138 3. COMPUTATIONAL METHODS FOR THE RELIABILITY OF A COMPONENT
P
L
Figure 3.7: A simple support beam with a concentrated loading.
Solution:
(1) e reliability of the beam by using the limit state function: g
S
y
; Z; P; L
D S
y
PL
4Z
.
Per this limit state function, we can establish the following equations before we compile
the program.
Per the surface of the limit state function: g
S
y
; Z; P; L
D S
y
PL
4Z
D 0, we can get
the explicit form of Equation (3.49) for this example to determine the value of the last variable
at the design point. It is:
L D g
1
S
y
; Z; P
D
S
y
4Z
P
: (a)
e Taylor series coefficients in this case are:
G
S
y
D
S
y
@g
S
y
; Z; P; L
@S
y
ˇ
ˇ
ˇ
ˇ
ˇ
atP
D
S
y
1
ˇ
ˇ
atP
D
S
y
(b)
G
Z
D
Z
@g
S
y
; Z; P; L
@Z
ˇ
ˇ
ˇ
ˇ
ˇ
atP
D
Z
PL
4Z
2
ˇ
ˇ
ˇ
ˇ
atP
D
Z
P
L
4
.
Z
/
2
(c)
G
P
D
P
@g
S
y
; Z; P; L
@P
ˇ
ˇ
ˇ
ˇ
ˇ
atP
D
P
L
4Z
ˇ
ˇ
ˇ
ˇ
atP
D
P
L
4Z
(d)
G
L
D
L
@g
S
y
; Z; P; L
@L
ˇ
ˇ
ˇ
ˇ
ˇ
atP
D
L
P
4Z
ˇ
ˇ
ˇ
ˇ
atP
D
P
P
4Z
: (e)
We can follow the program flowchart in Figure 3.6 to compile the MATLAB program for this
example. is program is listed as “A.1: e H-L method for Example 3.11” in Appendix A.
e data for the iterative process is shown in Table 3.1. e first column is the number
of iterative processes. e second to the fifth columns are the values of the design point. e
3.6. THE HASOFER–LIND (H-L) METHOD 139
Table 3.1: e iterative results for the limit state function (1) in Example 3.11
# S
y
*
Z
*
P
*
L
*
β
*
|∆ β
*
| < 0.0001
1 600000 0.0001 10 24 2.030685
2 496907.9 7.53E-05 12.47421 11.99154 2.885681 0.854996
3 443008.1 5.85E-05 12.5015 8.297361 2.953815 0.068134
4 448340.9 5.41E-05 12.1497 7.983943 2.947101 0.006714
5 457762.6 5.28E-05 12.0995 7.996524 2.945587 0.001514
6 462281 5.23E-05 12.08414 7.999504 2.94526 0.000327
7 464338.3 5.2E-05 12.07591 8.000162 2.945191 6.93E-05
sixth column is the reliability index ˇ, and the last column is the convergence condition. Per this
iterative results, the reliability index ˇ and corresponding reliability R of this component are :
ˇ D 2:9452 R D ˆ
.
ˇ
/
D ˆ
.
2:9452
/
D 0:9839:
(2) e reliability of the beam by using the limit state function: g
S
y
; Z; P; L
D S
y
Z
PL
4
.
Per the surface of the limit state function: g
S
y
; Z; P; L
D S
y
Z
PL
4
D 0, we can get
the explicit form of Equation (3.49) for this example to determine the value of the last variable
at the design point. It is:
L D g
1
S
y
; Z; P
D
S
y
4Z
P
: (f)
e Taylor series coefficients in this case are:
G
S
y
D
S
y
@g
S
y
; Z; P; L
@S
y
ˇ
ˇ
ˇ
ˇ
ˇ
atP
D
S
y
Z
ˇ
ˇ
atP
D
S
y
Z
(g)
G
Z
D
Z
@g
S
y
; Z; P; L
@Z
ˇ
ˇ
ˇ
ˇ
ˇ
atP
D
Z
S
y
ˇ
ˇ
atP
D
Z
S
Y
(h)
G
P
D
P
@g
S
y
; Z; P; L
@P
ˇ
ˇ
ˇ
ˇ
ˇ
atP
D
P
L
4
ˇ
ˇ
ˇ
ˇ
atP
D
P
L
4
(
i)
G
L
D
L
@g
S
y
; Z; P; L
@L
ˇ
ˇ
ˇ
ˇ
ˇ
atP
D
L
P
4
ˇ
ˇ
ˇ
ˇ
atP
D
P
P
4
(j)
We can follow the H-L method’s procedure and the program flowchart in Figure 3.6 to
compile a MATLAB program which is similar to the program for the limit state function (1).
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