# Insights on the experimental dynamic behaviour and energy losses of rocking blocks under free vibration

Carla Colombo, Georgios Vlachakis, Anastasios I. Giouvanidis, Nathanaël Savalle, Nuno Mendes, Paulo B. Lourenço

## ► To cite this version:

Carla Colombo, Georgios Vlachakis, Anastasios I. Giouvanidis, Nathanaël Savalle, Nuno Mendes, et al.. Insights on the experimental dynamic behaviour and energy losses of rocking blocks under free vibration. Bulletin of Earthquake Engineering, 2025, 10.1007/s10518-025-02224-8. hal-05322004

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# Insights on the experimental dynamic behaviour and energy losses of rocking blocks under free vibration

Carla Colombo<fnref n="1" />* · Georgios Vlachakis<fnref n="1" /> · Anastasios I. Giouvanidis<fnref n="2" /> · Nathanaël Savalle<fnref n="3" /> · Nuno Mendes<fnref n="1" /> · Paulo B. Lourenço<fnref n="1" />

Received: 7 February 2025 / Accepted: 6 July 2025

© The Author(s) 2025

## Abstract

This study focuses on the experimental response of free-standing hard limestone blocks under free vibration. The campaign includes 120 tests, varying the block's height-to-width ratio and considering multiple specimens to account for the aleatoric variability of the phenomenon. The study offers full reconstruction of the three-dimensional free-rocking motion, giving insights into the influence of unintended geometrical asymmetries, as well as material and interface irregularities on the response. The paper revisits the experimental estimation of the angular Coefficient of Restitution (CoR) through different methodologies based on the angular velocities, potential energy, and potential and frictional energies. The results are compared with Housner's theoretical model. Due to the sensitive nature of rocking motion, a statistical approach is employed. The findings indicate that Housner's model provides statistically accurate predictions of energy losses for blocks of medium-to-high slenderness (i.e. with aspect ratio $5 \le H/B \le 10$ while it becomes statistically inaccurate for very slender $H/B > 10$ and very stocky $H/B < 5$ blocks. Importantly, the study demonstrates that all three experimentally estimated CoRs statistically depend on the aspect ratio. Finally, the CoRs extracted from the potential energy and potential and frictional energies are found statistically dependent on the rocking amplitude, while the CoR extracted from the angular velocities show statistical independence. Overall, this study offers a pathway for more systematic analyses and improved predictions of energy losses during impacts.

**Keywords** Free-rocking · Coefficient of restitution · Unreinforced masonry · Out-of-plane · Experimental campaign · Digital image correlation

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<footnote>Carla Colombo
carla.colombo95@gmail.com</footnote>

<footnote n="1">¹ Department of Civil Engineering, University of Minho, ISISE, ARISE, Guimarães, Portugal</footnote>

<footnote n="2">² Department of Civil and Environmental Engineering, The University of Auckland, Auckland, New Zealand</footnote>

<footnote n="3">³ Clermont Auvergne INP, CNRS, Institut Pascal, Université Clermont Auvergne, Clermont-Ferrand F-63000, France</footnote>
# 1 Introduction

Rocking is a phenomenon frequently observed in slender and rigid structures during earthquakes. Under strong shaking, such structures may uplift and pivot about their base, as seen in bridge piers (Cheng 2007; Dimitrakopoulos and Giouvanidis 2015; Giouvanidis and Dong 2020; Makris and Vassiliou 2013; Reggiani Manzo and Vassiliou 2019), water tanks (Housner 1963), obelisks, columns, and unanchored equipment (Konstantinidis and Makris 2009; Makris and Vassiliou 2013; Psycharis 2018). Rocking also occurs in unreinforced masonry structures, affecting façades (Cappelli et al. 2020; Lagomarsino 2015; Mauro et al. 2015; Vlachakis et al. 2020) and/or non-structural elements, such as parapets and statues (Cocuzza Avellino et al. 2021; Giaretton et al. 2016). Specifically, if the restraints provided by diaphragms and lateral walls are insufficient, localised macroblocks can detach from the structure and experience a dynamic behaviour resembling that of a rocking block (Casapulla et al. 2017).

The work of Housner (1963) marked the first systematic analysis of the rocking motion of rigid structures, deriving the equations of motion and introducing an angular Coefficient of Restitution (CoR) to quantify energy losses. However, later experimental studies (Aslam et al. 1980; Priestley et al. 1978) revealed sensitivity of Housner’s model to input parameters, such as the geometric features of the block (Aslam et al. 1980; Lipscombe and Pellegrino 1993; Spanos and Koh 1985; Tso and Wong 1989; Yim et al. 1980). The idealised assumptions of perfect geometry, purely two-dimensional motion, continuous contact with the ground, and fully plastic impacts often fail to hold in practice, as real structures exhibit sliding, bouncing, and response variability due to interface imperfections and irregularities (Lipscombe and Pellegrino 1993; Peña et al. 2008). This has led to more advanced models incorporating three-dimensional motion (Bao and Konstantinidis 2020; Chatzis and Smyth 2012; D’Altri et al. 2024; Di Egidio et al. 2014; Várkonyi et al. 2022) and asymmetric geometries (Mathey et al. 2016; Purvance et al. 2008; Wittich and Hutchinson 2015).

Another challenging aspect of rocking motion is associated with the energy losses due to impacts. Housner’s model captures the energy dissipation at impact through the angular CoR, derived under the assumption of angular momentum conservation. However, later studies showed that the CoR is sensitive and relies on complex dynamic phenomena, thus questioning the assumptions underlying Housner’s model (Lipscombe and Pellegrino 1993). In response, several refined models have been developed. Some introduce modifications to the location of the impact point (Chatzis et al. 2017; Kalliontzis et al. 2016; Ther and Kollár 2017), while others relax the assumption of purely plastic impacts by incorporating the kinematic normal Coefficient of Restitution (CoR<sub>N</sub>) alongside Coulomb’s friction law (Brogliato et al. 2012; Giouvanidis and Dimitrakopoulos 2017; Shenton and Jones 1991). Additionally, compliant (Chatzis and Smyth 2012; Koh et al. 1986; Psycharis and Jennings 1983; Spanos et al. 2017; Vlachakis et al. 2021) and hybrid impact models (Bao 2024; Chatterjee et al. 2022; Yilmaz et al. 2009; Zhang et al. 2013; Zhao et al. 2019) are proposed in the literature, simulating the process in greater detail as a continuous phenomenon but requiring input parameters that are difficult to identify experimentally, such as interface stiffness and interface damping (Vlachakis et al. 2023).

Experimental free-rocking studies have played a key role in advancing the understanding of rocking motion and its associated energy dissipation mechanism. Early free-rocking tests (Aslam et al. 1980; Lipscombe and Pellegrino 1993; Priestley et al. 1978) compared

<figure><img src="image_4.png" /></figure>
observed CoR values with Housner's predictions. Later work explored a range of materials and geometries, including precast columns (Cheng 2007), granite blocks (Peña et al. 2008), steel blocks (Mathey et al. 2018, 2016), concrete walls (Kalliontzis and Sritharan 2018), masonry façades (Costa et al. 2013; Sorrentino et al. 2011) and aluminium blocks (Čeh et al., Čeh et al. 2018). Among them, Costa et al. (2013), ElGawady et al. (2011), Peña et al. (2008), and Sorrentino et al. (2008) extracted the CoR based on the preservation between potential and kinetic energies. A few other studies have examined the effect on the response of varying aspect ratios (Čeh et al. 2018; ElGawady et al. 2011), interface materials (Spanos et al. 2017), and imperfections (Al Shawa et al. 2012; Bachmann et al. 2018; Kafle et al. 2011; Mathey et al. 2018, 2016). Asymmetrical configurations have also been considered (Cocuzza Avellino et al. 2021; Huang et al. 2021; Purvance et al. 2008).

Overall, previous experimental studies have highlighted the high sensitivity of rocking motion to geometric irregularities and energy dissipation. As a matter of fact, significant variance is commonly observed in the CoR across different experimental campaigns. Moreover, the limited number of tests, small sample sizes, and narrow range of aspect ratios often hinder the ability to generalise findings. As a result, impact and damping models are typically evaluated partially, offering incomplete interpretations of the experimental data.

This paper presents an extensive experimental campaign on the free-rocking motion of limestone blocks, involving 120 tests across twelve aspect ratios. Ten repetitions per aspect ratio support uncertainty quantification and generalisation of the results. The study investigates the three-dimensional motion of the blocks and the influence of asymmetries and interface irregularities. A particular focus is placed on the experimental estimation of the angular CoR, providing alternative extraction strategies. Moreover, the work introduces a statistical analysis to compare the experimentally estimated CoR with the theoretical prediction of Housner's model and discusses dependencies on both the aspect ratio and the amplitude of rocking motion.

The paper is structured as follows. Section 2 outlines the key aspects of Housner's model. Section 3 details the experimental campaign and data acquisition system. Section 4 describes the post-processing methodology used to estimate key parameters related to the free-rocking motion. Section 5 presents the results of a representative free-rocking test, along with an overview of the experimental campaign. Section 6 offers a statistical analysis of the experimental CoR data and, finally, Section 7 provides the conclusions and key remarks.

## 2 Theoretical background

Consider the rigid block shown in Fig. 1a, freely standing on a rigid base and characterised by its height $H$, width $B$, mass $m$, moment of inertia $I_O$ about its corner pivot points $O$ and $O'$, radial distance $R$ between the centre of mass and the pivot points, and slenderness angle $\alpha = tan^{-1}(B/H)$. When tilted with an initial rotation smaller than $\alpha$ and let free, the block starts free-rocking around the pivot points on its base with a rotation angle $\theta_x$ (Fig. 1b). The equation of motion of the block is expressed as (Housner 1963):

$$ \ddot{\theta}_x = -p^2 [\sin(\alpha \operatorname{sgn}(\theta_x) - \theta_x)] \quad (1) $$
where the sgn() represents the signum function, $\ddot{\theta}_x$ is the angular acceleration of the block, and $p$ is the frequency parameter of the block defined as $p = \sqrt{mgR/I_0}$, where $g$ is the gravitational acceleration. For slender blocks with a low value of the slenderness angle $\alpha$, Housner suggested the linearisation of Eq. (1) and provided a closed-form solution of the differential equation. Consequently, the period of the free vibration during rocking is computed as:

$$T = \frac{4}{p} \cosh^{-1} \left[ \frac{1}{(1 - \theta_{x,peak}/\alpha)} \right] \quad (2)$$

where $\theta_{x,peak}$ corresponds to the peak rotation $\theta_x$ during the half-cycle of the free vibration. Fig. 1c shows that the period of the free vibration (Eq. (2)) is strongly and non-linearly dependent on the rotation amplitude $\theta_{x,peak}$, increasing from zero to infinite as the $\theta_{x,peak}/\alpha$ ratio goes from zero to one.

