128 3. COMPUTATIONAL METHODS FOR THE RELIABILITY OF A COMPONENT
B
D 36:3 (ksi) and a standard deviation
B
D 4:21 (ksi).
A
is a normally distributed ran-
dom variable with a mean
A
D 6:81 (ksi) and a standard deviation
A
D 2:65 (ksi). Calculate
the reliability of this component.
Solution:
In this example, the component strength index will be the ultimate material strength S
u
. e
component stress index will be the sum of normal stress
B
by bending moment and the normal
stress
A
by axial loading. erefore, the limit state function of this component will be:
g
.
S
u
;
B
;
A
/
D S
u
.
B
C
A
/
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
D 0 Limit state
< 0 Failure:
(a)
Since all three random variables S
u
,
B
, and
A
are normally distributed random variables, and
the limit state function is a linear function of them, we can use Equations (3.25) and (3.26) to
calculate the mean and standard deviation of the limit state function. ey are:
g
D
u
B
A
D 61:5 36:3 6:81 D 18:39 .ksi/ (b)
g
D
q
2
u
C
2
B
C
2
A
D
p
5:91
2
C 4:21
2
C 2:65
2
D 7:7248: (c)
Per Equation (3.27), we can calculate the reliability index ˇ:
ˇ D
g
g
D
18:39
7:7248
D 2:3806: (d)
Per Equation (3.28), we can calculate the reliability of the component. We can use Excel to
calculate the reliability of this component
R D ˆ
.
2:3806
/
D NORM:DIST
.
2:3806; 0; 1; true
/
D 0:9914:
3.5.2 THE FOSM METHOD FOR A NONLINEAR STATE FUNCTION
When a limit state function is a nonlinear function of all normally distributed random variable,
we can obtain an approximate value of the reliability by using the FOSM method. In the FOSM
method, the nonlinear limit state function will be simplified by the first-order Taylor series
expansion at the mean point where all random variables have their means. e first-order Taylor
3.5. THE FIRST-ORDER SECOND-MOMENT (FOSM) METHOD 129
series expansion at the mean point will be:
g
.
X
1
; X
2
; : : : ; X
n
/
D g
X
1
;
X
2
; : : : ;
X
n
C
n
X
iD1
G
xi
X
i
X
i
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
D 0 Limit state
< 0 Failure;
(3.29)
where,
X
i
is the mean of the normally distributed random variable X
i
; .i D 1; 2; : : : ; n/. e
mean point is the point where every variable takes its mean value, that is,
X
1
;
X
2
; : : : ;
X
n
.
G
xi
is a Taylor series constant-coefficient for Taylor expansion term
X
i
X
i
. It is
calculated by the following equation at the mean point:
G
xi
D
@g
.
X
1
; X
2
; : : : ; X
n
/
@X
i
ˇ
ˇ
ˇ
ˇ
at the mean point
i D 1; 2; : : : ; n: (3.30)
Now the simplified limit state function in Equation (3.29) is a linear function of all normal
distributed random variables. We can use Equations (3.25) and (3.26) to calculate the mean and
standard deviation of the simplified limit state function:
g
D g
X
1
;
X
2
; : : : ;
X
n
(3.31)
g
D
v
u
u
t
n
X
iD1
G
xi
X
i
2
: (3.32)
Per Equations (3.27) and (3.28), the reliability index ˇ, and the reliability R of the component
will be:
ˇ D
g
g
D
g
X
1
;
X
2
; : : : ;
X
n
q
P
n
iD1
G
xi
X
i
2
(3.33)
R D ˆ.ˇ/ D ˆ
0
B
@
g
X
1
;
X
2
; : : : ;
X
n
q
P
n
iD1
G
xi
X
i
2
1
C
A
: (3.34)
Here are some comments about the FOSM method for a nonlinear limit state function.
• e FOSM method will provide an approximate value of the reliability of a component
when the limit state function is a nonlinear function of all normally distributed random
variables.
130 3. COMPUTATIONAL METHODS FOR THE RELIABILITY OF A COMPONENT
• Since there might have different versions of a limit state function for the same problem,
we might get a different estimation of the reliability of the same problem [2]. We will use
the following example to demonstrate and to explain this situation.
Example 3.10
A simple support beam is under a concentrated loading at the middle of a beam, as shown in Fig-
ure 3.5. e yield strength S
y
, the concentrated load P , the beam span L and the section mod-
ulus Z of the beam are all normally distributed random variables. eir distribution parameters
are:
S
y
D 6 10
5
(kN/m
2
),
S
y
D 10
5
(kN/m
2
);
P
D 10 (kN),
P
D 2 (kN);
L
D 8 (m),
L
D 2:083 10
2
(m); and
Z
D 10
4
(m
3
),
S
y
D 2 10
5
(m
3
). Use the FOSM method
to calculate the reliability of the beam by using the following two different limit state functions.
1. e yield strength is the component strength index. So, the limit state function is:
g
S
y
; P; L; Z
D S
y
PL=.4Z/ D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
D 0 Limit state
< 0 Failure:
(a)
2. e allowable bending moment is the component strength index. So, the limit state func-
tion is:
g
S
y
; P; L; Z
D S
y
Z PL=4 D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
D 0 Limit state
< 0 Failure:
(b)
P
L
Figure 3.5: A simple support beam with a concentrated loading.
