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International Journal of Fatigue 23 (2001) 865–876www.elsevier.com/locate/ijfatigue

A structural stress definition and numerical implementation forfatigue analysis of welded joints

P. Dong*

Center for Welded Structures Research, Battelle, Columbus, OH 43016-2693, USA

Received 10 December 2000; received in revised form 11 May 2001; accepted 12 June 2001

Abstract

A mesh-size insensitive structural stress definition is presented in this paper. The structural stress definition is consistent withelementary structural mechanics theory and provides an effective measure of a stress state that pertains to fatigue behavior ofwelded joints in the form of both membrane and bending components. Numerical procedures for both solid models and shell orplate element models are presented to demonstrate the mesh-size insensitivity in extracting the structural stress parameter. Conven-tional finite element models can be directly used with the structural stress calculation as a post-processing procedure. To furtherillustrate the effectiveness of the present structural stress procedures, a collection of existing weld S-N data for various joint typeswere processed using the current structural stress procedures. The results strongly suggests that weld classification based S-N curvescan be significantly reduced into possibly a single master S-N curve, in which the slope of the S-N curve is determined by therelative composition of the membrane and bending components of the structural stress parameter. The effects of membrane andbending on S-N behaviors can be addressed by introducing an equivalent stress intensity factor based parameter using the structuralstress components. Among other things, the two major implications are: (a) structural stresses pertaining to weld fatigue behaviorcan be consistently calculated in a mesh-insensitive manner regardless of types of finite element models; (b) transferability of weldS-N test data, regardless of welded joint types and loading modes, can be established using the structural stress based parameters. 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Structural stress; Finite element analysis; Welded joints; Fatigue; Notch stress; Stress concentration; Mesh-size sensitivity

1. Introduction

At present, fatigue design of welded structures isprimarily based on a nominal stress or hot spot stressapproach with a series of classified weld S-N curves [1–4], although a local stress or initiation-based fatigue lifeapproaches [5,6] provide an alternative method forfatigue life predictions of welded joints. Without goinginto a detailed discussion of the merits of the two differ-ent approaches, the premise of this paper is that the nom-inal stress or hot spot stress approach has been wellaccepted by major industries, and recommended bynumerous national and international codes and standards(e.g. [3,4]). A series of S-N curves corresponding to eachclass of joint types and loading mode were well docu-mented in some of the codes and standards. With such an

* Tel.: +1 614-424-4908; fax:+1 614-424-3457.E-mail address: [email protected] (P. Dong).

0142-1123/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved.PII: S0142-1123 (01)00055-X

approach, nominal stresses with appropriate geometric orstructural stress concentration factor (SCF) for a parti-cular class of joints must be determined against the cor-responding S-N curve to calculate fatigue damage. Twocritical issues remain unresolved in this context. First,both nominal stresses and geometric SCFs cannot bereadily calculated from finite element models due to theirstrong dependence on element size at weld disconti-nuities. Secondly, the selection of an appropriate S-Ncurve for damage calculation can be very subjective,since the weld classifications were based on not onlyjoint geometry, but also dominant loading mode.

There are numerous on-going international efforts toaddress the above two issues. A majority of the efforthas been on developing effective hot-spot stress extra-polation procedures and an ability to correlate variousavailable S-N curves (e.g., [7,8]). However, the extrapol-ation procedures available to date still lack consistencyfor general applications [9]. This is in part due to thefact that extrapolation procedures are based on the

866 P. Dong / International Journal of Fatigue 23 (2001) 865–876

assumption that the surface stresses on a structural mem-ber provides an indication of the stress state at a weldfatigue prone location, such as a weld toe. This underly-ing assumption may become questionable if the struc-tural member is not a dominant load-transfer member ina joint. Under such circumstances, the surface stressesat some distance away from a weld toe may not be rel-evant to the stress state of concern. In addition, a refer-ence nominal stress in such a structural member may notbe readily identified for conventional SCF calculations.

