Most of in-service welded built-up structures, which contain many welded joints as fatigue crack initiation sites, are subjected to many types of loading and these loadings have different axial components with different phases. However, the structural integrities are evaluated according to design codes based on theoretical and experimental investigations under a uniaxial loading condition. Most of these codes are based on the S-N curves approach. On the other hand, authors proposed the numerical simulation method of fatigue crack propagation histories of a cracked plate subjected biaxial loadings with phase difference of each loading component. Fracture mechanics approach is applied to establish our method. In this study, fatigue crack growth behaviour of an out-of-plane gusset welded joint under biaxial loading with two different phase conditions were investigated. The phase difference effect for fatigue crack shape evolution under biaxial loading was confirmed by measured ones. Besides, comparisons of measured crack evolution with the numerical simulation results were performed to validate of our fatigue crack growth simulation for welded joints.

1. Procedia Materials Science 3 (2014) 1536 – 1541
Available online at www.sciencedirect.com
2211-8128 © 2014 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/3.0/).
Selection and peer-review under responsibility of the Norwegian University of Science and Technology (NTNU), Department of Structural Engineering
doi:10.1016/j.mspro.2014.06.248
ScienceDirect
20th European Conference on Fracture (ECF20)
Fatigue crack growth behaviour of an out-of –plane gusset welded
joints under biaxial tensile loadings with different phases
*Koji Gotoha
*, Toshio Niwab
and Yosuke Anaic
a
Department of Marine Systems Engineering Kyushu University, 744 Motooka, Nishi-ku, Fukuoka, 819-0395, Japan
b
Maintenance Research Technology Group, National Maritime Research Institute, 6-38-1, Shinkawa, Mitaka, Tokyo, Japan
c
Structural Analysis and Processing Research Group, National Maritime Research Institute, 6-38-1, Shinkawa, Mitaka, Tokyo, Japan
Abstract
Most of in-service welded built-up structures, which contain many welded joints as fatigue crack initiation sites, are subjected to
many types of loading and these loadings have different axial components with different phases. However, the structural
integrities are evaluated according to design codes based on theoretical and experimental investigations under a uniaxial loading
condition. Most of these codes are based on the S-N curves approach. On the other hand, authors proposed the numerical
simulation method of fatigue crack propagation histories of a cracked plate subjected biaxial loadings with phase difference of
each loading component. Fracture mechanics approach is applied to establish our method. In this study, fatigue crack growth
behaviour of an out-of-plane gusset welded joint under biaxial loading with two different phase conditions were investigated. The
phase difference effect for fatigue crack shape evolution under biaxial loading was confirmed by measured ones. Besides,
comparisons of measured crack evolution with the numerical simulation results were performed to validate of our fatigue crack
growth simulation for welded joints.
© 2014 The Authors. Published by Elsevier Ltd.
Selection and peer-review under responsibility of the Norwegian University of Science and Technology (NTNU), Department of
Structural Engineering.
Keywords: Fatigue; Biaxial loading with phase difference; Numerical simulation of fatigue crack growth; RPG stress; Out-of-plane gusset welded
joints
* Corresponding author. Tel.:+81-92-802-3457; fax:+81-92-802-3368.
E-mail address: gotoh@nams.kyushu-u.ac.jp
© 2014 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/3.0/).
Selection and peer-review under responsibility of the Norwegian University of Science and Technology (NTNU), Department
of Structural Engineering

2. 1537Koji Gotoh et al. / Procedia Materials Science 3 (2014) 1536 – 1541
1. Introduction
In general, most of in-service welded structures and vehicles are subjected to many types of loading which
contain different axial components with different phases. However, the structural integrity of most working
structures and vehicles are assessed by the design codes of each structure, which are based on theoretical and
experimental investigations under uniaxial loading conditions from a practical point of view.
Fatigue strength under biaxial loading has been investigated via the S–N curves approach and the fracture
mechanics approach. The researches based on the fracture mechanics approach (Hoshide et al. 1981, Yuuki et al.
1984 1988, Brown and Miller 1985) could not reach a definite conclusion regarding the effect of biaxial loading on
fatigue crack growth behavior because of conflicting results. The researches based on the S–N curves (Sonsino 1995,
Takahashi et al. 1997, 1999) merely gave measured S-N curves for individual loading conditions. Hence, it is very
important to establish a quantitative evaluation procedure for fatigue crack growth under biaxial loading conditions.
