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27 3.1 Introduction Bridge scour processes, including pier, abutment, and contraction scour, have been well researched over the past several decades and equations have been developed to estimate scour depths for each of the scour components. The vulnerability of a bridge to scour is due to the existence of a weakness or a design that can lead to an unexpected, undesirable event compro- mising the bridge safety. By assessing and quantifying all sources of uncertainty in the param- eters and equations used in the design estimation for scour, the reliability of a bridge scour estimate and the probability that the design estimate will be exceeded over the design life of the bridge can be determined, thus reducing the vulnerability to an undesirable event. This chapter describes an approach to evaluating the uncertainty of the three scour compo- nents. The approach is based on Monte Carlo simulation linked directly with the most common and widely accepted hydraulic model used in current practice, HEC-RAS. For each individual scour component, the parameters that were allowed to vary in the Monte Carlo simulations are discussed along with a matrix of other factors and/or considerations that were not addressed. The chapter provides a discussion of model uncertainty and the definitions of bias and the COV in relation to the scour equations. The chapter concludes with a discussion of the linkage between the hydraulic model HEC-RAS and the Monte Carlo simulation software, and the implementation and testing of the software. 3.2 Determining Individual Scour Component Uncertainty The current practice for determining the total scour prism at a bridge crossing generally involves summing individually calculated scour components. The scour components include local scour (pier and abutment), contraction scour (live-bed or clear-water), and long-term channel change (degradation, lateral migration, and channel widening). Uncertainty is not directly addressed in the determination of any of the scour components, so current practice establishes a design amount of scour that is generally recognized as conservative, although the level of conservatism is undefined. For scour at bridge abutments, HEC-18 (Arneson et al. 2012) now recommends a methodology developed under NCHRP Project 24-20 (Ettema et al. 2010), which provides an estimate of abutment and contraction scour combined. Chapter 4 provides a summary of the individual scour equations. Pier, abutment, and contraction scour each involve two types of uncertainty; parameter (aleatory) uncertainty and model (epistemic) uncertainty. This is because each type of scour is defined by an equation (model) that includes variables (parameters) that must be esti- mated. In the research that produced this reference guide, Monte Carlo simulation was used C H A P T E R 3 Evaluating Uncertainty Associated with Scour Prediction
28 Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction to assess parameter uncertainty and observed data (laboratory and field) was used to assess model uncertainty. Each of the variables (discharge, velocity, flow depth, particle size, etc.) used in scour calculations possesses a probability density function defined by the distribution type (normal, lognormal, etc.), and distribution properties (mean, standard deviation [SD], skew, etc.). Monte Carlo simulation was used to address the parameter uncertainty for the local and contraction scour equations or in the case of abutment scour, local scour and contraction scour combined. The Monte Carlo simulation included a hydraulic modeling step in which a hydrau- lic model was run for a large number of scenarios to develop the input variables for the scour computations. HEC-RAS was used for this step (see USACE 2010). Each run provided data to be used to compute both local and contraction scour. The HEC-RAS input parameters that were varied are discharge, boundary condition (energy slope), channel Manning n, and floodplain Manning n. HEC-RAS produced the hydraulic variables for the scour components, which are velocity, flow depth, and flow distribution between the channel and the overbank areas. 3.3 Parameter and Model Uncertainty 3.3.1 Parameter Uncertainty The Monte Carlo simulations included the following random variables: â¢ Hydraulic modeling â Hydrologic uncertainty (Log-Pearson Type III) â Channel Manning n â Floodplain Manning n â Boundary condition (energy slope) â¢ Pier scour â Equation (HEC-18 and Florida DOT [Sheppard et al. 2011]) â Velocity and flow depth â¢ Abutment scour â Equation and methodology for total scour (NCHRP Project 24-20 [Ettema et al. 2010]) â Obstructed flow area, discharge, velocity, and depth â¢ Contraction scour â Upstream flow distribution (Q1) â Bridge flow distribution (Q2, Qleft, Qright) â Flow depths (Y1, Yo) Two categories of factors were not included in the Monte Carlo simulation (see Table 3.1). One category is composed of parameters that would be known in a bridge design, such as pier dimen- sions or road elevation. Therefore, these parameters would be constants and thus be considered deterministic instead of random. The other category that was excluded from the Monte Carlo simulation includes factors that would overly complicate the analysis. As shown in Table 3.1, examples of these types of variables are multiple bridge openings and time rate of scour. 3.3.2 Model Uncertainty Model (equation) uncertainty depends on how well a given scour equation predicts scour. It can be evaluated by comparing observed scour to predicted scour, comparing simulated scour to predicted scour, or by expert knowledge. For this study, model uncertainty is represented by the statistical properties of the ratio of observed scour to predicted scour for a given scour equation. The mean of the ratios is the bias (l), and the standard deviation of the ratios divided by the bias is the coefficient of variation (COV).
