Numerical Block-Based Simulation of Rocking Structures Using a Novel Universal Viscous Damping Model

When simulating the rocking response using numerical block-based models, normal and tangential stiffness ($k_n$ and $k_s$) are introduced at the corresponding interfaces to capture the interaction of the contacting bodies. The introduction of the interface stiffness facilitates the modeling of additional macroscale characteristics of the rocking problem, such as rough-gap closure stiffness (Lourenço et al. 2005), material elasticity (Acikgoz and DeJong 2012; Avgenakis and Psycharis 2017; Vassiliou et al. 2014), foundation flexibility (Spanos et al. 2017), or even wall degradation (Griffith et al. 2004). Subsequently, the moment-rotation (or equivalently force-displacement) diagram has the form shown in Fig. 2 (Costa et al. 2013; Giordano et al. 2020; Godio and Beyer 2017; Mehrotra and Dejong 2020). More specifically, the nonlinear curve is characterized by an initial elastic part that corresponds to the full contact phase, a plateaulike part where rocking around a pivot point occurs, and a softening part that, due to the geometrically nonlinear nature of the problem, converges to the theoretically softening part of the rigid body.

Based on Fig. 2, the initial rotational contact stiffness ($k_{\mathrm{rot}·\mathrm{contact}}$) of the block can be expressedwhere ($2b$) = block’s width (Fig. 1); and ($2l$) = block’s depth in the general three-dimensional (3D) configuration. The normal stiffness $k_n$ is distributed over the area of the interface and consequently has units of Newtons per cubic meter.

Recall that the dynamic rocking response of a numerical model with finite stiffness is slightly different from the classical rocking theory (Housner 1963), specifically with regard to the treatment of impact. In contrast with the assumptions of instantaneous duration of impact and infinite normal stiffness during full contact, the numerical model presents finite stiffness, and thus, impact occurs over finite displacement and time. Hence, the rotational contact/impact frequency can be approximated as follows:where $I_{\mathrm{rot}·}$ = rotational mass moment of inertia about the center of the interface section. For brevity, influence of potential overburden weight at the section (i.e., stress stiffening effect), block deformability, and geometrical nonlinearity are neglected. Evidently, the finite stiffness model experiences a huge stiffness increase during impact (i.e., when contact occurs), which lasts for the short time interval until the block changes pivot point and continues rocking

To illustrate the overall similarity between the analytical rigid model and the numerical finite stiffness model, Fig. 3 plots the nonlinear dependency of the rocking period described by Eq. (2) with solid lines, and a set of points marks the response of the numerical finite stiffness model for different values of the scale $R$ and slenderness (i.e., aspect ratio) $h/b[=1/\arctan (\alpha )]$. Overall, Fig. 3 shows good agreement between the rigid and finite stiffness model. The small difference observed at higher amplitudes stems from the small shrinkage of the negative stiffness part in the moment-rotation diagram (Fig. 2). This is an additional attribute introduced by the finite stiffness model, which arises by a slight inward shift of the pivot point at the base during rocking.

Fig. 3(c) plots the analytical expression of Eq. (2) with respect to the top displacement (instead of rotation) of the block. Specifically, Fig. 3(c) provides a closer look at the bottom-left corner of Fig. 3(b), illustrating the difference of the examined models during contact and the influence of the finite stiffness of the numerical model compared with the theoretically infinite stiffness of the analytical rigid model. The response of the finite stiffness model does not converge to the axis’ origin, but stops on a finite period. Fig. 3(c) also plots the period of vibration ($T_{\mathrm{rot}·\mathrm{contact}}=1/f_{\mathrm{rot}·\mathrm{contact}}$) of Eq. (7) (dashed horizontal lines). Importantly, Fig. 3(c) illustrates the very good estimation of the contact frequency that Eq. (7) provides despite the adopted simplifications. The limit frequency shown in Fig. 3(c), and analytically expressed through Eq. (7), is in agreement with a recent experimental campaign on the (two-sided) rocking behavior of masonry walls (Cappelli et al. 2020).

Numerical models also treat energy dissipation in a different manner from the classical rocking theory (Housner 1963). Specifically, instead of the instantaneous event-based approach that the classical rocking theory assumes, energy loss is regarded as a continuous process throughout rocking motion, usually simulated via a viscous definition or a hysteretic constitutive relationship (Charney 2008). However, such damping models are not intended to describe the micromechanical nature of the phenomena taking place, but rather reproduce the overall dissipation in a phenomenological fashion. To this end, calibration methodologies are employed that tune the damping model to produce the desired response.

