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Assessment of stress intensity factors for load-carrying fillet welded cruciform joints using a digital camera
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International Journalof Fatigue
International Journal of Fatigue 30 (2008) 1861–1872
www.elsevier.com/locate/ijfatigue
Assessment of stress intensity factors for load-carrying fillet welded cruciform joints using a digital camera H.Y. Chung *, S.H. Liu, R.S. Lin, S.H. Ju Department of Civil Engineering, National Cheng Kung University, No. 1, University Road, Tainan City 701, Taiwan, ROC Received 12 December 2006; received in revised form 25 January 2008; accepted 29 January 2008 Available online 12 February 2008
Abstract This paper presents an experimental method to determine the stress intensity factors (SIFs) of load-carrying fillet welded cruciform joints. This experimental method measured the crack opening displacements (CODs) of the cruciform joint detail by a common digital camera, and utilized the acquired COD data to derive the SIF for the joint detail by a suggested least-squares procedure. A total of fifteen fillet welded cruciform joint specimens were tested and analyzed in this study. The test results were compared with the SIF formula provided in British Standard 7910, and showed that the demonstrated experimental method is able to give satisfactory SIF evaluation for fillet welded cruciform joints. In fact, this method can be applied to determine the SIF of any detail with surface crack where the Irwin’s series solution for crack tip displacement fields is valid. Moreover, the simplicity and portability of this method make the field measurement of SIF possible and provide a good means to acquire SIFs directly from details at the site. This is very useful for fatigue evaluation of details in structures, such as steel bridges or offshore structures. Ó 2008 Elsevier Ltd. All rights reserved. Keywords: Fillet weld; Cruciform joint; Digital camera; Stress intensity factor; Crack opening displacement
1. Introduction Fillet welded cruciform joints are widely used in metallic structures. When this kind of joints are subjected to tensile or cyclic loadings, the fillet weld roots or toes are easy to initiate cracks or result in fractures. Hence, computing the stress intensity factors (SIFs) at fillet weld roots or toes of cruciform joints was of great interests in the past and extensive works have been done. Previous studies on numerical evaluation of the SIFs at weld toes were carried out by Maddox [1], Hobbacher [2], Bowness and Lee [3] and Lie et al. [4], . . ., etc. Notably, Maddox [1] introduced the concept of magnification factor Mk in SIF to consider various geometric conditions that amplify the SIF of the detail of interest. Pang [5] utilized closedform integration and weight function methods to compute the SIFs of weld toe cracks in a full-penetration welded *
Corresponding author. Tel.: +886 6 2757575x63145; fax: +886 6 2358542. E-mail address: [email protected] (H.Y. Chung). 0142-1123/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2008.01.017
cruciform joint. For load-carrying fillet welded cruciform joints, the weld root in the un-penetrated region of the joint may act like a crack tip to initiate crack under fatigue loadings. Frank and Fisher [6,7] first proposed the polynomial expression of SIF magnification factor for the weld root in a load-carrying fillet welded cruciform joint by the compliance method using the results from finite element analysis. The SIF was found to be directly related to the size of the fillet weld leg and the crack size in the un-penetrated region as follows: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi paffi A1 þ A2 ð2a=wÞ KI ¼ rm pa sec ð1Þ 1 þ 2ðh=BÞ w and A1 ¼ 0:528 þ 3:287ðh=BÞ 4:361ðh=BÞ 3
2
4
5
þ 3:696ðh=BÞ 1:875ðh=BÞ þ 0:415ðh=BÞ A2 ¼ 0:218 þ 2:717ðh=BÞ 10:171ðh=BÞ2
þ 13:122ðh=BÞ3 7:755ðh=BÞ4 þ 1:783ðh=BÞ5
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where a is the half crack size in the un-penetrated region, B is the thickness of attachment plate, h is the size of fillet weld leg, w = B + 2h, and rm is the tensile stress applied on the attachment plate (see Fig. 1). BS PD-6493 [8] and BS 7910 [9,10] adopted Eq. (1) for SIF estimation of the weld root crack in a cruciform joint, and then improved it as follows: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pa K I ¼ M k rm pa sec ð2Þ w and Mk = A0 + A1(2a/w) + A2(2a/w)2 A0 ¼ 0:956 0:343ðh=BÞ A1 ¼ 1:219 þ 6:210ðh=BÞ 12:220ðh=BÞ
2
þ 9:704ðh=BÞ3 2:741ðh=BÞ4 2
A2 ¼ 1:954 7:938ðh=BÞ þ 13:299ðh=BÞ 9:541ðh=BÞ þ 2:513ðh=BÞ
3
4
Both Eqs. (1) and (2) are valid when 0.2 6 (h/B) 6 1.2 and 0.1 6 (2a/W) 6 0.7. Also, Usami and Kusumoto [11] implemented finite element method to compute the variation of SIF with respect to the weld leg size and the crack size in the un-penetrated region, and identified the mode I and mode II (shear mode) components of SIF in the weld root. From their study, the mode II component is very small compared to the mode I component. Khodadad Motarjemi et al. [12] determined the SIF for two-dimensional cruciform and T-welded joints by using linear-elastic finite element analysis and J-integral evaluation. Lie and Bian [13] and Kai et al. [14] employed a boundary element method to analyze the SIFs of load-carrying fillet welds. As can be seen, numerical approaches for evaluating the SIF
force
L1
B
α = 45º
h 2a α
2. SIF calculation using least-squares method Least-squares method has been successfully employed in finite element methods as a useful technique for computing stress intensity factors of sharp cracks in isotropic and composite standard specimens (Ju [15], Ju et al. [16], Ju [17] and Ju and Rowlands [18]). A least-squares method combining the measurement of the displacement field near a crack tip with a digital camera was applied in this study to calculate the stress intensity factors for load-carrying fillet welded cruciform joints. The main idea of this method is to obtain the stress intensity factors of the cracked detail by minimizing the sum of squares of the residuals that are the differences between the theoretical point displacements and the measured point displacements by the digital camera. For a linear-elastic and isotropic plate having a sharp crack (see Fig. 2), the in-plane displacement fields near the crack tip can be shown in the form of a series expansion (Irwin [19]) as follows: 1 X I u¼ cn f1n ðr; h; nÞ þ cII ð3Þ n f2n ðr; h; nÞ n¼1 1 X I v¼ cn g1n ðr; h; nÞ þ cII n g2n ðr; h; nÞ
α
t
w α
of the weld root in a load-carrying cruciform joint have been studied successfully in the past and good results were achieved. However, experimental approaches to acquire the SIFs of load-carrying cruciform joints or to verify the numerical SIF results are rarely reported in the literature. This paper presents an experimental approach to determine the stress intensity factors of load-carrying fillet welded cruciform joints. A commercial digital camera was utilized in this study for measuring the crack opening displacements (CODs) of the crack in the un-penetrated region of a load-carrying cruciform joint. The measured COD data were incorporated into a suggested least-squares procedure to derive the stress intensity factor. In order to verify this experimental approach, fifteen fillet welded cruciform joint specimens with five weld leg sizes and three unpenetrated crack sizes were tested and analyzed in this study, and the acquired SIF results were compared with the SIF values computed from the formula provided in the BS 7910 [10].
ð4Þ
n¼1
α
y
crack
fillet weld
v
Top Square Symbols
u
L2 Crack Surface
r
θ
Δy force Fig. 1. Geometry of the load-carrying fillet welded cruciform joint.
Crack Tip Bottom Square Symbols Fig. 2. Sharp crack in an infinite plate.
