Ocean Engineering 281 (2023) 114676

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Ocean Engineering
journal homepage: www.elsevier.com/locate/oceaneng

Effect of repair welding on the fatigue behavior of AA6082-T6 T-joints in

marine structures based on FFS and experiments
Zhihao Chen a, Yong Liu b, *, Hongliang Qian a, b, Ping Wang b, Yizhou Liu b
a
School of Civil Engineering, Harbin Institute of Technology, Harbin, 150001, China
b
School of Ocean Engineering, Harbin Institute of Technology at Weihai, Weihai, 264209, China

A R T I C L E I N F O A B S T R A C T

Handling Editor: Prof. A.I. Incecik In this study, the effect of repair welding on the fatigue behavior of AA6082-T6 welded T-joints in marine
structures was investigated based on experiments and the fitness-for-service (FFS) assessment in BS7910. Repair
Keywords: welding led to an increase in residual stress and a decline in fatigue life according to fatigue tests and FEA.
Fatigue behavior Meanwhile, repair welding changed the crack initiation site (CIS) and crack shape factor (CSF, the ratio of crack
Repaired aluminum T-joints
depth a to length 2c; i.e., a/2c), as indicated by fatigue fractography. In addition, when FFS assessment was
Fitness-for-service
performed, two methods for characterizing the CSF under different repair welding lengths during fatigue crack
Traction structural stress
Residual stress growth were presented: fixed and non-fixed a/2c. The results showed that repair welding led to a transition from
ductile fracture to brittle fracture. For a fixed CSF, the critical crack depth and crack opening area of T-joints with
repair welding lengths of 30 mm were greatly reduced by 63–66% and 86–89%, respectively, compared to T-
joints without repair welding. However, the corresponding reductions for the non-fixed a/2c method were
approximately 40% and 90%. The comparison indicated the results obtained with a non-fixed CSF were more
consistent with the experiments, with a deviation below 15%.

welded structures with defects, the rejection, down rating, or repairs

1. Introduction based on FFS assessment and experience with material, stress, and
environmental combinations also need to be considered (BS 7910,
AA6082-T6 aluminum alloy has the advantages of low density, high 2015). Repair welding is an excellent method for prolonging the service
strength, and good corrosion resistance. Therefore, this alloy has been life of failed structures that are recommended for repair. However, due
increasingly and widely used in the lightweight design of naval archi­ to the great influence of repair welding on failed structures, it is
tecture and marine engineering applications (Farajkhah and Liu, 2016; particularly significant to conduct FFS assessment on these structures
Hosseinabadi and Khedmati, 2021). Double-sided welded T-joints are after repair welding.
one of the most common joint forms, as shown in Fig. 1. For T-joints in The increasing interest in welded structures in service has increased
service, fatigue cracks occur at the weld toe due to the stress concen­ the need for reasonable and accurate safety analysis. Lu et al. (2021)
tration caused by geometric discontinuities. This does not necessarily provided a novel defect model of incomplete penetration defects at the
lead to component failure, because it cannot meet the economic and root of girth weld pipelines based on a stress function method. This
environmental protection requirements in actual production. In view of model was highly suitable for evaluating the residual strength of a
this principle, structural failure can only be judged when cracks exceed a submarine X80 pipeline well. Chen et al. (2015) safely and economically
specified size (Jacquemin et al., 2018; Revie, 2015; Gordon, 1993). addressed two critical thresholds (when to repair and replace) of com­
Onyegiri and Kashtalyan (2019) evaluated the maximum allowable posite structures in aircraft, and they evaluated the operational lifecycle
defect size for a swaged weld under load based on fracture mechanics of the structural strength considering impact damage. Popov and Kor­
methodologies. Wang et al. (2020) calculated the critical fracture length zunin (2009) estimated whether aboveground steel pipelines need to be
in a weld, HAZ, and base metal according to DIC technology and FAD repaired or not based on their stress state. Oh et al. (2007) calculated the
evaluation models. Reasonable methods for failure assessment should remaining life of a bridge deck based on a damage model, taking the
satisfy economic requirements and meet fitness-for-service (FFS) re­ traffic loads and environmental effects into consideration. Most of these
quirements (Jang et al., 2005, Simandjuntak and Shibli, 2011). For works performed FFS assessment according to the residual strength,

* Corresponding author.
E-mail addresses: hitliuy@hit.edu.cn (Y. Liu), qianhl@hit.edu.cn (H. Qian), nancywang@hit.edu.cn (P. Wang), 1524917963@qq.com (Y. Liu).

https://doi.org/10.1016/j.oceaneng.2023.114676
Received 28 December 2022; Received in revised form 13 April 2023; Accepted 23 April 2023
Available online 6 May 2023
0029-8018/© 2023 Elsevier Ltd. All rights reserved.
Z. Chen et al. Ocean Engineering 281 (2023) 114676

Nomenclature FADs Failure assessment diagrams

σ m, Pm Membrane stress due to external loads
Lr Load ratio α’’ The function of a, c, B, and W used in the calculation of
σref Reference stress used for creep and plastic consideration collapse stresses
σb, Pb Bending stress due to external loads KS1 Stress intensity factor at the current crack size due to
KP1 Stress intensity factor at the current crack size due to secondary stress
primary stress Kmat Material fracture toughness
V Plastic correction factor (Yσ)P Primary stress intensity correction function
Kr Fracture ratio μ, N Parameter used in constructing FAD line
(Yσ)S Secondary stress intensity correction function a Crack depth before repair welding
Lr,max Permitted limit of Lr t The thickness of base metal
σs Traction structural stress F External load
σn Nominal stress S The area of the grip end of test specimens
σR Residual stress Q Secondary stress
Z The measure of position through the thickness Qb Bending stress due to welding action
Qm Membrane stress due to welding action W Weld width
B Section thickness in-plane of flaw ar Crack depth after repair welding
c Half flaw length for surface or embedded flaw, a/c = 0.8 ar/a The ratio of crack depth
σu Tensile strength S Crack opening area
σs Traction structural stress without considering angular σ sm Corrected traction structural stress
misalignment t Base metal thickness
kα Angular misalignment-induced SCF α Angular misalignment
L The total length of a test specimen FFS Fitness-for-service
Lc Position of critical location CIS Crack initiation site
FEA Finite element analysis

Fig. 1. Partial welded structures in ships (Lillemäe et al., 2017; Qiu et al., 2021) (a) Superstructure of luxury yacht (b) Reinforced structure (Zhang et al., 2019).

