- Email: [email protected]
Surface flaw in a pressurised and thermally-shocked hollow cylinder
SURFACE FLAW IN A PRESSURISED A N D THERMALLYSHOCKED HOLLOW CYLINDER
A. S. KOBAYASHI,A. F. EMERY,N. POLVANICH& W. J. LOVE
University of Washington, Department of Mechanical Engineering, Seattle, Washington, USA (Received : 7 October, 1975)
ABSTRACT The first part of this paper deals with stress intensity factors of a semi-elliptical crack, with a third order polynomial pressure distribution in a finite thickness flat plate. These solutions are determined by the alternating technique in three-dimensional fracture mechanics where both the front and back surface effects of the flat plate are accounted for. The second part is concerned with the appfications of these solutions in deriving approximate solutions for a semi-elliptical crack in a pressurised cylinder and in a thermally-shocked cylinder. Curvature effects of the cylinder are estimated from two-dimensional finite element solutions of fix-gripped single-edge notched plates with prescribed crack pressure and single-edge notched cylinders with the same prescribed crack pressure. The stress intensity factors of a pressurised semi-elliptical crack in a flat plate are modified by this curvature correction factor to yield an estimate of the stress intensity factors of semi-elliptical cracks in a pressurised or thermally-shocked cylinder, with outer to inner diameter ratios ranging from 10:9 to 3:2 at crack depths of O.4 to 0.8.
INTRODUCTION A surface crack geometry commonly encountered in both the primary steam and the coolant system of a nuclear power plant is an internal, semi-elliptical crack in a cylinder subjected to pressure, as well as thermal loading. Due to the lack of an appropriate three-dimensional solution to the simplest problem of an internal surface crack in a pressurised straight pipe, however, the linear theory of fracture mechanics could not be used with confidence in the failure analysis of piping system. As a result, various approximate solutions have, in the past, been developed to meet the practical needs of design engineers. 103 Int. J. Pres. Ves. & Piping (5) (1977)--© Applied Science Publishers Ltd, England, 1976 Printed in Great Britain
104
A. S. KOBAYASHI, A. F. EMERY, N. POLVANICH, W. J. LOVE
A brief discussion on a two-dimensional solution--as well as on an approximate three-dimensional solution--to a simpler problem in fracture mechanics, i.e. a pressurised cylinder with an equally pressurised internal surface crack, was given by Underwood. 1 Another approximate solution for a surface cracked cylinder was obtained by Kobayashi 2 who used simple superposition of two known elliptical crack solutions subjected to constant and linearly varying pressures to estimate the stress intensity factors for semi-elliptical surface cracks in the internal and external surfaces of a pressurised cylinder. The former results were in qualitative agreement with those for pressurised cracks in pressurised cylinders obtained by Underwood. 1 Both Underwood's and Kobayashi's solutions were approximate in nature and neither considered deep cracks where the influence of back surface must be accounted for. Possibly the first three-dimensional finite element solution of a semi-elliptical crack in a pressurised cylinder was obtained by Marcal 3 who used a singular finite element with a l / r singularity. More recently, Hellen and Blackburn 4 and Ayres 5 used the 1/4-point, condensed isoparametric 3-D finite element with a l/(r) i singularity to analyse a semi-elliptical crack in a pressurised cylinder and in a thermally-shocked cylinder, respectively. The ease with which the I/4-point, condensed isoparametric 3-D finite element can be incorporated into the available 3-D finite element code has provided the practising engineer with a numerical tool with which 3-D problems in fracture mechanics can be readily analysed, as is evidenced by recent papers by Reynen 6 and Broekhoven. 7 Despite the availability of the 3-D finite element method in 3-D fracture mechanics, the high computer expenses involved in solving each problem have limited its use in parametric studies of problems involving semi-elliptical cracks. The writers feel that, at the present stage of development, the costly 3-D finite element method should be used to establish check points for approximate solutions which, in turn, will be used to generate the bulk of the design solutions in 3-D fracture mechanics involving semi-elliptical and quarter-elliptical cracks. The basis of the approximate solutions presented in this paper is a set of solutions for a semi-elliptical crack in a finite thickness plate and subjected to a polynomial pressure distribution. These solutions s- 10, are generated by the alternating method in 3-D fracture mechanics involving, in theory, no approximation and thus can also be used to check the accuracy of 3-D finite element analysis. The alternating technique, in its current state of development, however, cannot be used to solve surface crack problems in cylinders since one of the two solutions used in this alternating process is Love's half-space solution with a flat surface. 11'12 Comparable Love's solution for a cylindrical surface in an infinite solid or an infinite cylinder does not exist at present. In order to convert the above stress intensity factors of a semi-elliptical crack in flat plates to those in a curved cylinder, the former were corrected by curvature correction factors derived from twodimensional analogues of internally edge-cracked, pressurised cylinders and single
SURFACE F L A W IN A H O L L O W CYLINDER
105
edge-cracked plates with fixed-edge displacements and hoop stresses prescribed on the crack surface. Figure l shows schematically the logic behind this approximate procedure which combines the results of three-dimensional surface-cracked flat plates and the curvature corrections derived from two-dimensional analogues. The solution procedure used in this analysis thus consists of two major parts: solutions for a semi-elliptical crack in a flat plate and two-dimensional solutions for internal edge-cracks in cylinders and single edge-cracked plates. In the following, a brief description of the procedure is prescribed, followed by two applications involving semi-elliptical cracks in a pressurised cylinder and a thermally-shocked cylinder.
ELLIPTICAL C R A C K IN A FLAT PLATE
The alternating technique for solving three-dimensional problems in fracture mechanics is an iterative procedure which is well-documented in the papers by Shah s and Smith 9 and thus will not be repeated here. The numerical difficulties in computing the front-surface stress intensity magnification factor for a semielliptical crack, as described by Shah, s were reduced by prescribing appropriate fictitious pressure on the fictitious one-half part of the elliptical crack which protrudes into the free half space. 1a. 14 Such properly prescribed fictitious pressure enhanced the numerical convergence of the iterative procedures. The effectiveness of an appropriately prescribed fictitious pressure can be seen from the work of Shah ~5 and, more recently, that of Broekhoven, 16 who have used the fictitious pressure distribution to model directly the effects of a curved cylindrical surface which intersects the semi-elliptical crack. Although the computational simplicity of Shah and Broekhoven's approach is a definite advantage, such an approach was not used in the present paper since the boundary value problem, as illustrated in Fig. 1, must also incorporate the effects of the cylindrical front surface as well as the back surface. In addition to improving the numerical convergences of the three-dimensional alternating procedure, the efficiency of the numerical procedure was improved by partial replacement of the third order derivatives of the stress functions with numerical differentiation and judicious reduction of the number of repetitive computations in freeing the front and back surfaces. Thus, a workable numerical procedure, in which the stress intensity magnification factors of shallow to deep surface flaws in a structural member with free front and back surface could be computed, was developed. The pressure profile prescribed on the surface of a semi-elliptical crack in a flat plate is represented by the following third order polynomial: trzz(x,y ) = Boo + B o l ( 1
--
y/b) + B 0 2 ( 1 - - y/b) 2 + B o a ( 1 - y/b) 3
(1)
where the Cartesian co-ordinate system and the geometry of the semi-elliptical crack are given in Fig. 1. B~j's are the as-yet undetermined coefficients and b is the
106
A . S . KOBAYASHI, A. F. EMERY, N. POLVANICH, W. J. LOVE
K C YLINDER
iv, = "KcYLINDER ~. c KpLATE h : Ro-R i O'ZZ (y)= ° ' 0 8 ( r ~ _ ~ : ~
I
K
Kr/Ca2s,n2,.bos2,,,/4 . % ,/-e-
Fig. I.
