Comparison of displacement coefficient method and capacity spectrum method with experimental results of RC columns

For the performance‐based seismic design of buildings, both the displacement coefficient method used by FEMA‐273 and the capacity spectrum method adopted by ATC‐40 are non‐linear static procedures. The pushover curves of structures need to be established during processing of these two methods. They are applied to evaluation and rehabilitation of existing structures. This paper is concerned with experimental studies on the accuracy of both methods. Through carrying out the pseudo‐dynamic tests, cyclic loading tests and pushover tests on three reinforced concrete (RC) columns, the maximum inelastic deformation demands (target displacements) determined by the coefficient method of FEMA‐273 and the capacity spectrum method of ATC‐40 are compared. In addition, a modified capacity spectrum method which is based on the use of inelastic design response spectra is also included in this study. It is shown from the test specimens that the coefficient method overestimates the peak test displacements with an average error of +28% while the capacity spectrum method underestimates them with an average error of ‐20%. If the Kowalsky hysteretic damping model is used in the capacity spectrum method instead of the original damping model, the average errors become ‐11% by ignoring the effect of stiffness degrading and ‐1.2% by slightly including the effect of stiffness degrading. Furthermore, if the Newmark–Hall inelastic design spectrum is implemented in the capacity spectrum method instead of the elastic design spectrum, the average error decreases to ‐6.6% which undervalues, but is close to, the experimental results. Copyright © 2003 John Wiley & Sons, Ltd.     

A static predictor of seismic demand on frames based on a post‐elastic deflected shape

Predictors of seismic structural demands (such as inter‐storey drift angles) that are less time‐consuming than nonlinear dynamic analysis have proven useful for structural performance assessment and for design. Luco and Cornell previously proposed a simple predictor that extends the idea of modal superposition (of the first two modes) with the square‐root‐of‐sum‐of‐squares (SRSS) rule by taking a first‐mode inelastic spectral displacement into account. This predictor achieved a significant improvement over simply using the response of an elastic oscillator; however, it cannot capture well large displacements caused by local yielding. A possible improvement of Luco's predictor is discussed in this paper, where it is proposed to consider three enhancements: (i) a post‐elastic first‐mode shape approximated by the deflected shape from a nonlinear static pushover analysis (NSPA) at the step corresponding to the maximum drift of an equivalent inelastic single‐degree‐of‐freedom (SDOF) system, (ii) a trilinear backbone curve for the SDOF system, and (iii) the elastic third‐mode response for long‐period buildings. Numerical examples demonstrate that the proposed predictor is less biased and results in less dispersion than Luco's original predictor. Copyright © 2006 John Wiley & Sons, Ltd.     

Estimation of inelastic seismic deformations in asymmetric multistorey RC buildings

Four real buildings with three to six stories, strong irregularities in plan and little engineered earthquake resistance are subjected to inelastic response‐history analyses under 56 bidirectional EC8‐spectra‐compatible motions. The average chord rotation demand at each member end over the 56 response‐history analyses is compared to the chord rotation from elastic static analysis with inverted triangular lateral forces or modal response spectrum analysis. The storey‐average inelastic‐to‐elastic‐chord‐rotation‐ratio was found fairly constant in all stories, except when static elastic analysis is applied to buildings with large higher mode effects. Except for such buildings, static elastic analysis gives more uniform ratios of inelastic chord rotations to elastic ones within and among stories than modal response spectrum analysis, but generally lower than 1.0. With increasing EPA the building‐average inelastic‐to‐elastic‐chord‐rotation‐ratio decreases but scatter in the results increases. Static elastic analysis tends to overestimate the inelastic torsional effects at the flexible or central part of the torsionally flexible buildings and underestimate them at their stiff side. Modal response spectrum analysis tends to overestimate the inelastic torsional effects at the stiff or central part of the torsionally stiff buildings and underestimate them at the flexible side. Overall, for multistorey RC buildings that typically have fundamental periods in the velocity‐sensitive part of the spectrum, elastic modal response spectrum analysis with 5% damping gives on average unbiased and fairly accurate estimates of member inelastic chord rotations. If higher modes are not significant, elastic static analysis in general overestimates inelastic chord rotations of such buildings, even when torsional effects are present. Copyright © 2007 John Wiley & Sons, Ltd.     

