Generalized dynamic analysis of structural single rocking walls (SRWs)

The investigation of structural single rocking walls (SRWs) continues to gain interest as they produce self-centering lateral load responses with reduced structural damage. The simple rocking model with modifications has been shown to capture these responses accurately if the SRW and its underlying base are infinitely rigid. This paper advances previous rocking models by accounting for (1) the inelastic actions at or near the base of the SRW and (2) the flexural responses within the wall. Included in the proposed advancements are hysteretic and inherent viscous damping associated with these two deformation components so that the total dynamic responses of SRWs can be captured with good accuracy. A system of nonlinear equations of motion is developed, in which the rocking base is discretized into fibers using a zero-length element to locate the associated compressive deformations and damage. The flexural deformations of the rocking body are captured using an elastic term, while the impact events are modeled using impulse-momentum equations. Comparisons with experiments of structural precast concrete and masonry SRWs show that the proposed approach accurately estimates the dynamic responses of different SRWs with and without unbonded posttensioning, for various dynamic excitations and degrees of hysteretic action. Using the proposed approach, a numerical investigation employs different configurations of structural SRWs to quantify the various sources of energy loss, including hysteretic action and impact damping, during various horizontal ground motions.     

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.     

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.     

Hybrid simulation testing of a self‐centering rocking steel braced frame system

The self‐centering rocking steel frame is a seismic force resisting system in which a gap is allowed to form between a concentrically braced steel frame and the foundation. Downward vertical force applied to the rocking frame by post‐tensioning acts to close the uplifting gap and thus produces a restoring force. A key feature of the system is replaceable energy‐dissipating devices that act as structural fuses by producing high initial system stiffness and then yielding to dissipate energy from the input loading and protect the remaining portions of the structure from damage. In this research, a series of large‐scale hybrid simulation tests were performed to investigate the seismic performance of the self‐centering rocking steel frame and in particular, the ability of the controlled rocking system to self‐center the entire building. The hybrid simulation experiments were conducted in conjunction with computational modules, one that simulated the destabilizing P‐Δ effect and another module that simulated the hysteretic behavior of the rest of the building including simple composite steel/concrete shear beam‐to‐column connections and partition walls. These tests complement a series of quasi‐static cyclic and dynamic shake table tests that have been conducted on this system in prior work. The hybrid simulation tests validated the expected seismic performance as the system was subjected to ground motions in excess of the maximum considered earthquake, produced virtually no residual drift after every ground motion, did not produce inelasticity in the steel frame or post‐tensioning, and concentrated the inelasticity in fuse elements that were easily replaced. Copyright © 2014 John Wiley & Sons, Ltd.     

Modeling of energy dissipation in structural devices and foundation soil during seismic loading

Though rocking shallow foundations could be designed to possess many desirable characteristics such as energy dissipation, isolation, and self-centering, current seismic design codes often avoid nonlinear behavior of soil and energy dissipation beneath foundations. This paper compares the effectiveness of energy dissipation in foundation soil (during rocking) with the effectiveness of structural energy dissipation devices during seismic loading. Numerical simulations were carried out to systematically study the seismic energy dissipation in structural elements and passive controlled energy dissipation devices inserted into the structure. The numerical model was validated using shaking table experimental results on model frame structures with and without energy dissipation devices. The energy dissipation in the structure, drift ratio, and the force and displacement demands on the structure are compared with energy dissipation characteristics of rocking shallow foundations as observed in centrifuge experiments, where shallow foundations were allowed to rock on dry sandy soil stratum during dynamic loading. For the structures with energy dissipating devices, about 70–90% of the seismic input energy is dissipated by energy dissipating devices, while foundation rocking dissipates about 30–90% of the total seismic input energy in foundation soil (depending on the static factor of safety). Results indicate that, if properly designed (with reliable capacity and tolerable settlements), adverse effects of foundation rocking can be minimized, while taking advantage of the favorable features of foundation rocking and hence they can be used as efficient and economical seismic energy dissipation mechanisms in buildings and bridges.     

Seismic safety of reinforced concrete members and structures

Based on cyclic load tests of large-scale reinforced concrete elements and assemblages, a probabilistic model of member failure is developed. The model gives the probability of survival at time t as a functional of damage ratio and dissipated energy up to t. After extension to multivariate survival of several members with correlated resistance, the model is used to calculate the safety of reinforced concrete frames subjected to given input motions. Results are in terms of the probability of local failure and of no failure anywhere in the system.     

