A practical design method for reinforced concrete structures with viscous dampers

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Engineering Structures 39 (2012) 187–198
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Engineering Structures
journal homepage: www.elsevier .com/ locate /engstruct
A practical design method for reinforced concrete structures with viscous dampers
Ying Zhou ⇑, Xilin Lu, Dagen Weng, Ruifu Zhang
State 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 2011
Revised 7 February 2012
Accepted 9 February 2012
Available online 22 March 2012
Keywords:
Viscous damper
Design method
Reinforced concrete structure
Wenchuan Earthquake
0141-0296/$ - see front matter � 2012 Elsevier Ltd. A
doi:10.1016/j.engstruct.2012.02.014
⇑ Corresponding author. Tel.: +86 21 6598 6157; fa
E-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 code
with increased protection categories and seismic intensities. According to the new code, a lot of school
and 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 those
existing 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 engineers
and 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 the
preliminary stage, the quantity, mechanical parameters and configurations of the viscous dampers are
determined. In the next stage, the reduction of deformations, additional damping ratio, and connection
of the dampers to the structure are examined. An example is also given to demonstrate the application
of 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 of
research 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 hit
Wenchuan in China. The earthquake had a shallow focal depth of
approximately 19 km, with the epicenter located 80km WNW of
Chengdu, capital of Sichuan province. The entire country was dev-
astated by the mega earthquake. The death toll of the earthquake
was of over 69,000 killed people, with over 374,000 people injured
and nearly 5 million homeless; the total lost was estimated at
US$130 billion.
After the Wenchuan Earthquake, new Chinese seismic design
codes were issued with the modifications in attempt to increase
protection categories and seismic intensities. In Chinese code, the
protection 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), and
particular protection (PP), in the order of the increasing protection
requirement. 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 class
buildings, dormitories, and dining halls in kindergartens, primary
schools, and middle schools are increased from standard seismic
protection buildings to emphasized seismic protection buildings.
This modification means that the seismic forces of those school
buildings will be calculated commensurate with the design inten-
sity while the structural details will be checked one degree higher
than the design intensity. The purpose of this modification is to
protect young and valuable students in earthquakes. On the other
hand, the seismic intensities of cities are specified in the Code for
Seismic Design of Buildings (GB50011-2008) [3]. After Wenchuan
Earthquake, the seismic intensities of many cities in China are
increased by half degree or more. Lots of school buildings and
industrial 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 buildings
enables easier construction, and reduced labor and downtime.
1.2. Existing research on viscous dampers
Viscous dampers themselves are old technology, dating back
to more than a century ago to full-scale usage on US large cali-
http://dx.doi.org/10.1016/j.engstruct.2012.02.014
mailto:yingzhou@tongji.edu.cn
mailto:lxlst@tongji.edu.cn
mailto:wdg@tongji.edu.cn
mailto:zhangruifu@gmail.com
http://dx.doi.org/10.1016/j.engstruct.2012.02.014
http://www.sciencedirect.com/science/journal/01410296
http://www.elsevier.com/locate/engstruct
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 not
available for the public disclosure or usage until the Cold War
ended. In 1990, Taylor Devices received the permission to sell
this technology to the public. Despite the long history and
well-established usage of viscous damper, it is still a relatively
new building technology yet to be further developed and
studied.
Studies have been published regarding viscous dampers design
methodology. Constantinou and Symans [5] proposed a simplified
method for calculating the modal characteristics of structures
with added fluid dampers. The method was used to obtain esti-
mates of peak response of the tested structures by utilizing the
response spectrum approach. Gluck et al. [6] suggested a design
method for supplemental dampers in multi-story structures,
adapting the optimal control theory by using a linear quadratic
regulator (LQR) to design linear passive viscous (VS) or viscoelas-
tic (VE) devices depending on their deformation and velocity. Fu
and Kasai [7] compared frames dynamic behavior using VE or
pure VS dampers, where identical mathematical expressions were
derived 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 design
methodologies for passive energy dissipation devices based on ac-
tive control theories leading to the determination of VS and VE
dampers, defining different forms of performance functions. Lee
and Taylor [10] developed the energy dissipation technology
and suggested that approximately 15–25% of additional damping
is a desirable range in the damper designed buildings. Lin et al.
