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.     

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.     

Simplified seismic analysis of asymmetric building systems

The paper reviews the uncoupled modal response history analysis (UMRHA) and modal pushover analysis (MPA) procedure in the analysis of asymmetric structures. From the pushover curves in ADRS format, showing the relationships of base shear versus roof translation and base torque versus roof rotation, a bifurcating characteristic of the pushover curves of an asymmetric structure is observed. A two‐degree‐of‐freedom (2DOF) modal stick is constructed using lump mass eccentrically placed at the end of beam which is connected with the column by a rotational spring. By converting the equation of motion of a whole structure into 2DOF modal equations, all of the elastic properties in the 2DOF modal sticks can be determined accurately. A mathematical proof is carried out to demonstrate that the 2DOF modal stick is consistent with the single‐degree‐of‐freedom (SDOF) modal stick at elastic state. The bifurcating characteristic of modal pushover curves and the interaction of modal translation and rotation can be considered rationally by this 2DOF modal stick. In order to verify the effectiveness of this proposed 2DOF modal stick, a two‐storey asymmetric building structure was analysed by the UMRHA procedure incorporating this novel 2DOF modal sticks (2DMPA) and conventional SDOF modal sticks (SDMPA), respectively. The analytical results are compared with those obtained by nonlinear response history analysis (RHA). It is illustrated that the accuracy of the rotational response histories obtained by 2DMPA is much better than those obtained by SDMPA. Consequently, the estimations of translational response histories on flexible side (FS) and stiff side (SS) of the building structure are also improved. Copyright © 2006 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.     

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.     

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.     

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.     

Dispersions for the pushover‐based risk assessment of reinforced concrete frames and cantilever walls

The paper presents the results of an investigation into the dispersion values, expressed in terms of limit‐state spectral accelerations, which could be used for the pushover‐based risk assessment of low‐height to mid‐height reinforced concrete frames and cantilever walls. The results of an extensive parametric study of a portfolio of test structures indicated that the dispersion values due to record‐to‐record variability and modelling uncertainty (β_LS,RU) are within the range from 0.3 to 0.55 for the near collapse limit state, and between 0.35 and 0.60 for the collapse limit state. The dispersions β_LS,RU proposed for the code‐conforming and the majority of old (non code‐conforming) frames are in between these values. On the other hand, the dispersions proposed for the old frames with a soft storey and an invariant plastic mechanism, and for the code‐conforming cantilever walls, are at the lower and upper bounds of the presented values, respectively. The structural parameters that influence these dispersions were identified, and the influence of different ground motion sets, and of the models used for the calculation of the rotation capacities of the columns, on the calculated fragility parameters was examined and quantified. The proposed dispersion values were employed in a practice‐oriented pushover‐based method for the estimation of failure probability for eight selected examples. The pushover‐based risk assessment method, although extremely simple and economical when compared with more rigorous probabilistic methods, was able to predict seismic risk with reasonable accuracy, thus showing it to be a practical tool for engineers. Copyright © 2016 John Wiley & Sons, Ltd.     

Influence of deterioration modelling on the seismic response of steel moment frames designed to Eurocode 8

This paper assesses the influence of cyclic and in‐cycle degradation on seismic drift demands in moment‐resisting steel frames (MRF) designed to Eurocode 8. The structural characteristics, ground motion frequency content, and level of inelasticity are the primary parameters considered. A set of single‐degree‐of‐freedom (SDOF) systems, subjected to varying levels of inelastic demands, is initially investigated followed by an extensive study on multi‐storey frames. The latter comprises a large number of incremental dynamic analyses (IDA) on 12 frames modelled with or without consideration of degradation effects. A suite of 56 far‐field ground motion records, appropriately scaled to simulate 4 levels of inelastic demand, is employed for the IDA. Characteristic results from a detailed parametric investigation show that maximum response in terms of global and inter‐storey drifts is notably affected by degradation phenomena, in addition to the earthquake frequency content and the scaled inelastic demands. Consistently, both SDOF and frame systems with fundamental periods shorter than the mean period of ground motion can experience higher lateral strength demands and seismic drifts than those of non‐degrading counterparts in the same period range. Also, degrading multi‐storey frames can exhibit distinctly different plastic mechanisms with concentration of drifts at lower levels. Importantly, degrading systems might reach a “near‐collapse” limit state at ductility demand levels comparable to or lower than the assumed design behaviour factor, a result with direct consequences on optimised design situations where over‐strength would be minimal. Finally, the implications of the findings with respect to design‐level limit states are discussed.     

