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Notch stress intensity factors and fatigue strength of aluminium and steel welded joints
International Journal of Fatigue 23 (2001) 225–232 www.elsevier.com/locate/ijfatigue
Notch stress intensity factors and fatigue strength of aluminium and steel welded joints P. Lazzarin a
a,*
, P. Livieri
b
Department of Management and Engineering, University of Padova, Stradella S. Nicola 3, 36100 Vicenza, Italy b Department of Engineering, University of Ferrara, Via Saragat 1, 44100 Ferrara, Italy Received 12 June 2000; received in revised form 21 September 2000; accepted 21 September 2000
Abstract According to a recent and appropriate definition, stress field parameters, namely notch stress intensity factors (N-SIFs), can be used to predict the fatigue behaviour of mechanical components weakened by V-shaped re-entrant corners, where the singularity in the stress distribution makes any failure criterion based on elastic peak stress no longer applicable. Commonly thought of as parameters able to control the fatigue crack initiation life, N-SIFs are, under certain circumstances, also useful for predicting the component total fatigue life. The fatigue strength of aluminium welded joints with different geometries and thicknesses are summarised in a single scatter band by using an N-SIF-based approach. The statistical analysis is carried out taking into account experimental data already reported in the literature, referring to welded joints with a thickness ranging from 3 to 24 mm. Results of steel and aluminium welded joints are then compared: at high number fatigue life, the relative fatigue strength is slightly greater than 2, in agreement with the value previously reported in the literature for butt spliced bolted joints. The value of the theoretical exponent quantifying the scale effect (0.326 against 0.25 suggested by Eurocodes) is discussed. 2001 Published by Elsevier Science Ltd. Keywords: Fatigue; Welded joints; Notch stress intensity factors
1. Introduction Williams [1] was able to demonstrate that in the context of the elasticity theory, the asymptotic stress state near a re-entrant corner is singular and its degree of singularity is a function of the only notch opening angle. The stress field intensity depends on the overall geometry of the component and the far-field loading. In the context of a stress field theory, a field parameter called the “notch stress intensity factor” (hereafter NSIF) was explicitly defined by Nui et al. [2] and applied to the fracture toughness of brittle materials. Afterwards, the N-SIF concept was used by Boukharouba et al. and by Verreman and Nie for fatigue crack initiation estimates at notches [3] and weld toes [3,4]. Recently, an analytical background able to quantify different stress components, such as the influence of symmetric and
* Corresponding author. Tel.: +39-0444-998-711; fax: +39-0444998-888. E-mail address: [email protected] (P. Lazzarin). 0142-1123/01/$ - see front matter 2001 Published by Elsevier Science Ltd. PII: S 0 1 4 2 - 1 1 2 3 ( 0 0 ) 0 0 0 8 6 - 4
anti-symmetric stress fields, supported the N-SIF definition [5]. Due to weld geometry, both components are always present at the weld toe, also under a remote uniaxial load, and vary from case to case according to the global geometry of the joint. Thus, two notch stress intensity factors K N1 and K N2 (for opening and sliding modes) were determined by means of a finite element analysis and then plotted as a function of the main geometrical parameters of the joints [5]. So, in the highly stressed region in the neighbourhood of the weld toe, stress components can be predicted on the basis of a linear combination of K N1 and K N2; this could be useful, particularly along the virtual direction of fatigue crack propagation (when a well-established linear elastic fracture mechanics approach is used to predict the fatigue life of the joints [6–9]) and on the surface free edge (where strain gauges are generally placed in experimental approaches). It has already been shown in [5,10] that the total fatigue life of transverse non-load-carrying fillet welded joints could be efficiently predicted by using only the K N1 factor, the contribution due to the sliding mode being
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non singular for that type of joint. Note that most of the weld details classified by Eurocode 3 and other national standards in force exhibit mean values of the V-shaped notch opening angles close to 135°. Thus, neglecting the influence of K N2 in the fatigue failure criterion should be reasonable in all these cases, while comparing the K N1 of different units (due to variations of the opening angles) is not [5,10,11]. It was already highlighted [12] that a precise theoretical link exists (due to Bueckner’s superposition principle) between the conventional MFLE stress intensity factor KI and the N-SIFs K N1 and K N2. The aims of this paper are as follows: 앫 to extend the N-SIF-based approach, previously applied only to steel weldments, to aluminium welded joints; 앫 to compare the fatigue strength of steel and aluminium welded joints of different geometry on the basis of the relevant N-SIFs; 앫 to briefly discuss the exponent quantifying the size effect by comparing theoretical predictions (based only on Williams’ Mode I singularity) and results of a statistical re-analysis involving all the experimental data considered herein. The theoretical penalty exponent is more penalising than that suggested by Eurocodes. Fig. 1. Coordinate system and geometrical parameters for the analyses of the welded joints.
2. Singular stress fields due to sharp corners Williams [1] stated that, even in a re-entrant V-shaped corner, as happens in a crack, the Mode I (and often Mode II) stress field is singular close to the tip. Then, in a polar frame of reference (r, J) (see Fig. 1), the stress field is defined within two constants (a1 and a2) and can always be written as the sum of the symmetric field, with stress singularity of the 1/r1−l1 type, and the anti-symmetric field, with stress singularity of the 1/r1−l2 type:
冦冧 sJ
sr ⫽l1r trJ
冦 冧 f1,J(J)
l1−1
a1 f1,r(J)
f1,rJ(J)
⫹l2r
冦 冧
a2 f2,r(J)
sr
trJ
⫽
1
rl1−1K N1
冑2p(1+l )+c (1−l ) 1
1
冦
(1)
f2,rJ(J)
where l1 and l2 are, as is well known, the first eigenvalues for Mode I and Mode II, respectively, in Williams’ equations [1]. Obviously, when l2 is greater than 1.0, only Mode I is singular. This happens when 2a is greater than 102°. It is possible to present Williams’ formulae for stress components as explicit functions of the N-SIFs [5]. For Mode I stress distributions are:
1
r=0
cos(1+l1)J
⫹c1(1⫺l1) −cos(1+l1)J
f2,J(J)
l2−1
冦冧 sJ
sin(1+l1)J
冤冦
冧
(1+l1)cos(1−l1)J
(3−l1)cos(1−l1)J ⫹ (1−l1)sin(1−l1)J
冧冥
(2)
For Mode II:
冦冧 sJ sr
trJ
⫽
r=0
1
rl2−1K N2
冑2p(1−l )+c (1+l ) 2
2
2
冤冦
冧
−(1+l2)sin(1−l2)J
−(3−l2)sin(1−l2)J ⫹ (1−l2)cos(1−l2)J
P. Lazzarin, P. Livieri / International Journal of Fatigue 23 (2001) 225–232
冦
−sin(1+l2)J
⫹c2(1⫹l2)
sin(1+l2)J cos(1+l2)J
冧冥
(3)
Expressions for k1 and k2 have already been reported for transverse non-load carrying fillet welded joints subjected to tensile stresses [5] or bending stresses [10]. It is useful to report here such expressions, since most welded details considered herein refer to just such types of joints. Traction: k1⫽1.212⫹0.495e−0.985(2h/t)
General expressions of the coefficients in Eqs. (2) and (3) are reported in [5,10]. Since all the series of welded joints considered in the present analyses will be characterised by an opening angle 2a=135°, it is sufficient to give here only the parameter values associated with this particular angle: l1=0.674, l2=1.302, c1=4.153, c2=⫺ 0.569. Two convenient expressions of N-SIFs for welded joints are the following [5]: K N1⫽k1snt1−l1
(4a)
K N2⫽k2snt1−l2
(4b)
where ki are non-dimensional coefficients analogous to the theoretical stress concentration factors Kt, sn is the remotely applied nominal stress and t is the main plate thickness. Looking at Eqs. (2) and (3), it is worth noting that when q=0, sr and sq depend only on Mode I distribution while, on the contrary, the trq component is associated with Mode II. So, by plotting stress distributions along the bisector of a particular geometry, there is a zone in the neighbourhood of the weld toe where the sij /rli−1 ratios have a constant value (Fig. 2). As a consequence, K N1 and K N2 can be univocally determined since, on the basis of Eqs. (2) and (3), the intensities of the stress distributions are ruled just on such parameters. It is worth noting that only Mode I stress distribution is singular in Fig. 2 while, in contrast, Mode II stress components are null when r=0. As soon as K N1 and K N2 are known, the relevant nondimensional coefficients k1 and k2 can easily be computed by means of Eq. (4a,b).
