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Stress intensity factors for circumferential through cracks in hollow cylinders subjected to combined tension and bending loads
Engineering l"racture Mechanics Vol. 21, No. 3, pp. 563-571, 1985 Printed in the U.S.A.
0013-7944,'85 $3.00 + .00 ~ 1985 Pergamon Press Ltd.
STRESS INTENSITY FACTORS FOR CIRCUMFERENTIAL THROUGH CRACKS IN HOLLOW CYLINDERS SUBJECTED TO COMBINED TENSION AND BENDING LOADS R. G. FORMAN NASA, Lyndon B. Johnson Space Center, Houston, TX 77058, U.S.A. and J. C. HICKMAN and V. SHIVAKUMAR Lockheed-EMSCO, Houston, TX 77258, U.S.A. Abstract--Stress intensity factors are obtained for circumferential through-cracks in complete cylindrical shells subjected to combined tension and bending. A closed-form expression is derived from Sanders' energy release rate integral solution and compared with finite element results for different cylinder lengths and boundary conditions.
NOTATION G
C, B, S, Fo, Fb E F h ?, lo, lb 1, Io, l b
Kt L M
PI R U IIc X, y Q E
At ~t, ~q, g(ct) v
k 0", 0"0, O"b
0
crack half length stress intensity magnification factors, given in text modulus of elasticity axial force cylinder thickness energy release rate integrals, given in text dimensionless energy release rate integrals, given in text Mode I stress intensity factor cylinder length bending moment force applied at node i cylinder radius elastic energy crack displacement in axial direction dimensionless crack displacement in axial direction Cartesian coordinates half opening angle of crack shell parameter (eqn 4) displacement of node i parameters given in text relating to energy release rate Poisson's ratio shell parameter, given in text stresses circumferential coordinate of cylinder
INTRODUCTION THE DERIVATION of more applicable stress intensity factor solutions for circumferentially cracked cylindrical shells is important to safety analysis in numerous structural applications. The need for these solutions occurs in such fields as off-shore oil drilling and production, pressure vessels and piping and aircraft and aerospace structures and subsystems. For the space shuttle and payloads fracture control program, which is the basis for the present authors' work, the relevant problems pertain to control rods, tubing, landing gear axles and miscellaneous support structure members. The loading condition for most crack cases are either axial tension or bending, or a combination of the two, such as from applied axial loads and induced vibration. In all of these problems, more accurate or general stress intensity factor formulas are needed to calculate either the safe life or critical crack length. The earliest theoretical results for the title problem were derived by Folias in 1967 [1] using shallow shell theory. The Folias results and those by other early investigators were restricted to short-type cracks or to small crack angles. The first analytic results for long crack problems 563
564
R . G . FORMAN et al.
were obtained by Sanders [2] and Nicholson and Simmonds [3] using path-independent integrals to calculate the energy release rate. Also relevant to the problem are the finite element results of Barsoum, Loomis and Stewart [4], who made calculations for cracks as long as halfway around the circumference for different values of cylinder length and wall thickness. In the present paper, the energy release rate results of Sanders are reformulated into a nondimensional stress intensity factor solution and compared with finite element results for a wide range of cylinder lengths and crack angles. Instead of using special crack-tip finite elements as in Ref. [4], the strain energy release rate method is used to calculate the stress intensity factor values. Both the path-independent integral I of Sanders and the more conventional compliance method are used to calculate the energy release rates. The close accuracy of the analytic solution for both the bending and tension cases is confirmed for long cylinders and with crack angles exceeding two-thirds of the circumference. The finite element results also'indicate that the analytic solution is valid for practical short cylinder problems where loading is transmitted through a relatively stiff ring such as a tube fitting. These results provide a significant improvement in the capability of performing fracture control analysis of circumferential throughcracks in cylindrical shells loaded by combined tension and bending. ANALYTIC SOLUTION The solution for the energy release rate of an extending circumferential through-crack in a cylindrical shell subjected to axial tension was derived by Sanders[2]. In integral form, the energy release rate (with respect to crack angle ct) was shown to be aa
ltc dO,
(I)
where the integral was made dimensionless by the relation = (cr2RZhlE)I,
(2)
and llc was the dimensionless crack displacement in the axial direction equal to -ficEkroR. For the special case of a long crack or large ed2e, a solution was obtained for uc, and the resulting expression for I was I = e-I(x2g(cx),
(3)
where e2= (h)[12(1-v2)]-v2
(4)
1 - ~ cot c~ )2 1 + 2~tcottx + ~ c - ~ - [ ( ~ - et)/V~] "
(5)
and
g(ct) = 2~/2
For the case of a short crack, the energy release rate solution of Nicholson and Simmonds [3] was assumed to be applicable. In terms of stretching and bending intensity factors S and B, the solution was given in the form I = 2~ret[S2 + (1 - v)(3 + v)n2].
