Stress concentration factors and fatigue behavior of square bird-beak SHS T-joints under out-of-plane bending

Engineering Structures 99 (2015) 677–684

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Stress concentration factors and fatigue behavior of square bird-beak SHS T-joints under out-of-plane bending Bin Cheng a,b,⇑, Qin Qian a, Xiao-Ling Zhao c a

Department of Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, China c Department of Civil Engineering, Monash University, Melbourne, VIC 3168, Australia b

a r t i c l e

i n f o

Article history: Received 27 December 2013 Revised 20 May 2015 Accepted 22 May 2015 Available online 10 June 2015 Keywords: Square hollow section Bird-beak joint Hot spot stress Stress concentration factor Fatigue crack High-cycle fatigue test

a b s t r a c t In this study, fatigue tests of square bird-beak square hollow section (SHS) joints were conducted under out-of-plane bending. Nine orthogonally designed T-joints were fabricated. Static loading was first conducted, during which elastic strain distributions near the potential hot spots were measured. Stress concentration factors (SCFs) at weld toes were calculated using the quadratic extrapolation approach. Critical locations were also identified. Based on the high-cycle fatigue testing that followed, the fatigue crack initiation, propagation and fatigue failure mode were obtained. Variations of the surface crack length and out-of-plane rigidity against number of load cycles were also revealed. The cycle numbers that correspond to through-thickness cracks were observed to be greater than the numbers obtained from S–N curves provided by the IIW code, which indicates that the IIW curves/formulations for conventional joints could also be used to estimate the fatigue lives of square bird-beak joints if the hot spot stress ranges have been properly determined. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Square hollow section (SHS) constructions have extensive applications in many industries, such as buildings, bridges, offshore structures and lattice masts, where steel members are generally directly welded to each other [1]. Traditionally, the chord walls in a welded SHS–SHS connection are aligned parallel/perpendicular to the brace walls, which makes end cutting and fabricating extremely easy, as shown in Fig. 1(a). Furthermore, other configurations, known as bird-beak joints, are created by rotating the members 45° about their longitudinal axes. Fig. 1(b) and (c) demonstrate two types of bird-beak SHS T-joints, i.e. a square bird-beak joint generated by rotating only the chord and a diamond bird-beak joint generated by rotating both the chord and brace. These innovative connections were originally proposed for their aesthetic appeals. Subsequently, researchers and engineers determined that bird-beak joints could assist in relieving lateral wind loads and providing higher structural resistances. The improved mechanical behaviors were attributed to ability of the new orientation to assist in transferring forces between members through a natural in-plane action rather than the bending of walls. ⇑ Corresponding author at: Room A508, Ruth Mulan Chu Chao Building, 800 Dongchuan Road, Shanghai 200240, China. Tel.: +86 21 34207985. E-mail address: [email protected] (B. Cheng). http://dx.doi.org/10.1016/j.engstruct.2015.05.033 0141-0296/Ó 2015 Elsevier Ltd. All rights reserved.

For the safety of using these novel constructions, many researchers devoted their efforts to the ultimate resistances of bird-beak joints under static loads. Ono et al. [2–4] initially investigated the ultimate bearing capacities and failure modes of square bird-beak T-joints using a structural test and numerical analysis, where multiple load cases, including brace axial force, in-plane bending, and out-of-plane bending, were considered. Davies and Kelly [5] conducted a similar finite element analysis. Owen et al. [6] focused on the differences in structural behavior between bird-beak and traditional concrete-filled SHS T-joints. Davies et al. [7] described the effects of purlin loads on the capacities of over-lapped bird-beak K-joints. Owen et al. [8] also investigated the influences of structural parameters, such as brace-to-chord length ratio, chord slenderness ratio, sectional wall slenderness ratio and boundary conditions at chord ends, on the resistances of bird-beak X-joints loaded by pairs of brace axial compression forces. Christitsas [9] and Lei [10] conducted experiments and a parametric FEM analysis of square bird-beak X-joints subject to in-plane bending and proposed formulas for design resistance as a result. Zhu and Liu [11] studied the non-linear behavior of axially loaded diamond bird-beak XT-joints using a finite element method. The studies indicated that the ultimate strengths of bird-beak joints were higher than those of conventional joints with the same non-dimensional parameters.

