Welding in the World (2020) 64:2053–2074
https://doi.org/10.1007/s40194-020-00982-4
RESEARCH PAPER
Determination of notch factors for welded butt joints based
on numerical analysis and metamodeling
Markus Oswald 1
1
& Josef Neuhäusler & Klemens Rother
1
Received: 10 October 2019 / Accepted: 19 August 2020 / Published online: 6 September 2020
# International Institute of Welding 2020
Abstract
Fatigue evaluation based on the effective notch stress approach requires stress concentration factors for idealized notch geometries. In
this paper, stress concentration factors are calculated numerically for different idealizations of the weld geometry. The joints covered
here are one-sided Y-butt joints and two-sided DY-butt joints, each with partial and full penetration. The variable parameters of the
finite element models are the weld flank angle, the notch radius, the sheet thickness, the ratio of weld seam width to sheet thickness,
and the ratio of height of non-fused root face to sheet thickness. Existing estimation formulae for stress concentration factors will be
compared with new methods for stress concentration factor estimation: (a) polynomial regression using mixed terms and (b) artificial
neural networks. These two methods show similar or superior quality compared with the existing estimations which is expressed
through lower estimation errors with respect to the numerically calculated stress concentration factors. The database for the regression
analysis using methods (a) and (b) consists of a total of 11,871 design alternatives obtained by finite element analysis. In addition to
the improved quality of prognosis, the range of allowable geometrical parameters is significantly increased compared with the
existing formulae. The methods (a) and (b) provide a possibility to a fast estimation of stress concentration factors of sufficient
quality for a large range of geometrical weld seam parameters. The presented formulae could also be part of a programmed solution.
Keywords Elastic analysis . Finite element analysis . Mathematical models . Sampling . Notches . Butt joints
Nomenclature
Symbols, abbreviations
ANN [−]
Artificial neural network
β [°]
Flank angle
bi [−]
Bias vectors for artificial neural networks
ck [−]
Scalar multiplication parameter for the
PRC method
d [mm]
Total model depth
E [MPa]
Young’s modulus
errrel [%]
Relative error
F[N]
Force
f [−]
fk [−]
g [−]
h [mm]
Kf [−]
Kt [−]
Kt, EST [−]
Kt, FEM [−]
kt [−]
Kt, AKS [−]
Recommended for publication by Commission XIII - Fatigue of Welded
Components and Structures
* Markus Oswald
markus.oswald0@hm.edu
1
Department of Mechanical, Automotive and Aeronautical
Engineering, Munich University of Applied Sciences, Dachauer Str.
98b, 80335 Munich, Germany
Kt, ANN [−]
Kt, LER [−]
Kt, PRC [−]
Kt, RAI [−]
Kt, YL [−]
Ratio of weld seam width to sheet thickness
Value of geometric multiplication parameter
for the PRC method
Input vector for the ANN method
Height of non-fused root face
Fatigue notch factor
Stress concentration factor
Stress concentration factor, estimated
Stress concentration factor, calculated by
FEM
Stress concentration output vector of the
ANN method
Stress concentration factor of Anthes
et al. method
Stress concentration factor of the ANN method
Stress concentration factor of Lehrke’s method
Stress concentration factor of the PRC method
Stress concentration factor of Rainer’s method
Stress concentration factor of Yung
and Lawrence’s method
Kw [−]
Ratio of notch stress to structural stress
2054
Kw, min [−]
Minimum ratio of notch stress to structural
stress
L [mm]
Sheet length
M [N/mm] Moment
m [−]
Weld angle exponent in Lehrke’s method
ν [−]
Poisson ratio
Φi [−]
Artificial neural network layer potential
PRC [−]
Polynomial regression with coupling terms
r [mm]
Notch radius
Sb [MPa]
Nominal bending stress
St [MPa]
Nominal tension stress
σe [MPa]
Notch stress
σw [MPa] Structural stress
t [mm]
Sheet thickness
u [−]
Weld reinforcement
w [mm]
Width of weld seam
Wi [−]
Weight matrices of artificial neural networks
xi,gain [−]
Gain input vector for artificial neural networks
xi,offset [−] Offset input vector for artificial
neural networks
yo,gain [−]
Gain output vector of artificial
neural networks
yo,offset [−] Offset output vector of artificial
neural networks
z [−]
Ratio of height of non-fused root face to
sheet thickness
Indices, superscripts
b, bend Bending
t, tens
Tension
f. p.
Full penetration
k
PRC method index
p. p.
Partial penetration
r, root
Root
toe
Toe
Y
Y-butt joint
DY
DY-butt joint
1 Introduction
The IIW-recommendations [1] and associated literature, e.g.,
[2–4], provide a method for the estimation of analytical fatigue strength of welded components by using the effective
notch stress approach. That requires the knowledge of a stress
concentration factor which describes the stress increase
caused by notches at the weld seam. Further information can
be found in the corresponding paper on cruciform joints [5].
The notch stress concept as it is found in the IIWrecommendations [1] was motivated by early works of
Radaj [6]. Radaj derived a notch stress model with a fictitious
rounding of the notch at welded joints. That idea originates
from Neuber’s microstructural support effect [7] and assumes
worst case conditions of the actual notch radii in toes and roots
Weld World (2020) 64:2053–2074
of welded joints. In the 1980s, Seeger and co-workers at
Darmstadt University developed the effective notch stress
concept [8] as it is found in [1]. The method provides a modeling guideline for unified modeling of weld geometries using
discrete radii in weld toes and roots. The calculated stress
values from such standardized models can then be used with
a concept conforming S-N curve to estimate fatigue lives. The
S-N-data have been back calculated from experimental fatigue
testing of different welded joints with the respective rules for
modeling and consequently cover scatter of fatigue strength,
e.g., influence of geometrical details and material parameters.
Therefore, the stress concentration factor Kt can be interpreted
as an effective fatigue notch factor Kf.
Numerical calculation of stress concentration factors using
finite element analysis requires the creation of a suitable model with the respective idealized weld geometries. That effort
still deters practitioners from performing such detailed estimations. Consequently, existing formulae are used which can be
purely analytical, purely empirical, or both combined. Some
solutions for the stress concentration factor of DY-butt welded
joints have been listed in [2]. To mention are the methods by
Yung and Lawrence [9, 10], Rainer [11, 12], and Anthes et al.
[13, 14] as well as by Lehrke [15]. The database for singlesided Y-butt joints is very restricted.
In this paper, an excessive database of 11,871 design samples
is created and two methods, one based on polynomial regression
(PRC) and one on artificial neural networks (ANN), are introduced to predict stress concentration factors for a wide variety of
single-sided Y-butt joints and double-sided DY-butt joints. The
results show improved quality of prognosis as well as an extension of the applicable range of geometrical parameters which
increases the applicability of the notch stress method compared
with the nominal or the structural stress concepts.
2 Numerical simulation of welded butt joints
The joint types covered in this study are one-sided Y-butt
joints or double-sided DY-butt joints, both with full or partial
penetration. Axial or angular misalignment of the attached
sheets is not covered in this paper. Possible misalignment
must be considered in the stress analysis as secondary effect
or the allowable stresses have to be reduced.
