analysis of the local stresses at laserwelded lap...

WELDING RESEARCH

SEPTEMBER 2014 / WELDING JOURNAL 351-s

Introduction Using high-power lasers to weld lap joints has becomemore economically viable to create transportation body pan-els. Laser welding transmits a relatively low heat input andresults in a narrow weld bead compared to resistance spotwelding. The high-cycle fatigue strength of lap joints of thinmetal sheets depends on the joining process, types of alloys,geometrical features of the component or structure, surfaceroughness, and defects such as porosity and cracks.Although there have been many studies performed in thisarea, the high-cycle fatigue assessment of laser-welded lapjoints requires a more in-depth stress analysis and fatiguetests because high-power lasers, new applications, and ma-terials such as high-strength steels and aluminum alloyscontinue to develop. The fatigue analysis by Radaj et al. (Ref. 1) was based onlocal stress concepts, such as the structural or nominal

stress, local stress concentrationaround a notch, and crack initiationand propagation. Hsu and Albright(Ref. 2) presented the fatigue analysisof laser-welded lap joints using Golandand Reissner’s local stress equations,which incorporate the shear and peel-ing stresses. Zhang (Ref. 3) developed amethod that predicted the maximumlocal stress of a laser-welded overlapweld by measuring the strains of theouter surface near a joint.

Hobbacher (Ref. 4) separated anotch stress into membrane stress,bending stress, and nonlinear stresspeak and suggested formulas of stressconcentration factors for misalignmentbetween flat panels as a closed form interms of misalignment, thicknesses,and lengths of two plates of a buttjoint. Tovo and Lazzarin (Ref. 5) inves-tigated the relationship between thestructural stress and the local stress

field by characterizing a notch opening angle and expressingthe stress components as an equation of two notch stress in-tensity factors. They suggested local stress and structuralstress close to the toe of single fillet weld geometry and dou-ble fillet weld geometry. Li, Orme, and Yu (Ref. 6)established an elasticity model of a V-groove butt-joint weldby solving approximately the in-plane displacement filedwith the Ritz method and obtained a closed form relationbetween the overall Young’s modulus and those ofindividual zone. Because of the complexity of local stress fields near orwithin a laser weld, the finite element method (FEM) iswidely used to obtain detailed local stresses; however, it isdifficult to estimate the effect of the structural parameterson the fatigue life using an FEM model. Wang (Ref. 7)predicted the fatigue resistance of a stitch laser weld usingan FEM model and a J-integral around a root opening,which was treated as a crack-like notch front. Cho et al. (Ref.

Analysis of the Local Stresses at LaserWelded Lap Joints

Geometric eccentricity affects the tensile fatigue life of lap welds

BY K. D. LEE, K. I. HO, AND K. Y. PARK

ABSTRACT Theoretical and numerical stress analyses were performed to understandlocal stress effect due to geometric eccentricity to estimate the fatigue life oflaser-welded overlap joints of automotive thin steel sheets. Two thicknessesand a nonzero root opening were used. The two-dimensional problem of thestress function within a laser-welded bead was solved in the form of a gener-alized crosswise Fourier series. The nondimensional local stress factors withrespect to the remote tensile stress for twelve cases were compared at threelocations in a bead. The locations were chosen based on where localmaximum stresses occur; such points can cause failure during a fatigue test.

KEYWORDS • Local Stress Function • Laser Welded Lap Joint • Geometrical Eccentricity • Crosswise Fourier Series • Fatigue Strength • Automotive Steel Sheet

K. D. LEE ([email protected]) is director, Institute for Advanced Engineering, Younginsi, Gyeonggido, Republic of Korea. K. I. HO ([email protected]) is professor,Department of Mechanical Engineering, Suwon University, Hwaseongsi, Gyeonggido, Republic of Korea. K. Y. PARK ([email protected]) is senior researcher, Institute for Advanced Engineering, Younginsi, Gyeonggido, Republic of Korea.

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WELDING RESEARCH

WELDING JOURNAL / SEPTEMBER 2014, VOL. 93352-s

8) predicted the fatigue life curves of a laser-welded lap jointof steel sheets, where the FEM model included the effect ofthe thermal residual stresses due to the laser welding. Yeand Moan (Ref. 9) predicted the fatigue strength and initia-tion area of an aluminum box-stiffener lap joint by calculat-ing the stress concentration factors using FEM. Anand et al. (Ref. 10) experimentally obtained a fatigueratio (the ratio of fatigue limit to tensile strength) of 0.4 fora laser-welded butt joint that combined 1.5- and 0.9-mmsteel sheets. Moreover, Lee et al. (Ref. 11) experimentallyobtained a fatigue ratio of 0.073 for a laser-welded lap jointthat combined 2.0- and 1.5-mm stainless steel sheets.Despite the different materials and the differentthicknesses, the large difference between the fatigue ratiosof the butt and lap joints is readily apparent. To model such a large reduction in the fatigue limit oflaser-welded lap joints, many studies adapted the notchstress concept, which considers the root opening as afictitious root notch. Eibl et al. (Ref. 12) evaluated the notchstress by replacing the root opening by a fictitious notchwith a 0.05-mm radius to model the extenuation in thefatigue strength of a laser-welded lap joint with completepenetration. Zhang et al. (Ref. 13) predicted the fatigue lives of laser-welded lap welds using the shell theory and assuming theroot opening to be a fictitious root notch with a 0.05-mm ra-dius. Using the shell theory, the structural stress at theouter surface increased to four times greater than the nomi-nal (average) stress in the thick and thin sheet. Thecombined stress concentration factor due to the fictitiousnotch effect was then used as the basis of the structuralstress at the outer surface from the shell theory to reducethe relatively large scattering between the differentthickness combination and the same thickness combination. In this study, a more detailed theoretical stress analysis ona laser-welded lap joint was performed, and the local stressvalue and the high-cycle fatigue strength are presented. Thehigh-cycle fatigue strength was calculated because Lee et al.

