See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/268357345
Parametric Equations for Stress Intensity Factors of Cracked Tubular T&Y-Joints
Article · January 2003
CITATIONS
READS
6
541
4 authors, including:
Chi King Lee
UNSW Sydney
215 PUBLICATIONS 3,571 CITATIONS
SEE PROFILE
All content following this page was uploaded by Chi King Lee on 25 January 2015.
The user has requested enhancement of the downloaded file.
Proceedings of The Thirteenth (2003) International Offshore and Polar Engineering Conference
Honolulu, Hawaii, USA, May 25 –30, 2003
Copyright © 2003 by The International Society of Offshore and Polar Engineers
ISBN 1 –880653 -60 –5 (Set); ISSN 1098 –6189 (Set)
Parametric Equations for Stress Intensity Factors of Cracked Tubular T&Y-Joints
S P Chiew, S T Lie, C K Lee & Z W Huang
School of Civil and Environmental Engineering
Nanyang Technological University, Singapore
from the crown or the saddle of the joints, etc. This restriction limits the
wider and more general application of the current SIF solutions. A
reliable and practical method for the calculation of the SIFs of the weld
toe surface crack of the tubular T & Y-joints is still desirable.
ABSTRACT
Stress intensity factor (SIF) solutions are generally used for fatigue
assessment of the cracked tubular joints in steel offshore jacket
structures. Reliable SIF solutions are necessary for accurate prediction
of the residual fatigue life of such joints. For this purpose, more than a
thousand tubular T & Y-joint models with surface cracks located at
different locations around the brace-chord intersection are generated
using an automatic mesh generator proposed earlier by the authors. The
models cover a wide range of joint dimensions and crack profiles
commonly found in practice. Linear elastic analyses are carried out to
determine the SIFs for the T and Y-joints. Contact problems between
the crack surfaces are also considered. Based on the results from the
parametric study, regression analyses are carried out to develop the
equations to predict the SIFs of the welded tubular joints with surface
crack located at any positions around the joint intersection. A method
to calculate the SIFs of the cracked tubular joints subjected to basic and
combined loads using the proposed equations is also suggested. The
parametric equations are validated against experimental results and
other published solutions.
In this paper, 1360 models of these joints with surface cracks located at
arbitrary positions around the brace-chord intersections are generated
using an automatic mesh generator proposed earlier (Chiew et al,
2001a, 2001b and Lie et al, 2002). Parametric SIFs equations, in the
form similar to those for stress concentration factor, are derived. A
method to calculate the SIFs of the cracked tubular joints subjected to
basic and combined loads using the proposed equations is also given.
Finally, the validity of the equations is established by comparing
against the experimental results and other SIF solutions. The good
agreements show that the proposed equations are acceptable and
reasonably accurate for both basic and combined load cases.
NUMERICAL MODELING
Although large-scale steel joint fatigue tests can provide actual and
reliable information of the fatigue joint performance, they are very time
consuming and expensive to carry out. Due to limitations of the test
facilities, it is also difficult to implement all the possible test load cases.
On the other hand, the finite element method provides an inexpensive,
fast and flexible way to carry out fatigue study of the tubular joints.
However, its reliability depends on the accuracy of the numerical joint
modeling. It is not easy to use finite elements especially the threedimension (3D) elements to generate good quality meshes for most
cracked joints because of the complex joint geometry at the
intersections between the members (Cao et al, 1998). In order to solve
this difficulty, a modeling technique, which can change the complicated
3D mesh generation procedure for the tubular joint to a procedure of
mesh generation in a flat plate, was proposed earlier by the authors.
The weld and the crack are also modeled in the flat plate and then
mapped back to the 3D mesh of the cracked tubular T & Y-joint. A
mesh generator based on this technique was developed and it can mesh
the welded tubular T & Y-joints with surface crack automatically and
quickly. The contact problem is also considered. After the mesh model
is generated, contact surfaces are defined on the crack surfaces to
prevent the crack surfaces from penetrating each other under certain
INTRODUCTION
Fracture mechanics method is generally used for the fatigue assessment
of cracked tubular joints in offshore jacket structures. Fatigue cracks
found during service life must be assessed to establish the limits on its
operation conditions. The application of fracture mechanics is largely
based upon the stress intensity factor (SIF). Hence, the essential part of
the solution of a fatigue problem by fracture mechanics approach is the
accurate and reliable establishment of the stress intensity factor. In this
connection, some fracture mechanics models to predict the SIFs of the
cracked tubular joints have been proposed. These include semiempirical models (Dover and Dharmavasan, 1982 and Etube et al,
2000), flat plate solutions (Maddox, 1975, Burdekin et al, 1986, Dover
and Connolly, 1986, Thorpe, 1986, Van Delft et al, 1986 and Bowness
and Lee, 1996) and numerical models (Du & Hancock, 1987, Huang et
al, 1988, Haswell, 1991 and Chong Rhee et al, 1991). However, all the
current SIF solutions are based on some limiting assumptions and
cannot fully include the actual characteristics of the weld toe surface
cracks, for example, mixed mode conditions, crack position shifting
255
load cases. Further information of this technique can be found in Chiew
et al, 2001a, 2001b and Lie et al, 2002.
accordance with the minimum thickness suggested by AWS (1996).
Both chord ends of all the models are fixed during the analyses.
This modeling method was validated against experimental results
and good agreements were obtained. Experimental fatigue tests were
carried out on three tubular T-joints having the same geometrical
parameters and they were subjected to in-plane bending (IPB) only,
combination of IPB and out-of-plane bending (OPB), and combination
of axial loading (AX), IPB and OPB respectively. Further details can be
found in Chiew et al, 2002.
