Stress concentration factors of diamond bird-beak SHS T-joints under brace loading

Thin-Walled Structures 74 (2014) 201–212

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Stress concentration factors of diamond bird-beak SHS T-joints under brace loading Lewei Tong a,b, Yuguang Fu a,b, Yongqiang Liu b,c, Xiao-Ling Zhao d,n a

State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University, Shanghai 200092, China College of Civil Engineering, Tongji University, Shanghai 200092, China c Hunan Provincial Architectural Design Institute, Hunan 410011, China d Department of Civil Engineering, Monash University, Clayton, VIC 3800, Australia b

art ic l e i nf o

a b s t r a c t

Article history: Received 4 July 2013 Received in revised form 7 October 2013 Accepted 7 October 2013 Available online 30 October 2013

T-joints made of hollow structural sections (e.g. circular hollow section (CHS) or square hollow section (SHS)) are often found in welded trusses that may be subjected to fatigue loading. The brace member of conventional T-joints (e.g. CHS-to-CHS, SHS-to-SHS and CHS-to-SHS T-joints) is perpendicular to the chord member. When the brace member and the chord member both rotates 451 a conventional SHS-to-SHS T-joint becomes a so-called “diamond bird-beak” SHS T-joint. Most of the research work on bird-beak joints has been focused on the static strength. Very limited information is available with regard to the fatigue strength of diamond bird beak joints. This paper will fill the knowledge gap in understanding the stress concentration factor (SCF) of this type of T-joints under brace loading (i.e. axial load in the brace and in-plane bending in the brace). A series of tests were conducted to measure the SCF of diamond bird-beak T-joints with a reasonable range of three key parameters, i.e. β (ratio of brace diameter to chord diameter), 2γ (ratio of chord diameter to chord thickness) and τ (ratio of brace thickness to chord thickness). It was found that the SCFs for the diamond bird beak T-joints are much lower than those for SHS-to-SHS and CHS-to-SHS T-joints, but just slightly smaller than those for CHS-to-CHS T-joints. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Square hollow section (SHS) Bird-beak T-joints Hot spot stress Stress concentration factor (SCF) Welded joints

1. Introduction Welded trusses made of thin-walled hollow structural sections have been widely used in civil structural applications where fatigue loading may be critical [20,23,26,25]. A truss structure may consist of welded joints in various shapes, such as T-joints, K-joints and N-joints. Five types of T-joints were found in practice according to the shape (circular hollow section (CHS) or square hollow section (SHS)) of the brace and chord or the orientation of brace and chord as shown in Fig. 1. They are (a) CHS-to-CHS T-joint (e.g. [13]), (b) SHS-to-SHS T-joint (e.g. [24], [1,15]), (c) CHS-to-SHS T-joint (e.g. [14,21,22]), (d) square bird-beak SHS T-joint and (e) diamond bird-beak SHS T-joint (e.g. [19,10]). The difference between the last two bird-beak T-joints is that only chord is rotated 451 about the member axis in the square bird-beak SHS T-joint (Fig. 1(d)), whereas both the brace and the chord are rotated in the diamond bird-beak SHS T-joint (Fig. 1(e)). This paper will deal with the diamond bird-beak SHS T-joint shown in Fig. 1(e). Most of the research work on bird-beak joints has been focused on the static strength [16–19,9,5,4]. The main differences between

n

Corresponding author. Tel.: þ 61 3 9905 4972; fax: þ61 3 9905 4944. E-mail address: [email protected] (X.-L. Zhao).

0263-8231/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tws.2013.10.008

bird beak joints compared to conventional joints include higher joint stiffness, more uniform load distribution within the joint, less chance of local buckling of the chord member and increase of the length of the weld [11]. The results of the above mentioned research reveal that the static strength of bird beak joints is equal to or higher than those of conventional types of joints. However, limited information is available with regard to the fatigue strength of diamond bird beak joints. Ishida [8] carried out fatigue tests on 10 diamond bird-beak T-joints and demonstrated an increased fatigue life of such joints compared with conventional T-joints. Keizer et al. [10] and Liu et al. [12] attempted to study the stress concentration factor (SCF) of such T-joints using numerical analysis approach. There is a lack of experimental data on SCF of diamond bird-beak joints. This paper will fill the knowledge gap in understanding the SCF of this type of T-joints under brace loading (i.e. axial load in the brace and in-plane bending in the brace). A series of tests were conducted to measure the SCF of diamond bird-beak T-joints. The key parameters varied in the testing program included three non-dimensional geometric parameters, i.e. β (ratio of brace diameter to chord diameter), 2γ (ratio of chord diameter to chord thickness) and τ (ratio of brace thickness to chord thickness), as for conventional T-joints defined in Zhao et al. [26]. Both linear and quadratic extrapolation methods were applied to determine hot

