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Masonry infill walls in reinforced concrete frames as a source of structural
damping
Article in Earthquake Engineering & Structural Dynamics · June 2014
DOI: 10.1002/eqe.2380
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Earthquake Engineering and Structural Dynamics
Masonry infill walls in RC frames as a source of structural
damping
Journal:
Earthquake Engineering and Structural Dynamics
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Manuscript ID:
Wiley - Manuscript type:
Date Submitted by the Author:
10-Apr-2013
Ozkaynak, Hasan; Beykent University, Depertmant of Civil Engineering
Yüksel, Ercan; Istanbul Technical University, Faculty of Civil Engineering
Yalcin, Cem; Bogazici University, Civil Engineering
Dindar, Ahmet; Istanbul Kultur University, Department of Civil Engineering
Büyüköztürk, Oral; Massachusetts Institute of Technology, Department of
Civil Engineering
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Keywords:
Research Article
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Complete List of Authors:
EQE-12-0222.R1
Masonry Infilled Frames, CFRP Retrofitting, Damping, Equivalent Damping,
Energy Methods
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Page 1 of 24
EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS
Masonry infill walls in RC frames as a source of structural damping
H. Ozkaynak1, E.Yuksel2∗, C.Yalcin3, A.A.Dindar4, O. Buyukozturk5
1
Department of Civil Engineering, Beykent University, Istanbul, Turkey
Faculty of Civil Engineering, Istanbul Technical University, Istanbul, Turkey
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Department of Civil Engineering, Bogazici University, Istanbul, Turkey
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Department of Civil Engineering, Istanbul Kultur University, Istanbul, Turkey
5
Civil and Environmental Eng., Massachusetts Institute of Technology, MA, USA
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SUMMARY
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This paper presents the results of an experimental study on the determination of damping
characteristics of bare, masonry infilled, and carbon fiber reinforced polymer (CFRP)
retrofitted infilled reinforced concrete (RC) frames. It is well known that the masonry infills
are used as partitioning walls having significant effect on the damping characteristics of
structures as well as contribution to the lateral stiffness and strength. The main portion of the
input energy imparted to the structure during earthquakes is dissipated through hysteretic and
damping energies. The equivalent damping definition is used to reflect various damping
mechanisms globally. In this study, the equivalent damping ratio of CFRP retrofitted infilled
RC systems is quantified through a series of 1/3-scaled, one-bay one storey frames. QuasiStatic tests (QS) are carried out on 8 specimens with two different loading patterns: one and
three-cycled displacement histories, and the Pseudo-Dynamic tests (PsD) performed on 8
specimens for selected acceleration record scaled at three different PGA levels with two
inertia mass conditions. The results of the experimental studies are evaluated in two phases: i)
equivalent damping is determined for experimentally-obtained cycles from quasi-static and
pseudo-dynamic tests, ii) an iterative procedure is developed based on the energy balance
formulation to determine the equivalent damping ratio. Based on the results of these
evaluations, equivalent damping of levels of 5%, 12%, and 14% can be used for bare, infilled
and retrofitted infilled RC frames, respectively.
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Key words: Masonry Infilled Frames, CFRP Retrofitting, Damping, Equivalent Damping,
Energy Methods.
1. INTRODUCTION
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Past earthquakes and research demonstrated that the masonry infill walls have advantages in
the improvement of energy dissipation as well as increase of stiffness and strength properties
of RC structures when they are placed regularly throughout the structure and/or they do not
cause shear failures of columns, [1]. Damping in RC structures arises through energy
dissipation by various mechanisms such as cracking of concrete and sliding between
structural and non-structural elements etc. Since it is very difficult and also unpractical to
directly calculate the damping, experimental research is essential in order to determine a
range of such energy dissipation characteristic. A review is given in the following:
∗
Corresponding author. Tel/Fax:+90 212 285 6761.
E-mail address: yukselerc@itu.edu.tr (E.Yuksel)
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Earthquake Engineering and Structural Dynamics
Buttmann [2] conducted an experimental study on the specimens with the dimensions of
100×200×11.5 cm and 24 cm. The horizontal sinusoidal excitation applied to the specimens
was generated by a dynamic oscillator with a maximum power of 20 kN. The experimental
study yielded critical damping ratios of 11% for shear walls and 24% for masonry walls.
Farrar and Baker [3], performed an experimental study on 1/3-scaled low aspect ratio RC
shear walls. It was concluded that within elastic range of the testing, damping ratio was found
to be 2%, and when the damages increased and re-bars yielded, this value increased up to
22%. Fardis and Panagiotakos [4] evaluated PsD test results that were conducted in ELSA
Laboratory by Negro and Verzeletti, [5]. It was concluded that the infills resulted damping
after the first cracks observed. It was stated that the hysteretic energy dissipation occurred
through the masonry infills. Also, the response spectra of an elastic SDOF infilled frame,
despite infill’s apparent stiffening effect on the system, a reduction in the spectral
displacement and forces were obtained mainly through high level damping. Hashemi and
Mosalam [6], [7] conducted shake table tests on 3/4-scaled, three dimensional infilled RC
frames. The tests resulted nearly 4 times higher structural stiffness, shortened natural period
nearly 50%, increased damping coefficient from about 4% to 12% and also increased the
energy dissipation capacity of the system.
