Dynamic Analysis of Structures with Interval Uncertainty
Publication: Journal of Engineering Mechanics
Volume 140, Issue 4
Abstract
A new method for dynamic response spectrum analysis of a structural system with interval uncertainty is developed. This interval finite-element–based method is capable of obtaining the bounds on dynamic response of a structure with interval uncertainty. The present method is performed using a set-theoretic (interval) formulation to quantify the uncertainty present in the structure’s parameters, such as material properties and cross-sectional geometry. Independent and/or dependent variations for each element of the structure are considered. At each stage of analysis, the existence of variation is considered as presence of the perturbation in a pseudodeterministic system. Having this consideration, first, a linear interval eigenvalue problem is performed using the concept of monotonic behavior of eigenvalues for symmetric matrices subjected to nonnegative definite perturbations. Then, using the procedures for perturbation of invariant subspaces of matrices, the bounds on directional deviation (inclination) of each mode shape are obtained. Following this, the interval response spectrum analysis is performed considering the effects of input variation in terms of the structure’s total response, which includes maximum modal coordinates, modal participation factors, and mode shapes. Using this method, for the problems considered, it is shown that calculating the bounds on the dynamic response is more computationally efficient than the combinatorial or Monte Carlo–solution procedures. Several problems that illustrate the behavior of the method and comparison with combinatorial and Monte Carlo–simulation results are presented.
Get full access to this article
View all available purchase options and get full access to this article.
References
Alefeld, G., and Herzberger, J. (1983). Introduction to linear computation, Academic Press, New York.
Anderson, A. W., et al. (1952). “Lateral forces on earthquake and wind.” Trans. Am. Soc. Civ. Eng., 117(1), 716–754.
Biggs, J. M. (1964). Introduction to structural dynamics, McGraw Hill, New York.
Biot, M. A. (1932). “Transient oscillations in elastic systems—Chapter II: Vibrations of buildings during earthquake.” Ph.D. thesis, Aeronautics Dept., California Institute of Technology, Pasadena, CA.
Clough, R. W., and Penzien, J. (1993). Dynamics of structures, McGraw Hill, New York.
Dief, A. (1991). Advanced matrix theory for scientists and engineers, Abacus, Preston, U.K., 262–281.
Dwyer, P. S. (1951). Linear computations, Wiley, New York.
Heath, T. L., trans. (1897). The works of Archimedes, Cambridge University Press, Cambridge, U.K.
Housner, G. W. (1959). “Behavior of structures during earthquake.” Proc. Am. Soc. Civ. Eng., 85(4), 109–129.
Hudson, D. E. (1956). “Response spectrum techniques in engineering seismology.” Proc., World Conf. on Earthquake Engineering, Earthquake Engineering Research Institute, Berkeley, CA.
Looney, C. T. G. (1954). “Behavior of structures subjected to a forced vibration.” Proc. Am. Soc. Civ. Eng., 80(10), 451–462.
Modares, M., Mullen, R. L., and Muhanna, R. L. (2006). “Frequency analysis of a structure with bounded uncertainty.” J. Eng. Mech., 1363–1371.
Moens D., and Vandepitte D. (2007). “Interval sensitivity theory and its application to frequency response envelope analysis of uncertain structures.” Comput. Methods Appl. Mech. Eng., 196(21–24), 2486–2496.
Moore, R. E. (1966). Interval analysis, Prentice Hall, Englewood Cliffs, NJ.
Muhanna, R. L., and Mullen, R. L. (1999). “Formulation of fuzzy finite element methods for mechanics problems.” Comput. Aided Civ. Infrastruct. Eng., 14(2), 107–117.
Muhanna, R. L., and Mullen, R. L. (2001). “Uncertainty in mechanics problems: Interval-based approach.” J. Eng. Mech., 557–566.
Neumaier, A. (1990). Interval methods for systems of equations, Cambridge University Press, Cambridge, U.K.
Newmark, N. M. (1962). “A method of computation for structural dynamics.” Trans. Am. Soc. Civ. Eng., 127(1), 1406–1435.
Newmark, N. M., Blume, J. A., and Kapur, K. K. (1973). “Seismic design spectra for nuclear power plants.” J. Power Div., 99(2), 287–303.
Qiu, Z., and Elishakoff, I. (1998). “Antioptimization of structures with large uncertain but non-random parameters via interval analysis.” Comput. Methods Appl. Mech. Eng., 152(3–4), 361–372.
Rosenblueth, E., and Bustamente, J. I. (1962). “Distribution of structural response to earthquakes.” Proc. Am. Soc. Civ. Eng., 88(3), 75–106.
Stewart, G. W., and Sun, J.-G. (1990). Matrix perturbation theory, Academic Press, Boston.
Sunaga, T. (1958). “Theory of an interval algebra and its application to numerical analysis.” Jpn. J. Ind. Appl. Math., 26(2–3), 125–143.
Veletsos, A. S., and Newmark, N. M. (1957). “Natural frequencies of continuous flexural members.” Trans. Am. Soc. Civ. Eng., 122, 249–278.
Young, R. C. (1931). “The algebra of many-valued quantities.” Math. Ann., 104(1), 260–290.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 140 • Issue 4 • April 2014
Copyright
© 2013 American Society of Civil Engineers.
History
Received: Nov 8, 2010
Accepted: Apr 30, 2013
Published online: May 2, 2013
Published in print: Apr 1, 2014
Discussion open until: May 31, 2014
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.