Inelastic displacement demands in steel structures and their relationship with earthquake frequency content parameters

This paper deals with the estimation of peak inelastic displacements of SDOF systems, representative of typical steel structures, under constant relative strength scenarios. Mean inelastic deformation demands on bilinear systems (simulating moment resisting frames) are considered as the basis for comparative purposes. Additional SDOF models representing partially‐restrained and concentrically‐braced (CB) frames are introduced and employed to assess the influence of different force‐displacement relationships on peak inelastic displacement ratios. The studies presented in this paper illustrate that the ratio between the overall yield strength and the strength during pinching intervals is the main factor governing the inelastic deformations of partially‐restrained models and leading to significant differences when compared with predictions based on bilinear structures, especially in the short‐period range. It is also shown that the response of CB systems can differ significantly from other pinching models when subjected to low or moderate levels of seismic demand, highlighting the necessity of employing dedicated models for studying the response of CB structures. Particular attention is also given to the influence of a number of scalar parameters that characterise the frequency content of the ground motion on the estimated peak displacement ratios. The relative merits of using the average spectral period T_aver, mean period T_m, predominant period T_g, characteristic period T_c and smoothed spectral predominant period T_o of the earthquake ground motion, are assessed. This paper demonstrates that the predominant period, defined as the period at which the input energy is maximum throughout the period range, is the most suitable frequency content scalar parameter for reducing the variability in displacement estimations. Finally, noniterative equivalent linearisation expressions based on the secant period and equivalent damping ratios are presented and verified for the prediction of peak deformation demands in steel structures. Copyright © 2011 John Wiley & Sons, Ltd.     

Prediction of spatially distributed seismic demands in specific structures: Structural response to loss estimation

A companion paper has investigated the effects of intensity measure (IM) selection in the prediction of spatially distributed response in a multi‐degree‐of‐freedom structure. This paper extends from structural response prediction to performance assessment metrics such as probability of structural collapse; probability of exceeding a specified level of demand or direct repair cost; and the distribution of direct repair loss for a given level of ground motion. In addition, a method is proposed to account for the effect of varying seismological properties of ground motions on seismic demand that does not require different ground motion records to be used for each intensity level. Results illustrate that the conventional IM, spectral displacement at the first mode, S_de(T₁), produces higher risk estimates than alternative velocity‐based IM's, namely spectrum intensity, SI, and peak ground velocity, PGV, because of its high uncertainty in ground motion prediction and poor efficiency in predicting peak acceleration demands. Copyright © 2009 John Wiley & Sons, Ltd.     

Comparing response of SDF systems to near‐fault and far‐fault earthquake motions in the context of spectral regions

In spite of important differences in structural response to near‐fault and far‐fault ground motions, this paper aims at extending well‐known concepts and results, based on elastic and inelastic response spectra for far‐fault motions, to near‐fault motions. Compared are certain aspects of the response of elastic and inelastic SDF systems to the two types of motions in the context of the acceleration‐, velocity‐, and displacement‐sensitive regions of the response spectrum, leading to the following conclusions. (1) The velocity‐sensitive region for near‐fault motions is much narrower, and the acceleration‐sensitive and displacement‐sensitive regions are much wider, compared to far‐fault motions; the narrower velocity‐sensitive region is shifted to longer periods. (2) Although, for the same ductility factor, near‐fault ground motions impose a larger strength demand than far‐fault motions—both demands expressed as a fraction of their respective elastic demands—the strength reduction factors R_y for the two types of motions are similar over corresponding spectral regions. (3) Similarly, the ratio u_m/u₀ of deformations of inelastic and elastic systems are similar for the two types of motions over corresponding spectral regions. (4) Design equations for R_y (and for u_m/u₀) should explicitly recognize spectral regions so that the same equations apply to various classes of ground motions as long as the appropriate values of T_a, T_b and T_c are used. (5) The Veletsos–Newmark design equations with T_a=0.04 s, T_b=0.35 s, and T_c=0.79 s are equally valid for the fault‐normal component of near‐fault ground motions. Copyright © 2001 John Wiley & Sons, Ltd.     