(6)
A combined stress intensity factor, C, was then introduced, which in terms of eqn (6) reads I = 2~otC 2.
(7)
Stress intensity factors for circumferentialthrough cracks in hollow cylinders
565
Formulas that accurately fit the numerical solutions for C were given as Ir h2 - 0"0293h3 C = 1 + "i'6 = (-~h)
O'5 + ( P - ~ )
~" -< 1 ~
k>
(8)
I,
where ~. is a crack length parameter equal to a/2e. Finally, for a " c u r v e fit" to both the long and short crack solutions given by eqns (3) and (7), Sanders derived the formula I = e-lo.2[g(o0 + ' ~ . - I C 2 -- 2X/21.
(9)
The validity of eqn (9) was said to be limited to cylinders of sufficient length to avoid end effects. The minimum distance between the crack and any boundary or other disturbance was estimated to be approximately 2Rx/-R-~. Continuing in subsequent work to the case of combined tension and bending, Sanders extended his analysis for this problem and the results are reported in Ref. [5]. The solution was expressed in the form
2,rr2Ehe
rlaF + 2p. -~ sin a
,
(10)
where r g(oO ] uz rl =
[.2V~_]
(11)
and
I~ = 1 +
r + c~ cot z cx - cot cx 4 cot[Or - a)/X/2] + X / 2 c o t a "
(12)
This combined load solution is for a long crack without the " c u r v e fit" approximation for the complete range of crack lengths. The modification of eqn (I0) to cover both short and long cracks is straightforward and was performed by the present authors. With the assumption that F is equal to zero, and letting the maximum bending stress be given by Orb = M h r h R z,
(13)
the energy release rate equation has the form 2X/2hR z Ib = ~ (Iscrb sin a) 2.
(14)
In the dimensionless form where R2g~h
i~ - ---7-/t,,
(15)
the solution is given by (16)
566
R . G . FORMAN et aL
For the other limiting case, or the short crack solution, eqn (5) is still applicable with the substitution of trb for cro. Making this change, the curve fit-type formula for pure bending is Ib =
~-'a2 [ 2 V ~ (Li-i-~ ix)z + rrC2/h - 2 V ~ ] .
(17)
The dimensionless stress intensity factors for the individual cases of tension and bending can be determined from expressions equivalent to eqn (7), which are F o = (Io/2~r~) lr2
and Fb = (Ib/2~a) ~/2,
(18)
where Io = I in eqn (9). The resulting formula, then, for combined tension and bending stress intensity factors can be written as Kt = (~0Fo + crbFb)~/"~-a,
(19)
where a = oR. FINITE ELEMENT ANALYSIS The finite element analysis was conducted on a VAX 11/780 computer using the MSC/ NASTRAN computer program [6]. The cylinder was modeled using eight-node quadrilateral shell elements (QUAD8) and six-node triangular shell elements (TRIA6). The QUAD8 is an isoparametric-type element with a parabolic displacement field. The TRIA6 is a linear strain, linear curvature triangle that was used as a transition element when mesh sizes were changed. The computations were run utilizing all element stiffnesses, viz., membrane, bending and transverse shear. Since MSC/NASTRAN has superelement or substructuring capability, this option was also used and provided a significant reduction in computer central processing unit (CPU) time (e.g. 30 min versus 3 hr). A drawing of a cylinder model showing the element geometry and the superelement configuration is given in Fig. I. Two methods were used to calculate the stress intensity factor values. The first method was based on Sanders's energy release rate integral given by eqn (1). The second method was based
QUAD 8
TRIA 6
Fig. 1. Finite element model of cylinder.
