A discretely damped SDOF model for the rocking response of freestanding blocks

Liu Hanquan, Huang Yuli and Qu Zhe

1.Key Laboratory of Earthquake Engineering and Engineering Vibration, Institute of Engineering Mechanics, China Earthquake Administration; Key Laboratory of Earthquake Disaster Mitigation, Ministry of Emergency Management, Harbin 150080, China Sanhe 065201, China

2.Arup, San Francisco, CA 94105, USA

Abstract: This paper presents a single-degree-of-freedom (SDOF) constitutive model for assessing the performance of freestanding block contents of buildings.The model incorporates a bespoke damper to account for energy dissipation associated with rocking.It is advantageous in its direct correlation, via energy conservation, to the restitution coefficient for impact during rocking.A comparative study with the existing SDOF rocking models shows that the proposed model significantly improves the accuracy of free-rocking simulations, in which inherent damping predominantly affects response.It provides a promising and efficient tool for computationally intensive performance evaluation of nonstructural components.

Keywords: contents; rigid block; equivalent viscous damping; free rocking; forced rocking; single-degree-of-freedom system

1 Introduction

Block contents, such as furniture, facilities, and equipment, are among the most typical nonstructural components of buildings.Unanchored contents are typically treated as freestanding blocks.They may undergo sliding and rocking due to floor movement that occurs during earthquake events (Shenton, 1996).Excessive sliding, rocking, or overturning may result in nonstructural damage, business interruption, and casualties (Taghavi and Miranda, 2003).Recognizing sliding as an issue to be addressed separately (Newmark,1965; Lopez Garcia and Soong, 2003; Chaudhuri and Hutchinson, 2005; Konstantinidis and Makris, 2009;Gazetaset al., 2012; Nagaoet al., 2012; Konstantinidis and Nikfar, 2015; Nikfar and Konstantinidis, 2017), we focus on the seismic behavior of freestanding blocks dominated by rocking, a process that features a partial uplift from a base and a change in rotation center.

The seismic response of a solitary rigid block on a rigid base is typically evaluated following the framework proposed by Housner (1963), in which a restitution coefficient solely dependent on geometry is used to capture energy dissipation during rocking.Over the years, the rocking response of freestanding contents has attracted increasing attention from experimental(Filiatraultet al., 2004; Peñaet al., 2007; Kuoet al.,2011; Konstantinidis and Makris, 2010; Nasi, 2011;Bachmannet al., 2016; Di Sarnoet al., 2019; Huanget al., 2021; Vassiliouet al., 2021) and numerical standpoints (Ishiyama, 1982; Zhang and Makris, 2001;Chae and Kim, 2003; Di Egidioet al., 2014; Vassiliouet al., 2014; Dimitrakopoulos and Paraskeva, 2015;Vetret al., 2016; Petroneet al., 2017; Fragiadakiset al., 2017; Gesualdoet al., 2018; Diamantopoulos and Fragiadakis, 2019; Kasinoset al., 2020; Lindeet al.,2020; Reggiani Manzo and Vassiliou, 2021; D′Angelaet al., 2021).Rocking has been recognized as chaotic and highly sensitive to initial conditions and dynamic processes (Aslamet al., 1980; Yimet al., 1980; Acikgozet al., 2016; Ibarra, 2016).An efficient tool is needed for the computationally intensive probabilistic evaluation of rocking responses (Bachmannet al., 2018; Vassiliouet al., 2021).

The restitution coefficient,e, accounts for energy dissipation closely related to impact during rocking.It is numerically inconvenient to implement under the framework of finite element analysis, a pervasive tool used in engineering simulations.Priestleyet al.(1978) proposed using equivalent viscous damping to simulate energy loss due to impact.They derivedan equivalent viscous damping ratio,ζ, by applying the logarithmic decay rule to block rotation.Giannini and Masiani (1990) suggested a different equation evaluatingζas a function of the restitution coefficientevia energy conservation.Following Priestley′s approach, Makris and Konstantinidis (2003) proposed a simplified empirical equation in a logarithmic form to approximate the relationship betweenζande.Vassiliouet al.(2014) proposed estimatingζof a rocking rigidbody as a function of geometry and the inertial mass of a rigid body, as opposed toe, by comparing the energy loss per cycle of free rocking and viscously damped free vibrations.These methods assume energy will be continuously dissipated by viscous damping during the entire rocking process.This assumption appears inconsistent with physical mechanisms that occur only when a block impacts a floor in its neutral position at rest.

