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FHWA > Engineering > Hydraulics > HEC 15 
Design of Roadside Channels with Flexible Linings

(D.1) 
where,
D_{50}  = mean riprap size, m (ft) 
SF  = safety factor 
d  = maximum channel depth, m (ft) 
S  = channel slope, m/m (ft/ft) 
Δ  = function of channel geometry and riprap size 
F_{*}  = Shield's parameter, dimensionless 
SG  = specific gravity of rock (γ_{s}/γ), dimensionless 
The stability of riprap is determined by analyzing the forces acting on individual riprap element and calculating the factor of safety against its movements. The forces acting on a riprap element are its weight (W_{s}), the drag force acting in the direction of flow (F_{d}), and the lift force acting to lift the particle off the bed (F_{L}). Figure D.1 illustrates an individual element and the forces acting on it.
The geometric terms required to completely describe the stability of a riprap element include:
α = angle of the channel bed slope
β = angle between the weight vector and the weight/drag resultant vector in the plane of the side slope
δ = angle between the drag 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
As the element will tend to roll rather than slide, its stability is analyzed by calculating the moments causing the particle to roll about the contact point, c, with an adjacent riprap element as shown in Figure D.1. The equation describing the equilibrium of the particle is:
_{2 }W_{s} cosθ = _{1}W_{s} sinθ cosβ + _{3} F_{d} cosδ + _{4 }F_{L}  (D.2) 
The factor of safety against movement is the ratio of moments resisting motion over the moments causing motion. This yields:
(D.3) 
where,
SF  = Safety Factor 
Figure D.1. Hydraulic Forces Acting on a Riprap Element
Evaluation of the forces and moment arms for Equation D.3 involves several assumptions and a complete derivation is given in Simons and Senturk (1977). The resulting set of equations are used to compute the factor of safety:
(D.4) 
where,
η'  = side slope stability number 
The angles α and θ are determined directly from the channel slope and side slopes, respectively. Angle of repose, φ, may be obtained from Figure 6.1. Side slope stability number is defined as follows:
(D.5) 
where,
η  = stability number 
The stability number is a ratio of side slope shear stress to the riprap permissible shear stress as defined in the following equation:
(D.6) 
where,
τ_{s}  = side slope shear stress = K_{1}τ_{d}, N/m^{2} (lb/ft^{2}) 
F_{* }  = dimensionless critical shear stress (Shields parameter) 
γ_{s}  = specific weight of the stone, N/m^{3} (lb/ft^{3}) 
γ  = specific weight of water, N/m^{3} (lb/ft^{3}) 
D_{50 }  = median diameter of the riprap, m (ft) 
Finally, β is defined by:
(D.7) 
Returning to design Equation D.1, the parameter Δ can be defined by substituting equations D.5 and D.6 into Equation D.4 and solving for D_{50}. It follows that:
(D.8) 
Solving for D_{50} using Equations D.1 and D.8 is iterative because the D_{50} must be known to determine the flow depth and the angle β. These values are then used to solve for D_{50}. As discussed in Chapter 6, the appropriate values for Shields' parameter and Safety Factor are given in Table 6.1