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L. Tong et al. / Thin-Walled Structures 74 (2014) 201–212

in

CHS SCF SHS SNCF SNCFlin

in

SNCFquad

in

b0 b1 t0 t1 β 2γ τ

Notation B1 and B1′ weld toe locations at the crown in the brace, diamond bird-beak joints (see Fig. 4) B2 and B2′ weld toe locations near the crown in the brace, diamond bird-beak joints (see Fig. 4) B3 and B3′ weld toe locations near the saddle in the brace, diamond bird-beak joints (see Fig. 4) B4 and B4′ weld toe locations at the saddle in the brace, diamond bird-beak joints (see Fig. 4) C1 and C1′ weld toe locations at the crown in the chord, diamond bird-beak joints (see Fig. 4) C2 and C2′ weld toe locations near the crown in the chord, diamond bird-beak joints (see Fig. 4) C3 and C3′ weld toe locations near the saddle in the chord, diamond bird-beak joints (see Fig. 4) C4 and C4′ weld toe locations at the saddle in the chord, diamond bird-beak joints (see Fig. 4)

in

in in in in

spot stress on the basis of measured strain at a certain distance away from weld toes in accordance with Zhao et al. [26]. The measured SCF values are compared with those from conventional T-joints (CHS-toCHS, SHS-to-SHS and CHS-to-SHS). It was found that the SCFs for the diamond bird beak T-joints are much lower than those for SHS-toSHS and CHS-to-SHS T-joints, but just slightly smaller than those for CHS-to-CHS T-joints.

2. Test specimens Fig. 2 shows the dimensions and parameter definitions of a bird-beak SHS T-joint specimen. The joints consist of a SHS brace and a SHS chord. Seven bird-beak SHS T-joints were manufactured for testing, which were made up of low carbon steel grade Q235B, in accordance with GB/T700-2006 [2]. Grade Q235B steel has a

snom sh.s.

circular hollow section stress concentration factor square hollow section strain concentration factor strain concentration factor determined using the linear extrapolation method strain concentration factor determined using the quadratic extrapolation method width of a chord width of a brace wall thickness of a chord wall thickness of a brace ratio of brace width (b1) to chord width (b0) ratio of chord width (b0) to chord wall thickness (t0) ratio of brace wall thickness (t1) to chord wall thickness (t0) nominal stress hot spot stress

minimum specified yield stress of 235 MPa. Full penetration welds were adopted in all test specimens with gas metal arc welding method, in accordance with JGJ81-2002 [3]. Table 1 gives an overview of the test specimens, listing dimensions and parameters of bird-beak SHS T-joints. The seven specimens were designed on the basis of three non-dimensional geometric parameters, i.e. β (¼d1/b0), 2γ ( ¼b0/t0) and τ (¼t1/t0) where the dimensions are defined in Fig. 2. The ranges of the three parameters are listed as follows: 0:40 o β o 0:72 20:83 o 2γ o31:25 0:50 o τ o 0:83 Among these seven specimens, β varied for series T1, T2 and T3 with 2γ and τ kept constant, 2γ varied for series T2, T6 and T7 with β and τ kept constant, and τ varied for series T2, T4 and T5 with β

Fig. 1. Hollow steel section T-joints: (a) CHS-to-CHS T-joint, (b) SHS-to-SHS T-joint, (c) CHS-to-SHS T-joint, (d) square bird-beak SHS T-joint, and (e) diamond bird-beak SHS T-joint.

L. Tong et al. / Thin-Walled Structures 74 (2014) 201–212

203

and 2γ kept constant. The main reason for the design is to investigate the individual effect of the three parameters on SCF. The length of the members is chosen such that the strains to be measured were not influenced by the end conditions, e.g. the joint area should be taken to be at least 2.5b away from end restraints. Accordingly, the length of the chord member for tests was taken as 6b0, where b0 is the width of the chord member and the length of the brace was 700 mm.

3. Test setup and procedures

Fig. 2. Dimensions and parameter definitions of bird-beak SHS T-joint specimens.

Fig. 3 shows the test set up. Each specimen was pin-supported at two ends of the chord and tested under two loading cases, respectively: (1) axial tensile force in the brace and (2) in-plane bending in the brace. Static loads were applied in a specific range

Table 1 Dimensions and parameters of diamond bird-beak SHS T-joint specimens. Specimen

Chord

Brace

Parameter

Width b0 (mm) Thickness t0 (mm) Width b1 (mm) Thickness t1 (mm) β ¼ b1/b0 2γ ¼b0/t0 τ ¼t1/t0 T1 T2 T3 T4 T5 T6 T7

250 250 250 250 250 250 250

12 12 12 12 12 10 8

100 140 180 140 140 140 140

6 6 6 8 10 5 4

0.40 0.56 0.72 0.56 0.56 0.56 0.56

20.83 20.83 20.83 20.83 20.83 25.00 31.25

0.500 0.500 0.500 0.667 0.833 0.500 0.500

Fig. 3. Test setup of bird-beak SHS T-joints.

Fig. 4. Strain gauges arrangement on bird-beak SHS T-joints: (a) strip strain gauges, (b) strip strain gauges arrangement on brace, and (c) strip strain gauges arrangement on chord.

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based on preliminary finite-element analyses, so that all the stresses in specimens were kept in linear elastic range. The location and labeling of strain gauges on bird-beak SHS T-joints are shown in Fig. 4. Strip strain gauges, each consists of four strain gauges, were utilized to measure strain distributions close to weld toes. As illustrated in Fig. 4(b) and (c), for each specimen, there were eight extrapolation lines C1–C4 and C1′–C4′ symmetrically arranged in the chord, as well as the same number of lines B1–B4 and B1′–B4′ in the brace. For instance, C1 is symmetrical to C1′, thus strains were measured and averaged in these two positions. At these preselected measurement lines, strip strain gauges were placed within the extrapolation region recommended in CIDECT Design Guide No. 8 for measuring hot spot strains in welded hollow steel section joints [26]. The first strain gage should be located at a distance of 0.4 t but not less than 4 mm

from the toe of the weld, thus ensuring that geometric stress is determined and the weld toe effects are neglected. Fig. 5 shows a schematic view of extrapolation methods of hot spot stress. In Fig. 5 Lr,min is the minimum length for extrapolation and Lr,max is the maximum length for extrapolation. After the bracing load (either brace axial load and brace inplane bending) is applied, strain readings at specified locations were recorded for calculating the stress concentration factors.

