Shear lag effect on ultimate tensile capacity of high strength steel angles

Journal of Constructional Steel Research 145 (2018) 300–314

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Journal of Constructional Steel Research

Shear lag effect on ultimate tensile capacity of high strength steel angles Ke Ke a,b,⁎, Y.H. Xiong b, Michael C.H. Yam b,e, Angus C.C. Lam c, K.F. Chung d,e a

College of Civil Engineering, Hunan University, Changsha, China Department of Building and Real Estate, The Hong Kong Polytechnic University, Hong Kong, China c Department of Civil and Environmental Engineering, University of Macau, Macau, China d Department of Civil and Environmental Engineering, The Hong Kong Polytechnic University, Hong Kong, China e Chinese National Engineering Research Centre for Steel Construction (Hong Kong Branch), The Hong Kong Polytechnic University, Hong Kong, China b

a r t i c l e

i n f o

Article history: Received 5 December 2017 Received in revised form 1 February 2018 Accepted 16 February 2018 Available online xxxx Keywords: Shear lag High strength steel Tension angle Experiment Design method

a b s t r a c t This research investigates the shear lag effect on the behaviour and ultimate tensile capacity of high strength steel (HSS) tension angles with bolted and welded connections. Eighteen full-scale tests were conducted, including fourteen specimens with HSS tension angles and four specimens with normal steel (NS) tension angles. For these specimens, single tension angles were connected to the gusset plates either by bolted or welded connections. The main test parameters included steel grade, connection length and out-of-plane eccentricity. In general, the test observations showed that the shear lag effect was significant for the bolted HSS angle specimens connected by the short leg. The effectiveness of the design equation in the current design specifications for quantifying the shear lag (1−x=L rule, where x ¼ out‐of‐plane eccentricity and L = connection length) was evaluated using the test results. The comparison of the test results and the predictions by the design equations showed that the latter gave un-conservative estimates of the ultimate tensile capacity of the specimens with bolted HSS angles connected by the short leg. Based on the finite element models validated by the test results, a parametric study was carried out, and the results also indicated that the current design equation would lead to unsafe estimates of the ultimate tensile capacities of bolted HSS angles connected by the short leg. Finally, a modified design guideline was proposed based on the results of the numerical study. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction In steel construction, hot-rolled or welded sections such as I-sections, angles, and tees are often utilised as tension or compression members to resist axial forces. These steel members are usually connected to other structural components such as gusset plates using bolted or welded connections. Since it is common to connect only part of the section of a tension member (e.g. one leg of a single angle) to the connecting elements at the connection, therefore the unconnected part of the section near the connection may not be effectively mobilised to carry the applied tension force. Consequently, the tensile strength of the section cannot be fully developed in the vicinity of the connection, and this phenomenon is commonly known as “shear lag”. Taking a single angle with welded connection shown in Fig. 1 as an example, the applied tension load is transferred from the connected leg to the outstanding leg by shear, leading to stress lagging behind from the heel to the outstanding toe of the section. Thus, the unconnected leg is ⁎ Corresponding author at: College of Civil Engineering, Hunan University, Changsha, China. E-mail address: [email protected] (K. Ke).

https://doi.org/10.1016/j.jcsr.2018.02.015 0143-974X/© 2017 Elsevier Ltd. All rights reserved.

not completely effective in carrying the load. In addition, the applied axial load is generally transferred by bolts (for bolted connections) or welds (for welded connections), and hence the load transfer path which is in line with the bolt line or the weld plane may not align with the centroid of the cross-section. In this context, secondary bending produced by the loading eccentricity would also be expected. In general, the cascading effect of shear lag, secondary bending and stress concentration at the connection would compromise the ultimate tensile resistance of a tension member. In practical cases, since it is difficult to decouple the interdependence between the effect of shear lag and secondary bending, a modification factor is usually utilised to consider both. The research efforts dedicated to the shear lag effect on the behaviour and ultimate tensile capacity of tension members were initiated in the last century. For instance, Davis and Boomsliter [1] carried out an investigation to quantify the ultimate tensile strength of tension angles with welded or riveted connections. Gibson and Wake [2] conducted a series of tests of angles subjected to tension, and the influence of the weld detail on the ultimate tensile resistance of the specimens was studied. To provide a comprehensive insight into the behaviour and ultimate tensile strength of tension members of bolted or

K. Ke et al. / Journal of Constructional Steel Research 145 (2018) 300–314

Welded tension angle Shear lag effect Unconnected leg Critical section

Gusset plate

Weld

Fig. 1. Stress distributions in welded single angle due to shear lag effect.

riveted connections, Chesson and Munse [3,4] conducted experimental works on specimens covering a wide spectrum of parameters, and they proposed a strength reduction coefficient to quantify the shear lag effect of tension members, as reproduced in Eq. (1): URule ¼ 1−x=L

ð1Þ

where URule = design section efficiency, x ¼ out‐of‐plane eccentricity and L = connection length as illustrated in Fig. 2. Eq. (1) has been included in various design specifications [5–7] to account for the effect of shear lag in tension members. Regan and Salter [8] conducted an experimental study on single angles of welded connections with varied welding details. Utilising the test results of seventeen specimens, they proposed design equations for quantifying the ultimate tensile strength of tension angles with welded connections. In addition, Easterling and Gonzalez [9] investigated the behaviour of double members under tension, and the effect of the presence of the transverse welds on the shear lag effect was evaluated. Later, Kulak and Wu [10] initiated an experimental programme including twenty-four single and double angle tension members with bolted connections and developed a set of design recommendations for quantifying the shear lag effect of bolted tension angles. Abi-Saad and Bauer [11] developed an analytical model for predicting the ultimate tensile capacity of tension members using assumed stress distribution in the vicinity of the connection. Subsequently, more test data were provided by Zhu et al. [12] to characterise

S

L Centroid

x

S (a)

301

the behaviour of single tension angles with welded connections. More recently, Fang et al. [13] studied the effect of the shear lag on the behaviour and the ultimate tensile capacity of tension angles and tees with welded connections. Nonetheless, all of the research works described above and the corresponding research findings are limited to tension members made of normal steels (NSs) with a nominal yield stress lower than 460 MPa. Recently, developments and technological advances in metallurgical industry have improved the availability and economic viability of high strength steels (HSSs) with a specified minimum yield stresses equal to or higher than 460 MPa [14]. Research communities have also initiated explorations in behaviour of HSS material [15,16], HSS connections [17–20], components [21] and HSS structures [22,23]. In engineering practice, utilising HSS can achieve significant reductions in crosssection dimensions of components and overall weight of a structure because HSS possesses significantly higher yield and ultimate strengths compared with those of NS, and hence producing more economical and sustainable structures. In this respect, steel tension members, which are generally controlled by strength criteria, may be fabricated using HSS to satisfy the required load resistance with more efficient section sizes. On the other hand, recent research findings also indicate that HSSs generally possess limited ductility, and the characteristics of the stress-strain curves of HSSs are quite different from those of NSs that show adequate postyielding strength hardening and deformability, e.g. S275 steel and S355 steel. Typically for S690 steel, although its nominal yield strength could reach 690 MPa, the ultimate strength to yield strength ratio is close to unity, and the ultimate strain is only around 6–8% along with the fracture strain close to 15%. In this context, when using HSS in bolted or welded tension members, premature fracture of the material may occur in the vicinity of the connection before sufficient stress redistribution is achieved in the cross-section. This effect is particularly aggravated if only part of the section is connected in a connection such as a single angle tension member connected to a gusset plate. Hence, due to the reduced ductility of HSS, the shear lag effect on HSS angle tension members is expected to be more significant than that of the counterparts made of NS. Since the existing design rules for quantifying the shear lag effect were developed primarily based on the test results of NS tension members, therefore they may not be adequate for predicting the structural behaviour of HSS tension members. Hence, more experimental and analytical studies are needed to examine the influence of the shear lag on the behaviour and the ultimate tensile capacity of HSS tension members with bolted and welded connections. Based on the above, an experimental investigation consisted of fourteen HSS and four NS single angles with bolted and welded connections was carried out to examine the shear lag effects on the ultimate tensile capacity of HSS angle tension members. The main parameters were steel grade, connection length and out-of-plane eccentricity. The test results were compared with the predictions by the classical 1−x=L rule proposed by Munse and Chesson [3,4] to examine its applicability for quantifying the shear lag effect of HSS single angle. Subsequently, detailed finite element (FE) models simulating the structural behaviour of the specimens were established and verified by the test results, and a parametric study was conducted utilising the validated FE models covering a wide range of parameters. Finally, based on the experimental and numerical results, design considerations for quantifying the shear lag effect on the ultimate tensile capacity of HSS tension members were proposed. 2. Test programme 2.1. Test specimens

(b) Fig. 2. Definition of connection length and out-of-plane eccentricity: (a) specimens with bolted connections and (b) specimens with welded connections.

