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An Introductory Talk on Reliability
Analysis
Jeen-Shang Lin
University of Pittsburgh
With contribution from Yung Chia HSU
May 2007
Hua Fan University, Taipei
Supply vs. Demand
Failure takes place when
demand exceeds supply.
For an engineering system:
– Available resistance is the
supply, R
– Load is the demand, Q
– Margin of safety, M=R-Q
The reliability of a system
can be defined as the
probability that R>Q
represented as:
PM 0 PR Q
Risk
The probability of failure, or risk
pF P( M 0) 1 P( M 0) 1
How to find the risk?
– If we known the distribution of M;
– or, the mean and variance of M;
– then we can compute P(M<0) easily.
Normal distribution: the bell curve
For a wide variety of conditions, the
distribution of the sum of a large number of
random variables converge to Normal
distribution. (Central Limit Theorem)
0 1
1
( x )2
f ( x)
e(
)
2
2
x
F ( x)
1
( x )2
e(
)
2
2
0 1
When
x
F ( x)
1
( x )2
1
2
e(
)
e
(
x
) ( x )
2
2
2
x
IF M=Q-R is normal
pF P( M 0) (
M
M
)
Because of symmetry
( x ) 1 ( x )
Define reliability index
M
pF 1 (
)
M
M / M
Example: vertical cut in clay
M c H / 4
c 100 c 30 kPa
20 2 kN / m3
c 0.5
H 10
If all variables are normal,
M 50 M 27.83 1.796 pF 3.62 102
Some basics
y a1x a2 x
y a1x1 (a2 ) x2
y a1x1 a2 x2
y a1x (a2 )x
1
2
1
y 2 E[( y y ) 2 ]
E[ y 2 ] y
2
2
y ai xi
2
2
y 2 E[( y y ) 2 ]
E[ y 2 ] y
a1 x1 a2 x2 2a1a2 12 x1 x2
2
Negative
coefficient
2
2
a1 x1 a2 x2 2a1 ( a2 ) 12 x1 x2
2
2
2
2
y ai x
i
y 2 ai 2 x 2 ai a j ij x x
i
i
j i
i
j
Engineers like Factor of safety
F=R/Q, if F is normal
pF P( F 1) (
1 F
F
) 1 ( )
F 1
reliability index
F
Lognormal distribution
The uncertain variable can increase without
limits but cannot fall below zero.
The uncertain variable is positively skewed, with
most of the values near the lower limit.
The natural logarithm of the uncertain variable
follows a normal distribution.
F is also often treated as lognormal
In case of lognormal
Ln(R) and ln(Q) each is normal
1
ln[x] )2
f ( x)
e(
)
2
2
First order second moment method
The MFOSM method assumes that the
uncertainty features of a random variable
can be represented by its first two moments:
mean and variance.
This method is based on the Taylor series
expansion of the performance function
linearized at the mean values of the random
variables.
First order second moment method
Taylor series
expansion
g g ( x1, x2 ,.....xn )
g g( X ) ( x X )
g
g ( x1 , x2 ,...,xn ) g ( x1 , x 2 ,...,xn ) ( xi xi )
xi
g g( x1, x 2 ,...,xn )
2
2
g
g
g g
xixj xi xj
xi
xi x j
i i j
xi
2
dg
dx
Example: vertical cut in clay
F
4c
H
c 100 c 30 kPa
20 2 kN / m3
c 0.5
H 10
If all variables are normal,
F
4
c H
F 4c
2
H
F 1
F
F 2 F 0.5292 1.8896 pF 2.94 102
1-normcdf(1.8896,0,1) MATLAB
Slope stability
cx
n
FS
i 1
i
n
f R
FS
Q
-55
Wi u i xi tani 1
M
(
)
i
-35
W sin
i 1
i
25
i
2 (H): 1(V) slope
with a height of 5m
5
-15
5
-15
-35
25
45
Reliability Analysis
The reliability of a system can be defined as the
probability that R>Q represented as:
PQ R
cx W
n
FS
i 1
i
i
u i xi tani 1
M i ( )
n
W sin
i 1
i
i
FS FS FS
X i
2m X i
FS contour
10 2
c 10 c 2 kPa
4
2.17
2
2.24
2
05 8
2.1
7
3
37
2.0
1.6
94
9
9
94
1.6
8
68
1.9
19
83
1.
