On the Distribution in Height of Base Shear Forces in Linear Static Analysis of Base-Isolated Structures

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Article On the Distribution in Height of Base Shear Forces in Linear Static Analysis of Base-Isolated Structures

Adamo Zinco 1, Fernando Fraternali 1 , Gianmario Benzoni 2 and Enzo Martinelli 1,*

1 Department of Civil Engineering, University of Salerno, 84084 Fisciano, Italy; [email protected] (A.Z.); [email protected] (F.F.) 2 Department of Structural Engineering, University of California, San Diego, CA 92093-0085, USA; [email protected] * Correspondence: [email protected]; Tel.: +39-089-96-4098

Received: 28 September 2020; Accepted: 29 October 2020; Published: 1 November 2020

Abstract: Although base isolation is nowadays a well-established seismic-protection technique for both buildings and bridges, and several issues are still open and attract the interest of the research community. Among them, the formulation of computationally efficient and accurate analysis methods is a relevant aspect in structural design of seismic-isolated buildings. In fact, codes and guidelines currently in force in various parts of the world generally include the possibility for designers to utilize linear-elastic analysis methods based on equivalent linearization of the non-linear force-displacement response of isolators. This paper proposes a formula for defining the force distribution in height that should be considered in linear-static analyses to obtain a more accurate approximation of the actual structural response, supposedly simulated by means of non-linear time history analysis. To do that, it summarizes the results of a wide parametric analysis carried out on a batch of structures characterized by three different heights and various properties of base isolators. The reported results highlight that the equivalent static force distribution provided by both Italian and European codes tend to underestimate the actual seismic lateral forces acting on base-isolated buildings, whereas the inverted triangular distribution, proposed in various American codes and standards, is often conservative.

Keywords: seismic isolation; linear static analysis; vertical distribution of shear forces

1. Introduction Seismic isolation is based on well-established mechanical principles [1,2] and it is nowadays widely adopted worldwide both in the realization of new buildings [3,4] and retrofitting existing ones [5–7]. Several studies have demonstrated the effectiveness of base isolation in the case of historic structures with the aim to preserve and reuse the buildings or protect their non-structural elements and contents [8,9]. A wide variety of seismic isolators are currently available on the market [10,11] and further devices are being developed and tested in research laboratories [12,13]. Although they are based on various mechanical phenomena (i.e., elastic response of soft materials, yielding of metals either in bending or torsion, surface friction, sliding, etc.), the mechanical behavior of seismic isolators is generally characterized by non-linear behavior [14,15] and, hence, non-linear time history (NLTH) analyses are needed to consistently simulate the response of base-isolated (BI) buildings subjected to seismic excitations [16]. However, NLTH analyses are computationally demanding and time consuming, which makes them unfit for practice-oriented design activities. Therefore, modern codes and guidelines [17–20] consider both linear and non-linear analysis methods for seismic design of base-isolated buildings. Specifically, linear static analysis (LSA) is generally permitted with some limitations, depending on the

Buildings 2020, 10, 197; doi:10.3390/buildings10110197 www.mdpi.com/journal/buildings Buildings 2020, 10, 197 2 of 20 properties of both the isolation system (IS) and the building super-structure. It is based on defining a pattern of horizontal forces and equivalent stiffness and viscous damping for the base-isolation system [21–24]. According to various structural codes, IS may be modelled by an equivalent linear visco-elastic behavior with some limitations defined in terms of effective damping ratio, re-centering capability and invariability of the mechanical properties [15,20]. Various researchers have investigated accuracy and suitability of LSA in design-oriented simulations of base-isolated buildings [25–27]. Specifically, several studies were intended at assessing and possibly enhancing the definition of the properties of linearized structural models to be employed in LSA, especially in the case of bridges [28–30]. Moreover, the use of alternative lateral force patterns was assessed to perform pushover analyses on BI-structures [31]. However, setting an appropriate equivalent linearized model is only the first step in LSA procedures for multi-story buildings [32,33], as they also require a consistent definition of the distribution of equivalent horizontal forces in height. As a matter of principle, two typical behaviors can be distinguished. On the one hand, if it is assumed that the base-isolation system is significantly more flexible than the superstructure (the latter being basically assumed as a rigid body), a uniform distribution can be figured out where the equivalent static seismic forces Fi at the i-th story can be determined as a function of both the total design base shear (Vb) and the story masses mi [17,34]: m F = V i (1) i PN j=1 mj On the other hand, other seismic codes, including the US and Chinese ones [20,35,36], assume an inverted triangular distribution of the base shear Vb that can be described as follows:

m h F = V i i (2) i PN j=1 mjhj However, some studies [37,38] demonstrate that uniform distribution tends to neglect flexibility of the superstructure and the participation of the higher vibration modes. Conversely, the inverted triangular force distribution has been found to overestimate the maximum seismic responses of most BI-buildings, even when the IS exhibits a strong non-linear behavior [39,40]. Therefore, enhanced formulations are also available in the literature to describe the distribution in height of lateral forces with the aim to take into account relevant parameters, such as the frequency ratio between BI- and fixed-base structure [41,42], the fundamental period of the superstructure [43], the modal shape of the latter [42], the non-linearity factor [14] of seismic isolators [44,45] or combination of some of the above parameters [46–49]. Under the mathematical standpoint, the proposed formulations are based on either introducing an exponent to hi in Equation (2) [44,46] or combining Equations (1) and (2) [41,42,48]. This work aims at assessing several formulations available in the scientific literature [41–49], as well as in codes and standards [17,20,34], for the distribution in height of lateral earthquake-induced forces to be considered in LSA of base-isolated structures. Specifically, NLTH analyses are carried out on the numerical models of three reinforced-concrete (RC) frames with three, five and seven stories and various fundamental periods of vibration at the fixed-base configuration. Various ISs, characterized by elastic-hardening force-displacement curves, are considered. The parameters describing the isolation devices range within the interval of relevance in practical applications with the aim to reproduce the actual behavior of commercial ISs, which may either be realized by lead rubber bearings (LRB), high-damping rubber bearings (HDRB), low-damping rubber bearings (LDRB) and friction pendulum bearings (FPB). Regardless of the specific technological solution, the formulation proposed herein depends on the relevant mechanical properties of the IS under consideration and allows us to analyze BI-structures by means of LSA in lieu of more time-consuming NLTH analyses, which results in a more efficient design procedure. Buildings 2020, 10, 197 3 of 20

The paper is organized as follows. Section2 describes the types of analysis and the domain of variation of the relevant parameters considered in this study. Section3 summarizes the results of the seismicBuildings analyses 2020, 10 and, x FOR shows PEER REVIEW the comparisons between the outcome of NLTH and LSA, the latter3 of 20 being carried out with lateral forces distributed both uniformly and as an inverted triangle, according to two well-establishedcarried out with lateral practices. forces Section distributed4 discusses both un theiformly results and by as comparingan inverted thetriangle, pattern according of horizontal to forcestwo resulting well-established from NLTH practices. analyses Section 4 with discusses the relevantthe results formulations by comparing the available pattern inof horizontal the literature; a practice-orientedforces resulting formulafrom NLTH for lateralanalyses force with distribution the relevant to formulations be adopted availa in LSAble isin finally the literature; proposed a as a mainpractice-oriented finding of the present formula paper. for lateral The force main distribution results of to the be work adopted are in then LSA remarked is finally proposed upon in Sectionas a 5. main finding of the present paper. The main results of the work are then remarked upon in Section 2. Research5. Methodology: Parametric Analysis The2. Research present Methodology: study is based Parametric on comparing Analysis the results of accurate, yet computationally intensive, NLTH analysesThe present with study the is corresponding based on comparing ones the obtained results of via accurate, LSA underyet computationally lateral loads intensive, supposedly equivalentNLTH toanalyses the eff ectwith of the the corresponding seismic excitation. ones ob Thetained relevant via LSA modelling under lateral assumptions loads supposedly and analysis proceduresequivalent are to described the effect into of the details seismic in excitation. the following The sections.relevant modelling assumptions and analysis procedures are described into details in the following sections. 2.1. Superstructure 2.1. Superstructure The present study is based on considering a set of three-, five- and seven-story reinforced RC frame structures.The Specifically, present study the is attention based on hasconsidering been focused a set of on three-, regular five- buildings and seven-story with invariant reinforced horizontal RC stiffnessframe throughout structures. theSpecifically, height ofthe the attention structure has andbeen moderatelyfocused on regular variable buildings floor mass. with invariant The building horizontal stiffness throughout the height of the structure and moderately variable floor mass. The models have a rectangular plan and a symmetric distribution of resisting systems. Figure1 depicts the building models have a rectangular plan and a symmetric distribution of resisting systems. Figure 1 model of the five-story base-isolated building. depicts the model of the five-story base-isolated building.