During rocking, a change in the sign of $\theta_x$ marks the transition from smooth pivoting to abrupt impact, accompanied by sudden energy dissipation. Housner modelled this impact as an instantaneous event and introduced the angular CoR, defined as the ratio of angular velocities after ($\dot{\theta}_x^+$) and before ($\dot{\theta}_x^-$) impact. Moreover, Housner proposed an impact model to estimate the value of the CoR based on non-smooth impulsive dynamics. Under angular momentum conservation, the resulting expression shows that the CoR depends solely on the block's slenderness (i.e. height-to-width ratio), independent of its size, material, or interface properties:

$$CoR_{Housner} = 1 - \frac{3}{2}\sin^2\alpha \quad (3)$$

<figure><img src="image_5.png" /><figcaption>Fig. 1 Scheme of a planar rocking block: (a) resting, and (b) tilted configurations; and (c) period of vibration based on Eq. (2)</figcaption></figure>
## 3 Experimental campaign and acquisition system

The free-rocking experimental campaign is performed using 60 limestone blocks. A preliminary investigation was carried out to characterise the material properties of the limestone blocks and the limestone-to-limestone interfaces (Vlachakis et al. 2023). Table 1 summarises the average values of the characterisation campaign, including the density of the material $\rho$, the compressive strength $f_c$, the modulus of elasticity $E$, the static friction coefficient $\mu_{static}$, and the normal $k_n$ and tangential $k_s$ interface stiffness, which varied widely depending on the normal stress. The normal and tangential interface stiffness are mechanical parameters describing the dry contact behaviour, which are essential for researchers aiming to model the rocking behaviour of the tested blocks using compliant contact models. The results of this characterisation campaign suggest that it is reasonable to assume that the blocks behave as rigid bodies, with most of the flexibility concentrated at the interfaces. Further details on the characterisation campaign are available in Vlachakis et al. (2023).

The limestone blocks have nominally identical plan dimensions but varying heights (Fig. 2a). Specifically, each block nominally measures 50 mm in width $B$ and 150 mm in length $L$, while the height $H$ ranges from 200 mm to 750 mm, varying in 50 mm increments. This results in 12 aspect ratios $H/B$ ranging from 4 to 15. The broad variation of slenderness ($H/B$) of the specimens has been selected as it is representative of real-world rocking structures, as indicated in studies such as Casapulla et al. (2017), while the length-to-width ratio ($L/B$) equal to three is intended to minimise three-dimensional effects and promote a predominantly two-dimensional rocking response (Costa et al. 2013; Peña et al. 2008; Ther and Kollár 2017). Table 2 summarises the geometrical features of the blocks, including the measured dimensions with the Coefficient of Variation (CoV) shown in parentheses as a percentage, the geometrical ratios, and the number of blocks. Each aspect ratio group consists of five nominally identical blocks. Nevertheless, each block can be tested in two configurations by rotating it 180° around the x-axis, thereby allowing the two opposite horizontal surfaces ($B \times L$), top and bottom, to be considered as distinct testing interfaces. This is important because the rocking response is highly sensitive to slight variations of the contact interface, which can significantly impact the response of the blocks. Therefore, each unique interface accounts for an independent test, resulting in a total of 120 tests conducted during the free-rocking campaign.

Figure 2b provides an overview of the experimental setup. A free-standing block rests on a fixed bottom block made of the same limestone material, with the contact surface between them forming the interface (properties detailed in Table 1). To initiate a test, the top block is manually rotated and positioned at rest against an external retaining system. This retaining structure consists of a stiffened L-profile anchored to the substructure and it is designed to accommodate a threaded bar. As the bar is gradually screwed, it causes the block to rotate slowly until it reaches its slenderness angle, at which point the block begins to rock freely on its fixed base without any external excitation. This method is used in other experimental campaigns and has proven to be more effective than alternative approaches (Sorrentino et

<table><caption>**Table 1** Average mechanical properties of the limestone blocks and limestone-to-limestone interfaces</caption><thead><tr><th>Limestone block</th><th></th><th>Limestone-to-limestone interface</th><th></th></tr></thead><tbody><tr><td>ρ [kg/m³]</td><td>2240</td><td>μ<sub>static</sub> [-]</td><td>0.70</td></tr><tr><td>f<sub>c</sub> [MPa]</td><td>47.6</td><td>k<sub>n</sub> [MPa/mm]</td><td>15 (at 0.1 MPa) – 100 (at 1 MPa)</td></tr><tr><td>E [GPa]</td><td>32.7</td><td>k<sub>s</sub> [MPa/mm]</td><td>5 (at 0.1 MPa) – 30 (at 1 MPa)</td></tr></tbody></table>
<figure><img src="image_6.png" /><figcaption>**Fig. 2 (a)** Limestone blocks with varying heights, (b) side view of the experimental setup, (c) 3D scheme of the DIC acquisition system; and (d) displacement contour plot obtained from the DIC measurements</figcaption></figure>

al. 2011). The motion until rocking commences is disregarded, while the end of the motion is assumed when there is a higher than two orders of magnitude difference between the peak rotation amplitudes and the measurement noise.

The free-rocking motion is captured using a four-camera Digital Image Correlation (DIC) acquisition system (Fig. 2c). The DIC is an optical, contactless measurement technique, which allows the estimation of the field of displacement of objects by comparing a series of subsequent images relative to a reference image (Sutton et al. 2009). The benefit of using this monitoring strategy relies on the capability of tracking the specimen at a sufficiently high speed without physically interfering with the experimental setup and reaching a measurement accuracy of around 0.01 mm. Accurate image tracking relies on stable lighting conditions and the quality of the speckle pattern applied to the specimen. In this study, a high-contrast grey-scale pattern is applied to two side surfaces of the block (Fig. 2b). The DIC system features stereoscopic cameras with 25 mm lenses, capable of recording 145 frames per second. The four cameras are positioned around the perimeter of the experimental setup, with each pair focused on one surface of the block (Fig. 2c). The DIC data processing is performed using the ISTRA 4D commercial software from Dantec Dynamics.

<figure><img src="image_7.png" /></figure>
<table><caption>**Table 2** Geometrical features of the free-standing blocks. The blocks' dimensions are accompanied by the Coefficient Of Variation (CoV) reported in parentheses as a percentage. The listed aspect ratios refer to the nominal dimensions</caption><thead><tr><th>Block group</th><th>Height H [mm]</th><th>Width B [mm]</th><th>Length L [mm]</th><th>H/B [-]</th><th>L/B [-]</th><th>No. of blocks</th></tr></thead><tbody><tr><td>1</td><td>202.4 (0.24%)</td><td>50.1 (0.26%)</td><td>153.2 (0.26%)</td><td>4</td><td>3</td><td>5</td></tr><tr><td>2</td><td>248.0 (0.36%)</td><td>49.8 (0.68%)</td><td>150.2 (0.27%)</td><td>5</td><td>3</td><td>5</td></tr><tr><td>3</td><td>299.8 (0.13%)</td><td>50.3 (0.62%)</td><td>150.0 (0.00%)</td><td>6</td><td>3</td><td>5</td></tr><tr><td>4</td><td>349.6 (0.14%)</td><td>50.7 (0.38%)</td><td>150.2 (0.27%)</td><td>7</td><td>3</td><td>5</td></tr><tr><td>5</td><td>398.6 (0.12%)</td><td>49.3 (0.37%)</td><td>151.0 (0.42%)</td><td>8</td><td>3</td><td>5</td></tr><tr><td>6</td><td>448.8 (0.17%)</td><td>49.3 (0.66%)</td><td>151.6 (0.32%)</td><td>9</td><td>3</td><td>5</td></tr><tr><td>7</td><td>498.4 (0.24%)</td><td>48.9 (0.28%)</td><td>151.2 (0.49%)</td><td>10</td><td>3</td><td>5</td></tr><tr><td>8</td><td>551.4 (0.25%)</td><td>49.4 (0.68%)</td><td>152.2 (0.49%)</td><td>11</td><td>3</td><td>5</td></tr><tr><td>9</td><td>598.0 (0.22%)</td><td>49.5 (0.42%)</td><td>152.0 (0.31%)</td><td>12</td><td>3</td><td>5</td></tr><tr><td>10</td><td>652.2 (0.44%)</td><td>49.2 (0.56%)</td><td>152.0 (0.59%)</td><td>13</td><td>3</td><td>5</td></tr><tr><td>11</td><td>701.8 (0.14%)</td><td>50.8 (2.84%)</td><td>153.4 (0.32%)</td><td>14</td><td>3</td><td>5</td></tr><tr><td>12</td><td>750.6 (0.32%)</td><td>51.6 (3.10%)</td><td>151.6 (1.07%)</td><td>15</td><td>3</td><td>5</td></tr></tbody></table>

From the relative 2D perspective of the four cameras, each point of an object surface can be reconstructed into a 3D perspective using the principles of stereo-triangulation (Fig. 2d). For the purpose of the current investigation, the information in terms of coordinates and displacement along the x-, y-, and z-axes of two gauge points per surface are extracted. Fig. 2c clarifies the axes, gauge points' location and their denomination. Moreover, the displacement and coordinate values of each point are associated with an uncertainty parameter, which indicates the accuracy of the measurement.

# 4 Post-processing methodology

## 4.1 Three-dimensional motion

Assuming the limestone blocks behave as rigid bodies, their three-dimensional motion can be described by six degrees of freedom, shown schematically in Fig. 3: three translations $u_{x,GC}$, $u_{y,GC}$, $u_{z,GC}$, and three rotations $\theta_x$, $\theta_y$, $\theta_z$, around the x-, y-, and z-axes. The reconstruction of the three-dimensional motion is achieved using the Kabsch algorithm (Lawrence et al. 2019), which processes the raw data from the DIC in the form of coordinates and displacements of the four gauge points shown in Fig. 2c. The three translations ($u_{x,GC}$, $u_{y,GC}$ and $u_{z,GC}$) are referred to the Geometric Centroid (GC) of the four gauge
<figure><img src="image_8.png" /><figcaption>Fig. 3 Degrees of freedom of a rigid block: translations along the: (a) x-axis ($u_{x,GC}$) (b) y-axis ($u_{y,GC}$), and (c) z-axis ($u_{z,GC}$); and rotations over the (d) x-axis ($\theta_x$) (e) y-axis ($\theta_y$) and (f) z-axis ($\theta_z$) [Note: to resemble the actual rocking motion, the rotations $\theta_x$ and $\theta_y$ are represented as rotations around the pivot axes, i.e. coupled with translational components along the x and y axes, respectively]</figcaption></figure>

points and are calculated for each direction as the mean displacement of the four points. The computation of the rotations requires the estimation of the rotation matrix $\mathbf{R}$, which is reconstructed using the singular value decomposition method (Lawrence et al. 2019). The rotations are derived from the rotation matrix $\mathbf{R}$ following the convention of the Tait-Bryan Euler angles as follows:

$$ \theta_x = \tan^{-1}(-R_{23}/R_{33}) \qquad (4) $$

$$ \theta_y = \sin^{-1}(R_{13}) \qquad (5) $$

$$ \theta_z = \tan^{-1}(-R_{12}/R_{11}) \qquad (6) $$

where each element $R_{ij}$ corresponds to the components of the 3 × 3 rotation matrix $\mathbf{R}$. Consequently, the Euler angular velocities relative to each rotation ($\dot{\theta}_x$, $\dot{\theta}_y$ and $\dot{\theta}_z$), are
obtained by differentiating the rotations over time using the second-order accurate central differences. Similarly, the Euler angular accelerations ($\ddot{\theta}_x$, $\ddot{\theta}_y$ and $\ddot{\theta}_z$) are estimated by differentiating the Euler angular velocities.

