Keywords: rigid block, unilateral model, structural dynamics, theoretical set up, experimental tests.

The rocking response and the possibility of overturning of rigid bodies during earthquakes represent important features in seismic safety problems. During strong ground shaking, rigid structures (such as electrical equipment, retaining walls, liquid storage tanks, tall rigid buildings, tombstones, etc.) slide, bounce, rock or overturn, sometimes resulting in a final substantial damage [1,2,3,4].

Despite the apparent simplicity, the motion of rigid blocks poses difficult problems to solve. The analysis of the rocking response of rigid blocks has attracted the attention of many researchers, especially from a theoretical point of view, and the relevant dynamics are less understood than in many other non-linear vibration problems [5].

The cause of the interest in treating the dynamics of rigid blocks is mainly due to possible applications: for example a broadly similar response is actually exhibited, during earthquakes, by ancient stone temples and sculptures saved in museums or in archaeological open spaces. In almost every destructive earthquake, free standing columns of monuments have survived undamaged in earthquakes that caused spectacular destruction around them.

Moreover, one should mention that a broadly similar response may be also observed, during earthquakes, in a variety of other cases, for example in slender water tanks, petroleum cracking towers, stacks of graphite of nuclear reactors [6] so on. Starting from the consideration that the dynamic evolution of the rocking motion of a rigid block is strongly affected by the impacts, which occur after a rocking cycle and which represent one of the most interesting features of this study, this paper introduces an original theory that permits the inclusion of the impact problem in a single formulation by developing a distributional approach; the approach presented develops a theory in which solutions are modeled by means of distributions, such as the Dirac-delta distribution. In the analysis of dynamics of a rigid block hitting a rigid plane surface, it is very useful to adopt an approximation of the impact of the block hitting the surface by means of a delta-function, offering some advantages that are clearly emphasized in the paper.

1

G.W. Housner, "The behavior of inverted pendulum structures during earthquakes", Bull. Seism. Soc. Am., 53(2), 404-417, 1963.

2

C.S. Yim, A.K. Chopra, J. Penzien, "Rocking Response of Rigid Blocks to Earthquakes", J. Earthquake Engineering and Structural Dynamics, 8(6), 565-587, 1980. doi:10.1002/eqe.4290080606

3

J. Milne, "Experiments in observational seismology", Trans. Seism. Soc. Japan, 3, 12-64, 1994.

4

I. Corbi, R. Orefice, "Study of Rocking Response of Rigid Blocks using Shaking Table Experiments", in B.H.V. Topping, C.A. Mota Soares, (Editors), "Proceedings of the Seventh International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 270, 2004.

5

N. Makris, J. Zhang, "Response and Overturning of Anchored Equipment under Seismic Excitation", Report No. PEER-98/05,Pacific Earthquake Engineering Research Center, University of California, Berkeley, USA, 1999.

6

W.K. Tso, C.M. Wong, "Steady-state Rocking Response of Rigid Blocks", parts I and II, J. Earthquake Engineering and Structural Dynamics, 18(6), 89-120, 1989. doi:10.1002/eqe.4290180109

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