黄河阶地特征与地震动参数的相关性研究

在分析已有资料的基础上划分了兰州盆地与建设工程分布密切相关的T₀、T₁、T₂、T₃和T₄级黄河阶地,建立201个土层地震反应模型。通过一维等效线性化计算和反应谱分析,得出兰州盆地沉积阶地50年超越概率10%地表地震动参数,分析阶地高度和v_S≤500 m/s覆盖层厚度特征与地震动参数峰值加速度Am和加速度反应谱特征周期T_g的相关性。表明兰州盆地T₀~T₂阶地覆盖层厚度与50年超越概率10%A_m呈正相关,T₃及以上阶地覆盖层厚度对A_m增大有明显的减小作用。T_g值随T₀~T₃阶地覆盖层厚度的增加而变大,当覆盖层厚度进一步变大,T_g值不再同步增大,阶地覆盖层厚度对T_g的影响是有限的,阶地海拔高度与地表50年超越概率10%地震动参数没有关系。     

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.     

Nonlocal period parameters of frequency content characterization for near-fault ground motions

This paper focuses on the frequency property analysis of near-fault ground motions with and without distinct pulses, separately from the Chi-Chi and Northridge earthquakes. Ten scalar period parameters of ground motions, especially several nonlocal period parameters, are considered. Two new nonlocal parameters, namely the mean period of Hilbert marginal spectrum (T_mh) and the improved characteristic period (T_gi), are suggested. Moreover, comprehensive comparison and analysis indicate that T_mh, T_gi and T_avg (average spectral period) can distinguish the low-frequency components of near-fault ground motions; T_m (mean period of Fourier amplitude spectrum) and T_o (smoothed spectral predominant period) represent the moderate- and high-frequency components, respectively. The variance coefficient of predominant instantaneous frequency of Hilbert spectrum (H_cov) can be regarded as an alternative index to measure the non-stationary degree of near-fault ground motions. Finally, the velocity pulses and earthquake magnitude remarkably affect the frequency parameters of near-fault ground motions. Copyright © 2012 John Wiley & Sons, Ltd.     

Near‐fault ground motions,and the response of elastic and inelastic single‐degree‐of‐freedom (SDOF) systems

In order to investigate the response of structures to near‐fault seismic excitations, the ground motion input should be properly characterized and parameterized in terms of simple, yet accurate and reliable, mathematical models whose input parameters have a clear physical interpretation and scale, to the extent possible, with earthquake magnitude. Such a mathematical model for the representation of the coherent (long‐period) ground motion components has been proposed by the authors in a previous study and is being exploited in this article for the investigation of the elastic and inelastic response of the single‐degree‐of‐freedom (SDOF) system to near‐fault seismic excitations. A parametric analysis of the dynamic response of the SDOF system as a function of the input parameters of the mathematical model is performed to gain insight regarding the near‐fault ground motion characteristics that significantly affect the elastic and inelastic structural performance. A parameter of the mathematical representation of near‐fault motions, referred to as ‘pulse duration’ (T_P), emerges as a key parameter of the problem under investigation. Specifically, T_P is employed to normalize the elastic and inelastic response spectra of actual near‐fault strong ground motion records. Such normalization makes feasible the specification of design spectra and reduction factors appropriate for near‐fault ground motions. The ‘pulse duration’ (T_P) is related to an important parameter of the rupture process referred to as ‘rise time’ (τ) which is controlled by the dimension of the sub‐events that compose the mainshock. Copyright © 2004 John Wiley & Sons, Ltd.     

Ductility‐dependent intensity measure that accounts for ground‐motion spectral shape and duration

The calculated nonlinear structural responses of a building can vary greatly, even if recorded ground motions are scaled to the same spectral acceleration at a building's fundamental period. To reduce the variation in structural response at a particular ground‐motion intensity, this paper proposes an intensity measure (IM_comb) that accounts for the combined effects of spectral acceleration, ground‐motion duration, and response spectrum shape. The intensity measure includes a new measure of spectral shape that integrates the spectrum over a period range that depends on the structure's ductility. The new IM is efficient, sufficient, scalable, transparent, and versatile. These features make it suitable for evaluating the intensities of measured and simulated ground motions. The efficiency and sufficiency of the new IM is demonstrated for the following: (i) elastic‐perfectly plastic single‐degree‐of‐freedom (SDOF) oscillators with a variety of ductility demands and periods; (ii) ductile and brittle deteriorating SDOF systems with a variety of periods; and (iii) collapse analysis for 30 previously designed frames. The efficiency is attributable to the inclusion of duration and to the ductility dependence of the spectral shape measure. For each of these systems, the transparency of the intensity measure made it possible to identify the sensitivity of structural response to the various characteristics of the ground motion. Spectral shape affected all structures, but in particular, ductile structures. Duration only affected structures with cyclic deterioration. Copyright © 2015 John Wiley & Sons, Ltd.     