probabilistic approach (Jung et al., 2018), stress state, and remaining cracks in the staggered weld of a gas pipeline according to the FADs in
life. However, they rarely considered the critical crack dimension in API-579. Yang et al. (2008) calculated the critical defect size of the
failed welded structures. buckling region of damaged X65 pipeline based on the FFS assessment
In addition, countries around the world have customized corre­ procedure in ASME. It is worth noting that little attention has been paid
sponding standards according to FFS, and these standards have been to the evolution law of weld defect size (depth a, length 2c) under fa­
widely used by researchers. For example: the American Society of Me­ tigue load and the relationship of a/2c.
chanical Engineers (ASME) by Liu et al. (2021) and Mousavi and Mog­ The repair welding of welded structures leads to the deterioration of
haddam (2020); the Structure Integrity Assessment Procedure (SINTAP) their microstructure (Hu et al., 2022; Moreno et al., 2022); the decline of
by Jeyakumar and Christopher (2013) and Zerbst (2009); the French partial mechanical property at the repair welding zone, such as: fatigue
design code (RCC-MR) by Ma et al. (2021) and Moon (2020); British strength (Seo et al., 2021), fracture toughness (Moarrari et al., 2020),
standard BS7910 by Chen et al. (2022). Among these standards, failure stress corrosion cracking sensitivity (Luo et al., 2020), and the redistri­
assessment diagrams (FADs) are generally used to evaluate the safety bution of residual stress (Amadeus et al., 2022; Chen et al., 2022) have
and reliability of cracked structures by adopting the principle of double been investigated. Of course, the residual stress is also affected by the
criterion and considering the plastic and brittle failure of the structure. proximity weld. Bhardwaj et al. (2022a, 2022b) studied the effect of
Radu et al. (2018) determined the defect types and critical failure size of welding procedures and welding distances on the residual stress at the
a cylindrical steel bridge structure with welded joints based on the FADs proximity region between two welds of S355 structural steel pipe. They
in BS7910. Jacquemin et al. (2018) obtained the acceptable size of reported a significant difference in Charpy and hardness values, and the

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Z. Chen et al. Ocean Engineering 281 (2023) 114676

Table 1 mechanical properties of these materials are shown in Tables 1 and 2,

Chemical compositions of T-joints. respectively.
Material Si Mg Mn Fe Zn Cr Ti Al With regard to the dimensions of the test specimens, the width of
each test specimen was the most important parameter because width
AA6082-T6 1.08 0.50 0.54 0.14 0.02 0.06 0.02 Bal.
ER5356 0.08 5.26 0.12 0.19 0.01 0.1 0.08 Bal. greatly affects the distribution of the residual stress field. For medium
and thick plates, the plate width should be about 10 times the plate
thickness, and for test specimens with a plate thickness exceeding 16
mm, the plate width should be 6 times greater than the plate thickness,
Table 2
Mechanical properties of T-joints.
as verified by our previous study (Chen et al., 2022). The plate width and
thickness in this study were 110 and 12 mm, respectively. The length of
Material Yield strength Tensile strength Elongation
test specimens was reflected in the calculation process of traction
Rel/MPa Rm/MPa e/% structural stress. In this study, the specimen length was 300 mm. Fig. 2
AA6082-T6 240 280 10 shows the preparation process of the test specimens. The original welded
ER5356 120–190 250–300 15–25 T-plates were obtained by double-sided melt inert-gas (MIG) welding
aluminum alloy plates with sizes of 350 × 150 × 12 and 350 × 300 × 12
(length × width × thickness, mm). These plates were cut using a wire
residual axial stresses at a 5-mm proximity distance were increased
cutting machine to fabricate 110 mm wide test specimens. The 10 mm
beyond the yield strength of the steel. Regarding the research on welded
long welds at both ends of the T-plates were removed because the po­
T-joints with repair welding based on FADs, Chen et al. (2019) used
sitions where the weld started and stopped contained many welding
FADs to study the fracture of steel pipeline, finding that the repair
defects.
process significantly influenced the safety margin of strain damage due
The same welding procedure specification (WPS) was adopted to
to more significant residual stress. Heldt et al. (2014) conducted a full
prepare the original and repair welded T-joints and for the imple­
structural weld overlay repair of a dissimilar metal weld and found the
mentation of subsequent tests. Before the T-joints were welded, the
acceptable level of defects in the weld layer based on FADs. However,
rectangular panels were heated at 175 ◦ C for 5 h to eliminate the binding
the effect of repair welding size on the FFS assessment of T-welded joints
stress and ensure that no deformation was present (EI-Danaf et al.,
based on FADs has rarely been studied.
2013). The residual stress was retained in the post-weld T-plates, and
Therefore, in this study, the effect of repair welding on the fatigue
this was used to study the effect of repair welding on FFS assessment.
behavior of T-joints was investigated based on experimental tests and
FFS assessment. Fatigue tests of T-joints with and without repair welding
were first conducted, which determined the input parameters of the
BS7910 assessment: crack initiation site, direction, and shape factor a/
2c. Then, the effect of repair welding length on the FFS assessment of T-
joints was studied, with two methods adopted to characterize the rela­
tionship between cracking depth a and length 2c during fatigue crack
growth. Finally, these two methods were analyzed from the perspectives
of critical crack depth, length, and crack opening area, and they were
compared with the experimental tests.

2. Materials and methods

2.1. Preparation of test specimens

The base metal and weld wire materials used in this study were
AA6082-T6 aluminum alloy (Singh et al., 2017) and ER5356 aluminum
Fig. 3. Macroscopic observation of T-joints after repair welding.
alloy (Chen et al., 2019), respectively. The chemical compositions and

Fig. 2. Preparation process of original and repair welded T-joint specimens.

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Z. Chen et al. Ocean Engineering 281 (2023) 114676

Table 3
Welding parameters for the original and repair welding process.
Welding parameter Original welding (MIG welding) Repair welding (TIG welding)

Pass 1 Pass 2 Pass 3 Pass 4 Pass 5 Pass 1 Pass 2

Wire diameter mm 1.2 1.2 1.2 1.2 1.2 1.2 1.2

Avg. current A 220 220 220 220 210 280 265
Avg. voltage V 26 26 26 26 24 – –
Avg. travel speed mm/min 31 26 27 26 18 58 59
Wire feed speed mm/min 60 50 50 50 35 105 105
Inter. pass temp. ◦
C 100 100 100 100 100 100 100

Fig. 4. Flow chart of FFS assessment based on BS7910 (a) Flow chart of FFS assessment (b) Level 3A failure assessment diagram.

