Site-dependent design spectra and strength modification factors, based on records from Greece

Elastic and inelastic spectra are derived, based on a representative sample of acceleration records from Greece, carefully selected based on magnitude, distance and peak ground acceleration criteria, and grouped into three ground condition categories according to the 2004 Eurocode 8 (EC8) provisions. Using software developed in-house, elastic (pseudoacceleration, pseudovelocity and displacement), as well as inelastic (strength and displacement) spectra are computed for various critical damping ratios and ductility levels. After appropriate scaling, mean spectra are computed both irrespective of, as well as for each different, ground condition, and comparisons with EC8 provisions are made. As a further evaluation of the code spectra, three additional earthquake scenarios are considered representing ground-motion characteristics not reflected in the compiled dataset of records. Subsequently, modification factors for strength (q_μ) are derived from statistical analysis of constant ductility spectra, and corresponding empirical relationships, suitable for design purposes, are proposed.     

Characterization of displacement demand for elastic and inelastic SDOF systems

The results of a research concerning the characterization of elastic and inelastic displacement spectral demand as a function of magnitude, source-to-site distance, and soil type are presented. The displacement spectra were computed for single degree of freedom systems subjected to a large set of strong ground motion records.In the elastic case, design displacement spectra, modeled in a simplified way with a bilinear shape in the period range 0–4 s, are then proposed for the estimation of the displacement demand to structures located on different local soil condition, at different distance from the causative fault, and for different levels of magnitude. In order to evaluate the reliability of the proposed design displacement spectra, probabilistic displacement spectra corresponding to different levels of probability of non-exceedance were also carried out.The inelastic displacement demand to elasto-plastic systems was analyzed through the ratio between inelastic and elastic spectral displacements. Simplified relationships of the inelastic displacement ratio are then proposed as a function of displacement ductility, soil condition and period of vibration. Finally, as a comparison, the inelastic displacement ratios were also estimated considering other constitutive models.     

Consistent inelastic design spectra: Strength and displacement

A procedure for the determination of inelastic design spectra (for strength, displacement, hysteretic and input energy) for systems with a prescribed ductility factor has been developed. All the spectra are consistent (interrelated and based on the same assumptions). This is the first of two companion papers which deals with the ‘classical’ structural parameters: strength and displacement. The input data are the characteristics of the expected ground motion in terms of a smooth elastic pseudo-acceleration spectrum. Simple, approximate expressions for the strength reduction factor R are proposed. The value of R depends on the natural period of the system, the prescribed ductility factor, the hysteretic behaviour, damping and ground motion. Fairly accurate approximations to the inelastic spectra for strength and displacement can be derived from the elastic spectrum using the proposed values for R.     

Study on inelastic displacement ratio spectra for near-fault pulse-type ground motions

In displacement-based seismic design, inelastic displacement ratio spectra (IDRS) are particularly useful for estimating the maximum lateral inelastic displacement demand of a nonlinear SDOF system from the maximum elastic displacement demand of its counterpart linear elastic SDOF system. In this study, the characteristics of IDRS for near-fault pulse-type ground motions are investigated based on a great number of earthquake ground motions. The in? uence of site conditions, ratio of peak ground velocity (PGV) to peak ground acceleration (PGA), the PGV, and the maximum incremental velocity (MIV) on IDRS are also evaluated. The results indicate that the effect of near-fault ground motions on IDRS are signifi cant only at periods between 0.2 s - 1.5 s, where the amplifi cation can approach 20%. The PGV/PGA ratio has the most signifi cant in? uence on IDRS among the parameters considered. It is also found that site conditions only slightly affect the IDRS.     

Inelastic seismic spectra including a damage criterion: A stochastic approach

In this paper, a stochastic approach for obtaining damage-based inelastic seismic spectra is proposed. The Park and Ang damage model, which includes displacement ductility and hysteretic energy, is adopted to take into account the cumulative damage phenomenon in structural systems under strong ground motions. Differently from previous studies in this field, damage-based seismic spectra are obtained by means of peak theory of stochastic processes. The following stochastic inelastic seismic spectra are constructed and then analyzed: damage-based displacement and acceleration inelastic spectra, damage-based response modification factor spectra, damage-based yield strength demand spectra and damage-based inelastic displacement ratio spectra.     

