Finite element analysis of a circumferentially cracked cylindrical shell loaded in torsion

Engineering Fracrure Mechanics Vol. 32, No. I, pp. 123-136, 1989

0013-7944/89 $3.00 + .OO 0 1989 Pergamon Press plc.

Printed in Great Britain.

FINITE ELEMENT ANALYSIS OF A CIRCUMFERENTIALLY CRACKED CYLINDRICAL SHELL LOADED IN TORSION M. KUMOSAt Department

of Materials

Science

and D. HULL

and Metallurgy, University Cambridge CB2 342, U.K.

of Cambridge,

Pembroke

Street,

Abstract-Finite element methods are used to evaluate all possible fracture modes at the tip of a circumferential crack in an isotropic cylindrical shell subjected to torsion. The finite element results are compared with shallow-shell theory data for cylinders with different geometries. The results indicate that there is good agreement between the membrane stress intensity factors calculated from finite element displacements at the crack tip and the factors found using shallow-shell theory for small crack lengths. For longer cracks, disagreement between these two approaches is very pronounced. In particular, the bending stress intensity factors from the finite element analysis are significantly greater than the corresponding values determined from the shallow-shell theory. It is observed that the out-of-plane, antisymmetric deformation near the crack produces mixed mode fracture with crack opening and tearing displacements.

1. INTRODUCTION THE PRESENT paper contains a finite element analysis of isotropic cylindrical shells with a circumferential through crack under torsion. This problem has been investigated by, classical method using shallow-shell theory[ l] and perturbation techniques[2]. The approximation of these methods present several problems. Shallow-shell theory or perturbation techniques give only an

approximate solution valid for cracks of small length. The crack problems considered in plates and shells are three dimensional owing to the three dimensional nature of the highly localized stresses at the crack tip. Such problems have not yet been solved analytically even in their simplest form. In the two dimensional elasticity solutions[ 1,2] the stresses around the crack tips have the standard square root singularity. Moreover, the methods do not take into account the effect of transverse shear stress. The assumptions are a gross approximation of through cracks and are inadequate to determine the real 3-D stress distribution at the crack tip. The finite element solutions presented in this paper are obtained for moderately thick to very thin cylindrical shells assuming different crack lengths and wall thicknesses. Using 3-D isoparametric elements the membrane and bending stress intensity factors are evaluated. The effect of antisymmetric out-of-plane deformation on fracture modes at the crack tip is examined. 2. ANALYTICAL

WORK

The existing solution of this problem by Erodgan and Ratwani[l] takes into account the membrane loading and is based on an eight-order shallow-shell theory in which the effect of transverse shear stress is ignored. ‘The following equations were used to solve this problem Eta’ d2w 7S+V2F=0,

a2 h2F V4w -RD6x2=D,

qa4

D =

Et’

12(1 - 9)

where F is the stress function, w is the normal displacement, q is the normal traction, E and v the elastic constants with the dimensions a, t, R describing half crack length, shell thickness and shell radius, respectively. The coordinates are shown in Fig. 1. The Kirchhoff-type assumption was made regarding the crack surface boundary condition with, N,,=M,,=O

for

x=0

and

jz1>,0

and NO,,= No V,=O for x =O and Izl
leave from Wroclaw

Technical

University,

Wroclaw,

Poland. 123

shear

stress

resultants,

124

M. KUMOSA and D. HULL

Fig. 1. Finite element model of one haif of a tube with a = 20” and R/t = 25 (t = 1mm).

The above boundary conditions assume that the only external loading acting on the shell is the in-plane shear stress resultant No applied on the crack surface. The solution of this problem was reduced to a system of singlular integral equations of the first kind. For a circumferential crack in a cylindrical shell subjected to uniform torsion away from the crack, the membrane stress intensity factor Ic;r;land the bending stress intensity factor iu!i were obtained, with .rUkbased on the bending stresses at the outer layer of the shell. If the thickness to radius ratio t/R and the crack length a are sufficiently small, then the general solution of the stress intensity factor may be obtained by approximately superimposing the results of membrane and bending solution with K&r= K; + F K;i *

(2)

The coordinate y is equal to t/2 on the outer surface and -t/2 The stress intensity factors were expressed a@] K;/K,=G,=

-&2a,-(1

on the inner surface.

