Planar rocking response and stability analysis of an array of free‐standing columns capped with a freely supported rigid beam

This paper investigates the planar rocking response of an array of free‐standing columns capped with a freely supported rigid beam in an effort to explain the appreciable seismic stability of ancient free‐standing columns that support heavy epistyles together with the even heavier frieze atop. Following a variational formulation, the paper concludes to the remarkable result that the dynamic rocking response of an array of free‐standing columns capped with a rigid beam is identical to the rocking response of a single free‐standing column with the same slenderness yet with larger size, that is a more stable configuration. Most importantly, the study shows that the heavier the freely supported cap beam is (epistyles with frieze atop), the more stable is the rocking frame regardless of the rise of the center of gravity of the cap beam, concluding that top‐heavy rocking frames are more stable than when they are top light. This ‘counter intuitive’ finding renders rocking isolation a most attractive alternative for the seismic protection of bridges with tall piers, whereas its potential implementation shall remove several of the concerns associated with the seismic connections of prefabricated bridges. Copyright © 2012 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.     

Seismic response analysis of multidrum classical columns

This paper presents a numerical investigation on the seismic response of multidrum classical columns. The motivation for this study originates from the need to understand: (a) the level of ground shaking that classical multidrum columns can survive, and (b) the possible advantages or disadvantages of retrofitting multidrum columns with metallic shear links that replace the wooden poles that were installed in ancient times. The numerical study presented in this paper is conducted with the commercially available software Working Model 2D?, which can capture with fidelity the sliding, rocking, and slide‐rocking response of rigid‐body assemblies. This paper validates the software Working Model by comparing selected computed responses with scarce analytical solutions and the results from in‐house numerical codes initially developed at the University of California, Berkeley, to study the seismic response of electrical transformers and heavy laboratory equipment. The study reveals that relative sliding between drums happens even when the g‐value of the ground acceleration is less than the coefficient of friction, µ, of the sliding interfaces and concludes that: (a) typical multidrum classical columns can survive the ground shaking from strong ground motions recorded near the causative faults of earthquakes with magnitudes M_w=6.0–7.4; (b) in most cases multidrum classical columns free to dislocate at the drum interfaces exhibit more controlled seismic response than the monolithic columns with same size and slenderness; (c) the shear strength of the wooden poles has a marginal effect on the sliding response of the drums; and (d) stiff metallic shear links in‐between column drums may have an undesirable role on the seismic stability of classical columns and should be avoided. Copyright © 2005 John Wiley & Sons, Ltd.     

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.     

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.     

Seismic isolation of rigid cylindrical tanks using friction pendulum bearings

Storage tanks are vulnerable to earthquakes, as numerous major earthquakes have demonstrated. The trend of recent revisions to make seismic design criteria for large‐scale industrial storage tanks increasingly stringent has made development of cost‐effective earthquake‐resistant design and retrofit techniques for industrial tanks imperative. This study assesses the feasibility of seismic base isolation for making liquid‐filled storage tanks earthquake resistant. The sliding‐type friction pendulum seismic (FPS) bearings are considered rather than the elastomeric bearings because the dynamic characteristics of an FPS‐isolated tank remain unchanged regardless of the storage level. This work has devised a hybrid structural‐hydrodynamic model and solution algorithm, which would permit simple, accurate and efficient assessment of the seismic response of rigid cylindrical storage tanks in the context of seismic isolation. Extensive numerical simulations confirm the effectiveness of seismic base isolation of rigid cylindrical tanks using FPS bearings. Copyright © 2001 John Wiley & Sons, Ltd.     

