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.     

The rocking spectrum and the limitations of practical design methodologies

This paper is concerned with the superficial similarities and fundamental differences between the oscillatory response of a single‐degree‐of‐freedom (SDOF) oscillator (regular pendulum) and the rocking response of a slender rigid block (inverted pendulum). The study examines the distinct characteristics of the rocking spectrum and compares the observed trends with those of the response spectrum. It is shown that the rocking spectrum reflects kinematic characteristics of the ground motions that are not identifiable by the response spectrum. The paper investigates systematically the fundamental differences in the dynamical structure of the two systems of interest and concludes that rocking structures cannot be replaced by ‘equivalent’ SDOF oscillators. The study proceeds by examining the validity of a simple, approximate design methodology, initially proposed in the late 1970s and now recommended in design guidelines to compute rotations of slender structures by performing iteration either on the true displacement response spectrum or design spectrum. This paper shows that the simple design approach is inherently flawed and should be abandoned, in particular for smaller, less‐slender blocks. The study concludes that the exact rocking spectrum emerges as a distinct intensity measure of ground motions. Copyright © 2002 John Wiley & Sons, Ltd.     

Rocking response of SDOF systems on shallow improved sand: An experimental study

Recent studies have highlighted the potential advantages of allowing inelastic foundation response during strong seismic shaking. Such an alternative seismic design philosophy, in which soil failure is used as a “fuse” for the superstructure has recently been proposed, in the form of “rocking isolation”. Within this context, foundation rocking may be desirable as a means of bounding the inertia forces transmitted onto the superstructure, but incorporates the peril of unacceptable settlements in case of a low static factor of safety FS_v. Hence, to ensure that rocking is materialized through uplifting rather than sinking, an adequately large FS_v is required. Although this is feasible in theory, soil properties are not always well-known in engineering practice. However, since rocking-induced soil yielding is only mobilized within a shallow layer underneath the footing, “shallow soil improvement” is considered as an alternative approach to release the design from the jeopardy of unforeseen inadequate FS_v. For this purpose, this paper studies the rocking response of relatively slender SDOF structures (h/B ratio equals 3 and rocking dominates over sliding), with emphasis on the effectiveness of shallow soil improvement stretching to various depths below the foundation. A series of reduced-scale monotonic and slow-cyclic pushover tests are conducted on SDOF systems lying on a square surface foundation. It is shown that shallow soil improvement may, indeed, be quite effective provided that its depth is equal to the width of the foundation. For lightly-loaded systems, an even shallower soil improvement may also be considered effective, depending on design requirements. The effectiveness of shallow soil improvement is ameliorated with the increase of cyclic rotation amplitude, and with repeating cycles of loading.     

Dynamic response and impact energy loss in controlled rocking members

Unbonded posttensioning anchors a rocking structural member to its foundation and produces its controlled rocking response when the member undergoes seismic action. Unlike rocking of free-standing bodies, little attention has been given to the dynamic behavior of these controlled rocking members. This paper utilizes experiments of concrete structural members with unbonded posttensioning, varying member geometries, and levels of initial posttensioning force to (a) characterize the associated impact energy loss and (b) improve modeling of controlled rocking motions. Experimental results show that impact energy loss in controlled rocking members can be captured accurately using the coefficient of restitution (r) approach of the modified simple rocking model (MSRM). Based on the MSRM, a controlled rocking model (CRM) is developed that additionally accounts for the variations in contact length at the member-to-foundation (rocking) interface. The CRM reproduces the experimental responses of controlled rocking members with good accuracy and is used to investigate controlled rocking motions under horizontal base excitations.     

Effect of the spectral shape of ground motion records on the collapse fragility assessment of degrading SDOF systems

The main purpose of this paper is to study the collapse capacity of single degree of freedom(SDOF) systems and to produce fragility curves as well as collapse capacity spectra while considering a broad range of structural parameters, including system degradation, the P-Δ effect, ductility capacity and the post-capping stiffness ratio. The modified IbarraKrawinkler deterioration model was used to consider hysteretic behavior. A comprehensive study was conducted to extract the collapse capacity spectrum of SDOF systems with a wide range of periods, varying from 0.05 to 4 s, to cover short, intermediate and long periods. Incremental dynamic analysis(IDA) was performed for SDOF systems to identify the condition in which the collapse capacity of the system is determined. The IDAs were performed using different sets of seismic ground motions. The ground motion records were categorized into different sets based on three spectral shape parameters, including the epsilon, Sa Ratio and N_p. The collapse fragility curves of SDOF systems with different periods were extracted to illustrate the collapse capacity at different probability levels. The results show that structural degradation and ductility as well as the spectral shape parameters significantly affect the collapse capacity of SDOF systems. On the other hand, the post-capping stiffness ratio and small levels of the P-Δ effect do not remarkably change collapse capacity. Also, the collapse capacity of SDOF systems is more sensitive to the records categorized based on Sa Ratio and N_p than those classified based on epsilon.     