-v)b,,) (3)

I

and G,=$lirn/Y&fr~~= 0 r-

-[3(1 - ~~)]‘~~~b, I

for

y = t/2

where A, = Uoa, and B,, = U,b, are complex coefficients related to the Chebishev polynomials of the second kind and IJ, = No&R/Et, where A, is the shell parameter defined as follows: A*= [12(1 - ,2)1’@a. (Rt)“’

(5)

The stress intensity factor K. is given by

where t is the uniform shear stress in the shell, remote from the crack and T is the uniform axial torque. The approximate Kirchhoff boundary conditions imposed on the crack surface (transverse

125

Finite element analysis

shear stress neglected) are inadequate to establish the stress intensity factors at the crack tip. Comparing the two different solutions (finite element model and shallow-shell theory) of a circumferential crack in a cylindrical shell under axial tension[3], it has been found that the membrane stress intensity component G, was almost unaffected by the lack of the assumption regarding the transverse shear stress in ref.[4]. It has been found, however, that the bending stress intensity components were significantly different. G,, values obtained from the finite element analysis were very similar to the results found using the shallow-shell theory[5] when accounting for the effect of the transverse shear stress. Thus, it is expected that the transverse shear stress will have a similar influence on the stress conditions at the crack tip in a cylindrical shell subjected to uniform in-plane shear loading. Owing to the mathematical complexities of this problem, an analytical or numerical solution has not yet been proposed. A number of results related only to the membrane stress intensity factors for an inclined crack under uniform in-plane shear loading has been published[6]. The problem has been solved by using Reissner’s transverse shear theory and three membrane stress intensity factors for three different circumferential crack lengths and one particular wall thickness/cylinder radius ratio (R/t) have been proposed.

3. FINITE

ELEMENT

MODEL

A finite element package PAFEC[7] was used to determine displacements and stress intensity factors in cylindrical shells with circumferential cracks subjected to torsion. The problem considered in this paper is anti-symmetric. This requires half of the cylinder to be taken into account to obtain accurate results with the degrees of freedom allowed to model anti-symmetric boundary conditions along the crack plane. A 378 element idealization of the structure with a half angular crack length a = 20”, l-mm wall thickness and 25-mm mean radius is shown in Fig. 1. The elements used in the analysis were isoparametric 20-noded (brick type) and 15-noded (wedge type) 3-D elements with three translatory degrees of freedom at each node. Special 3-D crack tip elements[8] with the midside nodes moved to the quarter point, were placed at both crack tips. The stress in these elements has an inverse square root singularity at the corner of the elements adjacent to the quarter point nodes, making it possible to obtain accurate values of the stress intensity factors. The mesh in the regions close to the crack tips was fine enough to adequately model the stress singularity in those areas with the size of the crack tip elements of 2-3% of the crack length. All elements away from the crack tips have quadratic displacement functions in the axial (x), hoop (z) and the radial (r) directions. This type of 3-D element usually gives reasonable results for the radial stress and good results for the axial and hoop stresses. The aspect ratio of the elements used was always lower than 10 even for very thin shells. The idealization required 7835 degrees of freedom and the maximum frond size was 471. The partitioned banded method was used to solve the finite element equations. The finite element model in Fig. 1 was constrained such that along the cross-section x = 0 there are no translational degrees of freedom in the hoop direction and the structure could deform only in the axial and radial directions. The crack face was free from any constraints. The whole model was constrained against the rigid body motion. The finite element computations were performed on moderately thick to very thin cylindrical shells with the same mean radius R mean = 25 mm while the half angular crack length varied from 5 to 60”. Five different values of wall thicknesses were studied: R/t = 4.16, 6.25, 12.5, 25 and 35.7. The Poisson’s ratio was taken as v = 0.33. The shell parameter A2 was 0.77 < 2, < 9.23. The cylinders were chosen long enough to simulate the condition of an infinite cylinder with their lengths several times greater than the crack length. To investigate end effects, results for K,, from cylinders with CI= 20” and R/t = 25 mm for different cylinder lengths were compared. 4. COMPUTATION

OF STRESS

INTENSITY

FACTORS

The stress intensity factors K,, were calculated following the method proposed by Shih et a/.[91 using the following equation:

K=2fiP -”

K+l

J1

(4U, - Vc)

(7)