On the other hand, we have been using our developed numerical simulation code of fatigue crack propagation to
evaluate the fatigue crack growth under arbitrary loading histories. This code enables to consider the fatigue crack
opening / closing phenomena caused by crack wake over crack surfaces and implements the Re-tensile Plastic zone
Generating (RPG) stress criterion (Toyosada et al. 2004a) as the fatigue crack propagation law.
The validity of the estimated fatigue crack propagation histories for the following objects were confirmed by
comparing measured results.
A cracked plate under uniaxial variable loadings (Toyosada et al. 2004a), including
superimposed loadings with two different frequencies (Gotoh et al. 2012)
Welded joints under uniaxial variable loadings (Toyosada et al. 2004b, Nagata et al. 2009)
A cracked plate under biaxial loadings with phase difference (Gotoh et al. 2013)
The aim in this study is to verify the applicability of our numerical approach to estimate the fatigue crack growth
histories of welded joints under biaxial loadings with phase difference from a practical point of view.
We performed fatigue crack propagation tests using the out-of-plane gusset welded joints under biaxial loading
with two different phase conditions, highlighting the effect of the phase difference under biaxial loading on the
fatigue crack growth behavior. Applicability of the numerical simulation method of fatigue crack growth based on
the RPG criterion under biaxial loading with different loading phases is
verified by comparing the estimated fatigue crack growth histories with
measured ones.
2. Specimen configuration and loading condition
Cruciform-shaped out-of-plane gusset welded joints specimens shown
in Fig.1 were prepared for fatigue crack propagation tests under biaxial
loading conditions. All the specimens were formed from the same mild
steel plate (ClassNK grade KA). The chemical composition and
mechanical properties of the tested steel are listed in Table 1. In
general, the fatigue crack in out-of-plane gusset joints initiates at toes
of boxing fillet weld. Because of restricting the fatigue crack initiation
site, three toe regions of the boxing fillet shown in Fig.1 were
processed by grinding to relieve the stress concentration. The fatigue
crack in both specimens initiated from the as welded boxing fillet weld
toe.
We performed fatigue crack propagation tests under biaxial loading
conditions using a testing system that consisted of four independent
servo loading actuators that enabled controlled variable loading under
different phases, see Fig.2.
The applied loading conditions for each specimen are listed in
Table 2. Because our study focused on the effects of the phase difference between the applied stresses in the two
perpendicular directions, the stress amplitudes in both directions were kept equal.
500
10
as welded
toe dressed
by grinder
100
100100
z
x
Rolling direction
500
75757575
75 75 75
120
5050
120
400
200
200
80R120R
500500
400
160
Rolling direction
Stiffener
y
x
Fig.1 Configuration of the cruciform-shaped out-
of-plane gusset welded joint.

3. 1538 Koji Gotoh et al. / Procedia Materials Science 3 (2014) 1536 – 1541
(b) Enlarged view near the
crack initiation site
(a) Overview of fracture surface
Fig.3 Observed fracture surface (Specimen ID: G-1)
Table 1 Mechanical properties and Chemical composition of
tested material (ClassNK grade KA).
Mechanical properties
Yield strength
[MPa]
Tensile strength
[MPa]
Elongation
[%]
298 439 30
Chemical composition [%]
C Si Mn P S
0.14 0.23 0.86 0.018 0.005
Table 2 Loading conditions.
3. Results of fatigue crack propagation tests
Figures 3 and 4 shows observed fractured surface of each specimen.
In case of specimen G-1, plurality of fatigue cracks occur near the as welded boxing fillet weld regions. These
cracks grew while repeating coalescence and became a large single surface crack. Continue, this surface crack
penetrated the base plate thickness. Followed by the penetration, a fatigue crack continue to propagate as a through
thickness crack. In case of specimen G-2, the evolution process of plural surface fatigue cracks was the same to
specimen G-1. It is, however, two different fatigue cracks propagate through different cross section of base plate.
The step of fracture surface was confirmed from Fig.4.
