic tests.” Earthquake Eng. Struct. Dyn., 26, 917–929.
Chopra, A. N. (1997). Dynamics of structures, Prentice Hall, Upper Saddle River, N.J.
Clough, R. W., and Penzien, J. (1993). Dynamics of structures, McGraw–Hill, New York.
Hilber, H. M., and Hughes, T. J. R. (1978). “Collocation, dissipation, and ‘overshoot’ for time integration schemes in structural dynamics.” Earthquake Eng. Struct. Dyn., 6, 99–118.
Hilber, H. M., Hughes, T. J. R., and Taylor, R. L. (1977). “Improved numerical dissipation for time integration algorithms in structural dynamics.” Earthquake Eng. Struct. Dyn., 5, 283–292.
Houbolt, J. C. (1950). “A recurrence matrix solution for the dynamic response of elastic aircraft.” J. Aeronaut. Sci., 17, 540–550.
Hughes, T. J. R. (1987). The finite element method, Prentice–Hall, Inc., Englewood Cliffs, N.J.
Mugan, A., and Hulbert, G. M. (2001a). “Frequency-domain analysis of time-integration methods for semidiscrete finite element equations-Part I: Parabolic problems.” Int. J. Numer. Methods Eng., 51, 333–350.
Mugan, A., and Hulbert, G. M. (2001b). “Frequency-domain analysis of time-integration methods for semidiscrete finite element equations-Part II: Hyperbolic and parabolic-hyperbolic problems.” Int. J. Numer. Methods Eng., 51, 351–376.
Newmark, N. M. (1959). “A method of computation for structural dynamics.” J. Eng. Mech. Div., Am. Soc. Civ. Eng., 85, 67–94.
Park, K. C. (1975). “An improved stiffly stable method for direct integration of nonlinear structural dynamic equations.” J. Appl. Mech., 42, 464–470.
Sha, D., Zhou, X., and Tamma, K. K. (2003). “Time discretized operators. Part 2: Towards the theoretical design of a new generation of a generalized gamily of unconditionally stable implicit and explicit representations of arbitrary order for computational dynamics.” Comput. Methods Appl. Mech. Eng., 192, 291–329.
Strang, G. (1986). Linear algebra and its applications, Harcourt Brace–Jovanovich, San Diego.
Tamma, K. K., Sha, D., and Zhou, X. (2003). “Time discretized operators. Part 1: Towards the theoretical design of a new generation of a generalized gamily of unconditionally stable implicit and explicit representations of arbitrary order for computational dynamics.” Comput. Methods Appl. Mech. Eng., 192, 257–290.
Tamma, K. K., Zhou, X., and Sha, D. (2001). “A theory of development and design of generalized integration operators for computational structural dynamics.” Int. J. Numer. Methods Eng., 50, 1619–1664.
Zhou, X., and Tamma, K. K. (2004). “Design, analysis and synthesis of generalized single step solve and optimal algorithms for structural dynamics.” Int. J. Numer. Methods Eng., 59, 597–668.
Zienkiewicz, O. C. (1977). The finite element method, 3rd Ed., McGraw–Hill Ltd., London.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 132 • Issue 1 • January 2006
Pages: 34 - 45
Copyright
© ASCE.
History
Received: Dec 4, 2002
Accepted: Jul 14, 2005
Published online: Jan 1, 2006
Published in print: Jan 2006
Notes
Note. Associate Editor: Henryk Stolarski
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.