Determining Scour Depth Around Structures in Gravel-Bed Rivers (2023)

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39   3.1 Measured Controls on Bridge-Pier Scour in Laboratory Experiments To determine the variables that needed to be included in the system of new scour equations, the authors first analyzed the measured scour depths in the experiments from the large flume from Section 2.1. In this chapter, the authors present the variation in scour depths through time for all experiments, as well as a comparison of maximum scour depths between experiments as functions of different flow conditions, sand content, and gravel sorting parameters. To simplify the discussion, all discussions of scour depth variations with gravel sorting parameters σgc and sand contents Fsc use the composite values (average of armor and subsurface layer values) of these variables for each experiment. However, both sediment layers individually affected the measured scour depths and are individually included in the scour equations developed in Section 3.2. 3.1.1 Controls on Scour Depths for Single Sediment Mixtures In many of the experiments, the scour depth dramatically increased in the first few hours of the experiment and then either remained constant or decreased slightly to a roughly constant value after a certain elapsed time. These temporal changes in the scour depth were influenced both by the flow conditions and the bed grain size distribution. To highlight the importance of flow conditions, the researchers compared the scour depth between experiments that had the same Fsc and σgc but experienced three different flow conditions [flows that moved the D00 (clear water), D16, and D84] (Figure 33). For a given Fsc and σgc, the scour hole was more likely to refill in the D16 and D84 moving-flow experiments than in clear-water conditions because gravel mobilized from upstream gradually filled the scour hole and reduced the scour depth (Figure 33). For most experiments with clear-water flow conditions (D00), the scour depth remained relatively constant after reaching the maximum scour depth because no sediment was transported into the scour hole from upstream (Figures 33 and 34). The flow was higher during the D84 moving- flow conditions than in the D16 moving-flow conditions, which therefore caused the scour hole to be partly refilled with upstream sediment in many experiments (Figures 33 and 34). The researchers extracted the maximum scour depth in a given experiment (Table 10) from the time series of scour depths (Figures 33 and 34). Between experiments, the water depth varied based on the flow condition used (clear-water conditions, D16, or D84 moving flow), and the researchers calculated the dimensionless water depth as the water depth divided by pier width. Also following standard practices, the researchers calculated the dimensionless maximum scour depth as the maximum scour depth divided by pier width. Next examined is the influence of the dimensionless flow depth, σgc, and Fsc on the dimension- less maximum scour depth. C H A P T E R 3 Findings and Applications

40 Determining Scour Depth Around Structures in Gravel-Bed Rivers 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 2 4 6 8 10 M ax . S co ur D ep th (m ) Time (hr) σgc = 1.53, Fsc = 0 D00 D16 D84 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 2 4 6 8 10 M ax . S co ur D ep th (m ) Time (hr) σgc = 1.53, Fsc = 0.06 D00 D16 D84 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 2 4 6 8 10 M ax . S co ur D ep th (m ) Time (hr) σgc = 1.53, Fsc = 0.067 D00 D16 D84 Figure 33. The scour depth as a function of experiment time (in hours) for experiments with sgc 5 1.53 under flow conditions that move the D00 (clear water), D16, and D84, with different sand contents between experiments. For a given dimensionless flow depth, the maximum scour depth generally declined as σgc increased (Figure 35). However, D50c covaried with σgc in the experiments, and therefore the researchers cannot fully isolate whether the variation in maximum scour depth was caused by a higher median grain size or an increase in gravel sorting parameter (Figure 36). These results imply that coarser and wider gravel grain size distributions will result in less scour than finer and narrower gravel grain size distributions. The researchers hypothesize that this occurred partly because the wider gravel size distributions included more large rocks (e.g., larger D50c and D84c) that were difficult to move and limited the scour depth. Wide grain size distributions also likely resulted in a more interlocked, clustered, and packed together (lower porosity) bed than occurred for narrower grain size distributions. Such enhanced bed structure in wide grain size distributions would make all grain sizes more difficult to move, which would also result in lower maximum scour depths. For a given dimensionless flow depth but variable values of σgc, the dimensionless maximum scour depth did not systematically vary with Fsc, likely because σgc exerted a stronger control on the scour depth than did the sand content (Figure 35). When σgc was held constant at 1.53, the dimensionless maximum scour depth still did not consistently change with Fsc (Figure 37) for a given dimensionless flow depth. This is likely because two of the sand contents were very similar (Fsc = 6%, 6.7%). However, the dimensionless maximum scour depth was generally greater when beds had sand (Fsc = 6%, 6.7%) than when they did not (Fsc = 0%). Thus, for the same gravel

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 2 4 6 8 10 M ax . S co ur D ep th (m ) Time (hr) Clear water σgc = 1.16, Fsc = 0.029 σgc = 1.53, Fsc = 0.06 σgc = 1.94, Fsc = 0.06 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 4 8 12 16 M ax . S co ur D ep th (m ) Time (hr) Live Bed (D16) 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 4 8 12 16 M ax . S co ur D ep th (m ) Time (hr) Live Bed (D84) σgc = 1.16, Fsc = 0.029 σgc = 1.53, Fsc = 0.06 σgc = 1.94, Fsc = 0.06 σgc = 1.16, Fsc = 0.029 σgc = 1.53, Fsc = 0.06 σgc = 1.94, Fsc = 0.06 Composite Gravel Sorting Parameter σgc Composite Sand Fraction Fsc Flow Condition Flow Conditions Max. Scour Depth (cm) Discharge (m3/s) Water Depth (cm) Average Velocity* (m/s) ADV Velocity (m/s) 1.16 0.029 Clear water 0.35 17 1.04 1.26 10.41 1.16 0.029 Live bed (D16) 0.49 19 1.28 1.33 11.16 1.16 0.029 Live bed (D84) 0.56 21 1.32 1.32 11.99 1.53 0.00 Clear water 0.35 17 1.04 1.06 6.04 1.53 0.00 Live bed (D16) 0.44 19 1.16 1.18 7.68 1.53 0.00 Live bed (D84) 0.57 21 1.35 1.26 6.23 1.53 0.060 Clear water 0.35 17 1.04 1.08 8.68 1.53 0.060 Live bed (D16) 0.44 19 1.16 1.23 9.33 1.53 0.060 Live bed (D84) 0.57 21 1.35 1.27 11.5 1.53 0.067 Clear water 0.35 17 1.04 1.25 9.52 1.53 0.067 Live bed (D16) 0.44 19 1.16 1.26 8.5 1.53 0.067 Live bed (D84) 0.57 21 1.35 1.26 10.26 1.94 0.060 Clear water 0.34 16.5 1.03 1.12 5.07 1.94 0.060 Live bed (D16) 0.41 18 1.14 1.14 8.92 1.94 0.060 Live bed (D84) 0.76 24 1.58 1.23 9.51 Top mixture: 1.16 Top mixture: 0.029 Bottom mixture: 0.06 Clear water 0.35 17 1.26 1.26 8.83 Bottom mixture: 1.94 Top and bottom mixture: 1.53 Top mixture: 0.06 Clear water 0.35 19 1.08 1.08 9.34 Bottom mixture: 0 *Average velocity calculated as discharge/(water depth × flume width). ADV = acoustic doppler velocimeter. ADV velocity was measured upstream of the pier at one location within the flow depth. Figure 34. The scour depth as a function of experiment time (in hours) for a given flow condition [moved the D00 (clear water), D16, and D84] with different sand contents and gravel sorting parameters between experiments. Table 10. Measured maximum scour depths for all experiments.

42 Determining Scour Depth Around Structures in Gravel-Bed Rivers 0.2 0.4 0.6 0.8 1 1.2 1.4 1.2 1.4 1.6 1.8 2 2.2 2.4 D im en si ol es s M ax im um S co ur D ep th (y m ax /b ) Dimensionless Water Depth (h/b) σgc=1.16, Fsc = 0.029, D = 9.95 mm σgc=1.53, Fsc = 0.06, D = 10.4 mm σgc=1.94, Fsc = 0.06, D = 14 mm gc gc gc sc sc sc 50 50 50 0 0.5 1 1.5 2 2.5 8 9 10 11 12 13 14 15 C om po si te g ra ve l s or tin g pa ra m et er Composite D50c (mm) 0.2 0.4 0.6 0.8 1 1.2 1.4 1.2 1.4 1.6 1.8 2 2.2 D im en si on le ss M ax im um S co ur D ep th (Y m ax /b ) Dimensionless Water Depth (h/b) σgc=1.53 Fsc = 0.0 Fsc = 0.06 Fsc = 0.067 Figure 35. The dimensionless maximum scour depth as a function of dimensionless water depth and sediment mixture (different lines) for nine runs (each symbol) (ymax: maximum scour depth; h: water depth; b: pier width). Figure 36. Composite gravel sorting parameter as a function of composite D50c for all sediment mixtures. Figure 37. The dimensionless maximum scour depth as a function of dimensionless water depth and sand content (different lines) for nine runs (each symbol) with sgc 5 1.53 (ymax: maximum scour depth; h: water depth; b: pier width).

