Numerical Simulation of Rigid Blocks Subjected to Rocking Motion

This paper follows two approaches to formulate the equations that describe the free rocking motion of a rigid block in three dimensions. In the first approach, the orientation of the block is parametrized using a 3-1-2 set of Euler angles, {, , } θφψ . Lagrange's equations of motion, where the Euler angles {, , } θφψ serve as generalized coordinates, yield a system χχ χ χ () ( , )

8th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering Methods in Structural Dynamics and Earthquake Engineering, 2021

Masonry structures have been observed to display a high vulnerability to failure under seismic action. This stems from the fact that their structural configurations usually lack adequate connections among the distinct elements, resulting in the formation of local mechanisms experiencing Out-Of-Plane (OOP) collapse. In this context, rocking dynamics has proven to be a valuable methodology for the analysis of masonry walls. Classical rocking theory can provide a fast solution to the dynamic phenomena taking place if simple configurations are examined. Nevertheless, as the degrees of freedom and the boundary conditions increase, the complexity increases, and thus the classical rocking theory becomes impractical. In the meantime, recent developments in computational modelling of masonry structures are gaining significant attraction. This includes block-based models which inherently consider the complexity of the problem and enable the solution to be obtained easily in the discretised spatial and time domains. However, despite their widespread use, applications of such models usually lack a reliable treatment of damping. The present work attempts to bridge the gap between the well-established energy loss of the classical rocking theory and the treatment of damping of block-based computational models. To do so, the dynamics of the problem are reviewed and an equivalent viscous damping model is proposed. A unilateral dashpot formulation allows the replication of the impulsive nature of the energy loss at impact. Afterwards, a calibration methodology is adopted for the practical range of the problem's parameters and a ready-to-use equation is provided, which respects energy equivalence. The performance of the proposed damping model is also evaluated through comparisons with experimental results.

2021

In this paper, the underlying energy flow in rocking dynamics of rigid blocks is utilized to elucidate their response details and overturning conditions. Based merely on the law of the conservation of energy, the exact criteria for a rigid block to overturn in the free vibration regime are established for the first time in the context of both nonlinear and linear theory. That is, if the initial conditions of the free oscillation regime fall within the prescribed stable libration region, then it is a priori determined that the block will not overturn and vice versa. The analytical solution to the nonlinear free rocking problem is also derived, utilizing integration techniques originating from the solution of the nonlinear pendulum. Additionally, pulse-type overturning spectra are constructed easily by numerical integration of the equation of motion only during the pulse duration. If the block does not topple during the forced regime, the angle and angular velocity at the end of the e...

This work deals with the rocking response of a rigid free-standing block under the action of ground motion, as basic reference for the seismic analysis of masonry structures. A new computational strategy for representing the seismic input is suggested in order to define such an amplitude resonance as representing the most disadvantageous conditions for the block. The rocking response to artificial accelerograms is then analysed by means of a simplified equation of motion between two impacts. The coefficient of restitution is assumed as a variable of the problem to account also for other damping effects (e.g. local plastic deformations). A stabilised phase of the motion is identified for which an upper-bound of the maximum rotation angle of the block can be defined in closed form. The results are plotted in a kind of resonance spectra which point out the influence of such meaningful parameters as the coefficient of restitution and the size and slenderness of the block.

Acta Mechanica, 1999

The problem of rocking response of a rigid block in free and forced motion has been studied for a number of technical reasons. Apart from the technical interests, the problem of rigid block rocking is intrinsically of interest from a theoretical point of view. In fact, the problem is highly nonlinear in nature. Aim of this paper is to study the “contact-impact” problem of a rigid block colliding on a frictional base, by means of a numerical simulation, and to compare numerical results with analytical responses known from the literature. The influence of sliding and bouncing on the orbit type and stability is analyzed by a 3-DOF model of the system and by a new refined model of the contact forces between block and base. Furthermore, attention has been paid to two-dimensional free motion of the block with three degrees of freedom. Refined analytical stress-strain relations in either normal and tangential directions with respect to the contact surfaces are formulated which allow to account for (i) up-lifting and hysteretic damping in normal direction, (ii) coupling between shear strength and compression force, friction dissipation and cumulating damage in tangential direction.

