Dumpers Design

Bull Earthquake Eng (2013) 11:2429–2446DOI 10.1007/s10518-013-9474-z

ORIGINAL RESEARCH PAPER

On the dimensioning of viscous dampers for themitigation of the earthquake-induced effects inmoment-resisting frame structures

Michele Palermo · Saverio Muscio · Stefano Silvestri ·Luca Landi · Tomaso Trombetti

Received: 11 December 2012 / Accepted: 23 June 2013 / Published online: 5 July 2013© Springer Science+Business Media Dordrecht 2013

Abstract The effectiveness of viscous dampers in mitigating the seismic excitation impactsupon building structures has been widely proved. Recently, with reference to the specific caseof equal mass, equal stiffness, shear-type structures, the authors developed a direct practicalprocedure which gives the mechanical characteristics of the manufactured viscous damperscapable of providing the frame structure with a prescribed value of the first damping ratio. Inthis paper, a comprehensive rational framework is presented, which allows to formally extendthe validity of the proposed procedure to the more realistic case of a generic moment-resistingframe structure. Also the influence of various lateral stiffness distributions is investigated.

Keywords Moment-resisting frame structures · Added viscous dampers · Designprocedure · Seismic response · Target damping ratio

1 Introduction

Manufactured viscous dampers are hydraulic devices which can be installed in structures inorder to mitigate the seismic effects through dissipating the kinetic energy transmitted bythe earthquake to the structure (Soong and Dargush 1997; Constantinou et al. 1998; Hart

M. Palermo (B) · S. Muscio · S. Silvestri · L. Landi · T. TrombettiDepartment DICAM, University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italye-mail: [email protected]

S. Muscioe-mail: [email protected]

S. Silvestrie-mail: [email protected]

L. Landie-mail: [email protected]

T. Trombettie-mail: [email protected]

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2430 Bull Earthquake Eng (2013) 11:2429–2446

and Wong 2000; Chopra 1995; Christopoulos and Filiatrault 2006). These devices have beenthe object of several research works since the 1980s (Constantinou and Tadjbakhsh 1983;Constantinou and Symans 1993; Takewaki 1997, 2000 and 2009, Singh and Moreschi 2002;Trombetti and Silvestri 2004, 2007; Levy and Lavan 2006; Silvestri and Trombetti 2007; Dio-tallevi et al. 2012; Adachi et al. 2013). Most of these works basically develop sophisticatedalgorithms for dampers optimization, sometimes leading to complex design procedures. Nev-ertheless, even if all the above cited works are remarkable from a scientific point of view, theyrequire a computational effort which is often beyond the common capabilities of the practicalengineers. Indeed, the issue of developing a practical method (i.e. a direct and immediatehelp for the practitioners) in order to size the viscous dampers which are capable of achievinga target level of seismic performances is still open.

In this respect, in 1992, the report NCEER-92-0032 (Constantinou and Symans 1992)first investigated the problem of selecting the damping coefficient of linear viscous dampersin an elastic system to provide a specific damping ratio. In 2000, the report MCEER-00-0010 (Ramirez et al. 2000) proposed an analytical relationship between the viscous dampingratio in a given mode of vibration and the damping coefficients on the basis of an energeticapproach, assuming a given undamped mode shape. Starting from the fundamental resultsof this research work, other methods have been proposed. Among these, the most useful forthe practitioners are likely to be the following ones: (1) Lopez Garcia (2001) developed asimple algorithm for optimal damper configuration in Multi-Degree-Of-Freedom (MDOF)structures, assuming a constant inter-storey height and a straight-line first modal shape; (2)ASCE 7 (2005) absorbed the MCEER-00-0010 approach.

However, also alternative approaches leading to practical design procedures for the sizingof the viscous dampers have been proposed in the years. Among these, the following onesshould be mentioned: (1) Christopoulos and Filiatrault (2006) suggested a design approachfor estimating the damping coefficients of added viscous dampers consisting in a trial anderror procedure; (2) Silvestri et al. (2010) proposed a direct design approach, called “thefive-step procedure”. A recent work by Whittle et al. (2012) compares the effectiveness ofsome of the above mentioned design approaches.