Among the various extrapolation procedures proposedin the open literature (e.g., [7,8]), a typical one is basedon a linear extrapolation from stress values at both 0.4tand 1t from a weld toe [8,9], as shown in Fig. 1, wheret represents the plate thickness of a structural member.The drawback in such an extrapolation scheme becomesimmediately clear in view of Fig. 1 in which some ofthe well-studied joints in the research community areillustrated. The stress concentration behaviors can becategorized into two types [10]: one is rather localizedstress concentration behavior (Type I) at weld toe, whilethe other is more global in length-scale (Type II). Inorder to correlate the fatigue behavior in various jointtypes, stress concentration behavior at the weld toe ofvarious joint types must be captured. However, as shownin Fig. 1(b), any stress concentration effects in Type I

Fig. 1. Stress concentration behavior in welded joints.

joints cannot be captured in this extrapolation scheme,resulting in little stress concentration effects from thiscalculation. On the other hand, for Type II joints, Fig.1(b) shows that extrapolation from the two referencepositions (open circles) should provide some indicationof the concentrations at the weld toes. Then, one obviousquestion is if such calculation procedures provide areliable stress concentration measurement or hot spotstresses. As discussed in Neimi [9], the results are oftenquestionable due to the fact that these stresses can bestrongly dependent on mesh-size and loading modes.

To improve the S-N curve approach (using eithernominal stresses or hot spot stresses) for welded struc-tures, a relevant stress parameter must satisfy the twobasic requirements: (a) mesh-size insensitivity in finiteelement solutions; (b) ability to differentiate stress con-centration effects in different joint types (e.g., butt jointsversus T-fillet cruciform joints) in welded structures. Inthe following, such a stress parameter is presented andthe corresponding finite element procedures using bothsolid and shell element models are given. The validationof such a structural stress parameter is demonstrated byreprocessing a series of existing S-N data for joint typeslisted in Fig. 1(a).

2. Structural stress definition and formulation

As discussed in Dong [10] and Dong et al. [11], astructural stress definition that follows elementary struc-tural mechanics theory can be developed with follow-ing considerations:

(a) It can be postulated that for a given local through-thickness stress distribution as shown in Fig. 2(a)obtained from a finite element model, there exists acorresponding simple structural stress distribution asshown in Fig. 2(b), in the form of membrane andbending components that are equilibrium-equivalentto the local stress distributions in Fig. 2(a).

(b) The structural stress distribution must satisfy equi-librium conditions within the context of elementarystructural mechanics theory at both the hypotheticalcrack plane [e.g., at weld toe in Fig. 2(a)] and anearby reference plane, on which local stress distri-butions are known a priori from typical finiteelement solutions. The uniqueness of such a struc-tural stress solution can be argued by consideringthe fact that the compatibility conditions of the cor-responding finite element solutions are maintained atthis location in such a calculation.

(c) While local stresses near a notch are mesh-sizesensitive due to the asymptotic singularity behavioras a notch position is approached, the imposition ofthe equilibrium conditions in the context of elemen-tary structural mechanics with respect to this regime

867P. Dong / International Journal of Fatigue 23 (2001) 865–876

Fig. 2. Structural stresses definition for through-thickness fatiguecrack. (a) Local through-thickness normal and shear stress at weld toe,(b) Structural stress definition at weld toe.

should eliminate or minimize the mesh-size sensi-tivity in the structural stress calculations. This is dueto the fact that the local stress concentration closeto a notch is dominated by self-equilibrating stressdistribution, as discussed by Niemi [9].

Along this line, the following typical situations are con-sidered:

2.1. Solid model with monotonic through-thick stressdistributions

As shown in Fig. 2(a), the stress distribution at the T-fillet weld toe is assumed to exhibit a monotonicthrough-thickness distribution with the peak stressoccurring at the weld toe. It should be noted that in typi-cal finite element based stress analysis, the stress valueswithin some distance from the weld toe can change sig-nificantly as the finite element mesh design changes(e.g., [9]), referred to as mesh-size sensitivity in thispaper. The corresponding statically equivalent structuralstress distribution is illustrated in Fig. 2, in the form of

a membrane component (sm) and bending component(sb), consistent with elementary structural mechanicsdefinition:

ss�sm�sb. (1)

The normal structural stress (ss) is defined at a locationof interest such as Section A–A at the weld toe in Fig.2(b) with a plate thickness of t. In the above, the trans-verse shear (tm) of the structural stress components [tobe calculated based on local transverse stress distributionfrom Fig. 2(a)] is not considered in the structural stressdefinition in the present discussions. In practice, thetransverse shear component can play an important role incontrolling crack propagation path if the remote loadingimposes a significant transverse shear at the weld toe.