[11] presented a seismic displacement-based design method for
new and regular buildings equipped with passive energy dissipa-
tion systems. Using the substitute structure approach for the
building structure and simulating the mechanical properties of
the passive energy dissipation devices by the effective stiffness
and effective viscous damping ratio, a rational linear iteration
method was proposed. Uetani et al. [12] proposed a practical
method for optimum structural design of building frames with
viscous dampers. The method first did the stiffness design of a re-
ducedshear-building model with viscous dampers. Then the opti-
mum design for building frames was performed under static
design loads. Design examples were presented to demonstrate
the 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 the
structure under various earthquake scenarios. The damping ratio
was estimated by a method of inputting a series of sine waves
and 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 damped
structures. 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 devices
based 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-period
structures and assuming a harmonic response to compute the peak
velocity, story and base shear. In China, although damper technol-
ogies are specified in the 2001 version and 2008 modified version
of Code for Seismic Design of Buildings (GB50011) [18], only general
information is given in the codes. A standardized design method is
therefore needed for a more widespread and routine inclusion of
viscous damper in structural design practice.
In the following sections, a practical design method is proposed
for structures with viscous dampers. An example is also given to
demonstrate the application of the proposed method.
2. Design method for structures with viscous dampers
In the preliminary stage of designing viscous dampers in a
structure, the following tasks needed to be done: (1) determine
the number of viscous dampers, (2) choose the parameters of vis-
cous dampers, and (3) configure the layout of viscous dampers. In
the second stage of design, engineers should check the structural
deformations, 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 dampers
The 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 in
the whole process of design, which governs the number of dampers
and 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 �Ws
ð1Þ
where f is the effective damping ratio added by viscous damper
devices, Wc is the energy dissipated by viscous damper devices in
one cycle of expected displacement of a structure, i.e., the total area
of the force–displacement curves, and Ws is the total strain energy
of an energy dissipated structure under expected displacement.
Take one story for an example. The total strain energy Ws can be
expressed as Eq. (2), where F is the horizontal story shearing force;
and D is the relative story displacement.
Ws ¼
1
2
FD ð2Þ
Fig. 1 shows a typical force–displacement curve of a nonlinear
viscous damper, where the area is simplified to be a parallelogram
(Fig. 2). Thus, the energy dissipated by viscous dampers, Wc can be
calculated as Eq. (3).
Wc ¼ 4Fd D� Fd
Kd
� �
ð3Þ
where Fd the damping is force of viscous dampers and Kd is the
stiffness of viscous dampers.
0
Fig. 2. Simplified parallelogram of the force–displacement curve of a nonlinear
viscous damper.
Y. Zhou et al. / Engineering Structures 39 (2012) 187–198 189
Combining Eqs. (1) and (3) leads to a following expression for
the damping ratio f, as described in Eq. (4).
f ¼
4Fd D� Fd
Kd
� �
4p � 1
2 FD
¼
2Fd 1� Fd
Kd
=D
� �
p � F ð4Þ
with l ¼ Fd
Kd
=D,
f ¼ 2ð1� lÞ
p
� Fd
F
ð5Þ
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 additional
damping ratio f induced by the viscous dampers can be correlated
to the displacement ratio l and the force ratio Fd/F.
According to Eq. (5), the damping force is,
Fd ¼
p
2ð1� lÞ � f � F ð6Þ
An important parameter in determining the damping force is
the damping ratio f. There are several guidelines. Lee and Taylor
[10] suggested a practical upper limit for the combined viscous
and 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%, it
should be taken as 20%. Thereby, for preliminary design, f is taken
as 15%, which will be checked later for the accuracy of this
assumption.
As defined above, l is the ratio of the damper capacity to the
structural demand. In general, it is not desirable for a damper to
reach its capacity either too early under a minor earthquake or
too later under a major earthquake. It is thus assumed here that a
viscous damper reaches its capacity under a moderate earthquake.
Table 1 lists the inter-story drift objectives of reinforced
concrete (RC) structures under various earthquake levels [20]. It
Table 1
Inter-story drift objectives of RC structures.
