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Experimental investigation of cold-formed high strength steel tubular beams Jia-Lin Ma a, Tak-Ming Chan b, Ben Young a,⇑ a b

Dept. of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, China Dept. of Civil and Environmental Engineering, The Hong Kong Polytechnic University, Hong Kong, China

a r t i c l e

i n f o

Article history: Received 4 September 2015 Revised 4 July 2016 Accepted 15 July 2016

Keywords: Beam test Cold-formed steel High strength steel Slenderness limit Tubular section

a b s t r a c t This paper presents a test program on cold-formed high strength steel tubular beams. The nominal 0.2% proof stresses of the high strength steel were 700, 900 and 1100 MPa in this study. Twenty-five four-point bending tests on circular, rectangular and square hollow structural sections were conducted. Load-deformation histories and failure modes of the beams were reported. Experimental results were compared against design values calculated from European code, Australian standard and North American specification for hot-rolled and cold-formed steel structures. In addition, the test strengths were also compared with the Direct Strength Method predictions. The compactness criteria were assessed by comparing the section slenderness to the slenderness limits in codes. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction High Strength Steel (HSS) has considerable advantages over mild steel in terms of the strength-to-weight ratio and the material cost. Owning to the rapid development in material and manufacturing technologies, the yield strength of steel has increased in the last few decades. High strength steel tubes with nominal yield strengths (proof stresses) of 700, 900 and 1100 MPa are now commercially available. Their potential use should be exploited to achieve more economic design in steel structures. As part of a wider research program, this paper thus presents the experimental findings on the cross-sectional flexural behaviour of HSS tubes in 3 different grades: namely, H-Series, V-Series and S-Series (with nominal proof stresses of 700, 900 and 1100 MPa, respectively). High strength steel beams behave differently from mild steel beams because of the high material strength and relatively lower ductility. In the past decades, investigations have been conducted on HSS fabricated I-section beams. Beg and Hladnik [1] conducted 10 tests on welded HSS I-beams with nominal yield stress fy = 700 MPa (NIONICRAL 70) and results showed that the flange and web interaction can significantly affect the ultimate carrying capacities of the sections. Ricles et al. [2] and Green et al. [3] found there is less rotation capacity available in the HSS beams with nominal yield stress fy = 552 MPa than their mild-steel counter-

parts (fy = 250 MPa) through testing welded I beams. It was found that the rotation capacities for beams also vary from uniform moment loading conditions to moment gradient ones. Lee et al. [4] investigated the flexural behaviour of full-scale I-shaped beams built up from high-strength steels with nominal yield stresses of fy = 650 MPa and fy = 690 MPa and compared the results to the behaviour of ordinary steel beams fabricated from plates with nominal yield stress of fy = 490 MPa. The specimens showed sufficient strength for elastic design whereas their rotation capacities for plastic design were marginal. Generally HSS beams have low residual stress to 0.2% proof stress ratios, good flexural resistances but limited rotation capacities. The rotation capacities were found to decrease with the increase in material strength, flange slenderness and web slenderness. Jiao and Zhao [5] tested cold-formed HSS circular hollow sections (CHS) with nominal 0.2% proof stress as high as 1300 MPa. The nominal outer diameter of the test specimens ranged from 31.8 mm to 75 mm while the D/t ratios ranged from 16 to 48. It was suggested that different plastic slenderness limits and yield slenderness limits might apply for HSS circular hollow sections. Further investigation is needed to determine the effect of yield stresses on the section slenderness limits. 2. Experimental investigation 2.1. Test specimens

⇑ Corresponding author. E-mail address: [email protected] (B. Young). http://dx.doi.org/10.1016/j.engstruct.2016.07.027 0141-0296/Ó 2016 Elsevier Ltd. All rights reserved.

The beam specimens were cut from the same batch of cold-formed HSS tubular sections as described in Ma et al. [6]. Nine

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Nomenclature B b D dmax dmin E fy H h I k kp Ls Lm Mexp Mp My Mu MAISC MAS4100 MEC3 MAISI MDSM MDSM-IR P

overall width of cross-section width of flat portion in cross-section outer diameter of circular hollow section measured maximum internal diameter measured minimum internal diameter modulus of elasticity yield stress of steel (0.2% proof stress) overall depth of cross-section depth of flat portion in cross-section second moment of area curvature of moment span elastic curvature corresponding to the plastic moment length of shear span in four-point-bending test length of moment span in four-point-bending test experimental ultimate moment of cross-section plastic moment of cross-section yield moment of cross-section ultimate moment of cross-section nominal strength (unfactored design strength) from ANSI/AISC 360-10 nominal strength (unfactored design strength) from AS 4100 nominal strength (unfactored design strength) from EN 1993 nominal strength (unfactored design strength) from AISI S100 nominal strength (unfactored design strength) from Direct Strength Method without inelastic reserve nominal strength (unfactored design strength) from Direct Strength Method with inelastic reserve applied load

