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The influence of failure mode order on structural system reliability
Compvrers Printed
& Slrucrures
in Great
Vol. 34, No.
I, pp.
17-22,
1990
cKM5-7949po
Britain.
0
THE INFLUENCE OF FAILURE MODE ORDER STRUCTURAL SYSTEM RELIABILITY
s3.00 + 0.00
1989 Pcrgamon
Press
plc
ON
B. F. SONGand Y. S. FENG Department
of Aircraft Engineering, Northwestern
Polytechnical University, Xian, Shaanxi Province,
People’s Republic of China (Received 1 December 1988)
Abstract-The failure probability of a structural system solved by O’Ditilevsen’s narrow bound method or Y. S. Feng’s high accuracy method is influenced by the orders of the failure modes. In this paper, the O’Ditilevsen minimal upper bound and maximal lower bound can be solved by using a max-tree finding algorithm in a correlative weighted graph and an optimality criterion respectively, so that the interval width between the upper and lower bounds becomes more narrow. Using an optimality criterion similar to the above-mentioned one, the maximum failure probability of a structural system combined with the high accuracy method can be obtained. Two examples are given to illustrate the effectiveness of the method presented in this paper.
1. INTRODUCTION
less than 0.6 and the difference increases with increas-
There are two fundamental bound methods which have recently been used in the field of structural system reliability. One is the method of simple bounds. This can be expressed as
ing P,. Recently, Professor Y. S. Feng[2] presented a method with high accuracy. In this high accuracy method, the system reliability can be accurately expressed by first order probabilities, and second and third order joint probabilities. It can be expressed as
(1)
p,’
where Pi and P, indicate the failure probability corresponding to the ith failure mode and the failure probability of the structural system respectively, and the number m, in general, indicates the total number of significant failure modes. Though the expression of simple bounds is very straightforward and the computational result is accurate, the interval width is too wide for excluding the joint failure probabilities among the significant failure modes. Another method is O’Ditilevsen’s [l] narrow bounds. This can be expressed as
Q f Pi - f i=, j4
p, + PI - p,2 + p, - (p3, + p32)+ +.. .+P,-
1
,:,I
p312
P,,+max
where Piik indicates the third order joint failure probability among the ith, jth and kth modes. Besides considering the second order joint failure probabilities and the correlation of structural element strengths, the high accuracy method has also considered the third order joint probabilities so its accuracy is very high. Figure la shows the structural system of the illustrative example of [2]. The calculated results of [2] are shown in Fig. 1b, in which the two dotted lines represent O’Ditilevsen’s upper and lower bounds respectively and the solid line represents the calculation with the high accuracy method. In general, with formula (2), using O’Ditilevsen’s method, there are different bounds for different orders of structural system failure modes, and one has to find the minimal upper bound and the maximal lower bound. In this paper, the minimal O’Ditilevsen upper bound can be obtained by using a max-tree finding algorithm in a correlative weighted graph and the maximal O’Ditilevsen lower bound by using an optimality criterion. Therefore, the interval width between the upper and lower bounds becomes considerably narrower.
Fax Pii (2) 1<1
where Pii is the joint failure probability between the ith and the jth failure modes. Because O’Ditilevsen’s bounds regard the correlation among failure modes and structural element strengths respectively, the interval width between the upper and lower bounds is narrower than that of simple bounds. In [2], it has been shown that the difference between the upper and lower bounds using O’Ditilevsen’s method is quite small when Pv is 17
18
B. F.
SONG and PO(P)
(a)
