Effects of stiffness degradation on ductility requirements for multistorey buildings

The effect of stiffness degradation in reinforced concrete structural members on the inelastic response of multistorey buildings to earthquakes is investigated. In particular, the following question is examined. How do the ductility requirements for multistorey systems with degrading stiffness behaviour compare with those for structures with ordinary bilinear hysteretic property? Inelastic dynamic responses of two idealized multistorey buildings, one having a long and the other a relatively short fundamental period, to an ensemble of twenty simulated earthquakes representative of moderately intense ground motions in California at moderate epicentral distances on firm ground, are analysed for ordinary bilinear hysteretic behaviour and for bilinear hysteretic behaviour with stiffness degradation property. The conclusions deduced from the results of this investigation include the following (1) It is, in general, not possible to predict the maximum response of a degrading stiffness system from results for the corresponding ordinary bilinear system (2) The differences in ductility requirements due to stiffness degradation are generally smaller than those associated with probabilistic variability from one ground motion to another (3) Stiffness degradation has little influence on the ductility requirements for flexible buildings, but it leads to increased ductility requirements for stiff buildings.     

Non-linear seismic behavior of structures with limited hysteretic energy dissipation capacity

This paper investigates the non-linear seismic behavior of structures such as slender unreinforced masonry shear walls or precast post-tensioned reinforced concrete elements, which have little hysteretic energy dissipation capacity. Even if this type of seismic response may be associated with significant deformation capacity, it is usually not considered as an efficient mechanism to withstand strong earthquakes. The objective of the investigations is to propose values of strength reduction factors for seismic analysis of such structures. The first part of the study is focused on non-linear single-degree-of-freedom (SDOF) systems. A parametric study is performed by computing the displacement ductility demand of non-linear SDOF systems for a set of 164 recorded ground motions selected from the European Strong Motion Database. The parameters investigated are the natural frequency, the strength reduction factor, the post-yield stiffness ratio, the hysteretic energy dissipation capacity and the hysteretic behavior model (four different hysteretic models: bilinear self-centring, with limited or without energy dissipation capacity, modified Takeda and Elastoplastic). Results confirm that the natural frequency has little influence on the displacement ductility demand if it is below a frequency limit and vice versa. The frequency limit is found to be around 2 Hz for all hysteretic models. Moreover, they show that the other parameters, especially the hysteretic behavior model, have little influence on the displacement ductility demand. New relationships between the displacement ductility demand and the strength reduction factor for structures having little hysteretic energy dissipation capacity are proposed. These relationships are an improvement of the equal displacement rule for the considered hysteretic models. In the second part of the investigation, the parametric study is extended to multi-degree-of-freedom (MDOF) systems. The investigation shows that the results obtained for SDOF systems are also valid for MDOF systems. However, the SDOF system overestimates the displacement ductility demand in comparison to the corresponding MDOF system by approximately 15%.     

Determination of ductility factor considering different hysteretic models

In current seismic design procedures, base shear is calculated by the elastic strength demand divided by the strength reduction factor. This factor is well known as the response modification factor, R, which accounts for ductility, overstrength, redundancy, and damping of a structural system. In this study, the R factor accounting for ductility is called the ‘ductility factor’, R_μ. The R_μ factor is defined as the ratio of elastic strength demand imposed on the SDOF system to inelastic strength demand for a given ductility ratio. The R_μ factor allows a system to behave inelastically within the target ductility ratio during the design level earthquake ground motion. The objective of this study is to determine the ductility factor considering different hysteretic models. It usually requires large computational efforts to determine the R_μ factor. In order to reduce the computational efforts, the R_μ factor is prepared as a functional form in this study. For this purpose, statistical studies are carried out using forty different earthquake ground motions recorded at a stiff soil site. The R_μ factor is assumed to be a function of the characteristic parameters of each hysteretic model, target ductility ratio and structural period. The effects of each hysteretic model to the R_μ factor are also discussed. Copyright © 1999 John Wiley & Sons, Ltd.     

