As observed from Table 3, the drift ratio at axial failure in SC-1.7 and RC-1.7 Series specimens reduced by around 14 and 30%, respectively, as the column axial load ratio was increased
Trang 1Experimental Studies on the Backbone Curves
of Reinforced Concrete Columns with Light
Transverse Reinforcement Cao Thanh Ngoc Tran, Ph.D.1; and Bing Li, Ph.D.2
Abstract: This paper presents the investigations carried out on the backbone curves of reinforced concrete (RC) columns with light trans-verse reinforcement An experimental program consisting of five half-scale RC columns with light transtrans-verse reinforcement was carried out The specimens are tested to the point of axial failure to obtain the backbone curves of such columns Quasi-static cyclic loading simulating earthquake actions is applied The backbone curves obtained from the tested specimens are then compared with the existing seismic assess-ment guidelines The test results indicate that the initial column stiffness and ultimate displaceassess-ments (displaceassess-ments at axial failure) are overestimated and underestimated by some guidelines and provisions The existing initial stiffness and ultimate displacement models are briefly reviewed and compared with the experimental results The results show that the existing initial stiffness and ultimate displacement models produced better results than the existing seismic assessment guidelines.DOI: 10.1061/(ASCE)CF.1943-5509.0000626 © 2014 American Society of Civil Engineers
Author keywords: Reinforced concrete columns; Seismic loading; Backbone; Light transverse reinforcement
Introduction
A large number of existing reinforced concrete (RC) columns has
not been designed following the requirements of modern seismic
design codes Vital deficiencies in such columns include typical
reinforcement details such as (1) lightly, widely spaced, and poorly
anchored transverse reinforcement and (2) lap-splice details These
are generally termed as nonseismically detailed RC columns
Recent postearthquake investigations (ERRI 1999a,b,c) indicated
that extensive damage occurred as a result of excessive shear
de-formation in nonseismically detailed RC columns, thus leading to
shear failure, axial failure, and full collapse of structures
There-fore, a thorough evaluation of nonseismically detailed RC columns
is needed to understand the seismic behavior of these structures
Extensive experimental research studies carried out on ductile
columns in different countries throughout the last decades have
given a better understanding on the seismic behavior of ductile
col-umns However, there is relatively limited literature available for
nonseismically detailed columns with respect to ductile detailed
columns In addition, most tests of RC columns subjected to
seis-mic loading have been terminated shortly after loss of lateral load
resistance Few tests on RC columns have been carried out to the
point of axial failure (Yoshimura and Yamanaka 2000;Lynn 2001;
Sezen 2002; Nakamura and Yoshimura 2002; Yoshimura and
Nakamura 2003;Yoshimura et al 2003;Ousalem 2006;Henkhaus
et al 2009;Tran 2010;Wibowo et al 2014) This has resulted in the
limited understanding of failure and collapse mechanisms of non-seismically detailed structures
Therefore, further analytical and experimental studies are needed to understand the seismic behavior of nonseismically de-tailed columns better The main focus of this research is on the backbone curves of the RC columns with light transverse reinforce-ment This paper comprises of two parts The first part presents the test results obtained from an experimental program consisting of five half-scale RC columns with light transverse reinforcement The backbone curves obtained from these tests are the compared with the existing seismic assessment guidelines [FEMA 356 (FEMA 2000); Elwood et al 2007] Further comparisons with the existing initial stiffness (Elwood and Eberhard 2009; Tran and Li 2012) and ultimate displacement models (Elwood and Moehle 2005;Tran and Li 2013;Wibowo et al 2014) are presented
in the second part of the paper The recommended backbone curves are also presented in the second part of the paper
Previous Seismic Assessment Models
In this section, the backbone curves based on FEMA 356 (FEMA
2000) and the provisions of ASCE 41 (Elwood et al 2007) are reviewed briefly According to FEMA 356 (FEMA 2000) and ASCE 41 (Elwood et al 2007), the force-displacement relationship follows the general trend as shown in Fig.1
Flexural and shear rigidity are considered in the calculation of the initial stiffness of columns in both FEMA 356 (FEMA 2000) and ASCE 41 (Elwood et al 2007) Shear rigidity for rectangular cross sections is defined as 0.4EcAgin both FEMA 356 (FEMA
2000) and ASCE 41 (Elwood et al 2007) According to FEMA
356 (FEMA 2000) and ASCE 41 (Elwood et al 2007), flexural rigidity is related to applied column axial loads as tabulated in Table1
The deformation indexes (a, b) as illustrated in Fig.1are de-fined as flexural plastic hinge ratios which depend on column axial load, nominal shear stress, and details of columns The index c as
1 Lecturer, Dept of Civil Engineering at International Univ., Vietnam
National Univ., Ho Chi Minh 70800, Vietnam (corresponding author).
