HY-8: POLYNOMIAL COEFFICIENTS

Table 1. Polynomial Coefficients – Circular

EQ #’s: REFERENCE

• 1-9: Calculator Design Series (CDS) 3 for TI-59, FHWA, 1980, page 60
• 1-10: Hydraulic Computer Program (HY) 1, FHWA, 1969, page 18

Table 2. Polynomial Coefficients – Embedded Circular

EQ #’s: REFERENCE

• 1-12: NCHRP 15-24 report

Table 3. Polynomial Coefficients – Box

EQ #’s: REFERENCE

• 1-6: Hydraulic Computer Program (HY) 6, FHWA, 1969, subroutine BEQUA
• 1,4,5: Hydraulic Computer Program (HY) 3, FHWA, 1969, page 16
• 1,3,4,6: Calculator Design Series (CDS) 3 for TI-59, FHWA, 1980, page 16

Table 4. Polynomial Coefficients – Ellipse

EQ #’s: REFERENCE

• 27-30: Calculator Design Series (CDS) 4 for TI-59, FHWA, 1982, page 20
• 31-33: Calculator Design Series (CDS) 4 for TI-59, FHWA, 1982, page 22

Table 5. Polynomial Coefficients – Pipe Arch

EQ #’s: REFERENCE

• 12-23: Calculator Design Series (CDS) 4 for TI-59, FHWA, 1982, page 17
• 24-26: Calculator Design Series (CDS) 4 for TI-59, FHWA, 1982, page 24
• 12,16,20: Hydraulic Computer Program (HY) 2, FHWA, 1969, page 17

Table 6. Polynomial Coefficients – Concrete Open-Bottom Arch

Span:Rise Ratio	Wingwall Angle (Inlet Configuration)	KE	SR	A	BS	C	DIP	EE	F	Diagram/Notes
2:1	0 Degrees (Mitered to Conform to Slope)	0.7	0.0	0.0389106557	0.6044131889	-0.1966160961	0.0425827445	-0.0035136880	0.0001097816	2:1 Coefficients are used if the span:rise ratio is less than or equal to 3:1.
2:1	45 Degrees (45-degree Wingwall)	0.5	0.0	0.0580199163	0.5826504262	-0.1654982156	0.0337114383	-0.0026437555	0.0000796275	2:1 Coefficients are used if the span:rise ratio is less than or equal to 3:1.
2:1	90 Degrees (Square Edge with Headwall)	0.5	0.0	0.0747688320	0.5517030198	-0.1403253664	0.0281511418	-0.0021405250	0.0000632552	2:1 Coefficients are used if the span:rise ratio is less than or equal to 3:1.
4:1	0 Degrees (Mitered to Conform to Slope)	0.7	0.0	0.0557401882	0.4998819105	-0.1249164198	0.0219465031	-0.0015177347	0.0000404218	4:1 coefficients are used if the span:rise ratio is greater than 3:1
4:1	45 Degrees (45-degree Wingwall)	0.5	0.0	0.0465032346	0.5446293346	-0.1571341119	0.0312822438	-0.0024007467	0.0000704011	4:1 coefficients are used if the span:rise ratio is greater than 3:1
4:1	90 Degrees (Square Edge with Headwall)	0.5	0.0	0.0401619369	0.5774418238	-0.1693724912	0.0328323405	-0.0024131276	0.0000668323	4:1 coefficients are used if the span:rise ratio is greater than 3:1

References for Concrete Open-bottom Arch polynomial coefficients:

• Thiele, Elizabeth A. Culvert Hydraulics: Comparison of Current Computer Models. (pp. 121-126), Brigham Young University Master’s Thesis (2007).
• Chase, Don. Hydraulic Characteristics of CON/SPAN Bridge Systems. Submitted Study and Report (1999)

