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Evaluation of throughwall crack pipes under displacement controlled loading
Nuclear Engineering and Design 100 (1987) 11-19 North-Holland, Amsterdam
11
EVALUATION OF THROUGHWALL CRACK PIPES UNDER DISPLA CEMENT r CONTROLLED LOADING Akram ZAHOOR Novetech Corporation, 5 Choke Cherry Road, Roekville, AID 20850, USA Received 2 September 1986
Tearing modulus solutions are developed for flawed throughwall pipes subjected to displacement controlled loading. Two cases of loading were considered: (1) a displacement controlled bending loading, and (2) a displacement controlled axial tension loading. A revised version of the EPRI J-integral estimation scheme is used in the development of the solutions. These solutions can be used for the entire range of elastic-plastic loading, from linear elastic, contained yielding, to large scale yielding of the crack section. Experimental data from pipes in bending were used to assess the accuracy of the compliant loading solutions. The evaluations were performed using elastic plastic J-integral (J) and tearing modulus (T) analysis methods. These solutions are shown to have good accuracy when used to predict the experimental results. The methodology and procedure can also be applied to part-throughwall cracks. These solutions have application to the leak before break fracture mechanics analyses.
1. I n t r o d u c t i o n
Elastic-plastic fracture mechanics methods are now increasingly used in the evaluation of structural integrity of flawed light water reactor (LWR) piping [1-4]. Among several methods currently being used, the J-integral tearing modulus methodology ( J - T ) developed by Paris and co-workers [5,6] and later developed and applied to piping problems extensively by Zahoor [7-10] has received much attention. This progress has led to the development of J - T based flaw evaluation procedures [2], definition of the American Society of Mechanical Engineers Boiler and Pressure Vessel Code (ASME Code) allowable flaw sizes and loads for austenitic steel piping weldments [1], and new efforts aimed at verifying various J - T solutions and flaw evaluation procedures with flawed piping experimental data [11]. Since the introduction of the J-integral tearing modulus ( J - T ) concepts in 1977, a number of solutions have been developed for the applied tearing modulus for flawed piping. Initial work in this area, by Tada et al. [6] and by Zahoor et al. [7,8], considered a circumferential throughwall crack in a pipe under predominantly bend loading. Subsequently, the methodology was extended by Zahoor [9] to handle circumferential part-throughwall cracks. In this work stability of crack growth in the radial and circumferential direction was investigated. This work led to improved under-
standing of the potential for leak-before-break in LWR piping. Further work on J and tearing modulus by Zahoor considered axial loading on flawed throughwall pipes [10] and torsional loading on pipe with a circumferential part-throughwall crack [12]. Real loadings on LWR piping are complex and involve some combination of axial load, bending moment, and torsional loading. While fracture mechanics analyses for such a loading is very complex and solutions are not currently available, some progress has been made in this area by Zahoor and co-workers [13,14]. All of these efforts assumed a displacement controlled loading and involved the assumption that the crack section has fully yielded. Further, except in the work by Zahoor [7,8, 10, 12] the material stress-strain behavior was approximated by an elastic-perfectly plastic behavior. These assumptions allowed simple solutions for the applied J and T, but it was difficult to predict instability load accurately because of the perfectly plastic assumption for the stress-strain curve. The results were generally obtained in terms of pipe displacement or deflection which is not easy to implement in a flaw evaluation procedure. Typically, as in the ASME Code [1], J - T results are implemented by applying safety factors on instability load. All of the above work applies to piping materials that are sufficiently ductile and tough so that net-sec-
0 0 2 9 - 5 4 9 3 / 8 7 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)
12
A. Zahoor / Evaluation of throughwall crack pipes
tion yielding can be assumed. LWR piping welds (submerged arc welds and shielded metal arc welds) and certain ferritic base metals are known to have low toughness [2,3]. In these materials crack initiation and instability can occur before the crack section has fully yielded. Consequently, the methods and solutions referenced in the preceding cannot be applied accurately and new solutions are required that do not require full yielding of the crack section. Recently Zahoor [15] has developed tearing modulus solutions for flawed throughwall pipes subjected to combined tension and bending. This work is limited in its applicability because it can only handle contained yielding in the crack section. Kumar and co-workers have developed elastic-plastic J-integral solutions for throughwall pipes using finite element methods [16,17]. This work assumes Ramberg-Osgood stress-strain behavior for the material and is suitable for developing the tearing modulus solutions. The objective of the work presented in this paper is to develop applied tearing modulus solutions for flawed throughwall pipes that can be applied for a wide range of loading, from linear elastic, contained yielding, to large plastic yielding of the crack section. The paper considers two types of loadings: (1) a displacement controlled bending loading, and (2) a displacement controlled axial tension loading. Simple solutions are presented for the applied tearing modulus and these can be easily incorporated into a computer software for efficient computation of crack instability. A revised version of the EPRI J-integral estimation scheme [11] is used in the development of these solutions. Experimental data from pipes in bending were used to assess the accuracy of the compliant loading solutions. The evaluations were performed using elastic plastic J-integral ( J ) and tearing modulus (T) analysis methods. These solutions are shown to have good accuracy when used to predict the experimental results. Additional numerical results are presented to illustrate the influence of crack size, pipe size, and piping stiffness on crack stability.
where c is the true strain, o is the true stress, a and n are constants, and oo and c o are the reference stress and strain, respectively, with c o = oo/E. The reference stress, Oo, is generally assumed to be the 0.2% offset yield strength. Fig. 1 illustrates the geometry of the throughwall crack in a pipe, where 2 0 is the total crack angle and R and t are the mean radius and thickness, respectively, of the pipe. The EPRI J-integral estimation scheme for a circumferentially cracked throughwall pipe in bending [16,17] is
J =Je(aeef) + a a o % c ( a / b ) h l ( M / M o ) "+l,
where M o is a reference limit moment determined using %; J¢ is the linear elastic component of the J-integral corrected for plasticity using an Irwin type plastic zone size adjustment to the crack length; 2a and 2b are total crack length and pipe circumference; and c = b - a i~ half the uncracked portion of the pipe circumference. The second term in the above J estimation scheme is the plastic component of J which results from the strain hardening portion of the stress-strain curve. During initial J estimation scheme development [18] the form represented by eq. (2) was verified using the compact tension (CT) geometry, where good agreement was found between experimental data and predictions based on the estimation scheme solutions. Following these initial efforts, J estimation scheme solutions were generated for additional structures, flaws, and loads, including a pipe with pure bending loading and a circumferential throughwall flaw [16,17]. The circumferential throughwall flaw solutions were used to evaluate [4] previously performed austenitic and ferritic steel pipe experiments [8,19]. The results from this evaluation [4] indicate the J-estimation scheme can be overly conservative for predicting initiation and maximum load. The over-conservatism in the calculated J at first crack extension varied by factors of 3 to 7 times the experimentally inferred values for the stainless steel pipe tests, and by factors of up to 3 for the ferritic steel pipe tests [4].
2. J-integral estimation scheme for flawed pipes The application of the EPRI J-integral estimation scheme [16-18] requires that the stress-strain behavior of the material is expressed by the Ramberg-Osgood ( R - O ) stress-strain relation, , / % = 0/% + a ( o / o 0) ",
(1)
(2)
Fig. 1. Illustration of a throughwall crack in a pipe.
A. Zahoor / Evaluation of throughwall crack pipes Initially, the source of these differences was thought to be related to the region of the stress-strain curve used to define the R - O parameters and the definition of the reference stress, %. Consequently, parametric studies were performed to assess the effect of the R - O parameters on J computed from the EPRI J estimation scheme. The results from these [4,20] and other evaluations by this author did not fully explain the differences between the pipe experiments and estimation scheme predictions and attention then focused on the need for modification to the estimation scheme in eq. (2).
