The extended N2 method taking into account higher mode effects in elevation

The N2 method has been extended in order to take into account higher mode effects in elevation. The extension is based on the assumption that the structure remains in the elastic range when vibrating in higher modes. The seismic demand in terms of displacements and storey drifts can be obtained by enveloping the results of basic pushover analysis and the results of standard elastic modal analysis. The approach is consistent with the extended N2 method used for plan‐asymmetric buildings. The proposed procedure was applied to three variants of three steel frame buildings used in the SAC project. The structural response was investigated for two sets of ground motions. Different ground motion intensities were used in order to investigate the influence of the magnitude of plastic deformations. The N2 results were compared with the results of nonlinear response‐history analysis, two other pushover‐based methods (modal pushover analysis (MPA) and modified MPA (MMPA)), and pushover analysis without consideration of higher modes. It was found that a considerable influence of higher modes on storey drifts is present at the upper part of medium‐and high‐rise structures. This effect is the largest in the case of elastic behaviour and decreases with ground motion intensity. The higher mode effects also depend on the spectral shape. The approximate methods (extended N2, MPA and MMPA) are able to provide fair estimates of response in the case of the test examples. Accuracy decreases with the height of the building, and with the intensity of ground motion. The N2 results are generally conservative. Copyright © 2011 John Wiley & Sons, Ltd.     

Assessing the seismic response of existing RC buildings using the extended N2 method

Extensive studies have confirmed the good performance of the N2 method, recommended by Eurocode8, when performing pushover analyses in regular structures. However, this procedure shows lack of accuracy in predicting the torsional motion of plan-asymmetric buildings. In order to overcome this problem, Peter Fajfar and his team have proposed an extension of the method based on a combination of a pushover analysis and of an elastic response spectrum analysis. Since definitive answers about this topic have not yet been reached, this paper intends to proceed the study applying the extended N2 method to real existing RC buildings. Three real plan-asymmetric buildings with three, five and eight storeys were assessed. The results obtained with the extended N2 method were compared with the ones evaluated by means of the original N2 and with the nonlinear dynamic analysis through the use of semi-artificial ground motions. The analyses were performed for different seismic intensities in order to evaluate the torsional response of the building through different stages of structural inelasticity. The results obtained show that the extended N2 method generally reproduces in a very good fashion the real torsional behavior of the analyzed buildings. The conclusions herein outlined, added to the ones already published by the aforementioned authors, seem to confirm that the extended N2 method can be introduced in the next version of Eurocode8 as a nonlinear static procedure capable of accurately predicting the torsional response of plan-asymmetric buildings.     

Inelastic seismic response of code-designed multistorey frame buildings with regular asymmetry

Based on an asymmetric multistorey frame building model, this paper investigates the influence of a building's higher vibration modes on its inelastic torsional response and evaluates the adequacy of the provisions of current seismic building codes and the modal analysis procedure in accounting for increased ductility demand in frames situated at or near the stiff edge of such buildings. It is concluded that the influence of higher vibration modes on the response of the upper-storey columns of stiff-edge frames increases significantly with the building's fundamental uncoupled lateral period and the magnitude of the stiffness eccentricity. The application of the equivalent static torsional provisions of certain building codes may lead to non-conservative estimates of the peak ductility demand, particularly for structures with large stiffness eccentricity. In these cases, the critical elements are vulnerable to excessive additional ductility demand and, hence, may be subject to significantly more severe structural damage than in corresponding symmetric buildings. It is found that regularly asymmetric buildings excited well into the inelastic range may not be conservatively designed using linear elastic modal analysis theory. Particular caution is required when applying this method to the design of stiff-edge frame elements in highly asymmetric structures.     