2.1 Parametrization
Each finite element model, one for the Y-butt joint and one for
the DY-butt joint, with parametric geometry were modeled
using ANSYS Mechanical™ 18.11 (see Fig. 1 and Table 1)
[16]. The models were loaded under pure tension and under
1
ANSYS Mechanical™ is a trademark of ANSYS, Inc., Canonsburg, PA,
USA, see http://www.ansys.com
Weld World (2020) 64:2053–2074
2055
Fig. 1 Parametrized geometry of the Y-butt joint (full lines) and DY-butt joint (dotted lines)
pure bending for a wide variety of geometrical parameters. For
each joint type, fully and partially penetrating welds are
considered.
The following assumptions are used:
&
&
&
&
&
&
&
&
Symmetric geometry for the model of the DY-butt joint
Equal sheet thicknesses
No axial or angular misalignment
No nonlinear contact in the non-fused root face
Equal reference radii of weld toe and weld root, modeled
as fillet radii
The weld reinforcement is modeled by a B-spline
Plane strain condition
Constant parameters for linear elastic material:
&
&
Young’s modulus E = 210GPa
Poisson ratio ν = 0.3
Table 1
Range
Flank angle [°]
Reference radius [mm]
β
r
[5; 70]
see Table 2
Sheet thickness [mm]
Height of non-fused root face to
sheet thickness ratio
t
[0.5; 100]
[0] for fully penetrated
welds
[0.02; 0.7] for partially
penetrated welds
Weld seam width to thickness ratio
f ¼ wt
z ¼ ht
&
Uniform tension or bending nominal stress of St = Sb =
1MPa applied along the sheet end faces t (see Fig. 1)
Evaluation of maximum principle stress σ1
According to the IIW-recommendations [1] and DVSMerkblatt 0905 [3], the reference radius depends on the sheet
thickness when stress concentration factors are calculated for
the effective stress method. Three ranges of sheet thicknesses
were defined. For each range, an adequate reference radius
can be assigned which can be found in Table 2. The parameter
ranges in Table 2 exceed the ranges of almost all of the
existing empirical solutions for stress concentration factors.
Table 2
z
Parameter ranges of numerical model
Parameter
&
[0.5; 4]
Parameter ranges for space filling Latin hypercube samplings
t [mm]
Subsystem Y-butt joint
1 0
[0.5; 7.5]
2 0
[3.6; 25]
3 0
[10; 100]
4 [0.02; 0.65] [0.5; 7.5]
5 [0.02; 0.65] [3.6; 25]
6 [0.02; 0.65] [10; 100]
Subsystem DY-butt joint
1 0
[0.5; 7.5]
2 0
[3.6; 25]
3 0
[10; 100]
4 [0.02; 0.7]
[0.5; 7.5]
5 [0.02; 0.7]
[3.6; 25]
6 [0.02; 0.7]
[10; 100]
r [mm]
f
β [°]
Samples
0.05
0.3
1
0.05
0.3
1
[1; 4]
[1; 4]
[1; 4]
[1; 4]
[1; 4]
[1; 4]
[5; 65]
[5; 65]
[5; 65]
[5; 65]
[5; 65]
[5; 65]
991
996
998
999
996
993
0.05
0.3
1
0.05
0.3
1
[0.5; 4]
[0.5; 4]
[0.5; 4]
[0.5; 4]
[0.5; 4]
[0.5; 4]
[5; 70]
[5; 70]
[5; 70]
[5; 70]
[5; 70]
[5; 70]
964
996
992
956
994
996
2056
Weld World (2020) 64:2053–2074
Fig. 2 Finite element model for effective notch stress analysis including boundary conditions. Left side: notch areas with mapped mesh of hex element
with mesh size 0.05r. Right side: Evaluation of the first principle stress at the weld toe and non-fused root face
2.2 Discretization
analytically exact plane strain state and a real structure
with an actual depth.
Mesh refinement of the notch radii is essential in order to get
reliable stress results. A convergence study was conducted
and showed converged notch stresses for the following mesh
parameters in the notch: A mapped mesh of quadratic
PLANE183 elements [17], element lengths of 0.05 · r, and
refinement depths of 0.4 · r (see Fig. 2). The rest of the joint
consists of an unstructured mesh with PLANE 183 elements
with an adequate transition to the locally refined meshes in the
notches.
2.3 Applicability of 2D-modeling vs. 3D-structures
The finite element calculations in this paper are carried
out with 2D models and plane strain conditions.
Differences between plane stress and plane strain condition must be considered. The formation of a plane strain
state requires a minimum total depth of the geometry. The
designer is advised to consider the difference between the
Table 3
ranges
Each type of butt joint is split into six subsystems according to
the respective range of sheet thicknesses the existence or nonexistence of a weld root face (see Table 2). This is based on the
recommendations given in [3]. A total of 1000 design alternatives or samples were created for each subsystem using a
Space Filling Latin Hypercube Sampling with optiSLang®
6.1.02 [18]. A total of 5973 samples were created for the Ybutt joint and 5898 samples for the DY-butt joint automatically. A total of 129 samples failed because of extreme parameter
combinations.
The spatial distribution of the samples has already been
shown in the previous investigation of stress concentration
factors for welded cruciform joints [5].
2.5 Resulting stress concentration factors
Simulated radii and corresponding stress concentration factor
Y-butt joint
DY-butt joint
Table 4
2.4 Sampling
Radius r [mm]
Stress concentration factor range Kt
0.05
0.3
1
0.05
0.3
1
≤75.177
≤64.312
≤65.881
≤21.245
≤15.636
≤16.729
3 Known methods of notch factor estimation
of butt-welded joints
Restrictions of Yung and Lawrence’s method
Parameter combination
Restriction
t
r
1…60
For each radius and joint type, the stress concentration factors
are within about the same range (see Table 3). The recommended ranges, the ratio between sheet thickness and applied
notch radius, are therefore reasonable. The maximum stress
concentration factors in Table 3 are all at the root of partially
penetrated joints under tension loading.
β [°]
f
z
10 ° …45°
1
–
A choice of existing methods for stress concentration factor
estimation is described in the following. All of the existing
2
optiSLang® is a trademark of Dynardo GmbH, Weimar, Germany, see
https://www.dynardo.de/software/optislang.hmtl
Weld World (2020) 64:2053–2074
Table 5
2057
Method by Yung and Lawrence
Non-fused Root
face
Position
Loading
No
Weld
toe
Tension
Equation
1
K t;YL;toe;tens ¼ 1 þ 0:27ðtanβÞ4
Bending
and both new methods are based on maximum principle stress
σ1.
3.1 Method by Yung and Lawrence
Yung and Lawrence [9, 10] approximation formulae are
valid for DY-butt joints without non-fused root face under
tension and bending loading. The limiting ranges are given in Table 4. The method requires the sheet thickness t,
notch radius r, and the weld toe angle β for their formulae
given in Table 5.
