What is permissible shear stress?

Permissible shear stress is the largest boundary shear a lining can take before it starts to erode or its stones start to move. It is a property of the lining and the soil under it, not of the flow. HEC-15 design compares it with the applied shear of the design flow: the lining is adequate when permissible shear is at least the applied shear times a safety factor.

How do you calculate shear stress in a channel?

For uniform flow, HEC-15 takes the maximum shear on the bed as the unit weight of water times the maximum flow depth times the bed slope. A channel 0.922 m deep on a 6% slope carries 9810 × 0.922 × 0.06, about 543 N/m² (Pa), on its bed. The banks see a fraction of that, and the outside of a bend sees more, by the bend factor Kb of up to 2.0.

What rock size do I need for a riprap channel lining?

Rearrange the Shields relation: D50 must be at least the safety factor times the applied shear, divided by F* times the difference between the unit weights of rock and water. F* and the safety factor rise from 0.047 and 1.0 to 0.15 and 1.5 as the particle Reynolds number increases. Because larger stone also raises Manning's n and the flow depth, check the depth again once you pick a size.

Is there an HEC-15 spreadsheet or Excel version?

Many offices keep their own HEC-15 spreadsheets. The Channel Lining tool in Delta Hydro Studio does the same checks and exports a Word, Excel or PDF report with the inputs, applied and permissible shear, safety margin and verdict, so the Excel output can stand in for the spreadsheet and the report goes to the reviewer as it is.

Should I use HEC-15 or HEC-23 to size riprap?

They are for different structures. HEC-15 designs a flexible lining over the whole bed and banks of a roadside or other small prismatic drainage channel, sizing rock from shear stress; HEC-15 section 1.1 covers prismatic channels typically up to 1.5 m deep and refers non-prismatic channels to HEC-11. HEC-23 Design Guideline 4 sizes riprap revetment for streambanks, embankments and beds from the near-bank velocity, and its other guidelines cover bridge piers, abutments and spurs. The FHWA Hydraulic Toolbox keeps them as separate Channel Lining Design and Riprap calculators. Use this tool for a lined roadside channel and the riprap sizing calculator for revetment and scour protection.

HEC-15 Channel Lining Calculator: Permissible Shear Stress

HEC-15 shear check
Bed, τ_d = γdS	543Pa
Side slopes, τ_s = K_sτ_dKs = 0.76	413Pa
Permissible shear of the lining, τ_pF* = 0.15 at Re* = 2.28 × 10⁵ (HEC-15 Table 6.1)	845Pa
Safety factor used, SFRaised to the HEC-15 Table 6.1 value for this Re*	1.5
Allowable shear, τ_p/SF	563Pa
Margin, allowable ÷ largest appliedLargest applied shear is on the bed	1.04

Uniform flow with the lining's roughness
Manning's n of the liningBlodgett, HEC-15 Eq 6.1 (da/D50 = 1.91)	0.0802
Normal depth, d	0.922m
Mean velocity, V	2.24m/s
Top width, T	6.69m
Froude number	0.87
Angle of repose of the stoneBanks are 26.6°	34.3°
Recommended freeboard	0.718m

Using the calculator above

The calculator opens on the rock-lined chute worked further down this page: a 3.0 m bed, 2H:1V side slopes, a 6% slope, 10 m³/s and 350 mm angular rock at 25.9 kN/m³.

Channel. A trapezoid: bottom width, side slope as horizontal per 1 vertical on both banks, bed slope as a ratio (0.06 is 6%) and the design flow.
Lining. Rock riprap (D50, unit weight, stone shape), a gabion mattress (thickness, fill D50), grass (stand condition, growth form, stem height and the soil beneath) or a rolled erosion control product (type, Manning’s n, shear at 12 mm soil loss, soil). The safety factor is a minimum: rock and gabions use HEC-15’s own value when it is higher.
Bend. The centreline radius, or blank for a straight reach.

The calculator solves the normal depth with the lining’s own Manning’s n, so depth and roughness agree, then checks the bed, side-slope and bend shear against the allowable shear. It warns when the inputs leave the method’s range: rock on slopes over 5%, where HEC-15 also applies its steep-slope relation; banks too steep for the stone; gabions outside Eqs 7.1 and 7.2; and channels deeper than 1.5 m.

