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Progress Report No.1
DEVELOPMENT OF WEIGHT FUNCTIONS AND
COMPUTER INTEGRATION PROCEDURES FOR
CALCULATING STRESS INTENSITY FACTORS AROUND
CRACKS SUBJECTED TO COMPLEX STRESS FIELDS
Prepared for
Analytical Services& Materials, Inc.
107 Research Drive,
Hampton, VA 23666, USA
by
G. Glinka, Ph. D.
(on behalf of)
SaFFD, Inc.
Stress and Fatigue-Fracture Design
RR #2, Petersburg
Ontario, Canada N0B 2H0
March 1996
2
Summary
The report consist the first part of the project described in the proposal associated with
the Contract No. 554. The Progress Report No.1 consists of the theoretical background,
the integration methods and the collection of weight functions as described in Tasks 2.1,
2.2 and 2.3. The development of the coded numerical routines is being carried out and it
will be submitted in the next Report.
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1. INTRODUCTION 5
2. STRESS INTENSITY FACTORS AND WEIGHT FUNCTIONS - TECHNICAL
BACKGROUND 5
3. UNIVERSAL WEIGHT FUNCTIONS FOR ONE-DIMENSIONAL CRACKS 9
4. UNIVERSAL WEIGHT FUNCTIONS FOR TWO-DIMENSIONAL PART-
THROUGH SURFACE AND CORNER CRACKS 11
5. SEQUENCE OF STEPS FOR CALCULATING STRESS INTENSITY
FACTORS USING WEIGHT FUNCTIONS 13
6. DETERMINATION OF WEIGHT FUNCTIONS 13
7. NUMERICAL INTEGRATION OF THE WEIGHT FUNCTION AND
CALCULATION OF STRESS INTENSITY FACTORS 17
7.1 Integration by using the centroids of areas under the weight function curve 17
7.2 Analytical integration of the linearized piecewise stress distribution and the
weight function 23
8. WEIGHT FUNCTIONS FOR CRACKS IN PLATES AND DISKS 26
8.1 Through crack in an infinite plate subjected to symmetrical loading 26
8.2 Central through crack in a finite width plate subjected to symmetric loading 28
8.3 Edge crack in a semi-infinite plate 29
8.4 Edge crack in a finite width plate 31
8.5 Two symmetrical edge cracks in a finite width plate subjected to symmetrical
loading 32
8.6 Embedded penny-shape crack in an infinite body subjected to symmetric
loading 33
8.7 Shallow semi-elliptical surface crack in a finite thickness plate (a/c < 1) 36
8.8 Deep semi-elliptical surface crack in a finite thickness plate (a/c > 1) 41
4
8.9 Quarter-elliptical corner crack in a finite thickness plate 46
8.10 Radial edge crack in a circular disk 51
8.11 Edge crack in a finite thickness plate with the right angle corner 53
8.12 Edge crack in a finite thickness plate with an angular corner 55
9. WEIGHT FUNCTIONS FOR CRACKS IN CYLINDERS 57
9.1 Internal circumferential edge crack in a thick cylinder subjected to
axisymmetric loading (Ro/Ri=2.0) 58
9.2 Two internal symmetrical axial edge cracks in a thick cylinder subjected to
symmetrical loading (Ro/Ri=2.0) 61
9.3 External axial edge crack in a thick cylinder (Ro/Ri=2.0) 64
9.4 Internal circumferential semi-elliptical surface crack in a thick cylinder
(Ro/Ri=1.1) 67
9.5 Internal axial semi-elliptical surface crack in a thick cylinder (Ro/Ri=2.0) 71
9.6 Internal axial semi-elliptical surface crack in a thick cylinder (Ro/Ri=1.5) 76
9.7 Internal axial semi-elliptical surface crack in a thick cylinder (Ro/Ri=1.25) 80
9.8 Internal axial semi-elliptical surface crack in a thick cylinder (Ro/Ri=1.1) 84
9.9 External axial semi-elliptical surface crack in a thick cylinder (Ro/Ri=2.0) 88
9.10 External axial semi-elliptical surface crack in a thick cylinder (Ro/Ri=1.5) 93
9.11 External axial semi-elliptical surface crack in a thick cylinder (Ro/Ri=1.25) 98
9.12 External axial semi-elliptical surface crack in a thick cylinder (Ro/Ri=1.1) 102
10. NOTATION 106
11. LITERATURE 108
5
1. INTRODUCTION
Durability and strength evaluation of notched and cracked structural elements requires
calculation of stress intensity factors for cracks located in regions characterized by
complex stress fields. This is particularly true for cracks emanating form notches and
other stress concentration regions that might be frequently found in mechanical and
structural components. In the case of engine components complex stress distributions are
often due to temperature effects. The existing stress intensity factor handbook solutions
are not sufficient in such cases due to the fact that most of them have been derived for
simple geometry and loading configurations. The variety of notch and crack
configurations and the complexity of stress fields occurring in engineering practice
require more efficient tools for calculating stress intensity factors than the large but
nevertheless limited number of currently available ready made solutions, obtained for a
few specific geometry and load combinations.
Therefore, it is proposed to develop a methodology for efficient calculation of stress
intensity factors for cracks in complex stress fields by using the weight function
approach.
2. STRESS INTENSITY FACTORS AND WEIGHT FUNCTIONS -
TECHNICAL BACKGROUND
Most of the existing methods of calculating stress intensity factors require separate
analysis for each load and geometry configuration. The weight function method
developed by Bueckner [1] and Rice [2] simplifies considerably the determination of
stress intensity factors. If the weight function is known for a given cracked body, the
stress intensity factor due to any load system applied to the body can be determined by
using the same weight function. Therefore there is no need to derive ready made stress
intensity factor for each load system and resulting internal stress distribution. The stress
intensity factor can be obtained by multiplying the weight function m(x,a) and the internal
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stress distribution (x) in the prospective crack plane and integrating the product along
the crack length „a‟.
The success of the weight function technique for calculating stress intensity factors lies in
the possibility of using superposition. It can be shown [3] that the stress intensity factor
for a crack body (Fig. 1) subjected to the external loading S is the same as stress intensity
factor in geometrically identical body with the local stress field (x) applied to the crack
faces. The local stress field (x) in the prospective crack plane is due to the external load
S and it is determined for uncracked body by ignoring the presence of the crack.
The unique feature of the weight function method is that once the weight function for a
particular cracked body is determined, the stress intensity factor for any loading system
applied to the body can be calculated by a simple integration of the product of the stress
field, (x), and the weight function, m(x,a).
The idea of using weight functions is illustrated in Fig.2, where the through thickness
crack is used as an example. The weight function m(x,a) can be interpreted as the stress
intensity factor that results from a pair of splitting forces applied to the crack face at
position x. Since the stress intensity factors are linearly dependent on the applied loads,
the contributions from multiple splitting forces applied along the crack surface can be
superposed and the resultant stress intensity factor can be calculated as the sum of all
individual load contributions. This results in the integral (1) of the product of the weight
function m(x,a) and the stress field (x) for the case of continuously distributed stress
field.
(1) dxa,xmxKa
0
7
Fig. 2. a) Weight function and stress intensity factors for a through crack in an infinite
plate.
8
Fig.2. b) Weight function and stress intensity factors for a through crack in an infinite
plate subjected to multiple loads or/and continuously distributed stress.
9
3. UNIVERSAL WEIGHT FUNCTIONS FOR ONE-DIMENSIONAL
CRACKS
The weight function is dependent on the geometry only and in principle should be derived
individually for each geometrical configuration. However, Glinka and Shen 4 have
found that one general weight function expression can be used to approximate weight
functions for a variety of geometrical configurations of cracked bodies with one
dimensional cracks of Mode I.
The system of coordinates and the notation for internal through and edge cracks are given
in Fig. 3.
In order to determine the weight function m(x,a) of eq(2) for a particular cracked body it
is sufficient to determine [5] the three parameters M1, M2, and M3. Because the weight
function of form (2) is the same for all cracks then the same integration routine can be
used for calculating stress intensity factors from eq.(1). Moreover, only limited number
of generic weight functions is needed to enable the determination of stress intensity
factors for a large number of load and geometry configurations.
(2) a
x1M
a
x1M
a
x1M1
xa2
2a,xm
2
3
3
1
2
2
1
1
10
Figure 3. Single Edge Crack and Central Through Crack Under Symmetrical Loading
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4. UNIVERSAL WEIGHT FUNCTIONS FOR TWO-DIMENSIONAL
PART-THROUGH SURFACE AND CORNER CRACKS
In the case of 2-D cracks such as semi-elliptical and corner surface cracks in plates and
cylinders the stress intensity factor changes along the crack front. However, in most
practical cases the deepest point A (Fig. 4) and the surface point B are associated with the
highest and the lowest value of the stress intensity factor along the crack front. Therefore,
the weight functions for the points A and B have been derived [6] analogously to the
universal weight function (2).
For point A (Fig. 4)
For point B (Fig. 4)
The weight functions mA(x,a) and mB(x,a) given above corresponding to the deepest and
the surface point A and B respectively have been derived for one-dimensional stress
fields (Fig. 4), dependent on one variable , x , only.
(4) a
xM
a
xM
a
xM1
x
2ta,ca,a,xm
2
3
B3
1
B2
2
1
B1B
(3) a
x1M
a
x1M
a
x1M1
xa2
2ta,ca,a,xm
2
3
A3
1
A2
2
1
A1A
12
Fig. 4. Semi-elliptical surface crack in an infinite plate of finite thickness, t.
13
5. SEQUENCE OF STEPS FOR CALCULATING STRESS
INTENSITY FACTORS USING WEIGHT FUNCTIONS
In order to calculate stress intensity factors using the weight function technique the
following tasks need to be carried out:
Determine stress distribution (x) in the prospective crack plane using linear elastic
analysis of uncracked body (Fig. 1a), i.e. perform the stress analysis ignoring the
crack and determine the stress distribution (x) = 0 f(S,x);
where: 0 - nominal stress, x - coordinate, S - external load
Apply the “uncracked” stress distribution (x) to the crack surfaces (Fig. 1b) as
tractions
Choose appropriate generic weight function (i.e. choose appropriate M1, M2, and M3
parameters).
Integrate the product of the stress function (x) and the weight function m(x,a) over
the entire crack length or crack surface, eq.(1).
6. DETERMINATION OF WEIGHT FUNCTIONS
The derivation of the weight function of eq.(2) for a one-dimensional crack is essentially
reduced to the determination of parameters M1, M2, and M3. There are several methods
available depending on the amount of information available as described in reference [5].
The most frequently used method based on two reference stress intensity factors is
described below.
Supposing that two reference stress intensity solutions Kr1 and Kr2 for two different stress
fields r1(x) and r2(x) respectively are known, a set of two equations can be written by
using eqs.(1) and (2). The third equation necessary to determine parameters M1, M2, and
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M3 can be formulated based on the knowledge of the crack surface slope [5,6] at the crack
center (central through cracks) or from the crack surface curvature at the crack mouth
(edge cracks). The three pieces of information result in three independent equations
which can be used for the determination of unknown parameters M1, M2, and M3.
