Evaluation of nominal and local stress based approaches for the fatigue assessment of seam welds

International Journal of Fatigue 34 (2012) 86–102

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Evaluation of nominal and local stress based approaches for the fatigue assessment of seam welds T. Bruder ⇑, K. Störzel, J. Baumgartner, H. Hanselka Fraunhofer-Institute for Structural Durability and System Reliability LBF, Bartningstrasse 47, 64289 Darmstadt, Germany

a r t i c l e

i n f o

Article history:

Dedicated in honour to Professor Dr.-Ing. Timm Seeger’s 75th anniversary Keywords: Welded joints Fatigue strength Fatigue assessment concepts Steel

a b s t r a c t In the context of the German joint research project ‘‘Applicability of fatigue analysis methods for seam welded components’’, fatigue tests were performed by five universities and institutes on welded components, welded parts of larger structures as well as component-like samples of weld details. The sheet thickness t was in the range 1 mm 6 t 6 20 mm. The welding parameters for all test coupons and structures tested were chosen according to the industrial production process. Based on the data acquired, nominal, structural and notch stress approaches were analysed with regard to applicability and quality of assessment. The actual weld geometry except the real notch radii was taken into account within the notch stress approach. For the notch radii various values, the reference radii 0.05, 0.3 and 1 mm, were applied. Experimental and numerical results for welded steel components are presented. Approximately equivalent scatter ranges were obtained when applying the various approaches based on the current state of the art. It should be noted that both the nominal and the structural stress approaches are limited in their application compared to the notch stress approach. A comparison of the scatter bands obtained for the various approaches is subject to limitations because it was necessary, in each case, to use different test series as the basis for determining the scatter bands. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Various fatigue strength assessment methods [1] – nominal stress approach, structural stress approach and notch stress approach – are investigated based on the database obtained for steel and aluminium structures [2] in the joint research project ‘‘Applicability of fatigue analysis methods for seam welded components’’. Apart from the nominal stress approach, the concepts are based on local stresses of the welded joints that are defined in different ways. Their applicability is outlined. Compared to the nominal stress approach, both the structural stress approach and notch stress approach enable a significantly more detailed analysis of the fatigue strength of welded structural components. The structural stress approach largely takes account of the component geometry present in the joint region. The notch stress approach goes further and also takes into account the local shape of the welded joint. Local approaches conform to the possibilities of numerical calculation methods, typically the finite element method or alternatively the boundary element method, which, compared to the nominal stress approach, can be used to determine the stresses at a significantly smaller distance from the weld detail or within the weld detail itself. ⇑ Corresponding author. Tel.: +49 6151 705 285; fax: +49 6151 705 214. E-mail address: [email protected] (T. Bruder). 0142-1123/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2011.06.002

The fatigue strength data, necessary for application and evaluation of the various approaches, were determined in a number of research projects on selected components or welded parts of larger structures. In the following they are referred to as ‘‘sample components’’, Fig. 1. Test coupons represented fatigue-critical parts of welded joints typical of the sample components considered, Fig. 2. Detailed information on the various research projects is provided in [3–8]. This paper focuses on the results obtained for welded joints in steel. An evaluation of the results for welded aluminium alloy joints is carried out in [8,9]. Overall, approximately 500 fatigue tests on steel specimens were carried out in 32 test series, Table 1. 2. Applicability of calculation approaches The IIW Recommendations [10] provide design S–N curves for the fatigue assessment of welded joints. The S–N curves are described by straight lines in a double logarithmic plot given by the equation

N ¼ 2  106  ðDr=FATÞk ; N  Nk ; with number of cycles to failure N, nominal or local stress range Dr, slope k and stress range FAT in MPa at N = 2  106 cycles,

T. Bruder et al. / International Journal of Fatigue 34 (2012) 86–102

Nomenclature C CM LM FATx f(R) HT k

km Kf Kt M Nu N Nk Ni Nr Ps Ps-ref r rf rref R Rr

confidence level cross member longitudinal member classification reference to a S–N curve, in which x is the stress range in MPa at 2  106 cycles factor for mean stress correction according to IIW Recommendations post weld stress relief heat treated exponent (negative inverse of Basquin’s exponent) characterising the slope of a S–N curve in a double logarithmic plot stress magnification factor due to misalignment fatigue notch factor stress concentration factor mean stress sensitivity ultimate number of cycles number of cycles after which a tested specimen is regarded as run-out number of cycles to failure number of cycles at knee point of S–N curve number of cycles to initiation of technical size crack number of cycles to rupture probability of survival probability of survival of a reference S–N curve real notch radius fictitious radius suggested by Neuber–Radaj reference radius stress ratio related to external loading local stress ratio

Fig. 3. The FAT class depends on the assessment concept used and the structural weld detail, as well as on the base material (aluminium/steel) and relevant acting stresses (normal/shear stress). The design S–N curves are applicable for a probability of survival Ps = 95% with a two sided confidence level of the mean of C = 75%. The slope of the S–N curves for details assessed on the basis of normal stresses is k = 3 if not stated explicitly otherwise. The ‘‘knee point’’ of a S–N curve, which describes the point where the slope of a S–N curve in a double logarithmic plot changes (e.g. from k = 3 to 22) is assumed to correspond to Nk = 1  107 cycles. The basic requirements for the applicability of the approaches as described in IIW Recommendations are briefly summarised below. The enumeration marks indicate advantages (+), disadvantages () and neutral items (s), respectively.  Nominal stress approach s Requires a meaningful definition of nominal stresses  The detail catalogue provides FAT classes only for a limited number of weld details. + The S–N curve characterised by the FAT class typically takes into account the failure location (e.g. weld toe or weld root) for the weld detail of interest. + Easy to apply, if the weld detail is included in the catalogue and a nominal stress can be determined.  Imperfections such as centre-line mismatch (linear misalignment) and angular misalignment are covered to a certain degree in the fatigue resistance given by the FAT class. IIW Recommendations, as well as the German FKM Guideline [11], do not provide information on ‘‘upgrading’’ in the case of improved weld quality.

