**1. Introduction**

Despite the many progresses done in the modelling of rigid blocks, the grounding work for most of the research in this field remains [1], where a 2D model of the rigid block is obtained and the rocking and slide-rocking approximated conditions are written. Following papers on the dynamics of rigid bodies can be divided in two main groups, according to the kind of excitation used: earthquake excitation or sine-type pulse excitation (mainly one-sine). To the first group belong [2-5], in the second one, [6-10] can be found. In time, models of rigid blocks, very useful in many research fields, have been increased in complexity. Recently, for instance, sliding phenomena and the eccentricity of the center of mass have been considered (see [3, 11]). Some papers have been focused to specific problems, for example in [12] the behavior of two stacked rigid blocks has been considered, whereas in [13, 14] the attention has been pointed to blocks on flexible foundations. The dynamics and control of 2D blocks have also been analyzed in the framework of the bifurcation theory in [15, 16, 17].

The effects of the simultaneously presence of horizontal and vertical base excitations have been considered in some papers. For example, in [12, 18, 19] different problems related to the overturning of bi-dimensional rigid blocks have been studied in details.

Lately a large interest has been given to models of rigid bodies with base isolation systems, in order to improve the safety of art objects (see [20-22]). It has been proved that, in certain ranges of geometrical parameters of the rigid block, the effectiveness of base isolation can be amplified when coupling the isolating systems with devices able to limit the displacement of the oscillating base, so as to prevent the falling of the base of the body (see [23, 24]).

Recently, 3D models, mostly circular based, have been used in particular research fields, more precisely to study motions of a disk of finite thickness ([25, 26]), the wobbling of a frustum ([27]) or the sloshing in a tank ([28]).

© 2013 Contento et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Contento et al.; licensee InTech. This is a chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Contento et al.; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A three-dimensional model of rigid body with a rectangular base, able to rock around a side or a vertex of the base, already presented by the authors in [29], is used herein to further study the dynamic behavior of rigid blocks. In particular the effects of a vertical one-sine excitation, acting concurrently to the horizontal one, and the seismic response of rigid bodies are considered. The body can experience only rocking motion since it is herein assumed that it possesses a slender‐ ness for which bouncing is not triggered (see [22, 23]). Eccentricity of the center of mass, evaluat‐ ed with respect to the geometrical center of the parallelepiped that ideally circumscribes the body, is also considered. The equations of motion of the body are obtained making use of the balance of moments. Impacts between the base and the ground are treated by imposing the con‐ servation ofthe angular momentum before and after the impact. Starting conditions of rocking motion around a side or a vertex of the base are obtained by balancing the overturning moments and the resisting moments. Results are obtained by numerical integration of the equations of motion by using a IMSL routine developed in Fortran [30].

vertices with respect to the vertex *i* is indicated as **r**

**^**

allowed to rotate alternatively around one of the vertices, being this vertex in contact with the coordinate plane *z* =0. If the body is rocking about the *i*-th vertex (see Fig. 1e for *i* =*C*), the

where **R**(*t*) is the 3D finite rotation matrix which can be written in terms of three timedepending angles *ϑ*1(*t*), *ϑ*2(*t*), *ϑ*3(*t*) (see Appendix for a representation of the matrix **R**). Therefore the total acceleration of the generic point with respect to a fixed frame is written as

where **a***<sup>g</sup>* is the ground acceleration, **g** is the gravity acceleration and the dot stands for time

acting on the generic point of the block during the rocking motion around the vertex *i* is

**Figure 1.** Geometrical characteristics of the rigid block: (a) three-dimensional view; (b) x-z plane projection; (c) x-y

plane projection; (d-e) displacements of the rigid block: 3D rocking.

( ) ˆ *g i* **f a g Rr** = -+ r

position vector rotates about the vertex; its time evolution is described by

differentiation. If the mass per volume of the block is indicated as

**a**, which becomes, using Eq. (2),

*f* =ρ *ij*(*t*) (*i* = *A*, *B*, *C*, *D*, *j* ≠*i*). The body is

http://dx.doi.org/10.5772/54863

159

Seismic Behaviour of Monolithic Objects: A 3D Approach

() ()ˆ *i i* **r Rr** *t t* = (1)

<sup>ˆ</sup> *g i* **a a g Rr**= -+ && (2)

ρ

&& (3)

, the total volume force

Rocking and overturning curves that furnish the amplitude of the one-sine pulses able to uplift or to overturn the body, versus the angular direction of the horizontal excitation, are obtained. The role of the period of the excitations, the geometrical characteristics of the body and the eccentricity of the center of mass are also highlighted. Particular attention is focused to the relative phase between the horizontal and vertical excitations. The presence of the vertical pulse can strongly change the behavior of the system with respect to the case where only the horizontal excitation is considered.

Regarding the seismic excitation, three different registered Italian earthquakes, with different spectrum characteristics, are used in the analyses. Two type of analyses are performed in the pa‐ per: the first is conducted by varying the direction of the seismic input to point out if, for some di‐ rections, the 3D model of rigid block furnishes more accurate results than the classical 2D models; the second is performed by fixing the direction of the input with the aim to highlight the role of the mechanical and the geometrical characteristics of the rigid block in the seismic re‐ sponse. Also in this case, rocking and overturning curves, that furnish the amplitude of the seis‐ mic excitation able to uplift or to overturn the body versus the angular direction of the excitation, are obtained. The role of the type of spectrum of the seismic excitation, the geometrical and me‐ chanical characteristics of the body and the eccentricity of the mass center are also highlighted.

Finally, almost all the figures in the paper refer to a well-known statue. It is taken as example of the use of the model here discussed, but there are many other possible applications.