FigureFigure 1. Three-dimensional 1. Three-dimensional model model ofof aa prototype base base isolated isolated building building with with five ﬁvestories. stories.

The followingThe following main main assumptions assumptions are are made made inin modelling the the superstructure: superstructure: i. i. linear-elasticlinear-elastic behaviorbehavior of of the the superstructure, superstructure, which which is generally is generally accepted accepted in BI-structures; in BI-structures; ii. ii. rigidrigid diaphragm diaphragm constraints are are set set at all at allfloor; ﬂoor; iii. soil-structure interaction is neglected; iii. soil-structure interaction is neglected; iv. torsional effects are also neglected, as the superstructure is regular both in plan and height. iv. torsional eﬀects are also neglected, as the superstructure is regular both in plan and height. The plan dimensions are 16.50 m × 13.50 m. The story area is approximately 223 m2 and is divided Theinto three plan equal dimensions bays along are the 16.50 X-direction m 13.50 and m. three The equal story bays area along is approximately the Z-direction. 223 The m story2 and height is divided × into threeis 3.10 equal m (Figure bays 2). along the X-direction and three equal bays along the Z-direction. The story height is 3.10 mThe (Figure mass2 ).is the same at each level, except for base- and roof-levels. Uniform dead loads (G1 and G2) at the intermediate floor is equal to 7.17 kN/m2, at the base levels 8 kN/m2 and at the roof levels 4.30 Buildings 2020, 10, 197 4 of 20

The mass is the same at each level, except for base- and roof-levels. Uniform dead loads (G1 and 2 2 G2) atBuildings the intermediate 2020, 10, x FOR PEER floor REVIEW is equal to 7.17 kN/m , at the base levels 8 kN/m and at the roof4 of 20 levels 2 2 4.30 kN/m . Live load (Qk) is the same for each level and it is equal to 2 kN/m . Moreover, the infill wallskN/m are not2. Live considered load (Qk) is as the structural same for each elements level and and it areis equal modelled to 2 kN/m with2. Moreover, a uniform the load infill ofwalls 7.25 are kN /m. Therefore,not considered the total as masses structural of theelements three-, and five- are mode and seven-floorlled with a uniform superstructures load of 7.25 are kN/m. 1142 Therefore, t, 1817 t and the total masses of the three-, five- and seven-floor superstructures are 1142 t, 1817 t and 2555 t, 2555 t, respectively. respectively.

FigureFigure 2. 2.Standard Standard ﬂoor floor plan.plan. Dimensions are are in incentimeters. centimeters.

The superstructureThe superstructure is modelledis modelled by by elastic elastic elementselements whose whose transverse transverse sections sections are aredimensioned dimensioned in suchin such a way a way that that the corresponding corresponding fixed-base fixed-base structure structure has realistic has fundamental realistic fundamental vibration periods. vibration Specifically, the beam cross-sections are 0.40 m × 0.60 m in the longitudinal direction and 0.40 m × periods. Specifically, the beam cross-sections are 0.40 m 0.60 m in the longitudinal direction and 0.50 m in the transversal direction. At the base level, a greater× depth of the beam sections is adopted 0.40 m 0.50 m in the transversal direction. At the base level, a greater depth of the beam sections (0.70× m). Conversely, the sizes of column sections vary for the three prototype buildings. A square is adoptedshape is (0.70 always m). assumed: Conversely, 0.40 them × sizes 0.40 m of for column the 3-story sections building, vary for0.50 the m × three 0.50 m prototype for the 5-story buildings. A square shape is always assumed: 0.40 m 0.40 m for the 3-story building, 0.50 m 0.50 m for the building and 0.60 m × 0.60 m for the 7-story× building. The section dimensions are the same× from the 5-storybase building to the roof. and 0.60 m 0.60 m for the 7-story building. The section dimensions are the same × from the baseTable to 1 summarizes the roof. the cross-section sizes of the structural elements. Table1 summarizes the cross-section sizes of the structural elements. Table 1. Structural element sections for standard floor. Table 1. Structural element sections for standard floor. Building Beam Girder Column three storiesBuilding 40 cm Beam× 60 cm 40 Girdercm × 50 cm Column 40 cm × 40 cm five stories 40 cm × 60 cm 40 cm × 50 cm 50 cm × 50 cm three stories 40 cm 60 cm 40 cm 50 cm 40 cm 40 cm seven stories 40 cm × 60× cm 40 cm× × 50 cm 60 ×cm × 60 cm five stories 40 cm 60 cm 40 cm 50 cm 50 cm 50 cm × × × seven stories 40 cm 60 cm 40 cm 50 cm 60 cm 60 cm Table 2 reports the main properties× of the three-prototype× buildings, including× the total height and mass of the buildings, the superstructure-to-base-level mass ratio Ms/Mb and the superstructure Tablefundamental2 reports period the mainTbf. properties of the three-prototype buildings, including the total height and mass of the buildings, the superstructure-to-base-level mass ratio Ms/Mb and the superstructure Table 2. Main properties of the three structural model in fixed-base configuration. fundamental period Tbf. Superstructure Height Total Mass Ms/Mb Period Tbf threeTable stories 2. Main properties 9.3 of m the three structural 1142 modelt in fixed-base 3.40 configuration. 0.41 s five stories 15.5 m 1817 t 5.33 0.55 s Superstructureseven stories Height 21.7 m Total 2555 Mass t Ms/M 7.34b Period 0.70T sbf three stories 9.3 m 1142 t 3.40 0.41 s In fiveNLTH stories analyses, 2% of 15.5 the mcritical damping 1817 is tassumed by using 5.33 Rayleigh damping 0.55 s [50–52] for theseven superstructures, stories in addition 21.7 m to the dissipated 2555 hysteretic t energy 7.34 at the isolation level 0.70 due s to the non-linear behavior of the IS. In NLTH analyses, 2% of the critical damping is assumed by using Rayleigh damping [50–52] for 2.2. Seismic Isolation System the superstructures, in addition to the dissipated hysteretic energy at the isolation level due to the non-linear behavior of the IS. Buildings 2020, 10, 197 5 of 20

Buildings 2020, 10, x FOR PEER REVIEW 5 of 20 2.2. Seismic Isolation System Nowadays, several types of isolation devices are available in the market, which have been briefly Nowadays, several types of isolation devices are available in the market, which have been brieﬂy mentioned in Section 1. Although more sophisticated models can be found in the literature to mentioned in Section1. Although more sophisticated models can be found in the literature to simulate simulate the non-linear response of those devices [53–56], the present study is based on assuming a the non-linear response of those devices [53–56], the present study is based on assuming a general general elastic-hardening hysteretic force-displacement law (Figure 3), which, in principle, can elastic-hardening hysteretic force-displacement law (Figure3), which, in principle, can represent the represent the response of three main groups of isolation devices, such as LDRB (or HDRB), LRB and response of three main groups of isolation devices, such as LDRB (or HDRB), LRB and FPB. FPB.

FigureFigure 3. 3. Force-displacementForce-displacement curve curve of of a atypical typical isolation isolation device. device.

