N2 Method Research Articles

Influence of shear flexibility in structural shear walls on pushover analysis

The aim of this work is to show the main differences which exist, taking in to account the influence of the type of finite element used, when performing pushover analysis of reinforced concrete structures. The non-linear analysis was performed using FE software SAP2000, and the results were extracted from models including Frame and Shell elements, respectively. Several reinforced concrete structures were modelled with Frame elements and Shell elements, which will be further presented. Therefore, it was possible to validate the results obtained from the analysis, also to identify certain restrictions according to the type of finite element used in the modelling of the resistant walls. In the first phase, three isolated structural walls were modelled with distinct geometries. The first one presents a rectangular shape, the second – “L” shape and the third one “U” shape. The application of pushover analysis through the different examples presented in this document, intends to validate the results obtained for the Shell elements. Subsequently, the same kind of analysis was performed on a building. These examples intend to show that the performance of ductility is strongly dependent from the type of element, which is not taken into account in the pushover analysis nowadays. N2 method was applied to all examples, in order to understand the differences in the structures seismic design, according to the type of element used in the modelling. The results are compared, and the differences are identified. As well as, the limitations of applicability of Shell elements in the modelling of structural walls were determined.

Seismic retrofit of gravity load designed RC buildings using external steel concentric bracing systems

This paper discusses the seismic vulnerability of Reinforced Concrete (RC) buildings designed to gravity loads only, which are representative of the Italian building stock mainly erected between ‘70s and ‘80s, and investigates the use of external steel concentric bracing systems to upgrade their seismic performance. A case study building has been selected and designed according to the codes and the current practice in force at the time of construction. The seismic performance has been evaluated through the so-called N2 method in the format of the Capacity Spectrum Method applied to three different models of the building, namely bare frame, full infilled frame and pilotis frame, to highlight the role of the infill walls arrangement. The three models have been retrofitted with the external bracing systems designed with an adopted rational methodology. The external arrangement of the bracing systems has been chosen in order to both minimize the impact on the existing building and avoid the local interaction between the RC structural elements and the steel bracings, as well as the additional loads on the existing foundations. The results have shown the strong role of the infills in the structural behaviour of the existing building and the efficiency of the external steel bracing systems as retrofitting technique of existing RC buildings.

Displacement-based seismic performance evaluation and vulnerability assessment of buildings: The N2 method revisited

The so-called N2 method is a widely used nonlinear static method for the seismic design and/or assessment of different building typologies. In this paper, the idea of reversing the so-called N2 method currently adopted in the Annex B of Eurocode 8 (the code on which the present paper is based) in a direct displacement-based procedure is presented in a simple alternative format. In this way, it is possible to obtain the percentage of the seismic action associated to any performance displacement. By doing so, it is also possible to determine the target displacement without the need for an iterative procedure. The proposed method presents several advantages for professional engineering practice in respect to the seismic vulnerability assessment of new and existing buildings. The comparison of the original N2 method with the proposed one is illustrated by means of a test structure example.

Cost-benefit analysis of traditional seismic retrofitting strategies integrated in the renovation of stone masonry buildings

This study discusses the cost-benefit analysis resulting from the application of traditional seismic retrofitting strategies on four case studies considered representative of both rural and urban stone masonry building stock of Faial Island, in Azores (Portugal). The seismic performance-based assessment was carried out by applying the N2 Method procedure, and the global seismic capacity of each case study estimated by using a three-dimensional model based on the macroelement approach, which combines both the in-plane and out-of-plane response of masonry buildings, to perform nonlinear static analyses. Fragility and loss estimation was evaluated according to the HAZUS methodology. In general terms, the cost-benefit analysis has demonstrated that the studied retrofitting strategies are indeed capable of reducing substantially the seismic vulnerability of the considered case studies, and also that they represent, in fact, a limited amount of the total replacement cost, not compromising, therefore, their economic viability.

