Analysis of weld toe profiles and weld toe cracks

Int d Fatigue 15 No 1 (1993) pp 31-36

Analysis of weld toe profiles and weld toe cracks H.L.J. Pang Measurement of actual weld toe profiles showed that a wide combination of weld toe angles and toe radii are possible. Experimental and finite element analysis of stress concentration and stress distribution was carried out for a particular cruciform welded joint. Closed-form stress pattern equations were compared from different sources. Stress intensity factor calculations using closed-form integration and weight function methods were carried out for weld toe cracks. Key words: fillet weld; toe cracks; weight function

Nomenclature a

da dur dx E H'

KI K~ K~ m(x,a) Mk

Mk(,)

Crack depth of edge crack or weld toe crack Incremental change in crack depth Incremental change in crack opening displacement field Incremental change of stress on the crack plane Elastic modulus H' = E for plane stress and H ' = E/(1 - v 2) for plane strain conditions Mode I stress intensity factor Reference stress intensity factor Geometric stress concentration factor Weight function for edge crack Magnification factor for weld toe crack defined in Equation (9) Mk calculated by closed-form integration method, Equation (4)

The weld toe profile in a full penetration cruciform welded joint can be described by the toe radius p and fillet flank angle 0 as shown in Fig. 1. The geometry of the weld toe profile may be used to determine the elastic stress distribution and stress concentration factor in uncracked fillet welds. However, study of welding defects in fillet-welded joints I has shown that crack-like flaws were found at the weld toes. These crack-like flaws readily contribute to fatigue crack initiation. Therefore the remaining fatigue process is then taken up by crack propagation starting from such flaws. In fatigue crack growth analysis using linear elastic fracture mechanics it is necessary to calculate stress intensity factors for a wide range of weld toe crack depths. Solutions for weld toe cracks are not readily available and finite element techniques are often used. Some finite element studies have modelled the weld toe profile as a sharp pointed fillet notch without a crack and have used the elastic stress distribution along the anticipated crack path to calculate the stress intensity factor. 2'3 Other

Mk(,,) P S

$1,$2 S3,S4,Ss

T

Mk calculated by weight function method, Equation (6) Point opening force, Equation (4) Applied stress normal to crack plane Normalized stress perpendicular to crack plane, Equations (1), (2) Normalized stress perpendicular to crack plane, Equation (3) Plate thickness of fillet welded joint

Greek symbols

/3./32,/33 p

,,(x) Oryy

0

Constants in weight function, Equation (8) Radius of weld toe profile (see Fig. 1) Distributed stress function applied in weight function Finite element stress component normal to crack plane Fillet flank angle of weld toe profile (see Fig. 1) Poisson's ratio

/L

0

-~ . . . . . .

IIIII

t .....

a

II

b

Fig. 1 Cruciform welded joint section

0142-1123/93/010031-06 © 1993 Butterworth-Heinemann Ltd Int J Fatigue January 1993

31

studies have examined a range of weld toe radii and angles 4-6 and have shown that the stress concentration factor (for an uncracked fillet weld) and the stress intensity factor (for a cracked fillet weld) are sensitive to changes in the weld toe radius and angle. These studies are based on hypothetical geometries of weld toe radii and angles and may not be representative of the as-welded toe profile. Therefore this study deals, first, with measurement of weld toe profiles in a cruciform welded joint. 7 9 Experimental and finite element stress analyses results are discussed and proposals for closed-form stress distributions are made. Analytical techniques were used to calculate the stress intensity factor. The solution is provided in a non-dimensionalized form called the stress concentration magnification factor, Mk. Weld

toe

profile

measurements

A typical cruciform welded joint section is shown in Fig. l(a). Several slices were cut out from a specimen by spark erosion machining and polished. The definition of the weld toe profile is indicated in Fig. l(b) and measurements of the weld toe radius (p) and fillet flank angle (0) are given in Table 1. The definition of p and 0 describes the local shape of the weld bead deposited at the weld toe. The sharpest toe profile was given by p -- 0.127 mm and 0 = 115° while the bluntest case is given by p = 3.75 mm and 0 = 50 °. It is interesting to consider the implications of the sharpest and bluntest toe profile in terms of stress concentration factors. The finite element results by Engesvik and Larsen 6 provided graphical solutions of stress concentration factors for a wide range of toe radius and angle combinations. Using their solution, the stress concentration factors for the bluntest and sharpest weld toe profiles were 2.2 and 12.0 respectively. It can be seen that the toe profile depends significantly on the local shape of the weld bead deposited and this is subjected to wide variations.

