Journal of Constructional Steel Research 66 (2010) 850–862

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Journal of Constructional Steel Research

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

Suggested empirical models for the axial capacity of circular CFT stub columnsZhao-Hui Lu a, Yan-Gang Zhao b,∗a School of Civil Engineering and Architecture, Central South University, 22 Shaoshannan Road, Changsha 410075, Chinab Department of Architecture, Kanagawa University, 3-27-1 Rokkakubashi, Kanagawa-ku, Yokohama 221-8686, Japan

a r t i c l e i n f o

Article history:Received 4 August 2008Accepted 24 December 2009

Keywords:Composite columnsConcreteSteelCircular tubesDesignCapacity

a b s t r a c t

In this paper, a total of 250 experimental tests of axially loaded circular concrete-filled steel tube (CFT)stub columns, published in the literature was summarized. The applicability of the current design codessuch as ACI, Australian Standards, AISC, AIJ, Eurocode 4, DL/T and some available empirical modelsproposed by various researchers for calculating the axial capacity of circular CFT stub columns wasexamined using these experimental data. Based on the investigations, four new empirical models forpredicting the axial capacity of circular CFT stub columns are proposed. The comparisons betweenthe experimental results and the predictions of these models show that the proposed empiricalmodels provide a direct, compact, and efficient representation of the ultimate strength of circular CFTstub columns made with not only normal strength but also high strength steel tubes and concrete.Finally, the limiting values of the maximum effective length, the compressive strength of concrete,the yield strength of steel tubes and the diameter-to-thickness for circular CFT stub columns withrespect to the present empirical models are suggested. It is expected that engineers can easily use thepresent empirical models to estimate the axial capacities of circular CFT stub columns for engineeringdesigns.

© 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Concrete-filled steel tubular (CFT) columns are being morewidely used in the construction of high-rise buildings, bridges,subway platforms, and barriers. Their usage provides excellentstatic and earthquake-resistant properties, such as high strength,high ductility, high stiffness, and large energy-absorption capacity.CFT columns provide the benefits of both steel and concrete: a steeltube surrounding a concrete columnnot only assists in carrying theaxial load but also confines the concrete. Furthermore, it eliminatesthe permanent formwork, which reduces construction time, whilethe concrete core takes the axial load and prevents or delays localbuckling of the steel tube.With the advent of high-strength steel and the production

of high-strength concrete using conventional materials withcareful quality control, high-strength CFT columns are bothtechnically and economically feasible. However, they are scarcelyadopted in the construction industry, mainly due to the lack ofunderstanding of their structural behavior and reliable designrecommendations [1]. The present design codes, such as AIJ[2,3], AISC [4], Eurocode 4 [5], and DL/T [6] have some limitationsin applications concerning the material’s strength and thediameter (width)-to-thickness ratio of circular (square) tubes. For

∗ Corresponding author. Tel.: +81 45 481 5661x3483; fax: +81 45 481 5360.E-mail address: zhao@kanagawa-u.ac.jp (Y.-G. Zhao).

0143-974X/$ – see front matter© 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.jcsr.2009.12.014

illustration, details of the limitations for circular CFT columnsare shown in Table 1, where fcyl,100 and fcyl,150 correspond to theconcrete’s compressive strength obtained from cylinder tests of100× 200 mm and 150× 300 mm specimens, respectively; fcu,150denotes the concrete’s compressive strength with 150 mm cubetests; fy is the yield strength of the steel tube; F is the smaller ofyield stress and 0.7 times the tensile strength; and Es is the elasticmodulus of the steel tube (=2.1× 105 MPa). There is uncertaintyas to whether they can be used for circular CFT stub columns witha highermaterial strength and higher diameter-to-thickness ratiosthan those stipulated in the codes or not.On the other hand, a large number of theoretical and

experimental studies on circular CFT stub columns with high-strength concrete or high-strength steel or high diameter-to-thickness ratio has been done, and several empirical models havebeen proposed to predict the axial capacity of stub circular CFTcolumns in the last few years [e.g., [7–13]]. These models usuallyshow good agreement with some experiments, especially with theresults used for the model development, but there is uncertaintyas to whether they can provide satisfactory predictions or not inmany other cases.Therefore, themain objective of the present paper is to evaluate

the applicability of the methods described in the codes andempirical formulae proposed by various researchers for computingthe axial capacities of circular CFT stub columns and if necessary, topropose simple modifications. Available experimental data of stubcircular CFT columns made with normal strength as well as highstrength steel tubes and concrete published in the literature areused for this purpose.

Z.-H. Lu, Y.-G. Zhao / Journal of Constructional Steel Research 66 (2010) 850–862 851

Table 1Limitations of applications of concerning the material’s strength and diameter-to-thickness ratio of circular steel tubes in some available design rules for CFT columns.

AIJ [2,3] AISC [4] Eurocode 4 [5] DL/T [6]

Limitations of yield strength of steel tube (MPa) 235≤ fy ≤ 355 fy ≤ 525 235≤ fy ≤ 460 235≤ fy ≤ 390Ranges of compressive strength of normal weight concrete(MPa)

fcyl,100 ≤ 58.8 21≤ fcyl,150 ≤ 70 20≤ fcyl,150 ≤ 50 or25≤ fcu,150 ≤ 60

30≤ fcu,150 ≤ 80

Limitations of diameter-to-thickness ratio of circular steeltube

≤1.5 · (23500/F) ≤0.15 · (Es/fy) ≤90 · (235/fy) 20–100

2. Summary of experimental data

Recently, Goode [14] has compiled a series of databases,available on the Internet, documenting 1819 CFT tests. However,a careful comparison of the database of circular CFT stub columnswith those published in the original literature reveals that althoughthe compressive strength of concrete of some specimens wereobtained through 100 mm cube test or 150 mm cube test or100 × 200 mm cylinder test, the concrete strengths of all thespecimens are deemed by Goode [14] as obtained from the 150×300 mm cylinder test. This will have significant effects on thepredicted results. Therefore a database of circular CFT stub columnsis compiled in this section.The experimental data from a relatively large number of

specimens are considered in this study. These are:Gardner and Jacobson [15], 12 tests; Luksha and Nesterovich

[16], 10 tests; O’Shea and Bridge [9], 12 tests; Sakino andHayashi [17], 12 tests; Kato [18], 12 tests; Saisho et al. [8], 29tests; Yamamoto et al. [19], 13 tests; Sakino et al. [11], 36 tests;Yu et al. [20], 28 tests; Giakoumelis and Lam [10], 8 tests; Zhangand Wang [21], 36 tests; Han et al. [12], 26 tests; and Tan [13], 16tests.The type of loading application as illustrated in Fig. 1, was used

in all the 250 experimental specimens, in which the steel tubeand concrete are loaded simultaneously. In Fig. 1, D is the externaldiameter, t is thewall thickness of the steel tube, and L is the lengthof the CFT column. Table 2 summarizes the measured dimensions,material properties, and ultimate strength of the tested specimens,in which fcu,100 denotes the concrete’s compressive strengthobtained from 100 mm cube tests, the compressive yield stressof steel tube σscy is defined as the stress corresponding to anoffset strain of 0.2%, and the tensile yield stress of steel tube σsy isobtained from tensile coupon tests (for the case of the steel havingno clear yield plateau, it corresponds to the stress at 0.2% offset).The ultimate axial capacities of the circular CFT stub columnsobtained from experiments (Ntest) correspond to the maximum(peak or limit points) of the axial load-shortening curves, and arenot limited by a strain limitation. Important characteristics of theselected specimens are summarized as follows.

(1) The yield strength of the steel tube fy (defined by σscy orσsy) ranged from 185.7 to 853.0 MPa, while the compressivestrength (150 × 300 mm cylinder test) of concrete from 15.0to 130.2 MPa.