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Costa et al. [8], performed in-situ tests on masonry walls of abandoned traditional houses.
Five specimens were tested aiming at characterizing the out-of-plane behavior of stone
masonry walls and strengthening solutions recommended for post-earthquake interventions.
Even for small drift values as 0.1%, the hysteresis is significant leading to an equivalent
hysteretic damping value of 12% mainly explained by permanent deformations developed at
the joints already for small displacement levels. The evolution of hysteretic damping is almost
linear with the evolution of drift up to the formation of a complete diagonal crack to the
foundation which occurred for the drift cycle of 0.75%. This led to significant residual
deformations along the wall and an equivalent hysteretic damping level of 26%. The final part
of the test (drift of 1.0 and 1.25%) shows a constant hysteretic damping level close to 25% as
a result of the severe damage observed and permanent deformations of the wall.
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Sofronie [9] indicated that the masonry walls act as active dampers when strengthened with
FRP by increasing frictional forces between wall elements resulting in higher damping.
Santa-Maria et al. [10] conducted experimental studies on masonry walls under the effect of
monotonic and cyclic loadings. The masonry specimens were retrofitted by horizontal,
vertical and diagonally-braced FRP. Especially horizontally-retrofitted specimen displayed a
great increase in damping. Elgawady et al. [11] investigated the behaviors of seven specimens
of 1/2-scaled FRP retrofitted masonry walls under cyclic displacement reversals. FRP caused
a great increase in lateral stiffness, strength and energy dissipation capacity. The damping
values were also determined for each specimen at varying drift levels. FRP confinement
provided increase in damping ratios. Some of the specimens were retrofitted after tested and
these specimens produced higher values of structural damping.
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Although various experimental studies have been conducted on the determination of damping
characteristics of masonry walls and masonry infilled RC frames; for the quantification of
equivalent damping, there is an apparent gap in the literature on CFRP-retrofitted infilled
frames. There is a necessity about the damping characteristic which is particularly important
for the accurate estimation of seismic forces, of CFRP-retrofitted infilled RC frames for the
development of realistic structural models.
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In this paper, a new concept based on the energy balance is proposed to quantify the damping
characteristic of the specimens.
The scope of this experimental study is limited to sixteen specimens of 1/3-scaled one bayone storey RC frames subjected to Quasi-Static (QS) and PsD tests. The test results are used
in the quantification of equivalent damping ratios for bare, masonry infilled and CFRPretrofitted masonry infilled RC frames.
2. EXPERIMENTAL STUDY
The experimental study is conducted on sixteen 1/3-scaled one bay-one story RC frame which
is taken out from a three-span and five-storey RC building. The specimens were built
according to the old construction practice which had several variances with the current
seismic design code of Turkey, [12]. Dimensions and reinforcement details of the specimen
are illustrated in Fig.1a. Longitudinal reinforcement ratio in columns and beam is 1% while
transverse reinforcement ratio is around 0.4%. No confinement reinforcement in and around
beam-column connections are used. Compression strength of concrete is obtained 19 MPa
from the standard cylinder tests which corresponds to the strength at the day of testing. Yield
strength of reinforcements is 420 MPa and 500 MPa for 8 and 6 mm diameters, respectively.
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Section a-a
b
Section b-b
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4φ 8
φ 6/140
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5φ 12
100 200
933
1533
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(a) Geometry and reinforcement details
(b) Clay brick
(c) Anchorage detail of CFRP strips
Fig.1 Properties of the test specimens (All dimensions are in mm)
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Earthquake Engineering and Structural Dynamics
The clay type brick used in the infill wall has a dimension of 87×84×56 mm, Fig.1b. Both
sides of infill wall was plastered having a thickness of 10 mm. Compression tests of the
masonry wallets with the dimensions of 350×350×56 mm resulted compression strengths of
5.0 and 4.1 MPa in the two perpendicular directions. The diagonal tension (shear) test defined
in ASTM E519–02 [13] was applied and the shear strength of 0.95 MPa is reached, [14]. Unit
weight and fiber density of the used CFRP are 300 g/m2 and 1.79 g/cm3, respectively.
Modulus of elasticity, tensile strength and ultimate elongation capacity of CFRP are 230 GPa,
3900 MPa and 1.5%, correspondingly. Special anchorages were provided along the CFRP
sheets at approximately quarter distances of the diagonal with the length of the 24 cm which
will be enough to cover the CFRP strips applied on both sides of the infill. The CFRP sheet
was rolled with enough amount of epoxy and was installed in the infill through the bricks,
Fig.1c.