The actual geometry of each limestone block differs from the nominal geometry. This is due to several factors, such as variations in block dimensions resulting from the stone-cutting process, shifts in the centre of gravity due to material heterogeneity, surface irregularities at the contact interfaces that can affect the position of the pivot points or create unintended inclinations of the free-standing block, interface accumulated damage during testing, and others (Costa et al. 2013; Kalliontzis and Sritharan 2018; Peña et al. 2008). To account for these imperfections, this study performs an experimental determination of the parameters $p$ and $\alpha$ of each test, by using as reference the peak rotations and periods of each free-rocking half-cycle (Peña et al. 2008). Specifically, $p$ and $\alpha$ are estimated by using the analytical expression proposed by Housner (Eq. (2)) and through a fitting process that minimises the root mean square error between measured and computed periods. During this fitting process, the half-cycles with positive and negative rotations are treated separately, thus resulting in $p_+$ and $\alpha_+$ for $\theta_x > 0$, and $p_-$ and $\alpha_-$ for $\theta_x < 0$.

The post-processing methodology includes the computation of the potential $U$, kinetic $K$ and total $E$ energies of the rocking blocks as follows:

$$U = mgR_{\pm} [\cos(\alpha_{\pm} - |\theta_x|) - \cos\alpha_{\pm}] \quad (7)$$

$$K = 0.5 I_{o,\pm} \dot{\theta}_x^2 \quad (8)$$

$$E = U + K \quad (9)$$

where the subscript symbol “±” refers to the positive and negative signs of rotation. $R_{\pm}$ is obtained from $p_{\pm} = \sqrt{\frac{mgR_{\pm}}{I_{o,\pm}}}$, where $I_{o,\pm} = \frac{m}{12}(B^2 + H^2) + mR_{\pm}^2$, with $B$ and $H$ being the nominal values.

Furthermore, the post-processing methodology involves the estimation of the Instantaneous Axis of Rotation (IAR), which represents the imaginary axis around which a rigid body is rotating at a specific time instant. The IAR is calculated for each time step using the three-dimensional motion and is inherently orthogonal to the velocity vector of any point. Furthermore, the IAR is used to experimentally estimate the coordinates of the pivot axes at the block's base during rocking motion. Specifically, the y coordinate of the pivot axes, denoted as $y_{IAR}$, is estimated as the median value of the y coordinate of the IAR, separately for positive and negative rotations $\theta_x$. Finally, the estimation of sliding $u_{y,pp}$ (i.e. horizontal translation along the y-axis) and uplifting $u_{z,pp}$ (i.e. vertical translation along the z-axis) are calculated for the two points on the pivot axes at the centre of the block's length ($x_{IAR} = 0$). Sliding along the x-axis is considered negligible compared to the y-axis and thus it is ignored.

## 4.2 Energy losses

Following the model of Housner, the energy dissipation of the free-rocking response is described using the angular CoR. As a first approach, the experimental quantification of the CoR may be defined as the ratio of the angular velocities right after and before any impact $n$:
$$CoR_{vel} = \frac{\dot{\theta}_{x;n}^{+}}{\dot{\theta}_{x;n}^{-}} \qquad (10)$$

Identifying the exact time instants of each impact presents a significant challenge, as impacts are not instantaneous but physically continuous and occur over a finite time, contrary to the idealised treatment by Housner. To this end, the present study uses the vertical acceleration of the geometric centroid $\ddot{u}_{z,GC}$ to identify the onset and end time instants of each impact (Kalliontzis et al. 2016). Further details on this approach are given in Appendix A. However, differentiating the experimental displacement response twice introduces noise into the acceleration response $\ddot{u}_{z,GC}$. Additionally, during impact, the block experiences high velocities, and the number of measurements within this short time frame of impact is constrained by the adopted sampling rate. The combination of these factors introduces uncertainties and high variance in the quantification of the $CoR_{vel}$, as discussed further in Sect. 5.2 and demonstrated in previous studies (Costa et al. 2013; Giresini et al. 2018).

An alternative strategy to experimentally quantify the angular CoR relies on the assumption that energy is lost solely during impacts. As a result, the kinetic energy $K_n^-$ before any impact $n$ is equal to the potential energy $U_{peak;n}^-$ at the peak of the preceding half-cycle. Similarly, the kinetic energy $K_n^+$ after impact $n$ is equal to the potential energy $U_{peak;n}^+$ at the peak after the $n^{th}$ impact. This permits the estimation of the angular CoR from the potential energy of two consecutive peaks $CoR_U^{cons}$ as follows:

$$CoR_U^{cons} = \sqrt{\frac{U_{peak;n}^{+} I_{0,\mp}}{U_{peak;n}^{-} I_{0,\pm}}} \qquad (11)$$

where the subscript symbol $\pm$ refers to the positive and negative signs of rotation, whereas the superscripts “+” and “-” denote the post-impact and pre-impact states, respectively. Similarly, the aforementioned methodology can also be applied to estimate the angular CoR considering two successive impacts $n$ and $n+1$, i.e. accounting for the peaks of the potential energy from the same side of motion (refer to Appendix A for further details) (Sorrentino et al. 2011; Costa et al. 2013):

$$CoR_U^{side} = \sqrt{\sqrt{\frac{U_{peak;n+1}^{+}}{U_{peak;n}^{-}}}} \qquad (12)$$

A third method for estimating the angular CoR is also suggested, which relaxes the assumption that energy is lost only at impacts. The proposed method builds on the previous one, which is based on the potential energy, and also considers the frictional energy loss due to sliding $W$ during rocking (details for the estimation of $W$ are provided in Appendix A). As a result, the angular CoR considering both the potential and frictional energies for two consecutive peaks $CoR_{U+F}^{cons}$ before and after an arbitrary impact $n$ reads:

$$CoR_{U+F}^{cons} = \sqrt{\frac{U_{peak;n}^{+} + W_n^{+} I_{0,\mp}}{U_{peak;n}^{-} - W_n^{-} I_{0,\pm}}} \qquad (13)$$
where, again, the subscript symbol ± refers to the positive and negative signs of rotation, whereas the superscripts “+” and “−” denote the post-impact and pre-impact states, respectively. Applying the aforementioned method twice for two successive impacts n and n +1 and thus considering the peaks from the same side of motion gives (refer to Appendix A):

$$CoR_{U+F}^{side} = \sqrt{\frac{\left( \begin{array}{c} W_{imp \to peak; n}^{+} + W_{peak \to imp; n+1}^{-} \\ + \sqrt{\left( W_{imp \to peak; n}^{+} + W_{peak \to imp; n+1}^{-} \right)^2 + 4 \left( U_{peak; n}^{-} - W_{peak \to imp; n}^{-} \right)} \\ \left( U_{peak; n+1}^{+} + W_{imp \to peak; n+1}^{+} \right) \left( \frac{I_{o, \pm}}{I_{o, \mp}} \right)^2 \end{array} \right)}{2 \left( U_{peak; n}^{-} - W_{peak \to imp; n}^{-} \right) \frac{I_{o, \pm}}{I_{o, \mp}}} \quad (14)$$

where the subscript of the frictional energy W denotes the segment of the half-cycle that sliding occurs (e.g. imp → peak) and the corresponding impact (e.g. n).

Finally, an alternative method to describe the energy dissipation at any impact n is the kinematic normal CoR<sub>N</sub> (Corral et al. 2021; Giouvanidis and Dimitrakopoulos 2017; Goldsmith 1960; Pfeiffer and Glocker 2000), which is defined for any impact point as the ratio between the velocities normal to the contact interface after and before the n<sup>th</sup> impact:

$$CoR_N = - \frac{\dot{u}_{z; n}^{+}}{\dot{u}_{z; n}^{-}} \qquad (15)$$

The normal CoR<sub>N</sub> is estimated for a grid of points over the contact interface. Specifically, to obtain admissible values of the normal CoR<sub>N</sub> ranging between 0 and 1 (Goldsmith 1960; Pfeiffer and Glocker 2000), the grid points with positive y coordinate analyse the impacts with negative angular velocity $\dot{\theta}_x < 0$, and the grid points with negative y coordinate study the impacts with positive angular velocity $\dot{\theta}_x > 0$. Furthermore, the time instants before and after each impact are identified following the methodology used for the CoR<sub>vel</sub> (see Appendix A for further details).

## 5 Results

This section outlines the results of the free-rocking experimental campaign, applying the methodology detailed in Section 4. The results are divided into two main sections. Sec. 5.1 explores various aspects of the free-rocking motion, highlighting the effect of imperfections and asymmetries in the response. Sec. 5.2 revisits the topic of energy dissipation during impacts, comparing the three methodologies for estimating the angular CoR (Eqs. (10)-(14)), along with the computation of the normal CoR<sub>N</sub> (Eq. (15)). Each section demonstrates the results of a representative test, together with overviews that incorporate the results from all tests. The representative examples refer to a block with nominal dimensions H = 0.20 m, B = 0.05 m, L = 0.15 m, corresponding to an aspect ratio of H/B = 4 (α = 0.245 rad).
## 5.1 Three-dimensional motion

Figure 4 shows the response time history of the representative free-rocking test. Fig. 4a presents the three rotations of the block $\theta_x$, $\theta_y$, and $\theta_z$, over the x-, y-, and z-axes, respectively. $\theta_x$ governs the overall rocking motion of the block and shows a typical oscillatory response. It starts from a rotation approximately equal to the nominal slenderness angle $\alpha$, with its amplitude decreasing progressively after each impact until the block returns to its free-standing (rest) condition. The rotations around the y- and z-axes, i.e. $\theta_y$ and $\theta_z$ (Fig. 3), have nonzero but negligible amplitudes (up to 0.004 rad, or 0.2 degrees) compared to $\theta_x$. Fig. 4b plots the angular velocity $\dot{\theta}_x$ around the x-axis. $\dot{\theta}_x$ shows a rather linear trend in each half-cycle of rocking, with zero crossings at the peak of rotation and maximum amplitudes during impact and gradually decreasing amplitudes over time after consecutive impacts. The angular acceleration $\ddot{\theta}_x$ around the x-axis is reported in Fig. 4c. It remains nearly constant with a slight concavity during rocking, yet its sign changes abruptly at each impact. It is worth mentioning that consecutive differentiation of the acquired signal introduces noise and, for this reason, $\ddot{\theta}_x$ shows some noisy features.

Figure 4d shows the sliding response $u_{y,pp}$ along the y-axis of the Pivot Points (PP) at the block's base. The representative block exhibits sliding throughout the time history, with greater variation at the time of impacts. Nevertheless, the amplitudes remain minimal, with a maximum value of less than 0.1 mm. The noisy appearance of the sliding response is due to the comparable accuracy of the DIC acquisition, which is in the order of 0.01 mm. Fig. 4e shows the uplifting response $u_{z,pp}$ of the pivot points, i.e. the vertical translation of the two mid-points of the pivot axes. The uplifting of the two pivots alternates such that when pivoting occurs on one corner, that corner remains in contact with the base ($u_{z,pp} = 0$), while the other one experiences vertical translation ($u_{z,pp} > 0$). The potential, kinetic and total energies are presented in Fig. 4f. During a half-cycle, the potential energy $U$ reaches its peak value when the rotation becomes maximum and gradually decreases to zero when the rotation gradually becomes zero. Conversely, the kinetic energy $K$ becomes maximum during impact (when rotation is zero) and drops to zero when the rotation becomes maximum. Subsequently, the total energy $E$ remains practically constant during the rocking motion and decreases in a stepwise fashion at each impact. Note that the asymmetric nature of the response becomes evident from the different rates of decrease of the total energy $E$ when transitioning from positive to negative rotations or vice versa. A similar behaviour was already reported in the literature when analysing the dynamics of asymmetric objects (Cocuzza Avellino et al. 2021).

