Question 8 of 13: Dragline Side-Casting Geometry

Question 2: Dragline Side-Casting Geometry (15 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.

Given.

Find. Range-diagram definition; sketch with labelled geometry; tub-positioning offsets; swelled volume of a 10 m³ sample; cut and spoil areas, spoil height, operational stacking height and horizontal reach for a 1 m wide slice; and whether the dragline as specified can complete the cut.

Approach. Work through the geometry and mass-balance relations bench by bench: define the range diagram, size the tub-positioning offsets from the given percentage, apply the swell factor to a sample volume, then build the full 1 m-wide cross-section (cut, swelled spoil, pile height, operational stacking height, horizontal reach) and compare the results against the machine’s rated reach and stacking height to answer the capability question.

1. 2.1 – Range diagram. A range diagram is the manufacturer’s to-scale drawing of a dragline’s digging and dumping envelope – boom angle, operating radius, maximum digging depth and maximum dumping height/radius plotted as a set of concentric arcs about the machine centreline – used to check, for a specific pit geometry, whether every point that must be dug or every point where spoil must be placed actually falls inside the machine’s reach.
2. 2.4 – Tub positioning. Positioning is defined as (distance, highwall edge to tub centreline) $\div$ tub diameter. At 75% positioning: $$\text{highwall edge to centreline} = 0.75 \times 20\ \text{m} = 15.0\ \text{m}$$ $$\text{highwall edge to nearest tub edge} = 15.0 - \tfrac{20}{2} = \boxed{5.0\ \text{m}}$$ so the tub sits with its centreline 15.0 m back from the crest and its nearest edge only 5.0 m from the crest – a realistic working setback for a machine walking the edge of an active highwall.
3. 2.5 – Swelled sample volume. Swell factor relates broken (placed) volume to unbroken (in-situ) volume: $V_{broken}=V_0(1+SF)$. $$V_{broken} = 10\ \text{m}^3 \times (1+0.25) = \boxed{12.5\ \text{m}^3}$$
4. 2.6.1 – Cut area (1 m slice). Modelling the cut as the rectangular overburden block removed ahead of the coal (pit width × overburden depth, per the sketch, Fig. 2.1): $$A_{cut} = W \times D = 40\ \text{m}\times 25\ \text{m} = \boxed{1000\ \text{m}^2/\text{m} \;(=1000\ \text{m}^3\ \text{per 1 m slice})}$$
5. 2.6.2 – Spoil-pile area (1 m slice). Broken material occupies more volume than the in-situ block it came from, by the swell factor: $$A_{spoil} = A_{cut}(1+SF) = 1000\times1.25 = \boxed{1250\ \text{m}^2/\text{m}}$$
6. 2.6.3 – Spoil-pile height above the coal-seam floor. Free-dumped spoil settles at its angle of repose on both sides, forming a symmetric (isosceles) triangular pile of area $A_{spoil}$; with half-base $=h/\tan\phi$, $A_{spoil}=h\cdot(h/\tan\phi) = h^2/\tan\phi$: $$h_{spoil} = \sqrt{A_{spoil}\tan\phi} = \sqrt{1250 \times \tan 35^\circ} = \sqrt{1250\times0.7002} = \boxed{29.6\ \text{m above the coal floor}}$$
7. 2.6.4 – Spoil height above the dragline-tub base. The tub sits at the original ground surface, i.e. at overburden-depth elevation (25 m above the coal floor): $$h_{above\,tub} = h_{spoil} - D = 29.6 - 25 = \boxed{4.6\ \text{m}}$$ – the freshly swelled spoil crest pokes up 4.6 m above the machine’s own operating level.
8. 2.6.5 – Operational stacking height. Distinct from the manufacturer’s rated stacking height (a machine-reach spec, measured above the crest the dragline is working from), the operational stacking height is the actual total vertical rise the bucket must lift and cast through – from the pit floor up to the pile crest: $$h_{op} = h_{spoil} = \boxed{29.6\ \text{m}} \quad \text{vs. the machine’s rated 12 m stacking height}$$
9. 2.6.6 – Horizontal reach factor. Half-base of the symmetric spoil pile plus the pit width gives the horizontal distance from the highwall crest (dig point) to the pile crest (dump point): $$\text{half-base} = \frac{h_{spoil}}{\tan\phi} = \frac{29.6}{0.7002}= 42.3\ \text{m}, \qquad \text{reach factor} = W + \text{half-base} = 40 + 42.3 = \boxed{82.3\ \text{m}}$$
10. 2.7 – Capability check. Compare both results against the machine’s ratings:

    Check	Required	Rated	Adequate?
    Stacking height	29.6 m	12 m	No – exceeded by 17.6 m
    Reach (dig-to-dump)	82.3 m	50 m operating radius	No – exceeded by 32.3 m

    Both the pile-stacking capability and the operating reach are exceeded, so no, this single walking dragline cannot complete the planned 25 m-deep, 40 m-wide side-cast in one pass without modification to the mining method or machine.
11. 2.8 – Auxiliary methods. Because the shortfall is in both reach and stacking height, the standard fixes are: (a) an advanced/extended bench (Q1.2) worked by dozer or a smaller stripping shovel ahead of the dragline, cutting the overburden column the dragline must handle in one lift; (b) pullback re-handling of the spoil (Q1.2) to lower and lengthen the pile so a single cast does not need to reach the full 82 m/29.6 m envelope; or (c) blast-casting (higher powder factor, directional timing) to throw a portion of the overburden directly toward the spoil side before the dragline ever picks it up, cutting the effective cast distance the machine itself must supply.