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The four groups of specimens used in the study are shown schematically in Fig.2. Two
alternative CFRP retrofitting are applied to the infilled frames.
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100 315
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1533 mm
100 200 120 234 224 234 120 200 100
1533 mm
315 100
c) Cross-braced frame
d) Diamond cross-braced frame
Fig.2 Definition and the specimens
A servo-controlled 280 kN-capacity hydraulic jack is used for the lateral loading. The
specimens are fixed to the rigid steel beam of the test frame, Fig.3. No axial forces were
applied to the columns to attain fairly simple testing setup, particularly for the PsD tests. To
prevent the potential out of plane deformations, four restrainers were used in the testing set-
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up. There is small distance between the restrainers and the beam, and grease was applied to
the surface of the restrainers.
Steel Reaction Frame
Out of Plane Restrainers
Load Cell
Hinge
Hydraulic Actuator
Hinge
Loading Frame
Strong Wall
Strong Floor
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Fig.3 The testing set-up
In QS tests, eight specimens were tested in two groups using two different drift-based
reversed cyclic loading patterns, namely; one and three-cycled displacement history cases,
Fig.4.
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b) Three Cycle Loading Pattern
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a) One Cycle Loading Pattern
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Fig.4 Displacement patterns used in QS Tests
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The acceleration record used in the PsD tests were derived from the BOL090 component of
October 12, 1999 Düzce Earthquake which has PGA=0.822g. The part between 8 to 18 s of
the record was modified to comply the acceleration design spectrum defined in Turkish
Earthquake Code [12] for seismic Zone 1 and firm type soils (Z2), Fig. 5. The Oasis Sigraph
[15] software was used for this process. The target acceleration record of PGA=0.4g is called
as the design earthquake. The other two records which are PGA=0.2g and PGA=0.6g were
derived from the design earthquake by linear scaling.
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Spectra Sae [m/s2]
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TEC-07
Düzce-R Spectra
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Time (s)
Fig.5 Spectrum compatible acceleration record of PGA=0.4g
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3.0
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Earthquake Engineering and Structural Dynamics
The power spectra of the original and modified records are compared in Fig.6. The imparted
energies which are the area enclosed by the power spectra, are 0.109 and 0.156 units for the
original and modified records, respectively, Kuwamura et.al, [16]. So, the modified
acceleration record used in PsD tests imparted more energy to the specimen respect to the
original one.
Power Spectral Values
0.25
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Modified-Duzce
Original-Duzce
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Frequency
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Fig.6 Power spectra of the original and modified records
Mass intensity was one of the main parameters of PsD tests. The values practiced are
M1=0.0085 kNs2/mm and M2=0.0221 kNs2/mm. They are representing the lower and upper
storey masses which are scaled down from the prototype structure. The experimental details
can be found in Ozkaynak’s PhD dissertation [17] and Ozkaynak et al., [18].
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A high sensitive optical displacement transducer is used in the application of target
displacement to the specimens in PsD tests. Also, high sensitive load cell was instrumented
for all the experiments. The restoring force corresponding to a particular displacement
increment is evaluated within a number of buffering force.
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The PsD tests were conducted by slow mode. Application of the target displacement,
measuring of restoring force, measuring of displacements and deformations throughout the
specimen, solution of dynamic equilibrium equation for the next step elapsed 10-15 seconds
for each point of the acceleration record. No data filtering was used.
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The typical lateral load versus top displacement curves obtained in QS tests for bare, infilled
and retrofitted cases are shown in Fig.7. The increase of strength and stiffness from bare to
infilled and retrofitted specimens was clearly seen from the curves. Also, the level of damage
within the inelastic range is significant indicating progression of damping through energy
dissipation.
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a) Bare Frame
b) Infilled frame
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c) Cross-braced frame
d) Diamond cross-braced frame
Fig.7 Lateral load versus top displacement curves obtained in QS tests
The behavior of the test specimens for the PsD tests is given in Fig.8 for the parameters of
M2= 0.0221 kNs2/mm and PGA=0.4g.
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Story Drift(%)
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Fig.8 Lateral load versus top displacement curves obtained in PsD tests of M2=0.0221kNs2/mm
and PGA=0.4g
In case of the bare frame, the sources of energy dissipation mechanisms could be clearly
observed from the progression of damages through the increments of drifts. Flexural types of
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Earthquake Engineering and Structural Dynamics
cracks were observed at both ends of the columns. The crack distribution photos of bare frame
at the initial stage and end of the test are illustrated in Figs. 9a and 9b, respectively.
a) Damage condition for bare frame at the
initial stage of PGA=0.2g PsD test
b) Damage condition for bare frame at the
intermediary stage of PGA=0.4g PsD test
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Fig.9 Typical damages of bare frame specimens in PsD tests
In case of the infilled frame, flexural cracks were first formed on column ends, then slight
separation of infill wall from RC members were observed, Fig.10a. It was followed by infill
cracking in both diagonal directions. Finally, corner crushing in infill wall and corner
separations were observed, Fig.10b.
