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11 2.1 Introduction This chapter provides a discussion of the various types and sources of uncertainty that must be considered in the assessment of bridge scour. Citations from the literature provide relevant background information on the current state of practice. Hydrology and hydraulics both intro- duce uncertainties in the determination of the variables that are subsequently used as input to the various scour equations. That is, the three components of scour (pier, contraction, and abutment scour) are fundamentally linked to both the hydrologic estimation of the magnitude of a design flood event and the anticipated hydraulic conditions associated with that event. It must also be understood that the scour equations themselves have uncertainty, as evidenced by the fact that even under controlled laboratory conditions the equations do not precisely predict the observed scour. Current guidance from FHWA on incorporating risk in bridge scour analyses is summarized; and the scour problem is framed in the context of AASHTO LRFD statistical meth- ods and procedures used in bridge structural design from the perspective of a hydraulic engineer. 2.2 Hydrologic Uncertainty 2.2.1 Overview The majority of hydrologic phenomena (e.g., droughts and floods, precipitation, dewpoint, etc.) are stochastic processes, which can be characterized as processes governed by laws of chance. Strictly speaking, no pure deterministic hydrologic processes exist in nature; hydrologic phenomena have traditionally been understood and described using methods of probability theory (Yevjevich 1972). Scour prediction is typically associated with a design hydrologic event that has a given likeli- hood of recurrence (e.g., the 100-year flood). Hydraulic conditions from such an event, in terms of the depth and velocity of flow corresponding to the peak rate of flow, are used to predict local and contraction scour at the bridge using methods described in the 5th edition of HEC-18 (Arneson et al. 2012). This scour prediction is in turn used for determining structural stability for the case in which all the soil material in the scour prism is removed. Usually the time rate of scour is ignored and scour is assumed, in effect, to occur instantaneously in response to the peak hydraulic load for the event of interest. Practitioners understand that the 100-year flood is defined as the discharge rate that has a 1% chance of being equaled or exceeded in any given year; the 50-year flood has a 2% probability of exceedance, and so forth. Typically, the discharge is estimated based on flow records from stream gaging stations upstream or downstream of the bridge, and the discharge estimates are adjusted C H A P T E R 2 Uncertainty in Hydraulic Design
12 Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction to the bridge location using area-weighting and other techniques. Where gaging station records on the particular stream or river are unavailable, data are used from stations in nearby water- sheds of similar size and nature to the watershed of interest. In many cases, regional regression relationships are available for use, and these typically include watershed area and a rainfall index (such as the 2-year, 24-hour rainfall depth) as input values to the regression equations. Practitioners understand that the magnitude of any recurrence-interval event is an estimate, but the current state of practice in bridge scour prediction places no emphasis on quantifying the reliability of that estimate. However, it has been standard practice to report the 95% confi- dence limits as part of the methodology described in U.S. Geological Survey (USGS) Bulletin 17B for nearly half a century (USGS 1981). As with any probability-based estimate, confidence in the predicted value of the 100-year flood increases with the number of observations from the population of discharges. Regional regression relationships often include a measure of uncer- tainty about the predicted recurrence-interval values. NCHRP Report 717: Scour at Bridge Foundations on Rock (Keaton et al. 2012) provides an example to illustrate this issue. In the research for that project, four field sites were investigated where the erodibility of rock at bridge pier foundations was assessed. One site, SR-22 over Mill Creek in western Oregon, has exhibited approximately 7 ft of scour over the period from December 1945 to August 2008. Data are available from the USGS gaging station upstream of the bridge from the stationâs installation in 1958 until use of the station was discontinued in 1973, so only 15 years of mean daily flows and annual instantaneous peaks are available from that location. The time series was extended by regression analysis using data from stations on the South Yamhill River, located farther downstream. This technique provided additional data necessary to assess the cumulative hydraulic loading experienced by the bridge to the present time. The resulting time series of mean daily flows is shown in Figure 2.1. Mill Creek at SR 22 synthesized time series 0 1000 2000 3000 4000 5000 6000 7000 Jan-30 Jan-40 Jan-50 Jan-60 Jan-70 Jan-80 Jan-90 Jan-00 Jan-10 Date M ea n D ai ly D is ch ar ge , f t3 / s Extended via regression Extended via regression Mill Creek observed Figure 2.1. Mean daily flows, Mill Creek at SR-22 showing observed and extended records.