Eighteen full-scale specimens with bolted and welded single angles of varied steel grades, connection lengths and out-of-plane eccentricities were examined in the test programme. Out of the eighteen

K. Ke et al. / Journal of Constructional Steel Research 145 (2018) 300–314

Specimen with welded connections

Centre line

Specimen with bolted connections

Critical section

Critical section

Centre line

302

Mid-length section

Mid-length section

Strain gauge

Strain gauge

(a)

(b)

Fig. 3. Typical configuration and strain gauge layout of specimens: (a) specimens with bolted connections and (b) specimens with welded connections.

specimens, fourteen of them were fabricated with Grade S690 steel (nominal yield stress of 690 MPa). For comparison, the angles in the other four specimens were made of NS of Grade S275. The HSS angles were fabricated by welding two S690 steel plates using groove welds, while the NS counterparts were hot-rolled angles. Nine specimens were fabricated with bolted connections, and the other nine were with welded connections. For the specimens with bolted connections, five M22 Grade 12.9 high strength bolts were used on each side of the angle. The sizes of the bolt holes were designed 2 mm larger than the diameter of the bolts. For the specimens with welded connections, unbalanced welds configuration with equal lengths of the heel and toe welds were utilised, and transverse welds were also arranged for each specimen. Each end of the specimens was connected to a 16 mm thick gusset plate that was then connected to the end fixtures of the testing machine. Typical specimen configurations are shown in Fig. 3. Hereafter, the term “angle” refers to the single angle in a specimen, and “specimen” represents the angle together with the two gusset

plates. In the process of constructing the specimen matrix, the effect of the connection length was considered by changing the bolt spacing in the connection for the specimens with bolted connections, while varied longitudinal weld lengths were designed for the specimens with welded connections. Different section sizes combining with different connected legs were utilised to examine the effect of out-of-plane eccentricity. During the design of the test programme, preliminary FE analyses of the specimens were conducted in order to ensure that failure would be triggered at the tension angle sections before failure of the welds, bolts or gusset plates. For all the specimens, a clear length of 800 mm was arranged between the inside end of the two gusset plates, and the distance from the tension angle end to the external edge of the gusset plates for gripping was 170 mm as shown in Fig. 3. The total lengths of the specimens varied from 1580 mm to 2100 mm to accommodate various connection configurations. Table 1 shows the detailed information on the specimens. For easy reference, each specimen was assigned a test code. In particular, the first capital letter

Table 1 Specimen description and test results. Specimen code

Angle section (mm)

Steel grade

Bolt spacing (mm)

Longitudinal weld length (mm)

Failure mode

Ultimate strength (kN)

Nominal ultimate strength (kN)

Test efficiency, UTest

1− xL rule, URule

UTest/URule

A1-60L A1-75L A1-90L A1-75S A2-60S A2-75S A2-90S B1-75L B2-75S C1-220L C1-300L C1-380L C1-300S C2-220S C2-300S C2-380S D1-300L D2-300S

80 × 60 × 8 80 × 60 × 8 80 × 60 × 8 80 × 60 × 8 100 × 65 × 8 100 × 65 × 8 100 × 65 × 8 80 × 60 × 8 100 × 65 × 8 80 × 60 × 8 80 × 60 × 8 80 × 60 × 8 80 × 60 × 8 100 × 65 × 8 100 × 65 × 8 100 × 65 × 8 80 × 60 × 8 100 × 65 × 8

S690 S690 S690 S690 S690 S690 S690 S275 S275 S690 S690 S690 S690 S690 S690 S690 S275 S275

60 75 90 75 60 75 90 75 75 – – – – – – – – –

– – – – – – – – – 220 300 380 300 220 300 380 300 300

C C C C C+W C C C C C M C C C C C M C

632 656 660 542 590 613 625 395 311 796 839 850 748 836 823 925 488 511

687

0.92 0.95 0.96 0.79 0.70 0.72 0.74 1.01 0.64 0.95 1.00 1.01 0.89 0.84 0.82 0.93 1.02 0.90

0.94 0.95 0.96 0.92 0.86 0.89 0.91 0.95 0.89 0.93 0.95 0.96 0.92 0.85 0.89 0.91 0.95 0.89

0.98 1.00 1.00 0.83 0.81 0.81 0.81 1.06 0.72 1.02 1.05 1.05 0.97 0.99 0.92 1.02 1.07 1.01

846

392 483 840

999

479 570

Notes: “C” indicates the critical section failure mode; “C + W” indicates the mixed failure mode combined critical section failure followed by shear failure of weld after the ultimate strength was reached; “M” indicates the mid-length section failure mode.

K. Ke et al. / Journal of Constructional Steel Research 145 (2018) 300–314

1200

Engineering stress (MPa)

“A” and “B” represent specimens with bolted angle made of S690 and S275 steel, respectively. Letter “C” and “D” represent specimens with welded angle made of S690 and S275 steel, respectively. The numbers “1” and “2” following the capital letter (before the hyphen) stand for the section size of 80 × 60 × 8 (in mm) and 100 × 65 × 8 (in mm) of the angle, respectively. The bolt spacing for the bolted angles or the weld length for the welded angles is successively represented by a value following the hyphen in the test code. The last capital letter “L” and “S” represent tension angles connected by the long leg and the short leg at the connection, respectively. Taking specimen C1-220 L as an example, the code stands for a specimen with welded connections, grade S690 steel, 80 × 60 × 8 angle section and was connected to the gusset plate by the long leg with a longitudinal weld length of 220 mm. For all the specimens, tension coupon tests were carried out to obtain the actual material properties of the angles. For the HSS angles, tensile coupons (plate type) were cut from the S690 steel plates along the longitudinal direction, and the counterparts for the NS angles were extracted from two legs of the hot-rolled angles of S275 steel. The coupon tests were conducted according to ASTM 370 standard [24]. The coupon test results including the Young's modulus, the yield strength, the ultimate strength and the ultimate strain of the material are given in Table 2. Typical engineering stress-strain curves of coupon samples are shown in Fig. 4.

303

S275 sample S690 sample

900 600 300 0 0.00

0.05

0.10 0.15 0.20 Engineering strain

Fig. 4. Typical engineering stress-strain curves of coupons.

complete failure after fracture occurred in the angles, and the test was terminated.

