Residual Specimen

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Effect of residual stresses on the fatigue of butt joints using

thermal elasto-plastic and multiaxial fatigue theory

Tso-Liang Tenga,*, Chin-Ping Fungb, Peng-Hsiang Changb

aDepartment of Mechanical Engineering, Da-Yeh University, Da-Tsuen, Changhua 515, Taiwan, ROCbUniversity of National Defense Chung Cheng Institute of Technology, Ta-Shi, Taiwan, ROC

Received 19 April 2002; accepted 13 October 2002

Abstract

This investigation performs a thermal elasto-plastic analysis using finite element techniques to analyze thermo-

mechanical behavior and evaluate residual stresses in weldments. An effective procedure is also developed by combin-

ing finite elements and multiaxial fatigue theory while considering the welding residual stress as the initial conditions in

accurately predicting the fatigue life of welded joints. Herein, the fatigue lives of butt-welded joints are forecast using

the proposed procedure. The proposed procedure that followed the conventional strain-based method (maximum

principal strain and von Mises effective strain) to predict the fatigue life of the butt-welded joints was fairly sensitive to

welding residual stress. Furthermore, the maximum principal strain method led to conservative life estimates and the

von Mises effective strain method offered the best agreement with the experimental data of butt-welded joints.

# 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Thermal history; Residual stress; Multiaxial fatigue; Weld fatigue; Finite element analysis

1. Introduction

The safety and durability of welded structures are becoming important because the sudden failure of

complex systems such as boiling water reactor piping systems, ground vehicles, aircraft, offshore structures,

pipelines and pressure vessels may cause many injuries, much financial loss and environmental damage.

Many of these welded components are subjected to complicated states of stress and strain, due to complex

loadings and welding residual stresses [1]. Fatigue under these conditions, as governed by multiaxial fatigue

theory, is an important design consideration for reliable operation of many welded components.

For predicting the multi-axial fatigue life of weldments, design codes [25] include various strength

hypotheses, such as the distortion energy hypothesis according to von Mises, the shear stress hypothesis of

Tresca and the normal stress hypothesis according to Galilei, to evaluate stressstrain states by means of an

equivalent stress or equivalent strain. Moreover, Kang et al. [6] performed a set of experiments to determine

the effects of combined tension and shear loads on the fatigue life of spot welded joints. However, to the best

1350-6307/03/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved.P I I : S 1 3 5 0 - 6 3 0 7 ( 0 2 ) 0 0 0 6 8 - 7

Engineering Failure Analysis 10 (2003) 131151

www.elsevier.com/locate/engfailanal

* Corresponding author. Tel.: +886-4-8511221; fax: +886-4-8511224.

E-mail address: [email protected] or [email protected] (T.-L. Teng).
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of the authors knowledge, no literature considers the multiaxial fatigue life of welded joints under com-

bined welding residual stress and service loading. Therefore, an effective procedure for estimating the fati-

gue life of weldments is needed.

This investigation performs a thermal elasto-plastic analysis using finite element techniques to analyze

the thermomechanical behavior and assess residual stresses in weldments. An effective procedure combines

the finite element, multiaxial fatigue theory and considers the residual stress effect to predict fatigue crack

initiation (FCI) life in weldments. The residual stress is assumed to be one of the initial conditions in pre-

dicting fatigue life via the finite element method. The proposed procedure can determine the complete

distribution of structural residual stress and strain-time history at the weld toe using the finite element

method. Herein, the fatigue lives of the butt welded joints are predicted according to the proposed proce-

dure, and the effect of welding residual stress on predicted fatigue is also discussed. Furthermore, the pre-

dictions of the proposed procedure are compared with the BS 5400 [2], AASHTO [7] standard

specifications, and experimental results [8]. Comparative results demonstrate that the estimates of fatigue

life made by the novel procedure closely approximate to the experimental results.

