Engineering Structures 28 (2006) 1586–1598www.elsevier.com/locate/engstruct

New approach to AISC P–M interaction curve for square concrete filledtube (CFT) beam–columns

Young-Hwan Choi∗, Douglas A. Foutch, James M. LaFave1

Department of Civil and Environ. Eng., University of Illinois at Urbana-Champaign, 205 N. Mathews Ave., Urbana, IL 61801, USA

Received 9 July 2005; received in revised form 3 January 2006; accepted 20 February 2006Available online 19 April 2006

Abstract

This paper presents a new method to design concrete filled tube (CFT) beam–columns based on the current American Institute of SteelConstruction’s Load and Resistance Factor Design (AISC/LRFD) method that is known to provide an over-conservative estimation of strength.The over-conservativeness of the current AISC/LRFD method comes mainly from neglecting the contribution of the concrete in flexure andthe different behavior of composite beam–columns from pure steel beam–columns. The new method reported herein assumes full compositeaction and idealizes the P–M interaction curve with two lines, as in the current AISC method; however, the lines intersect at a point where amaximum moment occurs. A complete parametric study is performed using fiber analysis to determine the maximum moment (normalized bythe nominal moment strength) and the axial load ratio at the maximum moment with respect to tube width-to-thickness ratio (b/t) and relativeconcrete compressive strength to yield strength of the steel tube ( f ′

c/Fy). The strengths based on the modified AISC P–M interaction equationsfor square CFT columns subjected to axial load and single axis bending are compared with a wide range of experimental data, and they showgreatly improved results when compared with current AISC/LRFD design equations.c© 2006 Elsevier Ltd. All rights reserved.

Keywords: Square concrete filled tube; AISC; Beam–column; P–M interaction curve

1. Introduction

Substantial structural demands are typically imposed oncolumn members in mid- to high-rise buildings. To meet suchdemands effectively, the concept of a composite system isoften introduced to make the most of the construction materialsused. There are two common types of composite columns:concrete filled tube (CFT) and concrete encased structural steel(frequently referred to as steel reinforced concrete (SRC)).CFTs have many advantages over reinforced concrete, steel, orSRC members [1–3]. The key benefits of CFTs from a structuralpoint of view are the ability of the concrete to delay localbuckling of the steel tube and the confinement of the concreteprovided by the steel tube. Since the two materials interact witheach other, CFTs show better structural performance than thesimple summation of each element’s individual performance.

∗ Corresponding author. Tel.: +82 10 6460 5349; fax: +82 31 910 0361.E-mail addresses: choi4u@gmail.com (Y.-H. Choi), jlafave@uiuc.edu

(J.M. LaFave).1 Tel.: +1 217 333 8064; fax: +1 217 265 8039.

0141-0296/$ - see front matter c© 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2006.02.009

Cost benefits of using CFTs have also been evaluated by manyother researchers [4–7].

In spite of their many benefits, however, CFTs have not beenwidely used in the US, mainly because of a lack of knowledgeabout them, which seems to be attributed to their relativelyshort history of use in the US. Furthermore, current US designprovisions offer little guidance for use of CFTs in practice. Inthe US, the American Institute of Steel Construction (AISC)Load and Resistance Factor Design (LRFD) specification [8] isused for steel construction. However, the current AISC/LRFDmethod provides questionable (and potentially quite over-conservative) results for estimating the strength of CFTbeam–columns, as many other researchers have reported [2,9,10], although it does provide a reasonably accurate estimateof the pure axial compression strength of CFTs for designpurposes [11].

Attempts to develop new design equations for CFTbeam–columns have been pursued by other researchers in theUS [12–14]. Aho [12] established a thorough database ofcomposite column tests. Hajjar and Gourley [13] successfullycompiled polynomial equations to represent the three-

Y.-H. Choi et al. / Engineering Structures 28 (2006) 1586–1598 1587

dimensional sectional strength of square or rectangular CFTbeam–columns. Leon and Aho [14] proposed seven ideas toimprove the current AISC/LRFD method; they stressed thenecessity of a new form of design equation.

The main objectives of this study are to identify the causesfor the over-conservativeness of the current AISC/LRFD andto formulate new modified AISC/LRFD design equations forCFTs. A simple method for computing the nominal momentstrength considering the contribution of the concrete is alsopresented. Square CFTs are chosen for this study, simplybecause of the fact that a square shape has the possibility ofeasier connection details. It is also assumed herein that CFTsdo not include reinforcing bars.

2. Design standards for CFTs

2.1. Current design standards

The current AISC/LRFD specifications for compositecolumns are based on a report by Task Group 20 ofthe Structural Stability Research Council (SSRC) [15]. InAISC/LRFD, traditional linear axial force–bending momentinteraction equations for pure steel columns are maintained andapplied for CFT columns, as follows (simplified for single axisbending and with the strength reduction factors omitted):

forPu

Pn≥ 0.2; Pu

Pn+ 8

9

Mu

Mn≤ 1.0 (1)

forPu

Pn< 0.2; Pu

2Pn+ Mu

Mn≤ 1.0 (2)

where Pu is the required axial strength and Mu is the requiredflexural strength, defining the P–M interaction curve. Mu musttake into account the second-order effect for a slender column.Pn is the nominal axial compression strength for the compositesection, assuming that no bending moment is applied. Mn

is the nominal flexural strength determined from a plasticstress distribution on the composite section, assuming that noaxial force is applied. However, AISC/LRFD also prescribesthat, when the axial term, Pu/Pn , in Eqs. (1) and (2) is lessthan 0.3, Mn shall be determined by using a straight linetransition between the flexural strength of the composite sectionat Pu/Pn = 0.3 and the flexural strength of just the steel sectionat Pu = 0. This implies that, when the axial term is zero,no composite action is assumed, and Mn is determined froma plastic stress distribution on the steel section only (whichis very conservative). According to the Commentary of theAISC/LRFD specification [8], the transition of the nominalmoment strength based on the axial load ratio is included, sincethere was no test data available on the loss of bond strength inCFTs.