3.2 Method by Rainer
Rainer’s stress concentration factors are based on a study on
notch factors of welded joints in 1985 [11, 12]. The approximation formulae cover DY-butt joints and Y-butt joints, both
with fully penetrated welds under tension and bending loading. The parameter ranges are given in Table 6. The geometric
parameters required for the formulae in Table 7 are the sheet
thickness t, notch radius r, and the weld reinforcement u.
3.3 Method by Lehrke
Lehrke published stress concentration factors for DY-butt joints
with and without non-fused root face under tension loading [15].
The restrictions for his method are given Table 8. His formula for
the weld toe is derived using stress intensity factors at V-shaped
sharp notches as well as Neuber’s notch stress solution for hyperbolic notches. His formula for the weld root is also based on
stress intensity factors.
The formulae in Table 9 were verified using discrete parameter combinations calculated by BEM. The restrictions in
Table 8 are in accordance with his comparative investigations.
Note that the formula for the notch stress is restricted to joint
with equal sheet thickness and weld seam width between the
weld toes (t = w).
Table 6
pffit
(1)
r
1
K t;YL;toe;bend ¼ 1 þ 0:165ðtanβÞ6
pffit
(2)
r
Lehrke is using the parameters sheet thickness t, notch
radius r, and weld toe angle β for the stress concentration
factors at the weld toe. For the partially penetrated DY-butt
joint, the formula for the stress concentration factors at the
weld root contains the non-fused root face to thickness ratio
z and the non-fused root face length z ∙ t. The factor m is an
exponent dependent on β according to tables given in [15].
3.4 Method by Anthes et al.
The method by Anthes et al. [13, 14] is valid for the DY-butt
joint without non-fused root face exposed to tension and bending loading. His restrictions are given in Table 10. The formulae in Table 11 are based on boundary element simulation
results. Required parameters are sheet thickness t, notch radius
r, and the weld toe angle β.
4 New methods of notch factor determination
Recent investigations on the determination of notch factors for
welded cruciform joints [5] have already proven the adaptability of the two new regression methods, polynomial regression,
and artificial neural networks.
4.1 Polynomial regression with coupling terms (PRC
method)
The software optiSLang® 7.2.0 is used to perform a regression analysis on the numerically calculated finite element data.
Polynomial regression functions with quadratic order and coupling terms eventually lead to the formulation in Eq. (11). The
regression formulae for the PRC method had to be additionally restricted to the geometry parameter ranges in Table 12.
The four parameters (t, β, f, z), their squares, each combination
of two parameters, and an additional constant term lead to 15
terms in this formula. The factors according to Appendix
Restrictions of Rainer’s method
Parameter combination
Restriction
t
r
DY-butt joint: 0…400
Y-butt joint: 0…200
β [°]
f
z
45°
–
–
u
t
DY-butt joint: 0…2.5
Y-butt joint: 0…5
2058
Table 7
Weld World (2020) 64:2053–2074
Method by Rainer
DY-butt
joint
Non-fused root face
Position
Loading
No
Weld toe
Tension
Equation
K t;RAI;toe;tens;DY ¼ 1 þ
Bending
K t;RAI;toe;bend;DY ¼ 1 þ
Y-butt joint
No
Weld toe
0:8
þ 1:1
t
2r
K t;RAI;toe;bend;Y ¼ 1 þ
þ 0:2
2r
t
2r
!−12
ð2rt þurÞðurÞ
(3)
1:33
!−12
2:25
t
1þ2rt
2r
pffiffitffi
þ 0:2 t u u 1:33
0:66 þ 3:8
t
ðurÞ
ð2rþr Þðr Þ
2r
2r
Bending
pffiffitffi
2:2
(4)
0:4
Tension
K t;RAI;toe;tens;Y ¼ 1 þ
Table 8
0:55
ðurÞ
1þ2rt
0:55
ðurÞ
0:8
1þrt
þ 1:1 t pffit
r
2:2
r
t
r
þ 0:2 t u u 1:33
ðr þ r Þð r Þ
!−12
(5)
1
ðtþutÞ
!−12
2:25
t
1þrt
r
pffit
þ 0:2 t u u 1:33
0:66 þ 3:8
t
ðurÞ
ðr þ r Þð r Þ
r
r
1:1
(6)
0:4
Restrictions of Lehrke’s method
Parameter combination
Restriction
t
r
Toe: 20…200
Root: −
β [°]
f
z
Toe: 10 ° …60°
Root: −
Toe: 1
Root: −
Toe:−
Root: <0.7
Tables 21, 22, 23, 24, 25, and 26 can be chosen according to
the nomenclature in Table 13. Note that standard filter settings
of optiSLang are used, similar to [5].
Each combination of stress location (weld root or weld
toe), loading (tension or bending), and reference radius leads
to 18 regression formulae for the Y-butt joint and 18 for the
DY-butt joint.
15
K t;PRC ¼ ∑ ck f k ðt; β; f ; zÞ
2r
zt
t
w
Toe:−
Root: 0.01…1.0
1
More information about the neural network can be found in [5,
19] (Tables 13 and 14).
The restrictions of the ANN method are given in Table 14.
The mathematical expressions for the used network can be
found in Table 15,3 and the corresponding normalization vectors, weighting matrices and bias vectors can be found
Appendix Tables 27, 28, 29, and 30.
ð1Þ
k¼1
The regression formulae for the PRC method had to be
additionally restricted to the geometry parameter ranges in
Table 12.
4.2 Application of artificial neural networks
Matlab’s Neural Network Toolbox provides sufficient and
graphically guided tools to fit neural networks to sample data.
The feed-forward network employed in this paper consists of
three so called hidden layers. One hidden layer consists of five
neurons which is in accordance with the number of input
variables (t, β, f, z, r). In the case of partially penetrated welds,
the output layer consists of four neurons for the stress concentration factors at weld toe and root under bending and tension
loading (kt = (Kt, ANN, root, b, Kt, ANN, root, t, Kt, ANN, toe, b, Kt,
ANN, toe, t)). For fully penetrated welds the output layer consists
of two neurons (kt = (Kt, ANN, toe, b, Kt, ANN, toe, t)), see Fig. 3.
5 Comparison of notch factor determination
and quality
Due to all of the authors giving restrictions to their models for
comparison of the results, the following has to be considered:
–
–
–
3
The limitations of the compared methods are summarized
in Tables 16 and 17. Note that some of the existing
methods reviewed in this paper show significant restrictions of the design space.
The method by Lehrke was not furtherly considered since
resulting stress concentration factors differed by a factor
1.5 and higher from the finite element results.
Similar to [5], stress concentration factors below unity
were neglected. The designer is advised to check for a
minimum ratio of notch stress to structural stress
Kw = σe/σw , see [20]:
∘ indicates the elementwise Hadamard product, ⊘ the elementwise
Hadamard division
Weld World (2020) 64:2053–2074
Table 9
2059
Method by Lehrke
Non-fused root
face
Pos.
