**Suggested Citation:**"Chapter 6 - Calibration of Scour Factors for a Target Reliability." National Academies of Sciences, Engineering, and Medicine. 2013.

*Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction*. Washington, DC: The National Academies Press. doi: 10.17226/22477.

**Suggested Citation:**"Chapter 6 - Calibration of Scour Factors for a Target Reliability." National Academies of Sciences, Engineering, and Medicine. 2013.

*Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction*. Washington, DC: The National Academies Press. doi: 10.17226/22477.

**Suggested Citation:**"Chapter 6 - Calibration of Scour Factors for a Target Reliability." National Academies of Sciences, Engineering, and Medicine. 2013.

*Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction*. Washington, DC: The National Academies Press. doi: 10.17226/22477.

**Suggested Citation:**"Chapter 6 - Calibration of Scour Factors for a Target Reliability." National Academies of Sciences, Engineering, and Medicine. 2013.

*Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction*. Washington, DC: The National Academies Press. doi: 10.17226/22477.

**Suggested Citation:**"Chapter 6 - Calibration of Scour Factors for a Target Reliability." National Academies of Sciences, Engineering, and Medicine. 2013.

*Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction*. Washington, DC: The National Academies Press. doi: 10.17226/22477.

**Suggested Citation:**"Chapter 6 - Calibration of Scour Factors for a Target Reliability." National Academies of Sciences, Engineering, and Medicine. 2013.

*Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction*. Washington, DC: The National Academies Press. doi: 10.17226/22477.

**Suggested Citation:**"Chapter 6 - Calibration of Scour Factors for a Target Reliability." National Academies of Sciences, Engineering, and Medicine. 2013.

*Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction*. Washington, DC: The National Academies Press. doi: 10.17226/22477.

**Suggested Citation:**"Chapter 6 - Calibration of Scour Factors for a Target Reliability." National Academies of Sciences, Engineering, and Medicine. 2013.

*Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction*. Washington, DC: The National Academies Press. doi: 10.17226/22477.

**Suggested Citation:**"Chapter 6 - Calibration of Scour Factors for a Target Reliability." National Academies of Sciences, Engineering, and Medicine. 2013.

*Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction*. Washington, DC: The National Academies Press. doi: 10.17226/22477.

**Suggested Citation:**"Chapter 6 - Calibration of Scour Factors for a Target Reliability." National Academies of Sciences, Engineering, and Medicine. 2013.

*Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction*. Washington, DC: The National Academies Press. doi: 10.17226/22477.

**Suggested Citation:**"Chapter 6 - Calibration of Scour Factors for a Target Reliability." National Academies of Sciences, Engineering, and Medicine. 2013.

*Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction*. Washington, DC: The National Academies Press. doi: 10.17226/22477.

**Suggested Citation:**"Chapter 6 - Calibration of Scour Factors for a Target Reliability." National Academies of Sciences, Engineering, and Medicine. 2013.

*Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction*. Washington, DC: The National Academies Press. doi: 10.17226/22477.

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

80 6.1 Approach 6.1.1 Background The calculations performed for the probability-based scour estimates described in Chap- ter 5 are for a single discharge rate that corresponds to a design return period (e.g., the dis- charge rate having a return period of 100 years). Thus, the probability-based scour estimate obtained in Chapter 5 is a conditional probability of exceedance that is conditioned on the occurrence of the design discharge rate, which can be expressed symbolically as follows for a 100-year discharge rate: P 100 yr rate (6.1)ex â During its service life, Tn, a bridge might be exposed to a large range of possible discharge rates. Some of these discharge rates may exceed the one used for the design return period. Many others will be smaller, but they are still capable of causing scour at the bridge. Within the service life, each of these possible discharge rates will have a probability of occurrence, Pi. Therefore, the unconditional probability of exceedance should account for the probabilities of exceedance for all the possible discharge rates along with their probabilities of occurrence. 6.1.2 Reliability Analysis Several methods can be used to calculate the unconditional probability of exceeding the design scour depth within a service life, Tn. One method consists of performing the conditional probability-based scour estimates described in Chapter 5 for a whole set of return periods and associating each conditional probability of exceedance with the probability of occurrence, Piâthat is, the probability that the maximum discharge rate within the service life will equal that of the selected return period, which is labeled as Pi. The final unconditional probability of exceedance will be the sum of the products of the probability of exceedance for each discharge rate times the probability of the occurrence of the discharge rate for which the probability of exceedance is calculated. This can be expressed as: P T P i yr rate P (6.2)ex n ex th all return years iâ ( )( ) = â Ã Where Pex (Tn) is the probability of exceeding the design scour within a service life period Tn, (Pex/i th - yr rate) is the probability of exceeding the design scour given that the hydraulic event corresponds to that of a return period equal to i-years, and Pi is the probability that the maxi- mum discharge rate within the service life of the bridge has a probability of occurrence equal to that of the discharge rate having the return period i-years corresponding to the ith hydraulic C H A P T E R 6 Calibration of Scour Factors for a Target Reliability

