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59 5.1 Introduction Chapter 5 provides two approaches for assessing the conditional probability that the design scour depth will be exceeded for a given design flood event. Either approach can be used to estimate this probability for each of the three individual scour components. The first approach (Level I) assumes that the practitioner can categorize a bridge based on three general condi- tions: (1) the size of the bridge, channel, and floodplain; (2) the size of the piers; and (3) the hydrologic uncertainty. This Level I approach provides scour factors which can be used to mul- tiply the estimated scour depth to achieve a desired level of confidence based on the reliability index, commensurate with standard LRFD practice. When the practitioner cannot match a particular site to a category based on the general con- ditions described in the preceding paragraph, a Level II approach is necessary. The Level II approach is necessarily site-specific and is illustrated using data from a bridge on the Sacramento River. The discussion includes the results for pier, contraction, abutment, and total scour consid- ering hydrologic uncertainty, hydraulic uncertainty, and scour prediction (model) uncertainty. A step-by-step summary of the Level II procedure is also provided. 5.2 Approach 5.2.1 Background The primary objective of NCHRP Project 24-34 (Lagasse et al. 2013) was to develop a meth- odology that can be used to estimate the probability that the design scour level will be exceeded. The goal was to check whether the probability of design scour exceedance will meet an accept- able level of risk. The developed probabilistic procedures were to be consistent with LRFD approaches used by structural and geotechnical engineers. This objective was achieved by providing a set of tables of probability values and scour factors for a given design event that can be used to associate the estimated scour depth with a condi- tional probability of exceedance (i.e., that probability is conditional based on the hydrologic design event selected). The probability values and scour factors were calibrated for typical bridge foundations and river channel geometries and conditions. A 100-year return period was used as the design event. This approach is identified as Level I analysis. For complex foundation systems and channel conditions, or for cases requiring special consideration, a Level II approach that consists of a step-by-step procedure that hydraulic engineers can follow to provide site-specific probability estimates was developed. Providing the Level II option is similar to what the AASHTO Guide Manual for Condition Evaluation and Load and Resistance Factor Rating (LRFR) of Highway Bridges (2005) proposes when a refined evaluation is deemed necessary. C H A P T E R 5 Probability-Based Scour Estimates
60 Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction 5.2.2 Calibration of Level I Statistical Parameters The Level I approach provides an easy-to-apply method to allow the engineer to control the level of safety to use when designing a foundation for scour. The calibration of the probability values and scour factors requires knowledge of the appropriate bias and COV values, which may depend on the bridge foundation and channel geometric and site conditions. These two parameters must account for all the levels of uncertainties and conservative assumptions that are intentionally or unintentionally embedded in the scour estimation process. Two types of uncertainties need to be accounted for: 1. Aleatory uncertainties (natural uncertainties due to inherent parameter variability and ran- domness), and 2. Epistemic uncertainties (modeling uncertainties). Aleatory uncertainties are due to random variations in the variables that control the param- eter being estimated. For example, a 100-year river discharge rate used for design is only an estimated value that is calculated from previous discharge rates. Such estimates are associated with various levels of uncertainties. Similarly, even when measured in laboratory tests, esti- mated values of soil properties are associated with various levels of uncertainties that are due to local spatial variations in the soil profile and uncertainties in the accuracy of the test devices. The calibration of the probability values and scour factors accounts for the uncertainties inherent in the scour analysis process. These include modeling (epistemic) uncertainties as well as parametric (aleatory) uncertainties as described in the preceding paragraph. The availability of probability values and scour factors that represent typical (standard) conditions provides an engineer with the flexibility of selecting the level of scour risk appropriate for the particular bridge being analyzed for a given design event. That level of risk is represented by a reliability index, b. (See Section 2.5.2 for a discussion of the reliability index as a measure of structural safety.) Section 5.2.3 and Section 5.3 outline the development of the scour factor tables, describe a representative table, and summarize the bias and COV values for the individual scour com- ponents. Chapter 7 provides illustrative examples applying the Level I approach to determine the conditional probability of exceedance for estimated scour depths for bridges selected from different physiographic regions of the United States. 5.2.3 Level I Applications for Typical Site Conditions The Level I approach to providing probability values and scour factors for typical or standard bridge configurations is shown in Table 5.1. A 3 x 3 matrix based on bridge size (bridge length) and pier size is considered as shown in the table. The analysis includes a small, medium, and large bridge each with small, medium, and large piers. The size of the piers increases propor- tionately with each bridge. Bridge, channel, and floodplain size scale together and each must be represented by a Monte Carlo simulation. In addition, the typical bridge matrix was expanded by including three levels Bridge Length (ft) Pier Size (ft) Bridge Size Range Monte Carlo Small Medium Large Small < 100 50 1 2 3 Medium 100â300 180 1.5 3 4.5 Large > 300 1200 3 6 9 Table 5.1. Bridge and pier geometry for typical bridges.
Probability-Based Scour Estimates 61 of hydrologic uncertainty. The values in Table 5.2 show the 100-year discharges used for the typical bridges and correspond to the values shown in Table 3.2 and Table 3.4. Thus, a total of 27 scour permutations were considered for the Level I analysis. 5.3 Level I Analysis and Results The results of each of the 27 Monte Carlo scour simulations (3 bridge sizes à 3 pier sizes à 3 hydrologic uncertainties) were analyzed to compute pier scour (HEC-18 and Florida DOT), contraction scour (HEC-18), and abutment scour (NCHRP 24-20) for representative 100-year design events (see Table 5.2). Total scour, the sum of pier and contraction scour, was also com- puted using each of the pier scour equations. Each simulation included a computation of design scour for the base condition. With every Monte Carlo realization, the computed amounts of each scour component were adjusted with the laboratory bias and COV applied as normally dis- tributed random numbers. This produced data sets of 10,000 scour values that included model (equation) uncertainty and hydraulic uncertainty, where hydraulic uncertainty is the combina- tion of hydrologic, Manning n, and boundary condition uncertainties. From each Monte Carlo simulation (10,000 runs), the expected scour (mean of the data set), bias (expected/design), standard deviation (SD), and COV (standard deviation/expected) were computed. In total, more than 300,000 HEC-RAS/Monte Carlo simulations were required to produce the statistics on which the 27 tables in Appendix B are based. In addition, more than 300,000 scour calculations for each of the scour equations (i.e., more than 1.2 million off-line scour calculations) were completed off-line. For each of the types of scour, the bias from the Monte Carlo simulation was essentially equal to the model bias. This was expected because the hydraulic uncertainties result in scour conditions more and less severe than the base hydraulic condition. For pier scour (both HEC-18 and Florida DOT), the COV from the Monte Carlo simulations was also essentially the same as the model COV. This indicates that the model bias and COV are the primary factors for the extreme conditions represented by the Monte Carlo simulations, which were computed for 100-year events. For contraction and abutment scour, although the bias from the Monte Carlo simulations was essentially equal to the model bias from the laboratory data, COV was greater in the Monte Carlo simulations. Although the hydraulic conditions were both more and less severe than the base condition, the variability of hydraulic conditions produced highly variable contraction scour results. Because abutment scour depends on contraction scour, the increased variability was also seen in the abutment scour results. Table 5.3 shows the summary table from one Monte Carlo simulation. Appendix B presents 27 summary tables from the Monte Carlo simulations (see also Table 5.1 and Table 5.2). Table 5.3 represents a medium bridge with a medium pier size and medium hydrologic uncertainty, and corresponds to Table B.14 in the Appendix. Each of the types of scour is shown. For pier scour, the HEC-18 equation results in design scour of 7.20 ft. Design contraction scour is 8.02 ft, for Hydrologic Uncertainty Low Medium High Bridge Size Q100 (cfs) 5% 95% 5% 95% 5% 95% Small 1,840 1,610 2,100 1,520 2,230 1,430 2,370 Medium 29,800 24,800 35,700 22,800 38,900 21,000 42,200 Large 144,000 117,000 178,000 106,000 196,000 96,400 216,000 Table 5.2. Bridge discharges for typical bridges.