Central through cracks under symmetric loading (Fig. 3a)
Edge cracks
The two reference cases can be obtained from the literature or they can be derived using
finite element method. Most often available are the solutions for the uniform and linear
(7) 0
a
x1M
a
x1M
a
x1M1
xa2
2
x
(6) dxa
x1M
a
x1M
a
x1M1
xa2
2xK
(5) dxa
x1M
a
x1M
a
x1M1
xa2
2xK
0x
2
3
3
1
2
2
1
1
2
3
3
1
2
2
1
1
a
02r2r
2
3
3
1
2
2
1
1
a
01r1r
(10) 0
a
x1M
a
x1M
a
x1M1
xa2
2
x
(9) dxa
x1M
a
x1M
a
x1M1
xa2
2xK
(8) dxa
x1M
a
x1M
a
x1M1
xa2
2xK
0x
2
3
3
1
2
2
1
1
2
3
3
1
2
2
1
1
a
02r2r
2
3
3
1
2
2
1
1
a
01r1r
15
stress distributions induced by simple tension or bending load and those cases were
predominantly used to derive the weight functions listed below.
Semi-elliptical surface and corner cracks
The set of equations necessary for deriving MiA parameters for the weight function
corresponding to the deepest point A on the crack front (Fig. 3) was found to be identical
to the set of equations derived for the edge crack.
In the case of the surface point B the first two equations were the same as previously and
the additional equation was derived [6] by satisfying the condition that the weight
function must vanish for x=0.
(13) 0
a
x1M
a
x1M
a
x1M1
xa2
2
x
(12) dxa
x1M
a
x1M
a
x1M1
xa2
2xK
(11) dxa
x1M
a
x1M
a
x1M1
xa2
2xK
0x
2
3
A3
1
A2
2
1
A1
2
3
A3
1
A2
2
1
A1
a
02r
A2r
2
3
A3
1
A2
2
1
A1
a
01r
A1r
(16) MMM10
(15) dxa
xM
a
xM
a
xM1
x
2xK
(14) dxa
xM
a
xM
a
xM1
x
2xK
B3B21B
2
3
B3
1
B2
2
1
B1
a
02r
B
2r
2
3
B3
1
B2
2
1
B1
a
01r
B
1r
16
The three unknown parameters Mi can be determined by simultaneously solving one of
the set of equations presented above.
17
7. NUMERICAL INTEGRATION OF THE WEIGHT FUNCTION
AND CALCULATION OF STRESS INTENSITY FACTORS
The calculation of stress intensity factor from the weight function requires integration of
of the product “s(x)m(x,a)” along the crack length according to eq.(1). The weight
function can always be written in the general form of eq.(3) or eq.(4). However the stress
distribution “s(x)” can take any form depending on the problem of interest. If the stress
distribution is given in the form of a mathematical expression analytical integration can
be performed and closed form integrals of eq.(1) are sometimes feasible. However, very
often the stress distribution “s(x)” is obtained from finite element calculations and the
results are given as a series of stress values corresponding to a range of point of
coordinate “x”. Therefore, a numerical integration technique is needed for the integration
of equation (1) and the calculation of stress intensity factors. Two methods of efficient
integration of eq.(1) are described below.
7.1 Integration by using the centroids of areas under the weight function curve
The integration method using the area centroids is based on the following theorem:
If m(x,a) and (x) are monotonic and linear function respectively and both depend on
variable x only, (Fig. 5), then the integral (1) can be calculated from expression (14),
representing the product of the area S under the curve m(x,a) and the value of the function
(X) corresponding to the coordinate x=X of the centroid C.
The weight functions, m(x,a), are monotonic and non-linear. The stress functions,
(x), are usually nonlinear as well. Therefore, in order to apply the theorem above to the
integral (1), the integration interval is divided into “n” sub-intervals in such a way that,
the stress function (x) is approximated by the secant line drawn between the end points
of each sub-interval Fig. (6). Thus, the stress function (x) over the sub-interval “i”, may
be written in the form of Eq. (15),
(14) XSK
18
After substitution of expression (15) into eq. (1) and summation over all sub-intervals the
following expression for the stress intensity factor can be derived.
Each integral in eq.(16) can be computed by using the simplified (Fig. 5) integration
method given in the form of Eq.(14). Thus, the stress intensity factor, K, can be finally
written in the form of expression (17).
In order to calculate the stress intensity factor given in the form of Eq. (17), it is necessary
to calculate the areas Si under the weight function curve m(x,a), and the co-ordinates of
their centroids, Xi. Both the areas Si and the centroid co-ordinates Xi for each sub-interval
can be calculated once in a general form based on the generalised weight function (2).
However, they appear to be too lengthy for an easy hand calculation. Fortunately, further
simplification of the integration routine is possible due to the fact that the weight
functions are smooth within their ranges of integration. Therefore the procedure can be
reversed (Fig. 7) by calculating first the areas Si* under the stress function (x) and the
coordinates Xi* of their centroid. Then the appropriate values of the weight function
m(X*,a) can be calculated from expression (2). It is worth noting that in the case of the
piece-wise approximation of the stress function, (x), the areas Si* and the coordinates of
their centroids, Xi*, can be easily calculated from relations (18) and (19) respectively.
(15a) xAxB and
xx
xxA
:where
(15) xx1xfor BxAx
iiii
1ii
1iii
iiiii
(16) dxa,xmBxAKn
1i
x
x
ii
i
1i
(17) n 1,2,i here w,XSK i
n
i
i
19
Finally the stress intensity factor, K, is calculated from expression (20).
Thus the numerical procedure for calculating the stress intensity factor using the
integration method described above requires the calculation of appropriate parameters
using equations (2) and (18-20). The method described above is recommended for quick
approximate calculations with the help of a hand calculator.
(19) xx3
xx2xxxX
(18) xxxx2
1S
1ii
i1i1iii
*i
1ii1ii*i
(20) a,XmSK *i
n
1i
*i
20
Fig. 5.Graphical representation of simplified integration of the product of two one-
dimensional functions, a) monotonic non-linear weight function m(x,a), b) linear stress
function s(x).
21
Fig. 6. Application of the simplified integration method to the case of non-linear stress
distribution; a) weight function m(x,a), b) linearized stress function.
22
Fig. 7. Integration method based on the sub-areas and centroids of the linearized
piecewise stress function, a) weight function m(x,a), b) linearized piecewise stress
function.
23
7.2 Analytical integration of the linearized piecewise stress distribution and the weight
function
The integration technique described in Section 7.1 is convenient for hand calculator
calculation when the stress distribution can be approximated by a few linear segments.
However, the segments adjacent to the crack tip can not be large because the weight
function tends to infinity near the crack tip and if the stress is the highest near the crack
tip the method described above might be sometimes inaccurate. Moreover, in the case of
stress distributions characterized by high gradients accurate approximation of the stress
distribution requires relatively large number of linear pieces and the integration has to be
carried out with the help of a computer. However, the computer integration routine can
also be significantly simplified because closed form solution to the integral (16) can be
derived analytically for each linear piece (Fig. 7) of the stress function s(x).
The stress function s(x) over the linear segment “i” can be given in the form of the linear
equation (15). Thus the contribution to the stress intensity factor associated with the stress
segment “i” can be calculated from eq.(1) after substituting appropriate expressions for
the stress and the weight function. The solutions given below have been derived for the
deepest and for the surface point of semi-elliptical crack using the weight function (3) and
(4) respectively. The solution to one-dimensional cracks is the same as for the deepest
point of semi-elliptical cracks.
Deepest point A (Fig. 4)
Surface point B (Fig.4)
(21) dxa
x1M
a
x1M
a
x1M1
a
x1a2
2BxAK
2
3
A3
1
A2
2
1
A1
x
xii
Ai
i
1i
(22) dxa
xM
a
xM
a
xM1
x2
2BxAK
2
3
B3
1
B2
2
1
B1
x
xii
B
i
i
1i
24
The closed form expressions resulting from the integration of eq.(21) and eq.(22) are
given below.
Deepest point A (Fig. 4)
Surface point B (Fig.4)
3
i
3
1i16
2
5
i2
5
1i5i
2
i
2
1i14
2
3
i2
3
1i3i
1
i
1
1i12
2
1
i2
1
1i1i
iiiii
6iA35iA24iA13ii
4iA33iA22iA11iiAi
a
x1
a
x1
3
aC
a
x1
a
x1
5
a2C
a
x1
a
x1
2
aC
a
x1
a
x1
3
a2C
a
x1
a
x1aC
a
x1
a
x1a2C
aA and aAB
:where
(23) CMCMCMC
CMCMCMC
a
2K
25
Equations (23) and (24) can be used for calculating stress intensity contributions due to
each linear piece of the stress distribution function by substituting appropriate values for
a, xi-1, xi, Ai and Bi. The stress intensity factor K can be finally calculated as the sum of
allcontributions Ki associated with the linear pieces within the range of 0 = x = a.
Thus the integration can be reduced to the substitution of appropriate parameters into
equations (23 -24) and summation according to eq(25). This makes it possible to develop
very efficient numerical integration routine which is important in the case of lengthy
fatigue crack growth analyses.
(25) KKn
i
i
3
1i
3
i16
2
5
1i2
5
i5i
2
1i
2
i14
2
3
1i2
3
i3i
1
1i
1
i12
2
1
1i2
1
i1i
iiiii
6iB35iB24iB13ii
4iB33iB22iB11iiiB
i
a
x
a
x
3
a D
a
x
a
x
5
a2D
a
x
a
x
2
a D
a
x
a
x
3
a2D
a
x
a
xa D
a
x
a
xa2D
aA and aAB
:where
(24) DMDMDMD
DMDMDMD
a
2K
26
8. WEIGHT FUNCTIONS FOR CRACKS IN PLATES and DISKS
The weight functions are given in the form of expressions describing the Mi parameters as
functions of crack dimensions and geometry of the cracked body. Given are the range of
application, the accuracy and the source of the reference stress intensity factors for each
set of parameters Mi including the generic geometry of the crack body.
8.1 Through crack in an infinite plate subjected to symmetrical loading
WEIGHT FUNCTION (Fig. 8a)
PARAMETERS
M1= 0.0698747
M2= -0.0904839
M3= 0.427203
ACCURACY: Max. error less than 1% when compared to exact solution
REFERENCES:
weight function:
Glinka G., Shen G., 1991, “Universal Features of Weight Functions for Cracks in Mode I
“Engng. Fract. Mech”, Vol. 40, No. 6, pp. 1135-1146.
reference data:
Tada H., Paris P. C., Irwin G.R., 1985, The Stress Analysis of Cracks Handbook, 2nd ed.,
Paris Productions Inc., St. Louis, Missouri, pp. 5.11-5.11a.
2
3
3
1
2
2
1
1a
x1M
a
x1M
a
x1M1
xa2
2a,xm
27
Figure 8. Central through-crack in an infinite plate (a) and finite-width plate (b).
