Bridge Scour: Method

Equations, Conventions, Limits and Sources

Scope

The tool computes scour at a highway bridge over an alluvial stream by the equations of the Federal Highway Administration's Hydraulic Engineering Circular No. 18 (HEC-18, fifth edition, 2012), and the riprap sizes of its companion HEC-23 (2009). Seven analyses share one form: total scour at a bridge, contraction scour, pressure-flow scour under a submerged deck, the entrance of an open-bottom culvert, a single pier, a complex pier on a pile cap and piles, and an abutment. It is a calculator for the equations, not a hydraulic model: the depths, velocities and discharges of the scour flood come from a water-surface profile (one- or two-dimensional), and the stream-stability assessment of HEC-20 comes first.

Units and constants

HEC-18 prints each equation with one constant for U.S. customary units and one for SI units, and the two are rounded independently: the critical velocity constant is 11.17 in feet and 6.19 in metres, although 11.17 converts to 6.17. The tool uses the constant of the unit system selected on the page (and g = 32.2 ft/s² or 9.81 m/s²), so a printed example is reproduced in either system; switching the system therefore moves a result by up to about half a percent of a depth, and more on a small difference of two depths. Lengths, discharges, velocities and the critical shear stress go through the site's unit dictionary. Grain sizes are plain numbers in millimetres in both systems, slopes are ratios and angles are in degrees.

Total scour

Total scour at an element is the sum of long-term degradation, contraction scour and local scour at the pier or abutment (HEC-18 Section 3.2). The components are assumed independent, which is conservative. Degradation is an input: it is estimated from the stream's history and geomorphology (HEC-18 Chapter 5, HEC-20), not from an equation. The bed elevation of a zone is the water-surface elevation less the entered depth; the scour elevation is that bed lowered by the three components. The tool compares it with a foundation elevation that the user enters, by the owner's criterion (for a spread footing on soil, the bottom of the footing must lie below the total scour; for piles, the embedment left below the scour must carry the loads). A negative cover means the scour reaches below that elevation, and a bridge whose foundations are unstable for the computed scour is scour critical.

The cross-section drawing places the piers by their stations from the left abutment, lowers each zone by degradation and contraction scour, and cuts a hole at each pier and abutment with a top width of 2.0 y_s on each side (Section 7.8: the top width ranges from about 1 to 2.8 y_s with the angle of repose and the bottom width of the hole; 2.0 is the manual's practical value). Where the thalweg can shift or the channel migrate, HEC-18 sets every foundation for the deepest scour, as its Appendix D does.

Contraction scour

The critical velocity of the bed material is V_c = K_u y^1/6 D^1/3 (equation 6.1; K_u = 11.17 or 6.19). If the mean velocity of the approach channel exceeds it, bed material is carried into the bridge section and the scour is live-bed; otherwise it is clear-water.

Live bed, the modified Laursen equation 6.2: y₂/y₁ = (Q₂/Q₁)^6/7 (W₁/W₂)^k1, with y₁, Q₁, W₁ the depth, flow and bottom width of the upstream main channel, Q₂ the flow in the contracted channel and W₂ its bottom width less the pier widths. The exponent k₁ is 0.59, 0.64 or 0.69 as the ratio of the shear velocity V_* = (g y₁ S₁)^1/2 to the fall velocity of the D₅₀ is below 0.5, between 0.5 and 2, or above 2. The fall velocity may be entered; left blank it is read from the 20 °C curve of Figure 6.8, which the tool carries as 27 digitized points with log-log interpolation (the curve gives the manual's own readings, 0.10 m/s for 0.7 mm and 0.20 m/s for 2 mm; the 0 °C and 40 °C curves are not carried).

Clear water, equation 6.4: y₂ = [K_u Q² / (D_m^2/3 W²)]^3/7 with K_u = 0.0077 or 0.025 and D_m = 1.25 D₅₀. A D₅₀ below 0.2 mm is raised to 0.2 mm, the lower limit of Section 6.7, because cohesion increases the critical shear of finer material. In both cases y_s = y₂ − y₀, and a negative result means no contraction scour. Where coarse material may armor the bed, both equations are computed and the smaller scour used (Section 6.3, note 8). In the total-scour analysis the main channel takes whichever equation the critical velocity indicates (or the user's choice), and the overbank areas under the bridge take the clear-water equation with their own flow and width, as in Case 1c and Appendix D.

