Closed-form structural stress and stress intensity factor solutions for spot welds in commonly used...

Engineering Fracture Mechanics 75 (2008) 5187–5206

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Engineering Fracture Mechanics

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Closed-form structural stress and stress intensity factor solutions forspot welds in commonly used specimens

P.-C. Lin a, J. Pan b,*

a Mechanical Engineering, National Chung Cheng University, Chia-Yi 621, Taiwanb Mechanical Engineering, University of Michigan, 2350 Hayward Street, Ann Arbor, MI 48109-2125, USA

a r t i c l e i n f o

Article history:Received 28 February 2008Received in revised form 9 August 2008Accepted 19 August 2008Available online 27 August 2008

Keywords:Spot weldStructural stressStress intensity factorLap-shear specimenSquare-cup specimenU-shape specimenCross-tension specimenCoach-peel specimenFatigueFracture mechanics

0013-7944/$ - see front matter � 2008 Elsevier Ltddoi:10.1016/j.engfracmech.2008.08.005

* Corresponding author. Tel.: +1 734 764 9404; faE-mail address: [email protected] (J. Pan).

a b s t r a c t

Closed-form new structural stress and stress intensity factor solutions for spot welds inlap-shear, square-cup, U-shape, cross-tension and coach-peel specimens are obtainedbased on elasticity theories and fracture mechanics. The loading conditions for spot weldsin the central parts of the five types of specimens are first examined. The resultant loads onthe weld nugget and the self-balanced resultant loads on the lateral surface of the centralparts of the specimens are then decomposed into various types of symmetric and anti-sym-metric parts. Closed-form structural stress and stress intensity factor solutions for spotwelds under various types of loading conditions are then adopted from the recent workof Lin and Pan to derive new closed-form structural stress and stress intensity factor solu-tions for spot welds in the five types of specimens. The selection of a geometric factor forsquare-cup specimens and the decompositions of the loads on the central parts of theU-shape, cross-tension and coach-peel specimens are based on the corresponding three-dimensional finite element analyses of these specimens. The new closed-form solutionsare expressed as functions of the spot weld diameter, the sheet thickness, the width andthe length of the five types of specimens. The closed-form solutions are also expressedas functions of the angular location along the nugget circumference of spot welds in thefive types of specimens in contrast to the limited available solutions at the critical locationsin the literature. The new closed-form solutions at the critical locations of spot welds in thefive types of specimens are listed or can be easily obtained from the general closed-formsolutions for fatigue life predictions.

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1. Introduction

Resistance spot welding is widely used to join sheet metals in the automotive industry. These spot welds are subjectedto complex multiaxial loads under service or crash conditions. The fatigue lives of spot welds in various types of speci-mens have been investigated by many researchers, for example, see Zhang [1]. Since a spot weld provides a natural crackor notch along the nugget circumference, fracture mechanics has been adopted to investigate the stress intensity factors atthe critical locations of spot welds in order to investigate the fatigue lives of spot welds in various type of specimens [1–15]. The stress intensity factors usually vary point by point along the circumference of spot welds in various types of spec-imens. Pook [2,3] gave the maximum stress intensity factors for spot welds in lap-shear specimens, coach-peel specimens,circular plates, and other bending dominant plate and beam configurations. Radaj [4] and Radaj and Zhang [6–8] estab-lished the foundation to use the structural stresses to determine the stress intensity factors for spot welds under various

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Nomenclature

a nugget radiusb half specimen widthb0 equivalent radius for the equivalent circular plate modelE, m Young’s modulus and Poisson’s ratiofc, gc, hc geometric factorsF resultant force~F force per unit lengthKI, KII, KIII mode I, mode II and mode III stress intensity factorsL length of the upper or lower sheets of lap-shear specimens, half length of the bottom and top sheets of U-

shape and cross-tension specimens, or eccentricity distance between the applied force and the center ofthe spot weld in coach-peel specimens

M resultant bending momentMx0y0 ;My0x0 resultant twisting moment in the new x0 and y0 coordinate system~Mc bending moment per unit length caused by the flange constraint~Mb bending moment per unit length caused by the longer bottom and top sheets of the specimen~Mx; ~My bending moment per unit length in the x and y directions~Mx0y0 ; ~My0x0 twisting moment per unit length in the x0 and y0 coordinate systemr,h polar coordinatest sheet thicknessV overlap lengthx,y original Cartesian coordinatesx0,y0 new Cartesian coordinates rotated clockwise 45� from the original Cartesian x and y coordinatesX,Y,X0 functions of a, b and mr,s normal structural stress and shear structural stressrrr normal stress in the radial direction in the polar coordinate system

Superscripts and SubscriptsA, B, C, . . ., G different square plate models for different specimensAnti, Symm anti-symmetric and symmetric loading conditionsInf, Fin infinite plate model and finite plate modelO, CB, CtB opening, counter bending and central bending conditionsS, T, TW, UB shear, tension, twisting and uniform bending conditionsLap, Cross, Coach lap-shear, cross-tension and coach-peel specimensSquare, Ushape square-cup and U-shape specimensUpper, Lower upper half and lower half of the specimen

5188 P.-C. Lin, J. Pan / Engineering Fracture Mechanics 75 (2008) 5187–5206

types of loading conditions. Based on a strip model, the stress intensity factor solutions were determined in the form ofthe structural stress multiplied by the square root of the thickness. Numerical solutions were obtained for the coefficientsthat link the stress intensity factors and the structural stresses for the strip model. Zhang [11,12] presented closed-formstress intensity factor solutions for selected critical locations of spot welds in various types of specimens based on theanalytical stress solutions for a rigid inclusion in a plate and correlated the solutions with the experimental fatigue dataof spot welds in these specimens under cyclic loading conditions. Analytical solutions for various types of specimens wereobtained for the coefficients that link the stress intensity factors and the structural stresses from the strip model and theclosed-form analytical solutions for a plate with a rigid inclusion under various types of loading conditions.

Different formulae for stress intensity factor solutions were proposed in the literature, for example, by Pook [2], Radaj[4], Swellam et al. [10] and Zhang [11,12] for lap-shear specimens, by Pook [3] and Zhang [11,12] for coach-peel speci-mens and by Pook [2] for circular plates with connection and some bending dominant plate and beam configurations.Lin et al. [15] attempted to correlate the experimental fatigue data for spot welds in lap-shear, square-cup and coach-peelspecimens of low-carbon, high strength and dual phase steels based on a fracture mechanics crack growth model withkinked crack growth from spot welds. Lin et al. [15] found that the stress intensity factor solution of Zhang [11] canbe used to correlate well the fatigue data for lap-shear and coach-peel specimens. Lin et al. [15] also found that the stressintensity factor solution for circular plate with connection of Pook [3] can be used to correlate well the fatigue data forsquare-cup specimens. As mentioned in Zhang [11], computational results were available from Smith and Cooper [16]and Radaj [5] for lap-shear specimens and Wang and Ewing [17] for coach-peel specimens. However, these computationalresults are for their specimens of specific geometries and these results cannot be used for general applications to speci-mens of different designs. Formulae such as those provided by Pook [2], Yuuki et al. [18], Swellam et al. [10] and Zhang[11] are needed for different designs of specimens. However, no comprehensive study has been conducted to validatethese proposed formulae.

P.-C. Lin, J. Pan / Engineering Fracture Mechanics 75 (2008) 5187–5206 5189

Wang et al. [13] conducted a three-dimensional finite element analysis of the mode I stress intensity factor solutionsfor spot welds in circular plates with connection and square-cup specimens based on axisymmetric and three-dimensionalfinite element models. The computational results are benchmarked to the known solution of Pook [3]. Later, Wang et al.[14] launched a parametric study of the stress intensity factor solutions at the critical locations of spot welds in lap-shearspecimens based on the well benchmarked full three-dimensional finite element model of Wang et al. [13]. The compu-tational results indicated that only the solution of Zhang [11] can accurately account for the dependence of the mode IIstress intensity factors on the spot weld diameter and sheet thickness for the given range of the thickness and spot welddiameter. However, the computational results show a strong dependence of the mode I stress intensity factors on thespecimen geometric parameters, especially the specimen width. Wang et al. [14] found that a generic analytical solutionfor spot welds under counter bending conditions is needed to model the width dependence of the mode I stress intensityfactor in lap-shear specimens. Lin et al. [19] later developed a new analytical solution for a finite square plate under coun-ter bending conditions. The new analytical solution can be used successfully to model the model I stress intensity factorsolutions from the computational results of Wang et al. [14]. With this new analytical solution for a rigid inclusion in afinite square plate under counter bending conditions, new analytical solutions with size dependence for lap-shear, square-cup, U-shape, cross-tension and coach-peel specimens can therefore developed in this paper for engineering applicationsof different specimen designs.