(Ref. 11) had observed that the local stress effect at the weldbead of a laser-welded lap joint cannot be neglected, even ifthe flexural deformation from the gripping misalignment isnegligible. Both the geometric eccentricity at the lap joint andthe thickness ratio influence the local stress distributionwithin or around the weld bead during the tensile-shearfatigue loading. Numerical calculations were performed to ob-tain the detailed stress distribution within a weld bead by solv-ing the two-dimensional problem using crosswise Fourier se-ries for the stress functions. The boundary conditions of theweld bead were derived by assuming a linear deformationcaused by a rotation of the lap-jointed weld bead in the formof an explicit nondimensional equation. The distribution ofthe ratios of the local stresses relative to the remote uniformtensile stress [local stress factor (LSF)] in the weld bead wasobtained for 12 cases, where two parameters, the thicknessratio and the root opening ratio, were varied (four thicknessratios and three root opening ratios). The local maximum can-didates of the LSF at three locations were compared to selectthe maximum value of the LSF for each case; at theselocations, a failure was most likely to occur during the fatiguetesting (i.e., candidates for hot spots).

Fig. 1 — Tensileshear fatigue specimen.

Fig. 2 — Stress distribution at the boundary of a weld bead.

Fig. 3 — Freebody diagram depicting loading and three locations of A, B, and C.

Table 1—DOE for the Numerical Analysis

t2⁄t11.0 1.1 1.2 1.3

2g⁄t1 0.0 E1(A,B,C) E2(A,B,C) E3(A,B,C) E4(A,B,C)0.1 E5(A,B,C) E6(A,B,C) E7(A,B,C) E8(A,B,C)0.2 E9(A,B,C) E10(A,B,C) E11(A,B,C) E12(A,B,C)

Table 2 — Comparison of the Fatigue Stresses at 5 × 106 Cycles between aButtJoint Weld and a LapJoint Weld from Eibl et al. (Ref. 12) and TheirCalculated LSF

Material Joint Thickness Cycles (x106) Stress (MPa) LSF(mm)

GD AlSi10MgT6 butt 3.0/3.0 5.0 38.4 4.3

lap 3.0/3.0 5.0 9.0

AlMg4.5Mn butt 1.5/1.5 5.0 43.1 4.8lap 1.5/1.5 5.0 9.0


WELDING RESEARCH


Theoretical Analysis The two-dimensional problem of the stress function wassolved in the form of a crosswise Fourier series within aweld bead of a fatigue specimen because failure during a fa-tigue test was typically observed around the weld bead andthe heat-affected zone (HAZ).

Design of Experiment for Analysis

A welded (tensile-shear) fatigue specimen is shown inFig. 1. Two sheets with different thicknesses were welded bya laser into a lap joint. The sheet thicknesses are denoted byt1 and t2, where t2 is equal to or greater than t1. The rootopening size is denoted by 2g. The weld bead width isdenoted by 2w. In the numerical analysis, 2w was 1 mm andt1 was 1.5 mm. The design of experiment (DOE) for the numerical analy-sis is given in Table 1, where three parameters were used.Two of the parameters were t2 ⁄ t1 and 2g ⁄ t1, which are thenondimensional geometrical parameters of the weld bead.The third parameter was the location denoted by A, B, or C,which is shown in Fig. 2. Locations A and B are candidatesfor the occurrence of local maximum normal stress in theboundary of the thin sheet and the thick sheet, respectively.Location C is at the center of the root opening and the weldbead width, which is where a local maximum shear stresswill occur. Twelve numerical experiments (E1–E12) wereperformed, and the LSFs were calculated at the threelocations (A, B, C).

Boundary Conditions

Because the high-cycle fatigue was to be considered, alllocal stresses were calculated assuming elastic deformation.Geometrical defects, including notches and thermal resid-ual stresses, were not considered in this study. It wasassumed that the mechanical and metallurgical propertiesof the base materials and the weld bead were within thesame tolerance. Superposition of the stresses was applica-ble within the weld bead. The tensile-shear specimen was loaded in a two-stepprocess: gripping and tensile-shear loading. When aspecimen with a lap joint is gripped during a fatigue test, aflexural deformation generally occurs due to the geometrical

eccentricity of the two sheets at the weld bead. A free-body diagram around the weld bead during tensile-shear loading with a grip misalignment is shown in Fig. 3,where F is a remote load activated by a fatigue machine. Aremote load applies a uniform average stress (remoteuniform tensile stress) and a shear stress through the beadwidth in the load direction. When the tensile-shear loadingbegins, the transverse (vertical) shear force at the end gripscan be removed by adding spacers with the same thicknessat each grip. The relationship between the parameters, suchas the flexural moments, Mg1 and Mg2, the axial force, F, andthe eccentricities, e1 and e2, measured from the center of thethickness of each sheet at the end grips, is shown inEquation 1. The flexural moments, shear forces, and eccen-tricities at the end grips, such as Mg1, Mg2, Vg1, Vg2, e1, ande2, could be solved in terms of V, F, M1, M2, and l. However,because the distance between the end grip and the weldbead, l (40 mm), is so long that the specimen is flexible tobend, the Euler beam bending theory can be applied to thespecimen between the end grip and the weld bead.Therefore, the parameters at the end grips do not need to becalculated when the stress analysis at the weld bead isperformed and this paper focuses on the solution for thestresses at the weld bead, where fatigue failure would occur.

Fig. 4 — Distribution of Sx, Sy, Sxy in the case of Equation 9. A —Sx; B — Sy; C — Sxy.

A B

C

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WELDING RESEARCH


After the tensile-shear loading begins within the elasticrange without a grip misalignment, the geometric eccentricityaround a weld bead will cause the weld bead to rotate. This ro-tation, , creates the moments and vertical forces at the weldbead. The moments due to the rotation of the weld bead gen-erate the tensile and compressive local stresses in the verticalsections of the weld bead, as the remote load results in anaxial, nonuniform elongation through the thickness of thesheets according to the grip movement. The tensile and com-pressive stresses are superimposed on the uniform remotestress. From the force and moment equilibrium, Equation 2can be derived.