The weld material was removed by machining before repair welding, main input parameters in the FFS assessment: equivalent characteriza­
and the weld dimensions were measured by vernier calipers. The repair tion of welding defects, primary stress P, secondary stress Q, and fracture
weld (made by removing the existing welding layers and making new toughness KIC. The crack equivalent characterization can be determined
layers, see Fig. 2) was made in the mid-width of the original weld according to the crack initiation location and geometric features. The
without undergoing a fatigue load, and test specimens with a repair primary stress P caused by the external loads includes Pm (membrane
welding length of approximately 55 mm were obtained by stress) and Pb (bending stress); the secondary stress Q composed of Qm
tungsten-inert gas (TIG) welding (TIG welding is a better welding (membrane stress) and Qb (bending stress) is related to the residual
method for repair welds with short lengths). Pre-weld heat treatment stress generated by welding action; the fracture toughness KIC is the
and repair welding positions in this study were consistent with the works criterion for material fracture, reflecting the material resistance to crack
of Arnavaz et al. (2022), Nikfam et al. (2019), and Ghafoori-Ahangar instability propagation. This study mainly investigated the effect of
and Verreman (2019). The repair welding width and depth of the test primary stress P and secondary stress Q on the critical crack depth. The
specimens are shown in Fig. 3. Macroscopic observation of the T-joints influence of repair welding on fracture toughness was ignored in this
shows that the test specimens had one-layer and two-pass repair welds. work. The KIC value of T-joints with and without repair welding was set
The specific process parameters of each pass are presented in Table 3. to a fixed value of 36.3 MPa m0.5 (Nowotnik et al., 2006; Bergant et al.,
Test specimens were welded by qualified WPS and welders, and they 2016).
were determined to be free of defects after welding by visual inspection The abscissa and ordinate in the FADs are the load ratio Lr and
and ultrasonic flaw detection. Therefore, defects were caused by fatigue fracture ratio Kr, respectively. The load ratio Lr represents the driving
crack propagation during the subsequent fatigue tests. force of the plastic instability of structures, which is related to the yield
strength σ Y and primary stress P; the fracture ratio Kr is the driving force
of the brittle fracture of structures, which is related to fracture toughness
2.2. FFS assessment method KIC and secondary stress Q. The value of Lr can be calculated according
to Eqs. (1)–(3).
The FFS assessment method in BS 7910 (2015) was mainly used for
σref
service life judgement and the maximum defect size determination of Lr = (1)
σY
the welded structures based on fracture mechanics. In practical appli­
cations, FFS assessment can determine the rejection, down rating, ( )0.5
and/or repairs of welded structures. Thus, FFS can reduce costs by Pb + P2b + 9P2m (1 − α′′ )2
σ ref = (2)
eliminating unnecessary repairs. FFS assessment can also be used to 3(1 − α′′ )2
make run-repair-replace decisions to help determine if components in
a/
welded structures containing flaws (that have been identified by in­ α′′ = (B ), for W ≥ 2(c + B) (3)
spection) can continue to operate safely for some period of time. 1 + B/
c
The assessment method can be divided into three levels. Among
these, the level 3A failure assessment diagram suitable for evaluating where σref represents the reference stress used for creep and plastic
plastic materials was used in this study. As shown in Fig. 4, there are four consideration; a, B, c, and W are the crack depth, section thickness in-

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Z. Chen et al. Ocean Engineering 281 (2023) 114676

Fig. 5. Temperature-dependent thermal and mechanical parameters of AA6082-T6 and ER5356 aluminum alloy. (a) AA6082-T6 aluminum alloy (b) ER5356
aluminum alloy.

plane of the flaw, half-flaw length of a surface or embedded flaw, and ⎧

weld width, respectively. ⎨ ( 1 + 0.5L2 )− 0.5 [ ( )]
0.3 + 0.7 exp − μL6r , forLr ≤ 1;
f (Lr ) = r
(10)
According to the plasticity correction factor V, the stress intensity ⎩ f (Lr ) = f (1)L(N− 1)/(2N)
, for1 < Lr ≤ Lr,max ;
factors KP1 and KS1 at the current crack size are related to primary stress
r

and secondary stress, respectively. The value of Kr can be obtained with

f(Lr) = 0 when Lr > Lr,max, where Lr,max represents the permitted limit of
Eqs. (4)–(6).
Lr. Lr is the value of (σY + σ u)/2σ Y. μ and N are parameters used in
K1P + VK1s constructing the FADs line, and both parameters are related to the yield
Kr = (4) strength σY, as shown below:
Kmat
⎧ [ ( ) ] μ = min(0.001E∕σ Y , 0.6) (11)
⎨ min 1 + 0.2Lr + 0.02K s Lr (1 + 2Lr ); (3.1 − 2Lr ) ,
⎪
V=
1
K1p
for Lr < 1.05 ( )
σY
⎪
⎩ N = 0.3 1 − (12)
1, for L ≥ 1.05 σu
(5)
The value of (Lr, Kr) can be calculated according to the input of the
{ √̅̅̅̅̅ four parameters discussed above. If the positions of (Lr, Kr) are within
K1p = (Y σ )p πa
√̅̅̅̅̅ (6) the FAC curve, this means that the test specimen will not fracture;
K1s = (Y σ)s πa otherwise, the specimen will fail.

where (Yσ )p and (Yσ )s represent the primary and secondary stress in­
2.3. Welding simulation of T-joints
tensity correction functions, respectively. These functions are defined
as:
2.3.1. Governing equations
(Υ σ )p = MfW {ktm Mkm Mm Pm + ktb Mkb Mb [Pb + (km − 1)Pm ]} (7) The welding process of T-joints is very complicated, involving ther­
mal radiation, heat conduction, thermal convection, and mechanical
(Y σ )s = Mm Qm + Mb Qb (8) properties. This welding process results in welding residual stress. In this
study, heat transfer analysis and mechanical calculations were per­
{ [( )( ) ]}0.5 /
πc a 0.5 formed based on a weak coupled temperature-displacement FE model.
fw = sec , which equals1.0 if a 2c = 0 (9) Thermal conduction qcond was used to simulate the heat flow, thermal
W B
convection qconv and radiation qrad were considered to simulate the heat
where fw is the finite width correction factor; M is the ratio of the yield loss in FEA. These terms were calculated using Eqs. (13)-(15).
strength of the weld metal to that of the parent material, which equals Thermal conduction qcond conforms to Fourier’s law:
1.0 for surface flaws in plates; Ktm and Ktb are the membrane and
∂T
bending stress concentration factors, respectively; Mkm, Mm, Mkb, and Mb qcond = − λgradT = − λ n (13)
∂n
are the stress intensity magnification factors (the calculation method for
these factors is given in Appendix A). Thermal convection qconv conforms to Newton’s law:
The FAC curve in FADs is an assessment criterion of the FFS of T- qconv = − hf (Tsur − T0 ) (14)
joints, related to tensile strength σu and Young’s modulus E. FAC curves
can be drawn based on Eq. (10). Thermal radiation qrad follows Stefan-Boltzmann’s law:
[ 4 ]
qrad = − εb σ Tsur − T04 (15)

where Tsur and T0 are the surface temperature of the T-joints and the

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Z. Chen et al. Ocean Engineering 281 (2023) 114676

Fig. 6. Finite element modeling, meshing, and boundary conditions (a) Geometric model, (b) Meshing and (c) Boundary conditions.