Effects of torsion on the behavior of peripheral steel‐braced frame systems

In this study, the torsional response of buildings with peripheral steel‐braced frame lateral systems is evaluated. A three‐dimensional model of a three story braced frame with various levels of eccentricity is created and the effects of torsion on the seismic response is assessed for four hazard levels. The response history analysis results indicate that, unlike frame structures, the torsional amplifications in the inelastic systems exceed those of corresponding elastic systems and tend to increase with an increase in the level of inelasticity. The ability of two simplified procedures, elastic response spectrum analysis and pushover analysis, to capture the torsional amplifications in steel‐braced frames is evaluated. Copyright © 2010 John Wiley & Sons, Ltd.     

Evaluation of three‐dimensional modal pushover analysis for unsymmetric‐plan buildings subjected to two components of ground motion

The accuracy of the three‐dimensional modal pushover analysis (MPA) procedure in estimating seismic demands for unsymmetric‐plan buildings due to two horizontal components of ground motion, simultaneously, is evaluated. Eight low‐and medium‐rise structures were considered. Four intended to represent older buildings were designed according to the 1985 Uniform Building Code, whereas four other designs intended to represent newer buildings were based on the 2006 International Building Code. The median seismic demands for these buildings to 39 two‐component ground motions, scaled to two intensity levels, were computed by MPA and nonlinear response history analysis (RHA), and then compared. Even for these ground motions that deform the buildings significantly into the inelastic range, MPA offers sufficient degree of accuracy. It is demonstrated that PMPA, a variant of the MPA procedure, for nonlinear systems is almost as accurate as the well‐known standard response spectrum analysis procedure is for linear systems. Thus, for practical applications, the PMPA procedure offers an attractive alternative to nonlinear RHA, whereby seismic demands can be estimated directly from the (elastic) design spectrum. In contrast, the nonlinear static procedure specified in the ASCE/SEI 41‐06 Standard is demonstrated to grossly underestimate seismic demands for some of the unsymmetric‐plan buildings considered. Copyright © 2011 John Wiley & Sons, Ltd.     

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.     

An Incremental Response Spectrum Analysis Procedure Based on Inelastic Spectral Displacements for Multi-Mode Seismic Performance Evaluation

The so-called Nonlinear Static Procedure (NSP) based on pushover analysis has been developed in the last decade as a practical engineering tool to estimate the inelastic response quantities in the framework of performance-based seismic evaluation of structures. However NSP suffers from a major drawback in that it is restricted with a single-mode response and therefore the procedure can be reliably applied only to the two-dimensional response of low-rise, regular buildings. Recognizing the continuously intensifying use of the pushover-based NSP in the engineering practice, the present paper attempts to develop a new pushover analysis procedure to cater for the multi-mode response in a practical and theoretically consistent manner. The proposed Incremental Response Spectrum Analysis (IRSA) procedure is based on the approximate development of the so-called modal capacity diagrams, which are defined as the backbone curves of the modal hysteresis loops. Modal capacity diagrams are used for the estimation of instantaneous modal inelastic spectral displacements in a piecewise linear process called pushover-history analysis. It is illustrated through an example analysis that the proposed IRSA procedure can estimate with a reasonable accuracy the peak inelastic response quantities of interest, such as story drift ratios and plastic hinge rotations as well as the story shears and overturning moments. A practical version of the procedure is also developed which is based on the code-specified smooth response spectrum and the well-known equal displacement rule. This revised version was published online in July 2006 with corrections to the Cover Date.     

Predicting torsion‐induced lateral displacements for pushover analysis: Influence of torsional system characteristics

The increasing popularity of simplified nonlinear methods in seismic design has recently led to many proposals for procedures aimed at extending pushover analysis to plan asymmetric structures. In terms of practical applications, one particularly promising approach is based on combining pushover analysis of a 3D structural model with the results of linear (modal) dynamic analysis. The effectiveness of such procedure, however, is contingent on one fundamental requirement: the elastic prediction of the envelope of lateral displacements must be conservative with respect to the actual inelastic one. This paper aims at verifying the above assumption through an extensive parametric analysis conducted with simplified single‐storey models. The main structural parameters influencing torsional response in the elastic and inelastic range of behaviour are varied, while devoting special attention to the system stiffness eccentricity and radius. The analysis clarifies the main features of inelastic torsional response of different types of building structures; in this manner, it is found that the above‐mentioned method is generally suitable for structures characterized by moderate to large torsional stiffness, whereas it cannot be recommended for extremely torsionally stiff structures, as their inelastic torsional response almost always exceeds the elastic one. Copyright © 2010 John Wiley & Sons, Ltd.     