Findings and Applications 43   sorting parameter and flow condition, more scour will occur when sand is present, and sand content is important to include in a bridge-pier scour equation. For a given σgc, the dimensionless maximum scour depth generally increased with higher dimensionless water depths because of higher flows being able to transport more sediment (Figure 35). These results demonstrate that equations developed only for clear-water flow conditions are unlikely to accurately predict scour depths for live-bed conditions, which are often deeper. 3.1.2 Effects of Multiple Mixtures on Scour Depth Two scour experiments with multiple sediment mixtures were also completed, and the researchers compared the scour depths obtained from the single sediment mixtures (two layers: armor and subsurface, discussed previously) to those obtained with two combined sediment mixtures (four layers: two armor and two subsurface). For the first experiment with multiple sediment mixtures, the researchers selected sediment mixtures with σgc = 1.16 (placed on top) and σgc = 1.94 (placed on bottom), which were the two σgc that had some of the greatest differences in scour depths during the single-mixture experi- ments (Figure 38). The researchers set the sediment mixture thickness to about 0.07 m for the top sediment mixture (σgc = 1.16), which was less than the maximum scour depth measured for the single sediment mixture with σgc = 1.16 (scour depth over 0.10 m) but more than that for the single mixture with σgc = 1.94 (scour depth of ∼0.05 m). Therefore, the researchers aimed to determine if the same flow conditions (clear water) would result in lower scour depths for the multiple-mixture experiment than occurred for the equivalent single-mixture experiment, particularly when the underlying mixture was more difficult to erode than the top mixture. The increase in scour depth through time for the single mixture (σgc = 1.16) and double mixture was the same until the scour depth passed 0.07 m (Figure 38, left panel). After passing 0.07 m, scour halted and no longer deepened in the double mixture, whereas it continued in the single mixture (σgc = 1.16). This occurred because scour passed through the entire upper mixture (σgc = 1.16) and encountered the armored sediment layer of the bottom mixture (σgc = 1.94), which was more difficult to move than that of the upper mixture. 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 2 4 6 8 10 Sc ou r D ep th (m ) Time (hr) σgc = 1.16, Fsc = 0.029 σgc = 1.94, Fsc = 0.06 σgc 1.16 + σgc 1.94 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 2 4 6 8 10 Sc ou r D ep th (m ) Time (hr) Fsc = 0 Fsc = 0.06 Fsc 0.06 + Fsc 0 Figure 38. (Left) The scour depth as a function of experiment time for experiments under clear-water flow conditions but with different gravel sorting parameters between experiments. (Right) The scour depth as a function of experiment time for experiments with sgc 5 1.53 and clear-water flow conditions but with different sand contents between experiments. Experiments with single and multiple sediment mixtures are shown for comparison.

44 Determining Scour Depth Around Structures in Gravel-Bed Rivers For the second experiment with multiple sediment mixtures, the researchers used a σgc = 1.53 for both mixtures but with a 0.07-m-thick upper mixture with Fsc = 6% and a lower mixture with Fsc = 0%. These two mixtures produced different scour depths in the single-mixture experi- ments, with Fsc = 6% producing a greater scour depth (∼0.085 m) than Fsc = 0% (∼0.06 m). The researchers therefore expected scour in the double-mixture experiment to stop once it reached the bottom of the upper mixture (0.07 m). However, the researchers did not observe any signi- ficant changes in the scour depth between the single (Fsc = 6%) and multiple-mixture experi- ments for the same clear-water flow conditions (Figure 38, right panel). In the multiple-mixture experiments, the scour hole continued to deepen after reaching the less-erodible bottom layer, which was contrary to expectations. Sand from the top sediment mixture may have winnowed down into the lower sediment mixture during water working of the top mixture. This would have increased the sand content of the bottom mixture from what the researchers expected during the single-mixture experiments (Fsc = 0%), thereby making the bottom mixture more erodible. Based on these results, the researchers conclude that layers can significantly affect scour depths and need to be considered in bridge-pier scour equations. In particular, changes in gravel sorting parameters between sediment layers may affect scour depths more than a change in sand content between layers. 3.2 Development of New Bridge-Pier Scour Equations Using Laboratory Experiments As discussed in Chapter 1, the new set of bridge-pier scour equations for gravel-bedded rivers is a combination of (1) an equation to predict critical shear stresses as a function of the sand content Fs and gravel sorting parameter σg of a given sediment layer, and (2) equations to predict the decay of applied shear stress in the scour hole as the scour hole develops. The maximum scour depth is predicted as the shallowest scour depth at which the applied shear stress in the scour hole declines below the critical shear stress. The researchers developed the critical shear stress equation using grain size distribution information for each sediment layer (armor, subsurface) that was created after water working in the laboratory experiments. However, most field data do not have layer-specific grain size distributions, and the critical shear stress equation can also be applied in these circumstances. This is discussed in detail in Section 3.3. 3.2.1 Critical Shear Stress Equation from Laboratory Experiments 3.2.1.1 Variables That Control the Critical Shear Stress The previous section discussed the fact that the dimensionless maximum scour depth was controlled by the gravel sorting parameter, sand content, dimensionless flow depth, and presence of different sediment layers. The process mechanics that may explain some of these variations in scour depth are now examined. In particular, the onset of sediment transport will likely be influenced by gravel sorting parameters and the sand content of various layers. Therefore, the authors determined whether the gravel sorting parameter and bed sand content influenced the measured dimensionless critical shear stress (critical Shields stress) for the median grain size of the bed τ*c50, which was used as the representative bed grain size. The authors previously discussed composite (armor and subsurface combined) values of the gravel sorting parameter and sand content when discussing the maximum scour depth because the specific influence of each layer on scour depth could not be isolated. However, the influence of each layer on τ*c50 can be isolated because there are independent measurements of τ*c50a and τ*c50s for the armor and subsurface layers, respectively. The researchers also measured the gravel sorting parameters (σga and σgs) and sand contents (Fsa and Fss) of each layer, which allows the effects of these layer- specific variables on the critical Shields stress to be determined. Specifically, the researchers have

Findings and Applications 45   Armor Layer Subsurface Layer Mixture σga Fsa τ*c50 σgs Fss τ*c50 A 1.18 0 0.0397 1.14 0.058 0.0349 B 1.25 0 0.0413 1.48 0.071 0.0371 C 1.41 0 0.0503 1.65 0 0.0418 D 1.41 0 0.0478 1.65 0.12 0.0377 E 1.41 0.0034 0.0482 1.65 0.13 0.035 F 1.66 0 0.0512 1.87 0.09 0.0392 G 1.95 0.0032 0.0556 1.92 0.1173 0.0403 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0 0.05 0.1 0.15 τ* c5 0 Fs 0.01 0.02 0.03 0.04 0.05 0.06 0.07 1.00 1.20 1.40 1.60 1.80 2.00 τ* c5 0 σg Table 11. Critical Shields stress for the median grain size as a function of the gravel sorting parameter and sand content for two different sediment layers in each experiment. Figure 39. Variation of critical Shields stress for the median grain size of a given layer with: (left) different sand contents of each sediment layer, and (right) different gravel sorting parameters of each sediment layer. one paired (τ*c50, σg, and Fs) measurement for each layer (armor and subsurface) of each of seven experiments with different sediment mixtures placed on the bed (Table 11). These 14 paired measurements are combined in one analysis to obtain a general critical Shields stress equation that can be applied to any bed sediment layer. The generic symbols for variables τ*c50, σg, and Fs are employed because the authors are using both armor layer and subsurface layer measurements in this analysis but are not using the composite (average of the two layers) variables. The critical Shields stress for the median grain size decreased when the layer (armor or subsurface) sand content increased from 0% to 5%, but τ*c50 was largely constant when layer Fs was between 5% and 13% (Figure 39, left panel). The large number of data points at Fs of 0% was caused by the lack of sand in the armor layer (Table 11), whereas the data points with larger Fs were from the subsurface layer. Scatter in τ*c50 for a given sand content was caused by many of these experiments having different gravel sorting parameters, which was also a control on τ*c50 (see next paragraph). Relatively small changes in critical Shields stresses with sand content likely occurred because the researchers water worked each bed to create an armored bed surface prior to every experiment. During water working, the sand content on the bed surface decreased as most sand was washed out of the bed or went into the bed subsurface. Thus, changes in the critical Shields stress with sand content were less likely to occur than observed in previous studies because the armor layer actually contained very little sand and the majority of the data points with sand were from the subsurface layer. For the same sand content, the τ*c50

46 Determining Scour Depth Around Structures in Gravel-Bed Rivers was somewhat higher than what was measured in Wilcock and Crowe (2003). The researchers attribute these differences to different experimental conditions between the two studies; in this study, the researchers did not feed any sediment from upstream, included τ*c50 for different sediment layers, and also used different gravel sorting parameters, which can affect τ*c50 (see next paragraph). Different bed structure, and therefore τ*c50, between the experiments in these two studies may have also resulted from different sediment feeding conditions. The critical Shields stress for the median grain size also varied with the gravel sorting param- eter of different layers (Figure 39, right panel). Specifically, more poorly sorted gravel grain size distributions (higher gravel sorting parameters) experienced slightly higher τ*c50. Increases in τ*c50 with higher gravel sorting parameters were likely caused by greater hiding effects that occur in wider grain size distributions. These changes in τ*c50 could also be driven by greater bed inter- locking and packing (e.g., lower porosity) in the armor layer that occurs with a wider grain size distribution. However, significant scatter exists in the relation between τ*c50 and the gravel sorting parameter, which is partly caused by including individual measurements for both the armor and subsurface layers. These layers had similar gravel sorting parameters (Table 11) but very different sand contents, as mentioned previously. Given that sand content is a significant control on τ*c50 the different layer sand contents are partly obscuring the relation between τ*c50 and the gravel sorting parameter. In addition, the armor layer likely had greater interlocking and imbrication compared to the subsurface layer, which could produce higher τ*c50 in the armor layer. For example, the subsurface layer of Mixture C and the armor layer of Mixture F had similar gravel sorting parameters and sand contents, but the armor layer had a higher τ*c50 (Table 11). Such differences in bed structure between layers likely contributed to the observed scatter in τ*c50 for a given gravel sorting parameter or sand content. 3.2.1.2 Multiple Regression Equation for Critical Shear Stress These results imply that a multiple regression equation between τ*c50 and the gravel sorting parameter and sand content is needed to isolate the individual impact of each variable on the onset of sediment motion. These results are supported by previously published studies that have developed predictive equations for the critical Shields stress using either the sand content or the gravel sorting parameter. For example, Wilcock and Crow (2003) developed a relation for the critical Shields stress τ*csm of the mean surface grain size dsm as a function of the surface layer sand content Fs (Equation 11). exp F d d e exp d d csm s ci csm i sm e i sm ( )τ = + − τ τ =     = + −    * 0.021 0.015 20 0.67 1 1.5 (11) where τci and τcsm are the critical shear stress for each grain size and the mean grain size, respec- tively. A two-fraction transport model by Wilcock and Kenworthy (2002) also employed the subsurface sand content (Equation 12) to estimate the dimensionless critical shear stress for each grain size τ*ci using the dimensionless critical shear stress (τ*ci)0 and (τ*ci)1 at Fs = 0, Fs = 1, respectively. * * * * 1 exp 25 0.16 (12)0 0 1 Fci ci ci ci s ( ) ( ) ( )( )( )τ = τ − τ − τ + − −

