Stress intensity factors for semi-elliptical surface cracks in plate-to-plate
butt welds with parallel misalignment of same thickness
H. S. Zhao1, S. T. Lie1 and Y. Zhang1
1
School of Civil & Environmental Engineering
Nanyang Technological University
50 Nanyang Avenue
Singapore 639798
E-mail: [email protected], Tel.: +65-82850279
Abstract
A set of fatigue and fracture assessment equations had been incorporated into British Standard
7910:2013 (BS 7910:2013) to estimate the effect of misalignment for various geometric
configurations containing a surface crack. In this study, extensive 3-D finite element (FE)
analyses are carried out to determine the dimensionless stress intensity factors (Y) at the crack
deepest point and crack ends of plate-to-plate butt welds with parallel misalignment of same
thickness. In comparison to BS 7910:2013, it is observed that the assessment equations
underestimate the Y values for shallow cracks and overestimate the corresponding values for
deeper cracks with percentage differences in values being as high as 82% and 12%, respectively.
New Y equations for misaligned butt welds are proposed using multiple regression analyses.
Keywords: Finite element analyses; Misaligned butt welds; Regression analyses; Surface
crack
1. Introduction
When plates or shells are welded together, there is invariably some degree of misalignment such
as centreline offset, angular or both. The misalignment induces a local bending stress, increasing
the risk of brittle fracture and shortening the fatigue life of a welded joint [1]. Therefore, it is
very important to be able to assess the effect of misalignment on the fatigue and fracture
strength of welded joints. Maddox [2] investigated the influence of misalignment on the fatigue
strength of transverse butt welds. He found that fatigue test results for misaligned butt welds
-1-
could be associated with the corresponding results of aligned butt welds by taking into account
the stress concentration factor (SCF) induced by misalignment based on the superposition
principle. This approach was employed by Andrews [3] to analyze a more complex geometry,
the transverse load-carrying cruciform joint with centerline offset misalignment, and the
following specific assessment expression is used,
𝐾I = 𝑌a 𝜎a √𝜋𝑎 + (𝑘m − 1)𝑌b 𝜎a √𝜋𝑎
(1)
where KI is the Mode-I stress intensity factor (SIF), Ya and Yb are the geometry factors for the
aligned joints subjected to axial and bending loading, σa is the applied axial stress, a is the crack
depth, km is the SCF induced by misalignment, and it is defined as
𝑘m = 1 +
σs
σa
(2)
where σs is the maximum induced bending stress due to misalignment. Obviously, the effect of
misalignment on SIFs is introduced into Eq. (1) based on using km. Therefore, determining the
SCF equations for various geometric configurations and types of misalignment is significant. In
the past several decades, considerable research [4-12] has been conducted to develop the SCF
equations, and these equations have been incorporated into several codes of practice such as
DNV-RP-C203 [13] and BS 7910:2013 [14].
BS 7910:2013 [14] includes a set of fatigue and fracture assessment equations for cracked butt
welds with misalignment. In these equations, the Newman and Raju equation [15] for a semielliptical surface crack in a plain plate is chosen as the base stress intensity equation. The effect
of the weld toes in welded joints is considered using a weld toe magnification factor Mk, and the
SCF equations are taken into account for misalignment. Therefore, these assessment equations
are actually determined based on the superposition principle. In this study, finite element (FE)
models for plate-to-plate butt welds with parallel misalignment of same thickness are generated
by the mesh generator and extensive 3-D FE analyses are carried out to determine directly the
required SIFs. By comparing the FE results with BS7910:2013 [14], it is found that the
-2-
assessment equations underestimate the SIFs for shallow cracks and overestimate the
corresponding values for deeper cracks. Therefore, a set of new SIF equations for cracked butt
welds with misalignment are built up using multiple regression analyses.
2. Scope of parametric numerical study
The various parameters used in this study are chosen based on the works done by Lie et al. [16]
and the AWS D1.1:2000 recommendations [17]. The SIF equations mentioned in the
BS7910:2013 [14] are usually valid for a sharp weld toe where the weld toe radius is taken to be
zero, and Lazzarin and Livieri [18] proposed that the sharp weld toe assumption is more realistic
because the toe radius is difficult to measure and is affected by a large scatter. Therefore, the
present study solely focuses on the sharp weld toe cases. Considering that butt welds should be
made with minimum face reinforcement not exceeding 3 mm and the weld surface needs to be
flushed in many cases, butt welds with flushed surfaces are only analysed, which means that the
weld reinforcement (R) is zero. The surface crack located at the weld toe is perpendicular to the
main plate face as indicated in Fig. 1. Four basic parameters, namely, crack depth ratio (a/T),
crack aspect ratio (a/c), width ratio of weld bead (L/T) and centreline offset ratio (e/T) are
included in the parametric study (please refer to Fig. 1 for notations).