E-mail: tctngoc@hcmiu.edu.vn
2 Associate Professor, School of Civil and Environmental Engineering,
Nanyang Technological Univ., Singapore 639798.
Note This manuscript was submitted on October 4, 2013; approved on
May 6, 2014; published online on September 8, 2014 Discussion period
open until February 8, 2015; separate discussions must be submitted for
individual papers This paper is part of the Journal of Performance of
Constructed Facilities, © ASCE, ISSN 0887-3828/04014126(11)/$25.00.
Trang 2defined in FEMA 356 (FEMA 2000) is 0.2, whereas per ASCE 41
(Elwood et al 2007), this index ranges from 0 to 0.2 depending on
the column axial load, nominal shear stress, and detailing of the
columns
According to both FEMA 356 (FEMA 2000) and ASCE 41
(Elwood et al 2007) guidelines, the maximum shear force of
the column is limited by its shear strength The shear strength is
defined in both FEMA 356 (FEMA 2000) and ASCE 41 (Elwood
et al 2007) as
Vn¼ k1
Avfytd
s þ λk2
0
@0.5pffiffiffiffiffifc0
as=d
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
0.5 ffiffiffiffiffifc0
p
Ag
A0.8Ag ð1Þ
where k1is equal to 1 for transverse steel spacing less than or equal
to d=2, k1is equal to 0.5 for spacing exceeding d=2 but not more
that d, k1is equal to 0 otherwise; k2is taken as 1 for displacement
ductility less than 2, as 0.7 for displacement ductility more than 4,
and varies linearly for intermediate displacement ductility; as=d
shall not be taken greater than 3 or less than 2; and λ is equal
to 1 for normal-weight concrete
Experimental Studies
An experimental program on RC columns with light transverse
reinforcement subjected to seismic loading was conducted to study
the backbone curves of such columns Five half-scale RC columns
with light transverse reinforcement were tested up to the point of
axial failure
Specimen Details and Test Procedure Fig.2and Table2illustrate the schematic dimensions and detailing
of test specimens The variables in the test specimens included col-umn axial loads, aspect ratio, and cross-sectional shapes Longitu-dinal reinforcement which consisted of 8-T20 deformed bars were characterized by a yield strength fyof 408 MPa (59.2 ksi) This resulted in a ratio of longitudinal reinforcement area to the gross area of column of 2.05% The transverse reinforcement of all test specimens comprised of R6 mild steel bars with 135° bent spaced at
125 mm (4.92 in.) were characterized by a yield strength fy of 393MPa (57.0 ksi) The theoretical flexural strength Vu and yield force Vy of the test specimens were estimated using the material properties obtained through tests and in accordance with the recommendations provided by FEMA 356 guidelines (FEMA
2000) The nominal shear strengths Vn of the test specimens were calculated based on the suggestion of FEMA 356 (FEMA 2000) The values of Vu, Vy, and Vn of the test specimens are tabulated
in Table2
A schematic of the loading apparatus is shown in Fig 3 A reversible horizontal load was applied to the top of the column us-ing a 1,000-kN (224.8-kip.) capacity actuator which was mounted onto a reaction wall The actuator was pinned at both ends to allow rotation during the test The base of the column was fixed to
a strong floor with four posttensioned bolts The axial load was applied to the column using two 1,000-kN (224.8-kip.) capacity actuators through a transfer beam
The column axial load was increased slowly until the designated level was achieved During each test, the column axial load was maintained by manually adjusting the vertical actuators after each load step The lateral load was applied cyclically through the hori-zontal actuator in a quasi-static fashion as shown in Fig 3 The loading protocol consisting of displacement-controlled steps is illustrated in Fig.4
The test specimens had been extensively instrumented both internally and externally Among those measurements were lateral load and displacement imposed at the top of the column, shear, and flexure deformations at the critical regions of the specimen and also the strains in the steel reinforcing bars as shown in Fig.5
Experimental Results
Cracking Patterns The crack patterns of the test specimens at shear failure (a loss of more than 20% of obtained maximum shear force) are shown in Fig.6 Important features in crack development of the specimens are highlighted
