Engineering Structures 46 (2013) 526–534

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Engineering Structures

journal homepage: www.elsevier .com/locate /engstruct

Analytical modeling and axial load design of a novel FRP-encasedsteel–concrete composite column for various slenderness ratios

0141-0296/$ - see front matter � 2012 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.engstruct.2012.08.016

⇑ Corresponding author. Tel.: +1 905 525 9140; fax: +1 905 529 9688.E-mail addresses: karimk@mcmaster.ca (K. Karimi), taitm@mcmaster.ca

(M.J. Tait), eldak@mcmaster.ca (W.W. El-Dakhakhni).

Kian Karimi ⇑, Michael J. Tait, Wael W. El-DakhakhniDept. of Civil Engineering, McMaster University, Hamilton, Ontario, Canada L8S 4L7

a r t i c l e i n f o

Article history:Received 9 January 2011Revised 18 August 2012Accepted 19 August 2012Available online 27 September 2012

Keywords:Analytical techniquesBucklingComposite columnsConfinementDesignFiber reinforced polymer (FRP)GuidelinesSlenderness ratioTubes

a b s t r a c t

A novel composite column composed of steel, concrete and a fiber reinforced polymer (FRP) tube is pre-sented in this paper. The confinement and composite action between the constituent materials result inenhanced compressive strength, ductility and energy dissipation capacity of the proposed compositecolumn compared to a traditional reinforced concrete (RC) column. Due to the presence of the FRP tube,current design methods for concrete-filled steel tubes (CFSTs) or concrete-encased steel (CES) columnsare not directly applicable. An analytical model was developed to predict the behavior of the compositecolumn for various slenderness ratio values. Predicted values are found to be in good agreement with theexperimental results from tests of six columns ranging from 500 mm to 3000 mm in height. A parametricstudy is conducted to investigate the influence of column diameter, FRP tube thickness, axial compressivemodulus of the FRP tube and steel-to-concrete area ratio on the capacity relationships and slendernesslimits. Finally, a simplified design equation is proposed to predict the compressive load capacity of thistype of composite column.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Fiber reinforced polymer (FRP) materials are increasingly beingutilized in the construction of structural members due to their highstrength-to-weight ratio and corrosion resistance. In reinforcedconcrete (RC) column retrofit, FRP is primarily used as an externaljacket to provide confinement to the concrete core [1–3]. Confine-ment also enhances the compressive load capacity of a concrete-filled FRP tube (CFFT) column, and reduces the required columncross-section compared to that of a conventional RC column.Furthermore, replacing the external steel tube, utilized in a con-crete-filled steel tubular (CFST) column, with an FRP tube, whichis corrosion resistant, enhances the long-term column durability.

In an effort to integrate the advantages of various constructionmaterials, Karimi et al. [4] proposed a novel composite column. Thecomposite column consists of a steel section surrounded by a con-crete-filled FRP tube, which eliminates the need for lateral ties. Aschematic view of the composite column is shown in Fig. 1a. Theconfinement and composite action between the constituent mate-rials result in enhanced mechanical and durability behavior. TheFRP tube provides confinement to the concrete core and theconcrete increases the buckling capacity of the steel column.

Furthermore, the FRP protects the column from weathering andimproves its durability. Finally, the proposed composite columnhas higher ductility than a CFFT column due to the addition of steel[5].

A substantial amount of research focusing on modeling thebehavior of short composite columns has been conducted [6–9].However, few studies have concentrated on the development ofanalytical models for long composite columns where overall buck-ling is the governing failure mode [10–12]. The primary objectiveof this study is to develop an analytical model to predict the behav-ior of the proposed FRP encased steel–concrete composite column,with various slenderness ratios, under axial load.

Analytical model predictions are compared with the experi-mental test results conducted on six composite column specimens.The capacity curve obtained from the analytical model is utilized inestablishing slenderness limits and developing design expressionsfor the proposed composite column.

2. Experimental program

A total of six composite columns, ranging between 500 mm and3000 mm in height, were tested. The specimens were constructedby placing a glass FRP (GFRP) tube around an I-shaped steel columnand the voids between the steel and the FRP tube weresubsequently filled with concrete. Fig. 1b shows a photographand the cross-section dimensions of the tested columns.

