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N2-A Method for Nonlinear Seismic Analysis of Regular Buildings
... -Step 1: Identification and assessment of critical structural zones (e.g., weakest elements, non-seismically designed details, non-code conforming connections, etc.); -Step 2: Global assessment of the existing structure through non-linear analyses accounting for the response of the critical details; -Step 3: Design of local interventions in order to avoid premature failure of the critical zones; -Step 4: Estimation of the required lateral stiffness of the lateral-force resisting system [37,38]; -Step 5: Ductile design of new CBFs accounting for the capacity of the surrounding existing parts (i.e., members and connections); -Step 6: Design of the number and in-plan location of new CBFs to enhance the global performance and to mitigate torsional effects. ...
... The N2 method [37,38] is applied to estimate the lateral displacement demand δ PP (i.e., in correspondence of the so-called performance point PP) and the related capacity δ C , which is estimated modelling the local response of the critical details. To achieve this goal, pushover (PO) curves are first transformed into equivalent bi-linear (BL) capacity curves. ...
... It is also worth observing that, when the maximum base shear V b,Y,max is attained, the majority of compressive braces has already achieved their ultimate post-buckling resistance. The pushover response curves were bi-linearized, transposed into the ADRS domain and compared to the elastic spectrum [37,38], as shown in Fig. 11. The N2 method was used to check the lateral performance of the building, resulting in safety checks being unfulfilled in longitudinal direction due to poor displacement capacity i.e., δ cd,X,AB = 0.18 m < δ PP,X,AB = 0.20 m (see Fig. 11, black curve), which descends from the high rotational demand on third level MR joints due to the soft-storey mechanism. ...
... It is worth noting that the proposed beam model could provide a satisfactory representation of the seismic behaviour of multi-storey buildings both in terms of capacity and demand, especially when compared with the commonly used equivalent SDOF system [28][29][30][31][32][33][34]. Since the model is defined as an equivalent MDOF system it allows considering the contribution of higher modes and directly evaluating global engineering demand parameters such as inter-storey drifts and floor accelerations. ...
... The results, expressed in terms of capacity curves for several directions of the input, are considered for the definition of the seismic assessment of the building by referring to an equivalent inelastic SDOF system considered as representative of the nonlinear global seismic behaviour for each considered direction of the loading. The abovedescribed strategy inspired by the well-known N2 method [30][31][32][33] is at the base of several national and international seismic codes [35][36][37]. The term N2 is intended to emphasize the evaluation of the nonlinear (N) response of the multi-storey building by means of two (2) models: the 3D nonlinear FEM model and the SDOF equivalent system. ...
... For this latter reason, the nonlinear inter-storey constitutive law, obtained by the above-described calibration strategy has to be extended to a cyclic inelastic behaviour. Aiming at referring to simple inelastic constitutive laws, the nonlinear cyclic law can be substituted with a bi-linear curve following the same strategies already proposed for the SDOF equivalence system [30,31]. More sophisticated cyclic calibration procedures can also be adopted as reported in [51] where different strategies are suggested for the latter purpose. ...
... Once the pushover is converted, it is possible to proceed to the comparison with the seismic demand to compute the maximum displacement demand (denoted as d* max ). For this issue, different possible alternatives are investigated, as depicted in Figure 2. [8] (A1), [56] (A2) and [32] (A3); (B) procedure for the dmax estimate according to [34] (B1), [33] (B2), [57] (B3) and [32] (B4); (C) computation of the maximum intensity measure compatible with the fulfillment of NC (IMMC). ...
... More specifically, in the proposal introduced in the version of Eurocode 8-3 [56] published on September 2019 (in the following, it is briefly recalled as "Updating EC8-3 rel.2019" and is currently under review also with additional modifications), an elastoplastic relationship is adopted but with the following differences from NTC2018: the initial stiffness is determined by equating the areas beneath the curve up to the peak of the pushover curve ( * ); the strength * is assumed equal to * until the peak, while, after, it is computed by imposing the equivalence of the areas up to the d , * displacement. Finally, according to Marino's proposal, introduced in [32], the following adaptive criteria are adopted: until the shear peak, the stiffness is computed by imposing the area's equivalence up to the displacement associated with the LS under [8] (A1), [56] (A2) and [32] (A3); (B) procedure for the d max estimate according to [34] (B1), [33] (B2), [57] (B3) and [32] (B4); (C) computation of the maximum intensity measure compatible with the fulfillment of NC (IM MC ). ...
