• Structural Engineering and Mechanics
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• v.18 no.3
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• pp.343-362
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• 2004

Rocking response of rigid bodies with rectangular footprint, freely standing on horizontal rigid plane is studied analytically. Bodies are subjected to simulated single component of horizontal earthquakes. The effect of baseline correction, applied to simulated excitations, on the rocking response is first examined. The sensitiveness of rocking motion to the details of earthquakes and geometric properties of rigid bodies is investigated. Due to the demonstrated sensitivity of rocking response to these factors, prediction of rocking stability must be made in the framework of probability theory. Therefore, using a large number of simulated earthquakes, the effects of duration and shape of intensity function of simulated earthquakes on overturning probability of rigid bodies are studied. In the case when a rigid body is placed on any floor of a building, the corresponding probability is compared to that of a body placed on the ground. For this purpose, several shear frames are employed. Finally, the viability of the energy balance equation, which was introduced by Housner in 1963 and widely used by nuclear power industry to estimate the rocking stability of bodies, is evaluated. It is found that the equation is robust. Examples are also given to show how this equation can be used.

• Journal of Mechanical Science and Technology
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• v.17 no.9
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• pp.1249-1260
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• 2003

This present work focuses on the influence of nonlinearities associated with impact on the rocking behavior of a rigid body block subjected to a two-dimensional excitation in the horizontal and vertical directions. The nonlinearities in rocking system are found to be strongly dependent on the impact between the block and the base that abruptly reduces the kinetic energy. In this study, the rocking systems of the two types are considered : The first is an undamped rocking system model that disregards the energy dissipation during the impact and the second is a damped rocking system, which incorporates energy dissipation during the impact. The response analysis is carried out by a numerical method using a non-dimensional rocking equation in which the variations in the excitation levels are considered. Chaos responses are observed over a wide range of parameter values, and particularly in the case of large vertical displacements, the chaotic characteristics are observed in the time histories, Poincare sections, the power spectral density and the largest Lyapunov exponents of the rocking responses. Complex behavior characteristics of rocking responses are illustrated by the Poincare sections.

• Structural Engineering and Mechanics
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• v.22 no.3
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• pp.293-310
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• 2006

In seismic prone areas it is possible to meet very different objects (equipment components, on shelf artefacts, simple architectural elements) that can be modelled as a rigid body rocking on a rigid foundation. The interest in their behaviour can have different reasons: seismological, in order to estimate the ground motion intensity, or more strictly mechanical, in order to limit the response severity and to avoid overturning. The behaviour of many rigid bodies subjected to twenty wide ranging acceleration recordings is studied here. The response of the blocks is described using kinematic and energy parameters. A condition under which a so called scale effect is tangible is highlighted. The capacity of the signals to produce overturning is compared to different ground motion parameters, and a good correlation with the Peak Ground Velocity is unveiled.

• Structural Engineering and Mechanics
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• v.29 no.4
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• pp.433-453
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• 2008

The equation of motion of a one way (vertical) spanning strip wall, as an assembly of two rigid bodies, is presented. Only one degree of freedom is needed to completely describe the wall response as the bodies are assumed to be perfectly rectangular and are allowed to rock but not to slide horizontally. Furthermore, no arching action occurs since vertical motion of the upper body is not restrained. Consequently, the equation of motion is nonlinear, with non constant coefficients and a Coriolis acceleration term. Phenomena associated with overburden to self weight ratio, motion triggering, impulsive energy dissipation, amplitude dependency of damping and period of vibration, and scale effect are discussed, contributing to a more complete understanding of experimental observations and to an estimation of system parameters based on the wall characteristics, such as intermediate hinge height and energy damping, necessary to perform nonlinear time history analyses. A comparison to a simple standing, or parapet, wall is developed in order to better highlight the characteristics of this assembly.

• Structural Engineering and Mechanics
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• v.54 no.3
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• pp.491-500
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• 2015

Under seismic loading, underground station structures behave differently from above ground structures. Underground structures do not require designated energy dissipation system for seismic loads. These structures are traditionally designed with shear or racking deformation capacity to accommodate the movement of the soil caused by shear waves. The free-field shear deformation method may not be suitable for the design of shallowly buried station structures with complex structural configurations. Alternatively, a station structure can develop rocking mechanisms either as a whole rigid body or as a portion of the structure with plastic hinges. With a rocking mechanism, station structures can be tilted to accommodate lateral shear deformation from the soil. If required, plastic hinges can be implemented to develop rocking mechanism. Generally, rocking structures do not expect significant seismic loads from surrounding soils, although the mechanism may result in significant internal forces and localized soil bearing pressures. This method may produce a reliable and robust design of station structures.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 1997.10a
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• pp.279-286
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• 1997

The objective of this paper is to investigate the effects of the rocking motion and the vertical natural frequency of a seismically isolated structure using LRBs. The governing equations of motion of the rigid body structure, which consider a large rocking motion, are derived. For the mathematical model of LRB, the horizontal stiffness equation based on Haringx's theory is used. From this paper, as decreasing the vertical natural frequency, the rocking responses increase and the horizontal isolation frequency and its earthquake responses are severely affected.