Pressure flow (Section 6.10): when the water surface rises above the low chord the flow is also contracted vertically. A separation zone forms under the deck with thickness t = 0.5 h_b (h_b h_t/h_u²)^0.2 (1 − h_w/h_t)^−0.1 (equation 6.16), where h_b is the opening height, h_u the upstream depth, h_t = h_u − h_b and h_w = h_t − T the depth of weir flow over a deck of depth T. The scour is y_s = y₂ + t − h_b (equation 6.14) with y₂ from equation 6.2 or 6.4. With overtopping the live-bed equation uses the effective depth h_ue = h_b + T and the effective channel discharge Q_ue = Q₁ (h_ue/h_u)^8/7 (equation 6.15), and the bridge flow is the flow through the opening only.

Open-bottom culverts (Section 6.9): the depth at the upstream corner of the entrance, contraction and local scour together, is y_max = K_u Q_BI^0.28 [Q/(W_c D₅₀^1/3)]^0.26 with wing walls and K_u Q_BI^0.12 [Q/(W_c D₅₀^1/3)]^0.60 without (equations 6.10 and 6.12; K_u = 0.84 and 0.57 in feet, 1.16 and 0.88 in metres), for clear water only.

Pier scour

The HEC-18 equation 7.1, from the Colorado State University equation, serves live-bed and clear-water scour: y_s/y₁ = 2.0 K₁ K₂ K₃ (a/y₁)^0.65 Fr₁^0.43. K₁ is the nose shape factor of Table 7.1 (square 1.1, round, cylinder or group of cylinders 1.0, sharp 0.9), taken as 1.0 when the angle of attack exceeds 5 degrees; K₂ = (cos θ + (L/a) sin θ)^0.65 with L/a not above 12 (equation 7.4); K₃ is 1.1 for clear water, plane beds, antidunes and small dunes, up to 1.3 for large dunes (Table 7.3). The manual's rule of thumb for round-nose piers aligned with the flow, y_s not above 2.4 a for Fr ≤ 0.8 and 3.0 a above (equation 7.2), is reported next to the result but not imposed, because the manual keeps the computed depth in its own examples (9.9 ft against 9.6 ft in Section 7.10.1, 9.0 ft against 7.2 ft in Appendix D).

Wide piers in shallow flow (Section 7.4, Johnson and Torrico): where y/a is below 0.8, a/D₅₀ above 50 and the flow subcritical, K_w = 2.58 (y/a)^0.34 Fr^0.65 for V/V_c below 1 and (y/a)^0.13 Fr^0.25 otherwise. The factor is optional, rests on limited flume data, and the tool never lets it exceed 1.

Coarse beds (Section 7.11, clear water, D₅₀ of 20 mm or more, σ = D₈₄/D₅₀ of 1.5 or more): y_s = 1.1 K₁ K₂ a^0.62 y₁^0.38 tanh[H²/(1.97 σ^1.5)], H = V₁/[g (S_g − 1) D₅₀]^1/2.

Florida DOT methodology (Section 7.3, after NCHRP Report 682): the effective width is a* = K_sf (a cos θ + L sin θ), K_sf = 1 for round or circular piers and 0.86 + 0.97 |θ − π/4|⁴ (θ in radians) for square noses. With f₁ = tanh[(y₁/a*)^0.4], f₂ = 1 − 1.2 [ln(V₁/V_c)]² and f₃ = (a*/D₅₀)^1.13 / [10.6 + 0.4 (a*/D₅₀)^1.33], the scour is zero below 0.4 V_c, 2.5 f₁ f₂ f₃ a* up to V_c, rises linearly to 2.2 f₁ a* at the live-bed peak velocity V_lp = max(5 V_c, 0.6 (g y₁)^1/2) and stays there. Here V_c = 5.75 u*_c log(5.53 y₁/D₅₀) with u*_c from equations 7.13 and 7.14 (D₅₀ in mm, 0.1 to 100 mm). The length L is limited to 12 a, as the manual's example assumes.

Columns (Section 7.6): a bent of columns closer than 5 diameters acts as one pier of length (number of columns) × a with K₂; at 5 diameters or more, without debris, the scour is limited to 1.2 times that of one column. Debris (Section 7.7, NCHRP Report 653): the effective width is a*_d = [K₁ H W + (y − K₁ H) a]/y with K₁ = 0.79 for a rectangular and 0.21 for a triangular mass of thickness H and width W; the nose shape factor is then 1.0. The tool keeps K₂ of the pier with debris, which the manual does not state either way.