The design concept in ACI [16] for CFT beam–columns isessentially the same as that for an ordinary reinforced concretebeam–column. The continuous steel tube is transformed anddistributed into discrete equivalent “reinforcing bars” along theperimeter of the section. The area and location of the equivalentreinforcing bars are dependent on how the tube is divided. Oncethe tube is transformed and distributed, the transformed section

Fig. 1. Standards comparison (per Varma [19]).

is then treated as an ordinary reinforced concrete member. Thesame rules for ordinary reinforced concrete are applied to thetransformed section for generating a P–M interaction curve,such as a linear strain distribution and strain compatibility(perfect bond).

The P–M interaction curve for CFT beam–columns, as perthe AIJ [17], can be represented by superposition. The capacityof a CFT section can be predicted by superposition of thecapacity of each material. Since the strengths are computedindependently, the AIJ does not require strain compatibilitybetween the steel and concrete, which can result in aninconsistent location of the neutral axis.

The fundamental concept of Eurocode 4 [18] for designingCFT beam–columns is full composite action. The capacityof the members is computed based on an assumed perfectbond condition. Unlike the other standards, Eurocode 4 takesinto account the better condition of the confined concrete inCFT sections, by employing the full cylinder strength of theconcrete, f ′

c (while the other codes use a reduced strength, suchas 0.85 f ′

c).Fig. 1 illustrates P–M interaction curves generated by

Varma [19] for CFTs based on the four standards (along withsome experimental results). The steel section is a 305 mmsquare tube (A500 Grade B) with yield stress of 317 MPa and1500 mm long. The width-to-thickness ratio of the tube wallsis approximately 35. The steel tube is filled with 110 MPacompressive strength concrete. Although the concrete strengthis much higher than the maximum limit specified in theAISC/LRFD specification, it can still clearly be seen that thestrength estimation by the current AISC/LRFD is considerablyconservative compared with the experimental results, as well ascompared with all the other standards.

2.2. Causes for the over-conservativeness in AISC/LRFD

The axial force–bending moment (P–M) interaction curvefrom Eqs. (1) and (2) is illustrated schematically with solid linesin Fig. 2, where the solid dots represent typical experimentalresults for short CFT beam–columns. The x-axis and they-axis are normalized to the experimental moment strengthwith no axial load applied and the axial strength withno moment applied, respectively. Points A, B, and C aredetermined from AISC/LRFD and represent the nominalflexural strength of the steel section, the intersection point of

1588 Y.-H. Choi et al. / Engineering Structures 28 (2006) 1586–1598

Fig. 2. Comparison of P–M curve by the current AISC/LRFD andexperiments.

the two equations (Eqs. (1) and (2)), and the nominal axialcompression strength of the composite section, respectively.Point D represents the maximum moment from experimentalresults. Considering that the general range of the axial load ratioin practical beam–column design is typically from about 0.1 to0.3, the current AISC/LRFD method is somewhat problematicfor design guidance. The causes of the over-conservatism arediscussed below.

First, the current AISC/LRFD method ignores much ofthe contribution of the concrete. When the design equationsthat were used for steel members were first introduced forcomposite columns with transformed properties, no test datawere available for composite action; therefore, it was suggestedthat the contribution of the concrete be ignored for safetyreasons when the axial load ratio is less than 0.3. However,Furlong [2] long ago noted that the moment strength of the steeltube alone is undesirably low for estimating the pure bendingstrength of CFTs. Lu and Kennedy [20], who conducted pureflexure tests on square and rectangular CFTs, observed: (1)identical moment–curvature relationships irrespective of theshear span ratio; (2) negligible slip before the maximummoments and very small relative movements; and (3) closecorrespondence between steel strains on the flange and concretestrains. Based on these three independent observations, it wasconcluded that no degradation of the moment capacity occurreddue to shear transfer problems. Since Point A in Fig. 2 doesnot account for the contribution of the concrete, the momentstrength at Point A is likely to be less than the experimentalvalue. However, even when the contribution of the concrete isfully accounted for in determining the flexural strength at noaxial load, the curve per the current AISC/LRFD is still over-conservative, due to the following reason.

Before discussing the second reason in detail, Fig. 3 ispresented to better understand the P–M interaction curveof CFT beam–columns. Fig. 3(a) is drawn for an elasticmaterial that has the same strength in tension and compression,whereas Fig. 3(b) is for an elastic material whose strength incompression is greater than that in tension. While the maximummoment in Fig. 3(a) occurs when no axial load is applied, theone in Fig. 3(b) occurs when some amount of axial compressionload is applied. The typical shape of a P–M interactioncurve for steel is similar to Fig. 3(a), while that for ordinaryreinforced concrete is similar to Fig. 3(b). It can be reasoned

Fig. 3. P–M interaction curve for elastic column.

that the shape of a P–M interaction curve for CFTs would besimilar to Fig. 3(b), as observed in many experimental results.

As mentioned earlier, the current AISC/LRFD P–Mequations were derived from pure steel members and expandedto include CFT members. Although the contribution of theconcrete is taken into account by transforming concrete toequivalent steel, the overall shape of the P–M interactioncurve is still based on the “steel” section, which is similarto Fig. 3(a). Due to the fundamental difference between steeland reinforced concrete, the P–M interaction curve based onequivalent steel may not simulate the correct response of CFTmembers. Therefore, this problem would be observed evenwhen the moment strength (Point A in Fig. 2) is computedassuming full composite action. This can also be verifiedmathematically. Line 2 in Fig. 2 is represented by Eq. (2), whichcan be rearranged as:

forPu

Pn< 0.2; Pu

Pn= −2

(Mu

Mn

)+ 2. (3)

Notice that the line represented by Eq. (3) has a negativeslope, indicating that Point B most probably will not be locatedclose to Point D (which is on the right side of Point A). Onemay think that it is possible for Point B to be located nearPoint D, since the flexural strength, Mn , along Line 2 is nota constant but rather a variable changing from that based on thesteel section to that based on the composite section. Althoughthis is true, Point B is still likely to be located to the left of PointA, unless very high strength concrete is used.