Calibration of Scour Factors for a Target Reliability 81 event. Although there are an infinite number of hydraulic events, these can be combined into discrete segments where each segment has a probability of occurrence Pi. Note that the sum of all the hydraulic event probabilities, Pi, must add up to 1.0: P 1.0 (6.3)i all return years â = It is common in the probabilistic evaluation of bridge safety to use the reliability index, b, as a measure of safety. The reliability index, b, is inversely related to the probability of scour depth exceedance through the normal cumulative distribution function (CDF), F: P T (6.4)ex n ( )( ) = Î¦ âÎ² 6.1.3 Reliability Calculation Process The process for calculating the reliability for a given design scour depth can be summarized as follows: Step 1. Find the design scour for a bridge using current methods. Step 2. Divide the set of possible discharge rates that could occur within the service life, Tn, of the bridge into a limited number of representative discrete sets of discharge rates. These discharge rates can be identified based on the return period they are associated with. Step 3. Find the probability of occurrence, Pi, that the maximum discharge expected to occur within the service life will equal each of the discharge rates, i, selected in Step 2. Step 4. Use the approach described in Chapter 5 to find Pex/i th - yr, which gives the conditional probability of exceeding the design scour for each of the discharge rates, i, selected in Step 2. Step 5. Multiply Pex/i th - yr calculated in Step 4 by the probability Pi of Step 3. Step 6. Repeat steps 3, 4, and 5 to cover the entire set of representative discharge rates. Step 7. Add all the results from Step 6 to give Pex (Tn), which is the overall probability of exceed- ance in the service life Tn. Step 8. Find the reliability index, b, using the normal CDF, F. 6.1.4 Calibration of Design Equations A properly calibrated design scour methodology should provide a reliability index, b, that meets a target value as closely as possible for the range of applicable bridge geometries and channel conditions. If the current design methodology does not meet the target reliability level, adjustments to the scour design methodology must be made. One possible approach is to apply a scour factor on the results of the design scour calculations to ensure that the reliability levels obtained after adjustment meet the target reliability levels. 6.1.5 Simplified Example In this section, an example set of calculations is performed and the probabilities are obtained as shown in Table 6.1. For this simplified example, it is assumed that the current design method will stipulate a design scour depth of 15 ft. The table illustrates the application of Equation (6.2) when the probability of exceedance for a service period Tn = 1 year is desired. The calculations assume that the entire range of hydraulic events can be divided into seven discrete segments (represented by the seven return periods Tr = 5 years, Tr = 20 years, Tr = 50 years, Tr = 75 years, Tr = 100 years, Tr = 200 years, and Tr = 500 years). The probability of occurrence, Pi, that cor- responds to each segment is calculated to cover all the probabilities between the different return