62 Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction a total design scour of 15.22 ft. Considering the bias in the scour equations, the results of the Monte Carlo simulation indicate expected scour of 4.89 ft of pier scour, 7.42 ft of contraction scour, and 12.31 ft of total scour. Although the sum of the expected component scour values equals the total expected scour, the expected total scour was actually calculated as the average of the 10,000 computed total scour amounts. This very consistent result indicates that the expected total scour can be computed from the expected values of pier and contraction scour. Looking in Table 5.3 and using Equation (2.8), the HEC-18 pier scour equation reliability index, b, is calculated as (7.20 - 4.89)/0.77 = 3.0, which compares to the tableâs value of 2.99. The difference is due to the number of significant figures displayed in the table. Contraction scour has a very low reliability based on the expected scour being only slightly less than the design Pier Scour (HEC-18) Pier Scour (FDOT) Scour Total Scour (HEC-18) Total Scour (FDOT) Abutment Scour Design scour (ft) 7.20 5.94 8.02 15.22 13.95 15.12 Expected scour (ft) 4.89 4.45 7.42 12.31 11.87 11.35 Bias 0.68 0.75 0.93 0.81 0.85 0.75 Std. dev. (ft) 0.77 0.79 2.74 2.86 2.89 3.18 COV 0.16 0.18 0.37 0.23 0.24 0.28 Design scour 2.99 1.89 0.22 1.01 0.72 1.18 Non-exceedance 0.9986 0.9706 0.5857 0.8444 0.7648 0.8818 Scour Non-exceedance (ft) Based on Monte Carlo Results = 0.5 (0.6915) 5.29 4.85 8.60 13.58 13.13 12.77 = 1.0 (0.8413) 5.68 5.24 10.17 15.18 14.76 14.55 = 1.5 (0.9332) 6.05 5.63 11.89 16.90 16.47 16.38 = 2.0 (0.9772) 6.44 6.01 13.56 18.69 18.28 18.21 = 2.5 (0.9938) 6.73 6.37 15.50 20.73 20.21 20.54 = 3.0 (0.9987) 6.96 6.62 17.24 22.54 22.19 22.31 = 0.5 (0.6915) 0.73 0.82 1.07 0.89 0.94 0.84 = 1.0 (0.8413) 0.79 0.88 1.27 1.00 1.06 0.96 = 1.5 (0.9332) 0.84 0.95 1.48 1.11 1.18 1.08 = 2.0 (0.9772) 0.89 1.01 1.69 1.23 1.31 1.20 = 2.5 (0.9938) 0.94 1.07 1.93 1.36 1.45 1.36 = 3.0 (0.9987) 0.97 1.11 2.15 1.48 1.59 1.48 Scour Non-exceedance (ft) Based on Scour Mean and Standard Deviation = 0.5 (0.6915) 5.28 4.84 8.79 13.75 13.31 12.94 = 1.0 (0.8413) 5.66 5.23 10.16 15.18 14.75 14.53 = 1.5 (0.9332) 6.05 5.63 11.53 16.61 16.20 16.12 = 2.0 (0.9772) 6.43 6.02 12.91 18.04 17.64 17.72 = 2.5 (0.9938) 6.82 6.42 14.28 19.48 19.08 19.31 = 3.0 (0.9987) 7.20 6.81 15.65 20.91 20.53 20.90 Scour Factors Based on Scour Mean and Standard Deviation = 0.5 (0.6915) 0.73 0.82 1.10 0.90 0.95 0.86 = 1.0 (0.8413) 0.79 0.88 1.27 1.00 1.06 0.96 = 1.5 (0.9332) 0.84 0.95 1.44 1.09 1.16 1.07 = 2.0 (0.9772) 0.89 1.01 1.61 1.19 1.26 1.17 = 2.5 (0.9938) 0.95 1.08 1.78 1.28 1.37 1.28 = 3.0 (0.9987) 1.00 1.15 1.95 1.37 1.47 1.38 Scour Factors Based on Monte Carlo Results Contraction Table 5.3. Medium bridge, medium hydrologic uncertainty, medium pier (3 ft).
