Further development of a multimodal pushover analysis procedure for seismic assessment of bridges

An improvement is first suggested to the modal pushover analysis (MPA) procedure for bridges initially proposed by the writers (Earthquake Engng Struct. Dyn. 2006; 35 (11):1269–1293), the key idea being that the deformed shape of the structure responding inelastically to the considered earthquake level is used in lieu of the elastic mode shape. The proposed MPA procedure is then verified by applying it to two actual bridges. The first structure is the Krystallopigi bridge, a 638 m‐long multi‐span bridge, with significant curvature in plan, unequal pier heights, and different types of pier‐to‐deck connections. The second structure is a 100 m‐long three‐span overpass bridge, typical in modern motorway construction in Europe, which, although ostensibly a regular structure, is found to exhibit a rather unsymmetric response in the transverse direction, mainly due to torsional irregularity. The bridges are assessed using response spectrum, ‘standard’ pushover (SPA), and MPA, and finally using non‐linear response history analysis (NL‐RHA) for a number of spectrum‐compatible motions. The MPA provided a good estimate of the maximum inelastic deck displacement for several earthquake intensities. The SPA on the other hand could not predict well the inelastic deck displacements of bridges wherever the contribution of the first mode to the response of the bridge was relatively low. Copyright © 2009 John Wiley & Sons, Ltd.     

A comparison of single‐run pushover analysis techniques for seismic assessment of bridges

Traditional pushover analysis is performed subjecting the structure to monotonically increasing lateral forces with invariant distribution until a target displacement is reached; both the force distribution and target displacement are hence based on the assumption that the response is controlled by a fundamental mode, that remains unchanged throughout. However, such invariant force distributions cannot account for the redistribution of inertia forces caused by structural yielding and the associated changes in the vibration properties, including the increase of higher‐mode participation. In order to overcome such drawbacks, but still keep the simplicity of using single‐run pushover analysis, as opposed to multiple‐analyses schemes, adaptive pushover techniques have recently been proposed. In order to investigate the effectiveness of such new pushover schemes in assessing bridges subjected to seismic action, so far object of only limited scrutiny, an analytical parametric study, conducted on a suite of continuous multi‐span bridges, is carried out. The study seems to show that, with respect to conventional pushover methods, these novel single‐run approaches can lead to the attainment of improved predictions. Copyright © 2007 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.     

An improved modal pushover analysis procedure for estimating seismic demands of structures

The pushover analysis （POA） procedure is difficult to apply to high-rise buildings, as it cannot account for the contributions of higher modes. To overcome this limitation, a modal pushover analysis （MPA） procedure was proposed by Chopra et al. （2001）. However, invariable lateral force distributions are still adopted in the MPA. In this paper, an improved MPA procedure is presented to estimate the seismic demands of structures, considering the redistribution of inertia forces after the structure yields. This improved procedure is verified with numerical examples of 5-, 9- and 22-story buildings. It is concluded that the improved MPA procedure is more accurate than either the POA procedure or MPA procedure. In addition, the proposed procedure avoids a large computational effort by adopting a two-phase lateral force distribution..     

模态pushover分析方法的研究和改进

鉴于传统pushover方法不能考虑结构高阶振型的贡献,从而难以应用到高层结构中,有学者提出采用考虑多阶振型组合作用的模态pushover分析方法。本文通过考虑结构屈服后地震作用发生变化这一特性,对此方法进行了改进并通过算例进行了验证。结果表明本文改进的方法有很好的精度。     

Estimation of hysteretic energy demand using concepts of modal pushover analysis

Hysteretic energy dissipation in a structure during an earthquake is the key factor, besides maximum displacement, related to the amount of damage in it. This energy demand can be accurately computed only through a nonlinear time‐history analysis of the structure subjected to a specific earthquake ground acceleration. However, for multi‐story structures, which are usually modeled as multi‐degree of freedom (MDOF) systems, this analysis becomes computation intensive and time consuming and is not suitable for adopting in seismic design guidelines. An alternative method of estimating hysteretic energy demand on MDOF systems is presented here. The proposed method uses multiple ‘generalized’ or ‘equivalent’ single degree of freedom (ESDOF) systems to estimate hysteretic energy demand on an MDOF system within the context of a ‘modal pushover analysis’. This is a modified version of a previous procedure using a single ESDOF system. Efficiency of the proposed procedure is tested by comparing energy demands based on this method with results from nonlinear dynamic analyses of MDOF systems, as well as estimates based on the previous method, for several ground motion scenarios. Three steel moment frame structures, of 3‐, 9‐, and 20‐story configurations, are selected for this comparison. Bias statistics that show the effectiveness of the proposed method are presented. In addition to being less demanding on the computation time and complexity, the proposed method is also suitable for adopting in design guidelines, as it can use response spectra for hysteretic energy demand estimation. Copyright © 2008 John Wiley & Sons, Ltd.     