4. Test results Stress concentration factor (SCF) is defined as the ratio of hot spot stress (sh.s. to the nominal stress (snom), i.e. SCF¼sh.s./snom. The nominal stress is determined by simple beam theory for members subjected to brace axial forces and brace in-plane bending moments, respectively. Since SCF cannot be directly obtained in tests, strain concentration factor (SNCF) is measured and converted to SCF by means of relationship between SCF and SNCF, as indicated in the previous research of welded tubular joints [26]. Similar to SCF, SNCF is defined as the ratio of geometrical hot spot strain to the nominal strain. Both linear and quadratic extrapolation methods were adopted for the determination of the hot spot strain. 4.1. Diamond bird-beak T-joints subjected to axial force in the brace Table 2 lists the test results of SNCF for specimens under axial force in the brace. In the table, the SNCF highlighted in bold is the maximum measured value in brace and chord. The SNCF measured in B4 of T7 is invalid since the strain gauges were broken. Each SNCF was the average value of two symmetrical lines, such as C1 and C1′ and B1 and B1′. The difference between SNCF values by quadratic extrapolation (SNCFquad) and linear extrapolation (SNCFlin) can indicate the degree of non-linearity along different measurement lines where SNCF was obtained. As given in Table 2, the average ratio of SNCFquad to SNCFlin ranges from 1.1 to 1.2 in the brace and from 1.08 to 1.25 in the chord.

Fig. 5. Extrapolation methods of hot spot stress (schematic view).

Table 2 Measured SNCFs for diamond bird-beak SHS T-joints under axial force in the brace. Specimen

Extrapolation method

SNCF in brace

SNCF in chord

B1

B2

B3

B4

Quadratic Linear Quadratic/linear

2.59 1.94 1.34

1.86 1.63 1.14

1.68 1.48 1.14

2.92 2.57 1.14

T2

Quadratic Linear Quadratic/linear

3.88 3.05 1.27

1.18 1.15 1.03

3.30 2.83 1.17

4.32 3.82 1.13

T3

Quadratic Linear Quadratic/linear

2.71 2.35 1.15

2.73 2.36 1.16

4.66 4.00 1.17

4.20 3.63 1.16

T4

Quadratic Linear Quadratic/linear

2.75 2.49 1.10

2.02 1.83 1.10

2.46 2.17 1.13

4.80 4.57 1.05

T5

Quadratic Linear Quadratic/linear

3.55 3.10 1.15

2.81 2.48 1.13

4.49 4.08 1.10

5.45 5.05 1.08

T6

Quadratic Linear Quadratic/linear

2.20 1.88 1.17

1.95 1.82 1.07

2.44 2.08 1.17

5.50 3.96 1.39

Quadratic Linear Quadratic/linear

2.98 2.66 1.12

2.86 2.36 1.21

3.24 2.92 1.11

– – –

T1

T7

Avg.

C1

C2

C3

C4

Avg.

1.19

3.89 2.75 1.41

2.86 2.45 1.17

3.45 3.06 1.13

3.64 3.18 1.14

1.21

1.15

4.78 3.29 1.45

3.37 2.70 1.25

3.90 3.42 1.14

4.30 3.65 1.27

1.25

1.16

4.67 4.25 1.10

2.58 2.46 1.05

3.20 2.98 1.07

5.08 4.47 1.14

1.09

1.10

6.35 4.93 1.29

3.01 2.71 1.11

4.52 3.85 1.17

5.10 5.10 1.00

1.14

1.11

7.55 6.87 1.10

5.86 5.22 1.12

7.10 6.60 1.08

9.06 8.70 1.04

1.08

1.20

4.64 4.29 1.08

4.73 4.21 1.12

6.22 5.80 1.07

5.80 5.50 1.05

1.08

1.15

5.49 4.62 1.19

3.09 3.03 1.02

7.16 5.69 1.26

6.80 6.01 1.13

1.15

L. Tong et al. / Thin-Walled Structures 74 (2014) 201–212

types of joints. Accordingly, the relationship between SNCF and SCF in bird-beak T-joints should not be simply determined by Eqs. (1) and (2). Based on Hooke's Law, results from relevant finite element analysis in this investigation have indicated that there is a high correlation between the SNCFs and the SCFs in both brace and chord, when the joints are under each of two load types: axial force and in-plane bending. Through linear statistics regression via the finite element method, the relationship between SNCF and SCF in bird-beak T-joints can be given by following formulae:

Then, it can be concluded that strain distribution close to weld toes is strongly non-linear both in the brace and in the chord. Therefore, the quadratic extrapolation method will give more realistic values for SNCF at weld toes and should be adopted to analyze hot spot strains in diamond bird-beak T-joints under axial force. This is in agreement of the recommendation in Zhao et al. [26] that the quadratic extrapolation method should be used for welded SHS joints. Previous research has indicated that there is a linear relationship between SNCF and SCF in welded tubular joints [7,6], which can be expressed as follows: ð1Þ

SCFCHS ¼ 1:2 SNCFCHS for CHS joints

ð2Þ

SCF ¼ 1:1 SNCF in brace

ð3Þ

SCF ¼ 1:15 SNCF in chord

ð4Þ

Based on Eqs. (3) and (4), SCF is converted from the measured SNCF which was obtained by quadratic extrapolation method. These equations are also suitable to specimens under in-plane bending. It can be noted that the expression of Eqs. (3) and (4) is almost the same as that of Eqs. (1) and (2).

