Proceedings of the

Annual Stability Conference

Structural Stability Research Council

Toronto, Canada, March 25-28, 2014

Analysis and Design of Noncompact and Slender

Concrete-filled Steel Tube (CFT) Beam-columns

Z. Lai1, A. H. Varma

2

Abstract

Concrete-filled steel tube beam-columns are categorized as compact, noncompact or slender

depending on the slenderness ratio (width-to-thickness b/t ratio) of the steel tube wall. Current

AISC Specification (AISC 360-10) uses the bilinear axial force-bending moment (P-M)

interaction curve for bare steel members to design noncompact and slender CFT beam-columns.

This interaction curve is shown to be over-conservative by comparisons with results from the

experimental database compiled by the authors. The behavior of noncompact and slender CFT

beam-columns depends on three major parameters: the tube wall slenderness ratio, axial load

ratio, and member length. To explicitly investigate the effects of these three parameters, detailed

3D finite element models (FEM) are developed and benchmarked using experimental test results.

Based on the parametric analysis results, revisions of the AISC 360-10 interaction curve for

designing noncompact and slender CFT beam-columns in are proposed.

1. Introduction

CFT members consist of rectangular or circular steel tubes filled with concrete. These composite

members optimize the best use of both construction materials, as compared to bare steel or

reinforced concrete structures. The steel tube provides confinement to the concrete infill, while

the concrete infill delays the local buckling of the steel tube. The behavior of CFT members

under axial compression, flexure, and combined axial compression and flexure can be more

efficient than that of bare steel or reinforced concrete members. Moreover, the steel tube serves

as formwork for placing the concrete, which expedites and facilitates construction while

reducing labor costs.

CFT members are categorized as compact, noncompact or slender depending on the slenderness

ratio (width-to-thickness b/t ratio) of the steel tube wall. AISC 360-10 specifies the slenderness

limits for demarcating the members, as shown in Table 1. These slenderness limits are proposed

by Varma and Zhang (2009), based on the research of Schilling (1965), Winter (1968), Tsuda et

al. (1996), Bradford et al. (1998, 2002), Leon (2007) and Ziemian (2010). Developments of the

slenderness limits are discussed in detail by Lai et al. (2014a, 2014b).

1 Ph.D. Candidate, Purdue University, School of Civil Eng., West Lafayette, IN. laiz@purdue.edu

2 Assoc. Prof., Purdue University, School of Civil Eng., West Lafayette, IN. ahvarma@purdue.edu

2

For a CFT column or beam, if the governing slenderness ratio is less than or equal to p, the

member is classified as compact; if the governing slenderness ratio is greater than p but less

than or equal to r, the member is classified as noncompact; if the governing slenderness ratio is

greater than r, the member is classified as slender. The slenderness ratio is also limited to a

maximum permitted value limit due to: (i) the lack of experimental data for CFTs with such

slender steel tubes, and (ii) potential issues with deflections and stresses in the slender tube walls

due to concrete casting pressures and other fabrication processes.

For a CFT beam-column, the member is classified as compact if the governing slenderness ratio

is less than or equal to p for columns or beams (whichever is smaller); the member is classified

as noncompact if the governing slenderness ratio is less than or equal to r for columns or beams

(whichever is smaller); if the governing slenderness ratio is greater than r for columns or beams

(whichever is smaller), the member is classified as slender. The maximum permitted slenderness

ratio is also limited to limit of columns or beams (whichever is smaller).

Table 1: Slenderness limit for CFT members

Loading Description of Element

Width-to-

Thickness

Ratio

λ p

Compact/

Noncompact

λ r

Noncompact/

Slender

λ limit

Maximum

Permitted

Steel tube walls of

Rectangular CFT

Members

b/t

Steel tube wall of

Circular CFT MembersD/t

Flanges of Rectangular

CFT Membersb/t

Webs of Rectangular

CFT Membersh/t

Steel tube wall of

Circular CFT MembersD/t

Axial

compression

Flexure

y

s

F

E26.2

y

s

F

E00.3

y

s

F

E00.5

y

s

F

E26.2

y

s

F

E00.3

y

s

F

E00.5

y

s

F

E15.0

y

s

F

E19.0

y

s

F

E31.0

y

s

F

E00.3

y

s

F

E70.5

y

s

F

E70.5

y

s

F

E09.0

y

s

F

E31.0

y

s

F

E31.0

As an innovative and efficient structural component, CFT members are used widely around the

world in various types of structures. In China, CFT members are used in more than three hundred

composite bridges. Fig.1 (a) shows a typical application of CFT members in a half-through arch

bridge. The chords, webs, and bracings of the four-pipe truss are all made of CFTs. CFT

members are also used as columns in composite moment or braced frames, for example in: (1) 3

Houston Center in Houston, USA, and (ii) Wuhan International Financial Center in Wuhan,

China. Fig.1 (b) shows a typical application of CFT members as mega columns in composite

braced frames. Both the CFT chord members in bridges and CFT columns in frames are often

subjected to combined axial compression and flexure, and therefore should be designed as beam-

columns.

For CFT beam-columns, existing research focuses mostly on the members with compact sections.

For example, Furlong (1967), Bridge (1976), Wang and Moore (1997), Cai (1991), Matsui et al.

(1997), Elremaily and Azizinamini (2002), Seo et al. (2002), Varma et al. (2002, 2004), Han and

3

CFT columns

Four-pipe truss

Yao (2003), Hardika and Gardner (2004). However, the research on noncompact and slender

CFT beam-columns is limited.

(a) (b)

Fig.1: Typical application of CFT members: (a) in a half-through arch bridge; and (b) in composite braced frames

Due to this lack of research, AISC 360-10 provides Eq. (1) and (2) for developing P-M

interaction curves and designing noncompact and slender CFT beam-columns. These equations

are based on those for steel columns, and do not account for the expected beneficial effects of

axial compression on the flexural capacity, which is quite conservative.

When 2.0nc

r

P

P

0.1

9

8

nb

r

nc

r

M

M

P

P

(1)

When 2.0nc

r

P

P

0.1

2

nb

r

nc

r

M

M

P

P

(2)

where Pr is the required axial compressive strength, Mr is the required flexural strength, Pn is the

nominal axial capacity, Mn is the nominal flexural capacity, and c is the resistance factor for

compression (c=0.75). The calculations of Pn and Mn for CFT members are specified in the

AISC 360-10.