Viscous damping models are mathematical artifices employed to simulate sources of dissipation (such as damping radiation at impact) that are not modeled explicitly in their physical sense, yet are important enough at the structural system’s level to not be ignored (Hall 2006). By definition, viscous models assume that the damping forces are proportional to the velocity of the structural system through the damping coefficient $\mathbf{C}$. This provides a mathematical convenience to simulate the energy dissipation because velocity is out of phase with displacement and acceleration. Different viscous damping models have been proposed, which mainly differ based on (1) the degrees of freedom (DOFs) of the velocity vector upon which $\mathbf{C}$ acts, (2) the way $\mathbf{C}$ behaves during time-history analyses, and (3) the exact value of $\mathbf{C}$.

Damping can be treated as a material characteristic assigned at each DOF of the structure, where the dissipation process is distributed along the structure. Alternatively, damping can also be treated as a dashpot assigned at specific DOFs and act locally on the relative velocity between two nodes. Considering the localized nature of the rocking problem, the dashpot definition appears to be a more viable solution. In addition, several viscous damping definitions have been proposed to describe the energy dissipation during the response-history of a structure. Fig. 4 illustrates the most widely used.

In particular, the constant damping coefficient (CDC) model retains the same value of $\mathbf{C}$ throughout the response history. This is equivalent to a mass-proportional Rayleigh damping definition, thus damping out low-frequency content. Another well-established model associates a constant damping ratio (CDR) with all frequencies. This definition allows an update of the stiffness matrix but not the damping ratio $\xi$ during the response history, resulting in an equivalent dissipation of all the frequency content. Further, a stiffness-proportional damping ratio (SDR) allows the damping ratio $\xi$ to follow the changes of the stiffness matrix during the response history. As a result, the high-frequency content of the response is highly dissipated. Finally, a combination of both mass and stiffness proportional models acting at the same time, known as Rayleigh damping, damps out both the low and high range of the frequency content (Fig. 4).

During rocking, when impact occurs, a huge increase in the frequency content takes place [Eq. (7)] for a very short, yet finite, time and displacement. Therefore, the SDR definition presents a convenient approach to model the damping of rocking structures. Physically speaking, if the SDR definition is used together with a dashpot model, unilateral behavior of the dashpot is achieved, i.e., energy is dissipated only when the dashpot is in contact.

In addition, a proper treatment of impact requires a proper definition of the value of $\mathbf{C}$ (or equally, the value of $\xi$). Particularly for the rocking problem, a relationship between the coefficient of restitution $e$ and damping ratio $\xi$ is critical to ensure energy equivalence between the classical rocking theory (Housner 1963) and the numerical viscous damping model. Table 1 provides a summary of the $\xi –e$ relationships available in literature applicable to the rocking problem accompanied by their basic assumptions. More specifically, Priestley et al. (1978) first examined the energy loss equivalence of the classical impulsive dynamics theory and the viscous decay of an elastic oscillator. Later, Makris and Konstantinidis (2003) simplified the $\xi –e$ expression of Priestley et al. (1978), assuming damping independent of the rocking amplitude. Anagnostopoulos (2004) studied the energy equivalence between the COR and a spring-dashpot viscous model based on the two colliding masses scheme, and Giannini and Masiani (1990) and Imanishi et al. (2012) proposed equivalent formulations. DeJong (2009) simulated the rocking block with corner spring-dashpots adopting a SDR damping formulation to critically damp either the axial frequency, the corner frequency, or the rotational frequency. Recently, Tomassetti et al. (2019) followed a calibration process of a single degree of freedom (SDOF) analytical formulation of rocking structures adopting different formulations of viscous models (i.e., CDC, CDR, and SDR), and the influences on the response of additional parameters, such as the interface stiffness, rocking amplitude, and aspect ratio, were also investigated.

Importantly, the aforementioned proposals are based either on simplified SDOF analytical schemes (Anagnostopoulos 2004; Giannini and Masiani 1990; Imanishi et al. 2012; Priestley et al. 1978; Tomassetti et al. 2019) or their equivalence with the COR has been omitted (DeJong 2009). In fact, a meticulous investigation on the energy loss equivalence between the viscous damping model used in block-based numerical simulations and the classical rocking (COR) theory is still lacking from the literature, despite the widespread use of numerical models in the last decades.

