Flexural behaviour of concrete-filled steel tubes

Journal of Constructional Steel Research 60 (2004) 313–337 www.elsevier.com/locate/jcsr

Flexural behaviour of concrete-filled steel tubes Lin-Hai Han College of Civil Engineering and Architecture, Fuzhou University, Gongye Road 523, Fuzhou, Fujian Province 350002, PR China Received 3 January 2003; received in revised form 28 August 2003; accepted 28 August 2003

Abstract This paper develops a mechanics model that can predict the behaviour of concrete-filled hollow structural section (HSS) beams. A form of unified theory, where a confinement factor (n) was introduced [Steel Compos. Struct.—Int. J. (2001) 1(1) 51] to describe the composite action between the steel tube and filled concrete, is used in the analysis. A series of concretefilled square and rectangular tube beam tests were carried out. The main parameters varied in the tests were the depth-to-width ratio (b) from 1 to 2, and tube depth to wall thickness ratio from 20 to 50. The load vs. lateral deflection relationship was established for concretefilled HSS beams both experimentally and theoretically. The predicted curves of load vs. mid-span deflection are in good agreement with the presented test results. Formulas which should be suitable for incorporation into building codes are developed for calculating the moment capacity of concrete-filled HSS beams. Comparisons are made with predicted beam capacities and flexural stiffness using the existing codes, such as AIJ-1997, BS5400-1979, EC4-1994, and LRFD-AISC-1999. # 2003 Elsevier Ltd. All rights reserved. Keywords: Concrete-filled steel hollow sections; Beams; Design; Hollow sections; Mechanics model; Flexural strength; Flexural stiffness

1. Introduction Concrete-filled hollow structural sections (HSS) are widely used in building construction [1]. In the past, although there were a large number of research studies on the behaviour of concrete-filled HSS columns and beam-columns, such as Furlong [2], Gardner and Jacobson [3], Ge and Usami [4], Han [5,6], Han et al. [7], Kato

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L.-H. Han / Journal of Constructional Steel Research 60 (2004) 313–337

Nomenclature As Ac Asc B D Ec Es fcu fck 0 fc fscy fsy i Isc Ki Ks L M Mu Muc Mue t Wscm

a b r e n / cm

steel cross-sectional area concrete cross-sectional area cross-sectional area of the composite section, given by As þ Ac width of rectangular steel tube depth of rectangular steel tube, or outer diameter of circular steel tube concrete modulus of elasticity steel modulus of elasticity characteristic 28-day concrete cube strength characteristic concrete strength characteristic concrete cylinder strength yield strength of the composite section yield strength of the steel tube pffiffiffiffiffiffiffiffiffiffiffiffiffiffi radius of gyration of the composite section, given by Isc =Asc moment of inertia for composite cross-section initial section flexural stiffness of the composite beam serviceability-level section flexural stiffness of the composite beam effective buckling length of column in the plane of bending moment ultimate strength of composite beams predicted moment capacity maximum test moment wall thickness of the steel tube section modulus of the composite beams, given by B3/6 for composite beams with square sections; given by BD2/6 and B2D/6 about major (x–x) and minor (y–y) axes, respectively, for composite beams with rectangular sections; given by pD3/32 for composite beams with circular sections steel ratio (¼ As =Ac ) depth-to-width ratio of RHS sections, given by D/B stress strain confinement factor ð¼ ðAs fsy Þ=ðAc fck ÞÞ curvature flexural strength index

[8], Kilpatrick and Rangan [9], Knowles and Park [10], Matsui et al.[11], Neogi et al. [12], O’Shea and Bridge [13]; Prion and Boehme [14]; Rangan and Joyce [15]; Schneider [16]; Shakir-Khalil and Mouli [17], Tomii et al. [18], Uy [19,20]; Varma et al. [21], Wang [22], and etc., there is relatively little research reported on the flexural behaviour of concrete-filled HSS beams.


315

Furlong [2] tested one concrete-filled steel SHS beam with diameter-to-width ratio of 32. He found that the value of flexural strength from the capacity of steel tube alone was about 49% lower than that of the specimen tested. Prion and Boehme [14] conducted four concrete-filled steel CHS beams with diameter-to-thickness ratio of 89.4, and concrete cylinder strength of 73 MPa. It was found that the beam specimens failed in a very ductile manner. Uy [19] reported five beam tests on concrete-filled steel SHS. The test specimens were selected for examining the effects of different width-to-wall thickness ratios (from 40 to 100), and different concrete cylinder strength (38 and 50 MPa). The tests showed that all beam members had a significant yielding plateau, and thus exhibited adequate ductility. As part of a program on the strength of beam-columns, Uy [20] reported three beam tests on concrete-filled high-strength steel SHS. Lu and Kennedy [23] performed 12 beam tests on concrete-filled steel SHS and RHS. The test specimens were selected for examining the effects of different depth to width ratios, and different values of shear span to depth. The tests showed that the ultimate flexural strength of the composite beams is increased by about 10–30% over that of bare steel sections, depending on the relative proportions of steel and concrete. The flexural stiffness is also enhanced. It was found that the slip between the steel and concrete was not detrimental. Formulae for the flexural strength of concrete-filled steel SHS and RHS beams were suggested. Fully plastic stress blocks with the concrete at its maximum cylinder strength were used in the analysis. As part of a program on the strength of concrete-filled steel SHS beam-columns, Tomii and Sakino [24] performed eight specimens with zero axial loads. The widthto-thickness ratio ranged from 23.5 to 45. Elchalakani et al. [25] presented an experimental investigation of the flexural behaviour of circular concrete-filled steel tubes subjected to large deformation pure bending, where D=t ¼ 12 110. It was found that in general, void filling of the steel tube enhances strength, ductility and energy absorption especially for thinner sections. From the above review, it can be found that there is significant research work on the flexural behaviour of concrete-filled HHS beams; however, there still exists areas that need to be studied further: 1. A mechanics model for the analysis of the flexural behaviour for concrete-filled HHS beams need to be developed, the composite action between the steel tube and the concrete core should be considered. 2. The differences of the predicted beam capacities and flexural stiffness using the existing codes, such as AIJ [26], BS5400 [27], EC4 [28], and LRFD-AISC [29], need to be illustrated. 3. Simplified model with reasonable accuracy for the calculations on the flexural strength of the beams need to be developed. The main objectives of this paper were thus fourfold: firstly, to report a series of tests on composite beams. Secondly, to develop a mechanics model that can predict the behaviour of concrete-filled hollow structural section beams. Thirdly, to develop formulas for the calculation of the moment capacity of the concrete-filled