2. Fatigue-analysis procedure

The prediction of the fatigue life for weldments involves two steps. First, the welding residual stress dis-

tributions are calculated by a thermal elasto-plastic analysis using finite element method, as illustrated in

Fig. 1. Second, the finite element method and multiaxial fatigue theory are combined, and the residual

Nomenclature

density

C specific heat

T temperature

t time

{q} heat flux

Q the rate of internal heat generation

unit outward normal vector

hf film coefficient

TB bulk temperature of the adjacent fluid

TA temperature at the surface of the

model

[N] element shape functions

{Te} nodal temperature vector[C]

V

C N T N dV[K]

V

B T D B dV A

hf N N TdA{Fe}

V

Q N dV A

hfTB N dA{P} surface force vector

{f} body force vector

{u} displacement vector

{"} strain vector

{} stress vector

[B] strain-displacement matrix

[L] differential operator matrix

{R}

AN T Pf gdA

VN T f dV

{e} nodal stress increment matrix

{Dep} {De}+{Dp}

{De} elastic stiffness matrix

{Dp} plastic stiffness matrix

{Ue} nodal displacement vector

{T} temperature increment matrix

{Cth} thermal stiffness matrix

{Te} nodal temperature increment matrix

[M] temperature shape functionm+1{K1}

V

B T D epf g B dVm+1{K2}

V

B T Cth M dVY longitudinal residual stress

X transverse residual stressNI fatigue crack initiation life

0f uniaxial fatigue strength coefficientb uniaxial fatigue strength exponent

"0f uniaxial fatigue ductility coefficientc uniaxial fatigue ductility exponent

r residual stress

o mean stress

"1, "2, "3 principal strain

"e von Mises effective strain

132 T.-L. Teng et al. / Engineering Failure Analysis 10 (2003) 131151


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Fig. 1. Flow diagram of residual stress analysis.

T.-L. Teng et al. / Engineering Failure Analysis 10 (2003) 131151 133


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Fig. 2. Flow chart for predicting fatigue life for weldments.



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stress distributions determined in the first step is considered as an initial condition in predicting the fatigue

life of weldments, as shown in Fig. 2.

2.1. Residual stress analysis model

Welding residual stresses are calculated using the finite element method. Fig. 1 presents the analytical

procedures. During each weld pass, thermal stresses are calculated from the temperature distributions

determined by the thermal model. The residual stresses from each temperature increment are then added to

the nodal point location to update the behavior of the model before the next temperature increment.

Fig. 3. Geometry of multipass butt weld.



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2.1.1. Thermal model

During each weld pass, the temperature distributions are calculated from the thermal model. This

investigation simulates the increment of heat loading in the welding process via the lead temperature curve.

The convective heat coefficients on the surfaces were estimated (using engineering formulae for naturalconvection) to be 15 W/m2K. The initial temperature was taken to be 18 C.

2.1.2. Mechanical model

In the mechanical analysis, the temperature history obtained from the thermal analysis was entered into

the structural model as a thermal loading. The thermal strains and stresses can then be calculated at each

time increment, and the final state of the residual stresses will be accumulated by the thermal strains and

stresses. During each weld pass, thermal stresses are calculated from the temperature distributions deter-

mined by the thermal model. The residual stresses from each temperature increment are then added to the

nodal point location to update the behavior of the model before the next temperature increment. The

material was assumed to follow the von Mises yield criterion and the associated flow rules. Linear kine-

matic hardening was assumed. Free boundary conditions were used for the free surfaces except at the

centerline of the cross-section, where a symmetry condition was used. Initial stresses and strains were zero.Phase transformation effects were not considered herein, due to lack of material information, especially at

high temperatures, such as near the melting point.

Fig. 4. Transverse residual stress at the top surface of the plate.



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2.1.3. Verification

The proposed method was compared with ABAQUS finite element package and experimental results

taken from Shim et al. [9] to confirm its accuracy. In Shim et al.s investigation, a specimen was constructed

using multi-pass butt welding, as shown in Fig. 3. Figs. 4 and 5 portray the distribution of the transverseand longitudinal residual stress on the thick plate computed by Shim et al. and the present method. As

Fig. 4 indicates, the ABAQUS package result showed slightly lower tensile transverse stress near the weld

centerline. The present method tends to the experimental results near the surface. As Fig. 5 indicates, both

analysis results show tensile stress near the weld centerline. The residual stress calculated using the present

method correlates well with that determined using Shim et al.s experiments. Therefore, the procedure

proposed here is considered appropriate for analyzing residual stresses due to welding.

