Question 4 of 6: Crowned-Road Gutter Capacity Under Climate Change
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper format. National Exams, December 2016 — 98-Civ-A5 Hydraulic Engineering. Closed book, 3 hours, one aid sheet permitted. Six questions of equal value (20 marks each); candidates answer any five. All six are solved here as a study resource. Take water density ρ = 1000 kg/m³, kinematic viscosity ν = 1.31 × 10−6 m²/s; local losses and velocity head are negligible unless stated.
Reference texts. Mays, Water Resources Engineering (Wiley) — pipe/pump systems & distribution networks; Chow, Open-Channel Hydraulics (McGraw-Hill) — normal/critical depth, specific energy, unsteady flow; Transportation Association of Canada, Geometric Design Guide for Canadian Roads — gutter/roadway drainage. Permitted equation set (from the exam cover): Hazen–Williams $Q = 0.278\,C\,D^{2.63}\,S^{0.54}$ with $S = h_f/L$; Manning $Q = \tfrac{1}{n}A R^{2/3} S^{1/2}$; total dynamic head $\text{TDH} = H_s + H_f$.
Question 4: Crowned-Road Gutter Capacity Under Climate Change (20 marks)
Given. A symmetric crowned road drains to a curb-and-gutter on each side.
Given data — Question 4
Quantity
Value
Pavement width (edge to edge)
8 m → half-width $B = 4$ m each side of the crown
Crossfall (transverse) slope
$S_x = 2\% = 0.02$
Manning’s $n$ (asphalt)
0.013
Longitudinal slope
$S_o = 0.01$
Crown / edge-of-pavement / top-of-curb elevation
100.00 / 99.02 / 100.01 m
Flow (a) / flow (b)
1.0 m³/s / 1.15 m³/s
Find. (a) flow depth at 1.0 m³/s; (b) flow depth at the 15%-increased flow, and whether the roadway contains it below the top of curb.
Check / figure inconsistency. The edge-of-pavement elevation (99.02 m) is 0.98 m below the crown (100.00 m) over a 4 m half-width, which implies a crossfall of $0.98/4 = 24.5\%$ — irreconcilable with the stated 2%. Following exam Note 1, the stated $S_x = 2\%$ is adopted as the design crossfall; the gutter invert at the curb face is then $100.00 - 0.02(4) = 99.92$ m, and the top of curb (100.01 m) — only 0.01 m above the crown — is taken as the spill limit.
Figure 2 (re-drawn, adopting 2% crossfall). At 1 m³/s the water surface has risen above the crown, so the whole roadway conveys as one shallow flooded channel bounded by the two curbs.
Approach. First treat each half as a triangular gutter carrying half the road flow (Izzard/HEC-22). If the resulting spread exceeds the 4 m half-width, water has overtopped the crown and the section must be re-analysed as a single flooded channel spanning both curbs. Compare the flooded capacity at the top of curb with the design flows.
Triangular-gutter check (each side carries $Q/2$). The gutter form of Manning is
$$Q_{\text{side}} = \frac{0.376}{n}\,S_x^{5/3}\,S_o^{1/2}\,T^{8/3}, \qquad y = S_x\,T,$$
where $T$ is the spread. For $Q_{\text{side}} = 0.5$ m³/s this gives $T = 5.97$ m.
Spread exceeds the half-width. $T = 5.97\ \text{m} \gt 4\ \text{m}$: the water reaches the crown and spills over it, so a triangular gutter cannot exist — the roadway floods across its full 8 m width.
Flooded-section geometry. With the water surface a height $d$ above the crown, the area and wetted perimeter of the full 8 m section are
$$A = 8d + 0.32\ \text{m}^2, \qquad P = 2\sqrt{4^2+0.08^2} + 2(d+0.08) \approx 8.16 + 2d\ \text{m}.$$
Capacity to the top of curb. At the spill limit the surface is at 100.01 m, i.e. $d = 0.01$ m above the crown: $A = 0.40$ m², $P = 8.18$ m, $R = 0.0489$ m,
$$Q_{\text{cap}} = \frac{1}{n}A R^{2/3}S_o^{1/2} = \frac{1}{0.013}(0.40)(0.0489)^{2/3}(0.01)^{1/2} = \boxed{0.41\ \text{m}^3/\text{s}}.$$
The roadway can convey only about 0.41 m³/s before water tops the curb.
Depth at 1.0 m³/s (part a). Solving $\frac{1}{n}AR^{2/3}S_o^{1/2}=1.0$ for the flooded section gives $d = 0.046$ m above the crown, i.e. a maximum flow depth at the curb of $\boxed{0.126\ \text{m}}$ (water surface at 100.045 m). This already stands 0.035 m proud of the top of curb.
Depth at 1.15 m³/s (part b). Increasing the flow by 15% to 1.15 m³/s gives $d = 0.053$ m above the crown, a maximum flow depth at the curb of $\boxed{0.133\ \text{m}}$ (surface at 100.053 m).
Answer to “can the road contain the flow?”: No. The roadway’s containment capacity (below the top of curb) is only ~0.41 m³/s. Even the base flow of 1.0 m³/s overtops the curbs by ~0.035 m, and the climate-adjusted 1.15 m³/s overtops by ~0.043 m. The section cannot contain either flow; roughly 0.6–0.7 m³/s spills past the curbs onto the adjacent boulevard/sidewalk. (The reported depths use an idealised extension of the curb walls; physically the surplus simply floods beyond the curb.)
