Numerical block-based simulation of rocking structures using a novel universal viscous damping model

Numerical Block-Based Simulation of Rocking Structures Using a Novel Universal Viscous Damping Model Georgios Vlachakis1; Anastasios I. Giouvanidis2; Anjali Mehrotra3; and Paulo B. Lourenço4 Abstract: Unreinforced masonry structures, particularly façade walls, are seismically vulnerable due to their weak connections with adjacent walls, floors, and/or roofs. During an earthquake, such walls formulate local mechanisms prone to out-of-plane collapse. This behavior has been largely investigated using classical rocking theory, which assumes the structure responds as a rigid body undergoing rocking motion, with energy dissipation at impact. Due to the complexity of the problem, however, e.g., number of degrees of freedom or boundary conditions, numerical block-based modeling is gaining momentum. However, numerical models lack a consistent and reliable treatment of the energy loss at impact. This paper bridges the gap between the well-established energy loss of classical rocking theory and the treatment of damping in numerical modeling. Specifically, it proposes an equivalent viscous damping model through novel ready-to-use predictive equations that capture the dissipative phenomena during both one-sided and two-sided planar rocking motion. The results reveal a satisfactory performance of the proposed model through comparisons with experimental results from literature and highlight its universality and robustness through applications of the model in fundamentally different block-based numerical modeling software. DOI: 10.1061/(ASCE)EM.19437889.0001985.This work is made available under the terms of the Creative Commons Attribution 4.0 International license, https:// creativecommons.org/licenses/by/4.0/. Author keywords: Rocking; Coefficient of restitution; Viscous damping; Numerical modeling; Out-of-plane collapse. Introduction Masonry structures display high vulnerability to seismic action— threatening human lives, built assets, and a major part of our cultural heritage (Bruneau 1994;Penna et al. 2014). Among the observed types of failure, out-of-plane (OOP) collapse is the most frequent—particularly in the case of façade walls with poor connections to transversal elements or diaphragms (Ingham and Griffith 2011;Vlachakis et al. 2020, among others). Analysis of these OOP collapse mechanisms is typically conducted using simplified limit analysis procedures (D’Ayala and Speranza 2003;Vaculik et al. 2014), whereby the capacity of the mechanisms is evaluated through calculation of an equivalent static lateral force required to trigger each mechanism and eventually cause collapse of the structure. However, forced-based approaches can be overconservative because they tend to neglect the dynamic reserve of stability, particularly in the case of large-scale structures, which undergo significant displacements before overturning (Godio and Beyer 2019;Sorrentino et al. 2017). To that end, the employment of theoretical rocking dynamics has been proposed to evaluate the OOP dynamic stability of masonry walls (e.g., Casapulla et al. 2017 and references therein). Following Housner’s(1963) seminal work on the single rigid rocking block, the equations that describe the motion of masonry walls (simulated as rigid blocks) have been developed using rocking (Lagrangian) dynamics. Furthermore, the influence on the response of both seismological parameters (Giouvanidis and Dimitrakopoulos 2018) and certain structural characteristics, such as the presence of additional loads due to, e.g., masses from floor and/or roof elements, or thrust from vaults (Giresini et al. 2015;Mauro et al. 2015), tie bars (AlShawa et al. 2019;Mauro et al. 2015), and transverse walls (Giresini and Sassu 2017;Al Shawa et al. 2012; Sorrentino et al. 2011), as well as the formulation of a two-block mechanism, which occurs in the case of walls restrained by floors or a roof (Mauro et al. 2015;Sorrentino et al. 2011) has been extensively investigated. In such a methodological framework, energy dissipation is assumed to occur entirely at impact, and is accounted for by the coefficient of restitution (COR) which correlates the angular velocity of the structure before and after impact. The COR may be determined analytically through the assumption of conservation of angular momentum, after specifying the geometry of the block and the points of impact (Hogan 1992;Housner 1963). Experimental investigations have evaluated the accuracy of the COR calculated using the classical rocking theory, and despite some discrepancies observed, it