Mathematical Problems in Engineering

Volume 2016, Article ID 8934196, 17 pages

http://dx.doi.org/10.1155/2016/8934196

## Base Isolation for Seismic Retrofitting of a Multiple Building Structure: Evaluation of Equivalent Linearization Method

Department of Civil Engineering, Design, Building and Environment, Second University of Naples, Via Roma 29, 81031 Aversa, Italy

Received 22 April 2016; Revised 21 July 2016; Accepted 17 October 2016

Academic Editor: Alessandro Palmeri

Copyright © 2016 Massimiliano Ferraioli and Alberto Mandara. 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

Although the most commonly used isolation systems exhibit nonlinear inelastic behaviour, the equivalent linear elastic analysis is commonly used in the design and assessment of seismic-isolated structures. The paper investigates if the linear elastic model is suitable for the analysis of a seismically isolated multiple building structure. To this aim, its computed responses were compared with those calculated by nonlinear dynamic analysis. A common base isolation plane connects the isolation bearings supporting the adjacent structures. In this situation, the conventional equivalent linear elastic analysis may have some problems of accuracy because this method is calibrated on single base-isolated structures. Moreover, the torsional characteristics of the combined system are significantly different from those of separate isolated buildings. A number of numerical simulations and parametric studies under earthquake excitations were performed. The accuracy of the dynamic response obtained by the equivalent linear elastic model was calculated by the magnitude of the error with respect to the corresponding response considering the nonlinear behaviour of the isolation system. The maximum displacements at the isolation level, the maximum interstorey drifts, and the peak absolute acceleration were selected as the most important response measures. The influence of mass eccentricity, torsion, and high-modes effects was finally investigated.

#### 1. Introduction

The most commonly used isolation devices for buildings are lead rubber bearings, high damping rubber bearings, and friction pendulum bearings. Many studies in the literature focused on the performance of isolators, either made of rubber (with or without lead cores) or based on sliding surfaces (single, double, or triple) [1–4]. These seismic isolation devices are characterized by a nonlinear inelastic behaviour with high initial stiffness for minor horizontal loads and high hysteretic energy dissipation during cycles of loading and unloading. Thus, the behaviour of base-isolated buildings under seismic loading is, in general, both dynamic and nonlinear. However, the nonlinearity is generally confined in the isolation bearings. In fact, the design of isolated structures is carried out so that the isolation system is stable and capable of sustaining forces and displacements associated with maximum considered earthquake ground motions. The structure above the isolation system shall remain essentially elastic since excessively large drifts could result due to the nature of long-period vibration. According to the current code provisions, the analysis of a seismically isolated building with an expected inelastic behaviour of the superstructure may be carried out using two alternative analysis methods: nonlinear time-history analysis (NLTHA) and nonlinear static analysis (Pushover). FEMA 356 [5] admits using the pushover analysis also for base-isolated buildings. The New US building design codes chiefly use this analysis method as a tool to check existing buildings, while both the Eurocode 8 [6] and the Italian Seismic Code [7] do not mention any nonlinear static procedure for the design of base-isolated buildings. Especially for low-rise buildings, the inelastic behaviour of the superstructure has a small impact on the maximum seismic response, and the hysteretic damping and the energy dissipation capacity are controlled by the isolation system. Thus, the analysis of the seismically isolated building is usually carried out neglecting the inelastic behaviour of the superstructure. Moreover, under certain conditions prescribed in many codes, the force-deformation relationship of the isolation system may be defined assuming an equivalent linear viscoelastic model. Thus, the linear (static or dynamic) analysis is carried out, at least at the preliminary design and analysis phases. This means that the structure moves as a rigid body, the distribution of the inertia forces is considered to be almost uniform, and the higher oscillation modes are ignored. The approximate linear methods were generally assessed based on a large number of numerical simulations that are generally performed on simplified ideal systems [8–10] while their application to real 3D complex models is still lacking. Moreover, the linear dynamic analysis is in general expected to be limited in its ability to capture the influence of higher modes on the overall response only when using highly nonlinear isolation system while it is considered adequate for isolation system satisfying the limited conditions of equivalent linearization commonly suggested in seismic standards and codes. In this paper, the accuracy of the equivalent linearization was evaluated when applied to a 3D complex model of a multiple building structure with a full representation of bidirectional loading and torsional response. The isolation system meets all the conditions to be considered as being equivalent to linear according to both Eurocode 8 [6] and the Italian Seismic Code [7]. The multiple building structure is located in the hospital campus of Avellino (Italy) and is composed of three adjacent five-storey buildings separated to avoid pounding. The three original fixed-base structures were retrofitted creating a common isolation plane at ground level. It may be very interesting to investigate the appropriateness of using equivalent linear analysis methods when applied to this real case study. In fact, the approximate linear methods were generally assessed based on a single base-isolated structure modelled as a single-degree-of-freedom (SDOF) system, while the case study is a multiple building structure that is quite different from a single base-isolated structure. Thus, relevant issues regarding the higher-modes effects on the superstructure, accidental torsional, and poundings effects were discussed.

