Extended N2 Method Research Articles

Effects of the extended N2 method on non-linear static procedures of reinforced concrete frame structures

This paper intends to contribute to the validation of the extended N2 nonlinear static method of seismic analysis through the study of two existing reinforced concrete frames. The scope is to further investigate the limitations of the original N2 method in predicting the seismic response of plan asymmetric buildings and to assess the higher accuracy of the extended N2 method. The two case study buildings are representative of the Italian construction between the ’70s and ’90s. One is designed for gravity loads only, the other is designed according to an old seismic code. Two modelling strategies are used, one with distributed and the other with lumped plastic hinges. Three eccentricities eCM between centre of mass and plan geometric centre are considered (eCM = 0%, 5% and 15%). The pushover analyses are carried out following the N2 method, the N2 method plus the 100:30 directional combination rule and the extended N2 method. The results are compared with those obtained from nonlinear history analyses. For each building the seismic demand is assessed for three hazard levels. The results indicate that the application of the extended N2 method significantly enhances the shape of the demand prediction, particularly for buildings with higher plan eccentricities. On the other hand, the directional combination does not significantly change the results and should be neglected.

On the accuracy of enhanced pushover procedures for seismic performance evaluation of code-conforming RC moment-resisting frame buildings subjected to pulse-like and non-pulse-like excitations

In this paper, the applicability and validity of enhanced pushover procedures in estimating the seismic demands of mid- and high-rise reinforced concrete (RC) building frames under pulse-like near-fault ground motions having forward-directivity and fling-step effects as well as non-pulse-like near-fault and far-fault ground motions are assessed through the comparison with the nonlinear response history analysis (NL-RHA). The enhanced pushover procedures including the modal pushover analysis (MPA), force-based adaptive pushover (FAP), displacement-based adaptive pushover (DAP), extended N2 method (EN2), single-run multi-mode pushover (SMP), and non-adaptive displacement-based pushover (NADP) procedures, as well as a conventional pushover method based on the first mode of the structure, are performed. To this end, three plan-symmetric buildings including the 9-, 12-, and 18-story RC frames designed in accordance with the provisions of ASCE 710 and ACI 318-14 are examined. The engineering demand parameters considered are the interstory drift ratios and story shears, which are representatives of the deformation and force demands, respectively. Additionally, the seismic performance of the RC buildings is assessed using the conventional and the enhanced multi-modal pushover procedures in accordance with the systematic evaluation scheme of the ASCE 41-17 standard. The comparative evaluation demonstrates that the accuracy of the NSPs varies with the type of the ground motions. Some methods result in satisfactory performance for pulse-like ground motions, some for non-pulse-like ones, and some for both of them. All in all, the MPA, EN2, and SMP procedures provide a relatively better prediction of the seismic demands of RC building frames.

Higher Mode Effects in Pushover Analysis of Irregular Masonry Buildings

ABSTRACT The paper presents a study on the seismic behavior of irregular unreinforced masonry buildings, aiming at evaluating the effects of higher modes. First, the macroelement model implemented in the TREMURI software and used in this study was validated, based on shake table tests. Then, the extended N2 method, taking into account the in-plan and in-elevation irregularities, was applied to two case studies and compared with respect to the nonlinear dynamic analysis. The results showed that the method is able to simulate correctly the seismic response of both irregular masonry buildings.

Relevance of torsional effects on the seismic assessment of an old RC frame-wall building in Lisbon

Torsional effects may significantly modify the seismic response of a building and the distribution of damage throughout the structure. In this paper, the torsional response of a case study Reinforced Concrete (RC) wall-frame building built between 1960 and 1980 is presented. The building was constructed without considering appropriate seismic design criteria and without taking into account the forces induced by torsional vibrations. A numerical investigation on the seismic behaviour of the case-study building is carried out by means of nonlinear static pushover analyses performed on a 3-D numerical model of the building; the Extended N2 method is adopted to take into account the influence of higher mode effects in plan and in elevation. The results are then compared with the ones obtained from nonlinear Time-History (TH) analysis. It is concluded that the shear demand evaluated according to the Extended N2 method provides a conservative estimate in comparison to TH analysis. Thus, an improved procedure is proposed for the application of the Extended N2 method in the evaluation of shear demand.

Nonlinear Static Seismic Performance Assessment of Plan-Irregular Steel Structures

ABSTRACTThis paper focuses on the evaluation of a set of Nonlinear Static Procedures (NSPs), applied to groups of plan-regular and irregular steel structures with three stories. The NSPs selected are the conventional and the Extended N2, the Capacity Spectrum Method and the Adaptive Capacity Spectrum Method. In terms of translational response, all NSPs lead to adequate estimations of results. Concerning the torsional response, the results of the Extended N2 are the most appropriate to the studied irregular steel structures, except for torsionally stiff concentrically braced frame structures. An improvement version of the Extended N2 method is proposed for these structures.

