Heavy-Haul Transporter Lab: Axle Loads, Off-Tracking, Braking and Bridge Limits

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GVW - Kingpin - Worst axle -

The science

Physics teaching model. A generic civil heavy-haul combination (tractor with a multi-axle low-loader carrying an excavator-like machine) used to explain road-transport engineering. No real vehicle, brand or operator is modelled; limits and masses are illustrative and differ by country.

Statics. The trailer frame is a rigid body supported at the kingpin and by the axle groups: sum F = W, sum M = 0. With one hydraulic circuit every axle carries the same load, so the problem is statically determinate (two supports). With rigid or paired groups the frame sits on springs: F_j = k_j (y0 + th x_j + h_j), so a bump h_j changes the loads. The kingpin load then acts on the tractor, which is split between the steer and drive groups by moments about the drive group.
Bridge formula. The US Federal Bridge Formula (well-known): W = 500 (L N/(N-1) + 12 N + 36), W in lb, L (outer axle spacing) in ft, N axles; it is applied to every group of consecutive axles. Here it is converted to kg and m. Other countries use different rules.
Bridge moment. For a simply supported span the moment under load k is found with the influence line; the worst train position is found numerically and checked against Barre's theorem (maximum when a load and the resultant are symmetric about midspan). Capacity is an equivalent uniform load q L^2/8 (illustrative).
Off-tracking. Steady low-speed turn: R1 = sqrt(Rf^2 - Lw^2) (drive group), hitch Rh = sqrt(R1^2 + c^2), trailer axle group R2 = sqrt(Rh^2 - L2^2); off-tracking is each section's R - sqrt(R^2 - L^2). Steered trailer axle lines use a simple follow-steer law delta_r = -g psi (real systems compute a common centre of rotation).
Braking. Deceleration a = mu g at the limit; weight transfer dN = m a h / L, here solved for both units including the hitch force. Ideal distribution: brake force on each group proportional to its dynamic load. Stopping distance d = v t_r + v^2/(2 a). The planar simulation has two bodies, a hitch constraint, tyre side force Fy = -mu Fz sin(C atan(B alpha)) reduced by the friction ellipse when braking, and axle lock or ABS (holds 85 % of the limit).
Uphill. Traction F = mu_t N_drive against M g (sin a + c_rr cos a); the trailer pulls back at the hitch height, so a heavier ballast or a more forward fifth wheel loads the drive axles and raises the traction limit. Power limit P eta = F v.
Rollover. Static stability factor SSF = T/(2h), rigid threshold a_y/g = SSF; with roll stiffness K the CG moves outward by q a_y/g, giving a_y/g = (T/2)/(h + q), q = m g c^2/(K - m g c). Bend speed limits v^2 = g R (a + e)/(1 - a e) (e = bank).
Simplifications. Planar tyre model, no pitch dynamics, rigid trailer frame in roll, equalised groups treated as equal-load supports, no brake fade, level road except the test hill. Not for real route or permit planning.