x
H.Y. Chung et al. / International Journal of Fatigue 30 (2008) 1861–1872
where
r and h are located on the non-negative-y-coordinate region. In the experiment of this study, Dy in Fig. 2 (i.e., the shortest distance from the center of a square symbol to the crack surface) is controlled to be very small, so h values for the top and bottom square symbols in Eqs. (11) and (12) are very close to p. As result, the even terms for Un and Vn are approximately zero. In order to reduce the errors in the suggested least-squares method, one can disregard the even terms in Eqs. (9) and (10) as follows:
n i r h n n n j þ þ ð1Þn cos h cos 2 h ; f1n ¼ 2 2 2 2 2l i n=2 h r n n n n n f2n ¼ j þ ð1Þ sin h sin 2 h ; 2 2 2 2 2l i n=2 h r n n n n g1n ¼ j ð1Þn sin h þ sin 2 h 2 2 2 2 2l n=2
and n i rn=2 h n n n j þ þ ð1Þn cos h cos 2 h ; 2 2 2 2 2l
g2n ¼
II II II T II Du ¼ ½U 1 U 3 U 5 U N ½cII 1 c3 c5 cN ¼ ½Ufc g
u and v are the displacements in x and y-direction, respectively, and expressed by r and h in the polar coordinates; l = shear modulus; j = 3 4m for plane stress and j = (3 m)/(1 + m) for plane strain; m = Poisson’s ratio. cIn and cII n are parameters to be determined. Specifically, the first terms of cIn and cII n are related to the mode I and mode II stress intensity factors, respectively, as follows: KI cI1 ¼ pffiffiffiffiffiffi ; 2p K II cII 1 ¼ pffiffiffiffiffiffi : 2p
ð5Þ ð6Þ
Once the parameters, cIn and cII n , are determined, the stress intensity factors of the cracked detail can be derived. Eqs. (3) and (4) can be rewritten in matrix forms for a given N terms: T
II u ¼ ½f11 f1N f 21 f2N ½cI1 cIN cII 1 cN ¼ ½f½c
v¼
II T ½g11 g1N g21 g2N ½cI1 cIN cII 1 cN
¼ ½g½c
ð7Þ ð8Þ
In this study, crack opening displacements (CODs) rather than point displacements near the crack tip are measured by the digital camera in the experiment. Two rows of evenly spaced square symbols printed on the paper are glued on the specimen with the crack in the middle as shown in Fig. 2. Each row of the square symbols locates Dy away from the crack surface in the ydirection. The CODs of the crack are measured by the relative displacements between the top and bottom rows of square symbols. Rewriting Eqs. (7) and (8), the COD components in x and y-directions for a pair of top and bottom square symbols expanded in N terms can be expressed as T
II II II Du ¼ ½U 1 U 2 U N ½cII 1 c2 cN ¼ ½Ufc g T
Dv ¼ ½V 1 V 2 V N ½cI1 cI2 cIN ¼ ½VfcI g
ð10Þ
n i rn=2 h n n n n j þ ð1Þ sin h sin 2 h 2 2 2 2 l ð11Þ
Vn ¼
r
n=2
l
h
n j ð1Þn 2
n i n n sin h þ sin 2 h 2 2 2
Dv ¼ ½V 1 V 3 V 5 V
I I I N ½c1 c3 c5
cIN T
I
¼ ½Vfc g
ð13Þ ð14Þ
Suppose the CODs of m pairs of top and bottom square symbols near the crack tip are considered. The sum of error squares of these CODs resulted from experimental measurement or numerical simulation can be expressed as follows: m X P¼ ½ð½Ui fcII g Dui Þ2 þ ð½Vi fcI g Dvi Þ2 ð15Þ i¼1
where P is the sum of error squares; Dui and D vi are the COD components in x and y-directions resulted from experimental measurement or numerical simulation for the ith pair of square symbols; [U]i and [V]i are [U] and [V] matrixes in Eqs. (9) and (10) for the ith pair of square symbols. To perform least-squares method, the sum of error squares P can be minimized by letting oP/o{cII} = 0 and oP/o{cI} = 0. Therefore, the linear system equations can be rearranged as ½KU fcII g ¼ fFU g
ð16Þ
I
½KV fc g ¼ fFV g
ð17Þ
where ½KU ¼
m X
fFU g ¼
½UTi ½Ui ; ½KV ¼
i¼1 m X i¼1
m X
½VTi ½Vi ;
i¼1
Dui ½UTi
and
fFV g ¼
m X
Dvi ½VTi
i¼1
By solving Eqs. (16) and (17), the parameters in {cII} and {cI}, i.e., cIn and cII n , can be obtained. The stress intensity factors of the cracked detail under consideration can then be determined by using Eqs. (5) and (6).
ð9Þ
where Un ¼
1863
ð12Þ
3. Fabrication and preparation of the fillet welded cruciform The cruciform joint specimens tested in this study (see Fig. 1) were made of three A36 steel rectangular plates (width = 15 mm and thickness = 15 mm) and were fillet welded at the four corners using manual tungsten inert gas (TIG) arc welding by the ER70S-G filler metal which conforms to AWS A5.18 specifications. The chemical compositions and the mechanical properties of the ER70S-G filler metal are shown in Table 1 and Table 2, respectively.
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Table 1 Chemical composition of the ER70S-G weld metal C
Mn
Si
P
S
Cu
0.07
1.50
0.85
0.015
0.005
0.020
Table 2 Mechanical properties of the ER70S-G weld metal Yield strength (N/mm2)
Tensile strength (N/mm2)
Elongation (%)
CVN impact energy (29 °C) (Joule)
470
560
31
57
A total of fifteen fillet welded cruciform joint specimens were fabricated by the combination of five fillet weld leg sizes (h = 3 mm, 7.5 mm, 9 mm, 12 mm and 15 mm) and three types of edge preparations, which could result in three center crack sizes (2a = 15 mm, 12 mm and 9 mm) in the un-penetrated region due to lack of penetration after TIG welding (see Fig. 1). The fifteen specimens were then classified into three groups (A, B and C) by the three center crack sizes in the un-penetrated region of the specimens, and each group had five specimens of different fillet weld leg sizes. The design and measured geometrical dimensions of the fifteen fillet welded cruciform joint specimens are listed in Table 3. In order to ensure the specified uniform center crack size through the thickness in the cruciform joint specimen, the two center cracks in the un-penetrated regions of the cruciform joint were re-cut by the wire electrical discharge machining (EDM) process to remove any irregular weld penetration occurred at the weld roots through the thickness. Table 4 shows the fillet welded portions of the fifteen cruciform joint specimens after fabrication and wire EDM process. It is noted that the tensile forces applied on the test specimens were in the direction perpendicular to the crack direction, as shown in Fig. 1.
To measure the CODs near the crack tip (weld root), a piece of paper with two parallel rows of black square symbols printed on it and with a split cut in the central line of the two rows of symbols was glued on the specimen with the center crack of the specimen exposed in the split cut. The two rows of black square symbols were printed on the paper by a commercial laser printer with the resolution of 600 dpi. The size of a square symbol was around 0.16 mm, and the center-to-center length of the black square symbols was 0.25 mm. The thickness of the paper was around 0.11 mm which is commonly used for laser printers. Due to the resolution limit of the commercial laser printer, the tiny black square (about 0.16 mm 0.16 mm) could not be printed properly in square shape. As a result, the printed black square often looks like a circular black dot. Fig. 3 shows the A15 specimen prepared for the experiment. 4. Procedures of digital camera experiment 4.1. Digital camera, microscope, illumination and testing machine A Leica Z6 microscope lenses attached to the Fujifilm FinePix S3 Pro digital camera (highest resolution = 4256 2848 pixels) was employed in this experiment to record the subtle changes of CODs in the un-penetrated region while the specified tensile loads were applied on the specimens. The digital camera was fixed on a tripod that allowed movement in all three axes at the same time, and the camera was connected to a laptop computer by an IEEE 1394 firewire cable. The photo shooting and camera adjustment (such as shutter speed, aperture value and ISO number) were controlled by the shooting software in the laptop computer through the IEEE 1394 firewire cable, and digital image files (in TIFF format) were saved in the laptop computer. The illumination for the specimens while
Table 3 Design and measured geometric parameters of specimens Specimen number
2a
B
w
Design (mm)
Measured (mm)
Design (mm)
Measured (mm)
Design (mm)
Measured (mm)
Design (mm)
Measured (mm)
A3 A7.5 A9 A12 A15 B3 B7.5 B9 B12 B15 C3 C7.5 C9 C12 C15
15 15 15 15 15 12 12 12 12 12 9 9 9 9 9
15.07 14.98 14.93 14.93 15.00 12.06 10.09 12.03 11.97 12.02 9.13 9.03 9.17 9.07 9.17
15 15 15 15 15 15 15 15 15 15 15 15 15 15 15
15.05 14.99 14.99 15.03 15.00 15.04 15.08 15.02 15.01 14.99 15.06 15.03 15.06 15.03 15.04
21 30 33 39 45 21 30 33 39 45 21 30 33 39 45
21.97 28.19 30.01 35.56 45.01 21.51 31.50 31.11 37.63 44.56 23.07 30.30 32.51 36.62 42.56
3.00 7.50 9.00 12.00 15.00 3.00 7.50 9.00 12.00 15.00 3.00 7.50 9.00 12.00 15.00
3.46 6.60 7.51 10.27 15.01 3.24 8.21 8.04 11.31 14.78 4.01 7.64 8.72 10.79 13.76
L1 = 150 mm, L2 = 95 mm, t = 15 mm, specimen thickness = 15 mm.
h
H.Y. Chung et al. / International Journal of Fatigue 30 (2008) 1861–1872
1865
Table 4 Photos of the fifteen specimens
Fig. 3. Details of A15 specimen in the cruciform joint.
testing was provided by two gooseneck optical fiber lights connected to a 20 V–150 W halogen light source box. The Instron-8800 servo-hydraulic testing machine with load capacity of 100 kN was employed for load-control tension tests in this experiment. The instrumentation of this digital camera experiment for the load-carrying cruciform joint specimens is shown in Fig. 4. 4.2. Experimental procedures The procedures for the experiment are simply described as follows: (1) Clamp the specimen at the top and bottom attachment plates. Adjust the tripod, microscope and digital camera system and focus the camera on the area near the weld root as the region of interest (ROI) of
Fig. 4. Test instrumentation of the digital camera experiment.
the specimen. The ROI area shown in the image file is about 3 mm (x-direction) 2 mm (y-direction) and includes two rows of 10 black symbols for COD measurement. The first ROI image of each specimen includes the weld root. (2) Apply zero load on the specimen, and take a picture using the camera shooting software in the laptop computer to prevent the vibration of the camera body caused by pushing the shutter button. Add the tensile load applied on the specimen to the specified load (5, 10, 15 or 20 KN), and take a picture at each load. (3) Gradually and horizontally shift the digital camera away from the weld root (crack tip) to the next ROI area of the specimen for COD measurement and make sure there are two rows of 10 black symbols included in the ROI area. Repeat procedure (2).
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(4) Repeat procedure (3) until the ROI area covers the crack center.
