Improved response-spectrum analysis of base-isolated buildings: A substructure-based response spectrum method

Engineering Structures 162 (2018) 198–212

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Improved response-spectrum analysis of base-isolated buildings: A substructure-based response spectrum method

T

⁎

D. De Domenico , G. Falsone, G. Ricciardi Department of Engineering, University of Messina, 98166 Messina, Italy

A R T I C L E I N F O

A B S T R A C T

Keywords: Response-spectrum-method Base-isolated buildings Substructure approach Spectrum-compatible artificial earthquakes Modal superposition Nonclassically damped structures

Unsatisfactory numerical predictions may result from applying the classical modal analysis in conjunction with the response-spectrum-method (RSM) to nonclassically-damped systems such as base-isolated buildings. This inaccuracy is highlighted by comparing the conventional RSM outcomes with results from nonlinear time-history analyses consistent with that given spectrum. Indeed, some underlying assumptions of the conventional RSM are not really appropriate for base-isolated buildings, thus only approximate results are obtained, whereas either a complex-value modal analysis or the direct integration of the equations of motion should be undertaken to follow an exact approach to this problem. In an attempt to overcome the limitations of the conventional RSM as well as the mathematical difficulties and computational cost of the exact approach, in this paper an improved response-spectrum analysis procedure for base-isolated buildings is elaborated. Based upon the substructure approach, this procedure makes use of novel response spectra that quantify the effects of the base-isolationsystem (BIS) to the superstructure while accounting for the dynamic interaction between BIS and superstructure. The developed procedure improves the conventional RSM in two aspects: (1) the seismic response of the baseisolated building is computed by applying the modal analysis to the superstructure only, which is typically considered as a classically damped system, rather than to the overall structure having nonclassical damping; (2) the BIS can potentially be modeled as a nonlinear subsystem with its actual hysteretic characteristics. The effectiveness of the proposed procedure and the improvements over the conventional RSM are scrutinized against time-history analyses with Monte Carlo simulated spectrum-compatible accelerograms.

1. Introduction The strategy of seismic isolation has been increasingly adopted in earthquake-prone regions to mitigate or reduce damage potential due to the shaking ground [1,2]. Basically, some types of supports (typically, laminated rubber bearings or sliding elements) having low lateral stiffness and equipped with some inherent (viscous, hysteretic or frictional) damping mechanism are interposed between the superstructure and the foundation so as to decouple the building structure from the ground motion. The lengthening of the first-mode period combined with the damping features of the BIS, which provides additional energy dissipation, considerably reduce the earthquake-induced forces in the superstructure to such level that practically no damage will occur [3–5]. Consequently, the building can be designed to remain in the elastic range. While in conventional structural analysis an equivalent viscous damping ratio ζs = 5% is usually assumed, thus implying that some degree of structural and nonstructural damage will occur during a

⁎

strong ground motion, the aforementioned reduction of the expected damage in the superstructure justifies the adoption of a lower value of the damping ratio, say ζs = 2%. On the other hand, the BIS generally possesses an equivalent viscous damping ratio of about ζb ≈ 10–40% . When the difference in damping between the two substructures attains such very high values, the equations of motion in the modal subspace are far from being uncoupled, which is due to the (non-negligible) offdiagonal terms of the damping matrix. Strictly speaking, the BI building is a nonclassically damped system [6], its natural modes of vibration are not real-valued, consequently a complex modal analysis should be undertaken to correctly assess the structural response (unless direct integration of the equations of motion in the nodal coordinates is performed). Nevertheless, in the framework of the response-spectrum analysis, the base-isolated structure is often treated as a classically damped system, i.e., with modal equations decoupled and the off-diagonal terms of the modal damping matrix ignored, at least to obtain some preliminary estimates of the structural response. In this paper we will refer to this approach as “the conventional response-spectrum-method (RSM)”.

Corresponding author. E-mail address: [email protected] (D. De Domenico).

https://doi.org/10.1016/j.engstruct.2018.02.037 Received 18 April 2017; Received in revised form 8 February 2018; Accepted 12 February 2018 0141-0296/ © 2018 Elsevier Ltd. All rights reserved.

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superstructure as a classically damped system. Therefore, conventional combination rules for classically damped systems may be adopted, see e.g. [24], thus avoiding the need of introducing ad hoc cross-correlation coefficients specifically developed for nonclassically damped systems, e.g. [19,25–27]. The novelty and main advantage of this procedure over both the conventional RSM and the complex-mode-superposition approach is that the nonclassical damping nature of the system is taken into account without the evaluation of complex eigenmodes, which is possible by resorting to a substructure approach. Two main differences as compared to the conventional RSM are recognized: (1) the substructure approach makes it possible to apply the classical modal analysis to the superstructure only, which is typically considered as a classically damped system, rather than to the overall structure having nonclassical damping; (2) while the superstructure can reasonably be modeled as a linear subsystem (owing to the reduction of the expected structural damage induced by the base isolation), linearizing the BIS behavior may or may not be acceptable in practice. We propose a formulation in which incorporating a nonlinear model of the BIS does not imply any extra computational cost as compared to the linearized behavior. Potentialities and limitations of the proposed SB-RSM procedure are scrutinized by a few simple numerical applications. Comparison with (averaged) results from nonlinear RHA with Monte Carlo simulated spectrum-compatible accelerograms has shown remarkable improvements of the proposed formulation over the conventional RSM, especially with increasing (nonclassical) coupling of the modal equations.

In the simplified, conventional RSM, neglecting the off-diagonal terms in the modal damping matrix may lead to wrong structural response evaluations, especially when the non-classical damping nature of the system is pronounced [7]. The modal equations may still be uncoupled, but in a complex (not real) subspace. Therefore, a more appropriate response-spectrum-analysis (RSA) procedure would require the solution of a complex eigenvalue problem that incorporates the damping matrix, thus implying a complex mode superposition approach, see the pioneering works [8,9] and the subsequent applications and enhanced variants proposed by researchers in the last few decades [10–16]. Of particular relevance to the present paper, this complexmode procedure was used for the analysis of base-isolated structures [17,18]. It is worth noting that two response spectra are necessary to deal with nonclassically damped systems, as the seismic response of these systems depends not only on modal displacements (which requires the pseudo-acceleration response spectrum like in classically damped systems), but also on modal velocities (which needs an additional, relative velocity response spectrum [19,20], or the cosine spectrum [21], or again the more widely adopted pseudo-velocity response spectrum [22,23]), which further complicates the analysis. Additionally, besides being computationally more expensive, the complex modal analysis would make the engineer lose the physical insight and the intuitive grasp of the system’s behavior. This is why in engineering practice a real-value RSA procedure for base-isolated buildings that is as simple as the conventional RSM but more accurate than this procedure would be highly desirable, at least to capture, in a simplified manner and under some reasonable assumptions, preliminary estimates of the system response, to be checked subsequently via more sophisticated nonlinear response history analysis (RHA). In an attempt to overcome the limitations of the conventional RSM while at the same time preserving the simplicity and the attractive features of a real-value response-spectrum approach, the aim of this paper is to present an improved, real-value RSA procedure for baseisolated buildings. The key idea of the proposed procedure stems from the substructure approach wherein the primary system is the (linear) superstructure and the secondary system is the (potentially nonlinear) BIS. The two subsystems interact with each other due to the inertia forces of the BIS to the superstructure as well as the feedback of the superstructure to the BIS. Strictly speaking, such feedback would depend upon all the natural vibration modes of the superstructure, which characterize the actual nonclassical damping nature of the system. However, in most practical cases little error is made by truncating to the first mode, which is the main assumption made in the proposed procedure. In particular, from the overall set of m coupled modal equations (with m being a reasonable number of modes of vibration, up to a given cut-off frequency, that contribute to the structural response in a significant manner), a group of just two equations are extracted to characterize the first-mode response while accounting for the coupling terms between first mode and BIS but neglecting the higher-order-mode feedback contributions. In a similar fashion, a group of just three equations are extracted to evaluate the response of a generic higherorder mode of vibration, retaining the first-mode equation and BIS equation in addition to the jth modal equation of the analyzed modal oscillator. Substructuring the equations of motion suggests to denominate this procedure a “substructure-based response-spectrum-method (SBRSM)”. For the practical application of the proposed SB-RSM, a family of novel response spectra are constructed by averaging the peak values of the response obtained via direct integration of the equations of motion, for given dynamic properties of BIS and superstructure. Once such novel response spectra are constructed, the remainder of the proposed RSA procedure turns out to be nothing but a conventional RSM applied to the fixed-base structure only. Indeed, the presence of the isolation system and the coupling effects due to the nonclassical damping nature of the system are a priori incorporated in the definition of the above response spectra and one can straightforwardly deal with the