square hollow sections (SHS), two rectangular hollow sections (RHS) and six circular hollow sections (CHS) were investigated in this study. According to their nominal yield stress, fy, they are categorized into: H-Series (fy = 700 MPa), V-Series (fy = 900 MPa) and S-Series (fy = 1100 MPa). The nominal width and depth of the SHS and RHS ranged from 50 mm to 200 mm while the nominal outer diameter of the CHS varied from 89 mm to 139 mm. The nominal plate width-to-thickness and depth-to-thickness ratios of SHS and RHS ranged from 8 to 35 while the nominal outer diameterto-thickness ratios of CHS ranged from 22 to 34. All SHS and RHS specimens are labelled as ‘‘Series, flange, web, thickness”, and

R Req Rc r t Ur Wpl Wel y

a b d

ru r0:2 e25mm kp kpf kpw ky kyf kyw hmax hp

outer corner radius of square and rectangular hollow sections equivalent radius of curvature on moment span measured rotation capacity on moment span inner corner radius of square and rectangular hollow sections plate or wall thickness out-of-roundness parameter plastic section modulus elastic section modulus measured deflection in moment span obtained from transducers square/rectangular hollow section plate element slenderness circular hollow section slenderness measured local geometric imperfection static ultimate tensile strength static 0.2% tensile proof stress non-proportional elongation at fracture based on gauge length of 25 mm plastic slenderness limit plastic slenderness limit of flange plastic slenderness limit of web yield slenderness limit yield slenderness limit of flange yield slenderness limit of web total end rotation at plastic moment during unloading response of beam elastic end rotation corresponding to plastic moment

CHS specimens are labelled as ‘‘Series, outer diameter, thickness”. The flexural behaviour of the beams was investigated through four-point bending tests. Both major-axis and minor-axis flexural tests were conducted for the RHS beams. RHS beam specimen H50 100 4-B refers to major axis bending and the ‘B’ indicates that it is a beam specimen. Tables 1 and 2 show the measured cross-section dimensions using the nomenclature from Fig. 1. The symbol ‘#’ denotes that it is a repeated test. The calculated elastic section modulus Wel, the plastic section modulus Wpl, the shear span length Ls and the moment span length Lm are also tabulated in the last four columns of the tables.

Table 1 Measured SHS and RHS beam dimensions.

#

Specimen B H t (mm)

B (mm)

D (mm)

t (mm)

R (mm)

r (mm)

Wel (103 mm3)

Wpl (103 mm3)

Ls (mm)

Lm (mm)

H80 80 4-B H80 80 4-B# H100 100 4-B H120 120 4-B H140 140 5-B H140 140 5-B# H140 140 6-B H160 160 4-B H100 50 4-B H50 100 4-B H200 120 5-B H120 200 5-B

80.3 80.2 100.3 121.0 141.3 141.2 140.9 160.6 100.3 50.3 200.4 121.2

80.1 80.1 100.4 121.4 140.3 140.5 141.2 160.5 50.3 100.2 121.5 200.4

3.92 3.93 3.94 3.95 4.94 4.95 5.98 3.99 3.93 3.97 4.95 4.95

9.5 9.5 8.5 8.0 12.0 12.0 13.0 10.5 8.5 8.5 13.0 13.0

5.0 5.0 4.3 4.0 7.0 7.0 7.0 6.0 3.5 3.5 7.5 7.5

26.9 26.9 44.8 67.7 111.7 112.0 132.6 32.2 17.9 26.4 124.1 161.9

32.1 32.0 52.7 79.0 130.9 131.2 156.8 141.5 20.6 33.6 140.5 197.6

500 500 600 600 800 800 800 800 500 500 800 800

450 450 600 600 600 600 600 600 450 450 600 600

V80 80 4-B V100 100 4-B V120 120 4-B V120 120 4-B#

80.3 100.2 120.9 120.9

80.2 100.2 121.1 121.1

3.95 3.96 3.93 3.91

10.0 11.5 9.5 9.5

6.0 7.0 6.0 6.0

27.1 43.6 66.9 66.6

32.3 51.5 78.1 77.7

500 600 600 600

450 600 600 600

Repeated test.

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Table 2 Measured CHS beam dimensions.

#

Table 4 CHS tensile coupon test results.