Fig. 2.
Plots of stresses along the bisector.
227
(5a)
⫺1.259e−1.120(2h/t)−0.485(L/t) k2⫽0.508⫺0.797e−1.959(2h/t)
(5b)
⫹2.723e−1.126(2h/t)−0.769(L/t) Bending: k1⫽0.900⫹0.326e−5.289(2h/t)
(6a)
⫺0.474e−3.064(2h/t)−1.420(L/t) k2⫽0.818⫺1.760e−5.356(2h/t)
(6b)
⫹1.851e
−2.982(2h/t)−1.026(L/t)
According to symbols shown in Fig. 1, h is the height of the weld bead and L the transverse plate thickness. Estimates based on Eqs. (5a,b) are accurate when 0ⱕ L/tⱕ3.0 and 0.25ⱕ2h/tⱕ2.5 [5] while limits for Eqs. (6a,b) are 0.2ⱕL/tⱕ5.0 and 0.25ⱕ2h/tⱕ2.5 [10]. Out of these geometrical conditions, a finite element analysis should be carried out, according to the procedures detailed in [5].
3. Fatigue strength data in terms of N-SIFs Tables 1 and 2 summarise geometrical and fatigue strength data related both to steel and aluminium welded joints, respectively. Those pertinent to steel joints have already been partly analysed in [5,10]. Table 3 reports materials, welding processes and postwelding conditions for all the series considered. Original data are reported in the well-known books by Maddox [6] and Gurney [7,9] as well as in two papers by Kihl and Sarkani [13,14]. Most data refer to transverse nonload-carrying fillet joints; some series, however, consider welded joints of a different type. In all cases, the nominal value of the notch tip radius is to be considered null (as the weld toes are always represented by sharp Vshaped notches in the original papers), while the opening angle 2a is 135°. The relevant k1 coefficients are summarised in Tables 1 and 2. It is worth noting that the main plate thickness ranges between 6 and 100 mm in steel welded joints, and between 3 and 24 mm in aluminium welded joints. The variability of the transverse plates is even more pronounced (3–200 mm).
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Table 1 Geometrical and fatigue strength properties of steel welded joints (nominal load ratio R⬇0)a Series
St-1 St-2 St-3 St-4 St-5 St-6 St-7 St-8 St-9 St-10 St-11 St-12 St-13 St-14 St-15 St-16 St-17 St-18 St-19 St-20 St-21 St-22 St-23 St-24
Welded joint geometry [Ref. No.]
cruciform-nlc [6] cruciform-nlc [6] cruciform-nlc [6] cruciform-nlc [7] cruciform-nlc [7] cruciform-nlc [7] cruciform-nlc [7] cruciform-nlc [7] cruciform-nlc [7] cruciform-nlc [7] cruciform-nlc [7] cruciform-nlc [7] cruciform-nlc [7] cruciform-nlc [7] cruciform-nlc [7] cruciform-nlc [7] cruciform-nlc [13] cruciform-nlc [13] cruciform-nlc [13] cruciform-nlc [13] cruciform-nlc [14] cruciform-nlc [9] T-nlc [9] cruciform-lc [9]
Load type
T T T T T T T T T T T T B B B B T T T T T T B T
t [mm]
L/t
13 50 100 13 13 25 25 25 38 38 100 100 25 50 100 100 6 19 25 11 11 6 6 6
2h/t
0.769 1.000 0.500 0.231 0.769 0.120 1.280 8.800 0.342 5.789 0.030 2.200 0.120 0.060 0.030 0.130 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
1.231 0.640 0.320 0.769 1.231 0.400 0.720 1.200 0.421 0.789 0.100 0.300 0.400 0.200 0.100 0.160 0.750 0.750 0.750 0.750 0.750 2.000 2.000 2.000
N=5·106
⌬K N1,50% [MPa mm0.326] N=5·106
79.5 59.6 55.5 91.7 76.7 93.9 66.0 59.7 68.7 45.5 95.7 40.1 87.9 98.1 94.5 75.1 103.1 77.8 57.4 107.4 93.6 93.6 111.3 98.6
209.4 234.2 219.8 204.8 202.0 211.1 217.4 231.8 196.3 209.5 236.6 228.7 198.6 232.4 248.1 230.6 205.7 226.0 182.2 261.0 227.3 201.4 213.5 255.8
⌬sn,50% [MPa]
k1
1.14 1.10 0.88 0.97 1.14 0.79 1.15 1.36 0.87 1.41 0.55 1.27 0.79 0.66 0.59b 0.68 1.11 1.11 1.11 1.11 1.11 1.20 1.07b 1.45b
Type of test: T = traction; B = bending. Type of fillet: nlc = non-load carrying fillet weld; lc = load-carrying fillet weld. For the series 1–14, 16–22, k1 has been determined by means of Eqs. (5a)–(6a). In the remaining cases, k1 has been determined by means of an “ad hoc” finite element analysis. a
b
Table 2 Geometrical and fatigue strength properties of aluminium welded joints (nominal load ratio R⬇0.1)a
Series
AL1 AL2 AL3 AL4 AL5 AL6 AL7 AL8 AL9 AL10
Welded joint geometry [Ref. No.]
cruciform-nlc [8] cruciform-nlc [8] cruciform-nlc [8] cruciform-nlc [8] cruciform-nlc [8] cruciform-nlc [8] T-nlc [17] T-nlc [16] cruciform-lc [16] cruciform-lc [15]
Load type
T T T T T T T T T T
t [mm]
3 6 12 24 24 12 12 12 12 12
L/t
1.000 1.000 1.000 1.000 0.250 0.500 0.833 1.000 1.000 1.000
2h/t
3.000 2.333 1.667 1.708 0.583 1.167 1.333 1.333 1.333 1.060
Km
1.14 1.09 1.08 1.01 1.05 1.25 1.00 1.00 1.00 1.00
k1
1.22 1.21 1.19 1.19 0.91 1.10 0.93b 0.93b 1.73b 2.07b
N=5·106
⌬K N1,50% [MPa mm0.326] N=5·106
59.3 45.3 40.5 29.1 40.9 38.0 43.1 53.0 28.0 26.3
103.2 97.8 108.6 97.7 105.0 94.1 89.7 110.3 108.8 122.5
⌬sn,50% [MPa]
Type of test: T = traction; B = bending. Type of fillet: nlc = non-load carrying fillet weld; lc = load-carrying fillet weld. For the series AL1–AL6, k1 has been determined by means of Eqs. (5a)–(6a). In the remaining cases, k1 has been determined by finite element analyses. a
b
Fig. 3 summarises steel welded joint data, all referring to a nominal load ratio R⬇0, in a single scatter band (mean values ± two standard deviations), of which the top and bottom lines refer to a probability of survival equal to 2.3 and 97.7%, respectively. At Nref=5·106
cycles to failure, the mean value of ⌬K N1 is 211 MPa·mm0.326 while the TK scatter index (TK= K N1,Ps=2.3%/K N1,Ps=97.7%) is 1.85. As far as the aluminium welded joints are concerned, six series were reported in a well-documented contri-
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229
Table 3 Steel and aluminium welding material type and welding conditions Series
Material
Yield stress [MPa]
Welding process
Conditions
St-1÷St-3 St-4÷St-16 St-17÷St-20 St-21 St-22÷St-24 AL1÷AL6 AL7 AL8, AL9 AL10
BS4360:50 BS4360:50 HSLA-80 HSLA-80 Steel UTS 515 MPa 6061-T6 5083-H3 6061-T651 Al Zn Mg 1
360÷398 290÷405 598÷671 ⬇671 412 277÷298 255 ⬇250 304
Manual metal arc welding Manual metal arc welding Gas metal arc welding (pulse) Gas metal arc welding Metal inert gas Gas metal arc welding Metal inert gas Metal inert gas Metal inert gas
As-welded As-welded with spot-heated As-welded As-welded As-welded As-welded As-welded As-welded As-welded
Fig. 3. Fatigue strength of aluminium and steel welded joints as a function of Mode I N-SIFactor. Scatter band related to mean values ± 2 standard deviations.
