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Fatigue Damage Evaluation of Orthotropic Steel Deck
Considering Weld Residual Stress Relaxation Based on
Continuum Damage Mechanics
Chuang Cui
1
; Qinghua Zhang
2
; Yi Bao, Aff.M.ASCE
3
; Yizhi Bu
4
; and Zhongtao Ye
5
Abstract: Weld residual stresses (WRSs) in orthotropic steel decks (OSDs) contribute prominently to fatigue damage. WRSs relax during
the accumulation of fatigue damage. However, their effects are ignored in conventional methods of fatigue damage evaluation. This paper
presents the first evaluation approach to fatigue damage that considers of the relaxation of WRSs. WRSs were analyzed using a thermome-
chanical analysis that took into account the relaxation of WRS. An elastic–plastic fatigue damage model was developed using continuum dam-
age mechanics. The proposed model of fatigue damage was experimentally validated using fatigue tests on U-rib weld connection specimens
under different cyclic loading conditions. The test results indicated the feasibility of the elastic–plastic fatigue damage model for predicting
WRSs and fatigue damage in OSDs. Ignoring the effects of WRSs may result in the overestimation of fatigue life, resulting in structural
designs that are not conservative enough. DOI: 10.1061/(ASCE)BE.1943-5592.0001280.©2018 American Society of Civil Engineers.
Author keywords: Orthotropic steel deck (OSD); Fatigue evaluation; Weld residual stress; Fatigue damage model; Weld residual stress
relaxation.
Introduction
Orthotropic steel decks (OSDs) are widely employed in steel
bridges because of their superior properties, such as their high load-
bearing capacity, light weight, ease of assembly, and modular con-
struction. However, fatigue cracks caused by traffic loads, weld re-
sidual stresses (WRSs), weld defects, and other negative influences
have been observed for OSDs during the early operational stages of
the in-service period under traffic loads, and these have significantly
reduced the service life of such structures (Ya et al. 2011;Zhang
et al. 2015,2017;Wang et al. 2016;Cui et al. 2017;Heng et al.
2017;Fu et al. 2017). WRSs and external loads are two predominant
factors in the stress distributions that govern the fatigue life of steel
bridges (Webster and Ezelio 2001;Cheng et al. 2003;Barsoum and
Barsoum 2009;Cui et al. 2017,2018;Cheng et al. 2017a,b). WRSs
have been found to be related to types of welding (Grong 1994),
including gas shielded arc welding (GSAW), tungsten inert gas
welding (TIGW), submerged arc welding (SAW), metal inert gas
welding (MIGW), and others. The evaluation of fatigue damage
with due considerationfor the combined effect of WRS and vehicle-
induced stress remains a challenge.
Much research has been devoted to evaluation of fatigue damage
of OSDs under traffic loads. Various stress-based approaches have
been presented to assess the fatigue resistance of OSDs under exter-
nal loadings. Connor and Fisher (2006) proposed a nominal stress
approach to evaluate the fatigue resistance of OSDs. Stresses at
locations with a specific distance to a reference point are used to
evaluate the fatigue resistance of a weld joint in an OSD (Connor
and Fisher 2006). A hot-spot stress approach was proposed to evalu-
ate the fatigue life of the weld joints in OSDs and was implemented
using fatigue test data (Liu et al. 2014;Zhang et al. 2015). A notch
stress approach was proposed to consider the effects of the notch
shape of the weld joint and applied to evaluate the fatigue life of
OSDs (Zhang et al. 2015). Although stress-based approaches have
been successfully applied for fatigue problems induced by traffic
loads WRS have not been considered in this context.
Simultaneously, energy-based approaches have been proposed
and implemented in fatigue evaluation of OSDs that do take WRSs
into account. Recently, Cui et al. (2017) performed a thermome-
chanical analysis to analyze the combination effects of WRS and
traffic loading. Investigations of the influence of WRSs on fatigue
damage have been undertaken, based on the assumption of that
WRSs are constant throughout the whole process of development of
fatigue damage. However, as a matter of fact, WRSs change under
cyclic loadings (Barsoum and Barsoum 2009;Lee et al. 2015).