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Chord

Brace

(a) Chord

Chord

Brace

Brace

All specimens featured chords that measured 1200 mm in length and braces at least 600 mm in length. Therefore, the member ends were separated from the junction area by at least three times the member’s sectional width, so that the effects of boundary conditions can be ignored. The brace members were connected to the chords through 80% partial joint penetration (PJP) groove welds and fillet welds, which were produced by carbon dioxide gas arc welding. The cold-formed steel tubes used for fabrication were of Q420 grade, which strictly conforms to GB/T 6728-2002 [30]. Standard coupons were taken from the steel tubes and tested in uniaxial tension. The measured mechanical properties were determined as follows: yield stress Sy = 494 MPa, ultimate tensile strength Su = 597 MPa and Young’s modulus E = 202 GPa. 2.2. Test rig

(b)

(c)

Fig. 1. Welded SHS–SHS T-joints: (a) conventional; (b) square bird-beak; (c) diamond bird-beak.

However, limited references are available concerning the fatigue behavior of bird-beak joints. Ishida [12] performed the earliest fatigue tests of bird-beak T-joints for a brace axial force load case. Keizer et al. [13] investigated the stress concentration factors of diamond bird-beak joints under brace axial forces in his master’s degree thesis. Tong et al. [14,15] presented an experimental study of stress concentration factors for diamond bird-beak T-joints under axial force and in-plane bending on the brace. Cheng et al. [16] carried out tests to obtain stress concentration factors of square bird-beak T-joints under chord and brace axial forces. Compared with abundant products for fatigue of conventional CHS and SHS steel connections that have been presented in literature [17–27] and with detailed provisions in current design codes such as IIW Fatigue Design Recommendation [28] and CIDECT Design Guide No. 8 [29], existing fatigue studies of bird-beak SHS joints are far from systematic and complete. This lack of literature may prevents the applications of bird-beak joints in structures dominated by cyclic repeating live loads, e.g. bridges. In a tubular truss bridge, the joints that connect floor beams and chords are the most prone to fatigue failure because the floor beams are directly subject to cyclic vehicle or walking loads. Therefore, square bird-beak T-joints under out-of-plane bending, which are similar in configuration and loading conditions to chord-to-floor beam connections in real structures, are employed as research objects for fatigue considerations. Stress concentration factors and fatigue behaviors, including the failure mode, characteristic fatigue life, crack initiation and propagation, as well as the variation of joint rigidity were investigated using the experimental method. 2. Test setup 2.1. Specimens Nine square bird-beak joints (SBBJ) were fabricated. The nine specimens were orthogonally designed based on the three types of non-dimensional parameters, i.e. brace-to-chord width ratio b = b1/b0, chord wall slenderness ratio 2c = b0/t0 and brace-to-chord wall thickness ratio s = t1/t0. In this study, b0 and t0 represent the sectional width and wall thickness of the chord, and b1 and t1 correspond to the sectional width and wall thickness of the brace (Fig. 2). The parameter values used for the orthogonal experiment were originally selected as b = 0.45, 0.60 and 0.75; 2c = 16.67, 20 and 25; and s = 0.5, 0.65 and 0.8, and the real s values of certain specimens were slightly modified based on the actual modular dimensions of steel tubes provided by the manufacturing factory, as listed in Table 1.