2. Conventional RSM for base-isolated buildings Before proceeding with the formulation of the proposed RSA procedure, it is worth summarizing the basic steps of the conventional implementation of the RSM for base-isolated buildings. To this aim, first the equations of motion are briefly summarized and expressed in a format that will be useful for the next derivations, then the steps of the conventional RSM are carefully analyzed, underlining assumptions and limitations of this technique when applied to BI structures. 2.1. Equations of motion of a base-isolated building Let us consider a planar n-story building as sketched in Fig. 1, which is base-isolated and subject to the horizontal ground acceleration u¨ g (t ) . Following a typical approach to this problem, a preliminary static condensation method has already been applied to the structure in order to eliminate the (zero-mass) rotational degrees of freedom (DOFs). Consequently, with axial deformations in structural elements neglected, the mass of the n-story frame depicted in Fig. 1 is lumped at the floor level, with mj denoting the mass at the jth floor associated with the jth translational DOF, while cj and kj denote the viscous damping coefficient and the condensed (lateral) stiffness term at the jth floor. As a result, the superstructure has n dynamic DOFs, represented by the displacements of the n stories relative to the BIS (cf. again Fig. 1) which are collected in the array uTs (t ) = [u1 (t ),u2 (t ),…,un (t )]T . Note that the above assumptions do not necessarily imply that a one-bay shear frame is being analyzed since the rotational DOFs may be calculated once the translational uj DOFs are determined. On the other hand, the BIS is represented by an additional single degree of freedom (SDOF) system (i.e., the displacement ub (t ) relative to the ground) having mass mb and interconnected to the superstructure. For the sake of generality and as better clarified in the following Section 3, two schematic models of the hysteretic BIS behavior are postulated in Fig. 1(a): (1) an equivalent linearized behavior, featured by a combination of a spring and a dashpot element whose stiffness and viscous damping coefficients are denoted as k eff and ceff , respectively (adopting a common terminology of most building codes [28,29], the subscript eff stands for an effective variable that is the linearized counterpart of an intrinsically nonlinear variable); (2) a more realistic rigid-plastic behavior, characterized by a 199

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superstructure and the isolation floor masses. Since we have written the equations in relative displacements, the resulting Eqs. (1) are decoupled in terms of elastic and viscous forces but coupled in terms of inertia forces due to the off-diagonal terms of the M matrix in (2). The latter terms characterize the dynamic interaction of the two subsystems. Note also that the Eqs. (1) refer to the more general nonlinear behavior of the BIS as sketched in Fig. 1(a); the linearized behavior may be retrieved as a special case for the following choice of the BIS parameters: c b = ceff ,kb = k eff and μb = 0 . The conventional RSM assumes the latter linearized behavior of the BIS. As will be emphasized in Section 3, in so doing one may overestimate or, even worse, underestimate the response of the BIS especially in the range of typical periods and damping ratios adopted in base-isolated buildings.

2.2. Classical modal analysis approximation It is easy to observe that while the superstructure can realistically be considered a classically damped system (the Caughey and O’Kelly condition [30] Cs M−s 1K s = K sM−s 1Cs being generally satisfied), this is not the case for the coupled BI building whose mass, damping and stiffness matrices are reported in (2). In fact, due to the high contrast in damping between the superstructure and the BIS, the base-isolated building is a nonclassically damped system [6]. Unlike the classically damped systems, natural frequencies and modes of vibration of such systems are different in the undamped and damped cases; in other words, one should solve a complex-value eigenproblem (including the damping matrix) to decouple the equations of motion in the modal subspace. This significantly complicates the mathematical aspects making the derivation of the eigenvalues and eigenvectors computationally more demanding. The conventional RSM neglects this complication and indeed deals with the BI building as a classically damped structure, thus solving the following real-value eigenproblem associated with the stiffness and mass matrices reported in (2)

Fig. 1. Sketch of a base-isolated n-story building: (a) linearized and nonlinear alternative behaviors of the BIS; (b) idealization of the BIS hysteretic loop with interpretation of the coefficients.

stiffness term kb along with viscous and (Coulomb) frictional damping coefficients c b and μb , respectively. Interpretation of the coefficients of such two idealizations is given in Fig. 1(b) with reference to a schematic hysteretic behavior of the BIS. Further clarifications will be given in Section 3, we here limit ourselves to state that the proposed BIS hysteretic behavior may be representative of either friction pendulum devices, or (idealized) lead-rubber-bearing (LRB) isolators. In the spirit of the substructure approach, two subsystems can be identified from Fig. 1: the primary system is the (linear) superstructure, which is subject to the ground motion along with the inertia forces transferred by the secondary system, the (potentially nonlinear) BIS, which is in turn excited by a feedback of the primary system. Due to the above rationale, a dynamic interaction between the two subsystems arises, which is reflected by the following equations of motion of the BIS-superstructure combined (n + 1) -DOF system:

Mu ¨ (t ) + Cu̇ (t ) + Ku (t ) + va μb g sign(uḃ (t )) = −vu u¨ g (t )

(3)

KΦ = MΦΩ2

where Φ and Ω2 are the modal matrix and spectral matrix, respectively, collecting the first m (with m ⩽ n + 1) eigenvectors ϕi and eigenvalues ωi2 , i.e., Φ = [ϕ1…ϕm] and Ω2 = diag{ω12,…,ωm2 } , with diag{(·)} being the square diagonal matrix of elements (·) and ω1 ⩽ ω2 ⩽ …⩽ωm . Generally, the arbitrary multiplicative factor of the natural modes ϕi is chosen such that the modal matrix be orthonormal to the mass matrix, that is, ΦT MΦ = Im , with Im denoting the identity matrix of order m. 2.3. Mode superposition method and peak structural response via the RSM

(1)

Once the eigenproblem (3) is solved, the nodal displacements of the BI system are expressed as a linear combination of the first m modal coordinates as follows

in which a super-imposed dot denotes the derivative with respect to time, the sign function of the BIS velocity, associated with Coulomb friction, has been introduced, while g is the acceleration of gravity. The matrices entering Eq. (1) are defined as follows

m

u (t ) =

∑

ϕi γi di (t ) = ΦΓd (t )

(4)

i=1

Ks 0 ⎤ Ms Ms τ ⎤ Cs 0 ⎤ u s (t ) ⎤ ; C=⎡ T ; K=⎡ T ; u (t ) = ⎡ ; M=⎡ T ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 M τ m k u 0 c b s tot b ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎦ ⎣ b (t ) ⎥ n Mτ 0 va = ⎡ m ⎤; vu = ⎡ s ⎤; mtot = mb + τ T Ms τ = mb + ∑ mi ⎣ tot ⎦ ⎣ mtot ⎦ i=1

ϕiT vu

is the ith modal participation where Γ = diag{γi} , in which γi = factor corresponding to the ith mode SDOF system whose equation of motion is

d¨i (t ) + Ξi,j dj̇ + ωi2 di (t ) = −u¨ g (t ) (with i = 1,2,…,m),

= mb + mstot (2)