M1 + M2 + 2wV = (0.5(t1 + t2) + 2g)F (2)

Equation 3 can be obtained using the Euler beam bendingtheory. As for the boundary condition, it was assumed that thesum of the deflection at the left side of the weld bead (Section2) and the deflection at the right side of the weld bead(Section 1) is approximately zero because the bead moves upand down together, and the deflection caused by the rotationof the weld bead is negligible within the elastic limit.

Equation 4 can be obtained by assuming that the slope(absolute value) at the left side of the weld bead (Section 2)is equal to the slope at the right side of the weld bead(Section 1) because the elastic deformation is in the range ofthe fatigue limit.

By solving the three equations, M1, M2, and V can be ex-pressed by F and the geometric parameters.

VlEI

MEI

Vl3EI

MEI

− + − + =3

0 (3)2

2

1

1

12

2 2(4)

2

2

1

1

12

− + = − +VlEI

MEI

VlEI

MEI

V V V

M Fe M Vl

M Fe M Vl

M M Ft t

e e

g g

g 1

g

g g( )

= =

+ = −

+ = −

+ =+

− −⎛⎝⎜

⎞⎠⎟2

(1)

1 2

1 1

2 2 2

1 21 2

1 2

=+

⎛⎝⎜

⎞⎠⎟

+

− +⎛⎝⎜

⎞⎠⎟

⎛

⎝⎜⎞

⎠⎟+

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

=+

⎛⎝⎜

⎞⎠⎟

+⎛

⎝⎜⎞

⎠⎟−

⎛⎝⎜

⎞⎠⎟

− +⎛⎝⎜

⎞⎠⎟

⎛

⎝⎜⎞

⎠⎟+

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

=+

⎛⎝⎜

⎞⎠⎟

+⎛

⎝⎜⎞

⎠⎟−

⎛⎝⎜

⎞⎠⎟

− +⎛⎝⎜

⎞⎠⎟

⎛

⎝⎜⎞

⎠⎟+

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

V

tt

2gt

12II

II

wl

tl

F

M

ttt t

II

II

II

wl

F

M

ttt t

II

II

II

wl

F

g

g

(5)

12

1

56

12

12

12 5

121

12

56

112

2

12

12 5

121

12

56

112

2

2

1 1

2

1

1

2

1

1

12

1 1

1

2

2

1

1

2

2

12

1 1

2

1

2

1

1

2

A B

C D

Fig. 5 — The LSFs at three locations: A — Sx; B — Sy; C — Sxy; D — Sx, Sy, Sxy at A and C for the root opening ratio 0.2.


WELDING RESEARCH


As for the boundary conditions, the local stresses fromSections 1 and 2 are given in Equation 6 in an explicit formfrom the Euler beam bending theory. The normal stress at thevertical cross section can be obtained by superimposing thenormal stress from the bending moment due to the geometri-cal misalignment onto the uniform nominal stress from theremote axial force. The shear stresses at the vertical cross sec-tion can be approximated to be uniformly distributed throughthe vertical axis. Furthermore, it is assumed in this paper thatthese shear stresses are comparatively negligible because theywere within 10% of the normal stresses when the thicknessratio was less than 1.3.

Analytical Solution within a Weld Bead

The stress distribution around the weld bead was evalu-ated to identify where the maximum stress was located,which is where fatigue failure would occur. To evaluate the

stress distribution around the weld bead, the crosswiseFourier series solutions suggested by Timoshenko (Ref. 14)was used within the weld bead, whose cross section was as-sumed to be a rectangular narrow beam, as shown in Fig. 2. Equations 7 and 8 are the partial differential equationsrepresenting the stress function that can be satisfied by su-perimposing a single Fourier series in the x and y directionsfor finite rectangles.

The generalized crosswise Fourier series solution for thestress function is expressed as Equation A1 or expanded inEquation A2 in the Appendix. Thus, the stresses derived bysubstituting the stress function from Equation A1 to Equa-tion 8 are as in Equation A3. The above theoretical equations were applied to a weldbead, where Equation 6 was used as the boundary condition,which is isolated. A rectangular coordinate system was used, asshown in Fig. 2. Due to the bending effects from the eccentri-cally applied remote tensile forces at the ends of plates, thestresses at the upper-right vertical surface (Section 2) and thelower-left vertical surface (Section 1) were assumed to be acombination of the uniform normal stress and the linear nor-

0.512.5

0.512.5

12.5

12.5(6)

11

11

1

22

22

2

11

22

( )

( )

( )

( )

( )

( )

( )

( )

σ = + + +

σ = − + +

τ =

τ =

MI

y g tF

tfor the thin sheet

–MI

y g tF

tfor the thick sheet

v

tfor the thin sheet

v

tfor the thick sheet

x

x

xy

xy

x x y y

∂ ϕ∂

+ ∂ ϕ∂ ∂

+ ∂ ϕ∂

=2 0 (7)4

4

4

2 2

4

4

y x x yx y xyσ = ∂ ϕ∂

σ = ∂ ϕ∂

τ = − ∂ ϕ∂ ∂

(8)2

2

2

2

2

A B

C D

Fig. 6 — The LSF equivalents for the plane strain and plane stress conditions. A — Seq,1; B — Seq,2; C — Seq,1 and Seq,2 at three locationsfor a root opening ratio of 0.2; D — Seq,2.


WELDING RESEARCH


mal stress. The origin of the coordinate system is at the centerof the root opening. In the root opening surface between theupper and lower plates, the normal stresses are zero. At thetop and bottom horizontal surfaces, at y = ±p, the normal andshear stresses are zero and can be expressed as follows. The boundary conditions applied at the finite rectangleare given in Equation A4. From the boundary conditions, wederived all the coefficients of the stress components as inEquation A5. Because all the coefficients in the above stresscomponents are coupled to each other, they can bedetermined through a numerical iteration method with theinitial conditions.

