Evaluation of Bridge-Scour Research: Abutment and Contraction Scour Processes and Prediction (2011)

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APPENDIX A Table A-1. A selection of abutment scour formulas (revised and extended from Melville and Coleman 2000). Reference Formula Notes Γ and δ are given as functions of the  Bδ Garde et ds  F1 = Γ al. (1961) B−L drag coefficient of the sediment Y1   d Applies to live-bed scour at an abutment 17  d L  + 1 − 1 = 2.75 s s Laursen encroaching into the main channel     11.5Y1  Y1 Y1   (1960)  d  Applies to clear-water scour at an 7/6    + 1 Laursen* s abutment encroaching into the main d s   11.5Y1     L − 1 = 2.75  (1963) channel 05  τ1  Y1 Y1   τ1 = grain roughness component of bed    τ  shear stress; τc = critical shear stress  c   Liu et al. Applies to live-bed scour at spill- 04 L ds F1 .33 0 = 1.1  through abutments; F1=V1/(gY1)0 5 (1961)  Y1  Y1 Liu et al. Applies to live-bed scour at wing-wall 04 L ds F1 .33 0 = 2.15  Y  (1961) or vertical-wall abutments  1 Y1 Liu et al. Applies to clear-water scour at vertical- ds = 12.5F1 β wall abutments; β=contraction ratio (1961) Y1 Y2 = flow depth at bridge ≈ Y1 +ds −3 / 7  τc  τc  0.25 d  Y2 β 6 / 7  β 1 / m 1 − = 8.375  +  Gill (1972) Equation given at the threshold τ1  τ1   Y1   Y1  condition F1=Froude number of the approach flow upstream of the abutment; Fc=critical  F Sturm and ds = 7.7 1 − 0.35    MF Janjua Froude number for initiation of motion;   Yf (1994) M=Qo/Qtot, Qo=portion of approach flow c in bridge opening width, Qtot=total flowrate Applies to live-bed scour; Yf=average 0 43 L  Froehlich depth of flow in the floodplain; ds = 2.27 K s Kθ   F 0 61 F=Ve/(gYf)0 5, Ve=Qe/Ae, Qe=flow (1989)  Yf  Yf   (HEC-18) obstructed by the embankment, Ae=flow area corresponding to Qe Richardson Applies when L/Y1>25, and for ds = 7.27 K s Kθ F1 0 33 and Davis conditions similar to field conditions Y1 (2001) from which equation was derived (HIRE) ds = K yL K I K d K s Kθ K G Melville KYL=2L L/Y1<1 KYL=2(Y1L)0 5 1<L/Y1<25 (1992, 1997) KYL=10Y1 L/Y1>25 Lim (1997, Applies to clear-water scour ds = K s (0.9 Χ − 2)     L 1998b) 0.25 0.5 0 d Χ = 0.9 θ c−0.375 Fd .75  50   0.9  + 1 − 2 Y1 Y1    Y1        θc = Shields entrainment function Fd = densimetric Froude number 83