This section utilizes a calibration methodology of the proposed viscous damping model to achieve energy loss equivalence between the analytical model, based on the classical rocking theory of Housner (1963), and the numerical viscous damping model proposed in the previous section. Specifically, a phenomenological calibration methodology is adopted. To this end, free-rocking simulations are performed, where the analytical model is considered as a reference and the numerical (viscous damping) model is properly adjusted to fit it.

For the analytical model, the COR $e$ (i.e., $e_{2s}$ and $e_{tr}$ for two- and one-sided rocking behavior, respectively) is assumed as an independent parameter. The choice to keep $e$ independent of the geometry facilitates any correction to the assumptions of the classical rocking theory. Specifically, the COR $e$ [e.g., in Eq. (11) and/or Eq. (13)] can be replaced by either an experimental correction ratio (Costa et al. 2013; Sorrentino et al. 2011; Tomassetti et al. 2019), or any theoretically refined model (Kalliontzis et al. 2016; Ther and Kollar 2017). Thus, the present approach allows the final calibrated relationship to be easily adapted to any existing or new experimental and theoretical refinement of the classical rocking theory.

For the numerical model, the damping ratio $\xi$ (i.e., ${\xi }_b$ and ${\xi }_s$ for two- and one-sided rocking behavior, respectively) is considered the one that minimizes the root-mean square error (RMSE) metricwhere ${\theta }_{an,t}$ = rocking rotation of the reference analytical model; ${\theta }_{\mathrm{num},t}$ = rocking rotation of the numerical (viscous damping) model; $t$ = sampling step; and $N$ = total number of steps examined.

A sampling step of 0.01 s is considered adequately small for the two-sided rocking problem [Figs. 1(a) and 5(a)] because the whole response-history lasts for several seconds [Fig. 6(a)]. On the contrary, because the one-sided rocking problem [Figs 1(b) and 5(b)] experiences a fast decay after impacts [Fig. 6(c)], a sampling step of 0.001 s is preferred. The total number of steps is selected in each case up to the point where the response has experienced a relative decay ratio $r={\theta }_t/{\theta }_0$ of 95% for the two-sided and 99.95% for the one-sided case, respectively, i.e., $\forall t$, until $\exists {\theta }_t\ge (1-r){\theta }_0$.

Before conducting the calibration for each rocking problem (i.e., two-sided and one-sided rocking), the influence of the different parameters involved needs to be examined. In other words, only the most critical independent parameters will be retained during the calibration process. Table 2 presents the parameters to be investigated for each rocking problem and the corresponding range of values. These parameters include the slenderness (or aspect ratio) of the block ($h/b$), the scale of the block ($R$), the rocking amplitude (${\theta }_0/\alpha$), and the normal interface stiffness ($k_n$), as defined in the previous sections. Both the reference and range of values in Table 2 have been chosen to be representative of unreinforced masonry structures (Derakhshan et al. 2013; Doherty et al. 2002; Tomassetti et al. 2019). However, due to the wide range of the examined parameters, the calibrated $\xi –e$ expressions can also be applicable to various rocking structures that can be dynamically equivalent to block-based structures and whose parameters fall within the examined range. An overview flowchart of the calibration methodology is shown in Fig. 8, indicating the key steps of the process.

Recall that the classical rocking theory (Housner 1963) assumes that energy loss (captured through the COR) depends only on the slenderness of the block $\alpha$ [Eqs. (3) and (4)]. Therefore, the scale ($R$) and amplitude $({\theta }_0/\alpha )$ do not affect the energy loss process. At the same time, the rigidity of the structure–foundation interaction assumed in the classical rocking theory excludes the influence of the interface stiffness from the energy dissipation process. To this end, the analytical model adopted herein as reference is in accordance with the classical rocking theory and disregards any influence of the normal interface stiffness at the base ($k_{n,b}$) and/or the sidewall ($k_{n,s}$). However, recent experimental tests revealed that the interface stiffness affects the rocking response. In particular, ElGawady et al. (2011) and Spanos et al. (2017) highlighted that blocks rocking on a flexible foundation usually experience higher energy loss than that of the classical rocking theory. Nevertheless, the experimental rocking response can be statistically captured by the classical rocking theory (Bachmann et al. 2018), as long as the rigid block assumption is adequately respected (which is reasonable for masonry structures where monolithic behavior can be ensured). Hence, the numerical model adopts a material elasticity modulus of 50 GPa. Such a value, representative of, e.g., hard granite material, is considered high enough to ensure that the block’s deformation is negligible compared with the interface. A thorough investigation of the interaction of block’s flexibility with its rocking response merits further investigation and is beyond the scope of the present study; however, it will be addressed in detail in future work.