316


HSS beams, such formulas should be suitable for incorporation into building codes. And finally, to illustrate the differences of the predicted beam capacities and flexural stiffness using the existing codes, such as AIJ-1997 [26], BS5400-1994 [27], EC4-1994 [28], and LRFD-AISC-1999 [29]. 2. Experimental investigations A total of 16 concrete-filled steel SHS and RHS beam specimens were tested. Fig. 1 shows the sectional details and dimensions of the specimens. A summary of the specimens is presented in Table 1 where the section sizes and material properties are given. The specimens were designed with a wide range of width-to-depth ratio (from 1 to 2) was achieved. The tube depth to wall thickness ratio ranged from 20 to 50. All the specimens were 1100 mm in length. The tubes were manufactured from mild steel sheet. The tubes with four plates were cut from the sheet, tack welded into a square or rectangular shape and then welded with a single bevel butt weld at the corners. The ends of the steel tubes were cut and machined to the required length. The insides of the tubes were wire brushed to remove any rust and loose debris present. The deposits of grease and oil, if any, were cleaned away. Each tube was welded to a square steel base plate of 10 mm thickness. Strips of the steel tubes were tested in tension in accordance with the Chinese standard related to metal materials. Three coupons were taken from each of the

Fig. 1. Beam specimen details and dimensions.

27.3 35.2 35.2 35.2 31.3 31.3 40 40 34.5 34.5 34.5 34.5 34.5 34.5 34.5 34.5

1 1 1 1 1 1 1 1 1.25 1.25 1.33 1.33 1.67 1.67 2 2

&-120 120 3:84 &-120 120 3:84 &-120 120 3:84 &-120 120 3:84 &-120 120 5:86 &-120 120 5:86 &-120 120 5:86 &-120 120 5:86 &-150 120 2:93 &-150 120 2:93 &-120 90 2:93 &-120 90 2:93 &-150 90 2:93 &-150 90 2:93 &-120 60 2:93 &-120 60 2:93

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

RB1-1 RB2-1 RB2-2 RB2-3 RB3-1 RB3-2 RB4-1 RB4-2 RB5-1 RB5-2 RB6-1 RB6-2 RB7-1 RB7-2 RB8-1 RB8-2

b (D/B) fcu (MPa)

Sectional dimension D B t (mm)

Number Specimen label

Table 1 Specimen labels, material properties and moment capacities

330.1 330.1 330.1 330.1 321.1 321.1 321.1 321.1 293.8 293.8 293.8 293.8 293.8 293.8 293.8 293.8

fsy (MPa) 896 960 1002 856 1356 1409 1360 1184 1607 1746 749 722 1216 1269 489 485

890 840 894 852 1224 1265 1116 1160 1037 1106 598 613 879 953 469 457

29.34 30.16 32.25 31.69 40.90 41.54 41.43 42.61 31.4 31.4 21.1 20.2 28.4 29.4 18.4 17.8

28.3 28.8 28.8 28.8 39.8 39.8 40.3 40.3 28.7 28.7 16.4 16.4 24.1 24.1 13.4 13.4

0.966 0.955 0.893 0.909 0.972 0.957 0.972 0.945 0.914 0.914 0.777 0.812 0.849 0.847 0.728 0.753

