Scour at Wide Piers and Long Skewed Piers (2011)

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37 Introduction Several methods have been proposed for predicting local scour evolution rates at bridge piers. Some of these methods and, where possible, the associated equations are shown in Table 12. The methods/equations vary in complexity. Some require computer programs to evaluate and therefore cannot be displayed in the table. In all cases considered, the com- puter programs were obtained from the developer and used in the evaluation process. A brief description of each of the 10 methods is presented in the following paragraphs. Shen et al. (1966) Shen et al. (1966) fitted the Chabert and Engeldinger (1956) data to an exponential function, Equation 29, to obtain an expression for scour depth as a function of time. The equation is based on a narrow range of flow and sediment conditions and was developed for circular piers. Sumer et al. (1992) Sumer et al. (1992) used measurements from 18 experiments of local live-bed scour at cylindrical piers to determine the exponent of an exponential function, Equation 30, for pre- dicting scour evolution rates. The equilibrium scour depth must be known a priori. If this value is not known, the authors recommend a value of 1 to 1.5 times pier width based on the results of Breusers et al. (1977). Kothyari et al. (1992) Kothyari et al. (1992) based their method on the scouring potential of the horseshoe vortex, which decreases as the scour hole enlarges. They assumed that the shear stress under the vortex at time t is a function of the initial shear stress, the initial area of the vortex, and the area of the vortex at time t. The initial shear stress is assumed to be 4 times the shear stress on the approach flow bed. Sediment nonuniformity and stratification are shown to have a significant effect on the scour depth. The effects of these elements, as well as that of unsteadiness of flow, are taken into account in the proposed method. The method applies to scour development at circu- lar piers and is based on experiments with pier widths rang- ing from 0.21 to 0.56 ft, uniform sediments with D50 equal to 0.4 to 4 mm, and sediment mixtures with D50 equal to 0.5 and 0.71 mm and standard deviations up to 7.8. Melville and Chiew (1999) Melville and Chiew (1999) proposed Equation 32 for the temporal development of local scour at cylindrical piers in uniform sand beds. The method was based on the data of Ettema (1980), together with additional data collected at the University of Auckland and Nanyang Technological University. They defined an equilibrium time scale for the scour process and showed that the equilibrium time scale, t*, is subject to the same influences of flow and sediment parameters as the equilibrium scour depth, ys. That is, t* is dependent on y1/a, V1/Vc, and a/D50. According to their method, the maximum time (in days) for the development of equilibrium scour depth is equal to 28.96 a/V1. Miller and Sheppard (2002) The method comprises a semi-empirical mathematical model for the time rate of local scour at a circular pile located in cohesionless sediment and subjected to steady or unsteady water flow. The model is intended for use for both clear-water and live-bed scour conditions. A knowledge of the structure dimensions, flow conditions, sediment properties, and the equilibrium scour depth for the instantaneous flow conditions is required as input to the model. The scour hole is assumed to have the idealized geometry of an inverted frustum of a C H A P T E R 5 Local Scour Evolution Predictive Methods

38 Reference Equations Notes No. Shen et al. (1966) 2 s t 0 4 m E y 2 5 Fr 1 e a . . - 1 2 932 y 1 0 3 3 0 3 3 1 1 1 m 0 026e , y in ft V t a E F r y y . . . . ln sρ ρ 29 Sumer et al. (1992) t T s t s 2 2 2 2 1 3 s 5 0 s 5 0 y y 1 e y u a T 0 0005 a g S 1 Dg S 1 D . * . y st = time-dependent scour 30 Kothyari et al. (1992) Computer program [see the discussion in the section “Kothyari et al. (2007)”] Computer Program 31 Melville and Chiew (1999) s t t s y ( t) K y 1 6 c t 1 1 e V t K C V t . exp ln 1 1 1 e 2 1 c c 0 2 5 1 1 1 1 e 3 1 c c V y V a t d ays C 0 4 6 1 0 0 4 V V a V V y y V a t d ays C 0 4 6 1 0 0 4 V V a a V . ( ) - . , . . ( ) - . , . . y st = time-dependent scour y s = Melville’s equilibrium equatio n C 1 = –0.03 C 2 = 48.26 days/sec C 3 = 30.89 days/sec 32 Miller and Sheppard (2002) Computer program [see the discussion in the section “Miller and Sheppard (2002)”] Computer Program 33 Oliveto and Hagar (2002, 2005) Oliveto et al. (2007) Clear water: st 0 5 1 5 s g d 1 3 2 R 1 y t 0 0 68 K F t y a . . / . l og Live bed: 1 4 s t 0 5 g d di 1 3 2 R R 1 1 4 s t 0 5 5 g d di 1 3 2 R R 1 y t t 0 4 4 F F 300 t t y a y t t F F 0 8 0 0 12 300 10 t t y a / . / / . / . l og . . lo g 1 d s 5 0 1 3 2 1 R 1 3 g s 50 V F S 1 gD y a t S 1 gD / / F di = densimetric particle Froude number for inception of scour 34 Mia and Nago (2003) Computer program [see the discussion in the section “Mia and Nago (2003)”] Computer Program 35 Chang et al. (2004) Computer program [see the discussion in the section “Chang et al. (2004)”] Computer Program 36 Yanmaz (2006) 0 3 7 0 9 5 2 s s S 2 S 1 dS dT T S S . . cot co t s t 0 5 50 s 5 0 s 2 0 9 5 0 6 3 0 2 4 1 9 g 50 s 5 0 1 3 s 50 2 y S a t D S 1 gD T a u a 0 2 31 TD D S 1 g D S 1 g D D . . . . . * * / * . t an y st = time-dependent scour = angle of repose of bed sediment S s Ss = specific gravity of bed sediment g = geometric standard deviation of particle size distribution u * = shear velocity T = transport-stage parameter T s = dimensionless time = kinematic viscosity 37 Table 12. Equations/methods for estimating local scour depth evolution with time.