appears that the overall energy loss of rocking blocks can be adequately captured (Bachmann et al. 2018;Cappelli et al. 2020;Chatzis et al. 2017;Costa et al. 2013;Kalliontzis and Sritharan 2018;Lipscombe and Pellegrino 1993). However, as the complexity of the structure increases, e.g., more degrees of freedom, different boundary conditions, or introduction of flexible interfaces, among others, the classical rocking theory becomes complicated. Thus, alternative analytical and numerical models 1Ph.D. Candidate, Institute for Sustainability and Innovation in Structural Engineering, Dept. of Civil Engineering, Univ. of Minho, Guimarães 4800-058, Portugal. Email: [email protected] 2Postdoctoral Fellow, Institute for Sustainability and Innovation in Structural Engineering, Dept. of Civil Engineering, Univ. of Minho, Guimarães 4800-058, Portugal (corresponding author). ORCID: https://orcid .org/0000-0002-5850-3095. Email: [email protected] 3Postdoctoral Fellow, Institute for Sustainability and Innovation in Structural Engineering, Dept. of Civil Engineering, Univ. of Minho, Guimarães 4800-058, Portugal. ORCID: https://orcid.org/0000-0002-3453 -2879. Email: [email protected] 4Professor, Institute for Sustainability and Innovation in Structural Engineering, Dept. of Civil Engineering, Univ. of Minho, Guimarães 4800-058, Portugal. Email: [email protected] Note. This manuscript was submitted on January 11, 2021; approved on May 11, 2021; published online on August 24, 2021. Discussion period open until January 24, 2022; separate discussions must be submitted for individual papers. This paper is part of the Journal of Engineering Mechanics, © ASCE, ISSN 0733-9399. © ASCE 04021089-1 J. Eng. Mech. J. Eng. Mech., 2021, 147(11): 04021089 Downloaded from ascelibrary.org by 193.137.92.16 on 03/23/22. Copyright ASCE. For personal use only; all rights reserved. have been proposed to capture the transient nonlinear dynamic response of various rocking configurations (Giouvanidis and Dimitrakopoulos 2017a;MehrotraandDejong2020;Spanos et al. 2001). At the same time, recent developments in computational approaches used for modeling masonry structures are gaining momentum, particularly block-based models, which have been found capable of reproducing the dynamic response of masonry walls while also being able to model masonry texture and interaction with surrounding structural elements. These include the finiteelement (FE) method (D’Altri et al. 2019), the discrete-element (DE) method (Lemos 2019), and multibody dynamics (Portioli and Cascini 2018). However, despite their widespread use, applications of these models usually lack a reliable treatment of the energy loss due to the nonsmooth behavior of impacts during rocking motion (de Felice et al. 2017;Sarhosis et al. 2019;Vassiliou et al. 2021). Thus, viscous damping models are usually adopted. Viscous models are mathematical artifices with a continuous nature, seemingly in complete contrast with the impulsive energy loss suggested by the classical rocking theory. This characteristic appears to be a source of uncertainty in numerical simulations of rocking bodies (AlShawa et al. 2017;Lemos and Campos Costa 2017;Malomo et al. 2021), making damping one of the main parameters that needs to be adjusted to obtain a better fit to a reference response rather than a quite consistent method, such as the classical rocking theory (Housner 1963). The main objective of the present study is to bridge the gap between the well-established energy loss of the classical rocking theory (Housner 1963) and the treatment of damping of blockbased numerical models. This is conducted following a phenomenological calibration of a viscous damping model to mimic the classical rocking (COR) theory, and pertinent ready-to-use expressions are proposed. More specifically, two of the most common problems evident in masonry structures are investigated: two-sided and one-side rocking (Fig. 1). Two-sided rocking behavior represents the motion of a parapet, gable or boundary wall rocking over their foundation when subjected to ground excitation, whereas one-sided rocking behavior describes the motion of masonry façades with insufficient connections to transversal walls. Through a series of over a thousand numerical simulations, this work proposes ready-to-use expressions that agree with the well-established classical rocking theory and efficiently capture the energy loss during rocking motion of any numerical block-based structure. Importantly, the results of this study serve as the basis for a more rational and holistic approach to model (1) multi-degree-of-freedom rocking