(2) The tube diameter varied from 60 to 1020 mm and the tubethickness varied from 0.86 to 13.25 mm.

(3) The diameter-to-thickness (D/t) ratio of the specimens rangedfrom 16.7 to 220.9 and local buckling did not occur in anyspecimen.

(4) Relatively short length of specimens (the length-to-diameter:from 1.99 to 3.52) were used to ensure that they would be stubcolumns with a small slenderness effect and would not fail inoverall buckling.

It should be noted that different test standards are used todefine the compressive strength of concrete by the researchers,which are specified clearly in Table 2. The conversion relationsbetween fcyl,150 and fcu,150, proposed by Eurocode 2 [22] are shownin Table 3. The conversion relationship between fcyl,150 and fcyl,100

Fig. 1. Illustration of load applied to the entire section of circular CFT stub columns.

can be expressed as [23]:fcyl,150 = 0.96fcyl,100. (1)The similar conversion relationship between fcu,150 and fcu,100 is

used in this study, i.e., fcu,150 = 0.96fcu,100.

3. Available formulae for axial capacity of circular CFT stubcolumns

For completeness, a brief review of the determination of theaxial capacity of circular CFT stub columns using the methodsdescribed in the codes or empirical relations proposed by variousresearchers is presented as follows. In all the design calculations,the resistance factors and material partial factors are set to one.

3.1. The ACI [24] and Australian Standards [25,26]

The ACI [24] and Australian Standards (AS) [25,26] use the sameformula for calculating the ultimate axial capacity of the circularCFT stub columns. Neither of these specifications has taken intoaccount the concrete confinement and the interaction between theconcrete core and steel tube. The equation for axial capacity of acircular CFT stub column, NACI/AS is given as:

NACI/AS = 0.85fcyl,150Ac + fyAs (2)where Ac = the cross-sectional area of the concrete; and As = thecross-sectional area of the steel tube.

3.2. Giakoumelis and Lam [10]

A modification for the ACI [24] and AS [25,26] equation hasbeen proposed by Giakoumelis and Lam [10]. A new coefficientwas suggested to take into account the effect of the concrete’sconfinement on the axial load capacity of circular CFT stubcolumns. In this case, a revised equation is given as followsNGL = 1.3fcyl,150Ac + fyAs. (3)

3.3. The AISC [4]

The AISC [4] composite column design method has differentequations for the cross sectional strength depending on whetherthe columns are encased composite columns or filled compositecolumns. For the circular CFT columns, the AISC [4] allows for anincrease of the usable concrete stress to account for the beneficialeffects of the restraining hoop action arising from transverseconfinement. This is accomplished by increasing the multiplier ofthe first term of Eq. (2) from 0.85 to 0.95. Therefore, the cross-

852 Z.-H. Lu, Y.-G. Zhao / Journal of Constructional Steel Research 66 (2010) 850–862

Table 2Measured specimen dimensions, material properties and axial capacities (Circular CFT stub columns).

No. ofspecimens

Name ofspecimens

Dimensions Material properties Axialcapacity Ntest(kN)

Tested by

D (mm) t (mm) D/t L (mm) L/D Concretecompressivestrength (MPa)

Yield strengthof steel tubefy (MPa)

1 3 101.7 3.07 33.1 203.3 2.00 34.1 605.1 1112

Gardner andJacobson [15](12 tests)

2 4 101.7 3.07 33.1 203.3 2.00 31.2 605.1 10673 8 120.8 4.06 29.7 241.3 2.00 34.4 451.6 12004 9 120.8 4.09 29.5 241.4 2.00 34.1 451.6 12005 10 120.8 4.09 29.5 241.4 2.00 29.6 451.6 11126 13 152.6 3.18 48.1 304.8 2.00 25.9 415.1 12007 14 152.6 3.07 49.7 304.9 2.00 20.9 (fcyl, 150) 415.1 (σsy) 12008 15 152.6 4.93 31.0 304.9 2.00 42.0 633.4 29089 16 152.6 4.90 31.1 304.9 2.00 43.4 633.4 291310 19 76.4 1.70 44.9 152.3 1.99 25.0 363.3 35511 22 76.5 1.68 45.6 152.3 1.99 40.9 363.3 43412 23 76.4 1.70 44.9 152.4 1.99 25.9 363.3 372

13 SB1 159.0 5.07 31.4 447.0 3.00 41.5 381.5 2230

Luksha andNesterovich [16](10 tests)

14 SB2 630.0 7.00 90.0 1890.0 3.00 36.0 291.4 1665015 SB3 630.0 7.61 82.8 1890.0 3.00 35.0 349.5 1800016 SB4 630.0 8.44 74.6 1890.0 3.00 34.5 350.0 1860017 SB5 630.0 10.21 61.7 1890.0 3.00 38.4 (fcyl, 150) 323.3 (σsy) 2050018 SB6 630.0 11.60 54.3 1890.0 3.00 46.0 347.2 2440019 SB7 720.0 8.30 86.7 2160.0 3.00 15.0 312.0 1500020 SB8 820.0 8.93 91.8 2460.0 3.00 45.0 331.0 3360021 SB9 1020.0 9.64 105.8 3060.0 3.00 16.9 336.0 3000022 SB10 1020.0 13.25 77.0 3060.0 3.00 28.9 368.7 46000

23 S30CS50B 165.0 2.82 58.5 580.5 3.52 48.3 363.3 1662

O’Shea andBridge [9](12 tests)

24 S20CS50A 190.0 1.94 97.9 663.5 3.49 41.0 256.4 167825 S16CS50B 190.0 1.52 125.0 664.5 3.50 48.3 306.1 169526 S12CS50A 190.0 1.13 168.1 664.5 3.50 41.0 185.7 137727 S10CS50A 190.0 0.86 220.9 659.0 3.47 41.0 (fcyl, 150) 210.7 (σsy) 135028 S30CS80A 165.0 2.82 58.5 580.5 3.52 80.2 363.3 229529 S20CS80B 190.0 1.94 97.9 663.5 3.49 74.7 256.4 259230 S16CS80A 190.0 1.52 125.0 663.5 3.49 80.2 306.1 260231 S12CS80A 190.0 1.13 168.1 662.5 3.49 80.2 185.7 229532 S30CS10A 165.0 2.82 58.5 577.5 3.50 108.0 363.3 267333 S20CS10A 190.0 1.94 97.9 660.0 3.47 108.0 256.4 336034 S12CS10A 190.0 1.13 168.1 660.0 3.47 108.0 185.7 3058

35 L-20-1 178.0 9.00 19.8 360.0 2.02 22.2 283.0 2120

Sakino andHayashi [17](12 tests)

36 L-20-2 178.0 9.00 19.8 360.0 2.02 22.2 283.0 206037 H-20-1 178.0 9.00 19.8 360.0 2.02 45.4 283.0 272038 H-20-2 178.0 9.00 19.8 360.0 2.02 45.4 283.0 273039 L-32-1 179.0 5.50 32.5 360.0 2.01 22.1 (fcyl, 100) 249.0 (σsy) 141040 L-32-2 179.0 5.50 32.5 360.0 2.01 23.9 249.0 156041 H-32-1 179.0 5.50 32.5 360.0 2.01 43.7 249.0 208042 H-32-2 179.0 5.50 32.5 360.0 2.01 43.7 249.0 207043 L-58-1 174.0 3.00 58.0 360.0 2.07 23.9 266.0 122044 L-58-2 174.0 3.00 58.0 360.0 2.07 23.9 266.0 122045 H-58-1 174.0 3.00 58.0 360.0 2.07 45.7 266.0 164046 H-58-2 174.0 3.00 58.0 360.0 2.07 45.7 266.0 1710

47 C04LB 301.5 4.50 67.0 904.5 3.00 27.7 381.2 3851

Kato [18](12 tests)