ThisThis bilinear bilinear force-displacement force-displacement relationship relationship of the of theisolation isolation systems systems is identified is identified by three by main three parameters:main parameters: the initial the initialstiffness stiff nessK1 (before K1 (before yielding/sliding), yielding/sliding), the thepost-yield post-yield stiffness stiffness K2 K 2andand the the intersectionintersection Q Q between between the the post-elastic post-elastic branch branch and and the the vertical vertical axis axis (Figure (Figure 3).3). The The other other parameters parameters areare the the yield yield force force and and displacement, displacement, Fy Fandy and Xy, Xandy, andthe NLTH the NLTH design design displacement displacement demand demand Xd. The Xd . ratiosThe ratios α = Kα2/K= K1 and2/K1 μand = Xµd/X= yX ared/X ytheare post-yield the post-yield hardening hardening and and ductility ductility ratio ratio of ofthe the IS, IS, respectively. respectively. Specifically,Specifically, for for FPB, FPB, the the mechanical mechanical behavior behavior is is de describedscribed by by an an elastic-hardening elastic-hardening curve curve with with a a high high valuevalue of of the the initial initial stiffness, stiffness, K K2 2isis given given by by the the W/R W/R ratio ratio and and the the characteristic characteristic force force Q Q is is equal equal to to υW,υW, wherewhere the the W isis thethe weight weight supported supported by by the the bearings, bearin Rgs, is R the is e fftheective effective curvature curvature radius radius of the concaveof the concavesurface, surface, and υ is and the frictionυ is the friction coefficient. coefficient. SinceSince existing existing building building codes codes tend to propose, atat leastleast forfor preliminarypreliminary design design purposes, purposes, the the use use of ofsimplified simplified models models and and techniques techniques that that facilitate facilitate the the analysis analysis of of seismically seismicall isolatedy isolated structures structures in thein theLSA, LSA, an an equivalent equivalent linearization linearizatio ofn theof the IS non-linearIS non-linear hysteretic hysteretic behavior behavior is assumed,is assumed, as isolatorsas isolators are aremodelled modelled through through their their eff effectiveective sti ffstiffnessness Ke Kff effand and eff effectiveective damping damping ratio ratioξ .ξ The. The e ffeffectiveective sti stiffnessffness is isdefined defined as as the the secant secant sti ffstiffnessness to theto the design design displacement displacement Xd and Xd and is related is related to the to e theffective effective period period Teq of Ttheeq of BI-building. the BI-building. The hystereticThe hysteretic damping damping ratio isratio given is given by the by energy the energy dissipation dissipation principle, principle, based based on the onequivalence the equivalence between between the energy the energy dissipated dissipated by one by cycle one of cycle the bilinear of the bilinear model, EH,model, and EH, the dampingand the dampingenergy of energy the linear of the damped linear system,damped ED, system, related ED, to re thelated maximum to the maximum displacement displacement value [57 ].value [57]. TheThe effective effective stiffness stiffness K Keffeff andand the the effective effective damping damping ratio ratio ξξ, ,at at the the displacement displacement X Xdd areare calculated calculated byby the the following following equations: equations: K X +K (X X Fd F K1 Xy +K2(X d-X )y 1+α1+μ-1α(µ 1) K = =d 1 y 2 d − y = K − (3) eff KeffX= = X =K1 1 µ (3) d Xd Xd μ EH 4X(y(K1-K2))(X( d-Xy) 2(1-α)(μ-1) Eξ= 4X= y K 1 K2 X d Xy= 2(1 α)(µ 1) (4) ξ = H = − 2 − = − − (4) ED 2πKeffXd2 πμ 1+α(μ-1) ED 2πK X πµ[1 + α(µ 1)] eff d − In this study, the main IS properties are defined by three fixed a-priori effective values of the equivalent vibration period Teq, and various values of initial stiffness K1 of the isolators. For the sake of clarity, the ranges of α and ξ parameters assumed in this study are wider than those typically associated to the IS in order to get a better knowledge of the BI structural dynamics and evaluated Buildings 2020, 10, 197 6 of 20

In this study, the main IS properties are defined by three fixed a-priori effective values of the equivalent vibration period Teq, and various values of initial stiffness K1 of the isolators. For the sake of clarity, the ranges of α and ξ parameters assumed in this study are wider than those typically associated to the IS in order to get a better knowledge of the BI structural dynamics and evaluated the applicability of the formulae over a larger selection. In fact, the α values vary between 0 and 1 and the ξ values, with respect to the limit values of the previous parameter, can increase from 0 to 0.63. On the one hand, the lower effective damping value, related to α equal to 1, represents a linear behavior, on the other the upper limit, related to α equal to 0, describes an elastic perfectly plastic curve. It is worth noting that the typical values of ξ parameter range from 0.05 to 0.35. Table3 reports the main parameters of the isolation devices employed for the previously described buildings. In particular, the following parameters are expected to have the most influence on the vertical distribution of the base shear: effective period of isolated system (Teq), ductility ratio (µ), initial stiffness and period (K1,T1), the stiffness ratio (α) and, obviously, the hysteretic damping ratio (ξ).

Table 3. Mechanical properties of the isolation system.

Effective Period [s] 1.5; 2.0; 2.5 Ductility [-] 15; 20; 25 Initial Stiffness [N/mm] 2500; 5000; 7500; 10,000 Damping [-] 0.006–0.55 Stiffness ratio [-] 0.005–0.78

2.3. Ground Motions The Italian Seismic Building Code [34] defines the response spectra in each point of a network covering the entire Italian territory. Based on location of the building and the return period of the seismic events, three parameters, ag,F0, and Tc* are provided to generate the horizontal elastic acceleration and displacement spectra. More details (the shape of an acceleration spectrum and the meanings of the different parameters) can be found in Chapter 3 of NTC 2018 [34]. In this case, to determine the non-linear properties of the isolation system, the target IS deformation Xd, defined according to the simplified approach suggested by the Italian Building Code, is obtained from the elastic design spectrum for a reference return period Tr = 975 years. Only the limit state of collapse prevention (SLC) with a low probability of occurrence of the 5% in the design working life of 50 years is considered as seismic input. The seismic parameters for the design spectra are listed in Table4. As for geotechnical parameters, soil type B and topology type T 1 are considered.

Table 4. Parameters to deﬁne the elastic acceleration spectrum.

Limit State Tr [years] ag/g [-] F0 [-] Tc* [s] SLO 30 0.062 2.356 0.280 SLD 50 0.084 2.330 0.297 SLV 475 0.270 2.278 0.379 SLC 975 0.368 2.281 0.410

To perform NLTH analysis of the BI buildings under consideration, a set of seven natural ground motions has been selected by means of the REXELite software [58]. Figure4 shows the selected-accelerograms and the corresponding response spectra: the graph demonstrates the limited scatter of the single signal spectra with respect to the target design spectra assumed in this study. According to the Italian Code [34], spectro-compatibility is checked in the period range between 0.15 s and 1.2 Teq: the average spectrum is never lower than 10% and higher than 30% of the target response spectrum. Buildings 2020, 10, 197 7 of 20 Buildings 2020, 10, x FOR PEER REVIEW 7 of 20

4 1.6 Horizontal acceleration response spectra 0 1.4 -4 0 5 10 15 20 25 30 35 1.2 4 1 0

-4 0.8 0 5 10 15 20 25 30 35 Se(T) [1/g] Se(T) 0.6 4

0 0.4

-4 0.2 0 5 10 15 20 25 30 35 4 ] 0 2 0 0.0 1.0 2.0 3.0 4.0

a [m/s -4 Period T [s] 0 5 10 15 20 25 30 35 0.9 Displacement response spectra 4 0.8

0 0.7 -4 0 5 10 15 20 25 30 35 0.6 4 0.5

0 0.4

-4 [m] SDe(T) 0 5 10 15 20 25 30 35 0.3 4 0.2

0 0.1 -4 0 0 5 10 15 20 25 30 35 0.0 1.0 2.0 3.0 4.0 Time [s] Period T [s]

average spectra reference spectra lower and upper limits

Figure 4.4. Ground motion records and horizontalhorizontal elasticelastic displacementdisplacement (S(SDeDe)) and acceleration (S(Se)) response spectra.