Uncertainty in Hydraulic Design 13 The data from other gaging stations allowed the period of record to be extended from 1935 through 2008 (74 years) for purposes of quantifying the cumulative hydraulic loading from the time the bridge was built to the present. Figure 2.1 clearly shows that the single largest flood event in the entire period of record (mean daily flow of 5,980 ft3/s, with an instantaneous peak of 7,320 ft3/s) occurred during the period of time when the Mill Creek gaging station was active. All other mean daily flows recorded for the 74-year period were less than 4,000 ft3/s. The USGS flood frequency analysis software PKFQWin (Flynn et al. 2006) was used to esti- mate the magnitudes of various recurrence-interval floods using Bulletin 17B methodology, assuming a Log-Pearson Type III probability distribution. For both the 15- and 74-year periods of record, the generalized skew of 0.086 at this location was combined with the observed station skew to produce a weighted skew for use with this probability distribution. Table 2.1 presents the results of these flood frequency analyses. Figure 2.2 presents the predicted frequency curves and associated 95% confidence limits for the 15 years of observed annual peaks and for the entire 74-year extended period of record. As seen in Figure 2.2, the estimates of the recurrence-interval flood magnitude and the corresponding confidence limits are quite different for the two periods of record. Table 2.1 and Figure 2.2 illustrate how the confidence limits associated with a Log-Pearson Type III probability distribution are sensitive to the number of observations, and how the con- fidence interval becomes wider as the recurrence interval increases. For smaller, more frequent events, the reliability of the discharge estimate is greater than for larger, less frequent floods. In summary, the characteristics of the probability distribution typically used in flood fre- quency analyses (Log-Pearson Type III) are well known and described. This result is well suited to LRFD procedures for establishing a probability-based characterization of scour using stan- dard practices in hydrologic analysis. Clearly, an understanding of and ability to characterize sources of hydrologic uncertainty are central to probability-based bridge scour predictions. 2.2.2 Evaluating Hydrologic Uncertainty 2.2.2.1 Flood Frequency Estimates From Gaging Station Data As discussed in Section 2.2.1, characteristics of the probability distribution typically used in flood frequency analyses (Log-Pearson Type III) are well known and described. Uncertainty in hydrologic estimates can, therefore, be easily incorporated within the framework of exist- ing LRFD procedures to establish a probability-based characterization of scour using standard practices in hydrologic analysis. Recurrence Interval (yrs) 15-Year Period Weighted Skew = 0.159 74-Year Period Weighted Skew = 0.253 Discharge (ft3/s) 95% Confidence Discharge (ft3/s) 95% Confidence Lower Upper Lower Upper 1.5 3,142 2,633 3,629 2,806 2,653 2,954 2 3,630 3,116 4,220 3,138 2,981 3,302 5 4,858 4,182 5,995 3,946 3,734 4,205 10 5,686 4,811 7,384 4,472 4,197 4,827 25 6,752 5,562 9,344 5,132 4,761 5,634 50 7,561 6,102 10,940 5,622 5,171 6,247 100 8,384 6,633 12,650 6,113 5,575 6,871 500 10,380 7,860 17,120 7,275 6,514 8,383 Table 2.1. Flood frequency analyses for SR-22 over Mill Creek, Oregon.
14 Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction The USGS software package PKFQWin can be used to determine hydrologic uncertainty when dealing with data from gaged sites. The software is a public-domain, Windows-based program that allows the user to access annual peak flow records in standard USGS format. Flood frequency estimates, as well as the 95% confidence limits about the estimated values, are part of the PKFQWin output files. From the USGS gaging station identifier, PKFQWin identi- fies the generalized skew based on location (latitude and longitude) and computes the actual station skew using the observed record from the site. These values are then used to compute a weighted skew value in accordance with USGS Bulletin 17B procedures. 2.2.2.2 Flood Frequency Estimates From Regional Regression Equations The USGS has developed and published regression equations for every state, the Common- wealth of Puerto Rico, and a number of metropolitan areas in the United States. The National Streamflow Statistics (NSS) software compiles all current USGS regional regression equations for estimating streamflow statistics at ungaged sites in an easy-to-use interface that operates on computers with Microsoft Windows operating systems. NSS expands on the functionality of the National Flood Frequency program, which it replaces. The regression equations included in NSS (Ries 2007) are used to transfer streamflow statis- tics from gaged to ungaged sites through the use of watershed and climatic characteristics as explanatory or predictor variables. Generally, the equations were developed on a statewide or metropolitan-area basis as part of cooperative study programs. NSS output also provides indi- cators of the accuracy of the estimated streamflow statistics. The indicators may include any FLOOD FREQUENCY ESTIMATES MILL CREEK AT SR 22, OREGON 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 1 10 100 1000 Recurrence Interval (years) D is ch ar ge (f t3 /s ) 74 years 15 years 95% confidence limits for the 100-yr flood, 15 years of record 95% confidence limits for the 100-yr flood, 74 years of record Figure 2.2. Flood frequency estimates for 15- and 74-year periods of record, SR-22 over Mill Creek, Oregon.