Evaluating Uncertainty Associated with Scour Prediction 29 As discussed in Chapter 4, the bias and COV for each of the scour equations were evaluated based on available laboratory and field data, and the reliability index, b, was determined for each scour equation. Because the determination of bias and COV requires observed data, the limitations of each data source need to be addressed. Laboratory data have the disadvantages of small scale, inconsistent length scales (geometric and sediment), and a predominance of clear-water conditions. Field data have the disadvantages of being uncontrolled, large param- eter uncertainty, difficulties associated with measuring scour, difficulties in separating types of scour, unmeasured scour-hole refill, highly variable bed materials, and non-ultimate scour levels. Contraction scour is widely accepted as a sediment transport problem. However, finding reliable laboratory and field contraction scour data was a problem (see Section 4.3). Ultimate live-bed contraction scour is reached when the rate of sediment transport in the bridge opening matches the supply of sediment from the upstream channel. Ultimate clear-water contraction scour is reached when the flow can no longer erode the bed. Most bridge waterway openings are, in reality, short contractions. However, the HEC-18 contraction scour equations were derived using a long contraction, thus introducing additional uncertainty. Long-term channel changes are components of total scour that need to be considered in bridge design, although they cannot be addressed in the same manner as local and contrac- tion scour. Degradation and lateral migration often contribute significantly to total scour at bridges, although aggradation and channel widening may also cause problems. Future degra- dation and aggradation may be estimated in several ways, including bridge inspection profiles, rating curve shifts, equilibrium slope, sediment continuity, sediment transport modeling, and headcut analysis. Future amounts of channel migration can be estimated by comparing historic Topic Deterministic Variables Overcomplicating Factors Hydraulic modeling Bridge or embankment skew Pier size, shape, and skew Varying road elevation Abutment shape Non-stationary aspects of hydrologic uncertainty (climate change, sea level rise/fall) Multiple bridge openings 2-D modeling or complex hydraulic situations Pier scour Pier shape Pier width Pier length Skew angle Material erodibility (clay, rock) Complex pier geometry Debris or ice Time rate of scour Armoring Abutment scour Abutment shape Embankment skew Material erodibility (clay, rock) Time rate of scour Change in abutment shape during scour Contraction scour Embankment length Abutment setback Approach channel width Contracted channel width Material erodibility (clay, rock) Relief bridge scour Time rate of scour Pressure scour (vertical contraction scour) Channel bed forms for live-bed conditions Scour interaction NA Overlapping scour holes (pier-to- pier or abutment-to-pier) Long-term channel changes NA Aggradation, degradation, or headcuts Lateral migration Channel width adjustments Table 3.1. Deterministic variables and overcomplicating factors not considered in the Monte Carlo simulations.
30 Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction aerial photos as described in NCHRP Report 533: Handbook for Predicting Stream Meander Migration (Lagasse et al. 2004). Rather than developing uncertainty parameters related to long-term vertical and lateral channel change, standard design approaches were used. The standard approach currently used in bridge design is to establish a conservative estimate of future channel change (Arneson et al. 2012, Lagasse et al. 2012). Uncertainty and reliability approaches for predicting long-term channel change were not considered in this study. 3.4 Development of Supporting Software 3.4.1 HEC-RAS For each bridge type analyzed, a representative HEC-RAS model was developed to assess the hydraulic conditions at the bridge given the hydrologic and other input variable uncertainties. Models were developed representing small, medium, and large bridges to support the Level I and Level II analyses (see Chapter 5). Each model consists of: â¢ A single reach of four cross sections plus the (automatically-generated by HEC-RAS) bridge upstream and bridge downstream sections. These cross sections include ineffective flow areas and flow transition reach lengths, determined using standard engineering methods, appropriate to capture the full effect of flow contraction and expansion at the bridge. â¢ Manning roughness (n) values assigned with a channel Manning n value and a single over- bank Manning n value for all four cross sections for each realization. â¢ Design discharge (100-year) set for each bridge. â¢ The downstream boundary condition, determined by HEC-RAS using a normal depth computation, driven by friction slope. No supercritical simulations were performed; conse- quently, no upstream water surface elevation computation was required. Neither overtopping (relief) nor internal pier geometry was directly represented in the models. All flow was forced through the bridge opening. Pier and abutment geometry was considered in the (post-process) scour computations. 3.4.2 Integration of HEC-RAS and Monte Carlo To analyze the probability of scour depth exceedance, it was necessary to perform a large number of Monte Carlo realizations (cycles) using the HEC-RAS model. This precluded the use of HEC-RAS through its standard graphical user interface (GUI). Consequently, the HEC-RAS application programming interface (API) was used to integrate HEC-RAS simulations with the Monte Carlo simulation software. Research software (the rasToolÂ©) was developed to automate the running of HEC-RAS. This software included specifying input variables for HEC-RAS geometric and flow files. Results from each run were then appended to a summary output file. This section provides a descrip- tion of the final rasToolÂ© software and its application to scour risk analysis. The rasToolÂ© was developed using Microsoft Visual Studio 2010, using the VB.NET lan- guage. The program is compatible with Windows XP or Windows 7. The rasToolÂ© software is a research-level software engine requiring considerable insight on the part of the user for application. The application process used in this study is described in the following paragraphs.
