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Design of Roadside Channels with Flexible Linings
Hydraulic Engineering Circular Number 15, Third Edition

Appendix D: Riprap Stability on a Steep Slope

The design of riprap for steep gradient channels presents special problems. On steep gradients, the riprap size required to stabilize the channel is often of the same order of magnitude as the depth of flow. The riprap elements often protrude from the flow, creating a very complex flow condition.

Laboratory studies and field measurements (Bathurst, 1985) of steep gradient channels have shown that additional factors need to be considered when computing hydraulic conditions and riprap stability. The development of design procedures for this manual has, therefore, been based on equations that are more general in nature and account directly for several additional forces affecting riprap stability. The design equation used for steep slopes in Chapter 6 of this manual is as follows. This appendix provides additional information on the development of the equation.

D sub 50 equals SF times d times S times delta divided by F sub * divided by the quantity (SG minus 1) (D.1)


D50= 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 (γs/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 (Ws), the drag force acting in the direction of flow (Fd), and the lift force acting to lift the particle off the bed (FL). 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:

l2 Ws cosθ = l1Ws sinθ cosβ + l3 Fd cosδ + l4 FL (D.2)

The factor of safety against movement is the ratio of moments resisting motion over the moments causing motion. This yields:

SF equals l sub 2 times W sub s times cosine theta divided by the quantity (l sub 1 times W sub s times sine theta times cosine beta plus l sub 3 times F sub d times cosine delta plus l sub 4 times F sub L (D.3)


SF= Safety Factor

top section shows cross section definition, middle section shows force vectors, lower section shows forces on adjacent riprap elements
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:

SF equals cosine theta times tangent theta divided by the quantity (eta prime times tangent phi plus sine theta times cosine beta) (D.4)


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:

eta prime equals eta times (1 plus sine (alpha plus beta)) divided by 2 (D.5)


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:

eta equals tau sub s divided by F sub * divided by the quantity (gamma sub s minus gamma) divided by D sub 50 (D.6)


τs= side slope shear stress = K1τd, N/m2 (lb/ft2)
F* = dimensionless critical shear stress (Shields parameter)
γs= specific weight of the stone, N/m3 (lb/ft3)
γ= specific weight of water, N/m3 (lb/ft3)
D50 = median diameter of the riprap, m (ft)

Finally, β is defined by:

beta equals the inverse tangent of the quantity (cosine alpha divided by the quantity (2 times sine theta divided by eta divided by tangent phi plus sine alpha)) (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 D50. It follows that:

delta equals K sub 1 times the quantity (1 plus sine alpha plus beta) times tangent phi divided by 2 divided by the quantity (cosine theta times tangent phi minus SF times sine theta times cosine beta) (D.8)

Solving for D50 using Equations D.1 and D.8 is iterative because the D50 must be known to determine the flow depth and the angle β. These values are then used to solve for D50. As discussed in Chapter 6, the appropriate values for Shields' parameter and Safety Factor are given in Table 6.1

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Updated: 09/22/2014

United States Department of Transportation - Federal Highway Administration