This paper proposes an improved numerical model for freestanding blocks by introducing discrete viscous damping to simulate energy dissipation during rocking.The proposed model can capture energy dissipation precisely when the block impacts the floor at its neutral position at rest.Free rocking and shaking table tests are used to calibrate the proposed and existing models.The results show that the new energy loss model proposed in this paper has improved accuracy in simulating free and forced rocking responses in which inherent damping has a significant effect on response.

2 Discretely damped SDOF model for rigidbody rocking

2.1 Rocking responses of rigid blocks

Freestanding content in a building is represented by a rigid rectangular block on a horizontal, rigid, rough surface (Fig.1).Its breadth and height are 2band 2h,respectively.The total mass ismand the center-of-mass(CM) is situated at the geometric center.The geometry of the block can also be fully defined by the size parameter,, which is the distance from CM to the pivot point and the slenderness parameter,α=atan(b/h).Its moment of inertia about the pivot point O or O′ isIO=4(mR2)/3.

Fig.1 The geometry of a rectangular block

By assuming no jumping or sliding, the block motion is fully described by the rotationθaround the pivot point.The moment equilibrium about the pivot point O or O′provides the equation of motion of undamped rocking under a horizontal excitationü0as:

wheregis gravity acceleration,H(θ) andB(θ) are the vertical and horizontal transient distances from CM to the current pivot point, respectively (Eqs.(2) and (3)):

where sgn() is the sign function.

The restoring momentM=mgB(θ) exhibits a nonlinear elastic relationship with rotationθ, as shown by th e light solid line in Fig.2.When the block is at rest,˙θ==0,H=h,B=b, and Eq.(1) becomes:

Fig.2 Moment-rotation relationship of a rocking block

In other words, the block does not rock unless accelerationü0exceeds the threshold shown in Eq.(5).The restoring moment corresponding to threshold acceleration is denoted asM0(Eq.(6)).Once the rigid block begins to rock, the restoring momentMdecreases monotonically with the increase in rotationθ, and reaches zero whenθ=α.

2.2 SDOF model for undamped rocking

Diamantopoulos and Fragiadakis (2019) introduced an equivalent single-degree-of-freedom (SDOF) model to simulate the rocking response of freestanding blocks in which mass is lumped at the center of mass (CM) with a distance ofRaway from the pivot point (Fig.3(a)).The lumped mass is supported by a rigid cantilever beam above the floor, with a nonlinear elastic rotational spring at the base (Fig.3(b)).Massmis provided with an additional moment of inertiamR2/3 so that the total moment of inertia of the lumped mass about the pivot point equals that of the original rigid block.The zerolength rotational spring at the bottom of the rigid beam describes the elastic nonlinearM-θrelationship of the rocking block (Fig.2).The nonlinear descending branch of theM-θrelationship can be simplified to a linear relationship, as seen in Eq.(7) (dashed line in Fig.2),which exhibits a negative tangent stiffnesskr(Eq.(8)).

In a finite element analysis, a finite initial stiffnessk0=n|kr| is required to approximate the infinite stiffness in theM-θrelationship before the rigid block begins to rock.The system is assumed to oscillate linearly within a small range of ±δαon both sides of the neural positionθ=0, whereδ=1/(n+1).In Fig.2 the simplified bilinear elasticM-θrelationship is compared with the original nonlinear elastic relationship.

2.3 Energy dissipation in rigid-body rockings

Energy is dissipated whenever a rocking block returns to its original position (θ=0) and impacts the floor(Fig.4(a)).Newton′s experimental law quantifies energy loss due to impact by the use of restitution coefficiente,which is the ratio of the relative velocity of objects after or before a collision (Weir and McGavin, 2008).In the case of rockingare the angular velocities before and after the block impacts with the floor.Housner (1963) derived a rigid-body restitution coefficient, denoted aseRhereinafter (Eq.(9)), by conserving angular momentum before and immediately after impact, when the pivot point shifts from O to O′.This is solely dependent on the geometry of the block,while the restitution coefficienteof a real-world collision also depends on the localized nonlinearity of colliding materials.Therefore, it is usually smaller thaneR.