The relationship between $\theta_x$ with $\theta_y$, and $\theta_x$ with $\theta_z$ is presented in Fig. 5. Interestingly, Fig. 5a shows that $\theta_x$ and $\theta_y$ appear to be strongly correlated throughout the time-history, despite the different orders of magnitude. In fact, the representative behaviour of Fig. 5a is recurrent to all the free-rocking tests and shows a form of “trilinear” interaction: there are three clear distinct ranges of approximately linear correlation associated with significant amplitudes (herein of $\theta_x/\alpha > 2\%$ and $\theta_x/\alpha < -2\%$), and the range around impact (herein $-2\% \le \theta_x/\alpha \le 2\%$). Note that the similarity among the different tests lies in the trilinear form of this interaction, rather than the values or sign of the correlation. This random but recurrent correlation is due to the unique geometrical surface morphology of each contact interface between the free-standing block and its base, which directly affects the two rota-

<figure><img src="image_9.png" /></figure>
<figure><img src="image_10.png" /><figcaption>**Fig. 4** Free-rocking response of the representative test: (a) rotations around the x- ($\theta_x$), y- ($\theta_y$), and z- ($\theta_z$) axis, (b) angular velocity ($\dot{\theta}_x$) over the x-axis, (c) angular acceleration ($\ddot{\theta}_x$) over the x-axis, (d) translation of pivot points along the y-axis ($u_{y,pp}$) and (e) translation of Pivot Points (PP) along the z axis ($u_{z,pp}$), and (f) potential ($U$) kinetic ($K$) and total ($E$) energies</figcaption></figure>

tions. On the other hand, the interaction between $\theta_x$ and $\theta_z$ (Fig. 5b) is not consistent among the tests.

Further insight into the asymmetric rocking response is given in Fig. 6a, which examines the relationship between the absolute peaks of rotation $|\theta_{x,peak}|$ and the correspond-
<figure><img src="image_11.png" /><figcaption>**Fig. 5** Relationship between rocking rotations for the representative test: (a) around the x-axis ($\theta_x$) and y-axis ($\theta_y$), and (b) around the x-axis ($\theta_x$) and z-axis ($\theta_z$)</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Fig. 6** (a) Comparison of periods and amplitudes of the representative test: experimental data (dots and rhombus), fitted relationships of experimental data (solid lines) and nominal relationship (dashed line); (b) experimentally fitted slenderness angles normalised by the nominal ones ($\alpha_{exp}/\alpha_{nom}$) for each $H/B$</figcaption></figure>

ing half-periods $T/2$, doubled to $T$. The different markers (dots and rhombus) indicate the experimental data of the representative test shown in Fig. 4, the continuous lines represent the fitting of the $p$ and $\alpha$ using the experimental data and Eq. (2) (refer to Section 4.1), while the black dashed line corresponds to the period-amplitude relationship of Eq. (2) for the nominal $p$ and $\alpha$. The asymmetric feature of the response is evident in the periods $T$

<figure><img src="image_13.png" /></figure>
when comparing the experimental data for the positive and negative rotations, especially for high response amplitudes. Consequently, a distinct fitting process is performed for *p* and *α* of each side. In this representative case, the experimental data for positive rocking rotations (dots) closely resemble the nominal relation, while the data for negative rocking rotations (rhombus) differ notably for rotations greater than 0.08 rad. Fig. 6, b,c provide a global overview of the experimentally fitted *p*<sub>exp</sub> and *α*<sub>exp</sub> normalised by their nominal counterparts, for all the 120 tests grouped for different aspect ratios *H*/*B*. Fig. 6b shows that the experimentally extracted slenderness angles *α*<sub>exp</sub> are randomly scattered around their nominal values, with greater variance for higher aspect ratios *H*/*B*. Conversely, Fig. 6c shows a consistent underestimation of *p*<sub>exp</sub> in comparison with the nominal values for all the aspect ratios for both positive and negative rotations. This behaviour is attributed to the finite interface stiffness, imperfections, irregularities and accumulated damage after test repetitions at the contact interfaces and the pivoting edges. Finally, Fig. 6b,c shows that the experimental estimations of *α*<sub>exp</sub> and *p*<sub>exp</sub> for positive and negative rotations do not have significant differences between them, highlighting that the asymmetry of the response is associated with the aleatoric variability of each contact interface.

Fig. 7a,b report the absolute value of the residual sliding |u<sub>y,pp</sub>| and absolute residual torsional rotation |θ<sub>z</sub>| respectively, grouped for different aspect ratios H/B. The residual sliding |u<sub>y,pp</sub>| indicates negligible values, always below 1.5 mm, which is less than 3% of the width B of the block. The residual values of |θ<sub>z</sub>| for all the aspect ratios are also of low amplitudes, reaching a maximum of less than 0.015 rad (0.85 degrees). Considering all the tests, the residual rotation |θ<sub>z</sub>| corresponds to an average of less than 3% of the maximum rotation |θ<sub>x,peak</sub>| observed in each test. Overall, no clear dependency on the H/B is observed for any of the two cases.

Figure 8 presents the results of the IAR, starting with the representative test in Fig. 8a,b and an overview of all the tests in Fig. 8c. Specifically, Fig. 8a shows a time window zoom of the y-coordinate position of the IAR *y*<sub>IAR</sub> corresponding to the representative test of Fig. 4. The *y*<sub>IAR</sub> appears to be rather constant during rocking, albeit with noticeable noise, while sudden changes can be observed during impacts that alternate the pivoting edges of the block. The noise during rocking arises from the low velocity of the block, particularly at

<figure><img src="image_14.png" /><figcaption>**Fig. 7 (a)** Absolute residual translation of the pivot points over the y-axis (u<sub>y,pp</sub>), and (b) absolute residual rotation around the z-axis (|θ<sub>z</sub>|), for each H/B</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>**Fig. 8** Instantaneous Axis of Rotation (IAR): (a) y coordinate ($y_{IAR}$) over time of the representative test, (b) 3D representation for the whole time history of the representative test, and (c) histogram plot of the y coordinate normalised by the width of the block ($y_{IAR}/B$) for all the free-rocking tests</figcaption></figure>

the maximum rotation, which affects the accurate estimation of the IAR. Fig. 8b presents a three-dimensional coordinate plot of the IAR over the entire time history, i.e. each line represents the IAR of each time instant. Similar to Fig. 8a, abrupt lines correspond to the noisy and inaccurate estimation of the IAR for the time instants with negligible angular velocity (i.e. at the peaks of the rocking rotation). As expected, Fig. 8b highlights the concentration of the IAR at the edges of the block, resembling the contact imprint of the block. Nevertheless, the experimental evidence also indicates that pivoting occurs across a distributed area of the contact interface, and not at the idealised edge (or pivot point in a 2D view). This aspect is further highlighted in Fig. 8c, which shows a histogram of the distribution of the $y_{IAR}$ for all the tests, together with the nominal width of the blocks. The frequency of appearances of y-coordinate of the IAR reaches a peak very close to the nominal width of the block but slightly shifted inwards. Specifically, the majority of the counts lie around 0.42 to 0.50 and demonstrate a peak at 0.46. This discrepancy may be attributed to imperfections and irregularities due to heterogeneities in the body and/or the contact interface, deviations between the nominal and actual geometry of the block (as shown in Fig. 6), and/or the fact that contact occurs across an area rather than along the idealised edge of the block due to the finite stiffness of the interface (Vlachakis et al. 2023). Finally, the long tail distribution of y-coordinates of the IAR beyond the edge threshold does not have a physical interpretation but is rather attributed to the noisy estimation of the IAR during rocking with negligible velocities.

<figure><img src="image_16.png" /></figure>
## 5.2 Energy losses

Figure 9 presents a representative example of the three different methodologies for estimating the angular CoR, plotted against the normalised absolute rocking rotation at the peak after each corresponding impact $|\theta_{x,peak}|/\alpha$, where $\alpha$ is the nominal value. Specifically, while the CoR calculated from the angular velocities $CoR_{vel}$ yields a single value per impact, the estimation of the angular CoR from potential energy $CoR_U$, and potential and frictional energies $CoR_{U+F}$, varies depending on whether one impact ($CoR_U^{cons}$, $CoR_{U+F}^{cons}$) or two successive impacts ($CoR_U^{side}$, $CoR_{U+F}^{side}$) are considered (refer to Section 4.2). Additionally, Fig. 9 includes the angular CoR$_{Housner}$ of the Housner model estimated using Eq. (3), which is calculated for the nominal value of $\alpha$ for this specific test. $CoR_{vel}$ exhibits the most scattered behaviour. The same behaviour has been observed in previous research (Costa et al. 2013; Giresini et al. 2018), and it is attributed to the challenges associated with the accurate estimation of the angular velocities before and after each impact. Conversely, both $CoR_U^{side}$ and $CoR_{U+F}^{side}$ are nearly constant with negligible variance, showing almost identical values, which in this test, are generally higher than $CoR_{Housner}$. In contrast, $CoR_U^{cons}$ and $CoR_{U+F}^{cons}$ show scattered results, which diverge significantly for low amplitudes. This is a consequence of the asymmetries in the rocking response. Therefore, the subsequent analysis will focus exclusively on the side angular CoR data ($CoR_U^{side}$, $CoR_{U+F}^{side}$). Moreover, the fact that $CoR_U^{side}$ and $CoR_{U+F}^{side}$ (similarly $CoR_U^{cons}$ and $CoR_{U+F}^{cons}$) are almost identical implies that the frictional energy has a marginal effect on the angular CoR value.

Figure 10a presents an overview of the $CoR_{U+F}^{side}$ extracted from all 120 tests. Specifically, the $CoR_{U+F}^{side}$ is normalised by the $CoR_{Housner}$ (using the nominal $\alpha$ value in Eq. (3)), while the abscissa consists of the normalised absolute peak rocking rotation $|\theta_{x,peak}|/\alpha$ after each impact. The different aspect ratios are visually distinguished using colour varia-

<figure><img src="image_17.png" /><figcaption>**Fig. 9** Experimental angular CoR for the representative test using the different methodologies, against the absolute normalised peak rocking rotation after each impact ($|\theta_{x,peak}|/\alpha$)</figcaption></figure>
<figure><img src="image_18.png" /><figcaption>**Fig. 10 (a)** Experimental CoR of all the tests derived using the potential and frictional energies normalised by the Housner model's predictions $CoR_{U+F}^{side}/CoR_{Housner}$ versus the absolute normalised peak rocking rotation after each impact ($|\theta_{x,peak}|/\alpha$), and (b) histogram plot of the normalised CoR distribution</figcaption></figure>

tions. Fig. 10a shows that $CoR_{U+F}^{side}/CoR_{Housner}$ exhibits rather constant values at high rotation amplitudes, and significant scatter at lower amplitudes, similar to the representative test in Fig. 9. This trend, which was also observed in previous studies (ElGawady et al. 2011), is due to the greater influence of the irregularities of the block and contact interface at the low response amplitudes and/or the acquisition accuracy of the measurements. Nonetheless, 90% of the CoR data lie within 0.95 and 1.02, indicating a small difference compared to Housner's model. More specifically, it appears that blocks of medium-to-high aspect ratio closely follow $CoR_{Housner}$ (Eq. (3)), whereas, for stockier blocks, Housner theory tends to underestimate the experimental CoR and thus overestimate the energy losses. A histogram of the CoR distribution is provided in Fig. 10b, where most of the experimental results cluster around $CoR_{Housner}$ and have a marginal skewness towards lower values. Furthermore, the experimental data are predicted well by a Cauchy distribution with a median of 0.997 and a median absolute deviation of 0.009.

