Critical Shear Stress for Sediment Entrainment

This is Shields' relation used the way an engineer uses it: not to characterise a flow, but to set a number a channel must stay under. Choose a critical Shields value, multiply by the submerged weight of a grain, and you have the shear stress at which the bed starts to go. Compare it against $\rho g R S$ for the design flow, and you know whether the channel is stable. This is the tractive force method, and it is what replaced the older permissible-velocity approach in stable channel design.

The form written here uses the relative density $s = \rho_s/\rho$, which is how sediment work usually states it: $\tau_c = \theta_c (s-1)\rho g d$. For quartz sand in fresh water $s = 2.65$ and $s - 1 = 1.65$, and that 1.65 appears so often in sediment transport that it is worth committing to memory. Coal fines run near 1.3, magnetite near 5, and a plastic pellet may sit below 1 and simply float — at which point the equation stops applying and you have a flotation problem instead.

Why the tractive force method beat the permissible velocity method is worth understanding, because it is the same reason shear stress is the right variable throughout this shard. Two channels can run at the same mean velocity and apply completely different forces to their beds: a deep channel at 1 m/s exerts far more boundary shear than a shallow one at 1 m/s, because shear scales with depth and slope, not with velocity alone. Permissible-velocity tables tried to patch this with depth corrections. The tractive force approach starts from the quantity that grains actually feel and needs no patch.

Three limits to keep in view. First, this is a non-cohesive relation: for silt and clay it is meaningless, and a firm cohesive bank resists stresses that would move gravel while a soft one fails in blocks that no grain-scale criterion predicts. Second, on a side slope the grain's own weight is helping it move, so the threshold on a bank is lower than on a flat bed by a factor that depends on the bank angle relative to the material's angle of repose — a channel lining designed for the invert is under-designed for the sides. Third, riprap is not sediment. A graded, interlocked, properly filtered stone blanket resists far more than the same $d_{50}$ of loose bed material, and a purpose-built riprap method will beat this equation for that job. Where this relation earns its keep is in deciding whether a natural bed, a grassed waterway or an unlined ditch will hold.

One scaling worth noticing: $\tau_c$ is directly proportional to $d$, so the stone size needed rises in exact proportion to the shear stress — but stone MASS goes as $d^3$. Doubling the design depth doubles the required grain size and multiplies the stone weight by eight. That cubing is why armouring gets expensive so quickly, and why flattening the slope or widening the section is usually the cheaper answer.