Result and Discussion

Local Stress Factors in a Weld Bead

The local stress factors are defined by the ratios of thelocal stresses relative to the remote uniform tensile stress inthe thin sheet (t1) and denoted by Sx, Sy, Sxy, which were cal-culated for the twelve cases.

where ∞ is the remote uniform tensile stress in the thinsheet and is Poisson’s ratio. As a fatigue failure criterion,Seq,1 and Seq,2 are the von Mises equivalent stresses normal-ized by ∞ for the plane strain and plane stress conditions,respectively (Refs. 15–17). Each case in Table 1 has one numerical data set. The nu-merical calculations for the 12 cases produce 12 sets of thelocal stresses. The numerical results of the LSF distributionfor Equation 9 are shown in Fig. 4. The points A and B arethe local maximums of Sx, and point C is the local minimumof Sxy. The locations of the maximum and minimum of Sywere slightly different from those of Sx. Numerical results for the LSFs at the three locations aregiven in Fig. 5 for the various thickness and root opening ra-tios. When the thickness ratio is one, the LSFs at A and Bshould be equal to within a numerical error. A notation ofeach curve in Fig. 5 represents the location and root openingratio. For example, A0.1 represents the LSF curve forlocation A with a root opening ratio of 0.1. The normal stress in the x direction, Sx, is shown in Fig.5A. As the thickness ratio increased, Sx increased at A (in thethin sheet) but decreased at B (in the thick sheet). For exam-ple, A0.2 increased from 4.5 to 5.8 and B0.2 decreased from4.5 to 3.8. The larger root opening ratio resulted in a largerSx at both the A and B locations. For example, Sx at Aincreased from 5.0 to 5.8 when the TR was 1.3 and the GRwas zero, and Sx at C was between 1.0 and 1.9, which wasmuch lower than at A or B. When the TR was larger than 1.0,Sx at A was always greater than at B. The normal stress in the y direction, Sy, is shown in Fig.5B. As the thickness ratio increased, Sy increased at A butdecreased at B. For example, A0.2 increased from 3.6 to 5.4and B0.2 decreased from 3.6 to 2.3. As the root openingratio increased, Sy increased at both A and B, which indicatesthat the peeling stress was greater in the y direction.However, Sy at C was nearly zero and did not change much,even as the thickness or root opening ratio increased. Figure 5C shows Sxy, which is negative within a weldbead. A local minimum of Sxy was at location C, and at A andB, Sxy was almost zero. As the thickness ratio increased, theabsolute value of Sxy increased. For example, the absolutevalue of C0.2 increased from 2.1 to 2.4. As the root openingratio became larger, the absolute value of Sxy increased. Forexample, the absolute value of Sxy at C increased from 1.8 to2.4 when a thickness ratio of 1.3 was used. The maximum LSFs for the different thickness ratios areshown in Fig. 5D, where the root opening ratio was 0.2. Thesolid lines are the LSFs for A0.2, and the dotted lines are theLSFs for C0.2. For A0.2, Sx and Sy were positive, whereas Sxywas almost zero, but for C0.2, Sy was almost zero, Sx waspositive, and Sy was negative.

Equivalent LSF for Multiaxial Stress States

Both Seq,1 and Seq,2, as defined in Equation 9, are theequivalent local stress factors for the plane strain and planestress conditions, respectively, which were calculated fromthe LSFs and used as a criterion to estimate the fatigue lifeunder the multiaxial stress state in the weld bead. Figure 6A shows Seq,1 for the various thicknesses androot opening ratios from Table 1. The plane strain condition

1 1 2 2 3

0.5 3

(9)

12 2 2 2 2 0.5

22 2 2 2

0.5

( )( ) ( )

( )

= σ σ = σ σ = τ σ

= − μ + μ + − + μ − μ +⎡⎣

⎤⎦

= − + +⎛⎝⎜

⎞⎠⎟ +⎡

⎣⎢⎤⎦⎥

∞ ∞ ∞S / S / S /

S S S S S S

for the plane strain condition

S S S S S S

for the plane stress condition

x x y y xy xy

eq , x y x y xy

eq , x y x y xy

Fig. 7 — SN curves for mild steel, STS301L, GDAlSi10MgT6from Refs. 8, 11, 12, and 18. A — Mild steel (UWS): calculatedby Equation 11 with LSF = 1 (Ref. 8); B — mild steel (planestrain): calculated by Equation 11 with LSF = 2.0 (plane straincondition); C — mild steel (plane stress): calculated by Equation11 with LSF = 3.3 (plane stress condition); D — mild steel (butt):R = 0, 30 Hz, t1 = 0.9 mm, width of specimen = 7 mm (Ref. 18); E— mild steel (lap): R = 0, 10 Hz, t2/t1 = 1.0/1.0, width of specimen = 40 mm, 2w = 0.8~2.0 mm (Ref. 8); F — STS301L(lap): R =0, 20 Hz, t2/t1 = 2.0/1.5, width of specimen = 12.5 mm, 2w = 1.0mm (Ref. 11); G — GDAl (butt): R = 0, 25~35 Hz, t1 = 3.0 mm,width of specimen = 30.0 mm, 2w = 3 mm (Ref. 12); H — GDAl(lap): R = 0, 25~35 Hz, t2/t1 = 3.0/3.0, width of specimen = 30.0mm, 2w = 2~4 mm (Ref. 12).