2.4. Methods A totaltotal of of 108 108 diff erentdifferent bilinear bilinear curves curves to describe to thedescribe isolators the behavior isolators (3 buildingbehavior configurations (3 building 3 equivalent periods 4 initial stiffnesses 3 ductility values) have been considered in the NLTH ×configurations × 3 equivalent× periods × 4 initial× stiffnesses × 3 ductility values) have been considered analyses.in the NLTH A total analyses. number A total of 756 number NLTH of analyses 756 NLTH have analyses been. have been. In this work, work, one-directional one-directional seismic seismic input input is is considered considered in in X-direction X-direction and and no no amplification amplification is isconsidered considered for fortorsional torsional effects. effects. The considered The considered structures structures are regular are regular in plan in and plan elevation, and elevation, which whichmakes makestorsional torsional effects negligible, effects negligible, and the andstudy the of study bi-direction of bi-direction seismic input seismic is beyond input is the beyond scope the of scopethis paper. of this The paper. step-by-step The step-by-step procedure procedure followed followed in this in thisstudy study to derive to derive the the actual actual lateral lateral force distributions is summarized below. Step 1: NLTH analyses of base isolated building configurations by considering a set of seven Step 1: NLTH analyses of base isolated building configurations by considering a set of seven spectrum-compatible accelerograms; spectrum-compatible accelerograms; Step 2: Average of the maximum story shear values at the different levels derived from the Step 2: Average of the maximum story shear values at the different levels derived from the selected selected seven accelerograms; seven accelerograms; Step 3: Conversion of the median peak story shears Vi at the i-th level to median lateral force Fi, Step 3: Conversion of the median peak story shears V at the i-th level to median lateral force F , according to the following equations: i i according to the following equations: NColumn NColumnX(i) Vi= Fj (i) (5) Vi = Fj (5) j=1 j=1 (6) Fi=Vi-Vi+1 Buildings 2020, 10, 197 8 of 20

Buildings 2020, 10, x FOR PEER REVIEW 8 of 20

Fi= Vi Vi+1 (6) where the Fj are the shear forces recorded −at the column of each i-th level, Vi and Vi+1 are where the F are the shear forces recorded at the column of each i-th level, V and V are the the averagej of the maximum story shears at the i-th and (i + 1)-th levelsi and i+the1 Fi are averagethe oflateral the maximumforce applied story at shearsthe i-th at level; the i-th and (i + 1)-th levels and the Fi are the lateral force applied at the i-th level; Step 4:StepNormalization 4: Normalization of the of storythe story shear shear and and lateral lateral force force distributions distributions by by base base shearshear (assuming(assuming Vb = 1), in order to focus on relative distribution of force rather than their values. Vb = 1), in order to focus on relative distribution of force rather than their values.

3. R Resultsesults Displacement demand on the isolation system and story shear forces are the main factors for the design of BIBI structures:structures: thethe followingfollowing subsections subsections report report a a comparison comparison between between the the results results obtained obtained by byNLTH NLTH analyses analyses and and LSA LSA on theon the RC RC structures structures deﬁned defined in Section in Section2. 2.

3.1. Base-Isolated Base-Isolated (BI) Struct Structureure Response: Ground Ground Motions Various configurationsconfigurations ofof base-isolated buildings are presented in order to evaluate the dynamic response obtained for the didifferentfferent seismic inputs.inputs. Figure 55 showsshows thethe maximummaximum demandsdemands inin termsterms of inter-story drift ratio (IDR). It demonstrates that the maximum inter-story drift ratios obtained from NLTH analysesanalyses are generallygenerally belowbelow 0.45%0.45% inin all thethe consideredconsidered structures.structures. This confirmsconfirms that the demand on the isolated structures structures is lower lower than than the the corresponding corresponding limits limits of of the the elastic elastic range range [31], [31], which is consistent with assuming a linear-elasticlinear-elastic modelmodel forfor thethe superstructures.superstructures.

5 7 (a) (b) (c) 6 4 5

3 4 Level Level

3 2 2

1 1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.00.10.20.30.40.50.6 Drift [%] Drift [%] row data mean mean ± std

Figure 5. Distribution in in height height of of the the maximum maximum inter-st inter-storyory drift ratio obtained from from NLTH analyses for each base-isolated conﬁguration:configuration: three-story ( a), ﬁve-storyfive-story (b) and seven-story buildingbuilding ((cc).).

Figure6 6 reports reports the the distribution distribution of of story story shear shear V i Vnormalizedi normalized with with respect respect to the to meanthe mean of the of base the baseshear shear Vb obtained Vb obtained from from NLTH NLTH analysis. analysis. Since the actual hysteretic response of the vari variousous BI-structures depend on the different different seismic inputs, the the values values and and distribution distribution of of the the lateral lateral shear shear forces forces may may vary vary significantly. significantly. As Asshown shown in the in graphsthe graphs of the of first the column first column in Figure in Figure 6, when6, the when BI structure the BI structure has a low has equivalent a low equivalent damping dampingvalue, the value,envelopes the envelopesof shear offorces shear for forces each for selected each selected ground ground motion motion (and, (and, consequently, consequently, the the average distribution represented by the bold line) present an almost linear distribution as as a result of a base isolation system. By By increasing increasing the the damping damping ratio ratio of of the the isolation isolation systems, systems, more more seismic seismic force is transferred to the superstructure. The The contribution contributionss of of the the higher modes become significant significant in the upper levels so the normalized lateral shear envelope is more bulgedbulged (see(see thirdthird columncolumn ofof FigureFigure6 6).). Moreover, althoughalthough the the accelerograms accelerograms have have been been scaled scaled to be compatibleto be compatible with a selected with a spectrum, selected spectrum,earthquake earthquake excitation excitation characteristics characteristics and intensity and aintensityffect the affect seismic the response seismic response of the base of the isolated base isolatedbuildings buildings as well. Theas well. first The reason first is reason the significant is the significant difference difference in low- andin low- high- and period high-energy period contentenergy contentof the scaled of the spectrum. scaled spectrum. In fact, some In fact, of thesome scaled of th grounde scaled motions ground havemotions a large have amount a large of amount energy of in energythe high-frequency in the high-frequency range and thisrange can and trigger this thecan higher-modestrigger the higher-modes response. The response. second reasonThe second is the reasonequivalent is the damping equivalent ratio. damping For low ratio. damping For low values, damping in a classically values, in damped a classically vibrating damped linear vibrating system, linear system, all modes tend to be mutually orthogonal [2]. In these cases, the dynamic response becomes first-mode dominated and the influence of the seismic input becomes negligible. Buildings 2020, 10, 197 9 of 20 all modes tend to be mutually orthogonal [2]. In these cases, the dynamic response becomes first-mode dominatedBuildings 2020 and, 10, xthe FOR influence PEER REVIEW of the seismic input becomes negligible. 9 of 20

2 Level 1 ξ = 0.065 ξ = 0.127 ξ = 0.321

Tis = 2.0 s Tis = 1.5 s Tis = 2.5 s 0 0 0.25 0.5 0.75 1 1.25 1.5 0 0.25 0.5 0.75 1 1.25 1.5 0 0.25 0.5 0.75 1 1.25 1.5 V i / V b (a) three-story building 5

3 Level 2

1 ξ = 0.052 ξ = 0.128 ξ = 0.241 Tis = 2.0 s Tis = 1.5 s Tis = 2.5 s 0 0 0.25 0.5 0.75 1 1.25 1.5 0 0.25 0.5 0.75 1 1.25 1.5 0 0.25 0.5 0.75 1 1.25 1.5

V i / V b (b) five-story building 7

Level 3

1 ξ = 0.053 ξ = 0.136 ξ = 0.273 Tis = 1.5 s Tis = 2.0 s Tis = 2.5 s 0 0 0.25 0.5 0.75 1 1.25 1.5 00.250.50.7511.251.5 0 0.25 0.5 0.75 1 1.25 1.5

V i / V b (c) seven-story building

FigureFigure 6. 6. StoryStory shear shear at each at eachlevel levelobtained obtained from the from NLTH the analyses NLTH analysesunder the under seven thespectrum- seven spectrum-compatiblecompatible ground motions ground for motions different for base-isolated diﬀerent base-isolated configurations: conﬁgurations.