Uncertainty in Hydraulic Design 15 combination of the standard error of estimate, the standard error of prediction, the equivalent years of record, or 90% prediction intervals, depending on what was provided by the authors of the equations. NSS is a public-domain software program that can be used to: ⢠Obtain estimates of flood frequencies for sites in rural (non-regulated) ungaged basins. ⢠Obtain estimates of flood frequencies for sites in urbanized basins. ⢠Estimate maximum floods based on envelope curves. ⢠Create hydrographs of estimated floods for sites in rural or urban basins and manipulate the appearance of the graphs. ⢠Create flood frequency curves for sites in rural or urban basins and manipulate the appear- ance of the curves. ⢠Quantify the uncertainty of flood frequency estimates. ⢠Obtain improved flood frequency estimates for gaging stations by weighting estimates obtained from the systematic flood records for the stations with estimates obtained from regression equations. ⢠Obtain improved flood frequency estimates for ungaged sites by weighting estimates obtained from the regression equations with estimates obtained by applying the flow per unit area for an upstream or downstream gaging station to the drainage area for the ungaged site. 2.3 Hydraulic Uncertainty 2.3.1 Overview As discussed in Section 2.2, hydraulic conditions associated with the design event must be determined in order to estimate scour depths. At a particular location, such as a pier, the hydraulic parameters of flow depth and velocity are related through the Manning n resistance factor and the local energy slope. From these basic parameters, other hydraulic variables, such as Froude number, shear stress, shear velocity, stream power, and so forth, are calculated. The distribution of flow and velocity within the main channel or between the main channel and overbank (floodplain) is highly sensitive to the river reach geometry and the choice of Manning n used to characterize these areas. Figure 2.3 shows a bridge opening approach cross section and the associated velocity distri- butions from a HEC-RAS model (USACE 2010), which bases flow distribution on conveyance. A change in geometry or in Manning n would result in a different flow distribution between channel and floodplain (impacting the contraction scour) and the magnitudes of the computed velocities (impacting local pier and abutment scour). However, whether simple models (e.g., Manningâs equation) or more sophisticated approaches (e.g., HEC-RAS, FESWMS, etc.) are used to estimate the hydraulic conditions at a particular site and at a particular discharge, such estimates necessarily result from a simplification of the complex physical processes involved with open-channel flow. Several broad categories of uncertainty are common to any design process. These can be described as model uncertainty, parameter uncertainty, randomness, and human error. 2.3.1.1 Model Uncertainty Model uncertainty results from attempting to describe a complex physical process or phe- nomenon through the use of a simplified mathematical expression. Model uncertainty in scour analysis is the result of selecting a particular scour equation to estimate scour. Each equation has bias that causes it (on average) to over- or underpredict scour for certain situations.
16 Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction 0 1 2 3 4 Ve lo ci ty a t B rid ge (m /s) Left Embankment Right Embankment Bridge Cross Section Bridge Velocity Distribution Approach Cross Section Approach Velocity Distribution 0 1 2 3 0 200 400 600 800 1000 1200 1400 Distance, m A pp ro ac h Ve lo ci ty (m /s) 1600 Figure 2.3. Flow distribution from 1-D hydraulic modeling. 2.3.1.2 Parameter Uncertainty Parameter uncertainty results from difficulties in estimating model parameters. For example, Manningâs roughness coefficient and design discharge are two common parameters that can- not be measured directly; therefore, they must be estimated or assumed. The result is parameter uncertainty. Examples of hydraulic models used in bridge designs include HEC-RAS (USACE 2010) and FESWMS-FST2DH (Froehlich 2003). Each model has strengths and weaknesses that can lead to more or less parameter uncertainty based on the particular bridge, road embank- ment, and river conditions. Parameter uncertainty can be reduced by using more sophisticated
Uncertainty in Hydraulic Design 17 models (e.g., 2-D models) for more complex situations or by calibrating the model to measured conditions. 2.3.1.3 Randomness Natural (or inherent) randomness is a source of uncertainty that includes random fluctua- tion in parameters, such as flow discharges and velocities. Other types of randomness may be changes to floodplain vegetation that occur over time (seasonally or over the life of the bridge). 2.3.1.4 Human Error Potential always exists for human error in design and in implementation of a design. This type of uncertainty includes calculation and construction errors. Human errors are not usu- ally considered in current reliability-based calculation of load and resistance factors, but their possible occurrence may be considered during the selection of the target reliability levels in the code calibration process. 2.3.2 Evaluating Hydraulic Uncertainty Hydraulic parameters such as roughness coefficient, channel or energy slope, and critical shear stress, common to many hydraulic engineering problems, are known to contain con- siderable uncertainty. A common way to express the uncertainty associated with hydraulic parameters is through the coefficients of variation and associated distributions of parameters. Johnson (1996) quantified uncertainty in common hydraulic parameters based on data from the scientific literature, experiments, and field observations. The Johnson study yielded the data presented in Table 2.2, which provides the COV, distribution, and reference or method by which the data were determined. In Table 2.2, where two values are included, the values repre- sent either a different assumed probability distribution or a different situation. Of course, in a hydraulic model used for bridge design, several Manning n values will be used, including channel, left overbank, and right overbank. Each Manning n value will have its own uncer- tainty, which may differ from the others. The channel Manning n may be calibrated for fre- quent bankfull flows and the overbank Manning n values may be selected based on experience Variable Coefficient of Variation (COV) Distribution Reference or Method Manning n Manning n Manning n Manning n Manning n Manning n 0.1, 0.15 0.2, 0.053 0.08 0.10, 0.055 0.20-0.35 0.28, 0.18 Normal Normal Triangular Triangular, gamma Lognormal Uniform Cesare 1991 Mays and Tung 1992 Yeh and Tung 1993 Tung 1990 HEC 1986 Johnson 1996 Channel slope Channel slope Channel slope 0.3, 0.068 0.12, 0.164 0.25 Normal Triangular Lognormal Mays and Tung 1992 Tung 1990 Johnson 1996 Particle size Particle size 0.02 0.05 Uniform Uniform Yeh and Tung 1993 Johnson and Ayyub 1992 Friction slope 0.17 Uniform Yeh and Tung 1993 Sediment sp. Weight 0.12 Uniform Yeh and Tung 1993 Flow velocitya Flow velocity 0.008xb 0.12xb Triangular Uniform Meter manufacturer; Johnson 1996 aMeasured using electromagnetic meter bx = average velocity Table 2.2. Uncertainty of hydraulic variables (Johnson 1996).
