Chattering: an overlooked peculiarity of rocking motion

Complete chattering occurs when a structure undergoes a theoretically infinite sequence of impacts in finite time, that eventually bring the structure to the state of persistent (continuous) contact. This study investigates the conditions under which a rigid rocking block undergoes complete chattering when subjected to sinusoidal ground excitation. The analysis explains how the acceleration amplitude of the ground excitation affects the chattering time. It also proves that there exists a (sinusoidal) ground acceleration amplitude, below which rocking motion terminates even under a nonzero ground excitation, almost independently of the frequency of the ground excitation. Furthermore, the study adopts perturbation theory and proposes an asymptotic approximation of the time needed for chattering to be completed, i.e. chattering time. It then verifies the asymptotic approximation using an independent semi-analytical approach. Overall, the results highlight the importance of complete chattering on the dynamic rocking response; a feature of nonlinear dynamics which is often overlooked in earthquake engineering.This study has been funded by the STAND4-HERITAGE project (new STANDards FOR seismic assessment of built cultural HERITAGE) that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant No. 833123) as an Advanced Grant. Its support is gratefully acknowledged. The opinions and conclusions presented in this paper are those of the authors and do not necessarily reflect the views of the sponsoring organization

Dynamic simulation of one-sided rocking masonry façades using an energy-consistent viscous damping model

Unreinforced masonry façades are specifically vulnerable to seismic actions. Their weak connectivity with adjacent structural members results in their detachment during an earthquake, thus, forming local collapse mechanisms which exhibit one-sided rocking motion. Such mechanisms can accommodate considerable displacements before collapsing/overturning. Hence, their dynamic stability is of great interest. The dynamic response of such collapse mechanisms has been investigated using the classical rocking theory. This is a reliable and fast model that efficiently simulates the dynamic response and energy losses of rocking structures, yet limited to simple structural configurations. As the problem’s complexity increases (e.g. degrees of freedom, boundary conditions, and/or material nonlinearities) numerical modelling of such structures has been recently gaining momentum. However, despite the great advances of such numerical modelling techniques, simulation of energy losses still remains challenging. The present work proposes a novel numerical block-based model that efficiently simulates energy losses during one-sided rocking motion. Specifically, an equivalent viscous damping model is adopted and calibrated in a phenomenological fashion after the classical rocking theory. Importantly, the unilateral dashpot formulation of the proposed viscous damping model allows for an accurate replication of the impulsive nature of impacts. Ready-to-use predictive equations are presented, which are also validated against experimental results from literature

Experimental characterisation of dry-joint masonry structures: Interface stiffness and interface damping

The accurate description of the dynamics of dry-joint masonry structures strongly relies on the characterisation of the interaction at the units’ interfaces. Several experimental techniques are available for estimating the mechanical properties of the interface (i.e. stiffness and damping), yet, their reliability remains questionable given the lack of comprehensive comparative studies. This work presents an extensive experimental campaign on the meso-scale mechanics of dry-joint interfaces and quantifies both the interface stiffness and interface damping. Importantly, this paper reveals, for the first time, remarkable agreement of the interface stiffness estimated by inherently different experimental methods, namely deformation-based and vibration-based. Thus, it paves the way for the formulation of reliable constitutive laws that govern structural response in numerical modelling of dry-joint masonry structures

A multi-level approach to predict the seismic response of rigid rocking structures using artificial neural networks

This paper explores the use of Artificial Neural Networks (ANN) for the rocking problem. The paper adopts rigid rocking blocks of different sizes and slenderness, which undergo rocking motion without sliding and bouncing when subjected to recorded earthquakes. This research focuses on the cases where the blocks overturn or safely return to their initial (rest) position at the end of the groundshaking. An ANN model is trained to efficiently categorise the response into overturning or safe rocking using the structural parameters, ground motion characteristics, and the coefficient of restitution as input. The results show the substantial contribution of velocity and frequency characteristics of the ground motion to overturning. In addition, ANN is used to predict the response amplitude and identify the most critical input variables that govern safe rocking. Theanalysisrevealsthatrockingamplitudeisgovernedbyacombinationofduration, frequency, and intensity characteristics of the ground excitation. Interestingly, the maximum incremental velocity (MIV), a novel intensity measure for the rocking literature, shows a substantial correlation with the rocking amplitude. In this context, this paper proposes closed-form expressions using the most influential input variables to provide a quick, yet adequately accurate, response prediction. Finally, this study pays special attention to the contribution of the coefficient of restitution, which, in general, is less critical to the peak safe rocking response, while it becomes more important to the overturning response.This study has been partly funded by the STAND4HERITAGE project (new STANDards FOR seismic assessment of built cultural HERITAGE) that has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (Grant No. 833123) as an Advanced Grant. This work was also partly financed by FCT/MCTES through national funds (PIDDAC) under the R&D Unit Institute for Sustainability and Innovation in Structural Engineering (ISISE), under reference UIDB/04029/2020 (doi.org/10.54499/UIDB/04029/2020), and under the Associate Laboratory Advanced Production and Intelligent Systems ARISE under reference LA/P/0112/2020. The opinions and conclusions presented in this paper are those of the authors and do not necessarily reflect the views of the sponsoring organisations