Loading
Equation
No
Weld toe
Tension
m
2ð2tanβÞ
K t;LER;toe;tens ¼ βþ1sin2 ð2βÞ 2rt ð0:5Þm
Yes
Weld
root
Tension
1−2m
(7)
2
&
&
&
Kw, min = 1.6 for r = 1mm
Kw, min = 2.13 for r = 0.3mm
Kw, min = 3.56 for r = 0.05mm
Nevertheless, samples not exceeding these ratios were used
in regression as well, since these ratios always have to be
checked by the user himself, depending on parameter combination and loading.
As already shown in the investigation of welded cruciform
joints [5], the quadratic extrapolation method leads to the lowest structural stresses. This also leads to the most nonconservative results for Kw, min.
Due to the given limitations some of the results have to be
neglected.
5.1 Comparison of methods for Y-butt joints
All following figures show:
–
–
The resulting relative errors for all investigated methods
as boxplots on the left side: The blue boxes include 50%
of the data points, from the 25% to the 75% quartile. The
red line inside the blue box indicates the median relative
error. The black whiskers outside the blue box embed 1.5
times the interquartile range from the 25% to the 75%
quartile of the data. The remaining red crossed data points
are outside this range.
On the right side of each figure are probability plots for
normal distributions. The dashed straight lines indicate a
perfect normal distribution.
The relative errors are calculated by Eq. (17):
errrel ¼
K t;EST −K t;FEM
K t;FEM
Table 10
Restrictions of Anthes’ method
Parameter combination
Restriction
ð2Þ
t
r
0…200
β [°]
f
z
0 ° …90°
–
–
2
ffi
K t;LER;root;tens ¼ 1 þ pffiffiffiffiffiffiffiffiffiffi
cosðπz
2Þ
pffiffiffi
ffi
tz
(8)
2r
Figure 4 shows boxplots with simulation data fulfilling
all restrictions given by the different authors simultaneously for fully penetrated Y-joints. In Fig. 5, the
boxplots are generated with data fulfilling the restrictions
the authors have given individually for each method for
fully penetrated Y-joints. In Fig. 6, the boxplots are generated with data fulfilling the restrictions the authors have
given individually for each method with partially penetrated Y-joints. Further restrictions to the sample data in
Fig. 6 had to be set to obtain reasonable data statistics for
the PRC and ANN methods, see Table 18.
For the following the relative error is calculated by
errrel ¼
K t;EST −K t;FEM
K t;FEM
ð3Þ
Table 19 shows the percentage of data that had to be
neglected in comparison with the total data available for
evaluation. Remarkable are the low error values of fully
penetrated Y-butt joints for the PRC and ANN methods
that demonstrate the much larger range of application of
those metamodels. For partially penetrated welds, the
standard deviations in Table 19 are higher than those for
fully penetrated welds.
In Fig. 4 showing the simultaneous fulfillment of all restrictions, the method by Rainer and the two new methods cluster
the estimated stress concentration factors around the correct
values from finite element simulation. Rainer’s method is generally shifted to the safe side. The line for Rainer’s method on
the right side in Fig. 5 shows that most of the data is clustered
in the area from 7 to 31% relative error.
The ANN method shows the lowest scattering regarding
the 25% and 75% quantiles, followed by the PRC method and
the method by Rainer. The same applies to the 1% and 99%
quantiles.
Figure 6 generally shows a similar behavior in terms of
the performance of the ANN and PRC methods. Both
methods are clustering around a median of − 0.55% and
− 1.44% relative error with standard deviations of 7.08%
and 12.88%, respectively. The outliers in the boxplots on
the left show only samples outside the 1.5 times interquartile range from the 25 and 75% quantiles of all simulated
2060
Table 11
Weld World (2020) 64:2053–2074
Method by Anthes et al.
Non-fused root
face
Position Loading Equation
No
Weld
toe
0:2131 K t;AKS;toe;tens ¼ 1−0:138 rt
f1 þ ½0:169 þ 1:503sinβ−1:968ðsinβ Þ 2 þ 0:713ðsinβÞ 3 t 0:2491þ0:3556sinðβþ6:1937Þ
g
r
Bending
t 0:2070 K t;AKS;toe;bend ¼ 1−0:156 r
f1 þ ½0:181 þ 1:207sinβ−1:737ðsinβÞ 2 þ 0:689ðsinβÞ 3 t 0:2919þ0:3491sinðβþ3:2830Þ
g
r
Tension
samples and are caused by extreme parameter combinations. These outliers generally tend to the safe side. Nonfused root face to sheet thickness ratios higher than z = 0.4
should be avoided, especially in bending loading, where
the weld root is likely to fail (see Table 18).
5.2 Comparison of methods for DY-butt joints
Similar to the Y-butt joints, all figures show the resulting
relative errors for all investigated methods as boxplots on
the left side and probability plots for a normal distribution.
The errors are calculated again according to Eq. (18).
Figure 7 shows boxplots with simulation data fulfilling all
restrictions given by the different authors simultaneously. In
Fig. 8, the boxplots are generated with data fulfilling the restrictions the authors have given individually for each method.
Similar to the Y-butt joints, all figures show the resulting
relative errors for all investigated methods as boxplots on the
left side and probability plots for a normal distribution.
Table 20 shows the percentage of data that had to be
neglected in comparison with the total data available for evaluation. Remarkable are the low error values for the PRC and
ANN methods that demonstrate the much larger range of application of those metamodels.
Table 12
(9)
(10)
The errors are calculated again according to Eq. (17).
In Fig. 7 showing the simultaneous fulfillment of all restrictions, the ANN and PRC methods cluster the estimated stress
concentration factors around the correct values from finite
element simulation. The methods by Yung and Lawrence,
and Rainer and Anthes et al. are generally shifted to the unsafe
side, while the method by Anthes et al. has the strongest deviations from the simulation data.
The similar behavior is found in Fig. 8, where the
median values of the relative error for the ANN and
PRC methods deviate − 0.01% and − 0.29% from zero.
That is followed by Rainer’s method with a median value of 1.99%. The method by Anthes et al. is situated at
the unsafe side with a 1% quantile of − 82.61% and a
99% quantile of − 9.17%. The ANN method shows the
lowest scattering regarding the 25% and the 75%
quantiles, followed by the PRC method and the method
by Rainer, then the method by Yung and Lawrence and
Anthes et al.
Note that the ANN and PRC methods cover the whole
number of samples. The method by Anthes et al. neglect almost 50% of all samples. The methods by Yung and
Lawrence and Rainer neglect more than 90% of the data
(see Table 19).