How the HEC-15 tractive-force check works

A lining is adequate when the shear stress it can resist, divided by a safety factor, is at least the shear stress the flow applies:

\frac{\tau_p}{SF} \ge \tau_d \qquad\qquad \tau_d = \gamma\, d\, S_0

The applied shear $\tau_d$ is the HEC-15 maximum shear on the bed for uniform flow. HEC-15 uses the maximum depth rather than the hydraulic radius, which is the conservative choice for a narrow channel. The calculator checks three locations and compares the largest applied shear with the lining’s permissible shear:

Location	Applied shear	Notes
Bed	$\tau_d = \gamma d S_0$	Maximum depth on the centreline
Side slopes	$\tau_s = K_s\,\tau_d$	$K_s$ = 0.70 to 0.77, rising with the bed-width-to-depth ratio, after Lane’s tractive-force work
Outside of a bend	$\tau_b = K_b\,\tau_d$	$K_b$ from the bend radius and top width, table below

Symbol	Meaning	Units
$\tau_d$ , $\tau_s$ , $\tau_b$	Applied shear on the bed, the side slope and the outside of a bend	N/m² (Pa)
$\tau_p$	Permissible shear of the lining	N/m²
$\gamma$	Unit weight of water, 9810	N/m³
$d$	Maximum flow depth	m
$S_0$	Bed slope	m/m
$SF$	Safety factor	–

The depth comes from a uniform-flow calculation of the channel at the design flow. The calculator above solves it with the lining’s own Manning’s n. For another section shape, run it through the Manning’s equation calculator first, or import a saved Channel Analysis run into the Studio tool.

Permissible shear for rock riprap

Rock lining follows the Shields stability criterion:

\tau_p = F^{*}\,(\gamma_s - \gamma_w)\,D_{50} \qquad\qquad Re^{*} = \frac{V^{*} D_{50}}{\nu}, \quad V^{*} = \sqrt{g\, d\, S_0}

$F^{*}$ is the Shields parameter, $\gamma_s$ and $\gamma_w$ the unit weights of rock and water, $D_{50}$ the median stone size and $\nu$ the kinematic viscosity of water ( $1.131 \times 10^{-6}$ m²/s, HEC-15’s value at 15.5 °C). The shear velocity $V^{*}$ comes from the depth and slope, so the mean velocity does not enter. HEC-15 ties both $F^{*}$ and the safety factor to the particle Reynolds number $Re^{*}$ :

Particle Reynolds number $Re^{*}$	Shields parameter $F^{*}$	Safety factor $SF$
$\le 4 \times 10^{4}$	0.047	1.0
$4 \times 10^{4}$ to $2 \times 10^{5}$	Interpolated linearly	Interpolated linearly
$\ge 2 \times 10^{5}$	0.15	1.5

The calculator uses the larger of this safety factor and any you enter. It also reports the angle of repose for the stone size and shape you choose (crushed, angular or round), so you can confirm the side slopes are flatter than the rock will stand.

The four lining families

Lining	What sets its permissible shear	Inputs	Where it fits
Rock riprap	Shields relation above	$D_{50}$ , rock unit weight, stone shape	Steep or high-shear channels, culvert outlet channels
Vegetation	The soil’s own permissible shear, raised by how completely the grass covers it	Soil type ( $D_{75}$ for sands and gravels, plasticity index for clays), stand condition, growth form, stem height	Low-gradient roadside and residential channels with maintenance
Rolled erosion control product (RECP)	The product’s tested shear rating combined with the soil underneath	Open-weave textile, erosion control blanket or turf reinforcement mat, or a tested value	Protection while grass establishes; reinforced mats as permanent lining
Gabion mattress	The larger of a stone-fill relation ( $F^{*}$ = 0.10 on the fill $D_{50}$ ) and a mattress-thickness relation (HEC-15 Eqs 7.1 and 7.2), with a safety factor of at least 1.25	Thickness, fill $D_{50}$ , rock unit weight	Steep channels where loose stone would move, or where only smaller stone is available

Vegetation gives no protection until it is established, usually six to twelve months after seeding. Design for the bare soil, or cover the establishment period with an RECP.

Worked example: a rock-lined chute

A trapezoidal channel with a 3.0 m bed and 2H:1V side slopes runs at 6% and must carry 10 m³/s. The lining is angular rock with a unit weight of 25 900 N/m³, and the trial size is $D_{50}$ = 350 mm.