This section applies the calibration methodology illustrated in Fig. 8 and proposes a novel ${\xi }_b–e_{2s}$ relationship that captures the response history of any block-based numerical model that undergoes two-sided rocking motion [e.g., Figs. 1(a) and 5(a)]. As a first approach, the influence on the response of the main parameters involved in the problem, i.e., scale ($R$), slenderness (or aspect ratio) ($h/b$), rocking amplitude (${\theta }_0/\alpha$), and normal interface stiffness at the base ($k_{n,b}$), are studied independently. For each combination of the examined parameters, as the COR $e_{2s}$ varies, a unique ${\xi }_b$ is determined that minimizes the RMSE error metric [Eq. (9)]. Fig. 9 plots those ${\xi }_b–e_{2s}$ pairs identified with a unique point marker and reveals the influence of each independent parameter examined herein. For each combination, a logarithmic function is fitted to these points, having the form (Makris and Konstantinidis 2003; Tomassetti et al. 2019)where $c$ = constant computed by the fitting process.

Fig. 9 plots also the fitted functions identified with solid lines. Fig. 9 reveals that the logarithmic function of Eq. (10) provides a good estimation of the ${\xi }_b–e_{2s}$ correlation, at least for the specific range of values examined in Table 2.

Fig. 9(a) presents the results of three different scales $R$ (Table 2). Importantly, Fig. 9(a) unveils that the scale does not influence the performance of the numerical (viscous damping) model. Therefore, it can be disregarded from the two-sided rocking calibration process (thus, the reference value of $R=2.1\mathrm{m}$ is considered for the rest of the calibration process).

On the contrary, a significant influence is noticed when the slenderness (or aspect ratio) $h/b$ is varied, as illustrated by Fig. 9(b). More specifically, Fig. 9(b) reveals that the more slender the structure, the higher the damping ratio required by the numerical model to dissipate an equal amount of energy compared with the analytical model. Interestingly, such a trend is in agreement with the classical rocking theory (Housner 1963), which suggested that stockier rocking blocks experience higher energy dissipation than more slender ones.

Fig. 9(c) investigates the effect of the rocking amplitude on the numerical (viscous damping) model. Compared with the influence of the other independent parameters, the influence of the amplitude ${\theta }_0/\alpha$ is considered marginal. In addition, including this dependency would significantly complicate the calibration process because the rocking amplitude is unknown a priori to any forced-vibration analysis. Therefore, the median rocking amplitude of ${\theta }_0/\alpha =0.5$ is used for the rest of the calibration process. This choice is expected to be on the engineering safe side (i.e., on the conservative side, by dissipating less energy than dictated) for the large and critical oscillations, whereas higher energy loss is anticipated in low rocking amplitudes [Fig. 6(a)]. The implications of this choice are discussed in more detail subsequently.

Finally, Fig. 9(d) suggests a high sensitivity of the viscous damping model to the normal interface stiffness at the base $k_{n,b}$. A model with lower stiffness appears to inherently dissipate more energy for a given damping ratio. Thus, it requires a lower damping ratio $\xi$ to efficiently capture the analytical rocking response. This trend is in line with the experimental findings of ElGawady et al. (2011) and Spanos et al. (2017), where blocks rocking on flexible foundations (e.g., of rubber-type material) were observed to dissipate more energy than when they rock on less flexible foundations (e.g., marble or concrete). Nevertheless, as discussed in the previous section, because this influence has not yet been quantified in the literature, the normal interface stiffness is treated as an additional independent parameter during the ${\xi }_b–e_{2s}$ calibration process.