Table 7. Polynomial Coefficients – South Dakota Concrete Box

Description	KE	SR	A	BS	C	DIP	EE	F	Diagram/Notes
Sketch 1: 30 degree-flared wingwalls; top edge beveled at 45 degrees	0.5	0.5	0.0176998563	0.5354484847	-0.1197176702	0.0175902318	-0.0005722076	-0.0000080574
Sketch 2: 30 degree-flared wingwalls; top edge beveled at 45 degrees; 2, 3, and 4 multiple barrels	0.5	0.5	0.0506647261	0.5535393634	-0.1599374238	0.0339859269	-0.0027470036	0.0000851484
Sketch 3: 30 degree-flared wingwalls; top edge beveled at 45 degrees; 2:1 to 4:1 span-to-rise ratio	0.5	0.5	0.0518005829	0.5892384653	-0.1901266252	0.0412149379	-0.0034312198	0.0001083949
Sketch 4: 30 degree-flared wingwalls; top edge beveled at 45 degrees; 15 degrees skewed headwall with multiple barrels	0.5	0.5	0.2212801152	0.6022032341	-0.1672369732	0.0313391792	-0.0024440549	0.0000743575
Sketch 5: 30 degree-flared wingwalls; top edge beveled at 45 degrees; 30 degrees to 45 degrees skewed headwall with multiple barrels	0.5	0.5	0.2431604850	0.5407556631	-0.1267568901	0.0223638322	-0.0016523399	0.0000490932
Sketches 6 & 7: 0 degree-flared wingwalls (extended sides); square-edged at crown and 0 degree-flared wingwalls (extended sides); top edge beveled at 45 degrees; 0- and 6-inch corner fillets	0.5	0.5	0.0493946080	0.7138391179	-0.2354755894	0.0473247331	-0.0036154348	0.0001033337
Sketches 8 & 9: 0 degree-flared wingwalls (extended sides); top edge beveled at 45 degrees; 2, 3, and 4 multiple barrels and 0 degree-flared wingwalls (extended sides); top edge beveled at 45 degrees; 2:1 to 4:1 span-to-rise ratio	0.5	0.5	0.1013668008	0.6600937637	-0.2133066786	0.0437022641	-0.0035224589	0.0001078198
Sketches 10 & 11: 0 degree-flared wingwalls (extended sides); crown rounded at 8-inch radius; 0- and 6-inch corner fillets and 0 degree-flared wingwalls (extended sides); crown rounded at 8-inch radius; 12-inch corner fillets	0.5	0.5	0.0745605288	0.6533033536	-0.1899798824	0.0350021004	-0.0024571627	0.0000642284
Sketch 12: 0 degree-flared wingwalls (extended sides); crown rounded at 8-inch radius; 12-inch corner fillets; 2, 3, and 4 multiple barrels	0.5	0.5	0.1321993533	0.5024365440	-0.1073286526	0.0183092064	-0.0013702887	0.0000423592
Sketch 13: 0 degree-flared wingwalls (extended sides); crown rounded at 8-inch radius; 12-inch corner fillets; 2:1 to 4:1 span-to-rise ratio.	0.5	0.5	0.1212726739	0.6497418331	-0.1859782730	0.0336300433	-0.0024121680	0.0000655665

References for South Dakota Concrete Box polynomial coefficients:

• Thiele, Elizabeth A. Culvert Hydraulics: Comparison of Current Computer Models. (pp. 121-126), Brigham Young University Master’s Thesis (2007).
• Effects of Inlet Geometry on Hydraulic Performance of Box Culverts (FHWA Publication No. FHWA-HRT-06-138, October 2006)

Table 8. User Defined, Open Bottom Arch, Low-Profile Arch, High-Profile Arch, and Metal Box HW/D Values.

Reference for User-defined interpolation coefficients: FHWA HDS-5, Appendix D, Chart 52B

HEC-15 Section 6.1 – MANNING’S ROUGHNESS

Manning’s roughness is a key parameter needed for determining the relationships between depth, velocity, and slope in a channel. However, for gravel and riprap linings, roughness has been shown to be a function of a variety of factors including flow depth, D₅₀, D₈₄, and friction slope, S_f. A partial list of roughness relationships includes Blodgett (1986a), Limerinos (1970), Anderson, et al. (1970), USACE (1994), Bathurst (1985), and Jarrett (1984). For the conditions encountered in roadside and other small channels, the relationships of Blodgett and Bathurst are adopted for this manual.

Blodgett (1986a) proposed a relationship for Manning’s roughness coefficient, n, that is a function of the flow depth and the relative flow depth (d_a/D₅₀) as follows (Equation 6.1):

where,

• n = Manning’s roughness coefficient, dimensionless
• d_a = average flow depth in the channel, m (ft)
• D₅₀ = 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/D₅₀ ≤ 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):

where,

• d_a = average flow depth in the channel, m (ft)
• g = acceleration due to gravity, 9.81 m/s² (32.2 ft/s²)
• 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 semi-empirical relationship applicable for the range of conditions where 0.3<d_a/D₅₀<8.0. The three terms in the denominator represent functions of Froude number, roughness element geometry, and channel geometry given by the following equations:

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:

Equations 6.1 and 6.2 both apply in the overlapping range of 1.5 ≤ d_a/D₅₀ ≤ 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/D₅₀. The Bathurst equation (6.2) is recommended for 0.3<d_a/D₅₀<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.

