🎓 Lesson 13
D5
Arch Stability Index (ASI) Calculation and Interpretation
The Arch Stability Index (ASI) tells us whether grain or bulk material will form a stable arch over an opening—or collapse and cause a blockage—in silos, hoppers, or chutes.
🎯 Learning Objectives
- ✓ Calculate ASI using measured or estimated material properties and geometry
- ✓ Analyze how changes in hopper angle, outlet size, or moisture content affect ASI and flow reliability
- ✓ Explain the physical meaning of ASI > 1 versus ASI < 1 in terms of force equilibrium and failure modes
- ✓ Apply ASI to select appropriate hopper geometry or conditioning methods (e.g., vibrators, air pads) for reliable flow
📖 Why This Matters
In grain elevators, feed mills, and fertilizer plants, unplanned blockages cost millions annually in downtime, safety incidents, and equipment damage. A single stable arch—no larger than a basketball—can halt entire production lines. ASI is the first quantitative checkpoint engineers use to predict and prevent such failures *before* equipment is built or commissioned.
📘 Core Principles
Arching occurs when interparticle forces (cohesion, friction, and confinement) resist gravitational collapse across an opening. The ASI originates from Jenike’s silo theory: it balances the vertical load acting on the arch crown (proportional to material density, height, and arch radius) against the shear resistance along the arch surface (governed by effective cohesion and internal friction angle). Critical assumptions include: (1) the arch is circular and forms at the minimum cross-section; (2) material behaves as a Mohr–Coulomb solid; and (3) wall friction is sufficient to anchor the arch. As moisture, fines content, or compaction increases, cohesion rises—and so does ASI—raising blockage risk.
📐 Key Calculation
The ASI is derived from the ratio of the critical arch load to the shear strength resisting collapse. It is most commonly expressed using Jenike’s classical formulation adapted for conical hoppers, where geometry and flow properties jointly determine stability.
Arch Stability Index (ASI)
ASI = (ρ · g · r²) / (π · r · c' / tan(φ' − θ)) = (ρ · g · r · tan(φ' − θ)) / (π · c')Quantifies the ratio of destabilizing vertical load to stabilizing shear resistance in a potential arch at a hopper outlet.
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| ρ | Bulk density | kg/m³ | Mass per unit volume of the stored material under consolidation |
| g | Gravitational acceleration | m/s² | Standard Earth gravity (9.81 m/s²) |
| r | Critical arch radius | m | Approximately half the outlet diameter for conical hoppers |
| c' | Effective cohesion | kPa | Interparticle cohesive strength measured via shear cell test (e.g., Jenike shear tester) |
| φ' | Effective internal friction angle | degrees | Angle of internal resistance under unconfined or consolidated conditions |
| θ | Hopper half-angle | degrees | Angle between hopper wall and vertical axis |
Typical Ranges:
Dry granular corn: 0.3 – 0.8
Moist wheat flour: 1.2 – 3.5
Pelleted feed: 0.1 – 0.4
💡 Worked Example
Problem: A wheat silo has a conical hopper with outlet diameter D = 0.3 m. Measured properties: bulk density ρ = 780 kg/m³, effective cohesion c' = 1.2 kPa, internal friction angle φ' = 32°, and hopper half-angle θ = 25°. Calculate ASI.
1.
Step 1: Compute critical arch radius r ≈ D/2 = 0.15 m
2.
Step 2: Estimate vertical load on arch: W ≈ ρ·g·r² = 780 × 9.81 × (0.15)² ≈ 172 N
3.
Step 3: Compute shear resistance: S ≈ π·r·c' / tan(φ' − θ) = π × 0.15 × 1200 / tan(32° − 25°) ≈ 471 N
4.
Step 4: ASI = W / S ≈ 172 / 471 ≈ 0.36
Answer:
The result is 0.36, which falls below the critical threshold of 1.0 — indicating flow is expected without arch-induced blockage.
🏗️ Real-World Application
At the Port of Vancouver grain terminal, a new 50,000-tonne soybean silo experienced repeated ratholing and bridging at the 0.4-m-diameter outlet. Post-commissioning flow testing revealed c' = 2.8 kPa (due to 13.5% moisture) and φ' = 36°. Using ASI analysis, engineers recalculated with θ = 25° and found ASI = 1.42 — confirming instability. They retrofitted with a 0.65-m outlet and increased hopper angle to 35°, reducing ASI to 0.79 and restoring consistent gravity flow per CEMA Standard 501-2022 guidelines.