T a b l e 1. F i l l e t w e l d t o e g e o m e t r y

Measurement no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

measurements

Fillet weld toe radius p (mm)

Fillet weld toe angle 0 (deg)

Ratio TIp (T=26 mm)

0.127 0.127 0.127 0,127 0.190 0.254 0.254 0.390 0.390 0.390 0.390 0.600 0.600 0.600 0.600 0.600 0.635 0.790 0.790 0.790 1.580 1.580 2.380 2.380 2.380 3.750

70 85 105 115 105 60 70 40 45 50 55 45 45 50 60 60 50 35 55 70 40 60 50 6O 70 50

197 197 197 197 132 98 98 64 6~ 64 64 42 42 42 42 42 39 32 32 32 16 16 10 10 10 7

A closer examination of the weld toe profiles given in Table 1 indicates that the overall fillet flank angle of 0 = 45 ° is a realistic value to use for the weld toe angle. This, combined with the smallest weld toe radius of p -- 0. ! mm, will give the worst-case fillet notch profile. Using the graphical solution from Engesvik and Larsen 6 the stress concentration factor for p = 0.1 mm and 0 = 45 ° is approximately equal to 5.0. Experimental

stress

analysis

Measurement of the stress concentration factor in a cruciform welded joint was made using a strip of five strain gauges as shown in Fig. 2. The centre of the first strain gauge was located at a distance of 2 mm from the weld toe and a stress concentration of 1.57 was recorded for this gauge. The remaining strain gauges in the strip were spaced at 2 mm intervals and the stress concentration measured reduced to 1.33, 1.22, 1.12 and 1.05 respectively. Linear extrapolation, using the first and second strain gauge result, to the weld toe gave a stress concentration factor of Kt = 1.85. This is much lower than the projected value based on weld toe profiles as discussed in the previous section. It should be noted that strain-gauge measurements have an averaging effect over the gauge length and may not be sensitive enough to pick up steep stress gradients, especially at the weld toe. Linear extrapolation also underestimates the stress concentration factor because the stress distribution approaching the weld toe is not linear. This is the method used for calculating hot-spot stresses in tubular welded joints, and fatigue endurance curves and assessments using experimental data for stress concentration factors or hot-spot stresses can lead to non-conservative results. The presence of crack-like flaws at the toes of filletwelded joints makes it necessary to examine the problem in terms of fatigue crack growth. Short weld toe cracks grow on a plane normal to the maximum tensile principal stress and for this fillet geometry it was noted 9 that this is slightly skew from the idealized crack path. However, the difference in the results is unlikely to be significant because the skew angle is small, and as in other studies, 3-5 an idealized crack path was adopted. Therefore it is important to know the stress distribution into the thickness direction of the welded joint along the anticipated crack propagation path. The following sections will deal with this issue and methods of calculating the stress intensity factor solution for weld toe cracks. Finite

element

stress

analysis

Finite element stress analysis of a full-penetration fillet-welded joint with a sharp pointed weld toe p = 0 and a fillet flank

Fig. 2 Strain gauge measurement of stress concentration factor

32

Int J F a t i g u e J a n u a r y 1 9 9 3

plastic analysis was carried out to determine the extend of notch plastic zone sizes for applied stresses of 50, 100, 150, 200 and 250 N mm -2. For a material yield stress of 383 N mm -2, the applied stress-to-yield-stress ratios are between 0.13 and 0.65. The corresponding notch plastic zone sizes in the thickness direction were 0.022, 0.087, 0.196, 0.347 and 0.543 mm respectively. Based on this observation it can be seen that linear elastic stress analysis conditions can be used at sharp fillet notches provided the applied stress-toyield-stress ratio is low. For fillet-welded joints, the fatigue limit strength for a class F joint s is 46 N mm -2. Therefore linear elastic stress analysis and fracture mechanics concepts can be applied in fatigue studies.

