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International Journal of Fatigue 22 (2000) 691702

www.elsevier.com/locate/ijfatigue

Evaluation of multiaxial spot weld fatigue parameters forproportional loading

H. Kang a, M.E. Barkey a,*, Y. Lee b

a Aerospace Engineering and Mechanics Department, The University of Alabama, Tuscaloosa, AL 35487-0280, USAb Stress Laboratory and Durability Development, DaimlerChrysler Corporation, Auburn Hills, MI 48326-2757 USA

Received 30 August 1999; received in revised form 23 January 2000; accepted 26 March 2000

Abstract

The authors have conducted set of experiments to study the effects of combined tension and shear loads on the fatigue life ofspot welded joints. The fatigue life of the specimens depended on the applied load amplitude, the ratio of shear to normal loading,and spot weld nugget diameter. The lower load amplitudes had longer fatigue lives, as did the cases which contained a higheramount of shear loading and specimens with a larger nugget diameter. Based on the test results, Swellam and co-workers model,Sheppards model, Rupp and co-workers model, and an interpolation/extrapolation model are evaluated. The four approaches werecorrelated with the experimental fatigue life for the multiaxial test results with reasonable accuracy. The success of Swellam andco-workers method relies heavily on determining the appropriate parameters b and b0. Sheppards structural stress method agreedreasonably well for mutiaxial test results, although the maximum structural stress range is sensitive to the variation of the sheetthickness, and the determination ofM* is a complex procedure. Rupp and co-workers method is suitable for application to largestructural models because mesh refinement is not necessary for modeling the spot weld connection. 2000 Elsevier Science Ltd.All rights reserved.

Keywords: Fatigue life; Spot welded joint; Combined tension and shear; In-plane shear

1. Introduction

Spot welding is the primary method of joining sheetmetal for body and structural applications in the groundvehicle industry. Although most spot welded connec-tions are designed to primarily carry shear loads, in cer-tain applications a significant amount of peel force, orforce normal to the spot weld, must be carried by the

joint. This, in combination with the joint geometry, canlead to states of combined or multiaxial stress at or nearthe spot welded connection.

Very detailed finite element models of a spot weldedjoint can be constructed to calculate the stress states nearthe joint. However, such an approach is typically usedwhen analyzing the characteristics of a single joint [1],and not during the development phase of the ground

* Corresponding author. Tel.: +1-205-348-7300; fax: +1-205-348-

7240.

E-mail address: [email protected] (M.E. Barkey).

0142-1123/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved.

PII: S0 1 4 2 - 1 1 2 3 ( 0 0 ) 0 0 0 3 7 - 2

vehicle design process. Instead, forces and moments act-ing on each joint are usually determined by the linearelastic finite element method and these forces andmoments are used in the calculation of a fatigue damageparameter for the joint [25].

Recent researchers have proposed fatigue damageparameters for spot welded joints subjected to combinedloading based on loads that can be calculated by finiteelement analysis. These parameters have typically beenformulated using either fracture mechanics basedapproaches [610] or structural stress based approaches[25], and then correlated with an experimental databaseof fatigue life results. However, most of these previousexperimental databases have been for either in-planeshear or 90 out-of-plane tensile normal specimens, eventhough the methods have been proposed for generalstates of loading.

The authors have conducted a series of fatigue teststo study the effects of combined loading on the fatiguelife of a single spot welded joint. Based on these testresults, the fatigue life calculation approaches of Swel-


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693H. Kang et al. / International Journal of Fatigue 22 (2000) 691702

Xgk kth coordinate of gth interpolation pointj index indicating data pointk index indicating dimensiong index indicating interpolation point

lam and co-workers, Sheppard, and Rupp and co-work-ers are evaluated. The results are also compared to anumerical approach for calculating fatigue life of spotwelded joints applied to the data as described by Kangand Barkey [11]. In the following discussion, theseapproaches are described in detail and the models areapplied to the multiaxial test results.