2.2. Test setup, instrumentations and test procedures

3. Test results

The experimental work was conducted in a SATEC testing machine with a tensile capacity of 2000 kN. The arrangement of the test setup is shown in Fig. 5. In particular, the specimens were connected to the crossheads of the testing machine via two end-fixtures, and they were conservatively designed to stay in the elastic stage during the tests. Seven high strength bolts (diameter 24 mm, Grade 8.8) at the gusset plates were utilised to fix the specimens on to the end-fixtures. To focus on the examination of the behaviour of tension angles with bearing-type bolted connections, the effect of friction among contact surfaces of angles, gusset plates and bolts was minimised as the bolts were snug-tightened by hand during the specimen preparation. In the loading process, the applied quasi-static load exerted from the testing machine was continuously recorded by the built-in load cell, and the elongation of each specimen was monitored by the built-in displacement transducers of the testing machine. To capture the strain distributions and their development, strain gauges (i.e. bonded foil strain gauge with 10 mm gauge length) were mounted at the critical section and the mid-length section of the specimens, as schematically illustrated in Fig. 3. At the beginning of the test, the specimens were positioned in the testing machine and they were firstly aligned by a small preload. Subsequently, all the readings were reset and the formal test commenced using a load control process with an increment of 50 kN in each step before yielding of the specimens was detected. Then, after yielding behaviour of the angles was observed according to the strain readings, the loading process was converted to displacement control with the loading rate of 1 mm/min. All the specimens were loaded to

3.1. Overview The test results included the failure mode and the ultimate strength of all the specimens are given in Table 1. For the specimens with bolted connections, the nominal ultimate strength shown in Table 1 was determined by the product of the tensile strength of the material (fu) from coupon tests and the net-section area (An) of the critical section, i.e. gross cross-section area (Ag) subtracted the area lost due to the bolt hole. For the specimens with welded connections, the nominal ultimate strength was calculated by the product of fu and Ag. Typical failure modes of the specimens are illustrated in Fig. 6. For the specimens with bolted connections, seven of them developed net-section fracture at the critical section (Figs. 3 and 6) of the angles near the innermost bolt hole. However, specimen A2-60S failed in a mixed mode with net-section fracture followed by shear fracture of the welds connecting the two HSS plates, and the shear failure was observed after the ultimate load was reached. For the specimens with welded connections, seven specimens experienced gross-section fracture of the angles at the

Upper cross-head End-fixture Gusset plate End-fixure Gusset plate

Angle Table 2 Summary of tension coupon test results (mean value). Material

S690 (Specimen A/C series) S275 (Specimen B/D series) S960 (For parametric study)

0.25

Angle Gusset plate

Elastic modulus E (MPa)

Static yield stress fy (MPa)

Static ultimate stress fu (MPa)

Ultimate strain εu (%)

215,374

761

798

6.7

218,268

292

453

20.7

206,730

997

1038

5.9

End-fixture Lower cross-head

Test machine Fig. 5. Layout of test setup.

304

K. Ke et al. / Journal of Constructional Steel Research 145 (2018) 300–314

Fracture at critical section

Fracture at critical section

Shear fracture of welds Fracture at Mid-length section

Fracture at critical section

Broken critical section

Fracture surface Fig. 6. Typical failure modes.

critical section as shown in Fig. 6. The other two specimens C1-300L and D1-300L fractured near the mid-length of the angles. It should be noted that brittle failure was observed at the critical section of specimen B275S with a bolted NS angle when the applied load was much lower than the nominal ultimate strength of the angle. This observation will be further discussed in later sections. The test efficiency (UTest), which was determined by the ratio of the test ultimate strength of the specimens to the corresponding nominal ultimate strength (i.e. fuAn or fuAg), is illustrated in Table 1. The test efficiency of the specimens with bolted connections ranged from 0.64 to 1.01 (0.64 for specimen B2-75S), and the test efficiency of the specimens with welded connections ranged from 0.82 to 1.02. 3.2. Normalised load-elongation curves The normalised load-elongation curves of all the specimens are shown in Fig. 7. Specifically, the normalised load was obtained through dividing the applied load by the nominal ultimate strength of the specimens (i.e. fuAn or fuAg). For the specimens with bolted connections, the normalised load-elongation curve of specimen B1-75L with a NS angle achieved a long yield plateau, whereas such phenomenon was not observed in the responses of the counterparts with HSS angles. Fractures of the bolted HSS angle specimens occurred right after the ultimate loads of specimens were reached. For the specimens with welded connections, apart from the specimens with NS angles (i.e. D1-300L and D2-300S), the normalised load-elongation curves of the two specimens with HSS angles (i.e. C1-300L and C1-380L) also exhibited a small yield plateau before failure occurred. In general, the total elongations of the specimens with NS angles were significantly larger than those of the

HSS angles. As mentioned, specimen B2-75S did not show the expected ductile normalised load-elongation response as illustrated in Fig. 7b. The unexpected premature failure of this specimen was believed to be caused by the defects of the specimen possibly produced during the fabrication process such as damaged bolt hole. Therefore, it was decided to exclude this specimen from the discussions of test results. However, the structural behaviour of this specimen will be further examined in the finite element analysis study discussed in Section 5. 3.3. Strain distribution Representative strain distributions of the HSS specimens with bolted connections (A1-90L and A2-90S) and the specimens with welded connections (C1-220L and C2-220S) are illustrated in Fig. 8. The pattern of strain distributions for the specimens with bolted connections and welded connections were generally similar and hence only the results of the specimens with bolted connections are discussed herein. The positions of the numbered strain gauges in the sections are shown in Fig. 8. To illustrate the yielding behaviour of the section, the theoretical yield strain is also included in the figure. The strains of two cross-sections are discussed, i.e. the critical section and the mid-length section as illustrated in Fig. 3. In general, non-uniform strain distribution was observed over the cross-sections. In the critical section, the strains were generally in tension with tensile yielding firstly occurred near the bolt hole at the connected leg due to stress concentration. However, compressive strains were observed near the outstanding edge of the unconnected leg (strain gauge 5) due to the combined effects of shear lag and secondary bending during the initial loading stage. The strain evolution at the outstanding edge of the unconnected leg in the critical section

K. Ke et al. / Journal of Constructional Steel Research 145 (2018) 300–314

A1-75L B1-75L A1-75S

0.6 0.4 0.2 0.0

Normalised load

Normalised load

0.8

1.0

A1-90L

A1-60L

1.0

5

A2-90S

0.8

A2-75S A2-60S

0.6 B2-75S

0.4 0.2 0.0

0

10 15 20 25 30 35 40 Elongation (mm)

0

5

10 15 20 25 Elongation (mm)

(a) 1.2

C1-220L

1.0

C2-380S

0.6

C1-300S

C1-300L

D1-300L

Normalised load

Normalised load

0.8

0.8

0.6

C2-300S

C2-220S D2-300S

0.4

0.4

0.2

0.2 0.0

30

(b)

C1-380L

1.0

305

0

20 40 60 80 100 120 140 160 Elongation (mm)

(c)

0.0

0

10

20

30 40 50 60 Elongation (mm)

70

(d)

Fig. 7. Normalised load-elongation curves of all specimens: (a) A1/B1 series, (b) A2/B2 series, (c) C1/D1 series and (d) C2/D2 series.

depended on the out-of-plane eccentricity. Specifically, the compressive strain gradually turned to tensile strain with increasing applied load when the angle specimen was connected by the long leg. However, for the angle specimen connected by the short leg, compressive strains continued to increase until fracture of the critical section occurred. This observation can be explained by the fact that angles connected by the short leg had a much larger out-of-plane eccentricity, thus bringing about more significant combined effects of shear lag and secondary bending. For the angles connected by the long leg, which had a much smaller out-of-plane eccentricity, the in-plane rotation of the connection allowed the alignment of the loading line to the centroid of the angle, and hence reduced the secondary bending moment effect as the loading increased. Therefore, the compressive strain at the outer edge of the angle turned to tensile strain for these specimens as loading increased. 4. Discussion of test results 4.1. Effect of steel grade As illustrated in Fig. 7, satisfactory ductile behaviour characterised by an evident yield plateau in the normalised load-elongation response curves was achieved by all the NS angle specimens (ignoring specimen B2-75S which failed in brittle manner because of fabrication defects). However, only limited ductility was achieved by the HSS angle specimens, and there was no obvious yield plateau in the response curves of most of these specimens. This observation can be attributed to the difference in ductility and ultimate strength to yield strength ratio between NS and HSS. In particular, according to the coupon test results, the ultimate strength to yield strength ratios of S690 and S275 steels used for the angles were 1.05 and 1.56, respectively, and the corresponding ultimate strains were 6.7% and 20.7%, respectively. Due to the large ductility and the relatively high ultimate strength to yield strength ratio of the NS material, effective redistribution of stresses in