(5)

where Ξi,j is the ith row jth column element of the damping matrix Ξ = ΦT CΦ. From Eq. (5) we note that, unlike classically damped structures, the m equations of motion in the modal subspace are not uncoupled due to the (non-zero) off-diagonal terms of the modal damping matrix Ξ. To assess to what extent the BI building is nonclassically damped the following definition of the coupling index can be resorted to [31]

with Ms,Cs,K s being the n-dimensional matrices of mass, damping and stiffness of the superstructure as if it were on a fixed base, τ is the n × 1 influence vector of the substructure associated with the ground motion u¨ g (t ) and 0 is a n × 1 vector of zero terms. In the definitions (2) mstot represents the total mass of the superstructure, while mtot denotes the total mass of the base-isolated structure, including both the 200

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Inc = max i≠j

2 ⎧ Ξi,j ⎫ . ⎨ Ξi,i Ξj,j ⎬ ⎩ ⎭

(1) the procedure implicitly neglects the coupling of modal equations due to nonclassical damping, and this may be inaccurate because of the high difference in damping between the two subsystems, namely the (linear) superstructure and the (potentially nonlinear) BIS; (2) modal superposition is carried out by adopting combination rules that have been conceived for classically-damped systems [24], whereas cross-correlation coefficients specifically developed for nonclassically-damped systems should be adopted (see e.g. [19,25–27]); (3) most seismic codes provide the response spectrum just for a single, reference value of the viscous damping ratio (typically ζ 0 = 5%), which is quite different from the usual values assumed for the BIS and the superstructure, ζb and ζs respectively, and only approximated relationships are given in order to extrapolate the spectra for different ζ values (via the so-called damping correction factor (DCF)); (4) the actual nonlinear BIS hysteretic behavior is converted into an equivalent linear idealization, e.g. an equivalent (effective) stiffness and viscous damping ratio. As a matter of fact, there is no possibility of considering the actual hysteretic characteristics of the BIS.

(6)

Assuming that the modal damping matrix be diagonal may lead to rough results especially with increasing values of the above coupling index (Inc = 0 for classically damped systems, Inc → 1 for fully nonclassical damping). The conventional RSM does make such assumption and, in fact, considers the Eqs. (5) of the m modal SDOF systems decoupled, i.e., Ξi,j = 0 for i ≠ j and Ξi,i = 2 ζi ωi . Which equivalent viscous damping ratio to assume for each mode is another questionable aspect of the procedure. Normally, ζ1 = ζb is assumed for the first mode (the so-called isolation mode as the BIS undergoes deformation but the superstructure behaves as essentially rigid), and ζi = ζs is set for the remaining m−1 higher modes (i = 2,3,…,m ) which involve deformations in the superstructure thus dissipating less energy. Any response quantity r (t ) of interest (be it a node displacement, the base shear, an interstory drift, and so forth) can be expressed through a generalization of the superposition rule in Eq. (4) m

r (t ) =

m

∑

ri (t ) =

i=1

∑

ψi di (t )

(7)

i=1

Some mathematical aspects are outlined in Section 3 to more consistently modeling the BIS behavior based on the actual (experimentally recorded) hysteresis curves of the individual isolators.

ψi representing an influence coefficient quantifying the ith-mode contribution to the response quantity r (t ) (obviously, the shape of ψi depends on the sough response quantity: for instance, for the jth story displacement ψi = γi ϕji (cf. Eq. (4)), for the jth interstory drift ψi = γi (ϕji−ϕj − 1,i ) , for the base shear ψi = ωi2 γi2 .). The following step is to determine the peak value of the generic structural response of interest r ̂ = max t |r (t )| by combining the corresponding peak modal responses ri ̂ (i = 1,2,…,m ) given directly from the earthquake response-spectrum. For instance, adopting the complete quadratic combination (CQC) rule [24] yields

r ̂=

m

m

m

m

∑i =1 ∑ j=1 ρi,j ri ̂rj ̂ = ∑i =1 ∑ j=1 ρi,j ψi ψj

3. Experimentally consistent modeling of the isolation system As already sketched in Fig. 1(a), we can assume that the set of individual isolators located at the column bases are converted into one equivalent SDOF representing the BIS as a whole and whose dynamic properties depend on the isolator types and distribution across the structure. In this section, a few considerations are outlined regarding how to consistently modeling the BIS behavior based on the actual (experimentally recorded) hysteresis curves of the individual isolators.

(j ) (i) (ωj,ζ j ) Spa (ωi,ζi ) Spa

ωi2

ωj2 (8)

3.1. Analysis of the hysteretic behavior of a single isolator

where ρi,j is the correlation coefficient between the ith and jth mode (i) [24], while Spa (ωi,ζi ) = ωi2 Sd(i) (ωi,ζi ) ≡ ωi2 di ̂ (ωi,ζi ) is the ith ordinate of the earthquake pseudo-acceleration response spectrum, which depends on the natural frequency ωi and the assigned viscous damping ratio ζi . Two further sources of inaccuracy may be recognized: firstly, in the literature it has been recognized that the conventional CQC rule is not really suitable for nonclassically damped systems such as BI buildings, therefore alternative techniques have been proposed [19,25–27] that either make use of new expressions of the correlation coefficients based on the complex eigenproblem, or introduce some corrective terms to improve the accuracy of the results; secondly, in most building codes [28,29] the earthquake response spectrum is defined for just a single reference value of the viscous damping ratio (namely ζ 0 = 0.05, which is quite different from the ζb and ζs values relevant for BI buildings) and only approximated relationships are given in order to extrapolate the spectra for different ζ values via the DCF. As shown in the next section, the accuracy of the conventional RSM decreases as the viscous damping ratio differs markedly from the reference one, with discrepancies in the response prediction up to 35% (values obtained for periods and damping ratios commonly used for the BIS).

To clarify and interpret the hysteretic behavior shown in Fig. 1(b), first the analysis of a single isolator is discussed, then we present some relationships to determine the dynamic coefficients of the BIS based on the ones of the individual isolators. In Fig. 2 we report the experimental findings of dynamic tests on a real LRB isolator conducted at the Eurolab laboratory of the CERISI [32] in Messina, Italy. With the symbols defined in the foregoing Fig. 1, from a rudimental observation of the experimental curves one may easily identify the dynamic parameters, reported in Table 1, of both the linearized and the nonlinear SDOF model. The experimental curves, depicted in Fig. 2(b) along with the numerical fitting from the nonlinear model, refer to two test frequencies, namely 0.01 Hz and 0.33 Hz. The nonlinear μb and kb parameters are kept identical for the two test frequencies, the difference between the two hysteresis curves being mainly ascribed to the viscous damping contribution c b . Indeed, for high-velocity tests the c b u̇b term is not negligible as in the low-frequency limit. This is not the case for the linear SDOF, as emphasized in Table 1 where different (ζ eff ,k eff ) couples are found for the two test frequencies. Some interesting experimental features of this kind of isolators are discussed more in-depth in [33]. Nevertheless, the dynamic characteristics of the isolator are usually identified based upon a single, reference test frequency (e.g. the third cycle for a 0.5 Hz test as specified by the EN15129 [34]), consequently the resulting ζ eff and k eff values may not correctly reproduce the actual behavior of the isolator for lower frequencies (in which, as a matter of fact, the damping could be overestimated). Apart from the above remarks, one could accept a simplification of the nonlinear SDOF model proposed in Fig. 1(a) by slightly modifying

2.4. Main sources of inaccuracy of the conventional RMS for base-isolated buildings In all the above line of reasoning some misleading and contradictory aspects of the conventional RSM when applied to base-isolated buildings may easily be recognized. The following four sources of inaccuracy are identified: 201

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Fig. 2. Experimental investigation on a real LRB isolator: (a) photograph of the Eurolab laboratory where the test was conducted; (b) force-displacement curve recorded in the test.

corresponding circular frequencies of the linearized and nonlinear model of the BIS are ωeff = k eff / mtot and ω b = kb/ mtot , respectively, and the associated natural periods of vibration are Teff = 2π / ωeff and Tb = 2π / ω b (although Tb is just a mathematical, meaningless fictitious period). By inspection of Table 1 we note that ω b < ωeff or, equivalently, Tb > Teff ; for instance, for the above LRB isolator Teff ≃ 3.81s and Tb ≃ 4.68 s. From Fig. 1(a) we observe that when analyzing the behavior of the BIS only (without superstructure) mtot ≡ mb , however we keep this definition since it will be useful for the next derivations for the assembled base-isolated building.