Kie (kN m2) Kse (kN m2) Mue (kN m) Muc (kN m) Muc =Mue

L.-H. Han / Journal of Constructional Steel Research 60 (2004) 313–337 317

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steel sheets; from these tests, the average yield strength of the steel tube (fsy) of each specimen is listed in Table 1. The modulus of elasticity of the tubes was found to be approximately 200,000 MPa. A type of concrete, with a compressive cube strength (fcu) at 28 days of 30 MPa, was designed. The mix proportions of the concrete were as follows: cement: 457 kg/m3; water: 206 kg/m3; sand: 608 kg/m3; and coarse aggregate: 1129 kg/m3. The concrete was filled in layers and was vibrated by a poker vibrator. The specimens were placed upright to air-dry until testing. During curing, a very small amount of longitudinal shrinkage of 1.2 mm or so occurred at the top of the beams. A high-strength epoxy was used to fill this longitudinal gap so that the concrete surface was flush with the steel tube at the top. For each batch of concrete mixed, three 150 mm cubes were also cast and cured in conditions similar to the related specimens. The modulus of elasticity (Ec) of concrete is 26,700 MPa. The average cube strength of each specimen at the time of tests is listed in Table 1. The experimental study was able to determine not only the maximum moment capacity of the specimens, but also to investigate the failure pattern up to and beyond the ultimate load. It had been found by Lu and Kennedy [23] that the ratio of shear span to depth, varying from 1.03 to 5.05, had almost no effect on the moment–curvature diagram and the test-to-predicted moment ratio. Thus, the shear span to depth ratio is selected as 1.67–2.1 for the current tests. A four point bending rig was used to apply the moment (see Fig. 2a). The in-plane displacements were measured at locations along the specimen by three displacement transducers. Eight strain gauges were used for each specimen to measure strains at the mid-span. Fig. 2b gives a general view of the test setup. A load interval of less than one-tenth of the estimated load capacity was used. Each load interval was maintained for about 2–3

Fig. 2. Arrangement of beam tests.


319

Fig. 3. Typical failure mode of the composite beams.

min. At each load increment, the strain readings and the deflection measurements were recorded. All specimens were loaded to failure. The tested specimens failed in a very ductile manner. No tensile fracture was observed on the tension flange. Typical failure modes of the steel tube are shown in Fig. 3. The measured bending moment vs. mid-span deflections are given in Fig. 4. Typically measured bending moment vs. extreme fibre compressive and tensile strains are shown in Fig. 5. The measured strains were used to determine the bending curvature. Typical moment (M) vs. curvature (/) graphs are shown in Fig. 6. The moment vs. curvature diagrams show that there is an initial elastic response, then inelastic behaviour with gradually decreasing stiffness, until the ultimate moment is reached asymptotically. For practical considerations, the moment corresponding to the maximum fibre strain of 0.01 was defined as the moment capacity (Mu) of the composite beam in this paper. It was found that after

Fig. 4. Load vs. mid-span deflection of beam specimens.

320


Fig. 5. Moment vs. extreme fibre strains at mid-height of beam specimens.

the maximum fibre strain reaches 0.01, the moment tends to stabilize. The moment capacities (Mue) of the current specimens determined are listed in Table 1. A careful examination of the test results revealed that, in general, the moment vs. curvature relationship goes into an inelastic stage at 20% of the moment capacity (Mu), so the initial section flexural stiffness (Ki) was defined as the secant stiffness corresponding to a moment of 0.2 Mu. The moment vs. curvature response was also used to determine the serviceability-level section flexural stiffness (Ks). Ks was defined as the secant stiffness corresponding to the serviceability-level moment of 0.6 Mu [21]. The initial section flexural stiffness (Kie) and the serviceability-level

Fig. 6. Typical moment (M) vs. curvature (/) relations of the beam specimens.


321

Fig. 7. Specimen height vs. mid-span lateral deflection curves.

section flexural stiffness (Kis) of the tested specimens so determined are listed in Table 1. Because of the infill of concrete, the tested CFST beams behaved in a relatively ductile manner and testing proceeded in a smooth and controlled way. Fig. 7 shows a typical deflection curve of a tested specimen under flexure. The enhanced structural behaviour of the beams can be explained by the establishment of ‘‘composite action’’ between the steel tube and the concrete core. 3. Mechanics models 3.1. Material properties Stress vs. strain relations for the steel and concrete of concrete-filled steel HSS hollow sections presented in Han et al. [7,30] were used for the analysis of the composite beams in this paper. A typical stress–strain curve for steel can consist of five stages as shown in Fig. 8. Detailed expressions were given in Han et al. [7]. A typical stress–strain curve for the confined concrete with fck ¼ 41 MPa is shown in Fig. 9, where the confinement factor (n) is defined as: n¼

As fsy Ac fck

ð1Þ

Fig. 8. Typical stress–strain curves for steel.

322


Fig. 9. r vs. e relations of concrete core. (a) For concrete-filled CHS. (b) For concrete-filled SHS.

in which As is the cross-sectional area of the steel tube, Ac is cross-sectional area of the concrete core, fsy is the yield stress of the outer steel tube, and fck is the compression strength of concrete. The value of fck for normal strength concrete is determined using 67% of the compression strength of cubic blocks. Detailed expressions are given in Han et al. [7,30]. It can be seen from Fig. 9 that the higher the confinement factor (n), the higher the compression strength of confined concrete. It can also be seen from Fig. 9 that