the critical section occurred in the NS angle specimens such that the un-connected leg of the specimens was better mobilised before fracture of the section initiated. This allowed the development of an evident yield plateau in the normalised load-elongation curves of the specimens with bolted NS angles. On the other hand, for the specimens with bolted HSS angles, due to limited ductility and the much lower ultimate strength to yield strength ratio of the material, stress redistribution in the critical section was limited. Hence, fracture of the critical section was initiated before the un-connected leg was effectively mobilised to resist the applied load. Hence, inelastic deformations were concentrated in the critical section without evident yield plateau in the normalised load-elongation curves of the bolted HSS angle specimens. Similar observations as those of the bolted angle specimens were also recorded for the welded HSS angle specimens except for specimens C1-300L and C1-380L. For these two specimens, the HSS angle was connected by the long leg and with a relatively long connection length (length of the longitudinal welds). Hence, the combined shear lag and secondary bending effects on these specimens were reduced, which allowed the specimens to deform with a moderate yield plateau in the normalised load-elongation curves. The test efficiency (UTest) of the HSS angle specimens was generally lower than those of the NS angle specimens and this tendency was more evident for the specimens connected by the short leg as shown in Table 1. For instance, the test efficiency of specimen D2-300S with a NS angle was 10% higher than that of the corresponding specimen C2-300S with a HSS angle. However, such tendency was not evident for specimens connected by the long leg. The test efficiency of specimen C1-300L with welded HSS angle and the corresponding specimen with welded NS angle (specimen D1-300L) were 1.00 and 1.02, respectively. For the specimens with bolted connection, specimen B1-75L with a NS angle achieved the test efficiency of 1.01, while specimen A1-75L with the HSS angle of the identical section resulted in the test efficiency of 0.95. Therefore, it can be seen that the shear lag effect generally became significant as the steel grade increased, particularly

306

K. Ke et al. / Journal of Constructional Steel Research 145 (2018) 300–314

A1-90L (Critical section) 5

700 600

600 4

400

500

1

3

Load (kN)

Load (kN)

500

2

300 200 100

-1000

0

A1-90L(Mid-length section) 67

700

Yield strain

0

5 4 23

1

9

400 300 200 100 0

1000 2000 3000 4000 5000 Microstrain

1000 2000 3000 4000 5000 Microstrain

(b)

1

200

5 4 3 2

3 2

Yield strain

-2500

1

0 2500 Microstrain

Load (kN)

Load (kN)

300

10

200

0

1000 2000 3000 4000 5000 Microstrain

Load (kN)

Load (kN)

800

2 4 5 3

Yield strain

0

C1-220L (Mid-length section)

1000 1

0 -1000

6 5 1 2 34

12

600

12 11 7 8 9

0 -1000

Yield strain 10 1000 2000 3000 4000 5000 Microstrain

0

(e)

5

400

4 3

200 Yield strain

0

2500 Microstrain

C2-220S (Mid-length section) 12

800

Load (kN)

Load (kN)

1000

2

600

0 -2500

(f)

1 6

8

9

400

1000 2000 3000 4000 5000 Microstrain

800

7 11 10

200

C2-220S (Critical section)

1000

7

(d)

6

200

9 8

6

Yield strain

0 -1000

5000

800

400

7

300

(c)

600

6 8

400

100

C1-220L (Critical section)

1000

9

500

4

400

0 -5000

10

600

500

100

A2-90S (Mid-length section)

700

Yield strain

5

600

10 9 78

6

Yield strain

0

(a) A2-90S (Critical section)

700

8

10

6 5 1 2 3

11

600

10

400

5000

(g)

0

9

12

200 4

8 7

Yield strain

-1000 0

11 7 8 9

10

1000 2000 3000 4000 5000 Microstrain

(h)

Fig. 8. Typical strain distributions of the specimens: (a) critical section of A1-90L (b) mid-length section of A1-90L, (c) critical section of A2-90S, (d) mid-length section of A2-90S, (e) critical section of C1-220L, (f) mid-length section of C1-220L, (g) critical section of C2-220S and (h) mid-length section of C2-220S.

for the short leg connected specimens, and the effect of steel grades interacted with the influence of connection length and out-ofplane eccentricity, which will be further clarified in the following discussions. 4.2. Effect of connection length As shown in Table 1 and Fig. 7, the effect of connection length on the ultimate resistance and the test efficiency of the specimens can be observed. The results indicated that the test efficiency of the specimens with bolted connections increased slightly with increasing connection length for all angles connected by either the long leg or the short leg. For example, the test efficiencies of specimens A1-60L and A1-90L

were 0.92 and 0.96, respectively, which represented an increase of only 4% in test efficiency with a 50% increase in connection length. The test efficiency of specimen A2-90S connected by the short leg was 0.74, and was around 6% higher than that of specimen A2-60S. In general, since the connection lengths for all the specimens with bolted connections were conservatively designed to avoid failure of the bolts, therefore increasing the connection length might not have an appreciable influence on the ultimate load and the test efficiency. Similar observations were also found in the specimens with welded connections connected by the long leg. For specimens with welded angles connected by the short leg, a general increase of the ultimate tensile capacity of the specimens and the corresponding test efficiency were observed with increasing connection

K. Ke et al. / Journal of Constructional Steel Research 145 (2018) 300–314

length (length of the longitudinal welds). In particular, the test efficiency of specimen C2-380S was about 11% higher than that of specimen C2-220S. However, there was a very small decrease in test efficiency (2%) when the connection length was increased from 220 mm (C2-220S) to 300 mm (C2-300S). In general, the overall test results of the welded angle specimens indicated that the test efficiencies increased with increasing connection length.

4.4. Evaluation of 1−x=L rule To evaluate the applicability of the classical 1−x=L rule (URule) proposed by Munse and Chesson [3,4] for quantifying the shear lag effect on the ultimate capacity of tension angles, particularly for the HSS angle specimens, the test efficiency-to-predicted efficiency ratios (UTest/URule) of the specimens are provided in Table 1. It can be observed that the rule generally produced conservative predictions of the section efficiency of the NS angle specimens (excluding specimen B2-75S). For the HSS angle specimens connected by the long leg with either welded or bolted connections, the UTest/URule ratios were generally larger than or equal to unity except for specimen A1-60L which had a ratio of 0.98. However, for the HSS angle specimens connected by the short leg, the UTest/URule ratio varied from 0.81 to 0.83 for the specimens with bolted connections and from 0.92 to 1.02 for the specimens with welded connections. Therefore, it can be seen that the classical 1−x=L rule generally produced non-conservative predictions of the tensile capacity of the HSS angle specimens connected by the short leg. In particular, the URule significantly over-estimated the efficiency of the bolted HSS angles connected by the short leg. To further examine the shear lag effect on the tensile strength and behaviour of HSS angles and the applicability of the classical 1−x=L rule in quantifying the shear lag effect, finite element analysis of HSS angles was conducted and will be presented in the following sections. 5. Finite element analysis and design considerations 5.1. Finite element model Finite element (FE) models of all the test specimens were developed using ABAQUS [25] for a more comprehensive understanding of the shear lag effect on the ultimate tensile capacity of HSS tension angles. In general, four main components including steel angles, bolts, welds and gusset plates were modelled. All the components were discretized by three dimensional, eight-node linear brick elements with reduced integration and hourglass control (C3D8R). To eliminate the hourglass effect, at least four elements were used in the thickness direction for all the components [25]. The illustration of typical finite element models representing the specimens with bolted connections and welded connections are shown in Fig. 9a and b, respectively. For the bolted angle specimen models, a general “surface to surface” interaction was used to rationally account for the interaction among the components, i.e.