Prior to the test, a rigid steel platform was completely fixed upon the foundation of the testing machine. Fastening bolts, which were used to mount chord ends on the platform, ran through the platform and were then anchored into the threaded holes that had been reserved in the end plates at chord ends, as shown in Fig. 2(b). A concentrated force was applied at the brace end in the direction normal to the joint plane so that out-of-plane bending was produced for conjunction areas. The actuator had a maximum capacity of 200 kN. Fig. 3 displays the scenes of static test and fatigue test. 2.3. Measurement of strains Two types of strain gauges, i.e. regular gauges and strip gauges, were used in the test to record the nominal strains and hot spot strains, respectively. The section used for nominal strain monitoring, i.e. the brace section II-II in Fig. 2, was located 300 mm away from the brace end, which indicated that the strain distribution on the section would neither be affected by the junction details nor by the end boundaries. Twelve equally spaced regular gauges were symmetrically arranged around the outer surface of the section, as shown in Fig. 4(a). The strip gauges, each of which contained several regular base elements, were arranged along the potential hot lines to obtain the hot spot strains at the weld toes using an extrapolation approach. Finite element analyses were performed in advance to facilitate the determination of potential hot lines because relevant references for stress distribution regularities in published reports were lacking. The results indicate that stresses in crown areas are sufficiently low to be ignored for out-of-plane bending. Therefore, six saddle hot lines (i.e. Sa-B, Sa-C and Sa-D in the chord and Sa-E, Sa-F and Sa-A in the brace) were selected for the bird-beak specimens by adopting a similar criteria used for conventional joints provided in CIDECT Design Guide No. 8 [29], as shown in Fig. 4(b). Among these, four corner lines Sa-A, Sa-B, Sa-D and Sa-E were required to be aligned with the inner surface plane of the brace wall. Strip gauges of the same quantities were also arranged in symmetrical lines about the brace axis in the joint plane, which implied that strains in twelve hot lines were actually recorded for each T-joint. The larger SCFs of the two sides were used. By referring to the boundaries of the extrapolation region for conventional RHS joints as provided in the CIDECT Design Guide No. 8 [29], each extrapolation region was required to start from a length Lmin away from the weld toe and have a length of t in this study, where t represents the sectional wall thickness of the member on which the strains were measured; Lmin was taken to be the greater value between 0.4 t and 4 mm, as shown in Fig. 4(b). For every strip gauge, the uniform intervals between the regular base elements were precisely designed by the manufacturer to be

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t1

Butt weld

b1 b1

I

30

End plate Stiffener

L1+15+0.5b1 L1+15

300 II

II

315

L1 End plate

Concentrated load

Brace

End plate II

Chord

Brace

b0 End plate

t0

Threaded hole

Chord Butt weld

II

30

b0

230

Butt weld

L0

Stiffener

Rigid platform Fastening bolt

I

(a)

5x70

(b) Fig. 2. Specimens: (a) square bird-beak T-joints; (b) section I-I.

Table 1 Dimensions of T-joint specimens. Specimen

SBBJ-1 SBBJ-2 SBBJ-3 SBBJ-4 SBBJ-5 SBBJ-6 SBBJ-7 SBBJ-8 SBBJ-9

Chord

Brace

Non-dimensional para

b0 (mm)

t0 (mm)

L0 (mm)

b1 (mm)

t1 (mm)

L1 (mm)

b

2c

s

200 200 200 200 200 200 200 200 200

12 10 8 12 10 8 12 10 8

1200

150 150 150 120 120 120 90 90 90

6 6 6 10 5 5 8 8 4

600

0.75 0.75 0.75 0.60 0.60 0.60 0.45 0.45 0.45

16.67 20.00 25.00 16.67 20.00 25.00 16.67 20.00 25.00

0.500 0.600 0.750 0.833 0.500 0.625 0.667 0.800 0.500

Fig. 3. Test rig: (a) static loading; (b) fatigue loading.

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y

(a)

B7

B8

B12

B11

b1/3 B1

B6

B2

B5

b1 b1/3

x

b1/3 B9

B3 b1/3

B10

B4 b1/3

b1/3

b1

Center line of brace section

(b)

Crown area

Crown area

Right side

Left side

2mm t1 Saddle area

Lmin Sa-C

2mm

Brace wall Sa-E Sa-F Sa-A

Sa-D

Sa-E' Sa-F' Sa-A'

t1

t1 Sa-B

Toe of fillet weld

Sa-B'

t1

t1

Lmin t0

t0 45°

Sa-D'

Sa-C'

Saddle area

45°

Fig. 4. Arrangement of strain gauges: (a) regular gauges on brace section II-II; (b) strip gauges at hot spots (top view).