M. KUMOSA

126

and D. HULL

where lJ, and Uc are the edge sliding displacements calculated at the quarter point node (B) and the edge node (C) of the crack tip elements respectively. 1is the length of the singular crack element along the crack face, p is the shear modulus, K = 3 - 4v for plane strain and K = 3 - v/l + v for plane stress. Applying eqs 2 and 7 the stress intensity components G,, Gb can be extracted from the crack tip elements using only the singular part of their displacement shape function. It has been reported[lO] that the stress through the thickness of thin shells is usually not linear which could not be calculated using the model in Fig. 1 since there is only one layer of elements through the thickness. This implies that the stress intensity factors should be calculated for both plane stress and plane strain conditions. The factors determined for plane stress condition will give the lower bound and for plane strain the upper bound of the true values. 5. NUMERICAL 5.1. Out-of-plane

RESULTS AND DISCUSSION

deformation

The deformation of thin-walled cylindrical shells is highly dependent upon the wall thickness and crack length. Figures 2(a) and (b) represent the displaced shapes of two inner meshes containing circumferential cracks with M= 5 and 20”, respectively. Both inner meshes have the same R/t = 25 ratios. It is apparent that the inner mesh with the shorter crack is deformed less extensively than the second one. For a = 5, the edge sliding displacement U, and U, are predominant, describing almost perfect mode II loading at the crack tip. The radical displacement U, (out-of-plane) is negligible. In the second case (Fig. 2b), however, there is a significant out-of-plane deformation (large U,) in the regions around the crack. All of the crack tip displacements U,, U, and U, are of the same order of magnitude. The finite element deformation of the shell containing a long circumferential crack can be compared with the deformation of a

CRACK

TIPS

Fig. 2. Deformation of inner meshes containing circumferential cracks; (a) a = 5” and (b) a = 20‘. View along x.

Finite element

Fig. 3. Out-of-plane

deformation

of a paper

cylinder

127

analysis

with a circumferential

crack

under

torsion.

Finite element analysis

129

Gm

I-:~"" 0

1

2

3

”

4

”

5

”

6

”

7

6

" 9

J

10

A*

Fig. 4. Membrane stress intensity components G, for tube with R/r= 25 and a ranging from 5” to 60”.

paper cylinder with a similar crack under torsion. An example of the kind of deformation is shown in Fig. 3. Edge lighting was used to light the surface profile and the buckling of the regions near the crack faces can be clearly seen. This extremely simple experiment proved that the finite element solution predicts perfectly well the cylinder deformation near the crack. 5.2. Membrane and bending stress intensity components Figures 4 and 5 show the finite element results for membrane and bending stress intensity ratios G, and Gb, respectively, plotted against the shell parameter 1,. The results were computed for R/t = 25 with the half angular crack length ranging between 5 and 60”. The finite element data is compared with the shallow-shell theory results by Erdogan and Ratwani[l] for the same cylinder geometries. For 11*smaller than 3.5 the membrane stress intensity components computed for both plane stress and plane strain conditions, give a lower and upper bound to the values obtained from the shallow-shell theory. With increasing A2(up to 22”) the shallow-shell theory results approach the finite element values calculated under plane stress conditions. Increasing the half crack length over approximately 23” shows a significant increase in the finite element G,,, values as compared with those by Erdogan and Ratwani. For c1= 60”, the finite element G, is almost twice G,,, in ref.[l]. This indicates that for long cracks (1, > 3.5) and R/t = 25 shallow-shell theory is no longer

Fig. 5. Bending stress intensity components

G, for tubes with R/t= 25 and a ranging from 5” to 60”.

M. KUMOSA and D. HULL

130

0

1

2

3

4

L

Fig. 6. Membrane stress intensity ratios G, for a: = 20” and R/r in the range of 4.16 to 35.1.

valid. A similar discrepancy between the stress intensity factors for a circumferential crack under tension has been previously reported[3, 10, 111. According to Erdogan and Ratwani, the bending component G, is of the order of &4 x 10m6 in the AZranges considered. The finite element values of Gb are significantly greater than the values predicted by shallow-shell theory. The reason for this difference is that the bending component Gb is very sensitive to the transverse shear effect in a similar manner as in the tension case. G, and Gb are highly dependent on R/tratio and decrease significantly with increasing t (see Figs 6 and 7). The thickness effect was investigated for cylinders with 1, ranging between 1.28 and 3.75 for five different R/tratios and c1= 20”. The finite element G, values computed under plane stress conditions (Fig. 6) are very similar to those obtained from shallow-shell theory[l]. On the other hand, G,, values (Fig. 7) show enormous differences as compared to the data in ref.[l]. For very thick cylinders the G,, values are slightly negative and approach zero for t -+ co. Semi-membrane solutions decay very slowly away from the crack along the tube length. Therefore, the validity of shallow-shell theory is restricted to cylinders long enough to avoid any end effects. It has been found[lO] that circumferential through-cracks are significantly affected by end conditions especially in the case of a moment applied to the crack. In tension, for cracks longer than ~1= 90” the finite element analysis was performed with cylinders of length 6L, and 3OL,

