Calculating Stripping Ratios for
Irregular Geometries
©Dr. B. C. Paul Spring 2003
The problem of complex geometry
• Many mines are not strip coal mines
• Some hard to describe with cones and
cylinders
– Had conveniently level surface
• Consider the Mountain Top Removal
Operation
Numerical Calculation of
Stripping Ratios
Average End Area from Topo
• Use a digitizer or Autocad Digitizing
Function
• 1700 ft Contour Area is 53.452,860 ft2
• sits directly above coal seam - half way
down to next slice is 0
• sits 50 ft below 1750 ft Contour Area halfway up to next slice is 25 ft.
• Volume ftom slice #1 = 53,452,860ft2 *
(Oft + 25ft) = 1,336,321,500 ft3
Next Slice
• 1750 ft Contour Area is 37,540,860 ft2
• half way down to 1700 slice is 25 ft
halfway up to 1800 slice is 25 ft.
• Volume from slice #2 = 37,540,860ft2 *
(25ft + 25ft) = 1,877,043,000 ft3
Remaining Slices
• 1800 ft Contour Area is 18,390,654ft2 *
(50ft) = 919,532,700 ft3
• 1850 ft Contour Area is 6,176,782ft2 *
(50ft) = 308,839,100 ft3
• 1900 ft Contour Area is 360,049ft2
• halfway down to 1850 is 25ft
• half way up to 1950 is Zoicks! There is no
1950!!! Now What do I do?
Alternatives for handling the
missing top slice
• Approximation #1 - Do nothing - ignore the next slice up
• 360,049 ft2 * 25ft = 9,001,225 ft3
• Approximation #2 - Treat the next slice up as a slice of 0
area and go half way to it.
• 360,049ft2 * 50ft = 18,002,450 ft3
• Approximation #3 - Imagine a sort of pyramid above the
top slice going halfway up to slice of 0 area
volume is 1/3 * height* base
-
pyramid
• 9,001,225 ft3 + 360,049ft2 * 25 *0.333 =
11,998,632 ft3
More Approximations
• Approximation #4 - The height of the pyramid could have
been some other number including 50ft or the distance to the
peak.
• Great - You just gave me 5 different answers - which
one is right?
• Sum up the volumes
– 1,336,321,500
– + 1,877,043,000
– + 919,532,700
– + 308,839,100
–+
9,001,225
– Total - 4,450,737,525
Concept on Handling Ends
• This is an engineering approximation. You may
use your knowledge of the true geometry to
sharpen the approximation. You may choose an
approximation to keep the math simple and fast.
You may make a reasonable but arbitrary choice
in the presence of insufficient data. Often other
estimates and judgments that must be made will
have far more influence on the investors bottom
line $$$$ than how you approximated a peak.
Finishing the Stripping Ratio
• Convert OB volume to yards
– 4,450,737,525/ 27 = 164,842,131 yd3
• Get Weight of Coal
– 1700 ft slice 53,452,860 ft2
–
X
5 ft
–
267,264,300 ft3
–
X
80 lbs/ft3
–
/
2000 lbs/ton
–
10,690,572 tons
Finalize Stripping Ratio
• 164,842,131 yd3/ 10,690,572 tons =
15.42:1
• If Dragline Operations Cost 35 cents per
cubic yard
– 15.42 * $0.35 = $5.40/ton
– Minimum margin on coal to remove O.B.
The Concept
• When geometry gets to complex for
standard formulas – use a numeric
approximation to integrating what ever
shape you have
– That’s the way all the basic geometry
formulas were derived (they were simple
enough to work out analytic answers)
• With computers today who does analytic answers
• We’re engineers (how close do we have to be
before you don’t care anymore)
What if there is no Off-the shelf
contour map?
• Probably remember from calculus that you
can integrate horizontal or vertical slices
to get the same formula
• Take Multiple Cross-Sections of your pit
and do an average end area
Phosphate Mine Example
Do a series of cross-sections
This geometry is simple
Enough.
Then do average end area
With each cross section
Weighted by thickness
Half way to its neighbor
Four Basic Ways to Get Stripping
Ratio
• 1 Dimensional Formulas
– Works on coal strip pits
• Simple Geometric Calculations
• Average End Area off of Topo
• Average End Area off of Cross-Section
• Which One Do I Use?
• Simplest One that gives you the desired
accuracy