Figure 5 shows measured aspect ratio evolutions of each surface crack in the specimens. Aspect ratios were
identified by approximating the elliptical shape of a surface crack. Measurement was performed with respect to the
single surface crack after coalescence. It is confirmed from Fig.5 that the shape evolution of a surface crack is
affected by the phase difference of biaxial loading.
Specimen ID
Stress ratio:
R = x
max
/ x
min
Stress
amplitude:
[MPa]
x / y
Phase difference:
G-1
0.05 110 1
0
G-2
Fig.2 Set-up of fatigue test under biaxial tensile loading.
Fig.4 Observed fracture surface (Specimen ID: G-2)
(b) Enlarged view near the
crack initiation site
(a) Overview of fracture surface

4. 1539Koji Gotoh et al. / Procedia Materials Science 3 (2014) 1536 – 1541
Figures 6 (a) and (b) shows measured fatigue crack growth histories. It is also confirmed that fatigue crack
growth rate under biaxial loading is affected by the phase difference of loading components. Fatigue crack in
specimen G-2 with phase difference of biaxial loading component grew faster than specimen G-1 with zero phase
difference. This tendency related to the phase difference effect is the same as the measured results at the center
cracked cruciform shaped specimen (Gotoh et al. 2013).
4. Numerical simulation of fatigue crack growth
4.1. Overview of the Equivalent Distributed Stress method
We performed the numerical simulation of the fatigue crack propagation using our numerical simulation code
which models the fatigue crack opening/closing behavior and estimates the relationship between the fatigue crack
shape, e.g. the depth or the width of a planer shape defect, and the number of applied loading cycles. Details of the
theoretical background for our code can be found in reference (Toyosada et al. 2004a). On the other hand, our
numerical simulation code of fatigue crack propagation cannot be applied directly to arbitrary geometrical-shaped
joints with a crack because explicit expressions of the weight function of the stress intensity factor for such cracked
joints have not yet been established. An alternative method, the Equivalent Distributed Stress (EDS) method
(Toyosada et al. 2004b) for estimating the fatigue crack propagation histories was proposed. Calculation flow by
applying EDS method is as follows.
1) Obtaining the relationship between the fatigue crack length (a) and stress intensity factor (K).
2) Transformation of a-K relationship into the EDS as a function of crack length. EDS enables the stress
field near the crack tip of planar cracks in an infinite wide plate with a through thickness crack to
reproduce. EDS must satisfy the following equation,
(1).
Where,
EDS(x): EDS at crack length x and
Kobject(a): Stress intensity factor at crack
length a.
3) Performing the numerical simulation of fatigue crack growth for an infinite wide plate with a through
thickness crack the condition that EDS is distributed along a crack line
0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
2b
a
10mm
Aspectratio:a/b
Crack depth: a (mm)
G-1 ( = 0)
G-2 ( )
Fig.5 Measured aspect ratio evolutions
of each defect.
0 1 2 3
0
20
40
60
80
100
2b1
2b2
a
Crackdepth:a,Halfcracklength:b[mm]
Number of cycles: N [ x 10
6
cycle ]
Crack depth: a
Crack length: b1
Crack length: b2
Specimen G-1 ( =0
(a) Specimen G-1
0 1 2 3
0
20
40
60
80
100
2b1
2b2
a
Crackdepth:a,Halfcracklength:b[mm]
Number of cycles: N [ x 10
6
cycle ]
Crack depth: a
Crack length: b1
Crack length: b2
Specimen G-2 (
(b) Specimen G-2
Fig.6 Measured fatigue crack growth histories.
2
0
2 1
a
EDS objecta x x a dx K a

5. 1540 Koji Gotoh et al. / Procedia Materials Science 3 (2014) 1536 – 1541
Before applying the EDS method to estimate the fatigue crack
propagation in the welded joints to be evaluated, the relationship
between the fatigue crack shape and stress intensity factor must be
determined. The superposition procedure for a cracked body,
which is illustrated in Fig. 7, were applied to obtain SIF in this
research. Detailed explanation for these SIF estimation procedure
including the consideration method of the case where stress is
distributed in the crack width direction was introduced in the
references (Toyosada et al., 2004b, Nagata et al. 2009).
Approximated conversion method of biaxial loading condition to
uniaxial loading to the normal direction to the crack surface
(Gotoh et al., 2013) is applied for the SIF calculations. That is, the
biaxial loading effect is incorporated in the SIF evolutions.