2.1. Details of crack and weld parameters
As outlined in Table 1, 11 different crack depth ratios are analysed, varying from a very shallow
crack depth ratio of 0.05 to a deep crack depth ratio of 0.9. For all the analyses, the plate
thickness (T) is kept constant and the crack depth (a) is gradually changed to get the required
crack depth ratio. The crack aspect ratios vary from 0.1 to 1.0. Width ratios of weld beads of
0.2, 0.8, 1.4 and 2.0 are applied based on AWS D1.1:2000 recommendations [17]. The wide
range of parameters practically covers the most commonly used butt weld configurations. As for
the centreline offset, a typical value used in many fabrication standards is e = 0.15T or
maximum 3-4 mm [12]. Hence, centreline offset ratios (e/T) of 0.00, 0.05, 0.10 and 0.15 are
-3-
employed in the present study.
2.2. Finite element modelling
A new mesh generator has been developed to generate FE models of cracked plate-to-plate butt
welds with parallel misalignment of same thickness as shown in Fig. 2 automatically. This mesh
generator is based on the earlier one serving for aligned plate-to-plate butt weld joints [16]. All
FE analyses are performed using ABAQUS [19], the general purpose FE software, and a 20node brick element with reduced integration (C3D20R) is used throughout the models. Based on
symmetry, only half of the plate model is considered, and high mesh density around the crack
front is required to study the large deformation field (Fig. 3). The middle nodes of the elements
around the crack tip are shifted to quarter position to simulate the crack singularity. Fig. 4
presents the boundary conditions for the misaligned butt welds and the principal direction of the
applied stress at the crack deepest point. A symmetric boundary condition about Y-axis is
applied to the plate face of Y = 0 due to symmetry of the model analyzed. One end of the plate
is restrained in X-axis and Z-axis directions, and the other end is only constrained in the Z-axis
direction and a uniform axial stress is applied in the X-axis direction to provide the loading
condition. It is noted clearly from Fig. 4 that the crack face is perpendicular to the principal
direction of the stress at the crack deepest point. The Young’s modulus (E) and Poisson’s ratio
(v) used in the analyses are 210 kN/mm2 and 0.3, respectively. The SIFs are obtained from the Jintegral, and the virtual crack extension methodology implemented in ABAQUS [19] is used for
the numerical computation of the J-integral. For each crack front node, the SIF is determined
from the average J-integral value of contours 2, 3, 4 and 5, and the corresponding J-integral for
the first contour is ignored due to the numerical errors. For the cracked plate subjected to axial
loading, the crack-tip material tries to contract in Y-axis and Z-axis directions due to the large
stress normal to the crack face. However, the contraction is restricted by the surrounding
material, which induces a triaxial state of stress near the crack tip. Therefore, the plane strain
condition is used along the crack front except the crack ends [20], and the SIF is calculated by
-4-
the following expression:
𝐾I = √
𝐽𝐸
(1 − 𝜈 2 )
(3)
Owing to the effect of the free surface of the plate, the stress triaxiality is lower near the free
surface, and a state of pure plain stress exists at the free surface. Therefore, the plane stress
condition is assumed at the crack ends [20] and E/(1-v2) in Eq. (3) is replaced by E to determine
the corresponding SIF values. In the present study, dimensionless SIFs (Y) are used to
characterize the fatigue and fracture strength of the structural configurations due to SIF values
changing with specific crack sizes and loading conditions, and it is calculated by the following
expression:
𝑌=
𝐾I
σa √𝜋𝑎
(4)
By using multiple regression analyses, the relationship between Y and other independent
variables a/T, a/c, L/T and e/T are built up.
3. Verification of the mesh generator
This section details the validation of FE mesh generator before it is used for the parametric
study of cracked plate-to-plate butt welds with parallel misalignment of same thickness. At first,
the FE model of a plain plate containing a semi-elliptical surface crack generated by the mesh
generator is verified by comparing the results obtained using the classical Newman and Raju
equation [15]. Subsequently, the convergent tests for misaligned butt welds are carried out to
further demonstrate the accuracy of the mesh generator.
3.1. Classical Newman and Raju equation
Newman and Raju [15] proposed an empirical SIF equation for a plain plate with a surface crack
as a function of parametric angle, crack depth, crack length, plate thickness and plate width. In
this study, a plain plate model can be generated by the mesh generator by setting R = 0 and e/T =
0. The Y values obtained by FE analyses are compared with the corresponding values of the
-5-
classical Newman and Raju equation [15] at the crack deepest point (Fig. 5) and crack ends (Fig.
6). The percentage difference is introduced to characterize quantitatively the difference of the
FE results with the classical Newman and Raju equation [15] using the following expression:
% Difference =
|𝑌FEA −𝑌equation |
𝑌equation
× 100%
(5)
At the crack deepest point, the present Y values agree very well with the values of Newman and
Raju equation [15] for a/T ≤ 0.7 with the largest percentage difference being 4.3% (Table 2),
and the FE results largely deviate the variation tendency of Y versus a/T obtained from Newman
and Raju equation [15] only for very deep crack cases, especially for a/T = 0.9, a/c = 0.1 and a/T
= 0.9, a/c = 0.2. Newman and Raju [15] stated that this equation is within ±5% of the FE results
for a/T ≤ 0.8. Therefore, the FE data for a/T ≤ 0.7 calculated in this study are assumed to be
available and can be used in the multiple regression analyses. At the crack ends, the numerical
results are in good agreement with that of Newman and Raju equation [15] for the whole crack
depth ratios with the largest percentage difference being 4.5% (Table 2), and the regression
analyses include all the FE values with a/T varying from 0.05 to 0.9.