Effectiveness of damped braces to mitigate seismic torsional response of unsymmetric-plan buildings

The seismic retrofitting of unsymmetric-plan reinforced concrete (r.c.) framed buildings can be carried out by the incorporation of damped braces (DBs). Yet most of the proposals to mitigate the seismic response of asymmetric framed buildings by DBs rest on the hypothesis of elastic (linear) structural response. The aim of the present work is to evaluate the effectiveness and reliability of a Displacement-Based Design procedure of hysteretic damped braces (HYDBs) based on the nonlinear behavior of the frame members, which adopts the extended N2 method considered by Eurocode 8 to evaluate the higher mode torsional effects. The Town Hall of Spilinga (Italy), a framed structure with an L-shaped plan built at the beginning of the 1960s, is supposed to be retrofitted with HYDBs to attain performance levels imposed by the Italian seismic code (NTC08) in a high-risk zone. Ten structural solutions are compared by considering two in-plan distributions of the HYDBs, to eliminate (elastic) torsional effects, and different design values of the frame ductility combined with a constant design value of the damper ductility. A computer code for the nonlinear dynamic analysis of r.c. spatial framed structures is adopted to evaluate the critical incident angle of bidirectional earthquakes. Beams and columns are simulated with a lumped plasticity model, including flat surface modeling of the axial load-biaxial bending moment elastic domain at the end sections, while a bilinear law is used to idealize the behavior of the HYDBs. Damage index domains are adopted to estimate the directions of least seismic capacity, considering artificial earthquakes whose response spectra match those adopted by NTC08 at serviceability and ultimate limit states.

Nonlinear seismic analysis of r.c. framed buildings with setbacks retrofitted by damped braces

Current seismic codes allow the use of damped braces (DBs) for the seismic retrofitting of reinforced concrete (r.c.) framed buildings, but simplified design procedures need to be developed in the case of structures with in-elevation irregularities due to setbacks. In this work, an improved Displacement-Based Design (DBD) procedure of hysteretic damped braces (HYDBs) is proposed to take into account the higher vibration mode effects along the elevation. In line with the extended N2 method, this iterative procedure combines the usual nonlinear static analysis with an elastic modal analysis. Two in-elevation irregular six-storey r.c. framed buildings, characterized by one setback (at the third-storey) and two setbacks (at the second- and fourth-storey), are to be retrofitted by inserting of HYDBs to attain performance levels imposed by the Italian code (NTC08) in a high-risk zone. Local and global displacement-based demand parameters are considered to compare the nonlinear dynamic response of the original and retrofitted structures, considering sets of ground motions whose response spectra match those adopted by NTC08 at serviceability and ultimate limit states. R.c. frame members are idealized by a two-component model, assuming a bilinear moment-curvature law whose ultimate bending moment depends on axial load, while the response of the HYDBs is idealized by a bilinear force-displacement law to prevent yielding and buckling of steel braces.

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.

A pushover seismic analysis method for asymmetric and tall buildings

A pushover seismic analysis method based on the extended N2 method is proposed. Compared to the extended N2 method, the proposed method has four modifications that account for nonlinear higher mode effects. In the first modification, a lateral load based on the equivalent static forces from the dominant dynamic modes of a building combined using the effective mass ratio is used in pushover analysis to consider the higher mode effect in elevation. In the second modification, the total torsional mode effective mass factor is used to modify target displacement in pushover analysis to consider inelastic higher mode effect in plan. In the third modification, the coefficient of torsion (CT) is determined by combining the CT from pushover analysis with that from response spectrum analysis using a contribution factor depending on the ductility level. In the fourth modification, the maximum inter-story drift is derived based on target displacement while considering the effects of story height, ductility, and building period. These modifications are verified using a 14-story building with 10% eccentricity and medium to high structural ductility. Analytical results show that the proposed method predicts better target displacement, CT, and maximum inter-story drift than the N2 and extended N2 method.

Predicting displacement demand of multi-storey asymmetric buildings by nonlinear static analysis and corrective eccentricities

Nonlinear static methods are not always effective in the assessment of multi-storey asymmetric structures because of the errors committed in the evaluation of the deck rotation. To overcome this problem, some of the authors have recently proposed the use of two nonlinear static analyses, characterised by lateral forces applied to different points of the deck. For each of these two analyses, the distance between the point of application of the lateral force and the centre of mass of the deck (corrective eccentricity) is given by relations resulting from a parametric study on a large set of single-storey systems subjected to bidirectional ground motions. The corrective eccentricity depends on four parameters: the rigidity eccentricity, the strength eccentricity, the ratio of the uncoupled torsional to lateral frequencies and the ratio of the elastic strength demand to the actual strength of the system. In this paper, the effectiveness of the proposed method is evaluated for different values of the abovementioned parameters, considering two seismic excitation levels and multi-storey buildings with r.c. framed structure or steel structure equipped with buckling restrained braces. The seismic response of the structures is evaluated by nonlinear time-history analysis and compared with that resulting from the application of the proposed nonlinear static method. To highlight the accuracy of the proposed method, a comparison is also made with the response obtained from the standard application of the nonlinear static method, i.e. without corrective eccentricities, and with the response predicted by the extended N2 method.

