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## Design of Roadside Channels with Flexible Linings Hydraulic Engineering Circular Number 15, Third Edition

### Chapter 2: Design Concepts

The design method presented in this circular is based on the concept of maximum permissible tractive force. The method has two parts, computation of the flow conditions for a given design discharge and determination of the degree of erosion protection required. The flow conditions are a function of the channel geometry, design discharge, channel roughness, channel alignment and channel slope. The erosion protection required can be determined by computing the shear stress on the channel lining (and underlying soil, if applicable) at the design discharge and comparing that stress to the permissible value for the type of lining/soil that makes up the channel boundary.

#### 2.1 Open Channel Flow

##### 2.1.1 Type of Flow

For design purposes in roadside channels, hydraulic conditions are usually assumed to be uniform and steady. This means that the energy slope is approximately equal to average ditch slope, and that the flow rate changes gradually over time. This allows the flow conditions to be estimated using a flow resistance equation to determine the so-called normal flow depth. Flow conditions can be either mild (subcritical) or steep (supercritical). Supercritical flow may create surface waves whose height approaches the depth of flow. For very steep channel gradients, the flow may splash and surge in a violent manner and special considerations for freeboard are required.

More technically, open-channel flow can be classified according to three general conditions:

• uniform or non-uniform flow
• subcritical or supercritical flow

In uniform flow, the depth and discharge remain constant along the channel. In steady flow, no change in discharge occurs over time. Most natural flows are unsteady and are described by runoff hydrographs. It can be assumed in most cases that the flow will vary gradually and can be described as steady, uniform flow for short periods of time. Subcritical flow is distinguished from supercritical flow by a dimensionless number called the Froude number (Fr), which is defined as the ratio of inertial forces to gravitational forces in the system. Subcritical flow (Fr < 1.0) is characterized as tranquil and has deeper, slower velocity flow. In a small channel, subcritical flow can be observed when a shallow wave moves in both the upstream and downstream direction. Supercritical flow (Fr > 1.0) is characterized as rapid and has shallow, high velocity flow. At critical and supercritical flow, a shallow wave only moves in the downstream direction.

##### 2.1.2 Normal Flow Depth

The condition of uniform flow in a channel at a known discharge is computed using the Manning's equation combined with the continuity equation:

 (2.1)

where,

 Q = discharge, m3/s (ft3/s) n = Manning's roughness coefficient, dimensionless A = cross-sectional area, m2 (ft2) R = hydraulic radius, m (ft) Sf = friction gradient, which for uniform flow conditions equals the channel bed gradient, So, m/m (ft/ft) α = unit conversion constant, 1.0 (SI), 1.49 (CU)

The depth of uniform flow is solved by rearranging Equation 2.1 to the form given in Equation 2.2. This equation is solved by trial and error by varying the depth of flow until the left side of the equation is zero.

 (2.2)

##### 2.1.3 Resistance to Flow

For rigid channel lining types, Manning's roughness coefficient, n, is approximately constant. However, for very shallow flows the roughness coefficient will increase slightly. (Very shallow is defined where the height of the roughness is about one-tenth of the flow depth or more.)

For a riprap lining, the flow depth in small channels may be only a few times greater than the diameter of the mean riprap size. In this case, use of a constant n value is not acceptable and consideration of the shallow flow depth should be made by using a higher n value.

Tables 2.1 and 2.2 provide typical examples of n values of various lining materials. Table 2.1 summarizes linings for which the n value is dependent on flow depth as well as the specific properties of the material. Values for rolled erosion control products (RECPs) are presented to give a rough estimate of roughness for the three different classes of products. Although there is a wide range of RECPs available, jute net, curled wood mat, and synthetic mat are examples of open-weave textiles, erosion control blankets, and turf reinforcement mats, respectively. Chapter 5 contains more detail on roughness for RECPs.

Table 2.2 presents typical values for the stone linings: riprap, cobbles, and gravels. These are highly depth-dependent for roadside channel applications. More in-depth lining-specific information on roughness is provided in Chapter 6. Roughness guidance for vegetative and gabion mattress linings is in Chapters 4 and 7, respectively.