All of test specimens developed fine flexural cracks that were concentrated at both ends of the columns when loaded up to a drift ratio of 0.40% The lower the applied axial load, the more flexural cracks were observed in the columns Slight inclination was also observed in the flexural cracks of the test specimens at this stage
In loading to a drift ratio of 1.00%, whereas the specimens with a lower axial load developed severe shear cracking at both ends of the columns, the specimens with a higher axial load only showed a slight inclination in the flexural cracks In the subsequent loading cycles, the occurrence of a steep diagonal crack and the opening of the existing diagonal cracks resulted in a reduction in the shear-resisting capacity of the test specimens
There are two axial failure modes observed in the test specimens
as illustrated in Fig.7 In the first mode of axial failure, the steep diagonal crack developed in the column during the previous stages became wider This led to sliding between the crack surfaces as well
Displacement
cV A
Vy
Vu
u
Fig 1 Generalized force-displacement relationship in FEMA 356
Table 1 Flexural Rigidity in FEMA 356 ( FEMA 2000 ) and ASCE 41
( Elwood et al 2007 )
Column axial load
FEMA 356 ( FEMA 2000 )
ASCE 41 ( Elwood et al 2007 )
P ≥ 0.5f 0
P ≤ 0.3f 0
P ≤ 0.1f 0
Note: Linear interpolation between values listed in the table shall be
permitted.
Trang 38-T20 R6 350 350
135 degree hook
25
SC-2.4-0.20
R6 @ 125 T20
900
1700
350 350
8-T20 R6 900
1200
600 600
350 350
135 degree hook
R6 @ 125 T20
25
SC-1.7-0.20 SC-1.7-0.35
8-T20 R6 900
1700 350
490
135 degree hook 350
25
R6 @ 125 T20
RC-1.7-0.20 RC-1.7-0.35 Fig 2 Reinforcement details of test specimens (in mm)
Table 2 Summary of Test Specimens
Specimen
f 0 c (MPa)
b × h
c Ag
Aspect ratio
Vy (kN)
Vu (kN)
Vn (kN)
Note: 1 mm ¼ 0.04 in:; 1 kN ¼ 0.225 kip.
Actuator
Actuator Actuator
Strong Floor
Strong Wall
Specimen
L-shaped Steel Frame
Fig 3 Experimental setup
Trang 4as the buckling of longitudinal reinforcing bars and fracturing of
transverse reinforcing bars along this diagonal crack In the second
mode of axial failure, crushing of concrete as well as buckling of
longitudinal reinforcing bars and fracturing of transverse developed
across a damaged zone This type of axial failure was only observed
in Specimen RC-1.7-0.20, whereas the rest of the test specimens
exhibited the first mode of axial failure
Hysteretic Responses
Fig.8shows the hysteretic responses of the test specimens The
backbone curves based on FEMA 356 (FEMA 2000) and ASCE
41 (Elwood et al 2007) guidelines are also shown in Fig.8 Typical
brittle-failure hysteretic responses were observed in all test
speci-mens The hysteretic loops of the specimens show the degradation
of stiffness and load-carrying capacity during repeated cycles due
to the cracking of the concrete and yielding of the reinforcing bars
The pinching effect was observed in the hysteretic loops of all the
test specimens The shear failure in most test specimens occurred at
a drift ratio of less than 2.0% as shown in Table3
Shear Strengths Table3summarizes the shear strengths of the test specimens The shear strength of SC-1.7 Series specimens was enhanced by around 14%, as the column axial load was increased from 0.20 to 0.35f0
cAg An analogous trend was observed in the specimens of RC-1.7 Series, whose shear strengths experienced an enhancement
of around 13% as the applied axial load was increased from 0.20 to 0.35f0
cAg The previous discussion clearly indicates that the column axial load was beneficial to the shear strength of the test specimens whose applied axial load was in the range of 0.20 to0.35f0
cAg The shear strength of Specimens SC-2.4-0.20 and SC-1.7-0.20 obtained from the tests were 218.9 kN (49.2 kip.) and 294.2 kN (66.1 kip.), respectively The increase in shear strength between Specimens SC-2.4-0.20 and SC-1.7-0.20 was 34% Thus, it can
be concluded that the shear strength of the specimens in the current experimental program increased with a decrease in aspect ratio Initial Stiffness
