1. INTRODUCTION In recent years, earthquake design philosophy has shifted from a traditional force-based approach toward a displacement-based ideology. The assumed initial stiffness of reinforced concrete (RC) columns could affect the estimation of the displacement and displacement ductility, which are crucial in displacement-based design. In addition, the assumed initial stiffness properties of columns also affect the estimation of the fundamental period and distribution of internal forces of structures. Therefore, an accurate evaluation of the initial stiffness of columns becomes an inevitable requirement. Literature reviews show that there is a considerable amount of uncertainty regarding the estimation of the initial stiffness of columns when subjected to seismic loads. Current design codes often employ a stiffness reduction factor to deal with this uncertainty. In an attempt to address these uncertainties, the study presented within this paper is devoted to developing a rational method to determine the initial stiffness of RC columns when subjected to seismic loads. A comprehensive parametric study based on the proposed Advances in Structural Engineering Vol. 15 No. 2 2012 265 Initial Stiffness of Reinforced Concrete Columns with Moderate Aspect Ratios Cao Thanh Ngoc Tran 1 and Bing Li 2, * 1 Department of Civil Engineering, International University, Vietnam National University, Ho Chi Minh City, Vietnam 2 School of Civil and Environment Engineering, Nanyang Technological University, Singapore 639798 (Received: 1 December 2010; Received revised form: 17 May 2011; Accepted: 4 June 2011) Abstract: The estimation of the initial stiffness of columns subjected to seismic loadings has long been a matter of considerable uncertainty. This paper reports a study that is devoted to addressing this uncertainty by developing a rational method to determine the initial stiffness of RC columns when subjected to seismic loads. A comprehensive parametric study based on a proposed method is initially carried out to investigate the influences of several critical parameters. A simple equation is then proposed to estimate the initial stiffness of RC columns. The applicability and accuracy of the proposed method and equation are then verified with the experimental data obtained from literature studies. Key words: reinforced concrete, column initial stiffness, stiffness ratio. method was carried out to investigate the influences of several critical parameters. A simple equation to estimate the initial stiffness of RC columns is also proposed within this paper. The applicability and accuracy of the proposed method and equation are then verified with the experimental data obtained from the literature. 2. DEFINING INITIAL STIFFNESS OF RC COLUMNS There are two methods as illustrated in Figure 1(a) that are commonly utilized to determine the initial stiffness of RC columns (K i ). In the first method, the initial stiffness of RC columns are estimated by using the secant of the shear force versus lateral displacement relationship passing through the point at which the applied force reaches 75% of the flexural strength (0.75 V u ). In the second method, the column is loaded until either the first yield occurs in the longitudinal reinforcement or the maximum compressive strain of concrete reaches 0.002 at a critical section of the column. This corresponds to point A in Figure 1(a). Generally, the two approaches give similar values. In this study, the later approach was adopted. *Corresponding author. Email address: [email protected]; Tel: +65-6790-5292. Associate Editor: J.G. Dai.
Transcript
1. INTRODUCTIONIn recent years, earthquake design philosophy has shiftedfrom a traditional force-based approach toward adisplacement-based ideology. The assumed initialstiffness of reinforced concrete (RC) columns could affectthe estimation of the displacement and displacementductility, which are crucial in displacement-based design.In addition, the assumed initial stiffness properties ofcolumns also affect the estimation of the fundamentalperiod and distribution of internal forces of structures.Therefore, an accurate evaluation of the initial stiffness ofcolumns becomes an inevitable requirement.
Literature reviews show that there is a considerableamount of uncertainty regarding the estimation of theinitial stiffness of columns when subjected to seismicloads. Current design codes often employ a stiffnessreduction factor to deal with this uncertainty. In anattempt to address these uncertainties, the studypresented within this paper is devoted to developing arational method to determine the initial stiffness of RCcolumns when subjected to seismic loads. Acomprehensive parametric study based on the proposed
Advances in Structural Engineering Vol. 15 No. 2 2012 265
Initial Stiffness of Reinforced Concrete Columns with
Moderate Aspect Ratios
Cao Thanh Ngoc Tran1 and Bing Li2,*1Department of Civil Engineering, International University, Vietnam National University, Ho Chi Minh City, Vietnam
2School of Civil and Environment Engineering, Nanyang Technological University, Singapore 639798
(Received: 1 December 2010; Received revised form: 17 May 2011; Accepted: 4 June 2011)
Abstract: The estimation of the initial stiffness of columns subjected to seismicloadings has long been a matter of considerable uncertainty. This paper reports a studythat is devoted to addressing this uncertainty by developing a rational method todetermine the initial stiffness of RC columns when subjected to seismic loads. Acomprehensive parametric study based on a proposed method is initially carried out toinvestigate the influences of several critical parameters. A simple equation is thenproposed to estimate the initial stiffness of RC columns. The applicability andaccuracy of the proposed method and equation are then verified with the experimentaldata obtained from literature studies.
