Seismic protection of rocking structures with inerters

The seismic behaviour of a wide variety of structures can be characterized by the rocking response of rigid blocks. Nevertheless, suitable seismic control strategies are presently limited and consist mostly on preventing rocking motion all together, which may induce undesirable stress concentrations and lead to impractical interventions. In this paper, we investigate the potential advantages of using supplemental rotational inertia to mitigate the effects of earthquakes on rocking structures. The newly proposed strategy employs inerters, which are mechanical devices that develop resisting forces proportional to the relative acceleration between their terminals and can be combined with a clutch to ensure their rotational inertia is only employed to oppose the motion. We demonstrate that the inclusion of the inerter effectively reduces the frequency parameter of the block, resulting in lower rotation seismic demands and enhanced stability due to the well-known size effects of the rocking behaviour. The effects of the inerter and inerter-clutch devices on the response scaling and similarity are also studied. An examination of their overturning fragility functions reveals that inerter-equipped structures experience reduced probabilities of overturning in comparison with uncontrolled bodies, while the addition of a clutch further improves their seismic stability. The concept advanced in this paper is particularly attractive for the protection of rocking bodies as it opens the possibility of nonlocally modifying the dynamic response of rocking structures without altering their geometry.     

远场长周期地震下基于规范设计隔震结构的抗震性能研究

隔震结构具有较长自振周期,且容易受地震动长周期特性的影响,因此其在长周期地震动作用下的抗震性能值得研究。以某基于规范设计的基础隔震结构为例,通过对该结构在规范规定地震作用和远场长周期地震作用下的地震响应进行分析和对比,研究长周期地震动对结构地震响应的影响;通过对钢筋和混凝土的损伤状态进行定义和标识,探讨长周期地震作用下基础隔震结构的损伤分布规律。研究结果表明,长周期地震动作用下隔震结构发生破坏的概率远大于具有相同峰值地面加速度的普通地震动,其中长周期地震动反应谱的谱峰值"后移"被认为是造成这种情况的主要原因,且长周期地震动作用下隔震结构的损伤分布并不均匀,其主要集中在结构的底层。     

Seismic control of flexible rocking structures using inerters

Allowing flexible structures to uplift and rock during earthquakes can significantly reduce the force demands and residual displacements. However, such structures are still susceptible to large deformations and accelerations that can compromise their functionality. In this paper, we examine the dynamic response of elastic rocking oscillators and suggest that their lateral drifts and accelerations can be limited effectively by using inerter devices. To this end, we offer a detailed examination of the effects of structural flexibility on the efficiency of the proposed system. The analytical expressions governing the motion of deformable structures with base uplift are revisited to incorporate the effects of the supplemental rotational inertia. The proposed model is then used to study the structural demands of flexible rocking structures under coherent pulses as well as noncoherent real pulse-like ground motions. Our results show that combining rocking with inerters can be an efficient strategy to control the deformation and acceleration demands in uplifting flexible systems.     

A finite element model for seismic response analysis of deformable rocking frames

A new finite element model to analyze the seismic response of deformable rocking bodies and rocking structures is presented. The model comprises a set of beam elements to represent the rocking body and zero‐length fiber cross‐section elements at the ends of the rocking body to represent the rocking surfaces. The energy dissipation during rocking motion is modeled using a Hilber–Hughes–Taylor numerically dissipative time step integration scheme. The model is verified through correct prediction of the horizontal and vertical displacements of a rigid rocking block and validated against the analytical Housner model solution for the rocking response of rigid bodies subjected to ground motion excitation. The proposed model is augmented by a dissipative model of the ground under the rocking surface to facilitate modeling of the rocking response of deformable bodies and structures. The augmented model is used to compute the overturning and uplift rocking response spectra for a deformable rocking frame structure to symmetric and anti‐symmetric Ricker pulse ground motion excitation. It is found that the deformability of the columns of a rocking frame does not jeopardize its stability under Ricker pulse ground motion excitation. In fact, there are cases where a deformable rocking frame is more stable than its rigid counterpart. Copyright © 2016 John Wiley & Sons, Ltd.     