A second reference plane can be defined along SectionB–B in Fig. 3(a), along which both local normal andshear stresses can be directly obtained from a finiteelements solution. The distance, d, represents the dis-tance between Sections A–A and B–B (in local xdirection) at the weld toe. For convenience, a row ofelements with same length of d can be used in the finiteelement model. By imposing equilibrium conditionsbetween Sections A–A and B–B, the structural stresscomponents sb and sm must satisfy the following con-ditions:

sm�1t�

t

0

sx(y)·dy (2)

sm·t2

2�sb·

t2

6��

t

0

sx(y)·y·dy�d�t

0

txy(y)·dy. (3)

Eq. (2) represents the force balances in x direction,evaluated along B–B and Eq. (3) represents moment bal-

Fig. 3. Structural stresses calculation procedure for through-thicknessfatigue crack.


ances with respect to Section A–A at y =0. The integralterm on the right-hand side of Eq. (3) represents thetransverse shear force as an important component of thestructural stress definition. It is then follows that ifelement size (d) is small or transverse shear is negligible,the integral representations of sb and sm in Eqs. (2) and(3) can be directly evaluated at Section A–A in Fig. 3(a).

2.2. Solid model with finite fatigue crack depth

Often, a fatigue crack of a finite depth is used as afatigue failure criterion (e.g. [9]), the correspondingstructural stress can be then defined in a similar mannerto that in Fig. 3. In Fig. 4, the depth of the fatigue crack

Fig. 4. Structural stresses definition for partial thickness (t1) fatiguecrack weld toe. (a) Local normal stress distribution, (b) Structuralstress definition.

at failure is assumed to be t1. Without losing generality,the structural stress procedures [10,11] can be effectivelydemonstrated using the example in Fig. 4. Note that forconvenience, the local y coordinate is defined as shownin Fig. 4(a), different from that in Fig. 3, At a horizontalcross section of depth t1 from the top surface, both nor-mal stress (sy) and shear stress (tyx) are present in gen-eral.

By imposing equilibrium conditions between SectionsA–A and B–B, as well as the horizontal cross sectionin between, it can be shown that the structural stresscomponents (sb and sm) must satisfy the following equa-tions:

sm�1t1�t1

0

sx(y)·dy�1t1�d

0

tyx(x)·dx (4)

sm·t212

�sb·t21

6��

t1

0

sx(y)·y·dy�d�t1

0

txy(y)·dy (5)

��d

0

sy(x)·x·dx.

Additionally, by considering the bottom element(spanning t–t1) between Sections A–A and B–B, it canbe shown that

sm�sb�sm��sb� (6)

sm��1

t−t1�

t�t1

0

sx(y)·dy�1

t−t1�d

0

tyx(x)·dx. (7)

As in Eqs. (2) and (3), the integrals in the above can beaccurately evaluated using finite element solutions atcross sections away from the geometric discontinuity.The structural stress components sb and sm, includingsb� and sm� can then be solved.

2.3. Solid model with non-monotonic through-thickness stress distributions

In thick section joints and some joint configurations,non-monotonic through-thickness or in-plane distri-butions may develop, as shown in Fig. 5(a). The corre-sponding structural stress definition can be consistentlydefined in a similar manner as that in Fig. 4. Note thatthe parameter t1 can be determined based on the positionat which the transverse shear stress changes direction, ifthere is no specified crack depth as a failure criterion.Eqs. (4, 5) and (7) can be directly used for structuralstress calculations except a minor modification of Eq.(6) as follows:

sm�sb�sm��sb�. (8)


Fig. 5. Structural stresses definition for non-monotonic through-thickness stress distribution.

A special case in this category is that the joint configur-ation and loading are symmetric with respect the neutralaxis of the horizontal member in Fig. 5 such as JointsB and C in Fig. 1(a). In this case, the shear stress on thecross section along the symmetry line is zero as shown inFig. 6. Thus, Eqs. (4) and (5) can be directly used tocalculate the structural stress components sb and sm, bysubstituting t1=t/2 and tyx=0.