The joint area close to the weld toe is under multi-axial stresses, with principal stresses perpendicular to weld toes and other relatively small stresses in different directions. Therefore, the difference should not be neglected between stress distributions in this area in different

SCF

SCFRHS ¼ 1:1 SNCFRHS for RHS joints ðincluding SHSÞ

205

6

T1 Brace T1 Chord

5 4 3 2 1

location

6

3

T3 Brace T3 Chord

2

location

8

3

2

SCF

1

0

4

T4 Brace T4 Chord

6

location 1

2

SCF

1

10

3

4

T5 Brace T5 Chord

8 6

4

4

location 2

SCF

1

10

3

4

T6 Brace T6 Chord

8

2 0

location 1

2

10

3

4

T7 Brace T7 Chord

SCF

2

8

6

6

4

4

2 0

4

4

2

0

3

6

4

0

2

8

T2 Brace T2 Chord

5

1

SCF

SCF

0

location 1

2

3

4

2 0

location 1

2

3

Fig. 6. SCF distribution in bird-beak SHS T-joints T1–T7 under axial force in the brace.

4

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L. Tong et al. / Thin-Walled Structures 74 (2014) 201–212

Table 3 Measured SNCFs for diamond bird-beak SHS T-joints under in-plane bending in the brace. Specimen

Extrapolation method

SNCF in brace

SNCF in chord

B1

B2

B3

B4

T1

Quadratic Linear Quadratic/linear

1.38 1.25 1.10

1.02 1.00 1.02

0 0 –

0 0 –

T2

Quadratic Linear Quadratic/linear

1.91 1.49 1.28

0.82 0.73 1.12

0 0 –

0 0 –

T3

Quadratic Linear Quadratic/linear

1.88 1.70 1.11

2.01 1.71 1.18

0 0 –

0 0 –

T4

Quadratic Linear Quadratic/linear

1.82 1.62 1.12

1.27 1.23 1.03

0 0 –

0 0 –

T5

Quadratic Linear Quadratic/linear

1.81 1.62 1.11

1.25 1.19 1.05

0 0 –

0 0 –

T6

Quadratic Linear Quadratic/linear

2.10 1.78 1.18

1.21 1.13 1.07

0 0 –

0 0 –

T7

Quadratic Linear Quadratic/linear

2.24 2.14 1.05

1.98 1.78 1.11

0 0 –

0 0 –

Fig. 6 shows the SCF distribution which was obtained in seven test specimens under axial force. It revealed that the maximum SCF in the brace usually occurred at the saddle (B4), and the maximum SCF in the chord may appear either at the crown (C1) or at the saddle (C4). Therefore, hot spot stresses both at the crown and at the saddle of either braces or chords should be focused in order to assess fatigue properties of diamond bird-beak joints under axial force.

4.2. Diamond bird-beak T-joints subject to in-plane bending in the brace Table 3 lists the test results of SNCF for specimens under inplane bending. Each SNCF was the average value of two symmetrical lines, similar to the data in Table 2. Both linear and quadratic extrapolation methods were adopted to evaluate the degree of non-linearity along different lines of measurement. As given in Table 3, the average ratio of SNCFquad to SNCFlin ranges from 1.06 to 1.2 in the brace and from 1.12 to 1.35 in the chord, which indicates that strain distribution close to weld toes is quite non-linear both in the brace and in the chord. Accordingly, the quadratic extrapolation method is more suitable to determine hot spot strains at weld toes in diamond bird-beak T-joints under in-plane bending. Similar to the data in the case of axial force, SCF can also be obtained from SNCF based on Eqs. (3) and (4). Fig. 7 shows the SCF distribution which was obtained in seven test specimens under inplane bending. It can be concluded that the maximum SCF in the brace always larger than that in the chord in the case of in-plane bending, which was quite different to the situation under axial force. The maximum SCF both in the brace and in the chord usually occurred at the crown, namely B1 and C1, respectively. Since B3, B4, C3 and C4 are located at neutral axis in the case of in-plane bending, the SCFs measured at these positions were negligible and given as equal to zero both in the brace and in the chord. Therefore, hot spot stresses at the crown of both braces and chords should be highlighted in order to assess fatigue properties of diamond bird-beak joints under in-plane bending.

Avg.

C1

C2

C3

C4

Avg.