Stress intensity factors for circumferential through cracks in hollow cylinders
567
on the compliance approach, in which the potential energy of the external forces is given by
u = 89G P~ai.
(20)
In eqn (20), Ai is the node displacement parallel to each Pi force applied at the end boundary nodes. For the first method, the integral of the displacements (i.e. the crack opening area) was calculated numerically using Simpson's rule. Twenty-four elements of unit length were used to model the half circumference at the plane of the crack. The calculations were made for unit crack extensions from zero to a maximum length of 20 elements (i.e. 0 <-- a -< 150~ The derivatives of the integral values were calculated by using a spline fit with a fourth-degree polynomial equation through five points. Double precision was used to calculate the polynomial coefficients. For the tension case, the value of I was then calculated at the midpoint of each spline fit with the equation E
I-
,9 ~, B,,a2,,.
troROan=l
(21)
The stress intensity factors were then obtained from
KI tr~
( I'~ '/2 = \2--~a]
"
(22)
The calculations for the strain energy release rate based on the compliance approach were also made using fourth-degree polynomial spline fits. The stress intensity factors for each fit were obtained from the equation
(E OU'~I/2 K,, = \ T z ~ a ] '
~
h
(23)
= ~=11.78 h
(a) C Y L I N D E R W I T H E N D R I N G S
(b) C Y L I N D E R W I T H O U T E N D R I N G S
Fig. 2. Geometry of cylinder with and without end rings.
R. G. F O R M A N
568
et al.
where 4
U = E
Cna2n.
(24)
n=l
T h e cylinder length was varied by simply adding or subtracting superelements. Since the application of uniform stress or pure bending on the boundaries of short cylinders with long cracks is not physically realistic, additional computer analysis was conducted for which the boundary forces were applied through a stiff ring. The drawing in Fig. 2 shows the cylinder geometry with the ring and without the ring. RESULTS AND DISCUSSION The initial objectives for the finite element analysis were to compare the stress intensity factor results determined by the two different methods given by eqns (22) and (23) and to compare these results with the analytic solution expressed by eqn (19). The numerical results of comparing the two methods for calculating K~ using finite element analysis are listed in Table 1. The comparison shows that the results were identical up to almost six significant figures. This confirms the high accuracy of the modeling and indicates that the energy release rate integral approach does not give greater accuracy than the compliance approach for this particular problem. For this reason, all subsequent analysis was conducted using the compliance method because it was simpler, especially for the bending problem. The comparison of the finite element results with curves for the analytic solution given by eqn (19) is shown in Fig. 3. The accuracy of the fits between the data points and the curves for both tension and bending verifies the accuracy of the closed form expression for crack angles as large as 135~ Since eqn (19) should be equally accurate in general for long cylinders governed by thin shell theory, no additional calculations were made to compare the results for different RIh ratios. Instead, finite element analysis showing the results of varying the cylinder length and boundary conditions were considered to have more practical value. When the cylinder length was shortened to approximately the cylinder circumference or less, the stress intensity correction factors became increasingly larger for the constant tension or pure bending boundary conditions. This behavior is similar to the theoretical case of a centercracked strip in which the length is relatively short and the applied boundary stress is assumed to be constant. For the alternative problem, however, when the strip is given a prescribed uniform extension at the boundary, the infinite long solution provides an upper bound for the correction factors. This same result is found to occur for the circumferentially cracked cylinder. To more easily study the end boundary effects, a thick ring was connected to the end of the cylinder. The ring could not be made orders of magnitude stiffer than the shell because the global stiffness matrix would become ill-conditioned. The final ring size, therefore, was selected by comparing displacement results for variable ring thicknesses, starting with the ring and shell having equal thickness. By using the same element length as for the shell and increasing the Table I. Comparison of methods for calculating stress intensity factors from finite element results
KdcroV'~
30 ~ 45 ~ 60 ~ 75 ~ 90 ~ 105~ 120~ 135~ 9
= 0.1,
Sander's Integral
Compliance
!.71923 2.24741 2.91232 3.84372 5.25590 7.57664 11.80560 20.79159
!.71923 2.24741 2.91232 3.84372 5.25590 7.57664 11.80561 20.79161
RIh = 30, LIR = 12.3
Stress intensity factors for circumferential through cracks in hollow cylinders
J/
20 "- FINITE ELEMENT RESULTS E ] F o (TENSION) O F B (BENDING) 15 -~
2.