Nomenclature

Ac cross-section area of concreteAf cross-section area of FRP tubeAs cross-section area of steel columnAeq cross-section area of equivalent reinforced concrete col-

umnD inside diameter of FRP tubeea axial straineau ultimate axial strain of short composite columnseco axial strain in concreteeco axial strain corresponding to peak strength of uncon-

fined concreteel lateral strain in FRP tubeelu ultimate lateral strain in FRP tubeEc elastic modulus of concreteE2 slope of the linear branch in axial stress–strain relation-

ship of confined concreteEf axial compressive modulus of FRP tubeEf,l lateral tensile modulus of FRP tubeEs elastic modulus of the steelf 0co compressive strength of unconfined concretef 0cc compressive strength of confined concreteflu ultimate lateral confining pressurefy yield strength of the steelIc moment of inertia of concreteIeq moment of inertia of the equivalent reinforced concrete

columnIf moment of inertia of FRP tube

Is,yy moment of inertia of the steel column with respect tothe weak axis

k effective length factorL unbraced length of columnmla Poisson’s ratio of FRP tube when loaded in the axial

direction and contraction occurring in the lateral direc-tion

mal Poisson’s ratio of FRP tube when loaded in the lateraldirection and contraction occurring in the axial direc-tion

r radius of gyrationrc axial stress in concreterl lateral stress in FRP tuberl,u lateral stress in FRP tube at failurerf axial stress in FRP tuberf,u axial stress in FRP tube at failurers axial stress in steelSa,c axial compressive strength of FRP tubeSa,t axial tensile strength of FRP tubeSl,c lateral compressive strength of FRP tubeSl,t lateral tensile strength of FRP tubetf thickness of FRP tubeP axial loadPcs cross-section compressive load capacityPr factored compressive load capacityPu unfactored compressive load capacity

Fig. 1. (a) Schematic view of the proposed composite column (b) Photograph andcross-section dimensions (mm) of a test specimen.

Table 1Geometric and mechanical properties of the FRP tube and the composite material.a

Nominal pipe size (mm) 200Inside diameter (mm) 211Structural wall thickness (mm) 3.2Lateral tensile strength (MPa) 275Lateral tensile modulus (GPa) 15.9Axial compressive strength (MPa) 138Axial compressive modulus (GPa) 10.3Axial tensile strength (MPa) 138axial tensile modulus (GPa) 10.3Poisson’s mla 0.11Ratiob mal 0.19

a Manufacturer’s data (http://www.ameron-fpg.com).b The first subscript denotes the contraction direction and the second subscript

denotes direction of the applied force. ‘‘a’’ and ‘‘l’’ denotes axial and lateral direction,respectively.

K. Karimi et al. / Engineering Structures 46 (2013) 526–534 527

Table 1 shows the mechanical properties of the FRP tube. Theaverage compressive strength of the unconfined concrete at thetime of testing was 48.3 MPa. The yield and ultimate tensilestrength of the steel were 411 MPa and 526 MPa, respectively.

The composite columns were tested using a compression testapparatus capable of applying 5000 kN of axial load and the

boundary conditions were pin–pin. The compressive load capacityand ultimate axial strain values, obtained from test results, aresummarized in Table 2. It can be observed that both the compres-sive load capacity and ultimate axial strain decreased with in-creased column height. Details of the experimental program arereported elsewhere [5].

3. Analytical modeling

The analytical model is based on an incremental technique pro-posed by Mirmiran et al. [10]. In this technique the Euler bucklingload is evaluated based on the stiffness of the column at each axialstrain level and compared with the corresponding cross-sectioncompressive load capacity. If the Euler buckling load is greater thanthe cross-section compressive load capacity for all the strain levels,failure occurs due to loss of cross-section compressive load capac-

Table 2Analytical model predictions versus experimental results.

Specimen ID Height (H) Slenderness ratio (kL/r) Compressive capacity (Pu) Ultimate axial strain (eau)

(mm) Analytical (kN) Experimental (kN) Ana./exp. Analytical (le) Experimental (le) Ana./exp.

R-0.5 500 10 3440 3821 0.90 11,865 11,707 1.01R-1.0 1000 21 3179 3040 1.04 9160 9388 0.98R-1.5 1500 31 2387 2935 0.81 2330 3277 0.71R-2.0 2000 41 2372 2545 0.93 2208 2840 0.78R-2.5 2500 51 2264 2295 0.99 2059 2369 0.87R-3.0 3000 62 2020 2251 0.90 1751 1903 0.92

Fig. 2. Free body diagram of the composite column components (a) concrete-encased steel section (b) FRP tube.

