QRA Calculations: Frequency, Consequence and Risk Examples

Quantitative Risk Assessment calculations are not a single formula. A defensible QRA is a calculation chain that connects equipment failure, release conditions, detection and isolation, ignition outcomes, physical effects, exposure and vulnerability.

Core QRA Risk Equations

Scenario frequency

For a simplified outcome branch:

Outcome frequency = initiating-event frequency × conditional probabilities

Conditional probabilities can represent successful or failed isolation, immediate ignition, delayed ignition, weather category or other mutually consistent branches. Probabilities must not be multiplied blindly: the event tree must conserve probability and avoid double counting.

Individual risk

At a location x, individual risk is commonly expressed conceptually as:

IR(x) = Σ [f_i × P_fatality,i(x)]

Here, f_i is the annual frequency of scenario i, and P_fatality,i(x) represents the probability of fatality at the location for that outcome. The calculation may also include directional, weather and occupancy weighting defined in the study basis.

Societal risk

Societal risk considers the number of people affected. An F-N curve plots cumulative annual frequency F against N, the number of fatalities or another defined harm measure. Each point represents the combined frequency of scenarios capable of producing at least N consequences.

Individual Risk, Societal Risk, F-N Curves and ALARP

Risk measure	What it represents	Typical presentation	Essential inputs
Individual risk	Annual risk to a defined hypothetical person at location x	Risk contours over the plot plan and surrounding area	Scenario frequencies, fatality probability, weather direction and defined exposure assumptions
Location-specific individual risk	Risk assuming the defined person remains at the location according to the study convention	A value per year at a point or a contour	Study-specific presence and vulnerability conventions
Individual-specific risk	Risk to a named or representative worker considering time spent in different locations	Annual risk value by worker group	Occupancy, movement, work pattern and vulnerability assumptions
Societal risk	Frequency of accidents affecting groups of different sizes	Cumulative F-N curve	Scenario frequency, consequence footprint, population distribution, occupancy and time basis

How is an F-N curve calculated?

For each scenario, the analyst estimates the number of fatalities or other defined harm measure, N, and its annual frequency. The scenarios are ordered by consequence size. At each value of N, the cumulative frequency F is the sum of the frequencies of all scenarios causing N or more consequences. Both axes are normally logarithmic, and the curve must identify whether it represents onsite, offsite or combined populations.

How should a 10⁻⁶ per-year contour be interpreted?

A 10⁻⁶ per-year individual-risk contour represents a modelled annual probability of one in one million under the QRA’s stated assumptions for a hypothetical person at that location. It does not predict that one person will die every million years, and it should not be treated as a universal approval boundary without checking the applicable authority, endpoint and study basis.

What does ALARP mean in a QRA?

ALARP means As Low As Reasonably Practicable. It is not demonstrated merely because a calculated risk falls below a plotted line. The study should identify further risk-reduction options, estimate their safety benefit, test material measures where practicable, document good-practice requirements, and justify why any remaining options are not reasonably practicable within the governing legal framework.

Are QRA acceptance criteria universal?

No. Criteria can vary by country, authority, facility type, worker/public exposure, risk endpoint and project decision. UK HSE guidance, for example, discusses individual and societal risk within its own tolerability framework; those numerical examples must not be copied into an Indian or international project without confirming that the client and competent authority accept the same basis.

Minimum F-N curve quality checks

• State whether F is cumulative frequency of N or more consequences.
• Define the harm endpoint, population boundary, occupancy and day/night treatment.
• Use mutually consistent scenarios and avoid double counting event-tree branches.
• Show the effect of weather, ignition, isolation and vulnerability assumptions.
• Identify dominant scenarios and complete sensitivity checks for material uncertainties.
• Name and justify the project’s acceptance criteria instead of presenting unlabeled diagonal lines.

Authoritative background: UK HSE’s QRA guidance for explosives in ports explains individual risk, societal risk and the ALARP/tolerability framework. Its values are jurisdiction-specific examples, not universal QRA criteria.

Worked QRA Frequency Example

The following numbers are illustrative only and are not suitable for a real facility.

Input	Illustrative value
Initiating leak frequency	2 × 10−4 per year
Probability that rapid isolation fails	0.20
Probability of immediate ignition	0.10

The illustrative jet-fire branch frequency is:

2 × 10⁻⁴ × 0.20 × 0.10 = 4 × 10⁻⁶ per year

This does not yet equal risk at a person’s location. Consequence modelling must determine the thermal dose at that location, and the vulnerability relationship converts the exposure into a harm probability. Other event-tree branches must also be calculated and aggregated.