Fig. 7. Bending stress intensity ratios Gb for a = 20” and R/r ranging from 4.16 to 35.7.

Finite element analysis

131

(I;, = 2.5JRt[ 10,12]), however the end effects were still noticeable. For short cracks c1= 20” and R/t = 25 in cylinders under tension[3] the difference between G, and G, computed for cylinder with 60 and 140mm in length was about 2%. To study the end boundary effects, five cylinders with different lengths were analysed assuming constant crack length CI= 20” and R/t = 25. Table 1 presents the lengths of the cylinders analysed and the stress intensity components G,, G,, for plane stress conditions. The difference between G, and Gb computed for the shortest and longest cylinders is almost negligible (less than 0.7%) which indicates that the end effects do not have any significant effect on the results for the above cylinder geometry. 5.3. Crack opening and ieff~ing dis~l~&ement~ It has been reported[3, 51 that the transverse shear stress produces symmetric out-of-plane deformation (bulging) in cylindrical shells with circumferential cracks under tension. A shell around a circumferential crack bulges out, and some distance ahead of the crack the out-of-plane displacement component becomes negative. Displacements of two crack faces are symmetric across the crack plane. In torsion, however, the out-of-plane defo~ation is much more complex. To examine in detail the out-of-plane crack tip displacements, a substructure with an internal circumferential crack was modelled. The antisymmetric displacement boundary conditions imposed on the substructure were taken from displacements of a cylinder with the same crack geometry and wall thickness. The crack tip elements were 15 noded wedge type quarter-point elements. The difference in the stress intensity factors calculated from one half of the cylinder (Fig. 1) and the substructure was smaller than 0.75%. Two different views of the displaced substructure are presented in Fig. 8(a) and (b) showing very complicated deformation of two crack faces. It is clear from Fig. S(b) that the deformation is antisymmetric with two crack faces crossing each other at the middle point of the crack. The crack tip radial U, and axial U, displacements of two crack faces obtained from the finite element analysis of the substructure are shown in Fig. 9. The displacements, computed for 1 Nm torque, describe the relative positions of two crack faces very close to the crack tip after deformation. S, refers to the displacements of the crack tip on the outer, middle and inner surface. S_, indicates the displacements of corner nodes at a distance of one element ahead of the crack tip. s 1.2.3.4 refer to the displacements of corner nodes at a distance of one, two, three and four elements from the crack tip along the crack faces. All crack tip displacements were almost linear through the thickness. The size of all crack tip elements (distances between S, , S, . . . ) was 0.545 mm (on the middle surface). Additionally, the ratios of axial and radial displacements to the tangential displacement (edge sliding displacement) UXjUz and U,IU,, determined at a distance of one element along the crack face are presented in Table 2. It is apparent from the results in Fig. 9 and Table 2 that the influence of out-of-plane deformation on the crack tip displacements cannot be ignored. Moreover, an examination of the relative position of two crack faces reveals that significant crack opening U” and crack tearing UT displacements exist at the crack tip. The crack opening displacement Us was determined between two parallel lines; the first along the displaced crack tip line and the second passing through the middle surface point (indicated by X) of the displaced i-element along the crack face. Similarly, the crack tearing displacement UT was determined

Table L 60 80 100 120 140

1. (a =20”, R/t = 25, plane stress} Cl

1.2962 1.2914 I .2889

1.2880 1.2876

G, 0.5271 0.5252 0.5241 0.5237 0.5235

Table 2.

(a =

20”, R/I = 25)

Surface

u*PJ,

W-J:

Outer Middle Inner

0.24 0.85 2.49

1.00 1.12 1.69

132

M. KUMOSA

and

D. HULL

CRACK TIPS

/

CRACK FACES 0.))

