Prediction of post-elastic seismic response of structures by a mode superpostion technique
DOI:
https://doi.org/10.5459/bnzsee.16.3.222-233Abstract
For the determination of the dynamic response of a structural system to earthquake ground motion the mode superposition technique offers an alternative approach to the well established direct integration method. Whereas the principle of modal superposition is encountered commonly in elastic analyses, the response of a yielding structure is predicted almost universally by direct integration of the equation of motion.
In this paper the principles of modal superposition are extended into the post-elastic domain. Among the advantages which accrue from the modal based transformation are a potential reduction in the number of dynamic degrees-of-freedom considered in the solution, and a new insight into the structure response provided by the instantaneous dynamic properties of the structure.
References
Newmark, N.M., "A Method of Computation for Structural Dynamics", Proceedings, ASCE, Journal of the Engineering Mechanics Division, Vol. 85, No. EM3, July 1959.
Dunham, R.S., Nickell, R.E., Stickler, D.C., "Integration Operators for Transient Structural Response", Computers and Structures, Vol. 2, pp. 1-15, 1972. DOI: https://doi.org/10.1016/0045-7949(72)90019-3
Weeks, G., "Temporal Operators for Nonlinear Structural Dynamics Problems", Proceedings, ASCE, Journal of the Structural Division, Vol. 98, pp. 1087-1104, October, 1972.
Nickell, R.E., "Direct Integration Methods in Structural Dynamics", Proceedings, ASCE, Journal of the Engineering Mechanics Division, Vol. 99, No. EM2, April 1973.
Geradin, M., "A Classification and Discussion of Integration Operators for Transient Structural Response", American Institute of Aeronautics and Astronautics, Paper 74-105, 1974.
Hughes, T.J.R., "Stability, Convergence and Growth and Decay of . Energy of the Average Acceleration Method in Nonlinear Structural. Dynamics", Computers and Structures, Vol. 6, pp. 313-324, 1976. DOI: https://doi.org/10.1016/0045-7949(76)90007-9
Hilber, H.M. and Hughes, T.J.R., "collocation. Dissipation and 'Overshoot ' for Time Integration Scheme in Structural Dynamics", Earthquake Engineering and Structural Dynamics, Vol. 6, pp. 99-117, 1978. DOI: https://doi.org/10.1002/eqe.4290060111
Clough, R.W., "Analysis of Structural Vibrations of Dynamic Response", U.S.-Japan Seminar on Matrix Methods of Structural Analysis and Design, Tokyo, Japan, 1971.
Carmignani, C., Cella, A. and DePaulis, A., "Structural Dynamics by Finite Elements: Modal and Fourier Analysis", Conference on Structural Mechanics in Reactor Technology, 2nd, Berlin.
Bathe , K.J. and Wilson, E.L., "Numerical Methods in Finite Element Analysis", Prentice-Hall Inc., New Jersey, 1976.
Morris, N.F., "The Use of Mode Superposition in Nonlinear Dynamics", Computers and Structures, Vol. 7, pp. 65-72, 1977. DOI: https://doi.org/10.1016/0045-7949(77)90061-X
Nickell, R.E., "Nonlinear Dynamics by Mode Superposition", Computer Methods in Applied Mechanics and Engineering, Vol. 7, pp. 107-129, 1976. DOI: https://doi.org/10.1016/0045-7825(76)90008-6
Stricklin, J.A. and Haisler, W.E., "Formulations and Solution Procedures for Nonlinear Structural Analysis", Computers and Structures, Vol. 7, pp 125-136, 1977. DOI: https://doi.org/10.1016/0045-7949(77)90067-0
Nakao, Y. and Takano, S., "Elasto- plastic Seismic Response Analysis of Framed Structures by Mode Super- position" , Theory and Practice in Finite Element Structural Analysis, University of Tokyo Press, 1973.
Bathe, K.J., "Computational Methods in Structural Dynamics", Computing in Applied Mechanics AMD - Vol 18, pp 163-176, 1976 .
Cronin, D.L., "Numerical Integration of Uncoupled Equations of Motion using Recursive Digital Filtering", International Journal for Numerical Methods in Engineering, Vol. 6, pp. 137-152, 1973 . DOI: https://doi.org/10.1002/nme.1620060114
Dempsey, K. and Irvine, H.M., "A Note on the Numerical Evaluation of Duhamel's Integral", Earthquake Engineering and Structural Dynamics, Vol. 7, No. 1, 1979.
Caughey, T.K., "Classical Normal Modes in Damped Linear Dynamic Systems", A.S.M.E. Journal of Applied Mechanics, pp. 269-271 June 1960. DOI: https://doi.org/10.1115/1.3643949
Wilson, E.L. and Penzien, J., "Evaluation of Orthogonal Damping Matrices", International Journal for Numerical Methods in Engineering, Vol. 4, pp. 5-10, 1972. DOI: https://doi.org/10.1002/nme.1620040103
Warburton, G.B. and Soni, S.R., "Errors in Response Calculations for Non-Classically Damped Structures", Earthquake Engineering and Structural Dynamics, Vol. 5, pp. 365-376, 1977. DOI: https://doi.org/10.1002/eqe.4290050404
Clough, R.W. and Mojtahedi, S., "Earthquake Response Analysis Considering Non-Proportional Damping", Earthquake Engineering and Structural Dynamics, Vol. 4, pp. 489-496, 1976. DOI: https://doi.org/10.1002/eqe.4290040506
Sharpe, R.D. and Carr, A.J., "The Seismic Response of Inelastic Structures", Bulletin of the New Zealand National Society for Earthquake Engineering, Vol. 8 , No.3. 1975.
Wilkinson, J.H., "The Algebraic Eigenvalue Problem", Clarendon Press, Oxford, 1965.
Gupta, K.K., "Eigenproblem Solution by a Combined Sturm Sequence and Inverse Iteration Technique", International Journal for Numerical Methods in Engineering, Vol. 7 , pp. 17-42, 1973. DOI: https://doi.org/10.1002/nme.1620070103
Whiston, G.S., "On an Eigensystem Routine due to K.K. Gupta", International Journal for Numerical Methods in Engineering, Vol. 11, No. 4/ pp. 759-764, 1977. DOI: https://doi.org/10.1002/nme.1620110413
Dong, S.B. and Wolf, J.A., "A Direct-Iterative Eigensolution Technique", International Journal for Numerical Methods in Engineering, Vol. 4, pp. 155-161, 1972. DOI: https://doi.org/10.1002/nme.1620040202
Jennings, A. and Orr, D.R.L., "Application of the Simultaneous Iteration Method to Undamped Vibration Problems", International Journal for Numerical Methods in Engineering, Vol. 3, pp. 13-24, 1971. DOI: https://doi.org/10.1002/nme.1620030105
"Code of Practice for General Structural Design and Design Loadings for Buildings", NZS 4203, 1976. Standards Association of New Zealand.
Jury, R.D., "Seismic Load Demands on Columns of Reinforced Concrete Multistorey Frames", Research Report 78-12, Department of Civil Engineering, University of Canterbury.
Gillies, A.G., "Post-elastic Dynamic Analysis of Three-Dimensional Frame Structures", Unpublished Ph.D. Thesis, University of Auckland, New Zealand, 1979.