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.     

Tests on low‐ductility RC frames under high‐ and low‐frequency excitations

The response of low‐ductility reinforced concrete (RC) frames, designed typically for a non‐seismic region, subjected to two frequencies of base excitations is studied. Five half‐scaled, two‐bay, two‐storey, RC frames, each approximately 5 m wide by 3.3 m high, were subjected to both horizontal and/or vertical base excitations with a frequency of 40 Hz as well as a lower frequency of about 4 Hz (close to the fundamental frequency) using a shake table. The imposed acceleration amplitude ranged from 0.2 to 1.2g. The test results showed that the response characteristics of the structures differed under high‐ and low‐frequency excitations. The frames were able to sustain high‐frequency excitations without damage but were inadequate for low‐frequency excitations, even though the frames exhibited some ductility. Linear‐elastic time‐history analysis can predict reasonably well the structural response under high‐frequency excitations. As the frames were not designed for seismic loads, the reinforcement detailing may not have been adequate, based on the crack pattern observed. The effect of vertical excitation can cause significant additional forces in the columns and moment reversals in the beams. The ‘strong‐column, weak‐beam’ approach for lateral load RC frame design is supported by experimental observations. Copyright © 2001 John Wiley & Sons, Ltd.     

Application of MPA to estimate probability of collapse of structures

This paper develops a modal pushover analysis‐ (MPA) based approximate procedure to quantify the collapse potential of structural systems. The computationally demanding incremental dynamic analysis (IDA) of the structural system is avoided by MPA of the structure in conjunction with empirical equations for the collapse strength ratio for the first‐mode single‐degree‐of‐freedom (SDF) system; higher modes of vibration play essentially no role in estimating the ground motion intensity required to cause collapse of the structure. Presented are collapse fragility curves for 6‐, 9‐, and 20‐story regular special moment‐resisting teel frames computed by the exact and approximate procedures, demonstrating that the MPA‐based approximate procedure requires only a small fraction (1% in one example) of the computational effort inherent in exact IDA and still achieves highly accurate results. Copyright © 2010 John Wiley & Sons, Ltd.     

Effects of supplemental viscous damping on inelastic seismic response of asymmetric systems

This paper investigates the effects of supplemental viscous damping on the seismic response of one‐storey, asymmetric‐plan systems responding in the inelastic range of behaviour. It was found that addition of the supplemental damping reduces not only deformation demand but also ductility and hysteretic energy dissipation demands on lateral load resisting elements during earthquake loading. However, the level of reduction strongly depends on the plan‐wise distribution of supplemental damping. Nearly optimal reduction in demands on the outermost flexible‐side element, an element generally considered to be the most critical element, was realized when damping was distributed unevenly in the system plan such that the damping eccentricity was equal in magnitude but opposite in algebraic sign to the structural eccentricity of the system. These results are similar to those noted previously for linear elastic systems, indicating that supplemental damping is also effective for systems expected to respond in the inelastic range. Copyright © 2001 John Wiley & Sons, Ltd.     

Experimental–numerical investigation on the seismic behaviour of moment‐resisting timber frames with densified veneer wood‐reinforced timber joints and expanded tube fasteners

This paper investigates the seismic behaviour of moment‐resisting timber frames with beam‐column joints fastened with expanded tubes and reinforced with densified veneer wood. Laboratory experiments are carried out on single joints to investigate the cyclic behaviour and, more specifically, the impairment of strength, the ductility ratio and the equivalent viscous damping ratio. A phenomenological numerical model is proposed, where the beams and columns are schematized using linear‐elastic beam elements, and the joints with non‐linear hysteretic spring calibrated on the results of the experimental tests. The model is used to analyse some representative moment‐transmitting structures characterised by different number of bays and storeys. After an estimation of the lateral load‐carrying capacity using a pushover analysis, the numerical model is used to estimate the behaviour factor. An incremental dynamic analysis is performed using a set of accelerograms spectrum consistent with a chosen design spectrum. The analyses lead to an estimation of the behaviour factor of 3 and 6 for a portal frame and a five‐storey, three‐bay frame, respectively, which confirms the highly dissipative behaviour of this kind of moment connection. Copyright © 2017 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.     