2.3. Degree of over-conservativeness in AISC/LRFD

This section examines how much the current AISC/LRFDmethod underestimates the moment strength of CFTs at an axialload ratio of 0.2 compared with when a full composite actionis assumed, as in other standards (ACI and Eurocode 4). A254×254×6.35 mm square tube with a yield stress of 317 MPaand filled with 55 MPa compressive strength concrete is chosenas an example section. The effective length of the memberis assumed to be small (approaching zero). From Eq. (1) (orEq. (2)), the required moment at Pu/Pn = 0.2, Mu,0.2, iscomputed as

Mu,0.2 = 0.9Mn,0.2 (4)

Y.-H. Choi et al. / Engineering Structures 28 (2006) 1586–1598 1589

tCA

ssapAmcccpcc

scHconlaa

f

f

NAfd

3

3

ec

Fig. 4. Nominal moment strength by AISC/LRFD.

where Mn,0.2 represents the nominal moment strength when theaxial load ratio is 0.2 and can be computed as (see Fig. 4):

Mn,0.2 = 2

3Mcom + 1

3Ms (5)

where Mcom is the moment strength of the fully compositesection and Ms is the moment strength of the steel section only.For the section being considered, Mcom is equal to 1.185Ms .Substituting Mcom = 1.185Ms into Eq. (5) yields:

Mn,0.2 = 2

3(1.185Ms) + 1

3Ms = 1.123Ms. (6)

By inserting Eq. (6) into Eq. (4), Mu,0.2 can be computed as

Mu,0.2 = 0.9 × 1.123Ms = 1.011Ms. (7)

In order to determine analytically the moment strength ofthis same member, a fiber analysis can be performed with theassumption of a complete bond condition. (Details of the fiberanalysis are explained later.) Fiber analysis here is performedusing two different models: analysis model 1 with elasto-plasticsteel and unconfined concrete, and analysis model 2 with elasto-plastic steel and confined concrete. For the confined concretemodel, strength enhancement due to the confinement effect isnot taken into account (only increased ductility is considered).From fiber analysis, the following relation may be found forthis CFT:

Mu,0.2 ={

1.396Ms for unconfined concrete1.453Ms for confined concrete.

(8)

By comparing Eqs. (7) and (8), it can be observed that,when full composite action is assumed, the current AISC/LRFDmethod can ignore as much as nearly 40% of the momentcapacity of the CFT member being considered, even whenthe strength enhancement due to the confinement effect isnot considered. The comparisons for other sections have alsobeen performed and are summarized in Table 1, where theyield strength of steel is 317 MPa, the thickness of the squaresteel tube is 6.35 mm, and strength enhancement due to theconfinement effect is not considered.

2.4. New design approach for CFT beam–columns

Based on the reasons discussed in the previous section, anew method to cure the problems is presented in this section.In this study, full composite action is assumed, as in some otherstandards (ACI and Eurocode 4), which is also supported bythe experimental results on pure flexure strength [20]. With

Fig. 5. Concept of modified equations.

he assumption of perfect bond, the maximum moment forFTs occurs when some amount of axial load is applied.ccordingly, it is reasoned that the accurate shape of the

P–M interaction curve for CFTs is similar to the dotted curvehown in Fig. 5, as observed from experimental results. In thistudy, the dotted curve is idealized with two lines, Lines And B, intersecting at a point of maximum moment. Threeoints are required to define the two lines completely: Point

for nominal flexural strength, Point B for the maximumoment, and Point C for nominal axial strength. Point A is

omputed based on a plastic bending stress distribution for theomposite section, instead of just for the steel section as in theurrent AISC/LRFD method. Because it is assumed that theure compression strength is reasonably well-predicted by theurrent AISC/LRFD method [11], the same method as in theurrent AISC/LRFD is applied for Point C.

When both axes are normalized (with the nominal momenttrength and the nominal axial strength, respectively), theoordinates of Point A become (1, 0), and Point C is at (0, 1).owever, the coordinates for Point B move depending on the

ontribution of the concrete. In order to find the coordinatesf Point B, two factors are introduced: the maximum momentormalized by the nominal flexural strength (α) and the axiaload at the maximum moment normalized with the nominalxial strength (β). Including the unknowns α and β, Lines And B can be expressed as:

orPu

Pn≥ β: Pu

Pn+ 1 − β

α

Mu

Mn= 1 (9)

orPu

Pn< β: 1 − α

β

Pu

Pn+ Mu

Mn= 1. (10)

ote that Eqs. (9) and (10) are identical to the currentISC/LRFD equations when α = 0.9 and β = 0.2. Functional

orms need to be formulated to represent α and β that varyepending on the contribution of the concrete.

. Description of parametric study

.1. Related behavioral studies for parametric study

Researchers have agreed that CFTs exhibit significantlynhanced ductility in comparison with their constituentomponents [2,3,20–22]. The concrete delays local buckling

1590 Y.-H. Choi et al. / Engineering Structures 28 (2006) 1586–1598

Table 1Conservativeness of the current AISC/LRFD method

b (mm) f ′c (MPa) Mcom

MsMn,0.2 Mu,0.2 Differences

Eq. (5) (x Ms ) Eq. (4) (x Ms ) FA1 (x Ms ) FA2 (x Ms ) D1 (x Ms ) (%) D2 (x Ms ) (%)

152

21 1.071 1.047 0.943 1.056 1.101 11 1628 1.088 1.059 0.953 1.083 1.143 13 1934 1.103 1.068 0.962 1.120 1.182 16 2241 1.115 1.077 0.969 1.154 1.217 18 2548 1.127 1.084 0.976 1.189 1.251 21 2855 1.137 1.091 0.982 1.218 1.284 24 30

178

21 1.082 1.055 0.949 1.062 1.120 11 1728 1.100 1.067 0.960 1.109 1.167 15 2134 1.116 1.077 0.970 1.155 1.210 19 2441 1.129 1.086 0.978 1.194 1.250 22 2748 1.141 1.094 0.985 1.229 1.288 24 3055 1.151 1.101 0.991 1.261 1.324 27 33