Near‐fault effects on residual displacements of RC structures

Residual displacements of single‐degree‐of‐freedom systems due to ground motions with velocity pulses or fling step displacements are presented as a function of period T and of its ratio to the pulse period T_p. Four hysteretic behaviors are considered: bilinear elastoplastic, stiffness‐degrading with cycling, stiffness‐cum‐strength degrading, with or without pinching. When expressed in terms of T/T_p, peak inelastic and residual displacements due to motions with a pulse or fling appear similar to those due to far‐fault motions, if the response to far‐field records are expressed in terms of the ratio of T to the record's characteristic period. However, as the latter is usually much shorter than the pulse period of motions with fling, the range of periods of interest for common structures becomes a short‐period range under fling motions and exhibits very large amplification of residual and peak inelastic displacements. Similar, but less acute, are the effects of motions with a velocity pulse. Wavelets of different complexity are studied as approximations to near‐fault records. Simple two‐parameter wavelets for fling motions overestimate peak inelastic displacements; those for pulse‐type motions overestimate residual displacements. A more complex four‐parameter wavelet for motions with a velocity pulse predicts overall well residual and peak displacements due to either pulse‐ or fling‐type motions; a hard‐to‐identify parameter of the wavelet impacts little computed residual displacements; another significantly affects them and should be carefully estimated from the record. Even this most successful of wavelets overpredicts residual displacements for the periods of engineering interest. Copyright © 2016 John Wiley & Sons, Ltd.     

Influence of long‐period filter cut‐off on elastic spectral displacements

The effect of the long‐period filter cut‐off, T_c, on elastic spectral displacements is investigated using a strong ground‐motion database from Europe and the Middle East. The relation between the filter and oscillator responses is considered to observe the influence of T_c for both analogue and digital records, and the variations with site classification, magnitude, filter order and viscous damping. Robust statistics are derived using the re‐processed European data to generalize the effects of the long‐period filter cut‐off on maximum oscillator deformation demands as a function of these seismological and structural features. Statistics with a 95% confidence interval are derived to suggest usable period ranges for spectral displacement computations as a function of T_c. The results indicate that the maximum period at which spectral displacements can be confidently calculated depend strongly on the site class, magnitude and filter order. The period range where reliable long‐period information can be extracted from digital accelerograms is twice that of analogue records. Copyright © 2006 John Wiley & Sons, Ltd.     

Determination of first mode ground period and site-dependent design response spectrum for Taiwan High Speed Rail

The objective of this paper is to identify the first mode ground period T_g, at each interesting site along Taiwan High Speed Rail (THSR) route, and further, to determine the site-dependent design response spectrum for particular sites with T_g larger than 1 s. In this paper, a linear model of shear wave propagating in a multi-layered half-space is developed to identify the first mode ground period T_g of an interesting site on the basis of the field bore hole data. Furthermore, for each one of the strong ground motion observation stations within the Chiayi-Tainan area, the first mode ground period T_g was identified by either coda waves or pre-event ambient vibrations from the seismograms. In addition, the site-specific parameter C_v in defining the normalized response spectrum was also determined based on the observed strong ground motions at the specified observation station, and hence the empirical function between T_g and C_v can be regressed for this interesting area. Therefore, for a particular site along THSR route within the Chiayi-Tainan area, the site-specific parameter C_v can be determined from the regressed empirical function by the identified first mode ground period T_g, and further, used to develop the site-dependent design response spectrum.     