Y. S.
(b)
FENG
P&)/MO)
4.44.24.0
s
I I, I, I I I I 0 0.2 0.4 0.6 0.8 top Fig. 1. (a) The structural system of the illustrative example of [2]. (b) The calculated results of the structural system shown in Fig. la. Using formula (3), their solutions may be different from each other to a small extent corresponding to the different orders of failure modes. In general, the maximum value of P may be meaningful. In this paper, the maximum P is taken as the structural system failure probability, and it can be obtained by using a similar optimality criterion. 2. THEORY AND METHOD
2. I. Graph, weighted graph and tree A graph is the set of points and lines in a plane. The graph is said to be joined when there are direct or indirect lines between any two points. Figure 2 is an example of a joined graph. In a joined graph, weighted values are usually given to the points and lines respectively and this graph is then called a weighted joined graph. A joined graph that excludes any closed path is called a tree. There may be many trees in a joined graph. That of the maximum sum of line weighted values is called the max-tree. There are many kinds of methods used to find max-trees. A simple method is presented by the authors of this paper as foflows. (i) Ordering the lines in tetms of their weighted values. The greater the weighted value, the more forward the line is. For example, if the inequalities WI,) &&I,) 3 * . . II are true, then a sequence is obtained, that is I,, l,, 13. . . 1m,where lj and O(&) indicate the ith line and its weighted value respectively, and the number m is the total number of lines. (ii) Taking the most forward two lines as the branches of the max-tree. In this example, Ii and i2 are taken as the branches of the max-tree. b
C
a
@
e
d
Fig. 2. An example of a joined graph.
(iii) Choosing the line which cannot construct a closed path with the two branches chosen in (ii) and has a greater weighted value than the third branch of the max-tree. In this example, if iS does not construct a closed path with 1, and 1, it is taken as the third branch, and if it does, it cannot be taken as the third branch, and another must be chosen. (iv) Choosing the line which cannot construct a closed path with the three branches chosen in (ii) and (iii) and has a greater weighted value than the fourth branch of the max-tree. In this example, if I., does not construct a closed path with the branches chosen in (ii) and (iii), it is taken as the fourth branch, and if it does, it cannot be taken as the fourth branch, and another must be chosen. (v) Continuing the process until all points in the weighted graph are linked by the above-mentioned branches. 2.2, The minimal O’Ditilevsen upper bound In formula (2), the upper bound is expressed as
The order of the failure modes is not influenced by the first term on the right-hand side of formula (4). Therefore, the problem of finding the minimal upper bound can be transformed as follows:
This program problem may be solved by terms of the max-tree finding algo~thm of graph theory. The collapse caused by a single failure mode and two simultaneous failure modes are indicated by a point or a line respectively, and the two end points of this line correspond to the two failure modes. In Fig. 3, A, B, C, D and E denote the structural failure modes, P(A), P(E), P(C), P(D) and P(E) are the failure probabilities corresponding to A, B, C, D and E, and P(AB) indicates the second order joint failure probability between A and B and so on.
19
Failure mode order and system reliability
PM E)
/
(iv) Continuing the above process until any one of the remaining points x satisfies the inequality
where q is the last chosen point. 2.4. The maximum solution with the high accuracy method In formula (3), the term ‘P(C
0) P,-
Fig. 3. An example of a weighted graph. Finding the max-tree in this joined weighted graph and renewing the order of the points, the minimal O’Ditilevsen upper bound can be obtained by calculating formula (4).
2.3. The maximal O’Ditilevsen lower bound
C P .+max
,,’
m’
(i;;:,
pmu)
may be less than zero, so formula (3) can be modified into &= P, + Pz - P,, + P, - (Ps, + P& + P,,, + . . * P,-
+max
C Pml.+max (iiY5’, :I,,
(
pmv). O)-
C7)
In formula (2) the lower bound is expressed as
The order of failure modes has an influence on the value of pf. For example, when m equals 4 and the initial order of failure modes is 1,2,3,4, the following result is obtained
If the inequality
Q’ = P, + P2 - P,, + P, - (Ps, + P32) i-l
Pi-
1 P,>O
+ PX! + P4 - (P41+ P42+ P43)
j-I
is true for any i, then formula (6) can be changed into Pfl = 5 Pi - 5 ‘i’ Pg.
Then, the value of Pfl has no influence on the order of failure modes. The maxima1 O’Ditilevsen lower bound problem is in choosing some points in a correlative weighted graph to make the formula m
i-l
,F,pi- s2jIE,pii give a maximum value, where n, is the total number of points chosen. This paper suggests an optima1 criterion to judge whether a point is chosen or not. The concrete procedure is as follows. (i) Choosing the maximum weighted value point, and reordering it as a. (ii) In the remainder, choosing x to make the value of P, - P., maximum, and reordering it as b. (iii) In the other, choosing x to make the value of
px-pax--P,, maximum,
+
P42,,
If the new order is I, 2,4,3, obtained,
pd3,
+
another
pd32).
result is
i=2i=,
i=,
“8
+ maW4,2 + P4,3, p421
and reordering
it as c.