39 right circular cone that maintains a constant shape through- out the scour process. The slope of the sides of the scour hole is uniform and equal to the submerged angle of repose for the sediment. Removal of sediment from the scour hole is limited to a narrow band adjacent to the cylinder where the effective shear stress is greatest. The sediment transport function used in the model is based on commonly used functions for transport on a flat bed. The effective shear stress in the scour hole, used in the sediment transport equation, is a function of the normalized scour depth (scour depth/equilibrium scour depth) and the structure, flow, and sediment parameters. The function for the shape of the effective shear stress versus normalized scour depth and its dependency on structure, flow, and sediment parameters was determined empirically using data from a number of clear-water and live-bed scour experiments conducted at the Universities of Florida and Auckland and the USGS Laboratory in Turners Falls, Massachusetts. These experiments cover a wide range of structure, flow, and sediment conditions. This method must be programmed to obtain the time history of the scour depth. Oliveto and Hagar (2002, 2005) and Oliveto et al. (2007) Oliveto and Hager (2002, 2005) proposed a method for the evolution of clear-water scour, while Oliveto et al. (2007) gave a method for the development of scour under live-bed con- ditions. The methodology, which is given as Equation 34, is based on an extensive set of experiments conducted at ETH Zurich, Switzerland. In formulating the equations, Oliveto et al. assumed that scour depth varies logarithmically with time. The clear-water equation is based on data for uniform sediments ranging from D50 equal to 0.55 to 5.3 mm and cylindrical pier widths ranging from 0.066 to 1.64 ft. Their live-bed equation is based on Chabert and Engeldinger (1956) and Sheppard and Miller (2006) data. Mia and Nago (2003) This method comprises a mathematical model for the development of local scour at cylindrical piers in cohesionless sediment subjected to steady flows in the clear-water scour regime. Input parameters to the model are structure dimen- sions, flow conditions, and sediment properties. The scour hole is assumed to be the frustum of an inverted cone with the angle of the frustum being the angle of repose of the bed sediment. The change in shear stress at the nose of the pier with increasing scour depth is estimated using a modified form of an equation by Kothyari et al. (1992) for the temporal vari- ation of bed-shear velocity. The bed-load sediment transport function used is attributed to Yalin (1977). This method does not require knowledge of the equilibrium scour depth but rather, according to the authors, can compute the equilibrium scour depth and the time required to reach this depth. This model was able to predict the data obtained by the model developers but did not accurately predict the data from other researchers. Chang et al. (2004) This method comprises a mathematical model for the devel- opment of local scour at circular piers in cohesionless non- uniform diameter sediments subjected to steady or unsteady flows in the clear-water scour regime. Input parameters for this model are structure dimensions, sediment properties (size, size distribution, mass density), and flow parameters. A sediment mixing layer thickness is computed along with an equivalent sediment size in the mixed layer. The dimensionless scour depth is expressed in terms of time normalized by Melville and Chiew’s expression for time to equilibrium. The scour rates are divided into three normalized time intervals as the scour depth progresses toward an equilibrium time. This method was developed for steady flows, but according to the authors can be applied in a finite step-wise manner to unsteady flows. As with Reference Equations Notes No. Kothyari et al. (2007) 2 3s t 0 5 g d d1 32 R1 1 6 0 25 1 3 d di g 50 0 2e d R y t0 272 F F ty a RF F 1 26 D t 4 8F t / . / / . / . . log . log . = a/B B = rectangular channel width R = hydraulic radius te = time to “end scour” tR = reference time 1 d s 50 VF S 1 gD Fdi = Fd for inception of scour 38 Table 12. (Continued).