The five-step procedure, which has been proposed in 2010 by some of the authors, aimsat guiding the professional engineer from the choice of the target objective performance tothe identification of the mechanical characteristics (i.e. damping coefficient and oil stiff-ness, for given alpha exponent) of commercially available viscous dampers. The originalversion of the procedure (Silvestri et al. 2010) was developed with reference to the follow-ing assumptions: (1) a Shear-Type (referred hereafter to as ST) structure schematisation, (2)uniform lateral stiffness distribution along the height of the building, and (3) equal floormasses. On the other hand, in the same work (Silvestri et al. 2010), the authors addedtwo simple applications of the procedure developed on two moment-resisting frames (areinforced-concrete frame and a steel frame), which lead to overtake the assumption of STschematization.

The purpose of the present research work is to provide a comprehensive theoretical frame-work, which also allows to formally extend the validity of the proposed procedure to thegeneral case of a Flexible-Type (FT) structure schematisation, i.e. a structural model whichconsiders the actual stiffness of the beams. This objective will require a further insight(Cheng 2001; Occhiuzzi 2009) into the damping properties of systems characterized bynot proportional damping. Moreover, a parametric study will be presented to show theinfluence of the lateral stiffness distribution on the effectiveness of the proposed proce-dure.

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2 The rationale behind the five-step procedure

Most of the procedures proposed in the scientific literature are grounded on the energeticapproach suggested by the report MCEER-00-0010: the viscous damping ratio in a givenmode of vibration is analytically obtained as a function of the energy dissipated by thedamping system per cycle of motion in the given mode, and the maximum strain energy ofthe system. However, an assumption on the modal shape is required in order to numericallyevaluate the damping ratio.

In recent years, the authors faced the same problem in an alternative way: starting fromthe study of the equations of motion for damped ST structures (Trombetti and Silvestri2006), a direct five-step procedure has been proposed for the dimensioning of the dampingcoefficients of the viscous dampers (Silvestri et al. 2010). The core of this procedure (i.e.Step 2) lies in the simple analytical relationship between the fundamental damping ratio andthe total damping coefficient of the damper system which only requires the knowledge of thefloor masses and the fundamental period of vibration of the structure. No assumption on thefundamental modal shape is necessary, given that the effective modal shape of the first modeof vibration of an equal mass, equal stiffness, ST system is implicitly used. The interestedreader is referred to the discussion of the underlying eigenproblem presented in sections 6,7 and 8 of the paper by Trombetti and Silvestri (2006).

The five-step procedure has been formally developed assuming:

i Shear-type (ST) schematisation of the structure;ii Equal lateral stiffness distribution (along the height of the building);

iii Equal mass distribution (along the height of the building).

Provided that assumption (iii) is close to the actual characteristics of common frame struc-tures, the objective of this research work is twofold:

i To extend the procedure to generic moment-resisting frames (i.e. to remove the STassumption);

ii To investigate the effectiveness of the procedure with reference to other stiffness distri-butions.

The first objective will be reached in Sect. 3, by means of an original comprehensive frame-work and a parametric study carried out in the field of complex damping theory.

The second objective will be reached in Sect. 4, by means of a parametric study. Oncethe characteristics of the linear viscous dampers are obtained through the first two steps ofthe procedure, the designer can identify the characteristics of the actual non-linear viscousdampers (i.e. identification of a system of manufactured viscous dampers) capable of pro-viding the structure with actions (on the structural members) comparable to those obtainedusing the linear viscous dampers identified in Step 2. All the details, including applicationsof the procedure, can be found in the work by Silvestri et al. (2010).

3 Objective 1: the influence of the frame flexibility on the effectiveness of the proposedprocedure

3.1 Problem formulation

The “system” defined here is composed of a 2-D specific frame structure (characterized byfloor mass mi = m ∀ i = 1,…,N ; column moment of inertia Ji , with i = 1,. . . , N ; N indicates

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Fig. 1 Schematic representationof: a system A; b system B; csystem C

the total number of storeys) equipped with a specific dampers system (damping coefficientsc j , j = 1,. . . N x n, where n is the number of dampers per each floor).