During the derivations of the stress intensity factor solutions for lap-shear, square-cup, U-shape, cross-tension and coach-peel specimens, the structural stress solutions along the circumference of the rigid inclusion need to be derived first. Zhang[1,11,12] obtained the structural stress solutions at the critical locations of spot welds in various types of specimens andautomotive structures, where the spot welds were treated as rigid in the analytical or numerical solution procedures. Ruppet al. [20,21] used a beam element model whereas Salvini et al. [22,23] and Vivio et al. [24] used a spot weld assembly finiteelement model to represent a spot weld to obtain the resultant forces and moments through the spot weld and then used thestructural stresses as the damage parameters for fatigue life prediction. From the fracture mechanics viewpoint, the struc-tural stresses are local fatigue parameters that can be used to obtain the stress intensity factor solutions. Also, the crack ini-tiation and growth near the weld nugget circumference are controlled by the stress intensity factors of the main cracks orkinked cracks near the weld nuggets. Therefore, the structural stresses are possible candidates to characterize the fatiguelives of spot welds. The structural stress solutions are also useful in the sense that they can be used to validate the compu-tational models of specimens and structures where spot welds are modeled as rigid inclusions and sheet metals are modeledas plates or shells.

In this paper, closed-form structural stress and stress intensity factor solutions for spot welds in lap-shear, square-cup,U-shape, cross-tension and coach-peel specimens are obtained based on elasticity theories and fracture mechanics. The load-ing conditions for spot welds in the central parts of the five types of specimens are first examined. The resultant loads on theweld nugget and the self-balanced resultant loads on the lateral surface of the central parts of the specimens are thendecomposed into various types of symmetric and anti-symmetric parts. Closed-form structural stress and stress intensityfactor solutions for spot welds under various types of loading conditions are then adopted from Lin and Pan [25] to derivenew closed-form structural stress and stress intensity factor solutions for spot welds in the five types of specimens. The newclosed-form solutions are expressed as functions of the spot weld diameter, the sheet thickness, the width and the length ofthe five types of specimens. The closed-form solutions are also expressed as functions of the angular location along the nug-get circumference of spot welds in the five types of specimens in contrast to the limited available solutions at the criticallocations in the literature.

2. Closed-form solutions for five types of spot weld specimens

Fig. 1 shows five types of half spot weld specimens with the applied forces marked by the bold arrows. These specimensare commonly used to obtain the mechanical properties of spot welds. For clear demonstration of spot weld locations, onlyhalves of specimens are shown. The spot welds are idealized as circular cylinders and shown as half cylinders in the figure.The sheet metals used to make these specimens have the thickness t. As shown in the figures, these specimens have the spec-imen width 2b and the nugget diameter 2a. The bold arrows represent the resultant forces F or F/2 applied to the specimens.Fig. 1a schematically shows a half lap-shear specimen which is used to investigate the behavior of spot welds under dom-inant shear loading conditions. A set of doublers is used to align the loads to avoid the initial realignment of the specimenunder lap-shear loading conditions. Here, L represents the length of the upper or lower sheets of the specimen, and V rep-resents the overlap length. Fig. 1b schematically shows a half square-cup specimen recently adopted to investigate thebehavior of spot welds under pure opening loading conditions. Note that the square-cup specimen is also a good candidatefor investigation of the behaviors of spot welds under combined opening and shear loading conditions [26,27]. The details ofsquare-cup specimens are discussed in Wung and Stewart [28]. Fig. 1c shows a half U-shape specimen which is used toinvestigate the behavior of spot welds under combined opening and counter bending conditions. Fig. 1d shows a halfcross-tension specimen which is used to investigate the behavior of spot welds under dominant opening loading conditions.Fig. 1e shows a half coach-peel specimen which is used to investigate the behavior of spot welds under combined opening,central bending and counter bending conditions. For the U-shape and cross-tension specimens shown in Fig. 1c and d, thehalf lengths of the bottom and top sheets of the upper and lower halves are denoted as L. For the coach-peel specimen shownin Fig 1e, the eccentricity distance between the applied force and the center of the spot weld is also denoted as L. In general, L

F F

Doubler

tDoubler2a

b

VL

ADB

F/2 F/2

F/2 F/2

b

2b

2a

F/2 F/2

F/2 F/2

b

2a2b

2L

B A

F/2

F/2 F/2

2L

b

2a2b

B AD

F

F

2a

L

2b

b

B

a

b c

d e

Fig. 1. Five types of half spot weld specimens with the applied forces shown as the bold arrows. (a) lap-shear specimen, (b) square-cup specimen,(c) U-shape specimen, (d) cross-tension specimen, and (e) coach-peel specimen.


is larger than the half specimen width b. However, as suggested by the computational results of Wang et al. [14], L can beselected to be as small as 0.6 b so that the analytical solution for a large square plate with a rigid inclusion under counterbending conditions can be used to model the mode I stress intensity factor. In this paper, new analytical stress intensity fac-tor solutions for five types of specimens are developed based on the central square parts of the specimens with a rigid inclu-sion under various types of loading conditions. When the bottom and top plates are longer than the chosen square ones,additional counter bending moments should be considered in the analytical solutions. In the following derivations of theclosed-form analytical solutions, the spot welds in the specimens are treated as rigid inclusions. We will concentrate onthe upper halves of the five types of specimens to derive closed-form structural stress and stress intensity factor solutions.Fig. 2 represents a schematic top view of the rigid inclusion in the upper sheet of these spot weld specimens with the cylin-drical and Cartesian coordinate systems centered at the center of the inclusion on the mid-plane of the upper sheet. The loca-tions of points A, B, C and D are 0�, 180�, 90� and 270� with respect to the x axis, respectively.

r

x

y

zθ

AB

C

D

Fig. 2. A schematic top view of a rigid inclusion in the upper sheet of spot weld specimens with the cylindrical and Cartesian coordinate systems centered atthe center of the inclusion on the mid-plane of the upper sheet. The locations of points A, B, C and D are 0�, 180�, 90� and 270� with respect to the x axis,respectively.


2.1. Lap-shear specimen

According to the works of Radaj [4] and Radaj and Zhang [6–8], the loads of various types of specimens containing spotwelds can be decomposed into several symmetric and anti-symmetric parts based on the superposition principle of the lin-ear elasticity theory. Fig. 3 schematically shows the decomposition of the shear load of the central part of a lap-shear spec-imen. Based on the schematic of the lap-shear specimen shown in Fig. 1a, schematics of the overlap parts of the specimensunder various types of loads are shown as models A to F in Fig. 3. As shown in the figure, we select the overlap length V equalto the plate width 2b. The analytical solutions developed here can be valid for the overlap length V equal to 60% of the halfwidth b [14]. In these schematics, two square plates with the width 2b and the thickness t represent the upper or lower partsof the overlap sheets and the rigid circular cylinder with the diameter 2a represents the spot weld. Note that in this paper, asingle-headed arrow represents a force, a double-headed arrow represents a bending moment, and a dash line or a dash ar-row represents a hidden portion of a component or a force (or moment) applied to the hidden portion. As shown in Fig. 3,model A represents a spot weld under lap-shear loading conditions. For model A, F represents the resultant force of the uni-formly distributed force applied to the edges of the square plates along the interfacial surface of the rigid circular cylinder.For model B, the shear resultant force F applied along the interfacial plane is now changed to the membrane force F and thebending moment Ft/2 applied to the mid-planes of the upper and lower plates. The shear force F and the bending moment Ft/2 of a lap-shear specimen is then decomposed into four types of symmetric and anti-symmetric loads: counter bending(model C), central bending (model D), in-plane shear (model E), and tension (model F). The bending moments in modelsC and D have a magnitude of Ft/4, and the forces in models E and F have a magnitude of F/2. The decomposition of the shearload of lap-shear specimens has been presented in Lin et al. [19].

2.1.1. Structural stress solutions for lap-shear specimensThe structural stress solutions derived here is for the upper sheet of the specimen. For model C, the closed-form stress

solutions of the infinite and finite plate models under counter bending conditions listed Section 5.2 in Lin and Pan [25]are considered. The stress solutions are expressed in terms of the cylindrical coordinate system as shown in Fig. 3 for theupper plate. For the infinite plate model, the maximum bending stress rCB (which is rrr on the lower surface of the upperplate) along the nugget circumference at r = a is given as

ðrCBÞInf C ¼3F½1� mþ 2ð1þ mÞ cos 2h�

4btð1� m2Þ ð1Þ

For the finite plate model, the maximum bending stress rCB (which is rrr on the lower surface of the upper plate) along thenugget circumference at r = a is given as

ðrCBÞFin C ¼�3F

8btXY½2b2X þ 4Yða4b4 þ b8Þ cos 2h� ð2aÞ

where X and Y are defined as

X ¼ ð�1þ mÞða4 þ b4Þ2 � 4a2b6ð1þ mÞ ð2bÞY ¼ a2ð�1þ mÞ � b2ð1þ mÞ ð2cÞ

=

=

++

A (Lap-shear load) B (Equivalent lap-shear load)

C (Counter bending)

E (In-plane shear) F (Tension)

2a

2b

2b

t

t

F F FF

Ft/2 Ft/2

y z

x

Ft/4

Ft/4

Ft/4

Ft/4

D (Central bending)

y z

x

Ft/4Ft/4 Ft/4

Ft/4

y z

xF/2

F/2

F/2

F/2y z

xF/2

F/2

F/2

F/2

Load Decomposition for Lap-Shear Specimen

+

Fig. 3. Decomposition of the load for a lap-shear specimen. Model A represents a spot weld in the overlap part of a lap-shear specimen under lap-shearloading conditions. Model B represents a spot weld under the equivalent loading condition of model A. The forces and moments of model B are decomposedinto four types of symmetric and anti-symmetric loads: counter bending (model C), central bending (model D), in-plane shear (model E), and tension(model F).


