a displacement-based design method for seismic retrofit of rc...

Research ArticleA Displacement-Based Design Method for Seismic Retrofit of RCBuildings Using Dissipative Braces

Massimiliano Ferraioli and Angelo Lavino

Department of Engineering, University of Campania “Luigi Vanvitelli”, Via Roma 29, 81031 Aversa (CE), Italy

Correspondence should be addressed to Massimiliano Ferraioli; [email protected]

Received 3 August 2018; Revised 5 November 2018; Accepted 30 November 2018; Published 27 December 2018

Academic Editor: Junwon Seo

Copyright © 2018 Massimiliano Ferraioli andAngelo Lavino.This is an open access article distributed under theCreativeCommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

The paper proposes a displacement-based design method for seismic retrofit of RC buildings using hysteretic dissipative braces. Atfirst, a fully multimodal procedure based on an adaptive version of the capacity spectrummethod is applied to the 3Dmodel of thedamped braced structure. Then, the properties of an idealized bilinear model are defined using the seismic characteristics of thecompound system thus accounting for the frame-damped brace interaction. Finally, an iterative procedure is developed to providean optimal distribution of dampers. The proposed method overcomes the limitations of the design procedures in the literaturethat generally neglect the frame-damped braces interactions. Moreover, it addresses the main issues of seismic design of dampedbraces: effect of force demands applied to the frame due to the damper yielding and strain hardening, higher modes contribution,effect of soft-storey irregularities, and torsion effect in asymmetric buildings.The proposed design procedure is first validated usingnonlinear static and dynamic analyses of a numerical example. Then, it is implemented to a real case study of a RC school buildingto assess its applicability in current practice.

1. Introduction

Experience from recent severe earthquakes has confirmedthat RC buildings designed without earthquake-resistancerequirements (precode structures) or following outdatedstructural codes are extremely vulnerable to seismic exci-tation. From literature review, considerable effort has beenmade to evaluate the seismic vulnerability of structures onboth global and local levels [1–5]. This is especially true forpublic buildings such as hospitals or schools, whose seismicresistance is very important in view of the consequencesassociated with failure [6–9]. Low-rise RC school buildingsoften have large windows with a long rectangular floorplan thus producing a seismically weak direction alongthe internal corridor. Furthermore, the buildings designedwithout consideration on seismic loading often are irregularin plan and/or elevation. The consequent lateral-torsionalcoupling leads inevitably to considerable increase in theseismic vulnerability [10–12]. Finally, the seismic gap betweenadjacent structures often is not adequate to accommodatetheir relative motions, thus resulting in a significant seismic

hazard of pounding during earthquake excitation. Therefore,the existing RC building structures often need retrofit toresist the design earthquake demand. Many techniques areavailable for seismic upgrading of existing buildings andhave proved to be effective in increasing the capacity ofthe structure and/or reducing the seismic demand. Amongthese techniques, the hysteretic dissipative braces (buckling-restrained braces (BRBs) and steel hysteretic dampers) havebeen extensively applied for the seismic retrofitting of RCframed buildings and proved to give additional energydissipation capacity, control the interstorey drifts and lateraldisplacements, and, finally, encourage dissipative collapsemechanisms [13–19]. Steel bracings have also other advan-tages such as their relatively low weight, their suitability forprefabrication, and the possibility of allowing inner and outeropenings. Moreover, the braces may be directly connectedto concrete members without using steel frames fixed tothe concrete structure. Finally, these devices can be used asstructural fuses since they are easy to replace, concentratedamage while the rest of the structure remains undamaged,

HindawiMathematical Problems in EngineeringVolume 2018, Article ID 5364564, 28 pageshttps://doi.org/10.1155/2018/5364564

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https://creativecommons.org/licenses/by/4.0/

https://doi.org/10.1155/2018/5364564

2 Mathematical Problems in Engineering

and fix a limitation of the brace force that is transmitted tothe highly stressed anchorage.

There are many studies available in the literature focusingon experiment tests and analytical results that have provedtheir excellent hysteretic behaviour without any strengthand stiffness degradation [20–25]. However, although con-ceptually clear as general principle, the application of hys-teretic dissipative braces require more complex proceduresif compared with other retrofit strategies. For example, theseismic retrofit with base isolation may be studied using asimple linear elastic analysis [26, 27] while the traditionaltechniques to increase strength and/or ductility require theplastic section analysis. The design of hysteretic dissipativebraces is certainly more complex. Application of dampedbracing systems in Europe is seriously limited by the lack ofa standardized European design procedure. In fact, despitethis technique proved its effectiveness for seismic upgradingof RC structures, Eurocode 8 [28] does not give any rulesfor design of hysteretic dissipative braces. The traditionalmethods appropriate for conventional structures (such asthe use of a q-factor in Eurocode 8 [28] or the R-factorsas prescribed by ACI) become worthless. The nonlinearresponse history analysis is complex and computationallydemanding and, thus, its implementation is generally avoidedin current practice for design of hysteretic dissipative braces.Thus, many procedures have been proposed in the literaturefor designing the hysteretic dissipative braces. Some of themare based on the Direct Displacement-BasedDesign (DDBD)method [29–32]. Practically, the hysteretic damping providedby inelastic deformation of the damped braces is replacedby an equivalent viscous damping to convert the nonlinearsystem into an equivalent linear system. Bergami et al. [33]developed a displacement-based procedure based on twoperformance objectives: protect the structure against struc-tural damage or collapse and avoid nonstructural damageas well as excessive base shear. Mazza et al. [34] developeda design procedure that combines a proportional stiffnesscriterion (which assumes the elastic lateral storey-stiffnessdue to the braces proportional to that of the unbraced frame)to the displacement-based design. Choi et al. [18] proposed anenergy-based design method using hysteretic energy spectraand accumulated ductility spectra. Bosco et al. [35] proposeda design procedure for steel frames equipped with BRBsand proper values of the behaviour factor were developedthrough numerical investigation. Bai et al. [36] proposed aperformance-based plastic design (PBPD) method for dualsystem of buckling-restrained braced reinforced concretemoment-resisting frames. Guerrero et al. [37] developed amethod for seismic design of buildings equipped with BRBsthat uses seismic records to solve the dynamic equation ofmotion for dual oscillators. Barbagallo et al. [38] proposeda design method for seismic upgrading of existing RCframes by BRBs that allows a direct control of storey driftdemands. The equivalent viscous damping ratio to accountfor the energy dissipated by the damped braced frame is adominant parameter in procedures based on displacement-based design. Ghaffarzadeh et al. [39] presented results ofexperimental and numerical investigations performed for

estimating the equivalent viscous damping in DDBD proce-dure of two lateral resistance systems, moment frames andbracedmoment frames.Mazza et al. [40] andDwairi et al. [41]developed analytic expressions of the equivalent-dampingconsidering the energy dissipated by the hysteretic dampersand the framed structure.

The main drawbacks of the design methods based onDDBD [29–34] are that they neglect the frame-dampedbrace interaction and are based on the proportional stiffnesscriterion. In fact, the equivalent single-degree-of-freedom(SDOF) system of the RC bare frame is defined from thepushover analysis of the existing RC structure. The effectiveperiod and secant stiffness of the structure are then calculatedcombining the damping characteristics of the structuralcomponents. Finally, the same value of the stiffness ratiobetween damped braces and bare frame is assumed for eachstorey. Thus, the damper stiffness is distributed along theheight of the building according to the profile correspondingto the fundamental mode of the structure. This approach canproduce a nonuniform distribution of peak storey drift underearthquake ground motion and, therefore, it is not able toprevent soft-storey mechanisms.

As an alternative, Kasai et al. [42–44] developed amethodbased on the SDOF idealization of the multistorey buildingstructure and proposed a rule to arrange the damper stiffnessover the height of the building so to produce the uniformdistribution of drift angle and ductility demand under thedesign shear force, although those of the frame withoutdampers may be nonuniform. The damper design methodgiving optimal dampers distributions was initially proposedfor elastic steel structures as well as elastoplastic structuresincluding steel and timber structures [45] and then extendedfor retrofitting of existing RC structures with elastoplasticdampers [31, 46]. This design approach is very useful sinceit gives a closed form expression for the required damper toRC frame stiffness and the optimal damper stiffness for eachstorey. However, it suffers somemain limitations for applyingin current practice. First, all these damper design methodsgiving optimal dampers distributions are generally based ona shear beam model that consists of a mass and two springs(one for damper and the other for frame) for each storey.Thismodel represents the hysteretic characteristics of the dampersbut neglects some considerable interaction effects betweenthe damped braces and the RC frame that may influence theseismic characteristics of compound system. In fact, despitethe increase of strength and stiffness, the introduction ofdamped braces leads to an increase of the axial forces inthe columns. This effect reduces the deformation capacityof the RC columns and may lead to their premature failure.Thus, the design method should include the effects of frame-dampers interaction so to relate the damped brace capacitywith the deformation capacity of the RC columns. Moreover,the damper quantity obtained in the equivalent SDOF systemis distributed to each storey of MDOF system by using theconstraint condition that the equivalent stiffness of passivecontrolMDOF system at the target drift angle is proportionalto the storey shear force acting on each storey. This designapproach is very sensitive to the design shear force andframe stiffness distributions along the height. The existing

Mathematical Problems in Engineering 3

RC buildings typically show significant reduction in cross-section of the column along the height. Thus, the framestiffness may decrease rapidly along the height and thestorey drift at upper stories increases (upper-deformed typestructure). In this case, the required damper to RC framestiffness becomes high value at the storey expected to havelarge drifts of frame without dampers. On the contrary,the closed form expression of the optimal damper stiffnessgives negative values in the first stories of the structurewhich means that no damper should be inserted in thesestories. This optimal design solution obtained for an idealshear beam model cannot be easily implemented in currentpractice due to the effects of frame-damper interactions. Infact, in the retrofitted building all the lateral load resistingsystems (including the damped braces) should run withoutinterruption from their foundations to the top of the building.If no damper is inserted in the first storey, the uniformityin the development of the structure along the height of thebuilding is lessened, and this gives a concentration of shearforces in the first storey columns below the damped bracesthat might prematurely cause collapse.