Estimation of inelastic displacement ratio for base‐isolated structures

This study develops a straightforward approximate method to estimate inelastic displacement ratio, C₁ for base‐isolated structures subjected to near‐fault and far‐fault ground motions. Taking into account the inelastic behavior of isolator and superstructure, a 2 degrees of freedom model is employed. A total of 90 earthquake ground motions are selected and classified into different clusters according to the frequency content features of records represented by the peak ground acceleration to peak ground velocity ratio, A_p/V_p. A parametric study is conducted, and effective factors in C₁ (i.e., fundamental vibration period of the superstructure, T_s; postyield stiffness ratio of the superstructure, α_s; strength reduction ratio, R; vibration period of the isolator, T_b; strength of the isolator, Q; ratio of superstructure mass to total mass of the system, γ_m) are recognized. The results indicate that the practical range of C₁ values could be expected for base‐isolated structures. Subsequently, effective parameters are included in simple predictive equations. Finally, the accuracy of the proposed approximate equations is evaluated and verified through error measurement, and comparisons are made in the analyses.     

Characterization of forward-directivity ground motions in the near-fault region

Ground motions close to a ruptured fault resulting from forward-directivity are significantly different than other ground motions. These pulse-type motions can place severe demands on structures in the near-fault region. To aid in the characterization of these special type of ground motions, a simplified parameterization is proposed based on a representative amplitude, pulse period, and number of significant pulses in the velocity–time history. Empirical relationships were developed for estimating the peak ground velocity (PGV) and period of the velocity pulse (T_v) of available forward-directivity motions. PGV in the near-fault region varies significantly with magnitude and distance. Additionally, the PGV for soil sites are systematically larger than those at rocks sites. T_v is a function of moment magnitude and site conditions with most of the energy being concentrated within a narrow-period band centered on the pulse period. Hence, lower magnitude events, which produce lower pulse periods, might produce more damaging ground motions for the stiff structures more common in urban areas.     

Estimation of seismic demands on isolators in asymmetric buildings using non‐linear analysis

A procedure based on rigorous non‐linear analysis is presented that estimates the peak deformation among all isolators in an asymmetric building due to strong ground motion. The governing equations are reduced to a form such that the median normalized deformation due to an ensemble of ground motions with given corner period T_d depends primarily on four global parameters of the isolation system: the isolation period T_b, the normalized strength η, the torsional‐to‐lateral frequency ratio Ω_θ, and the normalized stiffness eccentricity e_b/r. The median ratio of the deformations of the asymmetric and corresponding symmetric systems is shown to depend only weakly on T_b, η, and Ω_θ, but increases with e_b/r. The equation developed to estimate the largest ratio among all isolators depends only on the stiffness eccentricity and the distance from the center of mass to the outlying isolator. This equation, multiplied by an earlier equation for the deformation of the corresponding symmetric system, provides a design equation to estimate the deformations of asymmetric systems. This design equation conservatively estimates the peak deformation among all isolators, but is generally within 10% of the ‘exact’ value. Relative to the non‐linear procedure presented, the peak isolator deformation is shown to be significantly underestimated by the U.S. building code procedures. Copyright © 2003 John Wiley & Sons, Ltd.     

Multifractal characteristic analysis of near-fault earthquake ground motions

This study aims to reveal the multi-scaling behavior and quantify the irregularity of near-fault earthquake ground motions from a new perspective of multifractal theory. Based on multifractal detrended fluctuation analysis, the multifractal characteristic parameters of acceleration time series for typical near-fault ground motions are calculated, and their correlations with two period parameters (i.e., mean period T_m and characteristic period T_c) and box-counting fractal dimensions are analyzed. Numerical results of strong nonlinear dependence of generalized Hurst exponents h(q) upon the fluctuation orders q indicate that near-fault ground motions present the multifractal properties and long-range correlation obviously. Furthermore, the scaling exponent h(2) of near-fault records has a strong correlation with their periods T_m and T_c, and strongly negative correlation with their box dimension. Moreover, h(2) can be regarded as a measure of frequency content and irregularity degree of strong earthquake ground motions. Finally, it is pointed out that the long-range correlation of small and large fluctuation is the major source of multifractality of near-fault ground motions.     