Angle

Angle Weld

Bolt Gusset plate

Gusset plate

(a)

U2

4.3. Effect of out-of-plane eccentricity The test results showed that the ultimate tensile capacity and the test efficiency of the specimens with either bolted or welded connections decreased considerably with increasing out-of-plane eccentricity. For instance, although specimen A1-75L and specimen A1-75S had identical cross-section and connection length, the test efficiency of specimen A1-75L connected by the long leg was 20% higher than that of specimen A1-75S connected by the short leg. For the specimens with welded connections, the same trend was observed. For example, the test efficiency of specimen C1-300L connected by the long leg was 12% higher than that of specimen C1-300S which was connected by the short leg. Hence, it can be seen that the increase of out-of-plane eccentricity, which amplified the combined effects of shear lag and secondary bending moment, decreased the efficiency of the tension angle.

307

(b)

U3 =0, UR1

UR 2 =UR 3

0

Fixed edge

Leading edge Spring element in U1

2

Load 1

3

U2

U3 =0, UR1

UR 2 =UR 3

0

(c)

Fig. 9. Typical finite element model: (a) models with bolted connections, (b) models with welded connections and (c) boundary conditions.

bolts, angles and gusset plates. In particular, the “hard contact” was utilised to simulate the contact in the normal direction of the two contacting surfaces avoiding surface penetration of the components, and the “penalty friction” was used to characterise the behaviour of contacted surfaces in the tangential direction. In addition, although a 2 mm clearance between the bolt shank and the bolt holes was used, the bolts were placed in contact with the bolt holes in the beginning stage to eliminate major slips, thus avoiding potential solution convergence failure during the analysis. As mentioned, the bolt clamping force was not significant since the bolts were snug- tightened by hand. Therefore, the effect of bolt clamping force was not included in the numerical analysis. This approach also avoided numerical instability when the bolts slid into bearing on the plates. As for the residual stress, since the angle specimens were primarily subjected to tension, the inelastic deformation in the longitudinal direction of the specimen would allow stress redistribution to occur after the angles yielded, and hence

Table 3 FE analysis results of the specimens with bolted and welded connections. Specimen

Ultimate load PTest (kN)

FE load PFEM (kN)

PTest/PFEM

A1-60L A1-75L A1-90L A1-75S A2-60S A2-75S A2-90S B1-75L B2-75S

632 656 660 542 590 613 625 395 311

C1-220L C1-300L C1-380L C1-300S C2-220S C2-300S C2-380S D1-300L D2-300S

796 839 850 748 836 823 925 488 511

664 677 680 580 611 637 659 360 408 Meana CoVa 765 830 836 696 878 919 925 480 531 Mean CoV

0.95 0.97 0.97 0.93 0.97 0.96 0.95 1.10 0.76 0.96 0.02 1.04 1.01 1.02 1.07 0.95 0.90 1.00 1.02 0.96 1.00 0.05

a The result of specimen B2-75S is excluded when determining the mean PTest/PFEM ratio and the corresponding CoV of bolted specimens.

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alleviated the influence of the residual stress on the angle. Therefore, the effect of residual stress was excluded from the analysis model. For the welded angle specimen models, similarly, the “surface to surface” interaction was employed between the angles and the gusset plates to avoid surface penetration of the components under applied loads. To simulate the welds, the “tie” constraint was utilised to connect the weld component to the angle and the gusset plate during modelling. To produce the restraint of the end-fixtures, all the translational and rotational

A1-75L

0.8

Fracture initiation

0.6 0.4 Test FE result

0.2 0

5

A1-75S

1.0

10 15 20 Elongation(mm)

25

0.8

Normalised load

Normalised load

1.0

0.0

displacements at both ends of the gusset plates were restrained except the translational displacement in the longitudinal direction (U1) as shown in Fig. 9c. Uniform displacement was applied to the leading edge of the gusset plate to stimulate the loading process during the test. At the end of the other gusset plate, linear spring elements in the longitudinal direction (elongation direction) were used (Fig. 9c) to simulate the flexibility contributed by the end-fixtures and the test machine. It should be noted that the stiffness of the spring elements was

0.4 0.2 0.0

30

Fracture initiation

0.6

Test FE result 0

5

10 15 20 25 Elongation (mm)

(a)

(b)

B1-75L

1.2

Normalised load

Normalised load

B2-75S

1.0 Fracture initiation

1.0 0.8 0.6 0.4 Test FE result

0.2 0.0

0

10

20 30 Elongation (mm)

0.8 Fracture initiation

0.6 0.4 0.2 0.0

40

Test FE result 0

10

(c)

0.6 0.4 Test FE result 0

10

20 30 40 50 Elongation (mm)

60

Normalised load

Normalised load

Fracture initiation

0.2

0.8 Fracture initiation

0.6 0.4 Test FE result

0.2 0.0

0

(e) 1.0

Fracture initiation

0.6 0.4 Test FE result

0.2 0.0

0

40

80 120 160 Elongation (mm)

(g)

200

Normalised load

Normalised load

1.0 0.8

5

10 15 20 Elongation (mm)

25

30

(f)

D1-300L

1.2

40

C2-380S

1.0

1.0 0.8

20 30 Elongation (mm)

(d)

C1-380L

1.2

0.0

30

D2-300S

0.8 Fracture initiation

0.6 0.4 0.2 0.0

Test FE result 0 10 20 30 40 50 60 70 80 90 100 Elongation (mm)

(h)

Fig. 10. comparisons of typical normalised load-elongation curves from test and FEMs: (a) A1-75L, (b) A1-75S, (c) B1-75L, (d) B2-75S, (e) C1-380L, (f) C2-380S, (g) D1-300L and (h) D-300S.

K. Ke et al. / Journal of Constructional Steel Research 145 (2018) 300–314

calibrated based on the slope of the load-elongation curves extracted from the test results in the linear stage. An isotropic-hardening elasticplastic material model with the von Mises yield criterion was employed in the models. The engineering stress-strain curves obtained from the coupon tests were firstly converted to the true stress-true strain responses and utilised as the input data. The true stress-true strain curve was divided into two stages, namely, before and after the necking. The peak load point in the engineering stress-strain curve [26] was used to quantify the boundary of the two stages. Prior to necking, the following converting equations were adopted:
 σ true ¼ σ eng 1 þ εeng

ð2Þ


 σ true εptrue ¼ ln 1 þ εeng − E

ð3Þ

where σtrue = true stress, σeng = engineering stress, εptrue = true plastic strain and εeng = engineering strain and E = the modulus of elasticity (Young's modulus). The two equations were derived based on the definition of true stress and true strain as shown in Eqs. (4) and (5), respectively, assuming that the volume of steel remains constant during the plastic stage. σ true ¼

F A Z

εtrue ¼

ð4Þ L

L0

    dL L A0 ¼ ln ¼ ln L L0 A

ð5Þ

where F = current applied load, A = current cross-section area, A0 = original cross-section area, L = current length and L0 = the original length. Note that Eqs. (2) and (3) were only valid where uniform strain was achieved along the gauge length of the tension coupon. After the initiation of necking of the tension coupons, significant non-uniform strain would be developed in the necking region. Thus, the true stress and the true strain could not be calculated directly by the Eqs. (2) and (3) [26]. For simplification, it was assumed that the true stress-true strain response was linear after necking. To determine the last point in the true stress-true strain curve, the applied force and the measured sectional area (obtained using a digital image correlation method) of the coupons at fracture were substituted into Eqs. (4) and (5). The

