A practical procedure for evaluating SIFs along fronts of semi-elliptical surface cracks at weld toes in complex stress fields

Int. J. Fatigue Vol. 18, No. 2, pp. 127-135, 1996 Copyright ~) 1996 Elsevier Science Limited Printed in Great Britain. All fights reserved 0142-1123/%/$15.00

ELSEVIER

0142-1123(95)00012-7

A practical procedure for evaluating SIFs along fronts of semi-elliptical surface cracks at weld toes in complex stress fields Yan-Lin Lu

Department of Mechanical Engineering, Zhejiang University of Technology, Hangzhou, Zhejiang 310014, P. R. China (Received 19 November 1994; revised 3 February 1995) Distributions of stress intensity factors (SIF) along crack fronts for 27 geometries of semi-elliptical surface cracks at weld toes subjected to eight types of selected elementary stress mode were analysed by a 3-D finite element method. SIF distributions for general geometries in complex stress fields can be calculated through interpolations and linear combinations of these elementary solutions. The surface effect of the thickness ratio of fillet-welded plates on SIF distributions can be taken into account. The procedure is very simple and fast. (Keywords: stress intensity factor; semi-elliptical surface crack; weld toe; welded joint)

Surface cracks at weld toes are thus usually subjected to very complex stress fields. In addition, the influence of geometry also becomes complicated. Geometrical parameters such as the thickness ratio of the welded plates, the weld toe angle and the shape of the weld bead not only influence the stress concentration - that is, they will influence the stress distribution at the weld toe when no crack exists in the joint - - but also have a surface effect on SIF distributions: that is, they will cause a change in the SIF distribution along crack fronts when a crack exists in the joint. In other words, even for cracks with the same crack shape and size subjected to the same crack face loading, the SIF distributions along crack fronts may be different owing to change in these geometrical parameters. The surface effects have been taken into attention by many investigators. Niu and Glinka 7 studied the influence of corner angle (corresponding to the weld toe angle) on SIF for through-thickness cracks and semi-elliptical surface cracks. A weight function method was used. Unfortunately, for surface cracks, only SIF values at the deepest point can be obtained by this method. In fatigue crack growth, the surface crack in a given stress field will have its natural crack pattern development path. In ref. 8, some e~amples show that the predicted fatigue life largely depends on the crack pattern development path. So, for accurate prediction of fatigue life, it is very important to obtain information about the SIF distribution along the whole crack front, or at least at two positions: the

Part-through surface cracks occur frequently at weld toes in ships, offshore platforms, pressure vessels and other welded structures. These cracks normally have a shape close to a semi-ellipse. In fracture mechanics, the fatigue crack growth rate da/dN can be estimated through Paris's equation: da ON - C(AK) m

( 1)

where C and m are usually considered as material constants, and AK is the stress intensity factor (SIF) range (or effective SIF range). For reliable prediction of fatigue lives of these structures, accurate estimation of SIF distributions along crack fronts is necessary. For the problem of SIF distributions along fronts of semi-elliptical surface cracks in plates, a lot of contributions have been made by many investigators 1-6. Among these, the numerical solutions given by Newman and Raju z-4 for cracked plates subjected to uniform tension and bending are usually considered as good results, and have been widely referenced. Solutions of SIF distributions for semi-elliptical surface cracks in plates subjected to linearly or cosine-wise distributed crack-face loading were given by Wu and Carlsson 5,6. When dealing with surface cracks at weld toes of butt or fillet-welded joints, the situation is more complicated than that in plates. Owing to stress concentration, the applied stress field caused by service loading is complex at weld toes. Furthermore, there are usually large values of residual stress at weld toes. The distribution of residual stress is also very complex.

127

128

Yan-Lin Lu

deepest point and the intersection point of the crack front with the plate surface. For engineering fatigue analysis, the crack pattern development path is integrated step by step from SIF solutions for a series of crack configurations. So, the method used to calculate SIFs should not be too timeconsuming. In this study, a very direct, very simple but very fast procedure was established to meet this need. The method comprises an SIF database for semi-elliptical surface cracks at weld toes for some elementary configurations and stress modes, and a program to calculate SIFs from the database for general configurations and stress fields. The surface effect of the thickness ratio of welded plates can be taken into account in the present method.