18 Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction or by comparison with published values. Since its original publication, this table has been cited numerous times in risk, reliability, and other studies, and it has been the basis for parameter input for bridge scour, levee and dam overtopping, and other hydrodynamic studies. Hydraulic parameters are typically input to hydraulic models such as HEC-RAS to estimate flood elevations and velocities. Uncertainty in the parameters will propagate through the model to create uncertainty in the resulting calculation. In addition, uncertainty in the model itself will combine with the parameter uncertainty to create additional uncertainty. Uncertainties in hydraulic conditions can be reduced when measured data are available to calibrate the model. For example, high water marks or other observations of water surface elevation, combined with a discharge measurement, can be extremely valuable in adjusting Manning n values in the main channel and overbank areas so that the hydraulic model matches observed conditions. Discharge measurements made using the velocity-area method also provide useful information on the velocity distribution across the channel. However, channels and floodplains change through time. These changes include matur- ing vegetation; land-use change; and channel aggradation, degradation, migration, and width adjustment. Though future conditions often are difficult to estimate, they introduce consider- able uncertainty and should not be neglected during the hydraulic analysis. They impact flow velocity and depth directly, and they impact the distribution of flow and velocity, each of which has an impact on scour estimates. As with hydrologic uncertainty, incorporating hydraulic uncertainty into bridge design is not a trivial matter. Considerable information is available on the subject of hydraulic uncer- tainty, however, and through the use of hydraulic models the levels of uncertainty in velocity, depth, and flow distribution can be quantified both in general and in any specific application. 2.4 Uncertainty in Bridge Scour Estimates 2.4.1 Background Scour at bridges is a very complex process. Scour and channel instability processes, including local scour at the piers and abutments, contraction scour, channel bed degradation, channel widening, and lateral migration, can occur simultaneously. The sum and interaction of all of these river processes create a very complex phenomenon that has, so far, eluded definitive mathematical modeling. To further complicate a mathematical solution, countermeasures such as riprap, grout bags, and gabions may be in place to protect abutments and piers from scour. A complete mathematical model would also have to account for these structures. Considerable uncertainty exists in estimating all components of scour at the piers and abut- ments. Sources of uncertainty include model, parameter, and data uncertainties. For some bridges, the uncertainty is much greater than for other bridges because of unusual circum- stances and difficulties in estimating parameters. For example, Oben-Nyarko and Ettema (2011) point out that scour depth at a pier located close to an abutment is determined predomi- nantly by scour at the abutment and, therefore, may substantially exceed the depth estimated for an isolated pier. When the prototype conditions differ significantly from the conditions under which the model was calibrated, the model uncertainty is increased and may over- shadow all other types of uncertainty. A number of studies have been aimed at developing probabilistic estimates of bridge scour, particularly for piers, for the purpose of design and mitigation. This section provides useful background information and summarizes several of these studies.
Uncertainty in Hydraulic Design 19 Jones (1984) compared numerous pier scour equations using laboratory data and limited field data. He found that the HEC-18 equation tended to give reasonable, although conservative, results. Johnson (1991) listed four primary concerns with bridge pier scour prediction methods, including the inability to determine the impact of future storms on the scour depth or on the probability that the bridge will fail or survive. In an effort to use probabilistic estimates as a tool in decision making, Johnson and Dock (1998) developed a probabilistic framework for estimating scour using deterministic methods given in HEC-18. Uncertainties in the HEC-18 model, in determination of the parameters, and in estimating the hydraulic variables for a large event storm were included in the analysis. The probabilistic framework was then used as the basis for determining the likelihood of achieving various scour depths, probabilities of failure for various foundation designs, the pile depths necessary to achieve a specified probability of failure for a design bridge life span, and for comparing designs based on various storm events. Johnson and Dock (1998) used the Bonner Bridge in North Carolina as an example. Pile depths appropriate for various design life spans were calculated based on both 100-year and 500-year storm events. The following assumptions were made: (1) the failure event was defined as the point at which the scour reached the base of the piles; (2) the arrival of hurricanes is a Poisson process (see glossary in Appendix A); and (3) the piles can be placed at a depth yp with a small COV and follow a normal distribution. The first assumption can be readily changed to reflect different design criteria. Figure 2.4 shows the resulting frequency histogram for 1,000 simulated scour depths. Based on this resulting normal distribution, a mean scour depth of 53.2 ft (16.21 m), and a standard deviation of 4.8 ft (1.46 m), the probability that a scour depth of less than 68.9 ft (21 m) will occur is 97.4%. The scour depth at which the probability of non- exceedance is 90% is 63.5 ft (19.36 m). The uncertainty in a scour estimate can be computed in different ways. One method is to use simulation techniques, such as Monte Carlo simulation or modifications of the Monte Carlo simulation technique. The benefit of using simulation is that the uncertainty in scour can be quantified as a function of the uncertainty in the hydraulic model and its parameters. The result is a probabilistic scour estimate (i.e., one that has a mean, standard deviation, and probability distribution associated with it). The drawback of using Monte Carlo simulations is the large number of calculations that are needed, particularly when dealing with large numbers of random variables and low probabilities. Johnson and Dock (1998) used Monte Carlo simulation to generate random samples of the parameters in the HEC-18 pier scour equation based on the associated coefficients of variation 0 20 40 60 80 100 120 140 160 29 .5 32 .8 36 .1 39 .4 42 .6 45 .9 49 .2 52 .5 55 .8 59 .0 62 .3 65 .6 68 .9 72 .2 75 .4 78 .7 82 .0 Fr eq ue nc y Scour Depth (ft) Figure 2.4. Frequency histogram for 1,000 simulated scour depths.