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.     

Self-centering seismic retrofit scheme for reinforced concrete frame structures: SDOF system study

This paper presents the results of a parametric study of self-centering seismic retrofit schemes for reinforced concrete (RC) frame buildings. The self-centering retrofit system features flag-shaped hysteresis and minimal residual deformation. For comparison purpose,an alternate seismic retrofit scheme that uses a bilinear-hysteresis retrofit system such as buckling-restrained braces (BRB) is also considered in this paper. The parametric study was carried out in a single-degree-of-freedom (SDOF) system framework since a multi-story building structure may be idealized as an equivalent SDOF system and investigation of the performance of this equivalent SDOF system can provide insight into the seismic response of the multi-story building. A peak-oriented hysteresis model which can consider the strength and stiffness degradation is used to describe the hysteretic behavior of RC structures. The parametric study involves two key parameters -the strength ratio and elastic stiffness ratio between the seismic retrofit system and the original RC frame. An ensemble of 172 earthquake ground motion records scaled to the design basis earthquake in California with a probability of exceedance of 10% in 50 years was constructed for the simulation-based parametric study. The effectiveness of the two seismic retrofit schemes considered in this study is evaluated in terms of peak displacement ratio,peak acceleration ratio,energy dissipation demand ratio and residual displacement ratio between the SDOF systems with and without retrofit. It is found from this parametric study that RC structures retrofitted with the self-centering retrofit scheme (SCRS) can achieve a seismic performance level comparable to the bilinear-hysteresis retrofit scheme (BHRS) in terms of peak displacement and energy dissipation demand ratio while having negligible residual displacement after earthquake.     

Self‐centering structural systems with combination of hysteretic and viscous energy dissipations

This paper presents an innovative set of high‐seismic‐resistant structural systems termed Advanced Flag‐Shaped (AFS) systems, where self‐centering elements are used with combinations of various alternative energy dissipation elements (hysteretic, viscous or visco‐elasto‐plastic) in series and/or in parallel. AFS systems is developed using the rationale of combining velocity‐dependent with displacement‐dependent energy dissipation for self‐centering systems, particularly to counteract near‐fault earthquakes. Non‐linear time‐history analyses (NLTHA) on a set of four single‐degree‐of‐freedom (SDOF) systems under a suite of 20 far‐field and 20 near‐fault ground motions are used to compare the seismic performance of AFS systems with the conventional systems. It is shown that AFS systems with a combination in parallel of hysteretic and viscous energy dissipations achieved greater performance in terms of the three performance indices. Furthermore, the use of friction slip in series of viscous energy dissipation is shown to limit the peak response acceleration and induced base‐shear. An extensive parametric analysis is carried out to investigate the influence of two design parameters, λ₁ and λ₂ on the response of SDOF AFS systems with initial periods ranging from 0.2 to 3.0 s and with various strength levels when subjected to far‐field and near‐fault earthquakes. For the design of self‐centering systems with combined hysteretic and viscous energy dissipation (AFS) systems, λ₁ is recommended to be in the range of 0.8–1.6 while λ₂ to be between 0.25 and 0.75 to ensure sufficient self‐centering and energy dissipation capacities, respectively. Copyright © 2010 John Wiley & Sons, Ltd.     

Characterization of rocking shallow foundations using centrifuge model tests

This paper presents new results of centrifuge model tests exploring the behavior of rocking shallow foundations embedded in dry sand, which provides a variety of factors of safety for vertical bearing. The results of slow (quasi‐static) cyclic tests of rocking shear walls and dynamic shaking tests of single‐column rocking bridge models are presented. The moment–rotation and settlement–rotation relationships of rocking footings are investigated. Concrete pads were placed in the ground soil to support some models with the objective of reducing the settlement induced by rocking. The behavior of rocking foundation was shown to be sensitive to the geometric factor of safety with respect to bearing failure, L_f/L_c, where L_f was the footing length, and the L_c was the critical soil‐footing contact length that would be required to support pure axial loading. Settlements were shown to be small if L_f/L_c was reasonably large. Placement of concrete pads under the edges of the footing was shown to be a promising approach to reduce settlements resulting from rocking, if settlements were deemed to be excessive and also had impacts on the energy dissipation and rocking moment capacity. A general discussion of the tradeoffs between energy dissipation and re‐centering of rocking foundations and other devices is included. Copyright © 2011 John Wiley & Sons, Ltd.     

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.