Probability-Based Scour Estimates 63 scour and a very large value of COV, which was 0.21 from the model (equation) and increased to 0.37 for this bridge associated with hydraulic uncertainty. Also included in Table 5.3 is an estimate of the design-equation non-exceedance b value and percentile computed from the design scour, expected scour, and scour standard deviation assuming a normal distribution. As indicated in Table 5.3, a b value of 0.5 (for example) results in a probability of scour depth non-exceedance of 69.15%, or conversely, an exceedance prob- ability of 30.85% for this bridge during a 100-year event. Notice that Table 5.3 provides scour non-exceedance depths and corresponding scour factors derived directly from the Monte Carlo simulation (based on Monte Carlo results), and also with the assumption that the 10,000 pre- dicted scour depths are normally distributed (based on scour mean and standard deviation). The fact that the scour depths and scour factors are similar but not identical indicates that the probability distribution based on Monte Carlo results is not precisely normal. The pier scour standard deviation for this simulation was 0.77 ft (COV = 0.16). Contraction scour was much more variable with a standard deviation of 2.74 ft (COV = 0.37). The total scour standard deviation from the Monte Carlo results was 2.86 ft (COV = 0.23) and can be estimated from the pier and contraction component values as the square root of the sum of the squares (0.772 + 2.742)0.5 = 2.85 ft (Equation [2.8]). As shown in Table 5.3, HEC-18 and Florida DOT pier scour results have the highest level of reliability, contraction scour has the lowest level of reliability, and abutment scour has an inter- mediate level of reliability. Because total scour is used in design at a pier, the high reliability of the pier scour compensates for the lower level of reliability in the contraction scour value. This cannot, however, be considered a general result because of cases where there is small pier scour and large contraction scour. Table 5.3 also shows non-exceedance scour amounts for b ranging from 0.5 to 3.0. These amounts are computed in two ways for comparison. The first method is to take the amount directly from the Monte Carlo results and the second method is to calculate the amount based on the expected scour and standard deviation. The results of the two methods generally fall within plus or minus 5% for all scour components; however, the contraction scour amounts tend to be greater with the Monte Carlo results than from the statistics for b of 2.0 to 3.0. From the non-exeedance scour values, the scour factors for each scour component are also shown. For this bridge, pier size, and hydrologic uncertainty, the Monte Carlo results show that the HEC-18 pier scour equation provides a b of 3.0 without any increase whereas the Florida DOT equation would require a small scour factor (1.11) to achieve a b of 3.0. Based on the Monte Carlo results, the current design values of contraction and abutment scour would have to be increased by factors of 2.15 and 1.48 to achieve this level of reliability. The scour factors for each component can be used for that component individually but can- not be combined individually to arrive at the scour factor for total scour. Abutment scour is total scour based on the development of the NCHRP Project 24-20 equation. Total scour at a pier includes pier and contraction scour. Although the scour factors for total scour (pier plus contraction) are shown, they depend on the relative amounts of the two types of scour. Therefore, the b value for total scour should include calculation of the design scour compo- nents and total scour, expected scour components and total scour, and the standard deviation of the scour components and total scour. Simply adding the scour components for a specific b value would be overly conservative. For example, using a b of 2.5 and the statistical results in Table 5.3, Florida DOT pier scour is 6.42 ft and contraction scour is 14.28 ft, which combines to 20.70 ft. The total scour for b = 2.5 is 19.08 ft. Using the expected scour and standard devia- tions of the scour components, the total scour for b = 2.5 is 19.0 ft, which is very close to the desired result. The value of 19.0 ft comes from expected scour of 11.87 ft (4.45 ft pier + 7.42 ft
64 Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction contraction) and standard deviation of 2.85 ft (0.792 + 2.742)0.5, with a 2.5 multiplier for b (11.87 + 2.5 Ã 2.85 = 19.00 ft). This approach is general in that it accounts for any relative range of pier and contraction scour. Figure 5.1 shows the scour factors for HEC-18 pier scour for all 27 bridge, pier, and hydrologic uncertainty combinations presented in Appendix B (see Figure B.1). In the legend SB, MB, and LB represent small, medium, and large bridges; LH, MH, and HH represent low, medium, and high hydrologic uncertainty; and SP, MP, and LP represent small, medium, and large piers. Figure 5.1(a) shows the scour factors obtained directly from the results of the Monte Carlo simulations and Figure 5.1(b) shows the scour factors obtained from the bias and COV of each (a) (b) 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 0.50 1.00 1.50 2.00 2.50 3.00 Sc ou rF ac to r Reliability Index ( ) HEC-18 Pier Scour - Monte Carlo Results SB LH SP SB LH MP SB LH LP SB MH SP SB MH MP SB MH LP SB HH SP SB HH MP SB HH LP MB LH SP MB LH MP MB LH LP MB MH SP MB MH MP MB MH LP MB HH SP MB HH MP MB HH LP LB LH SP LB LH MP LB LH LP LB MH SP LB MH MP LB MH LP LB HH SP LB HH MP LB HH LP 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 0.50 1.00 1.50 2.00 2.50 3.00 Sc ou rF ac to r Reliability Index ( ) HEC 18 Pier Scour Sta stical Results SB LH SP SB LH MP SB LH LP SB MH SP SB MH MP SB MH LP SB HH SP SB HH MP SB HH LP MB LH SP MB LH MP MB LH LP MB MH SP MB MH MP MB MH LP MB HH SP MB HH MP MB HH LP LB LH SP LB LH MP LB LH LP LB MH SP LB MH MP LB MH LP LB HH SP LB HH MP LB HH LP Figure 5.1. Scour factors for HEC-18 pier scour equation.
Probability-Based Scour Estimates 65 of the simulations. For pier scour, whether the HEC-18 or Florida DOT equation is used, there is very little difference in the scour factors among the 27 simulations. At a b of 3, the range obtained from the Monte Carlo results is 0.97 to 1.04 with an average of 0.99. From the sta- tistical results, the range is 1.00 to 1.03 with an average of 1.01. The two highest scour factors were computed for the large bridge, large pier, medium and high hydrologic uncertainty runs. Although the bias for these runs was consistent with the other runs, the COV for these runs was 0.17, compared with 0.16 for all the other runs. Figure 5.2 shows the scour factors for the Florida DOT equation. There is very little differ- ence in the scour factors among the 27 runs and very little difference between the Monte Carlo (a) (b) 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 0.50 1.00 1.50 2.00 2.50 3.00 Sc ou rF ac to r Reliability Index ( ) FDOT Pier Scour - Monte Carlo Results SB LH SP SB LH MP SB LH LP SB MH SP SB MH MP SB MH LP SB HH SP SB HH MP SB HH LP MB LH SP MB LH MP MB LH LP MB MH SP MB MH MP MB MH LP MB HH SP MB HH MP MB HH LP LB LH SP LB LH MP LB LH LP LB MH SP LB MH MP LB MH LP LB HH SP LB HH MP LB HH LP 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 0.50 1.00 1.50 2.00 2.50 3.00 Sc ou rF ac to r Reliability Index ( ) FDOT Pier Scour Sta s cal Results SB LH SP SB LH MP SB LH LP SB MH SP SB MH MP SB MH LP SB HH SP SB HH MP SB HH LP MB LH SP MB LH MP MB LH LP MB MH SP MB MH MP MB MH LP MB HH SP MB HH MP MB HH LP LB LH SP LB LH MP LB LH LP LB MH SP LB MH MP LB MH LP LB HH SP LB HH MP LB HH LP Figure 5.2. Scour factors for Florida DOT (FDOT) pier scour equation.
