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16 CHAPTER 4 Equilibrium Local Scour Predictive Equations Introduction tion includes a multiplying factor, Kw, to be applied to wide piers in shallow flows. The factor was developed by Johnson and A large number of equations has been proposed for estimat- Torrico (1994) using laboratory and field data for large piers. ing equilibrium local scour depths at bridge piers. Detailed Equation 18, presented by Gao et al. (1993), has been used descriptions of the data used in the development of the equa- by highway and railway engineers in China for more than tions are not usually available. However, the equations can be 20 years. The equation was developed from Chinese data for compared with each other for hypothetical, but practical, lab- local scour at bridge piers, including 137 live-bed data points oratory and field situations. This procedure was used for the and 115 clear-water data points. The equation was tested initial screening of the equilibrium scour equations. using field data obtained prior to 1964. A selection of the more widely used and recently published Several of the equations are based on field data, including equations is given in Table 4. A brief description of the equa- the regime equations (Inglis 1949, Ahmad 1953, Chitale 1962, tions is given in the following paragraphs. and Blench 1969) and the relations by Froehlich (1988), The regime equations of Inglis (1949), Ahmad (1953), Ansari and Qadar (1994) and Wilson (1995). Equation 15 by Chitale (1962), and Blench (1969) were derived from meas- Froehlich (1988) was fitted to 83 on-site measurements from urements in irrigation canals in India and are supposed to bridges in the United States and elsewhere. It includes a safety describe the conditions under which these canals are stable factor equal to one pier diameter. for the existing sediment supply. These equations can lead to Ansari and Qadar (1994) fitted envelope equations (num- negative estimates of local scour. bered 19 in Table 4) to more than 100 field measurements of Laursen (1958, 1963) extended the solutions for the long pier scour depth, derived from 12 different sources and sev- rectangular contraction to local scour at piers using the eral countries, including 40 measurements from India. Ansari observation that the depth of local scour does not depend on and Qadar (1994) also presented a comparison of the field the contraction ratio until the scour holes from neighboring data they used with estimates of scour depth obtained using piers start to overlap. For sand, the width of the scour hole Equation 6 by Larras (1963), Equation 7 by Breusers (1965), normal to the flow was observed to be about 2.75 times the Equation 12 by Neill (1973), Equation 17 by Breusers and scour depth (2.75ys). Laursen (1958, 1963) assumed the scour Raudkivi (1991), and an equation by Melville and Sutherland in the contraction, defined by this width, to be a fraction of (1988) that is the forerunner of Equation 21. the scour depth at the pier or abutment, leading to Equations 3 The May and Willoughby (1990) equation (numbered 16 and 5 in Table 4. in Table 4) was derived from data produced by a laboratory The equation in HEC-18 (Richardson and Davis 2001), study of local scour around large obstructions such as caissons Equation 22 in Table 4, was determined from a plot of labora- and cofferdams for the construction of bridges across rivers and tory data for circular piers. The data used were selected from estuaries. The laboratory study, which focused on cases where Chabert and Engeldinger (1956) and Colorado State University