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An innovative self-centering hybrid rocking braced steel frame (SCHRBF) with separate braced span and rocking span is proposed for improving seismic resilience. The braced span utilizes buckling-restrained braces (BRBs) to provide energy-dissipating capacity, and the rocking span consists of stiff rocking cores and self-centering braces (SCBs) to achieve a uniform inter-story drift distribution and low post-earthquake residual displacement under a strong earthquake. This study first describes the basic composition and nonlinear mechanical behavior of this novel system. Then, a force-based seismic design procedure for the SCHRBF system is proposed, including the determination and allocation of base shear, design of BRBs in braced span, design of SCBs in rocking span, and design of rocking core members. The influence of key design parameters on the seismic responses of the system is then explored through parametric analysis. And recommended values of the design parameters are provided according to the analysis results. Although the properly designed structure has significant partial re-centering behavior, its peak inter-story drifts, residual inter-story drifts, and deformation patterns can be effectively controlled under strong earthquakes. Finally, the superiority of the SCHRBF in controlling seismic displacement responses is verified by comparing with other structural systems.

In this paper, the exact frequency and mode shape expressions are derived for universal bellows type of expansion joint in lateral and rocking modes of vibration. The effect of equivalent support stiffness and mass on the natural frequencies and mode shapes are studied in detail and the results for a range of non-dimensional parameters are presented in graphical forms which should be useful for piping and bellow designers.

This paper deals with the rocking and sliding behavior of a monolithic stone block or a system of stone blocks under earthquakes, a problem commonly observed for ancient temples in Greece and Southern Italy. The analysis for the above monolithic or multi-drum columns is conducted by a simple process based on generally accepted simplifications. The effects of column geometry, earthquake characteristics and restitution ratio due to impact are also studied herein. Furthermore, an analytical approach for the solution of the complete nonlinear equations of motion (including the one for the vertical earthquake excitation) for the subject considered is proposed. Finally, characteristic representative examples are presented with useful conclusions drawn. It was found that the stability criteria based on static conditions are reliable, while the corresponding dynamic criteria may lead to erroneous results.

This paper investigates the rocking behavior of two free-standing columns capped with a freely supported superstructure in an effort to explain the seismic stability of Tang-Song timber frames subjected to pulse-type excitations. The geometric characteristics of the column determine the acceleration amplitude needed to create uplift of the rocking frame, which also affect the loss of energy during impact along with the mass ratio of the superstructure to the column. Two distinct rocking modes are analyzed according to whether the frame can be limited by the tenon-mortise joint. The expression that yields the minimum acceleration required to overturn the frame is derived within the limits of the linear approximation of the governing equation of motion. Different safety and overturning regions plotted on the stability diagrams indicates that the partially joint limitation may weaken the seismic behavior. The entirely limited frame will rock within the fixed amplitude caused by the joint gap and never topple down. Probability analysis results of the entirely limited frame suggest that a great majority of this type of traditional timber structures have excellent seismic capacity to resist the horizontal excitation.

A system of stacked rigid blocks can be found in many applications of non-structural components: a statue resting on the top of a table is such an example. Previous studies usually assume that the friction at contact interface is so large that only rocking motion can be activated. However, this assumption may not be realistic when assessing the seismic response of unanchored non-structural components. Motivated from this constraint, this paper contributes to the state-of-the-art research of the classical rocking problem by presenting a numerical model within which sliding, rocking and sliding-rocking response of stacked rigid blocks can be computed by the time-history analysis. The exact fully nonlinear equations of motion, transition criteria for different response modes and treatment to handle the impact are presented in detail. The accuracy of the developed model is validated. A case study is also provided to investigate the overall failure probability of the stacked rigid blocks with realistic friction coefficient. In this particular case study, it is also shown that increasing the friction coefficient makes the stacked rigid blocks more susceptible to failure.