4.3. Image processing for computing CODs near the crack tip A FORTRAN program CCD3 (http//myweb.ncku.edu.tw/~juju/index.htm) was employed for processing the image files obtained from the digital camera experiment. After the image processing, (1) the coordinates of the cen-
troid of each black symbol can be acquired, and (2) the displacements of each symbol from 0 kN to the specified tensile loads (5, 10, 15 or 20 kN) can be calculated, respectively. Based on the displacements of each symbol under the specified tensile load (5, 10, 15 or 20 kN), the CODs near the crack tip at each load condition can be computed from each top and bottom symbol pair by simple vector analysis. Fig. 5a and 5b show the images (in TIFF format) taken separately by the digital camera for the same specimen under the zero load and 20 kN tensile load. After the image processing of Fig. 5a and 5b by the CCD3
Fig. 5. The unprocessed and processed digital images for COD measurement. (a) Unprocessed image near the crack tip without load. (b) Unprocessed image near the crack tip under the 20 kN tensile load. (c) Processed image to show the crack opening from 0 kN to 20 kN tensile load.
H.Y. Chung et al. / International Journal of Fatigue 30 (2008) 1861–1872
1867
Fig. 6. Finite element (FE) model and meshing for fillet welded cruciform joint specimen under tensile load. (a) FE model and meshing for the joint detail of C15 specimen. (b) Magnified view of FE meshing close to the crack tip.
program, a new image file shown as Fig. 5c is generated and Fig. 5c utilizes Fig. 5a as the background. The blue and red symbols on Fig. 5c represent the symbol positions under zero load and 20 kN tensile load, respectively.1 The position change of each symbol from 0 kN to 20 kN can be clearly seen on Fig. 5c. For each specified tensile load, the measured CODs near the crack tip can be substituted into the suggested least-squares method as described in Eqs. (15)–(17) to derive the stress intensity factor of the cruciform joint specimen. The procedures of the program CCD3 are illustrated as follows: (1) Obtain the red, green and blue (RGB) values at each pixel from a TIFF file (totally, 4256 2848 pixels) in Subroutine RIMAGE (about 120 statements). (2) Find the region of each black symbol block in Subroutine GP (about 100 statements). Since the RGB values of the pixels in a black symbol block are much different from those of other place, a procedure of the 1 For interpretation of color in Fig. 5c, the reader is referred to the web version of this article.
polygon filling can be used to find the block region. The block area and size are known, so too large or too small regions that are noises or wrong regions will be skipped. (3) Calculate the x and y centers of each block region for this current picture. It must be noted that the black square symbol is too small (about 0.16 mm 0.16 mm), so it is often not a good square shape. This situation will not cause any trouble for the image processing, since the CCD3 program can find the region that has the certain RGB values, and the square shape is not necessary. (4) Go to step 1 for the next picture. Until finishing the last picture, stop the program.
5. Experimental results A total of fifteen load-carrying fillet welded cruciform joint specimens (see Table 4) were tested and analyzed in this study. The specified tensile loads applied on each specimen were carefully calculated and determined to prevent
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H.Y. Chung et al. / International Journal of Fatigue 30 (2008) 1861–1872
a
9.E-03 8.E-03
COD- y direction (mm)
7.E-03 6.E-03 FEM(5kN)
5.E-03
FEM(10kN) FEM(15kN)
4.E-03
FEM(20kN)
3.E-03
EXP(5kN)
2.E-03
EXP(10kN) EXP(15kN)
1.E-03 EXP(20kN)
0.E+00 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
Distance Away from the Crack Tip (mm) 9.E-03
b
8.E-03
COD- y direction (mm)
7.E-03 6.E-03
FEM(5kN)
5.E-03
FEM(10kN) FEM(15kN)
4.E-03
FEM(20kN)
3.E-03
EXP(5kN)
2.E-03
EXP(10kN) EXP(15kN)
1.E-03
EXP(20kN)
0.E+00 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
Distance Away from the Crack Tip (mm)
c
9.E-03 8.E-03
COD- y direction (mm)
7.E-03 6.E-03 FEM(5kN)
5.E-03
FEM(10kN) FEM(15kN)
4.E-03
FEM(20kN)
3.E-03 EXP(5kN)
2.E-03
EXP(10kN)
1.E-03
EXP(15kN) EXP(20kN)
0.E+00 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
Distance Away from the Crack Tip (mm) Fig. 7. Comparison of COD results from FE analysis and digital camera experiment. (a) CODs of A15 Specimen under Various Tensile Loads. (b) CODs of B15 Specimen under Various Tensile Loads. (c) CODs of C15 Specimen under Various Tensile Loads.
H.Y. Chung et al. / International Journal of Fatigue 30 (2008) 1861–1872
the specimens from damages. Except for the A3, B3 and C3 specimens, all the other specimens were tested under the four specified tensile loads (5 kN, 10 kN, 15 kN and 20 kN) to obtain the CODs at different load levels. The A3, B3 and C3 specimens with smaller fillet weld leg sizes were tested under 5 kN and 10 kN tensile loads only. After following the procedures of the digital camera experiment described in the previous section, each specimen can yield a series of COD data near the crack tip under the specified tensile load. With the measured COD data, the stress intensity factor of the specimen corresponding to the specified load can be computed by the least-squares method. 5.1. Accuracy of the measured COD data The COD data measured from the digital camera experiment were compared with the numerical results analyzed by a finite element program, Micro-SAP (Hsu and Ju [20]), for verification. Linear-elastic and plane stress conditions were employed in the finite element analyses. The Young’s modulus (E) and Poisson’s ratio (m) of A36 steel are 200 GPa and 0.3, respectively. Eight-node quadrilateral isoparametric elements were utilized in the finite element meshes. In addition, quarter-point singular isoparametric elements (Henshell and Shaw [21]) were employed around the crack tip. Fig. 6a and b show the finite element model
and meshing for the C15 specimen. Fig. 7a–c illustrate the comparison of the y-direction CODs measured from the digital camera experiment and computed from the finite element analyses for the A15, B15 and C15 specimens under 5 kN, 10 kN, 15 kN and 20 kN tensile loads. It can be seen that the COD results from the digital camera experiment and from the finite element analyses match well in all tensile load conditions, so the presented experimental procedure provides a good means for the COD measurement in the cruciform joint specimens. 5.2. Comparison of the SIF values from least-squares method and BS 7910 The stress intensity factor of each specimen can be obtained by substituting the measured COD data into the suggested least-squares method. In order to compare the KI values derived from the least-squares method with those computed from Eq. (2) provided in the British Standard 7910 [10], KI values were normalized as follows: Mk ¼
KI qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi rm pa sec pa w
2a/w
h/B
Mk(Eq. (2))
A3
0.230
0.686
0.9930
A7.5
0.440
0.532
0.8263
A9
0.501
0.497
0.7883
A12
0.683
0.420
0.6879
A15
0.333
1.000
0.5561
Average error
K I =rm
2a/w
h/B
Mk(Eq. (2))
B3
0.215
0.561
0.9244
B7.5
0.544
0.320
0.7522
B9
0.536
0.387
0.7595
B12
0.753
0.318
0.6564
B15
0.986
0.270
0.5671
Average error
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pa secðpa=wÞ (Absolute relative error %)
5 kN
10 kN
15 kN
20 kN
0.9860 (0.71%) 0.8177 (1.04%) 0.7776 (1.36%) 0.6907 (0.40%) 0.5533 (0.50%)
1.0005 (0.75%) 0.8624 (4.37%) 0.7793 (1.15%) 0.7074 (2.83%) 0.5299 (4.71%)
– – 0.8094 (2.05%) 0.8364 (6.10%) 0.6854 (0.36%) 0.5730 (3.03%)
– – 0.8240 (0.29%) 0.8455 (7.25%) 0.7048 (2.46%) 0.5551 (0.18%)
(0.80%)
(2.76%)
(2.89%)
(0.80%)
Table 5b Comparison of magnification factor for group B specimens Specimen number
ð18Þ
where Mk is defined as the magnification parameter of the detail [1]; rm is the applied tensile stress on the top and bottom attachment plates.