structures, (2) the interaction of complex geometries and boundary conditions with the rocking response, and (3) the inclusion of material nonlinearities. To this end, the main dynamic characteristics of both approaches, i.e., classical rocking theory and the numerical block-based modeling, are firstly outlined, paving the way for an appropriate selection of the viscous damping model. During the calibration process of the proposed viscous damping model, special attention is given to its universality via its application in fundamentally different numerical modeling software, as well as its capability of including modifications of the classical rocking theory suggested by previous experimental and theoretical studies (e.g., Kalliontzis et al. 2016;Sorrentino et al. 2011;Ther and Kollar 2017, among others). Finally, the performance of the proposed numerical viscous damping model is evaluated against experimental campaigns available in literature, highlighting both its merits and shortcomings. Analytical Modeling of Rocking Structures In principle, during a strong ground motion, rocking action activates the structure’s rotational inertia and offers a favorable seismic isolation effect, which relieves the structure from deformation and ultimately damage. Rocking becomes evident in a variety of structural configurations, e.g., from bridges (Giouvanidis and Dimitrakopoulos 2017b;Giouvanidis and Dong 2020;Routledge et al. 2020) and buildings (Bachmann et al. 2017) to cultural heritage structures (Mehrotra and DeJong 2018) and classical monuments (Di Egidio and Contento 2009;Psycharis et al. 2013). However, the main concern in rocking dynamics is the existence of large displacements (or equivalently rotations) during rocking motion, which might cause instability and eventually overturning of the structure. Therefore, the associated challenges behind this peculiar seismic behavior should be properly addressed with advanced analytical and numerical simulations that can capture with high fidelity the transient nonlinear dynamic behavior of rocking structures. Consider the rocking block of Fig. 1standing free on a rigid ground. During an earthquake, rocking commences when the seismic demand due to the horizontal ground excitation becomes equal to the seismic resistance due to the gravitational and inertial forces acting on the body. This condition yields the minimum ground acceleration necessary for rocking to initiate ¨ ug;min ¼g· tanðαÞ (Housner 1963), where gis the acceleration of gravity and αis the slenderness of the block α¼arctanðb=hÞ(Fig. 1). After rocking initiates, the equation that describes the motion of the rocking block of Fig. 1can be expressed as follows (Housner 1963): ¨ θ¼−p2sinðα−θÞþ¨ ug gcosðα−θÞð1Þ where ¨ ug= (horizontal) ground acceleration; positive and negative signs = clockwise and counterclockwise rotation θ, respectively; and p= frequency parameter of the block, defined as p¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mgR=I0 pwith mrepresenting the mass of the block, and I0representing the rotational moment of inertia with respect to the pivot points. In the case of a rectangular block, psimplifies to p¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi 3g=4R p, where Ris the diagonal distance from the center of mass of the block to the pivot point (Fig. 1). The importance of the parameter p, being inversely proportional to the square root of the size R, has been studied in the past (e.g., Makris 2014, among others). From a physical perspective, the frequency parameter prefers to the pendulum frequency of the block as if it is hanging from its pivot point (DeJong and Dimitrakopoulos 2014), and not the classical natural frequency, which usually measures cycles of vibration per second. This discrepancy stems from the fact that the natural frequency and period Fig. 1. Archetypal rocking block exhibiting (a) two-sided rocking motion; and (b) one-sided rocking motion when subjected to horizontal ground excitation. © ASCE 04021089-2 J. Eng. Mech. J. Eng. Mech., 2021, 147(11): 04021089 Downloaded from ascelibrary.org by 193.137.92.16 on 03/23/22. Copyright ASCE. For personal use only; all rights reserved. of the rocking motion is amplitude dependent, and thus, unsuitable for characterizing a structure (Housner 1963) Trock ¼4 pcosh−11 1−θ0 αð2Þ During rocking, the smooth motion of the block is interrupted by nonsmooth impacts at its contact/pivot points. The classical rocking theory (Housner 1963) considers impact as an instantaneous event, where the system is characterized by infinite stiffness (Fig. 2). When impact occurs, energy is lost. Ignoring