20 Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction and distributions described in Section 2.3.2 and Table 2.2. They also accounted for model uncertainty using a model correction factor and its COV and distribution. Using the example of a 500-year storm at the Bonner Bridge in North Carolina, they generated a scour distribu- tion with a mean scour depth of 53.2 ft (16.21 m), a COV of 0.090, and a normal distribution (Figure 2.4). Following this process, probabilistic statements can be made regarding the likeli- hood of obtaining a specified scour depth. These types of results, based on the uncertainty in the hydrologic input, hydraulic parameters, and model uncertainty, are the basic input for risk and reliability analyses. 2.4.2 FHWA GuidanceâIncorporating Risk in Bridge Scour Analyses As additional background, this section presents FHWAâs latest guidance on risk analysis as applied to bridge scour (see the 5th edition of HEC-18, published in April 2012). To ensure preci- sion, the section presents pertinent excerpts from HEC-18 Chapter 2 and Appendix B verbatim. Bridge foundations for new bridges should be designed to withstand the effects of scour caused by hydraulic conditions from floods larger than the design flood. In 2010, the U.S. Congress recommended that FHWA apply risk-based and data-driven approaches to infrastructure initiatives and other FHWA bridge program goals. This included the FHWA Scour Program. Risk-based approaches factor in the importance of the structure and are defined by the need to provide safe and reliable waterway crossings and consider the economic consequences of failure. For example, principles of economic analysis and experience with actual flood damage indicate that it is almost always cost-effective to provide a foundation that will not fail, even from very large events. However, for smaller bridges designed for lower frequency floods that have lower consequences of failure, it may not be necessary or cost-effective to design the bridge foundation to with- stand the effects of extraordinarily large floods. Prior to the use of these risk-based approaches, all bridges would have been designed for scour using the Q100 flood magnitude and then checked with the Q500 flood magnitude. [Table 2.3] presents recommended minimum scour design flood frequencies and scour design check flood frequencies based on hydraulic design flood frequencies (Arneson et al. 2012). The Hydraulic Design Flood Frequencies outlined in [Table 2.3] assume an inherent level of risk. There is a direct association between the level of risk that is assumed to be acceptable at a structure as defined by an agencyâs standards and the frequency of the floods they are designed to accommodate. 2.4.2.1 Discussion of Design Flood Frequencies The Scour Design Flood Frequencies presented in [Table 2.3] are larger than the Hydraulic Design Flood Frequencies because there is a reasonably high likelihood that the hydraulic design flood will be exceeded during the service life of the bridge. For example, using [Table 2.4] . . . it can be seen that during a 50-year design life there is a 39.5% chance that a bridge designed to pass the Q100 flood will experience that flood or one that is larger. Similarly, there is a 63.6% chance that a bridge that is designed to pass the Q50 flood will experience that or a larger flood during a 50-year design life. Using the larger values for the Scour Design Flood Frequency for the 200-year flood and a 50-year design life reduces the exceedance value to 22.2%. This is considered to be an acceptable level of risk reduction. In other words, a bridge must be designed to a higher level for scour than for the hydraulic design because if the hydraulic design flood is exceeded then a greater amount of scour will occur which could lead to bridge failure. Also, designing for a higher level of scour than the hydraulic design flood ensures a level of redundancy after the hydraulic design event occurs. Hydraulic Design Flood Frequency, QD Scour Design Flood Frequency, QS Scour Design Check Flood Frequency, QC Q10 Q25 Q50 Q25 Q50 Q100 Q50 Q100 Q200 Q100 Q200 Q500 Table 2.3. Hydraulic design, scour design, and scour design check flood frequencies (Arneson et al. 2012).