Barsoum and Barsoum (2009) presented a method using linear elas-
tic fracture mechanics to analyze the growth of fatigue cracks using
afinite element model. The research focus in that study was placed
on the evaluation of the effect of the relaxation of WRSs on
the growth and propagation of fatigue cracks. An elastic–plastic
fatigue damage model has been developed that is based on damage
mechanics and the constitutive model of elastic–plastic damage
(Rousselier and Chaboche 1983;Marmi et al. 2009;Shi et al. 2011;
Murakami and Kamiya 1997;Qian et al. 2013). Kim et al. (2013)
used X-ray diffraction to measure the WRSs of shot-peened carbon
1
Ph.D. Candidate, Dept. of Bridge Engineering, Southwest Jiaotong
Univ., 111 Section of Northbound 1, Second Ring Road, Chengdu
610031, China. Email: swjtu_cc@126.com
2
Professor, Dept. of Bridge Engineering, Southwest Jiaotong Univ.,
111 Section of Northbound 1, Second Ring Road, Chengdu 610031, China
(corresponding author). Email: swjtuzqh@126.com
3
Research Fellow, Dept. of Civil and Environmental Engineering,
Univ. of Michigan, Ann Arbor, MI 48109. Email: yibao@umich.edu
4
Professor, Dept. of Bridge Engineering, Southwest Jiaotong Univ.,
111 Section of Northbound 1, Second Ring Road, Chengdu 610031,
China. Email: yizhibu@163.com
5
Senior Engineer, State Key Laboratory for Bridge Structure Health
and Safety, 103 JianShe Road, Wuhan, Hubei 430034, China. Email:
lanyiyzt@foxmail.com
Note. This manuscript was submitted on October 18, 2017; approved
on March 30, 2018; published online on July 19, 2018. Discussion period
open until December 19, 2018; separate discussions must be submitted for
individual papers. This paper is part of the Journal of Bridge
Engineering, © ASCE, ISSN 1084-0702.
© ASCE 04018073-1 J. Bridge Eng.
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steel, producing empirical equations to estimate the relaxation of re-
sidual stresses. Lee et al. (2015) developed a three-dimensional
(3D) elastoplastic finite element model to investigate the redistribu-
tion of the residual stresses under cyclic mechanical loading, using
damage mechanics. That model incorporates the constitutive equa-
tion of cyclic plasticity based on the Chaboche nonlinear kinematic
hardening and the isotropic hardening rule (Chaboche and Lesne
1988), indicating the feasibility of a damage evaluation based on
elastic–plastic damage, which is constitutive, with consideration of
residual stress and its relaxation under cyclic mechanical loading.
At present, there is little understanding of the effects of WRS and
stress relaxation on fatigue damage to OSDs.
In this study, the relaxation of WRSs was investigated and the
fatigue damage of OSDs subjected to a combination of WRSs and
cyclic loadings was evaluated. The relaxation of WRSs was taken
into consideration by using an elastic–plastic fatigue damage
model, which was based on continuum damage mechanics. First, a
numerical model developed by the authors in a previous study was
used to analyze WRSs. The relaxation of WRSs was quantified
using a simplified model. Then, an elastic–plastic fatigue damage
model was developed to predict fatigue damage. The prediction
results of fatigue damage were validated using model tests. Finally,
fatigue life was predicted for the presented fatigue detail presented
for OSDs using the proposed method.
Generation and Relaxation of WRS
Generation
Inelastic strains induced in the welding process include transient ther-
mal strains, cumulative plastic strains, and final inherent-shrinkage
strains. A thermomechanical analysis was performed to analyze
WRSs (Dong and Brust 2000). This thermomechanical analysis
was conducted in two steps. First, a heat transfer analysis was car-
ried out to determine the welding temperature; then, based on the
temperature field, a mechanical analysis was performed to deter-
mine the residual stresses.
A uniform heat flux was applied over the cross section of the
specimen to simulate the effects of the welding heat source. A volu-
metric heat source with a double ellipsoidal distribution (Goldak
et al. 1984) was used in the heat transfer analysis to determine tem-
perature distribution. The distribution of WRSs was calculated
using the temperature distribution. The influence of solid-state
phase transformations on WRSs was not considered for mild carbon
steel (Goldak et al. 1984). The total strain rate (_
ɛt) was composed of
three components: elastic strain rate (_
ɛe), plastic strain rate (_
ɛp) and
thermal strain rate (_
ɛts), as shown in Eq. (1). This thermomechanical
analysis was elaborated by Cui et al. (2017).
