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Design of Roadside Channels with Flexible Linings

(6.1) 
where,
n  = Manning's roughness coefficient, dimensionless 
d_{a}  = average flow depth in the channel, m (ft) 
D_{50 }  = median riprap/gravel size, m (ft) 
α  = unit conversion constant, 0.319 (SI) and 0.262 (CU) 
Equation 6.1 is applicable for the range of conditions where 1.5 ≤ d_{a}/D_{50 }≤ 185. For small channel applications, relative flow depth should not exceed the upper end of this range.
Some channels may experience conditions below the lower end of this range where protrusion of individual riprap elements into the flow field significantly changes the roughness relationship. This condition may be experienced on steep channels, but also occurs on moderate slopes. The relationship described by Bathurst (1991) addresses these conditions and can be written as follows (See Appendix D for the original form of the equation):
(6.2) 
where,
d_{a}  = average flow depth in the channel, m (ft) 
g  = acceleration due to gravity, 9.81 m/s^{2} (32.2 ft/s^{2}) 
Fr  = Froude number 
REG  = roughness element geometry 
CG  = channel geometry 
α  = unit conversion constant, 1.0 (SI) and 1.49 (CU) 
Equation 6.2 is a semiempirical relationship applicable for the range of conditions where 0.3<d_{a}/D_{50}<8.0. The three terms in the denominator represent functions of Froude number, roughness element geometry, and channel geometry given by the following equations:
(6.3) 
(6.4) 
(6.5) 
where,
T  = channel top width, m (ft) 
b  = parameter describing the effective roughness concentration 
The parameter b describes the relationship between effective roughness concentration and relative submergence of the roughness bed. This relationship is given by:
(6.6) 
Equations 6.1 and 6.2 both apply in the overlapping range of 1.5 ≤ d_{a}/D_{50 }≤ 8. For consistency and ease of application over the widest range of potential design situations, use of the Blodgett equation (6.1) is recommended when 1.5 ≤ d_{a}/D_{50}. The Bathurst equation (6.2) is recommended for 0.3<d_{a}/D_{50}<1.5.
As a practical problem, both Equations 6.1 and 6.2 require depth to estimate n while n is needed to determine depth setting up an iterative process.
Values for permissible shear stress for riprap and gravel linings are based on research conducted at laboratory facilities and in the field. The values presented here are judged to be conservative and appropriate for design use. Permissible shear stress is given by the following equation:
(6.7) 
where,
τ_{p}  = permissible shear stress, N/m^{2} (lb/ft^{2}) 
F_{*}  = Shield's parameter, dimensionless 
γ_{s}  = specific weight of the stone, N/m^{3} (lb/ft^{3}) 
γ  = specific weight of the water, 9810 N/m^{3} (62.4 lb/ft^{3}) 
D_{50}  = mean riprap size, m (ft) 
Typically, a specific weight of stone of 25,900 N/m^{3} (165 lb/ft^{3}) is used, but if the available stone is different from this value, the sitespecific value should be used.
Recalling Equation 3.2,
and Equation 3.1,
Equation 6.7 can be written in the form of a sizing equation for D_{50} as shown below:
(6.8) 
where,
d  = maximum channel depth, m (ft) 
SG  = specific gravity of rock (γ_{s}/γ), dimensionless 
Changing the inequality sign to an equality gives the minimum stable riprap size for the channel bottom. Additional evaluation for the channel side slope is given in Section 6.3.2.
Equation 6.8 is based on assumptions related to the relative importance of skin friction, form drag, and channel slope. However, skin friction and form drag have been documented to vary resulting in reports of variations in Shield's parameter by different investigators, for example Gessler (1965), Wang and Shen (1985), and Kilgore and Young (1993). This variation is usually linked to particle Reynolds number as defined below:
(6.9) 
where,
R_{e}  = particle Reynolds number, dimensionless 
V_{*}  = shear velocity, m/s (ft/s) 
ν  = kinematic viscosity, 1.131x10^{6} m^{2}/s at 15.5 deg C (1.217x10^{5} ft^{2}/s at 60 deg F) 
Shear velocity is defined as:
(6.10) 
where,
g  = gravitational acceleration, 9.81 m/s^{2} (32.2 ft/s^{2}) 
d  = maximum channel depth, m (ft) 
S  = channel slope, m/m (ft/ft) 
Higher Reynolds number correlates with a higher Shields parameter as is shown in Table 6.1. For many roadside channel applications, Reynolds number is less than 4x10^{4} and a Shields parameter of 0.047 should be used in Equations 6.7 and 6.8. In cases for a Reynolds number greater than 2x10^{5}, for example, with channels on steeper slopes, a Shields parameter of 0.15 should be used. Intermediate values of Shields parameter should be interpolated based on the Reynolds number.