bution due, once again, to Maddox [8]. In this paper, all figures of the joint geometry are interested by sharp Vshaped notches, the total fatigue life being thought of as fatigue crack propagation life. The remaining four series are due to Jacoby [15], Ribeiro et al. [16] and Meneghetti [17]. Also, for such series it is not possible to give an upper bound for the real weld toes, since this information is not explicitly given in the original papers. All fatigue tests were carried out with a nominal load ratio R about equal to 0.1. Under the hypothesis of log-normal distributions of the number of cycles to failure, the mean values of ⌬sn,50% have been determined by a least-square method. One might note that ⌬sn,50% values pertinent to the series tested by Maddox are slightly different from the values tabled in [8]. This is because Maddox performed a bestfitting analysis of fatigue data by imposing a Wo¨hler curve slope equal to 4.0 for all the series tested by him. Maddox’ data also take into account secondary bending effects, quantified by the experimental coefficient Km, determined in [8] by means of strain gauge measurements. For the remaining series in Table 2, Km is equal to unity. Aluminium welded joints show a mean value of ⌬K N1=99 MPa·mm0.326 at 5·106 cycles to failure, while
the TK index practically coincides with the steel joint value (1.80 against 1.85). Due to the limited number of experimental data consistent with those shown in Fig. 3, for the time being it is not possible to extend the approach to different nominal load ratios, with the aim of quantifying the differences with respect to the R⬇0 case. However, two series of experimental data reported in [14] and not considered in Table 1 show that: (a) the mean ⌬K N1 curve goes down slightly when R=0.33 (with a decrease in strength of about 5% at 5 million cycles); (b) conversely, the mean ⌬K N1 curve goes up in the presence of negative mean stresses (R=⫺2), the difference in strength being more pronounced with respect to the former case. It is natural to think that these trends will be confirmed, being in agreement with the results obtained with the conventional nominal stress approach. Finally, it might be useful to note that the mean ⌬K N1 values for steel and aluminium joints are in a ratio of 2.1. The same ratio had been shown for butt splice bolted joints [19] (nominal stress amplitude equal to 88 MPa for steel; 41 MPa for aluminium), where such a geometry made a comparison still possible in terms of nominal stresses. 3.1. Size effect The size effect is predicted by Eq. (2), where the penalty exponent g=1⫺l1 is different from that reported in the Eurocodes (0.326 against 0.25, where, as far as the authors are aware, the latter value is empirical in nature). Due to the availability of experimental data and the accuracy of the sources considered, we have looked for the value of the exponent g which minimises q, q being defined as follows:
冘冋
q⫽
冉 冊册
⌬snj k1ref tref ⫺ ⌬snref k1j tj
g 2
(7)
As reference values, we have chosen t=25 mm for steel welded joints (and, in particular, the series St-6 has been taken as the reference series) and t=12 mm for alu-
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minium welded joints (series AL-3). Fig. 4 plots the ratio q/qmin against g. It is interesting to note that, at least for the joints considered herein, characterised by a nominal value of the notch tip radius equal to zero, the scatter is minimised corresponding to the theoretical value based on Williams’ eigenvalue for steel, at g=0.30 for aluminium. Support for the present analysis is given in a recent contribution by Macdonald and Haagensen [18] who emphasise the fact that assessment of recent research data has indicated the influence of a thickness stronger than g=0.25, so that in the latest HSE and API/ISO revision for offshore structures, a higher penalty factor of g=0.30 is imposed.
4. Further developments of the N-SIF-based approach and some answers to reviewers Due to lucky circumstances, the paper was reviewed by two anonymous referees whose suggestions and careful judgements (both those favourable and those quite critical) have been greatly appreciated by the authors. Some problems raised by the referees are intriguing, needing further investigation to be fully clarified and so they are surely of interest for many researchers engaged in fatigue design of components weakened by sharp stress raisers. For this reason, we have decided to report faithfully the referees’ opinions here. Reviewer A wrote that “the conservative assumption that the weld toe radius is equal to zero is helpful, because weld toe radius is not easy to measure, needs time and is affected by a large scatter”. Nevertheless, “this presentation of the fatigue resistance of the welded joints is of limited interest” for two reasons: (a) “it not easy to use the fatigue resistance curve for a weld toe geometry with an included angle different from 135°” (in that units for K N1 are no longer MPa·mm0.326); (b) “it is not possible to appreciate directly the fatigue reduction factor in presence of smooth welded joints”.
Fig. 4. Minimum values of q for aluminium and steel welded joints.
Reviewer A also wrote that the discussion “about the thickness effect is a more interesting contribution. It is based on the assumption that thickness effect is due to the stress gradient precisely described by the exponent of singularity”. In his opinion, the N-SIF-based analysis implies that: (c) “the thickness effect depends on the weld toe angle (so a universal value cannot be accepted)”; (d) “loading modes have no direct effect (which is not a true assumption)”. Only point (d) can be easily confuted. Different expressions are used in the paper for welded joints subjected to tensile loading and bending loading. This simply means that loading modes (and not only the global geometry) influence the intensity of the stress distributions in the neighbourhood of the sharp notch (but not the degree of singularity [20], which depends only on 2a). When given in terms of nominal stress ranges, the fatigue strength of welded joints subjected to bending loads is generally recognised as greater than that exhibited by the same joint under tensile loads. A reduction of 13% in averaged terms is reported by Hobbacher [21], a reduction ranging from 0% to 25% is shown in [20]. This scatter is no longer statistically significant if N-SIFs are used instead of nominal stresses for the simple reason that N-SIFs include the loading mode effect. Note that series St-11, 12, 15, 16 in Table 1 (t=100 mm) show a ⌬K N1 value that ranges, for bending and tensile loads, from 229 to 248 MPa·mm0.326. Point (c) reflects exactly the authors’ opinion when the geometry is weakened by sharp V-shaped notches. However, as soon as a notch with a tip radius r constant and different from zero is present, the situation becomes more complex. Obviously, stress distribution due to a rounded notch does not coincide with that of the sharp V-notch. A small zone exists in the close neighbourhood of the notch tip where the stress distribution substantially depends only on r, so that its features can be considered to a certain degree “universal”. Moreover, the stress gradient is not constant but varies as a function of the distance from the notch tip. Outside this limited zone, of which the dimensions are about 0.3 r, the influence of the opening angle becomes important and the stress gradient coincides with that related to the corresponding sharp V-shaped notch. The properties of the material determine whether the fatigue strength is controlled by the former or the latter zone. With regard to point (a), the authors agree with the reviewer. However, it is evident that an opening angle of about 135° represents the most common geometry and that small variations of the angle could be tolerated by engineers engaged in fatigue problems. One should also note that in a very accurate multi-parameter design optimisation of load-carrying fillet cruciform joints car-