=1 I

PRINCIPLE OF SUPERPOSITION TO C A L C U L A T E SIFs IN COMPLEX STRESS FIELDS The principle of superposition can be used to calculate SIFs of a crack in a complex stress field, which may be caused by both service loading and residual stress. Figure 1 illustrates the principle. Suppose that the stress distribution function along the crack position is o'q when the crack has not been introduced into the body; the SIF (K value) when the crack has been introduced into the body is equal to the K value of the cracked body subjected to crack face loading -trq. ELEMENTARY STRESS MODES Eight types of distribution of crack face loading were selected as elementary stress modes in the present study, as shown in Figure 2. In mode 1, the crack face is subjected to uniform pressure (or = 1), which corresponds to the case for a crack subjected to uniform tension. In mode 2, the whole crack face is subjected to a pressure loading that varies linearly in the thickness direction. In modes 3 and 4, only a part of the crack face is subjected to pressure loading, which varies linearly in the thickness direction. By interpolation and linear combination of modes 1-4, one can obtain approximations to complex stress situations that vary only in the thickness direction. Modes 5-8 represent similar cases to modes 2-4, but the pressures vary in the crack length direction and are symmetrical about their middle symmetry planes. Through interpolation and linear combination of modes 1 and 5-8, one can also obtain approximations to the cases in which stresses vary only in the crack length direction and symmetrically.

Mode 1

Mode 5

Mode 2

Mode 6-8

Mode 3-4

Figure 2

aly

2c

Elementary stress modes

COMPUTATION MODELS AND P R O C E D U R E S F O R CALCULATING SIFs

Figures 3 and 4 show the computation models in the study. The model in Figure 3 is intended to simulate butt-welded joints, but the influence of weld bead geometry is neglected at this stage. So the model in Figure 3 is indeed a semi-elliptical surface crack in a plate. The model in Figure 4 simulates T-type filletwelded joints. The crack is located at the fillet toe on the main plate. Here, the thickness ratio of the welded plates (T1/T) is considered to be the main parameter having a surface effect on SIF distribution, and only one typical weld bead geometry is selected as elementary. The toe angle is 55 ° in the model, and the weld bead has a step of h/Hf = 0.05 at the toe. Other geometry parameters are as follows for the cases in both Figure 3 and Figure 4: B >-5 , c

B />2 T '

H --=2 B

(2)

The finite element method (FEM) models corresponding to Figures 3 and 4 are shown in Figures 5 and 6 respectively. The meshes were formed by general

i

O'ij = fiapp + fires

K'=O+K Figure 1 Principle of superposition

I

Figure 3

Butt-welded joint

Evaluating SlFs at weld toes in complex stress fields

129

m

T_.,

a

2H

Figure 4 T-type fillet-welded joint

the following equation was used to calculate the apparent K values KAp at the quarter node A ( F i g u r e 7) and K~p at the adjacent node B: Kap=

X/27r E w 4 1-v2"~

(3)

where E and v were chosen as 210 GPa and 0.3 respectively. Then the SIF at the present crack front was calculated by extrapolation12: K = 2K A - KaBp

Figure 5 FEM mesh for butt-welded joint

3-D 20-node isoparametric elements, but 3-D 20node collapsed quarter-point prism singular elements proposed by B a r s o u m 9,1° and Henshell and Shaw 11 were used around the crack fronts (the first layer only) to simulate the r -1/2 singularity of the stress field near the crack tip. The crack face opening displacement method was used to evaluate K values along the crack front. From the opening displacement w of the node on the crack face and the normal distance r from the crack front,

Figure 6 FEM mesh for T-type fillet-welded joint

(4)

For fillet joints, the mean value of K calculated from Equation (4) for the upper and lower crack faces was taken as the SIF at the present crack front. The element sizes used in the FEM meshes have an effect on the results. To obtain accurate numerical results, the mesh parameters were optimized through computation tests for known solutions. The details will be neglected here. SIF RESULTS FOR ELEMENTARY MODELS In this study, a series of geometries of semi-elliptical surface cracks were selected as elementary crack geometries, which contain a series of crack aspect ratios a/c combined with a series of crack depth ratios

Yan-Lin Lu

130 (

¥

/,2A

_ B

\ Figure 9 Figure 7

Surface crack in bending load

Elements surrounding crack tip

a~T. In Tables 1-3, the F E M results for 27 elementary geometries are given (a/c = 0.2, 0.5, 0.8 combined with a/T = 0.2, 0.5, 0.8 for elementary crack geometries, which are further combined with plate thickness ratios T1/T = 0, 0.5, 1.0, where the case T~/ T = 0 corresponds to a butt-welded joint). Table 1 is for butt joints (in flat plates). Tables 2 and 3 are for fillet joints with TI/T = 0.5 and 1.0 respectively. The K results are represented by the engineering magnification factor M in the tables: K