Uncertainty in Hydraulic Design 21 The Scour Design Check Flood Frequencies are larger than the Scour Design Flood Frequencies using the same logic and for the same reasons as outlined above (Arneson et al. 2012). If there is a flood event greater than the Hydraulic Design Flood but less than the Scour Design Flood that causes greater stresses on the bridge, e.g., overtopping flood, it should be used as the Scour Design Flood. For this condition there would not be a Scour Design Check Flood since the overtopping flood is the one that causes the greatest stress on the bridge. Similarly, if there is a flood event greater than the Scour Design Flood but less than the Scour Design Check Flood that causes greater stresses on the bridge, it should be used as the Scour Design Check Flood. Balancing the risk of failure from hydraulic and scour events against providing safe, reliable, and economic waterway crossings requires careful evaluation of the hydraulic, structural, and geotechnical aspects of bridge foundation design (Arneson et al. 2012). 2.4.2.2 Flood Exceedance Probabilities A flood event with a recurrence interval of T years has a 1/T probability of being exceeded in any one year. The 100-year recurrence-interval flood is often used as a hydraulic design value and to establish other types of flooding potential. Regardless of the flood design level, there is a chance, or probability, that it will be exceeded in any one year and the probability increases depending on the life of the structure. The probability that a flood event frequency will be exceeded in N years depends on the annual probability of exceedance as defined by: )(= â âP 1 1 P (2.1)N a N where: PN = Probability of exceedance in N years Pa = Annual probability of exceedance (1/T) N = Number of years T = Flood event frequency of exceedance The number of years, N, can be assumed to equal the bridge design life or remaining life. [Table 2.4] shows the probability of exceedance of various flood frequencies for time periods (that may be assumed to equal the bridge design life) ranging from 1 to 100 years. For example a 100-year flood has an annual (N = 1) probability of exceedance of 1.0%, but has a 39.5% chance of exceedance in 50 years. A 200-year flood has a 22.2% chance of being exceeded in 50 years and a 31.3% chance of being exceeded in 75 years (Arneson et al. 2012). FHWA notes that the probability of exceedance may be applied to an individual bridge or for a population of similar bridges. Therefore, if a 200-year design flood condition is used for a population of bridges with expected design lives of 75 years, then that flood condition will be exceeded at approximately 31.3% of the bridges over their lives. Because design flood con- ditions are exceeded at many bridges during their useful lives, factors of safety, conservative design relationships, and LRFD are used to provide adequate levels of safety and reliability in bridge design. Flood Frequency Probability of Exceedance in N Years (or Assumed Bridge Design Life) Years N = 1 N = 5 N = 10 N = 25 N = 50 N = 75 N = 100 10 10.0% 41.0% 65.1% 92.8% 99.5% 100.0% 100.0% 25 4.0% 18.5% 33.5% 64.0% 87.0% 95.3% 98.3% 50 2.0% 9.6% 18.3% 39.7% 63.6% 78.0% 86.7% 100 1.0% 4.9% 9.6% 22.2% 39.5% 52.9% 63.4% 200 0.5% 2.5% 4.9% 11.8% 22.2% 31.3% 39.4% 500 0.2% 1.0% 2.0% 4.9% 9.5% 13.9% 18.1% Table 2.4. Probability of flood exceedance of various flood levels (Arneson et al. 2012).
22 Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction 2.5 LRFDâA Hydraulic Engineering Perspective 2.5.1 Introduction The LRFD methodologies for bridge design were initially developed using concepts derived from structural engineering procedures. LRFD incorporates state-of-the-art analysis and design methodologies with load and resistance factors based on the known variability of applied loads and material properties. These load and resistance factors are calibrated from actual bridge sta- tistics to ensure a uniform level of reliability. LRFD allows a bridge designer to focus on a design objective or limit state, which can lead to a similar probability of failure in each component of the bridge. Bridges designed with the LRFD specifications should have relatively uniform safety levels, which should ensure superior serviceability and long-term maintainability. Scour of earth materials from around bridge foundation elements does not represent a load, but a loss of resistance. Hydraulic engineers are tasked with estimating scour depths for different types of scour processes (e.g., pier, contraction, and abutment scour). Scour estimates typically are associated with a design flood event (e.g., a 100-year flood). Struc- tural and geotechnical engineers use this information for developing a bridge design to maintain structural stability that accommodates the loss of resistance due to scour. 2.5.2 Reliability The aim of reliability theory, as incorporated in LRFD methodology, is to account for the uncertainties encountered while evaluating the safety of structural systems or during the cali- bration of load and resistance factors for structural design codes. More detailed explanations of the principles discussed in this section can be found in published texts on structural reliability and risk such as those by Thoft-Christensen and Baker (1982); Nowak and Collins (2000); Melchers (1999); Ayyub (2003); and Ayyub and McCuen (2003). The uncertainties associated with predicting the load-carrying capacity of a structure, the intensities of the loads expected to be applied, and the effects of these loads may be represented by random variables. The value that a random variable can take is described by a probability distribution function. That is, a random variable may take a specific value with a certain prob- ability and the ensemble of these values and their probabilities is described by the probability distribution function. The most important characteristics of a random variable are its mean (or average) value, and the standard deviation that gives a measure of dispersion (or uncertainty) in estimating the variable. A dimensionless measure of the uncertainty is the coefficient of variation (COV), which is the ratio of standard deviation divided by the mean value. For example, the COV of the random variable R is defined as VR such that: V R (2.2)R R = Ï where: sR = Standard deviation R â = Mean value Codes often specify nominal values for the variables used in design equations. These nominal values are related to the means through bias values. The bias is defined as the ratio of the mean to the nominal value used in design. For example, if R is the resistance, then the mean of R, expressed as R â , can be related to the nominal (or design) value Rn using a bias factor such that: R b R (2.3)r n=
