Engineering Structures 39 (2012) 187–198

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Engineering Structures

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A practical design method for reinforced concrete structures with viscous dampers

Ying Zhou ⇑, Xilin Lu, Dagen Weng, Ruifu ZhangState Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University, 1239 Siping Road, Shanghai 200092, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 19 July 2011Revised 7 February 2012Accepted 9 February 2012Available online 22 March 2012

Keywords:Viscous damperDesign methodReinforced concrete structureWenchuan Earthquake

0141-0296/$ - see front matter � 2012 Elsevier Ltd. Adoi:10.1016/j.engstruct.2012.02.014

⇑ Corresponding author. Tel.: +86 21 6598 6157; faE-mail addresses: yingzhou@tongji.edu.cn (Y. Zhou

wdg@tongji.edu.cn (D. Weng), zhangruifu@gmail.com

As a result of the 2008 Wenchuan Earthquake, the Chinese government issued a modified seismic codewith increased protection categories and seismic intensities. According to the new code, a lot of schooland industrial buildings need to be seismically retrofitted to satisfy the new seismic requirements. Com-pared to the retrofitting technology of seismic isolation, the installation of viscous dampers to thoseexisting buildings is more realistic because of easy construction. However, the design of viscous dampers,which provides a high level of damping in a structure, is a relatively new application in China for a well-established and proven technology in other seismically active regions in the world. Only general informa-tion on the usage of viscous damper is given in Chinese code, which would potentially confuse engineersand researchers. Thus, the intent of this paper is to propose a practical design method for reinforced con-crete (RC) structures with viscous dampers. The proposed design process is divided into two stages. In thepreliminary stage, the quantity, mechanical parameters and configurations of the viscous dampers aredetermined. In the next stage, the reduction of deformations, additional damping ratio, and connectionof the dampers to the structure are examined. An example is also given to demonstrate the applicationof the proposed method to retrofit a RC frame structure by viscous dampers. It is concluded that the pro-posed design method satisfies the urgent requirement of design and pushes the further development ofresearch on viscous dampers.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

1.1. Background

On May 12th, 2008, a magnitude of MW 8.0 earthquake hitWenchuan in China. The earthquake had a shallow focal depth ofapproximately 19 km, with the epicenter located 80km WNW ofChengdu, capital of Sichuan province. The entire country was dev-astated by the mega earthquake. The death toll of the earthquakewas of over 69,000 killed people, with over 374,000 people injuredand nearly 5 million homeless; the total lost was estimated atUS$130 billion.

After the Wenchuan Earthquake, new Chinese seismic designcodes were issued with the modifications in attempt to increaseprotection categories and seismic intensities. In Chinese code, theprotection categories of buildings specified in the Standard for Clas-sification of Seismic Protection of Building Constructions (GB50223,2008) [1] are classified into four categories: moderate protection(MP), standard protection (SP), emphasized protection (EP), andparticular protection (PP), in the order of the increasing protectionrequirement. The design requirements for the four building types

ll rights reserved.

x: +86 21 6598 2668.), lxlst@tongji.edu.cn (X. Lu),(R. Zhang).

of protection categories are different in structural details and seis-mic forces [2]. After the quake, protection categories for all classbuildings, dormitories, and dining halls in kindergartens, primaryschools, and middle schools are increased from standard seismicprotection buildings to emphasized seismic protection buildings.This modification means that the seismic forces of those schoolbuildings will be calculated commensurate with the design inten-sity while the structural details will be checked one degree higherthan the design intensity. The purpose of this modification is toprotect young and valuable students in earthquakes. On the otherhand, the seismic intensities of cities are specified in the Code forSeismic Design of Buildings (GB50011-2008) [3]. After WenchuanEarthquake, the seismic intensities of many cities in China areincreased by half degree or more. Lots of school buildings andindustrial buildings in those cities are not complied with new seis-mic code requirements and thus need to be retrofitted.

Comparing to traditional retrofitting practices, such as enlarg-ing cross sections or adding steel plates to structural elements[4], the application of supplementary viscous dampers to buildingsenables easier construction, and reduced labor and downtime.

1.2. Existing research on viscous dampers

Viscous dampers themselves are old technology, dating backto more than a century ago to full-scale usage on US large cali-

200

400

600

800

20 40 60-20-40-60

-200

-400

-600

-800

0

Fig. 1. Force–displacement curve of a nonlinear viscous damper.

188 Y. Zhou et al. / Engineering Structures 39 (2012) 187–198

ber military cannons in the 1860s. This technology was notavailable for the public disclosure or usage until the Cold Warended. In 1990, Taylor Devices received the permission to sellthis technology to the public. Despite the long history andwell-established usage of viscous damper, it is still a relativelynew building technology yet to be further developed andstudied.

