**5. Description of the results**

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 obtain the following diagrams. To give an idea of the calculus time needed to get all the results, from which those shown in this paper have been selected, a calculator with 12 Gb ofRAM and a Intel-I7 quad-core CPU with 2.0 GHz clock has been running for about two months.

In the following figures (Figs. 4, 5, 7-13), the same line styles are used. Solid thick lines refer to overturning events (they furnish the amplitude of the excitation at which the first occurrence of an overturning event manifests itself for a specific direction of the pulse). Dashed thick lines refer to the uplift of the body on a side of the rectangular base. In particular, for a specific direction of the excitation, below this curves, the body remains in perfect contact with the ground (full-contact) whereas, above them, a 2D rocking motion occurs. Dotted curves furnish the amplitude of the excitation for which an uplift on a vertex of the base occurs. Above this amplitude, a 3D rocking motion takes place. Directions of the excitation where the body manifests an uplift directly on a vertex (where dashed and dotted curves touch each others) always exist. The analyses performed in this paper do not permit to obtain the so-called survival regions, that could exist also in 3D rocking motions above the first overturning occurrence, as found for 2D rocking motions (see [6, 11]).

#### **5.1. Rocking motion due to one-sine excitation**

### *5.1.1. Rigid block with square base and no eccentricity*

The first analysis, shown in Fig. 4, is conducted with the aim to check the influence of the phase *ϕ* of the one-sine vertical excitation. These results refer to a body with a square base and without any eccentricity of the center of mass, excited by a vertical pulse with fixed amplitude, when considered, and by a horizontal excitation with variable amplitude. The curves reported in the diagrams represent the value of the horizontal amplitude that causes the uplift or the overturning of the body. The angular sectors where a 3D rocking motion manifests itself are marked along the angular circle with solid thick lines. The diagrams shown in Fig. 4a refer to the case in absence of vertical excitation. It is a case already reported and discussed in [29]. When a vertical one-sine pulse is added to the system, the uplift and overturning curves change as shown in Figs. 4b-d.

**Figure 4.** Direction vs Horizontal Amplitude of the excitation: (a) *Av* =0;(b) *Av* =0.5*g*, <sup>ϕ</sup>=90*o*; (c) *Av* =0.5*g*, <sup>ϕ</sup>=180*o*; (d)

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With respect to the previous case, results shown in Fig. 5 refer to different periods of the excitations. In Fig. 5a, the case in absence of vertical excitation is shown ([29]). When a vertical excitation with fixed amplitude and *ϕ* =0 is considered (Fig. 5b), the uplift and the overturning curves strongly change. In particular, in many angular sectors, the horizontal amplitude able to overturn the body becomes smaller than the one in absence of vertical pulse (gray curve). Also a very critical condition takes place: in the gray angular sectors, contained into the 3D rocking regions, the horizontal amplitudes able to overturn the body become smaller than those obtained along the directions 0, 90, 180, 270 degrees, where the excitation is orthogonal to one of the four sides of the base (dash-dot circle). Since, in order to obtain the amplitudes along the directions 0, 90, 180, 270 degrees, a bi-dimensional model of rigid block is sufficient, the necessity to use a 3D model of rigid block to remain in favour of safety, is confirmed.

*Av* =0.5*g*, <sup>ϕ</sup>=270*o*; (*Th* =0.75*s*, *Tv* =0.75*s*, *<sup>b</sup>*x=*by* =0.3*m*, *<sup>h</sup>* =1.0*m*, <sup>ε</sup>x=ε*<sup>y</sup>* =0 ).

In particular when *ϕ* =90*<sup>o</sup>* (Fig. 4b) the sectors where a 3D rocking motion occurs are wider, the system uplifts for smaller horizontal amplitudes and overturns for higher amplitudes (gray curve represents the results reported in Fig. 4a in absence of vertical pulse). Therefore, with respect to the overturning collapse events, in this case, the presence of the one-sine pulse with this specific phase acts in favor of safety. A change of the phase (*ϕ* =180*<sup>o</sup>* , see Fig. 4c) produces a worsening of the situation in terms of overturning events. Many angular sectors, where a smaller horizontal amplitude is required to overturn the body, appear; they are contained in 2D or 3D rocking regions. When the phase is *ϕ* =270*<sup>o</sup>* (Fig. 4d) the amplitudes necessary to overturn the body become smaller than those obtained in absence of vertical excitation (gray curve) in almost all the angular space.

obtain the following diagrams. To give an idea of the calculus time needed to get all the results, from which those shown in this paper have been selected, a calculator with 12 Gb ofRAM and a Intel-I7 quad-core CPU with 2.0 GHz clock has been running for about

In the following figures (Figs. 4, 5, 7-13), the same line styles are used. Solid thick lines refer to overturning events (they furnish the amplitude of the excitation at which the first occurrence of an overturning event manifests itself for a specific direction of the pulse). Dashed thick lines refer to the uplift of the body on a side of the rectangular base. In particular, for a specific direction of the excitation, below this curves, the body remains in perfect contact with the ground (full-contact) whereas, above them, a 2D rocking motion occurs. Dotted curves furnish the amplitude of the excitation for which an uplift on a vertex of the base occurs. Above this amplitude, a 3D rocking motion takes place. Directions of the excitation where the body manifests an uplift directly on a vertex (where dashed and dotted curves touch each others) always exist. The analyses performed in this paper do not permit to obtain the so-called survival regions, that could exist also in 3D rocking motions above the first overturning occurrence, as found for 2D

