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1 Probabilistic Assessment of Local Scour in Bridge Piers under Changing 2 Environmental Conditions 3 A.N. Kallias 1, B. Imam 2, * 4 Department of Civil & Environmental Engineering, University of Surrey, Guildford, Surrey, 5 GU2 7XH, UK. 6 (Email: [email protected], 2 [email protected], Tel.: +44 1483 689679) 7 * Corresponding author 8 9 10 11 12 13 14 15 16 17 18 19 20 21 1 22 Probabilistic assessment of local scour in bridge piers under changing 23 environmental conditions 24 Scour is one of the most widespread causes of bridge failure worldwide. The magnitude 25 of the river flow at the bridge location is a key factor which directly affects the scour 26 hole depth. Climate change may cause changes in the flow characteristics in a river due 27 to changes in the precipitation patterns and catchment characteristics. In this paper, 28 statistical analysis of the expected maximum annual flow of rivers is combined with 29 Monte Carlo simulation to estimate the probability of local scour failure. Climate 30 change is assumed to manifest itself through gradual changes in the statistical 31 characteristics of the expected maximum annual flow distributions. Results are 32 presented from a case study using a bridge in the UK, which revealed that a time- 33 dependent increase in the mean of the expected maximum annual flow has a more 34 pronounced effect on scour performance as compared to an increase of its variability 35 alone. Amongst the cases examined, however, the most adverse effect on local scour 36 performance is observed from the simultaneous increase in both mean and variability of 37 the expected maximum annual flow. The results also highlighted the significance of the 38 foundation depth and local scour model parameter in relation to the changing flow 39 characteristics. 40 keywords: Climate change, local scour, bridge pier, probabilistic assessment, Monte 41 Carlo simulation, flow modelling. 42 43 Nomenclature 44 B is the river width (m) 45 D is the pier diameter (m) 46 DF is the foundation depth (m) 47 F0 is the Froude number 48 G(t) is the time-dependent safety margin 49 H2 is used to measure the heterogeneity of the pooling group 2 50 k is the shape parameter of the GEV distribution 51 K1 is a coefficient used to consider the influence of pier shape on the predicted scour 52 K2 is a coefficient used to consider the angle of attack on the predicted scour 53 K3 is a coefficient used to consider the streambed conditions on the predicted scour 54 K4 is a coefficient used to consider the material bed size on the predicted scour 55 N is the number of simulations in each year 56 n is the Manning’s coefficient 57 pf is the probability of scour failure. 58 Q is the flow magnitude (m3/s) 59 QMED is the expected maximum annual flow with return period T = 2 years 60 R is the resistance 61 s is the longitudinal slope of the channel 62 S(t) is the time-dependent load effect 63 V is the flow velocity (m/s) 64 y0 is the depth (m) of the flow upstream of the bridge pier 65 ymax is the maximum local scour depth (m) 66 α is the scale parameter of the GEV distribution 67 Γ is the gamma function 68 ε is the location parameter of the GEV distribution 69 λsc is a model parameter in the HEC-18 local scour equation 70 μ is the sample mean of the expected maximum annual flow 3 71 σ is the sample standard deviation of the expected maximum annual flow 72 1 Introduction 73 Scour is identified as one of most widespread causes of structural failure in bridges spanning 74 over rivers (Imam & Chryssanthopoulos, 2012; Imhof , 2004; Smith, 1976; Wardhana & 75 Hadipriono, 2003). Scour is characterised by the erosion/removal of (underwater) river bed 76 material in the vicinity of the piers and abutments (Melville & Coleman, 2000), leading to the 77 development of scour holes. Scour hole depths exceeding the foundation depth of the 78 piers/abutments can lead to structural instability and sudden bridge failure (Maddison, 2012; 79 Van Leeuwen & Lamb, 2014). A number of factors are associated with the scour depths 80 which may potentially develop at the pier and/or abutments of a bridge including, among 81 others, the geometrical characteristics of the pier/abutment, the river characteristics including 82 bed material and angle of attack as well as the flow magnitude at the bridge location. The 83 scour phenomenon has been extensively investigated and a number of – mainly empirical – 84 models are available which allow the quantification of scour depths considering the most 85 significant influencing factors such as pier geometry, river and flow characteristics (Breusers, 86 Nicollet & Shen, 1977; Melville, 1997; Melville & Coleman, 2000; Richardson & Davis, 87 2001; Sheppard, Melville & Demir, 2014). 88 The prediction of scour depths in practice involves significant uncertainty caused by 89 the variability associated with the different influencing factors, including bridge, river and 90 flow characteristics as well as the scour prediction models themselves, which have been 91 developed empirically through small-scale laboratory experiments. A source of uncertainty 92 which is becoming increasingly relevant when predicting future scour performance of bridges 93 is the potential influence of climate change. The increased risk of scour of bridges due to 94 climate change has been recognised worldwide, but only on a qualitative basis (DEFRA, 95 2012; DoT, 2005; TRB, 2008). The potential consequences of climate change on bridge scour 4 96 performance are currently unknown, on a quantitative basis, and need to be investigated to 97 assist the development and implementation of adaptation strategies (Meyer & Weigel, 2011). 98 Hence, capturing the aforementioned uncertainties in the analysis would allow a more 99 reliable performance assessment of scour prone bridges and assist towards more efficient 100 101 decision-making under conditions of uncertainty. Some previous studies examined the probability of scour failure, without, however, 102 considering the potential influence of climate change in performance assessment (Briaud, 103 Brandimarte, Wang, D’Odorico, 2007; Briaud, Gardoni & Yao, 2014; Deco & Frangopol, 104 2011; Johnson, 1992; Muzzammil & Siddiqui, 2009; NCHRP, 2003; Stein, Young, Trent & 105 Pearson, 1999). On the other hand, risk-based frameworks considering the impact of climate 106 risks on the long-term performance of civil engineering structures are becoming increasingly 107 relevant within the context of infrastructure management (Stewart & Deng, 2015). Very few 108 studies in the literature examine the potential effects of climate change on scour performance 109 of bridge piers. For instance, Kirshen et al. (2002) examined, on a deterministic basis, the 110 potential effects of climate change on scour performance by considering up to 30% assumed 111 increases in the magnitude of the 100 year flow. 112 An important parameter which directly affects the depth of the developing scour hole 113 is the magnitude of the flow which may potentially be encountered in a river at the bridge 114 location. The flow magnitude in a river is governed by a several factors such as catchment 115 characteristics, precipitation patterns, etc. Climate change is predicted to cause changes in the 116 river flow characteristics due to changes in the precipitation patterns and catchment 117 characteristics (IPCC, 2012; Robson, 2002; UKCP09, 2009). 