bouncing and assuming the block sustains pure rocking motion (i.e., no sliding at the contact interface), the energy loss at impact is captured by COR, e, acting as radiation damping. The COR connects the preimpact with the postimpact angular velocity e¼ ˙ θþ= ˙ θ−. When the rectangular block of Fig. 1(a) undergoes two-sided rocking motion (denoted as 2shenceforth), conservation of angular momentum yields (Housner 1963) e2s;an ¼1−3 2sin2αð3Þ From Eq. (3), COR depends entirely upon the geometry (i.e., slenderness) of the block. Thus, the material properties (e.g., mass or stiffness) are irrelevant to the damping phenomenon during impact. However, there are cases where the block during its smooth rocking motion, apart from the impact with the ground, also comes into contact with an adjacent wall representing masonry façades inadequately connected with the transversal walls. In such cases, the block exhibits one-sided rocking motion (denoted as 1shenceforth) as shown in Fig. 1(b). Bao and Konstantinidis (2020) treated the impact with the adjacent sidewall as an additional event characterized by a separate/independent COR. Similarly, Sorrentino et al. (2011) assumed three consecutive impacts taking place in close but distinct time instants. Specifically, an impact at the base of the block [i.e., Point 2 in Fig. 1(b)], followed by an impact at the upper corner [i.e., Point 3 in Fig. 1(b)], and finally an additional impact at the base [i.e., Point 1 in Fig. 1(b)], resulting in a postimpact rocking rotation around the same pivot point as the preimpact rocking rotation but in the opposite direction (i.e., ˙ θþ= ˙ θ−<0). Under these assumptions, COR, which captures the impact of the rectangular rocking block of Fig. 1(b) with the transverse wall utilizing conservation of angular momentum, becomes etr;an ¼1−3 2cos2αð4Þ Therefore the total energy loss when the block of Fig. 1(b) undergoes one-sided rocking motion can be expressed as a lumped COR considering all three impacts (Sorrentino et al. 2011) e1s¼e2 2s·etr ð5Þ Numerical Modeling of Rocking Structures From a numerical perspective, modeling the behavior of the rocking structure of Fig. 1is challenging, mainly due to the presence of finite interface stiffness (as opposed to the theoretically infinite interface stiffness of the classical rocking theory) and the nonsmooth nature of impacts. Interface Stiffness When simulating the rocking response using numerical blockbased models, normal and tangential stiffness (knand ks) 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 (krot·contact) of the block can be expressed krot:contact ¼1 12 knð2bÞ3ð2lÞð6Þ where (2b) = block’s width (Fig. 1); and (2l) = block’s depth in the general three-dimensional (3D) configuration. The normal stiffness knis 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: frot:contact ≃1 2πffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi krot:contact Irot· sð7Þ where Irot· = 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 Fig. 2. Moment-rotation relationship of the rocking block of Fig. 1 based on the classical rocking theory and more realistic nonlinear numerical modeling with finite stiffness. (Data from Housner 1963.) © ASCE 04021089-3 J. Eng. Mech. J. Eng. Mech., 2021, 147(11): 04021089 Downloaded from ascelibrary.org by 193.137.92.16 on 03/23/22. Copyright ASCE. For personal use only; all rights reserved. frot:contact ≫ 1 Trock ð8Þ To illustrate the overall similarity between the analytical rigid model and the numerical finite stiffness model, Fig. 3plots 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 Rand slenderness (i.e., aspect ratio) h=b½¼ 1=arctanðαÞ. Overall, Fig. 3shows 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 momentrotation 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)withrespectto 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 (Trot·contact ¼1=frot·contact)ofEq.(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). Energy Dissipation 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 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 Cacts, (2) the way Cbehaves during timehistory analyses, and (3) the exact value of 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. 