203

21 1.092 1.061 0.955 1.084 1.141 13 1928 1.111 1.074 0.967 1.137 1.193 17 2334 1.128 1.085 0.977 1.183 1.241 21 2641 1.142 1.094 0.985 1.231 1.286 25 3048 1.154 1.102 0.992 1.271 1.328 28 3455 1.164 1.109 0.998 1.307 1.368 31 37

229

21 1.101 1.067 0.960 1.104 1.163 14 2028 1.121 1.081 0.973 1.162 1.220 19 2534 1.138 1.092 0.983 1.218 1.272 24 2941 1.153 1.102 0.992 1.266 1.321 27 3348 1.165 1.110 0.999 1.309 1.367 31 3755 1.175 1.117 1.005 1.348 1.411 34 41

254

21 1.109 1.073 0.965 1.123 1.185 16 2228 1.131 1.087 0.978 1.190 1.246 21 2734 1.148 1.099 0.989 1.247 1.303 26 3141 1.162 1.108 0.997 1.298 1.356 30 3648 1.174 1.116 1.005 1.351 1.405 35 4055 1.185 1.123 1.011 1.396 1.453 39 44

FA1: fiber analysis with elasto-plastic steel and unconfined concrete; FA2: fiber analysis with elasto-plastic steel and confined concrete; D1: FA1 – Eq. (4); D2: FA2– Eq. (4).

of the steel tube by altering a key deformation mode [3]. Theinward buckling for CFTs is delayed by virtue of the presenceof the concrete core, which allows the steel to deform tosubstantially larger strains than those of hollow sections. Lu andKennedy [20] noted that the maximum curvature and rotationalcapacity of CFT beams was about three times that of hollowsteel beams.

The concrete in axially loaded CFTs is actually under a tri-axial state of stress [23]. The strength and ductility of concretesubjected to tri-axial compressive stress increases depending onthe degree of confinement provided [24]. However, Knowlesand Park [25] concluded from an experimental study that thesteel tube was not able to restrain radial deformation of theconcrete until it reached a strain of 0.002. Other researchers [2,11] noted that there is no interaction between the two materialsuntil about 90% of the maximum load has been reached.

Strength enhancement of concrete due to a confinementeffect is generally accepted for circular CFTs; however, ithas been suggested by a number of researchers that theconcrete strength enhancement due to confinement for squareor rectangular CFTs be ignored, although there is a benefit toconcrete ductility [2,22].

An important factor as to how much confinement can beattained for CFTs is the width-to-thickness ratio (b/t or d/t) ofthe steel tube. The confinement effect is a result of restrainingpressure from the steel tube on the concrete core, so theconfinement effect cannot be achieved if the steel tube is tooweak or flexible to restrain lateral dilation of the concrete.Furlong [2] compared the load vs. strain relationships of twoCFTs having different d/t ratios. For the stiffer CFT (smallerd/t), the strength of the concrete tends to increase even aftersignificant strains, due to the confinement effect, while theweaker section fails without any increase in strength.

3.2. Fiber analysis models

To establish the sectional response of CFTs, fiber analysisis adopted as an analytical tool. Fiber analysis has beenwidely acknowledged by many researchers as a powerful toolfor the sectional analysis of CFT beam–columns [19,26,27].The fiber analyses in this study are based on the followingassumptions: (1) plane sections remain plane after bending(linear strain variation); (2) the section satisfies equilibrium andcompatibility; (3) tensile strength of concrete is neglected; (4)

Y.-H. Choi et al. / Engineering Structures 28 (2006) 1586–1598 1591

Fig. 6. Stress–strain for local buckling.

Fig. 7. Stress–strain for confined concrete (per Tomii and Sakino [32]).

no slip occurs between the steel tube and the concrete core(perfect bond) [26], and (5) effects of shrinkage and creep areignored [28].

The steel tube is modeled using an elasto-plastic materialtype, three strain-hardening material types (by Burns andSiess [29], Varma [19], and Saenz [30]), and two other typesto account for local buckling. In this case, the Varma andSaenz models are used for tension fibers and elasto-plastic isused for compression fibers (Fig. 6). For the concrete core, theTodeschini et al. model [31] is used to represent unconfinedconcrete, while confined concrete is represented by the Tomiiand Sakino model [32] (Fig. 7), which is chosen for this studysince it corresponds well to the findings from a study of theliterature. Based on the material models for steel and concrete,then, seven overall analysis models are considered for theparametric study (Table 2). The CFT section is divided intoa sufficient number of layers (50) in order to minimize anypossible errors due to discretization — five layers each for thetop and bottom flanges and 40 layers for the web.

3.3. Parameters studied

There are several factors that affect the strength of squareCFT beam–columns. Other than the interaction between thesteel tube and the concrete core, geometric and materialproperties like area of steel (As), yield strength of steel (Fy),area of concrete (Ac), and/or compressive strength of concrete( f ′

c) play a key role. Since the goal of the parametric studyis to evaluate the contribution of the concrete quantitatively,the relative area and strength proportions of concrete to steelconstitute the main parameters, i.e., the ratio of concrete areato steel area (Ac/As) and the ratio of concrete compressivestrength to yield strength of the steel tube ( f ′

c/Fy). Also, the

Table 2Analysis models

Model # Steel ConcreteTension Compression

Mo. 1 Elasto-plastic Elasto-plastic UnconfinedMo. 2 Elasto-plastic Elasto-plastic ConfinedMo. 3 Burns and Siess Burns and Siess ConfinedMo. 4 Varma Varma ConfinedMo. 5 Saenz Saenz ConfinedMo. 6 Varma Elasto-plastic ConfinedMo. 7 Saenz Elasto-plastic Confined

compressive strength of concrete after a certain level of strainin CFTs varies due to the confinement effect depending onb/t of the steel tube, as shown in Fig. 7; therefore, b/t shouldbe included in the parameters in some way. Instead of addinganother parameter, it is accounted for by replacing the previousparameter Ac/As with b/t , since Ac/As can be expressed fromb/t as:

Ac

As= (b − 2t)2

4t (b − t)= (b/t − 2)2

4(b/t − 1). (11)

3.4. Selection of CFT sections

According to the AISC/LRFD specification [8] and ASTMstandards [33,34], there are several types of materialspotentially available for HSS. For rectangular shapes, however,ASTM A500 Grade B (minimum yield strength of 317 MPa andminimum ultimate strength of 400 MPa) is the most commonlyavailable without any special order in the US [8]. Sherman [35]noted that only A500 Grade B can realistically be obtained inthe US. Therefore, only A500 Grade B is considered for thisstudy.