Inelastic earthquake spectra

This paper presents a procedure for computing inelastic design spectra in terms of yield displacement. The procedure results in substantial computational savings as compared to the current procedures for computing the inelastic spectra in terms of displacement ductility. The proposed method is used to obtain inelastic design spectra for alluvium and rock. The results for alluvium are compared to those from the Newmark and Hall,¹⁰ Riddell and Newmark¹² and Lai and Biggs⁴ studies. The findings indicate that for given ductility and frequency one may de-amplify the elastic response or reduce the elastic forces more for a structure on rock than for a structure on alluvium. A technique is presented so that the inelastic design spectrum can be scaled to any desired yield displacement and acceleration. An example is presented to illustrate how the proposed inelastic spectra can be used to compute the required ductility for a given system.     

Inelastic spectra for infilled reinforced concrete frames

In two companion papers a simplified non‐linear analysis procedure for infilled reinforced concrete frames is introduced. In this paper a simple relation between strength reduction factor, ductility and period (R–µ–T relation) is presented. It is intended to be used for the determination of inelastic displacement ratios and of inelastic spectra in conjunction with idealized elastic spectra. The R–µ–T relation was developed from results of an extensive parametric study employing a SDOF mathematical model composed of structural elements representing the frame and infill. The structural parameters, used in the proposed R–µ–T relation, in addition to the parameters used in a usual (e.g. elasto‐plastic) system, are ductility at the beginning of strength degradation, and the reduction of strength after the failure of the infills. Formulae depend also on the corner periods of the elastic spectrum. The proposed equations were validated by comparing results in terms of the reduction factors, inelastic displacement ratios, and inelastic spectra in the acceleration–displacement format, with those obtained by non‐linear dynamic analyses for three sets of recorded and semi‐artificial ground motions. A new approach was used for generating semi‐artificial ground motions compatible with the target spectrum. This approach preserves the basic characteristics of individual ground motions, whereas the mean spectrum of the whole ground motion set fits the target spectrum excellently. In the parametric study, the R–µ–T relation was determined by assuming a constant reduction factor, while the corresponding ductility was calculated for different ground motions. The mean values proved to be noticeably different from the mean values determined based on a constant ductility approach, while the median values determined by the different procedures were between the two means. The approach employed in the study yields a R–µ–T relation which is conservative both for design and performance assessment (compared with a relation based on median values). Copyright © 2004 John Wiley & Sons, Ltd.     

Simplified non-linear seismic displacement demand prediction for low period structures

The prediction of non-linear seismic demand using linear elastic behavior for the determination of peak non-linear response is widely used for seismic design as well as for vulnerability assessment. Existing methods use either linear response based on initial period and damping ratio, eventually corrected with factors, or linear response based on increased equivalent period and damping ratio. Improvements to the original EC8 procedure for displacement demand prediction are proposed in this study. Both propositions may be graphically approximated, which is a significant advantage for practical application. A comparison with several other methods (equal displacement rule, EC8 procedure, secant stiffness and empirical equivalent period methods) is performed. The study is based on non-linear SDOF systems subjected to recorded earthquakes, modified to match design response spectra of different ground types, and focuses on the low frequency range that is of interest for most European buildings. All results are represented in the spectral displacement/fundamental period plane that highlights the predominant effect of the fundamental period on the displacement demand. This study shows that linearized methods perform well at low strength reduction factors but may strongly underestimate the displacement demand at strength reduction factors greater than 2. This underestimation is an important issue, especially for assessment of existing buildings, which are often related with low lateral strength. In such cases, the corresponding strength reduction factors are therefore much larger than 2. The new proposals significantly improve the reliability of displacement demand prediction for values of strength reduction factors greater than 2 compared to the original EC8 procedure. As a consequence, for the seismic assessment of existing structures, such as unreinforced masonry low-rise buildings, the current procedure of EC8 should be modified in order to provide accurate predictions of the displacement demand in the domain of the response spectrum plateau.     