Minor earthquake level Moderate earthquake le
Operational Immediate occupancy
Frame structures 1/550 1/250
Shear wall structures 1/1000 1/500
Hybrid structures 1/800 1/400
is shown that the average value of
p
2ð1� lÞ can be approximated
to 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 in
each story can be preliminarily taken as 30% of the story shearing
force. If the force capacity of each damper Fdi is chosen, then the
number of viscous dampers is determined as Eq. (8).
n ¼ Fd
Fdi
ð8Þ
2.1.2. Parameters of viscous dampers
Viscous 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 the
damping factor. v represents the velocity of the viscous damper
and its exponential parameter a determines the relationship
between 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 analysis
of the various parameters is shown in the next section to demon-
strate the reasoning behind choosing certain parameters for a
viscous damper.
2.1.2.1. Parameter analysis. It is assumed that the displacement and
the 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 x
2p ¼ 0:1 Hz, and C = 100 kN s/mm, the
force–displacement curves under various a values are shown in
Fig. 3a. Next, holding A and f as constants, the force–displacement
curves under various C values for a = 0.2 are shown in Fig. 3b. From
Fig. 3, it can be easily found that the area inside the curve, i.e. the
energy 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 for
structures.
Keeping the same displacement (A = 60 mm) and force
(Fdi = 3770 kN), the force–displacement curves at a = 0.2 and
a = 1.0 are given in Fig. 4. It can be seen that the shape of the curve
at a = 0.2 is closer to a rectangle while the shape of the curve at
a = 1.0 resembles more of an ellipse. Apparently, more energy dis-
sipation areawill be achieved when a is taken a smaller value.
Martinez-Rodrigo and Romero [21] compared the retrofitting effect
on a six-story steel structure by using linear dampers and by using
nonlinear dampers. It was concluded that a nonlinear damper force
vel Major earthquake level l p
2ð1� lÞ
Life safety Collapse prevention
1/100 1/50 0.20 2.0
1/250 1/120 0.24 2.1
1/200 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. An
excessively high damper stiffness would potentially create diffi-
culty in designing the structural elements connecting with the
dampers. Thus, selecting the appropriate a and C values deserve
special attention in the preliminary design stage.
2.1.2.2. Parameter determination. Eq. (9) could be transformed as
Eq. (13).
C ¼ Fdi
jv ja � signðvÞ
ð13Þ
A viscous damper catalogue is usually provided by the manufac-
turers for several given a values. Generally, the a value selected for
seismic design is smaller than that of wind resistant design. Using a
Taylor device as an example, for structures with seismic demand
dominating over wind demand, the a exponent value typically
ranges from 0.3 to 1.0. On the other hand, for structures with wind
controlled design, the a exponent value is usually between 0.7 and
1.0 [22].
A velocity of a damper can be preliminarily premised and
checked in the second design stage. It is suggested a velocity of
200–250 mm/s will be suitable for viscous dampers on buildings.
For example, researchers took 10 in/s (254 mm/s) as a design
velocity of viscous dampers [23].
Given the force capacity Fdi for each damper, all parameters in
Eq. (13) are now determined.
2.1.3. Configuration of viscous dampers
The main concept to keep in mind when determining the con-
figuration of the viscous dampers in a building is to place them
in those stories where inter-story drifts are relatively large. How-
ever, putting dampers only in the stories with excessively large
displacement actually has a counter effect of increasing inter-story
drift in the upper stories [24]. As a result, dampers should also be
placed in the adjacent stories to ensure a uniform deformation
shape for the building.
2.2. Second stage of design
In the preliminary design, viscous dampers have been selected
and configured in building structures. The following results should
be checked in the second stage of design: (1) structural deforma-
tions; (2) additional damping ratio; and (3) connecting structural
elements. Damper values and configuration determined in preli-
minary design of the viscous dampers may need to be modified
according 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
72
00
17
40
0
72
00
A
B
C
30
00
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 to
optimize energy dissipation effect.
2.2.1. Checking for structural deformations
In the Code for Seismic Design of Buildings (GB50011, 2008) [3], a
‘‘two-stages-and-three-levels’’ method is specified for seismic de-
Table 2
First six periods of the structure.