I O
2.0
I
e
E 0.8 o.6 L
ott/t"+')Oo, o c ( a / b ) h , ( M / M o ) "+1
(3) Examination of eq. (3) indicates that the original plastic component of J is modified by replacing a with a raised to the 1 / ( n + 1) power. This modification reduces the plastic J component compared to the original formulation. Fig. 2 shows the predictions [11,21] by the original and improved J estimation scheme for the flawed pipe tests in ref. [8,19]. Comparison of predictions made using the improved J estimation scheme with experimental results indicate that the improved method has much better accuracy than the original method and produces predictions that fall within the experimental scatter bands for pipes with throughwall flaws and pure bending loads. The improved J estimation scheme was used in the work that follows.
I t Q• /
mc
"o 0~
+
i
• Experimental Data • Original EPRI Estimation Scheme (~) Improved EPRI Estimation Scheme i
4.0
3. Improved J-integral estimation scheme
J = Je(aeff)
I
6.0
~ g .~
The original J-integral estimation scheme in eq. (2) was examined for possible modifications by Zahoor [11,21]. In this work the plastic portion of J was investigated carefully because the linear elastic solution, i.e. the first term without the plastic zone correction, has been shown to agree with the results from other investigators. In ref. [16-18] the second term in the J formula comes from a finite element analysis using a pure power law stress-strain relation with a = 1. It was assumed that J scales linearly with a; however, this assumption is correct for a pure power law material, but is not correct for the R - O type stress-strain curve. The details of the sources of error and possible modification to EPRI J estimation scheme can be found in Zahoor [21]. The improved J-integral estimation formula [11,21] is
13
Estimated Scatter Band
I I I I I
o.4
o.2
I
0.7
'
I I
I
I
0.8
0.9
I
1.0
Experimental Moment at Crack Initiation/Limit Moment
Fig. 2. A comparison of predicted to experimentally inferred J values at initiation for the original and improved J estimation scheme.
4. Applied tearing modulus for displacement controlled loading In this section the applied tearing modulus solutions are developed for flawed throughwall pipes subjected to displacement controlled loading. The solutions are presented first for displacement controlled bending loading and then for a displacement controlled axial tension loading. These solutions can be applied for the entire range of elastic-plastic loading, from linear elastic to large scale yielding in the crack section. The methodology and procedure developed below can be applied to part- throughwall cracks. 4.1. Displacement controlled bending loading Consider a circumferentially cracked pipe under displacement controlled bending loading as shown in fig. 3 where L and q0 are the length and the bending angle of the pipe, respectively, q0 consists of the elastic and plastic portions of the bending angle for the crack and uncracked pipe length. CM is the compliance of the piping system or the testing machine. The geometry of the throughwall crack is shown in fig. 1, where O is the crack half angle. Because crack growth and crack insta-
A. Zahoor / Evaluation of throughwall crack pipes
14
where, from [11,17,21]
cp,¢= M L / E I +
a,o(L/R)(M/4%RZtfl)"
,
(9)
Tc = f a ( a e , R / t ) M R / E I + a'/"%ah4( M / M o ) "+1, (10)
¢T =
O+ MC M
Fig. 3. A circumferentially cracked pipe under displacement controlled bending.
bility are governed by the applied displacement (controlled) loading, the applied tearing modulus formulation requires that the applied bending angle be included in the analysis development. The total bending angle of the piping system (fig. 3) is given as OPT= ¢P+ C M M ,
3 = (~f~/2) r(1 + 1 / 2 , ) / r ( 3 / 2 + 1/2.),
(11)
M o = 4 % R Z t [ c o s ( O / 2 ) - ½ sin O].
(12)
a, n, and oo are the parameters in the R - O representation for the stress-strain curve, and (0 = oo/E. R, t, and I are the pipe mean radius, wall thickness, and the moment of inertia, respectively. The function h 4 and the procedure for plastic zone adjustment to the crack length, a¢, are defined in Kumar et al. [17]. The function f4 may be conveniently expressed from Zahoor [15,22] as
f4 ( 0 / • ,
+ 33.433A (O/~)6.24
(4)
+ 166.83A2(0/~) s
where M is the bending moment and CM is the bending compliance in radians per unit moment. The applied tearing modulus for displacement controlled bending loading (fig. 3) is defined as
T = ( 3 J / 3 a ) ~T E/o~,
( OJ/Oa )w T = ( OJ/Oa ) g -- ( 3 J / O M ) . ( Ocp/aa ) g X[CM+(3~p/3M),]
+ 123.9A 2 ( O/or )7.74 + 26.298A 2( O/~r )~0 4s,
(5)
where E and of are the elastic modulus and flow stress, respectively. The flow stress is usually defined as onehalf the sum of yield and ultimate strengths. The partial derivative in the above is the change in J with crack length under the condition that the displacement is held fixed (¢PT = constant), and is given by Zahoor [7],
'.
R / t ) = 19.739(O/~)2 + 103.7A (O/~r)3s
A
f [O.lZ5(R/t) - 0.25] 0.25 \
[ [ 0 . 4 ( R / t ) - 3.0] 0.25
for 5 ~< R/t~< 10, for 10 ~< R / t <~20. (14)
The J integral is expressed as [11,17,21,22]
J = fl ( a e, R / t ) M 2 / E +al/("+')Oo, o C ( a / b ) h l ( M / M o )
(6)
(13)
"+1,
(15)
R/t),
(16)
fl(O/~r, R/t)=¢ra(R/I)2F~(O/~r,
When C M is very large compared to the pipe compliance, the imposed loading is load control, and eq. (6) reduces to
Fb ( O/~r, R / t ) = 1 + A [4.5967(O/~r) 15
( a J / a a )~ T = ( 3 J / 3 a ) g.
The partial derivatives of cp and J, required in eq. (6), are obtained from eqs. (8) through (17). These are summarized below. The partial derivative of J with respect to M is
(7)
Therefore, eq. (6) can be used for practical applications, ranging from displacement controlled loading to load controlled loading. With the above discussion, it is now appropriate to present solutions for J and q0, and their partial derivatives. The bending angle of the pipe, rp, can be split into two parts as ~ = ~nc + ,pc,
(8)
+ 2.6422( O/~r )424].
(3J/3M).
= ( 2 / M ) J 1 + [(n + 1 ) / M ] J 2,
(17)
(18)
where J1 and ./2 are the first and the second term in the sum in eq. (15). Similarly, if rp~ and ¢P2 denote the first and the second term in % (eq. 10), and ¢P3 and ¢P4 denote the
A. Zahoor/ Evaluationof throughwallcrackpipes
15
T P"~T
first and the second term in ~0nc (eq. 9), then
(Oep/3M), = (cpl + ~03)/M + n(~02 + ~4)/M.
(19)
The partial derivatives with respect to " a " are obtained from
(3J/3a)g=[J(a+Aa)--J(a)]/Zlalg
(20)
and
(3ep/3a)g=[~c(a+Aa)--~¢(a)]/Aalg,
(21)
where a is the original crack length a n d ' A a is an arbitrary but small amount of crack growth. Eq. (18) through (21) when substituted in eq. (6) give the necessary formulation required to compute the applied tearing modulus for displacement controlled bending loading. The above development is applicable to circumferential throughwall cracks. This same procedure can be used for part-throughwall cracks provided the crack length " a " is replaced by the crack area and appropriate solutions are used for J, qo,~, and ~0c.
4.2 Displacement controlled axial tension loading Consider a circumferential throughwall crack (fig. 1) in a pipe under axial tension as shown in fig. 4. The length of the pipe is L and the axial displacement of the pipe is denoted as za. The displacement consists of elastic and plastic portions of the axial displacement for the crack and the uncracked pipe length. A spring in series with the pipe is considered to simulate the additional compliance of the piping system or testing machine. The total axial displacement of the piping system is denoted as A TSimilar to the bending loading case, the applied tearing modulus for axial tension loading from Zahoor [10] is
T = (3J/3 a ) a TE/°f z,
(22)
=• + PCM
Fig. 4. A circumferentially cracked pipe subjected to displacement controlled axial tension. where the subscripts "nc" and "c" denote the displacement component in the absence of crack and with the crack present, respectively. The solutions taken from ref. [11,17,21] are Anc=Lo/E+~¢.oL(o/OO)
n,
o=P/2frRt,
( a J / a a ) a T = ( a J / a a ) p - (OJ//aP).(OA//aa)p X [C M + ( O A / O P ) . ] - 1
(23)
and
A T = A + CMP.