66 Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction results shown in Figure 5.2(a) and the statistics shown in Figure 5.2(b). The scour factors for the Florida DOT method are slightly higher than those for the HEC-18 equation, indicating slightly lower conservatism in the Florida DOT design equation. For a b of 2.5, the Florida DOT equation would require a scour factor of only 1.09. Table 5.4 and Table 5.5 show the bias and COV for HEC-18 and Florida DOT pier scour equations and all 27 Monte Carlo simulations. These tables demonstrate that three significant figures are required to discern any difference in these statistics, except for COV of the large bridge, large pier, medium and high hydrologic uncertainty conditions for the HEC-18 equa- tion. Therefore, the pier scour bias and COV can be summarized and were applied as shown in Table 5.6, which shows the same values as the laboratory data values. Pier Scour Bias (HEC-18) Small Bridge Medium Bridge Large Bridge S-Pier M-Pier L-Pier S-Pier M-Pier L-Pier S-Pier M-Pier L-Pier Hydrologic uncertainty Low 0.680 0.679 0.679 0.680 0.680 0.682 0.680 0.679 0.680 Medium 0.680 0.679 0.680 0.680 0.680 0.681 0.679 0.677 0.680 High 0.679 0.678 0.679 0.682 0.682 0.682 0.680 0.676 0.682 Pier Scour COV (HEC-18) Small Bridge Medium Bridge Large Bridge S-Pier M-Pier L-Pier S-Pier M-Pier L-Pier S-Pier M-Pier L-Pier Hydrologic uncertainty Low 0.159 0.160 0.160 0.159 0.159 0.162 0.158 0.161 0.162 Medium 0.159 0.160 0.160 0.157 0.157 0.161 0.158 0.163 0.166 High 0.158 0.160 0.161 0.159 0.159 0.163 0.157 0.164 0.169 Table 5.4. HEC-18 pier scour bias and COV from Monte Carlo analysis. Pier Scour Bias (Florida DOT) Small Bridge Medium Bridge Large Bridge S-Pier M-Pier L-Pier S-Pier M-Pier L-Pier S-Pier M-Pier L-Pier Hydrologic uncertainty Low 0.751 0.751 0.751 0.750 0.750 0.750 0.748 0.748 0.748 Medium 0.751 0.751 0.751 0.748 0.749 0.749 0.750 0.750 0.750 High 0.750 0.750 0.750 0.752 0.753 0.754 0.751 0.752 0.752 Pier Scour COV (Florida DOT) Small Bridge Medium Bridge Large Bridge S-Pier M-Pier L-Pier S-Pier M-Pier L-Pier S-Pier M-Pier L-Pier Hydrologic uncertainty Low 0.177 0.177 0.177 0.181 0.181 0.181 0.178 0.178 0.179 Medium 0.180 0.180 0.180 0.177 0.177 0.178 0.179 0.180 0.181 High 0.179 0.179 0.179 0.178 0.179 0.180 0.178 0.180 0.181 Table 5.5. Florida DOT pier scour bias and COV from Monte Carlo analysis. Equation Pier Scour Bias COV HEC-18 0.68 0.16 Florida DOT 0.75 0.18 Table 5.6. Pier scour equation bias and COV from Monte Carlo analysis.
Probability-Based Scour Estimates 67 Figure 5.3 shows the scour factors for contraction scour. Pier size was considered a second- ary influence with contraction scour; therefore, the nine conditions represent bridge size and hydrologic uncertainty. Because the contraction scour equation is a predictive equation and is significantly influenced by the variability of flow distribution resulting from hydraulic uncer- tainty, the scour factors are significantly greater than for pier scour. Figure 5.3(a) shows the scour factors obtained directly from the Monte Carlo results and Figure 5.3(b) shows the scour factors calculated from the statistics (bias and COV). Up to b of 1.5 there is little difference in the two plots, but the curves diverge for higher levels of b. This indicates that there is positive skew in the distribution, as is shown in Figure 5.9 (see (a) (b) 1.00 1.50 2.00 2.50 3.00 3.50 0.50 1.00 1.50 2.00 2.50 3.00 Sc ou rF ac to r Reliability Index ( ) Contraction Scour Monte Carlo Results SB LH SB MH SB HH MB LH MB MH MB HH LB LH LB MH LB HH 1.00 1.50 2.00 2.50 3.00 3.50 0.50 1.00 1.50 2.00 2.50 3.00 Sc ou rF ac to r Reliability Index ( ) Contrac on Scour Statis cal Results SB LH SB MH SB HH MB LH MB MH MB HH LB LH LB MH LB HH Figure 5.3. Scour factors for contraction scour.
68 Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction Section 5.4.3). Had a lognormal distribution been used, the degree of curvature would have exceeded what is shown in Figure 5.3(a). Also shown in Figure 5.9 is an example of the reduced extreme values of contraction scour when relief from road overtopping is included. Extreme flows are most likely to create overtopping, but also produce the greatest contraction scour in the Monte Carlo simulation (which excludes overtopping). Table 5.7 shows the bias and COV for contraction scour Monte Carlo runs and the labora- tory results. The bias is very consistent and similar to the laboratory results with the exception of the large bridge with medium to high hydrologic uncertainty, where the bias ranges from 0.96 to 0.99. A value of 0.93 is reasonable for all other cases. COV increases with bridge size and hydrologic uncertainty and is considerably greater than the laboratory value. Abutment scour results are very similar to the contraction scour results. Figure 5.4 shows the scour factors, which are less than those for contraction scour but greater than those for pier scour. Table 5.8 shows that the bias is similar to that of the laboratory results, with increased values for the large bridge. COV also increases with bridge size and hydrologic uncertainty. The level of bias is lower for abutment scour because the amplification factors developed for abutment scour in the NCHRP Project 24-20 method enveloped the data (see Section 4.4.2). 5.4 Level II Analysis and Results The application of the 27 tables calibrated for the Level I approach can be executed on a regular basis for probability-based analyses of typical or standard scour site conditions. How- ever, the calibration of the Level I statistical parameters will average the model biases for pier, abutment, and contraction scour (lp, la, and lc) and associated COV values and distributions for random variables at similar sites (see Section 3.3.2). When a bridge site does not fit any of the categories identified, or when the bridge is unique or is classified as being critically important for economic, societal, or security reasons, it may be necessary to execute site-specific probabilistic or reliability analyses of scour depths using site-specific statistical data for each variable that is used as input in the scour model. Site-specific (Level II) analysis may also be required if the hydraulic uncertainty parameters exceed the values used in Level I or if other parameters not considered in Level I are deemed to be significant in the design. The process described in detail in Section 5.4.1 would need to be followed to perform a Level II analysis. This process includes performing a Monte Carlo analysis using a hydraulic model with valid uncertainty parameters including, but not necessarily limited to, hydrologic uncertainty, flow resistance uncertainty, and boundary condition uncertainty. The scour equation bias and COV from the laboratory data as described in Chapter 4 would be used in conjunction with the hydraulic modeling results to develop the distribution of scour components and total scour. If other scour equations are used, then the individual bias and COV of these equations would also need to be determined. Contraction Scour Bias Contraction Scour COV Bridge Size Bridge Size Small Medium Large Small Medium Large Hydrologic uncertainty Low 0.92 0.92 0.93 0.26 0.30 0.39 Medium 0.93 0.93 0.96 0.29 0.37 0.50 High 0.93 0.92 0.99 0.35 0.44 0.60 Laboratory data 0.92 0.21 Table 5.7. Contraction scour bias and COV.