The first analysis, shown in Fig. 4, is conducted with the aim to check the influence of the phase *ϕ* of the one-sine vertical excitation. These results refer to a body with a square base and without any eccentricity of the center of mass, excited by a vertical pulse with fixed amplitude, when considered, and by a horizontal excitation with variable amplitude. The curves reported in the diagrams represent the value of the horizontal amplitude that causes the uplift or the overturning of the body. The angular sectors where a 3D rocking motion manifests itself are marked along the angular circle with solid thick lines. The diagrams shown in Fig. 4a refer to the case in absence of vertical excitation. It is a case already reported and discussed in [29]. When a vertical one-sine pulse is added to the system, the uplift and overturning curves change

In particular when *ϕ* =90*<sup>o</sup>* (Fig. 4b) the sectors where a 3D rocking motion occurs are wider, the system uplifts for smaller horizontal amplitudes and overturns for higher amplitudes (gray curve represents the results reported in Fig. 4a in absence of vertical pulse). Therefore, with respect to the overturning collapse events, in this case, the presence of the one-sine pulse with

a worsening of the situation in terms of overturning events. Many angular sectors, where a smaller horizontal amplitude is required to overturn the body, appear; they are contained in 2D or 3D rocking regions. When the phase is *ϕ* =270*<sup>o</sup>* (Fig. 4d) the amplitudes necessary to overturn the body become smaller than those obtained in absence of vertical excitation (gray

, see Fig. 4c) produces

this specific phase acts in favor of safety. A change of the phase (*ϕ* =180*<sup>o</sup>*

two months.

rocking motions (see [6, 11]).

as shown in Figs. 4b-d.

curve) in almost all the angular space.

**5.1. Rocking motion due to one-sine excitation**

166 Engineering Seismology, Geotechnical and Structural Earthquake Engineering

*5.1.1. Rigid block with square base and no eccentricity*

**Figure 4.** Direction vs Horizontal Amplitude of the excitation: (a) *Av* =0;(b) *Av* =0.5*g*, <sup>ϕ</sup>=90*o*; (c) *Av* =0.5*g*, <sup>ϕ</sup>=180*o*; (d) *Av* =0.5*g*, <sup>ϕ</sup>=270*o*; (*Th* =0.75*s*, *Tv* =0.75*s*, *<sup>b</sup>*x=*by* =0.3*m*, *<sup>h</sup>* =1.0*m*, <sup>ε</sup>x=ε*<sup>y</sup>* =0 ).

With respect to the previous case, results shown in Fig. 5 refer to different periods of the excitations. In Fig. 5a, the case in absence of vertical excitation is shown ([29]). When a vertical excitation with fixed amplitude and *ϕ* =0 is considered (Fig. 5b), the uplift and the overturning curves strongly change. In particular, in many angular sectors, the horizontal amplitude able to overturn the body becomes smaller than the one in absence of vertical pulse (gray curve). Also a very critical condition takes place: in the gray angular sectors, contained into the 3D rocking regions, the horizontal amplitudes able to overturn the body become smaller than those obtained along the directions 0, 90, 180, 270 degrees, where the excitation is orthogonal to one of the four sides of the base (dash-dot circle). Since, in order to obtain the amplitudes along the directions 0, 90, 180, 270 degrees, a bi-dimensional model of rigid block is sufficient, the necessity to use a 3D model of rigid block to remain in favour of safety, is confirmed.

vertical pulse, the body overturns whereas, when the vertical excitation is considered, the body does not overturn (Fig. 6b). Solid curves represent the case with vertical excitation, whereas dashed curves the case without vertical excitation. In both cases, where the body does not overturn, the time-histories *zA* touch and remain on the *zA*=0 axis in several time ranges. This happens when, during the 3D rocking motion, the point *A* hits the ground and becomes the

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**Figure 6.** Time-histories: (a) Vertical position of the vertex A and angle *<sup>y</sup>* vs time calculated at point H labeled in Fig. 5c (*Ah* =0, <sup>875</sup>*g*, *Av* =0.5*g*, <sup>α</sup> =30*o*, <sup>ϕ</sup>=180*o*); (b) Vertical position of the vertex A and angle *y*vs time calculated at point K labeled in Fig. 5c (*Ah* =1.74*g*, *Av* =0.5*g*, <sup>α</sup> =18*o*, <sup>ϕ</sup>=180*o*); (*Th* =0.5*s*, *Tv* =0.5*s*, *<sup>b</sup>*x=*by* =0.3*m*, *<sup>h</sup>* =1.0*m*, <sup>ε</sup>x=ε*<sup>y</sup>* =0;solid

Very interesting is the case shown in Fig. 7. The results refer to a value of the period of the horizontal one-sine pulse for which, in absence of vertical excitation, many sectors where the 3D model furnishes more accurate results with respect to the classical 2D model exist (gray sectors in Fig. 7a, see [29] for more details). The vertical excitation, also in this case, strongly changes the scenario. In particular, when the *ϕ* =0 (Fig. 7b), the critical gray sectors reduce but, in all the angular plane, the overturning amplitudes become smaller than those obtained without vertical pulse (gray curve). For *ϕ* =90*<sup>o</sup>* (Fig. 7c) in several angular sectors the amplitude able to overturn the body becomes higher than the one obtained without vertical excitation, but the critical gray sectors, where it is necessary the use of a 3D model to better evaluate the overturning of the body, completely disappear. These critical regions appear again for *ϕ* =180*<sup>o</sup>*

lines: with vertical excitation, dashed lines: without vertical excitation).

instantaneous rotation center.