AISC/LRFD specifies four limitations on compositecolumns. To qualify as a composite column and be designedaccordingly, the following limitations must be met for squareCFTs: (1) As ≥ 0.04Ag; (2) 21 MPa ≤ f ′

c ≤ 55 MPa fornormal weight concrete; (3) Fy ≤ 415 MPa; and (4) b/t ≤(3Es/Fy)

0.5. By solving the first limitation with respect to b/t(in part using Eq. (11)), it can be re-expressed as 1 ≤ b/t ≤ 99.On the other hand, the maximum value of b/t for ASTM A500Grade B is determined as 43.5 from the fourth limitation. Forsteel with yield strength of 317 MPa, therefore, any HSS sectionthat satisfies the b/t limit automatically satisfies the limitationon minimum steel area.

Based on the AISC/LRFD limitations and availability inthe US, five series of square HSS tubes are selected with b/tratios ranging from 24 to 40 (in b/t increments of 4). Eachseries has seven concrete strengths, ranging from 21 MPa to 55MPa, in f ′

c increments of 6.9 MPa. In addition, the followingproperties are also assumed for steel: modulus of elasticity(Es) = 200,000 MPa; ultimate strength (Fu) = 400 MPa;yield strain (εy) = 0.00159; strain at the onset of strainhardening (εsh) = 0.0186; and ultimate strain (εu) = 0.15.

1592 Y.-H. Choi et al. / Engineering Structures 28 (2006) 1586–1598

(a) α vs. f ′c/Fy . (b) α vs. b/t .

Fig. 8. α(=Mmax/Mn ) versus f ′c/Fy and b/t .

4. Formulation of new design equations

4.1. Nominal moment strength determination: Mn

In order to express the maximum moment in dimensionlessform, a definition of the nominal moment strength (in purebending) first needs to be established. In this study, the nominalmoment strength of CFTs is computed based on the fullcomposite section irrespective of the axial load ratio, followingother standards (such as ACI and Eurocode 4) [16,18] aswell as an experimental study [20] in which no degradationof moment strength due to bond related problems occurred.When computing the moment strength considering the concretecontribution, a concrete compressive strength of 0.85 f ′

c is usedfollowing the study by Aho [12].

CFT nominal moment strength is computed from a plasticstress distribution on the composite section, with the neutralaxis located at a distance “c” from the bottom of the top flange.To find the distance c, the section is divided into five segments:concrete in compression; steel top flange in compression; steelweb in compression; steel web in tension; and steel bottomflange in tension. The location of the neutral axial is found fromthe force equilibrium requirement as:

c = 2t (b − 2t)Fy

4t Fy + (0.85 f ′c)(b − 2t)

. (12)

Once the distance c is known, the moment is computedby summing the products of the forces multiplied by theirrespective moment arms to the neutral axis as:

Mn = c2(b − 2t)0.85 f ′

c

2+ bt (b − t)Fy + (c2 + (b − 2t − c)2)t Fy (13)

where the first term represents the moment from the concrete,the second term from the steel top and bottom flanges, and thethird term from the steel web.

4.2. Formulation of α for new design equations

In order to investigate the effects of the main parameters( f ′

c/Fy , b/t) on the normalized maximum moment (α), asused in Eqs. (9) and (10), several plots are generated withrespect to the different parameters. Fig. 8(a) illustrates a typical

variation of α with respect to f ′c/Fy for a b/t = 36 series.

The value of α increases almost linearly as f ′c/Fy increases;

this trend is also observed in other b/t series (not shown inthis paper). The overall trend found from Fig. 8(a) is that therelation between α and f ′

c/Fy can be represented by a simplelinear equation, although it may also be possible to express itwith more complicated equations for greater accuracy.

Fig. 8(b) illustrates a typical variation of α with respect tob/t for an f ′

c/Fy = 0.109 series ( f ′c = 34.5 MPa). It can

be observed that α increases as b/t increases, which mightseem to be opposite of the effect of b/t on concrete strengthas described in the confined concrete model shown in Fig. 7,where concrete strength decreases as b/t increases (after astrain of 0.005) due to a weaker confinement. It might thereforebe expected that the maximum moment would decrease as b/tincreases. However, it should be noted that b/t also indirectlyrepresents the relative area of concrete (Eq. (11)), so a largervalue of b/t indicates that the concrete area is larger and thepotential contribution of the concrete is larger, which results ina positive contribution. The overall trend found from Fig. 8(b)is that the maximum moment with respect to b/t may alsobe represented by a simple equation similar to a square rootrelation or a logarithm relation (taking b/t as the independentvariable).

The current AISC/LRFD beam–column design equationswere developed based on several guidelines [36–38]. One ofthe guidelines is that the equation should not be more than5% unconservative when compared with strengths obtainedfrom analytical solutions; the same constraint is applied inthis study. Emphasis is also placed on simplicity and accuracy.The constraints used for formulating α for use in the modifieddesign equations are as follows:

• any equations should be simple to use for design purposes;• the results should not be more than 5% different

(unconservative or conservative) from the analyticalsolution;

• when there is no concrete, Eqs. (9) and (10) would preferablyconverge to the current AISC/LRFD equations;

• the maximum α value from the equations should not be morethan 1.5.

Any complicated formulation is obviously excluded for thepurposes of routine design. In order to emphasize the accuracy

Y.-H. Choi et al. / Engineering Structures 28 (2006) 1586–1598 1593

(a) b/t = 24. (b) b/t = 36.

