Transcript Reliability computations

6. Reliability computations
Objectives
• Learn how to compute reliability of a component given the
probability distributions on the stress,S , and the strength,
Su.
• Given the probability distributions of all input random
variables, find the failure probability of a component
• Learn how to estimate failure probability of components or
systems using standard Monte-Carlo simulation and
Monte-Carlo simulation with variance reduction
techniques
– Generate sample random numbers given their probability
distributions
– Estimate failure probability and quantify accuracy of the estimate
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Finding probability of failure of component given
the probability distributions of strength and stress
• Definition: Performance function, z:
– z>0 survival
– z<0 failure
– z=limit state
Su
z=0
z>0
z<0 (failure region)
S
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Calculation of failure probability
Joint probability density of S and Su, fSUS(su,s)
Su
S
Failure region: z<0
Su=S
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Calculation of failure probability
P( F )    f Su ( su ) f S ( s )dsu ds
z 0
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Stress-strength interference
3
7.508 10
ultimate stress
0.01
stress
f s( s )
f Su ( s )
0.005
8
5.733 10
interference area
0
200
200
300
400
500
s
600
700
800
700


P( F )   FSu ( s ) f S ( s )ds   [1  FS ( s )] f Su ( s )ds


The integration limits must be adjusted if the stress or
strength assume values in a particular region only
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Examples
• Stress is normal, ultimate stress follows the
Weibull distribution
• Both stress and ultimate stress are normal
• Safety index = number of standard deviations of
Z=Su-S from E(Z) to zero.
fZ(z)
Failure
region
0
E(Z)
=E(Z)/Z
z
If stress and ultimate stress normal then P(F)=(-)
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General method for calculation of
failure probability
Failure probability = integral of joint probability
density function of random variables over failure
region
P( F )   ...  f X 1... X n ( x1 ,..., xn )dx1...dxn
z 0
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Monte-Carlo simulation
• Key idea: generate sample values of the
uncertain variables on the computer, test if
the system fails for each sample and
approximate the probability of failure by the
relative frequency of failure.
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Standard Monte-Carlo simulation
Define problem
Estimate probability distribution
of random variables
Generate N sets of sample values
of the random variables
Calculate the performance
function for each set
P(F) = number of failures/N
 P( F ) 
P ( F )[1  P ( F )]
N
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How to generate random numbers from
given probability distribution, FX(x)
( z)  FX ( x)
(z )
z
FX (x)
x
x, z
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Comments on standard Monte-Carlo
simulation
• Expensive, especially when failure
probability is small (i.e., 10-6)
• Often used to validate approximation of
failure probability or to validate optimum
design selected using approximate methods
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Importance sampling
• Reduces sample size required to estimate P(F)
with given accuracy
• Idea: generate random numbers from sampling
density, f s, instead of true density, f
• Sampling distribution is selected so as to generate
many failures
• Discount each failure according to ratio of true
probability of occurrence to probability of
occurrence based on sampling distribution
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Importance sampling
Define problem
Estimate probability distribution of random
variables, f
Generate N sets of sample values random
variables from sampling distribution f s
Calculate the performance function for each
set
f
1 N
P ( F )  i 1 ( I i
)*
N
fs
*Ii failure index function,
1 if failure occurs, 0 otherwise
 P( F )  {
1 N
f 2
)  P( F ) 2 }1 / 2
i 1 ( I i
N2
fs
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Suggested reading
• Ghiocel, D., M., “Stochastic Simulation
Methods for Engineering Predictions,”
Engineering Design Reliability Handbook,
CRC press, 2004, p. 20-1.
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