A Bord and Pillar panel is developed at a depth of 250 m in a flat coal seam. The vertical stress gradient is 0.027 MPa/m. If the strength of a square pillar is 12.5 MPa, the extra... A Bord and Pillar panel is developed at a depth of 250 m in a flat coal seam. The vertical stress gradient is 0.027 MPa/m. If the strength of a square pillar is 12.5 MPa, the extraction ratio of the pillar for a safety factor of 1.5 is _____ (round off up to 2 decimals).
Understand the Problem
The question describes a Bord and Pillar panel in a coal seam and provides information about depth, vertical stress gradient, and the strength of a square pillar. It asks to calculate the extraction ratio of the pillar given a safety factor, which involves using these parameters in a mathematical formula.
Answer
The extraction ratio of the pillar for a safety factor of 1.5 is approximately $0.81$.
Answer for screen readers
The extraction ratio of the pillar for a safety factor of 1.5 is approximately $0.81$.
Steps to Solve
- Calculate Vertical Stress To find the vertical stress (σ_v) at a depth of 250 m, use the vertical stress gradient.
The formula is: $$ \sigma_v = \text{depth} \times \text{vertical stress gradient} $$
Substituting the values: $$ \sigma_v = 250 , \text{m} \times 0.027 , \text{MPa/m} = 6.75 , \text{MPa} $$
- Calculate Allowable Stress Calculate the allowable stress using the safety factor (SF). The formula is: $$ \text{Allowable Stress} = \frac{\text{Strength of Pillar}}{\text{Safety Factor}} $$
Substituting the values: $$ \text{Allowable Stress} = \frac{12.5 , \text{MPa}}{1.5} = 8.33 , \text{MPa} $$
- Calculate Extraction Ratio The extraction ratio (ER) can be calculated by comparing the vertical stress to the allowable stress: $$ \text{Extraction Ratio} = \frac{\sigma_v}{\text{Allowable Stress}} $$
Substituting the calculated values: $$ \text{Extraction Ratio} = \frac{6.75 , \text{MPa}}{8.33 , \text{MPa}} \approx 0.81 $$
- Round Off the Ratio Finally, round the extraction ratio to two decimal places: $$ \text{Extraction Ratio} \approx 0.81 $$
The extraction ratio of the pillar for a safety factor of 1.5 is approximately $0.81$.
More Information
The extraction ratio indicates how much of the coal can be safely extracted without compromising the structural integrity of the pillars. An extraction ratio less than one suggests that the operation is within safe limits.
Tips
- Misunderstanding the Safety Factor: Ensure that the strength of the pillar is divided by the safety factor, not multiplied.
- Incorrectly Calculating Vertical Stress: Make sure to use the depth and the correct stress gradient without errors in unit conversion.
- Rounding Prematurely: Always do the calculations fully before rounding off to avoid losing precision.
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