ambient temperature, respectively (T0 was 20 ◦ C); λ, hf, and ε represent (Yaghi et al., 2006). To perform welding heat flow analysis more
the heat transfer coefficients of thermal conduction, thermal convection accurately, the prescribed temperature of the weld bead should be about
(237 W/(m2K)), and thermal radiation (0.8), respectively; σ is the 5–10% higher than the melting temperature of the weld wire (Song
Stefan-Boltzmann constant (σ = 5.67 × 10− 8 W m− 2⋅K− 4). The outer et al., 2014). At temperatures higher than the melting temperature, all
surface of the FE models in ABAQUS was used for thermal convection the thermal-mechanical properties of the material were set to be con­
and thermal radiation. stant except the thermal conductivity, which was assumed to be
According to the governing equations of heat transfer, the residual doubled. This was done to account for the enhanced convection asso­
strain was obtained due to the nonlinear and nonuniform temperature ciated with the molten weld pool (Song and Dong, 2016). The
field. In the process of mechanical analysis, the total strain Δεtotal con­ temperature-dependent thermal and mechanical parameters of the base
sists of three parts: elastic strain Δεe, plastic strain Δεp, and thermal metal and welding wire are shown in Fig. 5. (Lai et al., 2013). The linear
strain Δεth, as shown in Eq. (16) (Liu et al., 2021). heat input Q of FE simulation included the heat input Q1 used to bringing
a weld deposit to melting temperature from initial ambient temperature
Δεtatal = Δεe + Δεp + Δεth (16)
and the heat input Q2 involved in maintaining melting temperature of
the weld deposit for duration time thold. To ensure that linear heat input
2.3.2. Finite element modeling Q in FEA was consistent with the WPS, duration time thold can be
The birth-and-death element method is used to simulate the process determined according to the works of Dong (2005) and Song et al.
of solder filling, the instantaneous heat source model has the advantages (2015a, b and 2016).
of better calculation convergence and relatively lower requirements for The modeling procedure was investigated and validated by Dong and
mesh density compared to the distributed heat source model, and this colleagues (Song et al., 2012; Dong and Brust, 2000). Residual stress
model can guarantee computational efficiency during FE simulation. measurement was carried out on the surface of the test specimens using
The combination of a birth-and-death element and the instantaneous X-ray diffraction (McDonald et al., 2002; Hilson et al., 2009), in the
heat source model is suitable for complicated shape weld sections as well interior of the test specimens by neutron diffraction (Song et al., 2014;
as multi-pass and multi-layer welding. In this study, the FE modeling Withers, 2004), and along the through thickness direction using a
procedure for analyzing residual stress was characterized as a sequen­ deep-hole drilling (DHD) technique (Dong et al., 2014; Song et al.,
tially coupled thermal-mechanical analysis using ABAQUS. 2015). The residual stress extracted by the FE modeling method was in
Thermal analysis was performed with element nodes defined within good agreement with the results obtained by these three experimental
the weld bead heated to a prescribed temperature. Each bead in this methods. In the following chapter, residual stress will be further vali­
study was considered to be a pass. Therefore, the number of passes in the dated based on blind hole drilling method.
FE model was equal to the number of beads in the simulated weld. The solidus and liquidus temperatures of the ER5356 aluminum alloy

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Z. Chen et al. Ocean Engineering 281 (2023) 114676

Fig. 7. The penetration shapes of original and repair T-welded joints in FEA.

Fig. 8. Temperature history curves during welding (a) Whole-temperature history curve at weld toe (b) Temperature history curve at different locations during pass
2 of repair welding.

are 582 ◦ C and 638 ◦ C, respectively. Solid phase transition did not occur repair welding. Comparing Figs. 3 and 7 shows that the penetration
in the aluminum alloy. The latent heat of fusion of the aluminum alloy shapes of the original and repair T-welded joints in the FE simulation
was set to 329.1 kJ/kg in FEA to simulate the solid-liquid phase trans­ were basically consistent with the weld bead of the T-joints.
formation (Sun et al., 2007). The inter-pass temperature was about 100 ◦ C in the actual welding
As shown in Fig. 6, to simulate the welding process more accurately, process. Therefore, as shown in Fig. 8(a), each pass was heated to 690 ◦ C
three-layer and five-pass welding on both sides of the original T-joints, in FEA, then held for 2.5 s, and the next weld pass was started after the
and one-layer and two-pass repair welding on the right side of the T- current weld cooled to approximately 100 ◦ C. The last pass was required
joints in the FE models were performed. According to the macroscopic to cool to ambient temperature. According to Fig. 8(b), during pass 2 of
observation shown in Fig. 3, the weld bead profile was simplified to a repair welding, the maximum temperature gradually decreased with
relatively regular shape to facilitate calculation convergence. The layout increasing distance from the repair welding position.
of non-uniform elements was conducted with a weld seam and base In the process of thermal elastic-plastic (TEP) FE analysis, the ma­
metal sizes of 2 mm and 5 mm, respectively. There was a total of 67295 terial elastic behavior conformed to Hook’s law, and the perfectly plastic
elements and 77510 nodes. The boundary condition in FEM was model was adopted in the current welding simulation since the hard­
consistent with the actual fixities of the specimens during original or ening behavior of the AA6082 aluminum alloy was not obvious (Shu

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Z. Chen et al. Ocean Engineering 281 (2023) 114676

Fig. 9. Schematic diagram of FE model boundary conditions and nodal forces at weld toe.

et al., 2014). fatigue properties of joints, as reported by Dong (2005). The traction
The cutting process led to the release of residual stress to a certain structural stress presents the degree of stress concentration at the weld
extent. This was taken into account and modeled based on the birth and toe or root, which can be easy to be determined based on FE software
death elements in ABAQUS when the effect of the repair welding length Fe-safe and ABAQUS by considering factors such as the weld geometry,
on FFS was investigated. constraint, loading pattern (Xing et al., 2016; Wang et al. (2020); Mei
and Dong, 2017; Yang et al., 2020, 2022).
2.3.3. Mesh size and element type sensitivity analysis As shown in Fig. 9, when T-joints are subjected to external force, the
Mesh size and element type play a significant role in model accuracy stress near the welded toe or root is highly nonlinear. This stress can be
and computational time. Therefore, sensitivity analysis was performed divided into two parts according to stress linearization: traction struc­
to determine the proper mesh size and element type. tural stress σs (related to external forces) and notch stress (self-balanced
Mesh sensitivity analysis was used to evaluate FE models with mesh stress). The traction structural stress σ s is composed of membrane stress
sizes of 7.5 mm, 3.75 mm, 2.5 mm, and 1.25 mm in Ahmad’s work σ m and bending stress σb, and it is mesh-insensitive. According to the
(2022). In these models, all the material properties, heat source models, symmetry of the T-joints, half of the FE model with symmetric constraint
and thermal boundary conditions were identical. Comparing the was used for extracting the nodal stress, and the boundary condition of
maximum temperature and substrate deformation between FEA and the the FE model was consistent with the actual fatigue test, ensuring the
experiments showed that a mesh size of 2.5 mm was optimal for accuracy of stress analysis.
modeling the nearby and weld bead area in the FE simulation. There­ In FEA, the weldline and cracking direction were first determined
fore, in this study, a weld bead with a mesh size of 2.0 mm for FEA was with the C3D8 solid elements. Then, the nodal stress at the weld toe was
used. extracted, and the traction structural stress was finally calculated based
The influence of four element types (C3D8, C3D4, C3D20, and on Eqs. (17)-(19) (Chen et al., 2022).
C3D10) on welding simulation deformation was analyzed in the work of ∫ ′

Zhang et al. (2016). Compared with C3D20 (which had a more accurate 1∑ n
1 t
(17)
′
σm = Fx′ ,i = ′ σ ′ (y )dx
description of deformation displacement), the error of welding defor­ t 0 x
′
t i=1
mation using C3D8 was only 3%, and the use of C3D8 greatly reduced
mesh number and calculation time. Therefore, the element type used in 6 ∑
n ( t) 6
∫ t′ ( t)
′

(18)
′

this study was C3D8. σb = Fx′ ,i y − = ′ 2 σ x′ (y ) x − dx

(t′ )2 i=1
2 (t ) 0 2

2.4. Traction structural stress method σs = σm + σb (19)

2.4.1. Calculation procedure

The traction structural stress method can accurately characterize the

Fig. 10. FE models of T-joints with different mesh sizes.