(24)
Now, the axial displacement of the pipe, A, can be split into two parts as (25)
(26)
A¢ = f 3 ( a c , R / t ) P / E
+~I/",0ah3(o/,~, n,
R/t)(P/Po)",
(27)
where f3 as given by Zahoor [15,22] is
f3 ( @/~r, R / t ) = (1/Trt)
[4.9348(O/~')2
+ 30.0618A (O/or) 35 + 59.3854A (O/or) 6"24 + 56.083A 2 ( O/~r ) 5 + 255.2A 2 (O/~r)7.74
+ 331.83A2( O/~r)l°48].
where
a = a.c + a o
A T
(28)
The quantity A is defined in eq. (14). The procedure for the plastic zone adjustment to the crack length (ae) and the function h 3 are defined in ref. [18]. Similarly, the J integral for a throughwall crack pipe in axial tension with the improved J estimation [17,11,21] is
J=fl(ae, R/t)p2/E +al/t"+a)oo%C(a/b)hl(@/cr, R / t , n ) ( e / P o ) "+a
(29)
A. Zahoor / Evaluation of throughwall crack pipes
16
In the above, fl may be conveniently expressed in closed form, Zahoor [15,22], as
fl (O/~r, R / t ) = (O/~r) Ft2/4Rt 2 Ft = 1 + A [5.3303(O/~r) ls + 18.773(O/¢r)4 24].
(30)
The values of h 1 are given by Kumar et al. [17]., Following the procedure for the bending loading case, the various partial derivatives are obtained as
(OJ/3P)~ = ( 2 / P ) J , + [(n + 1 ) / P ] J 2,
(31)
(~A//~P)a=(AI q-A3)//P+tI(A2+A4)//P ,
(32)
(aJ/aa)p=[J(a+aa)-J(a)]/aale
(33)
and
(~)A/Oa)p=[Ac(a+Aa)--A.(a)]/Aa[p.
(34)
Substituting eqs. (31) through (34) into (23) gives the applied tearing modulus for displacement controlled axial loading.
5. Evaluation of displacement controlled pipe bending test results The solutions derived in the preceding section were used to evaluate the pipe test results described in refs. [8,19]. An objective of this evaluation was to assess the accuracy of the predictions from the improved estimation scheme. Ref. [19] presents data for nine throughwall crack pipes tested in displacement controlled fourpoint bending. The pipes were 8 inch in diameter and
made of A106 GrB ferritic steel. The throughwall crack lengths varied from 20% to 33% of the pipe circumference. One throughwall crack pipe test from ref. [8] was also evaluated; the pipe was 4 inch diameter and made of Type 304 austenitic steel. The throughwall crack length for the austenitic pipe test, Test 3T, was 29% of the pipe circumference. Test 3T was the first pipe test in which crack instability was successfully predicted from a pretest J / T analysis [7,8]. The R a m b e r g - O s g o o d parameters for the A106 GrB ferritic steel and Type 304 stainless steel were taken from Zahoor [11,21]. The constants for the A106 GrB steel are: a = 3 . 8 , n = 4 . 0 , o0 = 37,000 psi, and E = 27.0 x 106 psi. The constants for the stainless steel are a = 4.7, n = 3.8, oo = 34,500 psi, and E = 28.3 x 1 0 6 psi. These constants were obtained from the truestress-true-strain curve for the pipe material and represent best fit for the entire range of the available data [11]. Table 1 summarizes the test conditions and results. As indicated, three of the nine ferritic pipe tests were conducted with a stiff test system typical of laboratory testing. The remaining six ferritic steel pipe tests were conducted with various combinations of test system compliance and crack length. Pipe Test 13, 14, 15, and 3T showed unstable ductile crack extension. Table 1 also summarizes the nature of final crack extension and the maximum moment for each pipe test. Predictive analyses were performed for all ten pipe tests. The objective of the analyses was to predict (1) whether crack growth would occur unstably and (2) the
Table 1 Stability predictions for A106 GrB ferritic steel and type 304 austenitic steel pipe tests Experiment number
(9/~r
3 7 8 10 11 12 13 14 15 3T
0.300 0.328 0.302 0.262 0.248 0.232 0.215 0.192 0.210 0.290
Pipe length (inch)
48 48 48 48 42 42 42 42 42 60
Predictions (improved EPRI estimation)
Additional compliance (rad/in-lb)
Test results Stable/ unstable
Maximum moment (in-kip)
Stable/ unstable
2.47 x 10- 8 2.47x 10 8 2.47X 10 8 3.17 x 10 - 7 4.81 x 10 v 4.81x10 7 4.91× 10 7 4.81 x 10- 7 4.83 × 10 7 4.36 × 10 7
Stable Stable Stable Stable Stable Stable Unstable Unstable Unstable Unstable
999 851 830 1,095 1,133 1,178 1,236 1,367 1,273 205
Stable Stable Stable Unstable Unstable Unstable Unstable Unstable Unstable Unstable
Maximum moment (in-kip)
1,083 1,241 1,171 1,184 1,455 1,279 255
A. Zahoor / Evaluation of throughwall crack pipes
The i n p u t to the J / T analysis were the initial crack length, pipe length, additional (spring) compliance, pipe size, pipe material properties (tensile a n d fracture). The applied loading was increased a n d the corresponding applied J a n d T pairs were calculated. The J - T pairs a n d the material curve (fig. 5) were plotted o n a J / T space to determine the intersection of the applied a n d material curves. Crack growth was predicted to be stable for those tests where intersection was n o t found. Table 1 also summarizes the analysis predictions. As shown, the m o d e of failure is predicted correctly for all pipe tests except for tests 10, 11, a n d 12. F u r t h e r e x a m i n a t i o n of predictions for these three tests (table 2) showed that the a m o u n t of crack growth at the onset of instability was greater t h a n the final crack growth in the pipe test. This suggests that h a d these tests been continued, the pipe would have failed b y crack instability. Table 2 also shows that the predicted b e n d i n g m o m e n t s are in good agreement with the experiment for pipes that-failed b y crack instability. These test predictions provide additional validation of the improved J estimation scheme. Parametric analyses were also performed to determine the effect of system compliance, crack size, a n d pipe size o n crack stability. Because b e n d i n g is of p r i m a r y interest, numerical results are generated for b e n d i n g case; similar results are expected for the axial tension loading. Predictive analyses were performed first for the crack size used in pipe test n u m b e r 13. Fig. 5 shows the
20,000
O/1T= 0.215
16,000
(~) (~) (~
CM=O0 (Load Control) CM = 4.91 X 10 -7 rad/in-lb (Compliant Test Machine) CM = 2.47 X 10 -8 rad/inqb (Rigid Test Machine)
12,000 Without Crack Growth ~
--
With Crack Growth
8,000
4,000
01 -50
"
I
I
50
I
I
I
I
150 250 Tearing Modulus
I 350
17
I 450
Fig. 5. Illustration of the effect of compliance on crack stability.
b e n d i n g m o m e n t a n d crack growth at the onset of crack instability. The applied J a n d the tearing modulus were calculated using the formulation presented in the preceding section.
Table 2 Maximum moment and crack growth at maximum moment predictions for A106 GrB ferritic steel and Type 304 austenitic steel pipe tests Experiment number
3 7 8 10 11 12 13 a) 14 a) 15 a) 3T a)
Test results
Predictions using improved estimation scheme
Maxium moment (in-kip)
Crack growth at instability (inch)
Instability moment (in-kip)
Crack growth at instability (inch)
999 851 830 1,095 1,133 1,178 1,236 1,367 1,273 205
1.101 0.576 0.738 0.733 1.120 1.150 0.796 0.643 0.995 0.750
1,083 1,241 1,171 1,184 1,455 1,279 225
2.10 1.38 1.37 0.71 0.84 0.83 0.73
a) Test showed unstable crack extension. b) Percent error = (predicted m o m e n t - experimental moment) × lO0/experimental moment.
