National Academies Press: OpenBook

Scour at Wide Piers and Long Skewed Piers (2011)

Chapter: Chapter 4 - Equilibrium Local Scour Predictive Equations

« Previous: Chapter 3 - Data Acquisition and Analyses
Page 16
Suggested Citation:"Chapter 4 - Equilibrium Local Scour Predictive Equations." National Academies of Sciences, Engineering, and Medicine. 2011. Scour at Wide Piers and Long Skewed Piers. Washington, DC: The National Academies Press. doi: 10.17226/14426.
×
Page 16
Page 17
Suggested Citation:"Chapter 4 - Equilibrium Local Scour Predictive Equations." National Academies of Sciences, Engineering, and Medicine. 2011. Scour at Wide Piers and Long Skewed Piers. Washington, DC: The National Academies Press. doi: 10.17226/14426.
×
Page 17
Page 18
Suggested Citation:"Chapter 4 - Equilibrium Local Scour Predictive Equations." National Academies of Sciences, Engineering, and Medicine. 2011. Scour at Wide Piers and Long Skewed Piers. Washington, DC: The National Academies Press. doi: 10.17226/14426.
×
Page 18
Page 19
Suggested Citation:"Chapter 4 - Equilibrium Local Scour Predictive Equations." National Academies of Sciences, Engineering, and Medicine. 2011. Scour at Wide Piers and Long Skewed Piers. Washington, DC: The National Academies Press. doi: 10.17226/14426.
×
Page 19
Page 20
Suggested Citation:"Chapter 4 - Equilibrium Local Scour Predictive Equations." National Academies of Sciences, Engineering, and Medicine. 2011. Scour at Wide Piers and Long Skewed Piers. Washington, DC: The National Academies Press. doi: 10.17226/14426.
×
Page 20
Page 21
Suggested Citation:"Chapter 4 - Equilibrium Local Scour Predictive Equations." National Academies of Sciences, Engineering, and Medicine. 2011. Scour at Wide Piers and Long Skewed Piers. Washington, DC: The National Academies Press. doi: 10.17226/14426.
×
Page 21
Page 22
Suggested Citation:"Chapter 4 - Equilibrium Local Scour Predictive Equations." National Academies of Sciences, Engineering, and Medicine. 2011. Scour at Wide Piers and Long Skewed Piers. Washington, DC: The National Academies Press. doi: 10.17226/14426.
×
Page 22
Page 23
Suggested Citation:"Chapter 4 - Equilibrium Local Scour Predictive Equations." National Academies of Sciences, Engineering, and Medicine. 2011. Scour at Wide Piers and Long Skewed Piers. Washington, DC: The National Academies Press. doi: 10.17226/14426.
×
Page 23
Page 24
Suggested Citation:"Chapter 4 - Equilibrium Local Scour Predictive Equations." National Academies of Sciences, Engineering, and Medicine. 2011. Scour at Wide Piers and Long Skewed Piers. Washington, DC: The National Academies Press. doi: 10.17226/14426.
×
Page 24
Page 25
Suggested Citation:"Chapter 4 - Equilibrium Local Scour Predictive Equations." National Academies of Sciences, Engineering, and Medicine. 2011. Scour at Wide Piers and Long Skewed Piers. Washington, DC: The National Academies Press. doi: 10.17226/14426.
×
Page 25
Page 26
Suggested Citation:"Chapter 4 - Equilibrium Local Scour Predictive Equations." National Academies of Sciences, Engineering, and Medicine. 2011. Scour at Wide Piers and Long Skewed Piers. Washington, DC: The National Academies Press. doi: 10.17226/14426.
×
Page 26
Page 27
Suggested Citation:"Chapter 4 - Equilibrium Local Scour Predictive Equations." National Academies of Sciences, Engineering, and Medicine. 2011. Scour at Wide Piers and Long Skewed Piers. Washington, DC: The National Academies Press. doi: 10.17226/14426.
×
Page 27
Page 28
Suggested Citation:"Chapter 4 - Equilibrium Local Scour Predictive Equations." National Academies of Sciences, Engineering, and Medicine. 2011. Scour at Wide Piers and Long Skewed Piers. Washington, DC: The National Academies Press. doi: 10.17226/14426.
×
Page 28
Page 29
Suggested Citation:"Chapter 4 - Equilibrium Local Scour Predictive Equations." National Academies of Sciences, Engineering, and Medicine. 2011. Scour at Wide Piers and Long Skewed Piers. Washington, DC: The National Academies Press. doi: 10.17226/14426.
×
Page 29
Page 30
Suggested Citation:"Chapter 4 - Equilibrium Local Scour Predictive Equations." National Academies of Sciences, Engineering, and Medicine. 2011. Scour at Wide Piers and Long Skewed Piers. Washington, DC: The National Academies Press. doi: 10.17226/14426.
×
Page 30
Page 31
Suggested Citation:"Chapter 4 - Equilibrium Local Scour Predictive Equations." National Academies of Sciences, Engineering, and Medicine. 2011. Scour at Wide Piers and Long Skewed Piers. Washington, DC: The National Academies Press. doi: 10.17226/14426.
×
Page 31
Page 32
Suggested Citation:"Chapter 4 - Equilibrium Local Scour Predictive Equations." National Academies of Sciences, Engineering, and Medicine. 2011. Scour at Wide Piers and Long Skewed Piers. Washington, DC: The National Academies Press. doi: 10.17226/14426.
×
Page 32
Page 33
Suggested Citation:"Chapter 4 - Equilibrium Local Scour Predictive Equations." National Academies of Sciences, Engineering, and Medicine. 2011. Scour at Wide Piers and Long Skewed Piers. Washington, DC: The National Academies Press. doi: 10.17226/14426.
×
Page 33
Page 34
Suggested Citation:"Chapter 4 - Equilibrium Local Scour Predictive Equations." National Academies of Sciences, Engineering, and Medicine. 2011. Scour at Wide Piers and Long Skewed Piers. Washington, DC: The National Academies Press. doi: 10.17226/14426.
×
Page 34
Page 35
Suggested Citation:"Chapter 4 - Equilibrium Local Scour Predictive Equations." National Academies of Sciences, Engineering, and Medicine. 2011. Scour at Wide Piers and Long Skewed Piers. Washington, DC: The National Academies Press. doi: 10.17226/14426.
×
Page 35
Page 36
Suggested Citation:"Chapter 4 - Equilibrium Local Scour Predictive Equations." National Academies of Sciences, Engineering, and Medicine. 2011. Scour at Wide Piers and Long Skewed Piers. Washington, DC: The National Academies Press. doi: 10.17226/14426.
×
Page 36

Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

16 Introduction A large number of equations has been proposed for estimat- ing equilibrium local scour depths at bridge piers. Detailed descriptions of the data used in the development of the equa- tions are not usually available. However, the equations can be compared with each other for hypothetical, but practical, lab- oratory and field situations. This procedure was used for the initial screening of the equilibrium scour equations. A selection of the more widely used and recently published equations is given in Table 4. A brief description of the equa- tions is given in the following paragraphs. The regime equations of Inglis (1949), Ahmad (1953), Chitale (1962), and Blench (1969) were derived from meas- urements in irrigation canals in India and are supposed to describe the conditions under which these canals are stable for the existing sediment supply. These equations can lead to negative estimates of local scour. Laursen (1958, 1963) extended the solutions for the long rectangular contraction to local scour at piers using the observation that the depth of local scour does not depend on the contraction ratio until the scour holes from neighboring piers start to overlap. For sand, the width of the scour hole normal to the flow was observed to be about 2.75 times the scour depth (2.75ys). Laursen (1958, 1963) assumed the scour in the contraction, defined by this width, to be a fraction of the scour depth at the pier or abutment, leading to Equations 3 and 5 in Table 4. The equation in HEC-18 (Richardson and Davis 2001), Equation 22 in Table 4, was determined from a plot of labora- tory data for circular piers. The data used were selected from Chabert and Engeldinger (1956) and Colorado State University data (Shen et al. 1966); these data are the same as those used by Shen et al. (1969) in the derivation of Equation 9. Equation 22 has been progressively modified over the years and is currently recommended by the FHWA for estimating equilibrium scour depths at simple piers (Richardson and Davis 2001). The equa- tion includes a multiplying factor, Kw, to be applied to wide piers in shallow flows. The factor was developed by Johnson and Torrico (1994) using laboratory and field data for large piers. Equation 18, presented by Gao et al. (1993), has been used by highway and railway engineers in China for more than 20 years. The equation was developed from Chinese data for local scour at bridge piers, including 137 live-bed data points and 115 clear-water data points. The equation was tested using field data obtained prior to 1964. Several of the equations are based on field data, including the regime equations (Inglis 1949, Ahmad 1953, Chitale 1962, and Blench 1969) and the relations by Froehlich (1988), Ansari and Qadar (1994) and Wilson (1995). Equation 15 by Froehlich (1988) was fitted to 83 on-site measurements from bridges in the United States and elsewhere. It includes a safety factor equal to one pier diameter. Ansari and Qadar (1994) fitted envelope equations (num- bered 19 in Table 4) to more than 100 field measurements of pier scour depth, derived from 12 different sources and sev- eral countries, including 40 measurements from India. Ansari and Qadar (1994) also presented a comparison of the field data they used with estimates of scour depth obtained using Equation 6 by Larras (1963), Equation 7 by Breusers (1965), Equation 12 by Neill (1973), Equation 17 by Breusers and Raudkivi (1991), and an equation by Melville and Sutherland (1988) that is the forerunner of Equation 21. The May and Willoughby (1990) equation (numbered 16 in Table 4) was derived from data produced by a laboratory study of local scour around large obstructions such as caissons and cofferdams for the construction of bridges across rivers and estuaries. The laboratory study, which focused on cases where the width of the structure was large relative to the flow depth, is especially relevant to this study. May and Willoughby noted that existing design formulae tended to overestimate the amount of scour. Melville (1997) presented a physically justified method (Equation 21) to estimate local scour depth at piers based on C H A P T E R 4 Equilibrium Local Scour Predictive Equations

17 Reference Equation Notes No. Inglis (1949) 0 782 3 s 1y y q1 70 a a ./ . q = average discharge intensity upstream from the bridge (ft2/s) a = pier width 1 Ahmad (1953) 2 3s 1 s 2y y 0 45K q /. q2 = local discharge intensity in contracted channel (ft2/s) 2 Laursen (1958) 1 7 s s 1 1 1 y ya 5 5 1 1 y y 11 5 y . . . Applies to live-bed scour 3 Chitale (1962) 2s 1 y 6 65 Fr 0 51 5 49 Fr y . . . Fr = Froude Number of the approach flow = V1/(gy1)0.5 4 Laursen (1963) 7 6 s 1s 0 5 1 1 1 c y 1 11 5 yya 5 5 1 y y / . . . Applies to clear-water scour 1 = grain roughness component of bed shear c = critical shear stress at threshold of motion 5 Larras (1963) 0 75s sy 0 43K K a .. a is in ft 6 Breusers (1965) sy 1 4a. Derived from data for tidal flows 7 Blench (1969) 0 25 s 1 r r y y a1 8 y y . . yr = regime depth =1.48(q2/FB)a/3 where FB=1.9 (D)0.5, D is in mm and q in m2/s 8 Shen et al. (1969) 0 619 1 s V ay 0 000223 . . 0 619 1 s V ay 0 000223 . . = kinematic viscosity 9 Coleman (1971) 0 9 1 1 s V V0 6 a2gy . . 10 Hancu (1971) 1 32 s c1 c 2Vy V2 42 1 a V ga / . (2V1/Vc – 1) = 1 for live-bed scour 11 Neill (1973) s sy K a Ks = 1.5 for round-nosed and circular piers; Ks = 2.0 for rectangular piers 12 Breusers et al. (1977) s 1 1 1 2 c y V yf 2 0 K K a V a . tanh f(V1/Vc) = 0 for V1/Vc 0.5 = (2V1/Vc – 1) for 0.5 < V1/Vc < 1 = 1 for V1/Vc > 1 13 Jain (1981) 0 3 0 25s 1 c yy 1 84 Fr a a . . . Applies to maximum clear-water scour 14 Froehlich (1988) 0 080 62 0 46 p0 2s 1 s 50 ay y a0 32K Fr 1 a a a D .. . . . ap = projected width of pier 15 May and Willoughby (1990) s sc s s sc sm y yy 2 4f y y . For circular cylinder: fs = 1.0 1 76 s 1 1 sc c c 1 c y V V 1 3 66 1 0 52 1 0 y V V V 1 0 1 0 V . . . . . . 0 6 sc 1 1 sm 1 y y y 0 55 2 7 y a a y 1 0 1 0 a . .. . . . 16 Table 4. Equilibrium scour predictive equations. (continued on next page)

