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
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* 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
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Fig. 1. Flow diagram of residual stress analysis.
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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.
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