Uncertainty in Hydraulic Design 23 where: br = Resistance bias Rn = The nominal value as specified by the design code For example, A36 steel has a nominal design yield stress of 36 ksi (248,220 kPa), but coupon tests show an actual average value close to 40 ksi (275,800 kPa). Hence, the bias of the yield stress is 40/36 or 1.1. In addition to material properties, the bias and the COV in member resis- tance account for fabrication errors and modeling uncertainties reflecting the existing lack of precision in the ability to model the actual strength of structural members even when the material properties and dimensions are precisely known. In structural reliability, safety may be described as the situation in which capacity (e.g., strength, resistance, fatigue life, foundation depth) exceeds demand (e.g., load, moment, stress ranges, scour depth). Probability of failure (i.e., probability that capacity is less than load demand) may be formally calculated; however, its accuracy depends on detailed data on the probability distri- butions of load and resistance variables. Given that such data are often unavailable, approximate models are often used for calculation. The reserve margin of safety of a bridge component can be defined as Z, such that: Z R S (2.4)= â where: R = Capacity S = Total load demand The probability of failure, Pf, is the probability that R is less than or equal to the total applied load effect, S, which is equivalent to the probability that Z is less than or equal to zero. This relationship is symbolized by Equation (2.5): P P R S (2.5)f r [ ]= ⤠where: Pr is used to symbolize the term Probability. If R and S follow independent normal distributions, then: = Φ â Ï ï£« ï£ï£¬   = Φ â â Ï + Ï ï£« ï£ï£¬  P 0 Z R S (2.6)f Z R 2 S 2 where: F = Normal probability function that gives the probability that the normalized random variable is below a given value Z â = Mean safety margin sZ = Standard deviation of the safety margin Equation (2.6) gives the probability that Z is less than zero. The reliability index, b, is defined such that: P (2.7)f ( )= Φ âβ which for the normal distribution case gives: Z R S (2.8) Z R 2 S 2 β = Ï = â Ï + Ï
24 Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction Thus, the reliability index, î¢, which is often used as a measure of structural safety, gives in this instance the number of standard deviations that the mean margin of safety falls on the âsafeâ side. The reliability index, b, defined by Equation (2.8), provides an exact evaluation of risk (fail- ure probability) if R and S follow normal distributions. Although b was originally developed for normal distributions, similar calculations can be made if R and S are lognormally distributed (i.e., when the logarithms of the basic variables follow normal distributions). Other methods have been developed to obtain the reliability index, b, for cases when the basic variables are not normally distributed. More advanced techniques have also been developed to improve the estimates when the failure function is highly nonlinear. On the other hand, Monte Carlo simulations can be used to provide estimates of the probability of failure (e.g., the probability of exceeding a design value). Monte Carlo simulations are suitable for any random variable probability distribution type and failure equation. In essence, a Monte Carlo simulation involves the creation of a large number of âexperi- mentsâ through the random generation of sets of resistance and load variables. Estimates of the probability of failure, Pf, are obtained by comparing the number of experiments that produce failure (or exceedance of a design value) to the total number of generated experiments. Given values of the probability of failure, Pf, the reliability index, b, is calculated from Equation (2.7) and used as a measure of structural safety even for non-normal distributions. 2.5.3 LRFD Code Calibration The reliability index, b, has been used by many groups throughout the world to express structural risk. A value of b in the range of 2 to 4 is usually specified for different structural applications. For example, b = 3.5 was used for the calibration of the Strength I limit state in AASHTOâs LRFD specifications for the design of new bridges (AASHTO 2007). The calibra- tion process as described by Nowak (1999) and Kulicki et al. (2007) is based on the reliability of bridge members subject to random truck loads within a 75-year design life. On the other hand, the load and resistance factor rating (LRFR) provisions in the AASHTO Manual for Bridge Evaluation (2008) were calibrated to meet a target reliability index of b = 2.5 for checking the safety of existing bridges under random truck loads for a rating period of 5 years (Moses 2001). The difference between the two return periods and target reliability values in the AASHTO LRFD and LRFR is justified based on a strict inspection process for existing bridges and a qualitative cost-benefit analysis. Although demanding higher reliability levels for new designs will imply a marginal increase of bridge construction costs, the replacement of existing bridges would lead to major construction as well as other tangible and intangible economic and other costs associated with the disruption of traffic. In structural design and evaluation, the reliability index values used in the AASHTO LRFD and LRFR calibrations correspond to the failure of a single component. If there is adequate redundancy, however, overall system reliability indices will be higher, as indicated by Ghosn and Moses (1998) and Liu et al. (2001), who proposed the application of system factors cali- brated to meet system reliability criteria rather than component criteria. A slightly different approach taken in ASCE 7-10 (2010) recommends the use of different member reliability targets based on the consequences of a memberâs failure. Thus, the target reliability, btarget, to be used for the design of a connection must be higher than that of a beam in bending. Generally speaking, the reliability index, b, is not used in practice for making decisions regarding the safety of a particular design or an existing structure, but instead is used by
Uncertainty in Hydraulic Design 25 code-writing groups for recommending appropriate load and resistance safety factors for new structural design or evaluation specifications. One commonly used calibration approach is based on the principle that each type of structure should have uniform or consistent reliability levels over the full range of applications. For example, in the calibration of the AASHTO LRFD and LRFR codes, load and resistance factors were chosen to produce b values that uniformly match a target reliability level, btarget, for bridges of different span lengths, number of lanes, simple or continuous spans, roadway categories, strength, and so forth. Ideally, a single target b is achieved for all applications. Current reliability models do not account for the effects of material degradation under envi- ronmental factors or the expected changes in truck loading conditions over time. A significant amount of theoretical research work has been ongoing over the last 2 decades to develop time- dependent reliability models to account for the deterioration of concrete beams and the corrosion of steel bridge girders and their influence on member, as well as system strength. These efforts, however, have not yet matured to a level