Table 2.1. Typical Roughness Coefficients for Selected Linings
Manning's n1
Lining Category2 Lining Type Maximum Typical Minimum
Rigid Concrete 0.015 0.013 0.011
Grouted Riprap 0.040 0.030 0.028
Stone Masonry 0.042 0.032 0.030
Soil Cement 0.025 0.022 0.020
Asphalt 0.018 0.016 0.016
Unlined Bare Soil 0.025 0.020 0.016
Rock Cut (smooth, uniform) 0.045 0.035 0.025
RECP Open-weave textile 0.028 0.025 0.022
Erosion control blankets 0.045 0.035 0.028
Turf reinforement mat 0.036 0.030 0.024
1Based on data from Kouwen, et al. (1980), Cox, et al. (1970), McWhorter, et al. (1968) and Thibodeaux (1968).
2Minimum value accounts for grain roughness. Typical and maximum values incorporate varying degrees of form roughness.
Table 2.2. Typical Roughness Coefficients for Riprap, Cobble, and Gravel Linings
Manning's n for Selected Flow Depths1
Lining Category Lining Type 0.15 m (0.5 ft) 0.50 m (1.6 ft) 1.0 m (3.3 ft)
Gravel Mulch D50 = 25 mm (1 in.) 0.040 0.033 0.031
D50 = 50 mm (2 in.) 0.056 0.042 0.038
Cobbles D50 = 0.1 m (0.33 ft) -2 0.055 0.047
Rock Riprap D50 = 0.15 m (0.5 ft) -2 0.069 0.056
D50 = 0.1 m (0.33 ft) -2 -2 0.080
1Based on Equation 6.1 (Blodgett and McConaughy, 1985). Manning's n estimated assuming a trapezoidal channel with 1:3 side slopes and 0.6 m (2 ft) bottom width.
2Shallow relative depth (average depth to D50 ratio less than 1.5) requires use of Equation 6.2 (Bathurst, et al., 1981) and is slope-dependent. See Section 6.1.

#### 2.2 Shear Stress

##### 2.2.1 Equilibrium Concepts

Most highway drainage channels cannot tolerate bank instability and possible lateral migration. Stable channel design concepts focus on evaluating and defining a channel configuration that will perform within acceptable limits of stability. Methods for evaluation and definition of a stable configuration depend on whether the channel boundaries can be viewed as:

• essentially rigid (static)
• movable (dynamic)

In the first case, stability is achieved when the material forming the channel boundary effectively resists the erosive forces of the flow. Under such conditions the channel bed and banks are in static equilibrium, remaining basically unchanged during all stages of flow. Principles of rigid boundary hydraulics can be applied to evaluate this type of system.

In a dynamic system, some change in the channel bed and/or banks is to be expected due to transport of the sediments that comprise the channel boundary. Stability in a dynamic system is attained when the incoming supply of sediment equals the sediment transport rate. This condition, where sediment supply equals sediment transport, is referred to as dynamic equilibrium. Although some detachment and transport of bed and/or bank sediments occurs, this does not preclude attainment of a channel configuration that is basically stable. A dynamic system can be considered stable so long as the net change does not exceed acceptable levels. Because of the need for reliability, static equilibrium conditions and use of linings to achieve a stable condition is usually preferable to using dynamic equilibrium concepts.

Two methods have been developed and are commonly applied to determine if a channel is stable in the sense that the boundaries are basically immobile (static equilibrium): 1) the permissible velocity approach and 2) the permissible tractive force (shear stress) approach. Under the permissible velocity approach the channel is assumed stable if the mean velocity is lower than the maximum permissible velocity. The tractive force (boundary shear stress) approach focuses on stresses developed at the interface between flowing water and materials forming the channel boundary. By Chow's definition, permissible tractive force is the maximum unit tractive force that will not cause serious erosion of channel bed material from a level channel bed (Chow, 1979).

Permissible velocity procedures were first developed around the 1920's. In the 1950's, permissible tractive force procedures became recognized, based on research investigations conducted by the U.S. Bureau of Reclamation. Procedures for design of vegetated channels using the permissible velocity approach were developed by the SCS and have remained in common use.

In spite of the empirical nature of permissible velocity approaches, the methodology has been employed to design numerous stable channels in the United States and throughout the world. However, considering actual physical processes occurring in open-channel flow, a more realistic model of detachment and erosion processes is based on permissible tractive force which is the method recommended in this publication.

##### 2.2.2 Applied Shear Stress

The hydrodynamic force of water flowing in a channel is known as the tractive force. The basis for stable channel design with flexible lining materials is that flow-induced tractive force should not exceed the permissible or critical shear stress of the lining materials. In a uniform flow, the tractive force is equal to the effective component of the drag force acting on the body of water, parallel to the channel bottom (Chow, 1959). The mean boundary shear stress applied to the wetted perimeter is equal to:

 (2.3)

where,

 τo = mean boundary shear stress, N/m2 (lb/ft2) γ = unit weight of water, 9810 N/m3 (62.4 lb/ft3) R = hydraulic radius, m (ft) So = average bottom slope (equal to energy slope for uniform flow), m/m (ft/ft)

Shear stress in channels is not uniformly distributed along the wetted perimeter (USBR, 1951; Olsen and Florey, 1952; Chow, 1959; Anderson, et al., 1970). A typical distribution of shear stress in a prismatic channel is shown in Figure 2.1. The shear stress is zero at the water surface and reaches a maximum on the centerline of the channel. The maximum for the side slopes occurs at about the lower third of the side.

