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When subjected to earthquakes, many objects or structural elements behave like rocking rigid blocks. Computer servers, medical shelves, art objects, statues, and electrical transformers are frequently included in this category. Protection of these objects is an important task, considering that their value could be inestimable or their operation crucial during earthquakes; base isolation technology has been proven to be a viable option for this purpose. Initially, the dynamic model of a rocking rigid block placed on a base isolation device is reviewed. Then, two equivalent‐static displacement‐based procedures for designing the isolators for these types of objects are proposed, and the main steps are illustrated. The first procedure aims to determine isolator characteristics to prevent the initiation of rocking motion during the code‐level earthquake event. The second procedure is aimed at designing isolators that allow a specified maximum rotation of the block during seismic events. The proposed procedures are validated by means of time‐history analyses for a suite of spectrum‐compatible accelerograms. The first displacement‐based procedure appears particularly suitable for objects of small to medium size. The validation of the second procedure demonstrates that the equal displacement rule can be applied for this kind of systems, despite their softening. The results also indicate that the approach is particularly effective for medium to large structures/objects, if small oscillations are acceptable. The controlled rocking procedure offers a significant advantage by allowing for a reduction in the maximum displacement and period of the isolator, compared to situations where rocking motion must be prevented entirely. [ABSTRACT FROM AUTHOR]

This paper presents the results of a numerical study aimed at assessing the effectiveness of an innovative steel modular retrofit system on improving the in‐plane lateral capacity of unreinforced masonry (URM) load‐bearing structures. The investigated retrofit solution consists of modular steel frames fastened to the external surface of masonry piers through chemical anchors. Distinct element models are built and calibrated against experimental data coming from a series of in‐plane quasi‐static tests performed on piers in both bare and retrofitted conditions. Two different masonry typologies are considered herein: one made up with solid clay bricks and lime mortar, assembled in a header bond pattern, and the other with typical Italian hollow clay units and cement‐lime mortar, arranged in a Flemish bond pattern. A simplified microscale approach, combined with a mesh‐refinement strategy, is adopted to decrease computational demand without losing numerical accuracy. All in‐plane failure mechanisms, such as mortar and unit tensile failures, as well as masonry crushing for high compressive stresses, are included in the developed numerical models. Moreover, a methodology to explicitly account for the contribution of the investigated retrofit in the distinct element method (DEM) framework is presented and discussed. The proposed modeling strategy yielded a good prediction in terms of lateral stiffness, lateral strength, hysteretic response, and crack pattern for unreinforced and strengthened specimens. A novel set of metrics is proposed to quantitatively assess the retrofit performance benefits. [ABSTRACT FROM AUTHOR]

The rocking‐dominated seismic response of reinforced concrete (RC) columns is a complex mechanism that entails P‐Δ effects, concrete crushing and spalling, rupture and dislocation of steel bars and stirrups, as well as combined localized failures causing splitting and shear‐flexure cracking. In this paper, we present a new meso‐scale numerical modeling strategy for the detailed simulation of the rocking response of slender RC columns under seismic actions. This work advances the current state‐of‐the‐art by exploring the unprecedented use of the Distinct Element Method (DEM), originally conceived for geotechnical studies and whose applicability in earthquake engineering has been primarily limited to the assessment of unreinforced masonry sub‐structures. To decrease analysis time and enable the use of DEM for RC problems, concrete is herein idealized as an assembly of solid macro‐block layers of size directly proportional to their distance from expected plastic hinges. These layers are connected by nonlinear interface springs to simulate tensile and shear cracking via a simplified Mohr‐Coulomb criterion. Crushing of concrete in deformable blocks is modeled using the Feenstra and De Borst strain‐softening linearized compression law, while confinement and combined failures are accounted for numerically through the explicit representation of longitudinal and transversal reinforcement bars, idealized as link elements. Previous experimental static and dynamic tests on full‐scale RC column specimens and results from fiber‐based Finite Element and sectional analysis models are used to evaluate DEM results and quantify the influence of key modeling parameters, including block number and deformability, as well as the amount of damping. The agreement between simulated failure modes and force and displacement capacities and their experimental counterparts demonstrate the applicability of the proposed approach, while demonstrating the pros and cons of simplified numerical methods. [ABSTRACT FROM AUTHOR]