Probability-Based Scour Estimates 69 (a) (b) 0.50 1.00 1.50 2.00 2.50 3.00 0.50 1.00 1.50 2.00 2.50 3.00 Sc ou rF ac to r Reliability Index ( ) Abutment Scour Monte Carlo Results SB LH SB MH SB HH MB LH MB MH MB HH LB LH LB MH LB HH 0.50 1.00 1.50 2.00 2.50 3.00 0.50 1.00 1.50 2.00 2.50 3.00 Sc ou rF ac to r Reliability Index ( ) Abutment Scour Statis cal Results SB LH SB MH SB HH MB LH MB MH MB HH LB LH LB MH LB HH Figure 5.4. Scour factors for NCHRP abutment scour equation. Abutment Scour Bias Abutment Scour COV Bridge Size Bridge Size Small Medium Large Small Medium Large Hydrologic uncertainty Low 0.74 0.74 0.76 0.24 0.26 0.39 Medium 0.74 0.75 0.78 0.24 0.28 0.51 High 0.75 0.75 0.80 0.26 0.31 0.61 Laboratory data 0.74 0.23 Table 5.8. Abutment scour bias and COV.
70 Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction 5.4.1 Step-by-Step Procedure for Level II Analysis A Level II analysis involves developing the statistical distribution of each scour component and total scour at a particular bridge site. This type of analysis may be required if the site conditions differ significantly from the conditions used to develop the Level I tables pre- sented in Appendix B. A Level II analysis is useful if (1) the bridge has hydrologic or hydraulic uncertainties that are not reasonably represented by the range of Level I conditions; (2) site conditions require the use of other scour equations than were tested in Chapter 4; or (3) the bridge is considered to be significantly important and warrants a more detailed, site-specific analysis. Not all the steps outlined in this section would necessarily be required for every Level II analysis. For example, if the standard scour equations apply at the bridge site, then the model (equation) bias and COV developed in Chapter 4 would apply. The Level II steps follow the approach used in Sections 5.2 and 5.3 to develop the Level I scour factors. Therefore, familiarity with the rest of this document is useful if a Level II analysis is to be performed. For many of the steps, a prior or subsequent chapter or section in this reference guide can serve as reference material. Steps are provided to determine the statistical distribu- tion of scour for a specific event, such as the 100-year event, and therefore address conditional probabilities. A Monte Carlo simulation can be run for other events (as described in Chapter 6) to evaluate scour exceedance over the life of the bridge (unconditional probability). The steps of the Level II procedure are as follows: Step 1. Develop a site-specific hydraulic model. a. Develop a four cross-section HEC-RAS hydraulic model of the bridge site (see Section 3.4.1). The Monte Carlo analysis was developed for a four cross-section HEC-RAS model. If a large- extent model is required, then modification of the Monte Carlo software (e.g., rasTool©) would be required. b. Make best estimates of Manning n for the channel and overbank areas. Because the Monte Carlo analysis will vary Manning n around the starting estimate, it is important to not use conservative values (high or low) of Manning n, as doing so will bias the results. Calibrated values should be used if observed water surface data are available. c. Make a best estimate of the starting water surface boundary condition. It is recommended that the energy slope boundary condition be used, as doing so will vary the starting water surface for the various discharge values that will be applied in the Monte Carlo analysis. As with Manning n, a best estimate of the boundary condition should be used rather than a conservatively high or low value. d. Evaluate site-specific hydrologic uncertainty (see Section 3.5.2). The Level I analysis uses a range of hydrologic uncertainties for each bridge size. For a Level II analysis, the best estimate of hydrologic uncertainty should be developed and applied. The preferred approach is to per- form gage analysis and apply Bulletin 17B (Log-Pearson Type III) procedures to determine the target discharge and confidence limits. Notes: (1) When applying the HEC-RAS model to a wide range of conditions it may be necessary to limit road overtopping to produce more stable models. If the model is stable for road overtopping conditions, it is recommended that road overtopping be allowed, as doing so will provide more representative contraction scour results. (2) As described in Step 3 (perform Monte Carlo analysis), the model results should be evaluated to determine that the variability of water surface is reasonable for the site conditions. Step 2. Determine scour equation (model) uncertainty (bias and COV). a. If the standard scour equations are used (i.e., HEC-18 pier scour, Florida DOT pier scour, HEC-18 live-bed or clear-water contraction scour, or NCHRP Project 24-20 abutment scour),
Probability-Based Scour Estimates 71 then the model uncertainties (bias and COV) from the laboratory data analysis presented in Chapter 4 should be used. b. If another scour equation is used (e.g., vertical contraction scour, coarse-bed pier scour, scour in cohesive or erodible rock materials, etc.), then the model uncertainties (bias and COV) from these alternative equations should be developed following the procedure in Chapter 4. The laboratory data for developing these equations should be used as they are from controlled conditions. HEC-18 (Arneson et al. 2012) includes references to research reports describing the development of several alternative equations. Step 3. Perform Monte Carlo analysis. a. Test the Monte Carlo simulation software for the bridge site (see Section 3.5) using the target (best estimate) values of discharge, channel and overbank Manning n, and starting energy slope and the uncertainties (COV) associated with these three input parameters. Determine the COV of the discharge using Equation (3.1) through Equation (3.5). The COV for Man- ning n should be 0.015, and uncertainty related to Manning n should be determined using Equation (3.9) through Equation (3.11). The COV of starting slope should be 0.10. However, as described in Section 3.5.3, the hydraulic results of the simulations should be reviewed to determine if the results are representative for the site. The tests should include holding dis- charge constant and varying only Manning n, only starting slope, and both variables. If the water surface varies much more or less than is expected and reasonable, then adjust the COV for Manning n and starting slope to better represent the site conditions. Do not adjust the discharge COV, as this was determined through statistical analysis. b. Run the Monte Carlo simulation software using the target values of discharge, Manning n, and starting slope and the appropriate values of COV for these input variables. The number of cycles should be large enough to fully represent the range of possible hydraulic results. Because HEC-RAS executes quickly, a 10,000-cycle simulation can be achieved in less than 2 hours and should be sufficient. Notes: (1) The rasTool© used with the Monte Carlo simulation software developed for this project is a research tool. It was not developed for distribution, nor is it thoroughly docu- mented or supported for general use. It is, however, considered robust and could be applied to a range of bridge and/or open-channel applications. (2) If a different hydraulic model will be used, then a specific software