Table 5a Comparison of magnification factor for group A specimens Specimen number
1869
K I =rm
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pa secðpa=wÞ (Absolute relative error %)
5 kN
10 kN
15 kN
20 kN
0.9289 (0.49%) 0.7043 (6.37%) 0.8109 (6.76%) 0.6334 (3.51%) 0.5796 (2.20%)
0.9559 (3.40%) 0.7944 (5.60%) 0.7666 (0.93%) 0.6504 (0.91%) 0.5326 (6.09%)
– – 0.7585 (0.84%) 0.7965 (4.86%) 0.6695 (1.99%) 0.5927 (4.50%)
– – 0.7134 (5.16%) 0.7927 (4.37%) 0.6673 (1.67%) 0.5889 (3.83%)
(3.87%)
(3.39%)
(3.05%)
(3.76%)
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Table 5c Comparison of magnification factor for group C specimens Specimen number
2a/w
h/B
Mk (Eq. (2))
C3
0.266
0.396
0.8573
C7.5
0.508
0.298
0.7675
C9
0.579
0.282
0.7362
C12
0.718
0.248
0.6760
C15
0.915
0.215
0.6002
Average error
K I =rm 5 kN
10 kN
15 kN
20 kN
0.8337 (2.74%) 0.7948 (3.57%) 0.7439 (1.05%) 0.6296 (6.86%) 0.6422 (7.00%)
0.8267 (3.56%) 0.7659 (0.20%) 0.7676 (4.27%) 0.7010 (3.70%) 0.6324 (5.37%)
– – 0.7390 (3.70%) 0.7280 (1.12%) 0.6346 (6.13%) 0.6409 (6.77%)
– – 0.7256 (5.45%) 0.7457 (1.29%) 0.7180 (6.21%) 0.6227 (3.75%)
(4.24%)
(3.42%)
(4.43%)
(4.18%)
Percentage of All 54 Tests
a The normalized KI values of all fifteen specimens under different tensile load levels from the digital camera experiment that included 54 test results and the corresponding absolute errors relative to the results from Eq. (2) are listed in Tables 5a, 5b and 5c. Fig. 8a–c show the histograms that summarize the SIF error distribution of the digital camera experiment. For further analysis, the total average error of this digital camera experiment is around 3%, which means that the suggested digital camera experiment is acceptable for engineering applications.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pa secðpa=wÞ (Absolute relative error %)
24% 22%
Portion of "Group C" Specimens
20%
Portion of "Group B" Specimens
18%
Portion of "Group A" Specimens
16% 14% 12% 10% 8% 6% 4% 2% 0%
0%-0.99% 1%-1.99% 2%-2.99% 3%-3.99% 4%-4.99% 5%-5.99% 6%-6.99% 7%-7.99%
SIF Error
5.3. Experimental error analysis
b
24% Portion of the Specimens under 20kN
22%
where ts is the shutter speed (s) set in the digital camera; vmax is the maximum vibration velocity (mm/s) in the experimental environment; C is the actual unit (mm) per
Percentage of All 54 Tests
Portion of the Specimens under 15kN
20%
Portion of the Specimens under 15kN
18%
Portion of the Specimens under 5kN
16% 14% 12% 10% 8% 6% 4% 2% 0%
0%-0.99% 1%-1.99% 2%-2.99% 3%-3.99% 4%-4.99% 5%-5.99% 6%-6.99% 7%-7.99%
SIF Error
c
24% 22%
Percentage of All 54 Tests
Fig. 9 presents the average SIF error of each specimen group under various tensile load levels. For a given tensile load level, the group order of average SIF error from large to small is: C > B > A. This is related to the different crack sizes in the un-penetrated region of the three specimen groups. Group C specimens have a shorter design crack size (2a = 9 mm), and this crack size in Group C specimens produces smaller CODs than that of the other two groups. Due to the resolution limit in processing digital image files, even one pixel difference might involve larger errors in COD measurement for specimens having smaller CODs. This type of error due to the lack-of-resolution can be addressed by increasing the resolution of the digital camera. In addition, errors might be caused by subtle movements between the specimen and the digital camera during picture taking. Vibration in the experimental environment is the main cause of this type of errors. Hence, the possible error from the micro-vibration of the Instron machine in the digital camera experiment is considered. The maximum pixel error (epix) from pixel movement during taking a picture can be defined as ts vmax epix ¼ ð18Þ C
20% 18%
Portion of the Specimens with h=15mm Portion of the Specimens with h=12mm Portion of the Specimens with h=9mm Portion of the Specimens with h=7.5mm Portion of the Specimens with h=3mm
16% 14% 12% 10% 8% 6% 4% 2% 0%
0%-0.99% 1%-1.99% 2%-2.99% 3%-3.99% 4%-4.99% 5%-5.99% 6%-6.99% 7%-7.99%
SIF Error Fig. 8. Histograms of test errors summarized from all 54 tests.
H.Y. Chung et al. / International Journal of Fatigue 30 (2008) 1861–1872 5% Group A Specimens
Average SIF Error
Group B Specimens
4%
Group C Specimens
3% 2% 1% 0%
5kN
10kN
15kN
20kN
Applied Tensile Load Fig. 9. Histogram of average test errors for the 15 specimens.
pixel in a digital image file. In the digital camera experiment, three velocity meters were mounted on the Instron machine to measure the vibrations in three directions. As a result, the maximum vibration velocity (vmax) of 0.025 mm/s was acquired. The shutter speed used in the digital camera was 1/1000 s, and the actual unit per pixel (C) obtained from processing the digital image files ranges from 6.2 104 to 6.4 104 mm/pixel. Therefore, the maximum error epix due to micro-vibration in the digital camera experiment is around 0.04 pixel, which does not affect the accuracy of the experiment too much. In addition, the possible error from micro-vibration of the digital camera body was diminished by using a remote camera shooting software and a stabilized tripod for picture taking. For more detailed observation, the errors of the experimental results primarily come from two major sources: fabrication error of test specimens and image distortion from microscope lenses. It is noted that Eq. (2) is based on the perfect geometry profile of a fillet welded cruciform joint specimen, in which the four fillet welds are of the same isosceles right triangle shape with a flat weld face. However, the specimens tested in the digital camera experiment were fillet welded by manual TIG, so the shape of fillet welds is somewhat different from the ideal isosceles right triangle. Geometric flaws (convexity or concavity) of the fillet welds might involve computation errors in SIF evaluation using Eq. (2). In addition, image distortion caused by optical lenses in the microscope sometimes produces a one or two pixel difference in certain locations of the image files, which affects the experimental precision in a direct manner. To eliminate the errors from image distortion, a distortion calibration procedure is suggested for further study of image processing. Besides, good optical fiber lighting utilized in the digital camera experiment also helps reduce the errors from illumination. 6. Conclusion A procedure of digital camera experiment combined with least-squares analysis is presented in this paper to determine the SIFs of load-carrying fillet welded cruciform joints. Fifteen cruciform joint specimens of various weld
1871
and crack sizes were carefully tested and analyzed, and the accuracy of the measured COD data from the digital camera experiment was numerically verified by finite element results in this study. The SIF results of the fifteen specimens under various tensile loads from the digital camera experiment agree well with the corresponding SIF values computed from Eq. (2) provided in the British Standard 7910 [10]. Hence, the digital camera experiment presented in this study essentially verified the BS 7910 equation for load-carrying fillet welded cruciform joints experimentally. In fact, the suggested procedure for determining SIFs can be applied on any homogenous and isotropic detail/member with a surface crack where the Irwin’s equation [19] can be employed. Furthermore, the simplicity of the digital camera experiment and the portability of the experimental instrumentation make the field measurement of SIF possible and provide a good means to acquire SIFs directly from details at the site, which is very useful for fatigue evaluation of details in structures like steel bridges and offshore structures. Acknowledgements The authors gratefully acknowledge the assistance from Mr. Yi-Tzuo Lin and the financial support through a Research Grant (NSC95-2221-E-006-333) awarded by the National Science Council, Taiwan, ROC. References [1] Maddox SJ. An analysis of fatigue cracks in fillet welded joints. Int J Fract 1975;11(2):221–43. [2] Hobbacher A. Stress intensity factors of welded joints. Eng Fract Mech 1993;46(2):173–82. [3] Bowness D, Lee MMK. Stress intensity factor solutions for semielliptical weld toe cracks in T-butt geometries. Fatigue Fract Eng Mater Struct 1996;19(6):787–97. [4] Lie ST, Zhao Z, Yan SH. Two-dimensional and three-dimensional magnification factors, Mk, for non-load-carrying fillet welds cruciform joints. Eng Fract Mech 2000;65(4):435–53. [5] Pang HLJ. Analysis of weld toe profiles and weld toe cracks. Int J Fatigue 1993;15(1):31–6. [6] Frank KH. The fatigue strength of fillet welded connections. PhD thesis. PA, USA: Lehigh University; 1971. [7] Frank KH, Fisher JW. Fatigue strength of fillet welded cruciform joints. J Struct Div ASCE 1979;105(ST9):1727–40. [8] British Standard PD 6493. Guidance on methods for assessing the acceptability of flaws in fusion welded structures. Appendix E4. London, UK: British Standard Institution; 1991. [9] British Standard 7910. Guide on methods for assessing the acceptability of flaws in metallic structures. London, UK: British Standard Institution; 1999. [10] British Standard 7910. Guide on methods for assessing the acceptability of flaws in metallic structures. London, UK: British Standard Institution; 2005. [11] Usami S, Kusumoto S. Fatigue strength at root of cruciform, tee and lap joints (1st Report). Trans Jpn Weld Soc 1978;9(1):1–10. [12] Khodadad Motarjemi A, Kokabi AH, Ziaie AA, Manteghi S, Burdekin FM. Comparison of the stress intensity factor of T and cruciform welded joints with different main and attachment plate thickness. Eng Fract Mech 2000;65(4):455–66.
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[13] Lie ST, Bian C. Load-carrying fillet welds using dual boundary element method. J Struct Eng ASCE 1997;123(12):1603–13. [14] Kai X, Yong L, Lie ST. Load-carrying welds joints analysis using boundary element method. Int J Solids Struct 2005;42(9–10):2965–75. [15] Ju SH. Simulating stress intensity factors for anisotropic materials by the least-squares method. Int J Fract 1996;81:283–97. [16] Ju SH, Lesniak JR, Sandor BI. Finite element simulation of stress intensity factors via the thermoelastic technique. Exp Mech 1997;SEM 37:278–84. [17] Ju SH. Simulating three-dimensional stress intensity factors by the least-squares method. Int J Numer Meth Eng 1998;43:1437–51.
[18] Ju SH, Rowlands RE. Thermoelastic determination of KI and KII in an orthotropic graphite/epoxy composite. J Compos Mater 2003;37:2011–25. [19] Irwin GR. Analysis of stresses and strains near end of a crack transversing a plate. Trans ASME. J Appl Mech 1957;24:361. [20] Hsu DS, Ju SH. Micro SAP-finite element structural analysis package. Taiwan: Unalis Corporation; 1988. [21] Henshell RD, Shaw KG. Crack tip elements are unnecessary. Int J Numer Meth Eng 1975;9:495–509.
