Random Vibration of Flexible, Uncertain Beam Element
Publication: Journal of Engineering Mechanics
Volume 117, Issue 10
Abstract
A stochastic finite element method is developed to analyze the random responses of geometrically nonlinear beams and frames with combined uncertain material and geometric properties under simultaneous spatial and temporal random excitations. The method is based on an equivalent linearization scheme, combined with the mean‐centered second‐order perturbation technique and the modal expansion approach. The procedure can be straightforwardly extended to incorporate other existing finite elements. Examples include large‐amplitude free vibration and dynamic random response of three simply supported beams and a portal frame with combined uncertain material and geometric properties. For the case of beams and frame with no structural uncertainties, the results obtained are in good agreement with alternative solutions. For the case of beams and frame with structural uncertainties, alternative representative results are also obtained using Monte Carlo simulation to compare and validate the present solution and method.
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
Atalik, T. S., and Utku, S. (1976). “Stochastic linearization of multi‐degree‐of‐freedom nonlinear system.” Earthquake Engrg. Struct. Dyn. 4(4), 411–420.
2.
Bogdanoff, J. L., and Goldberg, J. E. (1960). “On the Bernoulli‐Euler beam theory with random excitation.” J. Aerosp. Sci. 27(5), 371–376.
3.
Branstetter, L., and Paez, T. (1986). “Dynamic response moments of random parametered structures with random excitation.” Proc. 3rd Conf. on Dynamic Response of Structures, Los Angeles, Calif., 668–675.
4.
Chang, C. C. (1989). “Random vibration of uncertain beam structures with geometrically nonlinearity,” thesis presented to the School of Aeronautics and Astronautics, Purdue University, at West Lafayette, Ind., in partial fulfillment of the requirements for the degree of Master of Science.
5.
Chen, C. T., and Yang, T. Y. (1988). “Random vibration of geometrically nonlinear finite element structures.” Proc. 5th ASCE Speciality Conf. on Probabilistic Methods in Civil Engineering, ASCE, New York, N.Y., 281–284.
6.
Clough, R. W., and Penzien, J. (1975). Dynamic of structures. McGraw‐Hill Company, New York, N.Y.
7.
Collins, J. D., and Thomas, W. J. (1969). “The eigenvalues problem for structural system with statistical properties.” AIAA J., 7(4), 642–648.
8.
Contreras, H. (1980). “The stochastic finite element.” Comput. Struct., 12(3), 341–348.
9.
Crandall, S. H., and Mark, W. D. (1963). “Random vibration in mechanical system.” Academic Press Inc., New York, N.Y.
10.
Crandall, S. H., and Zhu, W. Q. (1983). Random vibration: A survey of recent development.” J. Appl. Mech., 50(12), 953–962.
11.
Eringen, A. C. (1957). “Response of beams and plates to random loading.” J. Appl. Mech., ASME, 24(1), 46–52.
12.
Fox, R. L., and Kapoor, M. P. (1968). “Rate of change of eigenvalues and Eigenvectors.” AIAA J. 6(12), 2426–2429.
13.
Ghanem, R., and Spanos P. D. (1989). “Galerkin‐based response surface approach for reliability analysis.” Proc. 5th Int. Conf. on Structural Safety and Reliability, ASCE, New York, N.Y., 1081–1087.
14.
Handa, K., and Anderson, K. (1981). “Application of finite element methods in the stochastic analysis of structures.” Proc. 3rd Int. Conf. on Structural Safety and Reliability, ASCE, New York, N.Y., 409–417.
15.
Hasselman, T. K., and Hart, G. C. (1972). “Modal analysis of random structural system.” J. Engrg. Mech. Div., ASCE, 98(3), 561–579.
16.
Herbert, R. E. (1964). “Random vibration of a nonlinear elastic beam.” J. Acoust. Soc. Am., 36(11), 2090–2094.
17.
Herbert, R. E. (1965). “On the stresses in a nonlinear beam.” Int. J. Solids Struct., 1(2), 235–242.
18.
Hisada, T., and Nakagiri, S. (1981). “Stochastic finite element method developed for structural safety and reliability.” Proc. 3rd Int. Conf. on Structural Safety and Reliability, ASCE, New York, N.Y. 395–407.
19.
Hisada, T., and Nakagiri, S. (1985). “Role of stochastic finite element method in structural safety and reliability.” Proc. 4th Int. Conf. on Structural Safety and Reliability, ASCE, New York, N.Y., 385–394.
20.
Hoshiya, M., and Shah, H. C. (1969). “Free vibration of a beam‐column with stochastic properties.” Proc. ASCE‐EMD Speciality Conf. on Probabilistic Concepts and Method in Engineering, ASCE, New York, N.Y. 107–111.
21.
Iwan, W. D., and Mason, A. B., Jr. (1980). “Equivalent linearization for system subjected to non‐stationary random excitation.” Int. J. Non‐linear Mech., 15(2), 71–82.
22.