Findings and Applications 47   Value Standard Error t Stat P-Value Constant –3.5 0.13 –27.2 1.9E-11 σg 0.279 0.09 3.2 0.008 Fs –2.38 0.42 –5.7 0.00013 0.03 0.035 0.04 0.045 0.05 0.055 0.03 0.04 0.050.035 0.045 0.055 Pr ed ic te d τ* c5 0 Measured τ*c50 Table 12. Significance tests for developed critical shear stress equation. Figure 40. Measured critical Shields stresses versus predicted values. Conversely, Shvidchenko et al. (2001) suggested that the critical Shields stress for a given grain size di is a function of σg (Equations 13 and 14). * * (13) 50 50 d d ci c i eτ τ =     1.17 (14)0.24e g= − σ− For this study’s equation, several equation forms (e.g., linear, logarithmic) were tested to develop simple and accurate expressions. Among the equation forms tested, an equation using an exponential function had the highest R2 and the simplest form. It also allowed positive τ*c50 to be predicted at high sand contents, whereas a linear equation predicted negative τ*c50 when the sand content was outside of the data range. The proposed predictive relation for the layer (armor or subsurface) τ*c50 as a function of the layer (armor or subsurface) Fs and σg using multiple regression is: * exp 0.279 2.38 3.5 (15)50 Fc g s( )τ = σ − − The R2 of this equation is 0.75, and all of the coefficients are statistically significant by the t-test (Table 12). To further confirm that the equation is accurate, the researchers compared the measured and predicted τ*c50 values (Figure 40). Finally, the τ*c50values are within the range of values (e.g., 0.03–0.08) typically reported for gravel-bedded rivers, which are usually armored. This implies that the equation will provide τ*c50 that is representative of armored rivers to allow for accurate estimates of the onset of sediment transport.

48 Determining Scour Depth Around Structures in Gravel-Bed Rivers Equation 15 can now be written in dimensional terms by combing it with the Shields equation (Equation 16): * (16)50 50 50gDc c s( )τ = τ ρ − ρ obtaining: exp 0.279 2.38 3.5 (17)50 50F gDc g s s( )( )( )τ = σ − − ρ − ρ 3.2.2 Shear Stress Decay Equations Using Laboratory Experiments The critical shear stress equation was developed using the reach-averaged total shear stress at the onset of sediment motion. This total shear stress τT for undisturbed (far from the pier) conditions is: (18)ghSTτ = ρ Where ρ is the water density, g is the acceleration due to the gravity, h the undisturbed flow depth, and S the undisturbed bed slope. The new set of bridge-pier scour equations will calculate scour by comparing the critical shear stress to the applied shear stress at the pier. The applied total shear stress at the pier cannot be calculated using Equation 8 because this equation is based on the assumption of steady, uniform flow and can only be applied for reach-averaged conditions; it does not account for convective accelerations around the pier. Therefore, calculated mean (spatial and temporal average) near-bed Reynolds stresses at the pier from the numerical models in Chapter 2 were used to determine the applied shear stresses at the pier. However, total shear stresses in rivers are normally much larger than mean near-bed shear stresses (Robert, 1997; Biron et al., 2004; Kostaschuk et al., 2004), and this results in all mean near-bed shear stresses being automatically less than the critical shear stress, which was measured using a total shear stress in the laboratory experiments discussed in Chapter 2. When this occurs, zero scour will always be predicted by the set of new bridge-pier scour equations. Therefore, the researchers needed to convert the mean near-bed shear stresses at the pier into equivalent total shear stresses at the pier for comparison with critical shear stresses. This was accomplished through a series of transfer functions to indirectly estimate the total shear stresses immediately around the pier before any scour has occurred, which the researchers hereinafter call the zero-scour condition. The total shear stress at the zero-scour condition is needed because this (1) determines whether any scour will initiate when this applied shear stress is compared to the critical shear stress, and (2) defines the beginning of the shear stress decay curve if scour does initiate. Then a shear stress decay curve was developed to predict how the total shear stress at the pier will decline as the scour hole initiates and deepens. When this system of equations is combined with the critical shear stress equation (Equation 17), the maximum scour depth can be estimated. Here, the researchers focus on the detailed equation derivation but simplify this system of equations for practical application in estimating pier scour in Section 3.4. 3.2.2.1 Conversions Between Total and Near-Bed Shear Stresses at Zero-Scour Conditions The near-bed Reynolds stress at the pier for zero-scour conditions τp0 could not be directly related to the undisturbed total shear stress τT because this relation would incorporate two different and competing effects: (1) average near-bed shear stresses are normally lower than total

Findings and Applications 49   shear stresses at the same location, and (2) for zero-scour conditions, any (near-bed or total) shear stresses at the pier are normally higher than any (near-bed or total) shear stresses far from the pier. The researchers needed to account for these two effects in two different transfer functions to avoid obscuring their separate and competing influences. The first transfer function (Equation 19) accounts for the fact that total shear stresses are higher than average near-bed shear stresses in the same location. This transfer function therefore relates τT to the undisturbed (far from the pier) average near-bed Reynolds stress τb (Figure 41a). The second transfer function (Equation 20) accounts for the fact that near-bed shear stresses at the pier for zero-scour conditions are higher than near-bed shear stresses far from the pier. Therefore, this transfer function relates the average near-bed Reynolds stress at an undisturbed location far from the pier τb to the average near-bed Reynolds stress at the pier for zero-scour conditions τp0 (Figure 41a). For both transfer functions, the near-bed shear stresses are near-bed Reynolds stresses calculated from the numerical simulations conducted at zero-scour conditions (i.e., experiment time step of 0 hours). The experiments used were σgc = 1.16 with Fsc = 0.029 under clear-water and D16 moving-flow conditions, and σgc = 1.53 with Fsc = 0.060 under clear-water and D16 and D84 moving-flow conditions (Table 6). The Reynolds stresses were spatially and temporally averaged at 0.01 m above the bed surface around the pier (for τp0) or far from the pier (for τb). The sampling region near the pier corresponded to the bed area with altered topography when maximum scour occurred (in each respective experiment). The sampling region far from the pier was a circle having approximately the same area as that used near the pier and was located ∼1 m upstream of pier in the center of the channel width. The researchers first related τT/τb to the relative submergence (h/D84a, where D84a is the 84th percentile of the armored layer grain size distribution) (Figure 42), which is an easily measured variable that describes the bed roughness. Bed roughness is known to affect both near- bed shear stresses and total shear stresses, and it was therefore expected that τT/τb would vary with relative submergence. Given that the researchers were relating two shear stresses that were unaffected by the pier, they needed to correlate these shear stresses to easily measured roughness properties of the bed rather than to pier roughness (e.g., ratio of flow depth to pier diameter). The laboratory data of Yager et al. (2018) were added to this analysis because the researchers only had five modeled experiments, which represented a limited range of relative submergences from what might be expected to occur in the field. Combining both data sets, τT/τb increased Figure 41. Schematic definition of flow variables and different shear stresses involved in scour. All variables are time-averaged values, b is the pier width, and V and h are the average undisturbed velocity and flow depth. (a) The subscript 0 denotes zero-scour conditions. Here, tT and tb are the total shear stress and the near-bed Reynolds stress far upstream of the pier (i.e., undisturbed conditions). The variables tTp0 and tp0 are near the pier for zero-scour conditions and the total shear stress and the mean near-bed Reynolds stress, respectively. (b) Scoured conditions are denoted by the subscript y. The variables, tpy and tTpy are near the pier for a scour depth of y and are the total shear stress and the mean near-bed Reynolds stress, respectively.