82 Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction periods. The probability of exceedance within a 1-year period is calculated to be Pex(Tn = 1 - yr) = 1.681e-3. Notice that the return period, Tr, serves to specify the hydraulic event to be used. This is different from the service life, Tn, which defines the period for which the bridge will be in service. Using a similar approach for the case when Tn = 75 years, the probability of exceedance within a 75-year design life is Pex(Tn = 75 - yr) = 12.1%. The reliability index, b, for the 75-year design life is found to be 1.18. To obtain a reliability index, b, of 1.5, the design scour depth will have to be increased by a scour factor, SC, of 1.10. In other words, the design scour must be increased from 15 ft to 16.5 ft. The integration approach for calculating the reliability index as described in this section, based on Equation (6.2), Equation (6.3), and Equation (6.4), provides a simplified approach for calibrat- ing scour factors for a target reliability consistent with LRFD procedures used by structural and geotechnical engineers as discussed in Section 2.5. The example in Table 6.1 uses seven return periods. Next, Section 6.2 presents a discussion of the number of return periods that can be used for the integration to obtain an optimum balance between accuracy and calculation efficiency. 6.2 Validation of the Simplified Procedure 6.2.1 Overview of the Procedure This section describes an algorithm for the calculation of the reliability of design scour depth exceedance using a limited number of return periods. The validity of the proposed approach is verified by comparing the results from a full-fledged Monte Carlo simulation to those of the evaluation at discrete return periods. The comparison shows that it is sufficient to perform Monte Carlo simulations for five return periods or fewer to obtain good estimates of the mean and standard deviation (SD) of the actual scour depth. The statistics of the actual scour depth can subsequently be used to estimate the probability of exceeding the design scour depth. A list of suggested return periods to check for various service lives is provided in Table 6.2. As mentioned earlier, several methods can be used to find the reliability of a bridge that may be subject to scour. The most basic approach consists of performing a Monte Carlo simulation to find the probability that the maximum scour depth around a bridge foundation will exceed the scour design depth at any time within the service life of the bridge. However, the full- fledged Monte Carlo simulation requires a heavy computational effort. As outlined in Section 6.1, a simplified approach was developed whereby the Monte Carlo simulation is executed at Return Period (Tr) P = 1/Tr Probability of Occurrence (Pi) Conditional Probability of Exceeding Design Scour Product of Pi Times Conditional Probability 5 years 0.2 0.875 6.82e-4 5.97e-4 20 years 0.05 0.09 5.06e-3 4.55e-4 50 years 0.02 0.0183 1.22e-2 2.24e-4 75 years 0.0133 0.005 1.65e-2 0.83e-4 100 years 0.01 0.00567 2.02e-2 1.14e-4 200 years 0.005 0.0035 3.14e-2 1.10e-4 500 years 0.002 0.0025 3.92e-2 0.98e-4 Sum 0.1Pi =1.681e-3Pex Tn = 1 â yr Table 6.1. Example calculations for determining probability of design scour exceedance within a 1-year period.

Calibration of Scour Factors for a Target Reliability 83 only a limited number of discrete return periods and the results are integrated to obtain esti- mates of the reliability of the bridge over the entire service period. The objective of the reliability analysis is to find the reliability index, b, which as defined in Equation (6.4) is related to the probability of design scour depth exceedance within a service period, Pex(Tn). This relationship also can be expressed as: P T Pr y y (6.5)ex n max expected sc design( ) ( )( ) = â¥ = Î¦ âÎ² where ymax expected = the expected maximum scour depth during the service life of the bridge, ysc design = the design scour depth, and F = the CDF for the normal distribution. Notice also that ysc design is deterministic, computed from the HEC-18 equation (or any appro- priate design equation) and ymax expected is determined from the Monte Carlo simulation, based on uncertainty and the expected discharges occurring over the service life of the bridge. The process of finding the probability of design scour depth exceedance and the reliability index, b, involves the following steps. Step 1. Find the design scour for the bridge, ysc design, from the as-built conditions or by using typical design equations such as the HEC-18 equations for the 100-year discharge rate. Step 2. Use the discharge rate data to find the statistics of the maximum expected discharge rate within the remaining service life of the bridge. For example, knowing the probability dis- tribution for the yearly discharge rate, FQ(x), the maximum flood discharge in a service period, Tn, has a cumulative probability distribution, FQTn(x), related to the probability distribution of the 1-year maximum discharge by: ( ) ( )=F x F x (6.6)QTn Q Tn Step 3. Apply FQTn(x) and the bias and COV of the modeling variables into a Monte Carlo simu- lation to find the statistics of ymax expected for different possible values of the scour within a service period, Tn. Step 4. Determine the percentage of cases for which ymax expected exceeds ysc design and find the reli- ability index from Equation (6.5). Because of the numerical difficulties associated with covering the whole range of possible values of the cumulative distribution function, FQTn(x), a limited number of discharge rates were used to estimate the probability of scour depth exceedance. Through different comparisons between the full-fledged Monte Carlo simulation and simulations that used a limited number of discharge rates, it was determined that good accuracy can be achieved when the simulations are executed for five different return periods or fewer. The higher the service life, Tn, the lower the number of return periods that are needed for Q. This is because as Tn increases, QTn evaluated from Equation (6.6) Service Period (Tn) Return Period 1 Return Period 2 Return Period 3 Return Period 4 Return Period 5 5 years 3 years 5 years 8 years 15 years 50 years 20 years 10 years 20 years 30 years 60 years 200 years 75 years 50 years 100 years 500 years Table 6.2. Proposed return periods for use in estimating the scour reliability for different service lives.