Lin, Y. K. (1976). Probabilistic theory of structural dynamics. McGraw‐Hill, New York, N.Y.
23.
Lin, Y. K., Kozen, F., Wen, Y. K., Casciati, F., Schueller, G. I., Kiureghian, A., Der Ditlevsen, O., and Vanmarcke, E. H. (1986). “Methods of stochastic structural dynamics.” Struct. Safety, 3(3/4), 167–194.
24.
Liu, W. K., Belytschko, T., and Mani, A. (1985). “Probabilistic finite elements for transient analysis in nonlinear continua.” Advances in Aerospace Structural Analysis AD‐09, Burnside and Parr, eds. American Society of Mech. Engrs. (ASME), New York, N.Y., 9–24.
25.
Liu, W. K., Belytschko, T., and Mani, A. (1986a). “Probabilistic finite element for nonlinear structural dynamics.” Comput. Methods Appl. Mech. Engrg., 56(1), 61–81.
26.
Liu, W. K., Belytschko, T., and Mani, A. (1986b). “Random field finite element.” Int. J. Numer. Methods Engrg., 23(10), 1831–1845.
27.
Mei, C., and Paul, D. B. (1986). “Nonlinear multimode response of clamped rectangular plates to acoustic loading.” AIAA J. 24(4), 643–648.
28.
Mochio, T., Samaras, E., and Shinozuka, M. (1985). “Stochastic equivalent linearization for finite element‐based reliability analysis.” Proc. 4th Int. Conf. on Structural Safety and Reliability, ASCE, New York, N.Y., 375–384.
29.
Nemat‐Nasser, S. (1968). “On the response of shallow thin shells to random excitation.” AIAA J. 6(7), 1327–1331.
30.
Roberts, J. B. (1984). “Techniques for nonlinear random vibration problems.” Shock Vib. Dig., 16(2), 3–14.
31.
Roberts, J. B., and Dunne, J. F. (1988). “Nonlinear random vibration in mechanical system.” Shock Vibr. Dig., 20(5), 15–25.
32.
Seide, P. (1976). “Nonlinear stresses and deflection of beams subjected to random time dependent uniform pressure.” J. Engrg. Indust., ASME, 8, 1014–1020.
33.
Shinozuka, M. (1972). “Probability modeling of concrete structures.” J. Engrg. Mech. Div., ASCE, 98(6), 1433–1451.
34.
Shinozuka, M., and Astill, C. J. (1972). “Random eigenvalue problem in structural analysis.” AIAA J. 10(4), 456–462.
35.
Shinozuka, M., and Dasgupta, M. (1986). “Stochastic finite element methods in dynamics.” Proc. 3rd Conf. on Dynamic Response of Structures, ASCE, New York, N.Y., 44–54.
36.
Shinozuka, M., and Deodatis, G. (1988). “Response variability of stochastic finite element system.” J. Engrg. Mech., ASCE, 114(3), 499–519.
37.
Spanos, P.‐T. D., and Iwan, W. D. (1978). “On the existence and uniqueness of solutions generated by equivalent linearization.” Int. J. Non‐Linear Mech., 13(2), 71–78.
38.
Spanos, P.‐T. D. (1980). “Formulation of stochastic linearization for symmetric or asymmetric M.D.O.F. nonlinear system.” J. Appl. Mech., 47(3), 209–211.
39.
Spanos, P.‐T. D., and Ghanem, R. (1989). “Stochastic finite element expansion for random media.” J. Engrg. Mech., ASCE, 115(5), 1035–1053.
40.
To, C. W. S. (1984). “The response of nonline structures to random excitation.” Shock Vib. Dig., 16(1), 13–33.
41.
To, C. W. S. (1987). “Random vibration of nonlinear system.” Shock Vib. Dig., 19(3), 3–9.
42.
Vanmarcke, E. H. (1977). “Probabilistic modeling of soil profile.” J. Geotech. Engrg. Div., ASCE, 103(11), 1227–1246.
43.
Vanmarcke, E. H. (1983a). “Stochastic finite element analysis of simple beams.” J. Engrg. Mech., ASCE, 109(5), 1203–1214.
44.
Vanmarcke, E. H. (1983b). Random field. The MIT Press, Cambridge, Mass.
45.
Vanmarcke, E. H., Shinozuka, M., Nakagiri, S., Schueller, G. I., and Grigoriu, M. (1986). “Random fields and stochastic finite elements.” Struct. Safety, 3(3/4), 143–166.
46.
Woinowsky‐Krieger, S. (1950). “The effect of an axial force on the vibration of hinged bars.” J. Appl. Mech., ASME, 17(1), 35–36.
48.
Yang, T. Y., and Han, A. D. (1983). “Buckled plate vibrations and large amplitude vibration using high‐order triangular elements.” AIAA J. 21(5), 758–766.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 117 • Issue 10 • October 1991
Pages: 2329 - 2350
Copyright
Copyright © 1991 ASCE.
History
Published online: Oct 1, 1991
Published in print: Oct 1991
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.