1
Procedure for estimating stress intensity factor of an internal semi-elliptical crack in a thermally-shocked cylinder,
semi-minor diameter of the ellipse. This pressure profile is least square fitted to the hoop stress generated by the prescribed internal pressure or by internal loading on the uncracked cylindrical wall. The pressure distribution represented by eqn. (l) suggests that the stress intensity factors for crack pressure loadings of tx=z(x,y ) = 1, (1 - y / b ) , (1 - y / b ) 2 and (1 - y / b ) a are all that are needed to generate the solutions for the pressurised cylinder and thermally-shocked cylinder problems. As a result, stress intensity factors for a semi-elliptical crack in a finite thickness plate subjected to these four
SURFACE FLAW IN A HOLLOW CYLINDER
107
pressure loadings on the crack surface were obtained by the use of the threedimensional alternating technique. These solutions are listed by Emery et aL ~7 In addition, Kobayashi et a1.18 lists the stress intensity magnification factors for two elliptical cracks, b/a = 0.98 and 0'2 at a crack depth o f b / h = 0.8 in a plate of thickness, h, for four applied crack pressures of 1, (1 - y / b ) , (1 - y / b ) 2 and (1 - y/b) 3. Kobayashi et al. 19 list the corresponding stress intensity magnification factors for semi-elliptical cracks at crack depths ofb/h = 0.6 and 0.4. Five iterations in the alternating technique were executed for each loading condition, requiring approximately 1000,-~ 1500 sec of CPU time on a CDC 6400 computer. The residual surface tractions on the crack surface ranged from a high of 0.10 to an isolated region where the semi-elliptical crack periphery intersects the free front surface to 10-3 for b/h = 0.98 and tr~m = 1. Generally, the maximum residual tractions for most semi-elliptical crack problems were of the order of 0.02 at the isolated region where the crack periphery intersects the free front surface. The maximum residual surface tractions on the free front bounding surface were of the order of 10-2 and 10- 3 on the free back bounding surface for a uniform crack pressure of tr~z = 1. One-time eliminations of the isolated high residual surface tractions, following the procedure outlined by Kobayashi and Enetanya, 14 were then conducted. The relatively small values of residual surface tractions on the crack surfaces, however, did not produce a pronounced downward trend in the stress intensity magnification factors near the free front bounding surface, as reported by Kobayashi and Enetanya 14 and originally by Hartranft. 2o
TWO-DIMENSIONAL ANALOGUES
The flat plate approximation of an internally edge-cracked cylinder with a pressurised crack is a single edge-notched plate, which is restrained from end rotation, with a pressurised crack. These two sets of solutions were obtained by using conventional finite element programs. Figure 2 shows typical nodal breakdown for these two 2-D problems. This solution procedure differs from those described by Kobayashi et al. 18 and Kobayashi et al. 19 in that side restraints which model the side restraint imposed by the finite free surface boundary areas in the alternating technique are added to the fiat plate solution shown on the right-hand side of Fig. 2. Note also that particular care was taken to use the same element breakdown and element sizes in the cylindrical and fiat plate solutions in Fig. 2. Typically, the finite element analysis consists of 338 nodal points with 320 elements where a refined element spacing of 20 elements per cylinder thickness at an included angle spacing of 1 degree was used in the vicinity of the crack. Stress intensity factors were computed by using the more accurate but double-computation procedure based on the strain energy release rate method.21 The ratios of the stress intensity factors obtained from these two solutions yielded the curvature correction factor
108
A . S . KOBAYASHI, A. F. EMERY, N. POLVANICH, W. J. LOVE
~
,~.~
~
CRACKLENGT~
~---~"-. 7"-. NO. OF NODAL PTS = 32 ~ 7",,,.. ~ NO. OF ELEM : 31
~r-gmz:z:t:~ ~ / ~ ~. \ "x ~~b I : ~ / ~ ~ ~ CRACK• LENGTH,
c,.'-= (R°/rl2"
"~ \ ~ ' ~ \ \
o-=,. (-Ro~R'I2'' o
AXISOF'ELLIP ICAL CRACK
"F'~\ \ \ l
i (Ro/Ri)2 I
,,,=,NTERNAL PRESSORE
I/1/tl
NO. OF NODAL PTS. : 5 3 8
M R°- IRi' ~ [ ~ : ~ f / ~ . . . ~ .
_
L
~
LUS OF ELASTICITY =30x IO6
POISSONS RATIO=0.5
/7~
%
~,,-~ RO-Ri --- ~ Fig. 2.
Finite element b r e a k d o w n s o f a cylinder with a pressurised internal edge crack a n d o f an equivalent flat plate with a pressurised crack.
sought for converting the flat plate solution into that of a cylinder, as illustrated schematically in Fig. 1. THERMAL STRESSES 1N AN UNCRACKED CYLINDER SUDDENLY CHILLED AT ITS INSIDE SURFACE
For a hollow cylinder which is restrained against axial displacement, the hoop stress, ~ee, is given b y 22"
109
SURFACE FLAW IN A HOLLOW CYLINDER
0i
¥
g I l\\ I---
O|
~
,
\
\
\
\
~
,
,
,
t
--.
ooool
=
.00005
: . 0001
-.2
= .0005
I -.4
~
--.05
Fig. 3. Thermal stresses in a cylinder suddenly chilled at the internal surface.
--
~ . = r2 ~ o ~ - ~
.,
Tr dr +
I" .,
Tr dr -
Tr
)
(2)
where ~ is the linear coefficient of thermal expansion; E is the modulus of elasticity; R~ and Ro are the internal and external radii of the cylinder, respectively; T is the temperature and r is the radial distance from the cylinder axis. Emery 23 has shown that this expression is accurate to within 1/2~o for cylinders with thickness to external radius ratios smaller than one. The temperature distribution in a cylinder initially at To whose inner surface (r = Ri) is instantaneously reduced to T is thus given by24:
T - ,Too
,~
To -- Too = --re ~
where:
Jo(Riot,)Jo(Root,)Uo(rOt.) exp (--~c~2)n ' ~
-
-
~
In r/R i + Ro/R---~ In ~
(3)
1 lO
A . S . KOBAYASHI, A. F. EMERY, N. POLVANICH, W. J. LOVE
t
Z,
KpLATE I.d F,¢:I ..J n 3C nr td r,
1.4
FI
z
J >.
flll
h= Ro-R i
o~e ( r ) = O'zz(y)
2 ii f,j
z 0
1.2 /b =b.sinO
I.-
I.I
o~.~ bJ rr
1,0
I-n-" 0.9 0
.2
I
I
I
.4
.6
.8
I 1.0
A
CRACK LENGTH, b/(R o - R i )
Fig. 4. Curvature correction factor for Ro/R, between 5/4 and 10/9.
Uo(rOt.) = Jo(rO~.) Yo(Root.) - Jo(Root.) Yo(rOt.) J0 and Y0 are Bessel functions of the first and third kind, respectively c¢. are the roots of Uo(Riu.) = 0 (4) K is the thermal diffusivity Note that infinite surface heat transfer coefficients are not attainable in real systems and therefore the temperature response is slower than the values predicted by eqn. (3). However, the effect o f a delayed response only reduces the stress intensity factors 2 2 and does not change the temporal characteristics o f the fracture response. Figure 3 shows typical thermal stress distributions at several time intervals in a thermally-shocked cylinder o f R o / R i = 10/9, where the internal cylindrical surface is suddenly chilled. This situation models the emergency core cooling in reactor
SURFACE F L A W IN A H O L L O W CYLINDER
L"~
O'o8(r) = G.O(RO/r)2+l
(Ro/Ri)2+ I-
°'°p-~y
-'L/I
%= Pi
~- "L,...J/
(Ro/Ri)2+ I
(Ro/Ri )2- I
Pi = INTERNAL PRESSURE
-"
g
I 1l
+
b,o:o.
~
b/(Ro- Ri) =0.4
o
1~
.
.
.
i
I 20
.
.
Ro/ Ri
.
10/9
u.l
(,0
:~
0.9
0
i
I 40
i
[ 60
J
I 80
I
I
I00
CIRCULARANGLE,8DEGREES Fig. 5.
Stress intensity magnification factor of an internal semi-elliptical crack in a pressurised cylinder.
pressure vessels. At a normalised time of about r t / R ~ 2 = 0.05 the thermal stress distribution reaches a steady state condition for this particular vessel.
RESULTS
Me(0 ) Stress intensity factors for pressurised cylinders with outer to inner diameter ratios R o / R i = 10/9, 7/6, 5/4 and 3/2 and a two-dimensional, internal edge crack were determined previously by Kobayashi e t al. 1 a who used finite element analysis with nodal breakdown similar to that shown in the left-hand side of Fig. 2. These solutions were then used to derive the curvature correction factor, Me(n/2 ) corresponding to the deepest penetration, i.e. 0 -- 7r/2, of a semi-elliptical crack in a Curvature correction,
112
A.S. KOBAYASHI, A. F. EMERY, N. POLVANICH, W. J. LOVE ~z
~t .- "1.~/~ G° F" / ~_f ~ , j
er O -O
°-80(r)= Go (R°/r)2+1 (Ro/Ri)2+I y
GO= Pi
(Ro/Ri)Z+
I
(Ro/Ri)2- I Pi = INTERNAL PRESSURE
~
°-zzlY):
14 t
%
t.)
