304 5.6 Chapter 5. Design of Drainage Channels Scour at Bridges Bridges facilitate transportation from one side of a channel to the other. The presence of a bridge is usually accompanied by channel obstructions, such as bridge piers, and the scour of bed material around such obstructions can be a serious problem that affects the integrity of the bridge structure. Bridge sections sometimes require the contraction of the open channel in the approach section to the bridge, and such channel contractions are usually accompanied by increased flow velocities and associated vortices that increase the shear stresses on the bottom and sides of the channel; these effects can lead to erosion of the bed material. Since noncohesive bed material usually has a lower critical shear stress than cohesive bed material, erosion usually begins with the removal of the noncohesive (coarse) material, leaving finer material behind. The erosion of sediment from around bridge piers and abutments is called bridge scour, which is usually manifested by scour holes around bridge piers and abutments. Such scour effects can lead to bridge failure and collapse, particularly around pier foundations. Scouring around bridges can also cause the channel section to become deeper around the contracted bridge section. Scour at a bridge section will continue to occur until the shear stress on the bed is less than the critical shear stress, or until sediment is being added to scour holes at the same rate at which sediment is being scoured. Types of scour. The two main types of scour are clear-water scour and live-bed scour. Clear-water scour occurs under the condition that the approach flow is clear, which means the bed shear stress in the approach channel is insufficient to erode the upstream bed material. Live-bed scour occurs under the condition that the approach flow contains suspended sediment eroded from upstream of the bridge section. Under clear-water scour conditions, no sediment is being supplied to replace scoured sediment at the bridge section; while under live-bed scour conditions, there is an upstream supply of sediment. Usually, clear-water scour holes are deeper than live-bed scour holes. Due to natural variations in flow rate, both scour conditions can occur at the same bridge section. Practicalities. Guidance on the design, evaluation, and inspection of bridges for scour can be found in the latest editions of the following key documents published by the U.S. Federal Highway Administration: HEC-18, HEC-20, and HEC-23. These documents give particular guidance on procedures for estimating pier scour, design of abutments to minimize scour, estimation of pier scour with debris loading, and the design of counter measures to mitigate scour at bridges. Problems Section 5.2: Basic Principles 5.1. Show that the best hydraulic section for a rectangular-shaped section is one in which the bottom width is equal to twice the flow depth. 5.2. Show that the best hydraulic section for a triangular-shaped section is one in which the top width is equal to twice the flow depth. 5.3. How do the side slopes in the best trapezoidal section compare with the side slopes in the best triangular section? Problems 305 5.4. A highway median channel is to be sized to accommodate a peak flow rate of 2.4 m3 /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. 5.5. A power-law channel section is to be excavated to accommodate 15 m3 /s when flowing full. The channel is on a slope of 0.07%, and Manning’s n of the lining material is estimated as 0.015. If the channel is to have a top width of 1.50 m, what is the equation of the shape of the best hydraulic section? 5.6. A channel with a longitudinal slope of 0.5% is to be excavated in rock, and the channel must be of sufficient size to carry a design flow of 20 m3 /s. Two alternatives have been identified as shown in Figure 5.17. The first alternative is to use a compound section, and the second alternative is to use a trapezoidal section. Determine the required dimensions for (a) the compound section, and (b) the trapezoidal section. Identify the preferred alternative. Alternative 1: Compound section Alternative 2: Trapezoidal section Figure 5.17: Alternative channels 5.7. A trapezoidal channel is to be designed for a flow rate of 20 m3 /s, with a freeboard of 0.40 m, a slope of 0.05%, and side slopes of 3:1 (H:V). The channel is to be lined with a material having a Manning’s n of 0.020. What are the dimensions of the best hydraulic section? 5.8. The existing channel shown in Figure 5.18 is to be reshaped to be the most hydraulically efficient section having the same capacity as the existing channel. Both the existing and reshaped channel must have a freeboard of 0.3 m at the design flow, a Manning’s n of 0.016, and a longitudinal slope of 0.08%. Determine the dimensions of the reshaped channel, and compare the cross-sectional areas of the existing and reshaped channels. 