PFA
FV 3102 Probabilistic Risk Assessment (PRA) – Assignment (June 23, 2009) Past Paper of 2020/2021 Page 1 of 7 FV3102 Probabilistic Risk Analysis (PRA) Assignment in lieu of Exam The deadline for submission is at 23:59 (HKT) on 8th December 2020 (Tuesday) The answers of all questions should be typed or word-processed except for calculation and formulae which are allowed to be answered in handwriting, and submitted through the CityU Canvas Assignment folder. Aims of Assessment To encourage you to précis information from relevant books and lectures To assess your knowledge and understanding of a wide range of techniques used in probabilistic risk modelling and encourage you to start preparing for your end of semester exam. To enhance your ability of analysis, modelling and problem solving in the engineering problems. Learning Outcomes This piece of assessment will test your ability to meet learning outcomes 4 and 5 as described in your module booklet: - 4. Review Reliability Engineering Analysis of complex systems 5. Apply and assess the effectiveness of fault, success and event trees Assignment Details This is an individual assignment. All plagiarism will result in a mark of ZERO for the assignment. This assignment requires the students to complete ALL questions as attached; Round the results to FOUR decimal places as needed; All the ASSUMPTIONS and STEPS in your answers should be clearly stated due to be awarded; and This assignment will carry 60% weighting of the total mark for this module. Part A counts towards 55% and Part B counts towards 45% of this Assignment. All questions are short questions. Submission Details Answer ALL questions. All figures and curves must be properly drawn or drawn on graph papers. The assignment should be answered in your own handwriting and then scanned the completed assignment as pdf format. Turnitin report is not required for this assignment. Past Paper of 2020/2021 Page 2 of 7 Assignment should be submitted through the CityU Canvas Assignment folder by the deadline set out at the top of the page. Five days rule will be applied for any late submission, i.e. 40% mark is capped. Assignment Details – Answer ALL Questions Part A (55 Marks) (Learning Outcome 4) A.1 Test is performed to estimate the failure rate of a new designed component. Ten identical components were tested until they either fail or reach 1000 hours. The test data are shown in the following table. Component Hours Failure 1 1000 No failure 2 912 Failed 3 89 Failed 4 1000 No failure 5 855 Failed 6 1000 No failure 7 782 Failed 8 1000 No failure 9 972 Failed 10 1000 No failure a) Based on the empirical method, calculate the reliability ?(?ⅈ), failure probability ?(??), failure density ?(??) and hazard rate ℎ(??) in each 100-hour period; b) Plot the ‘bath-tub’ diagram using the hazard rate and estimate the duration of burn- in period, useful life period and time at which burn-out period starts. A.2 A system contains two subsystems in series. System 1 consists of four identical components and for this particular group it is necessary that two out of the four components functions satisfactorily for system success. System 2 has four identical components in parallel and system success requires that at least three of these components must function. a) Present a Reliability Block Diagram to represent the system. b) Assume each component has the same reliability in terms of ?, derive a general expression for the unreliability of the system (i.e. Failure probability of the system). c) Assume each component has an identical failure rate of λ = 0.0002 per hour. Estimate the reliability of the system at 2000-hours operation time. Past Paper of 2020/2021 Page 3 of 7 A.3 Components Following please find three designed systems which consisted of six identical components: System A: System B: System C: Assume the identical components are independently and exponential distributed with a constant failure rate 0.02 per hours. For each system, derive the average failure rate and mean time to failure (MTTF) of the system. A.4 A system is defined to have three states: (a) working; (b) under repair; (c) waiting for a new task. a) Suppose that if the system was working yesterday, today the probability to break is 0.1 and the probability to go to waiting is 0.2; if the system was under repair yesterday, then today the system will get repaired and become waiting state with