The initial stiffness was calculated based on a point obtained from the measured force-displacement envelope with a shear force that is equal to the theoretical yield force This theoretical yield force is defined as either the first yield that occurs within the longitudinal reinforcement or when the maximum compressive strain of the con-crete attains a value of 0.002 at any critical section of the column This definition would not apply for columns whose shear strength does not substantially exceed its theoretical yield force For such columns, defined as those whose maximum measured shear force was less than 1.07 of the theoretical yield force, the initial stiffness was defined based on a point on its measured force-displacement envelope with a shear force that equates to 0.80 of the obtained maximum shear force
The initial stiffness of all the test specimens is tabulated in Table3 The initial stiffness of SC-1.7 Series specimens was en-hanced by around 7% as the column axial load was increased from 0.20 to0.35f0
cAg A similar trend was observed in the specimens of RC-1.7 Series As compared with Specimen RC-1.7-0.20, Speci-men RC-1.7-0.35 experienced a 23% increase in its initial stiffness This clearly indicates that the column axial load was beneficial to the initial stiffness of the test specimens It should be noticed that
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34
Cycle number
DR=1/1000
DR=1/700 DR=1/500
DR=1/400 DR=1/300
DR=1/250 DR=1/200
DR=1/150 DR=1/125
DR=1/100 DR=1/80
DR=1/70 DR=1/65
DR=1/55 DR=1/50
Fig 4 Loading protocol
250 250
365
365
412
490
Fig 5 Typical strain gauge and linear variable displacement
transdu-cer (LVDT) locations (in mm)
Trang 5the reinforcement details and cross sections of the specimens in RC-1.7 Series are different from the ones in SC-1.7 Series This could attribute to the differences in the increase in the initial stiffness of the specimens in these series when the column axial load is increased
The initial stiffness of Specimens SC-2.4-0.20 and SC-1.7-0.20 obtained from the tests were 12.9 kN=mm (73.7 kip:=in:) and 26.9 kN=mm (153.6 kip:=in:), respectively There was an increase
in the initial stiffness of Specimens SC-1.7-0.20 and SC-2.4-0.20 of approximately 108.5% The only difference in the details of these specimens is the height of the specimens Specimen SC-2.4-0.20 is higher than Specimen SC-1.7-0.20; therefore, it is susceptible to deformation than Specimen SC-1.7-0.20
Drift Ratios at Axial Failure The drift ratios at axial failure of the test specimens are tabulated in Table3 An increase in the column axial load ratio reduced the drift ratio at axial failure As observed from Table 3, the drift ratio at axial failure in SC-1.7 and RC-1.7 Series specimens reduced by around 14 and 30%, respectively, as the column axial load ratio was increased from 0.20 to 0.35
The effects of aspect ratio on the drift ratio at axial failure can be noticed by comparing Specimens SC-2.4-0.20 and SC-1.7-0.20 At
a column axial load ratio of 0.20, the drift ratio at axial failure re-duced from 2.82 to 1.82% with a decrease in the aspect ratio from 2.4 to 1.7
Fig 6 Observed cracks at shear failure of test specimens: (a) SC-1.7-0.20; (b) SC-1.7-0.35; (c) RC-1.7-0.20; (d) RC-1.7-0.35; (e) SC-2.4-0.20
Fig 7 Typical modes of axial failure in test specimens: (a) Mode 1;
(b) Mode 2
Trang 6Strains in Reinforcing Bars
Strain profiles from Specimen RC-1.7-0.35 were selected to
illus-trate the distribution of strain in both transverse and longitudinal
reinforcing bars as it is not possible to present the results of all
the specimens in this paper Detailed strain profiles of all test
spec-imens have been reported elsewhere (Tran 2010)
The measured strains along the longitudinal reinforcing bars of Specimen RC-1.7-0.35 are shown in Fig.9 The general strain pro-files of Specimen RC-1.7-0.35 have a good agreement with the bending moment pattern The largest recorded tensile strain of 0.0027 was observed at Location L6 In loading to a drift ratio
of 1.44%, the compressive strain at Location L6 exceeded the com-pressive yield strain of−0.0025 During the test, both compressive
-400
(e)
-300 -200 -100 0 100 200 300 400
-24 -18 -12 -6 0 6 12 18 24
Lateral Displacement (mm)
-89.6 -67.2 -44.8 -22.4 0 22.4 44.8 67.2 89.6 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2
SC-1.7-0.20
Proposed Model
FEMA 356
ASCE 41
-400 -300 -200 -100 0 100 200 300 400
-24 -18 -12 -6 0 6 12 18 24
Lateral Displacement (mm)
-89.6 -67.2 -44.8 -22.4 0 22.4 44.8 67.2 89.6 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2
Drift Ratio (%)
SC-1.7-0.35
Proposed Model
FEMA 356
ASCE 41
-400
-300
-200