z 0
~
J.3
+
b/o = O. 2
b/(Ro-Ri)=0.6 --
u~
Z t~ t..9 o
Ro/R i = IO/9 = 7 /6
1.2
=5/4
Z
I la'l
b
=5/2
I.O
Or) II ~
0,9
1
O
I
20
I
J
40
I
I
60
I
I 80
I
I
IO0
C I R C U L A R A N G L E , 8 DEGREES
Fig. 6. Stress intensity magnification factor of an internal semi-elliptical crack in a pressurised cylinder. pressurised cylinder. 18 A recent comparison of this curvature correction factor with the single result from a 3-D finite element analysis 4 has led the authors not only to modify the curvature correction factor to account for the added side restraint mentioned previously but also to apply a variable curvature correction Me(O) along the entire semi-elliptical crack periphery. Figure 4 shows the curvature correction, Me(O), obtained by comparing the two stress intensity factors for an edge crack in a cylinder and in a flat plate with crack pressures of l, (1 - y / b ) , (l - ) ' / b ) 2 and (I - y / b ) 3. Although the individual curvature corrections for each of the above four pressures criss-crossed and differed at the most by l0 %, the approximate nature of curvature correction did not warrant separate corrections for each of the four pressure distributions. The crack depth, b, in Fig. 4 is the local crack depth along the periphery of the semi-elliptical crack. The modified curvature correction, Me(0), starts with no correction at the inner cylindrical surface or 0 = 0 and reaches its largest value at the deepest penetration
113
SURFACE FLAW IN A HOLLOW CYLINDER
|z o'88(r)--
~o/,.
y
2 o; (Ro/r) +1 o (Ro/Ri)2+I
(Ro/Ri)Z+ I % = Pi (Ro/Ri)Z_ I
L
Pi = INTERNAL PRESSURE O'zz(y) : obo (r)
n." ~
1.4
L~ e~
b/o =0.2
Ro/Ri
U
¢0 N ,7 .E
3/2
LLI ~ I--m m ~ 0.9
I
o
I
20
I
40
I
I
60
i
I
80
i
I
I00
CIRCULAR ANGLE, 0 DEGREES Fig. 7. Stress intensity magnification factor of an internal semi-elliptical crack in a pressurised cylinder. o f the elliptical crack surface at 0 = rr/2. Note also that the curvature correction is essentially unity for crack depths up to b/(R o - Ri) = 0.7, indicating that the effect o f the curved cylindrical surface is hardly noticed by the internal surface cracks up to these crack depths. In contrast, the curvature correction factor in reference 18 reached a m i n i m u m value o f about 0.9 ,,~ 0.8 in the vicinity o f 60 to 70 ~ crack penetration.
Surface flaw in a pressurised cylinder F o r a semi-elliptical crack in a pressurised cylinder, the crack pressure represented by eqn. (1) must be least square fitted to the normalised h o o p stress of:
114
A.S.
KOBAYASHI,
A . F. E M E R Y ,
N. POLVANICH,
W.
~'~
_
Z
_(2 v--
J. L O V E
o: ( R ° / r ) 2 + I
!//Y
(Ro/Ri)2*I Cro=Pi ('R'o/Ri)21 Pi: INTERNALPRESSURE
-~-~
~zz(Y) = o~8(r )
8u
oJ
-~ +
b / o = 0.98
I.J
to t~ U. .=-z ¢u~ o
I0
I--
0.9
~
~
~
Ro/Ri
Iz
b°l '''
09 ~,~b~°'8 U) bJ 0.7 0
t
I
20
I
I
40
J
I
60
=
I
80
t
I
I00
CIRCULARANGLE,8DEGREES Fig. 8. Stress intensity magnification factor of an internal semi-circular crack in a pressurised cylinder.
troo(X' y) = pi{Ro2/(Ri + y)2 + 1 }/{Ro2/Ri 2 - 1} (5) where Pl is the prescribed internal pressure for the cylindrical geometries of Ro/Ri = 10/9 to 3/2 considered in this paper. This least square fitting was accomplished within 5 % of the h o o p stresses represented by eqn. (5). The stress intensity factor for a semi-elliptical crack in a fiat plate can be computed by linearly superposing the various solutions listed in reference 17 using the coefficients Bo~ determined through the above least square fitting process. Figures 5 to 9 show the stress intensity magnification factor, Mxs, for a semielliptical crack in a fiat plate with a prescribed crack pressure equal to the h o o p stress in a pressurised cylinder. These magnification factors are normalised by the local stress intensity factor of a completely embedded elliptical crack. Thus, the
SURFACE FLAW IN A HOLLOW CYLINDER
l 13
~Z
°'es(r)
= %(Ro/t
)2+ I
(Ro/R i) + I (Ro/Ri)2+ I
°'o= "i CRo>R~)2.i E_..~ 0 0
N
~2
~
Pi = INTERNAL PRESSURE,
1.3
O'zz(y) = 0"88(r)
1.2
b/a = 0.98 .
=
,
RolRi = I019 =
716
=5/4
hi
H" =3/2
#
0.8
i
0
I
20
L
l
40
,.
I
60
L
I
80
J
I
I00
CIRCULAR ANGLE,0 DEGREES Fig. 9.
Stress intensity magnification factor of an internal semi-circular crack in a pressurised cylinder.
stress intensity factor in an oblong semi-elliptical crack with crack aspect ratio of
b/a = 0.2 reaches its maximum value at the deepest penetration, i.e. 0 = 90 °, for all crack depths. In a nearly semi-circular crack with crack aspect ratio of b/a = 0.98, however, the maximum stress intensity factor occurs near the free front surface, i.e. 0 = 0 ° for crack depths up to b/(Ro - Ri) = 0.8. The actual stress intensity magnification factor, Mr, for a semi-elliptical crack in a pressurised cylinder is obtained by: Mr = Mc(O)MKs(O) (6) Note that Mc is denoted as a function of 0 to indicate that it varies with the local crack depth, b, along the crack periphery, as shown in Fig. 4. Although the curvature correction, Me(O), is missing for the thick-walled cylinder of Ro/R~ = 3/2, the curve in Fig. 4 can be used as a rough approximation, par-
116
A . S . KOBAYASHI, A. F. EMERY, N. POLVAN1CH, W. J. LOVE
\ 1.4
T~\
TO
cr°~'~I Ro/Ri=i0/9 nO I--
Z 0 I-< u_
~-
/q /~I //I/'luI
1.2 ¸
i /
b/a =0.2
,,[,.J b/(Ro-Ri)=0.4
L ~
1.0
+ (:13 0.8 AT K t / R 2 = 0 . 0 5
W O~
O-o = °'O0]m*"
~
Z ll.l I-Z
(;
a E ( TO-To,=)
TIME, Kt/R~=
~.ooooJ II(/~ .0001 :00005
0.2
I
0
~
I
20
i
1
40
I
I
60
I
1
80
1
I
I00
CIRCULAR ANGLE, 8 DEGREES Fig. 10,
Stress i n t e n s i t y m a g n i f i c a t i o n
factor of an internal semi-elliptical shocked cylinder.
c r a c k in a t h e r m a l l y -
ticularly for a deeper crack, when b/(Ro - Ri) = 0-8. The previously-mentioned limited comparison between a finite element result ~ indicates that this rough approximation of curvature correction for a moderately deep crack, e.g. b/(Ro- Re) = 0.4 and 0.6, in such a thick-walled cylinder could be underestimated as much as 20 %. It is because o f this uncertainty in the accuracy o f the curvature correction for thick-walled cylinders that Figs. 5 onwards are plotted in terms of MKs instead of the resultant MK values. In the event that a more reliable Mc(O) is established, the MKs values, which will remain unchanged, can be used to obtain the final Mr values for various loading conditions on the cylinder.
117
SURFACE FLAW IN A HOLLOW CYLINDER
T~
TO
°"o[-~,B 'J Ro/IRi:10/9 /I /]
i",t/I
O
b/a :0.2 b/(Ro-R i)=0.6
0z ,.a 7-
+
Z
o4~
,2
AT
Kt/Ri2:0.05 % aE(To-T=,4,) TIME,Kt/R2= .00001
~
b°l L~
m ~. 0.2
.05
0:7 to
.0005 .00005
,,e
• 000 I
0
I
20
,
I
40
i
1
60
i
I
80
J
I I
I00
CIRCULAR ANGLE, 0 DEGREES Fig. 11.
Stress intensity magnification factor of an internal semi-elliptical crack in a thermallyshocked cylinder.
Surface flaw in a thermally-shocked cylinder The least square fitting of a third order polynomial equation (eqn. (1)) is particularly difficult in the regions of steep stress gradients. As a result appropriate weighting functions were used to reduce the influence of the steep stress gradient in determining the coefficients of B u to the thermal stress distributions shown in Fig. 3. The stress intensity magnification factors for each pressure distribution of Bo~(1 - y/b)j were then superposed to derive the stress intensity magnification, Mrs, shown in Figs. 10 to 14, for flat plates subjected to transient thermal stress distributions. These curves show the characteristic decrease in stress intensity magnification factors, Mrs, due to decreasing thermal stresses and front surface effect of the internal cylindrical surface. It is also interesting to note that the
] 18
h.s.
