**3. General formulation**

#### **3.1. Equations of motion**

The equations of motions have been already presented in [29], where they are obtained imposing the balance of the moments acting on the body. In particular, when the body is rocking around the *i*-th vertex, which is assumed to lie on the horizontal, coordinate plane *z* =0, and is identified by its initial position **x ^** *i* (*t*), *i* = *A*, *B*, *C*, *D* , the equations of motion read

$$\text{skw}\left(\hat{\mathbf{s}}\_{i}\otimes(\mathbf{a}\_{\boldsymbol{\varrho}}(t)-\ddot{\mathbf{R}}(t)(\hat{\mathbf{x}}\_{i}-\hat{\mathbf{x}}\_{A})-\mathbf{g})\mathbf{R}^{T}(t)-\ddot{\mathbf{R}}(t)\hat{\mathbf{j}}\_{i\boldsymbol{\Delta}}\mathbf{R}^{T}(t)\right)=\mathbf{O}\tag{4}$$

(1 ) (1 ) *G G x z y z y y x x h h h h bb bb*

where 2*b*x, 2*by*, 2*h* are the lengths of the three edges of the parallelepiped, respectively, *e*x, *ey*, *ez* are the components of the distance vector between the center of mass of the body and the center

An initial uplift around a side of the rectangular base leads to a 2D motion. In this case a rocking motion takes place when the overturning moment is equal or greater than the resisting

uplift around a side parallel to the x direction (AB in Fig. 2a) is considered. Similar conditions can be found for the orthogonal directions, since the mechanical system is symmetric. Thus

( ) *<sup>y</sup>*

1 *y y z x*

<sup>+</sup> = . &&

e

An uplift on a vertex can occur either during a 2D rocking motion or directly from the full

In order to uplift the body directly from the rest, the rocking conditions around the two adjacent sides of the base have to be simultaneously satisfied. For example, to uplift the body and, consequently, to get 3D motion around the vertex *A*, rocking conditions around AB and AD

When the body is rocking around a side of the base, AB as example, the uplift condition on the vertex *A* is similar to the one for the 2D rocking motion on the side AD. The overturning

( ) *<sup>x</sup>*

*M ma M mu b e o g r zx x* = = = +. *h*¢*<sup>G</sup>* && (9)

l

*u u*

is the component of the ground acceleration along *y*. Using nondimensional

e

*M ma h M mu b e o gG r zy y* = == + && (7)

&& (8)

(6)

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161

Seismic Behaviour of Monolithic Objects: A 3D Approach

*<sup>z</sup>* of the center of mass. An initial

 := = + ; := = + e l

l

moment, due to the vertical component of the acceleration *u*¨

of the parallelepiped, and *h <sup>G</sup>* =*h* + *ez*.

*3.2.1. Starting condition of 2D rocking*

the uplift occurs when:

quantities, Eq. (7) reads:

have to be satisfied.

*3.2.2. Starting condition of 3D rocking*

contact phase. In both cases a 3D motion is obtained.

moment *Mo* has to be at least equal to the resisting moment *Mr*:

where a*gy*

where *s* **^** *<sup>i</sup>* : =*∫ C* ^ ρ(**x ^** −**x ^** *i* )*dV* is the vector of (initial) static moment of the body with respect to the vertex *i* (**x ^** is the initial position of a generic point of the block); ⊗ is the tensor product; **a***g*(*t*) is the imposed base acceleration vector; **R**(*t*) is the 3D rotation matrix, which in turn depends on three angles *ϑ*1(*t*), *ϑ*2(*t*), *ϑ*3(*t*); *J* **^** *iA* : =*∫ C* ^ ρ(**x ^** −*bhi* ) ⊗ (**x ^** −**x ^** *<sup>A</sup>*)*dV* is the (initial) Euler tensor with respect to the vertices *i* and *A*; **g** is the gravity acceleration vector; the superscript *T* stands for transpose and the dot for time derivative; skw() is the skew part of the tensor in argument (see Fig. 1 and, for details, [29]). In Appendix A all the tensor and the vector quantities appearing in Eq. (4) are explicitly written, to make possible the reproduction of the numerical simulations reported in the following sections. Equations (4) reduce to the special case of 2D motion of the block around a side of the base (the same equation in [22]), when **R**(*t*) describes a planar rotation around one of the coordinate axes x or *y*.