2.2. Fatigue crack initiation analysis model

Fatigue cracks are initiated most readily at the surface of the weld toe, and are concentrated by

material or geometric stress raisers. Therefore, care must be taken in life prediction to account for pro-

cessing and other factors that alter the surface and create stress raisers. Accordingly, in this study, pre-dicting fatigue life for weldments involves structural and fatigue analysis of critical areas, as illustrated in

Fig. 2.

Fig. 5. Longitudinal residual stress at the top surface of the plate.



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2.2.1. Structural analysis

The structural analysis calculated the stresses and strains in a highly stressed region where slip con-

centrates from the input loads for a given material and geometry. In the structural analysis, the residual

stress was considered as an initial condition in predicting the fatigue life. The structural analysis allowsstrains and stresses to be calculated at each time increment following a finite element method, in which

loading history is the input of the welded structural model. The stress-strain field in these critical areas

within the weldments can also be found via the finite element method.

Fig. 6. Temperature-dependent material properties of ASTM A36 carbon steel.



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2.2.2. Fatigue analysis of critical areas

Fatigue analysis of the critical areas approach involves the following technique for converting welding

residual stresses, load history, weldment geometry, and material cyclic properties input into a prediction of

fatigue life. The operations involved in the prediction must be performed sequentially, as shown in Fig. 2.First, the stress and strain at the critical site are estimated by the finite element method. The finite element

method is then used to convert reduced load-time history into a strain-time history and calculate the stress

and strain in the highly stressed area. Then the multiaxial fatigue theory is used to incorporate the strain-

life approach to predict the fatigue life of the weldment. The simple linear damage hypothesis proposed by

Palmgren and Miner is used to accumulate the fatigue damage. Finally, the stress and strain at the critical

location are used to compute damage, and their historical values summed algebraically until a critical

damage sum (failure criteria) is reached. The point at which the failure criteria is met is the predicted life.

2.2.3. Predicting life

The fatigue resistance of metals can be characterized by a strain-life curve. These curves are derived from

polished laboratory specimens that are tested under completely reversed strain control. The relationship between

total strain amplitude, "=2, and reversals to failure, 2NI, can be expressed through the following form [10,11]:

"

2

0f

E2NI b"0f 2NI c; 1

The strain-life equation has been modified to account for mean stress effects. Morrow [12] suggested that

the mean stress effect could be considered by modifying the elastic term in the strain-life equation by mean

stress, o:

"

2

0f o

E2NI b"0f 2NI c: 2

Manson and Halford [13] modified both the elastic and plastic terms of the strain-life equation to

maintain the independence of the elasticplastic strain ratio from mean stress:

"

2

0f o

E2NI b"0f

0f o0f

c=b2NI c: 3

Meanwhile, Smith, Watson, and Topper (SWT) [14] proposed another equation to represent mean stress

effects:

Fig. 7. Geometry of the butt joints.



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max"

2

0f

2E

2NI 2b0f"0f 2NI bc; 4

where

Fig. 8. Weld thermal cycles of A36 carbon steel.

Table 1

Schematic of pass sequences along with welding parameters for each pass

Pass sequence Pass no. Welding parameter

Current (A) Voltage (V) Speed (mm/s)

1 190 25 3.34

23 215 26 4.70

4 190 25 3.34

56 215 26 4.70



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max 2

o:

Two parameters ("1,"e) of multi-axial damage are examined in predicting fatigue life.(a) The maximum principal strain parameter, "1. The maximum principal strain approach is analo-

gous to the traditional use of the applied strain amplitude in uniaxial analysis. For welding residual

stresses, geometries and loadings used in this study, and principal strain ("1, "2, "3) are determined by an

appropriate transformation according to the finite element method. In correlating multiaxial fatigue tests,

the range of the maximum principal strain on the plane that experiences the maximum principal strain is

considered to be the dominant parameter describing damage, and is included in the strain life equation:

Fig. 10. Maximum principal stress distribution along the X-direction for different finite element meshes.

Fig. 9. Finite element meshes for the butt weld joint with 706 elements.



