numerical evaluation of sif for radial cracks in thick annular ring using cyclic symmetry

Pergamon Engineering Fracture Mechanics Vol. 56, No. 2, pp. 141-153, 1997 Copyright © 1996 Elsevier Science Ltd

Printed in Great Britain. All rights reserved PII: S0013-7944(96)00097-5 0013-7944/97 $17.00 + 0.00

NUMERICAL EVALUATION OF SIF FOR RADIAL CRACKS IN THICK ANNULAR RING USING CYCLIC

SYMMETRY

K. RAMESH,t S. SHUKLA, P. M. DIXIT and N. KARUPPAIAH

Department of Mechanical Engineering, Indian Institute of Technology, Kanpur, India 208 016

Abstract--Mode I Stress Intensity Factor for periodic radial cracks emanating from inner and outer boundaries of an annular ring is evaluated using finite element technique and the concept of cyclic symmetry. Potter's method of solution algorithm is used to reduce the in-core memory requirements. The software and the solution methodology are validated by comparing the results of SIF for a large number of radial cracks emanating from the inner boundary of an internally pressurised cylinder with those available in the literature. Using the software developed, SIF is evaluated for a large number radial cracks for two g/a ratios emanating from the outer boundary of a pressurised cylinder. SIF is also evaluated for a large number of cracks emanating from inner and outer boundaries of a rotating disk. Further, SIF is evaluated as a function of t /(b - a) ratio for various b/a ratios for four and eight cracks emanating from inner and outer boundaries of a rotating disk. Copyright © 1996 Elsevier Science Ltd

1. INTRODUCTION

THE PROBLEM of determining the stress distribution in an annular plate containing radial cracks originating from one of the boundaries of the plate which is subjected to a prescribed loading is one of great importance. For instance, the plane strain problem is relevant to the study of the action of internal pressure in the interior of hollow cylinders of large wall thicknesses with radial cracks originating from the inner surface. Analysis of the radial cracks in an annular disc subjected to centrifugal loading is important to extend the use of fracture mechanics in designing practically important classes of mechanical engineering components such as bladed disc assem- blies, flywheels etc. One of the most important fracture mechanics parameters to be determined in such cases is the Stress Intensity Factor (SIF)[1].

The problem of cracks emanating from inner boundary of an annular disc with internal pressure has been previously studied for one crack[2], and for two cracks [3]. SIF for an array of 48 cracks originating from inner boundary of finite walled cylinder with internal pressure was calculated by Baratta[4] using load relief factors. In order to increase the maximum pressure the vessel can bear, techniques such as autofrettage have been developed to induce residual com- pressive stress at the internal boundary. However, these techniques can also induce residual ten- sile stress at the outer surface which combined with the stress due to internal pressure may cause the formation of cracks at the outer surface [5]. SIFs for one to four cracks emanating from the inner/outer surfaces of a ring subjected to internal pressure have been studied by Tracy [6] using modified mapping collocation method. Perl and Arone [7] calculated SIF for large arrays of radial cracks up to 1000 in a partially autofrettaged pressurized thick walled cylinder using finite element method. Though extensive data is available for SIF for a large array of cracks emanating from inner boundary of a pressurised vessel, the available data con- cerning radial cracks from outer boundary is limited to 4 cracks only[6].

The evaluation of SIF for cracks in a rotating disk has received wide attention [8-11] since the appearance of the work of Rooke and Tweed [8]. The applicability of linear elastic fracture mechanics and the validity of the brittle-fracture criterion in predicting fracture behaviour of rotating disks have been established by Blauel et al. [11] in 1977 for a single radial crack. The availability of data for SIF for a rotating annular disk is limited to two cracks emanating from the inner boundary of the disk[12].

tAuthor to whom all correspondence should be addressed.