Studies have been published regarding viscous dampers designmethodology. Constantinou and Symans [5] proposed a simplifiedmethod for calculating the modal characteristics of structureswith added fluid dampers. The method was used to obtain esti-mates of peak response of the tested structures by utilizing theresponse spectrum approach. Gluck et al. [6] suggested a designmethod for supplemental dampers in multi-story structures,adapting the optimal control theory by using a linear quadraticregulator (LQR) to design linear passive viscous (VS) or viscoelas-tic (VE) devices depending on their deformation and velocity. Fuand Kasai [7] compared frames dynamic behavior using VE orpure VS dampers, where identical mathematical expressions werederived in terms of two fundamental nondimensional parameters.Kasai et al. [8] proposed a simplified theory to predict and com-pare the seismic performance of VE and elastoplastic (EP) damp-ing devices. Yang et al. [9] proposed two optimal designmethodologies for passive energy dissipation devices based on ac-tive control theories leading to the determination of VS and VEdampers, defining different forms of performance functions. Leeand Taylor [10] developed the energy dissipation technologyand suggested that approximately 15–25% of additional dampingis a desirable range in the damper designed buildings. Lin et al.[11] presented a seismic displacement-based design method fornew and regular buildings equipped with passive energy dissipa-tion systems. Using the substitute structure approach for thebuilding structure and simulating the mechanical properties ofthe passive energy dissipation devices by the effective stiffnessand effective viscous damping ratio, a rational linear iterationmethod was proposed. Uetani et al. [12] proposed a practicalmethod for optimum structural design of building frames withviscous dampers. The method first did the stiffness design of a re-duced shear-building model with viscous dampers. Then the opti-mum design for building frames was performed under staticdesign loads. Design examples were presented to demonstratethe usefulness of the proposed design method. Chen et al. [13]performed elastic and elastoplastic analysis on Wenchuan Hospi-tal with VS damper to check the seismic performance of thestructure under various earthquake scenarios. The damping ratiowas estimated by a method of inputting a series of sine wavesand calculating the earthquake energy.

In the United States, ASCE/SEI 7-05 [14] and FEMA P-750 docu-ment [15] summarize design strategies for viscously dampedstructures. In addition, Sadeck et al. [16] also identified some lim-itations in the FEMA 273 [17] procedures for the design of struc-tures with velocity-dependent passive energy dissipation devicesbased on the analysis of single-degree-of-freedom structures.One of the major limitations includes an unconservative estima-tion of peak response and base shear when using a constant reduc-tion factor to obtain displacement response of short-periodstructures and assuming a harmonic response to compute the peakvelocity, story and base shear. In China, although damper technol-ogies are specified in the 2001 version and 2008 modified versionof Code for Seismic Design of Buildings (GB50011) [18], only generalinformation is given in the codes. A standardized design method istherefore needed for a more widespread and routine inclusion ofviscous damper in structural design practice.

In the following sections, a practical design method is proposedfor structures with viscous dampers. An example is also given todemonstrate the application of the proposed method.

2. Design method for structures with viscous dampers

In the preliminary stage of designing viscous dampers in astructure, the following tasks needed to be done: (1) determinethe number of viscous dampers, (2) choose the parameters of vis-cous dampers, and (3) configure the layout of viscous dampers. Inthe second stage of design, engineers should check the structuraldeformations, the additional damping ratio, and the dampers’ con-nection to other structural elements in order to ensure the work-ability of the damper systems.

2.1. Preliminary design

2.1.1. Number of viscous dampersThe essence of damper technology is to add additional damping

to structures to ‘‘eat up’’ the energy induced by wind or earth-quake. Thus the additional damping ratio is the key parameter inthe whole process of design, which governs the number of dampersand controls the effect of dissipation.

The effective damping ratio can be calculated using Eq. (1) [19],which is also specified in the Chinese code [3] and American code[14].

f ¼ Wc

4p �Wsð1Þ

where f is the effective damping ratio added by viscous damperdevices, Wc is the energy dissipated by viscous damper devices inone cycle of expected displacement of a structure, i.e., the total areaof the force–displacement curves, and Ws is the total strain energyof an energy dissipated structure under expected displacement.

Take one story for an example. The total strain energy Ws can beexpressed as Eq. (2), where F is the horizontal story shearing force;and D is the relative story displacement.

Ws ¼12

FD ð2Þ

Fig. 1 shows a typical force–displacement curve of a nonlinearviscous damper, where the area is simplified to be a parallelogram(Fig. 2). Thus, the energy dissipated by viscous dampers, Wc can becalculated as Eq. (3).

Wc ¼ 4Fd D� Fd

Kd

� �ð3Þ

where Fd the damping is force of viscous dampers and Kd is thestiffness of viscous dampers.

0

Fig. 2. Simplified parallelogram of the force–displacement curve of a nonlinearviscous damper.

Y. Zhou et al. / Engineering Structures 39 (2012) 187–198 189

Combining Eqs. (1) and (3) leads to a following expression forthe damping ratio f, as described in Eq. (4).

f ¼4Fd D� Fd

Kd

� �4p � 1

2 FD¼

2Fd 1� FdKd=D

� �p � F ð4Þ

with l ¼ Fd

Kd=D,

f ¼ 2ð1� lÞp

� Fd

Fð5Þ

In Eq. (5), l is the ratio of the damper displacement to the rel-ative story displacement (Fig. 2), which can also be seen as ductil-ity demand. In addition, it can be observed that the additionaldamping ratio f induced by the viscous dampers can be correlatedto the displacement ratio l and the force ratio Fd/F.

