6' Reliability computations

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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
• zgt0 survival
• zlt0 failure
• zlimit state

z0
Su
zgt0
zlt0 (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 zlt0
SuS
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Calculation of failure probability
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Stress-strength interference
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
  ZSu-S from E(Z) to zero.

If stress and ultimate stress normal then
P(F)?(-?)
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General method for calculation of failure
probabilityFailure probability integral of
joint probability density function of random
variables over failure region
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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
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How to generate random numbers from given
probability distribution, FX(x)
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Comments on standard Monte-Carlosimulation

• 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
Ii failure index function, 1 if failure occurs,
0 otherwise
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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.