     

Seismic response of self‐centring hysteretic SDOF systems

The seismic response of single‐degree‐of‐freedom (SDOF) systems incorporating flag‐shaped hysteretic structural behaviour, with self‐centring capability, is investigated numerically. For a SDOF system with a given initial period and strength level, the flag‐shaped hysteretic behaviour is fully defined by a post‐yielding stiffness parameter and an energy‐dissipation parameter. A comprehensive parametric study was conducted to determine the influence of these parameters on SDOF structural response, in terms of displacement ductility, absolute acceleration and absorbed energy. This parametric study was conducted using an ensemble of 20 historical earthquake records corresponding to ordinary ground motions having a probability of exceedence of 10% in 50 years, in California. The responses of the flag‐shaped hysteretic SDOF systems are compared against the responses of similar bilinear elasto‐plastic hysteretic SDOF systems. In this study the elasto‐plastic hysteretic SDOF systems are assigned parameters representative of steel moment resisting frames (MRFs) with post‐Northridge welded beam‐to‐column connections. In turn, the flag‐shaped hysteretic SDOF systems are representative of steel MRFs with newly proposed post‐tensioned energy‐dissipating connections. Building structures with initial periods ranging from 0.1 to 2.0s and having various strength levels are considered. It is shown that a flag‐shaped hysteretic SDOF system of equal or lesser strength can always be found to match or better the response of an elasto‐plastic hysteretic SDOF system in terms of displacement ductility and without incurring any residual drift from the seismic event. Copyright © 2002 John Wiley & Sons, Ltd.     

Evaluation of the structural response under seismic actions using non‐linear static methods

The paper investigates the degree of accuracy achievable when some non‐linear static procedures based on a pushover analysis are used to evaluate the seismic performance. In order to assess the significance of different sources of errors, three types of structural systems are analysed: (i) single‐degree‐of‐freedom (SDOF) systems with different hysteretic behaviour; (ii) shear‐type multi‐degree‐of‐freedom (MDOF) systems with elastic–perfect plastic (EPP) shear force–interstorey drift relationships; (iii) a steel moment‐resisting frame with rigid joints and EPP moment–curvature relationship. In SDOF systems, the source of approximation comes only from the calibration of the demand spectrum, while in MDOF systems some further errors are introduced by the schematization with an equivalent SDOF system. The non‐linear static procedures are compared with rigorous time‐history analyses carried out by considering ten generated earthquake ground motions compatible with the Eurocode 8 elastic spectra. It was found that SDOF systems with longer periods satisfy the equal displacement approximation regardless of the hysteretic model, while hysteresis loops with smaller energy dissipated indicate lower response for shorter periods. This is the opposite of what predicted by the ATC‐40 capacity spectrum method, which underestimates and overestimates, respectively, the actual response of low‐ and high‐ductility systems. Conversely, the inelastic spectrum method proposed by Vidic, Fajfar and Fischinger leads to the most accurate results for all types of structural systems. The analyses carried out on EPP shear‐type frames point out a large concentration of the ductility demand on some storeys. However, such a concentration markedly reduces when some hardening is accounted for. Copyright © 2006 John Wiley & Sons, Ltd.     

Mechanism and bounding of earthquake energy input to building structure on surface ground subjected to engineering bedrock motion

The mechanism of earthquake energy input to building structures is clarified by considering the surface ground amplification and soil–structure interaction. The earthquake input energies to superstructures, soil–foundation systems and total swaying–rocking system are obtained by taking the corresponding appropriate free bodies into account and defining the energy transfer functions. It has been made clear that, when the ground surface motion is white, the input energy to the swaying–rocking model is constant regardless of the soil property (input energy constant property). The upper bound of earthquake input energy to the swaying–rocking model is derived for the model including the surface ground amplification by taking full advantage of the above-mentioned input energy constant property and introducing the envelope function for the transfer function of the surface ground amplification. Extension of the theory to a general earthquake ground motion model at the engineering bedrock is also made by taking full advantage of the above-mentioned input energy constant property.     

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.     

Analytical modelling of multi‐mass flexible rocking structures

This paper presents a new analytical model for describing the large rocking response of an elastic multi‐mass structure resting on ideally rigid ground. Using the experimental results from a rocking steel column, the ability of the proposed analytical model to estimate the rocking and translational acceleration response under free vibration, pulse and earthquake excitations is evaluated. It is observed that the classical treatment of impact may result in an unrealistically large transfer of energy to vibrations. Therefore a new Dirac‐delta type impact model that spreads the effects of impact over time and space is proposed. The use of a Dirac‐delta model and accurate restitution factors play a pivotal role in prediction of rocking and acceleration responses. In order to characterize the nonlinear response better, a modal analysis of the linearized system is proposed. With this approach, the vibration mode frequencies and shapes during rocking action were determined. A comparison of analytical and experimental modal estimations suggests good agreement. The results emphasize that the vibration characteristics of several vibration modes are affected by rocking action, and these modes may be excited at impact. Copyright © 2016 John Wiley & Sons, Ltd.     

