Investigation on the accuracy of the N2 method and the equivalent linearization procedure for different hysteretic models

In this paper, an extensive parametric study was carried out to evaluate the dynamic response of single degree of freedom (SDOF) systems with elasto-plastic and flag-shape hysteretic behaviour for three different dissipation capacities. Three sets of natural accelerograms were used, each one composed by at least 7 records which are on average spectrum-consistent in pseudo-acceleration, spectral displacement or both of them. All sets were also employed to draw the inelastic spectra for different ductility values. Such rigorous spectra were then compared with the approximated curves calculated using the N2 method and the Equivalent Linearization Procedure (ELP) based on the use of overdamped elastic spectra.The analyses demonstrate a general accuracy of the N2 method, which is mostly based on the ductility of the system, even for the hysteretic behaviour characterised by reduced energy dissipation. Larger discrepancies were found for systems with lower damping ratios and reduced fundamental periods. The ELP, which mainly depends upon the dissipated energy, led instead to overall slightly larger discrepancies than the N2 method, particularly for not dissipative and ductile systems, whereas the approximation is generally acceptable for elasto-plastic systems.     

型钢高性能混凝土剪力墙恢复力模型研究

在前人研究成果的基础上,本文通过对6个型钢高性能混凝土剪力墙荷载-位移滞回曲线的分析,得出了简化的三折线型恢复力模型,并提出各个特征参数的计算公式,计算值与与试验值较为吻合。滞回规则采用考虑刚度退化和强度退化的剪切滞变模型,为型钢混凝土剪力墙的弹塑性分析提供了依据。     

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。     

Seismic damage assessment based on different hysteretic rules

The inelastic response of a single-degree-of-freedom (SDOF) system to different sets of earthquake records was analysed to study the significance of ground motion characteristics on structural damage. Six non-linear models were used. The ductility ratio and hysteretic energy index were chosen as two important damage indices to measure the structural damage. The dispersion phenomenon exhibited by yield spectra due to input ground motion was reduced by incorporating the ground motion a/v ratio into the two damage indices. Finally, empirical formulae for estimating two measures of structural damage were developed for each hysteretic model.     

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

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

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.     

预应力管式空心板柱结构抗震性能试验研究

通过一四层无粘结预应力现浇混凝土空心板柱结构的1/4比例模型的模拟地震振动台试验,对其在弹性和弹塑性阶段的抗震性能以及在El Centro地震动不同峰值加速度作用下的地震反应和受力过程进行了研究.对无黏结预应力现浇混凝土空心板柱结构的破坏形态、滞同性能、骨架曲线、恢复力模型、延性、刚度退化、耗能能力进行了比较深入的分析...     

Statistical insight into constant-ductility design using a non-stationary earthquake ground motion model

A recently developed earthquake ground motion model non-stationary in both intensity and frequency content is validated at the inelastic Single-Degree-Of-Freedom (SDOF) structural response level. For the purpose of this study, the earthquake model is calibrated for two actual earthquake records. The objective of a constant (or target) displacement ductility used in conventional earthquake-resistant design is examined from the statistical viewpoint using this non-stationary earthquake model. The non-linear hysteretic structural behaviour is modelled using several idealized hysteretic SDOF structural models. Ensemble-average inelastic response spectra corresponding to various inelastic SDOF response (or damage) parameters and conditioned on a constant displacement ductility response are derived from the two identified stochastic ground motion models. The effects of the type of hysteretic behaviour, the structural parameters, the target displacement ductility factor, and the ground motion model on the statistics of the inelastic response parameters are thoroughly investigated. The results of this parametric study shed further light on the proper interpretation and use of inelastic response or damage parameters in earthquake-resistant design in order to achieve the desirable objective of ‘constant-damage design’. © 1997 by John Wiley & Sons, Ltd.     

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.     

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.     

Estimating inelastic response spectra from elastic spectra

Inelastic displacement response spectra are determined for a broad class of single-degree-of-freedom hysteretic structures. Based on these spectra, effective linear period and damping parameters are defined as a function of ductility. A simple empirical formula is derived which may be used to estimate the mid-period range inelastic response spectrum of a general hysteretic structure given the linear response spectrum of the excitation. The estimates obtained from this formula are compared with those obtained by the Newmark-Hall method, the substitute–structure method and the ATC–3 tentative procedure. It is found that the empirical formula not only gives good estimates of the average behaviour of the inelastic spectrum, but also reproduces some of the details of the spectrum.     

