What is the HEC-18 CSU pier scour equation?

The CSU equation in FHWA HEC-18 gives local pier scour depth as ys = 2.0·y1·K1·K2·K3·(a/y1)^0.65·Fr1^0.43, where y1 is the approach flow depth, a is pier width, Fr1 is the approach Froude number, and K1, K2, K3 are correction factors for pier nose shape, flow angle of attack, and bed condition. Scour is capped at 2.4 times pier width for Fr1 at or below 0.8, and 3.0 times pier width above that.

What is the difference between live-bed and clear-water scour?

Live-bed scour occurs when the approach velocity exceeds the critical velocity for the bed material, so sediment is continuously transported into and out of the scour hole and depth oscillates about an equilibrium. Clear-water scour occurs when approach velocity is below critical, so no sediment is resupplied and the hole deepens asymptotically to a maximum. You determine which mode applies by comparing V1 to Vc and use the matching equation. You do not compute both and take the larger.

What is the Laursen contraction scour equation?

For live-bed conditions, Laursen gives y2/y1 = (Q2/Q1)^(6/7)·(W1/W2)^k1, where k1 is 0.59, 0.64 or 0.69 depending on the ratio of shear velocity to fall velocity. For clear-water conditions the form is y2 = [Ku·Q²/(Dm^(2/3)·W²)]^(3/7), with Ku = 0.025 in SI and 0.0077 in US customary units and Dm = 1.25·D50. Contraction scour depth is y2 minus the existing bed depth.

How do you calculate total scour at a bridge?

Total scour is the sum of three independent components: long-term aggradation or degradation of the channel bed, contraction scour through the bridge opening, and local scour at each pier or abutment. They are computed separately and added. HEC-18 requires evaluation at both the design flood and a check flood, typically the 100-year and 500-year events.

What are the K1 K2 and K3 factors in the CSU equation?

K1 corrects for pier nose shape: 1.1 square nose, 1.0 round nose or circular cylinder, 0.9 sharp nose. K2 corrects for the angle of attack, K2 = (cos θ + (L/a)·sin θ)^0.65, and grows rapidly with skew — a 30-degree skew on a pier with length-to-width ratio of 8 gives K2 = 2.75. K3 corrects for bed condition, 1.1 for clear-water and plane bed up to 1.3 for large dunes.

All tools

Riprap sizing reference →

Bridge Scour Equations — Reference

By Michael Flynn, PE · water resources & dam safety engineer

The FHWA HEC-18 equation set for pier and contraction scour: CSU and Froehlich for local pier scour, Laursen for contraction scour, with the K-factor tables. Equations below are dimensionally consistent — use one unit system throughout, and note that only the clear-water contraction coefficient K_u changes between SI and US customary.

Total Scour Is a Sum of Three Components

y_s,total = y_degradation + y_contraction + y_local

Component	What it is	Method
Long-term degradation	Channel-wide bed lowering over years, independent of the bridge	Geomorphic assessment, HEC-20
Contraction scour	Bed lowering across the whole opening from flow constriction	Laursen (live-bed or clear-water)
Local scour	The hole at an individual pier or abutment from vortex action	CSU / Froehlich (piers); Froehlich / HIRE (abutments)

They are computed independently and added. HEC-18 expects evaluation at both a design flood and a check flood — commonly the 100-year and 500-year events.

Live-Bed vs Clear-Water — Decide First

Condition	Criterion	Behaviour
Live-bed scour	V₁ > V_c	Sediment continuously resupplied; depth oscillates about an equilibrium
Clear-water scour	V₁ < V_c	No resupply; hole deepens asymptotically to a maximum

This is a fork, not a maximum. Compare approach velocity V₁ against the critical velocity V_c for the bed D₅₀, then use the matching contraction equation. Computing both and taking the larger is a common and wrong shortcut — clear-water often returns the bigger number in a case where live-bed physically governs.

Local Pier Scour

CSU equation (HEC-18 primary)

y_s = 2.0 · y₁ · K₁ · K₂ · K₃ · (a / y₁)^0.65 · Fr₁^0.43

y₁ = approach flow depth, a = pier width, Fr₁ = V₁/√(g·y₁). Upper limits: y_s ≤ 2.4·a for Fr₁ ≤ 0.8, and y_s ≤ 3.0·a for Fr₁ > 0.8.

Froehlich equation (design form)

y_s = 0.32 · φ · (a′)^0.62 · y₁^0.47 · Fr₁^0.22 · D₅₀^−0.09 + a

φ = 1.3 square nose, 1.0 round nose, 0.7 sharp nose. a′ = pier width projected normal to the flow. The trailing + a is a deliberate safety addition, which is why Froehlich is often used as a check rather than the primary. Unlike CSU, this form is sensitive to D₅₀.

Pier Correction Factors

K₁ — pier nose shape

Nose shape	K₁
Square nose	1.1
Round nose	1.0
Circular cylinder	1.0
Group of cylinders	1.0
Sharp (triangular) nose	0.9

K₂ — angle of attack

K₂ = ( cos θ + (L / a) · sin θ )^0.65

Angle θ	L/a = 4	L/a = 8	L/a = 12
0°	1.0	1.0	1.0
15°	1.5	2.0	2.5
30°	2.0	2.75	3.5
45°	2.3	3.3	4.3
90°	2.5	3.9	5.0

K₂ dominates everything else when flow is skewed. A 30° skew on a long pier multiplies scour by 3.5 — if K₂ > 1, nose shape stops mattering and HEC-18 directs you to use K₁ = 1.0.

