Seismic responses of postyield hardening single–degree-of-freedom systems incorporating high-strength elastic material

It is widely accepted that ductility design improves the seismic capacity of structures worldwide. Nevertheless, inelastic deformation allows serious damage to occur in structures. Previous studies have shown that a certain level of postyield stiffness may reduce both the peak displacement and residual deformation of a structure. In recent years, several high-strength elastic materials, such as fiber-reinforced polymer (FRP) and high-strength steel bars, have been developed. Application of these materials can easily provide a structure with a much higher and more stable postyield stiffness. Many materials, members, and structures that incorporate both high-strength elastic materials and conventional materials show significant postyield hardening (PYH) behaviors. The significant postyield stiffness of PYH structures can help effectively reduce both peak and residual deformations, providing a choice when designing resilient structures. However, the findings of previous studies of structures with elastic-perfectly plastic (EPP) behavior or small postyield stiffness may not be accurate for PYH structures. The postyield stiffness of a structure must be considered an important primary structural parameter, in addition to initial stiffness, yielding strength, and ductility. In this paper, extensive time history and statistical analyses are carried out for PYH single–degree-of-freedom (SDOF) systems. The mean values and coefficients of variation of the peak displacement and residual deformation are obtained and discussed. A new R-μ_p-T-α relationship and damage index for PYH structures are proposed. A theoretical model for the calculation of residual deformation is also established. These models provide a basis for developing the appropriate seismic design and performance evaluation procedures for PYH structures.     

Inelastic displacement ratios for evaluation of structures built on soft soil sites

This paper summarizes the results of a comprehensive statistical study aimed at evaluating peak lateral inelastic displacement demands of structures with known lateral strength and stiffness built on soft soil site conditions. For that purpose, empirical information on inelastic displacement ratios which are defined as the ratio of peak lateral inelastic displacement demands to peak elastic displacement demands are investigated. Inelastic displacement ratios were computed from the response of single‐degree‐of‐freedom systems having 6 levels of relative lateral strength when subjected to 118 earthquake ground motions recorded on bay‐mud sites of the San Francisco Bay Area and on soft soil sites located in the former lake‐bed zone of Mexico City. Mean inelastic displacement ratios and their corresponding scatter are presented for both ground motion ensembles. The influence of period of vibration normalized by the predominant period of the ground motion, the level of lateral strength, earthquake magnitude, and distance to the source are evaluated and discussed. In addition, the effects of post‐yield stiffness and of stiffness and strength degradation on inelastic displacement ratios are also investigated. It is concluded that magnitude and distance to the source have negligible effects on constant‐strength inelastic displacement ratios. Results also indicate that weak and stiffness‐degrading structures in the short spectral region could experience inelastic displacement demands larger than those corresponding to non‐degrading structures. Finally, a simplified equation obtained using regression analyses aimed at estimating mean inelastic displacement ratios is proposed for assisting structural engineers in performance‐based assessment of structures built on soft soil sites. Copyright © 2006 John Wiley & Sons, Ltd.     

Residual displacement ratios for assessment of existing structures

Results of an analytical study aimed at evaluating residual displacement ratios, C_r, which allow the estimation of residual displacement demands from maximum elastic displacement demands is presented. Residual displacement ratios were computed using response time‐history analyses of single‐degree‐of‐freedom systems having 6 levels of relative lateral strength when subjected to an ensemble of 240 earthquake ground motions recorded in stations placed on firm sites. The results were statistically organized to evaluate the influence of the following parameters: period of vibration, level of relative lateral strength, site conditions, earthquake magnitude, and distance to the source. In addition, the influence of post‐yield stiffness ratio in bilinear systems and of the unloading stiffness in stiffness‐degrading systems was also investigated. A special emphasis is given to the uncertainty of these ratios. From this study, it is concluded that mean residual displacement ratios are more sensitive to changes in local site conditions, earthquake magnitude, distance to the source range and hysteretic behaviour than mean inelastic displacement ratios. In particular, residual displacement ratios exhibit large levels of record‐to‐record variability and, therefore, this dispersion should be taken into account when estimating residual displacements. A simplified expression is presented to estimate mean residual displacements ratios for elastoplastic systems during the evaluation of existing structures built on firm soil sites. Copyright © 2005 John Wiley & Sons, Ltd.     

