## 1. Introduction

Despite the many progresses done in the modelling of rigid blocks, the grounding work for most of the research in this field remains [1], where a 2D model of the rigid block is obtained and the rocking and slide-rocking approximated conditions are written. Following papers on the dynamics of rigid bodies can be divided in two main groups, according to the kind of excitation used: earthquake excitation or sine-type pulse excitation (mainly one-sine). To the first group belong [2-5], in the second one, [6-10] can be found. In time, models of rigid blocks, very useful in many research fields, have been increased in complexity. Recently, for instance, sliding phenomena and the eccentricity of the center of mass have been considered (see [3, 11]). Some papers have been focused to specific problems, for example in [12] the behavior of two stacked rigid blocks has been considered, whereas in [13, 14] the attention has been pointed to blocks on flexible foundations. The dynamics and control of 2D blocks have also been analyzed in the framework of the bifurcation theory in [15, 16, 17].

The effects of the simultaneously presence of horizontal and vertical base excitations have been considered in some papers. For example, in [12, 18, 19] different problems related to the overturning of bi-dimensional rigid blocks have been studied in details.

Lately a large interest has been given to models of rigid bodies with base isolation systems, in order to improve the safety of art objects (see [20-22]). It has been proved that, in certain ranges of geometrical parameters of the rigid block, the effectiveness of base isolation can be amplified when coupling the isolating systems with devices able to limit the displacement of the oscillating base, so as to prevent the falling of the base of the body (see [23, 24]).

Recently, 3D models, mostly circular based, have been used in particular research fields, more precisely to study motions of a disk of finite thickness ([25, 26]), the wobbling of a frustum ([27]) or the sloshing in a tank ([28]).

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 slenderness for which bouncing is not triggered (see [22, 23]). Eccentricity of the center of mass, evaluated 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 conservation 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 motion by using a IMSL routine developed in Fortran [30].

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 horizontal excitation is considered.

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 paper: the first is conducted by varying the direction of the seismic input to point out if, for some directions, 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 response. Also in this case, rocking and overturning curves, that furnish the amplitude of the seismic 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 mechanical 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.

## 2. Description of the considered mechanical system

The base of the rigid block is supposed to be rectangular, with the four vertices indicated as

When the body is at rest, at the *i*-th vertex (*i* is indicated as *i*-th vertex (see Fig. 1_{e} for

where

where *i* is

## 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

where *i* (*i* and

### 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

The eccentricity of the mass center with respect to the geometrical center of the parallelepiped, when considered, is obtained by introducing a concentrated mass

and the slendernesses:

where

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

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 _{a}) is considered. Similar conditions can be found for the orthogonal directions, since the mechanical system is symmetric. Thus the uplift occurs when:

where

#### 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

When the body is rocking around a side of the base, AB as example, the uplift condition on the vertex

In this case, _{b}). The quantity

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 (

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

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 *i* (

where

## 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 (*x*-axis. The pulse-type acceleration used in the analyses are

where

### 4.2. Seismic excitation

In the seismic analysis, only the horizontal effects of the seismic source

### 4.3. Description of the simulation

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

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 one-sine 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 geometrical 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

## 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

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 _{a} 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. 4_{b-d}.

In particular when _{a} 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 (_{c}) 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 _{d}) 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.

With respect to the previous case, results shown in Fig. 5 refer to different periods of the excitations. In Fig. 5_{a}, the case in absence of vertical excitation is shown ([29]). When a vertical excitation with fixed amplitude and _{b}), 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.

It is interesting to note that the case in absence of vertical excitation (Fig. 5_{a}) 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 (_{c}) 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 (_{d}) 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 _{c}, are shown. Point _{a}); point _{b}). 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

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. 7_{a}, see [29] for more details). The vertical excitation, also in this case, strongly changes the scenario. In particular, when the _{b}), 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 _{c}) 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 _{d}) 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

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 _{a} (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 _{b} and 8_{d} 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 _{c}), the critical regions disappear.

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

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 (_{a} and
Fig. 9_{b}, 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. 9_{b}) 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 _{c}), two new critical regions appear whereas, for _{d}), these critical regions change their position and amplitude with respect to the case where the vertical excitation is null (Fig. 9_{b}).

### 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. 10_{a,b} refer to two different blocks (_{c,d}). Very different are the results obtained by exciting the body by the Calitri earthquake (Fig. 10_{e,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 overturning PGA (dash-dotted circle). However, smaller values of the PGA than the previous earthquakes 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.

In Fig. 11 the effect of the eccentricity of the mass center is outlined. In particular, a block of dimensions _{a,b} refer to the case without eccentricity and the case with eccentricity along the y-axis (

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 (dash-dotted 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. 11_{c,d}, that refer to Buia earthquake, confirm what previously said. The situation changes if the block is excited by the Calitri earthquake (Fig. 11_{e,f}). In this case the 3D overturning amplitudes remain always far enough from the 2D overturning PGA’s, also when an eccentricity is considered.

To summarize, when the block is excited by an earthquake with narrow spectrum (Brienza and Buia earthquakes, see Fig. 3_{a,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. 3_{c}) 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.

Finally, the case of blocks with a rectangular base is considered. Results shown in Fig. 12_{a,b} refer to rectangular based block (

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 interesting 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

The sequence of results shown in Fig. 13_{a-c} refers to cases with fixed slenderness _{b} (_{d-f} refers to cases with fixed mass (_{f} (

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.

## 6. Conclusion

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 of the angular momentum.

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

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 presence 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 accurate to correctly evaluate the occurrence of the overturning and, therefore, a three-dimensional model is needed.

The seismic response of the rigid body excited by three different Italian registered earthquakes has been analyzed, reporting rocking and overturning curves. Results show that, for narrow spectrum earthquakes, bi-dimensional models are not enough accurate to correctly 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 response of a rigid block in favour of safety.

## Appendix A. Vector and tensor quantities

The rotation

where, for

When the block is a parallelepiped of uniform mass density, with sides of length

The mass is

The representation of the Euler tensor with respect to the point

To get the generic static moment

where the tensor product