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Water-Resources Engineering

David A. Chin

Chapter 5

Design of Drainage Channels - all with Video Answers

Educators

Chapter Questions

01:02

Problem 1

Show that the best hydraulic section for a rectangularshaped section is one in which the bottom width is equal to twice the flow depth.

James Kiss
James Kiss
Numerade Educator
01:02

Problem 2

Show that the best hydraulic section for a triangularshaped section is one in which the top width is equal to twice the flow depth.

James Kiss
James Kiss
Numerade Educator
02:39

Problem 3

How do the side slopes in the best trapezoidal section compare with the side slopes in the best triangular section?

Trinity Steen
Trinity Steen
Numerade Educator
11:27

Problem 4

A highway median channel is to be sized to accommodate a peak runoff rate of $2.4 \mathrm{~m}^3 / \mathrm{s}$. Regulations require that the channel side slopes be 3:1 (H:V), and site conditions require a lining with an $n$ value of 0.020 and a longitudinal slope of $0.8 \%$. Determine the dimensions of the most efficient trapezoidal section.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:35

Problem 5

An existing trapezoidal channel has a bottom width of $3 \mathrm{~m}$, side slopes of 2:1 (H:V), a longitudinal slope of $0.5 \%$, and a lining material with a Manning's $n$ of 0.020 and a permissible shear stress of $25 \mathrm{~Pa}$. The channel is lined up to an elevation $1 \mathrm{~m}$ above the bottom of the channel. Determine the maximum flow rate for which the lining material will be stable.

James Kiss
James Kiss
Numerade Educator
01:24

Problem 6

Consider a channel with flexible lining in which the lining material has a permissible shear stress of $\tau_p$. Use the Manning equation to show that the permissible velocity, $v_p$, in the channel is given by
$$
v_p=\frac{0.010}{n}\left(\frac{R^4}{y^3}\right)^{\frac{1}{6}} \tau_p^{\frac{1}{2}}
$$
where $R$ is the hydraulic radius and $y$ is the depth of flow. Explain why it is better to design a channel based on maximum permissible shear stress rather than maximum permissible velocity.

Anand Jangid
Anand Jangid
Numerade Educator
02:34

Problem 7

The bottom sediments of an open channel consist of cohesionless particles with a median grain size of $2.5 \mathrm{~mm}$ and a specific gravity of 2.65 . If the temperature of the water is $20^{\circ} \mathrm{C}$, estimate the permissible shear stress on the bottom sediments.

Matt Just
Matt Just
Numerade Educator
03:01

Problem 8

Derive Equation 5.29 from Equations 5.25 and 5.28 .

Chai Santi
Chai Santi
Numerade Educator
02:35

Problem 9

The theoretical tractive-force ratio, $K$, in trapezoidal channels is given by Equation 5.29, and the bottom and side shear stresses are given by Equations 5.16 and 5.17. Use these relations to identify the side slopes under which the critical shear stress will be on the side of an unlined (trapezoidal) channel. If the side slopes of a particular channel are $25^{\circ}$ and the angle of repose is $30^{\circ}$, will the critical shear stress on the sides be reached before the critical shear stress on the bottom is exceeded?

James Kiss
James Kiss
Numerade Educator
02:35

Problem 10

A trapezoidal channel is to be laid on a slope of $1 \%$, have a bottom width of $4 \mathrm{~m}$, side slopes of 2.5:1 (H:V), and lined with angular stones of median size $250 \mathrm{~mm}$ and 85 -percentile size $300 \mathrm{~mm}$. The stone lining is derived from an open gradation. The permissible shear stress of the stone lining is estimated to be $200 \mathrm{~Pa}$. Determine the maximum flow depth in the channel for the lining to remain stable.

James Kiss
James Kiss
Numerade Educator
02:35

Problem 11

A trapezoidal drainage channel has a lining material with a permissible shear stress of $60 \mathrm{~Pa}$, a Manning's $n$ of 0.023 , a bottom width of $2 \mathrm{~m}$, side slopes of $4: 1(\mathrm{H}: \mathrm{V})$, a longitudinal slope of $0.5 \%$, and a design flow depth of $0.80 \mathrm{~m}$. Determine the maximum radius of curvature that can be used without having to provide additional armoring to the channel.

James Kiss
James Kiss
Numerade Educator
02:30

Problem 12

What type of material can withstand vertical side slopes? What is a typical side slope used in roadside channels?

Devin Lea
Devin Lea
Numerade Educator
01:55

Problem 13

Under what conditions would you use a freeboard greater than $30 \mathrm{~cm}$ in a drainage channel?