Computed pipe diameters should be increased to a larger nominal dimension to avoid pressure flow. Standard English and metric nominal sizes used for storm drains are given in the table below.

HEC-15 Section 3.1 & HDM Index 864.2

Shear Stress Equation

The maximum shear stress is given by:

τ = γ⋅H_n⋅S

where:

• τ = shear stress in channel at maximum depth (N/m² or psf)
• γ = unit weight of water
• H_n = depth of flow in channel (ft or m)
• S = channel bottom slope

Per the Caltrans Highway Design Manual Index 309.1(2), the roadside environment can and should be made as safe as practical. A clear recovery zone is an unobstructed, relatively flat (4:1 or flatter) or gently sloping area beyond the edge of the traveled way which affords the drivers of errant vehicles the opportunity to regain control.

The AASHTO Roadside Design Guide provides detailed design guidance for creating a forgiving roadside environment. Channels should be safe for vehicles accidentally leaving the traveled way. The figure below illustrates the preferred geometric cross section for ditches with gradual slope changes in which the front and back slopes are traversable (AASHTO 2002).

This figure (from HDS-4 Section 5.1) is applicable for rounded ditches with bottom widths of 8 ft (2.4 m) or more, and trapezoidal ditches with bottom widths equal to or greater than 4 ft (1.2 m).

HEC-15

Section 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.

Source: HEC-15
Author: RT Kilgore and K Cotton (FHWA)

CHAPTER 6: RIPRAP, COBBLE, AND GRAVEL LINING DESIGN

Riprap, cobble, and gravel linings are considered permanent flexible linings. They may be described as a non-cohesive layer of stone or rock with a characteristic size, which for the purposes of this manual is the D₅₀. The applicable sizes for the guidance in this manual range from 15 mm (0.6 in) gravel up to 550 mm (22 in) riprap. For the purposes of this manual, the boundary between gravel, cobble, and riprap sizes will be defined by the following ranges:

Other differences between gravels, cobbles, and riprap may include gradation and angularity. These issues will be addressed later.

Gravel mulch, although considered permanent, is generally used as supplement to aid in the establishment of vegetation (See Chapter 4). It may be considered for areas where vegetation establishment is difficult, for example, in arid-region climates. For the transition period before the establishment of the vegetation, the stability of gravel mulch should be assessed using the procedures in this chapter.

The procedures in this chapter are applicable to uniform prismatic channels (as would be characteristic of roadside channels) with rock sizes within the range given above. For situations not satisfying these two conditions, the designer is referred to another FHWA circular (No. 11) “Design of Riprap Revetment” (FHWA, 1989).

6.1 MANNING’S ROUGHNESS

Manning’s roughness is a key parameter needed for determining the relationships between depth, velocity, and slope in a channel. However, for gravel and riprap linings, roughness has been shown to be a function of a variety of factors including flow depth, D₅₀, D₈₄, and friction slope, S_f. A partial list of roughness relationships includes Blodgett (1986a), Limerinos (1970), Anderson, et al. (1970), USACE (1994), Bathurst (1985), and Jarrett (1984). For the conditions encountered in roadside and other small channels, the relationships of Blodgett and Bathurst are adopted for this manual.

Blodgett (1986a) proposed a relationship for Manning’s roughness coefficient, n, that is a function of the flow depth and the relative flow depth (d_a/D₅₀) as follows:

where

Equation 6.1 is applicable for the range of conditions where 1.5 ≤ d_a/D₅₀ ≤ 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):

Equation 6.2 is a semi-empirical relationship applicable for the range of conditions where 0.3 < d_a/D₅₀ < 8.0. The three terms in the denominator represent functions of Froude number, roughness element geometry, and channel geometry given by the following equations:

where,

The parameter b describes the relationship between effective roughness concentration and relative submergence of the roughness bed. This relationship is given by:

Equations 6.1 and 6.2 both apply in the overlapping range of 1.5 ≤ d_a/D₅₀ ≤ 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/D₅₀. The Bathurst equation (6.2) is recommended for 0.3 < d_a/D₅₀ <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.