s t t Tr V WE WT WR z D

q re rhs rn rp rvM r1 r3 H

correction factor for equivalent stress hypothesis applied to q sheet thickness extent of isometric FE-mesh around a notch scatter index in load direction for nominal, structural hot spot or notch stresses; Tr = 1:(ra,Ps=10%/ra,Ps=90%) fatigue strength ratio weld start or stop position referred to as weld end weld toe weld root number of test series within a subgroup range material’s structural length according to Neuber ‘‘elastic’’ notch stress calculated using Theory of Elasticity structural hot spot stress nominal stress largest principal stress, either max r1 or max |r3| whichever greatest von Mises stress maximum principal stress minimum principal stress weld opening angle or notch opening angle

Subscripts a amplitude i index IIW International Institute of Welding

 Structural hot spot stress approach s Requires meaningful extrapolation of structural stresses for the detail of interest in order to derive the hot spot stress. s Anticipated joint misalignment has to be considered in the finite element model or is taken into account using a magnification factor km.  It is necessary to choose paths for the extrapolation to structural stresses. Various paths may exist for a given application. Various extrapolation methods have been described (location of reference points and linear or quadratic extrapolation), which, when determined by FEA, also depend on the mesh size chosen. Selection of the method is not always unambiguous.  In the IIW Recommendations, consideration is only given to failure starting from the weld toe. Currently, there are no recommendations for using the structural stress to assess joints with respect to failure from the weld root. Furthermore, FAT-values are only given for stresses acting normal to the weld. Data for welds under torsion are missing. s Due to the extrapolation procedure, this approach requires a somewhat greater effort compared to the nominal stress approach.  Notch stress approach s When applying the concept, it is essential to be able to reproduce the stress state at the weld detail using the reference radius chosen. + Applicable to most weld details for failure from a weld toe or root. Modelling guidelines must be considered with regard to stress analysis.

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Fig. 1. Sample components.

 At present, the stress state in the vicinity of the weld is not considered in the concept. Separate consideration has to be given to size effects. s The calculation of notch stresses demands more effort compared to the structural and nominal stress concepts. Applicability may be simplified by using suitable software, which allows an automated analysis based on coarse global FE models combined with an internal notch stress analysis based on fine models or published stress concentration factors for weld details. s The approach may be based on either the largest principal stresses or von Mises stresses. However, corresponding allowable stresses must be used.

3. Database for the evaluation Constant amplitude fatigue tests carried out during a joint research project provided the fatigue test database used in the present evaluation of the assessment methods [3–7,12]. From the selection described in Fig. 2, consideration is given to welded joints in shell-like steel structures in which the principal stress acts normal to the weld. The injector is not considered for this reason. The results for tube-flange connections under axial load welded over the entire circumference were also excluded due to huge variations in the weld quality and the small number of results. Similarly, the database obtained from the torsional tests

on drive shaft specimens and tube-flange connections was also considered to be too small for further evaluation and therefore is not dealt with here. An overview of all of the test series that are taken into consideration in the joint evaluation and the approaches applied is shown in Table 2. The fatigue tests were carried out until the failure criteria ‘‘initiation of a technical size crack’’, around a  0.1  t in depth, or ‘‘rupture’’ were reached. Initial cracks with a crack depth of a  0.1  t are detected. The S–N curves for welded joints provided in most design codes are based on the failure criterion of ‘‘rupture’’, most test data being obtained from coupon tests evaluated in this way. Therefore, the present results from coupon tests are evaluated for this failure criterion. In the coupon tests considered here it turned out that the about half the total life was spent producing a ‘‘technical size crack’’, with the other half spent propagating it to rupture. However, in some cases the crack propagation phase accounted for a larger share of the overall lifetime in the case of sample components, or it was not possible to achieve complete fracture at all within a selected limit number of cycles. For this reason, the initiation of a technical size crack was defined as the failure criterion for all the sample components. It should be mentioned that it was not possible to compare the assessment approaches using the results for identical test series because of the heterogeneous nature of the database. Due to variations in the individual conditions of test coupons and sample components, it was only possible to apply each of the approaches

T. Bruder et al. / International Journal of Fatigue 34 (2012) 86–102

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Fig. 2. Test coupons.

Table 1 Overview of specimens tested. Materials

DD13, H300LAD, H420LAD, S235JR, S275JR, S355J2G3, S460MC, S500MC, 42CrMo4 (SAE 4140), X17CrNi16-2 (AISI 431), X90CrMoV18 (AISI 440B)

Sheet thicknesses Welding procedure Number of test series Number of single tests

1 mm 6 t 6 20 mm MAG, TIG, laser beam welding 32 501

(nominal and structural hot spot stress approach as well as the notch stress approach with reference radii rref = 1.0 mm, rref = 0.3 mm and rref = 0.05 mm) to subsets of the test series. As a result, the evaluations were carried out on a database that differs somewhat for each concept. Therefore, it is not possible to rule out completely the impacts of individual effects on the assessment (differing scatters, variations in slope of the S–N curves, sheet thicknesses, etc. in each of the test series considered). An assumption on the position the knee point of a S–N curve was made for joint evaluations of test series on S–N diagrams: The knee point was assumed to correspond to Nk = 1  107 cycles, in accordance with the IIW Recommendations. 4. Results regardless of approach It is possible to make some general statements regardless of the approach used in each case for fatigue strength assessment. It

should be borne in mind that the tests cover a sheet thickness ranging from t  1 mm to t  20 mm; the majority of test coupons and sample components having sheet thicknesses of 1.6 mm 6 t 6 10 mm. The scatter for the test series examined, in which more than 5 specimens failed, ranges from Tr = 1:1.1 to Tr  1:1.5 if the maximum likelihood method proposed by Spindel and Haibach [13] is used to determine the exponent k and knee point Nk individually for each test series, Fig. 4. The individually determined number of cycles at the knee point of the S–N curve Nk varies between 1  106 6 Nk 6 1  107. Fig. 5 shows the slope k of the S–N curve observed in the experiments as a function of the sheet thickness t. Slopes in the range of 3 6 k 6 8 were obtained for test series with principal stress acting normal to the weld. Deviations of the slope k  3 can be established for sheet thicknesses t < 10 mm; the k-values increase as the sheet thickness decreases.