50 Determining Scour Depth Around Structures in Gravel-Bed Rivers with greater relative submergence, and this relation was used to develop the first transfer function (Equation 19 and Figure 42): 0.7487 (19) 84 0.6976 h D T b a τ τ =     The R2 of Equation 19 is 0.72, and all the coefficients are statistically significant using the t-test (Table 13). The second transfer function converts between undisturbed mean near-bed shear stresses and mean near-bed shear stresses at the pier for zero-scour conditions. For this equation, a sigmoid function was used because two essential boundary conditions had to be met: (1) τb/τp0 can never be negative, and (2) τb/τp0 can never be larger than unity because obstacles such as piers cause convective accelerations that increase the local mean near-bed shear stress compared to the mean near-bed shear stress at locations without obstacles. In general, any activation function could be implemented here (e.g., Dirac delta, hyperbolic tangent, or error function) and the results would be similar. Given that the authors did not find any published data that relate τb/τp0 to easily measured flow properties at obstacles, only the numerical simulations could be used to obtain this function (Equation 20). It was assumed that τb/τp0 will vary with h/b because the ratio of the flow depth to the obstruction size is commonly used to represent the effects of Figure 42. Ratio of the undisturbed total shear stress to the undisturbed near-bed shear stress (tT/tb) as a function of the relative submergence. Data points include laboratory measurements from Yager et al. (2018) for h/D84a <– 10 and the numerical simulations for h/D84a > 10. The dashed line represents the fit given by Equation 19. Value Standard Error t Stat P-Value Coefficient 0.7487 0.3281 2.1416 0.0462 h/D84a 0.6976 0.0462 6.4019 5.00E-06 Table 13. Significance tests for developed equation (Equation 19).

Findings and Applications 51   obstacles on the flow field and is also often used in pier scour equations (Figure 43). Additional data collected by future studies would be useful to further validate this assumption. 0.95 0.9 1 exp 3.5 (20) 0 h b b p τ τ = − + δ −    In this equation, the constants 0.95 and 0.9 account for the fact that τb/τp0 must be between 0 and 1 by providing the range 0.05 ≤ τb/τp0 ≤ 0.95. The displacement coefficient δ = 7.983 is the parameter adjusted to fit the function to the researchers’ observations (Figure 43). A total shear stress at the pier for zero-scour conditions (τTp0), which is what is required for calculating scour depths, can now be determined by combining Equations 19 and 20. For simplicity, Equations 19 and 20 can be rewritten as: ,4 0 5f f T b eq b p eq τ τ = τ τ = Given that τb is in both equations, τp0 is equal to: i 0 4 5f f p T eq eq τ = τ It is assumed that the ratio of total shear stress to mean near-bed shear stress at any given location will result in Equation 19; that is, the same method to convert between total and mean near-bed shear stresses should apply regardless of whether the location is far from the pier or close to the pier. In Equation 19, a total undisturbed shear stress of τT produces a mean Note: [ ] denotes a dimensionless variable. Figure 43. Ratio of the undisturbed mean near-bed Reynolds stress and mean near-bed Reynolds stress at the pier for zero-scour conditions (tb/tp0) as a function of the dimensionless water depth. The dashed line represents the fit given by Equation 20.

52 Determining Scour Depth Around Structures in Gravel-Bed Rivers undisturbed near-bed shear stress of τb. Therefore, the total shear stress at the pier for zero-scour conditions of τTp0 produces a mean near-bed shear stress at the pier for zero-scour conditions of τp0, and these shear stresses are related using the same equation form as in Equation 19. With this assumption, the ratio of shear stresses at the pier for zero-scour conditions is: 0 0 4f Tp p eq τ τ = and by combining these last two equations: i 0 4 5f f p T eq eq τ = τ 0 0 4fTp p eqτ = τ i 0 4 5 4f f fTp T eq eq eqτ = τ 0 5f Tp T eq τ = τ Now, rewriting the terms, a simple expression can be obtained to calculate τTp0 from τT and h/b. 10 5f Tp T eq τ τ = 0.95 0.9 1 exp 3.5 (21)0 h b Tp Tτ = τ − + δ −    Equation 21 is the only equation that is used from this subsection to predict bridge-pier scour; all other equations presented were only used in the derivation of this equation. 3.2.2.2 Equations to Estimate the Total Shear Stress Decay The total shear stress decay curve for different scour depths γ is now developed. Again, the researchers do not have direct estimates of the total shear stress at the pier with different scour depths, and therefore the derivation of this shear stress decay equation begins with under- standing how the numerically modeled mean near-bed Reynolds stresses at the pier change with scour depth. Once the form of the shear stress decay equation from these mean near- bed stresses is known, the same general equation form can be applied to the decay of total shear stress at the pier (Figure 41b). In each numerically modeled experiment, the mean near-bed shear stress at the pier (τpy) decreased with larger scour depths (Figure 44a). The relation between scour depth and τpy varied between experiments and did not collapse onto one well-defined line (Figure 44b) because of the differences in undisturbed flow conditions (flow depths and Froude numbers) between experiments. Therefore, an equation was derived that collapsed all the near-bed shear stress decay curves into a single function based on the undisturbed flow properties and the pier width.

Findings and Applications 53   The proposed universal shear stress decay equation for the mean near-bed shear stress at the pier uses the ratio of the mean near-bed shear stress at the pier for any given scour depth τpy and τTp0 (Figure 41). As stated previously, τTp0 is the total shear stress at the pier for zero-scour conditions and is obtained from Equation 21. The shear stress decay equation was obtained using multiple regression for all numerically simulated experiments: 0.435 (22) 0 2.21 0.638 F y h b py Tp r τ τ = +    − where Fr = V/ gh is the Froude number and V is the undisturbed flow velocity (far upstream of the pier). The R2 of Equation 22 is 0.87, and all the coefficients are statistically significant using the t-test (Table 14). When h and V are known and τTp0 is estimated, Equation 22 can be applied to different time steps or scour depths y in the experiments to predict the change in τpy (Figure 45). Equation 22 is now used to inform the shape of the actual shear stress decay curve for scour predictions, which is based on the change in the total shear stress at the pier with scour depth. The researchers therefore needed to convert Equation 22 from using average near-bed Reynolds stresses at the pier to using total shear stresses at the pier. To do this, a new function was needed that converts between mean near-bed and total shear stresses at the pier for the same scour depth (e.g., zero scour, certain scour depth y). The researchers used the two end members of scour in the numerically modeled experiments to obtain these concurrent shear stresses for the con- version equation: shear stresses at zero scour and at the maximum scour depth. For zero scour, the researchers have already used these shear stresses in previous equations; the average near- bed shear stress at the pier for zero-scour conditions (τp0) was obtained from the numerical models, Figure 44. Variation of mean near-bed shear stress at the pier (tpy) with scour depth y in all simulated experiments. (a) All individual experiments (different symbols) show a decrease in the average near-bed shear stress with increasing scour depth. (b) Although all experiments show a similar magnitude change in average near-bed shear stress with scour depth, each experiment had its own relation between scour depth and shear stress because of different flow conditions between experiments. Value Standard Error t Stat P-Value Coefficient 0.435 0.091 –9.135 4.53E-07 Fr 2.21 0.244 9.060 1.06E-07 (y + h)/b –0.638 0.093 –6.854 3.87E-06 Table 14. Significance tests for developed equation (Equation 22).

54 Determining Scour Depth Around Structures in Gravel-Bed Rivers and the total shear stress at the pier for zero-scour conditions (τTp0) was obtained using Equa- tion 21. For the maximum scour depth conditions, the average near-bed shear stress at the pier (τpymax) was obtained from the numerical models with maximum scour conditions, whereas the total shear stress was assumed to be equal to the critical shear stress (τc50, in dimensional form) for that experiment. The total shear stress at the pier must equal the critical shear stress at the condition of maximum scour because, by definition, scour needs to stop once the total shear stress is less than the critical value. The near-bed shear stresses τp0 and τpymax were obtained from the numerical simulations of experiments with σgc = 1.16 and Fsc = 0.029 for flow conditions D00 and D16 and of experiments with σgc = 1.53 and Fsc = 0.060 for flow conditions D00, D16, and D84 (Table 6). In all cases, τp0 was for simulations using the experiment time step at 0 hours, and τpymax was for simulations using the time steps of maximum scour (varied between experiments, see Table 6). Therefore, the researchers had 10 concurrent mean near-bed and total shear stresses at the pier (five both at zero and maximum scour) to develop the function that converts the mean near-bed shear stress at the pier for a given scour depth τpy into a total shear stress at the pier for the same scour depth τTpy (Figure 46a): 26.239 (23)1.001 3.129 f F y h bc r = +    − To obtain the shear stress decay equation in terms of total shear stresses, simply multiply Equation 22 by fc, obtaining: 11.414 (24) 0 3.211 3.767 F y h b Tpy Tp r τ τ = +    − The researchers confirmed that the coefficients in the equation were statistically significant using a t-test (Table 15). To analyze whether this equation provided accurate total shear stresses at the pier in experiments other than those used in its calibration, the researchers tested Equa- tion 24 using all of the laboratory experiments. In these other experiments, the values of τTpy were Figure 45. Mean near-bed shear stress at the pier for a given scour depth calculated in the numerical simulations versus that predicted by Equation 22. Different symbols represent different experiments.

Findings and Applications 55   estimated independently from Equation 24 for two conditions: (1) for zero-scour conditions, where τTpy = τTp0 (obtained from Equation 21), and (2) for the maximum scour depth at which τTpy = τc50 (obtained from Equation 17). The researchers obtained an R2 of 0.96 in a linear fit between the calculated and predicted total shear stresses at the pier (Figure 46b). Total shear stresses from Equation 24 are relatively accurate, but they can be further improved by strictly forcing the equation to have a certain total shear stress prediction for zero-scour condi- tions. For zero-scour conditions, the estimated τTpy/τTp0 was equal to 1 for each simulated experi- ment, but this condition was not always strictly met for every experiment in the fit regression equation because regression involves optimizing a fit between many data points. Any incorrectly predicted τTpy for zero scour conditions can result in large errors in predicted maximum scour depths (when combining the entire system of scour equations), and therefore it was important to obtain the exact value of τTpy/τTp0 = 1 for each experiment. A correction function was added to Equation 24 to force the equation to always predict τTpy = τTp0 when y = 0. This corrected equation is simply written as: 11.414 * (25) 0 3.211 3.767 F y h b Tpy Tp r τ τ = +    + ∆τ − where Δτ* is the correction function that smoothly varies with y and is valid for the range y ≤ 1.4b: * 1.4 2 1.4 1 (26)0 0 2y b y b Tp Tp ∆τ = ∆τ τ     −     +     Value Standard Error t Stat P-Value Coefficient 11.414 0.218 11.170 4.91E-08 Fr 3.211 0.667 4.817 3.36E-04 (y + h)/b –3.763 0.221 –17.01 2.90E-10 Figure 46. Results of the conversion from mean near-bed shear stresses to total shear stresses at the pier. (a) Conversion between the mean near-bed shear stresses and the total shear stresses at the pier using values of these stresses at zero and maximum scour in each modeled experiment. The dashed line only represents the trend of the data and does not reflect the predictions of Equation 24. (b) Calculated and predicted total shear stresses at the pier for different scour depths. Table 15. Significance tests for the developed equation (Equation 24).