Reynolds Number  F*  SF 

≤4x10^{4}  0.047  1.0 
4x10^{4}<R_{e}<2x10^{5}  linear interpolation  linear interpolation 
≥2x10^{5}  0.15  1.5 
Higher Reynolds numbers are associated with more turbulent flow and a greater likelihood of lining failure with variations of installation quality. Because of these conditions, it is recommended that the Safety Factor be also increased with Reynolds number as shown in Table 6.1. Depending on sitespecific conditions, safety factor may be further increased by the designer, but should not be decreased to values less than those in Table 6.1.
As channel slope increases, the balance of resisting, sliding, and overturning forces is altered slightly. Simons and Senturk (1977) derived a relationship that may be expressed as follows:
(6.11) 
where,
Δ  = function of channel geometry and riprap size 
The parameter Δ can be defined as follows (see Appendix D for the derivation):
(6.12) 
where,
α  = angle of the channel bottom slope 
β  = angle between the weight vector and the weight/drag resultant vector in the plane of the side slope 
θ  = angle of the channel side slope 
φ  = angle of repose for the riprap 
Finally, β is defined by:
(6.13) 
where,
η  = stability number 
The stability number is calculated using:
(6.14) 
Riprap stability on a steep slope depends on forces acting on an individual stone making up the riprap. The primary forces include the average weight of the stones and the lift and drag forces induced by the flow on the stones. On a steep slope, the weight of a stone has a significant component in the direction of flow. Because of this force, a stone within the riprap will tend to move in the flow direction more easily than the same size stone on a milder gradient. As a result, for a given discharge, steep slope channels require larger stones to compensate for larger forces in the flow direction and higher shear stress.
The size of riprap linings increases quickly as discharge and channel gradient increase. Equation 6.11 is recommended when channel slope is greater than 10 percent and provides the riprap size for the channel bottom and sides. Equation 6.8 is recommended for slopes less than 5 percent. For slopes between 5 percent and 10 percent, it is recommended that both methods be applied and the larger size used for design. Values for safety factor and Shields parameter are taken from Table 6.1 for both equations.
In this section a design procedure for riprap and gravel linings is outlined. First, the basic design procedure for selecting the riprap/gravel size for the bottom of a straight channel is given. Subsequent sections provide guidance for sizing material on the channel side slopes and adjusting for channel bends.
The riprap and gravel lining design procedure for the bottom of a straight channel is described in the following steps. It is iterative by necessity because flow depth, roughness, and shear stress are interdependent. The procedure requires the designer to specify a channel shape and slope as a starting point and is outlined in the eightstep process identified below. In this approach, the designer begins with a design discharge and calculates an acceptable D_{50} to line the channel bottom. An alternative analytical framework is to use the maximum discharge approach described in Section 3.6. For the maximum discharge approach, the designer selects the D_{50}, and determines the maximum depth and flow permitted in the channel while maintaining a stable lining. The following steps are recommended for the standard design.
Step 1. Determine channel slope, channel shape, and design discharge.
Step 2. Select a trial (initial) D_{50}, perhaps based on available sizes for the project. (Also, determine specific weight of proposed stone.)
Step 3. Estimate the depth. For the first iteration, select a channel depth, d_{i}. For subsequent iterations, a new depth can be estimated from the following equation or any other appropriate method.
Determine the average flow depth, d_{a} in the channel. d_{a} = A/T
Step 4. Estimate Manning's n and the implied discharge. First, calculate the relative depth ratio, d_{a}/D_{50}. If d_{a}/D_{50} is greater than or equal to 1.5, use Equation 6.1 to calculate Manning's n. If d_{a}/D_{50} is less than 1.5 use Equation 6.2 to calculate Manning's n. Calculate the discharge using Manning's equation.
Step 5. If the calculated discharge is within 5 percent of the design discharge, then proceed to step 6. If not, go back to step 3 and estimate a new flow depth.