To provide a more robust representation against outliers and clarify the central tendency of each test, Fig. 11 presents an overview of the angular CoR using only the median value of each test. Choosing one value per test balances out the undesired dependency of the CoR data within each test, i.e. a slender block experiences more response cycles, more impacts, and thus more CoR data compared to a stockier block. The three subplots in Fig. 11 differentiate the results of the angular CoR as extracted from (a) the angular velocity ($CoR_{vel}$), (b) the potential energy ($CoR_{U}^{side}$), and (c) the potential and frictional energies ($CoR_{U+F}^{side}$), respectively. In each subplot, the black line indicates the $CoR_{Housner}$ of Housner's model (Eq. (3)). Moreover, a boxplot is presented for each aspect ratio $H/B$, where the central solid dot corresponds to the median of the data, while the vertical lines indicate the first and third quartiles. Fig. 11a shows that the $CoR_{vel}$ is characterised by high variance per aspect ratio $H/B$, yet the median values deviate slightly from Housner's model.

$CoR_{U}^{side}$ in Fig. 11b and $CoR_{U+F}^{side}$ in Fig. 11c share common features between them, indicating that for the cases of this experimental campaign, the frictional energy dissipation has a minor influence on the estimation of the angular CoR. In particular, for $H/B = 4$,

<figure><img src="image_19.png" /></figure>
<figure><img src="image_20.png" /><figcaption>**Fig. 11** Experimental CoR distribution with respect to the aspect ratio *H/B* considering the median value per test, extracted based on (a) the angular velocity (*CoR*<sub>vel</sub>), (b) the potential energy (*CoR*<sub>U</sub><sup>side</sup>), and (c) the potential and frictional energies (*CoR*<sub>U+F</sub><sup>side</sup>)</figcaption></figure>

Housner's model underpredicts the experimental CoR estimates of CoR<sub>U</sub><sup>side</sup> and CoR<sub>U+F</sub><sup>side</sup>, suggesting that the block dissipates less energy than the Housner model predicts (Kalliontzis et al. 2016). For medium aspect ratios 5 ≤ *H/B* ≤ 11, the median values of the experimental CoR closely align with the Housner model predictions, with the exception of *H/B* = 7, implying the accuracy of Housner's model to capture the energy losses. However, for higher aspect ratios (12 ≤ *H/B* ≤ 15), the experimental CoR shows consistently lower values than the Housner model, suggesting that the block dissipates more energy than the Housner model predicts (Čeh et al., 2018). From a physical perspective, the higher experimental CoR observed mainly in stockier blocks can be attributed to the inward shift of the impact point from the nominal edge, caused by compliance and convex irregularities at the contact interface, as indicated by the IAR results (Fig. 8) (Ther and Kollár 2017). On the other hand, the lower experimental CoR observed in more slender blocks may be linked to additional dissipative phenomena that are overlooked in the CoR estimation, such as plastic deformations, hysteresis at the interface (Vlachakis et al. 2023) or radiation damping.

Finally, Fig. 12 presents the experimental results of the normal CoR<sub>N</sub>. Fig. 12a shows a three-dimensional view of the representative test, where the normal CoR<sub>N</sub> is plotted across the block's interface and features two surfaces: the surface with positive y-coordinate corresponds to impacts with negative angular velocity (θ̇<sub>x</sub> < 0), and vice versa. Moreover, each point of the two surfaces represents the median value of all the impacts of the test at the specific x and y coordinates. Fig. 12a shows that the normal CoR<sub>N</sub> approaches zero near the pivot edges of the block and tends to one at the centre. Interestingly, the two surfaces are not symmetric along the x-axis, corroborating the presence of a three-dimensional component during the impacts due to irregularities and asymmetries at the interface. Fig. 12b,c show two section views of the normal CoR for three x and y coordinates respectively, while an absolute reference coordinate system is considered, i.e. all the CoR<sub>N</sub> surfaces are plotted using the positive values of y and x. Moreover, each line represents the median value of all the impacts for all 120 tests. The prediction of Housner's model is also shown with dashed lines, which assumes perfectly plastic corner impact and consequently the corresponding kinematic normal CoR<sub>N</sub> of any point at the contact interface is given
<figure><img src="image_21.png" /><figcaption>**Fig. 12** Normal CoR<sub>N</sub>: (a) 3D plot over the block's interface using the experimental median surfaces for the representative test; and experimental median surfaces for all the tests: (b) across the blocks' width *B*, at three x coordinates across the length *L*, and (c) across the blocks' length *L*, at three y coordinates across the width *B*</figcaption></figure>

as: CoR<sub>N</sub> = CoR<sub>Housner</sub> ((1 - 2|y|/B) / (1 + 2|y|/B)). Fig. 12b shows that the experimental normal CoR<sub>N</sub> surface follows a similar trend with respect to Housner's model across the width of the block, but consistently providing a slightly lower prediction. Additionally, the experimental normal CoR<sub>N</sub> appears to be negative at the nominal edges of the blocks (|y|/B = 0.50), which is not physically admissible. This inconsistency indicates once more that the actual impact point is not exactly at the idealised edge of the blocks (|y|/B = 0.50), but rather lies slightly inwards, on average, between |y|/B = 0.42 to |y|/B = 0.45. This observation corroborates the findings of Fig. 8c, where the distribution of the y/B coordinate of the IAR shows a peak between 0.42 and 0.5. Fig. 12c, in conjunction with Fig. 12b, shows that the normal CoR<sub>N</sub> presents a rather constant value along the length of the blocks |x|/L, thus highlighting that the asymmetry of the experimental normal CoR<sub>N</sub> surfaces observed in each unique test (such as Fig. 12a) vanishes when considering all 120 tests, hence, indicating an aleatoric uncertainty.

<figure><img src="image_22.png" /></figure>
# 6 Statistical analysis

Section 5 estimated the angular CoR from the free-rocking experiments using three methods and compared the results with the model of Housner. Nevertheless, a statistical analysis is presented in this section to draw more robust conclusions. This includes a series of statistical tests that address the following questions:

* Is the experimental CoR<sub>exp</sub> statistically equal to Housner's model prediction CoR<sub>Housner</sub> (Section 6.1)?

* Is CoR<sub>exp</sub>/CoR<sub>Housner</sub> statistically dependent on the aspect ratio H/B (Section 6.2)?

* Is CoR<sub>exp</sub>/CoR<sub>Housner</sub> statistically dependent on the rocking amplitude (Section 6.3)?

The analyses are guided by a research question involving the definition of a null hypothesis H₀ and an alternative hypothesis H₁. The null hypothesis H₀ is assumed to hold until there is statistical evidence against it, while the alternative hypothesis H₁ represents the scenario in case H₀ is rejected. The p-value is calculated to test the null hypothesis and compared with different confidence levels. It measures how strong the evidence is against the null hypothesis: the lower the p-value, the stronger the evidence against H₀ (Wasserman 2013). Table 3 illustrates the confidence levels and the corresponding p-values adopted in this investigation. All analyses are conducted using the experimental CoR<sub>exp</sub> normalised by the Housner model CoR<sub>exp</sub>/CoR<sub>Housner</sub> as a variable. This implicitly considers the dependency of the CoR on the aspect ratio and permits mutual comparisons among blocks with different aspect ratios.

Three experimental datasets are employed for the statistical analyses. The first one includes the angular CoRs at each impact of each test and counts 7430 values in total. It represents longitudinal data for each free-rocking test, as i) impacts within the same test are interdependent, and ii) the number of impacts per test varies. The second dataset uses the median value from each test, thus reducing the dataset to 120 values, and represents a cross-sectional dataset where the measurements are independent. Finally, a third dataset is used for grouping the data for each aspect ratio using their median, resulting in 12 counts. Specifically, Sec. 6.1 employs the second dataset (120 counts), Sec. 6.2 uses both the second (120 counts) and third (12 counts) datasets, while Sec. 6.3 uses the first dataset (7430 counts).

The investigation of the research questions requires the selection of appropriate statistical methods. In particular, a decision must be made between parametric or non-parametric methods, depending on whether the assumptions of normality, independence, and homogeneity of variance for the variables under study are met. More specifically, parametric methods can be applied if all these assumptions hold; otherwise, non-parametric methods are required. To assess the normality of the data, a Kolmogorov-Smirnov test (K-S) is conducted (Kolmogorov 1933; Smirnoff 1939). This test compares the sample distribution to an equivalent set of normally distributed data centred on the experimental median. The K-S test assumes as H₀ that the data are normally distributed and evaluates whether there is sufficient

<table><caption>**Table 3** Evidence classification of the p-value scale for different confidence levels (Wasserman 2013)</caption><thead><tr><td>Confidence level</td><td>p-value</td><td>Evidence</td></tr></thead><tbody><tr><td>99%</td><td>≤0.01</td><td>Very strong against H<sub>0</sub></td></tr><tr><td>95%</td><td>(0.01–0.05]</td><td>Strong against H<sub>0</sub></td></tr><tr><td>90%</td><td>(0.05–0.10]</td><td>Medium-weak against H<sub>0</sub></td></tr><tr><td>Less than 90%</td><td>>0.10</td><td>Small or none against H<sub>0</sub></td></tr></tbody></table>

For visual clarity, the p-values presented in the following sections adopt the font styles reported in this table.
<table><caption>**Table 4** Computed p-values of the Kolmogorov-Smirnov test of normality</caption><thead><tr><th>CoR from median per test</th><th>p-value</th><th>CoR from median per H/B</th><th>p-value</th></tr></thead><tbody><tr><td>CoR<sub>vel</sub> / CoR<sub>Housner</sub></td><td>&lt;0.01</td><td>CoR<sub>vel</sub> / CoR<sub>Housner</sub></td><td>&lt;0.01</td></tr><tr><td>CoR<sup>side</sup><sub>U</sub> / CoR<sub>Housner</sub></td><td>&lt;0.01</td><td>CoR<sup>side</sup><sub>U</sub> / CoR<sub>Housner</sub></td><td>&lt;0.01</td></tr><tr><td>CoR<sup>side</sup><sub>U+F</sub> / CoR<sub>Housner</sub></td><td>&lt;0.01</td><td>CoR<sup>side</sup><sub>U+F</sub> / CoR<sub>Housner</sub></td><td>&lt;0.01</td></tr></tbody></table>

evidence to reject this assumption. Table 4 presents the results of the K-S tests for the three experimental CoRs, based solely on the cross-sectional dataset of 120 values (second dataset) and 12 values (third dataset), as the K-S test method assumes data independence. Given that the p-values for all the analyses are lower than the assumed confidence levels (Table 3), it can be concluded that the data significantly deviates from the normal distribution. Since one of the key assumptions of the parametric methods is not satisfied, the subsequent statistical tests are conducted with non-parametric methods.

