CHAPTER 5: LOCAL SCOUR AT PIERS, continued

Figure 7. Scatterplots of computed versus observed scour, in meters (m), for selected pier scour equations.

Figure 7. Scatterplots of computed versus observed scour, in meters (m), for selected pier scour equations, continued.

Figure 7. Satterplots of computed versus observed scour, in meters (m), for selected pier scour equations, continued.

Figure 7. Scatterplots of computed versus observed scour, in meters (m), for selected pier scour equations, continued.

Figure 7. Scatterplots of computed versus observed scour, in meters (m), for selected pier scour equations, continued.

Ranking the performance of scour-prediction equations is difficult because of the tradeoff between accuracy and underpredictions. If only accuracy is considered, the sum of squared errors can be used to evaluate the equations' performance (table 5). This statistic shows the Froehlich equation to be the most accurate; however, the Froehlich is a regression expression and underpredicted the depth of scour for 129 of 266 field observations. If the smallest number of underpredictions is used to evaluate the equations, the Froehlich Design equation is the best equation because it underestimated only four observations. The Froehlich Design equation, however, ranked 19^th based on the sum of squared errors criteria. The magnitude of the underpredictions is just as important, if not more so, than the number of underpredictions; thus, the sum of squared errors for those observations that were underpredicted is another factor that should be considered. The Melville and Sutherland equation had the lowest sum of squared errors for the underpredicted observations, but this equation ranked 26^th in overall sum of squared errors. The Melville and Sutherland equation did not underestimate scour by much, but grossly overestimated scour for many cases (figure 7T). The Froehlich Design, HEC-18-K4, HEC-18, HEC-18-K4Mu, and HEC-18-K4Mo (>2 mm) equations all had low sum of squared errors for the underpredicted observations. If the all the ranks are totaled, the Froehlich Design equation appears to be the top equation, followed by the HEC-18-K4Mu, HEC-18-K4, HEC-18, Mississippi, and HEC-18-K4Mo (>2 mm) equations; however, the Froehlich Design equation had the largest sum of squared errors for this group. If only the ranks based on the two sum of squared error categories are used, the HEC-18-K4Mu equation is favored and the Froehlich Design equation drops to a rank of 8.5.No single equation is conclusively better than the rest, but the top six equations generally appear to be the Froehlich Design, HEC-18-K4,

HEC-18-K4Mu, HEC-18-K4Mo (>2 mm), Mississippi, and HEC-18 equations.

Table 5. Summary of the performance of the selected pier scour equations.

SSE-sum of squared errors

Sheppard's equations have been revised several times as reported by Alkhalidi, Sheppard et al., and Sheppard.^(65,66,67) The 2001 version of the equation as listed in table 5 was used for comparisons in their analysis because it was the version provided by Sheppard to the author at the time the analysis was conducted. That version of the Sheppard equations ranked 19^th based on the two sum of squared error categories.

Since no single equation was superior to the others and none of the equations accurately predicted the scour for all conditions, it is important to assess where the equations failed. Residuals of selected equations were plotted against Froude number (V_o/(gy_o)^0.5), relative velocity (V_c/V_o), median grain size (D₅₀), pier width (b), relative bed material size (b/D₅₀), and relative depth (y_o/b) to assess where the equations may fail to properly account for the scour processes. Figure 8 shows that the Froehlich equation, which is a regression equation, has no significant patterns; it fits the data reasonably well. However, to convert the Froehlich equation from a regression equation to a design equation, Froehlich added the pier width as a factor of safety. The factor of safety increases the scatter in the data significantly. The plot of residuals versus pier width shows that factor of safety becomes too large as the pier width increases (figure 9). The HEC-18-K4 equation shows patterns of increasing overprediction as Froude number (0-0.4), median grain size, and pier width increase (figure 10). The K4, proposed by Mueller, reduces the effect of the Froude number and median grain size, but patterns are still evident in the pier width (figure 11).⁽⁵⁾ Only pier width displays a pattern in the residuals of the Mississippi equation (figure 12). The revised HEC-18 equation, HEC-18-K4Mo, also shows patterns in the residuals with Froude number and median grain size, and the most dominant pattern is the bottom envelope on the pier width (figure 13).⁽¹⁾ Most underpredictions seem to occur for grain sizes less than 2 mm. Table 5 shows that two thirds of the underpredictions by HEC-18-K4Mo occur at grain sizes less than 2 mm. Thus, limiting the K_i and K₄ corrections to grain sizes greater than 2 mm improves the performance of the Molinas correction.