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a) Damage condition for infilled frame at
the initial stage of one-cycle QS test.
b) Damage condition for infilled frame at
the end of one-cycle QS test.
Fig.10 Typical damages of infilled specimens in PsD tests
In case of the retrofitted specimens, the first observed damages were the flexural cracks on
columns. The succeeding damages observed in further steps of the tests were diagonal cracks
occurred on infill walls. Towards end of the tests, crashing of plaster and infill damages were
localized around the near vicinity of CFRP anchorages, Fig.11.
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a) Damage condition for cross braced
frame at the initial stage of one-cycle QS
test.
b) Damage condition for cross braced
frame at the end of one-cycle QS test.
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c) Damage condition for diamond cross
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d) Damage condition for diamond cross
braced frame at the end of one-cycle QS
test.
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Fig.11 Typical damages of the retrofitted specimen in QS tests
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3. EVALUATION of QS and PsD TEST RESULTS IN TERMS OF
DISSIPATED ENERGY IN SUCCESSIVE CYCLES
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Typical behavior of a structure or structural member having energy dissipating capability,
subjected to cyclic loading is schematically illustrated in Fig.12. The energy dissipated in the
structure is given by the area EH enclosed in the hysteresis loop. Equivalent damping ratio is
defined according to Chopra [19] in terms of dissipated energy (EH) and strain energy (ES0),
hereafter referred to as Energy Ratio Method. The equivalent damping ratio ζeq is given by
Eq.1.
ζeq= (1/4π) (EH/ES0)
(1)
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Earthquake Engineering and Structural Dynamics
Force(kN)
EH
ES0
Displacement (mm)
Fig.12 Definition of the dissipated and strain energies, [19]
3.1 Evaluation of QS Test Results
The attained equivalent damping ratios with increasing drifts in QS tests are discussed here.
r
Fo
At the beginning of one cycle static tests, the obtained equivalent damping ratio for bare
frame was around 5%, whereas the resulting damping ratios of infilled and retrofitted
specimens were close to 10-12%, indicating that the effect of retrofitting was not apparent,
Fig.13.
Pe
25
Equivalent Damping (%)
30
35
30
25
20
15
10
20
15
10
Re
Equivalent Damping (%)
35
er
5
5
1 Cycle
3 Cycle
0
0
1
2
3
Drift Ratio (%)
4
5
0
30
Equivalent Damping (%)
35
30
25
20
15
10
1 Cycle
3 Cycle
0
1
2
3
Drift Ratio (%)
2
3
Drift Ratio (%)
4
5
b) Infilled frame
35
0
1
4
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Equivalent Damping (%)
a) Bare Frame
5
1 Cycle
3 Cycle
0
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25
20
15
10
5
1 Cycle
3 Cycle
0
5
0
1
2
3
Drift Ratio (%)
4
5
c) Cross-braced frame
d) Diamond cross-braced frame
Fig.13 Equivalent Damping Variations of QS Tests Results
For the three cycle QS tests, the equivalent damping ratio was calculated based on the average
of the results of three cycles. At the initial stage, the obtained equivalent damping ratio for
bare frame was around 5%, whereas the resulting damping ratios of infilled and retrofitted
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specimens were close to 9-10%, and again, indicating that the effect of retrofitting was not
apparent.
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
Ratio
Ratio
The ratio of the damping values calculated for the one-cycle tests to the one obtained from the
first cycle of the three-cycle test is given in Fig. 14 in terms of drift ratios. There are two
distinct regions in the graphics. Here up to 2.0% drift represents relatively undamaged stage
whereas beyond 2.0% is the damaged stage. Specifically for the cross-braced frame, this ratio
in the first region was around unity where it was determined as 1.31 for the second region.
This can be evaluated as in the damaged stage one cycle test results with 30% greater value
than that of three-cycle test.
r
Fo
rav=1.32
rav=0.96
0
1
2
3
Drift Ratio (%)
4
5
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
1
2
3
Drift Ratio (%)
4
5
4
5
b) Infilled frame
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
Ratio
er
Ratio
a) Bare Frame
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
rav=1.05
rav=1.38
0
Pe
rav=1.01
0
1
rav=1.31
4
5
rav=1.15
0
1
rav=1.0
2
3
Drift Ratio (%)
vi
2
3
Drift Ratio (%)
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c) Cross-braced frame
d) Diamond cross-braced frame
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Earthquake Engineering and Structural Dynamics
Fig.14 Equivalent damping ratios for the 1-Cycle to 1st loop of 3-Cycle QS test varying with
drift ratios.
Depending on the QS test results, one can be concluded that prior to inelastic range, damping
ratios for bare, infilled and retrofitted frames can be assigned to 5%, 9-12% and 9-12%,
respectively. However, within the inelastic range which is initiated beyond 2% drift, the
equivalent damping ratios for bare, infilled and retrofitted frames can be taken in the range of
8-10%, 10-13% and 13-15%, respectively.