18 Reference Equation Notes No. Breusers and Raudkivi (1991) s y s d y 2 3 K K K K K a . For an aligned pier, y s =2.3K s K d K b 17 Gao et al. (1993) 0 6 0 0 15 0 0 7 1 c s 1 c c V V y 0 46 K a y D V V ' . . . ' . 0 5 0 1 4 7 s 1 1 c 0 7 2 y 1 0 y V 1 7 6 D 6 05 x1 0 D D . . . . . 0 053 c c D V 0 645 V a . ' . where y s , a, y 1 , D, V 1 , V c , V c ' are in S.I. units. V c = incipient velocity for local scour at a pier K = shape and alignment factor = 1 for clear-water scour < 1 for live-bed scour i.e., 10 9 3 5 2 23 D c V V . . lo g 18 Ansari and Qadar (1994) 3 0 s p p 0 4 s p p y 0 024a a 7 2 ft y 2 238 a a 7 2 ft . . . . . . 19 Wilson (1995) 0 4 s 1 y y 0 9 a a . * * . a * = effective width of pier 20 Melville (1997) s y a s I D y K K K K K ya 1 0 5 ya 1 1 ya 1 1 1 l p c 1 l p c I c c I K 2 4a for a y 0 7 K 2 y a for 0.7 a y 5 K 4 5y for a y 5 V V V V V V K for 1 0 V V K 1 . . / . ( ) / . / ( . 1 l p c c D 1 0 50 50 D 50 V V V for 1 0 V a a K 0 57 2 2 4 for 25 D D a K 1 for 25 D ( ) ) . . l og . 21 Richardson and Davis (2001) 0 3 5 0 4 3 s 1 s w 3 4 y y 2K K K K K Fr a a . . K 3 = factor for mode of sediment transport K 4 = factor for armoring by bed ma terial K w = factor for very wide piers after Johnson and Torrico (1994) y s(m ax ) = 2.4b for Fr 0.8 y s(m ax ) = 3b for Fr > 0.8 22 Table 4. (Continued).

extensive sets of laboratory data from The University of Auckland and elsewhere (Chabert and Engeldinger 1956, Laursen and Toch 1956, Jain and Fischer 1979, Chee 1982, Chiew 1984, Ettema 1980, Hancu 1971, Shen et al. 1966). The method uses a number of multiplying factors (K-factors) for the effects of the various parameters, which influence scour. The values of the K-factors were determined from envelope curves fitted to the data. The method is, therefore, inherently conservative. The method defines wide piers as those having large values of the ratio a/y1 (>5). A similar rationally based method is given by Sheppard and Miller (2006) equations (numbered 23 in Table 4). The equa- tions are based principally on laboratory data, as well as a few field measurements. The equations include the important observation that normalized local scour depths’ dependence on a/D50 increases until the value of a/D50 equals approxi- mately 40, at which point dependence begins to decrease. One possible explanation for this behavior was given by Sheppard (2004). Ettema et al. (2006) conducted experiments for local scour at cylindrical piers placed in a sand bed. The authors contend that the experiments show the importance of considering similitude of large-scale turbulence structures when conduct- ing flume experiments on local scour at cylinders. They pro- posed a correction factor, ao, to adjust scour-depth estimates obtained from small-scale cylinders. Ettema et al. (2006) used the largest cylinder size (1.3 ft) as the reference size. It is not known if this equation can be applied to wider piers and, if so, how to select ao. Initial Screening of Equilibrium Scour Predictive Equations Twenty-three equations were assembled for evaluation and assessment. These equations are presented in Table 4. Some of these equations are a function of critical velocity. If the equation did not specify a method to calculate the critical velocity, it was calculated using Equation 24. The first screening procedure consisted of solving all of the equations for a range of input values and comparing the results. The values of the parameters used in Figures 13 through 20 are V u y D u ft s c c c * * . . . = ⎛⎝⎜ ⎞⎠⎟ = + 5 75 1685 0 377 0 1 50 log 410 0 1 1 24 0 0 50 1 4 50 50 0 5 D mm D mm u Dc . * . . ( ) . < < = − 213 1 100501 50 50 D mm D mm where D is in mm u c − < < , &* V are in ft s and y ftc , in1 19 Reference Equation Notes No. Sheppard and Miller (2006) s 1 1 2 3 c lp 1 1 l p s c c c 1 1 3 lp lp c c c c s 1 y V 2 5 f f f for 0 4 7 1 .0 a V V V V 1 V y V V V V f 2 2 2 5f for 1 V V a V V 1 1 V V y 2 2 f a * * * . . . . . l p 1 c c 0 4 1 1 2 1 2 c 50 3 1 2 0 1 3 50 50 V V for V V y f a V f 1 1 7 5 V a D f a a 0 4 10 6 D D . * * . . * * tanh . l n . . lp 1 0 lp 2 * c 1 0 1 90 lp 1 l p1 lp 2 l p lp 2 l p2 lp 1 V = 0. 8 g y V = 29.31 u 4 y D V for V V V V fo r V V log 4 a effective diameter projected width * shape factor Shape factor = 1, circular = 0 .8 6 + 0.97 square 4 = f lo w skew angle in radians * , 23 Table 4. (Continued).

01 2 3 4 5 y s /a V1/Vc=3, y1/a=0.33, D50=0.2 mm a= 2 in a= 3 ft a= 33 ft Ing lis 19 49 Ah ma d 1 95 3 La urs en 19 58 , 6 3 Ch ital e 1 96 2 La rra s 1 96 3 Bre us ers 19 65 Ble nc h 1 96 9 Sh en et al . 1 96 9 Co lem an 19 71 Ha nc u 19 71 Ne ill 1 97 3 Bre us ers e t a l. 1 97 7 Ja in 19 81 Fro eh lich 19 88 Ma y & W illo ug hb y 1 99 0 Bre us er s & Ra ud kiv i 1 99 1 Ga o e t a l. 1 99 3 An sa ri & Qa dar 19 94 Wi lso n 19 95 Me lvil le 19 97 Ric ha rds on & Da vis 20 01 Sh ep pa rd & M ille r 2 00 6 Note: The pier width is large compared to the water depth, and the sediment is fine sand. 0 1 2 3 4 5 y s /a V1/Vc=1, y1/a=0.33, D50=0.2 mm a= 2 in a= 3 ft a= 33 ft Ing lis 19 49 Ah ma d 1 95 3 La urs en 19 58 , 6 3 Ch ital e 19 62 La rra s 1 96 3 Bre us ers 19 65 Ble nc h 1 96 9 Sh en et al . 1 96 9 Co lem an 19 71 Ha nc u 19 71 Ne ill 1 97 3 Bre us er s e t a l. 1 97 7 Ja in 19 81 Fro eh lich 19 88 Ma y & W illo ug hb y 1 99 0 Bre us er s & Ra ud kiv i 1 99 1 Ga o e t a l. 1 99 3 An sa ri & Qa dar 19 94 Wi lso n 19 95 Me lvil le 19 97 Ric ha rds on & Da vis 20 01 Sh ep pa rd & M ille r 2 00 6 Note: The pier width is large compared to the water depth, and the sediment is fine sand. 20 Figure 13. Comparison of normalized local scour depth predictions using 22 different methods for transition from clear-water to live-bed scour conditions. Figure 14. Comparison of normalized local scour depth predictions using 22 different methods for a particular live-bed scour condition.

