How is total scour at a bridge computed?

HEC-18 treats total scour as the sum of three components: long-term aggradation or degradation, contraction scour, and local scour at the pier or abutment. The components are computed separately and added. The one exception is the NCHRP 24-20 abutment method, whose result already includes contraction scour, so only long-term degradation is added to it.

How do I know whether contraction scour is live-bed or clear-water?

Compare the mean velocity in the upstream approach section with the critical velocity for the bed material, Vc = Ku y^(1/6) D50^(1/3) (HEC-18 Eq. 6.1, Ku = 11.17 US or 6.19 SI). If V is below Vc the bed upstream is not moving and scour is clear-water (Eq. 6.4). If V is at or above Vc, bed material is being delivered to the bridge and scour is live-bed (Eq. 6.2). Overbank flow on a vegetated floodplain is usually clear-water even when the main channel is live-bed.

What pier width do I use for a skewed pier?

Use the actual pier width a and let K2 handle the skew. K2 = (cos θ + (L/a) sin θ)^0.65, with L/a capped at 12. HEC-18 also says to set K1 = 1.0 once the skew exceeds 5 degrees, because the angle of attack then dominates the nose shape. Apply K2 only when the whole pier length sees the skewed flow; if an abutment or another pier shields part of it, use the exposed length.

Is there a maximum pier scour depth?

For round-nose piers aligned with the flow, HEC-18 Eq. 7.2 gives a rule-of-thumb maximum of 2.4 times the pier width when the Froude number is 0.8 or less and 3.0 times the pier width above 0.8. This calculator applies that limit only to round-nose and circular piers with zero skew, which is the case HEC-18 states it for.

Froehlich or HIRE for abutment scour?

HEC-18 applies the HIRE equation when the projected embankment length is more than 25 times the flow depth, because HIRE came from field data at the ends of long Mississippi River spurs. For shorter embankments Froehlich's equation is used. Both give the local abutment scour component, which is added to contraction scour. HEC-18 5th edition also presents the NCHRP 24-20 approach, which gives total scour at the abutment directly.

Why does the NCHRP 24-20 section ask me for the amplification factor?

The amplification factors αA (live-bed) and αB (clear-water) are published only as design curves in HEC-18 Figures 8.9 to 8.12, as a function of the unit-discharge ratio q2/q1 and abutment type. This calculator does not digitize those curves. You read α from the figure for your q2/q1 ratio, which the calculator reports, and it does the rest.

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Bridge Scour Calculator (HEC-18)

Contraction, pier and abutment scour at a bridge per FHWA HEC-18, Evaluating Scour at Bridges, 5th edition (FHWA-HIF-12-003). Checks live-bed against clear-water conditions, applies the HEC-18 pier scour equation with shape, skew and bed-form factors, computes Froehlich and HIRE abutment scour, and adds the components into total scour at the pier and at the abutment.

Units

Long-term bed degradation (0 if none)ft

1 · Contraction scour

Upstream main-channel flow, Q₁cfs

Upstream main-channel width, W₁ft

Upstream average depth, y₁ft

Upstream energy slope, S₁—

Flow in contracted section, Q₂cfs

Contracted width less piers, W₂ft

Existing depth in contracted section, y₀ft

Bed material D₅₀mm

Fall velocity of D₅₀, ω (0 = estimate)ft/s

Coarse bed material that can armor (HEC-18 §6.3 Note 8)?

Approach velocity, V₁ = Q₁/(W₁y₁)—ft/s

Critical velocity, V_c (Eq. 6.1)—ft/s

Scour condition—

Shear velocity V_* / fall velocity ω——

Exponent k₁——

Live-bed contraction scour (Eq. 6.2)—ft

Clear-water contraction scour (Eq. 6.4)—ft

Contraction scour used—ft

2 · Pier scour

Pier nose shape

Pier width, aft

Pier length, Lft

Angle of attack (skew), θdegrees

Depth directly upstream of pier, y₁ft

Velocity directly upstream of pier, V₁ft/s

Bed condition (Table 7.3)

Froude number, Fr₁——

K₁ · K₂ · K₃——

Pier scour, Eq. 7.1—ft

Rule-of-thumb limit, Eq. 7.2—ft

Pier scour used—ft

3 · Abutment scour (Froehlich / HIRE)

Abutment shape (Table 8.1)

Embankment angle to flow, θ (90° = normal; <90° points downstream)degrees

Embankment length projected normal to flow, Lft

Length blocking active flow, L′ (0 = use L)ft

Flow obstructed by embankment, Q_ecfs

Average floodplain depth upstream, y_aft

Depth at abutment tip, y₁ (HIRE)ft

Velocity at abutment tip, V₁ (HIRE)ft/s

K₂ = (θ/90)^0.13——

L / y_a——

Froehlich scour (Eq. 8.1)—ft

HIRE scour (Eq. 8.2)—ft

Method used (HIRE when L/y_a > 25)—

Abutment scour used—ft

4 · Abutment scour (NCHRP 24-20, total at abutment)

