**4. Description of the excitations**

#### **4.1. One-sine excitation**

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 used in the analyses are

$$\begin{cases} a\_h(t) = A\_h \sin\left(\frac{2\pi}{T\_h}t\right) & 0 \le t \le T\_h\\ a\_h(t) = 0 & T\_h < t \le t\_{\text{max}}\\ a\_v(t) = -A\_v \sin\left(\frac{2\pi}{T\_v}t + \phi\right) & 0 \le t \le T\_v\\ a\_v(t) = 0 & T\_v < t \le t\_{\text{max}} \end{cases} \tag{12}$$

**4.3. Description of the simulation**

of the body.

ρ

=2000*kg* / *m*<sup>3</sup>

Results have been obtained by the numerical integration of the equations of motion. A Fortran code has been implemented by using the IMSL Math libraries [30]. In particular the DASPG routine, able to numerically integrate the equations of motion in implicit form, has been chosen. It uses the well known Gear's Backward-Differentiation-Formulas method. Special care has been devoted to the detection of impacts. The integration time step has been fixed for all the simulation to 1 / 2<sup>16</sup> sec. At each integration step, checks are made in order to verify if, under vertical excitation, the conditions of sliding or free-flight occur. Consequently the results of the evaluation have not been taken into account, since the model is not able to describe them.

For the one-sine excitation, the analyses are conducted by varying continuously the direction of the horizontal excitation and by evaluating the amplitudes of the horizontal or vertical onesine pulse at which an uplift or an overturning collapse event manifests itself. This type of analysis is performed for several values of other parameters, such as period of the excitations, phase between the horizontal and vertical pulses, eccentricity and geometrical characteristics

The seisimic analyses are performed by exciting rigid blocks with different mechanical and geometrical characteristics, by three different Italian registered earthquakes acting along different directions. Two type of analyses are performed in this study: the first is conducted with the aim to point out if for some directions of the escitation the 3D model of rigid block furnishes more accurate results than the classical 2D models; the other is performed by fixing the direction of the input, with the aim to highlight the role of the mechanical and the geo‐

In the following analyses, a rigid body in the shape of a parallelepiped with a volume equal to *V* =8(*b*x*byh* ) is always assumed. 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

ρ

Results are shown by using polar diagrams where, along the angle-axis (external circle), the angle that measures the direction of the horizontal excitation with respect to the xaxis, positive if counterclockwise, is reported. Along the radial-axis, the amplitude of the horizontal or vertical excitation able to uplift or to overturn the body is reported. These diagrams have been obtained by a massive use of calculator. Increments equal to 1.0*o* for the direction of the pulse and equal to 0.01*g* for the amplitudes have been adopted to

ρ¯ = ρ ρ

Seismic Behaviour of Monolithic Objects: A 3D Approach

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

165

*V* . As a consequence the mass density

¯*V* (1 + *β*) is taken equal to

<sup>1</sup> <sup>+</sup> *<sup>β</sup>* . The value of *β* =0.20 and

ρ¯ of a

metrical characteristics of the rigid block in the seismic response.

(Fig. 2c). This means that the mass of the body with eccentricity *m*=

are always taken in the following analyses.

the mass of the body without eccentricity *m*=

**5. Description of the results**

body when an eccentricity is considered leads to

where *h* and *v* stand for horizontal and vertical, respectively; *Th* and *Tv* are the periods of the two one-sine pulses, *Ah* and *Av* are their amplitudes and *T*max is the maximum time used in the numerical integrations. This is always taken at least five times the period *Th* . The phase of the one-sine horizontal excitation is assumed always equal to zero in this paper, although this parameter, in principle, can affect the behavior of the system. Only the phase of the vertical excitation will be taken into account, since the objective of the analysis is to point out the role of the difference of phase between the horizontal and vertical sine pulses.

#### **4.2. Seismic excitation**

In the seismic analysis, only the horizontal effects of the seismic source *u*¨ *<sup>g</sup>* =*γf* (*t*) are considered, where *f* (*t*) is the registered seismic acceleration normalized to a *PGA*=1 *g* (PGA stands for Peak Groung Acceleration) and *γ* is a variable coefficient used to scale the amplitude of the seismic ac‐ celerations. The time-histories and the elastic response spectrums of the three normalized seis‐ mic inputs used in the analyses are shown in Fig. 3. Brienza, Buia and Calitri earthquakes are choosen with the aim to perform a simplified analysis, able to evaluate the influence of the spec‐ tral characteristic of the earthquake in the dynamics of the three-dimensional rigid block.