Minimum cost performance‐based seismic design of reinforced concrete frames with pushover and nonlinear response‐history analysis

AbstractPrevious studies compare results of pushover and nonlinear response‐history analysis of predesigned reinforced concrete frames. The present study employs nonlinear response‐history analysis and pushover analysis with the N2 method in a computational framework for the minimum cost performance‐based seismic design of reinforced concrete frames according to the fib Model Code 2010 methodology and compares their obtained design solutions in terms of cost and structural performance. It is found that the minimum costs of pushover‐based designs are similar to the costs from response‐history analysis for regular frames but the pushover‐based designs can be more expensive for irregular frames. Furthermore, the pushover‐based designs are not guaranteed to satisfy performance objectives when subjected to response‐history analysis even when more than one lateral load distributions are applied.

A Target Displacement for Pushover Analysis to Estimate Seismic Demand of Eccentrically Braced Frames

A main challenge for performance-based seismic engineering is to develop simple, practical and precise methods for assessing existing structures to satisfy considerable performance objectives. Pushover analysis is a simplified nonlinear analysis technique that can be implemented for estimating the dynamic demands imposed on a structure under earthquake excitations. In this method, structure is subjected to specified load pattern to reach a target displacement. The present study provides a target displacement for estimating the seismic demand of eccentrically braced frames (EBFs). A parametric study is conducted on a group of 30 EBFs under a set of 15 accelerograms. The results of nonlinear dynamic analyses of EBFs have been post-processed by nonlinear regression analysis and a relation is proposed for target displacement. In order to verify the capability of the proposed procedure, three EBFs are assessed by the present method in which the results show that the proposed method is capable of reproducing the peak dynamic responses with relatively good accuracy. Additionally, the comparison of obtained results with those of other conventional target displacement methods such as N2 method, and displacement coefficient method confirms the efficiency of the suggested one.

Simplified and Refined Analyses for Seismic Investigation of Historical Masonry Clusters: Comparison of Results and Influence of the Structural Units Position

The goal of this work is the seismic vulnerability assessment of building aggregates within the historical centre of Arsita (Teramo, Italy), damaged by the 2009 L’Aquila earthquake, by comparing two different analysis procedures applied to a construction compound case study. First, the seismic vulnerability of structural units of the building compound, has been appraised by using a vulnerability evaluation quick form appropriately conceived for masonry aggregates. In particular, heading and intermediate structural units, having different geometrical configuration and seismic behaviour, have been inspected by using the before mentioned form, which allowed to calculate for them a synthetic vulnerability index. Starting from these indexes, the probable damages suffered by the examined structural units under different grade earthquakes have been estimated. Later on, both the single structural units and the whole construction compound have been modelled by using the macro-element refined method foreseen by the 3Muri non-linear analysis software. Static non-linear analyses performed on the above-mentioned structures have provided the related pushover curves, used to estimate, through the N2 method, the damages suffered under seismic actions expected in that site. Therefore, damages of single structural units have been compared to those experienced by the same structures within the building aggregate. Finally, the results deriving from the two analysis methodologies considered have been compared to each other, allowing to confirm the effectiveness of the simplest technique to predict the seismic damage and vulnerability of investigated structures.

Improved Urban Seismic Vulnerability Assessment Using Typological Curves and Accurate Displacement Demand Prediction

ABSTRACT This article addresses seismic vulnerability assessment at an urban scale using mechanical methods, more specifically potential improvements by involving typological curves instead of standard capacity curves and by using accurate displacement demand determination for reliable building damage prediction. The impact in terms of damage grades is computed for two typical Swiss cities. In Europe, seismic-vulnerability assessment is usually performed using the Risk-UE methodology, which involves an empirical approach (LM1 method) and a mechanical approach (LM2 method). The LM2 method contains standard capacity curves of conventional building types for computation of the corresponding damage grades. These capacity curves have been developed for southern Europe and are not optimal for describing features of other building stocks. New specific refined capacity curves are developed for northern Europe. Eurocode 8 prescribes the computation of displacement-demand for seismic assessment of existing building through a process based on the N2 method. However, the inaccuracy of the N2 method in certain conditions has already been studied. Therefore, modifications of N2 method have been proposed to improve the reliability of displacement-demand determination. In this study, the impact of typological capacity curves and the modified N2 method are investigated independently. Results show that both refinements contribute to improve the damage-grade distribution assessment.