According to Eq. (5), the damping force is,

Fd ¼p

2ð1� lÞ � f � F ð6Þ

An important parameter in determining the damping force isthe damping ratio f. There are several guidelines. Lee and Taylor[10] suggested a practical upper limit for the combined viscousand structural damping of 25%, and a desirable range of 15% -25%for typical building. In contrast, the Code for Seismic Design of Build-ings (GB50011) [3,18] specifies that whenever the effective damp-ing ratio f including the energy dissipation devices exceeds 20%, itshould be taken as 20%. Thereby, for preliminary design, f is takenas 15%, which will be checked later for the accuracy of thisassumption.

As defined above, l is the ratio of the damper capacity to thestructural demand. In general, it is not desirable for a damper toreach its capacity either too early under a minor earthquake ortoo later under a major earthquake. It is thus assumed here that aviscous damper reaches its capacity under a moderate earthquake.

Table 1 lists the inter-story drift objectives of reinforcedconcrete (RC) structures under various earthquake levels [20]. It

Table 1Inter-story drift objectives of RC structures.

Minor earthquake level Moderate earthquake le

Operational Immediate occupancy

Frame structures 1/550 1/250Shear wall structures 1/1000 1/500Hybrid structures 1/800 1/400

is shown that the average value ofp

2ð1� lÞ can be approximatedto be 2.0. Therefore,

Fd ¼ 2:0� 0:15� F ¼ 0:3F ð7Þ

That is to say, the damping force induced by viscous dampers ineach story can be preliminarily taken as 30% of the story shearingforce. If the force capacity of each damper Fdi is chosen, then thenumber of viscous dampers is determined as Eq. (8).

n ¼ Fd

Fdið8Þ

2.1.2. Parameters of viscous dampersViscous damper is one type of velocity-dependent dampers. The

theoretical formula of a viscous damper can be given as follows.

Fdi ¼ C � jv ja � signðvÞ ð9Þ

Fdi is the damping force of a single viscous damper, while C is thedamping factor. v represents the velocity of the viscous damperand its exponential parameter a determines the relationshipbetween force and velocity. It should be evident that when a = 1,Eq. (9) expresses the relationship of linear viscous dampers.

Since the values of the aforementioned parameters substan-tially affect the behavior of a viscous damper, a sensitivity analysisof the various parameters is shown in the next section to demon-strate the reasoning behind choosing certain parameters for aviscous damper.

2.1.2.1. Parameter analysis. It is assumed that the displacement andthe velocity of dampers are expressed in Eqs. (10) and (11).

d ¼ A sin xt ð10Þ

v ¼ _d ¼ Ax cos xt ð11Þ

Thus,

Fdi ¼ C � jv ja � signðvÞ ¼ C � jAx cos xtja � signðAx cos xtÞ ð12Þ

Given A = 60 mm, f ¼ x2p ¼ 0:1 Hz, and C = 100 kN s/mm, the

force–displacement curves under various a values are shown inFig. 3a. Next, holding A and f as constants, the force–displacementcurves under various C values for a = 0.2 are shown in Fig. 3b. FromFig. 3, it can be easily found that the area inside the curve, i.e. theenergy dissipation capacity of the damper, will be larger with an in-crease in both C and a values. However, this result does not neces-sarily indicate that the larger C and a values, the better it is forstructures.

Keeping the same displacement (A = 60 mm) and force(Fdi = 3770 kN), the force–displacement curves at a = 0.2 anda = 1.0 are given in Fig. 4. It can be seen that the shape of the curveat a = 0.2 is closer to a rectangle while the shape of the curve ata = 1.0 resembles more of an ellipse. Apparently, more energy dis-sipation area will be achieved when a is taken a smaller value.Martinez-Rodrigo and Romero [21] compared the retrofitting effecton a six-story steel structure by using linear dampers and by usingnonlinear dampers. It was concluded that a nonlinear damper force

vel Major earthquake level l p2ð1� lÞ

Life safety Collapse prevention

1/100 1/50 0.20 2.01/250 1/120 0.24 2.11/200 1/100 0.25 2.1

Average 2.0

-5000

-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

5000

-80 -60 -40 -20 0 20 40 60 80

α=0.00

α=0.10

α=0.20

α=0.30

α=0.50

α=0.75

α=1.00

(a) Force-displacement curves under various α (b) Force-displacement curves under various C

(mm)

(kN)

(mm)

(kN)

Fig. 3. Parameter analysis of a nonlinear viscous damper.

-5000

-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

5000

-80 -60 -40 -20 0 20 40 60 80

α=0.20

α=1.00

(mm)

(kN)

Fig. 4. Force–displacement curves at a = 0.2 and a = 1.0.

Fig. 5. Horizontal cracks of infilled walls.