_
ɛt¼_
ɛeþ_
ɛpþ_
ɛts (1)
Relaxation
WRSs are relived and redistributed when mechanical loading is
applied. The extent of relaxation is dependent on the stress range,
number of loading cycles, loading scenario, and properties of the
material (Lee et al. 2015). Several models have been proposed to
quantify the degree of relaxation (Kim et al. 2013;Yi and Lee 2017;
Xie et al. 2017), as listed in Table 1. Those predicted results are
compared to the experimental data in Fig. 1.
In this study, the stress-relaxation model recommended by Xie
et al. (2017) is adopted
s
R¼
s
R0
s
R0a1
s
a=
s
y
a2þa3
hi
ln Nþ1
ðÞ½
a4(2)
where
s
a
is the amplitude of the cyclic stress;
s
y
is the yielding
stress; Nis the number of loading cycles; and a
1
,a
2
,a
3
, and a
4
are
four material parameters obtained by residual stress measurement.
Elastic–Plastic Fatigue Damage Model
The fatigue damage that results from the combination of WRSs and
mechanical cyclic loading was analyzed using an elastic–plastic fa-
tigue damage model.
Constitutive Equation of Damaged Material
Based on the dissipation potential of a damaged material using the
generalized normality rule, and assuming decomposition of elastic
Table 1. Residual stress relaxation model
References Relaxation model Parameter Value
Kim et al. (2013)
s
R¼A
s
aþmlog NA−1.01
m18.03
Yi and Lee (2017)
s
R¼
s
R0Nkk−0.0045
Xie et al. (2017)
s
R¼
s
R0
s
R0a1
s
a=
s
y
a2þa3
hi
ln Nþ1
ðÞ½
a4a
1
4.625
a
2
6.456
a
3
6.480
a
4
0.08789
Note:
s
Ris the residual stress at Ncycles;
s
R0is the initial residual stress; and Aand mare the parameters related to the amplitude of the cyclic stress
s
a
and
the yielding stress
s
y
.
Fig. 1. Comparison between predicted results and experimental data.
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and plastic strain rate, the total strain-rate tensor includes within it
the elastic- and plastic-strain tensors. Thus, the changing rate of
total strain tensor can be written as follows (Lemaitre et al. 1999;
Do et al. 2015):
_
eij ¼_
ee
ij þ_
ep
ij (3)
where _
eij is the rate of total strain tensor; _
ee
ij is the rate of elastic
strain tensor; and _
ep
ij is the rate of plastic strain tensor. The changing
rate of the elastic-strain tensor is expressed as follows (Murakami
and Kamiya 1997):
_
ee
ij ¼1þv
ðÞ
_
rij
E1D
ðÞ
v_
rkk
d
ij
E1D
ðÞ (4)
where E,v,andDare Young’s modulus, Poisson ratio, and the dam-
age variable, respectively; _
rij is the rate of the stress components;
_
rkk is the rate of hydrostatic stress tensor; and
d
ij is the Kronecker
delta: if i=j,
d
ij =1;ifi=j,
d
ij =0.
The changing rate of the plastic strain tensor is written as
(Murakami and Kamiya 1997)
_
ep
ij ¼_
λ∂FD
∂
s
ij
¼3_
λ
21D
ðÞ
Aij=1D
ðÞ
Bij
re
(5)
where _
λis the plastic multiplier determined by the plastic-flow
consistency condition; FDis the plastic dissipation potential
(Rousselier and Chaboche 1983); Aij and Bij is the deviator of
equivalent stress tensor and back stress of the nonlinear kinematic
hardening; and
s
eis the equivalent stress tensor that is defined as
follows:
re¼3
2
Aij
1DBij
Aij
1DBij
1=2
(6)
Then, the accumulated plastic strain rate ( _
P) can be obtained as
follows (Rousselier and Chaboche 1983;Chaboche 1991):
_
P¼2
3
_
ep
ij _
ep
ij
1=2
(7)
The kinematic hardening rate ( _
Bij) is dependent on the isotropic
hardening variable (
g
k) and the kinematic hardening variable (
a
k),
which can be determined using experimental testing data (Lemaitre