118 A number of methods have been devised to predict the potential flow in a particular 119 river, for instance through statistical analysis of historic flow records or alternatively using 120 the rainfall-runoff method which requires knowledge of the precipitation patterns and the 5 121 catchment descriptors (Robson & Reed, 1999). The statistical analysis of historic flow 122 records through the widely-used WINFAP-FEH 3 software allows the estimation of the 123 statistical properties of the expected maximum annual flow for any river in the UK. 124 In this paper, a methodology is presented for the reliability assessment of scour prone 125 bridges considering the potential effects of climate change. In this paper, statistical analysis 126 of the expected maximum annual flow is combined with Monte Carlo simulation (MCS) to 127 compute the probability of scour failure, through scenario-based modelling. The Flood 128 Estimation Handbook (FEH) and the statistical software WINFAP-FEH 3 are used to 129 estimate the statistical properties of the expected maximum annual flow for the river where 130 the bridge is located. The potential influence of climate change on the flow characteristics is 131 considered through gradual changes in the mean and variability (i.e. standard deviation) of 132 the expected maximum annual flow distribution. The uncertainties associated with the 133 different factors which influence scour performance are taken into account through suitable 134 distributions. Results are presented from a case study using a bridge in the UK, in which 135 scour reliability profiles are presented for a number of scenarios which assume time- 136 dependent changes in the distribution of the expected maximum annual flow. Furthermore, 137 the influence of foundation depth and local scour model parameters are also investigated in 138 relation to the changing flow characteristics. 139 2 Climate change and river flow 140 It is broadly accepted in the literature that climate change due to anthropogenic emissions of 141 greenhouse gases (i.e. CO2) is taking place (IPCC, 2007; 2013). A noticeable increase in the 142 global temperature has been observed during the 20th century and it is expected to continue to 143 follow an increasing trend in the coming years (IPCC, 2013). The prediction of the future 144 temperature increase is associated with high uncertainty, since the future global CO2 6 145 emissions are determined by a number of factors, which themselves are highly volatile, 146 including among others, government policy, global population and economic growth and 147 emergence of new technologies (Climate Change Act, 2008; Lutz & Samir, 2010; Swiss Re, 148 2013a; WEF, 2012). 149 The built infrastructure, including buildings, transport systems and bridges can be 150 adversely affected by climate change (Kirshen, Ruth & Anderson, 2008; Meyer & Weigel, 151 2011; Posey, 2012; Kumar & Imam, 2013). Natural disasters such as hurricanes, extreme 152 precipitation and flooding can have major socioeconomic impacts, including among others 153 loss of life and damage to the built infrastructure (e.g. loss of service in transport and other 154 infrastructure systems, damage to buildings/bridges, etc.) leading to potentially significant 155 macro-economic effects (Lehner, Doll, Alcamo, Henrichs & Kaspar, 2006; Swiss Re, 2013b). 156 Alterations in the climatic and weather conditions due to climate change can potentially 157 increase the uncertainty associated with the magnitude as well as the prediction of extreme 158 weather events including extreme precipitation and river flows (IPCC, 2012). The available 159 evidence suggests that these changes may prevail as temporal changes in the statistical 160 properties and distribution of key climatic parameters such as temperature and precipitation 161 (IPCC, 2012; Katz, 1993). To enable both the quantification of the potential effects of climate 162 change and the development of future adaptation strategies, a number of potential emission 163 scenarios have been devised by IPCC which cover a period up to the end of the 21st Century 164 (IPCC, 2007; 2013). Increasing flood frequency and magnitude due to increasing 165 precipitation and/or changes to the catchment characteristics can have a significant effect on 166 the scour performance of bridges. 167 In the UK, during the years 1961-1995, an increase is observed in the extreme 168 precipitation events during winter (Osborn, Hulme, Jones & Basnett, 2000). A decreasing 169 annual trend of rainfall due to climate change in a particular area can include significant 7 170 increases in precipitation associated with seasonal trends (e.g. during winter). In general, the 171 climatic projections for the UK (UKCP09, 2009) indicate this situation. The impacts of 172 climate change, however, are catchment specific and their magnitude can differ remarkably 173 even for catchments located close to each other (Prudhomme, Kay, Crooks & Reynard, 174 2013). Climate change may also affect the vegetation type within an area (Walther et al., 175 2002); this in turn can influence the evapotranspiration and runoff characteristics of a 176 catchment (Peel, McMahon, Finlayson & Watson, 2002). Recently, increasing trends have 177 been observed in relation to flood frequency and magnitude in the UK (Prudhomme, Jakob & 178 Svensson, 2003); however, the findings of Robson (2002) suggest that these increasing trends 179 are associated with natural climatic variability rather than climate change. Despite these 180 observations, Robson (2002) suggested that climate change expressed as increases in rainfall 181 should not be ignored as a potential source of future increase of flooding in the UK. In his 182 study, the causes which limit the ability to identify trends in flow data are highlighted. 