28
8.2 Central through crack in a finite width plate subjected to symmetric loading
WEIGHT FUNCTION (Fig. 8b )
PARAMETERS
RANGE OF APPLICATION: 0 < a/w < 0.9.
ACCURACY: Better than 1% when compared to the reference solution.
REFERENCES:
- weight function:
Moftakhar A., Glinka G., 1992, “Calculation of Stress Intensity Factors by Efficient
Integration of Weight Functions,“ Engng. Fract. Mech., Vol. 43, No. 5, pp. 749-756.
Glinka G., Shen G., 1991, “Universal Features of Weight Functions for Cracks in Mode I,
“Engng. Fract. Mech., Vol. 40, No. 6, pp. 1135-1146.
- reference data:
Tada H., Paris P. C., Irwin G.R., 1985, The Stress Analysis of Cracks Handbook, 2nd ed.,
Paris Productions Inc., St. Louis, Missouri, pp. 2.1-2.2, 2.34.
2
3
3
1
2
2
1
1a
x1M
a
x1M
a
x1M1
xa2
2a,xm
765
432
3
65
432
2
765
432
1
w
a1575.68
w
a882.295
w
a128.574
w
a255.347
w
a40.138
w
a6349.29
w
a56001.2427216.0M
w
a128.111
w
a445.392
w
a599.210
w
a6553.89
w
a5325.22
w
a14886.209049.0M
w
a218.140
w
a939.377
w
a395.552
w
a699.200
w
a0886.50
w
a5407.5
w
a40117.006987.0M
29
Sih G., 1973, Handbook of Stress Intensity Factors, Institute of Fracture and Solid
Mechanics, Lehigh University, Bethlehem, P.A.
8.3 Edge crack in a semi-infinite plate
WEIGHT FUNCTION (Fig. 9a)
PARAMETERS
M1= 0.0719768
M2= 0.246984
M3= 0.514465
ACCURACY: Max. error less than 1% when compared to the reference solution
REFERENCES:
- weight function:
Glinka G., Shen G., 1991, “Universal Features of Weight Functions for Cracks in Mode
I,“ Engng. Fract. Mech., Vol. 40, No. 6, pp. 1135-1146.
- reference data:
Tada H., Paris P. C., Irwin G.R., 1985, The Stress Analysis of Cracks Handbook, 2nd ed.,
Paris Productions Inc., St. Louis, Missouri, pp. 8.3-8.3a.
Sih G., 1973, Handbook of Stress Intensity Factors, Institute of Fracture and Solid
Mechanics, Lehigh University, Bethlehem, P.A.
2
3
3
1
2
2
1
1a
x1M
a
x1M
a
x1M1
xa2
2a,xm
30
Fig 9: Edge crack solutions in semi-infinite and finite-width plates.
31
8.4 Edge crack in a finite width plate
WEIGHT FUNCTION (Fig.9b)
PARAMETERS
RANGE OF APPLICATION: 0 < a/w < 0.9.
ACCURACY: Better than 1% when compared to the reference solution.
REFERENCES:
- weight function:
Moftakhar A., Glinka G., 1992, “Calculation of Stress Intensity Factors by Efficient
Integration of Weight Functions,“ Engng. Fract. Mech., Vol. 43, No. 5, pp. 749-756.
Glinka G., Shen G., 1991, “Universal Features of Weight Functions for Cracks in Mode
I,“ Engng. Fract. Mech., Vol. 40, No. 6, pp. 1135-1146.
- reference data:
Tada H., Paris P. C., Irwin G.R., 1985, The Stress Analysis of Cracks Handbook, 2nd ed.,
Paris Productions Inc., St. Louis, Missouri, p. 2.27.
2
3
3
1
2
2
1
1a
x1M
a
x1M
a
x1M1
xa2
2a,xm
8765
432
3
109876
5432
2
109876
5432
1
w
a7.35780
w
a106246
w
a128746
w
a6.81388
w
a2.28592
w
a53.5479
w
a074.532
w
a3235.22529659.0M
w
a127291
w
a531940
w
a936863
w
a904370
w
a520551
w
a181755
w
a8.37303
w
a76.4058
w
a456.176
w
a47583.6246984.0M
w
a49544
w
a208233
w
a368270
w
a356469
w
a205384
w
a7.15907
w
a8.14583
w
a95.1554
w
a1001.61
w
a51346.10719768.0M
32
Kaya A. C.., Erdogan F., 1980, “Stress Intensity Factors and COD in an Orthotropic Strip,
“Int. J. Fracture, Vol. 16, pp. 171-190.
8.5 Two symmetrical edge cracks in a finite width plate subjected to symmetrical
loading
WEIGHT FUNCTION (Fig.9c)
PARAMETERS
RANGE OF APPLICATION: 0 < a/w < 0.9.
ACCURACY: Better than 1% when compared to the reference solution.
REFERENCES:
- weight function:
Moftakhar A., Glinka G., 1992, “Calculation of Stress Intensity Factors by Efficient
Integration of Weight Functions,“ Engng. Fract. Mech., Vol. 43, No. 5, pp. 749-756.
Glinka G., Shen G., 1991, “Universal Features of Weight Functions for Cracks in Mode I,
“Engng. Fract. Mech., Vol. 40, No. 6, pp. 1135-1146.
2
3
3
1
2
2
1
1a
x1M
a
x1M
a
x1M1
xa2
2a,xm
65
432
3
65
432
2
8765
433
1
w
a8893.59
w
a976.131
w
a933.115
w
a9251.47
w
a5597.10
w
a7512.04983.0M
w
a4731.86
w
a634.182
w
a937.151
w
a5326.56
w
a1687.11
w
a6146.02234.0M
w
a009.127
w
a591.336
w
a837.336
w
a266.481
w
a6924.19
w
a64559.4
w
a41028.1
w
a02230.008502.0M
33
- reference data:
Tada H., Paris P. C., Irwin G.R., 1985, The Stress Analysis of Cracks Handbook, 2nd ed.,
Paris Productions Inc., St. Louis, Missouri, p. 2.31.
8.6 Embedded penny-shape crack in an infinite body subjected to symmetric loading
WEIGHT FUNCTION (Fig. 10) - Point A
m x a
a xM
x
aM
x
aM
x
a
M
M
M
A A A A
A
A
A
( , )
.
.
.
/ /
2
21 1 1 1
0 646714
0 303783
0527654
1
1 2
2 3
3 2
1
2
3
PARAMETERS
WEIGHT FUNCTION (Fig. 10) - Point B
m x ax
Mx
aM
x
aM
x
a
M
M
M
B B B B
B
B
B
( , )
/ /
21
1
0
0
1
1 2
2 3
3 2
1
2
3
PARAMETERS
ACCURACY: Better than 1% when compared to the reference solution.
REFERENCES:
- weight function:
Glinka G., Shen G., 1991, “Universal Features of Weight Functions for Cracks in Mode I
,“Engng. Fract. Mech., Vol. 40, No. 6, pp. 1135-1146.
- reference data:
Gallin L. A., 1953, Contact Problems in the Theory of Elasticity, (in Russian),
translation: North Carolina State College Publications (1961).
34
Tada H., Paris P. C., Irwin G.R., 1985, The Stress Analysis of Cracks Handbook, 2nd ed.,
Paris Productions Inc., St. Louis, Missouri, pp. 24.2-24.3.
35
Fig. 10. Embedded penny-shape crack in an infinite body subjected to
symmetrical loading
36
8.7 Shallow semi-elliptical surface crack in a finite thickness plate (a/c < 1)
WEIGHT FUNCTION (Fig.11) - Deepest Point A
PARAMETERS
where:
and
2
3
A3
1
A2
2
1
A1Aa
x1M
a
x1M
a
x1M1
xa2
2a,xm
5
8YY2
Q2
6M
3M
5
24Y3Y2
Q2
2M
01A3
A2
10A1
071.8
3
953.9
2
688.0
2
1
32
0
6
3
4
2
2
100
65.1
c
a1211.24
c
a014.0
1459.1A
c
a10.22
c
a027.0
1995.0A
c
a147.0
1
c
a007.2
c
a045.3456.0A
c
a4394.0
c
a7703.0
c
a2581.00929.1A
t
aA
t
aA
t
aAAY
c
a464.11Q
37
WEIGHT FUNCTION (Fig.11) - Surface Point B
PARAMETERS
where:
2
3
B3
1
B2
2
1
B1Ba
xM
a
xM
a
xM1
xa2
2a,xm
203.9
3
286.9
2
846.0
2
1
32
0
6
3
4
2
2
101
c
a1924.21
c
a020.0
1
c
a643.3228.4B
c
a1259.17
c
a041.0
1
c
a126.3148.3B
c
a198.0
1
c
a534.0
c
a665.1652.1B
c
a460.0
c
a7412.0
c
a1231.04537.0B
t
aB
t
aB
t
aBBY
8F7F10Q
3M
15F3F2Q
15M
8F3F5Q
3M
01B3
10B2
01B1
38
and
RANGE OF APPLICATION: 0 = a/t =0.8 and 0 = a/c = 1.0
ACCURACY: Better than 3% when compared to the FEM data.
REFERENCES:
- weight function:
Wang X., Lambert S. B., 1995, “Stress Intensity Factors for Low Aspect Ratio Semi-
Elliptical Surface Cracks in Finite-Thickness Plates Subjected to Nonuniform Stresses,“
Engng. Fract. Mech., Vol. 51, No. 4, pp. 517-532.
Shen G., Glinka G., 1991, “Weight Functions for a Surface Semi-Elliptical Crack in a
Finite Thickness Plate,“ Theor. Appl. Fract. Mech., Vol. 15, No. 2, pp. 247-255 (this
paper contains the older version of the weight function).
c
a157.0
879.0101.1C
c
a5128.0
c
a3219.15083.1C
c
a0185.0
c
a1548.029782.1C
c
a
t
aC
t
aCCF
c
a464.11Q
2
2
1
2
0
4
2
2
100
65.1
c
a035.0
199.0
c
a608.0190.0D
c
a5876.0
c
a2289.11207.1D
c
a7250.0
c
a4646.1
c
a0642.12687.1D
c
a
t
aD
t
aDDF
2
2
1
32
0
4
2
2
101
39
Shen G., Plumtree A., Glinka G., 1991, “Weight Function for the Surface Point of Semi-
Elliptical Surface Crack in a Finite Thickness Plate,“ Engng. Fract. Mech., Vol. 40, No.
1, pp. 167-176 (this paper contains the older version of the weight function).
- reference data:
Shiratori M., Miyoshi T., Tanikawa K., 1986, “Analysis of Stress Intensity Factors for
Surface Cracks Subjected to Arbitrarily Distributed Surface Stress (2nd Report, Analysis
and Application of Influence Coefficients for Flat Plates with a Semielliptical Surface
Crack),“ Trans. JSME, Vol. 52, pp. 390-398.