TT

Closed-form stress distributions For linear elastic conditions, the stress distribution can be expressed in closed-form equations and this information can be used to calculate the stress intensity factor solution for weld toe cracks. For a fillet notch with a toe radius, the maximum stress at the weld toe will be finite and the stress distribution depends on the size of the weld toe radius. The finite element result by Verreman n is appropriate for this study because it models a small fillet weld toe radius of p = 0.05mm and flank angle of 0 = 45 °. The stress distribution equation is given by

Fig. 3 Finite e l e m e n t m o d e l of cruciform w e l d e d joint

angle of 0 = 45 ° was modelled as shown in Fig. 3. The finite element mesh is for one-quarter of the cruciform welded joint making use of the symmetry boundaries and consisted of 254 plane strain quadratic elements. The elastic stress distribution along the anticipated crack path from the weld toe is shown in Fig. 4. A stress singularity was observed ahead of the sharp fillet notch and this was curve-fitted to a power law expression: ~Y- = 0 . 5 ( x / T ) - ° 3

S

for

0.1

x/T<

(1)

The coefficient of the stress singularity is - 0 . 3 and this is close to the theoretical value of -0.333 for an angular notch with the same included angle. ~° Using the same finite element mesh an elastic-perfectly

S ,~..~

/Oyy

.o\

E" >,', i ¢ 1 1 , 1 r l

~

3

E

2

o Z

Symmetry

i

-0.333 A0 + Al

+

A2

S + A3

+ A4

(2)

where A0 = 0.414, A1 = 0.815, A2 - -1.71, A3 = 4.16, A4 = -3.7. This result is comparable to Equation (1) but is valid for a wide range of x / T values up to 0.5. For larger weld toe radii, the finite element results by Lawrence et al 4 showed that the stress distribution can be expressed as a function of the stress concentration factor, Kt:

(3, Equation (3) was calculated for three weld toe radii values of P = 0.1, 0.2 and 0.76 mm and the corresponding stress concentration factors calculated from Lawrence et aP were Kt = 4.92, 3.92 and 2.92 respectively. These were chosen to give a sample of the measured toe radii data shown in Table 1. The stress distributions for Equations (1)-(3) were compared over the range of normalized depths of 0.001 < x / T < 0.1. This is shown in Fig. 5 and Table 2. The notation $1 is for Equation (1), $2 is for Equation (2), $3 to Ss is for Equation (3) with three different weld toe radii. The result for Equation (2) was selected to represent the worst case stress distribution along the anticipated crack path. Stress intensity factors can be calculated using this stress distribution.

Stress intensity factor calculations

0

I 0.02

I 0.06

Normalized depth,

I 0.10

x/T

Fig. 4 Normalized stress variation with depth

Int J Fatigue January 1993

The principle of elastic superposition can be used to determine the stress intensity factor for a range of weld toe crack sizes using the stress distributions discussed earlier. A closed-form integration method and a weight function technique were used. The closed-form integration method employs the stress intensity factor solution for a single edge crack in a finite

33

I,-0-:t;j

where P is the discrete splitting force, x/a is the normalized distance from the crack mouth along the crack face and a/T is the normalized crack depth. For the same crack subjected to a distributed stress on the crack face, Equation (4) can be used by substituting a continuous series of splitting forces described by the distributed stress:

4 L/3 D

"O

kA/ \,T/

KI=

E

O Z

I 0.02

I 0.0q

I

I

I

0.06

0.08

0.1

0.12

Normalized depth, x/T Fig. 5 Normalized stress distributions for different toe radii. Stress identity and toe radius: 0, Sl, 0 mm; +, S2, 0.05 mm; x, S3, 0.1 mm; FI, S4, 0.2 mm; *, Ss, 0.76 mm Table 2. N o r m a l i z e d stress distributions

Normalized stress

Normalized

0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080 0.090 0.10

(5)

The resulting stress intensity factor can be calculated by numerical integration using a six-point Gauss-Legendre method. The weight function method by Petroski and Achenbach 13 for a single edge cracked plate (SECP) geometry was used in this study. This is an extension of the weight function method proposed by Bueckner. 14 The weight function m(x,a) for a mode I crack is a unique property of geometry and it enables stress intensity factors to be calculated for any stress distribution applied to that cracked body. Therefore for the SECP geometry subject to a distributed stress, the stress intensity factor can be calculated from a weight function by

K, = f] o'(x) m(x,a) dx

depth

x/T

(,,

Sl

S2

S3

S4

S5

3.972 3.226 2.856 2.620 2.451 2.320 2.215 2.128 2.050 1.950 1.617 1.431 1.313 1.228 1.162 1.110 1.066 1.020 1.0

4.138 3.292 2.881 2.623 2.440 2.301 2,109 2.098 2.022 1.955 1.581 1.404 1.297 1.223 1.168 1.124 1.090 1.061 1.038

4.417 3.979 3.597 3.264 2.974 2.721 2.500 2.308 2.140 1.994 1.219 1.056 1.014 1.004 1.0 1.0 1.0 1.0 1.0