Reference Equations Notes No. Melville/ Sheppard (Recommended) s t t s 1 6 c t 1 1 e 1 1 1 e 2 1 c c 0 25 1 1 1 1 e 3 1 c c 90 y (t) = K y V tK C V t V y Va t (days) C 0 4 6 0 4 V V a V V y y Va t (days) C 0 4 6 0 4 V V a a V t (d . . exp ln - . , . - . , . 1 e c V ays) 1 83 t V exp( . ) yst = time-dependent scour ys = S/M equilibrium scour equation C1 = –0.04 C2 = 200 days/sec C3 = 127.8 days/sec te = reference time t90 = time to reach 90% of equilibrium scour depth 39 40 the two previous models, this model must be programmed in order to produce scour depth time histories. Chang et al. (2004) provided the computer program for this model, but it yielded unreasonable results for a number of conditions in the data sets and, therefore, was eliminated in the final analysis. Yanmaz (2006) Yanmaz’s (2006) method for temporal variation of clear- water scour depth at cylindrical bridge piers is based on a common assumption that the shape of the scour hole can be approximated by an inverted cone having a circular base and slope equal to the angle of repose of the bed sediment. In applying the method, Yanmaz used initial measured scour depths as the starting point for the integration of Equation 37. The resulting equation has the form which renders the results very sensitive to the choice of initial conditions. If the equation is integrated from time equals zero, the results are quite different from those presented by Yanmaz (2006), which started the integration from measured values in his experiments. Kothyari et al. (2007) Kothyari et al. (2007) undertook additional experiments to extend the methods of Oliveto and Hagar (2002, 2005) for evo- lution of clear-water scour at bridge piers. They developed a new relationship for the temporal scour evolution at piers based on the similitude of Froude by relating the scour depth to the difference between the actual and the entrainment densimetric particle Froude numbers. The new relationship is validated by the complete ETH Zurich data set and verified using data from Chabert and Engeldinger (1956), Ettema (1980), and Melville and Chiew (1999). An expression is given for the time to “end scour,” which is equivalent to time to equilibrium scour. y tst 0.05∝ , ( )28 Initial Screening of Scour Evolution Predictive Methods An initial assessment of the selected scour evolution equations was performed to see if any of the methods yielded results that were clearly unreasonable. All of the methods were evaluated for combinations of values of the pertinent independent variables (and dimensionless groups). This analysis showed that the methods do not yield consis- tent results, leading to a wide range of predictions of scour depth development with time. The comparison yielded several interesting results. The method by Shen et al. (1966) can give very high, or very low, predictions relative to the other methods. The methods of Kothyari et al. (1992, 2007) lead to very large scour depths under live-bed conditions. Similarly, the methods of Oliveto et al. (2007) and Oliveto and Hager (2002, 2005) yield relatively deep scour predictions. At this point in the study, the various predictive methods were not tested for the conditions of the laboratory and field data, thus only those producing unrealistically large or small scour values were eliminated from further consideration. The methods eliminated were Shen et al. (1966), Yanmaz (2006), Chang et al. (2004) and Sumer et al. (1992) due to their unrealistic predictions. Modifications of Scour Evolution Predictive Methods The possibility of improving the accuracy of the better performing predictive equations was investigated. Some of the methods, such as the Miller and Sheppard (2002) model, are complex and modifications would require significant effort. The Melville and Chiew (1999) model is less complex and easy to use and modify. By adjusting the coefficients in Melville and Chiew’s model and replacing the equilibrium scour depth equation with the S/M equation, its accuracy was improved. The recommended equation is given as Equation 39 (Table 13) with ys being evaluated using the S/M equation. Table 13. Modified Melville and Chiew equation (M/S equation).