56 Determining Scour Depth Around Structures in Gravel-Bed Rivers In Equation 26, ΔτTp0/τTp0 is the difference between 1 and the ratio τTpy/τTp0 predicted by Equation 24 when y = 0. Specifically: 1 11.414 (27)0 0 3.211 3.767 F h b Tp Tp ∆τ τ = −         − Equation 25 smoothly corrects the shear stress decay curve given by Equation 24 (Figure 47) to always predict the correct total shear stress at the pier for zero scour. This type of correction improves estimates of the total shear stress when scouring starts occurring (close to y = 0) because it ensures that τTpy/τTp0 is equal to 1 for the zero-scour state under any given flow condition. 3.2.3 Predictions of Maximum Scour Depths in Laboratory Experiments Combining shear stress equations (Equations 21, 25–27) and the critical shear stress equation (Equation 17) allows estimation of the maximum scour depth in each experiment. The authors outline the practical details of these calculation steps in Section 3.4 and here focus on the accu- racy of the predicted maximum scour values. Given that these equations are valid for multiple sediment layers, the researchers iteratively solved for when the total shear stress at the pier at a given scour depth (τTpy , Equation 25) first equaled the critical shear stress from the critical shear stress equation (Figure 48). Several intersections of these two equations can occur in a multilayer experiment, and the location where the first intersection (lowest y value, smallest scour depth) occurs is the predicted maximum scour depth. When applying this system of equations to all the laboratory experiments, the researchers obtained a root mean square error (RMSE) in predicted scour depth of 0.015 m. The scatter around the 1:1 line between measured and predicted maximum scour depths was similar between the calibration experiments used in the development of the equations and the validation experi- ments that were not used in any equation development (Figure 49). This test verifies that the equations can accurately predict scour in experiments that were not used in their development. Figure 47. Example total shear stress decay curve based on the uncorrected Equation 24 and the corrected Equation 25. The exact value of tTp0 at y 5 0 is produced when Equation 25 is applied, whereas Equation 24 overpredicts this value. Data from experiments with sgc 5 1.16 and Fsc 5 0.029 under clear-water conditions are shown here.

Findings and Applications 57   Note: [ ] denotes a dimensionless variable. Figure 48. A graphical description of the method to estimate the maximum scour depth based on the total shear stress decay curve (Equation 25) and the critical shear stress equation (Equation 17). The estimated scour depth is 0.105 m and reaches the subsurface bed layer. The measured scour depth was 0.112 m. Data correspond to the experiment with sgc 5 1.16, Fsc 5 0.029, and a D16 moving-flow condition. Figure 49. Measured and predicted dimensionless maximum scour depths for all experiments (y 1 h)/b. The 1:1 line represents perfect agreement between observed and predicted values.

58 Determining Scour Depth Around Structures in Gravel-Bed Rivers 3.3 Testing and Modification of Scour Equations Using Field Data 3.3.1 Testing Original System of Scour Equations The researchers tested whether this system of equations, which were developed using only laboratory experiments, could accurately predict the observed maximum scour depths for bridge piers in the field. The researchers used the USGS BSDMS data set (Landers et al., 1996) that contains the needed flow properties (e.g., flow depth and flow velocity), bed properties (e.g., bed slope and grain size distribution from which sand content and gravel sorting parameters can be estimated), pier width, and maximum scour depth. The field sites were limited to only include locations dominated by coarse bed material (gravel and coarser); any location where the median grain size was sand sized or finer was discarded. In this first assessment of the equations, the researchers only considered cylindrical piers with a single column, given that the experiments used this pier shape (see next section for other pier shapes). The researchers did not use observations in which (a) the calculated dimensional critical shear stress was larger than the calculated total shear stress at the pier for zero-scour conditions (τTp0) because those cases predicted no scour, and (b) the site had a slope with a default value of 0.0001. The researchers identified these default slopes by comparing the shear stress estimated using the drag coefficient method (τT = 0.5 ρCdV2, where Cd is the drag coefficient) to that estimated using τT = ρghS. Assuming that the velocity was accurately measured in all cases, the researchers determined that if the ratio between these two total undisturbed shear stresses was larger than 20, the measured slope was incorrectly registered, and the field site location was discarded for all subsequent analyses. The objective of this first round of testing was to determine if the functional form of the equations could be used to accurately predict scour depths for field conditions. After the researchers confirmed that the predictions were relatively accurate, they subsequently were able to remove the critical shear stress restriction (i.e., they later included sites where τTp0 ≤ τc50 in a design scour equation; see next section) as well as to extend and modify the equations to account for different pier shapes, configurations (single column or a group of columns), and angles of attack (see next section). After removing these locations, the researchers had 68 data points (including 15 from the labora- tory experiments). The bed grain size distributions were not reported in different sediment layers in the BSDMS data set, and the researchers therefore assumed one layer of sediment at all field sites. Only the D16, D50, and D84 were provided in the BSDMS data set rather than the entire grain size distribu- tion, and therefore the researchers could not separate the grain size distribution into gravel- and sand-sized particles. The researchers calculated the gravel sorting parameter (see previous sections for this equation) using the reported D16 and D84, but some of the D16 values were for sand-sized particles, which will induce some errors in the estimated gravel sorting parameters. No sand content was reported for any of the locations, and therefore the researchers estimated that the sand content was zero if the D16 was gravel sized (greater than 0.002 m), and the sand content was 16% if the D16 was sand sized. This likely produced an underestimate of sand content given that (1) some locations could have sand content greater than 16% but less than 50% when the D16 was sand sized but the D50 was gravel sized, or (2) some locations could have sand content between 1% and 15% even though the D16 was gravel sized. Therefore, uncertainty exists in both the gravel sorting parameters (possibly overestimated) and sand fractions (possibly underestimated) for all field locations. These uncertainties could have the same effects on predicted scour depths to possibly produce underestimates of scour; overestimates of the gravel sorting parameter and underestimates of sand content could both reduce scour depths by producing higher critical shear stresses than those actually present for the bed. When applying the new set of scour equations to these field sites and the laboratory data, the predicted and measured dimensionless scour depths closely followed the 1:1 line (Figure 50),

Findings and Applications 59   suggesting that the scour equations provided relatively accurate estimates of the maximum scour depth even without any field calibration. The predicted maximum scour depth had a RMSE of 0.51 m. 3.3.2 Field Calibration and Development of a Design Scour Equation The equations from the previous sections were obtained solely from laboratory experiments. Given that it was demonstrated that these equations could predict maximum scour depths relatively accurately in both field and laboratory conditions, Equation 25 (shear stress decay curve) is now further calibrated directly using a subset of the field observations. Following the procedure that was used to recalibrate Equation 24 to obtain Equation 25, Equation 25 was recalibrated using cylindrical or round piers with a single column. For this recalibration, randomly chosen observations were used from 50% of the field and laboratory data that had these pier shapes and that met the condition that τTp0 was greater than τc50. The recalibration was based on two conditions: (1) the total shear stress at the pier at zero scour had to be τTp0 (from Equation 21) calculated for that field site, and (2) the total shear stress at the pier at maximum scour had to be equal to τc50 (from Equation 17) calculated for that field site. In this new calibration, the researchers included angle-of-attack effects on round piers by correcting τTp0 (from Equation 21) with an additional calibration factor. After calibrating, the new set of shear stress decay equations can be written as: 0.95 0.9 1 exp 7.98 3.5 (28)0 K ghS h b Tpτ = ρ − + −    θ Note: [ ] denotes a dimensionless variable. Figure 50. Measured and predicted dimensionless maximum scour depths for field and laboratory observations. The 1:1 line represents a perfect agreement between observed and predicted values.

60 Determining Scour Depth Around Structures in Gravel-Bed Rivers 1.65 * 1 1.65 (29)0 6.9 2.75 6.9 2.75 F y h b F h bTpy Tp r Tpy r τ = τ +        + ∆τ −             − − ∆τ =     −     +* 1.4 2 1.4 1 (30) 2y b y bTpy Where Kθ is a function that corrects the total shear stress at the pier to account for angle-of-attack (θ, in degrees) effects for round-shaped piers: 1.5 0.1 (31)K = + θθ The angle of attack did not affect cylindrical piers; therefore, Kθ = 1 for this pier shape. The RMSE in the predicted maximum scour depth using this new set of equations was 0.608 m. This RMSE was higher than that provided by the original version of the equations (see Section 3.31) because the researchers included more data points (increased from 68 to 112) when testing the new equations by using both cylindrical and round-shaped piers (Figure 51). Although Equations 28 through 31 were developed using cylindrical and round-shaped piers with a single column, they can predict the maximum scour depths for sharp-nosed and square- nosed piers with a single column (also from the BSDMS field database) reasonably well without any modi fications (Figure 52). Equation 31 is used for all pier shapes except for cylindrical piers. The scour equations were developed to reduce the difference between observed and predicted dimensionless scour depths (y + h)/b. However, this set of equations resulted in approximately 33% of the data set having underpredicted maximum scour depths (Figure 53). The difference in the amount of scatter between dimensionless and dimensional scour depths is caused by Note: [ ] denotes a dimensionless variable. Figure 51. Measured and predicted dimensionless maximum scour depths for single cylindrical and round-shaped piers with predictions being calibrated using a subset of all observations. The 1:1 line represents a perfect agreement between observed and predicted values.