P. Lazzarin, P. Livieri / International Journal of Fatigue 23 (2001) 225–232
ried out by Radaj and Zang [22] (see also Radaj and Sonsino [23]), the only geometrical parameter considered constant is the opening angle. Conversely, there is no doubt that, from a theoretical point of view, the complex units of K N1 do not allow a direct comparison between joints with different opening angle. The problem, already highlighted by Hasebe et al. [24], can be overcome either by introducing a virtual crack at the notch toe [24,12] or by using, perhaps by taking a small step forward, Eqs. (2) and (3) to determine the energy in a small sector of radius R surrounding the sharp notch [25]. Such energy is strictly correlated to K N1 and K N2 but it obviously has the merit to be expressed in Nmm/mm3. As regards point (b), Hasebe et al. [24] were able to demonstrate that a precise analytical link exists between the N-SIF (determined for sharp-V-shaped notches under Mode I conditions) and the elastic peak stress value of a rounded V-shaped notch, the notch tip radius being small but different from zero. More precisely, Hasebe et al. wrote [24]: K N1⫽limr→0C˜r1−l1sq,max
(8)
where symbols of the original paper have been upgraded to current symbols. The parameter C˜ was summarised for several notch opening angles in [24]. Eq. (8) makes it evident that using K N1 or sq,max (after having introduced a small notch tip radius r) results in exactly the same fatigue predictions. Obviously, in real cases, one would use Eq. (8) in the presence of a well-defined value of r (for example rf=1, according to Radaj’s fictitious weld toe/root radius [26]). By using a complex potential function and Neuber’s conformal mapping, an equation analogous to Eq. (8) was reported also in [27]: K N1⫽
冉 冊
冑2p[1+l +c (1−l )] 1
1
4
1
q−1 q
1−l1
r1−l1sq,max
(9)
⫽1.22·r1−l1·sq,max where, on the right side of Eq. (9), a coefficient valid for the 2a=135° case is introduced. In Eq. (9), due to the absence of the limit condition r→0, K N1 was not thought of as numerically coincident with the value pertinent to the r=0 case. Eq. (9), without upgrading K N1, results in a strong simplification in the peak stress evaluation, but also some degree of inaccuracy. The problem is that of defining this degree of inaccuracy. Table 4 gives a precise idea of the errors for two geometries with L/t=0.5 and 1.0. The differences between analytical and finite element results vary from case to case, their mean value being about 10%. Note that some formulae summarised in [28], and suitable for estimating peak stress in cruciform welded joints, are acknowledged as being able to provide a substantially equivalent degree of accuracy [23]. It is evident that Eqs. (8) and (9) provide a bridging
231
Table 4 Stress concentration factor Kt of transverse non-load-carrying fillet welded joints under tensile loads (in all FE analyses t=20 mm, 2h/t=1, see Fig. 1). Values of k1 determined according to Eq. (5a)) r/t
L/t
k1 Eq. (5a)
Kt Eq. (9)
Kt FEM
0.02 0.05 0.1 0.2 0.02 0.05 0.1 0.2
0.5 0.5 0.5 0.5 1 1 1 1
1.075 1.075 1.075 1.075 1.144 1.144 1.144 1.144
3.43 2.54 2.03 1.62 3.65 2.71 2.16 1.72
3.83 2.85 2.30 1.88 4.02 3.00 2.41 1.95
between the N-SIF approach and Radaj’s notch stress approach for welded joints [26,23], where fatigue predictions can be performed on the basis of sq,max (that is on the basis of the theoretical stress concentration factor Kt), but only after having introduced a precise value of the fictitious notch tip radius (rf=1 mm in most welded details of practical interest, but also rf=0.25 mm for spot-welded overlap joints made in rolled steels, rf=0.2 mm for cruciform joints in the presence of longitudinal shear loading [23]). Reviewer B “fully agrees with the N-SIF approach” since it “is based on a principle of similitude and overcomes problems encountered with predictions using a local strain approach and/or an integration of Paris’ relationship”. He has been “convinced for years that this is the best way for predicting the life of welded joint (as-welded) and that it should be included in codes”. Asking the authors to discuss the fatigue fracture of welded joints and to explain why the approach works, he helps them by providing the following convincing explanation: “A severe notch with a very small toe radius results in a short microstructural initiation life — even without toe ‘defects’ — and in an immediate crack propagation; this explains why the influence of a microstructure is weak. Furthermore, most of the life is consumed at short crack depth, within the singularity; that explains why good correlation is obtained with total fatigue life.” In addition, it might be useful to remember that LEFM stress intensity factor KI is analytically correlated to NSIFs [12]. A conventional evaluation of residual life from an initial crack value of 0.3 mm (the crack being thought of as through the thickness) turned out to be in a ratio of 1:3 to experimental total life, having assumed the exponent in the Paris law as being equal to 3.0 [12]. In conclusion, N-SIF is easy to calculate, and plays an essential role in short microstructural initiation life, short crack life and crack propagation life.
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5. Conclusions Notch stress intensity factors (N-SIFs) have been used, in a single scatter band, to summarise fatigue properties of aluminium welded joints (cruciform and T non-load-carrying or load-carrying fillet weld joints). More precisely, fatigue total life (and not only fatigue crack initiation life) has been correlated to the relevant Mode I N-SIFs, i.e. ⌬K N1. Welded joints were characterised by a thickness ranging between 3 and 24 mm, able to put in evidence any scale effect. Fatigue properties of aluminium welded joints have been compared with those related to steel welded joints (of which the main plate thickness varied from 6 to 100 mm). The mean ⌬K N1 values turn out to be in a ratio slightly greater than 2.0, while the scatter band size (mean value ± 2 standard deviations) practically coincides. Taking into account the only singular stress distribution (associated to Mode I fracture), the exponent quantifying the scale effect penalty is 0.326, which is quite different from the 0.25 value suggested by the Eurocodes. In the presence of a weld toe radius approaching zero, a statistical re-analysis of all fatigue data showed that a value equal to 0.3 was more realistic both for aluminium and steel welded joints. It is worth noting that in some recent standards on off-shore structures, the exponent 0.3 has already substituted the 0.25 value present in Eurocodes. Finally, the link between Mode I N-SIF and the peak value of the maximum linear elastic stress (determined in the presence of a small weld toe radius) was discussed. References [1] Williams ML. Stress singularities resulting from various boundary conditions in angular corners of plates in extension. J Appl Mech 1952;19:526–8. [2] Nui LS, Chehimi C, Pluvinage G. Stress field near a large blunted tip V-notch and application of the concept of the critical notch stress intensity factor (NSIF) to the fracture toughness of very brittle materials. Engng Fract Mech 1994;49:325–35. [3] Boukharouba T, Tamine T, Nui L, Chehimi C, Pluvinage G. The use of notch stress intensity factor as a fatigue crack initiation parameter. Engng Fract Mech 1995;52:503–12. [4] Verreman Y, Nie B. Early development of fatigue cracking at manual fillet welds. Fatigue Fract Engng Mater Struct 1996;19:669–81. [5] Lazzarin P, Tovo R. A notch stress intensity factor approach to the stress analysis of welds. Fatigue Fract Engng Mater Struct 1998;21:1089–103. [6] Maddox SJ. The effect of plate thickness on the fatigue strength of fillet welded joints. Abington, Cambridge: Abington Publishing, 1987.