528 K. Karimi et al. / Engineering Structures 46 (2013) 526–534

ity and the column is consequently classified as a short column. Inlonger composite columns, the stability and cross-section capacitydiagrams intersect (also referred to as the intersection point).Stability diagrams show the Euler buckling load at each strain le-vel. If a column fails due to overall buckling beyond the strain limitcorresponding to the intersection point it is consequently classifiedas a long column. If the two diagrams intersect within the elasticrange (corresponding axial strain of less than 0.002) the columnis classified as a slender composite column, which fails due to elas-tic overall buckling. If the intersection point is located in the inelas-tic range, the column fails due to inelastic buckling and is classifiedas an intermediate long column.

The methodology used to develop the cross-section compres-sive load capacity and stability relationships is explained in the fol-lowing sections.

3.1. Cross-sectional compressive load capacity

Fig. 2 shows a free body diagram of the composite column com-ponents. The compressive load corresponding to an axial strainlevel is calculated by superimposing the separate contributions ofthe steel, the confined concrete and the FRP tube to the load carry-ing capacity given by:

P ¼ rsAs þ rcAc þ rf Af ð1Þ

where rs, rc and rf are the axial stress in the steel, concrete and theFRP tube, respectively, and As, Ac and Af are their correspondingcross-section areas.

In this model, the steel is assumed to be an elastic–plastic mate-rial with a strain-hardening behavior from the yield to the ultimatestress. In the proposed composite column, the FRP tube is under abi-axial state of stress as shown in Fig. 2b. The FRP is assumed toremain elastic in the axial and lateral directions for all strain levels.The stresses in the FRP tube can be evaluated based on themechanics of composite materials [13] as:

rf ¼Ef

1� malmlaea þ

malEf

1� malmlael ð2Þ

rl ¼mlaEf ;l

1� malmlaea þ

Ef ;l

1� malmlael ð3Þ

where rf and rl are the axial and lateral stresses in the FRP tube,respectively, and ea and el are the corresponding strains. Ef and Ef,l

are the axial compressive and the lateral tensile moduli of the FRPtube material, respectively. mla and mal are Poisson’s ratios corre-sponding to loading in axial and lateral directions, respectively.

In evaluating the cross-section compressive load capacity, it isassumed that failure occurs due to rupture of the FRP tube, result-ing in a loss of confinement. Rupture of the FRP tube can beassessed using the Tsai-Wu criterion, which is commonly used inevaluating the failure of composite materials [14], given by:

1Sl;t� 1

Sl;c

� �rlu þ

1Sa;t� 1

Sa;c

� �rfu þ

1Sl;tSl;c

r2lu þ

1Sa;tSa;c

r2fu

� 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiSa;cSa;tSl;cSl;t

p rlurfu ¼ 1 ð4Þ

where Sa,c and Sl,c are the axial and lateral compressive strength ofthe FRP tube, respectively, and Sa,t and Sl,t are the axial and lateraltensile strength values, respectively.

Axial strain of the FRP tube at rupture can be evaluated byassuming strain compatibility between the FRP and the concreteand utilizing the expression proposed by Lam and Teng [15] forthe ultimate axial strain of FRP-confined concrete given by:

eau ¼ eco 1:75þ 12:0flu

f 0co

� �elu

eco

� �0:45" #

ð5Þ

where f 0co is the peak strength and eco is the corresponding axialstrain (commonly taken as 0.002) of the unconfined concrete. flu isthe ultimate lateral confining pressure, which can be evaluated con-sidering the equilibrium of the FRP tube (see Fig. 2b) as:

flu ¼2rlutf

Dð6Þ

The ultimate state of stress in the FRP tube (rfu,rlu) and the cor-responding ultimate strain values (eau,elu) can be evaluated bysubstituting Eq. (6) into Eq. (5) and solving Eqs. (2)–(5).

The constitutive relationship proposed by Lam and Teng [15]was used in evaluating the concrete behavior in the proposedcomposite column. The model assumes a parabolic stress–stainrelationship for the confined concrete up to an axial strain level

Fig. 3. Constitutive model for FRP-confined concrete.