#### 2. Equivalent Linearization of Seismic Isolation System

The use of equivalent linearization is limited by several requirements, which are usually involved in equivalent stiffness, damping ratio, and restoring force. Many methods proposed in the literature are based on equivalent linearization of the system through analytical or empirical formulas for effective lateral stiffness and equivalent damping ratio [1, 8–12]. The most frequently used is the method originally proposed by Rosenblueth and Herrera [13] and nowadays adopted by several seismic codes [6, 7, 14]. More recently, Matsagar and Jangid [10] studied the influence of isolator characteristics on the response of base-isolated structures. Mavronicola and Komodromos [15] provided an evaluation of equivalent linearized models assessed through parametric studies. Pant et al. [16] studied the accuracy of the equivalent lateral force (ELF) procedure for the analysis and design of seismically isolated structures. In Liu et al. [17, 18] limited conditions specified in the equivalent linearization of seismic isolation system were investigated when subjected to seismic loads. Zordan et al. [8] proposed an improved expression for equivalent linearization of structures supported on lead rubber bearings. According to Eurocode 8 [6] the use of linear dynamic analysis to avoid a large amount of computational time is allowed only when the isolation system may be modelled with equivalent linear (EL) viscoelastic behaviour. The occurrence of this event depends on a number of limitations that should be satisfied in order to allow the usage of equivalent linear elastic analysis. Specifically, the following limited conditions should be met: (a) the effective stiffness of the isolation system is at least 50% of the effective stiffness at 20% of design displacement ; (b) the effective damping ratio of the isolation system does not exceed 30%; (c) the force-displacement characteristics of the isolation system do not vary by more than 10% due to the rate of loading or due to the vertical loads; (d) the increase of the restoring force in the isolating system for displacement between and is at least 2.5% of the total gravity load above the isolating system. The equivalent linearization of nonlinear isolation systems is based on the determination of effective stiffness and the effective viscous damping ratio, in order to represent both the deformation forces and the energy dissipation during earthquake ground motions. In this paper, the method generally suggested in seismic standards and codes [6, 7] for estimating equivalent period and the equivalent viscous damping ratio was applied. This formulation is based on the method originally proposed by Rosenblueth and Herrera [13]. The effective stiffness of the isolating system is calculated as the sum of the effective stiffness of the isolating bearings. The effective stiffness is expressed as the secant stiffness of the force-displacement curve at the maximum displacement, , as follows: where is the elastic stiffness, is the displacement ductility ratio defined as the ratio of the maximum inelastic displacement to the yield displacement, and is the post-to-pre yield stiffness ratio. The energy dissipation in bearings is evaluated from the measured energy dissipated in cycles with frequency in the range of the natural frequencies of the modes considered and expressed in terms of an equivalent viscous damping (effective damping ). The effective hysteretic damping ratio, , of a linear elastic isolation is characterized according to the equal energy dissipation principle, as follows: where is the characteristic strength of the isolator. The effective stiffness and viscous damping ratio of the isolation system were calculated with reference to the design displacement that is determined according to the design earthquake. Since the effective stiffness and the effective damping depend, in general, on the maximum displacement of the isolation bearing, an iterative procedure was applied until the difference between assumed and calculated values of does not exceed 5% of the assumed value.