**Figure 5.** Direction vs Horizontal Amplitude of the excitation: (a) *Av* =0;(b) *Av* =0.5*g*, <sup>ϕ</sup>=0*o*; (c) *Av* =0.5*g*, <sup>ϕ</sup>=180*o*; (d) *Av* =0.5*g*, <sup>ϕ</sup>=270*o*; (*Th* =0.5*s*, *Tv* =0.5*s*, *<sup>b</sup>*x=*by* =0.3*m*, *<sup>h</sup>* =1.0*m*, <sup>ε</sup>x=ε*<sup>y</sup>* =0 ).

It is interesting to note that the case in absence of vertical excitation (Fig. 5a) does not manifest the necessity of the use of a 3D model. Therefore, it is possible to assert that this critical situation is only caused by the presence of vertical excitation. A change of the phase of the vertical pulse (*ϕ* =180*<sup>o</sup>* , see Fig. 5c) causes the enlargement of the critical sector where the amplitudes obtained by a 3D model are smaller than the amplitude of a 2D model (dash-dot circle). A further change of the phase (*ϕ* =270*<sup>o</sup>* , see Fig. 5d) strongly changes the scenario. The critical gray sectors disappear whereas, in all the angular plane, the overturning amplitudes become smaller than the case in absence of vertical excitation (gray curve).

In Fig. 6 the time-histories of vertical position *zA* of the base point *A* and of the angle *ϑy*, referring to the cases labeled with *H* and *K* in Fig. 5c, are shown. Point *H* refers to a case where, in absence of vertical pulse, the body does not overturn whereas, when the vertical excitation is considered, the body does overturn (Fig. 6a); point *K* refers to a case where, in absence of vertical pulse, the body overturns whereas, when the vertical excitation is considered, the body does not overturn (Fig. 6b). Solid curves represent the case with vertical excitation, whereas dashed curves the case without vertical excitation. In both cases, where the body does not overturn, the time-histories *zA* touch and remain on the *zA*=0 axis in several time ranges. This happens when, during the 3D rocking motion, the point *A* hits the ground and becomes the instantaneous rotation center.

**Figure 6.** Time-histories: (a) Vertical position of the vertex A and angle *<sup>y</sup>* vs time calculated at point H labeled in Fig. 5c (*Ah* =0, <sup>875</sup>*g*, *Av* =0.5*g*, <sup>α</sup> =30*o*, <sup>ϕ</sup>=180*o*); (b) Vertical position of the vertex A and angle *y*vs time calculated at point K labeled in Fig. 5c (*Ah* =1.74*g*, *Av* =0.5*g*, <sup>α</sup> =18*o*, <sup>ϕ</sup>=180*o*); (*Th* =0.5*s*, *Tv* =0.5*s*, *<sup>b</sup>*x=*by* =0.3*m*, *<sup>h</sup>* =1.0*m*, <sup>ε</sup>x=ε*<sup>y</sup>* =0;solid lines: with vertical excitation, dashed lines: without vertical excitation).

**Figure 5.** Direction vs Horizontal Amplitude of the excitation: (a) *Av* =0;(b) *Av* =0.5*g*, <sup>ϕ</sup>=0*o*; (c) *Av* =0.5*g*, <sup>ϕ</sup>=180*o*; (d)

It is interesting to note that the case in absence of vertical excitation (Fig. 5a) does not manifest the necessity of the use of a 3D model. Therefore, it is possible to assert that this critical situation is only caused by the presence of vertical excitation. A change of the phase of the vertical pulse

by a 3D model are smaller than the amplitude of a 2D model (dash-dot circle). A further change

disappear whereas, in all the angular plane, the overturning amplitudes become smaller than

In Fig. 6 the time-histories of vertical position *zA* of the base point *A* and of the angle *ϑy*, referring to the cases labeled with *H* and *K* in Fig. 5c, are shown. Point *H* refers to a case where, in absence of vertical pulse, the body does not overturn whereas, when the vertical excitation is considered, the body does overturn (Fig. 6a); point *K* refers to a case where, in absence of

, see Fig. 5c) causes the enlargement of the critical sector where the amplitudes obtained

, see Fig. 5d) strongly changes the scenario. The critical gray sectors

*Av* =0.5*g*, <sup>ϕ</sup>=270*o*; (*Th* =0.5*s*, *Tv* =0.5*s*, *<sup>b</sup>*x=*by* =0.3*m*, *<sup>h</sup>* =1.0*m*, <sup>ε</sup>x=ε*<sup>y</sup>* =0 ).

168 Engineering Seismology, Geotechnical and Structural Earthquake Engineering

the case in absence of vertical excitation (gray curve).

(*ϕ* =180*<sup>o</sup>*

of the phase (*ϕ* =270*<sup>o</sup>*

Very interesting is the case shown in Fig. 7. The results refer to a value of the period of the horizontal one-sine pulse for which, in absence of vertical excitation, many sectors where the 3D model furnishes more accurate results with respect to the classical 2D model exist (gray sectors in Fig. 7a, see [29] for more details). The vertical excitation, also in this case, strongly changes the scenario. In particular, when the *ϕ* =0 (Fig. 7b), the critical gray sectors reduce but, in all the angular plane, the overturning amplitudes become smaller than those obtained without vertical pulse (gray curve). For *ϕ* =90*<sup>o</sup>* (Fig. 7c) in several angular sectors the amplitude able to overturn the body becomes higher than the one obtained without vertical excitation, but the critical gray sectors, where it is necessary the use of a 3D model to better evaluate the overturning of the body, completely disappear. These critical regions appear again for *ϕ* =180*<sup>o</sup>*

(Fig. 7d) but the overturning amplitudes become higher than those obtained in absence of vertical pulse, in all the angular plane.