K. Karimi et al. / Engineering Structures 46 (2013) 526–534 529

of et followed by a linear branch until failure, as shown in Fig. 3.The proposed axial stress–strain relationship is defined as:

rc ¼Ecec � ðEc�E2Þ2

4f 0coe2

c ð0 6 ec 6 etÞf 0co þ E2ec ðet 6 ec 6 eauÞ

(ð7Þ

where ec and et are the axial strain and the strain at the transitionpoint in the stress–strain relationship of the unconfined concrete,respectively. Ec and E2 are the elastic moduli of the unconfined con-crete and the slope of the linear branch in the stress–strain relation-ship of the confined concrete shown in Fig. 3, respectively.

Based on a comprehensive database of experimental tests, Lamand Teng [15] expressed the compressive strength of confinedconcrete as a function of the lateral confining pressure using thefollowing relationship:

f 0cc ¼ f 0co þ 3:3f lu ð8Þ

The above model was utilized in developing the cross-sectioncapacity curve for the composite columns tested in this study.

3.2. Stability relationships

The buckling capacity of the composite column is estimatedusing the Euler buckling formula expressed as:

PE ¼ p2 EI

ðkLÞ2ð9Þ

where k is the effective length factor taken as 1.0 for the pin–pinboundary conditions and L is the unbraced column length. EI isthe equivalent tangent stiffness of the composite column, whichis obtained from superimposing the stiffness of its components as:

Fig. 4. Transforming the composite cross-se

EI ¼ EsIs þ EcIc þ Ef If ð10Þ

where Es is the modulus of elasticity of the steel; Ic and If are themoment of inertia of the concrete and the FRP tube, respectively.The concrete modulus of elasticity varies with the strain and isevaluated from the first derivative of Eq. (7) as:

Ec ¼drc

dec¼ Ec � ðEc�E2Þ2

2f 0coec ð0 6 ec 6 etÞ

E2 ðet 6 ec 6 eauÞ

(ð11Þ

The modulus of elasticity of the steel after yielding is taken asthe slope of the strain-hardening branch in the stress–strain rela-tionship. The FRP elastic modulus is assumed constant at all thestrain increments.

3.3. Definition of slenderness ratio

The slenderness ratio (kL/r) of the proposed composite columnwas estimated based on an equivalent radius of gyration given by:

r ¼ffiffiffiffiffiffiffiIeq

Aeq

sð12Þ

where Ieq and Aeq are the moment of inertia and cross-section areaof an equivalent RC column obtained by transforming the area ofthe steel and FRP to concrete as shown in Fig. 4. Ieq and Aeq are givenby:

Ieq ¼Es

Ec� 1

� �Is;yy þ

p64

Dþ 2tfEf

Ec

� �4

ð13Þ

Aeq ¼Es

Ec� 1

� �As þ

p4

Dþ 2tfEf

Ec

� �2

ð14Þ

where D and tf are the inside diameter and thickness of the FRP tubeand Is,yy and As are the moment of inertia with respect to the weakaxis and cross-section area of the steel section, respectively. Theestimated slenderness ratio values for the tested composite columnspecimens are presented in Table 2.

Fig. 5 shows the stability and cross-section compressive loadcapacity curves for the tested composite columns. The axial loadand strain are normalized with respect to the cross-section com-pressive load capacity and ultimate axial strain of short compositecolumns (Pcs and eau), respectively. As can be seen from Fig. 5, theshort composite specimen R-0.5 (kL/r = 10) would fail due to loss ofthe cross-section compressive load capacity, whereas the remain-ing specimens would fail as a result of overall buckling. The pre-dicted failure modes correspond to the failure modes observed inthe experimental testing program.

ction to an equivalent concrete section.

0

1

2

3

4

5

0.0 0.2 0.4 0.6 0.8 1.0

Normalized Axial Strain (ε /ε au )

Nor

mal

ized

Loa

d (P

u/P

cs)

41

31 21

51

10

62

kL/r

Column Stability Relationships

Cross-Section CapacityStrength Limit

Fig. 5. Predicted normalized axial load versus normalized axial strain of thecolumns.