PJr, = P, + P2 - P,, + P.l - (P4, + P42) +
p42,
+
m=W$,2
+
p3
-
(P3,
+
h,
+
p32
p32,
+
+
PM)
421,
h,
+
PM21
where p)‘, and py, are the outcomes of P corresponding to the two above-mentioned orders respectively. With the aid of calculation and analysis, it is found that the following formula is impossible,
In this paper, the maximum p, is taken as the structural system failure probability, that is P, = max PI.
Slightly modifying the above-mentioned optima1 criterion, it can be used to obtain the maximum p, (i) Choosing the maximum weighted value point, and reordering it as a. (ii) In the remainder, choosing x to maximize the P, - P,,, value, and reordering it as b.
20
B. F.
SONG
(iii) In the other, choosing x to maximize the formula P, - P,, - Ps.V+ PobX,and reordering it as (iv) k the remainder, choosing x to maximize the formula P, - (P,, + PbX+ P,.J + max(PXab+ P,,, Pxbo + Pxbc3p*, + Pxcb17 and reordering it as d. (v) Continuing the above process until any one of the remaining points x satisfies the inequality P,-
i
ng’ PXi/ GO,
PXi+max
i=l
( i.p1.n
where the number points chosen.
>
n is the total number
of
FENG
and Y. S.
is shown in Fig. 4a. Ordering the lines from the greatest weighted value to the least weighted value, the following sequence is obtained: 23,
24,
AD(O)=
Example 1 In a structural system there are five significant failure modes. The failure probabilities of a single mode and the second and third order joint failure probabilities are shown below. P = 6.85 x lo-‘,
p = 1.11 x 10-3,
P =0.95 x 10-3,
P = 2.00 x lo--‘,
P = 3.02 x 10-3,
15,
34,
12,
35,
14, 45,
Pjj’ - P)l”’
x 100% i24.8%,
f(Pp + Pf’)
P$’ - P.2’ AD’” = tcpJLj + pjJ) + 100% G 11.7%. (c) The maximum
P = 5.10 x 10-3,
P,
with the high accuracy
P = 4.21 x 10-3,
P = 3.91 x 10-3,
P = 1.00 x 10-3,
P = 2.13 x 10-3,
P = 4.10 x 10-3,
P = 1.12 x 10-3,
P = 1.01 x 10-3,
P = 0.99 x 10-3,
p = 0.475 x 10-3, P = 0.50 x lo-‘,
P = 0.555 x 10-3,
P = 0.475 x lo-),
P = 0.475 x 10-3,
P = 0.495 x 10-3, P = 0.56 x 10-3,
P = 0.505 x 10-3,
P = 0.475 x 10-3,
P = 0.475 X lo-‘.
(a) The minimal O’Ditilevsen upper bound. 1. P/t) with initial the order. By formula (4) P$) = 1.796 x 10-2. 2. P$) with the max-tree finding algorithms. The joined weighted graph corresponding to this problem
method. 1. @‘) with the initial order. By formula (7),
FY’ = 1.5535 x 10-2. 2. The maximum p,(P*). Using the optimal criterion shown in Sec. 2.3, the points chosen are 1, 2 and
(al 1.11 x
‘O-3\
3.02 x 10-a
1
2.13 x 10-S 1.0 x 10-S s
13.
Because the forward-most four lines not only have great weighted values but also do not construct a closed path, we can construct the max-tree shown in Fig. 4b. By formula (4), P$L)= 1.694 x 10e2. (b) The maximal O’Ditilevsen lower bound. 1. P$) with the initial order. By formula (6), Pjy = 14 x IO-2 2. P$; with the’optimal criterion. Using the optimal criterion shown in Sec. 2.2, the points chosen are 1, 2 and 4. Pj) = 1.507 x 10e2 (the process of calculation is shown in Table 1). 3. The relative interval widths AD(O) and AD(i),
3. EXAMPLES
P = 8.12 x 10-3,
25,
5
Fig. 4. (a) The weighted joined graph corresponding to Example 1. (b) The max-tree corresponding to Example 1.
