A. HOBBACHER Welding Laboratory, Fachhochschule Wilhelmshaven, D Wilhelmshaven, Germany

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1 ,%gineerhg Fracture Mechanics Vol. 46, NO. 2, pp , /93 $ Printed in Great Britain. Pqamon Press Ltd. STRESS INTENSITY FACTORS OF WELDED JOINTS A. HOBBACHER Welding Laboratory, Fachhochschule Wilhelmshaven, D Wilhelmshaven, Germany Ahatrae-Stress intensity factors of welded joints have been calculated for a variety of joint types and dimensional parameters. The following have been considered: non-loadcarrying transverse and longitudinal attachments, cruciform joints with K-butt and with lillet welds, and lap joints with fillet welds. The correction function of the stress intensity factor Y, has been split into a function for the general con8guration Y, and a function M,, which covers the local stress concentration field of the weld. Parametric formulae have been established for Mk. 1. INTRODUCTION FOR WELDED joints, only a few solutions for the calculation of stress intensity factors can be found in current publications or compilations. This might be due to the fact that structural details at welded joints differ widely from the geometries of standard test specimens used in material science. However, weld imperfections, such as cracks or cracklike planar defects, can be detected more and more precisely by modern non-destructive testing methods, and so the procedures of assessment by fracture mechanics become more developed and are widely used. One drawback is the absence of parametric formulae for the calculation of stress intensity factors at most geometric configurations of welded joints. In general, stress intensity factors are calculated with the basic formula [eq. (l)]. For a welded joint it proved to be useful to split the universal correction function Y,(a) into a function Y(a), which refers to a solution of a standard crack configuration, and a second function M&z), which gives a correction according to the local stress concentration of the weld. K=a.~(xa).Y,=a.~(rra).Y.M,. (1) Usually, stresses are not evenly distributed along the wall thickness. The total notch stress can be separated into different stress parts: the membrane stress o,,,, the shell bending stress ab, which is linearly distributed, and the nonlinear peak stress a,,(x), which is caused at the local stress concentration [l] (Fig. 1). For a given stress distribution a(x) for x = 0 at one surface to x = t at through thickness, an analytical separation can be performed by a,=-* 1 t I x=, x-0 a(x) * dx a,(x) = a(x) - a, - ah(x). (2) Fig. 1. Stress components in a welded joint. 173

2 174 A. HOBBACHER Membrane stress and shell bending stress, strictly seen, require separate correction functions. So, eq. (1) becomes K = J(m) (a,. y,,, Mk,m + uh yh Mk.h). (3) 2. CALCULATION OF STRESS INTENSITY FACTORS For the establishment of a parametric formula for a specific welded joint type, a variety of dimensional parameters have to be considered. For each combination of parameters, the stress intensity factors for least 10 different crack depths along the anticipated crack path have to be calculated. At a minimum of 20 combinations of parameters, about 200 results have to be determined, serving as the basis for a regressional analysis. One economical method of calculation of stress intensity factors is based on the weight function approach [2]. Here, a standard specimen configuration with a known solution is taken as a reference. Then, the stress concentration of the untracked welded joint along the anticipated crack path is calculated (Fig. 2). Finally, from both data the stress intensity factors can be derived. For one combination of dimensional parameters only one finite element calculation run is required. A simplified rapid calculation procedure of this method is described in ref. [3]. A pilot application to a welded joint application can be found in ref. [4]. The formula developed in ref. [3] can be written as K=&+ s p a(x) 0 J(a - x2) dx. (4 Equation (4) can be generalized by division by K = aj(mz), giving a direct formula for the Mk -value. The stress concentration factor K,(x) along the anticipated crack path can be easily calculated by finite elements. The integration can be carried out numerically. For each combination of dimensional parameters, an interpolation formula in the form of k has to be derived by a regression. The constant C and the exponent k are then dependent on the dimensional parameters. They can be determined by a multi-dimensional regression analysis. The accuracy of the method was checked by a comparison with results published in ref. [S]. There, the radius at the weld transition was set to zero, giving a sharp concave edge with a singularity. The basic plate thickness T was 40 mm. The stress intensity factors beginning at a crack depth of 0.02 mm were calculated by special crack tip elements. The very small crack tip element Fig. 2. Reference crack configuration and structural detail