84 Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction will have a lower COV. Figure 6.1 provides a flow chart for evaluating the reliability index, b, using the simplified procedure. 6.2.2 Case Studies for Validation To verify the validity of the simplified approach, several comparisons between the results obtained from the approach described in Figure 6.1 and a full-fledged Monte Carlo simulation were performed. To obtain realistic results for the effect of scour, different possible discharge rate data from a selected set of rivers are used and design scour depths are calculated for each of these river discharge rates. The simplified approach was shown to reproduce the full-fledged Monte Carlo simulation results quite well for the five rivers used to assess the procedure. The validation procedure is described in detail in the Contractorâs Final Report for NCHRP Project 24-34, available at www.trb.org. 6.3 Implementation of Reliability Analysis for Sacramento River Bridge Data In this section, the analysis procedure presented in Section 6.2 is implemented for a reli- ability analysis for the Sacramento River bridge that was analyzed in Chapters 3 and 5. This reliability analysis covers the following cases: â¢ Pier scour when the foundation is designed using the HEC-18 method. â¢ Pier scour when the foundation is designed using the Florida DOT method. â¢ Contraction scour using the HEC-18 equation. â¢ Combined pier scour and contraction scour when the foundation is designed using the HEC- 18 method for the pier scour component. â¢ Combined pier scour and contraction scour when the foundation is designed using the Flor- ida DOT method for the pier scour component. â¢ Abutment scour using the NCHRP Project 24-20 approach as recommended in the 5th edi- tion of HEC-18. A reliability analysis for the 75-year service life was executed in Section 6.1 using three return periods: Tr = 45 years, Tr = 110 years, and Tr = 400 years. However, during the implementation process it was decided to use the slightly modified set of typical return periods (Tr = 50 years, Tr = 100 years, and Tr = 500 years) because hydraulic engineers use these return periods on a regular basis and their values are more readily available. A sensitivity analysis on a random set of cases has shown that using the modified set of return periods does not lead to noticeable differences in the results. 6.3.1 Pier Scour Designed Using HEC-18 Method As a first step, the simulations are executed to find the pier scour that would be obtained if no modeling bias is assumed (i.e., assuming that the HEC-18 equations give on the average good estimates of the actual pier scour depth). Figure 6.2 presents the results, assuming that the bridge is subjected to the hydraulic event that corresponds to each of the three return periods (Tr = 50 years, Tr = 100 years, and Tr = 500 years). Given that the HEC-18 design scour for this bridge is 13.7 ft (see Section 5.4.2), the results show that if the 50-year event were to occur, the scour around the bridge pier would have a 27.64% prob- ability of exceeding the 13.7 ft design scour, corresponding to a reliability index of b = 0.59. If the 100-year event were to occur, the scour around the bridge pier would have a 58.84% probability of exceeding the 13.7 ft design scour (b = -0.22); and if the 500-year event were to occur, the scour around the bridge pier would have a 93.73% probability of exceeding the 13.7 ft design scour (b = -1.5). Using the combined results from the 50-year, 100-year, and 500-year return periods, the