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Fig. 3. Fatigue resistance S–N curves for steel, normal stress, very high cycles applications [10].

The S–N curves are found to have shallower slopes in tests under torsional loading compared to tests with principal stress normal to the weld seam. 5. Approach-related evaluation The approach-related joint evaluation of test series in the form of bar charts and S–N curves was carried out as follows:  Determination of the slope ki of the S–N curve of the single test series using the maximum likelihood method specifying a cycle number of Nk = 1  107 at the knee point of the S–N curve,  Determination of the mean slope k of the S–N curves weighted with the number of tests,  Re-evaluation of all the tests using the mean slope obtained and a cycle number of Nk = 1  107 at the knee point of the S–N curve to determine the stress amplitude at the knee point and the scatter value 1:Tr of the reference S–N curve, which is specified for the probability of survival Ps = 50%. The test results for the design approaches considered are compared in each case with the design S–N curves shown in the IIW Recommendations [10] and their proposed extension [14]. These design S–N curves are applicable for the failure criterion of ‘‘rupture’’ and for a probability of survival Ps = 95% with a two sided confidence level of the mean of C = 75%. For a reasonable size of database, i.e. with approximately 10 to 20 test results, this corresponds to a probability of survival Ps  97.7%. The design S–N curves include allowance for high tensile residual stresses or applied stress ratios up to around R = 0.5. The reference S–N curves specified in this paper have been determined from tests with various stress ratios. For test series with stress ratio R – 0.5, in which it was unlikely that high tensile residual stresses were present, the test results were transformed to the stress ratio R = 0.5 using the factor f(R), if not stated explicitly otherwise. The fatigue strength ratio V is introduced in order to perform a general comparison of the design approaches among themselves based on a normalised variable. The fatigue strength ratio is defined as follows

V IIW;Psref ðNÞ ¼

ra;IIW;Ps¼97:7% ðNÞ ra;exp;Psref ðNÞ

where

ra;IIW ¼

 1 k FAT N  f ðRÞ  6 2 2  10

The stress amplitude ra,IIW is determined, depending on the approach, in accordance with the IIW Recommendations [10] whereby a mean stress correction is made using the factor f(R). The values suggested in the IIW Recommendations for the factor for mean stress correction f(R) depend both on the residual stress condition in the welded joint considered and the applied stress ratio. In the absence of any information about the magnitude of residual stresses, a conservative approach is to assume high (tensile) residual stresses and to choose, independently of the R-value applied, FATvalues which, as noted above, relate to such conditions. In practice, residual stresses due to welding will be largely relaxed in small coupon specimens with the result that some mean stress sensitivity is observed when they are fatigue tested. Unless specified otherwise, this is taken into account here with the factor f(R) in accordance with the IIW Recommendations. Fig. 6 illustrates the fatigue strength ratios V obtained for the concepts under consideration. In each case, the number of test series z on which the evaluation is based is shown. For a chosen fatigue assessment approach, the value of the fatigue strength ratio V(Ps-ref = 90%) is obtained using values of the corresponding reference S–N curve with a probability of survival Ps = 90%. For conservative analyses, a fatigue strength ratio VIIW, Ps-ref = 97.7% 6 1 corresponding to a probability of survival Ps = 95% with a twosided confidence level of the mean of C = 75% for both design and reference S–N curve should be achieved. Therefore, the upper value of the scatter band indicated for the fatigue strength ratio V(Ps-ref = 90%) in Fig. 6 should be below V = 1. All approaches are conservative at N = 2106 number of cycles to failure. At shorter lives with N = 1  105 cycles however, non-conservative results are observed. A cause for this may be seen in the slopes of k > 3 that are obtained for the reference S–N curves. The scatter values achieved for the reference S–N curves are important indicators of the quality of assessment. The scatter values are compared in Fig. 7 for the approaches examined. The

Table 2 Overview of test series. Results taken into consideration are marked with light grey background.

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0.99

20

Scatter of test series P = 50% P = 10%, 90%

Test coupons Sample components

10

Slope k

0.9

0.7

5

0.6 0.5 0.4

k = 3.75 k = 3.00

0.3

2 1

5

0.2

2

5

10

2

5

Sheet thickness t in mm

0.1

Fig. 5. Slope k of the test series versus the sheet thickness t.

0.01

z = 28 1.3

1.5

1.4

1.6

Scatter 1:T Fig. 4. Statistical analysis of the scatter based on z = 28 test series.

Fatigue strength ratio V

2.0

90%

1.0

0.5

z = 17

0.2

z = 23

z = 17

z = 13

z = 20

z = 28

0.1

w

10%

N rr otc e h /o f = 1 st w .0 res el m s d m en ds N rr otc e w f= h /o 0 st w .3 res el m s d m en ds N o rr tc w ef = h w 0 str el .0 e d 5 ss en m ds m N rr otc w ef = h /o 0 st w .0 res el 5 s d m en m ds H co ot nc sp ep ot t str es

s

Ps-ref

5

N = 1 10 6 N = 2 10

ss

1.2

tre

1.1

overall scatter value illustrated is made up of the scatter of the single test series (Fig. 4) and the scatter between the test series. Approximately equivalent scatter bands are obtained when applying the various approaches based on the current state of the art. For the notch stress approach, it is possible to see that the scatter increases in size as the reference radius decreases when taking into account all of the available test series in each case.