48 C06LB 298.5 5.74 52.0 895.5 3.00 27.7 399.8 453749 C08LB 298.4 7.65 39.0 895.2 3.00 27.7 384.2 491950 C12LB 297.0 11.88 25.0 891.0 3.00 27.7 347.9 590951 C04MB 301.5 4.50 67.0 904.5 3.00 35.6 (fcyl, 100) 381.2 (σsy) 454752 C06MB 298.5 5.74 52.0 895.5 3.00 32.3 399.8 512553 C08MB 298.4 7.65 39.0 895.2 3.00 35.5 384.2 582154 C12MB 297.0 11.88 25.0 891.0 3.00 35.6 347.9 722255 C12MBH 301.3 11.59 26.0 903.9 3.00 35.6 471.4 859456 C06HB 298.5 5.74 52.0 895.5 3.00 82.4 399.8 793857 C08HB 298.4 7.65 39.0 895.2 3.00 82.4 384.2 838858 C12HB 297.0 11.88 25.0 891.0 3.00 82.4 347.9 9388

sectional strength, P0,AISC is given by

P0,AISC = 0.95fcyl,150Ac + fyAs. (4)

In reality, P0,AISC is the nominal, zero length strength (i.e., plasticcapacity of the section). Therefore, the nominal axial capacity ofa circular CFT stub column, NAISC included with length effects is

computed by

NAISC = P0,AISC

[0.658

( p0,AISCpe

)](Pe ≥ 0.44P0,AISC ) (5)

where Pe is the elastic buckling load. According to AISC [4], Pe isgiven by

Z.-H. Lu, Y.-G. Zhao / Journal of Constructional Steel Research 66 (2010) 850–862 853

Table 2 (continued)

No. ofspecimens

Name ofspecimens

Dimensions Material properties Axialcapacity Ntest(kN)

Tested by

D (mm) t (mm) D/t L (mm) L/D Concretecompressivestrength (MPa)

Yield strengthof steel tubefy (MPa)

59 S-30.1 101.6 3.00 33.9 304.8 3.00 117.0 377.3 1117

Saisho et al. [8](29 tests)

60 S-30.2 101.6 3.00 33.9 304.8 3.00 117.0 377.3 119561 S-30.3 101.6 2.99 34.0 304.8 3.00 135.6 377.3 118562 S-50.1 139.8 2.78 50.2 419.4 3.00 117.0 341.0 206763 S-50.2 139.8 2.78 50.2 419.4 3.00 117.0 341.0 196064 S-50.3 139.8 2.78 50.2 419.4 3.00 135.6 341.0 208765 S-50.4 139.8 2.78 50.2 419.4 3.00 135.6 341.0 204866 S-50.5 139.8 2.78 50.2 419.4 3.00 135.6 341.0 207767 S-60.1 139.8 2.37 59.1 419.4 3.00 135.6 462.6 217568 S-60.2 139.8 2.37 59.1 419.4 3.00 135.6 462.6 213669 S-60.3 139.8 2.37 59.1 419.4 3.00 135.6 462.6 216570 H-30.1 101.6 2.99 34.0 304.8 3.00 62.4 (fcyl, 100) 377.3 (σsy) 92171 H-30.2 101.6 2.99 34.0 304.8 3.00 62.4 377.3 92172 H-30.3 101.6 2.96 34.3 304.8 3.00 62.4 377.3 90173 H-50.1 139.8 2.78 50.2 419.4 3.00 57.3 341.0 132374 H-50.2 139.8 2.78 50.2 419.4 3.00 57.3 341.0 139175 H-50.3 139.8 2.78 50.2 419.4 3.00 57.3 341.0 131376 H-60.1 139.8 2.37 59.1 419.4 3.00 62.4 462.6 155877 H-60.2 139.8 2.37 59.1 419.4 3.00 70.8 462.6 157778 H-60.3 139.8 2.37 59.1 419.4 3.00 70.8 462.6 157779 H-60.4 139.8 2.37 59.1 419.4 3.00 70.8 462.6 162680 L-30.1 101.6 2.96 34.3 304.8 3.00 25.4 377.3 67681 L-30.2 101.6 2.99 34.0 304.8 3.00 27.7 377.3 71582 L-30.3 101.6 2.99 34.0 304.8 3.00 29.4 377.3 71583 L-50.1 139.8 2.78 50.2 419.4 3.00 25.4 341.0 93184 L-50.2 139.8 2.78 50.2 419.4 3.00 27.7 341.0 95085 L-60.1 139.8 2.37 59.1 419.4 3.00 27.7 462.6 109886 L-60.2 139.8 2.37 59.1 419.4 3.00 27.7 462.6 110787 L-60.3 139.8 2.37 59.1 419.4 3.00 27.7 462.6 1078

88 C10A-2A-1 101.4 3.02 33.6 304.2 3.00 23.2 371.0 660

Yamamotoet al. [19](13 tests)

89 C10A-2A-2 101.9 3.07 33.2 305.7 3.00 23.2 371.0 64990 C10A-2A-3 101.8 3.05 33.4 305.4 3.00 23.2 371.0 68291 C20A-2A 216.4 6.66 32.5 649.2 3.00 24.3 452.0 356892 C30A-2A 318.3 10.34 30.8 954.9 3.00 24.2 331.0 656593 C10A-3A-1 101.7 3.04 33.5 305.1 3.00 40.2 371.0 80094 C10A-3A-2 101.3 3.03 33.4 303.9 3.00 40.2 (fcyl, 100) 371.0 (σsy) 74295 C20A-3A 216.4 6.63 32.6 649.2 3.00 38.2 452.0 402396 C30A-3A 318.3 10.35 30.8 954.9 3.00 39.2 339.0 793397 C10A-4A-1 101.9 3.04 33.5 305.7 3.00 51.3 371.0 87798 C10A-4A-2 101.5 3.05 33.3 304.5 3.00 51.3 371.0 86299 C20A-4A 216.4 6.65 32.5 649.2 3.00 46.8 452.0 4214100 C30A-4A 318.5 10.38 30.7 955.5 3.00 52.2 339.0 8289

101 CC4-A-2 149.0 2.96 50.4 447.0 3.00 25.4 308.0 941

Sakino et al. [11](36 tests)

102 CC4-A-4-1 149.0 2.96 50.3 447.0 3.00 40.5 308.0 1064103 CC4-A-4-2 149.0 2.96 50.3 447.0 3.00 40.5 308.0 1080104 CC4-A-8 149.0 2.96 50.5 447.0 3.00 77.0 308.0 1781105 CC4-C-2 301.0 2.96 101.5 903.0 3.00 25.4 279.0 2382106 CC4-C-4-1 300.0 2.96 101.4 900.0 3.00 41.1 279.0 3277107 CC4-C-4-2 300.0 2.96 101.4 900.0 3.00 41.1 279.0 3152108 CC4-C-8 301.0 2.96 101.5 903.0 3.00 80.3 279.0 5540109 CC4-D-2 450.0 2.96 152.0 1350.0 3.00 25.4 279.0 4415110 CC4-D-4-1 450.0 2.96 152.0 1350.0 3.00 41.1 279.0 6870111 CC4-D-4-2 450.0 2.96 152.0 1350.0 3.00 41.1 279.0 6985112 CC4-D-8 450.0 2.96 152.0 1350.0 3.00 85.1 279.0 11665113 CC6-A-2 122.0 4.54 26.9 366.0 3.00 25.4 576.0 1509114 CC6-A-4-1 122.0 4.54 26.8 366.0 3.00 40.5 (fcyl, 100) 576.0 (σscy) 1657115 CC6-A-4-2 122.0 4.54 26.8 366.0 3.00 40.5 576.0 1663116 CC6-A-8 122.0 4.54 26.8 366.0 3.00 77.0 576.0 2100117 CC6-C-2 239.0 4.54 52.5 717.0 3.00 25.4 507.0 3035118 CC6-C-4-1 238.0 4.54 52.5 714.0 3.00 40.5 507.0 3583119 CC6-C-4-2 238.0 4.54 52.4 714.0 3.00 40.5 507.0 3647120 CC6-C-8 238.0 4.54 52.4 714.0 3.00 77.0 507.0 5578121 CC6-D-2 361.0 4.54 79.4 1083.0 3.00 25.4 525.0 5633122 CC6-D-4-1 361.0 4.54 79.4 1083.0 3.00 41.1 525.0 7260123 CC6-D-4-2 360.0 4.54 79.3 1080.0 3.00 41.1 525.0 7045124 CC6-D-8 360.0 4.54 79.4 1080.0 3.00 85.1 525.0 11505125 CC8-A-2 108.0 6.47 16.7 324.0 3.00 25.4 853.0 2275126 CC8-A-4-1 109.0 6.47 16.8 327.0 3.00 40.5 853.0 2446127 CC8-A-4-2 108.0 6.47 16.7 324.0 3.00 40.5 853.0 2402