Fig. 9. Difference of α from the fiber analysis.

of the resulting equation, a goal of only a 5% differencefrom the analytical solution is also applied on the conservativeside. The third constraint is preferably included in that α herehas been embedded into the current AISC/LRFD equationsto account for the contribution of the concrete; therefore,the proposed equations would be the same as the currentAISC/LRFD equations if there is no concrete in the section.However, this constraint cannot be included as a mandatorycriterion due to the fact that the definition of α is differentfor steel members and CFTs. The term α is used to representan inflection point for both steel and CFTs; however, themoment at the inflection point for just a steel beam–columnis not a maximum moment, while the term α is derivedfrom the maximum moment for CFTs. This discrepancy isunavoidable, since it comes from the actual different behaviorof steel members and CFT members. The final constraint isintentionally included to prevent excessive over-estimation forcertain combinations of parameters.

Based on the constraints mentioned above, a total of threepossible equations are developed (Eqs. (14) through (16)), allbased on results from analysis model 2 (though it may bepossible to develop other equations from other analysis modelsthat can accurately represent the variation in α, none of themwas found to be as simple for purposes of design):

α = 0.05

(2 + b

t

)f ′c

Fy+ 0.9 (14)

α =(

0.5

√b

t− 1.5

)f ′c

Fy+ 0.95 (15)

α =(

0.044b

t− 0.01 ln

b

t

)f ′c

Fy+ 0.95. (16)

Eq. (14) has the simplest form and also satisfies the“preferable” constraint. Eq. (15) provides a more accurateestimation than Eq. (14), while Eq. (16) provides the mostaccurate estimation. When developing and evaluating theequations, conservative estimation is given more emphasis thanover-estimation, although the constraint of a maximum 5%difference is applied to each. The differences between Eqs. (14)through (16) and the fiber analyses are shown in Fig. 9 (negativeindicates unconservative). The differences for Eq. (14) arelarger for lower f ′

c/Fy , due to the fact that the result from

Fig. 10. β versus f ′c/Fy .

Eq. (14) is forced to converge to a value of 0.9 when no concreteexists. Results from Eqs. (15) and (16) are matched well to theresults from the fiber analysis for all b/t ratios and f ′

c/Fy , andany small differences are evenly distributed across parameters.Eq. (16) is somewhat more complex to use compared with theother equations due to the logarithm function. In the end, Eq.(15) is proposed for the final formulation of α, as it provides asimple and accurate estimation without any bias with respect tothe analytical solutions.

4.3. Formulation of β for new design equations

Since α is formulated based on analysis model 2, only thatanalysis model is considered for formulation of the axial loadratio at the maximum moment (β). The variation of β withrespect to f ′

c/Fy is illustrated in Fig. 10; it can be observed thatβ increases almost linearly as f ′

c/Fy increases.The constraints used to formulate α are not applicable for

formulating β, due to the following reasons. First, estimation ofβ does not have any meaning with respect to conservativenessor over-estimation. While the formulation of α is to estimatestrength, and therefore a conservative approach may makesense, the procedure for β is simply to find the secondcoordinate of a specific point, not to estimate strength. Ifestimation of β is not accurate, it can incur conservativenessand over-estimation at the same time. Next, the difference inresults between the formulation and the exact solution is toosensitive to a small amount of variation to apply the samemaximum difference of 5% that is used in the formulation of α;

1594 Y.-H. Choi et al. / Engineering Structures 28 (2006) 1586–1598

Fig. 11. Differences of β to the fiber analysis.

the constraint for the maximum difference should be adjustedto account for this sensitivity. It is also not feasible to force β

to converge to 0.2 when there is no concrete, again due to thesensitivity. The constraints to develop the equation for β arethen as follows:

• the equation should be simple to use for design;• the equation should not be more than 10% different from the

analytical solution;• values from the equation should be neither more than 0.4 nor

less than 0.2.

A value of 0.2 is used to represent the lower limit and 0.4 isused to represent the upper limit. The value of β increases to ashigh as 0.32 for the sections considered in this study; it could behigher depending on the combination of different values of b/tand f ′

c/Fy . Based on the constraints, one equation is developedfrom a least square method as:

β = 0.34

(√b

t− 2

)f ′c

Fy+ 0.08. (17)

Differences between results from fiber analyses and Eq. (17) areillustrated in Fig. 11. The maximum differences from the fiberanalyses are 9.6% and −9.3%.

Since Eqs. (15) and (17) are based on the currentAISC/LRFD limitations whose maximum b/t is limited to43.5, it is recommended that Eqs. (15) and (17) be used forsquare CFTs with b/t less than 45.

5. Verification of the new design equations

In order to verify the suggested equations, comparisons witha wide range of existing experimental results are performedand illustrated in Fig. 12 and Table 3. Although the derivationsof the proposed equations (Eqs. (15) and (17) for use inEqs. (9) and (10)) are based on sections that satisfy theAISC/LRFD limitations, the comparisons are carried outwithout any limitations. A total of 87 specimens are foundfrom the literature: Tomii and Sakino [32] for short columnswith normal strength materials; Furlong [2] and Fujimotoet al. [39] for medium length columns with normal to highstrength steel; Grauers [40] for long columns; Varma [19] forcolumns with high strength concrete; and Lu and Kennedy [20]for pure flexure specimens. The comparisons are carried out

without considering the strength reduction factor. The momentsreported by Furlong [2] and Tomii and Sakino [32] includethe second-order moment, while the others [19,39,40] reportedthe end moment without the second-order moment. When thereported test results did not include the second-order moment,the P–M interaction curve is generated with Mnt instead ofMu , where Mu = B1Mnt ; the factor B1 is explained in theAISC/LRFD specification.

As shown in Fig. 12, the new method (solid lines) showsgreatly improved results compared to the current AISC/LRFDmethod (dashed lines). The improvement of the new methodcan be observed especially for the test specimens that havea higher degree of contribution from concrete, as expected.One or two test results may lead one to think that thestrength prediction by the new method could be over-estimated;however, even in these cases the degree of over-estimation isquite small. Therefore, the degree of over-estimation by the newmethod is thought to be acceptable. Moreover, the P–M curvesare generated without considering the strength reduction factor.Therefore, the strengths will be conservatively and accuratelypredicted when the reduction factor is used.