20
FS 1.436
1.626 4
2.2
42 7
1.558
1
2
FS
0.056
3
4
5
6
7
8
2.
1
2. 05 8
17
42
1
2. 1 .900
03 .9 3
7 3 68 8
5
89
1.4
1
1. .763 4
83
19
1.
1.6 626
94 4
9
5
0
2.1742
1.626 4
2.1058
2.0373
1.968 8
3
9
58
1.5
58
1.5
34
76
1.
1.8
1.9 31 9
1.9 00 3
68
8
FS
0.089
c
5
1.489 5
10
0
1.90
1. 48
9
1
1.83
1.6949
0.21.
9
94
1.6
4
26
1.6
15
4
63
1.7
1.558
,
1.6
2
1.6 6 4
94
9
,
1.7
63
4
03
90
1.
1.
76
34
25
2.079
2
11 79 748 1 6 6
2.32.32.42.51
9
FS 0.21
10
pF 0.0188
First Order Reliability Method
Hasofer-Lind (FORM)
Probability of failure can be found
obtained in material space
Approximate as distance to Limit
state
Distance to failure criterion
If F=1 or M=0 is a straight line
Reliability becomes the shortest
distance
Constraint Optimization:Excel
May get similar results with FOSM
1.796
FOSM
1-normcdf(1. 796,0,1)=0.0362 MATLAB
Monte Carlo Simulation
correlation=0
Monte Carlo=0.0495
Monte Carlo Simulation
correlation=0.5
{Y * } [CY ]1/ 2 [T ]T {X *}
1/ 2 1/ 2
[T ]
1 / 2 1 / 2
FORM=0.0362
0
1
[CY ]
1
0
1
( X X )T C X ( X X )
min
XLimit state
3
2
FS=1.0
*
1
Mean
FS=1.549
Mean + - S.D.
FS=1.259
FS=1.613
0
FS=1.436
-1
FS=1.324
-2
1
-3
y
2
x
-4
-5
-5
-4
-3
-2
-1
0
c*
1
cc
c
2
3
4
5
FS=1.0 (M=0)
The matrix form of the
Hasofer-Lind (1974)
c c
UNSAFE
Region
FS<1 or
M<0
c
The matrix form of the
Hasofer-Lind (1974)
Soil properties
FOS=1
Soil properties>0
25
25
2.2
2.1
5. 25
1.8
5
4.8
2
1.9
3.
75
1.7
1.8
20
3.3
5
3.15
3.3
5
1.6
1. 7
1.6
5
3.1
1.48
5
3.3
3. 15
2.2
2.75
5
2.9
1.7
1.8
1.6
1. 65
5
3.7
2.1
15
4. 85
5
2.9
2
1.9
4.55
4.15
5
1.6
1.55
5.25
1.
7
1.65
20
15
5
4.5
5
4.1
3.75
1. 55
1
2
3
4
4.15
5
2.9 15
3.
35
3.
5
5
0
2.7
5
5.
25
2.1
2.
2
8
1.
3.
15
3.7
5
1.9
2
1.6
4.
1
4.84.55 5
5
4
4.8 .55
5
1.65
1.48
7
1.
1.55
2. 75
1.6
2.1 2
2.55
5
2.5
95
2.
1.
65
5
3.3
5
3.7
7
1.
8
1.
1.
9
10
1.48
1.55
10
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
FS=1.0
c
c
Correlation=.99
UNSAFE
Region
FS<1 or
Correlation=-.99
Correlation=0
M<0
c
The distance
(c* ) ( * )
2
2
FS 1
2
2
(FS / c)2 c (FS / )2
FOSM maybe wrong
1
2
Y
Soil 1
c1
X
Soil 2
c2
2
9.10 m
6.14 m
FOSM
A projection Method
( xi ) p X i
( FS 1)(FS / X i ) X
n
2
(
FS
/
X
)
i X
i 1
2
I
2
i
Check the FOSM
Use the slope, projected to where the
failure material is
Use the material to find FS
If FS=1, ok
May 2007
Hua Fan University, Taipei