1m 1 0.6 m 3 Figure 5.18: Existing trapezoidal channel 5.9. An existing trapezoidal channel has a bottom width of 3 m, side slopes of 2:1 (H:V), a total depth of 1 m, a longitudinal slope of 0.5%, and a lining material with a Manning’s n of 0.020. The permissible shear stress on the bottom and sides of the channel is 25 Pa. Determine the maximum flow rate for which the lining material will be stable. 306 Chapter 5. Design of Drainage Channels 5.10. Consider a channel with a flexible lining in which the lining material has a permissible shear stress of tp . Use the Manning equation to show that the permissible velocity, vp , in the channel is given by 0.010 vp = n ( R4 y3 ) 16 1 tp2 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. 5.11. The bottom sediments of an open channel consist of noncohesive particles with a median grain size of 2.5 mm and a specific gravity of 2.65. If the temperature of the water is 20◦ C, estimate the permissible shear stress on the bottom sediments. 5.12. Derive Equation 5.46 from Equations 5.42 and 5.45. 5.13. (a) Identify the side-slope condition under which the critical shear stress will be on the side of an unlined trapezoidal channel. (b) If the side slopes of a particular trapezoidal channel are 25◦ , and the angle of repose of the surrounding earth material is 30◦ , will the critical shear stress on the sides be reached before the critical shear stress on the bottom is exceeded? 5.14. A drainage channel is to be designed to convey a peak flow rate of 5 m3 /s on a slope of 0.5%. It is expected that the channel will be lined with stones having a critical Shields parameter of 0.05. Design guidelines indicate that, for stones with a median size of d50 , the angle of repose, a (degrees), and Manning’s n are given by a = 33.09(d50 )0.00778 , 1 n = 0.047(d50 ) 6 where d50 is in meters. Design the most economical trapezoidal channel and lining. 5.15. A trapezoidal channel is to be laid on a slope of 1%, have a bottom width of 4 m, side slopes of 2.5:1 (H:V), and lined with angular stones of median size 250 mm and 85-percentile size 300 mm. The stone lining is derived from an open gradation. The permissible shear stress of the stone lining is estimated to be 200 Pa. Determine the maximum flow depth in the channel for the lining to remain stable. 5.16. A trapezoidal drainage channel is to be designed to accommodate a peak flow rate of 30 m3 /s, and the channel is to be laid on a longitudinal slope of 0.4%. To control erosion within the channel, it is proposed to line the channel with gravel that is characterized by the supplier as being angular with a median size of 10 mm. The local regulatory agency requires that a tractive-force method be used to design the channel. In the tractive-force method, the critical shear stress for the lining is derived from the Shields diagram, and the shape of the channel is such that the critical boundary shear stresses are not exceeded. Design the channel. 5.17. A trapezoidal drainage channel has a lining material with a permissible shear stress of 60 Pa, a Manning’s n of 0.023, a bottom width of 2 m, side slopes of 4:1 (H:V), a longitudinal slope of 0.5%, and a design flow depth of 0.80 m. Determine the maximum radius of curvature that can be used without having to provide additional armoring to the channel. Problems 307 5.18. What type of material can withstand vertical side slopes? What is a typical side slope used in roadside channels? 5.19. Under what conditions would you use a freeboard greater than 30 cm in a drainage channel? 5.20. A trapezoidal channel has a bottom width of 1 m, side slopes of 3:1 (H:V), and a longitudinal slope of 5%. Manning’s n for the channel can be taken as 0.013. Supercritical flow is expected in the channel when it is operational, and the freeboard is to be specified in accordance with Equation 5.53. (a) For what range of flows will a freeboard of 1 m be sufficient? (b) For what range of flows does the supercritical condition occur? 5.21. A trapezoidal channel has been designed with a bottom width of 10 m, side slopes of 2:1 (H: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 m3 /s, and the radius of curvature of the channel is 100 m. Determine the design depth of flow in the channel, and the minimum required freeboard. Section 5.3: Design of Channels with Rigid Linings 5.22. Design the most efficient (straight) lined trapezoidal channel to carry 30 m3 /s on a longitudinal slope of 0.2%. The lining of the channel is to be float-finished concrete. 5.23. You are being asked to design a canal that is to be lined with concrete, and have a trapezoidal cross section. Under design conditions, the canal is to convey a flow of 5 m3 /s. The longitudinal slope of the canal is to be 10%, and Manning’s n of the lining is estimated as 0.013. (a) Design the best hydraulic section, determine the corresponding Froude number, and assess whether this is a desirable flow condition. (b) If it is desired that the Froude number of the flow be equal to 0.8 under design conditions, design the channel dimensions. (c) If the flow contains suspended sediment, and the Shields parameter is taken as 0.1, what is the smallest size of sediment particle that can be expected to accumulate on the bottom of the channel? 