*5.1.2. Rigid block with square base and with eccentricity*

manifests itself. For *ϕ* =90*<sup>o</sup>*

*ϕ* =90*<sup>o</sup>*

*cm*<sup>3</sup>

0<sup>o</sup> , 90<sup>o</sup>

, 180<sup>o</sup>

*5.1.3. Rigid block with near-square base*

case where the vertical excitation is null (Fig. 9b).

**5.2. Rocking motion due to seismic excitation**

*5.2.1. The role of the direction of the input*

and 50×50×200*cm*<sup>3</sup>

, 270<sup>o</sup>

In Fig. 8 the role of a fixed vertical excitation applied to a square based, eccentric, rigid body is investigated. The case of absence of vertical excitation, horizontal pulse with period *Th* =0.75*s* and eccentricities *ε*<sup>x</sup> =*ε<sup>y</sup>* =0.25, is shown in Fig. 8a (see [29] for more details). Two critical regions (gray sectors) where a 3D model is necessary to better evaluate the overturning collapse events manifest themselves. When a vertical fixed excitation is considered, a change of the previous scenario occurs. In particular, for *ϕ* =0 and *ϕ* =270*<sup>o</sup>* (Fig. 8b and 8d respectively), a slightly modification of the critical sectors takes place: A diminution of the horizontal overturning amplitude with respect to the case in absence of vertical pulse (gray curve)

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(Fig. 8c), the critical regions disappear.

Finally, in Fig. 9, the role of a fixed vertical excitation applied to a near-square based, eccentric, rigid body is investigated. The case of absence of vertical excitation, without and with eccentricity (*ε*<sup>x</sup> =0, *ε* =0.5) are shown in Fig. 9a and Fig. 9b, respectively (see [29] for more details). The eccentricity of the mass center of the body is the cause of the apparition of the critical regions (gray sectors, Fig. 9b) where the overturning amplitude obtained during a 3D rocking motion is smaller than the overturning amplitude furnished by a 2D model (dash-dot circle). The vertical excitation, in addition to changing the overturning curves, acts also on thecritical regions. In particular, for *ϕ* =0 (Fig. 9c), two new critical regions appear whereas, for

(Fig. 9d), these critical regions change their position and amplitude with respect to the

In the first analysis, square based block with constant height are excited by the three different earthquakes. In particular results reported in Fig. 10a,b refer to two different blocks (30×30×200

serve a general increase of the PGA able to overturn the body when the dimension of the base in‐ creases. Also a slight increase of the sectors of 3D rocking motion manifests itself for higher bases of the block. The overturning amplitude in 3D regions is always larger than the amplitude able to overturn the body during a 2D rocking motion (the value observed along the

for larger bases of the body, the 3D overturning amplitude becomes very close to the 2D over‐ turning PGA, along some specific directions of the excitation. A similar behavior can be ob‐ served when the same previous blocks are excited by the Buia earthquake (Fig. 10c,d). Very different are the results obtained by exciting the body by the Calitri earthquake (Fig. 10e,f). The amplitude of the sectors where a 3D rocking motion manifests itself are a lot smaller than the previous cases and the 3D overturning amplitude remains always far enough from the 2D over‐ turning PGA (dash-dotted circle). However, smaller values of the PGA than the previous earth‐

, respectively) subject to the Brienza earthquake. It is possible to ob‐

directions) and marked in the graphs with a dash-dotted circle. However,

**Figure 7.** Direction vs Horizontal Amplitude of the excitation: (a) *Av* =0;(b) *Av* =0.5*g*, <sup>ϕ</sup>=0*o*; (c) *Av* =0.5*g*, <sup>ϕ</sup>=90*o*; (d) *Av* =0.5*g*, <sup>ϕ</sup>=180*o*; (*Th* =0.35*s*, *Tv* =0.5*s*, *<sup>b</sup>*x=*by* =0.3*m*, *<sup>h</sup>* =1.0*m*, <sup>ε</sup>x=ε*<sup>y</sup>* =0).

## *5.1.2. Rigid block with square base and with eccentricity*

In Fig. 8 the role of a fixed vertical excitation applied to a square based, eccentric, rigid body is investigated. The case of absence of vertical excitation, horizontal pulse with period *Th* =0.75*s* and eccentricities *ε*<sup>x</sup> =*ε<sup>y</sup>* =0.25, is shown in Fig. 8a (see [29] for more details). Two critical regions (gray sectors) where a 3D model is necessary to better evaluate the overturning collapse events manifest themselves. When a vertical fixed excitation is considered, a change of the previous scenario occurs. In particular, for *ϕ* =0 and *ϕ* =270*<sup>o</sup>* (Fig. 8b and 8d respectively), a slightly modification of the critical sectors takes place: A diminution of the horizontal overturning amplitude with respect to the case in absence of vertical pulse (gray curve) manifests itself. For *ϕ* =90*<sup>o</sup>* (Fig. 8c), the critical regions disappear.