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Z. Chen et al. Ocean Engineering 281 (2023) 114676

joints. As shown in Fig. 11, the maximum traction structural stress of the
T-joints under different mesh sizes was comparable, which validated the
mesh insensitivity of the traction structural stress method.

2.5. Consideration of angular misalignment

Angular misalignment emerged in the welded joints due to the

heating and cooling at the local area during the welding process. As
shown in Fig. 12, T- joints with angular misalignments α were fixed by
hydraulic wedges during the fatigue tests, which resulted in an addi­
tional moment M in the T-joints. Meanwhile, additional tensile stress
emerged at the weld toe. The weld toe was the weak area of the T-joints,
and the additional tensile stress further reduced the fatigue performance
and critical crack size. Therefore, the effect of angular misalignment was
considered when the FFS of the T-joints was performed.
According to the angular misalignment α, the total length of test
specimen L, the position of critical location LC, and the base metal
thickness t, the angular misalignment-induced SCF kα was obtained
using Eq. (20) (Xing et al., 2016). The traction structural stress was
Fig. 11. Traction structural stress of T-joints under different mesh sizes. calculated with Eq. (21) to consider angular misalignment (Xing et al.,
2017).
2.4.2. Validation of mesh insensitivity ( ( 4 ))
4 L − 9L3 Lc + 39L2 L2c − 60L3c L + 30Lc4
In this section, three mesh sizes were used for validation of mesh kα = α (20)
5 L3 t
insensitivity: 1 mm, 2 mm, and 4 mm, as shown in Fig. 10. The boundary
condition and element type were consistent with the setting in Section σ sm = σs ⋅(1 + kα ) (21)
2.4.1, and a nominal stress of 1 MPa was applied to one end of the T-

Fig. 12. T- joint constraint with angular misalignment in fatigue test (a) T-joint with angular misalignment (b) T-joint constraint.

Fig. 13. Schematic diagram of test specimens clamped by hydraulic wedges before fatigue tests.

9
Z. Chen et al. Ocean Engineering 281 (2023) 114676

Table 4
Fatigue test parameters and results.
Specimen series Repair welding Test specimen Thickness t/mm Griping length L’/mm Nominal stress σn/MPa Stress ratio R Fatigue life N/cycles

F NO F6 12 78 101.25 0.1 77214

F1 12 78 90 0.1 62202
F5 12 78 81 0.1 138609
RF YES RF4 12 78 101.25 0.1 29870
RF3 12 78 90 0.1 62752
RF2 12 78 81 0.1 61853

Fig. 14. Failure location of T-joints with and without repair welding (a) F test specimens (failure at weld toe on either side) (b) RF test specimens (failure at weld toe
on repair welding side).

on ASTM E837-13a in this section, and the accuracy of the numerical

3. Results and discussion
simulation was validated.
The principle of the blind hole drilling method is that the residual
3.1. Fatigue test results
stress is calculated according to the relieved strains near the drilled hole
on the surface of a T-joint. In the blind hole drilling tests, seven BSF120-
Fig. 13 shows a schematic diagram of the T-joint specimen fatigue
1.5CA-T rosette strain gauges were arranged on the longitudinal
tests. A sinusoidal cycle loading with a stress ratio of 0.1 was applied to
centerline of the T-plate with unequal spacing. These gauges had three
both ends of the test specimens, and a 25-ton MTS 647 universal testing
measuring directions (0◦ , 45◦ , and 90◦ ), which was convenient for
machine was used in the fatigue tests. The griping lengths were 78 mm
calculating the longitudinal and transverse residual stresses of the T-
and the test specimens were fixed by hydraulic wedges. The loading
joints, see Fig. 16(a). Then, a hole on the surface of the T-joints with a
frequency and the applied stress range in fatigue tests were 15 Hz and
depth of 1.5 mm was drilled using RSD1 equipment with a drilling bit
81–101.25 MPa, respectively. The fatigue test parameters and results are
(radius r = 1.0 mm). The residual strain (ε0◦ , ε45◦ , and ε90◦ ) was deter­
shown in Table 4, demonstrating that repair welding resulted in a sig­
mined near the surface of T-joints using an ASMB2-16 strain collector
nificant decrease in fatigue life.
(see Fig. 16(b)), and residual stress was calculated according to Eqs. (22)
The failure locations of the T-joints after the fatigue tests are shown
and (23).
in Fig. 14. The F test specimens failed from the weld toe on either side,
The specific operation steps of the blind hole drilling tests were re­
while the failure locations of the RF test specimens were at the weld toe
ported in Zeinoddini’s work (2013), which was very instructive for
on the repair welding side.
investigating residual stress. In addition, the residual stress at the cor­
According to the macro-fracture surface characteristics of the test
responding path in the FE model was extracted, as shown in Fig. 17.
specimens shown in Fig. 15, multiple cracks were initiated from the
√̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
whole length of the weld toe of the original test specimens. These cracks E ε0 + ε90 ( )
σ 1,2 = + (ε0 − ε90 )2 ± (2ε45 − (ε0 + ε90 ))2 (22)
propagated along the through thickness direction in the approximate 4000000B 4A
form of semi-ellipses. However, for the test specimens with repair
welding, the cracks were mainly concentrated in the repair welding tan 2θ =
2ε45 − ε0 − ε90
(23)
zone. Therefore, repair welding led to a change in CIS and CSF. The ε0 − ε90
outlines of the main cracks in Fig. 15 are depicted by red dotted lines,
and the measured dimensions are shown. where E is Young’s modulus, σ 1 and σ2 respectively represent the
maximum and minimum principal stress, A and B are strain calibration
3.2. Measurement of residual stress and validation of FEM constants, and θ is the angle of the strain gauge from the x-axis.
The longitudinal and transverse tensile residual stresses were the
Welding residual stress can be divided into macroscopic (Type I) and highest (275 and 137 MPa, respectively) near the weld toe of the T-
microscopic (Type II and Type III) stresses (Everaerts et al., 2018), and plates, as shown in Fig. 18. The longitudinal residual stress dropped to
BS7910 is concerned with macroscopic residual stress. Therefore, blind − 84 MPa with increasing distance from the weld toe and then slowly
hole drilling tests were used to measure the surface residual stress based rose to approximately − 50 MPa. However, the transverse residual stress

10
Z. Chen et al. Ocean Engineering 281 (2023) 114676

Fig. 15. Macro-fracture surface characteristics of test specimens (a) Nominal stress σn = 81 MPa (b) Nominal stress σ n = 90 MPa (c) Nominal stress σn = 101.25 MPa.