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"1

2

0f

E2NI b"0f 2NI c: 5

(b) von Mises effective strain parameter, "e. The von Mises effective strain may be thought of as the rootmean square of the maximum principal shear strains, normalized to axial loading. Also called the octahe-

dral shear strain, this parameter is given by:

"e 1ffiffiffi2

p1 "1 "2

2 "2 "3 2 "3 "1 2 1=2

:

The effective strain is used as an equivalent uniaxial strain amplitude with Eq. (1), in predicting fatigue

life. The von Mises effective strain parameter can be directly correlated to the uniaxial Coffin-Manson

strain -life equation:

"e

2

0f

E

2NI

b

"0f 2NI

c:

6

When the fatigue properties of a given metal are known and the service environment is defined, the

complicated problem of predicting fatigue life of weldments becomes a simple matter of determining the

Fig. 11. Maximum principal stresstime history for different finite element meshes.



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welding residual stress and local-strain amplitude for each reversal, so that Eqs. (1)(6) can be solved for

fatigue life.

3. Analytical model

To consider the influence of residual stress on the predicted fatigue life of welds and confirm the accuracy

of the present calculation procedure, this study develops an effective procedure for estimating the FCI life

of butt-weld joints.

Fig. 12. Maximum principal stress distribution along the X-direction for nominal stress, S=146.4 MPa.



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3.1. Specimen and material properties

The material used herein is ASTM A36 carbon steel. Its mechanical properties are dependent on tem-

perature, as illustrated in Fig. 6 [9]. As Fig. 6 indicates, mechanical properties of metals change under variousconditions when temperature increases, the modulus of elasticity, yield stress and thermal conductivity

decrease while the thermal expansion and specific heat increase. Furthermore, the width of weld zone was

assumed as that of the heat source. Fig. 7 displays the geometry and dimensions of two A36 plates, joined

by a multipass butt-weld. Meanwhile, Ref. [15] specifies the cyclic strain-life properties and stressstrain

curves of base metal (BM), weld-metal (WM), and the heat-affected zone (HAZ) for weldments of ASTM

A36 carbon steel. Linear kinematic hardening was assumed. Therefore, these data are used here for the

stress-strain and fatigue analysis of the butt-weld joints.

Fig. 13. Maximum principal stresstime history at point MX.



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3.2. Simulation of welding and fatigue loading

Heat sources are applied along the weld path for practical welds. Pass sequences and welding parameters

are shown in Table 1. However, this investigation simulates the increment of heat loading on the weldprocess via the weld thermal cycle curve, as shown in Fig. 8 [15]. A simple lumped pass model was devel-

oped to simulate the weld filler in butt-welded joints. Lump bead weld volumes for each pass in that layer

were added and distributed over the top surface of the layer. This is a very efficient method for reducing the

computational cost for both thermal and stress analysis, especially for thick plates. Following welding, this

study has been constructed for constant-amplitude uniaxial loading (see Fig. 7) with a stress ratio of R=0

for butt-welded joints.

3.3. Finite element model for the butt-weld joints

This investigation develops a two-dimensional symmetrical plane strain model to estimate the residual

stresses and converts a load-time history into a strain-time history of the weldments using the finite element

method. For predicting residual stress of weldments, a total of weld passes were lumped into 6 passes in allthick plate with a double V-groove. Fig. 9 demonstrates the finite element mesh for the welded joints, along

Fig. 14. Fatigue life of the presented procedure, combined with the maximum principal strain theories and strain-life equations

(without considering the weld residual stress as the initial conditions).



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with the refined meshes used in the weld toe. The symmetric model has 706 elements and 775 nodes after

meshing.

3.4. Mesh sensitivity study

The influence of mesh refinement on the highly stressed area was studied to examine the adequacy of

element sizes. The model with refined meshes consists of 810 elements and 886 nodes. Fig. 10 displays the

distributions of the maximum principal residual stress, 1 along the X-direction with 706 and 810 finite

element mesh models. Fig. 11 presents the maximum principal stress-time history of the weld toe for a

nominal stress range, S=146.4 MPa with 706 and 810 finite element mesh models. Figs. 10 and 11

summarize the results obtained using models with two mesh densities, but with identical material models

and geometries. Similar distributions of the results near the weld toe obtained from the simulations using

these two finite element meshes are observed, and we remark that the model is not sensitive to the finite

element mesh refinement when the number of elements is equal to or greater than 706. Therefore, the ori-

ginal finite element model without mesh refinement in the butt-welded joints can be worked for this study.

Fig. 15. Fatigue life of the presented procedure, combined with the maximum principal strain theories and strain-life equations

(considering the weld residual stress as the initial conditions).