Percent error b)
- 4.0 + 6.5 + 0.5 + 10.0
18
A. Zahoor / Evaluation of throughwall crack pipes
20,000
20,000
9 16,000
12,000 "¢
8,000
~
16,000 C M = 4.91 X 10 - 7 rad/in-lb
C M = 4.91 X 10 - 7 rad/in-lb
(~) 0 / i t = 0.215
O/'tI = 0.215
(~ O/l"r=0.30 (~) 0/1/=0.35 @ O/~" = 0.40
(~
28 inch pipe
k 8,000
4,000
4,000
13 --50
,7,22
12,000
-=
I 50
[
1 150
I
I 250
I
I 350
I
I 450
-50
Tearing Modulus
Fig. 6. Illustration of the effect of crack size on crack stability.
influence of the additional system compliance on the applied J - T behavior. On this figure, curve 1 represents the load control (C M ~ oo) condition, curve 2 is for the compliance used in the pipe test number 13, and curve 3 is for a rigid test machine. Two observations can be made from these results. First, as system stiffness increases, the likelihood of crack stability increases and larger amounts of crack growth would have to occur before instability. The second observation is that the crack growth correction shifts the applied J - T curves to the left, especially for reduced compliance. Consequently, the potential for stability increases when the analysis includes crack growth and compliance decreases. This is best illustrated for the curve 3. In this case, the applied J - T curve that does not include crack growth correction predicts crack instability, whereas the properly corrected curve shows that crack will be stable. Fig. 6 illustrates the effect of crack size on the stability of crack growth. Here, curve 1 is for the pipe test 13. As crack length increases, the crack tends to be more stable. Fig. 7 shows the influence of pipe size on the applied J - T curve, Here, crack length (O/,r) is held constant for all pipe sizes. In this case, the curve 1 is for the pipe test number 13. From these results it can be concluded
50
t
I 150
I
I 250
I
I 350
I 450
Tearing Modulus
Fig. 7. Illustration of the effect of pipe size on crack stability.
that as pipe size increases, the amount of stable crack growth increases. The effect of pipe size on crack stability is not as significant as the effect of crack length or system compliance. The trends of the parametric results are qualitatively similar to those in previous work [8,11]. In contrast to the previous work, the formulation of this paper allows quantitative results for the instability load for displacement controlled loading on flawed pipes.
5. Discussion Solutions for compliant bending and axial loading were developed for throughwaU flaws in piping. Experimental data from pipes in bending were used to confirm the compliant loading solutions. These solutions, which included the improved estimation scheme to compute J and T, were shown to have good accuracy when used to predict the experimental results. This comparison provides additional confirmation for the accuracy of the improved J estimation scheme. These solutions can be used for the entire range of elastic-plastic loading, from linear elastic, contained yielding, to large scale yielding of the crack section. Further the development provides a procedure that can be used to obtain accurate results for crack stability in
A. Zahoor / Et~aluation of throughwall crack pipes pipes u n d e r displacement controlled loading. A l t h o u g h the applied tearing modulus solutions are derived for circumferential throughwall cracks, the m e t h o d o l o g y a n d procedure developed here is also applicable to part-throughwall cracks.
Acknowledgement The work described in this p a p e r was supported in p a r t by the Electric Power Research Institute u n d e r research project R P 2457-8. The a u t h o r wishes to t h a n k Dr. Douglas M. Norris for support and e n c o u r a g e m e n t d u r i n g this work.
6. References [1] American Society of Mechanical Engineers Boiler and Pressure Vessel Code, 1983 Edition and Addenda, Section XI, Paragraph IWB-3640; see also 1986 Edition, pp. 116-123. [2] A. Zahoor, H.S. Mehta, S. Yukawa, R.M. Gamble and S. Ranganath, Flaw evaluation procedures and standards for ferritic piping, Final Report on Research Project RP175751, Electric Power Research Institute, Palo Alto, California (November 1985). [3] Evaluation of Flaws in Austenitic Steel Piping, prepared by Section XI Task Group for Piping Flaw Evaluation, ASME Boiler and Pressure Vessel Code Committee, EPRI Report NP 4690-SR (April 1986), ASME J. Pressure Vessel Technology 108 (1986) 352-367. [4] Piping Review Committee - Evaluation of potential for pipe breaks, NUREG/CR-1061, Vol. 3 (Appendix A), US Nuclear Regulatory Commission, Washington, DC (1984) [5] P.C. Paris, H. Tada, A. Zahoor and H. Ernst, A treatment of the subject of tearing instability, NUREG-0311, Nuclear Regulatory Commission, Washington, DC (1977). [6] H. Tada, P.C. Paris and R.M. Gamble, Stability analysis of circumferential cracks in reactor piping systems, NUREG/CR-0838, Nuclear Regulatory Commission, Washington, DC (1979). [7] A. Zahoor and M.F. Kanninen, A plastic fracture mechanics prediction of fracture instability in a circumferentially cracked pipe in bending - Part I: J-integral analysis, ASME J. Pressure Vessel Technology 103 (1981) 352-358. [8] A. Zahoor, G. Wilkowski, I. Abou-Sayed, C. Marschall, D. Broek, S. Sampath, H. Rhee and J. Ahmad, Instability predictions for circumferentially cracked type-304 stainless steel pipe under dynamic loading, ed. M.F. Kanninen, EPRI NP-2347, Vols. 1 and 2, Electric Power Research Institute, Palo Alto, California (April 1982).
19
[9] A. Zahoor and M.F. Kanninen, A plastic fracture instability analysis of wall breakthrough in circumferentially cracked pipe subjected to bending loads, ASME J. Pressure Vessel Technology 103 (1981) 194-200. [10] A. Zahoor and D.M. Norris, Ductile fracture of circumferentially cracked type-304 stainless steel pipes in tension, ASME J. Pressure Vessel Technology 106 (1984) 399-404. [11] A. Zahoor and R.M. Gamble, Evaluation of flawed pipe experiments, Final Report on EPRI Project RP 2457-8, Electric Power Research Institute, Palo Alto, California (July 1986). [12] A. Zahoor, J-integral and tearing modulus estimates for circumferentially cracked pipes in torsion, ASME J. Pressure Vessel Technology 108 (1986). [13] A. Zahoor, Advanced studies of the stability of circumferentially cracked pipes, Final Report on EPRI Project Tl18-9-1, Electric Power Research Institute, Palo Alto, California (December 1982). [14] A. Zahoor and R.M. Gamble, Leak-before-break analysis for BWR recirculation piping having cracks at multiple weld locations, EPRI NP-3522-LD, Electric Power Research Institute, Palo Alto, California (April 1984). [15] A. Zahoor, Fracture of circumferentially cracked pipes, ASME J. Pressure Vessel Technology 108 (1986). [16] M.D. German and V. Kumar, Elastic-plastic analysis of crack opening, stable growth and instability behavior in flawed 304 stainless steel piping, ASME PVP Vol. 58 (American Society of Mechanical Engineers, 1982) pp. 109-141. [17] V. Kumar, M.D. German, W.W. Wilkening, W.R. Andrews, H.G. deLorenzi and D.F. Mowbray, Advances in elastic plastic fracture analysis, EPRI NP 3607, Electric Power Research Institute, Palo Alto, California (August 1984), [18] V. Kumar, M.D. German and C.F. Shih, An engineering approach for elastic-plastic fracture analysis, EPRI NP 1931, Electric Power Research Institute, Palo Alto, California (July 1981). [19] M.G. Vassilaros, R.A. Hays, J.P. Gudas and J.A. Joyce, J integral tearing instability analyses for 8-inch diameter ASTM A106 steel pipe, NUREG/CR-3740, Nuclear Regulatory Commission, Washington, DC (April 1984). [20] A. Okamoto and D.M. Norris, Sensitivity analysis of flaw growth instability, Proc. CSNI/NRC Workshop on Ductile Piping Fracture Mechanics, Southwest Research Institute, San Antonio, Texas, 1984, to appear in a NUREG Report. [21] A. Zahoor, Evaluation of J-integral estimation scheme for flawed throughwall pipes, Nucl. Engrg. Des. 100 (1987) 1-9, in this issue. [22] A. Zahoor, Closed form expressions for fracture mechanics analysis of cracked pipes, ASME J. Pressure Vessel Technology 107 (1985) 203-205.
