🎓 Lesson 9
D5
Fatigue Life Prediction for Cyclic Pressure Vessels
Fatigue life prediction estimates how many pressure cycles a vessel can safely endure before cracks start to grow and cause failure.
🎯 Learning Objectives
- ✓ Calculate fatigue life using the Basquin equation for high-cycle fatigue conditions
- ✓ Apply linear elastic fracture mechanics (LEFM) to estimate crack growth per cycle using Paris’ law
- ✓ Analyze stress concentration effects on fatigue initiation using notch sensitivity factors
- ✓ Design inspection intervals for cyclic pressure vessels based on predicted crack growth rates
- ✓ Explain the influence of surface finish, residual stress, and environment on fatigue life
📖 Why This Matters
In mining and mineral processing, pressure vessels—including autoclaves, leach tanks, and compressed air receivers—experience thousands of daily pressure cycles due to batch operations, valve actuation, and thermal transients. Undetected fatigue damage has led to catastrophic failures: e.g., the 2017 autoclave rupture at a gold processing plant in Western Australia caused unplanned shutdowns and near-miss incidents. Predicting fatigue life isn’t just theoretical—it directly enables risk-based inspection planning, extends asset life, and fulfills regulatory requirements under API RP 581 and ASME BPVC Section VIII, Division 2.
📘 Core Principles
Fatigue failure occurs in three stages: (1) crack initiation at microstructural defects or stress concentrators (e.g., weld toes, pitting corrosion), (2) stable crack growth governed by cyclic plasticity or fracture mechanics, and (3) unstable final fracture when the remaining ligament can no longer support peak load. High-cycle fatigue (>10⁴ cycles) is stress-driven and modeled statistically via S–N curves; low-cycle fatigue (<10⁴ cycles) involves significant plastic strain and requires strain-life (ε–N) approaches like Manson–Coffin. For pressurized equipment with known flaws, fracture mechanics (e.g., ΔK-based Paris’ law) provides superior accuracy—especially when combined with non-destructive testing (NDT) data. Surface condition, mean stress (via Goodman or Gerber corrections), and environmental embrittlement (e.g., H₂S in hydrometallurgical vessels) must be explicitly accounted for.
📐 Paris’ Law for Crack Growth Prediction
Paris’ law quantifies the rate of fatigue crack growth per cycle (da/dN) as a function of the stress intensity factor range (ΔK). It is the cornerstone of fracture mechanics–based fatigue life prediction for components with known or detectable flaws—and is mandated in ASME BPVC Section VIII, Division 2, Part 5 for flaw tolerance assessment.
Paris’ Law
da/dN = C (ΔK)^mRelates fatigue crack growth rate per cycle to the stress intensity factor range.
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| da/dN | Crack growth rate | m/cycle | Incremental increase in crack length per loading cycle |
| C | Material constant | MPa⁻ᵐ·m^(1−m)/cycle | Empirically determined coefficient dependent on material, environment, and microstructure |
| ΔK | Stress intensity factor range | MPa·√m | Difference between maximum and minimum K values during a cycle: ΔK = K_max − K_min |
| m | Crack growth exponent | dimensionless | Typically 2–5; reflects material resistance to crack propagation |
Typical Ranges:
Austenitic stainless steels (air): 2.5–4.0
Carbon steels (corrosive aqueous): 3.0–5.0
💡 Worked Example
Problem: A 12 mm deep semi-elliptical surface crack is detected in a stainless steel (316L) autoclave shell during ultrasonic testing. The vessel operates between 0.5 MPa and 4.0 MPa internal pressure. Geometry factor β = 1.12; thickness t = 40 mm; crack depth a = 12 mm; C = 2.5 × 10⁻¹² MPa·m^(−m), m = 3.0 (for 316L in ambient aqueous service). Estimate da/dN at mid-cycle.
1.
Step 1: Compute nominal stress range Δσ = P_max × r/t − P_min × r/t. Assume radius r = 1.5 m → Δσ = (4.0 − 0.5) MPa × (1.5 m / 0.04 m) ≈ 131.25 MPa.
2.
Step 2: Calculate ΔK = β × Δσ × √(π × a) = 1.12 × 131.25 MPa × √(π × 0.012 m) ≈ 1.12 × 131.25 × 0.194 ≈ 28.3 MPa√m.
3.
Step 3: Apply Paris’ law: da/dN = C × (ΔK)^m = 2.5 × 10⁻¹² × (28.3)³ ≈ 2.5 × 10⁻¹² × 22,670 ≈ 5.67 × 10⁻⁸ m/cycle (≈ 57 nm/cycle).
Answer:
The crack grows ~57 nanometers per pressure cycle. At this rate, it would take ~17,600 cycles to grow from 12 mm to critical size (~24 mm, assuming K_IC = 85 MPa√m for 316L), implying ~48 years at 365 cycles/year—justifying 5-year NDT intervals per API RP 581.
🏗️ Real-World Application
At the Cannington lead–zinc mine (Australia), a 2.4 m diameter × 8 m long sulfuric acid leach autoclave experienced premature wall thinning near a nozzle weld. Post-failure fractography revealed beach marks and striations confirming fatigue. Engineers applied ASME BPVC Section VIII, Division 2, Part 5: they modeled the weld toe as a 0.5 mm initial flaw, used measured pressure cycling (0–1.8 MPa, 2×/day), incorporated surface roughness (Ra = 12.5 μm), and calculated remaining life as 11.2 years—validating their 3-year phased array UT inspection schedule. This case is documented in AusIMM’s 2021 ‘Process Equipment Integrity Guidelines’.
📋 Case Connection
📋 Hydrogen Sulfide Flare Stack Integrity Assessment at Gulf Coast Refinery
Unplanned shutdown due to wall thinning from sulfidic corrosion; no CUI monitoring program
📋 Nitric Acid Storage Tank MOC Failure Root Cause Analysis at Fertilizer Facility
Post-MOC leak occurred due to incompatible gasket material (EPDM vs. concentrated HNO₃)