Fig. 8. Deformations

of inner meshes showing relative positions of two crack faces for c( = 20”, R/r = 25; (a) view perpendicular to x, (b) along X.

between two parallel lines passing through the middle surface points of the crack tip and i-element. The lines are perpendicular to the displaced crack tip line. The separation of UT and UT from the axial and radial displacements in Fig. 10 was performed only for the right crack face. In the same way, the displacements can be extracted from U, and U, of the left side. The crack opening displacement Us are symmetric and the tearing displacement U;’ are antisymmetric across the displaced crack tip line. Because of mathematical difficulties, the theoretical determination of the nature of stress/strain singularities at the crack tip in a cylindrical shell has not been completely resolved. By using the quarter-point crack tip elements in the finite element analysis, it implies that there is a proper r -‘D stress singularity around the crack. This also implies that the displacements near the crack should be a function of r':*[U -f(r"2)J. Knowing the major crack tip displacements (i.e. U",UT and Uz) their distributions at the crack tip can be examined (Fig. 10) assuming the following relationship between U and r. log U = A + m(t log r).

(8)

For m = 1, the displacement should be related to r”* and the r -‘I2 stress singularity should exist at the crack tip. It is evident from the results in Fig. 10 that the shear stress singularity for mode II is very close to r-Ii2 with the factor m = 1.05. However, for U” and UT the distributions seem to be different from that expected in the Westergaard model. This is particularly so for the out-of-plane tearing mode (UT);the displacement distribution is significantly different from the classical prediction with m = 1.58. The crack opening Up and crack tearing Uf displacements computed at a corner node one element along the crack face are shown in Figs 11 and 12 for constant R/t= 25, various c1and

Finite element

133

analysis

(Y=20°

ux+ur . OUTER

R/t q25

x

MIDDLE

T=lNm

OUTER _._.-.-.

MIDDLE

_._.-.-.-

INNER SURFACE -.-.-.-.-.

LEFT

SIDE

)

\

\ ‘\

is, \

Fig.

9. Axial CJX and displacements

radial CJ, displacements of two crack faces and evaluated crack U” and crack tearing displacements UT for a = 20” and R/t = 25.

opening

constant a = 20”,

various R/t ratios, respectively. They show the magnitude of the out-of-plane deformation around the crack tip as a function of crack length and wall thickness. Both U” and UT increase dramatically with increasing a and decreasing t. Thus, it can be expected that for short cracks and thick walled cylinders mode II will dominate since U” and UT are almost negligible for these geometries. For longer cracks or thinner cylinders, mixed mode fracture should be expected with a very well defined transition from one mode to another if the critical crack displacements are significantly different for different fracture modes[ 131. 5.4. Crack propagation direction It is well known that mode II growth proceeds usually in a direction that forms an angle to the direction of maximum shear stress. Further crack growth occurs in mode I by branching. To investigate the crack branching effect at the tip of a circumferential crack under in-plane shear loading, the strain energy density theory was used as a suitable failure criterion[l4]. The strain energy density can be computed using the following equation dW dV =

s

(‘1 o oij dc,

(9)

where uij and cij are the rectangular components of the stress and strain tensor. The theory indicates that the crack grows in the direction of the element with the minimum value of the energy density while the element with the maximum strain energy density initiates tFM

32, I-,

M. KUMOSA

134

and D. HULL Q .20°

-\ r2.9

c

’

: -0.18

-0.12

-008

l\

0

-0.04

l/2 log

w 0.04

0.08

0.12

0.16

r

Fig. 10. Distribution of crack tip displacements.

UT6 U0 5 .I&

4

3

2

1

0

Fig. Il. Crack opening U” and tearing UT displacements for cylinders with R/t = 25 and half angular crack lengths ranging from a = 5” to 60”.

Fig. 12. Crack opening displacements U” and crack tearing displacements UT for ranging from 4.16 to 35.7.

a = 20” and

R/rratios

135

Finite element analysis

Fig. 13. Branching and yielding initiation for tl = 5” and R/r = 25.

yielding. Fracture initiation takes place when the element in the direction of minimum d W/d V at a distance of r,, from the crack tip absorbs a critical amount of strain energy density. As an example, two geometries are evaluated here with the same R/t = 25 but different crack lengths; u = 5” and a = 20” (which refers to Fig. 2(a) and (b)). Owing to the coarseness of the finite element mesh near the crack tip, an exact evaluation of the crack propagation direction could not be obtained. However, the results in Figs 13 and 14 present some general view about the nature of crack propagation in the cases considered. For c1= 5” the branching initiation is at an angle of -70” to the original crack plane, which is in good agreement with the direction prediction by the maximum principle stress theory for pure mode II fracture. In the second case, however, the branching will occur at an angle significantly smaller (Fig. 14). This can be explained by the presence of mixed mode fracture at the crack tip as a result of the extensive out-of-plane deformation. For both crack lengths, yielding will proceed along the original crack plane.