method was carried out to investigate the influences ofseveral critical parameters. A simple equation to estimatethe initial stiffness of RC columns is also proposedwithin this paper. The applicability and accuracy of theproposed method and equation are then verified with theexperimental data obtained from the literature.
2. DEFINING INITIAL STIFFNESS OF RCCOLUMNS
There are two methods as illustrated in Figure 1(a) that arecommonly utilized to determine the initial stiffness of RCcolumns (Ki). In the first method, the initial stiffness of RCcolumns are estimated by using the secant of the shearforce versus lateral displacement relationship passingthrough the point at which the applied force reaches 75%of the flexural strength (0.75 Vu). In the second method,the column is loaded until either the first yield occurs inthe longitudinal reinforcement or the maximumcompressive strain of concrete reaches 0.002 at a criticalsection of the column. This corresponds to point A inFigure 1(a). Generally, the two approaches give similarvalues. In this study, the later approach was adopted.
However, the above mentioned definition cannot beused for columns whose shear strengths do notsubstantially exceed its theoretical yield force. For thesecolumns, defined as those whose maximum measuredshear force was less than 107% of the theoretical yieldforce, the effective stiffness was defined based on apoint on the measured force-displacement envelopewith a shear force equal to 0.8 Vmax as illustrated inFigure 1(b) (Elwood et al. 2009).
Assuming the column is fixed against rotation at bothends and has a linear variation in curvature over theheight of the column, the measured effective moment ofinertia can be determined as:
(1)IL K
Eei
c=
3
12
The stiffness ratio (κ) is defined as follows:
(2)
where Ig is the moment of inertia of the gross section; Ki
is the initial stiffness of columns and L is the height ofcolumns and Ec is the elastic modulus of concrete.
3. REVIEW OF EXISTING INITIALSTIFFNESS MODELS
3.1. ACI 318-08 (2008)
ACI 318-08 (2008) recommends the following optionsfor estimating member stiffness for the determination oflateral deflection of building systems subjected tofactored lateral loads: (a) 0.35 EIg for members with anaxial load ratio of less than 0.10 and 0.70 EIg formembers with an axial load ratio of more than or equalto 0.10; or (b) 0.50 EIg for all members.
3.2. FEMA 356 (2000)
FEMA 356 (2000) suggests the variation of effectivestiffness values with the applied axial load ratio. Theeffective stiffness is taken as 0.50 EIg for members with anaxial load ratio of less than 0.30, while a value of 0.7 EIg
is adopted for members with an axial load ratio of morethan 0.50. This value varies linearly for intermediate axialload ratios as illustrated in Figure 2.
3.3. ASCE 41 (2007)
As shown in Figure 2, ASCE 41 (2007) recommends thatthe effective stiffness is taken as 0.30 EIg for members
κ = ×I
Ie
g100%
266 Advances in Structural Engineering Vol. 15 No. 2 2012
Initial Stiffness of Reinforced Concrete Columns with Moderate Aspect Ratios
(a)
A' A
Initial stiffness
She
ar fo
rce
0.75Vu
Vu
Vy
Lateral displacement
(b) (Elwood et al. 2009)
Initial stiffness
A
Vu
She
ar fo
rce
0.80Vmax
Vy
Lateral displacement
Figure 1. Methods to determine initial stiffness
0− 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.2
0.4
0.6
0.8
1
Axial load ratio f 'c Ag
Stif
fnes
s ra
tio k
(%
)
ACI 318-0.8 (a)ACI 318-0.8 (b)
FEMA 356ASCE 41
PP92EE09
Figure 2. Relationships between stiffness ratio and axial load ratio
of existing models
with an axial load ratio of less than 0.10, as 0.7 EIg formembers with an axial load ratio of more than 0.50 andvaries linearly for intermediate axial load ratios.