0.5 1 1.5 2 2.5 3 0 50 100 150 V1/Vc t 9 0 in D a ys a=3 ft a=7 ft a=15 ft D50=1 mm y1/a=2 This modified equation is referred to here as the Melville/ Sheppard (M/S) equation. Note that the original Melville and Chiew equation was developed for clear-water scour, while the M/S equation seems to work equally well for the live-bed scour data. Scour approaches an equilibrium value asymptotically; thus “time to equilibrium” is misleading at best and has no practical value. Time to a value like 90% of equilibrium is, however, useful and thus an expression for t90 is included with the M/S Equation. The expression for t90 in terms of the reference time t e and V1/Vc is given in Equation 40. Plots of t90 versus V1/Vc for one sediment size and normal- ized depth and three pier widths are shown in Figure 41. Final Evaluation of Scour Evolution Predictive Methods The scour evolution data are reported with different time steps. To calculate prediction errors, the following procedure was used. Each time series was divided into 100 equal time steps and the scour depths interpolated to these points in time. All of the predictive methods were then evaluated at each of the 100 times for the conditions of the experiment and compared with the measured values. The prediction error was computed using the following equations: The normalized SSE then becomes: SSE% y y y j i s measured j i s computed ji j i s me = −( )∑∑ 2 asured ji ( ) ×∑∑ 2 100 41( ) t V V t90 1 c e= − ⎛ ⎝⎜ ⎞ ⎠⎟exp 1 83 40. ( ) where i is the index for the time step and j is the index for the experiment. Seven different scour evolution predictive equa- tions were evaluated (six from the literature review plus the rec- ommended M/S equation). The total and underprediction errors for the various data sets were computed and the results are presented in Figures 42 through 45. Note that the scales for total and underprediction are substantially different in these plots in order to emphasize the underprediction. The Mia and Nago (2003) equation errors are not shown in the wide-pier plots (Figures 43 and 45), because they are very large. The computed scour evolution errors include the differences between the equilibrium scour depth that each experiment would achieve and the predicted value using the S/M equation, as well as the errors for the scour rate. However, not all of the scour evolution tests were conducted for a sufficient duration; therefore, differences between predicted and measured equi- librium values could not be determined and used to assess the quality of the data. Based on these results, the M/S equation performed the best of the seven in that it has the least total error and nearly the lowest underprediction error. However, as can be seen from the plots, all of the existing and modified methods have relatively large normalized errors. Note that the M/S equation has been optimized using the S/M equilibrium equation. The M/S equation should not be used with any other equilibrium scour equation. It is important to note that the only live-bed scour evolution data in the data set is for small laboratory structures (maximum 1.0 ft for 1.3 > V1/Vc >1.1 and 0.5 ft for V1/Vc >1.3). The SSEn% y a y a j i s measured j i s computed ji = − ⎛ ⎝⎜ ⎞ ⎠⎟∑ 2 ∑ ∑∑ ⎛⎝⎜ ⎞ ⎠⎟ × j i s measured ji y a 2 100 42( ) 41 Figure 41. Plot of t90 versus V1 /Vc for different pier diameters.

0 50 100 150 0 1 2 3 4 5 6 7 Melville 97 Oliveto 02-07 Kothyari 92 Kothyari 07 Miller 02 Mia 03 M/S Total Scour Error (%) Er ro r ( %) 0 5 10 15 20 25 30 35 40 0 0.5 1 1.5 2 2.5 3 3.5 Melville 97 Oliveto 02-07 Kothyari 92Kothyari 07 Miller 02M/S Total Normalized Scour Error (%) N o rm a liz e d Sc o u r E rr o r (% ) 0 20 40 60 80 100 0 0.5 1 1.5 2 2.5 3 3.5 4 Melville 97 Oliveto 02-07 Kothyari 92 Kothyari 07 Miller 02 Mia 03 M/S Total Normalized Scour Error (%) N o rm a liz e d Sc o u r Er ro r (% ) 42 Figure 42. Underprediction versus total normalized scour evolution error. Figure 43. Underprediction versus total normalized scour evolution error for wide piers (defined as y1/a < 0.5 and a/D50 > 100). Figure 44. Underprediction versus total scour evolution error.

80 85 90 95 100 0 1 2 3 4 5 Melville 97 Oliveto 02-07 Kothyari 92 Kothyari 07 Miller 02 M/S Total Scour Error (%) Sc ou r E rr or (% ) 43 Figure 45. Underprediction versus total scour evolution error for wide piers (defined as y1/a < 0.5 and a/D50 > 100). best-performing equation with existing data, the M/S equa- tion, yields, what appears to be, very conservative results for large prototype structures (conservative in the sense of pre- dicting scour rates much higher than seems reasonable); however, there are no large-structure data in the live-bed scour range with which to test the predictions. To illustrate the effects of sediment size on scour rates, plots of predicted time to reach 50%, 75%, and 90% of equilibrium scour depths versus flow velocity for a 30 ft diameter circular pier in 30 ft water depth are shown in Figure 46. An example problem with a large, long skewed pier, founded in fine sand and subjected to live-bed flow condi- tions is presented in Appendix D. This example illustrates, among other things, the conservativeness of the scour evo- lution equation.

44 0 1 2 3 4 5 6 7 8 10 0 10 2 V (ft/s) Ti m e in D a ys b = 30 ft D50= 0.4 mm y0 = 30 ft t90 t75 t50 0 1 2 3 4 5 6 7 8 10 0 10 2 V (ft/s) Ti m e in D a ys b = 30 ft D50= 1 mm y0 = 30 ft t90 t75 t50 0 1 2 3 4 5 6 7 8 10 0 10 2 V (ft/s) Ti m e in D a ys b = 30 ft D50= 3 mm y0 = 30 ft t90 t75 t50 Figure 46. Time to 50%, 75%, and 90% of equilibrium scour versus flow velocity for 0.4, 1.0, and 3.0 mm sediment diameters.