Finally, it should be highlighted that the design methodsavailable in literature generally include only the first modecontribution while the contributions of higher modes inevaluating the response of MDOF elastoplastic system isneglected. Moreover, the design methods based on optimaldampers distributions [31, 42–47] are generally tested on two-dimensional planar shear-bar models and their efficiencyfor 3D asymmetric building structures should be furtherinvestigated. Thus, they are fully reliable only when appliedto symmetric-in-plan buildings and this is a very strong lim-itation since many existing RC buildings (including schools,hospital and other public buildings) are asymmetric in planand/or in elevation. On the other side, the application ofdesign methods based on DDBD [29–34] to asymmetric-plan buildings creates some problems of accuracy. In fact,the dynamic properties of the damped braced structure arecalculated combining the damping characteristics of the RCbare frame and the damped braces. However, the responseof the RC bare frame is dominated by the torsional effects,while the response of damped braced structure is poorlyconditioned by these effects since they are mitigated by thedamped braces. Clearly, this would affect the effectiveness ofthese design methods.

In this paper, a design method is proposed to addressthe main issues of seismic design of damped braces: effectsof frame-damper interactions, higher modes contribution,effect of soft-storey irregularities, and torsion effect in asym-metric building.The proposed method is based on the DirectDisplacement-Based Design (DDBD) and is an evolutionand improvement of the procedure proposed by Mazza et al.[34]. The design procedure explicitly considers the frame-damper interaction (i.e., the force demands applied to theframe due to the damper yielding and strain hardening). Infact, the seismic design is carried out using the pushovercurve of the dual RC-brace system thus considering theyieldmechanismof compound systemand the correspondingnonlinear drift demand for achieving the expected seismicperformance. To solve the drawback of fixed-load pattern, the

pushover analysis is carried out by using the Displacement-based Adaptive Pushover (DAP) method [48] together withan adaptive version of the capacity spectrum method [49].Based on a 3D model and a fully multimodal procedure, theproposed method allows accounting for the higher modescontribution and torsion effect in asymmetric buildings. Theequivalent viscous damping of the dual RC-brace systemis characterized using a specific formulation calibrated onexperimental results for steel braced RC frames. The lateralforce distribution is based on the inelastic state of thestructure and this gives the stiffness and strength of the hys-teretic dissipative braces to prevent undesired failure modes(i.e., partial mechanisms and soft-storey mechanisms). Theeffectiveness and reliability of the proposed procedure isinvestigated using a case study example. Its practical applica-bility is investigated with reference to a real case study that isan asymmetric in plan school building. Nonlinear dynamictime history analyses have been carried out to evaluate theseismic performance of the retrofitted building and validatethe displacement-based design procedure.

2. Proposed Retrofit Design Method

2.1. Preliminary Design. A preliminary design of the dampedbraces is carried out with the displacement-based designprocedure proposed by Mazza et al. [34]. According to thisprocedure, the capacity curve of the framed RC structureis selected from the pushover analysis that is carried outunder constant gravity loads and increasing lateral forceswithdifferent distributions over the height of the building. Thelowest capacity curve (base shear V(F) versus top displace-ment d(F)) is used for the preliminary design. The designdisplacement 𝑑(𝐹)𝑝 is selected from the seismic performanceof the existing RC building. In particular, 𝑑(𝐹)𝑝 may beselected as the value of the top displacement correspondingto the Limit State of Life Safety (LS). Typically, this limitstate is defined from the ultimate chord rotation capacity ofductilemembers (beams, columns, andwalls), the interstoreydrift capacity of the building or, even, the width of seismicgaps to prevent the structural pounding between adjacentstructures during earthquakes.The capacity curve is idealizedas bilinear for design purposes. The original RC structureis then represented by an equivalent SDOF system [50]characterized by a bilinear curve with yield displacement 𝑑(𝐹)𝑦and corresponding base shear 𝑉(𝐹)𝑦 , and stiffness hardeningratio rF (Figure 1). The Jacobsen formulation [51] is used tocalculate the equivalent viscous damping due to hysteresisof the RC structure. The equivalent viscous damping of thedamped braced structure (𝜉𝐷𝐵𝐹) is estimated by summing theelastic viscous damping for RC structure (𝜉V=5%) and theequivalent viscous damping of the system composed of theframed structure (F) and the damped braces (DB) evaluatedas a weighted average as follows:

𝜉𝐷𝐵𝐹 = 𝜉V +𝜉(𝐹)𝑉(𝐹)𝑝 + 𝜉(𝐷𝐵)𝑉(𝐷𝐵)𝑝

𝑉(𝐹)𝑝 + 𝑉(𝐷𝐵)𝑝(1)


P

６∗＆＞∗Ｇ？

？

６(＆)Ｊ /Γ

６(＆)Ｓ /Γ

＋＆

＋(＆)？

＞Ｊ/Γ＞(＆)Ｓ /Γ

Ｌ＆＋＆

(a)

P

６∗DB ＞∗

DB

６(DB)Ｊ /Γ

６(DB)y /Γ

＋DB

＋(DB)？

＋(DB)？

＞Ｊ/Γ＞(DB)Ｓ /Γ

ＬDB＋DB

(b)

Figure 1: Bilinear SDOF systems: (a) bare frame; (b) damped brace (Mazza et al. [34]).

0.0000.0200.0400.0600.0800.1000.1200.1400.1600.1800.200

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00T (s)

Sd (5%‐damped)

３＞(Ｇ

)

＞∗Ｊ

＞∗Ｊ

４？

Sd(＄＂＆)

Figure 2: Calculation of effective period from damped spectrum.

The base shear 𝑉(𝐷𝐵)𝑝 in the hysteretic dissipative bracesis unknown since the effective strength properties of theequivalent damped brace is one of the parameters of thedesign procedure.Thus, an iterative procedure is required. Atfirst, an attempt value of the equivalent viscous damping 𝜉𝐷𝐵𝐹is imposed. Then, a damping reduction factor R𝜉 defined as

𝑅𝜉 = ( 0.070.02 + 𝜉𝐷𝐵𝐹)

0.5

(2)

multiplies the spectral ordinates to get the design displace-ment values. The effective period Te of the damped bracedstructure is estimated as the period of the 𝜉𝐷𝐵𝐹-dampeddisplacement spectrum corresponding to the performancedisplacement dp (Figure 2). The equivalent stiffness Ke of thedamped braced structure is calculated as follows:

𝐾𝑒 = 4𝜋2𝑚𝑒𝑇2𝑒 (3)

The equivalent stiffness of the damped braces is given by

𝐾(𝐷𝐵)𝑒 = 𝐾𝑒 − 𝐾(𝐹)𝑒 (4)

Finally, since the constitutive law of the equivalent dampedbrace is idealized as bilinear, the performance and yieldingbase shears (𝑉(𝐷𝐵)𝑝 and 𝑉(𝐷𝐵)𝑦 ) are calculated as follows:

𝑉(𝐷𝐵)𝑝 = 𝐾(𝐷𝐵)𝑒 ⋅ 𝑑𝑝 (5)

𝑉(𝐷𝐵)𝑦 = 𝑉(𝐷𝐵)𝑃1 + 𝑟𝐷𝐵 (𝜇𝐷𝐵 − 1) (6)

The base shear from (5) may be used as a new attempt valueof 𝑉(𝐷𝐵)𝑝 in (1) to calculate the equivalent viscous damping𝜉𝐷𝐵𝐹 of the damped braced frame. This iterative procedureprogresses very quickly to a converged solution. To completethe design procedure, it is necessary to define the distributionof base shear 𝑉(𝐷𝐵)y along the height. The preliminary designis based on the proportional stiffness criterion; that is, at eachstorey the same value of the stiffness ratio between the lateralstiffness values for damped braces and bare frame is assumed.Moreover, the mode shapes of the structure are consideredunchanged even after inserting the damped braces. Thus, thebase shear is distributed along the height of the buildingaccording to the profile corresponding to the fundamentalmode. This gives the yielding design shear force 𝑉(𝐷𝐵)𝑦,𝑖 ofdamped braces at i-th storey and allows defining the strength𝑁(𝐷𝐵)𝑦,𝑖 of the damper and the stiffness 𝐾(𝐷𝐵)𝑖 of the dampedbrace (i.e., brace + damper). In the case of one bracing in asingle bay, the strength𝑁(𝐷𝐵)𝑦,𝑖 and the stiffness𝐾(𝐷𝐵)𝑖 are givenby

𝑁(𝐷𝐵)𝑦,𝑖 = 𝑉(𝐷𝐵)𝑦,𝑖cos𝛼𝑖

(7)

𝐾(𝐷𝐵)𝑖 = 𝑉(𝐷𝐵)𝑦,𝑖(𝜙𝑖 − 𝜙𝑖−1) 𝑑(𝐷𝐵)𝑦 cos2 𝛼𝑖

(8)

where 𝜙i and 𝜙i-1 are, respectively, the i-th and (i-1)-thcomponents of the fundamental mode shape vector 𝜙.