323

the higher is n, the more ductile is the confined concrete. The confinement factor (n), to some extent, represents the ‘‘composite action’’ between steel tubes and concrete. 3.2. Mechanics model for the composite beams A member subjected to bending moment (M) is shown in Fig. 10, where um is the mid-span deflection. The fabrication of concrete-filled steel SHS or RHS beams involves welding which introduces residual stresses due to the cooling of the weld and the tensile stresses generated at the weld region. Fig. 11 shows a typical residual stress distribution for a steel plate of a beam that has been fabricated with four steel plates and a longitudinal fillet weld [31]. The test results of residual stresses on steel plates for concrete-filled steel SHS columns were summarized by Uy [31], where they were shown to be about 15–25% of the yielding stress in compression. The average value of 20% of the yielding stress in compression is selected in the analysis in this paper. The moment (M) vs. mid-span deflection (um) relations can be established based on the following assumptions: 1. The stress–strain relationship for steel given in Fig. 8 is adopted for both tension and compression. The stress–strain relationship for concrete given in Han et al. [7,30] is adopted for compression only. The contribution of concrete in tension is neglected. 2. Original plane cross-sections remain plane. 3. The effect of shear force on deflection of members is omitted. 4. The deflection curve of the member is assumed as a sine wave. 5. Residual stress distribution for a steel plate of a concrete-filled steel SHS or RHS beam as shown in Fig. 11 is used in the analysis. According to assumption no. 4, the deflection (u) of the member can be expressed as: p u ¼ um sin z ð2Þ L where, um is the mid-span deflection, L is the length of the member and z is the horizontal distance from the left support as defined in Fig. 10.

Fig. 10. A schematic view of a beam.

324


Fig. 11. Residual stress distribution across plate of steel box column.

The curvature (/) at the mid-span can be calculated as: /¼

p2 um L2

ð3Þ

The strain distribution is shown in Fig. 12, where e0 is the strain along the geometrical centre line of the section. The term ei is the strain at the location yi as defined in Fig. 12. Along the line with y ¼ yi , the section can be divided into two kinds of elements (dAsi for steel tube and dAci for concrete, respectively) with unit depth. The strain at the centre of each element can be expressed as: ei ¼ eo þ / yi

ð4Þ

The stress at the centre of each element (rsi for steel tube and rci for concrete) can be determined using the stress–strain relationship given in Han et al. [7,30]. The internal moment (Min) and axial force (Nin) can be calculated as: X Min ¼ ðrsi dAsi yi þ rci dAci yci Þ ð5Þ i

X ðrsi dAsi þ rci dAci Þ Nin ¼

ð6Þ

i

According to the equilibrium condition, Min ¼ Mapplied

ð7Þ

Nin ¼ 0

ð8Þ

From the above equations, the load vs. mid-span deflection relations can be established for the composite beams. To consider the effects of thin steel plate in excess of compact plate, local buckling could be incorporated in the current model by using a method adopted by Uy [19].


325

Fig. 12. Distribution of strains. (a) Rectangular section. (b) Circular section.

The model used for determining the effective width (be) of a steel plate is given as: rffiffiffiffiffiffiffi be rol ¼a ð9Þ b ry where b ¼ steel plate width; a ¼ factor to account for residual stresses and initial imperfections; rol ¼ local buckling stress; and ry ¼ steel yield stress [32]. The predicted moments are compared with the current experimental values in Table 1, where a mean of 0.885 and coefficient of variation (COV) of 0.08 obtained. The predicted curves of load vs. lateral deflection are compared in Figs. 4, 13–16 with experimental curves. A generally good agreement is obtained between the predicted and tested curves.

326


Fig. 13. Examples of comparisons: moment (M) vs. mid-span deflection (um) curves [19].

4. Simplified models 4.1. Flexural strength For convenience of analysis, flexural strength index (cm) is defined in this paper. It is expressed as: cm ¼

Mu Wscm fscy

ð10Þ

in which Mu is the moment capacity of the composite beams; Wscm is the section modulus of the compsite beams, given by B3 =6 for composite beams with square sections; given by BD2 =6 and B2 D=6 about major (x–x) and minor (y–y) axes, respectively, for composite beams with rectangular sections; given by (pD3)/32 for composite beams with circular sections; fscy is the ‘‘nominal yielding strength’’ of the composite sections [7,33], is given by For concrete-filled steel SHS and RHS :

fscy ¼ ð1:18 þ 0:85 nÞ fck

For concrete-filled steel CHS beams : fscy ¼ ð1:14 þ 1:02 nÞ fck

ð11aÞ ð11bÞ

The flexural strength index (cm) so determined according to the mechanics model described above is plotted in Fig. 17 against the confinement factor (n). It can be seen from Fig. 17 that cm increases when the confinement factor (n) increases.


327

Fig. 14. Examples of comparisons: moment (M) vs. curvature (/) curves [23].

Using the relations between the flexural strength index (cm) and the confinement factor (n) that determine it, the following formula for the flexural strength index (cm) can be obtained by using regression analysis method, i.e. For concrete-filled steel SHS and RHS beams: rm ¼ 1:04 þ 0:48 lnðn þ 0:1Þ

ð12aÞ

For concrete-filled steel CHS beams: rm ¼ 1:1 þ 0:48 lnðn þ 0:1Þ

ð12bÞ

328


Fig. 15. Examples of comparisons: moment (M) vs. mid-span deflection (um) curves [24].

According to Eq. (10), the flexural capacity of the composite beam (Mu) can be given by Mu ¼ cm Wscm fscy

ð13Þ

in which, Wscm can be given by DB2 =6 and BD2 =6 for concrete-filled steel RHS beams about major (x–x) and minor (y–y) axes, Wscm ¼ pD3 =32 for concrete-filled steel CHS beams.

Fig. 16. Comparisons of moment (M) vs. mid-span deflection (um) curves [14].


329

Fig. 17. cm vs. n relations. (a) Square and rectangular beams. (b) Circular beams.