International Journalof Fatigue
International Journal of Fatigue 30 (2008) 1861–1872
www.elsevier.com/locate/ijfatigue
Assessment of stress intensity factors for load-carrying fillet welded cruciform joints using a digital camera H.Y. Chung *, S.H. Liu, R.S. Lin, S.H. Ju Department of Civil Engineering, National Cheng Kung University, No. 1, University Road, Tainan City 701, Taiwan, ROC Received 12 December 2006; received in revised form 25 January 2008; accepted 29 January 2008 Available online 12 February 2008
Abstract This paper presents an experimental method to determine the stress intensity factors (SIFs) of load-carrying fillet welded cruciform joints. This experimental method measured the crack opening displacements (CODs) of the cruciform joint detail by a common digital camera, and utilized the acquired COD data to derive the SIF for the joint detail by a suggested least-squares procedure. A total of fifteen fillet welded cruciform joint specimens were tested and analyzed in this study. The test results were compared with the SIF formula provided in British Standard 7910, and showed that the demonstrated experimental method is able to give satisfactory SIF evaluation for fillet welded cruciform joints. In fact, this method can be applied to determine the SIF of any detail with surface crack where the Irwin’s series solution for crack tip displacement fields is valid. Moreover, the simplicity and portability of this method make the field measurement of SIF possible and provide a good means to acquire SIFs directly from details at the site. This is very useful for fatigue evaluation of details in structures, such as steel bridges or offshore structures. Ó 2008 Elsevier Ltd. All rights reserved. Keywords: Fillet weld; Cruciform joint; Digital camera; Stress intensity factor; Crack opening displacement
1. Introduction Fillet welded cruciform joints are widely used in metallic structures. When this kind of joints are subjected to tensile or cyclic loadings, the fillet weld roots or toes are easy to initiate cracks or result in fractures. Hence, computing the stress intensity factors (SIFs) at fillet weld roots or toes of cruciform joints was of great interests in the past and extensive works have been done. Previous studies on numerical evaluation of the SIFs at weld toes were carried out by Maddox [1], Hobbacher [2], Bowness and Lee [3] and Lie et al. [4], . . ., etc. Notably, Maddox [1] introduced the concept of magnification factor Mk in SIF to consider various geometric conditions that amplify the SIF of the detail of interest. Pang [5] utilized closedform integration and weight function methods to compute the SIFs of weld toe cracks in a full-penetration welded *
Corresponding author. Tel.: +886 6 2757575x63145; fax: +886 6 2358542. E-mail address: [email protected] (H.Y. Chung). 0142-1123/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2008.01.017
cruciform joint. For load-carrying fillet welded cruciform joints, the weld root in the un-penetrated region of the joint may act like a crack tip to initiate crack under fatigue loadings. Frank and Fisher [6,7] first proposed the polynomial expression of SIF magnification factor for the weld root in a load-carrying fillet welded cruciform joint by the compliance method using the results from finite element analysis. The SIF was found to be directly related to the size of the fillet weld leg and the crack size in the un-penetrated region as follows: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi paffi A1 þ A2 ð2a=wÞ KI ¼ rm pa sec ð1Þ 1 þ 2ðh=BÞ w and A1 ¼ 0:528 þ 3:287ðh=BÞ 4:361ðh=BÞ 3
2
4
5
þ 3:696ðh=BÞ 1:875ðh=BÞ þ 0:415ðh=BÞ A2 ¼ 0:218 þ 2:717ðh=BÞ 10:171ðh=BÞ2
þ 13:122ðh=BÞ3 7:755ðh=BÞ4 þ 1:783ðh=BÞ5
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where a is the half crack size in the un-penetrated region, B is the thickness of attachment plate, h is the size of fillet weld leg, w = B + 2h, and rm is the tensile stress applied on the attachment plate (see Fig. 1). BS PD-6493 [8] and BS 7910 [9,10] adopted Eq. (1) for SIF estimation of the weld root crack in a cruciform joint, and then improved it as follows: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pa K I ¼ M k rm pa sec ð2Þ w and Mk = A0 + A1(2a/w) + A2(2a/w)2 A0 ¼ 0:956 0:343ðh=BÞ A1 ¼ 1:219 þ 6:210ðh=BÞ 12:220ðh=BÞ
2
þ 9:704ðh=BÞ3 2:741ðh=BÞ4 2
A2 ¼ 1:954 7:938ðh=BÞ þ 13:299ðh=BÞ 9:541ðh=BÞ þ 2:513ðh=BÞ
3
4
Both Eqs. (1) and (2) are valid when 0.2 6 (h/B) 6 1.2 and 0.1 6 (2a/W) 6 0.7. Also, Usami and Kusumoto [11] implemented finite element method to compute the variation of SIF with respect to the weld leg size and the crack size in the un-penetrated region, and identified the mode I and mode II (shear mode) components of SIF in the weld root. From their study, the mode II component is very small compared to the mode I component. Khodadad Motarjemi et al. [12] determined the SIF for two-dimensional cruciform and T-welded joints by using linear-elastic finite element analysis and J-integral evaluation. Lie and Bian [13] and Kai et al. [14] employed a boundary element method to analyze the SIFs of load-carrying fillet welds. As can be seen, numerical approaches for evaluating the SIF
force
L1
B
α = 45º
h 2a α
2. SIF calculation using least-squares method Least-squares method has been successfully employed in finite element methods as a useful technique for computing stress intensity factors of sharp cracks in isotropic and composite standard specimens (Ju [15], Ju et al. [16], Ju [17] and Ju and Rowlands [18]). A least-squares method combining the measurement of the displacement field near a crack tip with a digital camera was applied in this study to calculate the stress intensity factors for load-carrying fillet welded cruciform joints. The main idea of this method is to obtain the stress intensity factors of the cracked detail by minimizing the sum of squares of the residuals that are the differences between the theoretical point displacements and the measured point displacements by the digital camera. For a linear-elastic and isotropic plate having a sharp crack (see Fig. 2), the in-plane displacement fields near the crack tip can be shown in the form of a series expansion (Irwin [19]) as follows: 1 X I u¼ cn f1n ðr; h; nÞ þ cII ð3Þ n f2n ðr; h; nÞ n¼1 1 X I v¼ cn g1n ðr; h; nÞ þ cII n g2n ðr; h; nÞ
α
t
w α
of the weld root in a load-carrying cruciform joint have been studied successfully in the past and good results were achieved. However, experimental approaches to acquire the SIFs of load-carrying cruciform joints or to verify the numerical SIF results are rarely reported in the literature. This paper presents an experimental approach to determine the stress intensity factors of load-carrying fillet welded cruciform joints. A commercial digital camera was utilized in this study for measuring the crack opening displacements (CODs) of the crack in the un-penetrated region of a load-carrying cruciform joint. The measured COD data were incorporated into a suggested least-squares procedure to derive the stress intensity factor. In order to verify this experimental approach, fifteen fillet welded cruciform joint specimens with five weld leg sizes and three unpenetrated crack sizes were tested and analyzed in this study, and the acquired SIF results were compared with the SIF values computed from the formula provided in the BS 7910 [10].
ð4Þ
n¼1
α
y
crack
fillet weld
v
Top Square Symbols
u
L2 Crack Surface
r
θ
Δy force Fig. 1. Geometry of the load-carrying fillet welded cruciform joint.
Crack Tip Bottom Square Symbols Fig. 2. Sharp crack in an infinite plate.