The three‐dimensional behavior of inverted pendulum cylindrical structures during earthquakes

In order to use rocking as a seismic response modification strategy along both directions of seismic excitation, a three‐dimensional (3D) rocking model should be developed. Since stepping or rolling rocking structural members out of their initial position is not a desirable performance, a rocking design should not involve these modes of motion. To this end, a model that takes the aforementioned constraint into account needs to be developed. This paper examines the 3D motion of a bounded rigid cylinder that is allowed to uplift and sustain rocking and wobbling (unsteady rolling) motion without sliding or rolling out of its initial position (i.e., a 3D inverted pendulum). Thus, the cylinder is constrained to zero residual displacement at the end of its 3D motion. This 3D dynamic model of the rocking rigid cylinder has two DOFs (three when damping is included), making it the simplest 3D extension of Housner's classical two‐dimensional (2D) rocking model. The development of models with and without damping is presented first. They are simple enough to perform extensive parametric analyses. Modes of motion of the cylinder are identified and presented. Then, 3D rocking and wobbling earthquake response spectra are constructed and compared with the classical 2D rocking earthquake response spectra. The 3D bounded rocking earthquake response spectra for the ground motions considered seem to have a very simple linear form. Finally, it is shown that the use of a 2D rocking model may lead to unacceptably unconservative estimates of the 3D rocking and wobbling seismic response. Copyright © 2017 John Wiley & Sons, Ltd.     

Wavelet‐based non‐stationary seismic rocking response of flexibly supported tanks

The non‐stationary rocking response of liquid storage tanks under seismic base excitations including soil interaction has been developed based on the wavelet domain random vibration theory. The ground motion has been characterized through statistical functionals of wavelet coefficients of the ground acceleration history. The tank–liquid–foundation system is modelled as a multi‐degree‐of‐freedom (MDOF) system with both lateral and rocking motions of vibration of the foundation. The impulsive and convective modes of vibration of the liquid in the tank have been considered. The wavelet domain coupled dynamic equations are formulated and then solved to get the expressions of instantaneous power spectral density function (PSDF) in terms of functionals of input wavelet coefficients. The moments of the instantaneous PSDF are used to obtain the stochastic responses of the tank in the form of coefficients of hydrodynamic pressure, base shear and overturning base moment for the largest expected peak responses. Parametric variations are carried out to study the effects of various governing parameters like height of liquid in the tank, height–radius ratio of the tank, ratio of total liquid mass to mass of foundation, and shear wave velocity in the soil medium, on the responses of the tank. Copyright © 2003 John Wiley & Sons, Ltd.     

Rocking amplification and strong‐motion duration

This paper characterizes the ability of natural ground motions to induce rocking demands on rigid structures. In particular, focusing on rocking blocks of different size and slenderness subjected to a large number of historic earthquake records, the study unveils the predominant importance of the strong‐motion duration to rocking amplification (ie, peak rocking response without overturning). It proposes original dimensionless intensity measures (IMs), which capture the total duration (or total impulse accordingly) of the time intervals during which the ground motion is capable of triggering rocking motion. The results show that the proposed duration‐based IMs outperform all other examined (intensity, frequency, duration, and/or energy‐based) scalar IMs in terms of both “efficiency” and “sufficiency.” Further, the pertinent probabilistic seismic demand models offer a prediction of the peak rocking demand, which is adequately “universal” and of satisfactory accuracy. Lastly, the analysis shows that an IM that “efficiently” captures rocking amplification is not necessarily an “efficient” IM for predicting rocking overturning, which is dominated by the velocity characteristics (eg, peak velocity) of the ground motion.     

Approximate formulas for rotational effects in earthquake engineering

The paper addresses the issue of researching into the engineering characteristics of rotational strong ground motion components and rotational effects in structural response. In this regard, at first, the acceleration response spectra of rotational components are estimated in terms of translational ones. Next, new methods in order to consider the effects of rotational components in seismic design codes are presented by determining the effective structural parameters in the rotational loading of structures due only to the earthquake rotational components. Numerical results show that according to the frequency content of rotational components, the contribution of the rocking components to the seismic excitation of short period structures can never be ignored. During strong earthquakes, these rotational motions may lead to the unexpected overturning or local structural damages for the low-rise multi-story buildings located on soft soil. The arrangement of lateral-load resisting system in the plan, period, and aspect ratio of the system can severely change the seismic loading of wide symmetric buildings under the earthquake torsional component.     

