The science
Specific energy and torque. Excavating soil costs energy per unit volume, the specific energy SE (J/m3 = Pa). For soft ground SE is of the order of 20 to 60 MJ/m3 (weak rock is higher; the 120 MJ/m3 used for mudstone is illustrative). The cutterhead power that reaches the ground is P = eta Pinst (eta = 0.7 assumed). It excavates a volume rate Q = P / SE, limited by the cutters' maximum penetration Qpen = (pi D2/4) pmax n/60 and by the muck system Qmuck = capacity / bulking factor (loose muck takes 1.25 to 1.55 times the space of the bank soil). The slowest of the three sets the rate Qeff.
Each revolution advances the head by p = Qeff / (A n/60) and excavates A p; the work per revolution is 2 pi T, so
SE = 2 pi T / (A p) => T = SE A p / (2 pi), P = T omega = SE Q
That is why more rpm does not help once the power is used up: the same power divided by a larger omega only gives a smaller torque and a thinner cut per revolution.
Thrust. FT = Fpen + pf A + pi D Ls qs: the force pressing the cutters in, the face support pressure acting on the face, and skin friction on the shield (Ls = 7 m, qs 10 to 20 kPa). With a mean cutter radius of D/3 and a rolling coefficient mu = 0.12 (illustrative), the cutting force follows from the torque: Fpen = 3 SE D p / (8 mu).
Ring cycle. Digging one ring of length L takes tdig = L / vdig, with vdig = Qeff / A. In stop-and-go operation the lining ring (tring) is erected while the machine waits:
vnet = L / (tdig + tring)
With continuous support (segments erected while digging, as in double-shield layouts) a share o of the erection overlaps with the next ring's excavation, and a single erector can only build one ring at a time: vnet = L / max(tdig + (1-o) tring, tring). At o = 1 the slower of the two jobs sets the pace, so the best possible gain is a factor (tdig + tring) / max(tdig, tring) of at most 2. Daily advance is the working-time rate times the availability.
Face stability. To keep the face from falling in, the support pressure at the axis must at least equal the active earth pressure (Rankine, plane strain):
sigmaa = (gamma z + q) Ka - 2 c sqrt(Ka), Ka = tan2(45 - phi/2)
with surcharge q = 20 kPa. A negative value means the soil can stand unsupported in this simple model. Too much pressure is also bad: the passive pressure sigmap = (gamma z + q) Kp + 2 c sqrt(Kp) is the blow-out limit. For clay the Broms-Bennermark stability number N = (gamma z + q - pf) / c is also shown; N above about 6 signals an unstable face. A real 3D face failure needs less support than this 2D wall, so sigmaa is conservative; groundwater is ignored.
Settlement trough. Gaussian curve of Peck (1969) and O'Reilly and New (1982): S(x) = Smax exp(-x2 / (2 i2)), with trough width i = K z (K = 0.5 clay, 0.35 sand, 0.25 to 0.45 for the other soils). Because the trough volume per metre equals the volume loss VL times the tunnel area, Smax = VL (pi D2/4) / (i sqrt(2 pi)). Along the drive the trough builds up behind the face as Phi((xface - x)/i) (Attewell and Woodman), half of Smax directly above the face. In the model the effective volume loss grows when the face pressure is below the active pressure: VL,eff = VL (1 + 2 max(0, 1 - pf/sigmaa)) (illustrative coupling). The drawing exaggerates settlement vertically by the stated factor; 10 mm and 25 mm are typical levels used for building damage screening.
Traffic capacity. In steady following each vehicle keeps a gap of at least the standstill gap s0 plus the distance travelled during the time headway h: spacing d = s0 + Lv + v h, so
q = v / (s0 + v h + Lv) capacity at the speed limit
With 60 km/h, h = 1.5 s, s0 = 2 m, Lv = 4.2 m this is about 1900 vehicles per hour per lane, the familiar order of magnitude. In a platoon of n vehicles the leader keeps h and the followers keep the short headway hp, so the mean headway is ((n-1) hp + h)/n; connected autonomous vehicles can in principle use a much shorter headway. The fundamental diagram is triangular: free flow q = k v up to the critical density kc = 1/d, then q = (1 - k (s0 + Lv)) / h down to the jam density 1/(s0 + Lv). Safety (braking distance behind a vehicle that stops instantly) is not modelled.
People per hour. Vehicle flow times occupancy. Assumptions used here: an urban lane at 50 km/h with 1.5 s headway, 4.5 m cars and 1.5 persons per car; a metro line with a 120 s headway and 6 cars of 150 passengers (27000 persons/h per direction). A tunnel lane with small vehicles is a road lane, not a rail line; the comparison shows the order of magnitude and how much headway and occupancy matter.
Sources and simplifications. Specific-energy and rolling-force relations after the Colorado School of Mines TBM model (Rostami) in simplified form; Peck (1969), O'Reilly and New (1982) and Attewell and Woodman (1982) for settlement; Broms and Bennermark (1967) for face stability; Pipes and Newell car-following and the triangular fundamental diagram for traffic. No groundwater, no cutter wear, no learning curve, homogeneous soil, no ventilation or emergency-egress limits for the tunnel. All soil, machine and traffic parameters are illustrative teaching values, not design data. The scene is inspired by publicly reported small-diameter urban tunnelling projects; every vehicle, building and machine shape is generic.
Context: publicly described small-diameter urban road-tunnel projects (for example those reported for The Boring Company) motivate the question of how fast such a tube can be bored and how many people a single lane can carry. Our parameters are illustrative, not project data. Different angle from the catalogue pages on tunnel boring and hard-rock TBMs: this lab is about performance (torque, advance rate, settlement) and the throughput of the finished tunnel.
Independent educational model, not affiliated with or endorsed by any company named here; trademarks belong to their owners. Numbers are textbook-level/publicly reported approximations.