### *5.1.3. Rigid block with near-square base*

(Fig. 7d) but the overturning amplitudes become higher than those obtained in absence of

**Figure 7.** Direction vs Horizontal Amplitude of the excitation: (a) *Av* =0;(b) *Av* =0.5*g*, <sup>ϕ</sup>=0*o*; (c) *Av* =0.5*g*, <sup>ϕ</sup>=90*o*; (d)

*Av* =0.5*g*, <sup>ϕ</sup>=180*o*; (*Th* =0.35*s*, *Tv* =0.5*s*, *<sup>b</sup>*x=*by* =0.3*m*, *<sup>h</sup>* =1.0*m*, <sup>ε</sup>x=ε*<sup>y</sup>* =0).

vertical pulse, in all the angular plane.

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Finally, in Fig. 9, the role of a fixed vertical excitation applied to a near-square based, eccentric, rigid body is investigated. The case of absence of vertical excitation, without and with eccentricity (*ε*<sup>x</sup> =0, *ε* =0.5) are shown in Fig. 9a and Fig. 9b, respectively (see [29] for more details). The eccentricity of the mass center of the body is the cause of the apparition of the critical regions (gray sectors, Fig. 9b) where the overturning amplitude obtained during a 3D rocking motion is smaller than the overturning amplitude furnished by a 2D model (dash-dot circle). The vertical excitation, in addition to changing the overturning curves, acts also on thecritical regions. In particular, for *ϕ* =0 (Fig. 9c), two new critical regions appear whereas, for *ϕ* =90*<sup>o</sup>* (Fig. 9d), these critical regions change their position and amplitude with respect to the case where the vertical excitation is null (Fig. 9b).

#### **5.2. Rocking motion due to seismic excitation**

#### *5.2.1. The role of the direction of the input*

In the first analysis, square based block with constant height are excited by the three different earthquakes. In particular results reported in Fig. 10a,b refer to two different blocks (30×30×200 *cm*<sup>3</sup> and 50×50×200*cm*<sup>3</sup> , respectively) subject to the Brienza earthquake. It is possible to ob‐ serve a general increase of the PGA able to overturn the body when the dimension of the base in‐ creases. Also a slight increase of the sectors of 3D rocking motion manifests itself for higher bases of the block. The overturning amplitude in 3D regions is always larger than the amplitude able to overturn the body during a 2D rocking motion (the value observed along the 0<sup>o</sup> , 90<sup>o</sup> , 180<sup>o</sup> , 270<sup>o</sup> directions) and marked in the graphs with a dash-dotted circle. However, for larger bases of the body, the 3D overturning amplitude becomes very close to the 2D over‐ turning PGA, along some specific directions of the excitation. A similar behavior can be ob‐ served when the same previous blocks are excited by the Buia earthquake (Fig. 10c,d). Very different are the results obtained by exciting the body by the Calitri earthquake (Fig. 10e,f). The amplitude of the sectors where a 3D rocking motion manifests itself are a lot smaller than the previous cases and the 3D overturning amplitude remains always far enough from the 2D over‐ turning PGA (dash-dotted circle). However, smaller values of the PGA than the previous earth‐ quakes are requested to cause a collapse event. Considering the results of these first analyses, it does not seem necessary the use of a 3D model to study the seismic behavior of a rigid block, since the most dangerous situations manifest themselves during a 2D rocking motion.

overturnig PGA's smaller than the minimum required during the 2D rocking motion (dashdotted circle) manifest themselves. In other words, inside the gray sectors, during a 3D rocking motion, an overturning collapse event occurs for a PGA smaller than the minimum obtained by using a 2D model of rigid block. Results shown in Fig. 11c,d, that refer to Buia earthquake, confirm what previously said. The situation changes if the block is excited by the Calitri earthquake (Fig. 11e,f). In this case the 3D overturning amplitudes remain always far enough

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**Figure 9.** Direction vs Horizontal Amplitude of the excitation: (a) *Av* =0, εx=ε*<sup>y</sup>* =0; (b) *Av* =0, εx=0, ε*<sup>y</sup>* =0.5; (c) *Av* =0.5*g*,

<sup>ε</sup>x=0, <sup>ε</sup>*<sup>y</sup>* =0.5, <sup>ϕ</sup>=0*o*; (d) *Av* =0.5*g*, <sup>ε</sup>x=0, <sup>ε</sup>*<sup>y</sup>* =0.5, <sup>ϕ</sup>=0*o*; (*Th* =0.75*s*, *Tv* =0.5*s*, *<sup>b</sup>*x=*by* =0.3*m*, *<sup>h</sup>* =1.0*m*).

from the 2D overturning PGA's, also when an eccentricity is considered.

**Figure 8.** Direction vs Horizontal Amplitude of the excitation: (a)*Av* =0; (b) *Av* =0.5*g*, <sup>ϕ</sup>=0*o*; (c) *Av* =0.5*g*, <sup>ϕ</sup>=90*o*; (d) *Av* =0.5*g*, <sup>ϕ</sup>=270*o*; (*Th* =0.75*s*, *Tv* =0.5*s*, *<sup>b</sup>*x=*by* =0.3*m*, *<sup>h</sup>* =1.0*m*,εx=ε*<sup>y</sup>* =0.25).