3.2. Convergent test for misaligned butt welds
The range of percentage difference between the FE results and the classical Newman and Raju
equation [15] is within 0.4%-4.5%. Hence, the mesh generator is considered to be well suited for
further analyses of more complex configurations where sufficient elements are used to obtain
good convergence. In this section, the mesh generator is employed to create the FE models of
plate-to-plate butt welds with parallel misalignment of same thickness. The convergent test is
then carried out to determine the optimal number of elements to be used in the parametric study.
Fig. 7 shows the comparison of SIF results from a convergent test for a semi-elliptical surface
crack (a/T = 0.2, a/c = 0.4, e/T = 0.10) in the misaligned butt weld. The number of elements is
varied from 6818 to 13328. It can be seen that the percentage differences of KI at the crack
deepest point and crack ends are 0.05% and 0.89%, showing a good convergence. More
-6-
convergent tests covering the range of parameters presented in Table 1 are performed to
determine the optimal number of elements ranging from 5,888 to 15,188 depending on the crack
size and plate geometry.
4. Validation and proposing new Y equations
The aim of this section is to compare and verify the SIF values obtained by FE analyses and BS
7910:2013 [14] before proposing new Y equations.
4.1. BS 7910:2013 assessment equations
The general form of the SIF solution has been presented in BS 7910:2013 [14] to assess the
effect of misalignment on the fatigue and fracture of cracked butt welds subjected to the axial
loading, and it is expressed in the following form:
𝐾I = (𝑌σa )√𝜋𝑎
(6)
𝑌 = 𝑀𝑓w [𝑘ta 𝑀ka 𝑀a + 𝑘tb 𝑀kb 𝑀b (𝑘m − 1)]
(7)
where
and M is the bulging correction factor, fw represents the finite width correction factor, kta and ktb
are the axial and bending SCFs due to gross structural discontinuities, km is the SCF induced by
weld misalignment, Mka and Mkb are the axial and bending stress intensity magnification factors
for a flaw or crack located in a region of local stress concentration, Ma and Mb are the stress
intensity magnification factors for a plain plate with a surface flaw or crack under the axial and
bending loadings, respectively. However, for the misaligned butt welds with zero weld
reinforcement in this study, kta, ktb, Mka and Mkb are all equal to 1.0, and the above complicated
equation can be simplified as
𝑌 = 𝑀𝑓w [𝑀a + 𝑀b (𝑘m − 1)]
(8)
The detailed expressions of M, fw and Ma in Eq. (8) are outlined in Appendix A because they are
served for the construction of the new Y equations in Sub-section 5.1. The expression of Mb is
not listed here and can be found in Annex M of BS 7910:2013 [14]. In the present study, km
-7-
indicates the SCF due to centreline offset misalignment, and it is expressed as follows:
𝑘m = 1 + 3
𝑒
𝑇
(9)
4.2. Comparison of Y values
Figs. 8-9 show the comparison of Y values obtained from 3-D FE analyses and Eq. (8), and the
representative plots cover the varying a/T for a/c = 0.2, L/T = 0.8, e/T = 0.10. For the crack
deepest point cases shown in Fig. 8, Eq. (8) underestimates the Y values for shallow cracks and
slightly overestimates the Y values for very deep cracks. The FE results are nearly consistent
with Eq. (8) in the region of intermediate crack depths (0.2 < a/T < 0.6). For the crack ends
cases (Fig. 9), the Y values obtained by FE analyses are underestimated by Eq. (8) for shallow
and intermediate cracks and overestimated for very deep cracks. In this study, the
underestimation/overestimation percentage difference is introduced to describe quantitatively
the difference between Y values obtained by FE analyses and that given by Eq. (8), which is
calculated by the following expression:
𝑌FEA −𝑌Eq.(8)
% Underestimation = (
𝑌Eq.(8)
) × 100%
(10)
% Overestimation = (
𝑌Eq.(8) −𝑌FEA
𝑌Eq.(8)
) × 100%
Table 3 presents the corresponding comparison results. It is noted clearly that at the crack
deepest point, the difference between FE results and Eq. (8) is marginal except in the case of
shallow cracks (a/T < 0.1). By taking into account all the Y values, the largest underestimation
percentage difference of Eq. (8) is 21% (a/T = 0.05, a/c = 0.1, L/T = 0.2, e/T = 0.15). For the
crack ends cases, a significant difference for Y values given by FE analyses and Eq. (8) is
observed, especially when a/T < 0.2. The largest underestimation and overestimation percentage
differences are 82% (a/T = 0.05, a/c = 0.1, L/T = 0.2, e/T = 0.15) and 12% (a/T = 0.9, a/c = 0.1,
L/T = 0.2, e/T = 0.15), respectively.