Step-by-Step QRA Calculation Workflow

1. Freeze the study basis: define boundaries, operating cases, weather categories, population treatment, risk criteria and required outputs.
2. Build the scenario register: connect each release case to equipment, material, hole size, pressure, temperature, isolation and release duration.
3. Assign initiating frequencies: use the project-approved database or study, with units and equipment population stated explicitly.
4. Construct event trees: represent detection, isolation, ignition and alternative outcomes without overlapping branches.
5. Calculate source terms: estimate release rate, duration, flashed fraction and inventory discharged.
6. Model physical effects: calculate toxic concentration, thermal radiation or explosion overpressure as applicable.
7. Apply exposure and vulnerability: account for occupancy, location and the project-approved harm relationships.
8. Aggregate risk: sum compatible scenario contributions into individual-risk and societal-risk outputs.
9. Test sensitivity: vary material assumptions such as weather, ignition, isolation time and population.
10. Recalculate material safeguards: demonstrate residual risk after feasible mitigation where the data allow it.

QRA Input Data and Units

Calculation area	Typical inputs	Common quality failure
Release modelling	Pressure, temperature, composition, inventory, hole size, line diameter	Using normal rather than credible upset conditions without explanation
Frequency analysis	Equipment count, service, failure-rate basis, operating time	Mixing per-item and per-metre-year data
Event tree	Detection, isolation, immediate and delayed ignition	Branches that do not sum consistently or duplicate outcomes
Weather	Wind speed, stability, direction and frequency	Using an unrepresentative single weather case for risk aggregation
Population	Occupancy, shift pattern, indoor/outdoor fraction, public receptors	Applying peak population continuously for all scenarios or omitting nearby receptors
Vulnerability	Thermal dose, toxic dose or overpressure relationship	Using a threshold as if it were a probability model

Consequence Calculations

QRA software may calculate toxic dispersion, flammable-cloud extent, jet fire, pool fire, fireball or BLEVE, and vapour-cloud explosion. The US EPA states that ALOHA can model these six broad hazard categories and display threat zones. More complex geometry, congestion, terrain or dense-gas behaviour may require a different validated tool and specialist review.

See the QRA software selection guide for PHAST, SAFETI and ALOHA considerations.

Uncertainty and Sensitivity

A risk value should not be presented as exact. Frequency data, ignition probabilities, human actions, weather, source terms, population and vulnerability all introduce uncertainty. The UK Health and Safety Executive cautions against assigning precise human-failure probabilities without documenting the assumptions and data sources.

At minimum, the report should identify the parameters that control the result and show how reasonable alternatives change risk contours, scenario rankings or F-N curves.

Why Data Quality Matters

The US Chemical Safety Board’s FY 2025 report states that, from March 2020 through 30 September 2025, it received reports of more than 555 serious chemical incidents across 43 US states. Those reports involved fatalities at 87 facilities, serious injuries at 308 facilities and/or substantial property damage at 260 facilities. These incident statistics do not provide universal failure rates, but they demonstrate why scenario data, safeguards and assumptions require evidence rather than generic templates.

QRA Calculation Review Checklist

• Are all frequencies traceable to a source, population and unit?
• Do event-tree branches cover the defined outcomes without double counting?
• Are release duration and isolation assumptions consistent with the design?
• Are weather and directional probabilities applied consistently?
• Can key consequence results be reproduced from the stated inputs?
• Are individual and societal risk calculated using the approved population basis?
• Are dominant scenarios and uncertainty drivers visible?
• Are recommendations linked to scenarios and recalculated where material?

Use the QRA report review checklist for the complete deliverable structure.

Frequently Asked Questions

What is the basic QRA formula?

The simplest expression is risk equals frequency multiplied by consequence. A full process-safety QRA aggregates many scenario frequencies, physical effects, exposure conditions and vulnerability probabilities.

What does 10⁻⁶ per year mean?

It represents one chance in one million per year under the defined model and assumptions. It is not automatically an acceptance criterion; the project must state the applicable risk criteria.

Are QRA calculations possible in a spreadsheet?

Spreadsheets can support frequency calculations, event trees and aggregation, but complex source-term and consequence modelling normally requires a validated specialist tool and QA process.

How are safeguards credited?

A safeguard should be credited only when its independence, reliability, detection, response time and operating context are justified. Unsupported credit can materially understate risk.

Official Sources

• US EPA — ALOHA Software
• UK HSE — Managing Human Failures
• UK HSE RR1186 — Failure Rates for Major-Accident Hazard Pipelines
• US CSB — FY 2025 Performance and Accountability Report

Before applying a numerical threshold: confirm its jurisdiction, endpoint and decision context in the international QRA standards and acceptance-criteria guide.