Nonlinear seismic analysis of unsymmetric-plan structures retrofitted by hysteretic damped braces

A displacement-based design procedure using hysteretic damped braces (HYDBs) is proposed for the seismic retrofitting of unsymmetric-plan structures. An expression of the viscous damping equivalent to the hysteretic energy dissipated by the damped braced frame is proposed under bidirectional seismic loads, where corrective factors are assumed as a function of design parameters of the HYDBs. To this end, the nonlinear dynamic analysis of an equivalent two degree of freedom system is firstly carried out on seven pairs of real ground motions whose displacement response spectra match, on average, the design spectrum proposed by the Italian seismic code for a high-risk seismic zone and a medium subsoil class. Then, the extended N2 method considered by the European seismic code, which combines the nonlinear static analysis along the in-plan principal directions of the structure with elastic modal analysis, is adopted to evaluate the higher mode torsional effects. The town hall of Spilinga (Italy), a reinforced concrete (r.c.) framed building with an L-shaped plan, is supposed to be retrofitted with HYDBs. Six structural solutions are compared considering two alternative in-plan distributions of the HYDBs, to eliminate (elastic) torsional effects, and three design values of the frame ductility combined with a constant design value of the damper ductility. To check the effectiveness and reliability of the DBD procedure, the nonlinear static analysis of the test structures is carried out, by evaluating the vulnerability index of r.c. frame members and the ductility demand of HYDBs for different in-plan directions of the seismic loads.     

The applicability of the N2 method to the estimation of torsional effects in asymmetric base‐isolated buildings

The paper deals with the applicability of the extended N2 method to base‐isolated plan‐asymmetric building structures. The results obtained by the usual pushover analysis of a 3D structural model are further combined with the aid of linear dynamic (spectral) analysis to account for the dynamic effects caused by structural asymmetry. In the paper, the method has been applied to the seismic analysis of a mass‐eccentric four‐storeyed RC frame building isolated with lead rubber bearings. Three different positions of the center of isolation system (CI) with respect to the center of mass (CM) and the center of stiffness of the superstructure (CS) were considered. The response was analyzed for three different eccentricities, three different torsional to lateral frequency ratios of the superstructure, and two ground motion intensities. The stiffness of the isolators was selected for three different protection levels, which resulted in elastic as well as moderately to excessively damaged superstructure performance levels. The results are presented in terms of the top, base and relative displacements, as well as the stiff/flexible side amplification factors. A more detailed insight into the nonlinear behavior of the superstructure is given in a form of ductility factors for the flexible and stiff side frames. The results of the extended N2 method for selected lateral load distributions are compared with the average results of nonlinear dynamic analyses. It was concluded that the extended N2 method could, with certain limitations, provide a reasonable prediction of the torsional influences in minor to moderately asymmetric base‐isolated structures. Copyright © 2010 John Wiley & Sons, Ltd.     

A modal pushover analysis procedure to estimate seismic demands for unsymmetric‐plan buildings

An Erratum has been published for this article in Earthquake Engng. Struct. Dyn. 2004; 33:1429. Based on structural dynamics theory, the modal pushover analysis (MPA) procedure retains the conceptual simplicity of current procedures with invariant force distribution, now common in structural engineering practice. The MPA procedure for estimating seismic demands is extended to unsymmetric‐plan buildings. In the MPA procedure, the seismic demand due to individual terms in the modal expansion of the effective earthquake forces is determined by non‐linear static analysis using the inertia force distribution for each mode, which for unsymmetric buildings includes two lateral forces and torque at each floor level. These ‘modal’ demands due to the first few terms of the modal expansion are then combined by the CQC rule to obtain an estimate of the total seismic demand for inelastic systems. When applied to elastic systems, the MPA procedure is equivalent to standard response spectrum analysis (RSA). The MPA estimates of seismic demand for torsionally‐stiff and torsionally‐flexible unsymmetric systems are shown to be similarly accurate as they are for the symmetric building; however, the results deteriorate for a torsionally‐similarly‐stiff unsymmetric‐plan system and the ground motion considered because (a) elastic modes are strongly coupled, and (b) roof displacement is underestimated by the CQC modal combination rule (which would also limit accuracy of RSA for linearly elastic systems). Copyright © 2004 John Wiley & Sons, Ltd.     