11
EVALUATION OF THROUGHWALL CRACK PIPES UNDER DISPLA CEMENT r CONTROLLED LOADING Akram ZAHOOR Novetech Corporation, 5 Choke Cherry Road, Roekville, AID 20850, USA Received 2 September 1986
Tearing modulus solutions are developed for flawed throughwall pipes subjected to displacement controlled loading. Two cases of loading were considered: (1) a displacement controlled bending loading, and (2) a displacement controlled axial tension loading. A revised version of the EPRI J-integral estimation scheme is used in the development of the solutions. These solutions can be used for the entire range of elastic-plastic loading, from linear elastic, contained yielding, to large scale yielding of the crack section. Experimental data from pipes in bending were used to assess the accuracy of the compliant loading solutions. The evaluations were performed using elastic plastic J-integral (J) and tearing modulus (T) analysis methods. These solutions are shown to have good accuracy when used to predict the experimental results. The methodology and procedure can also be applied to part-throughwall cracks. These solutions have application to the leak before break fracture mechanics analyses.
1. I n t r o d u c t i o n
Elastic-plastic fracture mechanics methods are now increasingly used in the evaluation of structural integrity of flawed light water reactor (LWR) piping [1-4]. Among several methods currently being used, the J-integral tearing modulus methodology ( J - T ) developed by Paris and co-workers [5,6] and later developed and applied to piping problems extensively by Zahoor [7-10] has received much attention. This progress has led to the development of J - T based flaw evaluation procedures [2], definition of the American Society of Mechanical Engineers Boiler and Pressure Vessel Code (ASME Code) allowable flaw sizes and loads for austenitic steel piping weldments [1], and new efforts aimed at verifying various J - T solutions and flaw evaluation procedures with flawed piping experimental data [11]. Since the introduction of the J-integral tearing modulus ( J - T ) concepts in 1977, a number of solutions have been developed for the applied tearing modulus for flawed piping. Initial work in this area, by Tada et al. [6] and by Zahoor et al. [7,8], considered a circumferential throughwall crack in a pipe under predominantly bend loading. Subsequently, the methodology was extended by Zahoor [9] to handle circumferential part-throughwall cracks. In this work stability of crack growth in the radial and circumferential direction was investigated. This work led to improved under-
standing of the potential for leak-before-break in LWR piping. Further work on J and tearing modulus by Zahoor considered axial loading on flawed throughwall pipes [10] and torsional loading on pipe with a circumferential part-throughwall crack [12]. Real loadings on LWR piping are complex and involve some combination of axial load, bending moment, and torsional loading. While fracture mechanics analyses for such a loading is very complex and solutions are not currently available, some progress has been made in this area by Zahoor and co-workers [13,14]. All of these efforts assumed a displacement controlled loading and involved the assumption that the crack section has fully yielded. Further, except in the work by Zahoor [7,8, 10, 12] the material stress-strain behavior was approximated by an elastic-perfectly plastic behavior. These assumptions allowed simple solutions for the applied J and T, but it was difficult to predict instability load accurately because of the perfectly plastic assumption for the stress-strain curve. The results were generally obtained in terms of pipe displacement or deflection which is not easy to implement in a flaw evaluation procedure. Typically, as in the ASME Code [1], J - T results are implemented by applying safety factors on instability load. All of the above work applies to piping materials that are sufficiently ductile and tough so that net-sec-
0 0 2 9 - 5 4 9 3 / 8 7 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)
12
A. Zahoor / Evaluation of throughwall crack pipes
tion yielding can be assumed. LWR piping welds (submerged arc welds and shielded metal arc welds) and certain ferritic base metals are known to have low toughness [2,3]. In these materials crack initiation and instability can occur before the crack section has fully yielded. Consequently, the methods and solutions referenced in the preceding cannot be applied accurately and new solutions are required that do not require full yielding of the crack section. Recently Zahoor [15] has developed tearing modulus solutions for flawed throughwall pipes subjected to combined tension and bending. This work is limited in its applicability because it can only handle contained yielding in the crack section. Kumar and co-workers have developed elastic-plastic J-integral solutions for throughwall pipes using finite element methods [16,17]. This work assumes Ramberg-Osgood stress-strain behavior for the material and is suitable for developing the tearing modulus solutions. The objective of the work presented in this paper is to develop applied tearing modulus solutions for flawed throughwall pipes that can be applied for a wide range of loading, from linear elastic, contained yielding, to large plastic yielding of the crack section. The paper considers two types of loadings: (1) a displacement controlled bending loading, and (2) a displacement controlled axial tension loading. Simple solutions are presented for the applied tearing modulus and these can be easily incorporated into a computer software for efficient computation of crack instability. A revised version of the EPRI J-integral estimation scheme [11] is used in the development of these solutions. Experimental data from pipes in bending were used to assess the accuracy of the compliant loading solutions. The evaluations were performed using elastic plastic J-integral ( J ) and tearing modulus (T) analysis methods. These solutions are shown to have good accuracy when used to predict the experimental results. Additional numerical results are presented to illustrate the influence of crack size, pipe size, and piping stiffness on crack stability.
where c is the true strain, o is the true stress, a and n are constants, and oo and c o are the reference stress and strain, respectively, with c o = oo/E. The reference stress, Oo, is generally assumed to be the 0.2% offset yield strength. Fig. 1 illustrates the geometry of the throughwall crack in a pipe, where 2 0 is the total crack angle and R and t are the mean radius and thickness, respectively, of the pipe. The EPRI J-integral estimation scheme for a circumferentially cracked throughwall pipe in bending [16,17] is
J =Je(aeef) + a a o % c ( a / b ) h l ( M / M o ) "+l,
where M o is a reference limit moment determined using %; J¢ is the linear elastic component of the J-integral corrected for plasticity using an Irwin type plastic zone size adjustment to the crack length; 2a and 2b are total crack length and pipe circumference; and c = b - a i~ half the uncracked portion of the pipe circumference. The second term in the above J estimation scheme is the plastic component of J which results from the strain hardening portion of the stress-strain curve. During initial J estimation scheme development [18] the form represented by eq. (2) was verified using the compact tension (CT) geometry, where good agreement was found between experimental data and predictions based on the estimation scheme solutions. Following these initial efforts, J estimation scheme solutions were generated for additional structures, flaws, and loads, including a pipe with pure bending loading and a circumferential throughwall flaw [16,17]. The circumferential throughwall flaw solutions were used to evaluate [4] previously performed austenitic and ferritic steel pipe experiments [8,19]. The results from this evaluation [4] indicate the J-estimation scheme can be overly conservative for predicting initiation and maximum load. The over-conservatism in the calculated J at first crack extension varied by factors of 3 to 7 times the experimentally inferred values for the stainless steel pipe tests, and by factors of up to 3 for the ferritic steel pipe tests [4].
2. J-integral estimation scheme for flawed pipes The application of the EPRI J-integral estimation scheme [16-18] requires that the stress-strain behavior of the material is expressed by the Ramberg-Osgood ( R - O ) stress-strain relation, , / % = 0/% + a ( o / o 0) ",
(1)
(2)
Fig. 1. Illustration of a throughwall crack in a pipe.
A. Zahoor / Evaluation of throughwall crack pipes Initially, the source of these differences was thought to be related to the region of the stress-strain curve used to define the R - O parameters and the definition of the reference stress, %. Consequently, parametric studies were performed to assess the effect of the R - O parameters on J computed from the EPRI J estimation scheme. The results from these [4,20] and other evaluations by this author did not fully explain the differences between the pipe experiments and estimation scheme predictions and attention then focused on the need for modification to the estimation scheme in eq. (2).
I O
2.0
I
e
E 0.8 o.6 L
ott/t"+')Oo, o c ( a / b ) h , ( M / M o ) "+1
(3) Examination of eq. (3) indicates that the original plastic component of J is modified by replacing a with a raised to the 1 / ( n + 1) power. This modification reduces the plastic J component compared to the original formulation. Fig. 2 shows the predictions [11,21] by the original and improved J estimation scheme for the flawed pipe tests in ref. [8,19]. Comparison of predictions made using the improved J estimation scheme with experimental results indicate that the improved method has much better accuracy than the original method and produces predictions that fall within the experimental scatter bands for pipes with throughwall flaws and pure bending loads. The improved J estimation scheme was used in the work that follows.