5.24. A trapezoidal channel is to be designed to fit within a 15-m wide right-of-way as shown in Figure 5.19. The geologist’s report indicates that there is a layer of solid rock at a depth of 2 m below the ground surface, and she recommends against trying to excavate any of the rock. The ground slope in the flow direction is 0.4%, and the channel is to be lined with concrete so as to minimize leakage from the channel, and maximize the capacity of the channel. Local regulations allow maximum side slopes of Right-of-way 15 m Soil Drainage channel 1 3 Solid rock Figure 5.19: Trapezoidal channel in right-of-way Soil 2m 308 Chapter 5. Design of Drainage Channels 3:1 (H:V). Size a channel that will provide the maximum flow capacity within the available space, and determine the corresponding safe flow capacity of the channel. 5.25. A lined rectangular channel is to be constructed on a slope of 0.042% to handle a design flow rate of 4.5 m3 /s. The lining of the channel is to be unfinished concrete, and the maximum radius of curvature is 150 m. Determine the minimum depth and width of the channel to be excavated. 5.26. A lined straight triangular channel is to be constructed on a slope of 0.032% to handle a design flow rate of 4.4 m3 /s. The lining of the channel is to be smooth asphalt. Determine the dimensions of the most efficient channel. 5.27. A parabolic-shaped channel is to be excavated and lined with mortar. If the channel is to carry 6 m3 /s on a slope of 0.05%, determine the minimum depth of channel to be excavated. Give the equation of the channel sides. Section 5.4: Design of Channels with Flexible Linings 5.28. A trapezoidal channel has a design peak flow rate of 0.42 m3 /s, a bottom width of 0.4 m, side slopes of 3:1, and a longitudinal slope of 0.8%. If an open-graded gravel-mulch lining is to be used, estimate an adequate median stone size for the lining. 5.29. A triangular channel is to be used to provide drainage for a peak flow rate of 0.2 m3 /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/m3 of channel volume. A Spanish firm proposes to use a locally available gravel-mulch lining, with round stones having a median size of 50 mm. The Spanish firm does not guarantee that they will use the best hydraulic section, and will charge $70/m3 of channel volume. Design both channel alternatives, determine the likely best alternative, and estimate the difference in cost per kilometer of roadway. 5.30. A channel is to be excavated in rock to carry a flow of 10 m3 /s on a slope of 0.08%. Proposed design alternatives are as follows: (a) the most efficient compound section; and (b) a channel that is semicircular below the design water elevation in the channel. For each of the two alternative designs: What channel lining would you use? What are the dimensions of the channel that you would provide to the construction contractor? Compare the excavation requirements of the two alternatives. Which alternative would you recommend? 5.31. Evaluate the adequacy of using a sod-grass lining of characteristic height 7.5 cm in good condition in a trapezoidal channel having a bottom width of 0.90 m, side slopes of 3:1 (H:V), and a longitudinal slope of 3%. The design flow rate is 0.5 m3 /s. The channel is underlain by clayey sand (SC classification) with a plasticity index of 16 and a void ratio of 0.5. 5.32. Design a triangular grassed channel to handle a design flow of 1 m3 /s on a longitudinal slope of 0.1%. Regulations for grass-lined channels require a density stiffness coefficient of 106, a cover factor of 0.75, and a grass height of 15 cm. The soil has a 75-percentile grain size of 2 mm, and a plasticity index of 14. 5.33. A trapezoidal channel is to be excavated on a slope of 0.1%, and have a design discharge of 35 m3 /s. Right-of-way constraints require the top width of the channel to be 17 m, and the channel is expected Problems 309 to consist of clean gravel (moderately angular) with a 75-percentile diameter of 3 cm. If freeboard is taken to be zero, determine the required bottom width and side slopes of the channel. Is the required channel practical? 5.34. 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 m3 /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 CH. Design the unlined median channel. Compare the designed channel to the best hydraulic section. 5.35. An existing channel is on a 0.3% slope in moderately angular coarse gravel that has a 75-percentile diameter of 3 cm. The channel is 3 m deep, with a bottom width of 6 m, and side slopes of 2.5:1 (H:V). Estimate the safe discharge capacity of the channel. 