Table 1 Identification of the coefficients of the LRB isolator whose hysteretic behavior is shown in Fig. 2(b). Linear SDOF (0.01 Hz)

Linear SDOF (0.33 Hz)

Nonlinear SDOF (both the frequencies)

k eff [kN/ mm]

ζ eff [–]

k eff [kN/ mm]

ζ eff [–]

kb [kN/ mm]

μb [–]

c b [kN s/ mm]

2.01

0.26

2.12

0.35

1.33

0.018

0.19

the μb and kb values so that a simpler rigid-plastic idealization (not including the c b term) accurately fits the experimental hysteresis curve at the reference frequency. In order to reduce the number of coefficients to two for both the linear and nonlinear SDOF, this assumption has been made throughout the paper. In other words, the nonlinear model will be defined by a rigid-plastic behavior through a spring stiffness kb and a Coulomb frictional element μb , while the linearized SDOF is characterized by the (k eff ,ζ eff ) couple. Although it is well known that the friction coefficient is a complicated function of bearing pressure, sliding velocity and temperature rise induced by friction heating [35–37], in the sequel of the paper the coefficient μb is assumed constant for simplicity.

3.3. Linear vs nonlinear modeling In the previous sections we have presented two idealizations of the BIS behavior, namely the linearized SDOF defined by the couple (k eff ,ζ eff ) and the nonlinear model characterized by the couple (kb,μb ) . The governing equations of motion of the two alternatives are reported below

Based on the simple identification of the dynamic parameters of the single isolator discussed before, we need to extend the above line of reasoning to the ensemble of isolators, that is, to the equivalent SDOF representing the BIS as a whole. The goal is to express the coefficients of such equivalent SDOF in terms of the dynamic characteristics of the individual isolators. For the sake of generality, we consider a set of M distinct isolators acting in parallel so that they undergo an equal lateral displacement. Following the previous section, the individual ith isolator, subject to (i) (i) ,ζeff ) or nonlinear normal forces Ni , is described by a couple of linear (keff (i) (i) coefficients (kb ,μ b ) . Through simple algebra, it can be demonstrated that the corresponding coefficients of the BIS are expressed as M

∑

M

kb(i); μb =

i=1

where αki =

∑

M

μ b(i) αNi; k eff =

i=1 (i) k eff

k eff

∑

(10a)

u¨b (t ) + ω b2 ub (t ) + μb g sign(uḃ (t )) = −u¨ g (t ).

(10b)

The big arising question is: how close are the results obtained by such two BIS models and to what extent the linearized SDOF may be a reasonable approximation? To give an appropriate answer to this question, first we develop some relationships that correlate the two couples of parameters univocally so that it is possible to switch between the linear and nonlinear models in a consistent fashion. With the aid of Fig. 3, indicating with ED the area enclosed by the nonlinear hysteresis loop and u 0 the related maximum displacement attained by the BIS, one can write

3.2. From the single isolator to the base-isolation system

kb =

2 u¨b (t ) + 2 ζ eff ωeff uḃ (t ) + ωeff ub (t ) = −u¨ g (t )

k eff = kb +

μb mtot g 2 μb mtot g 2 μb mtot g ED ; ζ eff = = = u0 π k eff u 0 π (kb u 0 + μb mtot g) 4πES 0 (11a)

π k eff ζ eff u 0 π kb = k eff ⎛1− ζ eff ⎞; μb = 2 mtot g ⎝ 2 ⎠

M (i) keff ; ζ eff =

i=1

∑

(i) ζeff αki

i=1 M

and αNi =

Ni , N

with N ≡

∑ i=1

Ni. (9)

From relationships (9) it emerges that the BIS can be modeled through an equivalent SDOF characterized by stiffness equal to the sum of the stiffness of the individual isolators, and having an equivalent viscous damping ratio and frictional coefficient that are the weighted mean of the ones of the individual isolators (with weights the stiffness coeffi(i) cients keff and the normal forces Ni of the isolators, respectively). The

Fig. 3. Relationship between the linear and nonlinear coefficients.

202

(11b)

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We observe that the nonlinear spectrum is lower than linear one, however this result pertains to ζ eff = 0.05, which is of little interest considering the BIS damping typically ranging from 10% to 30%. By inspection of Fig. 5 (referring to an arbitrary accelerogram and to the SDOF having Teff = 2.5 s) we realize that about 4 iterations are sufficient to achieve convergence in the nonlinear model (with tol = 10−3 ); the obtained time-histories of the displacement response and the corresponding force-displacement loops are depicted in Fig. 5(b) and (c), respectively. More realistic damping ratios of the BIS, namely ζ eff = 0.2 and 0.3, are then investigated, whose results are reported in Fig. 6. As already documented in the relevant literature (see e.g. [40]), the EC8 DCF [29] (the so-called η-factor) turns out to be inaccurate for high damping ratios. For instance, for a period T = 2 s and ζ eff = 0.3, the EC8 spectral ordinate is about 35% higher than the corresponding time-history response. Opposite trend is observed for the latter ζ eff value in the longperiod range, where the EC8 spectrum leads to underestimation of the displacement response up to 27% as compared to the nonlinear SDOF. The same issue arises for other code-based DCFs [41], e.g. the B-factor of the american building code [28]. Consequently, a wealth of research has been aimed at developing improvements to the existing approach, see e.g. [42] for an overview. Unfortunately, at present these improvements are not yet fully incorporated in the building codes, thus the conventional RSM applied to BI building is, as a matter of fact, further affected by the above simplifications due to the DCF.

that are the sought bijective functions expressing the linear (nonlinear) coefficients in terms of the nonlinear (linear) ones. It is worth noting that the above correspondence relationships involve u 0 , which is the unknown maximum displacement of the nonlinear model resulting from Eq. (10b), i.e., u 0 ≡ ub̂ = max t |ub (t )|. Among all the possible hysteresis loops characterized by (k eff ,ζ eff ), two possible examples of which are the loops depicted in red dash-dotted line in Fig. 3, only one, namely that displayed in blue continuous line, correctly reproduces the nonlinear behavior of Eq. (10b). Therefore, an iterative procedure is needed to detect the actual nonlinear couple (kb,μb ) associated with the linear one (k eff ,ζ eff ) : for a fixed (k eff ,ζ eff ) pair, a first-attempt value of u 0 is assumed, a reasonable choice could be taking the Eurocode 8 (EC8) [29] displacement response-spectrum u0(1) = Sd (ωeff ,ζ eff ) ; from the set (k eff ,ζ eff ,u0(1) ) the couple (kb,μ b(1) ) is computed via Eq. (11b); with the latter nonlinear parameters we solve the equations of motion (10b) and find a new maximum displacement u0(2) ≠ u0(1) ; then the procedure is carried on iteratively by adjusting μb according to Eq. (11b), in particular μ b(i + 1) = μ b(i) u0(i + 1) / u0(i) (kb being not affected by u 0 , cf. also the sketch of Fig. 3) until convergence is reached (i.e., |u0(i + 1)−u0(i) |
4. Improved response-spectrum analysis via the SB-RSM 1. according to the Monte Carlo method (MCM) we generate an ensemble of 400 artificial ground motion accelerograms consistent with the EC8 reference response spectrum [29] (ground type A) for peak ground acceleration (PGA) ag = 0.3 g . Such accelerograms, whose stationary duration is Ts = 20 s , are originated by the superposition of harmonic waves with random phase [38] from a parent spectrum-compatible power spectral density function [39]; 2. for a SDOF system with given dynamic properties (Teff and ζ eff for the linear system, the equivalent Tb and μb for the nonlinear counterpart) 400 complete time-histories of response (one for each accelerogram) are analyzed in order to pinpoint 400 peak values of the displacement, which are then averaged to obtain the mean extreme values of the displacement response Sd (T ,ζ eff ) (mean pseudo-accel2 ); erations Spa are computed by multiplying Sd by ωeff 3. the complete response spectrum is then constructed by varying the dynamic characteristics of the SDOF (more specifically, we vary T = Teff for fixed ζ eff ) and repeating step 2. Note that all the response spectra of the BIS are plotted in terms of the effective period, i.e., with T = Teff in abscissa.