(5)

where q~ = [1 + 1.464 (a/c)"6511/2

(a/c <~1) (6)

and q~ is defined in Figure 8. It has been mentioned above that stress mode 1 corresponds to the case for a semi-elliptical surface crack subjected to uniform tension. Also shown in Table 1 are Newman and Raju's results for this case, which were taken from ref. 2 directly or by interpolation. The agreement with the present results is very good for most geometries. For non-slender cracks (a/c = 0.8 or 0.5 for all a/T), the disagreement for M is - 3 . 4 to 1.0% at all q~ except for those at q~ = 7r/2, where the present result is 3.6-6.5% higher than Newman's. As the material at ~0 = 7r/2 is not subjected to plane strain but to plane stress, Equation (3) will yield higher estimates. However, it has been shown that the singularity of stress here is not r -a/2 yet, so no attempt has been made to correct the results. The agreement for slender cracks (a/c = 0.2 for all a/T) is still very good at positions for ~o < ~r/4, where the disagreement is - 3 . 7 to 3.3%. At the

positions for q~ > rr/4, the disagreement for long but not deep cracks is still not large ( - 8 . 6 to 2.9% for a~ T = 0.2, 0.3-6.8% for a/T = 0.5). Only for long and deep cracks (a/c = 0.2 and a/T = 0.8) and at the positions for q~ > 7r/4 does the disagreement become significantly large (8.6-19.8%). Through linear combination of modes 1 and 2, one can calculate the corresponding M factors subjected to bending load (Figure 9): 2a Mb = M2 • f

+

(2~) M~ • 1 -

(7)

where M~ and Mz corresponds to modes 1 and 2 respectively, and Mb corresponds to bending. The accuracy of the Mb values thus calculated show the same tendency as that in tension, but have a comparatively larger disagreement. The validity of the present results was also verified 2.0 1.8'

1.6 1.4 1.2 -

a/T = 0.25

1.0 0.8

¢

Ir/4

~r/2

1.2 Bending 1.0

(

/ a/T = 0.25 0.8

O

/

{0

{'1

fJ

I'l

('

0.6 0.4 0.2 2c Figure 8

Semi-elliptical surface crack

--~ C

0 Figure 10

I 'rr/4

I ¢

'rr/2

Comparison of present with benchmark problem

Evaluating SlFs at weld toes in complex stress fields

131

Table 1 Engineering magnification factor M for semi-elliptical surface cracks in a butt-welded joint (T1/T = O) Mode 1 uniform a/T

a/c

2~/~"

