A 12 ha commercial site — roofs, parking, almost no grass — drains to a single 2.4 m wide box culvert under the access road. The municipality's IDF curve for the 10-year storm fits Sherman's form with a = 800 and c = 0.65, and the site's time of concentration is 20 minutes, so the 20-minute intensity is the design intensity. The paved catchment carries a runoff coefficient of 0.85. Find the design rainfall intensity, the peak runoff it produces, and the critical depth in the culvert — the depth that tells you how much of the barrel the design storm actually uses.
Every number in this problem is editable — change any value below and the whole chain recalculates.
Given
a = 800 — — Sherman fitted constant (local curve)
c = 0.65 — — Sherman fitted exponent
t = 20 min — Storm duration = time of concentration
C = 0.85 — — Runoff coefficient, paved site
A = 12 ha — Drainage area
b = 2.4 m — Culvert width
Determine
(a)the 20-minute design rainfall intensity
(b)the peak runoff at the culvert
(c)the critical depth in the 2.4 m barrel
Step 1 of 3(a) · solve for Rainfall intensity
The duration is not a choice — it is the time of concentration, because the peak happens when the whole catchment is finally contributing at once. Twenty minutes into the local curve gives 114.1 mm/h: a rate the sky only holds briefly, which is exactly the point of an IDF curve.
Rearranged for i
i=tca
Your values, in your units
i=((20min))(0.65)(800mm/h⋅minc)
Answer
i=107.84in/day
Carried onward at full precision, not this rounded figure.
Q = CiA is bookkeeping, not hydrology: of what falls, C sheds, and the peak is what the whole area sheds at the concentrated moment. 85 % of 114.1 mm/h over 12 ha is 3.23 m³/s arriving at the culvert mouth.
Rearranged for Q
Q=CiA
107.84 in/daycarried from step 1
Your values, in your units
Q=(0.85)(0.0000317043m/s)(12ha)
Converted to base units
Q=(0.85)(114.135mm/h)(120,000m2)
Answer
Q=194.03m3/min
Carried onward at full precision, not this rounded figure.
Critical depth is the flow's own yardstick — the depth where this discharge carries itself with least energy, and the dividing line every culvert calculation is measured against. 3.23 m³/s in a 2.4 m barrel sits at y_c = 0.57 m, so a 1.2 m rise barrel has honest headroom at the design storm.
Rearranged for y_c
yc=(gb2Q2)1/3
194.03 m³/mincarried from step 2
Your values, in your units
yc=(9.80665((2.4m))2((3.23384m3/s))2)1/3
Answer
yc=569.94mm
Carried onward at full precision, not this rounded figure.
Therefore the 10-year, 20-minute storm rains at 114.1 mm/h, the paved site delivers a 3.23 m³/s peak, and the culvert's critical depth is 0.57 m — the barrel runs about half full at the moment the design storm peaks.
Why this order
The three steps are three different kinds of knowledge. The IDF curve is measured — decades of rain gauges compressed into two fitted constants that are only true for one municipality. The Rational Method is an assumption dressed as an equation: steady rain, uniform catchment, peak at the time of concentration — honest up to a square kilometre or so and increasingly fiction beyond. Critical depth is the one piece of real hydraulics, and it is the number to carry forward: above y_c the barrel flows subcritical and backwater rules; below it the flow is supercritical and the outlet controls.
The trap is the duration. Taking a 5-minute intensity because “it's more conservative” actually isn't — a 5-minute burst ends before the far corner of the site ever contributes, so the real peak never sees that rate. Duration equals time of concentration is not a convention; it is the mechanism that makes the peak a peak.
Carried values move at full precision, not the rounded figure shown — chaining rounded numbers compounds error.
