HEC-15 Section 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:

τ_{p} = F*.(γ_{s} – γ).D_{50} |
(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 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 D_{50} as shown below:

D_{50} ≥ (SF.d.S_{o})/(F*.(SG – 1)) |
(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:

R_{e} = V*.D_{50}/ν |
(6.9) |

where,

- R
_{e} = particle Reynolds number, dimensionless
- V* = shear velocity, m/s (ft/s)
- ν = kinematic viscosity, 1.131×10
^{-6} m^{2}/s at 15.5 deg C (1.217×10^{-5} ft^{2}/s at 60 deg F)

Shear velocity is defined as:

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 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.

Table 6.1. Selection of Shields’ Parameter and Safety Factor
**Reynolds number** |
**F*** |
**SF** |

≤ 4×10^{4} |
0.047 |
1.0 |

4×10^{4}<R_{e}<2×10^{5} |
Linear interpolation |
Linear interpolation |

≥ 2×10^{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 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:

D_{50} ≥ SF•d•S•Δ/(F*•(SG – 1)) |
(6.11) |

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):

Δ = (K_{1}•(1 + sin(α + β)•tan Φ)/(2•(cosθ•tanΦ – SF•sinθ•cosβ)) |
(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 of the riprap.

Finally, β is defined by:

β = tan^{-1}(cosα/(2•sinθ/(η•tanΦ) + sinα)) |
(6.13) |

where,

The stability number is calculated using:

η = τ_{s}/(F*•(Υ_{s} – Υ)•D_{50}) |
(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 (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.