1.06

0.71 0.50 1.42

0.68 0.53 1.28

0 0 –

0 0 –

1.35

1.20

0.74 0.54 1.37

0.53 0.45 1.18

0 0 –

0 0 –

1.28

1.15

0.84 0.73 1.15

0.67 0.62 1.08

0 0 –

0 0 –

1.12

1.08

0.84 0.63 1.33

0.60 0.59 1.02

0 0 –

0 0 –

1.18

1.08

0.96 0.82 1.17

0.89 0.77 1.16

0 0 –

0 0 –

1.17

1.13

1.34 0.94 1.43

0.89 0.74 1.20

0 0 –

0 0 –

1.32

1.08

1.76 1.09 1.61

0.95 0.88 1.08

0 0 –

0 0 –

1.35

5. A comparison of diamond SHS T-joints with other shapes of T-joints 5.1. General As stated in Section 1, the stress concentration factors of tubular joints are mainly determined by three non-dimensional geometric parameters (β, 2γ and τ). The effect of these three parameters on SCFmax is analyzed by comparing test results of different joints which have two parameters with the same value (or similar value) but the other one with different value, as illustrated in Figs. 8 and 9. Test data of three conventional T-joints are collected in previous research in Tables 4–9 [24,13,1,22]. They are CHS-to-CHS T-joint, SHS-to-SHS T-joint and CHS-to-SHS T-joint. Fig. 10 illustrates typical hot spot locations of various types of T-joints. Figs. 8 and 9 show the relationship between SCFmax and three parameters (β, 2γ and τ). The figures also show the comparison between SCFmax of diamond bird-beak SHS T-joint and that of three conventional T-joints when axial load and in-plane bending are applied separately. The original test data (SNCFs) is listed in Tables 4–9, while some SCFs are listed because of the unavailable SNCFs of some joints in literatures. Though the data points are relatively few and certain dispersion exits, the curve obtained from experiments outlines the general trend of the maximum SCF in brace and chord with the variation of parameters.

5.2. Parametric analysis of diamond bird-beak SHS T-joints From Figs. 8 and 9(a), one can find that SCFmax both in chord and brace increases as β varies from 0.40 to 0.72 under axial load and in-plane bending, based on the data of three specimens, T1, T2 and T3. Therefore, conclusion can be drawn that when the brace is under axial load or in-plane bending, the larger the β is, the larger the SCFmax. Likewise, in Figs. 8 and 9(c), based on test data of T2, T6 and T7, SCFmax in both chord and brace increases as 2γ varies from 20.83 to 31.25. It can also be concluded that there is a positive correlation between SCFmax and 2γ. Similarly, Figs. 8

L. Tong et al. / Thin-Walled Structures 74 (2014) 201–212

SCF

2

207

T1 Brace T1 Chord 1

SCF

3

3

T2 Brace T2 Chord

2

2

3

T3 Chord

1 location

location

SCF

2

3

4

1

3

T4 Brace T4 Chord

2

0

2

T5 Chord

1 location 2

3

location 0

4

1

SCF

3

2

3

SCF

1

3 2

T7 Chord

2

T6 Chord

4

T7 Brace

T6 Brace

1

1 0

4

T5 Brace

2

1

3

SCF

1

3

0

4

T3 Brace

2

1 0

location 1 SCF

0

location 1

2

3

4

0

location 1

2

3

4

Fig. 7. SCF distribution in bird-beak SHS T-joints T1–T7 under in-plane bending in the brace.

and 9(e) show the trend of SCFmax as τ increases from 0.5 to 0.833, when T2, T4 and T5 are tested under axial load and in-plane bending, respectively. When the brace is under axial load, SCFmax in both brace and chord is positive correlated with τ, and SCFmax grows much larger in chord than that in brace, with the variation of τ. On the other hand, when it comes to the case of in-plane bending, SCFmax and τ are positive correlated in chord, however, negative correlated in brace. 5.3. T-joints subjected to axial force in the brace The measured SCFs of CHS-to-CHS T-joints were extracted from the database reported in Lloyd's Register of Shipping [13] and listed in Tables 4 and 5. The measured SNCF of SHS-to-SHS T-joints given in Wingerde [24] are reproduced in Tables 6 and 7. The tables also include the measured SCF of four SHS-to-SHS specimens tested by Chiew et al. [1]. Tables 8 and 9 give the measured SNCF of CHS-to-SHS T-joints reported in Tong et al. [22]. Before these data are compared and drawn in Figs. 8 and 9, several modifications should be conducted

to these original data to make sure that they are under the same (or similar) conditons and are arranged in such a way to show the influence of the three key parameters (β, 2γ and τ). The modifications are listed below: (1) Since acrylic model tests do not include a weld fillet on tubular joint intersection, the measured SCFs should be applied with a reduction factor, if compared to steel model test results. The work of Lloyd's Register of Shipping [13] showed a reduction factor of 0.95 on the chord side and 0.88 on the brace side. Accordingly, the data obtained from acrylic models in Tables 4 and 5 is modified with the reduction factors. (2) The butt welded specimens in Wingerde [24] are selected for comparison. The data is averaged for those specimens having the same non-dimensional parameters (β, 2γ and τ). The measured SNCF is converted to SCF by multiplying a factor of 1.1, as suggested by Wingerde [24]. (3) The measured SNCF of CHS-to-SHS in Tables 8 and 9 is obtained by quadratic extrapolation method. The measured

