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An approximate method to assess the fracture failure probability of a spherical tank
Reliability Engineering and System Safety 24 (1989) 1-9
An Approximate Method to Assess the Fracture Failure Probability of a Spherical Tank
Cao Tianjie Taiyuan Heavy Machinery Institute, Taiyuan, Shanxi, People's Republic of China (Received 5 February 1988; accepted 21 March 1988)
ABSTRACT This paper presents an approximate method to assess the fracture failure probability of a spherical tank. The method derives the general formulae to calculate the first foi~r moments of the crack on the fatigue growth occasion by use of the power series of the crack. The paper then derives the first four moments of the crack tip opening displacement according to Japan WES2805 code and the probability distribution of the crack tip opening displacement is determined through the skewness-kurtosis curve. Lastly the fracture failure probability of a spherical tank is estimated on the interference theory and the presented method is compared with the Monte Carlo simulation technique.
NOTATION a ao
C,m
e
E
EE']
£(')
f ( x l , x2, ..., xn)
Effective size of a surface crack Initial effective size of a surface crack Material constants Strain Young's modulus Expected, or mean, value Probability density function; subscript indicates variable Joint probability density function of x 1, x 2..... x. 1
Reliability Engineering and System Safety 0951-8320/89/$03"50 © 1989 Elsevier Science Publishers Ltd, England. Printed in Great Britain
2
Cao Tianjie
gx(')
Cumulative distribution function; subscript indicates variable Coefficient of stress concentration caused by alternate edge Coefficient of stress concentration Coefficient of stress concentration caused by angle deformation. Stress intensity range N u m b e r of load repetition Fracture failure probability Probability o f event ['] Probability density function Crack tip opening displacement Critical crack tip opening displacement Constant = 3-141 59 ... M e m b r a n e stress Stress range during cyclic loading
G K, AK N
Pf P[] PDF 6
O"m
Aa
1 INTRODUCTION Calculation o f the fracture failure probability o f a spherical tank is that on the basis o f the fracture model of deterministic fracture mechanics we first investigate and collect data of every indeterminate parameter in the model, then fit a probability distribution suitable to the data, after that establish a failure model and assess the failure probability according to the interference theory. If the deterministic C O D failure criterion 8 > 6¢ is taken, for example, and Japan WES2805 code is adopted, we have 6 = 3"5ca
where e
= Ktffm/E,
Kt =
(1)
q + Kh+ Kw
~" 1"5, taking the height of a welding seam into account q = (1"0, not taking the height o f a welding seam into account According to the Paris law d a M N = C(AK)", crack size a can be determined as follows a = ao/[1 -- ((m/2) -- 1)CK~' Aa"rtm/2Na~om- 2)/212/(,.- 2) (2) I f a o, C, K h, Kw, Aa, o-m and 6¢ are considered r a n d o m variables, the accurate calculation of the fracture failure probability Pf = P[6 > 6¢] is to calculate a multidimensional probability integral fnf(ao,
C, K h, K w, Aa, am, 3¢) dfl
Fracture failure probability of a spherical tank
3
Because the integral is multidimensional, d e t e r m i n a t i o n o f ~, the domain o f the failed surface represented by the limit state, is quite difficult, evaluation of such an integral is impossible practically. So the fracture failure probability now is calculated by approaches. These approaches are mainly the linearization method, the m e t h o d o f moments, the method of the first two-order moments and the Monte Carlo method. Although the methods have merits, there are some disadvantages. The linearization m e t h o d and the method of the first two-order moments would bring bigger errors. Because the m e t h o d of moments still calculates multidimensional integrals, though the integral domain can be determined, the calculation of the integral is still very difficult. The Monte Carlo method, which is well known and widely employed, is more accurate, but it requires a lot o f time and computer costs become very great. Moreover, although, we can theoretically prove that, so long as there are randoms obeying a uniform distribution between 0 and 1, we can get other randoms obeying an arbitrary distribution function, we cannot obtain arbitrary randoms easily except for a few kinds of standard distribution functions. Because o f this, the applied range o f the Monte Carlo method is confined. There are some drawbacks to the above methods, so this paper will present an approximate m e t h o d to estimate the fracture failure probability. Examples prove that the method is feasible.