## 6.1 Is CoR<sub>exp</sub> statistically equal to CoR<sub>Housner</sub>?

This section conducts a statistical analysis to investigate whether Housner's CoR<sub>Housner</sub> can predict the experimental CoR<sub>exp</sub>. To statistically assess the potential equivalence between the predicted CoR<sub>Housner</sub> and the experimental CoR<sub>exp</sub>, the Wilcoxon signed-rank test is employed (Wilcoxon 1945). In this context, the null hypothesis H<sub>0</sub> states that CoR<sub>exp</sub>/CoR<sub>Housner</sub> = 1. The analysis considers the median CoR<sub>exp</sub> per test (i.e. the dataset with 120 counts), once for all H/B ratios (4 ≤ H/B ≤ 15) and then for every H/B ratio individually. Table 5 presents the results of the Wilcoxon signed-rank test showing p-values for all three methods of extracting the CoR<sub>exp</sub>. When considering all H/B ratios, there is strong evidence against the assumed equivalence between Housner's CoR<sub>Housner</sub> and the experimental CoR<sub>exp</sub>. However, interesting results emerge when the H/B ratios are considered individually, revealing negligible evidence against the null hypothesis that CoR<sub>Housner</sub> is statistically equal to CoR<sub>exp</sub> for 5 ≤ H/B ≤ 10, with the exception of H/B = 7, regardless of the method used to extract the experimental CoR<sub>exp</sub>. This statistically reinforces the efficiency of Housner's CoR<sub>Housner</sub> to capture the energy losses of rocking structures of medium-to-high aspect ratios. On the contrary, the statistical analysis indicates strong or very strong evidence against the H<sub>0</sub> for aspect ratios H/B ≥ 12, implying that for very slender rocking structures, CoR<sub>Housner</sub> is not a statistically accurate predictor of the energy losses. Finally, for H/B = 4 and H/B = 11, the three methods provide inconclusive results, with the CoR<sub>vel</sub> showing a better correlation with the Housner model. This outcome is in agreement with Fig. 11, showing that Housner's model tends to overpredict the energy losses for stocky blocks, while it underpredicts the energy losses for very slender blocks.

## 6.2 Is CoR<sub>exp</sub>/CoR<sub>Housner</sub> statistically dependent on the aspect ratio H/B?

This section examines whether the normalised CoR<sub>exp</sub>/CoR<sub>Housner</sub> has any statistical dependency on the H/B ratio, in addition to the dependency introduced by the Housner model. Since the data are not normally distributed, the non-parametric Spearman's rank correlation test is used (Spearman 1961). The Spearman's test provides an associated p-value and a correlation coefficient r. The null hypothesis H<sub>0</sub> indicates no correlation, thus stating
<table><caption>Table 5 Computed p-values of the wilcoxon signed-rank test, examining the ability of the housner model to predict the experimental CoR</caption><thead><tr><th rowspan="2">CoR from<br>median per test</th><th colspan="12">H/B ratios</th></tr><tr><th>4-15</th><th>4</th><th>5</th><th>6</th><th>7</th><th>8</th><th>9</th><th>10</th><th>11</th><th>12</th><th>13</th><th>14</th><th>15</th></tr></thead><tbody><tr><td>CoR<sub>uel</sub> / CoR<sub>Housner</sub></td><td>&lt;0.01</td><td>0.11</td><td>0.38</td><td>0.18</td><td><u>0.05</u></td><td>0.21</td><td>0.42</td><td>0.22</td><td>0.08</td><td><u>0.02</u></td><td>&lt;0.01</td><td>&lt;0.01</td><td><u>0.02</u></td></tr><tr><td>CoR<sub>side</sub> / CoR<sub>Housner</sub></td><td>&lt;0.01</td><td>&lt;0.01</td><td>0.85</td><td><u>0.06</u></td><td><u>0.05</u></td><td>0.07</td><td>0.42</td><td>0.13</td><td><u>0.02</u></td><td><u>0.01</u></td><td>&lt;0.01</td><td>&lt;0.01</td><td>&lt;0.01</td></tr><tr><td>CoR<sub>side</sub> / CoR<sub>Housner</sub></td><td>&lt;0.01</td><td>&lt;0.01</td><td>0.85</td><td>0.11</td><td><u>0.05</u></td><td>0.10</td><td>0.57</td><td>0.54</td><td><u>0.03</u></td><td><u>0.01</u></td><td>&lt;0.01</td><td>&lt;0.01</td><td>&lt;0.01</td></tr></tbody></table>
<table><caption>**Table 6** Computed p-values and correlation coefficient r of the Spearman's rank correlation test, examining the monotonicity of the normalised experimental CoR against the aspect ratio $H/B$</caption><thead><tr><th>CoR from median per test</th><th>p-value</th><th>r-value</th></tr></thead><tbody><tr><td>CoR<sub>vel</sub>/CoR<sub>Housner</sub></td><td><strong>&lt;0.01</strong></td><td>-0.25</td></tr><tr><td>CoR<sup>side</sup><sub>U</sub>/CoR<sub>Housner</sub></td><td><strong>&lt;0.01</strong></td><td>-0.46</td></tr><tr><td>CoR<sup>side</sup><sub>U+F</sub>/CoR<sub>Housner</sub></td><td><strong>&lt;0.01</strong></td><td>-0.46</td></tr></tbody></table>

the independence of the $CoR_{exp}/CoR_{Housner}$ to the aspect ratio $H/B$. The r-value measures the strength and direction of the monotonic relationship between two variables, ranging from -1 to +1, where 0 indicates no correlation. An r-value of -1 or +1 implies a perfect negative or positive monotonic relationship, respectively. Table 6 presents the results of Spearman's test for the median $CoR_{exp}$ data per test (i.e. the dataset with 120 counts) across all three extraction methods. In all cases, there is very strong evidence against $H_0$ as observed by the p-values, indicating a statistically significant correlation with the aspect ratio $H/B$. Moreover, the value of r reveals a negative monotonic relationship, with values ranging from -0.25, for the $CoR_{vel}$, to -0.46, for $CoR_{U}^{side}$ and $CoR_{U+F}^{side}$. In accordance with Fig. 11, these findings suggest that the normalised $CoR_{exp}/CoR_{Housner}$ is inversely dependent on the aspect ratio $H/B$, thus implying that Housner's $CoR_{Housner}$ shows a stronger dependency on the aspect ratio than the experimental $CoR_{exp}$.

The dependency between the $CoR_{exp}/CoR_{Housner}$ and the aspect ratio $H/B$ is further examined by investigating whether a linear regression model can adequately capture the normalised experimental data. The regression is conducted on two datasets: i) the median $CoR_{exp}$ from each test (dataset with 120 counts), and ii) their median value for each aspect ratio $H/B$ (dataset with 12 counts). Table 7 presents the results of a linear regression analysis, including the p-value and the coefficient of determination $R^2$. The null hypothesis $H_0$ posits that the slope of the relationship between $CoR_{exp}/CoR_{Housner}$ and $H/B$ ratio is zero, implying lack of linear correlation. The p-values in Table 7 reveal strong evidence against the lack of correlation for the dataset with the median of each test (dataset with 120 counts), with a coefficient of determination between 0.04 and 0.18. The coefficient of determination notably increases for the dataset that only considers the median value per aspect ratio $H/B$ (dataset with 12 counts). This increase occurs for the $CoR_{U}^{side}$ and $CoR_{U+F}^{side}$, up to 0.54 to 0.56 respectively, while the correlation is insignificant for the $CoR_{vel}$, i.e. 0.15. A closer examination of the data reveals that the correlation is heavily influenced by the results of the aspect ratio $H/B = 7$. When this aspect ratio is excluded (i.e. results in parenthesis in Table 7), the p-value drops to <0.01 for all $CoR_{exp}/CoR_{Housner}$, and the linear regression model better fits the remaining data. This results in notably higher $R^2$ values of 0.82, 0.67 and 0.68 for $CoR_{vel}$, $CoR_{U}^{side}$ and $CoR_{U+F}^{side}$, respectively. Fig. 13 illustrates the experimental data together with the results of the linear regression analysis from Table 7, highlighting that the fitted lines provide a similar estimate for both datasets (i.e. datasets considering the median value per test and the median value per $H/B$).

## 6.3 Is $CoR_{exp}/CoR_{Housner}$ statistically dependent on the rocking amplitude?

This section investigates statistically whether the experimental angular $CoR_{exp}$ depends on the rocking amplitude, in contrast to Housner's model, which assumes that $CoR_{Housner}$ (Eq. (3)) is constant and response-independent. Two rocking amplitudes are considered as candidate independent variables: i) the peak rotation of the succeeding half-cycle normalised by the slenderness angle $|\theta_{x,peak}|/\alpha$, and ii) the angular velocity at impact normalised by
<table><caption>**Table 7** Computed p-value, coefficient of determination R², intercept and slope, of the linear regression analysis between the CoR<sub>exp</sub>/CoR<sub>Housner</sub> and the aspect ratio H/B.
[note: the values in parenthesis exclude H/B = 1]</caption><thead><tr><th rowspan="2"></th><th colspan="3">CoR from<br>median per test</th><th colspan="3">CoR from<br>median per H/B</th></tr><tr><th>p-value</th><th>R<sup>2</sup> value</th><th>intercept</th><th>slope</th><th>p-value</th><th>R<sup>2</sup> value</th><th>intercept</th><th>slope</th></tr></thead><tbody><tr><td>CoR<sub>vel</sub>/CoR<sub>Housner</sub></td><td>0.02</td><td>0.04</td><td>1.001</td><td>-0.0014</td><td>0.21</td><td>0.15</td><td>1.003</td><td>-0.0013</td></tr><tr><td>CoR<sub>side</sub>/CoR<sub>Housner</sub></td><td>&lt;0.01</td><td>0.18</td><td>1.016</td><td>-0.0028</td><td><strong>0.01</strong></td><td>(0.82)</td><td>(1.011)</td><td>(-0.0019)</td></tr><tr><td>CoR<sub>side</sub>/CoR<sub>Housner</sub></td><td>&lt;0.01</td><td>0.18</td><td>1.017</td><td>-0.0028</td><td>&lt;0.01</td><td>(0.67)</td><td>(1.024)</td><td>(-0.0032)</td></tr></tbody></table>
<figure><img src="image_23.png" /><figcaption>**Fig. 13** Linear regression analysis on the CoR<sub>exp</sub>/CoR<sub>Housner</sub> data considering either the test medians (solid line) or a median value per H/B (dashed line). [note: the values in parenthesis and the dashed lines exclude H/B = 7]</figcaption></figure>

the maximum angular velocity $|\dot{\theta}_{x,impact}|/\dot{\theta}_{x,max}$, where $\dot{\theta}_{x,max}$ corresponds to the velocity at impact during a cycle of a free-rocking block with $\theta_{x,peak} = \alpha$ and is computed as $\dot{\theta}_{x,max} = 2p \sin(\alpha/2)$ (Charalampakis et al. 2022). The investigation requires the use of the longitudinal dataset (with 7430 counts) that includes all the impacts from each test. This is important since the median CoR value from each test disregards the amplitude of each impact. Consequently, for the complete dataset: i) the number of measurements per test varies, and ii) the CoR values from each test cannot be considered independent, as they pertain to the same block and interface. As a result, mixed effects statistical models must be employed to analyse the dataset. In this approach, a random effect is assumed for the CoR values of each test, alongside fixed effects that account for all the tests. Specifically, the Linear Mixed Model (LMM) is used (Verbeke and Molenberghs 1997), with the null hypothesis $H_0$ positing that there is no dependency between the independent rocking amplitude and the dependent CoR<sub>exp</sub>. The LMM provides estimations of the fixed effects and their associated p-values, the random effects for each test, and the Intraclass Correlation Coefficient (ICC), which quantifies the correlation within each test.