Seismic assessment of the historical third tower in San Marino based on a 3D laser scanner survey (3D-LSS)

Italian recent earthquakes demonstrated that historic and monumental buildings are extremely vulnerable to seismic motions. In particular, masonry towers are particularly affected by horizontal loads because of their typological features (slenderness and base shear area). In this regard, San Marino historical center was object of an extended case study to assess the seismic assessment of the monuments. This paper presents a seismic assessment of the historic third tower in San Marino based on several subsequent steps: a preliminary 3D laser scanner survey, a modal frequency identification and a final pushover analysis. The evaluation of the vulnerability of the tower is fundamental for decision makers in order to provide guidelines and establish priority of intervention among historic monuments. In particular, Level 1 and Level 3 were chosen as the approaches under which the structural parameters were calibrated. Following EC8, N2 method was applied in order to assess the vulnerability of the tower.

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.

Damage Assessment by Numerical Modeling of Sant'Agostino's Sanctuary in Offida During the Central Italy 2016–2017 Seismic Sequence

The subject of this work is the Sant’Agostino Sanctuary in Offida (Italy); we investigated both the dynamic behavior and the seismic vulnerability of the complex, used nowadays in its parts as school building, oratory, and church. Offida is in central Italy; the village has been severely damaged by the last seismic events of 2016. The sanctuary was heavily damaged by the earthquake of 24 August 2016. We recurred to finite elements to estimate the vulnerability of the sanctuary and its dynamic response, considering masonry’s nonlinear behavior by means of proper constitutive assumptions. To estimate how the monastery bears the lateral loads related to the expected demands resulting from seismic actions (N2 method) using nonlinear static analysis (Pushover), we recurred to a homogenized material and smeared cracking and crushing constitutive law. As may be remarked by observing buildings that share the same features of the sanctuary and, moreover, by comparing seismic demand vs. capacity, the structure is prone to massive damage leading to collapse. The paper underlines how advanced numerical analysis grants fundamental data on how historical masonry buildings behave under seismic action, providing a method that may easily be implemented at historic monasteries in Europe.

Accuracy Assessment of Nonlinear Seismic Displacement Demand Predicted by Simplified Methods for the Plateau Range of Design Response Spectra

The nonlinear seismic displacement demand prediction for low‐period structures, i.e., with an initial fundamental period situated in the plateau of design response spectra, is studied. In Eurocode 8, the computation of seismic displacement demands is essentially based on a simplified method called the N2 method. Alternative approaches using linear computation with increased damping ratio are common in other parts of the world. The accuracy of three methods for seismic displacement demand prediction is carefully examined for the plateau range of Type‐1 soil class response spectra of Eurocode 8. The accuracy is assessed through comparing the displacement demand computed using nonlinear time‐history analysis (NLTHA) with predictions using simplified methods. The N2 method, a recently proposed optimization of the N2 method, and the Lin and Miranda method are compared. Nonlinear single‐degree‐of‐freedom systems are subjected to several sets of recorded earthquakes that are modified to match design response spectra prescribed by Eurocode 8. The shape of Eurocode 8 response spectra after the plateau is defined by a constant pseudovelocity range (1/T). However, the slope of this declining branch may be specified using precise spectral microzonation investigation. However, the N2 method has been found to be particularly inaccurate with certain microzonation response spectra that are characterized by a gently decreasing branch after the plateau. The present study investigates the impact of the slope of the decreasing branch after the plateau of response spectra on the accuracy of displacement demand predictions. The results show that the accuracy domain of the N2 method is restricted to strength reduction factor values around 3.5. Using the N2 method to predict displacement demands leads to significant overestimations for strength reduction factors smaller than 2.5 and to significant underestimations for strength reduction factors larger than 4. Fortunately, the optimized N2 method leads to accurate results for the whole range of strength reduction factors. For small values of strength reduction factors, up to 2.5, the optimized N2 method and the Lin and Miranda method both provide accurate displacement demand predictions. However, the accuracy of displacement demand prediction strongly depends on the shape of the response spectrum after the plateau. A gently decreasing branch after the plateau affects the accuracy of displacement demand predictions. A threshold value of 0.75 for the exponent of the decreasing branch (1/Tα) after the plateau is proposed. This issue should be considered for the ongoing developments of Eurocode 8.