While the rigid-body impact that dissipates energy is not explicitly modeled, a damping forcefd()can be introduced to the simplified SDOF model to approximately account for energy dissipation that takes place during rocking.The equation of motion of the simplified SDOF model shown in Fig.3(b) when subjected to floor motionü0becomes:whereis thedamping force and is giveninEq.(11)by assuming initialstiffness-proportional damping;k(θ)is the tangent stiffness in a bilinearM-θrelationship,as shown in Fig.2.A non-zero damping force implies that energy is being dissipated throughout the entire rocking process whenever0.This is referred to as a continuous damping model (Fig.4(b)).

Fig.3 (a) Lumped mass representation of rigid block and(b) equivalent SDOF model of a rigid rocking block

Fig.4 Energy dissipation during rocking by (a) impact, (b) continuous damping, and (c) discrete damping

wherecis the equivalent viscous damping coefficient andζis the corresponding damping ratio.

Researchers have proposed various empirical equations to relate an equivalent viscous damping ratioζwith the restitution coefficiente.These equations are used with continuous damping in rocking simulations,attempting to justify the physical meaning of the equivalent damping ratio.Giannini and Masiani (1990)derived Eq.(12) to estimate an approximate damping ratio, denoted asζGMhereinafter.This equation satisfies the boundary condition of undamped rocking (ζ=0 whene=1) and offers a large damping of 2/π whene=0.

Makris and Konstantinidis (2003) proposed a logarithmic function (Eq.(13)) to estimate an approximate damping ratio, hereinafter denoted asζMK.It also satisfies the undamped rocking boundary condition but provides an infinitely large damping ratio wheneapproaches zero.In addition, Tomassettiet al.(2019) suggested that the empirical constant of 0.68 is not universal and should be subjected to experimental calibration.

Although the above models exhibit a similar negative correlation betweenζandefor the common range ofe, they present significantly different values of the equivalent damping ratio, given the samee(Fig.5).

Fig.5 Existing relationships of ζ and e

Vassiliouet al.(2014) proposed a model of estimating the equivalent viscous damping coefficient, denoted ascV,of a rocking rigid-body, as seen in Eq.(14).This model is distinct from the above ones by relating the damping coefficient to the geometry and inertial mass of a rigid body instead ofe.Therefore, it does apply to non-rigidbody cases, in which the in-elasticities of local material may dissipate energy during impact.In addition, it shares the same lack of rationality as the above-mentioned models because they introduce continual energy loss whenever a block is moving.This is inconsistent with how the restitution coefficient quantifies energy dissipation.

2.4 Discrete damping model

This section proposes a discrete damping model to correctly simulate the phenomenon by which energy is dissipated if—and only if—the block passes its original position during rocking (Fig.4(c)).This model imposes a viscous damping forcefd=cDwithin the small range of ±δαon both sides of the original positionθ=0,as described in Eq.(15):

wherecDis the discrete viscous damping coefficient.

The conservation of angular momentum when the SDOF model passes its original position during free rocking requires:

whereis the average angular velocity, ranged ±δα.

Substituting the restitution coefficientinto Eq.(16), one obtains the relationship between the equivalent viscous damping coefficientcDin the proposed discrete damping model, and the restitution coefficiente(Eq.(17)).Unlike the proportional damping model as represented in Eq.(11),cDis not constant during the rocking process but is proportional to angular velocity before each impact.

wherec′=IO(1-e)/(2δα) is the damping constant dependent only on the geometry of the block, the restitution coefficiente, and the assumed range of energy dissipationδα.

A user-defined materialElasticBilinDampedis implemented in OpenSees by introducing a separate term for the discrete damping forcefdto the restoring force of the default uniaxial materialElasticBilin.It is used to simulate the hysteresis behavior of the rotational spring at the bottom of the SDOF system, as seen in Fig.3(b).The core of the source code ofElasticBilinDampedis provided in Appendix A.

The total resisting moment-rotation and damping force-rotation relationships of damped free rocking SDOF systems with discrete and conventional continuous damping models are compared in Fig.6.They are significantly different regarding when the damping force is present.Significantly,cDis physically associated with the restitution coefficient during impact by the conservation of angular momentum.This approach is equally applicable to the rocking of either a rigid body or a deformable one for a given restitution coefficient,which can be determined by means of experimental tests.