183 The preceding discussion indicates that at present it is difficult to precisely quantify 184 the effect of climate change in terms of precipitation and temperature changes on fluvial 185 flood frequency and magnitude. However, the results of several studies suggest that in some 186 areas flood frequency and magnitude will increase in the future (i.e. occurrence of extreme 187 events will become more frequent). In view of these limitations, this paper aims to quantify 188 the effect of potential changes of flood characteristics (i.e. changes in the frequency and 189 magnitude of the expected maximum annual flow due to climate change) on the probability 190 of local scour failure. Initially, statistical analysis of existing flow records is used to obtain 191 the distribution of the expected maximum annual flow (see section 4.2). Thereafter, gradual 192 changes are introduced to the statistical properties of the flow to account for the potential 193 effect of climate change, i.e. indirectly accounting for changes in the precipitation patterns 194 and/or catchment characteristics for a specific location. Results are presented from a case 8 195 study using a bridge in the UK to investigate the potential effects of changing mean and 196 variability of the expected maximum annual flow on the probability of local scour failure in 197 bridge piers. 198 3 Local scour prediction model 199 A number of models have been proposed in literature for the estimation of local scour in 200 bridge piers (Breusers, Nicollet & Shen, 1977; Melville, 1997; Melville & Coleman, 2000; 201 Richardson & Davis, 2001; for a review of local sour models see Sheppard, Melville, and 202 Demir (2014)). In this paper, local scour is estimated using the HEC-18 design equation 203 (Arneson, Zevenbergen, Lagasse & Clopper, 2012), given by equation (1), which considers 204 scour as a time-independent process, i.e. temporal effects of local scour development are not 205 modelled (Chang, Lai & Yen, 2004; Melville & Chiew, 1999). 206 ymax D 2 y0 K1 K 2 K 3 K 4 y0 0.65 F0 0.43 (1) 207 where, ymax is the maximum scour depth (m), y0 is the depth (m) of the flow upstream of the 208 bridge pier, K1, K2, K3 and K4 are coefficients which allow for pier shape, angle of attack, 209 streambed conditions and the river bed material size, D is the pier diameter and F0 is the 210 Froude number given by equation (2): 211 212 213 F0 V gy0 0.5 (2) where, g is the gravity acceleration and V is flow velocity given by equation (3): V Q By0 (3) 9 214 and the flow depth y0 is given by equation (4) (BD 97/12, 2012): nQ y0 1 2 Bs 215 35 (4) 216 where, Q is the flow (m3/s), B is the river width (m), n is the Manning’s coefficient and s is 217 the longitudinal slope of the channel. 218 K1 depends on the pier nose shape and it can take the following values: 1.1 for a 219 square nose, 1.0 for round nose or circular cylinder, 0.9 for a sharp nose. Coefficient K2 is a 220 function of the angle of attack of the river flow with respect to the pier (Arneson, 221 Zevenbergen, Lagasse & Clopper, 2012). K3 depends on whether there is clear-water scour or 222 the river bed is plane (K3=1.1) as opposed to the case of having dune bed configurations with 223 different dune heights (K3 ranging between 1.1 to 1.3 depending on the dune height). K4 224 depends on the diameter of the river bed material and can range between 0.4 (fine material) to 225 1.0 (coarse material). 226 Equation (4) can be used for wide rivers with ratios B/y0 exceeding about 10, giving 227 conservative predictions for cases where this is less than about 10 (BD 97/12, 2012). In this 228 paper, the flow Q is estimated using the statistical analysis procedures implemented in the 229 software WINFAP-FEH 3 (for more details see section 4.2 and 5.1 in this paper and (Robson 230 & Reed, 1999). 231 4 Probabilistic assessment of local scour 232 4.1 Framework for probabilistic analysis 233 Reliability analysis allows the estimation of the failure probability, pf, of a structure for 234 different limit states. The performance function for the limit state of a bridge is given by 235 equation (5): 10 G t R S t 236 (5) 237 where, G(t) is the time-dependent safety margin, R and S(t) are the resistance (i.e. foundation 238 depth) and time-dependent load effects (i.e. maximum scour depth given as a function of flow 239 magnitude, river and pier characteristics, etc.), respectively. The performance function for 240 local scour in bridge piers is given by equation (6): 241 0.65 D 0.43 G t DF ymax t DF 2 y0 t K1K 2 K 3 K 4 F t 0 y t 0 (6) 242 where, DF is the foundation depth. G(t) ≤ 0 indicates the failure realization of the limit state. 243 Using the statistical properties, including distribution type, for the random variables of 244 equation (6) and assuming that the resistance DF and the load effects ymax(t), are statistically 245 independent, the instantaneous (annual) probability of failure, pf(t), is given by equation (7): 246 0.65 D 0.43 p f t P G t 0 P DF 2 y0 t K1K 2 K3 K 4 F t 0 0 y0 t 247 In this paper, equation (7) is evaluated using Monte Carlo simulation (MCS) 248 implemented in MATLAB software. Alternatively, the availability of a closed form 249 expression for the limit state function enables the use of other reliability analysis methods 250 (e.g. FORM). The use of MCS, however, allows to estimate the failure probability as well as 251 to obtain the distribution of the scour depth (Eq. 1) – which is not known a-priory – for the 252 analysis cases examined. The cumulative (time-dependent) probability of failure, at any point 253 within a time period, is given by equation (8), provided that the failures are statistically 254 independent. (7) 11 k 255 256 p f 0, tL 1 1 P Gti t 0 i 1 (8) The uncertainties associated with the random variables of equation (7), can be 257 separated as aleatory and epistemic uncertainties (Frangopol & Liu, 2007; Kiureghian & 258 Ditlevsen, 2009; Merz & Thieken, 2005). Within the context of scour assessment under 259 changing flow magnitude (i.e. expected maximum annual flow) due to climate change, the 260 variability of flow magnitude is associated with both natural variability and epistemic 261 uncertainty. Aleatory uncertainty in relation to flow modelling is due to the lack of a specific 262 emission scenario for future climate change since climate change itself is a function of 263 several variables which are not always possible to be objectively quantified such as future 264 global population, government policy, technological breakthroughs etc. (see also Section 2 of 265 this paper). On the other hand, epistemic uncertainty is caused by the lack of understanding 266 of how a specific climate change scenario (e.g. increased precipitation, etc.) will affect the 267 flow conditions of a particular catchment. Furthermore, the available models for scour are 268 associated with epistemic uncertainty, since the inherently complex nature of scour is 269 modelled through approximations developed from laboratory experiments of small-scale pier 270 models (e.g. Equation 1). This uncertainty type can be reduced by developing more accurate 271 models of the phenomenon through, for instance, additional experimentation (this is 272 discussed later in this paper). 