Table 8 summarises the results of the LMM applied for the two different amplitudes and the three methods of extracting the experimental CoR<sub>exp</sub>, while Fig. 14a–c visualises the results specifically for the rotation amplitude $|\theta_{x,peak}|/\alpha$. The findings indicate that the CoR<sub>vel</sub> has a very low R<sup>2</sup>=0.03, ICC=0.25 and a relatively high p-value (>0.3) for the fixed slope, revealing small or no evidence against $H_0$, i.e. no dependency between the experimental CoR<sub>vel</sub> for both rotation and velocity amplitudes. On the contrary, the results of the CoR<sub>U</sub><sup>side</sup> and CoR<sub>U+F</sub><sup>side</sup> indicate an excellent ICC=1, moderate predictability with 0.52 ≤ R<sup>2</sup> ≤ 0.55 and p-value<0.01. This provides very strong evidence against $H_0$ and supports the conclusion that the experimental CoR<sub>U</sub><sup>side</sup> and CoR<sub>U+F</sub><sup>side</sup> depend on the rocking amplitudes ($|\theta_{x,peak}|/\alpha$ and $|\dot{\theta}_{x,impact}|/\dot{\theta}_{x,max}$). Nonetheless, the LMM is a parametric

<figure><img src="image_24.png" /></figure>
<table><caption>**Table 8** Results of the linear mixed Model (LMM) between the normalised rocking amplitude ($|\hat{\theta}_{x,peak}|/\alpha$ or $|\hat{\theta}_{x,impact}|/\hat{\theta}_{xx,max}$) and the normalised experimental CoRexp (CoRvel/CoRHousner, CoRside/U+U or CoRHousner/CoRHousner): intraclass correlation coefficient (ICC), intercept and slope of the fixed effects, and computed p-values using the LMM and the permutations tests, respectively</caption><thead><tr><th>Amplitude (independent variable)</th><th>CoR (dependent variable)</th><th>ICC</th><th>R<sup>2</sup> value</th><th>Fixed intercept</th><th>Fixed slope</th><th>Fixed slope p-value (LMM)</th><th>Fixed slope p-value (permutations)</th></tr></thead><tbody><tr><td>|θ<sub>x,peak</sub>|/α</td><td>CoR<sub>vel</sub>/CoRHousner</td><td>0.25</td><td>0.03</td><td>0.960</td><td>0.128</td><td>0.31</td><td>0.21</td></tr><tr><td>|θ<sub>x,peak</sub>|/α</td><td>CoR<sub>side</sub>/CoRHousner</td><td>1.00</td><td>0.53</td><td>0.976</td><td>0.025</td><td>&lt;0.01</td><td>&lt;0.01</td></tr><tr><td>|θ<sub>x,peak</sub>|/α</td><td>CoR<sup>side</sup><sub>U+U</sub>/CoRHousner</td><td>1.00</td><td>0.52</td><td>0.977</td><td>0.032</td><td>&lt;0.01</td><td>&lt;0.01</td></tr><tr><td>|θ<sub>x,impact</sub>|/θ<sub>x,max</sub></td><td>CoR<sub>vel</sub>/CoRHousner</td><td>0.25</td><td>0.02</td><td>0.946</td><td>0.098</td><td>0.34</td><td>0.16</td></tr><tr><td>|θ<sub>x,impact</sub>|/θ<sub>x,max</sub></td><td>CoR<sup>side</sup><sub>U</sub>/CoRHousner</td><td>1.00</td><td>0.55</td><td>0.972</td><td>0.019</td><td>&lt;0.01</td><td>&lt;0.01</td></tr><tr><td>|θ<sub>x,impact</sub>|/θ<sub>x,max</sub></td><td>CoR<sub>side</sub><sub>U+U</sub>/CoRHousner</td><td>1.00</td><td>0.55</td><td>0.972</td><td>0.023</td><td>&lt;0.01</td><td>&lt;0.01</td></tr></tbody></table>
<figure><img src="image_25.png" /><figcaption>**Fig. 14 (a-c)** Prediction of the linear mixed Model (LMM) on the dependency of the normalised experimental CoR<sub>exp</sub> (CoR<sub>vel</sub>/CoR<sub>Housner</sub>, CoR<sub>U</sub><sup>side</sup>/CoR<sub>Housner</sub> or CoR<sub>U+F</sub><sup>side</sup>/CoR<sub>Housner</sub>) on the normalised rocking amplitude (|θ<sub>x,peak</sub>|/α); and (d-f) permutation tests on the fixed slope of the LMM</figcaption></figure>

test and assumes the normality of the residuals, an assumption that is not satisfied in any of the tests shown in Table 8.

To overcome the violation of this assumption and ensure more robust conclusions, this study employs non-parametric permutation tests of the LMM (Good 2013), using the p-value of the fixed slope as the statistical measure. Specifically, 10'000 permutations are generated for the six tests outlined in Table 8, by shuffling the CoR measurements and refitting the LMM a total of 60'000 times. The shuffling process disrupts any inherent relationship between the amplitude and the fixed slope, simulating the null hypothesis where no dependency exists. If the LMM consistently predicts a strong dependency on amplitude during permutations, the observed dependency in the original data is likely spurious. Conversely, if dependency is rarely predicted during permutations, the initial prediction gains statistical support. The p-value, indicating the likelihood of obtaining the observed fixed slope under the null hypothesis, is calculated as the proportion of permutations where the fixed slope exceeds the observed value. Fig. 14d-f depicts the permutation test outcomes for the fixed slope specifically for the rotation amplitude |θ<sub>x,peak</sub>|/α, while the last column of Table 8 presents the p-value across all permutation tests. The results consistently support the

<figure><img src="image_26.png" /></figure>
same conclusion: there is strong evidence that $CoR_U^{side}$ and $CoR_{U+F}^{side}$ depend on the rocking amplitudes, whereas no such dependency is observed for $CoR_{vel}$.

# 7 Conclusions

This study conducts a comprehensive experimental campaign on the free-rocking response of limestone blocks. The specimens cover a wide range of aspect ratios $4 \le H/B \le 15$, which is representative of a range of rocking structures. To allow for a thorough examination of aleatoric variations in the response, multiple blocks with identical aspect ratios are tested, accounting for the inherent material and interface irregularities that affect the dynamic response. The campaign counts 120 tests, which provides a robust dataset to reveal correlations.

The three-dimensional behaviour of the blocks is fully reconstructed. The results illustrate the influence of the system's irregularities and imperfections on the response. Furthermore, the assumption of purely planar motion is discussed, revealing the presence of additional types of motion. Specifically, sliding and torsional rotation are observed, though to a limited extent. The two geometrical parameters that characterise the planar motion, $p$ and $\alpha$, are experimentally identified and compared to their theoretical values. The results indicate that the experimental $p$ and $\alpha$ differ from their nominal counterparts, primarily due to inherent material and interface variability. Another aspect highlighted in this study is the location of the rocking pivot points/axes. The analysis indicates that the actual pivot location may differ from the block's corner/edge and cover an extended area of the interface.

One of the challenges of rocking motion is the energy dissipation, which occurs mainly during impacts. In this work, three methodologies for the extraction of the angular Coefficient of Restitution (CoR) from the experimental data are presented. The first methodology uses the ratio of angular velocities after and before impact. This leads to scattered results and limitations associated with the difficulty in identifying the exact time instant of impact. The second methodology assumes energy conservation during the rocking phase and utilises the potential energy at the peak rotation. It provides stable outcomes when the peaks have the same sign, since the method is strongly affected by the asymmetric nature of the response. Finally, the third methodology builds on the second by incorporating an additional source of energy dissipation, namely friction due to sliding. The results of these three methodologies were compared with each other, as well as with the predictions of Housner's model, highlighting their similarities and differences. Moreover, it is shown that the two methodologies based on potential energy and potential and frictional energies have negligible differences, indicating the minor influence of frictional dissipative phenomena in the response of these blocks.

The high sensitivity of the rocking response is reflected in the variability of the CoR estimation, indicating the importance of a statistical approach to draw more robust conclusions. To this end, statistical hypothesis testing is conducted on the experimental dataset. The findings of the analysis can be summarised as follows: i) Housner's model provides statistically accurate predictions of the experimental $CoR_{exp}$ for blocks with aspect ratios $5 \le H/B \le 10$ regardless of the extraction method; ii) the normalised $CoR_{exp}/CoR_{Housner}$ is statistically dependent on the aspect ratio showing a negative correlation, indicating that Housner's model underpredicts the CoR for stocky blocks ($H/B < 5$) and overpredicts it for slender blocks ($H/B > 10$); iii) the experimental $CoR_{exp}$ shows a statistical dependency on the
rocking amplitude when extracted using the two methods based on the potential energy and potential and frictional energies ($CoR_U^{side}$ and $CoR_{U+F}^{side}$), whereas it is statistically independent when extracted using the method based on the angular velocities ($CoR_{vel}$).

Finally, it is important to emphasise that the findings of this study are inherently specific to the investigated block configurations and their associated material properties. Any extrapolation of these conclusions to different materials or structural scales require further research for the generalisation of the findings, as variations may distinctly influence the dynamic response and associated energy dissipation mechanisms.

## Appendix A

This section provides a detailed description of the estimation of the angular CoR according to the three methodologies outlined in Sec. 4.2 Specifically, Sec. A.1 discusses the estimation of the angular CoR extracted from the angular velocities, including the procedure for the detection of the time instants prior and after impact, while Sec. A.2 and A.3 detail the derivation of the angular CoR extracted from the potential energy and the combined consideration of potential and frictional energies, respectively. The notation for each instance follows the scheme shown in Fig. 15, where key time instants along with the associated potential $U$ and kinetic $K$ energies are schematically highlighted in relation to two arbitrary consecutive impacts, i.e. $n$ and $n+1$. These instants are: i) “A” which corresponds to the peak rotation $\theta_{x,peak}$ before impact $n$; ii) “B” and “C” that represent the moments just before and after impact $n$, respectively; iii) “D” which is the instant of the peak rotation $\theta_{x,peak}$ after impact $n$ and before impact $n+1$; iv) “E” and “F” that denote the instants just before and after impact $n+1$; and (v) “G” that corresponds to the peak rotation $\theta_{x,peak}$ following impact $n+1$. Observe from Fig. 15, that although point “A” represents a positive rotation, the present formulation is also valid when point “A” represents a negative rotation. For conciseness, the latter case is not shown.