[7] Gurney TR. The fatigue strength of transverse fillet welded joints. Abington, Cambridge: Abington Publishing, 1991. [8] Maddox SJ. Scale effect in fatigue of fillet welded aluminium alloys. In: Proceedings of the Sixth International Conference on Aluminium Weldments; Cleveland, Ohio, 1995:77–93. [9] Gurney TR. Fatigue of thin walled joints under complex loading. Abington, Cambridge: Abington Publishing, 1997. [10] Atzori B, Lazzarin P, Tovo R. Stress field parameters to predict the fatigue strength of notched components. J Strain Anal 1999;34:437–53. [11] Dunn ML, Suwito W, Cunningham SJ, May CW. Fracture initiation at sharp notches under mode I, mode II, and mild mixed mode loading. Int J Fract 1997;84:367–81. [12] Atzori B, Lazzarin P, Tovo R. From the local stress approach to fracture mechanics: a comprehensive evaluation of the fatigue strength of welded joints. Fatigue Fract Engng Mater Struct 1999;22:369–81. [13] Kihl DP, Sarkani S. Thickness effects on the fatigue strength of welded steel cruciforms. Int J Fatigue 1997;19:S311–6. [14] Kihl DP, Sarkani S. Mean stress effects in fatigue of welded steel joints. Prob Engng Mech 1999;14:97–104. ¨ ber das verhalten von schweissverbindungen aus alu[15] Jacoby G. U miniumlegierungen bei schwingbeanspruchung. Dissertation, Technische Hochschule, Hannover; 1961. [16] Ribeiro AS, Gonc¸alves JP, Oliveira F, Castro PT, Fernandes AA. A comparative study on the fatigue behaviour of aluminium alloy welded and bonded joints. In: Proceedings of the Sixth International Conference on Aluminium Weldments, Cleveland, Ohio, 1995:65–76. [17] Meneghetti G. PhD Thesis, University of Padua; 1998. [18] Macdonald KA, Haagensen PJ. Fatigue design of welded aluminium rectangular hollow section joints. Engng Failure Anal 1999;6:113–30. [19] Lazzarin P, Milani V, Quaresimin M. Scatter bands summarizing the fatigue strength of aluminium alloy bolted joints. Int J Fatigue 1997;19:401–7. [20] Lazzarin P. Effect of bending loads on stress fields and fatigue strength of welded joints. Riv Ital Saldatura 1999;2(March/April): 137–43 [in Italian]. [21] Hobbacher A. Stress intensity factors of welded joints. Engng Fract Mech 1993;46:173–82 [and Corrigendum 1994;49:323]. [22] Radaj D, Zang S. Multiparameter design optimisation in respect of stress concentrations. In: Engineering optimisation design processes 1991. Berlin: Springer, 1991:181–9. [23] Radaj D, Sonsino CM. Fatigue assessment of welded joints by local approaches. Abington, Cambridge: Abington Publishing, 1998. [24] Hasebe N, Nakamura T, Iida J. Notch mechanics for plate and thin plate bending problems. Engng Fract Mech 1990;37:87–99. [25] Lazzarin P, Zambardi R. A finite-volume-energy based approach to predict the static and fatigue behavior of components with sharp V-shaped notches. Submitted for publication. [26] Radaj D. Design and analysis of fatigue resistant welded structures. Abington, Cambridge: Abington Publishing, 1990. [27] Lazzarin P, Tovo R. A unified approach to the evaluation of linear elastic stress fields in the neighborhood of cracks and notches. Int J Fract 1996;78:3–19. [28] Iida K, Uemura T. Stress concentration factor formulae widely used in Japan. Fatigue Fract Engng Mater Struct 1996;19(6):779–86.
Notch stress intensity factors and fatigue strength of aluminium and steel welded joints P. Lazzarin a
a,*
, P. Livieri
b
Department of Management and Engineering, University of Padova, Stradella S. Nicola 3, 36100 Vicenza, Italy b Department of Engineering, University of Ferrara, Via Saragat 1, 44100 Ferrara, Italy Received 12 June 2000; received in revised form 21 September 2000; accepted 21 September 2000
Abstract According to a recent and appropriate definition, stress field parameters, namely notch stress intensity factors (N-SIFs), can be used to predict the fatigue behaviour of mechanical components weakened by V-shaped re-entrant corners, where the singularity in the stress distribution makes any failure criterion based on elastic peak stress no longer applicable. Commonly thought of as parameters able to control the fatigue crack initiation life, N-SIFs are, under certain circumstances, also useful for predicting the component total fatigue life. The fatigue strength of aluminium welded joints with different geometries and thicknesses are summarised in a single scatter band by using an N-SIF-based approach. The statistical analysis is carried out taking into account experimental data already reported in the literature, referring to welded joints with a thickness ranging from 3 to 24 mm. Results of steel and aluminium welded joints are then compared: at high number fatigue life, the relative fatigue strength is slightly greater than 2, in agreement with the value previously reported in the literature for butt spliced bolted joints. The value of the theoretical exponent quantifying the scale effect (0.326 against 0.25 suggested by Eurocodes) is discussed. 2001 Published by Elsevier Science Ltd. Keywords: Fatigue; Welded joints; Notch stress intensity factors
1. Introduction Williams [1] was able to demonstrate that in the context of the elasticity theory, the asymptotic stress state near a re-entrant corner is singular and its degree of singularity is a function of the only notch opening angle. The stress field intensity depends on the overall geometry of the component and the far-field loading. In the context of a stress field theory, a field parameter called the “notch stress intensity factor” (hereafter NSIF) was explicitly defined by Nui et al. [2] and applied to the fracture toughness of brittle materials. Afterwards, the N-SIF concept was used by Boukharouba et al. and by Verreman and Nie for fatigue crack initiation estimates at notches [3] and weld toes [3,4]. Recently, an analytical background able to quantify different stress components, such as the influence of symmetric and
* Corresponding author. Tel.: +39-0444-998-711; fax: +39-0444998-888. E-mail address: [email protected] (P. Lazzarin). 0142-1123/01/$ - see front matter 2001 Published by Elsevier Science Ltd. PII: S 0 1 4 2 - 1 1 2 3 ( 0 0 ) 0 0 0 8 6 - 4
anti-symmetric stress fields, supported the N-SIF definition [5]. Due to weld geometry, both components are always present at the weld toe, also under a remote uniaxial load, and vary from case to case according to the global geometry of the joint. Thus, two notch stress intensity factors K N1 and K N2 (for opening and sliding modes) were determined by means of a finite element analysis and then plotted as a function of the main geometrical parameters of the joints [5]. So, in the highly stressed region in the neighbourhood of the weld toe, stress components can be predicted on the basis of a linear combination of K N1 and K N2; this could be useful, particularly along the virtual direction of fatigue crack propagation (when a well-established linear elastic fracture mechanics approach is used to predict the fatigue life of the joints [6–9]) and on the surface free edge (where strain gauges are generally placed in experimental approaches). It has already been shown in [5,10] that the total fatigue life of transverse non-load-carrying fillet welded joints could be efficiently predicted by using only the K N1 factor, the contribution due to the sliding mode being
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non singular for that type of joint. Note that most of the weld details classified by Eurocode 3 and other national standards in force exhibit mean values of the V-shaped notch opening angles close to 135°. Thus, neglecting the influence of K N2 in the fatigue failure criterion should be reasonable in all these cases, while comparing the K N1 of different units (due to variations of the opening angles) is not [5,10,11]. It was already highlighted [12] that a precise theoretical link exists (due to Bueckner’s superposition principle) between the conventional MFLE stress intensity factor KI and the N-SIFs K N1 and K N2. The aims of this paper are as follows: 앫 to extend the N-SIF-based approach, previously applied only to steel weldments, to aluminium welded joints; 앫 to compare the fatigue strength of steel and aluminium welded joints of different geometry on the basis of the relevant N-SIFs; 앫 to briefly discuss the exponent quantifying the size effect by comparing theoretical predictions (based only on Williams’ Mode I singularity) and results of a statistical re-analysis involving all the experimental data considered herein. The theoretical penalty exponent is more penalising than that suggested by Eurocodes. Fig. 1. Coordinate system and geometrical parameters for the analyses of the welded joints.