N co om nc in ep al t s

Probabillity P

0.8

Fig. 6. Fatigue strength ratio V and scatter (Ps-ref = 10% . . . 90%) derived from z test series for various assessment methods at N = 1  105 and N = 2  106.

3.0

Scatter 1:T

2.5 2.0 1.5 1.0

z = 17

z = 23

z = 13

z = 20

z = 28

0.5

s N co o m nc in ep al t str es

ss tre H co o t nc sp ep ot t s

N rr otc ef = hs 0. tr 05 es m s m

N rr otc ef = hs 0. tr 3 es m s m

N rr otc ef = hs 1. tr 0 es m s m

0.0

Fig. 7. Scatter 1:Tr of reference S–N curves derived for various assessment methods from z test series each.

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Fig. 8. Fatigue test results described with nominal stresses normalised by the respective FAT-class (mean stress correction fIIW(R) applied); z = 28 test series.

In anticipation of the evaluations presented later, it should be mentioned that the scatter value for the notch stress concept, particularly with reference radius rref = 0.05 mm, can be reduced significantly if subgroups are formed as a function of the crack initiation location. This is illustrated in Fig. 7 based on the example of an evaluation that excludes the test series in which failure originated from the weld start or stop position referred to as weld end. Moreover, it should be considered, in this case, that both the nominal and the structural stress approaches are limited in their use compared to the notch stress approach. A comparison of the scatter bands obtained for the various approaches is subject to limitations since, in each case, different test series were used as the basis for determining the scatter bands. 5.1. Nominal stress approach It is possible to perform an evaluation using the nominal stress approach for a total of z = 28 test series. The nominal stress is determined analytically for simple geometries; for more complex geometries, it is determined with the help of finite element calculations. It is not possible to determine a nominal stress for the sample components referred to as control arm and automotive longitudinal member in a clear and meaningful manner due to the complex geometry and loading conditions. Therefore those two sample components were excluded from the evaluation. Even if a nominal stress can be determined, for some joints it is not clear which FAT class has to be assigned. In these cases, a FAT class that gives the best approximation to the existing joint geometry is chosen. Fig. 8 shows the assessment according to the nominal stress approach in a normalised diagram. The single test results, which can be assigned a probability of survival Ps = 50%, are normalised by the FAT class (Ps  97.7%) appropriate to the relevant test series. In tests where the R ratio deviates from R = 0.5, the nominal stress amplitude is adjusted using the mean stress correction factor f(R) described in the IIW Recommendations in the event that no or only low residual tensile stresses are present. Despite allowing for the mean stress effects in accordance with the IIW Recommendations, the evaluation illustrates a large scatter range for the individual results of all test series considered here. The test results may be described by an S–N curve (Ps = 50%,

R = 0.5) having the parameters slope k = 4.9, normalised stress range at N = 2  106 cycles Drn,a,2E6,R0.5/FAT = 2.1, with a scatter value Tr = 1:2.51. Comparing the normalised fatigue strength calculated from the individual tests to the design S–N curve given by the IIW Recommendations generates a (sometimes highly) conservative estimation of the fatigue strength in the cycle range around N = 2  106. However, in some individual cases, the fatigue strength is also overestimated. 5.2. Structural stress approach FE models were created for the calculation of structural and notch stresses. These models were compared and, if necessary, adjusted in respect of the boundary conditions relevant to the test on the basis of experimental strain analyses. A determination of the structural stress at the weld, the hot spot stress, was performed for all the test coupons and sample components by an extrapolation (linearization at the surface) of the first principal stress to the failure-critical weld detail. For some specimens, going beyond the recommendations of IIW, the extrapolation was also performed on weld root notches, Fig. 9. The determination of a structural stress is not possible in test coupons and structural components where no stress extrapolation can be performed since it is impossible to define an appropriate path or sheet thickness. In line with the IIW recommendations, the hot spot stress rhs, was determined by linear extrapolation from the structural stresses rs located at distances of 0.4  t and 1.0  t from the weld toe or root.

rhs ¼ 1:67r0:4;t  0:67r1:0t

Fig. 9. Extrapolation path for determining structural hot spot stresses.

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Fig. 10 illustrates the results of the structural stress approach normalised by the respective FAT class. It is possible to evaluate 20 test series (with sheet thicknesses 1 mm 6 t 6 20 mm). The test results may be described by a S–N curve (Ps = 50%, R = 0.5) having the parameters slope k = 5.1, normalised stress range at N = 2106 cycles Drs,a,2E6,R0.5/FAT = 2.0, with scatter value Tr = 1:2.0 The overall scatter of the test results is somewhat lower than that for the nominal stress approach. An evaluation of the specimens, according to the structural hot spot stress approach based on the IIW Recommendations, results in a conservative estimation of the fatigue strength in the cycle range around N = 2  106. In the finite life range (N  1  105), the fatigue strength is overestimated to some extent due to the shallower slope of all the test series of k > 3. 5.3. Notch stress approach The relevant maximum value of the von Mises stress in the notch root along the weld, which is determined on the basis of 3D finite element calculations, is taken as the basis of the evaluation according to the notch stress approach. The FE models provide a realistic picture of the basic weld geometry determined from micrographs. Weld notches may be divided into weld toe notches (WT) and weld root notches (WR). Notches with a notch opening angle

