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.     

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.     

Dynamics and seismic performance of rocking bridges accounting for the abutment-backfill contribution

The present study explores analytically the concept of rocking isolation in bridges considering for the first time the influence of the abutment-backfill system. The dynamic response of rocking bridges with free-standing piers of same height and same section is examined assuming negligible deformation for the substructure and the superstructure. New relationships for the prediction of the bridge rocking motion are derived, including the equation of motion and the restitution coefficient at each impact at the rocking interfaces. The bridge structure is found to be susceptible to a failure mode related to the failure of the abutment-backfill system, which can occur prior to the well-known overturning of the rocking piers. Thus, a new failure spectrum is proposed called Failure Minimum Acceleration Spectrum (FMAS) which extends the overturning spectrum put forward in previous studies, and it differs in principle from the latter. The comparison with the dynamic response of bridges modelled as rocking frames without abutments reveals not only that seat-type abutments and their backfill have a generally beneficial effect on the seismic performance of rocking pier bridges by suppressing the free rocking motion of the frame system, but also that the simple frame model cannot capture all salient features of the rocking bridge response as it misses potential failure modes, overestimating the rocking bridge's safety when these modes are critical.     

Seismic response of rocking isolated railway bridge piers with sacrificial components

In this study, sacrificial components were incorporated into self-centering railway bridge piers to improve the lateral stiffness. The seismic response of this new detail was investigated. First, the method to compute the initial uplift moment of the self-centering pier is given. In addition, shaking table tests were conducted on a free-rocking pier without sacrificial components, which was used to validate a two-spring numerical model. Good agreement was obtained between the numerical results and experimental data. Furthermore, the validated model was employed to investigate the influence of sacrificial components on the seismic response of rocking piers. For this purpose, two models were developed, with and without sacrificial components. Nonlinear response history analysis was then performed on both models under three historical motions. The results showed that compared to the one without sacrificial components, the rocking pier with sacrificial components has comparable displacement at the top of the pier, and maximum uplift moment at high amplitude motion. Therefore, incorporating sacrificial components into the rocking pier can increase the lateral stiffness at service load and low amplitude frequent earthquakes but can produce comparable response at high seismic excitation. These results provide support for performance-based seismic design of self-centering rocking piers.

     

高承台下自由桩长对双薄壁墩连续刚构桥地震响应的影响

为探究高承台下自由桩长对双薄壁墩连续刚构桥地震响应的影响,基于OpenSees程序建立了实桥有限元模型并进行弹塑性时程分析,通过对比不同自由桩长模型的时程曲线、峰值响应及滞回特性,分析了自由桩长对桥梁地震响应的影响.结果表明:自由桩长增加会减小桥梁刚度;地震作用下,随自由桩长增加,主梁、支座及自由桩顶的水平位移增大,且...     

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.     

Centrifuge modeling of rocking‐isolated inelastic RC bridge piers

Experimental proof is provided of an unconventional seismic design concept, which is based on deliberately underdesigning shallow foundations to promote intense rocking oscillations and thereby to dramatically improve the seismic resilience of structures. Termed rocking isolation, this new seismic design philosophy is investigated through a series of dynamic centrifuge experiments on properly scaled models of a modern reinforced concrete (RC) bridge pier. The experimental method reproduces the nonlinear and inelastic response of both the soil‐footing interface and the structure. To this end, a novel scale model RC (1:50 scale) that simulates reasonably well the elastic response and the failure of prototype RC elements is utilized, along with realistic representation of the soil behavior in a geotechnical centrifuge. A variety of seismic ground motions are considered as excitations. They result in consistent demonstrably beneficial performance of the rocking‐isolated pier in comparison with the one designed conventionally. Seismic demand is reduced in terms of both inertial load and deck drift. Furthermore, foundation uplifting has a self‐centering potential, whereas soil yielding is shown to provide a particularly effective energy dissipation mechanism, exhibiting significant resistance to cumulative damage. Thanks to such mechanisms, the rocking pier survived, with no signs of structural distress, a deleterious sequence of seismic motions that caused collapse of the conventionally designed pier. © 2014 The Authors Earthquake Engineering & Structural Dynamics Published by John Wiley & Sons Ltd.     

Biaxial testing of unbonded post‐tensioned rocking bridge piers with external replacable dissipaters

The biaxial response of two bridge piers is experimentally investigated. A post‐tensioned precast bridge pier with external replaceable mild‐steel dissipaters is tested under biaxial loading. The performance of the post‐tensioned bridge pier is compared with a conventionally reinforced monolithic bridge pier. The experimental biaxial response is then compared with previous uniaxial experimental testing of identical bridge piers to understand the influence of biaxial loading, specifically concerning post‐tensioned rocking sections. A 3‐dimensional moment–curvature and moment–rotation analysis program is created to generate the monotonic section response of a conventional and post‐tensioned bridge pier. After comparing the accuracy of the section analysis program to the experimental testing of the monolithic pier, the program is validated against the experimental testing of the post‐tensioned bridge pier. This section analysis program is then used in the calibration of a macro‐model to capture the entire cyclic response of the post‐tensioned bridge pier. The macro‐model adopts multiple linear‐elastic compression‐only springs at the rocking interface, combined with non‐linear inelastic springs for each of the mild‐steel dissipaters and returns encouraging results at both local and global levels. The paper concludes with a number of biaxial moment‐interaction design charts for monolithic and post‐tensioned bridge piers as a function of mechanical and geometric section properties. The design charts define the biaxial yield surface at nominal yield and at the design section capacity defined by one of three material limit states. Copyright © 2011 John Wiley & Sons, Ltd.     