The rocking response of large flexible structures to earthquakes

The rocking response of structures subjected to strong ground motions is a problem of ‘several scales’. While small structures are sensitive to acceleration pulses acting successively, large structures are more significantly affected by coherent low frequency components of ground motion. As a result, the rocking response of large structures is more stable and orderly, allowing effective isolation from the ground without imminent danger of overturning. This paper aims to characterize and predict the maximum rocking response of large and flexible structures to earthquakes using an idealized structural model. To achieve this, the maximum rocking demand caused by different earthquake records was evaluated using several ground motion intensity measures. Pulse-type records which typically have high peak ground velocity and lower frequency content caused large rocking amplitudes, whereas non-pulse type records caused random rocking motion confined to small rocking amplitudes. Coherent velocity pulses were therefore identified as the primary cause of significant rocking motion. Using a suite of pulse-type ground motions, it was observed that idealized wavelets fitted to velocity pulses can adequately describe the rocking response of large structures. Further, a parametric analysis demonstrates that pulse shape parameters affect the maximum rocking response significantly. Based on these two findings, a probabilistic analysis method is proposed for estimating the maximum rocking demand to pulse-type earthquakes. The dimensionless demand maps, produced using these methods, have predictive power in the near-field provided that pulse period and amplitude can be estimated a priori. Use of this method within a probabilistic seismic demand analysis framework is briefly discussed.     

Dynamic response analysis of solitary flexible rocking bodies: modeling and behavior under pulse‐like ground excitation

Results obtained for rigid structures suggest that rocking can be used as seismic response modification strategy. However, actual structures are not rigid: structural elements where rocking is expected to occur are often slender and flexible. Modeling of the rocking motion and impact of flexible bodies is a challenging task. A non‐linear elastic viscously damped zero‐length spring rocking model, directly usable in conventional finite element software, is presented in this paper. The flexible rocking body is modeled using a conventional beam‐column element with distributed masses. This model is verified by comparing its pulse excitation response to the corresponding analytical solution and validated by overturning analysis of rocking blocks subjected to a recorded ground motion excitation. The rigid rocking block model provides a good approximation of the seismic response of solitary flexible columns designed to uplift when excited by pulse‐like ground motions. Guidance for development of rocking column models in ordinary finite element software is provided. Copyright © 2014 John Wiley & Sons, Ltd.     

Dynamics of rocking podium structures

A rocking podium structure is a class of structures consisting of a superstructure placed on top of a rigid slab supported by free‐standing columns. The free‐standing columns respond to sufficiently strong ground motion excitation by uplifting and rocking. Uplift works as a mechanical fuse that limits the forces transmitted to the superstructure, while rocking enables large lateral displacements. Such ‘soft‐story’ system runs counter to the modern seismic design philosophy but has been used to construct several hundred buildings in countries of the former USSR following Polyakov's rule‐of‐thumb guidelines: (i) that the superstructure behave as a rigid body and (ii) that the maximum lateral displacement of the rocking podium frame be estimated using elastic earthquake displacement response spectra. The objectives of this paper are to present a dynamic model for analysis of the in‐plane seismic response of rocking podium structures and to investigate if Polyakov's rule‐of‐thumb guidelines are adequate for the design of such structures. Examination of the rocking podium structure response to analytical pulse and recorded ground motion excitations shows that the rocking podium structures are stable and that Polyakov's rule‐of‐thumb guidelines produce generally conservative designs. Copyright © 2017 John Wiley & Sons, Ltd.     

Response spectrum analysis for non-classically damped linear system with multiple-support excitations

A new response spectrum method, which is named complex multiple-support response spectrum (CMSRS) method in this article, is developed for seismic analysis of non-classically damped linear system subjected to spatially varying multiple-supported ground motion. The CMSRS method is based on fundamental principles of random vibration theory and properly accounts for the effect of correlation between the support motions as well as between the modal displacement and velocity responses of structure, and provides an reasonable and acceptable estimate of the peak response in term of peak seismic ground motions and response spectra at the support points and the coherency function. Meanwhile, three new cross-correlation coefficients or cross covariance especially for the non-classically damped linear structures with multiple-supports excitations are derived under the same assumptions of the MSRS method of classically damped system. The CMSRS method is examined and compared to the results of time history analyses in two numerical examples of non-classically damped structures in consideration of the coherences of spatially variable ground motion. The results show that for non-classically damped structure, the cross terms representing the cross covariance between the pseudo-static and dynamic component are also quite small just as same as classically damped system. In addition, it is found that the usual way of neglecting all the off-diagonal elements in transformed damping matrix in modal coordinates in order to make the concerned non-classically damped structure to become remaining proportional damping property will bring some errors in the case of subjected to spatially excited inhomogeneous ground motion.     