In this paper, the experimental database of tests conducted on noncompact and slender CFT

beam-columns is presented and discussed first. The degree of conservatism of the AISC 360-10

beam-column design equations is evaluated by using them to predict the strength of the tests in

the database. Finite element models (FEMs) are then developed and benchmarked using

experimental test results. The benchmarked models are used to perform comprehensive analyses

that explicitly investigate the effects of slenderness ratio, axial load ratio, and member length on

the behavior of CFT beam-columns. Based on the analysis results, revisions of the AISC 360-10

beam-column interaction curve for designing noncompact and slender CFT beam-columns are

proposed.

2. Experimental database and comparisons with design equations

2.1 Experimental database

Several experimental databases have been developed for tests conducted on CFT members, for

example, Nishiyama et al. (2002), Kim (2005), Gourley et al. (2008), and Hajjar (2013). These

databases usually include tests on CFT columns, beams, beam-columns, connections, and

frames. The database compiled by Gourley et al. (2008) and Hajjar (2013) are the most

4

comprehensive, and include the material, geometric, and loading parameters, and brief

description of all tests. As part of the research by Lai et al. (2014a, 2014b), a comprehensive

database of tests conducted on noncompact and slender CFT members subjected to axial

compression, flexure, and combined axial compression and flexure was compiled. The database

included tests from the database of Gourley et al. (2008) and Hajjar (2013), and additional tests

from other databases and literature as applicable. A total of one hundred and eighty-seven tests

were included in the database, including eighty-eight column tests, forty-six beam tests, and

fifty-three beam-column tests. In this paper, only the part focusing on beam-column tests is

presented.

Seventeen rectangular and thirty-six circular noncompact and slender CFT beam-column tests

were compiled into the experimental database. Three types of loading schemes (Type-1, Type-2

and Type-3) were used in the beam-column tests, as shown in Fig.2. In Type-1 loading,

concentric axial load was applied first and kept constant, and then lateral loads were applied to

produce moment. In Type-2 loading, concentric axial loading was applied first and maintained

constant, and then end moments were applied incrementally. In Type-3 loading, eccentric axial

load was applied incrementally, so that both axial force and bending moment increased

proportionally. Type-1 and Type-2 loading are fundamentally the same, therefore they are called

Type-A loading hereafter. Type-3 loading is called Type-B loading hereafter. Table 2

summarizes the noncompact and slender CFT beam-column tests that were compiled into the

experimental database.

Fig.2: Typical loading schemes for CFT beam-column tests

Table 2(a) includes the length (L), width (B), depth (H), flange thickness (tf), web thickness (tw),

governing tube slenderness ratio (bi/tf and H/tw), and the slenderness coefficient (coeff ) obtained

by dividing the governing slenderness ratio by ys FE for rectangular CFT beam-columns.

Table 2(b) includes the length (L), diameter (D), tube thickness (t), tube slenderness ratio (D/t),

and the slenderness coefficient (coeff ) obtained by dividing the governing slenderness ratio by

Es/Fy for circular CFT beam-columns. These tables also include the measured steel yield stress

(Fy) and concrete strength (f’c) where reported by the researchers. The experimental axial load

capacity (Pexp) and moment capacity (Mexp) are included along with the nominal strength (Pn and

Mn).

5

Table 2(a): Noncompact and slender rectangular CFT beam-column tests

ER4-D-4-06 969 323 4.38 71.7 323 4.38 71.7 2.56 3.0 262.0 41.1 30.34 4520.1 3306.0 0.73 209.7 201.0 0.96

ER4-D-4-20 969 323 4.38 71.7 323 4.38 71.7 2.56 3.0 262.0 41.1 30.34 4520.1 1479.0 0.33 209.7 297.0 1.42

ER6-D-4-10 954 318 6.36 48.0 318 6.36 48.0 2.63 3.0 618.0 41.1 30.34 7725.2 4100.0 0.53 614.3 408.0 0.66

ER6-D-4-30 957 319 6.36 48.2 319 6.36 48.2 2.64 3.0 618.0 41.1 30.34 7752.8 1967.0 0.25 617.6 593.0 0.96

BR4-3-10-02 600 200 3.17 61.1 200 3.17 61.1 2.41 3.0 310.0 119 51.62 4330.3 1049.0 0.24 72.0 136.0 1.89

BR4-3-10-04-1 600 200 3.17 61.1 200 3.17 61.1 2.41 3.0 310.0 119 51.62 4330.3 2108.0 0.49 72.0 136.0 1.89

BR4-3-10-04-2 600 200 3.17 61.1 200 3.17 61.1 2.41 3.0 310.0 119 51.62 4330.3 2108.0 0.49 72.0 139.0 1.93

BR8-3-10-02 600 200 3.09 62.7 200 3.09 62.7 3.92 3.0 781.0 119 51.62 4242.7 1294.0 0.30 144.2 195.0 1.35

BR8-3-10-04 600 200 3.09 62.7 200 3.09 62.7 3.92 3.0 781.0 119 51.62 4242.7 2569.0 0.61 144.2 143.0 0.99

BRA4-2-5-02 600 200 2.04 96.0 200 2.04 96.0 3.42 3.0 253.0 47.6 32.65 1594.2 380.0 0.24 31.9 62.7 1.96

BRA4-2-5-04 600 200 2.04 96.0 200 2.04 96.0 3.42 3.0 253.0 47.6 32.65 1594.2 761.0 0.48 31.9 69.1 2.16

BRA4-6-5-04-C 600 200 2.04 96.0 200 2.04 96.0 3.42 3.0 253.0 47.6 32.65 1594.2 380.0 0.24 31.9 63.5 1.99

BRA4-2-5-04-C 600 200 2.04 96.0 200 2.04 96.0 3.42 3.0 253.0 47.6 32.65 1594.2 761.0 0.48 31.9 71.5 2.24

HSS16 630 210 5.00 40.0 210 5.00 40.0 2.45 3.0 750.0 32.0 26.77 4077.3 3106.0 0.76 249.1 77.7 0.31

HSS17 630 210 5.00 40.0 210 5.00 40.0 2.45 3.0 750.0 32.0 26.77 4077.3 2617.0 0.64 249.1 130.9 0.53

SH-C210 2416 220 5.00 42.0 220 5.00 42.0 2.59 11.0 761.0 20.0 21.16 3539.3 2939.0 0.83 265.5 58.8 0.22

SH-C260 2817 270 5.00 52.0 270 5.00 52.0 3.21 10.4 761.0 20.0 21.16 4122.9 3062.0 0.74 368.3 76.6 0.21

Type-A

Mursi and Uy

(2004)Type-B

Uy, B. (2001) Type-B

Nakahara and

Sakino (1998,

2000a)

Ec

(GPa)

Pn

(kN)

Pexp

(kN)

tf

(mm)Pexp/Pn

Mn

(kN-m)

Mexp

(kN-m)Mexp/Mn

Morino et al.