141

142 K. RAMESH et al.

All the previous investigators have assumed that the radial cracks occur with certain periodicity. This aspect is taken advantage of here in devising the solution methodology. The concept of cyclic symmetry [13] is used to reduce the problem size and the Potter's method [14] of solution algorithm is used to solve for the displacements. The computer software developed is veri- fied by comparing the results of SIF of the present approach to that of Perl and Atone [7] for the problem of radial cracks emanating from the inner boundary of a pressurised cylinder. The results are found to be in good agreement. Using the software developed, radial cracks up to 90 emanating from the outer boundary of a pressurised cylinder are calculated. SIF is also evaluated for different crack lengths and up to 36 radial cracks emanating from inner/outer surfaces of rotating annular disk. Further, SIF is evaluated as a function of U(b - a) ratio for various b/ a ratios for four and eight cracks emanating from inner and outer boundaries of a rotating disk.

2. PROBLEM FORMULATION AND SOLUTION METHODOLOGY

Annular disk without cracks can be analysed easily as an axi-symmetric problem. But, as soon as cracks are formed, the structure loses its axisymmetric nature. Modelling of a crack involves the use of a large number of small elements near the crack-tip. As the number of cracks increases, analysing the problem as a whole becomes prohibitive and one would require large computer resources to solve the problem. However, if the cracks are periodic and of equal lengths, then this periodicity can be utilised in solving the problem. The assumption of periodic cracks of equal lengths has been used by all earlier investigators [4-7] and it is also used here. In view of the periodic appearance of cracks (Fig. l), it is enough that one analyses only one sector. In the present investigation, displacement formulation of finite element (FE) is used. Eight- noded isoparametric element is used to model the plate and the crack-tip is modelled by col- lapsed eight noded quarter-point element. The recommendations of Saouma and Schwemmer [15] are followed in modelling the crack-tip using quarter-point elements. Potter's method of sol-

® @ @ . . . . ® S.cto no . 1 2 3 . . . . . n P a r t i t i o n n o s .

Fig. 1. An example of a cyclically symmetric structure.

Numerical evaluation of SIF for radial cracks 143

ution is adopted to solve for the displacement. This method helps in drastically reducing the requirement of in-core memory•

In order to apply the Potter's method of solution, a proper scheme of element and node numbering is needed to get a global stiffness matrix such that it is identifiable as an assembly of sub-matrices [A]i, [B]i and [C]i (eq. (1)). A simple scheme of element and node numbering is to increase the element numbering radially and also circumferentially from a reference section• Since, [A];, [B]i and [C]i form a set of rows of the global stiffness matrix, the corresponding nodes in the actual structure are termed to constitute a partition for convenience• The governing equation of the cyclically symmetric structure as a whole shown in Fig. 1 is given as,

- [B]l [A]I 0 0 . . . 0 0 [C]1 1 {z}l ! f } l [C ]2 [B]2 [A]2 0 . . . 0 0 0 ] {z}2 / f}2

0 [C]3 [B]3 [A]3 "'" 0 0 0 {z}3 {f}3 . . . . . . . . = . (1)

0 0 0 0 "'" [C ]m-I [Blm-I [A]m-1 {Z}m-1 { f}m-1 [Alto 0 0 0 ' ' ' 0 [C ]m [n]m " (Z}m ( f } , ,

where, rn is the total number of partitions and is equal to N x n, {z}; is the displacement vector and { f } i is the force vector•

In terms of the partition concept, the sub-matrix [B]i is the effect of partition on itself, [C]; is the effect of earlier partition on the present one and [A]i is the effect of next partition on the present one. An FE modelling of a linear structure leads to a banded matrix• However, when the structure closes upon itself as in Fig. 1, the last partition namely the mth partition and the first partition are connected. Hence, by the definition of submatrices [A]i, [B]i and [C]i, in the present case matrices [Ch and [A]m exist• In view of this, the bandwidth is equal to the size of the matrix and the banded-solution algorithms are not attractive•

Since the structure is cyclically symmetric, in the static case the displacement vector of each sector is the same i.e.

(Z}n+i = {Z}i; (Z}2 n = (Z}n; ... (2)

Using the above condition the overall problem can be reduced to governing equation takes the form

[Bh [~1]~ o 0 . . . 0 0 [c12 [B]2 [A]2 0 . . . 0 0

0 [C]3 [B]3 [A]3 .-- 0 0

o o o o . . - [ C l , _ ~ [B],_~ [A]. 0 0 0 .-- 0 [C].