... The N2 method, initially introduced by Fajfar and Fischinger in their work [33], is explicitly recommended in the Italian Structural Code (as Method A [8]) and Eurocode 8-1 [49], besides the Updating EC8-3 rel.2019 document. Equation (4), presented in Table 3, recalls how the assessment of the inelastic displacement demand is computed according to this approach. ...
This paper presents, firstly, an overview of the nonlinear static procedures (NSPs) given in different codes and research studies available in the literature, followed by the results achieved by the authors to evaluate the reliability of the safety level that they guarantee. The latter is estimated by adopting the fragility curve concept. In particular, 125 models of a masonry building case study are generated through a Monte Carlo process to obtain numerical fragility curves by applying various NSPs. More specifically, among the NSPs, the N2 method (based on the use of inelastic response spectra) with different alternatives and the capacity spectrum method (CSM)-based on the use of overdamped response spectra-are investigated. As a reference solution to estimate the reliability of the nonlinear static approach, nonlinear dynamic analyses (NLDAs) are carried out using the cloud method and a set of 125 accelerograms; the results are post-processed to derive fragility curves under the assumption of a lognormal distribution. The focus of this investigation is to quantify the influence that the NSP method's choices imply, such as the criteria adopted to calculate the displacement demand of a structure or those for the bilinearization of the pushover curve. The results show that the N2 methods are all non-conservative. The only method that provides a good approximation of the capacity of the analyzed URM structures as derived from NLDAs is the CSM. In particular, bilinearization is proven to have a relevant impact on the results when using the N2 method to calculate displacement capacities, whereas the CSM method is not affected at all by such an assumption. The results obtained may have a significant impact on engineering practice and in outlining future directions regarding the methods to be recommended in codes.
... There are also many other NSPs, including adaptive and multi-mode approaches, available in the literature other than the aforementioned codified procedures. Some of them comprise the N2 method [7,8], the MPA [9,10], the upper-bound pushover analysis (UBPA) [11], the adaptive modal combination (AMC) procedure [12], the adaptive capacity spectrum method (ACSM) [13], the consecutive modal pushover (CMP) [14], the generalized pushover analysis (GPA) procedure [15], and the normalized multi-mode nonlinear static (NMP) procedure [16]. Kalkan and Kunnath [17] evaluated four NSPs, namely MPA, UBPA, AMC, and FEMA 356 NSP methods, for seismic evaluation of buildings using NTHAs with different ground motions, and found that the AMC procedure was the most consistent and accurate. ...
... Since cannot exceed 0.25, = 0.2 was chosen to represent buildings with high stability problems. The stability coefficient in Eq. (8), which represents the impact of gravitational forces on stability, is linked to the building's story stiffness, represented by the story stiffness ratio / . By adjusting the columns' moments of inertia, this coefficient can be manipulated to highlight stability concerns at various building levels. ...
Classical design procedures are less advantageous than performance-based seismic design (PBSD) of buildings, which is included in existing standards such as ASCE 41-23 for new buildings or retrofitting. PBSD requires accurate assessment of building seismic responses. Such assessments can be done using either faster nonlinear static procedures (NSPs) or more time-intensive nonlinear time-history analyses (NTHAs). However, the reliability of NSPs can be questionable, as shown by previous research. Practitioners need to conduct further investigations to determine safety margins and the applicability scope of these methods. This is especially important for irregular buildings and near-fault zones. This problem is investigated in this paper by first using 1250 single-degree-of-freedom (SDOF) systems to evaluate the ASCE 41-23’s coefficient method and performing 25000 NTHAs for near- and far-fault records. Second, the responses obtained from two alternative approaches, the modal pushover analysis (MPA) and FEMA 440’s capacity spectrum method (CSM), are compared with NTHA responses for buildings with significant higher-mode effects. American standards are used to design 96 3D symmetric and asymmetric steel moment-resisting frame (MRF) buildings with different characteristics such as lateral, lateral-torsional, and torsional modes of vibration dominance as well as different stability conditions, which are considered in this paper. The MPA and CSM are compared with NTHAs in this paper. The results show that the ASCE 41-23’s coefficient method is unreliable for near-fault zones and that the MPA and CSM are unreliable for seismic evaluation of buildings with dominant lateral-torsional modes of vibration or significant P-Δ effects. The results also revealed that MPA is a conservative approach for seismic evaluation of torsionally dominant buildings while CSM is not.
