1

6. Reliability computationsObjectives

• 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

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>0z=0

z<0 (failure region)

3

Calculation of failure probability

S

Su

Su=SFailure region: z<0

Joint probability density of S and Su, fSUS(su,s)

4

Calculation of failure probability

dsdssfsfFP uSuSu0z

)()()(

5

Stress-strength interference

2003004005006007008000

0.005

0.017.508103

5.733108

fss( )

fSus( )

700200 s

ultimate stress

stress

interference area

dssfsFdssfsFFP SuSSSu )()](1[)()()(

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=Su-S from E(Z) to zero.

z

fZ(z)

E(Z)0=E(Z)/Z

Failureregion

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

n1n1XX0z

dxdxxxfFP n1 ...),...,(...)( ...

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 simulationDefine 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

NFP1FP

FP)]()[(

)(

10

How to generate random numbers from given probability distribution, FX(x)

)()( xFz X

x, zxz

)(z )(xFX

11

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

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

212N1i

2si2FP FP

f

fI

N

1 /)( })()({

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

*)()( N

1i sif

fI

N1

FP

*Ii 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. 20-1.