#### **3.2. Starting and ending conditions**

The rigid block is assumed to be in a full-contact condition with the ideal horizontal support at the beginning of the base excitation. The rocking phase begins when the rigid block uplifts. An uplift occurs when the resisting moment *Mr* due to the weight of the body and to the vertical external acceleration is smaller than the overturning moment *Mo* due to the horizontal inertial forces. The uplift can occur around a side of the rectangular base (2D rocking motion) or around one of the four lower vertices of the base (3D rocking motion).

The eccentricity of the mass center with respect to the geometrical center of the parallelepiped, when considered, is obtained by introducing a concentrated mass *mE* =*βm* and, however, always keeping the total mass *m* of the body constant. Referring to Fig. 1, the following nondimensional eccentricities are introduced to characterize the system:

$$
\varepsilon\_x = \frac{e\_x}{b\_x}; \varepsilon\_y = \frac{e\_y}{b\_y}; \varepsilon\_z = \frac{e\_z}{h} \tag{5}
$$

and the slendernesses:

Seismic Behaviour of Monolithic Objects: A 3D Approach http://dx.doi.org/10.5772/54863 161

$$\mathcal{A}\_{\chi} := \frac{h\_G}{b\_y} = \frac{h}{b\_y} (1 + \varepsilon\_z); \mathcal{A}\_y := \frac{h\_G}{b\_x} = \frac{h}{b\_x} (1 + \varepsilon\_z) \tag{6}$$

where 2*b*x, 2*by*, 2*h* are the lengths of the three edges of the parallelepiped, respectively, *e*x, *ey*, *ez* are the components of the distance vector between the center of mass of the body and the center of the parallelepiped, and *h <sup>G</sup>* =*h* + *ez*.

#### *3.2.1. Starting condition of 2D rocking*

**3. General formulation**

*z* =0, and is identified by its initial position **x**

160 Engineering Seismology, Geotechnical and Structural Earthquake Engineering

on three angles *ϑ*1(*t*), *ϑ*2(*t*), *ϑ*3(*t*); *J*

**3.2. Starting and ending conditions**

and the slendernesses:

rotation around one of the coordinate axes x or *y*.

one of the four lower vertices of the base (3D rocking motion).

nondimensional eccentricities are introduced to characterize the system:

The equations of motions have been already presented in [29], where they are obtained imposing the balance of the moments acting on the body. In particular, when the body is rocking around the *i*-th vertex, which is assumed to lie on the horizontal, coordinate plane

*<sup>i</sup> <sup>g</sup> i A iA* **<sup>s</sup>** Ä - -- - = **a R** *tt tt t* **x x gR R R O <sup>J</sup>** && && (4)

)*dV* is the vector of (initial) static moment of the body with respect to the

is the initial position of a generic point of the block); ⊗ is the tensor product; **a***g*(*t*)

) ⊗ (**x ^** −**x ^**

is the imposed base acceleration vector; **R**(*t*) is the 3D rotation matrix, which in turn depends

respect to the vertices *i* and *A*; **g** is the gravity acceleration vector; the superscript *T* stands for transpose and the dot for time derivative; skw() is the skew part of the tensor in argument (see Fig. 1 and, for details, [29]). In Appendix A all the tensor and the vector quantities appearing in Eq. (4) are explicitly written, to make possible the reproduction of the numerical simulations reported in the following sections. Equations (4) reduce to the special case of 2D motion of the block around a side of the base (the same equation in [22]), when **R**(*t*) describes a planar

The rigid block is assumed to be in a full-contact condition with the ideal horizontal support at the beginning of the base excitation. The rocking phase begins when the rigid block uplifts. An uplift occurs when the resisting moment *Mr* due to the weight of the body and to the vertical external acceleration is smaller than the overturning moment *Mo* due to the horizontal inertial forces. The uplift can occur around a side of the rectangular base (2D rocking motion) or around