Pushover分析方法的发展及其在桥梁结构中的应用

非线性静力分析方法（Pushover分析方法）可以较好地检验结构的变形能力,找到结构的薄弱环节,控制强烈地震作用下结构破坏程度,对工程设计有很强的指导意义。但目前Pushover分析方法的种类很多,各自有着不同的优缺点和适用范围,针对桥梁结构,阐述静力非线性分析方法（Pushover）的原理及其研究发展概况,评述了各种Pushover方法的优缺点,并分析了Pushover方法用于桥梁结构的基本原理和评价方法,指出Pushover方法用于桥梁结构存在的问题。     

Modified consecutive modal pushover procedure for seismic investigation of one-way asymmetric-plan tall buildings

The effects of higher modes and torsion have a significant impact on the seismic responses of asymmetric-plan tall buildings.A consecutive modal pushover(CMP) procedure is one of the pushover methods that have been developed to consider these effects.The aim of this paper is to modify the(CMP) analysis procedure to estimate the seismic demands of one-way asymmetric-plan tall buildings with dual systems.An analysis of 10-,15-and 20-story asymmetric-plan buildings is carried out,and the results from the modified consecutive modal pushover(MCMP) procedure are compared with those obtained from the modal pushover analysis(MPA) procedure and the nonlinear time history analysis(NLTHA).The MCMP estimates of the seismic demands of one-way asymmetric-plan buildings demonstrate a reasonable accuracy,compared to the results obtained from the NLTHA.Furthermore,the accuracy of the MCMP procedure in the prediction of plastic hinge rotations is better than the MPA procedure.The new pushover procedure is also more accurate than the FEMA load distribution and the MPA procedure.     

框架剪力墙结构模态静力非线性抗震分析方法研究

本文在模态pushover分析方法基础上推导建立了模态静力非线性分析方法,对一栋10层框架剪力墙结构进行了静力非线性分析,提出了目标位移求解的等效单自由度体系弹塑性时程分析迭代法,计算结果与相应时程分析结果进行了比较,表明两者吻合较好,验证了本文计算方法的有效性。另外,对同一结构,计算分析了在不同水平荷载模式下的静力非线性分析结果,比较不同荷载模式对计算结果的影响,为静力非线性分析方法的推广使用提供参考。     

Evaluation of modal pushover analysis using generic frames

An Erratum has been published for this article in Earthquake Engineering and Structural Dynamics 2003; 32:1795. The recently developed modal pushover analysis (MPA) has been shown to be a significant improvement over the pushover analysis procedures currently used in structural engineering practice. None of the current invariant force distributions accounts for the contribution of higher modes—higher than the fundamental mode—to the response or for redistribution of inertial forces because of structural yielding. By including the contributions of a sufficient number of modes of vibration (generally two to three), the height‐wise distribution of responses estimated by MPA is generally similar to the ‘exact’ results from non‐linear response history analysis (RHA). Although the results of the previous research were extremely promising, only a few buildings were evaluated. The results presented below evaluate the accuracy of MPA for a wide range of buildings and ground motion ensembles. The selected structures are idealized frames of six different heights: 3, 6, 9, 12, 15, and 18 stories and five strength levels corresponding to SDF‐system ductility factor of 1, 1.5, 2, 4, and 6; each frame is analysed for 20 ground motions. Comparing the median values of storey‐drift demands determined by MPA to those obtained from non‐linear RHA shows that the MPA predicts reasonably well the changing height‐wise variation of demand with building height and SDF‐system ductility factor. Median and dispersion values of the ratios of storey‐drift demands determined by MPA and non‐linear‐RHA procedures were computed to measure the bias and dispersion of MPA estimates with the following results: (1) the bias and dispersion in the MPA procedure tend to increase for longer‐period frames and larger SDF‐system ductility factors (although these trends are not perfect); (2) the bias and dispersion in MPA estimates of seismic demands for inelastic frames are usually larger than for elastic systems; (3) the well‐known response spectrum analysis (RSA), which is equivalent to the MPA for elastic systems, consistently underestimates the response of elastic structures, e.g. up to 18% in the upper‐storey drifts of 18‐storey frames. Finally, the MPA procedure is simplified to facilitate its implementation in engineering practice—where the earthquake hazard is usually defined in terms of a median (or some other percentile) design spectrum for elastic systems—and the accuracy of this simplified procedure is documented. Copyright © 2003 John Wiley & Sons, Ltd.     