5.36. Evaluate the adequacy of a vegetation lining with Class D retardance in a trapezoidal channel having a bottom width of 0.90 m, side slopes of 3:1 (H:V), and a longitudinal slope 3%. The design flow rate is 0.5 m3 /s. The channel is underlain by clayey sand (SC classification) with a plasticity index of 16 and a void ratio of 0.5. 5.37. A large housing development has an existing trapezoidal swale with a bottom width of 5 m, side slopes of 3:1 (H:V), and a total depth of 1 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 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. 5.38. A trapezoidal drainage channel is to be constructed in an arid region that will not support a vegetative lining. Right-of-way constraints limit the top width of the channel to 10 m, serviceability of the channel requires side slopes of 4:1 (H:V), and the longitudinal slope of the channel is required to be 0.05% to match the existing ground slope. After excavation of the channel, it is anticipated that the channel perimeter will consist of bare soil, with a 75-percentile grain size of 5 mm. (a) Design the channel for maximum capacity, and state the capacity of the channel. (b) If persistent rainfall and unwitting irrigation in the area cause Bermuda grass to grow to a height of 20 cm in the channel, determine the corresponding flow capacity of the channel. Assume water at 20◦ C. 5.39. A roadside channel is to be lined with Bermuda grass, and regularly mowed to a height of 4 cm. The channel is to be designed for a peak flow rate of 1.1 m3 /s and a longitudinal slope of 1.4%, and it is to have a triangular cross section with side slopes of 3:1 (H:V). Using the alternative retardance-based approach, determine the depth of flow in the channel and assess the adequacy of the channel lining. 5.40. A trapezoidal drainage channel on a golf course is lined with Kentucky bluegrass, and is expected to convey 10 m3 /s under design conditions. The longitudinal slope of the drainage channel is 0.1%. If the channel has a bottom width of 3 m, side slopes of 2:1 (H:V), and a total depth of 2 m, use the alternative retardance-based approach to assess the adequacy of the existing channel. 5.41. An existing triangular channel has a depth of 2 m, 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 C, and maximum 310 Chapter 5. Design of Drainage Channels permissible shear stress of 98.9 Pa. Use the alternative retardance-based approach to determine the maximum flow rate that can be handled by this channel. 5.42. A planned trapezoidal channel is to have a bottom width of 3 m, side slopes of 3:1 (H:V), a total depth of 2 m, and a longitudinal slope of 0.4%. It is proposed to line the channel with grass that has a retardance classification of C. Under this condition, what is the safe flow capacity of the channel? Local regulations allow you to use the alternative retardance-based design. 5.43. A trapezoidal channel is to be designed to accommodate a flow rate of 4 m3 /s on a longitudinal slope of 0.8%. Excavation constraints limit the flow depth to 1 m, and it is proposed that the lining of the channel be centipede grass with an average height of 15 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 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 retardance-based approach. 5.44. A triangular channel has a grass lining in good condition, and it is estimated that Cs = 106 and Cf = 0.75. The longitudinal slope of the channel is 1%, the side slopes of the channel are 3:1 (H:V), the 75-percentile size of the underlying soil is 2 mm, and the channel is expected to handle a design flow rate of 0.04 m3 /s. Determine the maximum height of grass in the channel for which the channel lining will be adequate. 5.45. A grass-lined drainage channel is to be designed to accommodate a peak flow rate of 2 m3 /s. The longitudinal slope of the channel is to match a ground slope of 1%. Over the course of a typical year, the grass is expected to maintain Class B retardance. (a) Determine whether using the best hydraulic section with a grass lining would work and, if so, design the channel dimensions. You may assume that for Class B retardance, Cs = 81, Cn = 0.418, and the maximum permissible stress is 100 Pa. (b) If there is no feasible best hydraulic section, list the steps and associated equations that you would follow to find a Class B grassed section that will work. [Note: There is no need to do Part (b) if you can find a workable section in Part (a).] 5.46. A site investigation for a new farming development shows that the local soil is classified as loam, has a 75-percentile diameter of 1 mm, a plasticity index of 7, and an angle of repose of 30◦ . A main drainage channel is to be excavated in this soil, and the channel must be able to accommodate a peak flow rate of 1 m3 /s, and the longitudinal slope of the channel is to be 0.1%. (a) Design the channel if no lining is provided. (b) Design the channel if a lining of Class C grass is provided, where the appropriate design parameters are Cn = 0.220 and Cf = 0.90. Compare the channel designs in terms of practicality. 