A statistical summary of the comparisons from Table 3 ispresented in Table 4. It includes whether the specimens satisfythe AISC/LRFD limitations. The term “All” is used to representall of the sections from Table 4, while “Ok” only representsthose sections that satisfy the current specification limitations.Comparisons are performed for moment strength at a givenaxial load in the tests. For high axial load, at which the momentstrength is quite small, a little deviation of the moment strengthmay lead to a large percentage difference for this type ofcomparison. To avoid such a problem, axial load ratio less than0.4 is also included for a criterion in some rows of the table.From the statistical summary, it can be seen that the proposeddesign method provides greatly improved moment strengthestimations. For sections that satisfy the current AISC/LRFDlimitations and subjected to axial load less than 0.4Pn, theratio of the estimation by the new method to the experimentalresults varies from 0.80 to 1.05 with an average of 0.93, whilethe current AISC/LRFD method provides 0.44 to 0.91 with anaverage of 0.76. There are a few instances where the proposedequations provide over-estimated results, with a maximum of9%. However, when the strength reduction factor is included,the strength is still safely estimated even in these extremecases.

6. Summary and conclusions

1. The over-conservatism of the current AISC/LRFD methodfor designing CFT beam–columns comes from the followingreasons. The current method ignores much of the contributionof the concrete, since the nominal moment strength is computedbased only on the steel section. However, the majority of theover-conservativeness comes from the design method; sincethe current AISC/LRFD beam–column equations for CFTmembers are based on equivalent “steel” sections, there is aninherent discrepancy from the response of “composite” CFTs.

Y.-H. Choi et al. / Engineering Structures 28 (2006) 1586–1598 1595

(a) Furlong [2]. (b) Tomi and Sakino [32].

(c) Grauers [40]. (d) Varma [19].

(e) Fujimoto et al. [39].

Fig. 12. Verification of new design method.

2. A new method for designing CFT beam–columns isproposed to account for the contribution of the concrete. Thenew method idealizes the real response of CFTs with two linesintersecting at a point of maximum moment. The fundamentalidea of the modified P–M equations is a full composite actionand the maximum moment at the intersection point is greaterthan the nominal moment. Furthermore, the degree of increasein the moment varies depending on the contribution of theconcrete, which in turn varies with b/t and f ′

c/Fy . The newequation that can be applied for either circular or square CFTsis expressed as

Pu

φc Pn≥ β

Pu

φc Pn+ 1 − β

α

Mu

φb Mn≤ 1

Pu

φc Pn< β

1 − α

β

Pu

φc Pn+ Mu

φb Mn≤ 1.

3. For square CFT beam–columns with b/t less than 45subjected to single axis bending plus axial force, the normalizedmaximum moment (α) and the axial load ratio at the maximummoment (β) are proposed as:

α =(

0.5

√b

t− 1.5

)f ′c

Fy+ 0.95 (α ≤ 1.5)

β = 0.34

(√b

t− 2

)f ′c

Fy+ 0.08 (0.2 ≤ β ≤ 0.4).

4. The modification of current AISC/LRFD P–M equationscan be completed with additional research on rectangularshapes and circular shapes. Research on circular CFTs isespecially recommended, since they can typically exhibit evenbetter performance than square-type CFTs.

1596 Y.-H. Choi et al. / Engineering Structures 28 (2006) 1586–1598

Table 3Verification of new method

References Fy (MPa) f ′c (MPa) b (mm) t (mm) K L (mm) b/t Pexp (kN) Mexp (kN m) Mn (kN m) α β

McurMexp

MnewMexp

[32] 193 29 99 2.29 305 44 0 7 8 1.218 0.31 0.89 1.05193 46 99 2.29 305 44 76 10 8 1.376 0.40 0.64 0.86193 46 99 2.29 305 44 160 11 8 1.376 0.40 0.54 0.88193 46 99 2.29 305 44 191 11 8 1.376 0.40 0.49 0.93193 46 99 2.29 305 44 267 12 8 1.376 0.40 0.37 0.76193 44 99 2.29 305 44 329 9 8 1.359 0.40 0.34 0.69193 44 99 2.29 305 44 329 8 8 1.359 0.40 0.37 0.75303 26 99 2.29 305 44 0 11 11 1.104 0.21 0.90 1.03303 26 99 2.29 305 44 44 12 11 1.104 0.21 0.80 0.96338 26 99 2.29 305 45 93 13 12 1.093 0.20 0.81 1.01338 26 99 2.29 305 45 138 13 12 1.093 0.20 0.77 0.94290 26 99 2.29 305 45 182 12 11 1.116 0.22 0.59 0.76290 26 99 2.29 305 45 231 11 11 1.116 0.22 0.54 0.68290 26 99 2.29 305 45 276 9 11 1.116 0.22 0.50 0.63290 25 99 3.05 305 34 0 14 14 1.069 0.20 0.87 0.97290 25 99 3.05 305 34 49 15 14 1.069 0.20 0.82 0.96290 25 99 3.05 305 34 102 15 14 1.069 0.20 0.77 0.95290 25 99 3.05 305 33 151 14 14 1.069 0.20 0.74 0.88290 25 99 3.05 305 33 205 13 14 1.069 0.20 0.69 0.82290 25 99 3.05 305 33 254 12 14 1.069 0.20 0.62 0.74290 25 99 3.05 305 33 307 11 14 1.069 0.20 0.58 0.68283 22 99 4.32 305 24 0 18 18 1.023 0.20 0.91 0.98283 22 99 4.32 305 24 62 19 18 1.023 0.20 0.85 0.95283 22 99 4.32 305 24 125 19 18 1.023 0.20 0.84 0.97283 22 99 4.32 305 24 187 18 18 1.023 0.20 0.78 0.88283 23 99 4.32 305 24 249 17 18 1.027 0.20 0.70 0.80283 23 99 4.32 305 24 316 16 18 1.027 0.20 0.65 0.74290 23 99 4.32 305 23 374 14 18 1.026 0.20 0.60 0.68