[Ch {zh 0 {z}2 0 {z}3

[a]._~ {z}._~ [B], {z},

Further due to symmetry of the stiffness matrix,

[C1] = [An] r and [Ci+I] = [Ai] r for i = 1, 2 ...... n - 1

only one sector and the

{f}l {f}2 {f}3

= (3)

{f l , - I {f}n

(4)

Perl and Arone [7] reported SIF by analysing only one sector using FEM. They used col- lapsed eight noded isoparametric element to model the crack-tip and used four noded quadrilat- eral elements elsewhere• Though, the computational effort is considerably reduced by considering only one sector, the demand of in-core computer memory is very high if conven- tional banded-solution algorithms are used which is probably the method adopted by Perl and Arone.

Potter's method is an out of core solution technique. At a time, it requires only submatrices of the ith partition in the core memory of the computer• All other matrices are stored on sec- ondary storage devices such as tapes, disks etc. This makes optimum use of the computer in core memory. Ramamurti and his co-workers [16, 17] have extensively used the Potter's method of solution for solving a large variety of complex problems in mechanical engineering. In all their studies, they have used three noded triangles to model the problem. In such a case, a par-


tition is identified as the corner nodes of the triangles forming a radial line. However, in the present study, since eight-noded isoparametric elements are used, the partition constitutes both the corner nodes and mid-side nodes as shown in Fig. 2. This is a very important and subtle point in the application of Potter's solution methodology while using eight-noded isoparametric elements. The solution methodology to evaluate the displacements can be found in ref. [13].

2.1. Discretisation details As large amount of input data is required to solve the problem, it is difficult to prepare the

complete input file manually and a semi-automatic discretisation procedure is adopted. In order to obtain accurate results, close partitions are formed near the crack line and it becomes less dense as one goes away from the crack line. Typical meshes for a sector containing crack emanating from inner boundary and outer boundary are shown in Figs 2 and 3 respectively. As the number of cracks increases the sector angle becomes smaller and smaller. Thus, contrary to the total size of the problem, the CPU time and memory decreases as the number of cracks increases. This is a tremendous advantage as the fracture mechanics problem requires large number of elements in the vicinity of the crack-tip.

2.2. Evaluation of SIF Crack-tip opening displacement (CTOD) method has been used to evaluate SIF. In CTOD

method, the displacement Vqp at the quarter point of the singular element on the face of the crack is noted and the following relation[7] is used to calculate the SIF.

Evqp 2~ 7 KI -- (1 q- v)(1 + k) (5)

®

(D, (~ ) . . . . . . . . . Partition numbers.

Fig. 2. Typical finite element mesh for a sector with a radial crack emanating from the inner boundary. The figure also mentions the partitions.


Fig. 3. Typical finite element mesh for a sector with a radial crack emanating from the outer boundary.

where, r is the distance from the crack-tip and

k - (3 - v) for plane stress and k = (3 - 4v) for plane strain (1 + v )

(6)

3. VALIDATION OF THE SOFTWARE

The problem of radial cracks originating from inner boundary of a pressurised cylinder is studied. Instead of applying uniform internal pressure p, uniform tension p is applied at the outer boundary as it has been shown that both cases bear the same SIF[7]. This is done since it is easier to impose displacement boundary conditions than stress boundary conditions on the crack faces. The typical mesh is shown in Fig. 2. Mode I SIF for number of cracks k = 4, 6, 9, 12, 15, 20, 30, 40, 60, 90 is calculated. The results of KI are normalised by dividing it by pff-d. The curve for normalised SIF as a function of number of cracks is shown in Fig. 4. The results are compared with that of Perl and Arone[7] and are found to be in good agreement. Normalised SIF tends to decrease monotonically as the number of cracks increases.