Commercial FE software MSC Nastran 2012 was applied to
calculate the stress distribution for no cracked specimen under
external applied biaxial loading conditions. Inherent stress method
(Matsuoka and Yoshii, 1998) for the out-of-plane gusset joint was
applied to calculate the residual stress distribution for no cracked
specimen.
Figure 8 shows the relation of the reference crack length (X)
(Toyosada et al., 2004b) defined by Eq. (2) and the stress intensity factor (K) of specimen G-1 as an example.
Several weld toe radus conditions were set to estimate SIFs under external loading conditions because the exact
value of the toe radii for each specimen had not been measured. Figure 9 shows the EDS distributions derived from
Fig.8. Definition of the reference crack length is shown in Eq. (2). EDS of specimen G-2 was obtained according to
the same procedure for specimen G-1.
(2)
4.2. Numerical simulation of fatigue crack growth
Fig.10 shows a comparison between the fatigue crack growth histories obtained by the numerical simulation and
the measured ones. Note the horizontal axis (Number of cycles) is drawn by the normal scale, not logarithmic one.
fora surface crack
for a through thickness crack ,
where
:crack depth, :plate thickness, :half crack length and
: at crack penetrating through thickness.
penetration
penetration
a
X
c c t
a t c
c c
0 5 10 15
0
5
10
15
Reference crack length X [mm]
StressintensityfactorK[N/mm
1.5
]
: Radius of weld toe
Under unit external loading:
x= y=1 N/mm
2
Specimen G-1
: = 0.2mm
: = 0.3mm
: = 0.7mm
: = 1.0mm
0 5 10 15
0
500
1000
1500
2000
2500
Reference crack length X [mm]
StressintensityfactorK[N/mm
1.5
]
Under residual stress field
Specimen G-1
Fig.8 Relationship between Stress intensity factor (K) and
Reference crack length (X) for specimen G-1.
0 5 10 15
0
1
2
3
4
5
6
7
8
9
10
0 0.5 1
0
1
2
3
4
5
6
7
8
9
10
Reference cracklenght: X [mm]
EquivalentdistributedstressEDS[N/mm
2
]
Reference crack lenght: X [mm]
EquivalentdistributedstressEDS[N/mm
2
]
Specimen G-1
Under unit external
loading:
x= y=1 N/mm
2
: = 0.2mm
: = 0.3mm
: = 0.7mm
: = 1.0mm
0 5 10 15
0
100
200
300
400
500
Reference crack lenght: X [mm]
EquivalentdistributedstressEDS[N/mm
2
]
Specimen G-1
Under residual stress field
Fig.9 Relationship between Equivalent Distributed Stress ( EDS) and
Reference crack length (X) for specimen G-1.
(a) (b) (c) (d)
(a) Surface crack in stress concentration field.
(b) Surface crack in a plate subjected to tensile loading
and out-of bending.
(c) Through thickness edge crack in stress concentration
field.
(d) Through thickness edge crack in plate subjected to
tensile loading and out-of bending.
Fig.7 Approximate method to estimate the SIF at the
deepest point of a surface crack in an arbitrary stress
distributed field.

6. 1541Koji Gotoh et al. / Procedia Materials Science 3 (2014) 1536 – 1541
Considering the ambiguity in the SIF estimation procedure shown in Fig.7, we could conclude that the numerical
simulation method proposed in this paper could give fair estimation result.
It is also confirmed from Fig.10 that the fatigue life is strongly affected the phase difference of biaxial loading
and that the effect on the fatigue crack growth through the total life of weld toe radius could be negligible.
5. Concluding remarks.
Fatigue tests for the out-of-plane gusset joint under biaxial loading
with two different phase conditions were performed. It is confirmed that
the shape evolution of surface fatigue is affected by the phase difference
of the biaxial loading. A numerical simulation method of fatigue crack
growth for welded joints under biaxial load with different loading phases,
which is based on the Equivalent Distributed Stress (EDS) method, is
proposed and the method was validated by comparing the estimated
fatigue crack growth histories with the measured ones.
Acknowledgements
This research was funded by a Grant-in-Aid for Young Scientists (S)
(No. 21676007) from the Japan Society for the Promotion of Science.