Complex piers

Section 7.5 (Jones and Sheppard) adds three components, each from equation 7.1 on an equivalent pier and each computed on a bed lowered by half the scour of the components before it. Pier stem: the scour of a full-depth pier of the stem's width times K_{h pier}, the equation printed on Figure 7.6 in h₁/a and f/a (h₁ the height of the stem base above the bed, f the cap overhang). Pile cap above the bed (Case 1): an equivalent full-depth width a*_pc from the equation printed on Figure 7.7 in T/y₂ and h₂/y₂. Cap or footing at or below the bed (Case 2): the full footing width with the depth y_f to the top of the footing and the velocity V_f = V₂ ln(10.93 y_f/k_s + 1)/ln(10.93 y₂/k_s + 1) of the flow below it; the pile group is then not added. Pile group: projected width a_proj times the spacing factor K_sp (Figure 7.11) and the rows factor K_m (Figure 7.12, constant beyond six rows, 1.0 for skewed flow), and the height factor K_{h pg} of Figure 7.13. All four figures print their equations, which the tool uses; it holds them at the ends of the plotted ranges and says so. The tool takes an aligned group (piles across × pile width); for skewed or staggered groups enter the projected width of Figure 7.10 yourself.

Abutment scour

Froehlich's regression of laboratory data (equation 8.1): y_s/y_a = 2.27 K₁ K₂ (L′/y_a)^0.43 Fr^0.61 + 1. HIRE, from field data at spurs on the Mississippi River, for L/y₁ above 25 (equation 8.2): y_s/y₁ = 4 Fr^0.33 (K₁/0.55) K₂. K₁ is 1.00 for a vertical wall, 0.82 with wing walls and 0.55 for a spill-through abutment (Table 8.1), and K₂ = (θ/90)^0.13. Both give local scour, to be added to contraction scour, and both are known to be conservative.

NCHRP 24-20 (Section 8.6.3) computes the total scour at the abutment, contraction included: y_max = α y_c and y_s = y_max − y₀. For an abutment near the channel (embankment length at least 75 percent of the floodplain width) y_c = y₁ (q_2c/q₁)^6/7; for one set back on the floodplain y_c = [q_2f/(K_u D₅₀^1/3)]^6/7, or (γ/τ_c)^3/7 (n q_2f/K_u)^6/7 when the critical shear stress of the soil is known. The unit discharges are entered as velocity times depth. The amplification factors α_A and α_B are the solid design curves of Figures 8.9 to 8.12, which the manual gives only as graphs; the tool carries them as 21 to 25 digitized points each, within about 0.02 of the figures except on the steep rising limbs just above a ratio of 1.0, holds the end values outside the plotted range, and accepts an entered value in their place. The result is limited in practice by the geotechnical stability of the embankment or bank, which the tool does not check.

Riprap

HEC-23 Design Guideline 11, piers: d₅₀ = 0.692 (K₁ K₂ V)²/[(S_g − 1) 2g], K₁ = 1.5 for round-nose and 1.7 for square-faced piers, K₂ from 0.9 to 1.7 for the position of the pier when V is a channel average; thickness at least 3 d₅₀, extent 2 a. Design Guideline 14, abutments: d₅₀/y = [K/(S_s − 1)] (V²/g y) with K = 0.89 (spill-through) or 1.02 (vertical wall) for Fr ≤ 0.80, and [K/(S_s − 1)] (V²/g y)^0.14 with K = 0.61 or 0.69 above. The stone is then taken as the next larger of the ten classes of Design Guideline 4 (nominal d₅₀ 6 to 42 in). HEC-18 does not accept riprap in place of foundation depth at new piers; it is a countermeasure for existing bridges and for abutments.

Limits

The equations are for cohesionless beds and are envelopes of laboratory and field data: they give the scour that a long enough flood can produce, not a most likely value. Not included: scour in cohesive soil and erodible rock (HEC-18 Sections 6.7, 6.8, 7.12 and 7.13, which need erosion testing), tidal waterways (Chapter 9), time-dependent scour, the 0 °C and 40 °C fall velocity curves, skewed or staggered pile groups, guide banks and the other countermeasures of HEC-23, and the hydraulic model itself.

Checks

The library reproduces the printed examples of HEC-18 Sections 6.6, 6.9.3, 6.10.2, 7.10, 7.11 and 8.7 and of Appendix D, and the riprap examples of HEC-23 Design Guidelines 11 and 14, within the rounding of the manual's intermediate values, and agrees with a separately written numpy implementation of every equation on seeded cases in both unit systems. Three slips of the manual were found: Section 7.10.4 adds 3.7 + 12.8 + 21.24 = 37.74 ft although its pier stem component is 3.2 ft (37.24 ft); Section 6.10.2 Example 4 quotes 2.3 ft for the scour without the deck where equation 6.4 gives 0.73 ft (its SI value, 0.7 m, has the right digits); and Section 6.6.2 rounds D_m up to 0.0030 ft, which lowers y₂ by 1.3 percent.