The interaction integral method proposed by Shih and Asaro (1988)
is used to calculate the SIFs of the cracks. The Mode I, Mode II and
Mode III stress intensity factors along the crack front in different
counters can be obtained directly from the ABAQUS (2001) output
files. The SIFs along the crack front are then calculated by averaging
the stress intensity factors in different counters except those in counter
1. Based on these SIFs along the crack front, the non-dimensional SIF,
K i /(σ n πa ) are computed where Ki is the mode i stress intensity factor
, i = I, II and III, a is the crack depth and σn is the nominal stress on the
brace defined as,
PARAMETRIC STUDY
Validity range
Axial tension (AT):
For data generation purpose, parametric SIF study is carried out
involving 95 basic joint geometrical cases of the cracked tubular T &
Y-joints. The chord diameter D is kept constant at 1016mm, the brace
diameter d varies from 203.2 to 914mm. The chord thickness T varies
from 16 to 100mm, and the brace thickness t varies from 5 to 100mm.
The semi-elliptical crack depth, a varies from 0.8 to 80mm and the
semi-elliptical crack half-length, c varies from 10.16 to 685.5mm. For
the angle between the brace and chord axes as shown in Fig. 1, θ = 31°,
45°, 60°, 90°, 120°, 135° and 149° are used. The crack location is
denoted by φ in Fig. 1. In the models generated, φ = 0, 15°, 22.5°, 30°,
45°, 60°, 67.5°, 75° and 90° are used. A total of 1360 models of tubular
T & Y-joints with surface cracks located in different positions along the
brace-chord intersection are then carefully selected. These models
cover a wide range of geometrical parameters. The objective is to
capture the relationships among the stress intensity factors, geometrical
parameters and crack profiles. The model parameter ranges generated
and analyzed can be summarized as: α = 15.7, 0.2 ≤ β ≤ 0.9, 5.08 ≤ γ ≤
31.75, 0.2 ≤ τ ≤ 1.0, 31° ≤ θ ≤ 149°, 0.05 ≤ a/T ≤ 0.8, 0.05 ≤ c/d ≤ 0.75,
a/c ≤ 1, and 0° ≤ φ ≤ 90°, where, α is relative chord length, β is
diameter ratio, γ is half chord diameter to thickness ratio, and τ is wall
thickness ratio. The value of parameter α is set at 15.7 which are
considered large enough to neglect the chord boundary effect. Thus, the
parameter α will be excluded from the proposed equations.
4P /(π [d 2 − (d − 2t b ) 2 ])
Out-of-plane bending (OPB):
32dM o /(π [d 4 − (d − 2t b ) 4 ])
32dM i /(π [d 4 − (d − 2t b ) 4 ])
(3)
where P, Mo and Mi are the brace axial force, out-of-plane and in-plane
bending moments respectively, d is diameter of brace and tb is brace
wall thickness. Then, all the stress intensity factor results are used to
calculated the equivalent stress intensity factor, Ke, which is based on
the energy release rate and is defined as:
Ke = [KI2+KII2+KIII2/(1-ν)]1/2
(4)
For all load cases, the proposed stress intensity factor equations are
developed for Ke.
During the analyses, all the models without contact surface
definition on the crack surfaces are first analyzed under the three basic
load cases, i.e. AT, IPB and OPB respectively because the contact
algorithm takes much more computing effort. If the KI solutions along
the crack front in the model are found to be negative, this means the
crack surfaces are in contact and they penetrated each other. The model
was then analyzed again including the contact surface definition on the
crack surfaces. In this case, the SIF solutions in the open area of the
model with contact surface definition are kept for equation
development. The negative values in the contact area in the model
without contact surface definition are also kept because these negative
values may contribute to the final SIF results in some cases under the
combined loads.
Crack tip 1
Crack deepest point
(2)
In-plane bending (IPB):
φ
Crack tip 2
(1)
Crack
Parametric equations
In most load cases, the crack on the tubular joint generally initializes at
the crown or saddle location because the highest hot spot stress is
located at these positions. Thus, the equations of the tubular joints with
surface crack at the crown or the saddle are proposed first. Based on the
Ke solution results, parametric regression analyses are carried out using
the multi-variable non-linear regression curve fitting program called
DATAFIT (Oakdale Engineering, 1998). During the analyses, the
power law curve fitting of Ke is performed using the logarithmic values
of the respective parameters as given below,
Figure 1. Tubular joint with surface crack
Data generation
The semi-elliptical shape was assumed for the surface crack. Along the
crack front, a number of collapsed brick element clusters were
modeled. Each of these clusters consists of eight elements surrounding
the crack front. The mid-side nodes of these collapsed brick elements
are moved to the quarter points near the crack front to create the square
root singularity. Reduced integration was used for all the brick
elements in the models. The weld thickness was included and in
 K e  = f(ln(β), ln(γ), ln(τ), ln(a/T), ln(c/d), ln(sinθ))

ln
 σ πa 
 n

256
(5)
especially when the crack depth is small. The cracks in Specimens 1
and 3 are almost at the crown location, whereas in Specimen 2 it is
located between the crown and saddle locations. The proposed
equations can be used to calculate the SIFs of the cracked tubular joints
under combined loads and also in the case where the crack is located
away from the crown or saddle location.
This equation is equivalent to
Ke = σ n πa e f(ln(β), ln(γ), ln(τ), ln(a/T), ln(c/d), ln(sinθ))
= Y(β, γ, τ, a/T, c/d, sinθ) σ n πa
(6)
where Y is the stress intensity modification factor, and σn is the
nominal stress on the brace as defined in Eqs. 1 to 3. After the f
functions in Eq. 5 are obtained by DATAFIT, they are converted to the
Y factor in Eq. 6. In this study, the Y factor is taken to be the product
of the four factors as follows:
K e (obtained from the SIF equations)
40
35
30
1/2
(7)
where Yg is the joint geometry factor and only considers the effect of β,
γ and τ, Ys is the crack size factor and only considers the effect of a/T
and c/d, Yi is the joint and crack coupling factor, and Yθ is the
influence factor of the angle between the brace and the chord axes. The
Y factors obtained are then grouped into these four individual factors
by hand calculation.