273 4.2 Procedure for probabilistic assessment local scour in piers 274 The Monte Carlo based procedure for the probabilistic assessment of local scour considering 275 the potential effects of climate change (see Figure 2) is implemented in MATLAB using a 276 sample size of N = 2×106 per year. Flow events in different years are assumed to be 277 independent. 12 278 The expected maximum annual flow, which is denoted here as QMED, is modelled by 279 fitting a suitable distribution to flood data using the statistical procedures implemented in 280 WINFAP-FEH 3 (Robson & Reed, 1999). This approach is based on the creation and analysis 281 of a pooling group of several catchments of similar hydrological characteristics, with the 282 available years of flow records for each station (in the different catchments) contributing to 283 the total number of station years. It should be noted that each water year in the UK starts on 284 the 1st October (Robson & Reed, 1999). Initially, the location of the bridge is established 285 using the maps of Flood Estimation Handbook (FEH) CD-ROM and the availability of 286 nearby stations is examined. In the case that a nearby station exists it can be used by the 287 WINFAP-FEH software to create the pooling group. In the case that no station is available, 288 the catchment descriptors of the selected area can be used instead (Fig. 1). A pooling group is 289 then created in the WINFAP-FEH software by specifying the required number of station 290 years (i.e. number of flow records in terms of years) to be analysed. The pooling group is 291 checked for heterogeneity using the heterogeneity measure H2. A pooling group is 292 considered homogeneous when the individual sites in the group follow the same distribution 293 when standardised by QMED (Robson & Reed, 1999). For H2 > 2, a revision of the pooling 294 group is recommended prior to the fitting of a suitable distribution (for more details see 295 Robson and Reed (1999)). 296 Equations (2)-(4) are used to calculate the Froude number, flow velocity and flow 297 depth, respectively, while equation (1) is used to compute the depth of local scour, for each 298 set of the N randomly generated values for the variables (e.g. expected maximum annual 299 flow, pier width, river slope, etc., see Table 1) in each year, equations (5)-(8) are used to 300 calculate the annual (instantaneous) and time-dependent (cumulative) failure probability for a 301 specific value of foundation depth. The foundation depth and the load effects y(t), i.e. the 302 scour depth, are assumed to be statistically independent. In actual bridges, the foundation 13 303 depth is governed by several factors, one of which is the maximum expected scour depth. 304 However, this correlation is not considered in the present analysis due to the lack of sufficient 305 information. The procedure (Figure 2) is demonstrated through a case study; the different 306 analysis cases examined are discussed in subsequent sections of this paper. 307 5 Bridge case study 308 5.1 Modelling of expected maximum annual flow 309 The bridge considered in this case study is assumed to be located on the river Earn in 310 Scotland, UK, assuming alluvial riverbed conditions (Gilvear & Black, 1999). Initially, the 311 location of the examined bridge is established on the maps of FEH CD-ROM 3 software and 312 the available nearby stations are identified. This station is then used in WINFAP-FEH 3 313 software to create the pooling group using stations from similar catchments with a total of 314 1000 station years (annual maxima series). During the revision of the pooling group the total 315 number of station years reduced to 827 due to removal of stations with unreliable records. 316 The value of the heterogeneity measure H2 indicates that the revised pooling group is 317 sufficiently homogeneous and no review is needed (i.e. standardised test value H2=0.0901). 318 WINFAP-FEH 3 provides a number of options for estimating QMED, which is the 319 maximum annual flow with a return period T = 2 years, for instance using the catchment 320 descriptors or annual maxima (AM) series (for more details see Robson and Reed (1999)). In 321 this paper, a QMED of 250.2 m3/s is estimated from AM series of the station. 322 Analysis of pooling group flood data (i.e. annual maxima series) using the WINFAP- 323 FEH 3 software indicates that the generalized extreme value (GEV) distribution, given by 324 equation (9) (Kottegoda & Rosso,1997), is the most suitable distribution for modelling the 325 magnitude of the expected maximum annual flow; the cumulative distribution function is 326 expressed as follows: 14 327 FX max 1k k x exp 1 (9) 328 where, k is the shape parameter, ε is the location parameter and α is the scale parameter. The 329 GEV distribution parameters obtained from the statistical analysis of stations in similar 330 catchments (i.e. pooling group) in WINFAP are α = 0.222, ε = 0.919 and k = 0.002. The scale 331 and location parameters of the GEV distribution are given by equations (10) and (11), 332 respectively (Kottegoda & Rosso,1997). 333 k 2 2 1 2k 2 1 k 334 1 1 k k (10) (11) 335 where, μ is the sample mean, σ is the sample standard deviation and k, ε and α are the shape, 336 location and scale parameters of the generalized extreme value distribution, respectively 337 (Kottegoda & Rosso,1997), and Γ is the gamma function which is defined by the following 338 integral: 339 340 x t x1et dt 0 for x > 0, otherwise x 0 (12) The potential effects of climate change on scour are examined through a parametric 341 study in which the scale and location parameters are gradually changed for increasing values 342 in the mean μ and variability (i.e. standard deviation σ) of the flood magnitude in equations 343 (10) and (11). The random variables and analysis scenarios considered are described next. 344 5.2 Random variables and analysis cases 345 Within the probabilistic framework developed, several variables associated with the bridge 15 346 and river characteristics are treated as random. Table 1 summarises the statistical properties 347 of the random variables considered in this case study. In this study, the bed material is 348 assumed to be deterministic and time invariant; however, over time, changing flow properties 349 may change the size of bed material. The river and pier dimensions (i.e. widths of the river 350 and pier, respectively) are also treated as random. Examination of historic drawings from 351 existing bridges have assisted the selection of the mean values for these variables, with the 352 selected bridge being considered as a representative example for this area. Actual 353 measurements taken during a site visit could allow for deterministic values to be used for 354 these variables. To this end, actual measurements on several (similar) piers within the same 355 bridge may be associated with some degree of variability. Similarly, the river width is not 356 constant and its value depends on the location that the measurement is obtained. 