Seismic vulnerability mitigation of liquefied gas tanks using concave sliding bearings

The aim of this paper is to evaluate the effectiveness of a concave sliding bearing system for the seismic protection of liquefied gas storage tanks through a seismic fragility analysis. An emblematic case study of elevated steel storage tanks, which collapsed during the 1999 ?zmit earthquake at Habas Pharmaceutics plant in Turkey, is studied. Firstly, a fragility analysis is conducted for the examined tank based on a lumped-mass stick model, where the nonlinear shear behaviour of support columns is taken into account by using a phenomenological model. Fragility curves in terms of an efficient intensity measure for different failure modes of structural components demonstrate the inevitable collapse of the tank mainly due to insufficient shear strength of the support columns. A seismic isolation system based on concave sliding bearings, which has been demonstrated a superior solution to seismically protect elevated tanks, is then designed and introduced into the numerical model, accounting for its non-linear behaviour. Finally, a vulnerability analysis for the isolated tank is performed, which proves a high effectiveness of the isolation system in reducing the probability of failure within an expected range of earthquake intensity levels.     

The existence of ‘complete similarities’ in the response of seismic isolated structures subjected to pulse‐like ground motions and their implications in analysis

In this paper the seismic response of isolated structures supported on bearings with bilinear and trilinear behavior is revisited with dimensional analysis in an effort to better understand the relative significance of the various parameters that control the mechanical behavior of isolation systems. An isolation system that consists of lead rubber bearings or of single concave spherical sliding bearings exhibits bilinear behavior; whereas, when a double concave configuration is used the behavior is trilinear. For the case of bilinear behavior it is well known that the value of the normalized yield displacement is immaterial to the response of the isolated superstructure—or, in mathematical terms, that the response of the bilinear oscillator exhibits complete similarity in the dimensionless yield displacement. Similarly, for the case of trilinear behavior the paper shows that the presence of the intermediate slope is immaterial to the peak response of most isolated structures—a finding that shows the response of the trilinear oscillator exhibits a complete similarity in the difference between the coefficients of friction along the two sliding surfaces as well as in the ratio of the intermediate to the final slope. This finding implies that even when the coefficients of friction of the two sliding surfaces are different, the response of isolated structures for most practical configurations can be computed with confidence by replacing the double concave spherical bearings with single concave spherical bearings with an effective radius of curvature and an effective coefficient of friction. Copyright © 2010 John Wiley & Sons, Ltd.     

Planar investigation of the seismic response of ancient columns and colonnades with epistyles using a custom-made software

In this paper, the dynamic behavior of multi-drum columns and colonnades with epistyles under earthquake excitations is examined through planar numerical simulations. A specialized software application, developed utilizing the discrete element methods (DEM), is used to investigate the influence of certain parameters on the seismic response of such multi-body structural systems. First, this custom-made software is extensively validated by comparing the computed responses of various problems, such as sliding, rocking and free vibration dynamics of rigid bodies, with the corresponding analytical solutions. Then, the developed software is used to study the influence of the frequency content and amplitude of the ground motions on the columns and colonnades, as well as the geometric characteristics of these structures. Parameters such as the number of drums that assemble each column and the number of columns of a colonnade appear to be defining parameters that affect the seismic response of colonnades with epistyles. For ground motions with relatively low predominant frequencies, rocking is the dominant effect in the response, while with the increase of the excitation frequency the response becomes even more complex involving both sliding and rocking phenomena. The numerical simulations show that earthquakes with relatively low predominant frequencies seem to endanger both standalone columns and colonnades with epistyles more than earthquakes with higher predominant frequencies.     

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.     

Seismic control of rocking structures via external resonators

Tall rigid blocks are prevalent in ancient historical constructions. Such structures are prone to rocking behaviour under strong ground motion, which is recognizably challenging to predict and mitigate. Our study is motivated by the need to provide innovative nonintrusive solutions to attenuate the rocking response of historical buildings and monuments. In this paper, we examine a novel scheme that employs external resonators buried next to the rocking structure as a means to control its seismic response. The strategy capitalizes on the vibration absorbing potential of the structure-soil-resonator interaction. Furthermore, the benefits of combining the resonators with inerters in order to reduce their gravitational mass without hampering their motion-control capabilities are also explored. Advanced numerical analyses of discrete models under coherent acceleration pulses with rocking bodies of different slenderness ratios under various ground motion intensities highlight the significant vibration absorbing qualities of the external resonating system. The influence of key system parameters such as the mass, stiffness, and damping of the resonator and those of the soil-structure-resonator arrangement are studied. Finally, a case study on the evaluation of the response of rocking structures with external resonators under real pulse-like ground-motion records confirms the important reductions in peak seismic rotational demands obtained with the proposed arrangement.     