3.2 Evaluation of PsD Test Results
The results of PsD tests were used for the determination of damping properties of bare,
infilled and CFRP retrofitted infilled specimens. Complete closed cyclic loops were extracted
from the load-displacement relations of PsD test results. Using the energy ratio method,
average equivalent damping ratios for M1 and M2, and also for different PGA intensities were
determined.
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Earthquake Engineering and Structural Dynamics
In Eq.1, the term involving the area of hysteresis loop is directly affected by the fluctuation of
lateral load within a small load range coupled with high number of loops during the
experiment. This may cause sudden increase or decrease of the area of the loops compared to
other successive loops. Consequently, the obtained damping values diverge in a wide range of
scatterness. Histogram analysis of the scattered data showed the distribution of damping ratios
in terms of recurrence number. This statistical analysis method let us to define the average
equivalent damping ratio using geometric-mean in lieu of arithmetic mean.
The drift vs. equivalent damping ratio graphics were plotted in the drift ranges of ±-1.5% in
order to make good comparison with high mass and intensity cases. Especially, for M1 case of
the infilled specimen, the points gravitate within ±0.1% drift region. So, the values are seen as
if they were concentrated in the vicinity of origin.
3.2.1 Bare frame
r
Fo
The analyses results for bare frame are shown in Fig. 15. It is seen that for increasing PGA
levels, the average equivalent damping ratio shown with a broken line is also increasing. The
equivalent damping ratio scatters between 5 to 20% with an average value of 8 to 13% for two
mass conditions, M1 and M2.
Pe
M1 Inertia Mass
Equivalent Damping Ratio (%)
25
20
15
10
5
Geo-Mean
10
5
Geo-Mean
0
-1.5
-1.0
-0.5 0.0 0.5
Drift Ratio (%)
1.0
1.5
-1.5
Equivalent Viscous Damping (%)
20
15
10
5
Geo-Mean
0
-1.5 -1.0 -0.5 0.0 0.5
Drift Ratio (%)
1.0
1.5
-1.0
-0.5 0.0 0.5
Drift Ratio (%)
1.0
1.5
1.0
1.5
ew
25
25
Equivalent Damping Ratio (%)
15
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0
0.4g
20
Re
0.2g
25
M2 Inertia Mass
Equivalent Damping Ratio (%)
PGA
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10
5
Geo-Mea n
0
-1.5
-1.0
-0.5 0.0 0.5
Drift Ratio (%)
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Equivalent Damping Ratio (%)
0.6g
25
20
15
NA
10
5
Geo-Mean
0
-1.5
-1.0
-0.5 0.0 0.5
Drift Ratio (%)
1.0
1.5
Fig.15 Equivalent damping ratio variation for bare frame
3.2.2 Infilled frame
r
Fo
Equivalent damping ratio distributions and the average values are illustrated in Fig. 16. For
various PGA intensities, the observed equivalent damping ratios are between 5 to 25%,
whereas the average damping is in the vicinity of 12% for the two mass cases, M1 and M2.
Pe
M1 Inertia Mass
Equivalent Damping Ratio (%)
20
M2 Inertia Mass
25
15
10
5
Geo-Mean
-0.5 0.0 0.5
Drift Ratio (%)
10
1.0
-1.5
1.5
Equivalent Damping Ratio (%)
25
20
15
10
5
Geo-Mean
0
Geo-Mea n
-1.0
-0.5
0.0
0.5
Drift Ratio (%)
1.0
1.5
ew
Equivalent Damping Ratio (%)
25
0.4g
5
vi
-1.0
15
0
0
-1.5
20
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0.2g
25
Equivalent Damping Ratio (%)
PGA
er
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15
10
5
Geo-Mean
0
-1.5
-1.0
-0.5 0.0 0.5
Drift Ratio (%)
1.0
1.5
-1.5
-1.0
-0.5 0.0 0.5
Drift Ratio (%)
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1.0
1.5
Earthquake Engineering and Structural Dynamics
Equivalent Damping Ratio (%)
0.6g
25
20
15
NA
10
5
Geo-Mea n
0
-1.5
-1.0
-0.5 0.0 0.5
Drift Ratio (%)
1.0
1.5
Fig.16 Equivalent damping variation for infilled frame
3.2.3 Cross-braced frame
r
Fo
Equivalent damping ratio variations and their average values are plotted in Fig.17. For various
PGA intensities the observed equivalent damping ratio was between 5 to 25%, whereas the
mean damping values range 10-13%, for two mass cases, M1 and M2.