Although straightforward and simple, the conventional RSM applied to BI buildings is affected by the aforementioned sources of inaccuracy. A novel approximated RSA procedure, based on the substructure approach and therefore named substructure-based response-spectrum method (SB-RSM), is here presented to improve the conventional RSM by circumventing some of these underlying misleading assumptions. Starting from the equations of motion of the coupled BIS-supestructure (n + 1) -DOF system, Eq. (1), we perform the classical modal analysis of the supestructure only, i.e., associated with the n-dimensional K s and Ms submatrices reported in (2) [43,44]:

K sΦs = Ms Φs Ω2s with ΦTs Ms Φs = Im

(12)

where Φs = [ϕs1…ϕsm] and Ω2s = diag{ωs12,…,ωs2m} , the subscript s emphasizing that the first considered m eigenvectors and eigenvalues refer to the supestructure as if it were on a fixed-base. It is worth noting that the use of the fixed-base eigenmodes shows similarities to other fields of the seismic response evaluation, for instance they are invoked to evaluate the floor response spectrum [45,46], to perform response spectrum analysis of light secondary substructures and subsystems [47,48], to study the primary-secondary structure interaction [49–51], and to perform modal analysis considering soil-structure interaction [52–54]. Here, in the spirit of the substructure approach, the fixed-base

In Fig. 4 the response spectrum for the reference viscous damping ratio ζ eff = 0.05 is illustrated. The spectrum-compatibility conditions specified in the EC8 [29] are met throughout the periods investigated.

Fig. 4. Response spectrum for ζ eff = 0.05 of the linear SDOF, Eq. (10a), and the nonlinear SDOF, Eq. (10b), in terms of: (a) pseudoacceleration; (b) displacement.

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Fig. 5. Results for an arbitrary accelerogram and for Teff = 2.5 s: (a) convergence process to identify the nonlinear parameters; (b) time-histories of the displacement response; (c) forcedisplacement loops.

Fig. 6. Displacement response spectrum of the linear and nonlinear SDOF: (a) ζ eff = 0.2 ; (b) ζ eff = 0.3 .

u¨b (t ) + 2 ζb ω b uḃ (t ) + ω b2 ub (t ) + μb g sign(uḃ (t ))

eigenproperties are useful to quantify the feedback term of the supestructure to the BIS due to the coupling of the equations of motion. Following Eq. (4), the nodal superstructure displacements are then expressed through the mode superposition method that is given by

m

= −u¨ g (t )−λ ∑ ∊i d¨si (t ). i=1

The m + 1 general Eqs. (15) can be specified both for the nonlinear BIS model (ζb = 0 ) and for the linear idealization (ζb = ζ eff ,ω b = ωeff and μb = 0 ). Apart from the truncation of higher-modes (if m < n ), the set of modal Eqs. (15) are perfectly equivalent to the original nodal Eqs. (1). It is worth noting that so far the combined BIS-superstructure system is being treated as nonclassically damped since no assumption is made to decouple the equations of motion of the assembled structure.

m

u s (t ) =

∑

ϕsi γsi d si (t ) = Φs Γs d s (t )

i=1

(13)

where Γs = diag{γsi} , in which γsi = ϕsTi Ms τ is the ith modal participation factor corresponding to the ith superstructure mode. Therefore, the first n nodal equations of Eq. (1) are rewritten into a set of m modal equations in the modal subspace that are decoupled since the superstructure can realistically be considered as a classically damped system, i.e., Ξs = ΦTs Cs Φs = diag{2 ζsi ωsi} . Combining with the BIS equation, the complete set of m + 1 equations of motion read

4.1. First-mode and BIS response spectrum

d¨si (t ) + 2 ζsi ωsi d ṡ i + ωs2i d si (t ) = −u¨ g (t )−u¨b (t ) (with i = 1,2,…,m)

The m + 1 Eqs. (15) are the starting point for the construction of the novel response spectra on which the proposed RSA procedure is based. We observe that if the feedback terms on the rhs of Eq. (15b) were neglected, a very simple cascade approach could be resorted to: first the BIS response is evaluated from the Eq. (15b), and in a second step the BIS response in terms of acceleration is applied to the superstructure, Eqs. (15a). The dynamic interaction between the two subsystems would be only partly accounted for as the BIS response might be understimated or overestimated. In reality, the superstructure does exert a feedback to the BIS, consequently it does influence the BIS response to some extent, thus the equations of motions should be solved simultaneously. An approximated solution of the Eqs. (15) may be obtained by truncating the feedback terms on the rhs of Eq. (15b) to the first-mode of the superstructure, which normally contributes more in the summation due to the higher value of the participating mass ratio ∊1. With this assumption, the first out of the overall m equations of the superstructure is extracted from Eq. (15a) along with the Eq. (15b) ruling the dynamics of the BIS, which yields

(14a)

mtot u¨b (t ) + c b uḃ (t ) + kb ub (t ) + mtot μb g sign(uḃ (t )) ¨ s (t ). = −mtot u¨ g (t )−τ T Ms u

(15b)

(14b)

The last term on the right-hand-side (rhs) of Eq. (14b) represents the socalled feedback effect of supestructure to the BIS. This term can be manipulated by exploiting Eq. (13), which yields τ T Ms u ¨ s (t ) = γsT Γs d¨ s (t ) = γs2 T d¨ s (t ) , with γs2 T = [γs12,…,γs2m] being the mdimensional row-vector collecting the squared modal participation factors, or equivalently, the “effective modal masses”. We remind the definition of the ith participating mass ratio ∊i = γs2i / mstot , where mstot is the total mass of the superstructure (cf. Eq. (2)). Therefore, introducing λ = mstot / mtot , the ratio of the mass of the superstructure and the total mass, and dividing Eq. (14b) by mtot leads to

d¨si (t ) + 2 ζsi ωsi d ṡ i + ωs2i d si (t ) = −u¨ g (t )−u¨b (t ) (with i = 1,2,…,m) (15a) 204

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Fig. 7. First-mode and BIS response spectrum for Teff = 3 s, ζ eff = 0.2 : (a) and (c) linear; (b) and (d) nonlinear.

̇ + ωs12 d s1 (t ) = −u¨ g (t )−u¨b (t ) d¨s1 (t ) + 2 ζs1 ωs1 d s1

BIS leads to an underestimation of the first-mode response in the lowperiod range, and an overestimation for high-periods as compared to the nonlinear counterpart. Although from Fig. 7(c) and (d) the participation coefficient λ ∊1 seems to have little influence in the BIS displacement response spectrum Sub , it does affect the BIS acceleration u¨b (1) as demonstrated by the variability of Spa with λ ∊1, cf. Fig. 7(a) and (b). It is worth making some remarks and discussing a few interesting limit cases: (i) the case λ ∊1 = 0 would lead to a cascade approach as the superstructure would have no effect on the dynamics of the isolation system and the two equations could be solved sequentially. This case is not reported in the plots as it would disproportionately overestimate the first-mode response, thus leading to completely wrong outcomes; (ii) for T → 0 , i.e., for an extremely stiff superstructure, d s1 → 0 , d¨s1 → 0 in Eq. (16b) and the resulting BIS displacement response spectrum Sub |T = 0 would be that of the SDOF analyzed in Section 3.3, furthermore (1) 2 Spa |T = 0 = ag + ωeff Sub |T = 0 regardless of the λ ∊1 value (cf. Figs. 7 and 8); (iii) for Teff → ∞, i.e., for an extremely flexible BIS, one would expect ub (t ) → −ug (as the total BIS displacement ubt = ub + ug would tend to zero), the two terms on the rhs of Eq. (16a) would accordingly vanish and d s1 → 0 , i.e., the higher displacements of the BIS are the price to pay for reducing the superstructure response, cf. Fig. 9(a) and (b); (iv) in order to reduce the BIS displacements to some acceptable values a