km is introduced into Eq. (8) to represent the effect of misalignment on the fracture resistance of
-8-
butt welds. Therefore, the final evaluation values of Eq. (8) for misaligned butt weld with a
surface crack are determined using km. In fact, km indicates the hot spot SCF, showing a much
lower value than the actual SCF at the weld toe (Fig. 10). The real stress concentration at the
weld toe is characterized by the notch stress concentration factor [21], ks, and it is defined as
𝑘s =
σls
σa
(11)
where σls is the local notch stress. The SIFs for a surface crack in the misaligned butt weld are
magnified by the presence of notch stress at the weld toe, especially for shallow cracks.
Therefore, Eq. (8) substantially underestimates the Y values for shallow cracks. The gradual
increase of crack depth reduces the effect of notch stress caused by misalignment on the SIFs,
which accounts for the decreased underestimation difference between Eq. (8) and FE results
with the increase of crack depth.
The ks values calculated by FE analyses are substituted into Eq. (8) instead of km to demonstrate
the main reason why BS 7910:2013 [14] greatly underestimates the Y values for shallow cracks.
The overestimated Y values for the entire range of crack depth are expected. The mesh design of
the FE models are based on the recommendations for the notch stress methodology used in
Appendix D of DNV-RP-C203 [13]. The notch at the weld toe is modelled using a radius of 1.0
mm. In the analyses, the element type used is 8-node plain strain element with complete
integration (CPE8). A total of 1260 elements are used in the analyses and the convergent test
shows that the mesh is adequate. The calculated ks values for different L/T and e/T are listed in
Table 4, and Fig. 11 shows the comparison of ks with corresponding km. Then, ks is substituted
into Eq. (8) instead of km and the following expression is obtained,
𝑌 = 𝑀𝑓w [𝑀a + 𝑀b (𝑘s − 1)]
(12)
Figs 12-13 depict the comparison of Y values from FE analyses with Eq. (12). It is observed
clearly that Eq. (12) overestimates the Y values for the entire range of crack depth as expected
from the above. The largest overestimation percentage differences are 42% (a/T = 0.15, a/c =
-9-
0.1, L/T = 0.2, e/T = 0.15) at the crack deepest point and 44% (a/T = 0.9, a/c = 0.1, L/T = 0.2,
e/T = 0.15) at the crack ends, respectively.
The effect of local geometry of width of weld bead on Y values is investigated to further verify
the need for proposing new Y equations, and the geometrical details of weld configurations with
varying L/T but constant e/T are presented in Fig. 14. Figs. 15-16 show the comparison on the
effect of varying L/T with constant e/T on Y values from FE analyses and Eq. (8). It is observed
that Y versus a/T curves obtained by Eq. (8) do not present any difference for varying L/T with
constant e/T, which means that the local geometry of width of weld bead is not considered in Eq.
(8). However, the representative plots based on FE analyses illustrate that the Y values increase
with decreasing L/T for constant e/T. It is because a smaller L/T for constant e/T results in a
higher steepness of the butt weld, raising a higher stress concentration effect especially for
shallow cracks. All the above comparison results reconfirm the need for having separate Y
equations for plate-to-plate butt welds with parallel misalignment of same thickness.
5. Multiple regression analyses
In this section, multiple regression analyses are applied to determine the new Y equations for
misaligned butt welds. The numerical data used in the multiple regression analyses are listed in
the following:

At crack deepest point: The data for a/T > 0.7 are not used in the multiple regression
analyses.

At crack ends: The regression analyses include all the Y values with a/T ranging from the
shallowest case of 0.05 to the deepest one of 0.9.
5.1. Regression analyses
The new Y equations are developed in stages to account for the effects of four parameters (a/T,
a/c, e/T and L/T) progressively. The first development stage allows for the effect of varying a/c
for a constant e/T (=0) and L/T (=0.2), which is the case of a plain plate. Therefore, the classical
-10-
Newman and Raju equation [15] is used as the first function of the Y equations as presented in
Appendix A. In the second development stage, the Y data for varying e/T but a constant L/T
(=0.2) are involved in the multiple regression analyses. The final stage makes allowance for
varying e/T and varying L/T by including all the Y data in the regression analyses. The new
parametric Y equations are built up, in stages, from the following three basic functions:
𝑎 𝐶12

The crack depth to a power 𝐶11 (𝑇)

One minus the crack depth to a power 𝐶21 [1 − (𝑇)]

Polynomial functions 𝐶31 + 𝐶32 (𝑇) + 𝐶33 (𝑇) + ⋯
𝑎
𝐶22
𝑎 2
𝑎
where C11, C12, C21, C22, C31, C32, C33 etc. are coefficients. The newly proposed Y equations are
outlined to determine the SIF values at the crack deepest point and the crack ends.