Pe
PGA
M1 Inertia Mass
15
10
5
Geo-Mean
0
Equivalent Damping Ratio (%)
Equivalent Damping Ratio (%)
20
M2 Inertia Mass
25
10
5
Geo-Mea n
-0.5 0.0 0.5
Drift Ratio (%)
1.0
vi
-1.0
1.5
-1.5
Equivalent Damping Ratio (%)
25
20
15
10
5
Geo-Mea n
0
-1.0
-0.5 0.0 0.5
Drift Ratio (%)
1.0
1.5
ew
25
Equivalent Damping Ratio (%)
15
0
-1.5
0.4g
20
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0.2g
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15
10
5
Geo-Mean
0
-1.5
-1.0
-0.5 0.0 0.5
Drift Ratio (%)
1.0
1.5
-1.5
-1.0
-0.5 0.0 0.5
Drift Ratio (%)
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1.0
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25
Equivalent Damping Ratio (%)
Equivalent Damping Ratio (%)
0.6g
25
20
15
10
5
Geo-Mea n
20
15
10
5
Geo-Mean
0
0
-1.5
-1.0
-0.5 0.0 0.5
Drift Ratio (%)
1.0
1.5
-1.5
-1.0
-0.5 0.0 0.5
Drift Ratio (%)
1.0
1.5
Fig.17 Equivalent damping ratio variation for cross-braced frame
3.2.4 Diamond cross-braced frame
r
Fo
Equivalent damping ratio variations and their mean values are plotted in Fig.18. For various
PGA intensities, the observed equivalent damping ratio was between 5 to 25%, whereas the
mean damping value is in the vicinity of 13%, for two mass cases, M1 and M2.
Pe
M1 Inertia Mass
Equivalent Damping Ratio (%)
20
M2 Inertia Mass
25
15
10
5
Geo-Mean
-1.5
-1.0
-0.5 0.0 0.5
Drift Ratio (%)
5
1.0
1.5
-1.5
25
Equivalent Damping Ratio (%)
20
15
10
5
Geo-Mean
-1.0
-0.5 0.0 0.5
Drift Ratio (%)
20
15
-1.0
-0.5 0.0 0.5
Drift Ratio (%)
1.0
1.5
1.0
1.5
10
5
Geo-Mea n
0
0
-1.5
Geo-Mea n
ew
Equivalent Damping Ratio (%)
10
0
25
0.4g
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vi
0
20
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0.2g
25
Equivalent Damping Ratio (%)
PGA
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1.0
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-1.5
-1.0
-0.5 0.0 0.5
Drift Ratio (%)
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Earthquake Engineering and Structural Dynamics
25
Equivalent Viscous Damping (%)
Equivalent Damping Ratio (%)
0.6g
25
20
15
10
5
Geo-Mean
0
-1.5
-1.0
-0.5 0.0 0.5
Drift Ratio (%)
1.0
20
15
10
5
Geo-Mea n
0
-1.5
1.5
-1.0
-0.5 0.0 0.5
Drift Ratio (%)
1.0
1.5
Fig.18 Equivalent damping ratio variation for diamond cross-braced frame
Displacement responses obtained from low mass (M1) and low intensity conditions are
somewhat less than that of high mass (M2) and high intensity conditions. For this reason, the
obtained equivalent damping ratios are gravitated around the low drift ratios as seen in Figs.
16,17 and 18.
r
Fo
From the results of tests conducted with two different methods, it can be concluded that the
equivalent damping ratios of the retrofitted infilled frames were greater than those of bare and
infilled frames. In all tests, the diamond cross-braced frame had higher damping ratios than
the cross-braced frames. Furthermore, an average value of equivalent damping ratio for the
retrofitted infilled frames can be proposed as 13%.
er
Pe
The equivalent damping ratios that were obtained from PsD and QS tests are compared in
Table 1.
Re
Table 1.Equivalent damping ratios (%) obtained from QS and PsD tests
QS
Specimen Type
One
Cycle
11
13
13
14
Three
Cycle
8
13
14
13
PsD
M1
Mass
ew
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Infilled frame
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Diamond cross-braced frame
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11
13
13
13
M2
Mass
10
13
12
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4. PROPOSED APPROACH TO DETERMINE THE EQUIVALENT DAMPING RATIO
BY USING ENERGY BALANCE METHOD
Early studies on energy-based evaluation methods initiated with the work of Zahrah [20] who
performed numerical analyses to show how the energy terms in a simple structure is extracted
from the displacement response of the structure under earthquake excitation. The energy
considerations on structural systems and evaluation of structural performance based on energy
concepts have been the scope of several studies including those works that were conducted by
Shen and Akbas [21], Chou and Uang [22] and Nurtug and Sucuoglu [23].
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Negro and Verzeletti [5] investigated a four story infilled RC structure by subjecting it to PsD
tests on the basis of which the structural performance evaluation was accomplished using
energy considerations. Mosalam et al. [24] performed PsD tests on a two bay-two storey steel
structure. The test results were evaluated in terms of energy concepts. It was concluded that
the infill walls increased the structural damping from 2% to 12%.