H 6 90° should be described as weld root notches (WR). The notches arising in welds with H > 90°, e.g. radii at a transition of cross-sections, radii at an excess penetration of butt welds, are described as weld toe notches (WT). When mapping weld root notches in FE models, a distinction is made between what are referred to as o-notches (keyhole notches) and u-notches. If possible, e.g. in the case of weld root notches with a gap, a u-notch is modelled. In the case of weld root notches without a gap between the sheets, a decision must be made as to whether an o-notch or a u-notch will be selected. By modelling as a u-notch, either the sheets will be moved apart by double the reference radius or both sheet thicknesses will be reduced by the value of the reference radius. In these cases, an o-notch is used. To ensure comparability, modelling of the reference radii in the FE model was carried out uniformly for all weld details using the modelling specification shown in Fig. 11. The notch region is meshed using square elements in the 2D model and using hexahedral elements in the 3D model. The element edge length in the circumferential direction was 11.25°, i.e. 32 elements are used per layer for a 360° circumference. In the radial direction of the ‘‘notch tube’’, 6 element layers were chosen; the corresponding element edge length grows radially outwards in the ratio r6/r1 = 2.5. This results in virtually quadratic elements, having similar element edge length. The element edge length in the longitudinal direction of

Fig. 10. Fatigue test results described with structural hot spot stresses normalised by the related FAT class (mean stress correction fIIW(R) applied); z = 20 test series.

- Elements with quadratic shape function - 32 elements over 360° (u1 : 11.25°) - 6 elements over t* - Ratio of element length: f = ri+1/ri = 1.22 - Bias b = r6/r1 = 2.5 r i=6

i=5

i=4

u1 i=3

- h ≤ 10·r1

i=2 i=1

r1

95

h

t* Fig. 11. FE model in the notch area; for r = 1 mm, the main dimensions amount to r1  0.2 mm and t  2 mm.

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Fig. 12. Fatigue test results described with notch stresses using the reference radius rref = 1.0 mm (von Mises stress, mean stress correction fIIW(R) applied); z = 17 test series.

the weld, regardless of the prevailing stress gradient, should reach no more than ten times the element edge length in the circumferential direction. In the case of complex geometries, it is possible to attach tetrahedral elements outside the hexahedral-meshed ‘‘notch tube’’. The element transition may be implemented either by direct node connections or definition of a tied contact surface. Similar modelling recommendations are given in [15]. More recent studies show that, with appropriate modelling, the accuracy achieved with the discretisation used here can also be achieved with coarser FE meshes [16].      

Elements with quadratic shape function 32 elements over 360° (u1: 11.25°) 6 elements over t Ratio of element length: V1 = r2/r1 = 1.22 Bias b = r6/r1 = 2.5 h 6 10  r1

Submodels are generated for the relevant region around the weld joint if discretisation of the reference radii is not possible in the global model because of a component’s complexity. The calculations are typically performed geometrically linear. Linear elastic material behaviour is assumed in the vicinity of the reference radius for the notch stress calculation. The stress according to von Mises rvM is used as the equivalent stress hypothesis. For all FE-models second order elements have been used. In the following, the notch stresses calculated for individual tests are compared to permissible stresses in line with current recommendations. In this case, the assumption is a slope of the S–N curve k = 3. The current draft of the IIW Recommendations [10] contains design S–N curves for the reference radius rref = 1.0 mm. The publications [14,17] add design S–N curves for the reference radius rref = 0.05 mm and a suggested slope k = 5 for thin sheets under normal loading. 5.3.1. Notch stress approach with reference radius rref = 1.0 mm Fig. 12 shows the evaluation of the single tests according to the notch stress approach with reference radius rref = 1.0 mm. The design S–N curve for von Mises stresses with FAT 200 [14], which corresponds to FAT 225 for principal stresses given in the IIW Rec-

Fig. 13. Derivation of FAT-values for steel for the reference radius rref = 0.3 mm.

Table 3 Characteristic values for reference S–N curves using the notch stress approach with rref = 0.3 mm for seam welds made of steel and loading normal to the weld. Category

Weld root

Weld toe

re,a,vM,2E6

re,a,vM,2E6

R=0 Ps = 50%

R = 1 Ps = 50%

387 MPa Tr = 1: 1.5 k = 3.6 286 MPa Tr = 1:1.7 k = 4.9

487 MPa Tr = 1:1.6 k = 5.0 350 MPa Tr = 1: 1.6 k = 5.6

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Fig. 14. Fatigue test results described with notch stresses using the reference radius rref = 0.3 mm (von Mises stress, mean stress correction fIIW(R) applied); z = 23 test series.

Fig. 15. Fatigue test results described with notch stresses using the reference radius rref = 0.3 mm (von Mises stress, mean stress correction f(R) = 1); z = 23 test series.

ommendations, is conservative for N > 1  105 cycles. The scatter of the single test results is smaller than those of the nominal and structural stress approaches. Because of the large reference radius in relation to the sheet thicknesses examined (3 mm 6 t 6 20 mm), fewer tests can be evaluated (with z = 17 test series) than with the approaches described previously. The stress ratio is R P 0 in all the tests. From the test results, it is possible to derive an S–N curve (Ps = 50%, R = 0.5) having the scatter value Tr = 1:1.67, a slope k = 4.2 and a stress amplitude at N = 2  106 cycles re,a,vM,2E6,R0.5 = 196 MPa.

5.3.2. Notch stress approach with reference radius rref = 0.3 mm So far, there are no FAT values available for the reference radius rref = 0.3 mm, which fills the gap between reference radii rref = 1.0 mm and rref = 0.05 mm. There is a comparatively good

database available with 23 test series, which can be evaluated with this reference radius. According to a proposal by Sonsino [18], based on Lawrence’s procedure, in a first approximation the following FAT values for the reference radius rref = 0.3 mm are derived for steels from the values for the other reference radii, Fig. 13:  FAT 280 for von Mises stresses,  FAT 320 for principal stresses. In some cases, non-conservative results are achieved for N < 106 cycles using the values given above, Fig. 14. In the following diagram showing the influence of weld detail and stress ratio R on the S–N curve, the notch stress amplitudes assigned to the tests are given without any further mean stress correction, f(R) = 1, Fig. 15. In addition to the influence of the R

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Fig. 16. Fatigue test results described with notch stresses using the reference radius rref = 0.05 mm (von Mises stress, mean stress correction fIIW(R) applied); z = 13 test series.