Inelastic design spectra accounting for soil conditions

This paper is concerned with the effect of soil conditions on the response of single-degree-of-freedom inelastic systems subjected to earthquake motions. The ground motions considered are 72 horizontal components of motion, most of them recorded during the 3 March, 1985 Chile earthquake (M_s = 7·8) and two main aftershocks; among these records are some of the strongest and longer duration earthquake motions ever recorded. The recording station sites were classified in one of three soil types, which can be generically referred to as rock, firm ground, and medium stiffness soil. Response results for each group were analysed statistically to obtain factors for deriving inelastic design spectra of the Newmark-Hall type, as well as alternative simplified spectral shapes suitable for code formulation. Particular attention was given to the response modification factors (R) that are commonly used in seismic codes to reduce the ordinates of the elastic spectrum to account for the energy dissipation capacity of the structure. The response modification factors, known to be function of both the natural period of vibration and the ductility factor, are found to be dependent on soil conditions, particularly in the case of medium stiffness soils. It is also shown that the indirect procedure of applying R to the elastic design spectrum is less accurate than directly using functions that represent the inelastic design spectrum.     

Consistent inelastic design spectra: Hysteretic and input energy

This is the second of two companion papers on inelastic design spectra (for strength, displacement, hysteretic and input energy) for systems with a prescribed ductility factor. All the spectra are consistent (interrelated and based on the same assumptions). This paper deals with two quantities related to cumulative damage: hysteretic and input energy. The input data for the procedure are the characteristics of the expected ground motion in terms of a smooth elastic pseudo-acceleration spectrum and the time integral of the square of the ground acceleration ∫a² dt. Simple, approximate expressions for two dimensionless parameters (the parameter γ and the hysteretic to input energy ratio E_HE_I) have been proposed. The parameter 7, which controls the reduction of the deformation capacity of structures due to low-cycle fatigue, depends on the natural period of the system, the prescribed ductility factor, the hysteretic behaviour and the ground motion characteristics. The ratio E_H/E_I is influenced by damping, the ductility factor and the hysteretic behaviour. Very good approximations to the inelastic spectra for hysteretic and input energy can be derived from the elastic spectrum using the spectra for the reduction factor R, proposed in the companion paper, and the proposed values for γ and E_H/E_I     

Aseismic design implications of near-fault san fernando earthquake records

Near-fault records of the 1971 San Fernando earthquake contain severe, long duration acceleration pulses which result in unusually large ground velocity increments. A review of these records along with the results of available theoretical studies of near-fault ground motions indicates that such acceleration pulses may be characteristic of near-fault sites in general. The results of an analytical study of a building severely damaged during the San Fernando earthquake indicate that such severe, long duration acceleration pulses were the cause of the main features of the observed structural damage. The implications of such pulses on current aseismic design methods, particularly those used to establish design earthquakes, are examined for buildings located near potential earthquake faults. Analytical studies of the non-linear dynamic response of single and multiple degree-of-freedom systems to several near-fault records, as well as to a more standard accelerogram, indicate that at near-fault sites: (a) very large displacement ductilities may result for current levels of code design forces; (b) smoothed elastic design response spectra should reflect the larger ground velocities that may occur; and (c) peak inelastic response cannot reliably be inferred from elastic response predictions.     

