ML20027E525
| ML20027E525 | |
| Person / Time | |
|---|---|
| Site: | La Crosse File:Dairyland Power Cooperative icon.png |
| Issue date: | 11/08/1982 |
| From: | Paris P, Tada H DEL RESEARCH CORP. |
| To: | NRC |
| Shared Package | |
| ML20027E516 | List: |
| References | |
| CON-FIN-A-6456 NUDOCS 8211150466 | |
| Download: ML20027E525 (10) | |
Text
....-.
s
...4*
Acpendix 2
}
ESTIMATION OF STRESS INTENSITY FACTORS AND THE CRACX OPENING AREA 0F A CIRCUMFERENTIAL AND A LONGITUDINAL THROUGH-CRACK IN A PIPE by H. Tada and P. Paris Del Research Corporation St. Louis, Missouri i
Introduction Fomulas for estimating the crack opening area are developed for a circumferential and a longitudinal through-crack in a pipe subjected to several types of loading. For the circumferential crack, estimation for-mulas are presented for axial force and bending moment applied to the pipe Tar from the cracked section and for internal ' pressure loading.,fpr the longitudinal crack, an estimation formula for the case of internal pres-sure is presented.
Estimation is based on the method of linear elastic fracture mechanics, which requires the knowledge of the solution of stress intensity factor, i
K, for each problem. For the internal pressure leading, K-solutions are readily availab'le for both circumferential,and longitudinal cracks as func-e tions of a single geometric parameter, A(= a/4?), relating crack size and pipe geometry.
Consequently, the crack opening area fomulas are also formulated as functions cf this single parameter.
For the case of tension and bending of circumferential ' crack, however, the stress intensity factors are not formulated as functions of a single parameter and no simple formula is readily available. Therefore, in this discussion, a typical value of 8211150466 821108 PDR ADOCK 05000409 P
n
- i,
'Y mean radius to thickness ratio, R/t = 10, is specifically selected and for-mulation is mace for this value.
Estimation formulas are expected to yield.
a slight overestimate for R/t = 10. For smaller R/t ratios, degree of overestimate would increase. The formulas presented here may'be used with a reasonable accuracy when R/t ratio is about 10. Formulas for the crack opening for these cases are not available in simple closed forms, but here -
moderately long power series approximations based directly on the estimating fonnulas for K are given.
A Circumferential Through-Crack in Tension and Bendfnc t
The K formulas are first developed here based on the results recently obtained by Sanders D, 2]. As stated above,s the K solutions for these
.-v loadings are not expressed as functions of a single geometric parameter.
Sanders presented approximate formulas for the energy release rate for these loadings, which are readily converted into K formulas. The formulas are, in essence, functions of two geometric parameters for given elastic constants, which may be written in either of the following forms.
I K=ev7{Ke7F(A,e)~
(1)
K = e/-(Re) F(e,f) where e is an applied stress, 2Re is the total circumferential length of through-crack.
9 e
wm w.
m.
,<,v-,
,g,y
-e.,
A*-~'
.._-n.
.1 In this discussion, e and R/t are chosen as geanetric parameters and the second form of Eq.(1) is employed for the stress intensity expression.
Approximate K' formulas and the subsequent estimation formulas for 'the crack opening areas are developed spec'ifically 'for R/t = 10, which is con-sidered to be a typical value of interest in the present study. That is, the function F(e) in the subsequent discussion represents F(e,10).
Let P and M be the axial tensile force and bending moment, respec-tively, applied to the pipe far from the crack location and let subscripts t and b represent respectively tension and bending. The nominal stresses due to tension and bending cre defined by P
't " 2nRt f
(2)
M
- 'b
- 2 wR t The stress intensity factors are expressed in the following forms.
K w(Re) F (e) t t
t (3) b * 'b *(Re) F (e)
/
K b
where F (e) and F (e) are no.n-dimensional functions. The nanerical values t
b of the functions F (e) and F (e) are calculated from Sanders' approximate g
b formulas for R/t = 10, which are tabulated as follows.
9
4 N'
I (F (e) and F (e) for R/t = 10) t b
e F (e)
F (e) t b
0*
1.000 1.000 9
1.039 1.037 18 1.151 1.140 27 1.314 1.278 36 1.505 1.425 45 1.725 1.580 54 1.'987 1.747 63 2.305 1.934 72 2.702 2.154 81 3.209
'2.406 90 3.872 2.760 99 4.764 3.209
..e -
108 6.003 3.827 These values repres'ent slight overestimates of F (e) and F (e) [1,2].
t b
The following approximate expressions of the functions F (e) and F (e) t b
represent the values of the table with a reasonable accuracy (within a few percent).