2 I N T R O D U C T I O N TO T H E M E T H O D It m a y be seen from eqns (1) and (2) that the key to calculate the first four moments o f the r a n d o m variable 6 is whether the first four moments o f the crack a can be determined. For simplicity, the paper only considers m = 4 in eqn (2). If m = 4, eqn (2) can be simplified as follows a -- ao/(1 - CKt4 Aa4zc2Nao)
(3)
We may see from eqn (3) that although a is a function ofao, C, K t, Air, and N, CKt4Aa*zr2Nao comes out in one product. A n d so we can temporarily consider the product as one r a n d o m variable x, this is a = ao/(1 - x)
0 _< x < 1
(4)
where x = CKt4 A a * n 2 N a o = CNzr2(q + K h + Kt) 4 Aa4ao . Expressing eqn (4) in an infinite series, we have oo
a=ao(l+x+x
z + x 3 + - ' ' ) = ao ) ' x i /......,..d
i=0
(5a)
4
Cao Tianjie
Similarly, we can express the ith power of a in an infinite series as below
aj
=(j-
2
aJo
(j + i-- 1)! xi
1)r
ir
(5b)
j = 2 , 3 ..... n
i=0
Combination of eqns (5a) and (5b) is as follows
aj_
aJo
(j + i - 1)! xi
( ~
i!
j = 1, 2, 3, ..., n
(5)
i~0
If we only take the first M powers of x into account, we have M
at_ a ~
2
(j+i-1)!xi
(j-l)!
i! i=0 M
~ - ~ ( j + i-1)! CiKt4iAcr4ircZiNiaio+j ir
1
- ( j - 1)! ,._,
j = 1, 2, 3 , . . . , n
(6)
i=0
Assuming that random variables C, K h, Kw, Aa, and ao are independent of each other and their moments all exist, we can easily determine the first four moments of the crack a from eqn (6) as follows M
V (j E[aJ] - (j--1 1)~/__a
+ ii!- 1)! E[Ci]E[Ktgi]E[Aa4i]rcZiNiE[aio+j
]
i=0
j = 1, 2, 3, 4
(7)
where E[K~ i] = E[(q + K h + K~']
~(4i)'q =
4i-s
(4i~St)tt!
E[K~-']E[K~]
s=Ot=O
From eqns (1) and (7) we can obtain the first four moments of the crack tip opening displacement 6 as follows
El,5 j] = 3"5JE[ei]E[a j] = (3"5/E)iE[KJt]E[aJm]E[aJ ]
j = 1, 2, 3, 4
(8)
If the critical crack tip opening displacement fie is considered a random variable independent of 6, the first four moments of6~ - 6 can be determined as below J
E[(6~- 6)j] = "--'~' ( - 1)~/! E[6jc_,]E[6, ] ,/ ~( j - i)!i! i=0
j = 1, 2, 3, 4
(9)
Fracture failure probability of a spherical tank
5
Then we can determine an approximate probability density function g(6¢ - 6) of 6~ - 3 which belongs to Pearson distributions from eqn (10) 1'2 6 c -- 6 + b 3 g'(6~ -- 6) = bo + b l ( 6 c - 6) + b 2 ( 6 ~ -
;o
6) 2 g(6~ - 6)
(10)
and calculate the failure probability from the following equation
Pf= P[6¢-6
_
g(6~-6)d(6~-6)
(11)
But in our experience, calculation of the failure probability in this way is often very difficult. The main reasons are that the approximate probability density function g(6~ - 6) determined by the first four moments of 6~ - 3 sometimes cannot pledge the existence of integral ~_+~ g(3~ - 6) d(6¢ - 6) and that although the integral sometimes is in existence, it cannot be calculated easily (for example, g(6~- 6) has single points and at the same time the integral must be calculated by means of numerical methods). So this paper confines the probability distribution function of 6 within the range of the exponential distribution, the Weibull distribution, the normal distribution, the log-normal distribution and the Gamma distribution and then calculates the skewness and kurtosis of 3 by use of the first four moments of 6 from the following formulae %/~i =
f12
113
#33/2
(12)
= #__.3" #2
(13)