Seismic design and testing of the bottom vertical boundary elements in steel plate shear walls. Part 1: design methodology

This research investigates the seismic design method and the cyclic inelastic behavior of the bottom column, also called the vertical boundary element (VBE), in steel plate shear walls (SPSWs). This study consists of two parts. This Part 1 paper discusses the anticipated pushover responses for properly designed SPSWs and the possible inelastic responses of the bottom VBE at various levels of inter‐story drift. Considering both the tension field action of the infill panel and the sway action of the boundary frame, this study develops a simplified method to compute the flexural and shear demands in the bottom VBE. Based on the superposition method, this approach considers various plastic hinge forming locations at different levels of inter‐story drift. One of the key performance‐based design objectives is to ensure that the top ends of the bottom VBEs remain elastic when the SPSWs are subjected to the maximum considered earthquake. This paper presents the comprehensive design procedures for the bottom VBE. Furthermore, this study conducted cyclic performance evaluation tests of three full‐scale two‐story SPSWs at the Taiwan National Center for Research on Earthquake Engineering in 2011 to validate the effectiveness of the proposed design methods. The experimental program, cyclic inelastic responses of the SPSWs and bottom VBEs, and numerical simulations are presented in Part 2. Copyright © 2014 John Wiley & Sons, Ltd.     

Evaluation of predictors of non‐linear seismic demands using ‘fishbone’ models of SMRF buildings

Predictors (or estimates) of seismic structural demands that are less computationally time‐consuming than non‐linear dynamic analysis can be useful for structural performance assessment and for design. In this paper, we evaluate the bias and precision of predictors that make use of, at most, (i) elastic modal vibration properties of the given structure, (ii) the results of a non‐linear static pushover analysis of the structure, and (iii) elastic and inelastic single‐degree‐of‐freedom time‐history analyses for the specified ground motion record. The main predictor of interest is an extension of first‐mode elastic spectral acceleration that additionally takes into account both the second‐mode contribution to (elastic) structural response and the effects of inelasticity. This predictor is evaluated with respect to non‐linear dynamic analysis results for ‘fishbone’ models of steel moment‐resisting frame (SMRF) buildings. The relatively small number of degrees of freedom for each fishbone model allows us to consider several short‐to‐long period buildings and numerous near‐ and far‐field earthquake ground motions of interest in both Japan and the U.S. Before doing so, though, we verify that estimates of the bias and precision of the predictor obtained using fishbone models are effectively equivalent to those based on typical ‘full‐frame’ models of the same buildings. Copyright © 2003 John Wiley & Sons, Ltd.     

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.     

Three‐dimensional modal pushover analysis of buildings subjected to two components of ground motion,including its evaluation for tall buildings

The modal pushover analysis (MPA) procedure, presently restricted to one horizontal component of ground motion, is extended to three‐dimensional analysis of buildings—symmetric or unsymmetric in plan—subjected to two horizontal components of ground motion, simultaneously. Also presented is a variant of this method, called the practical modal pushover analysis (PMPA) procedure, which estimates seismic demands directly from the earthquake response (or design) spectrum. Its accuracy in estimating seismic demands for very tall buildings is evaluated, demonstrating that for nonlinear systems this procedure is almost as accurate as the response spectrum analysis procedure is for linear systems. Thus, for practical applications, the PMPA procedure offers an attractive alternative whereby seismic demands can be estimated directly from the (elastic) design spectrum, thus avoiding the complications of selecting and scaling ground motions for nonlinear response history analysis. Copyright © 2010 John Wiley & Sons, Ltd.     