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.     

Influence of the mean period of ground motion on the inelastic dynamic response of single and multi degree of freedom systems

This paper focuses on examining the effects of frequency content of the ground motion on the inelastic demands imposed on both single degree of freedom (SDF) and multi degree of freedom (MDF) steel‐framed systems. A detailed literature review is conducted to identify the indicator that best represents the frequency content of ground motion. The mean period (T_m) of ground motion is selected owing to its ability to distinguish between various spectral shapes of ground motion, and its relationship with magnitude, distance and site characteristics. Inelastic displacement demands on SDF systems for target ductility levels are first studied in the light of T_m, using a suite of 128 ground motion records. The study is then extended to MDF systems with the help of incremental dynamic analysis by employing the same ground motion ensemble to assess the influence of T_m on various engineering demand parameters. The results obtained indicate that, for SDF systems, the amplification of displacements occurs when the period ratio between elastic period (T_e) and T_m is lower than unity. For MDF systems, the results demonstrate that the influence of higher modes on the base shear and maximum storey drift profile becomes more pronounced, as T_m approaches the higher mode periods of the structure. These observations, for both SDF and MDF systems, tend to be more evident for higher levels of inelasticity. The significance of the results, with particular reference to European seismic design procedures, is highlighted. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

Direct use of PGV for estimating peak nonlinear oscillator displacements

A predictive model is presented for estimating the peak inelastic oscillator displacements (S_d,ie) from peak ground velocity (PGV). The proposed model accounts for the variation of S_d,ie for bilinear hysteretic behavior under constant ductility (µ) and normalized lateral strength ratio (R) associated with postyield stiffness ratios of α=0 and 5%. The regression coefficients are based on a ground‐motion database that contains dense‐to‐stiff soil site recordings at distances of up to 30 km from the causative fault. The moment magnitude ( M ) range of the database is 5.2? M ?7.6 and the ground motions do not exhibit pulse‐dominant signals. Confined to the limitations imposed by the ground‐motion database, the model can estimate S_d,ie by employing the PGV predictions obtained from the attenuation relationships (ground‐motion prediction equations). In this way, the influence of important seismological parameters can be incorporated to the variation of S_d,ie in a fairly rationale manner. This feature of the predictive model advocates its implementation in the probabilistic seismic hazard analysis that employs scalar ground‐motion intensity indices. Various case studies are presented to show the consistent estimations of S_d,ie by the proposed model. The error propagation in the S_d,ie estimations is also discussed when the proposed model is associated with attenuation relationships. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

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.     

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.     

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.     

Inelastic displacement ratios for evaluation of existing structures

Results of a detailed statistical study of constant relative strength inelastic displacement ratios to estimate maximum lateral inelastic displacement demands on existing structures from maximum lateral elastic displacement demands are presented. These ratios were computed for single‐degree‐of‐freedom systems with different levels of lateral strength normalized to the strength required to remain elastic when subjected to a relatively large ensemble of recorded earthquake ground motions. Three groups of soil conditions with shear wave velocities higher than 180m/s are considered. The influence of period of vibration, level of lateral yielding strength, site conditions, earthquake magnitude, distance to the source, and strain‐hardening ratio are evaluated and discussed. Mean inelastic displacement ratios and those associated with various percentiles are presented. A special emphasis is given to the dispersion of these ratios. It is concluded that distance to the source has a negligible influence on constant relative strength inelastic displacement ratios. However, for periods smaller than 1s earthquake magnitude and soil conditions have a moderate influence on these ratios. Strain hardening decreases maximum inelastic displacement at a fairly constant rate depending on the level of relative strength for periods of vibration longer than about 1.0s while it decreases maximum inelastic displacement non‐linearly as the period of vibration shortens and as the relative‐strength ratio increases for periods of vibration shorter than 1.0s. Finally, results from non‐linear regression analyses are presented that provide a simplified expression to be used to approximate mean inelastic displacement ratios during the evaluation of existing structures built on firm sites. Copyright © 2003 John Wiley & Sons, Ltd.     