Solution:
In this example, all related random variables are normal distributions. However, both limit state
functions are a nonlinear function of these random variables. We can use Equations (3.30)–
(3.34) to conduct the calculations.
3.5. THE FIRST-ORDER SECOND-MOMENT (FOSM) METHOD 131
(1) e reliability of the beam based on the limit state function: g
S
y
; P; L; Z
D S
y
PL
4Z
.
Per Equation (3.31), we can calculate the mean of the limit state function:
g
D g
S
y
;
P
;
L
;
Z
D
S
y
P
L
4
Z
D 6 10
5
10 8
4 10
4
D 4 10
5
: (c)
Per Equation (3.30), we can calculate the Tylor series constant coefficients:
G
S
y
D
@g
S
y
; P; L; Z
@S
y
ˇ
ˇ
ˇ
ˇ
ˇ
at the mean point
D 1
G
P
D
@g
S
y
; P; L; Z
@P
ˇ
ˇ
ˇ
ˇ
ˇ
at the mean point
D
L
4Z
ˇ
ˇ
ˇ
ˇ
at the mean point
D
8
4 10
4
D 2 10
4
G
L
D
@g
S
y
; P; L; Z
@L
ˇ
ˇ
ˇ
ˇ
ˇ
at the mean point
D
P
4Z
ˇ
ˇ
ˇ
ˇ
at the mean point
D
10
4 10
4
D 2:5 10
4
G
Z
D
@g
S
y
; P; L; Z
@Z
ˇ
ˇ
ˇ
ˇ
ˇ
at the mean point
D
PL
4Z
2
ˇ
ˇ
ˇ
ˇ
at the mean point
D
10 8
4
10
4
2
D 2 10
9
:
Per Equation (3.32), we can calculate the standard deviation of the limit state function:
g
D
q
G
S
y
S
y
2
C
.
G
P
P
/
2
C
.
G
L
L
/
2
C
.
G
Z
Z
/
2
D
v
u
u
t
1 10
5
2
C
2 10
4
2
2
C
2:5 10
4
2:083 10
2
2
C
2 10
9
2 10
5
2
D 114892:4331: (d)
Per Equation (3.33) and data in Equations (c) and (d), the reliability index ˇ is:
ˇ D
g
g
D
4 10
5
114892:4331
D 3:4815: (e)
Per Equation (3.34) and data in Equation (e), we have the reliability R with the use of MATLAB
command:
R D ˆ
.
ˇ
/
D ˆ
.
3:4815
/
D normcdf
.
3:4815; 0; 1
/
D 0:9998:
(2) e reliability of the beam based on the limit state function: g
S
y
; P; L; Z
D S
y
Z
PL
4
.
132 3. COMPUTATIONAL METHODS FOR THE RELIABILITY OF A COMPONENT
We can repeat the above steps.
Per Equation (3.31), the mean of the limit state function is
g
D g
S
y
;
P
;
L
;
Z
D
S
y
Z
P
L
4
D 6 10
5
10
4
10 8
4
D 40: (f)
Per Equation (3.30), we can calculate the Taylor series constant coefficients:
G
S
y
D
@g
S
y
; P; L; Z
@S
y
ˇ
ˇ
ˇ
ˇ
ˇ
at the mean point
D Z
j
at the mean point
D 10
4
G
P
D
@g
S
y
; P; L; Z
@P
ˇ
ˇ
ˇ
ˇ
ˇ
at the mean point
D
L
4
ˇ
ˇ
ˇ
ˇ
at the mean point
D
8
4
D 2
G
L
D
@g
S
y
; P; L; Z
@L
ˇ
ˇ
ˇ
ˇ
ˇ
at the mean point
D
P
4
ˇ
ˇ
ˇ
ˇ
at the mean point
D
10
4
D 2:5
G
Z
D
@g
S
y
; P; L; Z
@Z
ˇ
ˇ
ˇ
ˇ
ˇ
at the mean point
D S
y
ˇ
ˇ
at the mean point
D 6 10
5
:
Per Equation (3.32), we can calculate the standard deviation of the limit state function
g
D
q
G
S
y
S
y
2
C
.
G
P
P
/
2
C
.
G
L
L
/
2
C
.
G
Z
Z
/
2
D
v
u
u
t
10
4
10
5
2
C
.
2 2
/
2
C
2:5 2:083 10
2
2
C
6 10
5
2 10
5
2
D 16:1246: (g)
Per Equation (3.33) and data in Equations (f) and (g), the reliability index ˇ is:
ˇ D
g
g
D
40
16:1246
D 2:48068: (h)
Per Equation (3.34) and data in Equation (e), we have the reliability R with the use of MATLAB
command:
R D ˆ
.
ˇ
/
D ˆ
.
2:48068
/
D normcdf
.
2:48068; 0; 1
/
D 0:9934:
Based on the data in Equations (e) and (h), the reliability index ˇ of the same component are
different when different limit state functions are used even though both limit state functions
specify the same event. e reason for this difference is because the FOSM method for a non-
linear limit state function provides an approximate result.
..................Content has been hidden....................
You can't read the all page of ebook, please click here login for view all page.