An analytical model of a deformable cantilever structure rocking on a rigid surface: development and verification

This paper extends previously developed models to account for the influence of the column and the foundation masses on the behavior of top‐heavy deformable elastic cantilever columns rocking on a rigid support surface. Several models for energy dissipation at impact are examined and compared. A novel Vertical Velocity Energy Loss model is introduced. Rocking uplift and overturning spectra for the deformable elastic cantilever model excited by sinusoidal ground motions are constructed. The effects of non‐dimensional model parameter variations on the rocking spectra and the overturning stability of the model are presented. It is shown that the remarkable overturning stability of dynamically excited large cantilever columns is not jeopardized by their deformability. Copyright © 2015 John Wiley & Sons, Ltd.     

A discretely damped SDOF model for the rocking response of freestanding blocks

This paper presents a single-degree-of-freedom (SDOF) constitutive model for assessing the performance of freestanding block contents of buildings. The model incorporates a bespoke damper to account for energy dissipation associated with rocking. It is advantageous in its direct correlation, via energy conservation, to the restitution coefficient for impact during rocking. A comparative study with the existing SDOF rocking models shows that the proposed model significantly improves the accuracy of free-rocking simulations, in which inherent damping predominantly affects response. It provides a promising and efficient tool for computationally intensive performance evaluation of nonstructural components.

     

Collapse risk of controlled rocking steel braced frames with different post‐tensioning and energy dissipation designs

Controlled rocking steel braced frames (CRSBFs) are low‐damage self‐centring lateral force resisting systems. Previous studies have shown that designing the energy dissipation (ED) and post‐tensioning (PT) in CRSBFs using a response modification factor of R=8 can prevent collapse of structures during earthquakes beyond the design level. However, designers have unique control over the hysteretic behaviour of the system, even after the response modification factor is selected. Additionally, recent studies have suggested that CRSBFs could also be designed using R>8 while still satisfying performance limits. This paper examines how the response modification factor and the design of the ED and PT influence the collapse performance of CRSBFs with three and six storeys where collapse occurs because of over‐rotation of the base rocking joint. In addition, the influence of using an additional rocking joint above the base to mitigate higher‐mode forces is evaluated for a 12‐storey frame. A total of 18 different designs are considered for the three buildings using different ED and PT design parameters, including different response modification factors. A suite of 44 ground motions is scaled until at least 50% of the records cause collapse, and fragility curves are generated using the truncated incremental dynamic analysis curves. The results from two different assessment methodologies show that the parameters selected have a marked influence on the collapse performance of a CRSBF. Nevertheless, even CRSBFs designed using R>8 or without supplemental ED can have acceptably low probabilities of collapse, provided that the frame members are designed to remain elastic. Copyright © 2017 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.     

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.     

Efficient 3D nonlinear Winkler model for shallow foundations

This paper presents a new practical modeling approach, based on the beam-on-a-nonlinear Winkler foundation (BNWF) model, to simulate the 3D rocking, vertical and horizontal responses of shallow foundations using structural elements that are readily available in the element library of commercially available structural analysis programs. An assemblage of a moment-rotation hinge, shear hinge connected in series with an elastic frame member attached to the bottom end of ground story columns was proposed to model the response of the footing under combined action of vertical, horizontal and moment loading. To couple the responses of these hinges, two bounding surfaces equations were introduced and derived mathematically: a surface that defines the interaction between the rocking and vertical capacities of the footing along its width and length; and a surface that defines the interaction between the horizontal capacities of the footing along its width and length. Simple calculation steps to evaluate the geometric and mechanical properties of the proposed assemblage of structural elements are provided. The proposed modeling approach was verified using experimental results from large scale model foundations subjected to cyclic loading. Based on this study, it was found that the proposed assemblage can be reliably used in modeling the rocking and horizontal responses of shallow foundations under cyclic loading.     

Single precast concrete rocking walls as earthquake force‐resisting elements

Precast concrete walls with unbonded post‐tensioning provide a simple self‐centering system. Yet, its application in seismic regions is not permitted as it is assumed to have no energy dissipation through a hysteretic mechanism. These walls, however, dissipate energy imparted to them because of the wall impacting the foundation during rocking and limited hysteretic action resulting from concrete nonlinearity. The energy dissipated due to rocking was ignored in previous experimental studies because they were conducted primarily using quasi‐static loading. Relying only on limited energy dissipation, a shake table study was conducted on four single rocking walls (SRWs) using multiple‐level earthquake input motions. All walls generally performed satisfactorily up to the design‐level earthquakes when their performance was assessed in terms of the maximum transient drift, maximum absolute acceleration, and residual drift. However, for the maximum considered earthquakes, the walls experienced peak lateral drifts greater than the permissible limits. Combining the experimental results with an analytical investigation, it is shown that SRWs can be designed as earthquake force‐resisting elements to produce satisfactory performance under design‐level and higher‐intensity earthquake motions. Copyright © 2016 John Wiley & Sons, Ltd.     

