Riprap sizing calculator

Using the calculator above

Start by choosing what the stone protects, because each application has its own equation and inputs. The calculator opens on the bend example worked below: bank revetment on the outside of a bend, a channel average velocity of 2.5 m/s, a toe depth of 2.0 m, a 2H:1V bank, angular quarried rock with a specific gravity of 2.65 and a safety factor of 1.0, on a bend of 60 m centreline radius with a 20 m water-surface width upstream. Enter the bank slope as horizontal per 1 vertical; the angle it makes with the horizontal is shown under the field.

The culvert outlet apron opens on the second example: 2.5 m³/s from a 1200 mm pipe with 0.3 m of tailwater and a subcritical outlet. A bridge pier needs the velocity and whether it is a channel average or already local at the pier; an abutment needs the velocity and depth at its toe.

Each result gives the required D50, the smallest FHWA class whose lower D50 bound meets it, the layer or apron thickness, and the class’s D15 to D100 band plotted against the required size. The US customary switch converts every input and result.

One equation per failure mode

Riprap sizing has no single formula. FHWA HEC-23 and HEC-14 give a different relationship for each place stone is used, because a different force moves the stone in each place. Studio’s riprap tool covers seven applications; the free calculator above runs channel revetment, bridge piers, bridge abutments and culvert outlets:

Every result goes through the same last step: the computed D50 is matched to a standard FHWA gradation class and given a minimum layer thickness, so the output is a specification a quarry can supply.

The HEC-23 bank revetment equation

Channel revetment is the application most designs start with. HEC-23 Design Guideline 4 sizes the stone from a Froude-type group at the toe of the bank:

On the outside of a bend the channel-average velocity is raised to the toe velocity with HEC-23 eqn 4.4, $V_{des} = V\,[1.74 - 0.52\log_{10}(R_c/T)]$, where $R_c$ is the centreline bend radius and $T$ the water-surface width upstream. Both the bend factor and $C_V$ reach 1.0 at about $R_c/T = 26$. Below $R_c/T = 2$ the calculator evaluates them at 2 and reports that it did, rather than extrapolating.

Two features of the equation drive most design decisions. The 2.5 exponent means a 20% error in velocity becomes about a 50% error in stone size, so the velocity deserves more care than any other input. And $K_1$ falls to zero as the bank angle approaches the angle of repose. A bank at or steeper than the angle of repose is unstable before any water reaches it, so the calculator refuses it and tells you to flatten the slope. Because $\phi$ itself depends on $D_{50}$, the two are iterated until the size settles.

The equation is dimensionally homogeneous. The constants apply unchanged in SI and in US customary units, so there is nothing to convert.

The other six applications

Bridge piers

HEC-23 eqn 11.1 gives $D_{50} = 0.692\,(KV)^2 / [(S_g - 1)\,2g]$, with $K$ = 1.5 for a round-nose pier and 1.7 for a square-faced pier. $K$ converts a channel-average velocity to the local velocity at the pier face. If your velocity already came from a 2D model at the pier, flag it as local: applying $K$ twice overstates the stone by a factor of 2.3 to 2.9.

Bridge abutments

HEC-23 switches equation at a Froude number of 0.80. Below it, $D_{50}/y = K/(S_g - 1)\,[V^2/(g y)]$ with $K$ = 0.89 (spill-through) or 1.02 (vertical wall). Above it, $D_{50}/y = K\,[V^2/((S_g - 1) g y)]^{0.14}$ with $K$ = 0.61 or 0.69. The two forms do not meet at 0.80. Within 0.1 of the threshold the calculator also reports the size from the other form, so you can choose the larger one deliberately.

Culvert outlet riprap aprons

HEC-14 eqn 10.4 sizes the apron stone:

where $Q$ is the design flow (m³/s), $D$ the barrel diameter (m) and $TW$ the tailwater depth (m). Tailwater sits in the denominator, so a free outfall would demand infinite stone. HEC-14 caps this by evaluating at $TW = 0.4D$ when the actual tailwater is lower.

Open-bottom culverts, spurs and wave attack

These three run in the Studio tool rather than the free calculator. Open-bottom culverts use HEC-23 eqn 18.1 with the envelope coefficient (0.68) by default, or the best-fit curve (0.53) for checking an existing structure. Spurs reuse the revetment equation with the channel velocity multiplied by a spur-nose factor, which you enter from HEC-23 DG-2 (typically 1.3 to 1.5). Wave attack uses the Hudson equation with $K_d$ = 2.2 for breaking waves and 3.5 for non-breaking; the smaller $K_d$ gives the larger stone.