01 2 3 4 5 y s /a V1/Vc=3, y1/a=3, D50=0.2 mm a= 2 in a= 3 ft a= 33 ft Ing lis 19 49 Ah ma d 1 95 3 La urs en 19 58 , 6 3 Ch ital e 1 96 2 La rra s 1 96 3 Bre us ers 19 65 Ble nc h 1 96 9 Sh en et al . 1 96 9 Co lem an 19 71 Ha nc u 19 71 Ne ill 1 97 3 Bre us ers e t a l. 1 97 7 Ja in 19 81 Fro eh lich 19 88 Ma y & W illo ug hb y 1 99 0 Bre us er s & Ra ud kiv i 1 99 1 Ga o e t a l. 1 99 3 An sa ri & Qa dar 19 94 Wi lso n 19 95 Me lvil le 19 97 Ric ha rds on & Da vis 20 01 Sh ep pa rd & M ille r 2 00 6 Note: The water depth is deep relative to pier width, and the sediment is fine sand. 0 1 2 3 4 5 y s /a V1/Vc=1, y1/a=3, D50=0.2 mm a= 2 in a= 3 ft a= 33 ft Ing lis 19 49 Ah ma d 1 95 3 La urs en 19 58 , 6 3 Ch ital e 1 96 2 La rra s 1 96 3 Bre us ers 19 65 Ble nc h 1 96 9 Sh en et al . 1 96 9 Co lem an 19 71 Ha nc u 19 71 Ne ill 1 97 3 Bre us ers e t a l. 1 97 7 Ja in 19 81 Fro eh lich 19 88 Ma y & W illo ug hb y 1 99 0 Bre us er s & Ra ud kiv i 1 99 1 Ga o e t a l. 1 99 3 An sa ri & Qa dar 19 94 Wi lso n 19 95 Me lvil le 19 97 Ric ha rds on & Da vis 20 01 Sh ep pa rd & M ille r 2 00 6 Note: The water depth is deep relative to pier width, and the sediment is fine sand. 21 Figure 15. Comparison of normalized local scour depth predictions using 22 different methods for transition from clear-water to live-bed scour conditions. Figure 16. Comparison of normalized local scour depth predictions using 22 different methods for a particular live-bed scour condition.

01 2 3 4 5 y s /a V1/Vc=3, y1/a=0.33, D50=3 mm Ing lis 19 49 Ah ma d 1 95 3 La urs en 19 58 , 6 3 Ch ital e 1 96 2 La rra s 1 96 3 Bre us ers 19 65 Ble nc h 1 96 9 Sh en et al . 1 96 9 Co lem an 19 71 Ha nc u 19 71 Ne ill 1 97 3 Bre us ers et al. 19 77 Ja in 19 81 Fro eh lich 19 88 Ma y & W illo ug hb y 1 99 0 Bre us er s & Ra ud kiv i 1 99 1 Ga o e t a l. 1 99 3 An sa ri & Qa dar 19 94 Wi lso n 1 99 5 Me lvil le 19 97 Ric ha rds on & Da vis 20 01 Sh ep pa rd & M ille r 2 00 6 Note: The pier width is large relative to the water depth and the sediment is very coarse sand. a= 2 in a= 3 ft a= 33 ft 0 1 2 3 4 5 y s /a V1/Vc=1, y1/a=0.33, D50=3 mm a= 2 in a= 3 ft a= 33 ft Ing lis 19 49 Ah ma d 1 95 3 La urs en 19 58 , 6 3 Ch ital e 1 96 2 La rra s 1 96 3 Bre us ers 19 65 Ble nc h 1 96 9 Sh en et al . 1 96 9 Co lem an 19 71 Ha nc u 19 71 Ne ill 1 97 3 Bre us er s e t a l. 1 97 7 Ja in 19 81 Fro eh lich 19 88 Ma y & W illo ug hb y 1 99 0 Bre us er s & Ra ud kiv i 1 99 1 Ga o e t a l. 1 99 3 An sa ri & Qa dar 19 94 Wi lso n 19 95 Me lvil le 19 97 Ric ha rds on & Da vis 20 01 Sh ep pa rd & M ille r 2 00 6 Note: The pier width is large compared to the water depth, and the sediment is very coarse sand. 22 Figure 17. Comparison of normalized local scour depth predictions using 22 different methods for transition from clear-water to live-bed scour conditions. Figure 18. Comparison of normalized local scour depth predictions using 22 different methods for a particular live-bed scour condition.

01 2 3 4 5 y s /a V1/Vc=3, y1/a=3, D50=3 mm a= 2 in a= 3 ft a= 33 ft Ing lis 19 49 Ah ma d 1 95 3 La urs en 19 58 , 6 3 Ch ital e 1 96 2 La rra s 1 96 3 Bre us ers 19 65 Ble nc h 1 96 9 Sh en et al . 1 96 9 Co lem an 19 71 Ha nc u 19 71 Ne ill 1 97 3 Bre us ers et al. 19 77 Ja in 19 81 Fro eh lich 19 88 Ma y & W illo ug hb y 1 99 0 Bre us er s & Ra ud kiv i 1 99 1 Ga o e t a l. 1 99 3 An sa ri & Qa dar 19 94 Wi lso n 1 99 5 Me lvil le 19 97 Ric ha rds on & Da vis 20 01 Sh ep pa rd & M ille r 2 00 6 Note: The water depth is deep relative to the pier width, and the sediment is very coarse sand. 0 1 2 3 4 5 y s /a V1/Vc=1, y1/a=3, D50=3 mm a= 2 in a= 3 ft a= 33 ft Ing lis 19 49 Ah ma d 1 95 3 La urs en 19 58 , 6 3 Ch ital e 1 96 2 La rra s 1 96 3 Bre us ers 19 65 Ble nc h 1 96 9 Sh en et al . 1 96 9 Co lem an 19 71 Ha nc u 19 71 Ne ill 1 97 3 Bre us ers et al. 19 77 Ja in 19 81 Fro eh lich 19 88 Ma y & W illo ug hb y 1 99 0 Bre us er s & Ra ud kiv i 1 99 1 Ga o e t a l. 1 99 3 An sa ri & Qa dar 19 94 Wi lso n 1 99 5 Me lvil le 19 97 Ric ha rds on & Da vis 20 01 Sh ep pa rd & M ille r 2 00 6 Note: The water depth is deep relative to the pier width, and the sediment is very coarse sand. 23 Figure 19. Comparison of normalized local scour depth predictions using 22 different methods for transition from clear-water to live-bed scour conditions. Figure 20. Comparison of normalized local scour depth predictions using 22 different methods for a particular live-bed scour condition.