... Hence, three sets of target displacements were determined for the nonlinear static analyses, while for the nonlinear dynamic analyses, seven real accelerograms were selected from a database and were scaled to define three sets of signals, each being compatible with the demand spectrum of the respective LS. The nonlinear static procedure (NSP) recommended in EC8-1 [49]-i.e., the N2 method [50]-was used to determine the target displacements for the nonlinear static analyses. As the structure lacks symmetry with regard to any axis perpendicular to the direction of the seismic action, the analysis had to be conducted for both senses of that direction (left-to-right and right-to-left). ...
... The nonlinear static procedure (NSP) recommended in EC8-1 [49]-i.e., the N2 method [50]-was used to determine the target displacements for the nonlinear static analyses. As the structure lacks symmetry with regard to any axis perpendicular to the direction of the seismic action, the analysis had to be conducted for both senses of that direction (left-to-right and right-to-left). ...
This paper examines how the decision to include (or exclude) masonry infill walls in the modelling of non-seismically designed RC framed structures can affect the results of the EC8-3 seismic assessment process. A frequently used macro-modelling technique for the simulation of infill panels within bounding RC members is first reviewed. A case-study application follows in which the seismic assessment of a sample structure is carried out, with and without considering the effect of its infill walls, using nonlinear static and dynamic analysis models. The obtained results are then discussed according to the applicable limit states’ performance requirements, and conclusions are drawn regarding the overall outcome. The study indicates that, when low and medium seismic input motions constitute the base demand for the assessment of older-type RC framed buildings, the protection provided to the RC members by the confined masonry infill panels should not be neglected. Moreover, it shows that the identification of the most likely collapse mechanism might also be significantly influenced by the modelling decision in question. As such, the default recommendation is to include masonry infill walls in the modelling of such structures.
... The horizontal seismic forces at each floor can be converted into an equivalent triangular distribution of horizontal forces. The principle of conversion is that the bending moment effect produced by the horizontal seismic forces F at each floor is equivalent to that produced by the triangular distribution of horizontal forces [8]. ...
Earthquakes, as one of the primary natural disasters, cause significant destruction to building structures. Historically, the collapse of buildings due to earthquakes has resulted in numerous casualties and economic losses. This paper, through a method of literature review, explores the mechanisms of damage to concrete structures under the action of an earthquake and presents the principles of earthquake-resistant design. It analyzes how factors such as the mechanical properties of structural materials, the form of the structure, the strength of connection nodes, and the conditions of the foundation affect structural damage. It proposes measures to enhance the seismic performance of concrete structures through ductility design, strength design, uniform design, and the strategy of multiple lines of defense. This study holds significant implications for the construction industry by offering a systematic set of earthquake-resistant design principles and strategies which can enhance the seismic resilience of existing and future structures, mitigating the damage caused by earthquakes. Moreover, for society at large, this research underscores the importance of reducing seismic disaster risks through scientific design, contributing to the protection of people's lives and property, and promoting sustainable societal development.
... Other well-established methods include the N2-method (Fajfar and Fischinger 1988), which established the concepts for the definition of the load pattern under a single-component earthquake; these were then incorporated into the Eurocode 8 (Annex B, (CEN 2004)). ...