Evaluating Uncertainty Associated with Scour Prediction 31 When rasToolÂ© is started, rasToolÂ© initializes a double-precision random-number generator (RNG), seeded with the computer clock time, to generate a large (~1055), uniformly distributed pseudo-random number string. The uniformly distributed pseudo-random number string values generated by the RNG are transformed as necessary during the Monte Carlo realizations into Gaussian-distributed Z-values using the polar form of the Box-Muller transform and used for all subsequent random numbers required by the simulation (eight per realization). Four of the random numbers are used in the HEC-RAS modeling (discharge, channel Manning n, floodplain Manning n, and energy slope) and four are used in computing scour (HEC-18 pier scour, Florida DOT pier scour, contraction scour, and abutment scour). Once the Monte Carlo realizations are launched, the rasToolÂ© performs Monte Carlo realiza- tions using the following steps: Step 1. Randomized input variables are determined for a realization using the input probability density function type, summary statistics, and generated randomized Z-values for each input variable. Step 2. These input variables are assigned to the HEC-RAS model using the Interop.RAS41 API for geometric variables and direct assignment to input text files for flow and boundary condition variables. The direct assignment of flow variables proved necessary as the Interop.RAS41 API allowed asynchronous updates of flow variables, geometric vari- ables, and simulations, resulting in interleaved updates and inconsistent hydraulic simu- lation of the desired input variables. Direct assignment of flow and boundary condition variables eliminated this conflict and ensured fully synchronous simulations. Step 3. HEC-RAS is run for the given geometric, flow, and boundary condition variables assigned. Step 4. Input variables and detailed hydraulic results are retrieved from the completed HEC- RAS model using the Interop.RAS41.dll API and stored in a results matrix. These results are sufficiently detailed to support contraction scour, abutment scour, and pier scour computations. Step 5. Steps 1 through 4 are iterated until the user-assigned number of realizations has been performed. From testing, it was determined that 10,000 realizations were sufficient to generate a fully-descriptive data set. Step 6. The Monte Carlo hydraulic results matrix is written to a text file (OutputMC.txt), along with four standard-normal (Gaussian) random variables per realization to support randomization of the scour results. For this study, scour computations (using HEC-18 and Florida DOT pier methods, contraction, and abutment scour) were performed as a post-processing step in a spreadsheet. A 10,000-realization simulation requires between 1 and 2 hours of computer time, depend- ing on the machine used. The rasToolÂ© requires input data from the user to perform its simulation. For each indepen- dent input variable, summary statistics and assumed distributions about an expected value are required. This effort randomized discharge, channel Manning n, overbank Manning n, and friction slope for normal depth boundary condition computation. The rasToolÂ© supports normal and lognormal distributions. The rasToolÂ© requires a representative HEC-RAS model of the bridge simulated. Simulation parameters (number of realizations, Z limit) also are required. Four assumed-independent random geometric and hydraulic variables for each bridge type analyzed were considered for this effort. They were discharge, Manning roughness (overbanks and main channel), and friction slope. The application of these variables is described in Sections 3.4.3, 3.4.4, and 3.4.5.