Energy loss mechanisms of rocking blocks: experimental observations

A variety of different structures experience rocking motion when subjected to dynamic actions, making rocking dynamics a fundamental problem of earthquake engineering. Rocking motion presents peculiar dynamic characteristics, such as negative stiffness during pivoting and non-smooth phenomena during impacts. Hence, modelling of the rocking problem faces significant challenges. One of the most significant is related to the energy losses that occur during impacts, commonly represented by the coefficient of restitution. Despite the numerous theoretical attempts to accurately estimate the coefficient of restitution, it is apparent that experimental observations are essential in providing a direct insight into the complex and non-smooth phenomena of rocking motion. To this end, the present work conducts an extended experimental campaign on the free-rocking motion of limestone blocks. More specifically, a total of 36 blocks are tested, corresponding to 12 different geometrical aspect ratios. The free-rocking motion is thoroughly analysed, while attention is also given to three-dimensional effects. Finally, the coefficient of restitution is experimentally quantified and compared with both previous theoretical and experimental results gathered from the literature.- (undefined

Macro vs Micro Limit Analysis models for the seismic assessment of in-plane masonry walls made with quasi-periodic bond types

Masonry bond patterns can considerably affect the seismic performance of in-plane walls. Although several numerical and experimental works addressed this topic, few attempts tried to investigate such an issue using analytical formulations. This paper aims to compare macro and micro limit analysis models investigating masonry walls arranged with different bond types, namely Running, Flemish and English. A dataset involving 81 combinations is generated by varying geometrical (panel aspect ratio, block aspect ratio, bond type) and mechanical (friction coefficient) parameters. Finally, one-way and two-way factor interactions are used to evaluate how each parameter affects the horizontal load multiplier and assess matching among the two adopted formulations.This work was partly financed by FCT/MCTES through national funds (PIDDAC) under the R&D Unit ISISE under reference UIDB/04029/2020. This study has been partly funded by the STAND4HERITAGE project that has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No. 833123), as an Advanced Gran

Tangential interface stiffness estimation of dry-joint masonry structures through an extended experimental campaign

Many monumental masonry structures, such as aqueducts and public or military constructions, have been built using regular units neatly dressed without mortar. Detailed numerical modeling is commonly utilized to simulate the behavior of such dry-joint structures, which necessitates the proper definition of various physical and mechanical input parameters to enhance the reliability of the results. Among them, the normal and tangential interface stiffness play a key role in simulating masonry (either mortared or dry-joint) structures. Despite their paramount importance, the existing literature lacks established experimental studies for their characterization, and importantly their comparative validation. To this end, this paper presented an extensive experimental campaign on limestone blocks focusing on the estimation of the tangential interface stiffness. Two intrinsically different methodologies were employed for the tangential interface stiffness description, aiming to obtain reliable and cross-validated results. The first methodology, namely deformation-based, used direct shear-box tests and measured the interface shear deformation upon shear stresses for different levels of normal stress. The second methodology, namely vibration-based, utilized ambient vibration noise to measure the natural frequencies of the dry-stack assembly, which was correlated with the tangential interface stiffness through an inverse dynamic analysis. The dependence of the tangential interface stiffness with respect to the normal stress acting at the interface was discussed and the two methodologies were compared and validated.This work has been partly financed by FCT/MCTES through national funds (PIDDAC) under the R&D Unit Institute for Sustainability and Innovation in Structural Engineering (ISISE), under reference UIDB/04029/2020. This work is financed by national funds through FCT - Foundation for Science and Technology, under grant agreements 2020.07325.BD and PRT/BD/152830/2021 attributed to the first and second authors, respectively. This study has also been partly funded by the STAND4HERITAGE project (new STANDards FOR seismic assessment of built cultural HERITAGE) that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant No. 833123) as an Advanced Grant

A semi-analytical approach to approximate chattering time of rocking structures

This paper investigates the seismic behaviour of a freestanding rigid rocking block when subjected to low amplitude sinusoidal ground excitations. An important scenario in such cases is the complete chattering the block might exhibit. Complete chattering occurs when the block undergoes a theoretically infinite sequence of decaying impacts that converge to the state of persistent (continuous) contact in finite time, even under a nonzero ground excitation. This study proposes a semi-analytical approach that approximates the (finite) time required for this to happen, i.e. chattering time. Specifically, this paper provides a detailed description of the semi-analytical scheme and shows the influence of the amplitude of the ground acceleration on the approximation of the chattering time. Importantly, the proposed scheme is based on the realisation that, during chattering, and after a sufficiently large number of impacts, the ratio of the time-intervals of every two consecutive impacts becomes constant and equal to the square of the coefficient of restitution. The proposed semi-analytical approach efficiently approximates chattering time providing a state-of-the-art mathematical formulation of the chattering phenomenon for the rocking problem.- (undefined

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

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.This study has been funded by the STAND4HERITAGE project (New Standards for Seismic Assessment of Built Cultural Heritage) that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant No. 833123) as an Advanced Gran