The eccentricity of the mass center with respect to the geometrical center of the parallelepiped, when considered, is obtained by introducing a concentrated mass *mE* =*βm* and, however, always keeping the total mass *m* of the body constant. Referring to Fig. 1, the following

*y x z*

=;=;= (5)

*e e e bb h*

*xyz x y*

eee

(*t*), *i* = *A*, *B*, *C*, *D* , the equations of motion read

*<sup>A</sup>*)*dV* is the (initial) Euler tensor with

**^** *i*

( ) skw ( ( ) ( )( ) ) ( ) ( ) ( ) ˆ ˆ ˆ ˆ *T T*

**^** *iA* : =*∫ C* ^ ρ(**x ^** −*bhi*

**3.1. Equations of motion**

where *s* **^** *<sup>i</sup>* : =*∫ C* ^ ρ(**x ^** −**x ^** *i*

vertex *i* (**x ^**

An initial uplift around a side of the rectangular base leads to a 2D motion. In this case a rocking motion takes place when the overturning moment is equal or greater than the resisting moment, due to the vertical component of the acceleration *u*¨ *<sup>z</sup>* of the center of mass. An initial uplift around a side parallel to the x direction (AB in Fig. 2a) is considered. Similar conditions can be found for the orthogonal directions, since the mechanical system is symmetric. Thus the uplift occurs when:

$$M\_o = ma\_{\mathcal{g}\_y} h\_G = M\_r = m\ddot{u}\_z (b\_y + e\_y) \tag{7}$$

where a*gy* is the component of the ground acceleration along *y*. Using nondimensional quantities, Eq. (7) reads:

$$\frac{\ddot{\boldsymbol{m}}\_y}{\ddot{\boldsymbol{m}}\_z} = \frac{1 + \boldsymbol{\varepsilon}\_y}{\lambda\_x}. \tag{8}$$

#### *3.2.2. Starting condition of 3D rocking*

An uplift on a vertex can occur either during a 2D rocking motion or directly from the full contact phase. In both cases a 3D motion is obtained.

In order to uplift the body directly from the rest, the rocking conditions around the two adjacent sides of the base have to be simultaneously satisfied. For example, to uplift the body and, consequently, to get 3D motion around the vertex *A*, rocking conditions around AB and AD have to be satisfied.

When the body is rocking around a side of the base, AB as example, the uplift condition on the vertex *A* is similar to the one for the 2D rocking motion on the side AD. The overturning moment *Mo* has to be at least equal to the resisting moment *Mr*:

$$M\_o = m a\_{\mathcal{g}\_x} \mathbb{H}\_G = M\_r = m \ddot{u}\_z (b\_x + e\_x). \tag{9}$$

In this case, *h* ′ *<sup>G</sup>* is the actual height of the center *G* of the body during a rocking around the x-axis (see Fig. 2b). The quantity *h* ′ *<sup>G</sup>* can be evaluated as in [22] by the relation *h* ′ *<sup>G</sup>* =*h <sup>G</sup>cosϑ*<sup>x</sup> + (*by* + *ey*)*sinϑ*x. By using the nondimesional quantities of Eq. (5), Eq. (9) reads:

$$\frac{\ddot{\mu}\_x}{\ddot{\mu}\_z} = \frac{1 + \varepsilon\_x}{\lambda\_y} \frac{1}{\left(\frac{1 + \varepsilon\_y}{\lambda\_x} \sin \mathcal{G}\_x + \cos \mathcal{G}\_x\right)}\tag{10}$$

*3.2.3. Rocking termination and collapse conditions*

the same than those just before the impact (instant*t* <sup>−</sup>

(*t*)=**R**<sup>−</sup>

**3.3. Impact conditions**

and therefore **R**<sup>+</sup>

conditions read

(*t*) and *ϑ*˙ <sup>3</sup>

**4.1. One-sine excitation**

used in the analyses are

+

**4. Description of the excitations**

where *J* **^** *<sup>j</sup>* : =*∫ C* ^ ρ(**x ^** −**x ^** *j* ) ⊗ (**x ^** −**x ^** *j*