Findings and Applications 61   Note: [ ] denotes a dimensionless variable. Figure 52. Measured and predicted dimensionless maximum scour depths for piers with a single column in coarse-bedded rivers. Cylindrical and round pier shapes were used in calibrating the scour equations, but the equations predict accurate scour depths for piers with sharp and square shapes as well. The 1:1 line represents a perfect agreement between observed and predicted values. Figure 53. Measured and predicted maximum scour depths for piers with a single column in coarse-bedded rivers. The 1:1 line represents a perfect agreement between observed and predicted values.

62 Determining Scour Depth Around Structures in Gravel-Bed Rivers each data point (scour measurement or prediction) having different possible h and b values. Two data points with similar dimensionless scour depths can have very different dimensional scour depths because of these variable h and b values used in the conversion between dimensionless and dimensional scour depths. The same change in scatter between dimensional and dimension- less scour depths occurs regardless of the scour equation employed (see next section) and is not specifically caused by this set of new scour equations. Although this set of equations provides accurate scour predictions, such equations could be unsuitable for practitioners because under- estimating scour depths can have dangerous consequences. Considering the potential problems with underestimated scour depths, the researchers now propose a design equation in which the calculated scour depths are scaled such that scour is never underestimated. The researchers scaled the predictions of Equation 17 and Equations 28 through 31 using a new equation (Equa- tion 32) to ensure that all scour depths were never underestimated. The researchers also now incorporate field locations with flow conditions where τTp0 ≤ τc50 and those piers with a group of columns into the calibration of this equation. The design equation correction modifies the pre- dicted maximum scour depth (hereinafter called ymax) from the previous equations (Equation 17 and Equations 28 through 31) to yield a design scour depth ydes according to: (32)maxy b y h b K hdes g= + + φ    − Where Kg is equal to one for piers with a single column and Kg = 0.8 for piers with a group of columns. The scale parameter ϕ varies with the Froude number and the pier shape as follows: e e F F r r φ = − + φ = − + − + − + 0.91 0.82 1 for cylindrical and round piers 0.385 0.77 1 for sharp and square-nosed piers (33) 8 6 11 1 The results of the design set of equations (Equation 17 and Equations 28 through 33) are shown in Figure 54, demonstrating that the design scour depth is never underpredicted regard- less of pier shape or configuration. 3.3.3 Comparison of Predicted Design Scour Depths Between New Equations and HEC-18 Equations To further evaluate the performance of this new system of scour equations, the researchers compared its predictions of design scour depths to those produced by various versions of the fifth edition of HEC-18 equations. The researchers used three different versions of the HEC-18 equations: the general bridge-pier scour equation (Equation 1), the coarse-bed material equation (Equation 3), and the HN/GC HEC-18 equation for bridge-pier scour (HN/GC is the hanger number/gradation coefficient–based pier scour equation) (Equation 34) (Shan et al., 2016). The HN/GC equation is given by: 1.32 tanh 1.97 (34)1 2 3 0.62 10.38 2 1.5 y K K K a y H s = σ     The correction factor K2 for the flow angle of attack θ is calculated using (cosθ + L/a sinθ)0.65 and requires knowing the pier length L. However, some field data in the researchers’ database

Findings and Applications 63   Figure 54. Measured and predicted design scour depths using the proposed design equations. Design scour depths are consistently above the 1:1 line, indicating that design scour depths are never underestimated. 0 1 2 3 4 5 6 7 8 9 10 0 2 4 6 8 10 Pr ed ic te d (y +h )/b Observed (y+h)/b USGS Univ. Idaho 0 1 2 3 4 5 6 0 2 4 6 Pr ed ic te d sc ou r d ep th (m ) Observed scour depth (m) USGS Univ. Idaho Figure 55. Observed versus predicted scour depths (left: dimensionless, right: dimensional) using new set of bridge-pier scour equations. do not have lengths reported, and it was assumed that the pier length L and pier width a were the same. The measured and predicted design scour depths were compared for the new design scour equations (Equation 17 and Equations 28 through 33) developed here (Figure 55), general HEC-18 equation (Figure 56), coarse-bed HEC-18 equation (Figure 57), and HN/GC HEC-18 equation (Figure 58) using a total of 198 data points (USGS BSDMS data: 183 field locations, University of Idaho: 15 experiments) for coarse-bed conditions (D50 > 0.002 m) with all available pier types and shapes. The researchers also tested the HEC-18 coarse-bed material equation (Figure 59) and HEC-18 HN/GC equation (Figure 60) using a filtered data set based on the criteria for the equa- tion calibration and application described in HEC-18. The HEC-18 coarse-bed equation is only applicable to clear-water flow conditions and to coarse-bed materials with D50 ≥ 0.020 m and gradation coefficients σ ≥ 1.5. When all three of these criteria are met, it is recommended that

64 Determining Scour Depth Around Structures in Gravel-Bed Rivers 0 1 2 3 4 5 6 7 8 9 10 0 2 4 6 8 10 Pr ed ic te d (y +h )/b Observed (y+h)/b 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Pr ed ic te d sc ou r d ep th (m ) Observed scour depth (m) 0 1 2 3 4 5 6 7 8 9 10 0 2 4 6 8 10 Pr ed ic te d (y +h )/b Observed (y+h)/b 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Pr ed ic te d sc ou r d ep th (m ) Observed scour depth (m) 0 1 2 3 4 5 6 7 8 9 10 0 2 4 6 8 10 Pr ed ic te d (y +h )/b Observed (y+h)/b 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Pr ed ic te d sc ou r d ep th (m ) Observed scour depth (m) Figure 56. Observed versus predicted scour depths (left: dimensionless, right: dimensional) using the fifth edition HEC-18 general pier scour equation. Figure 57. Observed versus predicted scour depths (left: dimensionless, right: dimensional) using the fifth edition HEC-18 coarse-bed material equation. Figure 58. Observed versus predicted scour depths (left: dimensionless, right: dimensional) using the fifth edition HEC-18 HN/GC scour equation.

Findings and Applications 65   0 1 2 3 4 5 6 7 8 9 10 0 2 4 6 8 10 Pr ed ic te d (y +h )/b Observed (y+h)/b 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Pr ed ic te d y Observed y 0 1 2 3 4 5 6 7 8 9 10 0 2 4 6 8 10 Pr ed ic te d (y +h )/b Observed (y+h)/b 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Pr ed ic te d y Observed y Figure 59. Observed versus predicted scour depths (left: dimensionless, right: dimensional) using the fifth edition HEC-18 coarse-bed material equation with a filtered data set based on the criteria listed for applying this equation. Figure 60. Observed versus predicted scour depths (left: dimensionless, right: dimensional) using the fifth edition HEC-18 HN/GC scour equation with a filtered data set based on the criteria listed for applying this equation. the coarse-bed pier scour equation of HEC-18 be used rather than the general pier scour equa- tion of HEC-18. The HEC-18 HN/GC equation can be used for a broader range of conditions and is applicable to clear-water and live-bed conditions with Fr < 1.7 and to bed materials with 0.00021 m (0.21 mm) < D50 < 0.127 m and σ < 7.5. There were 92 data points that met the criteria for applying the coarse-bed HEC-18 equation, and 181 data points that met the criteria for applying the HN/GC equation. The performance of each equation was assessed by visual inspection of Figures 55 through 60 and by quantitative measures of reliability and accuracy (Shan et al., 2016). Reliability was defined by the reliability index (RI), which measures the risk of underpredicting the actual value of some property, which in this case is the design scour depth. A higher RI generally indicates a lower risk of underpredicting the scour depth. RI is defined in general terms as: 1 (35)RI M S x x = −

66 Determining Scour Depth Around Structures in Gravel-Bed Rivers where Mx is the mean of a set of x values and Sx is the standard deviation of a set of x values. For this analysis, x is defined as: (36), , x y y s m s p = and is the ratio of the measured to predicted scour depth, where ys,m is the measured scour depth and ys,p is the predicted scour depth. To facilitate direct comparisons between predicted and measured scour depths, accuracy was characterized by the relative error (RE) and the relative root mean square error (RRMSE). RE and RRMSE are defined as: (37), , , RE y y y s p s m s m = − ∑ ( )= −            −1 (38), , , 2 RRMSE y y y ns p s m s m where n is the number of observations. The HEC-18 general pier scour equation (RI = 3.28) and the new set of scour equations (RI = 2.47) never underpredicted the measured scour depths and were more reliable than the coarse-bed (RI = –0.11) and HN/GC (RI = 0.18) equations, which often underpredicted scour (Table 16). However, the most accurate equation was generally the opposite of whichever equation was the most reliable. The HEC-18 coarse-bed equation had the lowest RRMSE, reflecting the most accurate scour predictions, and the HEC-18 HN/GC equation was the second most accurate (Table 16). The new set of scour equations was more accurate than the HEC-18 general equation by having a lower RRMSE. Given that reliability is the primary concern with predicting bridge-pier scour and that underpredicted scour depths can be dangerous, the general HEC-18 equation and the new scour equation are preferred over the coarse-bed HEC-18 and the HN/GC equations. The new scour equation increased predicted scour depth accuracy over the general HEC-18 equation without sacrificing reliability. In addition, the same results occurred when analyzing the reliability and accuracy of each equation for each different pier shape (Table 17), with the possible exception of square-shaped piers, but this pier shape only had six data points tested. Unlike any of the other tested scour equations, the new set of scour equations is based on the mechanistic processes that control bridge-pier scour such as the onset of sediment motion through a critical shear stress and the decay in applied shear stress in the scour hole as scour progresses. All Data Filtered Data HEC-18 General HEC-18 Coarse Bed HEC-18 HN/GC New Scour Equations (Univ. Idaho) HEC-18 Coarse Bed HEC-18 HN/GC n 198 198 198 198 92 181 RI 3.28 –0.03 0.16 2.47 –0.11 0.18 RRMSE 3.80 2.21 3.11 3.53 1.54 3.08 Table 16. Performance of tested bridge-pier scour equations.