At the crack deepest point:
𝑎 𝑒
𝑎 𝑒 𝐿
𝑌deep = 𝑀𝑓w 𝑀a 𝑓1 ( , ) 𝑓2 ( , , )
𝑇 𝑇
𝑇 𝑇 𝑇
(13)
𝑎 𝑒
𝑎 𝐴1
𝑎 4.044778
𝑓1 ( , ) = 0.985562 ( ) − 0.411288 (1 − )
𝑇 𝑇
𝑇
𝑇
(14)
𝑒
𝐴1 = −0.799242 ( ) − 0.107472
𝑇
(15)
where
𝑒
𝑎 𝑒 𝐿
𝑎 [𝐴2 (𝑇)+𝐴3 ]
𝑒
𝑎 0.106319
𝑓2 ( , , ) = −0.107638 ( )
+ [𝐴4 ( ) + 1.212403] (1 − )
𝑇 𝑇 𝑇
𝑇
𝑇
𝑇
(16)
𝐿
𝐴2 = −0.420833 ( ) − 0.258863
𝑇
(17)
𝐿
𝐴3 = −0.008472 ( ) − 0.256433
𝑇
(18)
𝐿 2
𝐿
𝐴4 = −0.092674 ( ) + 0.353952 ( ) + 0.076272
𝑇
𝑇
(19)
𝑎 𝑒
𝑎 𝑒 𝐿
𝑌ends = 𝑀𝑓w 𝑀a 𝑔1 ( , ) 𝑔2 ( , , )
𝑇 𝑇
𝑇 𝑇 𝑇
(20)
𝑎 𝑒
𝑎 𝐵2
𝑎 2.797931
𝑔1 ( , ) = 𝐵1 ( ) − 0.152234 (1 − )
𝑇 𝑇
𝑇
𝑇
(21)
At the crack ends:
where
-11-
𝑒
𝐵1 = 2.531216 ( ) + 1.019271
𝑇
(22)
𝑒 2
𝑒
𝐵2 = 4.499114 ( ) − 2.005011 ( ) − 0.032428
𝑇
𝑇
(23)
𝑒 2
𝑎 𝑒 𝐿
𝑎 [𝐵3 (𝑇)
𝑔2 ( , , ) = 0.897496 ( )
𝑇 𝑇 𝑇
𝑇
𝑒
𝑇
+𝐵4 ( )−0.195383]
𝑒 2
𝑒
𝑎 5.786906
+ [𝐵5 ( ) + 𝐵6 ( ) − 0.810763] (1 − )
𝑇
𝑇
𝑇
(24)
𝐿 3
𝐿 2
𝐿
𝐵3 = −3.016201 ( ) + 12.919036 ( ) − 18.097022 ( ) + 4.539506
𝑇
𝑇
𝑇
(25)
𝐿 2
𝐿
𝐵4 = −0.707222 ( ) + 2.548314 ( ) − 0.245403
𝑇
𝑇
(26)
𝐿
𝐵5 = −8.110312 ( ) + 5.506911
𝑇
(27)
𝐿
𝐵6 = 1.609112 ( ) + 1.510203
𝑇
(28)
Obviously, the equations are an extension to the Newman and Raju equation [15] by directly
considering the effect of misalignment on SIFs, not by the superposition principle.
5.2. Goodness of fit of proposed new equations
The newly proposed Y equations are complex due to various influencing parameters. Therefore,
percentage error frequency histograms are used to validate the goodness of fit of the equations,
which is calculated using the following expression:
% Error =
[𝑌equation −𝑌FEA ]
𝑌FEA
× 100%
(29)
In the histograms, the percentage error is plotted along the abscissa and the percentage of data
points inside the range of the particular percentage error is plotted along the ordinates. Figs. 1718 present the histograms describing the relative difference between the proposed Y equations
and the FE values used in the regression analyses. Overall, the Y equations are a good fit to these
regression values due to the error histograms showing a good normal distribution. Table 5 is
given to quantitatively represent the distribution of the percentage of data points for various
percentage errors depicted in Figs. 17-18. For the crack deepest point cases, 720 FE values are
include in the regression analyses, and 63% of these values are located in the percentage error
ranging from -2% to 2% in comparison with the Y equations, indicating that the proposed
-12-
equations agree very well with most of the values obtained by FE analyses. It is also noted from
Table 5 that the percentages of data points inside -10% to -8% and 8% to 10% are 2% and 1%,
respectively, which means that only very few data points are include inside the large range of
percentage errors. A similar situation can also be observed for the crack ends cases as shown in
Table 5. In order to demonstrate more vividly the accuracy of the proposed Y equations, the
comparisons of the Y values from the FE analyses and the proposed equations are depicted in
Figs. 19-20; showing a visual assessment of goodness of fit of the proposed equations. All the
above analyses confirm the validity of the proposed Y equations for the misaligned butt welds,
and the validity range of the equations for crack depth ratio (a/T) is from 0.05 to 0.7 at the crack
deepest point and from 0.05 to 0.9 at the crack ends.