Inelastic torsion of multistorey buildings under earthquake excitations

The inelastic earthquake response of eccentric, multistorey, frame‐type, reinforced concrete buildings is investigated using three‐ and five‐storey models, subjected to a set of 10, two‐component, semi‐artificial motions, generated to match the design spectrum. Buildings designed according to the EC8 as well as the UBC‐97 code were included in the investigation. It is found that contrary to what the simplified one‐storey, typical, shear‐beam models predict, the so‐called ‘flexible’ side frames exhibit higher ductility demands than the ‘stiff’ side frames. The substantial differences in such demands between the two sides suggest a need for reassessment of the pertinent code provisions. This investigation constitutes one of the first attempts to study the problem of inelastic torsion by means of realistic, multistorey inelastic building models. Additional studies with similar or even more refined idealizations will certainly be required to arrive at definite results and recommendations for possible code revisions. Copyright © 2005 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.     

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.     

Evaluation of the structural response under seismic actions using non‐linear static methods

The paper investigates the degree of accuracy achievable when some non‐linear static procedures based on a pushover analysis are used to evaluate the seismic performance. In order to assess the significance of different sources of errors, three types of structural systems are analysed: (i) single‐degree‐of‐freedom (SDOF) systems with different hysteretic behaviour; (ii) shear‐type multi‐degree‐of‐freedom (MDOF) systems with elastic–perfect plastic (EPP) shear force–interstorey drift relationships; (iii) a steel moment‐resisting frame with rigid joints and EPP moment–curvature relationship. In SDOF systems, the source of approximation comes only from the calibration of the demand spectrum, while in MDOF systems some further errors are introduced by the schematization with an equivalent SDOF system. The non‐linear static procedures are compared with rigorous time‐history analyses carried out by considering ten generated earthquake ground motions compatible with the Eurocode 8 elastic spectra. It was found that SDOF systems with longer periods satisfy the equal displacement approximation regardless of the hysteretic model, while hysteresis loops with smaller energy dissipated indicate lower response for shorter periods. This is the opposite of what predicted by the ATC‐40 capacity spectrum method, which underestimates and overestimates, respectively, the actual response of low‐ and high‐ductility systems. Conversely, the inelastic spectrum method proposed by Vidic, Fajfar and Fischinger leads to the most accurate results for all types of structural systems. The analyses carried out on EPP shear‐type frames point out a large concentration of the ductility demand on some storeys. However, such a concentration markedly reduces when some hardening is accounted for. Copyright © 2006 John Wiley & Sons, Ltd.     

Strengthening of moment‐resisting frame structures against near‐fault ground motion effects

Near‐fault ground motions with forward directivity are characterized by a large pulse. This pulse‐like motion may cause a highly non‐uniform distribution of story ductility demands for code‐compliant frame structures, with maximum demands that may considerably exceed the level of code expectations. Strengthening techniques for multi‐story frame structures are explored with the objective of reducing maximum drift demands. One option is to modify the code‐based SRSS distribution of story shear strength over the height by strengthening of the lower stories of the frame. The modified distribution reduces the maximum story ductility demand, particularly for weak and flexible structures. However, this strengthening technique is less effective for stiff structures, and is almost ineffective in cases in which the maximum demand occurs in the upper stories, i.e. strong and flexible structures. As an alternative, the benefits of strengthening frames with elastic and inelastic walls are evaluated. The effects of adding walls that are either fixed or hinged at the base are investigated. It is demonstrated that strengthening with hinged walls is very effective in reducing drift demands for structures with a wide range of periods and at various performance levels. Wall inelastic behavior only slightly reduces the benefits of strengthening with hinged walls.Copyright © 2004 John Wiley & Sons, Ltd.