Stiffness Ratio

• Stiffness Ratio = BH/ B

• where BH is Bench Height

• B is the Burden

• Target Stiffness Ratios• 3 to 4 is normal range• > 4 does not improve fragmentation (may risk

cut-offs much above 6)• 2 - redesign if you can• < 2 - Get someone you don’t like near shot

Stiffness Ratio Example

• What is the Stiffness Ratio for a 40 foot bench with 6.5 feet of burden on each hole• 40 / 6.5 = 6.15

• Shot won’t crater• No real advantage to such a high bench

and small hole• May need to watch for cut-off problems in

later practice or other equations

The Problem of Interrelationships

• Burden is a function of Hole Size and explosive

• Acceptable Bench Height is a function of burden and thus hole size and explosive

• Hole Size is a function of drilling equipment - but so is bench height

• Solution is usually to find what parameter is fixed and work out from there

Suppose Bench Height is Fixed

• In a quarry thickness of rock layers may set• May have been set in a previous mine plan• May be fixed by grade control constraints in

metal mine• need to be able to selectively mine• can’t scramble ore face with explosives

• Working Height of Equipment may Set

Solving for Maximum Allowable Hole Size on a

Fixed Bench• Combining the Burden and Stiffness Ratio

formulas and backsolving hole size• For SR = 2 , L = Bench Height = 40, E = 100,

Sgr = 2.6

• L / (2 * { E/ Sgr } 0.33) = 6 inches

• Holes larger than 6 inches will likely cause cratering - smaller size is desirable

Solving with a Fixed Hole Size

• Can Occur for Fixed Drilling Equipment• Can Determine a minimum acceptable bench

height with an approximation - Rule of 5• Lmin = 5 * De

• If De = 3 inch hole then Lmin = 15 ft

• Check• 15 / 6.37 = 2.35 ft (6.37 from burden formula with

ANFO and 2.6 Sgr)

What if I have no constraints and I’m Lost

• Most Real Problems Have Hidden Constraints

• Equipment may fix• A Truck size usually constrains loader and

loaders have digging height that may fix the bench

• A given type of drill can only drill holes of specific size range

• Hole loading technique may limit bench height

Finding Hidden Constraints

• Geology may Constrain• Need to mine certain intervals

• quarry rock quality control• toxic or substitute topsoil rock in coal mine

• Grade Control• height of bench averages out high and low grades

and may limit selective mining

• Need to Manipulate Problem Layers• hard layer on top won’t fragment in stemming zone• In middle may cause cut-offs

Finding Constraints

• Performance may constrain• May require certain fragmentation characteristics

or costs• fragmentation usually more uniform with smaller holes

and benches• fragmentation usually cheaper with larger holes and

benches

• Regulations may constrain• Some localities limit bench height for safety• Blast Vibration may limit charge length