The solution results of the models with crack located between the
crown and saddle are considered next. The equations for joints with
surface cracks between the crown and the saddle are built up by relating
it to the SIF results at the crown and the saddle and the crack position
(φ). The regression analyses are also used to obtain these equations.
All the equations at the crown, saddle and between the crown and the
saddle are given in Appendix I.
25
20
15
10
5
0
-5
-10
-15
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
a/tc
Figure 2. Comparison of SIFs from equations and test results of Specimen 1
50
45
40
35
Application of proposed equations
30
1/2
SIF (M pa*m )
The proposed equations are derived for basic load cases. However, it
can also be used to calculate the SIF of the cracked tubular joints under
combined loads. For the joints under combined load, the following
equation is proposed to predict the SIF using the equations given in
Appendix I,
Ke,com = Ke,AT + Ke,IPB + Ke,OPB
Experimental results (at Pn20)
45
SIF (M Pa *m )
Y = YgYsYiYθ
50
25
20
15
10
5
0
(8)
Experimental results (at P50)
-5
K e (obtained from the SIF equations)
-10
where Ke,AT, Ke,IPB, and Ke,OPB are the equivalent stress intensity factors
of axial tension, IPB and OPB calculated from Appendix I. In
summary, the following procedure can be used to calculate the SIFs for
the basic and combined load cases:
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
a/t c
Figure 3. Comparison of SIFs from equations and test results of Specimen 2
i. Decompose the complex loading into basic brace end load
cases;
ii. Determine the nominal stress of each individual basic load case
using Eqns. 1 to 3;
iii. Calculate the equivalent SIFs for each individual basic load
case using the proposed equations;
iv. Obtain the total SIFs of the crack by superposing each
equivalent SIF component of all the basic load cases.
v. Check the results. If negative values of the SIFs are obtained,
the SIFs at these points should be set to zero.
50
Experimental results (at Pn10)
45
Experimental results (at P0)
40
Experimental results (at P20)
35
K e (obtained from the SIF equations)
1/2
SIF (MPa*m )
30
25
20
15
10
5
VALIDITY OF PROPOSED EQUATIONS
0
-5
In order to test the reliability of the proposed SIF equations, validation
is carried out by comparing the results against existing fatigue
experimental test results and those from the fracture mechanics models
developed by other researchers.
-10
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 .0
a/t c
Figure 4. Comparison of SIFs from equations and test results of Specimen 3
The validity of the proposed equations is also assessed against the
test results of Ritchie & Huijskens (1989) and Myers (1998). In Fig. 5,
the numerical results analyzed by Ritchie & Huijskens (1989) are also
included. Except the first point, the SIF results predicted by the
Figures 2 to 4 show the comparisons of the SIFs obtained from tests
with those from the proposed equations. It can be seen that the SIFs
predicted by the proposed equations agree well with the test results,
257
proposed equations are all higher than the numerical results analyzed
by Ritchie and Huijskens (1989) and they are very close to the set 5
experimental results. In Fig. 6, the test results of the Specimen T1 by
Myers (1998) are used for comparison. His specimen T1 had the
following geometric parameters: α = 7.26, β = 0.71, γ = 14.28, τ = 1, θ
= 90°. The wall thickness of the chord and brace is 16mm. Specimen
T1 was tested under axial load and the fatigue crack was found at the
saddle of the joint. It should be pointed out that the non-dimension SIF,
Yexp, in Fig. 6 is different from that in proposed SIF equations of
Appendix I. The Yexp by Myers (1998) is obtained from the
experimental crack growth rates (da/dNexp) as shown below,
5.0
Results predicted by the equations
4.5
Test results from M yers (1998)
4.0
3.5
Y exp
3.0
2.5
2.0
1.5
1.0
1/m
Yexp
 1  da  
 

 C  dN exp  



=
SCFσ n πa
0.5
(9)
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
a/t c
Thus, the different between the Yexp by Myers (1998) and the Y in the
equations is that the Yexp by Myers (1998) does not include the effect of
the SCF of the joint. The Y predicted by proposed SIF equations should
be divided by the SCF of the joints before comparing with the Yexp by
Myers (1998). The comparison in Fig. 6 shows that the SIFs predicted
by the equations can fit the test results of T1 quite well for the small
crack depth. When the crack depth is larger, the SIFs predicted by the
equations are conservative but still acceptable.
Figure 6. Comparison of SIFs from equations and test results of Myers (1998)
5.0
Results predicted by the equations
4.5
Results from AVS model (1982)
4.0
Results from Chong Rhee's model (1991)
Results from Bowness & Lee's model (1996)
3.5
Test results from M yers (1998)
3.0
Y exp
The validity of the proposed SIF equations is also assessed by
comparing the results obtained from the fracture mechanics models
developed by others, i.e. AVS model (Dover and Dharmavasan, 1982),
empirical SIF equations by Chong Rhee et al (1991) and flat plate
model by Bowness and Lee (1996). Because some of these models can
only be used to predict the SIF of the cracked tubular T-joints under
basic loads, the geometry and crack propagation information of the
Specimen T1 by Myers (1998) is used to calculate the SIF using these
fracture mechanics models for comparison as shown in Fig. 7. It can be
seen that the SIFs predicted by the proposed equations are close to
those from the AVS model, especially when 0.2 ≤ a/tc ≤ 0.5. When a/tc
≤ 0.2, the proposed equations are better than those of the AVS model.