In Fig. 11 the effect of the eccentricity of the mass center is outlined. In particular, a block of dimensions 40×40×200*cm*<sup>3</sup> , with and without eccentricity, is excited by the three different earthquakes. Results shown in Fig. 12a,b refer to the case without eccentricity and the case with eccentricity along the y-axis (*ε<sup>y</sup>* =0.25, *ey* =5 cm) respectively. The presence of an eccentricity sensibly changes the overturning curve, that loses one symmetry axis and becomes more irregular than the case without eccentricity.

However, a very interesting phenomenon occurs in presence of the eccenticity: in some directions inside the 3D rocking regions (marked with thick lines along the external circle), overturnig PGA's smaller than the minimum required during the 2D rocking motion (dashdotted circle) manifest themselves. In other words, inside the gray sectors, during a 3D rocking motion, an overturning collapse event occurs for a PGA smaller than the minimum obtained by using a 2D model of rigid block. Results shown in Fig. 11c,d, that refer to Buia earthquake, confirm what previously said. The situation changes if the block is excited by the Calitri earthquake (Fig. 11e,f). In this case the 3D overturning amplitudes remain always far enough from the 2D overturning PGA's, also when an eccentricity is considered.

quakes are requested to cause a collapse event. Considering the results of these first analyses, it does not seem necessary the use of a 3D model to study the seismic behavior of a rigid block,

**Figure 8.** Direction vs Horizontal Amplitude of the excitation: (a)*Av* =0; (b) *Av* =0.5*g*, <sup>ϕ</sup>=0*o*; (c) *Av* =0.5*g*, <sup>ϕ</sup>=90*o*; (d)

In Fig. 11 the effect of the eccentricity of the mass center is outlined. In particular, a block of

earthquakes. Results shown in Fig. 12a,b refer to the case without eccentricity and the case with eccentricity along the y-axis (*ε<sup>y</sup>* =0.25, *ey* =5 cm) respectively. The presence of an eccentricity sensibly changes the overturning curve, that loses one symmetry axis and becomes more

However, a very interesting phenomenon occurs in presence of the eccenticity: in some directions inside the 3D rocking regions (marked with thick lines along the external circle),

, with and without eccentricity, is excited by the three different

*Av* =0.5*g*, <sup>ϕ</sup>=270*o*; (*Th* =0.75*s*, *Tv* =0.5*s*, *<sup>b</sup>*x=*by* =0.3*m*, *<sup>h</sup>* =1.0*m*,εx=ε*<sup>y</sup>* =0.25).

dimensions 40×40×200*cm*<sup>3</sup>

irregular than the case without eccentricity.

since the most dangerous situations manifest themselves during a 2D rocking motion.

172 Engineering Seismology, Geotechnical and Structural Earthquake Engineering

**Figure 9.** Direction vs Horizontal Amplitude of the excitation: (a) *Av* =0, εx=ε*<sup>y</sup>* =0; (b) *Av* =0, εx=0, ε*<sup>y</sup>* =0.5; (c) *Av* =0.5*g*, <sup>ε</sup>x=0, <sup>ε</sup>*<sup>y</sup>* =0.5, <sup>ϕ</sup>=0*o*; (d) *Av* =0.5*g*, <sup>ε</sup>x=0, <sup>ε</sup>*<sup>y</sup>* =0.5, <sup>ϕ</sup>=0*o*; (*Th* =0.75*s*, *Tv* =0.5*s*, *<sup>b</sup>*x=*by* =0.3*m*, *<sup>h</sup>* =1.0*m*).

**Figure 10.** Direction vs Amplitude of the excitation. Brienza earthquake: (a)30×30×200; (b) 50×50×200*cm*<sup>3</sup> . Buia earthquake: (c)30×30×200; (d) 50×50×200*cm*<sup>3</sup> . Calitri earthquake: (e)30×30×200; (f) 50×50×200*cm*<sup>3</sup> ,(εx=ε*<sup>y</sup>* =0).

**Figure 11.** Direction vs Amplitude of the excitation. Brienza earthquake: (a)εx=ε*<sup>y</sup>* =0; (b)εx=0, ε*<sup>y</sup>* =0.25. Buia earth‐

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quake: (c)εx=ε*<sup>y</sup>* =0; (d)εx=0, <sup>ε</sup>*<sup>y</sup>* =0.25. Calitri earthquake: (e)εx=ε*<sup>y</sup>* =0; (f) εx=0, <sup>ε</sup>*<sup>y</sup>* =0.25 (40×40×200)*cm*<sup>3</sup>

**Figure 11.** Direction vs Amplitude of the excitation. Brienza earthquake: (a)εx=ε*<sup>y</sup>* =0; (b)εx=0, ε*<sup>y</sup>* =0.25. Buia earth‐ quake: (c)εx=ε*<sup>y</sup>* =0; (d)εx=0, <sup>ε</sup>*<sup>y</sup>* =0.25. Calitri earthquake: (e)εx=ε*<sup>y</sup>* =0; (f) εx=0, <sup>ε</sup>*<sup>y</sup>* =0.25 (40×40×200)*cm*<sup>3</sup> .

**Figure 10.** Direction vs Amplitude of the excitation. Brienza earthquake: (a)30×30×200; (b) 50×50×200*cm*<sup>3</sup>

. Calitri earthquake: (e)30×30×200; (f) 50×50×200*cm*<sup>3</sup>

earthquake: (c)30×30×200; (d) 50×50×200*cm*<sup>3</sup>

174 Engineering Seismology, Geotechnical and Structural Earthquake Engineering

. Buia

,(εx=ε*<sup>y</sup>* =0).