-100
0 100 200 300 400
-68 -51 -34 -17 0 17 34 51 68
Lateral Displacement (mm)
-89.6 -67.2 -44.8 -22.4 0 22.4 44.8 67.2 89.6
Drift Ratio (%)
RC-1.7-0.20
Proposed Model
FEMA 356
ASCE 41
-400 -300 -200 -100 0 100 200 300 400
Lateral Displacement (mm)
-89.6 -67.2 -44.8 -22.4 0 22.4 44.8 67.2 89.6
Drift Ratio (%)
RC-1.7-0.35
Proposed Model
FEMA 356
ASCE 41
-300 -200 -100 0 100 200 300
-51 -34 -17 0 17 34 51
Lateral Displacement (mm)
-67.2 -44.8 -22.4 0 22.4 44.8
67.2
Drift Ratio (%)
SC-2.4-0.20
Proposed Model
FEMA 356
ASCE 41
Fig 8 Hysteretic responses of test specimens
Trang 7and tensile yielding were observed in the longitudinal reinforcing
bars
Strains in Transverse Reinforcing Bars
The measured strains in the transverse reinforcing bars of Specimen
RC-1.7-0.35 are illustrated in Fig 10 It was observed that the
measured strains varied considerably as drift ratios increased
The largest strain was recorded at Location T4
The strains in the transverse reinforcing bars have not reached
the yield strain of 0.002 up to a drift ratio of 1.44% The largest
recorded strain up to this stage was only 0.0013 In loading to a
drift ratio of 1.58%, the strains in the transverse reinforcing bars
increased drastically because of the growth and opening of diagonal
shear cracks along the specimen Yielding of the transverse steel bars was also observed at this stage
Displacement Decompositions The contribution of deformation components expressed as percent-ages of the total lateral displacements at the peak displacements during each displacement cycle of Specimen RC-1.7-0.35 is shown
in Fig.11 Detailed displacement decompositions of all test spec-imens had been reported elsewhere (Tran 2010)
Approximately 51.4–58.4% of the total lateral displacement was contributed by the flexural deformation component, whereas only up to 45% was accounted for the shear deforma-tion component The shear deformadeforma-tion component initially grew gradually to approximately 16.3% of the total lateral dis-placement up to a drift ratio of 1.44% As the drift ratio was increased up to 1.73%, the corresponding shear deformation component drastically grew to approximately 45% of the total displacement
Cumulative Energy Dissipation Table3shows the comparison between the maximum cumulative energy dissipation obtained from the test specimens There was a decrease in the maximum cumulative energy dissipation recorded from both SC-1.7 and RC-1.7 Series specimens as the column axial load was increased The maximum cumulative energy dissipations obtained from SC-1.7 Series specimens was reduced by around 33%, as the column axial load was increased from 0.20 to 0.35f0
cAg An analogous trend was observed in the specimens of RC-1.7 Series, whose maximum cumulative energy dissipations experienced a drop of around 40% as the applied axial load was increased from 0.20 to 0.35f0
cAg It can therefore be concluded based on the test results that column axial load decreases the
Table 3 Summary of Test Results
Specimen
Shear strength (kN)
Initial stiffness (kN =mm) shear failure (%)Drift ratio at
Drift ratio at axial failure (%)
Maximum cumulative energy dissipation (kN · m)
-4000
-3000
-2000
-1000
0
1000
2000
3000
4000
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
Drift Ratio (%)
εy
εy
-6 )
250 250
250 250 L6 L5 L4
L3 L2 L1
Fig 9 Local strains in longitudinal reinforcing bar of specimen
RC-1.7-0.35
0
1000
2000
3000
4000
Drift Ratio (%)
T6 T5 T4
T3 T2 T1 38
125
Fig 10 Local strains in transverse reinforcing bars of specimen
RC-1.7-0.35
0 20 40 60 80 100
Drift Ratio (%)
Shear
Unaccounted
Flexure
Fig 11 Displacement decompositions of Specimen RC-1.7-0.35
Trang 8maximum cumulative energy dissipation of the test specimens The
reinforcement details and cross sections of the specimens in RC-1.7
Series are different from the ones in SC-1.7 Series This could
attribute to the differences in the decrease in the maximum
cumu-lative energy dissipations of the specimens in these series when the
column axial load is increased
Backbone Curves
Comparison with FEMA 356 and ASCE 41
Table4summarizes all the indexes (a, b, c) for all the test
spec-imens calculated based on FEMA 356 (FEMA 2000) and ASCE 41
(Elwood et al 2007) Fig.8compares the backbone curves of the
test specimens with analytical results obtained from FEMA 356
(FEMA 2000) and ASCE 41 (Elwood et al 2007) models The test