I t Q• /
mc
"o 0~
+
i
• Experimental Data • Original EPRI Estimation Scheme (~) Improved EPRI Estimation Scheme i
4.0
3. Improved J-integral estimation scheme
J = Je(aeff)
I
6.0
~ g .~
The original J-integral estimation scheme in eq. (2) was examined for possible modifications by Zahoor [11,21]. In this work the plastic portion of J was investigated carefully because the linear elastic solution, i.e. the first term without the plastic zone correction, has been shown to agree with the results from other investigators. In ref. [16-18] the second term in the J formula comes from a finite element analysis using a pure power law stress-strain relation with a = 1. It was assumed that J scales linearly with a; however, this assumption is correct for a pure power law material, but is not correct for the R - O type stress-strain curve. The details of the sources of error and possible modification to EPRI J estimation scheme can be found in Zahoor [21]. The improved J-integral estimation formula [11,21] is
13
Estimated Scatter Band
I I I I I
o.4
o.2
I
0.7
'
I I
I
I
0.8
0.9
I
1.0
Experimental Moment at Crack Initiation/Limit Moment
Fig. 2. A comparison of predicted to experimentally inferred J values at initiation for the original and improved J estimation scheme.
4. Applied tearing modulus for displacement controlled loading In this section the applied tearing modulus solutions are developed for flawed throughwall pipes subjected to displacement controlled loading. The solutions are presented first for displacement controlled bending loading and then for a displacement controlled axial tension loading. These solutions can be applied for the entire range of elastic-plastic loading, from linear elastic to large scale yielding in the crack section. The methodology and procedure developed below can be applied to part- throughwall cracks. 4.1. Displacement controlled bending loading Consider a circumferentially cracked pipe under displacement controlled bending loading as shown in fig. 3 where L and q0 are the length and the bending angle of the pipe, respectively, q0 consists of the elastic and plastic portions of the bending angle for the crack and uncracked pipe length. CM is the compliance of the piping system or the testing machine. The geometry of the throughwall crack is shown in fig. 1, where O is the crack half angle. Because crack growth and crack insta-
A. Zahoor / Evaluation of throughwall crack pipes
14
where, from [11,17,21]
cp,¢= M L / E I +
a,o(L/R)(M/4%RZtfl)"
,
(9)
Tc = f a ( a e , R / t ) M R / E I + a'/"%ah4( M / M o ) "+1, (10)
¢T =
O+ MC M
Fig. 3. A circumferentially cracked pipe under displacement controlled bending.
bility are governed by the applied displacement (controlled) loading, the applied tearing modulus formulation requires that the applied bending angle be included in the analysis development. The total bending angle of the piping system (fig. 3) is given as OPT= ¢P+ C M M ,
3 = (~f~/2) r(1 + 1 / 2 , ) / r ( 3 / 2 + 1/2.),
(11)
M o = 4 % R Z t [ c o s ( O / 2 ) - ½ sin O].
(12)
a, n, and oo are the parameters in the R - O representation for the stress-strain curve, and (0 = oo/E. R, t, and I are the pipe mean radius, wall thickness, and the moment of inertia, respectively. The function h 4 and the procedure for plastic zone adjustment to the crack length, a¢, are defined in Kumar et al. [17]. The function f4 may be conveniently expressed from Zahoor [15,22] as
f4 ( 0 / • ,
+ 33.433A (O/~)6.24
(4)
+ 166.83A2(0/~) s
where M is the bending moment and CM is the bending compliance in radians per unit moment. The applied tearing modulus for displacement controlled bending loading (fig. 3) is defined as
T = ( 3 J / 3 a ) ~T E/o~,
( OJ/Oa )w T = ( OJ/Oa ) g -- ( 3 J / O M ) . ( Ocp/aa ) g X[CM+(3~p/3M),]
+ 123.9A 2 ( O/or )7.74 + 26.298A 2( O/~r )~0 4s,
(5)
where E and of are the elastic modulus and flow stress, respectively. The flow stress is usually defined as onehalf the sum of yield and ultimate strengths. The partial derivative in the above is the change in J with crack length under the condition that the displacement is held fixed (¢PT = constant), and is given by Zahoor [7],
'.
R / t ) = 19.739(O/~)2 + 103.7A (O/~r)3s
A
f [O.lZ5(R/t) - 0.25] 0.25 \
[ [ 0 . 4 ( R / t ) - 3.0] 0.25
for 5 ~< R/t~< 10, for 10 ~< R / t <~20. (14)
The J integral is expressed as [11,17,21,22]
J = fl ( a e, R / t ) M 2 / E +al/("+')Oo, o C ( a / b ) h l ( M / M o )
(6)
(13)
"+1,
(15)
R/t),
(16)
fl(O/~r, R/t)=¢ra(R/I)2F~(O/~r,
When C M is very large compared to the pipe compliance, the imposed loading is load control, and eq. (6) reduces to
Fb ( O/~r, R / t ) = 1 + A [4.5967(O/~r) 15
( a J / a a )~ T = ( 3 J / 3 a ) g.
The partial derivatives of cp and J, required in eq. (6), are obtained from eqs. (8) through (17). These are summarized below. The partial derivative of J with respect to M is
(7)
Therefore, eq. (6) can be used for practical applications, ranging from displacement controlled loading to load controlled loading. With the above discussion, it is now appropriate to present solutions for J and q0, and their partial derivatives. The bending angle of the pipe, rp, can be split into two parts as ~ = ~nc + ,pc,
(8)
+ 2.6422( O/~r )424].
(3J/3M).
= ( 2 / M ) J 1 + [(n + 1 ) / M ] J 2,
(17)
(18)
where J1 and ./2 are the first and the second term in the sum in eq. (15). Similarly, if rp~ and ¢P2 denote the first and the second term in % (eq. 10), and ¢P3 and ¢P4 denote the
A. Zahoor/ Evaluationof throughwallcrackpipes
15
T P"~T
first and the second term in ~0nc (eq. 9), then
(Oep/3M), = (cpl + ~03)/M + n(~02 + ~4)/M.
(19)
The partial derivatives with respect to " a " are obtained from
(3J/3a)g=[J(a+Aa)--J(a)]/Zlalg
(20)
and
(3ep/3a)g=[~c(a+Aa)--~¢(a)]/Aalg,
(21)
where a is the original crack length a n d ' A a is an arbitrary but small amount of crack growth. Eq. (18) through (21) when substituted in eq. (6) give the necessary formulation required to compute the applied tearing modulus for displacement controlled bending loading. The above development is applicable to circumferential throughwall cracks. This same procedure can be used for part-throughwall cracks provided the crack length " a " is replaced by the crack area and appropriate solutions are used for J, qo,~, and ~0c.
4.2 Displacement controlled axial tension loading Consider a circumferential throughwall crack (fig. 1) in a pipe under axial tension as shown in fig. 4. The length of the pipe is L and the axial displacement of the pipe is denoted as za. The displacement consists of elastic and plastic portions of the axial displacement for the crack and the uncracked pipe length. A spring in series with the pipe is considered to simulate the additional compliance of the piping system or testing machine. The total axial displacement of the piping system is denoted as A TSimilar to the bending loading case, the applied tearing modulus for axial tension loading from Zahoor [10] is
T = (3J/3 a ) a TE/°f z,
(22)
=• + PCM
Fig. 4. A circumferentially cracked pipe subjected to displacement controlled axial tension. where the subscripts "nc" and "c" denote the displacement component in the absence of crack and with the crack present, respectively. The solutions taken from ref. [11,17,21] are Anc=Lo/E+~¢.oL(o/OO)
n,
o=P/2frRt,
( a J / a a ) a T = ( a J / a a ) p - (OJ//aP).(OA//aa)p X [C M + ( O A / O P ) . ] - 1
(23)
and
A T = A + CMP.