The validity limits of Eq. (13) are: D ¼ 100 2000 mm; a ¼ 0:04 0:2; fsy ¼ 200 500 MPa and fck ¼ 20 80 MPa. It should be noted that Eq. (13), the tube diameter range being 2000 mm was based on numerical simulation due to the difficulties in testing such a large size specimen.

330


4.2. Comparison of moment capacity of the composite section The moment capacities predicted using the following five design methods are compared with the beam test results obtained in the current tests and those from Lu and Kennedy [23], Uy [19,20], Tomii and Sakino [24], Prion and Boehme [14], and Elchalakani et al.[25]: – – – – –

AIJ [26] BS5400 [27] LRFD [28] Eurocode 4, part 1 [29] The proposed method in this paper.

In all design calculations, the material partial safety factors were set to unity. Predicted section capacities (Muc) using the different methods are compared with 51 experimental results (Mue) in Table 2. Table 2 shows both the mean value and the standard deviation (COV) of the ratio of Muc/Mue for the different design methods. The results in this table clearly show that all the methods are conservative. Overall, AIJ (1997) and LRFD-AISC (1999) gave a moment capacity about 20% lower than that of test, BS5400 (1979) gave a moment capacity about 12% lower than that of test. EC4 (1994) and the proposed method in this paper gives a mean value of 0.903 and 0.903, a COV of 0.124 and 0.106, respectively, predicted about 10% lower capacity than the test results, is the best predictor, and thus are acceptable for the calculation of the moment capacity of concrete-filled HSS beams. 4.3. Flexural stiffness The formulae for the flexural stiffness of the composite sections in the different codes described above are listed as follows: 1. AIJ [26] Ke ¼ Es Is þ 0:2 Ec Ic

ð14Þ

Ke ¼ Es Is þ Ec Ic

ð15Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where Es ¼ 205; 800 MPa; Ec ¼ 21; 000 fc0 =19:6 MPa. 2. BS5400 [27] where Es ¼ 206; 000 MPa;Ec ¼ 450 fcu MPa. 3. Eurocode 4, part 1 [28] Ke ¼ Es Is þ 0:6Ec Ic where Es ¼ 206; 000 MPa;Ec ¼ 9500 ðfck þ 8Þ 4. LRFD [29] Ke ¼ Es Is þ 0:8Ec Ic

ð16Þ 1=3

pffiffiffiffi where Es ¼ 199; 000 MPa; Ec ¼ 4733 fc0 MPa.

MPa. ð17Þ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

&-120 120 3:84 &-120 120 3:84 &-120 120 3:84 &-120 120 3:84 &-120 120 5:86 &-120 120 5:86 &-120 120 5:86 &-120 120 5:86 &-150 120 2:93 &-150 120 2:93 &-120 90 2:93 &-120 90 2:93 &-150 90 2:93 &-150 90 2:93 &-120 60 2:93 &-120 60 2:93 &-152 152 4:8 &-152 152 4:8 &-152 152 4:8 &-152 152 9:5 &-254 152 6:4 &-254 152 6:4 &-254 152 6:4 &-254 152 9:5 &-254 152 9:5 &-152 254 6:4 &-152 254 6:4 &-152 254 6:4

Number Sectional dimension (mm)

18.3 23.6 23.6 23.6 22 22 26.8 26.8 23.1 23.1 23.1 23.1 23.1 23.1 23.1 23.1 63.2 41 35.8 62.4 60.8 51.2 47.2 56.8 44.9 64 38.8 33.5

29.34 30.16 32.25 31.69 40.90 41.54 41.43 42.61 31.4 31.5 21.1 20.2 28.4 29.4 18.4 17.8 73.6 75.1 71.3 146.5 210.7 210.7 207.6 283.8 282.2 144.7 146.7 142.9

25.7 25.7 25.66 25.66 37.83 37.83 37.83 37.83 24.13 24.13 14.68 14.68 20.34 20.34 11.65 11.65 60.71 60.71 60.71 125.2 161 161 161 232 232 112.6 112.6 112.6

0.875 0.851 0.796 0.810 0.925 0.911 0.913 0.888 0.768 0.766 0.696 0.727 0.716 0.692 0.633 0.654 0.825 0.808 0.851 0.855 0.764 0.764 0.775 0.817 0.822 0.778 0.768 0.788

Mue LRFD (1999) fck (MPa) (kN m) Muc = Muc (kN m) Mue

Table 2 Comparisons between predicted beam strengths and test results

25.82 25.82 25.82 25.82 37.99 37.99 37.99 37.99 24.4 24.4 14.81 14.81 20.6 20.6 11.79 11.79 61.07 61.07 61.07 113.1 162.6 162.6 162.6 244.2 244.2 113 113 113

0.880 0.856 0.801 0.815 0.929 0.915 0.917 0.892 0.777 0.775 0.702 0.733 0.725 0.701 0.641 0.662 0.830 0.813 0.857 0.772 0.772 0.772 0.783 0.860 0.865 0.781 0.770 0.790

27.82 28.26 28.26 28.26 40.32 40.32 40.72 40.72 27.8 27.8 16.58 16.58 23.36 23.36 13.04 13.04 68.22 67.83 67.68 133.2 187.6 187 186.7 270.5 269.4 124.7 124.1 123.8