#### 3. Dynamic Analysis Procedures

The seismic response of base-isolated buildings is generally evaluated using three different linear methods: (1) simplified linear static analysis (LSA), (2) response spectrum analysis (RSA), and (3) response history analysis (RHA). The simplified linear analysis method assumes that the superstructure is a rigid solid translating above the isolating system and superimposes static torsional effects. The response spectrum analysis takes into account the total eccentricity of the mass of the superstructure and considers both the horizontal displacements and the torsional movement about the vertical axis. Different damping ratios were considered during analysis: % for the three lower mode shapes that are dominated by the isolation system; % for the higher mode shapes that are dominated by the superstructure. The main sources of inaccuracy in the practical application of the RSA for base-isolated buildings are the response spectrum for different viscous damping ratios and the combination rule for nonconventional structures. In fact, the damping correction factor (DCF), even though largely applied in the current state of practice (e.g., the *η*-factor in Eurocode 8 [6] and Italian Code [7]), may provide inaccurate predictions for seismically isolated buildings [19, 20]. Moreover, the square root of sum of squares rule neglects the statistical correlation between modes of vibration. Thus, it may lead to unacceptable inaccuracies when the modal frequencies of the structure are close to each other (e.g., in the first three rigid-body modes for base-isolated buildings). In order to overcome these problems, Muscolino et al. [21] proposed an improved Response Spectrum Method based on a two-stage transformation of coordinates in parallel with a Damping-Adjusted Combination (DAC) rule. As far as the response history analysis is concerned, it must be observed that the base-isolated structures are typically nonlinear and nonclassically damped systems. Thus, the standard linear modal analysis based on a linear relationship between damping, mass, and stiffness and modal uncoupling cannot be applied. Thus, as an alternative to direct integration methods, the dynamic equilibrium equation may be uncoupled in the complex modal space (exact complex-valued modal analysis) [22]. This approach can give several advantages for the evaluation of seismic response, and it was found competitive in terms of computational effort with direct integration methods. In general, the response history analysis may be carried out with methods for directly integrating the governing nonlinear equations of motion. The nonlinear behaviour under seismic loading generally takes place only in the isolation bearings. In fact, the base isolation is a way of mitigating the seismic demand on the structure, and thus the superstructure should behave elastically or almost rigid. Thus, the nonlinearity is generally confined to the isolators and is relatively well known. Because of the nonlinearity present in the structural systems, the time domain method should be employed, although it is more time consuming if compared with frequency domain method. As an alternative, FEMA 356 [5] and UBC 97 [23] tolerate the use of nonlinear static procedures (pushover) also for the seismic analysis of isolated structures. On the contrary, both Eurocode 8 [6] and Italian Seismic Code [7] allow applying this procedure only for fixed-base structures. Therefore, the nonlinear time-history analysis (NLTHA) of base-isolated structures is not uncommon, even though it requires the modelling of the constitutive laws of the devices that are able to adequately reproduce the behaviour of the system in the range of deformations and velocities anticipated in the seismic design situation. This approach requires the solution of a system of nonlinear differential equations whose size is equal to the total number of degrees of freedom of the structure. However, in the case of isolated structures a very restricted number of points in which nonlinear behaviour takes place when subjected to seismic loading occur. This situation, with all nonlinearities restricted to the link elements, allows the application of a simplified nonlinear dynamic procedure for the solution of equilibrium equations: the fast nonlinear analysis (FNA) method [24]. The computational speed of FNA method is very high if compared with the traditional method of nonlinear analysis in which the complete equilibrium equations are formed and solved at each increment of load. Moreover, the use of Ritz vectors allows giving more accurate results than the use of the same number of exact mode shapes. The reason is that the Ritz vectors are generated by taking into account the spatial distribution of the dynamic loading, while the traditional exact mode shapes neglect this information.

#### 4. Case Study: Hospital Building in Avellino (Italy)

The case study is a 5-storey hospital building of the new hospital campus of Avellino in Campania (Italy). The construction of the building was never completed, and only the reinforced concrete skeleton structures were erected. The building was composed of three adjacent reinforced concrete frame structures (named A, B, and C) that were linked functionally but structurally disconnected by separation gaps to avoid poundings. The seismic retrofit was carried out by eliminating this separation gap at ground level, thus creating a common isolation plane. The isolation bearings were installed after cutting the bottom portion of the columns under the ground floor. Thus, the three existing reinforced concrete structures were transformed into a single base-isolated multiple building structure (Figures 1 and 2). The isolation system is composed of 25 circular shaped high damping rubber bearings (HDRBs) (with three diameters, namely, 600, 650, and 800 mm) and plane surface steel-teflon (PTFE) Sliding Devices (SDs). The plan layout of HDRBs and SDs was selected to minimize the torsion effects due to any eccentricity between the centre of mass of the superstructure and the centre of the rigidity of the isolation system (Figure 3). The design displacement was mm. The mechanical properties of the isolation rubber bearings used for the sample structure are summarized in Table 1. According to Italian Code [7] and Eurocode 8 [6], the behaviour factor is taken as being equal to for the design of the foundation substructure and the isolation system, for the design of the superstructure [25]. The seismic performance evaluation was carried out with the procedure reported in Annex B of EN 1998-3 [6] and in Italian Code [7]. This procedure is equivalent to the capacity spectrum method based on inelastic demand spectra [26, 27]. The details of the seismic performance evaluation and design process are extensively described in Ferraioli et al. [28, 29].