Optimal seismic design of steel frame buildings based on life cycle cost considerations

A multi‐objective optimization procedure is presented for designing steel moment resisting frame buildings within a performance‐based seismic design framework. Life cycle costs are considered by treating the initial material costs and lifetime seismic damage costs as two separate objectives. Practical design/construction complexity, important but difficult to be included in initial cost analysis, is taken into due account by a proposed diversity index as another objective. Structural members are selected from a database of commercially available wide flange steel sections. Current seismic design criteria (AISC‐LRFD seismic provisions and 1997 NEHRP provisions) are used to check the validity of any design alternative. Seismic performance, in terms of the maximum inter‐storey drift ratio, of a code‐verified design is evaluated using an equivalent single‐degree‐of‐freedom system obtained through a static pushover analysis of the original multi‐degree‐of‐freedom frame building. A simple genetic algorithm code is used to find a Pareto optimal design set. A numerical example of designing a five‐storey perimeter steel frame building is provided using the proposed procedure. It is found that a wide range of valid design alternatives exists, from which a decision maker selects the one that balances different objectives in the most preferred way. Copyright © 2003 John Wiley & Sons, Ltd.     

Damage‐dependent vulnerability curves for existing buildings

Seismic behavior of damaged buildings may be expressed as a function of their REsidual Capacity (REC), which is a measure of seismic capacity, reduced by damage. REC can be interpreted as the median value of collapse vulnerability curves. Its variation owing to damage is a useful indication of increased building vulnerability. REC reduction, indicating the lowering of seismic safety after an earthquake (performance loss, PL), represents an effective index for assessing the need of seismic repair/strengthening after earthquakes. The study investigates the applicability of a pushover‐based method in the analysis of damaged structures for the case of existing under‐designed RC buildings. The paper presents a systematization of the procedure in an assessment framework that applies the capacity spectrum method based on inelastic demand spectra; furthermore, the vulnerability variation of a real building is investigated with a detailed case study. The behavior of damaged buildings is simulated with pushover analysis through suitable modification of plastic hinges (in terms of stiffness, strength and residual drift) for damaged elements. The modification of plastic hinges has been calibrated in tests on nonconforming columns. The case study analysis evidenced that, for minor or moderate damages, the original structural displacement capacity was only slightly influenced, but the ductility capacity was significantly reduced (up to 40%) because of the increased structure deformability. This implied performance loss in the range 10%–20%. For severe damages the PL ranged between 41% and 56%. Local mechanism types exhibit PL nearly double with respect to global mechanism types. Copyright © 2012 John Wiley & Sons, Ltd.     

Experimental tests on full‐scale RC unretrofitted frame and retrofitted with buckling‐restrained braces

The results of experimental tests carried out on reinforced concrete (RC) full‐scale 2‐storey 2‐bays framed buildings are presented. The unretrofitted frame was designed for gravity loads only and without seismic details; such frame was assumed as a benchmark system in this study. A similar RC frame was retrofitted with buckling‐restrained braces (BRBs). The earthquake structural performance of both prototypes was investigated experimentally using displacement‐controlled pushover static and cyclic lateral loads. Modal response properties of the prototypes were also determined before and after the occurrence of structural damage. The results of the dynamic response analyses were utilized to assess the existing design rules for the estimation of the elastic and inelastic period of vibrations. Similarly, the values of equivalent damping were compared with code‐base relationships. It was found that the existing formulations need major revisions when they are used to predict the structural response of as‐built RC framed buildings. The equivalent damping ratio ξ_eq was augmented by more than 50% when the BRBs was employed as bracing system. For the retrofitted frame, the overstrength Ω and the ductility µ are 1.6 and 4.1, respectively; the estimated R‐factor is 6.5. The use of BRBs is thus a viable means to enhance efficiently the lateral stiffness and strength, the energy absorption and dissipation capacity of the existing RC substandard frame buildings. The foundation systems and the existing members of the superstructure are generally not overstressed as the seismic demand imposed on them can be controlled by the axial stiffness and the yielding force of the BRBs. Copyright © 2011 John Wiley & Sons, Ltd.     

Seismic safety of low ductility structures used in Spain

The most important aspects of the design, seismic damage evaluation and safety assessment of structures with low ductility like waffle slabs buildings or flat beams framed buildings are examined in this work. These reinforced concrete structural typologies are the most used in Spain for new buildings but many seismic codes do not recommend them in seismic areas. Their expected seismic performance and safety are evaluated herein by means of incremental non linear structural analysis (pushover analysis) and incremental dynamic analysis which provides capacity curves allowing evaluating their seismic behavior. The seismic hazard is described by means of the reduced 5% damped elastic response spectrum of the Spanish seismic design code. The most important results of the study are the fragility curves calculated for the mentioned building types, which allow obtaining the probability of different damage states of the structures as well as damage probability matrices. The results, which show high vulnerability of the studied low ductility building classes, are compared with those corresponding to ductile framed structures.     