4. PERIODIC RADIAL CRACKS ORIGINATING FROM OUTER BOUNDARY OF A PRESSURISED CYLINDER

Using the software developed, mode I SIFs for array of cracks k = 4, 6, 9, 12, 15, 20, 30, 40, 60, 90 are evaluated. The typical mesh is shown in Fig. 3. Results are obtained for two aspect ratios £ / a = 0.5 and 0.25. Tracy[6] has obtained SIF for two and four cracks emanating from outer boundary using modified boundary collocation method. The results of the present method are compared with those obtained by Tracy and are shown in Table 1. The results show good agreement and this again validates the software.

For both the aspect ratios, normalised SIF is plotted as a function of number of cracks and is shown in Fig. 5. It is found that for the same number of cracks, normalised SIF for outer


6 i

v - ~ . . . . . . Perl & Arone cur~

. . . . . . . . . . . w

I I I 1 I I I I I 0 20 40 60 80

N u m b e r of Cracks

Fig. 4. Comparison of non-dimensional SIF as a function of number of radial cracks emanating from the inner boundary of a pressurised cylinder with that of Perl and Arone[7].

crack is smaller than the inner crack. Further smaller the aspect ratio, smaller is the normalised SIF. For both the aspect ratios, SIF decreases monotonically as the number of cracks increases.

5. EVALUATION OF SIF FOR RADIAL CRACKS IN ROTATING DISKS

The element body force vector is calculated by the following equation.

{ f}e = pw2 J J[N ]T[N ]{r} dA (7)

where p is the density, 09 is the angular velocity, [N] is the vector of shape functions and {r} is the nodal co-ordinate vector of the element e with respect to the disk centre. The integral (eq. (7)) is evaluated by Gauss-quadrature technique to yield the consistent loads at each node.

5.1. Periodic radial cracks originating from inner boundary First, SIFs as a function of number of cracks k = 3, 4, 6, 12, 20, 24 and 36 are calculated

for three crack length ratios e / ( b - a) namely 0.1, 0.5 and 0.7 and for two different b/a ratios viz. 2 and 10. Next, SIFs as a function of crack length ratios e / ( b - a) from 0.1 t o 0.8 in steps of 0.1 are calculated for three b/a ratios namely 2, 3 and 10 and for two different numbers of cracks viz. 8 and 4. Figures 6 and 7 show the absolute value of SIF as a function of number of cracks with e/(b - a) as parameter for two b/a ratios namely 2 and 10. The SIF is calculated for a material with Poisson's ratio of 0.3. The results are non-dimensionalised by dividing it by Ko = ((3 + v ) /8 )x (/gw2(b 2 - a 2 ) ~ ) and are shown in Figs 8 and 9 respectively. Results have been reported in the literature for two cracks emanating for the inner boundary[12]. In ref. [12]

Table 1. Comparison of SIF of the present approach with that of Tracy

£/a No. of cracks Hi from [6] Hi (present work)

0.25 2 1.48 1.42 4 1.36 1.32

Kl(b 2 - a2). 14, -- 2pa------2-~ , b/a = 2


30 t ~ bla=2

2.o-~ \ ~ / ~/a=o.s

o . ~ . ~ ~ . _ . ~ _ o.~.~..~..~... " " ~ ' ~ " ~ " ~ " - ' - - - ~ . o . - . . . ~ .

I I 1, I 1 .... 1 I 0 20 40 60

Number of Crocks I 8O

$ O I

Fig. 5. Non-dimensional SIF as a function of number of radial cracks emanat ing from the outer boundary of a pressurised cylinder with (e /a) ratio as a parameter.

the results are non-dimensionalised by taking Ko = ((3 + v ) /8 )x (po92b2vr~) and they have expressed the crack length non-dimensionally as 2 where 2 = e/a. The results from the present approach are compared with those obtained in ref. [12] and these are shown in Table 2. The results show a very good agreement. In ref. [12] it is further reported that their results are accu-

50

4 0

~ 30 n

ac 20-

~4 I 0

I I I

~ b / a R. R M. = 2 0 0 0

= 2

} /,t/(b-o)=O."r 01

0

O I I I 0 I 0 20 30 40

No. of Cracks(inner) Fig. 6. SIF as a function of number of radial cracks emanat ing from the inner boundary of a rotating

annular disk with {£/ (b - a)} ratio as a parameter for b/a = 2.