4illustrates the most widely used. In particular, the constant damping coefficient (CDC) model retains the same value of Cthroughout 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 ξduring the response history, resulting in an equivalent dissipation of all the frequency content. Further, a stiffness-proportional damping ratio (SDR) allows the damping ratio ξto follow the changes of the stiffness matrix during the response history. As a result, the high-frequency content of the Fig. 3. Period of vibration with respect to the rocking amplitude for rocking blocks with different (a) scale R; (b) slenderness (or aspect ratio) h=b; and (c) period of vibration with respect to maximum top displacement for the area denoted in plot b. Fig. 4. Different viscous damping models with respect to the frequency content. © ASCE 04021089-4 J. Eng. Mech. J. Eng. Mech., 2021, 147(11): 04021089 Downloaded from ascelibrary.org by 193.137.92.16 on 03/23/22. Copyright ASCE. For personal use only; all rights reserved. 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.,energyisdissipatedonlywhenthedashpotisincontact. In addition, a proper treatment of impact requires a proper definition of the value of C(or equally, the value of ξ). Particularly for the rocking problem, a relationship between the coefficient of restitution eand damping ratio ξis critical to ensure energy equivalence between the classical rocking theory (Housner 1963) and the numerical viscous damping model. Table 1provides a summary of the ξ–erelationships 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 ξ–eexpression 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. Adopted Viscous Damping Model The main objective of the present work is to adopt a viscous damping model that is able to reproduce with high fidelity the impulsive dynamics’energy loss characteristics and subsequently ensure dynamic equivalence between the classical rocking theory (Housner 1963) and the numerical viscous damping model in a universal manner, i.e., applicable to different rocking structures of different materials simulated in different finite-element/discrete-element software. Recall that a unilateral dashpot definition provides the most convenient basis to meet this goal. This section provides a closer look at the performance of the adopted viscous damping model through a series of free-rocking simulations of the structure shown in Fig. 1undergoing both two-sided and one-sided rocking motion. The analytical model adopted herein is based on the classical rocking theory (Housner 1963). Specifically, an event-based approach is adopted, according to which the motion of the rocking block of Fig. 1can be decomposed into a smooth motion interrupted by nonsmooth contact events (i.e., impacts). The smooth rocking motion of the block is obtained through the solution of the differential equation of Eq. (1) using mathematical programming in MATLAB version 2017a (MathWorks 1992). Whenever impact occurs, the integration stops and the coefficient of restitution [Eq. (3) for the two-sided rocking case or Eq. (5) for the one-sided rocking case] is applied to determine the new initial conditions for the next iteration. The same problem is also formulated in a finite element environment, i.e., ABAQUS CAE version 2019 (Simulia 2012), a widely used FEM software well-suited for block-based simulations that include contact phenomena. A schematic representation of the Table 1. Summary of available ξ-erelationships in literature applicable to the rocking problem Reference Relationship Basic assumptions Priestley et al. (1978)ξ¼1 n·πln8 < : θ0 α2 41−ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1−e2n1−1−θ0 α2 s3 5 −19 = ; Equivalence between COR theory and viscous decay of elastic oscillator Makris and Konstantinidis (2003)ξ¼−0.68 ·lnðeÞSimplified approximation of the Priestley et al. (1978) relationship Giannini and Masiani (1990)ξ¼2·ð1−eÞ π·ð1þeÞEquivalence between COR theory and viscous energy loss of unilateral dashpot Anagnostopoulos (2004)ξ¼−lnðeÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi π2þðlnðeÞÞ2 pEquivalence between COR theory and viscous energy loss of unilateral dashpot Imanishi et al. (2012)ξ¼−lnðeÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4·π2þðlnðeÞÞ2 pEquivalence between COR theory and viscous energy loss of unilateral dashpot Cheng (2007)ξ¼ð1−e2Þ π·ð1−Δ=bÞEquivalence between COR theory and viscous energy loss DeJong (2009)ξ¼b 2·h;ξ¼b R;ξ¼100% Alternative proposals for spring dashpots at corners of discrete model Tomassetti et al. (2019)ξ¼−0.218 ·a1−0.195 ·lnðeÞEquivalence between COR theory and a SDOF analytical formulation with viscous energy loss Note: n= number of impacts; Δ=b= amplitude of motion; SDOF = single degree of freedom; and a1= alternative definition of system’s stiffness based on a trilinear force-displacement relationship. © ASCE 04021089-5 J. Eng. Mech. J. Eng. Mech., 2021, 147(11): 04021089 Downloaded from ascelibrary.org by 193.137.92.16 on 03/23/22. Copyright ASCE. For personal use only; all