2. Singular stress fields due to sharp corners Williams [1] stated that, even in a re-entrant V-shaped corner, as happens in a crack, the Mode I (and often Mode II) stress field is singular close to the tip. Then, in a polar frame of reference (r, J) (see Fig. 1), the stress field is defined within two constants (a1 and a2) and can always be written as the sum of the symmetric field, with stress singularity of the 1/r1−l1 type, and the anti-symmetric field, with stress singularity of the 1/r1−l2 type:
冦冧 sJ
sr ⫽l1r trJ
冦 冧 f1,J(J)
l1−1
a1 f1,r(J)
f1,rJ(J)
⫹l2r
冦 冧
a2 f2,r(J)
sr
trJ
⫽
1
rl1−1K N1
冑2p(1+l )+c (1−l ) 1
1
冦
(1)
f2,rJ(J)
where l1 and l2 are, as is well known, the first eigenvalues for Mode I and Mode II, respectively, in Williams’ equations [1]. Obviously, when l2 is greater than 1.0, only Mode I is singular. This happens when 2a is greater than 102°. It is possible to present Williams’ formulae for stress components as explicit functions of the N-SIFs [5]. For Mode I stress distributions are:
1
r=0
cos(1+l1)J
⫹c1(1⫺l1) −cos(1+l1)J
f2,J(J)
l2−1
冦冧 sJ
sin(1+l1)J
冤冦
冧
(1+l1)cos(1−l1)J
(3−l1)cos(1−l1)J ⫹ (1−l1)sin(1−l1)J
冧冥
(2)
For Mode II:
冦冧 sJ sr
trJ
⫽
r=0
1
rl2−1K N2
冑2p(1−l )+c (1+l ) 2
2
2
冤冦
冧
−(1+l2)sin(1−l2)J
−(3−l2)sin(1−l2)J ⫹ (1−l2)cos(1−l2)J
P. Lazzarin, P. Livieri / International Journal of Fatigue 23 (2001) 225–232
冦
−sin(1+l2)J
⫹c2(1⫹l2)
sin(1+l2)J cos(1+l2)J
冧冥
(3)
Expressions for k1 and k2 have already been reported for transverse non-load carrying fillet welded joints subjected to tensile stresses [5] or bending stresses [10]. It is useful to report here such expressions, since most welded details considered herein refer to just such types of joints. Traction: k1⫽1.212⫹0.495e−0.985(2h/t)
General expressions of the coefficients in Eqs. (2) and (3) are reported in [5,10]. Since all the series of welded joints considered in the present analyses will be characterised by an opening angle 2a=135°, it is sufficient to give here only the parameter values associated with this particular angle: l1=0.674, l2=1.302, c1=4.153, c2=⫺ 0.569. Two convenient expressions of N-SIFs for welded joints are the following [5]: K N1⫽k1snt1−l1
(4a)
K N2⫽k2snt1−l2
(4b)
where ki are non-dimensional coefficients analogous to the theoretical stress concentration factors Kt, sn is the remotely applied nominal stress and t is the main plate thickness. Looking at Eqs. (2) and (3), it is worth noting that when q=0, sr and sq depend only on Mode I distribution while, on the contrary, the trq component is associated with Mode II. So, by plotting stress distributions along the bisector of a particular geometry, there is a zone in the neighbourhood of the weld toe where the sij /rli−1 ratios have a constant value (Fig. 2). As a consequence, K N1 and K N2 can be univocally determined since, on the basis of Eqs. (2) and (3), the intensities of the stress distributions are ruled just on such parameters. It is worth noting that only Mode I stress distribution is singular in Fig. 2 while, in contrast, Mode II stress components are null when r=0. As soon as K N1 and K N2 are known, the relevant nondimensional coefficients k1 and k2 can easily be computed by means of Eq. (4a,b).
Fig. 2.
Plots of stresses along the bisector.
227
(5a)
⫺1.259e−1.120(2h/t)−0.485(L/t) k2⫽0.508⫺0.797e−1.959(2h/t)
(5b)
⫹2.723e−1.126(2h/t)−0.769(L/t) Bending: k1⫽0.900⫹0.326e−5.289(2h/t)
(6a)
⫺0.474e−3.064(2h/t)−1.420(L/t) k2⫽0.818⫺1.760e−5.356(2h/t)
(6b)
⫹1.851e
−2.982(2h/t)−1.026(L/t)
According to symbols shown in Fig. 1, h is the height of the weld bead and L the transverse plate thickness. Estimates based on Eqs. (5a,b) are accurate when 0ⱕ L/tⱕ3.0 and 0.25ⱕ2h/tⱕ2.5 [5] while limits for Eqs. (6a,b) are 0.2ⱕL/tⱕ5.0 and 0.25ⱕ2h/tⱕ2.5 [10]. Out of these geometrical conditions, a finite element analysis should be carried out, according to the procedures detailed in [5].
3. Fatigue strength data in terms of N-SIFs Tables 1 and 2 summarise geometrical and fatigue strength data related both to steel and aluminium welded joints, respectively. Those pertinent to steel joints have already been partly analysed in [5,10]. Table 3 reports materials, welding processes and postwelding conditions for all the series considered. Original data are reported in the well-known books by Maddox [6] and Gurney [7,9] as well as in two papers by Kihl and Sarkani [13,14]. Most data refer to transverse nonload-carrying fillet joints; some series, however, consider welded joints of a different type. In all cases, the nominal value of the notch tip radius is to be considered null (as the weld toes are always represented by sharp Vshaped notches in the original papers), while the opening angle 2a is 135°. The relevant k1 coefficients are summarised in Tables 1 and 2. It is worth noting that the main plate thickness ranges between 6 and 100 mm in steel welded joints, and between 3 and 24 mm in aluminium welded joints. The variability of the transverse plates is even more pronounced (3–200 mm).
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Table 1 Geometrical and fatigue strength properties of steel welded joints (nominal load ratio R⬇0)a Series
St-1 St-2 St-3 St-4 St-5 St-6 St-7 St-8 St-9 St-10 St-11 St-12 St-13 St-14 St-15 St-16 St-17 St-18 St-19 St-20 St-21 St-22 St-23 St-24
Welded joint geometry [Ref. No.]
cruciform-nlc [6] cruciform-nlc [6] cruciform-nlc [6] cruciform-nlc [7] cruciform-nlc [7] cruciform-nlc [7] cruciform-nlc [7] cruciform-nlc [7] cruciform-nlc [7] cruciform-nlc [7] cruciform-nlc [7] cruciform-nlc [7] cruciform-nlc [7] cruciform-nlc [7] cruciform-nlc [7] cruciform-nlc [7] cruciform-nlc [13] cruciform-nlc [13] cruciform-nlc [13] cruciform-nlc [13] cruciform-nlc [14] cruciform-nlc [9] T-nlc [9] cruciform-lc [9]
Load type
T T T T T T T T T T T T B B B B T T T T T T B T
t [mm]
L/t
13 50 100 13 13 25 25 25 38 38 100 100 25 50 100 100 6 19 25 11 11 6 6 6
2h/t
0.769 1.000 0.500 0.231 0.769 0.120 1.280 8.800 0.342 5.789 0.030 2.200 0.120 0.060 0.030 0.130 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
1.231 0.640 0.320 0.769 1.231 0.400 0.720 1.200 0.421 0.789 0.100 0.300 0.400 0.200 0.100 0.160 0.750 0.750 0.750 0.750 0.750 2.000 2.000 2.000
N=5·106
⌬K N1,50% [MPa mm0.326] N=5·106
79.5 59.6 55.5 91.7 76.7 93.9 66.0 59.7 68.7 45.5 95.7 40.1 87.9 98.1 94.5 75.1 103.1 77.8 57.4 107.4 93.6 93.6 111.3 98.6
209.4 234.2 219.8 204.8 202.0 211.1 217.4 231.8 196.3 209.5 236.6 228.7 198.6 232.4 248.1 230.6 205.7 226.0 182.2 261.0 227.3 201.4 213.5 255.8
⌬sn,50% [MPa]
k1
1.14 1.10 0.88 0.97 1.14 0.79 1.15 1.36 0.87 1.41 0.55 1.27 0.79 0.66 0.59b 0.68 1.11 1.11 1.11 1.11 1.11 1.20 1.07b 1.45b
Type of test: T = traction; B = bending. Type of fillet: nlc = non-load carrying fillet weld; lc = load-carrying fillet weld. For the series 1–14, 16–22, k1 has been determined by means of Eqs. (5a)–(6a). In the remaining cases, k1 has been determined by means of an “ad hoc” finite element analysis. a
b
Table 2 Geometrical and fatigue strength properties of aluminium welded joints (nominal load ratio R⬇0.1)a
Series
AL1 AL2 AL3 AL4 AL5 AL6 AL7 AL8 AL9 AL10
Welded joint geometry [Ref. No.]