Gross B., Srawley J. E., 1965, “Stress Intensity Factor for Single-Edge-Notch Specimens
in Bending or Combined Bending and Tension by Boundary Collocation of a Stress
Function,“ NASA TN, D-2603.
Gross B., Srawley J. E., 1966, Plane Strain Crack Toughness Testing of High Strength
Metallic Materials, ASTM STP 410.
Raju I. S., Newman J. C., 1979, “Stress Intensity Factors for a Wide Range of Semi-
Elliptical Surface Cracks in Finite Thickness Plate, “Engng. Fracture Mech., Vol. 11, pp.
817-829.
Newman J. C., Raju I. S., 1981, “An Empirical Stress Intensity Factor Equation for the
Surface Crack, “Engng. Fracture Mech., Vol. 15, pp. 185-192.
40
Fig. 11. Shallow semi-elliptical surface crack in a finite thickness plate.
41
8.8 Deep semi-elliptical surface crack in a finite thickness plate (a/c > 1)
WEIGHT FUNCTION (Fig. 12) - Deepest Point A
PARAMETERS
where:
and
2
3
A3
1
A2
2
1
A1Aa
x1M
a
x1M
a
x1M1
xa2
2a,xm
5
8YY2
Q2
6M
3M
5
24Y3Y2
Q2
2M
01A3
A2
10A1
2
2
2
1
2
0
4
2
2
100
265.1
65.1
c
a0934.0
c
a5517.07855.0A
c
a2454.0
c
a01106.108461.1A
c
a03526.0
c
a12945.013047.1A
t
aA
t
aAAY
0.1c
afor
c
a
a
c464.11
0.1c
a0for
c
a464.11
Q
42
WEIGHT FUNCTION (Fig. 12) - Surface Point B
PARAMETERS
where:
2
2
2
1
2
0
4
2
2
101
c
a0731.0
c
a4177.06459.0B
c
a1492.0
c
a6352.07259.0B
c
a0529.0
c
a2609.05044.0B
t
aB
t
aBBY
2
3
B3
1
B2
2
1
B1Ba
x1M
a
x1M
a
x1M1
xa2
2a,xm
8F7F10Q
3M
15F3F2Q
15M
8F3F5Q
3M
01B3
10B2
01B1
43
and
RANGE OF APPLICATION: 0 = a/t = 0.8 and 0.6 = a\c = 2.0
ACCURACY: Better than 2% when compared to the reference FEM data.
REFERENCES:
- weight function:
Wang X., 1995, “Weight Functions for High Aspect Ratio Semi-Elliptical Surface Cracks
in Finite-Thickness Plates”, University of Waterloo, to be published.
- reference data:
Shiratori M., Miyoshi T., Tanikawa K., 1986, “Analysis of Stress Intensity Factors for
Surface Cracks Subjected to Arbitrarily Distributed Surface Stress (2nd Report, Analysis
and Application of Influence Coefficients for Flat Plates with a Semielliptical Surface
Crack),“ Trans. JSME, Vol. 52, pp. 390-398.
2
2
2
1
2
0
4
2
2
100
265.1
65.1
c
a1894.0
c
a4423.008429.0C
c
a37185.0
c
a5275.1757673.1C
c
a08125.0
c
a29091.033469.1C
c
a
t
aC
t
aCCF
0.1c
afor
c
a
a
c464.11
0.1c
a0for
c
a464.11
Q
2
2
2
1
2
0
4
2
2
101
c
a1864.0
c
a4784.02246.0D
c
a23315.0
c
a98743.015312.1D
c
a0781.0
c
a2065.011855.1D
c
a
t
aD
t
aDDF
44
Murakami Y., et al., 1992, Stress Intensity Factors Handbook, Vol. 3, Pergamon Press,
pp. 588-590.
45
Fig. 12. Deep semi-elliptical surface crack in a finite thickness plate
46
8.9 Quarter-elliptical corner crack in a finite thickness plate
WEIGHT FUNCTION (Fig. 13) - Deepest Point A
PARAMETERS
where:
and
2
3
A3
1
A2
2
1
A1Aa
x1M
a
x1M
a
x1M1
xa2
2a,xm
5
8YY2
Q2
6M
3M
5
24Y3Y2
Q2
2M
01A3
A2
10A1
32
4
32
3
32
2
32
1
32
0
4
4
3
3
2
2100
65.1
c
a393.8
c
a797.10
c
a073.3572.0A
c
a441.13
c
a906.12
c
a857.1293.1A
c
a918.1
c
a747.6
c
a216.13972.4A
c
a028.0
c
a310.1
c
a953.1599.0A
c
a111.0
c
a186.0
c
a016.0041.1A
t
aA
t
aA
t
aA
t
aAAY
c
a464.11Q
47
WEIGHT FUNCTION (Fig. 13) - Surface Point B
PARAMETERS
where:
32
4
32
3
32
2
32
1
32
0
4
4
3
3
2
2101
c
a804.5
c
a250.8
c
a977.2162.0B
c
a370.6
c
a357.5
c
a731.1359.1B
c
a135.0
c
a349.6
c
a028.9468.3B
c
a184.0
c
a740.0
c
a373.1507.0B
c
a052.0
c
a213.0
c
a323.0500.0B
t
aB
t
aB
t
aB
t
aBBY
2
3
B3
1
B2
2
1
B1Ba
xM
a
xM
a
xM1
x
2a,xm
8F7F10Q
3M
15F3F2Q
15M
8F3F5Q
3M
01B3
10B2
01B1
48
and
RANGE OF APPLICATION: 0 = a\t =0.8 and 0.2= a\c = 1.0
ACCURACY: Better than 1.5% when compared to the FEM data.
REFERENCES:
32
4
32
3
32
2
32
1
32
0
4
4
3
3
2
2100
65.1
c
a92.99
c
a6.212
c
a9.14280.30C
c
a9.145
c
a0.302
c
a1.19416.39C
c
a92.72
c
a2.152
c
a9.10095.22C
c
a504.1
c
a658.1
c
a2261.02318.0C
c
a7278.0
c
a016.3
c
a495.4340.3C
c
a
t
aC
t
aC
t
aC
t
aCCF
c
a464.11Q
32
4
32
3
32
2
32
1
32
0
4
4
3
3
2
2101
c
a195.19
c
a228.30
c
a176.9141.1D
c
a910.13
c
a848.8
c
a932.15682.9D
c
a677.2
c
a624.12
c
a057.15019.4D
c
a226.13
c
a001.29
c
a498.20600.4D
c
a511.0
c
a477.2
c
a840.3831.2D
c
a
t
aD
t
aD
t
aD
t
aDDF
49
- weight function:
Zheng X. J., Glinka G., Dubey R. N., 1994, “Stress Intensity Factors and Weight
Functions for a Corner Crack in a Finite Thickness Plate,“ Engng. Fracture Mech.,(to be
published, 1996)
- reference data:
Shiratori M., Miyoshi T., 1985, “Analysis of Stress Intensity Factors for Surface Cracks
Subjected to Arbitrarily Distributed Surface Stress (Analysis and Application of Data
Base of Influence Coefficients Kij),“ Proc. Third Conf. Fract. Mech., Soc. Materials
Science Japan, pp 82-86.
Murakami Y., et al., 1992, Stress Intensity Factors Handbook, Vol. 3, Pergamon Press,
pp. 591-597.
Raju I. S., Newman J. C., 1988, “Stress Intensity Factors for Corner Cracks in
Rectangular Bars,“ in Fracture Mechanics: Nineteenth Symposium, ASTM STP 969,
T.A.Cruse, ed., pp. 43-55.
50
Fig. 13. Quarter-elliptical corner crack in a finite thickness plate
51
8.10 Radial edge crack in a circular disk
WEIGHT FUNCTION (Fig. 14)
PARAMETERS
RANGE OF APPLICATION: 0 < a/D< 0.9.
ACCURACY: Better than 1.5% compared to the reference solution.
REFERENCES:
- weight function:
Kiciak A., 1995, University of Waterloo, to be published.
Glinka G., Shen G., 1991, “Universal Features of Weight Functions for Cracks in Mode I,
“Engng. Fract. Mech., Vol. 40, No. 6, pp. 1135-1146 - this paper contains the original
version of the weight function with coefficients Mi given in tabular form.
- reference data:
Wu X. R., Carlsson J. A., 1991, Weight Functions and Stress Intensity Factor Solutions,
Pergamon Press.
2
3
3
1
2
2
1
1a
x1M
a
x1M
a
x1M1
xa2
2a,xm
7654
32
3
7654
32
2
654
32
1
D
a1270.36
D
a35817.61
D
a6.91042
w
a61627.54
D
a14897.48
D
a85709.16
D
a06605.309836.1
expM
D
a34267.125
D
a66155.356
D
a48183.428
D
a25303.280
D
a21153.112
D
a90993.30
D
a48276.858602.0
expM
D
a87969.9exp
D
a74080.0
D
a83026.2
D
a56109.4
D
a96175.3
D
a94141.1
D
a49586.004732.0
M
52
Fig. 14. Radial edge crack in a circular disk
53
8.11 Edge crack in a finite thickness plate with the right angle corner
WEIGHT FUNCTION (Fig. 15)
m x aa x
Mx
aM
x
a,
2
21 1 11 2
2
PARAMETERS
Ma
t
a
t
a
t
a
t
a
t
a
t
Ma
t
a
t
a
t
a
t
a
t
a
t
1
1 5 3 4 5 6
7 5 9
2
1 5 3 4 5 6
7 5
0 6643 12 7438 397 8081 3285.181 14162 587
30127 158 258119 535
0 1117 3 857 30127 158 285.4393 647 6118
934 4538 596 8319
. . . .
. .
. . . .
. .
. .
.
. .
.
9
RANGE OF APPLICATION: 0 = a\t = 0.5, a = 90o.
ACCURACY: Better than 3% compared to the reference data.
REFERENCES:
- weight function:
Niu X., Glinka G., 1990, “Weight Functions for Edge and Surface Semi-Elliptical Crack
in Flat Plates and Plates with Corners,“ Engng. Fract. Mech., Vol. 36, No. 3, pp. 459-
475.
- reference data:
Hasebe N., Ueda M., 1981, “A Crack Originating from an Angular Corner of a Semi-
Infinite Plate with a Step,“ Bull. JSME, Vol. 24, pp. 483-488.
Hasebe N., Matsuura S., 1984, “Stress Analysis of a Strip with a Step and a Crack,“
Engng. Fract. Mech., Vol. 20, pp. 447-462.
54
Fig. 15. Edge crack in a finite thickness plate with the right angle corner
55
8.12 Edge crack in a finite thickness plate with an angular corner
WEIGHT FUNCTION (Fig. 16)
m x a
a xM
x
aM
x
aF( , )
2
21 1 11 2
2
PARAMETERS
Ma
t
a
t
a
t
a
t
a
t
a
t
Ma
t
a
t
a
t
a
t
a
t
a
t
1
1 5 3 4 5 6
7 5 9
2
1 5 3 4 5 6
7 5
0 6643 12 7438 397 8081 3285.181 14162 587
30127 158 258119 535
0 1117 3 857 30127 158 285.4393 647 6118
934 4538 596 8319
. . . .