tool will need to be developed to control the random number generation for the input parameters and to run the number of required cycles in the Monte Carlo simulation. Given the relatively longer simulation times for 2-D models, it is unlikely that the number of cycles could be large enough for their application with standard office computers, and high-performance (supercomputer) technology would need to be used. Step 4. Compute component scour and total scour. a. The output from the Monte Carlo simulation software is a text file table that includes the number of requested cycles of the hydraulic variables needed to perform scour calculations. This table is intended to be imported into a spreadsheet for calculating scour components and total scour. Alternatively, the results could be read by other software to calculate scour. b. For each scour component, the computed scour should be determined by directly apply- ing the appropriate equation. This scour value includes any level of conservatism (bias) included in the development of the equation. The variability of scour results in this step is based on the variability of the hydraulic results. (See the pier scour example and Figure 5.5 in Section 5.4.2.) c. The computed scour from Step 4(b) is then adjusted to determine expected scour distribu- tion by multiplying the computed scour by a random number with mean equal to the model bias (0.68 in the case of HEC-18 pier scour) and standard deviation (SD) equal to the model bias times COV (0.16 in the case of HEC-18 pier scour, resulting in a standard deviation of
72 Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction 0.68 x 0.16 = 0.109). The Monte Carlo simulation software includes four normally distributed random numbers (R) of mean equal to zero and standard deviation equal to 1.0, so the desired random number set for a specific scour equation is (R x SD) + Bias. The results of the component scour (pier, contraction, and abutment) are then multiplied by the random number to provide the component scour distribution. (See the pier scour example and Figure 5.6 in Section 5.4.2.) d. At a pier, total scour is contraction plus local scour. The distribution of total scour is com- puted by adding the individual contraction and pier scour values including the bias and COV adjustments from Step 4(c). For abutment scour using the NCHRP Project 24-20 method, the result is total scour at the abutment. If a different scour equation is used, the evaluation of total scour must be consistent with the development of the equation. e. Based on the distribution of total scour, the designer selects the level of scour that achieves the desired probability of scour exceedance. The results of Step 4 are the distributions of scour for a given return period event (condi- tional probability). Steps 3 and 4 can be repeated for several events to evaluate the uncon- ditional probability of scour exceedance over the life of a bridge. As described in Chapter 6, performing the Monte Carlo analysis for the 50-year, 100-year, and 500-year events and com- bining the scour results will provide data to evaluate scour reliability for a 75-year bridge life. Notice that the 50-year hydrologic uncertainty would be less than the 100-year hydrologic uncertainty because the 90% confidence limits would be closer to the expected value for the smaller event. Conversely, the uncertainty would be greater for the 500-year return period event. As described in Chapter 6, other sets of events would need to be evaluated for other bridge design lives. The Level II process is illustrated in the following sections using the same Sacramento River bridge that was used to validate the HEC-RAS/Monte Carlo software in Section 3.5.3. The Level I application for this bridge is illustrated in Chapter 7 (Section 7.4, Example Bridge No. 3). 5.4.2 HEC-RAS/Monte Carlo Simulation Results for Pier Scour The HEC-RAS model for the Sacramento River bridge was run for 20,000 cycles to evalu- ate the range of hydraulic conditions and scour that result from the parameter uncertainty as described in Section 3.5.3. For this application, 20,000 cycles were run to fully test the Monte Carlo application and to produce results at the extremes of the input parameters. Subsequent evaluations revealed that 10,000 Monte Carlo simulation cycles provide virtu- ally identical probability distributions. Pier scour was evaluated using both the HEC-18 and Florida DOT procedures as described in Section 4.2. The design condition of Q = 140,000 cfs, channel Manning n of 0.025, floodplain Manning n of 0.09, and starting energy slope of 0.00035 produced a design depth and velocity at the bridge of 24.5 ft and 12.1 ft/s. The computed HEC-18 scour for the 6 ft diameter circular pier was 13.7 ft, and the Florida DOT equation resulted in 11.2 ft of scour for a 2.0-mm bed material size. The sediment transport condition is live-bed for these conditions. In the 20,000-cycle Monte Carlo simulation, discharge ranged from 87,000 cfs to 245,000 cfs and dominated the hydraulic conditions at the bridge. Energy slope, which ranged from 0.00022 to 0.00049, had the smallest impact on hydraulic conditions. Manning n ranged from 0.021 to 0.030 for the channel and from 0.074 to 0.108 for the floodplain. At the bridge, the design depth ranged from 19.5 ft to 30.2 ft, and design velocity ranged from 9.4 ft/s to 16.4 ft/s. The results for pier scour in the Monte Carlo simulation are summarized in Table 5.9. Although the simulated discharge varied by more than a factor of 2.5 and velocity varied
Probability-Based Scour Estimates 73 significantly, the computed range of pier scour was 4 ft for the HEC-18 equation and was only 1.6 ft for the Florida DOT equation. For this range of hydraulic conditions, the range of computed scour from the Florida DOT equation is very small, indicating that the Florida DOT equation is less sensitive to hydraulic conditions. Notice that the maximum computed scour from the HEC-18 equation exceeds 2.4 times the pier width, which is an expected upper limit based on a circular pier and Froude number less than 0.8. The pier scour results are also shown in Figure 5.5 and Figure 5.6. In Figure 5.5, the direct results of the Florida DOT and HEC-18 equations are shown for the computed velocity and depth from the HEC-RAS models. The design value for each of these equations is shown, and Pier Scour without Bias and COV 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% 0 500 1000 1500 2000 2500 3000 3500 4000 10 .5 10 .8 11 .1 11 .4 11 .7 12 .0 12 .3 12 .6 12 .9 13 .2 13 .5 13 .8 14 .1 14 .4 14 .7 15 .0 15 .3 15 .6 15 .9 Cu m ul a ve Pe rc en t Fr eq ue nc y Scour Depth ( ) HEC 18 Frequency FDOT Frequency HEC 18 Cumula ve % FDOT Cumula ve % HEC 18 Design 13.7 FDOTDesign 11.2 Figure 5.5. Direct pier scour results from HEC-RAS Monte Carlo simulations. 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% 0 200 400 600 800 1000 1200 1400 2. 6 3. 4 4. 1 4. 9 5. 6 6. 4 7. 1 7. 9 8. 6 9. 4 10 .1 10 .9 11 .6 12 .4 13 .1 13 .9 14 .6 15 .4 M or e Cu m ul a ve Pe rc en t Fr eq ue nc y Scour Depth ( ) HEC 18 Frequency FDOT Frequency HEC 18 Cumula ve % FDOT Cumula ve % HE C 18 De sig n 13 .7 ( = 2. 75 ) FD O TD es ig n 11 .2 ( = 1. 81 ) Pier Scour with Bias and COV Figure 5.6. Pier scour results from HEC-RAS Monte Carlo simulations after including equation bias and COV.