0

1

2

3

4

5

0.0 0.2 0.4 0.6 0.8 1.0

Normalized Axial Strain (ε /εau)

Nor

mal

ized

Loa

d ( P

/ Pcs

)

19.8 = Slenderness Limit, (kL/r )limit

Column Stability Relationship

Cross-Section Capacity

Intersection

55.3Slender

Inte

rmed

iate

Lon

g

Short

Fig. 6. Evaluating slenderness limit for the composite columns using the analyticalmodel.

530 K. Karimi et al. / Engineering Structures 46 (2013) 526–534

Slenderness limit is defined as the slenderness ratio belowwhich the composite columns are classified as short columns. Itcorresponds to the slenderness ratio of a column whose stabilityand cross-section compressive load capacity curves intersect at

0

1,000

2,000

3,000

4,000

0 3,000 6,000 9,000 12,000

Axial Strain (με)

Axi

al L

oad

(kN

)

ExperimentalAnalytical

0

1,000

2,000

3,000

4,000

Axi

al L

oad

(kN

)

ExperimentalAnalytical

0

1,000

2,000

3,000

4,000

Axi

al L

oad

(kN

)

ExperimentalAnalytical

(a) R-0.5

(c) R-1.5

(e) R-2.5

0 3,000 6,000 9,000 12,000

Axial Strain (με)

0 3,000 6,000 9,000 12,000

Axial Strain (με)

Fig. 7. Analytically predicted axial load versus axial strain relationship

the strain level equal to eau. A slenderness ratio limit of 19.8 wasobtained for the tested composite columns as shown in Fig. 6. Aslenderness ratio of 55.3, corresponding to a composite columnwhose stability and cross-section compressive load capacity curves

0

1,000

2,000

3,000

4,000A

xial

Loa

d (k

N)

ExperimentalAnalytical

0

1,000

2,000

3,000

4,000

Axi

al L

oad

(kN

)

ExperimentalAnalytical

0

1,000

2,000

3,000

4,000

Axi

al L

oad

(kN

)

ExperimentalAnalytical

(b) R-1.0

(d) R-2.0

(f) R-3.0

0 3,000 6,000 9,000 12,000

Axial Strain (με)

0 3,000 6,000 9,000 12,000

Axial Strain (με)

0 3,000 6,000 9,000 12,000

Axial Strain (με)

s for the composite columns in comparison with the test results.

1.2

Pu =Pcse

K. Karimi et al. / Engineering Structures 46 (2013) 526–534 531

intersect at an axial strain of 0.002, separates the intermediate longand slender composite column specimens (see Fig. 6).

0.0

0.2

0.4

0.6

0.8

1.0

0 20 40 60 80 100

Slenderness Ratio (kL/r)

ExperimentalAnalytical

kL/r=19.8

Short Intermediate Long

kL/r =55.3

Slender

Nor

mal

ized

Com

pres

siv

Cap

acity

(P u

/Pcs

)

Fig. 8. Predicted capacity curve versus experimental results.

1.2D = 211 mm, t = 3.2 mm

4. Comparison of analytical and experimental results

The compressive load capacity and ultimate axial strain of thecomposite columns calculated using the analytical model are sum-marized in Table 2 along with the ratio of analytical predictions toexperimental results. From the table, it can be seen that overall theanalytical predictions are in close agreement with the experimen-tal results. The axial load versus axial strain relationships for thesix tested composite columns obtained from the analytical modelare shown in Fig. 7 along with the experimental test results. Fromthis figure, it can be seen that the analytical model predictions arein good agreement with the overall load–deformation behavior ofthe tested columns. The lower compressive load capacity valuespredicted by the analytical model are partially attributed to theeffects of the shrinkage reducing admixture in the concrete mixutilized in the tested specimens [4].

0.0

0.2

0.4

0.6

0.8

1.0

0 20 40 60 80 100

Slenderness Ratio (kL/r )

D=D0

D=2D0

D=3D0

D=4D0

0 0

D = D0

D = 2D0

D = 3D0

D = 4D0

Ef0 = 10.3 GPa, As0 = 1,730 mm2

Nor

mal

ized

Com

pres

sive

C

apac

ity (P

u/P

cs)

(a)

30

35

40

ross

-Sec

tion

cs @

D/D

0=1)

8

10

12t0 = 3.2 mm, Ef0 = 10.3 GPa, As0 = 1,730 mm2

erne

ss R

atio

5. Capacity curve

A column capacity curve gives the compressive load capacity ofthe columns as a function of the slenderness ratio. Fig. 8 shows thetheoretical capacity curve for the tested columns obtained fromthe proposed analytical model versus the experimental data. Thecapacity predictions were within 11% of the experimental results.Based on Fig. 8, specimens R-1.0 to R-2.5 are classified as interme-diate long composite columns and the specimen R-3.0 as a slendercomposite column, respectively.