In the seismic analysis of unreinforced masonry (URM) structures, the modeling of out‐of‐plane (OOP) modes and their mechanical interaction with in‐plane (IP) loaded components are typically neglected when using simplified numerical methods. Although this may result in unconservative predictions, the high computational expense entailed by more refined approaches is often prohibitive for applied researchers and practitioners. To overcome these limitations, the demonstrated capabilities of a recently developed low‐cost Macro‐Distinct Element Model (M‐DEM) to simulate IP and OOP modes are extended in this work towards the modeling of IP/OOP interaction and combined failure mechanisms of URM assemblies. In the M‐DEM framework, shear and flexural damage are accounted for by zero‐thickness interface spring layers, whose layout is determined a priori as a function of the masonry texture, while crushing failure is modeled through homogenized finite element macro‐blocks. To adapt and validate this M‐DEM scheme to model IP/OOP interaction, past experiments on full‐scale C‐, U‐, and I‐shaped URM specimens tested under quasi‐static loading were simulated. The shake‐table response of a full‐scale C‐shaped URM assembly with openings was also numerically simulated up to collapse, representing a major improvement over previous macro‐element methods. After the comparison with experimental tests, a parametric investigation of the response of reference URM walls under combined IP/OOP actions was conducted, and the influence of previous IP damage on one‐ and two‐way OOP bending capacity is quantified. This aspect, despite being widely identified as of relevant interest, has only been marginally investigated in previous research, both experimentally and numerically. [ABSTRACT FROM AUTHOR]

Allowing structures to rock during an earthquake can effectively provide base isolation at a relatively small cost. Rocking limits the base shear demand and provides self‐centering, but the rocking response depends on energy dissipation caused by interaction with the soil and impacts during re‐centering. This paper addresses the computational modeling of buildings that have either been designed to rock on the soil beneath their foundation (foundation rocking) or at the foundation–structure interface (structural rocking). Within OpenSees, foundation and structural rocking were modeled using a beam‐on‐a‐nonlinear‐Winkler‐foundation model (BNWF) combined with flat‐slider elements for footing–soil and superstructure–footing interactions, respectively. The modified with flat‐slider elements BNWF model (mBNWF) involves an uplift‐dependent stiffness and viscous damping for both vertical and horizontal directions, and a friction–vertical force coupling. The proposed computational model was used to simulate an extensive set of centrifuge tests involving both structural rocking and foundation rocking with sequential excitations. Generally, the proposed modeling approach, without calibration of built‐in parameters, adequately captured the response observed in centrifuge experiments. More specifically, the modeling captured the response amplitude and waveform of story accelerations and building rocking angle in most cases, but including potential nonlinear behavior caused by previous ground excitations was in some cases critical to obtain reasonable predictions. This was more profound for foundation rocking due to its inherent dependency on the soil strength and energy dissipation; for structural rocking previous nonlinear response primarily affected the transition time between full contact and rocking, but had a smaller effect on the prediction of maximum response. [ABSTRACT FROM AUTHOR]

A blind prediction contest was organized to evaluate the ability of different modeling approaches to simulate the seismic rocking response of a full‐scale four‐column podium structure. The structure was tested on a shake table, and was subjected to two bidirectional ground motion ensembles comprising 100 synthetic records each. This short communication presents the main assumptions and results from the model, developed using the distinct element method, which provided the second‐best prediction of the experimental results. A comparison of the model predictions and the experimental results demonstrates that the numerical model was generally able to reproduce the large displacements induced by the more intense ground motion ensemble, while tending to overestimate the displacements of the less intense earthquake ensemble. This overestimation of the response was reduced through the inclusion of damping in the system. However, the addition of damping greatly increased the solve time, which is problematic for a competition, and in the case of the more intense ground motion ensemble also resulted in an underprediction of the maximum response of the structure. [ABSTRACT FROM AUTHOR]

Summary: The in‐plane capacity of unreinforced masonry (URM) elements may vary considerably depending on several factors, including boundary conditions, aspect ratio, vertical overburden, and masonry texture. Since the overall system resistance mainly relies on the in‐plane lateral capacity of URM components when out‐of‐plane modes are adequately prevented, the structural assessment of URM structures could benefit from advanced numerical approaches able to account for these factors simultaneously. This paper aims at enhancing and optimising the employment of the distinct element method, currently confined to the analysis of local mechanisms of reduced‐scale dry‐joint blocky assemblies, with a view to simulate the experimentally observed responses of a series of URM full‐scale specimens with mortared joints subjected to quasi‐static in‐plane cyclic loading. To this end, a mesoscale modelling approach is proposed that employs a simplified microscale modelling approach to effectively capture macroscale behaviour. Dynamic relaxation schemes are employed, in combination with time, size, and mass‐scaling procedures, to decrease computational demand. A new methodology for numerically describing both unit, mortar and hybrid failure modes, also including masonry crushing due to high‐compression stresses, is proposed. Empirical and homogenisation formulae for inferring the elastic properties of interface between elements are also verified, enabling the proposed approach to be applied more broadly. Using this modelling strategy, the interaction between stiffness degradation and energy dissipation rate was accounted for numerically. Although the models marginally underestimate the energy dissipation in the case of slender piers, a good agreement was obtained in terms of lateral strength, hysteretic response, and crack pattern. [ABSTRACT FROM AUTHOR]