To summarize, when the block is excited by an earthquake with narrow spectrum (Brienza and Buia earthquakes, see Fig. 3a,b), the presence of a small eccentricity makes possible the existence of angular sectors inside the 3D rocking regions, where the use of the 3D model of rigid block is necessary to obtain results in favour of safety. On the contrary, a wide spectrum earthquake (Calitri earthquake, see Fig. 3c) does not require the use of the 3D model of rigid block, since the eccentricity of the mass center never causes the existence of these critical sectors inside the 3D rocking regions.

body); the second analysis is performed by fixing the mass of the body (i.e. fixing the volume of the body) and varying its slenderness. Results of these analyses are reported in Fig. 13. In all the graphs, solid curves refer to PGA able to overturn the body when the excitation angle

Dotted curves, when reported, refer to overturning amplitude when the direction of the

directions permit to compare the overturning seismic response of the body during the 2D

**Figure 13.** Overturning curves under Brienza earthquake: (a)λx=λ*<sup>y</sup>* =5; (b)λx=λ*<sup>y</sup>* =6; (c)λx=λ*<sup>y</sup>* =7; (d)*m*=500 *kg*; (e)

The sequence of results shown in Fig. 13a-c refers to cases with fixed slenderness *λ*<sup>x</sup> =*λ<sup>y</sup>* =5, 6, 7). As it is possible to observe the two curves (solid and dashed curves) approach to each other at different values of the dimension of the base, depending on the slenderness used in the analysis. An increasing slenderness requires a decreasing base dimension to make possible the approach of the two curves. Very interesting is the case shown in Fig. 13b (ax =*λ<sup>y</sup>* =6) where the two curves touch each other. For this particular dimension of the base of the block, the same PGA is required to overturn the body during the 2D rocking motion and

eccentricity changes the 3D overturning curve (dotted curve) by making possible the existence of a region where the overturning PGA during the 3D motion is smaller than the one during the 2D motion. On the contrary, the sequence of results shown in Fig. 13d-f refers to cases with fixed mass (*m*=500, 640, 800 *kg*). The two curves (solid and dashed curves) approach to each other at different values of the slenderness, depending on the mass of the body used in the

each other. Also in this case the presence of a small eccentricity (dotted curve) makes possible

. The presence of a small

(*m*=800 *kg*) where the two curves touch

the 3D rocking motion, when the direction of the excitation is *α* =40<sup>o</sup>

analysis. Very interesting is the case shown in Fig. 13f

rocking motion (*α* =0) and the 3D rocking motion (*α* =40<sup>o</sup>

, while dashed curves refers to overturning events when the excitation angle is *α* =0.

and in presence of a very small eccentricity (*ε*<sup>x</sup> =0, *ε<sup>y</sup>* =0.1). The chosen

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is *α* =40<sup>o</sup>

excitation is *α* =40<sup>o</sup>

*m*=640 *kg*; (f)*m*=800 *kg*.

Finally, the case of blocks with a rectangular base is considered. Results shown in Fig. 12a,b refer to rectangular based block (30×40×200*cm*<sup>3</sup> ), subject to Brienza earthquake, without and with eccentricity (*ε*<sup>x</sup> =0, *ε<sup>y</sup>* =0.25), respectively.

**Figure 12.** Direction vs Amplitude of the excitation. Brienza earthquake: (a) εx=ε*<sup>y</sup>* =0; (b) <sup>ε</sup>x=0, <sup>ε</sup>*<sup>y</sup>* =0.25 (30×40×200)*cm*<sup>3</sup> .

As it is possible to observe, also in blocks with rectangular base, subject to a narrow spectrum earthquake (Brienza), sectors inside 3D rocking regions where the overturnig PGA's are smaller than the minimum required during the 2D rocking appears. This case is very interest‐ ing since, when a rectangular base of a rigid body occurs, the rocking motion is usually analysed by a 2D model in the plane of the smaller dimension of the base. As a consequence, the eccentricity of the mass center in the direction orthogonal to the plane of the analyzed motion is not taken into account. These results highlight the fact that also in this case, to evaluate the behavior of the system in favour of safety, it is necessary the use a 3D model of rigid body.

#### *5.2.2. The role of the mechanical and geometrical characteristics of the body*

In this section the analyses are performed by fixing the direction of the excitation and varying other geometrical and mechanical characteristic of the block, to point out their role in the seismic response of a square based body. In particular a first analysis is conducted by fixing the slenderness of the body and varying the base of the body (i.e. varying the volume of the body); the second analysis is performed by fixing the mass of the body (i.e. fixing the volume of the body) and varying its slenderness. Results of these analyses are reported in Fig. 13. In all the graphs, solid curves refer to PGA able to overturn the body when the excitation angle is *α* =40<sup>o</sup> , while dashed curves refers to overturning events when the excitation angle is *α* =0. Dotted curves, when reported, refer to overturning amplitude when the direction of the excitation is *α* =40<sup>o</sup> and in presence of a very small eccentricity (*ε*<sup>x</sup> =0, *ε<sup>y</sup>* =0.1). The chosen directions permit to compare the overturning seismic response of the body during the 2D rocking motion (*α* =0) and the 3D rocking motion (*α* =40<sup>o</sup> ).