2. Swellam and co-workers Ki parameter

Swellam and co-workers [6,7] developed a fatiguedesign parameter (Ki) that accounts for the effects ofgeometric factors and R-ratios (ratio of minimum tomaximum applied load) of spot-welded joints. It isassumed that resistance spot welds could experiencecombinations of Mode I and Mode II loadings by axialloads, bending moments, and shearing loads. The Kiparameter is a combination of superposed stress intensityfactors for Mode I and Mode II loadings.

They modeled a spot welded joint as two half-spacesjoined by a circular area under axial load, moment, andshearing load. Tada and co-workers equation [12] is

used to calculate stress intensity factors at the edge ofspot weld nugget as below:

KaxialP

2rpr, (1)

Kmoment3M

2r2pr, (2)

KshearQ

2rpr, (3)

where r is the nugget radius, P is the normal componentof the applied load, Q is the shear component of theapplied load, and M is the bending moment at the centerof the nugget due to applied load. These equations arederived based on linear elastic fracture mechanics(LEFM) assumptions. The basic assumptions of LEFMcontain the theory of elasticity including small displace-ments and general linearity between stresses and strains.It is often common to assume that these assumptions arevalid for a larger loading range for cyclic loading thanfor monotonic loading because of the reduced size of thecyclic plastic zone as compared to monotonic loading[13].

Fig. 1 shows resolved force components P, Q, and Mat the weld nugget for a general applied load F. Thestress intensity factors at the edge of the spot weld arederived by linear superposition:

KIKaxialKmoment, (4)

KIIKshear, (5)

KIeqK2I+bK2II. (6)

Here KIeq is equivalent stress intensity factor of Mode I,

and b is a material constant to account for the influenceof Mode II loading. The material constant b can bedetermined by plotting two sets of total fatigue life dataversus equivalent stress intensity factor (Keq) for speci-mens with the same geometric conditions. One set ofdata should be for only Mode I loading, while the otherset of data should be for combined Mode I and II load-ing. The material constant b was 2 for low carbon steelspecimens and 3 for high strength and low alloy steelspecimens in [6,7].

KIeq is calculated for the maximum applied load with-out consideration of the effects of the geometric factors

and load ratios. Therefore, Ki was proposed by applyinga geometrical correction factor (G) and load ratio effectas shown below:

Fig. 1. Resolved components P, Q and M at the nugget for a general

applied load F.


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694 H. Kang et al. / International Journal of Fatigue 22 (2000) 691702

G9Wt44r5

Wt2

r31/2, (7)

Kimax(K2Imax+bK2IImax)

G, (8)

KiKimax(1R)b0, (9)

where r is the nugget radius, t is the sheet thickness, W

is the specimen width, R is the load ratio (Pmin

Pmax), b is a

material constant, and Kimax is the equivalent stress inten-sity factor of Mode I at maximum applied load. Ki isvery sensitive to the variation of specimen thickness ascompared to the effect of the variation of the nuggetdiameter and specimen width. The constant b0 is determ-ined by trial and error to present a better plot of totalfatigue life (Nf) versus Ki on a loglog scale. A para-meterlife relationship can be derived based on this plotusing a power law equation:

KiA(Nf)h, (10)

where A and h are constants from the curve-fitting thatdepend on the test data.

3. Sheppards approach

Sheppard [2,3] assumed that the structural stress rangeat a spot weld has a very close relationship with the

fatigue life of the spot weld. This approach assumes thatfatigue cracks propagate in the sheet thickness directionand that nugget rotation is negligible. The structuralstress is an elastically calculated nominal stress determ-ined by bending moments, membrane forces, and axialforces by

SQ/(tw)6M/(t2W)P/t2, (11)

where Q/tw is a membrane stress term, 6M*/t2W is abending stress term, and P/t2 is the stress at the edgeof the weld nugget in longitudinal direction due to anaxial load. In these relations, the effective width in shearw=pd/3, t is the thickness of the sheet, d is the nuggetdiameter, and W is the width of the piece. Finite elementanalysis is used to calculate forces and moments at theweld nugget using plate elements and a beam elementto represent the spot welded connection. As shown in[7,8], the determination of M* is based on the con-ditions of the nodal force and moment ranges at the spotwelded connections. The structural stress range is fairlysensitive to the variation of the sheet thickness due tothe third term of Eq. (11).