Calibration of Scour Factors for a Target Reliability 85 Start 1. Assign scour design parameters 2. Assemble statistical data for all random variables: Q, n, Sf, sc 3. Determine design scour depth, ysc-design 4. Set return period, Tr, number i=1 Find Q for i Begin MCS Set j=1 i = max no. of Tr? End MCS End 1. Determine average value of ysc from set of ysc-ij 2. Determine standard deviation of ysc 3. Determine the probability that ysc > ysc-design by counting the number of cycles for which the calculated scour exceeds the design scour, or use the mean and standard deviation of the appropriate distribution (normal or lognormal) 4. Determine the reliability index No No Yes Yes Set i = i + 1 Set j = j + 1 j = max no. of cycles? 1. Generate random samples for Q, n, Sf, sc 2. Compute hydraulic conditions using HEC-RAS 3. Determine scour depth, ysc-ij Figure 6.1. Flow chart of simplified method for determining the reliability index, b, for scour depth exceedance.

86 Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction bridge would have a probability of 60.07% of exceeding the design scour within a 75-year service period (b = -0.25). Such reliability levels are certainly very low compared to acceptable levels. Fortunately, as demonstrated in Chapter 4, the HEC-18 pier scour equation is not a predictive model of scour depth but instead contains on average some level of conservatism with an aver- age bias of 0.68. In other words, based on laboratory and field data, the actual scour for a given hydraulic discharge rate is 0.68 times the scour depth predicted by the HEC-18 equation. On the other hand, Chapter 4 has shown a large level of variability in the bias around the 0.68 value, with a spread around the mean represented by a SD equal to 0.109 (COV = 16%). This spread around the bias offsets some of the conservatism of the HEC-18 equations by a level that can be evaluated using the simulation described in this section while accounting for the modeling bias and its COV. The results of the simulation accounting for bias = 0.68 and the COV = 16% are presented in Figure 6.3. The results in Figure 6.3 demonstrate the dominance of the bias, which tends to pull the histograms for the three return periods closer together. The combination of the three histograms also is illustrated in Figure 6.3, which also shows that the maximum scour depth expected within the 75-year service life approaches that of a normal distribution. The effect of the bias leads to a significant increase in the reliability of the bridge design such that the prob- ability that the actual scour will exceed the HEC-18 design scour depth of 13.7 ft is 0.38% with a reliability index of b = 2.67. This value is more in line with the reliability index that has been deemed acceptable for some bridges under extreme events such as earthquakes or for the rating of existing bridges under vehicular loading as discussed in Section 2.5.3. 0% 5% 10% 15% 20% 25% HEC 18 Scour without Bias 50 yr 100 yr 500 yr Figure 6.2. Pier scour depth histogram without bias calculated based on HEC-18. 0% 5% 10% 15% 4. 50 5. 50 6. 50 7. 50 8. 50 9. 50 10 .5 0 11 .5 0 12 .5 0 13 .5 0 14 .5 0 15 .5 0 M or e HEC 18 Scour with Bias 50 yr 100 yr 500 yr 0% 5% 10% 15% 4. 00 5. 00 6. 00 7. 00 8. 00 9. 00 10 .0 0 11 .0 0 12 .0 0 13 .0 0 14 .0 0 15 .0 0 16 .0 0 Combined HEC 18 Scour with Bias Data Figure 6.3. Pier scour depth histogram with bias calculated based on HEC-18.