m LL lO
L -
1.12
;30
-
d
=0.1
=
J
#
0
l 30
0
t 60
I 90
I 120
I 150
a, DEG Fig. 3. Comparisonof finite element results with closed-formsolution for stress intensity factors.
3
~=0.1 R/h = 30 I_/~R = 0.92 a= 150;
WITHOUT END R t
<1 1
f
0
-1
:
/
J~
N
~
~WITH
t
-
~ET
END RING
'
I
I
I
I
-0.5
0 y/R
0.5
1
Fig. 4. Comparison of end displacements for cylinders with and without end rings.
569
570
R . G . FORMAN et al.
....
WITH END RINGS - WITHOUT END RINGS
F=o= KI I o~ltJ.~R=
7 / 5.1
( =0.1
R/h = 30
~R 8
~2 u.
2.67 - - ~
90.29
L = 0.60 wR
0
I
I
I
45 ~
90*
135 ~
cr
Fig. 5. Dimensionless stress intensity factor versus crack angle for cylinder subjected to constant end tension.
- - - - - - WITH END RINGS WITHOUT END RINGS
Foo~p ~=0.1 R/h = 30
8
~R 2
ii
1.42"-~
L TrR
0
= 0.60
45 ~
-0.29
90='
135~
r162
Fig. 6. Dimensionless stress intensity factor versus crack angle for cylinder subjected to pure bending moment.
Stress intensity factors for circumferential through cracks in hollow cylinders
571
ring thickness to give a square cross section, the displacement results became very nearly linear. This geometry was then used for the remaining analysis. A comparison of the end displacements for the stiffened and unstiffened cylinder configurations is shown in Fig. 4. The final comparisons of the effect of the end conditions for different cylinder lengths are shown in Fig. 5 for tension and Fig. 6 for bending. The results show that forces applied through a stiff ring make the problem similar to that of a very long, unstiffened cylinder. At most, if the ring is relatively stiff, the infinite long cylinder without a ring gives an upper bound to the stress intensity factor. Equation (19), therefore, should give satisfactory results for most engineering applications. CONCLUSIONS Sanders's energy release rate integral for a circumferentially cracked cylindrical shell loaded in both tension and bending was used to derive the solution for the stress intensity factor. A curve fit approach proposed by Sanders for comb!ning long and short crack solutions in the tension case was extended by the present authors to the bending case. Finite element analysis showed that short cylinders with loads applied through ring-stiffened ends had nearly the same K1 values as infinite long cylinders. The analytic solution for infinite long cylinders is satisfactory, therefore, for most engineering calculations on short cylinders. REFERENCES [1] E. S. Folias, A circumferential crack in a pressurized cylindrical shell, Int. J. Fract. Mech. 3, 1-11 (1967). [2] J. Lyell Sanders, Jr., Circumferential through-cracks in cylindrical shells under tension, ASMEJ. Appl. Mech. 49, 103-107 (1982). [3] J. W. Nicholson and J. G. Simmonds, Sanders' energy-release rate integral for arbitrarily loaded shallow shells and its asymptotic evaluation for a cracked cylinder, ASME J. Appl. Mech. 47, 363-369 (1980). [4] R. S. Barsoum, R. W. Loomis and B. D. Stewart, Analysis of through cracks in cylindrical shells by the quarterpoint elements, Int. J. Fract. 15, 259-280 (1979). [5] J. L. Sanders, Jr., Circumferential through-crack in a cylindrical shell under combined bending and tension, ASME J. Appl. Mech. 50, 221 (1983). [6l MSC/NASTRAN, Handbook for Linear Static Analysis. The Macneal-Schwendler Corporation, Los Angeles, Califomia (Dec. 1981).