KOBAYASHI, A. F. EMERY, N. POLVANICH, W. J. LOVE
location of maximum stress intensity factor shifts from the vicinity of the inner cylindrical surface to the location of maximum crack penetration in the oblong crack, i.e. b/a = 0"2, while Fig. 12 shows that this location remains in the vicinity of the inner cylindrical surface for the nearly semi-circular crack, i.e. b/a = 0.98. Again, the resultant stress intensity magnification factor is obtained by eqn. (6), with the appropriate curvature correction, M~(O), from Fig. 4.
I /I R0/Ri =10/9 L b/a =0.98 C ~ /1" b/(Ro- Ri) =0.6 12t7 O -~"
'°[
O ¢J
Z ¢~ O .m I.-+
0.8
AT ~t/
¢,.) ea
17
.c
¢z~
o
05
O.E
i IVE
z
b°l Lu
I.LI CO
~
0
20
40
60
80
I00
CIRCULAR ANGLE, 8 DEGREES Fig. 12,
Stress intensity magnification factor of an internal semi-circular crack in a thermallyshocked cylinder.
1 19
SURFACE FLAW IN A HOLLOW CYLINDER
T
0.7
a;~,Ii1 Ro/R i =5/4 ' b/o = 0.2 // b/(Ro-Ri)=0.6 r-~
rr0
_~ o.6
Z 0
%
TO
s'
/
El,J,
I.0
U
o.5
+ od¢~
0.4
~>
O.B
N
8
AT K t / R ~ = O . 0 5 o': °'OO]m°x
o = a E (To_T~ ) TIME, KtlR 2=,
>-
.0001 z_ m m w t--m
~
0.2
•.O5
0 Fig. 13.
I,
I
[
I
I
I
~
I
20 40 60 80 CIRCULAR ANGLE ,8 DEGREES
~
I
I00
Stress intensity magnification factor of an internal semi-elliptical crack in a thermallyshocked cylinder.
CONCLUSIONS
(1) A procedure to estimate stress intensity magnification factors for semielliptical cracks in a pressurised and thermally-shocked cylinder was presented. (2) Using this procedure, stress intensity magnification factors for semi-elliptical cracks of b/a = 0.2 and 0.98 at a crack depth of b/h = 0-4, 0,6 and 0.8 in a
120
A. S. KOBAYASHI, A. F. EMERY, N. POLVANICH, W. J. LOVE
0.8
Cr°~l
Ro/Ri = 5 / 4
/I/t
0.7 I--
Z 0
=0.2
bla
//./~.J
b/(Ro-R i) = 0 . 4
, /
A
N ° "'
F-
+
u_
c:
0.5
Z (.9
°-° : a E (To-T,~:,) 2 T, ME .0001
hi F"
_z rJ)
hi
V
,~-i
.05
0.2
0.11 0
I
I
20
t
I
40
I
I
60
J
I
80
i
I
100
CIRCULAR ANGLE,8 DEGREES Fig. 14.
Stress intensity magnification factor o f an internal semi-elliptical crack in a thermallyshocked cylinder.
pressurised cylinder of Ro/Ri = 10/9, 7/6 and 5/4 were derived. In addition, estimates of corresponding M r in a thick-walled cylinder of Ro/R i = 3/2 were made. (3) Stress intensity magnification factors for the corresponding semi-elliptical cracks in a thermally-shocked cylinder were also obtained.
SURFACE FLAW IN A HOLLOW CYLINDER
121
ACKNOWLEDGEMENT Th e w o r k reported in this paper is sponsored by the Electric P o w e r Research Institute under C o n t r a c t N o . RP231-0-0. The authors wish to thank D r s C o n w a y C h a n and A. G o p a l a k r i s h n a n o f E P R I for t h ei r e n c o u r a g e m e n t t h r o u g h o u t the course o f this research p r o g r a m m e .
REFERENCES l. UNDERWOOD, J. H. Stress intensity factors for internally pressurized thick-wall cylinders, Stress Analysis and Growth of Crack, ASTM STP 513 (1972) pp. 59770. 2. KOBAYASHI,A. S. A simple procedure for estimating stress intensity factor in region of high stress gradient. In: Significance of defects in welded structures (ed. by T. Kanazawa and A. S. Kobayashi), University of Tokyo Press, 1974, pp. 127-43. 3. MARCAL, P. V. Three-dimensional finite element analysis for fracture mechanics. In: The surface crack: Physical problems and computational solutions (ed. by J. L. Swedlow), ASME, 1972, pp. 187-202. 4. HELLEN,T. K. and BLACKBURN,W. S. The calculation of stress intensity factors in two and three dimensions using finite elements. In: Computational fracture mechanics (ed. by E. F. Rybicki and S. E. Bengley), ASME, 1975. 5. AYRES,D. J. Three-dimensional elastic analysis of semi-elliptical surface cracks subjected to thermal shock. Ibid. Ioc. cir., pp. 133-43. 6. REYNEN,J. Analysis of cracked pressure vessel nozzels by finite element, Trans. of 3rd Int. Conf. on Reactor Technology, CECA, CEE, CEEA, Luxembourg, 1975. Paper GS/1. 7. BROEKHOVEN,M. J. G. Computation of stress intensity factors for nozzle corner cracks by various finite element procedures. Ibid., Ioc. cir. Paper G4/6. 8. SHAH, R. C. and KOBAYASHI,A. S. On the surface flaw problem. In: The surface crack: Physicalproblems and computational solutions (ed. by J. L. Swedlow), AS ME, 1972, pp. 79-124. 9. SMITH,F. W. The elastic analysis of the part-circular surface flaw problem by the alternating method. Ibid., loc. cir., pp. 125-52. 10. SMITH, F. W. and SORENSEN,D. R. Mixed mode stress intensity factors for semi-elliptical surface cracks. NASA CR-134684 prepared under NAS LeRC Grant NGL-06-002-063, June, 1974. 1I. LOVE,A. E. H. On stress produced in a semi-infinite solid by pressure on part of the boundary, Philosophical Transactions of the Royal Society, Series A, 288 (1929) pp. 387-93. 12. LOVE,A. E. H. A treatise on the mathematical theory of elasticity, Dover Publications, New York, 1944, pp. 241-5. 13. KOBAYASm,A. S., ENETANYA,A. N. and SHAH, R. C. Stress intensity factors of elliptical cracks. In: Prospects of fracture mechanics (ed. by G. C. Sih, H. C. van Elst and D. Brock), Noordhoff Int. Publishing, Leyden, The Netherlands, 1975, pp. 525-44. 14. KOaAYASm,A. S. and ENETANYA,A. N. Stress intensity factor of a corner crack. Mechanics of Crack Growth, ASTM STP 590, 1975, pp. 477-95. 15. SHAH, R. C. Stress intensity factors for through- and part-through cracks originating at fastener holes. Ibid., Ioc. cir., pp. 429-59. 16. BROEKHOVEN,M. J. G. Fatigue crack extension in nozzle junctions: Comparison of analytical approximations with experimental data, Trans. of 3rd Int. Conf. on Reactor Technology, CECA, CEE, CEEA, Luxembourg, 1975. Paper G4/7. 17. EMERY,A. F., KOBAYASm,A. S. LOVE,W. J. Theoretical investigation of reactor primary coolant pipe, welds and fittings. Electric Power Research Institute Research Project 231 Annual Report, June, 1975. 18. KOBAYASHI,A. S., POLVANICH,N., EMERY,A. F. and LOVE,W. J., Stress intensity factor of a surface crack in a pressurized cylinder. In: Computational fracture mechanics (ed. by E. F. Rybicki and S. E. Bengley), ASME, 1975, pp. 121-32. 19. KOBAYASHI,A. S., POLVANICH,N., EMERY,A. F. and LOVE,W. J. Surface flaw in a thermally shocked cylinder, Trans. of 3rd Int. Conf. on Structural Mechanics in Reactor Technology, CECA, CEE, CEEA, Luxembourg, 1975. Paper G4/3.
122
A.S. KOBAYASHI, A. F. EMERY, N. POLVANICH, W. J. LOVE
20. HARTRANFT,R. J. and Sin, G. C. Alternating method applied to edge and surface crack problems. In : Mechanics of fracture l--Methods of analysis and solutions of crack problems (ed. by G. S. Sih), Noordhoff Intl, 1973, pp. 179-238. 21. WATWOOD,V. B. The finite element method for prediction of crack behavior, Nuclear Engineering and Design, 11 (1969) pp. 323-32. 22. BOLEY, B. A. and WIENER, J. H. Theory of thermal stresses, John Wiley & Sons, New York, 1960. 23. EMERY, A. F., WALKER, G. F., JR. and WILLIAMS,J. A. A Green's function for the stress intensity factor and its applications to thermal stresses, J. of Basic Engineering, Trans. of ASME, Series D, 91 (Dccember, 1969) pp. 618-24. 24. CARSLAW, H. S. and JAEGER, J. C. Conduction of heat in solids, Oxford Press, Oxford, 1959.