Calibration of Scour Factors for a Target Reliability 87 6.3.2 Pier Scour Designed Using Florida DOT Method The approach was executed to find the pier scour that would be obtained if the bridge foun- dation is designed for the scour depth determined using the Florida DOT pier scour equation. The Florida DOT method leads to a design scour depth equal to 11.2 ft (see Section 5.4.2). For the Florida DOT equation, the average bias = 0.75 and the COV = 18%. The results of the simu- lation are presented in Figure 6.4. The results in Figure 6.4 show that the maximum scour depth expected within the bridgeâs 75-year service life approaches that of a normal distribution. The probability that the actual scour will exceed the Florida DOT design scour depth of 11.2 ft is 3.80% with a reliability index of b = 1.77. This value is somewhat on the low side compared to typical reliability indexes that have been deemed acceptable for bridges under extreme events. 6.3.3 Contraction Scour Designed Using HEC-18 Method The approach was executed to find the contraction scour that would be obtained if the bridge foundation is designed for the scour depth determined using the HEC-18 method. The HEC-18 method leads to a contraction design scour depth equal to 5.3 ft (see Section 5.4.3). For the contraction scour, the average bias = 0.916 and the COV = 20.9%. This high bias reflects the fact that the HEC-18 contraction scour equations were developed to be predictive equations rather than more conservative design equations. This high bias, in combination with the high COV, will lead to low reliability levels. The results of the simulation presented in Figure 6.5 show that the maximum scour depth expected within the 75-year service life approaches that of a lognor- mal distribution. The probability that the actual scour will exceed the contraction design scour depth of 5.3 ft is 47.1% with a reliability index of b = 0.07. This value is very low compared to typical reliability indexes that have been deemed acceptable for bridges under extreme events. 0% 5% 10% 15% 3. 50 4. 50 5. 50 6. 50 7. 50 8. 50 9. 50 10 .5 0 11 .5 0 12 .5 0 13 .5 0 14 .5 0 M or e Florida DOT Scour with Bias 50 yr 100 yr 500 yr 0% 5% 10% 15% 3. 00 4. 00 5. 00 6. 00 7. 00 8. 00 9. 00 10 .0 0 11 .0 0 12 .0 0 13 .0 0 14 .0 0 15 .0 0 Combined Florida DOT Scour with Bias Data Figure 6.4. Pier scour depth histogram with bias calculated based on Florida DOT method. 0% 5% 10% 15% 20% 25% 30% 1. 00 3. 00 5. 00 7. 00 9. 00 11 .0 0 13 .0 0 15 .0 0 17 .0 0 19 .0 0 21 .0 0 23 .0 0 M or e HEC 18 Scour with Bias 50 yr 100 yr 500 yr 0% 5% 10% 15% 20% 0. 00 2. 00 4. 00 6. 00 8. 00 10 .0 0 12 .0 0 14 .0 0 16 .0 0 18 .0 0 20 .0 0 22 .0 0 24 .0 0 Combined HEC 18 Scour with Bias Data Figure 6.5. Contraction scour depth histogram with bias calculated based on HEC-18.

88 Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction 6.3.4 Total Pier and Contraction Scour Using HEC-18 Methods The simulations were performed to find the combined (total) pier and contraction scour that would be obtained if the bridge foundation is designed for the scour depth determined using the HEC-18 methods for pier and contraction scour. The HEC-18 methods lead to a design total scour depth equal to 19 ft. The results of the simulation are presented in Figure 6.6, which shows that the maximum scour depth expected within the 75-year service life approaches that of a lognormal distribution but is not too different from a normal distribution. The prob- ability that the actual scour will exceed the total design scour depth of 19 ft is 13.6% with a reliability index of b = 1.10. This value is low compared to typical reliability indexes that have been deemed acceptable for bridges under extreme events. 6.3.5 Total Pier and Contraction Scour Using Florida DOT Method The simulations were performed to find the combined (total) pier and contraction scour that would be obtained if the bridge foundation were designed for the scour depth determined using the Florida DOT method for pier scour. Given that the Florida DOT method does not provide an equation for the contraction scour, the analysis looks at the design pier scour using the Florida DOT equation, whereas the contraction scour is obtained using the HEC-18 method. This leads to a design total scour depth equal to 16.5 ft. The results of the simulation presented in Figure 6.7 show that the maximum scour depth expected within the 75-year service life approaches that of a log- normal distribution. The probability that the actual scour will exceed the total design scour depth 0% 5% 10% 15% 20% 25% 30% 6. 50 9. 50 12 .5 0 15 .5 0 18 .5 0 21 .5 0 24 .5 0 27 .5 0 30 .5 0 33 .5 0 36 .5 0 39 .5 0 M or e HEC 18 Scour with Bias 50 yr 100 yr 500 yr 0% 5% 10% 15% 20% 5. 00 8. 00 11 .0 0 14 .0 0 17 .0 0 20 .0 0 23 .0 0 26 .0 0 29 .0 0 32 .0 0 35 .0 0 38 .0 0 41 .0 0 Total HEC 18 Scour with Bias Data Figure 6.6. Total pier and contraction scour depth histogram calculated using HEC-18. 0% 5% 10% 15% 20% 25% 30% 6. 50 9. 50 12 .5 0 15 .5 0 18 .5 0 21 .5 0 24 .5 0 27 .5 0 30 .5 0 33 .5 0 36 .5 0 39 .5 0 M or e Florida DOT Scour with Bias 50 yr 100 yr 500 yr 0% 5% 10% 15% 20% 25% 5. 00 8. 00 11 .0 0 14 .0 0 17 .0 0 20 .0 0 23 .0 0 26 .0 0 29 .0 0 32 .0 0 35 .0 0 38 .0 0 41 .0 0 Total Florida DOT Scour with Bias Data Figure 6.7. Total pier and contraction scour depth histogram calculated using Florida DOT.