(1996)Type-B

Fy

(MPa)

f'c

(MPa)bi/tf

H

(mm)

tw

(mm)H/tw λ coeff L/HReference Load Type Specimen ID

L

(mm)

B

(mm)

Table 2(b): Noncompact and slender circular CFT beam-column tests

C04F5M 2000 300 4.25 70.6 0.15 6.7 438.0 66.2 38.51 5262.6 3069.5 0.58 197.8 329.0 1.66

C06F3M 2000 300 5.83 51.5 0.11 6.7 420.0 64.3 37.95 5706.9 1932.0 0.34 258.2 348.0 1.35

C06F5M 2000 300 5.83 51.5 0.11 6.7 420.0 61.9 37.24 5583.1 3216.6 0.58 257.2 326.0 1.27

C06F5MA 2000 300 5.65 53.1 0.11 6.7 403.0 61.9 37.24 5445.5 2981.0 0.55 241.1 318.0 1.32

C06F7M 2000 300 5.65 53.1 0.11 6.7 419.0 66.2 38.51 5736.2 4560.0 0.79 250.5 256.0 1.02

C06F3C 2000 300 6.23 48.2 0.10 6.7 436.0 66.2 38.51 6021.6 1961.0 0.33 283.7 418.7 1.48

C06F5C 2000 300 5.65 53.1 0.11 6.7 419.0 66.2 38.51 5736.2 3255.8 0.57 250.5 402.0 1.60

C06S5M 2000 300 6.16 48.7 0.10 6.7 406.0 66.2 38.51 5854.3 3285.0 0.56 264.2 405.0 1.53

C06S5C 2000 300 5.70 52.6 0.11 6.7 431.0 66.2 38.51 5806.1 3305.0 0.57 258.5 391.0 1.51

C06SVC 2000 300 5.70 52.6 0.11 6.7 431.0 66.2 38.51 5806.1 4628.7 0.80 258.5 327.0 1.26

C06SVC 2000 300 5.70 52.6 0.11 6.7 431.0 66.2 38.51 5806.1 1981.0 0.34 258.5 416.0 1.61

DC04S5M 2000 165 4.35 38.0 0.11 12.1 588.0 66.2 38.51 1856.1 1285.0 0.69 76.0 99.0 1.30

DC04S5C 2000 165 4.35 38.0 0.11 12.1 588.0 66.2 38.51 1856.1 1285.0 0.69 76.0 99.0 1.30

BP11 2120 152 1.70 89.4 0.15 13.9 328.0 92.0 45.39 1341.1 470.0 0.35 16.6 29.7 1.79

BP12 2120 152 1.70 89.4 0.15 13.9 328.0 92.0 45.39 1341.1 570.0 0.43 16.6 32.1 1.93

BP13 2120 152 1.70 89.4 0.15 13.9 328.0 92.0 45.39 1341.1 670.0 0.50 16.6 28.5 1.71

BP14 2120 152 1.70 89.4 0.15 13.9 328.0 92.0 45.39 1341.1 820.0 0.61 16.6 29.2 1.76

BP15 2120 152 1.70 89.4 0.15 13.9 328.0 92.0 45.39 1341.1 970.0 0.72 16.6 30.5 1.83

BP17 2120 152 1.70 89.4 0.15 13.9 328.0 92.0 45.39 1341.1 270.0 0.20 16.6 30.1 1.81

BP18 2120 152 1.70 89.4 0.15 13.9 328.0 92.0 45.39 1341.1 270.0 0.20 16.6 30.8 1.85

BP19 2120 152 1.70 89.4 0.15 13.9 328.0 92.0 45.39 1341.1 670.0 0.50 16.6 34.8 2.09

BP20 1071 152 1.70 89.4 0.15 7.0 328.0 92.0 45.39 1655.1 1273.0 0.77 16.6 21.4 1.29

BP21 1071 152 1.70 89.4 0.15 7.0 328.0 92.0 45.39 1655.1 1451.0 0.88 16.6 13.8 0.83

BP22 1071 152 1.70 89.4 0.15 7.0 328.0 92.0 45.39 1655.1 1309.0 0.79 16.6 15.9 0.96

O'shea and Bridge

(1997c)Type-B S12E210B 662 190 1.13 168.1 0.18 3.5 185.7 113.9 32.30 2731.1 2438.0 0.89 10.8 20.7 1.92

S12E250A 664 190 1.13 168.1 0.18 3.5 185.7 41.0 17.81 1073.4 1229.0 1.14 10.2 10.5 1.02

S10E250A 662 190 0.86 220.9 0.26 3.5 210.7 41.0 17.81 886.6 1219.0 1.37 8.7 9.0 1.04

S16E150B 662 190 1.52 125.0 0.18 3.5 306.1 48.3 21.21 1270.2 1260.0 0.99 21.0 19.5 0.93

S12E150A 664 190 1.13 168.1 0.18 3.5 185.7 41.0 17.81 1073.4 1023.0 0.95 10.2 19.3 1.90

S10E150A 663 190 0.86 220.9 0.26 3.5 210.7 41.0 17.81 886.5 1017.0 1.15 8.7 14.1 1.63

S10E280B 666 190 0.86 220.9 0.26 3.5 210.7 74.4 27.58 1521.0 1910.0 1.26 9.0 16.4 1.82

S16E180A 664 190 1.52 125.0 0.18 3.5 306.1 80.2 28.45 1921.6 1925.0 1.00 21.9 27.5 1.26

S10E180B 665 190 0.86 220.9 0.26 3.5 210.7 74.7 27.58 1526.6 1532.0 1.00 9.1 27.4 3.03

S10E210B 661 190 0.86 220.9 0.26 3.5 210.7 112.7 31.47 2238.7 2112.0 0.94 9.3 8.5 0.91

S16E110B 661 190 1.52 125.0 0.18 3.5 306.1 112.7 31.47 2578.3 2420.0 0.94 22.5 31.2 1.39

S12E110B 662 190 1.13 168.1 0.18 3.5 185.7 112.7 31.47 2702.2 1925.0 0.71 10.8 32.9 3.05

O'shea and Bridge

(2000)Type-B

Pn

(kN)