6. Conclusions
In this study, the Y values obtained by FE analyses are compared with the values from Eq. (8) of
BS 7910:2013 [14]. It is noted that Eq. (8) underestimates the Y values of misaligned butt welds
for shallow cracks with percentage difference in values being as high as 82%. An overestimation
percentage difference of 12% for deeper cracks is produced by Eq. (8). Furthermore, Eq. (8)
does not consider the influence of different widths of weld beads on Y values. Therefore, a set of
new Y equations at the crack deepest point and crack ends of misaligned butt welds are
established using multiple regression analyses. Percentage error frequency histograms are
applied to verify the goodness of fit of the proposed equations. The normal distribution of the
percentage error indicates a good correlation between FE results and the proposed equations.
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-13-
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31, European Offshore Steel Research Seminar, Cambridge. 1978.
[7] Efthymiou M. Development of SCF formulae and generalised influence functions for use
in fatigue analysis. Recent developments in tubular joint technology, OTJ'88, London.
1988.
[8] Smedley S, Fischer P. Stress concentration factors for ring-stiffened tubular joints. In:
Proceedings of the 1st international offshore and polar engineering conference, Edinburgh.
1991.
[9] Lotsberg I. Stress concentration factors at circumferential welds in tubulars. Mar Struct
1998;11:203-230.
[10] Cui WC, Wan ZQ, Mansour AE. Stress concentration factor in plates with transverse buttweld misalignment. J Constr Steel Res 1999;52:159-170.
[11] Lotsberg I. Stress concentration factors at welds in pipelines and tanks subjected to internal
pressure and axial force. Mar Struct 2008;21:138-159.
[12] Lotsberg I. Stress concentrations due to misalignment at butt welds in plated structures and
at girth welds in tubulars. Int J Fatigue 2009;31:1337-1345.
[13] DNV-RP-C203. Fatigue design of offshore steel structures. Det Norske Veritas,
Recommended Practice DNV-RP-C203, Høvik. 2011.
[14] BS7910-Amendment 1. Guide to methods for assessing the acceptability of flaws in
metallic structures. British Standards Institution, UK. 2013.
[15] Newman JC, Raju IS. An empirical stress-intensity factor equation for the surface crack.
Eng Fract Mech 1981;15:185-192.
[16] Lie ST, Vipin SP, Li T. New weld toe magnification factors for semi-elliptical cracks in
double-sided T-butt joints and cruciform X-joints. Int J Fatigue 2015;80:178-191.
[17] AWS D1.1-00. Structural welding code: Steel. American Welding Society (AWS), USA.
2000.
[18] Lazzarin P, Livieri P. Notch stress intensity factors and fatigue strength of aluminium and
steel welded joints. Int J Fatigue 2001;23:225-232.
[19] ABAQUS. Standard user's manual, Version 6.11. Hibbett, Karlsson & Sorensen Inc, USA.
-14-
2011.
[20] Bowness D, Lee MMK. Prediction of weld toe magnification factors for semi-elliptical
cracks in T-butt joints. Int J Fatigue 2000;22:369-387.
[21] Lie ST, Lan S. A boundary element analysis of misaligned load-carrying cruciform welded
joints. Int J Fatigue 1998;20:433-439.
Appendix A
The expressions of M, fw and Ma are presented in the following, and these solutions are used to
calculate the SIF values for a plain plate containing a surface crack.
𝑀=1
𝑓w = {sec [(
π𝑐 𝑎 0.5 0.5
) ( ) ]}
𝐵 𝑇
𝑎 2
𝑎 4 𝑔𝑓θ
𝑀a = [𝑀1 + 𝑀2 ( ) + 𝑀3 ( ) ]
𝑇
𝑇
Φ
where
𝑎
𝑀1 = 1.13 − 0.09 ( )
𝑐
0.89
𝑀2 = [
] − 0.54
0.2 + (𝑎⁄𝑐 )
𝑀3 = 0.5 −
1
𝑎 24
+ 14 (1 − )
0.65 + (𝑎⁄𝑐 )
𝑐
Φ is the complete elliptic integral of the second kind, and can be determined from the following
solution,
𝑎 1.65 0.5
Φ = [1 + 1.464 ( ) ]
𝑐
a) At the crack deepest point:
𝑔=1
𝑓θ = 1
b) At the crack ends:
𝑎 2
𝑔 = 1.1 + 0.35 ( )
𝑇
𝑎 0.5
𝑓θ = ( )
𝑐
-15-
Fig. 1 Nomenclature of parameters of butt welds with parallel misalignment of same thickness
-16-
Fig. 2 Typical FE mesh of butt welds with parallel misalignment of same thickness
-17-
Fig. 3 Mesh detail at the crack region
-18-
Fig. 4 (a) Boundary conditions of the misaligned butt welds; (b) principal direction of the
applied stress at the crack deepest point.