Findings and Applications 67   All of the other tested scour equations are largely based on empirical fits between measured scour depths and assumed controlling variables but lack the variables that are more closely linked to the mechanics that control scour, such as critical shear stresses and near-bed shear stresses in the scour hole. The new set of scour equations also incorporate the fact that the critical shear stress depends on the gravel sorting parameter and sand content and could vary between layers. Such mechanics are important components that many bridge-pier scour equations are currently missing, and the researchers recommend inclusion of these variables (or similar variables) in any new developed scour equation. Given the increase in accuracy and the broader mechanistic basis of the new set of equations, the researchers recommend their use as an alternative or as a complement to the general version of the HEC-18 equation. Despite the inclusion of these detailed mechanics, the new set of scour equations did not perfectly predict scour depths, and the researchers attribute this to a number of important factors. First, like the other bridge-pier scour equations tested, this new set of equations was developed to never underpredict scour, which generally decreases scour prediction accuracy at the expense of safely predicting maximum possible scour magnitudes. Second, the field measure- ments of scour that were used to calibrate and validate this set of equations had very limited bed grain size data available. In particular, no measurements of sand content or detailed gravel grain sizes were available, and these grain size data were for a single layer rather than for multiple layers. Although more detailed grain size information is not needed to predict scour using this new set of equations, the researchers hypothesize that the new set of scour equations would be more accurate if such data were available for each field site. Therefore, collection of more detailed grain size information, when possible, is recommended for future studies. Third, the researchers have assumed that the near-pier shear stresses can be easily translated into reach-averaged shear stresses using simple and easily measured reach-averaged flow variables that were available in the field data. Although this assumption is generally valid, each bridge pier in the field will experi- ence complex local flow conditions that will likely depend on more than these simple variables, such as the local and overall channel morphology and the position of the pier in a cross-section. Currently, such local complexities of each pier cannot be included in the new set of pier scour equations because this information is not available. However, as state departments of trans- portation (DOTs) move to conduct more 2D flow modeling around piers to help inform their analyses, some information about the detailed flow complexity at each pier will be available. Near-pier shear stresses could be determined for such complex flow fields and could be used to modify the new set of scour equations to better represent the complexities of each pier. Chapter 4 discusses these next steps for building on the results and equations. All Data Filtered Data Pier Shape HEC-18 General HEC- 18 Coarse Bed HEC-18 HN/GC New Scour Equations (Univ. Idaho) HEC-18 Coarse Bed HEC-18 HN/GC All RI 3.28 –0.03 0.16 2.47 –0.11 0.18 RRMSE 3.80 2.21 3.11 3.53 1.54 3.08 Round and cylindrical RI 4.46 0.27 0.57 2.51 –0.11 0.18 RRMSE 4.03 2.30 3.25 3.54 2.06 3.22 Sharp RI 2.56 –0.29 –0.16 2.35 -0.45 –0.14 RRMSE 3.79 1.94 2.67 3.79 0.58 2.66 Square RI 2.93 3.97 6.21 4.01 – 6.21 RRMSE 3.76 2.13 2.94 3.81 – 2.93 Table 17. Performance of tested bridge-pier scour equations as a function of pier shape.

68 Determining Scour Depth Around Structures in Gravel-Bed Rivers 3.4 Example Steps of Application of New Scour Equations for Practitioners The new set of design scour equations requires (1) solving equations that contain different shear stress definitions that may be confusing, (2) a series of detailed steps to apply the equations, and (3) iteration to solve the equations. To help explain how these equations are applied, the researchers provide a detailed example that goes through all the required steps and solves for the design scour depth using all combined equations. The researchers used data from one of the experiments (σgc = 1.16, Fsc = 0.029, and D16 moving-flow condition) because it is an example of a more complex possible scenario, which is a bed with multiple sediment layers. Many field cases will only involve a single sediment layer because these may be the only grain size data that are available to practitioners. In such a case, the calculations are the same except that no iterations for different layers are required. Notice that the calculated scour depth will be slightly different than that provided in Figure 48 because, in that case, the original set of equations developed only for the laboratory experiments (Equation 17, Equation 21, and Equations 25 through 27) were used to calculate scour rather than the final design equations that use laboratory and field data (Equation 17 and Equations 28 through 33). The researchers rewrite Equation 17 and Equations 28 through 33 in the steps that follow so that the reader does not need to go through the entire report to apply the equations. The example is given first in SI units and then is replicated in U.S. customary units. 3.4.1 Step-by-Step Use of Scour Equations with a Worked Example In the worked example, the pier is cylindrical and its configuration is single. All required variables are summarized in Tables 18 and 19 with example values from the experiment. “Layers” denotes the number of sediment layers for which data are available; if only one layer is available, then N = 1. In this example, the researchers have two layers. If only one layer is present or measured, the provided layer thickness must be much larger than the expected scour depth. If multiple layers are measured, the provided thickness of all layers combined must be much larger than the expected scour depth. The flow depth h, flow velocity V, and bed slope S are undisturbed values. The sediment density ρs, water density ρ, and gravitational acceleration g are also needed. The required grain size information for each layer (or a single layer) includes the grain sorting parameter σsg, the fraction of sand content on the bed Fs, and the median grain size D50. The grain sorting parameter is defined as (D84/D16)0.5, where D84 and D16 are the 84th and 16th percentiles of the grain size distribution. In the experiments used to develop the scour equations, D50, D84, and D16 were defined using only gravel-sized particles (diameter > 0.002 m) to isolate the influ- ences of gravel and sand on scour depths. However, the field data available to further calibrate the scour equations did not provide information on gravel versus sand content, and therefore the D50, D84, and D16 values were defined from the entire measured grain size distribution (including sand). Thus, practitioners can use D50, D84, D16, and σsg defined using just the gravel portion of the layer distribution or using the entire layer grain size distribution (including sand and gravel) for application of the new set of scour equations. However, the equations are only valid for coarse-bedded rivers dominated by gravel rather than rivers dominated by finer material (e.g., sand and finer). Recommendations for sediment sampling procedures are provided in Appendix B. Example in SI Units Step 1: Calculate tTp0 To calculate the total shear stress at the pier for zero-scour conditions (τTp0), use:

Findings and Applications 69   K ghS h b Tpτ = ρ − + −    =θ 0.95 0.9 1 exp 7.98 3.5 25.621Pa0 Where Kθ is an angle-of-attack correction factor. In this example case, Kθ = 1 is used because it is a cylindrical pier, but for all other pier shapes use Kθ = 1.5 + 0.1θ. Step 2: Calculate the Shear Stress Decay Curve Use τTp0 and the shear stress decay equation to calculate the change in the total shear stress at the pier, τTpy, with scour depth y. The shear stress decay equation is given by: F y h b F h bTpy Tp r Tpy r τ = τ +        + ∆τ −             = − − 1.65 * 1 1.65 10.1Pa0 6.9 2.75 6.9 2.75 where 0.94,F V ghr = = and * 1.4 2 1.4 1 0.37 2y b y bTpy ∆τ =     −     + = and where the final equation is only valid when y ≤ 1.4b and is 0 otherwise. The results for each of these equations are provided for this specific example pier when y = 0.06 m. These equations need to be used for a range of y values to produce the shear stress decay curve for each field location, which is better represented graphically using the specific example experi- ment (Figure 61). Step 3: Calculate the Critical Shear Stress for All Individual Sediment Layers Calculate the critical shear stress for the median grain size τc50 as: ( )( )( )τ = σ − − ρ − ρexp 0.279 2.38 3.550 50F gDc g s s For the first layer in this example, σg = 12.81 9.2 = 1.18 and τc50 = 7.473 Pa, while for the second layer, σg = 10.007 7.7 = 1.14 and τc50 = 5.208 Pa. It was assumed that ρs = 2,650 kg/m3. Variables Flow depth h [m] 0.190 Layer 1 Layer 2 Flow velocity V [m/s] 1.278 16th percentile grain size D16 [mm] 9.200 7.700 Bed slope S [–] 0.0115 Median grain size D50 [mm] 11.000 8.900 Pier width b [m] 0.110 84th percentile grain size D84 [mm] 12.810 10.007 Angle of attack θ [°] 0 Sand content Fs [–] 0.000 0.058 Layer thickness z [m] 0.011 0.200 Table 18. Required field measurements and variables for worked example with SI units.