The capacity curve approach can be utilized as an effective de-sign tool in predicting the compressive resistance of columns.However, the analytical model shows that the capacity curvedeveloped in this study is influenced by the cross-section dimen-sions and mechanical properties of the constituent materials.Consequently, in the following section a study is carried out toinvestigate the effect of various design parameters on the capacitycurve and slenderness limit previously developed for the compos-ite columns tested in this study.

10

15

20

25

1 2 3 4

Normalized Column Diameter, D/D0

Nor

mal

ized

CC

apac

ity (P

cs/P

0

2

4

6

Slenderness LimitCross-Section CapacityL

imiti

ng S

lend

(b) Fig. 9. Influence of column diameter on the (a) capacity curve (b) cross-sectioncapacity and slenderness limit.

6. Influence of column parameters on the capacity curve, cross-section compressive load capacity and slenderness limit

A study was conducted using the proposed analytical model toinvestigate the effects of cross-section dimensions and mechanicalproperties of the constituent materials on the capacity curve andthe slenderness limit. These parameters included the columndiameter (D), thickness of the FRP tube (t), axial compressivemodulus of the FRP tube (Ef) and the steel-to-concrete area ratioin order to cover columns with various geometry and materialproperties. Four different cases were examined for each of theaforementioned parameters.

Fig. 9 shows influence of column diameter on the capacitycurve, cross-section compressive load capacity and slendernesslimit. The selected range for D is between one to four times thediameter of the tested columns. Other cross-section dimensionsand materials properties were held constant. Increasing the col-umn diameter results in reduced confinement, which affects thecross-section compressive load capacity (Pcs). However, stabilityof the composite column is enhanced with increased diameter,which results in a higher normalized compressive load capacity(Pu/Pcs) and slenderness limit, although the effect on the slender-ness limit is not as significant. The slenderness limit increasedfrom 19.8 to 25.6 with increased column diameter.

A similar trend was observed in Fig. 10a with the reduced thick-ness of the FRP tube for t/t0 ratios varying between 1.00 and 0.25,while all remaining parameters were held constant. The slender-ness limit increased from 19.8 to 24.7 with the reduced FRP thick-ness as shown in Fig. 10b. From Eq. (6) it can be seen that theultimate lateral confining pressure is proportional to the thicknessof the FRP tube. Consequently, increasing the thickness of the FRPtube enhances cross-section compressive load capacity (seeFig. 10b).

Fig. 11 shows the effect of the axial compressive modulus ofthe FRP tube (Ef) on the composite column capacity curve,

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 20 40 60 80 100

Slenderness Ratio (kL/r)

D0,t0

D0,t0/2

D0,t0/3

D0,t0/4

D0 = 211 mm, t0 = 3.2 mm

t = t0

t = t0/2

t = t0/3

t = t0/4

Ef0 = 10.3 GPa, As0 = 1,730 mm2

Nor

mal

ized

Com

pres

sive

C

apac

ity (P

u/P

cs)

(a)

10

15

20

25

30

35

40

0.25 0.50 0.75 1.00

Normalized FRP Tube Thickness,

Nor

mal

ized

Cro

ss-S

ectio

nC

apac

ity (P

cs/ P

cs @

t/t0

=1)

0.6

0.7

0.8

0.9

1.0

1.1

1.2

Slenderness LimitCross-Section CapacityL

imiti

ng S

lend

erne

ss R

atio

D0 = 211 mm, Ef0 = 10.3 GPa, As0 = 1,730 mm2

0tt /

(b) Fig. 10. Influence of thickness of the FRP tube on the (a) capacity curve (b) cross-section capacity and slenderness limit.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Ef=0

Ef0

2Ef0

3Ef0

D0 = 211 mm, t0 = 3.2 mm

Ef0 = 10.3 GPa, As0 = 1,730 mm2

Ef = 0

Ef = Ef0

Ef = 2Ef0

Ef = 3Ef0

Nor

mal

ized

Com

pres

sive

C

apac

ity (P

u/P

cs)

