1 2. Reliability measures Objectives: Learn how to quantify reliability of a system Understand and learn how to compute the following measures –Reliability function –Expected life –Failure rate and hazard function Learn some common probability density functions of time to failure and learn when to apply them –Exponential –Normal –Weibull Learn how to estimate hazard functions from data Learn how to select a reliability function for a given problem

2 Reliability function Assumption: New equipment T=Failure time, random variable because we do not know when a system will fail Probability density function of failure time, f T (t). Units: # of failures per unit time Reliability function, R(t)= probability that system will work properly at time t Failure distribution function, F T (t)= probability that a system will fail by time t

3 Notation Probability density function, f T (t) T random variable (in this case it is the component life) t value that the random variable assumes f T (t)=lim  t  0 P(t<T  t+  t)/  t

4 Expected life of a component or system, E(T) t f T (t) E(T) Expected life

5 Hazard function Hazard function: h(t)=probability that, given that a system has survived until time t, it will fail between times t and t+  t, divided by  t. Units of h(t): 1/unit time h(t)= f T (t)/R(t) Example start with N=1000 light bulbs, at T=1000 hrs, 300 light bulbs are still working. After 10 hrs 5 more bulbs fail. The hazard function is approximately: h(1005)=5/(300*10)

6 Shape of hazard function of most real-life systems: bathtub function t h(t) Debugging, or infant mortality Aging

7 Relation between reliability measures

8 Reliability and hazard functions for well known distributions Exponential –Good choice for systems or components whose strength does not change with time and which are subjected to extreme disturbances occurring completely at random and independently. –f T (t)=1/  *exp(-t/  ) –R(t)= exp(-t/  ) –h(t)=1/ 

9 Shape of exponential distribution t f T (t) 1/  E(T)=θ

10 Normal distribution t f T (t)  Standard deviation,  Two parameter distribution

11 No closed form analytical expression for cumulative distribution Cumulative distribution of standard normal,  (z), has been tabulated. We also have excellent polynomial approximations. Standard normal has zero mean and unit standard deviation. Very easy to do reliability computations with normal distributions Finding F T (t) if T is normal. Transform T into standard normal. F T (t)=P(T  t)=P[(T-  )/   (t-  )/  ]=  (z)

12 Cumulative distribution of standard normal variable

13 If we model the time to failure using a normal distribution then there is small probability of the time to failure being negative. This does not make sense. Always check that the probability of the time being negative is small compared to the probabilities we are calculating in the problem at hand. For example, if the we are working with systems whose failure probabilities are about 10 -3, then the probability of the time to failure being negative should be about or less. t f T (t)  The area under the curve to the left of zero is the probability of t being negative 0

14 Weibull distribution Good choice for systems or components whose strength deteriorates with time and which are subjected to extreme disturbances occurring completely at random and independently. Consider a building in Greece that is expected to be sustain a very strong earthquake (say above 6.5 in the Richter scale) once every ten years. Like any real life system, the strength of the building deteriorates with time. A Weibull distribution is a good candidate for modeling the time to failure (or length of the life) of the building. Very popular for describing strength and life length Generalizes the exponential distribution

15 Reliability function – for t greater than  –Three parameter distribution: use shape parameter, , to control shape  is the scale parameter, affects dispersion use location parameter, , to shift the mean value shape parameter=1, Weibull reduces to the exponential distribution

16 Shape of Weibull probability density function if shape parameter less than 3.6, density is skewed to the right if shape parameter is greater than 4, density is skewed to the left

17 Shape of Weibull probability density function

18 Shape of Weibull probability density function

19 Shape of Weibull probability density function

20 Effect of shape parameter Consider building exposed to earthquakes: The larger the value of the shape parameter, the larger the rate of deterioration in strength If the shape parameter is one then there is no deterioration in the strength

21 Statistics Median: 50% probability lower than median, 50% higher than median

22 Other common distributions Lognormal; If x is normal then exp(x) is lognormal Gamma: quite similar to Weibull

23 Estimating hazard function, failure density function and reliability function from data Case 1: Large sample of data about failures (N greater than 30) Start with N systems.

24 Case 2: Small samples Study homework 3

25 Selecting a probability distribution on the basis of knowledge of the particular physical situation causing failures In most real life problems, we do not have enough data to estimate probability distributions. Therefore, we rely on experience or on analytically obtained associations of physical situations causing failure and probability distributions to select type of probability distribution to failure.

26 Weibull and exponential models Extreme disturbances occurring completely at random and independently. Example: time of occurrence and intensity of a strong earthquake does not affect the time of occurrence and intensity of the next. Probability of occurrence of one earthquake during [t, t+dt] is dt. Average rate of occurrence of extreme disturbances is disturbances/unit time Probability of a system failing because of a disturbance, p(t)

27 Earthquake intensity versus time t Intensity Severe earthquakes severe earthquakes per year return period, 1/

28 Reliability

29 Suggested reading Fox, E., “The Role of Statistical Testing in NDA,” Engineering Design Reliability Handbook, CRC press, 2004, p