et al. 1999)
_
Bij ¼X
3
k¼1
_
Bij
k¼X
3
k¼1
g
k
3
2
a
k_
ep
ij Bij
k
_
P
(8)
Elastic–Plastic Fatigue Damage Model
The total fatigue damage is composed of the elastic damage (D
e
)
and inelastic damage (D
p
)(Chaboche and Lesne 1988;Marmi
et al. 2009). The elastic damage results from total stresses that are
less than the yielding strength of steel. The total stress is the sum
of the WRSs and mechanical stresses. The inelastic damage
results from total stresses that are higher than the yielding
strength of steel. The rate of the elastic damage of steel under
multiaxial cyclic stresses can be written as (Chaboche and Lesne
1988;Marmi et al. 2009)
_
De¼
d
De
d
N¼11D
ðÞ
b1þ1
hi
b2C
M1D
ðÞ
b1
(9)
b2¼1c1HCC
s
y
s
e;max
(10)
M¼M013c2
s
H;m
ðÞ (11)
where
s
e;max ¼max 1:5Aij :Aij
ðÞ
1=2
hi
, and it is could be replaced
by
s
max, which is the sum of the WRS
s
Rand the maximum cyclic
mechanical stress
s
app;max; H() donates that: H(x)¼xif x≥0;
H(x)¼0ifx<0. Mis a variable denoting the effect of the mean
stress;
s
yis the yielding tensile stress; and
s
H;mis the mean hydro-
static stress, defined as follows:
s
H;m¼1
6max
x
ðÞ
þmin
x
ðÞ
(12)
where
x
¼Prkk,andCis the octahedral shear stress range which
is defined as follows:
C¼1
2
3
2Aij;max Aij;min
ðÞ
Aij;max Aij;min
ðÞ
1=2
(13)
where Aij;max and Aij;min are the deviator of maximum and minimum
stress tensor under the cyclic loading. Cis the damage variable,
which is defined as follows (Sines 1959):
C¼
s
013c3
s
H;m
ðÞ (14)
Then, integrating Eq. (9), assuming that the failure damage vari-
able D
c
= 1, the fatigue failure cycles N
fe
under cyclic stress is as
follows:
Nfe ¼Cb1
c11þb1
ðÞ
Mb1
H
s
u
s
e;max
ðÞ
HCC
ðÞ (15)
The elastic damage parameters, including b
1
,M
0
,c
1
,c
2
,and
c
3
, can be determined from the fatigue test. To keep consistence
with the Palmgren–Miner rule for the linear model, Eq. (15) can
be expressed by the linear logarithmic function of N
fe
versus
s
e;max
log Nfe ¼log Cb1
c11þb1
ðÞ
Mb1
!
þlog Hð
s
y
s
e;maxÞ
HðCCÞ(16)
When the combination of WRS and the maximum cyclic stress
is greater than the yield stress of material, the damage process
evolved becomes low-cycle fatigue behavior, for which the inelastic
damage could be calculated using Lemaitre’s ductile damage model
(Lemaitre and Desmorat 2005)
_
Dp¼
s
2
eRv
2ES 1D
ðÞ
2
!
d
_
P(17)
where Sand dare the material parameters obtained from fatigue
test, and R
v
is the triaxial stress function
Rv¼2=3
ðÞ
1þv
ðÞ
þ312v
ðÞ
s
H=
s
e
ðÞ (18)
where
s
His the hydrostatic stress. By integrating Eq. (17), assum-
ing D
c
= 1, the fatigue failure cycles N
fp
under cyclic stress become
the following:
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Nfp ¼2ES
ðÞ
d
22dþ1
ðÞ
s
2d
eDɛp
(19)
where Dɛpis defined as Dɛp¼ɛmax
s
e=E. Similarly, Eq. (19)
can be rewritten as the linear logarithmic function of N
fp
versus
ɛmax
log Nfp ¼log 2ES
ðÞ
d
22dþ1
ðÞ
s
2d
e
!
log ɛmax
s
e=EðÞ(20)
WRS is time-dependent and changes when cyclic loading is
applied. The total stress decreases with the number of loading
cycles. The rate of fatigue damage _
Dwith the increase of cycles can
be expressed as follows:
_
D¼
_
Defor
s
Rþ
s
app;max <
s
y
_
Dpfor
s
Rþ
s
app;max
s
y
((21)
where
s
app;max is the maximum cyclic stress under external cyclic
loading. Thus, the increment of fatigue damage DDfrom N
0
to
N
0
þDNcycles are approximately expressed as Eq. (22) when the
number increment of loading cycles DNis small.