32 Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction 3.4.3 Discharge A lognormal discharge distribution about its expected value (mean in logarithm transform) was assumed. The expected value discharge was constant for all hydrologic uncertainty sce- narios for a given annual exceedance probability and bridge type (small, medium, or large) as presented in Table 3.2 and Table 3.3 (see also Section 5.2.3). The expected value discharge parameter (in natural logarithm space, A) was determined for each bridge type using Bulletin 17B methods for the relevant period of record and normalized to a N=50-year period of record (see Section 2.2). Notice that the Bulletin 17B predicted discharge for a given exceedance prob- ability represents the mode in linear space, not the mean (expected) value in logarithmic space. The expected value discharge (in natural logarithmic space) is the statistical mean discharge parameter of interest for the Monte Carlo realizations. COV values for a given hydrologic uncertainty scenario were based on a qualitative review of Bulletin 17B flood frequency analyses performed at eight USGS gaged sites to assess the observed range of discharge COV as a function of period of record and regional variation. COV was constant for a given hydrologic uncertainty and annual exceedance probability, as presented in Table 3.4. Bridge A [In (Q)] B [In (Q)] Hydrologic Uncertainty Low Medium High Large 11.8791 0.1282 0.1865 0.2448 Medium 10.3015 0.1111 0.1617 0.2123 Small 7.5175 0.0811 0.1180 0.1549 Table 3.2. 100-year discharge parameters for lognormal distribution (natural log space). Event A B p(X > x) T (yrs) [In (Q)] [In (Q)] 0.04 25 11.64920308 0.115282339 0.02 50 11.77682701 0.125057456 0.01 100 11.88793137 0.133901695 0.002 500 12.10459348 0.151400894 Table 3.3. Illustrative example: low hydrologic uncertainty; A and B based on gage analysis (N 5 49 years). Annual Exceedance Discharge COV (lognormal) p(X > x) T (yrs) Low Medium High 0.04 25 0.009 0.014 0.018 0.02 50 0.010 0.015 0.019 0.01 100 0.011 0.016 0.021 0.005 200 0.012 0.017 0.022 0.002 500 0.013 0.018 0.023 Table 3.4. Hydrologic uncertainty as a function of annual exceedance probability.
Evaluating Uncertainty Associated with Scour Prediction 33 The COV values in Table 3.4 were multiplied by the expected value discharge in natural logarithm space to determine discharge lognormal standard deviation values for each bridge type (small, medium, or large) and hydrologic uncertainty scenario (low, medium, or high). Natural log space expected value discharge (A) and standard deviation (B) were input to the Monte Carlo realizations. Input parameters to the Monte Carlo simulation were constant for each bridge type and hydrologic uncertainty scenario, and are presented in Table 3.2. 3.4.4 Manning Roughness Coefficient Manning roughness values were randomized assuming a lognormal distribution. Overbank roughness and main channel roughness were considered independent random variables for this analysis. They were held constant for a given Monte Carlo realization (e.g., all cross sections were assigned the same, independently randomized, overbank roughness and main channel roughness). The linear space mean values were estimated for each bridge type using standard engineering methods for estimating Manning roughness coefficients and were converted into natural logarithmic input variables using the variable transforms presented in Equation (3.10) and Equation (3.11). A constant COV was assumed for all bridge types and hydrologic sce- narios. The final natural log space variables are presented in Table 3.5. (See Section 3.5.3 for a discussion of initial estimates, testing, and refinement of these variables.) 3.4.5 Downstream Boundary Friction Slope The downstream boundary friction slope was assumed to be normally distributed about the expected (mean) value. Expected values were estimated in the field for each bridge type (see Section 5.2.3). Standard deviation values were determined using COV values devel- oped as shown in Table 3.6. The final values of downstream boundary friction slopes for each of three bridge types (small, medium, or large) as defined in Table 5.1 are presented in Table 3.6. (Section 3.5.3 provides a discussion of initial estimates, testing, and refinement of these variables.) Linear Natural Log Space Manning n COV A [In (n)] B [In (n)] 0.025 0.015 -3.690411607 0.055356174 0.035 0.015 -3.353672518 0.050305088 0.045 0.015 -3.102175432 0.046532631 0.09 0.015 -2.40859826 0.036128974 0.1 0.015 -2.303181866 0.034547728 0.12 0.015 -2.120769523 0.031811543 Table 3.5. Manning roughness coefficients assuming lognormal distribution. Linear COV 0.0048 0.1 0.00048 0.0024 0.1 0.00024 0.005 0.1 0.0005 Table 3.6. Friction slopes assuming normal distribution.