Pexp

(kN)Pexp/PnReference

Load

TypeSpecimen ID

L

(mm)

D

(mm)

t

(mm)

Ichinohe et al. (1991) Type-A

Prion and Boehme

(1993)

Type-A

Type-B

Mn

(kN-m)

Mexp

(kN-m)Mexp/MnD/t λ coeff L/D

Fy

(MPa)

f'c

(MPa)

Ec

(GPa)

2.2 Comparisons and discussions

The AISC beam-column design equations (Eq. 1 and Eq. 2) were used to predict the strength of

the fifty-three test specimens in the database. Comparisons of the results are shown in Fig.3. In

Fig. 3(a) and Fig. 3(b), the comparisons are shown separately for rectangular CFT beam-columns

with normal strength and high strength steel (Fy>=525 Mpa, as specified in the AISC 360-10).

6

As shown, the design equations are very conservative for all specimens with normal strength

steel, and slender specimens (with coeff = 3.92) with high strength steel. For example, the ratio

of experimental moment capacity (Mexp) to nominal moment capacity (Mn) is 2.24 for the

specimen with coeff = 3.42 and axial load ratio of 0.48. For other specimens with high strength

steel, the design equations match the test results closely. Fig.3 (c) shows that the design

equations are conservative for all circular beam-column specimens. The maximum Mexp/Mn ratio

is 3.05 for some specimens with nominal axial load ratio of 0.7.

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.50 1.00 1.50 2.00 2.50

Pex

p/P

n

Mexp/Mn

2.41

2.56

2.60

3.420.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.50 1.00 1.50 2.00 2.50

Pex

p/P

nMexp/Mn

2.45

2.59

2.63

2.64

3.21

3.92

(a) Rectangular CFT beam-columns (b) Rectangular CFT beam-columns

with normal strength steel with high strength steel

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

0.00 1.00 2.00 3.00 4.00

Pex

p/P

n

Mexp/Mn

0.1

0.11

0.15

0.18

0.26

0.11, high

strength steel

(c) Circular CFT beam-columns

Fig.3: Comparisons of the AISC 360-10 interaction curve and experimental results

The confinement factor (as will be discussed later in Eq.3) of CFT beam-columns with high

strength steel is greater than that of CFT beam-columns with normal strength steel. This is the

reason that design equations are less conservative for CFT beam-columns with high strength

steel. This paper is limited to CFT beam-columns with normal strength steel, since AISC does

not permit the use of CFT members with high strength steel (Fy>=525 Mpa) due to the lack of

comprehensive research.

For CFT members with normal strength steel, there are two primary reasons for the over-

conservatism of the AISC beam-column design equations. The first reason is that the AISC 360-

10 is conservative in evaluating the axial and flexural capacity of noncompact and slender CFT

columns and beams. As mentioned in Section 2.1, eighty-eight column tests and forty-six beam

tests on noncompact and slender CFT members were included in the database. The results of

these tests are compared with the Pn and Mn calculated according to AISC 360-10, as shown in

Fig.4. The ordinate represents the ratio of experiment to calculated value (Pexp/Pn or Mexp/Mn),

7

while the abscissa represents the normalized nondimensional slenderness coefficient (coeff ). As

shown, the Pexp/Pn and Mexp/Mn ratios are greater than 1.0 for all the specimens. The maximum

Pexp/Pn ratio is 1.74, and the maximum Mexp/Mn ratio is 1.72. The comparisons indicate that the

AISC design equations for calculating the strength of noncompact and slender columns and

beams are very conservative.

The second reason is that AISC 360-10 uses the bilinear interaction curve for bare steel members

to design CFT beam-columns. Similar to steel beam-columns, the slenderness ratio governs the

local buckling behavior of the steel tubes. Similar to steel or reinforced beam columns, the

behavior of CFT beam-columns is dependent on the axial load ratio and member length. The

axial load ratio (=P/Pn, where P and Pn is the applied axial load and nominal axial capacity

of the CFT columns, respectively) governs the axial load-bending moment (P-M) interaction

behavior of the CFT beam-column. When the axial load ratio (α) is low, i.e., below the balance

point on the P-M interaction curve, flexural behavior dominates the response. When α is high,

i.e., above the balance point, axial compression behavior dominates the response. The member

length determines the effects of secondary moments and imperfections.

However, the effects of slenderness ratio, axial load ratio and member length on the behavior of

CFT beam-columns are different than that of bare steel beam-columns or reinforced concrete

columns. The shape of the P-M interaction curve for CFT beam-columns is influenced by the

confinement factor ξ defined in Eq. 3, where As and Ac are the total areas of the steel tube and

concrete infill, respectively. CFT beam-columns with larger ξ values have P-M interaction

curves that are more similar to those for steel columns, while beam-columns with smaller ξ have

P-M interaction curves that are more similar to those for reinforced concrete columns, as shown

in Fig. 5.

'cc

ys

fA

FA (3)

Therefore, two tasks should be completed in order to improve the AISC 360-10 design equations

for noncompact and slender CFT beam-columns. The first task is to improve the design

equations for calculating Pn and Mn, and the second task is to improve the bilinear interaction

diagram. The first task is discussed in detail by the authors in separate artiles. This paper focuses

on the second task.

Attempts are made by several researchers to improve the interaction curve for CFT beam-

columns, namely, Hajjar and Gourley (1996), Choi (2004) and Han (2007). However, the

findings from those researchers are based on the research on CFT members with compact

sections. Therefore they may not be applicable to noncompact and slender CFT members. In

order to complete the second task, the effects of slenderness ratio, axial load ratio and member

length on the behavior of noncompact and slender CFT beam-columns, as well as the effect of

the confinement factor on the shape of the interaction curve should be investigated explicitly.