限制位移桥墩的连续刚构桥抗震性能研究

沿着摇摆桥墩的概念提出一种限制位移桥墩连续刚构桥体系。该体系通过对连续刚构桥墩底和承台之间采取一定措施,使桥梁在地震发生时能够在限制的位移量内活动,减小输入到桥梁结构中的能量,达到减震的目的。通过对一座铁路连续刚构桥的分析,发现这种限制位移桥墩连续刚构桥体系能大幅减小桥墩的延性和强度地震需求,减震效果明显,在选择合适的限制位移量的情况下,可保证桥墩在高烈度罕遇地震作用下几乎保持弹性工作状态,震后经简单处理即可保证使用功能,为震后救援工作带来极大便利,也大大减少了修复成本。     

Seismic retrofitting of braced frame buildings by RC rocking walls and viscous dampers

A design procedure for seismic retrofitting of concentrically and eccentrically braced frame buildings is proposed and validated in this paper. Rocking walls are added to the existing system to ensure an almost uniform distribution of the interstorey displacement in elevation. To achieve direct and efficient control over the seismic performance, the design procedure is founded on the displacement‐based approach and makes use of overdamped elastic response spectra. The top displacement capacity of the building is evaluated based on a rigid lateral deformed configuration of the structure and on the ductility capacity of the dissipative members of the braced frames. The equivalent viscous damping ratio of the braced structure with rocking walls is calculated based on semi‐empirical relationships specifically calibrated in this paper for concentrically and eccentrically braced frames. If the equivalent viscous damping ratio of the structure is lower than the required equivalent viscous damping ratio, viscous dampers are added and arranged between the rocking walls and adjacent reaction columns. The design internal forces of the rocking walls are evaluated considering the contributions of more than one mode of vibration. The proposed design procedure is applied to a large set of archetype braced frame buildings and its effectiveness verified by nonlinear dynamic analysis.     

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 performance of hybrid ductile‐rocking braced frame system

A new hybrid ductile‐rocking seismic‐resistant design is proposed which consists of a code‐designed buckling‐restrained braced frame (BRBF) that yields along its height and also partially rocks on its foundation. The goal of this system is to cost‐effectively improve the performance of BRBFs, by reducing drift concentrations and residual deformations, while taking advantage of their large ductility and their reliable limit on seismic forces and accelerations along a building's height. A lock‐up device ensures that the full code‐compliant lateral strength can be achieved after a limited amount of column uplift, and supplemental energy dissipation elements are used to reduce the rocking response. This paper outlines the mechanics of the system and then presents analyses on rocking frames with both ductile and elastic braces in order to highlight the large higher mode demands on elastic rocking frames. A parametric study using nonlinear time‐history analysis of BRBF structures designed according to the proposed procedure for Los Angeles, California is then presented. This study investigates the system's seismic response and the effect of different energy dissipation element properties and allowable base rotation values before the lock‐up is engaged. Finally, the effect of vertical mass modeling on analysis results was investigated. These studies demonstrated that the hybrid ductile‐rocking system can in fact improve the global peak and residual deformation response as well as reduce brace damage. This enhanced performance could eliminate the need for expensive repairs or demolition that are otherwise to be expected for conventional ductile fixed base buildings that sustain severe damage.     

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.     

罕遇地震作用下的框架-摇摆刚架结构体系动力可靠度研究

针对钢筋混凝土框架结构的受力特点,采用增设摇摆刚架的抗震设计方法,以提高罕遇地震下建筑结构的安全性。建立了框架一摇摆刚架结构体系的计算模型,结合状态空间法与虚拟激励法,求解结构的平稳随机响应,并根据计算所得随机响应对框架一摇摆刚架体系的动力可靠度进行分析。以西部地区某已建成的6层框架结构为算例,探讨了罕遇地震作用下不同刚度比的摇摆刚架对新结构体系动力可靠度的影响。结果表明,通过增设不同刚度比的摇摆刚架,可以有效协调结构体系的变形模式,充分发挥结构的耗能能力,降低整体结构的条件失效概率。     