results showed that both FEMA 356 (FEMA 2000) and ASCE 41
(Elwood et al 2007) guidelines provide a good prediction of the
shear strength of the test specimens However, the column initial
stiffness and ultimate displacements (displacements at axial failure)
were overestimated and underestimated by both FEMA 356
(FEMA 2000) and ASCE 41 (Elwood et al 2007) guidelines,
re-spectively Therefore, further work on the backbone curves is
needed to accurately capture the behavior of RC columns tested
to the point of axial failure
Research on the initial stiffness and the ultimate displacements
of RC columns had been done by Elwood and Moehle (2005),
Elwood and Eberhard (2009), Tran and Li (2012, 2013), and
Wibowo et al (2014) These models will be reviewed in the
fol-lowing part of the paper
Existing Initial Stiffness Models
Model of Elwood and Eberhard
Elwood and Eberhard (2009) recommend the following equation
for estimating the initial stiffness of reinforced concrete columns
subjected to seismic loading
k¼0.45 þ 2.5P=Agfc0
1 þ 110ðd b
hÞðh
a sÞ ≤ 1 and ≥ 0.2 ð2Þ where dbis the diameter of longitudinal reinforcing bars; asis the
shear span; and h is the column depth
Model of Tran and Li
Tran and Li (2012) developed an analytical model for the initial
stiffness of RC columns In the model of Tran and Li (2012),
the yield displacement is the sum of the displacement because
of flexure, bar slip, and shear The flexural deformations are calcu-lated by using moment-curvature analysis The bar slips are ac-counted for in the model using the simplified concept of the effective length of the member by Priestley et al (1996) The shear deformation of RC columns at yield force is derived by applying a method that is similar to the analogous truss model of Park and Paulay (1975) Details of the derivation of the model of Tran and Li (2012) have been reported elsewhere (Tran and Li 2012) Based on the model, Tran and Li (2012) performed a parametric study to investigate the effects of various parameters on the initial stiffness of RC columns The parameters investigated in the study
of Tran and Li (2012) include transverse reinforcement ratios (ρst), longitudinal reinforcement ratios (ρl), yield strength of lon-gitudinal reinforcing bars (fyl), concrete compressive strength (fc0), aspect ratio (as=d), and axial load ratio (P=f0
cAg) Tran and Li (2012) concluded that the stiffness ratio apparently in-creases with an increase in aspect ratios (Ra) and axial load ratio (Rn) The transverse and longitudinal reinforcement ratios, yield strength of longitudinal bars, and concrete compressive strength insignificantly influenced the stiffness ratio of RC columns Based
on the parametric study, Tran and Li (2012) has developed an equa-tion to estimate the stiffness ratio of RC columns as follows:
κ ¼ ð2.043R2
nþ 2.961Rnþ 1.739Þð3.023Raþ 2.573Þ ð3Þ where Ra and Rn are the aspect ratio (as=d) and axial load ratio (P=f0
cAg)
The stiffness ratio (κ) is defined as follows:
κ ¼Ie
where the measured effective moment of inertia can be determined as
Ie¼L3Ki
where Kiis the initial stiffness of columns; Igis the moment of in-ertia of the gross section; L is the height of columns; and Ecis the elastic modulus of concrete
Existing Ultimate Displacement Model
Model of Elwood and Moehle The model of Elwood and Moehle (2005) proposed the following equation for the drift ratio at axial failure based on the shear friction model:
Δ L
a
¼1004 1 þ ðtan θÞ2 tanθ þ Pð s
A st f yt d c tan θÞ ð6Þ where dcis the depth of the core (centerline to centerline of ties);
Astis the total transverse reinforcement area within spacing s;θ is the angle of diagonal crack; and fyt is the yield strength of trans-verse reinforcement
Model of Tran and Li Tran and Li (2013) developed an analytical model for the ultimate displacement of RC columns with light transverse reinforcement based on the energy analogy and the experimental data of 47
RC columns tested to the point of axial failure Details of the derivation of the model of Tran and Li (2013) had been re-ported elsewhere (Tran and Li 2013) In this model, the ultimate
Table 4 Modeling Parameters
Specimen
FEMA 356 ( FEMA 2000 ) ASCE 41 ( Elwood et al 2007 )
RC-1.7-0.35 0.0025 0.0087 0.2 ii 0.0062 0.0104 0.0470
Note: Condition iii = shear failure; Condition ii = shear-flexure failure.