Evaluation of seismic energy in structures

Research engineers use two types of energy equations to study single-degree-of-freedom (SDOF) systems subject to earthquake induced ground motions. The first method uses an absolute energy formulation; the second method uses a relative energy formulation. While the relative energy formulation has been used in the majority of previous investigations, this study shows that the absolute energy equation is physically more meaningful. For a given ductility ratio, the input energy demands calculated by both methods are significantly different for both the short and long period ranges although the results are similar in the intermediate period range. A comparison between the analytically predicted absolute input energy of a SDOF system with the experimentally measured input energy of a six-storey braced steel frame shows good correlation.     

An equivalent SDOF system model for estimating the response of R/C building structures with proportional hysteretic dampers subjected to earthquake motions

For the purpose of estimating the earthquake response, particularly the story drift demand, of reinforced concrete (R/C) buildings with proportional hysteretic dampers, an equivalent single‐degree‐of‐freedom (SDOF) system model is proposed. Especially in the inelastic range, the hysteretic behavior of an R/C main frame strongly differs from that of hysteretic dampers due to strength and stiffness degradation in R/C members. Thus, the proposed model, unlike commonly used single‐spring SDOF system models, differentiates the restoring force characteristics of R/C main frame and hysteretic dampers to explicitly take into account the hysteretic behavior of dampers. To confirm the validity of the proposed model, earthquake responses of a series of frame models and their corresponding equivalent SDOF system models were compared. 5‐ and 10‐story frame models were studied as representative of low‐ and mid‐rise building structures, and different mechanical properties of dampers—yield strength and yield deformation—were included to observe their influence on the effectiveness of the proposed model. The results of the analyses demonstrated a good correspondence between estimated story drift demands using the proposed SDOF system model and those of frame models. Moreover, the proposed model: (i) led to better estimates than those given by a single‐spring SDOF system model, (ii) was capable of estimating the input energy demand and (iii) was capable of estimating the total hysteretic energy and the participation of dampers into the total hysteretic energy dissipation, in most cases. Results, therefore, suggest that the proposed model can be useful in structural design practice. Copyright © 2010 John Wiley & Sons, Ltd.     

Quantification of ground-motion parameters and response spectra in the near-fault region

This study focuses on the characteristics of near-fault ground motions in the forward-direction and structural response associated with them. These ground motions are narrow-banded in nature and are characterized by a predominant period at which structures excited by them are severely affected. In this work, predominant period is defined as the undamped natural period of a single-degree-of-freedom (SDOF) oscillator at which its 5% damped linear elastic pseudo-spectral velocity (PSV) contains a clear and dominant peak. It is found that a linear relationship exists between predominant period and seismic moment. An empirical equation describing this relationship is presented by using a large set of accelerograms. Attenuation equations are developed to estimate peak ground velocity (PGV) as a function of earthquake magnitude and source-to-site distance. In addition, a predictive equation for spectral shapes of PSV (i.e., PSV normalized by PGV) is presented as a continuous function of the undamped natural period of SDOF oscillators. The model is independent of PGV, and can be used in conjunction with any available PGV attenuation relation applicable to near-fault ground motion exhibiting forward-directivity effects. Furthermore, viscous damping of the SDOF is included in the model as a continuous parameter, eliminating the use of so-called damping correction factors. Finally, simple equations relating force reduction factors and displacement ductility of elasto-plastic SDOF systems are presented.     

Effect of variation of normal force on seismic performance of resilient sliding isolation systems in highway bridges

In this study, a series of shaking table tests are carried out on scaled models of two seismically isolated highway bridges to investigate the effect of rocking motion and vertical acceleration on seismic performance of resilient sliding isolators. In addition, performance of RSI is compared with system having solely natural rubber bearings. Test results show that variation of normal force on sliders due to rocking effect and vertical acceleration makes no significant difference in response of RSI systems. In addition, analytical response of prototype isolated bridge and the model used in experiments is obtained analytically by using non‐linear model for isolation systems. It is observed that for seismically isolated bridges, dynamic response of full‐scale complex structures can be predicted with acceptable accuracy by experiments using a simple model of the structure. Copyright © 2005 John Wiley & Sons, Ltd.     

A noniterative design procedure for supplemental brace–damper systems in single‐degree‐of‐freedom systems

In this paper, a method for designing supplemental brace–damper systems in single‐degree‐of‐freedom (SDOF) structures is presented. We include the effects of the supporting brace stiffness in the dynamic response by using a viscoelastic Maxwell model. On the basis of the study of an SDOF under ground excitation, we propose a noniterative design procedure for simultaneously specifying both the damper and the brace while assuring a desired structural performance. It is shown that to increase the damper size beyond the value delivered by the proposed criteria will not provide any improvement but actually worsen the structural response. The design method presented here shows excellent agreement with the FEMA 273 design approach but offers solutions closer to optimality. Copyright © 2013 John Wiley & Sons, Ltd.