*ϑ*˙ 1 + (*t*), *ϑ*˙ <sup>2</sup> +

No particular conditions are assumed to describe the termination of the motion and the return to the full-contact phase. This means that the rocking motion finishes when the energy associated to this phase is completely dissipated. A collapse event occurs when the body overturns. In the analyses, this condition conventionally manifests itself when one of the four

The impact conditions are taken from [29] too. They model the process of changing of the vertex around which the rocking occurs, and take place when the base of the block hits the ideal, horizontal, coordinate plane *z* =0. Positions of the body underneath this plane are not allowed. For instance, if the body is rocking around the vertex *i* (*i* = *A*, *B*, *C*, *D*) and, at some special time *t*, another vertex, say *j* ( *j* ≠*i*), hits the horizontal plane, then the impact process happens

three angles after the impact is made by equating the angular momentum around the (new) center of rotation just after and before the impact (see [29], [31]). In particular, the impact

ˆ ˆ () () () () *T T*

Equations (11) provide a linear non-homogeneous algebraic system in the unknowns

The three-dimensional rigid body is excited by a one-sine pulse acceleration applied to the base of the body and acting along the horizontal direction (a*<sup>h</sup>* (*t*)) or both along the horizontal and the vertical directions (a*<sup>h</sup>* (*t*), a*v*(*t*)). The analyses are performed by varying the direction of the horizontal excitation, the period of the sine-pulses and their amplitudes. The direction is measured by a counterclockwise angle starting from the *x*-axis. The pulse-type acceleration

(*t*) and *ϑ*˙ <sup>3</sup>

− (*t*).

)*dV* and *J*

− (*t*), *ϑ*˙ <sup>2</sup> −

(*t*), in terms of *ϑ*˙ <sup>1</sup>

**^** *ji* : =*∫ C* ^ ρ(**x ^** −**x ^** *j* ) ⊗ (**x ^** −**x ^** *i*

):*ϑ*<sup>1</sup> + (*t*)=*ϑ*<sup>1</sup> − (*t*), *ϑ*<sup>2</sup> + (*t*)=*ϑ*<sup>2</sup> − (*t*), *ϑ*<sup>3</sup> + (*t*)=*ϑ*<sup>3</sup> − (*t*),

(*t*). On the other hand, the evaluation of the time-derivatives of the

Seismic Behaviour of Monolithic Objects: A 3D Approach

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

163

*<sup>j</sup> ji ttt t* + - **R JR R J R** <sup>=</sup> (11)

) are exactly

)*dV* are (initial) Euler tensors.

and *j* becomes the new center or rotation. The angles just after the impact (instant*t* <sup>+</sup>

upper vertices of the parallelepiped containing the body hits the ground.

**Figure 2.** rigid block: (a) forces acting during the full-contact phase; (b) forces acting during a 2D rocking around the AB side; (c) shape of the body used in the numerical simulations.

The natural symmetry of the mechanical system leads to similar conditions for an uplift on one of the other three vertices of the base.

For a square based body with no eccentricity and under an excitation directed along the diagonal, the 3D motion around one of the vertices along the diagonal is directly triggered. This fact highlights how plausible is the occurrence of a 3D rocking motion for a square or near-square based body. An uplift on a vertex could easily manifest itself also in the case of excitation close to the diagonal, just after the occurrence of an uplift on a side and, therefore, during a 2D rocking motion. In fact the vertical position of the center of mass can increase enough (*h* ′ *<sup>G</sup>* >*h <sup>G</sup>*, see Fig. 2b ) to satisfy Eq. (9).

On the contrary, for bodies with rectangular base and a side significantly larger than the other, an overturn around the largest side of the base is much more likely to occur before the uplift on a vertex.

#### *3.2.3. Rocking termination and collapse conditions*

No particular conditions are assumed to describe the termination of the motion and the return to the full-contact phase. This means that the rocking motion finishes when the energy associated to this phase is completely dissipated. A collapse event occurs when the body overturns. In the analyses, this condition conventionally manifests itself when one of the four upper vertices of the parallelepiped containing the body hits the ground.