2 mm, which was regarded to be highly consistent with the extrapolation regions and accurate enough for strain extrapolations. 2.4. Measurement of fatigue data Three types of indicators were monitored during the fatigue test as follows: (1) Fatigue life – Visible crack initiation life (N2). Appearances of initial cracks were determined by human eyes with the help of a magnifying glass. A very thin layer of white paint was sprayed on the junction surface to facilitate the crack detection. Human inspections were performed every 2 h. Furthermore, two HD cameras, the lens centers of which had been focused on the potential locations of crack initiations (i.e. hot spots Sa-B and Sa-B0 ), were also introduced to identify the accurate time of crack initiation, as shown in Fig. 3(b). – Through-thickness crack life (N3). Through-thickness cracks were identified by monitoring the internal pressures of airtight hollow sections. Two barometers, one for the chord and one for the brace, were connected to the member ends, and the internal air pressures were pumped to 2 bar (0.2 MPa) before testing. The internal pressure rapidly decreased when the crack penetrated the wall thickness. The camera focusing on the barometer recorded the variations of barometer indicators. – Failure or end of test life (N4). As reported in the existing literature, the fatigue failure or the end of fatigue test for tubular joints is normally determined using the surface crack length criterion. In this test, two cracks, which initiated at different brace corners, propagated toward each other and then combined into one before penetrating the

wall thickness. Therefore, N4 was considered to be the number of cycles corresponding to the ultimate total length of surface crack (Lmax) in this research, which equals 1.5 times the sum of the brace width and twice the weld size on the chord (i.e. Lmax = 1.5(b1 + 2w0,Sa)). Here, w0,Sa is the average projected length on the chord surface in the saddle area. (2) Surface crack length. To measure the crack length, visible crack tips were marked every 2 h, as shown in Fig. 7. Furthermore, two HD cameras were always focused on the area near the crack tips during the entire test process to ensure that the crack propagations between human markings were captured. (3) Out-of-plane displacement. The out-of-plane displacement ranges at brace ends, which are identical in value to the actuator travels, were automatically recorded in real time by the fatigue-loading machine. These displacement ranges are direct reflections of the out-of-plane rigidities of joints because the fatigue loads are constant for each joint. 2.5. Test procedure Static load and fatigue load were successively applied to the same specimen during the test. In the static test, specimens were loaded within the elastic range, and the monotonic load was gradually increased to its maximum value by several equal-sized load increments/steps. All strains at gauge locations were recorded at each load level, and the average values derived from effective load increments were then taken as the experimental results. In the fatigue test that followed, the joints were subjected to the constant stress amplitude fatigue loads that had been determined from the specified hot spot stress ranges and the previously measured SCFs. All stress ratios (R) were set to 0.1, and the loading frequencies

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varied between 1 and 5 Hz. The fatigue tests were terminated once the failure criteria of the total crack length were reached. 3. Discussion on stress concentration factors 3.1. Comparison of experimental and beam theory nominal strains The experimental nominal strain increments (Denom,exp) were compared to the theoretical nominal strain increments (Denom,theory), which were calculated using beam theory: P315 Denom;theory ¼ DEW , where DP is the concentrated load increment 1 at the brace end, 315 mm is the distance from the concentrated load to the brace section II-II (Fig. 2), W1 is the elastic section modulus of the brace, and E is Young’s modulus of steel. Fig. 5 shows that the experimental and theoretical nominal strains agree well. Fig. 6. Typical stress distributions along hot lines Sa-B and Sa-A of specimen SBBJ-2.