Although many equations have been proposed for predicting the depth of local pier scour, no equation presented here accurately predicts the depth of scour for the wide variety of conditions represented in the field data set. Six equations are identified as being better than the others when assessed for their value as design equations; however, even these display patterns in their residuals that indicate they are not properly accounting for the scour processes. Additional research and analysis are needed to develop a better local pier scour prediction equation.

(A) FROUDE NUMBER	(B) RELATIVE VELOCITY
(C) 50% FINER GRAIN SIZE, IN MM	(D) PIER WIDTH, IN M
(E) RELATIVE BED MATERIAL SIZE	(F) RELATIVE DEPTH

Figure 8. Evaluation of residuals for the Froehlich equation.

(A) FROUDE NUMBER	(B) RELATIVE VELOCITY
(C) 50% FINER GRAIN SIZE, IN MM	(D) PIER WIDTH, IN M
(E) RELATIVE BED MATERIAL SIZE	(F) RELATIVE DEPTH

Figure 9. Evaluation of residuals for the Froehlich design equation.

(A) FROUDE NUMBER	(B) RELATIVE VELOCITY
(C) 50% FINER GRAIN SIZE, IN MM	(D) PIER WIDTH, IN M
(E) RELATIVE BED MATERIAL SIZE	(F) RELATIVE DEPTH

Figure 10. Evaluation of residuals for the HEC-18-K4 equation.

(A) FROUDE NUMBER	(B) RELATIVE VELOCITY
(C) 50% FINER GRAIN SIZE, IN MM	(D) PIER WIDTH, IN M
(E) RELATIVE BED MATERIAL SIZE	(F) RELATIVE DEPTH

Figure 11. Evaluation of residuals for the HEC-18-K4Mu equation.

(A) FROUDE NUMBER	(B) RELATIVE VELOCITY
(C) 50% FINER GRAIN SIZE, IN MM	(D) PIER WIDTH, IN M
(E) RELATIVE BED MATERIAL SIZE	(F) RELATIVE DEPTH

Figure 12. Evaluation of residuals for the Mississippi equation.

(A) FROUDE NUMBER	(B) RELATIVE VELOCITY
(C) 50% FINER GRAIN SIZE, IN MM	(D) PIER WIDTH, IN M
(E) RELATIVE BED MATERIAL SIZE	(F) RELATIVE DEPTH

Figure 13. Evaluation of residuals for the HEC-18-K4Mo equation.

EVALUATION OF LABORATORY RESEARCH

General

Laboratory research has been the primary tool for defining the relations among variables affecting the depth of pier scour. The validity of these relations has not been proven in the field. Landers and Mueller evaluated many relations developed in the laboratory by use of transformed data (to obtain a more normal distribution) and smoothing techniques to assess general trends in the data.⁽²¹⁾ They found only minimal agreement between the field data and laboratory-based relations. The assessment presented here investigates the relations in the field data for variable combinations commonly reported by laboratory investigations. Unlike the data set used by Landers and Mueller, all data at skewed piers were removed to prevent bias by these data, as previously discussed.⁽²¹⁾ No transformations were applied unless necessary for consistency with published relations. Using the basic data without transformation results in a less uniform distribution of the data, but it provides for a more direct comparison with laboratory work.

Pier Geometry

For piers aligned with the approach flow, laboratory research indicates that streamlining the pier nose reduces the depth of scour. Many shapes have been tested in the laboratory (table 6) to determine what effect the pier shape has on the depth of scour, relative to a circular pier. The pier shapes for the field data are classified as unknown, cylinder, round-nose, sharp, or square. Landers and Mueller did not account for variables known to affect the depth of scour other than pier shape and found no correlation between pier shape and scour depth.⁽²¹⁾ It is necessary to remove the effects of pier width, velocity, depth, and bed material before comparing the depth of scour among different pier shapes. Figures 14 and 15 show the effect of pier nose shape on scour. First, only the effect of pier width is accounted for by dividing the depth of scour (y_s) by the pier width (b). Figure 14 shows that the median of the relative depth of scour decreases as the pier is more streamlined; however, there is overlap in the interquartile range, and the differences are not significant. Second, the effects of pier width, velocity, depth, and bed material are removed by regressing these variables with the depth of scour and by plotting the residuals of a regression against pier shape, which is not included in the regression. The residuals from the regression analysis (figure 15) show the same trend as observed in figure 14. The pier shape does not affect the depth of scour in the field as much as in the laboratory. In the field, flow directions are variable, pier shapes vary with depth, and the effect of submerged debris is not easily accounted for; these combine to reduce the effect of pier shape on the depth of scour.

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