Mechanisms to limit higher mode effects in a controlled rocking steel frame. 1: Concept,modelling, and low‐amplitude shake table testing

Controlled rocking steel frames have been proposed as an efficient way to avoid the structural damage and residual deformations that are expected in conventional seismic force resisting systems. Although the base rocking response is intended to limit the force demands, higher mode effects can amplify member design forces, reducing the viability of the system. This paper suggests that seismic forces may be limited more effectively by providing multiple force‐limiting mechanisms. Two techniques are proposed: detailing one or more rocking joints above the base rocking joint and providing a self‐centring energy dissipative (SCED) brace at one or more levels. These concepts are applied to the design of an eight‐storey prototype structure and a shake table model at 30% scale. A simple numerical model that was used as a design tool is in good agreement with frequency characterization and low‐amplitude seismic tests of the shake table model, particularly when multiple force‐limiting mechanisms are active. These results suggest that the proposed mechanisms can enable better capacity design by reducing the variability of peak seismic force demands without causing excessive displacements. Similar results are expected for other systems that rely on a single location of concentrated nonlinearity to limit peak seismic loads. Copyright © 2012 John Wiley & Sons, Ltd.     

Seismic response of rocking frames with top support eccentricity

The seismic response of rocking frames that consist of a rigid beam freely supported on rigid freestanding rectangular piers has received recent attention in the literature. Past studies have investigated the special case where, upon planar rocking motion, the beam maintains contact with the piers at their extreme edges. However, in many real scenarios, the beam‐to‐pier contact lies closer to the center of the pier, affecting the overall stability of the system. This paper investigates the seismic response of rocking frames under the more general case which allows the contact edge to reside anywhere in‐between the center of the pier and its extreme edge. The study introduces a rocking block model that is dynamically equivalent to a rocking frame with vertically symmetric piers of any geometry. The impact of top eccentricity (ie, the distance of the contact edge from the pier's vertical axis of symmetry) on the seismic response of rocking frames is investigated under pulse excitations and earthquake records. It is concluded that the stability of a top‐heavy rocking frame is highly influenced by the top eccentricity. For instance, a rocking frame with contacts at the extreme edges of the piers can be more seismically stable than a solitary block that is identical to one of the frame's piers, while a rocking frame with contacts closer to the centers of the piers can be less stable. The concept of critical eccentricity is introduced, beyond which the coefficient of restitution contributes to a greater reduction in the response of a frame than of a solitary pier.     

An analytical model of a deformable cantilever structure rocking on a rigid surface: experimental validation

This paper describes an experimental program to examine the dynamic response of deformable cantilevers rocking on a rigid surface. The primary goal of the tests is to verify and validate a dynamic rocking model that describes the behavior of these structures. The benchmark response data was obtained from shaking‐table tests on deformable rocking specimens with different natural vibration frequencies and different aspect ratios excited by analytical pulses and recorded ground motions. The responses computed using the model are found to be in good agreement with the benchmark test results. Widely used impact, restitution and damping assumptions are revisited based on the experiment results and the analytical model findings. Copyright © 2015 John Wiley & Sons, Ltd.     

Seismic performance of buildings with structural and foundation rocking in centrifuge testing

Rocking motion, established in either the superstructure in the form of a 2‐point stepping mechanism (structural rocking) or resulting from rotational motion of the foundation on the soil (foundation rocking), is considered an effective, low‐cost base isolation technique. This paper unifies for the first time the 2 types of rocking motion under a common experimental campaign, so that on the one hand, structural rocking can be examined under the influence of soil and on the other, foundation rocking can be examined under the influence of a linear elastic superstructure. Two building models, designed to rock above or below their foundation level so that they can reproduce structural and foundation rocking respectively, were tested side by side in a centrifuge. The models were placed on a dry sandbed and subjected to a sequence of earthquake motions. The range of rocking amplitude that is required for base isolation was quantified. Overall, it is shown that the relative density of sand does not influence structural rocking, while for foundation rocking, the change from dense to loose sand can affect the time‐frequency response significantly and lead to a more predictable behaviour.     

基于能量的新型消能摇摆结构体系抗震性能研究

利用能量法分析新型消能摇摆结构体系在地震动过程中的耗能机制,并对其抗震性能进行系统研究。以某已建成的6层框架结构为研究对象,建立新型消能摇摆结构体系的计算模型,利用Opensees软件获得新型消能摇摆结构体系在不同地震动激励下的动力响应,分析其各子结构的耗能部位及所占比例。结果表明:新型消能摇摆结构体系的耗能机制更为合理,其抗震性能优于传统结构体系,为工程设计和研究提供一些参考。