Fault Tree Analysis (FTA): AND/OR Gates and Worked Example

What fault tree analysis (FTA) is

Fault tree analysis (FTA) starts from an unwanted failure, called the top event, and works down through causes and combinations of causes to basic events such as a bearing failure or an operator error. By linking those events with AND and OR logic gates, the tree shows which combinations lead to the top event and lets you calculate its probability.

It is a deductive, top-down analysis. FMEA goes the other way: it starts from each component failure mode and asks what the effect is. Use FMEA to scan a design or process, and FTA to investigate one serious failure in depth.

Visual map of the fault tree

The map shows the example tree, the probability at each gate, the cut sets and the probability of the top event.

Symbols and logic gates

The OR gate adds failure paths; the AND gate requires coincidence and therefore represents a protection or redundancy. The more AND gates in the tree, the more robust the system.

How to build an FTA in 6 steps

1. Define the top event specifically and measurably: "cooling pump with no flow", not "cooling failure".
2. Set the scope: system boundaries, operating conditions and mission time (for example, 1 year).
3. Go down one level at a time: ask "what is necessary and sufficient for this event to occur?" and join the answers with AND or OR.
4. Stop at basic events, where failure data exists or further breakdown makes no sense.
5. Assign probabilities to basic events from maintenance history or manufacturer data.
6. Calculate from the bottom up and derive the minimal cut sets.

Gate formulas

With independent events, probabilities combine like this:

AND gate: P = P1 × P2 × … × Pn  |  OR gate: P = 1 − (1 − P1) × (1 − P2) × … × (1 − Pn)

For a constant failure rate λ over period t, the probability of a basic event is P = 1 − e^−λt. For rare events, the OR gate can be approximated by the sum of probabilities, which always slightly overestimates the result. The link between failure rate and equipment life is in the bathtub curve article, and rates come from indicators like MTBF.

Worked example: cooling pump with no flow

Fictional example, mission time of 1 year, independent events. Top event: cooling pump with no flow. Structure:

• Top = OR (G1, G2, E5).
• G1 "Motor without power" = AND (E1, E2).
• G2 "Pump mechanical failure" = OR (E3, E4).

The top event probability is 0.0636, or 6.36% per year. With the rare-event approximation, the sum 0.005 + 0.02 + 0.03 + 0.01 = 0.065 (6.5%) is close, with an error of about 2%.

Minimal cut sets

A cut set is a combination of basic events that, occurring together, causes the top event; a minimal one cannot be reduced without losing that property. The example has four:

First-order sets are single points of failure: one event takes the system down. Here, the mechanical seal and bearing account for 77% of the probability and deserve priority action, such as a preventive maintenance plan and a spare part.

Test an improvement: a lockout on the suction valve reduces E5 from 0.01 to 0.002. New top = 1 − 0.995 × 0.9506 × 0.998 = 0.0560, or 5.60%. The 0.76 percentage-point drop shows the gain before you spend on the change.

When to use FTA

• Investigating serious safety, environmental or long-downtime failures, alongside the 5 Whys and the fishbone diagram.
• Designing redundant systems, to show that the combination of failures is unlikely.
• Prioritizing actions from RCM when the failure consequence is high.

Common mistakes and limitations

• Vague top event. "Machine failure" produces an endless tree.
• Skipping levels, linking a basic event straight to the top without the mechanism.
• Treating events as independent when they share a cause, such as two pumps on the same electrical panel. The result is optimistic.
• Made-up probabilities. Without data, work with ranges and run a sensitivity analysis.
• Forgetting human error and procedural failure, which weigh 15% in the example.
• Mixing time bases: all probabilities must refer to the same period.

Frequently asked questions

What is a top event?

It is the unwanted failure the tree analyzes, defined specifically, such as "cooling pump with no flow".

What is the difference between an AND gate and an OR gate?

In an OR gate one input is enough for the output to occur; in an AND gate all inputs must occur at the same time.

How do you calculate the top event probability?

Work from the bottom up: multiply probabilities at AND gates and use 1 − ∏(1 − P) at OR gates. In the example, the result is 6.36%.

What is a minimal cut set?

It is the smallest combination of basic events that causes the top event. If it has a single event, it is a single point of failure.

What is the difference between FTA and FMEA?

FTA is deductive: it starts from the final failure and looks for causes. FMEA is inductive: it starts from each component failure and assesses the effect.

Does FTA work for any system?

It works for systems whose failures can be described with AND/OR logic. For systems with event sequences or repair, additional methods are needed.

About the author

Vagner Soares

Lean Manufacturing & Behavioral Management Specialist

Over 20 years in the automotive and metalworking industries (GM and Dana), Lean Manufacturing practitioner since 2006. SENAI instructor and mentor in Brazil’s Brasil Mais Produtivo program, delivering consulting, training and audits for 50+ companies, combining quality, productivity and people development.

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