2.2. Final Design. The preliminary design procedure basedon the proportional stiffness criterion may not give theexpected results when applied to real case studies. The maindrawbacks are as follows:

(1) The mode shapes of the structure are consideredunchanged even after inserting the damped braces.


(2) The dampers are distributed along the height accord-ing to the proportional stiffness criterion.

(3) The frame-damped braces interaction is neglected.

These drawbacks can affect the accuracy of the designmethodwhen applied to buildingswith soft stories, plan irregularities,or nonductile columns. Many existing RC buildings weredesigned without earthquake-resistance requirements andpartial collapsemechanisms, such as soft-storeymechanisms,are not necessarily avoided. Thus, if the mode shapes ofthe structure are maintained unchanged even after insertingthe damped braces, then even the retrofitted building maydevelop undesired partial collapse mechanisms under thedesign seismic action. Furthermore, it may be necessary touse the damped braces to improve the seismic behaviourof the building. In fact, the buildings designed withoutconsideration on seismic loading often are irregular inplan and elevation. The consequent lateral-torsional cou-pling leads inevitably to nonuniform displacement demandsamong resisting planes during the pushover analysis of theexisting RC building. Thus, the mode shapes and, in general,the seismic response of the building can vary greatly afterinserting the damped braces. For example, the torsionaleffects are greatly reduced in braced frame buildings since thedamped braces are positioned in plan to increase the torsionalstiffness and strength and minimize the eccentricity betweenthe centres of mass and stiffness. As a result, also the choiceof the design displacement𝑑(𝐹)𝑝 from the seismic performanceof the existing RC building becomes questionable.

Moreover, it should be highlighted that the damperdistribution along the height of the building according to theproportional stiffness criterionmay be not able to assure verylarge plastic strains in all the dampers.

Some authors [31, 42–47] have tried to overcome thislimitation by developing damper design methods that giveoptimal dampers distributions, so that the distribution ofductility demand becomes uniform, although that of theframe without damper is nonuniform. However, these meth-ods are generally based on a simplified shear beam model,while applications to 3D buildings models accounting forframe-damper interaction, higher modes effects, and planirregularities are still lacking. Moreover, these optimal designmethods provide that no damper must be inserted in the firststorey of “upper-deformed type” frames that are the framesin which the storey drift at upper stories increases. This givesa concentration of shear forces in the first storey columnsbelow the damped braces that might prematurely cause thecollapse of the structure. This effect cannot be highlightedby the shear beam model that consists of two springs inparallel (one for damper and the other for frame). This isa general drawback of many design procedures that neglectthe effect of frame-damped braces interactions (i.e., the forcedemands applied to the frame due to the damper yieldingand strain hardening), which may significantly affect thefailure mechanism. In fact, the internal force demands of RCbeams and columns intersected by braces are underestimatedif interaction is neglected. In particular, the increase of theaxial forces in the RC columns decreases their deformation

capacity (nonductile columns) and this may lead to theirpremature failure.

In order to overcome these drawbacks, a retrofit designmethod is proposed in this paper. In the first step the dampedbraces are defined from the preliminary design developedin Section 2.1. Then, the pushover analysis is carried outon the dual RC-damped brace system thus accounting forthe effects of frame-damped braces interactions, the yieldmechanism of compound system, and the correspondingnonlinear drift demands. This allows accounting for theincrease in the axial forces of the columns given by thedamped braces and its effects on the ultimate chord rotationcapacity of the RC columns. The pushover analysis is carriedout by using the Displacement-based Adaptive Pushover(DAP) method proposed by Antoniou and Pinho [48]. Thismethod overcomes the assumption that the structure vibratespredominantly in a single mode and that the dynamicproperties of the structure remain unchanged, but it providesno solution to determine the target displacement. Thus,an adaptive version of the capacity spectrum method isdeveloped in this paper. The classical formulation proposedby Fajfar [50] is useless to define the equivalent SDOF systemin adaptive pushover analysis. In fact, the capacity spectrummethod assumes that the response of the multiple degree-of-freedom (MDOF) system is entirely in the fundamentalmode, while no contribution is considered fromothermodes.This allows defining an idealized bilinear SDOF system fromthe transformed capacity curve (𝑉/Γ versus 𝑑/Γ, where Γis participation factor of the fundamental mode shape). Incase of adaptive pushover, the lateral force pattern and, thus,the equivalent SDOF system changes during the analysis.At each step of the pushover analysis, a different equivalentSDOF system is defined as a function of the actual lateraldisplacement pattern. In particular, the equivalent mass𝑀𝑖𝑒𝑞of the SDOF system at the i-th step of the pushover analysisis expressed as a function of the j-th storey displacement 𝛿𝑖𝑗as follows [49]:

𝑀𝑖𝑒𝑞 =(∑𝑁𝑗=1𝑚𝑗𝛿𝑖𝑗)2∑𝑁𝑗=1𝑚𝑗𝛿𝑖𝑗2

(9)

where N is the number of stories, mj is the mass of the j-thstorey, 𝛿𝑖𝑗 is the lateral displacement of the j-th storey at the i-th step of pushover analysis.The corresponding participationfactor is defined as follows:

Γ𝑖𝑒𝑞 =∑𝑁𝑗=1𝑚𝑗𝛿𝑖𝑗∑𝑁𝑗=1𝑚𝑗𝛿𝑖𝑗2

(10)

The capacity curve (base shear 𝑉 versus top displacement d)of the damped braced structure is transformed step by stepinto the capacity curve (𝑉∗ versus𝑑∗) of the equivalent SDOFsystem, as follows:

𝑉∗𝑖 = 𝑉∗(𝑖−1) + Δ𝑉∗𝑖 (11)

𝑑∗𝑖 = 𝑑∗(𝑖−1) + Δ𝑑∗𝑖 (12)


where Δ𝑉∗i and Δ𝑑∗i are the base shear and the correspond-ing top displacement increments at the i-th step of pushoveranalysis, defined as follows:

Δ𝑉∗𝑖 = Δ𝑉𝑖Γ𝑖𝑒𝑞

Δ𝑑∗𝑖 = Δ𝑑𝑖Γ𝑖𝑒𝑞

(13)

Finally, according to the principle of energy equivalence, thecapacity curve is idealized as bilinear.

It should be highlighted that the capacity curve (𝑉∗versus 𝑑∗) referred to the dual RC-brace system and thusexplicitly considers the frame-damped braces interaction.Moreover, (10)-(12) that control the transformation fromMDOF to SDOF model account for the dynamic behaviourof the damped braced structure. This overcomes the mainlimitations of the preliminary design procedure that do notconsider the frame-damped braces interaction and definethe equivalent SDOF system of the RC bare frame from thepushover analysis of the existing RC structure. This includesmany effects (such as lateral-torsional coupling, torsionaleccentricity, and partial or soft-storey mechanisms) that arepresent in the existing RC building but may disappear inthe dynamic response of the building after retrofit. Finally, itshould be observed that the pushover analysis of the dampedbraced structure allows calculating the capacity curve of theRC bare structure and the capacity curve of the dampedbraces, separately. In fact, the top floor displacement is thesame in both cases while the base shear V(F) of the RC barestructuremay be calculated as the difference between the totalbase shear 𝑉 and the base shear 𝑉(𝐷𝐵) in the damped braces.

The equivalent viscous damping of the damped bracedstructure (𝜉𝐷𝐵𝐹) is estimated using the equation proposed byGhaffarzadeh et al. [39] for steel braced RC frames:

𝜉𝐷𝐵𝐹 = 𝜉V + 𝑎𝜋 (1 −

𝑏𝜇4 − 𝑐𝜇

4) (14)

where 𝜉V=5% is the elastic viscous damping for the RCstructure while the coefficients a, b, and c are calibratedmatching experimental and numerical data [39], which givesa=70, b=43, and c=4.7x10−5.

Equations (2)-(6) give the equivalent stiffness Ke of thedamped braced structure and the performance base shear𝑉(𝐷𝐵)𝑝 and the yielding base shear𝑉(𝐷𝐵)𝑦 in the damped braces.The yielding base shear𝑉(𝐷𝐵)𝑦 should be distributed along theheight of the building. The conventional displacement-baseddesign procedures [29–34] are based on the proportionalstiffness criterion and distribute the yielding base shear𝑉(𝐷𝐵)𝑦along the height according to the profile corresponding to thefundamental mode of the RC bare structure. As aforemen-tioned this approach may fail when the bare structure showslateral-torsional coupling effects and/or undesired failuremodes. Moreover, it is not able to assure large plastic strainsin all the dampers.