(continued on next page)

854 Z.-H. Lu, Y.-G. Zhao / Journal of Constructional Steel Research 66 (2010) 850–862

Table 2 (continued)

No. ofspecimens

Name ofspecimens

Dimensions Material properties Axialcapacity Ntest(kN)

Tested by

D (mm) t (mm) D/t L (mm) L/D Concretecompressivestrength (MPa)

Yield strengthof steel tubefy (MPa)

128 CC8-A-8 108.0 6.47 16.7 324.0 3.00 77.0 853.0 2713129 CC8-C-2 222.0 6.47 34.3 666.0 3.00 25.4 843.0 4964130 CC8-C-4-1 222.0 6.47 34.3 666.0 3.00 40.5 843.0 5638131 CC8-C-4-2 222.0 6.47 34.3 666.0 3.00 40.5 843.0 5714132 CC8-C-8 222.0 6.47 34.4 666.0 3.00 77.0 843.0 7304133 CC8-D-2 337.0 6.47 52.1 1011.0 3.00 25.4 823.0 8475134 CC8-D-4-1 337.0 6.47 52.0 1011.0 3.00 41.1 823.0 9668135 CC8-D-4-2 337.0 6.47 52.0 1011.0 3.00 41.1 823.0 9835136 CC8-D-8 337.0 6.47 52.0 1011.0 3.00 85.1 823.0 13776

137 G4-1a 165.0 1.00 165.0 500.0 3.03 91.8 338.0 1773

Yu et al. [20](28 tests)

138 G4-1b 165.0 1.00 165.0 500.0 3.03 91.8 338.0 1430139 G4-1c 165.0 1.00 165.0 500.0 3.03 91.8 338.0 1372140 G4-1d 165.0 1.00 165.0 500.0 3.03 91.8 338.0 2038141 G2-2a 151.0 2.00 75.5 500.0 3.31 87.1 405.0 2132142 G2-2b 151.0 2.00 75.5 500.0 3.31 87.1 405.0 1933143 G4-2a 165.0 2.00 82.5 500.0 3.03 91.8 338.0 2244144 G4-2b 165.0 2.00 82.5 500.0 3.03 91.8 338.0 2381145 G4-2c 165.0 2.00 82.5 500.0 3.03 91.8 338.0 2077146 G4-2d 165.0 2.00 82.5 500.0 3.03 91.8 338.0 1930147 G4-2e 165.0 2.00 82.5 500.0 3.03 91.8 338.0 1920148 G2-3a 149.0 3.00 49.7 500.0 3.36 87.1 (fcu, 100) 438.0 (σsy) 2337149 G2-3b 149.0 3.00 49.7 500.0 3.36 87.1 438.0 2394150 G2-3c 149.0 3.00 49.7 500.0 3.36 87.1 438.0 2361151 G4-3a 165.0 3.00 55.0 500.0 3.03 91.8 338.0 2567152 G4-3b 165.0 3.00 55.0 500.0 3.03 91.8 338.0 2714153 G4-3c 165.0 3.00 55.0 500.0 3.03 91.8 338.0 2734154 G2-4.5a 151.0 4.50 33.6 500.0 3.31 87.1 438.0 2743155 G2-4.5b 151.0 4.50 33.6 500.0 3.31 87.1 438.0 2572156 G2-4.5c 151.0 4.50 33.6 500.0 3.31 87.1 438.0 2727157 G4-4a 165.0 4.00 41.2 500.0 3.03 91.8 338.0 2704158 G4-4b 165.0 4.00 41.2 500.0 3.03 91.8 338.0 2773159 G4-4c 165.0 4.00 41.2 500.0 3.03 91.8 338.0 2832160 G2-6a 159.0 6.00 26.5 500.0 3.14 87.1 405.0 2957161 G2-6b 159.0 6.00 26.5 500.0 3.14 87.1 405.0 3099162 G2-8a 159.0 8.00 19.9 500.0 3.14 87.1 438.0 3173163 G2-8b 159.0 8.00 19.9 500.0 3.14 87.1 438.0 3267164 G2-8c 159.0 8.00 19.9 500.0 3.14 87.1 438.0 3330

165 C3 114.4 3.98 28.7 300.0 2.62 31.4 343.0 948

Giakoumelis andLam [10](8 tests)

166 C4 114.6 3.99 28.7 300.0 2.62 93.6 343.0 1308167 C7 114.9 4.91 23.4 300.5 2.62 34.7 365.0 1380168 C8 115.0 4.92 23.4 300.0 2.61 104.9 (fcu, 150) 365.0 (σsy) 1787169 C9 115.0 5.02 22.9 300.5 2.61 57.6 365.0 1413170 C11 114.3 3.75 30.5 300.0 2.62 57.6 343.0 1067171 C12 114.3 3.85 29.7 300.0 2.62 31.9 343.0 998172 C14 114.5 3.84 29.8 300.0 2.62 98.9 343.0 1359

173 L-A-1-92h 167.4 3.32 50.4 503.0 3.00 50.8 354.0 1704

Zhang andWang [21](36 tests)

174 L-A-2-99h 167.3 3.35 49.9 502.0 3.00 50.8 354.0 1668175 L-A-3-98h 167.5 3.33 50.3 503.0 3.00 50.8 354.0 1700176 L-B-1-85h 138.9 3.29 42.2 419.0 3.02 44.6 332.0 1140177 L-B-2-88h 139.0 3.29 42.2 419.0 3.01 44.6 332.0 1220178 L-B-3-89h 139.5 3.37 41.4 419.0 3.00 44.6 332.0 1180179 L-C-1-87h 139.9 3.58 39.1 416.0 2.97 44.6 325.0 1222180 L-C-2-101h 139.9 3.54 39.5 421.0 3.01 44.6 325.0 1242181 L-C-3-30h 139.9 3.48 40.2 419.0 2.99 44.6 325.0 1300182 L-E-1-15h 133.4 5.21 25.6 396.0 2.97 46.5 351.0 1612183 L-E-2-25h 133.2 5.06 26.3 397.0 2.98 46.5 351.0 1580184 L-E-3-13h 133.4 5.23 25.5 398.0 2.98 46.5 351.0 1640185 M-A-1-97h 167.0 3.37 49.6 503.0 3.01 70.6 354.0 2075186 M-A-2-100h 167.1 3.33 50.2 503.0 3.01 70.6 354.0 2105187 M-A-3-95h 167.8 3.33 50.4 504.0 3.00 70.6 354.0 2055188 M-B-1-20h 138.6 3.31 41.9 418.0 3.02 62.5 332.0 1480189 M-B-2-26h 138.9 3.36 41.3 420.0 3.02 62.5 332.0 1520190 M-B-3-90h 138.6 3.30 42.0 420.0 3.03 62.5 (fcu, 100) 332.0 1500191 M-C-1-120h 140.3 3.62 38.8 418.0 2.98 61.4 325.0 (σsy) 1582192 M-C-2-96h 140.0 3.60 38.9 418.0 2.99 61.4 325.0 1582193 M-C-3-86h 139.7 3.61 38.7 420.0 3.01 61.4 325.0 1540194 M-E-1-21h 133.4 5.17 25.8 396.0 2.97 70.6 351.0 1810195 M-E-2-27h 133.2 5.03 26.5 396.0 2.97 70.6 351.0 1770