0.948 0.937 0.876 0.892 0.986 0.971 0.983 0.956 0.885 0.883 0.786 0.821 0.823 0.795 0.709 0.733 0.927 0.903 0.949 0.909 0.890 0.888 0.899 0.953 0.955 0.862 0.846 0.867

28.03 28.52 28.52 28.52 39.19 39.19 39.78 39.78 28.28 28.28 23.81 23.81 16.83 16.83 13.24 13.24 69.08 68.64 68.46 120.2 191.4 190.8 190.5 273.7 272.4 125.5 124.9 124.6

0.955 0.946 0.884 0.900 0.958 0.943 0.960 0.934 0.901 0.898 1.128 1.179 0.593 0.572 0.720 0.744 0.939 0.914 0.960 0.820 0.908 0.906 0.917 0.964 0.965 0.868 0.851 0.872

Muc (kN m)

Test data resources

26.62 0.907 Current 27.06 0.897 test 27.06 0.839 27.06 0.854 41.46 1.014 41.46 0.998 41.65 1.005 41.65 0.977 26.6 0.847 26.6 0.844 16.27 0.771 16.27 0.805 23.36 0.823 23.36 0.795 13.99 0.760 13.99 0.786 66.16 0.899 [23] 65.44 0.871 65.17 0.914 152.1 1.038 197.2 0.936 196.2 0.931 195.6 0.942 304.5 1.073 303.2 1.075 118.2 0.817 116.2 0.792 115.6 0.809 (continued on next page)

Muc Muc Muc =Mue Muc Muc = (kN m) (kN m) (kN m) Mue

Formula (13)

Muc Muc = (kN m) Mue

EC4 (1994)

BS5400 (1979)

AIJ (1997)


40.3 40.3 26.1 31 31 24 21.6 20.6 18.6 24.5 24.5 26.1 58.4 58.4 58.4 58.4 19.5 19.5 19.5 19.5 19.5 19.5 19.5 19.5 19.5 19.5 19.5 19.5

29 &-126 126 3 30 &-156 156 3 31 &-186 186 3 32 &-246 246 3 33 &-306 306 3 29 &-100 100 2:29 30 &-100 100 2:27 31 &-100 100 2:98 32 &-100 100 4:25 33 &110 110 5 34 &160 160 5 35 &210 160 5 36 v-152 1:65 37 v-152 1:65 38 v-152 1:65 39 v-152 1:65 40 v-109:9 1 41 v-110:4 1:25 42 v-110:9 1:5 43 v-101:83 2:53 44 v-88:64 2:79 45 v-76:32 2:45 46 v-89:26 3:35 47 v-60:65 2:44 48 v-76:19 3:24 49 v-60:67 3:01 50 v-33:66 1:98 51 v-33:78 2:63 Mean value Standard deviation (COV) 27.9 20.43 0.732 42.4 31.61 0.746 62.6 45.2 0.722 103.5 79.72 0.770 153 124 0.810 7.15 6.36 0.890 10.98 9.91 0.903 14.01 12.17 0.869 18.33 16.62 0.907 66 62.06 0.940 141 135.2 0.959 228 236.4 1.037 19.9 9.77 0.491 17.9 9.77 0.546 20.8 9.77 0.470 19 9.77 0.514 7.6 4.74 0.624 9.1 5.96 0.655 11 7.18 0.653 11.33 9.11 0.804 10.86 8.89 0.819 6.92 5.55 0.802 10.47 10.19 0.973 3.78 3.58 0.947 9.87 7.87 0.797 4.75 4.09 0.861 0.9 0.88 0.978 1.17 1.18 1.009 0.794 0.125

Mue LRFD (1999) fck (MPa) (kN m) Muc Muc = (kN m) Mue

Table 2 (continued )


20.59 0.738 31.91 0.753 45.73 0.731 80.91 0.782 126.2 0.825 6.42 0.898 10 0.911 12.25 0.874 16.7 0.911 62.33 0.944 136.0 0.965 238.3 1.045 10.02 0.504 10.02 0.560 10.02 0.482 10.02 0.527 4.89 0.643 6.1 0.670 7.33 0.666 9.21 0.813 8.97 0.826 5.6 0.809 10.27 0.981 3.61 0.955 7.92 0.802 4.11 0.865 0.88 0.978 1.18 1.009 0.800 0.123

23.99 0.860 37.72 0.890 53.11 0.848 96.64 0.934 152.4 0.996 7.45 1.042 11.19 1.019 13.46 0.961 17.85 0.974 64.52 0.978 142.9 1.014 254.5 1.116 12.22 0.614 12.22 0.683 12.22 0.588 12.22 0.643 5.36 0.705 6.73 0.740 8.04 0.731 9.83 0.868 9.59 0.883 5.99 0.866 10.8 1.032 3.79 1.003 8.33 0.844 4.33 0.912 0.93 1.033 1.24 1.060 0.886 0.115

24.33 0.872 38.27 0.903 53.95 0.862 98.01 0.947 154.4 1.009 7.55 1.056 11.43 1.041 13.6 0.971 17.91 0.977 64.31 0.974 143.6 1.019 257.1 1.128 13.81 0.694 13.81 0.772 13.81 0.664 13.81 0.727 5.81 0.764 7.14 0.785 8.47 0.770 10.25 0.905 9.71 0.894 6.07 0.877 11.03 1.053 3.85 1.019 8.4 0.851 4.34 0.914 0.92 1.022 1.22 1.043 0.903 0.124