As an alternative, in this paper the damper quantity of theequivalent SDOF system is distributed to each storey of the

MDOF system by using the following rule to distribute theyielding base shear 𝑉(𝐷𝐵)𝑦 along the height of the structure:

𝐹(𝐷𝐵)𝑖 = 𝑚𝑖𝛿𝑝,𝑖∑𝑁𝑗=1𝑚𝑗𝛿𝑝,𝑗

𝑉(𝐷𝐵)𝑦 (15)

where 𝛿𝑝,𝑖 is the i-th storey displacement at the performancepoint calculated from the pushover analysis. The dampersand the braces are dimensioned from the storey shears usinghorizontal equilibrium.

Practically, the distribution of the lateral loads carriedby the damped braces at the yielding point is based on theinelastic state of the structure calculated from the adaptivepushover analysis of the dual RC-brace system.

The distribution of the base shear according to (15) isfocused on optimizing the seismic response of the buildingafter retrofit. In fact, it increases the stiffness of the dampedbraces for the stories with higher interstorey drifts (thuspreventing undesired soft-storey mechanisms) and tends toassure very large plastic strains in all the dampers. In thecase of one bracing in a single bay, the yield strength of thedampers𝑁𝑦,𝑖 is given by (7), while the stiffness of the dampedbrace 𝐾(𝐷𝐵)𝑖 at i-th storey is calculated as follows:

𝐾(𝐷𝐵)𝑖 = 𝑉(𝐷𝐵)𝑦,𝑖(𝛿𝑝,𝑖 − 𝛿𝑝,𝑖−1) 𝑑(𝐷𝐵)𝑦 cos2 𝛼𝑖

(16)

where 𝛿𝑝,𝑖 and 𝛿𝑝,𝑖-1 are, respectively, the i-th and (i-1)-thstorey displacements at the performance point. The dampedbraces defined by (16) have varied with respect to the pre-vious iteration. Thus, the procedure should be iterated untilconvergence is achieved, that is until the damped braces and,therefore, the equivalent viscous damping do not vary withrespect to the previous iteration with the prefixed toleranceof 5%.

3. Case Study Example

The proposed procedure is applied to a regular 5-storeyRC moment frame (Figure 3) that is considered one of theelements of the lateral force resisting system of a regular RCframed building. The storey height is 3.5m for all floors. Thebay length is 5.00 m in both orthogonal directions. The steelmaterial used is B450C with tensile strength value of 450MPa.The concrete is assumed to have a nominal compressivestrength 𝑓𝑐𝑘=25N/mm2 (compressive strength class C25/30).The building is designed for vertical loads only, and thenretrofitted to Eurocode 8 [28] requirements for soil class A,damping ratio 5% and design Peak Ground Acceleration forLife Safety Limit State is PGALS=0.35g. The SeismoStructprogram [52] is used in the simulations presented in sec-tion. Distributed plasticity beam-column elements are usedto account for material nonlinearity. Specifically, inelasticforce-based fibre elements (infrmFB) are used for modellingcomponents. The sectional stress-strain state is obtainedby integrating the nonlinear uniaxial material response ofthe singlefibres in which the section is subdivided, fully


5

0,5

0,4

0,3

0,3

0,3

0,6

0,8

0,45

0,3

0,30,5

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553,

53,

53,

53,

53,

5(a)

5

3,5

3,5

3,5

3,5

3,5

55

35 ∘35 ∘

35 ∘35 ∘

35 ∘

(b)

Figure 3: (a) Front view of 5-storey bare frame. (b) Five-storey frame model with damped braces.

accounting for the spread of inelasticity along the memberlength and across the section depth. The reinforcing steelbars are modelled with a bilinear hysteretic model. Theconcrete is modelled accounting for the amount of confiningwith the well-known Mander model [53]. While the materialnonlinearity is accounted for flexural and axial degrees offreedom, sections are assumed to behave elastically undershear and torsion. The bracing in a single bay of the framedstructure is considered for retrofit. The dampers used havethe following mechanical parameters: stiffness hardeningratio rD=0.020, design ductility 𝜇𝐷=10, lateral stiffness ratiobetween damper, and brace 𝐾∗𝐷 = 𝐾𝐷/𝐾𝐵 = 0.2. Theseparameters may correspond to different hysteretic dissipativebraces (such as buckling-restrained braces (BRB) or steelhysteretic dampers (HBF)).

Figure 4(a) shows the pushover curves of the RC barestructure under three different lateral force distributionsalong the height: (a) uniform distribution, (b) equivalentstatic force distribution, and (c) first mode distribution.Figure 4(b) shows the lowest capacity curve of the RCbare structure and the corresponding performance pointsfor three limit states: Damage Limitation (DL), Life Safety(LS), andCollapse Prevention (CP). A soft-storeymechanismoccurs at the fourth level of the structure during pushoveranalysis. The limit states for the ductile and brittle failuremodes are determined according to Annex A of EC8-3 [28].The capacity of the ductile and brittle members is estimatedin terms of chord rotation and shear strength, respectively.The value of the capacity of both the ductile and brittlecomponents and mechanisms are then compared to thecorresponding demand for the safety verification.

The damped braces are designed applying the proposeddesign method. Figure 5 shows the pushover curve of thestructure with damped braces at each step of the iterative

procedure. The points corresponding to yielding and failureof each device are evidenced on each pushover curve. Thefirst step (Step 0) corresponds to the structure retrofittedby damped braces sized by the preliminary design. In thiscase, the failure of the 4th storey damper occurs before thetarget displacement dp is reached. This is due to the soft-storey mechanism that is formed at the fourth level of thestructure. The other dampers are very far from failure. Thisbehaviour derives from the proportional stiffness criterionthat is applied for the preliminary design. In fact, the modeshapes of the structure are maintained unchanged even afterinserting the damped braces and, thus, also the retrofittedbuilding develops a soft-storey mechanism at the 4th floorlevel of the structure. The last step (Step 7) correspondsto the structure retrofitted by damped braces sized by thefinal design. In this case, the failure in the first damperoccurs when the target displacement is reached and theother dampers are very near to failure. Figure 6 shows thevariation of the properties of the damped brace with heightat each step of the iterative procedure. The stiffness of thedamped brace 𝐾(𝐷𝐵)𝑖 is plotted in Figure 6(a), while thestrength𝑁𝑦,𝑖 of the dampers is plotted in Figure 6(b). Figure 7shows the variation during the iterative procedure of theequivalent-damping ratio 𝜉𝐷𝐵𝐹 of the damped braced frame.Results show that the trend tends to stabilize after someiterations giving the final value of the equivalent-dampingratio.

Finally, it should be underlined that the proposed designmethod allows accurately estimating the target displacementdp accounting for the real seismic demands of the dampedbraced frame. In fact, the pushover analysis of the dualRC-damped brace system explicitly considers the effects offrame-damped braces interactions and, particularly, the forcedemands applied to the frame due to the damper yielding and


Base

She

ar (k

N)

Top Displacement (m)

400350300250200150100

5000.00

Uniform DistributionEquivalent Static Force DistributionFirst Mode Distribution

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11

(a)


DL - RotationDL - DriftLSCP

0.000 0.020 0.040 0.060 0.080 0.100

Base

She

ar (k

N)

250

200

150

100

50

0

(b)

Figure 4: (a) Pushover curves under different lateral load distributions. (b) Lowest capacity curve and performance points.

Table 1: SDOF system of bare frame.

me∗ 0.142 kNs2/mmD 1.422 -dp 86.0 kNs2/mmdp∗ 60.5 mmK(F)e 1.721 kN/mmT(F)∗e 1.806 s

Table 2: SDOF system of damped braced frame.

𝜉DBF 15.93 %R𝜉 62% -TD 2 sSDC 173 mmTeq 1.12 sKeq 4.48 kN/mmrDB 0.06 -K(DB)e = Ke − K(F)e 2.76 kN/mm

strain hardening. This allows evaluating the effects of frame-damped braces interactions on the deformation capacity ofbeams and columns in terms of chord rotation (Appendix Aof EN 1998-3 [28]) and, thus, on the target displacement dp tobe considered in the analysis. This is a significant advantageover other procedures available in literature since they neglectthe frame-damped braces interactions and their effects on thedeformation capacity of the structure.

Tables 1 and 2 show the final parameters of the idealizedbilinear SDOF systems for the bare frame (Table 1) andthe damped braced frame (Table 2). Table 3 defines theparameters of MDOF system for the damped braced frame.Table 4 shows the design parameters of the damped braces ateach storey.

The seismic response of the RC structure with dampedbraces is finally evaluated by means of nonlinear time historyanalysis. For this purpose, according to the Italian Code [54]

Table 3: MDOF system of damped braced frame.

dp 86.00 mmK(DB)e 2.76 kN/mmV(DB)p = K(DB)e xdp 238 kNV(DB)y 201 kN𝜇DB 8.53 -d(DB)y = dp/𝜇DB 10.08 mm

a group of seven time histories is applied and the averageof the response quantities from all the analyses is used asthe design value of the seismic effect. The description ofthe seismic motion is made using recorded accelerogramsand this allow accounting for characteristics like frequency,duration, and energy of real earthquake ground motions.The European Strong-motion Database (ESD), the SIMBADdatabase (Selected Input Motions for displacement-BasedAssessment and Design), and the Italian Accelerometricarchive (ITACA) [55, 56] are used for selecting the recordedaccelerograms.The suite of accelerograms observed the rulesrecommended in the seismic standards. In fact, the meanof the zero period spectral response acceleration values(calculated from the single time histories) is not smallerthan the value of agS for the site in question, where S is thesoil factor and ag is the design ground acceleration on typeA ground. Furthermore, in the range of periods of interestno value of the mean elastic spectrum, calculated from alltime histories, is less than 90% of the corresponding valueof the target elastic spectrum. The recorded accelerogramsconsidered in the numerical analysis are summarized inTable 5. In Figure 8, the spectrum compatibility for theselected acceleration records is represented.