For model D, the central bending moment on the rigid inclusion is statically in equilibrium with the resultant moment onthe lateral plate surface. Therefore, as in Zhang [11], the closed-form stress solutions of the infinite plate model under centralbending conditions listed Section 4.3 in Lin and Pan [25] are considered and the maximum bending stress rCtB (which is rrr

on the lower surface of the upper plate) along the nugget circumference at r = a is given as

ðrCtBÞD ¼3F cos h

2patð3Þ

The three-dimensional finite element analysis of Wang et al. [14] showed that the solution for the infinite plate model undercentral bending conditions, used by Zhang [11], can fit the computational results quite well.

For model E, the closed-form stress solutions of the infinite plate model under in-plane shear loading conditions listed inSection 4.1 in Lin and Pan [25] are considered and the in-plane shear stress rS (which is rrr of the upper plate) along thenugget circumference at r = a is given as

ðrSÞE ¼F cos h2pat

ð4Þ

The out-of-plane shear stress sS (which is rrh of the upper plate) along the nugget circumference at r = a is given as

ðsSÞE ¼�F sin h

2patð5Þ

For model F, the closed-form stress solutions of the infinite plate model under tensile loading conditions listed in Section5.1 in Lin and Pan [25] are considered and the tensile stress rT (which is rrr of the upper plate) along the nugget circumfer-ence at r = a is given as

ðrTÞF ¼F½3� mþ 2ð1þ mÞ cos 2h�

4btð3þ 2m� m2Þ ð6Þ

Further explanations of rCB, rCtB, rS, sS and rT can be found in Lin and Pan [25].


The structural stresses on the lower surface of the upper plate at r = a are the sum of the corresponding closed-form stresssolutions of models C, D, E and F as listed above. The structural normal stress rInf

Lap and the structural out-of-plane shear stresssInf

Lap on the lower surface of the upper plate at r = a for lap-shear specimens with large size are

rInfLap ¼

3F½1� mþ 2ð1þ mÞ cos 2h�4btð1� m2Þ þ 4F cos h

2patþ F½3� mþ 2ð1þ mÞ cos 2h�

4btð3þ 2m� m2Þ ð7aÞ

sInfLap ¼

�F sin h2pat

ð7bÞ

Similarly, the structural normal stress rFinLap and the out-of-plane shear stress sFin

Lap on the lower surface of the upper plate atr = a for lap-shear specimens with finite size are

rFinLap ¼

�3F8btXY

½2b2X þ 4Yða4b4 þ b8Þ cos 2h� þ 4F cos h2pat

þ F½3� mþ 2ð1þ mÞ cos 2h�4btð3þ 2m� m2Þ ð8aÞ

sFinLap ¼

�F sin h2pat

ð8bÞ

where X and Y are given in Eqs. (2b) and (2c), respectively. Note that sFinLap is the same as sInf

Lap since they both are from thesolution for a rigid inclusion in an infinite plate under in-plane shear loading conditions. The structural stresses at the criticallocations A and B for lap-shear specimens as shown in Fig. 1a appear to be good engineering parameters to predict the fa-tigue lives of spot welds or friction spot welds in lap-shear specimens since cracks or kinked cracks are initiated at theselocations. The structural stresses at the critical location A in Fig. 1a for the upper plate can be obtained from the closed-formstress solutions for the infinite and finite plate models listed above by setting h = 0�.

2.1.2. Stress intensity factor solutions for lap-shear specimensThe stress intensity factor KI, KII and KIII solutions for spot welds in various types of specimens show strong dependence on

the structural stresses near the spot weld [4,6–8,11,12]. For example, the KI solution for lap-shear specimens is a function ofthe structural stresses near the spot weld under counter bending conditions (model C shown in Fig. 3). For model C, theclosed-form stress solutions of the infinite and finite plate models under counter bending conditions listed in Section 5.2in Lin and Pan [25] are considered. Although the procedure for deriving the stress intensity factor solutions for lap-shearspecimens was detailed in Lin et al. [19], we still summarize the procedure to include the general structural stress solutionsin this paper for completeness and to provide new insights to set up the procedure to derive closed-form solutions for theother four types of specimens.

2.1.2.1. KI solution for infinite plate. Substituting Eq. (1) into the expression of the KI solution in Section 6.1 in Lin and Pan [25]gives the KI solution for lap-shear specimens with large size as a function of h as

K I ¼ffiffitpðrCBÞInf Cffiffiffi

3p ¼

ffiffiffi3p

F½1� mþ 2ð1þ mÞ cos 2h�4b

ffiffitpð1� m2Þ

ð9Þ

The KI solution at critical locations (points A and B as shown in Fig. 1a) is given as

K I ¼ffiffiffi3p

Fð3þ mÞ4b

ffiffitpð1� m2Þ

ð10Þ

2.1.2.2. KI solution for finite plate. As discussed in Wang et al. [14], when the ratio is b/a less than 10, the closed-form solutionfor a finite plate with a rigid inclusion under counter bending conditions is needed to model accurately the mode I stressintensity factor. For the finite plate model, the derivation of the KI solution is similar to that for the infinite plate model.Substituting Eq. (2) into the expression of the KI solution in Section 6.1 in Lin and Pan [25] gives the KI solution for lap-shearspecimens with finite size as a function of h as

K I ¼ffiffitpðrCBÞFin Cffiffiffi

3p ¼

ffiffiffi3p

F8b

ffiffitp

XY½�2b2X � 4Yða4b4 þ b8Þ cos 2h� ð11Þ

where X and Y are given in Eqs. (2b) and (2c), respectively. The KI solution at critical locations (points A and B as shown inFig. 1a) is given as

K I ¼�

ffiffiffi3p

F8b

ffiffitp

XY½2b2X þ 4Yða4b4 þ b8Þ� ð12Þ

The detailed derivation of the KI solution for lap-shear specimens has been presented in Lin et al. [19] and is not repeatedhere. The solutions presented in Eqs. (11) and (12) are the same as but more compact than those presented in Lin et al.[19]. Note that Zhang [11,12] approximated the solution for an inclusion in a finite plate under counter bending conditionsby that for an inclusion in an infinite plate under central bending conditions to obtain the structural stress solutions for the


derivation of his KI solution. Consequently, the analytical KI solution given in Zhang [11,12] has no dependence on b/a andtherefore does not always agree with for the KI solutions for lap-shear specimens with different b/a’s according to the com-putational results of Wang et al. [14].

2.1.2.3. KII solution. The KII solution for lap-shear specimens is a function of the structural stresses near the spot weld undercentral bending and in-plane shear loading conditions (models D and E shown in Fig. 3, respectively). Substituting Eqs. (3)and (4) into the KII solution in Sections 6.2 and 6.3 in Lin and Pan [25] gives the KII solution for lap-shear specimens as afunction of h as

K II ¼ffiffitp½ðrCtBÞD þ ðrSÞE�

2¼ F cos h

paffiffitp ð13Þ

The KII solution at the critical location (point A as shown in Fig. 1a) is given as

K II ¼F

paffiffitp ð14Þ

Note that the KII solution at the other critical location (point B as shown in Fig. 1a) has the same magnitude as that at point Abut with a negative sign. The KII solutions at the critical locations of spot welds in lap-shear specimens were originally pre-sented in Zhang [11] and were shown to be valid for b/a as small as 5 according to the computational results of Wang et al.[14].

2.1.2.4. KIII solution. The KIII solution for lap-shear specimens is a function of the out-of-plane shear stress sS (in reference tothe two-dimensional strip model in Lin and Pan [25]) near the spot weld in model E (as shown in Fig. 3) under in-plane shearloading conditions. Therefore, substituting Eq. (5) into the expression of the KIII solution in Section 6.4 in Lin and Pan [25]gives the KIII solution for lap-shear specimens as a function of h as

K III ¼ffiffiffiffiffi2tpðsSÞE ¼

�F sin hffiffiffi2p

paffiffitp ð15Þ

The KIII solution at critical location (point D as shown in Fig. 1a) is given as

K III ¼�Fffiffiffi

2p

paffiffitp ð16Þ

The KIII solution at the critical locations of spot welds in lap-shear specimens was originally presented in Zhang [11] and wasshown to be valid for b/a as small as 5 according to the computational results of Wang et al. [14].

Note that the structural stresses near the spot weld under tensile loading conditions (model F shown in Fig. 3) have nocontribution to any stress intensity factor for lap-shear specimens. Also note that the stress intensity factor solutions at thecritical locations of spot welds in lap-shear specimens were presented in Zhang [11,12] and Lin et al. [19]. Here, we presentthe full stress intensity factor solutions along the circumference of spot welds in lap-shear specimens. Note that based on thecomputational results in Lin and Pan [29], portions of the sheets along the nugget circumference in lap-shear and coach-peelspecimens contact each other under loading conditions. It is out of the scope of this paper to derive closed-form structuralstress solutions for spot welds with consideration of contact using the plate and shell theory. In this investigation, contact isnot considered in both analytical and finite element analyses for the five types of the specimens. Therefore, a negative valueof the mode I stress intensity factor can be seen along some portions of the inclusion circumference in lap-shear and coach-peel specimens. Further research on the effects of contact on the stress intensity factor solutions should be conducted in thefuture.