Seismic responses of bridges with rocking column-foundation: A dimensionless regression analysis

Rocking column-foundation system is a new design concept for bridges that can reduce overall seismic damage, minimize construction and repair time, and achieve lower cost in general. However, such system involves complex dynamic responses due to impacts and highly nonlinear rocking behavior. This study presents a dimensionless regression analysis to estimate the rocking and shaking responses of the flexible column-foundation system under near-fault ground motions. First, the transient drift and rocking responses of the system are solved numerically using previously established analytical models. Subsequently, the peak column drifts and uplift angles are derived as functions of ground motion characteristics and the geometric and dynamic parameters of column-foundation system in regressed dimensionless forms. The proposed response models are further examined by validating against the numerical simulations for several as-built bridge cases. It is shown that the proposed model not only physically quantifies the influences of prominent parameters, but also consistently reflects the complex dynamics of the system. The seismic demands of rocking column-foundation system can be realistically predicted directly from structural and ground motion characteristics. This can significantly benefit the design of bridges incorporating this new design concept.     

A brief theory and computing of seismic ground rotations for structural analyses

The seismic ground rotations are important with respect to spatial structural models, which are sensitive to the wave propagation. The rotational ground motion can lead to significant increasing of structural response, instability and unusual damages of buildings. Currently, the seismic analyses often take into account the rocking and torsion motions separately using artificial accelerograms. We present an exact analytical method, proposed by Nazarov [15] for computing of three rotational accelerograms simultaneously from given translational records. The method is based on spectral representation in the form of Fourier amplitude spectra of seismic waves, corresponding to the given three-component translational accelerogram. The composition, directions and properties of seismic waves are previously determined in the form of a generalized wave model of ground motion. It is supposed that seismic ground motion can be composed by superposition of P, SV, SH- and surface waves. As an example, the dynamic response analysis of 25-story building is presented. Here recorded (low-frequency) and artificial (high-frequency) accelerograms were used; each of them includes three translational and three rotational components. In this structural analysis, we have clarified primarily conditions under which rotational ground motion should be taken into account. Next, we have calculated three rotational components of seismic ground motion. Then they were taken as additional seismic loads components for further seismic analysis of the building. Note, soil–structure interaction (SSI) is not considered in this study. For computing, we use the special software for structural analyses and accelerogram processing (FEA Software STARK ES and Odyssey software, Eurosoft Co., Russia). It was developed and is used in engineering practice in the Central Research Institute of Building Constructions (TsNIISK, Moscow, Russia).     

Verification of precarious rock methodology using shake table tests of rock models

Precariously balanced rocks in seismically active regions are effectively upper-limit strong motion seismoscopes that have been in place for thousands of years. Thus, estimates of the dynamic toppling acceleration of these rocks (through rigid body rocking) can provide constraints on the peak ground accelerations experienced during past earthquakes. We have developed a methodology that uses a two-dimensional numerical code to calculate the dynamic rocking response of precarious rocks to realistic ground acceleration time histories. Statistical analyses of the dynamic response of these rocks to a range of synthetic seismograms, as well as strong motion records, can provide important information about the ground motion attenuation curves and seismic hazard maps. We use shake table tests to investigate the dynamic rocking response of 13 wooden rectangular blocks of various sizes and aspect ratios subjected to realistic seismograms and compare the results with those of numerical tests. Our results indicate good agreement between the shake table and numerical results.     

Numerical study of characteristics of underground blast induced surface ground motion and their effect on above-ground structures. Part II. Effects on structural responses

Studies of structural responses and damage to high-frequency blast motion are very limited. Current practice uses some empirical allowable ground vibration limits in assessing structural performance. These empirical limits overlook the physical parameters that govern structural response and damage, such as the ground motion characteristics and inherent structural properties. This paper studies the response of RC frame structures to numerically simulated underground blast-induced ground motions. The structural response and damage characteristics of frame structures to ground motions of different frequencies are investigated first. The effects of blast ground motion spatial variations and soil–structure interaction on structural responses are also studied. A suitable discrete model that gives accurate response prediction is determined. A damage index defined based on the accumulated plastic hinge rotation is used to predict structural damage level. Numerical results indicated that both the low structural vibration modes (global modes) and the first elemental vibration mode (local) might govern the dynamic structural responses depending on the ground motion frequency and structural response parameters under consideration. Both ground motion spatial variations and soil–structure interaction effects are prominent. Neglecting them might yield inaccurate structural response prediction. The overall structural response and damage are highly ground motion frequency dependent. Numerical results of structural damage are also compared with some test results obtained in a previous study and with code specifications. Discussions on the adequacy of the code allowable ground vibration limits on RC frame structures are also made.     