rights reserved. numerical models is depicted in Fig. 5. Within ABAQUS CAE, the explicit solution scheme is adopted instead of an implicit scheme because it is preferable to proceed with the solution over small time increments. The main reason behind this choice is the reduced computational cost that the explicit scheme offers. Moreover, the contacting bodies of the problem are only forced to interact when they come into contact. The normal behavior of the contact area follows a linear stiffness pressure-overclosure relationship (i.e., soft contact approach). The tangential behavior adopts a penalty friction formulation with elastic stiffness and a friction coefficient that defines the slipping criterion. An artificially high value is given to the friction coefficient to avoid possible sliding at the contact interface, and dilatancy effects are neglected. For the case of the one-sided rocking problem of Fig. 5(b), distinct contact properties are set to govern the different body interactions, i.e., contact with the base (kn;b) and with the sidewall (kn;s). Finally, a viscous damping ratio ξis directly assigned to the contact interfaces following the unilateral dashpot definition discussed previously, i.e., ξbat the base and ξs at the sidewall. Fig. 6shows the response of the analytical and numerical models through a series of free-rocking simulations for both two-sided [Figs. 6(a and b)] and one-side [Figs. 6(c and d)] rocking motions. Fig. 6compares the two modeling approaches in terms of the rocking response history [Figs. 6(a and c)] and the total energy content expressed as the sum of the potential and kinetic energies of the system [Figs. 6(b and d)]. In Fig. 6, the initial rotation of the block is set to θ0=α¼0.5, and the block has height 2h¼4.2m and base width 2b¼0.6m. For the analytical model, the COR that characterizes impact with the base is taken as e2s¼0.97 [Eq. (3)], and the additional impact with the transversal wall, for the case of one-sided rocking, is captured through the etr ¼−0.47 [Eq. (4)]. For the numerical model, the damping ratio at the base is considered ξb¼5.3% [based on Eq. (11)], whereas for the one-sided rocking case, the damping ratio during contact with the sidewall is ξs¼0.74% [based on Eq. (13)]. Fig. 6reveals the remarkable agreement between the two modeling approaches. Observe that only for lower rocking amplitudes and only for the two-sided rocking case [Figs. 6(a and b)] do the two examined models differ, with the numerical model dissipating energy slightly faster than the analytical model. Most importantly, from Figs. 6(b and d), the numerical (viscous damping) model shows a similar behavior with the analytical model, i.e., negligible energy loss during the smooth rocking motion phase, whereas steplike energy loss appears at each impact due to the unilateral dashpot formulation adopted herein. At the same time, the finite amount of energy dissipation at each impact is controlled by the damping ratio ξ. Therefore, by adjusting the damping ratio of the viscous damping model, energy equivalence between the two modeling approaches can be achieved. To this end, a calibration methodology is presented in the next section, with the aim of providing generalized predictive ξ–erelationships applicable to a variety of block-based (numerically simulated) structures that undergo either one-sided or two-sided rocking motion. Importantly, even though the proposed numerical viscous damping model is presented using ABAQUS CAE, its applicability is universal and extended in other commonly used software for the numerical analysis of block-based rocking models. Fig. 7presents a comparative investigation on the response of the rocking block of Fig. 5(a) derived by two additional commonly used block-based software, DIANA FEA version 10.4 (DIANA 2017), and 3DEC version 7.0 (Itasca 2013) through free-rocking simulations. Fig. 5. Schematic representation of the numerical models adopted for (a) two-sided rocking motion; and (b) one-sided rocking motion. Fig. 6. Behaviour of the viscous damping model for (a and b) two-sided; and (c and d) one-sided rocking problems of Figs. 1and 5. Comparison with the analytical model are conducted in terms of rocking angle in plots (a) and (c) and total energy content in plots (b) and (d). Details of the examined structure are 2h¼4.2m, 2b¼0.6m, e2s¼0.97,etr ¼−0.47,kn;b¼5×108N=m3,ξb¼5.3%,kn;s¼5×108N=m3, and ξs¼0.74%. © ASCE 04021089-6 J. Eng. Mech. J. Eng. Mech., 2021, 147(11): 04021089 Downloaded from ascelibrary.org by 193.137.92.16 on 03/23/22. Copyright ASCE. For personal use only; all rights reserved. The three