Narayan Hari
Narayan Hari
Numerade Educator
05:18

Problem 14

A trapezoidal channel has been designed with a bottom width of $10 \mathrm{~m}$, side slopes of 2:1 ( $\mathrm{H}: \mathrm{V})$, and a longitudinal slope of $0.053 \%$. The Manning's $n$ of the channel material is estimated to be 0.030 . The design flow rate is $28 \mathrm{~m}^3 / \mathrm{s}$ and the radius of curvature of the channel is $100 \mathrm{~m}$. Determine the design depth of flow in the channel and the minimum required freeboard.

Rory Naguib
Rory Naguib
Numerade Educator
05:33

Problem 15

Design the most efficient (straight) lined trapezoidal channel to carry $30 \mathrm{~m}^3 / \mathrm{s}$ on a longitudinal slope of 0.002 . The lining of the channel is to be float-finished concrete.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:29

Problem 16

A lined rectangular channel is to be constructed on a slope of $0.042 \%$ to handle a design flow rate of $4.5 \mathrm{~m}^3 / \mathrm{s}$. The lining of the channel is to be unfinished concrete, and the maximum radius of curvature is $150 \mathrm{~m}$. Determine the minimum depth and width of the channel to be excavated.

Penny Riley
Penny Riley
Numerade Educator
05:33

Problem 17

A lined (straight) triangular channel is to be constructed on a slope of $0.032 \%$ to handle a design flow rate of $4.4 \mathrm{~m}^3 / \mathrm{s}$. The lining of the channel is to be smooth asphalt. Determine the dimensions of the most efficient channel.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:35

Problem 18

A parabolic-shaped channel is to be excavated and lined with mortar. If the channel is to carry $6 \mathrm{~m}^3 / \mathrm{s}$ on a slope of $0.05 \%$, determine the minimum depth of channel to be excavated. Give the equation of the channel sides.

James Kiss
James Kiss
Numerade Educator

Problem 19

A trapezoidal channel has a design peak flow rate of $0.42 \mathrm{~m}^3 / \mathrm{s}$, a bottom width of $0.4 \mathrm{~m}$, side slopes of $3: 1$, and a longitudinal slope of $0.8 \%$. If an open-graded gravelmulch lining is to be used, estimate an adequate median stone size for the lining.

Check back soon!
02:35

Problem 20

A triangular channel is to be used to provide drainage for a peak flow rate of $$0.2 \mathrm{~m}^3 / \mathrm{s}$$, and the channel is to be located adjacent to a roadway that has a longitudinal slope of $0.05 \%$. Informal proposals from two engineering design/construction firms have been received. A Brazilian firm proposes to use an unfinished concrete lining with the best hydraulic section and will charge $$\$ 100 / \mathrm{m}^3$$ of channel volume. A Spanish firm proposes to use a locally available gravel-mulch lining, with round stones having a median size of $$50 \mathrm{~mm}$$. The Spanish firm does not guarantee that they will use the best hydraulic section and will charge $$\$ 70 / \mathrm{m}^3$$ of channel volume. Design both channel alternatives, determine the likely best alternative, and estimate the difference in cost per $\mathrm{km}$ of roadway.

James Kiss
James Kiss
Numerade Educator
04:55

Problem 21

Evaluate the adequacy of using a sod-grass lining of characteristic height $7.5 \mathrm{~cm}$ in good condition in a trapezoidal channel having a bottom width of $0.90 \mathrm{~m}$, side slopes of 3:1 (H:V), and a longitudinal slope of $3 \%$. The design flow rate is $0.5 \mathrm{~m}^3 / \mathrm{s}$. The channel is underlain by clayey sand (SC classification) with a plasticity index of 16 and a void ratio of 0.5 .

Rory Naguib
Rory Naguib
Numerade Educator
03:21

Problem 22

A trapezoidal channel is to be excavated on a slope of $0.1 \%$ and have a design discharge of $35 \mathrm{~m}^3 / \mathrm{s}$. Right-ofway constraints require the top width of the channel to be $17 \mathrm{~m}$, and the channel is expected to consist of clean gravel (moderately angular) with a 75-percentile diameter of $3 \mathrm{~cm}$. If freeboard is taken to be zero, determine the required bottom width and side slopes of the channel. Is the required channel practical?

Narayan Hari
Narayan Hari
Numerade Educator

Problem 23

A roadway median channel in Georgia is to be excavated in compact clay and is expected to convey a peak flow rate of $0.5 \mathrm{~m}^3 / \mathrm{s}$ on a slope of $0.2 \%$. The clay has a plasticity index of 25 , a void ratio of 0.4 , and has the ASTM classification $\mathrm{CH}$. Design the unlined median channel. Compare the designed channel to the best hydraulic section.