Rolling and rocking of rigid uplifting structures

Allowing structures to uplift modifies their seismic response; uplifting works as a mechanical fuse and limits the forces transmitted to the superstructure. However, engineers are generally reluctant to construct an unanchored structure because the system could overturn due to lacking redundancy. Using a safety factor for the design of a flat rocking foundation, ie, designing it wider, goes against the main idea of this seismic modification method as the force demand for the structure increases. We propose to extend the flat base of a rocking block with curved extensions to better protect the block from overturning, yet not prevent its uplifting. After investigating the seismic response of such rocking blocks, we extend the study to investigate the seismic response of rolling and rocking frames comprising columns with curved base extensions. The equations of motion are derived, time history analyses are performed, and rocking spectra are constructed. We draw two important conclusions: (a) the response of a class of rocking oscillators with curved base extensions is equivalent to the response of a flat-base rocking oscillators of the same slenderness, yet larger size; (b) the rotation demand on two negative stiffness rocking and rolling oscillators with the same uplifting acceleration and the same size is roughly the same as long as the rocking oscillators are not close to overturning. The above findings can serve as a basis for the rational seismic design of structures supported on rocking columns with curved bases, a system that has been used since the 1960s.     

最不利设计地震动研究

实际记录到的真实地震动在工程结构的抗震研究、分析和设计中往往作为一种施加到结构上使结构振动，直至破坏的地震荷载．如何合理选择真实的地震动记录作为研究结构地震反应的输入，一直是国内外抗震研究和设计中引人关注的重要问题．本文首先提出了最不利设计地震动的概念；然后在收集到的国内外5000余条被认为有重要意义的地震动记录基础上，利用综合估计地震动潜在破坏势的方法，对４种场地类型分别给出了长周期、短周期和中周期结构的国内外最不利设计地震动；最后通过几类不同结构的地震反应分析，初步验证了本文所确定的最不利设计地震动的可靠性和合理性．      

Study on the severest real ground motion for seismic design and analysis

How to select the adequate real strong earthquake ground motion for seismic analysis and design of trucures is an essential problem in earthquake engineering research and practice.In the paper the concept of the severest design ground motion is proposed and a method is developed for comparing the severity of the recorded strong ground motions.By using this method the severest earthquake ground motions are selected out as seismic inputs to the structures to be designed from a database that consists of more than five thousand significant strong ground moton records collected over the world.The selected severest ground motions are very likely to be able to drive the structures to their critical response and thereby result in the highest damage potential.It is noted that for different structures with diffferent predominant natural periods and at different sites where structures are located the severest design ground motions are usually different.Finally.two examples are illustrated to demonstrate the rationality of the concept and the reliability of the selected design motion.     

北京地区中硬场地对地震动加速度反应谱的放大效应研究

当前,合理确定地震动峰值加速度与反应谱特征周期是工程场地地震动参数确定工作的主要内容。本文以北京地区典型中硬场地为研究对象,分析场地条件对不同周期地震动反应谱值的影响。首先,计算不同震级、震中距条件下的基岩地震动加速度反应谱,合成基岩输入地震动时程;再利用110个工程场地的钻孔资料进行土层地震反应计算,分析中硬场地条件对不同输入环境下的地震动加速度反应谱值的放大效应。结果表明,中硬场地对高、中频震动放大效应明显,尤其是对0.2-0.5s周期段地震动加速度反应谱值的放大倍数大多在1.3以上;场地覆盖层厚度变化对不同频段地震动加速度反应谱值的放大倍数所产生的影响是不同的,与场地自振周期的相关性很强;在不同的地震动输入环境下,中硬场地对不同频段地震动加速度反应谱的影响是不同的,这一结论对实际的抗震设防工作具有一定参考价值。