where/~i (i = 2, 3, 4) is the ith order central moment of 6. According to the values of the sknewness ~ and the kurtosis f12, we can decide the probability distribution function of 6 through the skewness-kurtosis curve,3 and determine the distribution parameters by the use of the first two moments of 6. Lastly the fracture failure probability Pf can be estimated from eqn (14) Pf = P[6¢ -- 6 < O] ~
fa(6)F6o(6)d6
(14)
3 EXAMPLES
3.1 Example 1 In this example, we will calculate" the fracture failure probability varying with time of a spherical tank by the use of the presented method and the
6
Cao Tianfie TABLE l R a n d o m Variables a n d their P D F with P a r a m e t e r s
Random variables (l 0
PDF
Ao-
;t = 0'65 m m
2e -'~X/(e ~.:,l_ e - '~:'")
1
O"m
Parameters
x/~a
exp
[ fx- ) :]
--exp L[
lge
j
2a2
J
/~ = 144-2 N / m m 2, tr = 19.6 N / m m 2
/1 = 86"3 N / m m 2, a = 19.6 N / m m 2
x/2gtr
----exp/
20.2
I tlg -.t2]
C
x/2xx~
K.
flkflj
exp
Kw
fl\fl]
6c
fl \ f l /
l, xt = 0, x u = 34 m m
d
L
[(TJ
/~ = - 1 " 1 7 2 x 10 -3, or= 7-023 x 10 - 6
-
~
:t=1'16, fl=O.134
exp
-
~
:t=2'98, fl=0'475
exp
-
:~= 1"91, fl = 0-138 m m
Monte Carlo method and compare the two methods. Basic data are as follows.
Constants: Young's modulus E = 2 . 0 6 x 105 N/mm2; wall thickness t = 34 mm; number of cycles every year is 252; the height of welding seams is not to be taken into account. Random: Probability density functions and their parameters of the random variables are shown in Table 1; the probability for a crack of which the effective size is x not to be found is Po(x)=exp(-A(x-Xo)) (A = 0.1575 m m - 1, Xo = 1.0 mm); in addition, assuming that surface cracks are of half circle during fatigue growth, that is, the depth of a surface crack is equal to one half of its length. The fracture failure probabilities varying with time estimated by the presented method and the Monte Carlo method are plotted in Fig. 1. The computer time taken by the presented method is approximately seven times less than that taken by the Monte Carlo method. 3.2 Example 2 All constants and probability density functions and parameters of the random variables in example 1 are not changed except
Fracture failure probability of a spherical tank
7
1_~
>10
.Q
2
"
U.
1
0
I
5
I
10
I
I
15 20 Time (years)
I
25
I
30
Fig. 1. Failure probability varying with time: - - - - - , Monte Carlo method; method.
10
o
07
tl .t3
o t3.
2 it
0
Fig. 2.
I
5
I
10
I
I
15 20 Time (years)
I
25
I
30
Failure probability varying with time.
, presented
8
Cao 7-'ianfie
Po(x) = exp ( - A(x - xo)) (A = 0"1575 m m - 1 Xo = 1-0 mm) is changed into Po(x) = 1 - (1 - Xo/X) r (xo = 2-5 mm, r = 0"5). In this example, when a r a n d o m of initial surface crack a o of which the depth is more than 2"5 m m is produced, ao must be solved from the equation below 2.5 e" ao R = ~x, xV ( - )~x)dx + ~2.5 [ 1 - (1 - Xo/X)r] exp ( - )ox)dx 2.5 ~xl exp ( - 2x) dx + ~75 [1 - (1 - Xo/XY ] exp ( - 2x) dx
where R is a r a n d o m obeying uniform distribution between 0 and 1. Because the equation is more difficult to solve and to calculate the failure probability a host of randoms have to be produced if the Monte Carlo method is employed, calculation of the failure probability by the Monte Carlo method is almost impossible though P ( x ) = 1 - ( 1 - Xo/X) ~ is very simple and common. But if the presented method is employed, we can get the failure probability easily. The failure probability varying with time estimated by the presented method is plotted in Fig. 2.