Present

Newman

0.2

0.2

0.000 0.125 0.250 0.375 0.500 0.625 0.750 0.875 0.9375 1.000

1.170 1.159 1.126 1.075 1.006 0.907 0.772 0.616 0.567 0.564

1.173 1.161 1.128 1.072 0.990 0.882 0.754 0.650

0.000 0.125 0.250 0.375 0.500 0.625 0.750 0.875 0.9375 1.000

1.105 1.097 1.076 1.041 0.994 0.939 0.885 0.853 0.868 0.883

1.124 1.117 1.095 1.059 1.011 0.953 0.892 0.850

0.000 0.125 0.250 0.375 0.500 0.625 0.750 0.875 0.9375 1.000

1.068 1.065 1.057 1.045 1.032 1.021 1.020 1.037 1.066 1.083

1.080 1.077 1.070 1.059 1.046 1.032 1.024 1.032

0.000 0.125 0.250 0.375 0.500 0.625 0.750 0.875 0.9375 1.000

1.537 1.521 1.474 1.406 1.318 1.200 1.046 0.867 0.813 0.836

1.501 1.486 1.439 1.362 1.253 1.123 0.982 0.864

0.000 0.125 0.250 0.375 0.500 0.625 0.750 0.875 0.9375 1.000

1.243 1.237 1.218 1.188 1.146 1.097 1.050 1.031 1.061 1.093

1.243 1.236 1.216 1.183 1.142 1.096 1.053 1.033

0.000 0.125 0.250 0.375 0.500 0.625 0.750 0.875 0.9375 1.000

1.145 1.144 1.140 1.135 1.130 1.131 1.144 1.181 1.225 1.256

1.136 1.136 1.136 1.134 1.132 1.133 1.141 1.170

0.2

0.2

0.5

0.5

0.5

0.5

0.8

0.2

0.5

0.8

0.617

0.842

1.045

0.807

1.041

1.193

Mode 2 t/a = 1.0

Mode 3

Mode 4

Mode 7

Mode 8

t/a = 0.4

Mode 5 b/c = 1,0

Mode 6

t/a = 0.8

b/c = 0.75

b/c = 0.5

b/c = 0.25

0.451 0.459 0.482 0.516 0.554 0.575 0.558 0.501 0.486 0.504

0.348 0.351 0.360 0.382 0.442 0.493 0.505 0.473 0.466 0.489

0.173 0.175 0.179 0.188 0.203 0.223 0.265 0.349 0.379 0.429

1.037 0.902 0.715 0.529 0.366 0.234 0.140 0.084 0.069 0.066

0.993 0.817 0.579 0.350 0.161 0.077 0.051 0.036 0.032 0.032

0.918 0.662 0.331 0.110 0.061 0.040 0.029 0.022 0.020 0.020

0.768 0.310 0.080 0.042 0.026 0.019 0.013 0.010 0.009 0.009

0.385 0.394 0.421 0.461 0.509 0.557 0.605 0.660 0.708 0.755

0.291 0.294 0.304 0.325 0.388 0.463 0.535 0.612 0.669 0.724

0.143 0.144 0.147 0.153 0.164 0.184 0.247 0.418 0.513 0.607

0.928 0.830 0.692 0.550 0.422 0.318 0.243 0.201 0.194 0.187

0.871 0.742 0.567 0.391 0.239 0.160 0.129 0.113 0.111 0.109

0.777 0.592 0.348 0.171 0.119 0.090 0.074 0.066 0.065 0.065

0.614 0.279 0.108 0.072 0.051 0.040 0.033 0.030 0.030 0.030

0.326 0.337 0.372 0.426 0.497 0.581 0.679 0.791 0.861 0.923

0.237 0.241 0.-254 0.282 0.365 0.473 0.595 0.730 0.811 0.884

0.112 0.113 0.118 0.126 0.142 0.171 0.256 0.488 0.615 0.738

0.889 0.803 0.687 0.571 0.467 0.384 0.326 0.296 0.293 0.279

0.831 0.717 0.566 0.417 0.284 0.215 0.194 0.184 0.184 0.176

0.738 0.572 0.355 0.196 0.154 0.130 0.118 0.113 0.113 0.109

0.582 0.272 0.116 0.088 0.070 0.061 0.056 0.054 0.054 0.052

0.691 0.696 0.712 0.737 0.764 0.773 0.744 0.674 0.657 0.694

0.558 0.559 0.562 0.577 0.627 0.668 0.670 0.626 0.618 0.660

0.296 0.297 0.298 0.303 0.313 0.328 0.364 0.443 0.474 0.536

1.283 1.141 0.939 0.737 0.555 0.408 0.298 0.225 0.205 0.212

1.201 1.018 0.765 0.519 0.313 0.214 0.175 0.147 0.138 0.146

1.071 0.809 0.465 0.229 0.166 0.134 0.114 0.097 0.092 0.097

0.852 0.389 0.150 0.103 0.080 0.066 0.056 0.048 0.045 0.048

0.476 0.486 0.515 0.559 0.610 0.663 0.716 0.781 0.840 0.900

0.371 0.374 0.386 0.411 0.478 0.556 0.634 0.719 0.785 0.853

0.190 0.191 0.196 0.203 0.217 0.239 0.306 0.483 0.584 0.686

1.016 0.918 0.781 0.641 0.515 0.413 0.342 0.307 0.309 0.312

0.943 0.815 0.639 0.465 0.315 0.237 0.209 0.198 0.203 0.208

0.829 0.643 0.399 0.222 0.172 0.144 0.130 0.126 0.130 0.134

0.641 0.305 0.134 0.098 0.078 0.067 0.061 0.060 0.062 0.064

0.376 0.389 0.426 0.485 0.563 0.655 0.763 0.889 0.969 1.041

0.282 0.287 0.302 0.335 0.423 0.538 0.669 0.817 0.907 0.989

0.138 0.140 0.146 0.157 0.176 0.210 0.301 0.540 0.673 0.802

0.937 0.851 0.738 0.626 0.528 0.451 0.402 0.383 0.389 0.384

0.870 0.757 0.608 0.461 0.333 0.269 0.255 0.254 0.261 0.260

0.766 0.600 0.384 0.227 0.188 0.168 0.161 0.163 0.167 0.168

0.596 0.286 0.131 0.104 0.087 0.080 0.078 0.079 0.081 0.082

Yan-Lin Lu

132 Table 1 Continued Mode 1 uniform a/T

a/c

2~/~r

Present

Newman

0.8

0.2

0.5

1.852 1.840 1.756 1.709 1.644 1.633 1.584 1.370 1.289 1.426 1.309 1.308 1.304 1.301 1.294 1.278 1,266 1.291 1.363 1.423

1.851 1.846 1.824 1.759 1.657 1.504 1.345 1.217

0.8

0.000 0.125 0.250 0.375 0.500 0.625 0.750 0.875 0.9375 1.000 0.000 0.125 0.250 0.375 0.500 0.625 0.750 0.875 0.9375 1.000

0.8

0.8

0.000 0.125 0.250 0.375 0.500 0.625 0.750 0.875 0,9375 1.000

1.162 1.166 1.175 1.188 1.208 1.235 1.280 1.359 1.443 1.476

1.188 1.191 1.197 1.198 1.218 1.237 1.290 1.357

1.190 1.356 1,355 1.349 1.327 1.309 1.284 1.281 1.295 1.336

1.409

Mode 2 t/a = 1.0

Mode 4' Mode 5 Mode 3' t/a = 0.625 t/a = 0.3125 b/c = 1.0

Mode 6 b/c = 0.75

Mode 7 b/c = 0.5

Mode 8 b/c = 0.25

0.846 0.862 0.873 0.935 0.991 1.086 1.143 1.044 1.003 1.123 0.501 0.520 0.568 0.640 0.720 0.798 0.875 0.971 1.061 1.141