A. S. KOBAYASHI,A. F. EMERY,N. POLVANICH& W. J. LOVE
University of Washington, Department of Mechanical Engineering, Seattle, Washington, USA (Received : 7 October, 1975)
ABSTRACT The first part of this paper deals with stress intensity factors of a semi-elliptical crack, with a third order polynomial pressure distribution in a finite thickness flat plate. These solutions are determined by the alternating technique in three-dimensional fracture mechanics where both the front and back surface effects of the flat plate are accounted for. The second part is concerned with the appfications of these solutions in deriving approximate solutions for a semi-elliptical crack in a pressurised cylinder and in a thermally-shocked cylinder. Curvature effects of the cylinder are estimated from two-dimensional finite element solutions of fix-gripped single-edge notched plates with prescribed crack pressure and single-edge notched cylinders with the same prescribed crack pressure. The stress intensity factors of a pressurised semi-elliptical crack in a flat plate are modified by this curvature correction factor to yield an estimate of the stress intensity factors of semi-elliptical cracks in a pressurised or thermally-shocked cylinder, with outer to inner diameter ratios ranging from 10:9 to 3:2 at crack depths of O.4 to 0.8.
INTRODUCTION A surface crack geometry commonly encountered in both the primary steam and the coolant system of a nuclear power plant is an internal, semi-elliptical crack in a cylinder subjected to pressure, as well as thermal loading. Due to the lack of an appropriate three-dimensional solution to the simplest problem of an internal surface crack in a pressurised straight pipe, however, the linear theory of fracture mechanics could not be used with confidence in the failure analysis of piping system. As a result, various approximate solutions have, in the past, been developed to meet the practical needs of design engineers. 103 Int. J. Pres. Ves. & Piping (5) (1977)--© Applied Science Publishers Ltd, England, 1976 Printed in Great Britain
104
A. S. KOBAYASHI, A. F. EMERY, N. POLVANICH, W. J. LOVE
A brief discussion on a two-dimensional solution--as well as on an approximate three-dimensional solution--to a simpler problem in fracture mechanics, i.e. a pressurised cylinder with an equally pressurised internal surface crack, was given by Underwood. 1 Another approximate solution for a surface cracked cylinder was obtained by Kobayashi 2 who used simple superposition of two known elliptical crack solutions subjected to constant and linearly varying pressures to estimate the stress intensity factors for semi-elliptical surface cracks in the internal and external surfaces of a pressurised cylinder. The former results were in qualitative agreement with those for pressurised cracks in pressurised cylinders obtained by Underwood. 1 Both Underwood's and Kobayashi's solutions were approximate in nature and neither considered deep cracks where the influence of back surface must be accounted for. Possibly the first three-dimensional finite element solution of a semi-elliptical crack in a pressurised cylinder was obtained by Marcal 3 who used a singular finite element with a l / r singularity. More recently, Hellen and Blackburn 4 and Ayres 5 used the 1/4-point, condensed isoparametric 3-D finite element with a l/(r) i singularity to analyse a semi-elliptical crack in a pressurised cylinder and in a thermally-shocked cylinder, respectively. The ease with which the I/4-point, condensed isoparametric 3-D finite element can be incorporated into the available 3-D finite element code has provided the practising engineer with a numerical tool with which 3-D problems in fracture mechanics can be readily analysed, as is evidenced by recent papers by Reynen 6 and Broekhoven. 7 Despite the availability of the 3-D finite element method in 3-D fracture mechanics, the high computer expenses involved in solving each problem have limited its use in parametric studies of problems involving semi-elliptical cracks. The writers feel that, at the present stage of development, the costly 3-D finite element method should be used to establish check points for approximate solutions which, in turn, will be used to generate the bulk of the design solutions in 3-D fracture mechanics involving semi-elliptical and quarter-elliptical cracks. The basis of the approximate solutions presented in this paper is a set of solutions for a semi-elliptical crack in a finite thickness plate and subjected to a polynomial pressure distribution. These solutions s- 10, are generated by the alternating method in 3-D fracture mechanics involving, in theory, no approximation and thus can also be used to check the accuracy of 3-D finite element analysis. The alternating technique, in its current state of development, however, cannot be used to solve surface crack problems in cylinders since one of the two solutions used in this alternating process is Love's half-space solution with a flat surface. 11'12 Comparable Love's solution for a cylindrical surface in an infinite solid or an infinite cylinder does not exist at present. In order to convert the above stress intensity factors of a semi-elliptical crack in flat plates to those in a curved cylinder, the former were corrected by curvature correction factors derived from twodimensional analogues of internally edge-cracked, pressurised cylinders and single
SURFACE F L A W IN A H O L L O W CYLINDER
105
edge-cracked plates with fixed-edge displacements and hoop stresses prescribed on the crack surface. Figure l shows schematically the logic behind this approximate procedure which combines the results of three-dimensional surface-cracked flat plates and the curvature corrections derived from two-dimensional analogues. The solution procedure used in this analysis thus consists of two major parts: solutions for a semi-elliptical crack in a flat plate and two-dimensional solutions for internal edge-cracks in cylinders and single edge-cracked plates. In the following, a brief description of the procedure is prescribed, followed by two applications involving semi-elliptical cracks in a pressurised cylinder and a thermally-shocked cylinder.
ELLIPTICAL C R A C K IN A FLAT PLATE
The alternating technique for solving three-dimensional problems in fracture mechanics is an iterative procedure which is well-documented in the papers by Shah s and Smith 9 and thus will not be repeated here. The numerical difficulties in computing the front-surface stress intensity magnification factor for a semielliptical crack, as described by Shah, s were reduced by prescribing appropriate fictitious pressure on the fictitious one-half part of the elliptical crack which protrudes into the free half space. 1a. 14 Such properly prescribed fictitious pressure enhanced the numerical convergence of the iterative procedures. The effectiveness of an appropriately prescribed fictitious pressure can be seen from the work of Shah ~5 and, more recently, that of Broekhoven, 16 who have used the fictitious pressure distribution to model directly the effects of a curved cylindrical surface which intersects the semi-elliptical crack. Although the computational simplicity of Shah and Broekhoven's approach is a definite advantage, such an approach was not used in the present paper since the boundary value problem, as illustrated in Fig. 1, must also incorporate the effects of the cylindrical front surface as well as the back surface. In addition to improving the numerical convergences of the three-dimensional alternating procedure, the efficiency of the numerical procedure was improved by partial replacement of the third order derivatives of the stress functions with numerical differentiation and judicious reduction of the number of repetitive computations in freeing the front and back surfaces. Thus, a workable numerical procedure, in which the stress intensity magnification factors of shallow to deep surface flaws in a structural member with free front and back surface could be computed, was developed. The pressure profile prescribed on the surface of a semi-elliptical crack in a flat plate is represented by the following third order polynomial: trzz(x,y ) = Boo + B o l ( 1
--
y/b) + B 0 2 ( 1 - - y/b) 2 + B o a ( 1 - y/b) 3
(1)
where the Cartesian co-ordinate system and the geometry of the semi-elliptical crack are given in Fig. 1. B~j's are the as-yet undetermined coefficients and b is the
106
A . S . KOBAYASHI, A. F. EMERY, N. POLVANICH, W. J. LOVE
K C YLINDER
iv, = "KcYLINDER ~. c KpLATE h : Ro-R i O'ZZ (y)= ° ' 0 8 ( r ~ _ ~ : ~
I
K
Kr/Ca2s,n2,.bos2,,,/4 . % ,/-e-
Fig. I.