4 CONCLUSIONS 1.
2.
3.
Because the presented m e t h o d can calculate the arbitrary order moments of the crack tip opening displacement accurately and avoid the complicated multidimensional integral as carried out in the method o f moments, the first four moments o f the crack tip opening displacement can be used to approximately estimate the failure probability. The presented method takes less time than the Monte Carlo method to estimate the failure probability. Particularly the case where failure probability is smaller (such as 10- 5 ,,~ 10-6), the presented method is very meaningful. It may be seen from example 2 that the presented method can treat more kinds of probability distribution function than the Monte Carlo method.
REFERENCES 1. Cramer, H., Mathematical Methods of Statistics, ch. 19. Princeton University Press, NJ, 1946. 2. Kendall, M. & Stuart, A., The Advanced Theory of Statistics, ch. 6. Charles Griffin, London, 1977. 3. Dai Shusen et al., Reliability Test and its Statistical Analyses [in Chinese], Vol. 2. National Defence and Industry Publishing House, Beijing, 1984, pp. 333-9.
Fracture failure probability of a spherical tank
9
EDITOR'S NOTE We would find it interesting to develop the paper further by means of an example: 'Let us consider a real spherical tank with two nozzles and a manhole with a foundation. Can the technique be applied?' We have asked the authors to consider this question, to give examples or to debate any reason for or against. We would welcome any c o m m e n t from readers. R. W. van Otterloo
AUTHOR'S REPLY There are some papers such as those by Urabe & Yoshitake 1 and Nakano, 2 in which the Monte Carlo m e t h o d is used to assess the failure probability of a pressure vessel. In m y opinion, analysis o f the failure probability of a spherical tank can at least be taken as a kind of reference. Cao Tianjie References 1. Urabe, N. & Yoshitake, A., Reliability analysis in welded structure---case study for brittle fracture of high pressure gas storage tank [in Japanese]. Pressure Engineering, 19(2) (1981) 87-97. 2. Nakano, M., Simulation of hydrostatic test of a pressure vessel containing flaws [-in Japanese]. Pressure Engineering, 23(1) (1985) 15-20.
An Approximate Method to Assess the Fracture Failure Probability of a Spherical Tank
Cao Tianjie Taiyuan Heavy Machinery Institute, Taiyuan, Shanxi, People's Republic of China (Received 5 February 1988; accepted 21 March 1988)
ABSTRACT This paper presents an approximate method to assess the fracture failure probability of a spherical tank. The method derives the general formulae to calculate the first foi~r moments of the crack on the fatigue growth occasion by use of the power series of the crack. The paper then derives the first four moments of the crack tip opening displacement according to Japan WES2805 code and the probability distribution of the crack tip opening displacement is determined through the skewness-kurtosis curve. Lastly the fracture failure probability of a spherical tank is estimated on the interference theory and the presented method is compared with the Monte Carlo simulation technique.