3.2. Non-linearity of strain distribution along hot line Fig. 6 illustrates the typical stress distributions along the hot spot location lines Sa-A and Sa-B of specimen SBBJ-2 as expressed by SCFs; both linear and quadratic extrapolation curves have been hs plotted. All SCFs were calculated as SCF ¼ 1:1  DDeenom , in which Dehs

is the extrapolated hot spot strain increment at the weld toe; Denom is the nominal strain increment at the saddle area that was converted from the measured nominal strain increment on   1 section II-II (i.e. Denom,exp) using Denom ¼ Denom;exp  L1 þ15þ0:5b ; a 315 factor of 1.1 is employed to describe the configuration induced relationship between stress concentration factor (SCF) and strain concentration factor (SNCF) [31]. Non-linear variations in the strains indicate that quadratic extrapolation will produce more accurate hot spot strains at weld toes. SCFl This conclusion is confirmed by introducing the ratio SCF , where q

SCFl and SCFq correspond to the strain concentration factors derived from linear and quadratic extrapolation, respectively. The mean values of the

SCFl SCFq

0.90. The minimum

ratios in the chord and brace were both

SCFl SCFq

Fig. 7. Typical failure mode of specimen SBBJ-5.

ratios in the chord and brace were

observed to be 0.68 and 0.59, respectively. In other words, linear extrapolation may underestimate the SCFs of square bird-beak RHS joints by up to 41%. Therefore, only the SCFs generated from quadratic extrapolation were used for the discussion in this study. 3.3. Magnitude of SCFs at hot spots The extrapolated magnitudes of SCFs at selected hot spots are provided in Table 2. Due to symmetry, the stress distributions on the left side and right side are identical. For the saddle spots in

the chord (i.e. Sa-B, Sa-C and Sa-D), the SCF is always maximized at spots Sa-B. The lowest SCFs at spots Sa-D are only approximately 30% of the maximum SCFs at spots Sa-B. Similar characteristics were observed for the three saddles spots in the brace, i.e. SCF decreases gradually as the location is shifted from Sa-A, through Sa-F, and to Sa-E. The differences among the three saddle spots in the chord are more significant than those in the brace, which could be attributed to the diversities in the directions of measured strains among chord spots. As a result of the asymmetrical weld sizes and experimental errors, the mean SCF differences between the left side and right side were measured as 6.4% and 17.1%, respectively, for chord spots and brace spots. In this study, the larger SCF of the two sides was taken as the experimental SCF of the hot spot. Table 2 also provides the maximum SCF in the chord (SCFch) and the maximum SCF in the brace (SCFbr). Except for specimen SBBJ-9, whose SCFch are 18% lower than SCFbr, the SCFch of the other eight bird-beak joints are greater than their SCFbr by up to 68%. Therefore, the maximum SCF of the entire square bird-beak joint is more likely to occur in the chord for out-of-plane bending.

3.4. Influence of non-dimensional parameters

Fig. 5. Experimental nominal strains versus beam theory nominal strains.

Influencing factor analyses based on the experimental results of nine orthogonally designed specimens were conducted to reveal the relative influences of three non-dimensional parameters on

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B. Cheng et al. / Engineering Structures 99 (2015) 677–684