3.636 3.380 3.149 2.940 2.752 2.582 2.430 2.289 2.164 2.051 1.341 1.123 1.044 1.016 1.006 1.0 1.0 1.0 1.0

2.795 2.678 2.569 2.467 2.372 2.283 2.200 2.120 2.048 1.981 1.468 1.239 1.122 1.062 1.032 1.016 1.0 1.0 1.0

In order to derive a solution for the weight function m(x,a), a reference solution Kr for a SECP loaded in uniform tension is used (see Fig. 6(a)). Substituting the reference stress intensity factor and an approximate crack opening displacement function ur into Equation (6) gives the weight function

rn(x,a) = ~

3.52 1 -

1

rn(x,a)=

(7)

[a

[

x~½

[a

where 131, /32, /33 are constants depending on a/T and are given in Table 3.

["

y

t

t

Ur

4.35 - 5.28

t o (x) Kr ~--~/

a a ( 1 - ~)1/2 /x W z

Knew :- x

_7.2 ~l

~1"30 - 0'3Ia )

x~

x~l

/a

a'T

F ( x a ) _ _-_ _ a , T a ( 1 - ~)3/2

da

where H' = E for plane stress and E / ( 1 - 1.,2)for plane strain conditions. Following through the derivations as described by Wu, ts the resulting weight function is given by

plate loaded by a pair of splitting forces on the crack face. This was taken from the handbook by Tada et al, 12 and the stress intensity factor solution is given by: KI = 2 ~ F

(6)

[

T

-i

,

r

(x)]

a

b

Fig. 6 Weight function method for SECP subjected to distributed stress: (a) reference solution; (b) stress at fillet weld toe

34

Int J Fatigue January 1993

Table 3. W e i g h t function m e t h o d coefficients a/ T

0,001 0,005 0.01 0,05 0,10 0,20 0.30 0.40 0.50

F(a/7)

(31

~2

~3

1.12 1.12 1.12 1.132 1.184 1.373 1.665 2.113 2.842

0.0708 0,1583 0.2237 0.5065 0.7490 1.2281 1.8243 2.6733 4.0199

47.04 20.96 14.80 7.20 6.60 8.81 13.59 22.67 40.15

7903.13 704.31 247.62 20.93 8.01 7.15 10.61 18.07 39.59

Using the principle of elastic superposition, the distributed stress at the fillet weld toe can be applied as shown in Fig. 6(b). The stress intensity factor for this distributed stress can be calculated using the unique property of the weight function m(x,a) and solving the integral in Equation (6). The same six-point Gauss-Legendre numerical integration method was used as before. The resulting stress intensity factor solution for a weld toe crack was normalized with that for a corresponding SECP case to factor out the stress concentration effect of the fillet weld on an edge crack. This is called the stress concentration magnification factor, Mk: Mk = Kx (. . . . . . . k) KI (SECP)

for crack depths less than a / T = 0.2 but underestimates the Mk factors for larger a / T values. For a / T -- 0.5 the difference was - 10.2%. The weight function method gave good agreement for all a / T ratios investigated with the largest difference of - 5 . 4 % . The reference solution by Gurney et aP was not as accurate for small a / T ratios; particularly at a / T =.0.001 the Mk factor was out by - 9 % . Conclusions

1)

2)

3)

Measurement of weld toe profiles shows a wide combination of weld toe radii and angles (P and 8). Estimated elastic stress concentration factors for the bluntest and sharpest weld toe profile are Kt -- 2.2 and 12.0 respectively. However, a more realistic value of Kt = 5.0 was proposed for a toe radius of p = 0.1 mm and fillet flank angle of 0 = 45 °. Experimental measurement of the stress concentration factor using strain gauges gave an extrapolated value of Kt = 1.85 at the weld toe. This is low when compared to the proposed value. Closed-form integration and weight function methods were used to calculate stress intensity factors for weld toe cracks and satisfactory comparisons were made with finite element results. Both methods underestimate the Mk factors by up to - 10.2% and - 5.4% respectively for crack sizes in the range of 0.001 < a / T < 0.5.

(9) References

Therefore a distribution of Mk values for a range of crack depths can be readily calculated. This was carried out using the stress distribution given by Equation (2) for both the closed-form integration and weight function methods discussed earlier. The calculated results are given in Table 4. The closedform integration method is given as Mk(,), while the weight function method is given as Mk(ii). Two reference cases were included for comparison. The first case by Gurney and Johnston 3 employs a simple superposition technique for a centre crack with a pair of splitting forces. This is often called the Albrecht and Yamada method, after the authors who used it. 2 The second case is a finite element study by Smith and Hurworth 16 where cubic isoparametric elements were used in modelling the correct crack tip stress singularity. This solution was expected to be more accurate and was therefore adopted as a benchmark for comparison. The closed-form integration method gives good results

1. 2. 3.