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4. Results and discussion

4.1. Residual stresses

Welding residual stresses are calculated using thermal and mechanical analysis. Fig. 10 depicts the dis-

tributions of the top surface maximum principal residual stress, 1 along the X-direction. Owing to the

locally concentrated heat source, the temperature near the weld bead and the heat-affected zone changes

rapidly with distance from the heat source, i.e., the highest temperature is limited to the domain of the heat

source, from which the lower temperature zones fan out. Owing to the temperature nonuniformity the

shrinkage varies through the weldment thickness during cool-down and, consequently a high tensile resi-

dual stress occurs on the surface of the weld toes. As Fig. 11 indicates, a high tensile residual stress occurs

near the weld toes, and its value of 219 MPa approaches the yield stress of the material.

4.2. Analysis of critical areas

The residual stress distributions from Section 4.1 are considered as initial conditions in predicting fatiguelife of weldments. Furthermore, the critical areas of the stressstrain field of the weldments were found by

Fig. 16. Fatigue life of the presented procedure, combined with the von Mises effective strain theories and strain-life equations

(without considering the weld residual stress as the initial conditions).



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the finite element method. Fig. 12 illustrates the contours of maximum principal stress, 1 for the analytical

model in which nominal stress S=146.4 MPa. As Fig. 12 shows, a high tensile stress occurred at point

MX near the weld toes. Fig. 13 displays the maximum principal stress-time history at point MX for

various nominal stress ranges (S). The figure reveals that the mean stress increases with the nominalstress range. The proposed procedure can predict fatigue life for the weldments because the cyclic strain

(stress)-time history and the strain (stress) range of the weldments on point MX are determined.

4.3. Predicting fatigue life for the butt-weld joints

Results of the procedure proposed here are compared to experimental results to consider the influence of

residual stress on the predicted fatigue life of the weldments and confirm the accuracy of the present cal-

culation procedure (Fig. 2).

Fig. 14 illustrates the results of applying the proposed procedure (without considering the weld residual

stress as initial conditions), combined with the multiaxial theories (maximum principal strain), Manson-

Halford and SWT strain-life equations to predict the fatigue life of the weldments. The analytical results

are compared with experimental results taken from Lawrence. As Fig. 14 indicates, no results were con-servative. Fig. 15 shows the results of applying the proposed procedure, combined with multiaxial theories

Fig. 17. Fatigue life of the presented procedure, combined with the von Mises effective strain theories and strain-life equations (con-

sidering the weld residual stress as the initial conditions).



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(maximum principal strain), Coffin-Manson, Manson-Halford and SWT strain-life equations to predict the

fatigue life of the weldments and consider the influence of weld residual stress. As the figure indicates,

combining the proposed procedure (considering the residual stress as the initial condition) with the max-

imum principal strain parameter, "1, and the SWT strain-life equation above, yields results consistent withthe experimental data.

Fig. 16 presents the results of applying the proposed procedure (without considering the weld residual

stress as initial conditions), combined with the multiaxial theories (von Mises effective strain), Manson-

Halford and SWT strain-life equations, to predict the fatigue life of the weldments. As Fig. 16 indicates, no

results were conservative. Fig. 17 illustrates the results of applying the proposed procedure, combined with

the multiaxial theories (von Mises effective strain), Coffin-Manson, Manson-Halford and SWT strain-life

equations to predict the fatigue life of the weldments and to consider the effect of weld residual stress.

According to Fig. 17, combining the proposed procedure (considering the residual stress as initial condi-

tions) with the von Mises effective strain parameter, "e, and the SWT strain-life equation above, yields

results that closely correspond to the experimental data. According to Figs. 1417, the proposed procedure

following the conventional strain based method (maximum principal strain and von Mises effective strain),

for predicting the fatigue life of the weldments was fairly sensitive to welding residual stress.Fig. 18 shows estimates of fatigue life by multiaxial fatigue theory (maximum principal strain and von

Mises effective strain), and correlations of experimental data with such estimates. This figure reveals that

Fig. 18. Fatigue life estimates by multiaxial fatigue theory (maximum principal strain and von Mises effective strain).



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predictions of fatigue life by the maximum principal strain method were conservative, while predictions

using the von Mises effective strain closely correspond to the experimental data.