This is accomplished in this paper by performing comprehensive finite element analysis using

ABAQUS (6.12).

8

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

1.80

2.00

2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.50

Pex

p/P

n

b/t(Fy/Es)0.5

NONCOMPACT SLENDER

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

1.80

0.10 0.15 0.20 0.25 0.30 0.35

Pexp/P

n

D/t(Fy/Es)

NO

NC

OM

PA

CT

SLENDER

(a) (b)

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

1.80

2.00

2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.50

Mexp/M

n

b/t(Fy/Es)0.5

NO

NC

OM

PA

CT

SLENDER

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

1.80

0.09 0.15 0.20 0.26 0.31M

exp/M

n

D/t(Fy/Es)

NONCOMPACT

(c) (d)

Fig.4: Comparisons of the calculated (based on AISC 360-10) and experimental results for noncompact and slender:

(a) rectangular CFT columns; (b) circular CFT columns; (c) rectangular CFT beams; and (d) circular CFT beams

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2

Interaction curve for

steel columns

Interaction curve for reinforced

concrete columns

increases

Fig.5: Effect of the confinement factor on the shape of CFT beam-column interaction curve

3. Development of FEM

In this part, detailed 3D finite element models (FEM) are developed and benchmarked using

experimental test results from the database presented in Section 2. The FEM accounts for steel

plasticity and local buckling, concrete cracking and compression plasticity, geometric

imperfections, contact between steel tube and concrete infill, as well as interaction between local

buckling and global column buckling.

3.1 FEM details

3.1.1 Material model

The steel material behavior was defined using Von Mises yield surface, associated flow rule, and

kinematic hardening. An idealized bilinear curve as shown in Fig.6 (a) was used to specify the

9

steel stress-strain behavior. Steel yield stress as listed in Table2 were used for each specimen.

The elastic modulus was assumed to be 200 Gpa, and the tangent modulus Et was assumed to be

Es/100.

The concrete material was modeled using the damaged plasticity (CDP) material model

developed by Lee and Fenves (1998). This model accounts for multiaxial behavior using a

special compression yield surface developed earlier by Lubliner et al. (1998) and modified by

Lee and Fenves (1998) to account for different evolution of strength in tension and compression.

It is well known that concrete has non-associated flow behavior in compression (Chen and Han

1995). The CDP model accounts for this non-associated flow using the Drucker-Prager

hyperbolic function as the flow potential G, a dilation angle ψ, and an eccentricity ratio e.

Increasing the dilation angle will result in larger volumetric dilation of the concrete, and

potentially better confinement, and resulting increase in strength and ductility. The behavior of

confined concrete has a significant effect on the behavior of CFT columns and beam-columns.

The value of the dilation angle is important from the perspective of modeling. For rectangular

CFT beam-columns, the dilation angle was defined as 15o. For circular CFT beam-columns,

extensive concrete dilation occurs in the compressive region. Changing the dilation values alone

in the FEM analysis was incapable of providing sufficient dilation. Instead, different

compressive stress-strain curve was used, as will be discussed later. The dilation angle was also

assumed to be 15o. The selected value of the dilation angle and concrete stress-strain curves are

reasonable for all specimens, as shown later in the comparisons of the FEM and experimental

results.

Other parameters required to define the plasticity are: the eccentricity ratio e, the ratio of biaxial

compressive strength to uniaxial strength f’bc/f’c, and the ratio of compressive to tensile

meridians of the yield surface in Π (deviatoric stress) space Kc. The default value of 0.1 was

specified for the eccentricity ratio. The default value of e indicates that the dilation angle

converges to ψ reasonably quickly with increasing value of compression dominant pressure (p).

The ratio of the biaxial-to-uniaxial compressive strength is assumed to be equal to 1.16 based on

Kupfer and Gerstle (1971). The ratio of the tensile-to-compressive meridian is assumed to be

equal to 0.67 based on Chen and Han (1995).

Two types of stress-strain curve were used to specify the compressive behavior of the concrete

infill, depending on the concrete strength. According to AISC 360-10, concrete with the

compressive strength f’c greater than 68.5 MPa is defined as high strength concrete. Under axial

compression, high strength concrete develops the compressive stress linearly up to f’c. There is

no significant dilation before the compressive strength is reached. Therefore, the empirical

stress-strain curve proposed by Popovics (1973) was used to define the compressive behavior of

the concrete infill for CFT beam-columns with high strength concrete, as shown in Fig.6 (b). For

CFT beam-columns filled with normal strength concrete, the concrete infill has significant

volumetric significant amount of dilation after the compressive stress exceeds 0.70f’c. Thus,

elastic perfectly plastic curve was used.

The tension behavior was specified using a stress-crack opening displacement curve that is based

on fracture energy and empirical models, as proposed by Wittmann (2002), as shown in Fig.6(c).

10

0

50

100

150

200

250

300

350

400

450

0 0.005 0.01 0.015 0.02

0

5

10

15

20

25

30

35

40

45

0 0.001 0.002 0.003 0.004 0.005 0.006

(a) (b)

Ten

sile

str

ess,

Mpa

Crack opening displacement, m

0

0.5

1

1.5

2

2.5

0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006

(c)

Fig.6: (a) idealized bilinear curve for the steel tube; (b) compressive stress-strain curve for concrete (40 Mpa); and (c)

tensile stress-crack opening displacement curve for concrete (40 Mpa)

3.1.2 Contact

General contact was used to simulate the normal and tangential interaction between concrete

infill and steel tube. For interaction in the normal direction, the hard contact pressure-overclosure

relationship with penalty constraint method was used. This enforced no penetration between steel

tube wall and concrete infill. For interaction in the tangential direction, penalty friction

formulation was used, with a Coulomb friction coefficient of 0.55, and maximum interfacial

shear stress of 0.41 Mpa (60 psi, as suggested by AISC 360-10). There was no additional bond

(adhesive or chemical) between the steel tube and the concrete infill in the model. The steel tube

and concrete infill are free to slip relative to each other if the applied interfacial shear stress (app)

is greater than 0.55 times the contact pressure (p).