NOTATION a ao
C,m
e
E
EE']
£(')
f ( x l , x2, ..., xn)
Effective size of a surface crack Initial effective size of a surface crack Material constants Strain Young's modulus Expected, or mean, value Probability density function; subscript indicates variable Joint probability density function of x 1, x 2..... x. 1
Reliability Engineering and System Safety 0951-8320/89/$03"50 © 1989 Elsevier Science Publishers Ltd, England. Printed in Great Britain
2
Cao Tianjie
gx(')
Cumulative distribution function; subscript indicates variable Coefficient of stress concentration caused by alternate edge Coefficient of stress concentration Coefficient of stress concentration caused by angle deformation. Stress intensity range N u m b e r of load repetition Fracture failure probability Probability o f event ['] Probability density function Crack tip opening displacement Critical crack tip opening displacement Constant = 3-141 59 ... M e m b r a n e stress Stress range during cyclic loading
G K, AK N
Pf P[] PDF 6
O"m
Aa
1 INTRODUCTION Calculation o f the fracture failure probability o f a spherical tank is that on the basis o f the fracture model of deterministic fracture mechanics we first investigate and collect data of every indeterminate parameter in the model, then fit a probability distribution suitable to the data, after that establish a failure model and assess the failure probability according to the interference theory. If the deterministic C O D failure criterion 8 > 6¢ is taken, for example, and Japan WES2805 code is adopted, we have 6 = 3"5ca
where e
= Ktffm/E,
Kt =
(1)
q + Kh+ Kw
~" 1"5, taking the height of a welding seam into account q = (1"0, not taking the height o f a welding seam into account According to the Paris law d a M N = C(AK)", crack size a can be determined as follows a = ao/[1 -- ((m/2) -- 1)CK~' Aa"rtm/2Na~om- 2)/212/(,.- 2) (2) I f a o, C, K h, Kw, Aa, o-m and 6¢ are considered r a n d o m variables, the accurate calculation of the fracture failure probability Pf = P[6 > 6¢] is to calculate a multidimensional probability integral fnf(ao,
C, K h, K w, Aa, am, 3¢) dfl
Fracture failure probability of a spherical tank
3
Because the integral is multidimensional, d e t e r m i n a t i o n o f ~, the domain o f the failed surface represented by the limit state, is quite difficult, evaluation of such an integral is impossible practically. So the fracture failure probability now is calculated by approaches. These approaches are mainly the linearization method, the m e t h o d o f moments, the method of the first two-order moments and the Monte Carlo method. Although the methods have merits, there are some disadvantages. The linearization m e t h o d and the method of the first two-order moments would bring bigger errors. Because the m e t h o d of moments still calculates multidimensional integrals, though the integral domain can be determined, the calculation of the integral is still very difficult. The Monte Carlo method, which is well known and widely employed, is more accurate, but it requires a lot o f time and computer costs become very great. Moreover, although, we can theoretically prove that, so long as there are randoms obeying a uniform distribution between 0 and 1, we can get other randoms obeying an arbitrary distribution function, we cannot obtain arbitrary randoms easily except for a few kinds of standard distribution functions. Because o f this, the applied range o f the Monte Carlo method is confined. There are some drawbacks to the above methods, so this paper will present an approximate m e t h o d to estimate the fracture failure probability. Examples prove that the method is feasible.
2 I N T R O D U C T I O N TO T H E M E T H O D It m a y be seen from eqns (1) and (2) that the key to calculate the first four moments o f the r a n d o m variable 6 is whether the first four moments o f the crack a can be determined. For simplicity, the paper only considers m = 4 in eqn (2). If m = 4, eqn (2) can be simplified as follows a -- ao/(1 - CKt4 Aa4zc2Nao)
(3)
We may see from eqn (3) that although a is a function ofao, C, K t, Air, and N, CKt4Aa*zr2Nao comes out in one product. A n d so we can temporarily consider the product as one r a n d o m variable x, this is a = ao/(1 - x)
0 _< x < 1
(4)
where x = CKt4 A a * n 2 N a o = CNzr2(q + K h + Kt) 4 Aa4ao . Expressing eqn (4) in an infinite series, we have oo
a=ao(l+x+x
z + x 3 + - ' ' ) = ao ) ' x i /......,..d
i=0
(5a)
4
Cao Tianjie
Similarly, we can express the ith power of a in an infinite series as below
aj
=(j-
2
aJo
(j + i-- 1)! xi
1)r
ir
(5b)
j = 2 , 3 ..... n
i=0
Combination of eqns (5a) and (5b) is as follows
aj_
aJo
(j + i - 1)! xi
( ~
i!
j = 1, 2, 3, ..., n
(5)
i~0
If we only take the first M powers of x into account, we have M
at_ a ~
2
(j+i-1)!xi
(j-l)!
i! i=0 M
~ - ~ ( j + i-1)! CiKt4iAcr4ircZiNiaio+j ir
1
- ( j - 1)! ,._,
j = 1, 2, 3 , . . . , n
(6)
i=0
Assuming that random variables C, K h, Kw, Aa, and ao are independent of each other and their moments all exist, we can easily determine the first four moments of the crack a from eqn (6) as follows M
V (j E[aJ] - (j--1 1)~/__a
+ ii!- 1)! E[Ci]E[Ktgi]E[Aa4i]rcZiNiE[aio+j
]
i=0
j = 1, 2, 3, 4
(7)
where E[K~ i] = E[(q + K h + K~']
~(4i)'q =
4i-s
(4i~St)tt!