楔形挡块对中小跨径梁桥横向抗震性能的影响

在地震作用下中小跨径梁桥横向易出现落梁以及桥墩破坏。为了防止桥梁出现上述震害,提出以楔形挡块作为限位装置来提升桥梁的横向抗震性能。以一座3×20 m连续混凝土梁桥为例,通过OpenSees软件来建立有限元模型,在考虑板式橡胶支座的摩擦滑移效应、钢筋混凝土桥墩的非线性等力学效应的情况下,对其进行动力时程分析。引入主梁位移响应、桥墩顶部最大位移响应等作为指标,用柔性挡块、刚性挡块两种工况来与楔形挡块进行对比分析,并且分析楔形挡块不同角度对位移响应的影响。结果表明:楔形挡块角度设置合适时能够有效约束梁体位移响应,并且不显著提高桥墩顶部的位移响应。     

Hybrid simulation of a multi‐span RC viaduct with plain bars and sliding bearings

This paper deals with the seismic response assessment of an old reinforced concrete viaduct and the effectiveness of friction‐based retrofitting systems. Emphasis was laid on an old bridge, not properly designed to resist seismic action, consisting of 12 portal piers that support a 13‐span bay deck for each independent roadway. On the basis of an OpenSEES finite element frame pier model, calibrated in a previous experimental campaign with cyclic displacement on three 1:4 scale frame piers, a more complex experimental activity using hybrid simulation has been devised. The aim of the simulation was twofold: (i) to increase knowledge of non‐linear behavior of reinforced concrete frame piers with plain steel rebars and detailing dating from the late 1950s; and (ii) to study the effectiveness of sliding bearings for seismic response mitigation. Hence, to explore the performance of the as built bridge layout and also of the viaduct retrofitted with friction‐based devices, at both serviceability and ultimate limit state conditions, hybrid simulation tests were carried out. In particular, two frame piers were experimentally controlled with eight‐actuator channels in the as built case while two frame piers and eight sliding bearings were controlled with 18‐actuator channels in the isolated case. The remaining frame piers were part of numerical substructures and were updated offline to accurately track damage evolution. Copyright © 2015 John Wiley & Sons, Ltd.     

Dynamic and equivalent static procedures for capacity design of controlled rocking steel braced frames

Controlled rocking steel braced frames (CRSBFs) have been proposed as a low‐damage seismic force resisting system with reliable self‐centring capabilities. Vertical post‐tensioning tendons are designed to self‐centre the system after rocking, and energy dissipation may be provided to limit the peak displacements. The post‐tensioning and energy dissipation can be designed using simple methods that rely primarily on the first‐mode response. However, the frame member forces are highly influenced by the higher‐mode response, resulting in more complex methods to design the frame members. This paper examines previous proposals and also proposes two new capacity design methods for CRSBFs. The first is a dynamic procedure that requires a truncated response spectrum analysis on a model of the frame with modified boundary conditions to consider the rocking behaviour. The second is an equivalent static method that does not require any modifications to the elastic frame model, instead using theory‐based lateral force distributions to consider the higher modes of the rocking structure. Neither method requires empirical calibration. The dynamic procedure is used to design two sets of CRSBFs with three, six, nine, twelve and eighteen stories, one set using a response modification factor of R = 8 and the other using up to R = 20. Based on the results of 800 nonlinear time history analyses, both methods are generally more accurate than the previous capacity design methods and at least as simple to implement. Finally, the displacement results suggest that taller CRSBFs designed using could still limit interstorey drifts to approximately 2.5% at the maximum considered earthquake level in the cases considered. Copyright © 2016 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.     

汶川大地震曲线梁桥震害及破坏机理分析

以汶川大地震中严重破坏的回澜立交桥为例,基于数值模拟手段并结合现场震害调查,分析了回澜立交桥的地震破坏机理。数值分析表明,地震时设有支座的最矮的1号桥墩支座发生滑移破坏,以致刚度较大（次矮）的2号刚构桥墩承受很大的地震惯性力,2号墩首先发生弯曲屈服,此后随延性发展因抗剪能力不足最终发生剪切破坏直至倒塌损毁,呈现典型的弯剪破坏特征。现场震害调查发现,回澜立交桥震害集中于抗弯刚度较大的刚构墩上,而其余桥墩震害相对较轻,主要表现为混凝土保护层的脱落、混凝土开裂以及墩顶支座的滑移破坏等。数值分析结果与震害调查呈现出较好的一致性。     

采用SMA拉索的摇摆自复位桥墩试验研究与动力分析

为了提升桥梁结构的抗震韧性,提出了一种采用超弹性形状记忆合金（SMA）拉索的摇摆自复位桥墩。首先对单根SMA拉索进行变幅循环拉伸试验,随后以SMA拉索的预张拉应力为试验参数,针对新型桥墩开展缩尺模型试验,详细考察了试件的力学行为与损伤模式,讨论了桥墩的滞回曲线和复位能力;提出了新型桥墩的初步设计建议,并利用OpenSees进行非线性时程分析验证。研究表明:得益于摇摆机制以及SMA拉索良好的可恢复变形性能,桥墩试件在4%滑移率内几乎不产生损伤;与传统梁桥相比,采用SMA拉索桥墩的新型梁桥可有效降低结构残余变形以及桥墩本身的损伤。虽然最大变形有所放大,但仍处在可控范围。