Specimen D t (mm)

D (mm)

t (mm)

Wel (103 mm3)

Wpl (103 mm3)

Ls (mm)

Lm (mm)

V89 3-B V89 3-B# V89 4-B

88.8 89.0 89.0

2.96 2.95 3.90

16.6 16.6 21.3

21.8 21.9 28.3

500 500 500

450 450 450

S89 4-B S89 4-B# S108 4-B S133 4-B S133 4-B# S139 6-B

88.9 89.0 108.2 133.4 133.4 139.4

3.86 3.90 3.90 3.93 3.91 5.93

21.0 21.2 32.2 50.3 50.0 79.6

27.9 28.2 42.5 65.9 65.6 105.6

500 500 500 600 600 600

450 450 450 600 600 600

Repeated test.

E (GPa)

r0:2 (MPa)

ru (MPa)

e25mm (%)

V89 3 V89 4

209 210

1054 1053

1124 1108

10 12

S89 4 S108 4 S133 4 S139 6

205 215 204 194

1180 1180 1159 1014

1317 1292 1291 1382

11 10 10 10

δ

δ Weld

Weld

H

Section D t (mm)

Flat coupon t

t

Curved coupon

r

Section nominal profile Section profile with local imperfection

R

D

B

Fig. 2. Definition of local imperfection.

Fig. 1. Definition of symbols and location of tensile coupons.

2.2. Tensile coupon tests and local imperfection measurements High strength steel normally has no yield plateau and displays insignificant strain hardening [7]. The 0.2% proof stress r0:2 of HSS is normally taken as the yield stress fy. To examine the material properties of the HSS cold-formed structural hollow sections, tensile coupons were extracted from tubes and tested in a 50 kN capacity MTS machine. Results from tensile coupon tests are summarized in Tables 3 and 4. It can be seen that the coupon specimens of S-series and V-series are generally higher in strength but lower in ductility than those from H-series. The 0.2% proof stresses of H-Series, V-Series and S-Series ranged from 663 MPa to 1180 MPa. A more detailed discussion on the material properties of these specimens can be found in Ma et al. [6]. Geometric local imperfections influence buckling behaviour of structural members. The local imperfection magnitudes of different sections are normally measured as part of the experimental investigation [8,9]. The maximum geometric local imperfections d (Fig. 2) were measured on 8 of the stub column specimens cut from the same batch of tubes with the beam specimens. The methodology applied in the study is explained in Ma et al. [10].

Table 5 Out-of-roundness and fabrication tolerance quality classes of tubes. Specimen D t (mm)

dmax (mm)

dmin (mm)

Ur

Class

V89 3-S V89 4-S

83.4 81.5

82.5 80.6

0.010 0.011

A A

S89 4-S S108 4-S S133 4-S S139 6-S

81.4 101.2 125.8 128.0

80.7 99.9 124.6 125.9

0.008 0.013 0.010 0.017

A A A B

The local geometric imperfections were assumed to be proportional to either the plate element slenderness a = (B-2R)/t for SHS/RHS or b = D/t for CHS, therefore the d values were further normalized to the element and section slenderness. The averaged maximum local imperfection to plate element slenderness ratios d/a were reported to be 0.0119 and 0.0146 for H-Series and VSeries, respectively. The averaged maximum local imperfections to section slenderness ratios d/b for CHS were reported to be 0.0058 and 0.0051 for V-Series and S-Series, respectively. To evaluate the cross-section flexural capacity of slender CHS, the out-ofroundness Ur and tolerance classes for all the CHS sections have also been evaluated in accordance with EN 1993-1-6 [11] and are summarized in Table 5.

Table 3 SHS, RHS tensile coupon test results. Section B H t (mm)

E (GPa)

r0:2 (MPa)

ru (MPa)

e25mm (%)

H80 80 4 H100 100 4 H120 120 4 H140 140 5 H140 140 6 H160 160 4 H100 50 4 H200 120 5

218 218 212 206 201 220 208 207

792 735 689 708 663 744 724 738

888 841 813 844 808 869 831 846

14 14 17 20 18 14 17 18

V80 80 4 V100 100 4 V120 120 4

210 207 204

1005 978 960

1187 1115 1153

11 11 13

2.3. Four-point bending tests To study the flexural behaviour of HSS beams, Fig. 3 shows the schematic set up for the four-point bending tests. The length of the shear spans was long enough to ensure that the section bending capacity would be reached first before shear failures. The special bearing shown in Fig. 3 eliminated possible gaps between the spreader beam and the two loading points, and was locked by four bolts after the pre-load. Three transducers were arranged below the specimen within the moment span to capture the deflection. The readings from the transducers can also be used to calculate the curvature. As shown in Fig. 4, assuming that the tensile side