Seismic response of bridge piers on pile groups for different soil damping models and lumped parameter representations of the foundation

This paper presents a wide parametric study aimed at elucidating the influence, on the computed seismic response of bridge piers, of two related aspects of the model: (1) the adoption of the classical hysteretic or the causal Biot's damping models for the soil and (2) the use of two different lumped parameter models of different complexity and accuracy to approximate the impedances of the pile foundation. A total of 2072 cases, including different superstructures, pile foundations, soil deposits, and seismic input signals, are studied. The results are presented so that the influence of the different parameters involved in the analysis can be assessed. From an engineering point of view, both lumped parameter models provide, in general, sufficiently low errors. The choice of the most adequate model for each case will depend not only on the configuration of the structure and the soil-foundation system but also on the assumed soil damping model, whose influence on the computed seismic responses is relevant in many cases. The nonphysical behaviour provided by the classical hysteretic damping model for the soil at zero frequency generates issues in the process of fitting the impedance functions. It is also found that larger deck displacements are predicted by Biot's model due to the higher damping at low frequencies provided by the classical hysteretic damping model.     

A nonlinear SDOF model for the simplified evaluation of the displacement demand of low-rise URM buildings

The determination of displacement demands for masonry buildings subjected to seismic action is a key issue in the performance-based assessment and design of such structures. A technique for the definition of single-degree-of-freedom (SDOF) nonlinear systems that approximates the global behaviour of multi-degree-of-freedom (MDOF) 3D structural models has been developed in order to provide useful information on the dependency of displacement demand on different seismic intensity measures. The definition of SDOF system properties is based on the dynamic equivalence of the elastic properties (vibration period and viscous damping) and on the comparability with nonlinear hysteretic behaviour obtained by cyclic pushover analysis on MDOF models. The MDOF systems are based on a nonlinear macroelement model that is able to reproduce the in-plane shear and flexural cyclic behaviour of pier and spandrel elements. For the complete MDOF models an equivalent frame modelling technique was used. The equivalent SDOF system was modelled using a suitable nonlinear spring comprised of two macroelements in parallel. This allows for a simple calibration of the hysteretic response of the SDOF by suitably proportioning the contributions of flexure-dominated and shear-dominated responses. The comparison of results in terms of maximum displacements obtained for the SDOF and MDOF systems demonstrates the feasibility and reliability of the proposed approach. The comparisons between MDOF and equivalent SDOF systems, carried out for several building prototypes, were based on the results of time-history analyses performed with a large database of natural records covering a wide range of magnitude, distance and local soil conditions. The use of unscaled natural accelerograms allowed the displacement demand to be expressed as a function of different ground motion parameters allowing for the study of their relative influence on the displacement demand for masonry structures.     

采用非线性纤维梁单元法的钢管混凝土组合框架动力响应分析

钢管混凝土结构已经广泛应用于我国高层和超高层建筑中,为研究该类结构的抗震性能,分别采用分离模量法和统一模量法对某13层钢管混凝土框架结构进行抗震性能分析,研究不同地震波作用下组合框架的模态和多遇地震下的弹性动力时程,对比组合框架顶点位移反应、加速度反应、层间侧移及动力放大系数等。研究结果表明,分离模量法和统一模量法建模方法在分析钢管混凝土框架抗震动力特性上总体相差不大,但前者可以考虑材料弹塑性,从而对结构弹塑性进行分析,而后者在弹塑性阶段需要用全曲线表达式,尚需进一步的研究。     

几条地震波的归一化时-频反应谱分析

时-频反应谱是地震动幅值、频谱和持时三要素构成的空间三维谱.基于此,本文提出归一化时-频反应谱定义与计算方法,计算三条典型地震波的归一化时-频反应谱,对比分析表明:不同地震记录的时-频反应谱的幅值在时间和周期两个轴上的分布差别很大,具有不同时-频反应谱的地震波可能对结构地震响应产生不同影响;对12层钢筋混凝土框架结构模型进行弹塑性时程分析,通过结构地震反应、结构损伤曲线与输入地震动的归一化时-频反应谱对比,发现结构最大反应并不一定是造成结构倒塌的直接原因,用结构弹性阶段反应最大值进行抗震设计存在一定局限性,从而证明归一化时-频反应谱可有效用来分析地震动特性和结构破坏机理.     

不同跨高比对多层钢框架内填混凝土深梁结构抗震性能影响

为了研究不同跨高比多层钢框架内填混凝土深梁结构的抗震性能,在钢框架内填混凝土深梁滞回性能试验的基础上,利用ABAQUS对六层纯钢框架（结构一）、钢框架内填跨高比为2混凝土深梁结构（结构二）和钢框架内填跨高比为0.75混凝土深梁结构（结构三）进行弹塑性时程分析。结果表明:内填混凝土深梁使结构整体刚度明显增大;在地震波的作用下,结构二的最大顶点位移降低可达58.3%,结构三的最大顶点位移降低可达89.3%,内填混凝土深梁,结构的抗侧移性能得到了极大改善,且随着深梁跨高比的减小而大幅度提升;结构二滞回曲线饱满,呈现纺锤形,混凝土深梁充分起到了第一道防线作用。经济合理的钢框架内填混凝土深梁结构具有一定的工程应用价值。     