2.2. Square-cup specimen

Fig. 4 schematically shows the decomposition of the load of the central square part of a square-cup specimen. Based onthe schematic of the square-cup specimen shown in Fig. 1b, schematics of the bottom and top square sheets of the upper andlower cups of the specimen under various types of loads are shown as models A to D in Fig. 4. The top and bottom squaresheets have the width 2b. As shown in Fig. 4, model A represents a spot weld in the central top and bottom square sheets ofthe specimen under opening loading conditions. ~F represents the uniformly distributed transverse shear force applied to thefour edges of the square plates. ~Mc represents the uniformly distributed bending moment applied to the four edges ofthe square plates. Note that under opening loading conditions, the square-cup flange walls constrain the deformation ofthe square plates. Based on the observations of the experimental results and computational results, the flange provides anearly clamped boundary condition to the four edges of the top and bottom square plates. Therefore, a uniformly distributedbending moments ~Mc is assumed to apply to the four edges of the top and bottom square plates. Note that ~Mc is caused bythe constraint of the flanges of the square-cup specimen. As shown in the figure, the geometry and loading conditions ofmodel A are symmetric with respect to the interfacial surface of the circular cylinder. Therefore, only the upper half of modelA is considered. Model B represents the upper half of model A. In model B, Fð¼ 8~FbÞ represents the resultant opening force

Mc

~

Mc

~

Mc

~

Mc

~

Mc

~

Mc

~

Mc

~

Mc

~Mc

~

=

=

A (Opening/bending) B (Upper half model)

D (Bending)

2a

2b

2b

t

t

F~

F~

F~

F~

F

C (Opening)

y z

x

Mc

~

F~

F~

F~

F~

F~

F~

F

F~

F~

F~

F~

F~

F~

y z

x

Load Decomposition for Square-Cup Specimen

Mc

~

Mc

~

Mc

~

Mc

~

Mc

~

Mc

~

+

Fig. 4. Decomposition of the load for a square-cup specimen. Model A represents a spot weld in the central square part of a square-cup specimen underopening loading conditions. Model B represents the upper half of model A. The forces and moments of model B are decomposed into two types of symmetricloads: opening (model C) and equal-biaxial counter bending (model D).


applied to the center of the circular interfacial surface of the spot weld. Finally, the forces and moments of model B aredecomposed into two types of symmetric loads: opening (model C) and equal bi-axial counter bending (model D). Note thatin this paper, a force or moment with a tilde represents a uniform distributed force or moment applied to the lateral edge ofthe square plate. The unit of a force or moment with a tilde is force or moment per unit length.

2.2.1. Structural stress solutions for square-cup specimensThe structural stress solutions derived here is for the upper half of the square-cup specimen. Note that model C has square

plate geometry. When the spot weld diameter is small compared to the plate width, the loading around the cylinder circum-ference is relatively uniform as shown in Wang et al. [13]. Therefore, the square plate model can be idealized to be an equiv-alent circular plate model as shown in Fig. 4c in Lin and Pan [25]. Here, b0 is denoted as the equivalent radius for theequivalent circular plate model and is approximately equal to 2b=

ffiffiffiffipp

using the simple area equivalence rule. For modelC, the opening force on the rigid inclusion is statically in equilibrium with the transverse shear force on the lateral plate sur-face. Therefore, the closed-form stress solutions of the circular plate model with a simply supported edge under openingloading conditions listed in Section 4.2 in Lin and Pan [25] are considered and the maximum bending stress rO (which isrrr on the lower surface of the upper plate) along the nugget circumference at r = a is given as

ðrOÞC ¼�3F½ða2 � b02Þð�1þ mÞ þ 2b02ð1þ mÞ lnðb0=aÞ�

2pt2½a2ð�1þ mÞ � b02ð1þ mÞ�ð17Þ

For model D, the uniformly distributed bending moment ~Mc is equivalent to a combination of the counter bending mo-ments ~Mx and ~My where ~Mx ¼ ~My ¼ ~Mc. Note that ~Mc is caused by the constraint of the flanges of the square-cup specimen.For example, when the walls have a full constraint on the bending of the plate edges, ~Mc is identical to the bending momentð ~MrÞr¼b0 which is defined as the moment along the clamped edge at r = b0 of a finite circular plate model with a clamped edgeunder opening loading conditions in Section 4.2 in Lin and Pan [25]. When the walls have no constraint on the bending of theplate edges, ~Mc is identical to 0. Therefore, ~Mc along the edges of the square plate can be written as

~Mc ¼ fc � ð ~MrÞr¼b0 ¼�fcF½a2 � b02 þ 2a2 lnðb0=aÞ�

4pða2 � b02Þð18Þ

where fc is a geometric factor which is a function of the geometric parameters of the square-cup specimen such as the detailedgeometry of the flange, the specimen width and the plate thickness. Here, fc is defined as 0 for the simply supported edge and 1for the clamped edge based on the earlier discussions. For example, the geometric factor fc is determined as 0.837 from thethree-dimensional finite element analysis of Lin and Pan [29] for the square-cup specimens used in Lin et al. [15].


For model D, the closed-form stress solutions of the infinite and finite plate models under counter bending conditionslisted in Section 5.2 Lin and Pan [25] are considered. For the infinite plate model under the uniform bending moment ~Mc,the maximum bending stress rUB (which is rrr on the lower surface of the upper plate) along the nugget circumferenceat r = a is given as

ðrUBÞInf D ¼6 ~Mc

t2ð1þ mÞ¼ �3f cF½a2 � b02 þ 2a2 lnðb0=aÞ�

2pt2ð1þ mÞða2 � b02Þð19Þ

For the finite plate model under the uniform bending moment ~Mc, the maximum bending stress rUB (which is rrr on the low-er surface of the upper plate) along the nugget circumference at r = a is given as

ðrUBÞFin D ¼�6b2 ~Mc

t2Y¼ 3b2fcF½a2 � b02 þ 2a2 lnðb0=aÞ�

2pt2Yða2 � b02Þð20Þ

where Y is in given Eq. (2c). Further explanations of rO and rUB can be found in Lin and Pan [25].The structural stresses on the lower surface of the upper plate at r = a are the sum of the corresponding closed-form stress

solutions of models C and D as listed above. The structural normal stress rInfSquare on the lower surface of the upper plate at

r = a for square-cup specimens with large size is

rInfSquare ¼

�3F½ða2 � b02Þð�1þ mÞ þ 2b02ð1þ mÞ lnðb0=aÞ�2pt2½a2ð�1þ mÞ � b02ð1þ mÞ�

� 3f cF½a2 � b02 þ 2a2 lnðb0=aÞ�2pt2ð1þ mÞða2 � b02Þ

ð21Þ

Similarly, the structural normal stress rFinSquare on the lower surface of the upper plate at r = a for square-cup specimens with

finite size is

rFinSquare ¼

�3F½ða2 � b02Þð�1þ mÞ þ 2b02ð1þ mÞ lnðb0=aÞ�2pt2½a2ð�1þ mÞ � b02ð1þ mÞ�

þ 3b2fcF½a2 � b02 þ 2a2 lnðb0=aÞ�2pt2Yða2 � b02Þ

ð22Þ

where Y is in given Eq. (2c). Note that both rInfSquare and rFin

Square are uniform along the nugget circumference since their corre-sponding stress solutions are all axisymmetric with respect to the z axis. Therefore, the structural stress on the lower surfaceof the upper plate at r = a appears to be a good engineering parameter to predict the fatigue lives of spot welds or frictionspot welds in square-cup specimens.

2.2.2. Stress intensity factor solutions for square-cup specimens2.2.2.1. KI solution for infinite plate. Substituting Eq. (21) into the expression of the KI solution in Section 6.1 in Lin and Pan[25] gives the KI solution for square-cup specimens of large size as

K I ¼ffiffitp

rInfSquareffiffiffi3p ¼

ffiffitpffiffiffi

3p �3F½ða2 � b02Þð�1þ mÞ þ 2b02ð1þ mÞ lnðb0=aÞ�

2pt2½a2ð�1þ mÞ � b02ð1þ mÞ�� 3f cF½a2 � b02 þ 2a2 lnðb0=aÞ�

2pt2ð1þ mÞða2 � b02Þ

( )ð23Þ

where fc is the geometric factor and b0 (¼ 2b=ffiffiffiffipp

) is the equivalent radius.

2.2.2.2. KI solution for finite plate. Substituting Eq. (22) into the expression of the KI solution in Section 6.1 in Lin and Pan [25]gives the KI solution for square-cup specimens of finite size as

K I ¼ffiffitp

rFinSquareffiffiffi3p ¼

ffiffitpffiffiffi


2pt2½a2ð�1þ mÞ � b02ð1þ mÞ�þ 3b2fcF½a2 � b02 þ 2a2 lnðb0=aÞ�

2pt2Yða2 � b02Þ

( )ð24Þ

where Y is in given Eq. (2c). Note that the KII and KIII solutions for square-cup specimens are both equal to zero. The KI solu-tions for square-cup specimens in Eqs. (23) and (24) are different from those presented in Wang et al. [13].