Z.-H. Lu, Y.-G. Zhao / Journal of Constructional Steel Research 66 (2010) 850–862 855

Table 2 (continued)

No. ofspecimens

Name ofspecimens

Dimensions Material properties Axialcapacity Ntest(kN)

Tested by

D (mm) t (mm) D/t L (mm) L/D Concretecompressivestrength (MPa)

Yield strengthof steel tubefy (MPa)

196 M-E-3-23h 133.2 5.07 26.3 397.0 2.98 70.6 351.0 1835197 H-B-1-310h 138.9 3.28 42.3 420.0 3.02 77.1 332.0 1688198 H-B-2-309h 138.7 3.28 42.3 418.0 3.01 77.1 332.0 1680199 H-B-3-312h 139.0 3.29 42.2 418.0 3.01 77.1 332.0 1628200 H-D-1-311h 159.3 5.36 29.7 477.0 2.99 77.1 356.0 2480201 H-D-2-308h 160.2 5.01 32.0 476.0 2.97 77.1 356.0 2440202 H-D-3-324h 159.3 5.07 31.4 478.0 3.00 77.1 356.0 2460203 H-E-1-322h 133.3 5.10 26.1 396.0 2.97 77.1 351.0 1930204 H-E-2-306h 133.4 5.20 25.7 396.0 2.97 77.1 351.0 1955205 H-E-3-323h 133.1 5.04 26.4 397.0 2.98 77.1 351.0 1955206 H-F-1-307h 133.3 5.43 24.5 397.0 2.98 77.1 392.0 1820207 H-F-2-313h 133.1 5.44 24.5 397.0 2.98 77.1 392.0 1915208 H-F-3-314h 133.1 5.43 24.5 397.0 2.98 77.1 392.0 1930

209 CA1-1 60.0 1.87 32.1 180.0 3.00 85.2 282.0 312

Han et al. [12](26 tests)

210 CA1-2 60.0 1.87 32.1 180.0 3.00 85.2 282.0 320211 CA2-1 100.0 1.87 53.5 300.0 3.00 85.2 282.0 822212 CA2-2 100.0 1.87 53.5 300.0 3.00 85.2 282.0 845213 CA3-1 150.0 1.87 80.2 450.0 3.00 85.2 282.0 1701214 CA3-2 150.0 1.87 80.2 450.0 3.00 85.2 282.0 1670215 CA4-1 200.0 1.87 107.0 600.0 3.00 85.2 282.0 2783216 CA4-2 200.0 1.87 107.0 600.0 3.00 85.2 282.0 2824217 CA5-1 250.0 1.87 133.7 750.0 3.00 85.2 282.0 3950218 CA5-2 250.0 1.87 133.7 750.0 3.00 85.2 282.0 4102219 CB1-1 60.0 2.00 30.0 180.0 3.00 85.2 404.0 427220 CB1-2 60.0 2.00 30.0 180.0 3.00 85.2 (fcu, 150) 404.0 415221 CB2-1 100.0 2.00 50.0 300.0 3.00 85.2 404.0 (σsy) 930222 CB2-2 100.0 2.00 50.0 300.0 3.00 85.2 404.0 920223 CB3-1 150.0 2.00 75.0 450.0 3.00 85.2 404.0 1870224 CB3-2 150.0 2.00 75.0 450.0 3.00 85.2 404.0 1743225 CB4-1 200.0 2.00 100.0 600.0 3.00 85.2 404.0 3020226 CB4-2 200.0 2.00 100.0 600.0 3.00 85.2 404.0 3011227 CB5-1 250.0 2.00 125.0 750.0 3.00 85.2 404.0 4442228 CB5-2 250.0 2.00 125.0 750.0 3.00 85.2 404.0 4550229 CC1-1 60.0 2.00 30.0 180.0 3.00 90.0 404.0 432230 CC1-2 60.0 2.00 30.0 180.0 3.00 90.0 404.0 437231 CC2-1 150.0 2.00 75.0 450.0 3.00 90.0 404.0 1980232 CC2-2 150.0 2.00 75.0 450.0 3.00 90.0 404.0 1910233 CC3-1 250.0 2.00 125.0 750.0 3.00 90.0 404.0 4720234 CC3-2 250.0 2.00 125.0 750.0 3.00 90.0 404.0 4800

235 GH1-1 125.0 1.00 125.0 438.0 3.50 116.0 232.0 1275

Tan [13](16 tests)

236 GH1-2 125.0 1.00 125.0 438.0 3.50 116.0 232.0 1239237 GH2-1 127.0 2.00 63.5 445.0 3.50 116.0 258.0 1491238 GH2-2 127.0 2.00 63.5 445.0 3.50 116.0 258.0 1339239 GH3-1 133.0 3.50 38.0 465.0 3.50 116.0 352.0 1995240 GH3-2 133.0 3.50 38.0 465.0 3.50 116.0 352.0 1991241 GH3-3 133.0 3.50 38.0 465.0 3.50 116.0 352.0 1962242 GH4-1 133.0 4.70 28.3 465.0 3.50 116.0 (fcu, 150) 352.0 (σsy) 2273243 GH4-2 133.0 4.70 28.3 465.0 3.50 116.0 352.0 2158244 GH4-3 133.0 4.70 28.3 465.0 3.50 116.0 352.0 2253245 GH5-1 127.0 7.00 18.1 445.0 3.50 116.0 429.0 3404246 GH5-2 127.0 7.00 18.1 445.0 3.50 116.0 429.0 3370247 GH5-3 127.0 7.00 18.1 445.0 3.50 116.0 429.0 3364248 GH6-1 108.0 4.50 24.0 378.0 3.50 106.0 358.0 1535249 GH6-2 108.0 4.50 24.0 378.0 3.50 106.0 358.0 1578250 GH6-3 108.0 4.50 24.0 378.0 3.50 106.0 358.0 1518

Table 3Conversion relations between fcyl,150 and fcu,150 [22].

fcyl,150 (MPa) 12 16 20 25 30 35 40 45 50 55 60 70 80 90fcu,150(MPa) 15 20 25 30 37 45 50 55 60 67 75 85 95 105

Pe =π2(EI)eff 1(KALA)2

(6a)

in which

(EI)eff 1 = EsIs + C3Ec1Ic (6b)

C3 = 0.6+ 2(

AsAc + As

)≤ 0.9. (6c)

KA = the effective length factor; LA = laterally unbraced lengthof the column; Is and Ic= moment of inertia of steel tube and

856 Z.-H. Lu, Y.-G. Zhao / Journal of Constructional Steel Research 66 (2010) 850–862

concrete core, respectively; Ec1 = the modulus of elasticity ofconcrete = 4730 (fcyl,150)1/2 MPa (normal weight concrete); and(EI)eff 1 = effective stiffness of composite section.

3.4. AIJ [2,3]

According to AIJ [2,3], the ultimate compressive strength ofan axially loaded circular CFT stub column, NAIJ is calculated byEq. (7).

NAIJ = 0.85fcyl,100Ac + (1.0+ η)fyAs (lk/D ≤ 4) (7)

where lk = effective length of a CFT column; and η = theconfinement factor= 0.27.According to Eq. (7), the improvement of load carrying capacity

due to the composite action is denoted by the confinement factorη, which is independent of the strength of the materials and thedimensions of the columns.