3.2.3.2. Equivalent Distribution of Lateral Forces TheThe equivalentequivalent static story forces resulting fromfrom thethe LSALSA codecode proceduresprocedures areare comparedcompared withwith thosethose obtained from from NLTH NLTH analysis. analysis. According According to toEuroCode EuroCode 8 and 8 and Italian Italian Building Building Code Code [17,34], [17, 34the], theLSA LSA equivalent equivalent lateral lateral forces forces are distributed are distributed over overthe height the height of the of building the building in proportion in proportion to the to story the storymasses. masses. Conversely, Conversely, in US codes in US a codesdifferent a di lateralfferent pattern lateral is pattern assigned is assignedbased on basedan inverted on an triangular inverted triangulardistribution distribution of story accelerations. of story accelerations. The results The of resultsthe NLTH of the analysis NLTH analysisexecuted executed in this study in this can study be canutilized be utilized to assess to the assess actual the accuracy actual accuracy of the two of thedistributions two distributions described described by Equation by Equation(1) and Equation (1) and Equation(2) in terms (2) of in lateral terms force of lateral distribution force distribution employed employed in LSA. Specifically, in LSA. Specifically, the maximum the maximum values of valuesVi and ofFi, Vanalyticallyi and Fi, analytically defined by defined Equations by Equations (5) and (6), (5) andrespectively, (6), respectively, can be determined can be determined for each for NLTH each NLTHanalysis analysis and the and corresponding the corresponding average average value value can canbe bedetermined determined for for the the seven seven accelerograms consideredconsidered inin thisthis study.study. TheThe basebase shearshear VVbb = VVi|i=1i|i= 1cancan also also be be determined. determined. TheThe diagramsdiagrams reportedreported inin FigureFigure7 7show show (on (on the the x-axes) x-axes) the the ratios ratios V Vi/iV/Vbb (left(left column)column) andand FFi/i/VVbb (right(right column)column) forfor thethe threethree buildingsbuildings withwith 3,3, 5,5, andand 77 stories.stories. It is worth worth highlighting that the resulting FFii/V/Vb gives a clear picture of the resulting seismicseismic laterallateral forceforce distribution,distribution, whichwhich cancan bebe deducteddeducted fromfrom NLTHNLTH analyses. analyses. The The dashed dashed and theand dash-dotted the dash-d linesotted reported lines reported in the same in diagramsthe same correspond, diagrams correspond, respectively, to the uniform acceleration pattern of Equation (1) and the inverted triangular distribution of Equation (2). Buildings 2020, 10, 197 10 of 20

Buildingsrespectively, 2020, 10 to, x theFOR uniform PEER REVIEW acceleration pattern of Equation (1) and the inverted triangular distribution10 of 20 of Equation (2).

ξ = 0.194 ξ = 0.194 1.00 1.00 ξ = 0.031 ξ = 0.031 0.80 0.80

tot 0.60 0.60 / h i 0.40 0.40 h ξ = 0.031 ξ = 0.065 0.20 ξ = 0.110 0.20 ξ = 0.155 ξ = 0.194 0.00 0.00 00.20.40.60.81 0 0.1 0.2 0.3 0.4 0.5 V i / V b F i / V b (a) three-story building

1.00 ξ = 0.190 1.00 ξ = 0.190 ξ = 0.048 ξ = 0.048 0.80 0.80

tot 0.60 0.60 / h / h i

h 0.40 ξ = 0.048 0.40 ξ = 0.088 0.20 ξ = 0.130 0.20 ξ = 0.145 ξ = 0.190 0.00 0.00 00.20.40.60.81 0 0.1 0.2 0.3 0.4 0.5 V i / V b F i / V b (b) five-story building

1.00 ξ = 0.160 1.00 ξ = 0.160

ξ = 0.033 ξ = 0.033 0.80 0.80

tot 0.60 0.60 / h / h i

h 0.40 0.40 ξ = 0.033 0.20 ξ = 0.055 0.20 ξ = 0.110 ξ = 0.160 0.00 0.00 00.20.40.60.81 0 0.1 0.2 0.3 0.4 0.5

V i / V b F i / V b (c) seven-story building

triangular distribution uniform distribution

FigureFigure 7 7.. StoryStory shear shear envelopes envelopes (left) (left )and and lateral lateral force force distributions distributions (right (right) )obtained obtained from from the the NLTH NLTH analysisanalysis compared compared with with the the equivalent equivalent distributi distributionon employed employed for for the the linear linear static static analysis analysis (LSA) (LSA) (uniform(uniform and and inverted inverted triangular). triangular).

TheThe synoptic synoptic view view of of the diagrams depicteddepicted inin FigureFigure7 demonstrates7 demonstrates that that the the distributions distributions of theof themaximum maximum story story shear-forces shear-forces obtained obtained from NLTHfrom analysisNLTH stronglyanalysis divergestrongly from diverge the conventional from the conventionaluniform distribution uniform distribution as the equivalent as the damping equivalent in damping the isolators in th increases.e isolators A increases. marked bulgeA marked at the bulgemid-height at the ofmid-height the building, of the due building, to the higher-mode due to the higher-mode contribution, contributi is observedon, with is observed a relatively with high a relativelyvalue of ξhigh. Compared value of with ξ. theCompared results of with the non-linear the results analysis, of the thenon-linear uniform distributionanalysis, the approach uniform is distribution approach is substantially unconservative. Conversely, the inverted triangular distribution shows a relatively good correspondence with the shear at the higher stories for medium- high damping values although it is too conservative for the lower stories. Buildings 2020, 10, 197 11 of 20 substantially unconservative. Conversely, the inverted triangular distribution shows a relatively good correspondence with the shear at the higher stories for medium-high damping values although it is too conservative for the lower stories. The results of NLTH analysis conﬁrm that both the uniform and the inverted triangular distributions are unable to characterize the dynamic behavior of base-isolated building. In fact, the uniform distribution underestimates the maximum shear forces, especially at the upper stories of the building, even for relatively low equivalent damping ratio values. On the other hand, the inverted triangular distribution results too conservative for the typical values of common applications (ξ < 0.20). However, for very high value of equivalent damping ratio (ξ > 0.30) the triangular distribution leads to an underestimation of lateral force at the top and the mid height of the building. Therefore, for typical values of ξ the average of nonlinear results under ground motion records shows that the real distribution falls between these two extremes.

3.3. Effective Height To better understand the dynamic behavior of an isolated building, the effective height can return an appropriate indication about the shear force profile, providing the location of the resultant equivalent force along the height of the building [37,41,49]:

PNstory = Fihi h = i 1 (7) eﬀ PNstory j=1 Fj

Figure8 presents the e ffective height related to 3-, 5- and 7-story buildings. It is compared with that calculated from uniform and inverted triangular distribution. The different configurations are described in terms of equivalent damping ratio of the isolation systems. Figure8 confirms that the dynamic response of base-isolated buildings significantly depends on the degree of non-linearity of the isolation system. For the three superstructures, higher values of the equivalent damping ratio cause an increase of the effective height: the resulting horizontal force is applied at a higher point due to the contribution of higher modes that increase the shear forces at the top level. The effective height for the uniform distribution is equal to 0.436 h, 0.460 h and 0.471 h for the 3-, 5-, and 7-story building respectively, while for the inverted triangular distribution it is equal to 0.724 h, 0.693 h and 0.682 h, in the same order. The values highlight that the uniform distribution of lateral forces over the height of the three structures always underestimates the inertial forces at the upper levels. By contrast, the inverted triangular distribution provides conservative values of the shear forces for values of the damping ratio smaller than 0.30 for 3-story buildings, 0.20 for the 5-story buildings and 0.16 for the 7-story buildings. It should be noted that structures with more levels and a longer period in fixed-base configurations show effective heights even greater than the values associated with a uniform distribution. Indeed, the shear envelope is closer to a triangular distribution than to the uniform one, even for small values of equivalent damping ratio. Nevertheless, for relatively high values of ξ also the triangular distributions are not suitable. The greater effective height implies the base shear distribution is more bulged than triangular distribution. For completeness, if the damping ratio tends to extreme values, non-linear analyses provide anomalous values of the effective height that becomes greater than the total height. Buildings 2020, 10, 197 12 of 20 Buildings 2020, 10, x FOR PEER REVIEW 12 of 20

1.2 1.1 1 0.9 0.8 0.7

height 0.6 0.5 0.4 0.3 Effective 0.2 0.1 0 0.03 0.03 0.04 0.06 0.08 0.08 0.10 0.11 0.12 0.13 0.13 0.14 0.16 0.16 0.18 0.19 0.19 0.21 0.22 0.25 0.26 0.26 0.30 0.31 0.32 0.34 0.40 0.41 0.42 0.43 0.50 ξ [-]

(a) three-story building 1.2 1.1 1 0.9 0.8 0.7 0.6 height 0.5 0.4 0.3

Effective 0.2 0.1 0 0.01 0.01 0.01 0.03 0.04 0.04 0.05 0.05 0.06 0.06 0.07 0.08 0.09 0.09 0.10 0.11 0.11 0.12 0.13 0.14 0.15 0.15 0.17 0.18 0.19 0.20 0.24 0.24 0.25 0.25 0.30 0.33 0.34 0.40 0.41 ξ [-]

(b) five-story building 1.2 1.1 1 0.9 0.8 0.7 0.6

height 0.5 0.4 0.3

Effective 0.2 0.1 0 0.01 0.02 0.02 0.02 0.02 0.03 0.04 0.04 0.05 0.05 0.06 0.06 0.07 0.07 0.07 0.08 0.09 0.10 0.10 0.11 0.12 0.13 0.14 0.16 0.16 0.17 0.17 0.20 0.23 0.23 0.27 0.28 0.38 ξ [-]

FigureFigure 8. 8. NormalizedNormalized effective eﬀective height height of of the the base base isolated isolated configurations. conﬁgurations.