Analytical modelling of near‐source pulse‐like seismic demand for multi‐linear backbone oscillators

Nonlinear static procedures, which relate the seismic demand of a structure to that of an equivalent single‐degree‐of‐freedom oscillator, are well‐established tools in the performance‐based earthquake engineering paradigm. Initially, such procedures made recourse to inelastic spectra derived for simple elastic–plastic bilinear oscillators, but the request for demand estimates that delve deeper into the inelastic range, motivated investigating the seismic demand of oscillators with more complex backbone curves. Meanwhile, near‐source (NS) pulse‐like ground motions have been receiving increased attention, because they can induce a distinctive type of inelastic demand. Pulse‐like NS ground motions are usually the result of rupture directivity, where seismic waves generated at different points along the rupture front arrive at a site at the same time, leading to a double‐sided velocity pulse, which delivers most of the seismic energy. Recent research has led to a methodology for incorporating this NS effect in the implementation of nonlinear static procedures. Both of the previously mentioned lines of research motivate the present study on the ductility demands imposed by pulse‐like NS ground motions on oscillators that feature pinching hysteretic behaviour with trilinear backbone curves. Incremental dynamic analysis is used considering 130 pulse‐like‐identified ground motions. Median, 16% and 84% fractile incremental dynamic analysis curves are calculated and fitted by an analytical model. Least‐squares estimates are obtained for the model parameters, which importantly include pulse period T_p. The resulting equations effectively constitute an R ? μ ? T ? T_p relation for pulse‐like NS motions. Potential applications of this result towards estimation of NS seismic demand are also briefly discussed. Copyright © 2016 John Wiley & Sons, Ltd.     

Dynamic response of bilinear asymmetric structures

The inelastic behaviour of eccentric single-storey building structures subjected to sinusoidal ground excitation is examined. The Kryloff-Bogoliuboff method is employed to provide approximate solutions in the amplitude-frequency domain. Structural resisting elements are assumed to exhibit bilinear hysteretic behaviour and coupled response is investigated in terms of both system response as well as individual element ductility requirements. In addition to demonstrating the well-known softening property inherent in yielding systems, the importance of the principal parameters governing coupled response is evaluated in a consistent parametric fashion. Within the context of earthquake resistant building design, the results indicate the absence of amplified response when torsional and translational frequencies are close, in contrast to the much emphasized observation of internal resonance for linear elastic structures. Equally important, structural elements located on the stiff edge of eccentric buildings are found to be only marginally affected by the magnitude of the eccentricity, thus indicating that seismic building codes which reduce design requirements for these elements underestimate actual behaviour substantially.     

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.     

Base isolation systems for earthquake protection of multi-storey shear structures

This paper is a study of the effectiveness of a wide range of bilinear hysteretic isolation systems in shielding multistorey 2-D shear structures from earthquake excitations. Important parameters of the isolation system are identified and their effect on structure response noted. It is shown that isolation systems can be constructed which allow the structure proper to remain purely elastic even during very strong ground motions. It is further shown that the shear responses and base displacements of structures on these isolation systems can be accurately estimated from elastic response spectra of the forcing earthquakes. The philosophy of structure isolation is discussed and an introduction given to the physical devices currently available to provide it.     

Effect of peak ground acceleration to velocity ratio on ductility demand of inelastic systems

A statistical analysis is performed to investigate the significance of peak ground acceleration to velocity ratio (a/v) on the displacement ductility demand of simple bilinear hysteretic systems. Three groups of earthquake records representative of low, normal and high<a/v ranges are used as input ground motions. The design yield strength of the inelastic systems is specified from the base shear formula in the 1980 National Building Code of Canada (NBCC 1980) and that in NBCC 1985 respectively. The former case represents the common practice of specifying seismic design base shear based on a peak site acceleration, while in the latter case the base shear is specified based on peak ground velocity and a/v ratio. Mean displacement ductility demands are obtained for the three groups of ground motions; and the corresponding dispersion characteristics are examined. The results show that the ground motion<a/v range has a significant effect on the displacement ductility demand, and it should be accounted for in design strength specification.     

