Chapter 6 — When is a Rock Engineering Design Acceptable?¶
6.1 The Fallacy of a Single Factor of Safety¶
In classical structural steel or reinforced concrete design, a deterministic Factor of Safety (\(FS\)) between \(1.5\) and \(2.0\) guarantees safety because material properties are manufactured under strict quality standards with small coefficients of variation (\(COV < 10\%\)).
In geotechnical and rock engineering, however: - Rock mass properties exhibit vast spatial scatter (\(COV\) of intact strength often \(20 - 40\%\); joint persistence and GSI often \(15 - 35\%\)). - An engineered slope with a calculated deterministic \(FS = 1.3\) based on mean parameters can have a Probability of Failure (\(P_f\)) exceeding \(20\%\) if the standard deviation of joint friction or pore pressure is high. - Conversely, a slope with \(FS = 1.15\) where geotechnical uncertainty is minimal may have \(P_f < 1\%\).
A single deterministic Factor of Safety is meaningless without specifying the associated uncertainty and consequences of failure.
6.2 Deterministic Factor of Safety (\(FS\)) vs. Probability of Failure (\(P_f\))¶
RESISTANCE R (Capacity)
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X <--- OVERLAP REGION: Probability of Failure Pf
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LOAD S (Demand)
- Deterministic Margin of Safety:
- Reliability Index (\(\beta\)): Assuming capacity \(R\) and demand \(S\) are normally distributed variables:
The Probability of Failure is directly related through the standard normal cumulative distribution:
- Monte Carlo Simulation: Instead of calculating a single \(FS\), input parameters (\(c', \phi', \text{GSI}, r_u\)) are sampled from statistical probability distributions (Normal, Lognormal, Beta) over \(10,000\) iterations:
6.3 Acceptable Risk Criteria Across Industries¶
Dr. Hoek established target criteria balancing economic investment against consequence of failure:
| Facility Type | Consequence Category | Minimum Deterministic \(FS\) (Static) | Maximum Acceptable \(P_f\) |
|---|---|---|---|
| Temporary Mine Open Pit Benches | Low (localized bench spill, no personnel exposure) | \(1.1 - 1.2\) | \(15\% - 30\%\) |
| Main Mine Haulage Ramps / Highwalls | Moderate (haulage disruption, economic loss) | \(1.3\) | \(5\% - 10\%\) |
| Permanent Civil Highway Slopes | High (public traffic, disruption to commerce) | \(1.5\) | \(1\%\) |
| Dam Abutments & Hydro Caverns | Extreme (potential catastrophic dam breach or loss of life) | \(1.5 - 2.0\) | \(< 0.1\%\) |
6.4 The Role of Trigger Action Response Plans (TARP)¶
Because rock engineering accepts residual uncertainty, operations must link real-time monitoring to explicit TARP action matrices:
- Level 1 (Green — Normal Operation): Deformation rate \(< 1.0\text{ mm/day}\); pore pressure below design phreatic line. Normal mining/tunneling continues.
- Level 2 (Amber — Advisory / Heightened Vigilance): Velocity increases to \(2 - 5\text{ mm/day}\) or accelerating trend detected. Double sensor polling frequency; install supplementary cablebolts; inspect tension cracks.
- Level 3 (Red — Immediate Evacuation): Inverse velocity analysis (\(1/v \to 0\)) indicates onset of tertiary creep failure. Sound alarms; extract all personnel and equipment; isolate area.
6.5 Canonical Terminology¶
| English Term | Canonical Vietnamese Translation | Definition |
|---|---|---|
| Factor of Safety (FS) | Hệ số an toàn (FS) | Ratio of total resisting forces to driving forces |
| Probability of failure (\(P_f\)) | Xác suất phá hoại (\(P_f\)) | Statistical likelihood that demand exceeds capacity |
| Reliability index (\(\beta\)) | Chỉ số độ tin cậy (\(\beta\)) | Number of standard deviations separating mean margin from failure |
| Monte Carlo simulation | Mô phỏng Monte Carlo | Computational method evaluating probabilistic distributions |
| Trigger Action Response Plan (TARP) | Kế hoạch ứng phó ngưỡng kích hoạt (TARP) | Documented matrix linking sensor thresholds to operational responses |