Fig. 16. Strain collector and triaxial strain rosettes used in this study (a) BSF120-1.5CA-T rosette strain gauges (b) ASMB2-16 strain collector.

11
Z. Chen et al. Ocean Engineering 281 (2023) 114676

Fig. .17. Extraction path of residual stress and layout of the strain gauges (a) Extraction path of residual stress (b) Layout of the strain gauges.

Fig. 20. Primary stress distribution of T-joints.

Fig. 18. Comparison of residual stresses obtained by FEA and experi­
mental tests.
3.3. Effect of repair welding on the fatigue behavior

In this section, the FFS assessment in BS7910 was used to investigate

the effect of repair welding on the fatigue behavior based on FADs.
According to the fractography analysis, the crack in this study was
equivalent to a semi-elliptical surface crack when FFS assessment was
performed, and the crack propagation direction θ was 90◦ . A schematic
diagram of the surface crack is shown in Fig. 19.

3.3.1. Calculation of primary stress P

According to nominal stress σn applied to T-joints, the maximum
force Fmax and the minimum force Fmin were determined based on Eq.
(24). The primary bending stress values were obtained after calculating
the traction structural stress σ s, as shown in Eq. (25).
Fig. 19. Schematic diagram of the surface crack (BS 7910, 2015).
Fmax − Fmin
Pm = σ n = (24)
S
slowly declined to about 0 MPa. The FEA and blind hole drilling test
results showed that the residual stress values obtained by FEA were in Pb = σ s − Pm (25)
good agreement with those measured by the experimental tests in terms
The T-joints all failed from the weld toe, so the traction structural
of the numerical values and variation trends. Therefore, the accuracy of
stress at the weld toe was extracted and analyzed. The structural stress
FEA was validated.
centration factor (SSCF) was defined as the ratio of traction structural
stress σ s to nominal stress σ n. According to FEA, this ratio was 1.13, as
shown in Fig. 20.

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Z. Chen et al. Ocean Engineering 281 (2023) 114676

Table 5 ∑ ∫ t
σ (x)⋅Δx 1
Traction structural stress, membrane stress, and bending stress. Qm = = σ(x)dx (27)
t t 0
Test Traction Angular Traction Membrane Bending
specimen structural misalignment structural stress Pm/ stress ∑ ∫ (
σ(x)⋅Δx⋅(x − t/2) 6 t
t)
stress σs/ α/rad stress σsm/ MPa Pb/MPa Qb = = σ(x) x − dx (28)
MPa MPa
t2 /6 t2 0 2

F6 114.41 0.049 145.89 104.29 41.60 The secondary stress Q values are exhibited in Fig. 23. The mem­
F1 101.70 0.046 127.96 92.70 35.26 brane stress Qm and bending stress Qb values of the T-joints significantly
F5 91.53 0.046 115.17 83.43 31.74 increased as the repair welding length shortened.
RF4 114.41 0.056 150.38 104.29 46.06
RF3 101.70 0.058 134.81 92.70 42.11
RF2 91.53 0.053 118.76 83.43 35.33 3.3.3. FFS assessment with fixed CSF
BS7910 does not provide a recommendation for the value of a/2c
when the fatigue crack dimension is unknown. Therefore, the CSF was
Using the SSCF, the nominal stress σ n, and the angular misalignment set at a fixed value of 0.4 in this section (i.e., a/2c = 0.4).
α of the test specimens, the corrected traction structural stress σsm was The FFS assessment of the T-joints with different repair welding
calculated using Eqs. (20) and (21). The calculated traction structural lengths is revealed in Fig. 24. The critical crack depth of the T-joints
stress, membrane stress, and bending stress values are shown in Table 5. presented a downward trend with increasing nominal stress. The frac­
ture ratio Kr of the as-welded T-joints slowly grew with increasing load
3.3.2. Calculation of secondary stress Q ratio Lr and intersected the BC segment in the FAC. The load ratio Lr
As shown in Fig. 21, four different FE models were used to investi­ varied with the crack depth a and width 2c, which reflected the degree of
gate the effect of repair welding length on residual stress: one as-welded plastic failure of the structures with defects. Therefore, for as-welded T-
FEM and three FEMs with repair welding lengths of 80 mm, 55 mm (half joints, the load-carrying area decreased with the crack propagation in
the width of the test specimens), and 30 mm. The Mises stress contours the depth and width directions, and ductile fracture occurred in the T-
of the FEMs in Fig. 22 show that the plastic zone gradually increased as joints when the load-carrying area was reduced to a certain value.
the repair welding length shortened. According to the stress lineariza­ However, the fracture ratio Kr of the T-joints with repair welding lengths
tion principle in BS7910, the residual stress σ R at the weld toe was of 55 mm and 30 mm promptly rose with increasing load ratio Lr,
decomposed into membrane stress Qm and bending stress Qb by Eq. (26) passing through the AB segment in the FAC. Welding promoted the
-(28). brittleness of the joints, and the welded joints had a higher triaxial stress
[ ( ) ( )2 ( )3 ] state after repair welding, which further increased the brittleness. (Tian
Z Z Z
σ R = σ Y 0.9415 − 0.0319 − 8.3394 + 8.660 (26) et al., 2017). Therefore, with decreasing repair welding length, the
B B B

Fig. 21. Schematic diagram of FEMs with different repair welding lengths.

Fig. 22. Mises stress contours of FEMs with different repair welding lengths.

13
Z. Chen et al. Ocean Engineering 281 (2023) 114676

rapidly increasing residual stress promoted the brittle fracture of the

T-joints.
According to the equivalent characterization of cracks (Fig. 19) and
critical crack depths (Fig. 24), the fracture morphologies obtained by
FFS assessment with different loading levels and repair welding lengths
are shown in Fig. 25. The crack opening area S was calculated by Eq.
(29). The crack opening area of T-joints with a repair welding length of
80 mm was basically the same as that of the as-welded T-joints. How­
ever, the crack opening area rapidly reduced when the repair welding
length was lower than 80 mm. The critical crack depth and crack
opening area of T-joints with a repair welding length of 30 mm were
significantly reduced by 63–66% and 86–89%, respectively, compared
with the as-welded T-joints.
π ⋅a⋅2c
S= = π⋅a⋅c (29)
2

3.3.4. FFS assessment with non-fixed CSF

According to the macro-fracture surface characteristics of test spec­
imens, the CSF was related to the repair welding length and plate
thickness. Therefore, the CSF of test specimens without repair welding
Fig. 23. Membrane and bending stresses at different repair welding lengths.
was set to 0.11 according to the plate thickness and width. For repair
welding lengths of 80, 55, and 30 mm, the values of a/2c were deter­
mined to be 0.15, 0.22, and 0.4, respectively, as shown in Table 6.