To summarize, Fig. 9 shows that the slenderness (or aspect ratio) $h/b$ and normal interface stiffness at the base $k_{n,b}$ are the most critical parameters that influence the energy dissipation during the whole response history of the two-sided rocking problem [Fig. 5(a)]. Hence, after conducting multivariable nonlinear regression analysis on the results of Fig. 9, the novel ${\xi }_b–e_{2s}$ relationship that connects the damping ratio of any block-based numerical model with the COR $e_{2s}$ is expressed

The adequacy of the proposed ${\xi }_b–e_{2s}$ relationship of Eq. (11) to predict the measured ${\xi }_b$ is visualized in Fig. 10. A coefficient of determination of $R^2=0.978$ characterizes all the data points, indicating a remarkable estimation capability of the proposed ${\xi }_b–e_{2s}$ relationship [Eq. (11)]. In more detail, whereas the slenderness variation appears to capture the measured ${\xi }_b$ quite well, with all points lying close to the reference line, the stiffness variation shows a slightly higher discrepancy, yet with the tendency to underestimate the damping ratio, thus giving a response on the conservative side.

This section focuses on the one-sided rocking problem of Figs. 1(b) and 5(b) and offers a novel ${\xi }_s–e_{tr}$ relationship which, together with Eq. (11), captures the response history of any block-based numerical model that undergoes one-sided rocking motion. From an analytical perspective, the one-sided rocking problem can be analyzed assuming two distinct CORs [Eq. (5)]: (1) $e_{2s}$ to capture the energy dissipation for impacts against the base, and (2) $e_{tr}$ for impacts against the transversal wall [Fig. 1(b)]. Numerically, the impacting/contacting areas are governed by distinct contact/interface properties [Fig. 5(b)]: $k_{n,b}$ and ${\xi }_b$ for normal interface stiffness and damping ratio at the base, and $k_{n,s}$ and ${\xi }_s$ for normal interface stiffness and damping ratio at the side (i.e., the transversal wall), respectively. Therefore, the expression between ${\xi }_b–e_{2s}$ of Eq. (11) is directly applied at the base, and the ${\xi }_s–e_{tr}$ relationship is investigated. Similarly, the scale ($R$), slenderness (or aspect ratio) ($h/b$), rocking amplitude (${\theta }_0/\alpha$), normal interface stiffness at the base ($k_{n,b}$), and the normal interface stiffness at the side ($k_{n,s}$) are considered as the independent parameters, and their influence on the whole response-history of the numerical model are examined herein.

Fig. 11 plots the $e_{tr}–{\xi }_s$ pairs that minimize the RMSE error metric [Eq. (9)]. Similar to the two-sided rocking problem, a logarithmic function of the following form is fitted on each combinationwhere $d$ = constant computed by the fitting process. Fig. 11 presents also the fitting functions identified as solid lines. Again, Fig. 11, the adopted logarithmic function of Eq. (12) satisfactorily fits the ${\xi }_s–e_{tr}$ variation for all combinations.

Fig. 11(a) illustrates the influence of the scale $R$, and it shows that, as in the case of two-sided rocking [Fig. 9(a)], the viscous damping model is not affected by the scale variation. Thus, the influence of the scale is disregarded from the one-sided rocking calibration process (using as a reference the value of $R=2.1\mathrm{m}$ for the rest of the calibration process).

Conversely, varying the slenderness (or aspect ratio) $h/b$ appears to have a significant influence on the response given by the numerical model [Fig. 11(b)]. Specifically, more slender rocking blocks suggest the use of a higher damping ratio in order to ensure the same energy dissipation (i.e., equal COR $e_{tr}$) as the analytical model. Similarly, this trend follows the theoretical indication that slender rocking blocks dissipate less energy than stockier blocks.

Fig. 11(c) investigates the effect of the rocking amplitude ${\theta }_0/\alpha$ on the numerical (viscous damping) model. It shows that the influence is almost negligible, especially when compared with the pertinent influence of the other parameters, i.e., slenderness $h/b$ [Fig. 11(b)] and/or normal interface stiffness at the base $k_{n,b}$ [Fig. 11(e)]. Therefore, the rocking amplitude can be omitted for the rest of the calibration process, with the median value of ${\theta }_0/\alpha =0.5$ to be considered for the rest of the calibration process.