Check back soon!
02:35

Problem 24

An existing channel is on a $0.3 \%$ slope in moderately angular coarse gravel that has a 75-percentile diameter of $3 \mathrm{~cm}$. The channel is $3 \mathrm{~m}$ deep with a bottom width of $6 \mathrm{~m}$ and side slopes of 2.5:1 (H:V). Estimate the safe discharge capacity of the channel.

James Kiss
James Kiss
Numerade Educator
04:55

Problem 25

Evaluate the adequacy of a vegetation lining with class D retardance in a trapezoidal channel having a bottom width of $0.90 \mathrm{~m}$, side slopes of $3: 1(\mathrm{H}: \mathrm{V})$, and a longitudinal slope $3 \%$. The design flow rate is $0.5 \mathrm{~m}^3 / \mathrm{s}$. The channel is underlain by clayey sand (SC classification) with a plasticity index of 16 and a void ratio of 0.5 .

Rory Naguib
Rory Naguib
Numerade Educator
02:35

Problem 26

A large housing development has an existing trapezoidal swale with a bottom width of $5 \mathrm{~m}$, side slopes of $3: 1$ (H:V), and a total depth of $1 \mathrm{~m}$. The swale has a longitudinal slope of $0.1 \%$ and is lined with a dense (sodded) cover of centipede grass in good condition with a retardance classification of $\mathrm{C}$. The channel is underlain by native compacted sandy-clay soil with a plasticity index of 14, a void ratio of 0.4 , and an ASTM classification of CL. (a) Determine the flow rate that can be safely accommodated by the swale. (b) If construction activity degrades the grass in the swale such that the lining is transformed back to the native compacted sandy-clay soil, determine the adjusted flow rate that could be safely accommodated by the channel.

James Kiss
James Kiss
Numerade Educator
02:35

Problem 27

A roadside channel is to be lined with Bermuda grass and regularly mowed to a height of $4 \mathrm{~cm}$. The channel is to be designed for a peak flow rate of $1.1 \mathrm{~m}^3 / \mathrm{s}$ and a longitudinal slope of $1.4 \%$, and it is to have a triangular cross section with side slopes of $3: 1(\mathrm{H}: \mathrm{V})$. Using the alternative retardance-based approach, determine the depth of flow in the channel and assess the adequacy of the channel lining.

James Kiss
James Kiss
Numerade Educator
03:07

Problem 28

A trapezoidal drainage channel on a golf course is lined with Kentucky bluegrass and is expected to convey $10 \mathrm{~m}^3 / \mathrm{s}$ under design conditions. The longitudinal slope of the drainage channel is $0.1 \%$. If the channel has a bottom width of $3 \mathrm{~m}$, side slopes of $2: 1(\mathrm{H}: \mathrm{V})$, and a total depth of $2 \mathrm{~m}$, use the alternative retardance-based approach to assess the adequacy of the existing channel.

Prabhat Tyagi
Prabhat Tyagi
Numerade Educator
02:35

Problem 29

An existing triangular channel has a depth of $2 \mathrm{~m}$ and side slopes of 4:1 (H:V), a longitudinal slope of $1 \%$, and is lined with a new type of Bermuda grass which has a retardance classification of $\mathrm{C}$ and maximum permissible shear stress of $98.9 \mathrm{~Pa}$. Use the alternative retardance-based approach to determine the maximum flow rate that can be handled by this channel?

James Kiss
James Kiss
Numerade Educator
02:35

Problem 30

A trapezoidal channel is to be designed to accommodate a flow rate of $4 \mathrm{~m}^3 / \mathrm{s}$ on a longitudinal slope of $0.8 \%$. Excavation constraints limit the flow depth to $1 \mathrm{~m}$ and it is proposed that the lining of the channel be centipede grass with an average height of $15 \mathrm{~cm}$, which is classified as sod grass in good condition. The grass lining is underlain by a sandy soil with a void ratio of 0.45 , a 75 percentile grain size of $0.8 \mathrm{~mm}$, and a plasticity index of 8 . (a) Design the bottom width and side slopes of the channel; (b) assess the adequacy of the grass lining using the conventional approach; and (c) assess the adequacy of the lining using the alternative retardancebased approach.