Scour condition

Upstream unit discharge, q₁ft²/s

Unit discharge at abutment, q₂ft²/s

Upstream flow depth, y₁ft

Flow depth at abutment before scour, y₀ft

Floodplain D₅₀ (clear-water only)mm

Amplification factor α, read from HEC-18 Fig. 8.9–8.12—

q₂/q₁ (enter the figure with this)——

Depth with contraction scour, y_c—ft

y_max = α·y_c—ft

NCHRP abutment scour, y_s = y_max − y₀—ft

Total scour

At pier = degradation + contraction + pier—ft

At abutment (Froehlich/HIRE) = degradation + contraction + abutment—ft

At abutment (NCHRP 24-20) = degradation + NCHRP y_s—ft

$$ V_c = K_u\, y^{1/6} D_{50}^{1/3}, \qquad K_2 = \left(\cos\theta + \tfrac{L}{a}\sin\theta\right)^{0.65} $$

$$ \frac{y_s}{y_a} = 2.27\,K_1K_2\left(\frac{L'}{y_a}\right)^{0.43} Fr^{0.61} + 1, \qquad \frac{y_s}{y_1} = 4\,Fr^{0.33}\,\frac{K_1}{0.55}\,K_2 $$

K_u = 11.17 (US) or 6.19 (SI) in Eq. 6.1; 0.0077 (US) or 0.025 (SI) in Eq. 6.4 · D_m = 1.25·D₅₀ · k₁ = 0.59, 0.64 or 0.69 for V_*/ω < 0.5, 0.5–2, > 2 · V_* = (g y₁ S₁)^½ · Froehlich Fr = V_e/(g y_a)^½ with V_e = Q_e/(y_aL).

How the calculator works

Contraction scour. The approach velocity V₁ = Q₁/(W₁y₁) is compared with the critical velocity V_c of the D₅₀ (Eq. 6.1). Below V_c the upstream bed is not moving, so scour is clear-water and Eq. 6.4 governs. At or above V_c it is live-bed (Eq. 6.2). The live-bed exponent k₁ depends on the ratio of shear velocity to fall velocity. If you leave the fall velocity at 0, the page estimates it from Ferguson and Church (2004) for quartz (SG 2.65) in 20 °C water. That formula tracks HEC-18 Figure 6.8, which gives 0.10 m/s for 0.7 mm sand where the formula gives 0.096 m/s. Enter your own ω to override it. HEC-18 treats 0.2 mm as a reasonable lower limit for D₅₀ in Eqs. 6.1 and 6.4, so finer inputs are raised to 0.2 mm, and the notes line says so when that happens. If the bed has coarse material that can armor, HEC-18 Note 8 says to compute both equations and use the smaller. Pick "Yes" in the armoring row to do that.

Pier scour. This is HEC-18 Eq. 7.1 with the shape factor K₁ (Table 7.1), the angle-of-attack factor K₂ (Eq. 7.4, with L/a capped at 12) and the bed-condition factor K₃ (Table 7.3). When the skew is more than 5°, K₁ is set to 1.0 as Table 7.3 Note 1 requires. For a circular pier L = a, so K₂ = 1. For round-nose and circular piers with zero skew, the rule-of-thumb limit of Eq. 7.2 caps the result: 2.4a when Fr ≤ 0.8 and 3.0a when Fr > 0.8. The 5th edition dropped the K₄ armoring factor, and this page does not use it.

Abutment scour. Froehlich (Eq. 8.1) and HIRE (Eq. 8.2) are both computed. The page uses HIRE when L/y_a > 25, which is the applicability limit HEC-18 gives, and Froehlich otherwise. L′ is the embankment length that blocks live flow. HEC-18 §8.2.2 recommends taking it from conveyance tubes as L′ = Q_e/q_tube, because using the full embankment length over-predicts scour on wide, shallow floodplains. NCHRP 24-20 (Eqs. 8.3 to 8.6) is in its own section. Its amplification factor α only exists as a design curve, so you read it from HEC-18 Figures 8.9 to 8.12 using the q₂/q₁ the page reports. For condition (a), live-bed, the NCHRP section uses Eq. 8.5, y_c = y₁(q₂/q₁)^6/7. For condition (b), clear-water, it uses Eq. 8.6, y_c = [q₂/(K_uD₅₀^1/3)]^6/7. The NCHRP result already includes contraction scour, so the page does not add contraction scour to it again.

