In a bord and pillar panel, a square pillar of size 35 m \( \times \) 35 m (centre to centre) is extracted to form four equal square-shaped stooks as shown. The width of each gallery and crosscut is 5 m. The height of the working seam is 3 m. The reduction in safety factor after pillar splitting by using Bieniawski’s pillar strength formula, in %, is _______ (rounded off to 2 decimal places).
Bieniawski's pillar strength formula is given by \( S_p = S_1 \left( 0.64 + 0.36 \frac{w}{h} \right) \), where \( S_1 \) is the strength of a 0.9 m\(^3\) coal block, \( w \) is the pillar width, and \( h \) is the mining height.
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Bieniawski’s pillar strength formula helps estimate the safety factor by considering the width and height of the pillar, and the strength of the coal. After pillar splitting, the safety factor typically reduces due to a decrease in pillar width.
Solution: 1. Initial Pillar Dimensions and Area:
Centre to centre dimension of initial pillar = 35 m × 35 m
Width of gallery/crosscut = 5 m
Actual width of initial pillar (w₁) = 35 m - 5 m = 30 m
Area of initial pillar (A₁) = w₁² = (30 m)² = 900 m² 2. Dimensions and Area of the Stooks:
Width of each stook (w₂) = (30 m - 5 m) / 2 = 25 m / 2 = 12.5 m
Area of each stook (Aₛₜₒₒₖ) = w₂² = (12.5 m)² = 156.25 m²
Total area of four stooks (A₂) = 4 × Aₛₜₒₒₖ = 4 × 156.25 m² = 625 m² 3. Mining Height:
Height of the working seam (h) = 3 m 4. Bieniawski's Pillar Strength:
Formula: Sₚ = S₁ (0.64 + 0.36 × w / h) 5. Initial Pillar Strength (Sₚ₁):
Sₚ₁ = S₁ (0.64 + 0.36 × 30 / 3) = S₁ (0.64 + 0.36 × 10) = S₁ (0.64 + 3.6) = 4.24 S₁ 6. Strength of Each Stook (Sₚ₂):
Sₚ₂ = S₁ (0.64 + 0.36 × 12.5 / 3) = S₁ (0.64 + 0.36 × 4.1667) = S₁ (0.64 + 1.5000) = 2.14 S₁ 7. Safety Factor:
Initial Safety Factor (SF₁) ∝ Sₚ₁ / A₁ = 4.24 S₁ / 900
Safety Factor after Splitting (SF₂) ∝ Sₚ₂ / Aₛₜₒₒₖ = 2.14 S₁ / 156.25 (for one stook) 8. Reduction in Safety Factor:
Ratio of safety factors: SF₂ / SF₁ = (2.14 S₁ / 156.25) / (4.24 S₁ / 900) = (2.14 / 156.25) × (900 / 4.24) = 1926 / 662.5 ≈ 2.9072
This ratio indicates an increase in the value of Sₚ / Area for a single stook compared to the initial pillar. However, we need to consider the overall stability of the panel. Let's consider the safety factor in terms of average stress assuming constant load on the panel.
Initial average stress (σ₁) ∝ 1 / A₁ = 1 / 900
Average stress after splitting (σ₂) ∝ 1 / A₂ = 1 / 625 (total area of stooks)
SF₁ ∝ Sₚ₁ × A₁ = 4.24 S₁ × 900 (considering capacity)
SF₂ ∝ Sₚ₂ × Aₛₜₒₒₖ × 4 = 2.14 S₁ × 156.25 × 4 = 1337.5 S₁ (total capacity of stooks)
SF₂ / SF₁ = 1337.5 S₁ / (4.24 × 900 × S₁) = 1337.5 / 3816 ≈ 0.3505
Percentage reduction in safety factor = (1 - 0.3505) × 100 = (1 - 0.3505) × 100 = 64.95 % Final Answer: The reduction in safety factor is 64.95%.
