1 6. Reliability computations Objectives Learn how to compute reliability of a component given the probability distributions on the stress,S, and the strength, S u. 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

2 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 S Su z>0 z=0 z<0 (failure region)

3 Calculation of failure probability S SuSu S u =S Failure region: z<0 Joint probability density of S and Su, f S U S (s u,s)

4 Calculation of failure probability

5 Stress-strength interference The integration limits must be adjusted if the stress or strength assume values in a particular region only

6 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=S u -S from E(Z) to zero. z f Z (z) E(Z) 0  =E(Z)/  Z Failure region If stress and ultimate stress normal then P(F)=  (-  )

7 General method for calculation of failure probability Failure probability = integral of joint probability density function of random variables over failure region

8 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.

9 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

10 How to generate random numbers from given probability distribution, F X (x) x, z x z

11 Comments on standard Monte-Carlo simulation Expensive, especially when failure probability is small (i.e., ) Often used to validate approximation of failure probability or to validate optimum design selected using approximate methods

12 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

13 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 *I i failure index function, 1 if failure occurs, 0 otherwise

14 Suggested reading Ghiocel, D., M., “Stochastic Simulation Methods for Engineering Predictions,” Engineering Design Reliability Handbook, CRC press, 2004, p