DD¼ðN0
N0þDN
_
DdN ¼_
DDN(22)
Thus, the total accumulated fatigue damage (D) can be calcu-
lated using Eq. (22) by the jump-in-cycles method. Assuming the
rate of fatigue damage is _
Diat DN
i
cycles, the total accumulated fa-
tigue damage can be expressed as follows:
D¼XDD¼X
k
i
_
DiDNi(23)
Correspondingly,
N¼X
k
i
DNi(24)
In particular, when Dreaches D
c
, the accumulated loading cycle
is N
f
, which can be calculated by Eq. (22) to Eq. (24).
Model Tests for Method Validation
The material parameters introduced in Eq. (8) were obtained using
the experimental data reported in existing studies. Fatigue testing of
welded steel plates was carried out to evaluate the fatigue damage
parameters in Eqs. (9)–(18). The distribution of WRS was analyzed
by a thermomechanical analysis. WRS was considered as an initial
loading in the elastic–plastic damage fatigue analysis of weldments
to validate the proposed method.
Parameter Determination of the Elastic–Plastic Fatigue
Damage Model
Fig. 2shows a specimen of an OSD that has U-rib connections with
weld backing bases. The U-rib connections are typically prone to
Fig. 2. The detail of U-rib weld connection with weld backing.
Table 2. Chemical composition of Q345 steel
Chemical Wt. %
C 0.18
Si 0.55
Mn 1.70
P 0.025
S 0.025
V 0.02
Nb 0.015
Ti 0.02
Al (min.) 0.015
Fe 97.45
Table 3. Mechanical and material parameters
Parameter Value
E206 GPa
v0.3
s
y
395 MPa
a
1
236.0
a
2
87.6
a
3
12.3
g
1
323.2
g
2
56.6
g
3
2.3
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fatigue damage. In this study, a U-rib connection specimen made of
Q345 steel was considered. This is a high-strength (nominal yield-
ing stress = 345 MPa), low-alloy structural steel, typically used in
bridges (Zhang et al. 2015). The chemical composition of Q345
steel is listed in Table 2.
According to the cyclic constitutive model reported by Shi et al.
(2011), the mechanical parameters Eand vin Eq. (4) and the mate-
rial hardening parameters
a
k
, and
g
k
of Q345 steel in Eq. (8) can be
obtained using the damage–constitutive relationship from the fa-
tigue test, as listed in Table 3. The elastic–plastic analytical model
of fatigue damage used in this study could be determined by the pa-
rameters gained from fatigue tests of weldment specimens. The pa-
rameters (b
1
,M
0
,c
1
,c
2
, and c
3
) of Q345 steel in Eqs. (10)–(16)
were obtained for high-cycle and low-cycle loading, based on the
nonlinear regression analysis of fatigue results (He et al. 2014;Hu
et al. 2017), as listed in Table 4. The variables Sand dcan be deter-
mined based on regression analysis of low-cycle fatigue test data
reported by Shi et al. (2011), as listed in Table 4.
Fatigue Test
In the Pelikan–Esslinger method (Vlasic et al. 2009), an assumption
is introduced to simplify the test specimen. The OSD is considered
to be continuously supported by diaphragms that are fixed on the
longitudinal web. The fatigue cracks in the butt weld connection of
the U-rib are caused by longitudinal bending moments under exter-
nal loading. The torsion and shear moments have little contribution
to the fatigue damage of the connection weld (Lugger 1995).
Therefore, the U-ribs can be considered to be beams that are sup-
ported elastically on the diaphragms, as illustrated in Fig. 3.The
maximum tensile stress of a section of the U-rib occurs at the bot-
tom surface, where cracks are initiated. Thus, the bottom plate of
Table 4. Elastic-plastic fatigue damage model parameters
Parameter Value
b
1
3.212
M
0
2.063 10
5
c
1
0.02413
c
2
0.008512
c
3
0.0125
S114.9
d1.036
Fig. 3. Mechanical simplification of U-rib weld connection.
Fig. 4. Butt weld specimen and test setup (unit: mm).
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the U-rib weld connection is selected to evaluate the fatigue resist-
ance of the weld connection for the U-rib.
In this study, steel plates that were connected using butt weld
joints were tested under cyclic loading to investigate fatigue dam-
age to weld joints in OSDs. Seven identical butt weldments were
designed to be tested under cyclic loading to validate the elastic–
plastic fatigue damage model. Fig. 4depicts the specimens and test
setup. For each specimen, two steel plates and one welding back
base were welded. All steel plates were 8-mm thick. The cross sec-
tion size A
o
of the specimen was 8 mm 48 mm, as shown in Fig. 4.