Structure, equipment, statue or storage cask may experience free-standing rocking on slopes because of the nonuniform settlement of the base, construction error or foundation failure. The free-standing rigid block on a slope subjected to one-sine pulse excitation and earthquake motions is examined in this paper. First, considering the free-standing rigid block on a slope, the overturning acceleration spectrum under one-sine pulse is built. The spectrum covers two overturning modes, i.e. overturning with one impact either before or after the excitation and overturning without impact. Then the influences of the slope on the overturning acceleration spectrum are discussed in depth. The analytical solution is compared with the numerical solution based on both linear and nonlinear formulations. It reveals that the safe region of the minimum overturning acceleration spectrum between mode 1 and mode 2 depends on the angle of the slope. Further, the block on a slope is subjected to the Tianjin, Kobe and Northbridge earthquake motions. By simplifying the main pulse of the motions to one-sine pulse, although the overturning acceleration spectrum fails to predict the overturning acceleration of the block subjected to real earthquake motions, it can provide some guidelines to predict the practical performance of the rocking structures.

The relatively uncontrolled dynamic behavior of a rigid block resting on a flat surface under ground motion has been well studied. In contrast, a block with a gently inclined V- or W-shaped sliding key would ensure dynamic self-centering or resetting performance under various seismic conditions. To better understand the nonlinear responses of rigid blocks having this type of interface where both rocking and sliding movements are possible, an event-based algorithm is put forward to cover various types of motions, including the transition stages between different types of motions. A comprehensive study is carried out to examine the dynamic responses of blocks of different sizes and aspect ratios. Moreover, the simplified pure rocking model and pure sliding model are also adopted to obtain rough estimates of dynamic response for comparison. The results show that the simplified models are reliable for some special cases only, e.g. squat blocks and slender blocks. The study also evaluated the influence of the coefficient of friction and slope inclination on the resetting capabilities of typical sliding-prone and rocking-prone systems. For those cases that are prone to toppling or excessive sliding under strong earthquakes, the provision of vertical post-tensioning can effectively ensure stability and resetting performance. Such findings provide insight into the performance of precast segmental bridge columns with resettable sliding joints.

It has been observed that after some earthquakes a number of structures resting on spread footings responded to seismic excitation by rocking on their foundation and in some cases this enabled them to avoid failure. Through application to a standard bridge supported by direct foundations, this paper discusses the major differences in response when foundation uplift is taken into consideration. Special focus is given on the modifications of rocking response under biaxial and triaxial excitation with respect to uniaxial excitation. It is found that inelastic rocking has a significant isolation effect. It is also shown that this effect increases under biaxial excitation while it is less sensitive to the vertical component of the earthquake. Finally, parametric analyses show that the isolation effect of foundation rocking increases as the size of the footing and the yield strength of the underlying soil decreases.

The displacement-based modelling methodology which has been applied extensively to buildings and bridges is extended herein to model the over-turning behaviour of rigid free-standing objects. The acceleration-displacement relationship associated with the overturning motion is linearised in order that the maximum displacement experienced by the object can be estimated using the elastic displacement response spectrum of the building floor. Whilst overturning motion is characterised by highly nonlinear acceleration-displacement properties, it was observed that modelling errors arising from nonlinear behaviour can be effectively controlled through limiting the maximum displacement of the object to some 50% of the ultimate displacement for overturning. The 50% safety margin is one of the key features in the proposed model.

Three rigid rectangular objects with depths of 100 mm, 300 mm and 500 mm were used initially to illustrate the use of the model. The height of these objects was 0.5 m, 1.5 m and 2.5 m respectively in order that every object has a common aspect ratio of 1:5. Despite that the aspect ratios of the objects were the same, they have very different levels of vulnerability to overturning. The proposed model was evaluated by nonlinear time-history analyses involving pulse-type excitations, recorded earthquake excitations and computer simulated earthquake excitations. Linear elastic models of buildings have also been used to simulate floor motions at the upper levels in the building. Predictions using the proposed linearised model based on the use of elastic response spectrum of the building floor was found to be very consistent with results obtained from nonlinear time-history analyses. Sufficient verification analyses have been carried out to provide the initial indications that the proposed linearised model seems to work well despite its simplicity.