| name | fault-tree-analysis |
| description | Fault tree analysis (FTA) — top-down failure deduction, Boolean logic, minimal cut sets, probability calculation, qualitative and quantitative FTA, IEC 61025, MIL-STD-1629. |
| metadata | {"priority":7,"promptSignals":{"phrases":["fault tree","FTA","fault tree analysis","minimal cut set","top event probability","Boolean fault tree"],"minScore":3}} |
Fault Tree Analysis (FTA) — Complete Skill
Purpose and Methodology
FTA is a top-down deductive analysis: start from undesired top event → identify causes
Steps:
- Define top event (undesired system failure)
- Identify immediate causes → connect with AND/OR gates
- Continue decomposing until basic events (component failures) reached
- Solve for minimal cut sets (qualitative)
- Quantify with failure probabilities (quantitative)
Gate Types and Symbols
| Gate | Symbol | Logic | Output occurs when |
|---|
| AND | flat bottom | Conjunction | ALL inputs occur |
| OR | curved bottom | Disjunction | ANY input occurs |
| INHIBIT | hexagon | Conditional AND | Input occurs AND condition met |
| Priority AND | AND + arc | Sequential | Inputs in specified order |
| Exclusive OR | XOR | | Exactly one input occurs |
Event Types
Basic event (circle): lowest-level cause; assigned failure probability
Undeveloped event (diamond): not further analyzed (lack of data or scope)
House event (house): external event assumed occurred or not; probability = 0 or 1
Intermediate event (rectangle): result of gate logic; developed further
Qualitative Analysis — Minimal Cut Sets
A cut set = set of basic events whose simultaneous occurrence causes top event
A minimal cut set (MCS) = smallest set; no event can be removed and still cause top event
Boolean reduction: MOCUS algorithm or BDD (Binary Decision Diagram)
MCS of size 1 → single point of failure
MCS of size 2 → two simultaneous failures needed
Higher-order MCS → less critical (lower probability)
Qualitative importance: single MCS (order 1) most critical → eliminate first
Quantitative Analysis
Failure probability:
- For independent basic events:
- AND gate: Q_AND = ∏ Q_i
- OR gate: Q_OR = 1 - ∏(1-Q_i) ≈ ΣQ_i (for small Q_i)
Top event probability (from MCS, inclusion-exclusion):
Q_top = P(MCS₁ ∪ MCS₂ ∪ ...)
Rare event approximation: Q_top ≈ Σ P(MCS_k)
Example:
MCS₁ = {A, B}; MCS₂ = {C}
Q_top ≈ Q_A × Q_B + Q_C (rare event)
Failure Rate vs. Probability
For constant failure rate λ and mission time t:
F(t) = 1 - e^(-λt) ≈ λt (for λt << 1)
Use failure rate databases: MIL-HDBK-217, OREDA, T-Book (nuclear)
Typical failure rates:
Electronic control unit: λ ≈ 10⁻⁶/hr
Valve (automatic, fail-safe): λ ≈ 10⁻⁵/hr
Check valve: λ ≈ 3×10⁻⁵/hr
Relief valve (fail to open): λ ≈ 10⁻⁵/hr
Importance Measures
Birnbaum importance (structural):
I_B = ∂Q_sys/∂Q_i (sensitivity of top event probability to component i reliability)
Risk Reduction Worth (RRW):
RRW_i = Q_sys / Q_sys|Q_i=0 (how much does top event decrease if component i is perfect?)
Risk Achievement Worth (RAW):
RAW_i = Q_sys|Q_i=1 / Q_sys (how much does top event increase if component i always fails?)
High RAW = critical reliability; high RRW = large improvement potential
Uncertainty Analysis
Component failure rates have uncertainty → propagate to top event
Monte Carlo: sample Q_i from distributions; compute Q_top many times → get distribution
Analytical: lognormal approximation with error factors (EF = 95th/5th percentile ratio^0.5)
GRS method: treat point estimates as median; assume lognormal with EF = 3–10
Common Format Requirements
IEC 61025: Standard for FTA methodology; defines symbols, analysis requirements
MIL-STD-1629A: Requirements for FMECA (companion to FTA)
Nuclear (10 CFR 50): PRA (probabilistic risk assessment) requirements
Aerospace (ARP4761): Safety assessment for civil aircraft (with FMEA)
Common Mistakes
- Confusing states with events (use active voice: "valve fails to open", not "valve open")
- Double-counting repeated events without proper treatment
- Ignoring common-cause failures (CCF) — can dominate redundant system risk
- Not validating cut sets against physical system behavior
Output
Provide: fault tree diagram (textual if needed), all minimal cut sets (with order), top event probability Q_top, component importance rankings (Birnbaum), single points of failure identified, uncertainty range (5th–95th percentile).