(24)
Now, the axial displacement of the pipe, A, can be split into two parts as (25)
(26)
A¢ = f 3 ( a c , R / t ) P / E
+~I/",0ah3(o/,~, n,
R/t)(P/Po)",
(27)
where f3 as given by Zahoor [15,22] is
f3 ( @/~r, R / t ) = (1/Trt)
[4.9348(O/~')2
+ 30.0618A (O/or) 35 + 59.3854A (O/or) 6"24 + 56.083A 2 ( O/~r ) 5 + 255.2A 2 (O/~r)7.74
+ 331.83A2( O/~r)l°48].
where
a = a.c + a o
A T
(28)
The quantity A is defined in eq. (14). The procedure for the plastic zone adjustment to the crack length (ae) and the function h 3 are defined in ref. [18]. Similarly, the J integral for a throughwall crack pipe in axial tension with the improved J estimation [17,11,21] is
J=fl(ae, R/t)p2/E +al/t"+a)oo%C(a/b)hl(@/cr, R / t , n ) ( e / P o ) "+a
(29)
A. Zahoor / Evaluation of throughwall crack pipes
16
In the above, fl may be conveniently expressed in closed form, Zahoor [15,22], as
fl (O/~r, R / t ) = (O/~r) Ft2/4Rt 2 Ft = 1 + A [5.3303(O/~r) ls + 18.773(O/¢r)4 24].
(30)
The values of h 1 are given by Kumar et al. [17]., Following the procedure for the bending loading case, the various partial derivatives are obtained as
(OJ/3P)~ = ( 2 / P ) J , + [(n + 1 ) / P ] J 2,
(31)
(~A//~P)a=(AI q-A3)//P+tI(A2+A4)//P ,
(32)
(aJ/aa)p=[J(a+aa)-J(a)]/aale
(33)
and
(~)A/Oa)p=[Ac(a+Aa)--A.(a)]/Aa[p.
(34)
Substituting eqs. (31) through (34) into (23) gives the applied tearing modulus for displacement controlled axial loading.
5. Evaluation of displacement controlled pipe bending test results The solutions derived in the preceding section were used to evaluate the pipe test results described in refs. [8,19]. An objective of this evaluation was to assess the accuracy of the predictions from the improved estimation scheme. Ref. [19] presents data for nine throughwall crack pipes tested in displacement controlled fourpoint bending. The pipes were 8 inch in diameter and
made of A106 GrB ferritic steel. The throughwall crack lengths varied from 20% to 33% of the pipe circumference. One throughwall crack pipe test from ref. [8] was also evaluated; the pipe was 4 inch diameter and made of Type 304 austenitic steel. The throughwall crack length for the austenitic pipe test, Test 3T, was 29% of the pipe circumference. Test 3T was the first pipe test in which crack instability was successfully predicted from a pretest J / T analysis [7,8]. The R a m b e r g - O s g o o d parameters for the A106 GrB ferritic steel and Type 304 stainless steel were taken from Zahoor [11,21]. The constants for the A106 GrB steel are: a = 3 . 8 , n = 4 . 0 , o0 = 37,000 psi, and E = 27.0 x 106 psi. The constants for the stainless steel are a = 4.7, n = 3.8, oo = 34,500 psi, and E = 28.3 x 1 0 6 psi. These constants were obtained from the truestress-true-strain curve for the pipe material and represent best fit for the entire range of the available data [11]. Table 1 summarizes the test conditions and results. As indicated, three of the nine ferritic pipe tests were conducted with a stiff test system typical of laboratory testing. The remaining six ferritic steel pipe tests were conducted with various combinations of test system compliance and crack length. Pipe Test 13, 14, 15, and 3T showed unstable ductile crack extension. Table 1 also summarizes the nature of final crack extension and the maximum moment for each pipe test. Predictive analyses were performed for all ten pipe tests. The objective of the analyses was to predict (1) whether crack growth would occur unstably and (2) the
Table 1 Stability predictions for A106 GrB ferritic steel and type 304 austenitic steel pipe tests Experiment number
(9/~r
3 7 8 10 11 12 13 14 15 3T
0.300 0.328 0.302 0.262 0.248 0.232 0.215 0.192 0.210 0.290
Pipe length (inch)
48 48 48 48 42 42 42 42 42 60
Predictions (improved EPRI estimation)
Additional compliance (rad/in-lb)
Test results Stable/ unstable
Maximum moment (in-kip)
Stable/ unstable
2.47 x 10- 8 2.47x 10 8 2.47X 10 8 3.17 x 10 - 7 4.81 x 10 v 4.81x10 7 4.91× 10 7 4.81 x 10- 7 4.83 × 10 7 4.36 × 10 7
Stable Stable Stable Stable Stable Stable Unstable Unstable Unstable Unstable
999 851 830 1,095 1,133 1,178 1,236 1,367 1,273 205
Stable Stable Stable Unstable Unstable Unstable Unstable Unstable Unstable Unstable
Maximum moment (in-kip)
1,083 1,241 1,171 1,184 1,455 1,279 255
A. Zahoor / Evaluation of throughwall crack pipes
The i n p u t to the J / T analysis were the initial crack length, pipe length, additional (spring) compliance, pipe size, pipe material properties (tensile a n d fracture). The applied loading was increased a n d the corresponding applied J a n d T pairs were calculated. The J - T pairs a n d the material curve (fig. 5) were plotted o n a J / T space to determine the intersection of the applied a n d material curves. Crack growth was predicted to be stable for those tests where intersection was n o t found. Table 1 also summarizes the analysis predictions. As shown, the m o d e of failure is predicted correctly for all pipe tests except for tests 10, 11, a n d 12. F u r t h e r e x a m i n a t i o n of predictions for these three tests (table 2) showed that the a m o u n t of crack growth at the onset of instability was greater t h a n the final crack growth in the pipe test. This suggests that h a d these tests been continued, the pipe would have failed b y crack instability. Table 2 also shows that the predicted b e n d i n g m o m e n t s are in good agreement with the experiment for pipes that-failed b y crack instability. These test predictions provide additional validation of the improved J estimation scheme. Parametric analyses were also performed to determine the effect of system compliance, crack size, a n d pipe size o n crack stability. Because b e n d i n g is of p r i m a r y interest, numerical results are generated for b e n d i n g case; similar results are expected for the axial tension loading. Predictive analyses were performed first for the crack size used in pipe test n u m b e r 13. Fig. 5 shows the
20,000
O/1T= 0.215
16,000
(~) (~) (~
CM=O0 (Load Control) CM = 4.91 X 10 -7 rad/in-lb (Compliant Test Machine) CM = 2.47 X 10 -8 rad/inqb (Rigid Test Machine)
12,000 Without Crack Growth ~
--
With Crack Growth
8,000
4,000
01 -50
"
I
I
50
I
I
I
I
150 250 Tearing Modulus
I 350
17
I 450
Fig. 5. Illustration of the effect of compliance on crack stability.
b e n d i n g m o m e n t a n d crack growth at the onset of crack instability. The applied J a n d the tearing modulus were calculated using the formulation presented in the preceding section.
Table 2 Maximum moment and crack growth at maximum moment predictions for A106 GrB ferritic steel and Type 304 austenitic steel pipe tests Experiment number
3 7 8 10 11 12 13 a) 14 a) 15 a) 3T a)
Test results
Predictions using improved estimation scheme
Maxium moment (in-kip)
Crack growth at instability (inch)
Instability moment (in-kip)
Crack growth at instability (inch)
999 851 830 1,095 1,133 1,178 1,236 1,367 1,273 205
1.101 0.576 0.738 0.733 1.120 1.150 0.796 0.643 0.995 0.750
1,083 1,241 1,171 1,184 1,455 1,279 225
2.10 1.38 1.37 0.71 0.84 0.83 0.73
a) Test showed unstable crack extension. b) Percent error = (predicted m o m e n t - experimental moment) × lO0/experimental moment.