2.4. Shell/plate element models

Shell or plate element models of complex structuresare widely used for performing stress analyses forfatigue evaluation. The underlying principles of struc-tural stress calculations are the same between the

Fig. 6. Structural stresses procedure for non-monotonic through-thickness stress distribution: (a) symmetry with respect to plate mid-thickness (t1=t/2); (b) structural stress definition.

shell/plate and solid element models, while special con-sideration will be given to shell/plate structural theoryand its finite element implementation. It should be notedthat shell/plate element solutions at geometric disconti-nuities (e.g., weld toe) converge only to the solutionsdescribed by the corresponding shell/plate theory.Consequently, the local stresses at a weld toe in actualstructures are forced to obey the shell/plate theory usedin the finite element model and are not the structuralstresses sought in Eq. (1), even though a linear through-thickness stress distribution is maintained. Two generalmethods for structural stress calculations are presentedbelow:


2.4.1. Using stresses and stress resultantsAs shown in Fig. 7, stresses and nodal quantities from

shell or plate element models are often defined in a glo-bal coordinate system (x, y, z), depending on the finiteelement codes used. Given the definition of the structuralstress components in Eq. (1), it is the local coordinatesystem (x�, y�, z�) that is convenient for calculating thestructural stresses with respect to a weld, with local x�and y� being perpendicular and parallel to the weld direc-tion, respectively. Consistent with the solid elementmodel approach (e.g., see Fig. 3), three components ofthe stress resultants (sectional forces and moments), i.e.,fx�, fz�, and my�, at Section B–B in Fig. 7 can be used tocalculate the structural stress components at SectionA–A:

ss�sm�sb�fx�

t�

6(my�+d·fz�)t2

. (9)

In the above, a finite element formulation with sixdegrees of freedom at each node is assumed, i.e., sixcomponents of generalized forces at each node (threetranslational and three rotational). If stresses in the glo-bal coordinate system (x, y, z) are used, they must betransformed to the local coordinate system (x�, y�, z�)before Eqs. (2) and (3) can be used for the structuralstress calculations.

2.4.2. Using element nodal forcesIn some applications, the reference section B–B in

Fig. 7 may not be available. This situation arises if weldsare rather close to each other or load transfer at a weldof interest is very localized. If the element sectionalforces and moments (with respect to the referenceelement in Fig. 7) at Section A–A are available froma finite element solution, the equilibrium requirements

Fig. 7. Structural stress procedures for a shell/plate element adjacentto a weld.

described by Eq. (9) are automatically satisfied withinthe accuracy of the finite element solutions. In view oftypical finite element procedures in commercial codes,a general structural stress calculation procedure ispresented below:

With respect to the global coordinate system (x, y, z),the element stiffness matrix {Ke} can be obtained eitherdirectly from a finite element solution or formulated sep-arately afterwards. The nodal displacements at a nodewithin the reference element are typically described inthe form of:

{u}Ti �{uxi, uii, uzi, qxi, qyi, qzi}

where uxi, uyi, uzi represent the three translational dis-placements in x, y, and z directions at node i and qxi,qyi, qzi three rotational displacements, respectively. Thesubscript i takes 1, 2, …, n, with n being the number ofthe nodes in the element. The element nodal force vector,

{Fe}Ti �{Fxi, Fyi, Fzi, Mxi, Myi, Mzi, …}, i�1,2,…,n

can be obtained by:

{Fe}�{Ke}{u}. (10)

The element nodal forces in the local coordinate system(x�, y�, z�) can then be computed as

{Fe�}�{T}{Fe} (11)

where the matrix {T} is the coordinate transformationmatrix built up of directional cosines of angles formedbetween the two sets of axes in Fig. 7.

Once the element nodal forces are obtained for thenodal positions along Section A–A in Fig. 7, the corre-sponding sectional forces and moments (fx�, fz�, andmy�) can be calculated using appropriate shape functions.Then, Eq. (9) simply becomes at Section A–A:

ss�sm�sb�fx�

t�

6my�

t2 . (12)

Note that transverse shear effects in Eq. (9) are alreadytaken into account in the finite element solution in thisinstance.