3/2
' s/2 7/2 F(e)=1+7.5(f) 15(e) + 33(e) g (4) 13.6(h)
+ 20(e)
F (e) = 1 + 6.8( )
b (0 < e < 100* )
G e
..m n,-
u n
..-m n
~6~
- t e
I (e) = 4
.e{F (e2de (9)' t t Substituting F (e) given by Eq. (4), I (s) is written as t t I (e) = 2e 1 + (8) {8.6 - 13.3(e) + 24(e)2} t + (8) { 22.5 - 75(e) ~+ 205.7(f) (10) ~ -247.5(8)'+24Ti(e)"} (0 < e < 100') 4 The crack opening area for bending load, A, however, can not be. obtained b as readily because the " crack absent stress distribution" is not uniform .. y along the crack (direct application of the energy method is difficult). Therefore, A 0# I (s) will be estimated in the following way. b b First, comparison of the crack absent stress distributions for tensile and bending loads, the following bounds are imposed on Ab cose) < A I"b) < A I't *'b) A I# * 'b t t b t (11) or ~ (cose)l (e) < I (e) < I (s) t b g Where A ("b) is the crack opening area by bending, and A ('t * *bcosejand b t A (eb " *b) are the crack opening area due to axial force with tension stress g cose and e, respectively. The first approximation would be to take the b b 9 I f
average uniform stress between these extremes and + A I'b) 1 A I'b t 2 ) = A ("bICOS )) b t or (12) I(e)=(cosf)*I(e) b t Since the function I (e) given by Eq. (12) may yield underestimated values b of the crack opening by bending, the stress intensity factors X and Xb are compared in a similar manner. Corresponding to Eq. (11), it is obvious that ~ K I't " 'b ose ) < K {'b) ' K ('t " 'b) c t b b or ,e_- (13) (cose)F (e) < F (e) < F (e) t b t 1 Averaging the extremes F(e)=(cosf)F(e) (14) b t Comparison of the numerical values of F.(e) and F (s), however, shows that t b Eq. (14) always underestimates F (e) and that the val,ues of F (e) lie be-b b tween' the following two bounds 1+(cosf) (cosf)2F (e) < F (e) < t(e) (15) t b 2 Therefore, taking the following expression for I (e) instead of Eq. (14), b e e L.
..'*l. -8 i W the risk of excessive underestimation of the crack opening area caused by i bending load may be avoided 1+(cosf) I (e) = I I8) " I I') III) b 2 t 4 t where I (e) is given by Eq. (10). t The total crack opening area caused by axial tension and bending can be written as Atotal = At+Ab (wR*)l (e) 1+ 3 + ") g 4 or (17) 2 + cose .-f (wR )I (e) + s-E t o 4 ,b e l The effect of the yielding near the crack tip may be incorporated by the customary method of plastic zone corrections in which e in these formulas , is replaced by e,ff, e,ff is cbtained by using 2 Ktota e,ff = e + (18) 2nRey l l for plane stress (maximum) plastic corrections. Repeated iterative proce-dures may be necessary for obtaining e,ff. F l' i e t I J m
s- ~ Circumferential Through-Crack Subjected to Internal Pressure i e For a pipe subjected to internal pressure, p, the membrane stress, e, in the axial direction is estimated by e= (19) The stress intensity factor for a circumferential through-crack is normally ~ expressed in the following form. K = c /Ta"
- F ( A )
(20) p p where 2a = 2Re is the total circumferential length of the crack, F (A) is p, nondimensional function of A = a/E and the subscript p repfisentspres-sure loading. Contrary to the cases of axial force and bending load, the geo-metric factor F (1) for this case is a function of a single geometric para-p meter as mentioned earlier. The following formula empirically represents the curve of F (A) presented p in Rooke-Cartwright's work [3]. The approximate formula is, for convenience, expressed in a form consistent with the formula for longitudinal crack which will be subsequently discussed. Accuracy of the formula is within a few per-cent over the range specified. a: F (A) = (1 + 0.32251 )1/2 (0 < A 1,1) 2 p (21) 0.9 + 0.25A (1 L A L ) 5 = where A =' a/E, O "-*rv-4
- M
--4 -r-w --vw
e n
?' Applying the energy method again, the crack opening area, Ap, is i. readily obtained as follows = { (2sRt) G (A) (22) A p p where G (A) is given by an integral p G (A) = 2 A{F (A)} dA p p Corresponding to Eq. (21), G (A) is evaluated as, p 2 4 G (A) = A- + 0.16A (0 < A < 1) P '3 ~4 (23) = 0.02 + 0.8112 + 0.301 + 0.03A (1 < -X<_ 5) The effect of yielding near the ' crack tip may be similarly incorporated using the effective (plastic zone corrected) crack size which is calculated from the iterative relation K D a,ff = a + 2 (24) 2xey Lonoitudinal Throuch-Crack Subjected to Internal Pressure For a pipe subjected to internal pressure, p, the hoop stress, e, is' estimated by a=8 (25) t O e _n. m
L-- -11. t The stress intensity factor for a longitudinal through-crack of length 2a is given by K = o E F(A) (26) where again A = a/ M. The geometric factor F(A) can be empirically expressed over the range of interest by F(A) = (1 + 1.25A ) (01 A1 ) 1 (27) = 0.6 + 0.91 (1 1 AL ) 5 .-f Eq. (27) provides a good approximation for the shell factor F( A) wi th accuracy of the order of one percent [3, 4, 5, 6]. The crack opening area, A, can be obtained by the method in the previous l discussion. l l A={(2nRt)G(1) (28) where G(A) corresponding to Eq. (27) is given by G(A) = A* + 0.625A" (O L A s 1) ,(29) 2 + 0.72A3 + 0.405A" (1 1 A 15) = 0.14 + 0.361 Iteration with a plastic zone correction ~ similar to' Eq.(24) can be applied to account for the yielding effect near the crack tip. S
== .o O References [1] J. L. Sanders, Jr., "Circumferential Through-Cracks in Cylindrical Shells Under Tension, " to be published in Journal of Applied Mechanics. 1 [2] J. L. Sanders, Jr., Under Bending, Private Communication, November,1981. [3] D. P. Rooke and D. J. Cartwright, " Compendium of Stress Intensity Factors," Her Majesty's Stationary Office, London,1976. [4] F. S. Folias, "An Axial Crack in a Pressurized Cylindrical Shell," Int. J. of Fracture Mechanics, Vol.1,1965, pp.104-113. [5] F. Erdogan and J. J. Kibler, " Cylindrical and Spherical Shells with Cracks," Int. J. of Fracture Mechanics, Vol.5, 1969, pp. 229-237. 06 ] S. Krenk, "Ir. fluence of Transverse Shear on an Asial Crack in a Cylindrical Shell," Inti J. of Fracture, Vol. 14, 1978, pp. 123-143. l}}