U n

LE

uJ

35

30

25

20

15

10

5

0 0 10 2 0 30 40

I I I

R. P. M. = 2000

/ t , / l b - a ) = 0 . 7

/ t,/lb-o)=o.5 m --" . I

I I I

No. of Cracks (inner)

Fig. 7. SIF as a function of number of radial cracks emanat ing from the inner boundary of a rotating annular disk with { U ( b - a)} ratio as a parameter for b/a = 10.

I I I

3 +V P ( g 2 ( b 2 - 8 2 ) ~ " ~ ; v = 0.3 Ko= 8 4.2 --i~ 4 -- b / o = 2

0

v 3 -

~ 2 - o ~

o 1 - ~ / ( b - o ) - . . z ~ / (b -o ) = o. I /

t , / l b - a ) = 0 .5

0 i i i 0 2 tO 20 3O 40

No. of Cracks(inner) Fig. 8. Non-dimensional SIF as a function of number of radial cracks emanating from the inner bound-

ary of a rotating annular disk with { l / ( b - a)} ratio as a parameter for b/a = 2.

Numerical evaluation o f SIF for radial cracks 149

:o

06

Z

2 . 0

1.6

1.2

0 . 8

0 . 4

0 0 2 10 20 3 0 4 0

I I I

K ° = 3 ~ V - - p o 2 ( b 2 - a 2 ) v / - ~ ; v = 0.3

t/¢b-o> = o.7 t+/lb-a) -- o.i" / , / (b-a) --0.51

I I I

No. of Cracks (inner) Fig. 9. Non-dimensional SIF as a function of number of radial cracks emanat ing f rom the inner bound-

ary of a rotat ing annular disk with {£/(b - a)} ratio as a parameter for b/a = 10.

rate up to 1% and in view of a very good agreement of the present approach, the present results are also within 1% of the correct values. In Figs 8 and 9 the non-dimensionalised graphs cross each other and do not represent a simple family of curves whereas Figs 6 and 7 represent clearly a family of curves. This is because the factor Ko is non-linear and distinctly different for different curves. Figures 10 and 11 show the non-dimensional SIF as a function of £/(b- a) with b/a as a parameter for two different situations of (i) four cracks emanating from the inner boundary and (ii) eight cracks emanating from the inner boundary.

It is observed from the graphs that,

(i) Non-dimensional SIF decreases as the number of cracks increases for constant crack length and b/a ratio.

(ii) Non-dimensional SIF decreases as b/a increases keeping b constant for the same crack length ratio and number of cracks.

Table 2. Compar i son of results of SIF for 2 cracks emanat ing f rom the inner boundary of a rotat ing disk

S. no

From[12] K~

Problem geometry J] K~ (present work)

b/a = 2 e l ( b - a ) = 0.5 3.14 ([3 = 0.5;X = 0.5) 4.19 4.20 b/a = 2 e/(b - a) = o. 1 2.25 (1~ = 0.5;~, = 0.1) 3.00 3.00 b/a = 10 e/(b-a) = 0.5 1.45 (13 = 0.1;~, = 4.5) 1.46 1.49

(upon extrapolation) b/a = 10 e/(b - a) = O.l 1.43 (1~ = 0.1;;~ = 0.9) 1.44 1.45

KI a KI ; K ~ = 3 + ; f l = b ; ~ ' - - a

f l = 3 + vpc.o2b2~'awJ,~, vty..o2"b2~ - a2)V'-~ - 8 8


5

x- " " 3

06

g Z I

I I I I

. ~ blo=2

~ b / o = 3

b/a =4 _ blo =5

b/a = I0

3 + v P ~ ' ( b 2 - a 2 ) f ~ ; v = 0.3 K o = 8

0 t t i I o 0 .2 0 .4 0.6 o.8 .o

~ / ( b - o )