3.5. Sakino et al. [11]

Sakino et al. [11] introduced a reduction factor γU to considerthe scale effect on the compressive strength of concrete and themodification for AIJ [2,3] equation was proposed as:

NSK = γU fcyl,100Ac + (1.0+ η)fyAs (8)

where γU = a reduction factor introduced to take the scale effectinto consideration = 1.67D−0.112c ; and Dc = the diameter of theconcrete core.

3.6. Eurocode 4 [5]

The Eurocode 4 [5] determines the resistant capacity of acircular CFT stub column by adding the contribution of thesteel tube and the concrete core and the confinement effectis considered. The strength of the concrete is increased by thecoefficient ηc because the concrete has a higher strength whena triaxial state of stress occurs. The strength of the steel tube isdecreased by ηa because the effective yield stress of the steel isreduced by the hoop stresses. The axial load capacity is given by

NEC4 =(1+ ηc

tDfyfcyl,150

)fcyl,150Ac + ηafyAs (9)

in which

ηc = 4.9− 18.5λ̄+ 17λ̄2 (ηc ≥ 0) (10a)

ηa = 0.25(3+ 2λ̄) (ηa ≤ 1.0) (10b)

λ̄ =

√NplRNcr

(10c)

NplR = fyAs + fcyl,150Ac (10d)

Ncr =π2(EI)eff 2l2

(10e)

(EI)eff 2 = EsIs + KeEc2Ic (10f)

where ηc = the coefficient of confinement for the concrete;ηa = the coefficient of confinement for the steel tube; λ̄= relativeslenderness; l= buckling length of the CFT column; Ec2 = elasticmodulus of concrete= 22000 [(fcyl,150 + 8)/10]0.3 MPa; (EI)eff 2 =the effective flexural stiffness for calculation of relative slender-ness; and Ke = a correction factor= 0.6.Particularly, when λ̄= 0, the sectional capacity (the zero length

strength), P0,EC4 can be simplified as

P0,EC4 =(1+ 4.9

tDfyfcyl,150

)fcyl,150Ac + 0.75fyAs. (11)

3.7. Goode and Narayanan [7]

At the viewpoint of the use of the two coefficients (i.e., ηc andηa) seems rather arbitrary [14], Goode andNarayanan [7] proposedthe following equation to consider the confining effect of circularCFT stub columns.

NGN = 0.85fcyl,150Ac +6t

(D− 2t)fyAc . (12)

3.8. The Chinese code DL/T [6]

The Chinese codeDL/T [6] for axially loaded concrete-filled steelcircular hollow-section stub columns treats the composite sectionas one ‘‘material’’ with a total area of Asc and the corresponding‘‘nominal yielding strength’’ of fscy. A confinement factor, ξ isintroduced to describe the composite action between the steel tubeand the filled concrete. The axial load capacity of circular CFT stubcolumns, NDL/T is given by:

NDL/T = fscyAsc (13)

in which

Asc = As + Ac (14a)

fscy = (1.212+ Bξ + Cξ 2)fck (14b)

B = 0.1759fy235+ 0.974 (14c)

C = −0.1038fck20+ 0.0309 (14d)

ξ =AsfyAc fck

(14e)

fck = 0.67fcu,150 (14f)

where ξ = the confinement factor; Asc = the area of compositesection; fscy = the ‘‘nominal yielding strength’’ of the compositesection; and fck = the characteristic concrete strength.

3.9. Han et al. [12]

To make the Chinese code DL/T [6] easier to use for calculatingthe section capacity of circular CFT stub columns, a simplifiedmodel is proposed by Han et al. [12].

NHAN = (1.14+ 1.02ξ)fckAsc (15)

in which Asc , ξ , and fck are given by Eqs. (14a), (14e) and (14f),respectively, and the validity range of Eq. (15) is: 0.1 < ξ < 5.

4. Comparative studies

4.1. Slenderness effects of AISC and Eurocode 4 specifications

In order to understand the effect of slenderness on the predictedultimate axial strength of circular CFT stub columns using AISCand Eurocode 4 provisions, a comparative study of predictions ofthe 250 experimental specimenswith andwithout considering theeffect of slenderness is conducted. Two cases i.e., both ends havingfull fixity or no fixity are considered as the follows:Case 1: The effective length of the column is taken as one-half ofthe column length for full fixity;Case 2: The effective length of the column is taken as one time ofthe column length for no fixity.The ratios of the predicted results by using the AISC provisions

considering the effect of slenderness for the 250 experimental tests

Z.-H. Lu, Y.-G. Zhao / Journal of Constructional Steel Research 66 (2010) 850–862 857

(a) The ratio of predictions of Case 1 to zero length strength. (b) The ratio of predictions of Case 2 to zero length strength.

Fig. 2. Illustration of effect of slenderness on the predicted ultimate axial strength of circular CFT stub columns using AISC provisions.

to those without considering the effect of slenderness (zero lengthstrength) are shown in Fig. 2(a) and (b) respectively for Case 1 andCase 2. It can be observed from Fig. 2 that there is little effect ofslenderness on the predicted ultimate axial strength of circular CFTstub columns using AISC provisions (the relative differences areless than 1%) and thereinafter for the AISC specification, the axialcapacity of a circular CFT stub column is approximated by using thezero length strength i.e., Eq. (4).Fig. 3(a) and (b) illustrate the ratios of predictions of Eurocode

4 considering the effect of slenderness for the 250 experimentaltests to those without considering the effect of slenderness (zerolength strength). One can see from Fig. 3(a) and (b) that there is asignificant effect of the slenderness on the predicted ultimate axialstrength of circular CFT stub columns using Eurocode 4 provisions(the relative differences are about 7.4% for case 1 and 13.5% forCase 2). The ratios of predictions of Case 1 to those of Case 2 isdepicted in Fig. 3(c), the relative difference is about 7.2% betweenthe two cases. Nevertheless, because the complete informationwith respect to the manner in which the loads were applied to theends of the entire 250 test specimens (such as the columns loadedwith both ends fixed or hinged and thickness of end plate), arenot available in the literature, the effective length of the columnsis assumed to be the same as the column length in the followingcomparison studies.

4.2. Comparison of tests results with those predicted by the availableformulations

The ratios of axial capacities obtained from the stub columntests to the predictions using the different methods as mentionedin the previous section for the 250 experimental tests are depictedin Fig. 4, together with the concrete’s compressive strength,yield strength of steel tube and D/t ratio of the experimentaltests. In Figs. 4 (d),(g),(i), and (k), the specimens with a materialstrength or D/t ratio beyond the limitations of the correspondingdesign codes are clearly marked. For overall comparisons, themean, standarddeviation,maximum,minimum, and the differencebetween maximum and minimum of the ratios for the differentdesign methods are listed in Table 4.Fig. 4 and Table 4 reveal the following:

(1) Generally, for the CFT specimens with a material strength orD/t ratio beyond the limitations of the design codes such asAIJ [2,3], AISC [4], Eurocode [5] and DL/T [6], the predictionsby the corresponding design methods have almost the sametrend as those in the limitations of design codes. However,the variations of the predictions apparently become larger.The predictions of all the four codes for the CFT stub columnsmade with higher D/t ratio than the upper limit of the AIJspecifications (e.g., specimens No. 138 and 139) are generallymuch larger than the test results. This may be due to theeffect of lock buckling, which on the other hand means thatthe limiting values for circular CFT stub columns mentioned

in AIJ [2,3] are appropriate. Great differences between thepredictions of all the four codes for some specimens madewith very high-strength concrete and test results are found(e.g., the predictions are larger than test results for specimensNo. 65, 66, and 67; while for specimens No. 245, 246, and 247,the predictions are much smaller than the test results). Thereason may be that, there is a larger randomness for high-strength concrete and the effects of steel hardening may existwhen the D/t ratio is very small. The differences betweenthe test results of specimens with very high-strength steeltubes (e.g., specimen No. 125–136) and the predictions ofthe codes are relative small, which means the effects of steelhardening in these cases are slightly less significant. Althoughthe composite action is considered in the DL/T [6] equations,it gave smaller predictions than those of ACI [24] for somespecimens, particularly with very high-strength steel tube(e.g., specimen No. 125, 126, 127, and 128), which seems to bea shortcoming of the DL/T [6] equations.