190 Y. Zhou et al. / Engineering Structures 39 (2012) 187–198

was 35% less than that of a linear damper with a equaling 1.0. Nev-ertheless, there is a fundamental trade-off between a and C. Mean-ing in order to achieve a certain amount of force using a smaller a,a larger C has to be chosen at the same time. This inverse relation-ship between a and C is also evident in Eq. (9). The value C, how-ever, is a parameter correlating to the stiffness of dampers. Anexcessively high damper stiffness would potentially create diffi-

culty in designing the structural elements connecting with thedampers. Thus, selecting the appropriate a and C values deservespecial attention in the preliminary design stage.

2.1.2.2. Parameter determination. Eq. (9) could be transformed asEq. (13).

C ¼ Fdi

jv ja � signðvÞð13Þ

A viscous damper catalogue is usually provided by the manufac-turers for several given a values. Generally, the a value selected forseismic design is smaller than that of wind resistant design. Using aTaylor device as an example, for structures with seismic demanddominating over wind demand, the a exponent value typicallyranges from 0.3 to 1.0. On the other hand, for structures with windcontrolled design, the a exponent value is usually between 0.7 and1.0 [22].

A velocity of a damper can be preliminarily premised andchecked in the second design stage. It is suggested a velocity of200–250 mm/s will be suitable for viscous dampers on buildings.For example, researchers took 10 in/s (254 mm/s) as a designvelocity of viscous dampers [23].

Given the force capacity Fdi for each damper, all parameters inEq. (13) are now determined.

2.1.3. Configuration of viscous dampersThe main concept to keep in mind when determining the con-

figuration of the viscous dampers in a building is to place themin those stories where inter-story drifts are relatively large. How-ever, putting dampers only in the stories with excessively largedisplacement actually has a counter effect of increasing inter-storydrift in the upper stories [24]. As a result, dampers should also beplaced in the adjacent stories to ensure a uniform deformationshape for the building.

2.2. Second stage of design

In the preliminary design, viscous dampers have been selectedand configured in building structures. The following results shouldbe checked in the second stage of design: (1) structural deforma-tions; (2) additional damping ratio; and (3) connecting structuralelements. Damper values and configuration determined in preli-minary design of the viscous dampers may need to be modifiedaccording to the three values above. Thus, the process of adding

Fig. 6. Damage at the beam and column ends.

50400

72007200 7200 7200 7200 7200 7200

1 2 3 4 5 6 7 8

7200

1740

0

7200

A

B

C

3000

Fig. 7. Plan layout of Story 3.

Fig. 8. Analytical model of the structure.

Fig. 9. Response spectra in Chinese code.

Y. Zhou et al. / Engineering Structures 39 (2012) 187–198 191

viscous dampers in a building is an iterative process in order tooptimize energy dissipation effect.

2.2.1. Checking for structural deformationsIn the Code for Seismic Design of Buildings (GB50011, 2008) [3], a

‘‘two-stages-and-three-levels’’ method is specified for seismic de-

Table 2First six periods of the structure.

No. Period (s) Modal shape

1 1.32 Translation in Y2 1.25 Torsion3 1.22 Translation in X4 0.43 Translation in Y5 0.42 Torsion6 0.41 Translation in X

192 Y. Zhou et al. / Engineering Structures 39 (2012) 187–198

sign of building structures. ‘‘Three levels’’ correspond to the minor,moderate, and major earthquake scenarios. The main performanceobjectives are to ensure structures immediate occupancy withoutdamage under minor earthquakes, operational with repairabledamage under moderate earthquakes, and functional withoutsevere collapse under major earthquakes. These objectives arefulfilled by checking forces and elastic displacements under minorearthquakes, and by checking elastoplastic displacements undermajor earthquakes, which is so called ‘‘two stages’’. The require-ment for the moderate earthquake level is only satisfied by thedesign of structural details. Thus, inter-story drifts are usually usedas the engineering demand parameter (EDP) for checking the effectof viscous dampers.

2.2.2. Checking for additional damping ratioIn the preliminary design, the additional damping ratio is first

employed to estimate the number of viscous dampers. However,in the second stage of design, the real force–displacement curvesof dampers have been obtained and should be used to check theadditional damping ratio. In Fig. 2, the displacement demand Dshould be replaced by the real damper displacement Ddmax andthe force Fd should be Fdmax. There is,

4Fd D� Fd

Kd

� �¼ 4c � Fd max � Dd max ð14Þ

where c is the shape coefficient, which denotes the shape differencebetween the parallelogram and the rectangular. c is taken as0.6–0.9.

According to Eqs. (4) and (14) for multi-story buildings,

f ¼PP

4cFd max � Dd max

4pP� 12 FD

ð15Þ

As mentioned before, when the checking result exceed 20%, itshould be taken as 20% [3].

Fig. 10. Structural inter-story drifts

2.2.3. Checking for connecting structural membersThe results of Uriz and Whittaker [25] showed that although the

retrofitted structural global seismic performance was improved bydampers, the original beams, columns and foundations also need tobe strengthened to ensure enough force transfer strength. Gener-ally, all elements on the force transfer path of viscous dampersshould be checked.

In the Code for Seismic Design of Buildings (GB50011), the stiff-ness of the energy-dissipating components in the direction of en-ergy-dissipating device may be calculated with the followingequation.