[2] 483 45 127 4.83 914 26 1112 35 57 1.049 0.20 0.53 0.62483 45 127 4.83 914 26 667 41 57 1.049 0.20 0.88 1.03483 45 127 4.83 914 26 667 49 57 1.049 0.20 0.75 0.87483 45 127 4.83 914 26 445 51 57 1.049 0.20 0.89 1.04331 23 102 2.03 914 48 374 5 12 1.088 0.20 0.42 0.51331 23 102 2.03 914 48 374 5 12 1.088 0.20 0.42 0.51331 23 102 2.03 914 48 240 10 12 1.088 0.20 0.57 0.68331 23 102 2.03 914 48 89 12 12 1.088 0.20 0.85 1.07331 23 102 2.03 914 48 89 13 12 1.088 0.20 0.78 0.98

[39] 262 41 150 4.32 660 34 267 54 42 1.172 0.28 0.66 0.87262 41 147 4.32 660 34 823 39 41 1.170 0.28 0.44 0.64262 26 216 4.32 660 49 503 101 88 1.144 0.24 0.71 0.96262 26 213 4.32 660 49 1139 69 88 1.143 0.24 0.55 0.74262 41 216 4.32 660 49 578 115 91 1.263 0.35 0.65 0.93262 41 216 4.32 660 49 1028 103 91 1.263 0.35 0.57 0.98262 41 216 4.32 660 49 1370 83 91 1.263 0.35 0.54 0.93262 79 213 4.32 660 49 1450 146 94 1.500 0.40 0.44 0.94262 79 216 4.32 660 49 2015 121 95 1.500 0.40 0.41 0.90262 41 323 4.32 660 74 1477 297 215 1.387 0.40 0.57 0.93262 41 323 4.32 660 74 3305 201 215 1.387 0.40 0.39 0.80621 41 145 6.35 660 23 609 124 119 1.008 0.20 0.82 0.93621 41 145 6.35 660 23 1699 84 119 1.008 0.20 0.58 0.66621 26 211 6.35 660 33 1041 253 263 1.007 0.20 0.86 0.97621 26 211 6.35 660 33 2393 146 261 1.006 0.20 0.80 0.90621 41 211 6.35 660 33 859 257 268 1.041 0.20 0.91 1.08621 41 211 6.35 660 33 2091 214 268 1.041 0.20 0.75 0.87621 41 208 6.35 660 33 2696 164 265 1.041 0.20 0.72 0.83621 79 211 6.35 660 33 1486 299 279 1.124 0.24 0.76 1.02621 79 211 6.35 660 33 3394 206 279 1.124 0.24 0.62 0.81621 41 320 6.35 660 50 4043 408 651 1.085 0.20 0.90 1.09

Y.-H. Choi et al. / Engineering Structures 28 (2006) 1586–1598 1597

Table 3 (continued)

References Fy (MPa) f ′c (MPa) b (mm) t (mm) K L (mm) b/t Pexp (kN) Mexp (kN m) Mn (kN m) α β

McurMexp

MnewMexp

[40] 303 47 119 5.08 3200 24 609 12 34 1.097 0.23 0.42 0.53441 46 119 5.08 3200 24 698 14 48 1.050 0.20 0.43 0.50324 96 119 5.08 3200 24 712 14 38 1.228 0.37 0.50 0.87441 96 119 5.08 3200 24 832 17 50 1.157 0.30 0.38 0.56324 39 119 8.13 3200 15 738 15 52 1.003 0.20 0.65 0.73303 46 119 8.13 3200 15 770 15 49 1.017 0.20 0.52 0.58379 47 119 8.13 3200 15 872 18 61 1.004 0.20 0.48 0.54324 103 119 8.13 3200 15 818 16 55 1.089 0.28 0.79 1.07379 103 119 8.13 3200 15 1001 20 64 1.068 0.25 0.46 0.58379 39 119 8.13 3200 15 818 16 61 0.995 0.20 0.59 0.65379 93 119 8.13 3200 15 1032 21 63 1.058 0.24 0.35 0.43365 93 119 8.13 3200 15 961 19 61 1.061 0.24 0.47 0.59365 80 119 8.13 3200 15 1161 12 61 1.046 0.22 0.20 0.23393 96 119 8.13 3277 15 1041 21 66 1.056 0.23 0.32 0.40407 92 119 8.13 3200 15 1010 20 68 1.049 0.22 0.44 0.53393 92 119 8.13 3200 15 961 19 66 1.051 0.23 0.54 0.66379 31 119 8.13 1702 15 1081 22 60 0.986 0.20 0.67 0.73379 92 119 8.13 1702 15 1299 26 64 1.056 0.23 0.70 0.86379 33 249 8.13 3200 31 3398 68 296 1.063 0.20 0.67 0.80379 91 249 8.13 3200 31 5302 106 316 1.261 0.37 0.42 0.75

[19] 262 110 323 8.64 1499 38 2504 606 409 1.500 0.40 0.56 0.85262 110 323 8.64 1499 37 5009 543 410 1.500 0.40 0.47 1.04469 110 302 5.84 1499 52 2313 597 443 1.443 0.40 0.61 0.91469 110 302 5.84 1499 52 2544 629 443 1.443 0.40 0.57 0.88

[20] 389 47 152 4.43 1975 34 0 74 65 1.123 0.24 0.77 0.88389 43 152 4.43 2430 34 0 75 64 1.108 0.22 0.75 0.86389 41 152 4.43 3040 34 0 71 64 1.102 0.22 0.79 0.90434 47 152 8.95 1976 17 0 147 128 1.011 0.20 0.82 0.88

Table 4Summary of verification

AISC limitations Pu/Pn Count McurMexp

MnewMexp

Fy f ′c b/t Average Minimum Maximum Average Minimum Maximum

All All All All 87 0.63 0.20 0.91 0.81 0.23 1.09All All All 0.4 38 0.75 0.44 0.91 0.94 0.80 1.08Ok Ok Ok All 30 0.65 0.34 0.91 0.82 0.53 1.05Ok Ok Ok 0.4 17 0.76 0.49 0.91 0.93 0.80 1.05Ok Ok All 0.4 20 0.74 0.49 0.91 0.93 0.80 1.05Ok All Ok 0.4 18 0.75 0.49 0.91 0.92 0.80 1.05All Ok Ok 0.4 24 0.78 0.49 0.91 0.92 0.80 1.05

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