Buildings 2020, 10, 197 13 of 20

4. Discussion The alternative lateral force distributions, available in the literature for the base isolated buildings, are compared to those used by building codes (uniform and inverted triangular) and to the shear force proﬁles obtained from NLTHA, in order to investigate their applicability and accuracy.

4.1. Comparison between Different Lateral Distribution As reported in Section1, most of the di fferent formulae, obtained for BI structures, are functions of the non-linearity properties of the IS and superstructure to adapt to various configurations of BI Systems [43,44,46,47]. As seen in Section3, a proper prediction can be obtained only by including the IS characteristics in the lateral force distributions. In addition, since the main problem in the triangular distribution is that the entire base shear is transferred to the superstructure and that the story force at the base level is equal to zero, some distributions consider the influence of the inertia of the base floor [41–43,46,48]. Indeed, ignoring this story force in distributing the base-shear forces leads to overestimating the forces at higher levels and overturning the moment on the structure. Therefore, some authors propose distributions given by a combination of different modes of vibration or by the union of uniform and triangular shear profiles, while others include concentrated force at the base level. In particular, Khoshnoudian et al. [49] and Cardone et al. [47] propose a distribution given by the combination of the first three simplified modal shape accounting for the superstructure period of vibration and for the non-linearity of the isolation system. Khoshnoudian et al. [48], Lee et al. [41], and Tsai et al. [42] proposed a combination of the uniform and linear distributions not considering the IS characteristics but only the superstructure properties. Since the last formulae, developed for configurations with almost linear isolation system and isolation periods well-separated from the superstructure period, provide similar values of story shear, in the following, only the Lee’s formula will be considered as reference. In the non-linear distribution proposed by Ryan et al. [43], York et al. [46] and Andriono et al. [44], the exponent “p” [44] or “k” [43], depending on IS parameters, predict with more accuracy the bulging shape of the shear envelopes due to the several damping values. In fact, the predicted superstructure force distribution is strongly influenced by effective damping: as the damping increases, the exponent value becomes greater and the lateral shear envelope becomes more bulged. For completeness, the distribution is nearly uniform (exponent tends to zero) for a lightly damped system and more than linear (exponent is greater than the unit) for a highly damped system. For the sake of simplicity, the improved force distributions are compared to the existing force distribution used by the codes in Figure9. The comparison is carried out for the 3-, 5- and 7-story prototype buildings, equipped with different isolation systems. In order to make it more meaningful, any of these IS parameters are compatible with those investigated by other authors in their studies. The charts in Figure9 refer to a 3-story (a), a 5-story (b) and a 7-story (c) buildings. For every type of superstructure two different configurations of IS are considered, the first one with a low value of the equivalent damping ratio, the second one with a medium-to-high value. As can be observed, for the base isolated configurations analyzed in this study, only some formulae can accurately predict the effective distribution of the inertial forces over the height of the superstructure. In their studies, Lee et al. [41] and Tsai et al. [42], for instance, refer to an equivalent viscous-elastic model that describes the isolation system with low degree of non-linearity. For this reason, their formulae underestimate the shear values at the upper level, especially for highly damped systems. The lateral force distribution proposed by Andriono and Carr [44,45], instead, tends to overestimate the shear force at the different levels because at the base story the inertial force is equal to 0. Also the first formula developed by Khoshnoudian et al. [48] provides values too conservative for slightly damped systems. It becomes even less accurate for the configurations with a greater number of stories. For a highly damped system, by contrast, it predicts unsafe values of the story Buildings 2020, 10, 197 14 of 20

Buildings 2020, 10, x FOR PEER REVIEW 14 of 20 shears. The same considerations hold also for the second formula [49]. For the structures under consideration,shears. The same the considerations formula by Cardone hold also et al. for [47 th]e does second not formula generally [49]. lead For to accuratethe structures predictions. under Conversely,consideration, more the recent formula formulations by Cardone [43 ,et46 al.] are [47] able does to predict not generally very accurately lead to accurate the shear predictions. force at the diConversely,fferent levels more of the recent superstructure. formulations The [43,46] distribution are able proposedto predictby very them accurately fits theshear the shear profile force even at the for highdifferent values levels of equivalent of the superstructu coefficientre. damping The distribution ratio. However, proposed while by for them three- fits and the five-story shear profile structures even afor perfect high overlapvalues of can equivalent be observed coefficient between thedamping results ratio. of the However, NLTH analysis while and for that three- of the and LSA, five-story for the seven-storystructures a buildingperfect overlap their distribution can be observed is less accuratebetween andthe results underestimates of the NLTH the shear analysis forces and at that the upperof the levels.LSA, for These the resultsseven-story may restrict building their their applicability distribution for buildingis less accurate with a number and underestimates of stories in accordance the shear withforces the at internationalthe upper levels. regulations These results [17,20 ].may restrict their applicability for building with a number of stories in accordance with the international regulations [17,20].

3 3

KHOSHNONDIKhoshnoudian et al. [48] AN et al.2 CARDONECardone et et al. al. [47]

YORKYork et etal. al. [46] 2 2 RYANRyan et et al. al. [43]

ANDRIONOAndriono et et al. [44] al. KHOSHNONDIKhoshnoudian et al. [49] Level AN et al.1 LEELee et etal. al. [41] 1 1

0 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.00.20.40.60.81.01.2

Vi / Vb Vi / Vb (a) three-story building 5 5

KHOSHNONDIANKhoshnoudian et al. [48] et al.2 4 4 CARDONECardone et et al. al. [47] YORKYork et et al. al. [46] RYANRyan et et al. al. [43] 3 3 ANDRIONOAndriono et et al. al. [44]

KHOSHNONDIANKhoshnoudian et al. [49] Level et al.1 2 2 LEELee et et al. al. [41]

1 1

0 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

Vi / Vb Vi / Vb (b) five-story building 7 7

KHOSHNONDIA 6 6 NKhoshnoudian et al.2 et al. [48] CARDONECardone et et al. al. [47] 5 5 YORKYork et et al. al. [46] RYAN et al. 4 4 Ryan et al. [43] ANDRIONOAndriono et et al. al. [44]

Level 3 3 KHOSHNONDIAKhoshnoudian et al. [49] N et al.1 LEELee et et al. al. [41] 2 2

1 1

0 0 0.00.20.40.60.81.01.20.0 0.2 0.4 0.6 0.8 1.0 1.2

Vi / Vb Vi / Vb (c) seven-story building mean ASCE 7-10 [20] EuroCode [17]

FigureFigure 9. ComparisonComparison between between the the several several lateral lateral force force distributions distributions available available in the inthe literature literature for forthe theprototype prototype BI buildings BI buildings of ofthree-story three-story (a), (a ),five-story ﬁve-story (b (b),), seven-story seven-story ( (cc)) equipped equipped with didifferentﬀerent isolationisolation systems.systems.