2.3. U-shape specimen

Fig. 5 schematically shows the decomposition of the load of the central square part of a U-shape specimen. Based on theschematic of the U-shape specimen shown in Fig. 1c, schematics of the bottom and top sheets of the upper and lower U partsof the specimen under various types of loads are shown as models A to F in Fig. 5. As shown in Fig. 5, model A represents aspot weld in the central square part of the specimen under opening loading conditions. Here, ~F represents the uniformly dis-tributed transverse shear force applied to the right and left edges of the square plates. ~Mb and ~Mc represent the counterbending moments applied to the right and left edges of the square plates. For a U-shape specimen with a longer bottom sheetof the upper U part or a longer top sheet of the lower U part, the contribution from the additional moment ~Mb (¼ ~F � ðL� bÞ)due to the opening force needs to be considered. Note that under opening loading conditions, the flanges of a U-shape spec-imen have similar effects as the flanges of a square-cup specimen. Therefore, a counter bending moment ~Mc is assumed toapply to the right and left edges of the square plate. As shown in the figure, the geometry and loading conditions of model A

Mb-Mc

~ ~

Mb-Mc

~ ~

y z

x

=

=


2a

2b

2b

t

t

F~

F~

F~

F~

F

C (Counter bending)

F~

F~

Load Decomposition for U-Shape Specimen

Mb-Mc

~ ~

Mb-Mc

~ ~ Mb-Mc

~ ~

Mb-Mc

~ ~

Mb-Mc

~ ~

Mb-Mc

~ ~

D (Opening)

Fy z

xF/2~

F/2~

F/2~

F/2~

+

F/2~

-F/2~

My'x'

45o

45o

(Free body diagram)-F/2~

-F/2~

E (Cross opening/closing)

y z

x

F/2~ F/2

~

y'x'

+

=

=

My'x'

~Mx'y'

~

y z

xy'

x'

Mx'y'

~

My'x'

~

F (Twisting)

=F/16=Fb/4~

2b

2b

-F/2~

-F/2~


y z

x

F/2~ F/2

~

y'x'

G (Cross Counter Bending)

Fy z

x

Mx=~

My

~

Mx

~My

~

Mx'y'

~

2b

2b

Fig. 5. Decomposition of the load for a U-shape specimen. Model A represents a spot weld in the central square part of a U-shape specimen under openingloading conditions. Model B represents the upper half of model A. The forces and moments of model B are approximately decomposed into three types ofloads: counter bending (model C), opening (model D) and cross opening/closing (model E). Model E under cross opening/closing conditions is approximatedby model F under twisting conditions or model G under cross counter bending conditions.



are symmetric with respect to the interfacial surface of the spot weld. Therefore, only the upper half of model A is consid-ered. Model B represents the upper half of model A. In model B, Fð¼ 4~Fb) represents the resultant opening force applied tothe center of the circular interfacial surface of the spot weld.

The forces and moments of model B can be decomposed into three types of loads: counter bending (model C), opening(model D) and cross opening/closing (model E) as shown in Fig. 5. We use the cross opening/closing conditions to describethe loading conditions of model E in the sense that the two pairs of the transverse shear forces for model E are similar tothose of a cross-tension specimen. However, a closed-form solution for a square plate with a rigid inclusion under crossopening/closing conditions is not available. Therefore, model F under twisting conditions or model G under cross counterbending conditions is proposed to approximate model E under cross opening/closing conditions. As shown in the schematicof model E, model F is the central square portion of model E as marked by dashed lines. Model F has the plate width

ffiffiffi2p

b.Both the original x and y coordinate system and a new x0 and y0 coordinate system are centered at the center of the rigidinclusion located on the middle plane of the sheet. The x0 and y0 coordinates are rotated 45� clockwise from the original xand y coordinates, respectively.

It should be noted that the development of all models in this paper is based on the assumption that the radius of the rigidinclusion, a, is much smaller than the half width of the square plate, b. According to the free body diagram of the equilateralright triangular plate next to the schematic of model E, the resultant twisting moment My0x0 ð¼ ~Fb2

=2ffiffiffi2pÞ is needed at the

middle of the hypotenuse edge to balance the uniform distributed forces �~F=2 and ~F=2 along the upper and left edges ofmodel E, respectively. Therefore, for the four triangles shown in model E, two pairs of the resultant moments ~My0x0 and~My0x0 acting at the middles of the hypotenuse edges are needed to satisfy the equilibrium conditions. For model F, uniformdistributed twisting moments ~Mx0y0 and ~My0x0 where ~Mx0y0 ¼ ~My0x0 ¼ Mx0y0=

ffiffiffi2p

b ¼ My0x0=ffiffiffi2p

b are assumed to applied to the fouredges to obtain an approximate closed-form solution for model E under cross opening/closing conditions. Note that ~Mx0y0 and~My0x0 are statically in equilibrium with the resultant twisting moment Mx0y0 and Mx0y0 . The three-dimensional finite elementanalysis of Lin and Pan [29] indicates that the analytical solutions based on the approximation of model F for model E appearto agree well with the computational results.

Since the closed-form solution for a square plate with a rigid inclusion under twisting conditions is available in Lin andPan [25], the solution for model E under cross opening/closing conditions is denoted as the twisting solution in this paper.Note that the closed-form solution for a square plate with a rigid inclusion under twisting conditions was obtained from thetransformation rule for a square plate with a rigid inclusion under cross counter bending conditions (Lin and Pan [25]). Thecross counter bending conditions which are defined as two pairs of counter bending moments to open and close the rigidinclusion as a spot weld. Therefore, model F under twisting conditions is statically equivalent to model G under cross counterbending conditions as shown in Fig. 5. We use the free body diagram to determine the magnitude of ~Mx0y0 and ~My0x0 which isequal to the magnitude of ~Mx and ~My of model G under cross counter bending conditions. Conceptually, it is much more intu-itive to use model G under cross counter bending conditions to approximate model E under cross opening/closing.

2.3.1. Structural stress solutions for U-shape specimensThe structural stress solutions derived here is for the upper U part of the specimen. For model C, ~Mc is a counter bend-

ing moment caused by the constraint of the flanges of a U-shape specimen. The flanges of a U-shape specimen have sim-ilar effects as those of a square-cup specimen. When L = b and the flanges a U-shape specimen have the similar geometryas those of a square-cup specimen, ~Mc for the square bottom and top sheets of the upper and lower U parts should beclose to that of model D for square-cup specimens in Eq. (18) when the load decomposition for the square plate modelas shown in Fig. 5 is adopted. When L >> b and the flanges have a full constraint on the bending of the sheet edges,~Mc is assumed to be ~FL=2 (� ~Mb=2) according to the elementary beam theory. When L > > b and the flanges give no con-straint on the bending of the sheet edges, ~Mc is 0. However, when the flanges do not give a full constraint, geometric fac-tors such as fc in Eq. (18) are needed. In general, L is larger than b but has a value close to that of b. Therefore, ~Mc along theright and left edges of the central square plates for L > b is expressed in terms of the solution at L = b and the solution at L> > b as

~Mc ¼ gcF½a2 � b02 þ 2a2 lnðb0=aÞ�

4pða2 � b02Þþ hc

~Mb

2ð25Þ

where gc is a geometric factor that has a value close to that of fc in Eq. (18) for square-cup specimens, hc is another geometricparameter which is a function of the geometry of the flange, the plate thickness, the specimen width, and the specimenlength when L > b and ~Mb ¼ F � ðL� bÞ=4b. Detailed computations are needed to determine the value of gc for L = b andthe value of hc for L > b.

For model C, the closed-form stress solutions of the infinite and finite plate models under counter bending conditionslisted in Section 5.2 in Lin and Pan [25] are considered. For the infinite plate model under the uniform bending moment~Mb � ~Mc, the maximum bending stress rCB (which is rrr on the lower surface of the upper plate) along the nugget circum-ference at r = a is given as

ðrCBÞInf C ¼6ð ~Mb � ~McÞ½1� mþ 2ð1þ mÞ cos 2h�

t2ð1� m2Þð26Þ


where ~Mb=F�(L-b)/4b and ~Mc is given in Eq. (25). For the finite plate model under the uniform bending moment ~Mb � ~Mc, themaximum bending stress rCB (which is rrr on the lower surface of the upper plate) along the nugget circumference at r = a isgiven as

ðrCBÞFin C ¼�3ð ~Mb � ~McÞ

t2XY½2b2X þ 4Yða4b4 þ b8Þ cos 2h� ð27Þ

where X and Y are given in Eqs. (2b) and (2c), respectively.For model D, the maximum bending stress (rO)D along the nugget circumference at r = a is identical to that of model C

for square-cup specimens in Eq. (17). For model E (or F and G), the closed-form stress solutions of the infinite and finiteplate models under twisting conditions listed in Section 5.4 in Lin and Pan [25] are adopted with substitution of h + p/4for h. For the infinite plate model under the uniform twisting moments ~Mx0y0 and ~My0x0 where ~Mx0y0 ¼ ~My0x0 , the maximumbending stress rTW (which is rrr on the lower surface of the upper plate) along the nugget circumference at r = a is givenas