(2) ACI [24] and AS [25,26] give a sectional capacity about 31.1%lower than the experimental results,mainly due to the fact thatthe composite action between the steel tube and the concretecore was not considered. As an improved model for ACI [24]and AS [25,26], the equation proposed by Giakoumelis andLam [10] predicts better results in the mean value, but thescatter becomes larger.

(3) Although the beneficial confining effect has been taken intoconsideration in the design code of AISC [4], still it providesconservative results. The sectional capacity is about 23.2%lower than the experimental results.

(4) AIJ [2,3] predicts a sectional capacity about 14.3% lower thanthe measured ultimate strengths. A reduction factor wasintroduced to consider the scale effect in the improved versionsuggested by Sakino et al. [11], but still it underestimates thetest results.

(5) Although DL/T [6] gives section capacity about 6.8% lower thanthemeasured ultimate strengths, the scatter of the predictionsis very large (standard deviation = 0.123 and the differencebetween maximum and minimum = 0.817). The scatter ofthe predictions of the simplified formula proposed by Hanet al. [12] becomes smaller, but it gives more conservativeresults (about 11.5% lower than the test results).

(6) Eurocode 4 [5] gives the best results with a mean andstandard deviation of the ratios of 1.020 and0.105 respectively.However, the computation is relative complex. AlthoughGoode and Narayanan [7] proposed a simplified equation,the predictions become worse (mean = 1.084 and standarddeviation= 0.118).

In view of the foregoing, it is desirable to propose new andsimple analytical formulae to predict the axial capacity of circularCFT stub columns made with not only normal strength but alsohigh strength concrete and steel tubes.

858 Z.-H. Lu, Y.-G. Zhao / Journal of Constructional Steel Research 66 (2010) 850–862

(a) The ratio of predictions of Case 1 to zerolength strength.

(b) The ratio of predictions of Case 2 to zerolength strength.

(c) The ratio of predictions of Case 1 tothose of Case 2.

Fig. 3. Illustration of the effect of slenderness on the predicted ultimate axial strength of circular CFT stub columns using Eurocode 4 provisions.

Table 4Comparisons of circular CFT stub column strengths obtained from experiments with the predictions of existing models and new proposals.

The ratios of test results to the predictions of existing models and new proposalsMean Standard deviation Maximum Minimum Maximum–Minimum

AIJ [2,3] 1.143 0.113 1.488 0.871 0.617Sakino et al. [11] 1.084 0.102 1.406 0.795 0.611New proposal 1 1.003 0.100 1.317 0.747 0.570

AISC [4] 1.232 0.134 1.665 0.844 0.821ACI [24]; AS [25,26] 1.311 0.134 1.750 0.933 0.817Giakoumelis and Lam [10] 1.025 0.146 1.468 0.636 0.832New proposal 2 1.002 0.099 1.299 0.771 0.528

Eurocode 4 [5] 1.020 0.105 1.419 0.779 0.640Goode and Narayanan [7] 1.084 0.118 1.411 0.816 0.595New proposal 3 1.022 0.100 1.353 0.773 0.580

DL/T [6] 1.068 0.123 1.603 0.786 0.817Han et al. [12] 1.115 0.106 1.371 0.844 0.527New proposal 4 1.002 0.096 1.241 0.745 0.496

5. New proposals for the axial capacity of circular CFT stubcolumns

The following conditions are considered in the new proposalsfor predicting the axial capacity of circular CFT stub columns:(1) The equations should represent the experimental data asaccurately as possible.

(2) The formulae should be as simple as possible and easily usablein any analysis.

(3) The expressions are similar to the existing ones in thecorresponding current design codes so that engineers canreadily make use of them in the practice of engineering design.

From the existing formulations for axial capacity of circular CFTstub columns, presented in the earlier sections we can see thatthree different definitions of compressive strength of concrete, i.e.,fcyl,150, fcyl,100, and fcu,150 are used. For this reason and consideringall the desirable conditions above, four new and simple empiricalmodels, developed from AIJ [2,3], ACI [24], AS [25,26], AISC [4],Eurocode 4 [5], and DL/T [6], are proposed in this study.

5.1. New proposal 1: Empirical formula improved from AIJ [2,3]

Assuming the plastic resistance of the composite section, NPR, isgiven by Eq. (16).

NPR = fcyl,100Ac + fyAs. (16)

Due to the composite action between the concrete and steeltubes, the axial capacity of a circular CFT stub column is generallydifferent from its plastic resistance. Similar to the design methodof AIJ [2,3], the difference between the axial capacity predicted bythe new proposal, NNew−1 and the plastic resistance is assumed asa function of the axial strength of the steel tube (=fyAs), i.e.,

NNew−1 = fcyl,100Ac + (1.0+ k1)fyAs (17)

where k1 is an augmentation factor.

Here, the value of the augmentation factor k1 is determinedas 0.4 by a regression analysis based on the 250 experimentaltests described previously, as shown in Fig. 5. Therefore, the newproposal improved from AIJ [2,3], NNew−1 is expressed as

NNew−1 = fcyl,100Ac + 1.4fyAs. (18)

5.2. New proposal 2: Empirical formula improved from ACI [24](AS [25,26], AISC [4])

Similar to the new proposal 1, the new proposal for the ACI/ASequation and the AISC’s zero length strength, NNew−2 can beexpressed as

NNew−2 = fcyl,150Ac + (1.0+ k2)fyAs (19)

where k2 is an augmentation factor.The value of the augmentation factor, k2 is determined by a

regression analysis based on the 250 experimental tests, as shownFig. 6 (k2 = 0.47). Therefore, the revised equation is proposed asfollows:

NNew−2 = fcyl,150Ac + 1.47fyAs. (20)

5.3. Newproposal 3: Empirical formula developed fromEurocode 4 [5]

According to Eq. (9), the axial capacity of a circular CFT stubcolumn predicted by Eurocode 4 [5], NEC4 can be rewritten as

NEC4 =[1+ ηc

tDfyfcyl,150

+ (ηa − 1)AsAc

fyfcyl,150

]fcyl,150Ac + fyAs.

(21)

Since

AsAc=πD2/4− π [(D− 2t)/2]2

π [(D− 2t)/2]2=4(D− t)t(D− 2t)2

≈ 4tD. (22)

Z.-H. Lu, Y.-G. Zhao / Journal of Constructional Steel Research 66 (2010) 850–862 859

Fig. 4. Comparison of experimental results with results predicted by the existing models.

Then, Eq. (21) can be expressed as

NEC4 =[1+ (ηc + 4ηa − 4)

tDfyfcyl,150

]fcyl,150Ac + fyAs. (23)

Substituting Eqs. (10a) and (10b) in Eq. (23), one obtains

NEC4 =[1+ (17λ̄2 − 16.5λ̄+ 3.9)

tDfyfcyl,150

]fcyl,150Ac + fyAs. (24)

According to Eqs. (10c)–(10f), there is a variety of factorsaffecting the relative slenderness λ̄ of circular CFT stub columns,such as the effective length of the column, the effective flexural

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Fig. 5. The augmentation factor k1 for the experimental specimens.

Fig. 6. The augmentation factor k2 for the experimental specimens.