Kb ¼ ð6p=T1Þ � Cv ð16Þ

where Kb is the stiffness of the supporting component in the direc-tion of the energy dissipating device; Cv is the linear damping factorof the energy-dissipating device, which corresponds to thefundamental vibration period of the structure and is determinedby testing; and T1 is the fundamental vibration period of theenergy-dissipated structure.

As introduced before, Cv in fact is a factor correlated to the stiff-ness of dampers. The physical significance of Eq. (16) is to build anequation between the stiffness of components and the stiffness ofdampers. Since Cv is hard to be accurately determined based on thefundamental vibration period of the structure, according to Eq. (9),

Fd0 ¼ Cv � jvja � signðvÞ ¼ Kc � Dmax ð17Þ

where Fd0 is the damping force when the displacement is zero; Kc isthe loss of stiffness for dampers (Fig. 1), which is defined as Kc = Fd0/Ddmax; and Ddmax is the maximum displacement of dampers.

There is,

Cv � jx � Dmaxja ¼ Kc � Dmax ð18Þ

when a = 1,

Cv �x ¼ Kc ð19Þ

Eq. (16) can be transformed to the equation as below.

Kb ¼ 3 �x � Cv ¼ 3 � Kc ð20Þ

Eq. (20) shows that the stiffness of energy-dissipating supportcomponents should be three times of the loss stiffness of dampersto ensure the serviceability of the system, which forms the basis ofchecking support components.

by response spectrum analysis.

Fig. 11. Time histories and response spectra of three input waves.

Y. Zhou et al. / Engineering Structures 39 (2012) 187–198 193

3. Example of a RC frame structure retrofitted with viscousdampers

3.1. Building description

The target building is an office building of a Power Gas Com-pany in Dujiangyan, Sichuan Province. It is made of reinforced con-crete (RC). Though the structure was originally designed based on aseismic intensity of 7 in 1997, it was damaged during 2008 Wench-uan Earthquake.

Most visible damages are the horizontal infilled wall cracks inthe longitudinal direction (Fig. 5). Cracks with a width of 0.1 mmto 3 mm were observed at the structural beam ends in longitudinaldirection. In Stories 3 and 4, minor cracks (0.1–0.5 mm) were alsoobserved at the ends of the columns (Fig. 6). Because of the limitedstructural elements damage found, the structure was evaluated as‘‘minor damage’’ grade by the seismic evaluation team.

To retrofit the structure, which is now designed under a seismicintensity level of 8 according to the modified requirement forDujiangyan City, two retrofit strategies are considered. First,

Fig. 12. Structural inter-story drifts by time history analysis.

Table 3Shear forces and preliminary damping forces of time history analysis.

Story Max. shear force (kN) Preliminary damping force (kN) Number of dampers

X direction Y direction X direction Y direction X direction Y direction

7 142 129 43 39 0 06 1430 1556 429 467 0 05 2237 2062 671 619 2x500kN 2x500kN4 2694 2282 808 685 2x500kN 2x500kN3 3022 2261 907 678 2x500kN 2x500kN2 3464 2701 1039 810 2x700kN 2x700kN1 3722 3238 1117 971 2x700kN 2x700kN

Lead rubber isolator

Viscous damper

Viscous damper

Joint strengthening

Fig. 13. Configuration of viscous dampers.

194 Y. Zhou et al. / Engineering Structures 39 (2012) 187–198

engineers will strengthen all the damaged joints by sticking steelplates, and then they will add viscous dampers.

3.2. Structural analytical parameters

The structure has a plan dimension of 50.4 m by 17.4 m (Fig. 7).There are seven stories with a story height of 4.6 m, 4.2 m,3 � 3.6 m, 4.2 m and 3.6 m from Story 1 to 7, respectively. The totalheight of the structure is 27.4 m. RC frame structural system isapplied to undertake the gravity loads and lateral forces. The crosssections of frame beams are 350 mm by 600 mm and those ofcolumns changed along the structural height from 800 mm by800 mm to 500 mm by 500 mm. The thickness of the slab is100 mm. All concrete design grades are C30 that has a cubic

compressive strength of 14.3 MPa. The infilled wall is made byair brick with a thickness of 200 mm. An analytical structural mod-el is built up by ETABS, as shown in Fig. 8. The frame elements areused to simulate structural beams and columns, and the slab ele-ments are applied for slabs. Later, link elements are used for vis-cous dampers.

First the response spectra analysis is carried out. The responsespectra in Chinese code is shown in Fig. 9 (GB50011-2008). Inthe analysis, the seismic coefficient under intensity 8 of Dujiang-yan is 0.16 and the site characteristic period is 0.4 s. A periodreduction coefficient is taken as 0.85 to consider the stiffness con-tribution of infilled walls to the structure. Table 2 lists the first sixperiods of the structure and the inter-story drift is shown in Fig. 10.One can easily find that the structural inter-story drifts in bothdirections are beyond the code limitation of 1/550 under minorearthquake of intensity 8. Adding viscous dampers are requiredto control the structural responses.