Sheppard assumed that the crack propagation life isequal to the total fatigue life of the spot weld. Then, acurve fitting equation was derived from a plot of

maximum structural stress range (Smax) versus meas-ured fatigue life (Nf/(1R)) on a loglog scale as below:

Nf

(1R)C(Smax)

b. (12)

Nf is defined as the total life spent propagating the crack

through the sheet thickness t. The coefficient C andexponent b are from a power law relation of maximumstructural stress range versus measured fatigue life forcrack propagation.

4. Rupp and co-workers approach

Rupp and co-workers [4] observed that spot weldedconnections experienced considerable local plastic defor-mation during the first load cycles. Therefore, exactstress calculation of stresses at spot welds using the lin-ear elastic finite element method may not be effective,even if the model contains a high number of elements.(An exception to this may occur in situations where thematerial near the spot weld nugget experiences shake-down to an elastic state after initial deformation.) Theydetermined local structural stresses at the spot weldinstead of determining elastic notch-root stresses orstress intensities to correlate with the fatigue life of thespot weld. The local structural stresses are calculatedbased on the cross-sectional forces and moments usingbeam, sheet, and plate theory.

To determine cross-sectional forces and moments atthe spot welded joints, a stiff beam element is used on

the finite element model to connect both sheet metals.The length of this beam element is recommended to beone-half of the thickness of both sheets [5].

A central idea of Rupps approach is to calibrate thestructural stress parameter with the fatigue life results ofspot welded structures, and not just with single spotwelded coupons. For the sheet steels St1403 (300 HV),St1403 (200 HV), StW24, and St1203, they have shownthat the mean stress corrected fatigue life results of spotwelds in various structures consolidated the data reason-ably well in a loglog plot of their structural stress para-meter versus fatigue life, as shown in Fig. 2 [4].

In Rupps approach, two types of failure modes areidentified for spot welded joints: cracking in the sheetmetal or cracking through the weld nugget. The determi-nation of the cracking mode is predicted with a rule ofthumb that is generally accepted in industry [4]. In therule of thumb, failure modes are plotted by nugget diam-eter versus sheet thickness. The transition value of crack-ing in the sheet and through the nugget are 3.5t, wheret is the sheet thickness. When the nugget diameter isabove the transition value, the crack will be in the sheet.Alternatively, when the nugget diameter of the spot weldis below this, the crack will be through the weld nugget.

In the case of cracking in the sheet metal, the formulae


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Fig. 2. Various spot welded specimens and fatigue test results [4].

of circular plate with central loading are applied to cal-culate local structural stresses. For deriving these formu-lae, a spot welded specimen is assumed as a circularplate with a rigid circular kernel at the center. The ratioof the kernel radius to the plate radius is assumed as 0.1.The circular plate is also assumed to have fixed outeredges. Fig. 3(a) presents the schematic diagram of theseassumptions and the equations of the maximum radialstress resulting from lateral forces, normal force, andmoments. For the equations given in the figure, 1 wastaken as 1.744 and 2 as 1.872. These parameters dependon the ratio of the nugget radius and specimen span [14].

The equivalent stresses for the damage parameter arecalculated by combination and superposition of the localstructural stress. The equivalent stresses are calculatedas a function of angle q around the circumference of thespot weld. Here, q is the angle measured from a refer-ence axis. The equivalent stresses for cracking in thesheet are calculated using superposition of formulae forthe plate subjected to central loading as below:

seq1(q)smax(Fx)cos qsmax(Fy)sin qs(Fz) (13)

smax(Mx)sin qsmax(My)cos q

where:

smax(Fx)Fx

pdt, (14)

smax(Fy)Fy

pdt, (15)

s(Fz)1.744Fzt2

for Fz0, (16)s(Fz)0 for Fz0, (17)

smax(Mx)1.872Mxdt2 , (18)

Fig. 3. The assumed models to determine structural stresses at the

nugget for the two possible failure modes. (a) Circular plate model for

sheet metals; (b) beam model for nugget subjected to tension, bending

and shear.

smax(My)1.872Mydt2 , (19)0.6t. (20)


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The parameter is a material dependent geometry factorapplied to the stress terms calculated from the bendingmoment. It effectively reduces the sensitivity of thesestress terms to the sheet thickness.

