This chapter addresses the numerical modeling of freestanding rigid blocks by means of a semi-discrete approach. The pure rocking motion of single rigid bodies can be easily studied with the differential equation of motion, which can be solved by numerical integration or by linearization. However, when we deal with sliding and jumping motion of rigid bodies, the mathematical formulation becomes quite complex. In order to overcome this complexity, a Semi-Discrete Model (SMD) is proposed for the study of rocking motion of rigid bodies, in which the rigid body is considered as a mass element supported by springs and dashpots, in the spirit of deformable contacts between rigid blocks. The SMD can detect separation and sliding of the body; however, initial base contacts do not change, keeping a relative continuity between the body and its base. Extensive numerical simulations have been carried out in order to validate the proposed approach.
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The study of the dynamic behavior of rigid bodies is an important task in the seismic assessment of structures, since selected structures or structural elements can be modeled as rigid bodies. Typical examples, among others, are: constructions formed by large stone blocks (Lemos, 2007; Papaloizou & Komodromos, 2012; Park & Kim, 2013); simple masonry structures which often fail forming large macro-blocks under seismic loadings (Sorrentino et al., 2008; Mohammadi & Yasrebi, 2010; Costa et al., 2013; Lagomarsino, 2015); machines, furniture, equipment, nuclear reactors, statues and art objects (Di Egidio & Contento, 2009; Erdik et al., 2010; Konstantinidis & Makris, 2009; Sharif et al., 2009); and base isolated buildings (Hoseini & Alavi, 2014; Komodromos et al., 2007, Palmeri & Makris, 2008).
Initial studies are from the late 19th century (Milne, 1881; Perry, 1881); however, Housner’s work (Housner, 1963) is considered as the first systematic study about the dynamics of rigid bodies. Housner has dealt only with pure rocking motion. After Housner, many authors have studied the dynamic motion of rigid bodies considering complex motion, as for example rocking plus sliding or bouncing (Anooshelpoor et al., 2004; Boroschek & Iruretagoyena, 2006; Hogan, 1990; Ishiyama, 1982; Plaut et al., 1996; Purvance et al., 2008; Taniguchi, 2002; Taniguchi & Miwa, 2007; Tso & Wong, 1989).
The pure rocking motion of single rigid bodies can be easily studied with the differential equations of motion proposed by Housner (1963) which can be solved by numerical integration or by linearization. However, when we deal with complex motion, the mathematical formulation becomes quite complex (Hogan, 1992; 1994; Nozaki et al., 2009; Spanos et al., 2001; Di Egidio & Contento, 2010; Voyagaki et al., 2014). In order to overcome this complexity, several authors have used other mathematical formulations, as the Discrete Element Method (Lemos, 2007; Peña et al., 2007; Komodromos et al., 2008) or have proposed novel analytical and numerical models (Andreus and Casini, 1999; Michaltsos and Raftoyiannis 2008; Nozaki et al., 2009; Prieto & Lourenço, 2005; Zulli et al., 2012). Thus, despite significant advances from past research, the study of complex motion of rigid bodies remains a challenging task.
In this chapter, a Semi-Discrete Model (SDM) is proposed for the study of freestanding blocks, in which the block is considered as a mass element supported by springs and dashpots, in the spirit of deformable contacts between rigid blocks (Andreaus & Casini, 1999).