Date post: | 18-Dec-2015 |
Category: |
Documents |
Upload: | alexander-flowers |
View: | 213 times |
Download: | 0 times |
May 2007May 2007 Hua Fan University, TaipeiHua Fan University, Taipei
An Introductory Talk on Reliability An Introductory Talk on Reliability AnalysisAnalysis
With contribution from Yung Chia HSUWith contribution from Yung Chia HSU
Jeen-Shang Lin
University of Pittsburgh
Supply vs. DemandSupply vs. Demand Failure takes place when dFailure takes place when d
emand exceeds supply.emand exceeds supply. For an engineering system:For an engineering system:
– Available resistance is the Available resistance is the supply, Rsupply, R
– Load is the demand, QLoad is the demand, Q– Margin of safety, M=R-QMargin of safety, M=R-Q
The reliability of a system The reliability of a system can be defined as the probcan be defined as the probability that R>Q representeability that R>Q represented asd as: :
QRPMP 0
RiskRisk
The probability of failure, or riskThe probability of failure, or risk
1)0(1)0( MPMPpF
How to find the risk?How to find the risk?– If we known the distribution of M;If we known the distribution of M;– or, the mean and variance of M;or, the mean and variance of M;– then we can compute P(M<0) easily.then we can compute P(M<0) easily.
Normal distribution: the bell curveNormal distribution: the bell curve
10
))(
(2
1)(
2
2
x
exf
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)
x xexF )
)((
2
1)(
2
2
IF M=Q-R is normalIF M=Q-R is normal
)()(2
1)
)((
2
1)( 2
2
2
xxex
exFxx
)()0(M
MF MPp
10 When
Because of symmetryBecause of symmetry)(1)( xx )(1
M
MFp
Define Define reliability indexreliability indexMM /
Example: vertical cut in clayExample: vertical cut in clay
kPacc 30100
4/HcM
3/220 mkN
5.0c
If all variables are normal,If all variables are normal,
21062.3796.183.2750 FMM p
10H
Some basicsSome basics
21 21 xxy aa
iixay
2121 122122
222
1
22
22
2
][
])[(
xxxx
y
yy
aaaa
yE
yE
21)( 21 xxy aa
2211 )( xaxay
2121 122122
222
1
22
22
)(2
][
])[(
xxxx
y
yy
aaaa
yE
yE
Negative coefficient
2211 xaxay
ixiy a
jii xxijji ij
ixiy aaa
222
Engineers like Factor of safetyEngineers like Factor of safety
F=R/Q, if F is normal F=R/Q, if F is normal
)(1)1
()1(
F
FF FPp
reliability indexreliability indexF
F
1
Lognormal distributionLognormal distribution The uncertain variable can increase without The uncertain variable can increase without
limits but cannot fall below zero. limits but cannot fall below zero. The uncertain variable is positively skewed, with The uncertain variable is positively skewed, with
most of the values near the lower limit. most of the values near the lower limit. The natural logarithm of the uncertain variable The natural logarithm of the uncertain variable
follows a normal distribution. follows a normal distribution.
F is also often treated as lognormal
The MFOSM method assumes that the unceThe MFOSM method assumes that the uncertainty features of a random variable can be rtainty features of a random variable can be represented by its first two moments: mean represented by its first two moments: mean and variance. and variance.
This method is based on the Taylor series eThis method is based on the Taylor series expansion of the performance function linearixpansion of the performance function linearized at the mean values of the random variazed at the mean values of the random variables. bles.
First order second moment methodFirst order second moment method
First order second moment methodFirst order second moment method
Taylor series Taylor series expansionexpansion
),.....,( 21 nxxxgg dx
dgXxXgg )()(
i
xiixnxxn x
gxgxxxg )(),...,,(),...,,( 2121
),...,,( 21 xnxxg g
i ji ji
xjxixixji
xigx
g
x
g
x
g 2
22
Example: vertical cut in clayExample: vertical cut in clay
kPacc 30100 H
cF
4
3/220 mkN
5.0c
If all variables are normal,If all variables are normal,
21094.28896.15292.02 FFF p
10H
Hc
F
4
H
cF2
4
F
F
1
1-normcdf(1.8896,0,1) MATLAB
Slope stabilitySlope stability
-35
-15
5
25
-55 -35 -15 5 25 45
n
iii
n
i iiiiii
W
MxuWxcFS
1
1
sin
)(1tan
Q
RFS f
2 (H): 1(V) slope with a height of 5m
Reliability AnalysisReliability Analysis
The reliability of a system can be defined as the pThe reliability of a system can be defined as the probability that R>Q represented asrobability that R>Q represented as: :
RQP
n
iii
n
i iiiiii
W
MxuWxcFS
1
1
sin
)(1tan
kPacc 210
210
0188.0Fp0 1 2 3 4 5 6 7 8 9 10
5
10
15
20
25
FS contouriXi m
FSFS
X
FS
2
436.1FS
089.0
c
FS
056.0
FS 21.0FS
,
,
0.21.
079.2
First Order Reliability Method First Order Reliability Method Hasofer-Lind Hasofer-Lind (FORM)(FORM)
Probability of failure can be found Probability of failure can be found obtained in material spaceobtained in material space
Approximate as distance to Limit Approximate as distance to Limit statestate
Distance to failure criterion Distance to failure criterion
If F=1 or M=0 is a straight lineIf F=1 or M=0 is a straight line Reliability becomes the shortest Reliability becomes the shortest
distancedistance
May get similar results with FOSMMay get similar results with FOSM
796.1 FOSM
1-normcdf(1. 796,0,1)=0.0362 MATLAB
Monte Carlo SimulationMonte Carlo Simulationcorrelation=0.5correlation=0.5
FORM=0.0362
}{][][}{ 21 *T/Y
* XTCY
2/12/1
2/12/1][T
10
01][ YC
-5
-4
-3
-2
-1
0
1
2
3
-5 -4 -3 -2 -1 0 1 2 3 4 5
Mean
Mean + - S.D.
FS=1.436
FS=1.0
FS=1.259
FS=1.324
FS=1.549
FS=1.613
c
ccc
*
*
xy 2
1
)XX(C)XX( 1X
Xmin
T
stateLimit
The distanceThe distance
unsaferegion
c*
*
c
ccc
*
)0,0( ** c
*FS
*cFS
*
1
c
FS
FS/),( ** c
2222
22
)/()/(
1)()( **
FScFS
FSc
c
A projection MethodA projection Method
Check the FOSMCheck the FOSM Use the slope, projected to where the Use the slope, projected to where the
failure material isfailure material is Use the material to find FSUse the material to find FS If FS=1, okIf FS=1, ok
n
iXi
Xiipi
i
I
XFS
XFSFSXx
1
22
2
)/(
)/)(1()(