To summarize, when the block is excited by an earthquake with narrow spectrum (Brienza and Buia earthquakes, see Fig. 3a,b), the presence of a small eccentricity makes possible the existence of angular sectors inside the 3D rocking regions, where the use of the 3D model of rigid block is necessary to obtain results in favour of safety. On the contrary, a wide spectrum earthquake (Calitri earthquake, see Fig. 3c) does not require the use of the 3D model of rigid block, since the eccentricity of the mass center never causes the existence of these critical sectors

Finally, the case of blocks with a rectangular base is considered. Results shown in Fig. 12a,b

**Figure 12.** Direction vs Amplitude of the excitation. Brienza earthquake: (a) εx=ε*<sup>y</sup>* =0; (b)

As it is possible to observe, also in blocks with rectangular base, subject to a narrow spectrum earthquake (Brienza), sectors inside 3D rocking regions where the overturnig PGA's are smaller than the minimum required during the 2D rocking appears. This case is very interest‐ ing since, when a rectangular base of a rigid body occurs, the rocking motion is usually analysed by a 2D model in the plane of the smaller dimension of the base. As a consequence, the eccentricity of the mass center in the direction orthogonal to the plane of the analyzed motion is not taken into account. These results highlight the fact that also in this case, to evaluate the behavior of the system in favour of safety, it is necessary the use a 3D model of

In this section the analyses are performed by fixing the direction of the excitation and varying other geometrical and mechanical characteristic of the block, to point out their role in the seismic response of a square based body. In particular a first analysis is conducted by fixing the slenderness of the body and varying the base of the body (i.e. varying the volume of the

), subject to Brienza earthquake, without and

inside the 3D rocking regions.

<sup>ε</sup>x=0, <sup>ε</sup>*<sup>y</sup>* =0.25 (30×40×200)*cm*<sup>3</sup>

rigid body.

.

*5.2.2. The role of the mechanical and geometrical characteristics of the body*

refer to rectangular based block (30×40×200*cm*<sup>3</sup>

176 Engineering Seismology, Geotechnical and Structural Earthquake Engineering

with eccentricity (*ε*<sup>x</sup> =0, *ε<sup>y</sup>* =0.25), respectively.

**Figure 13.** Overturning curves under Brienza earthquake: (a)λx=λ*<sup>y</sup>* =5; (b)λx=λ*<sup>y</sup>* =6; (c)λx=λ*<sup>y</sup>* =7; (d)*m*=500 *kg*; (e) *m*=640 *kg*; (f)*m*=800 *kg*.

The sequence of results shown in Fig. 13a-c refers to cases with fixed slenderness *λ*<sup>x</sup> =*λ<sup>y</sup>* =5, 6, 7). As it is possible to observe the two curves (solid and dashed curves) approach to each other at different values of the dimension of the base, depending on the slenderness used in the analysis. An increasing slenderness requires a decreasing base dimension to make possible the approach of the two curves. Very interesting is the case shown in Fig. 13b (ax =*λ<sup>y</sup>* =6) where the two curves touch each other. For this particular dimension of the base of the block, the same PGA is required to overturn the body during the 2D rocking motion and the 3D rocking motion, when the direction of the excitation is *α* =40<sup>o</sup> . The presence of a small eccentricity changes the 3D overturning curve (dotted curve) by making possible the existence of a region where the overturning PGA during the 3D motion is smaller than the one during the 2D motion. On the contrary, the sequence of results shown in Fig. 13d-f refers to cases with fixed mass (*m*=500, 640, 800 *kg*). The two curves (solid and dashed curves) approach to each other at different values of the slenderness, depending on the mass of the body used in the analysis. Very interesting is the case shown in Fig. 13f (*m*=800 *kg*) where the two curves touch each other. Also in this case the presence of a small eccentricity (dotted curve) makes possible the existence of a region where the overturning PGA during the 3D motion is smaller than the one during the 2D motion.

2 3 123 12 123 13 23 13 123 123 13 2 12 1 2


**<sup>e</sup> <sup>R</sup>** (13)

:= (14)

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179

ˆ ( ) *<sup>A</sup> mb b h xx yy y* **s** = ++ **e ee** (16)

= **J <sup>e</sup>** (17)

**^** *ji*

, the transport rules read:

(18)

(15)

cc ssc cs csc ss

æ ö ç ÷

è ø

*t t* J

() cs cc sss css sc s sc cc *xyz*

> c cos( ( )) s sin( ( )) *k k k k*

:=

ˆ 0 ˆ ˆ 2 ˆ ˆ 2 2

= = + =+ +

*A*

**x**

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

*xyz*

**^** *i*

ˆ ˆ ( ) ˆ ˆ

=+ -

**s s x x**

*m*

*i A A i*

*m*

, ,

ˆ ˆ 2

= +

**e x x**

*B A x x C A xx yy*

**x x e e e x x**

*b b b*

*b*x*byh* . The static moment with respect to the point *A* is

2

4 3

*A x y y y*

ˆ ˆ ˆ ˆ ( )( ) ˆˆ ˆˆ

**J J s sxx xx**

=+Ä -+ - Ä

*A A Ai Aj ji A*

( )( ) ˆˆ ˆˆ

**xx xx**

+ -Ä -

*Aj Ai*

*x y*

<sup>4</sup> [ ] <sup>ˆ</sup> <sup>3</sup>

2

*mbb b bh*

*bh bh h*

ë û

and the generic Euler tensor *J*

*x xy x*

*b bb bh*

é ù ê ú

2

4 3

*D A y y*

*b*

J

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

é ù =+ + ë û

*t*

respectively, the positions of the base vertices are

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

The mass is *m*=8

ρ

To get the generic static moment *s*

, ,


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 favour of safety.