Envelope‐based pushover analysis procedure for the approximate seismic response analysis of buildings

An envelope‐based pushover analysis procedure is presented that assumes that the seismic demand for each response parameter is controlled by a predominant system failure mode that may vary according to the ground motion. To be able to simulate the most important system failure modes, several pushover analyses need to be performed, as in a modal pushover analysis procedure, whereas the total seismic demand is determined by enveloping the results associated with each pushover analysis. The demand for the most common system failure mode resulting from the ‘first‐mode’ pushover analysis is obtained by response history analysis for the equivalent ‘modal‐based’ SDOF model, whereas demand for other failure modes is based on the ‘failure‐based’ SDOF models. This makes the envelope‐based pushover analysis procedure equivalent to the N2 method provided that it involves only ‘first‐mode’ pushover analysis and response history analysis of the corresponding ‘modal‐based’ SDOF model. It is shown that the accuracy of the approximate 16th, 50th and 84th percentile response expressed in terms of IDA curves does not decrease with the height of the building or with the intensity of ground motion. This is because the estimates of the roof displacement and the maximum storey drift due to individual ground motions were predicted with a sufficient degree of accuracy for almost all the ground motions from the analysed sets. Copyright © 2013 John Wiley & Sons, Ltd.     

Generalized force vectors for multi‐mode pushover analysis

A generalized pushover analysis (GPA) procedure is developed for estimating the inelastic seismic response of structures under earthquake ground excitations. The procedure comprises applying different generalized force vectors separately to the structure in an incremental form with increasing amplitude until a prescribed seismic demand is attained for each generalized force vector. A generalized force vector is expressed as a combination of modal forces, and simulates the instantaneous force distribution acting on the system when a given response parameter reaches its maximum value during dynamic response to a seismic excitation. While any response parameter can be selected arbitrarily, generalized force vectors in the presented study are derived for maximum interstory drift parameters. The maximum value of any other response parameter is then obtained from the envelope of GPAs results. Each nonlinear static analysis under a generalized force vector activates the entire multi‐degree of freedom effects simultaneously. Accordingly, inelastic actions develop in members with the contribution of all ‘instantaneous modes’ in the nonlinear response range. Target seismic demands for interstory drifts at the selected stories are calculated from the associated drift expressions. The implementation of the proposed GPA is simpler compared with nonlinear response history analysis, whereas it is less demanding in computational effort when compared with several multi‐mode adaptive nonlinear static procedures. Moreover, it does not suffer from the statistical combination of inelastic modal responses obtained separately. The results obtained from building frames have demonstrated that GPA is successful in estimating maximum member deformations and member forces with reference to the response history analysis. When the response is linear elastic, GPA and response spectrum analysis produce identical results. Copyright © 2010 John Wiley & Sons, Ltd.     

适用于结构时程分析法的输入地震波

本文对建筑结构抗震设计时程法所需的输入地震波提出一种基于规范反应谱的、与设防烈度、场地条件、设计近震或远震、结构自振特性有关的选择原则和方法。按远、近震和四类场地标定了反应谱、延性谱和积累损伤谱。通过对六幢不同高度的剪切和弯曲型结构模型的弹塑性分析,表明离差很小。建议在进行结构时程法分析时,按地震加速度反应谱特定的分布规律选择4条加速度记录作为输入计算。两个实际的例子表明按上述方法计算的结果与按底部剪力法计算的结果基本相符。     

Non-linear seismic response analysis using vector superposition methods

Computational algorithms based on the pseudo-force method (PFM) and the tangent spectrum method (TSM) for the seismic analysis of elasto-plastic MDOF structures by mode superposition are presented. The emphasis is put on the effect of the truncation of higher modes on the convergence of the ductility demand and energy dissipated during the earthquake. Eigenvectors and load-dependent vectors have been used in comparative analyses. Applications on a flexible 25-storey building and a stiff, 5-storey shear building indicate that, for a flexible structure, the ductility demand computed from the PFM is more sensitive to basis truncation than that computed from the TSM. However, for the stiff structure, the opposite behaviour is observed. The results indicate that the use of load-dependent vectors in inelastic analyses maintains the computational advantages found for elastic analyses in previous investigations. Although the PFM is more stable and computationally more effective than the TSM, it does not provide any information on the evolution of tangent modal properties in time that reflects dynamic response modification as the structure becomes inelastic.     