James Kiss
James Kiss
Numerade Educator
02:35

Problem 31

A temporary RECP lining is being considered for a trapezoidal roadside channel with a bottom width of $0.9 \mathrm{~m}$, side slopes of $3: 1(\mathrm{H}: \mathrm{V})$, and a longitudinal slope of $3 \%$. The channel is excavated in clayey sand (SC classification) with a plasticity index of 16 and a void ratio of 0.5 . The design flow rate in the channel is $0.3 \mathrm{~m}^3 / \mathrm{s}$. Assess the adequacy of a lining for which the shear stress at $12.5 \mathrm{~mm}$ soil loss is $100 \mathrm{~Pa}$, and the roughness rating is as follows:
$$
\begin{array}{cc}
\hline \begin{array}{c}
\text { Applied shear } \\
\text { (Pa) }
\end{array} & \text { Manning's } n \\
\hline 50 & 0.040 \\
100 & 0.036 \\
200 & 0.033 \\
\hline
\end{array}
$$

James Kiss
James Kiss
Numerade Educator
02:35

Problem 32

A riprap lining is being proposed for a trapezoidal drainage channel having a bottom width of $0.6 \mathrm{~m}$, side slopes of 3:1 (H:V), and a longitudinal slope of $2 \%$. The design flow rate is $1.13 \mathrm{~m}^3 / \mathrm{s}$. Assess the adequacy of riprap that has a median diameter of $150 \mathrm{~mm}$ and the stone is mostly subangular.

James Kiss
James Kiss
Numerade Educator
02:31

Problem 33

Design a trapezoidal channel to carry $0.2 \mathrm{~m}^3 / \mathrm{s}$ on a longitudinal slope of $0.2 \%$. The channel is to be excavated in coarse alluvium with a 50-percentile diameter of $25 \mathrm{~mm}$, with the grains on the perimeter of the channel moderately rounded.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:21

Problem 34

Gabion baskets are proposed as a lining for a trapezoidal channel with a bottom width of $0.60 \mathrm{~m}$, side slopes of 3:1 (H:V), and longitudinal slope of $9 \%$. The design flow rate is $0.28 \mathrm{~m}^3 / \mathrm{s}$. The gabion baskets have a thickness of $0.23 \mathrm{~m}$ and contain stones with a median size of $150 \mathrm{~mm}$ and specific weight of $25.9 \mathrm{kN} / \mathrm{m}^3$. Is the proposed lining adequate?

Narayan Hari
Narayan Hari
Numerade Educator
02:35

Problem 35

A trapezoidal channel is to be constructed with a composite lining in which the bottom of the channel is made of concrete to accommodate low flows, and the sides of the channel are lined with grass. The grass lining is to be a mixture of sod and bunch grasses, with good coverage and average height of $20 \mathrm{~cm}$. The soil underlying the grass is a clayey sand (SC classification) with a plasticity index of 16 and a void ratio of 0.5 . The bottom width of the channel is $0.90 \mathrm{~m}$, the side slopes are $3: 1(\mathrm{H}: \mathrm{V})$, and the longitudinal slope is $2 \%$. The design flow rate is $0.28 \mathrm{~m}^3 / \mathrm{s}$. Assess the adequacy of the proposed lining.

James Kiss
James Kiss
Numerade Educator
02:35

Problem 36

A median channel is to have a low-flow concrete bottom and grass sides. The bottom width of the channel is $1.2 \mathrm{~m}$, the longitudinal slope of the channel is $1.5 \%$, and the sides of the channel have a slope of $4: 1(\mathrm{H}: \mathrm{V})$. The
sides of the channel are lined with Class B mixed vegetation in good condition underlain by very fine sand with a plasticity index of 12 , a void ratio of 0.3 , and an ASTM classification of ML. For a design flow rate of $0.7 \mathrm{~m}^3 / \mathrm{s}$, assess the adequacy of the channel lining.

James Kiss
James Kiss
Numerade Educator
01:47

Problem 37

Show that the flow area, $A$, and wetted perimeter, $P$, of a channel with a triangular bottom section are given by
$$
\begin{aligned}
A= & \frac{T_b}{2}\left(\frac{T_b}{2 m_1}\right) \\
& +\left[T_b+\left(y-\frac{T_b}{2 m_1}\right) m_2\right]\left(y-\frac{T_b}{2 m_1}\right) \\
P= & 2 \sqrt{1+m_1^2}\left(\frac{T_b}{2 m_1}\right) \\
& +2 \sqrt{1+m_2^2}\left(y-\frac{T_b}{2 m_1}\right)
\end{aligned}
$$
where $T_b$ is the top width of the bottom section, $m_1$ is the side slope of the bottom section, $y$ is the depth of flow, and $m_2$ is the side slope of the channel.

Dominador Tan
Dominador Tan
Numerade Educator