System A, graphically represented in Fig. 1a, is defined as a FT frame structure (floormass m A,i , column’s moment of inertia JA,i ) equipped with Inter-Storey viscous dampers(damping coefficients cA, j ; total damping coefficient cA,tot , defined as the sum of thecA, j ).

The first objective of this research work is the identification of the total damping coefficientcA,tot of system A in order to obtain a target value, ξ̄ , of the first damping ratio of system A,ξA:

cA,tot = f (ξA) with ξA = ξ̄ (1)

For this purpose, it is necessary to introduce the following auxiliary systems:

• System B, that is graphically represented in Fig. 1b, is defined as the ST frame structure,with the same properties (m B,i = m A,i , JB,i = JA,i ) as defined in system A, but differenttranslational stiffness (kB,i �= kA,i , due to the restrained rotations of the nodes), equippedwith Inter-Storey viscous dampers, characterised by the same total damping coefficientof the damping system of system A (cB,tot = cA,tot ). It should be mentioned that thefundamental frequencies of the two systems, ωB and ωA, are different.

• System C, that is graphically represented in Fig. 1c is defined as the ST frame structure,with the same properties (mC,i = m B,i , JC,i = JB,i , and same restrained rotations of the

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Fig. 2 Flowchart of the scheme adopted to reach the first objective of the research work

nodes) as defined in system B, equipped with Fixed-Point viscous dampers, characterisedby the same total damping coefficient of the damping system of systems B and A (cC,tot =cB,tot = cA,tot ). Clearly, systems B and C are characterized by the same fundamentalfrequency (ωB is equal to ωC ). According to structural dynamics (Chopra 1995), it is wellknown that the total damping coefficient of system C can be expressed as a function ofthe corresponding damping ratio by the following equation (mass proportional dampingschematization):

cC,tot = 2 · ξC · ωC · mtot (2)

Since no analytical relationships are currently available in order to express the total dampingcoefficient of system A as a function of its fundamental damping ratio (i.e. Eq. (1)), then theprocedure schematically illustrated in the flowchart reported in Fig. 2 is introduced.

Provided that the three systems (A, B, and C) have been chosen so that:

cA,tot = cB,tot = cC,tot (3)

Equation (2) may be rewritten as follows:

cA,tot = 2 · ξC · ωC · mtot (4)

At this point, in order to achieve the objective stated by Eq. (1), ξC should be expressed as afunction of ξA. This is obtained in two stages through the introduction of system B.

The first stage consists in the derivation of the relationship between the fundamentaldamping ratios of systems C and B, and also represents the fundamental result of a previousresearch work developed by the authors (Trombetti and Silvestri 2006).

The second stage consists in the derivation of the relationship between the fundamentaldamping ratios of systems B and A, and represents the core of the present research work.Details are given in the next sections.

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Fig. 3 Effectiveness of the approximation km·ω2

C

∼= N (N+1)2 for N varying between 1 and 20

3.2 The relationship between the fundamental damping ratios of systems B and C

In a previous research work (Trombetti and Silvestri 2006), under the assumption of equallateral stiffness and floor mass at every storey (mi = m,∀i and ki = k,∀i) and under theequal total damping coefficient constraint, the authors demonstrated that the fundamentaldamping ratio of system C can be expressed as a function of the fundamental damping ratioof system B, as follows:

ξC = k

m · ω2C

· ξB (5)

that can be well approximated by (Trombetti and Silvestri 2006):

ξC ∼= N (N + 1)

2· ξB (6)

where N is the total number of storeys.Figure 3 shows the effectiveness of the approximation k

m·ω2C

∼= N (N+1)2 for N varying

between 1 and 20. The quality of the approximation increases as the total number of storeysdecreases.

3.3 The relationship between the fundamental damping ratios of systems A and B

In the previous sections it has been shown that, while for systems B and C it is possible todefine the damping matrix on the basis of the Stiffness Proportional Damping (SPD) or theMass Proportional Damping (MPD) limiting cases respectively, for system A it is necessaryto resort to the complex damping theory (Cheng 2001; Occhiuzzi 2009). In detail, insteadof searching for an exact analytical relationship between the fundamental damping ratios ofsystems A and B, an approximated numerical one has been searched.