examined packages of modeling software are fundamentally different; however, the responses predicted by the adopted software are in almost excellent agreement. In particular, DIANA FEA adopts an implicit scheme to solve dynamic problems (compared with the explicit scheme adopted in ABAQUS CAE). Even though implicit schemes allow the use of larger time steps, the relatively high frequency at impact [Eq. (7)] imposes a strict upper limit to the adopted time step. At the same time, even if this restriction is disregarded, to control spurious high-frequency oscillations, artificial numerical dissipation is introduced, e.g., through the use of the Hilbert-Hughes-Taylor (HHT) algorithm (Hilber et al. 1977). Thus, relatively small time steps are eventually used in order for the viscous dissipation to dominate the artificial one. To treat contact, DIANA FEA uses a zero-thickness interface element formulation, characterized by an elastic normal and tangential stiffness. The constitutive model of the interface is based on plasticity, following a nonassociative flow rule (with zero dilatancy) and a classical Mohr-Coulomb failure criterion. The physical damping is assigned to the interface elements based on a Rayleigh definition input. Because a unilateral SDR viscous damping is of interest, the alpha factor is set to zero and the beta factor is assigned accordingly to provide the desired ξvalue at the frot·contact of Eq. (7). The software 3DEC, on the other hand, adopts an explicit solution scheme, which makes use of a central-difference algorithm to solve the equations of motion (Lemos 2019). This software is based on the discrete-element method whereby the structure is modeled as an assembly of discrete rigid or deformable blocks. Rigid blocks are used in this paper, as a result of which system deformability is assumed to be entirely concentrated at the interfaces between blocks. Contact between blocks is modeled using zero-thickness nonlinear interface springs (point contacts); thus, no joint or interface elements are defined. As in the case of ABAQUS CAE and DIANA FEA, the soft contact approach is also used here, with the extent of interpenetration between blocks being controlled by the interface stiffness, and the nonlinear response is controlled via the tensile strength in the normal direction (set to zero herein) and the Coulomb-slip joint model in the tangential direction, which depends in turn on the values specified for the cohesion (zero in this case) and the friction angle (set artificially high) (Pulatsu et al. 2019). Finally, stiffness proportional damping is used, which is specified through the damping ratio ξand the frequency (in this case, frot·contact) at which it acts. Fig. 7illustrates a good agreement among the three simulation methodologies despite being fundamentally different (as explained previously). This conclusion highlights the universal applicability of the proposed numerical viscous damping model. The models used are readily available within each software package, and no additional implementations are required to reproduce the current results. Proposed Numerical Model Calibration Methodology 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., e2sand etr for twoand one-sided rocking behavior, respectively) is assumed as an independent parameter. The choice to keep eindependent 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 ξ(i.e., ξband ξsfor twoand one-sided rocking behavior, respectively) is considered the one that minimizes the root-mean square error (RMSE) metric RMSE ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 N·X N t¼1 ðθan;t−θnum;tÞ2 v u u tð9Þ where θan;t= rocking rotation of the reference analytical model; θnum;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 r¼θt=θ0of 95% for the two-sided and 99.95% for the one-sided case, respectively, i.e., ∀t, until ∃θt≥ð1−rÞθ0. 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 2presents 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 (θ0=α), and the normal interface stiffness (kn), 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 ξ–eexpressions can also be applicable to various rocking structures that can be dynamically equivalent to block-based structures and whose parameters fall within the Fig. 7. Free-rocking response of the structure of Fig. 1(a) through three different numerical simulation software, i.e., ABAQUS CAE, DIANA FEA, and 3DEC. Details of the examined structure are 2h¼4.2m, 2b¼0.6m, kn;b¼5×108N=m3, and ξb¼5.3%. © ASCE 04021089-7 J. Eng. Mech. J. Eng. Mech., 2021, 147(11): 04021089 Downloaded from ascelibrary.org by 193.137.92.16 on 03/23/22. Copyright ASCE. For personal use only; all rights reserved. 