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FHWA > Engineering > Hydraulics > HEC 14 > Chapter 5 
Hydraulic Design of Energy Dissipators for Culverts and Channels

α  β  θ  

Depth, h_{s}  2.27  0.39  0.06 
Width, W_{s}  6.94  0.53  0.08 
Length, L_{s}  17.10  0.47  0.10 
Volume, V_{s}  127.08  1.24  0.18 
The time of scour is estimated based upon knowledge of peak flow duration. Lacking this knowledge, it is recommended that a time of 30 minutes be used in Equation 5.1. The tests indicate that approximately 2/3 to 3/4 of the maximum scour depth occurs in the first 30 minutes of the flow duration. The exponents for the time parameter in Table 5.1 reflect the relatively flat part of the scourtime relationship (t > 30 minutes) and are not applicable for the first 30 minutes of the scour process.
Installation of a perpendicular headwall at the culvert outlet moves the scour hole downstream (Ruff, et al., 1982). However, the magnitude of the scour geometry remains essentially the same as for the case without the headwall. If the culvert is installed with a headwall, the headwall should extend to a depth equal to the maximum depth of scour.
The scour hole dimensions will vary with the height of the culvert invert above the bed. The scour hole shape becomes deeper, wider, and shorter, as the culvert invert height is increased (Doehring, 1994). The coefficients, C_{h}, are derived from tests where the pipe invert is adjacent to the bed. In order to compensate for an elevated culvert invert, Equation 5.1 can be modified to where C_{h}, expressed in pipe diameters, is a coefficient for adjusting thecompound scour hole geometry. The values of C_{h} are presented in Table 5.2.
H_{d}^{1}  Depth  Width  Length  Volume 

0  1.00  1.00  1.00  1.00 
1  1.22  1.51  0.73  1.28 
2  1.26  1.54  0.73  1.47 
4  1.34  1.66  0.73  1.55 
^{1}H_{d} is the height above bed in pipe diameters. 
The scour hole dimensions will vary with culvert slope. The scour hole becomes deeper, wider, and longer as the slope is increased (Abt, 1985). The coefficients presented are derived from tests where the pipe invert is adjacent to the bed. In order to compensate for a sloped culvert, Equation 5.1 can be adjusted with a coefficient, C_{s}, adjusting for scour hole geometry. The values of C_{s} are shown in Table 5.3.
Slope %  Depth  Width  Length  Volume 

0  1.00  1.00  1.00  1.00 
2  1.03  1.28  1.17  1.30 
5  1.08  1.28  1.17  1.30 
>7  1.12  1.28  1.17  1.30 
Step 1. Determine the magnitude and duration of the peak discharge. Express the discharge in m^{3}/s (ft^{3}/s) and the duration in minutes.
Step 2. Compute the full flow hydraulic radius, R_{c}
Step 3. Compute the culvert invert height above the bed ratio, H_{d}, for slopes > 0%.
Step 4. Determine scour coefficients from Table 5.1 and coefficients for culvert drop height, C_{h}, from Table 5.2 and slope, C_{s}, from Table 5.3.
Step 5. Determine the material standard deviation, σ = (D_{84}/D_{16})^{0.5} from a sieve analysis of a soil sample at the proposed culvert location.
Step 6. Compute the scour hole dimensions using Equation 5.1.
Step 7. Compute the location of maximum scour, L_{m} = 0.4 L_{s}.
Design Example: Estimating Scour Hole Geometry in a Cohesionless Soil (SI)
Determine the scour geometrymaximum depth, width, length and volume of scour. Given:
Solution
Step 1. Determine the magnitude and duration of the peak discharge: Q = 0.764 m^{3}/s and the peak flow duration is estimated to be 30 minutes.
Step 2. Compute the full flow hydraulic radius, R_{c}:
Step 3. Compute the height above bed ratio, H_{d}, for slopes > 0%:
Step 4. The Coefficients of scour obtained from Table 5.1, Table 5.2, and Table 5.3 are:
α  β  θ  C_{s}  C_{h}  