The SIFs obtained from the empirical equations by Chong Rhee et al
(1991) seems to be conservative and larger than others. The SIF results
obtained from Bowness and Lee (1996) are also more conservative than
those predicted using the proposed equations.
1.0
0.5
0.0
0.0
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Figure 7. Comparison of SIFs from equations and other researchers
CONCLUSIONS
A SIF database is built up from which a set of SIF equations is
developed using regression analyses. A method used to predict the SIFs
of cracked tubular joints under combined loads using the proposed
equations is also given. The accuracy of the proposed equations is
validated against experimental results and results from other fracture
mechanics models.
30
1/2
0.1
a/t c
35
in the brace (mm ) ∆ K/σ brace
2.0
1.5
40
Stress intensity/average axial stress
2.5
25
20
REFERENCES
Experimental results - set 1
Experimental results - set 2
15
ABAQUS User Manual, version 6.2, (2001), Hobbit, Karlsson &
Sorensen Inc., USA.
American Welding Society, (1996), “ANSI/AWS D1.1-96 Structural
Welding Code-Steel,” Miami, USA.
Bowness, D. and Lee, M.M.K., (1996), “Stress Intensity Factor
Solutions for Semi-Elliptical Weld-Toe Cracks in T-Butt
Geometries,” Fatigue Fract. Engng Mater. Struct. Vol. 19, No. 6,
pp. 787-797.
Burdekin, F.M., Chu, W.H., Chan, W.T.W. and Manteghi, S., (1986),
“Fracture Mechanics Analysis of Fatigue Crack Propagation in
Tubular Joints,” International Conference on Fatigue and Crack
Growth in Offshore Structures, IMechE, London, UK, C133/86.
Experimental results - set 3
Experimental results - set 4
10
Experimental results - set 5
Line spring result
5
3D finite element result
Results by proposed SIF equations in this study
0
5
10
15
20
25
30
35
Maximum depth (mm)
Figure 5. Comparison of SIFs from equations and experimental & numerical
results by Ritchie & Huijskens (1989)
258
Cao, J.J., Yang, G.J., Packer, J.A., Burdekin, F.M., (1998), “Crack
modeling in FE analysis of circular tubular joints,” Engineering
Fracture Mechanics, pp. 537-553.
Chiew, S.P., Lie, S. T., Lee, C.K., and Huang, Z.W., (2001a), “Stress
Intensity Factors of Through-Thickness and Surface Cracks in
Tubular Joints,” Proceedings of the 11th International Offshore
and Polar Engineering Conference (ISOPE 2001), Stavanger,
Norway, Vol. IV, pp 30-34.
Chiew, S.P., Lie, S.T., Lee, C.K., and Huang, Z.W., (2001b), “Stress
Intensity Factors for a Surface Crack in a Tubular T-Joint,”
International Journal of Pressure Vessels and Piping, 78, pp. 677685.
Chiew, S.P., Lie, S.T. and Huang, Z.W., (2002), “Multi-Axes Fatigue
Tests of Tubular T-joints under Complex Loads,” Proc. Of the 12th
International Offshore and Polar Engineering Conference, ISOPE2002, Kitakyushu, Japan, CD-Rom.
Chong Rhee, H., Han, S., and Gipson, G. S., (1991), “Reliability of
Solution Method and Empirical Formulas of Stress Intensity
Factors for Weld Toe Cracks of Tubular Joints,” Proc. 10th
Offshore Mechanics and Arctic Engineering Conference, ASME,
Vol. 3-B, pp 441-452.
DATAFIT (1998), Version 7.1, Oakdale Engineering.
Dover, W.D. and Dharmavasan, S., (1982), “Fatigue Fracture
Mechanics Analysis of T and Y joints,” Proc. Int. Offshore
Technology Conference, Houston, paper OTC 4404.
Dover W.D. and Connolly, (1986), “Fatigue Fracture Mechanics
Assessment of Tubular Welded Y and K-Joints,” International
Conference on Fatigue and Crack Growth in Offshore Structures,
IMechE, London, UK, C141/86.
Du, Z.Z. and Hancock J.W., (1987), “Stress Intensity Factors of Curved
Semi-Elliptical Cracks in a Tubular Welded Joint Using Line
Springs and 3D Finite Elements,” J. Press. Vess. Technol., 111,247
Etube, L.S., Brennan, F.P. and Dover, W.D., (2000), “A New Method
for Predicting Stress Intensity Factors in Cracked Welded Tubular
Joints,” International Journal of Fatigue, 22 (2000) pp. 447-456.
Haswell, J., (1991), “Simple models for Predicting Stress intensity
Factors for Tubular Joints,” Fatigue Fract. Engng. Mater. Struct.
v14, n5. pp. 499-513.
Huang, X., Du, Z.Z. and Hancock, J.W., (1988), “A Finite Element
Evaluation of the Stress Intensity Factors of Surface Cracks in a
Tubular T Joint,” Offshore Technology Conference, OTC 5165.
Lie, S.T., Chiew, S.P. and Huang, Z.W., (2002), “Finite Element Model
for the Fatigue Analysis of Cracked Tubular T-Joints under
Complex Loads” Proceedings of the 12th International Offshore
and Polar Engineering Conference, ISOPE-2002, Kitakyushu,
Japan, CD-Rom.
Maddox, S.J., (1975), “An Analysis of Fatigue Cracks in Fillet Welded
Joints,” International Journal of Fracture, II, pp. 221-243.
Myers, P.T., (1998), “Corrosion Fatigue and Fracture Mechanics of
High Strength Jack Up Steels,” Ph.D. Thesis, University College
London.