0.527 0.539 0.545 0.592 0.645 0.776 0.896 0.864 0.846 0.958 0.292 0.302 0.330 0.374 0.428 0.526 0.656 0.793 0.894 0.986

0.198 0.209 0.225 0.263 0.300 0.363 0.439 0.521 0.561 0.668 0.108 0.115 0.135 0.162 0.191 0.227 0.279 0.456 0.591 0.715

1.535 1.376 1.122 0.932 0.755 0.663 0.608 0.532 0.504 0.569 1.071 0.974 0.843 0.719 0.614 0.532 0.482 0.470 0.490 0.510

1,433 1.226 0.916 0.680 0.472 0.417 0,421 0.392 0.378 0,429 0.994 0.866 0,693 0,531 0,398 0,335 0,320 0,328 0,347 0,366

1,271 0,974 0,572 0,341 0,276 0,272 0,280 0,262 0.253 0.287 0.872 0,685 0.443 0.274 0.230 0.212 0.208 0.215 0.229 0.243

1.000 0.475 0.199 0.164 0.133 0.133 0.139 0.130 0.124 0.142 0.667 0.336 0.161 0.124 0.109 0.102 0.100 0.104 0.111 0.118

0.379 0.397 0.449 0.527 0.624 0.736 0.865 1.020 1.130 1.202

0.204 0.213 0.238 0.279 0.337 0.454 0.632 0.830 0.956 1.051

0.074 0.080 0.096 0.119 0.147 0.186 0.250 0.472 0.641 0.788

0.954 0.872 0.767 0.667 0.584 0.525 0.495 0.500 0.523 0.525

0,887 0.777 0.635 0.498 0.384 0.333 0.329 0.346 0.367 0.372

0.779 0.617 0.408 0.256 0.219 0.207 0.211 0.224 0.238 0.243

0.602 0.305 0.150 0.117 0.104 0.101 0.103 0.110 0.117 0.120

by comparing the interpolated M factors for the present results with the 'best estimate values' for benchmark problems (for a/T = 0.25, 0.75 and a/c -- 0.5 cracks) in ref. 1. The best-estimate M factors with 3% (for tension) or 0.03 (for bending) confidence bands for benchmark problems are shown in Figure 10 (solid lines). The corresponding interpolated M factors for the present results are plotted in the figure. It can be seen that, except for those at ~ = rr/2, all data for a/T = 0.25 and the data for tension when a~ T = 0.75 fall into the confidence bands. However, the data for bending when a/T = 0.75 fall away from the confidence bands slightly. The largest deviation is about 0.08.

CALCULATION OF SIF IN ACTUAL GEOMETRY AND STRESS FIELDS

Interpolated modes The results in Tables 1-3 are elementary solutions for some elementary geometries and stress modes. They were used to form a database of SIFs for semielliptical surface cracks at weld toes. In practical analyses, the geometry or stress mode may not be the same as that for elementary cases. In this situation, we first calculate approximations for some interpolated modes (or extrapolated modes), and then calculate actual K values through linear combination of these elementary or interpolated modes. Lagrange's interpolation formula or other interpolation formulae can be used to generate interpolated modes. For example,

Figure II Calculation of SIF in a complex stress field

Evaluating SIFs at weld toes in complex stress fields Table 2 Engineering magnification factor M for semi-elliptical surface cracks in a T-type fillet-welded joint Mode 1

133

(TJT

= 0.5)