3.4. Paulay and Priestley (1992)
According to Paulay and Priestley’s recommendation(1992), the effective stiffness is taken as 0.40 EIg formembers with an axial load ratio of less than −0.05, as0.8 EIg for members with an axial load ratio of morethan 0.50 and varies linearly for intermediate axial loadratios as illustrated in Figure 2.
3.5. Elwood and Eberhard (2009)
Elwood and Eberhard (2009) recommend the followingequation for estimating the initial stiffness of reinforcedconcrete columns subjected to seismic loading:
(3)
where db is the diameter of longitudinal reinforcingbars; a is the shear span and h is the column depth; Ag isthe gross sectional area of columns and f′c is thecompressive strength of concrete.
Figure 2 illustrates the variation of stiffness ratiobased on Elwood and Eberhard’s model (2009) versus
kP A f
d
hha
g c
b=
+ ′
+
≤ ≥0 45 2 5
1 1101 0
. . /.and 22
the axial load ratio for specimens with db and a equal to25 mm and 850 mm respectively.
4. EXPERIMENTAL INVESTIGATION ONINITIAL STIFFNESS OF RC COLUMNS
In this section, the experimental results obtained fromtesting of six RC columns conducted by Tran et al. (2009)are briefly discussed with respect to the initial stiffness ofthe test specimens. Four column axial loads of 0.05, 0.20,0.35, 0.50 f′c Ag and two aspect ratios of 1.71 and 2.43were investigated in this experimental program. Table 1summarizes all the details of the test specimens. It is to benoted that only a brief summary of important test featuresthat are relevant to this study are presented within thispaper. Detailed information has been documented inanother publication (Tran et al. 2009).
The relationships between initial stiffness and thecolumn axial load ratio obtained from all the testspecimens are tabulated in Table 2. The initial stiffness ofSC-1.7 Series specimens enhanced by around 9.8%,17.6%, and 40.4% as the column axial load was increasedfrom 0.05 to 0.20, 0.35, and 0.50 f′cAg, respectively. Ananalogous trend was observed in the specimens of RC-1.7Series, whose initial stiffness experienced anenhancement of around 33.9%, 64.3% and 86.1% with anincrease in the column axial load from 0.05 to 0.20, 0.35and 0.50 f′cAg, respectively. As compared to SpecimenSC-2.4-0.20, Specimen SC-2.4-0.50 experienced an
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Cao Thanh Ngoc Tran and Bing Li
P
f Ac g'
Table 1. Summary of test specimens (Tran et al. 2009)
Mean 0.735 0.242 0.301 0.262 0.312 0.232 0.588Coefficient of Variation 0.141 0.060 0.076 0.054 0.084 0.046 0.104
K
Ki
i EE
−−
−−
expK
Ki
i PP
−−
−−
expK
Ki
i ASCE
−−
−−
expK
Ki
i FEMA
−−exp
−
K
Ki
i ACI b
−−
−−
exp
( )
K
Ki
i ACI a
−−
−−
exp
( )
K
Ki
p
−−
−−
exp
iKi−−exp
increase in the initial stiffness of 20.2%. Theaforementioned discussion clearly indicated that columnaxial load was beneficial to the initial stiffness of testspecimens.
The initial stiffness of Specimens SC-2.4-0.20, SC-1.7-0.20, SC-2.4-0.50 and SC-1.7-0.50 obtained from the testswere 12.9 kN/mm, 26.9 kN/mm, 15.5 kN/mm and 34.4kN/mm respectively. The increase in the initial stiffnesswhen comparing between Specimens SC-1.7-0.20 andSC-2.4-0.20 was 108.5%. Similarly, an enhancement inthe initial stiffenss of 121.9% was observed in SpecimenSC-1.7-0.50 as compared to Specimen SC-2.4-0.50.
The initial stiffness of test columns calculated basedon ACI 318-2008 (2008), FEMA 356 (2000), ASCE 41(2007), Paulay and Priestley (1992), and Elwood andEberhard (2009) are also all tabulated in Table 2. Allthese models tend to overestimate the initial stiffness ofthe test columns. Amongst all of these existing models,Elwood and Eberhard (2009) provides the best meanratio of the experimental to predicted initial stiffness.However none of these models are accurate.