x
H.Y. Chung et al. / International Journal of Fatigue 30 (2008) 1861–1872
where
r and h are located on the non-negative-y-coordinate region. In the experiment of this study, Dy in Fig. 2 (i.e., the shortest distance from the center of a square symbol to the crack surface) is controlled to be very small, so h values for the top and bottom square symbols in Eqs. (11) and (12) are very close to p. As result, the even terms for Un and Vn are approximately zero. In order to reduce the errors in the suggested least-squares method, one can disregard the even terms in Eqs. (9) and (10) as follows:
n i r h n n n j þ þ ð1Þn cos h cos 2 h ; f1n ¼ 2 2 2 2 2l i n=2 h r n n n n n f2n ¼ j þ ð1Þ sin h sin 2 h ; 2 2 2 2 2l i n=2 h r n n n n g1n ¼ j ð1Þn sin h þ sin 2 h 2 2 2 2 2l n=2
and n i rn=2 h n n n j þ þ ð1Þn cos h cos 2 h ; 2 2 2 2 2l
g2n ¼
II II II T II Du ¼ ½U 1 U 3 U 5 U N ½cII 1 c3 c5 cN ¼ ½Ufc g
u and v are the displacements in x and y-direction, respectively, and expressed by r and h in the polar coordinates; l = shear modulus; j = 3 4m for plane stress and j = (3 m)/(1 + m) for plane strain; m = Poisson’s ratio. cIn and cII n are parameters to be determined. Specifically, the first terms of cIn and cII n are related to the mode I and mode II stress intensity factors, respectively, as follows: KI cI1 ¼ pffiffiffiffiffiffi ; 2p K II cII 1 ¼ pffiffiffiffiffiffi : 2p
ð5Þ ð6Þ
Once the parameters, cIn and cII n , are determined, the stress intensity factors of the cracked detail can be derived. Eqs. (3) and (4) can be rewritten in matrix forms for a given N terms: T
II u ¼ ½f11 f1N f 21 f2N ½cI1 cIN cII 1 cN ¼ ½f½c
v¼
II T ½g11 g1N g21 g2N ½cI1 cIN cII 1 cN
¼ ½g½c
ð7Þ ð8Þ
In this study, crack opening displacements (CODs) rather than point displacements near the crack tip are measured by the digital camera in the experiment. Two rows of evenly spaced square symbols printed on the paper are glued on the specimen with the crack in the middle as shown in Fig. 2. Each row of the square symbols locates Dy away from the crack surface in the ydirection. The CODs of the crack are measured by the relative displacements between the top and bottom rows of square symbols. Rewriting Eqs. (7) and (8), the COD components in x and y-directions for a pair of top and bottom square symbols expanded in N terms can be expressed as T
II II II Du ¼ ½U 1 U 2 U N ½cII 1 c2 cN ¼ ½Ufc g T
Dv ¼ ½V 1 V 2 V N ½cI1 cI2 cIN ¼ ½VfcI g
ð10Þ
n i rn=2 h n n n n j þ ð1Þ sin h sin 2 h 2 2 2 2 l ð11Þ
Vn ¼
r
n=2
l
h
n j ð1Þn 2
n i n n sin h þ sin 2 h 2 2 2
Dv ¼ ½V 1 V 3 V 5 V
I I I N ½c1 c3 c5
cIN T
I
¼ ½Vfc g
ð13Þ ð14Þ
Suppose the CODs of m pairs of top and bottom square symbols near the crack tip are considered. The sum of error squares of these CODs resulted from experimental measurement or numerical simulation can be expressed as follows: m X P¼ ½ð½Ui fcII g Dui Þ2 þ ð½Vi fcI g Dvi Þ2 ð15Þ i¼1
where P is the sum of error squares; Dui and D vi are the COD components in x and y-directions resulted from experimental measurement or numerical simulation for the ith pair of square symbols; [U]i and [V]i are [U] and [V] matrixes in Eqs. (9) and (10) for the ith pair of square symbols. To perform least-squares method, the sum of error squares P can be minimized by letting oP/o{cII} = 0 and oP/o{cI} = 0. Therefore, the linear system equations can be rearranged as ½KU fcII g ¼ fFU g
ð16Þ
I
½KV fc g ¼ fFV g
ð17Þ
where ½KU ¼
m X
fFU g ¼
½UTi ½Ui ; ½KV ¼
i¼1 m X i¼1
m X
½VTi ½Vi ;
i¼1
Dui ½UTi
and
fFV g ¼
m X
Dvi ½VTi
i¼1
By solving Eqs. (16) and (17), the parameters in {cII} and {cI}, i.e., cIn and cII n , can be obtained. The stress intensity factors of the cracked detail under consideration can then be determined by using Eqs. (5) and (6).
ð9Þ
where Un ¼
1863
ð12Þ
3. Fabrication and preparation of the fillet welded cruciform The cruciform joint specimens tested in this study (see Fig. 1) were made of three A36 steel rectangular plates (width = 15 mm and thickness = 15 mm) and were fillet welded at the four corners using manual tungsten inert gas (TIG) arc welding by the ER70S-G filler metal which conforms to AWS A5.18 specifications. The chemical compositions and the mechanical properties of the ER70S-G filler metal are shown in Table 1 and Table 2, respectively.
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Table 1 Chemical composition of the ER70S-G weld metal C
Mn
Si
P
S
Cu
0.07
1.50
0.85
0.015
0.005
0.020
Table 2 Mechanical properties of the ER70S-G weld metal Yield strength (N/mm2)
Tensile strength (N/mm2)
Elongation (%)
CVN impact energy (29 °C) (Joule)
470
560
31
57
A total of fifteen fillet welded cruciform joint specimens were fabricated by the combination of five fillet weld leg sizes (h = 3 mm, 7.5 mm, 9 mm, 12 mm and 15 mm) and three types of edge preparations, which could result in three center crack sizes (2a = 15 mm, 12 mm and 9 mm) in the un-penetrated region due to lack of penetration after TIG welding (see Fig. 1). The fifteen specimens were then classified into three groups (A, B and C) by the three center crack sizes in the un-penetrated region of the specimens, and each group had five specimens of different fillet weld leg sizes. The design and measured geometrical dimensions of the fifteen fillet welded cruciform joint specimens are listed in Table 3. In order to ensure the specified uniform center crack size through the thickness in the cruciform joint specimen, the two center cracks in the un-penetrated regions of the cruciform joint were re-cut by the wire electrical discharge machining (EDM) process to remove any irregular weld penetration occurred at the weld roots through the thickness. Table 4 shows the fillet welded portions of the fifteen cruciform joint specimens after fabrication and wire EDM process. It is noted that the tensile forces applied on the test specimens were in the direction perpendicular to the crack direction, as shown in Fig. 1.
To measure the CODs near the crack tip (weld root), a piece of paper with two parallel rows of black square symbols printed on it and with a split cut in the central line of the two rows of symbols was glued on the specimen with the center crack of the specimen exposed in the split cut. The two rows of black square symbols were printed on the paper by a commercial laser printer with the resolution of 600 dpi. The size of a square symbol was around 0.16 mm, and the center-to-center length of the black square symbols was 0.25 mm. The thickness of the paper was around 0.11 mm which is commonly used for laser printers. Due to the resolution limit of the commercial laser printer, the tiny black square (about 0.16 mm 0.16 mm) could not be printed properly in square shape. As a result, the printed black square often looks like a circular black dot. Fig. 3 shows the A15 specimen prepared for the experiment. 4. Procedures of digital camera experiment 4.1. Digital camera, microscope, illumination and testing machine A Leica Z6 microscope lenses attached to the Fujifilm FinePix S3 Pro digital camera (highest resolution = 4256 2848 pixels) was employed in this experiment to record the subtle changes of CODs in the un-penetrated region while the specified tensile loads were applied on the specimens. The digital camera was fixed on a tripod that allowed movement in all three axes at the same time, and the camera was connected to a laptop computer by an IEEE 1394 firewire cable. The photo shooting and camera adjustment (such as shutter speed, aperture value and ISO number) were controlled by the shooting software in the laptop computer through the IEEE 1394 firewire cable, and digital image files (in TIFF format) were saved in the laptop computer. The illumination for the specimens while
Table 3 Design and measured geometric parameters of specimens Specimen number
2a
B
w
Design (mm)
Measured (mm)
Design (mm)
Measured (mm)
Design (mm)
Measured (mm)
Design (mm)
Measured (mm)
A3 A7.5 A9 A12 A15 B3 B7.5 B9 B12 B15 C3 C7.5 C9 C12 C15
15 15 15 15 15 12 12 12 12 12 9 9 9 9 9
15.07 14.98 14.93 14.93 15.00 12.06 10.09 12.03 11.97 12.02 9.13 9.03 9.17 9.07 9.17
15 15 15 15 15 15 15 15 15 15 15 15 15 15 15
15.05 14.99 14.99 15.03 15.00 15.04 15.08 15.02 15.01 14.99 15.06 15.03 15.06 15.03 15.04
21 30 33 39 45 21 30 33 39 45 21 30 33 39 45
21.97 28.19 30.01 35.56 45.01 21.51 31.50 31.11 37.63 44.56 23.07 30.30 32.51 36.62 42.56
3.00 7.50 9.00 12.00 15.00 3.00 7.50 9.00 12.00 15.00 3.00 7.50 9.00 12.00 15.00
3.46 6.60 7.51 10.27 15.01 3.24 8.21 8.04 11.31 14.78 4.01 7.64 8.72 10.79 13.76
L1 = 150 mm, L2 = 95 mm, t = 15 mm, specimen thickness = 15 mm.
h
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Table 4 Photos of the fifteen specimens
Fig. 3. Details of A15 specimen in the cruciform joint.
testing was provided by two gooseneck optical fiber lights connected to a 20 V–150 W halogen light source box. The Instron-8800 servo-hydraulic testing machine with load capacity of 100 kN was employed for load-control tension tests in this experiment. The instrumentation of this digital camera experiment for the load-carrying cruciform joint specimens is shown in Fig. 4. 4.2. Experimental procedures The procedures for the experiment are simply described as follows: (1) Clamp the specimen at the top and bottom attachment plates. Adjust the tripod, microscope and digital camera system and focus the camera on the area near the weld root as the region of interest (ROI) of
Fig. 4. Test instrumentation of the digital camera experiment.
the specimen. The ROI area shown in the image file is about 3 mm (x-direction) 2 mm (y-direction) and includes two rows of 10 black symbols for COD measurement. The first ROI image of each specimen includes the weld root. (2) Apply zero load on the specimen, and take a picture using the camera shooting software in the laptop computer to prevent the vibration of the camera body caused by pushing the shutter button. Add the tensile load applied on the specimen to the specified load (5, 10, 15 or 20 KN), and take a picture at each load. (3) Gradually and horizontally shift the digital camera away from the weld root (crack tip) to the next ROI area of the specimen for COD measurement and make sure there are two rows of 10 black symbols included in the ROI area. Repeat procedure (2).
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(4) Repeat procedure (3) until the ROI area covers the crack center.