L. Tong et al. / Thin-Walled Structures 74 (2014) 201–212

Bird-Beak chord CHS-CHS brace CHS-CHS chord SHS-SHS brace SHS-SHS chord

8

CHS-SHS brace CHS-SHS chord

4 β

0 0.20

16

0.40

SCFmax

20

16

0.60

12

2γ=20.83 τ=0.50

CHS-CHS brace CHS-CHS chord

12

SHS-SHS brace

8

CHS-SHS brace

SHS-SHS chord

CHS-CHS SHS-SHS CHS-SHS

4

Bird-Beak brace Bird-Beak chord

Bird-Beak

8

β

0 0.20

0.80

β=0.56 τ=0.50

SCFmax

12

Bird-Beak brace

2γ=20.83 τ=0.50

20 16

0.40

SCFmax

16

SCFmax

208

0.60

0.80

Bird-Beak CHS-CHS SHS-SHS CHS-SHS

β=0.56 τ=0.50

12 8

CHS-SHS chord

4

4 2γ

20

SCFmax

24

20

30

Bird-Beak brace

β=0.56 2γ=20.83

Bird-Beak chord

24

CHS-CHS brace

20

CHS-CHS chord

16

SHS-SHS brace

12

SHS-SHS chord CHS-SHS brace

8

CHS-SHS chord

4 0 0.20

2γ

0 10

40

τ

0.45

0.70

0.95

1.20

20

SCFmax

0 10

30

40

Bird-Beak CHS-CHS SHS-SHS CHS-SHS

β=0.56 2γ=20.83

16 12 8 4 0 0.20

τ

0.45

0.70

0.95

1.20

Fig. 8. Effect of non-dimensional parameters on SCFmax for T-joints under axial force in the brace: (a) effect of β on SCFmax in the brace and chord, (b) effect of β on SCFmax in the joint, (c) effect of 2γ on SCFmax in the brace and chord, (d) effect of 2γ on SCFmax in the joint, (e) effect of τ on SCFmax in the brace and chord, and (f) effect of τ on SCFmax in the joint.

SNCF is converted to SCF by multiplying a factor of 1.15 on the chord side and 1.2 on the brace side, in accordance with the recommendation in Tong et al. [22]. Fig. 8(a), (c) and (e) shows the maximum SCFs in chord and brace, respectively, with the variation of three key parameters, in diamond bird-beak T-joints and three conventional T-joints under axial load. It can be concluded that SHS-to-SHS joints have the largest SCF in both chord and brace. The CHS-to-CHS and diamond bird-beak joints have smaller SCF in both chord and brace than other two types of joints. Furthermore the SCF of diamond birdbeak joints is close to but just slightly lower than that of CHS-toCHS joints. Fig. 8(b), (d) and (f) shows the maximum SCFs in chord and brace, with the variation of three key parameters, in four joints under axial load. Again it is clear that the SCFs for the diamond bird-beak T-joints are much lower than those for SHS-toSHS and CHS-to-SHS T-joints, and near to those for CHS-to-CHS T-joints. 5.4. T-joints subject to in-plane bending in the brace Similar modifications mentioned in Section 5.2 are conducted to the data in the case of in-plane bending. Fig. 9(a), (c) and (e) shows the maximum SCFs in chord and brace, respectively, with the variation of three key parameters, in diamond bird-beak T-joints and three conventional T-joints under in-plane bending.

It can be concluded that the CHS-to-CHS and diamond bird-beak joints have smaller SCF in both chord and brace than other two joints. The SCF of diamond bird-beak joints is near to and slightly lower than that of CHS-to-CHS joints. Fig. 9(b), (d) and (f) shows the maximum SCFs in chord and brace, with the variation of three key parameters, in four joints under in-plane bending. It can be concluded that the SCFs for the diamond bird-beak T-joints are much lower than those for SHS-to-SHS and CHS-to-SHS T-joints, and near to those for CHS-to-CHS T-joints. The lower SCF values in the diamond bird-beak T-joints match the observation of increased fatigue life for such joints [8] compared with conventional T-joints.

6. Conclusions A series of tests were conducted on bird-beak SHS T-joints to measure the stress concentration factors. The measured SCFs are compared with those for conventional T-joints obtained in the literature. The following conclusions are made based on the limited data available. (1) For both axial force and in-plane bending in the brace, the quadratic extrapolation method is more suitable to determine the SCF at weld toes of diamond bird-beak T-joints than the

SCFmax

12

SCFmax

L. Tong et al. / Thin-Walled Structures 74 (2014) 201–212

Bird-Beak brace

2γ=20.83 τ=0.50

Bird-Beak chord

12

CHS-CHS brace CHS-CHS chord

8

SHS-SHS brace

209

Bird-Beak CHS-CHS SHS-SHS CHS-SHS

2γ=20.83 τ=0.50

8

SHS-SHS chord CHS-SHS brace

4

CHS-SHS chord

4

β

SCFmax

12

0.40

0.60

β

0 0.20

0.80

Bird-Beak brace Bird-Beak chord CHS-CHS brace CHS-CHS chord SHS-SHS brace SHS-SHS chord CHS-SHS brace CHS-SHS chord

β=0.56 τ=0.50

8

4

0.40

SCFmax

0 0.20

12

0.60

Bird-Beak CHS-CHS SHS-SHS CHS-SHS

β=0.56 τ=0.50

8

4 2γ

2γ

SCFmax

16

20

30

0 10

40 Bird-Beak brace

β=0.56 2γ=20.83

Bird-Beak chord

16

CHS-CHS brace CHS-CHS chord

12

20

SCFmax

0 10

0.80

30

40

Bird-Beak CHS-CHS SHS-SHS CHS-SHS

β=0.56 2γ=20.83

12

SHS-SHS brace SHS-SHS chord

8

8

CHS-SHS brace CHS-SHS chord

4

4

τ 0 0.20

0.45

0.70

0.95

0 0.20

1.20

0.45

0.70

0.95

1.20

Fig. 9. Effect of non-dimensional parameters on SCFmax for T-joints under in-plane bending in the brace: (a) effect of β on SCFmax in the brace and chord, (b) effect of β on SCFmax in the joint, (c) effect of 2γ on SCFmax in the brace and chord, (d) effect of 2γ on SCFmax in the joint, (e) effect of τ on SCFmax in the brace and chord, and (f) effect of τ on SCFmax in the joint.