0013-7944,'85 $3.00 + .00 ~ 1985 Pergamon Press Ltd.
STRESS INTENSITY FACTORS FOR CIRCUMFERENTIAL THROUGH CRACKS IN HOLLOW CYLINDERS SUBJECTED TO COMBINED TENSION AND BENDING LOADS R. G. FORMAN NASA, Lyndon B. Johnson Space Center, Houston, TX 77058, U.S.A. and J. C. HICKMAN and V. SHIVAKUMAR Lockheed-EMSCO, Houston, TX 77258, U.S.A. Abstract--Stress intensity factors are obtained for circumferential through-cracks in complete cylindrical shells subjected to combined tension and bending. A closed-form expression is derived from Sanders' energy release rate integral solution and compared with finite element results for different cylinder lengths and boundary conditions.
NOTATION G
C, B, S, Fo, Fb E F h ?, lo, lb 1, Io, l b
Kt L M
PI R U IIc X, y Q E
At ~t, ~q, g(ct) v
k 0", 0"0, O"b
0
crack half length stress intensity magnification factors, given in text modulus of elasticity axial force cylinder thickness energy release rate integrals, given in text dimensionless energy release rate integrals, given in text Mode I stress intensity factor cylinder length bending moment force applied at node i cylinder radius elastic energy crack displacement in axial direction dimensionless crack displacement in axial direction Cartesian coordinates half opening angle of crack shell parameter (eqn 4) displacement of node i parameters given in text relating to energy release rate Poisson's ratio shell parameter, given in text stresses circumferential coordinate of cylinder
INTRODUCTION THE DERIVATION of more applicable stress intensity factor solutions for circumferentially cracked cylindrical shells is important to safety analysis in numerous structural applications. The need for these solutions occurs in such fields as off-shore oil drilling and production, pressure vessels and piping and aircraft and aerospace structures and subsystems. For the space shuttle and payloads fracture control program, which is the basis for the present authors' work, the relevant problems pertain to control rods, tubing, landing gear axles and miscellaneous support structure members. The loading condition for most crack cases are either axial tension or bending, or a combination of the two, such as from applied axial loads and induced vibration. In all of these problems, more accurate or general stress intensity factor formulas are needed to calculate either the safe life or critical crack length. The earliest theoretical results for the title problem were derived by Folias in 1967 [1] using shallow shell theory. The Folias results and those by other early investigators were restricted to short-type cracks or to small crack angles. The first analytic results for long crack problems 563
564
R . G . FORMAN et al.
were obtained by Sanders [2] and Nicholson and Simmonds [3] using path-independent integrals to calculate the energy release rate. Also relevant to the problem are the finite element results of Barsoum, Loomis and Stewart [4], who made calculations for cracks as long as halfway around the circumference for different values of cylinder length and wall thickness. In the present paper, the energy release rate results of Sanders are reformulated into a nondimensional stress intensity factor solution and compared with finite element results for a wide range of cylinder lengths and crack angles. Instead of using special crack-tip finite elements as in Ref. [4], the strain energy release rate method is used to calculate the stress intensity factor values. Both the path-independent integral I of Sanders and the more conventional compliance method are used to calculate the energy release rates. The close accuracy of the analytic solution for both the bending and tension cases is confirmed for long cylinders and with crack angles exceeding two-thirds of the circumference. The finite element results also'indicate that the analytic solution is valid for practical short cylinder problems where loading is transmitted through a relatively stiff ring such as a tube fitting. These results provide a significant improvement in the capability of performing fracture control analysis of circumferential throughcracks in cylindrical shells loaded by combined tension and bending. ANALYTIC SOLUTION The solution for the energy release rate of an extending circumferential through-crack in a cylindrical shell subjected to axial tension was derived by Sanders[2]. In integral form, the energy release rate (with respect to crack angle ct) was shown to be aa
ltc dO,
(I)
where the integral was made dimensionless by the relation = (cr2RZhlE)I,
(2)
and llc was the dimensionless crack displacement in the axial direction equal to -ficEkroR. For the special case of a long crack or large ed2e, a solution was obtained for uc, and the resulting expression for I was I = e-I(x2g(cx),
(3)
where e2= (h)[12(1-v2)]-v2
(4)
1 - ~ cot c~ )2 1 + 2~tcottx + ~ c - ~ - [ ( ~ - et)/V~] "
(5)
and
g(ct) = 2~/2
For the case of a short crack, the energy release rate solution of Nicholson and Simmonds [3] was assumed to be applicable. In terms of stretching and bending intensity factors S and B, the solution was given in the form I = 2~ret[S2 + (1 - v)(3 + v)n2].