Figure 9 shows height-wise distribution of axial displace-ment of damped braces obtained from the nonlinear time his-tory analysis. The results for the selected records are plottedin Figure 9(a) while Figure 9(b) compares the mean valueto the ultimate value (corresponding to failure). The results


Base

She

ar (k

N)


400450500

350300250200150100

500

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Step 0 Step 1

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Step 7Step 6

sy,2 sy,2

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dpsy,2 ; sy,3 ; sy,4

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dp

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su,4

su,3su,1

su,1su,2su,3

su,4su,5dp

su,2

dp

0.000 0.025 0.050 0.075 0.100＞p

Top Displacement (m)0.000 0.025 0.050 0.075 0.100＞p



Base

She

ar (k

N) 400

450

550500

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ar (k

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Base

She

ar (k

N) 400

450500

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500


Base

She

ar (k

N) 400

450500

350300250200150100

500


Figure 5: Pushover curve and performance points for each damped brace. Variation during the iterative procedure.


Table 4: Distribution of base shear along height and design parameters of damped braces.

Floor Mass Displacement Inclination Fi V(DB)y,i Ny,i K(DB)i

[kNs2/mm] 𝛿p,i brace [𝛼] [kN] [kN] [kN] [kN/mm]1 0.0646 0.073 35 6.69 201.3 245.8 407.4182 0.0631 0.240 35 21.4 194.6 237.6 172.5493 0.0620 0.467 35 41.1 173.2 211.4 112.4244 0.0616 0.772 35 67.4 132.1 161.3 64.1025 0.0457 1.000 35 64.8 64.76 79.05 42.004

0

1

2

3

4

5

0 50 100 150 200 250 300

Stor

ey L

evel

Step 0Step 1Step 2Step 3


K(DB)i (kN/mm)

(a)

0

1

2

3

4

5

0 50 100 150 200 250 300

Stor

ey L

evel



N(DB)y,i (kN)

(b)

Figure 6: Height-wise distribution of damped brace properties during the iterative procedure. (a) Damper Stiffness. (b) Damper Strength.

15.4%

15.6%

15.8%

16.0%

16.2%

16.4%

16.6%

16.8%

17.0%

1 2 3 4 5 6 7Step

＄＂＆

Figure 7: Equivalent-damping ratio of damped braced frame.Variation during the iterative procedure.

highlight that the proposed design procedure is effective incontrolling deformations and avoiding undesired soft-storeymechanisms of failure. Figure 10 shows the hysteresis loops ofdampers under the Izmit earthquake ground motion. It canbe noted that the proposed design method is able to assurevery large plastic strains in all the dampers thus increasingthe energy dissipation capacity during the earthquake groundmotion.

Figure 11 compares the height-wise distributions of storeydrift ratio obtained in the following cases: (1) existing RCstructure; (2) structure retrofitted with hysteretic dissipa-tive braces sized by the preliminary design; (3) structureretrofitted with hysteretic dissipative braces sized by the finaldesign. The average value from seven nonlinear time historyanalyses at the Life Safety Limit State is plotted. Results showthat the existing structure has a soft-storey mechanism at the4th floor level. The structure retrofitted with damped bracessized by the preliminary design maintains unchanged thefundamental mode shape even after retrofit thus evidencinga nonuniform distribution of storey drift along the height.On the contrary, the final design procedure uses the dampedbraces to give a more uniform height-wise distribution ofstorey drift thus improving the seismic behaviour of thestructure.

4. Application to a RC School Building

4.1. Description, In Situ Measurements, and Laboratory Tests.The real case study is a three-storey school building in ViboValentia (Calabria-Italy). The building is composed of threereinforced concrete framed structures named “A”, “B”, and“C” in Figure 12.The building was designed in 1962 accordingto the provisions of an Italian Code dating back to 1937


Table 5: Set of earthquake natural records.

Waveform ID Earthquake Name Date Direction PGA Magnitudo[m/s2] MW

1243X Izmit 13.09.99 NS 0.714 5.86326Y South Island 21.06.00 EW 1.142 6.44676X South Island 17.06.00 NS 3.920 6.5292X Campano-Lucano 23.11.80 NS 0.588 6.9368X Lazio-Abruzzo 07.05.84 NS 0.628 5.97142Y Bingol 01.05.03 EW 2.918 6.36331X South Island 21.06.00 NS 0.513 6.4

0.00

0.20

0.40

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1.00

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00T (s)

Target SpectrumUpper tolerance

Lower toleranceMean Spectrum

３；(g

)

Figure 8: Spectrum compatibility for the selected records.

0

1

2

3

4

5

Stor

e Lev

el

0 5

Su, DBAcc.1Acc.2

Acc.3Acc.4Acc.5

Acc.6Acc.7

10 15 20 3025SDB [mm]

(a)

0

1

2

3

4

5

Stor

e Lev

el

0 5 10 15 20 25

UltimateMean

SDB (mm)

(b)

Figure 9: Height-wise distributions of axial displacement of damped braces. (a) Results for each record. (b)Mean value compared to ultimatevalue. Collapse Prevention Limit State.


−300

−200

−100

0

100

200

300

−6 −4 −2 0 2 4 6N (k

N)

S (mm)

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N)

S (mm)

4th Floor

−100−80−60−40−20

020406080

100

−20 −15 −10 −5 0 5 10 15 20N (k

N)

S (mm)

5th Floor

Figure 10: Hysteresis loops of dampers under Izmit earthquake ground motion. Collapse Prevention Limit State.

[57]. The site belonged to the first seismic category zone,whose seismic intensity coefficient was C=0.07.The allowablestress design method was used in design. Structure A hasbeen selected as case study (Figure 13). The building hasan L-shaped floor plan with dimensions of 17.70 x 35.50 m(Figure 14). The following investigations were carried out: (1)soil investigations including sampling and testing, (2) geo-metricalmeasurements, and (3) determination ofmechanicalproperties of materials by testing of samples taken from thestructure. The soil deposits primarily comprise silt sandyloam, sand slightly silty clay, and micaceous sand. Themechanical properties of soil and the ground type accordingto soil classification of Eurocode 8 [28] give a soil classified asground type B. Due to the extensive measuring and testing,the full knowledge level KL3 [28, 54] is attained, whichallows a Confidence Factor CF=1. The geometry is knownfrom original outline construction drawings integrated bydirect visual survey. The structural details are obtainedfrom original construction drawings together with in situinspection. Information on the mechanical properties of theconstruction materials is taken from a comprehensive in situtesting. Collected results give the following mean values ofstrength: 𝑓cm=35.2 MPa for concrete and 𝑓ym=408 MPa forsteel.

4.2. Seismic Assessment. The seismic performance evaluationis developed using the procedure reported in current ItalianCode [54] and Annex B of EN 1998-3 [28]. The RC framedstructure is modelled in SAP2000 finite element computerprogram [58]. Figure 15 shows the fundamental mode shapesof the existing structure and the corresponding dynamicproperties. The parameters of the elastic design responsespectra used for seismic assessment are plotted in Table 6.The Limit States of Immediate Occupancy (IO), DamageLimitation (DL), Life Safety (LS), and Collapse Prevention(CP) are defined according to Eurocode 8 [28].The nonlinearanalysis is carried out using a fibre hinge model implementedin the SAP2000 [58]. The concrete is modelled with thestress-strain relationship originally proposed byMander et al.[53]. The steel is modelled with an elastic-plastic-hardeningrelationship. The rigid elements are placed at beam-columnconnections to prevent the development of plastic hingesinside the connections. The capacity of ductile and brittlemembers is estimated in terms of chord rotation and shearstrength, respectively.The deformation capacity of beams andcolumns is defined in terms of the chord rotation according toAppendix A of EN 1998-3 [28]. Figure 16 shows the pushovercurves for two directions (X and Y), two lateral force distri-butions (first mode and uniform), and accidental eccentricity


Table 6: Parameters of elastic design response spectra.

Limit State IO DL LS CPProbability of exceedance 𝑃VR 0.81 0.63 0.10 0.05Return Period 𝑇R (years) 120 201 1898 2475Peak ground acceleration PGA/g 0.086 0.112 0.315 0.418Amplification factor 𝐹o 2.276 2.276 2.448 2.485Transition Period 𝑇C (s) 0.293 0.315 0.380 0.412

Table 7: Capacity peak ground acceleration and risk index (IO, DL, and LS Limit States).

