**Appendix A. Vector and tensor quantities**

The rotation **R** is the composition of three planar rotations: if {**e**x, **e***y*, **e***z*} is the canonical basis, the first rotation, indicated as **R**1, of angle *ϑ*1, is around the axis **e**x; the second, indicated as **R**2, of angle *ϑ*2, around the axis **R**1**e***y*; the third, indicated as **R**3, of angle *ϑ*3, around the axis **R**2**R**1**e***z*. The representation of **R** on the canonical basis is

#### Seismic Behaviour of Monolithic Objects: A 3D Approach http://dx.doi.org/10.5772/54863 179

$$
\begin{bmatrix}
\mathbf{R}(t)\end{bmatrix}\_{\mathbf{e}\_{x,y,z}} = \begin{pmatrix}
\mathbf{c}\_2\mathbf{c}\_3 & \mathbf{s}\_1\mathbf{s}\_2\mathbf{c}\_3 - \mathbf{c}\_1\mathbf{s}\_2 & \mathbf{c}\_1\mathbf{s}\_2\mathbf{c}\_3 + \mathbf{s}\_1\mathbf{s}\_3 \\
\mathbf{c}\_2\mathbf{s}\_3 & \mathbf{c}\_1\mathbf{c}\_3 + \mathbf{s}\_1\mathbf{s}\_2\mathbf{s}\_3 & \mathbf{c}\_1\mathbf{s}\_2\mathbf{s}\_3 + \mathbf{s}\_1\mathbf{c}\_3 \\
\end{bmatrix} \tag{13}
$$

where, for *k* =1, 2, 3,

the existence of a region where the overturning PGA during the 3D motion is smaller than the

Finally, when the body is excited by a narrow spectrum earthquake, it is always possible to find cases where a 3D model of rigid block is necessary to evaluate the seimic responce in

The rocking motion around a side or a vertex of a rectangular based rigid body has been deeply studied, making use of a three-dimensional model already proposed by the same authors. Starting conditions of motion have been found by means of equilibrium between overturning and resisting moments, whereas the impact has been described considering the conservation

The dynamics of the rigid body excited by one-sine pulse horizontal and vertical excitations and horizontal seismic excitation has been analyzed. Rocking and overturning curves versus the angular direction of the horizontal pulse have been obtained. The influence on the motion of several parameters, such as the period of the excitations, the geometrical characteristics of

The vertical one-sine pulse strongly modifies the behavior of the system with respect to the case where only horizontal excitation acts on the body. Results show that, in pres‐ ence of vertical excitation and in significant ranges of the parameters, as happens when just horizontal base acceleration is considered, bi-dimensional models are not enough ac‐ curate to correctly evaluate the occurrence of the overturning and, therefore, a three-di‐

The seismic response of the rigid body excited by three different Italian registered earth‐ quakes has been analyzed, reporting rocking and overturning curves. Results show that, for narrow spectrum earthquakes, bi-dimensional models are not enough accurate to cor‐ rectly evaluate the occurrence of the overturning since, in significant sectors inside the 3D rocking regions, the overturning amplitudes are smaller than the ones given by the 2D models. Hence a 3D model of rigid block is necessary to evaluate the seismic re‐

The rotation **R** is the composition of three planar rotations: if {**e**x, **e***y*, **e***z*} is the canonical basis, the first rotation, indicated as **R**1, of angle *ϑ*1, is around the axis **e**x; the second, indicated as **R**2, of angle *ϑ*2, around the axis **R**1**e***y*; the third, indicated as **R**3, of angle *ϑ*3, around the axis

the body and of the eccentricity of the mass center has been pointed out.

one during the 2D motion.

178 Engineering Seismology, Geotechnical and Structural Earthquake Engineering

of the angular momentum.

mensional model is needed.

sponse of a rigid block in favour of safety.

**Appendix A. Vector and tensor quantities**

**R**2**R**1**e***z*. The representation of **R** on the canonical basis is

favour of safety.

**6. Conclusion**

$$\begin{aligned} \mathbf{c}\_k &:= \cos(\mathcal{G}\_k(t)) \\ \mathbf{s}\_k &:= \sin(\mathcal{G}\_k(t)) \end{aligned} \tag{14}$$

When the block is a parallelepiped of uniform mass density, with sides of length 2*b*x, 2*by*, 2*h* , respectively, the positions of the base vertices are

$$\begin{aligned} \hat{\mathbf{x}}\_A &= 0\\ \hat{\mathbf{x}}\_B &= \hat{\mathbf{x}}\_A + 2b\_x \mathbf{e}\_x\\ \hat{\mathbf{x}}\_C &= \hat{\mathbf{x}}\_A + 2b\_x \mathbf{e}\_x + 2b\_y \mathbf{e}\_y\\ \hat{\mathbf{x}}\_D &= \hat{\mathbf{x}}\_A + 2b\_y \mathbf{e}\_y \end{aligned} \tag{15}$$

The mass is *m*=8ρ*b*x*byh* . The static moment with respect to the point *A* is

$$
\hat{\mathbf{s}}\_A = m(\mathbf{b}\_x \mathbf{e}\_x + \mathbf{b}\_y \mathbf{e}\_y + h \mathbf{e}\_y) \tag{16}
$$

The representation of the Euler tensor with respect to the point *A* is

$$\begin{aligned} \left[\hat{\mathbf{J}}\_A\right]\_{\mathbf{e}\_{x,y,z}} &= m \begin{bmatrix} \frac{\mathbf{4}}{3} b\_x^2 & b\_x b\_y & b\_x h \\ b\_x b\_y & \frac{\mathbf{4}}{3} b\_y^2 & b\_y h \\ b\_x h & b\_y h & \frac{\mathbf{4}}{3} h^2 \end{bmatrix} \end{aligned} \tag{17}$$

To get the generic static moment *s* **^** *i* and the generic Euler tensor *J* **^** *ji* , the transport rules read:

$$\begin{aligned} \hat{\mathbf{s}}\_{i} &= \hat{\mathbf{s}}\_{A} + m(\hat{\mathbf{x}}\_{A} - \hat{\mathbf{x}}\_{i}) \\ \hat{\mathbf{J}}\_{ji} &= \hat{\mathbf{J}}\_{A} + \hat{\mathbf{s}}\_{A} \otimes (\hat{\mathbf{x}}\_{A} - \hat{\mathbf{x}}\_{i}) + (\hat{\mathbf{x}}\_{A} - \hat{\mathbf{x}}\_{j}) \otimes \hat{\mathbf{s}}\_{A} \\ &+ m(\hat{\mathbf{x}}\_{A} - \hat{\mathbf{x}}\_{j}) \otimes (\hat{\mathbf{x}}\_{A} - \hat{\mathbf{x}}\_{i}) \end{aligned} \tag{18}$$

where the tensor product ⊗ is defined such that (*u* ⊗ *v*)*w* =(*u* ⋅*w*)*v* for any vectors *u*, *v*, *w* of the same vector space.

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