(a)

10

15

20

25

30

35

40

0 1 2 3

Normalized Axial Modulus of the FRP

Nor

mal

ized

Cro

ss-S

ectio

n C

a pac

ity (P

cs/P

cs @

t/t0

=1)

0.6

0.7

0.8

0.9

1.0

1.1

1.2

Slenderness LimitCross-Section CapacityL

imiti

ng S

lend

erne

ss R

atio

D0 = 211 mm, t0 = 3.2 mm, As0 = 1,730 mm2

0ff EE /

(b)

0 20 40 60 80 100

Slenderness Ratio (kL/r )

Fig. 11. Influence of axial modulus of the FRP tube on the (a) capacity curve (b)cross-section capacity and slenderness limit.

532 K. Karimi et al. / Engineering Structures 46 (2013) 526–534

cross-section compressive load capacity and slenderness limit.Ef/Ef0 ratios were selected between 0 (corresponding to a tube withno axial stiffness) and 3, where Ef0 is the axial modulus of the FRPtube used in the tested composite columns. The normalized com-pressive load capacity was found to increase with increased FRPaxial modulus. However, the difference between the capacitycurves reduced for slender columns. Fig. 11 shows the significanceof the axial stiffness of the FRP tube on the slenderness limit. Theslenderness limit increased from 14.8 to 36.5 with the increasedaxial modulus of the FRP from 0.0 to 30.9 GPa. Eq. (2) shows thatEf influences the bi-axial state of stress in the FRP tube. Maximumcross-section compressive load capacity was obtained for Ef/Ef0

equal to 1. Increasing Ef advances rupture of the FRP tube underbi-axial stresses which reduces ultimate confinement. Thisresulted in smaller cross-section compressive load capacity forcolumns with Ef/Ef0 greater than unity (see Fig. 11b).

The effect of steel-to-concrete area ratio on the capacity curvesis shown in Fig. 12a. The capacity curves were obtained for steel-to-concrete area ratios of 0.0%, 2.5%, 5.0% and 10.0%. As can be seenfrom Fig. 12a, steel-to-concrete area ratio had a minor influence onthe capacity curves for short or slender composite columns (kL/r > 80), which also resulted in negligible influence on the slender-ness limit (see Fig. 12b). However, the normalized compressiveload capacity increased with increased steel-to-concrete area ratiofor intermediate long columns. Fig. 12b shows enhancement in thecross-section compressive load capacity with increased steel-to-concrete area ratio. However, an increase in the steel-to concretearea ratio may result in a less economical composite column dueto the typically higher cost of steel compared to concrete.

The analytical model showed that the additional steel inside thecomposite section increased the buckling load and cross-sectioncompressive load capacity of the composite column. Enhancementof the buckling load was more significant compared to the cross-section compressive load capacity for steel-to-concrete area ratiosof less than 2.5%. This resulted in a greater slenderness limit for acomposite column consisting of a steel section surrounded by aconcrete-filled FRP tube compared to a concrete-filled FRP tubewithout steel. The slenderness limit decreased with increasedsteel-to-concrete area ratios greater than 2.5% (see Fig. 12b).

7. Proposed design expression

The design equation for evaluation of compressive load capacityin concrete-filled steel tube columns provided in the Canadiansteel code, CAN/CSA-S16-09 [16] is modified and utilized in designof the composite columns in this study. The unfactored compres-sive load capacity of the column is expressed as:

Pu ¼ Pcsð1þ k2nÞ�1=n ð15Þ

where n is a constant and k is defined as:

k ¼

ffiffiffiffiffiffiPcs

PE

sð16Þ

where PE is evaluated from Eq. (9) at the stage of initial loading. Pcs

can be found from the analytical model.The coefficient n in Eq. (15) was obtained as 0.97 through

regression analysis using the capacity curve developed from the

0.0

0.2

0.4

0.6

0.8

1.0

1.2

A=0

A=A0/2

A=A0

A=2A0

D0 = 211 mm, t0 = 3.2 mm

Ef0 = 10.3 GPa, As0 = 1,730 mm2

As = 0, As/Ac = 0

As = As0, As/Ac = 2.5%

As = 2As0, As/Ac = 5.0%

As = 4As0, As/Ac = 10.0%

Nor

mal

ized

Com

pres

sive

C

apac

ity (P

u/P

cs)