(6)
A combined stress intensity factor, C, was then introduced, which in terms of eqn (6) reads I = 2~otC 2.
(7)
Stress intensity factors for circumferentialthrough cracks in hollow cylinders
565
Formulas that accurately fit the numerical solutions for C were given as Ir h2 - 0"0293h3 C = 1 + "i'6 = (-~h)
O'5 + ( P - ~ )
~" -< 1 ~
k>
(8)
I,
where ~. is a crack length parameter equal to a/2e. Finally, for a " c u r v e fit" to both the long and short crack solutions given by eqns (3) and (7), Sanders derived the formula I = e-lo.2[g(o0 + ' ~ . - I C 2 -- 2X/21.
(9)
The validity of eqn (9) was said to be limited to cylinders of sufficient length to avoid end effects. The minimum distance between the crack and any boundary or other disturbance was estimated to be approximately 2Rx/-R-~. Continuing in subsequent work to the case of combined tension and bending, Sanders extended his analysis for this problem and the results are reported in Ref. [5]. The solution was expressed in the form
2,rr2Ehe
rlaF + 2p. -~ sin a
,
(10)
where r g(oO ] uz rl =
[.2V~_]
(11)
and
I~ = 1 +
r + c~ cot z cx - cot cx 4 cot[Or - a)/X/2] + X / 2 c o t a "
(12)
This combined load solution is for a long crack without the " c u r v e fit" approximation for the complete range of crack lengths. The modification of eqn (I0) to cover both short and long cracks is straightforward and was performed by the present authors. With the assumption that F is equal to zero, and letting the maximum bending stress be given by Orb = M h r h R z,
(13)
the energy release rate equation has the form 2X/2hR z Ib = ~ (Iscrb sin a) 2.
(14)
In the dimensionless form where R2g~h
i~ - ---7-/t,,
(15)
the solution is given by (16)
566
R . G . FORMAN et aL
For the other limiting case, or the short crack solution, eqn (5) is still applicable with the substitution of trb for cro. Making this change, the curve fit-type formula for pure bending is Ib =
~-'a2 [ 2 V ~ (Li-i-~ ix)z + rrC2/h - 2 V ~ ] .
(17)
The dimensionless stress intensity factors for the individual cases of tension and bending can be determined from expressions equivalent to eqn (7), which are F o = (Io/2~r~) lr2
and Fb = (Ib/2~a) ~/2,
(18)
where Io = I in eqn (9). The resulting formula, then, for combined tension and bending stress intensity factors can be written as Kt = (~0Fo + crbFb)~/"~-a,
(19)
where a = oR. FINITE ELEMENT ANALYSIS The finite element analysis was conducted on a VAX 11/780 computer using the MSC/ NASTRAN computer program [6]. The cylinder was modeled using eight-node quadrilateral shell elements (QUAD8) and six-node triangular shell elements (TRIA6). The QUAD8 is an isoparametric-type element with a parabolic displacement field. The TRIA6 is a linear strain, linear curvature triangle that was used as a transition element when mesh sizes were changed. The computations were run utilizing all element stiffnesses, viz., membrane, bending and transverse shear. Since MSC/NASTRAN has superelement or substructuring capability, this option was also used and provided a significant reduction in computer central processing unit (CPU) time (e.g. 30 min versus 3 hr). A drawing of a cylinder model showing the element geometry and the superelement configuration is given in Fig. I. Two methods were used to calculate the stress intensity factor values. The first method was based on Sanders's energy release rate integral given by eqn (1). The second method was based
QUAD 8
TRIA 6
Fig. 1. Finite element model of cylinder.