5.47. An existing channel has a depth of 1.8 m, side slopes of 2:1 (H:V), a bottom width of 2 m, and a longitudinal slope of 1%. The channel is excavated in a sandy soil that has a 75-percentile grain size of 1 mm. It is desired to line the channel with a mixed (sod and bunch) grass in good condition. Grasses in retardance classes A–C are being considered for the lining, and the desired grass-lined channel is illustrated in Figure 5.20. Determine the channel capacity for grasses in each of the three retardance classes using both the more-state-of-the-art retardance based design, and the widely used alternative retardance-based design. Compare your results. Based on your results, what retardance-class grass would you specify for a design flow of 2 m3 /s? Problems 311 Figure 5.20: Design of grass-lined channel 5.48. A drainage channel is to be designed to convey a peak flow rate of 5 m3 /s on a longitudinal slope of 0.5%. It is expected that the channel will be lined with grass. Local regulations require that the alternative retardance based design approach be used. Design the most economical channel and the retardance class of the lining. 5.49. A roadside channel is to be lined with Bermuda grass, and regularly mowed to a height of 4 cm. The channel is to be designed for a peak flow rate of 1.1 m3 /s and a longitudinal slope of 1.4%, and it is to have a triangular cross section with side slopes of 3:1 (H:V). (a) Use the maximum-velocity criteria, assess the adequacy of the proposed lining. (b) Compare your assessment to the result obtained in Problem 5.39. 5.50. A temporary RECP lining is being considered for a trapezoidal roadside channel with a bottom width of 0.9 m, side slopes of 3:1 (H: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 m3 /s. Assess the adequacy of a lining for which the shear stress at 12.5 mm soil loss is 100 Pa, and the roughness rating is as follows: Shear stress (Pa) Manning’s n 50 100 200 0.040 0.036 0.033 5.51. A riprap lining is being proposed for a trapezoidal drainage channel having a bottom width of 0.6 m, side slopes of 3:1 (H:V), and a longitudinal slope of 2%. The design flow rate is 1.13 m3 /s. Assess the adequacy of riprap that has a median diameter of 150 mm and the stone is mostly subangular. 5.52. A trapezoidal channel is to be excavated in rock with a total depth of 2 m and a longitudinal slope of 1%. The channel is to be optimally sized for a design flow rate of 5 m3 /s, and is to have a Froude number of 0.4 under design conditions. (a) Determine the required Manning’s n of the lining. (b) If the required Manning’s n is to be attained by lining the channel with “fixed-in-place” riprap (i.e, where the riprap can be considered immovable), what should be the median rock size of the riprap lining? 5.53. Design a trapezoidal channel to carry 0.2 m3 /s on a longitudinal slope of 0.2%. The channel is to be excavated in coarse alluvium with a 50-percentile diameter of 25 mm, with the grains on the perimeter of the channel moderately rounded. 312 Chapter 5. Design of Drainage Channels 5.54. Consideration is being given to developing a drainage system that uses the existing channel shown in Figure 5.21. The riprap lining is angular, and the longitudinal slope of the channel is 2%. What flow rate can be safely handled by the channel? Riprap, d50 5 150 mm 2m 1 3.5 5m Figure 5.21: Existing channel lined with riprap 5.55. Gabion baskets are proposed as a lining for a trapezoidal channel with a bottom width of 0.60 m, side slopes of 3:1 (H:V), and longitudinal slope of 9%. The design flow rate is 0.28 m3 /s. The gabion baskets have a thickness of 0.23 m, and contain stones with a median size of 150 mm and specific weight of 25.9 kN/m3 . Is the proposed lining adequate? Section 5.5: Composite Linings 5.56. 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 an average height of 20 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 m, the side slopes are 3:1 (H:V), and the longitudinal slope is 2%. The design flow rate is 0.28 m3 /s. Assess the adequacy of the proposed lining. 5.57. A median channel is to have a low-flow concrete bottom and grass sides. The bottom width of the channel is 1.2 m, the longitudinal slope of the channel is 1.5%, and the sides of the channel have a slope of 4:1 (H: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 m3 /s, assess the adequacy of the channel lining. 5.58. Show that the flow area, A, and wetted perimeter, P, of a channel with a triangular bottom section are given by ( ) [ ( ) ]( ) Tb Tb Tb Tb A= + Tb + y m2 y 2 2m1 2m1 2m1 ( ) ( ) √ √ Tb Tb 2 2 P = 2 1 + m1 + 2 1 + m2 y 2m1 2m1 (5.121) (5.122) where Tb is the top width of the bottom section, m1 is the side slope of the bottom section, y is the depth of flow, and m2 is the side slope of the upper section.