5. PROPOSED METHOD 5.1. Yield Force (Vy)
The initial stiffness of columns is determined byapplying the second method as described in the previoussection. The yield force (Vy) corresponding to point A inFigure 1(a) is obtained from the yield moment (My)when the reinforcing bar closest to the tension edge ofcolumns has reached its yield strain. Moment-curvatureanalysis is adopted to determine this moment.
5.2. Displacement at Yield Force (∆′y)
The displacement of a column at yield force (Vy) can beconsidered as the sum of the displacement due toflexure, bar slip and shear.
(4)
where ∆′y is the displacement of a column at yield force;∆′flex is the displacement due to flexure and bar slip atyield force; and ∆′shear is the displacement due to shearat yield force
5.2.1. Flexure deformations (∆′flex)
In this proposed method, the simplified concept of aneffective length of the member suggested by Priestley etal. (1996) was used to account for the displacement dueto bar slip in flexure deformations. Assuming a linearvariation in curvature over the height of the column, thecontribution of flexural deformations and bar slips to thedisplacement at the yield force for RC columns with afixed condition at both ends can be estimated as follows:
′ = ′ + ′∆ ∆ ∆y flex shear
(5)
where φ′y is the curvature at the yield force determinedby using moment-curvature analysis and L is the clearheight of columns.
The strain penetration length (Lsp) is given by:
(6)
where fyl is the yield strength of longitudinal reinforcingbars; and db is the diameter of longitudinal reinforcingbars.
5.2.2. Shear deformations (∆′shear)
The idea of utilizing the truss analogy to model crackedRC elements has been around for many years. The trussanalogy is a discrete modeling of actual stress fieldswithin RC members. The complex stress fields withinstructural components resulting from applied externalforces are simplified into discrete compressive andtensile load paths. The analogy utilizes the general ideaof concrete in compression and steel reinforcement intension. The longitudinal reinforcement in a beam orcolumn represents the tensile chord of a truss while theconcrete in the flexural compression zone is consideredas part of the longitudinal compressive chord. Thetransverse reinforcement serves as ties holding the longitudinal chords together. The diagonal concretecompression struts, which discretely simulate theconcrete compressive stress field, are connected to the ties and longitudinal chords at rigid nodes to attainstatic equilibrium within the truss. The truss analogy isa very promising way to treat shear because it provides avisible representation of how forces are transferred in aRC members under an applied shear force.
Park and Paulay (1975) derived a method todetermine the shear stiffness by applying the trussanalogy for short or deep rectangular beams of unitlength. The shear stiffness is the magnitude of the shearforce, when applied to a beam of unit length that willcause unit shear displacement at one end of the beamrelative to the other. This model is reliable in estimatingshear deformations of short or deep beams in which theinfluences of flexure are negligible. The behaviors ofRC columns under seismic loading are much morecomplex because of the interaction between shear andflexure. The influences of axial strain due to flexure inestimating shear deformations of RC columns should beconsidered to accurately predict the initial stiffness ofRC columns. By applying a method that is similar toPark and Paulay’s analogous truss model (1975), the
L f dsp yl b= 0 022.
′ =′ +( )
∆ flexy spL Lφ 2
6
2
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Initial Stiffness of Reinforced Concrete Columns with Moderate Aspect Ratios
shear stiffness of RC columns is derived in this part ofthe paper. The effects of flexure in shear deformationsare incorporated in the proposed model through the axialstrains at the center of columns (εy,CL).
Assuming that transverse reinforcing bars startresisting the applied shear force when the shear crackingstarts occurring, the stress in transverse reinforcing barsat the yield force is calculated as:
(7)
where d is the distance from the extreme compressionfiber to centroid of tension reinforcement; s is thespacing of transverse reinforcement; Ast is the totaltransverse steel area within spacing s; and θ is the angleof diagonal compression strut. Hence the strain intransverse reinforcing bars is:
(8)
where εyt is the yield strain of transverse reinforcingbars; Es is the elastic modulus of steel.