. .
. . . .
. .
. .
.
. .
.
9
1 2 3 2 2
5 2 3
16
2 0 0355 3 3324 21 5999 58 8513 81 6246
56 9396 15.8784
Fa
t
a
t
a
t
a
t
a
t
a
t
. . . . .
.
/ /
/
RANGE OF APPLICATION: 0 = a/t = 0.5 and /6 = a = /3 for a in radians.
ACCURACY: unknown.
REFERENCES:
- weight function:
Niu X., Glinka G., 1990, “Weight Functions for Edge and Surface Semi-Elliptical Crack
in Flat Plates and Plates with Corners,“ Engng. Fract. Mech., Vol. 36, No. 3, pp. 459-
475.
Niu X., Glinka G., 1987, “The Weld Profile Effect on Stress Intensity Factors in
Weldments,” Int. J. Fract., Vol. 35, pp. 3-20.
- reference data:
Hasebe N., Ueda M., 1981, “A Cracks Originating from an Angular Corner of a Semi-
Infinite Plate with a Step,“ Bull. JSME, Vol. 24, pp. 483-488.
Hasebe N., Matsuura S., 1984, “StressAnalysis of a Strip with a Step and a Crack,“
Engng. Fract. Mech., Vol. 20, pp. 447-462.
56
Fig. 16. Edge crack in a finite thickness plate with an angular corner
57
9. WEIGHT FUNCTIONS FOR CRACKS IN CYLINDERS
The weight functions are given in the form of expressions describing the Mi parameters as
functions of crack dimensions and geometry of the cracked body. Given are the range of
application, the accuracy and the source of the reference stress intensity factors for each
set of parameters Mi including the generic geometry of the crack body. The geometry of a
cylinder is characterized by the ratio of the external to internal radius, Ro/Ri.
58
9.1 Internal circumferential edge crack in a thick cylinder subjected to axisymmetric
loading (Ro/Ri=2.0)
WEIGHT FUNCTION (Fig. 17)
m x aa x
Mx
aM
x
aM
x
a,
2
21 1 1 11
1
2
2
1
3
3
2
PARAMETERS
MY
at
Y
M
M YY
at
1
1
0
2
3 0
1
2
2
3 24
5
3
6
2
2 296
15
where:
t = R0 - Ri (cylinder wall thickness)
Y A Aa
tA
a
tA
a
tA
a
tA
a
t
A
A
A
A
A
A
0 0 1 2
2
3
3
4
4
5
5
0
1
2
3
4
5
1 033505
2 41397
15 34436
46 42214
22 49592
31 75198
26 94214
exp
.
.
.
.
.
.
.
RANGE OF APPLICATION: R0/Ri = 2 and 0 = a/t = 0.8
ACCURACY: Better than 5% when compared to the FEM data.
REFERENCES:
- weight function:
59
Liebster T. D., Glinka G., Burns D. J., Mettu S. R., 1994, “Calculating Stress Intensity
Factors for Internal Circumferential Cracks by Using Weight Functions,“ ASME PVP-
Vol. 281, High Pressure Technology, J. A. Kapp, ed., pp.1-6.
- reference data:
Mettu S. R., Forman R. G., 1993, “Analysis of Circumferential Cracks in Cylinders Using
the Weight Function Method,“ Fracture Mechanics: 23rd Symposium,ASTM STP 1189,
R. Chona, ed., American Society for Testing and Mateials, pp. 417-440.
60
Figure 17. Internal circumferential edge crack in a thick cylinder subjected to
axisymmetric loading.
61
9.2 Two internal symmetrical axial edge cracks in a thick cylinder subjected to
symmetrical loading (Ro/Ri=2.0)
WEIGHT FUNCTION (Fig. 18)
PARAMETERS
MY
s
Y
sY
MY
s
Y
s
Y
MY
s
Y
sY
1
1 2
2 0
2
1 2
2
0
3
1 2
2 0
6
2
13 142
48
5
105
2
3 3
221
12
2
22 214
64
5
where:
s = a/( R0 - Ri), and
for 0 s 0.1
Y s s s s s
s s s s s
0
2 3 5 4 7 5
9 6 10 7 11 8 12 9 12 10
111198 0 978998 360 49 42949 2 261945 10 9 5658 10
219014 10 31481 10 2 74822 10 132888 10 2 72711 10
. . . . . .
. . . . .
Y s s1
20 000291 0 66207 0 224015 . . .
Y s s2
6 4 28 01761 10 7 74379 10 0534838 . . .
for 0.1 s 0.8
Y s s s s s0
2 3 4 51071 0 424314 120826 511629 9 74362 6 08975 . . . . . .
Y s s s s s1
2 3 4 50 009535 0866583 135905 589469 7 68059 4 25993 . . . . . .
Y s s s s s2
2 3 4 50 007826 0161374 0 63532 375368 4 85997 2 85235 . . . . . .
RANGE OF APPLICATION: R0/Ri = 2.0 and 0.1 = s = 0.8
2
3
3
1
2
2
1
1a
x1M
a
x1M
a
x1M1
xa2
2a,xm
62
ACCURACY: Better than 0.2% when compared to the FEM data.
REFERENCES:
- weight function:
Shen. G., Glinka G. ,1993, “Stress Intensity Factors for Internal Edge and Semi-Elliptical
Cracks in Hollow Cylinders,“ ASME PVP-Vol. 263, High Pressure - Codes, Analysis, and
Applications, A. Khare, ed., pp.73-79.
Glinka G., Shen G., 1991, “Universal Features of Weight Functions for Cracks in Mode I,
“Engng. Fract. Mech., Vol. 40, No. 6, pp. 1135-1146.
- reference data:
Andrasic C. P., Parker A. P., 1984, “Dimensionless Stress Intensity Factors for Cracked
Thick Cylinder Under Polynomial Crack Face Loading,“ Engng. Fract. Mech., Vol. 19,
1984, pp. 187-193.
Murakami Y., et al., 1986, Stress Intensity Factor Handbook, Vol. 2, Pergamon Press,
pp. 309-317.
63
Fig. 18. Two internal symmetrical axial edge cracks in a thick cylinder subjected
to symmetrical loading
64
9.3 External axial edge crack in a thick cylinder (Ro/Ri=2.0)
WEIGHT FUNCTION (Fig. 19)
m x aa x
Mx
aM
x
aM
x
a,
2
21 1 1 11
1
2
2
1
3
3
2
PARAMETERS
MY
s
Y
sY
MY
s
Y
s
Y
MY
s
Y
sY
1
1 2
2 0
2
1 2
2
0
3
1 2
2 0
6
2
13 142
48
5
105
2
3 3
221
12
2
22 214
64
5
where:
s a R Ro i / ( ) , and
for 0 = s = 0.1
Y s s s s s
s s s
0
2 3 5 4 5 5
6 6 8 7 8 8
111201 167679 7 38413 354383 10534 10 7 33 10
101865 10 168865 10 652672 10
. . . . . .
. . .
Y s s1
4 20 907986 10 0 676241 0 310302 . . .
Y s s2
6 4 28 69582 10 9 8064 10 054827 . . .
for 0.1 = s = 0.8
Y s s s s s
s s
0
2 3 4 5
6 7
164 12 2614 121488 574 313 1542 3 2349 04
189584 628 971
. . . . . .
. .
Y s s s s s1
2 3 4 50 0243751 0 200288 342782 887021 12 4676 589425 . . . . . .
Y s s s s2
2 3 40 00172167 0 0261316 0 42169 0 230129 0 342948 . . . . .
RANGE OF APPLICATION: R0/Ri = 2.0 and 0.1 = s = 0.8
65
ACCURACY: Better than 0.2% when compared to the FEM data.
REFERENCES:
- weight function:
Shen. G., Liebster T. D., Glinka G. ,1993, “Calculation of Stress Intensity Factors for
Cracks in Pipes,” Proc. of the 12th Int. Conf. on Offshore Mech. and Arctic Engng..,
ASME,M. Salama et al., ed., Vol. III-B, Materials Engineering, pp.847-854.
Glinka G., Shen G., 1991, “Universal Features of Weight Functions for Cracks in Mode I,
“Engng. Fract. Mech., Vol. 40, No. 6, pp. 1135-1146.
- reference data:
Andrasic C. P., Parker A. P., 1984, “Dimensionless Stress Intensity Factors for Cracked
Thick Cylinder Under Polynomial Crack Face Loading,“ Engng. Fract. Mech., Vol. 19,
1984, pp. 187-193.
Murakami Y., et al., 1986, Stress Intensity Factor Handbook, Vol. 2, Pergamon Press, pp.
309-317.
66
Fig. 19. External axial edge crack in a thick cylinder
67
9.4 Internal circumferential semi-elliptical surface crack in a thick cylinder
(Ro/Ri=1.1)
WEIGHT FUNCTION (Fig. 20) - Deepest Point A
m x a
a xM
x
aM
x
aM
x
aA A A A( , )
/ /
2
21 1 1 11
1 2
2 3
3 2
PARAMETERS
MQ
Y Y
M
MQ
Y Y
A
A
A
1 0 1
2
3 1 0
2
22 3
24
5
3
6
22
8
5
where:
Qa
c
1 1464
1 65
.
.
, and t R Ro i
Y A Aa
tA
a
t
Aa
c
a
c
Aa
c
a
c
Aa
c
a
c
a
c
0 0 1 2
2
0
2
1
2
2
2 3
11378 0 4259 01299
01223 1502 08013
2 0572 9 4342 138825 65333
. . .
. . .
. . . .
Y B Ba
tB
a
t
Ba
c
a
c
Ba
c
a
c
a
c
Ba
c
a
c
a
c
1 0 1 2
2
0
2
1
2 3
2
2 3
05116 0 373 01129
0 3724 2 3922 33694 15589
13148 5771 7 9406 36193
. . .
. . . .
. . . .
WEIGHT FUNCTION (Fig. 20) - Surface Point B
68
m x ax
Mx
aM
x
aM
x
aB B B B( , )
/ /
21 1
1 2
2 3
3 2
PARAMETERS
MQ
F F
MQ
F F
MQ
F F
B
B
B
1 1 0
2 0 1
3 1 0
35 3 8
152 3 15
310 7 8
where:
Qa
c
1 1464
1 65
.
.
and t R Ro i
F Ca
c
Ca
t
a
t
Ca
t
a
t
a
t
a
t
C
0 0
0
2
1
2 3 4
1
0 9242 0 6172 01379
05437 21302 8 0279 118896 58708
. . .
. . . . .
and
F Da
c
Da
t
a
t
Da
t
a
t
a
t
a
t
D
1 0
0
2
1
2 3 4
1
0 7631 0 4891 01164
0 6287 388279 155542 241589 12 6441
. . .
. . . . .
RANGE OF APPLICATION: R0/Ri = 1.1, 0.1 = a/t = 0.8, and 0.2 = a/c = 1.0
ACCURACY: unknown.