<figure><img src="image_27.png" /><figcaption>**Fig. 15** Schematic representation of the free-rocking response illustrating the peak rotations before and after arbitrary consecutive impacts</figcaption></figure>

<figure><img src="image_28.png" /></figure>
## A.1 Angular CoR from angular velocities

The quantification of the angular CoR extracted from the angular velocities before and after the n^th^ impact (Fig. 15) follows the definition given by Housner (1963):

$$CoR_{vel} = \frac{\dot{\theta}_{x;C}}{\dot{\theta}_{x;B}} = \frac{\dot{\theta}_{x;n}^{+}}{\dot{\theta}_{x;n}^{-}} \qquad (A.1)$$

where superscripts “+” and “-” denote the post-impact and pre-impact states, respectively. This approach relies on the accurate estimation of the time instants “B” and “C” (Fig. 15). To achieve this, a methodology similar to Kalliontzis et al. (2016) is adopted. First, the vertical acceleration of the GC $\ddot{u}_{z,GC}$, which is obtained by computing the second derivative of the vertical translation $u_{z,GC}$, is filtered by applying a Savitzky-Golay filter with the minimal configuration of first-order polynomial and with a time window length of three steps. Then, impacts are detected as distinct peaks in the filtered vertical acceleration $\ddot{u}_{z,GC}$ (Fig. 16). For each (local) peak, the start (point B in Fig. 16) and end (point C in Fig. 16) of the impact interval are identified as the instants immediately below the 40% of the corresponding peak. This threshold value has been chosen empirically and has shown to provide results that are not affected by the inherent noisy nature of the acceleration signal. Nonetheless, this method inherently entails a degree of subjectivity due to the continuous nature of impacts and the challenges in identifying accurately the time instants of their start and end.

## A.2 Angular CoR from potential energy

An alternative strategy to experimentally quantify the angular CoR relies on the assumption that energy is lost solely during impacts. As a result, the kinetic energy before the n^th^ impact $K_B$ is equal to the potential energy $U_A$ at the peak of the preceding half-cycle $K_B = U_A$ (Fig. 15). Similarly, the kinetic energy after the n^th^ impact $K_C$ is equal to the potential energy $U_D$ at the peak after the n^th^ impact. Therefore:

<figure><img src="image_29.png" /><figcaption>**Fig. 16** Schematic representation of the acceleration response $\ddot{u}_{z,GC}$ for the identification of the time instants before (“B”) and after (“C”) impact n</figcaption></figure>
$$ \frac{K_C}{K_B} = \frac{U_D}{U_A} \qquad (A.2) $$

Considering that $K_C = 0.5 \cdot I_{0,\pm} \cdot \dot{\theta}_{x,C}^2$ and $K_B = 0.5 \cdot I_{0,\mp} \cdot \dot{\theta}_{x,B}^2$, Eq. (A.2) yields:

$$ CoR_{U}^{\text{cons}} = \frac{\dot{\theta}_{x,C}}{\dot{\theta}_{x,B}} = \sqrt{\frac{U_D I_{0,\mp}}{U_A I_{0,\pm}}} = \sqrt{\frac{U_{\text{peak};n}^{+} I_{0,\mp}}{U_{\text{peak};n}^{-} I_{0,\pm}}} \qquad (A.3) $$

where the subscript symbol “±” refers to the positive and negative signs of rotation, whereas the superscripts “+” and “-” denote the post-impact and pre-impact states, respectively.

The aforementioned methodology can also be applied to estimate the angular CoR considering two successive impacts $n$ and $n+1$, i.e. accounting for the peaks of the potential energy from the same side of motion. Similarly (see Fig. 15):

$$ \frac{K_F}{K_E} = \frac{U_G}{U_D} \qquad (A.4) $$

Substituting $U_D$ from Eq. (A.2) into Eq. (A.4), considering $K_F = 0.5 \cdot I_{0,\mp} \cdot \dot{\theta}_{x,F}^2$, $K_E = 0.5 \cdot I_{0,\pm} \cdot \dot{\theta}_{x,E}^2$ and assuming that the angular CoR of the two impacts $n$ and $n+1$ is the same, gives:

$$ CoR_{U}^{\text{side}} = \sqrt{\sqrt{\frac{U_G}{U_A}}} = \sqrt{\sqrt{\frac{U_{\text{peak};n+1}^{+}}{U_{\text{peak};n}^{-}}}} \qquad (A.5) $$

<figure><img src="image_30.png" /><figcaption>Fig. 17 Schematic representation of the frictional energy losses estimation: (a) plan view of the block interface, (b) block displacement due to torsional rotation $\theta_z$, (c) block displacement due to translation $u_{\text{trans}}$ along the y-axis, and (d) bilinear elasto-plastic sliding law</figcaption></figure>
### A.3 Angular CoR from potential and frictional energies

The third methodology for estimating the angular CoR builds on the approach presented in Sec. A.2 by accounting for both the energy dissipated during impact and the frictional energy losses during the rocking phase (see Fig. 17a). The estimation of the frictional energy dissipation from instant “A” to instant “B” $W_{A→B}$ (Fig. 15) relies on the calculation of the work due to sliding along the contact interface. Since sliding along the x-axis is considered negligible compared to sliding along the y-axis, the frictional work solely accounts for the latter and is computed as follows:

$$W_{A→B} = N_{FRIC} \int_{-L/2}^{L/2} [\max(|du_{\theta} + du_{trans}| - u_{y,elastic}; 0)]dx \quad (A.6)$$

where $N_{FRIC} = mg\mu_{dyn}$ is the sliding friction force at the contact interface, with $\mu_{dyn}$ representing the dynamic friction coefficient, herein assumed to be equal to the static friction coefficient $\mu_{dyn}=\mu_{static}=0.70$ (Vlachakis et al. 2023). $du_{\theta} = \sin(\theta_{z,B} - \theta_{z,A})x$ is the displacement of a point with coordinate x that experiences a finite rotation over the z-axis $\theta_{z,B} - \theta_{z,A}$ (Fig. 17b). $du_{trans} = u_{y,B} - u_{y,A}$ is the displacement of any point due to the finite translation $u_{y,B} - u_{y,A}$ (Fig. 17c), and $u_{y,elastic} = \frac{N_{FRIC}}{LBk_s}$ is the elastic relative displacement at the contact interface before sliding occurs. The definition of $u_{y,elastic}$ assumes a bilinear elastoplastic sliding law (Fig. 17d), with $k_s$ defining the elastic tangential interface stiffness (Vlachakis et al. 2023) and $\tau_{fric}$ being the sliding shear stress. Note that Eq. (A.6) assumes that the block is constantly in contact with the base during sliding and has a uniform distribution of reaction forces.

Considering the half-cycle before and after impact n, the balance of the potential, kinetic and frictional energies reads (see Fig. 15):

$$\frac{K_C}{K_B} = \frac{U_D + W_{C→D}}{U_A - W_{A→B}} \quad (A.7)$$

Substituting the kinetic energy expressions, Eq. (A.7) yields:

$$CoR_{U+F}^{cons} = \frac{\dot{\theta}_{x,C}}{\dot{\theta}_{x,B}} = \sqrt{\frac{U_D + W_{C→D} I_{o,\mp}}{U_A - W_{A→B} I_{o,\pm}}} = \sqrt{\frac{U_{peak;n}^{+} + W_{imp→peak;n}^{+} I_{o,\mp}}{U_{peak;n}^{-} - W_{peak→imp;n}^{-} I_{o,\pm}}} \quad (A.8)$$

Similarly, for impact n +1, the balance of the potential, kinetic and frictional energies gives (Fig. 15):

$$\frac{K_F}{K_E} = \frac{U_G + W_{F→G}}{U_D - W_{D→E}} \quad (A.9)$$

Substituting $U_D$ from Eq. (A.8) into Eq. (A.9) and assuming that the angular CoR of the two impacts n and n +1 is the same implies that:
$$
CoR_{U+F}^{side} = \sqrt{\frac{\left( \frac{W_{C \to D} + W_{D \to E} + \sqrt{(W_{C \to D} + W_{D \to E})^2 + 4(U_A - W_{A \to B})(U_G + W_{F \to G}) \left( \frac{I_{o,\pm}}{I_{o,\mp}} \right)^2}}{2(U_A - W_{A \to B}) \frac{I_{o,\pm}}{I_{o,\mp}}} \right)} = \\
= \sqrt{\frac{\left( W_{imp \to peak; n}^{+} + W_{peak \to imp; n+1}^{-} + \sqrt{(W_{imp \to peak; n}^{+} + W_{peak \to imp; n+1}^{-})^2 + 4(U_{peak; n}^{-} - W_{peak \to imp; n}^{-}) (U_{peak; n+1}^{+} + W_{imp \to peak; n+1}^{+}) \left( \frac{I_{o,\pm}}{I_{o,\mp}} \right)^2} \right)}{2(U_{peak; n}^{-} - W_{peak \to imp; n}^{-}) \frac{I_{o,\pm}}{I_{o,\mp}}}} \quad (A.10)
$$

**Acknowledgements** This work is financed by national funds through FCT– Foundation for Science and Technology, under grant agreements PRT/BD/152830/2021, 2020.07325.BD attributed to the first and second authors, respectively. This study has been partly funded by the STAND4HERITAGE project (new STANDards FOR seismic assessment of built cultural HERITAGE) that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant No. 833123) as an Advanced Grant. This study was also partly funded by FCT/MECI through project 2022.05425.PTDC (RESISTANCE) and project national funds (PIDDAC) under the R&D Unit Institute for Sustainability and Innovation in Structural Engineering (ISISE), under reference UIDB/04029/2020 (http://doi.org/10.54499/UIDB/04029/2020), and under the Associate Laboratory Advanced Production and Intelligent Systems ARISE under reference LA/P/0112/2020. The opinions and conclusions presented in this paper are those of the authors and do not necessarily reflect the views of the sponsoring organisations. The authors also greatly acknowledge the technical support from the staff of the University of Minho laboratory during the experimental campaign.

**Author contributions** All authors contributed to the conceptualisation and design of this study. Material preparation, data collection and analysis were performed by Carla Colombo and Georgios Vlachakis. The first draft of the manuscript was written by Carla Colombo, and all authors revised previous versions of the manuscript. All authors read and approved the final manuscript.

**Funding** Open access funding provided by FCT|FCCN (b-on). This work is financed by national funds through FCT — Foundation for Science and Technology, under grant agreements PRT/BD/152830/2021, 2020.07325.BD attributed to the first and second authors, respectively. This study has been partly funded by the STAND4HERITAGE project (new STANDards FOR seismic assessment of built cultural HERITAGE) that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant No. 833123) as an Advanced Grant. This study was also partly funded by FCT/MECI through project 2022.05425.PTDC (RESISTANCE) and project national funds (PIDDAC) under the R&D Unit Institute for Sustainability and Innovation in Structural Engineering (ISISE), under reference UIDB/04029/2020 (doi.org/10.54499/UIDB/04029/2020), and under the Associate Laboratory Advanced Production and Intelligent Systems ARISE under reference LA/P/0112/2020. The opinions and conclusions presented in this paper are those of the authors and do not necessarily reflect the views of the sponsoring organisations.

**Data availability** The datasets generated and analysed in this study are available from the corresponding author upon reasonable request.

**Declarations**

**Competing interests** The authors have no relevant financial or non-financial interests to disclose.
**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

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