Restrictions of the PRC method
Parameter combination
Y-butt joint
Restriction
DY-butt joint
Restriction
t
r
10…100
for r = 1.00mm
12…83.33
for r = 0.30mm
10…150
for r = 0.05mm
10…100
for r = 1.00mm
12…83.33
for r = 0.30mm
10…150
for r = 0.05mm
β [°]
f
z
5 ° …65°
1…4
0; 0.02…0.65
5 ° …70°
0.5…4
0; 0.02…0.7
Weld World (2020) 64:2053–2074
2061
Fig. 3 Schematic structure of the artificial network
Table 13
Stress concentration factor for maximum principle stress formulae according to new method
Non-fused weld root face ratio z
Restrictions
Loading
Sheet thickness t [mm]
Radius r [mm]
Full penetration:
z=0
Dimensions acc. to 1
Tension
1: [0.5; 7.5]
0.05
2: [3.6; 25]
0.3
3: [10; 100]
1
1: [0.5; 7.5]
0.05
2: [3.6; 25]
0.3
3: [10; 100]
1
1: [0.5; 7.5]
0.05
K p:p:
t;PRC;t;1
K p:p:;r
t;PRC;t;1
2: [3.6; 25]
0.3
K p:p:
t;PRC;t;2
K p:p:;r
t;PRC;t;2
3: [10; 100]
1
K p:p:
t;PRC;t;3
K p:p:;r
t;PRC;t;3
1: [0.5; 7.5]
0.05
K p:p:
t;PRC;b;1
K p:p:;r
t;PRC;b;1
2: [3.6; 25]
0.3
K p:p:
t;PRC;b;2
K p:p:;r
t;PRC;b;2
3: [10; 100]
1
K p:p:
t;PRC;b;3
K p:p:;r
t;PRC;b;3
Bending
Partial penetration: z = [0.02; 0.7]
Dimensions acc. to 1
Tension
Bending
Table 14
Weld toe
Weld root
f :p:
K t;PRC;t;1
f :p:
K t;PRC;t;2
f :p:
K t;PRC;t;3
f :p:
K t;PRC;b;1
f :p:
K t;PRC;b;2
f :p:
K t;PRC;b;3
Restrictions of the ANN method
Parameter combination
Y-butt joint
Restriction
DY-butt joint
Restriction
t
r
10…100
for r = 1.00mm
12…83.33
for r = 0.30mm
10…150
for r = 0.05mm
10…100
for r = 1.00mm
12…83.33
for r = 0.30mm
10…150
for r = 0.05mm
β[°]
f
z
5 ° …65°
1…4
0; 0.02…0.65
5 ° …70°
0.5…4
0; 0.02…0.7
2062
Weld World (2020) 64:2053–2074
Table 15
Formulae of the ANN method
ϕ1 = b1 + W1 · (((g − xi, offset) ∘ xi, gain) − 1)
ϕ2 = b2 + W2 · tanh(ϕ1)
ϕ3 = b3 + W3 · tanh(ϕ2)
ϕ4 = b4 + W4 · tanh(ϕ3)
kt = ((ϕ4 − yo, offset) ⊘ yo, gain) − 1
factors were calculated with respect to maximum principle
effective notch stress under the assumption of a plane stress
condition.
The investigated existing methods to calculate stress concentration factors show significant restrictions of the ranges of
application, whereas the ANN and PRC methods cover almost
all of the calculated design alternatives. In accordance with the
effective notch stress concept, three different notch radii, varying flank angles, and weld seam width for thin, medium, and
thick walled butt joints are covered. The ANN and PRC
methods show comparable predictive qualities of stress concentration factors and a significantly low scatter with respect
to relative errors.
The ANN and PRC methods provide means for quickly
estimating stress concentration factors without the lack of accuracy. Time-consuming finite element model generation can
be avoided. Both methods are implemented as a user friendly
programmed solution and can be accessed by http://rother.
userweb.mwn.de/scf-predictor.html.
(12)
(13)
(14)
(15)
(16)
6 Conclusion
A broad range of parameter variations were investigated for
Y-butt and DY-butt welded joints with full and partial penetration (with and without non-fused root face) under tension
and bending loading. The numerically calculated stress concentration factors of 11,871 design alternatives were used as
reference to calculate relative errors of the predicted stress
concentration factors of the presented existing and the newly
introduced ANN and PRC methods. The stress concentration
Table 16
Overview of all restrictions given by the authors, Y-butt joint
β[°]
f
z
Rainer 0…200
45°
–
–
PRC and 10…100
ANN for
r = 1.00mm
12…83.33
for
r = 0.30mm
10…150
for
r = 0.05mm
5°…
65°
1… 0; 0.02…
4
0.65
t
r
Table 17
u
t
2r
zt
0… –
5
–
–
Funding information The IGF project 19450 N of FOSTA Forschungsvereinigung Stahlanwendung e. V., Düsseldorf, is funded by
the Federal Ministry of Economic Affairs and Energy via the AiF within
the framework of the program for the promotion of the Industrielle
Gemeinschaftsforschung (IGF) based on a resolution of the German
Bundestag.
r
[mm]
–
0.05;
0.3;
1
Overview of all restrictions given by the authors, DY-butt joint
t
r
Yung and Lawrence
Rainer
Lehrke
Anthes et al.
PRC and ANN
1…60
0…400
Toe:
20…200
Root:
−
0…200
10…100
for r = 1.00mm
12…83.33
for r = 0.30mm
10…150
for r = 0.05mm
β [°]
f
z
10 ° …45°
45°
Toe:
10 ° …60°
Root:
−
0 ° …90°
5 ° …70°
1
–
Toe:
1
Root:
−
–
0.5…4
–
–
Toe:
−
Root:
<0.7
–
0; 0.02…0.7
r [mm]
u
t
2r
zt
–
0…2.5
–
–
–
–
–
Toe:
−
Root:
0.01…1
–
–
–
–
–
–
0.05;
0.3;
1
Weld World (2020) 64:2053–2074
2063
Fig. 4 Boxplot and probability plot of the relative errors for normal distribution—generated with data fulfilling all restrictions simultaneously, Y-butt
joint, full penetration
Fig. 5 Boxplot and probability plot of the relative errors for normal distribution—generated with data fulfilling the individual restrictions given for each
method, Y-butt joint, full penetration
2064
