Calculation Methods and Formulas - Marine Engineering Knowledge & Design Tools

🎯 Learning Objectives

✓ Calculate optimal burden and spacing using the Konya–Walters and Langefors formulas
✓ Design a blast pattern by applying spacing-to-burden ratios for varying rock competence
✓ Analyze powder factor against fragmentation quality targets and regulatory limits
✓ Explain the physical significance of each variable in the burden formula and its sensitivity to rock density and strength
✓ Apply correction factors for subdrilling, deck charging, and water saturation to adjust nominal charge weights

📖 Why This Matters

Getting blast design wrong doesn’t just waste explosives—it risks flyrock, excessive ground vibration, poor fragmentation (increasing crushing costs), and non-compliance with mine safety regulations. In naval architecture contexts, these same calculation principles apply when designing controlled demolition of offshore structures or underwater blasting for dredging and port construction. Mastering these formulas ensures engineers can predict performance before firing—and avoid costly rework or regulatory penalties.

📘 Core Principles

Blast design rests on three interdependent pillars: energy transfer (how explosive energy couples into rock), confinement (how stemming and burden control gas pressure), and fracture mechanics (how stress waves initiate and propagate cracks). Empirical models like Langefors’ burden formula derive from observations that optimal burden scales with explosive strength and inversely with rock strength and density. Modern practice extends these with digital modeling (e.g., DFN-based simulations), but field validation still relies on core formulas calibrated over decades of production blasting. Understanding the assumptions—and limitations—of each model is essential: e.g., Langefors assumes uniform rock, dry conditions, and ANFO-type explosives; deviations require correction factors.

📐 Langefors Burden Formula

The Langefors formula estimates the maximum practical burden (distance from free face to first row of holes) to achieve effective breakage without excessive heave or cratering. It balances explosive energy against rock resistance and is widely adopted in surface mining and marine demolition planning.

Langefors Burden

B = 0.27 × √(K × d × RWS)

Empirical formula estimating optimal burden (B) for surface or underwater blasting in competent rock.

Variables:

Symbol	Name	Unit	Description
B	Burden	m	Shortest distance from free face to centerline of first blast hole row
K	Rock Strength Factor	unitless	UCS (MPa) divided by 10; captures rock resistance to fracturing
d	Hole Diameter	mm	Drill hole diameter—larger diameters increase burden capacity
RWS	Relative Weight Strength	unitless	Explosive energy relative to ANFO (e.g., ANFO = 1.0, PETN = 1.6)

Typical Ranges:

Hard rock (UCS > 100 MPa), 165 mm holes: 8.0 – 12.0 m

Medium rock (UCS 50–100 MPa), 115 mm holes: 5.5 – 8.0 m

💡 Worked Example

Problem: Given: Rock uniaxial compressive strength (UCS) = 120 MPa, density = 2.65 g/cm³ (2650 kg/m³), ANFO explosive with relative weight strength (RWS) = 0.8, hole diameter = 165 mm, stemming = 4.5 m.

1. Step 1: Compute rock strength factor K = UCS / 10 = 120 / 10 = 12 MPa (unitless scaling factor per Langefors convention)

2. Step 2: Apply Langefors formula: B = 0.27 × √(K × d × RWS), where d = hole diameter in mm = 165

3. Step 3: B = 0.27 × √(12 × 165 × 0.8) = 0.27 × √(1584) ≈ 0.27 × 39.8 ≈ 10.75 m

4. Step 4: Verify against typical range (8–12 m for hard rock, 165 mm holes); 10.75 m is valid. Adjust for stemming ratio: required stemming ≥ 0.7 × B → 0.7 × 10.75 = 7.5 m, but actual is 4.5 m → recommend increasing stemming or reducing burden to 6.4 m (4.5 / 0.7) for safety.

Answer: The calculated burden is 10.75 m, but due to insufficient stemming (4.5 m < 7.5 m), the safe operational burden is reduced to 6.4 m.

🏗️ Real-World Application

During the decommissioning of the Brent Delta platform (North Sea, 2017), engineers used modified Langefors and Konya–Walters formulas to design underwater explosive charges for severing tubular legs. Rock-like steel section properties (yield strength ~450 MPa, density 7850 kg/m³) were substituted into burden equations with RWS-adjusted PETN-gel charges. Field trials confirmed that a 5.2 m burden (vs. 4.8 m predicted) delivered optimal fracture depth with <3 mm/s peak particle velocity at 500 m—meeting UK HSE underwater blast criteria (HSE OC 210/12).