Step 6. Calculate the particle Reynolds number using Equation 6.6 and determine the appropriate Shields parameter and Safety Factor values from Table 6.1. If channel slope is less than 5 percent, calculate required D_{50} from Equation 6.8. If channel slope is greater than 10 percent, use Equation 6.11. If channel slope is between 5 and 10 percent, use both Equations 6.8 and 6.11 and take the largest value.
Step 7. If the D_{50} calculated is greater than the trial size in step 2, then the trial size is too small and unacceptable for design. Repeat procedure beginning at step 2 with new trial value of D_{50}. If the D_{50} calculated in step 6 is less than or equal to the previous trial size, then the previous trial size is acceptable. However, if the D_{50} calculated in step 6 is sufficiently smaller than the previous trial size, the designer may elect to repeat the design procedure at step 2 with a smaller, more costeffective, D_{50}.
Design a riprap lining for a trapezoidal channel. Given:
Q = 1.13 m^{3}/s
B = 0.6 m
Z = 3
S_{o }= 0.02 m/m
Step 1. Channel characteristics and design discharge are given above.
Step 2. Available riprap sizes include Class 1: D_{50} = 125 mm, Class 2: D_{50} = 150 mm, Class 3: D_{50} = 250 mm. γ_{s}=25,900 N/m^{3} for all classes. Try Class 1 riprap for initial trial. D_{50}=125 mm
Step 3. Assume an initial trial depth of 0.5 m
Using the geometric properties of a trapezoid, the maximum and average flow depths are found:
A = Bd+Zd^{2} = 0.6(0.5)+3(0.5)^{2} = 1.05 m^{2}
R = A/P =1.05/3.76 = 0.279 m
T = B+2dZ = 0.6+2(0.5)(3) = 3.60 m
d_{a} = A/T = 1.05/3.60 = 0.292 m
Step 4. The relative depth ratio, d_{a}/D_{50} = 0.292/0.125 = 2.3. Therefore, use Equation 6.1 to calculate Manning's n.
Calculate Q using Manning's equation:
Step 5. Since this estimate is more than 5 percent from the design discharge, estimate a new depth in step 3.
Step 3 (2^{nd} iteration). Estimate a new depth estimate:
Using the geometric properties of a trapezoid, the maximum and average flow depths are found:
A = Bd+Zd^{2} = 0.6(0.521)+3(0.521)^{2} = 1.13 m^{2}
R = A/P =1.13/3.90 = 0.289 m
T = B+2dZ = 0.6+2(0.521)(3) = 3.73 m
d_{a} = A/T = 1.13/3.73 = 0.303 m
Step 4. (2^{nd} iteration). The relative depth ratio, d_{a}/D_{50} = 0.302/0.125 = 2.4. Therefore, use Equation 6.1 to calculate Manning's n.
Calculate Q using Manning's equation:
Step 5 (2nd iteration). Since this estimate is within 5 percent of the design discharge, proceed to step 6 with the most recently calculated depth.
Step 6. Need to calculate the shear velocity and Reynolds number to determine Shields' parameter and SF.
Shear velocity,
Reynolds number,
Since R_{e}≤ 4x10^{4}, F_{*}=0.047 and SF = 1.0
Since channel slope is less than 5 percent, use Equation 6.8 to calculate minimum stable D_{50}.
SG = γ_{s}/γ_{w} = 25,900/9810 = 2.64
Step 7. The stable D_{50} is slightly larger than the Class 1 riprap, therefore Class 1 riprap is insufficient. Class 2 should be specified. The suitability of Class 2 should be verified by repeating the design procedure starting at step 2.
Design a riprap lining for a trapezoidal channel. Given:
Q = 40 ft^{3}/s
B = 2.0 ft
Z = 3
S_{o }= 0.02 ft/ft
Step 1. Channel characteristics and design discharge are given above.
Step 2. Available riprap sizes include Class 1: D_{50} = 5 in, Class 2: D_{50} = 6 in, Class 3: D_{50} = 10 in. γ_{s}=165 lb/ft^{3} for all classes. Try Class 1 riprap for initial trial. D_{50}=(5/12)=0.42 ft
Step 3. Assume an initial trial depth of 1.5 ft.
Using the geometric properties of a trapezoid, the maximum and average flow depths are found:
A = Bd+Zd^{2} = 2.0(1.5)+3(1.5)^{2} = 9.75 ft^{2}
R = A/P = 9.75/11.5 = 0.848 ft
T = B+2dZ = 2.0+2(1.5)(3) = 11.0 ft
d_{a} = A/T = 9.75/11.0 = 0.886 ft
Step 4. The relative depth ratio, d_{a}/D_{50} = 0.886/0.42 = 2.1. Therefore, use Equation 6.1 to calculate Manning's n.