Effect of hysteretic smoothness on inelastic response spectra with constant‐ductility

This paper presents an efficient methodology for computing constant‐ductility inelastic response spectra. The computation of constant‐ductility spectra involves numerical root‐finding algorithms to find the strongest structure providing a desired ductility response. Smooth inelastic structural behavior is modeled using a first‐order nonlinear differential equation and the transient structural response is solved using an implicit algorithm requiring Newton iterations at each time step. For structural models with smooth hysteretic behavior (not piece‐wise linear), a simple root‐finding method involving a combination of hyperbolic fits, linear interpolation, and Newton's method converges upon the highest strength (conservative) solution with a small number of iterations. The effect of the hysteretic smoothness on the occurrence of multiple roots is examined for two near‐fault and two far‐fault earthquake records, and for two measures of ductility and for normalized hysteretic energy. The results indicate how the smoothness of the hysteretic behavior affects ductility demand and constant‐ductility response spectra. Copyright © 2010 John Wiley & Sons, Ltd.     

钢框架加劲肋壁板结构的受力失稳性能研究

钢框架壁板由于在宏观上参与了整体建筑结构的受力,为钢框架构件承担部分外荷载,使得壁板结构在受力性能方面需要进一步改善。加劲肋壁板可以保证其与外框架的连接状况较好,改善整体构件的受力性能。为此,提出钢框架加劲肋壁板结构的受力失稳性能研究。采用动力学分析方法计算壁板结构位移值,依照位移值大小判断壁板结构是否处于稳定状态,获取壁板结构稳定性。利用实例测试分析钢框架加劲肋壁板结构的受力失稳性能,发现滞回曲线形状拥有梭形滞回曲线的特点,整个结构受力后具有塑性变形性能与抗震性能;骨架曲线反映出实验试件承载力支持第一阶段壁板结构弹性设计所需且提供第二阶段弹塑性抗震设计所需以及延性要求,本文检测模型能够判断钢结构建筑壁板结构失稳性。     

Mechanical formulations for bilinear and trilinear hysteretic models used in base isolators

The best known model for numerically simulating the hysteretic behavior of various structural components is the bilinear hysteretic system. There are two possible mechanical formulations that correspond to the same bilinear model from a mathematical viewpoint. The first one consists of a linear elastic spring connected in series with a parallel system comprising a plastic slider and a linear elastic spring, while the second one comprises a linear elastic spring connected in parallel with an elastic-perfectly plastic system. However, the bilinear hysteretic model is unable to describe either softening or hardening effects in these components. In order to account for this, the bilinear model is extended to a trilinear one. Thus, two trilinear hysteretic models are developed and numerically tested, and the analysis shows that both exhibit three plastic phases. More specifically, the first system exhibits one elastic phase, while the second one exhibits two elastic phases according to the level of strain amplitude. Next, the change of slope between the plastic phases in unloading does not occur at the same displacement level in the two models. Furthermore, the dissipated energy per cycle in the first trilinear model, as proven mathematically and explained physically, decreases in the case of hardening and increases in the case of softening, while in the second trilinear model the dissipated energy per cycle remains unchanged, as is the case with the bilinear model. Numerical examples are presented to quantify the aforementioned observations made in reference to the mechanical behavior of the two trilinear hysteretic models. Finally, a set of cyclic shear tests over a wide range of strain amplitudes on a high damping rubber bearing is used in the parameter identification of the two different systems, namely (a) trilinear hysteretic models of the first type connected in parallel, and (b) trilinear hysteretic models of the second type also connected in parallel. The results show that the complex nonlinear shear behavior of high damping rubber bearings can be correctly simulated by a parallel system which consists of only one component, namely the trilinear hysteretic system of the first type. The second parallel system was not able to describe the enlargement of the dissipated hysteresis area for large strain amplitudes.     

Equivalent ductility factors,taking into account low-cycle fatigue

During strong earthquakes, the deformation capacity of structures is reduced due to the cumulative damage caused by cyclic load reversals. In the paper, equivalent (reduced) ductility factors have been proposed, which take into account this effect. They are based on different failure hypotheses. Ductility reduction due to low-cycle fatigue is controlled by a dimensionless parameter γ, which is a function of dissipated hysteretic energy, maximum displacement and the natural frequency of the structural system, and which has been proved to be a relatively stable quantity in the whole period range. If approximate values for γ are used, the determination of equivalent ductility is very simple, and thus appropriate for design purposes. The formulae for equivalent ductility factors include damage indices, and permit the designer to choose acceptable level of structural damage explicitly. As an example, equivalent ductility factors have been used to construct inelastic acceleration spectra, which are proportional to strength demand, for the El Centro 1940 SOOE record. The results have been compared with the ‘exact’ spectra obtained by non-linear dynamic analysis.