34 Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction 3.4.6 Summary Each Monte Carlo realization generated a set of randomized input variables based on the underlying input variables and summary statistics discussed in this section. These variables were assigned to the HEC-RAS model and a HEC-RAS model run was performed for each realization. Once this run was complete, input variables and hydraulic results for the realiza- tion to support pier scour (HEC-18 and Florida DOT methods), contraction scour (HEC- 18 methods), and abutment scour (NCHRP Project 24-20 method as presented in HEC-18) were accessed using the rasToolÂ© software and exported in a tab-delimited text file for post- processing. Four additional double-precision, normally distributed, random variable values were recorded for each realization to support randomization of the (post-processed) scour predictions based on the scour prediction component bias and COV from the data analysis in Chapter 4. 3.5 Implementing the Software 3.5.1 Approach A four cross-section HEC-RAS bridge hydraulic model was developed for each Monte Carlo simulation. The base-model input parameters, including discharge, Manning n (chan- nel and overbank), and downstream energy slope and the corresponding uncertainties in these parameters were specified in the input file of the rasToolÂ© program. The rasToolÂ© output included hydraulic results for the base condition (expected values) and output for each of the randomly generated input parameter values. RasToolÂ© does not make scour calculations but does create a table of output. This output table is then copied and pasted into an Excel spreadsheet that performs the scour calculations. For each simulation, pier (HEC-18 and Florida DOT), contraction (HEC-18), and abutment scour (NCHRP Project 24-20 method) are computed from the Monte Carlo simulation output. The rasToolÂ© software also includes four normally distributed random numbers (m = 0.0 and s = 1.0) for each simulation. The model (equation) bias and COV from the data analysis in Chapter 4 are then applied to each of the computed scour values to compute the expected distribution of each scour component for the specified event. 3.5.2 Hydraulic Parameter Uncertainty The HEC-RAS Monte Carlo analysis requires that uncertainty in the input parameters be quantified to compute the range of hydraulic conditions and scour that can occur at a bridge. The input parameters selected for HEC-RAS simulations were discharge, channel Manning n, overbank Manning n, and the energy slope downstream boundary condition. Each of these parameters has a value that is either determined or selected during the bridge hydraulic design process, which then results in the design value of scour. By incorporating the hydraulic param- eter uncertainty and the model (scour equation) uncertainty, the statistical characteristics of the individual scour components (pier, contraction, and abutment) and total scour can be evaluated. 22.214.171.124 Hydrologic Uncertainty (Discharge) Flood frequency analysis provides estimates of discharge versus exceedance probability. The Bulletin 17B procedure uses the Log-Pearson Type III distribution to develop the âBulletin 17B estimateâ over the range of annual exceedance probabilities ranging from 0.95 to 0.002, which are the recommended discharges for flood mapping, hydraulic structure design, and other types of analysis (see Section 2.2). The results of the Bulletin 17B procedure also include 95%
Evaluating Uncertainty Associated with Scour Prediction 35 confidence limits and an âexpected probabilityâ estimate of discharge. As defined by Bul- letin 17B, the confidence limits are one-sided, meaning that 95% of the estimates of discharge are greater than the lower bound and 95% less than the upper bound. The probability distribution of a discharge estimate is established by the expected probability value and the two confidence limits of the log-transformed values. Therefore: ( ) = Âµ + Ïln Q 1.645 (3.1)p-ex,0.95 ( ) = Âµ â Ïln Q 1.645 (3.2)p-ex,0.05 Ï = ï£« ï£ï£¬ ï£¶ ï£¸ï£· â â 1 3 29 30 95 0 05. ln ( . ), . , . Q Q 3p ex p ex Âµ = ( ) â â 0 5 40 95 0 05. ln ( . ), . , .Q Q 3p ex p ex COV (3.5)ln = Ï Âµ where Qp-ex,0.05 and Qp-ex,0.95 are the lower and upper one-sided 95% confidence limits for a par- ticular exceedance probability (p-ex), m is the log-transformed expected probability discharge value, and s is the standard deviation of the normally distributed probability density function of the particular exceedance probability. For example, the 100-year Bulletin 17B flow estimate (p-ex = 0.01) for the test Monte Carlo/ HEC-RAS analysis of the Sacramento River bridge (see Section 3.5.3) is 140,000 cfs with expected probability flow of 145,500 cfs and 95% confidence limits of 115,800 cfs and 179,900 cfs. These values result in s = 0.134 and m = 11.888, which are entered as the discharge values for the Monte Carlo simulation flagged as a log-transformed variable. 