The effective height is a parameter that makes the shape of the lateral force distribution immediately understandable. Figure 10 shows a comparison in terms of effective heights for a configuration of each prototype building considering similar values of the equivalent damping ratio. Buildings 2020, 10, 197 15 of 20

Buildings 2020, 10, x FOR PEER REVIEW 15 of 20 The eﬀective height is a parameter that makes the shape of the lateral force distribution immediately

understandable.(a) Effective Figure height 10 shows a comparison(b) inEffective terms height of eﬀective heights(c) for aEffective conﬁguration height of each

BuildingsKhoshnoudianKhoshnondian 2020et al. [48]et al.2, 10, x FOR PEER REVIEW Khoshnoudian et al. [48] Khoshnoudian et al. [48] 15 of 20 prototypeCardoneCardone et al. [47] et building al. considering similarCardone valuesCardone et al. et [47] al. of the equivalent dampingCardoneCardone etratio. al. et [47] al. York et Yorkal. [46] et al. York et al. [46] York et al. [46] Ryan etRyan al. [43] et al. RyanRyan et al. et [43] al. RyanRyan et al. et [43] al. AndrionoAndriono et al. [44] et al. Andriono et al. [44] Andriono et al. [44] KhoshnoudianKhoshnondian et(a) al. [49]et al.1 Effective height KhoshnoudianKhoshnondian(b) et al. et [49]al.1 Effective height KhoshnoudianKhoshnondian(c) et al. et [49]al.1 Effective height Lee et al.Lee [41] et al. Lee et al. [41] Lee et al. [41] KhoshnoudianKhoshnondianASCE 7-10 ASCEet al. [20] 7-10[48]et al.2 KhoshnoudianASCEASCE 7-10 et al.7-10 [20] [48] KhoshnoudianASCEASCE 7-10 et al.7-10 [20] [48] CardoneEurocodeCEN 98;Cardone et NTC 8al. [17] [47]2018 et al. CardoneEurocodeCardone et 8al. [17]et [47] al. CardoneEurocodeCardone et 8al. [17]et [47] al. York Medianet Yorkal.Median [46] et al. York Medianet al. [46] York Medianet al. [46] Ryan etRyan al. [43] et al. RyanRyan et al. et [43] al. RyanRyan et al. et [43] al. AndrionoAndriono et al. [44] et al.0 0.2 0.4 0.6 0.8 Andriono et al. [44]0 0.2 0.4 0.6 0.8 Andriono et al. [44]0 0.2 0.4 0.6 0.8 KhoshnoudianKhoshnondian et al. [49]et al.1 KhoshnoudianKhoshnondian et al. et [49]al.1 KhoshnoudianKhoshnondian et al. et [49]al.1 Lee et al.Lee [41] et al. Lee et al. [41] Lee et al. [41] 3 storey building; ξ = 0.127 5 storey building; ξ = 0.147 7 storey building; ξ = 0.126 ASCE 7-10ASCE [20] 7-10 ASCEASCE 7-10 7-10 [20] ASCEASCE 7-10 7-10 [20] EurocodeCEN 98; NTC 8 [17] 2018 Eurocode 8 [17] Eurocode 8 [17] MedianMedian Median Median

Figure 10.0 Effective 0.2 height 0.4 of 0.6 the 0.8different lateral0 distribution 0.2 0.4 of 0.6 shear 0.8 force available0 in 0.2 literature 0.4 for 0.6 0.8 three different base-isolated configurations: three-story (a), five-story (b) and seven-story building 3 storey building; ξ = 0.127 5 storey building; ξ = 0.147 7 storey building; ξ = 0.126 (c). Figure 10. EffectiveEffective height of the different different lateral distribution of shear force available in literature for Bythree comparing didifferentfferent base-isolated base-isolatedthe effective configurations: configurations: heights, for three-storymedium three-story values (a (),a), five-story five-story of equivalent (b )(b and) and seven-story damping seven-story buildingratio, building it (canc). be noted( cthat). the distribution proposed by Lee et al. [41] provides an underestimation of the effective By comparing the effective heights, for medium values of equivalent damping ratio, it can be noted height, while the formulations suggested by Khoshnoudian et al. [49], York at al. [46] and Andriono that theBy comparing distribution the proposed effective by heights, Lee et al.for [ 41medium] provides values an underestimationof equivalent damping of the e ratio,ffective it height,can be et al. [44] overestimate the shear forces at the upper levels, especially in the cases of three- (a) and notedwhile thethat formulations the distribution suggested proposed by Khoshnoudian by Lee et al. [41] et al. provides [49], York an atunderestimation al. [46] and Andriono of the eteffective al. [44] five-story (b) buildings. It can be observed again that the equivalent distribution of static forces height,overestimate while the formulations shear forces atsuggested the upper by levels, Khoshnoudian especially et in al. the [49], cases York of at three- al. [46] (a) and Andriono five-story proposed by Ryan et al. [43] better predicts the seismic response of the base-isolated buildings. et(b) al. buildings. [44] overestimate It can be the observed shear forces again thatat the the uppe equivalentr levels, distributionespecially in of the static cases forces of three- proposed (a) and by Ryan et al. [43] better predicts the seismic response of the base-isolated buildings. 4.2.five-story Statistical (b) Analysis buildings. and ItConsiderations can be observed again that the equivalent distribution of static forces proposed by Ryan et al. [43] better predicts the seismic response of the base-isolated buildings. 4.2.The Statistical results Analysis of the NLTH and Considerations analysis have demonstrated that, as expected, the superstructure force distribution4.2. StatisticalThe results is Analysisstrongly of the and NLTHinfluenced Considerations analysis by the have effectiv demonstratede damping. that, In fact, as expected, an apparent the correlation superstructure between force the damping ratio and the shape of the shear envelope can be observed in the previous charts (Figures distributionThe results is strongly of the NLTH influenced analysis by the have eff ectivedemonstrat damping.ed that, In fact,as expected, an apparent the superstructure correlation between force 7 and 8). distributionthe damping is ratiostrongly and influenced the shape by of the the effectiv sheare envelope damping. can In fact, be observed an apparent in thecorrelation previous between charts Figure 11 shows the correlation between the relative error of base shear distribution, obtained the(Figures damping7 and ratio8). and the shape of the shear envelope can be observed in the previous charts (Figures from a uniform distribution formula and non-linear dynamic analysis results, and the equivalent 7 andFigure 8). 11 shows the correlation between the relative error of base shear distribution, obtained from damping ratio. For the sake of simplicity, the relative error is reported in terms of story shear force a uniformFigure distribution 11 shows the formula correlation and non-linear between the dynamic relative analysis error of results, base shear and the distribution, equivalent dampingobtained (Equation 8): fromratio. a For uniform the sake distribution of simplicity, formula the relative and errornon-linear is reported dynamic in terms analysis of story results, shear and force the (Equation equivalent (8)): i i damping ratio. For the sake of simplicity, theF relatiNLTHAve -error F LSA is reported in terms of story shear force ∆Fi = i i (8) i F NLTHAi F LSA (Equation 8): ∆F = F LSA− (8) Fi i LSA i where i indicates the ith level and FNLTH and FLSAF represent, - F respectively, the peak of the story shear ∆Fi = NLTHA LSA (8) forcewhere obtained i indicates from the NLTHA ith level and FtheNLTH staticand force FLSA providedrepresent,i by respectively, the uniform the distribution peak of the for story the shear i-th F LSA level.force obtained from NLTHA and the static force provided by the uniform distribution for the i-th level. where i indicates the ith level and FNLTH and FLSA represent, respectively, the peak of the story shear th force4.0 obtained from NLTHA and(a) the5.0 static force provided by(b) the uniform4.0 distribution for the(c) i- level.3.0 4.0 3.0 Level 7 3.0 2.0 2.0 Level 5 Level 6 2.0 1.04.0 Level(a) 3 5.0 Level(b) 4 1.04.0 Level(c) 5 1.0 F [-] F [-] F [-] Δ Δ 0.03.0 Level 2 4.0 Level 3 Δ 0.03.0 Level 4 0.0 Level 7 Level 1 3.0 Level 2 Level 3 -1.02.0 -1.02.0 -1.0 Level 5 Level 6 Level 0 2.0 Level 1 Level 2 -2.01.0 Level 3 -2.0 Level 4 -2.01.0 Level 5 1.0 Level 0 Level 1 F [-] F [-] F [-] Δ Δ Level 2 Level 3 Δ Level 4 -3.00.0 -3.0 -3.00.0 Level 0 0.0 0.00 0.20 0.40 0.60 Level 1 0.00 0.20 0.40 0.60 Level 2 0.00 0.20 0.40 0.60 Level 3 -1.0 -1.0 ξ [-] -1.0 ξ [-] Level 0 ξ [-] Level 1 Level 2 -2.0 -2.0 -2.0 Level 0 Level 1 -3.0Figure 11. Relationship between relative-3.0 error of shear force at the varying-3.0 levels and the equivalent Figure 11. Relationship between relative error of shear force at the varying levels and the equivalent Level 0 damping0.00 0.20 ratio for 0.40 three-story 0.60 (a), five-story0.00 (b 0.20), and seven-story0.40 0.60 (c) buildings.0.00 0.20 0.40 0.60 damping ratioξ [-]for three-story (a), five-story (b), andξ [-] seven-story (c) buildings. ξ [-]

