How do you calculate shear stress in a channel?

The design boundary shear stress on the channel bottom is the tractive force τ = γ·d·S, with γ the unit weight of water (62.4 lb/cu ft), d the maximum flow depth in feet and S the channel bed slope. The result is in lb per sq ft. For a 1.5 ft depth on a 3 percent slope, τ = 62.4 × 1.5 × 0.03 = 2.8 lb/sq ft.

What is permissible velocity for a grass-lined channel?

For a good stand of Bermudagrass on erosion-resistant soil, NRCS permits about 8 ft/s on slopes under 5 percent, 7 ft/s on 5 to 10 percent, and 6 ft/s over 10 percent; on easily eroded soil those drop to 6, 5 and 4 ft/s. Bunch grasses and grass-legume mixtures are limited to about 5 ft/s or less, and annuals used as a temporary cover to about 3.5 ft/s.

Is permissible velocity or permissible shear the better design method?

Permissible shear (the tractive-force method) is the more defensible of the two and is what FHWA HEC-15 uses. Velocity is a proxy: the same velocity produces very different boundary stress in a shallow steep channel versus a deep flat one, and permissible-velocity tables are tied to the specific channel geometries they were developed from. Use velocity for a quick screen and for compliance with a criteria manual that is written that way, and shear for the actual lining decision.

How much does a channel bend increase shear stress?

HEC-15 applies a bend factor Kb to the straight-reach shear. Kb is 2.00 when the ratio of centerline radius to top width is 2 or less, drops through the relation Kb = 2.38 − 0.206(Rc/T) + 0.0073(Rc/T)² for ratios between 2 and 10, and is 1.05 for ratios of 10 or more. A tight bend therefore roughly doubles the design shear, and the higher stress persists for a protection length downstream of the bend exit.

What is the permissible shear stress for riprap?

A good working rule is permissible shear in lb per sq ft equals about 4 times D50 in feet: 0.33 for 1 in gravel, 0.67 for 2 in gravel, 2.0 for 6 in riprap and 4.0 for 12 in riprap. That reproduces the HEC-15 table values and is the inverse of the usual sizing step — take the computed design shear, divide by 4, and that is the required D50 in feet.

What are vegetation retardance classes?

Retardance classes A through E rank a vegetal lining by how much flow resistance it provides, which depends on species, stand quality and mowing height. Class A is very high retardance (tall excellent stands such as weeping lovegrass at 30 in.), Class C is moderate (a good Bermudagrass stand mowed to about 6 in.) and Class E is very low (Bermudagrass cut to 1.5 in. or burned stubble). Permissible shear falls with the class, from about 3.7 lb/sq ft for A to 0.35 for E, and Manning's n falls with it too.

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Permissible Velocity & Shear Stress — Reference

By Michael Flynn, PE · water resources & dam safety engineer

What a channel lining can take before it moves — stated both ways engineers are asked for it. The velocity tables are the older, simpler screen; the tractive-force tables are what actually governs. Both are here because criteria manuals still cite both.

The Tractive Force

τ_d = γ · d · S (lb/ft²; γ = 62.4 lb/ft³, d = max depth ft, S = bed slope)
Side slope: τ_s = K₁ · τ_d Bend: τ_b = K_b · τ_d
Design check: τ_permissible ≥ SF · τ_d

Use the maximum depth, not the hydraulic radius, for the bed shear in a lining check. HEC-15 recommends a safety factor of 1.0 to 1.5, toward the upper end where failure of the lining would endanger a road embankment, a dam, or a downstream structure.

Side-slope and bend factors (HEC-15)

Factor	Relation
K₁, side slope (z:1 horizontal:vertical)	0.77 for z ≤ 1.5 · 0.066z + 0.67 for 1.5 < z < 5 · 1.0 for z ≥ 5
K_b, bend (R_c = centerline radius, T = top width)	2.00 for R_c/T ≤ 2 · 2.38 − 0.206(R_c/T) + 0.0073(R_c/T)² for 2 < R_c/T < 10 · 1.05 for R_c/T ≥ 10

Bend protection extends a length L_p downstream of the bend exit, not just through the curve — the secondary circulation takes distance to decay. A common failure is a correctly sized bend lining that stops at the point of tangency.

Permissible Velocity & Tractive Force — Unlined Channels

Fortier & Scobey values for straight channels of small slope, after aging. “Clear water” vs. “water carrying colloidal silts” — sediment-laden flow permits more because the deposited fines armor the boundary.