4.3. Image processing for computing CODs near the crack tip A FORTRAN program CCD3 (http//myweb.ncku.edu.tw/~juju/index.htm) was employed for processing the image files obtained from the digital camera experiment. After the image processing, (1) the coordinates of the cen-
troid of each black symbol can be acquired, and (2) the displacements of each symbol from 0 kN to the specified tensile loads (5, 10, 15 or 20 kN) can be calculated, respectively. Based on the displacements of each symbol under the specified tensile load (5, 10, 15 or 20 kN), the CODs near the crack tip at each load condition can be computed from each top and bottom symbol pair by simple vector analysis. Fig. 5a and 5b show the images (in TIFF format) taken separately by the digital camera for the same specimen under the zero load and 20 kN tensile load. After the image processing of Fig. 5a and 5b by the CCD3
Fig. 5. The unprocessed and processed digital images for COD measurement. (a) Unprocessed image near the crack tip without load. (b) Unprocessed image near the crack tip under the 20 kN tensile load. (c) Processed image to show the crack opening from 0 kN to 20 kN tensile load.
H.Y. Chung et al. / International Journal of Fatigue 30 (2008) 1861–1872
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Fig. 6. Finite element (FE) model and meshing for fillet welded cruciform joint specimen under tensile load. (a) FE model and meshing for the joint detail of C15 specimen. (b) Magnified view of FE meshing close to the crack tip.
program, a new image file shown as Fig. 5c is generated and Fig. 5c utilizes Fig. 5a as the background. The blue and red symbols on Fig. 5c represent the symbol positions under zero load and 20 kN tensile load, respectively.1 The position change of each symbol from 0 kN to 20 kN can be clearly seen on Fig. 5c. For each specified tensile load, the measured CODs near the crack tip can be substituted into the suggested least-squares method as described in Eqs. (15)–(17) to derive the stress intensity factor of the cruciform joint specimen. The procedures of the program CCD3 are illustrated as follows: (1) Obtain the red, green and blue (RGB) values at each pixel from a TIFF file (totally, 4256 2848 pixels) in Subroutine RIMAGE (about 120 statements). (2) Find the region of each black symbol block in Subroutine GP (about 100 statements). Since the RGB values of the pixels in a black symbol block are much different from those of other place, a procedure of the 1 For interpretation of color in Fig. 5c, the reader is referred to the web version of this article.
polygon filling can be used to find the block region. The block area and size are known, so too large or too small regions that are noises or wrong regions will be skipped. (3) Calculate the x and y centers of each block region for this current picture. It must be noted that the black square symbol is too small (about 0.16 mm 0.16 mm), so it is often not a good square shape. This situation will not cause any trouble for the image processing, since the CCD3 program can find the region that has the certain RGB values, and the square shape is not necessary. (4) Go to step 1 for the next picture. Until finishing the last picture, stop the program.
5. Experimental results A total of fifteen load-carrying fillet welded cruciform joint specimens (see Table 4) were tested and analyzed in this study. The specified tensile loads applied on each specimen were carefully calculated and determined to prevent
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a
9.E-03 8.E-03
COD- y direction (mm)
7.E-03 6.E-03 FEM(5kN)
5.E-03
FEM(10kN) FEM(15kN)
4.E-03
FEM(20kN)
3.E-03
EXP(5kN)
2.E-03
EXP(10kN) EXP(15kN)
1.E-03 EXP(20kN)
0.E+00 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
Distance Away from the Crack Tip (mm) 9.E-03
b
8.E-03
COD- y direction (mm)
7.E-03 6.E-03
FEM(5kN)
5.E-03
FEM(10kN) FEM(15kN)
4.E-03
FEM(20kN)
3.E-03
EXP(5kN)
2.E-03
EXP(10kN) EXP(15kN)
1.E-03
EXP(20kN)
0.E+00 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
Distance Away from the Crack Tip (mm)
c
9.E-03 8.E-03
COD- y direction (mm)
7.E-03 6.E-03 FEM(5kN)
5.E-03
FEM(10kN) FEM(15kN)
4.E-03
FEM(20kN)
3.E-03 EXP(5kN)
2.E-03
EXP(10kN)
1.E-03
EXP(15kN) EXP(20kN)
0.E+00 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
Distance Away from the Crack Tip (mm) Fig. 7. Comparison of COD results from FE analysis and digital camera experiment. (a) CODs of A15 Specimen under Various Tensile Loads. (b) CODs of B15 Specimen under Various Tensile Loads. (c) CODs of C15 Specimen under Various Tensile Loads.
H.Y. Chung et al. / International Journal of Fatigue 30 (2008) 1861–1872
the specimens from damages. Except for the A3, B3 and C3 specimens, all the other specimens were tested under the four specified tensile loads (5 kN, 10 kN, 15 kN and 20 kN) to obtain the CODs at different load levels. The A3, B3 and C3 specimens with smaller fillet weld leg sizes were tested under 5 kN and 10 kN tensile loads only. After following the procedures of the digital camera experiment described in the previous section, each specimen can yield a series of COD data near the crack tip under the specified tensile load. With the measured COD data, the stress intensity factor of the specimen corresponding to the specified load can be computed by the least-squares method. 5.1. Accuracy of the measured COD data The COD data measured from the digital camera experiment were compared with the numerical results analyzed by a finite element program, Micro-SAP (Hsu and Ju [20]), for verification. Linear-elastic and plane stress conditions were employed in the finite element analyses. The Young’s modulus (E) and Poisson’s ratio (m) of A36 steel are 200 GPa and 0.3, respectively. Eight-node quadrilateral isoparametric elements were utilized in the finite element meshes. In addition, quarter-point singular isoparametric elements (Henshell and Shaw [21]) were employed around the crack tip. Fig. 6a and b show the finite element model
and meshing for the C15 specimen. Fig. 7a–c illustrate the comparison of the y-direction CODs measured from the digital camera experiment and computed from the finite element analyses for the A15, B15 and C15 specimens under 5 kN, 10 kN, 15 kN and 20 kN tensile loads. It can be seen that the COD results from the digital camera experiment and from the finite element analyses match well in all tensile load conditions, so the presented experimental procedure provides a good means for the COD measurement in the cruciform joint specimens. 5.2. Comparison of the SIF values from least-squares method and BS 7910 The stress intensity factor of each specimen can be obtained by substituting the measured COD data into the suggested least-squares method. In order to compare the KI values derived from the least-squares method with those computed from Eq. (2) provided in the British Standard 7910 [10], KI values were normalized as follows: Mk ¼
KI qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi rm pa sec pa w
2a/w
h/B
Mk(Eq. (2))
A3
0.230
0.686
0.9930
A7.5
0.440
0.532
0.8263
A9
0.501
0.497
0.7883
A12
0.683
0.420
0.6879
A15
0.333
1.000
0.5561
Average error
K I =rm
2a/w
h/B
Mk(Eq. (2))
B3
0.215
0.561
0.9244
B7.5
0.544
0.320
0.7522
B9
0.536
0.387
0.7595
B12
0.753
0.318
0.6564
B15
0.986
0.270
0.5671
Average error
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pa secðpa=wÞ (Absolute relative error %)
5 kN
10 kN
15 kN
20 kN
0.9860 (0.71%) 0.8177 (1.04%) 0.7776 (1.36%) 0.6907 (0.40%) 0.5533 (0.50%)
1.0005 (0.75%) 0.8624 (4.37%) 0.7793 (1.15%) 0.7074 (2.83%) 0.5299 (4.71%)
– – 0.8094 (2.05%) 0.8364 (6.10%) 0.6854 (0.36%) 0.5730 (3.03%)
– – 0.8240 (0.29%) 0.8455 (7.25%) 0.7048 (2.46%) 0.5551 (0.18%)
(0.80%)
(2.76%)
(2.89%)
(0.80%)
Table 5b Comparison of magnification factor for group B specimens Specimen number
ð18Þ
where Mk is defined as the magnification parameter of the detail [1]; rm is the applied tensile stress on the top and bottom attachment plates.