3.3. Preliminary design

3.3.1. Number of viscous dampersThe time history procedure is selected for the design of viscous

dampers. According to Chinese code, two ground motion recordsand one artificial accelerogram are necessary for the analysis(GB50011). Here 2008 Wenchuan Earthquake ground motionrecord (Wolong Station N-S), 1940 El Centro ground motion record,and XIN1 artificial accelerogram are selected. Their time historiesand response spectra are shown in Fig. 11. The peak ground accel-erations (PGA) are scaled down to 0.07 g, 0.2 g, 0.4 g to commensu-rate with the PGA under minor, moderate, and major earthquakesof intensity 8.

1 2 3 4 5 6 7 8

A

B

C

VD VDV

D

VD

1 2 3 4 5 6 7 8

A

B

C

VD VD

VD

VD

(a) Story 3~5 (b) Story 1~2

Story 1~2: four 700 kN VD in X and YStory 4~5: two 500 kN VD in X and Y

Story 3: two 700 kN VD in X and Y

Fig. 14. Plan layouts of viscous dampers.

Y. Zhou et al. / Engineering Structures 39 (2012) 187–198 195

Fig. 12 gives the inter-story drifts of the structure under threeinputs. It can be found that the responses under El Centro in bothdirections and under XIN1 in X direction exceed the code limita-tion. The maximum shear forces under three inputs are tabulatedin Table 3. As introduced in Section 2.1.1, the preliminary designdamping forces are taken as 30% of the shear forces of stories.Those forces for the target building are also listed in Table 3.

Dampers can be installed as diagonal members, as part of achevron brace, horizontally at the top of a chevron brace, or as atoggle brace [26,27]. The horizontal chevron configuration isapplied here as shown in Fig. 13. This system was proposed byLu and Zhou and tested on a shaking table in 2002 [28]. Twoviscous dampers are installed in parallel and supported by a steelchevron brace. Lead rubber bearings are installed at the top ofthe brace to keep the stability of the brace and to dissipate theenergy under minor earthquake. The viscous dampers withmaximum damping forces of 700 kN and 500 kN are first selectedand the estimated installation number is given in Table 3.

3.3.2. Parameters of viscous dampersViscous dampers produced in Shanghai Research Institute of

Materials (SHRIM) are chosen in the target building. In Eq. (9), takev = 200 mm/s and a = 0.2. When C is 250 kNs/mm and 200 kNs/mm, the final damping force is 721 kN and 577 kN, respectively.A 120 mm free movement displacement is required for the SHRIMviscous dampers.

3.3.3. Configuration of viscous dampersSteel bracings with a section of H400 � 250 � 9 � 14 are de-

signed, whose stiffness will be checked later. The final plan layoutsof viscous dampers in the structure are shown in Fig. 14.

3.4. Second stage of design

3.4.1. Checking for structural deformationsFig. 15 shows the maximum inter-story drifts of the structure

with viscous dampers under three earthquake scenarios. It canbe seen that with dampers, the structural deformation curvesapparently satisfy the code limitation of 1/550 for minor earth-quakes and 1/50 for major earthquakes. Table 4 lists the effect ofthe viscous dampers on the story shear forces. The story forcereductions under various earthquake levels are around 30%, whichaccommodate the vested objectives. Fig. 16 gives the roof acceler-ation with and without viscous dampers under major earthquakeof El Centro. It can be seen that the accelerations in both directionsare effectively reduced. The PGA in X direction reduced from 0.82 gto 0.65 g, and Y direction from 0.68 g to 0.60 g. Here the structuralresponses under three ground motions are checked. If the dynamic

characteristics of earthquakes vary, more analysis is needed tocheck the effect of adding dampers.

3.4.2. Checking for additional damping ratioThe additional damping ratios calculated by Eq. (15) are tabu-

lated in Table 5. It has been concluded that the average additionaldamping ratios decrease with increasing peak ground accelerationsfrom 0.07 g, 0.2 g, to 0.4 g under various earthquake levels. Alldamping ratios are less than the code limitation of 20% and neednot regulate the dampers. Fig. 17 shows the force–displacementcurves of one damper under minor, moderate, and major earth-quakes, respectively. It does not reach its force capacity under min-or level; however, under moderate earthquakes its full capacity isexerted. The basic assumption in Section 2.1.1 that the dampersshould work at the full until moderate level is verified.

3.4.3. Checking for connecting structural members3.4.3.1. Stiffness of the damping system. The stiffness of the dampingsystem is checked according to Eq. (20). One should note that forthe horizontal chevron configuration the damper and the chevronare connected in series. Thus, their combined stiffness should beconsidered. Kd is 140,000 N/mm as suggested by damper manufac-turer. According to the analytical results, there is

K 0b ¼ 284;000kN=mmKc ¼ 22;511kN=mm

ð21Þ

Thus,

11

K 0bþ 1

Kd

¼ 93;774 > 3 � Kc ¼ 67;533 ð22Þ

The stiffness of the damping system is enough to provide thestiffness for the serviceability of dampers.