Ritchie, D. and Huijskens, H., (1989), “Fracture Mechanics Based
Predictions of the Effects of the Size of Tubular Joint Test
Specimens on Their Fatigue Life,” The Eight International
Conference on Offshore Mechanics and Arctic Engineering, Hague.
Shih, C.F. and Asaro, R.J., (1988), “Elastic-Plastic Analysis of Cracks
on Bimaterial Interfaces: Part I—Small Scale Yielding,” Journal of
Applied Mechanics, pp 299-316.
Thorpe, T.W., (1986), “A Simple Model of Fatigue Crack Growth in
Welded Joints,” Offshore Technology Report, OTH 86 255.
van Delft, D.R.V., Dijkstra, O.D. and Snijder, H.H., (1986), “The
calculation of Fatigue Crack Growth in Welded Tubular Joints
Using Fracture Mechanics,” Offshore Technology Conference,
OTC 5352.
APPENDIX I - PROPOSED EQUATIONS FOR BASIC LOAD
Ke = Ya σ n πa for the deepest crack front point
Ke = Yc σ n πa for the crack tip point on chord surface
(1) Equations for Brace Axial Tension (AT)
(i) Crack located at the crown (φ=0°)
Deepest crack front point:
Ya (φ=0°, AT) = YgYsYiYθ
Yg = 5.8509βG1γG2τG3
G1 = -1.1403lnβ+0.9325lnτ+0.447ln2τ-0.4327lnγlnτ
G2 = 0.0648lnγ-0.1185lnβ+0.2512 ln2β-0.6524lnτ
G3 = 2.8749+0.581lnτ-0.313 ln2β
Ys= βQ1γQ2τQ3
Q1 = 0.3016A2-0.4208C-0.3325AC-0.2902Alnβ
Q2 = 0.2118A2-0.278C-0.1989AC+0.1048Alnγ
Q3 = 0.2446C+0.1619C2-0.0856AC
Yi = (a/T)S1(c/d)S2
S1 = 0.0848A+1.0747C-0.1789C2
S2 = 0.6685-0.2403C+0.7099A2+0.0925A3
Yθ = (sinθ)P
If 31° ≤ θ ≤ 90°
P = 4.1-0.38lnβ-0.66lnγ+0.3A+0.03C-0.1ln(sinθ)
If 90° ≤ θ ≤ 149°
P = -0.034+0.087lnβ+0.13lnγ-0.05A+0.14C-0.7ln(sinθ)
A = ln(a/T) and C = ln(c/d)
(θ is in degree)
Crack tip points on chord surface:
Yc1 (φ=0°, AT) = Yc2 (φ=0°, AT) = YgYsYiYθ
(Yc1 and Yc2 are the Y factor for crack tip 1 and crack tip 2 of the crack
respectively (refer to Figure 1))
Yg = 0.2612βG1γG2τG3
G1 = -3.1967-5.8443lnβ-2.0113 ln2β-0.7366lnγ
G2 = 1.0093-0.1783lnγ+0.2484lnτ
G3 = 0.6407+0.5884 lnβ+0.1725ln2β
Ys= βQ1γQ2τQ3
Q1 = 0.079A2-1.1499C-0.2442Alnβ-0.4628Clnβ
Q2 = 0.2A+0.6392C+0.1995C2
Q3 = -0.2025A-0.079A2-0.2618C-0.0899C2
Yi = (a/T)S1(c/d)S2
S1 = -1.8118-2.0221A-0.8424 A2-0.121A3
S2 = -2.5363-0.6839C+0.1968A
Yθ = (sinθ)P
If 31° ≤ θ ≤ 90°
P=4.1-0.02lnβ-0.6lnγ-0.5lnτ+0.006A+0.43C-0.56ln(sinθ)
If 90° ≤ θ ≤ 149°
P=1.6-0.32lnβ-0.22lnγ+0.32lnτ-0.2A+0.45C-0.7ln(sinθ)
A = ln(a/T) and C = ln(c/d)
(θ is in degree)
(ii) Crack located at the saddle (φ=90°)
Deepest crack front point:
Ya (φ=90°, AT) = YgYsYiYθ
259
Yg = 0.2402βG1γG2τG3
G1 = -0.5692-0.6852lnβ
G2 = 1.6643+0.3462lnτ
G3 = -0.1518lnτ-0.687 ln2β
For crack tip 2 (refer to Figure 1)
Yc2(φ, AT) = W1+(W2-W1) (sinφ)1.8
W1=1.16(c/d)0.016×exp(-0.14β-0.023β(c/d))×Yc2(φ=0°, AT)
W2=1.114(c/d)0.016×exp(-0.146β+0.3β(c/d))×Yc2(φ=90°, AT)
Ys= βQ1γQ2τQ3
Q1=1.5177A+0.2111A2+0.1407C2-0.2502AC+0.4639Alnβ
Q2 = -0.6725A+0.144Alnγ
Q3 = 0.0548A+0.3309C+0.1233C2
(φ is in degree)
(2) Equations for In-Plane Bending (IPB)
Yi = (a/T)S1(c/d)S2
S1 = 2.4674+0.96A+0.136 A2+1.0636C-0.1186C2
