Section 4: Soil and Water Conservation Engineering
Soil and Water Erosion
Subtopic: Biological and engineering measures to control erosion
Question Statement:
A trapezoidal grassed waterway with side slope (H:V) of 1:1 carries a design discharge of 1 m³ s⁻¹. The bed slope and Manning's roughness coefficient of this channel are 1% and 0.04, respectively. The design depth of the best hydraulic trapezoidal grassed waterway section in m is ______________ (rounded off to 2 decimal places).
Official Answer Key:
0.66 to 0.69
Step-by-Step Solution & Explanation
For the best (most efficient) hydraulic trapezoidal section with side slope z=1 (H:V=1:1): bottom width b = 2y(√(1+z²)−z) = 2y(√2−1) = 0.8284y, and hydraulic radius R = y/2 (a standard property of the most-efficient trapezoidal section).
Area A = y(b+zy) = y²(2√(1+z²)−z) = 1.8284y².
Manning's equation: Q = (1/n)×A×R^(2/3)×S^(1/2)
1 = (1/0.04)×1.8284y²×(y/2)^(2/3)×(0.01)^(0.5)
Solving this nonlinear equation numerically for y gives y ≈ 0.673 m. Within official range 0.66–0.69.
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