Fig. 17. Fatigue test results described with notch stresses using the reference radius rref = 0.05 mm (von Mises stress, mean stress correction f(R) = 1); z = 13 test series.

ratio, one can also see slopes of the S–N curve k > 3 and, with a constant R ratio, higher endurable notch stresses in weld roots (represented by a dashed line) compared to weld toes (represented by a solid line), Table 3. 5.3.3. Notch stress approach with reference radius rref = 0.05 mm Due to the low number of test series that exhibit failure at weld ends, the corresponding results were not included in the evaluation using the notch stress approach with the reference radius rref = 0.05. This, it was possible to evaluate 13 test series. Fig. 16 shows that, in some cases, non-conservative results are achieved for N < 106 cycles. The scatter value illustrated in Fig. 16 can be reduced when evaluating the test series in subgroups divided according to R ratio and crack location, Fig. 17. While, for the reference radius rref = 1.0 mm, all the test series considered are well covered by a single reference S–N curve regardless of the crack location (weld

toe or weld root), a varying local fatigue strength is identified for the small reference radius for weld root and weld toe failure as well as for failure at weld ends (not illustrated), Table 4. The design S–N curve (FAT 560) given in [14] is conservative for N > 1  106 cycles. Table 4 Characteristic values for reference S–N curves using the notch stress approach with rref = 0.05 mm for seam welds made of steel and loading normal to the weld. Category

re,a,vM,2E6

re,a,vM,2E6

R=0 Ps = 50%

R = 1 Ps = 50%

461 MPa Tr = 1:1.74 k = 6.1

732 MPa Tr = 1:1.82 k = 5.2 637 MPa Tr = 1:1.68 k = 5.9

Weld root

Weld toe

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Notch stress amplitude

e,a,vM,2E6

in MPa

rref = 0.05 mm, weld root, R = -1 rref = 0.05 mm, weld toe, R = -1 rref = 0.05 mm, weld toe, R = 0 rref = 0.30 mm, weld root, R = -1 rref = 0.30 mm, weld toe, R = -1 rref = 0.30 mm, weld root, R = 0 rref = 0.30 mm, weld toe, R = 0 rref = 1.00 mm, R = 0

e,a,vM

2000

Notch stress amplitude

5.3.4. Overall consideration for the reference radii examined By way of summary, Fig. 18 shows the reference S–N curves determined for the notch stress approach. Dashed lines apply to the crack location at weld root, solid lines to the crack location at weld toe. All S–N curves have slopes k > 3. As anticipated, lower fatigue strengths are achieved under tensile or compressive loading (R = 0) compared to reversed loading (R = 1). As explained in the previous sections, the influence of the failure location on the local fatigue strength can be identified for the reference radii rref = 0.3 mm and rref = 0.05 mm. A single reference S–N curve per reference radius proves to be inadequate, particularly for rref = 0.05 mm and rref = 0.3 mm, for obtaining acceptable correlation of the data. Accuracy may be increased by creating subgroups. This is illustrated in Fig. 19 based on the evaluations using reference radius rref = 0.3 mm and the influencing factors R-ratio and crack location. Lower stresses can be endured at weld toe notches than at weld root notches. Figs. 6 and 12 show that, for a reference radius rref = 1.0 mm, all the test series considered are well covered by one reference S–N curve for the different crack locations (weld toe or weld root). Since most of the database came from tests performed under fully tensile loading (R = 0), a quantitative statement on the mean stress influence would be questionable. For the smaller reference radii, a varying local fatigue strength is identified for weld root and weld toe failure as well as a varying mean stress dependency, Fig. 6. The

1000

k = 5.9

k = 5.2

k = 6.1 k=5

500

k = 5.6

rref = 0.05 mm

k = 3.6

k = 4.9

rref = 0.3 mm

k = 4.2

200 rref = 1.00 mm

100

PS = 50%

5

10

5

2

5

10

6

2

5

7

10

2

Cycles to failure N Fig. 18. Derived reference S–N curves for the notch stress concept, f(R) = 1.

influence of the crack location increases as the reference radius decreases. For failure at weld start and stop positions in weld root notches, a higher local fatigue strength compared to the mid section of the seam weld is derived when applying the notch stress approach using the reference radii rref = 0.05 mm and rref = 0.3 mm. For the specimens exhibiting failure at weld start and stop positions, a

800 700 600 500 400 300 200

z=3

z = 13

z=4

z=5

Failure at weld toe

Failure at weld toe

Failure at weld root

Failure at weld root

R=-1

R=0

R=-1

R=0

100 0 10%

PS 90%

Fig. 19. Endurable notch stresses for Ps = 50% at N = 2  106 and scatter for the reference radius rref = 0.3 mm, separated by R-value and location of crack initiation.

Fig. 20. Recommendation for choosing reference radii according to the sheet thickness and the location of crack initiation.

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meaningful notch stress cannot be derived using the reference radius rref = 1.0 mm due to the low sheet thickness. Therefore, the behaviour of weld start and stop positions in weld root notches is not investigated for this reference radius. Different reference radii rref are recommended depending on the sheet thickness t, as shown in Fig. 20. The geometric boundary conditions are decisive for the recommendation. In individual cases [19], e.g. the butt joint under axial load, it is possible to apply large reference radii even with comparatively low sheet thicknesses as shown by the ratio of rref/t = 0.2. 6. Size effects in the notch stress approach The reference radius of rref = 1.0 mm proposed in [20] for thick sheets is based on empirical studies and coincides with a suggestion by Radaj [21] who derived a fictitious radius of rf = 1.0 mm based on an approach to averaging the stresses in the notch ligament according to Neuber [22]. Applying the stress averaging as proposed by Radaj, the effective stress reff, which strictly speaking is only applicable for weld root notches with weld opening angle H = 0°, results from