Percent error b)
- 4.0 + 6.5 + 0.5 + 10.0
18
A. Zahoor / Evaluation of throughwall crack pipes
20,000
20,000
9 16,000
12,000 "¢
8,000
~
16,000 C M = 4.91 X 10 - 7 rad/in-lb
C M = 4.91 X 10 - 7 rad/in-lb
(~) 0 / i t = 0.215
O/'tI = 0.215
(~ O/l"r=0.30 (~) 0/1/=0.35 @ O/~" = 0.40
(~
28 inch pipe
k 8,000
4,000
4,000
13 --50
,7,22
12,000
-=
I 50
[
1 150
I
I 250
I
I 350
I
I 450
-50
Tearing Modulus
Fig. 6. Illustration of the effect of crack size on crack stability.
influence of the additional system compliance on the applied J - T behavior. On this figure, curve 1 represents the load control (C M ~ oo) condition, curve 2 is for the compliance used in the pipe test number 13, and curve 3 is for a rigid test machine. Two observations can be made from these results. First, as system stiffness increases, the likelihood of crack stability increases and larger amounts of crack growth would have to occur before instability. The second observation is that the crack growth correction shifts the applied J - T curves to the left, especially for reduced compliance. Consequently, the potential for stability increases when the analysis includes crack growth and compliance decreases. This is best illustrated for the curve 3. In this case, the applied J - T curve that does not include crack growth correction predicts crack instability, whereas the properly corrected curve shows that crack will be stable. Fig. 6 illustrates the effect of crack size on the stability of crack growth. Here, curve 1 is for the pipe test 13. As crack length increases, the crack tends to be more stable. Fig. 7 shows the influence of pipe size on the applied J - T curve, Here, crack length (O/,r) is held constant for all pipe sizes. In this case, the curve 1 is for the pipe test number 13. From these results it can be concluded
50
t
I 150
I
I 250
I
I 350
I 450
Tearing Modulus
Fig. 7. Illustration of the effect of pipe size on crack stability.
that as pipe size increases, the amount of stable crack growth increases. The effect of pipe size on crack stability is not as significant as the effect of crack length or system compliance. The trends of the parametric results are qualitatively similar to those in previous work [8,11]. In contrast to the previous work, the formulation of this paper allows quantitative results for the instability load for displacement controlled loading on flawed pipes.
5. Discussion Solutions for compliant bending and axial loading were developed for throughwaU flaws in piping. Experimental data from pipes in bending were used to confirm the compliant loading solutions. These solutions, which included the improved estimation scheme to compute J and T, were shown to have good accuracy when used to predict the experimental results. This comparison provides additional confirmation for the accuracy of the improved J estimation scheme. These solutions can be used for the entire range of elastic-plastic loading, from linear elastic, contained yielding, to large scale yielding of the crack section. Further the development provides a procedure that can be used to obtain accurate results for crack stability in
A. Zahoor / Et~aluation of throughwall crack pipes pipes u n d e r displacement controlled loading. A l t h o u g h the applied tearing modulus solutions are derived for circumferential throughwall cracks, the m e t h o d o l o g y a n d procedure developed here is also applicable to part-throughwall cracks.
Acknowledgement The work described in this p a p e r was supported in p a r t by the Electric Power Research Institute u n d e r research project R P 2457-8. The a u t h o r wishes to t h a n k Dr. Douglas M. Norris for support and e n c o u r a g e m e n t d u r i n g this work.
6. References [1] American Society of Mechanical Engineers Boiler and Pressure Vessel Code, 1983 Edition and Addenda, Section XI, Paragraph IWB-3640; see also 1986 Edition, pp. 116-123. [2] A. Zahoor, H.S. Mehta, S. Yukawa, R.M. Gamble and S. Ranganath, Flaw evaluation procedures and standards for ferritic piping, Final Report on Research Project RP175751, Electric Power Research Institute, Palo Alto, California (November 1985). [3] Evaluation of Flaws in Austenitic Steel Piping, prepared by Section XI Task Group for Piping Flaw Evaluation, ASME Boiler and Pressure Vessel Code Committee, EPRI Report NP 4690-SR (April 1986), ASME J. Pressure Vessel Technology 108 (1986) 352-367. [4] Piping Review Committee - Evaluation of potential for pipe breaks, NUREG/CR-1061, Vol. 3 (Appendix A), US Nuclear Regulatory Commission, Washington, DC (1984) [5] P.C. Paris, H. Tada, A. Zahoor and H. Ernst, A treatment of the subject of tearing instability, NUREG-0311, Nuclear Regulatory Commission, Washington, DC (1977). [6] H. Tada, P.C. Paris and R.M. Gamble, Stability analysis of circumferential cracks in reactor piping systems, NUREG/CR-0838, Nuclear Regulatory Commission, Washington, DC (1979). [7] A. Zahoor and M.F. Kanninen, A plastic fracture mechanics prediction of fracture instability in a circumferentially cracked pipe in bending - Part I: J-integral analysis, ASME J. Pressure Vessel Technology 103 (1981) 352-358. [8] A. Zahoor, G. Wilkowski, I. Abou-Sayed, C. Marschall, D. Broek, S. Sampath, H. Rhee and J. Ahmad, Instability predictions for circumferentially cracked type-304 stainless steel pipe under dynamic loading, ed. M.F. Kanninen, EPRI NP-2347, Vols. 1 and 2, Electric Power Research Institute, Palo Alto, California (April 1982).
19
[9] A. Zahoor and M.F. Kanninen, A plastic fracture instability analysis of wall breakthrough in circumferentially cracked pipe subjected to bending loads, ASME J. Pressure Vessel Technology 103 (1981) 194-200. [10] A. Zahoor and D.M. Norris, Ductile fracture of circumferentially cracked type-304 stainless steel pipes in tension, ASME J. Pressure Vessel Technology 106 (1984) 399-404. [11] A. Zahoor and R.M. Gamble, Evaluation of flawed pipe experiments, Final Report on EPRI Project RP 2457-8, Electric Power Research Institute, Palo Alto, California (July 1986). [12] A. Zahoor, J-integral and tearing modulus estimates for circumferentially cracked pipes in torsion, ASME J. Pressure Vessel Technology 108 (1986). [13] A. Zahoor, Advanced studies of the stability of circumferentially cracked pipes, Final Report on EPRI Project Tl18-9-1, Electric Power Research Institute, Palo Alto, California (December 1982). [14] A. Zahoor and R.M. Gamble, Leak-before-break analysis for BWR recirculation piping having cracks at multiple weld locations, EPRI NP-3522-LD, Electric Power Research Institute, Palo Alto, California (April 1984). [15] A. Zahoor, Fracture of circumferentially cracked pipes, ASME J. Pressure Vessel Technology 108 (1986). [16] M.D. German and V. Kumar, Elastic-plastic analysis of crack opening, stable growth and instability behavior in flawed 304 stainless steel piping, ASME PVP Vol. 58 (American Society of Mechanical Engineers, 1982) pp. 109-141. [17] V. Kumar, M.D. German, W.W. Wilkening, W.R. Andrews, H.G. deLorenzi and D.F. Mowbray, Advances in elastic plastic fracture analysis, EPRI NP 3607, Electric Power Research Institute, Palo Alto, California (August 1984), [18] V. Kumar, M.D. German and C.F. Shih, An engineering approach for elastic-plastic fracture analysis, EPRI NP 1931, Electric Power Research Institute, Palo Alto, California (July 1981). [19] M.G. Vassilaros, R.A. Hays, J.P. Gudas and J.A. Joyce, J integral tearing instability analyses for 8-inch diameter ASTM A106 steel pipe, NUREG/CR-3740, Nuclear Regulatory Commission, Washington, DC (April 1984). [20] A. Okamoto and D.M. Norris, Sensitivity analysis of flaw growth instability, Proc. CSNI/NRC Workshop on Ductile Piping Fracture Mechanics, Southwest Research Institute, San Antonio, Texas, 1984, to appear in a NUREG Report. [21] A. Zahoor, Evaluation of J-integral estimation scheme for flawed throughwall pipes, Nucl. Engrg. Des. 100 (1987) 1-9, in this issue. [22] A. Zahoor, Closed form expressions for fracture mechanics analysis of cracked pipes, ASME J. Pressure Vessel Technology 107 (1985) 203-205.