For damped Single-Degree-Of-Freedom (SDOF) systems, an analytical relationshipbetween the fundamental damping ratios of systems A and B can be drawn starting fromthe basic concepts of structural dynamics (Sect. 3.3.1).

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Fig. 4 a Shear-Type SDOFsystem equipped with viscousdamper; b Flexible-Type SDOFsystem equipped with viscousdamper

Fig. 5 Schematic representation of the MDOF frame systems used to perform the numerical analysis

For damped MDOF systems, a numerical analysis performed in the field of complexdamping has been conducted in order to verify if the same relationship still holds (Sect.3.3.2).

3.3.1 The basic idea

Two equivalent (same mass m, same column moment of inertia J and same damping coef-ficient c) SDOF systems, represented in Fig. 4, are considered. The first one represents aone-storey one-bay ST structure equipped with an Inter-Storey viscous damper system; andthe second one represents a one-storey one-bay FT structure equipped with an Inter-Storeyviscous damper system. Due to the difference in the beam stiffness, the fundamental circularfrequencies (ωST and ωFT ) and periods (TST and TFT ) of the two systems are not equal.Under the above mentioned assumptions, it is easy to show that the ratio of the modal damp-ing ratios pertained to the two systems is exactly the same as the corresponding ratio of thefundamental periods:

ρξ = ξFT

ξST= c/ (2 · m · ωFT )

c/ (2 · m · ωST )= ωST

ωFT= TFT

TST= ρT (7)

3.3.2 Numerical analyses: main results

The numerical analysis has been carried out on the FT structures schematized in Fig. 5, thatare characterized by the following main properties:

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0 1 2 3 4 5 infinity1

1.2

1.4

1.6

1.8

2

2.2

ρR

ρξ

2-storey systems3-storey systems4-storey systems5-storey systems6-storey systems

Fig. 6 ρξ versus ρR

– Total number of storeys, N , variable from 2 to 6;– Number of bays equal to 1;– Bay width equal to 6 m;– Inter-storey height equal to 3 m;– Square columns with constant cross section 40 cm × 40 cm at each storey (fixed for all

the models);– Beams with constant cross section for all storeys (different for each model);– Floor mass m equal to 80,000 kg;– Beam–column stiffness ratio, ρR = kbeam/kcolumn , variable from 0.5 to ∞ (ST system);– Elastic material with Young’s modulus, E , equal to 20,000 MPa.

In these analyses, where the main purpose is to investigate the influence of the flexural flexi-bility of beams, the axial flexibility of the structural elements has been neglected. Consideringthe kth FT system, each single analysis included the following phases: (1) evaluation of thefundamental modal damping ratio, ξFT,k , by means of the complex damping theory (Cheng2001; the interested reader can find additional details in Muscio 2009); (2) evaluation ofthe first modal damping ratio of the equivalent ST system by means of the Rayleigh theory,ξST,k ; (3) evaluation of the ratio between the fundamental modal damping ratio of the FTstructure and the one of the equivalent ST structure, ρξK ; (4) evaluation of the ratio betweenthe fundamental period of the FT structure and the one of the equivalent ST structure, ρTK .The main results are briefly illustrated through Figs. 6, 7, 8.

Figures 6 and 7 display the ratios ρξ and ρT versus the ratio ρR , respectively. Inspectionof the graphs leads to the following deductions:

• For all values, ρξ is always higher than the corresponding ρT ;• As it might be reasonably expected, for all ρR values, both ρξ and ρT increase with the

increase of the total number of storeys N ;• As it might be reasonably expected, both ρξ and ρT decrease with the increase of ρR .

Figure 8 renders the graphical representation of the correlation between ρξ and ρT . Inspectionof the graph leads to the following deductions:

• ρξ and ρT exhibit a strong linear correlation, with correlation coefficient equal to 0.998;

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0 1 2 3 4 5 infinity1

1.2

1.4

1.6

1.8

2

2.2

ρR

ρT


Fig. 7 ρT versus ρR

Fig. 8 ρξ versus ρT

1 1.2 1.4 1.6 1.8 2 2.21

1.2

1.4

1.6

1.8

2

2.2

ρT

ρξ


• ρξ values are larger than the corresponding ρT values; thus, ρT can be assumed as alower bound for ρξ ;

• For practical applications, ρξ can be approximately assumed equal to ρT ; the assumptionρξ

∼= ρT is conservative.