cruciform-nlc [8] cruciform-nlc [8] cruciform-nlc [8] cruciform-nlc [8] cruciform-nlc [8] cruciform-nlc [8] T-nlc [17] T-nlc [16] cruciform-lc [16] cruciform-lc [15]
Load type
T T T T T T T T T T
t [mm]
3 6 12 24 24 12 12 12 12 12
L/t
1.000 1.000 1.000 1.000 0.250 0.500 0.833 1.000 1.000 1.000
2h/t
3.000 2.333 1.667 1.708 0.583 1.167 1.333 1.333 1.333 1.060
Km
1.14 1.09 1.08 1.01 1.05 1.25 1.00 1.00 1.00 1.00
k1
1.22 1.21 1.19 1.19 0.91 1.10 0.93b 0.93b 1.73b 2.07b
N=5·106
⌬K N1,50% [MPa mm0.326] N=5·106
59.3 45.3 40.5 29.1 40.9 38.0 43.1 53.0 28.0 26.3
103.2 97.8 108.6 97.7 105.0 94.1 89.7 110.3 108.8 122.5
⌬sn,50% [MPa]
Type of test: T = traction; B = bending. Type of fillet: nlc = non-load carrying fillet weld; lc = load-carrying fillet weld. For the series AL1–AL6, k1 has been determined by means of Eqs. (5a)–(6a). In the remaining cases, k1 has been determined by finite element analyses. a
b
Fig. 3 summarises steel welded joint data, all referring to a nominal load ratio R⬇0, in a single scatter band (mean values ± two standard deviations), of which the top and bottom lines refer to a probability of survival equal to 2.3 and 97.7%, respectively. At Nref=5·106
cycles to failure, the mean value of ⌬K N1 is 211 MPa·mm0.326 while the TK scatter index (TK= K N1,Ps=2.3%/K N1,Ps=97.7%) is 1.85. As far as the aluminium welded joints are concerned, six series were reported in a well-documented contri-
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229
Table 3 Steel and aluminium welding material type and welding conditions Series
Material
Yield stress [MPa]
Welding process
Conditions
St-1÷St-3 St-4÷St-16 St-17÷St-20 St-21 St-22÷St-24 AL1÷AL6 AL7 AL8, AL9 AL10
BS4360:50 BS4360:50 HSLA-80 HSLA-80 Steel UTS 515 MPa 6061-T6 5083-H3 6061-T651 Al Zn Mg 1
360÷398 290÷405 598÷671 ⬇671 412 277÷298 255 ⬇250 304
Manual metal arc welding Manual metal arc welding Gas metal arc welding (pulse) Gas metal arc welding Metal inert gas Gas metal arc welding Metal inert gas Metal inert gas Metal inert gas
As-welded As-welded with spot-heated As-welded As-welded As-welded As-welded As-welded As-welded As-welded
Fig. 3. Fatigue strength of aluminium and steel welded joints as a function of Mode I N-SIFactor. Scatter band related to mean values ± 2 standard deviations.
bution due, once again, to Maddox [8]. In this paper, all figures of the joint geometry are interested by sharp Vshaped notches, the total fatigue life being thought of as fatigue crack propagation life. The remaining four series are due to Jacoby [15], Ribeiro et al. [16] and Meneghetti [17]. Also, for such series it is not possible to give an upper bound for the real weld toes, since this information is not explicitly given in the original papers. All fatigue tests were carried out with a nominal load ratio R about equal to 0.1. Under the hypothesis of log-normal distributions of the number of cycles to failure, the mean values of ⌬sn,50% have been determined by a least-square method. One might note that ⌬sn,50% values pertinent to the series tested by Maddox are slightly different from the values tabled in [8]. This is because Maddox performed a bestfitting analysis of fatigue data by imposing a Wo¨hler curve slope equal to 4.0 for all the series tested by him. Maddox’ data also take into account secondary bending effects, quantified by the experimental coefficient Km, determined in [8] by means of strain gauge measurements. For the remaining series in Table 2, Km is equal to unity. Aluminium welded joints show a mean value of ⌬K N1=99 MPa·mm0.326 at 5·106 cycles to failure, while
the TK index practically coincides with the steel joint value (1.80 against 1.85). Due to the limited number of experimental data consistent with those shown in Fig. 3, for the time being it is not possible to extend the approach to different nominal load ratios, with the aim of quantifying the differences with respect to the R⬇0 case. However, two series of experimental data reported in [14] and not considered in Table 1 show that: (a) the mean ⌬K N1 curve goes down slightly when R=0.33 (with a decrease in strength of about 5% at 5 million cycles); (b) conversely, the mean ⌬K N1 curve goes up in the presence of negative mean stresses (R=⫺2), the difference in strength being more pronounced with respect to the former case. It is natural to think that these trends will be confirmed, being in agreement with the results obtained with the conventional nominal stress approach. Finally, it might be useful to note that the mean ⌬K N1 values for steel and aluminium joints are in a ratio of 2.1. The same ratio had been shown for butt splice bolted joints [19] (nominal stress amplitude equal to 88 MPa for steel; 41 MPa for aluminium), where such a geometry made a comparison still possible in terms of nominal stresses. 3.1. Size effect The size effect is predicted by Eq. (2), where the penalty exponent g=1⫺l1 is different from that reported in the Eurocodes (0.326 against 0.25, where, as far as the authors are aware, the latter value is empirical in nature). Due to the availability of experimental data and the accuracy of the sources considered, we have looked for the value of the exponent g which minimises q, q being defined as follows:
冘冋
q⫽
冉 冊册
⌬snj k1ref tref ⫺ ⌬snref k1j tj
g 2
(7)
As reference values, we have chosen t=25 mm for steel welded joints (and, in particular, the series St-6 has been taken as the reference series) and t=12 mm for alu-
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minium welded joints (series AL-3). Fig. 4 plots the ratio q/qmin against g. It is interesting to note that, at least for the joints considered herein, characterised by a nominal value of the notch tip radius equal to zero, the scatter is minimised corresponding to the theoretical value based on Williams’ eigenvalue for steel, at g=0.30 for aluminium. Support for the present analysis is given in a recent contribution by Macdonald and Haagensen [18] who emphasise the fact that assessment of recent research data has indicated the influence of a thickness stronger than g=0.25, so that in the latest HSE and API/ISO revision for offshore structures, a higher penalty factor of g=0.30 is imposed.
4. Further developments of the N-SIF-based approach and some answers to reviewers Due to lucky circumstances, the paper was reviewed by two anonymous referees whose suggestions and careful judgements (both those favourable and those quite critical) have been greatly appreciated by the authors. Some problems raised by the referees are intriguing, needing further investigation to be fully clarified and so they are surely of interest for many researchers engaged in fatigue design of components weakened by sharp stress raisers. For this reason, we have decided to report faithfully the referees’ opinions here. Reviewer A wrote that “the conservative assumption that the weld toe radius is equal to zero is helpful, because weld toe radius is not easy to measure, needs time and is affected by a large scatter”. Nevertheless, “this presentation of the fatigue resistance of the welded joints is of limited interest” for two reasons: (a) “it not easy to use the fatigue resistance curve for a weld toe geometry with an included angle different from 135°” (in that units for K N1 are no longer MPa·mm0.326); (b) “it is not possible to appreciate directly the fatigue reduction factor in presence of smooth welded joints”.
Fig. 4. Minimum values of q for aluminium and steel welded joints.