Large stone in shallow flow is rough. The Blodgett relative-roughness equation used in HEC-15 gives n = 0.080 for 350 mm stone at this depth, and the uniform-flow solution is:

Hydraulics at 10 m³/s	Value
Maximum depth $d$	0.922 m
Velocity	2.24 m/s
Top width	6.69 m
Froude number	0.87

Applied shear:

Bed: $\tau_d$ = 9810 × 0.922 × 0.06 = 543 Pa.
Side slopes: $K_s$ = 0.76 for a bed-width-to-depth ratio of 3.3, so $\tau_s$ = 413 Pa.

Permissible shear:

$V^{*} = \sqrt{9.81 \times 0.922 \times 0.06}$ = 0.737 m/s, so $Re^{*}$ = 0.737 × 0.35 / 1.131 × 10⁻⁶ = 2.3 × 10⁵. That is above 2 × 10⁵, so $F^{*}$ = 0.15 and $SF$ = 1.5.
$\tau_p$ = 0.15 × (25 900 − 9 810) × 0.35 = 845 Pa, and $\tau_p / SF$ = 563 Pa.

Verdict: 563 Pa against 543 Pa, so 350 mm stone is adequate on the straight reach with a margin of 1.04. The tool reports an angle of repose of 34° for this stone, steeper than the 26.6° side slope, and a freeboard of 0.72 m.

Two checks show how close this is. With 300 mm stone, n falls to 0.074, the depth to 0.886 m and the bed shear to 522 Pa. $Re^{*}$ drops to 1.9 × 10⁵, inside the interpolation band, so $F^{*}$ = 0.145 and $SF$ = 1.47, and the permissible value is 474 Pa: the lining fails. And on the outside of a bend with a 40 m centreline radius, $R_c/T$ = 6.0 gives $K_b$ = 1.41, the applied shear rises to 765 Pa, and the 350 mm stone fails there too. The bend needs heavier stone, a larger radius, or both.

Bend shear multiplier Kb

HEC-15 raises the applied shear on the outside of a bend by $K_b$ , a function of the centreline radius $R_c$ over the top width $T$ :

K_b = 2.38 - 0.206\frac{R_c}{T} + 0.0073\left(\frac{R_c}{T}\right)^2, \qquad 2 < \frac{R_c}{T} < 10

$R_c/T$	$K_b$
2 or less	2.00
3	1.83
4	1.67
5	1.53
6	1.41
8	1.20
10 or more	1.05

The tool also reports the super-elevation of the water surface across the bend, $\Delta y = V^2 T / (g R_c)$ , which belongs in the freeboard.

Limitations

The check uses uniform-flow depth. Near a culvert outlet, weir or change of slope the depth differs; compute it with the gradually varied flow calculator and enter that depth.
One permissible value is compared with the largest of the bed, side and bend shear. The tool does not reduce rock’s permissible shear on the side slopes for the slope angle, so keep side slopes well flatter than the reported angle of repose.
HEC-15 sizes rock on slopes steeper than 10% with a separate steep-slope relation (Eq 6.11), and between 5% and 10% checks it alongside the Shields relation and keeps the larger stone. The tool does not include Eq 6.11, so the 6% example above also needs that check.
Rigid linings such as concrete and grouted stone are outside HEC-15.
Riprap needs a granular or geotextile filter under it. The tool does not design the filter.
For bank revetment sized from velocity, bridge scour protection or a culvert outlet apron, use the riprap sizing calculator, which applies HEC-23 and HEC-14.

What a free Studio account adds

A free Delta Hydro Studio account includes the full Channel Lining tool, which runs the same engine:

Rectangular, triangular and circular sections as well as trapezoids, with the section, depth, velocity and slope imported from a saved Channel Analysis calculation, or typed in from HEC-RAS output, a gradually varied flow profile or a hand calculation.
All four lining families (rock, grass, a turf reinforcement mat and a gabion mattress) run against the same channel, with their Manning’s n, permissible shear, margin and verdict side by side.
Rock gradation class and RECP product lookups that fill D50, Manning’s n and the tested shear.
A lined cross-section with freeboard and an applied-versus-permissible shear chart, saved to a project and exported as a Word, Excel or PDF report.

Culvert outlets are a common place for a lined channel to start: the culvert calculator gives the outlet velocity and depth that set the first reach.

HEC-15 channel lining calculator