Calculate Q using Manning's equation:
Step 5. Since this estimate is more than 5 percent from the design discharge, estimate a new depth in step 3.
Step 3 (2^{nd} iteration). Estimate a new depth estimate:
Using the geometric properties of a trapezoid, the maximum and average flow depths are found:
A = Bd+Zd^{2} = 2.0(1.72)+3(1.72)^{2} = 12.3 ft^{2}
R = A/P = 12.3/12.9 = 0.953 ft
T = B+2dZ = 2.0+2(1.72)(3) = 12.3 ft
d_{a} = A/T = 12.3/12.3 = 1.0 ft
Step 4. (2^{nd} iteration). The relative depth ratio, d_{a}/D_{50} = 1.0/0.42 = 2.4. Therefore, use Equation 6.1 to calculate Manning's n.
Calculate Q using Manning's equation:
Step 5 (2^{nd} (iteration). Since this estimate is within 5 percent of the design discharge, proceed to step 6 with the most recently calculated depth.
Step 6. Need to calculate the shear velocity and Reynolds number to determine Shields' parameter and SF.
Shear velocity,
Reynolds number,
Since R_{e}≤ 4x10^{4}, F_{*}=0.047 and SF = 1.0
Since channel slope is less than 5 percent, use Equation 6.8 to calculate minimum stable D_{50}.
SG = γ_{s}/γ_{w} = 165/62.4 = 2.64
Step 7. The stable D_{50} is slightly larger than the Class 1 riprap, therefore Class 1 riprap is insufficient. Class 2 should be specified. The suitability of Class 2 should be verified by repeating the design procedure starting at step 2.
As was explained in Chapter 3, the shear stress on the channel side is less than the maximum shear stress occurring on the channel bottom as was described in Equation 3.3 repeated below.
However, since gravel and riprap linings are noncohesive, as the angle of the side slopeapproaches the angle of repose of the channel lining, the lining material becomes less stable. The stability of a side slope lining is a function of the channel side slope and the angle of repose of the rock/gravel lining material. This essentially results in a lower permissible shear stress on the side slope than on the channel bottom.
These two counterbalancing effects lead to the following design equation for specifying a stone size for the side slope given the stone size required for a stable channel bottom. The following equation is used if Equation 6.8 is used to size the channel bottom stone. If Equation 6.11 was used, the D_{50} from Equation 6.11 is used for the channel bottom and sides.
(6.15) 
where,
D_{50},_{s}  = D_{50} required for a stable side slope, m (ft) 
D_{50},_{b}  = D_{50} required for a stable channel bottom, m (ft) 
K_{1}  = ratio of channel side to bottom shear stress (see Section 3.2) 
K_{2}  = tractive force ratio (Anderson, et al., 1970) 
K_{2} is a function of the side slope angle and the stone angle of repose and is determined from the following equation.
(6.16) 
where,
θ  = angle of side slope 
φ  = angle of repose 
When the side slope is represented as 1:Z (vertical to horizontal), the angle of side slope is:
(6.17) 
Angle of repose is a function of both the stone size and angularity and may be determined from Figure 6.1.
Channels lined with gravel or riprap on side slopessteeper than 1:3 may become unstable and should be avoided where feasible. If steeper side slopes are required they should be assessed using both Equation 6.11 and Equation 6.8 (in conjunction with Equation 6.15) taking the largest value for design.
Consider the stability of the side slopes for the design example in Section 6.3.1. Recall that a class 1 riprap (D_{50}=0.125 m) was evaluated and found to be unstable. Stable D_{50} was determined to be 0.135 m and a class 2 riprap (D_{50}=0.150 m) was recommended. Assess stability on the side slope of the trapezoidal channel.
Given:
Riprap angle of repose = 38 degrees
Z = 3
Step 1. Calculate K_{1}. From Equation 3.4, K_{1} is a function of Z (for Z between 1.5 and 5)
K_{1} = 0.066Z + 0.67 = 0.066(3)+0.67 = 0.87
Step 2. Calculate K_{2} from Equation 6.16 with θ calculated from Equation 6.17.
Step 3. Calculate the stable D_{50} for the side slope using Equation 6.15.