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 α[Eqs. (3) and (4)]. Therefore, the scale (R) and amplitude ðθ0=αÞdo 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 (kn;b) and/or the sidewall (kn;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. Two-Sided Rocking Motion This section applies the calibration methodology illustrated in Fig. 8 and proposes a novel ξb–e2srelationship that captures the response history of any block-based numerical model that undergoes twosided 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 (θ0=α), and normal interface stiffness at the base (kn;b), are studied independently. For each combination of the examined parameters, as the COR e2svaries, a unique ξbis determined that minimizes the RMSE error metric [Eq. (9)]. Fig. 9 plots those ξb–e2spairs 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) ξb¼c·lne2sð10Þ where c= constant computed by the fitting process. Fig. 9plots also the fitted functions identified with solid lines. Fig. 9reveals that the logarithmic function of Eq. (10) provides a good estimation of the ξb–e2scorrelation, 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 R¼2.1m is considered for the rest of the calibration process). On the contrary, a significant influence is noticed when the slenderness (or aspect ratio) h=bis 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 θ0=αis 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 θ0=α¼0.5is 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. Table 2. Independent parameters examined for the calibration process (Fig. 8) of the two-sided and one-sided rocking Parameter Range Reference value Scale, R(m) 1.4–2.8 2.1 Slenderness, h=b4.0–15.0 7.0 Amplitude, θ0=α0.3–0.8 0.5 Normal interface stiffness at base, kn;b(N=m3) 0.5×108−30 ×1085×108 Normal interface stiffness at side, kn;sa(N=m3) 0.5×108−30 ×1085×108 aOnly for the one-sided rocking case. Fig. 8. Flowchart of the adopted phenomenological calibration process. © ASCE 04021089-8 J. Eng. Mech. J. Eng. Mech., 2021, 147(11): 04021089 Downloaded from ascelibrary.org by 193.137.92.16 on 03/23/22. Copyright ASCE. For personal use only; all rights reserved. Finally, Fig. 9(d) suggests a high sensitivity of the viscous damping model to the normal interface stiffness at the base kn;b. A model with lower stiffness appears to inherently dissipate more energy for a given damping ratio. Thus, it requires a lower damping ratio ξto 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 ξb–e2scalibration process. To summarize, Fig. 9shows that the slenderness (or aspect ratio) h=band normal interface stiffness at the base kn;bare 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 ξb–e2srelationship that connects the damping ratio of any block-based numerical model with the COR e2sis expressed ξb¼−0.000292 ·H B0.935 ·k0.343 n;b·lne2sð11Þ The adequacy of the proposed ξb–e2srelationship of Eq. (11)to predict the measured ξbis visualized in Fig. 10. A coefficient of determination of R2¼0.978 characterizes all the data points, indicating a remarkable estimation capability of the proposed ξb–e2srelationship [Eq. (11)]. In more detail, whereas the slenderness variation appears to capture the measured ξbquite 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. One-Sided Rocking Motion This section focuses on the one-sided rocking problem of Figs. 1(b) and 5(b) and offers a novel ξs–etr 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) e2sto capture the energy dissipation for impacts against the base, and (2) etr for impacts against the transversal wall [Fig. 1(b)]. Numerically, the impacting/ contacting areas are governed by distinct contact/interface properties [Fig. 5(b)]: kn;band ξbfor normal interface stiffness and damping ratio at the base, and kn;sand ξsfor normal interface stiffness and damping ratio at the side (i.e., the transversal wall), respectively. Therefore, the expression between ξb–e2sof Eq. (11) is directly applied at the base, and the ξs–etr relationship is investigated. Similarly, the scale (R), slenderness (or aspect ratio) (h=b), rocking amplitude (θ0=α), normal interface stiffness at the base (kn;b), and the normal interface stiffness at the side (kn;s) are considered as the independent parameters, and their influence on the Fig. 9. Influence of (a) scale R; (b) slenderness (or aspect ratio) h=b; (c) amplitude θ0=α; and (d) interface stiffness kn;bat the base on the response history of the block of Figs. 1(a) and 5(a) when it undergoes two-sided rocking motion. Fig. 10. 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