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Fig. 6 Boxplot and probability plot of the relative errors for normal distribution—generated with data fulfilling the individual restrictions given for each
method, Y-butt joint, partial penetration
Table 18
Further restrictions for partially penetrated Y-butt welds
PRC and ANN
f
z
1…3
0; 0.02…0.4
Table 19 Statistical data of all evaluated parameter combination, sorted by evaluation method, quantiles counted, and % of samples used for the
respective method, Y-butt weld
Y-butt joints, full penetration, full
sample range
Y-butt joints, partial
penetration,z < 0.4; f < 3
By restrictions neglected results
Rainer
90.05%
PRC
0.00%
ANN
0.00%
PRC
20.10%
ANN
20.10%
Total number of samples used by this method
Mean
Standard deviation
1% quantile
10% quantile
Median
90% quantile
99% quantile
297
15.56%
3.89%
9.16%
11.20%
14.64%
20.95%
26.84%
2985
0.61%
2.87%
−6.82%
−2.83%
−0.04%
3.66%
8.70%
2985
0.00%
0.8%
−2.13%
−0.83%
−0.14%
−0.86%
−2.44%
2385
−0.80%
12.88%
−30.76%
−11.74%
−1.44%
9.23%
48.76%
2385
−0.34%
7.08%
−15.30%
−5.88%
−0.55%
7.82%
26.21%
Statistical data: relative error
Weld World (2020) 64:2053–2074
2065
Fig. 7 Boxplot and probability plot of the relative errors for normal distribution—generated with data fulfilling all restrictions simultaneously, DY-butt
joint
Fig. 8 Boxplot and probability plot of the relative errors for normal distribution—generated with data fulfilling the individual restrictions given for each
method, DY-butt joint
2066
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Table 20 Statistical data of all evaluated parameter combination, sorted by evaluation method, quantiles counted, and % of samples used for the
respective method, DY-butt weld
Yung and Lawrence Rainer
Statistical data: relative error By restrictions neglected results
98.44%
Total number of samples used by this method 92
Mean
−4.86%
Standard deviation
13.37%
1% quantile
−25.31%
10% quantile
−20.60%
Median
−7.91%
90% quantile
15.60%
99% quantile
28.88%
Anthes et al. PRC
95.34%
49.95%
275
2952
−0.92% −54.17%
10.58%
17.48%
−39.50% −82.61%
−14.22% −74.97%
1.99% −57.12%
8.72% −28.53%
13.62%
−9.17%
ANN
0.00%
0.00%
5898
5898
0.08%
−0.10%
4.25%
3.86%
−12.47% −12.77%
−3.44%
−2.88%
−0.29%
−0.01%
4.66%
3.64%
13.41%
12.84%
Appendix
Table 21
k
Regression formulae for PRC method for fully penetrated welds, Y-butt joint
fk
ck
f :p:
K t;PRC;t;1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
1
β
t
f
z
β2
t2
f2
z2
tβ
tf
tz
βf
βz
fz
f :p:
K t;PRC;t;2
f :p:
K t;PRC;t;3
f :p:
K t;PRC;b;1
f :p:
K t;PRC;b;2
f :p:
K t;PRC;b;3
0.661293
0.0575027
0.195294
0.741951
0.0490416
0.0434367
0.718577
0.0509534
0.0123124
0.713017
0.0517815
0.194359
0.771148
0.045794
0.0428935
0.756609
0.0470005
0.0121168
− 0.000618949
− 0.0252762
− 0.000559861
− 0.00146674
− 0.000574768
− 0.000109845
− 0.000446401
− 0.0284219
− 0.000439067
− 0.00165066
− 0.000440664
− 0.000122831
0.00706477
0.00150921
0.000425281
0.00859824
0.00185433
0.000521435
Weld World (2020) 64:2053–2074
Table 22
k
2067
Regression formulae for PRC method for partially penetrated welds at weld toe, Y-butt joint
ck
fk
K p:p:
t;PRC;t;1
K p:p:
t;PRC;t;2
K p:p:
t;PRC;t;3
K p:p:
t;PRC;b;1
K p:p:
t;PRC;b;2
p:p:
K t;PRC;b;3
0.410249
1.05919
1.07822
1.07051
0.0553706
0.195288
− 0.628827
2.29206
− 0.000426015
− 0.0293261
0.211841
0.828583
0.00958136
− 0.0144487
0.0754203
− 0.00473305
0.0273906
− 1.27938
0.0507735
0.0411134
− 0.573193
2.19835
− 0.000444603
− 0.00156409
0.178054
0.693032
0.00196678
− 0.00225828
0.0138402
− 0.00319661
0.0193688
− 1.09865
0.0518318
0.012609
− 0.600986
2.21624
− 0.000442787
− 0.000127437
0.190742
0.698863
0.0005628
− 0.000755825
0.00426811
− 0.00364676
0.0213756
− 1.14271
1
1
0.342074
0.362813
2
3
4
5
6
7
8
9
10
11
12
13
14
15
β
t
f
z
β2
t2
f2
z2
tβ
tf
tz
βf
βz
fz
0.0382746
0.110529
0.976182
− 3.78715
− 0.000636108
− 0.0201658
− 0.349906
− 0.940128
0.00531195
0.0483609
− 0.155677
0.0125732
− 0.0477396
2.0309
0.0331639
0.0312127
0.878727
− 3.3539
− 0.000535031
− 0.00152857
− 0.300115
− 0.895495
0.00113612
0.0130386
− 0.0452976
0.00885256
− 0.0349919
1.76178
Table 23
Regression formulae for PRC method for partially penetrated welds at non-fused root face, Y-butt joint
k
*
fk
*
0.0345634
0.00612723
0.905068
− 3.53925
− 0.000587583
− 8.56103e − 05
− 0.313094
− 0.78778
0.000329096
0.00317992
− 0.0100774
0.0102513
− 0.0385882
1.80983
ck
K p:p:;r
t;PRC;t;1
K p:p:;r
t;PRC;t;2
K p:p:;r
t;PRC;t;3
K p:p:;r
t;PRC;b;1
K p:p:;r
t;PRC;b;2
K p:p:;r
t;PRC;b;3
1
2
3
4
5
6
7
1
β
t
f
z
β2
t2
− 7.4965
0.115205
2.49959
2.4443
26.2697
0.00120545
− 0.211193
− 5.87872
0.0953848
0.503034
1.96638
22.936
0.000886349
− 0.0111477
− 6.20142
0.103194
0.145898
1.90416
23.6283
0.00107672
− 0.000862967
− 0.0739694
− 0.0461878
1.9686
− 0.731923
23.0682
0.00199754
− 0.08105
0.156067
− 0.0335248