Similar to Park and Paulay’s model (1975), theconcrete compression stress at the yield force is given as:
(9)
where b is the width of columns; Lcs = d sinθ is theeffective depth of the diagonal strut as shown in Figure 3.
Hence the strain in the concrete compression strut isgiven as:
fV
bLy
cs2 =
cosθ
ε εxsy
syt
f
E= ≤
fV V s
A dsyy cr
st=
−( )tanθ
(10)
where Ec is the elastic modulus of concrete given as:
(11)
Based on Vecchio and Collins’s model (1986), theeffective compressive strength of concrete is calculatedas follows:
(12)
By applying Mohr’s circle transformation for themean strains at the center of Section C-C as shown inFigure 4, it gives:
(13)
(14)
(15)
For the axial mean strains, compatibility requires thatthe plain sections remain plane. Hence the mean strainat the center of section C-C is given as:
(16)
where εy, top, εy, bot are the axial strains at the extremetension and compression fibers, respectively as shownin Figure 4(b).
There are six variables, namely εx , εy,CL, γxy, ε1, ε2
and θ; and six independent Eqns 8, 10, 13, 14, 15 and16. By solving these six independent equations, theshear strain (γxy) at the center of section C-C could bedetermined.
The column is divided into several segments along itsheight of the column to determine the total sheardeformation at the top of the column. The mean axialstrain at the center of the section is determined based on
εε ε
y CLy top y bot
,, ,=
+
2
tan,
2θγ
ε ε=
−xy
x y CL
εε ε ε ε γ
2
2 2
2 2 2=
+−
−
+
x y CL x y CL xy, ,
εε ε ε ε γ
1
2 2
2 2 2=
++
−
+
x y CL x y CL xy, ,
ff
fcec
c=+
≤'
'
.0 8 170 1ε
E fc c= 5000
ε22=
f
Ec
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Cao Thanh Ngoc Tran and Bing Li
Diagonal strut
LCS
d
θθ
Figure 3. Diagonal strut of RC columns (Park and Paulay1975)
the moment-curvature analysis. The shear strains at thelower and upper section of the segment are calculatedusing the above equations. Hence, the total sheardisplacement caused by the yield force can be calculatedas follows:
(17)
where γ ixy and γ xy
i+1 are the shear strains at the lower andupper section of the segment i; hi is the height ofsegment i and n is the number of segments.
5.3. Initial Stiffness
Once the flexural and shear deformations at the top ofcolumns under yield force are obtained, the initialstiffness of columns can be determined as:
(18)
6. VALIDATION OF THE PROPOSEDMETHOD
The proposed method is validated by comparing itsresults to the initial stiffness of six columns obtainedfrom the experimental study previously conducted byTran et al. (2009).
It was found that the average ratio of experimentalto predicted initial stiffness by the proposed methodwas 0.735 as tabulated in Table 2. It shows a relativelygood correlation between the analytical and
KV
iy
flex shear
=′ + ′∆ ∆
′ =+
+
=∑∆shear
xyi
xyi
ii
n
hγ γ 1
1 2
experimental results. The initial stiffness of the testedcolumns calculated based on ACI 318-2008 (2008),FEMA 356 (2000), ASCE 41 (2007), Paulay andPriestley (1992), and Elwood and Eberhard (2009) arealso tabulated in Table 2. The mean ratio of theexperimental to predicted initial stiffness and itscoefficient of variation were 0.242 and 0.060, 0.301and 0.076, 0.262 and 0.054, 0.312 and 0.084, 0.232and 0.046, and 0.588 and 0.104 for ACI 318-2008(2008a), ACI 318-2008 (2008b), FEMA 356 (2000),ASCE 41 (2007), Paulay and Priestley (1992), andElwood and Eberhard (2009) respectively. Comparisonof available models with experimental data indicatedthat the proposed method produced a better mean ratioof the experimental to predicted initial stiffness thanother models. The proposed method may be suitable asan assessment tool to calculate the initial stiffness ofRC columns.
7. PARAMETRIC STUDIESA parametric study conducted to improve theunderstanding of the effects of various parameters onthe initial stiffness of RC columns is presented withinthis section. The parameters investigated are transversereinforcement ratios (ρv), longitudinal reinforcementratios (ρl), yield strength of longitudinal reinforcingbars (fyl), concrete compressive strength (f′c), aspectratio (a/d) and axial load ratio (P/f′c Ag). In theparametric study, the effects of the parameters thatwere investigated on the initial stiffness of RC columnsare presented by the dimensionless stiffness ratio (k).