(16a)

u¨b (t ) + 2 ζb ω b uḃ (t ) + ω b2 ub (t ) + μb g sign(uḃ (t )) ≈ −u¨ g (t )−λ ∊1 d¨s1 (t ). (16b) In other words, only the first-mode contribution to the feedback effect is actually accounted for. This assumption is reasonably justified in most buildings, where values of ∊1 ≃ 80–90% and ∊j ≤ 5–10% ( j = 2,…,m ) are generally observed. The coupled Eqs. (15) can be solved for given dynamic (linear or nonlinear) properties of the BIS and for given firstmode SDOF parameters (ζs1,ωs1,λ ∊1). With ζs1 = 0.02 being assumed throughout, the mean extreme value of the first-mode SDOF response is and computed for variable T1 = 2π / ωs1, i.e., Sd(1) (T1) ≡ d s1̂ (1) Spa (T1) = ωs12 Sd(1) (T1) , together with the peak value of the BIS displacement, Sub (T1) ≡ ub̂ . Considering that λ < 1 and that λ increases with increasing number of stories, a variety of reasonable values of λ ∊1 are investigated that may be encountered in practical applications, namely λ ∊1 = 0.50–0.90 . Figs. 7–9 collect a few (linear and nonlinear) response spectra, constructed using the above 400 artificial accelerograms with ag = 0.3g , for typical (Teff ,ζ eff ) parameters of the BIS. Similar results are obtained for other combinations of parameters but are omitted for the sake of brevity. By comparing Fig. 7(a) and (b) we observe that linearizing the

Fig. 8. Linear first-mode response spectrum for some typical BIS parameters.

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Fig. 9. Linear first-mode and BIS response spectrum for λ ∊1 = 0.75 : influence of Teff and ζ eff .

improved RSM and how to use the novel response spectra developed in the previous sections for a response-spectrum analysis of a generic multi-story building structure.

common strategy is to increase damping, nevertheless in the limit as ζ eff → ∞ one would have ub (t ) → 0 but also u¨b (t ) → 0 on the rhs of Eq. (16a), consequently the superstructure would be subject to the entire ground acceleration u¨ g as if it were on a fixed base. Interestingly, from Fig. 9(d) it seems that increasing damping beyond a certain ζ eff value is counterproductive, rather than beneficial, for reducing the superstructure response.

4.4. Conventional vs improved RSM Potentialities and limitations of the proposed RSA procedure as well as the main differences as compared to the conventional RSM are outlined below:

4.2. Higher-order modes response spectrum

(i) unlike the conventional RSM, in the proposed RSA procedure the BI building is not treated as a classically damped system: indeed, the constructed response spectra arise from the classical modal analysis carried out on the superstructure only. Therefore, only the superstructure is assumed a classically damped system, which is realistic since similar damping mechanism are distributed throughout. On the contrary, in the conventional RSM the classical modal analysis is applied to the overall nonclassically damped system (i.e., the combined BIS-superstructure system), decoupling the equations of motion in the modal subspace thus neglecting the off-diagonal terms of the Ξ matrix, which may lead to wrong predictions especially with increasing values of the above coupling index Inc ; (ii) in the proposed RSA the calculation of the exact complex-valued eigenproperties of the BI building, which is a concept the professional engineer is not accustomed to, is avoided altogether. Once the above response spectra for common BIS parameters are constructed, the procedure turns out to be nothing but a conventional RSM applied to the superstructure only, as if it were a fixed-base structure, i.e., by adopting conventional combination rules; (iii) for what said in (i) and (ii), there is no need of formulating alternative combination rules with corrective terms for dealing with nonclassically damped systems (e.g. [19,25–27]); (iv) linearizing the BIS behavior is not compulsory in the proposed procedure, as one can potentially incorporate a more realistic nonlinear behavior, one example of which is that postulated in this paper. Note that this possibility is not permitted in the conventional RSM since the BIS is to be represented by an equivalent linear idealization necessarily; (v) two limitations and drawbacks may be recognized in the proposed procedure: first, the feedback effect has been truncated to the firstmode contribution, consequently one would expect the method is not really accurate for large λ ∊j values of the higher modes; second, each response spectrum is limited to a given choice of

The key idea presented in the previous section can easily be extended to analyze the response of an arbitrary jth higher-order mode ( j = 2,…,m ). In this case, beyond the two coupled Eqs. (16), an additional equation, governing the sought response of the jth modal SDOF, is extracted from the overall system (15) so that the following set of three coupled equations are to be solved:

̇ + ωs12 d s1 (t ) = −u¨ g (t )−u¨b (t ) d¨s1 (t ) + 2 ζs1 ωs1 d s1

(17a)

d¨sj (t ) + 2 ζsj ωsj d ṡ j + ωs2j d sj (t ) = −u¨ g (t )−u¨b (t )

(17b)

u¨b (t ) + 2 ζb ω b uḃ (t ) + ω b2 ub (t ) + μb g sign(uḃ (t )) ≈ −u¨ g (t )−λ ∊1 d¨s1 (t ). (17c) In principle, a further term −λ ∊j d¨sj (t ) could be added on the rhs of Eq. (17c), but for the sake of simplicity and to keep the same level of approximation as above this term has been neglected. The coupled Eqs. (17) are solved for given dynamic (linear or nonlinear) properties of the BIS, and for given parameters of both the first-mode SDOF (ζs1,ωs1,λ ∊1) and the considered jth mode SDOF (ζsj,ωsj ). With ζs1 = ζsj = 0.02 being assumed throughout, the mean extreme value of the jth mode SDOF response is computed for variable Tj = 2π / ωsj , i.e., Sd(j) (Tj ) ≡ d sĵ and (j ) Spa = ωs2j Sd(j) (Tj ) . Figs. 10–12 collect a few (linear and nonlinear) higher-order modes response spectra for typical (Teff ,ζ eff ) parameters of the BIS and T1 values. Other combinations of parameters have been investigated but are here omitted for the sake of brevity. 4.3. Application of the improved SB-RSM to base-isolated buildings With the above response spectra constructed, the application of the improved RSA procedure to a base-isolated multi-story building is quite simple. In order to summarize the key aspects and novelties of the proposed SB-RSM procedure, a schematic flow-chart has been constructed as reported in Fig. 13 that clarifies the main steps of the 206

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Fig. 10. Higher-order modes response spectrum for Teff = 3 s, ζ eff = 0.2 : (a), (c) linear; (b), (d) nonlinear.

consists in the way of incorporating a substructure-based approach within the application of an improved response-spectrum analysis for base-isolated buildings, in such a way as to avoid the need of complex eigenmodes or ad hoc cross-correlation coefficients specifically developed for nonclassically damped systems. The proposed SB-RSM significantly improves the conventional application of the response-spectrum method especially when the coupling index Inc defined in Eq. (6) is relatively high, say Inc > 0.3, as will be demonstrated in the next section by a few numerical applications.

parameters, therefore an extensive numerical investigation should be conducted to cover the entire possible variety of parameters normally encountered in practice. Nevertheless, the constructed response spectra lend themselves to idealizations by a series of straight lines, power-law functions or other simple formats depending on the above parameters, which would make the procedure more straightforward. The derivation of such “design expressions” will be discussed in a forthcoming study. As already pointed out at the beginning of Section 4, the proposed method shows similarities to other formulations proposed for the analysis of secondary systems, soil-structure interaction, etc., with regard to the presence of interaction terms pertinent to any other substructurebased approach. Nevertheless, to the authors’ best knowledge, the fundamental aspects of the proposed SB-RSM are new, especially with regard to the way of handling the coupled equations (arising from the nonclassical damping nature of the system) by splitting them into a set of two equations for characterizing the first-mode response, and into a set of three equations for describing the generic higher-order mode response, and afterwards combining them according to traditional combination rules relevant to response-spectrum analysis, as sketched in the flow-chart of Fig. 13. Therefore, the novelty of the method

5. Numerical examples In this section some numerical applications are presented to validate the proposed RSA procedure. Let us consider a five-story symmetricplan base-isolated building that can be analyzed through the equivalent planar frame depicted in Fig. 14. Without loss of generality, we assume constant bay width L and column height H throughout the superstructure, Two examples are studied in order to cover a broad variety of practical cases: example #1 includes the dotted part on the fifth story, i.e., the system is a quite regular five-story, five-bay frame, whereas example #2 is non-regular in elevation since the fifth story is composed of only one bay whose mass is much smaller than the other floors,

Fig. 11. Linear higher-order modes response spectrum for Teff = 2 s: (a), (b) ζ eff = 0.1; (c), (d) ζ eff = 0.3 .