a/T

a/c

2~/Ir

uniform

Mode 2 t/a = 1.0

t/a

Mode 3 = 0.8

Mode 4 t/a = 0.4

b/c

Mode 5 = 1.0

b/c

Mode 6 = 0.75

b/c

Mode 7 = 0.50

b/c =

0.2

0.2

0.000 0.250 0.500 0.750 1.000

1.067 1.019 0.871 0.612 0.388

0.399 0.424 0.468 0.444 0.361

0.314 0.313 0,366 0.408 0.358

0.153 0.155 0.152 0.192 0.328

0.966 0.656 0.332 0.127 0.105

0.955 c 0,618 c 0.274 c 0.081 c 0.092 c

0.920 d 0.4930 0.107 d 0.052 d 0.063 d

0.791 ~ 0.122 ¢ 0,045 ~ 0.024 ~ 0.039 ~

0.2

0.5

0.000 0.250 0.500 0.750 1.000

1.021 0.990 0.900 0.767 0.738

0.335 0.369 0.449 0.521 0.617

0.254 0.257 0.336 0.463 0.588

0.118 0.122 0.132 0.199 0.486

0.878 0.638 0.377 0.205 0.201

0.831 0.522 0.207 0.109 0.132

0.758 0.314 0.095 0.067 0.080

0.609 0.108 0.058 0,031 0.042

0.2

0.8

0.000 0.250 0.500 0.750 1.000

1.012 0.998 0.960 0,917 0.910

0.291 0.334 0.449 0.605 0.760

0.211 0.219 0.321 0.529 0.723

0.093 0.098 0.118 0.213 0.596

0.854 0.647 0.427 0.279 0.266

0.802 0.532 0.255 0.161 0.178

0.722 0.328 0.129 0.099 0.111

0.561 0.108 0.068 0.045 0.055

0.5

0.2

0.000 0.250 0.500 0.750 1.000

1.423 1.358 1.170 0.870 0,781

0.633 0.650 0.671 0.622 0.637

0.518 0.512 0.547 0.565 0.605

0.272 0.272 0.262 0.291 0.479

1.209 0.874 0,503 0.267 0.308

1.187 c 0.823 ~ 0.432 ~ 0.209 ~ 0.276 ~

1.118 d 0.662 d 0.231 ° 0.149 ° 0.203 °

0.941 * 0.224 e 0.110 ~ 0.075 ~ 0.112 *

0.5

0.5

0.000 0.250 0.500 0.750 1.000

1.155 1.125 1.045 0.934 1.050

0.425 0.460 0.546 0.633 0.835

0.333 0.338 0.421 0.562 0.783

0.165 0.169 0.184 0.258 0.609

0.960 0,722 0.467 0.305 0.377

0.897 0.591 0.281 0.188 0.270

0.801 0.363 0,148 0.120 0.174

0.634 0.135 0.086 0.059 0.092

0.5

0.8

0.000 0.250 0.500 0.750 1.000

1.086 1.076 1.055 1.037 1.156

0.341 0.387 0.513 0.685 0.932

0.255 0.265 0.377 0.600 0.877

0.119 0,125 0.152 0.255 0.692

0.899 0.695 0.486 0.353 0.410

0.839 0.572 0.303 0,221 0,293

0.749 0.356 0.163 0.140 0.190

0.576 0.122 0.086 0.066 0.096

0.8

0.2

0.000 0.250 0.500 0.750 1.000

1.747 1.694 1.508 1.128 1.150

0.854 0.898 0.943 0.853 0.991

0.564" 0.595" 0.629 ~ 0.674 a 0.873 ~

0.229 ~ 0.259 b 0.301 b 0.306 ~ 0.575 b

1.470 1.114 0.731 0.466 0.612

1,440 c 1.052 c 0.646 ~ 0.397 ~ 0.565 ~

1.347 ° 0.8560 0.400 ° 0.298 ° 0.4390

1.107 ~ 0.331 e 0.199 ~ 0.160 ~ 0,249 ~

0.8

0.5

0.000 0.250 0.500 0.750 1.000

1.228 1.201 1.120 1.035 1.359

0.487 0.534 0.628 0.731 1.087

0.292" 0.318" 0.369" 0.549 a 0.933 ~

0.109 ~ 0.128 ~ 0.164 b 0.218 b 0.643 b

1.031 0.794 0.544 0.410 0.599

0.963 0.656 0,350 0.275 0.447

0.855 0.417 0.203 0.183 0.301

0.687 0.184 0.124 0.106 0.172

0.8

0.8

0.000 0.250 0.500 0.750 1.000

1.106 1.099 1.090 1.113 1.407

0.370 0.424 0.562 0.756 1.131

0.202" 0.229 ~ 0.297 ~ 0.549 ~ 0.979"

0.073 b 0.087 b 0.128 b 0.204 b 0.705 ~

0.927 0.729 0.534 0,434 0,587

0.867 0.606 0.350 0.290 0.433

0.774 0,391 0.202 0.188 0.286

0.618 0.161 0.112 0.095 0.146

aMode 3'

(t/a =

0.625); bMode 4'

(t/a =

0.3125); ~Mode 6'

(b/c =

using Lagrange's formula for two parameters x and y (x and y may be T1/T, a/c, a/T, t/a, b/c or ~¢), the approximate M factor can be calculated by

M = ~

fi(X--Xk]fi(Y--Ylt k=#i

0.91); dMode 7'

where M~j is the M factor for a mode with parameter x~ and Yi. Calculation o f S I F in c o m p l e x stress fields

The stress field at the weld toe is usually complex, owing to both stress concentration and residual stress. So the calculation of SIF in an actual structure is usually concerned with a complex stress field. We can calculate the SIF approximations in these complex stress fields through linear combination of the above-

0.69); and eMode 8'

(b/c =

0.37), respectively

mentioned elementary modes or interpolated modes. Figure 11 illustrates the superposition method to calculate the SIF in a complex stress field, which varies in the thickness direction. Suppose that the stress distribution along the thickness direction can be represented by values at several depths ti:

l~j

(8)

(b/c =

Mode 8 0.25

cr

----

tr(ti)

(i = O, 1, 2, ..., n -- 1, n, ...)