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Special bearing Spreader beam Half-round support Stiffening plate

Inclinometers

Roller

Transducers Shear span Ls

Moment span Lm

Shear span Ls

(a) Special bearing Spreader beam

Inclinometers

Half-round support CHS Sitting Roller

Transducers Shear span Ls

Moment span Lm

Shear span Ls

(b) Stiffening plate

Rubber sheet CHS sitting

Stiffening woods

(c) Fig. 3. Schematic arrangement for beam tests of (a) SHS/RHS, (b) CHS and (c) stiffening at supports and loading points only.

of the beam members gives a smooth circular arch, the averaged curvature of moment span can be calculated using Eq. (1), in which Req is the equivalent radius and y is the deflection in the moment span obtained from transducer readings. Two inclinometers were attached at the ends of the moment span to capture the rotations.

k¼

1 8y ¼ Req 4y2 þ L2m

ð1Þ

Different stiffening systems were applied at the loading points and supports depending on the section shapes to allow uniform load transfers and avoid local premature failures. As shown in Fig. 3(a) and (c), for SHS and RHS, four sets of stiffening plates with width of 90 mm were applied on the webs to prevent web crippling at loading points and supports. Stiffening wood with length of 100 mm were also used inside the tubes. As shown in Fig. 3 (b) and (c), a set of special sitting and wood stiffeners were used to spread the forces onto the CHS beam specimens. Plastic rubber

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k = 1/Req

q

Req-y

Re

moments Mp and the curvatures normalized to kp = Mp/EI. The failure of the beams occurred in the moment spans for all test specimens. It shows that the test setup was able to obtain the moment capacities of the beam specimens. The failure of the specimens H140 140 5-B, H160 160 4-B, H200 120 5-B and S133 4-B are shown in Figs. 6–9, respectively. The experimental ultimate moments Mexp are summarized and compared to section elastic bending moments My and plastic bending moments Mp in Tables 6 and 8. The comparison of the test strengths with the predicted strengths calculated from European standards EN 19931-1 [12], EN 1993-1-5 [13], EN 1993-1-6 [11] (MEC3), American standards ANSI/AISC 360-10 [14] (MAISC), AISI S100 [15] (MAISI), Australian standard AS 4100 [16] (MAS4100) and the Direct Strength Method (MDSM, MDSM-IR) are also given in the tables. MDSM-IR and MDSM stand for the predictions using Direct Strength Method (DSM) [15] with and without inelastic reserve consideration, respectively. A reliability analysis was also carried out for the comparison between test results and code predictions. The obtained reliability indexes are summarized in Tables 6 and 8. The rotation capacities Rc were also evaluated using Eq. (2), in which hmax is the total end rotation at plastic moment Mp during the unloading response of the beams and hp is the elastic end rotation at Mp [17], as shown in Fig. 10. For sections failed to reach the plastic moment Mp, the rotation capacity Rc is equal to zero (Rc = 0). In general, the available rotation capacity is related to flange and web slenderness, lateral bracing, moment gradient and other factors [17]. The beams tested in this study were provided with sufficient lateral restraints, and the investigation focuses on the cross-

y

Lm/2 Lm Fig. 4. Calculation of curvature using transducers.

sheets were applied between sittings and CHS surfaces to ensure uniform contact. 3. Test results 3.1. General

1.4

1.4

1.2

1.2

1.0

1.0

0.8

0.8 H140×140×5-B H140×140×6-B H160×160×4-B H100×50×4-B H50×100×4-B H200×120×5-B H120×200×5-B

0.6

0.6 H80×80×4-B V80×80×4-B H100×100×4-B V100×100×4-B H120×120×4-B V120×120×4-B

0.4 0.2

0.4 0.2 0.0

0.0

0

1

2

k/k p

3

0

2

(a)

k/k p

(b)

1.4 1.2 1.0

M/Mp

M/Mp

The moment-curvature curves of the beam tests are plotted in Fig. 5(a)–(c), with the moments normalized to the section plastic

0.8 V89×3-B V89×4-B S89×4-B S108×4-B S133×4-B S139×6-B

0.6 0.4 0.2 0.0

0

1

2

3

4

5

6

k/k p

(c) Fig. 5. Normalized moment versus curvature curves of the beam tests.

4

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J.-L. Ma et al. / Engineering Structures 126 (2016) 200–209

Fig. 6. Failure of specimen H140 140 5-B in moment span.