IO X-Dir. DL X-Dir. LS X-Dir. IO Y-Dir. DL Y-Dir. LS Y-Dir.PGA/g IR PGA/g IR PGA/g IR PGA/g IR PGA/g IR PGA/g IR

+Modal + Ecc 0.143 1.402 0.197 1.470 0.351 1.023 0.094 0.922 0.128 0.955 0.432 1.259- Modal + Ecc 0.151 1.480 0.199 1.485 0.332 0.968 0.090 0.882 0.126 0.940 0.434 1.265+Modal - Ecc 0.148 1.451 0.197 1.470 0.321 0.936 0.090 0.882 0.126 0.940 0.435 1.268- Modal - Ecc 0.143 1.402 0.199 1.485 0.325 0.948 0.090 0.882 0.128 0.955 0.431 1.257+Uniform + Ecc 0.155 1.520 0.204 1.522 0.370 1.079 0.094 0.922 0.128 0.955 0.443 1.292-Uniform + Ecc 0.152 1.490 0.197 1.470 0.237 0.691 0.090 0.882 0.126 0.940 0.445 1.297+Uniform - Ecc 0.149 1.461 0.197 1.470 0.340 0.991 0.090 0.882 0.126 0.940 0.446 1.300-Uniform - Ecc 0.151 1.480 0.202 1.507 0.342 0.997 0.090 0.882 0.126 0.940 0.442 1.289

0

1

2

3

4

5

0.00% 0.50% 1.00% 1.50%

Stor

ey L

evel

Storey Drift Ratio

Bare FramePreliminary DesignFinal Design

Figure 11: Height-wise distribution of storey drift ratio. Averagevalue from 7 nonlinear time history analyses. Life Safety Limit State.

of 5%. The comparison between capacity and demand forthe Life Safety Limit State is carried out with the procedureimplemented in Annex B of Eurocode 8 [28] and ItalianCode [54]. The synthesis of the seismic safety verification isshown in Table 7, in which are plotted both the capacity ofthe existing building in terms of peak ground acceleration(PGA) for different limit states and the corresponding safetyindex IR (ratio between capacity and demand in terms of

PGA). The main deficiencies of the existing building may besummarized as follows: (1) insufficient stiffness for IO andDLLimit States, (2) poor shear capacity of brittle components, (3)torsional effects in X-direction that activate a partial failuremechanisms, (4) inadequate member chord rotation capacityfor the Life Safety Limit State, and (5) inadequate seismic gapfrom adjacent building structures.

4.3. Retrofit Design and Seismic Assessment. The seismicretrofit of the RC building is carried out using hysteretic dis-sipative braces that are allocated so to limit the underpinningarea of the existing foundation,minimize the torsional effects,and keep most areas of the building operational during theretrofit construction (Figures 17 and 18). Some external andinternal views of the building after retrofit are shown inFigures 19 and 20. The seismic retrofit has required somelocal interventions including strengthening of columns nextto the steel braces by steel angles and strips (Figure 18),shear strengthening of the unconfined joints, fibre reinforcedpolymer (FRP) shear, and bending reinforcement of somebeams at the first floor. The size of the seismic gap betweenstructures A and B has been increased up to 3 cm to avoidstructural pounding.

At first, the preliminary design of the damped braces iscarried out using the procedure described in Section 2.1. Thelowest capacity curve of the RC bare structure is selectedamong those plotted in Figure 16. A design displacement of 35mm is chosen in X-direction to avoid seismic pounding withthe adjacent structure B (Figure 12). A design displacement of62mm is selected inY-direction. Both values are checked and,if necessary, modified as the design progresses consideringthe response of the braced RC building. The seismic retrofitdesign is carried out using two different design solutions,one based on buckling-restrained braces (BRB) and the otherbased on steel hysteretic dampers (HBF). The steel hysteretic


Building C

Building D

Seismic Retrofit

N

of Building A

Building ABuilding B

Node 1

Node 3

Node 2

Node 4

Figure 12: Plan of the school buildings.

Figure 13: External view of Building A.


First Level Floor Plan +3.61m

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x65

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Level 2-3 Floor Plan +7.24m / +10.82m

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Roof Floor Plan at the stairs +13.55m

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０6 [90X40] ０12[90X40] ０21[80X40] ０6 [90X40]

０5 [70X40]

０4 [70X40]

０9 [70X40]０3 [70X40]

０2 [70X40] ０8 [70X40]

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０15[70X40]

０1 [70X40] ０7 [70X40] ０13[70X40]

０11[70X40]

０18[90X40]

０16[90X40]

０15[90X40]

０14[90X40] ０22[90X40]

０23[90X40]

０24[90X40]

０25[90X40]

０13[80X40]

０17[80X40]

Figure 14: Plan views of Building A.

First mode shapeFlexural Y

Period 0.681 sec

Second mode shapeFlexural X-Y and Torsional

Period 0.485 sec

Third mode shapeFlexural X and Torsional

Period 0.455 sec(Ｒ=0.03%; Ｓ=96.88%) (Ｒ=57.93%; Ｓ=0.22%) (Ｒ=35.16%; Ｓ=0.09%)

Figure 15: Fundamental mode shapes of the existing RC building.

dampers were finally used during the construction phase ofthe retrofit project (Figures 19 and 20). Tables 8–12 show allthe results of the preliminary design procedure described inSection 2.1.

The fundamental modal shapes and correspondingdynamic properties of the damped braced structure are

plotted in Figure 21. It can be noticed that the first twomodes are two flexural mode shapes with more than 85%modal mass participation. Practically, the seismic designretrofit with damped braces reduces the lateral-torsionalcoupling effects and, thus, themode shapes vary considerablyif compared with the existing RC building. That is one of the


0

1000

2000

3000

4000

5000

6000

−0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 0.15 0.20

Base

She

ar (K

N)


Fx,ecc(+5%)Fx,ecc(-5%)Fx,ecc(+5%)Fx,ecc(-5%)

POUNDING BUILDING A-B

DRIFT - IODRIFT - DLCHORD ROTATION - DLCHORD ROTATION - LSSHEAR FAILURE - LS15% STRENGTH REDUCTION

Fy,ecc(+5%)Fy,ecc(+5%)Fy,ecc(-5%)Fy,ecc(-5%)



Fy,ecc(+5%)Fy,ecc(+5%)Fy,ecc(-5%)Fy,ecc(-5%)



Fx,ecc(+5%)Fx,ecc(-5%)Fx,ecc(+5%)Fx,ecc(-5%)



First Mode X0

1000

2000

3000

4000

5000

6000

−0.30 −0.25 −0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30

BASE

SH

EAR

[KN

]


Uniform X

0

500

1000

1500

2000

2500

3000

3500

−0.30 −0.20 −0.10 0.00 0.10 0.20 0.30

Base

She

ar (K

N)


First Mode Y0

500

1000

1500

2000

2500

3000

3500

−0.30 −0.20 −0.10 0.00 0.10 0.20 0.30

Base

She

ar (K

N)


Uniform Y

Figure 16: Pushover curves of the existing building.

reasons why the design procedures based on the pushoveranalysis of the RC bare frame [29–34] may fail in this casestudy. On the other side, the optimal design procedures basedon an equivalent shear-bar model [31, 42–47] neglect thetorsional effects and, thus, their application to asymmetric inplan buildings deserves further study.

Figure 22 shows the pushover curves of the dampedbraced structure and the performance points for the differentlimit states for two directions (X and Y) and two lateral forcedistributions (first mode and uniform). When comparingFigures 16 and 22, it becomes clear the increase of stiffness,strength, and ductility due to the hysteretic dissipative braces.However, the structure is not ductile enough to sustain largeplastic deformation under the first mode distribution in Y-direction. In fact, the premature failure occurs since thechord rotation capacity at Life Safety Limit State is reached

in the first storey columns. The damped braces increase theaxial forces in the columns and this effect reduces theirdeformation capacity and prevents the collapsemechanism tofully develop. This result may be evidenced only if the effectsof frame-dampers interactions are included in the analysis.

However, it should be highlighted that the pushoveranalysis is not the appropriate method of analysis in thiscase. In fact, the code procedure [28, 54] based on pushoveranalysis is equivalent to the capacity spectrum method [50].The Elastic Demand Response Spectrum (EDRS) is generallyrepresented by the 5%-damped response spectrum. Thisapproach is too much conservative since the hysteretic steeldampers have low yield displacement (2 mm) and highenergy dissipation capacity. On the other side, the use ofhighly damped spectrumwith equivalent viscous damping inthe capacity spectrum method may not give accurate results.


Diss

ipat

ive B

race

1

Dissipative Brace 7

Diss

ipat

ive B

race

2

Diss

ipat

ive B

race

3

Diss

ipat

ive B

race

4

Dissipative Brace 5 Dissipative Brace 6

Dissipative Brace 8

P20

Figure 17: Plan view of the RC building after retrofit with damped braces.

P5 P4

CHS168.3/6

CHS193.7/8

CHS193.7/10

Brad 27/40-b

Brad 34/40-b

Brad 48/40-b

Pilefoundation

GroundLevel(0.00)

Pile foundation

CHS168.3/6

CHS193.7/8

CHS193.7/10

Brad 27/40-b

Brad 34/40-b

Brad 48/40-b

P4 P10 P19P3 P2 P1P6

0.00

Brad 27/40-bCHS168.3/6

Brad 34/40-bCHS193.7/8

Brad 48/40-bCHS193.7/10

-0.10

Pile foundation

Figure 18: Section view of the school building after retrofit with damped braces.