Trang 9displacement of RC columnsΔa can be found by solving the
fol-lowing equations:
Δa¼ 2Δyþ δ
P¼ ρlbh
yl 0.2874 × δ
aþ 1
1 sinθþ
dfytAst s
þ kpffiffiffiffiffifc0
whereρl is the longitudinal reinforcement ratio; fyl is the yield
strength of longitudinal reinforcement, respectively; k is a
param-eter that depends on the displacement ductility demand;Δyis the
yield displacement of columns; and Agis the cross-sectional area
There are two variables, namelyδ
a andΔa, and two indepen-dent equations [Eqs (7) and (8)] By solving these two independent
equations, the ultimate displacement,Δa, can be determined
Model of Wibowo et al
Wibowo et al (2014) used the curve fitting method to derive the
drift ratio at axial failure as follows:
δa¼ 5ð1 þ ρlÞ−ð 1
whereρstis transverse reinforcement area ratio (Ast=bs); β is
cal-culated as n=nb; n is the axial load ratio; and nb is the axial load
ratio at the balance point of the interaction diagram
Comparison with the Existing Initial Stiffness and
Ultimate Displacement Models
As shown in Table5, it was found that the initial stiffness models
developed by Elwood and Eberhard (2009) and Tran and Li (2012)
produced better results than the existing seismic assessment
guide-lines [FEMA 356 (FEMA 2000);Elwood et al 2007] Comparing
the models between Elwood and Eberhard (2009) and Tran and Li
(2012), the model of Tran and Li (2012) produced a better mean
ratio of the experimental to predicted initial stiffness and its coef-ficient of variation than the one of Elwood and Eberhard (2009)
As shown in Table6, the mean ratios of the experimental to the predicted displacement at axial failure and its coefficient of varia-tion are 0.959 and 0.200 for the model of Elwood and Moehle (2005), 1.085 and 0.283 for the model of Tran and Li (2012), and 0.952 and 0.161 for the model of Wibowo et al (2014), respec-tively Comparing the existing models with the experimental data indicates that the existing models produced better results than the existing seismic assessment guidelines [FEMA 356 (FEMA 2000); Elwood et al 2007] Among the existing models, the model of Elwood and Moehle (2005) produced a better mean ratio of the experimental to predicted initial stiffness than other models (Tran and Li 2013;Wibowo et al 2014)
Incorporating Existing Initial Stiffness and Ultimate Displacement Models to FEMA 356 Guidelines
As discussed in the previous part, the existing initial stiffness (Elwood and Eberhard 2009;Tran and Li 2012) and ultimate dis-placement models (Elwood and Moehle 2005;Tran and Li 2013; Wibowo et al 2014) produced better results than the existing seis-mic assessment guidelines [FEMA 356 (FEMA 2000); Elwood
et al 2007] Therefore, in this part of the paper, the backbone curve
of FEMA 356 (FEMA 2000) is modified based on the existing models (Elwood and Eberhard 2009;Tran and Li 2012;Elwood and Moehle 2005; Tran and Li 2013; Wibowo et al 2014) The modified backbone curve of FEMA 356 is shown in Fig.12
In this paper, the models proposed by Tran and Li (2012,2013) are incorporated in the modified backbone curve of FEMA 356 (FEMA 2000) as a sample evaluation of the proposed modified FEMA 356 backbone curve Other models (Elwood and Eberhard
2009;Elwood and Moehle 2005;Wibowo et al 2014) could be
Table 5 Comparison with the Existing Initial Models
Specimen
Ki- exp (kN =mm) (kNKi−Elwood=mm) (kNKi-Tran=mm) KKi-Elwoodi- exp=
K i- exp=
Ki-Tran
Table 6 Comparison with the Existing Ultimate Displacement Models
Specimen ðΔa=LÞexp ðΔa=LÞElwood ðΔa=LÞTran ðΔa=LÞWibowo ðΔa=LÞElwoodðΔa=LÞexp= ðΔa=LÞexp=ðΔa=LÞTran ðΔa=LÞWibowoðΔa=LÞexp=
a
A
E Displacement
Fig 12 Proposed backbone curves
Trang 10used instead of the models of Tran and Li (2012,2013) to verify
the accuracy of the proposed modified backbone curve of
FEMA 356