Fig.6 Hysteretic curves of a free rocking SDOF system of α=0.2 and R=0.38 m: (a) total resisting moment with discrete damping,(b) total resisting moment with continuous damping, (c) discrete damping force, (d) continuous damping force

3 Experimental verifications

SDOF models are established in OpenSees to simulate the experimental results of free and forced rocking of freestanding blocks, as reported in the literature.The proposed discrete damping model is compared with existing continuous damping (Eqs.(11)to (14)) to assess its performance.

3.1 Free rocking

The free rocking experiments taken from three different sources are simulated, including: (1) a rectangular concrete block (RCB) (Aslamet al.,1980), (2) an unreinforced masonry parapet (URMP)(Giarettonet al., 2016), and (3) an aluminum frame (AF)(Bachmannet al., 2018).Their geometric parameters are summarized in Table 1, in whicheRis calculated by Eq.(9) and the actualeis assumed to be no greater thaneRto account for possible energy dissipation by the discrete nonlinearity of the colliding materials (Sorrentinoet al.,2011).In all cases, we assume thatδ=0.05.

The response histories of rotation angleθ, angularvelocity˙and angular acceleration˙by use of the different damping models are compared with the experimental results displayed in Figs.7, 8 and 9.The three response quantities are normalized byα,Pαandgα, respectively, whereαis the slenderness parameter andP=is the rocking frequency parameter(rad/s).Table 2 compares the mean relative errors (MRE)in either the rotation amplitudeθpeakor the simultaneous periodT(see Fig.7(a)) in the first ten cycles of rocking as predicted by all the damping models.While continuous damping with eitherζGMorζMKsignificantly overestimates energy dissipation, which leads to a great discrepancy from the experimental results and large negative relative errors, the continuous damping withcVprovides a much slower decay of amplitudes for the RCB and URMP specimens and faster decay for the AF specimen.The proposed discrete damping outperforms all the continuous models and successfully simulates the amplitude decay of all three cases with the smallest relative errors in terms of both the rotation amplitudes and simultaneous periods.

Fig.7 Comparison of response histories of an RCB specimen (Aslam et al., 1980) by SDOF models with (a) discrete damping,(b) continuous damping of cV, (c) continuous damping of ζMK and (d) continuous damping of ζGM

Table 1 Shape parameters of free rocking experiments

In the approximate SDOF model for rocking blocks,the selection ofδis somehow arbitrary and may an effect on the simulation results.Figure 10 compares the free rocking histories of RCB for the different damping models by using variousδ′s.Unlike the continuous damping models, the proposed discrete damping provides results that are much less sensitive to the value ofδ.Because too small aδwill lead to spikes in the damping force in the discrete damping model and thus may cause numerical difficulties,δ=0.05 is the recommended choice.

3.2 Forced rocking in shake table tests

Nasi (2011) conducted an extensive experiment regarding forced rocking response by subjecting 312 unanchored blocks to unidirectional shake table motions.All blocks consisted of normal-weight concrete and all test data, including input and response histories, are accessible online (Klaboeet al., 2017, 2018).Six runs of the forced rocking of four types of blocks are selected to justify the SDOF models with the proposed damping model.The parameters of the four blocks are summarized in Table 3.All runs used the same waveform as the tableacceleration input, referred to as Motion 2 in Klaboeet al.(2018), but in different amplitudes.The time history and spectral accelerations of the input are depicted in Fig.11.

Fig.11 Table acceleration input of Motion 2: (a) time history (PGA=1.458 m/s2), (b) spectrum (ζ=5%)

Table 2 Peak rotation angles and errors of different experiments (%)

Fig.8 Comparison of response histories of URMP specimen (Giaretton et al., 2016) by SDOF models with (a) discrete damping,(b) continuous damping of cV, (c) continuous damping of ζMK and (d) continuous damping of ζGM

Fig.9 Comparison of response histories of an AF specimen (Bachmann et al., 2018) by SDOF models with (a) discrete damping,(b) continuous damping of cV, (c) continuous damping of ζMK and (d) continuous damping of ζGM