126.96.36.199 Regional Regression Equations Where gaging station data is unavailable, the use of regression relationships is a common method for estimating flood magnitudes for various return-period events. These relationships utilize watershed and climatologic characteristics specific to a physiographic region to estimate the 2-year up to the 500-year peak discharge at any location within the region of interest. Typi- cal relationships often take the form Qi = A(X1) b(X2) c, . . . , (Xn) n. In these equations, Qi is the estimated discharge for an i-year flood, A is a region-specific coefficient, X1, X2, . . . , Xn are water- shed and climatologic characteristics such as drainage area, mean annual precipitation, percent forest cover, mean basin elevation, and so forth, and b, c, . . . , n are region-specific exponents. The standard error of prediction (SE) in percent is typically reported for each equation and is a measure of the predictive accuracy of the equation for each return period Q2, Q5, . . . , Q500 as com- pared to actual streamflow measurements and gaging station data in that physiographic region. Approximately two-thirds of the estimates obtained from a regression equation for ungaged sites will have errors less than the standard error of prediction (Helsel and Hirsch, 1992). For purposes of assigning a level of hydrologic uncertainty to ungaged sites where regional regression equations are used to estimate flood magnitudes, the following standard error limits are suggested for the applications in this document: â¢ Low hydrologic uncertainty: SE < 15% â¢ Moderate hydrologic uncertainty: 15% < SE < 30% â¢ High hydrologic uncertainty: 30% < SE
36 Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction 188.8.131.52 Manning n Uncertainty As described in Johnson (1996), numerous methods have been used to describe the uncer- tainty in Manning n estimates. The âdataâ provided in USACE (1986) are the most comprehen- sive and are used in this study. The USACE study included nine natural channels with a wide range of conditions in locations throughout the United States. A group of 77 engineers were shown pictures of the channels and asked to estimate the Manning n for a 100-year flow at each location. The engineers could base their estimates on experience, tables, or pictures found in the scientific literature. Outliers were removed from the estimates so that the individual num- ber of estimates ranged from 71 to 77 at each site for a total of 675 estimates and an average of 75 estimates per site. The USACE concluded that the distribution of Manning n was lognormal but did not provide the statistical properties of the log-transformed data. For this study the USACE estimates were normalized by dividing by the mean estimate for each site and grouping the data into a single data set. The 675 normalized data were then log- transformed to evaluate the suitability of using a lognormal distribution. The results are shown in Figure 3.1 and indicate the suitability of using the lognormal distribution on Manning n for these data. Figure 3.2 shows the complete probability distribution function (PDF) and cumulative dis- tribution function (CDF) of the normalized Manning n data. With this distribution, 93% of the Manning n values fall between 0.5 and 1.5 of the expected Manning n. The assumption for applying the results in the HEC-RAS Monte Carlo simulation is that the mean estimate of 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 20 40 60 80 100 120 140 160 0.2 0.35 0.5 0.65 0.8 0.95 1.1 1.25 1.4 1.55 1.7 1.85 2 2.15 2.3 More Pr ob ab ili ty D en si ty Fu nc ti on O b se rv ed F re q u en cy (N = 6 75 ) Normalized Manning n (n/n-expected) Histogram and Ln normal PDF forManning n Frequency Ln normal PDF Figure 3.1. USACE (1986) Manning n data.
Evaluating Uncertainty Associated with Scour Prediction 37 the 77 engineers corresponds well to the expected Manning n for the nine rivers at 100-year flood stage. The COV of the log-transformed data was 0.082. Therefore, an estimate of a channel (or overbank) Manning n (n used in design) can be used to estimate m and s of the log-transformed variable using the following equations: n exp 0.5 exp 0.5 COV (3.6)2 2( ) ( )= Âµ + Ï = Âµ + Âµï£®ï£° ï£¹ï£» ( ) ( )Âµ = â + + = â + +1 1 2ln n COV COV 1 1 2ln n 0.082 0.082 (3.7) 2 2 2 2 COV 0.082 (3.8)Ï = Âµ = Âµ The values of s and m are entered for the Monte Carlo simulation flagged as a log-transformed variable. Because the value of COV of the log-transformed data is 0.082, only the expected channel or overbank Manning n value is required to develop the input for the HEC-RAS/Monte Carlo simulation (see Section 3.5.3). 184.108.40.206 Energy Slope Uncertainty Although energy slope and channel slope appear to be relatively simple parameters to esti- mate, Johnson (1996) found this variable to have relatively significant uncertainty that should 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 2 2.5 Cu m u la tiv e D en si ty Fu nc tio n Pr o ba bi lit y D en s ity Fu nc tio n Normalized Manning n (n/n-expected) Figure 3.2. Normalized Manning n probability distribution function and cumulative distribution function.