III European Conference on Computational Mechanics, 2006

This paper addresses the numerical modeling of masonry walls as rigid blocks. Models based on rigid block assemblies provide a suitable frame work for understanding their dynamic behavior under seismic actions. In this context, the problem is primarily concerned with Rocking Motion dynamics. The numerical tool is based on the Discrete Element Method (DEM) specially effective for the numerical modeling of rigid blocks. Some authors have been used successfully the DEM in the study of block structures. However, they have used experimental test to calibrate their models and to obtain the parameters used in the DEM; because these parameters cannot be obtained directly form the characteristics of the stones. In this context, a new methodology is proposed to find the parameters of the DEM by using the parameters of the classical theory. Special attention regards about the damping factor, since in the DEM a viscous damping is considered, but in reality the damping is due to impulsive forces that they can be considered as a type of Dirac-δ forces. An extensive comparison between numerical and experimental data has been carried out to verified the proposed methodology. Very good agreement between the numerical model and experimental data is achieved.

This paper presents the preliminary results of an experimental investigation on the behaviour of blocks under rigid body rocking. Nine blocks with different aspect ratios were tested with varying initial amplitudes and different materials at the contact interface. Three different materials, namely concrete, timber, and steel were used to construct the base on which the blocks could rock. The rocking characteristics of the blocks were compared to the predictions by Housner's simple rocking model (SRM). Preliminary results show that the rocking response is strongly dependent upon the aspect ratio of the block, in general accordance to SRM. In addition, different materials at the contact interface play an essential role on the block's rocking responses.

The paper deals with the nonlinear rocking response of rigid multi-block columns when their base is subjected to dynamic excitations. Through a wide parametric study, the influence of the friction coefficient which exists in the interface of the rigid multi-block columns when the latter are allowed to slide and rock, is examined. In order to accomplish this task, the study is divided into two major parts. Initially, 2D finite element models-which refer to a single rigid block-, were constructed in the advanced MSC Marc nonlinear finite element analysis software. These models are able to reproduce all the nonlinear phenomena which can be developed during the rocking and sliding motion through the appropriate contact and friction algorithms which are provided by the MSC Marc software. In order to create reliable numerical models, the series of analyses of the first part of the paper targets to the accurate simulation of the response of the free standing rigid column under free rocking and the response of the columns subjected to various types of dynamic excitations in their base, for which experimental results are available in the bibliography [1]. The comparison revealed very good agreement between the experimental and the corresponding numerical ones. In the second part of the paper, similar models of rigid multi-block columns were constructed. For these models a parametric study was carried out regarding the slenderness of the column, the different values of the friction coefficient and the dynamic excitation of their base (long and short sine harmonic excitations). Through this extended study, several conclusions were drawn concerning the stability of the multi-block rigid columns and how the latter is influenced by the friction coefficient when it is related to the slenderness and the dynamic excitation of the column.

Engineering Structures, 2018

In recent years only a very small number of studies analysed the use of active or semi-active control methods in order to avoid the overturning of rigid block-like structures under base excitation. In this paper, an active control algorithm, based on the pole placement method, is developed for rigid blocks starting from a description of the rocking motion through linearised equations. Such an approach is an excellent approximation for slender rigid blocks for which linearised equations can fully describe the rocking motion due to the smallness of the inclination angle. The control method has two objective: the first one is related to the possibility of vanishing the external excitation at each instant, the second one is to make the rest position of the block a stable point. The first analyses revealed the good robustness of the control algorithm to a variation of the sampling time and of the time-delay of the real control devices. Furthermore, the further parametric analyses pointed out the good effectiveness of the proposed control algorithm both in the reduction of the rocking angle and in the protection from the overturning, also in the case of blocks with low slenderness. Overturning spectra are obtained in the case of rigid block with and without active control.

Earthquake Engineering & Structural Dynamics, 1980

This investigation deals with the rocking response of rigid blocks subjected to earthquake ground motion. A numerical procedure and computer program are developed to solve the non‐linear equations of motion governing the rocking motion of rigid blocks on a rigid base subjected to horizontal and vertical ground motion.The response results presented show that the response of the block is very sensitive to small changes in its size and slenderness ratio and to the details of ground motion. Systematic trends are not apparent: The stability of a block subjected to a particular ground motion does not necessarily increase monotonically with increasing size or decreasing slenderness ratio. Overturning of a block by a ground motion of particular intensity does not imply that the block will necessarily overturn under the action of more intense ground motion.In contrast, systematic trends are observed when the problem is studied from a probabilistic point of view with the ground motion modelle...