3. Numerical examples

The structural stress procedures presented above [e.g.,Eqs. (2–5, 9) and (12)] can be implemented as post-pro-cessing procedures to the finite element results for astructure using commercially available finite elementcodes. In what follows, a series of numerical exampleswill be presented to demonstrate the effectiveness of thestructural stress procedures. Some typical joint detailsdiscussed in the open literature are considered here.


3.1. Plate lap fillet weld

A typical lap fillet joint is shown in Fig. 8(a). A planestrain eight-node solid element model is used as shownin Fig. 8(b) and (c), illustrating two representativemeshes with drastically different element sizes at theweld toes, among those used for mesh-size sensitivityinvestigation. Since large element sizes are to be investi-gated, parabolic elements with reduced integration areused for this joint, due to shear lock considerations if astrong bending action is present [11].

As the element size (a/b) varies from a/b=0.16t/0.1tto a/b=2t/1t, the structural stresses calculated at the weldtoe according Eqs. (2) and (3) remains essentially ident-ical. Fig. 8(d) summarizes the structural stress basedSCF values calculated with different element mesh sizes.The SCF values were calculated using the structuralstresses calculated using the present methods normalized

Fig. 8. Structural stress and mesh insensivity–single plate lap joint:(a) model definition; (b) a representative FE model with fine mesh(0.16t/0.1t) at weld toe; (c) a representative FE model with coarsemesh (0.8t/t) at weld toe; and (d) structural stress SCF calculated fromsix FE models.

with respect to remote nominal stress, i.e., F/A, whereF is remote loading and A the area of loaded member.

It is important to note that once the mesh-size insensi-tivity is ensured, the structural stress should serve as anintrinsic stress parameter for a given geometry andboundary conditions, regardless of numerical proceduresused. Then, it is natural to expect a similar structuralstress value from a shell/plate element model for a geo-metrically similar joint. Indeed, this is the case, as evi-denced in Fig. 9. The structural stress values obtainedusing the two shell element procedures, i.e., Eqs. (9) and(12), give the same results. The slightly higher structuralstress values than those obtained by solid element mod-els reflect, to a large extent, the simple representation ofthe fillet weld geometry in the shell model, as shown inFig. 9(a). A proper definition of the shell element thick-ness in the weld area should further improve the shellelement results. Attempts were not made here to optim-ize some of the detailed modeling issues.

3.2. Double plate lap fillet weld

The double plate lap fillet weld shown in Fig. 10serves as a special case of the proposed structural stressdefinition for non-monotonic through-thickness stressdistributions, as shown in Fig. 5. The calculation pro-cedures are illustrated in Fig. 6 by imposing the sym-metry conditions at the base plate mid-thickness, as

Fig. 9. Comparison of structural stresses calculated from both solidand shell models as a function of element size–single plate lap joint:(a) model definition; (b) structural stress results.


Fig. 10. Structural stress and mesh insensivity–double plate lap filletweld: (a) model definition; (b) structural stresses calculated from fourFE models with various mesh-sizes at weld toe.

shown in Fig. 10(a). For this case, shell or plate elementmodels cannot be not directly used. Instead, plane-strainsolid element models (same as those in Fig. 8) are usedfor demonstrating the mesh-size insensitivity in thestructural stress calculations. The structural stress resultsbased on Eqs. (4) and (5) are shown in Fig. 10(b). Themesh-size insensitivity can be demonstrated from verysmall element size up to a/b=0.4t/0.5t. Unlike the singleplate lap weld, the mesh-insensitivity of the structuralstress results will be limited to less than 1/2t since alarger element size will not be able to capture thelocalized stress distribution in this joint type.

3.3. Rectangular hollow section joint

Fig. 11(a) shows a typical rectangular hollow section(RHS) joint under the given loading of 1000 N. Four-node linear shell elements were used with a full inte-gration scheme. The procedure discussed earlier [Eq.(9)] is used for calculating the structural stresses at theweld toe on the vertical member. The results are summa-rized in Fig. 11(b). The local stress values directlyobtained from the shell element model are also shownfor comparison purposes. Again, the mesh-size insensi-tivity of the current structural stress procedures is evi-dent. It should be noted that the curved geometry of the

Fig. 11. Structural stress at weld toe for RHS joint (shell elementmodel). (a) Stress distribution - Shell Element model, (b) comparisonof local stress and structural stresses.