Fig. 7. The relative slenderness of the experimental specimens.

stiffness of composite section, the strength of the concrete andthe steel tubes, the thickness of the steel tube and the externaldiameter of the column. In this paper, λ̄ is taken as 0.152. Thisvalue results from regression analysis of the 250 experimental testsdescribed previously, as shown in Fig. 7. Therefore, the proposedempirical formula for predicting the ultimate strength of circularCFT stub column developed from Eurocode 4 [5], NNew−3 is givenas

NNew−3 =(1+ 1.8

tDfyfcyl,150

)fcyl,150Ac + fyAs. (25)

5.4. New proposal 4: Empirical formula improved from DL/T [6]

Based on the researches of DL/T [6] and Han et al. [12], the newequation for predicting the axial capacity of a circular CFT stubcolumn, NNew−4 may be expressed as the following equation.

NNew−4 = g(ξ)fckAsc (26)

in which g(ξ) is a function of ξ .

Fig. 8. The regression analysis of the function of the confinement factor ξ .

Similarly, g(ξ) is determined by a regression analysis based onthe 250 experimental tests described previously, as shown in Fig. 8and it is expressed as

g(ξ) = 1.3+ 1.1ξ . (27)

Therefore, a new empirical formula improved fromDL/T [6] andHan et al. [12], NNew−4 is proposed as

NNew−4 = (1.3+ 1.1ξ)fckAsc . (28)

5.5. Verification of the proposed empirical models

In order to verify the present formulae, the ratios of the axialcapacities obtained from the stub column tests to the predictionsusing the proposed empirical models are shown in Fig. 9 and themean, standarddeviation,maximum,minimum, and the differencebetween maximum and minimum of the ratios for the differentnew proposals are also listed in Table 4. It can be observed fromFig. 9 and Table 4 that the new proposals provide a direct, compact,and efficient representation of the ultimate strength of circular CFTstub columns.Due to the fact that the models of AIJ [2,3], ACI [24] (AS [25,26],

AISC [4]), and Han et al. [12] predict conservative results comparedwith the test results, new coefficients obtained from the regressionanalysis are proposed for the AIJ equation in the new proposal 1,for the ACI/AS equation and the AISC’s zero length strength in thenew proposal 2 and for Han et al.’s model in the new proposal 4,respectively. Therefore, the predictions of the new proposal 1, 2,and 4 are effectively improved which can be observed from Figs. 4and 9 and Table 4.From Eqs. (9) and (25), Figs. 4 and 9 and Table 4, one can

easily observe that although the new proposal 3 is much moresimple than the Eurocode 4 [5] provisions, it predicts almost thesame results as those predicted by Eurocode 4 provisions. Thismay be explained as follows: although there is significant effectof slenderness on the predicted ultimate axial strength of circularCFT stub columns using Eurocode 4 provisions, but as the meanvalue of the slenderness effect (λ̄) of the entire 250 stub columnspecimens is adopted in the new proposal and the scatter of theslenderness effect is very small (standard deviation = 0.033, seeFig. 7), the new proposal can provide almost the same predictionsas those of Eurocode 4 provisions.

5.6. Suggested limitations of material strength and D/t ratios for theproposed empirical models

It can be observed from Eqs. (18), (20), (25) and (28) that, theultimate axial strength of circular CFT stub columns predicted bythe proposed empirical models can be considered as the sectional

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Fig. 9. Comparison of experimental results with predictions of the new proposals.

capacities (zero length strength), i.e., the slenderness effect isnot included. It is therefore necessary to give a definition of a‘‘stub column’’ in the present models, i.e., the maximum effectivelength of a circular CFT stub column should be limited so thatthe length effect (overall buckling effect) may be ignored. As thenominal ratios of effective length to diameter of the 250 CFT stubcolumns varied from 2 to 3.5 and the present empirical models aredeveloped based on these experimental specimens, therefore, theeffective length (buckling length) of a circular CFT stub column, lkis limited to:

lk/D ≤ 3.5. (29)

From the comparisons of predictions of the proposed empiricalmodels with the test results, it can be observed that the newproposals provide a direct, compact, and efficient representationof the ultimate strength of circular CFT stub columns madefrom normal- or high-strength steel tubes filled with normal-or high-strength concrete. Thus, the following limitations for thecompressive strength of normal weight concrete (fcyl,150, or fcu,150,or fcyl,100, MPa) and yield strength of steel tube (fy, MPa) aresuggested in the proposed empirical models.

20 MPa ≤ fcyl,150 ≤ 130 MPa;or 25 MPa ≤ fcu,150 ≤ 150 MPa; or

21 MPa ≤ fcyl,100 ≤ 135 MPa. (30)

235 MPa ≤ fy ≤ 850 MPa. (31)

Although the proposed empirical models can provide goodpredictions for all the 250 circular CFT stub columns, thepredictions of the experimental specimens with D/t ratios beyondthe upper limits of D/t ratios predicted by AIJ [2,3] are generallylarger than those obtained from tests. This can be observed fromFig. 4(c) and Fig. 9. From a conservative viewpoint, the suggestedlimiting values ofD/t ratios for circular CFT columns in the presentmodels are similar as those in AIJ [2,3] and for simplicity, thevariable F (see Table 1) is replaced by the yield strength of steeltube. The revised formula is then given by

D/t ≤ 1.5 · (23500/fy). (32)

6. Summary and conclusions

Using the available experimental data, the applicability of thecurrent design codes such asACI [24], AS [25,26], AISC [4], Eurocode4 [5], AIJ [2,3], DL/T [6] and some available empirical models pro-posed by various researchers for calculating the axial capacity of

circular CFT stub columnswas evaluated. The database used hereincovered cases with normal- or high-strength steel tubes filled withnormal- or high-strength concrete. From the investigation of thepresent study, the following conclusions can be drawn:(1) For the circular CFT stub columns with a material strengthor diameter-to-thickness ratio beyond the limitations of thedesign codes such as AIJ, AISC, Eurocode 4 and DL/T, thepredictions by the corresponding design methods have almostthe same trend as those in the limitations of the design codes.However, the variations of the predictions apparently becomelarger.

(2) The upper limits of the strength of steel tubes and the concretecompressive strength for circular CFT stub columns in AIJ, AISC,Eurocode 4, and DL/T can be improved. The upper limits of thediameter-to-thickness ratio in AISC, Eurocode 4, and DL/T mayalso be improved as those in AIJ.

(3) Four newempiricalmodels corresponding toAIJ, AISC (ACI, AS),Eurocode 4, and DL/T specifications for calculating the axialcapacity of circular CFT stub columns are proposed. Compar-isons between the experimental results and the predictions ofthese formulae show that the proposed empirical models pro-vide a direct, compact, and efficient representation of the ulti-mate strength of circular CFT stub columnsmadewith not onlynormal strength but also with high strength steel tubes andconcrete.

(4) The limiting values of the maximum effective length, thecompressive strength of the concrete, the strength of thesteel tube and the diameter-to-thickness for circular CFT stubcolumns with respect to the present empirical models aresuggested. It is expected that engineers can easily use thepresent empirical models to estimate the axial capacities ofcircular CFT stub columns for engineering design.

Acknowledgements

This study is partially supported by the start-up funds fromCentral South University, the ‘‘Grant-in-Aid for Scientific Research(Tokubetsu Kenkyuin Shorei-hi)’’ from Japan Society for the Pro-motion of Science (JSPS) (No: 19.07399) and the Joint ResearchFund for Overseas Chinese, Hong Kong and Macao Young Scholars(No. 50828801) from the National Natural Science Foundation ofChina. The support is gratefully acknowledged. Beneficial discus-sion with Prof. X.F. Wang at the Shenzhen University, Prof. H.B. Geat Meijo University and Dr. G.D. Hatzigeorgiou at Democritus Uni-versity of Thrace are gratefully acknowledged. Finally, the writerswish to thank the reviewers of this paper for their critical com-ments and suggestions.

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