Seismic performance of irregular bridges – comparison of different nonlinear static procedures

The adequate seismic performance of transportation infrastructures is important for the functioning of the economy and society. This paper focuses on the seismic assessment and analysis of one of the most important components of these infrastructures, the bridges. In this field, nonlinear static procedures (NSPs) have gained significant attention, resulting in different proposals to improve the accuracy of the procedures while keeping their simplicity. The main goal of this study is focused on the evaluation of the applicability of NSPs for irregular reinforced concrete viaducts. A comparative approach is pursued by resorting to (1) the analyses of the performance of three well-known NSPs (N2 method, modal pushover analysis and adaptive capacity spectrum method) and (2) the extension of the scope of previous studies in this field to a more recent method, the extended N2. As such, a set of bridges with different levels of irregularity, configurations and lengths is investigated. The accuracy of different NSPs is evaluated by comparing the results of NSPs with the ones obtained by means of nonlinear dynamic analyses. The comparison of results confirms the acceptable performance of the multi-modal NSPs and highlights the effectiveness of extended N2 method with respect to its simplicity.

The Extended Adaptive Capacity Spectrum Method for the Seismic Assessment of Plan-Asymmetric Buildings

The use of nonlinear static procedures (NSPs) on the seismic assessment of real existing plan-irregular structures has so far been studied by a limited number of authors. This fact limits the application of such methods to assess current existing structures, the majority of which are irregular in plan. An extended version of the adaptive capacity spectrum method (ACSM) for the seismic assessment of plan-irregular buildings is presented in this paper. The novelty of this proposal is to comprise the most accurate features of commonly used NSPs in order to overcome the problems that subsist in three-dimensional pushover analyses. The accuracy of the procedure is tested in three plan-irregular real buildings. The results are compared with the capacity spectrum method (CSM) with the features proposed in FEMA 440, with the extended N2 method for plan-asymmetric structures, with the adaptive capacity spectrum method (ACSM) and with the most exact nonlinear dynamic analyses. Several seismic intensities are tested, in order to evaluate the performance of the procedure in different stages of structural inelasticity.

Bidirectional Pushover Analysis of Irregular Structures

AbstractPushover analysis is one of the most-used nonlinear static procedures for the seismic assessment of structures; therefore, nowadays it is extensively used by practicing engineers for the seismic analysis of virtually every type of building. This paper proposes a bidirectional pushover analysis (BPA) method to overcome the limitations of current pushover methods to assess the seismic response of irregular (both in plan and in elevation) buildings subjected to bidirectional ground motions. The extended N2 method and the proposed BPA method were applied to estimate the nonlinear response of six highly irregular reinforced concrete frame structures designed according to the requirements of Eurocode 8. Results in terms of interstory drifts and floor rotations are compared with those given by nonlinear response history analysis (NRHA). For the latter, a suite of twenty Italian bidirectional seismic ground motions was selected. It was found that results given by the proposed BPA method match those given ...

The extended N2 method considering higher mode effects in both plan and elevation

The extended N2 method has been developed, which takes into account higher mode effects both in plan and 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 drift can be obtained by combining the results of basic pushover analysis and those of standard elastic modal analysis. In the paper, the proposed procedure was summarized and applied to a test example which represents an actual 8-storey reinforced concrete building. The results obtained by the extended N2 method were compared with the results of nonlinear response history analysis and basic N2 analysis without the consideration of higher modes. The extended N2 method was able to provide fair, conservative estimates of response in the case of the test example. In comparison to the basic N2 method, the prediction of seismic demand was greatly improved in the upper part of the building and at the flexible edges.

Seismic analysis of an asymmetric fixed base and base-isolated high-rack steel structure

Since, in the case of plan-eccentric, externally-braced, high-rack storage structures made of steel, damage to the central part of the structure can cause damage to the stored goods, and thus higher earthquake-induced costs, an investigation has been performed into the seismic performance of such structures. An actual structure was analysed both as a fixed base variant, and as a base-isolated variant. The extended N2 method (pushover analysis) was used, and the results obtained were compared with selected results obtained by nonlinear-dynamic analysis. The response of several different mass-asymmetric structural variants, corresponding to different occupancy levels of the structure, was analysed, and the results obtained are presented as floor plan projection envelopes of the top, relative and base displacements, as well as the storey drifts. An interesting result obtained in the research was that asymmetry can increase the damage in the supporting structure on the flexible side, and that the central part of the rack structure remains in the elastic region only when the eccentricity is small. In fact, from the seismic point of view, full occupancy is not the most critical condition, but rather lower occupancy, which could cause eccentricities ranging up to 10% or 15% of the larger floor plan dimension; this could lead to damage propagation in some of the columns. The application of base isolation has a positive effect on the management of seismic performance, even in the case of higher levels of occupancy and larger mass eccentricities. It is pointed out that an eccentricity of 5%, which is prescribed in Eurocode 8, might not be sufficient in such structural types, and that similar concerns could be present in other types of industrial structures with a similar ratio between live and dead loads.

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

AbstractThe 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.

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

AbstractThe 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.