4.1 – Range diagram. A range diagram is a to-scale plot of a dragline’s working envelope – digging depth, dumping radius and dumping height, all measured from the machine’s centreline/tub – used to check, for a given pit geometry (overburden depth, pit width, spoil pile height), whether that specific machine’s reach and stacking capability can physically complete the planned cut, exactly the check performed numerically in 4.6 below.
Given. Tub diameter $=20$ m; positioning $=75\%$ of tub diameter from centreline to high-wall edge; operating radius $=90$ m; pit width $W=40$ m; overburden depth $D=25$ m; swell factor $SF=0.25$; angle of repose $\phi=35^\circ$; rated (max) stacking height $=12$ m; high-wall slope $=63^\circ$; coal seam $=3$ m.
Find. The tub-positioning offsets (4.2.2, 4.2.3), the swelled volume of a unit sample (4.3.1), the cut/spoil cross-section geometry for a 1 m wide slice (4.5.1–4.5.6), and whether this dragline can complete the cut as specified (4.6–4.7).
Fig. 4.1 – side-casting cross-section (1 m slice): tub positioning at 75% of diameter from the high-wall edge, cut width 40 m, overburden depth 25 m, spoil pile at 35° angle of repose (schematic, not to exact scale for legibility).
4.2.2 – Tub-positioning offsets. Positioning $=75\%$ of the 20 m tub diameter, measured from the dragline centreline to the high-wall edge:
$$d_{centreline}=0.75\times20=\boxed{15\text{ m}}$$
Distance from the high-wall edge to the NEAREST tub edge (tub radius $=10$ m):
$$d_{edge}=15-10=\boxed{5\text{ m}}$$
4.2.3 – Dragline reach. The 90 m operating radius is measured from the centreline; the reach beyond the high-wall edge is:
$$\text{reach}=90-15=\boxed{75\text{ m}}$$
4.3.1 – Swelled volume. $$V_{broken}=V_0(1+SF)=1\times(1+0.25)=\boxed{1.25\text{ m}^3}$$ – swell inflates volume (or, per unit length, area), not linear dimensions directly.
4.5.1 – Cut area (1 m slice). Modelling the cut as the rectangular overburden block removed ahead of the coal (pit width × overburden depth):
$$A_{cut}=W\times D=40\times25=\boxed{1000\ \text{m}^2\text{/m}}$$
4.5.2 – Spoil pile area. The swell factor inflates the cross-sectional area of the cast material by $(1+SF)$:
$$A_{spoil}=A_{cut}(1+SF)=1000\times1.25=\boxed{1250\ \text{m}^2\text{/m}}$$
4.5.3 – Spoil pile height above coal seam floor. Modelling the swelled spoil as a symmetric triangular pile standing at its natural angle of repose $\phi$, with area $A=h^2/\tan\phi$:
$$h_{spoil}=\sqrt{A_{spoil}\tan\phi}=\sqrt{1250\times\tan35^\circ}=\boxed{29.6\text{ m}}$$
4.5.4 – Height of spoil above the dragline tub base. The tub sits on the original ground surface, i.e. a level $D=25$ m above the coal floor:
$$h_{above\ tub}=h_{spoil}-D=29.6-25=\boxed{4.6\text{ m}}$$
4.5.5 – Operational stacking height. Distinct from the manufacturer’s RATED stacking height (a boom-geometry spec, 12 m, measured above the machine’s own working level), the OPERATIONAL stacking height is the actual total vertical rise the bucket must lift and cast through, from the pit floor to the pile crest:
$$h_{op}=h_{spoil}=\boxed{29.6\text{ m}}\quad\text{vs. the machine's rated 12 m}$$
4.5.6 – Horizontal reach factor. Half-base of the symmetric spoil triangle, plus the pit width, gives the horizontal distance from the high-wall crest (dig point) to the pile crest (dump point):
$$\text{half-base}=\dfrac{h_{spoil}}{\tan\phi}=\dfrac{29.6}{\tan35^\circ}=42.3\text{ m}$$
$$\text{reach factor}=W+\text{half-base}=40+42.3=\boxed{82.3\text{ m}}$$
4.6 – Capability check. The 82.3 m horizontal reach factor is comfortably within the machine’s 90 m operating radius (unlike the 50 m-radius machine considered for a similar geometry elsewhere in this course, this 90 m dragline CAN physically reach the required dump point). The operational stacking height, however, is 29.6 m against a rated maximum of only 12 m – exceeded by 17.6 m. The dragline can reach the required dump point, but cannot stack the swelled spoil to the height this single-pass cross-section demands, so it CANNOT complete the mining plan as specified without modification.
4.7 – Auxiliary methods/equipment. Since the shortfall is stacking height, not reach, the standard fixes are: (i) an extended/advanced bench (Question 1.4.3) – pre-strip part of the overburden column with a dozer/scraper so the dragline only casts the lower portion, reducing the spoil volume (and hence pile height) it must place in a single pass; (ii) a pullback (re-handle) pass – the dragline retreats and re-handles previously cast material to push the pile further back and lower its effective working height, at the cost of extra cycle time; or (iii) a secondary machine (dozer or a smaller stacking conveyor/spreader) dedicated to redistributing the upper part of the spoil pile so the dragline’s own single-lift stacking height stays within its 12 m rating.
Item
Value
4.2.2 Edge→centreline / edge→tub
15 m / 5 m
4.2.3 Dragline reach
75 m
4.3.1 Swelled volume of 1 m³
1.25 m³
4.5.1 Cut area
1000 m²/m
4.5.2 Spoil pile area
1250 m²/m
4.5.3 Spoil height above coal floor
29.6 m
4.5.4 Spoil height above tub base
4.6 m
4.5.5 Operational stacking height
29.6 m (vs. 12 m rated)
4.5.6 Horizontal reach factor
82.3 m (vs. 90 m operating radius)
4.6 Capable without modification?
No – stacking height exceeded by 17.6 m (reach is adequate)
Check: the cut/spoil cross-section is modelled as a simple rectangular cut and a symmetric triangular spoil pile at the angle of repose – the standard simplified textbook treatment for this class of problem; a detailed design would use the machine’s actual dumping-radius/height range diagram (Question 4.1) bucket-pass by bucket-pass.