Fig. 10. Non-dimensional SIF as a function of {£ / (b - a)} ratio with b / a as a parameter for four periodic radial cracks emanating from the inner boundary of a rotating annular disk.

m

Y v

i," .=~

° ~

nO

c~ 0 Z

4

3.5

3.0

2.5

2.0

1.5

1 , 0 - -

0.5

I I I I

b / a =2

Ko = 3 + V p e Z ( b 2 _ a 2 ) f ~ ; v = 0 . 3 8

0 I I I I 0 0 . 2 0 . 4 0 . 6 0 . 8

b/a = 3 J

b/a =I0-

{ / ( b - a )

.O

Fig. 11. Non-dimensional SIF as a function of { t / ( b - a)} ratio with b / a as a parameter for eight periodic radial cracks emanating from the inner boundary of a rotating annular disk.


0 Y

:Z v

IJ.:

if)

d 0 Z

3 . 5

3.0

2 . 5

2 . 0

1 . 5

1.0

0.5

I I I

KO = 3-S-v p=2~2-a2)f~ ; v = 0.3

=2

~ "'o.. / ~ / ( b - a ) = O . t

0 I I I 0 10 20 30 40

No. of Cracks(outer) Fig. 12. Non-dimensional SIF as a function of number of radial cracks emanat ing from the outer

boundary of a rotating annular disk with { e / ( b - a)} ratio as a parameter for b / a = 2.

1.4

1.2

• 1.0

~,- 0.8

r~ o.e

~£ 0 .4

Z 0 .2

I I I

Ko 3 + v ~ 2 ( b 2 - a 2 ) ~ - ~ • v = 0.3 = - - - ~ - - p

b/a = I0

/ / , / ( b - a) = O. I

0.0 = J i 0 I0 20 30 40

No. of Crack(outer) Fig. 13. Non-dimensional SIF as a function of number of radial cracks emanat ing from the outer

boundary of a rotating annular disk with { e / ( b - a)} ratio as a parameter for b / a = 10.

152 K. RAMESH e t al.

3 . 5 I I I I

b lo = 2

3 . 0

% 2.5

2 . 0 b l a = 3 , . / "~ ~ ,,,0 b/a -- 4

~ b / a =5

I . 0 b /a =I0

Z 0 . 5 - Ko = 3 * _ _ v pl~2(b 2 _a2)~-~ ~ ; v = 0.3

8

0 I i I i 0 0 .2 0.4 0 .6 0.8 1.0

?,/(b-o) for outer cracks

Fig. 14. Non-dimensional SIF as a function of {e / (b - a)} ratio with b /a as a parameter for four periodic radial cracks emanating from the outer boundary of a rotating annular disk.

3 . 0 I I I I

'~ b / a = 2

2 . 5

0

2 . 0 m

l ," 1.5 l o b / a = 3 _

1~ 1.0

_ j , , O b l a =10 ,= o o-----o--._o

Z 0 . 5 - Ko = 3 +___~v p~2(b2 - a : ) f ~ ; v 0.3

8 0 I I I I

0 0.2 0 .4 0.6 0.8 1.0

* / ( b - o ) f o r o u l e r c r a c k s

Fig. 15. Non-dimensional SIF as a function of {e l (b - a)} ratio with b l a as a parameter for eight periodic radial cracks emanating from the outer boundary of a rotating annular disk.


5.2. Periodic radial cracks originating f rom outer boundary

SIFs are ca lcu la ted s imilar ly for all the pa rame te r s as tha t o f inner cracks and the results have been shown graphica l ly in Figs 12, 13, 14 and 15. I t is observed f rom the graphs tha t the t rend for ou te r c racks is s imilar to tha t o f inner c racks except tha t the magn i tude o f non- d imens iona l S I F is a lit t le greater for inner cracks as c o m p a r e d to ou te r cracks o f the same d imensions .

