Understanding the Pressure Arch Theory for Coal Pillar Load Calculations
The Pressure Arch Theory is a key principle used to determine the load a coal pillar can withstand before it collapses. By using a mathematical formula, engineers can assess the stability of coal pillars in mines to ensure safe extraction practices.
Formula for Calculating Load on a Coal Pillar
The formula to calculate the load P on a coal pillar is:
P = K * (σ * h)2 / (2 * (1 - sin2 θ))
Where:
- P = Load on the pillar (in pounds)
- K = Constant (typically around 0.6)
- σ = Stress on the pillar (in pounds per square foot)
- h = Height of the pillar (in feet)
- θ = Angle of internal friction of the coal (in degrees)
Step-by-Step Example: Calculating the Load on a Coal Pillar
Let’s walk through an example to see how this formula works.
Given:
- Height of the coal pillar (h) = 200 feet
- Angle of internal friction (θ) = 30°
- Stress on the pillar (σ) = 500 pounds per square foot
- Constant (K) = 0.6
Now, we’ll plug these values into the formula to calculate the load:
P = K * (σ * h)2 / (2 * (1 - sin2 θ))
P = 0.6 * (500 * 200)2 / (2 * (1 - sin2 30°))
Step 1: Calculate the stress and height term
500 * 200 = 100,000
(100,000)2 = 10,000,000,000
Step 2: Calculate the sine of 30 degrees
sin 30° = 0.5 → sin2 30° = (0.5)2 = 0.25
Step 3: Apply the values into the denominator
1 - 0.25 = 0.75
Step 4: Calculate the load P
P = 0.6 * 10,000,000,000 / (2 * 0.75)
P = 6,000,000,000 / 1.5
P = 8,000,000,000 pounds
So, the load that the coal pillar can withstand before it collapses is approximately 8,000,000,000 pounds.
Important Notes:
- Rough Calculation: This is a simplified calculation. Real-world conditions require professional expertise to account for various other factors such as geological conditions, pillar shape, and more.
- Safety First: Always consult with mining engineers for precise load calculations in real-world mining operations.
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