3.1.3 Geometric imperfection, boundary conditions, element type, and analysis method

The shape of the geometric imperfection was developed by conducting eigenvalue buckling

analysis. The first mode shape as shown in Fig.7 (a) with the magnitude of 0.1 t (thickness of the

steel tube) was incorporated into the model as the imperfection. The boundary conditions and

constraints used for the FEM were designed to simulate those achieved in the experiments, i.e.,

by using coupling constrains as shown in Fig.7 (b).

The steel tube was modeled using a fine mesh of 4-node S4R shell elements with six degrees of

freedom per node, and reduced integration in the plane of the elements. These elements have a

11

general thick-shell formulation that reduces mathematically to thin-shell (discrete Kirchhoff)

behavior as the shell thickness becomes small. Transverse shear deformation is allowed in these

elements. Three section points through the thickness were used to compute the stresses and

strains through the thickness. Gauss quadrature was used to calculate the cross-sectional

behavior of the shell element. The concrete infill was modeled using eight-node brick (solid)

elements with reduced integration (C3D8R).This is a computational effective element for

modeling the concrete infill.

(a) (b)

Fig.7: (a) first mode imperfection from eigenvalue buckling analysis; and (b) example of boundary conditions,

couplings and constraints, and mesh in the FEM

The fracture behavior of concrete in tension makes it impossible to obtain converged results in

standard (predictor-corrector) analysis methods like full Newton or modified Newton-Raphson.

Even the arc-length based techniques like modified-Riks methods developed to predict the

behavior of structures undergoing large deformations and instability cannot provide converged

results. Implicit dynamic analysis based analysis methods also become unstable and cannot

provide results after significant cracking. Therefore, the explicit dynamic analysis method was

used. The primary advantage and reason for using this method is that it can find results close or

even slightly beyond failure, particularly when brittle materials (like concrete in tension) and

failures are involved. The explicit dynamic analysis method was used to perform quasi-static

analyses simulating the experiments.

3.2 Benchmarking the FEM

The developed FEM was used to model the experimental tests in the database. The authors

considered modeling all the tested specimens. However, this was not possible because: 1) some

specimens were tested under cyclic loading (which is beyond the scope of this paper); or 2) some

key information was not reported in the papers (i.e., specimen length, loading protocol, boundary

conditions, etc.) for some specimens. As a result, thirty-nine beam-column specimens were

modeled.

Comparisons of the strength from the FEM analysis and experimental tests are shown in Fig.8,

with ordinate being the ratio of experiment to FEM value, and abscissa being the slenderness

coefficient. For CFT beam-columns with Type-A loading where the axial load was kept constant,

only comparisons of the flexural strength were required, as shown in Fig.8 (a) and Fig. 8 (c). The

FEM flexural capacity was defined as the maximum moment value in the analysis. For CFT

beam-columns with Type-B loading, comparisons of both flexural and axial capacity were

shown. In this case, flexural capacity was defined as the moment value corresponding to

12

maximum applied load, including second order effect; and the axial capacity was defined as the

maximum applied axial load in the FEM analysis. Fig.9 shows the comparisons of moment-

curvature or moment-midspan deflection curve for selected beam-column tests. As shown in

Fig.8 and Fig.9, the finite element model is able to predict behavior of rectangular and circular

CFT beam-columns quite reasonably.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

2 2.5 3 3.5 4 4.5 5

Mexp/M

FE

M

b/t(Fy/Es)0.5

Type A

Type B

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

2.00 2.50 3.00 3.50

Pex

p/P

FE

M

b/t(Fy/Es)0.5

Type B

(a) Comparisons of the flexural strength of rectangular (b) Comparisons of the axial compressive strength of

beam-columns, Type A and Type B loading rectangular beam-columns, Type B loading

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

0.09 0.15 0.20 0.26 0.31

Mex

p/M

FE

M

D/t(Fy/Es)

Type A

Type B0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

0.100 0.150 0.200 0.250 0.300 0.350

Pexp/P

FE

M

D/t(Fy/Es)

Type B

(c) Comparisons of the flexural strength of circular (d) Comparisons of the axial compressive strength of

beam-columns, Type A and Type B loading circular beam-columns, Type B loading

Fig.8: Comparisons of the strength for CFT beam-columns from the FEM analysis and experimental tests

0

10

20

30

40

50

60

70

0 0.5 1 1.5 2 2.5 3 3.5

Mom

ent,

kN

-m

Curvature (10-2 rad)

FEM

EXP

0

10

20

30

40

50

60

70

80

0 1 2 3 4

Mom

ent,

kN

-m

Curvature (10-2 rad)

FEM

EXP

(a) BRA4-2-5-02 (b) BRA4-2-5-04

Fig.9: Comparisons of moment-curvature or moment rotation curves for selected beam-column tests from FEM and

experimental tests

13

0

50

100

150

200

250

300

350

400

450

0 0.1 0.2 0.3 0.4 0.5

Mom

ent,

kN

-m

Curvature, 1/m

FEM

EXP

0

50

100

150

200

250

300

350

400

0 0.1 0.2 0.3 0.4 0.5

Mom

ent,

kN

-m

Curvature, 1/m

FEM

EXP

(c) C06F3M (d) C06F5M

0

50

100

150

200

250

300

350

400

0 0.1 0.2 0.3 0.4 0.5

Mom

ent,

kN

-m

Curvature, 1/m

FEM

EXP

0

50

100

150

200

250

300

350

400

450

0 2 4 6 8 10 12

Mom

ent,

kN

-m

Rotation, %

FEM

EXP

(e) C06F7M (f) C06S5M

Fig.9: Continued

4. Parametric studies

The steel tube slenderness ratio and axial load ratio determine the cross section strength of CFT

beam-columns. These two parameters combined with the member length control the strength of a

CFT beam-column. In this part, the effects of these parameters are investigated by conducting

comprehensive analyses using the benchmarked FEM. Prototype specimens from the database

were selected first; then parametric studies were performed by changing the thickness, applied

axial load, and member length of the prototype specimens. For rectangular CFT beam-columns,

the test specimen BRA4-2-5-02 by Nakahara and Sakino (1998, 2000a) was selected as the

prototype specimen. The steel yield strength Fy and the concrete compressive strength f’c of this

specimen is 253 Mpa and 47.6 Mpa, respectively. Type-2 loading was used to load the specimen.