Stress intensity factors for circumferential through cracks in hollow cylinders
567
on the compliance approach, in which the potential energy of the external forces is given by
u = 89G P~ai.
(20)
In eqn (20), Ai is the node displacement parallel to each Pi force applied at the end boundary nodes. For the first method, the integral of the displacements (i.e. the crack opening area) was calculated numerically using Simpson's rule. Twenty-four elements of unit length were used to model the half circumference at the plane of the crack. The calculations were made for unit crack extensions from zero to a maximum length of 20 elements (i.e. 0 <-- a -< 150~ The derivatives of the integral values were calculated by using a spline fit with a fourth-degree polynomial equation through five points. Double precision was used to calculate the polynomial coefficients. For the tension case, the value of I was then calculated at the midpoint of each spline fit with the equation E
I-
,9 ~, B,,a2,,.
troROan=l
(21)
The stress intensity factors were then obtained from
KI tr~
( I'~ '/2 = \2--~a]
"
(22)
The calculations for the strain energy release rate based on the compliance approach were also made using fourth-degree polynomial spline fits. The stress intensity factors for each fit were obtained from the equation
(E OU'~I/2 K,, = \ T z ~ a ] '
~
h
(23)
= ~=11.78 h
(a) C Y L I N D E R W I T H E N D R I N G S
(b) C Y L I N D E R W I T H O U T E N D R I N G S
Fig. 2. Geometry of cylinder with and without end rings.
R. G. F O R M A N
568
et al.
where 4
U = E
Cna2n.
(24)
n=l
T h e cylinder length was varied by simply adding or subtracting superelements. Since the application of uniform stress or pure bending on the boundaries of short cylinders with long cracks is not physically realistic, additional computer analysis was conducted for which the boundary forces were applied through a stiff ring. The drawing in Fig. 2 shows the cylinder geometry with the ring and without the ring. RESULTS AND DISCUSSION The initial objectives for the finite element analysis were to compare the stress intensity factor results determined by the two different methods given by eqns (22) and (23) and to compare these results with the analytic solution expressed by eqn (19). The numerical results of comparing the two methods for calculating K~ using finite element analysis are listed in Table 1. The comparison shows that the results were identical up to almost six significant figures. This confirms the high accuracy of the modeling and indicates that the energy release rate integral approach does not give greater accuracy than the compliance approach for this particular problem. For this reason, all subsequent analysis was conducted using the compliance method because it was simpler, especially for the bending problem. The comparison of the finite element results with curves for the analytic solution given by eqn (19) is shown in Fig. 3. The accuracy of the fits between the data points and the curves for both tension and bending verifies the accuracy of the closed form expression for crack angles as large as 135~ Since eqn (19) should be equally accurate in general for long cylinders governed by thin shell theory, no additional calculations were made to compare the results for different RIh ratios. Instead, finite element analysis showing the results of varying the cylinder length and boundary conditions were considered to have more practical value. When the cylinder length was shortened to approximately the cylinder circumference or less, the stress intensity correction factors became increasingly larger for the constant tension or pure bending boundary conditions. This behavior is similar to the theoretical case of a centercracked strip in which the length is relatively short and the applied boundary stress is assumed to be constant. For the alternative problem, however, when the strip is given a prescribed uniform extension at the boundary, the infinite long solution provides an upper bound for the correction factors. This same result is found to occur for the circumferentially cracked cylinder. To more easily study the end boundary effects, a thick ring was connected to the end of the cylinder. The ring could not be made orders of magnitude stiffer than the shell because the global stiffness matrix would become ill-conditioned. The final ring size, therefore, was selected by comparing displacement results for variable ring thicknesses, starting with the ring and shell having equal thickness. By using the same element length as for the shell and increasing the Table I. Comparison of methods for calculating stress intensity factors from finite element results
KdcroV'~
30 ~ 45 ~ 60 ~ 75 ~ 90 ~ 105~ 120~ 135~ 9
= 0.1,
Sander's Integral
Compliance
!.71923 2.24741 2.91232 3.84372 5.25590 7.57664 11.80560 20.79159
!.71923 2.24741 2.91232 3.84372 5.25590 7.57664 11.80561 20.79161
RIh = 30, LIR = 12.3
Stress intensity factors for circumferential through cracks in hollow cylinders
J/
20 "- FINITE ELEMENT RESULTS E ] F o (TENSION) O F B (BENDING) 15 -~
2.