-19-
3.6
3.2
2.8
Y
2.4
2.0
1.6
FE. a/c = 0.1
FE. a/c = 0.2
FE. a/c = 0.4
FE. a/c = 0.7
FE. a/c = 1.0
Eq. a/c = 0.1
Eq. a/c = 0.2
Eq. a/c = 0.4
Eq. a/c = 0.7
Eq. a/c = 1.0
1.2
0.8
0.4
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
a/T
Fig. 5 Comparison of Y values of FE analyses and Newman and Raju [15] equation at the crack
deepest point
-20-
2.2
2.0
1.8
1.6
Y
1.4
1.2
1.0
FE. a/c = 0.1
FE. a/c = 0.2
FE. a/c = 0.4
FE. a/c = 0.7
FE. a/c = 1.0
Eq. a/c = 0.1
Eq. a/c = 0.2
Eq. a/c = 0.4
Eq. a/c = 0.7
Eq. a/c = 1.0
0.8
0.6
0.4
0.2
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
a/T
Fig. 6 Comparison of Y values of FE analyses and Newman and Raju [15] equation at the crack
ends
-21-
4.2
Number of elements
6818
13328
4.0
3.8
3/2
KI (N/mm )
3.6
3.4
3.2
3.0
2.8
2.6
2.4
0
20
40
60
80
100
120
140
Angle
Fig. 7 Convergent test of SIF results
-22-
160
180
2.10
FE results
Eq. (8)
1.95
1.80
Y
1.65
1.50
1.35
1.20
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
a/T
Fig. 8 Comparison of Y values from FE analyses and Eq. (8) at the crack deepest point
(a/c = 0.2, L/T = 0.8, e/T = 0.10)
-23-
1.8
FE results
Eq. (8)
1.6
Y
1.4
1.2
1.0
0.8
0.6
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
a/T
Fig. 9 Comparison of Y values from FE analyses and Eq. (8) at the crack ends
(a/c = 0.2, L/T = 0.8, e/T = 0.10)
-24-
Fig. 10 Schematic stress distribution at the weld toe
-25-
3.6
km, L/T = 0.2
ks, L/T = 0.2
3.0
km, L/T = 0.8
ks, L/T = 0.8
km, L/T = 1.4
2.4
ks, L/T = 1.4
SCF
km, L/T = 2.0
ks, L/T = 2.0
1.8
1.2
0.6
0.00
0.05
0.10
0.15
e/T
Fig. 11 Comparison of ks values with corresponding km values
-26-
2.4
FE results
Eq. (12)
2.2
Y
2.0
1.8
1.6
1.4
1.2
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
a/T
Fig. 12 Comparison of Y values from FE analyses and Eq. (12) at the crack deepest point
(a/c = 0.2, L/T = 0.8, e/T = 0.10)
-27-
2.4
2.1
FE results
Eq. (12)
Y
1.8
1.5
1.2
0.9
0.6
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
a/T
Fig. 13 Comparison of Y values from FE analyses and Eq. (12) at the crack ends
(a/c = 0.2, L/T = 0.8, e/T = 0.10)
-28-
Fig. 14 Local geometry details of width of weld bead with varying L/T but constant e/T (α is the
slope at the surface from the weld to the base material)
-29-
2.10
1.95
Y
1.80
1.65
FE. L/T = 0.2
FE. L/T = 0.8
FE. L/T = 1.4
FE. L/T = 2.0
Eq. (8) L/T = 0.2
Eq. (8) L/T = 0.8
Eq. (8) L/T = 1.4
Eq. (8) L/T = 2.0
1.50
1.35
1.20
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
a/T
Fig. 15 Effect of L/T on Y obtained by FE analyses and Eq. (8) at the crack deepest point
(a/c = 0.2, e/T = 0.10)
-30-
1.7
1.6
1.5
1.4
1.3
Y
1.2
1.1
FE. L/T = 0.2
FE. L/T = 0.8
FE. L/T = 1.4
FE. L/T = 2.0
Eq. (8) L/T = 0.2
Eq. (8) L/T = 0.8
Eq. (8) L/T = 1.4
Eq. (8) L/T = 2.0
1.0
0.9
0.8
0.7
0.6
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
a/T
Fig. 16 Effect of L/T on Y obtained by FE analyses and Eq. (8) at the crack ends
(a/c = 0.2, e/T = 0.10)
-31-
40
35
30
% of Values
25
20
15
10
5
0
-10 to -8 -8 to -6 -6 to -4 -4 to -2 -2 to 0
0 to 2
2 to 4
4 to 6
6 to 8 8 to 10
Error of Prediction (%)
Fig. 17 Error histogram for the misaligned butt welds at the crack deepest point
-32-
40
35
30