Pushover analysis is a common nonlinear static procedure, performed to evaluate seismic response of civil structures. This approach allows considering the structural nonlinear behavior under earthquake loading in a simplified yet effective way; however, in its classic formulation, it is limited by restrictive and unrealistic assumptions. Thus, the basic method is inadequate for the accurate analysis of buildings without a vertical plane of symmetry and/or for multi-component earthquakes, where torsion effects cannot be neglected. Furthermore, standard pushover analysis only comprises one dominant mode of vibration; higher modes, and therefore their effects, are not accounted for. For all these reasons, the standard technique is generally not considered suitable for the assessment of buildings at high risk of natural-technological (NaTech) events, such as nuclear facilities in areas of high seismic activity. To overcome these limitations, a multi-modal pushover analysis procedure is here tested and validated on the data from a non-symmetric structure (the SMART case study) under multi-component earthquakes. In detail, the procedure applies a linear combination of modal load patterns, defined according to the well-established Direct Vectorial Addition (DVA) method. With respect to other existing multi-modal pushover analysis techniques, elliptical response envelopes are employed to calculate the corresponding combination factors. Innovatively, the identification of the dominant modes for the load pattern is not limited to the classic maximization of the forces at the basis of a structure but rather generalized to shear and displacement maximization at different heights and locations throughout the whole structural frame.
In this study, a novel equivalent damping ratio model that is suitable for reinforced concrete (RC) structures considering cyclic degradation behavior is developed, and a new equivalent linearization analysis method for implementing the proposed equivalent damping ratio model for use in seismic damage evaluation is presented. To this end, Ibarra’s peak-oriented model, which incorporates an energy-based degradation rule, is selected for representing hysteretic behavior of RC structure, and the optimized equivalent damping for predicting the maximum displacement response is presented by using the empirical method, in which the effect of cyclic degradation is considered. Moreover, the relationship between the hysteretic energy dissipation of the inelastic system and the elastic strain energy of the equivalent linear system is established so that the proposed equivalent linear system can be directly integrated with the Park-Ang seismic model to implement seismic damage evaluation. Due to the simplicity of the equivalent linearization method, the proposed method provides an efficient and reliable way of obtaining comprehensive insight into the seismic performance of RC structures. The verification demonstrates the validity of the proposed method.
The aim of this paper is based on the simple nonlinear procedure founded on the Park-Ang damage index model at different constant damage levels to evaluate the target displacement and performance point (P.P.). The mentioned procedure represents the intersection of the pushover capacity curve with the seismic hazard demand curve according to the equivalent period of vibration and damping in ADRS format. Hence, the damage-based response acceleration ratio is determined at different damage levels and periods for developing inelastic spectra from elastic ones. Then, the relation between the strength reduction factor (R-factor) and damage was extended at different damage levels and periods of vibration to predict the target displacement at the desired damage level, known as the performance point. It is worth mentioning that the cited procedure is represented for four hysteresis models, including Elastic-Perfectly-Plastic (EPP), Modified Clough (MC), moderate stiffness-strength deterioration (MSD), and severe stiffness-strength deterioration (SSD) models, to consider the theory for well-design and not well-designed systems in both steel and concrete structures. The mentioned procedure is the N2 theory development based on the damage model called the DN2 method in this investigation. Two experimental reinforced concrete bridge piers and three steel moment resisting frame structures with different periods, from low to high duration, are considered to verify the suggested procedure. Statistical results show that the target displacement and P.P. are evaluated appropriately regarding presented equations and proposed methodology compared to previous studies.
This study analyzes the progression, utilization, and inherent challenges of traditional non-linear static procedures (NSPs) such as the capacity spectrum method, the displacement coefficient method, and the N2 method for evaluating seismic performance in structures. These methods, along with advanced versions such as multi-mode, modal, adaptive, and energy-based pushover analysis, help determine seismic demands, enriching our grasp on structural behaviors and guiding design choices. While these methods have improved accuracy by considering major vibration modes, they often fall short in addressing intricate aspects such as bidirectional responses, torsional effects, soil-structure interplay, and variations in displacement coefficients. Nevertheless, NSPs offer a more comprehensive and detailed analysis compared to rapid visual screening methods, providing a deeper understanding of potential vulnerabilities and more accurate predictions of structural performance. Their efficiency and reduced computational demands, compared to the comprehensive nonlinear response history analysis (NLRHA), make NSPs a favored tool for engineers aiming for swift seismic performance checks. Their accuracy and application become crucial when gauging seismic risks and potential damage across multiple structures. This paper underscores the ongoing refinements to these methods, reflecting the sustained attention they receive from both industry professionals and researchers.
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