RHS limits the increase in element sizes along the welddirection without losing a proper representation of theactual geometry, particularly at corner positions. Theelement size represents the variations in the directionperpendicular to the weld.

4. Implications and discussions

4.1. Properties of the structural stresses

As discussed in the previous sections, the structuralstress parameter presented in this paper demonstratesthat it can be robustly computed from typical finiteelement models used in practice. The mesh-size insensi-tivity in the structural stress calculations can be generallymaintained using rather coarse mesh (up to multiplethicknesses) for monotonic through thickness stress dis-tributions (Fig. 3) and up to 1/2t for non-monotonicthrough-thickness distributions such as those shown inFigs. 5 and 6. In former, other considerations in finalfinite element mesh design should be given in practicesince interactions of welds and a proper representationof structural geometry should be considered. In the latter,the element size limitation can be removed if the struc-tural stress based SCF can be used in conjunction withthe structural stresses calculated from shell/plate element


models (nominal structural stresses in this context) inactual applications.

By its very definition, the structural stress provides aglobal stress measure at a location of interest such asweld toe, which satisfies equilibrium conditions in thecontext of elementary structural mechanics. Conse-quently, the structural stress parameter is solely determ-ined by joint geometric parameters, such as joint type,plate thickness (or reference fatigue crack depth), andweld shape under a given load transfer mode. Once it isdetermined in a mesh-insensitive manner, the structuralstress parameter can be viewed as an equivalent struc-tural stress state with respect to a hypothetical crackorientation such as a weld toe crack in a simplegeometry. Both the stress concentration and load modeeffects in the actual structure are fully contained in thestructural stress parameter. It should also be noted thatthe average transverse shear stress [e.g., Eq. (3)] shouldalso be viewed as a component of the structural stressparameter. The effects of the transverse shear componentof the structural stress parameter are currently underinvestigation [14] for joints loaded with significant trans-verse shear loading.

4.2. Correlation between structural stresses and S-Ndata

Structural stress calculations were performed for a ser-ies of selected weld details upon which well-documentedS-N fatigue data can be found from a database at Bat-telle. These joint configurations are shown in Fig. 1(a)[Joints (A), (B), (C), (E), and (F)]. For comparison pur-poses, nominal stress and hot-spot stress extrapolationprocedure based on 0.4t and 1.0t positions [1–4] werealso used with the same finite element models, inaddition to the structural stress procedures discussed inthe above. A relatively small element size (about t/6) atthe weld toe were used in all these calculations due tothe hot spot stress extrapolation requirements, eventhough the present structural stress procedures are cap-able of generating the same results with much coarsermodels.

Fig. 12 summarizes the results processed using thethree stress definitions. To facilitate an one-to-one com-parison, all plots in Fig. 12 are plotted with same lengthscale and ranges both for the abscissa and ordinate. Asshown in Fig. 12(a), the nominal stress range versus lifeplot shows that each of the joint types essentially retainsits own S-N data trend, as expected. The hot spot stressbased plot in Fig. 12(b) did not provide any noticeableimprovement. Note that some of the data points such asthose for Joint E and Joint F have a lower nominal stressvalue (F/A) than 100 MPa and are outside of the plot inFig. 12(a), but having higher hot spot stress and struc-tural stress values than 100 MPa in Fig. 12(b) and (c).As discussed earlier, due to the highly localized stress

Fig. 12. Comparison of S-N data representations using various stressparameters published in the literature: (a) nominal stresses; (b) IIWhot spot stresses (0.4t/1.0t); (c) structural stresses.

concentration behavior in most of the joints (see Fig. 1),the stress extrapolations from positions at 0.4t and 1tfrom weld toe yield essentially the nominal stresses foralmost of all the joint types, except Joint (F). However,the structural stress based plot in Fig. 12(c) for the sameS-N data shows that the structural stress parameter pro-vides more effective consolidation of the S-N data fromthe different joint types studied. The structural stressbased SCFs (sm and sb values corresponding to unitremote tension) are listed in the parenthesis followingeach legend in Fig. 12(c). The results suggest that thestructural stress parameter can be used to establish thetransferability of the S-N data among the joint types.Further results will be reported separately [14] for more


detailed considerations of various other joint types andcorresponding S-N data.