V1/Vc = 1 and 3; y1/a = 0.33, 1, and 3; D50 = 0.2 and 3 mm; and a = 2 in., 3 ft, and 33 ft. The results from this evaluation are presented in the form of bar charts in Figures 13 though 20. Negative scour depth predictions were set equal to zero in the charts. Figures 13 and 15 show scour depth predictions for scenar- ios comprising clear-water to live-bed transition (threshold) flows (V1/Vc = 1), fine sand (D50 = 0.2 mm), and two different flow depth to pier width ratios (y1/a = 3, 0.33). Figures 14 and 16 are a parallel set of results for live-bed conditions (V1/Vc = 3). Similarly, Figures 17 through 20 apply to situations with coarser sediment (D50 = 3 mm). The equations used in producing the results shown in Figures 13 through 20 span the period from 1949 to 2006 in their development and publication. Improvements in the understanding of local scour processes and scour hole development during this time period resulted in improve- ments to the equilibrium scour predictive equations/methods. For example, several of the earlier equations predicted negative scour depths for some of the input conditions. Also, the differences between the predictions become less with time. Variations in the predictions of local scour for different pier sizes (“laboratory” to “typical field” to “very large field”) are reported. Some methods predict scour depth ratios decreas- ing with increasing pier size; others show constant values of scour depth ratio from laboratory to field, with one equation by Coleman (1971) showing larger normalized scour depths in the field than in the laboratory. These plots help identify those equations that produce un- realistic results for prototype-scale piers and thus aid in elimi- nating such equations from further consideration. The regime equations of Inglis (1949), Ahmad (1953) and Chitale (1962) yield negative scour depths in some cases. The Coleman (1971) equation yields an unrealistic trend with increasing pier size and therefore was eliminated. Several other equations predict unreasonably high normalized scour depths (Inglis 1949, Ahmad 1953, Chitale 1962, Hancu 1971, and Shen et al. 1969) and were eliminated. This process left 17 methods/equations for the final analysis. Modifications to Equilibrium Scour Predictive Methods One of the objectives of this study was to determine if any of the predictive equations could be modified to improve their accuracy. The overall accuracy of most of the equations could be improved by adjusting one or more of their coefficients. However, in almost every case this adjustment increased underprediction, which for design equations is not accept- able. The Sheppard and Miller (2006) equation was modified to create the new S/M equation. Sheppard/Melville (S/M) Equation The Sheppard and Miller (2006) and Melville (1997) equa- tions were melded and slightly modified to form a new equa- tion referred to here as the S/M equation. The modifications consisted of: 1. Changing the 1.75 coefficient to 1.2 in the term for f2 in Equation 23, 2. Changing the value of V1/Vc where local scour is initiated from 0.47 to 0.4, and 3. Modifying/simplifying the manner in which the live-bed peak velocity is computed. The resulting equation is presented in Table 5. These changes improved the accuracy of the predictions for both laboratory and field data. However, this equation underpredicts some of the measured field data at very low velocities (i.e., low values of V1/Vc), most likely due to rela- tively large sediment size distributions (large σg). The under- predictions are illustrated in Figures 21 and 22, which show before- and after-modification upper bound curves for labo- ratory and field data, respectively. The scour depths in this range of flow velocities are, however, very small and therefore are not likely to affect prediction of design scour depths. Also, the reported scour depths in this range of V1/Vc seem large for the magnitude of the flow velocities (i.e., the accuracy of these data is questionable). Equation in HEC-18 No attempt was made to modify the scour equation in the current version of HEC-18 (Richardson and Davis 2001) because it does not properly account for the physics of the local scour processes. That is, all of the known local scour mechanisms are not accounted for with the dimensionless groups in this equation. The equation does, however, contain a wide-pier correction factor developed by Johnson and Torrico (1994). Predicted versus measured scour depth plots using this equation are shown in Figure 23 (laboratory data) and Figure 24 (field data). In general, the wide-pier correc- tion decreases the magnitude of the predicted scour depths. The wide-pier correction does, however, increase the number of underpredictions in both the laboratory and field data. The overall error for the dimensional scour is reduced with the wide-pier correction factor, but the error for the normalized scour is increased. The wide-pier correction factor also creates 24

Reference Equation Notes No. S/M s 1 1 2 3 c lp 1 1 l p s c c c 1 1 3 lp lp c c c c s 1 y V 2 5 f f f for 0 4 1.0 a V V V V 1 V y V V V V f 2 2 2 5f for 1 0 V V a V V 1 1 V V y 2 2 f a * * * . . . . . . l p 1 c c 0 4 1 1 2 1 2 c 50 3 1 2 0 1 3 50 50 V V for V V y f a V f 1 1 2 V a D f a a 0 4 10 6 D D . * * . . * * tanh . l n . . lp 1 c lp 2 1 lp 1 l p1 lp 2 l p lp 2 l p2 lp 1 V 5 V V 0 6 g y V for V V V V fo r V V . 4 a e ffect iv e di ameter pr ojected width * shape factor Sh ape fact or =1 , circu la r = 0 .8 6 + 0.97 squar e 4 = f lo w skew angle in radians * , 25 25 Table 5. The S/M local scour equations. 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 V1/Vc y s /a Data Sheppard & Miller 2006 S/M Figure 21. Measured laboratory data at low velocities compared to the upper limit of Sheppard and Miller (2006) and S/M equations.

0 0.5 1 1.5 2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Measured y s /a Pr e di c te d y s /a With Wide-Pier Corr. W/O Wide-Pier Corr 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Measured y s /a* Pr e di c te d y s /a * With Wide-Pier Corr. W/O Wide-Pier Corr 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 V1/Vc y s /a Data Sheppard & Miller 2006 S/M 26 Figure 22. Measured field data at low velocities compared to the upper limit of Sheppard and Miller (2006) and S/M equations. Figure 23. Effect of wide-pier correction on HEC-18 normalized scour depth predictions for laboratory data. Figure 24. Effect of wide-pier correction on HEC-18 normalized scour depth predictions for field data. a discontinuity in predicted scour depth with increasing pier width as shown in Figure 25. Evaluation of Equilibrium Scour Predictive Methods Evaluating predictive equations using laboratory data is rel- atively straightforward because, in most cases in the database, either the scour has reached equilibrium or the measured depth has been extrapolated to an equilibrium value. In addi- tion, the input values (water depth, flow velocity, sediment properties, etc.) are all accurately known. Obtaining accurate measurements of input parameters in the field is much more difficult and the maturity of the scour hole is almost never known. Several of the scour depths in the field data set are extremely small for the pier size, water depth, and flow veloc- ity. This observation suggests that the duration of the flow at the reported velocity was short; that the bed material was cohesive although reported as cohesionless; or perhaps both. With the uncertainty associated with even the most reliable field data, it is not appropriate to use it directly to evaluate the predictive equations. However, with only a few exceptions, the predicted values should not be less than the measured val- ues, assuming the measured values are accurate and do not include other types of scour (contraction, degradation, etc.). One exception is the case where the measured scour depth is due to a previous, more severe, flow event. Figure 26 gives some insight into the flow duration problem for field

Note: The lines are best linear fits to the data. -2 0 2 0 0.5 1 1.5 2 m ea s u re d/ co m pu te d y s Jain 1981 -2 0 2 0 0.5 1 1.5 2 m ea s u re d/ co m pu te d y s Froehlich 1988 -2 0 2 0 0.5 1 1.5 2 m e as u re d/ c o m pu te d y s Melville 1997-mod. -2 0 2 0 0.5 1 1.5 2 m e as u re d/ c o m pu te d y s HEC-18 -2 0 2 0 0.5 1 1.5 2 m e a s u re d/ c o m pu te d y s HEC-18-no wp corr -2 0 2 0 0.5 1 1.5 2 m e a s u re d/ c o m pu te d y s S/M log( t90 in days) log( t90 in days) 0 5 10 15 20 25 30 0 5 10 15 20 25 30 35 Pier Width (ft) Eq u ili br iu m Sc o u r (ft) W/O Wide-Pier Corr With Wide-Pier Corr. 27 Figure 25. An example of HEC-18 scour predictions (y1 = 15 ft, D50 = 0.3 mm, V1 = 8 ft/s). Figure 26. Measured/predicted versus predicted time to reach 90% of equilibrium scour depth for the field data.