Depth of scour  2.27  0.39  0.06  1.03  1.26 
Width of scour  6.94  0.53  0.08  1.28  1.54 
Length of scour  17.10  0.47  0.10  1.17  0.73 
Volume of scour  127.08  1.24  0.18  1.30  1.47 
Step 5. Determine the material standard deviation. σ = 1.87
Step 6. Compute the scour hole dimensions using Equation 5.1:
Similarly,
Step 7. Compute the location of maximum scour. L_{m} = 0.4 L_{s} = 0.4 (7.06) = 2.82 m downstream of the culvert outlet.
Design Example: Estimating Scour Hole Geometry in a Cohesionless Soil (CU)
Determine the scour geometrymaximum depth, width, length and volume of scour. Given:
Solution
Step 1. Determine the magnitude and duration of the peak discharge: Q = 27 ft^{3}/s and the peak flow duration is estimated to be 30 minutes.
Step 2. Compute the full flow hydraulic radius, Rc:
Step 3. Compute the height above bed ratio, H_{d}, for slopes > 0%
Step 4. The coefficients of scour obtained from Table 5.1, Table 5.2, and Table 5.3 are:
α  β  θ  C_{s}  C_{h}  

Depth of scour  2.27  0.39  0.06  1.03  1.26 
Width of scour  6.94  0.53  0.08  1.28  1.54 
Length of scour  17.10  0.47  0.10  1.17  0.73 
Volume of scour  127.08  1.24  0.18  1.30  1.47 
Step 5. Determine the material standard deviation. σ = 1.87
Step 6. Compute the scour hole dimensions using Equation 5.1:
Similarly,
Step 7. Compute the location of maximum scour. L_{m} = 0.4 L_{s} = 0.4 (23.2) = 9.2 ft downstream of the culvert outlet.
If the soil is cohesive in nature, Equation 5.2 should be used to determine the scour hole dimensions. Shear number expressions, which relate scour to the critical shear stress of the soil, were derived to have a wider range of applicability for cohesive soils besides the one specific sandy clay that was tested. The sandy clay tested had 58 percent sand, 27 percent clay, 15 percent silt, and 1 percent organic matter; had a mean grain size of 0.15 mm (0.0059 in); and had a plasticity index, PI, of 15. The shear number expressions for circular culverts are:
(5.2)and for other shaped culverts:
(5.3)where,
D = culvert diameter, m (ft)
y_{e} = equivalent depth (A/2)^{1/2}, m (ft)
A = crosssectional area of flow, m^{2} (ft^{2})
V = mean outlet velocity, m/s (ft/s)
τ_{c} = critical tractive shear stress, N/m^{2} (lb/ft^{2})
ρ = fluid density of water, 1000 kg/m^{3} (1.94 slugs/ft^{3})
(ρV^{2})/τ_{c} is the modified shear number
α_{e} = α_{e} = α/0.63 for h_{s}, W_{s,} and L_{s} and α_{e} = α/(0.63)^{3} for V_{s}
α, β, θ, and α_{e} are coefficients found in Table 5.4
Use 30 minutes for t in Equation 5.2 and Equation 5.3 if it is not known.
The critical tractive shear stress is defined in Equation 5.4 (Dunn, 1959; Abt et al., 1996). Equations 5.2 and 5.3 should be limited to sandy clay soils with a plasticity index of 5 to 16.
(5.4)τ_{c} = 0.001 (S_{ν}+ α_{u}) tan (30 + 1.73 PI)
where,
τ_{c} = critical tractive shear stress, N/m^{2} (lb/ft^{2})
S_{ν} = the saturated shear strength, N/m^{2} (lb/ft^{2})
α_{u} = unit conversion constant, 8630 N/m^{2} (SI), 180 lb/ft^{2} (CU)
PI = Plasticity Index from the Atterberg limits
α  β  θ  α_{e}  

Depth, h_{s}  0.86  0.18  0.10  1.37 
Width, W_{s}  3.55  0.17  0.07  5.63 
Length, L_{s}  2.82  0.33  0.09  4.48 
Volume, V_{s}  0.62  0.93  0.23  2.48 
The design procedure for estimating scour in cohesive materials with PI from 5 to 16 may be summarized as follows.
Step 1. Determine the magnitude and duration of the peak discharge, Q. Express the discharge in m^{3}/s (ft^{3}/s) and the duration in minutes.
Step 2. Compute the culvert average outlet velocity, V.
Step 3. Obtain a soil sample at the proposed culvert location.
Step 4. Compute the critical tractive shear strength, τ_{c}, from Equation 5.4.
Step 5. Compute the modified shear number, S_{nm}, at the peak discharge and height above bed ratio, H_{d}, for slopes > 0%.
and 
Step 6. Determine scour coefficients from Table 5.4 and, if appropriate, coefficients for culvert drop height, C_{h}, from Table 5.2 and slope, C_{s}, from Table 5.3.
Step 7. Compute the scour hole dimensions using Equation 5.2 for circular culverts and Equation 5.3 for other shapes.
Step 8. Compute the location of maximum scour. L_{m} = 0.4 L_{s}.
Design Example: Estimating Scour Hole Geometry in a Cohesive Soil (SI)
Determine the scour geometry: maximum depth, width, length and volume of scour. Given:
Solution
Step 1. Determine the magnitude and duration of the peak discharge: Q = 1.133 m^{3}/s and the peak flow duration is estimated to be 30 minutes.
Step 2. Compute the culvert average outlet velocity, V:
Step 3. Obtain a soil sample at the proposed culvert location: The sandyclay soil was tested and found to have:
Step 4. Compute the critical tractive shear strength, τ_{c}, from Equation 5.4.
τ_{c} = 0.001 (S_{ν}+ α_{u}) tan (30 + 1.73 PI)
τ_{c} = 0.001 (23970 + 8630) tan [30 + 1.73(12)]
τ_{c} = 0.001 (32600) tan (50.76) = 39.9 N/m^{2}
Step 5. Compute the modified shear number, S_{nm}, at the peak discharge and height above bed ratio, H_{d}, for slopes > 0%.
Step 6. Determine scour coefficients from Table 5.4 and coefficients for culvert drop height, C_{h}, from Table 5.2 and slope from Table 5.3: C_{h} = 1 and C_{s} = 1
α  β  θ  