4.3. Effects of bending and membrane components

With a S-N curve approach, fatigue lives (in cycles)of welded joints are described as a function of eithernominal or hot spot stress ranges. In the presentapproach as shown in Fig. 12(c), structural stress rangesare used as a stress parameter to relate to fatigue livesof the welded joints under remote tension. For all thejoints analyzed in Fig. 12(c) under remote tension, thebending content (sb) of the structural stresses at weldtoe varies from 0.214 to in Joint A to 0.522 to Joint F,normalized by remote tension stress. It is conceivablethat the peak surface stress range parameter (sm+sb),although it can be calculated in a mesh-insensitive man-ner, may not be adequate to consolidate S-N data fromdrastically different remote loading modes, such as pureremote tension versus pure remote bending. In such situ-ations, the corresponding structural stress componentssm and sb will be drastically different as well. Theeffects of the membrane and bending effects on S-Nbehavior can be inferred from fracture mechanics con-siderations since stress intensity factor provides an effec-tive one-parameter characterization of cracking drivingforce under linear elastic conditions. A large amount ofpublished work demonstrates that crack growth rate canbe related to the stress intensity factor (�K) range in aform of the Paris law.

Once a structural stress is available for a jointgeometry in terms of sm and sb in Eq. (1), the calculatedstructural stress transforms the actual geometry and load-ing effects for a given weld detail on to a simplegeometry as shown in Fig. 13, where a represents afatigue crack length and tr stands for a reference lengthcorresponding to the final crack depth at failure. For sucha simple crack geometry in Fig. 13, stress intensity factorsolutions are readily available in the open literature. Forthe hypothetical crack plane in Fig. 13, the two structuralstress components serve as an equivalent remote loadingwith respect to the hypothetical crack face spanning froma=0 to a=tr. Then, the corresponding stress intensity fac-tor solution a for welded joint with structural stresscomponents of sm and sb can be directly constructed

Fig. 13. Fracture mechanics interpretation of the structural stressesas fatigue propagation parameter.

using existing solutions such as those given by Tada etal., [15]. It can be shown that the Mode I stress intensityfactor range for the crack geometry in Fig. 13 can beexpressed as a function of the ranges of the structuralstress components using superposition principle:

�K��Km��Kb��tr��smfm�atr��sbfb�a

tr�� (13)

The fm(a/tr) and fb(a/tr) are dimensionless functions ofa/tr for the membrane and bending components,expressed as follows [15]:

fm�atr��0.752�2.02�a

tr��0.37�1�sin

pa2tr�3��2tan

pa2tr

cospa2tr

(14)

fb�atr��0.923�0.199�1�sin

pa2tr�4��2tan

pa2tr

cospa2tr

. (15)

For a given combination of �sm and �sb, fm(a/tr) andfb(a/tr) are functions of a/tr. It should be pointed out thatthe K solutions using Eqs. (13)–(15) are strictly valid forjoint types without symmetry with respect to the mid-plane of the horizontal member in Fig. 1(a), such asJoints D, G, and G�. For joints with symmetry, such asJoints A–C, E, and F, the K solutions using Eqs. (13)–(15) should be viewed as an approximation. Theadequacy of the approximation for the current appli-cations can be justified from the fact that the structuralstress calculations have already taken into account of thestress gradient effects in the symmetrical joints (A–C,E, and F). Similar assumptions have been used in variousearly publications, e.g., [1,2]. A more refined approachby incorporating more appropriate functions fm(a/tr) andfb(a/tr) in place of Eqs. (14) and (15) is given by Donget al. [16] for joints with horizontal symmetry.

In typical fatigue testing of welded joints for S-N datageneration, crack size information such as a/tr is oftennot available. For S-N data interpretation purposes, as areasonable approximation, it may be assumed that acrack size at the beginning of a test is infinitesimallysmall (a/tr�0) and reaches about a/tr�1 (completeseparation) at the final failure. (It is recognized that inmost of fatigue tests, a final failure often occurs wellbefore a/tr=1. More detailed considerations are discussedin [16].) Then, an effective �K can be defined by con-sidering an averaged stress intensity factor integratingover a=0�tr. The average stress intensity factor range�K can be obtained by integrating the expressions inEqs. (14) and (15) over a/tr=0 to 1 as:


�K��tr �a/tr�1

a/tr�0

��smfm�atr��sbfb�a

tr��d�a

tr�.

