---
title: "Trapezoidal channel design with Chezy's formula"
subject: "Hydro Power"
chapter: 1
key: "semester_6/hydro_power:trapezoidal-channel-chezy-design"
source: https://www.bctengineer.com/q/semester_6/hydro_power/trapezoidal-channel-chezy-design
site: BCT Engineering
---

# Trapezoidal channel design with Chezy's formula

Hydro Power · Chapter 1: Introduction to Basic Fluid Mechanics and Hydraulics

- Asked in: 2082 Kartik, 6 marks
- Video: [PROBLEM ON CHEZYS FORMULA FOR TRAPEZOIDAL SECTION](https://www.youtube.com/watch?v=ECgDxNyQDzY) — e Tution
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/hydro_power/trapezoidal-channel-chezy-design
- Whole chapter: https://www.bctengineer.com/md/semester_6/hydro_power/chapter-1

## Question

A trapezoidal channel carries a discharge of 5 m³/s. Design the channel section given a slope of 1 in 1200 and side slopes of 1:1, using Chezy's formula with C = 56.

## Short answer

Assuming the most economical trapezoidal section (half top width = sloping side, R=y/2), Chezy's formula with C=56 gives flow depth y ≈ 1.42 m and bottom width b ≈ 1.17 m for a discharge of 5 m³/s on a slope of 1/1200 with 1:1 side slopes. Top width is approximately 4.01 m.

## Answer

Only $Q$, $S$, side slope and $C$ are given, so one more condition is needed. The channel is designed as the **most economical trapezoidal section** (standard assumption): $R = y/2$ and half the top width = one sloping side.

#### Proportions (side slope $n = 1$)
$$
\begin{aligned}
\frac{b + 2ny}{2} &= y\sqrt{1+n^2} \;\Rightarrow\; b = 2y(\sqrt{2} - 1) = 0.8284y \\
A &= (b + ny)y = (0.8284 + 1)y^2 = 1.8284y^2 \\
R &= y/2
\end{aligned}
$$

#### Chezy's equation
$$
\begin{aligned}
Q &= AC\sqrt{RS} \\
5 &= 1.8284y^2 \times 56 \times \sqrt{\frac{y}{2} \times \frac{1}{1200}} \\
5 &= 1.8284 \times 56 \times 0.020412\, y^{2.5} = 2.0901\, y^{2.5} \\
y^{2.5} &= 2.392 \;\Rightarrow\; y = 1.418\ \text{m}
\end{aligned}
$$

#### Dimensions
$$
\begin{aligned}
b &= 0.8284 \times 1.418 = 1.174\ \text{m} \\
A &= 1.8284 \times 1.418^2 = 3.674\ \text{m}^2, \quad R = 0.709\ \text{m} \\
V &= 5/3.674 = 1.361\ \text{m/s}
\end{aligned}
$$
Check: $3.674 \times 56 \times \sqrt{0.709/1200} = 5.00$ m³/s.

```text
  |<------- T = 4.01 m ------->|
  \                            /
   \  y = 1.42 m         1:1  /
    \________________________/
         b = 1.17 m
```

**Answer:** Bottom width ≈ 1.17 m, depth of flow ≈ 1.42 m, side slopes 1:1 (top width ≈ 4.01 m). In practice a freeboard of about 0.3–0.5 m is added.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Why must an extra assumption be made in this design?
   A. Only Q, S, side slope and C are given, so the most economical section is assumed
   B. Chezy's C is unknown
   C. The slope is unknown
   D. The discharge is unknown
   Answer: A. Two unknowns (b and y) need a second condition: the most economical trapezoid, with R = y/2. (Key point: Extra condition)

2. For side slope 1:1, the economical bottom width is:
   A. b = 2y
   B. b = 0.8284 y
   C. b = y
   D. b = 1.414 y
   Answer: B. b = 2y(sqrt2 - 1) = 0.8284 y, giving A = 1.8284 y^2. (Key point: Proportions)

3. Which equation is used to solve for y?
   A. Q = A C sqrt(R S) with R = y/2
   B. Q = A R^(2/3) S^(1/2)/n
   C. Q = C A S
   D. Q = A sqrt(2gy)
   Answer: A. Chezy: Q = AC sqrt(RS), giving y^2.5 = 2.392. (Key point: Chezy equation)

4. The design gives approximately:
   A. b = 1.17 m, y = 1.42 m
   B. b = 2.84 m, y = 1.42 m
   C. b = 1.42 m, y = 1.17 m
   D. b = 0.83 m, y = 1.00 m
   Answer: A. y = 2.392^0.4 = 1.418 m and b = 0.8284 x 1.418 = 1.174 m. (Key point: Result)

## Related in Hydro Power

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