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Pressure tank rupture of the NUREG-0492 Fault Tree Handbook: minimal cut sets, top-event probability and importances

ToolFault Tree Analysis
SourceVesely, W. E., Goldberg, F. F., Roberts, N. H., and Haasl, D. F. (1981). Fault Tree Handbook. NUREG-0492, U.S. Nuclear Regulatory Commission, Washington, DC. Chapter VIII (Figure VIII-14, Table VIII-1); repeated as Chapter 10 of Stamatelatos, M., et al. (2002). Fault Tree Handbook with Aerospace Applications. NASA, Washington, DC.
LevelUndergraduate

Problem

A pressure tank is filled by a pump whose motor is switched by relay K2; a timer relay R, relay K1, pressure switch S and switch S1 control the circuit. The top event is rupture of the tank after the start of pumping. With the handbook's reduced tree TOP = T + E2, E2 = K2 + E3, E3 = S · E4, E4 = S1 + E5, E5 = K1 + R and the failure probabilities of Table VIII-1, find the minimal cut sets, their probabilities, the top-event probability and the importance of each cut set.

Inputs to Enter

  • Gates: TOP: OR(T, E2); E2: OR(K2, E3); E3: AND(S, E4); E4: OR(S1, E5); E5: OR(K1, R)
  • Basic events: T = 5e-6 (tank), K2 = 3e-5, S = 1e-4 (pressure switch), K1 = 3e-5, R = 1e-4 (timer relay), S1 = 3e-5
  • Method: exact when the cut-set count allows; no truncation; path sets on
Open the Fault Tree Analysis tool (loaded with this example)

Answers

  • Five minimal cut sets, two singles and three doubles: K2 (3e-5), T (5e-6), S·R (1e-8), S·K1 (3e-9), S·S1 (3e-9) — the handbook's list
  • Rare-event sum 3.5016e-5 (the handbook prints 3.4e-5, an arithmetic slip: its own five terms add to 3.5016e-5); min-cut upper bound 3.5016e-5; exact probability of the union 3.50158e-5 (numpy truth table 3.5015848750e-5)
  • Cut-set importances K2 85.7 %, T 14.3 % (handbook 86 % and 14 %), the three doubles below 0.1 %
  • Fussell–Vesely K2 0.8568, T 0.1428; Birnbaum K2 0.999995, T 0.99997, S 1.600e-4; RAW of T and K2 28,558; RRW of K2 6.981
  • Minimal path sets {K2, S, T} and {K1, K2, R, S1, T}; the probability that no path set is intact equals the exact value from the cut sets

Notes

The tool's default inputs are this tree; press Submit. Every printed cut set, cut-set probability and importance is checked in test_fault_tree.php with the NASA handbook's monopropellant system (Chapter 11, seven cut sets, 2.3e-4) and three-seal common-cause example (Chapter 12, 1.0e-9 to 1.0e-4 with a beta factor of 0.1), the three-motor and detailed-tree cut-set counts of NUREG-0492 Chapters IX and XI, and the numpy enumeration References/Fault-Tree/checks/fta_truth_table.py.