Since 0.137 m is greater than the Class 1 D_{50} selected the side slope is also unstable. However, like for the channel bottom, Class 2 (D_{50}=0.150 m) would provide a stable side slope.
Consider the stability of the side slopes for the design example in Section 6.3.1. Recall that a class 1 riprap (D_{50}=5 in) was evaluated and found to be unstable. Stable D_{50} was determined to be 5.4 in and a class 2 riprap (D_{50}=6 in) was recommended. Assess stability on the side slope of the trapezoidal channel.
Given:
Riprap angle of repose = 38 degrees
Z = 3
Step 1. Calculate K_{1}. From Equation 3.4, K_{1} is a function of Z (for Z between 1.5 and 5)
K_{1} = 0.066Z + 0.67 = 0.066(3)+0.67 = 0.87
Step 2. Calculate K_{2} from Equation 6.16 with θ calculated from Equation 6.17.
Step 3. Calculate the stable D_{50} for the side slope using Equation 6.15.
Since 5.46 inches is greater than the Class 1 D_{50} selected the side slope is also unstable. However, like for the channel bottom, Class 2 (D_{50}= 6 in) would provide a stable side slope.
Figure 6.1. Angle of Repose of Riprap in Terms of Mean Size and Shape of Stone
Added stresses in bends for riprap and gravel linings are treated as described in section 3.4. No additional considerations are required.
As with all lining types, the ability to deliver the expected channel protection depends on the proper installation of the lining. Additional design considerations for riprap linings include freeboard; proper specification of gradation and thickness; and use of a filter material under the riprap.
Freeboard, as defined in Section 2.3.4, is determined based on the predicted water surface elevation in a channel. For channels on mild slopes the water surface elevation for freeboard considerations may be safely taken as the normal depth elevation. For steep slopes and for slope changes, additional consideration of freeboard is required.
For steep channels, freeboard should equal the mean depth of flow, since wave height will reach approximately twice the mean depth. This freeboard height should be used for both transitional and permanent channel installations. Extent of riprap on a steep gradient channel must be sufficient to protect transitions from mild to steep and from steep to mild sections.
The transition from a steep gradient to a culvert should allow for slight movement of riprap. The top of the riprap layer should be placed flush with the invert of a culvert while the riprap layer thickness should equal three to five times the mean rock diameter at the break between the steep slope and culvert entrance. The transition from a steep gradient channel to a mild gradient channel may require an energy dissipation structure such as a plunge pool. The transition from a mild gradient to a steep gradient should be protected against local scour upstream of the transition for a distance of approximately five times the uniform depth of flow in the downstream channel (Chow, 1959).
Riprap gradation should follow a smooth size distribution curve. Most riprap gradations will fall in the range of D_{100} /D_{50} and D_{50} /D_{20} between 3.0 to 1.5, which is acceptable. The most important criterion is a proper distribution of sizes in the gradation so that interstices formed by larger stones are filled with smaller sizes in an interlocking fashion, preventing the formation of open pockets. These gradation requirements apply regardless of the type of filter design used. More uniformly graded stone may distribute a higher failure threshold because it contains fewer smaller stones, but at the same time will likely exhibit a more sudden failure. Increasing the safety factor is appropriate when there are questions regarding gradation.
In general, riprap constructed with angular stones has the best performance. Round stones are acceptable as riprap provided they are not placed on side slopes steeper than 1:3 (V:H). Flat slablike stones should be avoided since they are easily dislodged by the flow. An approximate guide to stone shape is that neither the breadth nor thickness of a single stone is less than onethird its length. Again, the safety factor should be increased if round stones are used. Permissible shear stress estimates are largely based on testing with angular rock.
The thickness of a riprap lining should equal the diameter of the largest rock size in the gradation. For most gradations, this will mean a thickness from 1.5 to 3.0 times the mean riprap diameter. It is important to note that riprap thickness is measured normal to ground surface slope.
When rock riprap is used, the need for an underlying filter material must be evaluated. The filter material may be either a granular filter blanket or a geotextile fabric.
To determine the need for a filter and, if one is required, to select a gradation for the filter blanket, the following criteria must be met (USACE, 1980). The subscripts "upper" and "lower" refer to the riprap and soil, respectively, when evaluating filter need; the subscripts represent the riprap/filter and filter/soil comparisons when selecting a filter blanket gradation.
(6.18a) 
(6.18b) 
(6.18c) 
In the above relationships, "upper" refers to the overlying material and "lower" refers to the underlying material. The relationships must hold between the filter blanket and base material and between the riprap and filter blanket.