357 Climate change is likely to impact the precipitation patterns and catchment 358 characteristics of a specific area which in turn may affect the magnitude and frequency of 359 expected maximum annual flow. In this study, it is assumed that climate change will cause a 360 certain amount of temporal changes in the statistical properties of the flood predicted using 361 the FEH (e.g. 20% increase of sample mean). The different analysis cases examined in 362 relation to changing flow conditions are summarised in Table 2. These changes are assumed 363 to evolve linearly with time over a 60 year period. In this way, the effects of climate change 364 on the precipitation patterns and catchment descriptors and hence on the flood frequency and 365 magnitude are implicitly considered in the analysis of the different scenarios examined. A 366 foundation depth (FD) of 4.5m is assumed to facilitate the estimation of the scour failure 367 probability; the effect of this variable is investigated later in this paper. 16 368 6 Results and discussion 369 6.1 Scour depth evolution under changing expected maximum annual flow 370 Figures 3a to 3f show the distribution evolution in terms of their pdf and cdf of the expected 371 maximum annual flow normalised using the initial QMED (=250.2 m3/s) for the examined 372 scenarios; these cover the cases of increasing the flow mean up to 60% alone, increasing the 373 standard deviation (i.e. variability) up to 60% alone and simultaneously increasing the mean 374 and standard deviation up to 60%, respectively. Current design guidelines suggest a 20% 375 increase in the design flood, which is associated with a 200-year return period, as a means of 376 capturing the effects of climate change (BD 97/12). However, in the present study, the annual 377 maximum flow, which is associated with a much lower return period than the design flood, is 378 used. For this reason, a wide range of scenarios associated with the increase in the mean and 379 variability of the latter are being considered, i.e. up to 60% gradual increase in the long-term. 380 The parameters of the fitted GEV distribution for each scenario examined are also shown in 381 this figure. These results indicate that potential increase in the mean of the expected 382 maximum annual flow has a greater overall effect on the distribution compared to increasing 383 variability, see Figures 3a & b and 3c & d, respectively. This is because the gradual increase 384 of mean (equation (11)) causes the distribution area to gradually shift towards higher flow 385 values compared to the cases of increasing variability. This effect appears to be higher for the 386 case where the mean and variability of the expected maximum annual flow are assumed to 387 increase simultaneously (Figures 3e and 3f) compared to the individual effects of increasing 388 mean or increasing variability (see Figures 3a to 3d). 389 Further examination of the results in Figure 3 indicates that the gradual increase of 390 mean is accompanied by an increase of QMED (i.e. flow magnitude with 50% annual 391 probability of occurrence), while a slight reduction of QMED is observed for the cases of 392 increasing variability (see Figures 3a & b and 3c & d, respectively). It is interesting to note 17 393 that for the case of simultaneous increase of mean and variability, the increase of QMED is 394 slightly lower compared to the increase of QMED observed for the cases of increasing mean 395 alone (Figures 3a & b and 3e & f). 396 Figures 4a to 4f show the effect of the changing flow characteristics (shown in Figure 397 3) on the distribution evolution of the predicted scour depths (equations (1)-(4) and for the 398 scenarios of Table 2), with gradually increasing flow mean, standard deviation and the 399 simultaneous increase of both, respectively. The results indicate that in all cases examined the 400 variability associated with the predicted scour depths can be modelled using a lognormal 401 distribution (the fitted distributions are shown in Figure 4); this implies that the predicted 402 scour values are always on the positive side which is a meaningful result from a physical 403 point of view. 404 The results in Figure 4a & b indicate that the gradually increasing mean of the 405 expected maximum annual flow (Figures 3a & b) causes a gradual shift of the predicted scour 406 depth distribution towards higher (and more unfavourable in relation to scour performance) 407 values compared to the initial distribution of the base line case which assumes no changes in 408 the mean and variability of the flow. In the case where a gradual increase of the flow 409 variability is assumed (Figures 3c & d), the results in Figure 4c & d indicate that the 410 predicted scour depths (as expected) also exhibit increased variability. More specifically, an 411 increase of the areas of the distribution tails is observed, which indicates an increased 412 probability of observing larger scour depths. It is interesting to note that the results in Figure 413 4c & d indicate that the increased variability also causes an increase in the probability of 414 observing smaller scour depths compared to the base line case which assumes no changes in 415 the mean and variability. The results in Figure 4e & f indicate that the simultaneous gradual 416 increase of mean and variability causes (as expected) a shift in the distribution towards higher 417 scour depth values. In this case, the magnitude of the predicted maximum scour depths (i.e. 18 418 upper tail of the distribution) is higher compared to the case of gradually increasing mean 419 alone (Figures 4a & b). The effect of gradually changing statistical properties of the expected 420 maximum annual flow on the probability of pier scour failure is discussed next. 421 6.2 Effect of changing flow characteristics on pier scour failure probability 422 Figures 5, 6 and 7 show the annual (instantaneous) and time-dependent (cumulative) 423 probabilities of pier scour failure for the scenarios examined with increasing mean, increasing 424 variability and simultaneous increase of mean and variability in the expected maximum 425 annual flow, respectively, for a given foundation depth equal to 4.5m. In all figures, (A) and 426 (C) refer to annual and cumulative failure probability, respectively. The effect of foundation 427 depth on the predictions is investigated in a later section of this paper. 