Pulse-like versus non-pulse-like ground motion records: Spectral shape comparisons and record selection strategies

Pulse-like records are well recognized for their potential to impose higher demands on structures when compared with ordinary records. The increased severity of the structural response usually caused by pulse-like records is commonly attributed to the spectral increment around the pulse period. By comparing the building response to sets of spectrally equivalent pulse-like and ordinary records, we show that there are characteristics of pulse-like records beyond the shape of the acceleration response spectrum that affect the results of nonlinear dynamic analysis. Nevertheless, spectral shape together with the ratio of pulse period to the first-mode structural period, T_p/T₁, are confirmed as “sufficient” predictors for deformation and acceleration response metrics in a building, conditioned on the seismic intensity. Furthermore, the average spectral acceleration over a period range, AvgSA, is shown to incorporate to a good proxy for spectral shape, and together with T_p/T₁, form an efficient and sufficient intensity measure for response prediction to pulse-like ground motions. Following this latter route, we propose a record selection scheme that maintains the consistency of T_p with the hazard of the site but uses AvgSA to account for the response sensitivity to spectral shape.     

Ground motion scaling methods for different site conditions and structure characteristics

Non‐linear dynamic time‐history analyses conducted as part of a performance‐based seismic design approach often require that the ground motion records are scaled to a specified level of seismic intensity. Recent research has demonstrated that certain ground motion scaling methods can introduce a large scatter in the estimated seismic demands. The resulting demand estimates may be biased, leading to designs with significant uncertainty and unknown margins of safety, unless a relatively large ensemble of ground motion records is used. This paper investigates the effectiveness of seven ground motion scaling methods in reducing the scatter in estimated peak lateral displacement demands. Non‐linear single‐degree‐of‐freedom systems and non‐linear multi‐degree‐of‐freedom systems are considered with different site conditions (site soil profile and epicentral distance) and structural characteristics (yield strength, period, and hysteretic behavior). It is shown that scaling methods that work well for ground motions representative of stiff soil and far‐field conditions lose their effectiveness for soft soil and near‐field conditions for a wide range of structural characteristics. Copyright © 2003 John Wiley & Sons, Ltd.     

Fractal characterization and frequency properties of near-fault ground motions

This study explores the irregularity and complexity of strong earthquake ground motions from the perspective of fractal geometry, and constructs a relation with the frequency content of the ground motions. The box-counting fractal dimensions and five representative period parameters of near-fault ground motions from the Chi-Chi and Northridge earthquakes are calculated and compared. Numerical results indicate that the acceleration and velocity time histories of ground motions present the statistical fractal property, and the dominant pulses of near-fault ground motions have a significant influence on their box dimensions and periods. Further, the average box dimension of near-fault impulsive ground motions is smaller, and their irregular degree of wave forms is lower. Moreover, the box dimensions of ground motions reflect their frequency properties to a large extent, and can be regarded as an alternative indicator to represent their frequency content. Finally, the box dimension D of the acceleration histories shows a considerably negative correlation with the mean period T. Meanwhile, the box dimension of the velocity histories Dye is negatively correlated with the characteristic period T and improved characteristic period Tgi.     

Influence of deterioration modelling on the seismic response of steel moment frames designed to Eurocode 8

This paper assesses the influence of cyclic and in‐cycle degradation on seismic drift demands in moment‐resisting steel frames (MRF) designed to Eurocode 8. The structural characteristics, ground motion frequency content, and level of inelasticity are the primary parameters considered. A set of single‐degree‐of‐freedom (SDOF) systems, subjected to varying levels of inelastic demands, is initially investigated followed by an extensive study on multi‐storey frames. The latter comprises a large number of incremental dynamic analyses (IDA) on 12 frames modelled with or without consideration of degradation effects. A suite of 56 far‐field ground motion records, appropriately scaled to simulate 4 levels of inelastic demand, is employed for the IDA. Characteristic results from a detailed parametric investigation show that maximum response in terms of global and inter‐storey drifts is notably affected by degradation phenomena, in addition to the earthquake frequency content and the scaled inelastic demands. Consistently, both SDOF and frame systems with fundamental periods shorter than the mean period of ground motion can experience higher lateral strength demands and seismic drifts than those of non‐degrading counterparts in the same period range. Also, degrading multi‐storey frames can exhibit distinctly different plastic mechanisms with concentration of drifts at lower levels. Importantly, degrading systems might reach a “near‐collapse” limit state at ductility demand levels comparable to or lower than the assumed design behaviour factor, a result with direct consequences on optimised design situations where over‐strength would be minimal. Finally, the implications of the findings with respect to design‐level limit states are discussed.     

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.