• Earthquakes and Structures
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• v.20 no.1
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• pp.71-86
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• 2021

This paper focuses on the development and assessment of the expected damage for the rocking response of rigid anchored blocks, with irregular geometry and non-uniform mass distribution, considering the site conditions and the seismicity of Mexico City. The non-linear behavior of the restrainers is incorporated to evaluate the pure tension and tension-shear failure mechanisms. A probabilistic framework is performed covering a wide range of block sizes, slenderness ratios and eccentricities using physics-based ground motion simulation. In order to incorporate the uncertainties related to the propagation of far-field earthquakes with a significant contribution to the seismic hazard at study sites, it was simulated a set of scenarios using a stochastic summation methods of small-earthquakes records, considered as Empirical Green's Function (EGFs). As Engineering Demand Parameter (EDP), the absolute value of the maximum block rotation normalized by the body slenderness, as a function of the peak ground acceleration (PGA) is adopted. The results show that anchorages are more efficient for blocks with slenderness ratio between two and three, while slenderness above four provide a better stability when they are not restrained. Besides, there is a range of peak intensities where anchored blocks located in soft soils are less vulnerable with respect to those located in firm soils. The procedure used in here allows to take decisions about risk, reliability and resilience assessment of different types of contents, and it is easily adaptable to other seismic environments.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.33 no.11
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• pp.1288-1293
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• 2009

This paper discusses about modeling method to simulate a nonlinear behavior like sliding or rocking of two stacked body system under earthquake condition. A double body system design can be an option to reduce seismic response of a component in comparison to a single body system for free standing structures. Therefore, according to the priority of components, the structure is to be designed by proper ratio of partition in their height for improvement of seismic capability and structural integrity. Nonlinear modeling and analysis using simple rigid body and dynamic system has been performed to check the trend in such cases. As a result, one of the two bodies can be chosen to reduce the seismic response from energy absorption of the other one by appropriate application of friction ratios not only in slip-slip condition but in slip-rock condition.

• Earthquakes and Structures
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• v.9 no.1
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• pp.127-143
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• 2015

In conventional seismic design, structures are assumed to be fixed at the base. To reduce the impact of earthquake loading, while at the same time providing an economically feasible structure, minor damage is tolerated in the form of controlled plastic hinging at predefined locations in the structure. Uplift is traditionally not permitted because of concerns that it would lead to collapse. However, observations of damage to structures that have been through major earthquakes reveal that partial and temporary uplift of structures can be beneficial in many cases. Allowing a structure to move as a rigid body is in fact one way to limit activated seismic forces that could lead to severe inelastic deformations. To further reduce the induced seismic energy, slip-friction connectors could be installed to act both as hold-downs resisting overturning and as contributors to structural damping. This paper reviews recent research on the concept, with a focus on timber shear walls. A novel approach used to achieve the desired sliding threshold in the slip-friction connectors is described. The wall uplifts when this threshold is reached, thereby imparting ductility to the structure. To resist base shear an innovative shear key was developed. Recent research confirms that the proposed system of timber wall, shear key, and slip-friction connectors, are feasible as a ductile and low-damage structural solution. Additional numerical studies explore the interaction between vertical load and slip-friction connector strength, and how this influences both the energy dissipation and self-centring capabilities of the rocking structure.

• Journal of the Earthquake Engineering Society of Korea
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• v.23 no.1
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• pp.19-29
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• 2019

Recent earthquakes in Korea caused some damages to stone pagodas and thereby awakened the importance of earthquake preparedness. Korean stone pagodas which have been built with very creative style of material use and construction method are worthy of world heritage. Each stone pagoda consists of three parts: top; body; and base. However each tower is uniquely defined by its own features, which makes it more difficult to generalize the seismic assessment method for stone pagodas. This study has focused on qualitative preliminary evaluation of stone pagodas that enables us to compare the relative seismic performance across major aspects among many various Korean pagodas. Specifically an analytical model for multi-block stone pagodas is to be proposed upon the investigation of structural characteristics of stone pagoda and their dynamic behavior. A strategy for seismic evaluation of heritage stone pagodas is to be established and major evaluation factors appropriate for the qualitative evaluation are identified. The evaluation factors for overall seismic resisting behavior of stone pagodas are selected based on the dynamic motions of a rigid block and its limit state. Numerical simulation analysis using discrete element method is performed to analyze the sensitivity of each factor to earthquake and discuss some effects on seismic performance.