(9)

and the crack depth a of the semi-elliptical surface crack is between t,-1 and t,: t,-1 < a ~< t,, (10) Then through linear combination of mode 1, mode 2 and the interpolated mode for x = t/a = ti/a (i = 1, 2 . . . . . n - 1), the K value can be calculated from K = S, M1 + (S,,_~ - S,)M2 n--I

-

(sii=l

S,_,)M;

(11)

Yan-Lin Lu

134

Table 3 Engineering magnification factor M for semi-elliptical surface cracks in a T-type fillet-welded joint ( T J T = 1.0)

a/T

a/c

2~/7r

Mode 1

uniform

Mode 2 t/a = 1,0

Mode 3 t/a = 0.8

t/a = 0.4

Mode 4

Mode 5 b/c = 1.0

b/c = 0.75

Mode 6

Mode 7 b/c = 0.50

Mode 8 b/c = 0.25

0.2

0.2

0,000 0.250 0.500 0.750 1.000

1.020 0,975 0.836 0.580 0.345

0.368 0,394 0.444 0.422 0,328

0.287 0.287 0,345 0.389 0,327

0.138 0.140 0.139 0.178 0,305

0.938 0.630 0.310 0.110 0.085

0.929 c 0.593 c 0.253 c 0.065 c 0.075 ~

0.900 d 0.472 d 0.090 d 0.041 d 0,050 d

0.806 e 0,114 e 0.035 e 0.019 e 0.030 ~

0,2

0.5

0.000 0.250 0.500 0.750 1,000

1.000 0.968 0.875 0.736 0.673

0.322 0.356 0.433 0,500 0.566

0.242 0,246 0.321 0.444 0.540

0.112 0.115 0.123 0.187 0.451

0.865 0.625 0.362 0.189 0.177

0.821 0,512 0.195 0.097 0.114

0.751 0.307 0.086 0,059 0.068

0.603 0.106 0.053 0.027 0,036

0.2

0.8

0.000 0,250 0.500 0.750 1.000

0.998 0,983 0.940 0.897 0.855

0,282 0.325 0.437 0.591 0.717

0.203 0.211 0.310 0.517 0.683

0,089 0.094 0.112 0,207 0.566

0.843 0.637 0.416 0.267 0.245

0,793 0.524 0.246 0.153 0.163

0.715 0.323 0,123 0,094 0.101

0.555 0.106 0,065 0.042 0,050

0.5

0.2

0.000 0.250 0,500 0.750 1.000

1.314 1.256 1,090 0.821 0.700

0.559 0.579 0.614 0,586 0.577

0.453 0.449 0.495 0.534 0.550

0.234 0,235 0.228 0.268 0.447

1.128 0,807 0,463 0.245 0.280

1.109 ~ 0.759 ¢ 0.396 ~ 0.190 ~ 0.254 ~

1.048 d 0.609 d 0,205 d 0.137 a 0.191 d

0.894 c 0.194 e 0.100 c 0.070 ~ 0.108 ~

0.5

0.5

0.000 0,250 0.500 0,750 1.000

1.107 1.078 0,998 0.878 0.941

0,391 0.427 0.514 0.596 0.758

0,302 0.308 0.392 0.529 0.713

0.146 0.151 0.165 0.238 0.564

0.932 0.693 0.436 0.271 0.318

0,875 0.566 0.255 0.161 0.224

0.788 0.344 0.128 0,103 0.143

0.626 0,124 0.076 0,050 0.076

0.5

0.8

0,000 0.250 0.500 0.750 1,000

1.064 1.052 1.026 1.000 1,071

0:324 0,369 0,493 0,661 0.872

0.239 0.249 0.359 0.578 0,823

0.109 0.116 0.140 0.243 0.658

0,886 0.681 0,467 0.329 0.360

0.829 0.560 0.287 0,201 0.253

0.742 0.347 0.151 0.127 0.163

0.571 0.118 0,080 0.059 0,082

0.8

0.2

0.000 0.250 0.500 0.750 1.000

1.623 1,547 1.346 0.960 1.127

0.765 0,794 0.826 0.729 0.981

0.497 ~ 0.516 ~ 0,535 ~ 0.573 ~ 0.869 ~

0.199 b 0.223 ° 0.251 b 0.245 b 0,580 b

1,381 1,019 0.647 0,393 0.600

1,354 c 0.963 c 0.571 c 0.333" 0.557"