Approximate incremental dynamic analysis using the modal pushover analysis procedure

Incremental dynamic analysis (IDA)—a procedure developed for accurate estimation of seismic demand and capacity of structures—requires non‐linear response history analysis of the structure for an ensemble of ground motions, each scaled to many intensity levels, selected to cover the entire range of structural response—all the way from elastic behaviour to global dynamic instability. Recognizing that IDA of practical structures is computationally extremely demanding, an approximate procedure based on the modal pushover analysis procedure is developed. Presented are the IDA curves and limit state capacities for the SAC‐Los Angeles 3‐, 9‐, and 20‐storey buildings computed by the exact and approximate procedures for an ensemble of 20 ground motions. These results demonstrate that the MPA‐based approximate procedure reduces the computational effort by a factor of 30 (for the 9‐storey building), at the same time providing results to a useful degree of accuracy over the entire range of responses—all the way from elastic behaviour to global dynamic instability—provided a proper hysteretic model is selected for modal SDF systems. The accuracy of the approximate procedure does not deteriorate for 9‐ and 20‐storey buildings, although their dynamics is more complex, involving several ‘modes’ of vibration. For all three buildings, the accuracy of the MPA‐based approximate procedure is also satisfactory for estimating the structural capacities for the limit states of immediate occupancy, collapse prevention, and global dynamic instability. Copyright © 2006 John Wiley & Sons, Ltd.     

Seismic response predictions of multi‐span steel bridges through pushover analysis

In the new trend of seismic design methodology, the static pushover analysis is recommended for simple or regular structures whilst the time‐history analysis is recommended for complex structures. To this end, the applicable range of the pushover analysis has to be clarified. This study aims at investigating the applicability of pushover analysis to multi‐span continuous bridge systems with thin‐walled steel piers. The focus is concentrated on the response demand predictions in longitudinal or transverse directions. The pushover analysis procedure for such structures is firstly summarized and then parametric studies are carried out on bridges with different types of superstructure‐pier bearing connections. The considered parameters, such as piers' stiffness distribution and pier–0.5ptdeck stiffness ratio, are varied to cover both regular and irregular structures. Finally, the relation of the applicability of pushover analysis to different structural formats is demonstrated and a criterion based on the higher modal contribution is proposed to quantitatively specify the applicable range. Copyright © 2003 John Wiley & Sons, Ltd.     

模态静力非线性分析中模态选择的研究

本文应用模态静力非线性分析方法,对多个算例进行了静力非线性分析。比较分析了不同振型组合对计算精度的影响,对模态选择的参数进行了研究,提出采用模态质量参与系数进行模态选择的控制,为模态静力非线性分析方法的推广使用提供参考。另外,分析了结构基本周期对该方法计算精度的影响。     

Extended consecutive modal pushover procedure for estimating seismic responses of one-way asymmetric plan tall buildings considering soil-structure interaction

Performance based design becomes an effective method for estimating seismic demands of buildings. In asymmetric plan tall building the effects of higher modes and torsion are crucial. The consecutive modal pushover (CMP) procedure is one of the procedures that consider these effects. Also in previous studies the influence of soil-structure interaction (SSI) in pushover analysis is ignored. In this paper the CMP procedure is modified for one-way asymmetric plan mid and high-rise buildings considering SSI. The extended CMP (ECMP) procedure is proposed in order to overcome some limitations of the CMP procedure. In this regard, 10, 15 and 20 story buildings with asymmetric plan are studied considering SSI assuming three different soil conditions. Using nonlinear response history analysis under a set of bidirectional ground motion; the exact responses of these buildings are calculated. Then the ECMP procedure is evaluated by comparing the results of this procedure with nonlinear time history results as an exact solution as well as the modal pushover analysis procedure and FEMA 356 load patterns. The results demonstrate the accuracy of the ECMP procedure.     

Applicability of pushover methods for the seismic analysis of single‐column bent viaducts

An overview of the applicability of a typical single‐mode pushover method (the N2 method) and two typical multi‐mode pushover methods (the modal pushover analysis (MPA) and incremental response spectrum analysis (IRSA) methods) for the analysis of single column bent viaducts in the transverse direction is presented. Previous research, which was limited to relatively short viaducts supported by few columns, has been extended to longer viaducts with more bents. The single‐mode N2 method is accurate enough for bridges where the effective modal mass of the fundamental mode is at least 80% of the total mass. The applicability of this method depends on (a) the ratio of the stiffness of the superstructure to that of the bents and (b) the strength of the bents. In short bridges with few columns, the accuracy of the N2 method increases as the seismic intensity increases, whereas in long viaducts (e.g. viaducts with lengths greater than 500 m) the method is in general less effective. In the case of the analyzed moderately irregular long viaducts, which are common in construction design practice, the MPA method performed well. For the analysis of bridges where the modes change significantly, depending on the seismic intensity, the IRSA method is in principle more appropriate, unless a viaduct is torsionally sensitive. In such cases, all simplified methods should be used with care. Copyright © 2008 John Wiley & Sons, Ltd.     