Figure 2.1. Typical Distribution of Shear Stress

The maximum shear stress on a channel bottom, τd, and on the channel side, τs, in a straight channel depends on the channel shape. To simplify the design process, the maximum channel bottom shear stress is taken as:

 (2.4)

where,

 τd = shear stress in channel at maximum depth, N/m2 (lb/ft2) d = maximum depth of flow in the channel for the design discharge, m (ft)

For trapezoidal channels where the ratio of bottom width to flow depth (B/d) is greater than 4, Equation 2.4 provides an appropriate design value for shear stress on a channel bottom. Most roadside channels are characterized by this relatively shallow flow compared to channel width. For trapezoidal channels with a B/d ratio less than 4, Equation 2.4 is conservative. For example, for a B/d ratio of 3, Equation 2.4 overestimates actual bottom shear stress by 3 to 5 percent for side slope values (Z) of 6 to 1.5, respectively. For a B/d ratio of 1, Equation 2.5 overestimates actual bottom shear stress by 24 to 35 percent for the same side slope values of 6 to 1.5, respectively. In general, Equation 2.4 overestimates in cases of relatively narrow channels with steep side slopes.

The relationship between permissible shear stress and permissible velocity for a lining can be found by considering the continuity equation:

 (2.5)

where,

 V = flow velocity, m/s (ft/s) A = area of flow, m2 (ft 2)

By substituting Equation 2.4 and Equation 2.5 into Equation 2.1:

 (2.6)

where,

 Vp = permissible velocity, m/s (ft/s) τp = permissible shear stress, N/m2 (lb/ft2) α = unit conversion constant, 1.0 (SI), 1.49 (CU)

It can be seen from this equation that permissible velocity varies with the hydraulic radius. However, permissible velocity is not extremely sensitive to hydraulic radius since the exponent is only 1/6. Furthermore, n will change with hydraulic conditions causing an additional variation in permissible velocity.

The tractive force method has a couple of advantages compared to the permissible velocity method. First, the failure criteria for a particular lining are represented by a single permissible shear stress value that is applicable over a wide range of channel slopes and channel shapes. Second, shear stresses are easily calculated using Equation 2.4. Equation 2.4 is also useful in judging the field performance of a channel lining, because depth and gradient may be easier to measure in the field than channel velocity. The advantage of the permissible velocity approach is that most designers are familiar with velocity ranges and have a " feel" for acceptable conditions.

##### 2.2.3 Permissible Shear Stress

Flexible linings act to reduce the shear stress on the underlying soil surface. For example, a long-term lining of vegetation in good condition can reduce the shear stress on the soil surface by over 90 percent. Transitional linings act in a similar manner as vegetative linings to reduce shear stress. Performance of these products depends on their properties: thickness, cover density, and stiffness.

The erodibility of the underlying soil therefore is a key factor in the performance of flexible linings. The erodibility of soils is a function of particle size, cohesive strength and soil density. The erodibility of non-cohesive soils (defined as soils with a plasticity index of less than 10) is due mainly to particle size, while fine-grained cohesive soils are controlled mainly by cohesive strength and soil density. For most highway construction, the density of the roadway embankment is controlled by compaction rather than the natural density of the undisturbed ground. However, when the ditch is lined with topsoil, the placed density of the topsoil should be used instead of the density of the compacted embankment soil.

For stone linings, the permissible shear stress, τp, indicates the force required to initiate movement of the stone particles. Prior to movement of stones, the underlying soil is relatively protected. Therefore permissible shear stress is not significantly affected by the erodibility of the underlying soil. However, if the lining moves, the underlying soil will be exposed to the erosive force of the flow.

Table 2.3 provides typical examples of permissible shear stress for selected lining types. Representative values for different soil types are based on the methods found in Chapter 4 while those for gravel mulch and riprap are based on methods found in Chapter 7. Vegetative and RECP lining performance relates to how well they protect the underlying soil from shear stresses so these linings do not have permissible shear stresses independent of soil types. Chapters 4 (vegetation) and 5 (RECPs) describe the methods for analyzing these linings. Permissible shear stress for gabion mattresses depends on rock size and mattress thickness as is described in Section 7.2.