Kinematic soil-structure interaction effects on maximum inelastic displacement demands of SDOF systems

This paper presents a statistical study of the kinematic soil-foundation-structure interaction effects on the maximum inelastic deformation demands of structures. Discussed here is the inelastic displacement ratio defined as the maximum inelastic displacement demands of structures subjected to foundation input motions divide by those of structures subjected to free-field ground motions. The displacement ratio is computed for a wide period range of elasto-plastic single-degree-of-freedom (SDOF) systems with various levels of lateral strength ratios and with different sizes of foundations. Seventy-two earthquake ground motions recorded on firm soil with average shear wave velocities between 180 m/s and 360 m/s are adopted. The effects of period of vibration, level of lateral yielding strength and dimension of foundations are investigated. The results show that kinematic interaction will reduce the maximum inelastic displacement demands of structures, especially for systems with short periods of vibration, and the larger the foundation size the smaller the maximum inelastic displacement becomes. In addition, the inelastic displacement ratio is nearly not affected by the strength ratio of structures for systems with periods of vibration greater than about 0.3 s and with strength ratios smaller than about 3.0. Expressions obtained from nonlinear regression analyses are also proposed for estimating the effects of kinematic soil-foundation-structure interaction from the maximum deformation demand of the inelastic system subjected to free-field ground motions.     

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.     

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。     

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 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.     

单自由度自复位体系设计能量谱研究

本文对具有旗帜型滞回模型的单自由度自复位体系提出了设计能量谱的构造方法,包括设计输入能量谱和设计滞回耗能比谱。首先按中国规范场地类别选取360条实际强震记录进行时程分析,对影响单自由度自复位体系输入能量谱和滞回耗能比谱的参数,包括地震波类型、滞回模型、阻尼比、延性系数等进行研究。在此基础上分别建议了设计输入能量谱和设计滞回耗能比谱及其曲线参数的确定方法,并与实际强震记录计算结果进行比较。结果表明结构滞回模型对能量谱影响明显;阻尼比和延性系数对输入能量谱的影响在整个周期范围内有显著差异,但均有明显的削峰作用。建议的两种设计能量谱综合考虑了结构参数、地震波参数和中国场地类别的影响,可以较好的拟合实际情况,并对弹塑性单自由度自复位体系在地震作用下的耗能需求做出较准确的估计。     

Ductility demands of tall buildings subjected to base rocking induced by Rayleigh waves

Performance-based earthquake engineering requires accurate estimation of structural response associated with different damage states because of strong ground motion. In recent work (Meza-Fajardo and Papageorgiou, 2018, EESD), we demonstrated that a significant contribution to the response of elastic soil-structure systems for high-rise buildings is attributed to base rocking associated with Rayleigh waves. The present paper presents results of a study investigating the effects of Rayleigh waves on the response of soil-structure systems with nonlinear behavior at the level of the superstructure. By introducing a rigid-elastic rotational spring at the base of the building, we take into account the stiffness reduction due to damage to the lateral load-resisting system at its root, and with it, increased displacement demands. Considering different levels of ductility and post-yield stiffness, we investigate the impact of rocking because of Rayleigh waves on maximum and residual interstory drift ratios. Our results indicate that rocking due to surface waves should be an important consideration for design and evaluation of tall buildings, as inelastic action elongates their effective natural period, and consequently, they are more prone to be damaged by resonance and excitation of extended duration because of Rayleigh waves.