Table 2 Experimental SCFs. Specimen

Chord Sa-D

SBBJ-1 SBBJ-2 SBBJ-3 SBBJ-4 SBBJ-5 SBBJ-6 SBBJ-7 SBBJ-8 SBBJ-9 a

1.00 2.00 3.15 1.85 1.31 2.31 1.85 1.91 1.83

Brace Sa-C

Sa-B

3.03 5.50 10.09 4.87 3.16 N/A 3.52 4.64 4.54

4.75 6.80 10.30 6.75 5.57 7.51 4.87 6.01 4.87

Sa-B

0

Sa-C

5.30 6.70 10.77 6.13 5.14 7.24 4.49 6.64 4.85

0

3.77 2.98 7.72 5.30 3.09 8.12 3.54 N/A 4.43

Sa-D

0

N/A 1.76 2.92 2.38 1.69 2.99 1.51 1.57 1.88

SNCFch a

5.30 6.80a 10.77a 6.75a 5.57* 7.51a 4.87a 6.64a 4.87

Sa-E

Sa-F

Sa-A

Sa-A0

Sa-F0

Sa-E0

SNCFbr

2.45 3.87 4.81 N/A 3.18 3.38 3.31 1.98 2.29

3.53 5.07 4.86 3.37 4.00 5.83 3.21 3.26 3.36

3.82 5.05 6.53 3.96 5.53 7.08 3.44 3.95 5.96

4.81 5.29 8.81 4.44 N/A 5.08 3.11 3.95 4.70

4.49 4.21 4.96 2.85 2.38 4.20 2.63 4.15 3.60

2.66 2.59 3.71 N/A 1.83 2.96 1.49 3.47 2.68

4.81 5.29 8.81 4.44 5.53 7.08 3.44 3.95 5.96a

Represents the maximum SNCF within the entire joint.

the maximum SCFs of square bird-beak joints. An intuitive analysis and variance analysis were both performed in the study. In the current study, three non-dimensional parameters were considered, each of which contained three levels, i.e. b = 0.45, 0.60 and 0.75; 2c = 16.67, 20 and 25; and s = 0.5, 0.65 and 0.8. In the intuitive analysis, the three mean values of the SCFs that correspond to the three levels (i.e., K1, K2 and K3) were first calculated for each factor, followed by the calculation of the range R (i.e. the maximum mean value minus the minimum mean value), which was used to describe the influencing degree of each factor. A higher R value signified a more significant influence. However, for the variance analysis in which the random errors of test were considered, the sums of the squares of deviations (DEVSQ) were further calculated for each factor using the K1, K1 and K3 values provided in the intuitive analysis; furthermore, the degree of freedom (DF), the F-value, and the critical F-value (Fc) were also introduced. The relative influences of the factors were assumed to be reflected by their F-values, i.e. a greater F-value represented a more significant influence. Table 3 provides the analysis results for spots Sa-A and Sa-B. The two approaches provide similar conclusions. For the chord spot Sa-B, the R (or F) values of s are clearly greater compared to those of b and 2c, which indicates that s has the most notable effects on the maximum SCFs in the chord. For the brace spot Sa-A, the SCFs are the most sensitive to 2c; however, the influences of s are quite limited. 4. Discussion on fatigue observations 4.1. Failure mode The fatigue failure modes of nine specimens are similar, as shown in Fig. 7. Fatigue cracks are observed only in the chord walls on the tensile side of the joint, although the maximum SCFs of two specimens, i.e. SBBJ-5 and SBBJ-9, occurred in the brace. The cracking process can be normally described by the following three stages: Table 3 Results of influencing factor analysis for orthogonally designed bird-beak specimens. Spot