Based on the main result of these numerical simulations (ρξ∼= ρT ), the fundamental damping

ratio of system B can be expressed as a function of the fundamental damping ratio of systemA, as follows:

ξB ∼= ξA · TB

TA(8)

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or:

ξB ∼= ξA · ωA

ωB(9)

which represents the extension (for the case of MDOF systems) of the result given by Eq.(7) for the case of SDOF systems.

3.4 The total damping coefficient for system A

Substitution of Eq. (9) into (6) leads to:

ξC ∼= N (N + 1)

2· ωA

ωB· ξA (10)

Substitution of Eq. (10) into (4) leads to:

cA,tot = 2 · N (N + 1)

2· ωA

ωB· ξA · ωC · mtot (11)

Given that ωB = ωC , Eq. (11) can be finally simplified as follows:

cA,tot = N (N + 1) · ωA · ξA · mtot (12)

Imposing that the first modal damping ratio of system A is equal to a target damping ratio(ξA = ξ̄ ) leads to the following fundamental result:

cA,tot = N (N + 1) · ωA · ξ̄ · mtot (13)

which represents a simple formula for the dimensioning of the total damping coefficient ofsystem A (in a given direction) in order to achieve the target damping ratio ξ̄ .

It is possible to observe that Eq. (13) is formally coincident with Eq. (27) of (Silvestriet al. 2010). However, while the latter was derived assuming a ST structure schematization,Eq. (13) keeps its validity also for a generic FT frame structure.

3.5 The dimensioning of each viscous damper

Once the total damping coefficient cA,tot has been obtained, the damping coefficient of eachsingle damper should be evaluated.

Several studies have been carried out in the past in order to estimate the optimal dampersdistribution for inter-storey dampers placement. In a recent work, Takewaki (1997) provided acomprehensive literature review on this topic. Basically, to date, the design methods of addedviscous dampers to buildings may be subdivided into two main categories. The first is themost commonly used method by practical engineers, which has focused on the developmentof simple design formulas for calculating the added damping ratio to the building, likethe approach proposed in this paper. However, adopting these design formulas, a limitednumber of methods (Lee et al. 2008; Hwang et al. 2013) have been provided on how todistribute the total required damping coefficients to each storey. The second category includesseveral design methods based on iterative procedures aimed at obtaining the “optimal” damperdistribution which satisfies a certain target objective (a complete list of the works on thissubject may be found in the introduction of the book by Takewaki (2009)).

Almost all these studies highlighted that, for regular stiffness and mass distributions, auniform added dampers distribution usually leads to a building performance which is closeto the “optimal” solution. Thus, for design purposes, in the case of regular buildings, once thetotal damping coefficient ctot is evaluated, a uniform dampers distribution can be adopted.

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1

2

3

4

5

6

7

1 3 5 7 9 11 13 15 17

Stor

ey

Normalized stiffness

SD-IISD-I

Fig. 9 The two stiffness distributions (SD-I and SD-II) adopted for the parametric study

Under this assumption, the storey damping coefficient cstorey is given by the followingrelationship:

cstorey = (N + 1) · ωA · ξ̄ · mtot (14)

If n equal dampers are installed at each storey (diagonal inter-storey dampers), the dampingcoefficient c of each viscous damper is given by the following simple formula:

c = (N + 1) · ωA · ξ̄ · mtot

n · cos2 θ(15)

where the term cos2θ accounts for the inclination of the damper with respect to the horizontal.A further insight into the assumption of different stiffness distributions is provided in the

next section.

4 Objective 2: the influence of the lateral stiffness distribution on the effectiveness ofthe proposed procedure

4.1 Problem formulation

Let us consider a reference “system”, referred to as System 0, composed of a ST structure,characterised by floor mass mi = m, with i = 1,…,N ; column moment of inertia Ji , = J ,with i = 1,. . . , N (where N indicates the total number of storeys), and equipped with auniform distribution of inter-storey viscous dampers with damping ratio evaluated accordingto Eq. (14) of the present paper.