WELDING RESEARCH


can be applied to the center sections of the weld beadsections. For the plane strain condition in the centersection, Seq,1 had a local maximum at C, which was between3.2 and 4.2; another local maximum occurred at A or B,which was between 1.7 and 2.4. Figure 6B shows Seq,2 for the various thicknesses androot opening ratios from Table 1. The plane stress conditioncan be applied to the edge side sections of the weld bead sec-tions. For the plane stress condition in the edge-sidesection, Seq,2 at A and C increased as the thickness ratio in-creased, but the value at A was always larger than at C. How-ever, Seq,2 at B decreased as the thickness ratio increased.For example, A0.2 increased from 4.0 to 5.6, C0.2 increasedfrom 3.7 to 4.3, and B0.2 decreased from 4.0 to 3.2. Figure 6C gives Seq,1 and Seq,2 when the root openingratio was 0.2. Both A0.2Seq,2 and B0.2Seq,2 were much largerthan A0.2Seq,1 and B0.2Seq,1; however, C0.2Seq,1 andC0.2Seq,2 were essentially equal and close to A0.2Seq,2, whichwas the largest value. When the thickness ratio was one, thedifference between A0.2Seq,2 and C0.2Seq,2 (or C0.2Seq,1) wasless than 0.3. Therefore, fatigue failure would initiate at A inthe edge-side sections if it was assumed that no defect waspresent in the weld bead. However, when the thickness ratiois near one, the probability of fatigue failure in the shearmode at C in all the sections should be checked. For the plane stress condition in the edge-side sections,Fig. 6D gives Seq,2 for all the thicknesses and root openingratios. As the thickness ratio or root opening ratioincreased, Seq,2 increased. At A, Seq,2 increased more rapidlythan Seq,2 at C. When the root opening ratio was 0 or 0.1and the thickness ratio was one, the difference between Seq,2at A and Seq,2 at C was so small that fatigue failure would ini-tiate at both A and C in the edge-side sections. When thethickness ratio was larger than one, fatigue failure wouldinitiate only at A in the edge-side sections because A0Seq,2,A0.1Seq,2, and A0.2Seq,2 were larger than C0Seq,2, C0.1Seq,2,and C0.2Seq,2, respectively. If there are no additional stress concentrations due to aroot notch around the weld bead or other defects inside theweld bead, then the local stress factors and the locationswhere fatigue failure might initiate can be predicted, whichcan then be used in predicting the fatigue life.

Fatigue Strength of a Lap Joint

Without any defects in the weld bead, fatigue failurewould occur at either location A, B, or C, wherever the LSFhas the local maximum value. When the von Mises equiva-lent stress reaches a criterion for fatigue failure, the locationand its LSF value at the fatigue failure can be found in Fig. 6.The fatigue strength, SN, lap, of a lap joint can be calculatedfrom Equation 10 in terms of the fatigue strength, SN, of thesheet material and the maximum of the equivalent LSFs(such as Seq,2 in Fig. 6D). Afterward, SN, lap can be modifiedby the fatigue notch factor or by other correction factors.

According to Fig. 6C, fatigue failure occurred at locationA when the thickness ratio was not one. When the thicknessratio was 1 and the root opening ratio was 0.2, themaximum value of Seq,2 was 4.1, which indicates that the lapjoint will fail at the fatigue limit 4.1 times lower than itsbase material’s if there is no notch effect. When thethickness ratio was 1.3 and the root opening ratio was 0.2,the maximum Seq,2 was 5.6 at A. If the fatigue strength of a butt joint, SN, butt, of the sheetmaterial is known, Equation 11 is useful for getting thefatigue strength of a lap joint because the experimental fa-tigue strength of a butt joint already has the fatigue notchfactor or other correction factors, including the metallurgi-cal change.

For example, Fig. 7 shows typical S-N curves obtainedfrom Refs. 8, 11, 12, and 18. The vertical axis is the fatiguestress ratio (FR), which is defined as the ratio of theactivated fatigue stress at the number of cycles to theultimate tensile strength of the specimen, and the parallelaxis is the number of reversals (2N), where both axes aredisplayed on a logarithmic scale. The S-N curve of mild steel is shown as ‘mild st (UWS)’ inFig. 7, which is calculated by the relation between fatiguelife and ultimate tensile strength in Ref. 8. It depicts the fa-tigue strength, SN, of mild steel (the unwelded specimen)when FR = 0.9 at 2N = 103 and FR = 0.5 at 2N = 106. For lap-joint mild steel, Equation 12 is derived from Equation 10and the relation in Ref. 8. The new S-N curves of the lap-joint specimens without defects were obtained for twoequivalent LSF results, 2.0 (at A0 or B0 of the plane straincondition in Fig. 6A) and 3.3 (at A0 or B0 or C0 of the planestress condition in Fig. 6B) by calculating the fatiguestrength, SN, lap, from Equation 12. They are shown as ‘mildst (plane strain)’ and ‘mild st (plane stress)’ in Fig. 7.

An experimental S-N curve of lap joint mild steel (Ref. 8) isgiven as ‘mild st (lap)’ in Fig. 7 when TR = 1 and GR = 0.0 witha strip specimen of 40 mm width. Another experimental S-Ncurve of a butt joint mild steel for R = 0 and 15 Hz (Ref. 18) isshown as ‘mild st(butt)’ in Fig. 7 when a tensile specimen has0.9 mm thickness and 7-mm specimen width. Calculated LSFresults were between 2.0 and 2.8, which exhibited a good cor-relation with the S-N curve of equivalent LSF 2.0 at A0 or B0for TR = 1 and plane strain condition in Fig. 6A. An experimental S-N curve of a lap joint of STS301L (Ref.11) gives very low FR (0.07 at 106 cycles) when R = 0 at 20Hz using a tensile-shear specimen with TR = 1.3 and GR =0.1 in Fig. 1. This corresponds to the equivalent LSF 5.0 atA0.1 of plane stress condition in Fig. 6B. The experimental S-N curves of butt and lap joints ofGD-AlSi10Mg-T6 show very low fatigue stress ratio when R

SS

max LSFN ,lap

N

eq( )= (10)

SS

max LSFN ,lap

N ,butt

eq( )= (11)

