Robust Modeling of the Rocking Problem
Publication: Journal of Engineering Mechanics
Volume 138, Issue 3
Abstract
The rocking motion of a solid block on a moving deformable base is a dynamic problem that, despite its apparent simplicity, involves a number of complex dynamic phenomena such as impacts, sliding, geometric and material nonlinearities and, under some circumstances, chaotic behavior. For this reason, since the first model proposed by G.W. Housner in 1963, a number of alternative models have been proposed for its mathematical simulation. In this work, two new models are developed for the simulation of a rigid body experiencing a 2D rocking motion on a moving deformable base. The first model, the concentrated springs model, simulates the ground as tensionless vertical springs with vertical dampers placed at each of the two bottom corners of the body, whereas the second, the Winkler model, simulates the ground as a continuous medium of tensionless vertical springs with vertical dampers. Both models take into consideration sliding (with the use of both a penalty method and an analytical formulation for friction) and uplift and both are geometrically nonlinear. The models are used for simple free vibrational problems in which the effects of the ground deformability, sliding, and uplift are noted. In addition, the stability diagram for various parameters of the system, under excitation by ground motions that correspond to one full cycle sine pulses with varying amplitude and frequency, is created. The behavior of the two models is discussed and compared with the classic theory proposed by Housner.
Get full access to this article
View all available purchase options and get full access to this article.
References
Andreaus, U., and Casini, P. (1999). “On the rocking-uplifting motion of a rigid block in free and forced motion: Influence of sliding and bouncing.” Acta Mech., 138(3-4), 219–241.
Apostolou, M., Gazetas, G. G., and Garini, E. (2007). “Seismic response of slender rigid structures with foundation uplifting.” Soil Dyn. Earthquake Eng., 27(7), 642–654.
Cundall, P. A. (1971). “A computer model for simulating progressive, large scale movements in blocky rock systems.”Proc., Symp. of the Int. Society of Rock Mechanics, Vol. 2, No. 8, Nancy, France.
Housner, G. W. (1963). “The behavior of inverted pendulum structures during earthquakes.” Bull. Seismol. Soc. Am., 53(2), 403-417.
Ishiyama, Y. (1982). “Motion of rigid bodies and criteria for overturning by earthquake excitations.” Earthquake Eng. Struct. Dyn., 10(5), 635–650.
Koh, A., Spanos, P., and Roesset, J. (1986). “Harmonic rocking of rigid block on flexible foundation.” J. Eng. Mech., 112(11), 1165.
Lipscombe, P. R., and Pellegrino, S. (1991). “Free rocking of prismatic blocks.” J. Eng. Mech., 119(7), 1387–1410.
Olsson, H., Astrom, K. J., de Wit, C. Canudas, Gafvert, M., and Lischinsky, P. (1998). “Friction models and friction compensation.” Eur. J. Control, 4(3), 176–195.
Palmeri, A., and Makris, N. (2008). “Response analysis of rigid structures rocking on viscoelastic foundation.” Earthquake Eng. Struct. Dyn., 37(7), 1039–1063.
Pompei, A., Scalia, A., and Sumbatyan, M. A. (1998). “Dynamics of rigid block due to horizontal ground motion.” J. Eng. Mech., 124(7), 713–717.
Prieto, F., and Lourenco, P. B. (2005). “On the rocking behavior of rigid objects.” Meccanica, 40(2), 121–133.
Psycharis, I. N., and Jennings, P. C. (1983). “Rocking of slender rigid bodies allowed to uplift.” Earthquake Eng. Struct. Dyn., 11(1), 5776.
Rabinowicz, E. (1951). “The nature of the static and kinetic coefficients of friction.” J. Appl. Phys., 22(11), 1373–1379.
Shenton, H., and Jones, N. (1991). “Base excitation of rigid bodies. 1: Formulation.” J. Eng. Mech., 117(10), 2286–2306.
Wolf, J. P. (1994). Foundation Vibration Analysis Using Simple Physical Models, Prentice Hall, Upper Saddle River, NJ.
Wriggers, P. (1991). Computational Contact Mechanics, Springer, New York.
Yuan, L. (2005). “The effect of foundation flexibility on the dynamic response of a rigid block.” Ph.D. thesis, Notre Dame, Notre Dame, IN.
Zhang, J., and Makris, N. (2001). “Rocking response of free-standing blocks under cycloidal pulses.” J. Eng. Mech., 127(5), 473–483.
Zienkiewicz, O. C., and Taylor, R. L. (2005). The Finite Element Method for Solid and Structural Mechanics Sixth edition, Elsevier, Burlington, MA.
Information & Authors
Information
Published In
Copyright
© 2012 American Society of Civil Engineers.
History
Received: Jul 9, 2010
Accepted: Aug 18, 2011
Published online: Aug 20, 2011
Published in print: Mar 1, 2012
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.