309

rationale behind this simplification is supported by the research findings illustrated in [27–30]. To simulate the fracture observed in the physical tests of the specimens, the “Damage for Ductile Metals” module in ABAQUS was used in the material model. In particular, two essential parameters were utilised to simulate damage initiation and damage evolution of the material, i.e. a threshold value describing the damage initiation criterion and a damage evolution index. The damage initiation criterion quantifies the maximum strain at which the damage is triggered, and it corresponds to the beginning of the descending branch of the stress-strain curve. For the damage evolution index, it quantifies the degradation behaviour of the material subsequent to reaching the damage initiation criterion. It is worth pointing out that this damage evolution index does not appreciably influence the ultimate load, but may slightly affect the descending branch of the load-displacement response curve after the peak load is achieved. In this study, a linear law was utilised to allow the progressive damage evolution of the material from the inception of damage (damage initiation criterion) to complete loss of the element stiffness (damage evolution index). Since the fracture mechanism is complex and recent studies generally used various methods [27–31] to account for this issue, the damage initiation criterion and the damage evolution index in the damage model were calibrated using the coupon test curves based on an iteration process [30,32]. The rationale and limitations of this simplification were discussed in other research works [31–34]. Also, when the material reaches the strain softening stage, the standard stress-strain response may not precisely quantify the material behaviour, and the FE model will introduce a pronounced mesh dependency, which is directly related to the mesh size. In the current research, the mesh of the region where fracture was expected to occur was refined with an approximate size of 1.0 mm based on a sensitivity analysis. More information about the damage model can be found in [25]. 5.2. Finite element results The comparisons of the test and the FE analysis results of the specimens are shown in Table 3. For the bolted angle specimen models, the ratio of the test ultimate strength (PTest) to the ultimate strength obtained from the FE analysis (PFEM) ranged from 0.93 to 1.10, with a mean value of 0.96 and a CoV of 0.02 (excluding specimen B2-75S). For the welded angle specimen models, the PTest/PFEM ratio ranged

Fracture

Crack initiation Fig. 11. Comparison of test and FE failure mode of typical specimens.

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K. Ke et al. / Journal of Constructional Steel Research 145 (2018) 300–314

from 0.90 to 1.07, with a mean value of 1.00 and a CoV of 0.05. Hence, it can be seen that the FE analysis provided good predictions of the ultimate tensile capacity of the specimens. In addition, typical normalised load-elongation curves extracted from the test results compared well with those from the FE analysis results (i.e. specimen A1-75L, A1-75S, B1-75L, B2-75S, C1-380L, C2-380S, D1-300L and D2-300S), as illustrated in Fig. 10. As mentioned, specimen B2-75S failed in a brittle manner and

could not reach the expected section capacity. As shown in Fig. 10, the normalised load-elongation curves of specimen B2-75S from the test and the analysis were in good agreement before failure of the specimen occurred. Typical failure modes of representative test specimens, i.e. specimen A1-90L with bolted connections and specimen C1-220L with welded connections, were compared with those obtained from the FE analysis results, as shown in Fig. 11. It can be seen from the figure that

A1-90L(Critical section)

Microstrain

800 400 0 -400

5 4 23

1 0

1

2 3 4 5 Strain gauge number

6

1200

Microstrain

1200

A1-90L(Mid-length section)

1600

Test P=100kN FEM P=100kN Test P=200kN FEM P=200kN

800 400 0 -400 -800

7

5

A2-90S(Critical section)

2800 2400 2000 1600 1200 800 400 0 -400

5 4 3 2

1 1

2 3 4 5 Strain gauge number

6

1200 800 400 0 -400 -800

7

5

(c)

800

400 0 -400 0

Test P=100kN FEM P=100kN Test P=200kN FEM P=200kN 1 2 3 4 5 Strain gauge number

1

6 5 4 23

6

7

Microstrain

Microstrain

C1-220L(Mid-length section)

1200

800

-800

400 0 -400 -800

6

(e) 1200 1000 800 600 400 200 0 -200 -400 -600

Test P=100kN FEM P=100kN Test P=200kN 7 FEM P=200kN 7 8 9 10 11 12 Strain gauge number

12 11 8 10

13 9

(f)

C2-220S(Critical section)

C2-220S(Mid-length section)

1600 1200

Test P=100kN FEM P=100kN Test P=200kN FEM P=200kN 0

1

2 3 4 5 Strain gauge number

(g)

6 5 4 1 2 3 6

7

Microstrain

Microstrain

10 Test P=100kN FEM P=100kN 9 Test P=200kN 8 6 7 FEM P=200kN 6 7 8 9 10 11 Strain gauge number

(d)

C1-220L(Critical section)

1200

11

A2-90S(Mid-length section)

1600

Test P=100kN FEM P=100kN Test P=200kN FEM P=200kN

0

10 9 78

(b)

Microstrain

Microstrain

(a)

Test P=100kN FEM P=100kN Test P=200kN 6 FEM P=200kN 6 7 8 9 10 Strain gauge number

800 400 0 -400 6

12 Test P=100kN 11 FEM P=100kN 10 Test P=200kN 78 9 FEM P=200kN 7 8 9 10 11 12 13 Strain gauge number

(h)

Fig. 12. Typical comparisons of strain distributions from tests and FE analyses: (a) critical section of A1-90L (b) mid-length section of A1-90L, (c) critical section of A2-90S, (d) mid-length section of A2-90S, (e) critical section of C1-220L, (f) mid-length section of C1-220L, (g) critical section of C2-220S and (h) mid-length section of C2-220S.

K. Ke et al. / Journal of Constructional Steel Research 145 (2018) 300–314

the FE analysis produced good prediction of the fracture failure mode of the test specimens. Typical strain readings at the critical section and the mid-length section from the FE analysis results were compared with those from the tests (i.e. specimens A1-90L, A2-90S, C1-220L and C220S) as shown in Fig. 12. The comparisons were based on the strain readings at the applied load of 100 kN and 200 kN. It can be seen from the figure that reasonable agreements between the analysis results and the test results were achieved. In general, the FE analysis results were able to capture the overall trend of the strain distributions across the critical and the mid sections of the specimens. 5.3. Parametric study The validated FE models were employed to conduct a parametric study for an in-depth understanding of the shear lag effect on the ultimate tensile capacity of HSS angles. As discussed, the 1−x=L rule provided reasonable predictions of the ultimate tensile strength of the welded HSS angle specimens but overestimated the ultimate tensile strength of the bolted HSS angle specimens. In this context, this parametric study focused on examining the structural performance of bolted tension angles. The study consisted of thirty-nine bolted angles covering a reasonable spectrum of essential parameters, and the details of all the models are given in Table 4. In particular, the steel grades consisting of S275, S690 and S960 steel were considered. For the parameter of the connection length, it was considered by changing the numbers of bolts at the connection, and the bolt pitch was set as 60 mm. In addition,

311

the out-of-plane eccentricity was taken as a parameter by connecting the angles with either the long leg, the equal leg (for angles with an equal leg cross-section), or the short leg. For easy reference, each model was assigned a code. The code starts with a capital letter followed by a number, i.e. “M1”, “M2” and “M3”, representing the material of S275, S690 and S960 for the tension angle, respectively. The following letter ‘L’, ‘E’ and ‘S’ refer to the leg of connection: the long leg (L), the equal leg (E) and the short leg (S), respectively, which can be used to identify the out-ofplane eccentricity. The following number (after hyphen) describes the number of bolts at the connection and the letter at last refers to various cross-sections of angles (i.e. “a” = 75 × 50 × 8, “b” = 75 × 75 × 8, “c” = 100 × 100 × 8, “d” = 75 × 100 × 8, “e” = 75 × 125 × 8, “f” = 75 × 75 × 6, “g” = 75 × 75 × 10, “h” = 75 × 100 × 6 and “i” = 75 × 100 × 10). For instance, the code M2-E-4b refers to the models of equal leg tension angles 75 × 75 × 8 of S690 steel and connected by four bolts, and M3-L-5a describes the model of an unequal leg angle (i.e. 75 × 50 × 8) of S960 steel connected by the long leg with 5 bolts at the connection. The material properties for S275 and S690 steels were consistent with those used in Section 5.1. For S960 steel, the material property input in the model was obtained from coupon tests conducted by the authors in an ongoing research work (see Table 2). To maintain consistency with the test programme, the same dimension of the gusset plates was used in all the models, but failure was ensured to occur in the angles.