For the nugget failure mode, structural stresses forcracking through nugget are calculated based on the elas-

tic formulae of a beam subjected to tension, bending andshear as shown in Fig. 3(b). This type of cracking canoccur when a relatively small diameter spot weld is usedto connect relatively thick sheets. The equations of thenormal stress (sn), bending stress (sb), and the maximumshear stress (tmax) are given in Fig. 3(b).

The resolved tensile stress on the critical plane aretaken as the damage parameter, and are calculated fromthe state of combined tension and shear:

t(q)tmax(Fx)cos qtmax(Fy)cos q (21)

s(q)s(Fz)smax(Mx)sin qsmax(My)cos q (22)

where:

tmax(Fx)16Fx

3pd2, (23)

tmax(Fy)16Fy

3pd2, (24)

s(Fz)4Fzpd2

for Fz0, (25)

s(Fz)0 for Fz0, (26)

smax(Mx)32Mx

pd3

, (27)

smax(My)32Mypd3

. (28)

To account for mean stress, the equivalent stressamplitude at R=0 is calculated by

S0S+MsSmMs+1

, (29)

where S is the stress amplitude for the cycle, Ms is meanstress sensitivity, and Sm is the mean stress for the cycle.

The value for the mean stress sensitivity is the down-ward slope of the line on a Goodman diagram (meanstress vs. stress amplitude) that connects the endurancelimit stress at R=0 to the endurance limit stress at theother value ofR-ratio. The total fatigue life is then corre-lated with the calculated maximum equivalent stressamplitude.

5. A numerical approach

Kang and Barkey [11] applied a numericalinterpolation/extrapolation technique to estimate fatigue

life of spot welded specimens. This technique can calcu-late the fatigue life of spot welded specimens (a depen-dent variable) for the effects of different test conditionsand specimen geometries (independent variables). A setof independent variables for which fatigue life is esti-mated in the variable space is determined based on the

relative distance from points that compose the experi-mental database. The value of the dependent variable(fatigue life) is estimated at interpolation points as illus-trated in Fig. 4 for the case of two independent variables.The minimum and maximum point on the predictionvector in Fig. 4 are the extremes of the projections ofthe data onto the prediction vector.

The characteristic length of influence, D, is assumedto be the same for all the data points. The angle, ajg, inFig. 4 increases as the distance, Ljg, decreases since Dis fixed. The effect of each data point on the determi-nation of the value of the dependent variable at theinterpolation points depends on the angle a

jg

. Thus, ajgcan be considered as a weighting factor of the influence

for the magnitude of the dependent variable. The valueof the dependent variable at the interpolation point, Vg,is calculated from the weighting factor, ajg, and the valueof the dependent variable, Vj, of each data point in thedatabase by

Vg

n

j1

Vj

LN+1jg

n

j1

1

LN+1jg

. (30)

The distance Ljg is determined by using the distance for-mula,

LjgN

k1

(XjkXgk)2, (31)

Fig. 4. The estimation scheme of a dependent variable for an interp-

olation point in the database.


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and N is the number of independent variables, Xjk is thekth coordinate of the jth data point, and Xgk is the kthcoordinate of the gth interpolation point.

The final step of this technique is to establish a poly-nomial equation using the least squares fitting methodwhich represents the relationship between the interp-

olation points and the values of the dependent variable.In addition, the estimation of the dependent variable atthe target point is needed from the polynomial equation.The value of the dependent variable at the target pointcan be interpolated or extrapolated by the polynomialequation, depending on the location of the target point.As with any extrapolation technique, if a target point islocated out of the boundary of the database, the pro-cedure should be used with caution. In these cases, theinfluence of the database on the target point is very weaksince the polynomial equation is established based onthe distance from the point in the database.