205

Fig. 9. Failure of specimen S133 4-B in moment span.

sections according to the readings obtained from the two inclinometers.

Rc ¼

hmax 1 hp

ð2Þ

3.2. Square and rectangular hollow sections

Fig. 7. Failure of specimen H160 160 4-B in moment span.

Fig. 8. Failure of specimen H200 120 5-B in moment span.

sectional behaviour within the constant moment span in the fourpoint bending tests. Thus, the rotation capacity is only evaluated for the constant moment span to assess the ductility of the cross-

For square and rectangular hollow sections (SHS and RHS) in pure bending, four specimens (H160 160 4-B, H200 120 5-B, V120 120 4-B, V120 120 4-B#) failed earlier than reaching the plastic bending moment, while the rest of the specimens reached moment capacities that were higher than the plastic bending moment, as shown in Table 6. The specifications give conservative predictions for both compact and slender sections. The ANSI/AISC 360-10 [14] provides the closest prediction on average. The average Mexp/MAISC ratio equals to 1.17, the COV value is 0.047 and the corresponding reliability index is 3.2 for the resistance factor of 0.9. The cold-formed specifications AISI S100 [15] and the Direct Strength Method (without inelastic reserve) predict the capacities most conservatively, because the bending moment capacity is evaluated up to the section elastic bending moment. The inelastic reserve method in Direct Strength Method improves the prediction by around 5% on average, because it predicts the capacities more accurately for sections with low slenderness values. In EN 1993-1-1 [12], both class 1 and 2 sections should attain higher moment capacities than Mp whereas class 1 sections can form a plastic hinge with the rotation capacity required from plastic analysis without reduction of the resistance. To verify the class 1 limit for EN 1993-1-1 [12], the rotation capacity obtained from the tests are plotted against the normalized flange and web slenderness in Fig. 11(a) and (b), respectively. Galambos [18] and Neal [19] suggested a rotation capacity Rc of 2 (dotted line in Fig. 11). Yura et al. [20] later suggested a rotation capacity Rc of 3 (dashed line in Fig. 11), which is now widely accepted by various researchers [4,21–24] and adopted in EN 1993-1-1 [12], ANSI/AISC 360-10 [14]. In this study, most of the sections are square hollow sections, and the webs for the two rectangular hollow sections are not slender, thus the slenderness of the flanges are governing the classification for all the specimens. The majority of the specimens failed to reach the rotation capacity of 3. The test of H50 100 4-B specimen was terminated before the moment dropped to the plastic moment due to large curvature of the specimen and safety reason. However, the specimen had reached a rotation capacity larger

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Table 6 Comparison of SHS and RHS beam test strengths with design strengths. Specimen B H t (mm)

Mexp (kNm)

M exp My

M exp Mp

M exp M AISC

M exp M AS4100

M exp MEC3

M exp M AISI

M exp M DSM

M exp M DSMIR

Rc

H80 80 4-B H80 80 4-B# H100 100 4-B H120 120 4-B H140 140 5-B H140 140 5-B# H140 140 6-B H160 160 4-B H100 50 4-B H50 100 4-B H200 120 5-B H120 200 5-B

28.2 28.1 42.5 56.1 97.6 97.6 121.7 84.4 16.9 30.9 90.3 157

1.33 1.32 1.29 1.20 1.24 1.23 1.36 0.93 1.32 1.62 0.99 1.31

1.11 1.11 1.10 1.03 1.05 1.05 1.15 0.80 1.13 1.27 0.87 1.08

1.11 1.11 1.17 1.23 1.23 1.22 1.15 1.08 1.23 1.27 1.19 1.10

1.11 1.11 1.22 1.27 1.23 1.23 1.18 1.16 1.26 1.27 1.28 1.15

1.11 1.11 1.10 1.27 1.24 1.23 1.15 1.10 1.32 1.27 1.21 1.08

1.33 1.32 1.29 1.26 1.27 1.26 1.36 1.10 1.32 1.62 1.21 1.31

1.33 1.32 1.29 1.20 1.24 1.23 1.36 1.10 1.32 1.62 1.17 1.31

1.24 1.24 1.25 1.19 1.22 1.21 1.30 1.10 1.27 1.40 1.17 1.26

3.9 4.0 2.0 1.1 1.3 1.1 2.4 0.0 3.7 >10.3 0.0 1.4

V80 80 4-B V100 100 4-B V120 120 4-B V120 120 4-B#

37.5 50.9 68.6 68.5

1.38 1.19 1.07 1.07

1.15 1.01 0.92 0.92

1.15 1.16 1.16 1.17

1.17 1.17 1.20 1.21

1.15 1.19 1.18 1.18

1.38 1.21 1.18 1.19

1.38 1.19 1.17 1.18

1.31 1.18 1.17 1.18

2.7 1.1 0.0 0.0

1.17 0.047 0.90 3.20

1.20 0.046 0.90 3.07

1.18 0.061 1.00 2.60

1.29 0.089 0.90 3.39

1.28 0.095 0.90 3.32

1.23 0.056 0.90 3.37

Mean COV / b Repeated test.