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Fig. 12. Linear higher-order modes response spectrum for Teff = 3 s: (a), (b) ζ eff = 0.1; (c), (d) ζ eff = 0.3 .

case (3), have been performed by removing the linear behavior assumption of the BIS and considering a nonlinear BIS model, i.e., described by the (kb,μb ) values corresponding to Teff = 3.0 s and ζ eff = 0.2 via expressions (11b).

therefore it may be considered as a four-story building with a light appendage. With axial deformations in beams and columns neglected, the frames of example #1 and #2 have 35 and 31 DOFs, respectively, with 5 dynamic DOFs represented by the displacements of the 5 stories relative to the BIS. The elastic modulus is E = 26000 MN/m2 and E = 19500 MN/m2 for example #1 and #2 , respectively (typical of reinforced concrete frames [55]), and the second-moment of the crosssectional area is computed according to the column and beam sections reported in Fig. 14. The mass of the superstructure is lumped at the floor level, with mj = 40,000 kg denoting the mass for the first 4 floors ( j = 1,…,4 ), and m5 = 30,000 kg for the fifth floor in example #1, while the mass of the light appendage in example #2 is 1210 kg. The choice of such an irregular distribution of stiffness and mass in the second example activates certain special response features representative of a system with two natural frequencies that are close and serves to investigate a practical case with a high value of the second-mode participating mass ratio ∊2 , thus jeopardizing one of the main assumptions of the proposed procedure. Applying the classical modal analysis to the superstructure leads to the natural periods and participating mass ratios of Table 2. The baseisolation system is an ensemble of 6 isolators as shown in Fig. 14. The design considerations of BIS are beyond the scope of this paper, therefore we suppose the distribution of the individual isolators is given and the resulting BIS characteristics are determined through the above relationships (9). The BIS mass is assumed to be mb = 32,500 kg, so resulting in a mass ratio λ = 0.8533 and 0.8322 for example #1 and #2, respectively (the mass and stiffness data of the BI building have been chosen such that λ ∊1 = 0.7 and 0.5 for example #1 and #2, respectively). Eight different cases of BIS properties are investigated with a selection of Teff and ζ eff values as documented in Tables 3 and 4: in Table 3 we list the periods resulting from the classical modal analysis applied to the assembled structure via Eq. (3), i.e., according to the conventional RSM approach (we emphasize that the natural periods are not affected by the ζ eff value of the BIS). As expected, the first-mode period of the assembled BI building is almost coincident with Teff of the BIS. Assuming ζ1 = ζ eff for the (first) isolation mode and a constant value of the viscous damping ratio for the structural modes ζi = ζs = 0.02 (i = 2,…,6), a wide spectrum of coupling indexes Inc is covered by the above selection of parameters as reported in Table 4. Overall, 16 different structural cases have been explored that, considering 400 simulations for each case in the MCM, correspond to 6400 response history analyses. Other parameters have also been investigated but are here omitted for the sake of brevity. Additionally, two further analyses, namely example #1 and #2 isolated with the BIS of

5.1. Linearized behavior of the isolation system The predictive performance of the improved SB-RSM is investigated against that of the conventional RSM for each of the structural cases reported in Table 4. It is reminded that the conventional RSM is based on the EC8 response spectrum with the DCF as illustrated in Fig. 6, while the SB-RSM is based on the novel response spectra developed in Sections 4.1 and 4.2. The mean value of the dynamic response obtained with the MCM on the full structural model (with nonclassical damping) has been assumed as reference value. For each structural case, attention is focused on three arbitrary response quantities, namely the displacement of the fourth floor u4 , the third interstory drift Δu3 = u3−u2 and the BIS displacement ub , which are computed via the two RSMs and compared to the results of the MCM. The relative errors with sign (i.e., a negative error indicating an underestimation whereas a positive sign denoting an overestimation as compared to the MCM reference value) are computed for the three response quantities and reported in Tables 5 and 6. First, we focus on the conventional RSM predictions. As can be observed, apart from cases (1) and (2) with a relatively low coupling index due to the choice of ζ eff = 0.1, the errors of the conventional RSM (with all the inherent limitations discussed in this paper) are in most cases higher than 20% on the nonconservative (unsafe) side. The striking results are those found for the realistic combinations of BIS parameters associated with case (5), (7) and (8), namely Teff = 3.0 s and ζ eff = 0.3, 0.35 and 0.4 , respectively: in these cases the superstructure response is underestimated up to more than 50% (cf. errors related to u4 and Δu3 for the two examples). On the contrary, the BIS displacement is overestimated (rather than underestimated) up to more than 30%: this result is consistent with Fig. 6 wherein the curves associated with the EC8 with DCF for the range of Teff values investigated, namely Teff = 2.0 s and 3.0 s, lie above the MCM curves. On the other hand, the numerical predictions of the SB-RSM are in all the examined cases significantly more accurate than the conventional RSM ones. As expected, the improvements of the proposed RSM are remarkable for example #1 where the first-mode contribution is high, and become a bit less apparent for example #2 due to the particular choice of stiffness and mass data that resulted in a second-mode participating mass ratio ∊2 ≈ 25%. Nevertheless, by carefully analyzing the relative errors of 208

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Fig. 13. Schematic flow-chart summarizing the key aspects and novelties of the improved SB-RSM.

nonlinear model of the BIS, the previously discussed case (3) is here reanalyzed wherein the BIS is not described by the linear couple (Teff ,ζ eff ) but rather by the nonlinear parameters (kb,μb ) associated with Teff = 3.0 s and ζ eff = 0.2 through expressions (11b). After 8 iterations, the following BIS nonlinear parameters are identified: Tb = 3.62 s and μb = 0.012 . In other words, in this case the reference solution of the MCM is the mean value from 400 nonlinear time-history analyses, the nonlinearity of the equations being concentrated in the BIS behavior, cf. Eqs. (15). In Table 7 the relative errors of the above response quantities are reported. Since the reference solution has been changed, the relative errors for both the conventional and improved SB-RSM are different from those of the linearized case discussed in the previous section. The

example #2 one can notice that the SB-RSM approximtes the MCM reference results much better than the conventional RSM, the associated relative errors being in all but one case less than 10%. Therefore, even in this extreme case the proposed SB-RSM performs considerably better than the conventional RSM, which confirms the effectiveness and reliability of the developed procedure. 5.2. Nonlinear behavior of the isolation system A further advantage of the proposed SB-RSM procedure is the possibility of removing the hypothesis of linear idealization of the BIS in order to incorporate a more realistic nonlinear behavior model. To assess the accuracy of the improved SB-RSM when dealing with a 209

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Fig. 14. Sketch of the base-isolated building analyzed in examples #1 and #2 (all dimensions in m).

Table 2 Results from classical modal analysis applied to the superstructure only, according to Eq. (12).