No. Period (s) Modal shape
1 1.32 Translation in Y
2 1.25 Torsion
3 1.22 Translation in X
4 0.43 Translation in Y
5 0.42 Torsion
6 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 performance
objectives are to ensure structures immediate occupancy without
damage under minor earthquakes, operational with repairable
damage under moderate earthquakes, and functional without
severe collapse under major earthquakes. These objectives are
fulfilled by checking forces and elastic displacements under minor
earthquakes, and by checking elastoplastic displacements under
major earthquakes, which is so called ‘‘two stages’’. The require-
ment for the moderate earthquake level is only satisfied by the
design of structural details. Thus, inter-story drifts are usually used
as the engineering demand parameter (EDP) for checking the effect
of viscous dampers.
2.2.2. Checking for additional damping ratio
In 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 curves
of dampers have been obtained and should be used to check the
additional damping ratio. In Fig. 2, the displacement demand D
should be replaced by the real damper displacement Ddmax and
the 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 difference
between the parallelogram and the rectangular. c is taken as
0.6–0.9.
According to Eqs. (4) and (14) for multi-story buildings,
f ¼
PP
4cFd max � Dd max
4p
P
� 12 FD
ð15Þ
As mentioned before, when the checking result exceed 20%, it
should be taken as 20% [3].
Fig. 10. Structural inter-story drifts
2.2.3. Checking for connecting structural members
The results of Uriz and Whittaker [25] showed that although the
retrofitted structural global seismic performance was improved by
dampers, the original beams, columns and foundations also need to
be strengthened to ensure enough force transfer strength. Gener-
ally, all elements on the force transfer path of viscous dampers
should 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 following
equation.
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 factor
of the energy-dissipating device, which corresponds to the
fundamental vibration period of the structure and is determined
by testing; and T1 is the fundamental vibration period of the
energy-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 an
equation between the stiffness of components and the stiffness of
dampers. Since Cv is hard to be accurately determined based on the
fundamental 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 is
the 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 support
components should be threetimes of the loss stiffness of dampers
to ensure the serviceability of the system, which forms the basis of
checking 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 viscous
dampers
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 a
seismic intensity of 7 in 1997, it was damaged during 2008 Wench-
uan Earthquake.
Most visible damages are the horizontal infilled wall cracks in
the longitudinal direction (Fig. 5). Cracks with a width of 0.1 mm
to 3 mm were observed at the structural beam ends in longitudinal
direction. In Stories 3 and 4, minor cracks (0.1–0.5 mm) were also
observed at the ends of the columns (Fig. 6). Because of the limited
structural 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 seismic
intensity level of 8 according to the modified requirement for
Dujiangyan City, two retrofit strategies are considered. First,
Fig. 12. Structural inter-story drifts by time history analysis.
Table 3
Shear 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 0
6 1430 1556 429 467 0 0
5 2237 2062 671 619 2x500kN 2x500kN
4 2694 2282 808 685 2x500kN 2x500kN
3 3022 2261 907 678 2x500kN 2x500kN
2 3464 2701 1039 810 2x700kN 2x700kN
1 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 steel
plates, 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 total
height of the structure is 27.4 m. RC frame structural system is
applied to undertake the gravity loads and lateral forces. The cross
sections of frame beams are 350 mm by 600 mm and those of
columns changed along the structural height from 800 mm by
800 mm to 500 mm by 500 mm. The thickness of the slab is
100 mm. All concrete design grades are C30 that has a cubic
compressive strength of 14.3 MPa. The infilled wall is made by
air 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 are
used 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 response
spectra in Chinese code is shown in Fig. 9 (GB50011-2008). In
the analysis, the seismic coefficient under intensity 8 of Dujiang-
yan is 0.16 and the site characteristic period is 0.4 s. A period
reduction coefficient is taken as 0.85 to consider the stiffness con-
tribution of infilled walls to the structure. Table 2 lists the first six
periods of the structure and the inter-story drift is shown in Fig. 10.
One can easily find that the structural inter-story drifts in both
directions are beyond the code limitation of 1/550 under minor
earthquake of intensity 8. Adding viscous dampers are required
to control the structural responses.
3.3. Preliminary design
3.3.1. Number of viscous dampers
The time history procedure is selected for the design of viscous
dampers. According to Chinese code, two ground motion records
and one artificial accelerogram are necessary for the analysis
(GB50011). Here 2008 Wenchuan Earthquake ground motion
record (Wolong Station N-S), 1940 El Centro ground motion record,
and XIN1 artificial accelerogram are selected. Their time histories
and 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 earthquakes
of intensity 8.