Factor

Intuitive analysis

Variance analysis

K1

K2

K3

R

DEVSQ

DF

F

Fc

Sa-B

A (b) B (2c) C (s) Error

7.62 5.64 5.25

6.61 6.34 6.39

5.46 7.72 8.05

2.16 2.08 2.81

7.03 6.70 11.95 1.33

2 2 2 2

5.30 5.05 9.00 a = 0.1

9 9 9

Sa-A

A (b) B (2c) C (s) Error

6.30 4.23 5.43

5.68 4.92 5.27

4.45 7.28 5.73

1.85 3.05 0.46

5.34 15.37 0.33 0.91

2 2 2 2

5.89 16.97 0.37 a = 0.1

9 9 9

Stage I: Due to symmetry, two cracks initiated at weld toes near spots Sa-B and Sa-B0 . The locations of crack initiation are consistent with the points of maximum SCF. Stage II: As cyclic loading continued, two cracks propagated toward each other until they met and combined. The crack growth paths were always restricted along the weld toes on the chord. The cracks are more easily observed by the naked eye because their opening widths continued to be enlarged during this stage. Stage III: Two tips of the integrated crack, which are located near the spot Sa-C and spot Sa-C0 , further propagated away from the weld toes until the end of the test. As a result, two oblique crack parts appeared on the chord surface, and the angles between oblique crack segments and the chord’s longitudinal axes are measured to be 20–30°. 4.2. Fatigue life The experimental numbers of cycles (N2, N3, N4) are summarized in Table 4. The maximum SCF in the chord (SCFch), nominal stress range (Snom), and hot spot stress range (Srhs) are also provided. The crack initiation lives, N2, of all specimens are only 21– 56% of the through-thickness lives, N3, which indicates a very long fatigue life between the initial crack and through-thickness crack. The ratio of the fatigue failure life to the through-thickness life, N4/N3, takes on values between 1.07 and 1.35, except for specimen SBBJ-2, whose N4/N3 ratio reaches 1.64. The mean N4/N3 value of seven joints excluding SBBJ-2 is only 1.21. This finding indicates that the fatigue life between the through-thickness crack and fatigue failure are not long enough to be used. Therefore, the through-thickness life is commonly adopted as the design fatigue life for tubular applications.

4.3. Srhs–N design curve In Fig. 8, Srhs–N data obtained from the fatigue tests are also compared with the IIW design curves, which are proposed for conventional SHS joints. Evidently, the N3 of all specimens exceed the design fatigue lives as obtained from IIW curves. Therefore because sufficient experimental data for the establishment of new S–N design curves for bird-beak joints are not available at the current stage, the IIW curves can also be used to estimate the fatigue life for square bird-beak joints subject to out-of-plane bending on the premise that the stress concentration factors and hot spot stress ranges have been accurately calculated. The proposed approximation is expected to yield conservative results. With respect to the crack initiation life, some N2 data points are located below the IIW curves, which indicates that square bird-beak joints could crack before design fatigue life is reached.

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B. Cheng et al. / Engineering Structures 99 (2015) 677–684 Table 4 Measured fatigue life N2, N3 and N4. Specimen

SCFch

Snom (MPa)

Srhs (MPa)

N2

N3

N4

N2/N3

N4/N3

SBBJ-1 SBBJ-2 SBBJ-3 SBBJ-4 SBBJ-5 SBBJ-6 SBBJ-7 SBBJ-8 SBBJ-9

5.30 6.80 10.77 6.75 5.57 7.51 4.87 6.64 4.87

53 48 29 44 54 43 74 67 41

283 330 313 299 301 324 359 448 197

380,000 57,600 494,312 N/A 199,049 123,258 47,437 331,700 >950,100

826,100 265,781 1,048,093 558,998 384,659 306,802 225,413 591,260 N/A

987,918 436,040 1,371,349 752,554 508,668 354,003 251,048 630,424 N/A

0.46 0.22 0.47 N/A 0.52 0.40 0.21 0.56 N/A

1.20 1.64 1.31 1.35 1.32 1.15 1.11 1.07 N/A

Hot Spot Stress Range, Srhs (MPa)

1000

100

t=8mm t=10mm IIW Curves t=12mm N2 N3

10 10 3

10 4

10 5

10 6

10 7

10 8

10 9

Number of Cycles, N Fig. 8. Comparison between fatigue lives and IIW design curves.

4.4. Crack length growth The growths in the surface crack length during the fatigue test are plotted in Fig. 9. The crack length was normalized against the maximum crack length Lmax that was used to determine the fatigue failure of the specimen. The normalized crack length is defined as anom = measured crack length/{1.5(b1 + 2w0,Sa)}. The number of cycles is normalized against the through-thickness crack life, N3. The approximate linear increases in crack length the numbers of cycles progress are evident after the meeting of two separate cracks. The normalized number of cycles that corresponds to the crack meeting is between 0.8 and 1.0, which indicates that the crack will run through the wall thickness quickly once two separate cracks have been combined. Furthermore, except for specimen

Fig. 10. Normalized out-of-plane displacement at brace end versus normalized number of cycles.