This method only relates the independent variables(load, spot weld nugget diameter, etc.) to the fatigue life(the dependent variable). As such, there is no need forexplicit calculation of stresses or other local parameters,in the same manner that a traditional SN curve is usedto related nominal stresses (i.e. load) to fatigue life. Themethod is included in the comparison of the semi-empirical methods as a base-line calculation method.Any spot weld parameter with a meaningful physicalinterpretation should provide better results. Note, how-ever, that the use of load and geometric dimensions inthe numerical method would not preclude its use withmore physically meaningful parameters such as any of

the damage parameters discussed previously.

6. Experimental procedure and results

The four approaches were evaluated using the experi-mental data for combined tension and shear loads. Thedata include 140 fatigue test results of high strength steelspecimens subjected to combined tension and shearloads. The spot welded specimens used in this studywere made of two strips of high strength galvanizedsheet steel (410 MPa yield strength) bent into C shapesand welded together in the center with a single electricalresistance spot weld. The thickness of the specimens was1.6 mm and the nominal diameters of specimens were5.4 and 8 mm. The typical shape and dimensions of thespecimen are shown in Fig. 5. The three loading direc-tions, 30, 50, and 90, were used to apply the com-bined proportional, or in-phase, tension and shear loadson the weld nugget as shown in Fig. 6. Three mean loadsof 1110, 2220, and 3340 N were applied at each loadingdirection with non-negative R-ratios from 0 to 0.76.Applied load amplitudes ranged from 445 N to 3336 N.The test fixture and test procedures for multiaxial testingof spot welded specimens are described in detail in Leeet al. [15] and Barkey and Kang [16].

Fig. 5. A typical specimen subjected to the combined tension and

shear loads (d=8 mm for large diameter, and d=5.4 mm for smalldiameter).

The definition of failure for the fatigue tests was takenas separation of the coupon, or excessive deformation ofthe joint (approximately 15 mm). This definition allowedfor crack initiation and some crack propagation life with-out excessive deformation that would cause a largechange in the nugget orientation.

The experimental data are presented in Figs. 79,where fatigue life versus applied load amplitude is plot-ted at each loading angle for the two nugget diameters.The fatigue life of the specimens depended on theapplied load amplitudes, the loading angles (ratio ofnominal tension to shear loading), and nugget diameters.The higher load amplitudes had lower fatigue lives, asdid the higher loading angles, where there was a higherproportion of tensile loading applied to the specimen. Asindicated on the figures, the larger nugget diametershowed the longer fatigue life. In general, the resultswere independent of the R-ratio or mean stress level.

The failure modes of the coupons also varied withload level and loading angle. Fatigue cracks typicallyinitiated at the edge of the weld nugget on the centerlineof the faying surface and progressed through the thick-


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Fig. 6. The test fixture to apply the combined tension and shear loads

on the spot welded specimens.

Fig. 7. Experimental results of applied axial load amplitude versus

fatigue life for 30 loading direction.

ness of the sheet. After the through-thickness cracks hadformed, the early stage of crack propagation was in thewidth direction of the specimen and in one or bothsheets. For the 30 and 50 loading directions, the crackdirection changed to the specimen length direction whenthe crack size was about the half of the specimen width.For the 90 loading case, the fatigue crack initiated atthe edge of weld nugget and advanced around nuggetresulting in the weld nugget pulling out from one of themetal strips.





7. Comparison of the approaches with multiaxial

test results

For Swellam and Lawrences method, the Ki para-meter was first calculated using b=3 and b0=0.85 as forthe high strength steel in their study [6,7]. However, thisresulted in an unsatisfactory correlation between calcu-lated life and measured life, and the constants weredetermined for the new experimental data. By the trialand error method, b=0.5 and b

0=1.5 were determined for

multiaxial data. Fig. 10 shows fatigue life (Nf) versus Kithat were calculated using the constants determined fromthe data. The equation of the power law relation fromFig. 10 became the governing equation to estimatefatigue life, and the correlation coefficient (cc) of thebest fit line is indicated in this figure. Fig. 11 shows thecalculated versus measured fatigue life for the specimenssubjected to the combined tension and shear loads. Asshown in Fig. 11, the Ki parameter was reasonably corre-lated with measured fatigue life. For all of the compari-sons, the dotted line in the figure represents a perfectcorrelation between the measurements and the calcu-


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Fig. 10. Experimental fatigue life versus Ki for the specimens sub-

jected to the combined tension and shear loads.