地震动永久位移对结构地震反应影响

本文推导了基于位移激励计算单自由度体系拟速度谱公式,通过构造的脉冲位移时程对公式精度进行了验证;之后利用小波变换去除强震记录噪声而保留地震动永久位移,再基于去趋势项方法和滤波方法去除永久位移后,计算拟速度谱。算例结果表明:短周期段内,不保留永久位移的位移激励拟速度谱值与保留永久位移的位移激励拟速度谱值相差很小;中长周期段内,不保留永久位移的位移激励拟速度谱值总体上小于保留永久位移的位移激励拟速度谱值,且不保留永久位移时,滤波方法引起的拟速度谱降幅大于去趋势项方法所引起的拟速度谱降幅。因此,基于位移激励计算中长周期结构的地震反应时,应保留地震动永久位移,或基于去趋势项方法去除永久位移。      

Effects of hanging wall and forward directivity in the 1999 Chi-Chi earthquake on inelastic displacement response of structures

The characteristics of the inelastic response of structures affected by hanging wall and forward directivity in the 1999 Chi-Chi earthquake are investigated. Inelastic displacement ratios (IDRs) for ground motions impacted by these near- field effects are evaluated and comprehensively compared to far-field ground motions. In addition, the inelastic displacement responses to hanging wall and footwall ground motions are compared. It is concluded that the inelastic displacement response is significantly affected in the short period range by hanging wall and in the long period range by footwall. Although high peak ground acceleration was observed at hanging wall stations, the IDRs for structures on hanging wall sites are only larger than footwall sites in the very long period range. Forward directivity effects result in larger IDRs for periods longer than about 0.5s. Adopting statistical relationships for IDRs established using far-field ground motions may lead to either overestimation or underestimation in the seismic evaluation of existing structures located in near-field regions, depending on their fundamental vibration periods.     

Evaluation of site‐dependent constant‐damage design spectra for reinforced concrete structures

An investigation on the validity of the conventional design approach known as constant displacement ductility is carried out. The hysteretic behaviour described by the Modified Takeda model is taken to represent the characteristics of reinforced concrete structural systems. The results presented in the form of seismic damage spectra indicate that the conventional design approach may not be valid because cumulative damage is excessively high. The inelastic design spectra based on the constant‐damage concept are proposed in terms of simplified expressions. The expressions are derived from constant‐damage design spectra computed by non‐linear response analysis for SDOF systems subjected to ground motions recorded on rock sites, alluvium deposits, and soft‐soil sites. The proposed expressions, which are dependent on the local soil conditions, are functions of target seismic damage, displacement ductility ratio and period of vibration. The seismic damage of structures that have been designed based on this new design approach is also checked by a design‐and‐evaluation approach. The results are found to be satisfactory. Copyright © 2004 John Wiley & Sons, Ltd.     

An energy-based methodology for the assessment of seismic demand

A methodology for the assessment of the seismic energy demands imposed on structures is proposed. The research was carried out through two consecutive phases. Inelastic design input energy spectra for systems with a prescribed displacement ductility ratio were first developed. The study of the inelastic behavior of energy factors and the evaluation of the response modification in comparison with the elastic case were performed by introducing two new parameters, namely: (1) the Response Modification Factor of the earthquake input energy (R_E), representing the ratio of the elastic to inelastic input energy spectral values and (2) the ratio α of the area enclosed by the inelastic input energy spectrum in the range of periods between 0.05 and 4.0 s to the corresponding elastic value. The proposed design inelastic energy spectra, resulting from the study of a large set of strong motion records, were obtained as a function of ductility, soil type, source-to-site distance and magnitude.Subsequently, with reference to single degree of freedom systems, the spectra of the hysteretic to input energy ratio were evaluated, for different soil types and target ductility ratios. These spectra, defined to evaluate the hysteretic energy demand of structures, were described by a piecewise linear idealization that allows to distinguish three distinct regions as a function of the vibration period. In this manner, once the inelastic design input energy spectra were determined, the definition of the energy dissipated by means of inelastic deformations followed directly from the knowledge of hysteretic to input energy ratio.The design spectra of both input energy and hysteretic to input energy ratio were defined considering an elasto-plastic behavior. Nevertheless, other constitutive models were taken into account for comparison purposes.     

