Energy Engineering - RELIABILITY, SAFETY AND RISK ANALYSIS C

Chapter 04 - Exercises collection

Divided by topic

Chapter 4 Basics of probability theory for applications to reliability and risk analysis 4.1 Compressors failure Ten compressors, each one with a failure probability of 0.1, are tested independently. 1. What is the expected number of compressors that are found failed? 2. What is the variance of the number of compressors that are found failed? 3. What is the probability that none will fail? 4. What is the probability that two or more will fail? 4.2 Spare unit allocation The allocation of the proper number of spare units for a single­component system is of concern. Assume that if the operating component fails, it is instantaneously replaced by a spare unit, if available. Once the component has failed, it cannot be repaired . The rate of occurrence of components' failures is 1.67y ­1, if the component is operating. Assume that the component cannot fail while in the spares depot. The system designer's goal is to achieve a probability of system operation (reliability) of at least 0.95 at the mission time TM of one year. How many spare units should be allocated to achieve this goal? 4.3 Misprints occurrence in a book Consider the occurrence of misprints in a book, and suppose that they occur at the rate of 2 per page. 1. What is the probability that the f irst misprint will not occur in the first page? 2. Assuming a Poisson process, what is the expected number of pages until the first misprint appears? 3. Comment on the applicability of the Poisson assumptions to the occurrence of misprints or typing errors. 4.4 Peak stresses on a component Suppose that peak stresses ( i.e. stresses which exceed a certain value) occur randomly at an average rate Λ. The probability that a component will survive the application of a peak stress is (1 ­ p) (constant). Show that the reliability of the component is R(t) = e xp( ­pλt). 4.5 Traflic controll Suppose that, from a previous traffic count, an average of 60 cars per hour was observed to make left turns at an intersection. What is the probability that exactly 10 cars will be making left turns in a 10 minute interval? Discretize the time interval of interest to approach the problem with the binomial distribution. Show that the solution of the problem tends to the exact solution obtained with the Poisson distribution as the time discretization gets finer. 4.6 Traffic control 2 Suppose that it is observed that, on average, 100 cars per hour reach an intersection. Also, it has been estimated that the probability for a car to make a left turn is 0.6. What is the probability that exactly 10 cars will be making left turns in a 10 minute interval? 4.7 Aircraft flight panel An aircraft flight panel is fitted with two types of artificial horizon indicators. The times to failure of each indicator from the start of a flight follow an exponential distribution with a mean value of 15 hours for one and 30 hours for the other. A fli ght lasts for a period of 3 hours. 1. What is the probability that the pilot will be without an artificial horizon indication by the end of a flight? 2. The mean time to this event, if the flight is of a long duration? 4.8 Simple system Consider a system of two independent components with exponentially distributed failure times. The failure rates are λ1 and λ2 , respectively. Determine the probability that component l fails before component 2. 4.9 Machine survive period A machine has been observed to survive a period of 100 hours without failure with probability 0.5. Assume that the machine has a constant failure rate λ. 1. Determine the failure rate λ. 2. Find the probability that the machine will survive 500 hours without failure. 3. Determine the probability that the machine fails within 1000 hour, assuming that the machine has bee n observed to be functioning at 500 hours. 4.10 Television picture tubes The television picture tubes of manufacturer A have a mean lifetime of 6.5 years and a standard deviation of 0.9 years, while those of manufacturer B have a mean lifetime of 6.0 years and a standard deviation of 0.8 years. What is the probabilit y that a random sample of 36 tubes from manufacturer A will have a mean lifetime that is at least l year more than the mean lifetime of a sample of 49 tubes from manufacturer B? 4.11 Building safety In considering the safety of a building, the total force acting on the columns of the building must be examined. This would include the effects of the dead load D (due to the weight of the structure), the live load L (due to human occupancy, movable furnit ure, and the like), and the wind load W . Assume that the load effects on the individuai columns are statistically independent Gaussian variates with μD = 4.2 kips σD =0.3 kips μL = 6.5 kips σL =0.8 kips μW = 3.4 kips σW =0.7 kips 1. Determine the mean and standard deviation of the total load acting on a column. 2. lf the strength R of a column is also Gaussian with a mean equal to 1.5 times the total mean force, what is the probability of failure of the column? Assume that the coefficient of variation of the strength δR 15% and that the strength and load effects are statistically independent. 4.12 Capacitor A capacitor is placed across a power source. Assume that surge voltages occur on the line at a rate of one per month and they are normally distributed with a mean value of l00 volts and a standard deviation of 15 volts. The breakdown voltage of the capac itor is 135 volts. 1. Find the mean time to failure ( MTTF ) for this capacitor 2. Find its reliability for a time period of one month 4.13 Hazard function The hazard function for a device which has a performance characteristic x is ℎ(������)= 1 2√������ Derive the expression for: 1. The probability density function 2. The cumulative distribution function 3. Calculate the mean ofthe PDF 4. Which part of the bathtub curve 1s this hazard function approximating?