In addition, Figs. 11(d and e) illustrate the role of the normal interface stiffness at the side and the base, respectively. Specifically, Fig. 11(d) shows that the normal interface stiffness at the side $k_{n,s}$ does not influence the performance of the viscous damping model. On the contrary, the normal interface stiffness at the base $k_{n,b}$ significantly affects the performance of the viscous damping model. Therefore, only the latter is included in the calibration process (using as a reference value $k_{n,s}={5\times 10}^8\mathrm{N}/{\mathrm{m}}^3$ for the rest of the calibration process).

To summarize, Fig. 11 shows that the slenderness (or aspect ratio) $h/b$ and the normal interface stiffness $k_{n,b}$ at the base are the most critical independent parameters that influence the energy dissipation during the whole response history of the one-sided rocking problem [Figs. 1(b) and 5(b)]. Similarly, following a multivariable nonlinear regression analysis on the results of Fig. 11, the novel ${\xi }_s–e_{tr}$ relationship is expressed

Fig. 12 assesses the accuracy of the proposed relationship of Eq. (13) through a predicted-versus-measured plot. Fig. 12 reveals the excellent estimation capability of the proposed ${\xi }_s–e_{tr}$ relationship of Eq. (13), which translates into a coefficient of determination of $R^2=0.994$.

A comprehensive experimental campaign of rocking granite blocks has been conducted by Peña et al. (2008). The two-sided free- and forced-rocking motion of three specimens presented by Peña et al. (2008) are adopted to validate the proposed numerical (viscous damping) model. Table 3 provides the structural characteristics of the examined specimens. Peña et al. (2008) calculated an experimental COR (Table 3), which is slightly higher than the theoretical proposal [Eq. (3)]. Therefore, to avoid biased conclusions, the experimentally calculated COR is adopted here as input for the viscous damping model of Eq. (11), thus, illustrating its ability to be adapted to refined and/or experimentally measured COR proposals.

In addition to the classical material parameters, numerical modeling requires the use of a normal interface stiffness. However, this parameter was not measured experimentally by Peña et al. (2008). Thus, an estimated value for the normal interface stiffness at the base $k_{n,b}=5\times {10}^8\mathrm{N}/{\mathrm{m}}^3$ is adopted herein, which falls within the range of values adopted in previously conducted investigations on rocking masonry walls and could be considered representative of moderately stiff foundations. However, to facilitate the better understanding of the interface stiffness $k_{n,b}$ on the overall energy dissipation, additional $k_{n,b}$ values (i.e., $1\times {10}^8$ and $10\times {10}^8\mathrm{N}/{\mathrm{m}}^3$) are also examined and plotted in Fig. 13.

As a first approach, the free-rocking response of the three specimens is examined in Figs. 13(a–c). Figs. 13(a–c) show that, in general, the proposed viscous damping model is able to reproduce relatively well the overall dissipation phenomena of the different specimens. Even when varying the interface stiffness $k_{n,b}$, the overall response-history follows a similar response envelope despite varying slightly in phase. Specifically, for some cases, alternative $k_{n,b}$ values might capture the phase of the response slightly better [e.g., Fig. 13(a)] or worse [e.g., Fig. 13(b)] than the reference value of $5\times {10}^8\mathrm{N}/{\mathrm{m}}^3$ without affecting the peak response. However, to avoid confusion, the value of $k_{n,b}=5\times {10}^8\mathrm{N}/{\mathrm{m}}^3$ is considered for the remaining part of this study. Results of Figs. 13(a–c) can be further appreciated if one also considers that the different specimens are characterized by different values of slenderness and thus different CORs [Eq. (3)] which can significantly influence the response.

However, the proposed numerical model behaves in a robust way, adequately capturing the whole response history. In particular, the rocking motion of Specimen 1 [Fig. 13(a)] with an experimentally measured $e_{2s}=0.936$ damps out after approximately 3 s, whereas the numerical model damps out after around 2.5 s. For Specimens 2 [Fig. 13(b)] and 3 [Fig. 13(c)] with experimentally measured $e_{2s}=0.973$ and $e_{2s}=0.978$, the rocking response lasts approximately 13 and 14 s, respectively, whereas the numerical reproduction indicates a response of 13 s in both cases. Nevertheless, it is also clear from Figs. 13(a–c) that for Specimens 1 and 3 the numerical damping model dissipates energy faster than the experiments. More specifically, a very good agreement is achieved in the first cycles of response. However, for smaller rocking angles, the two response-histories start diverging. This shortcoming of the damping model (i.e., to dissipate more energy at lower amplitudes) is anticipated already by the calibration process. Specifically for the two-sided rocking problem, Fig. 9(c) highlights the dependency of dissipation on the rocking amplitude. Given this amplitude sensitivity, one could suggest its inclusion in the damping proposal [Eq. (11)] to better capture the complete response decay. However, this would not be of practical interest because the rocking amplitude is unknown a priori in the case of random ground excitations. Regardless of that limitation, the viscous damping model proposed in Eq. (11) performs quite well for rocking amplitudes $0.3<{\theta }_0/\alpha <0.8$, whereas for higher or lower amplitudes, less and more dissipation is expected, respectively. Ideally, a damping model that adapts ad hoc depending on the motion’s amplitude could tackle the issue. This issue, however, merits further investigation because such an attribute is not readily available in commonly used viscous damping models and it is beyond the scope of the present study.