REFERENCES:
- weight function:
69
Shen. G., Glinka G. ,1993, “Stress Intensity Factors for Internal Edge and Semi-Elliptical
Cracks in Hollow Cylinders,“ ASME PVP-Vol. 263, High Pressure - Codes, Analysis, and
Applications, A. Khare, ed., pp.73-79.
- reference data:
Shiratori M., 1989, “Analysis of Stress Intensity Factors for Surface Cracks in Pipes by an
Influence Function Method,“ ASME PVP-Vol. 167, pp. 45-50.
70
Figure 20. Internal circumferential semi-elliptical surface crack in a thick cylinder
71
9.5 Internal axial semi-elliptical surface crack in a thick cylinder (Ro/Ri=2.0)
WEIGHT FUNCTION (Fig. 21) - Deepest Point A
PARAMETERS
where:
and
2
3
A3
1
A2
2
1
A1Aa
x1M
a
x1M
a
x1M1
xa2
2a,xm
5
8Y2Y
Q2
6M
3M
5
24Y3Y
Q2
2M
10A3
A2
10A1
432
3
5432
2
5432
1
5432
0
4
3
2
2100
io
65.1
c
a084.52
c
a941.118
c
a243.85
c
a828.18140.0A
c
a532.131
c
a920.343
c
a535.323
c
a935.132
c
a265.20498.1A
c
a219.35
c
a89741
c
a649.83
c
a455.36
c
a205.7111.0A
c
a933.0
c
a007.2
c
a305.1
c
a153.0
c
a207.012.1A
t
aA
t
aA
t
aAAY
R-R tand c
a464.11Q
72
WEIGHT FUNCTION (Fig. 21) - Surface Point B
PARAMETERS
where:
432
3
5432
2
5432
1
5432
0
3
3
2
2101
c
a841.16
c
a425.33
c
a172.20
c
a742.3087.0B
c
a196.15
c
a961.44
c
a993.54
c
a313.33
c
a481.7821.0B
c
a554.23
c
a282.56
c
a963.41
c
a725.7
c
a496.1163.0B
c
a015.5
c
a482.12
c
a671.10
c
a617.3
c
a377.0687.0B
t
aB
t
aB
t
aBBY
2
3
B3
1
B2
2
1
B1Ba
xM
a
xM
a
xM1
x
2a,xm
8F10F3Q
3M
15FF3Q
15M
8F5F2Q
3M
10B3
01B2
10B1
73
and
RANGE OF APPLICATION: R0/Ri = 2.0, 0 = a/t = 0.8, a/c = 0 and 0.2 = a/c = 1.0
ACCURACY: Better than 3% when compared to the FEM data.
REFERENCES:
- weight function:
Zheng X. J., Glinka G., 1995, “Weight Functions and Stress Intensity Factors
for Longitudinal Semi-Elliptical Cracks in Thick-Wall Cylinders,“ J. Press.
Vess.Technol., ASME, vol. 117, 1995, pp 383 - 389
432
3
432
2
432
1
432
0
4
3
2
2100
io
65.1
c
a124.48
c
a822.144
c
a449.151
c
a389.64586.9C
c
a149.28
c
a904.86
c
a643.112
c
a902.7214.19C
c
a774.3
c
a935.9
c
a138.14
c
a686.11607.3C
c
a37.10
c
a634.31
c
a937.36
c
a55.20923.5C
c
a
t
aC
t
aC
t
aCCF
R-R tand c
a464.11Q
432
3
432
2
432
1
432
0
4
3
2
2101
c
a236.11
c
a036.27
c
a603.21
c
a54.6666.0D
c
a055.29
c
a735.82
c
a828.87
c
a608.42504.8D
c
a244.6
c
a881.17
c
a195.19
c
a399.9797.1D
c
a567.0
c
a981.1
c
a718.2
c
a821.1687.0D
c
a
t
aD
t
aD
t
aDDF
74
- reference data:
Mettu S. R., Raju I. S., Forman R. G., 1988, “Stress Intensity Factors for Part-Through
Surface Cracks in Hollow Cylinders,“ NASA Technical Report, No. JSC 25685 LESC
30124.
Andrasic C. P., Parker A. P., 1984, “Dimensionless Stress Intensity Factors for Cracked
Thick Cylinder Under Polynomial Crack Face Loading,“ Engng. Fract. Mech., Vol. 19,
1984, pp. 187-193.
Murakami Y., et al., 1986, Stress Intensity Factor Handbook, Vol. 2, Pergamon Press,
pp. 309-317.
75
Figure 21. Internal axial semi-elliptical surface crack in a thick cylinder (Ro/Ri=2.0).
76
9.6 Internal axial semi-elliptical surface crack in a thick cylinder (Ro/Ri=1.5)
WEIGHT FUNCTION (Fig. 21) - Deepest Point A
PARAMETERS
where:
and
2
3
A3
1
A2
2
1
A1Aa
x1M
a
x1M
a
x1M1
xa2
2a,xm
5
8Y2Y
Q2
6M
3M
5
24Y3Y
Q2
2M
10A3
A2
10A1
c
a165.4exp46.1140.0A
c
a386.3exp161.1498.1A
c
a393.3exp665.0111.0A
c
a051.5exp07.0044.1A
t
aA
t
aA
t
aAAY
R-R tand c
a464.11Q
3
2
1
0
4
3
2
2100
io
65.1
77
WEIGHT FUNCTION (Fig. 21) - Surface Point B
PARAMETERS
where:
c
a079.0exp284.1398.1B
c
a025.4exp753.0307.0B
c
a574.5exp265.0225.0B
c
a035.0exp16.2825.2B
t
aB
t
aB
t
aBBY
3
2
1
0
3
3
2
2101
2
3
B3
1
B2
2
1
B1Ba
xM
a
xM
a
xM1
x
2a,xm
8F10F3Q
3M
15FF3Q
15M
8F5F2Q
3M
10B3
01B2
10B1
78
and
RANGE OF APPLICATION: R0/Ri = 1.5, 0 = a/t = 0.8, a/c = 0 and 0.2 = a/c = 1.0
ACCURACY: Better than 3% when compared to the FEM data.
REFERENCES:
- weight function:
Zheng X. J., Kiciak A., Glinka G., 1996, “Determination of Weight Functions and Stress
Intensity Factors for Semi-Elliptical Cracks in Thick-wall Cylinders,“ Nucl. Engng.
Design, to be published.
- reference data:
2
3
2
2
2
1
2
0
4
3
2
2100
io
65.1
c
a817.9
c
a959.9exp33.28104.0C
c
a092.1
c
a081.4exp784.8119.0C
c
a009.4
c
a053.46exp239.10199.0C
c
a568.1
c
a061.5exp163.5972.0C
c
a
t
aC
t
aC
t
aCCF
R-R tand c
a464.11Q
432
3
432
2
432
1
432
0
4
3
2
2101
c
a433.30
c
a555.76
c
a546.65
c
a677.21243.2D
c
a636.3
c
a05.25
c
a644.45
c
a622.30535.6D
c
a099.13
c
a498.42
c
a221.50
c
a231.24448.3D
c
a690.2
c
a397.8
c
a708.9
c
a842.4033.1D
c
a
t
aD
t
aD
t
aDDF
79
Mettu S. R., Raju I. S., Forman R. G., 1988, “Stress Intensity Factors for Part-Through
Surface Cracks in Hollow Cylinders,“ NASA Technical Report, No. JSC 25685 LESC
30124.
Andrasic C. P., Parker A. P., 1984, “Dimensionless Stress Intensity Factors for Cracked
Thick Cylinder Under Polynomial Crack Face Loading,“ Engng. Fract. Mech., Vol. 19,
1984, pp. 187-193.
Murakami Y., et al., 1986, Stress Intensity Factor Handbook, Vol. 2, Pergamon Press,
pp. 309-317.
80
9.7 Internal axial semi-elliptical surface crack in a thick cylinder (Ro/Ri=1.25)
WEIGHT FUNCTION (Fig. 21) - Deepest Point A
PARAMETERS
where:
and
2
3
A3
1
A2
2
1
A1Aa
x1M
a
x1M
a
x1M1
xa2
2a,xm
5
8Y2Y
Q2
6M
3M
5
24Y3Y
Q2
2M
10A3
A2
10A1
c
a354.1exp061.0149.0A
c
a859.3exp269.3057.0A
c
a17.31exp366.0055.0A
c
a15.13exp0998.0010.1A
t
aA
t
aA
t
aAAY
R-R tand c
a464.11Q
3
2
1
0
4
3
2
2100
io
65.1
81
WEIGHT FUNCTION (Fig. 21) - Surface Point B
PARAMETERS
where:
c
a0434.0exp538.1552.1B
c
a562.4exp787.0269.0B
c
a663.8exp436.0136.0B
c
a012.0exp944.5594.6B
t
aB
t
aB
t
aBBY
3
2
1
0
3
3
2
2101
2
3
B3
1
B2
2
1
B1Ba
xM
a
xM
a
xM1
x
2a,xm
8F10F3Q
3M
15FF3Q
15M
8F5F2Q
3M
10B3
01B2
10B1
82
and
RANGE OF APPLICATION: R0/Ri = 1.25, 0 = a/t = 0.8, and a/c = 0 and 0.2 = a/c = 1.0
ACCURACY: Better than 3% when compared to the FEM data.
REFERENCES:
- weight function:
Zheng X. J., Glinka G., Dubey R. N., 1995, “Calculation of Stress Intensity Factors for
Semielliptical Cracks in a Thick-Wall Cylinder,“ Int. J.Pres.Ves. & Piping, Vol. 62, pp.
249-258.
- reference data:
432
3
432
2
432
1
432
0
4
3
2
2100
io
65.1
c
a507.35
c
a289.93
c
a789.83
c
a49.29468.3C
c
a55.9
c
a216.45
c
a59.72
c
a067.49497.12C
c
a616.1
c
a44.10
c
a618.16
c
a514.975.1C
c
a507.11
c
a705.33
c
a335.37
c
a583.19566.5C
c
a
t
aC
t
aC
t
aCCF
R-R tand c
a464.11Q
432
3
432
2
432
1
432
0
4
3
2
2101
c
a853.10
c
a37.26
c
a548.21
c
a605.6569.0D
c
a826.1
c
a235.12
c
a358.22
c
a985.15116.4D
c
a333.0
c
a999.0
c
a412.3
c
a626.2533.0D
c
a212.0
c
a793.0
c
a161.1
c
a879.0486.0D
c
a
t
aD
t
aD
t
aDDF
83
Mettu S. R., Raju I. S., Forman R. G., 1988, “Stress Intensity Factors for Part-Through
Surface Cracks in Hollow Cylinders,“ NASA Technical Report, No. JSC 25685 LESC
30124.
Andrasic C. P., Parker A. P., 1984, “Dimensionless Stress Intensity Factors for Cracked
Thick Cylinder Under Polynomial Crack Face Loading,“ Engng. Fract. Mech., Vol. 19,
1984, pp. 187-193.