428 The results in Figure 5 show the effect of increasing mean (equation (10)) in the 429 expected maximum annual flow on the annual and cumulative probabilities of failure for the 430 scenarios with mean increases of 20%, 40% and 60% over a 60 year period. The results in 431 this figure indicate that the effect of increasing mean is relatively small for the initial 10 years 432 and it gradually becomes noticeable between 10 and 20 years and significant beyond the 433 initial 20 year period. At the end of the examined period the cumulative probabilities of 434 failure for the scenarios with 20%, 40% and 60% increase in mean (i.e. see Figure 5 for cases 435 45M2000, 45M4000 and 45M6000; in this nomenclature, 45 stands for foundation depth of 436 4.5m, M stands for Model and 2000 stands for 20% increase in mean and 0% increase in 437 variability) are predicted to be 59%, 176% and 337%, respectively, higher than the base line 438 scenario which assumes no changes in the statistical properties of the expected maximum 439 annual flow over time. 440 The results in Figure 6 show the effect of increasing variability (equation (10)) in the 441 expected maximum annual flow on the annual and cumulative probabilities of failure for the 19 442 scenarios with variability increases of 20%, 40% and 60% over a 60 year period. The results 443 in this figure indicate that the effect of increasing variability is relatively small for the initial 444 15 years and it gradually becomes significant beyond the initial 20 year period. For the 445 analysis cases examined, the effect of increasing variability in the expected maximum annual 446 flow is predicted to have a relatively smaller effect on the failure probabilities compared to 447 the previous case of assuming a gradually increasing mean. In general, the results in Figure 6 448 follow a similar trend to the results obtained for increasing mean (Figure 5); that is as the 449 variability of the expected maximum annual flow increases, the probability of failure also 450 increases (cumulative failure probabilities at the end of the 60 year period are predicted to 451 increase by approximately 41%, 92% and 153% for cases 45M0020, 45M4040 and 452 45M0060, respectively relative to the base line case 45M0000, see Figure 6). 453 The results in Figure 7 show the effect of simultaneously increasing mean and 454 standard deviation up to 60% (see equations (10) and (11)) of the expected maximum annual 455 flow on the annual and cumulative probabilities of failure for scenarios 45M2020, 45M4040 456 and 45M6060 (see Table 2). The results in this figure indicate that the combined effect is 457 relatively small for the initial 10 year period and it gradually becomes significant beyond 15 458 years. For the analysis cases examined, the effect of simultaneous increase of mean and 459 variability has the highest relative effect on the predicted probabilities of local scour failure. 460 More specifically, the results for cases 45M2020, 45M4040 and 45M6060 show that at the 461 end of the 60 year period, an increase in the cumulative failure probabilities by approximately 462 117%, 332% and 633%, respectively, compared to the base-line case 45M0000. It is 463 interesting to note that the combined effects of increasing mean and variability associated 464 with the maximum expected annual flow (i.e. analysis cases 45M2020, 45M4040 and 465 45M6060) is greater than the sum of the individual effects (i.e. cases where the increasing 466 mean and variability are considered separately), see Figures 5-7. Based on these observations 20 467 it can be concluded that the combined effects of increasing mean and variability have the 468 greatest effect on the probability of local scour failure. When considering the impact of 469 increasing mean or variability individually, increasing mean in the maximum expected annual 470 flow is predicted to have a more significant (adverse) effect on the predictions compared to 471 the increasing variability. 472 The cumulative failure probability profiles have been developed assuming that the 473 flow characteristics occur gradually (linearly) over a 60 year time period. It would be 474 interesting to investigate the effect of potentially sudden changes in the statistical properties 475 of the expected maximum annual flow – due to climate change – on the scour probability of 476 failure. Furthermore, the preceding analyses assume that the distribution type (i.e. generalised 477 extreme value) of the expected maximum annual flow remains the same throughout the 478 examined period. At present, however, it is not possible to confidently predict the timing and 479 magnitude of potentially sudden changes in the flow characteristics that may occur in the 480 future (IPCC. 2012; Robson, 2002). One of the key challenges that still remain is to establish 481 a link between the altered precipitation patterns and/or catchment properties (due to climate 482 change) with the flow (i.e. expected maximum annual flow) characteristics. To this end, 483 significant uncertainty still exists as to which climate change scenario will be realised. 484 Recently developed quantitative tools on climatic projections (for example the UKCP09 485 interactive tool in (UKCP09, 2009) provides climatic projections covering the entire UK) can 486 assist the ongoing, e.g. (Dikanski, 2014) and future studies in establishing a more direct link 487 between the expected climatic conditions in the future and the expected flow characteristics 488 at a specific location. 489 6.3 Effect of foundation depth on scour probability of failure 490 A common challenge with existing bridges is that the foundation depth is often unknown 21 491 even if the original drawings of the superstructure are available (RSSB, 2005). The results 492 presented in the previous sections have been obtained by assuming, conservatively, a 493 foundation depth of 4.5m. A number of additional analysis cases are considered to investigate 494 the influence of assuming foundation depth, due to having unknown foundations, on the 495 cumulative probability of scour failure. The foundation depths examined are 4m and 5m. 496 These foundations depths are examined for a number of cases which assume increasing mean 497 or/and variability of the expected maximum annual flow distribution; these are: (a) a case of 498 40% increase in mean, (b) 40% increase of variability and (c) simultaneous increase of mean 499 and variability by 40%. 