The authors would like to express their appreciation to Nippon Steel &
Sumitomo Metal Corporation for supplying the steel plates for the
specimens. Mr. Kouki Sugino (Master course student, Kyushu
University) and Mr. Yoshihisa Tanaka, (Maintenance Research
Technology Group, National Maritime Research Institute, Japan)
contributed to the numerical and experimental works for this research.
References
Brown, M.W. and Miller, K.J., 1985. Mode I Fatigue Crack Growth under Biaxial Stress at
Room and Elevated Temperature, ASTM STP 853, 135-152.
Gotoh, K., Matsuda, K. and Kitamura, O, 2012. Numerical Simulation of Fatigue Crack
Propagation under Superposed Loading Histories with Two Different Frequencies, Proc.
of Hydroelasticity in Marine Technology, 287-297.
Gotoh, K., Niwa, T., Anai, Y. Omori, T., Tanaka, Y. and Murakami, K., 2013. Fatigue Crack
Propagation under Biaxial Tensile Loading, Proc. of OMAE 2013, OMAE 2013-10980.
Hoshide, T., Tanaka, K. and Yamada, A. 1981. Stress-Ratio Effect of Fatigue Crack Propagation in a Biaxial Stress Field, Fatigue Eng. Mater.
Struc. , 4, 4, 355-366.
Matsuoka, K. and Yoshii, T., 1998. Weld Residual Stress in Corner Boxing Joints, ClassNK technical bulletin, 16, 1-10.
Nagata, Y., Gotoh, K. and Toyosada, M., 2009. Numerical Simulations of Fatigue Crack Initiation and Propagation Based on Re-tensile Plastic
Zone Generating (RPG) Load Criterion for In-plane Gusset Welded Joints, J. Marine Sci. Tech., 14, 1, 104-114.
Sonsino, C.M., 1995, Multiaxial fatigue of welded joints under in-phase and out-of-phase local strains and stresses, Int. J. Fatigue, 17, 1, 55-70.
Takahashi, I., Takada, A., Akiyama, S., Ushijima, M. and Maenaka, H., 1998. Fatigue behavior of box welded joint under biaxial cyclic loads, J
SNAJ, 184, 321-327, (in Japanese).
Takahashi, I., Ushijima, M., Takada, A., Akiyama, S. and Maenaka, H., 1999. Fatigue behavior of a box-welded joint under biaxial cyclic loads,
Fatigue Eng. Mat. Struc., 22, 10, 869-877.
Toyosada, M., Gotoh, K. and Niwa, T., 2004a. Fatigue crack propagation for a through thickness crack, Int. J. Fatigue, 26, 9, 983-992.
Toyosada, M., Gotoh, K. and Niwa, T., 2004b. Fatigue life assessment for welded structures without initial defects, Int. J. Fatigue, 26, 9, 993-
1002.
Yuuki, R., Kitagawa, H., Tohgo, K. and Tanabe, M., 1984. Effect of Bi-Axial Stress on Fatigue Crack Growth, Mat. Sci. Research, 33, 373, 1271-
1277, (in Japanese)
Yuuki, R., Akita, K. and Kishi, N., 1988. Effect of Biaxial Stress Condition and its Change on Fatigue Crack Growth Properties, Mat. Sci.
Research, 37, 420, 1084-1089, (in Japanese).
0 1 2 3
0
20
40
60
80
100
Referencecracklength:X[mm]
Number of cycles: N [ x10
6
cycles]
: Before penetration
: After penetration
Specimen G-1
(Phase difference : =0)
Symbols: measurement
Curves: Numerical simulation
Crack penetrates
base plate
: = 0.2mm
: = 0.3mm
: = 0.7mm
: = 1.0mm
0 2 4 6 8
0
20
40
60
80
100
Referencecracklength:X[mm]
Number of cycles: N [ x 10
5
cycles]
: Before penetration
: After peneteration
Specimen G-2
(Phase difference:
: = 0.2mm
: = 0.3mm
: = 0.7mm
: = 1.0mm
Curves: Numerical simulation
Symbols: Measurement
Crack penetrates
base metal.
Fig.9 Comparison between estimated fatigue
crack growth histories and measured
values.