Fig. 14. Branching and yielding initiation for

a =

20” and R/f = 25.

I36

M. KUMOSA and D. HULL

CONCLUSIONS The finite element analysis has been used to evaluate displacements and stress intensity factors at the tip of a circumferential crack in a cylindrical shell under torsion. The results obtained in this study lead to the following conclusions: 1. Deformation is highly dependent on shell thickness and crack length. Short cracks produce almost perfect mode II loading at the crack tip whereas for longer cracks there is an antisymmetric out-of-plane deformation. 2. The membrane stress intensity factors obtained from the finite element analysis are in good agreement with the shallow-shell theory results only for short cracks. The shell parameter A2has to be smaller than 3.5 for R/t = 25. For longer cracks, increasing deviations from sha]low-shell theory occur. 3. The bending stress intensity factor is strongly influenced by the effect of transverse shear stress and should not be ignored in fracture analysis. The factors derived from shallow-shell theory are significantly smaller than the finite element data. 4. The antisymmetric out-of-plane deformation produces mixed mode loading at the crack tip with crack opening and tearing displacements which are dependent on shell geometry and crack length. 5. The strain energy density criterion was used to investigate the crack branching effect at the tip of a circumferential crack. For small CI(a z 5’) the circumferential crack will propagate at 70” angle to the original crack plane. For long cracks, however, the out-of-plane deformation changes the angle and the crack extension occurs at an angle significantly smaller than 70”. Acknowledgements-The authors are grateful to the Science and Engineering Research Council for financial support and to their colleagues for valuable advice.

REFERENCES [I] F. Erdogan and M. Ratwani. A circumferential crack in a cylindrical shell under torsion.

Int. J. Fracture Mech. 8,

87-95 (1972). [2] H. V. Lakshminarayana and M. V. V. Murthy, On stress around an aribitrarily oriented crack in a cylindrical shell. Int. J. Fracture 12, 547-566 (1976). [3] M. Kumosa and D. Hull, Finite element analysis of a circumferentially cracked cylindrical shell under uniform tensile loading. Engng Fracture Me&. 31, 817-826 (1988). [4] F. Erdogan and M. Ratwani, Fatigue and fracture of cylindrical shells containing a circumferential crack. Inr. J. Fracfure Mech. 6, 379-392 (1970). [S] F. Delale and F. Erdogan, Transverse shear effect in a circumferentially cracked cylindrical shell, Q. appl. Math. 239-258 (1979). [6] 0. S. Yahsi and F. Erodgan, A cylindrical shell with an arbitrarily oriented crack. Znt. J. Soliok Strucr. 19, 955-972 (1983). [7] Theory, Pafec Ltd, Nottingham (1984). [S] R. S. Barsoum. On the use of isoparametric finite elements in linear fracture mechanics. fnr. J. numer. Meth. Engng 10, 25-37 (1976). [9] C. F. Shih, H. G. deLorenzi and M. D. German, Crack extension modeling with singular quadratic isoparametric elements. Inr. J. Fracture 12, 647-651 (1976). [IO] R. S. Barsoum, R. W. Loomis and B. D. Stewart, Analysis of through cracks in cylindrical shells by the quarter-point elements. Inr. J. Fracture 15, 259-280 (1979). [I I] R. Ehlers, Stress intensity factors and crack opening displacements for circumferentially cracked cylindrical shells. Trans. 8th Int. Conf, on Structural Mechanics in Reacfor Technology, Brussels, Belgium, pp. 255-260 (1985). [12] H. Kraus, Thin Elastic Shells. Wiley, New York (1967). [13] M. Kumosa, J. A. Barnes and D. Hull, Crack growth in hoop wound tubes. To be published (1988). 1141G. C. Sih, Strain-energy density factor applied to mixed mode crack problems. Int. J. Fruerure 10, 305-321 (1974). (Received 4 December

1987)