S2 = 1.2438+0.3282C+0.075 C2+0.8013A2+0.1235A3
(i) Crack located at the crown (φ=0°)
Yθ = (sinθ)P
P = 1.3+0.3lnβ+0.13lnγ-0.06A-0.15C-0.4ln(sinθ)
Ya (φ=0°, IPB) = YgYsYiYθ
Deepest crack front point:
Yg = 0.4404βG1γG2τG3
G1 = 0.7804-0.3291lnβ
G2 = 0.5964-0.3604lnβ
G3 = 0.5117-0.1185lnβ
A = ln(a/T) and C = ln(c/d)
(θ is in degree)
Crack tip points on chord surface:
Yc1 (φ=90°, AT) = YgYsYiYθ1 for crack tip 1 (refer to Figure 1)
Yc2 (φ=90°, AT) = YgYsYiYθ2 for crack tip 2 (refer to Figure 1)
Ys= βQ1γQ2τQ3
Q1=0.595A+0.316A2-0.6148C-0.3603AC
Q2=0.3535A+0.2029A2-0.7967C-0.2437AC+0.0895Clnγ
Q3=0.0348A-0.0887C
Yg = 0.0672βG1γG2τG3)
G1 = 1.0358-0.5425lnγ+0.4165lnτ
G2 = 2.832-0.5251lnγ+1.6098lnτ
G3 = -1.5561-0.3676lnτ-0.2043ln2γ
Yi = (a/T)S1(c/d)S2
S1 = -0.9929-0.6647A-0.1768 A2-0.2313C2
S2 = 0.5548-0.4778C+0.7951A+0.3404A2
Ys= βQ1γQ2τQ3
Q1 = 0.2422A+0.8146C-0.063AC+0.6168Clnβ
Q2 = 0.2056A+0.7185C+1.0708C2+0.2294C3
Q3 = 0.0925A2+0.1567C-0.1593Alnτ
Yθ = (sinθ)P
If 31° ≤ θ ≤ 90°
P=0.52-0.25lnβ-0.03lnγ-0.71lnτ+0.12A+0.02C-0.34ln(sinθ)
If 90° ≤ θ ≤ 149°
P=1.84-0.3lnβ-0.26lnγ+0.54lnτ-0.2A+0.18C-0.2ln(sinθ)
Yi = (a/T)S1(c/d)S2
S1 = 0.4448+0.086A+0.3184C
S2 = -3.4687-3.6655C-0.7216C2+0.0539A2
A = ln(a/T) and C = ln(c/d)
(θ is in degree)
Crack tip points on chord surface:
Yθ1 = (sinθ)P1
If 31° ≤ θ ≤ 90°
P1 = 2.7+0.24lnβ-0.6lnγ-0.7lnτ-0.56C-0.57ln(sinθ)
If 90° ≤ θ ≤ 149°
P1 = 0.34+0.08lnβ-0.12lnγ-0.25lnτ-0.76C-0.73ln(sinθ)
Yc1 (φ=0°, IPB) = Yc2 (φ=0°, IPB) = YgYsYiYθ
(Yc1 and Yc2 are the Y factor for crack tip 1 and crack tip 2 of the crack
(refer to Figure 1) respectively.)
Yg = 0.0063βG1γG2τG3
G1 = -2.0676-5.2697lnβ-1.9359ln2β
G2 = 1.1288-0.1941lnγ-0.81lnβ+0.5573lnτ
G3 = -1.2019-1.2459lnτ-0.4234ln2τ+0.479lnβ
Yθ2 = (sinθ)P2
If 31° ≤ θ ≤ 90°
P2 = 0.34+0.08lnβ-0.12lnγ-0.25lnτ-0.76C-0.73ln(sinθ)
If 90° ≤ θ ≤ 149°
P2 = 2.7+0.24lnβ-0.6lnγ-0.7lnτ-0.56C-0.57ln(sinθ)
Ys= βQ1γQ2τQ3
Q1 = -0.6098C-0.0827C2-0.1161Alnβ
Q2 = 0.1979A+0.1319C+0.08C2
Q3 = 0.0192A-0.4007C-0.1048C2
A = ln(a/T) and C = ln(c/d)
(θ is in degree)
(iii) Crack located between the crown and saddle (φ)
Yi = (a/T)S1(c/d)S2
S1 = -1.082-1.9695A-0.8782 A2-0.1325A3+0.0988C2
S2 = -8.7955-7.0757C-2.4067C2-0.3322C3+0.6698A
Deepest crack front point:
Ya (φ, AT) = Ya (φ=0°, AT) + (Ya (φ=90°, AT) - Ya(φ=0, AT))(sinφ)H
H = 3.1-0.08γ+0.235βγ
Yθ = (sinθ)P
If 31° ≤ θ ≤ 90°
P=-0.15+0.6lnβ+0.32lnγ-1.37lnτ-0.24A+0.23C-0.89ln(sinθ)
If 90° ≤ θ ≤ 149°
P=2.45-0.56lnβ-0.5lnγ+0.93lnτ-0.29A+0.06C-0.41ln(sinθ)
Crack tip points on chord surface:
For crack tip 1 (refer to Figure 1)
A = ln(a/T) and C = ln(c/d)
(θ is in degree)
Yc1(φ, AT) = W1+(W2-W1) (sinφ)3.3
W1=0.185β-0.75(c/d)-0.09×exp(1.46β-0.135(c/d)+1.045β(c/d))×Yc1(φ=0°,
AT)
W2=0.71β-0.08(c/d)-0.08×exp(0.05β-0.013(c/d)+0.5β(c/d))×Yc1(φ=90°,
AT)
(ii) Crack located at the saddle (φ=90°)
Deepest crack front point:
260
W2=0.384γ1.17(a/T)-1.37(c/d)2.29×exp(1.277+5.66(a/T)-5.92(c/d)0.114γ(c/d)-3.1(a/T)(c/d))