The performance of the numerical (viscous damping) model is also evaluated against forced vibration tests (Peña et al. 2008) in Figs. 13(d–f). More specifically, Specimens 1 [Fig. 13(d)] and 2 [Fig. 13(e)] are subjected to harmonic sinusoidal excitations with different characteristics, and Specimen 3 [Fig. 13(f)] is subjected to a synthetic earthquake excitation (Table 3). Figs. 13(d–f) reveal the remarkable agreement of the proposed numerical model with the experimental results reported by Peña et al. (2008). More specifically, the numerical model effectively replicates the different phases of the experimental response-history, i.e., the transient, stationary, and free decay phase. Only when the rocking block reaches high rocking amplitudes critical to its stability (i.e., ${\theta }_0/\alpha >0.8$) does the proposed numerical model overpredict the rocking response resulting in earlier collapse [Fig. 13(f)]. Overall, the numerical model is capable of capturing the experimental response with adequate accuracy, particularly with respect to the phase of oscillations, peak response, and potential overturning.

This section evaluates the proposed numerical viscous damping model against one-sided free- and forced-rocking experimental tests. To that end, this work adopts the experimental campaign of Sorrentino et al. (2011) for the one-sided free-rocking simulations and the tests conducted by Al Shawa et al. (2012) for the one-sided forced-rocking evaluation.

In terms of one-sided free-rocking tests, a variety of geometrical configurations, materials, and boundary conditions were tested by Sorrentino et al. (2011). Table 4 presents the cases adopted herein together with their structural characteristics to test the performance of the proposed viscous damping model. In Table 4, ${\alpha }_{\mathrm{inst},\exp }$ denotes the rocking angle at which the free rocking initiates (Sorrentino et al. 2011), and $e_{1s,\exp }$ is the experimentally measured (total) COR [equivalent to the COR of Eq. (5)]. Subsequently, the COR that characterizes solely the impact with the transversal wall $e_{tr,\exp }$ can be analytically calculated using Eq. (5), i.e., $e_{tr,\exp }=e_{1s,\exp }/{(e_{2s,an})}^2$ where $e_{2s,an}$ is given by Eq. (3) using ${\alpha }_{\mathrm{inst},\exp }$. At the same time, the impact with the base is characterized by the analytical value of the COR $e_{2s,an}$ of Eq. (3).

Similar to the previous subsection and due to the absence of data regarding the normal stiffness at the interfaces, this subsection assumes values of $k_{n,b}=5\times {10}^8\mathrm{N}/{\mathrm{m}}^3$ for the stiffness at the base and $k_{n,s}=5\times {10}^7\mathrm{N}/{\mathrm{m}}^3$ for the stiffness at the side. The stiffness at the side is assumed to be one order of magnitude lower than the stiffness at the base due to the absence of acting axial load at the vertical interface (e.g., Lourenço et al. 2005, among others). The corresponding damping ratios at the base and the side are then computed using Eqs. (11) and (13), respectively.

Fig. 14 plots the response-histories of the examined cases of Table 4. In general, Fig. 14 shows that the numerical (viscous damping) model can capture the experimental response quite well if, again, one considers that due to the universal behavior of the proposed numerical model a single value is adopted to characterize the interface stiffness at the base and the side, respectively. Also, the numerical (viscous damping) model fails to capture the initial rotation ${\theta }_0/\alpha$ due to the way the experimental test was conducted (Sorrentino et al. 2011), which causes some discrepancies during the first cycle of rotation. In particular, the initial rotation is controlled by a mechanical system that prevents overturning of the specimen. In this way, it is possible to assign a limit value of the initial rotation (i.e., ${\alpha }_{\mathrm{inst},\exp }$ in Table 4) at which the experimental free-rocking motion commences. However, that initial value of rotation causes overturning to the numerical model because it is too close to the instability rotation. Therefore, numerically, a slightly lower value of the initial rotation is assigned to ensure free-rocking motion without causing overturning.