Seismic displacement demand prediction in non-linear domain: Optimization of the N2 method

In Europe, computation of displacement demand for seismic assessment of existing buildings is essentially based on a simplified formulation of the N2 method as prescribed by Eurocode 8 (EC8). However, a lack of accuracy of the N2 method in certain conditions has been pointed out by several studies. This paper addresses the assessment of effectiveness of the N2 method in seismic displacement demand determination in non-linear domain. The objective of this work is to investigate the accuracy of the N2 method through comparison with displacement demands computed using non-linear time-history analysis (NLTHA). Results show that the original N2 method may lead to overestimation or underestimation of displacement demand predictions. This may affect results of mechanical model-based assessment of seismic vulnerability at an urban scale. Hence, the second part of this paper addresses an improvement of the N2 method formula by empirical evaluation of NLTHA results based on EC8 ground-classes. This task is formulated as a mathematical programming problem in which coefficients are obtained by minimizing the overall discrepancy between NLTHA and modified formula results. Various settings of the mathematical programming problem have been solved using a global optimization metaheuristic. An extensive comparison between the original N2 method formulation and optimized formulae highlights benefits of the strategy.

Simplified Seismic Collapse Evaluation of Code-Conforming Reinforced Concrete Structures in Bucharest, Romania

A total of four code-conforming reinforced concrete structures designed for the seismic conditions of Bucharest, the capital city of Romania, are analysed in this study from the point of view of collapse assessment using a simplified approach. The height regime of the buildings is in the range 5–11 stories, thus covering the common range of both residential and office buildings in Bucharest. The simplified collapse fragility assessment applied in the study involves the seismic performance assessment through the N2 method instead of time-consuming nonlinear time–history analyses. The seismic demand is derived from a Monte Carlo simulated earthquake catalogue coupled with a ground motion model specifically developed for Bucharest through which the ground motion amplitudes are obtained. The results show mean annual collapse rates ranging from 4 × 10−4 to 10−5 with the largest values corresponding to reinforced concrete frames.

Feasibility of Using Neural Networks to Obtain Simplified Capacity Curves for Seismic Assessment

The selection of a given method for the seismic vulnerability assessment of buildings is mostly dependent on the scale of the analysis. Results obtained in large-scale studies are usually less accurate than the ones obtained in small-scale studies. In this paper a study about the feasibility of using Artificial Neural Networks (ANNs) to carry out fast and accurate large-scale seismic vulnerability studies has been presented. In the proposed approach, an ANN was used to obtain a simplified capacity curve of a building typology, in order to use the N2 method to assess the structural seismic behaviour, as presented in the Annex B of the Eurocode 8. Aiming to study the accuracy of the proposed approach, two ANNs with equal architectures were trained with a different number of vectors, trying to evaluate the ANN capacity to achieve good results in domains of the problem which are not well represented by the training vectors. The case study presented in this work allowed the conclusion that the ANN precision is very dependent on the amount of data used to train the ANN and demonstrated that it is possible to use ANN to obtain simplified capacity curves for seismic assessment purposes with high precision.

A Simple Analysis Method for Estimation of Required Damping Ratio for Seismic Response Reduction

This paper proposes a simple analysis method to estimate the additional damping ratio required to achieve a target seismic performance level. The proposed method employs the Newmark and Hall′s design response spectrum and the simple version of the N2 method. With the help of combining the characteristic of the two methods, it is possible to approximately estimate additional damping ratios required to achieve target elastic or inelastic seismic responses of a building structure without iteration. To examine the validity and effective range of the proposed method, time history response analysis is conducted, and the results are compared with those of a numerical analysis in which the post-yield stiffness ratio is considered. The analytical results show that the post-yield stiffness ratio and the yield displacement are important factors that affect the validity of the proposed analysis method; it is also found that the proposed simple analysis procedure may be quite efficient in evaluating the performance point and the damping requirements for seismic response reduction of building structures.

Metoda N2 za projektiranje zgrada

The N2 method is a fast nonlinear static method for the seismic design of buildings. This paper presents main properties and limitations of the method, its development over time, and possibilities of its application for different types of buildings. Differences between the traditional N2 method and its extensions are highlighted. Comparison of the N2 method with similar design approaches is also provided. The results obtained by calculations based on the N2 method are found to be in good agreement with the results obtained by much more complex methods. Guidelines for further research are given in the final part of the paper.