500 Figure 8 shows the cumulative probabilities of scour failure for the analysis cases 501 with varying assumed foundation depths. As expected, the results in this figure clearly show 502 the influence of foundation depth on the predicted cumulative pf; with smaller foundation 503 depths having higher probabilities of failure. For the case of FD = 4m the effect of changes in 504 the statistical properties of the flow (i.e. increasing mean or/and variability) have a very small 505 effect on the cumulative probabilities of scour failure. The results in Figure 8 indicate that, 506 for the FD=4m case, scour failure becomes almost certain during the last 5 years (of the 60 507 year period) for the cases with increasing mean and simultaneous increase of mean and 508 variability of the expected maximum annual flow distribution. 509 As the foundation depth increases the probabilities of scour failure decrease as shown 510 in Figure 8. The results in figure further indicate that the effect of increasing mean and/or 511 variability is not constant when different foundations depths are considered. For example, for 512 the case of FD = 5m, the increasing variability of the expected maximum annual flow has a 513 greater effect than the increasing mean. Conversely, the increasing mean of the expected 514 maximum annual flow distribution has a greater effect compared to the effect of increasing 515 variability for decreasing foundation depths. As shown in Figure 8, the simultaneous increase 22 516 of mean and variability of the flow distribution produces the highest probabilities of failure in 517 all cases examined. The influence of the simultaneous increase of mean and variability of the 518 expected maximum annual flow reduces for decreasing foundation depths. 519 These results indicate that foundation depth has a significant effect on the predictions. 520 In practice, this variable is deemed with high uncertainty while in many cases no data is 521 available on the actual foundation type and depth of a particular bridge (JBA, 2004). In such 522 cases, conservative values of FD are recommended in assessing scour performance (JBA, 523 2004). To this end, the systematic collection of actual foundation depth measurements of 524 piers in scour prone bridges would reduce the uncertainty and hence improve the accuracy of 525 the scour failure predictions during assessments. 526 6.4 Effect of model parameter λsc on predicted scour depths and failure probabilities 527 It has been shown that the scour equation (equation (1)) of HEC-18 leads to conservative 528 predictions of local scour (Landers & Mueller, 1996; NCHRP, 2003). This is due to the fact 529 that the scour prediction models used in codes of practice have been developed through 530 small-scale laboratory experiments. Comparisons of these prediction models with field 531 measurements of scour depths on real bridges have shown that there is a discrepancy between 532 them. It has been suggested that a model parameter λsc can be introduced in equation (1) to 533 reduce its conservatism (NCHRP, 2003) inherently taking into account epistemic uncertainty. 534 The local scour model is now given by equation (13): 535 536 D ymax t 2sc y0 t K1K 2 K3 K 4 y0 t 0.65 F0 t 0.43 (13) Several values have been proposed for the statistical properties of λsc; for a summary 537 see (NCHRP, 2003). To investigate the influence of this parameter a number of scenarios are 538 considered using values for the statistical properties of λsc suggested in (NCHRP, 2003). The 23 539 analysis cases in this section assume a 40% gradual increase in mean, standard deviation and 540 the simultaneous increase of both for the expected maximum annual flow considering two 541 foundation depths of 4.5m and 5m. The model parameter λsc is modelled using a normal 542 distribution with mean = 0.55 and COV = 52% (NCHRP, 2003). In this way comparisons can 543 be made with results presented in previous sections. 544 Figure 9a and 9b show the predicted time-dependent failure probabilities for the 545 analysis cases with foundation depths 4.5m and 5m, respectively. The results in these figures 546 indicate that although the mean value of λsc is less than 1 (i.e. mean value of λsc = 0.55), the 547 introduction of this model parameter causes an increase of the predicted time-dependent 548 failure probabilities. This observation can be explained by considering the influence of λsc on 549 the predicted scour depths in Figures 10 and 11. More specifically, the results in Figures 10 550 and 11 indicate that the high variability associated with model parameter λsc results in an 551 increase of the scour depth values exceeding 4.5m and 5m (generated using MCS in 552 MATLAB). As expected, the mean value of the predicted scour depths is lower than the 553 predicted scour depths when λsc = 1. The results in Figures 9a and 9b also indicate that the 554 very high variability of the model parameter λsc (COV = 0.52) overshadows the effect of the 555 assumed increasing variability in the expected maximum annual flow on the predicted time- 556 dependent probabilities of failure (i.e. in Figure 9 the increasing variability of flow is 557 predicted to have an insignificant effect on cumulative pf). 558 Figures 10 and 11 show the influence of different statistical properties and distribution 559 types for λsc on the predicted scour depths. The results indicate that negative scour depths are 560 predicted in the cases where λsc is modelled using normal distribution, which is not 561 acceptable from a physical point of view. In contrary, λsc remains positive when a lognormal 562 distribution is used for its modelling; in this case however the magnitudes of the predicted 24 563 maximum scour depths (near the upper tail) are significantly larger than the cases where a 564 normal distribution is used. Based on the results presented in this section it can be concluded that the use of λsc, 565 566 which is a source of epistemic uncertainty, has a significant effect on the predictions. 567 However, at present the available statistical properties for this variable are not consistent (e.g. 568 negative or very large scour depths) with actual observations. To this end, further research is 569 needed to obtain accurate statistical properties including the distribution type for this variable 570 through additional field measurements and comparison with the code predictions. 