Advances in Civil Engineering

Volume 2019, Article ID 1396019, 16 pages

https://doi.org/10.1155/2019/1396019

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

^{1}EPFL-ENAC-IIC-IMAC, Lausanne, Switzerland^{2}DICEA, Università degli studi di Napoli Federico II, Naples, Italy

Correspondence should be addressed to Pierino Lestuzzi; hc.lfpe@izzutsel.onireip

Received 28 February 2019; Accepted 9 August 2019; Published 19 September 2019

Guest Editor: Francisco López Almansa

Copyright © 2019 Pierino Lestuzzi and Lorenzo Diana. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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 (). 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 () after the plateau is proposed. This issue should be considered for the ongoing developments of Eurocode 8.

#### 1. Introduction

Structures do not remain elastic under extreme ground motions. Nonlinear behavior is therefore crucial in seismic response of structures. However, to avoid the use of more elaborated analyses that are suitable for strategic buildings only, structural-engineering approaches are generally based on simplified methods to determine seismic actions. The prediction of nonlinear seismic demand using linear elastic behavior for the determination of peak nonlinear response is widely used for seismic design as well as for vulnerability assessment. Existing methods involve either a linear response based on initial period and damping ratio, eventually corrected with factors, or a linear response based on increased equivalent period and damping ratio.

It is well established that for medium-to-high period structures, the displacements of elastic and inelastic systems are approximately the same. This empirical finding known as the equal displacement rule (EDR) is nowadays widely used for seismic design purposes, e.g., in Eurocode 8 [1]. The basic assumption of the EDR is to predict the seismic performance of an inelastic system using the equivalent elastic system with the same initial period and damping coefficient. For low-period structures, the EDR loses its validity, since inelastic displacements are larger than elastic displacements. Other methods are therefore needed for this range of periods. The plateau range of the design spectra is particularly important since it provides the highest spectral acceleration. Furthermore, most buildings in Europe have less than 5 stories and thus have a natural period of less than 1 s. As a consequence, the natural period of a large part of the European building stock is located on the plateau of design spectra, i.e., out of the assumed standard range of application of the EDR. Reliable displacement demand predictions are therefore crucial for seismic vulnerability assessment of existing buildings. This is a key issue in vulnerability assessment at urban scale, where such displacement demand predictions are used for determining building damage using mechanical methods [2]. On the plateau of design spectra, Riddell et al. [3] and later Vidic et al. [4] proposed to compute the displacement demand based on a linear variation of the strength reduction factor as a function of the period. Alternative methods using equivalent damping approaches [5] were also developed and included in the design code in the US [6, 7].

The N2 method developed by Fajfar [8] is based on the equal displacement rule associated to a correction for the plateau range of the response spectrum. A simplified version of the N2 method was proposed in Eurocode 8 [1, 8]. This simplified version of the N2 method is investigated in this study. However, the lack of accuracy of the simplified version of the N2 method especially for low-period structures has already been pointed out by several research studies [9, 10]. In case of high-strength reduction factors, the simplified N2 method leads to unconservative results, since displacement demand predictions are underestimated with respect to nonlinear time-history analysis. By contrast, in case of low-strength reduction factors, displacement demand tends to be overestimated when using the simplified N2 method. Therefore, an optimized version of the simplified N2 method has recently been proposed in order to improve the reliability of displacement demand predictions [11]. The objective of this paper is to investigate the accuracy domain of the simplified N2 method and that of the optimized N2 method for displacement demand prediction in the plateau range of seismic design spectra. For comparison, the Lin and Miranda [5] method using equivalent period and damping ratio approaches is also investigated. The accuracy assessment is based on a comparison of the displacement demand computed using nonlinear time-history analysis (NLTHA) with the ones predicted by the three simplified methods.

#### 2. Seismic Displacement Demand Prediction

The simplified version of the N2 method (according to Eurocode 8 and called “N2 method” in the following for simplicity), an optimized version of the simplified N2 method and the Lin and Miranda method are investigated in this study. These methods are briefly described in the following sections.

##### 2.1. N2 Method

Since Veletsos and Newmark [12], it has been widely acknowledged that the displacements of elastic and inelastic systems are approximatively the same (Equal Displacement Rule, EDR). This empirical rule was confirmed by numerous numerical and experimental investigations (e.g., [13]) except for low-period structures, for which inelastic displacements are higher than elastic displacements. Another method is therefore needed to replace the EDR for the plateau period range, where EDR is no longer valid. The basic assumption of the EDR is to model an inelastic system using the equivalent elastic system with the same period and the same damping ratio. As illustrated in Figure 1(a), the EDR states that inelastic peak displacements (*y*_{p}) are approximately equal to elastic peak displacements (*y*_{el}) whatever the selected yield strength ( or yield displacement ) of the structure. Note that when assuming the stiffness to be independent of the strength, the EDR leads to a strength reduction factor () equal to the global displacement ductility.