弹性能量半径演化谱及其在多自由度体系地震反应计算中的应用

目前用于结构抗震设计的反应谱仅能反映峰值反应,无法体现反应值随时间的变化。文中提出一种弹性能量半径演化谱,可反映线性单自由度体系弹性能量(即动能与弹性势能之和)随地震持时的变化,且其峰值近似等于结构峰值位移。文中给出了利用地震动演化功率谱得到该演化谱的方法并进一步发展了一种计算线性多自由度体系地震位移反应的新方法。通过两座框架结构的地震反应计算,将新方法与传统振型组合法及时程分析法的计算结果进行对比,发现对于振型稀疏结构,新方法计算结果与SRSS法接近;而对于振型密集结构,新方法计算结果较CQC法更精确,且避免了CQC法相关系数的复杂计算。     

Estimation of the behavior factor of existing RC-MRF buildings

In recent years, several research groups have studied a new generation of analysis methods for seismic response assessment of existing buildings. Nevertheless, many important developments are still needed in order to define more reliable and effective assessment procedures. Moreover, regarding existing buildings, it should be highlighted that due to the low knowledge level, the linear elastic analysis is the only analysis method allowed. The same codes (such as NTC2008, EC8) consider the linear dynamic analysis with behavior factor as the reference method for the evaluation of seismic demand. This type of analysis is based on a linear-elastic structural model subject to a design spectrum, obtained by reducing the elastic spectrum through a behavior factor. The behavior factor (reduction factor or q factor in some codes) is used to reduce the elastic spectrum ordinate or the forces obtained from a linear analysis in order to take into account the non-linear structural capacities. The behavior factors should be defined based on several parameters that influence the seismic nonlinear capacity, such as mechanical materials characteristics, structural system, irregularity and design procedures. In practical applications, there is still an evident lack of detailed rules and accurate behavior factor values adequate for existing buildings. In this work, some investigations of the seismic capacity of the main existing RC-MRF building types have been carried out. In order to make a correct evaluation of the seismic force demand, actual behavior factor values coherent with force based seismic safety assessment procedure have been proposed and compared with the values reported in the Italian seismic code, NTC08.     

强震区非对称连续梁桥地震响应及性能评价

为研究高烈度地区不等跨连续梁桥的抗震性能，依托某高速公路上一座主跨为（40+60+35）m的典型不等跨连续梁桥，建立其动力分析有限元模型，获得该桥的模态特性。在E1概率和E2概率两种地震水平作用下，同时采用反应谱分析和时程分析法，对不等跨桥梁结构的地震响应进行分析。最后根据桥墩验算截面的弯矩-曲率关系曲线，探讨该桥梁的抗震性能。研究结果表明：动态时程反应分析与反应谱分析所得的结果基本吻合，由于反应谱分析假定结构线弹性状态而时程反应分析考虑了材料的弹塑性，在E2概率水平下，两者个别响应值有较大差别；由于反应谱法是对各阶模态下最大响应的组合，动态时程反应分析是同一时刻各地震波引起的结构响应的组合，因而时域和频域计算结果会存在一些误差，频域结果偏于保守；E1、E2概率地震作用下，主桥桥墩检算截面仍然在弹性范围内工作，满足弹性设计要求。     

Prediction of the largest peak nonlinear seismic response of asymmetric buildings under bi-directional excitation using pushover analyses

A simplified procedure is proposed to predict the largest peak seismic response of an asymmetric building to horizontal bi-directional ground motion, acting at an arbitrary angle of incidence. The main characteristics of the proposed procedure is as follows. (1) The properties of two independent equivalent single-degree-of-freedom models are determined according to the principal direction of the first modal response in each nonlinear stage, rather than according to the fixed axis based on the mode shape in the elastic stage; the principal direction of the first modal response in each nonlinear stage is determined based on pushover analysis results. (2) The bi-directional horizontal seismic input is simulated as identical spectra of the two horizontal components, and the contribution of each modal response is directly estimated based on the unidirectional response in the principal direction of each. (3) The drift demand at each frame is determined based on four pushover analyses considering the combination of bi-directional excitations. In the numerical example, nonlinear time-history analyses of six four-story torsionally stiff (TS) asymmetric buildings are carried out considering various directions of seismic inputs, and these results are compared with the predicted results. The results show that the proposed procedure satisfactorily predicts the largest peak response displacement at the flexible-side frame of a TS asymmetric building.     