23.8 37.79 50.42 96.59 156.4 7.32 10.37 12.76 18.13 70.07 138.8 233.6 14.2 14.2 14.2 14.2 5.88 7.22 8.57 11.71 9.68 6.3 11.69 3.91 8.17 4.74 0.93 1.12 0.903 0.106

Muc Muc Muc =Mue Muc Muc = (kN m) (kN m) (kN m) Mue 0.853 0.891 0.805 0.933 1.022 1.024 0.944 0.911 0.989 1.062 0.985 1.025 0.714 0.793 0.683 0.747 0.774 0.793 0.779 1.034 0.891 0.910 1.117 1.034 0.828 0.998 1.033 0.957

Muc (kN m)

Formula (13)

Muc Muc = (kN m) Mue

EC4 (1994)

BS5400 (1979)

AIJ (1997)

– –

[25]

[14]

[20]

[24]

[19]

Test data resources

332 L.-H. Han / Journal of Constructional Steel Research 60 (2004) 313–337


333

The predicted initial section flexural stiffness (Kic) of the composite beams using the different methods is compared with the current experimental results and the results reported by Lu and Kennedy [23] (Kie) in Table 3. Table 3 shows both the mean value and the standard deviation (COV) of the ratio of Kic/Kie for the different design methods. The results in this table clearly show that all the methods of BS5400 (1979), EC4 (1994) and LRFD-AISC (1999) are un-conservative. Overall, BS5400 (1979), EC4 (1994) and LRFD-AISC (1999) gave an initial section flexural stiffness about 15–18% higher than that of test. The AIJ (1997) method gives a mean of 0.962 and a COV of 0.133, predicted a slightly lower stiffness than the test results, is the best predictor. Predicted serviceability-level section flexural stiffness (Ksc) of the composite beam using different methods is compared with the current experimental results (Kse) in Table 4. Table 4 shows both the mean value and the standard deviation (COV) of the ratio of Ksc/Kse for the different design methods. The results in this table show that all the methods of BS5400 (1979), EC4 (1994) and LRFD-AISC (1999) gave the serviceability-level section flexural stiffness about 20–25% higher than that of test. The AIJ (1997) method gives a mean value of 1.035 and a COV of 0.078, predicted a slightly higher stiffness than the test results, and is the best predictor.

5. Conclusions The following observations and conclusions can be drawn based on the limited research reported in the paper. (1) Because of the infill of concrete, the tested concrete-filled steel SHS and RHS beams behaved in a relatively ductile manner and testing proceeded in a smooth and controlled way. The enhanced structural behaviour of the columns can be explained in terms of ‘‘composite action’’ between the steel tube and the concrete core. (2) The predicted load vs. lateral deflection curves for the composite beams have been found in good agreement with experimental values. The predicted maximum strength of beams agrees well with the tested values. (3) Simplified methods for the calculations of the moment capacities of the composite beams have been proposed based on the mechanics model in this paper. (4) The moment capacity of concrete-filled HSS beams could be conservatively predicted by using the recommendations of AIJ (1997), BS5400 (1979), EC4 (1994), LRFD-AISC (1999), and the proposed model in this paper. Overall, AIJ (1997) and LRFD-AISC (1999) gave a moment capacity about 20% lower than that of test, BS5400 (1979) gave a moment capacity about 12% lower than that of test. EC4 (1994) and the proposed method predicted about 10% lower capacity than the test results, are the best predictor. (5) The initial section flexural stiffness (Ki) and the serviceability-level section flexural stiffness (Ks) of the composite beams were defined in this paper. It was found that all the methods of BS5400 (1979), EC4 (1994) and LRFD-AISC

Sectional dimension (mm)

1 &-120 120 3:84 2 &-120 120 3:84 3 &-120 120 3:84 4 &-120 120 3:84 5 &-120 120 5:86 6 &-120 120 5:86 7 &-120 120 5:86 8 &-120 120 5:86 9 &-150 120 2:93 10 &-150 120 2:93 11 &-120 90 2:93 12 &-120 90 2:93 13 &-150 90 2:93 14 &-150 90 2:93 15 &-120 60 2:93 16 &-120 60 2:93 17 &-152 152 4:8 18 &-152 152 4:8 19 &-152 152 4:8 20 &-152 152 9:5 21 &-254 152 6:4 22 &-254 152 6:4 23 &-254 152 6:4 24 &-254 152 9:5 25 &-254 152 9:5 26 &-152 254 6:4 27 &-152 254 6:4 28 &-152 254 6:4 Mean value Standard deviation (COV)

Number

330.1 330.1 330.1 330.1 321.1 321.1 321.1 321.1 293.8 293.8 293.8 293.8 293.8 293.8 293.8 293.8 389 389 389 432 377 377 377 394 394 377 377 377

fsy (MPa)

18.3 23.6 23.6 23.6 21 2 26.8 26.8 23.1 23.1 23.1 23.1 23.1 23.1 23.1 23.1 47 42.8 41.2 46.9 46.7 45.2 44.3 46.2 43.8 47.1 42.1 40.5

fck (MPa)

896 960 1002 856 1356 1409 1360 1184 1607 1746 749 722 1216 1269 489 485 2073 2209 2332 3478 9506 9763 10639 12017 12719 4177 4575 4578