3 Stress intensity factors of welded joints 175 Transverse non-loadcarrying attachment Cruciform Joint K-butt weld - Membrane Stress 5 I + I- Membrane Stress ai 1 10 Crack depth parameter a [mm] Cfack depth parameter a Imml Fig. 3. Comparison of methods. was surrounded by a relatively coarse overall meshing. In this study the FEM mesh width was 0.05 mm, so valid results can be expected starting from the 3rd node, i.e. from a crack depth of about 0.1 mm. The overall meshing was relatively fine, giving 1100 nodes per quarter model. The comparison is shown in Fig. 3. The results are in full conformity at the transverse non-loadcarrying attachment. At the cruciform joint with K-butt welds there is a difference of about 3% in membrane stress and 5% in bending. This might be due to different boundary conditions or restrictions in displacements, which have not been reported. The differences between membrane stress and bending are roughly 13%. Using only membrane stress solutions is conservative. 3. TYPICAL STRUCTURAL DETAILS AT WELDED JOINTS A special problem when modelling welded joints is the irregularity of the weld toe transition radius. In ref. f6] the effect of different toe radii has been studied. It was shown that in the interesting area of application, at a technical crack, the radius is of minor significance. So, for simplicity of modelling and conservativeness of result a radius of zero was chosen. The mesh refinement is such that crack depths a/t > give valid results. 3.1 Transverse non-loadcarrying attachment The effect of the ratio of wall thickness t/t is small. Within the practical range of variation of the dimensional parameters (Table 1) t/t < 2 it is less than 5%. Almost the same applies for the weld throat A (Figs 4 and 5). This is due to the fact that no forces are to be transmitted by the weld. At very small welds, not considered in this study, there might be a bigger effect due to the constraint between plate and attachment, which has to be carried over by the weld. An effect of Table 1. Variation of dimensions and validity range (transv. att.) Dimension Min Max HIT WIT 0.2 e 15 6& AIT rlt @) Table 2. Variation of dimensions and validity range (cruc. K-butt) Dimension Min MZLX HIT IT I; e 15 6L.W tlt

4 .I 176 A. HOBBACHER Fig. 4. Transverse non-loadcarrying attachment. the attachment height is non-existent. The only considerable effect is the transition angle 8 at the weld toe. In the regressional formula it is expressed by the weld legs H and W. C = k= g g *+00815!.! (T) * (T) * (T) Cruciform joints with K-butt we& Cruciform joints with K-butt welds usually contain a fillet weld at both sides. At a complete penetration, as considered here, the only variation in weld dimensions can be at the fillet (71 Transv. att: Effect of wall thickn. t/t Transv. attachm: Effect of height V/T Or-_ Mk-lactor - t/t Mk-tllctc, l-----l - V/L V/T. t + V/L. 15 / \I Crack depth a/t 0.01 Crack depth a/t Q1 Transv.attach: Effect of weld throat A/T Transv. att: Effect of weld angle theta IO,,- MI k-factor + A/T AIT. 0.6 a Oack depth a/t It CII)ck depth n/t Fig. 5. Effect of dimensional parameters at a non-loadcarrying transverse attachment.

5 Stress intensity factors of welded joints 177 Fig. 6. Cruciform joint K-butt weld. (Table 2). There is almost no effect of the weld throat A at the dimensions studied (Figs 6 and 7). At very small fillets, which are smaller than the size of the stress concentration field, a more pronounced effect is expected. The ratio of the wall thicknesses t/t also gives no effect. The same applies with the length of the transverse plate V. The only considerable effect is due to the variation of transition angle at the weld toe 8. In the regressional formula it is represented by the weld legs Hand W. C = O/W(;) + O.l596(;y + 0.,7,,(;) - O.l,,($~ k = (;) ($) (F) - o.o!$;~. (8) Cruc. K-butt: Effect of wall thlckn. t/t Cruc. K-butt: Eff. of member lengths V/T t-r : : : :u::: : : : ::i+tl : Crack depth s/t 0.01 CWck death a/t t Cruc. K-butt: Effect of weld throat A/T Cruc. K-butt: Effect of weld angle theta Mk-factov,n, 10, sol QmktWpth a/t O.Ol Cmckdwtn UT Fig. 7. FSect of variation of dimensional parameters at cruciform joint K-butt weld.