Failure mode order and system reliability
21
Table 1. The process of calculating P( x lo-‘) in Example 1 x
PX
1 2 3 4 5
pi - p, -
px-p,
8.12t 6.85 5.10 4.21 3.91
5.7q 4.15 3.21 1.78
pr
pbx
Px-P,-Pbx-P,
0.05 1.21t -
-
8.12+ 5.74+ 1.21 = 15.07
t Indicates the points to be chosen. Table 2. The process of calculating P( x lo-)) in Example 1 pi - (pi, + + x
PX
1 2 3 4 5
8.12.l 6.85 5.10 4.21 3.91
px-pa
px
-
5.74t 4.15 3.21 1.78
p,
-
pbx
+
pxob
maWxd
+pxbc,
0.525 1.71t -
pxca
pxb
+
pxc)
+
p,,,
, pxba
+
pxcb
)
-
8.12 + 5.74 f 1.71 = 15.57
p,
t Indicates the points to be chosen.
4. P/* = 1.557 x lo-’ (the process of calculation shown in Table 2). 3. The relative error AD,
is
x 100% &0.225%. Example 2 In this example, the probabilities of failure modes are equal to those of Example 1, and second and third order joint probabilities are half of those of Example 1. (a) The minimal O’Ditilevsen upper bound. 1. Pf? with the initial order. By formula (4), P$)= 2.3075 x lo-*. The max-tree of this problem is shown in Fig. 5. It is similar to that of Example 1 except the weighted values of the lines are half of those in Example 1. By formula (4), P$) = 2.2565 x lo-*. (b) The maximal O’Ditilevsen lower bound. 1. Pj’) with the initial order. By formula (6), P$) = 1.9475 x 10-2. 2. P$) with the optimal criterion. Because the inequality ,-I pi--c P,.20 j-l
is true for any i (i = 1, 2, 3, 4 and S), then PJ must not be influenced by the order of failure modes. That is, P$) = P$) = 1.9475 x lo-*.
Fig. 5. The max-tree corresponding
to Example 2.
3. The relative interval tidths
AD(‘) and AD(*),
AD”‘& 1.69% , AD( - 14.7%. (c) The @, with the high accuracy method. Because the inequality i-1 / i-7 Pi - ‘C P, + max j=l
\
(z~;_,q~o
is true for any i (i = 1, 2, 3, 4 and 5), the order of the
failure modes has no influence on P,. That is Pf=maxP,=P,;-2.101
x lo-*.
Comparing the results of the two examples, it is found that when the correlation coefficients among failure modes are rather great, the order of failure modes influences O’Ditilevsen’s upper and lower bounds greatly; otherwise, the effect is small. The influence of the order of failure modes on P with the high accuracy method is consistently small. 4. CONCLUSIONS 1. The theory and examples in this paper show that the optimal order of failure modes can reduce the interval width bounded by O’Ditilevsen’s bounds. In Example 1, the interval with I, corresponding to the initial order of failure modes equals 0.396 x lo-*, whilst the I, corresponding to the optimal order is 0.182 x IO-*; 1,/I, equals 0.46. 2. It is reasonable to take the maximum as the failure probability of a structural system. In this paper’s example, the maximum values of P, are within the intervals bounded by O’Ditilevsen’s bounds and the influence of the order of failure modes on P is very small.
B. F. SONGand Y. S. FENG
22
3.To obtain the optimal order by using the methods of this paper, only a small number of addition and subtraction calculations are needed. Therefore, the runtime of the computer is rather small. 4. In this paper, the method used to obtain the minimal O’Ditilevsen upper bound is accurate and the results
obtained
for the maximal
O’Ditilevsen
lower bound and P with the high accuracy method are approximate. REFERENCES
1. O’Ditilevsen, Narrow reliability bounds for structural systems. J. Struct. Mech. 7, 453472 (1979). 2. Y. S. Feng, A method for computing structural system reliability with high accuracy. Comput. Struct. 33, l-5 (1989).