than 10.3 as specified in Table 6, and the value 10.3 was plotted in Fig. 11. For the cold-formed HSS beams in this study, the rotation capacity of beams reduces significantly as the flange slenderness qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ b=t f y =E increases for both H-series and V-series. It is shown that the class 1 limit is suitable for the rotation capacity requirement of 2, but it is slightly unsafe for 3. Thus cold-formed high strength steel sections with grades higher than 700 MPa should be used with care in design, especially for structures where moment redistribution is needed. The results obtained from high strength steel in this study reveal that low material ductility reduces the rotation capacity of the beam members. Further investigation is needed to study the required rotation capacity and the proper class 1 limit for high strength steel beam members. The plastic and yield slenderness limits for SHS and RHS in bending are summarized in Table 7 for different specifications, in qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ which the normalized plate slenderness b=t f y =E is used to harmonize the comparison between different specifications. kpf and kpw are the plastic slenderness limits for flange and web

Fig. 10. Definition of rotation capacity.

respectively, while kyf and kyw are the yield slenderness limits for

Rotation capacity Rc

#

12 11 10 9 8 7 6 5 4 3 2 1 0

12 11 10 9 8 7 6 5 4 3 2 1 0

H-series V-series Yura et al. [18] Galambos [17] Class 1 limit

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0

0.5

1.0

1.5

2.0

(b/t)√( fy /E)

(h/t)√( fy /E)

(a)

(b)

Fig. 11. Comparison with the class 1 limit in EN 1993-1-1 for (a) flange and (b) web.

2.5

3.0

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J.-L. Ma et al. / Engineering Structures 126 (2016) 200–209

yield slenderness limit. As the web slenderness is not governing in the classifications, all the test results lie on the left hand side of the codified limits shown in Figs. 12(b) and 13(b). The plastic slenderness limits are generally conservative for the flanges. The stub column test results reported in Ma et al. [10] indicated that the codified yield slenderness limits are generally applicable to the compression internal elements in the case that all four sides of the sections are in compression. However, Fig. 13 shows that the yield slenderness limits are conservative for compression internal elements (flanges) when the webs are compact and under the stress-gradient. To summarize, the results shown that the class 1 slenderness limit and plastic slenderness limit for flanges from EN 1993-1-1 [12], yield slenderness limit from ANSI/AISC 360-10 [14] are recommended to be used for the studied range of cold-formed high strength steel SHS and RHS beams, although the plastic slenderness limits and yield slenderness limits are slightly conservative. The slenderness limits for webs of SHS and RHS cannot be verified in this study as the flange slenderness is more slender and governing the design of the beam members.

Table 7 Normalized slenderness limits in codes for plate elements in SHS and RHS subjected to bending. References

Flange

EN 1993-1-1 [12] EN 1993-1-5 [13] ANSI/AISC 360-10 [14] AS 4100 [16] AISI S100 [15] / AS/NZS 4600 [25]

Web

kpf

kyf

kpw

kyw

1.271 N/A 1.120 1.061 N/A

1.405 1.279 1.400 1.414 1.280

2.777 N/A 2.420 2.899 N/A

4.148 4.060 5.700 4.066 N/A

Mexp/M p

flange and web respectively. The section should be able to reach the section plastic moment Mp = Wplfy if both the web and flange slenderness are smaller than the plastic slenderness limits specified in Table 7. Similarly, if the web and flange slenderness are lower than the yield slenderness limits, the moment capacities of the sections should be higher than the section elastic moments My = Welfy. Plastic slenderness limits are not applicable for EN 1993-1-5 [13] and AISI S100 [15]. It should be noted that the yield slenderness limits in EN 1993-1-1 [12] are less stringent than those in EN 1993-1-5 [13]. To assess the plastic slenderness limits, the ultimate test moment capacities Mexp are normalized to Mp and then plotted against the normalized flange and web slenderness in Fig. 12. Similarly, Mexp is normalized to My and plotted in Fig. 13 to verify the

3.3. Circular hollow sections Table 8 compares the ultimate bending moment capacity of the circular hollow sections to the elastic bending moment My, plastic bending moment Mp and predictions from different specifications.