6 178 A. HOBBACHER Fig. 8. Cruciform joint fillet weld Cruciform joints with fillet weld At cruciform joints with fillet welds two loci of crack initiation exist, firstly the weld transition toe at the surface of the main plate and secondly at the weld root at the end of the root gap (Fig. 8). The first crack will propagate through the main plate whereas the second one propagates through the weld throat. In this study, only toe cracks have been considered. Some solutions for root cracks have already been publish~ elsewhere, but will be left to a more detailed future study. The variation of dimensional parameters (Table 3) shows almost no effect of the ratio of the wall thickness t ft (Fig. 9). No effect also is encountered for different lengths of the transversing plate V/T. Weld throat A/T has a strong effect, which is obvious, because all of the forces have to be transmitted through the throat. At small throats the graph of!he M,-values deviates considerably from a straight line in the double-log diagram. So, it cannot be represented by a simple Cruc. fillet: Effect of wall thickn. t/t Cfucfillet: Effect of member length V/T 10 Mk-lactot A Mk-Iactor CXick depth a I T G ack depth 8,s Cruc. fillet: Mk-faClOr Effect of weld throat A/T ICI C&x. fillet: Effect of weld angle theta &sick depth B/T WaCk aepth a/t 1 Fig. 9. Effect of dimensional parameters at cruciform joint with fillet welds.

7 Stress intensity factors of welded joints 179 Table 3. Variation of dimensions and validity range (cruc. fillet) Dimensions Min Max HIT 0.2 I WIT :,T lT Table 4. Variation of dimensions and validity range (lap) Dimensions Min Max HIT WIT U/T AIT r/t exponential formula. In order to maintain the simplicity of the regression, the area of the graph has been divided into three portions. The variation of the transition angle at the weld toe 8 reveals a strong effect. Both dimensional parameters, weld throat A and weld angle 6, can be expressed in terms of the weld legs H and W,.of which the interpolation formulae make use. k hfk> 1. If 0.2 < HIT < 0.5 and 0.2 < W/T < 0.5 and a/t < 0.07 then: C = (;) -,.,,56(;) k = ($) +,.4654(F). (10) If 0.2 < HIT < 0.5 and 0.2 < WIT -K 0.5 and a/t > 0.07 then: C= @ ($) k= ($)+.4319(;). (11) If 0.5 < H/T < 1.5 or 0.5 < W/T c 1.5 then: C = (T) g + 0 * 1753 (T) g (T > k = (12) 3.4. Lap joints with fillet wel& The variation of the dimensional parameters (Fig. 10 and Table 4) suggests an effect of the wall thickness ratio t/t. However, wall thickness cannot be varied independently from the weld legs H and W. Small thicknesses at the overlapping plate require small weld legs H and put the variation parameters out of balance. The multivariate regressional analysis eliminates this dependence. There is a clear effect of the weld throat (Fig. 11). In the interpolation formula it is Fig. 10. Lap joint.

8 180 A. HOBBA~HER Lap Joint: Effect of wall thickness t/t Lap Joint: Effect of overlap length U/T Mk-iactor Ml I + t/t-o.5 + U/T * U/l = U/T U/T-O6 -+ U/T - CJ 76 - A/T - t ,ot 0.1 Cack depth a/t Back depth a/t Lap Joint: Effect of weld throat A/T Lap Joint: Effect of weld angle theta 10 Mk-faCtOr Mk-feckx - A/T * A/T met.9-58 dsg * A/T-O theta - 46 d q + #T-Q28 -e- thete - 30 defj OfJO O.Of at Crack depth e/t Crack d&&h a/t 1 Fig. Il. Effect of dimensional parameters at lap joints. represented in terms of the weld legs H and W. The same applies to the effect of the weld angle 8. The overlap length at the main plate gives an ad~tional support to the lap plates and restricts the bending displacement by contact. This is effective at a small overlap. C = 1.02fO ($) ~(~~ - O OI87(~~ ~~) ~~~~ k = *($) +,.3409(,w) - 0.0**4(;) + 0.0*77(F) (;). (13) 35. Longitudinal non-loadcarrying attachments The three-dimensional geometry of the lon~tudinal non-load&a~ing at~chment (Fig. 12) requires a corresponding finite element analysis. So, isoparametric solid elements have been used. At the same mesh refinement as for the other two-dimensional structures, a finite element model of about 4000 nodes was created and varied in dimensional parameters (Table 5). Results have been compared with a study carried out elsewhere [7]. The results differ by 8% to the conservative side. After request, it was communicated that the published results [7] had been calibrated to strain gauge measurement and fatigue tests by about this amount. In pilot studies the height of the attachment V/T was varied from 2.5 to 5. In this range only a negligible effect was found. Several design codes recommend bevelling of the attachment. Within the variation of the bevel angle from 90 down to 45 almost no effect was found. This indicates that a reassessment is necessary.