Figure 10 shows an almost linear correlation between the relative error of shear forces and the FigureFigure 11.10 showsRelationship an almost between linear relative correlation error of shearbetw forceeen the at the relative varying error levels of and shear the forcesequivalent and the eﬀective damping. Passing through the origin, the relationship highlights the suitability of the uniform effectivedamping damping. ratio for Passing three-story through (a), five-story the origin, (b), and the seven-story relationship (c) buildings. highlights the suitability of the distribution for value of ξ near zero and the inadequacy to describe the seismic response of BI structures uniform distribution for value of ξ near zero and the inadequacy to describe the seismic response of evenFigure for relatively 10 shows low an values almost of linear the eﬀ ectivecorrelation damping. between the relative error of shear forces and the BI structures even for relatively low values of the effective damping. effective damping. Passing through the origin, the relationship highlights the suitability of the It can be observed that the slope of the curves is steeper for the upper levels. The negative slopes uniform distribution for value of ξ near zero and the inadequacy to describe the seismic response of of the lower levels, instead, underline that the uniform distribution overestimates the shear force of BI structures even for relatively low values of the effective damping. these levels while underestimating those of the upper levels with the increase of the damping. It can be observed that the slope of the curves is steeper for the upper levels. The negative slopes of the lower levels, instead, underline that the uniform distribution overestimates the shear force of these levels while underestimating those of the upper levels with the increase of the damping. Buildings 2020, 10, 197 16 of 20

It can be observed that the slope of the curves is steeper for the upper levels. The negative slopes of the lower levels, instead, underline that the uniform distribution overestimates the shear force of these levels while underestimating those of the upper levels with the increase of the damping. Specifically, for slightly and moderately damped systems, the effective shear envelopes fall between the two distributions recommended by the building codes, though it should be noted that the inverted triangular distribution is unconservative for highly damped systems, which tends to increase the superstructure forces. For this reason, in order to provide an accurate lateral distribution of base shear over the height of superstructure for LSA procedures and preserve the intrinsic simplicity and practicality of the method, a simplified formula is proposed by Equation (9), given by the combination of the two formulae used by building codes [15,18]:

mihi mi Fi = δ V + (1 δ) V (9) PN − PN j=1 mjhj j=1 mj

The accuracy of this force distribution can be related to an easily interpretable variable: the effective height of the shear forces. Figure8 has documented a strong correlation between the e ffective height and the degree of non-linearity of the IS, expressed in terms of equivalent damping ratio ξ. For practical purpose, a linear correlation between the effective height and ξ can be detected:

heff = heff,u + θ ξ (10) where the heff, u is the effective height assessed for uniform distribution. Thus, the δ parameters can be derived as a function of the equivalent damping ratio in order to provide an appropriate effective height for the force distribution. Since the effective height of the proposed distribution can be easily expressed by Equation (11), by equalizing Equations (10) and (11), δ can be derived from Equation (12):

h = h (1 δ) + h δ (11) eff eff,u − eff,t where heff,t is the effective height assessed for triangular distribution.

θ δ = ξ (12) h h eﬀ,u − eﬀ,t

In an idealized configuration, where the story mass is the same at the different levels, heff,u and heff,t reduces to 0.50 htot and 0.67 htot, respectively. Conversely, the expression of h in Equation (11) covers the case of more general distribution of lateral forces resulting from the variability of relevant structural parameters. As shown in Figure8, the θ coefficient, instead, depends on the number of stories of the superstructure. It assumes values ranging from 0.93 (for 3-story buildings) to 1.34 (for 7-story buildings).

5. Conclusions This study investigated the validity of linear static analysis (LSA) for BI-structures. Specifically, equivalent lateral force distributions recommended in structural building codes and proposed in the literature have been examined to evaluate realistic seismic loads in the preliminary analysis and design of seismic-isolated structures using simplified procedures. Based on the comparisons between the story-shear envelopes, obtained from NLTH analyses and the corresponding shear profiles proposed for LSA, considering the different lateral force distributions, the following conclusions can be drawn: Buildings 2020, 10, 197 17 of 20 i. the lateral force distribution currently recommended in EuroCode and NTC 2018, which neglects the contribution of higher modes, significantly underestimates the shear forces at the upper levels of the superstructures, even when the IS exhibits weakly non-linear response; ii. the lateral force distribution proposed by ASCE 7–10 provides results that are too conservative compared with those of dynamic analyses for low and medium equivalent damping ratios; iii. the degree of non-linearity of the isolation system strongly influences the seismic response of the base-isolated buildings. As the equivalent damping ratio increases, the shear envelope increasingly bulges because of more significant higher mode effects; iv. some formulations available in the literature provide more accurate predictions of the peak seismic forces throughout the height of buildings and hence of the lateral force distributions because of their explicit dependence on IS parameters; v. an accurate vertical distribution can be achieved as a function of the relevant parameters of the superstructures and isolation systems; vi. a simplified formula for the vertical distribution of the base shear, combining both uniform and linear distributions, is proposed as a function of the equivalent damping ratio (Equation (9)); vii. the formula provides slightly conservative seismic story forces, resulting in a more economic design compared to the procedure of the ASCE and in a safer method than that proposed in NTC 2018 for buildings that comply with the codes’ limitation.

Finally, the simplicity of the proposed formulation for determining lateral force distribution in LSA makes them a valid alternative to more computationally intensive NLTH analyses, with a significant reduction in time and an easier way to check the physical consistency of the analysis results. However, further research is needed to verify the general applicability of the proposed equations for different configurations of base-isolated buildings, including different structural types (e.g., shear-wall systems, masonry building), and different distributions of structural stiffness over height, as well as for a more diverse set of seismic records.

Author Contributions: Conceptualization, G.B. and E.M.; Data curation, A.Z.; Funding acquisition, F.F.; Investigation, A.Z.; Methodology, F.F. and G.B.; Resources, E.M.; Supervision, G.B.; Validation, E.M.; Visualization, A.Z.; Writing—original draft, A.Z.; Writing—review & editing, F.F. and E.M. All authors have read and agreed to the published version of the manuscript. Funding: The first author’s Ph.D. scholarship is granted by the Italian Ministry for Education, University and Research (MIUR) as part of the program “Dottorati Innovativi a caratterizzazione industriale”, ID DOT 1328490-2, funded by the European Union (Structural Funding ERDF-ESF for “Research and Innovation” 2014-2020). The authors are grateful for the support. Conflicts of Interest: The authors declare no conflict of interest.

Abbreviations

Fi equivalent lateral force at the i-th level mi story mass of the i-th floor Vb total design base shear force N number of stories hi, hj height of the i-th and j-th stories -from the base level Vi shear force at the i-th level/median peak story shear at the i-th level G1,G2 self-weight and permanent loads, respectively Qk live loads Ms mass of the superstructure Mb mass of the base level Tbf fundamental period of vibration of the superstructure assumed fixed at the base Teq effective fundamental period of the base isolated building K1,K2 initial stiffness of the isolators, post yield stiffness α stiffness ratio K2/k1 - post yield hardening Q intersection force of hysteresis cycle with vertical axis Buildings 2020, 10, 197 18 of 20

υ friction coefficient of sliding bearings or friction pendulum bearings (FPB) W weight supported by the bearings R effective curvature radius of the concave surface of FPB Xy,Fy yield displacement and force of the isolators Xd design displacement demand of the IS effective stiffness center µ ductility ratio Xd/Xy of the isolators Keff effective stiffness of the isolation system at a displacement Xd ξ equivalent damping ratio Tr reference return period of the design spectrum ag reference peak ground acceleration on type A ground g acceleration of gravity F0 amplification factor of the design spectrum Tc* period at the end of the constant acceleration branch of the elastic spectrum Se(T) elastic horizontal ground acceleration response spectrum SDe(T) elastic displacement response spectrum heff effective height heff,u effective height for the uniform lateral forces distribution heff,t effective height for the inverted triangular lateral forces distribution

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