1.6

1.6

1.4

1.4

1.2

1.2

1.0

1.0

0.8

0.8 H-series V-series EN 1993-1-1 ANSI/AISC 360-10 AS 4100

0.6 0.4 0.2 0.0 0

1

2

3

0.6 0.4 0.2 0.0 4

0

1

2

3

(b/t)√( fy /E)

(h/t)√( fy /E)

(a)

(b)

4

Mexp/My

Fig. 12. Comparison with the plastic slenderness limits for (a) flange and (b) web.

1.6

1.6

1.4

1.4

1.2

1.2

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

H-series V-series EN 1993-1-1 EN 1993-1-5 ANSI/AISC 360-10 AS 4100

0.0 0.0

0.5

1.0

1.5

2.0

2.5

0

1

2

3

4

(b/t)√( fy /E)

(h/t)√( fy /E)

(a)

(b)

Fig. 13. Comparison with the yield slenderness limits for (a) flange and (b) web.

5

6

208

J.-L. Ma et al. / Engineering Structures 126 (2016) 200–209

Table 8 Comparison of CHS beam test strengths with design strengths. Mexp (kNm)

M exp My

M exp Mp

M exp M AISC

M exp M AS4100

M exp M EC3

M exp M AISI

M exp M DSM

M exp M DSMIR

Rc

V89 3-B V89 3-B# V89 4-B

23.8 23.8 30.9

1.36 1.36 1.38

1.04 1.03 1.04

1.20 1.19 1.17

1.40 1.40 1.24

1.47 1.47 1.43

1.24 1.23 1.21

1.36 1.36 1.38

1.15 1.15 1.14

1.3 1.8 2.7

S89 4-B S89 4-B# S108 4-B S133 4-B S133 4-B# S139 6-B

35.1 35.9 53.6 79.0 77.9 137.1

1.41 1.43 1.40 1.34 1.33 1.70

1.06 1.08 1.06 1.03 1.02 1.28

1.22 1.24 1.23 1.21 1.20 1.45

1.34 1.35 1.47 1.55 1.54 1.52

1.50 1.51 1.51 1.50 1.49 1.78

1.26 1.28 1.27 1.25 1.24 1.50

1.41 1.43 1.40 1.34 1.33 1.70

1.18 1.19 1.18 1.16 1.15 1.41

2.2 2.8 1.9 1.1 1.1 2.6

Mean COV / b

1.23 0.068 0.90 3.30

1.42 0.073 0.90 3.61

1.52 0.066 1.00 3.56

1.28 0.068 0.95 3.21

1.41 0.079 0.90 3.76

1.19 0.072 0.90 3.13

Repeated test.

4.0

1.6

3.5

1.4

3.0

1.2

2.5

1.0

Mexp/Mp

Rotation capacity Rc

#

Specimen D t (mm)

2.0 V-series S-series Yura et al. [18] Galambos [17] Class 1 limit

1.5 1.0 0.5 0.0

0

0.056 0.1

0.8

V-series S-series EN 1993-1-1 ANSI/AISC 360-10 AS 4100

0.6 0.4 0.2 0.0

0.2

0.3

0.4

0.0

0.1

0.2

0.3

0.4

(D/t)( fy /E)

(D/t)( fy /E) Fig. 14. Comparison with the class 1 limit in EN 1993-1-1 for CHS.

Table 9 Summary of slenderness limits in codes for CHS under bending. References

kp

ky

EN 1993-1-1 [12] ANSI/AISC 360-10 [14] AS 4100 [16] AISI S100 [15] & AS/NZS 4600 [25]

0.078 0.070 0.063 N/A

0.101 0.310 0.150 0.328

The rotation capacities are also reported in the last column of the table. Generally, this batch of CHS beams have higher yield stress than 1000 MPa (Table 4) and the specifications tend to underpredict the capacities by 19–52%. Based on the inelastic reserve method, the Direct Strength Method was improved by 22% (from 1.41 to 1.19) and gave the closest predictions for HSS CHS beams. Since EN 1993-1-1 [12] uses the same yield slenderness limit for compression and bending for CHS, the specimens in this study are still classified as class 4 although they all have ultimate moment capacities higher than Mp and My. The capacity predictions are based on the shell buckling theory in EN 1993-1-6 [11]. The results from EN 1993-1-6 [11] underestimate the capacities of the sections by 52% on average. The closest prediction for cold-formed high strength steel CHS beams is from the Direct Strength Method coupled with inelastic reserve method [15]. The average Mexp/MDSM-IR ratio is 1.19, the COV value equals to 0.072 and the corresponding reliability index is 3.13 for the resistance factor of 0.9.