1 2 3 4 5 6 7 8
A
B
C
VD VD
V
D
V
D
1 2 3 4 5 6 7 8
A
B
C
VD VD
V
D
V
D
(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 three
inputs. It can be found that the responses under El Centro in both
directions and under XIN1 in X direction exceed the code limita-
tion. The maximum shear forces under three inputs are tabulated
in Table 3. As introduced in Section 2.1.1, the preliminary design
damping 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 a
chevron brace, horizontally at the top of a chevron brace, or as a
toggle brace [26,27]. The horizontal chevron configuration is
applied here as shown in Fig. 13. This system was proposed by
Lu and Zhou and tested on a shaking table in 2002 [28]. Two
viscous dampers are installed in parallel and supported by a steel
chevron brace. Lead rubber bearings are installed at the top of
the brace to keep the stability of the brace and to dissipate the
energy under minor earthquake. The viscous dampers with
maximum damping forces of 700 kN and 500 kN are first selected
and the estimated installation number is given in Table 3.
3.3.2. Parameters of viscous dampers
Viscous dampers produced in Shanghai Research Institute of
Materials (SHRIM) are chosen in the target building. In Eq. (9), take
v = 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 SHRIM
viscous dampers.
3.3.3. Configuration of viscous dampers
Steel bracings with a section of H400 � 250 � 9 � 14 are de-
signed, whose stiffness will be checked later. The final plan layouts
of viscous dampers in the structure are shown in Fig. 14.
3.4. Second stage of design
3.4.1. Checking for structural deformations
Fig. 15 shows the maximum inter-story drifts of the structure
with viscous dampers under three earthquake scenarios. It can
be seen that with dampers, the structural deformation curves
apparently satisfy the code limitation of 1/550 for minor earth-
quakes and 1/50 for major earthquakes. Table 4 lists the effect of
the viscous dampers on the story shear forces. The story force
reductions under various earthquake levels are around 30%, which
accommodate the vested objectives. Fig. 16 gives the roof acceler-
ation with and without viscous dampers under major earthquake
of El Centro. It can be seen that the accelerations in both directions
are effectively reduced. The PGA in X direction reduced from 0.82 g
to 0.65 g, and Y direction from 0.68 g to 0.60 g. Here the structural
responses under three ground motions are checked. If the dynamic
characteristics of earthquakes vary, more analysis is needed to
check the effect of adding dampers.
3.4.2. Checking for additional damping ratio
The additional damping ratios calculated by Eq. (15) are tabu-
lated in Table 5. It has been concluded that the average additional
damping ratios decrease with increasing peak ground accelerations
from 0.07 g, 0.2 g, to 0.4 g under various earthquake levels. All
damping ratios are less than the code limitation of 20% and need
not regulate the dampers. Fig. 17 shows the force–displacement
curves of one damperunder 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 is
exerted. The basic assumption in Section 2.1.1 that the dampers
should work at the full until moderate level is verified.
3.4.3. Checking for connecting structural members
3.4.3.1. Stiffness of the damping system. The stiffness of the damping
system is checked according to Eq. (20). One should note that for
the horizontal chevron configuration the damper and the chevron
are connected in series. Thus, their combined stiffness should be
considered. Kd is 140,000 N/mm as suggested by damper manufac-
turer. According to the analytical results, there is
K 0b ¼ 284;000kN=mm
Kc ¼ 22;511kN=mm
ð21Þ
Thus,
1
1
K 0b
þ 1
Kd
¼ 93;774 > 3 � Kc ¼ 67;533 ð22Þ
The stiffness of the damping system is enough to provide the
stiffness for the serviceability of dampers.
3.4.3.2. Internal forces of the columns. There are two objectives to
check the internal forces of the columns. One is to see if the
columns originally designed as intensity 7 could undertake loads
under seismic intensity 8. Another purpose is to make sure the
columns could transfer the additional forces induced by viscous
dampers. Table 6 gives the internal forces of a typical column
connected with the viscous dampers. The axial force, shear force
and the bending moment under minor earthquakes of intensity 7
without dampers are compared with those under minor earth-
quakes of intensity 8 with dampers. It is shown that the column
must have the capacity to sustain 16% additional internal forces.
