Closed-Form Solutions for the Response of Linear Systems to Fully Nonstationary Earthquake Excitation
Publication: Journal of Engineering Mechanics
Volume 124, Issue 6
Abstract
New explicit closed-form solutions are derived for the evolutionary correlation and power spectral–density (PSD) matrices characterizing the nonstationary response of linear elastic, both classically and nonclassically damped, multi-degree-of-freedom (MDOF) systems subjected to a fully nonstationary earthquake ground motion process. The newly developed earthquake ground motion model considered represents the temporal variation of both the amplitude and the frequency content typical of real earthquake ground motions. To illustrate the analytical results obtained, a three-dimensional unsymmetrical building equipped with viscous bracings is considered with a single-component ground motion acting obliquely with respect to the building principal directions. These new analytical solutions for structural response statistics are very useful in gaining more physical insight into the nonstationary response behavior of linear dynamic systems subjected to realistic stochastic earthquake ground motion models. Furthermore, the evolution in time of the cross-modal correlation coefficients is examined and compared with the classical stationary solution for white-noise ground motion excitation. The effects of cross-modal correlations on various mean-square response quantities also are investigated using the analytical solutions obtained.
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
Caughey, T. K., and O'Kelly, M. E. J.(1965). “Classical normal modes in damped linear dynamic systems.”J. Appl. Mech., 32(3), 583–588.
2.
Conte, J. P., and Peng, B.-F.(1996). “An explicit closed-form solution for linear systems subjected to nonstationary random excitation.”Probabilistic Engrg. Mech., Essex, U.K., 11(1), 37–50.
3.
Conte, J. P., and Peng, B.-F.(1997). “Fully nonstationary analytical earthquake ground motion model.”J. Engrg. Mech., ASCE, 123(1), 15–24.
4.
DebChaudhury, A., and Gasparini, D. A. (1980). “State space random vibration theory.”Res. Rep. ENG 77-19364, Dept. of Civ. Engrg., Case Inst. of Technol., Cleveland, OH.
5.
Der Kiureghian, A.(1980). “Structural response to stationary excitation.”J. Engrg. Mech., ASCE, 106(6), 1195–1213.
6.
Foss, K. A.(1958). “Coordinates which uncouple the equations of motion of damped linear dynamic systems.”J. Appl. Mech., 25(3), 361–364.
7.
Hurty, W. C., and Rubinstein, M. F. (1964). Dynamics of structures. Prentice-Hall, Englewood Cliffs, N.J.
8.
Igusa, T.(1992). “Nonstationary response of structures with closely spaced frequencies.”J. Engrg. Mech., ASCE, 118(7), 1387–1405.
9.
Igusa, T., Der Kiureghian, A., and Sackman, J. L.(1984). “Modal decomposition method for stationary response of non-classically damped systems.”Earthquake Engrg. and Struct. Dynamics, 12(1), 121–136.
10.
Lin, Y. K. (1967). Probabilistic theory of structural dynamics. McGraw-Hill Inc., New York, N.Y.; Krieger Publishing Co., Melbourne, Fla., 1976.
11.
Lutes, L. D.(1986). “State space analysis of stochastic response cumulants.”Probabilistic Engrg. Mech., 1(2), 94–98.
12.
Lutes, L. D., and Chen, D. C. K.(1992). “Stochastic response moments for linear systems.”Probabilistic Engrg. Mech., 7(3), 165–173.
13.
Mau, S. T.(1988). “A subspace modal superposition method for non-classically damped systems.”Earthquake Engrg. and Struct. Dynamics, 16(6), 931–942.
14.
Papadimitriou, C., and Lutes, L. D. (1994). “Approximate analysis of higher cumulants for multi-degree-of-freedom vibration.”Probabilistic Engrg. Mech., 9(1-2), 71–82.
15.
Peng, B.-F. (1996). “Nonstationary ground motion model and applications to analysis and design of earthquake-resistant structures,” PhD thesis, Rice University, Dept. of Civ. Engrg., Houston, Tex.
16.
Pradlwarter, H. J., and Wenlung, L.(1991). “On the computation of the stochastic response of highly nonlinear large MDOF systems modeled by finite elements.”Probabilistic Engrg. Mech., 6(2), 109–116.
17.
Priestley, M. B. (1987). Spectral analysis and time series, Volume 1: univariate series, Volume 2: multivariate series, prediction and control. Academic Press, Inc., San Diego.
18.
Reid, J. G. (1983). Linear system fundamentals: continuous and discrete, classic and modern. McGraw-Hill, Inc., New York, N.Y.
19.
Roberts, J. B., and Spanos, P. D. (1990). Random vibration and statistical linearization. John Wiley & Sons, West Sussex, U.K.
20.
Singh, M. P.(1980). “Seismic response by SRSS for non-proportional damping.”J. Engrg. Mech., ASCE, 106(6), 1405–1419.
21.
Spanos, P. D. (1983). “Spectral moment calculation of linear system output.”J. Appl. Mech., 50(4a), 901–903.
22.
Veletsos, A. S., and Ventura, C. E.(1986). “Modal analysis of non-classically damped structures.”Earthquake Engrg. and Struct. Dynamics, 14(2), 217–243.
23.
Wen, Y.-K.(1980). “Equivalent linearization for hysteretic systems under random excitation.”J. Appl. Mech., 47(1), 150–153.
Information & Authors
Information
Published In
Copyright
Copyright © 1998 American Society of Civil Engineers.
History
Published online: Jun 1, 1998
Published in print: Jun 1998
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.