Calibration of Scour Factors for a Target Reliability 89 of 16.5 ft is 21.75% with a reliability index of b = 0.78. This value is very low compared to typical reliability indexes that have been deemed acceptable for bridges under extreme events. 6.3.6 Total Scour at an Abutment Using NCHRP Project 24-20 Method This approach was executed to find the abutment scour that would be obtained if the bridge foundation were designed for the scour depth determined using the NCHRP Project 24-20 method as described and recommended in the latest edition of HEC-18. Notice that this method includes both the effect of the abutment scour and the contraction scour at the end of the abutment, and therefore is an estimate of total scour at that location. The method leads to a design abutment scour depth equal to 11 ft (see Section 5.4.4). For the abutment scour, the aver- age bias = 0.74 and the COV = 23%. The results of the simulation are presented in Figure 6.8 and show that the maximum scour depth expected within the 75-year service life approaches that of a lognormal distribution. The probability that the actual scour will exceed the abutment design scour depth of 11 ft is 30.58% with a reliability index of b = 0.51. This value is very low compared to typical reliability indexes that have been deemed acceptable for bridges under extreme events. 6.3.7 Summary The results of the reliability analysis for a 75-year service life of the Sacramento River bridge are summarized in Table 6.3. The results demonstrate how the reliability index values vary con- siderably for the different types of scour and the different equations that can be used to deter- mine the design scour depth. The results also demonstrate the dominant effect of the bias and 0% 5% 10% 15% 20% 25% 1. 50 4. 50 7. 50 10 .5 0 13 .5 0 16 .5 0 19 .5 0 22 .5 0 25 .5 0 28 .5 0 31 .5 0 34 .5 0 M or e Abutment Scour with Bias 50 yr 100 yr 500 yr 0% 5% 10% 15% 20% 0. 00 3. 00 6. 00 9. 00 12 .0 0 15 .0 0 18 .0 0 21 .0 0 24 .0 0 27 .0 0 30 .0 0 33 .0 0 36 .0 0 Total Abutment Scour with Bias Data Figure 6.8. Total abutment scour depth histogram using NCHRP Project 24-20 method (HEC-18, 5th Ed.). Scour Type Design Scour (ft) Bias COV Probability of Exceedance Reliability Index ( Pier scour (HEC-18) 13.7 0.68 16% 0.38% 2.67 Pier scour (Florida DOT) 11.2 0.75 18% 3.80% 1.77 Contraction scour 5.3 0.92 21% 47.1% 0.07 Combined HEC-18 pier and contraction scour 19 As shown in Section 6.3.4 13.6% 1.10 Combined Florida DOT pier and contraction scour 16.5 As shown in Section 6.3.5 21.8% 0.78 Abutment scour 11.0 0.74 23% 30.6% 0.51 Table 6.3. Summary of reliability analysis results for 75-year service life based on Sacramento River bridge data.