E[K~-']E[K~]
s=Ot=O
From eqns (1) and (7) we can obtain the first four moments of the crack tip opening displacement 6 as follows
El,5 j] = 3"5JE[ei]E[a j] = (3"5/E)iE[KJt]E[aJm]E[aJ ]
j = 1, 2, 3, 4
(8)
If the critical crack tip opening displacement fie is considered a random variable independent of 6, the first four moments of6~ - 6 can be determined as below J
E[(6~- 6)j] = "--'~' ( - 1)~/! E[6jc_,]E[6, ] ,/ ~( j - i)!i! i=0
j = 1, 2, 3, 4
(9)
Fracture failure probability of a spherical tank
5
Then we can determine an approximate probability density function g(6¢ - 6) of 6~ - 3 which belongs to Pearson distributions from eqn (10) 1'2 6 c -- 6 + b 3 g'(6~ -- 6) = bo + b l ( 6 c - 6) + b 2 ( 6 ~ -
;o
6) 2 g(6~ - 6)
(10)
and calculate the failure probability from the following equation
Pf= P[6¢-6
_
g(6~-6)d(6~-6)
(11)
But in our experience, calculation of the failure probability in this way is often very difficult. The main reasons are that the approximate probability density function g(6~ - 6) determined by the first four moments of 6~ - 3 sometimes cannot pledge the existence of integral ~_+~ g(3~ - 6) d(6¢ - 6) and that although the integral sometimes is in existence, it cannot be calculated easily (for example, g(6~- 6) has single points and at the same time the integral must be calculated by means of numerical methods). So this paper confines the probability distribution function of 6 within the range of the exponential distribution, the Weibull distribution, the normal distribution, the log-normal distribution and the Gamma distribution and then calculates the skewness and kurtosis of 3 by use of the first four moments of 6 from the following formulae %/~i =
f12
113
#33/2
(12)
= #__.3" #2
(13)
where/~i (i = 2, 3, 4) is the ith order central moment of 6. According to the values of the sknewness ~ and the kurtosis f12, we can decide the probability distribution function of 6 through the skewness-kurtosis curve,3 and determine the distribution parameters by the use of the first two moments of 6. Lastly the fracture failure probability Pf can be estimated from eqn (14) Pf = P[6¢ -- 6 < O] ~
fa(6)F6o(6)d6
(14)
3 EXAMPLES
3.1 Example 1 In this example, we will calculate" the fracture failure probability varying with time of a spherical tank by the use of the presented method and the
6
Cao Tianfie TABLE l R a n d o m Variables a n d their P D F with P a r a m e t e r s
Random variables (l 0
Ao-
;t = 0'65 m m
2e -'~X/(e ~.:,l_ e - '~:'")
1
O"m
Parameters
x/~a
exp
[ fx- ) :]
--exp L[
lge
j
2a2
J
/~ = 144-2 N / m m 2, tr = 19.6 N / m m 2
/1 = 86"3 N / m m 2, a = 19.6 N / m m 2
x/2gtr
----exp/
20.2
I tlg -.t2]
C
x/2xx~
K.
flkflj
exp
Kw
fl\fl]
6c
fl \ f l /
l, xt = 0, x u = 34 m m
d
L
[(TJ
/~ = - 1 " 1 7 2 x 10 -3, or= 7-023 x 10 - 6
-
~
:t=1'16, fl=O.134
exp
-
~
:t=2'98, fl=0'475
exp
-
:~= 1"91, fl = 0-138 m m
Monte Carlo method and compare the two methods. Basic data are as follows.