Fig. 11. Experimental life versus calculated life using the Ki para-

meter that was calculated with b0=0.5 and b=1.5 for multiaxial testdata.

lations, and the solid lines represent a factor three vari-ation from a perfect correlation.

The success of this method relies heavily on determin-ing the appropriate parameters b and b0 such that thescatter of the Ki vs. Nf plot is minimized. This scattercorresponds directly to the scatter in the plot of measuredlife versus calculated life, because the same test dataused to determine b and b0 were also used in the com-parison.

For Sheppards and Rupp and co-workersapproaches, a finite element model was made containinglinear elastic shell elements for the coupons, a stiff beamelement for the spot weld, and rigid body elements forthe loading fixtures. The length of the beam element wasequal to the thickness of the sheet. ABAQUS Version

Fig. 12. Experimental fatigue life versus Sheppards maximum struc-

tural stress range for the specimens subjected to tension and shear

loads.

Fig. 13. Experimental fatigue life versus calculated fatigue life using

Sheppards method for the specimens subjected to the combined ten-

sion and shear loads.

5.8 was used to calculate forces and moments at eachconnecting node of the beam element.

In Sheppards approach as described in [3], crackgrowth through the width of the sheet was not includedin Eq. (12). However, the multiaxial test data collectedin this study did not distinguish between the through-thickness and width propagation lives. Therefore, Eq.(12) was used to calculate fatigue life since Sheppardsbasic idea was the correlation of the fatigue parameterand test data. A relationship was determined between themaximum structural stress ranges (MPa) and multiaxialtest results (cycles to failure) as shown in Fig. 12, withthe correlation coefficient (cc) of the best fit line indi-cated in the figure. Fig. 13 shows the measured fatigue


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life versus the calculated fatigue life using Eq. (12) forhigh strength steel specimens under multiaxial loads.Finite element mesh density and element types may pro-duce a variation in calculated life as mentioned by Shep-pard [2,3]. The maximum structural stress range is sensi-tive to the variation of sheet thickness. In this study, the

specimens have one thickness and this results in a goodcorrelation. However, the scatter tends to increase assheets of different thickness are considered as shownin [11].

Rupp and co-workers approach was developed to usefatigue results generated by examining several spotwelded structures, and not just a single weld. However,the evaluation of their method in this paper was conduc-ted for single spot welded specimens because structuraltests were not performed. This also allows for a directcomparison of the spot weld fatigue parameter to theother approaches.

First, the failure mode was determined using the ruleof thumb (3.5t) to apply Rupp and co-workers method.All specimens were anticipated to fail in the sheet metalsince all nugget diameters were above the transitionvalue. Therefore, plate theory was applied to calculatelocal structural stresses at the center plane of the sheetmetal. The mean stress sensitivity of this test data wasset equal to zero since the effects of mean stress werenegligible. Fig. 14 shows total fatigue life (Nf) versusmaximum equivalent stress amplitude. The equation ofthe power law relation from Fig. 14 was the governingequation to calculate fatigue life. The correlation coef-ficient (cc) of the best fit line is again indicated in the

figure. Fig. 15 shows the calculated versus measuredfatigue life for multiaxial fatigue test data. As shownin Fig. 15, Rupp and co-workers approach can also becorrelated with measured fatigue life for the multiaxialresults with reasonable accuracy.

This method is also very sensitive to the variation of

Fig. 14. Experimental life versus Rupp and co-workers maximum

equivalent stress amplitude for the specimens subjected to multiaxial

loads.