& Slrucrures
in Great
Vol. 34, No.
I, pp.
17-22,
1990
cKM5-7949po
Britain.
0
THE INFLUENCE OF FAILURE MODE ORDER STRUCTURAL SYSTEM RELIABILITY
s3.00 + 0.00
1989 Pcrgamon
Press
plc
ON
B. F. SONGand Y. S. FENG Department
of Aircraft Engineering, Northwestern
Polytechnical University, Xian, Shaanxi Province,
People’s Republic of China (Received 1 December 1988)
Abstract-The failure probability of a structural system solved by O’Ditilevsen’s narrow bound method or Y. S. Feng’s high accuracy method is influenced by the orders of the failure modes. In this paper, the O’Ditilevsen minimal upper bound and maximal lower bound can be solved by using a max-tree finding algorithm in a correlative weighted graph and an optimality criterion respectively, so that the interval width between the upper and lower bounds becomes more narrow. Using an optimality criterion similar to the above-mentioned one, the maximum failure probability of a structural system combined with the high accuracy method can be obtained. Two examples are given to illustrate the effectiveness of the method presented in this paper.
1. INTRODUCTION
less than 0.6 and the difference increases with increas-
There are two fundamental bound methods which have recently been used in the field of structural system reliability. One is the method of simple bounds. This can be expressed as
ing P,. Recently, Professor Y. S. Feng[2] presented a method with high accuracy. In this high accuracy method, the system reliability can be accurately expressed by first order probabilities, and second and third order joint probabilities. It can be expressed as
(1)
p,’
where Pi and P, indicate the failure probability corresponding to the ith failure mode and the failure probability of the structural system respectively, and the number m, in general, indicates the total number of significant failure modes. Though the expression of simple bounds is very straightforward and the computational result is accurate, the interval width is too wide for excluding the joint failure probabilities among the significant failure modes. Another method is O’Ditilevsen’s [l] narrow bounds. This can be expressed as
Q f Pi - f i=, j4
p, + PI - p,2 + p, - (p3, + p32)+ +.. .+P,-
1
,:,I
p312
P,,+max
where Piik indicates the third order joint failure probability among the ith, jth and kth modes. Besides considering the second order joint failure probabilities and the correlation of structural element strengths, the high accuracy method has also considered the third order joint probabilities so its accuracy is very high. Figure la shows the structural system of the illustrative example of [2]. The calculated results of [2] are shown in Fig. 1b, in which the two dotted lines represent O’Ditilevsen’s upper and lower bounds respectively and the solid line represents the calculation with the high accuracy method. In general, with formula (2), using O’Ditilevsen’s method, there are different bounds for different orders of structural system failure modes, and one has to find the minimal upper bound and the maximal lower bound. In this paper, the minimal O’Ditilevsen upper bound can be obtained by using a max-tree finding algorithm in a correlative weighted graph and the maximal O’Ditilevsen lower bound by using an optimality criterion. Therefore, the interval width between the upper and lower bounds becomes considerably narrower.
Fax Pii (2) 1<1
where Pii is the joint failure probability between the ith and the jth failure modes. Because O’Ditilevsen’s bounds regard the correlation among failure modes and structural element strengths respectively, the interval width between the upper and lower bounds is narrower than that of simple bounds. In [2], it has been shown that the difference between the upper and lower bounds using O’Ditilevsen’s method is quite small when Pv is 17
18
B. F.
SONG and PO(P)
(a)