S mild steelS N

max LSFN ,lap

u

eq( ) ( )= 1.62

(12)-0.08509


WELDING RESEARCH


= 0 at 25~35 Hz using a 30-mm-wide specimen with TR =3.0/3.0 and GR = 0.2 (Ref. 12). Although the welds of GD-AlSi10Mg-T6 seem to have many defects or degradation, theLSF result between FRs of a lap joint and a butt joint isabout 4.2~4.4, which exhibited a good correlation withLSF4.1 at A0.2 of TR = 1 and the plane stress condition inFig. 6D. In Eibl (Ref. 12), the fatigue strength results of the buttand lap joints with complete penetration of two aluminumsheets were obtained. Comparisons between the fatiguestrength of a laser-welded butt joint in Appendix A and alaser-welded lap joint in Appendix B are shown in Table 2.Calculated LSF results (LSF4.3 and LSF4.8) exhibited a goodcorrelation with LSF4.1 (at A0 of TR = 1 in Fig. 6B) of thesame thicknesses if a small root opening is assumed. In Zhang et al. (Ref. 13), the discrepancy of theexperimental fatigue data between the different thicknesscombinations and the same thickness combinations could beexplained by introducing the LSF before the fictitious rootnotch radius because the notch effect was added byintroducing the mixed parameter, Pa, as the combined stressintensity factor. Afterward, the notch effect due to weldingdefects could be superimposed on the LSF results.

Summary and Conclusions 1. The local stress factors in the weld bead of a lap jointcreated by laser beam welding were numerically obtainedand displayed in graphs according to the nondimensionalvariations of the thickness ratio, root opening ratio, andthree candidate locations. 2. The two-dimensional differential equation of thestress functions was solved in a generalized crosswiseFourier series. The boundary condition of the weld beadafter tensile loading was derived theoretically in the nondi-mensional form by assuming a weld bead rotation caused bythe geometrical eccentricity and a linear deformationthrough the thickness. 3. The stress fields within the weld bead of a lap jointwere obtained according to the thickness ratio and the rootopening ratio. The maximum stress in the loading directionor in the transverse direction occurred at the corner, nearthe root opening of the thin sheet, but the maximum shearstress occurred at the center of the weld bead. All the maxi-mum stress components increased as the thickness ratioand/or the root opening ratio became larger. 4. Note that when a fatigue evaluation is under considera-tion, the tensile-shear effect in a lap-joint weld dramatically in-creases both the maximum normal stresses in the loading andtransverse directions as well as the shear stresses near the rootopening, which should not be neglected. 5. The equivalent local stress factors were taken as a cri-terion for the fatigue life estimation, and the comparisonsof the LSFs with the experimental results in other studieson the fatigue life resulted in excellent correlations.

The authors would like to thank the Institute for AdvancedEngineering (IAE) for providing partial financial support.

1. Radaj, D., Sonsino, C. M., and Fricke, W. 2009. Recent devel-opments in local concepts of fatigue assessment of welded joints.International Journal of Fatigue (31): 2–11. 2. Hsu, C., and Albright, C. E. 1991. Fatigue analysis of laserwelded lap joints. Engineering Fracture Mechanics 39(3): 575–580. 3. Zhang, S. 2002. Stresses in laser-beam-welded lap joints de-termined by outer surface strains. Welding Journal 81(1): 14-s to18-s. 4. Hobbacher, A. 1996. Fatigue design of welded joints andcomponents. pp. 115, 116. Abington Publishing. 5. Tovo, R., and Lazzarin, P. 1999. Relationship between localand structural stress in the evaluation of the weld toe stress distri-bution. International Journal of Fatigue 21: 1063–1078. 6. Li, L., Orme, K., and Yu, W. 2005. Effect of joint design onmechanical properties of Al7075 weldment. Journal of Material En-gineering and Performance 14(3): 322–326. 7. Wang, P. C. 1995. Fracture mechanics parameter for thefatigue resistance of laser weld. International Journal of Fatigue17(1): 25–34. 8. Cho, S. K., Yang, Y. S., Son, K. J., and Kim, J. Y. 2004. Fatiguestrength in laser welding of the lap joint. Finite Elements in Analysisand Design 40: 1059–1070. 9. Ye, N., and Moan, T. 2002. Fatigue and static behavior of alu-minum box-stiffener lap joint. International Journal of Fatigue 24:581–589. 10. Anand, D., Chen, D. L., Bhole, S. D., Andreychuk, P., andBoudreau, G. 2006. Fatigue behavior of tailor (laser)-welded blanksfor automotive applications. Materials Science and Engineering A420: 199–207. 11. Lee, K. D., Park, K. Y., and Ho, K. I. 2008. Stress analysis ofstainless steel overlap joining panel by laser welding for fatigue lifeestimation. Materials Science Forum, pp. 515–518. 12. Eibl, M., Sonsino, C. M., Kaufmann, H., and Zhang, G. 2003.Fatigue assessment of laser welded thin sheet aluminum. Interna-tional Journal of Fatigue 25: 719–731. 13. Zhang, G., Eibl, M., Singh, S., Hahn, O., and Kurzok Jr.2002. Methods of predicting the fatigue lives of laser-beam weldedlap welds subjected to shear stresses. Welding and Cutting 2:96–103. 14. Timoshenko, S. P., and Goodier, J. N. 1970. Theory of Elastic-ity. pp. 35–39, McGraw-Hill Book Co. 15. Kurath, P. 1997. Multiaxial fatigue criteria for spot welds.SAE PT-67. pp. 143–153. 16. Findley, W. N. 1959. A theory for the effect of mean stresson fatigue of metal under combined torsion and axial loading orbending. Journal of Engineering for Industry, 301–306. 17. Stephans, R. I., Fatemi, A., Stephans, R. R., and Fuchs, H.2001. Metal Fatigue in Engineering. pp. 318–328. John Wiley &Sons Inc. 18. Oh, T. Y., Kwon, Y. K. Lee, C. J., and Kwak, D. S. 2000. The fa-tigue behavior of tailored welded blank sheet metal by laser beam.Journal of the Korean Society of Machine Tool Engineers 9(4): 48–55.