Table 4 Coupling index computed via Eq. (6) for the analyzed cases. Case

Mode i

1 2 3 4 5

Example #1

BIS properties

Example #1

Example #2

Example #2

Ti [s]

∊i [% ]

Ti [s]

∊i [% ]

0.401 0.151 0.091 0.065 0.047

82.03 10.47 3.88 2.56 1.06

0.401 0.351 0.133 0.078 0.054

60.33 24.50 9.76 4.09 1.32

(1) (2) (3) (4) (5) (6) (7) (8)

general trend is, however, confirmed by the numerical values, with a predictive performance of the proposed RSM that is excellent for example #1 and reasonably good for example #2 as well. In contrast, by the conventional RSM the structural engineer would design the BI building considering some response quantities underestimated of 30–50% as compared to the actual value from nonlinear analysis. 6. Concluding remarks The main contents and findings of this research work are summarized as follows:

•

• the application of the conventional RSM to BI buildings may give rise to inadequate numerical predictions since the underlying assumptions are not really met, more specifically: (1) the equations of

•

ζ eff [–]

Teff [s]

Inc

Inc

0.1 0.1 0.2 0.2 0.3 0.3 0.35 0.4

3.0 2.0 3.0 2.0 3.0 2.0 3.0 3.0

0.262 0.336 0.422 0.514 0.525 0.617 0.564 0.600

0.264 0.340 0.424 0.518 0.527 0.620 0.566 0.599

motions are coupled in the modal subspace; (2) the cross-correlation coefficients for the combination rules should be corrected based on the complex eigenvalue problem; (3) extrapolating the response spectrum for damping ratios different from the reference value (ζ 0 = 0.05) via the DCF is a simplistic method that may lead to wrong conclusions for high damping ratios such as those involved in the BI buildings; (4) the behavior of the BIS is actually nonlinear but the conventional RSM allows an equivalent linear idealization only; from a practical point of view, the above simplifications may lead to considerably underestimate the dynamic response up to more than 50%, thus implying that the design of the BI building is carried out in a nonconservative (unsafe) manner; aimed at improving the conventional RSM but at the same time

Table 3 Results from classical modal analysis applied to the assembled BI building, according to Eq. (3). Mode i

1 2 3 4 5 6

BIS properties

Example #1

Example #2

BIS properties

Example #1

Example #2

Teff [s]

ζ eff [–]

Ti [s]

Ti [s]

Teff [s]

ζ eff [–]

Ti [s]

Ti [s]

2.0

∀

2.029 0.233 0.124 0.082 0.059 0.045

2.027 0.367 0.218 0.111 0.070 0.052

3.0

∀

3.019 0.235 0.124 0.082 0.059 0.045

3.018 0.367 0.219 0.111 0.070 0.052

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• the effectiveness of the proposed RSA procedure as well as the im-

Table 5 Relative errors with sign, expressed in % , of three different response quantities for example #1. Case

(1) (2) (3) (4) (5) (6) (7) (8)

err [%] related to u 4

err [%] related to Δu3

provements over the conventional RSM have been proven by comparing the numerical predictions with results of (linear and nonlinear) RHA for a variety of superstructure properties and characteristics of the BIS that may be encountered in practice. Within the reasonably acceptable level of inaccuracy arisen from the truncation of the higher-order-mode feedback terms, the proposed procedure is capable of approximating the RHA results much better than the conventional RSM.

err [%] related to ub

Conv. RSM

SB-RSM

Conv. RSM

SB-RSM

Conv. RSM

SB-RSM

−9.68 6.74 −23.62 −1.41 −39.36 −15.11 −46.01 −51.74

0.70 0.60 0.56 0.40 0.63 0.71 0.63 0.55

−11.73 4.82 −27.39 −5.09 −43.46 −19.74 −49.96 −55.45

6.10 6.28 3.28 −1.01 1.56 2.77 1.01 0.57

0.81 15.58 5.58 25.96 8.39 32.59 9.49 10.56

0.00 −0.01 0.00 −0.01 −0.01 −0.02 −0.01 −0.01

With regard to the equivalent linearization of the nonlinear BIS, cf. Section 3.3, it is worth noting that the effective stiffness and damping ratio implicitly depend upon the maximum response of the system. Therefore, unlike the linear BIS, the response of the nonlinear BIS and the validity of the response spectra reported in Section 4 are confined to the severity of the ground motion considered in this paper (i.e., an EC8based seismic input for ground type A with PGA ag = 0.3 g ). It is expected that these response spectra for the nonlinear BIS may be quite different depending on the type of seismic motion, e.g. far-fault versus near field, etc.

Table 6 Relative errors with sign, expressed in % , of three different response quantities for example #2 .

err [%] related to u 4

Case

(1) (2) (3) (4) (5) (6) (7) (8)

err [%] related to Δu3

err [%] related to ub

Conv. RSM

SB-RSM

Conv. RSM

SB-RSM

Conv. RSM

SB-RSM

−9.67 6.90 −23.94 −1.56 −40.11 −15.98 −46.78 −52.50

−9.96 −11.39 −1.46 −5.29 5.17 1.08 7.64 9.55

−12.47 4.33 −29.03 −6.71 −45.54 −22.25 −51.99 −57.41

−3.55 −4.63 1.65 −1.01 5.70 3.19 7.30 8.48

0.88 15.80 5.66 26.21 8.50 32.88 9.59 10.66

0.06 −0.02 0.01 −0.18 −0.02 −0.28 −0.06 −0.10

6.1. Future developments For the sake of simplicity and to better illustrate the fundamentals of the proposed procedure, attention has been purposely focused on symmetric-plan buildings, which simplifies the formulation since equivalent planar systems can be examined. This is the case of most newly designed base-isolated buildings whose centers of mass and stiffness tend to coincide. Extension of the proposed RSA procedure to unsymmetric-plan buildings slightly modifies the formulation and will be discussed in a forthcoming study. Based on the encouraging results of this paper, the authors’ opinion is that the proposed procedure should be simplified and made more straightforward so that it can easily be applied for the analysis of baseisolated buildings by professional engineers. To this aim, the novel response spectrum curves presented in Section 4 should be converted into some easy-to-use “design expressions” that depend on the main BIS and superstructure parameters. Idealization of the spectrum curves by a series of straight lines, power-law functions, or other simple formats the professional engineer is accustomed to (similar to the existing response spectra reported in the building codes, e.g. [28,29]) is indeed the object of an ongoing research. Extension to hysteretic behaviors other than the (Coulomb) frictional one, for instance the widely used Bouc Wen model, also deserves further investigation.

Table 7 Relative errors with sign, expressed in % , for case (3) and nonlinear behavior of the BIS. Example

#1 #2

•

•

err [%] related to u 4

err [%] related to Δu3

err [%] related to ub

Conv. RSM

SB-RSM

Conv. RSM

SB-RSM

Conv. RSM

SB-RSM

−30.50 −34.30

3.25 12.64

−41.35 −47.69

−1.26 2.32

9.15 9.23

−0.13 −0.18

preserving the simplicity and the attractive features of a real-value response-spectrum analysis, an improved RSA procedure, which is based on the substructure approach and thus termed SB-RSM, has been developed in this paper; the proposed procedure circumvents some of the underlying misleading assumptions of the conventional RSM: (1) the modal analysis is applied to the superstructure only, which can realistically be considered as a classically damped system, and not to the overall BI building, consequently the equations are actually decoupled in the modal subspace; (2) the effects of the BIS are incorporated in the definition of some novel response spectra that quantify the superstructure response based on the BIS characteristics; (3) with such novel response spectra developed, one can deal with the superstructure as a classically damped system, therefore the need of solving a complex eigenproblem is avoided altogether, and the conventional combination rules (without any correction for nonclassical damping) can be employed; (4) a more realistic nonlinear behavior of the BIS can be incorporated in the procedure without any additional computational cost, one example of which is that postulated in this paper; from a practical point of view, once such novel response spectra are defined, the proposed RSA procedure turns out to be nothing but the classical RSM applied to the superstructure as if it were on a fixedbase, i.e., frequencies and mode shapes are those of the fixed-base building;

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