Y. S.
(b)
FENG
P&)/MO)
4.44.24.0
s
I I, I, I I I I 0 0.2 0.4 0.6 0.8 top Fig. 1. (a) The structural system of the illustrative example of [2]. (b) The calculated results of the structural system shown in Fig. la. Using formula (3), their solutions may be different from each other to a small extent corresponding to the different orders of failure modes. In general, the maximum value of P may be meaningful. In this paper, the maximum P is taken as the structural system failure probability, and it can be obtained by using a similar optimality criterion. 2. THEORY AND METHOD
2. I. Graph, weighted graph and tree A graph is the set of points and lines in a plane. The graph is said to be joined when there are direct or indirect lines between any two points. Figure 2 is an example of a joined graph. In a joined graph, weighted values are usually given to the points and lines respectively and this graph is then called a weighted joined graph. A joined graph that excludes any closed path is called a tree. There may be many trees in a joined graph. That of the maximum sum of line weighted values is called the max-tree. There are many kinds of methods used to find max-trees. A simple method is presented by the authors of this paper as foflows. (i) Ordering the lines in tetms of their weighted values. The greater the weighted value, the more forward the line is. For example, if the inequalities WI,) &&I,) 3 * . . II are true, then a sequence is obtained, that is I,, l,, 13. . . 1m,where lj and O(&) indicate the ith line and its weighted value respectively, and the number m is the total number of lines. (ii) Taking the most forward two lines as the branches of the max-tree. In this example, Ii and i2 are taken as the branches of the max-tree. b
C
a
@
e
d
Fig. 2. An example of a joined graph.
(iii) Choosing the line which cannot construct a closed path with the two branches chosen in (ii) and has a greater weighted value than the third branch of the max-tree. In this example, if iS does not construct a closed path with 1, and 1, it is taken as the third branch, and if it does, it cannot be taken as the third branch, and another must be chosen. (iv) Choosing the line which cannot construct a closed path with the three branches chosen in (ii) and (iii) and has a greater weighted value than the fourth branch of the max-tree. In this example, if I., does not construct a closed path with the branches chosen in (ii) and (iii), it is taken as the fourth branch, and if it does, it cannot be taken as the fourth branch, and another must be chosen. (v) Continuing the process until all points in the weighted graph are linked by the above-mentioned branches. 2.2, The minimal O’Ditilevsen upper bound In formula (2), the upper bound is expressed as
The order of the failure modes is not influenced by the first term on the right-hand side of formula (4). Therefore, the problem of finding the minimal upper bound can be transformed as follows:
This program problem may be solved by terms of the max-tree finding algo~thm of graph theory. The collapse caused by a single failure mode and two simultaneous failure modes are indicated by a point or a line respectively, and the two end points of this line correspond to the two failure modes. In Fig. 3, A, B, C, D and E denote the structural failure modes, P(A), P(E), P(C), P(D) and P(E) are the failure probabilities corresponding to A, B, C, D and E, and P(AB) indicates the second order joint failure probability between A and B and so on.
19
Failure mode order and system reliability
PM E)
/
(iv) Continuing the above process until any one of the remaining points x satisfies the inequality
where q is the last chosen point. 2.4. The maximum solution with the high accuracy method In formula (3), the term ‘P(C
0) P,-
Fig. 3. An example of a weighted graph. Finding the max-tree in this joined weighted graph and renewing the order of the points, the minimal O’Ditilevsen upper bound can be obtained by calculating formula (4).
2.3. The maximal O’Ditilevsen lower bound
C P .+max
,,’
m’
(i;;:,
pmu)
may be less than zero, so formula (3) can be modified into &= P, + Pz - P,, + P, - (Ps, + P& + P,,, + . . * P,-
+max
C Pml.+max (iiY5’, :I,,
(
pmv). O)-
C7)
In formula (2) the lower bound is expressed as
The order of failure modes has an influence on the value of pf. For example, when m equals 4 and the initial order of failure modes is 1,2,3,4, the following result is obtained
If the inequality
Q’ = P, + P2 - P,, + P, - (Ps, + P32) i-l
Pi-
1 P,>O
+ PX! + P4 - (P41+ P42+ P43)
j-I
is true for any i, then formula (6) can be changed into Pfl = 5 Pi - 5 ‘i’ Pg.
Then, the value of Pfl has no influence on the order of failure modes. The maxima1 O’Ditilevsen lower bound problem is in choosing some points in a correlative weighted graph to make the formula m
i-l
,F,pi- s2jIE,pii give a maximum value, where n, is the total number of points chosen. This paper suggests an optima1 criterion to judge whether a point is chosen or not. The concrete procedure is as follows. (i) Choosing the maximum weighted value point, and reordering it as a. (ii) In the remainder, choosing x to make the value of P, - P., maximum, and reordering it as b. (iii) In the other, choosing x to make the value of
px-pax--P,, maximum,
+
P42,,
If the new order is I, 2,4,3, obtained,
pd3,
+
another
pd32).
result is
i=2i=,
i=,
“8
+ maW4,2 + P4,3, p421
and reordering
it as c.