6. C O N C L U S I O N S

It is successfully d e m o n s t r a t e d tha t the concept o f cyclic symmet ry is very helpful in deter- min ing the S I F in the case o f mul t i c racked annu la r plates subjected to var ious load ing con- di t ions. The results are c o m p a r e d for mul t i c racked pressur ised vessel with the avai lab le da t a on S I F and the correctness o f the sof tware deve loped is val idated. S I F for large number o f per iod ic radia l c racks emana t ing f rom the ou te r b o u n d a r y o f a pressur ised cyl inder is then obta ined . S IFs have also been de te rmined for a ro ta t ing annu la r d isk with radia l cracks as a funct ion o f c rack lengths, n u m b e r o f c racks and b/a ra t ios for bo th inner and outer cracks. The results o f S I F for the p r o b l e m of two cracks emana t ing f rom the inner b o u n d a r y o f a ro ta t ing disk is ava i lab le in the l i te ra ture [12]. W h e n the results o f the present a p p r o a c h are c o m p a r e d with tha t the results show a good agreement . In ref. [12] it is fur ther r epor ted tha t their results are accurate up to 1% and in view of a very good agreement o f the present app roach , the results ob ta ined are also within 1% o f the correct values.

REFERENCES

1. Broek, D. The Practical Use of Fracture Mechanics, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1989.

2. Wang, C. T., Applied Elasticity, McGraw-Hill, New York, 1953. 3. Bowie, O. L. and Freese C. E., Elastic analysis for a radial crack in a circular ring. U.S. Ammre Monograph, MS-

70-3, Watertown, MA, 1970. 4. Baratta, F. I., Stress intensity factors for internal multiple cracks in thick-walled cylinders stressed by internal press-

ure using load relief factors. Engineering Fracture Mechanics, 1978, 10, 691-697. 5. Kapp, J. A., The effect of autofrettage on fatigue crack propagation in externally flawed thick walled disks. U.S.

ARADCOM Tech. Rep. ARCLB-TR-77025, Watervliet, NY 1977. 6. Tracy, P. G., Elastic analysis of radial cracks emanating from the outer and inner surfaces of a circular ring.

Engineering Fracture Mechanics, 1979, 11, 291-300. 7. Perl, M. and Arone, R., Stress intensity factors for a radially multicracked partially autofrettaged pressurized thick

cylinder (Trans ASME). Journal of Pressure Vessel Technology, 1988, 110, 147-154. 8. Rooke, D. P. and Tweed, J., The stress intensity factors for a radial crack in a finite rotating elastic disk.

International Journal of Engineering Science, 1972, 10, 709-714. 9. Owen, D. R. J. and Griffiths, J. R., Stress intensity factors for cracks in a plate containing a hole in a spinning disk.

International Journal of Fracture, 1973, 9(4), 471-476. 10. Murakami, Y. and Nisitani, H., The stress intensity factors for the cracked hollow spin disk. Transactions of the

JSME, 1975, 41(348), 2255-2264. I 1. Blauel, J. G., Beinert, J. and Wenk, M., Fracture mechanics investigations of cracks in rotating disks. Experimental

Mechanics, 1977, 17, 106-112. 12. Murakami, Y., Stress Intensity Factors Handbook, Vol. 1, Pergamon Press, 1987, pp. 328-333. 13. Ramamurti, V., Computer Aided Design in Mechanical Engineering, Tata McGraw Hill Publication, New Delhi,

1984. 14. Ramamurti, V. and Balasubramanian, P., Static analysis of circumferentially periodic structures with Potter's

scheme. Computers and Structures, 1986, 22, 427-431. 15. Saouma, V. E. and Schwemmer, D., Numerical evaluation of the quarter-point crack tip element. International

Journal for Numerical Methods in Engineering, 1984, 20, 1920--1941. 16. Ramamurti, V. and Balasubramanian, P., Steady state stress analysis of centrifugal fan impellers. Computers and

Structures, 1987, 25, 129-135. 17. Omprakash, V. and Ramamurti, V., Dynamic stress analysis of rotating turbo-machinery bladed-disk systems.

Computers and Structures, 1989, 32, 477-488.

(Received 22 December 1995)

numerical evaluation of sif for radial cracks in thick annular ring using cyclic symmetry

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