Fig. 15. Experimental life versus calculated life using Rupp and co-

workers method for multiaxial load test data.

the sheet thickness as Rupp and co-workers tried toreduce the effects by using the geometry correction fac-tor. As in the other methods, the parameter correlatedwell because the test data was of one sheet thickness. Itis expected, however, that this method is less sensitiveto the sheet thickness because of the geometry factor.

The interpolation/extrapolation technique was applied

to this test data. For this test data, six independent vari-ablesthe sheet thickness, nugget diameter, specimenwidth, load amplitude, applied loading angle, and R-ratiowere considered. At the location of each datapoint for which fatigue life was calculated, the data pointwas removed from the database, and then the calculationwas made for that point. This was done so that thenumerical technique would not be biased by the pointfor which the calculation was made. Plots of calculatedversus measured values are presented in Fig. 16. Asshown in the figure, the interpolation/extrapolation tech-nique can be correlated with measured fatigue life for themultiaxial results with reasonable accuracy. The pointsoutside the factor of three boundary were based on extra-polations.

8. Summary and conclusions

Fatigue tests of the spot welded specimens subjectedto combined tension and shear loads were conducted.The fatigue life of the specimens depended most signifi-cantly on the applied load amplitudes and the spot weldnugget diameters. The initiation of fatigue cracks wereobserved at the edge of the weld nugget in the sheet


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Fig. 16. Experimental life versus calculated life using

interpolation/extrapolation for multiaxial load test data.

metal, and propagation of the cracks depended on theloading directions and amplitudes.

Four techniques for calculation of fatigue life of spotwelded joints were evaluated with multiaxial test data.

Swellam and co-workers, Sheppards, and Rupp and co-workers approaches were based on the physicalinterpretation of the spot welded joints. Additionally, anumerical technique was evaluated that used only thebasic loading conditions and specimen geometries todetermine the fatigue life.

The four approaches were evaluated with experi-mental fatigue life of the multiaxial tests, and allapproaches were fairly reasonable at consolidating thedata to within a factor of three of the actual test life,even though each method assumed linearity between theapplied load and the stresses or stress intensity factors.From this study, the following conclusions can bedrawn:

1. In Swellam and co-workers approach, the parametersb and b0 may be changed depending on materials andtesting. Therefore, the success of Swellam and co-workers method relies heavily on determining theappropriate parameters b and b0 such that the scatterof the Ki vs. Nf plot is minimized.

2. Sheppards method performed reasonably well forthese multiaxial test results that included only onethickness. However, the maximum structural stressrange is sensitive to the variation of the sheet thick-

ness. As shown in [7,8], the determination ofM* iscomplex procedure since the value of M* is basedon the conditions of the nodal force and momentranges at the spot welded connections.

3. As in Sheppards approach, sheet metal stresses inRupp and co-workers method were calculated from

elastic stress equations. The parameter , a materialdependent geometry factor effectively reduced thethickness effect on the equivalent stress.

4. The interpolation/extrapolation technique does notrequire a detailed mechanics model of the local stressstate or material constants for semi-empirical equa-tions since it only depends on the test data. However,this technique should be used with caution forextrapolated data as mentioned in [11].

Finally, each of the semi-empirical methods can besensitive to the variation of the sheet thickness because

of the bending stress or stress intensity terms if ageometry factor is not included. The experimental resultspresented here do not highlight this behavior becausethis series of tests were conducted using a single sheetthickness. This issue will be a topic of future work.

Acknowledgements

This work was sponsored by the DaimlerChryslerUniversity Challenge Fund and supported by the StressLaboratory and Material Engineering at the ChryslerTechnology Center in Auburn Hills, Michigan. Theauthors of this paper would like to thank all of the Daim-lerChrysler Spot Weld Design and Evaluation Commit-tee members, including Eric Pakalnins (Chairman),Thomas W. Morrissett, Dr Ming-Wei Lu, Robert W. Jud,Tim J. Whehner, and Anthony Kittrell for technical dis-cussions and for their efforts to provide the test fixtureand welded specimens.

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