m LL lO
L -
1.12
;30
-
d
=0.1
=
J
#
0
l 30
0
t 60
I 90
I 120
I 150
a, DEG Fig. 3. Comparisonof finite element results with closed-formsolution for stress intensity factors.
3
~=0.1 R/h = 30 I_/~R = 0.92 a= 150;
WITHOUT END R t
<1 1
f
0
-1
:
/
J~
N
~
~WITH
t
-
~ET
END RING
'
I
I
I
I
-0.5
0 y/R
0.5
1
Fig. 4. Comparison of end displacements for cylinders with and without end rings.
569
570
R . G . FORMAN et al.
....
WITH END RINGS - WITHOUT END RINGS
F=o= KI I o~ltJ.~R=
7 / 5.1
( =0.1
R/h = 30
~R 8
~2 u.
2.67 - - ~
90.29
L = 0.60 wR
0
I
I
I
45 ~
90*
135 ~
cr
Fig. 5. Dimensionless stress intensity factor versus crack angle for cylinder subjected to constant end tension.
- - - - - - WITH END RINGS WITHOUT END RINGS
Foo~p ~=0.1 R/h = 30
8
~R 2
ii
1.42"-~
L TrR
0
= 0.60
45 ~
-0.29
90='
135~
r162
Fig. 6. Dimensionless stress intensity factor versus crack angle for cylinder subjected to pure bending moment.
Stress intensity factors for circumferential through cracks in hollow cylinders
571
ring thickness to give a square cross section, the displacement results became very nearly linear. This geometry was then used for the remaining analysis. A comparison of the end displacements for the stiffened and unstiffened cylinder configurations is shown in Fig. 4. The final comparisons of the effect of the end conditions for different cylinder lengths are shown in Fig. 5 for tension and Fig. 6 for bending. The results show that forces applied through a stiff ring make the problem similar to that of a very long, unstiffened cylinder. At most, if the ring is relatively stiff, the infinite long cylinder without a ring gives an upper bound to the stress intensity factor. Equation (19), therefore, should give satisfactory results for most engineering applications. CONCLUSIONS Sanders's energy release rate integral for a circumferentially cracked cylindrical shell loaded in both tension and bending was used to derive the solution for the stress intensity factor. A curve fit approach proposed by Sanders for comb!ning long and short crack solutions in the tension case was extended by the present authors to the bending case. Finite element analysis showed that short cylinders with loads applied through ring-stiffened ends had nearly the same K1 values as infinite long cylinders. The analytic solution for infinite long cylinders is satisfactory, therefore, for most engineering calculations on short cylinders. REFERENCES [1] E. S. Folias, A circumferential crack in a pressurized cylindrical shell, Int. J. Fract. Mech. 3, 1-11 (1967). [2] J. Lyell Sanders, Jr., Circumferential through-cracks in cylindrical shells under tension, ASMEJ. Appl. Mech. 49, 103-107 (1982). [3] J. W. Nicholson and J. G. Simmonds, Sanders' energy-release rate integral for arbitrarily loaded shallow shells and its asymptotic evaluation for a cracked cylinder, ASME J. Appl. Mech. 47, 363-369 (1980). [4] R. S. Barsoum, R. W. Loomis and B. D. Stewart, Analysis of through cracks in cylindrical shells by the quarterpoint elements, Int. J. Fract. 15, 259-280 (1979). [5] J. L. Sanders, Jr., Circumferential through-crack in a cylindrical shell under combined bending and tension, ASME J. Appl. Mech. 50, 221 (1983). [6l MSC/NASTRAN, Handbook for Linear Static Analysis. The Macneal-Schwendler Corporation, Los Angeles, Califomia (Dec. 1981).