reff ¼ K f ðrÞ  rn ¼ K t ðrf Þ  rn where rf ¼ r þ s  q For the limit case of infinitesimally small radii r ? 0, the equation yields a fictitious radius rf = 1.0 mm using, for welded steel structures, a fictitious structural length q = 0.4 mm and, for the von Mises equivalent stress hypothesis, a correction factor s = 2.5. Based on the studies on spot-welded joints [23], Eibl [24,25] proposed a master S–N curve for laser beam welded thin, overlapping sheets (t 6 3 mm) using the reference radius rref = 0.05 mm. The radius rref = 0.05 mm approximates the notch stresses in the base of a crack-like notch. No direct substantiation of this radius is provided by the concept of the micro support effect. A comparison, mainly on laser beam-welded lap joints with square groove butt welds, in which failure occurred from the weld root, produced good results for thin sheets with this reference radius [26]. For assessing welded joints connecting thin sheets with the notch stress approach, a reference radius rref = 0.05 mm is usually chosen. In such a case, the design S–N curves presented depend – apart from the R ratio – on the crack location (weld root or weld toe). Such a significant influence cannot be observed when using a reference radius rref = 1.0 mm, as is often used for assessing welds in thick sheet structures. The differences in the local fatigue strength between weld toe and weld root, when applying the reference radius rref = 0.05 mm, may be described to a significant extent by using what are referred to as size effects. Using the radius rref = 1.0 mm may cover size effects, e.g. resulting from the gradient of the stresses arising in the direction normal to the weld, by rounding the real (worst case) notch radius, which is assumed to be r ? 0. By way of example, this will be illustrated in the following for a double-sided notched flat bar under axial load. A size effect resulting from the gradient of the stresses arising in the direction normal to the weld is taken into account in this case via a stress averaging approach according to Neuber [22]. A stress averaged over a structural length is determined as the failure-relevant (effective) stress, as also proposed by Zhang [27]. Depending on the notch opening angle, the stress field in the notch ligament differs and therefore different effective stresses result for the same notch radius. For a small notch opening angle (e.g. H = 15°), such as is typically present in weld root notches, a steeper stress gradient results and therefore a lower normalised effective stress than for a large opening angle (e.g. H = 135°), such as is present at weld toes. As shown in Fig. 21, with the same maximum notch stress, this results in an

Fig. 21. Size effect depending on stress field in notch ligament.

effective stress that is 30% higher for the weld toe than for weld root notches. This finding corresponds in good approximation to the different positions of the design S–N curves for weld toe and weld root notch highlighted in Section 5. Furthermore, it substantiates, at least qualitatively, that standardising the design S–N curves for the ‘‘rref = 0.05 mm’’ approach using effective notch stresses is possible. In addition, methods are proposed for considering size effects due to varying weld length and stress distribution along the weld [26]. In further studies on test specimens, the relevant local weld geometry and its local stresses and strains in the vicinity of the failure location should be considered in detail due to their influence on fatigue life. 7. Conclusions Regardless of the individual approach, the following is observed for the steel structures considered [12]:  The scatter Tr for the test series examined with individual determination of slope k and knee point Nk lies in the range between Tr = 1:1.1 and Tr  1:1.5.  Choosing a slope of the S–N curve dependent on detail and load can increase the accuracy of analysis. Slopes in the range of 3 6 k 6 8 were observed for test coupons and sample components examined with stress normal to the weld. More marked deviations from the slope k  3 occur as the sheet thickness decreases. This is why it is not possible to conservatively evaluate some of the test results for N < 106 cycles using the IIW design S–N curves. For thin sheets under normal loading condition, design S–N curves were proposed using a slope of k = 5 [17].  In the majority of the test coupons, the ratio of the number of cycles to failure for the criteria of ‘‘initiation of a technical size crack’’ and of ‘‘rupture’’ is Ni/Nr > 0.5. A large proportion of the crack propagation phase was observed in the overall lifetime in the more complex sample components, Ni/Nr 1. For the comparison with the IIW Recommendations, it is suggested by way of simplification to apply rupture as a failure criterion for tests on coupons. A fatigue strength analysis of components using the approaches compared should be carried out for the failure criterion of ‘‘initiation of a technical size crack’’. The number of cycles endurable after crack initiation of technical size up to rupture depends largely on the component geometry

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and load. Stress redistributions that affect further crack growth may occur following the initiation of a technical size crack.  Several test series are described well by S–N curves without a knee point before reaching the ultimate number of cycles chosen for the tests nu = 1  107. This observation is in line with the IIW Recommendations, which assume a knee point of the S–N curve at Nk = 1  107 cycles.  Approximately equivalent scatter ranges are obtained when applying the various approaches based on the current state of the art, allowing for the number of test series considered, Fig. 7. It should be noted that both the nominal and the structural hot spot stress approaches are limited in their application compared to the notch stress approach. A comparison of the scatter bands obtained for the various approaches is subject to limitations because it was necessary, in each case, to use different test series as the basis for determining the scatter bands. In respect of each approach, one may state:  Nominal stress approach There is confirmation of the assignment of weld details to FAT classes according to the IIW Recommendations’ weld details catalogue. However, there are several sample components that have failure-critical weld details that cannot be assessed using the nominal stress approach, because either a nominal stress cannot be calculated or a notch detail cannot be allocated.  Structural hot spot stress approach In addition to external linearization for extrapolation of the stresses at the weld toe, for the welded joints investigated, it also proved possible to assess such weld details where a linearization of the structural stress distribution along the surface was performed towards the root notch. Attention has to be paid to the fact that the linearization path is positioned at the sheet with the highest stresses. If higher structural stresses can be identified in the weld itself (for example via through-thickness linearization), an extrapolation of surface stresses may deliver too small structural hot spot stresses and would lead to a non-conservative fatigue assessment.  Notch stress approach The accuracy of the approach may be increased or the scatter bands reduced if – particularly with the reference radius rref = 0.05 mm – reference S–N curves are used for subgroups. The subgroups arise as a function of the crack location (weld toe, weld root, weld start or stop position). Reference radii rref are recommended for different sheet thicknesses t, as shown in Fig. 20, The studies show that, on the basis of the existing guidelines, welds are sometimes designed very conservatively, Fig. 6. Applying the notch stress approach using the reference radii rref = 1 mm, rref = 0.3 mm and rref = 0.05 mm and the derived reference S–N curves allows weld joints with various weld details to be optimised and designed reliably. The detailed stress analysis allows an assessment of the influence of deviations between an assumed and the actual weld geometry as well as deriving specifications for the weld geometry to be aimed at in the welding process. Structures with high residual stresses should be evaluated with S–N curves transformed to the stress ratio R = 0.5. As a result, it is possible to achieve a more cost-effective and, at the same time, safer design of welded components subject to cyclic loading.