74 Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction in each case the design value is very close to the mean of the calculated values. This is expected because in the Monte Carlo simulation velocity and depth are distributed around the base- model results. Figure 5.5 illustrates that for this particular bridge hydraulic condition, the Flor- ida DOT equation and the HEC-18 equation have no overlap, although the actual magnitude of scour is not significantly different. The Florida DOT equation is less sensitive to the hydraulic conditions, resulting in a spread of only 1.6 ft versus a spread of 4 ft for the HEC-18 equation. Figure 5.6 shows the results after each equationâs bias and COV are introduced. From the analysis of laboratory pier sour data, the bias and COV of the observed versus computed scour is 0.68 and 0.16 for the HEC-18 equation and 0.75 and 0.18 for the Florida DOT equation. Assuming a normal distribution, these values result in an estimated conditional reliability (b) of 2.92 and 1.78 for the HEC-18 and Florida DOT equations (see Section 2.5.2 for a discussion of the reliability index as a measure of structural safety). The computed scour was then multiplied by normally distributed random values with mean equal to the bias, and SD based on the COV for each equation. As shown in Figure 5.6, because the HEC-18 equation has a smaller bias and COV than the Florida DOT equation, the resulting distributions are similar with only a small offset. From these results the value of b can be determined for each equation. The computed values of b from the Monte Carlo analysis (the HEC-18 b = 2.75 and the Florida DOT b = 1.81) are essentially the same as those originally estimated from the live-bed laboratory data bias and COV, assuming a normal distribution, which indicates that the implementation of the Monte Carlo simulation is reliable. If a target b of 2.5 is desired, then the Florida DOT design value of 11.2 ft would need to be increased to 12.4 ft (multiplied by a factor of 1.11) and the HEC-18 equa- tion design value of 13.7 ft would need to be decreased to 13.4 ft (multiplied by a factor of 0.98). Table 5.9 also shows the pier scour results after applying the bias and COV for each equa- tion based on live-bed laboratory data. For this 100-year flow condition, the HEC-18 equation provides a b of 2.75, whereas the Florida DOT equation yields results that would need to be increased to provide the same level of reliability. The scour factors to achieve a b of 3.0 are shown, and in this case both equations would require greater design scour to achieve this level of reliability. Use of the Florida DOT equation for this bridge and hydraulic condition does Variable HEC-18 Equation Florida DOT Equation Design scour (ft) 13.7 11.2 Mean scour (ft) 13.8 11.3 SD (ft) 0.49 0.21 COV 0.036 0.019 Minimum computed scour (ft) 12.1 10.6 Maximum computed scour (ft) 16.0* 12.2 Results After Applying Bias and COV Mean scour (ft) 9.4 8.5 SD (ft) 1.56 1.59 COV 0.166 0.189 Minimum computed scour (ft) 3.5 2.6 Maximum computed scour (ft) 15.8* 14.1 (design result) 2.75 1.81 Scour factor for = 3.0 1.04 1.17 Scour required for = 3.0 (ft) 14.2 13.1 *Computed scour greater than 2.4 times the circular pier width. Table 5.9. Pier scour results from 20,000-cycle Sacramento River bridge HEC-RAS.
Probability-Based Scour Estimates 75 result in less required scour (11.2 x 1.17 = 13.1 ft) to achieve the same reliability as the HEC-18 equation (13.7 x 1.04 = 14.2 ft), as shown in Table 5.9 for a b of 3.0. This is due primarily to the fact that the Florida DOT equation is less sensitive over a wide range of velocity and depth. 5.4.3 HEC-RAS/Monte Carlo Simulation Results for Contraction Scour Contraction scour is caused by a change in flow distribution from upstream of the bridge (approach cross section) to the bridge. At the approach, flow is distributed throughout the overall cross section among the channel, left, and right floodplains based on the conveyance of these sub-areas. At the bridge, flow is concentrated in the bridge opening entirely in the chan- nel if the abutments are set at the channel bank or into the channel. If the abutments are set back from the channel banks, then some of the flow is conveyed in the setback areas between the channel banks and the abutments. The Monte Carlo simulations vary discharge, starting energy slope (downstream boundary condition), and channel and overbank Manning n values. Each of these parameters affects flow distribution at the approach and at the bridge. As with pier scour, contraction scour was computed for the 20,000-cycle simulation of the Sacramento River bridge to fully accommodate the extremes of the input parameters. The design condition produced a design contraction scour of 5.3 ft. Although the largest computed contraction scour was generated from the highest discharges, other combinations of conditions also produced significantly more (or less) contraction scour than the design value. For example, if the channel Manning n value is high and the floodplain Manning n is low, then more flow is conveyed in the floodplain. This condition results in a much greater amount of flow constriction and much greater contraction scour. Conversely, a low channel Manning n combined with a high floodplain Manning n concentrates flow in the channel, resulting in less flow constriction at the bridge and much less contraction scour. The range of computed con- traction scour was from 0.55 ft to 14.0 ft. Another process that affects contraction scour is road overtopping. It has been standard practice to limit scour analyses to flow up to the point of road overtopping (Arneson et al. 2012). The rationale is that once road overtopping commences, flow through the bridge will not increase because of the significant amount of relief provided by the weir flow over the road. To keep the HEC-RAS model stable over the full range of flow and other input parameters, road overtopping was eliminated from the model and all flow was conveyed through the bridge opening. It is also better to exclude road overtopping for the general Monte Carlo analyses because the elevation where road overtopping initiates will be specific to the bridge. In the spreadsheet used to compute scour, however, adjustments were made to assess the impacts of road overtopping for the Sacramento River bridge. To develop Figure 5.7, the road elevation was set at a reasonable height relative to the design water surface elevation. The lower limit of com- puted contraction scour was, of course, unchanged. The upper limit was 9.2 ft and occurred with slight road overtopping flow (3,000 cfs of a total 181,000 cfs in that run). A comparison of contraction scour estimates with and without road overtopping is shown in Figure 5.7. The majority of the simulations (16,907 cycles) did not generate road overtopping flow. The remaining simulations (3,093 cycles) generated up to 77,400 cfs of road overtopping flow. In Figure 5.7, the computed contraction scour is plotted versus the road overtopping discharge whether or not road overtopping was considered. This illustrates that for small amounts of road overtopping the scour is relatively unaffected by the relief flow, but that for the largest amounts road overtopping, scour can be minimal (2 ft versus 14 ft). Generally, road overtopping is undesir- able, but from this analysis it is clear that it can greatly reduce contraction scour potential. Contraction scour results from the Monte Carlo simulation are shown in Figure 5.8 and Figure 5.9. Figure 5.8 shows the contraction scour computed directly from the hydraulic results
76 Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 10000 20000 30000 40000 50000 60000 70000 80000 Co nt ra ct io n Sc ou r( ) RoadOvertoppingDischarge (cfs) All flow through bridge RoadOvertopping Figure 5.7. Contraction scour with and without road overtopping. 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% 0 500 1000 1500 2000 2500 0. 0 0. 5 1. 0 1. 5 2. 0 2. 5 3. 0 3. 5 4. 0 4. 5 5. 0 5. 5 6. 0 6. 5 7. 0 7. 5 8. 0 8. 5 9. 0 9. 5 10 .0 10 .5 11 .0 11 .5 12 .0 12 .5 13 .0 13 .5 14 .0 14 .5 M or e Cu m ul a ve Pe rc en t Fr eq ue nc y Contraction Scour (ft) Contracon Scour without Bias and COV Frequency Frequencywith Road Overtopping Cumulative % Cumulative % with Road Overtopping Figure 5.8. Computed live-bed contraction scour results from HEC-RAS Monte Carlo simulations.