the probability of 0.1. A broken system will never be brought directly to work in one step. If the system was waiting yesterday, there will be 0.9 probability to get into Comp 1 Comp 2 Comp 4 Comp 5 Comp 6 Comp 3 Comp 1 Comp 2 Comp 4 Comp 5 Comp 6 Comp 3 Comp 1 Comp 2 Comp 3 Comp 4 Comp 5 Comp 6 Past Paper of 2020/2021 Page 4 of 7 working. A waiting system will never break directly. Describe the system as a Markov process and find out the probability of the system at working status after a long time? b) Consider the same system in continuous time. Assume the state transition takes an exponential amount of time with constant rate. Suppose the failure rate (i.e. rate from working to repair) is 0.02 per day, the rate from waiting to operating is 1 per day, the rate from operating to waiting is 0.1 per day and the repair rate (i.e. rate from repair to waiting) is denoted as r. Describe the system as a Markov Chain and find out the repair rate r so that the system in steady state will be in the repair state less than 5% of the time. A.5 Consider the pumping system shown in the figure below. The purpose of the system is to pump water from point A to point B. The time to failure of all the valves and pump can be represented by the exponential distributions with failure rate ?? and ?? respectively. a) Calculate the reliability function of the pumping system in terms of ?? and ??; b) Given ?? = 0.002 failures per hour and ?? = 0.001 failures per hour, if we know that the pumping system has operated for 20 hours. What is the probability that the pumping system will survive for not more than 40 hours? c) If ?? = 0.002 failures per hour and ?? = 0.001 failures per hour, what is the MTTF of the pumping system? Point A Point B Past Paper of 2020/2021 Page 5 of 7 Part B (45 Marks) (Learning Outcome 5) B.1 Use fault tree analysis to study the failure of the top event (T). The system is illustrated as below: a) Find the minimum cut sets of this fault tree and identify the component importance of the basic components. b) Assume the failure probability for the basic components is ??, what is the failure probability of the top event (T). B.2 Assume an atrium has a headroom of 25m. As a performance-based fire safety design, beam detectors will be provided at level 14.0m and 24.5m to act as the smoke detection system for the atrium. To minimize the possible false alarm, cross zoned detection is applied to the beam detectors, that is, two beam detectors will be provided in the same detection zone. Due to the excessive ceiling height, long-throw sprinkler system actuated by infra-red flame detectors is installed at level 3.0m to provide adequate water coverage to the atrium base as the fire suppression system for the atrium. Cross zoned detection is also applied to the infra-red flame detectors, that is, two infra-red T E1 G1 E2 G2 G3 G4 G5 E1 G6 E4 E5 E6 E7 E5 E6 Past Paper of 2020/2021 Page 6 of 7 flame detectors will be provided in the same detection zone. Therefore, the fire protection system for the atrium include: Beam detectors Two at level 14.0m Two at level 24.5m Infra-red flame detectors Two at level 3.0m Long-throw sprinkler system One set (One pump in duty, One pump in standby and One control valve) a) Use fault tree or event tree to illustrate the failure of the fire protection system serving for the atrium; b) Given the probability of the following events, evaluate the failure probability of the fire protection system serving for the atrium. Basic Event Probability Beam detector fails to send fire signal to the fire safety control panel in case of fire 0.01 Infra-red flame detector fails to send fire signal to the fire safety control panel in case of fire 0.015 The fire safety control panel will not deliver a signal to actuate the pumps and valve 0.01 Pump fails to start upon receipt of the actuation signal 0.005 Valve fails to open upon receipt of the actuation signal 0.001 No water in the sprinkler tank 0.0001 B.3 The initiating event is a fire, for which we identify two possible consequences: activate the sprinkler system and activate the fire alarm. a) Use event tree analysis to develop the fire consequence b) What is the average risk of fire damage based on the following information: P(Sprinkler system success): 0.9 P(Fire alarm success | Sprinkler system success): 0.95 P(Fire alarm success | Sprinkler system failure): 0.6 Cost (fire handled OK):