Thus, the seismic assessment requires the nonlinear responsehistory analysis (NRHA). For this purpose, the well-knownBouc-Wen model [59] is used for the hysteretic dampers.The seismic motion consisted of two simultaneously actingaccelerograms along both horizontal directions. A groupof three pairs of time histories is considered for the LifeSafety Limit State and another group of three pairs of timehistories is applied for the Collapse Prevention Limit State.The envelope of the response quantities from all the analysesis used as the design value of the seismic effect. The artificialaccelerograms are used to define the input ground motion[60]. The choice of accelerograms follows the rules recom-mended in the seismic standards [28, 54]. The Life Safetyverification compares the member chord rotation demandin beams and columns to the corresponding chord rotation

capacity. The Collapse Prevention verification compares thedisplacement ductility demand of the hysteretic dampers tothe corresponding displacement ductility capacity.

Figure 23 shows the hysteresis loops of some steeldampers at the Collapse Prevention Limit State. The damperin the hysteretic dissipative brace 7 (see Figure 17) at thethird storey does not reach the inelastic region. This result iscommon to all the dampers of the third storey thus implyingthat these dampers are useless since their energy dissipationcapacity is not activated.

Table 13 shows the results obtained with the proceduredescribed in Section 2.2 (Final Design). In Figure 24 thehysteresis loops of the dampers in the dissipative brace 7 at theCollapse Prevention Limit State are plotted. Similar results areobtained for all the dissipative braces. It can be noticed that


Figure 19: External view after retrofit with damped braces.

Table 8: Preliminary Design. SDOF systems: (a) bare frame; (b)damped brace.

(a) SDOF system of bare frame

Direction x-x y-yme∗ 1.138 1.476 kNs2/mmD 1.417 1.183 -dp 50.00 73.85 kNs2/mmVF

u/D 3959 2535 kNVF

y/D 3880 2454 kNdFu/D 92 158 mmdFy/D 31 62 mmdp∗ 35.29 62.43 mmVF

p∗ 3886 2454 kN

(b) SDOF system of damped braces

Direction x-x y-ydmax 20 20 mmrD 5% 5% -dy 2 2 mm𝜇D 10 10 -KD∗ = KD/KB < 1 0.2 0.2 -𝜇DB = dp/d(DB)y 8.575 8.575 -rDB 0.06 0.06 -𝜉DB(%) 16% 16% -

Table 9: Preliminary design. Damped braced frame: (a) equivalentSDOF system; (b) MDOF system.

(a) SDOF system. Bare frame + damped braces

Direction x-x y-yK(DB)e 20.86 24.73 kN/mmV(DB)p = K(DB)e ∗ dp 1043 1826 kNV(DB)y = V(DB)p /(1 + rDB(𝜇DB − 1) 719 1260 kN𝜉e 9.63% 11.14% -R𝜉 0.78 0.73 -TD 2.86 2.86 secSDC 303 303 mmTeq 0.61 0.95 sec

(b) MDOF system

Direction x-x y-ydp 50.00 73.85 mmK(DB)e 20.85 24.73 kN/mmV(DB)p = K(DB)e x dp 1043 1826 kNV(DB)y 719 1260 kN𝜇DB 8.58 8.58 -d(DB)y = dp/𝜇DB 5.83 8.61 mm


Table10:P

relim

inarydesig

n.Seism

icdesig

nparameters.

Distrib

utionof

base

sheara

long

theh

eight

Floo

rMass

[kNs2/m

m]

Dire

ctionx-x

Dire

ctiony-y

Eigenvector

Braceinclin

ation

F iV(D

B)y,i

V(D

B)y,i/co

s𝛼K(

DB)

iEigenvector

Braceinclin

ation

F iV(D

B)y,i

V(D

B)y,i/co

s𝛼K(

DB)

i[Φ 1

][𝛼]

[kN]

[kN]

[kN]

[kN/m

m]

[Φ 1]

[𝛼][kN

][kN

][kN

][kN

/mm]

10.622

0.331

55136

719

1253

1132.80

0.331

47239

1260

1847

950.54

20.582

0.665

49256

583

888

694.00

0.665

4644

91021

1457

721.3

13

0.429

1.000

49284

326

497

388.68

1.000

46498

572

816

403.98


Table11:P

relim

inarydesig

n.Solutio

n1:Bu

cklin

g-restrained

braces

(BRB

).

(a)Distrib

utionof

BRBalon

gtheh

eight,directionx-x

Floo

rBR

Btype

K(D)

iN

y,i

L BRB

L TOT.

L Brace

SteelB

race

ABrace

K Brace

K(DB)

iK(

DB)

iTO

T[kN

/mm]

[kN]

mm

mm

mm

[mm2]

[kN/m

m]

[kN/m

m]

(kN/m

m]

148/40

210

417

1640

3400

1760

Φ193.7/10

5770

688.47

160.92

1287.33

234/40

153

301

1625

3660

2035

Φ193.7/8

4670

481.9

2116.13

929.0

43

27/40

123

239

1585

3700

2115

Φ168.3/6

3060

303.83

87.55

700.44

(b)Distrib

utionof

BRBalon

gtheh

eight,directiony-y

Floo

rBR

Btype

K(D)

iN

y,i

L BRB

L tot.1

L tot.2

L Brace1

L Brace2

SteelB

race

ABrace

K Brace1

K Brace2

K(DB)1

iK(

DB)2

iK(

DB)

iTO

T[kN

/mm]

[kN]

mm

mm

mm

mm

mm

[mm2]

[kN/m

m]

[kN/m

m]

[kN/m

m]

[kN/m

m]

[kN/m

m]

148/40

210

417

1640

4220

3911

2580

2271

Φ193.7/10

5770

469.6

5533.55

145.11

150.69

1172.06

234/40

153

301

1625

4310

4000

2685

2375

Φ193.7/8

4670

365.25

412.93

107.8

3111.6

4870.26

327/40

123

239

1585

4310

4000

2725

2415

Φ168.3/6

3060

235.82

266.09

80.84

84.12

653.25


Table12:P

relim

inarydesig

n.Solutio

n2:ste

elhyste

retic

dampers(H

BF).

(a)Distrib

utionof

HBF

alon

gtheh

eight,directionx-x

Floo

rDiss

ipativeB

races

SteelB

race

ABrace

L Brace

K Brace

HBF

type

Ny,i

L HBF

K HBF

L tot.

K(DB)

iTO

T[m

m2]

mm

[kN/m

m]

[kN]

mm

[kN/m

m]

mm

[kN/m

m]

1(1-2-3-4)

Φ193.7/20

10908

2.035

1126

420/40

418

0.58

22.35

2.62

21.915

2(1-2-3-4)

Φ193.7/14.2

8004

2.29

734

300/40

300

0.575

15.90

2.87

15.563

3(1-2-3-4)

Φ168.3/10

4981

2.44

428

240/40

238

0.575

12.45

3.02

12.098

(b)Distrib

utionof

HBF

alon

gtheh

eight,directiony-y

Floo

rDiss

ipativeB

races

SteelB

race

ABrace

L Brace

K Brace

HBF

type

Ny,i

L HBF

K HBF

L tot.

K(DB)

iTO

T[m

m2]

mm

[kN/m

m]

[kN]

mm

[kN/m

m]

mm

[kN/m

m]

1(5-6-8)

Φ193.7/16

8928

2.71

692

420/40

418

0.58

22.35

3.29

21.651

2(5-6-8)

Φ193.7/12.5

7112

2.96

505

300/40

300

0.575

15.90

3.54

15.414

3(5-6-8)

Φ168.3/8.8

4407

3.045

304

240/40

238

0.575

12.45

3.62

11.960

1(7)

Φ193.7/17.5

9682

2.42

840

420/40

418

0.58

22.35

3.00

21.771

2(7)

Φ193.7/12.5

7112

2.655

562

300/40

300

0.575

15.90

3.23

15.463

3(7)

Φ168.3/8.8

4407

2.73

339

240/40

238

0.575

12.45

3.31

12.009


Figure 20: Internal view of damped braces.

First mode shapeFlexural Y

Period 0.433 sec

Second mode shapeFlexural X

Period 0.393 sec

Third mode shapeTorsional

Period 0.350 sec(Ｒ=0.77%; Ｓ=86.57%) (Ｒ=85.18%; Ｓ=2.43%) (Ｒ=6.59%; Ｓ=5.77%)

Figure 21: Fundamental mode shapes of the building retrofitted with damped braces.

the proposed design method is able to give very large plasticstrains in all the dampers during the earthquake groundmotion thus increasing the energy dissipation capacity of thestructure.

Figures 25 and 26 show the height-wise distributionsof storey drift ratio at the Life Safety Limit State along Xand Y-direction, respectively. The envelope value from threenonlinear time history analyses under bidirectional ground


Table13:Finaldesig

n.Bu

cklin

g-restrained

braces

(BRB

).

(a)Distrib

utionof

BRBalon

gtheh

eight,directionx-x

Floo

rBR

Btype

K(D)

iN

y,i

L BRB

L TOT.

L Brace

SteelB

race

ABrace.