Fig. 15. Comparison with the plastic slenderness limits for CHS.

The rotation capacities Rc for CHS are plotted against the normalized section slenderness and compared to the requirements as stated in Galambos [18] and Yura et al. [20] (Fig. 14). Clearly, all the specimens had larger slenderness than the limit as stated in EN 1993-1-1 [12] and all of them do not possess sufficient rotation capacities of 3 or higher. However, four beam specimens had rotation capacities larger than 2. The rotation capacity of CHS drops as the section slenderness ðD=tÞðf y =EÞ increases. The highest rotation capacity from tests is 2.8 with cross-section slenderness ðD=tÞðf y =EÞ of 0.131. The class 1 slenderness limit seems to be conservative for both rotation capacity requirements of 2 and 3. Further investigation is required on sections with smaller D/t ratio to determine the suitable class 1 limit for high strength steel CHS under bending and to evaluate the rotation capacity requirement for HSS materials. Table 9 summarizes the plastic and yield slenderness limits for circular tubes. Section plastic moment and elastic yield moment can be reached if the section slenderness is smaller than the plastic slenderness limit kP and yield slenderness limit ky respectively. The normalized ultimate moment capacities Mexp/Mp are plotted in Fig. 15. The Mexp/Mp values are approaching 1.0 as the section slenderness increases. The normalized slenderness limits for EN 19931-1 [12], ANSI/AISC 360-10 [14] and AS 4100 [16] are around 0.07 whereas the limit for the specimens appears to be larger than 0.2, which shows that the plastic slenderness limits in the specifications are conservative for HSS circular tubes. Similarly, the Mexp/My values are plotted against the normalized section slenderness in Fig. 16. EN 1993-1-1 [12] adopts the same

J.-L. Ma et al. / Engineering Structures 126 (2016) 200–209

209

1.8 1.6

Mexp /My

1.4 1.2 1.0 V-series S-series EN 1993-1-1 ANSI/AISC 360-10 AS 4100 AISI S100 & AS/NZS 4600

0.8 0.6 0.4 0.2 0.0

0.0

0.1

0.2

0.3

0.4

(D/t)( fy /E) Fig. 16. Comparison with the yield slenderness limits for CHS.

yield slenderness limit of 0.101 for both bending and compression while other specifications relax the limits for bending. AS 4100 [16] gives the normalized yield slenderness limit as 0.15, whereas ANSI/ AISC 360-10 [14], AISI S100 [15] and AS/NZS 4600 [25] have the normalized yield slenderness limits more than 0.3, which is three times the limit in EN 1993-1-1 [12], and also twice the limit in AS 4100 [16]. Fig. 16 shows that the yield slenderness limits in EN 1993-1-1 [12] and AS 4100 [16] are conservative. However, more data on HSS tubes with larger D/t ratio are necessary to verify the yield slenderness limits in the codes. To summarize, for the studied range of cold-formed high strength steel CHS beams, the plastic slenderness limit from EN 1993-1-1 [12] and the yield slenderness from AS 4100 [16] can be used. Further experimental and numerical tests are needed to cover a wider range of cross-section slenderness.

4. Conclusions This paper has presented the experimental investigation on the flexural behaviour of cold-formed high strength steel (HSS) tubular beams. Different tubular sections (CHS, RHS and SHS) were examined with three series of high strength steel grades 700 MPa, 900 MPa and 1100 MPa. The ultimate cross-sectional bending moment capacities were compared with design strengths predicted by ANSI/AISC 360-10 [14], AS 4100 [16], EN 1993-1-1 [12], AISI S100 [15] and the Direct Strength Method [15]. Results showed that the current codes of practice are mostly conservative for the design of HSS tubes subjected to uniform bending moment. The rotation capacities, ultimate moment capacity to plastic moment ratios and ultimate moment capacity to yield moment ratios were also summarized to verify the codified slenderness limits for SHS, RHS and CHS subjected to bending. Results showed that low material ductility reduces the available rotation capacities of the sections and subsequently the class 1 limit is no longer applicable for HSS tubular sections. The plastic slenderness limit and yield slenderness limit were found to be conservative for the HSS tubular sections. Acknowledgements The authors would like to thank the final year undergraduate student, Mr. Tsz-On Wu and the technicians from the Department of the Civil Engineering at The University of Hong Kong for their assistance in the experimental investigation. The authors are also grateful to Rautaruukki Corporation for supplying the coldformed high strength tubular test specimens. The research work described in this paper was supported by a grant from the Research

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