Material	n	V, clear (ft/s)	V, colloidal silts (ft/s)	τ_p, clear (lb/ft²)	τ_p, colloidal (lb/ft²)
Fine sand, colloidal	0.020	1.50	2.50	0.075	0.15
Sandy loam, noncolloidal	0.020	1.75	2.50	0.075	0.15
Silt loam, noncolloidal	0.020	2.00	3.00	0.11	0.22
Alluvial silts, noncolloidal	0.020	2.00	3.50	0.11	0.22
Ordinary firm loam	0.020	2.50	3.50	0.15	0.22
Volcanic ash	0.020	2.50	3.50	0.15	0.22
Stiff clay, very colloidal	0.025	3.75	5.00	0.26	0.46
Alluvial silts, colloidal	0.025	3.75	5.00	0.26	0.46
Shales and hardpans	0.025	6.00	6.00	0.67	0.67
Fine gravel	0.020	2.50	5.00	0.075	0.32
Graded loam to cobbles, noncolloidal	0.030	3.75	5.00	0.38	0.66
Graded silts to cobbles, colloidal	0.030	4.00	5.50	0.43	0.80
Coarse gravel, noncolloidal	0.025	4.00	6.00	0.30	0.67
Cobbles and shingles	0.035	5.00	5.50	0.91	1.10

Grass-Lined Channels — Permissible Velocity (NRCS)

Cover	Slope (%)	Erosion-resistant soil (ft/s)	Easily eroded soil (ft/s)
Bermudagrass	0–5	8	6
	5–10	7	5
	> 10	6	4
Buffalograss, Kentucky bluegrass, smooth brome, blue grama	0–5	7	5
	5–10	6	4
	> 10	5	3
Grass-legume mixture	0–5	5	4
Grass-legume mixture	5–10	4	3
Lespedeza sericea, weeping lovegrass, kudzu, alfalfa, crabgrass	0–5	3.5	2.5
Annuals used as temporary cover	0–5	3.5	2.5

Grass-legume mixtures and the bunch/annual covers are not recommended on slopes steeper than 10% and 5% respectively. All values assume a uniform, well-established stand — the design condition for the first storm after seeding is bare soil plus whatever temporary lining you specified.

Permissible Shear Stress by Lining (HEC-15)

Lining	τ_p (lb/ft²)	Notes
Temporary / rolled erosion control products
Woven paper net	0.15	Very short service life
Jute net	0.45
Fiberglass roving, single	0.60
Fiberglass roving, double	0.85
Straw with net	1.45	The common construction-phase lining
Curled wood mat	1.55
Synthetic mat	2.00
Vegetative, by retardance class
Class A — very high retardance	3.70	Excellent stand, tall (~30 in.): weeping lovegrass, yellow bluestem
Class B — high	2.10	Good stand mowed 12–24 in.: smooth brome, Bermudagrass
Class C — moderate	1.00	Good stand mowed ~6 in.; grass-legume mixture
Class D — low	0.60	Good stand mowed ~2.5 in.; buffalograss
Class E — very low	0.35	Cut to 1.5 in.; burned or sparse stubble
Gravel & riprap
Gravel, D₅₀ = 1 in.	0.33	All four follow τ_p ≈ 4 · D₅₀(ft)
Gravel, D₅₀ = 2 in.	0.67
Rock riprap, D₅₀ = 6 in.	2.00
Rock riprap, D₅₀ = 12 in.	4.00

Required D₅₀ (ft) ≈ SF · τ_d / 4 D₅₀ (in) ≈ 3 · SF · τ_d

Retardance class is not a species — it is a species plus a stand condition plus a mowing height, and it changes seasonally. Design the lining for the retardance class that will exist at the worst time of year, and check capacity (Manning's n) for the class that will exist at the best time of year; those are two different classes and two different checks.

Rigid Linings

Cast-in-place concrete, grouted riprap, soil cement and articulated block are not shear-limited within the range of ordinary channel design. They fail three other ways, and those are what the design has to address:

Failure mode	What to detail
Undermining at the downstream terminus	Cutoff wall / toe-down keyed below the expected scour depth
Uplift from groundwater or from flow beneath the slab	Weep holes, underdrain, filter or geotextile bedding
Loss of subgrade support through joints and cracks	Joint sealing, reinforcement, and a graded filter under the lining

Worked check. A trapezoidal ditch runs 1.5 ft deep on a 3.0% grade, 2:1 side slopes, straight reach. τ_d = 62.4 × 1.5 × 0.03 = 2.81 lb/ft². Side slope: K₁ = 0.066(2) + 0.67 = 0.80, so τ_s = 2.25 lb/ft². Vegetated Class C (τ_p = 1.00) fails. Straw-with-net (1.45) fails. With SF = 1.0 the required riprap is D₅₀ = 2.81/4 = 0.70 ft ≈ 8.4 in., so an available 9 in. class works — and if this reach were in a bend with R_c/T = 3, K_b = 1.83 would push τ to 5.1 lb/ft² and D₅₀ to 15 in.

Sources: FHWA HEC-15, Design of Roadside Channels with Flexible Linings (permissible shear by lining, K₁ and K_b factors); Fortier & Scobey (1926) as tabulated in Chow, Open-Channel Hydraulics, Table 7-3; NRCS/SCS grass-lined channel permissible velocities. Manufacturer-tested values for a specific RECP or TRM supersede the generic product-class values above.

Need a D₅₀? Open the riprap sizing calculator → · Depth and velocity first? Manning's equation.

Related cheat sheets and tools

Get depth and velocity from Manning's equation with n from the roughness card and geometry from the open-channel geometry card, then bring τ_d here. Size the stone with riprap sizing and the riprap methods card, bed the stone per filter criteria, and check the outlet with stilling basin and the hydraulic jump card. For full channel and watershed design, see HydroComplete.