1
Procedure for estimating stress intensity factor of an internal semi-elliptical crack in a thermally-shocked cylinder,
semi-minor diameter of the ellipse. This pressure profile is least square fitted to the hoop stress generated by the prescribed internal pressure or by internal loading on the uncracked cylindrical wall. The pressure distribution represented by eqn. (l) suggests that the stress intensity factors for crack pressure loadings of tx=z(x,y ) = 1, (1 - y / b ) , (1 - y / b ) 2 and (1 - y / b ) a are all that are needed to generate the solutions for the pressurised cylinder and thermally-shocked cylinder problems. As a result, stress intensity factors for a semi-elliptical crack in a finite thickness plate subjected to these four
SURFACE FLAW IN A HOLLOW CYLINDER
107
pressure loadings on the crack surface were obtained by the use of the threedimensional alternating technique. These solutions are listed by Emery et aL ~7 In addition, Kobayashi et a1.18 lists the stress intensity magnification factors for two elliptical cracks, b/a = 0.98 and 0'2 at a crack depth o f b / h = 0.8 in a plate of thickness, h, for four applied crack pressures of 1, (1 - y / b ) , (1 - y / b ) 2 and (1 - y/b) 3. Kobayashi et al. 19 list the corresponding stress intensity magnification factors for semi-elliptical cracks at crack depths ofb/h = 0.6 and 0.4. Five iterations in the alternating technique were executed for each loading condition, requiring approximately 1000,-~ 1500 sec of CPU time on a CDC 6400 computer. The residual surface tractions on the crack surface ranged from a high of 0.10 to an isolated region where the semi-elliptical crack periphery intersects the free front surface to 10-3 for b/h = 0.98 and tr~m = 1. Generally, the maximum residual tractions for most semi-elliptical crack problems were of the order of 0.02 at the isolated region where the crack periphery intersects the free front surface. The maximum residual surface tractions on the free front bounding surface were of the order of 10-2 and 10- 3 on the free back bounding surface for a uniform crack pressure of tr~z = 1. One-time eliminations of the isolated high residual surface tractions, following the procedure outlined by Kobayashi and Enetanya, 14 were then conducted. The relatively small values of residual surface tractions on the crack surfaces, however, did not produce a pronounced downward trend in the stress intensity magnification factors near the free front bounding surface, as reported by Kobayashi and Enetanya 14 and originally by Hartranft. 2o
TWO-DIMENSIONAL ANALOGUES
The flat plate approximation of an internally edge-cracked cylinder with a pressurised crack is a single edge-notched plate, which is restrained from end rotation, with a pressurised crack. These two sets of solutions were obtained by using conventional finite element programs. Figure 2 shows typical nodal breakdown for these two 2-D problems. This solution procedure differs from those described by Kobayashi et al. 18 and Kobayashi et al. 19 in that side restraints which model the side restraint imposed by the finite free surface boundary areas in the alternating technique are added to the fiat plate solution shown on the right-hand side of Fig. 2. Note also that particular care was taken to use the same element breakdown and element sizes in the cylindrical and fiat plate solutions in Fig. 2. Typically, the finite element analysis consists of 338 nodal points with 320 elements where a refined element spacing of 20 elements per cylinder thickness at an included angle spacing of 1 degree was used in the vicinity of the crack. Stress intensity factors were computed by using the more accurate but double-computation procedure based on the strain energy release rate method.21 The ratios of the stress intensity factors obtained from these two solutions yielded the curvature correction factor
108
A . S . KOBAYASHI, A. F. EMERY, N. POLVANICH, W. J. LOVE
~
,~.~
~
CRACKLENGT~
~---~"-. 7"-. NO. OF NODAL PTS = 32 ~ 7",,,.. ~ NO. OF ELEM : 31
~r-gmz:z:t:~ ~ / ~ ~. \ "x ~~b I : ~ / ~ ~ ~ CRACK• LENGTH,
c,.'-= (R°/rl2"
"~ \ ~ ' ~ \ \
o-=,. (-Ro~R'I2'' o
AXISOF'ELLIP ICAL CRACK
"F'~\ \ \ l
i (Ro/Ri)2 I
,,,=,NTERNAL PRESSORE
I/1/tl
NO. OF NODAL PTS. : 5 3 8
M R°- IRi' ~ [ ~ : ~ f / ~ . . . ~ .
_
L
~
LUS OF ELASTICITY =30x IO6
POISSONS RATIO=0.5
/7~
%
~,,-~ RO-Ri --- ~ Fig. 2.
Finite element b r e a k d o w n s o f a cylinder with a pressurised internal edge crack a n d o f an equivalent flat plate with a pressurised crack.
sought for converting the flat plate solution into that of a cylinder, as illustrated schematically in Fig. 1. THERMAL STRESSES 1N AN UNCRACKED CYLINDER SUDDENLY CHILLED AT ITS INSIDE SURFACE
For a hollow cylinder which is restrained against axial displacement, the hoop stress, ~ee, is given b y 22"
109
SURFACE FLAW IN A HOLLOW CYLINDER
0i
¥
g I l\\ I---
O|
~
,
\
\
\
\
~
,
,
,
t
--.
ooool
=
.00005
: . 0001
-.2
= .0005
I -.4
~
--.05
Fig. 3. Thermal stresses in a cylinder suddenly chilled at the internal surface.
--
~ . = r2 ~ o ~ - ~
.,
Tr dr +
I" .,
Tr dr -
Tr
)
(2)
where ~ is the linear coefficient of thermal expansion; E is the modulus of elasticity; R~ and Ro are the internal and external radii of the cylinder, respectively; T is the temperature and r is the radial distance from the cylinder axis. Emery 23 has shown that this expression is accurate to within 1/2~o for cylinders with thickness to external radius ratios smaller than one. The temperature distribution in a cylinder initially at To whose inner surface (r = Ri) is instantaneously reduced to T is thus given by24:
T - ,Too
,~
To -- Too = --re ~
where:
Jo(Riot,)Jo(Root,)Uo(rOt.) exp (--~c~2)n ' ~
-
-
~
In r/R i + Ro/R---~ In ~
(3)
1 lO
A . S . KOBAYASHI, A. F. EMERY, N. POLVANICH, W. J. LOVE
t
Z,
KpLATE I.d F,¢:I ..J n 3C nr td r,
1.4
FI
z
J >.
flll
h= Ro-R i
o~e ( r ) = O'zz(y)
2 ii f,j
z 0
1.2 /b =b.sinO
I.-
I.I
o~.~ bJ rr
1,0
I-n-" 0.9 0
.2
I
I
I
.4
.6
.8
I 1.0
A
CRACK LENGTH, b/(R o - R i )
Fig. 4. Curvature correction factor for Ro/R, between 5/4 and 10/9.
Uo(rOt.) = Jo(rO~.) Yo(Root.) - Jo(Root.) Yo(rOt.) J0 and Y0 are Bessel functions of the first and third kind, respectively c¢. are the roots of Uo(Riu.) = 0 (4) K is the thermal diffusivity Note that infinite surface heat transfer coefficients are not attainable in real systems and therefore the temperature response is slower than the values predicted by eqn. (3). However, the effect o f a delayed response only reduces the stress intensity factors 2 2 and does not change the temporal characteristics o f the fracture response. Figure 3 shows typical thermal stress distributions at several time intervals in a thermally-shocked cylinder o f R o / R i = 10/9, where the internal cylindrical surface is suddenly chilled. This situation models the emergency core cooling in reactor
SURFACE F L A W IN A H O L L O W CYLINDER
L"~
O'o8(r) = G.O(RO/r)2+l
(Ro/Ri)2+ I-
°'°p-~y
-'L/I
%= Pi
~- "L,...J/
(Ro/Ri)2+ I
(Ro/Ri )2- I
Pi = INTERNAL PRESSURE
-"
g
I 1l
+
b,o:o.
~
b/(Ro- Ri) =0.4
o
1~
.
.
.
i
I 20
.
.
Ro/ Ri
.
10/9
u.l
(,0
:~
0.9
0
i
I 40
i
[ 60
J
I 80
I
I
I00
CIRCULARANGLE,8DEGREES Fig. 5.
Stress intensity magnification factor of an internal semi-elliptical crack in a pressurised cylinder.
pressure vessels. At a normalised time of about r t / R ~ 2 = 0.05 the thermal stress distribution reaches a steady state condition for this particular vessel.
RESULTS
Me(0 ) Stress intensity factors for pressurised cylinders with outer to inner diameter ratios R o / R i = 10/9, 7/6, 5/4 and 3/2 and a two-dimensional, internal edge crack were determined previously by Kobayashi e t al. 1 a who used finite element analysis with nodal breakdown similar to that shown in the left-hand side of Fig. 2. These solutions were then used to derive the curvature correction factor, Me(n/2 ) corresponding to the deepest penetration, i.e. 0 -- 7r/2, of a semi-elliptical crack in a Curvature correction,
112
A.S. KOBAYASHI, A. F. EMERY, N. POLVANICH, W. J. LOVE ~z
~t .- "1.~/~ G° F" / ~_f ~ , j
er O -O
°-80(r)= Go (R°/r)2+1 (Ro/Ri)2+I y
GO= Pi
(Ro/Ri)Z+
I
(Ro/Ri)2- I Pi = INTERNAL PRESSURE
~
°-zzlY):
14 t
%
t.)