PJr, = P, + P2 - P,, + P.l - (P4, + P42) +
p42,
+
m=W$,2
+
p3
-
(P3,
+
h,
+
p32
p32,
+
+
PM)
421,
h,
+
PM21
where p)‘, and py, are the outcomes of P corresponding to the two above-mentioned orders respectively. With the aid of calculation and analysis, it is found that the following formula is impossible,
In this paper, the maximum p, is taken as the structural system failure probability, that is P, = max PI.
Slightly modifying the above-mentioned optima1 criterion, it can be used to obtain the maximum p, (i) Choosing the maximum weighted value point, and reordering it as a. (ii) In the remainder, choosing x to maximize the P, - P,,, value, and reordering it as b.
20
B. F.
SONG
(iii) In the other, choosing x to maximize the formula P, - P,, - Ps.V+ PobX,and reordering it as (iv) k the remainder, choosing x to maximize the formula P, - (P,, + PbX+ P,.J + max(PXab+ P,,, Pxbo + Pxbc3p*, + Pxcb17 and reordering it as d. (v) Continuing the above process until any one of the remaining points x satisfies the inequality P,-
i
ng’ PXi/ GO,
PXi+max
i=l
( i.p1.n
where the number points chosen.
>
n is the total number
of
FENG
and Y. S.
is shown in Fig. 4a. Ordering the lines from the greatest weighted value to the least weighted value, the following sequence is obtained: 23,
24,
AD(O)=
Example 1 In a structural system there are five significant failure modes. The failure probabilities of a single mode and the second and third order joint failure probabilities are shown below. P = 6.85 x lo-‘,
p = 1.11 x 10-3,
P =0.95 x 10-3,
P = 2.00 x lo--‘,
P = 3.02 x 10-3,
15,
34,
12,
35,
14, 45,
Pjj’ - P)l”’
x 100% i24.8%,
f(Pp + Pf’)
P$’ - P.2’ AD’” = tcpJLj + pjJ) + 100% G 11.7%. (c) The maximum
P = 5.10 x 10-3,
P,
with the high accuracy
P = 4.21 x 10-3,
P = 3.91 x 10-3,
P = 1.00 x 10-3,
P = 2.13 x 10-3,
P = 4.10 x 10-3,
P = 1.12 x 10-3,
P = 1.01 x 10-3,
P = 0.99 x 10-3,
p = 0.475 x 10-3, P = 0.50 x lo-‘,
P = 0.555 x 10-3,
P = 0.475 x lo-),
P = 0.475 x 10-3,
P = 0.495 x 10-3, P = 0.56 x 10-3,
P = 0.505 x 10-3,
P = 0.475 x 10-3,
P = 0.475 X lo-‘.
(a) The minimal O’Ditilevsen upper bound. 1. P/t) with initial the order. By formula (4) P$) = 1.796 x 10-2. 2. P$) with the max-tree finding algorithms. The joined weighted graph corresponding to this problem
method. 1. @‘) with the initial order. By formula (7),
FY’ = 1.5535 x 10-2. 2. The maximum p,(P*). Using the optimal criterion shown in Sec. 2.3, the points chosen are 1, 2 and
(al 1.11 x
‘O-3\
3.02 x 10-a
1
2.13 x 10-S 1.0 x 10-S s
13.
Because the forward-most four lines not only have great weighted values but also do not construct a closed path, we can construct the max-tree shown in Fig. 4b. By formula (4), P$L)= 1.694 x 10e2. (b) The maximal O’Ditilevsen lower bound. 1. P$) with the initial order. By formula (6), Pjy = 14 x IO-2 2. P$; with the’optimal criterion. Using the optimal criterion shown in Sec. 2.2, the points chosen are 1, 2 and 4. Pj) = 1.507 x 10e2 (the process of calculation is shown in Table 1). 3. The relative interval widths AD(O) and AD(i),
3. EXAMPLES
P = 8.12 x 10-3,
25,
5
Fig. 4. (a) The weighted joined graph corresponding to Example 1. (b) The max-tree corresponding to Example 1.