Laboratory (441 points) Field (760 points) Reference SSE% SSEn% SSE% SSEn% Total Under Total Under Under Under Inglis (1949) 77.0 30.1 297.0 57.3 32.7 106.1 Ahmad (1953) 167.7 56.2 957.1 19.8 70.3 127.8 Laursen (1958, 1963) 24.0 21.4 24.3 12.6 4.1 7.2 Chitale (1962) 123.3 31.1 921.1 19.2 25.8 67.1 Larras (1963) 21.1 1.8 27.2 0.5 8.0 4.2 Breusers (1965) 31.0 1.8 10.9 5.7 0.2 0.2 Blench (1969) 18.6 18.3 60.6 60.0 5.7 7.8 Shen et al. (1969) 19.1 6.4 55.5 0.9 3.0 1.9 Coleman (1971) 87.2 87.2 97.7 97.7 25.2 49.8 Hancu (1971) 130.2 20.8 250.9 7.7 3.1 6.6 Neill (1973) 38.7 1.2 11.1 4.0 0.1 0.1 Breusers et al. (1977) 18.9 13.0 12.5 6.2 4.9 7.2 Jain (1981) 9.1 0.2 24.5 0.6 2.2 1.1 Froehlich (1988) 21.0 0.6 10.3 1.7 0.0 0.0 May & Willoughby (1990) 17.9 6.5 18.4 2.6 4.1 6.2 Gao et al. (1993) 75.7 16.2 158.5 8.1 7.4 12.9 Ansari & Qadar (1994) 52.9 52.5 97.8 97.8 0.6 12.0 Wilson (1995) 9.1 6.9 15.6 5.3 2.8 1.9 Melville (1997) 34.9 0.3 24.8 0.1 0.2 0.1 Melville (1997)–mod. 34.9 0.2 24.8 0.1 0.2 0.1 HEC-18 7.9 1.8 21.0 0.3 1.1 1.1 HEC-18–no wp corr 9.5 1.2 21.1 0.3 0.4 0.5 Sheppard & Miller (2006) 5.9 0.1 11.7 0.3 1.9 2.6 S/M 6.8 0.1 13.0 0.3 0.3 0.2 conditions. In this figure, the ratio of measured divided by pre- dicted scour is plotted versus t90 (time required to reach 90% of equilibrium) for six equilibrium scour equations. Best-fit lines to the data are also shown. The time to reach 90% of equi- librium scour depth is computed using the M/S equation dis- cussed later in this report. Note that t90 is more than 100 days for some situations. Underpredictions increase for all equa- tions with increasing t90. This result suggests that a large por- tion of the field cases have not reached equilibrium. The errors associated with predicting the measured equilib- rium scour depths were computed using the following equation for dimensional scour depths, ys: The corresponding equation for normalized scour depths, ys/a, is SSEn% y a y a y s measured s model s measure = − ⎛⎝⎜ ⎞⎠⎟∑ 2 d a ⎛⎝⎜ ⎞⎠⎟ × ∑ 2 100 27( ) SSE% y y y s measured s computed s measured = −( ) ( ) ∑ 2 2∑ × 100 26( ) The desired method/equation is the one that has the least overall error and least underprediction. Because these equations are being recommended for design, underpre- diction errors must be weighted heavier. That is, greater emphasis must be given to reducing underprediction than to total error. Table 6 gives percentage errors (SSE and SSEn) for the labo- ratory and field data sets. Underpredictions are much smaller compared to the total error for all predictive equations because they are all conservative by design. For the reasons discussed previously, total errors for the field data are not presented because the maturity of these data are not known. The fact that all the equilibrium scour equations overpredict many of the measured depths, especially for the larger structures, indicates the need to account for the design flow event dura- tion in the prediction. However, even the best scour evolu- tion equations are not sufficiently accurate for this task at this time. Large-scale, live-bed scour evolution tests and the development of improved predictive scour evolution equa- tions are definitely needed. Six different measures of error were used for each method/ equation. The overall fit of all equations can be improved by modifying their coefficients, but, in general, such modifica- tion increases their underpredictions as can be seen between Sheppard and Miller (2006) and its modified form, S/M. The 28 Table 6. Absolute and normalized errors for predictive equilibrium scour equations.

Laboratory (441 points) Field (760 points) Reference SSE% SSEn% SSE% SSEn% Total Under Total Under Under Under Jain (1981) 9.1 0.2 24.5 0.6 2.2 1.1 Froehlich (1988) 21.0 0.6 10.3 1.7 0.1 0.0 Melville (1997)–mod. 34.9 0.2 24.8 0.1 0.2 0.1 HEC-18 7.9 1.8 21.0 0.3 1.1 1.1 HEC-18–no wp corr 9.5 1.2 21.1 0.3 0.4 0.5 S/M 6.8 0.1 13.0 0.3 0.3 0.2 Laboratory (441 points) Field (760 points) Reference SSE SSEn SSE SSEn Total Under Total Under Overall Under Under Overall Inglis (1949) 20 20 22 21 20 23 23 23 Ahmad (1953) 24 23 24 20 24 24 24 24 Laursen (1958, 1963) 13 19 11 18 16 15 17 16 Chitale (1962) 22 21 23 19 22 22 22 22 Larras (1963) 12 9 15 7 13 20 13 17 Breusers (1965) 14 11 2 14 12 5 5 5 Blench (1969) 8 17 17 22 17 18 18 19 Shen et al. (1969) 10 12 16 9 15 13 11 12 Coleman (1971) 21 24 18 23 23 21 21 21 Hancu (1971) 23 18 21 16 19 14 15 14 Neill (1973) 17 7 3 12 10 2 2 2 Breusers et al. (1977) 9 15 5 15 14 17 16 18 Jain (1981) 5 4 12 8 6 11 8 9 Froehlich (1988) 11 6 1 10 5 1 1 1 May & Willoughby (1990) 7 13 8 11 11 16 14 15 Gao et al. (1993) 19 16 20 17 18 19 20 20 Ansari & Qadar (1994) 18 22 19 24 21 8 19 13 Wilson (1995) 4 14 7 13 9 12 10 10 Melville (1997) 15 5 13 2 8 4 4 4 Melville (1997)–mod. 16 3 14 1 7 3 3 3 HEC-18 3 10 9 4 3 9 9 8 HEC-18–no wp corr 6 8 10 3 4 7 7 7 Sheppard & Miller (2006) 1 2 4 6 1 10 12 11 S/M 2 1 6 5 2 6 6 6 prediction of the absolute scour depth is important because the design scour depth will be computed using the dimen- sional form of the equation. The results of the error statistics are presented in Table 6 for all equations. The error statistics confirm the screening process in that the equations eliminated are the ones with the greatest errors. Table 7 gives the ordering of the equations according to the values given in Table 6. There are also two columns for overall error order for field and laboratory data. Each overall error order was calculated by averaging the columns to its left and ordering the results. Field data errors are not very informative by themselves because they do not include total errors. Total errors for the field data are also not meaningful because so many of the reported scour depths are not equilibrium values. Note that an equa- tion that grossly overpredicts can have small underpredic- tion errors. Six equations were chosen for final evaluation based on their performances listed in Tables 6 and 7: Jain (1981), Froehlich (1988), modified Melville (1997), HEC-18, HEC-18 without wide-pier correction, and S/M. Only the modified/final versions of the equations were considered. However, both forms of HEC-18 are presented because they are currently widely used in the United States. The error statistics and rankings for these six equations, based on performance with all data, are given in Tables 8 and 9, respectively (sta- tistics are the same as those given in Table 6, but they are repeated for easier access). Tables 10 and 11 give the error 29 Table 7. Ranking of all predictive equilibrium scour equations. Table 8. Errors for selected predictive equations.

