Trapezoidal Channel Capacity and Freeboard Under Backwater
A 4.0 m bottom-width trapezoidal channel has 2H:1V side slopes, a longitudinal slope of 0.0015, and Manning n=0.030. The design peak flow is 8.0 m3/s. The channel banks are 1.75 m above the invert, and the design drawings show 0.60 m of freeboard above normal depth. A downstream culvert is being replaced, and a backwater analysis indicates that the 100-year water surface at the reach outlet will be 0.45 m above normal depth. Future development is expected to increase the peak to 10.0 m3/s. The channel has no levees. The reviewer must determine whether the existing banks and freeboard are adequate and whether riprap is needed at a bend.
1. Compute normal depth and average velocity at the current design flow of 8.0 m3/s.
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Manning equation: Q=(1/n)A R^(2/3) S^0.5, with A=4y+2y^2, P=4+2y sqrt(5)=4+4.472y, and R=A/P. For y=1.15 m: A=7.245 m2, P=9.143 m, R=0.792 m, R^(2/3)=0.856; Q=(1/0.030)(7.245)(0.856)(0.0015^0.5)=8.01 m3/s. So normal depth is about 1.15 m. V=Q/A=8.0/7.245=1.10 m/s.
2. Evaluate freeboard at the current flow and at the future 10.0 m3/s flow with the 0.45 m backwater. Is the channel adequate?
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At Q=10 m3/s, trial y=1.29 m gives A=8.49 m2, P=9.77 m, R=0.869 m, and Q about 10.0 m3/s. With the 0.45 m backwater, water surface is 1.29+0.45=1.74 m above invert. The bank is 1.75 m, leaving only 0.01 m freeboard, far below the 0.60 m design freeboard and generally unacceptable. Even at current Q=8, backwater gives 1.15+0.45=1.60 m, leaving 0.15 m. Recommend raising banks or increasing culvert capacity, and rechecking the 100-year profile. Maintain at least 0.3 to 0.6 m freeboard, with more where waves or debris are possible.
3. At the bend, is 150 mm riprap stable at the calculated flow? Show the shear-stress check and state a practical recommendation.
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At Q=8, tau0=gamma R S=9810(0.792)(0.0015)=11.7 Pa; at Q=10, tau0 is about 12.8 Pa. For 150 mm riprap, a Shields-type critical stress is about 0.047(rho_s-rho)gD=0.047(1650)(9.81)(0.15)=114 Pa, so the calculated bed stress is well below the threshold. A bend increases local shear, but a factor of 1.5 to 2 still leaves margin. Use properly graded riprap with a filter or geotextile, extend it through the bend and transitions, and check toe scour. Do not reduce to fines based only on the average shear.