唐山地震桥梁震害回顾

对唐山桥梁震害加以回顾分析,并参照国内外地震有关桥梁的震害规律,为今后灾区破坏桥梁的修复和重建工作提供一些参考。     

Lateral load pattern in pushover analysis

The seismic capacity curves of three types of buildings including frame, frame-shear wall and shear wall obtained by pushover analysis under different lateral load patterns are compared with those from nonlinear time history analysis. Based on the numerical results obtained a two-phase load pattern; an inverted triangle (first mode) load pattern until the base shear force reachesβ times its maximum value, V_max, followed by a (x/H)^α form, hereβ and α being some coefficients depending on the type of the structures considered, is proposed in the paper, which can provide excellent approximation of the seismic capacity curve for low-to-mid-rise shear type buildings. Furthermore, it is shown both the two-phase load pattern proposed and the invariant uniform pattern can be used for low-to-mid-rise shear-bending type and low-rise bending type of buildings. No suitable load patterns have been found for high-rise buildings.     

An improved displacement-based adaptive pushover procedure based on factor modal combination rule

An accurate estimation of the applied load pattern is an essential component in each pushover procedure. Recently, a number of adaptive pushover methods have been proposed in which the effects of the higher modes as well as the progressive changes in the dynamic characteristics of structures are taken into account to compute the applied load pattern. The basic shortcoming of these advanced pushover methods is related to employing the quadratic modal combination rule, whereby the sign reversals of the modal load vectors are suppressed. In this study, an improved displacement-based adaptive pushover method is developed in which the applied load pattern is computed using the factor modal combination rule(FMC). In the proposed procedure, multiple load patterns, depending on the number of the modes considered, are determined in order to take into account the sign reversals of different modal load vectors. The accuracy of the proposed method is verifi ed for seven moment resisting frame buildings of 3, 9 and 20 stories with regularity or vertically geometric and mass irregularities subjected to 60 earthquake ground motion records. The results showed that the proposed methodology is capable of reproducing the peak dynamic responses with very good accuracy.     

基于推覆分析的RC框架地震倒塌易损性预测

基于倒塌率的结构倒塌易损性分析是目前评价结构抗倒塌能力最合理的方法.但是,目前基于增量动力分析(IDA)的倒塌率分析方法,工作量和实施难度大,很难直接用于工程设计,因此有必要研究便于工程应用的新方法.本文基于18个典型多层RC框架结构的IDA倒塌率分析和静力推覆分析,发现RC框架在大震下的倒塌率及抗倒塌安全储备(CMR)与静力推覆得到的结构位移安全储备之间存在较好的相关关系.依据此关系,建议了保证大震倒塌率的推覆位移安全储备,并通过9个RC框架结构算例进行了验证.本文方法简单易行,可供规则多层RC框架结构抗倒塌设计参考.     

Long‐term seismic performance of reinforced concrete bridges under steel reinforcement corrosion due to chloride attack

This work presents a new seismic evaluation methodology for corroded reinforced concrete bridges on the basis of nonlinear static pushover analysis. Corrosion of steel reinforcement by chloride attack is considered. At the material level, the effects of corrosion are considered by modeling the degradation of the mechanical properties of steel reinforcement, softening of cover concrete under compression, degradation of core concrete due to confinement steel corrosion, and reduction of bond strength between concrete and steel reinforcement. At the structural level, the effects of corrosion on both flexural behavior and shear behavior, and their interaction are considered. Eleven bridges of various structural types in Taiwan that are located within 6.5 km of their nearest coastline are analyzed to identify their long‐term seismic performance. Relationships between the yield and collapse peak ground accelerations (PGAs), and service time and corrosion level are established for each bridge. Analysis results show that chloride corrosion starts in 2–32 years. The transverse steel reinforcement typically starts corroding before the longitudinal steel reinforcement, as the former has a thicker cover. Research results show that collapse PGA reduces by 0.94% or 1.23% per 10 years when the mean value plus 1 or 2 standard deviation of the collapse PGA values are considered, respectively. Therefore, we suggest increasing the design PGA from 4.70% to 6.15% for a bridge adjacent to a coastline to ensure adequate long‐term seismic performance for 50 years, the typical design life span of a regular bridge. Copyright © 2013 John Wiley & Sons, Ltd.