🎓 Lesson 18
D5
Lab: Weibull Fit & B10 Calculation for Crusher Bearings (Using Chilean Copper Data)
Weibull fit is a math tool that helps predict how long crusher bearings will last before failing, and B10 life tells you when 10% of them are expected to break down.
🎯 Learning Objectives
- ✓ Calculate Weibull shape (β) and scale (η) parameters from field failure data using linearized rank regression
- ✓ Determine B10 life (in hours or cycles) from fitted Weibull parameters and interpret its operational meaning
- ✓ Analyze right-censored bearing runtime data to adjust for surviving units in reliability estimation
- ✓ Explain how Weibull β values indicate dominant failure mechanisms (e.g., β < 1 = infant mortality, β ≈ 1 = random failures, β > 3.5 = wear-out)
- ✓ Apply Chilean copper mine crusher duty-cycle data to estimate realistic B10 life and compare against OEM specifications
📖 Why This Matters
In large-scale copper mines like Escondida or Collahuasi, primary gyratory crushers operate 24/7 under extreme loads—often with >10,000 tonnes/hour throughput. When a main shaft bearing fails unexpectedly, downtime can cost over USD $500,000/hour. Traditional 'mean time between failures' (MTBF) misleads because bearing life isn’t normally distributed—it’s skewed and highly variable. Weibull analysis reveals *how* and *when* failures cluster, letting engineers set proactive maintenance intervals, justify spare parts stocking, and validate supplier warranties—directly impacting mine availability and OPEX.
📘 Core Principles
Bearing life follows a three-phase reliability pattern: early failures (infant mortality), stable operation (random failures), then wear-out (increasing failure rate). The Weibull distribution captures this via two parameters: shape (β, also called 'Weibull slope') governs failure mode physics; scale (η, 'characteristic life') is the time at which ~63.2% of units fail. Unlike normal distributions, Weibull handles censored data—critical when 70% of bearings in a fleet are still running at study end. Rank regression on transformed data (using median ranks and double-log axes) linearizes the CDF to extract β and η. Once fitted, B10 life is computed as t_B10 = η × (−ln(0.9))^(1/β), representing the 10th percentile of the failure distribution.
📐 Weibull B10 Life Calculation
The B10 life is derived from the Weibull cumulative distribution function (CDF): F(t) = 1 − exp[−(t/η)^β]. Solving for t when F(t) = 0.10 yields the standard B10 formula. This is essential for comparing field performance against ISO 281 rated L10 life and adjusting for actual load, lubrication, and contamination conditions observed in Chilean porphyry copper operations.
B10 Life (Weibull-Derived)
t_B10 = η × [−ln(0.90)]^(1/β)Time at which 10% of the population has failed
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| t_B10 | B10 life | hours | Field-observed 10th percentile life |
| η | Scale parameter | hours | From Weibull fit |
| β | Shape parameter | dimensionless | From Weibull fit |
Typical Ranges:
Primary crusher bearings (Chilean porphyry): 8,000 – 14,000 h
Secondary cone crusher bearings: 12,000 – 20,000 h
💡 Worked Example
Problem: Chilean copper mine collected runtime-to-failure data (hours) for 12 identical SKF 241/1000 CA spherical roller bearings in primary crusher main shafts: [12,450, 14,820, 9,630, 18,210, 11,370, 16,950, 13,780, 21,440, 10,260, 15,190, 17,330, 19,870]. Two bearings were right-censored at 22,000 h (still operating). Fit Weibull and compute B10 life.
1.
Step 1: Sort failure times ascending and assign median ranks (Benard’s approximation): i-th failure rank = (i − 0.3)/(n + 0.4); for n=12 failures, rank for first failure = (1−0.3)/(12+0.4) = 0.0565.
2.
Step 2: Compute y = ln[−ln(1 − F_i)] and x = ln(t_i); perform linear regression → slope = β = 2.38, intercept = −ln(η) → η = exp(−intercept/β) = 16,840 h.
3.
Step 3: Apply B10 formula: t_B10 = 16,840 × (−ln(0.9))^(1/2.38) = 16,840 × (0.10536)^(0.4202) = 16,840 × 0.637 = 10,730 h.
Answer:
The B10 life is 10,730 hours (~1.22 years at 92% uptime), significantly lower than the ISO 281 rated L10 of 14,200 h—indicating harsher-than-assumed field conditions (e.g., abrasive dust ingress, thermal cycling).
🏗️ Real-World Application
At Codelco’s Chuquicamata expansion (2021–2023), vibration-monitored bearing runtime data from six 5,500 kW GP700 primary crushers was analyzed. Field Weibull β = 1.8 ± 0.3 indicated mixed failure modes (lubrication breakdown + micro-pitting), while η = 13,200 h implied premature wear. Engineers traced root cause to inadequate grease re-lubrication intervals under high-dust conditions (>25 mg/m³ airborne silica). Revised maintenance protocol—increased grease volume by 40%, added ultrasonic monitoring, and installed sealed labyrinth guards—raised β to 2.6 and extended field B10 life to 15,600 h, aligning with OEM predictions.
✏️ Reliability Diagnostic Exercise
Using the provided Excel dataset (simulated Chilean crusher bearing runtimes: 8 failures + 4 right-censored units), students will: (1) compute median ranks, (2) perform linear regression on Weibull plotting positions, (3) extract β and η, (4) calculate B10 and B50 lives, and (5) write a 150-word reliability recommendation for maintenance planning—including whether current spares policy (based on OEM L10) is justified.
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