In order to investigate the effectiveness of the proposed approach for the cases of not-uniform lateral stiffness distributions, a parametric study has been developed. In detail, twodifferent not-uniform stiffness distributions (Fig. 9) are considered:

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Table 1 The “systems” defined for the parametric analysis

Stiffness distribution

SD-0 (uniform) SD-I (area-based) SD-II (inertia-based)

Dampers distribution

DD-a uniform System 0(referencesystem)

System 1 System 3

DD-b stiffnessproportional

System 2 System 4

• Stiffness distribution I (referred to as SD-I): columns cross section area linearly increasesstarting from a minimum value at the top storey. This case is representative of r/c framestructures designed for vertical loads;

• Stiffness distribution II (referred to as SD-II): lateral stiffness is assumed proportionalto the storey shear due to a triangular horizontal forces distribution (typically of anequivalent static seismic analysis).

The uniform stiffness distribution is referred to as SD-0. It should be noted that SD-I cor-responds to a constant static axial stress criterion (area-based distribution), while SD-IIcorresponds to a constant inter-storey drift criterion (inertia-based distribution).

Also two different damper distributions are considered:

• Dampers distribution “a” (referred to as DD-a): uniform dampers distribution. In thiscase, the storey damping coefficient is given by Eq. (14);

• Dampers distribution “b” (referred to as DD-b): dampers distribution (along the height)proportional to the lateral stiffness distribution (Christopoulos and Filiatrault (2006)proposed an alternative procedure in order to evaluate a dampers distribution proportionalto the stiffness distribution of the unbraced structure). In this case, the generic storeydamping coefficient at the kth storey, cstorey,k , can be calculated with the followingformula:

cstorey,k = Jk

Jtot· N (N + 1) · ω · ξ̄ · mtot (16)

which is obtained assuming a total damping coefficient ctot as per Eq. (13).Based on the above introduced lateral stiffness and viscous dampers distributions, in

order to investigate the effectiveness of the procedure for different stiffness distributions, thefollowing classes of “systems” are introduced as (see Table 1):

• System 1: the system characterised by the same mass distribution of System 0, SD-I andDD-a;

• System 2: the system characterised by the same mass distribution of System 0, SD-I andDD-b;

• System 3: the system characterised by the same mass distribution of System 0, SD-II andDD-a;

• System 4: the system characterised by the same mass distribution of System 0, SD-II andDD-b.

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Table 2 Effectiveness of the procedure in terms of achieved fundamental damping ratio given a target dampingratio equal to 0.30

Total number of storeys System 0 System 1 System 2 System 3 System 4

7 0.39 0.51 0.31 0.39 0.43

6 0.39 0.53 0.34 0.41 0.43

5 0.39 0.48 0.37 0.44 0.44

4 0.39 0.50 0.39 0.46 0.44

3 0.38 0.51 0.40 0.46 0.43

Average 0.39 0.51 0.36 0.43 0.43

4.2 Parametric study

With reference to the 4 classes of systems introduced in the previous section (see Table 1),a parametric analysis has been carried out aimed at investigating the effectiveness of theproposed procedure. The characteristics of the systems are summarized as follows:

– Total number of storeys, N , variable from 3 to 7;– Number of bays equal to 1;– Bay width equal to 6 m;– Inter-storey height equal to 3 m;– Beams with square cross section calibrated in order to provide a ratio ρR =

kbeam/kcolumn = 1.0;– Floor mass m equal to 80,000 kg;– Elastic material with Young’s modulus, E , equal to 20,000 MPa.– Damping target ratio ξ̄ = 0.30

Numerical time-history analyses have been performed. In these analyses, both the flexuraland the axial flexibility of the structural elements have been considered. Each system has beensubjected to a base seismic input. A set of ten artificial accelerograms generated using thesoftware SIMQKE (Vanmarcke et al. 1990), compatible with the design spectrum prescribedby the Italian seismic code (NTC 2008), and corresponding to an average PGA equal to0.25 g have been used. The following response parameters have been considered: the top-storey absolute acceleration, a, the top-storey displacement, δ, and the base shear, V . Theresults of the time-history analyses are presented in terms of reduction factors ηa, ηδ and ηV .For instance, the mean value (over the ten seismic records) of the displacement reductionfactor, ηδmean , of ηδi , is computed as: ηδmean = 1

10

∑10i=1 ηδi , with ηδi = δi(ξ)

δi (ξ=0.05), where

δi (ξ=0.05) is the maximum value of the displacement response obtained for the bare system(i.e. the structure without added dampers) under the i th seismic record.