571 7 Conclusions 572 In this paper, statistical analysis of the expected maximum annual flow of rivers is combined 573 with Monte Carlo simulation (MCS) to estimate the probability of failure due to local scour 574 exceeding the foundation depth of bridge piers. Climate change is assumed to manifest itself 575 through gradual changes in the statistical properties (i.e. changes in the mean and variability) 576 of the expected maximum annual flow. Suitable distributions are used to model the 577 uncertainties associated with the different factors (i.e. pier and river characteristics) 578 influencing local scour performance. Results are presented from a case study using a bridge 579 in the UK, in which a number of scenarios to investigate the potential effect of changing flow 580 characteristics on the probability of scour failure are examined using the probabilistic 581 analysis procedure. The salient conclusions of this study are summarised as follows: 582 An increase on the mean of the expected maximum annual flow has a greater effect 583 on its distribution compared to an increase in its variability (standard deviation). The 584 gradual increase of mean results in a gradual shift of the distribution area towards 585 higher flow values compared to the case of increasing the variability. This effect 25 586 appears to be higher for the case where the mean and variability of the expected 587 maximum annual flow are assumed to increase simultaneously. 588 A gradual increase in the mean of the expected maximum annual flow was found to 589 result in a gradual shift of the predicted scour depth distribution towards higher 590 values. In the case where a gradual increase of the flow variability is assumed, the 591 results indicate that the predicted scour depths also exhibit increased variability. In the 592 case of simultaneous gradual increase of mean and variability, the magnitude of the 593 predicted maximum scour depths (i.e. upper tail of the distribution) is higher 594 compared to the case of gradually increasing the mean alone. 595 The results indicate that the effects of gradually changing statistical properties of the 596 expected maximum annual flow (i.e. increasing mean, variability and combined mean 597 and variability) on the predicted probabilities of scour failure is relatively small for 598 the initial 10-15 years. Beyond this initial period their effect on the predicted 599 probabilities of failure becomes significant, with the cases of simultaneous increase in 600 mean and variability of the flow having the greatest impact on the predictions. 601 The foundation depth was found to have a significant effect on the probability of 602 scour failure; that is as the foundation depth increases the probability of failure 603 decreases. 604 The results of the case study presented in this paper indicate that the effects of 605 changing flow characteristics on the scour failure probabilities are predicted to reduce 606 with reducing foundation depths. More specifically, when considering the effect of 607 foundation depth in conjunction with changing flow characteristics on the scour 608 failure probability, the results showed that the effects of increasing mean and/or 609 variability of the maximum expected annual flow was more significant in the cases 610 with deeper foundation depths are assumed in the analysis. 26 611 The use of scour model parameter λsc was found to have a significant effect on scour 612 depth predictions and associated probabilities of failure. However, at present the 613 available statistical properties for this variable appear to be unreliable since the use of 614 this variable can lead in some cases to negative or very large scour depths. 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K2) 1.0 - - - Pier width, D (m) 2 0.05 Normal Assumed 801 802 803 804 805 806 807 808 809 810 811 812 33 Table 2: Analysis cases for assessing the influence of assumed changes in the expected maximum annual flow. Foundation Mean, Standard deviation, depth (m) μ (Eq. 11) σ (Eq. 10) Analysis case 45M0000 No change 45M2000 20% increase no change 45M4000 40% increase no change 45M6000 60% increase no change No change 20% increase 45M0040 No change 40% increase 45M0060 No change 60% increase 45M2020 20% increase 20% increase 45M4040 40% increase 40% increase 45M6060 60% increase 60% increase 45M0020 4.5 813 814 815 816 817 818 819 820 821 822 823 824 34 825 Figure captions 826 Fig. 1: Procedure for obtaining the statistical properties of the expected maximum annual 827 flow using the FEH and WINFAP-FEH 3. 828 Fig. 2: Procedure for probabilistic assessment of local scour in bridge piers. 829 Fig. 3: Monte Carlo simulation results showing the PDF and CDF of the expected maximum 830 annual flow distribution normalised using the initial QMED (= 250.2 m3/s) for: (a) & (b) 831 increasing mean, (c) & (d) increasing variability (standard deviation, σ) and (e) & (f) 832 simultaneous increase of mean and variability. 833 Fig. 4: Monte Carlo simulation results showing the PDF and CDF of the predicted local scour 834 depths for changing flow characteristics: (a) & (b) increasing mean, (c) & (d) increasing 835 standard deviation (σ) and (e) & (f) simultaneous increase of mean and variability (standard 836 deviation, σ). 837 Fig. 5: Effect of increasing mean in the expected maximum annual flow on the annual (A) 838 and cumulative time-dependent (C) probabilities of scour failure. 839 Fig. 6: Effect of increasing variability (standard deviation, σ) in the expected maximum 840 annual flow on the annual (A) and cumulative time-dependent (C) probabilities of scour 841 failure. 842 Fig. 7: Effect of simultaneous increase of mean and variability in the expected maximum 843 annual flow on the annual (A) and cumulative time-dependent (C) probabilities of scour 844 failure. 845 Fig. 8: Effect of foundation depth on time-dependent probability of scour failure under 846 changing expected maximum annual flow. 847 Fig. 9: Effect of model parameter λsc on the time-dependent probabilities of local scour failure 848 for up to 40% increase in the mean or/and variability of the expected maximum annual flow 849 distribution: (a) foundation depth = 4.5m and (b) foundation depth = 5m. 850 Fig. 10: Effect of model parameter λsc (equation (13)) on the predicted scour depths for cases 851 with no changes in the distribution of the expected maximum annual flow. 35 852 Fig. 11: Effect of model parameter λsc (equation (13)) on the predicted scour depths for cases 853 with 40% increase in the mean of the expected maximum annual flow distribution. 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 36