0.40123
− 0.589315
18.3067
0.00153204
− 0.00410641
0.212545
− 0.0341946
0.113559
− 0.748549
19.4068
0.00168014
− 0.000332579
8
9
10
11
12
13
14
15
f2
z2
tβ
tf
tz
βf
βz
fz
0.145337
63.7339
− 0.00950572
− 0.160509
8.07645
− 0.0221683
− 0.575691
− 9.10014
0.0904765
50.3633
− 0.00225726
− 0.0338226
1.68678
− 0.0156066
− 0.472397
− 7.31363
0.140411
53.7678
− 0.000681599
− 0.00865977
0.475935
− 0.0190566
− 0.50668
− 7.65221
0.401186
8.62295
− 0.0126736
− 0.246488
2.35144
− 0.0166517
− 0.301697
− 4.47287
0.303461
7.01467
− 0.00276928
− 0.0520558
0.491481
− 0.0127909
− 0.246217
− 3.42091
0.351864
7.50722
− 0.000780135
− 0.013865
0.141029
− 0.0147975
− 0.265122
− 3.66698
2068
Weld World (2020) 64:2053–2074
Table 24
k
Regression formulae for PRC method for fully penetrated welds, DY-butt joint
ck
fk
f :p:
K t;PRC;t;1
f :p:
K t;PRC;t;2
f :p:
K t;PRC;t;3
f :p:
K t;PRC;b;1
1
2
3
4
5
6
1
β
t
f
z
β2
7
8
9
10
11
12
13
14
15
t2
f2
z2
tβ
tf
tz
βf
βz
fz
Table 25
Regression formulae for PRC method for partially penetrated welds at weld toe, DY-butt joint
k
fk
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
1
β
t
f
z
β2
t2
f2
z2
tβ
tf
tz
βf
βz
fz
f :p:
K t;PRC;b;2
f :p:
K t;PRC;b;3
0.535763
0.04162
0.142173
0.383688
0.585494
0.0378353
0.0333702
0.332074
0.575102
0.0385519
0.00935499
0.341048
0.637998
0.0429146
0.158895
0.168517
0.69273
0.0363052
0.0382322
0.144939
0.673695
0.0376007
0.0107202
0.151838
− 0.000530059
− 0.000494215
− 0.000506465
− 0.000471661
− 0.000420435
− 0.000433583
− 0.0296493
− 0.128055
− 0.00181038
− 0.106158
− 0.00013366
− 0.109984
− 0.0239666
− 0.0502292
− 0.001481
− 0.0406721
− 0.000109588
− 0.0422432
0.00878585
0.0289277
0.00188465
0.0068692
0.000536688
0.00183967
0.00668176
0.0118327
0.00143081
0.00263871
0.000407578
0.000677947
0.0089518
0.00694228
0.00737173
0.00242608
0.00173329
0.00186309
ck
K p:p:
t;PRC;t;1
K p:p:
t;PRC;t;2
K p:p:
t;PRC;t;3
0.579423
0.0463585
0.133916
0.0960076
1.341
− 0.000556359
− 0.0296386
0.011342
1.62451
0.00954031
0.0131078
0.0918635
0.00538108
0.0194837
−1.11951
0.64694
0.0410995
0.0326377
0.0547476
1.35768
− 0.000528549
− 0.00179462
0.013827
1.45522
0.0020502
0.00314948
0.0166176
0.004821
0.0145042
−0.990893
0.610542
0.0421909
0.00975511
0.0620194
1.34186
− 0.000538031
− 0.000136019
0.012158
1.63076
0.000568676
0.000972143
0.00428656
0.00517298
0.0157793
−1.0392
K p:p:
t;PRC;b;1
K p:p:
t;PRC;b;2
p:p:
K t;PRC;b;3
0.724746
0.0487632
0.178828
0.789812
0.0401548
0.042756
0.763754
0.0421057
0.0121641
− 0.000469739
− 0.0231246
− 0.00042199
− 0.00145484
− 0.000436125
− 0.000108907
0.00664961
0.00145257
0.000407871
Weld World (2020) 64:2053–2074
Table 26
k
2069
Regression formulae for PRC method for partially penetrated welds at non-fused root face, DY-butt joint
fk
ck
K p:p:
t;PRC;t;1
K p:p:
t;PRC;t;2
K p:p:
t;PRC;t;3
K p:p:
t;PRC;b;1
K p:p:
t;PRC;b;2
K p:p:
t;PRC;b;3
1
1
0.137089
0.277891
0.169294
− 0.0869425
− 0.0723055
− 0.100055
2
3
4
5
6
7
8
9
10
11
12
13
14
15
β
t
f
z
β2
t2
f2
z2
tβ
tf
tz
βf
βz
fz
− 0.0134725
1.25035
− 0.335881
17.3636
0.000856004
− 0.0559083
0.267537
− 3.25381
− 0.00415721
− 0.160279
1.50856
− 0.0159572
− 0.084388
− 2.92894
− 0.0145662
0.269044
− 0.250282
14.16
0.00070488
− 0.00329255
0.206861
− 2.49006
− 0.000849261
− 0.0357097
0.316341
− 0.0122399
− 0.0689383
− 2.38708
− 0.0130638
0.0779434
− 0.272115
14.9147
0.000715133
− 0.000261847
0.216908
− 2.7086
− 0.000248159
− 0.00978245
0.0889831
− 0.0127085
− 0.0725595
− 2.50506
− 0.00916037
0.141022
− 0.3555
4.73156
0.000311973
− 0.00390019
0.153101
1.705
− 0.00125477
− 0.0419178
0.330529
− 0.00230295
− 0.0420997
− 1.32993
− 0.00856331
0.0338065
− 0.29321
4.05481
0.000262836
− 0.000327648
0.123905
1.25084
− 0.000270591
− 0.00912879
0.069412
− 0.00182915
− 0.0339442
− 1.09771
− 0.00817973
0.00975432
− 0.312618
4.26628
0.000267369
− 2.59601e − 05
0.13023
1.3217
−7.89566e − 05
− 0.00245836
0.0191027
− 0.00184818
− 0.0360112
− 1.15195
2070
Weld World (2020) 64:2053–2074
Table 27
Neural network data for full penetration joints, Y-butt joints
2
3
5:03
6 1:0015 7
7
xi;offset ¼ 6
4 0:05 5
0:5035
yo;offset ¼
2
3
0:03337
6 0:66733 7
7
xi;gain ¼ 6
4 2:10526 5
0:02011
1:29125
1:28872
yo;gain ¼
2
3
2
3
−1:18352
−0:97407
6 −0:17554 7
6 0:90347 7
6
7
6
7
7
6 −0:73252 7
−1:01665
b
¼
b1 ¼ 6
2
6
7
6
7
4 0:56676 5
4 −2:07033 5
1:47174
1:16739
2
3
0:20859 −0:02916 −0:19403 −0:46674
6 0:21943 −0:02077 −0:09008 0:00947 7
6
7
7
W1 ¼ 6
6 0:39322 −0:21130 −0:02908 −0:05845 7
4 0:74769 0:26726
5:27578 −0:45294 5
0:38841 −0:00426 1:80405 −0:90168
2
−0:32893
−0:48836
−7:21039
0:25768
−1:43761
1:17452
0:07817
1:62273
−0:21866
0:55955
3
0:42391 1:39724
1:31322 −0:50270 7
7
1:17096 4:99165 7
7
3:71150 −0:06023 5
0:53231 −0:68854
7:15931
3:82317
6 −2:52707 −1:96178
6
W3 ¼ 6
6 4:31372 −1:03153
4 −0:17604 −0:93174
−2:77427 0:79608
−4:69479
1:85948
0:33885
0:51994
2:01416
