Hydraulic Design of Energy Dissipators for Culverts and Channels Hydraulic Engineering Circular Number 14, Third Edition
Appendix D: Riprap Apron Sizing Equations
A variety of relationships for sizing riprap aprons have been developed. Six are summarized and compared in this appendix. The first is from the Urban Drainage and Flood Control District in Denver Colorado (UD&FCD, 2004). These equations consider tailwater in addition to a measure of flow intensity.
(D.1a)
(D.1b)
where,
D_{50} = riprap size, m (ft)
Q = design discharge, m^{3}/s (ft^{3}/s)
D = culvert diameter (circular) or culvert rise (rectangular), m (ft)
B = culvert span (rectangular), m (ft)
TW = tailwater depth, m (ft)
α = unit conversion constant, 1.811 (SI) and 1.0 (CU)
An equation in Berry (1948) and Peterka (1978) has been used for apron riprap sizing. It is only based on velocity.
(D.2)
where,
V = culvert exit velocity, m/s (ft/s)
α = unit conversion constant, 0.0413 (SI) and 0.0126 (CU)
A relationship used in the previous edition of HEC 14 from Searcy (1967) and also found in HEC 11 (Brown and Clyde, 1989) for sizing riprap protection for piers is based on velocity.
(D.3)
where,
S = riprap specific gravity
Bohan (1970) developed two relationships based on laboratory testing that considered, among other factors, whether the culvert was subjected to "minimum" tailwater (TW/D < 0.5) or "maximum" tailwater (TW/D > 0.5). The equations for minimum and maximum tailwater, respectively, are as follows:
(D.4a)
(D.4b)
where,
Fr_{o} = Froude number at the outlet defined as V_{o}/(gD)^{0.5}
Fletcher and Grace (1972) used the laboratory data from Bohan and other sources to develop a similar equation to Equation D.1.
(D.5)
where,
α = unit conversion constant, 0.55 (SI) and 1.0 (CU)
Finally, the USDA/SCS has a series of charts for sizing riprap for aprons. These charts appear to be based on Bohan (Equation D.4a and D.4b).
Equation D.2 (Berry) and Equation D.3 (Searcy) are similar in their exclusive reliance on velocity as the predictor variable and differ only in terms of their coefficient. Equation D.1 (UD&FCD), Equation D.4 (Bohan), and Equation D.5 (Fletcher and Grace) incorporate some sort of flow intensity parameter, i.e. relative discharge or Froude number, as well as relative tailwater depth. (Bohan incorporates tailwater by having separate minimum and maximum tailwater equations.) UD&FCD and Fletcher and Grace have identical forms but differ in their coefficient and exponents.
These equations and the USDA charts were compared based on a series of hypothetical situations. A total of 10 scenarios were run with HY8 to generate outlet velocity conditions for testing the equations. The 10 scenarios included the following variations:
 Two culvert sizes, 760 and 1200 mm (30 to 48 in) metal pipe culverts
 Discharges ranging from (1.1 to 4.2 m^{3}/s) (40 to 150 ft^{3}/s)
 Slope and tailwater changes resulting in 5 inlet control and 5 outlet control cases
Figures D.1, D.2, and D.3 compare the recommended riprap size, D_{50}, relative to the outlet velocity, V, discharge intensity, Q/D^{2.5}, and relative tailwater depth, TW/D. The recommended D50 varies widely, but it is clear that the Berry equation (Equation D.2) results in the highest values for the range of conditions evaluated.
Equations D.2 and D.3 are not recommended because they do not consider tailwater effects. Equation D.4 is not further considered because it treats tailwater only as two separate conditions, minimum and maximum. Equations D.1 and D.5 are similar in their approach and are based on laboratory data. Both would probably both generate reasonable designs. For the ten hypothetical cases evaluated Equation D.1 produced the higher recommendation 3 times and the lower recommendation 7 times. Therefore, Equation D.5 is included in Chapter 10 of this manual.
Figure D.1. D_{50} versus Outlet Velocity
Figure D.2. D_{50} versus Discharge Intensity
Figure D.3. D_{50} versus Relative Tailwater Depth