Specially, the type of welding was MIGW, which matched that of a
previous study (Qian et al. 2013), to ensure the consistency of the
relaxation parameters of the residual stress.
Uniaxial loading was loaded on the end of steel plate with
bolt connection, as shown in Fig. 4.Table5shows the applied
external cyclic loading F
max
and F
min
and the stress amplitude
(D
s
) of the seven specimens, along with the test results of the
failure loading cycles (N
f
). The stress amplitude was defined as
D
s
=(F
max
−F
min
)/A
o
. Different loadings were applied to S1 to
S5, while S5 to S7 were subjected to the same loading for three
duplications.
Afinite element model for butt weld specimens was established
using ANSYS codes to analyze WRSs. The mesh refinement
scheme for a 3D finite -lement model with eight-node isoperimetric
solid elements is shown in Fig. 5. A minimal element size of
0.5 mm in the vicinity of the welded joint was adopted to meet the
demands of computational accuracy. Because the welding position
of the butt weldment is overhead manual arc welding, only the
x-axis translational DOF of nodes at the side section (x=0,
500 mm) were constrained to simulate the boundary conditions in
the welding process. The temperature-dependent properties of
Q345 are shown in Figs. 6(a and b) (Cui et al. 2017).
The initial WRS distribution of butt weldment was determined
through the thermomechanical analysis, as introduced in Section
2.1. The residual stress relaxation was determined, as introduced in
Section 2.2. Fatigue damage analysis was conducted using Matlab
code and the proposed elastic–plastic damage model in Section 3.2.
In fatigue-damage analysis, the geometry and materials were the
same as those used in the WRS simulation. The finite-element
meshing in the analysis of fatigue damage was the same as that in
WRS analysis, so the residual stress could be used as an input in the
finite-element model. Since the distribution of the damage of the
weldment varied with the number of cycles, fatigue damage was
iteratively solved, using Eqs. (22)–(24).
Results and Discussions
Figs. 7(a and b) show the distributions of WRS along the centerline
(y= 50 mm) of the specimen S1 at the top side (z= 0 mm), in the
longitudinal (xaxis) and transverse (yaxis) directions, respectively.
There were significant WRSs in the vicinity of the weld joint. As
the number of loading cycle increased, the values for WRS were
reduced, indicating a relaxation of the residual stress. Similarly, the
results of WRS on the bottom side (z= 8 mm) of the specimen are
plotted in Figs. 8(a and b).
Fig. 9shows the change in peak WRSs with the loading cycles
for the specimen S1. The relaxation ratio is defined as the absolute
ratio of the change value to the initial value of WRS. “Top_x”and
“Bottom_x”respectively represent the peak WRS at the top (z=
0 mm) and bottom (z= 8 mm) sides in the longitudinal direction;
“Top_y”and “Bottom_y”respectively represent the peak WRS at
the top and bottom sides in the transverse direction.
In the longitudinal direction, the most relaxation of WRSs
occurred in the first loading cycle. On the top side, WRSs were
reduced by 36% in the first loading cycle and by 42% after 100 load-
ing cycles. At the bottom side, WRSs were reduced by 18% in the
first loading cycle and by 20% after 100 loading cycles. WRSs on
the bottom side amounted to about 50% of those on the top side,
possibly due to the use of the backing base.
In the transverse direction, the relaxation ratios of the WRSs on
the top and bottom sides were very close at the same loading cycles,
with differences of 5% or less. WRSs were reduced by 7% in the
first loading cycle and by 20% after 100 loading cycles. These
trends were different from the trends in the relaxation ratio for the
longitudinal direction. This is probably because the external loading
was along the longitudinal direction. Both the stress amplitude and
the stress range in the longitudinal direction were significantly
larger than those in the transverse direction.
Fig. 10 shows the total accumulated fatigue damage, which
increased nonlinearly with the loading cycle, until the total accumu-
lated fatigue damage index reached the level where cracking occurs.
The elastic–plastic fatigue damage model predicted a total of
Table 5. Fatigue test results
Specimen
Cyclic loading (kN)
Constant amplitude stress D
s
app (MPa) Loading Frequency (Hz) N
f
(Number of cycles)F
max
F
min
S1 106.0 10 250 4.5 170,000
S2 98.3 230 5.0 444,700
S3 90.6 210 5.0 462,800
S4 79.1 180 5.0 983,500
S5 67.6 150 5.0 850,600
S6 67.6 150 5.0 581,700
S7 67.6 150 5.5 725,300
Fig. 5. Finite-element model.