ðrTWÞInf F ¼24 ~Mx0y0 cos 2h

t2ð1� mÞð28Þ

where ~Mx0y0 ð¼ F=16Þ is the uniform distributed twisting moment. For the finite square plate model under the uniform twist-ing moments ~Mx0y0 and ~My0x0 , the maximum bending stress rTW (which is rrr on the lower surface of the upper plate) along thenugget circumference at r = a is given as

ðrTWÞFin F ¼�24 ~Mx0y0

t2X0½a4ðb=

ffiffiffi2pÞ4 þ ðb=

ffiffiffi2pÞ8� cos 2h ð29Þ

where

X0 ¼ �½a4 þ ðb=ffiffiffi2pÞ4�2ð1� mÞ � 4a2ðb=

ffiffiffi2pÞ6ð1þ mÞ ð30Þ

Further explanations of rCB, rO and rTW can be found in Lin and Pan [25].The structural stress on the lower surface of the upper plate at r = a is the sum of the corresponding closed-form stress

solutions of models C, E and F as listed above. The structural normal stress rInfUshape on the lower surface of the upper plate at

r = a for U-shape specimens with large size is

rInfUshape ¼

6ð ~Mb � ~McÞ½1� mþ 2ð1þ mÞ cos 2h�t2ð1� m2Þ

� 3F½ða2 � b02Þð�1þ mÞ þ 2b02ð1þ mÞ lnðb0=aÞ�2pt2½a2ð�1þ mÞ � b02ð1þ mÞ�

þ 24 ~Mx0y0 cos 2h

t2ð1� mÞ

¼ �3F½ða2 � b02Þð�1þ mÞ þ 2b02ð1þ mÞ lnðb0=aÞ�2pt2½a2ð�1þ mÞ � b02ð1þ mÞ�

þ 6ð ~Mb � ~McÞt2ð1þ mÞ

þ 12ð ~Mb � ~Mc þ 2 ~Mx0y0 Þ cos 2h

t2ð1� mÞð31Þ

Similarly, the structural normal stress rFinUshape on the lower surface of the upper plate at r = a for U-shape specimens with fi-

nite size is

rFinUshape ¼

�3ð ~Mb � ~McÞt2XY

½2b2X þ 4Yða4b4 þ b8Þ cos 2h�

� 3F½ða2 � b02Þð�1þ mÞ þ 2b02ð1þ mÞ lnðb0=aÞ�2pt2½a2ð�1þ mÞ � b02ð1þ mÞ�

� 24 ~Mx0y0

t2X 0½a4ðb=


ffiffiffi2pÞ8� cos 2h

ð32Þ

where X and Y are given in Eqs. (2b) and (2c), respectively, and X0 is given in Eq. (30). Note that ~Mc is given in Eq. (25),~Mx0y0 ð¼ F=16Þ is the uniform distributed twisting moment and ~Mbð¼ FðL� bÞ=4bÞ is the additional moment due to longer bot-tom and top sheets of the upper and lower U parts. The structural stress at the critical locations A and B for U-shape spec-imens as shown in Fig. 1c appears to be a good engineering parameter to predict the fatigue lives of spot welds or frictionspot welds in U-shape specimens since cracks or kinked cracks are initiated at these locations. The structural stresses at thecritical location A in Fig. 1c for the upper plate based on the closed-form stress solutions for the infinite and finite plate mod-els can be obtained by setting h = 0� in the solutions listed above.

2.3.2. Stress intensity factor solutions for U-shape specimens2.3.2.1. KI solution for infinite plate. Substituting Eq. (31) into the expression of the KI solution in Section 6.1 in Lin and Pan[25] gives the KI solution for U-shape specimens of large size as a function of h as

K I ¼ffiffitp

rInfUshapeffiffiffi3p ð33Þ


The KI solution at the critical locations (points A and B as shown in Fig. 1c) is given as

Fig. 6.openincounter

K I ¼ffiffitpffiffiffi


2pt2½a2ð�1þ mÞ � b02ð1þ mÞ�þ 6ð ~Mb � ~McÞ

t2ð1þ mÞþ 12ð ~Mb � ~Mc þ 2 ~Mx0y0 Þ

t2ð1� mÞ

( )ð34Þ

Note that ~Mc is given in Eq. (25), ~Mx0y0 (=F/16) is the uniform distributed twisting moment and ~Mb (=F(L-b)/4b) is the addi-tional moment due to longer bottom and top sheets of the upper and lower U parts.

2.3.2.2. KI solution for finite plate. Substituting Eq. (32) into the expression of the KI solution in Section 6.1 in Lin and Pan [25]gives the KI solution for U-shape specimens of finite size as a function of h as

K I ¼ffiffitp

rFinUshapeffiffiffi3p ð35Þ

The KI solution at the critical locations (points A and B as shown in Fig. 1c) is given as


3p �3ð ~Mb � ~McÞ

t2XY½2b2X þ 4Yða4b4 þ b8Þ� � 3F½ða2 � b02Þð�1þ mÞ þ 2b02ð1þ mÞ lnðb0=aÞ�

2pt2½a2ð�1þ mÞ � b02ð1þ mÞ�

(

�24 ~Mx0y0

t2X 0½a4ðb=


ffiffiffi2pÞ8�)

ð36Þ

where X and Y are given in Eqs. (2b) and (2c), respectively, and X0 is given in Eq. (30). Note that the KII and KIII solutions forU-shape specimens are both equal to zero. Also note that the KI solutions in Eqs. (34) and (36) are functions of h .

Mb-Mc

~ ~

Mb-Mc

~ ~

Mb-Mc

~ ~

Mb-Mc

~ ~

=


2a

2b

2b

t

tF

F~

F~F

~F~

F~

F~

Load Decomposition for Cross-Tension Specimen

y z

x=

+

D (Opening)C (Counter bending)

Fy z

x

F/2~

F/2~

F/2~

F/2~

Mb-Mc

~ ~

Mb-Mc

~ ~

Mb-Mc

~ ~

Mb-Mc

~ ~

-F/2~

-F/2~


y z

x

F/2~ F/2

~

+

Decomposition of the load for a cross-tension specimen. Model A represents a spot weld in the central square part of a cross-tension specimen underg loading conditions. Model B represents the upper half of model A. The forces of model B are approximately decomposed into three types of loads:

bending (model C), opening (model D) and cross opening/closing (model E).


2.4. Cross-tension specimen

Fig. 6 schematically shows the decomposition of the load of the central square part of a cross-tension specimen.Based on the schematic of the cross-tension specimen shown in Fig. 1d, schematics of the bottom and top sheets ofthe upper and lower U parts of the specimens under various types of loads are shown as models A to E in Fig. 6. Asshown in Fig. 6, model A represents a spot weld in the central square part of the cross-tension specimen under openingloading conditions. Here, ~F represents the uniformly distributed transverse shear force applied to the right and left edgesof the upper plate and the front and back edges of the lower plate. ~Mb and ~Mc represent the counter bending momentsapplied to the right and left edges of the upper plate and the front and back edges of the lower plate. Similar to theU-shape specimen, the shift of the transverse shear force ~F to the edges of the central square plates causes the additionalcounter bending moment ~Mbð¼ ~F � ðL� bÞÞ along the edges of the square plates. Note that the flanges of a cross-tensionspecimen have similar effects as those of a U-shape specimen and cause a similar counter bending moment ~Mc along theright and left edges of the upper square plate and along the front and back edges of the lower square plate. Note thatthe upper and lower plates of model A are both under combined counter bending and counter opening conditions buttheir loading conditions are shifted by 90�. Therefore, only the upper half of model A is considered for simplicity. Asshown in the figure, model B is the upper half of model A. In model B, Fð¼ 4~FbÞ represents the resultant opening forceapplied to the center of the circular interfacial surface of the spot weld. Note that the loading condition of model B isidentical to that of the upper half of U-shape specimens as shown in Fig. 5. Therefore, the forces and moments of modelB can be approximately decomposed into three types of loads: counter bending (model C), opening (model D), and coun-ter opening/closing (model E). Note that for model E, a rigid inclusion in a smaller square plate subjected to uniformtwisting moments ~Mx0y0 and ~My0x0 is used to approximate the cross opening/closing conditions of model E as for U-shapespecimens as shown in Fig. 5.

2.4.1. Structural stress solutions for cross-tension specimensThe structural stress solutions derived here is for the upper U part of the specimen. Since model B is identical to the upper

part of U-shape specimens as shown in Fig. 5, the derivations of the structural stresses near spot welds in cross-tension spec-imens are similar to those for U-shape specimens. For model C, the closed-form stress solutions of the infinite and finite platemodels under counter bending moment ð ~Mb � ~McÞ listed in Section 5.2 in Lin and Pan [25] are considered. For the infiniteplate model, the maximum bending stress ðrCBÞUpper

Inf C (which is rrr on the lower surface of the upper plate) along the nuggetcircumference at r = a is identical to that of model C for U-shape specimens in Eq. (26). For the finite plate model, the max-imum bending stress ðrCBÞUpper

Fin C (which is rrr on the lower surface of the upper plate) along the nugget circumference at r = ais identical to that of model C for U-shape specimens in Eq. (27).