Appendix

List of Symbols

t1, t2: thickness of the thin and thick sheets, respectivelyg: half of the root opening sizew: half of the width of the weld beadl: distance between the end grip and the weld beadF: applied remote load

References

Acknowledgments


WELDING RESEARCH


V: transverse load at the weld beadM1, M2: moment at the weld bead: rotation angle of the weld beadI1, I2: moment of inertia at the cross section of the twosheets,t1 and t2, respectivelyx1, x1: local stresses in sections 1 and 2: stress functionx, y, xy: normal and shear stresses in the two-dimensional x-y spaceLSF: local stress factor defined by the ratio of the local nor-mal (or shear) stress to the remote uniform tensile stress(∞) for a thin thicknessSx=x ⁄∞: LSF for the normal stress in the x axisSy=y ⁄∞: LSF for the normal stress in the y axisSxy=xy ⁄∞: LSF for the shear stress in the x-y planeSeq,1: LSF for the equivalent stress for the plane strain conditionSeq,2: LSF for the equivalent stress for the plane stressconditionSN,lap: Fatigue strength at the number of cycles N for a lapjointSu: Ultimate tensile strengthTR: thickness ratio defined by t2 (thick sheet) divided by t1(thin sheet)GR: root opening ratio defined by 2g (root opening size) di-vided by t1 (thin sheet thickness)LO: location where fatigue failure may occurm: Poisson’s ratio

Equations

2 2

(A1)

0 20

0 2

1 21

1 11

∑

∑

( ) ( )

( ) ( )( ) ( )

( ) ( )

ϕ = + +

+ α +⎡⎣ ⎤⎦

+ β + β⎡⎣ ⎤⎦

=

∞

=

∞

Ax C xy

By

cos y f x sin a y f x

cos x g y sin x g y

n n n nn

m m m mm

2 2

and and are integers

2 (A2)

0 20

0 2

1 2

3 4

1 2

3 4

1

1 2

3 4

1 2

3 4

1

1 2 1 2

∑

∑

( ) ( ) ( )( ) ( )

( ) ( ) ( )( ) ( )

( ) ( ) ( )( ) ( )

( ) ( ) ( )( ) ( )

( ) ( )

ϕ = + +

+

α +

+ α +

⎛

⎝⎜

⎞

⎠⎟

α + α

+ α + α

⎛

⎝⎜

⎞

⎠⎟

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

+

ββ + β

+ β + β

⎛

⎝⎜

⎞

⎠⎟

+ ββ + β

+ β + β

⎛

⎝⎜

⎞

⎠⎟

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

= π β = π

= + = + + +

=∞

=∞

Ax C xy

By

cos a yA cosh x A sinh a x

A x cosh x A xsinh a x

sin a yB cosh x B sinh x

B x cosh x B xsinh x

cos xC cosh y C sinh y

C ycosh y C y sinh y

sin xD cosh y D sinh y

D ycosh y D y sinh y

where anp

,m

p, n m

p p p t g t g

nn n n n

n n n n

nn n n n

n n n n

n

mm m m m

m m m m

mm m m m

m m m m

m

n m

(A3)

02

12

21

1 21

0 1

2 221

0 1

1 21

1 2

1 2

∑∑

∑∑

∑∑

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

σ = + −α α − α α⎡⎣

⎤⎦

+ − β + β⎡⎣

⎤⎦

σ = + α + α⎡⎣

⎤⎦

+ −β β −β β⎡⎣

⎤⎦

τ = − −α α + α α⎡⎣

⎤⎦

− −β β +β β⎡⎣

⎤⎦

=∞

=∞

=∞

=∞

=∞

=∞

B cos y f sin y f

cos x g sin x g

A cos y f sin y f

cos x g sin x g

–C sin y f cos y f

sin x g cos x g

x n n n n n nn

m mII

m mII

m

y nII

nII

n

m 1m m mm

xy n nI

n nI

n

m m mI

m m mI

m

n n

m m

n n

0

0

0

0

0 0

0 0 (A4)

1 1

2

2

1

1 2

1 2

( )

( )

( )( )( ) ( )( ) ( )

σ = σ − − ≤ ≤ −

= − < ≤ +

σ − = σ ≤ ≤ +

= − − ≤ <

τ − = − − ≤ ≤ +

τ = − − ≤ ≤ +

σ − = τ − = − ≤ ≤

σ = τ = − ≤ ≤

w,y for t g y g

for g y g t

w,y for g y g t

for g t y g

w,y for g t y g t

w,y for g t y g t

x , p , x , p for w x w

x ,p , x ,p for w x w

x x

x x 2

xy

xy

y xy

x xy

2

2 2

2

2 2

2

2 2

2 2

1

52 5

2

2

2 52 1

3

2 52 1

4

2 52

( )

( )

( ) ( )

( )

( ) ( ) ( )

( )

( ) ( )

( )

( )

( )

=− +

α

⎛

⎝⎜⎞

⎠⎟α + α α

α + α

=− −

α

⎛

⎝⎜⎞

⎠⎟α + α − α

α − α

=

−α

⎛

⎝⎜⎞

⎠⎟α + α

α α

=

+α

−⎛

⎝⎜⎞

⎠⎟α α

α + α

A

RK K

sinh c c cosh c

sinh c c

A

K Kcosh c a c sinh c R c cosh c

sinh c c

A

K Kcosh c R sinh c

sinh c – c

A

K K2R sinh c

sinh c c

n

nn n

nn n n

n n

n

n n

nn n n n n

n n

n

n n

nn n n

n n

n

n n

n5n n

n n


2

2 2

2

2 2

2 2

2 2

1

63

2

2

3 62 2

3

3 62

4

3 62 6

( )

( )

( ) ( )

( )

( ) ( ) ( )

( )

( ) ( )( )

( )

( )