(a)

10

15

20

25

30

35

40

0.0 2.5 5.0 7.5 10.0

Steel-to-Concrete Ratio, As /Ac (%)

Nor

mal

ized

Cro

ss-S

ectio

n

Cap

acity

(Pcs

/Pcs

@ A

s/A

c=5%

)

0.6

0.7

0.8

0.9

1.0

1.1

1.2

Slenderness LimitCross-Section CapacityL

imiti

ng S

lend

erne

ss R

atio

D0 = 211 mm, t0 = 3.2 mm, Ef0 = 10.3

(b)

0 20 40 60 80 100

Slenderness Ratio (kL/r )

Fig. 12. Influence of steel-to-concrete ratio on the (a) capacity curve (b) cross-section capacity and slenderness limit.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0

Nor

mal

ized

Com

pres

sive

Cap

acity

(Pu

/Pcs

)

Experimental

Design Equation

Ecs PP /=λ

( )2csu 1PP λ+=

Fig. 13. Predicted unfactored compressive capacity of the composite columns fromthe proposed design equation in comparison with the experimental results.

K. Karimi et al. / Engineering Structures 46 (2013) 526–534 533

analytical model in Fig. 8. Eq. (15) can be simplified by approximat-ing n equal to unity as:

Pu

Pcs¼ 1þ k2 ð17Þ

The normalized compressive load capacity predicted using theabove equation was plotted against k for composite columns withvarious slenderness ratios and compared favorably with the exper-imental data as shown in Fig. 13. Finally, by introducing resistancefactors corresponding to each constituent material into Eq. (1) thefactored axial load capacity of the proposed composite column isgiven as follows in compliance with the expression provided inCSA-S806-02 [17] for FRP-confined reinforced concrete columns:

Pr ¼ b½a1/cf 0cAc þ /sfyAs�ð1þ k2Þ ð18Þ

where a1 = 0.85 � 0.0015f 0c P 0:39, /c = 0.60 for concrete and/s = 0.90 for the steel section. fy is the steel yield strength. The factorb is introduced to account for the capacity reduction due to unex-pected eccentricities. The axial load-carrying capacity of the FRPtube is conservatively ignored.

8. Conclusions

An analytical model was developed to predict the axial loadbehavior of a novel composite column consisting of steel, concreteand FRP. The model utilized an incremental technique, whichcompared the stability and cross-section compressive loadcapacity curves at each strain level. The stability curves wereevaluated using the Euler buckling formula accounting for the

strain softening of the composite columns. The cross section capac-ity curve was obtained by superposing separate contributions ofthe constituent materials in carrying the axial load. The compres-sive behavior of the concrete was simulated using an FRP-confinedconcrete model.

To verify the analytical model, six composite columns, between500 mm and 3000 mm in height, were constructed and testedunder axial load. The analytical model yielded close predictionsto experimental test result values. Over the wide range of kL/r val-ues investigated, less than 11% error was observed between thenormalized compressive load capacity predictions from the analyt-ical model and the experimental results.

Based on the analytical model, a slenderness limit of 19.8 wasestablished for the tested composite columns below which theslenderness effects are negligible and the columns are classifiedas short composite columns.

A study was conducted to investigate the effects of various col-umn design parameters such as column diameter, FRP tube thick-ness, FRP tube axial stiffness and steel-to-concrete area ratio on thecapacity curve and slenderness limit of the proposed compositecolumn. The study showed an increase in the normalized compres-sive load capacity of the columns with increased column diameterto FRP thickness ratio, FRP tube axial stiffness and steel-to-con-crete area ratio. The slenderness limit also increased significantlywith increased FRP tube axial stiffness.

A design expression was proposed based on the capacity curveobtained for the composite columns from the analytical model.This simplified expression can be used by structural designers topredict compressive load capacity of the proposed composite col-umn. It is recommended that additional experimental tests, cover-ing a wide range of column parameters including the columnparameters covered in this study, be conducted on this type ofcomposite column.

Acknowledgements

This study was carried out as part of ongoing research atMcMaster University Centre for Effective Design of Structuresfunded through Ontario Research and Development ChallengeFund of the Ministry of Research and Innovation. Funding was alsoprovided by Natural Sciences and Engineering Research Council ofCanada (NSERC).

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