the width of the structure was large relative to the flow depth, data (Shen et al. 1966); these data are the same as those used by is especially relevant to this study. May and Willoughby noted Shen et al. (1969) in the derivation of Equation 9. Equation 22 that existing design formulae tended to overestimate the has been progressively modified over the years and is currently amount of scour. recommended by the FHWA for estimating equilibrium scour Melville (1997) presented a physically justified method depths at simple piers (Richardson and Davis 2001). The equa- (Equation 21) to estimate local scour depth at piers based on
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17 Table 4. Equilibrium scour predictive equations. Reference Equation Notes No. 0.78 q = average discharge intensity ys y1 q2/ 3 upstream from the bridge (ft2/s) Inglis (1949) 1.70 1 a a a = pier width 2/ 3 q2 = local discharge intensity in Ahmad (1953) ys y1 0.45K s q 2 2 contracted channel (ft2/s) 1.7 a ys ys Laursen (1958) 5.5 1 1 Applies to live-bed scour 3 y1 y1 11.5 y1 Fr = Froude Number of the approach ys Chitale (1962) 6.65 Fr 0.51 5.49 Fr 2 flow 4 y1 = V1/(gy1)0.5 7/ 6 Applies to clear-water scour ys 1 = grain roughness component of a ys 11.5 y1 1 Laursen (1963) 5.5 0.5 1 bed shear 5 y1 y1 1 c = critical shear stress at threshold of c motion Larras (1963) ys 0 .43K s K a 0.75 a is in ft 6 Breusers (1965) ys 1. 4 a Derived from data for tidal flows 7 yr = regime depth 0.25 ys y1 a =1.48(q2/FB)a/3 Blench (1969) 1.8 8 yr yr where FB=1.9 (D)0.5, D is in mm and q in m2/s 0.619 V1a = kinematic viscosity ys 0.000223 Shen et al. (1969) 9 0.619 V1a ys 0.000223 0.9 V1 V1 Coleman (1971) 0.6 10 2gys a 1/ 3 ys 2V1 Vc2 (2V1/Vc 1) = 1 for live-bed scour Hancu (1971) 2.42 1 11 a Vc ga Ks = 1.5 for round-nosed and circular Neill (1973) ys K sa piers; 12 Ks = 2.0 for rectangular piers f(V1/Vc) = 0 for V1/Vc 0.5 Breusers et al. ys V1 y1 f 2.0 tanh K1 K 2 = (2V1/Vc 1) for 0.5 < V1/Vc < 1 13 (1977) a Vc a =1 for V1/Vc > 1 0.3 ys y1 Applies to maximum clear-water Jain (1981) 1.84 Frc0.25 14 a a scour 0.62 0.46 0.08 ys ap y1 a ap = projected width of pier Froehlich (1988) 0.32 K s Fr 0.2 1 15 a a a D50 For circular cylinder: fs = 1.0 1.76 ys V1 V1 1 3.66 1 0.52 1.0 y sc Vc Vc May and ys ysc V1 1.0 1.0 Willoughby ys 2.4 fs Vc 16 (1990) ysc ysm 0.6 y sc y1 y1 0.. .55 2. 7 y sm a a y1 1.0 1 .0 a (continued on next page)
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18 Table 4. (Continued). Reference Equation Notes No. Breusers and ys 2.3 K y Ks K d K K For an aligned pier, ys=2.3KsKdK b 17 Raudkivi (1991) a V1 Vc' Vc = incipient velocity for local scour ys 0.46 K a 0.60 y10.15 D 0.07 at a pier Vc Vc' K = shape and alignment factor 0 . 14 0 .5 y1 s 7 10 y1 = 1 for clear-water scour Vc 1 7 .6 D 6.05x10 Gao et al. (1993) D D 0.72 < 1 for live-bed scour 18 0.053 9.35 2.23 log10 D D Vc Vc ' 0.645 Vc i.e., a V where ys, a, y1, D, V1, Vc, Vc' are in S.I. units. Ansari and ys 0.024a 3 p .0 ap 7.2 ft 19 Qadar (1994) ys 2.238a 0.4 p ap 7.2 ft 0.4 ys y1 Wilson (1995) 0.9 a* = effective width of pier 20 a* a* ys K ya K I K D K s K K ya 2.4a for a / y1 0.7 0 .5 K ya 2( y1a ) for 0.7 a / y1 5 K ya 4.5y1 for a / y1 5 V1 Vlp Vc V1 ( Vlp Vc ) KI for 1.0 Vc Vc 21 Melville (1997) V1 ( Vlp Vc ) KI 1 for 1 .0 Vc a a KD 0.57 log10 2.24 for 25 D50 D50 a KD 1 for 25 D50 K3 = factor for mode of sediment transport K4 = factor for armoring by bed ys y1 0.35 material Richardson and 2 Ks K K3 K 4 K w Fr 0.43 Kw = factor for very wide piers after 22 Davis (2001) a a Johnson and Torrico (1994) ys(max) = 2.4b for Fr 0.8 ys(max) = 3b for Fr > 0.8