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.     

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.     

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: 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     

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.     

Probabilistic estimation of maximum inelastic displacement demands for performance‐based design

A probabilistic approach to estimate maximum inelastic displacement demands of single‐degree‐of‐freedom (SDOF) systems is presented. By making use of the probability of exceedance of maximum inelastic displacement demands for given maximum elastic spectral displacement and the mean annual frequency of exceedance of elastic spectral ordinates, a simplified procedure is proposed to estimate mean annual frequencies of exceedance of maximum inelastic displacement demands. Simplifying assumptions are thoroughly examined and discussed. Using readily available elastic seismic hazard curves the procedure can be used to compute maximum inelastic displacement seismic hazard curves and uniform hazard spectra of maximum inelastic displacement demands. The resulting maximum inelastic displacement demand spectra provide a more rational way of establishing seismic demands for new and existing structures when performance‐based approaches are used. The proposed procedure is illustrated for elastoplastic SDOF systems having known‐lateral strength located in a region of high seismicity in California. Copyright © 2007 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.     

Inclusion of P–Δ effect in displacement‐based seismic design of steel moment resisting frames

A procedure for treating the P– Δ effect in the direct displacement‐based seismic design of regular steel moment resisting frames with ideal elastoplastic material behaviour is proposed. A simple formula for the yield displacement amplification factor as a function of ductility and the stability coefficient is derived on the basis of the seismic response of an inelastic single degree‐of‐freedom system taking into account the P– Δ effect. Extensive parametric seismic inelastic analyses of plane moment resisting steel frames result in a simple formula for the dynamic stability coefficient as a function of the number of stories of a frame and the column to beam stiffness ratio. Thus, the P– Δ effect can be easily taken into account in a direct displacement‐based seismic design through the stability coefficient and the yield displacement amplification factor. A simple design example serves to illustrate the application of the proposed method and demonstrate its merits. Copyright © 2007 John Wiley & Sons, Ltd.     

INELASTIC SEISMIC RESISTANCE OF REINFORCED CONCRETE STACK-LIKE STRUCTURES

A method is presented to quantify the inelastic seismic resistance of reinforced concrete stack-like structures by non-linear earthquake analysis. The deformed configuration of stack is idealized as an assemblage of beam elements and actual stress–strain relationships of concrete and reinforcing steel are used to evaluate element matrices. Repeated non-linear analyses are performed by gradually increasing the intensity of acceleration time histories to a level where collapse of the stack is observed in primary stresses. The set of time histories thus obtained are then used to define the ultimate intensity of ground motion that the stack can sustain if inelastic deformations are permitted. A procedure is presented to quantify the difference between inelastic seismic resistance and elastic seismic resistance in terms of displacement ductility capacity factors. For seismic design using available inelastic resistance, values of curvature ductility factor demand for the cross-sections of stacks are also presented. © 1997 by John Wiley & Sons, Ltd.     

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.     

延性需求谱在基于性能的抗震设计中的应用

基于性能的抗震设计理论涉及如何简便而合理地确定结构在指定强度地震下的弹塑性位移需求。本文给出了利用延性需求谱求解结构位移需求的一般步骤:借助模态Pushover分析将多自由度体系分解为几个非线性单自由度体系,以考虑各阶振型的影响;利用延性需求谱计算对应模态的等效单自由度体系的延性及位移需求,并以一定方式组合转化为多自由度体系位移需求。最后,通过算例分析表明:利用延性需求谱求解结构位移需求是一种具有一定精度可为工程接受的简便方法,在基于性能的抗震设计中具有较好的应用前景。     

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.     

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.     

Constant-ductility strength demand spectra for seismic design of structures

In displacement-based seismic design, constant-ductility strength demand spectra （CDSDS） are very useful for preliminary design of new structures where the global displacement ductility capacity is known. The CDSDS can provide the required inelastic lateral strength of new structures from the required elastic lateral strength. Based on a statistical study of nonlinear time-history for an SDOF system, the mean CDSDS corresponding to four site conditions are presented and approximate expressions of the inelastic spectra are proposed, which are functions of the structural period and ductility level. The effects of site conditions, structural period, level of ductility, damping and post-yield stiffness of structures on CDSDS are also investigated. It is concluded that site conditions, ductility level and structural period have important effects on the CDSDS and damping, post-yield stiffness effects are rather complex and of minor importance. The damping, post-yield stiffness effects depend on both the level of ductility and the natural period of structures.     

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。     

Displacement spectra and displacement modification factors,based on records from Greece

Elastic and inelastic displacement spectra (for periods up to 4.0 s) are derived, using a representative sample of acceleration records from Greece, carefully selected based on magnitude, distance and peak ground acceleration criteria, and grouped into three ground type categories according to the Eurocode 8 (EC8) provisions. The modification factor for the elastic design spectrum adopted in EC8 for accounting for damping is verified herein and is found to be satisfactory in the short to medium period range and less so in the long period range. The equivalent viscous damping ratio concept is also evaluated and is found to lead to underestimation of inelastic displacement spectra. Finally, based on the previously derived elastic and inelastic spectra, equations suitable for design and/or assessment purposes, are proposed for the corresponding displacement modification factors.     

工程结构等延性地震抗力谱研究

研究结构的非弹性反应谱对改进现有的结构抗震设计、发展基于性态的抗震设计以及了解复杂地面运动特性与结构动力特性之间的关系具有重要的意义。利用大量的单自由度在强震记录作用下的弹塑性动力时程分析,对等延性地震抗力谱这一重要的非弹性反应谱进行了较为详尽的研究,给出了四类场地条件(基岩、硬土、一般土和软土)下的平均等延性地震抗力谱,总结了对工程结构的抗震设计和研究具有实际意义的规律和特征,并分析了场地条件、结构的延性系数以及周期等对等延性地震抗力谱的影响,提出了新的拟合公式,其成果可供抗震研究和设计直接应用。     

结构抗震设计中的强度折减系数研究

借助于单自由度弹塑性动力时程分析程序，对延性结构的强度折减系数进行了研究，在统计平均和回归分析的基础上，建立了平均强度折系数的函数形式，本文所建立的平强度折减系数函数，从理论上明确了结构具有延性对弹性地震力的折减关系，研究成果可供结构抗震设计规范采纳应用。     

Sectional response of T-shaped RC walls

Deformation quantities such as strain, curvature and displacement are of paramount importance in seismic design within a performance-based procedure that aims to control the structural response at predefined levels of inelastic action. Given the importance of curvature expressions independent of strength for the design process, and for the particular case of T-shaped walls, the curvature trends at yield, serviceability and ultimate limit state are determined in graphical and analytical form. The comprehensive set of equations proposed in this work are strength independent and allow the reliable computation of limit-state curvatures, essential in a displacement-based design approach, and thus the realistic estimation of appropriate ductility factors in the design of T-shaped walls. Furthermore, results regarding the section properties of T-shaped walls, such as the elastic stiffness and the moment capacity for opposite directions of loading, offer additional information on T-shaped walls.     

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。