Laboratory (17 points) Field (142 points) Reference SSE SSEn SSE SSEn Total Under Total Under Overall Under Under Overall Jain (1981) 5 1 5 1 1 6 5 5 Froehlich (1988) 6 2 6 2 6 1 1 1 Melville (1997)–mod. 1 5 3 5 3 4 3 3 HEC-18 2 6 1 6 5 5 6 6 HEC-18–no wp corr 4 4 2 4 4 2 2 2 S/M 3 3 4 3 2 3 4 4 Laboratory (17 points) Field (142 points) Reference SSE% SSEn% SSE% SSEn% Total Under Total Under Under Under Jain (1981) 15.4 0.0 10.9 0.0 5.6 5.4 Froehlich (1988) 94.8 0.0 55.2 0.0 0.0 0.0 Melville (1997)–mod. 3.3 1.3 7.8 0.4 0.7 1.0 HEC-18 4.2 3.9 2.4 1.9 3.6 6.2 HEC-18–no wp corr 8.7 0.6 4.9 0.2 0.3 0.9 S/M 5.1 0.0 8.5 0.0 0.5 1.5 Laboratory (441 points) Field (760 points) Reference SSE SSEn SSE SSEn Total Under Total Under Overall Under Under Overall Jain (1981) 3 3 5 5 5 6 5 5 Froehlich (1988) 5 4 1 6 6 1 1 1 Melville (1997)–mod. 6 2 6 1 3 2 2 2 HEC-18 2 6 3 3 2 5 6 6 HEC-18–no wp corr 4 5 4 2 4 4 4 4 S/M 1 1 2 4 1 3 3 3 statistics and ordering among these six equations based on performance for wide piers. Figures 27 through 30 show underprediction error ver- sus total error for the six selected methods. These plots show the performance of the different methods and how their errors can be modified by a multiplicative constant. The symbols indicate the errors produced by the method with a multiplier of one. As the multiplier changes from one, the symbol simply moves along the curve on which the symbol lies. Note that shifting the position of the symbol along its line for one data set does, however, shift its position on the other plots. For example, if the HEC-18 equation is multiplied by 1.2, its position moves from that shown with the left-pointing open triangle to that with the closed tri- angle. Note that even though there is improved performance for the dimensional scour (Figure 27), the normalized scour performance is significantly reduced (Figure 28). Figures 31 through 34 are plots of predicted versus mea- sured scour depths for six of the predictive equations for the various data sets. Based on these analyses, the best-performing equation and the one that attempts to account for the most important local scour mechanisms is the S/M equation. To illustrate this equation’s dependence on the various dimensionless groups, three example plots are presented in Figures 35 through 37. The HEC-18 equation is also presented in these equations. Figures 38 through 40 are additional comparisons between the S/M and HEC-18 equations for three different prototype design conditions (σ = 2.7, a/D50 = 10,000). 30 Table 9. Ranking of selected predictive equations based on performance with all data. Table 10. Errors for selected predictive equations for wide piers (a/D50 > 100, a/y1 > 2). Table 11. Ranking of selected predictive equations based on performance for wide piers (a/D50 > 100, a/y1 > 2).

0 5 10 15 20 25 30 35 40 45 0 0.5 1 1.5 2 Jain 1981 Froehlich 1988 Melville 1997-mod. HEC-18 HEC-18-no wp corr S/M Total Lab Error (%) Fi el d Un de rp re di ct io n E rr or (% ) Jain 1981 Froehlich 1988 Melville 1997-mod. HEC-18 HEC-18-no wp corr S/M 0 10 20 30 40 50 0 0.5 1 1.5 2 Jain 1981 Froehlich 1988 Melville 1997-mod. HEC-18 HEC-18-no wp corrS/M Total Error (%) Un de rp re di ct io n Er ro r ( %) Jain 1981 Froehlich 1988 Melville 1997-mod. HEC-18 HEC-18-no wp corr S/M 0 5 10 15 20 25 30 35 40 45 0 0.5 1 1.5 2 Jain 1981 Froehlich 1988 Melville 1997-mod. HEC-18 HEC-18-no wp corr S/M Total Error (%) Un de rp re di ct io n Er ro r ( %) Jain 1981 Froehlich 1988 Melville 1997-mod. HEC-18 HEC-18-no wp corr S/M 31 Figure 27. Underprediction scour error versus total error for laboratory data. Figure 28. Underprediction normalized scour error versus total error for laboratory data. Figure 29. Underprediction scour error for field data versus total error for laboratory data.

U.P. = underpredicted 0 1 2 3 4 5 0 1 2 3 4 5 Jain 1981 %cases U.P.=17.2 Measured y s (ft) Pr e di c te d y s (ft) 0 1 2 3 4 5 0 1 2 3 4 5 Froehlich 1988 %cases U.P.=30.3 Measured y s (ft) Pr e di c te d y s (ft) 0 1 2 3 4 5 0 1 2 3 4 5 Melville 1997-mod. %cases U.P.=7.25 Measured y s (ft) Pr e di c te d y s (ft) 0 1 2 3 4 5 0 1 2 3 4 5 HEC-18 %cases U.P.=14.7 Measured y s (ft) Pr e di c te d y s (ft) 0 1 2 3 4 5 0 1 2 3 4 5 HEC-18-no wp corr %cases U.P.=13.8 Measured y s (ft) Pr e di c te d y s (ft) 0 1 2 3 4 5 0 1 2 3 4 5 S/M %cases U.P.=8.16 Measured y s (ft) Pr e di c te d y s (ft) 0 5 10 15 20 25 30 35 0 0.5 1 1.5 2 Jain 1981 Froehlich 1988 Melville 1997-mod. HEC-18 HEC-18-no wp corr S/M Total Lab Error (%) Fi el d Un de rp re di ct io n E rr or (% ) Jain 1981 Froehlich 1988 Melville 1997-mod. HEC-18 HEC-18-no wp corr S/M 32 Figure 31. Predicted versus measured equilibrium scour depths for six predictive equations for laboratory data. Figure 30. Underprediction normalized scour error for field data versus total error for laboratory data.

