Question 6 of 6: Crowned Roadway Drainage Capacity
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper: National Exams — 98-Civ-A5 Hydraulic Engineering, May 2014. 3 hours, closed book (one aid sheet). Six questions of equal value (20 marks each); candidates complete any five, and all six are solved here as a study resource.
Reference texts. Mays, Water Resources Engineering, 3rd ed. (pipe networks, valves, pumps); Chow, Open-Channel Hydraulics (Manning uniform flow, hydraulic jump, gutter flow); Crowe, Elger & Roberson, Engineering Fluid Mechanics (pipe force balance, wall shear). Exam-supplied relations used throughout: Hazen–Williams $Q=0.278\,C\,D^{2.63}\,S^{0.54}$ with $S=h_f/L$ (SI), Manning $Q=\tfrac{1}{n}A\,R^{2/3}\,S^{1/2}$, Darcy–Weisbach $\Delta h=0.0826\,\tfrac{fL}{D^5}Q^2$, and $\text{TDH}=H_s+H_f$. Unless stated, local losses and velocity head are neglected and water has $\rho=1000\ \text{kg/m}^3$, $\nu=1.31\times10^{-6}\ \text{m}^2/\text{s}$.
Given. Crowned road, 8 m edge-to-edge (half-width 4 m each side), crossfall $S_x=2\%=0.02$, Manning $n=0.013$, longitudinal slope $S=0.02$; from the figure the centreline/crown is at 100.0 m, the top of curb at 100.01 m, and (consistent with the 2% crossfall) the edge of pavement at 99.92 m; part (a) $Q=1.3\ \text{m}^3/\text{s}$, part (b) $Q=1.15\times1.3=1.495\ \text{m}^3/\text{s}$.
Find. Water depth at each flow, and whether the 15%-increased flow stays within the curbs.
Figure 3. Crowned roadway cross-section (vertical scale exaggerated). Water drains both ways to the curbs; the crown (100.0 m) and top of curb (100.01 m) set the containment limit.
Approach. Treat each half of the crowned road as a triangular gutter carrying half the flow by Manning; if the computed spread exceeds the 4 m half-width the water has climbed over the crown, so re-evaluate the full flooded 8 m section against the top-of-curb limit to judge containment.
Triangular gutter, half the flow (a). Each side carries $Q/2=0.65\ \text{m}^3/\text{s}$. For a triangle of curb depth $d$ and spread $T=d/S_x$, Manning $Q=\tfrac1n A R^{2/3}S^{1/2}$ with $A=\tfrac12Td$, $R=A/P$. Solving gives $d=0.124\ \text{m}$ at the curb and spread $T=d/S_x=6.21\ \text{m}$.
Spread exceeds the half-width. $T=6.21\ \text{m}\gt4\ \text{m}$, so the water surface has risen above the crown—the two gutters merge and the simple triangle no longer applies. The nominal depth at the curb is $\boxed{d\approx0.12\ \text{m}}$, but the water is already over the crown.
True containment capacity. Evaluate the full 8 m section with the water surface at the top of curb (100.01 m, i.e. 0.01 m above the crown and 0.09 m at the curb). Manning on that section gives a maximum contained discharge of only $Q_{\text{cap}}\approx0.58\ \text{m}^3/\text{s}$.
Part (a) verdict. Since $1.3\ \text{m}^3/\text{s}\gg0.58\ \text{m}^3/\text{s}$, the roadway cannot hold the flow—water overtops the curbs; the 0.12 m is the depth the gutter would need if the curb did not spill.
Climate-adjusted flow (b). $Q=1.15\times1.3=1.495\ \text{m}^3/\text{s}$; the per-side triangular solution gives $\boxed{d\approx0.13\ \text{m}}$ (spread 6.55 m), even deeper.
Containment (b). The capacity to the top of curb is still $\approx0.58\ \text{m}^3/\text{s}$, far below 1.495, so the road $\boxed{\text{cannot contain the flow}}$—it overtops the curb on both sides.
Question 6 results
Quantity
Value
a) Nominal curb depth at 1.3 m3/s
≈ 0.12 m (spread 6.2 m)
b) Nominal curb depth at 1.495 m3/s
≈ 0.13 m (spread 6.5 m)
Containment capacity to top of curb
≈ 0.58 m3/s
b) Can the road contain the flow?
No — overtops the curbs
Check: the figure’s edge-of-pavement elevation of 99.02 m would imply a $\sim24.5\%$ crossfall over the 4 m half-width, contradicting the stated 2%. Following NOTE 1, the 2% crossfall is adopted for the hydraulics (edge of pavement then at 99.92 m) and the top of curb (100.01 m) is used as the spill limit. Both interpretations reach the same conclusion: the flows are an order of magnitude above the roadway’s drainage capacity.