38 Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction not be ignored. Johnson found that several types of distributions have been used to describe channel and friction slope, including uniform, normal, triangular, and lognormal. For this study, a normal distribution with COV = 0.17 was used initially (see Section 3.5.3). This distri- bution and value was selected such that plus or minus three standard deviations would result in 99.8% of the starting energy slope values between 0.5 and 1.5 times the expected value. 3.5.3 Testing and Adjusting the Software The Sacramento River bridge (Example Bridge No. 3 in Chapter 7) was used to refine the HEC-RAS/Monte Carlo software. For the initial runs, the hydraulic parameter uncertainty values for discharge, Manning n (channel and overbank), and energy slope as discussed in Section 3.4 were used. The Monte Carlo analysis was compared with data from a gage near the Sacramento River bridge site for flows in the range of the discharges in the Monte Carlo analysis. The comparison was done to test the reasonableness of the Monte Carlo runs. It was determined that the HEC-RAS modeling compared well with the discharge variation at the gage, and that the recommended energy slope parameter uncertainty (normal distribution and COV = 0.17) produced slightly greater variation in water surface and depth as compared to the gage data. The recommended Manning n parameter uncertainty produced extreme vari- ability in water surface. Therefore, the energy slope and Manning n parameter uncertainties were reduced until the combined effects of Manning n and energy slope produced variability similar to that of the water surface measurements at the gage. Given the large range in discharge as represented by the 5% and 95% confidence limits (see the discussion of hydrologic uncertainty in Section 3.4.3), a large range in water surface elevation and flow depth was expected at the bridge in the Monte Carlo simulation. The Monte Carlo simulation includes flows much further out on the tails of the distribution, so the small- est and largest simulated flows for the 100-year discharge were well under 100,000 cfs to well over 200,000 cfs. Over this range of flows, the HEC-RAS model computed water surface varied by nearly 7 ft when Manning n and energy slope were held constant. Figure 3.3 shows gage heights for extreme flows at the Butte City gage (11389000) on the Sacramento River, which is approximately 11 miles downstream of the bridge site. For flows in the range of 100,000 cfs to 144,000 cfs, the gage water surface varies by approximately 3.0 ft and the HEC-RAS water surface varies by 3.3 ft. Therefore, the variations in water surface and flow depth with changing discharge in the HEC-RAS model are reasonable. Sacramento River at Butte City - 11389000 90 90.5 91 91.5 92 92.5 93 93.5 94 94.5 80000 90000 100000 110000 120000 130000 140000 150000 W at er S ur fa ce E le va tio n (ft) Discharge, cfs Gage WS Elevation Linear (Gage WS Elevation) Figure 3.3. Gage heights versus discharge at Sacramento River, Butte City gage.
Evaluating Uncertainty Associated with Scour Prediction 39 The data in Figure 3.3 also illustrate the variability in water surface measurements at the Butte City gage. The standard deviation of the differences in the observed values versus the trend line is 0.49 ft. Therefore, this value was used to assess the reasonableness of the parameter uncertainties of Manning n and energy slope because these parameters will create variability in water surface for a given discharge. 220.127.116.11 Energy Slope Uncertainty As noted in Section 3.5.2, Johnson (1996) found that several types of distributions have been used to describe channel and friction slope, including uniform, normal, triangular, and lognor- mal. Initially, a normal distribution with COV = 0.17 was used for this study. This produced a standard deviation in water surface of 0.66 ft, which is greater than the observed value of 0.49 ft at Butte City gage. Therefore, COV was reduced to 0.10, which resulted in a water surface stan- dard deviation of 0.37 ft. 18.104.22.168 Manning n Uncertainty When the 675 data points of the USACE (1986) study were evaluated, a COV of 0.082 for the log-transformed data fit the data well (see Section 3.5.2). However, when this COV was used in the HEC-RAS Monte Carlo simulation, the standard deviation in water surface was 2.5 ft, which was twice the standard deviation created by discharge uncertainty and much greater than the observed variability in water surface for a given discharge. Therefore, the COV was adjusted until the variability in water surface was more consistent with observed amounts. The COV of the log-transformed Manning n variable of 0.015 yielded a standard deviation in water surface of 0.47. Therefore, an estimate of a channel (or overbank) Manning n can be used to estimate m and s of the log-transformed variable using the following equations: n exp 0.5 exp 0.5 COV (3.9)2 2( ) ( )= Âµ + Ï = Âµ + Âµï£®ï£° ï£¹ï£» ( ) ( )Âµ = â + + = â + +1 1 2ln n COV COV 1 1 2ln n 0.015 0.015 (3.10) 2 2 2 2 COV 0.015 (3.11)Ï = Âµ Ã = Âµ Ã The large difference in COV (0.082 based on selection of Manning n versus 0.015 based on impacts on water surface variability) indicates that Manning n is an important parameter that may often be difficult to reliably estimate. Therefore, calibration of Manning n to observed conditions is an important practice whenever possible. 