Question 7 of 11: Dragline coal-mining geometry (20 marks)

Question 2: Dragline coal-mining geometry (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

2.1 Range diagram

A range diagram is a scaled cross-section (or family of cross-sections at increasing overburden depth) that plots the dragline's own working envelope — its maximum digging radius/depth and its maximum dump radius/height, both measured from the machine's own tub centre — superimposed on the overburden/coal geometry being mined. It is the standard tool used to check, for a GIVEN cut geometry, whether a specific dragline model can reach the digging face and still reach far enough (and high enough) to place the spoil where the mine plan requires, exactly the check carried out numerically in 2.5–2.6 below.

2.2 Dragline positioning (75%, tub 20 m)

Given. Tub diameter D = 20 m; positioning factor P = 75% of tub diameter, measured from the dragline centreline to the highwall edge; operating radius R = 90 m.

Find. (2.2.1) sketch; (2.2.2) distance from highwall edge to nearest tub edge, and from highwall edge to dragline centreline; (2.2.3) dragline reach in terms of R and P.

Approach. The positioning percentage is defined directly on the tub diameter, so distances follow from simple subtraction of the tub radius and the operating radius.

1. Centreline-to-highwall distance. $$P_{dist} = 0.75 \times 20\ \text{m} = \boxed{15\ \text{m}}$$ this answers 2.2.1's sketch requirement directly (highwall edge sits 15 m from the tub centreline, i.e. 5 m inside the tub's own 10 m radius).
2. Highwall edge to nearest tub edge (2.2.2). $$d_{edge} = P_{dist} - \tfrac{D}{2} = 15 - 10 = \boxed{5\ \text{m}}$$ (the highwall edge sits 5 m short of reaching the tub's near side); the highwall-edge-to-centreline distance is simply $P_{dist}=15$ m (already found above).
3. Dragline reach (2.2.3). Reach is measured from the highwall edge to the end of the operating radius, both taken from the same centreline: $$\text{Reach} = R - P_{dist} = 90 - 15 = \boxed{75\ \text{m}}$$

2.3 Spoil swell (0.25)

Given. Swell factor SF = 0.25 (decimal). Find. Volume of 1 m³ unbroken overburden after digging and placing on the spoil pile.

1. Apply swell. $$V_{spoil} = V_{bank}(1+SF) = 1\times(1+0.25) = \boxed{1.25\ \text{m}^3}$$ every bank (in-situ) cubic metre expands to 1.25 loose cubic metres once broken and cast — the multiplier used again throughout 2.5 below.

2.4 Sketch to scale & 2.5 Section calculations

Given. Cut depth = overburden depth D_c = 25 m; pit width w = 20 m; highwall slope θ_hw = 63° from horizontal; spoil angle of repose θ_r = 35°; swell factor SF = 0.25; coal seam thickness = 3 m; operating radius R = 90 m; max stacking height (rated) = 12 m.

Find. (2.5.1) cut area; (2.5.2) spoil pile area; (2.5.3) spoil height above coal floor; (2.5.4) spoil height above dragline tub base; (2.5.5) operational stacking height; (2.5.6) horizontal reach factor.

Approach. Model the cut as a rectangle (pit width × depth) plus the triangular highwall-batter allowance, then bulk that area by the swell factor and re-shape it as a symmetric angle-of-repose triangle to find the spoil pile's own height and footprint; check: the dragline's own tub sits at the ORIGINAL ground surface (simple side-casting per 1.2.a), i.e. 25+3 = 28 m above the coal-seam floor, and the spoil pile is assumed dumped on that same floor with the SAME angle of repose on both faces (only one repose angle is given in the source).

1. Highwall horizontal setback. $$run = \dfrac{D_c}{\tan(63^\circ)} = \dfrac{25}{1.9626} = 12.74\ \text{m}$$
2. Cut area (2.5.1) — rectangle + highwall triangle. $$A_{cut} = w\,D_c + \tfrac12\,run\,D_c = (20)(25) + \tfrac12(12.74)(25) = 500 + 159.2 = \boxed{659.2\ \text{m}^2}$$ (per 1 m wide slice, so this is also the bank volume in m³/m).
3. Spoil pile area (2.5.2) — bulked by swell. $$A_{spoil} = A_{cut}(1+SF) = 659.2 \times 1.25 = \boxed{824.0\ \text{m}^2}$$
4. Spoil pile height above coal floor (2.5.3). Modelling the pile as a symmetric triangle with BOTH faces at the angle of repose, $A = h^2/\tan\theta_r$, so $$h_{spoil} = \sqrt{A_{spoil}\tan(35^\circ)} = \sqrt{824.0 \times 0.7002} = \boxed{24.0\ \text{m}}$$
5. Height above dragline tub base (2.5.4). The tub sits at the original surface, 25+3 = 28 m above the coal floor, so relative to the tub: $$h_{tub} = h_{spoil} - 28 = 24.0-28.0 = \boxed{-4.0\ \text{m}}$$ — the pile crest actually sits about 4 m BELOW the dragline's own standing level.
6. Operational stacking height (2.5.5). The true vertical excursion the boom must swing through is from the CUT FLOOR (28 m below the tub) up to the pile crest (4 m below the tub): $$\Delta z = (-4.0)-(-28.0) = \boxed{24.0\ \text{m}}$$ — well inside the rated 12 m ABOVE-TUB stacking height, because almost all of this 24 m of lift is spent climbing back up OUT of the cut, not stacking above grade.
7. Horizontal reach factor (2.5.6). Distance from the highwall crest to a vertical line through the spoil pile's own crest = highwall setback + pit width + half the spoil pile's base width. Spoil half-base $= h_{spoil}/\tan(35^\circ) = 24.0/0.7002 = 34.3$ m. $$X_{reach} = 12.74 + 20 + 34.3 = \boxed{67.0\ \text{m}}$$

2.6 Is the dragline capable without modification?

Yes. Both governing checks clear with margin: the horizontal reach factor (67.0 m, highwall crest to spoil crest) is comfortably inside the 90 m operating radius, and the spoil pile crest (2.5.4) sits 4.0 m BELOW the dragline's own tub level, i.e. nowhere near the 12 m rated maximum stacking height above grade. The single side-casting method of 1.2.a is therefore geometrically adequate for this cut — no walk-up to an advanced/extended bench is required.

2.7 Auxiliary methods if inadequate

Had either check failed — e.g. a deeper cut pushing the reach factor beyond the operating radius, or a swell/repose combination pushing the pile crest above the rated stacking height — the standard remedies are: (i) switch to advanced/extended bench mining (1.2.b/c) to gain both reach and stacking height by working from a self-built elevated platform; (ii) pre-strip part of the overburden with a truck-and-shovel fleet ahead of the dragline so the dragline only handles the portion within its own envelope; (iii) use a bulldozer to push/rehandle spoil further from the crest, flattening the pile and trading dozer cost for dragline reach; or (iv) re-cast with drill-and-blast pattern changes that reduce the swell factor (tighter fragmentation control), directly lowering the required spoil pile height.