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Manning's Equation — Trapezoidal Grass Channel Worked Example

Sizing check on a grass-lined trapezoidal outfall channel carrying a detention pond's 10-year release plus bypass flow. We iterate Manning's equation to normal depth for the design discharge, then run the checks a reviewer expects: velocity against the lining, Froude number, tractive shear per FHWA HEC-15, and freeboard. Sources: Chow, Open-Channel Hydraulics (1959); FHWA HEC-15, Design of Roadside Channels with Flexible Linings.

The channel

A 600-ft grass-lined swale conveys the combined 10-yr design flow of 60 cfs from a pond outfall to the receiving stream. Geometry from the grading plan:

Bottom width b	4.0 ft
Side slopes z	3H:1V (maintainable, mowable)
Longitudinal slope S	0.8% = 0.008 ft/ft
Manning's n	0.035 (established grass, moderate flow depth; see n reference)
Design discharge Q	60 cfs

Step 1 · Geometry functions

A, P, R for a trapezoid

A = y(b + zy) = y(4 + 3y)
P = b + 2y√(1 + z²) = 4 + 2y√10 = 4 + 6.325y
R = A/P

Manning's (US units): Q = (1.486/n) · A · R^2/3 · S^1/2
1.486/0.035 = 42.46, √S = √0.008 = 0.0894

Geometry formulas for all standard shapes are on the open channel geometry card.

Step 2 · Iterate to normal depth

Bracket and refine y until Q = 60 cfs

Trial y (ft)	A (ft²)	P (ft)	R (ft)	Q (cfs)	Bracket
1.50	12.75	13.49	0.945	46.6	Low
1.80	16.92	15.38	1.100	68.5	High
1.70	15.47	14.75	1.049	60.6	High (close)
1.69	15.33	14.69	1.044	59.9	Match

Normal depth y_n ≈ 1.70 ft
Check at 1.70: A = 1.70(4 + 5.10) = 15.47 ft², P = 4 + 6.325(1.70) = 14.75 ft, R = 1.049 ft
Q = 42.46 · 15.47 · 1.049^2/3 · 0.0894 = 42.46 · 15.47 · 1.032 · 0.0894 = 60.6 cfs ≈ 60 ✓

n is depth-dependent in grass. Retardance drops as grass submerges and lays over — n = 0.035 assumes the design depth well exceeds the mowed grass height. At shallow depths in the same channel (early storm, low flow), effective n can be 0.05–0.10 and depths run deeper than a constant-n model predicts. HEC-15's retardance-class curves handle this; a single n is a design-depth simplification.

Step 3 · Velocity and Froude checks

Is the flow erosive or unstable?

V = Q/A = 60/15.47 = 3.88 ft/s
Top width T = b + 2zy = 4 + 6(1.70) = 14.2 ft
Hydraulic depth D_h = A/T = 15.47/14.2 = 1.089 ft
Fr = V/√(gD_h) = 3.88/√(32.2 · 1.089) = 3.88/5.92 = 0.66 — subcritical ✓

3.9 ft/s is within the customary 4–6 ft/s permissible range for established grass on erosion-resistant soils; on easily eroded soils, the classical permissible-velocity guidance drops toward 3–4 ft/s. Modern practice checks shear instead — next step.

Step 4 · Tractive shear (HEC-15)

The modern lining check

τ = γ · y · S = 62.4 · 1.70 · 0.008 = 0.85 lb/ft²

Compare against the permissible shear for the specified lining and soil from HEC-15 — healthy established grass linings commonly carry permissible shear on the order of 1 lb/ft² and up depending on retardance class and underlying soil, so 0.85 is workable but not generous. During establishment (bare or matted soil), permissible shear is far lower — specify a turf reinforcement mat or erosion control blanket rated for ≥ 0.85 lb/ft² until the stand matures.

Design the establishment period, not just the end state. The most common grass-channel failure is the first season: design storm arrives before the grass does. The lining spec (ECB/TRM class) must carry the design shear unvegetated, per HEC-15 Chapter 5.

Step 5 · Freeboard and total section

Finished geometry

Normal depth 1.70 ft + freeboard 0.5 ft (typical minimum for small channels) = 2.2 ft total depth
Top width at freeboard elevation: 4 + 6(2.2) = 17.2 ft

Agency criteria vary (0.5 ft minimum, 1 ft common, more where a berm impounds). Superelevation and standing-wave allowances apply on bends. The grading footprint — 17+ ft of top width for a 4-ft-bottom channel — is why 3H:1V grass channels eat right-of-way, and why steeper-sided lined channels get chosen where space is tight.

Tools used in this example

Reproduce every step: Manning's equation tool (trapezoidal mode) for the normal-depth iteration, the Manning's n card for the grass value, the channel geometry card for A/P/R, and the riprap sizing tool if the shear check fails and the lining hardens.