3.6 Summary and Preview of Applications This chapter provided an approach to evaluating the uncertainty of the three scour compo- nents: pier, contraction, and abutment scour. The methodology is based on software, referred to as the rasToolÂ©, developed specifically for the purpose of linking the most widely accepted 1-D hydraulic model, HEC-RAS, with Monte Carlo simulation techniques. To reduce the com- plexity of the model runs (which will require on the order of 10,000 Monte Carlo realizations per run), only a limited number of hydraulic parameters can be allowed to vary. Other fac- tors are identified as either deterministic or overcomplicating; that is, they are user-defined bridge characteristics and scour variables that would be determined during design, or they are secondary considerations in the analysis of bridge scour that, if allowed to vary, would compli- cate the Monte Carlo simulations to the point that application of the rasToolÂ© would become impractical.
40 Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction The statistical properties describing model uncertainty, bias and COV, were defined. In addi- tion, model uncertainty in relation to the key hydraulic parameters of discharge (hydrologic uncertainty), Manning roughness, and downstream boundary friction slope were defined, and an approach to using regional regression equations to define hydrologic uncertainty was devel- oped. An application of the HEC-RAS/Monte Carlo linkage to a bridge on the Sacramento River was used to test and refine the software and provide a reality check on the assumptions driving the uncertainty statistics for the key hydraulic parameters. With the supporting software in place, tested, and adjusted to reflect typical hydraulic con- ditions in the field, the bias and COV for the equations used to estimate pier, contraction, and abutment scour can now be investigated. Chapter 4 accomplishes this using, primarily, care- fully screened hydraulic laboratory data for each of the scour components. Chapter 4 also sum- marizes the fundamental equations for the three scour components as presented in the current edition of HEC-18 (Arneson et al. 2012). Based on more than 300,000 HEC-RAS/Monte Carlo simulations, Chapter 5 presents two different approaches to assessing the conditional probability that the design scour depth will be exceeded for a given design flood event. The first (Level I) approach assumes that the practitioner can categorize a bridge based on (1) the size of the bridge, channel, and floodplain; (2) the size of the bridge piers; and (3) the hydrologic uncertainty. If so, a 27-element matrix is used to deter- mine scour factors that can be used to multiply the estimated scour depth to achieve a desired level of confidence for pier, contraction, abutment, and total scour based on a reliability index that is commensurate with standard LRFD practice When the practitioner cannot match a bridge to the categories established in the 27-element matrix, a site-specific (Level II) approach to the probability evaluation is required. For com- plex foundation systems and channel conditions, or for cases requiring special consideration, the Level II approach involves the application of HEC-RAS/Monte Carlo analyses for the site- specific conditions. A step-by-step procedure for developing probability-based estimates of scour factors for site-specific conditions is outlined in Chapter 5 and illustrated by application to the Sacramento River bridge (see Section 3.5.3). In some cases it may be necessary to determine the unconditional probability that a scour estimate will not be exceeded over the remaining service life of an existing bridge or the design life of a new bridge. To estimate this unconditional probability of scour depth exceedance, Chap- ter 6 provides an abbreviated LRFD approach using the conditional probabilities for a limited number of return period flood events. This approach integrates the conditional probabilities of three or more flood events to determine the unconditional probability of exceedance over the entire service life of a bridge. Chapter 7 provides illustrative examples that demonstrate the application of the 27-element matrix to determine the conditional probability of exceedance of estimated scour depths for a 100-year event at a bridge that fits one of the categories established in the matrix. For five bridges, representing a range of bridge configurations and physiographic regions across the continental United States, the examples guide the practitioner through the steps required to identify appro- priate scour factors for a desired level of confidence using the results of Chapter 5 and the matrix presented in Appendix B. Analysis of the Sacramento River bridge is included in this section. The user of this reference guide is strongly encouraged to cross reference the material presented in Chapters 4 and 5 with the illustrative examples of Chapter 7.