Many countries have standardized weldment design stresses, but since most national standards deal

specifically with the design of bridges and tubular structures, they are difficult to apply to the design of

other types of welded structures. Fig. 19 presents the BS 5400, AASHTO standard specifications for pre-

dicting the endurance of weldments under zero stress ratios. The figure indicates that predictions of fatigue

life using the AASHTO standard method were excessively conservative, while predictions by the BS 5400

standard method were not conservative. The closest agreement with experimental data was achieved by the

procedure herein presented, combined with the finite element method, multiaxial theories (von Mises

effective strain), SWT strain-life relationship, and consideration of the effect of welding residual stress.

5. Conclusion

This study combined the finite element method and multiaxial fatigue theory, while considering the

welding residual stress as the initial conditions, to develop a simple and effective procedure for predicting

the fatigue crack initiation life of butt-welded joints. Based on the results herein, the following conclusions

are reached:

Fig. 19. Fatigue life prediction using the BS 5400, AASHTO standard specifications method and the novel technique.



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1. Localized heating caused by welding and subsequent rapid cooling can cause tensile residual stresses

at the weld toe of butt-welded joints. These tensile residual stresses were considered one of the major

influences on fatigue strength.

2. The proposed procedure that followed the conventional strain-based method (maximum principalstrain and von Mises effective strain) to predict the fatigue life of the butt-welded joints was fairly

sensitive to welding residual stress.

3. The maximum principal strain method led to conservative life estimates and the von Mises effective

strain method offered the best agreement with the experimental data.

4. Combining the novel procedure with three different strain-life equations to evaluate the fatigue life

and the SWT equation achieved the best agreement with the experimental data.

References

[1] Maddox SJ. Fatigue strength of welded structures. Abington: Cambridge University Press; 1991.[2] Specification for steel. Concrete and Composite Bridges. Code of Practice for Fatigue, BS 5400, British Standards Institution,

London, 1980.

[3] Eurocode Nr. 3, Gemeinsame einheitliche Regeln fu r Stahlbauten Kommission der Europa ischen Gemeinschaften, Bericht Nr,

EUR 8849, DE, EN, FR, Stahlbau-Verlagsge-sellschaft mbH, Ko ln, 1984.

[4] ASME Boiler and Pressure Vessel Code, section III, Division 1, Subsection NA, Article XIV-1212, Subsection NB-3352.4, New

York, 1984.

[5] AD-Merkblatt S 2, Berechnung gegen Schwingbeanspru-chung; Beuth-Verlag, Berlin/Ko ln; 1982.

[6] Kang H, Barkey ME, Lee Y. Evaluation of multiaxial spot weld fatigue parameters for proportional loading. International

Journal of Fatigue 2000;22:691702.

[7] Standard Specifications for Highway Bridges. American Association of State Highway and Transportation Officials, 13th ed.,

1983.

[8] Lawrence FV, Burk JD, Yung JY. Influence of residual stress on predicted fatigue life of weldments. Residual Stress Effects in

Fatigue, ASTM STP 776. American Society for Testing and Materials, 1982. p. 3343.

[9] Shim Y, Feng Z, Lee S, Kim D, Jaeger J, Papritan JC, et al. Determination of residual stresses in thick-section weldments.

Welding Journal 1992:305s12s.

[10] Coffin LF. A study of effects of cyclic thermal stresses on a ductile metal. Transactions of the American Society of Mechanical

Engineers 1954;76:93150.

[11] Manson SS. Behavior of materials under conditions of thermal stress. National Advisory Commission on Aeronautics, Report

1170, Cleveland: Lewis Flight Propulsion Laboratory, 1954.

[12] Morrow J. Fatigue design handbook, advances in Engineering, Vol. 4. Warendale, Pa: Society of Automotive Engineers; 1968

Section 3.2, p. 21-29.

[13] Manson SS, Halford GR. Practical implementation of the double linear damage rule and damage curve approach for treating

cumulative fatigue damage. Int J Fract 1981;17(2):16972.

[14] Smith KN, Watson P, Topper TH. A stressstrain function for the fatigue of materials. J Mater 1970;5(4):76778.

[15] Higashida Y, Burk JD, Lawrence FV. Strain-controlled fatigue behavior of ASTM A36 and A514 grade F steels and -0 alumi-

num weld materials. Welding Research Supplement 5083;1978:334s44s.


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