弹塑性地震反应谱的长周期特性研究

在基于性能抗震设计中弹塑性反应谱在计算结构地震位移反应方面越来越受到重视。利用统计分析方法研究了等强度的延性需求谱和等延性的强度折减系数谱的长周期(至5 s)区段的特性,关注的重点是等位移准则和场地条件影响。给出了若干具有工程价值的结论:一是周期介于1.5Tg(地震动特征周期)和2.5 s之间的结构可近似认为等位移准则成立且与场地条件关系不大,这样确定的强度折减系数当位移延性系数小于等于4时结果将是偏于安全的;二是结构周期大于2.5 s后以硬土场地等延性强度折减系数谱或等强度延性需求谱代替软土场地谱求解系统强度需求或延性需求,将会得到偏于安全的结果。     

Evaluation of seismic displacement demands from the September 19, 2017 Puebla‐Morelos (Mw = 7.1) earthquake in Mexico City

This short communication presents the assessment of seismic inelastic and elastic displacement demands computed from earthquake ground motions (EQGMs) recorded in Mexico City during the intermediate‐depth intraslab Puebla‐Morelos earthquake on 19 September 2017 (Mw = 7.1). Evaluation is conducted by means of peak elastic and inelastic displacement demand spectra, inelastic displacement ratio, C_R, spectra, and generalized interstory drift spectra computed for selected recording stations located in different soil sites of Mexico City, including those located in areas of reported collapsed buildings. Results of this study confirm previous observations made from interplate (subduction) EQGMs that peak inelastic displacement demands are greater than corresponding elastic counterparts for short‐to‐medium period structures, while the opposite is true for medium‐to‐long period structures. Possible basin site effects were identified from generalized interstory drift spectra. It is also shown that an equation introduced in the literature to obtain estimates of C_R developed from interplate EQGMs provides also a good estimate for mean C_R computed from the intermediate‐depth intraslab EQGMs.     

双向地震动作用的拟等延性系数谱

首先建立了以强度折减系数表述的恢复力特性满足二维屈服面模型的理想弹塑性单质点系统(它在2个相互垂直的主轴方向上分别具有水平平动自由度)在双向地震动作用下的归一化运动方程。然后引入单向地震动作用下等延性系数的强度折减系数谱,给出了双向地震动作用的拟等延性系数谱(定义为系统分别承受双向和单向地震动作用,在同一主轴方向上的最大位移反应之比)最后通过硬土场地10组双向地震动记录拟等延性系数谱的统计平均结果,分析了结构周期、位移延性系数和阻尼等因素对谱值及结构双向地震反应的影响。结果表明,双向地震动作用与单向地震动作用相比主要增加结构较长周期方向的最大位移反应。若在基于位移的抗震设计中降低结构较短周期方向的设计位移延性系数,可在一定程度上降低双向地震动的不利影响。因定义的谱为比值形式,阻尼对其影响不大。     

Investigation of ‘equal displacement’ rule for bridges subjected to differential support motions

The ‘equal displacement’ rule is employed in seismic design practice to predict inelastic displacements from analyses of the corresponding linear elastic structural models. The accuracy and limitations of this rule have been investigated for ordinary structures but not for bridges subjected to spatially varying ground motions. The present study investigates this rule for moderate levels of inelastic behavior for four highway bridges in California accounting for the effects of spatial variability of the support motions due to incoherence, wave passage and differential site response. The bridge models vary significantly as to their fundamental periods and their overall configurations. Statistical analyses of pier‐drift responses are performed using as input simulated arrays of nonstationary ground motions in accordance with prescribed coherency models. It is found that the ‘equal displacement’ rule is fairly accurate for cases when the fundamental period of the bridge is longer than the transition period between the acceleration‐controlled and velocity‐controlled ranges of the response spectrum. Otherwise, the rule is non‐conservative for cases with large ductility factors and conservative for cases with small ductility factors. Wave passage and incoherence tend to reduce ratios of mean peak inelastic to elastic pier drifts, whereas incorporation of the differential site‐response effect by locating piers on softer soils tends to increase the same ratios. Mild or moderate positive correlation between these ratios and ductility demands is observed in most cases. Effects of spatial variability are more pronounced for longer and stiffer bridges. Copyright © 2013 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.