Inputs and where they come from

• Velocity and depth. For a prismatic channel, take normal depth and mean velocity from the Manning’s equation calculator. Where the reach is controlled by a downstream structure, use a backwater profile from the gradually varied flow calculator instead, because normal depth will be wrong.
• Bend geometry. Measure $R_c$ along the channel centreline and $T$ as the water-surface width just upstream of the bend, both at the design flow.
• Bank angle. From the design cross-section. 2H:1V is 26.6°; 1.5H:1V is 33.7°.
• Culvert flow, diameter and tailwater. From the culvert design. The culvert calculator gives the outlet velocity and flags when outlet protection is needed.
• Specific gravity. 2.65 unless the quarry has tested the rock.

Worked example: revetment on an outside bend

A channel carries its design flow at a mean velocity of 2.5 m/s and a toe depth of 2.0 m. The bank is 2H:1V (26.6°), protected with angular quarried rock ($S_g$ = 2.65). The reach is an outside bend with $R_c$ = 60 m and $T$ = 20 m, so $R_c/T$ = 3.0. Safety factor and thickness coefficient are 1.0.

1. Design velocity: $V_{des} = 2.5 \times [1.74 - 0.52 \log_{10} 3.0] = 2.5 \times 1.492 = 3.73$ m/s.
2. $C_V = 1.283 - 0.2 \log_{10} 3.0 = 1.188$; $C_S = 0.30$.
3. After iteration, the angle of repose is 36.5° and $K_1 = 0.660$.
4. $d_{30}$ = 417 mm, so $D_{50} = 1.2 \times 417 = 501$ mm.
5. The smallest class with a lower D50 bound of at least 501 mm is Class VI (508 mm). Class VII is offered as the conservative alternative.
6. Layer thickness = max(1.5 × 610, 1067) = 1067 mm, governed by the Class VI maximum D100.

The same channel on a straight reach needs a D50 of 206 mm (Class II, 457 mm layer). The bend raises the toe velocity by half and, through the 2.5 power, more than doubles the stone.

For a culvert outlet apron: $Q$ = 2.5 m³/s from a 1.2 m pipe with 0.3 m tailwater on a subcritical outlet. The tailwater is below 0.4D, so the equation is evaluated at 0.48 m. The discharge group is 0.506, giving $D_{50}$ = 242 mm and Class III. The apron is 9.6 m long (8D), flaring from 3.6 m to 6.0 m wide, and 711 mm thick.

FHWA riprap gradation table (Class I to X)

The calculator matches each result against the ten FHWA standard classes of HEC-23 Volume 2, Design Guideline 4, Table 4.1, which follows NCHRP Report 568. Each class is named by its nominal D50 and limited by a minimum and maximum D15, D50 and D85 and a maximum D100. Values are the published inch sizes converted at 25.4 mm/in, shown as minimum–maximum in millimetres.

Selection uses the lower D50 bound, the smallest stone the class guarantees. Thickness uses the upper bound, because the layer has to hold the largest stone the class allows. A required D50 above 1016 mm returns no class, with the reason stated, rather than an extrapolated Class XI.

Limitations and when to use something else

• The revetment equation applies to bank revetment in channels with a bed slope below about 2%, the limit the FHWA Hydraulic Toolbox puts on this application. A roadside or drainage channel lined over its whole bed and banks is a lining design, whatever its slope: use the HEC-15 channel lining calculator.
• The calculator sizes stone; it does not compute scour. Pier and abutment riprap must be placed below the computed scour elevation, which needs a separate scour analysis.
• The HEC-14 apron equation is fitted for barrels up to 1.524 m (60 in). Above that the result is flagged as indicative and a designed energy dissipator is the better answer.
• Filter design (HEC-23 DG-16), embankment overtopping and wind-generated design waves are not yet available. Their coefficient tables are not in the tool, and the calculator does not substitute estimates for them.
• Spur-nose multipliers are not defaulted. Read yours from HEC-23 DG-2 for the spur length and orientation.
• Results are only as good as the velocity. Where flow concentrates at a structure or around a bend, a 2D model velocity is worth the effort.

What a free Studio account adds

• All seven applications. Spurs, open-bottom culverts and wave attack join the four above, in the same workspace.
• Checks and derivation. A checklist (bank stability, fitted range, tailwater floor, gradation) and a step-by-step derivation for every result.
• Linked inputs. Velocity and depth can be sent straight in from the Manning’s equation tool (Channel Analysis), and the Culvert Designer sends its outlet velocity across.
• Sample designs. Worked cases to load and adapt, from a bend revetment to a pier.
• Saved and reported. Designs save to a project alongside the hydrology and culvert work they depend on, and export as a Word, Excel or PDF report that carries the derivation and checks for a design review.