Kie (kN m2)

886 894 894 894 1253 1254 1261 1261 1227 1227 574 574 993 993 431 431 2392 2335 2318 4007 10565 10454 10405 14266 14137 4601 4497 4470

0.989 0.931 0.892 1.044 0.924 0.890 0.927 1.065 0.764 0.703 0.768 0.798 0.817 0.783 0.881 0.889 1.154 1.057 0.994 1.152 1.111 1.071 0.978 1.187 1.111 1.101 0.983 0.976 0.962 0.133

992 1038 1038 1038 1370 1370 1406 1406 1527 1527 684 684 1215 1215 502 502 3161 3068 3034 4594 14221 14068 13969 17497 17272 5819 5646 5590

Kic (kN m2)

Kic (kN m2) 1.107 1.081 1.036 1.213 1.010 0.972 1.034 1.188 0.950 0.875 0.916 0.951 0.999 0.957 1.027 1.035 1.525 1.389 1.301 1.321 1.496 1.441 1.313 1.456 1.358 1.393 1.234 1.221 1.171 0.198

Kic =Kie

BS5400 (1999)

AIJ (1997) Kic =Kie

Table 3 Comparisons between predicted initial section flexural stiffness (Kic) and test results

1052 1067 1067 1067 1401 1401 1413 1413 1595 1595 709 709 1265 1265 518 518 2914 2820 2793 4406 13029 12846 12761 16422 16206 5430 5258 5214 1.147 0.140

Kic (kN m2)

EC4 (1994)

1.174 1.111 1.065 1.246 1.033 0.994 1.039 1.193 0.993 0.914 0.949 0.986 1.040 0.997 1.059 1.068 1.406 1.276 1.198 1.267 1.371 1.316 1.199 1.367 1.274 1.300 1.149 1.139

Kic =Kie 1038 1071 1071 1071 1382 1382 1412 1412 1623 1623 715 715 1283 1283 520 520 3195 2968 2902 4550 14323 13880 13684 17323 16809 5856 5442 5334 1.182 0.177

Kic (kN m2) 1.158 1.116 1.069 1.251 1.019 0.981 1.038 1.193 1.010 0.930 0.957 0.994 1.055 1.011 1.063 1.072 1.541 1.344 1.244 1.308 1.507 1.422 1.286 1.442 1.322 1.402 1.190 1.165

Kic =Kie

LRFD–AISC (1999)

– –

[23]

Current test

Test data resources

334 L.-H. Han / Journal of Constructional Steel Research 60 (2004) 313–337

1 &-120 120 3:84 2 &-120 120 3:84 3 &-120 120 3:84 4 &-120 120 3:84 5 &-120 120 5:86 6 &-120 120 5:86 7 &-120 120 5:86 8 &-120 120 5:86 9 &-150 120 2:93 10 &-150 120 2:93 11 &-120 90 2:93 12 &-120 90 2:93 13 &-150 90 2:93 14 &-150 90 2:93 15 &-120 60 2:93 16 &-120 60 2:93 Mean value Standard deviation (COV)


330.1 330.1 330.1 330.1 321.1 321.1 321.1 321.1 293.8 293.8 293.8 293.8 293.8 293.8 293.8 293.8

18.3 23.6 23.6 23.6 21.98 21.98 26.8 26.8 23.1 23.1 23.1 23.1 23.1 23.1 23.1 23.1

890 840 894 852 1224 1265 1116 1160 1037 1106 598 613 879 953 469 457

886 894 894 894 1253 1254 1261 1261 1227 1227 574 574 993 993 431 431 1.035 0.078

Ksc (kN m2)

fsy fck Ksc AIJ (1997) (MPa) (MPa) (kN m2)

0.995 1.064 1.000 1.049 1.024 0.991 1.130 1.087 1.183 1.109 0.959 0.936 1.130 1.042 0.919 0.943

Ksc =Kse 992 1038 1038 1038 1370 1370 1406 1406 1527 1527 684 684 1215 1215 502 502 1.209 0.120

1.115 1.236 1.161 1.218 1.119 1.083 1.260 1.212 1.473 1.381 1.144 1.116 1.382 1.275 1.070 1.098

1052 1067 1067 1067 1401 1401 1413 1413 1595 1595 709 709 1265 1265 518 518 1.247 0.129

1.182 1.270 1.193 1.252 1.144 1.107 1.266 1.218 1.538 1.442 1.185 1.156 1.439 1.328 1.105 1.134

Ksc =Kse

Ksc (kN m2)

Ksc (kN m2) Ksc =Kse

EC4 (1994)

BS5400 (1999)

Table 4 Comparisons between predicted serviceability-level section flexural stiffness (Ksc) and test results

1038 1071 1071 1071 1382 1382 1412 1412 1623 1623 715 715 1283 1283 520 520

1.166 1.275 1.198 1.257 1.129 1.093 1.265 1.217 1.565 1.467 1.196 1.167 1.459 1.346 1.109 1.138 1.253 0.147

Ksc Ksc =Kse (kN m2)

LRFD–AISC (1999)


336


(1999) are un-conservative in predicting the values of Ki and Ks. The AIJ (1997) method predicted a slightly lower stiffness value of Ki and a slightly higher value of Ks than those of test results, and thus is acceptable for the calculation of the flexural stiffness of concrete-filled steel SHS and RHS beams.

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