3.4.3.2. Internal forces of the columns. There are two objectives tocheck the internal forces of the columns. One is to see if thecolumns originally designed as intensity 7 could undertake loadsunder seismic intensity 8. Another purpose is to make sure thecolumns could transfer the additional forces induced by viscousdampers. Table 6 gives the internal forces of a typical columnconnected with the viscous dampers. The axial force, shear forceand the bending moment under minor earthquakes of intensity 7without dampers are compared with those under minor earth-quakes of intensity 8 with dampers. It is shown that the columnmust have the capacity to sustain 16% additional internal forces.The checking of the column indicates that the original design couldsatisfy the requirement without further strengthening.

Fig. 15. Maximum inter-story drifts of the structure with viscous dampers.

196 Y. Zhou et al. / Engineering Structures 39 (2012) 187–198

4. Conclusions and discussions

In this paper, a practical design method of reinforced concretestructures with viscous dampers is put forward. The design process

is divided into two stages. In the preliminary stage, the quantity,mechanical parameters and configurations of the viscous dampersare determined. Then check on the structural deformations, theadditional damping ratio, and the connecting structural elements

Table 4Effect of the viscous dampers on the story shear forces.

Earthquake scenario Floor Shear force without dampers (Q0, kN) Shear force with dampers (Qd, kN) (Qd � Q0)/Q0 (%)

X direction Y direction X direction Y direction X direction Y direction

Minor earthquake level 7 142 129 165 153 16 186 1430 1556 1444 1369 1 �125 2237 2062 1738 1566 �22 �244 2694 2282 2091 1833 �22 �203 3022 2261 2282 1950 �24 �142 3464 2701 2070 1884 �40 �301 3722 3238 2091 1895 �44 �41

Moderate earthquake level 7 407 371 350 330 �14 �116 4087 4455 3379 3206 �17 �285 6390 5902 4332 3774 �32 �364 7697 6519 5090 4672 �34 �283 8635 6461 6195 5489 �28 �152 9898 7716 6575 5556 �34 �281 10,630 9252 6626 5376 �38 �42

Major earthquake level 7 813 741 627 597 �23 �196 8173 8910 6426 6035 �21 �325 12,780 11,800 8832 7326 �31 �384 15,390 13,040 11,230 9812 �27 �253 17,270 12,920 13,360 11,380 �23 �122 19,800 15,430 14,130 11,630 �29 �251 21,270 18,500 14,220 12,340 �33 �33

Fig. 16. Roof acceleration with and without viscous dampers under major earthquake of El Centro.

Table 5Checking for the average damping ratio.

Earthquake scenario X direction (%) Y direction (%)

Minor earthquake level 19.4 19.8Moderate earthquake level 12.0 13.4Major earthquake level 7.4 7.6

-500

-400

-300

-200

-100

0

100

200

300

400

500

-6 -4 -2 0 2 4 6

X1

Y1-600

-400

-200

0

200

400

600

-15 -10 -5 0

(a) Minor earthquake level (b) Moderate ear

(mm)

Nk()Nk(

Fig. 17. Force–displacement cu

Y. Zhou et al. / Engineering Structures 39 (2012) 187–198 197

are performed in the second stage of design. An example is also gi-ven to demonstrate the application of the proposed method to ret-rofit a RC frame structure by viscous dampers. It is concluded thatthe damping forces could be estimated as 30% of the story forces inthe preliminary design. With viscous dampers designed by the

5 10 15

X2

Y2-800

-600

-400

-200

0

200

400

600

800

-30 -20 -10 0 10 20 30

X3

Y3

thquake level (c) Major earthquake level

)

(mm)

(kN)

(mm)

rves of a viscous damper.

Table 6Internal forces of one column connected to the viscous dampers.

Internal force Minor earthquakes of seismic intensity 7 (withoutdampers)

Minor earthquakes of seismic intensity 8 (withdampers)

Change of the internal force

X direction Y direction X direction Y direction X direction Y direction

Axial force (kN) 4556 5059 4713 5326 1.03 1.05Shear force (kN) 138 96 157 112 1.14 1.16Bending moment (kNm) 561 347 618 404 1.10 1.16

198 Y. Zhou et al. / Engineering Structures 39 (2012) 187–198

method proposed here, the structure could satisfy the seismicrequirements of intensity increase from 7 to 8 after WenchuanEarthquake. Yet the following points should be noted:

(1) In the proposed method, the additional damping ratio isobtained by estimating it for each story and sums those ofall stories. It implies the proposed design method is onlysuitable for a regular structure where superposition applies.

(2) When estimating the damping force, the vested performanceobjectives of RC structures are referred to predict thedisplacement ratio of dampers. For wider engineered appli-cation, further research on the performance-based seismicdesign of structures with viscous dampers is needed.

(3) Generally there are four types of damper installations andthe horizontal chevron configuration is applied in this paper.The effect of different damper configurations on structuralretrofitting could be an interesting research topic.