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159,990 cycles, which was similar to the test results (170,000
cycles). Three damage levels were selected for visualization of the
fatigue damage process. Fig. 11 shows the development of fatigue
damage in the tested specimen S1. The fatigue damage distributions
in the vicinity (in purple color) of the weld joint at three loading
cycles are plotted. As the loading cycle increased, the fatigue dam-
age was initiated and developed. The results of the numerical sim-
ulation of the location of the fatigue cracks is in agreement with
those of the tested specimen, as shown in Fig. 12. The distribution
of the transverse WRSs at the weld line of the back of the
Fig. 7. Residualstress relaxation at the top of the crosssection perpendicular to the weld line: (a)longitudinal WRS; and (b) transverse WRS.
Fig. 8. Residual stress relaxation at the bottom of the crosssection perpendicular to the weld line: (a) longitudinal WRS; and (b) transverse WRS.
Fig. 6. Temperature-dependent material properties: (a) thermal properties; and (b) mechanical properties.
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crosssection was not symmetrical, because of the effect of weld-
ing direction and boundary condition in the welding process. It is
worth noting that the distribution of damage was not symmetrical
with respect to the center line, due to the discrepancy in the distri-
bution and relaxation of WRSs. The locations of fatigue cracks
were also similar to the locations of the peak stresses of transverse
WRSs, demonstrating that fatigue damage was in direct relation
to the WRSs and that tensile WRSs had a negative effect on fa-
tigue resistance.
Comparisons of failure cycles using testing and numerical simu-
lation are plotted in Fig. 13 with a semilog coordinate system, which
was also used by Do et al. (2015). “Analysis data_1”represents
analyses without consideration of WRS; “Analysis data_2”denotes
the analysis data with consideration of WRS and its relaxation;
“Analysis data_3”represents analyses performed with considera-
tion of WRSs but not including the effects of its relaxation. The
dashed line was used to fit and show the trend for “Analysis
data_2.”
When WRSs are taken into account in fatigue analysis, the rel-
ative error of the predicted results was from 0.5% to 12.9%, rela-
tive to the experimental data. The analysis and testing data of the
seven specimens are plotted in Fig. 13 with a semilog coordinate
system, as was used by Do et al. (2015). The analysis method that
was presented in this study to consider WRS and the residual
stress relaxation provided a reasonable prediction of the loading
cycle for fatigue life. A lack of consideration for WRSs leads to
overestimations of fatigue life. A failure to consider consider the
relaxation of WRSs results in the underestimation of fatigue life.
Fig. 10. Damage accumulation curve of S1 under cyclic loading.
Fig. 11. Damage distribution in S1 at different loading cycles.
Fig. 12. Comparison of crack-initiation location.
Fig. 13. Semilog S-N curves of butt-weld joint.
Fig. 9. The relationships between WRS relaxation ratio and the load-
ing cycles.
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Conclusions
The main conclusions are summarized as follows:
•Different phenomena related to the relaxation of WRSs
occurred in the parallel and perpendicular directions of the
cyclic loading of butt-weld specimens. Overall, the relaxation
of WRS was more significant in the longitudinal than that in
the transverse direction; the relaxation rate of the longitudinal
direction was larger than that of the transverse direction, likely
due to the greater tensile stress magnitude and stress range in
the longitudinal direction. For a butt-weld specimen under
cyclic loading in the longitudinal direction, the relaxation ratio
was up to 42% in the longitudinal direction and 20% in the
transverse direction.
•According to comparisons with the results of fatigue testing,
the proposed method, which considers WRSs and the relaxa-
tion of residual stress, provides reasonable predictions for the
fatigue life of butt-weld connections of U-ribs in OSDs with a
maximum relative error of 12.9%. Fatigue life may be signifi-
cantly overestimated if WRSs are not considered, as well as
being underestimated where the relaxation of WRSs are not
considered.
Acknowledgments
This research was funded by the China National Science Foundation
[Grants 51778533 and 51578455], China Fundamental Research
Funds for the Central Universities [Grant 2682014CX078]. The first
author acknowledges the Chinese Scholarship Council (CSC) [Grant
201607000082] for providing financial support.
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