Specimen SC-2.4-0.20 with an aspect ratio of 2.4 isconsidered as the reference specimen in the parametric
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Initial Stiffness of Reinforced Concrete Columns with Moderate Aspect Ratios
V
C
(a)
Transversereinforcement
Tension chord
Diagonal strut
Compression chord
C
θ
C
x
y
ε
C
z
(b)
θ
(c)
y,CL
εy,CL
εy,CL
γxy
γxy
εx
ε1
ε1
ε2
ε2
εx
CL
Figure 4. Influences of flexure in estimating shear deformations
study. An axial load of 0.2 was applied to thespecimen. The concrete compressive strength of thespecimen (f′c ) at 28 days was 25.0 MPa. Thelongitudinal reinforcement consisted of 8-T20 (20 mmdiameter). This resulted in the ratio of longitudinalsteel area to the gross area of column to be 2.05%. Thetransverse reinforcement consisted of R6 bars (6 mmdiameter) with 135° bent spaced at 125 mm,corresponding to a transverse reinforcement ratio of0.129%.
7.1. Influence of Transverse Reinforcement
Ratio
The analyses as illustrated in Figure 5 were conducted toassess the influence of transverse reinforcement oneffective moment of inertia. Two column axial loads of0.05 f′cAg and 0.20 f′cAg were considered. Five types oftransverse reinforcement, R6-125 mm, R8-125 mm, R8-100 mm, R10-125 mm and R10-100 which correspondto five transverse reinforcement ratios ρv of 0.129%,0.230%, 0.287%, 0.359% and 0.449% respectively,were investigated.
Figure 5 shows that with an increase in transversereinforcement content from 0.129% to 0.230%, 0.287%,0.359% and 0.449%, stiffness ratios rose slightly byapproximately 3.4%, 4.5%, 5.5%, 6.4%, respectively forcolumns under an axial load of 0.20 f′cAg. The stiffnessratios increased by approximately 2.3%, 3.6%, 4.9%,6.1% for columns under an axial load of 0.05 f′cAg withan increase in transverse reinforcement content from0.129% to 0.230%, 0.287%, 0.359% and 0.449%,respectively. This suggested that the effect of transversereinforcement ratios on stiffness ratios is insignificant.In addition, Figure 5 shows a clear indication that
stiffness ratio increases with an increase in column axialload.
7.2. Influence of Longitudinal Reinforcement
Ratio
The influence of longitudinal reinforcement ratios onstiffness ratios is presented in Figure 6 for two differentcolumn axial loads of 0.05 f′cAg and 0.20 f′cAg. Four typesof longitudinal reinforcement, 8T16, 8T20, 8T22 and8T25 corresponding to longitudinal reinforcement ratiosρl of 1.66%, 2.05%, 2.48% and 3.21% respectively,were considered.
As shown in Figure 6, the stiffness ratios forcolumns under an axial load of 0.05 f′c Ag wereobserved to rise slightly with an increase inlongitudinal reinforcement ratio; while for columnsunder an axial load of 0.20 f′cAg the stiffness ratiosalmost remained the same. This suggested that forsimplicity the influence of longitudinal reinforcementratio on the initial stiffness of RC columns could beignored.
7.3. Influence of Yield Strength of Longitudinal
Reinforcing Bars
Four yield strengths of longitudinal reinforcing bars,362 MPa, 412 MPa, 462 MPa and 512 MPa were chosento investigate the influences of this variable on stiffnessratios. As shown in Figure 7, with a decrease in yieldstrength of longitudinal reinforcing bars from 512 MPato 462 MPa, 412 MPa and 362 MPa; the stiffness ratiosincreased slightly by approximately 3.1%, 4.3%, and5.0%, respectively for columns under an axial load of0.05 f′cAg; whereas stiffness ratios almost remains thesame for column under an axial load of 0.20 f′cAg. The
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Cao Thanh Ngoc Tran and Bing Li
25
20
15
10
5
00.1 0.2 0.3 0.4 0.5
Stif
fnes
s ra
tio k
(%
)
0.20 f 'c Ag0.05 f 'c Ag
Transverse reinforcement ratio v (%)ρ
Figure 5. Influences of transverse reinforcement ratios
on stiffness ratio
25
20
15
10
5
01.5 2 2.5 3 3.5
Stif
fnes
s ra
tio k
(%
)
Longitudinal reinforcement ratio l (%)ρ
0.20 f ′c Ag0.05 f ′c Ag
Figure 6. Influences of longitudinal reinforcement ratio
on stiffness ratio
analytical results suggested that the influences of yieldstrength of longitudinal reinforcing bars on stiffnessratios are negligible.