3
−4:64923 −1:81067
−0:00362 4:75732 7
7
−0:34529 3:27514 7
7
−1:59689 1:98197 5
−1:24937 4:48497
−3:26810
6 1:96285
6
W2 ¼ 6
6 0:61389
4 1:61694
0:94432
2
W4 ¼
5:07308 −2:28452
6:27123 0:41461
0:80311 −3:72046 1:91597
1:04051 −4:60977 −0:24447
0:40229
0:52808
2
3
−1:17925
6 2:34813 7
6
7
7
b3 ¼ 6
6 0:45257 7
4 −0:90407 5
3:57222
b4 ¼
−3:41138
−4:62925
Weld World (2020) 64:2053–2074
Table 28
2071
Neural network data for partial penetration joints, Y-butt joints
2
3
5:03
6 1:0015 7
7
xi;offset ¼ 6
4 0:05 5
0:5035
2
3
0:03337
6 0:66733 7
7
xi;gain ¼ 6
4 2:10526 5
0:02011
2
2
3
1:27394
6 0:00026 7
7
yo;offset ¼ 6
4 0:73557 5
2:38884
2
3
−0:11493
6 4:47114 7
6
7
7
b1 ¼ 6
6 0:18491 7
4 −1:27435 5
1:79329
2
0:00177 −0:00421
6 −0:01183 0:03538
6
W1 ¼ 6
6 −0:01844 0:03421
4 −0:27897 0:14065
0:26290 −0:09533
3
0:23812
6 0:40389 7
7
yo;gain ¼ 6
4 0:06551 5
0:02750
2
3
−3:67889
6 8:98778 7
6
7
7
b2 ¼ 6
6 −8:23804 7
4 −9:13761 5
−12:51035
3
0:40043
0:00204 −0:38174
0:21406 −0:00786 1:99910 7
7
0:09005 −0:01188 −0:08559 7
7
−0:01393 −0:07245 0:01179 5
0:01290
0:18908 −0:01067
2
1:32297
10:51147 −6:68859
6 1:14316
−3:81744 −4:71716
6
W2 ¼ 6
−1:23227 −13:08340
6 3:00316
4 −82:60633 0:24406 −14:67736
−2:16996
7:89387
8:90001
2
−3:76345
6 −2:31653
6
W3 ¼ 6
6 3:75352
4 6:26686
2:86043
2
2:62955
−3:06025
−2:50612
−5:69986
−3:38900
1:49929
1:28376
11:44487
3:32671
−5:70040
3
−3:45379
−3:69945 7
7
21:56658 7
7
3:33244 5
−2:00737
3
−0:28791 −0:04194 −1:08120
−7:99058 −0:22003 −3:99765 7
7
0:52286 −0:00942 1:18855 7
7
1:01254
0:20585 −2:45066 5
7:43315
0:21304 0:844235
−4:11956 −6:36577 −3:45711
6 −16:95793 6:37336 −14:61643
W4 ¼ 6
4 −2:87890
4:98598
−2:63725
−5:34813 11:08086 −4:78819
−3:70815
−4:30975
11:43757
6:72832
3
−1:12831
1:17022 7
7
4:80660 5
13:96681
2
3
4:10582
6 −4:49857 7
6
7
7
b3 ¼ 6
6 −3:92894 7
4 −7:08145 5
1:59769
2
3
1:14941
6 −7:88007 7
7
b4 ¼ 6
4 10:46236 5
9:11077
2072
Weld World (2020) 64:2053–2074
Table 29
Neural network data for full penetration joints, DY-butt joints
2
3
5:0325
6 0:50175 7
7
xi;offset ¼ 6
4 0:05 5
0:5035
yo;offset ¼
3
0:0308
6 0:572 7
7
xi;gain ¼ 6
4 2:10526 5
0:02011
1:2501
1:29897
2
3
0:67611
6 1:71935 7
6
7
7
b1 ¼ 6
6 5:01756 7
4 1:23506 5
−1:64965
2
−0:00260 0:00014
6 −0:06570 0:72056
6
W1 ¼ 6
6 −0:00053 0:00012
4 −0:00198 −0:00001
−0:76825 −0:00098
yo;gain ¼
2
3
−1:15629
6 0:17774 7
6
7
7
b2 ¼ 6
6 −0:90092 7
4 −2:37489 5
1:89842
3
0:28679
0:15740
0:00168 −0:00162 7
7
2:54825
1:71757 7
7
1:33435 −0:09168 5
−0:00410 −0:00296
2
3
−0:07456 −1:65604 0:80328
−5:41325 4:23919 −4:37225 7
7
−0:43262 0:31791
3:16259 7
7
2:36157 −1:85544 0:15021 5
−3:47536 2:82957 −2:32589
2
3
2:94004
3:51065
1:94951
−0:35988 6:08275
4:54147 7
7
0:28396 −9:34964 3:37224 7
7
−0:78954 −2:25065 −1:38860 5
−1:07873 8:68807 −3:66042
0:27939
0:08379
6 −4:91097 0:98452
6
W2 ¼ 6
6 −0:35002 6:98580
4 2:14206 −0:01273
−3:28278 0:01768
0:93376
0:29728
6 1:48686
0:59129
6
W3 ¼ 6
2:17338
−5:08179
6
4 −0:75495 −0:01129
−6:23681 0:02958
W4 ¼
2
−1:21981 3:27933
1:19900 2:50698
−0:41940
−3:50937
−6:40025 6:25745
−3:68060 5:06514
0:49419
0:33387
2
3
−3:44458
6 −0:94522 7
6
7
7
b3 ¼ 6
6 0:02752 7
4 1:31034 5
4:08112
b4 ¼
1:09726
4:47292
Weld World (2020) 64:2053–2074
Table 30
2073
Neural network data for partial penetration joints, DY-butt joints
2
3
5:0325
6 0:50175 7
6
7
7
xi;offset ¼ 6
6 0:05 7
4 0:02034 5
0:5035
2
3
0:0308
6 0:572 7
6
7
7
xi;gain ¼ 6
6 2:10526 7
4 2:94412 5
0:02011
2
2
3
1:23995
6 1:28513 7
7
yo;offset ¼ 6
4 0:00526 5
1:10605
3
−2:43447
6 −2:91803 7
7
6
7
b1 ¼ 6
6 −1:42358 7
4 −2:58923 5
−13:26457
2
0:00262 −0:02568
6 0:06320 −0:56680
6
W1 ¼ 6
6 0:03748 −0:01815
4 −0:43478 0:01296
−0:04204 0:01538
3
0:48784
6 0:26392 7
7
yo;gain ¼ 6
4 0:40314 5
0:09931
3
8:04536
6 −1:47131 7
7
6
7
b2 ¼ 6
6 −11:44685 7
4 2:14675 5
0:46747
2
2
−16:46882
6 14:55718
6
W2 ¼ 6
6 31:34601
4 −6:31820
11:63645
−8:96363 5:69841
6 39:43733 30:54644
6
W3 ¼ 6
6 −8:83898 −1:07113
4 −2:85478 14:60919
5:99425 −2:74630
2
8:51077 −0:08108
6 6:75875
0:09382
6
W4 ¼ 4
−4:43253 0:14785
−2:60143 −0:08071
2
3
14:65910
6 12:97637 7
7
b4 ¼ 6
4 12:6824 5
16:34654
2
3
−0:05974
0:17359 −0:05724
−0:08035
0:17285
0:01640 7
7
−1:68724
0:07551 −0:27948 7
7
−0:08478
0:15038
0:03563 5
−18:89527 −0:07659 4:77317
11:25071 −0:29295
−2:81294 −2:04073
−6:77946 −33:40802
−0:61696
1:45967
−7:90124
0:64391
2
3
0:01565
6 −0:07368 7
7
6
7
b3 ¼ 6
6 −0:50029 7
4 0:78705 5
−0:10573
2
3
13:60453
0:14777
−12:19188 1:00168 7
7
−18:38190 15:97441 7
7
8:36302
−0:72016 5
−3:62433 −0:32448
3
5:08581
23:33091 −18:10042
−7:64162 −14:68090 31:96772 7
7
−0:89132
6:83046
−22:50252 7
7
−2:53580 17:20299
−6:77219 5
−5:00440 −17:91866 13:51605
−0:51755
−0:71109
−0:98707
−0:95013
3
−18:17154 9:61484
−16:32149 7:73723 7
7
−15:99147 −2:46695 5
−20:29961 −1:34616
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