For model D, the maximum bending stress (rO)D (which is rrr on the lower surface of the upper plate) along the nug-get circumference at r = a is identical to that of model C for square-cup specimens in Eq. (17). For model E, the closed-form stress solutions of the infinite and finite plate models subjected to twisting moments ~Mx0y0 and ~My0x0 listed in Section5.4 in Lin and Pan [25] are adopted. For the infinite plate model, the maximum bending stress ðrTWÞUpper

Inf E (which is rrr onthe lower surface of the upper plate) along the nugget circumference at r = a is identical to that of model E for U-shapespecimens in Eq. (28). For the finite plate model, the maximum bending stress ðrTWÞUpper

Fin E (which is rrr on the lower sur-face of the upper plate) along the nugget circumference at r = a is identical to that of model E for U-shape specimens inEq. (29). For the lower part of cross-tension specimens, the maximum bending stresses ðrCBÞLower

Inf C ; ðrCBÞLowerFin C ; ðrTWÞLower

Inf E andðrTWÞLower

Fin E along the nugget circumference at r = a can be derived by substituting h + p/2 for h in Eqs. (26)–(29),respectively.

For cross-tension specimens, the loading conditions of the upper part are different from those of the lower part. The struc-tural stresses near the spot weld in the upper and lower sheets are not symmetric with respect to the interfacial surface ofthe nugget. Therefore, the structural stress distributions at each point along the nugget circumference in the upper and lowersheets should be decomposed into several symmetric or anti-symmetric distributions along the sheet thickness. Based on thedecomposition procedure shown in Fig. 8 of Lin and Pan [25], the maximum value of the bending stress rSymm

Cross under sym-metric loading conditions is given as

rSymmCross ¼

12ðrCBÞUpper

C þ ðrCBÞLowerC þ ðrOÞUpper

D þ ðrOÞLowerD þ ðrTWÞUpper

E þ ðrTWÞLowerE

h i¼ 1

2ðrCBÞUpper

C þ ðrCBÞLowerC

h iþ ðrOÞD þ

12ðrTWÞUpper

E þ ðrTWÞLowerE

h ið37Þ

The maximum value of the bending stress rAntiCross under anti-symmetric loading conditions is given as

rAntiCross ¼

12ðrCBÞUpper

C � ðrCBÞLowerC þ ðrOÞUpper

D � ðrOÞLowerD þ ðrTWÞUpper

E � ðrTWÞLowerE

h i¼ 1

2ðrCBÞUpper

C � ðrCBÞLowerC

h iþ 1

2ðrTWÞUpper

E � ðrTWÞLowerE

h ið38Þ

Note that in Eqs. (37) and (38), we use ðrOÞUpperD ¼ ðrOÞLower

D ¼ ðrOÞD.


2.4.2. Stress intensity factor solutions for cross-tension specimens2.4.2.1. KI and KII solutions for infinite plate. For the infinite plate model, combining Eqs. (17), (26), (28), and (37) with theexpression of the KI solution in Section 6.1 in Lin and Pan [25] gives the KI solution for cross-tension specimens of large sizeas a function of h as

Fig. 7.openinfour typ


3p 1

2ðrCBÞUpper

Inf C þ ðrCBÞLowerInf C

h iþ ðrOÞD þ

12ðrTWÞUpper

Inf E þ ðrTWÞLowerInf E

h i� �ð39Þ

Note that ðrCBÞLower and ðrTWÞLower are obtained by substituting h + p/2 for h in ðrCBÞUpper. The angular variation of ðrCBÞUpper

and ðrTWÞLower are in terms of cos 2h. Therefore, all terms containing cos 2h in Eq. (37) are cancelled out. The KI solution is nota function of h and only a function of b/a. Therefore, the KI solution is uniform along the nugget circumference. CombiningEqs. (26), (28), and (38) with the expression of the KII solution in Section 6.2 in Lin and Pan [25] gives the KII solution forcross-tension specimens of large size as

K II ¼ffiffitp

4ðrCBÞUpper

Inf C � ðrCBÞLowerInf C þ ðrTWÞUpper

Inf E � ðrTWÞLowerInf E

h ið40Þ

Note that the KII solution for cross-tension specimens is a function of h.

Mb-Mc

~ ~

F

=

=

A (Opening/peeling load) B (Upper half model)

C (Central bending) D (Opening)

2a

2b

2b

t

t

F

y z

x

y z

x

+

F (Counter bending)

M

F~

y z

x

F~

F~

F/2~

F/4~

F/4~

F/4~

F/4~

F/2~

Load Decomposition for Coach-Peel Specimen

Mb-Mc

~ ~

Mb-Mc

~ ~M+2b(Mb-Mc)

~ ~

(Mb-Mc)/2~ ~

(Mb-Mc)/2~ ~

+

-F/4~

-F/4~


y z

x

F/4~ F/4

~

+

G (Central bending)

y z

x

(Mb-Mc)/2~ ~

(Mb-Mc)/2~ ~

2b(Mb-Mc)~ ~

+

Decomposition of the load for a coach-peel specimen. Model A represents a spot weld in the right square part of a coach-peel specimen underg/peeling loading conditions. Model B represents the upper half of model A. The forces and moments of model B are approximately decomposed into

es of loads: central bending (model C and model G), opening (model D), cross opening/closing (model E) and counter bending (model F).


2.4.2.2. KI and KII solutions for finite plate. For the finite plate model, combining Eqs. (17), (27), (29), and (37) with the expres-sion of the KI solution in Section 6.1 in Lin and Pan [25] gives the KI solution for cross-tension specimens of finite size as afunction of h as


3p 1

2ðrCBÞUpper

Fin C þ ðrCBÞLowerFin C

h iþ ðrOÞD þ

12ðrTWÞUpper

Fin E þ ðrTWÞLowerFin E

h i� �ð41Þ

For the same reason discussed earlier, the KI solution is uniform along the nugget circumference. Combining Eqs. (27), (29),and (38) with the expression of the KII solution in Section 6.2 in Lin and Pan [25] gives the KII solution for cross-tension spec-imens of finite size as a function of h as

K II ¼ffiffitp

4ðrCBÞUpper

Fin C � ðrCBÞLowerFin C þ ðrTWÞUpper

Fin E � ðrTWÞLowerFin E

h ið42Þ

Note that the structural stresses near the spot weld in models C, D and E have no contributions to the KIII solution for cross-tension specimens based on the analytical structure stress solutions derived from the assumption of rigid inclusions for spotwelds. Therefore, the analytical KIII solution for cross-tension specimens is equal to zero along the nugget circumference.Since the detailed expressions in terms of the applied force in Eqs. (39)–(42) and those at the critical locations (points A,B and D as shown in Fig. 1d) are quite complex, we do not list the details of the solutions.

2.5. Coach-peel specimen

Fig. 7 schematically shows the decomposition of the load of the right square part of a coach-peel specimen shown inFig. 1e. The distance from the center of the nugget to the right edge of the coach-peel specimen shown in Fig. 1e is selectedto be the half width b. Note that this distance for the coach-peel specimen can be selected to be as small as 0.6 b such that theanalytical solution derived here can be valid as suggested by the computational results of Wang et al. [14] for lap-shear spec-imens. Based on the schematic of the coach-peel specimen shown in Fig. 1e, schematics of the right square part of the spec-imens near the spot welds under various types of loads are shown as models A to G in Fig. 7. As shown in Fig. 7, model Arepresents the right square part of a coach-peel specimen under opening/peeling loading conditions. Here, ~F representsthe uniformly distributed transverse shear force applied to the left edge of the square plates. ~Mb and ~Mc represent the bend-ing moments applied to the left edge of the square plates. Similar to the U-shape and cross-tension specimens, the shift ofthe transverse shear force ~F to the left edge of the square plate causes the bending moment ~Mbð¼ ~F � ðL� bÞÞ along the leftedge of the square plates. Note that the flanges of the of a coach-peel specimen have similar effects as those of U-shape spec-imens and cause a similar counter bending moment ~Mc along the left edges of the square plates.

As shown in Fig. 7, the geometry and loading conditions of model A are symmetric with respect to the interfacial surfaceof the spot weld. Therefore, only the upper half of model A is considered. Model B represents the upper half of model A.Fð¼ 2~FbÞ represents the resultant opening force applied to the center of the circular interfacial surface of the weld nugget.Additional resultant bending moments Mð¼ Fb ¼ 2~Fb2Þ and 2bð ~Mb � ~McÞ are also required for equilibrium. The forces andmoments of model B are approximately decomposed into four types of loads: central bending (model C and model G), open-ing (model D), cross opening/closing (model E) and counter bending (model F). Note that models C, D, and E take care of ~F onthe left edge of model B, and models F and G take care of ~Mb and ~Mc on the left edge of model B.