1.270 J 0.781 d 0.346 d 0.252 d 0.439 d

1,050 c 0.293 ~ 0,176 ~ 0.133 c 0.251 ~

0.8

0.5

0.000 0,250 0.500 0.750 1.000

1.192 1,149 1.056 0.935 1.166

0.457 0.494 0.580 0.661 0.947

0,269 ~ 0.286 ~ 0.329 ~ 0.494 ~ 0.823 ~

0,100 h 0.114 b 0,143 h 0,186 h 0,581 b

1.011 0.762 0.505 0,350 0.495

0.948 0.630 0.317 0,228 0.370

0.848 0,397 0.178 0.156 0,251

0.677 0.171 0.116 0.092 0.149

0.8

0.8

0.000 0.250 0.500 0.750 1.000

t.092 1.076 1.056 1.043 1.250

0.356 0,404 0.535 0.708 1.017

0.19P 0.213 a 0.274 a 0.512 a 0.890 ~

0,069 b 0,081 ° 0,115 h 0A83 ° 0.655 b

0.923 0.716 0.511 0,388 0.495

0.866 0.595 0.329 0.252 0,362

0,779 0.381 0.184 0,165 0.238

0.619 0.155 0,104 0.082 0.122

aMode 3' (t/a = 0.625); bMode 4' (t/a = 0.3125); cMode 6' (b/c = 0.91); dMode 7' (b/c = 0.69); and eMode 8' (b/c = 0.37), respectively

where M1 and M2 correspond to elementary modes 1 and mode 2, M~ corresponds to the interpolated mode for x = t/a = ti/a, and Si can be determined from So = ¢r ( 0 )

Si = cr(ti) +

t~[o(ti) ti+t - ti

(i = 1, 2, . . . , n - 1)

- o(ti+l)] (12)

When stress varies in the crack length direction and symmetrically, a similar method can be used based on mode 1 and modes 5-8. Consideration o f influence o f weld bead geometry In actual calculation, the weld bead geometry may be different from those defined in the elementary

geometry models. For example, the weld bead has a certain height and toe angle in butt-welded joints, but the toe angle may not be 55 ° in fillet-welded joints. Change in these parameters may cause two effects: a change in the stress concentration, and a change in the surface effect on the SIF distribution. In this study, we take only the former effect into account and neglect the latter effect. Thus, when calculating, we can first determine the stress distribution at the crack position when no crack exists in the joint, and then introduce the crack and calculate the SIF of the crack subjected to the corresponding crack face pressure loading from the above-mentioned method through the database. The accuracy of the method can be increased by selecting more elementary geometries and stress modes than those in the present paper. The great advantage

Evaluating SlFs at weld toes in complex stress fields of the present method is its simplicity and speed in determining of SIF distributions. The work can be done by a small program or even by hand. In fatigue analysis, the crack pattern development way must be determined by a series of calculations of SIF distributions for different crack geometries: hence this advantage makes it acceptable to use in fatigue prediction. CONCLUSION A series of FEM solutions has been given for SIF distributions along crack fronts of 27 elementary geometries of semi-elliptical surface cracks at weld toes subjected to eight types of elementary stress mode. Through interpolation and linear combination of these elementary solutions, one can calculate SIF distributions for general geometries of butt- and filletwelded joints subjected to complex stress fields, which may be caused by service loading or residual stress. The procedure is very simple and fast. The surface effect of the thickness ratio of the welded plates can be taken into account. The method is especially suitable for engineering fatigue analysis. REFERENCES

5 6 7 8 9 10

Benchmark Editorial Committee of the SESA Fracture Committee, Exp. Mech. 1980, 20, Aug, 253 Raju, I.S. and Newman, J.C. Jr Eng. Fract. Mech. 1979, 11,817 Newman, J.C. Jr and Raju, I.S. Analysis of surface cracks in finite plates under tension or bending loads, NASA TP1578, 1979 Newman, J.C. Jr and Raju, I.S. Eng. Fract. Mech. 1981, 15, 185 Wu, X.R. Eng. Fract. Mech. 1984, 19, 387 Wu, X.R. and Carlsson, J. Eng. Fract. Mech. 1984, 19, 407 Niu, X. and Glinka, G. Eng. Fract. Mech. 1990, 36, 459 Lu, Y.L. Int. J. Fatigue 1995, 17, 000 Barsoum, R.S. Int. J. Numer. Methods Eng. 1976, 10, 25 Barsoum, R.S. Int. J. Numer Methods Eng. 1977, 10, 85

11 12

135

Henshell, R.D. and Shaw, K.G. Int. J. Numer. Methods Eng. 1975, 9, 495 Barsoum, R.S. Int. J. Numer. Methods Eng. 1977, 11,402

NOMENCLATURE a Crack depth of a semi-elliptical surface crack b Characteristic dimension in crack length direction in elementary stress modes C Material constant in Paris's equation c Half-crack length of a semi-elliptical surface crack da/dN Crack growth rate E Young's modulus K Calculated stress intensity factor AK Stress intensity factor range gap Apparent K value calculated from nodal displacement M Engineering magnification factor m Material exponent in Paris's equation r Normal distance from crack front Si Ordinate for stress T Thickness of the main plate of a welded joint t Characteristic dimension in thickness direction in elementary stress modes T1 Thickness of the branch plate of a Ttype fillet-welded joint ti Abscissa in thickness direction w Opening displacement of node on crack face Greek letters v Poisson ratio or Characteristic stress o-ij Stress distribution at the position of crack in an uncracked body ~0 Angular parameter for position on semiellipse