Constant-ductility inelastic displacement ratios for displacement-based seismic design of self-centering structures

This paper focuses on constant-ductility inelastic displacement ratios of self-centering single-degree-of-freedom (SDF) systems with two different levels of energy dissipation capacity, in the presence of 5% viscous damping ratio. A statistical analysis is developed considering an earthquake database composed of 228 ground motions recorded in California with magnitudes greater than six and organized for NEHRP soil class, ground motion duration, and peak ground acceleration. The response of self-centering SDF systems with large variability of initial periods, ductility levels, and postyield stiffness ratios is investigated and compared with the responses of SDF systems with bilinear plastic, Clough, and Takeda hysteresis. The inelastic demand variation with soil class, initial period, postyield stiffness ratio, unloading stiffness degradation, ductility level, and hysteretic behavior is highlighted. Simple and conservative analytical estimates of constant-ductility inelastic displacement ratios for mean and 90th percentile values in terms of initial period, ductility level, and postyield stiffness ratio are proposed to allow the extension of the Displacement-Based Design via Inelastic Displacement Ratio (C_μDBD) to self-centering structural systems.     

双向近场地震作用下HDR隔震梁桥地震响应研究

以HDR隔震梁桥多自由度（MDOF）模型和等效双线性单自由度（SDOF）模型为研究对象，以典型近场地震动作为输入，研究HDR支座双向耦合效应对HDR隔震梁桥地震响应的影响。研究结果表明:不考虑双向耦合效应的HDR支座滞回曲线呈典型双线性；考虑双向耦合效应的HDR支座滞回曲线面积小于不考虑双向耦合效应的HDR支座滞回曲线面积。不考虑双向耦合效应的顺桥向HDR支座位移峰值d_b大于考虑双向耦合效应时，但横桥向的结果相反。近场地震作用下，对梁桥进行HDR支座隔震设计时，忽略双向耦合效应计算得到的墩底剪力峰值和弯矩峰值均偏于保守。可忽略HDR支座双向耦合效应对HDR隔震梁桥近场地震能量的影响。     

配置自复位耗能跨低多层钢框架体系的能量系数

利用超弹性SMA螺栓梁柱节点的耗能能力和自复位特性,将其引入到耗能跨而构建"自复位耗能跨",基于既有的节点试验研究结果对结构体系的滞回性能进行了探讨。在此基础上,以具有旗形滞回特征的单自由度体系为工具,对配置自复位耗能跨低多层钢框架体系的能量系数进行推导。能量系数可以合理量化具有旗形滞回规则结构的峰值响应需求,能量系数越低,表明地震动下结构的峰值响应越低。为了阐明滞回参数对能量系数的影响,对具有不同滞回参数组合可代表低多层结构的等效SDOF体系进行了非线性动力分析,参数组合包括周期、屈服后刚度比、延性系数及能量比。同时对能量系数的离散性也进行了分析。结果表明:能量系数及能量系数的离散性受结构周期、屈服后刚度比及延性系数影响较大,受能量比的影响较小。     

脉冲型近断层地震动作用单自由度体系响应分析

近断层前方向性效应地震动含有高幅值,短持时的速度脉冲,与远场地震动相比存在显著差异。本文根据所选取的40条近断层地震波记录,用小波分析方法将原始记录分解为脉冲波部分和高频波部分,对弹性和非弹性单自由度体系进行分析,得出了以下结论:对于弹性体系,大约0.484倍的速度脉冲周期可以作为临界周期,脉冲波部分将对固有周期大于临界周期的结构的响应起主导作用,反之,高频波部分将会产生显著影响;对于非弹性体系,仅仅用等效速度脉冲方法模拟近断层地震动的计算精度将会受到延性系数?的影响,随着延性系数的增加,脉冲波部分满足精度要求的结构固有周期范围将明显缩小,并且向较低周期范围偏移;仅用等效速度脉冲模型来模拟近断层地震动具有一定的局限性。     