Worked examples (HEC-18 5th edition)

The default inputs reproduce the HEC-18 example problems, so you can check each section against the manual.

Contraction — §6.6.1, live-bed

Given: Q = 27,300 cfs all in the channel; W₁ = 322 ft; y₁ = 8.6 ft; S = 0.004; bridge width 122 ft less 3 × 1.25 ft piers = 118.25 ft; y₀ = 7.1 ft; D₅₀ = 0.7 mm (0.0023 ft).

V = 27,300/(8.6 × 322) = 9.86 ft/s; V_c = 11.17 (8.6)^1/6(0.0023)^1/3 = 2.11 ft/s → live-bed

V_* = (32.2 × 8.6 × 0.004)^0.5 = 1.05 ft/s; ω ≈ 0.32 ft/s; V_*/ω ≈ 3.3 > 2 → k₁ = 0.69

y₂ = 8.6 (322/118.25)^0.69 = 8.6 × 2.00 = 17.2 ft

y_s = 17.2 − 7.1 = 10.1 ft (HEC-18: 10.1 ft)

Pier — §7.10.1, simple solid pier

Given: round nose, a = 4.0 ft, L = 59 ft, 0° skew, y₁ = 10.2 ft, V₁ = 11.02 ft/s, plane bed.

Fr = 11.02/(32.2 × 10.2)^0.5 = 0.61

y_s/y₁ = 2.0 × 1.0 × 1.0 × 1.1 × (4.0/10.2)^0.65 × 0.61^0.43 = 0.97 → y_s = 9.9 ft

Limit 2.4a = 9.6 ft governs → y_s = 9.6 ft. With 20° skew, K₂ = (cos 20° + 12 sin 20°)^0.65 = 2.86 and y_s = 28.3 ft (§7.10.2).

Abutment — §8.7.1, Froehlich

Given: vertical wall with wing walls, θ = 70°, L = 75 ft, Q_e = 960 cfs, y_a = 3.5 ft, conveyance tube at the tip 4.6 ft/s × 5.0 ft.

L/y_a = 21.4 < 25 → Froehlich; K₂ = (70/90)^0.13 = 0.97

V_e = 960/(3.5 × 75) = 3.66 ft/s; Fr = 0.34; L′ = 960/(4.6 × 5.0) = 42 ft

y_s = 3.5 [2.27 × 0.82 × 0.97 × (42/3.5)^0.43 × 0.34^0.61 + 1] = 13.1 ft (HEC-18: 13.0 ft)

Abutment — §8.7.3, NCHRP 24-20 live-bed

Given: wingwall abutment, L/B_f = 0.85 (condition a), q₁ = 57.0 ft²/s, q₂ = 78.6 ft²/s, y₁ = y₀ = 10.0 ft, α_A = 1.7 from Fig. 8.10.

y_c = 10.0 (78.6/57.0)^6/7 = 13.2 ft; y_max = 1.7 × 13.2 = 22.4 ft

y_s = 22.4 − 10.0 = 12.4 ft. HEC-18 shows 12.6 ft because it rounds q₂/q₁ to 1.4 before raising it to the 6/7 power.

What this page does not cover

Complex piers (pile cap or footing in the flow, HEC-18 §7.5). Break the pier into stem, cap and pile-group components by hand.
Pressure-flow scour when the low chord is submerged (§6.10), and debris effective width (§7.9).
The FDOT / Sheppard-Melville pier method (§7.3). HEC-18 recommends it as an alternative for wide piers in shallow flow over fine sand.
Cohesive soils and rock (§6.7, §7.13), where scour depends on erodibility testing, not D₅₀.
Countermeasures. Riprap at piers and abutments is designed per HEC-23. The riprap tool covers channel and embankment riprap, not the HEC-23 pier and abutment equations.

References: Arneson, L.A., Zevenbergen, L.W., Lagasse, P.F., Clopper, P.E. (2012). Evaluating Scour at Bridges, 5th ed., Hydraulic Engineering Circular No. 18, FHWA-HIF-12-003 — Eqs. 6.1–6.5, 7.1–7.4, 8.1–8.6; Tables 7.1–7.3, 8.1; Example Problems 6.6.1, 7.10.1–7.10.2, 8.7.1–8.7.4. Froehlich, D.C. (1989). "Local Scour at Bridge Abutments," Proc. ASCE National Hydraulic Conference. NCHRP (2010). Estimation of Scour Depth at Bridge Abutments, Project 24-20 draft final report. Ferguson, R.I., Church, M. (2004). "A simple universal equation for grain settling velocity," J. Sedimentary Research 74(6), 933–937.

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Pile capacity — check pier and abutment piles with the scoured bed removed
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