Depth, h_{s}  0.86  0.18  0.10 
Width, W_{s}  3.55  0.17  0.07 
Length, L_{s}  2.82  0.33  0.09 
Volume, V_{s}  0.62  0.93  0.23 
Step 7. Compute the scour hole dimensions using Equation 5.2 for circular culverts:
h_{s} = (1.0)(1.0)(0.86)(377.3)^{0.18}(0.09)^{0.10}(0.61) = 1.2 m
Similarly,
W_{s} = (1.0)(1.0)(3.55)(377.3)^{0.17}(0.09)^{0.07}(0.61) = 5.02 m
L_{s} = (1.0)(1.0)(2.82)(377.3)^{0.33}(0.09)^{0.09}(0.61) = 9.81 m
V_{s} = (1.0)(1.0)(0.62)(377.3)^{0.93}(0.09)^{0.23}(0.61)^{3 }= 20.15 m^{3}
Step 8. Compute the location of maximum scour. L_{m} = 0.4 L_{s} = 0.4(9.81) = 3.92 m downstream of culvert outlet.
Design Example: Estimating Scour Hole Geometry in a Cohesive Soil (CU)
Determine the scour geometry: maximum depth, width, length and volume of scour. Given:
Solution
Step 1. Determine the magnitude and duration of the peak discharge: Q = 40 ft^{3}/s and the peak flow duration is estimated to be 30 minutes.
Step 2. Compute the culvert average outlet velocity, V:
Step 3. Obtain a soil sample at the proposed culvert location: The sandyclay soil was tested and found to have:
Step 4. Compute the critical tractive shear strength, τ_{c}, from Equation 5.4.
τ_{c} = 0.001 (S_{ν}+ α_{u}) tan (30 + 1.73 PI)
τ_{c} = 0.001 (500 + 180) tan [30 + 1.73(12)]
τ_{c} = 0.001 (680) tan (50.76) = 0.83 lb/ft^{2}
Step 5. Compute the modified shear number, S_{nm}, at the peak discharge and height above bed ratio, H_{d}, for slopes > 0%.
Step 6. Determine scour coefficients from Table 5.4 and coefficients for culvert drop height, C_{h}, from Table 5.2 and slope from Table 5.3: C_{h} = 1 and C_{s} = 1
α  β  θ  

Depth, h_{s}  0.86  0.18  0.10 
Width, W_{s}  3.55  0.17  0.07 
Length, L_{s}  2.82  0.33  0.09 
Volume, V_{s}  0.62  0.93  0.23 
Step 7. Compute the scour hole dimensions using Equation 5.2 for circular culverts:
h_{s} = (1.0)(1.0)(0.86)(379.4)^{0.18}(0.09)^{0.10}(2) = 3.9 ft
Similarly,
W_{s} = (1.0)(1.0)(3.55)(379.4)^{0.17}(0.09)^{0.07}(2) = 16.5 ft
L_{s} = (1.0)(1.0)(2.82)(379.4)^{0.33}(0.09)^{0.09}(2) = 32.2 ft
V_{s} = (1.0)(1.0)(0.62)(379.4)^{0.93}(0.09)^{0.23}(2)^{3} = 713.7 ft^{3}
Step 8. Compute the location of maximum scour. L_{m} = 0.4 L_{s} = 0.4(32.2) = 12.9 ft downstream of culvert outlet.
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