Table 4 Results of parametric study. Model code

Section size (mm)

Steel grade

Bolt number

PFEM (kN)

UFEM

Failure mode

1−x=L (URule)

UFEM/URule

M1-L-3aa M1-L-4aa M1-L-5aa M1-E-3ba M1-E-4ba M1-E-4c M1-E-5ba M1-S-3da M1-S-4da M1-S-5da M1-S-5ea M2-L-3aa M2-L-4aa M2-L-5aa M2-E-3ba M2-E-3c M2-E-4ba M2-E-4c M2-E-4fa M2-E-4g M2-E-5ba M2-E-5c M2-S-3da M2-S-4da M2-S-4h M2-S-4i M2-S-5da M2-S-5e M3-L-3aa M3-L-4aa M3-L-5aa M3-E-3ba M3-E-4ba M3-E-4c M3-E-5ba M3-S-3da M3-S-4da M3-S-5da M3-S-5e

75 × 50 × 8 75 × 50 × 8 75 × 50 × 8 75 × 75 × 8 75 × 75 × 8 100 × 100 × 8 75 × 75 × 8 75 × 100 × 8 75 × 100 × 8 75 × 100 × 8 75 × 125 × 8 75 × 50 × 8 75 × 50 × 8 75 × 50 × 8 75 × 75 × 8 100 × 100 × 8 75 × 75 × 8 100 × 100 × 8 75 × 75 × 6 75 × 75 × 10 75 × 75 × 8 100 × 100 × 8 75 × 100 × 8 75 × 100 × 8 75 × 100 × 6 75 × 100 × 10 75 × 100 × 8 75 × 125 × 8 75 × 50 × 8 75 × 50 × 8 75 × 50 × 8 75 × 75 × 8 75 × 75 × 8 100 × 100 × 8 75 × 75 × 8 75 × 100 × 8 75 × 100 × 8 75 × 100 × 8 75 × 125 × 8

S275 S275 S275 S275 S275 S275 S275 S275 S275 S275 S275 S690 S690 S690 S690 S690 S690 S690 S690 S690 S690 S690 S690 S690 S690 S690 S690 S690 S960 S960 S960 S960 S960 S960 S960 S960 S960 S960 S960

3 4 5 3 4 4 5 3 4 5 5 3 4 5 3 3 4 4 4 4 5 5 3 4 4 4 5 5 3 4 5 3 4 4 5 3 4 5 5

314 316 316 362 380 530 382 390 424 440 482 565 570 570 590 741 635 798 474 487 661 843 610 655 786 810 687 711 708 737 742 744 797 1077 834 777 833 873 908

0.95 0.95 0.95 0.85 0.90 0.87 0.90 0.76 0.82 0.85 0.79 0.99 1.00 1.00 0.81 0.71 0.87 0.76 0.85 0.88 0.90 0.80 0.69 0.74 0.72 0.74 0.77 0.68 0.95 0.99 0.99 0.78 0.84 0.79 0.87 0.67 0.72 0.75 0.66

C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C

0.91 0.94 0.96 0.86 0.90 0.91 0.93 0.79 0.86 0.90 0.86 0.91 0.94 0.96 0.86 0.81 0.90 0.88 0.91 0.90 0.93 0.91 0.79 0.86 0.86 0.86 0.90 0.86 0.91 0.94 0.96 0.86 0.90 0.91 0.93 0.79 0.86 0.90 0.86

1.04 1.01 0.99 0.99 1.00 0.96 0.97 0.96 0.95 0.94 0.92 1.09 1.06 1.04 0.94 0.88 0.97 0.86 0.93 0.98 0.97 0.88 0.87 0.86 0.84 0.86 0.86 0.79 1.04 1.05 1.03 0.91 0.93 0.87 0.94 0.85 0.84 0.83 0.77

a

The results are shown in Fig. 13 to clarify the effect of influential parameters.

K. Ke et al. / Journal of Constructional Steel Research 145 (2018) 300–314

5.3.1. Parametric study: effect of steel grade The ultimate loads of the models are presented in Table 4, and the ratio of the FE efficiency (UFEM) defined by the ratio of the ultimate tensile capacity of angles from the FE analysis results to the nominal ultimate strength of the angles (i.e. fuAn) is given in the table. The effect of steel grade on the shear lag of the tension angles is illustrated by plotting the FE efficiency against the steel grades, as shown in Fig. 13a. Nine series of data with three different connection lengths and three different out-of-plane eccentricities were compared. For angles connected by the long leg, all the efficiencies were close to unity (also see Table 4), indicating that in these cases the shear lag effect was not appreciable. Thus, the material property did not have significant influence on the FE efficiency of the angles connected by the long leg which was in agreement with the test results. However, for the cases of equal leg and short leg connections, the FE efficiencies decreased with increasing steel grade. In particular, the efficiencies decreased significantly for short leg connected angles with higher steel grades. Hence, it can be seen that the shear lag effect was significant for HSS angles with either equal or short leg connections.

5.3.2. Parametric study: effect of connection length The effect of connection length on the shear lag of the tension angles is illustrated in Fig. 13b. Nine series of data that consist of three different steel grades and three different out-of-plane eccentricities are presented. It can be seen that the FE efficiencies were not significantly affected by the connection length for the long leg connected angles. However, for angles with equal leg connections, improvement of the FE efficiency was observed with increasing connection length. The average increase in efficiency by adding one more bolt was approximately 3% for angles of S275 steel and 5% for angles of both S690 and S960 steels. Comparatively, the average increase in the FE efficiency for the short leg connected angles by adding one more bolt was approximately 5% for all cases.

Long-3 bolts Long-4 bolts Long-5 bolts Equal-3 bolts Equal-4 bolts Equal-5 bolts Short-3 bolts Short-4 bolts Short-5 bolts

FE efficiency

1.0 0.9 0.8 0.7 0.6 0.5

S275

S690 Steel grade

5.3.3. Parametric study: effect of out-of-plane eccentricity To illustrate the effect of out-of-plane eccentricity on the shear lag of tension angles, the FE efficiencies are plotted against the type of connected leg as shown in Fig. 13c. The results indicated that the FE efficiency of the angles was greatly affected by the out-of-plane eccentricity. When the out-of-plane eccentricity was increased from 12.9 mm to 31.0 mm, the decrease of FE efficiency ranged from 23% to 30% and 24% to 29% (depending on the number of bolts) for S690 steel angles and S960 steel angles, respectively. Comparatively, for the angles with S275 steel, the decrease in the FE efficiency ranged from 10% to 20% for different connection lengths. In general, all of these findings matched the observations from the tests. 5.4. Design considerations The parametric study results were used to examine the applicability of the 1−x=L rule for predicting the ultimate tensile capacity of HSS bolted angles. The UFEM/URule ratios of all models are given in Table 4. It can be seen from the table that for the HSS bolted angles connected by the long leg, the UFEM/URule ratios were all larger than unity. Hence, the 1−x=L rule generally provided conservative predictions of the ultimate tensile capacity of long leg connected HSS bolted angles. However, for the equal leg and the short leg connected angles, the predictions by the URule were generally non-conservative as shown by the overestimations of the ultimate tensile capacity of the angles of S690 steel and S960 steel. All the analysis and test data of bolted tension angles in terms of UFEM/URule ratio or UTest/URule ratio are shown in Fig. 14a, and the mean value of UFEM/URule ratio with the corresponding CoV are also indicated in the figure. As can be seen from the figure, the efficiency ratios were generally below unity with a mean value of 0.94 and CoV of 0.09. Hence, it was proposed to modify the 1−x=L rule for improving the prediction of the ultimate tensile capacity of HSS bolted equal leg angles or short leg connected angles. In particular, two adjusting coefficients, βm and βt were proposed considering the effects of material properties

0.9 0.8 0.7 0.6 0.5

S960

(a)

3

4 5 Number of bolts

(b) S275-3 bolts S275-4 bolts S275-5 bolts S690-3 bolts S690-4 bolts S690-5 bolts S960-3 bolts S960-4 bolts S960-5 bolts

1.0

FE efficiency

S275-Long S690-Long S960-Long S275-Equal S690-Equal S960-Equal S275-Short S690-Short S960-Short

1.0

FE efficiency

312

0.9 0.8 0.7 0.6 0.5

Long leg

Equal leg Short leg Connection leg

(c) Fig. 13. Effect of essential parameters on the FE efficiency (UFEM): (a) effect of steel grades, (b) effect of number of bolts and (c) effect of connection leg.