能力谱方法在桥梁抗震性能评估中的应用研究

位移延性是桥梁抗震性能的重要指标之一,以Pushover分析为基础的能力谱方法能够考察结构在地震下的弹塑性位移响应,是抗震性能评估的一种有效手段.文中阐述了能力谱法的基本原理,说明了基于弹塑性反应谱的能力谱方法在求解性能点时不需要进行迭代计算;基于弹性设计反应谱建立了相对应的弹塑性反应谱,结合某实桥,将能力谱方法和增量动力分析方法进行了对比,并根据不同的地震基本烈度和场地土类型进行了抗震性能评估.分析认为,能力谱方法计算简便,对结构1阶振型的地震响应占主导时,具有较好的精度,并能够基于设计反应谱来考察结构的弹塑性抗震性能,可用于桥梁抗震性能的评估.     

Rayleigh波场的数值模拟及其应用

本文总结了Rayleigh波场数值模拟的4种思路和各自特点,根据Lamb问题的理论分析成果提出了基于地表激振的Rayleigh波场数值模拟技术,论述了地表集中震源作用下引起地表波动场的特点及主要影响因素。在此基础上,利用Plaxis 2D有限元软件实现了Rayleigh波场的数值模拟,并结合算例验证了所提方法的可行性和结果的合理性。最后,针对多层建筑结构,研究了不同地震动输入模式下结构动力反应的特点。结果表明,Rayleigh波作用下结构的动力反应特性明显区别于在底部输入剪切波时的结果,不同地震动输入模式对结构的振动形态和破坏模式有着明显的影响。     

扩孔螺栓连接型消能梁段的力学模型及参数研究

当前国家对建筑结构的抗震性能和震后功能恢复能力提出了更高要求。基于短剪切型消能梁段的受剪屈服特性和剪切扩孔型螺栓连接的受剪滑移性能,提出一种新型扩孔螺栓连接型消能梁段,可有效增大消能梁段的延性和耗能能力并同时减小消能梁段的损伤,使带扩孔螺栓连接型消能梁段的新型Y形偏心支撑结构更好地适应当前要求。采用有限元方法详细分析扩孔螺栓连接型消能梁段的滞回性能、破坏模式和耗能机理,由此得到其骨架曲线和力学模型,并阐述其力学模型的影响参数,为相应偏心支撑结构的设计和分析提供理论依据。     

非经典振型分解法求混凝土房屋及其上钢塔的地震响应

钢结构与混凝土结构阻尼比不同,混凝土房屋与其顶上钢塔组成了非比例阻尼结构系统。本文用非经典振型分解法求解该类结构系统的线弹性地震响应,发现只用前几阶振型响应迭加的结果即可逼近直接积分法的精确度。     

Pushover analysis of asymmetric-plan buildings based on distribution of the combined modal story shear and torsional moment

A pushover procedure with a load pattern based on the height-wise distribution of the combined modal story shear and torsional moment is proposed to estimate the seismic response of 3D asymmetric-plan building frames. Contribution of the higher modes and torsional response of asymmetric-plan buildings are incorporated into the proposed load pattern. The proposed pushover method is a single-run procedure, which enables tracing the nonlinear response of the structure during the analysis and averts the elusiveness of conducting multiple pushover analyses. The proposed method has been used to estimate the response of two moment-resisting building frames with 9 and 20 stories. The obtained results indicate the appropriate accuracy and efficiency of the proposed procedure in estimating the trend of the drift profiles of the structures resulted from nonlinear time history analyses.     

Simplified non‐linear seismic analysis of infilled reinforced concrete frames

The N2 method for simplified non‐linear seismic analysis has been extended in order to make it applicable to infilled reinforced concrete frames. Compared to the simple basic variant of the N2 method, two important differences apply. A multi‐linear idealization of the pushover curve, which takes into account the strength degradation which occurs after the infill fails, has to be made, and specific reduction factors, developed in a companion paper, have to be used for the determination of inelastic spectra. It is shown that the N2 method can also be used for the determination of approximate summarized IDA curves. The proposed method was applied to two test buildings. The results were compared with the results obtained by non‐linear dynamic analyses for three sets of ground motions, and a reasonable accuracy was demonstrated. A similar extension of the N2 method can be made to any structural system, provided that an appropriate specific R–µ–T relation is available. Copyright © 2004 John Wiley & Sons, Ltd.