For circular CFT beam-columns, the test specimen C06F3M by Ichinohe et al. (1991) was

selected as the prototype specimen. For this specimen, the steel yield strength Fy is 420 Mpa and

the concrete compressive strength f’c is 64.3 Mpa. Type-1 loading was used to load the specimen.

Details of these two specimens are shown in Table 2.

4.1 Effect of slenderness ratio and axial load ratio

Seven different slenderness ratios were selected for both prototype specimens. For specimen

BRA4-2-05-02, the slenderness coefficients are 2.49, 2.68, 2.90, 3.17, 3.87, 4.36 and 4.98. For

specimen C06F3M, the slenderness coefficients are 0.11, 0.16, 0.20, 0.22, 0.24, 0.28 and 0.31.

These slenderness coefficients cover the whole slenderness range for noncompact and slender

sections. The designated slenderness ratio was implemented by changing the tube wall thickness.

14

For every beam-column with the same slenderness ratio, analyses with different axial loads ratios

were conducted. Beam-columns with axial load ratio of 0 were subjected to bending moments

only to get the flexural capacity Mf. Beam-columns with axial load ratio of 1.0 were subjected to

axial compressive force only to get the axial capacity Pf. The applied axial force was calculated

as the product of the axial load ratio and the nominal axial strength Pn (calculated by the AISC

design equations). Therefore, the designated axial load ratios were nominal values, since Pn may

not be the same as actual axial strength Pf. For specimen BRA4-2-05-02, five nominal axial load

ratios were used (0, 0.2, 0.4, 0.6 and 1.0). For specimen C06F3M, six nominal axial load ratios

were used (0, 0.2, 0.4, 0.6, 0.8 and 1.0). As a result, thirty-five rectangular and forty-two circular

beam-column analyses were performed.

Fig.10 (a) and Fig.10 (b) shows the effect of steel tube slenderness ratio (with axial load ratio α =

0.2) on the behavior of rectangular and circular CFT beam-columns, respectively. As the steel

tube thickness decreases, the stiffness and strength of the CFT beam-columns decrease also.

This conclusion also applies to CFT beam-columns with other axial load ratios. The

corresponding figures are not shown in this paper for brevity.

(a) (b)

Fig.10: Effect of steel tube slenderness ratio (with α = 0.2) on the behavior of CFT beam-columns: (a) rectangular;

and (b) circular

Fig.11 (a) and Fig.11 (b) shows the effect of axial load ratio on the behavior of rectangular and

circular CFT beam-columns, respectively. The thickness for the rectangular and circular beam-

columns was 2.8mm and 5.7mm, respectively. When the axial load ratio is less than 0.4, the

stiffness and strength increase as axial load ratio increases. When the axial load ratio is greater

than 0.4, the stiffness and strength decreases as axial load ratio increases. The stiffness and

strength of the members with axial load ratio of 0.4 are greater than the members with other axial

load ratios. Also, the ductility decreases with increasing axial load ratio for rectangular beam-

columns (i.e., the curve drops more rapidly after failure occurs). These conclusions are also

applicable to CFT beam-columns with other thickness. Again, the corresponding figures are not

shown in this paper for brevity.

Fig.12 shows the comparisons of the analysis results with interaction curves calculated using

Eq.1 and Eq.2. These comparisons indicate that the axial and flexural strength of noncompact

and slender CFT members are overestimated by the AISC 360-10, and that the AISC interaction

curve is too conservative. The degree of conservatism increases as slenderness ratio increases.

Thickness decreases from 2.8 mm

( 49.2coeff ) to 1.4 mm ( 98.4coeff ) Thickness decreases from 5.7 mm

( 11.0coeff ) to 2.0 mm ( 31.0coeff )

15

(a) (b)

Fig.11: Effect of axial load ratio on the behavior of CFT beam-columns: (a) rectangular; and (b) circular

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0.000 1.000 2.000 3.000

PF

EM

/Pn

MFEM/Mn

2.49

2.68

2.9

3.17

3.87

4.36

4.98

AISC 360-10 interaction curve

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 0.5 1.0 1.5 2.0 2.5

PF

EM

/Pn

MFEM/Mn

0.11

0.16

0.2

0.22

0.24

0.28

0.31

AISC 360-10 interaction curve

(a) (b)

Fig.12: Comparisons of the analysis results with AISC 360-10 interaction curve for CFT beam-columns: (a)

rectangular; and (b) circular

(a) (b)

Fig.13: Effect of length on the behavior of CFT beam-columns: (a) rectangular; and (b) circular

4.2 Effect of member length

The member length determines the effect of second-order moments on the behavior of beam-

columns. This effect is more significant for CFT members with greater slenderness ratio.

Therefore, it is investigated for the rectangular and circular prototype beam-columns with the

minimum permitted thickness of 1.4 mm (coeff = 4.98) and 2.0 mm (coeff = 0.31), respectively.

Three different length-to-depth ratios (L/D) were used for both rectangular (L/B=3.0, 10.0, and

20.0) and circular members (L/D=6.7, 13.3, and 20.0).

Fig.13 shows the moment-curvature response for members with different length. As shown, the

stiffness of both rectangular and circular CFT beam-columns is not influenced by the length. For

16

rectangular members, the strength is not influenced by the length; however, failure occurs earlier

as length increases due to the earlier occurrence of local buckling. For circular members, the

strength increases slightly as length increases due to the steel strain hardening (local buckling did

not occur in those circular beam-columns). The effects of length for CFT members with other

axial load ratios (0.2, 0.4, and 0.6) are similar to the behavior shown in Fig.13.

The flexural capacity of CFT members without any axial load applied (α = 0) is not influenced

by the length. The axial capacity of CFT members without any moment applied (α = 1.0)

decreases as length increases. For example, the axial capacity for the rectangular members with

L/B=3.0, 10.0 and 20.0 is 2153.1 kN, 2020.2 kN and 1986.5 kN, respectively; and the axial

capacity for the circular members with L/D=6.7, 13.3 and 20.0 is 4225.2 kN, 4107.2 kN and

3825.6 kN, respectively. Fig.14 shows the comparisons of the analysis results along with AISC

360-10 interaction curve. As shown, the AISC interaction curve is too conservative.