In addition, observe from Fig. 14 that even though the experiments were designed without a gap between the specimen and the transversal walls, slight negative oscillations were observed [i.e., Figs. 14(c and f)]. Such negative oscillations might be a result of a presence of a gap due to e.g., debris accumulation and/or construction imperfections during the test, or a manifestation of the reduced effective base width of the masonry wall, and thus, inward-shifted pivoting. The experimental response is better captured when a gap between the rocking façade and the transversal wall is introduced in the numerical model [Figs. 14(c and f)]. This rather simple comparison highlights the sensitivity of the response of such tests, and, thus, explains small discrepancies between the experimental and numerical model.

Despite the aforementioned discrepancies coupled with the uncertainties regarding the CORs computed analytically and the appropriate value of the normal interface stiffness, it is clear that the overall dissipative phenomena are still very well captured by the numerical (viscous damping) model: significant amount of energy is lost at impact, resulting in the damping out of the oscillation after two or three impacts.

To further evaluate the proposed damping model against one-sided forced-rocking tests, this study adopts the experimental campaign of real-scale one-sided rocking tuff-masonry walls performed by Al Shawa et al. (2012). The façade tested has a width of $2b=0.25\mathrm{m}$ and height $2h=3.0\mathrm{m}$. Specific details regarding the shaking table excitation input can be found in Table 5. Before each test reported by Al Shawa et al. (2012), an initial out-of-plumb rotation was measured ($\eta /\alpha$ in Table 5) due to the sequential testing and mortar debris accumulation within the cracked joint. Moreover, a small gap of 4 mm between the rocking façade and the transversal wall was also noticed (Al Shawa et al. 2012).

Furthermore, the normal interface stiffness at the base of the block is again assumed to be $k_{n,b}=5\times {10}^8\mathrm{N}/{\mathrm{m}}^3$, which according to Eq. (7) results in $f_{\mathrm{contact}}=2.5\mathrm{Hz}$, close to the experimentally measured frequencies prior to testing (i.e., around 1–2 Hz). This assumption considers zero influence of the transversal walls on the frequency of the rocking façade when at rest, which is reasonable given the initial out-of-plumb rotation. As in the case of the free-rocking tests, the normal interface stiffness at the side is assumed to be $k_{n,s}=5\times {10}^7\mathrm{N}/{\mathrm{m}}^3$. Both COR values ($e_{2s}$ and $e_{tr}$), and the pertinent damping ratios (${\xi }_b$ and ${\xi }_s$), required for these tests were computed by the analytical Eqs. (5) and (6) and the proposed Eqs. (11) and (13), respectively (i.e., $e_{2s}=0.989$, $e_{tr}=-0.489$, ${\xi }_b=2.98\%$, and ${\xi }_s=0.84\%$).

Fig. 15 presents a comparison of the proposed numerical (viscous damping) model with experimental one-sided forced-rocking tests from Al Shawa et al. (2012). Tests 22 and 31 (i.e., TH 22 and TH 31 in Table 5) are examined in more detail here, as shown in Fig. 15. In the case of TH 22, Fig. 15(a) illustrates a good agreement between the prediction of the numerical model and the experimental response in terms of the peak response and amplitude decay for the assumed gap of 4 mm (based on Al Shawa et al. 2012). However, the numerical model fails to capture the phase of the experimental response. In contrast, Fig. 15(b) reveals a poor agreement in terms of both the amplitude and phase of oscillations for the dictated gap of 4 mm. The repetition of the test might have led to a progressive increase in the magnitude of the gap due to, e.g., debris accumulation. In this context, better agreement of the response is achieved in terms of both the peak response and amplitude decay when increasing the gap at the interface between the façade and the sidewall to 6 mm. This conclusion highlights the sensitivity of the response to specific characteristics of the experimental setup. Nevertheless, the proposed numerical (viscous damping) model shows adequate adaptability and robustness to such case-specific experimental characteristics.