7.4. Influence of Concrete Compressive
Strength
Figure 8 illustrates the influence of concrete compressivestrength on stiffness ratios for two different axial loads of0.05 f′cAg and 0.20 f′cAg. The concrete compressivestrengths investigated were 25 MPa, 35 MPa, 45 MPa,and 55 MPa. For both axial loads, with an increase inconcrete compressive strength, no significant changes onstiffness ratios were observed.
7.5. Influence of Aspect Ratio
Figure 9 and Table 3 show the influence of aspect ratioon stiffness ratios of RC columns. Six aspect ratios of
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Table 3. Stiffness ratio for various aspect ratios and axial load ratios
8. PROPOSED EQUATION FOR EFFECTIVEMOMENT OF INERTIA OF RC COLUMNS
It is observed that the stiffness ratio apparentlyincreased with an increase in aspect ratios (Ra) andaxial load ratio (Rn). The transverse and longitudinalreinforcement ratios, yield strength of longitudinalbars and concrete compressive strength insignificantlyinfluenced the stiffness ratio of RC columns. Forsimplicity, the influences of these factors wereignored. Based on the results of the parametric study,the stiffness ratio (κ) is given by the followingequation:
(19)
Berry et al. (2004) collected a database of 400 tests ofRC columns, which contained the hysteretic response,geometry, column axial load and material properties oftest specimens. This database provided the data neededto evaluate the accuracy of the proposed equation for thestiffness ratio. The verification was limited to the rangeof the parametric study. The axial load was limited from0 to 0.60 f′cAg, and the aspect ratio was limited from 1.5to 3.0. Only rectangular columns tested in the double-curvature configuration under unidirectional quasi-staticcyclic lateral loading were chosen. Details of the chosenRC columns are tabulated in Table 4.
It was found that the average ratio of theexperimental to predicted stiffness ratio by the proposedequation is 0.945 as shown in Figure 11 and Table 4,showing a good correlation between the proposedequation and experimental data. Therefore, the proposedequation may be suitable as an assessment tool tocalculate the stiffness ratio of RC columns within therange of the parametric study.
The stiffness ratio of columns calculated based on ACI318-2008 (2008), FEMA 356 (2000), ASCE 41 (2007),Paulay and Priestley (1992), and Elwood and Eberhard(2009) are also shown in Table 4. The mean ratio of theexperimental to predicted stiffness ratio and its coefficientof variation were 0.406 and 0.136, 0.409 and 0.095, 0.399and 0.097, 0.571 and 0.151, 0.380 and 0.096, and 0.855and 0.202 for ACI 318-2008 (2008a), ACI 318-2008(2008b), FEMA 356 (2000), ASCE 41 (2007), Paulay andPriestley (1992), and Elwood and Eberhard (2009)respectively. Comparison of available models withexperimental data indicated that the proposed equationproduced a better mean ratio of the experimental topredicted stiffness ratio than other models. It is to be notedthat the proposed equation gives slightly conservativeestimation of stiffness ratio in some cases and acceptablesmall underestimation in other cases.
κ = + +( ) +( )2 043 2 961 1 739 3 023 2 5732. . . . .R R Rn n a
1.50, 1.80, 2.10, 2.43, 2.70, and 3.00 were investigated.In general, the stiffness ratio increased with an increasein aspect ratio.
Figure 9 shows that with an increase in aspect ratiofrom 1.50 to 1.80, 2.10, 2.43, 2.70, and 3.00; thestiffness ratios of columns without axial loads rose byapproximately 18.5%, 39.8%, 62.8%, 83.6%, 109.4%,respectively. Similar trends were observed for thecolumns with an axial load ratio of 0.20. The stiffnessratios increased by approximately 15.6%, 27.4%,37.8%, 45.2% and 52.3% for columns under an axialload of 0.60 f′cAg with an increase in aspect ratio from1.50 to 1.80, 2.10, 2.43, 2.70, and 3.00, respectively.This suggested that the aspect ratio significantlyinfluences the stiffness ratio.