at which they can be applied in the LRFD specifications to help extend the useful life of the next generation of bridges and obtain good estimates of the safety and reliability of existing bridges subjected to harsh environments. The same is true with regard to the design for extreme events other than live loads. A proba- bilistic model for the consideration of ship collisions is based on calculating a nominal annual probability of failure that should not exceed 0.001 (see AASHTO 2009). However, the design criteria for limit states associated with other types of extreme events are based on previous generations of codes that were not based on reliability principles. In these cases, emphasis was placed on the hazard analysis of the load events without explicitly considering the uncertainties in the response of the bridge to these events and the ability of the bridge to withstand their effects. For example, recent proposals recommend using for design the earthquakes corresponding to a 1,000-year return period without explicitly accounting for the uncertainties associated with estimating the dynamic bridge response or the ability of a bridge system to resist the applied seismic ground motions (Imbsen 2007). Threats from floods are based on probabilistic models of flood occurrence without considering other modeling uncertainties (e.g., the bias and COV of scour prediction equations) and the parametric uncertainties associated with estimating discharge, flow depth, flow velocity, and so forth as discussed in Sections 2.2 and 2.3. Furthermore, existing bridge design codes propose different return periodsâand conse- quently various levels of conservatism (or safety factors)âfor different hazards. For example, the calibration of the live load factors in the AASHTO LRFD is based on the 75-year maximum load effect, the wind maps use 50-year maximum wind speeds, and a 1,000-year return period has been proposed for seismic hazards, whereas scour predictions are based on flood events having various return periods based on bridge size and level of service (see Table 2.3). Given that structural safety is related to both the magnitude of the hazard and the vulner- ability of the bridge elements to that hazard, the discrepancies in the current methods may not lead to consistent levels of reliability for the different hazards that a bridge may be sub- jected to. To account for many of these uncertainties, some specification writers have recom- mended the design of bridges for hazard levels corresponding to very high return periods. For example a 2,500-year return period was recommended for seismic hazards when using the traditional force-based design methods, whereas recent proposals have suggested the use of a 1,000-year return period in conjunction with a performance-based design approach (ATC/ MCEER 2002, Imbsen 2007). Although the use of different return periods to account for the different levels of conservatism and uncertainties associated with the analysis and design approaches is a valid approach for developing design codes, the determination of the code- specified design return period must be supported using probabilistic analyses of the overall safety of the structure.
26 Reference Guide for Applying Risk and Reliability-Based Approaches for Bridge Scour Prediction During the calibration of a new design code, the average reliability index from typical âsafeâ designs is used as the target reliability value for the new code. That is, a set of load and resistance factors as well as the nominal loads (or return periods for the design loads) are chosen for the new code such that bridges designed with these factors will provide reliability index values (b values) equal to the target value as closely as possible. For example, Nowak (1999) used a reliability index of b = 3.5 for the design of new bridge members. On the other hand, Moses (2001) used a reliability index of b = 2.5 for the evaluation of load capacity of existing bridges. Both targets are based on a generic set of load- and member-capacity statistical data bases that are believed to represent the most typical loading conditions and material properties. The differences between b = 3.5 (new bridges) and b = 2.5 (existing bridges) have been justified based on cost implications, given that the design of new bridges to higher safety standards would only marginally increase the cost of construction, whereas increasing the load capacity criteria for an existing bridge may require its replacement and lead to considerable costs. The lower safety cri- teria for existing bridges are, however, associated with strict requirements for regular inspection. Ghosn et al. (2003) found that existing design criteria for extreme events (other than scour) are associated with reliability index values that typically vary between b = 2.0 and b = 3.5. The calibration process discussed in this section does not contain any pre-assigned numeri- cal values for the target reliability index. This traditional approach to the calibration of LRFD criteria (e.g., AISC, AASHTO) has led code writers to choose different target reliabilities for different types of structural elements or for different types of loading conditions. For example, in the AISC LRFD, a target b equal to 3.5 was chosen for the reliability of beams in bending under the effect of dead and live loads, whereas a target b equal to 4.0 was chosen for the con- nections of steel frames under dead and live loads, and a target b equal to 2.5 may be chosen for the main members of a structure that is subject to earthquakes. A reliability index of b = 3.5 corresponds to a probability of limit state exceedance equal to 2.3 Ã 10-4, whereas a reliability index of b = 2.5 corresponds to a probability of exceedance equal to 6.2 Ã 10-3. Similarly, the USACE (1992) determined that probabilities of unsatisfactory conditions greater than 0.001 for inland navigation structures suggest that frequent outages for repair may occur and at a probability of 0.07 a need for extensive rehabilitation may be required. For struc- tures with even greater probabilities of limit state exceedance, emergency action is required to alleviate risks. Such differences in the target reliability index and associated probabilities clearly reflect the economic costs associated with the selection of the target b and the consequences of exceeding a limit state. Much progress has been made over the last 3 decades to apply reliability methods during the development of bridge design and evaluation specifications. However, current bridge design specifications and the equations used to model bridge behavior and member/system capaci- ties under various threats are inconsistent and are presented in a format that blurs the implicit levels of conservatism to the end users. Existing discrepancies in the design return periods and the methods used by the specifications to treat the different types of hazards, including scour, have to be overcome in order to address issues related to multi-hazard risk management and life cycle engineering principles (Ghosn et al. 2003).