Acknowledgements

The authors are grateful for the financial support in part fromthe National Natural Science Foundation of China (Grant Nos.90815029, 5102114006 and 51078274), and National BasicResearch of China (Grant No. 2007CB714202). China Strong MotionNetwork Center is much appreciated for their support on groundmotion earthquake records in Wenchuan Earthquake.

References

[1] China Ministry of Construction (CMC). Standard for classification of seismicprotection of building constructions (GB50223-2008). Beijing, China: ChinaArchitecture & Building Press; 2008 [in Chinese].

[2] Lu XL, Zhou Y. Modification of seismic design code after the SichuanEarthquake of May 12, 2008. In: Proceedings of international association forbridge and structural engineering, Cavtat, Croatia; 3–5 May 2010.

[3] China Ministry of Construction (CMC). Code for seismic design of buildings(GB50011-2001), modified version. Beijing, China: China Architecture &Building Press; 2008 [in Chinese].

[4] Lu XL. Retrofitting design of building structures. Boca Raton, USA: CRC Press;2010.

[5] Constantinou MC, Symans MD. Experimental study of seismic response ofbuildings with supplemental fluid dampers. Struct Des Tall Spec1993;2(2):93–132.

[6] Gluck N, Reinhorn AM, Gluck J, Levy R. Design of supplemental dampers forcontrol of structures. J Struct Eng 1996;122(12):1394–9.

[7] Fu Y, Kasai K. Comparative study of frames using viscoelastic and viscousdampers. J Struct Eng 1998;124(5):513–22.

[8] Kasai K, Fu Y, Watanabe A. Passive control system for seismic damagemitigation. J Struct Eng 1998;124(5):501–12.

[9] Yang JN, Lin S, Kim JH, Agrawal AK. Optimal design of passive energydissipation systems based on H1 infinity and H2 performances. Earthq EngStruct D 2002;31(4):921–36.

[10] Lee D, Taylor DP. Viscous damper development and future trends. Struct DesTall Spec 2001;10(5):311–20.

[11] Lin YY, Tsai MH, Hwang JS, Chang KC. Direct displacement-based design forbuilding with passive energy dissipation systems. Eng Struct2003;25(1):25–37.

[12] Uetani K, Tsuji M, Takewaki I. Application of an optimum design method topractical building frames with viscous dampers and hysteretic dampers. EngStruct 2003;25(5):579–92.

[13] Chen XW, Li JX, Cheang J. Seismic performance analysis of Wenchuan Hospitalstructure with viscous dampers. Struct Des Tall Spec 2010;19(4):397–419.

[14] American Society of Civil Engineering – ASCE. minimum design loads forbuildings and other structures. ASCE/SEI 7–05, USA; 2006.

[15] Federal Emergency Management Agency – ASCE. FEMA P-750, NEHRPrecommended seismic provisions for new buildings and other structures.Washington, DC: Federal Emergency Management Agency – ASCE; 2009.

[16] Sadek F, Mohraz B, Riley MA. Linear procedures for structures with velocity-dependent dampers. J Struct Eng 2000;126(8):887–95.

[17] Federal Emergency Management Agency – ASCE. FEMA 273, NEHRP guidelinesfor the seismic rehabilitation of buildings. Washington, DC: Federal EmergencyManagement Agency – ASCE; 1997.

[18] China Ministry of Construction (CMC). Code for seismic design of buildings(GB50011-2001). Beijing, China: China Architecture & Building Press; 2001.

[19] Chopra AK. Dynamics of structures: theory and application to earthquakeengineering. International series in civil engineering and engineeringmechanics. USA: Prentice-Hall; 2006.

[20] Lu XL. Seismic design guidelines for tall buildings beyond the scope of designcodes. Shanghai, China: Shanghai Construction Engineering StandardAdministration Office; 2009.

[21] Martinez-Rodrigo M, Romero ML. An optimum retrofit strategy for momentresisting frames with nonlinear viscous dampers for seismic applications. EngStruct 2003;25(7):913–25.

[22] Taylor DP. Smart buildings and viscous dampers – a design engineer’sperspective. Struct Des Tall Spec 2010;19(4):369–72.

[23] Sinclair M. Viscous damping devices: design and implementation issues. In:Proceedings of the 100th anniversary conference commemorating the 1906,San Francisco Earthquake, San Francisco; 18–22 April 2006.

[24] Hong MV. Retrofitting of steel moment-frame building using fluid viscousdampers. Thesis, University of California at Berkeley; 2009.

[25] Uriz PS, Whittaker A. Retrofit of pre-Northridge steel moment-resisting framesusing fluid viscous dampers. Struct Des Tall Spec 2001;10(5):371–90.

[26] Hwang JS, Tsai CH, Wang SJ, Huang YN. Experimental study of RC buildingstructures with supplemental viscous dampers and lightly reinforced walls.Eng Struct 2006;28(13):1816–24.

[27] Li ZY, Albermani F, Chand R, Kitipornchai S. Pinching hysteretic response ofyielding shear panel device. Eng Struct 2011;33(3):993–1000.

[28] Lu XL, Zhou Q. Dynamic analysis method of a combined energy dissipationsystem and its experimental verification. Earthq Eng Struct D2002;31(6):1251–65.