Behavior of moment‐resisting frame structures subjected to near‐fault ground motions

Near‐fault ground motions impose large demands on structures compared to ‘ordinary’ ground motions. Recordings suggest that near‐fault ground motions with ‘forward’ directivity are characterized by a large pulse, which is mostly orientated perpendicular to the fault. This study is intended to provide quantitative knowledge on important response characteristics of elastic and inelastic frame structures subjected to near‐fault ground motions. Generic frame models are used to represent MDOF structures. Near‐fault ground motions are represented by equivalent pulses, which have a comparable effect on structural response, but whose characteristics are defined by a small number of parameters. The results demonstrate that structures with a period longer than the pulse period respond very differently from structures with a shorter period. For the former, early yielding occurs in higher stories but the high ductility demands migrate to the bottom stories as the ground motion becomes more severe. For the latter, the maximum demand always occurs in the bottom stories. Preliminary regression equations are proposed that relate the parameters of the equivalent pulse to magnitude and distance. The equivalent pulse concept is used to estimate the base shear strength required to limit story ductility demands to specific target values. Copyright © 2004 John Wiley & Sons, Ltd.     

双向速度脉冲地震下偏心重力柱核心筒结构的弹塑性地震响应研究

研究速度脉冲地震和结构质量偏心综合不利条件下新型重力柱-核心筒结构体系的弹塑性反应规律。选取速度脉冲和非速度脉冲地震加速度记录各10条,进行地震动双向输入,采用结构非线性分析软件CANNY进行有限元数值分析,研究脉冲型地震和结构质量偏心对新型体系弹塑性地震反应的影响。分析结果表明,速度脉冲型地震作用下各结构的层间位移角、层间剪力和层间扭转角显著高于非速度脉冲地震下的相应值。质量偏心对结构弹塑性抗震需求影响显著,层间位移角和层间扭转角都随着偏心率的增大而增大,而层间剪力则随偏心率的增大呈减小趋势。建议在重力柱-核心筒结构设计中应重视速度脉冲地震和结构偏心的耦合不利影响。     

Inelastic seismic response of code-designed multistorey frame buildings with regular asymmetry

Based on an asymmetric multistorey frame building model, this paper investigates the influence of a building's higher vibration modes on its inelastic torsional response and evaluates the adequacy of the provisions of current seismic building codes and the modal analysis procedure in accounting for increased ductility demand in frames situated at or near the stiff edge of such buildings. It is concluded that the influence of higher vibration modes on the response of the upper-storey columns of stiff-edge frames increases significantly with the building's fundamental uncoupled lateral period and the magnitude of the stiffness eccentricity. The application of the equivalent static torsional provisions of certain building codes may lead to non-conservative estimates of the peak ductility demand, particularly for structures with large stiffness eccentricity. In these cases, the critical elements are vulnerable to excessive additional ductility demand and, hence, may be subject to significantly more severe structural damage than in corresponding symmetric buildings. It is found that regularly asymmetric buildings excited well into the inelastic range may not be conservatively designed using linear elastic modal analysis theory. Particular caution is required when applying this method to the design of stiff-edge frame elements in highly asymmetric structures.     

Collapse period of degrading SDOF systems

Seismic demand estimation for a structure is a critical issue for seismic performance assessment so that the potential damage can be estimated realistically. Many researchers proposed simplified methods to estimate the demand of a structure under strong ground motions. However, most of them did not consider degradation and collapse potential of the structures. Even some of theme considered the degradation effect, stiffness and strength degradation effects were considered separately without collapse potential caused by dynamic instability. In this study, collapse potential of SDOF systems caused by dynamic instability with stiffness and strength degradation has been investigated. Nonlinear time history analyses were performed, using an energy-based, strength and stiffness degraded hysteretic model that considers the collapse potential, with 160 earthquake acceleration time histories. An equation was proposed for the estimation of collapse period of SDOF systems as a function of certain strength reduction factor, ductility level and post-capping stiffness ratio. Finally, effects of parameters of the considered hysteretic model and local site conditions on the collapse period were investigated.     