=− +

α

⎛

⎝⎜⎞

⎠⎟α + α α

α + α

=− −

α

⎛

⎝⎜⎞

⎠⎟α + α − α

α − α

=

−α

⎛⎝⎜

⎞⎠⎟

α + α

α − α

=

+α

−⎛

⎝⎜⎞

⎠⎟α α

α + α

B

RK K

sinh c c cosh c

sinh c c

B

K Kcosh c a c sinh c R c cosh c

sinh c c

B

K Kcosh c 2R sinh c

sinh c c

B

K K2R sinh c

sinh c c

n

nn 6n

nn n n

n n

n

n n

nn n n n n

n n

n

n n

nn n n

n n

n

n n

nn n n

n n

2

2 2

2

2 2

2

2 2

2

2 2

114

29

39

414

( )( ) ( )( )( )

( )( )

( )( )

( )

=β +β β

β + β

=− β

β − β

=β

β − β

=− β

β + β

CR sinh p p cosh p

sinh p p

CR p cosh p

sinh p p

CR sinh p

sinh p p

CR sinh p

sinh p p

mm m m m

m m

mm m

m m

mm m

m m

mm m

m m

2

2 2

2

2 2

2

2 2

2

2 2

113

210

310

413

)( ) ( )( )

( )( )

( )( )

( )( )

(=

β +β ββ + β

=− β

β − β

=β

β − β

=− β

β + β

DR sinh p p cosh p

sinh p p

DR p cosh p

sinh p p

DR sinh p

sinh p p

DR sinh p

sinh p p

mm m m m

m m

mm m

m m

mm m

m m

mm m

m m

1

1

1

1

2

11

11 1

12 1

3

11

11 1

12 1

5

22

22 2

22 2

6

22

22 2

22 2

( )( )

( )

( )( )

( )

( ) ( )( )

( )( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( ) ( )

( )

( )

( ) ( )

( )

) ( )

( ) ( ) ( )

( )

( ) ( )

( ) ( ) ( )

( ) ( )

( )

( ) ( )

( )

( )

( ) ( )

( )

( )

=

α α − + − α −⎡⎣ ⎤⎦

+ α − + α − + − − α −⎡⎣ ⎤⎦

+α

α − + − α −⎡⎣ ⎤⎦

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

=

− α α − + − α −⎡⎣ ⎤⎦

− α − + α − + − − α −⎡⎣ ⎤⎦

+α

α − + − α −⎡⎣ ⎤⎦

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

=

α α − α −⎡⎣ ⎤⎦

+ α α − − α −⎡⎣

⎤⎦

+α

α − α −⎡⎣ ⎤⎦

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

=

α α − α −⎡⎣ ⎤⎦

− α α − − α −⎡⎣

⎤⎦

+α

α − α −⎡⎣ ⎤⎦

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

Kp

a sin p t sin p

b p t sin p t p sin p

b cos p t cos p

Kp

a cos p t cos p

b p t cos p t p cos p

b sin p t sin p

Kp

a sin p sin p t

b p sin p p t sin p t

b cos p cos p t

Kp

a cos p cos p t

b p sin p p t cos p t

b sin p sin p t

n

n n n

n n n

nn n

n

n n n

n n n

nn n

n

n n n

n n n

nn n

n

n n n

n n n

nn n

1 4

1 2

1 4

1 2

1 4

1 2

1 4

1 2

4

2 2 21

13

2

2 2 21

10

4

2 2 21

152

2 2 21

2

1

2

5

2

∑

∑

∑

∑

( )

( )

( )

( )

( )( )

( )( )

( )( )

( )( )

( )

( )( )

( )( )

( )

( )

( )

= −− β

β + α

β+ β β

⎡

⎣⎢

⎤

⎦⎥

= −

= −− β α

β + α

β− β β

⎡

⎣⎢

⎤

⎦⎥

= −

= −− β

β + α

β+ β β

⎡

⎣⎢

⎤

⎦⎥

= − +α

= −− β

β + α

β− β β

⎡

⎣⎢

⎤

⎦⎥

= − +α

+

=

∞

+

=

∞

+

=

∞

+

=

∞

p

R tanh p

p / sinh p

f c

p

R tanh p

p / sinh p

f c

p

R tanh p

p / sinh p

f cK

p

R tanh p

p / sinh p

f cK

Rm n

m

m nm

m m

m m

1nI

Rm n

m n

m nm

m m

m m

2nI

Rm n

m

m nm

13m m

m m

n

Rm n

m

m nm

13m m

m m

6

n

n

n

n

6n

WELDING RESEARCH



WELDING RESEARCH


14

21 2

14

21 2

1

2

2

1

2

2

1 2

1

2

2

tanh1

2

2

1 2(A5)

2

2 2 21

62 3 6

2 2 21

52 2 5

2

2 2 2

3 6

2 22

22

2 2 21

2 5

2 22

21

2

9

1

14

1

10

2

13

∑

∑

∑

∑

( )

( )

( )

( )

( )

( )

)

)

( )( )

( )( )

( ) ( )

( )( )

( ) ( )

( )( )

( )

( )

( )

( )

( )

( )

( )

( )

= −− α

β + α

α − +⎛⎝⎜

⎞⎠⎟ α

+ α α

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

= −

= −− α

β + α

α − +⎛⎝⎜

⎞⎠⎟ α

+ α α

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

=

= −− α

β + α

−⎛⎝⎜

⎞⎠⎟

α⎛

⎝⎜−

α− α

α +β + βα

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

+ β α− α α

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

= −

= −− α

β β + α

−⎛⎝⎜

⎞⎠⎟

α⎛

⎝⎜−

α− α

α +β + βα

⎞

⎠⎟

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

+ β α− α α

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

=

+

=

∞

+

=

∞

β

+

=

∞

+

=

∞

c

RK K

tanh c

c / sinh c

g p

c

RK K

tanh c

c / sinh c

g p

K K

ctanh c

ctanh c

R tanh c

c / sinh cg p

c

K K

cc

ctanh c

R tanh c

c / sinh cg p

Rm n

n

m nn

n nn n

n

n n

I

Rm n

n

m nn

n nn n

n

n n

Rm n

n

m nn 1

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