Ya (φ=90°, IPB) = YgYsYiYθ
Yg = 46.0441βG1γG2τG3
G1 = 3.3938-0.922lnβ+0.2086 ln2γ
G2 = -3.25+0.6174lnγ-2.0492lnβ-0.4286lnβlnτ
G3 = 0.5638-0.4504lnτ-0.8133ln2β
For crack tip 2 (refer to Figure 1)
Yc2(φ,IPB)=(1-(sinφ)W1)×Yc2(φ=0°,IPB)+(sinφ)W2×Yc2(φ=90°, IPB)
W1=1.098γ1.2(a/T)0.54(c/d)0.3×exp(1.112-0.021γ-2.18(a/T)6.42(c/d)+0.012γ(a/T)-0.121γ(c/d)+ 5.23(a/T)(c/d))
W2=0.516γ-1.12(a/T)-1.24(c/d)-1.5×exp(0.573+0.079γ+ 3.4(a/T)-0.43(c/d)0.057γ(c/d)+0.47(a/T)(c/d))
Ys= βQ1γQ2τQ3
Q1 = -0.3315A-0.0694C
Q2 = -0.0985A+0.2A2-1.367C-0.2249AC+0.2022Clnγ
Q3 = 0.295A+0.1142C
(φ is in degree)
Yi = (a/T)S1(c/d)S2
S1 = 0.2902-0.2984A-0.2186C2
S2 = 1.2044-0.271C-0.2883A
(3) Equations for Out-of-Plane Bending (OPB)
(i) Crack located at the crown (φ=0°)
Yθ = (sinθ)P
P=1.7+0.42lnβ-0.11lnγ+0.14lnτ+0.49A-0.25C-0.19ln(sinθ)
Deepest crack front point:
Ya (φ=0°, OPB) = YgYsYiYθ
A = ln(a/T) and C = ln(c/d)
(θ is in degree)
Yg = 0.1088βG1γG2τG3
G1 = 20.7997+10.2322lnβ-6.6994lnγ
G2 = 1.825-0.3044lnγ-3.5457ln2β
G3 = 0.9397+4.6079lnβ+2.8277ln2β
Crack tip points on chord surface:
Yc1 (φ=90°, IPB) = YgYsYiYθ1 for crack tip 1 (refer to Figure 1)
Yc2 (φ=90°, IPB) = (-1.0)×YgYsYiYθ2 for crack tip 2 (refer to Figure 1)
Ys= βQ1γQ2τQ3
Q1=-2.19A+4.2636C+0.6715C2-1.271Alnβ+ 0.9735Clnβ
Q2=0.6294A+1.1223C+0.4846C2
Q3=-1.4171A-1.1502C-0.3331C2-1.0473Alnτ
Yg = 0.47βG1γG2τG3
G1 = 0.7883-0.6241lnβ-0.7698lnγ+0.4983lnτ
G2 = 1.4361-0.3351lnγ+2.4187lnτ+0.4528ln2τ
G3 = -3.7322-1.7258lnτ-0.223ln2γ
Yi = (a/T)S1(c/d)S2
S1 = -2.344-0.2533A
S2 = -1.0095-0.984C-0.0611A2
Ys= βQ1γQ2τQ3
Q1=1.2014A-0.5449C+0.4179Alnβ
Q2=0.0939A-0.1088A2-1.0819C-0.2829C2+0.1321AC
Q3=0.1551A2+0.4976C+0.1406C2-0.0838AC-0.1802Alnτ
Yθ = (sinθ)P
If 31° ≤ θ ≤ 90°
P=-5.97+0.92lnβ+2.53lnγ-0.085lnτ+0.59A-0.02C-1.17ln(sinθ)
If 90° ≤ θ ≤ 149°
P=2.55+0.27lnβ-0.73lnγ+0.06lnτ-0.43A+0.02C-1.36ln(sinθ)
Yi = (a/T)S1(c/d)S2
S1 = 0.7814+0.2812A+0.0468C2
S2 = 2.5447+0.2202C-0.1117C2
A = ln(a/T) and C = ln(c/d)
(θ is in degree)
Yθ1 = (sinθ)P1
If 31° ≤ θ ≤ 90°
P1=3.1+0.83lnβ-0.5lnγ-0.49lnτ-0.045A-0.004C-2.08ln(sinθ)
If 90° ≤ θ ≤ 149°
P1=1.0-0.24lnβ+0.09lnγ+0.47lnτ-0.28A-0.35C-0.07ln(sinθ)
Crack tip points on chord surface:
Yc1 (φ=0°, OPB) = (-1.0)×Yc2 (φ=0°, OPB)
Yc2 (φ=0°, OPB) = YgYsYiYθ2
(Yc1 and Yc2 are the Y factor for crack tip 1 and crack tip 2 of the crack
(refer to Figure 1) respectively)
Yθ2 = (sinθ)P2
If 31° ≤ θ ≤ 90°
P2=1.0-0.24lnβ+0.09lnγ+0.47lnτ-0.28A-0.35C-0.07ln(sinθ)
If 90° ≤ θ ≤ 149°
P2=3.1+0.83lnβ-0.5lnγ-0.49lnτ-0.045A-0.004C-2.08ln(sinθ)
Yg = 0.4257βG1γG2τG3
G1 = -3.3923-12.4084lnβ-5.1538ln2β-2.792lnγ
G2 = 1.3315-0.4487lnγ-0.7358ln2β+0.6258lnτ
G3 = -1.071-0.6543lnτ-0.3486ln2β
A = ln(a/T) and C = ln(c/d)
(θ is in degree)
Ys==βQ1γQ2τQ3
Q1 = -0.1452A2-0.8788C+0.1461C2-0.5116Clnβ
Q2 = -0.2421A2+0.9621C+0.228AC-0.1935Clnγ
Q3 = -0.6793A+0.2258C-0.3731Alnτ
(iii) Crack located between the crown and saddle (φ)