Murakami Y., et al., 1986, Stress Intensity Factor Handbook, Vol. 2, Pergamon Press,
pp. 309-317.
84
9.8 Internal axial semi-elliptical surface crack in a thick cylinder (Ro/Ri=1.1)
WEIGHT FUNCTION (Fig. 21) - Deepest Point A
PARAMETERS
where:
and
2
3
A3
1
A2
2
1
A1Aa
x1M
a
x1M
a
x1M1
xa2
2a,xm
5
8YY2
Q2
6M
3M
5
24Y3Y2
Q2
2M
01A3
A2
10A1
983.0
2
2
2
1
32
0
4
2
2
100
io
65.1
c
a061.0
1
c
a027.13
c
a456.20519.8A
c
a931.6
c
a531.1084.3A
c
a5202.0
c
a0464.1
c
a6699.01449.1A
t
aA
t
aAAY
R-R tand c
a464.11Q
85
WEIGHT FUNCTION (Fig. 21) - Surface Point B
PARAMETERS
where:
92.0
2
2
32
1
32
0
4
2
2
101
c
a090.0
1
c
a097.8
c
a282.13251.6B
c
a291.1
c
a928.5
c
a901.6415.2B
c
a4417.0
c
a7576.0
c
a4967.04732.0B
t
aB
t
aBBY
2
3
B3
1
B2
2
1
B1Ba
xM
a
xM
a
xM1
x
2a,xm
8F7F10Q
3M
15F3F2Q
15M
8F3F5Q
3M
01B3
10B2
01B1
86
and
RANGE OF APPLICATION: R0/Ri = 1.1, 0 = a/t = 0.8, and 0 = a/c = 1.0
ACCURACY: Better than 5% when compared to the FEM data.
REFERENCES:
- weight function:
Wang X. J., Lambert S. B., 1996, “Stress Intensity Factors and Weight Functions for
Longitudinal Semi-Elliptical Surface Cracks in Thin Pipes,“ Int. J.Pres.Ves. & Piping,
Vol. 65, pp 75 -87.
Shen. G., Liebster T. D., Glinka G. ,1993, “Calculation of Stress Intensity Factors for
Cracks in Pipes,”Proc. of the 12th Int. Conf. on Offshore Mech. and Arctic Engng..,
05.12
5432
1
2
0
4
2
2
100
io
65.1
c
a2.0
1
c
a15.1065.2C
c
a502.92
c
a492.270
c
a879.293
c
a547.143
c
a96.271256.0C
c
a1203.0
c
a2935.02959.1C
c
a
t
aC
t
aCCF
R-R tand c
a464.11Q
05.12
5432
1
2
0
4
2
2
101
c
a299.0
1
c
a087.1879.1D
c
a309.52
c
a789.153
c
a357.167
c
a361.81
c
a433.153311.0D
c
a4901.0
c
a8104.02959.1D
c
a
t
aD
t
aDDF
87
ASME,M. Salama et al., ed., Vol. III-B, Materials Engineering, pp.847-854. (this paper
contains the older version of the weight function, limited to 0 = a/t = 0.8 and 0.2 =
a/c = 1.0)
- reference data:
Raju I. S., Newman J. C., 1982, “Stress Intensity Factors for Internal and External Surface
Cracks in Cylindrical Vessels,“ J. Press. Ves. Technol., Vol. 104, pp. 293-298.
Andrasic C. P., Parker A. P., 1984, “Dimensionless Stress Intensity Factors for Cracked
Thick Cylinder Under Polynomial Crack Face Loading,“ Engng. Fract. Mech., Vol. 19,
1984, pp. 187-193.
88
9.9 External axial semi-elliptical surface crack in a thick cylinder (Ro/Ri=2.0)
WEIGHT FUNCTION (Fig. 22) - Deepest Point A
PARAMETERS
where:
and
2
3
A3
1
A2
2
1
A1Aa
x1M
a
x1M
a
x1M1
xa2
2a,xm
5
8Y2Y
Q2
6M
3M
5
24Y3Y
Q2
2M
10A3
A2
10A1
432
3
432
2
432
1
432
0
4
3
2
2100
io
65.1
c
a64254.63
c
a12933.118
c
a77065.71
c
a84462.23exp78423.1707482.0A
c
a88622.55
c
a30990.118
c
a48907.87
c
a34798.22exp41917.041663.0A
c
a1922.85
c
a87345.185
c
a51103.138
c
a69496.35exp04469.002208.0A
c
a48712.21
c
a45540.53
c
a74122.44
c
a93568.11exp10409.098463.0A
t
aA
t
aA
t
aAAY
R-R tand c
a464.11Q
89
WEIGHT FUNCTION (Fig. 22) - Surface Point B
PARAMETERS
where:
6
5432
3
432
2
432
1
432
0
4
3
2
2101
c
a23.75335
c
a66299.84
c
a96622.118
c
a96909.85
c
a59049.35
c
a68986.800553.1B
c
a96618.58
c
a18837.103
c
a68409.66
c
a79057.14exp46420.012875.0B
c
a15246.87
c
a27997.158
c
a67649.103
c
a45619.24exp08062.002433.0B
c
a09569.16
c
a63312.39
c
a30859.82
c
a42681.25exp00525.071344.0B
t
aB
t
aB
t
aBBY
2
3
B3
1
B2
2
1
B1Ba
xM
a
xM
a
xM1
x
2a,xm
8F10F3Q
3M
15FF3Q
15M
8F5F2Q
3M
10B3
01B2
10B1
90
and
RANGE OF APPLICATION: R0/Ri = 2.0, 0 = a/t = 1.0, a/c = 0 and 0.2 = a/c = 1.0
ACCURACY: Better than 3% when compared to the FEM data.
REFERENCES:
- weight function:
Kiciak A., Glinka G., Burns D. J., 1996, “Weight Functions and Stress Intensity Factors
for Longitudinal Semi-Elliptical Cracks in Thick-wall Cylinders,“ J. Press. Vess.
Technol., ASME (to be published)
432
4
432
3
432
2
432
1
5432
0
5
4
4
3
2
2100
c
a35426.14
c
a66380.26
c
a30584.19
c
a87203.9exp
c
a88925.01
c
a95222.16C
c
a66400.2
c
a52675.7
c
a20568.8
c
a21823.10exp
c
a17520.11
c
a90204.55C
c
a51091.7
c
a10918.17
c
a42291.9
c
a60488.0exp111589.1C
c
a78266.6
c
a55459.13
c
a78270.5
c
a64659.3exp
c
a17520.11
c
a12461.15C
c
a80402.8
c
a68609.28
c
a14294.36
c
a48633.21
c
a93070.6exp140290.1C
t
aC
t
aC
t
aC
t
aCCF
32
2
432
1
432
0
2
2101
c
a33890.1
c
a71158.1
c
a27930.5exp
c
a90366.01
c
a67302.14D
c
a84826.4
c
a82767.10
c
a99914.4
c
a22628.4exp
c
a09108.11
c
a73349.9D
c
a86369.9
c
a62707.30
c
a07640.28
c
a38873.11exp120478.0D
t
aD
t
aDDF
91
- reference data:
Mettu S. R., Raju I. S., Forman R. G., 1988, “Stress Intensity Factors for Part-Through
Surface Cracks in Hollow Cylinders,“ NASA Technical Report, No. JSC 25685 LESC
30124.
Andrasic C. P., Parker A. P., 1984, “Dimensionless Stress Intensity Factors for Cracked
Thick Cylinder Under Polynomial Crack Face Loading,“ Engng. Fract. Mech., Vol. 19,
1984, pp. 187-193.
Murakami Y., et al., 1986, Stress Intensity Factor Handbook, Vol. 2, Pergamon Press,
pp. 309-317.
92
Figure 22. External axial semi-elliptical surface crack in a thick cylinder (Ro/Ri=2.0).
93
External axial semi-elliptical surface crack in a thick cylinder (Ro/Ri=1.5)
WEIGHT FUNCTION (Fig. 22) - Deepest Point A
PARAMETERS
where:
2
3
A3
1
A2
2
1
A1Aa
x1M
a
x1M
a
x1M1
xa2
2a,xm
5
8Y2Y
Q2
6M
3M
5
24Y3Y
Q2
2M
10A3
A2
10A1
94
5
c
a43104.150
4
c
a88103.419
3
c
a31551.400
2
c
a59759.148
c
a66325.1218341.0
exp024225.03
A
c
a50662.23
c
a40776.28
c
a41241.3
c
a46411.338467.0
exp25833.0A
c
a02799.32
c
a84164.109
c
a22714.142
c
a13521.84
c
a36041.2189775.1
exp83400.0A
c
a92014.0
c
a38695.2
c
a88533.1
c
a14072.085208.0
exp69716.0A
t
aA
t
aA
t
aAAY
R-R tand c
a464.11Q
4
32
2
54
32
1
4
32
0
4
3
2
2100
io
65.1
95
and
WEIGHT FUNCTION (Fig. 22) - Surface Point B
PARAMETERS
2
3
B3
1
B2
2
1
B1Ba
xM
a
xM
a
xM1
x
2a,xm
4
c
a67589.24
3
c
a51119.25
2
c
a30479.53
c
a17795.044818.0
exp02704.03
B
c
a080007.37
c
a92487.74
c
a49509.55
c
a46249.1311085.1
exp14247.0B
c
a43167.43
c
a35299.132
c
a08510.154
c
a19018.84
c
a51173.2027787.2
exp43259.0B
c
a25050.0
c
a04996.0
c
a19984.0
c
a07367.078477.0
exp22856.0B
t
aB
t
aB
t
aBBY
4
32
2
54
32
1
4
32
0
4
3
2
2101
96
where:
and
8F10F3Q
3M
15FF3Q
15M
8F5F2Q
3M
10B3
01B2
10B1
322
322
2
432
1
322
0
4
3
2
2100
io
65.1
c
a53042.7
c
a22175.21
c
a21310.17exp
c
a89888.12
c
a91620.1452544.27
c
a
3C
c
a62398.144
c
a84981.84
c
a29526.23exp
c
a72855.0
c
a50789.1170539.2C
c
a25396.27
c
a27126.55
c
a49788.38
c
a47111.16exp
c
a89759.68208999.55
c
aC
c
a37581.382
c
a14830.139
c
a87363.29exp
c
a34206.0
c
a31674.0169519.0C
t
aC
t
aC
t
aCCF
R-R tand c
a464.11Q
97
RANGE OF APPLICATION: R0/Ri = 1.5, 0 = a/t = 1.0, a/c = 0 and 0.2 = a/c = 1.0
ACCURACY: Better than 3% when compared to the FEM data.
REFERENCES:
- weight function:
Kiciak A., Glinka G., Burns D. J., 1996, University of Waterloo, to be published.
- reference data:
Mettu S. R., Raju I. S., Forman R. G., 1988, “Stress Intensity Factors for Part-Through
Surface Cracks in Hollow Cylinders,“ NASA Technical Report, No. JSC 25685 LESC
30124.