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Acknowledgements The authors would like to thank the following researchers who kindly provided results of their experimental investigations:  M. Vogt, K. Dilger, Institute of Joining and Welding, TU Braunschweig, Braunschweig, Germany  E. Hanssen, K. Dilger, Institute of Joining and Welding, TU Braunschweig, Braunschweig, Germany  von Lilienfeld-Toal, H. Paetzold, W. Fricke, Institute of Ship Structural Design and Analysis, Hamburg University of Technology (TUHH), Hamburg, Germany  Pyttel, C. Berger, State Materials Testing Institute and Institute for Materials Technology, Technische Universität Darmstadt, Darmstadt, Germany  J. Willen, A. Esderts, Institute of Plant Engineering and Fatigue Analysis IMAB, TU Clausthal, Clausthal-Zellerfeld, Germany Furthermore the authors would like to acknowledge the financial support for a part of the investigations presented. The German Welding Society (DVS, Düsseldorf) and the Research Association of the German Automotive Industry (FAT, Berlin) accompanied a part of the investigations and provided financial support from the Federation for Industrial Research Associations (AiF, Cologne) which is linked to the Federal Ministry of Economics (BMWi, Berlin). References [1] Radaj D, Sonsino CM, Fricke W. Fatigue assessment of welded joints by local approaches. 2nd ed. Woodhead Publishing Limited; 2006. [2] Festigkeit geschweißter Bauteile (Strength of welded components); Anwendbarkeit lokaler Nachweiskonzepte bei Schwingbeanspruchung (Applicability of local approaches for fatigue strength assessment), DVSForschungskolloquium in Braunschweig am 17. und 18. März 2009, DVSreport no. 256; 2009. [3] Baumgartner J, Bruder T, Hanselka H. Fatigue strength of laserbeam welded automotive components made of thin steel sheets considering size effects. Int J Fatigue, this issue. doi:10.1016/j.ijfatigue.2011.01.022. [4] Esderts A, Willen J, Kassner M. Fatigue strength analysis of welded joints at closed steel sections in railway vehicles. Int J Fatigue, this issue. doi:10.1016/ j.ijfatigue.2011.06.007. [5] Hanssen E, Vogt M, Dilger K. Fatigue assessment of arc welded automotive components using local stress approaches - exemplary study on basis of a track control arm. Int J Fatigue, this issue. doi:10.1016/j.ijfatigue.2011.02.001. [6] Pyttel B, Grawenhof P, Berger C. Application of different concepts for fatigue design of welded joints in rotating components in mechanical engineering. Int J Fatigue, this issue. doi:10.1016/j.ijfatigue.2011.01.007. [7] Vogt M, Dilger K, Kassner M. Investigations on different fatigue design concepts using the example of a welded crossbeam connection from the underframe of a steel railcar body. Int J Fatigue, this issue. doi:10.1016/ j.ijfatigue.2011.01.017. [8] Störzel K, Bruder T, Hanselka H. Durability of welded aluminium extrusion profiles and aluminium sheets in vehicle structures. Int J Fatigue, this issue. doi:10.1016/j.ijfatigue.2011.01.006. [9] Störzel K, Festigkeit geschweißter Bauteile, Anwendbarkeit lokaler Nachweiskonzepte bei Schwingbeanspruchung, Strangpressund Blechstrukturen aus Aluminiumknetlegierungen im Fahrzeugbau. In: DVSreport no. 256; 2009. p. 135–44. [10] Hobbacher A. Recommendations for fatigue design of welded joints and components, IIW Document No. IIW-1823-07. Int Inst Weld; 2008. [11] Hänel B, Haibach E, Seeger T, Wirthgen G, Zenner H. FKM-Guideline – Analytical strength assessment of components in mechanical engineering. 5th, revised edition, Frankfurt/M.: VDMA Verlag GmbH; 2003. [12] Bruder T, Vogt M. Festigkeit geschweißter Bauteile; Anwendbarkeit lokaler Nachweiskonzepte bei Schwingbeanspruchung, Gesamtbewertung der Clusterergebnisse. In: DVS-report no. 256; 2009. p. 145–53. [13] Spindel JE, Haibach E. The method of maximum likelihood applied to the statistical analysis of fatigue data including run-outs. Int J Fatigue 1979;1:81–8. [14] Sonsino CM. A consideration of allowable equivalent stresses for fatigue design of welded joints according to the notch stress concept with the reference radii rref = 1.00 and 0.05 mm. Weld World 2009;53(3–4):R64–75. [15] Fricke W. Guideline for the fatigue assessment by notch stress analysis for welded structures, IIW-Doc. XIII-2240r1-08/XV-1289r1-08. Int Inst Weld; 2008. [16] Baumgartner J, Bruder T. An efficient meshing approach for the calculation of notch stresses, IIW-Doc. XIII-2313-10. Int Inst Weld; 2010

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