U.P. = underpredicted 0 10 20 30 40 0 10 20 30 40 Jain 1981 %cases U.P.=3.68 Pr e di c te d y s (ft) 0 10 20 30 40 0 10 20 30 40 Froehlich 1988 %cases U.P.=0.78 0 10 20 30 40 0 10 20 30 40 Melville 1997-mod. %cases U.P.=1.44 0 10 20 30 40 0 10 20 30 40 HEC-18 %cases U.P.=9.34 Pr e di c te d y s (ft) Measured y s (ft) 0 10 20 30 40 0 10 20 30 40 HEC-18-no wp corr %cases U.P.=6.31 Measured y s (ft) 0 10 20 30 40 0 10 20 30 40 S/M %cases U.P.=2.89 Measured y s (ft) U.P. = underpredicted 0 1 2 3 0 0.5 1 1.5 2 2.5 3 Measured y s /a Pr e di c te d y s /a Jain 1981 %cases U.P.=17.2 0 1 2 3 0 0.5 1 1.5 2 2.5 3 Measured y s /a Pr e di c te d y s /a Froehlich 1988 %cases U.P.=30.3 0 1 2 3 0 0.5 1 1.5 2 2.5 3 Measured y s /a Pr e di c te d y s /a Melville 1997-mod. %cases U.P.=7.25 0 1 2 3 0 0.5 1 1.5 2 2.5 3 Measured y s /a Pr e di c te d y s /a HEC-18 %cases U.P.=14.7 0 1 2 3 0 0.5 1 1.5 2 2.5 3 Measured y s /a Pr e di c te d y s /a HEC-18-no wp corr %cases U.P.=13.8 0 1 2 3 0 0.5 1 1.5 2 2.5 3 Measured y s /a Pr e di c te d y s /a S/M %cases U.P.=8.16 Figure 32. Predicted versus measured normalized equilibrium scour depths for six predictive equations for laboratory data. Figure 33. Predicted versus measured equilibrium scour depths for six predictive equations for field data.

10 1 10 2 10 3 10 4 0.5 1 1.5 2 2.5 D50=0.4 mm, y1/a= 3, V1/Vc=1 a/D50 y s /a S/M HEC-18 U.P. = underpredicted 0 1 2 3 0 0.5 1 1.5 2 2.5 3 Pr e di c te d y s /a * Jain 1981 %cases U.P.=6.82 0 1 2 3 0 0.5 1 1.5 2 2.5 3 Froehlich 1988 %cases U.P.=1.13 0 1 2 3 0 0.5 1 1.5 2 2.5 3 Melville 1997-mod. %cases U.P.=1.39 0 1 2 3 0 0.5 1 1.5 2 2.5 3 Pr e di c te d y s /a * Measured y s /a* HEC-18 %cases U.P.=11.1 0 1 2 3 0 0.5 1 1.5 2 2.5 3 Measured y s /a* HEC-18-no wp corr %cases U.P.=6.95 0 1 2 3 0 0.5 1 1.5 2 2.5 3 Measured y s /a* S/M %cases U.P.=5.05 34 Figure 34. Predicted versus measured normalized equilibrium scour depths for six predictive equations for field data. Figure 35. Comparison of S/M and HEC-18 predictions of normalized scour depth versus a/D50.

0 10 20 30 40 50 60 70 0 5 10 15 20 25 30 35 a (ft) y s (ft) y1=30 ft, V1=3Vc, D50=0.2 mm S/M HEC-18 0 1 2 3 4 5 6 0 0.5 1 1.5 2 D50=0.4 mm, y1/a= 3, a/D50=10000 V1/Vc y s /a S/M HEC-18 0 1 2 3 4 5 0 0.2 0.4 0.6 0.8 1 D50=0.4 mm, a/D50=10000, V1/Vc=1 y1/a y s /a S/M HEC-18 35 Figure 36. Comparison of S/M and HEC-18 predictions of S/M normalized scour depth versus y1/a. Figure 37. Comparison of S/M and HEC-18 predictions of normalized scour depth versus V1 /Vc. Figure 38. Comparison of S/M and HEC-18 equation predictions of equilibrium scour depth as a function of pier diameter (for D50 = 0.2 mm).

0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 a (ft) y s (ft) y1=30 ft, V1=3Vc, D50=5 mm S/M HEC-18 0 10 20 30 40 50 60 70 0 10 20 30 40 50 a (ft) y s (ft) y1=30 ft, V1=3Vc, D50=0.8 mm S/M HEC-18 36 Figure 39. Comparison of S/M and HEC-18 equation predictions of equilibrium scour depth as a function of pier diameter (for D50 = 0.8 mm). Figure 40. Comparison of S/M and HEC-18 equation predictions of equilibrium scour depth as a function of pier diameter (for D50 = 5 mm).

Next: Chapter 5 - Local Scour Evolution Predictive Methods »
Scour at Wide Piers and Long Skewed Piers Get This Book
×
MyNAP members save 10% online.
Login or Register to save!
Download Free PDF

TRB’s National Cooperative Highway Research Program (NCHRP) Report 682: Scour at Wide Piers and Long Skewed Piers explores recommendations for a predictive equation for equilibrium local scour and a potential equation for predicting scour evolution rates at wide piers and skewed piers.

The equations are designed to help provide better estimates of local scour and scour evolution rates than those predicted by currently available equations. Such estimates have the potential to reduce over-predictions and the unwarranted need for countermeasures.

Appendixes A through E for NCHRP Report 682 provide further elaboration on the work performed in this project. The appendixes are only available online.

  1. ×

    Welcome to OpenBook!

    You're looking at OpenBook, NAP.edu's online reading room since 1999. Based on feedback from you, our users, we've made some improvements that make it easier than ever to read thousands of publications on our website.

    Do you want to take a quick tour of the OpenBook's features?

    No Thanks Take a Tour »
  2. ×

    Show this book's table of contents, where you can jump to any chapter by name.

    « Back Next »
  3. ×

    ...or use these buttons to go back to the previous chapter or skip to the next one.

    « Back Next »
  4. ×

    Jump up to the previous page or down to the next one. Also, you can type in a page number and press Enter to go directly to that page in the book.

    « Back Next »
  5. ×

    To search the entire text of this book, type in your search term here and press Enter.

    « Back Next »
  6. ×

    Share a link to this book page on your preferred social network or via email.

    « Back Next »
  7. ×

    View our suggested citation for this chapter.

    « Back Next »
  8. ×

    Ready to take your reading offline? Click here to buy this book in print or download it as a free PDF, if available.

    « Back Next »
Stay Connected!