UFEM/UMo or UTest/UMo

K. Ke et al. / Journal of Constructional Steel Research 145 (2018) 300–314

1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5

Test result (S690) Test result (S275)

FE results S275 S690 S960

UMO ¼ βm βt URule for equal leg and short leg connection

+15% +10%

Mean=0.94 CoV=0.09

Equal leg

Short leg

UFEM/UMo or UTest/UMo

(a) 1.5 Test result (S690) 1.4 Test result (S275) 1.3 1.2 1.1 1.0 0.9 0.8 0.7 Mean-2σ 0.6 0.5 Long leg Equal leg

FE results S275 S690 S960

+15% +10%

6. Summary and conclusions

-10% -15% Mean=1.03 CoV=0.04

Short leg

(b) Fig. 14. Comparison of test data or numerical data with design predictions: (a) equation developed by Munse and Chesson [3,4] and (b) the proposed modified equation.

(ratio of the ultimate strength to the yield strength) and the connection type, respectively. In light of the discussion in Section 4, the ratio of the ultimate strength to the yield strength and the ductility of the material could influence the effectiveness of stress redistribution of the sections in the angle and compromise the ultimate tensile capacity. Based on this observation and statistical analyses, the following preliminary equation of βm accounting for the ratio of the ultimate strength to yield strength ratio of the material for either equal leg angles or short leg connection was proposed using curve fitting of the FE analysis data: βm ¼ −0:34η−1 þ 1:14

ð6Þ

where η is the ratio of the ultimate strength to the yield strength of the material. Moreover, recognising the potential effect of the connection type on the shear leg of bolted angles, βt was preliminarily determined based on the available FE analysis results, given as follows: βt ¼ 0:4 S=S0 þ 0:7

ð7Þ

where S = length of the connected leg and S′ = length of the unconnected leg (Fig. 2a). Incorporating the above modification factors into the 1−x=L rule, the ultimate tensile capacity of HSS bolted angles can be determined by the following modified equations: UMO ¼ URule for long leg connection

ð9Þ

It is worth pointing out that the equations are preliminary since the modification factors were obtained based on the limited numerical data reported in this paper, and the least square rule was applied in the curve fitting procedure aiming to match the data points. The comparison of the numerical analysis results and the proposed modified equation are illustrated in Fig. 14b. It can be seen that the discrepancies in the efficiencies of all the data were within 10% and the maximum observed error was 9% on the conservative side. The mean UFEM/UMo ratio determined by the proposed modified design method is 1.03, and the corresponding CoV is only 0.04. The mean minus twice standard deviation (σ) is also indicated in the figure to show the conservatism of the proposed equations. To help verify the validity of the proposed equations, the ratios of UTest/UMO were also included in Fig. 14b. It can be seen from the figure that the proposed equations were able to produce good predictions of the test results. Thus, the improved accuracy of the modified approach for quantifying the ultimate tensile capacity of bolted angles is demonstrated. The proposed design equations can be further refined and verified by more experimental and numerical data covering a wider spectrum of parameters (i.e. steel grades and angle details). Designers are encouraged to use the design equations for the cases within the spectrum of parameters considered.

-10% -15%

Long leg

313

ð8Þ

An investigation consisting of experimental and numerical analyses were conducted to examine the shear lag effect on the behaviour and ultimate tensile capacity of HSS single tension angles with bolted and welded connections. Eighteen full-scale tests of tension angles including test parameters of steel grade, connection length and out-of-plane eccentricity were conducted. Out of fourteen specimens with HSS angles, seven of them were HSS angles with bolted connections and the other seven were HSS angles with welded connections. In general, the test observations showed that the test efficiency of the angle section reduced with higher steel grade, but this tendency was not evident for specimens with HSS angles connected by the long leg. For the effect of connection length, it was observed that increasing connection length resulted in the improvement of test efficiency, and such trend was more pronounced for specimens with short leg connected specimens. As expected, an increase of out-of-plane eccentricity led to a substantial reduction of test efficiency. Based on the test data, the applicability of the classical 1−x=L rule for quantifying the shear lag effect of HSS angles was also examined. Good agreements were found between the test results and the predictions for specimens with welded HSS angles and for the bolted HSS angles connected by the long leg. However, the 1−x=L rule generally overestimated the ultimate tensile capacity of the bolted HSS angle specimens connected by the short leg. Finite element analyses of the specimens were performed utilising ABAQUS. In general, satisfactory agreement between the test ultimate tensile capacities of the specimens and those determined by the finite element analyses was achieved. The test-to-analysis ratio of the ultimate tensile capacity ranged from 0.93 to 1.10 and 0.90 to 1.07 for the specimens with bolted connections and welded connections, respectively. The validated finite element models were employed to conduct a parametric study to provide an insight into the behaviour and ultimate tensile capacity of bolted tension angles. The analysis results were generally in line with the experimental findings. In general, the classical 1−x=L rule produced non-conservative predictions of the ultimate tensile capacity of bolted HSS angles connected by either the equal leg (for equal leg connections) or short leg. Recognising the limitation of the 1−x=L rule, a set of modified equations for predicting the ultimate tensile capacity of bolted HSS tension angles was proposed. Satisfactory agreements between the ultimate

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tensile capacity predicted by the modified equations and the FE analyses results were achieved with a mean analysis-to-predicted ratio of 1.03 and the corresponding CoV of 0.04. The tests results from this study were also utilised to further justify the accuracy of the modified equations for predicting the ultimate tensile capacity of bolted HSS tension angles, and satisfactory agreement between the test results and predictions by the modified equations was also obtained. Currently, tests covering a wider spectrum of parameters are being conducted. Numerical works based on a more comprehensive parametric study along with reliability analysis and statistical consideration are also underway. The research findings will fully reveal the shear lag effect on the behaviour and ultimate tensile capacity of HSS tension angles. Acknowledgement The work in this paper is partially supported by an MPhil studentship provided to the second author by the Hong Kong Polytechnic University. Funding support received from the Chinese National Engineering Research Centre For Steel Connection, The Hong Kong Polytechnic University (Project No. 1-BBYQ) for the study is also acknowledged. The First Author Would also like to thank the National Natural Science Foundation of China (Grant No. 51708197) for providing funding support. References [1] R.P. Davis, G.P. Boomsliter, Tensile tests of welded and riveted structural members, J. Am. Weld. Soc. 13 (4) (1934) 21–27. [2] G. Gibson, B. Wake, An investigation of welded connections for angle tension members, Weld. J. 21 (1) (1942) 44–49. [3] E. Chesson, W.H. Munse, Riveted and bolted joints: truss type tensile connection, J. Struct. Eng. ASCE 89 (1) (1963) 67–106. [4] W.H. Munse, E. Chesson, Riveted and bolted joints: net section design, J. Struct. Eng. ASCE 89 (1) (1963) 107–126. [5] ANSI/AISC 360–10, Specification for Structural Steel Buildings, American Institute of Steel Construction, Chicago, IL, USA, 2010. [6] CSA-S16-14, Limit States Design of Steel Structures, Canadian Standards Association, Toronto, 2001. [7] GB 50017-2003, Code for Design of Steel Structures, China Architecture and Building Press, Beijing, 2006. [8] P. Regan, P. Salter, Tests on welded-angle tension members, Struct. Eng. 62 (1984) 25–30. [9] W.S. Easterling, L. Gonzales, Shear lag effects in steel tension members, Eng. J. 3 (1993) 77–89.

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