**2. Description of the considered mechanical system**

The base of the rigid block is supposed to be rectangular, with the four vertices indicated as *A*, *B*, *C*, *D* (see Fig. 1). The block is circumscribed by an ideal parallelepiped with upper vertices *E*, *F* , *H* , *I*, and lower vertices coinciding with *A*, *B*, *C*, *D*. The point *M* is the the centroid of the parallelepiped, while *G* is the centroid of the block.

When the body is at rest, at the *t* =*t*0 initial time, the position vector of a generic point with respect the *i*-th vertex (*i* = *A*, *B*, *C*, *D*) is indicated as **r ^** *i* and the position vector of the other three vertices with respect to the vertex *i* is indicated as **r ^** *ij*(*t*) (*i* = *A*, *B*, *C*, *D*, *j* ≠*i*). The body is 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 position vector rotates about the vertex; its time evolution is described by

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

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

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.

The base of the rigid block is supposed to be rectangular, with the four vertices indicated as *A*, *B*, *C*, *D* (see Fig. 1). The block is circumscribed by an ideal parallelepiped with upper vertices *E*, *F* , *H* , *I*, and lower vertices coinciding with *A*, *B*, *C*, *D*. The point *M* is the the

When the body is at rest, at the *t* =*t*0 initial time, the position vector of a generic point with

**^** *i*

and the position vector of the other three

motion by using a IMSL routine developed in Fortran [30].

158 Engineering Seismology, Geotechnical and Structural Earthquake Engineering

**2. Description of the considered mechanical system**

centroid of the parallelepiped, while *G* is the centroid of the block.

respect the *i*-th vertex (*i* = *A*, *B*, *C*, *D*) is indicated as **r**

horizontal excitation is considered.

$$\mathbf{r}\_i(t) = \mathbf{R}(t)\hat{\mathbf{r}}\_i \tag{1}$$

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

$$\mathbf{a} = \mathbf{a}\_{\frac{\alpha}{\beta}} - \mathbf{g} + \ddot{\mathbf{R}}\hat{\mathbf{r}}\_{l} \tag{2}$$

where **a***<sup>g</sup>* is the ground acceleration, **g** is the gravity acceleration and the dot stands for time differentiation. If the mass per volume of the block is indicated as ρ, the total volume force acting on the generic point of the block during the rocking motion around the vertex *i* is *f* =ρ**a**, which becomes, using Eq. (2),

$$\mathbf{f} = \rho(\mathbf{a}\_{\mathcal{g}} - \mathbf{g} + \ddot{\mathbf{R}}\hat{\mathbf{r}}\_i) \tag{3}$$

**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.
