🎓 Lesson 5 D3

Calculation Methods and Formulas

Calculation methods and formulas are step-by-step math tools engineers use to plan safe, efficient, and cost-effective blasting patterns in mining.

🎯 Learning Objectives

✓ Calculate optimal burden and spacing using the Konya–Walter and Langefors formulas
✓ Design a blast pattern by applying powder factor and stemming ratios to meet fragmentation and cost targets
✓ Analyze blast performance data to adjust burden-to-spacing ratios for improved muck pile uniformity
✓ Explain the trade-offs between explosive energy input (kg/m³) and freight cost per tonne of excavated material
✓ Apply the modified Friedland formula to estimate blast-induced ground vibration and verify compliance with regulatory limits

📖 Why This Matters

In freight cost optimization, every tonne of oversized boulders increases haul cycle time, fuel consumption, and loader downtime—driving up transport costs by 12–25% per incident. Accurate blast design calculations prevent this at the source: they ensure consistent fragmentation *before* loading begins. This lesson equips you to translate geotechnical data and cost constraints into precise, auditable blast parameters—turning blasting from an art into a repeatable, cost-controlled engineering process.

📘 Core Principles

Blast design rests on three interdependent physical principles: (1) energy transfer—how much explosive energy is delivered per unit volume of rock; (2) confinement—how rock resistance and stemming control gas pressure buildup and fracture propagation; and (3) timing—how delay intervals govern stress wave interaction and fragment size distribution. Empirical models (e.g., Langefors, Konya–Walter) link these to measurable inputs like rock strength (UCS), density, and joint spacing. Modern practice overlays these with cost-driven constraints: powder factor must be minimized without compromising fragmentation, because explosive cost is ~30% of total blast cost—and freight cost scales directly with rehandling requirements from poor breakage.

📐 Langefors Burden Formula

The Langefors formula estimates the maximum practical burden (B) based on rock resistance and explosive power. It balances confinement pressure against rock tensile strength and is widely used for initial bench blast layout in open-pit mines. Valid for ANFO and emulsion explosives in competent to moderately jointed rock.

Langefors Burden

B = (f × e × √ρ) / 10

Empirical formula estimating optimal burden (m) based on rock resistance (f), explosive energy factor (e), and rock density (ρ in kg/m³).

Variables:

Symbol	Name	Unit	Description
B	Burden	m	Shortest distance from borehole to free face
f	Rock Factor	dimensionless	Empirical coefficient representing rock strength and jointing (0.4–1.2)
e	Explosive Energy Factor	dimensionless	Relative energy output vs. TNT (ANFO = 0.8–1.0; emulsion = 1.0–1.1)
ρ	Rock Density	kg/m³	Bulk density of in-situ rock

Typical Ranges:

Medium-strength granite (UCS 80–120 MPa): 3.5 - 4.5 m

Weak sedimentary rock (UCS < 40 MPa): 2.0 - 3.0 m

💡 Worked Example

Problem: Given: Rock density = 2.65 g/cm³ (2650 kg/m³), rock factor f = 0.8 (for medium-strength granite), explosive strength factor e = 1.0 (ANFO relative to TNT), bench height H = 12 m, desired stemming ratio = 0.3 × B.

1. Step 1: Convert rock density to kg/m³ → 2650 kg/m³

2. Step 2: Apply Langefors formula: B = (f × e × √ρ) / 10, where ρ = density in kg/m³ → B = (0.8 × 1.0 × √2650) / 10 ≈ (0.8 × 51.48) / 10 ≈ 4.12 m

3. Step 3: Verify against bench height constraint: B ≤ H/2 → 4.12 ≤ 6.0 → OK. Also check stemming: 0.3 × 4.12 ≈ 1.24 m, leaving 10.76 m for charge length — acceptable for 12-m bench.

Answer: The calculated burden is 4.1 m, which falls within the safe range of 3.5–4.5 m for medium-strength granite with ANFO.

🏗️ Real-World Application

At the Antamina Copper Mine (Peru), engineers reduced haul truck cycle time by 18% after recalibrating burden using Langefors + site-specific RQD and P-wave velocity data. Initial designs assumed uniform f = 0.9, but core logging revealed highly variable joint spacing (0.2–1.8 m). By segmenting the pit into four geotechnical domains and applying domain-specific f values (0.65–0.85), they achieved >92% <0.8 m fragments—cutting secondary breaking costs by $1.42/tonne and lowering freight-related OPEX by $0.38/tonne.