Effects of supplemental viscous damping on inelastic seismic response of asymmetric systems

This paper investigates the effects of supplemental viscous damping on the seismic response of one‐storey, asymmetric‐plan systems responding in the inelastic range of behaviour. It was found that addition of the supplemental damping reduces not only deformation demand but also ductility and hysteretic energy dissipation demands on lateral load resisting elements during earthquake loading. However, the level of reduction strongly depends on the plan‐wise distribution of supplemental damping. Nearly optimal reduction in demands on the outermost flexible‐side element, an element generally considered to be the most critical element, was realized when damping was distributed unevenly in the system plan such that the damping eccentricity was equal in magnitude but opposite in algebraic sign to the structural eccentricity of the system. These results are similar to those noted previously for linear elastic systems, indicating that supplemental damping is also effective for systems expected to respond in the inelastic range. Copyright © 2001 John Wiley & Sons, Ltd.     

Scaling of ductility and damage‐based strength reduction factors for horizontal motions

The conventional approach of obtaining the inelastic response spectra for the aseismic design of structures involves the reduction of elastic spectra via response modification factors. A response modification factor is usually taken as a product of (i) strength factor, R_S, (ii) ductility factor, R_μ, and (iii) redundancy factor, R_R. Ductility factor, also known as strength reduction factor (SRF), is considered to primarily depend on the initial time period of the single‐degree‐of‐freedom (SDOF) oscillator and the displacement ductility demand ratio for the ground motion. This study proposes a preliminary scaling model for estimating the SRFs of horizontal ground motions in terms of earthquake magnitude, strong motion duration and predominant period of the ground motion, geological site conditions, and ductility demand ratio, with a given level of confidence. The earlier models have not considered the simultaneous dependence of the SRFs on various governing parameters. Since the ductility demand ratio is not a complete measure of the cumulative damage in the structure during the earthquake‐induced vibrations, the existing definition of the SRF is sought to be modified with the introduction of damage‐based SRF (in place of ductility‐based SRF). A parallel scaling model has been proposed for estimating the damage‐based SRFs. This model considers damage and ductility supply ratio as parameters instead of ductility demand ratio. Through a parametric study on ductility‐based SRFs, it has been shown that the hitherto assumed insensitivity of earthquake magnitude and strong motion duration may not be always justified and that the initial time period of the oscillator plays an important role in the dependence of SRF on these parameters. Further, the damage‐based SRFs are found to show similar parametric dependence as observed in the case of the ductility‐based SRFs. Copyright © 2000 John Wiley & Sons, Ltd.     

Yielding oscillator subjected to simple pulse waveforms: numerical analysis & closed‐form solutions

Numerical and analytical solutions are presented for the elastic and inelastic response of single‐degree‐of‐freedom yielding oscillators to idealized ground acceleration pulses. These motions are typical of near‐fault earthquake recordings generated by forward rupture directivity and may inflict damage in the absence of substantial structural strength and ductility capacity. Four basic pulse waveforms are examined: (1) triangular; (2) sinusoidal; (3) exponential; and (4) rectangular. In the first part of the article, a numerical study is presented of the effect of oscillator period, strength, damping, post‐yielding stiffness and number of excitation cycles, on inelastic response. Results are presented in the form of dimensionless graphs and regression formulas that elucidate the salient features of the problem. It is shown that conventional R–µ relations may significantly underestimate ductility demand imposed by near‐fault motions. The second part of the article concentrates on elastic‐perfectly plastic oscillators. Closed‐form solutions are derived for post‐yielding response and associated ductility demand. It is shown that all three ground motion histories (i.e. acceleration, velocity, and displacement) control oscillator response—contrary to the widespread view that ground velocity alone is of leading importance. The derived solutions provide insight on the physics of inelastic response, which is often obscured by the complexity of numerical algorithms and actual earthquake motions. The model is evaluated against numerical results from near‐field recordings. A case study is presented. Copyright © 2006 John Wiley & Sons, Ltd.