The Point B in Fig.12is defined based on the values of Kiand
Vp The initial stiffness Ki is calculated based on Eq (3); Vp is
equal to the minimum value of the theoretical yield force Vy
and the nominal shear strength based on FEMA 356 model Vn
[FEMA 356 (FEMA 2000)]
The Point C in Fig.12is defined based on the values of a and
Vm; a is defined similarly to that in the model of FEMA 356
(FEMA 2000); Vmis equal to the minimum value of the theoretical
flexural strength Vu and the nominal shear strength based on
FEMA 356 model Vn [FEMA 356 (FEMA 2000)]
The Point E in Fig 12 is defined based on the values of c
andΔa; a is defined similarly to the model of FEMA 356 (FEMA
2000); the ultimate displacement Δa is calculated based on
Eqs (7) and (8)
Comparison of available models with the test results obtained
from the current experimental investigation as illustrated in Fig.8
indicated that the modified FEMA 356 provided a better prediction
of the behavior of the test specimens than FEMA 356 (FEMA
2000) and ASCE 41 (Elwood et al 2007) models The initial
stiff-ness and the ultimate displacement were captured well by the
modi-fied model The modimodi-fied method may be suitable as an assessment
tool to model the backbone curves of RC columns with light
trans-verse reinforcement
Conclusions
The backbone curves of the reinforced concrete columns with
light transverse reinforcement were investigated using the
exper-imental and analytical studies The conclusions drawn from the
experimental and analytical investigations of the five reinforced
concrete columns with light transverse reinforcement are as
follows:
1 The column axial load was found having a detrimental effect
on the drift ratio at axial failure and maximum energy
dissipa-tion capacity of test specimens However, the shear strength
and initial stiffness increased with an increase in column axial
load
2 Both FEMA 356 (FEMA 2000) and ASCE 41 (Elwood et al
2007) guidelines provided a good prediction of the shear
strength of the test specimens However, the column initial
stiffness and ultimate displacements were overestimated and
underestimated by both FEMA 356 (FEMA 2000) and ASCE
41 (Elwood et al 2007) guidelines, respectively
3 An analytical method is developed in this paper to model the
backbone curves of test specimens Comparison of available
models with the test results obtained from the current
experi-mental investigation indicated that the proposed method
pro-vided a better prediction of the behavior of the test specimens
than FEMA 356 (FEMA 2000) and ASCE 41 (Elwood et al
2007) models The initial stiffness and the ultimate
displace-ment were captured well by the proposed method The
pro-posed method may be suitable as an assessment tool to
model the backbone curves of RC columns with light
trans-verse reinforcement
Acknowledgments
This research is funded by Vietnam National Foundation for
Science and Technology Development (NAFOSTED) under grant
number 107.01-2013.12
Notation
The following symbols are used in this paper:
Ag= cross-sectional area;
Ast= total transverse reinforcement area within spacing s;
as=d = aspect ratio;
b= width of columns;
d= distance from the extreme compression fiber to centroid
of tension reinforcement;
fc0 = compressive strength of concrete;
fyl = yield strength of longitudinal reinforcement;
fyt = yield strength of transverse reinforcement;
h= depth of columns;
k2 = parameter depends on the displacement ductility demand;
P= applied axial load;
s= spacing of transverse reinforcement;
Vc = shear force carried by concrete;
Vn = nominal shear strength of columns;
Vu = theoretical flexural strength of columns;
Vy = theoretical yield force of columns;
θ = angle of shear crack; and
Δa = horizontal displacement of columns at the point of axial failure
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