Deepest crack front point:
Ya (φ,IPB) = Ya (φ=0°, IPB)+(Ya (φ=90°, IPB)-Ya (φ=0, IPB))(sinφ)H
H = 2.3+0.23γ-0.28βγ
Yi = (a/T)S1(c/d)S2
S1 = 0.8982A+0.239A2-(0.5269A+0.1944A2)C2
S2 = -0.0908C+1.4055A+3.2929A2+1.7153A3+0.249A4
Crack tip points on chord surface:
Yθ2 = (sinθ)P
If 31° ≤ θ ≤ 90°
P=0.01+1.15lnβ+1.3lnγ-0.42lnτ-0.34A+1.48C-0.06ln(sinθ)
If 90° ≤ θ ≤ 149°
P=2.55-0.71lnβ-0.52lnγ+0.27lnτ-0.04A+0.16C-0.94ln(sinθ)
For crack tip 1 (refer to Figure 1)
Yc1(φ,IPB)=(1-sinφ)W1)×Yc1(φ=0°,IPB)+(sinφ)W2× Yc1(φ=90°, IPB)
W1=4.53γ1.07(a/T)0.31(c/d)1.48×exp(2.93-0.01γ-1.29(a/T)-4.55(c/d)0.174γ(c/d) +1.565(a/T)(c/d))
261
A = ln(a/T) and C = ln(c/d)
(θ is in degree)
For crack tip 1 (refer to Figure 1)
Yc1(φ,OPB)=(1-(sinφ)W1)×Yc1(φ=0°,OPB)+(sinφ)W2×Yc1(φ=90°, OPB)
W1=2.3γ0.25(a/T)0.11(c/d)5.03×exp(1.9+3.62(a/T)+0.024γ(c/d)5.77(a/T)(c/d))
W2=0.019γ1.37(a/T)0.43(c/d)-1.55×exp(-2.66-0.046γ0.64(a/T)+16.32(c/d)+0.024γ(a/T)-2.0(a/T)(c/d))
(ii) Crack located at the saddle (φ=90°)
Deepest crack front point:
Ya (φ=90°, OPB) = YgYsYiYθ
Yg = 2.6159βG1γG2τG3
G1 = 2.7758-1.0791lnβ+0.311ln2γ+0.1755ln2τ
G2 = -0.546+0.3823lnγ-2.1178lnβ-0.3836lnβlnτ
G3 = 0.8101+0.9525lnβ-0.2734ln2β
For crack tip 2 (refer to Figure 1)
Yc2(φ,OPB)=(1-(sinφ)W1)×Yc2(φ=0°,OPB)+(sinφ)W2×Yc2(φ=90°, OPB)
W1=0.041γ1.25(a/T)3.39(c/d)-0.88×exp(-1.736+0.343γ+
15.49(c/d)+0.16γ(a/T)-0.68γ(c/d)-13.12(a/T)(c/d))
W2=0.44γ0.31(a/T)0.43(c/d)-0.42×exp(0.31+0.02γ-0.614(a/T)-7.24(c/d)0.064γ(c/d)+8.33(a/T)(c/d))
Ys= βQ1γQ2τQ3
Q1 = 0.5214A+0.3263A2+0.219C2-0.4776AC
Q2 = 0.3881A+0.1643A2-1.1C-0.1451AC+0.1806Clnγ
Q3 = 0.2943C+0.1044C2
(φ is in degree)
Yi = (a/T)S1(c/d)S2
S1 = -0.3605+0.2244A+0.9898C-0.2372C2
S2 = 1.9536-0.1366C+0.7932A2+0.0916A3
Yθ = (sinθ)P
P = 1.4+0.14lnβ+0.05lnγ-0.044A-0.13C-0.31ln(sinθ)
A = ln(a/T) and C = ln(c/d)
(θ is in degree)
Crack tip points on chord surface:
Yc1 (φ=90°, OPB) = YgYsYiYθ1 for crack tip 1 (refer to Figure 1)
Yc2 (φ=90°, OPB) = YgYsYiYθ2 for crack tip 2 (refer to Figure 1)
Yg = 0.0308βG1γG2τG3
G1 = -0.6075-1.6247lnβ+0.4413lnτ-0.3896lnγlnτ
G2 = 1.9253-0.3712lnγ+0.4739ln2β+1.2814lnτ
G3 = -0.9607-0.1699lnτ-0.4986ln2β-0.2279ln2γ
Ys= βQ1γQ2τQ3
Q1 = 0.3622A+0.5624C+0.5127Clnβ
Q2 = 0.2154A-0.4098C
Q3 = 0.1853A+0.0513A2
Yi = (a/T)S1(c/d)S2
S1 = 1.0236-0.0364A+1.2801C+0.5121C2+0.0807C3
S2 = -4.1764-3.7487C-1.2386C2-0.1559C3
Yθ1 = (sinθ)P1
If 31° ≤ θ ≤ 90°
P1=1.05+0.2lnβ-0.31lnγ-0.95lnτ-0.15A-0.61C-0.5ln(sinθ)
If 90° ≤ θ ≤ 149°
P1=1.2-0.43lnβ-0.33lnγ+0.008lnτ+0.095A-0.58C-0.3ln(sinθ)
Yθ2 = (sinθ)P2
If 31° ≤ θ ≤ 90°
P2=1.2-0.43lnβ-0.33lnγ+0.008lnτ+0.095A-0.58C-0.3ln(sinθ)
If 90° ≤ θ ≤ 149°
P2=1.05+0.2lnβ-0.31lnγ-0.95lnτ-0.15A-0.61C-0.5ln(sinθ)
A = ln(a/T) and C = ln(c/d)
(θ is in degree)
(iii) Crack located between the crown and saddle (φ)
Deepest crack front point:
Ya(φ,OPB)=Ya(φ=0°,OPB)+(Ya(φ=90°,OPB)-Ya(φ=0,OPB))(sinφ)H
H = 3.0-3.51β-0.087γ+0.35βγ
Crack tip points on chord surface:
262
View publication stats