Failure mode order and system reliability
21
Table 1. The process of calculating P( x lo-‘) in Example 1 x
PX
1 2 3 4 5
pi - p, -
px-p,
8.12t 6.85 5.10 4.21 3.91
5.7q 4.15 3.21 1.78
pr
pbx
Px-P,-Pbx-P,
0.05 1.21t -
-
8.12+ 5.74+ 1.21 = 15.07
t Indicates the points to be chosen. Table 2. The process of calculating P( x lo-)) in Example 1 pi - (pi, + + x
PX
1 2 3 4 5
8.12.l 6.85 5.10 4.21 3.91
px-pa
px
-
5.74t 4.15 3.21 1.78
p,
-
pbx
+
pxob
maWxd
+pxbc,
0.525 1.71t -
pxca
pxb
+
pxc)
+
p,,,
, pxba
+
pxcb
)
-
8.12 + 5.74 f 1.71 = 15.57
p,
t Indicates the points to be chosen.
4. P/* = 1.557 x lo-’ (the process of calculation shown in Table 2). 3. The relative error AD,
is
x 100% &0.225%. Example 2 In this example, the probabilities of failure modes are equal to those of Example 1, and second and third order joint probabilities are half of those of Example 1. (a) The minimal O’Ditilevsen upper bound. 1. Pf? with the initial order. By formula (4), P$)= 2.3075 x lo-*. The max-tree of this problem is shown in Fig. 5. It is similar to that of Example 1 except the weighted values of the lines are half of those in Example 1. By formula (4), P$) = 2.2565 x lo-*. (b) The maximal O’Ditilevsen lower bound. 1. Pj’) with the initial order. By formula (6), P$) = 1.9475 x 10-2. 2. P$) with the optimal criterion. Because the inequality ,-I pi--c P,.20 j-l
is true for any i (i = 1, 2, 3, 4 and S), then PJ must not be influenced by the order of failure modes. That is, P$) = P$) = 1.9475 x lo-*.
Fig. 5. The max-tree corresponding
to Example 2.
3. The relative interval tidths
AD(‘) and AD(*),
AD”‘& 1.69% , AD( - 14.7%. (c) The @, with the high accuracy method. Because the inequality i-1 / i-7 Pi - ‘C P, + max j=l
\
(z~;_,q~o
is true for any i (i = 1, 2, 3, 4 and 5), the order of the
failure modes has no influence on P,. That is Pf=maxP,=P,;-2.101
x lo-*.
Comparing the results of the two examples, it is found that when the correlation coefficients among failure modes are rather great, the order of failure modes influences O’Ditilevsen’s upper and lower bounds greatly; otherwise, the effect is small. The influence of the order of failure modes on P with the high accuracy method is consistently small. 4. CONCLUSIONS 1. The theory and examples in this paper show that the optimal order of failure modes can reduce the interval width bounded by O’Ditilevsen’s bounds. In Example 1, the interval with I, corresponding to the initial order of failure modes equals 0.396 x lo-*, whilst the I, corresponding to the optimal order is 0.182 x IO-*; 1,/I, equals 0.46. 2. It is reasonable to take the maximum as the failure probability of a structural system. In this paper’s example, the maximum values of P, are within the intervals bounded by O’Ditilevsen’s bounds and the influence of the order of failure modes on P is very small.
B. F. SONGand Y. S. FENG
22
3.To obtain the optimal order by using the methods of this paper, only a small number of addition and subtraction calculations are needed. Therefore, the runtime of the computer is rather small. 4. In this paper, the method used to obtain the minimal O’Ditilevsen upper bound is accurate and the results
obtained
for the maximal
O’Ditilevsen
lower bound and P with the high accuracy method are approximate. REFERENCES
1. O’Ditilevsen, Narrow reliability bounds for structural systems. J. Struct. Mech. 7, 453472 (1979). 2. Y. S. Feng, A method for computing structural system reliability with high accuracy. Comput. Struct. 33, l-5 (1989).