(a) (b)

Fig.14: Comparisons of the analysis results with AISC 360-10 interaction curve for CFT beam-columns with

different length: (a) rectangular; and (b) circular

5. Proposed interaction equations

As discussed in Section 2.2, the over-conservatism of the AISC beam-column equations for

designing noncompact and slender rectangular and circular CFT beam columns is due to: (i) the

conservative estimation of the axial and flexural strength (Pn and Mn) by the AISC 360-10 design

equations; and (ii) the bilinear interaction curve which is the same as that used for steel beam-

columns. The primary focus of this paper is to improve the bilinear interaction curve. Therefore,

the conservatism of the interaction curve due to (i) were eliminated by using the actual axial and

flexural strength (Pf and Mf) from the finite element analysis to normalize the results from the

parametric studies shown in Section 4, as shown in Fig.15 and Fig.16.

In Fig.15, the results are identified by the confinement factor ξ instead of the slenderness ratio.

This is because the confinement factor includes the effects of both slenderness ratio and material

strength ratio (Fy/f’c), and it determines the shape of the interaction curve for CFT beam-

columns. Fig.16 shows that the shape of interaction curves is not influenced significantly by the

member length up to length-to-depth ratio of 20.0 (which is good for practical design). Therefore,

the interaction curve could be improved by focusing on the effect of ξ only.

Several design methods could be used to improve the bilinear interaction curve used by AISC

360-10, as shown in Fig.17. Method A uses polynomial equation to represent the shape of the P-

M curve defined by analysis and tests. Method B simplifies Method A by using a trilinear curve

17

(Line ACDB), which is similar to the approach discussed in the commentary of AISC 360-10,

Section I5. Method C further simplifies Method B by using a bilinear curve (Line ADB). Method

C is recommended because: i) it captures the basic P-M behavior of noncompact and slender

CFT beam-columns; and ii) the bilinear form of Method C is similar to the current AISC 360-10

interaction curve, and is convenient for design. For Method C, Point A is the axial strength,

Point B is the flexural strength, and Point D corresponds to the largest increase in moment

capacity, i.e., MFEM/Mf ratio.

AISC 360-10 interaction curve

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

PF

EM

/Pf

MFEM/Mf

0.31

0.29

0.26

0.24

0.2

0.18

0.15

AISC 360-10 interaction curve

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8P

FE

M/P

fMFEM/Mf

0.11

0.16

0.2

0.22

0.24

0.28

0.31

(a) Rectangular (b) Circular

Fig.15: Comparisons of the proposed interaction curve with analysis results for CFT beam-column with different

axial load ratios and confinement factor

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2

PF

EM

/Pf

MFEM/Mf

L/B=3.0

L/B=10.0

L/B=20.0

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2

PF

EM

/Pf

MFEM/Mf

L/D=6.7

L/D=13.3

L/D=20.0

(a) Rectangular (b) Circular

Fig.16: Effect of length on the interaction curve

To use Method C, factors β1 and β2 for the ordinate and abscissa of Point D need to be

determined. Fig.15 shows that the maximum MFEM/Mf ratio increases as ξ increases. For

rectangular CFT beam-columns, the axial load ratio corresponding to the maximum MFEM/Mf

ratio decreases from 0.35 to 0.30 as ξ increases. For circular CFT beam-columns, the axial load

ratio corresponding to the maximum MFEM/Mf ratio increases from 0.40 to 0.50 as ξ increases.

Therefore, 1 for rectangular and circular beam-columns is selected conservatively as 0.30 and

0.40. Statistical analysis on the results from parametric studies shows that the MFEM/Mf ratio is

related to ξ as:

For rectangular beam-columns: 1.26.22 fFEM MM (4)

For circular beam-columns: 7.10.12 fFEM MM (5)

with the minimum value of β2 limited to 1.0.

18

The AISC 360-10 interaction curve (Eq.1 and Eq.2) for noncompact and slender CFT beam-

columns can be now updated as:

When 1

nc

r

P

P 0.1

1

2

1

nb

r

nc

r

M

M

P

P

(6)

When 1

nc

r

P

P 0.1

1

1

2

nb

r

nc

r

M

M

P

P

(7)

where β1 is 0.30 and 0.40 for rectangular and circular CFT beam-columns, respectively; and β2 is

calculated using Eq.4 and Eq.5.Comparisons of the interaction curve (dashed line) using Eq.6

and Eq.7 with analysis results are shown in Fig.15. As shown, the updated interaction curve

compares favorably with the analysis results.

A

C

B

D

Method A

Method BMethod C

Fig.17: Different design methods for defining beam-column interaction curve

6. Conclusions

AISC 360-10 specifies over-conservative equations for the design of noncompact and slender

rectangular and circular CFT beam-columns, as shown by the comparisons with tests results

from the experimental database complied by the authors. The over-conservatism of the design

equations is due to the conservative evaluations of axial and flexural strength, and the use of

bilinear P-M interaction curve for steel beam-columns. This paper focuses on improving the

bilinear interaction curve by conducting analytical parametric studies using finite element

models benchmarked against experimental data.

The bilinear interaction curve was improved by including factors β1 and β2 in the updated

versions (Eq.6 and Eq.7). In these equations, β1 is defined conservatively as 0.30 and 0.40 for

rectangular and circular CFT beam-columns, respectively. β2 is calculated using Eq.4 and Eq.5.

The values of β1 and the equations for β2 were determined using the results from parametric

studies focusing on the effects of axial load ratio, confinement factor, and member length. Of

these, the member length was found to have negligible effect on the shape of the interaction

curve, up to L/B or L/D ratio of 20. The updated interaction curve preserves the bilinear form of

the current AISC 360-10 interaction curve, while capturing the basic behavior of noncompact

and slender CFT beam-columns. Comparisons with the analysis results showed that the updated

interaction curve was able to predict the strength of the CFT beam-columns quite well.

19

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University of Sydney, Sydney, Australia, 1976

Cai, S.H. (1991), “Influence of Moment Distribution Diagram on Load-Carrying Capacity of Concrete-Filled Steel

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Wakabayashi, M. (ed.), Fukuoka, Japan, September 26-29, 1991, Association for International Cooperation and

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