4. 5. 6.

Smith, I.F.C. and Smith, R.A. 'Defects and crack shape development in fillet welded joints' Fatigue Eng Mater Struct 5 2 (1982) pp 151-165 Albrecht, P. end Yamada, K. 'Rapid calculation of stress intensity factor' J Struct Div Proc Am Soc Civ Eng 103 ST2 (1977) pp 377-389 Gurney, T.R. and Johnston G.O. 'A revised analysis of the influence of toe defects on the fatigue strength of transverse non-load carrying fillet welds' Weld Res /nt 19 3 (1979) pp 43-80 Lawrence,R.V., Ho, N.-J. and Mazumadar, P.K. 'Predicting the fatigue resistance of welds' Ann Rev Mater Sci 11 (1981) pp 401-425 Niu, X. and Glinka, G. 'The weld profile effect on stress intensity factors in weldments'/ntJ Fract35 1 (1987) pp 3-20 Engesvik,K. and Larsen, T. 'The effect of weld geometry on fatigue life' Proc Seventh Int Conf on Offshore Mechanics and Arctic Engineering, Houston, 1988 3

Table 4. Stress concentration magnification factor a/T

0.001 0,005 0.01 0.05 0.1 0.2 0.3 0.4 0.5

Mk(i)

% Diff

Mk(ii)

% Diff

Mk(Rof Z)

% Diff

Mk(Ref 16)

5.412 3.181 2.034 1.545 1.273 1.062 0,958 0.898 0.866

2.0% 1.6% 2.1% 0.3% 2.4% -5.3% -8.4% -9.9% - 10.2%

5.201 3.117 2.434 1.511 1.232 1.064 1.024 1.004 0.994

2.0% 1.3% - 5.4% -2.0% - 5.3% -5.0% -2.0% 1.0% 3.0%

4.890 2.961 2.388 1.546 1.301 1.107 1.015 1.0 1.0

-9.0% -7.0% -4.0% 0.5% 0.5% -1.3% -3.0% 0.5% 0.5%

5.304 3.130 2.488 1.540 1.297 1.122 1.046 0,997 0.964

Int J Fatigue January 1993

35

(American Society of Mechanical Engineers, 1988) pp 441-446 7.

Pang, H.L.J. 'A literature review of stress intensity factor solutions for a weld toe crack in a fillet welded joint' Internal NEL report DE/5/88 (National Engineering Laboratory, East Kilbride, Glasgow, 1988)

8.

Pook, L.P. 'Fatigue crack growth in cruciform-welded joints under non-stationary narrow-band random loading' Residual Stress Effects in Fatigue, ASTM STP 776 (American Society for Testing and Materials, 1982) pp 97-114

9.

Pang, H.L.J. 'Fracture mechanics analysis of fatigue failure in cruciform welded joints' PhD Thesis (University of Strathclyde, 1989)

lg.

Williams, M.L. 'Stress singularities resulting from various boundary conditions in angular corners of plates in extension' J Appl Mech 19 (1952) pp 526-528

11.

Verreman, Y., Bailon, J.-P. and Masounave, J. 'Fatigue life prediction of welded jointsman assessment' Fatigue Fract Eng Mater Struct 10 1 (1987) pp 17-36

12.

Tada, H., Paris, P.C. and Irwin, G.R. Stress Analysis of

36

Cracks Handbook (Del Research Corporation, Hellertown, PA, 1973)

13.

14. 15.

16.

Petroski, H.J. and Achenbach, J.D. 'Computation of the weight function from a stress intensity factor' Eng Fract Mech 10 (1978) pp 257-266 Bueckner, H. 'A novel principle for the computation of stress intensity factors' ZAMM (1970) 50 pp 529-546 Wu, X.R. 'Approximate weight functions for centre and edge cracks in finite bodies' Eng Fract Mech 20 1 (1984) pp 35-49 Smith, LJ. and Hurworth, S.J. 'The effect of geometry changes upon the predicted fatigue strength of welded joints' Research Report 244/1984 (The Welding Institute, Cambridge, UK, July 1984)

Author The author is with the School of Mechanical and Production Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 2263. Received 20 January 1992; accepted in revised form 10 August 1992.

Int J Fatigue J a n u a r y 1993