6.2 – PERMISSIBLE SHEAR STRESS

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:

where,

Typically, a specific weight of stone of 25,900 N/m³ (165 lb/ft³) is used, but if the available stone is different from this value, the site-specific value should be used.

Recalling Equation 3.2,

τ_p ≥ SF.τ_d

and Equation 3.1,

τ_d = γ.d.S_o

Equation 6.7 can be written in the form of a sizing equation for D₅₀ as shown below:

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:

where,

• R_e = particle Reynolds number, dimensionless
• V* = shear velocity, m/s (ft/s)
• ν = kinematic viscosity, 1.131×10^-6 m²/s at 15.5 deg C (1.217×10^-5 ft²/s at 60 deg F)

Shear velocity is defined as:

where,

• g = gravitational acceleration, 9.81 m/s² (32.2 ft/s²)
• 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 4×104 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 2×105, 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.

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 site-specific 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:

where,

• Δ = function of channel geometry and riprap size.

The parameter Δ can be defined as follows (see HEC-15 Appendix D for the derivation):

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 of the riprap.

Finally, β is defined by:

where,

• η = stability number.

The stability number is calculated using:

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 (not shown) 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.

Caltrans HDM Table 851.2

Suggested values for Manning’s Roughness coefficient (n) for design purposes are given in the tabe below.

HDS-4 Table B-3

Other: Variation of n with Flow Depth in Pipe

From “Scattergraph’s Principles and Practice, by Kevin L Enfinger, P.E. and James S Schutsbach, ADS Environmental Services, 2003.

A fourth order polynomial approximation of Camp’s varying roughness coefficient:

f(d)=1.04+2.30*(d/D)-6.86*(d/D)²+7.79*(d/D)³-3.27*(d/D)⁴

From http://www.engineeringexceltemplates.com, Manning Equation Partially Filled Circular Pipes:

The Manning equation was developed for flow in open channels with rectangular, trapezoidal, and similar cross-sections. It works very well for those applications using a constant value for the Manning roughness coefficient, n. Better agreement with experimental measurements is obtained for partially full pipe flow, however, by using the variation in Manning roughness coefficient developed by Camp …

The equations to calculate n/_nfull, in terms of (y/D) for y < (D/2) are as follows:>/p>

• n/n_full = 1 + (y/D)*(1/3) for 0 < (y/D) < 0.03
• n/n_full = 1.1 + ((y/D) – 0.03)*(12/7) for 0.03 < y/D < 0.1
• n/n_full = 1.22 + ((y/D) – 0.1)*(0.6) for 0.1 < (y/D) < 0.2
• n/n_full = 1.29 for 0.2 < (y/D) < 0.3
• n/n_full = 1.29 – ((y/D) – 0.3)*(0.2) for 0.3 < (y/D) < 0.5

The equation used for n/n_full for 0.5 < (y/D) < 1 is:

• n/n_full = 1.25 – [((y/D) – 0.5)/2]

Source: Caltran Highway Design Manual, Chapter 870
These supersede the previous Caltrans Riprap Sizes.

Caltrans requires the use of these new RSP gradations starting August 1st, 2016 in new projects and those not yet at 30 percent PS&E stage.

Caltrans Highway Design Manual Table 873.3A

NOTES:

1. Rock grading and quality requirements per Standard Specifications.

2. RSP-fabric Type of geotextile and quality requirements per Section 96 Rock Slope Protection Fabric of the Standard Specifications. For RSP Classes I thru VIII, use Class 8 RSP-fabric which has lower weight per unit area and it also has lower toughness (tensile x elongation, both at break) than Class 10 RSP-fabric. For RSP Classes IX thru XI, use Class 10 RSP-fabric.

3. Intermediate, or B dimension (i.e., width) where A dimension is length, and C dimension is thickness.

Caltrans Highway Design Manual Table 873.3B

RSP Class by Median Particle Weight⁽³⁾

NOTES:

1. Rock grading and quality requirements per Standard Specifications.
2. RSP-fabric Type of geotextile and quality requirements per Section 96 Rock Slope Protection Fabric of the Standard Specifications. For RSP Classes I thru VIII, use Class 8 RSP-fabric which has lower weight per unit area and it also has lower toughness (tensile x elongation, both at break) than Class 10 RSP-fabric. For RSP Classes IX thru XI, use Class 10 RSP-fabric.
3. Values shown are based on Table 873.3A dimensions and an assumed specific gravity of 2.65. Weight will vary based on density of rock available for the project.