70 Determining Scour Depth Around Structures in Gravel-Bed Rivers Step 4: Calculate the Maximum Scour Depth by Iterating for the Solution in Each Layer This can be done graphically by superimposing the critical shear stress calculated for each sediment layer from Step 3 and the total stress decay curve from Step 2, as seen in Figure 62. The location where both curves first intersect (lowest scour depth) corresponds to the maximum scour depth. Mathematically, the shear stress decay equation can be evaluated at the maximum depth of each individual layer and τTpy compared at that depth to τc50. If τTpy > τc50, one must proceed to Figure 61. Example of the total shear stress decay curve for the example experiment. Figure 62. Example of total shear stress decay curve (solid line) superimposed with the critical shear stress calculated for each sediment layer (dashed line), where the maximum scour depth is the first depth (lowest y value) at which the two curves cross.

Findings and Applications 71   the next layer. For this example, in the first layer with Δτ*Tpy = 0.862 (see equation in Step 2) and using the layer depth of 0.011 m in the shear stress decay equation, the resulting τTpy at the bottom of the first layer in this experiment is: F y h b F h bTpy Tp r Tpy r y Tpy τ = τ +        + ∆τ −             τ = − − = 1.65 * 1 1.65 22.06 Pa 0 6.9 2.75 6.9 2.75 0.011m Therefore, τTpy > τc50 (22.06 Pa > 7.472 Pa), and one must continue to the maximum depth (0.211 m) of the next layer using the shear stress decay equation again as: F y h b F h bTpy Tp r Tpy r y Tpy τ = τ +        + ∆τ −             τ = − − = 1.65 * 1 1.65 0.787 Pa 0 6.9 2.75 6.9 2.75 0.211m where Δτ*Tpy = 0 in this layer because y > 1.4b (see Step 2). Now, τTpy < τc50 (0.787 Pa < 5.208 Pa) of this layer, meaning that the maximum scour depth must be within the second sediment layer. To find the exact maximum scour depth ymax, set τTpy = τc50 in the shear stress decay equation and use the specific conditions of this second sediment layer to solve for ymax = y. However, this nonlinear equation cannot be solved analytically and needs some type of solver or root-finding algorithm (see spreadsheet implementation in Section 3.4.2). The calculated maximum scour depth in this example is ymax = 0.092 m and has reached the second layer. Step 5: Calculate the Design Scour Depth The design scour depth ydes is given by: maxy b y h b K hdes g= + + φ    − Where Kg is equal to one for piers with a single column and Kg = 8 for piers with a group of columns. The scale parameter ϕ varies with the Froude number and the pier shape as follows: e e F F φ = − + φ = − + − + − + 0.91 0.82 1 for cylindrical and round piers 0.385 0.77 1 for sharp and square-nosed piers 8 6 11 1 Given that this example has a cylindrical pier with a single column, Kg = 1 and ϕ is: φ = − + − + 0.91 0.82 1 = 0.241 8 6e F Finally, ydes = 0.118 m in this example.

72 Determining Scour Depth Around Structures in Gravel-Bed Rivers Example in U.S. Customary Units Step 1: Calculate tTp0 To calculate the total shear stress at the pier for zero-scour conditions (τTp0), use: 0.95 0.9 1 exp 7.98 3.5 0.535 psf0 K ghS h b Tpτ = ρ − + −    =θ where Kθ is an angle-of-attack correction factor. In this example case, Kθ = 1 is used because it is a cylindrical pier, but for all other pier shapes, use Kθ = 1.5 + 0.1θ. Step 2: Calculate the Shear Stress Decay Curve Use τTp0 and the shear stress decay equation to calculate the change in the total shear stress at the pier τTpy with scour depth y. The shear stress decay equation is given by: 1.65 * 1 1.65 0.211 psf0 6.9 2.75 6.9 2.75 F y h b F h bTpy Tp r Tpy r τ = τ +        + ∆τ −             = − − where 0.94,F V ghr = = and * 1.4 2 1.4 1 0.37 2y b y bTpy ∆τ =     −     + = and where the final equation is only valid when y > 1.4b and is 0 otherwise. The results for each of these equations are provided for the specific example pier when y = 0.197 ft. These equations need to be used for a range of y values to produce the shear stress decay curve for each field location, which is better represented graphically using the specific example experiment (Figure 63). Step 3: Calculate the Critical Shear Stress for All Individual Sediment Layers Calculate the critical shear stress for the median grain size τc50 as: ( )( )( )τ = σ − − ρ − ρexp 0.279 2.38 3.550 50F gDc g s s Variables Flow depth h [ft] 0.623 Layer 1 Layer 2 Flow velocity V [ft/s] 4.193 16th percentile grain size D16 [in] 0.362 0.303 Bed slope S [ft/ft] 0.0115 Median grain size D50 [in] 0.433 0.350 Pier width b [ft] 0.361 84th percentile grain size D84 [in] 0.504 0.394 Angle of attack θ [°] 0 Sand content Fs [–] 0.000 0.058 Layer thickness z [ft] 0.036 0.656 Table 19. Required field measurements and variables for worked example with U.S. customary units.

Findings and Applications 73   For the first layer in this example, σg = 0.504 0.361 = 1.18 and τ50 = 0.156 psf while for the second layer, σg = 0.394 0.303 = 1.4 and τ50 = 0.109 psf. It is assumed that ρs = 165.43 lb/ft3. Step 4: Calculate the Maximum Scour Depth by Iterating for the Solution in Each Layer This can be done graphically by superimposing the critical shear stress calculated for each sediment layer from Step 3 and the total stress decay curve from Step 2, as seen in Figure 64. The location where both curves first intersect (lowest scour depth) corresponds to the maxi- mum scour depth. Figure 63. Example of the total shear stress decay curve for the example experiment in U.S. customary units. Figure 64. Example of total shear stress decay curve (solid line) superimposed with the critical shear stress calculated for each sediment layer (dashed line), where the maximum scour depth is the first depth (lowest y value) at which the two curves cross.

74 Determining Scour Depth Around Structures in Gravel-Bed Rivers Mathematically, the shear stress decay equation can be evaluated at the maximum depth of each individual layer and τTpy compared at that depth to τc50. If τTpy > τc50, one must proceed to the next layer. For this example, in the first layer with Δτ*Tpy = 0.862 (see equation in Step 2) and using the layer depth of 0.036 ft in the shear stress decay equation, the resulting τTpy at the bottom of the first layer in this experiment is: F y h b F h bTpy Tp r Tpy r y Tpy τ = τ +        + ∆τ −             τ = − − = 1.65 * 1 1.65 0.46 psf 0 6.9 2.75 6.9 2.75 0.036 ft Therefore, τTpy > τc50 (0.46 psf > 0.156 psf), and one must continue to the maximum depth (0.692 ft) of the next layer using the shear stress decay equation again as: F y h b F h bTpy Tp r Tpy r y Tpy τ = τ +        + ∆τ −             τ = − − = 1.65 * 1 1.65 0.0164 psf 0 6.9 2.75 6.9 2.75 0.692 ft where Δτ*Tpy = 0 in this layer because y > 1.4b (see Step 2). Now, τTpy < τc50 (0.0164 psf < 0.109 psf) of this layer, meaning that the maximum scour depth must be within the second sediment layer. To find the exact maximum scour depth ymax, set τTpy = τc50 in the shear stress decay equation and use the specific conditions of this second sediment layer to solve for ymax = y. However, this nonlinear equation cannot be solved analytically and needs some type of solver or root-finding algorithm (see spreadsheet implementation Section 3.4.2). The calculated maximum scour depth in this example is ymax = 0.301 ft and has reached the second layer. Step 5: Calculate the Design Scour Depth The design scour depth ydes is given by: maxy b y h b K hdes g= + + φ    − where Kg is equal to one for piers with a single column and Kg = 0.8 for piers with a group of columns. The scale parameter ϕ varies with the Froude number and the pier shape as follows: e e F F φ = − + φ = − + − + − + 0.91 0.82 1 for cylindrical and round piers 0.385 0.77 1 for sharp and square-nosed piers. 8 6 11 1 Given that this example has a cylindrical pier with a single column, Kg = 1, and ϕ is: φ = − + − + 0.91 0.82 1 = 0.241 8 6e F Finally, ydes = 0.388 ft in this example.

Findings and Applications 75   3.4.2 Implementation of Scour Equations in a Simple Spreadsheet Given that estimating the maximum scour depth requires solving nonlinear equations and numerous calculation steps, the authors have developed a spreadsheet that solves the set of equations automatically for practitioners. The development of the new scour equations and this spreadsheet incorporated comments received during meetings with members of seven state DOTs (see Appendix A). A visual basic code is embedded within the spreadsheet; therefore, only the input variables are required to calculate the design scour depths (Figure 65). The spread- sheet can be found on the on the National Academies Press website (nap.nationalacademies.org) by searching for NCHRP Research Report 1031: Determining Scour Depth Around Structures in Gravel-Bed Rivers. A drop-down menu allows the user to select SI or U.S. customary units. Pier configuration and shape are also selected using a drop-down menu. All other input variables are as previously detailed in the calculation steps section. Once the input variables are entered, the user presses “Execute,” and the main results are shown (Figure 66). When using U.S. customary units, the same workflow is required, and the results are displayed consistently with the selected unit system (Figure 67). The user should select the units for bed slope to choose between ft/ft and ft/mile.

Figure 65. Implementation of the scour equations in an Excel spreadsheet. Only the input variables are required to estimate the design scour depth.

Figure 66. Results displayed in the Excel spreadsheet. All results are automatically displayed after the equations have been solved.

Figure 67. Results displayed in the Excel spreadsheet for the same case as Figure 66 but in U.S. customary units. All results are automatically displayed after the equations have been solved.