Table 5a Comparison of magnification factor for group A specimens Specimen number
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K I =rm
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pa secðpa=wÞ (Absolute relative error %)
5 kN
10 kN
15 kN
20 kN
0.9289 (0.49%) 0.7043 (6.37%) 0.8109 (6.76%) 0.6334 (3.51%) 0.5796 (2.20%)
0.9559 (3.40%) 0.7944 (5.60%) 0.7666 (0.93%) 0.6504 (0.91%) 0.5326 (6.09%)
– – 0.7585 (0.84%) 0.7965 (4.86%) 0.6695 (1.99%) 0.5927 (4.50%)
– – 0.7134 (5.16%) 0.7927 (4.37%) 0.6673 (1.67%) 0.5889 (3.83%)
(3.87%)
(3.39%)
(3.05%)
(3.76%)
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Table 5c Comparison of magnification factor for group C specimens Specimen number
2a/w
h/B
Mk (Eq. (2))
C3
0.266
0.396
0.8573
C7.5
0.508
0.298
0.7675
C9
0.579
0.282
0.7362
C12
0.718
0.248
0.6760
C15
0.915
0.215
0.6002
Average error
K I =rm 5 kN
10 kN
15 kN
20 kN
0.8337 (2.74%) 0.7948 (3.57%) 0.7439 (1.05%) 0.6296 (6.86%) 0.6422 (7.00%)
0.8267 (3.56%) 0.7659 (0.20%) 0.7676 (4.27%) 0.7010 (3.70%) 0.6324 (5.37%)
– – 0.7390 (3.70%) 0.7280 (1.12%) 0.6346 (6.13%) 0.6409 (6.77%)
– – 0.7256 (5.45%) 0.7457 (1.29%) 0.7180 (6.21%) 0.6227 (3.75%)
(4.24%)
(3.42%)
(4.43%)
(4.18%)
Percentage of All 54 Tests
a The normalized KI values of all fifteen specimens under different tensile load levels from the digital camera experiment that included 54 test results and the corresponding absolute errors relative to the results from Eq. (2) are listed in Tables 5a, 5b and 5c. Fig. 8a–c show the histograms that summarize the SIF error distribution of the digital camera experiment. For further analysis, the total average error of this digital camera experiment is around 3%, which means that the suggested digital camera experiment is acceptable for engineering applications.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pa secðpa=wÞ (Absolute relative error %)
24% 22%
Portion of "Group C" Specimens
20%
Portion of "Group B" Specimens
18%
Portion of "Group A" Specimens
16% 14% 12% 10% 8% 6% 4% 2% 0%
0%-0.99% 1%-1.99% 2%-2.99% 3%-3.99% 4%-4.99% 5%-5.99% 6%-6.99% 7%-7.99%
SIF Error
5.3. Experimental error analysis
b
24% Portion of the Specimens under 20kN
22%
where ts is the shutter speed (s) set in the digital camera; vmax is the maximum vibration velocity (mm/s) in the experimental environment; C is the actual unit (mm) per
Percentage of All 54 Tests
Portion of the Specimens under 15kN
20%
Portion of the Specimens under 15kN
18%
Portion of the Specimens under 5kN
16% 14% 12% 10% 8% 6% 4% 2% 0%
0%-0.99% 1%-1.99% 2%-2.99% 3%-3.99% 4%-4.99% 5%-5.99% 6%-6.99% 7%-7.99%
SIF Error
c
24% 22%
Percentage of All 54 Tests
Fig. 9 presents the average SIF error of each specimen group under various tensile load levels. For a given tensile load level, the group order of average SIF error from large to small is: C > B > A. This is related to the different crack sizes in the un-penetrated region of the three specimen groups. Group C specimens have a shorter design crack size (2a = 9 mm), and this crack size in Group C specimens produces smaller CODs than that of the other two groups. Due to the resolution limit in processing digital image files, even one pixel difference might involve larger errors in COD measurement for specimens having smaller CODs. This type of error due to the lack-of-resolution can be addressed by increasing the resolution of the digital camera. In addition, errors might be caused by subtle movements between the specimen and the digital camera during picture taking. Vibration in the experimental environment is the main cause of this type of errors. Hence, the possible error from the micro-vibration of the Instron machine in the digital camera experiment is considered. The maximum pixel error (epix) from pixel movement during taking a picture can be defined as ts vmax epix ¼ ð18Þ C
20% 18%
Portion of the Specimens with h=15mm Portion of the Specimens with h=12mm Portion of the Specimens with h=9mm Portion of the Specimens with h=7.5mm Portion of the Specimens with h=3mm
16% 14% 12% 10% 8% 6% 4% 2% 0%
0%-0.99% 1%-1.99% 2%-2.99% 3%-3.99% 4%-4.99% 5%-5.99% 6%-6.99% 7%-7.99%
SIF Error Fig. 8. Histograms of test errors summarized from all 54 tests.
H.Y. Chung et al. / International Journal of Fatigue 30 (2008) 1861–1872 5% Group A Specimens
Average SIF Error
Group B Specimens
4%
Group C Specimens
3% 2% 1% 0%
5kN
10kN
15kN
20kN
Applied Tensile Load Fig. 9. Histogram of average test errors for the 15 specimens.
pixel in a digital image file. In the digital camera experiment, three velocity meters were mounted on the Instron machine to measure the vibrations in three directions. As a result, the maximum vibration velocity (vmax) of 0.025 mm/s was acquired. The shutter speed used in the digital camera was 1/1000 s, and the actual unit per pixel (C) obtained from processing the digital image files ranges from 6.2 104 to 6.4 104 mm/pixel. Therefore, the maximum error epix due to micro-vibration in the digital camera experiment is around 0.04 pixel, which does not affect the accuracy of the experiment too much. In addition, the possible error from micro-vibration of the digital camera body was diminished by using a remote camera shooting software and a stabilized tripod for picture taking. For more detailed observation, the errors of the experimental results primarily come from two major sources: fabrication error of test specimens and image distortion from microscope lenses. It is noted that Eq. (2) is based on the perfect geometry profile of a fillet welded cruciform joint specimen, in which the four fillet welds are of the same isosceles right triangle shape with a flat weld face. However, the specimens tested in the digital camera experiment were fillet welded by manual TIG, so the shape of fillet welds is somewhat different from the ideal isosceles right triangle. Geometric flaws (convexity or concavity) of the fillet welds might involve computation errors in SIF evaluation using Eq. (2). In addition, image distortion caused by optical lenses in the microscope sometimes produces a one or two pixel difference in certain locations of the image files, which affects the experimental precision in a direct manner. To eliminate the errors from image distortion, a distortion calibration procedure is suggested for further study of image processing. Besides, good optical fiber lighting utilized in the digital camera experiment also helps reduce the errors from illumination. 6. Conclusion A procedure of digital camera experiment combined with least-squares analysis is presented in this paper to determine the SIFs of load-carrying fillet welded cruciform joints. Fifteen cruciform joint specimens of various weld
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and crack sizes were carefully tested and analyzed, and the accuracy of the measured COD data from the digital camera experiment was numerically verified by finite element results in this study. The SIF results of the fifteen specimens under various tensile loads from the digital camera experiment agree well with the corresponding SIF values computed from Eq. (2) provided in the British Standard 7910 [10]. Hence, the digital camera experiment presented in this study essentially verified the BS 7910 equation for load-carrying fillet welded cruciform joints experimentally. In fact, the suggested procedure for determining SIFs can be applied on any homogenous and isotropic detail/member with a surface crack where the Irwin’s equation [19] can be employed. Furthermore, the simplicity of the digital camera experiment and the portability of the experimental instrumentation make the field measurement of SIF possible and provide a good means to acquire SIFs directly from details at the site, which is very useful for fatigue evaluation of details in structures like steel bridges and offshore structures. Acknowledgements The authors gratefully acknowledge the assistance from Mr. Yi-Tzuo Lin and the financial support through a Research Grant (NSC95-2221-E-006-333) awarded by the National Science Council, Taiwan, ROC. References [1] Maddox SJ. An analysis of fatigue cracks in fillet welded joints. Int J Fract 1975;11(2):221–43. [2] Hobbacher A. Stress intensity factors of welded joints. Eng Fract Mech 1993;46(2):173–82. [3] Bowness D, Lee MMK. Stress intensity factor solutions for semielliptical weld toe cracks in T-butt geometries. Fatigue Fract Eng Mater Struct 1996;19(6):787–97. [4] Lie ST, Zhao Z, Yan SH. Two-dimensional and three-dimensional magnification factors, Mk, for non-load-carrying fillet welds cruciform joints. Eng Fract Mech 2000;65(4):435–53. [5] Pang HLJ. Analysis of weld toe profiles and weld toe cracks. Int J Fatigue 1993;15(1):31–6. [6] Frank KH. The fatigue strength of fillet welded connections. PhD thesis. PA, USA: Lehigh University; 1971. [7] Frank KH, Fisher JW. Fatigue strength of fillet welded cruciform joints. J Struct Div ASCE 1979;105(ST9):1727–40. [8] British Standard PD 6493. Guidance on methods for assessing the acceptability of flaws in fusion welded structures. Appendix E4. London, UK: British Standard Institution; 1991. [9] British Standard 7910. Guide on methods for assessing the acceptability of flaws in metallic structures. London, UK: British Standard Institution; 1999. [10] British Standard 7910. Guide on methods for assessing the acceptability of flaws in metallic structures. London, UK: British Standard Institution; 2005. [11] Usami S, Kusumoto S. Fatigue strength at root of cruciform, tee and lap joints (1st Report). Trans Jpn Weld Soc 1978;9(1):1–10. [12] Khodadad Motarjemi A, Kokabi AH, Ziaie AA, Manteghi S, Burdekin FM. Comparison of the stress intensity factor of T and cruciform welded joints with different main and attachment plate thickness. Eng Fract Mech 2000;65(4):455–66.
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[13] Lie ST, Bian C. Load-carrying fillet welds using dual boundary element method. J Struct Eng ASCE 1997;123(12):1603–13. [14] Kai X, Yong L, Lie ST. Load-carrying welds joints analysis using boundary element method. Int J Solids Struct 2005;42(9–10):2965–75. [15] Ju SH. Simulating stress intensity factors for anisotropic materials by the least-squares method. Int J Fract 1996;81:283–97. [16] Ju SH, Lesniak JR, Sandor BI. Finite element simulation of stress intensity factors via the thermoelastic technique. Exp Mech 1997;SEM 37:278–84. [17] Ju SH. Simulating three-dimensional stress intensity factors by the least-squares method. Int J Numer Meth Eng 1998;43:1437–51.
[18] Ju SH, Rowlands RE. Thermoelastic determination of KI and KII in an orthotropic graphite/epoxy composite. J Compos Mater 2003;37:2011–25. [19] Irwin GR. Analysis of stresses and strains near end of a crack transversing a plate. Trans ASME. J Appl Mech 1957;24:361. [20] Hsu DS, Ju SH. Micro SAP-finite element structural analysis package. Taiwan: Unalis Corporation; 1988. [21] Henshell RD, Shaw KG. Crack tip elements are unnecessary. Int J Numer Meth Eng 1975;9:495–509.