9 Stress intensity factors of welded joints 181 Table 5. Variation of dimensions and validitv range tlona. att.) Dimensions Min Max LIT 5 40 BIT e/4s [IT Table 6. Principal significant dimensional parameters No. Joint type Principal significant parameters 1 Non-loadcarrying transverse Weld throat, weld angle attachment 2 Cruciform joint K-butt weld Weld angle 3 Cruciform joint fillet weld Weld throat, weld angle 4 Lap joint transverse fillet weld Plate thicknesses, overlap length, weld throat, weld angle 5 Longitudinal non-loadcarrying Plate thicknesses, attachment attachment length, plate width, weld angle The variation of the ratio of the wall thicknesses t/t has a clear effect (Fig. 13). The thicker the attachment, the higher the stress concentration and so the &-value. The well known effect of the attachment length also appears from the calculations. From a length L/T = 2.5 the stress concentration rises constantly, but at an asymptotic effect,is encountered. In order to reduce the number of variations, the weld throat A was kept constant at A = 0.7t. As at the transverse attachment, no major effect is expected, provided the weld is not very small. The variation of the transition angle at the weld toe 8 gives an effect comparable to that of the other structural details. The variation of the plate with B/T shows a significant effect. The wider the plate, the more local and sharper the stress concentration. The extreme will be reached at an infinitely wide plate. Whether or not there is an asymptotic effect near a width of B/T = 10 cannot be decided from the results available. At the other side, at small widths the joint type more and more approaches that of a thick transverse attachment with a lower stress concentration. This gives an indication that at longitudinal attachments near the plate edge or at flat side gussets, the larger distance to the plate edge might be the decisive parameter. However, a more detailed study is required. C=O f L E (T) CT> (T> (T) * (:) k = (+) - 0.3,,,(&$ (-&). (14) Fig. 12. Longitudinal non-loadcarrying attachment.

10 182 A. HOBBACHER Long. attach: Effect of wall thickn. t/t Mk-Ii3CtOr t t/t j j j ; I I i IO- Long. attach: Effect of att. length L/l LOE-03 l.oe oE*oo CTack depth.,t J +:I! I I I lfl+h : i : t:-++ 1.oE Oack depth a,t 10- Long. attachm: Effect of plate width B/l Long. attach: Effect of weld angle theta Mk-fBCfOr oE-02 Crack depth I.OE-01 a/t NE-03 l.ce-02 1.oE-01 CIM;k depth ait Fig. 13. Effect of dimensional parameters at non-loadcarrying longitudinal attachment. 4. CONCLUSIONS Parametric formulae for the calculation of stress intensity factor at welded joints have been established for several typical welded joints under membrane stress. Parametric formulae have been established which cover the principal significant dimensional parameters (Table 6). The formulae have been checked against published results. They are in good conformity. The formulae have been calculated for membrane stress. An application to bending stress is conservative by roughly 13%. The formulae cover a wider range than those used up to now [S]. They are simple and ready for application to design codes. REFERENCES VI E. Niemi, Recommendations concerning stress determination for fatigue analysis of welded components. IIW dot. XIII /NV , International Institute of Welding (1992). VI H. F. Bueckner, A novel principle for computation of stress intensity factors. Z. angew. Math. Mech. 50, (1970). [31 P. Albrecht and K. Yamada, Rapid calculation of stress intensity factors. J. Strucf. Div., AXE 103(ST2), (1977). 141 A. Hobbacher, Stress intensity factors of plates under tensile load with welded-on flat side gussets. Engng Fracture Mech. 41, (1992). PI S. J. Maddox, J. P. Lechocki and R. M. Andrews, Fatigue analysis for the revision of PD 6493: Report 3873/l/86, The Welding Institute, Cambridge, U.K. (1986). PI X. Niu and G. Glinka, The weld profile effect on stress intensity factors in weldments. Znt. J. Fracture 35, 3-20 (1987). [71 C. A. Castiglioni, Analisi parametrica della conzentratione delle tensioni al piede di saldatura in attacchi longitudinale. Politechnico di Milano, Dipartimento di Ingegneria Strutturale (1991). PI A. Hobbacher, Recommendations for assessment of weld imperfections in respect to fatigue. IIW dot. X /XV , International Institute of Welding (1988). (Received 11.Zunuary 1993)

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