Fig. 24. Critical crack depth of T-joints with fixed CSF (a) σ n = 81 MPa (b) σn = 90 MPa (c) σ n = 101.25 MPa.

14
Z. Chen et al. Ocean Engineering 281 (2023) 114676

Fig. 25. Fracture morphology obtained by FFS assessment with fixed CSF.

when L < 80 mm. The non-fixed CSF T-joints showed the opposite trend.
Table 6
CSF values of test specimens with and without repair welding.
3.3.6. Validity analysis of FSS assessment procedure
Welding Plate thickness t/ Plate (repair welding) length L/ CSF Based on the critical crack depths and lengths measured in the
status mm mm
fracture surfaces of test specimens shown in Fig. 15, the deviation be­
As-welded 12 110 0.11 tween the FSS assessment with a fixed CSF and the experimental tests is
Repair length 12 80 0.15
shown in Table 7. The deviations of critical crack depth a and CSF were
L = 80 mm
Repair length 12 55 0.22
both more than 50%. This was because the fracture surface character­
L = 55 mm istics of T-joints with and without repair welding were not considered.
Repair length 12 30 0.4 Therefore, the fixed CSF values were not applicable to the FFS assess­
L = 30 mm ment of the T-joints with different repair welding lengths.
After fully considering and analyzing the fracture surface charac­
The critical crack depth values of T-joints with a non-fixed CSF are teristics of the T-joint test specimens with and without repair welding,
depicted in Fig. 26. The fracture ratio Kr of the T-joints rapidly grew with the deviations between the FSS assessment results with a non-fixed CSF
increasing load ratio Lr and intersected the AB segment in the FAC. The and experimental results were less than 15% and 8%, respectively. These
critical crack depth of T-joints with a repair welding length of 55 mm deviations were significantly lower than those of the fixed CSF result
was slightly lower than that of the T-joints with a repair welding length (see Table 8). Compared with the deviation of the as-welded T- joints,
of 30 mm. However, according to the fracture morphology obtained by the T-joint with repair welding had poorer quality and more imperfec­
FFS assessment in Fig. 27, the crack opening area of T-joints with a tions, and the critical crack size measured from the fatigue fracture
repair welding length of 50 mm was approximately 1.7 times that of T- surface characteristics more significantly deviated from the FFS assess­
joints with a repair welding length of 30 mm due to the longer crack. In ment results. To some extent, the FFS assessment with a non-fixed CSF
addition, compared with as-welded T-joints, the critical crack depth and was highly accurate. Therefore, the validity of the FSS assessment pro­
crack opening area of T-joints with a repair welding length of 30 mm cedure with a non-fixed CSF was verified.
significantly decreased by approximately 40% and 90%, respectively. According to the FFS assessment results, welded structures with
repair welding in practical applications should be critically examined
3.3.5. Comparative analysis of FFS assessment results due to the smaller failure crack dimensions.
In this section, the FFS assessments with a fixed and non-fixed CSF
were compared. As shown in Fig. 28(a), the critical crack depth rapidly 4. Conclusions
decreased when the repair welding length L was less than 80 mm, but for
non-fixed CSF T-joints, critical crack depth increased inversely when L In this study, the effect of repair welding on the fatigue behavior of T-
< 55 mm. The critical crack depth with a fixed CSF was 1.6–2.2 times joints was investigated based on the FFS assessment in BS7910 and
higher than that with a non-fixed a/2c, except for the repair welding experimental testing. Some of the key findings in this study are as
length of 30 mm (which had the same a/2c value). According to Fig. 28 follows:
(b), for as-welded T-joints, the critical crack length with a non-fixed CSF
was approximately 2.2 times that with a non-fixed CSF. The critical (1) Repair welding changed the CIS and CSF. Multiple cracks initi­
crack length rapidly declined with decreasing repair welding length and ated from the whole length of the weld toe of the original welded
was slightly lower than that with a non-fixed CSF when the repair T-joints and propagated along the thickness direction in the form
welding length was lower than half the plate width (i.e., 55 mm). of an approximate semi-ellipse. However, for the T-joints with
Sr/S was defined as the ratio of the crack opening area with repair repair welding, the cracks were mainly concentrated in the repair
welding to that of the as-welded state. According to Fig. 29, for fatigue welding zone.
crack growth with a fixed and non-fixed CSF, the crack opening area and (2) For a fixed CSF, the critical crack depth rapidly decreased due to a
the ratio Sr/S of the T-joints showed different downward trends. As the more significant increase in residual stress when the repair
repair welding length decreased, the crack opening area and the ratio Sr/ length L < 55 mm. The critical crack depths and crack opening
S of the T-joints with a fixed CSF slowly decreased and then dropped areas of T-joints with a repair welding length of 30 mm were

15
Z. Chen et al. Ocean Engineering 281 (2023) 114676

Fig. 26. Critical crack depth of T-joints with non-fixed CSF (a) σ n = 81 MPa (b) σn = 90Mpa (c) σ n = 101.25 MPa.

Fig. 27. Fracture morphology of T-joints with non-fixed CSF.

significantly reduced by 63–66% and 86–89%, respectively, (3) For a non-fixed CSF, T-joints with a repair welding length of 30
compared with T-joints without repair welding. mm had a higher crack depth, but the crack opening area was
only 59% of that for repair welding length of 55 mm, and the

16
Z. Chen et al. Ocean Engineering 281 (2023) 114676

Fig. 28. Critical crack depth and length of T-joints with fixed and non-fixed CSF (a) Critical crack depth (b) Critical crack length.

Fig. 29. Crack opening area and the ratio Sr/S of the T-joints with fixed and non-fixed CSF (a) Crack opening area S (b) Sr/S ratio.

Table 7 Table 8
Deviation of FSS assessment procedure with fixed CSF. Deviation of FSS assessment procedure with non-fixed CSF.
Test specimen FSS assessment Experiment Deviation Test specimen FSS assessment Experiment Deviation

a CSF a CSF a CSF a CSF a CSF a CSF

F6 8.8 0.4 4.9 0.101 79.6% 75.5% F6 5.1 0.109 4.9 0.101 5.1% 7.3%
F1 9.5 0.4 5.5 0.105 72.7% 73.8% F1 5.6 0.109 5.5 0.105 1.8% 3.7%
F5 9.8 0.4 6.3 0.108 55.6% 73.0% F5 6.0 0.109 6.3 0.108 4.8% 0.9%
RF4 5.4 0.4 3.3 0.184 63.6% 54.0% RF4 2.9 0.218 3.3 0.184 12.1% 15.6%
RF3 6.5 0.4 3.4 0.187 91.2% 53.3% RF3 3.1 0.218 3.4 0.187 8.8% 14.2%
RF2 7.4 0.4 3.8 0.188 94.7% 53.0% RF2 3.4 0.218 3.8 0.186 10.5% 14.7%

critical crack depth and crack opening area were approximately that for the non-fixed CSF showed a linear decrease first and then
60% and 10% of the corresponding values of the as-welded T- a slow decline.
joints, respectively. (5) The validity of FSS assessment with a non-fixed CSF was verified.
(4) The two methods for characterizing crack size presented in this For T-joints with and without repair welding, the deviations be­
paper show different trends in the FFS assessment. The crack tween the FSS assessment with a non-fixed CSF and the experi­
opening area and critical crack length of the T-joints with a fixed mental results were less than 15% and 8%, respectively. These
CSF slowly declined and then decreased linearly, but the trend of deviations were significantly lower than those obtained with a
fixed CSF.