Dynamic analysis of stacked rigid blocks

The dynamic behavior of a structural model of two stacked rigid blocks subjected to ground excitation is examined. Assuming no sliding, the rocking response of the system standing free on a rigid foundation is investigated. The derivation of the equations of motion accounts for the consecutive transition from one pattern of motion to another, each being governed by a set of highly nonlinear differential equations. The system behavior is described in terms of four possible patterns of response and impact between either the two blocks or the base block and the ground. The equations governing the rocking response of the system to horizontal and vertical ground accelerations are derived for each pattern, and an impact model is developed by conservation of angular momentum considerations. Numerical results are obtained by developing an ad hoc computational scheme that is capable of determining the response of the system under an arbitrary base excitation. This feature is demonstrated by using accelerograms from the Northridge, CA, 1994, earthquake. It is hoped that the two-blocks model used herein can facilitate the development of more sophisticated multi-block structural models.     

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.     

Seismic response assessment of rocking systems using single degree-of-freedom oscillators

A new modeling for the seismic response assessment of free-standing, rigid or flexible, pure rocking systems is presented. The proposed modeling is based on equivalent single degree-of-freedom (SDOF) oscillators that can be implemented with common engineering software or user-made structural analysis codes. The SDOF models adopted use beam elements that are connected to a nonlinear rotational spring with negative stiffness that describes the self-centering capacity of the rocking member. The loss of energy at impact is treated with an “event-based” approach consistent with Housner's theory. Different variations pertinent to rigid blocks are first presented, and then the concept is extended to the flexible case. The implementation of the method requires some minor programming skills, while thanks to the versatility of the finite element method, it is capable to handle a variety of rocking problems. This is demonstrated with two applications: (a) a vertically restrained block equipped with an elastic tendon and (b) a rigid block coupled with an elastic SDOF oscillator. The accuracy and the efficiency of the proposed modeling is demonstrated using simple wavelets and historical ground motion records.     

Modeling of rocking frames under seismic loading

A novel modeling approach for the seismic response assessment of rocking frames is presented. Rocking frames are systems with columns that are allowed to fully, or partially, uplift. Despite the apparent lack of a mechanism to resist lateral forces, they have a remarkable capacity against earthquake loading. Rocking frames are found in old structures, for example, ancient monuments, but it is also a promising design concept for modern structures such as bridges or buildings. The proposed modeling can be implemented in a general-purpose structural analysis software, avoiding the difficulties that come with the need of formulating and solving specifically tailored differential equations, or the use of detailed computational models. Different configurations of a rocking portal frame problem are examined. The model is based on rigid, or flexible, beam elements that describe the members of the frame. Negative-stiffness rotational springs are smartly positioned at the rocking interfaces in order to simulate the rocking restoring moment, while the mass and the rotational moment of inertia are considered either lumped or distributed. Both the cases of rigid and flexible piers/columns are discussed, while it is shown that frames with restrained columns can be considered in a straightforward manner. A simple alternative based on an equivalent oscillator that follows the generalized rocking equation of motion is also investigated. The efficiency and the accuracy of the proposed modeling is demonstrated with the aid of carefully chosen case studies.     

Numerical investigation on seismic performance of a shallow buried underground structure with isolation devices

A design procedure for improving the seismic performance of unequal-span underground structures by installing isolation devices at the top end of columns is proposed based on the seismic failure mode of frame-type underground structures and the design concept of critical support columns. A two-dimensional finite element model (FEM) for a soil-underground structure with an unequal-span interaction system was established to shed light on the effects of a complex subway station with elastic sliding bearings (ESB) and lead rubber bearings (LRB) on seismic mitigation. It was found that the stiffness and internal force distribution of the underground structure changed remarkably with the installation of isolation devices at the top end of the columns. The constraints of the beam-column joints were significantly weakened, resulting in a decrease in the overall lateral stiffness and an increase in the structural lateral displacement. The introduction of the isolation device effectively reduces the internal force and seismic damage of the frame column; however, the tensile damage to the isolation structure, such as the roof, bottom plate, and sidewall, significantly increased compared to those of the non-isolation structure. Although the relative slip of the ESB remains within a controllable range under strong earthquake excitation as well as frame columns with stable vertical support and self-restoration functions, the LRB shows a better performance during seismic failure and better lateral displacement response of the unequal-span underground structure. The analysis results provide new ideas and references for promoting the application of seismic isolation technology in underground structures.     