K₃ — bed condition

Bed condition	Dune height H	K₃
Clear-water scour	—	1.1
Plane bed and antidune flow	—	1.1
Small dunes	0.6–3 m (2–10 ft)	1.1
Medium dunes	3–9 m (10–30 ft)	1.1–1.2
Large dunes	≥ 9 m (30 ft)	1.3

Contraction Scour — Laursen

Live-bed

y₂ / y₁ = (Q₂ / Q₁)^6/7 · (W₁ / W₂)^k₁

V_* / ω	Mode of bed-material transport	k₁
< 0.50	Mostly contact bed-material discharge	0.59
0.50 – 2.0	Some suspended bed-material discharge	0.64
> 2.0	Mostly suspended bed-material discharge	0.69

V_* = √(g·y₁·S₁) is shear velocity and ω is the fall velocity of the bed D₅₀ — see the settling velocity card.

Clear-water

y₂ = [ K_u · Q² / ( D_m^2/3 · W² ) ]^3/7

Term	SI	US customary
K_u	0.025	0.0077
D_m (effective diameter)	1.25 · D₅₀	1.25 · D₅₀

In both forms, contraction scour depth is y₂ minus the existing bed depth in the contracted section, not y₂ itself.

Worked Example

Circular pier, a = 4 ft. Approach y₁ = 10 ft, V₁ = 8 ft/s, flow aligned. Bed D₅₀ = 2 mm, plane bed (K₃ = 1.1). Opening contracts from W₁ = 400 ft to W₂ = 250 ft, all flow through the bridge, live-bed with k₁ = 0.64.

Step	Result
Fr₁ = 8 / √(32.2 × 10)	0.446
CSU pier scour = 2.0(10)(1.0)(1.0)(1.1)(0.4)^0.65(0.446)^0.43	8.57 ft
CSU limit check, Fr ≤ 0.8 → 2.4a = 9.6 ft	OK, not governed
Froehlich pier scour (φ = 1.0, a′ = 4 ft)	6.94 ft
Laursen live-bed y₂ = 12(1)^6/7(400/250)^0.64	16.21 ft
Contraction scour = 16.21 − 12	4.21 ft
Total (contraction + CSU local)	12.78 ft

CSU returns 8.57 ft against Froehlich's 6.94 ft — about 23% apart, which is typical. CSU is the HEC-18 primary; Froehlich is the check. Long-term degradation would be added on top from a geomorphic assessment.

Scour equations are conservative by construction. They were regressed largely on flume data at equilibrium, and field-measured scour is frequently shallower than predicted. That conservatism is deliberate — the failure mode is a bridge in a river. Do not "calibrate it out" without measured site data and a very clear rationale in the record.

Practice Notes

Issue	Guidance
Which pier equation governs	CSU is the HEC-18 primary; run Froehlich as an independent check
Skewed flow	K₂ swamps every other factor. If K₂ > 1, set K₁ = 1.0
Debris	Effective pier width increases — HEC-18 gives a debris-width procedure; do not ignore it on small streams
Design vs check flood	Evaluate both; the check flood (often 500-yr) may govern the foundation
Countermeasures	Riprap sizing at piers follows HEC-23, not the channel riprap methods
Abutments	Not covered here — use Froehlich or HIRE per HEC-18, with the NCHRP 24-20 approach where applicable

Sources: Arneson, L.A., Zevenbergen, L.W., Lagasse, P.F. & Clopper, P.E. (2012), Evaluating Scour at Bridges, 5th ed., FHWA HEC-18 (FHWA-HIF-12-003) — CSU equation, K-factor tables, Laursen contraction relations. Froehlich, D.C. (1988), "Analysis of Onsite Measurements of Scour at Piers," ASCE Hydraulic Engineering. Laursen, E.M. (1960), "Scour at Bridge Crossings," J. Hydraulics Division 86(HY2), and (1963), "An Analysis of Relief Bridge Scour," 89(HY3). See also FHWA HEC-20 (stream stability) and HEC-23 (countermeasures). Worked-example values above were computed directly from the equations as written.

Sizing protection? Riprap sizing methods → · Settling / fall velocity? Stokes card · Riprap calculator.

Related cheat sheets and tools

Fall velocity ω for the Laursen k₁ selection comes from the settling velocity card, with fluid properties from the water properties table. Approach hydraulics come from Manning's equation and the open-channel geometry card; check the flow state on the specific energy card. For scour protection see riprap sizing. For full watershed and channel modeling, see HydroComplete.

Bridge Scour Equations — Reference

Total Scour Is a Sum of Three Components

Live-Bed vs Clear-Water — Decide First

Local Pier Scour

CSU equation (HEC-18 primary)

Froehlich equation (design form)

Pier Correction Factors

K1 — pier nose shape

K2 — angle of attack

K3 — bed condition

Contraction Scour — Laursen

Live-bed

Clear-water

Worked Example

Practice Notes

Related cheat sheets and tools

Get future cheat sheets in your inbox

K₁ — pier nose shape

K₂ — angle of attack

K₃ — bed condition