Mining Haul Truck Lab: Rimpull, Retarding on Grades, Tyre Loads and Haul Cycles

Loading Blender-built truck, pit and shovel...

speed - road - gear - rimpull - brakes - -

The science

A generic teaching model. The truck is a generic 6x4 rigid-frame open-pit haul truck (about 62 t empty, 100 t rated payload, 950 kW at the flywheel). It is not any manufacturer's product; every parameter is a textbook-order or illustrative value, not data for a real model. Educational vehicle dynamics only.

Haul-road resistance. Rolling resistance RR = C_rr W cosθ with typical haul-road values of roughly 2 % (hard, smooth, maintained), 3 % (firm, average), 4 % (wet clay), 5 % (loose gravel) and 8 % or more on soft or muddy ground. A common rule of thumb is about 2 % plus 0.6 % for every centimetre the tyre sinks. Grade resistance GR = W sinθ ≈ W × grade. The mining shorthand effective (total) grade is TR % = grade % + RR %: an 8 % ramp on a 3 % road feels like an 11 % hill. The model also includes air drag ½ ρ C_dA v² (small at haul speeds).
Rimpull. The tractive force at the tyre patch is F = η P / v, a hyperbola of constant power, capped at low speed by the transmission torque of each gear (the plateaus) and by traction F ≤ μ W_drive. The torque converter slips below the lock-up speed (efficiency 0.75 rising to 0.98), the 7 gear curves are drawn thin and the best-gear envelope thick. Axle loads include the longitudinal load transfer W sinθ h/L on a grade.
Speed on a grade is where available rimpull meets total resistance, F(v) = W(sinθ + C_rr cosθ) + ½ρC_dA v², found by bisection and checked against an RK4 integration of m_eff dv/dt = F(v) − F_res(v) (m_eff = 1.06 m for rotating parts). Power-to-weight: a loaded truck has only about 6 kW/t, so on an 11 % effective grade it crawls at about 15 km/h.
Retarding. On a descent the force to absorb is F_b = −(W sinθ + RR + drag). The retarder (hydraulic or electric, here F_ret = min(F_cap, P_ret/v)) takes the load first; the service brakes take the rest. The potential energy to dissipate on a descent is E = m g Δh (minus rolling and drag work). Brake discs are a lumped thermal mass: C dT/dt = P_sb − hA (T − T_amb), with closed form T(t) = T_eq + (T₀ − T_eq) e^{−t hA/C}, T_eq = T_amb + P/hA. One stop from speed v adds ΔT = ½ m v² / C before cooling. Above the fade temperature friction falls (effectiveness 1 → 0.45), so a hot brake stops less: the runaway risk when brake power exceeds cooling. The planner never plans a descent faster than the speed whose steady disc temperature stays at or below the safe value.
Tyre loads and TKPH. Static tyre load comes from the axle split (front steer vs rear tandem duals). The tyre heat criterion is TKPH_site = Q_m V_m with Q_m = (Q_empty + Q_loaded)/2 (tonnes per tyre) and V_m = L_cycle N_cycles / H (mean speed over a working hour including waiting), compared with the manufacturer rating after correction factors K1 (long cycle) and K2 (hot ambient). The K1/K2 forms and the deflection law (load capacity proportional to p^0.6, 13 % deflection at rated load) are illustrative; use the tyre maker's tables for real work.
Cornering. Quasi-static rigid-axle lateral load transfer ΔW/W = a_y h / (g T); rollover when one side unloads, a_y = g T / (2h); sliding when a_y = μ g. A loaded truck has its centre of gravity high (about 2.6 m), so on a wet road it skids long before it rolls; the sim reports which comes first.
Haul cycle. A speed profile is built on a 2 m grid by a forward pass (acceleration limited by rimpull minus resistance) and a backward pass (braking), with the steady-speed limits per segment, site limits and hairpin limits. Cycle time = Σ ds/v + load time (spot + passes × pass time, passes = ⌈payload/bucket⌉) + dump time. Fuel = ∫F_drive v/η dt × bsfc + accessories + idle. Cost per tonne = (truck cost/h × cycle + shovel share + fuel)/payload.
Truck-shovel matching. The closed system of N trucks and one shovel is the machine-repairman queue: with travel time T_t and shovel service time T_s, P_n ∝ N!/(N−n)! (T_s/T_t)ⁿ, throughput X = (1 − P₀)/T_s and queue wait W_q = N/X − T_t − T_s (an M/M/1 queue overestimates waiting for a small closed fleet). A discrete-event simulation cross-checks it.
Optimal ramp grade. For a fixed lift H, ramp length is H/grade. Steeper ramps are shorter but slower, so cycle time flattens and then turns up once the converter slips and the descent is limited by retarding; the curve is shallow near the optimum. Road construction cost per tonne falls as 1/grade. Real designs are also limited by safety, brake and retarder capacity and tyre heating; many mines use 8–10 %.
Simplifications. Quasi-static drivetrain, constant efficiencies, single lumped brake temperature, no engine-cooling or hydraulic limits, no bench geometry effects on the haul distance, 3D pit is a schematic illustration (its road geometry does not change when you edit the profile; the truck is placed on it by progress along the profile), and queueing assumes exponential times.

References: Caterpillar Performance Handbook (rimpull, retarding charts, resistance rule of thumb); Kaufman & Ault, Design of Surface Mine Haulage Roads (USBM IC 8758, 1977); Hartman & Mutmansky, Introductory Mining Engineering; Wong, Theory of Ground Vehicles; Gillespie, Fundamentals of Vehicle Dynamics; tyre-maker earthmover TKPH technical notes.

Independent educational model, not affiliated with any equipment or tyre manufacturer. Numbers are textbook-level approximations and illustrations, not data for any product or site.