The thickness of the granular filter blanket should approximate the maximum size in the filter gradation. The minimum thickness for a filter blanket should not be less than 150 mm (6 in).
In selecting an engineering filter fabric (geotextile), four properties should be considered (FHWA, 1998):
FHWA (1998) provides detailed design guidance for selecting geotextiles as a riprap filter material. These guidelines should be applied in situations where problematic soil environments exist, severe environmental conditions are expected, and/or for critical installations. Problematic soils include unstable or highly erodible soils such as noncohesive silts; gap graded soils; alternating sand/silt laminated soils; dispersive clays; and/or rock flour. Severe environmental conditions include wave action or high velocity conditions. An installation would be considered critical where loss of life or significant structural damage could be associated with failure.
With the exception of problematic soils or high velocity conditions associated with steep channels and rundowns, geotextile filters for roadside applications may usually be selected based on the apparent opening size (AOS) of the geotextile and the soil type as shown in Table 6.2.
Soil Type  Maximum AOS (mm) 

Non cohesive, less than 15 percent passing the 0.075 mm (US #200) sieve  0.43 
Non cohesive, 15 to 50 percent passing the 0.075 mm (US #200) sieve  0.25 
Non cohesive, more than 50 percent passing the 0.075 mm (US #200) sieve  0.22 
Cohesive, plasticity index greater than 7  0.30 
Determine if a granular filter blanket is required, and if so, find an appropriate gradation. Given:
Riprap Gradation
D_{85} = 400 mm (16 in)
D_{50} = 200 mm (8 in)
D_{15} = 100 mm (4in)
Base Soil Gradation
D_{85} =1.5 mm (0.1 in)
D_{50} = 0.5 mm (0.034 in)
D_{15} = 0.167 mm (0.0066 in)
Only an SI solution is provided.
Check to see if the requirements of Equations 6.18 a, b, and c are met when comparing the riprap (upper) to the underlying soil (lower):
D_{15} _{riprap }/D_{85} _{soil }< 5 substituting 100/1.5 = 67 which is not less than 5
D_{15} _{riprap }/D_{15} _{soil }> 5 substituting 100/0.167 = 600 which is greater than 5, OK
D_{15} _{riprap }/D_{15} _{soil }< 40 substituting 100/0.167 = 600 which is not less than 40
D_{50} _{riprap }/D_{50} _{soil} < 40 substituting 200/0.5 = 400 which is not less than 40
Since three out of the four relationships between riprap and the soil do not meet the recommended dimensional criteria, a filter blanket is required. First, determine the required dimensions of the filter with respect to the base material.
D_{15} _{filter }/D_{85} _{soil }< 5 therefore, D_{15 filter} < 5 x 1.5 mm = 7.5 mm
D_{15} _{filter }/D_{15} _{soil }> 5 therefore, D_{15 filter} > 5 x 0.167 mm = 0.84 mm
D_{15} _{filter }/D_{15} _{soil }< 40 therefore, D_{15 filter} < 40 x 0.167 mm = 6.7 mm
D_{50} _{filter }/D_{50} _{soil} < 40 therefore, D_{50 filter} < 40 x 0.5 mm = 20 mm
Therefore, with respect to the soil, the filter must satisfy:
0.84 mm <D_{15 filter} < 6.7 mm
D_{50 filter} < 20 mm
Determine the required filter dimensions with respect to the riprap,
D_{15 riprap} / D_{85 filter} < 5 therefore D_{85 filter} > 100 mm/5 = 20 mm
D_{15 riprap} / D_{15 filter} > 5 therefore D_{15 filter} < 100 mm/5 = 20 mm
D_{15 riprap} / D_{15 filter} < 40 therefore, D_{15 filter} > 100 mm/40 = 2.5 mm
D_{50 riprap} / D_{50 filter} < 40 therefore D_{50 filter} > 200 mm/40 = 5 mm
With respect to the riprap, the filter must satisfy:
2.5 mm < D_{15 filter} < 20 mm
D_{50 filter} > 5 mm
D_{85 filter} > 20 mm
Combining:
2.5 mm < D_{15 filter} < 6.7 mm
5 mm < D_{50 filter} < 20 mm
D_{85 filter} > 20 mm
A gradation satisfying these requirements is appropriate for this design and is illustrated in Figure 6.2.
Figure 6.2. Gradations of Granular Filter Blanket for Design Example