Summary: Failure of masonry structures generally occurs via specific collapse mechanisms which have been well documented. Using rocking dynamics, equations of motion have been derived for a number of different failure mechanisms ranging from the simple overturning of a single block to more complicated mechanisms. However, most of the equations of motion derived thus far assume that the structures can be modelled as rigid bodies rocking on rigid interfaces with an infinite compressive strength—which is not always the case. In fact, crushing of masonry—commonly observed in larger scale constructions and vertically restrained walls—can lead to a reduction in the dynamic capacity of these structures. This paper rederives the rocking equation of motion to account for the influence of flexible interfaces, characterized by a specific interface stiffness as well as finite compressive strength. The interface now includes a continually shifting rotation point, the location of which depends not only on the material properties of the interface but also on its geometry. Expressions have thus also been derived for interfaces of different geometries, and parametric studies conducted to gauge their influence on dynamic response. The new interface formulations are also implemented within a new analytical modelling tool that provides a novel approach to the dynamic analysis of masonry collapse mechanisms. Finally, this tool is exemplified, along with the importance of the interface formulation, by evaluating the collapse of the Dharahara Tower in Kathmandu, which was almost completely destroyed during the 2015 Gorkha earthquake. [ABSTRACT FROM AUTHOR]

Summary: Rocking motion, established in either the superstructure in the form of a 2‐point stepping mechanism (structural rocking) or resulting from rotational motion of the foundation on the soil (foundation rocking), is considered an effective, low‐cost base isolation technique. This paper unifies for the first time the 2 types of rocking motion under a common experimental campaign, so that on the one hand, structural rocking can be examined under the influence of soil and on the other, foundation rocking can be examined under the influence of a linear elastic superstructure. Two building models, designed to rock above or below their foundation level so that they can reproduce structural and foundation rocking respectively, were tested side by side in a centrifuge. The models were placed on a dry sandbed and subjected to a sequence of earthquake motions. The range of rocking amplitude that is required for base isolation was quantified. Overall, it is shown that the relative density of sand does not influence structural rocking, while for foundation rocking, the change from dense to loose sand can affect the time‐frequency response significantly and lead to a more predictable behaviour. [ABSTRACT FROM AUTHOR]

This paper presents a new analytical model for describing the large rocking response of an elastic multi-mass structure resting on ideally rigid ground. Using the experimental results from a rocking steel column, the ability of the proposed analytical model to estimate the rocking and translational acceleration response under free vibration, pulse and earthquake excitations is evaluated. It is observed that the classical treatment of impact may result in an unrealistically large transfer of energy to vibrations. Therefore a new Dirac-delta type impact model that spreads the effects of impact over time and space is proposed. The use of a Dirac-delta model and accurate restitution factors play a pivotal role in prediction of rocking and acceleration responses. In order to characterize the nonlinear response better, a modal analysis of the linearized system is proposed. With this approach, the vibration mode frequencies and shapes during rocking action were determined. A comparison of analytical and experimental modal estimations suggests good agreement. The results emphasize that the vibration characteristics of several vibration modes are affected by rocking action, and these modes may be excited at impact. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY Predicting the rocking response of structures to ground motion is important for assessment of existing structures, which may be vulnerable to uplift and overturning, as well as for designs which employ rocking as a means of seismic isolation. However, the majority of studies utilize a single rocking block to characterize rocking motion. In this paper, a methodology is proposed to derive equivalence between the single rocking block and various rocking mechanisms, yielding a set of fundamental rocking parameters. Specific structures that have exact dynamic equivalence with a single rocking block, are first reviewed. Subsequently, approximate equivalence between single and multiple block mechanisms is achieved through local linearization of the relevant equations of motion. The approximation error associated with linearization is quantified for three essential mechanisms, providing a measure of the confidence with which the proposed methodology can be applied. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY Numerous structures uplift under the influence of strong ground motion. Although many researchers have investigated the effects of base uplift on very stiff (ideally rigid) structures, the rocking response of flexible structures has received less attention. Related practical analysis methods treat these structures with simplified 'equivalent' oscillators without directly addressing the interaction between elasticity and rocking. This paper addresses the fundamental dynamics of flexible rocking structures. The nonlinear equations of motion, derived using a Lagrangian formulation for large rotations, are presented for an idealized structural model. Particular attention is devoted to the transition between successive phases; a physically consistent classical impact framework is utilized alongside an energy approach. The fundamental dynamic properties of the flexible rocking system are compared with those of similar linear elastic oscillators and rigid rocking structures, revealing the distinct characteristics of flexible rocking structures. In particular, parametric analysis is performed to quantify the effect of elasticity on uplift, overturning instability, and harmonic response, from which an uplifted resonance emerges. The contribution of stability and strength to the collapse of flexible rocking structures is discussed. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]