90 Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction its COV on the reliability index, which varies from an acceptable value of 2.67 (obtained when the HEC-18 pier scour equation is used to design the foundation) to the very low value of 0.07 (obtained when the HEC-18 equations are used for designing the foundation for contraction scour). These results are based on the bias and COV obtained by comparing the results from different equations to laboratory data. Results from the field may produce slightly different biases and COV; however, field data are generally considered less reliable because of the various difficulties discussed in Chapter 4. 6.4 Calibration of Scour Factors The reliability analysis performed in Section 6.3 and summarized in Table 6.3 reveals large variations in the reliability levels obtained for the different types of scour. In most cases the reli- ability index obtained for the bridge is low compared to the level obtained for bridges designed for other extreme events. This low reliability is primarily due to the bias and COV of the exist- ing contraction and abutment scour equations. Other causes for the variability include the hydrologic uncertainty of the discharge rates expected over the service life of the bridge, vari- ability in soil and sediment properties, and the geometric and roughness conditions of the channel and overbank areas. One approach that can be used to increase the reliability of existing scour equations is to apply a safety factor on the design scour calculated from current procedures so that bridges designed using the safety factor produce reliability levels that meet an acceptable target reliabil- ity index, b. The target reliability index must be set by the code-writing authorities and bridge owners to provide a balance between safety and cost. As indicated earlier, most current bridge LRFD specifications have used a target reliability level that varies between b = 2.5 and b = 4.0, depending on the types of loads, the consequences of exceeding the target reliability levels, the construction costs, and past histories of successful designs (see Section 2.5.3). In this section, a set of scour factors is calibrated to reach different reliability levels for each scour type. The final decision regarding which target reliability should be used must be made by the appropriate code-writing authorities. A trial-and-error process is used to find the scour factors required to reach different target reliability levels (see Table 6.4). The analyses performed in Table 6.4 are based on the scour depths generated directly from the Monte Carlo simulations for the Sacra- mento River bridge referenced in Section 6.3. The calibration of the scour factors performed in this section assumes a 75-year service life, is based on the data for the Sacramento River bridge, and assumes that these data are representative of typical bridge conditions. Before actual implementation into a design code, similar analyses should be performed for numerous and varied bridges to confirm the consistency of the results. Table 6.4. Scour factors to meet different target reliability levels for 75-year service life based on Sacramento River bridge data. Target Reliability Index (Î²) Scour Factor Pier Scour Using HEC-18 Pier Scour Using Florida DOT Contraction Scour Using HEC-18 Total Scour Using HEC-18 Total Scour Using Florida DOT Abutment Scour 1.50 N/A N/A 1.95 1.10 1.18 1.60 2.00 N/A 1.03 2.35 1.23 1.33 1.95 2.50 N/A 1.10 2.77 1.37 1.47 2.31 3.00 1.04 1.15 3.20 1.50 1.60 2.75

Calibration of Scour Factors for a Target Reliability 91 For the case analyzed, the scour factors shown in Table 6.4 indicate that no additional safety factors would be required for the HEC-18 pier scour equation if the target reliability index is set at 2.50 or lower. A scour factor equal to 1.04 would be needed to reach a target reliability index of b = 3.0. Similarly, only modest scour factors need to be applied to the Florida DOT pier scour equation to achieve reasonable target reliabilities. Table 6.4 also shows that the current contraction scour equations would need significant additional safety factors to reach acceptable reliability levels. A modest target reliability index of b = 1.50 would require an additional safety factor equal to 1.95. Only slightly lower safety factors would be needed to improve the reliability of bridges designed using the NCHRP Project 24-20 abutment scour equation. The safety factors obtained in Table 6.4 are quite modest for the HEC-18 and Florida DOT pier scour equations. However, larger factors are needed to offset the large variability observed between the scour depths measured in the laboratory compared to those predicted from the current contraction and abutment scour equations. Additional analyses are recommended to confirm the consistency of the results for different bridge and channel configurations and hydraulic conditions.