**Figure 3.** Normalized time-history and spectrum of the registered Italian earthquakes used in the numerical simula‐ tions: (a) Brienza earthquake (*PGA*=2.2 *m* / *s* <sup>2</sup> , *lenght* =34.012 *s*); (b) Buia earthquake (*PGA*=2.3 *m* / *s* <sup>2</sup> , *lenght* =20.252 *s*); (c) Calitri earthquake (*PGA*=1.2 *m* / *s* <sup>2</sup> , *lenght* =35.019 *s*).

### **4.3. Description of the simulation**

max

(12)

,

*<sup>g</sup>* =*γf* (*t*) are considered,

<sup>2</sup> ( ) sin <sup>0</sup>

*h h h h h h*

æ ö <sup>=</sup> ç ÷ £ £ ç ÷ è ø = < £ æ ö =- + £ £ ç ÷ è ø = < £

*at A t t T T a t T tt*

p

*v v v v v v*

where *h* and *v* stand for horizontal and vertical, respectively; *Th* and *Tv* are the periods of the two one-sine pulses, *Ah* and *Av* are their amplitudes and *T*max is the maximum time used in the numerical integrations. This is always taken at least five times the period *Th* . The phase of the one-sine horizontal excitation is assumed always equal to zero in this paper, although this parameter, in principle, can affect the behavior of the system. Only the phase of the vertical excitation will be taken into account, since the objective of the analysis is to point out the role

where *f* (*t*) is the registered seismic acceleration normalized to a *PGA*=1 *g* (PGA stands for Peak Groung Acceleration) and *γ* is a variable coefficient used to scale the amplitude of the seismic ac‐ celerations. The time-histories and the elastic response spectrums of the three normalized seis‐ mic inputs used in the analyses are shown in Fig. 3. Brienza, Buia and Calitri earthquakes are choosen with the aim to perform a simplified analysis, able to evaluate the influence of the spec‐

**Figure 3.** Normalized time-history and spectrum of the registered Italian earthquakes used in the numerical simula‐

, *lenght* =35.019 *s*).

, *lenght* =34.012 *s*); (b) Buia earthquake (*PGA*=2.3 *m* / *s* <sup>2</sup>

tral characteristic of the earthquake in the dynamics of the three-dimensional rigid block.

*at A t t T T a t T tt*

p f

<sup>2</sup> ( ) sin <sup>0</sup>

of the difference of phase between the horizontal and vertical sine pulses.

In the seismic analysis, only the horizontal effects of the seismic source *u*¨

**4.2. Seismic excitation**

tions: (a) Brienza earthquake (*PGA*=2.2 *m* / *s* <sup>2</sup>

*lenght* =20.252 *s*); (c) Calitri earthquake (*PGA*=1.2 *m* / *s* <sup>2</sup>

() 0

ì ï ïï í ï ï ïî ì ï ïï í ï ï ïî

164 Engineering Seismology, Geotechnical and Structural Earthquake Engineering

() 0

max

Results have been obtained by the numerical integration of the equations of motion. A Fortran code has been implemented by using the IMSL Math libraries [30]. In particular the DASPG routine, able to numerically integrate the equations of motion in implicit form, has been chosen. It uses the well known Gear's Backward-Differentiation-Formulas method. Special care has been devoted to the detection of impacts. The integration time step has been fixed for all the simulation to 1 / 2<sup>16</sup> sec. At each integration step, checks are made in order to verify if, under vertical excitation, the conditions of sliding or free-flight occur. Consequently the results of the evaluation have not been taken into account, since the model is not able to describe them.

For the one-sine excitation, the analyses are conducted by varying continuously the direction of the horizontal excitation and by evaluating the amplitudes of the horizontal or vertical onesine pulse at which an uplift or an overturning collapse event manifests itself. This type of analysis is performed for several values of other parameters, such as period of the excitations, phase between the horizontal and vertical pulses, eccentricity and geometrical characteristics of the body.

The seisimic analyses are performed by exciting rigid blocks with different mechanical and geometrical characteristics, by three different Italian registered earthquakes acting along different directions. Two type of analyses are performed in this study: the first is conducted with the aim to point out if for some directions of the escitation the 3D model of rigid block furnishes more accurate results than the classical 2D models; the other is performed by fixing the direction of the input, with the aim to highlight the role of the mechanical and the geo‐ metrical characteristics of the rigid block in the seismic response.

In the following analyses, a rigid body in the shape of a parallelepiped with a volume equal to *V* =8(*b*x*byh* ) is always assumed. 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 (Fig. 2c). This means that the mass of the body with eccentricity *m*=ρ¯*V* (1 + *β*) is taken equal to the mass of the body without eccentricity *m*=ρ*V* . As a consequence the mass density ρ¯ of a body when an eccentricity is considered leads to ρ¯ = ρ <sup>1</sup> <sup>+</sup> *<sup>β</sup>* . The value of *β* =0.20 and ρ =2000*kg* / *m*<sup>3</sup> are always taken in the following analyses.