K Brace

K(DB)

iK(

DB)

iTO

T[kN/m

m]

[kN]

mm

mm

mm

[mm2]

[kN/m

m]

[kN/m

m]

(kN/m

m]

148/40

210

417

1640

3400

1760

Φ193,7/10

5770

688.47

160.92

1287.33

227/40

123

239

1585

3660

2075

Φ193,7/6

3540

358.27

91.56

732.51

314/40

60119

1560

3700

2140

Φ168,3/4

2060

202.15

46.27

370.14

(b)Distrib

utionof

BRBalon

gtheh

eight,directiony-y

Floo

rBR

Btype

K(D)

iN

y,i

L BRB

L tot.1

L tot.2

L brace1

L brace2

SteelB

race

ABrace.

K Brace1

K Brace2

K(DB)1

iK(

DB)2

iK(

DB)

iTO

T[kN/m

m]

[kN]

mm

mm

mm

mm

mm

[mm2]

[kN/m

m]

[kN/m

m]

[kN/m

m]

[kN/m

m]

(kN/m

m]

148/40

210

417

1640

4220

3911

2580

2271

Φ193,7/10

5770

469.6

5533.55

145.11

150.69

1172.06

227/40

123

239

1585

4310

4000

2725

2415

Φ193,7/6

3540

272.81

307.8

384.78

87.88

684.43

314/40

60119

1560

4310

4000

2750

2440

Φ168,3/4

2060

157.31

177.3

043.43

44.83

350.26


0100020003000400050006000700080009000

10000

−0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 0.15 0.20

Base

She

ar (K

N)

Displacement (m)


First Mode X0

100020003000400050006000700080009000

10000

−0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 0.15 0.20

Base

She

ar (K

N)

Displacement (m)


Uniform X

0

1000

2000

3000

4000

5000

6000

7000

8000

−0.30 −0.20 −0.10 0.00 0.10 0.20 0.30

Base

She

ar (K

N)

Displacement (m)


First Mode Y0

1000

2000

3000

4000

5000

6000

7000

8000

−0.30 −0.20 −0.10 0.00 0.10 0.20 0.30

Base

She

ar (K

N)

Displacement (m)

Unifom Y


Figure 22: Pushover curves of the retrofitted building and performance points for each limit state.

−500−400−300−200−100

0100200300400500

−15 −10 −5 0 5 10 15 20

Forc

e (kN

)

Displacement [mm]

1st Floor−400

−300

−200

−100

0

100

200

300

400

−10 −5 0 5

Forc

e (kN

)

Displacement [mm]

2nd Floor−250

−200−150−100−50

050

100150200

−5 0 5

Forc

e (kN

)

Displacement [mm]

3rd Floor

Figure 23: Hysteresis loops of the dampers (Dissipative Brace 7), Preliminary Design. Collapse Prevention Limit State.

motions is plotted. The following cases are compared: (1)existing RC structure; (2) structure retrofitted with hystereticdissipative braces sized by the preliminary design; (3) struc-ture retrofitted with hysteretic dissipative braces sized by thefinal design. The existing structure shows higher values of

first storey drift due to the soft-storey mechanism failure.The structure retrofitted by damped braces sized by thepreliminary design shows lower values of storey drift butmaintains unchanged their nonuniform distribution alongthe height. The proposed design method (Final design) is


−600−500−400−300−200−100

0100200300400500

−30 −25 −20 −15 −10 −5 0 5 10 15 20

Forc

e (kN

)

Displacement [mm]

1st Floor−300

−200

−100

0

100

200

300

−10 −5 0 5 10

Forc

e (kN

)

Displacement [mm]

2nd Floor−150

−100

−50

0

50

100

150

−5 0 5

Forc

e (kN

)

Displacement [mm]

3rd Floor

Figure 24: Hysteresis loops of the dampers (Dissipative Brace 7), Final Design. Collapse Prevention Limit State.

0

1

2

3

0.00% 0.50% 1.00% 1.50%

Stor

ey L

evel

Storey Drift Ratio (X-Direction) Storey Drift Ratio (X-Direction) Storey Drift Ratio (X-Direction)

A

B

EXISTING STRUCTURE

0

1

2

3

0.00% 0.50% 1.00% 1.50%

Stor

ey L

evel

A

B

PRELIMINARY DESIGN

0

1

2

3

0.00% 0.50% 1.00% 1.50%

Stor

ey L

evel

COLUMN ACOLUMN B

COLUMN ACOLUMN B

COLUMN ACOLUMN B

A

B

FINAL DESIGN

Figure 25: Height-wise distribution of storey drift ratio (X-Direction). Envelope value from 3 nonlinear time history analyses. Life SafetyLimit State.

Storey Drift Ratio (Y-Direction) Storey Drift Ratio (Y-Direction) Storey Drift Ratio (Y-Direction)

COLUMN ACOLUMN B

COLUMN ACOLUMN B

COLUMN ACOLUMN B

0

1

2

3

0.00% 0.50% 1.00% 1.50%

Stor

ey L

evel

EXISTING STRUCTURE

A

B

0

1

2

3

0.00% 0.50% 1.00% 1.50%

Stor

ey L

evel

A

B

PRELIMINARY DESIGN

0

1

2

3

0.00% 0.50% 1.00% 1.50%

Stor

ey L

evel

A

B

FINAL DESIGN

Figure 26: Height-wise distribution of storey drift ratio (Y-Direction). Envelope value from 3 nonlinear time history analyses. Life SafetyLimit State.


successful in giving a more uniform height-wise distributionof storey drift thus improving the seismic behaviour of thestructure.

5. Conclusions

Thepaper proposes a designmethod for seismic retrofit of RCbuildings with hysteretic dissipative braces. This method isbased on the displacement-based design and on the adaptiveversion of the capacity spectrum method and explicitlyaccounts for the effects of frame-damped brace interactions.The lateral forces for seismic design are distributed alongthe height of the building according to the inelastic stateof the structure. The optimal distribution of dampers isselected through an iterative procedure. The accuracy of theproposed method is validated by comparison with nonlineartime history analysis. This study has led to the followingconclusions:

(1) The proposed method has been applied to two com-prehensive case studies (a plane RC frame and a realschool building with RC framed structure) showingits effectiveness to address the main issues of seismicdesign of damped braces: effects of frame-damperinteractions, higher modes contribution, effect ofsoft-storey irregularities, and torsion effect in asym-metric building.

(2) The effects of frame-damped braces interactionshave been addressed through the adaptive pushoveranalysis of the dual RC-damped brace system. Thisallowed accurately estimating the target displace-ment accounting for the real seismic demands ofthe damped braced frame. These effects have shownto be particularly important since the introductionof damped braces leads to an increase of the axialforces in the columns that reduces their deformationcapacity. In one of the cases here examined this hasled to the premature failure of the structure since thechord rotation capacity at the Life Safety Limit State isexceeded. Clearly, these results cannot be highlightedby design procedures neglecting the frame-dampedbraces interactions.

(3) The issues of local weaknesses of certain stories(weak storey) have been discussed in detail. It hasbeen demonstrated that the proposed design methodis successful to arrange the damper stiffness andstrength along the height of building so as to givea more uniform height-wise distributions of storeydrift, reduce the storey drift demand below the driftcapacity and assure very large plastic strains in allthe dampers. The design methods based on optimaldampers distributions can produce the uniform dis-tribution of peak storey drift. However, they suggestthat no damper should be inserted in case of so called“upper-deformed type” frames. This gives a concen-tration of shear forces in the first storey columnsbelow the damped braces that might prematurelyproduce the collapse of the structure. This effect

cannot be evidenced by these methods since they arebased on an ideal shear beam model neglecting theframe-damped braces interactions. On the other side,the conventional design procedures based on the pro-portional stiffness criterion used for the preliminarydesign has evidenced a non-uniform distribution ofstorey drifts and local ductility demands along theheight.

(4) The location of the damped braces in the buildingplan is of key importance in current practice. Manyexisting buildings are designed without considerationon seismic loading and, thus, they are often irreg-ular in plan and/or elevation with the consequentlateral-torsional coupling. The proposed procedurehas been applied also to buildings with plan irregu-larities (represented by a L-shaped in plan RC schoolbuilding). Many design procedures in the literatureare based on lumped-mass models, and, thus, theymay be rigorously applied only to symmetric-in-plan buildings. On the other side, the conventionaldesign procedures based on the proportional stiffnesscriterion uses the hypothesis that the mode shapesof the structure may be considered unchanged evenafter inserting the damped braces. This hypothesis isnot well verified in case of irregular buildings sincethe damped braces are positioned in plan to improvethe seismic behaviour of the building and significantlyreduce the lateral-torsional coupling effect.

(5) Results of the nonlinear time history analyses haveshown that the proposed method can be used incurrent practice as a substitute to the traditionalforce-based design method for seismic retrofit of RCbuildings with damped braces.

Finally, it is important to emphasize that the equivalentviscous damping of the damped braced structure is a veryimportant design parameter. Thus, future developments arerequired to calibrate this parameter on the basis of hystereticresponses obtained from nonlinear cyclic analyses of dampedbraced frames.

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors wish to express their gratitude to Eng. GennaroDi Lauro and all the professional team belonging to AiresIngegneria for their important support during the on-siteactivities.


References

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