#### **3.3. Impact conditions**

In this case, *h* ′

*<sup>G</sup>* =*h <sup>G</sup>cosϑ*<sup>x</sup> + (*by*

*h* ′

x-axis (see Fig. 2b). The quantity *h* ′

*<sup>G</sup>* is the actual height of the center *G* of the body during a rocking around the

+ *ey*)*sinϑ*x. By using the nondimesional quantities of Eq. (5), Eq. (9) reads:

 J

&& (10)

1 1 1

*x x*

<sup>+</sup> <sup>=</sup>

e

l

*u u*

162 Engineering Seismology, Geotechnical and Structural Earthquake Engineering

AB side; (c) shape of the body used in the numerical simulations.

*<sup>G</sup>* >*h <sup>G</sup>*, see Fig. 2b ) to satisfy Eq. (9).

one of the other three vertices of the base.

enough (*h* ′

on a vertex.

&&

*x*

l

e

*<sup>z</sup> <sup>y</sup> x x*

sin cos *<sup>y</sup>*

+

 æ ö <sup>+</sup> ç ÷ è ø

J

**Figure 2.** rigid block: (a) forces acting during the full-contact phase; (b) forces acting during a 2D rocking around the

The natural symmetry of the mechanical system leads to similar conditions for an uplift on

For a square based body with no eccentricity and under an excitation directed along the diagonal, the 3D motion around one of the vertices along the diagonal is directly triggered. This fact highlights how plausible is the occurrence of a 3D rocking motion for a square or near-square based body. An uplift on a vertex could easily manifest itself also in the case of excitation close to the diagonal, just after the occurrence of an uplift on a side and, therefore, during a 2D rocking motion. In fact the vertical position of the center of mass can increase

On the contrary, for bodies with rectangular base and a side significantly larger than the other, an overturn around the largest side of the base is much more likely to occur before the uplift

*<sup>G</sup>* can be evaluated as in [22] by the relation

The impact conditions are taken from [29] too. They model the process of changing of the vertex around which the rocking occurs, and take place when the base of the block hits the ideal, horizontal, coordinate plane *z* =0. Positions of the body underneath this plane are not allowed. For instance, if the body is rocking around the vertex *i* (*i* = *A*, *B*, *C*, *D*) and, at some special time *t*, another vertex, say *j* ( *j* ≠*i*), hits the horizontal plane, then the impact process happens and *j* becomes the new center or rotation. The angles just after the impact (instant*t* <sup>+</sup> ) are exactly the same than those just before the impact (instant*t* <sup>−</sup> ):*ϑ*<sup>1</sup> + (*t*)=*ϑ*<sup>1</sup> − (*t*), *ϑ*<sup>2</sup> + (*t*)=*ϑ*<sup>2</sup> − (*t*), *ϑ*<sup>3</sup> + (*t*)=*ϑ*<sup>3</sup> − (*t*), and therefore **R**<sup>+</sup> (*t*)=**R**<sup>−</sup> (*t*). On the other hand, the evaluation of the time-derivatives of the three angles after the impact is made by equating the angular momentum around the (new) center of rotation just after and before the impact (see [29], [31]). In particular, the impact conditions read

$$\mathbf{R}^+(t)\hat{\mathbf{J}}\_{\rangle}\mathbf{R}^T(t) = \mathbf{R}^-(t)\hat{\mathbf{J}}\_{\neq}\mathbf{R}^T(t) \tag{11}$$

where *J* **^** *<sup>j</sup>* : =*∫ C* ^ ρ(**x ^** −**x ^** *j* ) ⊗ (**x ^** −**x ^** *j* )*dV* and *J* **^** *ji* : =*∫ C* ^ ρ(**x ^** −**x ^** *j* ) ⊗ (**x ^** −**x ^** *i* )*dV* are (initial) Euler tensors. Equations (11) provide a linear non-homogeneous algebraic system in the unknowns *ϑ*˙ 1 + (*t*), *ϑ*˙ <sup>2</sup> + (*t*) and *ϑ*˙ <sup>3</sup> + (*t*), in terms of *ϑ*˙ <sup>1</sup> − (*t*), *ϑ*˙ <sup>2</sup> − (*t*) and *ϑ*˙ <sup>3</sup> − (*t*).