z 0
~
J.3
+
b/o = O. 2
b/(Ro-Ri)=0.6 --
u~
Z t~ t..9 o
Ro/R i = IO/9 = 7 /6
1.2
=5/4
Z
I la'l
b
=5/2
I.O
Or) II ~
0,9
1
O
I
20
I
J
40
I
I
60
I
I 80
I
I
IO0
C I R C U L A R A N G L E , 8 DEGREES
Fig. 6. Stress intensity magnification factor of an internal semi-elliptical crack in a pressurised cylinder. pressurised cylinder. 18 A recent comparison of this curvature correction factor with the single result from a 3-D finite element analysis 4 has led the authors not only to modify the curvature correction factor to account for the added side restraint mentioned previously but also to apply a variable curvature correction Me(O) along the entire semi-elliptical crack periphery. Figure 4 shows the curvature correction, Me(O), obtained by comparing the two stress intensity factors for an edge crack in a cylinder and in a flat plate with crack pressures of l, (1 - y / b ) , (l - ) ' / b ) 2 and (I - y / b ) 3. Although the individual curvature corrections for each of the above four pressures criss-crossed and differed at the most by l0 %, the approximate nature of curvature correction did not warrant separate corrections for each of the four pressure distributions. The crack depth, b, in Fig. 4 is the local crack depth along the periphery of the semi-elliptical crack. The modified curvature correction, Me(0), starts with no correction at the inner cylindrical surface or 0 = 0 and reaches its largest value at the deepest penetration
113
SURFACE FLAW IN A HOLLOW CYLINDER
|z o'88(r)--
~o/,.
y
2 o; (Ro/r) +1 o (Ro/Ri)2+I
(Ro/Ri)Z+ I % = Pi (Ro/Ri)Z_ I
L
Pi = INTERNAL PRESSURE O'zz(y) : obo (r)
n." ~
1.4
L~ e~
b/o =0.2
Ro/Ri
U
¢0 N ,7 .E
3/2
LLI ~ I--m m ~ 0.9
I
o
I
20
I
40
I
I
60
i
I
80
i
I
I00
CIRCULAR ANGLE, 0 DEGREES Fig. 7. Stress intensity magnification factor of an internal semi-elliptical crack in a pressurised cylinder. o f the elliptical crack surface at 0 = rr/2. Note also that the curvature correction is essentially unity for crack depths up to b/(R o - Ri) = 0.7, indicating that the effect o f the curved cylindrical surface is hardly noticed by the internal surface cracks up to these crack depths. In contrast, the curvature correction factor in reference 18 reached a m i n i m u m value o f about 0.9 ,,~ 0.8 in the vicinity o f 60 to 70 ~ crack penetration.
Surface flaw in a pressurised cylinder F o r a semi-elliptical crack in a pressurised cylinder, the crack pressure represented by eqn. (1) must be least square fitted to the normalised h o o p stress of:
114
A.S.
KOBAYASHI,
A . F. E M E R Y ,
N. POLVANICH,
W.
~'~
_
Z
_(2 v--
J. L O V E
o: ( R ° / r ) 2 + I
!//Y
(Ro/Ri)2*I Cro=Pi ('R'o/Ri)21 Pi: INTERNALPRESSURE
-~-~
~zz(Y) = o~8(r )
8u
oJ
-~ +
b / o = 0.98
I.J
to t~ U. .=-z ¢u~ o
I0
I--
0.9
~
~
~
Ro/Ri
Iz
b°l '''
09 ~,~b~°'8 U) bJ 0.7 0
t
I
20
I
I
40
J
I
60
=
I
80
t
I
I00
CIRCULARANGLE,8DEGREES Fig. 8. Stress intensity magnification factor of an internal semi-circular crack in a pressurised cylinder.
troo(X' y) = pi{Ro2/(Ri + y)2 + 1 }/{Ro2/Ri 2 - 1} (5) where Pl is the prescribed internal pressure for the cylindrical geometries of Ro/Ri = 10/9 to 3/2 considered in this paper. This least square fitting was accomplished within 5 % of the h o o p stresses represented by eqn. (5). The stress intensity factor for a semi-elliptical crack in a fiat plate can be computed by linearly superposing the various solutions listed in reference 17 using the coefficients Bo~ determined through the above least square fitting process. Figures 5 to 9 show the stress intensity magnification factor, Mxs, for a semielliptical crack in a fiat plate with a prescribed crack pressure equal to the h o o p stress in a pressurised cylinder. These magnification factors are normalised by the local stress intensity factor of a completely embedded elliptical crack. Thus, the
SURFACE FLAW IN A HOLLOW CYLINDER
l 13
~Z
°'es(r)
= %(Ro/t
)2+ I
(Ro/R i) + I (Ro/Ri)2+ I
°'o= "i CRo>R~)2.i E_..~ 0 0
N
~2
~
Pi = INTERNAL PRESSURE,
1.3
O'zz(y) = 0"88(r)
1.2
b/a = 0.98 .
=
,
RolRi = I019 =
716
=5/4
hi
H" =3/2
#
0.8
i
0
I
20
L
l
40
,.
I
60
L
I
80
J
I
I00
CIRCULAR ANGLE,0 DEGREES Fig. 9.
Stress intensity magnification factor of an internal semi-circular crack in a pressurised cylinder.
stress intensity factor in an oblong semi-elliptical crack with crack aspect ratio of
b/a = 0.2 reaches its maximum value at the deepest penetration, i.e. 0 = 90 °, for all crack depths. In a nearly semi-circular crack with crack aspect ratio of b/a = 0.98, however, the maximum stress intensity factor occurs near the free front surface, i.e. 0 = 0 ° for crack depths up to b/(Ro - Ri) = 0.8. The actual stress intensity magnification factor, Mr, for a semi-elliptical crack in a pressurised cylinder is obtained by: Mr = Mc(O)MKs(O) (6) Note that Mc is denoted as a function of 0 to indicate that it varies with the local crack depth, b, along the crack periphery, as shown in Fig. 4. Although the curvature correction, Me(O), is missing for the thick-walled cylinder of Ro/R~ = 3/2, the curve in Fig. 4 can be used as a rough approximation, par-
116
A . S . KOBAYASHI, A. F. EMERY, N. POLVAN1CH, W. J. LOVE
\ 1.4
T~\
TO
cr°~'~I Ro/Ri=i0/9 nO I--
Z 0 I-< u_
~-
/q /~I //I/'luI
1.2 ¸
i /
b/a =0.2
,,[,.J b/(Ro-Ri)=0.4
L ~
1.0
+ (:13 0.8 AT K t / R 2 = 0 . 0 5
W O~
O-o = °'O0]m*"
~
Z ll.l I-Z
(;
a E ( TO-To,=)
TIME, Kt/R~=
~.ooooJ II(/~ .0001 :00005
0.2
I
0
~
I
20
i
1
40
I
I
60
I
1
80
1
I
I00
CIRCULAR ANGLE, 8 DEGREES Fig. 10,
Stress i n t e n s i t y m a g n i f i c a t i o n
factor of an internal semi-elliptical shocked cylinder.
c r a c k in a t h e r m a l l y -
ticularly for a deeper crack, when b/(Ro - Ri) = 0-8. The previously-mentioned limited comparison between a finite element result ~ indicates that this rough approximation of curvature correction for a moderately deep crack, e.g. b/(Ro- Re) = 0.4 and 0.6, in such a thick-walled cylinder could be underestimated as much as 20 %. It is because o f this uncertainty in the accuracy o f the curvature correction for thick-walled cylinders that Figs. 5 onwards are plotted in terms of MKs instead of the resultant MK values. In the event that a more reliable Mc(O) is established, the MKs values, which will remain unchanged, can be used to obtain the final Mr values for various loading conditions on the cylinder.
117
SURFACE FLAW IN A HOLLOW CYLINDER
T~
TO
°"o[-~,B 'J Ro/IRi:10/9 /I /]
i",t/I
O
b/a :0.2 b/(Ro-R i)=0.6
0z ,.a 7-
+
Z
o4~
,2
AT
Kt/Ri2:0.05 % aE(To-T=,4,) TIME,Kt/R2= .00001
~
b°l L~
m ~. 0.2
.05
0:7 to
.0005 .00005
,,e
• 000 I
0
I
20
,
I
40
i
1
60
i
I
80
J
I I
I00
CIRCULAR ANGLE, 0 DEGREES Fig. 11.
Stress intensity magnification factor of an internal semi-elliptical crack in a thermallyshocked cylinder.
Surface flaw in a thermally-shocked cylinder The least square fitting of a third order polynomial equation (eqn. (1)) is particularly difficult in the regions of steep stress gradients. As a result appropriate weighting functions were used to reduce the influence of the steep stress gradient in determining the coefficients of B u to the thermal stress distributions shown in Fig. 3. The stress intensity magnification factors for each pressure distribution of Bo~(1 - y/b)j were then superposed to derive the stress intensity magnification, Mrs, shown in Figs. 10 to 14, for flat plates subjected to transient thermal stress distributions. These curves show the characteristic decrease in stress intensity magnification factors, Mrs, due to decreasing thermal stresses and front surface effect of the internal cylindrical surface. It is also interesting to note that the
] 18
h.s.