Table 4 Measured SCFs for CHS-to-CHS T-joints under axial force in the brace [13]. Specimen

CHS-CHS-T19 CHS-CHS-T20 CHS-CHS-T39 CHS-CHS-TG1 CHS-CHS-3U/2 CHS-CHS-11AU/1 CHS-CHS-14U/1 CHS-CHS-14U/2 CHS-CHS-14U/3 CHS-CHS-14U/5

Parameters

Brace

Chord

β

2γ

τ

α

Saddle

Crown

Side

Saddle

Crown

Side

0.53 0.53 0.25 0.50 0.50 0.50 0.26 0.50 0.80 0.50

26.80 26.60 28.60 24.00 24.00 24.00 24.00 24.00 24.00 48.00

0.86 0.51 0.40 0.52 0.75 1.00 0.40 0.40 0.40 0.40

10.50 10.00 10.20 13.50 10.00 10.00 10.00 10.00 10.00 10.00

6.50 4.90 3.00 6.30 8.00 8.40 4.90 6.40 4.50 10.40

0.30 0.40 0.80 1.90 2.00 1.50 2.60 2.20 1.90 1.60

6.50 4.90 4.40 6.30 8.00 8.40 4.90 6.40 4.50 10.40

11.40 6.50 4.20 5.90 9.30 10.30 4.30 4.60 3.50 9.20

5.20 2.80 2.40 3.30 3.70 6.70 2.80 2.60 2.60 2.20

11.40 6.50 4.20 5.90 9.30 10.70 4.30 4.60 3.50 9.20

Note: Hot spot locations are shown in Fig. 10(b).

linear extrapolation method, because the stress distribution in the extrapolation region was found to be quite non-linear. (2) Under axial force in the brace, maximum SNCFs occurred either at the saddle or at the crown, whereas, in the case of in-plane bending, maximum SNCFs only appeared at the crown.

(3) For bird-beak SHS T-joints under axial force or in-plane bending, the ratio of SCF to SNCF was about 1.1 in the brace and 1.15 in the chord, according to statistics regression analysis. (4) The maximum SCF in the brace is always larger than that in the chord in the case of in-plane bending, however, the

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Table 5 Measured SCFs for CHS-to-CHS T-joints under in-plane bending in the brace [13]. Specimen

Non-dim. parameters

CHS-CHS-T19 CHS-CHS-T20 CHS-CHS-T39 CHS-CHS-TG1 CHS-CHS-3U/2 CHS-CHS-11AU/1 CHS-CHS-14U/1 CHS-CHS-14U/2 CHS-CHS-14U/3 CHS-CHS-14U/5

Brace

Chord

β

2γ

τ

α

Crown

Side

Crown

Side

0.53 0.53 0.25 0.50 0.50 0.50 0.26 0.50 0.80 0.50

26.80 26.60 28.60 24.00 24.00 24.00 24.00 24.00 24.00 48.00

0.86 0.51 0.40 0.52 0.75 1.00 0.40 0.40 0.40 0.40

10.50 10.00 10.20 13.50 10.00 10.00 10.00 10.00 10.00 10.00

1.70 2.30 1.30 1.76 1.76 1.50 1.67 1.76 1.58 2.02

2.10 2.40 1.50 1.76 2.02 1.85 1.67 1.85 1.94 2.38

2.50 1.70 0.80 1.71 2.57 3.23 1.52 1.71 1.43 2.00

2.50 1.70 1.00 1.71 2.57 3.23 1.52 1.71 1.43 2.00

Note: Hot spot locations are shown in Fig. 10(b).

Table 6 Measured SNCFs or SCFs for SHS-to-SHS T-joints under axial force in the brace [24,1]. Specimen

Paper ref.

SNCF/SCF

Parameters

Brace

Chord

β

2γ

τ

A

E

B

C

D

SHS-to-SHS-T9 SHS-to-SHS-T10 SHS-to-SHS-T11 SHS-to-SHS-T12 SHS-to-SHS-T13 SHS-to-SHS-T14 SHS-to-SHS-T15 SHS-to-SHS-T16 SHS-to-SHS-T21 SHS-to-SHS-T22 SHS-to-SHS-T23 SHS-to-SHS-T24

[24]

SNCF

0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70

16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0

0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.64

6.08 6.95 – 7.77 – – – – 6.82 – – –

– – – – – – – – – – – –

3.86 4.18 – 4.00 – – – – 3.50 – – –

5.51 4.67 5.51 5.34 5.49 5.87 6.49 5.88 6.22 7.08 6.12 5.71

4.49 4.47 – 5.10 – – – – 6.45 – – –

SHS-to-SHS-I SHS-to-SHS-II SHS-to-SHS-III SHS-to-SHS-IV

[1]

SCF

0.71 0.57 0.57 0.57

21.88 21.88 21.88 21.88

1.00 1.00 0.75 0.62

12.48 9.39 10.48 11.85

6.26 2.00 2.94 5.36

15.25 21.84 13.06 14.06

17.52 21.74 15.40 13.31

15.26 12.06 10.52 11.64

Note: Hot spot locations are shown in Fig. 10(c).