Also snap-back tests have been performed in order to calculate the equivalent first-modaldamping ratio, by means of the method of the logarithmic decrement (Chopra 1995).

4.3 The main results of the parametric study

The results of the parametric analysis are summarized through Tables 2, 3, 4, 5.First, the results of the snap-back tests are commented. Table 2 provides a summary of the

obtained damping ratios for all analyzed systems. Inspection of these results clearly showsthat the proposed procedure leads to conservative results: in all cases, the obtained dampingratio is higher than the target one (which was equal to 0.30). The average damping ratio over

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Table 3 Effectiveness of the procedure in terms of top-storey absolute acceleration reduction factor (ηamean )


7 0.30 0.24 0.21 0.23 0.25

6 0.31 0.28 0.25 0.25 0.26

5 0.32 0.32 0.29 0.26 0.27

4 0.33 0.32 0.29 0.29 0.28

3 0.35 0.35 0.33 0.33 0.33

Average 0.32 0.30 0.27 0.27 0.28

Table 4 Effectiveness of the procedure in terms of top-storey displacement reduction factor (ηδmean )


7 0.44 0.35 0.43 0.44 0.41

6 0.44 0.37 0.45 0.43 0.44

5 0.44 0.40 0.47 0.44 0.45

4 0.45 0.43 0.48 0.45 0.46

3 0.41 0.40 0.43 0.47 0.48

Average 0.44 0.39 0.45 0.45 0.45

Table 5 Effectiveness of the procedure in terms of base shear reduction factor (ηVmean )


7 0.40 0.66 0.47 0.61 0.53

6 0.41 0.65 0.46 0.60 0.54

5 0.41 0.65 0.47 0.59 0.51

4 0.43 0.59 0.44 0.55 0.54

3 0.41 0.47 0.37 0.51 0.46

Average 0.41 0.60 0.44 0.57 0.52

all systems is equal to 0.41. It has to be noted that, if only the reference cases are considered(i.e. ST systems with uniform stiffness distribution), an average damping ratio equal to 0.39 isobtained. On the contrary, if only the remaining systems (i.e. those systems with not-uniformstiffness distributions) are considered, an average damping ratio equal to 0.42 is obtained. Thissuggests that stiffness distribution does not significantly affect, on average, the effectivenessof the procedure, leading only to slightly more conservative results. Moreover, as expected, thetotal number of storeys (i.e. the fundamental period of the system) does not significantly affectthe response, provided that the influence of system flexibility is already taken into accountin the formula of the ctot (Eq. 13). The confirmation of the effectiveness of the procedurehighlights also the negligible influence of the axial flexibility of the structural elements.

As far as the single classes of systems are concerned, the following observations may begiven:

• If the SD-I is considered (i.e. the one farther from the uniform distribution with respectto the SD-II), the adoption of the two dampers distributions leads to different responses:the average damping ratio jumps from 0.50 (very conservative) for the uniform dampers

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distributions to 0.36 (slightly conservative) for the stiffness proportional dampers distri-bution.

• If the SD-II is considered, it is possible to observe that the two dampers distributions leadto very close values (i.e. “robust” behaviour) of average damping ratios: 0.42 and 0.43for the uniform dampers distributions and the stiffness proportional dampers distribution,respectively.

In order to check the snap-back tests results, a further verification has been developed: eachbare system has been subjected to a modal time history analysis with an imposed modaldamping ratios (for all modes) as obtained from the snap-back test (as also suggested byChristopoulos and Filiatrault 2006). The results of these time history analyses have beencompared with those obtained for the systems equipped with diagonal inter-storey viscousdampers having damping coefficients calculated with the proposed procedure (these resultswill be commented later in the present section). The comparison showed a good agreement,thus providing a further confirmation of the effectiveness of the damping ratios as evaluatedfrom the snap-back tests. For sake of conciseness, Fig. 10 reports only the comparison, interms of average inter-storey drift profiles, of the 3- and 7-storey systems.