17
Z. Chen et al. Ocean Engineering 281 (2023) 114676

CRediT authorship contribution statement interests or personal relationships that could have appeared to influence
the work reported in this paper.
Zhihao Chen: Conceptualization, Methodology, Investigation,
Writing – original draft, Writing – review & editing, Visualization. Yong Data availability
Liu: Conceptualization, Methodology, Validation, Supervision. Yizhou
Liu: Writing – review & editing. Ping Wang: Conceptualization, Data will be made available on request.
Methodology, Validation, Resources, Writing – review & editing, Visu­
alization, Supervision, Project administration, Funding acquisition. Acknowledgments
Hongliang Qian: Conceptualization, Visualization, Supervision, Project
administration, Funding acquisition. This research was funded by the National Key R & D Program of
China (No. 2019YFB1600702).
Declaration of competing interest

The authors declare that they have no known competing financial

Appendix. A

1. The calculation procedure of the parameter Mm

[ ( a )2 ( a )4 ] gf
(A1)
θ
Mm = M1 + M2 + M3
B B φ
When 0 < a/2c ≤ 0.5;
(a)
M1 = 1.13 − 0.09 (A2)
c
⎡ ⎤
⎢ 0.89( )⎥
M2 = ⎣ ⎦ − 0.54 (A3)
0.2 + a/c

1 ( a)24
M3 = 0.5 − ( a) + 14 1 − (A4)
0.65 + c c
[ ( a )2 ]
g = 1 + 0.1 + 0.35 (1 − sin θ)2 (A5)
B
[( ) ]0.25
a 2 2
fθ = cos θ + sin2 θ (A6)
c
[ (a)1.65 ]0.5
φ = 1 + 1.464 (A7)
c
M1, M2, M3, g, fθ and φ are Parameters in calculating stress intensity factor solution, θ is the parametric angle to identify position along an elliptical
flaw front.

2. The calculation procedure of the parameter Mb

Mb = HMm (A8)

H = H1 + (H2 − H1 )sinq θ (A9)

When 0 < a∕2c ≤ 0.5;
( ) ( / )2
H2 = 1 + G1 a/B + G2 a B (A10)

( )
G1 = − 1.22 − 0.12 a/ (A11)
c
( / )0.75 ( / )1.5
G2 = 0.55 − 1.05 a c + 0.47 a c (A12)

In which H1, H2, G1 and G2 are parameters in calculating stress intensity factor solutions; When θ = 90o, H1 is eliminated, and only H2 needs to be
solved.

3. The calculation procedure of the parameter Mkm

When Mkm <1, Mkm = 1;

When Mkm ≥1,

18
Z. Chen et al. Ocean Engineering 281 (2023) 114676

( a a) (a) ( )
a L
Mkm = f1 , + f2 + f3 , (A13)
B c B B B
[ { ( )}g3 ]
( a a) ( a ) g1 + g2 Ba {( ) }
a − 0.050966
f1 , = 0.43358 + 0.93163 exp + g4 (A14)
B c B B
()
(a) [ ( a )]176.4199 ( a )− 0.10740 a

(A15)
B
f2 = − 0.21521 1 − + 2.8141
B B B
( ) ( ) ( ) [ ( ) (a) ]
a L a 5g a 0.23003 a 2
f3 , = 0.33994 + 1.9493 + g6 + g7 + g8 (A16)
B B B B B B
(a)2 (a)
g1 = − 1.0343 − 0.15657 + 1.3409 (A17)
c c
(a)− 0.61153
g2 = 1.3218 (A18)
c
(a)
g3 = − 0.87238 + 1.2788 (A19)
c
(a)3 (a)2 (a)
g4 = − 0.46190 + 0.67090 − 0.37571 + 4.6511 (A20)
c c c
( )3 ( )2 ( )
L L L
g5 = − 0.015647 + 0.090889 − 0.17180 − 0.24587 (A21)
B B B
( )2 ( )
L L
g6 = − 0.20136 + 0.93311 − 0.41496 (A22)
B B
( )2 ( )
L L
g7 = 0.20188 − 0.97857 + 0.068225 (A23)
B B
( )2 ( )
L L
g8 = − 0.027338 + 0.12551 − 11.218 (A24)
B B

Where g1, g2, g3, g4, g5, g6, g7, g8 are parameters in calculating stress intensity factor solutions.

4. The calculation procedure of the parameter Mkb

If 0.5 <a/B ≤ 0.9, Mkb = 1; If 0.005 ≤a/B ≤ 0.5, then the following expression applies:
( a a) (a) ( )
a L
Mkb = f1 , + f2 + f3 , (A25)
B c B B B

Where:
{ [ ( )]g3 }
( a a) ( a ) g1 + g2 Ba [( ) ]
a − 0.10364
f1 , = 0.065916 + 0.52086 exp + g4 (A26)
B c B B
(a) [ ( a )]2.8086 ( a )g5
f2 = − 0.02195 1 − + 0.021403 (A27)
B B B
( ) ( a )g6 ( a )− 0.20077 [ ( a )2 (a) ]
a L
f3 , = 0.23344 − 0.14827 + g7 + g8 + g9 (A28)
B B B B B B

Where:
(a)2 (a)
g1 = − 0.014992 − 0.021401 − 0.23851 (A29)
c c
(a)− 1.0278
g2 = 0.61775 (A30)
c
(a)
g3 = 0.00013242 − 1.4744 (A31)
c
(a)3 (a)2 (a)
g4 = − 0.28783 + 0.58706 − 0.37198 − 0.89887 (A32)
c c c

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Z. Chen et al. Ocean Engineering 281 (2023) 114676

( a )2 (a)
g5 = 17.195 + 12.468 − 0.51662 (A33)
B B
( )3 ( )2 ( )
L L L
g6 = − 0.059798 + 0.38091 − 0.8022037 + 0.31906 (A34)
B B B
( )2 ( )
L L
g7 = − 0.35848 + 1.3975 − 1.7535 (A35)
B B
( )2 ( )
L L
g8 = 0.31288 − 1.3599 + 1.6611 (A36)
B B
( )2 ( )
L L
g9 = − 0.0014701 − 0.0025074 − 0.0089846 (A37)
B B

Where g1、g2、g3、g4、g5、g6、g7、g8、g9 are parameters in calculating stress intensity factor solutions.

References Gordon, J., 1993. A review of fracture assessment procedures and their applicability to
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