Table 7 Measured SNCFs or SCFs for SHS-to-SHS T-joints under in-plane bending in the brace [24,1]. Specimen

Paper ref.

SNCF/SCF

Parameters

Brace

Chord

β

2γ

τ

A

E

B

C

D

SHS-SHS-T37 SHS-SHS-T38

[24]

SNCF

0.70 0.70

16.0 16.0

0.64 0.64

4.20 4.49

4.01 –

1.66 –

3.24 3.28

2.28 –

SHS-to-SHS-I SHS-to-SHS-II SHS-to-SHS-III SHS-to-SHS-IV

[1]

SCF

0.71 0.57 0.57 0.57

21.88 21.88 21.88 21.88

1.00 1.00 0.75 0.62

8.16 4.44 4.43 6.65

5.88 2.75 2.89 5.09

9.93 9.86 5.24 8.98

10.45 12.56 8.25 8.74

5.32 5.94 4.69 6.57

Note: Hot spot locations are shown in Fig. 10(c).

situation was reversed under axial force. The SCF under axial force is quite higher than that under in-plane bending. To be specific, the former is usually larger than 2 and the latter less than 2. (5) Under axial load or in-plane bending, SCFmax in both chord and brace is positively correlated with β and 2γ. Similarly, under axial load, SCFmax in both brace and chord is positively correlated with τ. However, when it comes to the case of

in-plane bending, SCFmax and τ are positively correlated in the chord and negatively correlated in the brace. (6) In the condition of the same key parameters (β, 2γ and τ), the SCF for the diamond bird beak T-joints is much lower than the SCF for SHS-to-SHS and CHS-to-SHS T-joints. It is near and slightly smaller, when comparing to SCF of CHS-to-CHS T-joints. This matches the observation of increased fatigue life for such joints [8] compared with conventional T-joints.

L. Tong et al. / Thin-Walled Structures 74 (2014) 201–212

211

Table 8 Measured SNCFs for CHS-to-SHS T-joints under axial force in the brace [22]. Specimen

CHS-SHS-T1 CHS-SHS-T2 CHS-SHS-T3 CHS-SHS-T4 CHS-SHS-T5 CHS-SHS-T6 CHS-SHS-T7 CHS-SHS-T8

Parameters

Brace

Chord

β

2γ

τ

B0

B45

B60

B90

C0

C45

C60

C90

0.532 0.532 0.532 0.532 0.532 0.665 0.443 0.900

20.834 20.834 20.834 25.000 31.250 20.000 25.000 16.666

0.500 0.333 0.833 0.500 0.500 0.500 0.500 0.667

4.05 2.73 0.55 2.10 4.68 1.09 3.92 0.47

5.25 6.53 – 3.47 3.89 4.16 4.01 3.54

6.78 5.76 – 6.13 7.49 4.45 5.23 3.84

8.07 6.54 8.56 8.58 11.69 6.39 10.33 4.10

3.78 3.33 5.85 3.97 8.56 4.02 3.48 5.47

5.32 3.39 8.22 5.99 8.30 3.19 5.04 4.27

5.51 3.69 9.98 7.10 10.52 2.90 6.00 4.15

5.403). 3.68 10.18 8.29 14.18 1.21 8.10 0.00

Note: Hot spot locations are shown in Fig. 10(d).

Table 9 Measured SNCFs for CHS-to-SHS T-joints under in-plane bending in the brace [22]. Specimen

CHS-SHS-T1 CHS-SHS-T2 CHS-SHS-T3 CHS-SHS-T4 CHS-SHS-T5 CHS-SHS-T6 CHS-SHS-T7 CHS-SHS-T8

Parameters

Brace

Chord

β

2γ

τ

B0

B45

B60

B90

C0

C45

C60

C90

0.532 0.532 0.532 0.532 0.532 0.665 0.443 0.900

20.834 20.834 20.834 25.000 31.250 20.000 25.000 16.666

0.500 0.333 0.833 0.500 0.500 0.500 0.500 0.667

4.46 4.02 1.95 3.20 4.04 1.53 3.24 1.00

2.83 2.36 – 1.83 2.30 2.14 1.99 3.85

2.06 2.21 – 2.10 2.26 2.18 1.77 4.87

0 0 0 0 0 0 0 0

2.28 1.57 3.41 2.72 6.70 1.69 2.11 1.86

2.13 1.70 4.08 2.98 5.15 1.42 2.58 2.77

2.18 1.52 3.84 2.45 3.88 1.16 2.16 2.75

03). 0 0 0 0 0 0 0

Note: Hot spot locations are shown in Fig. 10(d).

Fig. 10. Comparison of typical hot spot locations of various T-joints under brace loading: (a) diamond bird-beak (this paper), (b) CHS–CHS [13], (c) SHS–SHS ([24,1]), and (d) CHS–SHS [22].

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