% of Values
25
20
15
10
5
0
-10 to -8 -8 to -6 -6 to -4 -4 to -2 -2 to 0
0 to 2
2 to 4
4 to 6
6 to 8
8 to 10
Error of Predition (%)
Fig. 18 Error histogram for the misaligned butt welds at the crack ends
-33-
2.2
e/T
Eqn. e/T 
e/T 
Eqn. e/T 
e/T 
Eqn. e/T 
e/T 
Eqn. e/T 
2.0
1.8
Y
1.6
1.4
1.2
1.0
0.8
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
a/T
Fig. 19 Comparison of proposed equations with the FE data at the crack deepest point
(a/c = 0.4, L/T = 0.8)
-34-
3.0
2.7
2.4
2.1
Y
1.8
e/T 
Eqn. e/T 
e/T 
Eqn. e/T 
e/T 
Eqn. e/T 
e/T 
Eqn. e/T 
1.5
1.2
0.9
0.6
0.3
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
a/T
Fig. 20 Comparison of proposed equations with the FE data at the crack ends
(a/c = 0.4, L/T = 0.8)
-35-
Table 1 Table of analyses
Loading
L/T
a/T
a/c
e/T
m = axial
0.2
0.05,0.1,0.15,0.2,0.3,0.4,
0.5,0.6,0.7,0.8,0.9
0.1,0.2,0.4,0.7,1.0
0.00,0.05,0.10,0.15
m = axial
0.8
0.05,0.1,0.15,0.2,0.3,0.4,
0.5,0.6,0.7,0.8,0.9
0.1,0.2,0.4,0.7,1.0
0.00,0.05,0.10,0.15
m = axial
1.4
0.05,0.1,0.15,0.2,0.3,0.4,
0.5,0.6,0.7,0.8,0.9
0.1,0.2,0.4,0.7,1.0
0.00,0.05,0.10,0.15
m = axial
2.0
0.05,0.1,0.15,0.2,0.3,0.4,
0.5,0.6,0.7,0.8,0.9
0.1,0.2,0.4,0.7,1.0
0.00,0.05,0.10,0.15
-36-
Table 2 Y values of present FE analyses and Newman and Raju [15] equation
a/T
a/c
0.05
0.1
1.136
0.05
0.4
0.05
FE
Deepest point
Newman and
Raju
Crack ends
Newman and
Raju
Diff. (%)
FE
1.110
2.3
0.398
0.387
2.8
0.981
0.953
2.9
0.643
0.664
3.2
1.0
0.684
0.663
3.2
0.751
0.730
2.9
0.3
0.1
1.407
1.361
3.4
0.480
0.487
1.4
0.3
0.4
1.057
1.039
1.7
0.716
0.744
3.8
0.3
1.0
0.704
0.675
4.3
0.793
0.764
3.8
0.7
0.1
2.547
2.516
1.2
1.006
1.012
0.6
0.7
0.4
1.302
1.353
3.8
1.106
1.088
1.7
0.7
1.0
0.737
0.734
0.4
0.957
0.934
2.5
0.9
0.1
/
3.540
/
1.480
1.549
4.5
0.9
0.4
/
1.492
/
1.288
1.306
1.4
0.9
1.0
/
0.779
/
1.046
1.078
3.0
-37-
Diff. (%)
Table 3 Y values from FE analyses and BS 7910:2013 [14] assessment equation
Deepest point
a/T
a/c
0.05
Crack ends
FE
BS 7910:2013
Diff. (%)
FE
BS 7910:2013
Diff. (%)
0.2
1.479
1.363
-8.5
0.877
0.678
-29.4
0.1
0.2
1.411
1.361
-3.7
0.821
0.686
-19.7
0.15
0.2
1.401
1.375
-1.9
0.819
0.702
-16.7
0.2
0.2
1.411
1.407
-0.3
0.821
0.729
-12.6
0.3
0.2
1.479
1.477
-0.1
0.868
0.792
-9.6
0.4
0.2
1.585
1.577
-0.5
0.936
0.880
-6.4
0.5
0.2
1.710
1.699
-0.6
1.022
0.991
-3.1
0.6
0.2
1.829
1.837
+0.4
1.127
1.124
-0.3
0.7
0.2
1.948
1.980
+1.6
1.245
1.276
+2.4
0.8
0.2
/
2.119
/
1.362
1.443
+3.5
0.9
0.2
/
2.242
/
1.485
1.617
+8.2
Note: “-” and “+” denote underestimation and overestimation, respectively.
-38-
Table 4 Notch stress concentration factors of misaligned butt welds
e/T
L/T
0.00
0.05
0.10
0.15
0.2
1.0
1.783
2.300
2.676
0.8
1.0
1.383
1.766
2.162
1.4
1.0
1.286
1.579
1.890
2.0
1.0
1.237
1.480
1.737
-39-
Table 5 Distribution of data points for varying percentage errors in Fig. 17-18
Percentage
error (%)
Deepest point
Number of data Percentage of data
points
points (%)
Crack ends
Number of data
Percentage of data
points
points (%)
-10 to -8
14
2
9
1
-8 to -6
14
2
18
2
-6 to -4
36
5
35
4
-4 to -2
72
10
70
8
-2 to 0
239
33
265
30
0 to 2
217
30
273
31
2 to 4
79
11
123
14
4 to 6
29
4
35
4
6 to 8
14
2
26
3
8 to 10
7
1
26
3
Total
720
100
880
100
-40-