Probability-Based Scour Estimates 77 with and without road overtopping flow. The design scour is 5.3 ft, which is centered within the distributions. Road overtopping shifts the most extreme amounts of contraction scour to lower values. Unlike the HEC-18 and Florida DOT equations, the contraction scour equations are predic- tive and do not include conservative factors for design. The clear-water contraction scour equa- tion is developed from sediment incipient motion criteria and the live-bed contraction scour equation is developed from sediment transport relationships. The HEC-18 and Florida DOT pier scour equations have bias values of 0.68 and 0.75 based on comparisons to the laboratory data, which indicates a level of conservatism. The clear-water contraction scour equation has a bias of 0.92 based on comparisons with laboratory data (see Chapter 4), which indicates very lit- tle bias (i.e., no built-in conservatism as expected in a predictive equation). Contraction scour laboratory data has a higher COV than pier scour (0.16 for HEC-18 and 0.18 for Florida DOT). This indicates greater variability in contraction scour. Although the bias and COV are for clear- water conditions, these values were applied to the live-bed equation to produce Figure 5.9. Both equations are derived based on sediment transport relationships and are predictive, so this is justifiable though not ideal. From a practical standpoint, there is insufficient live-bed data to develop independent bias and COV for the live-bed equation. Therefore, clear-water values were applied to the live-bed results. With bias close to 1.0, the contraction scour equation has very low reliability, with b close to zero. As shown in Figure 5.9, for the larger COV the range of computed contraction scour increases significantly as compared to Figure 5.8, though the mean scour is relatively unchanged from the design value of 5.3 ft. It also made relatively little difference whether road overtopping was included. Table 5.10 summarizes results from this set of bridge-specific simulations. Based on these results, scour factors are shown for various target levels of reliability. For example, a b of 2 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% 0 500 1000 1500 2000 2500 0. 0 0. 5 1. 0 1. 5 2. 0 2. 5 3. 0 3. 5 4. 0 4. 5 5. 0 5. 5 6. 0 6. 5 7. 0 7. 5 8. 0 8. 5 9. 0 9. 5 10 .0 10 .5 11 .0 11 .5 12 .0 12 .5 13 .0 13 .5 14 .0 14 .5 15 .0 15 .5 16 .0 16 .5 M or e Cu m ul a ve Pe rc en t Fr eq ue nc y Contraction Scour ( ) Contracon Scour with Bias and COV Frequency Frequencywith Road Overtopping Cumula ve % Cumula ve % with Road Overtopping Figure 5.9. Live-bed contraction scour results from HEC-RAS Monte Carlo simulations after including equation bias and COV.
78 Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction would require that contraction scour be multiplied by a factor of 1.8, resulting in a design scour of 9.7 ft if road overtopping is not considered. With road overtopping at this bridge, a b of 2 would require multiplying contraction scour by 1.6, giving 8.5 ft of scour to be used for design. This is considerably greater than the 5.3 ft that would currently be used. The level of bias for the contraction scour equations is quite reasonable considering their origin. 5.4.4 HEC-RAS/Monte Carlo Simulation Results for Abutment Scour As described in Section 4.4, abutment scour is both a contraction and local scour process. The constriction of flow in the bridge opening that produces contraction scour also concen- trates flow at the abutments. Therefore, the starting point for abutment scour is a contrac- tion scour calculation. The obstruction of the abutment produces vortices and turbulence that amplify the contraction scour. The equations and figures in Section 4.4 present this approach to computing abutment scour for various hydraulic and sediment conditions and abutment configurations (Ettema et al. 2010). The 20,000-cycle Monte Carlo simulation results were used to compute abutment scour at the Sacramento River bridge. The computed abutment scour for the base condition was 11.0 ft, but ranged from less than 1.0 ft to more than 30 ft depending on the hydraulic conditions com- puted in HEC-RAS. This variability is similar to the variability of computed contraction scour. This was expected because of the similarities of the two scour processes. As with the other scour components, the abutment scour equation bias and COV were applied to the computed scour values to determine the reliability of the design scour amount. For abutment scour, the bias and COV values are 0.74 and 0.23 from the data analysis in Chapter 4. The bias is lower than the contraction scour bias because the amplification values were developed to envelop the laboratory results. Figure 5.10 shows the distributions of computed abutment scour and abutment scour after including equation bias and COV. Table 5.11 summarizes the results and shows scour factors Variable All Flow Through Bridge Road Overtopping Design scour (ft) 5.4 5.3 Mean scour (ft) 5.5 5.2 SD (ft) 1.85 1.53 COV 0.338 0.293 Minimum computed scour (ft) 0.55 0.55 Maximum computed scour (ft) 14.0 9.2 Results After Applying Bias and COV Mean scour (ft) 5.00 4.78 SD (ft) 2.02 1.74 COV 0.404 0.364 Minimum computed scour (ft) 0.41 0.41 Maximum computed scour (ft) 16.3 11.7 (design scour) 0.26 0.33 Target 1 1.5 2 2.5 3 1 1.5 2 2.5 3 Scour factor for target 1.3 1.6 1.8 2.1 2.4 1.2 1.4 1.6 1.0 2.0 Scour required for target (ft) 7.0 8.3 9.7 11.3 12.9 6.6 7.5 8.5 9.4 10.4 Table 5.10. Contraction scour results from 20,000-cycle Sacramento River bridge HEC-RAS.
Probability-Based Scour Estimates 79 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% 0 500 1000 1500 2000 2500 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 M or e Fr eq ue nc y Abutment Scour () Frequency of Computed Scour Frequency after Bias and COV Cumula ve % of Computed Scour Cumula ve % after Bias and COV Figure 5.10. Abutment scour results from the HEC-RAS Monte Carlo simulations. Variable Value Design scour (ft) 11.0 Mean scour (ft) 11.3 SD (ft) 3.7 COV 0.33 Minimum computed scour (ft) 0.35 Maximum computed scour (ft) 30.4 Results After Applying Bias and COV Mean scour (ft) 8.3 SD (ft) 3.9 COV 0.46 Minimum computed scour (ft) -1.4 (0.0) Maximum computed scour (ft) 30.4 (design scour) 0.78 Target 1.0 1.5 2.0 2.5 3.0 Scour factor for target 1.1 1.3 1.6 1.9 2.2 Scour required for target (ft) 12.1 14.6 17.5 20.9 24.0 Table 5.11. Abutment scour results from 20,000-cycle Sacramento River bridge HEC-RAS. needed to achieve various levels of reliability (b). For example, to achieve a target b value of 2.0, the design abutment scour of 11.0 ft (rounded from 10.94) would have to be increased by a factor of 1.6 to 17.5 ft. These results are based on the hydraulic variables computed without adjusting for road overtopping. As with contraction scour, it is expected that abutment scour potential would be reduced when road overtopping occurs.