SBBJ-7, a through-thickness crack occurs when the normalized crack length is within the range of 0.7–0.8. This number range facilitates the determination of the fatigue life of practical connection where surface crack length is measurable; however, crack depth in the thickness direction is unobtainable. 4.5. Out-of-plane rigidity Fig. 10 shows the variations of out-of-plane displacement ranges at brace ends as a function of the number of cycles. The normalized displacement ranges against the displacement range correspond to the through-thickness life, and the normalized numbers of cycles against the through-thickness life are also shown. During the early stage, where the normalized number of cycles was below 0.7, the increases in the displacement range were smaller than 5%, which indicates that the variations in joint rigidities can be ignored. Subsequently, the curve slope significantly improves, which indicates a notable degradation in the out-of-plane rigidity of the joint. When the normalized number of cycles exceeds 1.2, slight decreases in increasing rate of the displacement range (i.e. degradation rate of joint rigidity) are also expected. 5. Conclusions This study focuses on the stress concentration factors and fatigue behaviors of square bird-beak T-joints subject to out-of-plane bending. Based on the experimental data, the following can be summarized:

Fig. 9. Normalized surface crack length versus normalized number of cycles.

(1) A quadric extrapolation approach is commonly required to calculate hot spot SCFs due to the significant non-linearity of stress distributions within extrapolation regions.

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(2) The maximum SCFs in the chord occur at saddle spot Sa-B/Sa-B0 , and the maximum SCFs in the brace occur at saddle spot Sa-A/Sa-A0 . The maximum SCFs in the chord are more likely to be greater than that of the brace. (3) Parameter s exhibits more significant effects on the maximum chord SCFs than b and 2c. However, the influence of s on the maximum brace SCFs is negligible; instead, the influence of 2c is the most significant. (4) During the fatigue test, two fatigue cracks initiated near the hot spots Sa-B and Sa-B0 in the chord and then propagated toward each other until meeting. The combined crack further extended away from the weld corners and continued propagating along the chord surface until fatigue failure. (5) The number of cycles that correspond to crack union are approximately 0.8–1.0 times the through-thickness fatigue life, and the crack lengths that correspond to through-thickness fatigue life are approximately 0.7–0.8 times the specified ultimate length. (6) The number of cycles between the initial crack and through-thickness crack are considerable, while the fatigue life between the through-thickness crack and fatigue failure are not long enough to be used. (7) The out-of-plane rigidity of the joint remains nearly constant during the early stage of the test and then decreases rapidly when the number of cycles is close to or exceeds through-thickness fatigue life. (8) The S–N curves provided in the IIW code could be used for the fatigue design of square bird-beak joints if the hot spot stress ranges have been properly determined. The SCF formulations used for the calculation of hot spot stress ranges are expected to be proposed in the authors’ future research work. Acknowledgment The present research was undertaken with support from the Science and Technology Research and Development Plan funded by the Ministry of Railway of P.R.C. (No. J2011G002). References [1] Wardenier J, Packer JA, Zhao XL, van der Vegte GJ. Hollow sections in structural applications. The Netherlands: Bouwen met Staal; 2010. [2] Ono T, Iwata M, Ishida K. An experimental study on joints of new truss system using rectangular hollow sections. In: Proc. of 4th int. symposium on tubular structures; 1991. p.344–53. [3] Ono T, Iwata M, Ishida K. Local failure of joints of new truss system using rectangular hollow sections subjected to in-plane bending moment. In: Proc. of 5th int. symposium on tubular structures; 1993. p. 503–10. [4] Ono T, Iwata M, Ishida K. Local failure of joints of new truss system using rectangular hollow sections subjected to out-of-plane bending moment. In: Proc. of 6th int. symposium on tubular structures; 1994. p. 441–8. [5] Davies G, Kelly RB. Bird beak T-joints in square hollow sections: a finite element investigation. In: Proc. of 5th int. offshore and polar engineering conference; 1995. p. 1–6.

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