7.6. Influence of Axial Load
It is generally recognized that the presence of columnaxial load can effectively increase the flexural strengthof columns and thus lead to larger initial flexuralstiffness, which results in a higher stiffness ratio. Theanalyses as illustrated in Figure 10 and tabulated inTable 3 were carried out to assess the influence of axialload ratio on stiffness ratio The axial load ratio wasvaried from 0 to 0.60.
In general, the stiffness ratio increased with anincrease in axial load ratio. Figure 10 showed thatwith an increase in axial load ratio from 0 to 0.20,0.40, and 0.60; the stiffness ratios for specimens withan aspect ratio of 1.5 rose by approximately 35.2%,98.7% and 167.9%, respectively. Similar trends wereobserved for other aspect ratios. It can thus beconcluded that the axial load ratio significantly affectsthe stiffness ratio.
Advances in Structural Engineering Vol. 15 No. 2 2012 273
Cao Thanh Ngoc Tran and Bing Li
0.00 f ′ c Ag 0.05 f ′ c Ag 0.10 f ′ c Ag0.15 f ′ c Ag 0.20 f ′ c Ag 0.25 f ′ c Ag0.30 f ′ c Ag 0.35 f ′ c Ag 0.40 f ′ c Ag0.45 f ′ c Ag 0.50 f ′ c Ag 0.55 f ′ c Ag0.60 f ′ c Ag
01.5 1.8 2.1 2.4 2.7 3
5
10
15
20
25
30
35
40
45
50
Aspect ratio a/h
Stif
fnes
s ra
tio k
(%
)
Figure 10. Influences of axial load ratio on stiffness ratio
274 Advances in Structural Engineering Vol. 15 No. 2 2012
Initial Stiffness of Reinforced Concrete Columns with Moderate Aspect Ratios
Ta
ble
4.
Ex
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Esa
ki e
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.(19
85)
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HT
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/52.
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/32.
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n0.
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Coe
ffic
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of
Var
iatio
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202
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095
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70.
151
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60.
202
κκ κκix
p
EE
κκ κκix
p
PP
κκ κκix
p
ASC
E
κκ
κκix
p
FE
MA
κκ
κκix
p
AC
Ib(
)
κκ
κκix
p
AC
Ia(
)
κκ κκixp p
κκex
pκκ
pR
nR
a
9. CONCLUSIONS This paper presents an analytical method to estimate theinitial stiffness of RC columns. A comprehensiveparametric study is carried out based on the proposedmethod to investigate the influences of several criticalparameters. A simple equation to estimate the initialstiffness of RC columns is also proposed. The followingprovides specific findings of the paper:
Comparisons made between the analytical resultsand the experimental results of the six specimens testedin Tran et al.’s study (2009) show relatively goodagreement. This shows the applicability and accuracyof the proposed method to estimate initial stiffness ofRC columns.
The parametric study based on the proposedmethod shows that the stiffness ratio (κ) increasesalong with aspect ratios (Ra) and axial load ratio (Rn).The transverse and longitudinal reinforcement ratios,yield strength of longitudinal bars and concretecompressive strength showed a negligible impact onthe stiffness ratio.
It was found that by the proposed equation, theaverage ratio of the experimental to predicted stiffnessratio is 0.945, showing a good correlation between theproposed equation and the experimental data. Theproposed equation may be suitable as an assessmenttool to calculate the stiffness ratio of RC columnswithin the range of the parametric study, where theaxial load was limited from 0 to 0.60 f′cAg, and theaspect ratio limited from 1.5 to 3.0. Only rectangularcolumns tested in the double-curvature configurationunder unidirectional quasi-static cyclic lateral loadingwere chosen.
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ASCE 41 (2007). Seismic Rehabilitation of Existing Buildings,
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Arakawa, T., Arai, Y., Mizoguchi, M. and Yoshida, M. (1989).
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Bett, B.J., Klingner, R.E. and Jirsa J.O. (1985). Behavior of
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