Vertical stratification of redox conditions, denitrification and recharge in shallow groundwater on a volcanic hillslope containing relict organic matter

Natural denitrification in groundwater systems has been recognised as an ecosystem service that reduces the impact of agriculturally-derived nitrate inputs to surface waters. Identification of this ecosystem service within the landscape would permit spatially differentiated land management and legislation. However, spatial variation in groundwater redox conditions poses a significant challenge to such a concept. To gain understanding of the small-scale mosaic of biogeochemical and hydrological controls on denitrification, we established a well field consisting of 11 multilevel well (MLW) clusters on a hillslope containing relict organic matter buried by volcanic deposits 1.8 ka before present. Based on site-specific redox classification thresholds, vertical redox gradients and denitrification potentials were detected at 7 of the 11 sites. Palaeosols or woody debris, which had previously been identified in laboratory experiments as resident electron donors fuelling denitrification, were visually recognisable at 4 of the 7 MLW sites with vertical redox gradients. Moderately enhanced groundwater dissolved organic carbon (DOC) concentrations occurred where resident electron donors were evident. DOC concentrations were lower where anoxic and nitrate-depleted groundwater was found but with an absence of resident electron donors. In these instances, it was assumed that nitrate reduction had occurred somewhere upgradient of the sampled well screen along the lateral groundwater flow path, with the proximate electron donor (DOC) largely consumed in the process, since no evidence was found for denitrification being fuelled by inorganic electron donors. Due to high variability in the isotopic signature of nitrate in oxidised groundwater, the nitrate dual isotope method did not yield firm evidence for denitrification. However, realistic vertical patterns were obtained using the excess N2 method. Tritium-based age dating revealed that oxic conditions were restricted to young groundwater (mean residence time ≤ 3 y), while anoxic conditions were observed across a wider age range (3–25 y).

Impact of displacement demand reliability for seismic vulnerability assessment at an urban scale

This paper addresses seismic vulnerability assessment at an urban scale by focusing on the displacement demand determination for building damage prediction.The study is based on the comparison of urban seismic damage distributions obtained by the displacement demand computed using non-linear time-history analysis (NLTHA) with three simplified methods. These methods include the N2 method, the Lin & Miranda proposal and an optimized version of the N2 method. Comparing the different damage distributions from the three simplified methods with the one obtained by time-history analysis helps understanding the reliability of displacement demand determination. The study is carried out on Sion and Martigny, two typical Swiss cities.For the case of Sion, results clearly show that using N2 method may lead to significant overestimation of damage grade distribution. The use of Lin & Miranda method and optimized version of N2 improves the damage prediction in both cases. For the other studied case of Martigny, N2 method and Lin & Miranda proposal are not accurate. The optimized version of N2 method provides stable and reliable results.

Extended spectrum-based pushover analysis for predicting earthquake-induced forces in tall buildings

Achievements have been made in the development of effective and accurate simplified methods for predicting the seismic demand of tall buildings. However, most of these methods are primarily focused on the estimation of seismic deformation of buildings, while no analysis and calculation of earthquake-induced forces in the structures have been presented. Based on the spectrum-based pushover analysis (SPA) for estimating the seismic demand of tall buildings, where the complicated vibration mode coupling in the nonlinear vibration of structures is simplified, an extended spectrum-based pushover analysis (ESPA) is developed for the fast prediction of earthquake-induced forces of tall buildings. In ESPA, a linear response spectrum analysis procedure is incorporated into SPA to consider the modal combination in the linear stage of vibration. A case study of two high-rise, special steel moment-resisting frame buildings subjected to different levels of ground-motion intensity is conducted to investigate the applicability and accuracy of the ESPA method. Comparisons of the results of earthquake-induced forces are made among ESPA, nonlinear response time history analysis (NLRHA), N2 method, modal pushover analysis (MPA) and consecutive modal pushover analysis (CMP). It is seen that the earthquake-induced forces, including bending moments and shear forces, predicted by the proposed ESPA method agree very well with those of NLRHA.