Reviewer A also wrote that the discussion “about the thickness effect is a more interesting contribution. It is based on the assumption that thickness effect is due to the stress gradient precisely described by the exponent of singularity”. In his opinion, the N-SIF-based analysis implies that: (c) “the thickness effect depends on the weld toe angle (so a universal value cannot be accepted)”; (d) “loading modes have no direct effect (which is not a true assumption)”. Only point (d) can be easily confuted. Different expressions are used in the paper for welded joints subjected to tensile loading and bending loading. This simply means that loading modes (and not only the global geometry) influence the intensity of the stress distributions in the neighbourhood of the sharp notch (but not the degree of singularity [20], which depends only on 2a). When given in terms of nominal stress ranges, the fatigue strength of welded joints subjected to bending loads is generally recognised as greater than that exhibited by the same joint under tensile loads. A reduction of 13% in averaged terms is reported by Hobbacher [21], a reduction ranging from 0% to 25% is shown in [20]. This scatter is no longer statistically significant if N-SIFs are used instead of nominal stresses for the simple reason that N-SIFs include the loading mode effect. Note that series St-11, 12, 15, 16 in Table 1 (t=100 mm) show a ⌬K N1 value that ranges, for bending and tensile loads, from 229 to 248 MPa·mm0.326. Point (c) reflects exactly the authors’ opinion when the geometry is weakened by sharp V-shaped notches. However, as soon as a notch with a tip radius r constant and different from zero is present, the situation becomes more complex. Obviously, stress distribution due to a rounded notch does not coincide with that of the sharp V-notch. A small zone exists in the close neighbourhood of the notch tip where the stress distribution substantially depends only on r, so that its features can be considered to a certain degree “universal”. Moreover, the stress gradient is not constant but varies as a function of the distance from the notch tip. Outside this limited zone, of which the dimensions are about 0.3 r, the influence of the opening angle becomes important and the stress gradient coincides with that related to the corresponding sharp V-shaped notch. The properties of the material determine whether the fatigue strength is controlled by the former or the latter zone. With regard to point (a), the authors agree with the reviewer. However, it is evident that an opening angle of about 135° represents the most common geometry and that small variations of the angle could be tolerated by engineers engaged in fatigue problems. One should also note that in a very accurate multi-parameter design optimisation of load-carrying fillet cruciform joints car-
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ried out by Radaj and Zang [22] (see also Radaj and Sonsino [23]), the only geometrical parameter considered constant is the opening angle. Conversely, there is no doubt that, from a theoretical point of view, the complex units of K N1 do not allow a direct comparison between joints with different opening angle. The problem, already highlighted by Hasebe et al. [24], can be overcome either by introducing a virtual crack at the notch toe [24,12] or by using, perhaps by taking a small step forward, Eqs. (2) and (3) to determine the energy in a small sector of radius R surrounding the sharp notch [25]. Such energy is strictly correlated to K N1 and K N2 but it obviously has the merit to be expressed in Nmm/mm3. As regards point (b), Hasebe et al. [24] were able to demonstrate that a precise analytical link exists between the N-SIF (determined for sharp-V-shaped notches under Mode I conditions) and the elastic peak stress value of a rounded V-shaped notch, the notch tip radius being small but different from zero. More precisely, Hasebe et al. wrote [24]: K N1⫽limr→0C˜r1−l1sq,max
(8)
where symbols of the original paper have been upgraded to current symbols. The parameter C˜ was summarised for several notch opening angles in [24]. Eq. (8) makes it evident that using K N1 or sq,max (after having introduced a small notch tip radius r) results in exactly the same fatigue predictions. Obviously, in real cases, one would use Eq. (8) in the presence of a well-defined value of r (for example rf=1, according to Radaj’s fictitious weld toe/root radius [26]). By using a complex potential function and Neuber’s conformal mapping, an equation analogous to Eq. (8) was reported also in [27]: K N1⫽
冉 冊
冑2p[1+l +c (1−l )] 1
1
4
1
q−1 q
1−l1
r1−l1sq,max
(9)
⫽1.22·r1−l1·sq,max where, on the right side of Eq. (9), a coefficient valid for the 2a=135° case is introduced. In Eq. (9), due to the absence of the limit condition r→0, K N1 was not thought of as numerically coincident with the value pertinent to the r=0 case. Eq. (9), without upgrading K N1, results in a strong simplification in the peak stress evaluation, but also some degree of inaccuracy. The problem is that of defining this degree of inaccuracy. Table 4 gives a precise idea of the errors for two geometries with L/t=0.5 and 1.0. The differences between analytical and finite element results vary from case to case, their mean value being about 10%. Note that some formulae summarised in [28], and suitable for estimating peak stress in cruciform welded joints, are acknowledged as being able to provide a substantially equivalent degree of accuracy [23]. It is evident that Eqs. (8) and (9) provide a bridging
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Table 4 Stress concentration factor Kt of transverse non-load-carrying fillet welded joints under tensile loads (in all FE analyses t=20 mm, 2h/t=1, see Fig. 1). Values of k1 determined according to Eq. (5a)) r/t
L/t
k1 Eq. (5a)
Kt Eq. (9)
Kt FEM
0.02 0.05 0.1 0.2 0.02 0.05 0.1 0.2
0.5 0.5 0.5 0.5 1 1 1 1
1.075 1.075 1.075 1.075 1.144 1.144 1.144 1.144
3.43 2.54 2.03 1.62 3.65 2.71 2.16 1.72
3.83 2.85 2.30 1.88 4.02 3.00 2.41 1.95
between the N-SIF approach and Radaj’s notch stress approach for welded joints [26,23], where fatigue predictions can be performed on the basis of sq,max (that is on the basis of the theoretical stress concentration factor Kt), but only after having introduced a precise value of the fictitious notch tip radius (rf=1 mm in most welded details of practical interest, but also rf=0.25 mm for spot-welded overlap joints made in rolled steels, rf=0.2 mm for cruciform joints in the presence of longitudinal shear loading [23]). Reviewer B “fully agrees with the N-SIF approach” since it “is based on a principle of similitude and overcomes problems encountered with predictions using a local strain approach and/or an integration of Paris’ relationship”. He has been “convinced for years that this is the best way for predicting the life of welded joint (as-welded) and that it should be included in codes”. Asking the authors to discuss the fatigue fracture of welded joints and to explain why the approach works, he helps them by providing the following convincing explanation: “A severe notch with a very small toe radius results in a short microstructural initiation life — even without toe ‘defects’ — and in an immediate crack propagation; this explains why the influence of a microstructure is weak. Furthermore, most of the life is consumed at short crack depth, within the singularity; that explains why good correlation is obtained with total fatigue life.” In addition, it might be useful to remember that LEFM stress intensity factor KI is analytically correlated to NSIFs [12]. A conventional evaluation of residual life from an initial crack value of 0.3 mm (the crack being thought of as through the thickness) turned out to be in a ratio of 1:3 to experimental total life, having assumed the exponent in the Paris law as being equal to 3.0 [12]. In conclusion, N-SIF is easy to calculate, and plays an essential role in short microstructural initiation life, short crack life and crack propagation life.
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5. Conclusions Notch stress intensity factors (N-SIFs) have been used, in a single scatter band, to summarise fatigue properties of aluminium welded joints (cruciform and T non-load-carrying or load-carrying fillet weld joints). More precisely, fatigue total life (and not only fatigue crack initiation life) has been correlated to the relevant Mode I N-SIFs, i.e. ⌬K N1. Welded joints were characterised by a thickness ranging between 3 and 24 mm, able to put in evidence any scale effect. Fatigue properties of aluminium welded joints have been compared with those related to steel welded joints (of which the main plate thickness varied from 6 to 100 mm). The mean ⌬K N1 values turn out to be in a ratio slightly greater than 2.0, while the scatter band size (mean value ± 2 standard deviations) practically coincides. Taking into account the only singular stress distribution (associated to Mode I fracture), the exponent quantifying the scale effect penalty is 0.326, which is quite different from the 0.25 value suggested by the Eurocodes. In the presence of a weld toe radius approaching zero, a statistical re-analysis of all fatigue data showed that a value equal to 0.3 was more realistic both for aluminium and steel welded joints. It is worth noting that in some recent standards on off-shore structures, the exponent 0.3 has already substituted the 0.25 value present in Eurocodes. Finally, the link between Mode I N-SIF and the peak value of the maximum linear elastic stress (determined in the presence of a small weld toe radius) was discussed. References [1] Williams ML. Stress singularities resulting from various boundary conditions in angular corners of plates in extension. J Appl Mech 1952;19:526–8. [2] Nui LS, Chehimi C, Pluvinage G. Stress field near a large blunted tip V-notch and application of the concept of the critical notch stress intensity factor (NSIF) to the fracture toughness of very brittle materials. Engng Fract Mech 1994;49:325–35. [3] Boukharouba T, Tamine T, Nui L, Chehimi C, Pluvinage G. The use of notch stress intensity factor as a fatigue crack initiation parameter. Engng Fract Mech 1995;52:503–12. [4] Verreman Y, Nie B. Early development of fatigue cracking at manual fillet welds. Fatigue Fract Engng Mater Struct 1996;19:669–81. [5] Lazzarin P, Tovo R. A notch stress intensity factor approach to the stress analysis of welds. Fatigue Fract Engng Mater Struct 1998;21:1089–103. [6] Maddox SJ. The effect of plate thickness on the fatigue strength of fillet welded joints. Abington, Cambridge: Abington Publishing, 1987.
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