KOBAYASHI, A. F. EMERY, N. POLVANICH, W. J. LOVE
location of maximum stress intensity factor shifts from the vicinity of the inner cylindrical surface to the location of maximum crack penetration in the oblong crack, i.e. b/a = 0"2, while Fig. 12 shows that this location remains in the vicinity of the inner cylindrical surface for the nearly semi-circular crack, i.e. b/a = 0.98. Again, the resultant stress intensity magnification factor is obtained by eqn. (6), with the appropriate curvature correction, M~(O), from Fig. 4.
I /I R0/Ri =10/9 L b/a =0.98 C ~ /1" b/(Ro- Ri) =0.6 12t7 O -~"
'°[
O ¢J
Z ¢~ O .m I.-+
0.8
AT ~t/
¢,.) ea
17
.c
¢z~
o
05
O.E
i IVE
z
b°l Lu
I.LI CO
~
0
20
40
60
80
I00
CIRCULAR ANGLE, 8 DEGREES Fig. 12,
Stress intensity magnification factor of an internal semi-circular crack in a thermallyshocked cylinder.
1 19
SURFACE FLAW IN A HOLLOW CYLINDER
T
0.7
a;~,Ii1 Ro/R i =5/4 ' b/o = 0.2 // b/(Ro-Ri)=0.6 r-~
rr0
_~ o.6
Z 0
%
TO
s'
/
El,J,
I.0
U
o.5
+ od¢~
0.4
~>
O.B
N
8
AT K t / R ~ = O . 0 5 o': °'OO]m°x
o = a E (To_T~ ) TIME, KtlR 2=,
>-
.0001 z_ m m w t--m
~
0.2
•.O5
0 Fig. 13.
I,
I
[
I
I
I
~
I
20 40 60 80 CIRCULAR ANGLE ,8 DEGREES
~
I
I00
Stress intensity magnification factor of an internal semi-elliptical crack in a thermallyshocked cylinder.
CONCLUSIONS
(1) A procedure to estimate stress intensity magnification factors for semielliptical cracks in a pressurised and thermally-shocked cylinder was presented. (2) Using this procedure, stress intensity magnification factors for semi-elliptical cracks of b/a = 0.2 and 0.98 at a crack depth of b/h = 0-4, 0,6 and 0.8 in a
120
A. S. KOBAYASHI, A. F. EMERY, N. POLVANICH, W. J. LOVE
0.8
Cr°~l
Ro/Ri = 5 / 4
/I/t
0.7 I--
Z 0
=0.2
bla
//./~.J
b/(Ro-R i) = 0 . 4
, /
A
N ° "'
F-
+
u_
c:
0.5
Z (.9
°-° : a E (To-T,~:,) 2 T, ME .0001
hi F"
_z rJ)
hi
V
,~-i
.05
0.2
0.11 0
I
I
20
t
I
40
I
I
60
J
I
80
i
I
100
CIRCULAR ANGLE,8 DEGREES Fig. 14.
Stress intensity magnification factor o f an internal semi-elliptical crack in a thermallyshocked cylinder.
pressurised cylinder of Ro/Ri = 10/9, 7/6 and 5/4 were derived. In addition, estimates of corresponding M r in a thick-walled cylinder of Ro/R i = 3/2 were made. (3) Stress intensity magnification factors for the corresponding semi-elliptical cracks in a thermally-shocked cylinder were also obtained.
SURFACE FLAW IN A HOLLOW CYLINDER
121
ACKNOWLEDGEMENT Th e w o r k reported in this paper is sponsored by the Electric P o w e r Research Institute under C o n t r a c t N o . RP231-0-0. The authors wish to thank D r s C o n w a y C h a n and A. G o p a l a k r i s h n a n o f E P R I for t h ei r e n c o u r a g e m e n t t h r o u g h o u t the course o f this research p r o g r a m m e .
REFERENCES l. UNDERWOOD, J. H. Stress intensity factors for internally pressurized thick-wall cylinders, Stress Analysis and Growth of Crack, ASTM STP 513 (1972) pp. 59770. 2. KOBAYASHI,A. S. A simple procedure for estimating stress intensity factor in region of high stress gradient. In: Significance of defects in welded structures (ed. by T. Kanazawa and A. S. Kobayashi), University of Tokyo Press, 1974, pp. 127-43. 3. MARCAL, P. V. Three-dimensional finite element analysis for fracture mechanics. In: The surface crack: Physical problems and computational solutions (ed. by J. L. Swedlow), ASME, 1972, pp. 187-202. 4. HELLEN,T. K. and BLACKBURN,W. S. The calculation of stress intensity factors in two and three dimensions using finite elements. In: Computational fracture mechanics (ed. by E. F. Rybicki and S. E. Bengley), ASME, 1975. 5. AYRES,D. J. Three-dimensional elastic analysis of semi-elliptical surface cracks subjected to thermal shock. Ibid. Ioc. cir., pp. 133-43. 6. REYNEN,J. Analysis of cracked pressure vessel nozzels by finite element, Trans. of 3rd Int. Conf. on Reactor Technology, CECA, CEE, CEEA, Luxembourg, 1975. Paper GS/1. 7. BROEKHOVEN,M. J. G. Computation of stress intensity factors for nozzle corner cracks by various finite element procedures. Ibid., Ioc. cir. Paper G4/6. 8. SHAH, R. C. and KOBAYASHI,A. S. On the surface flaw problem. In: The surface crack: Physicalproblems and computational solutions (ed. by J. L. Swedlow), AS ME, 1972, pp. 79-124. 9. SMITH,F. W. The elastic analysis of the part-circular surface flaw problem by the alternating method. Ibid., loc. cir., pp. 125-52. 10. SMITH, F. W. and SORENSEN,D. R. Mixed mode stress intensity factors for semi-elliptical surface cracks. NASA CR-134684 prepared under NAS LeRC Grant NGL-06-002-063, June, 1974. 1I. LOVE,A. E. H. On stress produced in a semi-infinite solid by pressure on part of the boundary, Philosophical Transactions of the Royal Society, Series A, 288 (1929) pp. 387-93. 12. LOVE,A. E. H. A treatise on the mathematical theory of elasticity, Dover Publications, New York, 1944, pp. 241-5. 13. KOBAYASm,A. S., ENETANYA,A. N. and SHAH, R. C. Stress intensity factors of elliptical cracks. In: Prospects of fracture mechanics (ed. by G. C. Sih, H. C. van Elst and D. Brock), Noordhoff Int. Publishing, Leyden, The Netherlands, 1975, pp. 525-44. 14. KOaAYASm,A. S. and ENETANYA,A. N. Stress intensity factor of a corner crack. Mechanics of Crack Growth, ASTM STP 590, 1975, pp. 477-95. 15. SHAH, R. C. Stress intensity factors for through- and part-through cracks originating at fastener holes. Ibid., Ioc. cir., pp. 429-59. 16. BROEKHOVEN,M. J. G. Fatigue crack extension in nozzle junctions: Comparison of analytical approximations with experimental data, Trans. of 3rd Int. Conf. on Reactor Technology, CECA, CEE, CEEA, Luxembourg, 1975. Paper G4/7. 17. EMERY,A. F., KOBAYASm,A. S. LOVE,W. J. Theoretical investigation of reactor primary coolant pipe, welds and fittings. Electric Power Research Institute Research Project 231 Annual Report, June, 1975. 18. KOBAYASHI,A. S., POLVANICH,N., EMERY,A. F. and LOVE,W. J., Stress intensity factor of a surface crack in a pressurized cylinder. In: Computational fracture mechanics (ed. by E. F. Rybicki and S. E. Bengley), ASME, 1975, pp. 121-32. 19. KOBAYASHI,A. S., POLVANICH,N., EMERY,A. F. and LOVE,W. J. Surface flaw in a thermally shocked cylinder, Trans. of 3rd Int. Conf. on Structural Mechanics in Reactor Technology, CECA, CEE, CEEA, Luxembourg, 1975. Paper G4/3.
122
A.S. KOBAYASHI, A. F. EMERY, N. POLVANICH, W. J. LOVE
20. HARTRANFT,R. J. and Sin, G. C. Alternating method applied to edge and surface crack problems. In : Mechanics of fracture l--Methods of analysis and solutions of crack problems (ed. by G. S. Sih), Noordhoff Intl, 1973, pp. 179-238. 21. WATWOOD,V. B. The finite element method for prediction of crack behavior, Nuclear Engineering and Design, 11 (1969) pp. 323-32. 22. BOLEY, B. A. and WIENER, J. H. Theory of thermal stresses, John Wiley & Sons, New York, 1960. 23. EMERY, A. F., WALKER, G. F., JR. and WILLIAMS,J. A. A Green's function for the stress intensity factor and its applications to thermal stresses, J. of Basic Engineering, Trans. of ASME, Series D, 91 (Dccember, 1969) pp. 618-24. 24. CARSLAW, H. S. and JAEGER, J. C. Conduction of heat in solids, Oxford Press, Oxford, 1959.