Table 2.3. Typical Permissible Shear Stresses for Bare Soil and Stone Linings
Permissible Shear Stress
Lining Category Lining Type N/m2 lb/ft2
Bare Soil Cohesive (PI=10)1 Clayey sands 1.8-4.5 0.037-0.095
Inorganic silts 1.1-4.0 0.027-0.11
Silty sands 1.1-3.4 0.024-0.072
Bare Soil Cohesive1 (PI≥20) Clayey sands 4.5 0.094
Inorganic silts 4.0 0.083
Silty sands 3.5 0.072
Inorganic clays 6.6 0.14
Bare Soil Non-cohesive2 (PI<10) Finer than coarse sand D75<1.3 mm (0.05 in) 1.0 0.02
Fine gravel D75=7.5 mm (0.3 in) 5.6 0.12
Gravel D75=15 mm (0.6 in) 11 0.024
Gravel Mulch3 Coarse gravel D50=25 mm (1 in) 19 0.4
Very coarse gravel D50=50 mm (2 in) 38 0.8
Rock Riprap3 D50=0.15 m (0.5 ft) 113 2.4
D50=0.30 m (1.0 ft) 227 4.8
1Based on Equation 4.6 assuming a soil void ratio of 0.5 (USDA, 1987).
2Based on Equation 4.5 derived from USDA (1987)
3Based on Equation 6.7 with Shields' parameter equal to 0.047.

#### 2.3 Design Parameters

##### 2.3.1 Design Discharge Frequency

Design flow rates for permanent roadside and median drainage channel linings usually have a 5 or 10-year return period. A lower return period flow is allowable if a transitional lining is to be used, typically the mean annual storm (approximately a 2-year return period, i.e., 50 percent probability of occurrence in a year). Transitional channel linings are often used during the establishment of vegetation. The probability of damage during this relatively short time is low, and if the lining is damaged, repairs are easily made. Design procedures for determining the maximum permissible discharge in a roadway channel are given in Chapter 3.

##### 2.3.2 Channel Cross Section Geometry

Most highway drainage channels are trapezoidal or triangular in shape with rounded corners. For design purposes, a trapezoidal or triangular representation is sufficient. Design of roadside channels should be integrated with the highway geometric and pavement design to insure proper consideration of safety and pavement drainage needs. If available channel linings are found to be inadequate for the selected channel geometry, it may be feasible to widen the channel. Either increasing the bottom width or flattening the side slopes can accomplish this. Widening the channel will reduce the flow depth and lower the shear stress on the channel perimeter. The width of channels is limited however to the ratio of top width to depth less than about 20 (Richardson, Simons and Julien, 1990). Very wide channels have a tendency to form smaller more efficient channels within their banks, which increase shear stress above planned design range.

It has been demonstrated that if a riprap-lined channel has 1:3 or flatter side slopes, there is no need to check the banks for erosion (Anderson, et al., 1970). With side slopes steeper than 1:3, a combination of shear stress against the bank and the weight of the lining may cause erosion on the banks before the channel bottom is disturbed. The design method in this manual includes procedures for checking the adequacy of channels with steep side slopes.

Equations for determining cross-sectional area, wetted perimeter, and top width of channel geometries commonly used for highway drainage channels are given in Appendix B.

##### 2.3.3 Channel Slope

The slope of a roadside channel is usually the same as the roadway profile and so is not a design option. If channel stability conditions are below the required performance and available linings are nearly sufficient, it may be feasible to reduce the channel slope slightly relative to the roadway profile. For channels outside the roadway right-of-way, there can be more grading design options to adjust channel slope where necessary.

Channel slope is one of the major parameters in determining shear stress. For a given design discharge, the shear stress in the channel with a mild or subcritical slope is smaller than a channel with a supercritical slope. Roadside channels with gradients in excess of about two percent will usually flow in a supercritical state.

##### 2.3.4 Freeboard

The freeboard of a channel is the vertical distance from the water surface to the top of the channel at design condition. The importance of this factor depends on the consequence of overflow of the channel bank. At a minimum, the freeboard should be sufficient to prevent waves or fluctuations in water surface from washing over the sides. In a permanent roadway channel, about 0.15 m (0.5 ft) of freeboard should be adequate, and for transitional channels, zero freeboard may be acceptable. Steep gradient channels should have a freeboard height equal to the flow depth. This allows for large variations to occur in flow depth for steep channels caused by waves, splashing and surging. Lining materials should extend to the freeboard elevation.

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### Contact:

Dan Ghere
Resource Center (Matteson)
708-283-3557
dan.ghere@dot.gov

Updated: 04/07/2011

United States Department of Transportation - Federal Highway Administration