The following simple expression can then be obtained:

�K��tr(33.17�sm�11.87�sb). (16)

For a given combination of �sm and �sb, Eq. (16) pro-vides the stress intensity factor based measurement oftheir relative contributions to the total �K. Note that Eq.(16) is rather general since the structural stress compo-nents already contain the geometry and three dimen-sional loading effects from an actual welded joint, asdiscussed earlier.

Naturally, if �K serves as an effective parameter forcrack propagation behavior, a single �K�N log–log plotshould be expected regardless of the relative magnitudeof the membrane and bending components. However, ina log–log based S-N plot, if �sm and �sb are signifi-cantly different from those in Fig. 12(c), a slope changeis anticipated since Eq. (1) provides only a peak struc-tural stress measure at the surface with an equal weighton both components. To substantiate this augment, twosets of S-N data for Joints G and G� by Huther et al.[12] and Maddox [13] were processed using both Eqs.(1) and (16). Joint G� is the same as Joint G in geometry,but under horizontal tension instead of pure bending.The structural stress based SCFs for the bend component(sb) for Joints G and G� are 1.206 and 0.383, respect-ively, as indicated in Fig. 14(a), drastically differentfrom each other. The structural stress based S-N plot isshown in Fig. 14(a). It can be seen that the two sets of S-N data exhibit two distinct bands separating membrane-dominant structural stress from the bending-dominant.

However, if Eq. (16) is used, Fig. 14(b) shows thatthe fatigue data from the two drastically different mem-brane and bending combinations fall into essentially thesame band. Fig. 14(c) summarizes the stress intensityfactor ranges versus cycles at failure for all joints [JointsA though G in Fig. 1(a) and G�]. The results in Fig. 14(c)clearly suggest that the S-N data for Joints A through G�can be effectively consolidated into a single band, eventhough both the joint types and loading are significantlydifferent from one another. In calculating the structuralintensity factor ranges for each of the joint types, Eq.(16) was used throughout the process once the mesh-insensitive structural stresses are obtained for all jointtypes and loading mode. This can also be viewed as avalidation of the structural stress based transformationprocess shown in Fig. 13 and its potential usefulness forgeneral applications.

Further discussions on the use of the relation in Eq.(13) or Eq. (16) and more refined definition of �K para-meters for consolidating existing S-N data be reportedseparately [16], in which structural stresses at notches

Fig. 14. Effects of bending and membarane components on S-Ncurve slope: (a) structural stress range; (b) stress intensity factor range;(c) stress intensity factor range for joints A–G�.

and detailed K solutions were also used [16]. It shouldbe noted that the structural stress based �K�N interpret-ation for fatigue behavior has been successfully used forconsolidating S-N data of spot welds under drasticallydifferent loading conditions (e.g., [17,18]), where �K,however, can be expressed in a simple form withoutexplicit crack size dependency, as initially proposed byZhang [17]. A recent independent evaluation on itseffectiveness can be found in Gao et al. [19].


5. Summary

1. A structural stress definition has been presented as aneffective measure of the stress state at welded joints.The structural stress parameter in terms of both mem-brane and bending components is consistent withelementary structural mechanics theory and can bereadily calculated in a mesh-insensitive manner fromconventional finite element models. Its applications incharacterizing weld toe cracking for a series of jointtypes were demonstrated.

2. The structural stress based interpretation of existingS-N data considered here suggests that transferabilityof S-N data among different joint types and loadingconditions can be established.

3. The relative bending to membrane ratio of the struc-tural stress components calculated at weld toe canprovide additional insight on the S-N curve behavior,particularly for those joints tested under drasticallydifferent remote loading conditions. A simplifiedstress intensity factor range parameter (an integratedaverage) proposed in this study has demonstrated itspotential in correlating such S-N curve behaviors interms of the membrane and bending effects.

4. The structural stress procedure can be viewed as atransformation process that maps a complex stressstate at a joint situated in an actual structure subjectedto complex loading mode onto a simple structuralstress state represented by membrane and bendingcomponents. Consequently, stress intensity factorscan be readily calculated using existing solutions fora simple crack problem for various welded joints oncethe structural stress components become available.

References

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