Social Risk & F-N Curve

1. Introduction#

Social Risk quantifies the hazard that an industrial facility poses to the surrounding community. Unlike Individual Risk (which evaluates the probability that a specific person will die), Social Risk expresses the relationship between the frequency of an accident and the number of fatalities it could cause.

The primary tool for visualizing and evaluating Social Risk is the F-N Curve (frequency-number chart):

• X-axis (N): Number of fatalities in a single event
• Y-axis (F): Cumulative frequency $F(N)$ = annual frequency of events causing at least N fatalities [events/year]

Both axes use a logarithmic scale.

2. Key Concepts#

2.1 Formal Definitions#

3. Calculation Sequence#

flowchart TD
  A["Risk Models<br/>(PoolFire, JetFire, VCE, FlashFire, FireBall)<br/>with frequency P(As) per scenario"] --> B["Step 1: Model Fatalities<br/>FatalityUtils.js<br/>Concentric rings 5m wide, up to 10 km<br/>E[F(As)] = Σ Fatal_i"]
  B --> C["Step 2: Exposed Population Ng<br/>ng-calculator.ts<br/>Probit: Ng = π×r₁%²×ρ_pop<br/>Flash Fire: Ng = A_LFL×ρ_pop<br/>Wind rose: Ng = Σ Ng_i×P(dir_i)"]
  C --> D["Step 3: F-N Curve Construction<br/>social-risk-table.tsx<br/>F(N) = Σ fs for all s where Ns ≥ N<br/>Decreasing step function on log-log axes"]
  D --> E["Step 4: ALARP Criteria Evaluation<br/>fn-curve-analysis.ts<br/>Upper limit: F_lim(N) = C₁/N^|n|<br/>Excess: ΔF(N) = F(N) − F_obj(N)"]

  style A fill:#e1f5fe
  style E fill:#c8e6c9
  style D fill:#fff3e0

1. Model Fatalities — For each scenario, FatalityUtils.js integrates the death probability over concentric rings of 5 m width (up to 10 km), computing $E[F(A_s)] = \sum_i F_i$ using the applicable Probit function.

2. Exposed Population $N_g$ — ng-calculator.ts computes $N_g$ using the 1% Probit radius for thermal/overpressure models, the LFL polygon area for Flash Fire, or the weighted wind-rose sum for multi-directional scenarios.

3. F-N Curve Construction — calculateFnCurve() collects all $(f_s, N_s)$ pairs and computes $F(N_k) = \sum_{s:\,N_s \geq N_k} f_s$ for each unique $N$ value, producing a decreasing log-log step function.

4. ALARP Evaluation — analyzeUnacceptableScenarios() compares each curve point against the upper limit $F_{lim}(N) = C_1/N^{|n|}$, computes excess frequency $\Delta F$, and generates frequency reduction recommendations per scenario.

4. Fatality Calculation by Concentric Rings#

4.1 Ring Geometry and Thermal Dose#

The study area is divided into concentric rings of 5 m width, from the emission point to the maximum radius (default 10 km):

The accumulated thermal dose in each ring:

Code: FatalityUtils.js → calculateFatalities().

4.2 Fatality Probit Functions#

4.3 Probit to Fatality Percentage#

4.4 Fatalities per Ring#

5. Exposed Population — $N_g$#

6. F-N Curve Construction#

6.1 Formal Definition#

Given $S$ accidental scenarios, each with frequency $f_s$ [events/year] and fatalities $N_s$ [persons]:

$F(N)$ is the sum of frequencies of all scenarios that produce at least $N$ fatalities.

Code: social-risk-table.tsx → calculateFnCurve().

6.2 Algorithm#

1. Collect all scenarios with their pairs $(f_s, N_s)$: frequency and fatalities for each

2. Extract the unique set of $N$ values, sorted in descending order

3. For each $N_k$: compute $F(N_k) = \sum f_s$ where $N_s \geq N_k$

4. Plot the pairs $(N_k,\, F(N_k))$ as a decreasing step function on log-log axes

7. Tolerance Zones and ALARP Criteria#

7.1 Limit Lines#

The tolerability lines are power functions on the log-log plane:

Code: fn-curve-analysis.ts → calculateUpperLimit(); fn-curve-chartjs.tsx → drawLimitLines().

7.2 ALARP Zones#

7.3 International Standard Criteria#

8. Unacceptable Scenario Analysis#

A point $(N, F(N))$ is unacceptable when $F(N) > C_1 / N^{|n|}$.

1. Identify unacceptable point — Find all $(N_k, F(N_k))$ where $F(N_k) > F_{upper}(N_k)$

2. Apply ALARP target — Set objective with 10% safety margin below upper limit: $F_{target}(N) = F_{upper}(N) \cdot (1 - m), \quad m = 0.10$

3. Compute excess frequency: $\Delta F(N) = F(N) - F_{target}(N)$

4. Required reduction per scenario — For contributing scenario $s$: $f_{required,s} = f_{current,s} - \Delta F(N), \quad RF_s = \frac{f_{current,s}}{f_{required,s}}$ If $f_{required,s} \leq 0$, the scenario alone cannot fix the problem — combined reductions are required.

Code: fn-curve-analysis.ts → analyzeUnacceptablePoint(), lines 60–127.

9. Individual Risk#

9.1 Calculation#

9.2 Classification#

Code: individual-risk-summary.tsx → evaluateRiskStatus().

9.3 Criteria by Country / Standard#

Code: individual-risk-presets.ts.

10. How to Use in the System#

1. Define scenarios — For each source, create RiskModels with event type (PoolFire, JetFire, FlashFire, VCE, FireBall) and assign frequency $P(A_s)$.

2. Run consequence calculations — Execute each model to obtain fatalidades.sumaFatalidades and fatalidades.resultados.

3. Configure criteria — In Project Settings → Social Risk Criteria, select the international standard or enter custom $C_1$, $C_2$, and $n$ values.

4. Visualize the F-N Curve — In the Social Risk tab → F-N Curve sub-tab, the table automatically computes $N$ and $F(N)$ for each scenario and plots them.

5. Interpret zones — Points above the upper line ($F > C_1/N^{|n|}$) are unacceptable. The system generates automatic frequency reduction recommendations.

6. Individual Risk — In the Individual Risk sub-tab, enter $N_g$ or let the system calculate it automatically, then verify $RI_g$ against the country criterion.

11. Applications of the F-N Curve#

12. Analysis Limitations#

13. Bibliographic References#