1

2

3

4

5

6

7

0 0.005 0.01 0.015 0.02

stor

ey

inter-storey drift ratio

Bare

SD-II & DD-a

SD-II Modal

1

2

3

0 0.005 0.01 0.015 0.02 0.025

stor

ey


Bare

SD-I & DD-a

SD-I Modal

(a)

(b)

Fig. 10 Seismic inter-storey drift profiles for a 7-storey systems and b 3-storey systems, with and withoutadded dampers

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The results of the time-history analyses performed on the systems equipped with diagonalinter-storey dampers, summarized through Tables 3, 4, 5, lead to the following observations:

• The average top-absolute acceleration reduction factor (Table 3) is equal to 0.29. Asexpected, the top-absolute acceleration reduction is affected by the fundamental period:the reduction increases (i.e. the reduction factor decreases) as the total number of storeys(i.e. the fundamental period) increases. Reference systems exhibit an higher averagereduction factor, 0.32, with respect to the other cases, 0.28, since the former (ST struc-tures) are characterized by fundamental period quite lower than the latter. Despite theinfluence of the fundamental period, the variability of the absolute acceleration reductionfactor is very limited;

• The average top-displacement reduction factor (Table 4) is equal to 0.43 (which leads toan equivalent damping ratio equal to 0.52 calculated inverting the formula by Bommeret al. 2000). All classes of systems exhibit almost the same average reduction factor

1

2

3

4

5

6

0 0.005 0.01 0.015 0.02 0.025

stor

ey


Bare

DD-a

DD-b

1

2

3

4

5

6

0 0.005 0.01 0.015 0.02 0.025

stor

ey


Bare

DD-a

DD-b

(a)

(b)

Fig. 11 The seismic inter-storey drift profiles for the 6-storey system: a with SD-I stiffness distribution andb SD-II stiffness distribution

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(0.44–0.45 with a limited dispersion), except System 1 which shows the smallest averagereduction factor (0.39);

• The average base shear reduction factor (Table 5) is equal to 0.50. If only the referencecases are considered, an average base shear reduction factor of 0.41 is obtained. On thecontrary, if only the remaining systems are considered, an average base shear reductionfactor approximately equal to 0.52 is obtained. It can be noted that the variability of thebase shear reduction is larger than the variability of the other parameters.

In order to provide a further insight into the seismic response of the studied systems, Fig. 11displays the graphs of the average (over the ten seismic records) inter-storey drift profilesfor the specific case of the 6-storey structure. It can be observed that the DD-b (damper sizeproportional to lateral stiffness, i.e. large size at the base) reduces the first inter-storey driftmore than the DD-a (uniform dampers distribution), which on the contrary is more effectiveat the higher storeys.

5 Conclusions

This research work investigates the effectiveness of a five-step design procedure recentlyproposed by some of the authors for sizing the damping coefficients of added viscous dampersto be inserted in moment-resisting frame systems.

The procedure provides a simple design formula for the evaluation of the total dampingcoefficient of the added dampers systems leading to a specific target damping ratio, whichis only based on the knowledge of the total building mass and fundamental period. Theprocedure was originally developed for the specific case of shear-type structures with uniformdistribution of floor mass and lateral stiffness.

In this paper, a comprehensive framework is presented, which allows to formally extendthe validity of the proposed procedure to the more realistic case of a generic moment-resistingframe structure with uniform distribution of floor mass and lateral stiffness. For this purpose,numerical analyses in the field of complex damping theory have been performed.

Moreover, the effectiveness of the five-step procedure has been investigated removing theassumption of uniform stiffness distribution, through the development of a parametric study.The results show that the application of the proposed procedure for the dimensioning of theadded viscous dampers in the case of not-uniform stiffness distribution leads to slightly moreconservative results with respect to the application of the procedure in the case of uniformstiffness distribution.

Acknowledgments Financial supports of Department of Civil Protection (Reluis 2010–2013 Grant—Thematic Area 2, Research line 3, Task 2: “Development and analysis of new technologies for the seismicretrofit”) is gratefully acknowledged.

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