Estimation of seismic drift and ductility demands in planar regular X‐braced steel frames

This paper summarizes the results of an extensive study on the inelastic seismic response of X‐braced steel buildings. More than 100 regular multi‐storey tension‐compression X‐braced steel frames are subjected to an ensemble of 30 ordinary (i.e. without near fault effects) ground motions. The records are scaled to different intensities in order to drive the structures to different levels of inelastic deformation. The statistical analysis of the created response databank indicates that the number of stories, period of vibration, brace slenderness ratio and column stiffness strongly influence the amplitude and heightwise distribution of inelastic deformation. Nonlinear regression analysis is employed in order to derive simple formulae which reflect the aforementioned influences and offer a direct estimation of drift and ductility demands. The uncertainty of this estimation due to the record‐to‐record variability is discussed in detail. More specifically, given the strength (or behaviour) reduction factor, the proposed formulae provide reliable estimates of the maximum roof displacement, the maximum interstorey drift ratio and the maximum cyclic ductility of the diagonals along the height of the structure. The strength reduction factor refers to the point of the first buckling of the diagonals in the building and thus, pushover analysis and estimation of the overstrength factor are not required. This design‐oriented feature enables both the rapid seismic assessment of existing structures and the direct deformation‐controlled seismic design of new ones. A comparison of the proposed method with the procedures adopted in current seismic design codes reveals the accuracy and efficiency of the former. Copyright © 2007 John Wiley & Sons, Ltd.     

Evaluation of ASCE-41, ATC-40 and N2 static pushover methods based on optimally designed buildings

Alternative static pushover methods for the seismic design of new structures are assessed with the aid of advanced computational tools. The current state-of-practice static pushover methods as suggested in the provisions of European and American regulations are implemented in this comparative study. In particular the static pushover methods are: the displacement coefficient method of ASCE-41, the ATC-40 capacity spectrum method and the N2 method of Eurocode 8. Such analysis methods are typically recommended for the performance assessment of existing structures, and therefore most of the existing comparative studies are focused on the performance of one or more structures. Therefore, contrary to previous research studies, we use static pushover methods to perform design and we then compare the capacity of the outcome designs with reference to the results of nonlinear response history analysis. This alternative approach pinpoints the pros and cons of each method since the discrepancies between static and dynamic analysis are propagated to the properties of the final structure. All methods are implemented in an optimum performance-based design framework to obtain the lower-bound designs for two regular and two irregular reinforced concrete building configurations. The outcome designs are compared with respect to the maximum interstorey drift and maximum roof drift demand obtained with the Incremental Dynamic Analysis method. To allow the comparison, also the life-cycle cost of each design is calculated; i.e. a parameter that is used to measure the damage cost due to future earthquakes that will occur during the design life of the structure. The problem of finding the lower bound designs is handled with an Evolutionary type optimization algorithm.     

Generalized force vectors for multi‐mode pushover analysis of torsionally coupled systems

A generalized multi‐mode pushover analysis procedure was developed for estimating the maximum inelastic seismic response of symmetrical plan structures under earthquake ground excitations. Pushover analyses are conducted with story‐specific generalized force vectors in this procedure, with contributions from all effective modes. Generalized pushover analysis procedure is extended to three‐dimensional torsionally coupled systems in the presented study. Generalized force distributions are expressed as the combination of modal forces to simulate the instantaneous force distribution acting on the system when the interstory drift at a story reaches its maximum value during seismic response. Modal contributions to the generalized force vectors are calculated by a modal scaling rule, which is based on the complete quadratic combination. Generalized forces are applied to the mass centers of each story incrementally for producing nonlinear static response. Maximum response quantities are obtained when the individual frames attain their own target interstory drift values in each story. The developed procedure is tested on an eight‐story frame under 15 ground motions, and assessed by comparing the results obtained from nonlinear time history analysis. The method is successful in predicting the torsionally coupled inelastic response of frames responding to large interstory drift demands. Copyright © 2014 John Wiley & Sons, Ltd.