Locating Events in Step‐by‐Step Integration of the Equations of Motion
Publication: Journal of Structural Engineering
Volume 117, Issue 2
Abstract
The calculation of time‐step sizes to prevent interstep events in multilinear problems is addressed. The treatment is restricted to the Newmark‐Beta family of implicit integration methods. Closed‐form equations in the fractional steps are derived for single‐degree‐of‐freedom (SDOF) systems. Explicit solutions are, however, not feasible because the equations are cubic. Numerical results indicate that the location of events can lead to marked improvements in accuracy, when compared to solutions that handle the nonlinear error at the ends of the time steps. It is contended that the derivation of closed‐form expressions for the substep sizes, in general multidegree‐of‐freedom (MDOF) systems, is precluded by coupling. Identification of the steps that contain events is prerequisite to any event location technique. In this regard, it is shown that a customary test to detect unloading, based on incremental hinge rotations, sometimes fails. An approach that does not suffer from the identified shortcoming is presented. The numerical stability characteristics of an approximate event location strategy based on interpolation is examined using an energy approach. It is shown that the constant‐average‐acceleration method does not satisfy the requirements for unconditional stability when the events are interpolated.
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References
1.
Allahabadi, R. (1987). “Drain‐2DX seismic response and damage assessment for 2D structures,” dissertation presented to the University of California, at Berkeley, Calif., in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
2.
Belytschko, T., and Schoeberle, D. F. (1975). “On the unconditional stability of an implicit algorithm for nonlinear structural dynamics.” J. Appl. Mech., 97, 865–869.
3.
Esteva, L., and Guerra, O. R. (1976). “Efectos del Comportamiento no Lineal en la Respuesta Sismica de Estructuras.” Report to the Secretaria de Obras Publicas, Instituto de Ingenieria, UNAM, Mexico (in Spanish).
4.
Ghose, A. (1974). “Computational procedures for inelastic dynamic analysis,” dissertation presented to the University of California, at Berkeley, Calif., in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
5.
Golafshani, A. (1982). “A computer program for inelastic seismic response of structures,” dissertation presented to the University of California, at Berkeley, Calif., in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
6.
Hilber, H. M. (1976). “Analysis and design of numerical integration methods in structural dynamics.” Report No. EERC 76‐29, Earthquake Engrg. Res. Ctr., Univ. of California, Berkeley, Calif.
7.
Hughes, T. J. R., and Belytschko, T. (1983). “A precis of developments in computational methods for transient analysis.” J. Appl. Mech., 50, (Dec.), 1033–1041.
8.
Kannan, A. E., and Powell, G. H. (1973). “DRAIN‐2D, a general purpose computer program for dynamic analysis of inelastic plane structures.” Report No. UCB/EERC 73‐6, Earthquake Engrg. Res. Ctr., Univ. of California, Berkeley, Calif.
9.
Nau, J. M. (1983). “Computation of inelastic spectra.” J. Engrg. Mech. Div., ASCE, 109(1), 279–288.
10.
Newmark, N. M. (1959). “A method of computation for structural dynamics.” J. Engrg. Mech. Div., ASCE, 85(3), 67–94.
11.
Nigam, N. C., and Jennings, P. C. (1968). “Digital calculation of response spectra from strong motion earthquake records.” Earthquake Engrg. Res. Lab., California Inst. of Tech., Pasadena, Calif.
12.
Nigam, N. C., and Jennings, P. C. (1969). “Calculation of response spectra from strong motion earthquake records.” Bulletin of the Seismological Soc. of America, 59(2), 909–922.
13.
Powell, G. H. (1970). “Computer evaluation of automobile barrier system.” Report No. UC SESM 70‐17, Federal Highway Admin., Univ. of California, Berkeley, Calif.
14.
Richtmyer, R. D., and Morton, K. W. (1967). Difference methods for initial‐value problems. 2nd Ed., Interscience Publishers, Division of John Wiley and Sons, New York, N.Y.
15.
Villaverde, R. (1984). Discussion of “Computation of inelastic response spectra,” by J. M. Nau, J. Engrg. Mech. Div., ASCE, 109(1), 279–288.
16.
Villaverde, R., and Lamb, R. C. (1989). “Scheme to improve numerical analysis of hysteretic dynamic systems.” J. Struct. Engrg., ASCE, 115(1), 228–233.
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Copyright © 1990 ASCE.
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Published online: Feb 1, 1991
Published in print: Feb 1991
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