2.5.1. Structural stress solutions for coach-peel specimensThe structural stress solutions derived here is for the upper part of the coach-peel specimen. For model C, the central

bending moment on the rigid inclusion is statically in equilibrium with the resultant transverse shear force on the lateralplate surface. Therefore, the closed-form solutions of the circular plate model with a simply supported edge under centralbending conditions listed in Section 4.3 in Lin and Pan [25] are adopted and the maximum bending stress rCtB (which isrrr on the lower surface of the upper plate) along the nugget circumference at r = a is given as

ðrCtBÞC ¼3Mða2 � b02Þ½a2ð�1þ mÞ � b02ð3þ mÞ� cos h

pat2½a4ð�1þ mÞ � b04ð3þ mÞ�ð43Þ

Here, b0 is denoted as the equivalent radius for the equivalent circular plate model as discussed for the opening model ofsquare-cup, U-shape and cross-tension specimens in the previous sections. b0 is approximately equal to 2b=

ffiffiffiffipp

using thesimple area equivalence rule. M(=Fb) is an additional resultant bending moment required for equilibrium. We select the solu-tion for simply supported boundary conditions is to be consistent with the solution selected for model D as discussed later.

For model G, the central bending moment on the rigid inclusion is statically in equilibrium with the resultant moment onthe lateral plate surface. Therefore, the closed-form solutions of the circular plate model with a clamped edge under centralbending moment 2bð ~Mb � ~McÞ listed in Section 4.3 in Lin and Pan [25] are adopted and the maximum bending stress rCtB

(which is rrr on the lower surface of the upper plate) along the nugget circumference at r = a is given as

ðrCtBÞG ¼6bð ~Mb � ~McÞða2 � b02Þ cos h

pat2ða2 þ b02Þð44Þ


where ~Mb ¼ FðL� bÞ=2b and ~Mc is given in Eq. (25). It should be noted that the solution listed in Eq. (43) is based on thecircular plate model with a simply supported edge, and the distributed transverse shear force and bending moment alongthe edges of the equivalent square plate model are not exactly uniform and zero, respectively, as schematically indicatedin Fig. 7. The solution listed in Eq. (44) is based on the circular plate model with a clamped edge, and the distributed bendingmoment and transverse shear force along the edges of the equivalent square plate model are not exactly uniform and zero,respectively, as schematically indicated in Fig. 7. Therefore, the solutions listed Eqs. (43) and (44) for square plates areapproximate in nature. However, from the three-dimensional finite element analysis of Lin and Pan [29], the approximatesolutions appear to work well when compared to the computational solutions.

For model D, the maximum bending stress (rO)D (which is rrr on the lower surface of the upper plate) along the nuggetcircumference at r = a is identical to that of model C for square-cup specimens in Eq. (17). For model E, a rigid inclusion in asmaller square plate subjected to uniform twisting moments ~Mx0y0 and ~My0x0 is used to approximate the cross opening/closingconditions as for U-shape specimens as shown in Fig. 5. The twisting moment ~Mx0y0 ð¼ ~My0x0 ¼ F=16Þ has the same expressionas that for U-shape specimens and the maximum bending stresses (rTW)Inf_E and (rTW)Fin_E (which is rrr on the lower surfaceof the upper plate) of model E along the nugget circumference at r = a are identical to those of model F for U-shape specimensin Eqs. (28) and (29) based on the infinite and finite plate models, respectively.

For model F, both the infinite and finite plate models under counter bending conditions shown in Section 5.2 in Lin andPan [25] are adopted. The maximum bending stress rCB (which is rrr on the lower surface of the upper plate) of model Falong the nugget circumference at r = a is given as

ðrCBÞInf F ¼3ð ~Mb � ~McÞ½1� mþ 2ð1þ mÞ cos 2h�

t2ð1� m2Þð45Þ

where ~Mb ¼ FðL� bÞ=2b and ~Mc is given in Eq. (25). The maximum bending stress rCB (which is rrr on the lower surface ofthe upper plate) of model F along the nugget circumference at r = a is given as

ðrCBÞFin F ¼3ð ~Mb � ~McÞ

2t2XY½�2b2X � 4Yða4b4 þ b8Þ cos 2h� ð46Þ

where X and Y are given in Eqs. (2b) and (2c), respectively. Further explanations of rCtB, rCB, rO and rTW can be found in Linand Pan [25].

The structural stresses on the lower surface of the upper plate at r = a are the sum of the corresponding closed-form stresssolutions of models C, D, E, F and G as listed above. The structural normal stress rInf

Coach on the lower surface of the upper plateat r = a for coach-peel specimens with large size is

rInfCoach ¼

3Mða2 � b02Þ½a2ð�1þ mÞ � b02ð3þ mÞ� cos h

pat2½a4ð�1þ mÞ � b04ð3þ mÞ�� 3F½ða2 � b02Þð�1þ mÞ þ 2b02ð1þ mÞ lnðb0=aÞ�

2pt2½a2ð�1þ mÞ � b02ð1þ mÞ�

þ 24 ~Mx0y0 cos 2h

t2ð1� mÞþ 3ð ~Mb � ~McÞ½1� mþ 2ð1þ mÞ cos 2h�

t2ð1� m2Þþ 6bð ~Mb � ~McÞða2 � b02Þ cos h


Similarly, the structural normal stress rFinCoach on the lower surface of the upper plate at r = a for coach-peel specimens with

finite size is

rInfCoach ¼

3Mða2 � b02Þ½a2ð�1þ mÞ � b02ð3þ mÞ� cos h

pat2½a4ð�1þ mÞ � b04ð3þ mÞ�� 3F½ða2 � b02Þð�1þ mÞ þ 2b02ð1þ mÞ lnðb0=aÞ�

2pt2½a2ð�1þ mÞ � b02ð1þ mÞ�� 24 ~Mx0y0

t2X 0

� ½a4ðb=ffiffiffi2pÞ4 þ ðb=

ffiffiffi2pÞ8� cos 2hþ 3ð ~Mb � ~McÞ

2t2XY½�2b2X � 4Yða4b4 þ b8Þ cos 2h�

þ 6bð ~Mb � ~McÞða2 � b02Þ cos h


where M = Fb, b0 ¼ 2b=ffiffiffiffipp

; ~Mx0y0 ¼ F=16, X, Y, and X0 are given in Eqs. (2b), (2c), and (30), respectively. Note that ~Mc is the mo-ment due to flange constraint and is given in Eq. (25). ~Mbð¼ FðL� bÞ=2bÞ is the moment due to longer bottom and top sheetsof the upper and lower parts. The structural stress at the critical location A for coach-peel specimens as shown in Fig. 1eappears to be a good engineering parameter to predict the fatigue lives of spot welds or friction spot welds in coach-peelspecimens since cracks or kinked cracks are initiated at this locations. The structural stress at the critical location A inFig. 1e for the upper plate based on the closed-form stress solutions for the infinite and finite plate models can be obtainedby setting h = 0� in the solutions listed above.

2.5.2. Stress intensity factor solutions for coach-peel specimens2.5.2.1. KI solutions for infinite plate. Substituting Eq. (47) into the KI solution in Section 6.1 in Lin and Pan [25] gives the KI

solution for coach-peel specimens with large b ’s as a function of h as

K I ¼ffiffitp

rInfCoachffiffiffi3p ð49Þ


2.5.2.2. KI solutions for finite plate. Substituting Eqs. (48) into the KI solution in Section 6.1 in Lin and Pan [25] gives the KI

solution for coach-peel specimens with finite b ’s as a function of h as

K I ¼ffiffitp

rFinCoachffiffiffi3p ð50Þ

Note that the KII and KIII solutions for coach-peel specimens are both equal to zero along the nugget. The KI solutions at thecritical locations for the finite and infinite plate models can be obtained from Eqs. (47)–(50) by setting h = 0.

3. Conclusions

Closed-form stress intensity factor solutions for spot welds in lap-shear, square-cup, U-shape, cross-tension and coach-peel specimens are obtained based on elasticity theories and fracture mechanics. The loading conditions for spot welds in thecentral parts of the five types of specimens are first examined. The resultant loads on the weld nugget and the self-balancedresultant loads on the lateral surface of the central square parts of the specimens are then decomposed into various types ofsymmetric and anti-symmetric parts. Closed-form stress intensity factor solutions for spot welds under various types ofloading conditions are then adopted from Lin and Pan [25] to derive new closed-form stress intensity factor solutions forspot welds in the five types of specimens. The new closed-form solutions are expressed as functions of the spot weld diam-eter, the sheet thickness, the width and the length of the five types of specimens. The closed-form solutions are also ex-pressed as functions of the angular location along the nugget circumference of spot welds in the five types of specimensin contrast to the limited available solutions at the critical locations in the literature. The new closed-form solutions atthe critical locations of spot welds in the five types of specimens are listed or can be easily obtained from the generalclosed-form solutions for fatigue life predictions. The closed-form solutions for the five types of specimens are selectivelyvalidated in normalized forms by the three-dimensional finite element computational results which will be reported in de-tails in Lin and Pan [29].

Acknowledgements

The support of an Army/Ford IMPACT project, a Ford University Research Program, a NSF grant under Grant No. DMI-0456755 at University of Michigan and a NSC grant under Grant No. 96-2221-E-194-047 at National Chung Cheng Universityis greatly appreciated. Helpful discussions with Dr. S. Zhang of DaimlerChrysler are appreciated.

References

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