Constants: Young's modulus E = 2 . 0 6 x 105 N/mm2; wall thickness t = 34 mm; number of cycles every year is 252; the height of welding seams is not to be taken into account. Random: Probability density functions and their parameters of the random variables are shown in Table 1; the probability for a crack of which the effective size is x not to be found is Po(x)=exp(-A(x-Xo)) (A = 0.1575 m m - 1, Xo = 1.0 mm); in addition, assuming that surface cracks are of half circle during fatigue growth, that is, the depth of a surface crack is equal to one half of its length. The fracture failure probabilities varying with time estimated by the presented method and the Monte Carlo method are plotted in Fig. 1. The computer time taken by the presented method is approximately seven times less than that taken by the Monte Carlo method. 3.2 Example 2 All constants and probability density functions and parameters of the random variables in example 1 are not changed except
Fracture failure probability of a spherical tank
7
1_~
>10
.Q
2
"
U.
1
0
I
5
I
10
I
I
15 20 Time (years)
I
25
I
30
Fig. 1. Failure probability varying with time: - - - - - , Monte Carlo method; method.
10
o
07
tl .t3
o t3.
2 it
0
Fig. 2.
I
5
I
10
I
I
15 20 Time (years)
I
25
I
30
Failure probability varying with time.
, presented
8
Cao 7-'ianfie
Po(x) = exp ( - A(x - xo)) (A = 0"1575 m m - 1 Xo = 1-0 mm) is changed into Po(x) = 1 - (1 - Xo/X) r (xo = 2-5 mm, r = 0"5). In this example, when a r a n d o m of initial surface crack a o of which the depth is more than 2"5 m m is produced, ao must be solved from the equation below 2.5 e" ao R = ~x, xV ( - )~x)dx + ~2.5 [ 1 - (1 - Xo/X)r] exp ( - )ox)dx 2.5 ~xl exp ( - 2x) dx + ~75 [1 - (1 - Xo/XY ] exp ( - 2x) dx
where R is a r a n d o m obeying uniform distribution between 0 and 1. Because the equation is more difficult to solve and to calculate the failure probability a host of randoms have to be produced if the Monte Carlo method is employed, calculation of the failure probability by the Monte Carlo method is almost impossible though P ( x ) = 1 - ( 1 - Xo/X) ~ is very simple and common. But if the presented method is employed, we can get the failure probability easily. The failure probability varying with time estimated by the presented method is plotted in Fig. 2.
4 CONCLUSIONS 1.
2.
3.
Because the presented m e t h o d can calculate the arbitrary order moments of the crack tip opening displacement accurately and avoid the complicated multidimensional integral as carried out in the method o f moments, the first four moments o f the crack tip opening displacement can be used to approximately estimate the failure probability. The presented method takes less time than the Monte Carlo method to estimate the failure probability. Particularly the case where failure probability is smaller (such as 10- 5 ,,~ 10-6), the presented method is very meaningful. It may be seen from example 2 that the presented method can treat more kinds of probability distribution function than the Monte Carlo method.
REFERENCES 1. Cramer, H., Mathematical Methods of Statistics, ch. 19. Princeton University Press, NJ, 1946. 2. Kendall, M. & Stuart, A., The Advanced Theory of Statistics, ch. 6. Charles Griffin, London, 1977. 3. Dai Shusen et al., Reliability Test and its Statistical Analyses [in Chinese], Vol. 2. National Defence and Industry Publishing House, Beijing, 1984, pp. 333-9.
Fracture failure probability of a spherical tank
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EDITOR'S NOTE We would find it interesting to develop the paper further by means of an example: 'Let us consider a real spherical tank with two nozzles and a manhole with a foundation. Can the technique be applied?' We have asked the authors to consider this question, to give examples or to debate any reason for or against. We would welcome any c o m m e n t from readers. R. W. van Otterloo
AUTHOR'S REPLY There are some papers such as those by Urabe & Yoshitake 1 and Nakano, 2 in which the Monte Carlo m e t h o d is used to assess the failure probability of a pressure vessel. In m y opinion, analysis o f the failure probability of a spherical tank can at least be taken as a kind of reference. Cao Tianjie References 1. Urabe, N. & Yoshitake, A., Reliability analysis in welded structure---case study for brittle fracture of high pressure gas storage tank [in Japanese]. Pressure Engineering, 19(2) (1981) 87-97. 2. Nakano, M., Simulation of hydrostatic test of a pressure vessel containing flaws [-in Japanese]. Pressure Engineering, 23(1) (1985) 15-20.