橡胶砂芯组合砌块垫层隔震性能试验研究

构造简单、成本低廉的简易隔震方法是提高经济欠发达农村地区的低矮房屋抗震性能的有效措施.文中提出一种新的简易隔震方法:橡胶砂芯组合砌块隔震垫层(RSMCBL).其主要原理为:上部结构通过离散的刚性盖板支承在一系列离散的由废旧轮胎颗粒-砂混合物构成的土柱上,橡胶砂柱被刚性砌块所侧向约束,以保证隔震层的强度和稳定性.针对此种...     

隔震防倒塌支座及楼梯间位置对框架结构抗震性能的影响研究

为研究梯段板下端设置隔震防倒塌支座和楼梯间位置对钢筋混凝土框架结构抗震性能的影响,利用ETABS软件建立不包括、包括隔震防倒塌支座的3种楼梯间布置方案,6个框架结构计算模型。通过模态分析、反应谱分析和Pushover分析,研究隔震防倒塌支座和楼梯间位置对框架结构的振型、内力及破坏机制的影响。结果表明:梯段板下端设置隔震防倒塌支座后,楼梯间位置对钢筋混凝土框架结构的扭转效应影响较小,且框架结构在两个主轴方向的动力特性比较接近;楼梯间框架柱内力均显著降低,但楼梯间布置在最边跨时,在垂直于梯跑方向地震作用下,框架结构边柱内力较大;框架梁对整体框架结构的耗能贡献较多,增强了框架结构的抗震性能,大震时楼梯构件严重破坏较晚,设置隔震防倒塌支座可保证楼梯整体稳定性。     

基于柱顶隔震的3层3跨地铁地下车站结构抗震性能研究

针对3层3跨框架式地铁地下车站结构抗震薄弱构件,采用在柱顶不同位置设置铅芯橡胶隔震支座的方法,建立土-地下连续墙-主体结构非线性静动力耦合相互作用的二维整体时域有限元分析模型,分析柱顶隔震支座对车站主体结构的侧向变形、地震损伤和动应力反应等结构地震反应特性的影响.结果表明,仅在抗震薄弱的顶层和底层中柱柱顶设置2层隔震支...     

Seismic response of torsionally coupled base isolated structures

An analytical study of the seismic response of typical base isolated structures mounted on rubber bearings is presented. Isolated buildings are liable to have closely spaced lower modes of vibration with small eccentricity between centres of mass and rigidity. The isolated structure is modelled as a rigid deck with lumped masses supported on axially inextensible elastomeric rubber bearings. This simplified system has three degrees of freedom (dof), two translations and one rotation in the horizontal plane. The Green's functions for the displacement response of the 3 dof system are derived for both undamped and damped cases with small and large eccentricities. The small eccentricity case is taken from a specific isolated building, while the large eccentricity case arises from the 5 per cent accidental eccentricity which is required by various seismic codes. An interaction equation for normalized displacements is established for an idealized flat velocity spectrum or hyperbolic acceleration spectrum. An isolated building on rubber bearings would have its fundamental period fall into this range of a design spectrum. Numerical results for the specific building subjected to the El Centro earthquake of 1940 are presented. Both the time history and the response spectrum modal superposition analysis were performed. In the response spectrum analysis, the Complete Quadratic Combination (CQC) showed superiority over the Square Root of the Sum of Squares (SRSS) in estimating maximum responses. It is concluded that the effect of torsional coupling on the transient response of base isolated structures is insignificant, due to the combined effect of the time lag between the maximum translational and torsional responses and the influence of damping in the isolation system which for elastomeric bearings can be as high as 8 to 10 per cent.