Andrasic C. P., Parker A. P., 1984, “Dimensionless Stress Intensity Factors for Cracked
Thick Cylinder Under Polynomial Crack Face Loading,“ Engng. Fract. Mech., Vol. 19,
1984, pp. 187-193.
Murakami Y., et al., 1986, Stress Intensity Factor Handbook, Vol. 2, Pergamon
Press, pp. 309-317.
25.032
2
2
32
1
32
0
4
3
2
2101
c
a19501.52exp1
c
a861145.2
c
a64517.4
c
a02421.223183.0
3D
c
a38520.1
c
a31235.21
c
a57311.3exp107021.2D
c
a58371.0
c
a66328.0
c
a30088.21
c
a75502.11exp1
c
a66304.0D
c
a55715.27exp
c
a79576.0
c
a27723.1
c
a51333.1109646.0D
t
aD
t
aD
t
aDDF
98
9.10 External axial semi-elliptical surface crack in a thick cylinder (Ro/Ri=1.25)
WEIGHT FUNCTION (Fig. 22) - Deepest Point A
PARAMETERS
where:
2
3
A3
1
A2
2
1
A1Aa
x1M
a
x1M
a
x1M1
xa2
2a,xm
5
8Y2Y
Q2
6M
3M
5
24Y3Y
Q2
2M
10A3
A2
10A1
54
32
3
432
2
432
1
432
0
4
3
2
2100
io
65.1
c
a71664.98
c
a74017.354
c
a99358.497
c
a20961.340
c
a58315.11294324.15
exp198498.0A
c
a95069.5
c
a48057.11
c
a515307.4
c
a16636.358221.1exp57088.0A
c
a23850.4
c
a96283.11
c
a06481.12
c
a66494.482502.0exp38656.0A
c
a62456.10
c
a59266.26
c
a08395.23
c
a56013.868499.1exp94235.0A
t
aA
t
aA
t
aAAY
R-R tand c
a464.11Q
99
and
WEIGHT FUNCTION (Fig. 22) - Surface Point B
PARAMETERS
4
c
a64059.4
3
c
a85419.25
2
c
a70843.33
c
a85207.710419.2
exp01828.04
B
4
c
a50684.8
3
c
a45236.2
2
c
a45912.13
c
a88270.055548.0
exp02569.03
B
c
a32259.1
c
a84029.304497.0exp04652.0B
c
a37007.1
c
a08396.3
c
a21570.2
c
a59120.031448.0
exp53122.1B
c
a71914.4
c
a00652.10
c
a58393.7
c
a14909.271519.1
exp050249B
t
aB
t
aB
t
aB
t
aBBY
2
2
4
32
1
4
32
0
4
3
3
2
2101
2
3
B3
1
B2
2
1
B1Ba
xM
a
xM
a
xM1
x
2a,xm
100
where:
and
8F10F3Q
3M
15FF3Q
15M
8F5F2Q
3M
10B3
01B2
10B1
22
322
2
2
1
2
0
4
3
2
2100
io
65.1
c
a79275.3
c
a18934.3exp
c
a01701.1
c
a08060.2124024.0
3C
c
a88988.78
c
a86421.34
c
a39742.11exp
c
a55332.0
c
a27898.1180687.1C
c
a11736.14exp
c
a47068.1
c
a79131.1222958.0C
c
a02505.14exp
c
a19842.0
c
a58454.1147251.0C
t
aC
t
aC
t
aCCF
R-R tand c
a464.11Q
101
RANGE OF APPLICATION: R0/Ri = 1.25, 0 = a/t = 1.0, a/c = 0 and 0.2 = a/c = 1.0
ACCURACY: Better than 3% when compared to the FEM data.
REFERENCES:
- weight function:
Kiciak A., Glinka G., Burns D. J., 1996, University of Waterloo, to be published.
- reference data:
Mettu S. R., Raju I. S., Forman R. G., 1988, “Stress Intensity Factors for Part-Through
Surface Cracks in Hollow Cylinders,“ NASA Technical Report, No. JSC 25685 LESC
30124.
Andrasic C. P., Parker A. P., 1984, “Dimensionless Stress Intensity Factors for Cracked
Thick Cylinder Under Polynomial Crack Face Loading,“ Engng. Fract. Mech., Vol. 19,
1984, pp. 187-193.
Murakami Y., et al., 1986, Stress Intensity Factor Handbook, Vol. 2, Pergamon Press,
pp. 309-317.
54
32
2
2
22
1
0
4
3
2
2101
c
a09600.18
c
a62440.60
c
a95158.77
c
a80652.46
c
a47981.1208880.1
c
a
3D
c
a96829.0
c
a89058.11
c
a21697.1exp117158.2D
c
a35842.2
c
a24795.5exp
c
a03870.1
c
a30358.2115562.0D
c
a19737.3exp
c
a54116.0112603.0D
t
aD
t
aD
t
aDDF
102
9.11 External axial semi-elliptical surface crack in a thick cylinder (Ro/Ri=1.1)
WEIGHT FUNCTION (Fig. 22) - Deepest Point A
PARAMETERS
where:
and
5
8YY2
Q2
6M
3M
5
24Y3Y2
Q2
2M
01A3
A2
10A1
2
3
A3
1
A2
2
1
A1Aa
x1M
a
x1M
a
x1M1
xa2
2a,xm
094.1
2
2
2
1
2
0
4
2
2
100
io
65.1
c
a066.0
1
c
a305.8
c
a559.1444.7A
c
a29.5
c
a98.800.4A
c
a2984.0
c
a4322.01492.1A
t
aA
t
aAAY
R-R tand c
a464.11Q
103
WEIGHT FUNCTION (Fig. 22) - Surface Point B
PARAMETERS
where:
2
3
B3
1
B2
2
1
B1Ba
xM
a
xM
a
xM1
x
2a,xm
8F7F10Q
3M
15F3F2Q
15M
8F3F5Q
3M
01B3
10B2
01B1
006.1
2
2
2
1
32
0
4
2
2
101
c
a097.0
1
c
a062.5
c
a653.969.5B
c
a850.2
c
a0937.54478.2B
c
a453.0
c
a788.0
c
a5211.04840.0B
t
aB
t
aBBY
104
and
RANGE OF APPLICATION: R0/Ri = 1.1, 0 = a/t = 1.0, and 0 = a/c = 1.0
ACCURACY: Better than 5% when compared to the FEM data.
REFERENCES:
- weight function:
Wang X. J., Lambert S. B., 1996, “Stress Intensity Factors and Weight Functions for
Longitudinal Semi-Elliptical Surface Cracks in Thin Pipes,“ Int. J.Pres.Ves. & Piping,
Vol. 65, pp 75 -87.
- reference data:
Raju I. S., Newman J. C., 1982, “Stress Intensity Factors for Internal and External Surface
Cracks in Cylindrical Vessels,“ J. Press. Ves. Technol., Vol. 104, pp. 293-298.
2
2
5432
1
2
0
4
2
2
100
io
65.1
c
a262.2
c
a746.3405.1C
c
a8339.56
c
a848.177
c
a832.210
c
a107.115
c
a953.251110.0C
c
a0895.0
c
a2532.02964.1C
c
a
t
aC
t
aCCF
R-R tand c
a464.11Q
2
2
5432
1
2
0
4
2
2
101
c
a58529.0
c
a3817.17704.0D
c
a6732.38
c
a252.115
c
a93.129
c
a574.67
c
a4065.142128.0D
c
a4668.0
c
a7814.02531.1D
c
a
t
aD
t
aDDF
105
Andrasic C. P., Parker A. P., 1984, “Dimensionless Stress Intensity Factors for Cracked
Thick Cylinder Under Polynomial Crack Face Loading,“ Engng. Fract. Mech., Vol. 19,
1984, pp. 187-193.
106
10. NOTATION
a - crack length for one-dimensional cracks or crack depth (minor semi-
axis) for semi-elliptical cracks
Ai, Bi - coefficients of the linearized stress function in the sub-interval “i”
c - crack length (major semi-axis) for semi-elliptical crack
Cij - coefficients of the integrated weight function mA(x,a) in association with
the linearised stress field s(x)
Dij - coefficients of the integrated weight function mB(x,a)in association with the
linearised stress field s(x)
K - stress intensity factor
KI - mode I stress intensity factor (general)
K0A - mode I reference stress intensity factor for the deepest point A of a crack
under the uniform stress field
K0B
- mode I reference stress intensity factor for the surface point B of a crack
under the uniform stress field
KA - mode I stress intensity factor for the deepest point A on the crack front
KB
- mode I reference stress intensity factor for the surface point B on the crack
front
Mi - parameters of the weight function (i = 1, 2, 3)
MiA - coefficients of the weight functions for the deepest point A (i= 1,2,3)
MiB - coefficients of the weight functions for the surface point B (i= 1,2,3)
m(x,a) - weight function
mA(x,a) - weight function for the deepest point A of a crack
mB(x,a) - weight function for the surface point B of a crack
S - area under a monotonic curve m(x,a)
Si*
- area under the linearized stress function s(x) corresponding to the
sub-interval “i”
Si - area under the weight function curve m(x,a) corresponding to the sub-
107
interval “i”
Q - the semi-elliptical crack shape factor: Q = 1 +1.464(a/c)1.65
for a/c < 1
t - wall thickness of a cylinder or plate thickness
x - the local, through the thickness co-ordinate along the crack depth
X - co-ordinate x of the centroid of the area S
Xi - co-ordinate x of the centroid of the area Si corresponding the sub-
interval “i”
Xi*
- co-ordinate x of the centroid of the area Si* corresponding to the sub-
interval “i”
(x) - stress distribution
(xi) -value of stress function at x = xi
(X) - value of stress function at x = X
(x) - a reference stress distribution
YnA - the geometric stress intensity correction factor for the deepest point A
YnB - the geometric stress intensity correction factor for the surface point B
108
11. LITERATURE
1. H. F. Bueckner, A novel principle for the computation of stress intensity
factors. Zeitschrift fur Angewandte Mathematik und Mechanik, 50, 529-546,
1970
2. J. R. Rice, Some remarks on elastic crack-tip stress field. International Journal of
Solids and Structures, 8, 751-758, 1972
3. D. Broek, The Practical Use of Fracture Mechanics, Kluwer, 1988.
4. G.Glinka and G.Shen, Universal features of weight functions for cracks in mode I.
Engineering Fracture Mechanics, 40, 1135-1146, 1991
5. G. Shen and G.Glinka, Determination of weight functions from reference stress
intensity factors. Theoretical and Applied Fracture Mechanics, 15, 237-245, 1991
6. Shen G., Glinka G., 1991, “Weight Functions for a Surface Semi-Elliptical Crack in a
Finite Thickness Plate,“ Theor. Appl. Fract. Mech., Vol. 15, No. 2, pp. 247-255
7. T. Fett, C. Mattheck and D. Munz,”On the Evaluation of Crack opening Displacement
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