Three of the runs involve rocking without overturning.Figure 12 compares the simulated response histories of the rotation with experimental results, and Table 4 compares the relative errors in the maximum rotation |θmax| of the three runs by the use of different damping models.Each run is named according to the “Block ID-Motion ID-amplitude scale factor”convention, as per Klaboeet al.(2018).The results show that the discrete damping model can correctly approximate the response history and the maximum rotation angle of the rocking with the smallest relative error for Runs 4.1-2-80 and case 4.2-2-100, and thesecond smallest relative error for Run 6.1-2-60.Among the three continuous damping models,ζGMperforms similarly well as the discrete damping model in these runs (Figs.12(a) and 12(d)).ζMKleads to a significantly smaller response than the experimental results (Fig.12(c)), while, in contrast,cVtends to overestimate the response and predicts overturning for Runs 6.1-2-60 and 4.2-2-100 that did not occur in the experiment (Fig.12(b)).

Fig.12 Comparison of rotation histories of non-overturning runs by SDOF models with (a) discrete damping, (b) continuous damping of cV, (c) continuous damping of ζMK and (d) continuous damping of ζGM

Table 3 Parameters of shaking table tests

Fig.10 Comparison of rotation histories of RCB by SDOF models with (a) discrete damping, (b) continuous damping of cV,(c) continuous damping of ζMK and (d) continuous damping of ζGM

The other three runs saw overturning of the block.In these cases, it is essentially important for the numerical models to predict whether or not the overturning takes place.Both the discrete damping and the continuous damping (ζGM) successfully estimate the occurrence of overturning for all three runs (Figs.13(a) and 13(d)).Additionally,cVcorrectly predicts the occurrence of overturning in two of the three runs (Fig.13(b)).As in the non-overturning runs,ζMKstill gives conservative estimates by introducing too much energy dissipation and fails to predict the overturning of all three runs(Fig.13(c)).The performance of the damping models in predicting the occurrence of overturning also is compared in Table 5.

Table 4 Relative errors in |θmax| of non-overturning runs (%)

Table 5 Detection of overturning of overturned runs (√=overturned, ×=non-overturning)

Fig.13 Comparison of rotation histories of overturned runs by SDOF models with (a) discrete damping, (b) continuous damping of cV, (c) continuous damping of ζMK and (d) continuous damping of ζGM

4 Conclusions

Simplified SDOF models can well represent the dynamics of rocking rigid blocks and provide a highly efficient tool for extensive numerical simulation of freestanding objects.However, the simulation ofenergy loss during impact when a block goes through its original position has been subjected to the arbitrary choice of a global damping ratio without clear physical interpretation.This paper proposes a discrete damping model to solve this problem.In the proposed model,the block receives damping force only at the vicinity of its original position, consistent with the actual energy dissipating process that occurs during rocking.The equivalent viscous damping coefficient is proportional to the angular velocity before the block impacts the floor and is derived based on angular momentum conservation.The accuracy and the applicability of the proposed discrete damping model are demonstrated by simulating experimental tests of free and forced rocking in the literature and by comparing these results with three existing models of continuous damping.The following observations are made from the results.

(1) The proposed discrete damping outperforms all the existing models in all the free rocking tests and most forced rocking tests.It results in much smaller relative errors in the decaying of both the rotation amplitude and the simultaneous period in free rocking cases and can successfully predict overturning for all the forced rocking tests.

(2) Among the three continuous damping models,cVperforms acceptably well for free rocking but fails to predict overturning for half of the forced rocking runs.ζMKsignificantly underestimates the responses in all the tests of either free or forced rocking.ζGMperforms similarly well as the discrete damping for the forced rocking tests but performs badly for the free rocking tests.

(3) When implemented in an SDOF model with finite initial stiffness, the proposed discrete damping is least sensitive among the four models regarding the arbitrary choice of the initial stiffness, which is governed by theδfactor in the SDOF model used in this study.

It is worth mentioning that, in this study, the assessment of the damping models was conducted in a deterministic manner.It neglected the inherent uncertainty in the rocking and overturning response of freestanding blocks to arbitrary motion.The proposed discrete damping model is yet to be calibrated against more experimental tests that adequately address such uncertainty.

Acknowledgment

This work was sponsored by the Key Program of the CEA Key Laboratory for Earthquake Engineering and Engineering Vibration (2019EEEVL0304) and the Heilongjiang Touyan Innovation Team Program, China.