The checking of the column indicates that the original design could
satisfy 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 concrete
structures 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 dampers
are determined. Then check on the structural deformations, the
additional damping ratio, and the connecting structural elements
Table 4
Effect 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 18
6 1430 1556 1444 1369 1 �12
5 2237 2062 1738 1566 �22 �24
4 2694 2282 2091 1833 �22 �20
3 3022 2261 2282 1950 �24 �14
2 3464 2701 2070 1884 �40 �30
1 3722 3238 2091 1895 �44 �41
Moderate earthquake level 7 407 371 350 330 �14 �11
6 4087 4455 3379 3206 �17 �28
5 6390 5902 4332 3774 �32 �36
4 7697 6519 5090 4672 �34 �28
3 8635 6461 6195 5489 �28 �15
2 9898 7716 6575 5556 �34 �28
1 10,630 9252 6626 5376 �38 �42
Major earthquake level 7 813 741 627 597 �23 �19
6 8173 8910 6426 6035 �21 �32
5 12,780 11,800 8832 7326 �31 �38
4 15,390 13,040 11,230 9812 �27 �25
3 17,270 12,920 13,360 11,380 �23 �12
2 19,800 15,430 14,130 11,630 �29 �25
1 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 5
Checking for the average damping ratio.
Earthquake scenario X direction (%) Y direction (%)
Minor earthquake level 19.4 19.8
Moderate earthquake level 12.0 13.4
Major 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 that
the damping forces could be estimated as 30% of the story forces in
the 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 6
Internal forces of one column connected to the viscous dampers.
Internal force Minor earthquakes of seismic intensity 7 (without
dampers)
Minor earthquakes of seismic intensity 8 (with
dampers)
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 1.05
Shear force (kN) 138 96 157 112 1.14 1.16
Bending 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 seismic
requirements of intensity increase from 7 to 8 after Wenchuan
Earthquake. Yet the following points should be noted:
(1) In the proposed method, the additional damping ratio is
obtained by estimating it for each story and sums those of
all stories. It implies the proposed design method is only
suitable for a regular structure where superposition applies.
(2) When estimating the damping force, the vested performance
objectives of RC structures are referred to predict the
displacement ratio of dampers. For wider engineered appli-
cation, further research on the performance-based seismic
design of structures with viscous dampers is needed.
(3) Generally there are four types of damper installations and
the horizontal chevron configuration is applied in this paper.
The effect of different damper configurations on structural
retrofitting could be an interesting research topic.
Acknowledgements
The authors are grateful for the financial support in part from
the National Natural Science Foundation of China (Grant Nos.
90815029, 5102114006 and 51078274), and National Basic
Research of China (Grant No. 2007CB714202). China Strong Motion
Network Center is much appreciated for their support on ground
motion earthquake records in Wenchuan Earthquake.
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	A practical design method for reinforced concrete structures with viscous dampers
	1 Introduction
	1.1 Background
	1.2 Existing research on viscous dampers
	2 Design method for structures with viscous dampers
	2.1 Preliminary design
	2.1.1 Number of viscous dampers
	2.1.2 Parameters of viscous dampers
	2.1.2.1 Parameter analysis
	2.1.2.2 Parameter determination
	2.1.3 Configuration of viscous dampers
	2.2 Second stage of design
	2.2.1 Checking for structural deformations
	2.2.2 Checking for additional damping ratio
	2.2.3 Checking for connecting structural members
	3 Example of a RC frame structure retrofitted with viscous dampers
	3.1 Building description
	3.2 Structural analytical parameters
	3.3 Preliminary design
	3.3.1 Number of viscous dampers
	3.3.2 Parameters of viscous dampers
	3.3.3 Configuration of viscous dampers
	3.4 Second stage of design
	3.4.1 Checking for structural deformations
	3.4.2 Checking for additional damping ratio
	3.4.3 Checking for connecting structural members
	3.4.3.1 Stiffness of the damping system
	3.4.3.2 Internal forces of the columns
	4 Conclusions and discussions
	Acknowledgements
	References