The energy imparted into the structure during earthquake is divided into several mechanisms
namely elastic strain, kinetic, hysteretic and damping energies. The equation of motion of a
single degree of freedom system (SDOF) represents the excitation of ground and the response
of structure in terms of forces in time domain, Eq.2. Integration of the equation of motion
respect to the relative displacement of the system, u(t), yields the following equation, [25];
∫ m.u&&(t )du + ∫ c.u& (t )du + ∫ f du = − ∫ m.u&& (t )du
s
g
r
Fo
(2)
where the terms are named as Kinetic Energy (EK), Damping Energy (ED), Absorbed Energy
that is composed of Elastic Strain (ES) and Hysteretic (EH) Energies and Input Energy (EI),
respectively. Thus, Eq.2 can be re-expressed in the following form;
EK + ED + ES + EH = EI
(3)
Pe
Care should be paid to the third term on the left hand side of Eq.2 which includes the energy
of elastic and plastic strains. The elastic strain energy (Es) is formulated with respect to the
initial stiffness (k) of the structure, as follows;
er
ES =
( f s (t ) )2
(4)
2k
Re
The plastic strain energy namely hysteretic energy (EH) is calculated as cumulative area
enclosed by the load-displacement curves.
The displacement history is directly obtained as the response of test specimen due to the
applied input acceleration in PsD tests. The velocity and acceleration response histories are
then derived from the displacement history by using 7-point stencil central-differences.
ew
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Earthquake Engineering and Structural Dynamics
For an assumed equivalent damping ratio, the energy terms defined in Eq. 3 are calculated and
the global energy balance is checked. This iterative process is carried out till the energy
balance is satisfied with a predefined tolerance. Consequently EK, ED, ES and EH energy terms
are discretely determined. A general flow chart of the algorithm is illustrated in Fig. 19.
The well-known relation given in Eq. 5 is used to calculate the damping coefficient c as a
basis for this iterative scheme.
c=2ζmω
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(5)
Earthquake Engineering and Structural Dynamics
ζ
u
τ
0
0
E D (t ) = ∫ cu& (t )du = ∫ cu& (τ )dτ
∫
u
0
u
u
u
0
0
0
(ζ +∆ζ.)
m u&&(t )du + ∫ cu& (t ) du + ∫ f s (u , u& ) du = − ∫ m u&&g (t ) du
u
u
u
r
Fo
u
∫0 E K + ∫0 E D + ∫0 E S + ∫0 E H
?= −
u
∫0 E I
No
Yes
Equivalent Damping Ratio
Pe
Fig.19. Evaluation of equivalent damping ratio by Energy Balance Method.
The proposed energy balance methodology was carried out to PsD tests for M1 and M2 cases
and acceleration intensities of 0.2g, 0.4g and 0.6g. The energy distribution graphics and the
obtained equivalent damping ratios are presented in Figs. 20 and 21 for M1 and M2 cases,
respectively.
er
Re
PGA=0.4g
10000
8000
Energy (kN.mm)
Energy (kN.mm)
6000
EI
EK+ES+ED+EH
EK+ES+ED
EK+ES
EK
4000
ew
Bare Frame
10000
EI
EK+ES+ED+EH
EK+ES+ED
EK+ES
EK
8000
PGA=0.6g
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6000
4000
2000
2000
0
0
0
2
4
6
8
10
0
2
4
ζ
6
Time (s)
Time (s)
=15%
ζ =16%
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2500
EI
EK+ES+ED+EH
EK+ES+ED
EK+ES
EK
2000
Energy (kN.mm)
Energy (kN.mm)
2000
Infilled frame
2500
EI
EK+ES+ED+EH
EK+ES+ED
EK+ES
EK
1500
1000
1500
1000
500
500
0
0
0
2
4
6
8
0
10
2
4
ζ =%8.5
EI
EK+ES+ED+EH
EK+ES+ED
EK+ES
EK
1500
1000
10
8
10
1000
500
500
0
0
0
2
4
6
8
0
10
2
4
1500
EI
EK+ES+ED+EH
EK+ES+ED
EK+ES
EK
2000
1500
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Energy (kN.mm)
2000
ζ =%11
2500
er
EI
EK+ES+ED+EH
EK+ES+ED
EK+ES
EK
6
Time (s)
ζ =%10
2500
Energy (kN.mm)
8
1500
Time (s)
1000
500
1000
500
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0
Time (s)
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Time (s)
ew
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EI
EK+ES+ED+EH
EK+ES+ED
EK+ES
EK
2000
Energy (kN.mm)
Energy (kN.mm)
2000
8
ζ =%11
2500
Pe
Cross-braced frame
2500
6
Time (s)
Time (s)
r
Fo
ζ =%10
ζ =%11
Fig. 20 Energy distributions and obtained equivalent damping ratios for M1 Case
PGA=0.2g
5000
EI
EK+ES+ED+EH
EK+ES+ED
EK+ES
EK
4000
Energy (kN.mm)
Energy (kN.mm)
PGA=0.4g
5000
EI
EK+ES+ED+EH
EK+ES+ED
EK+ES
EK
4000
Bare Frame
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3000
2000
3000
2000
1000
1000
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Time (s)
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12000
EI
EK+ES+ED+EH
EK+ES+ED
EK+ES
EK
9000
Energy (kN.mm)
Energy (kN.mm)
Infilled frame
12000
6000
3000
0
EI
EK+ES+ED+EH
EK+ES+ED
EK+ES
EK
9000
6000
3000
0
0
2
4
6
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0
2
4
Time (s)
ζ =%9
30000
Energy (kN.mm)
Energy (kN.mm)
15000
10000
5000
20000
10
8
10
10000
5000
0
0
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2
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10
0
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4
20000
15000
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EI
EK+ES+ED+EH
EK+ES+ED
EK+ES
EK
25000
Energy (kN.mm)
20000
ζ =12%
30000
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EK+ES+ED+EH
EK+ES+ED
EK+ES
EK
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Time (s)
ζ =11%
30000
Energy (kN.mm)
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Time (s)
10000
5000
10000
5000
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Time (s)
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2
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6
Time (s)
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Diamond cross-braced frame
10
EI
EK+ES+ED+EH
EK+ES+ED
EK+ES
EK
25000
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Cross-braced frame
20000
8
ζ =%11
30000
EI
EK+ES+ED+EH
EK+ES+ED
EK+ES
EK
25000
6
Time (s)
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ζ =10%
ζ =12%
Fig. 21 Energy Distributions and obtained equivalent damping ratios for M2 Case
The average equivalent damping ratios that are determined by using both energy ratio and
energy balance methods are summarized in Table 2. The results given for PsD tests belongs to
the design earthquake (PGA=0.4g). In the energy ratio method, equivalent damping ratio of
the specimens were originated as the geometric mean of damping ratio obtained for each loop
as indicated in Figs.15, 16, 17, 18. However, the energy balance method takes the frequency
content and the entire duration of the record into account as per Figs. 20 and 21. Hence, in the
evaluation of the PsD test results, the developed energy balance method is more reliable. For
the design earthquake (PGA=0.4g), the energy ratio method yields relatively higher values
compared to the energy balance method.
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Table 2. Equivalent damping ratios (%) obtained from QS and PsD tests
Energy Balance
Method
Energy Ratio Method
Specimens
QS Tests
PsD Tests
PsD Tests
One
cycle
Three
cycle
M1
M2
M1
M2
Bare frame
11
8
10
12
15
10
Infilled frame
13
13
11
11
9
11
Cross-braced frame
13
14
10
12
10
11
Diamond cross-braced frame
14
13
12
11
10
10
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According to all of the performed QS and PsD test results, the minimum value for the
equivalent damping ratio of infilled and CFRP-retrofitted infilled frames would be 10% to
13%.
5. CONCLUSIONS
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The results of the experimental study have been evaluated for bare, infilled and retrofitted
infilled RC frames in order to quantify the equivalent damping ratios in terms of experimental
parameters that were used in QS tests such as one and three cycle loading patterns and in PsD
tests such as inertia masses and various PGA levels. In the analysis, the Energy Ratio Method
and Energy Balance Method have been used to determine the equivalent damping ratios for
each parameter and also their trends were discussed. The equivalent damping ratios obtained
from the Energy Balance Method, are smaller than those derived from Energy Ratio Method.
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The following conclusions are drawn from the QS Tests:
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1. The equivalent damping for bare frame varies between 8 to 11% depending on the
damage level.
2. The equivalent damping ratio derived for infilled frame is 13%. Therefore the effect of
infill walls on damping can be clearly seen from the results.
3. The cross-braced and diamond cross-braced frames has the biggest damping ratio which
is up to 14%.
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The following conclusions are drawn for the PsD Tests:
1. In the comparison made for the design earthquake (PGA=0.4g), equivalent damping
characteristics of all the specimens are around 10-15%. The value for bare frame accounts
for the hysteretic cycles prior to the collapse.
2. The average damping characteristics of the retrofitted frames are consistently stable for
all masses and PGA levels. Although bare frame failed during the design earthquake
(PGA=0.4g), both types of the retrofitted frames could endure at the end of the
PGA=0.6g earthquake. So, for the retrofitted frames, equivalent damping ratio turns out
to be sustainable through the seismic action.
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The equivalent damping ratio for infilled frames is obtained in the range of 10 to 13%. This
result is consistent with the existing literature.
Based on the overall evaluation, equivalent damping ratio of 10-13% is recommended as a
sustainable range for the CFRP retrofitted RC frames.
Acknowledgments
This study was conducted at the Structural and Earthquake Engineering Laboratory of
Istanbul Technical University. It was sponsored by research projects 106M050 of the
Scientific and Technological Research Council of Turkey (TUBITAK) and 31966 of Istanbul
Technical University (ITU) Research Funds. The contributions of M.Sc. H. Saruhan, E.S.
Tako and I. Bastemir to the experimental works are gratefully acknowledged.
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