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Probabilistic Assessment of Local Scour in Bridge Piers under Changing
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Environmental Conditions
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A.N. Kallias 1, B. Imam 2, *
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Department of Civil & Environmental Engineering, University of Surrey, Guildford, Surrey,
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GU2 7XH, UK.
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(Email: [email protected], 2 [email protected], Tel.: +44 1483 689679)
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*
Corresponding author
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Probabilistic assessment of local scour in bridge piers under changing
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environmental conditions
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Scour is one of the most widespread causes of bridge failure worldwide. The magnitude
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of the river flow at the bridge location is a key factor which directly affects the scour
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hole depth. Climate change may cause changes in the flow characteristics in a river due
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to changes in the precipitation patterns and catchment characteristics. In this paper,
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statistical analysis of the expected maximum annual flow of rivers is combined with
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Monte Carlo simulation to estimate the probability of local scour failure. Climate
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change is assumed to manifest itself through gradual changes in the statistical
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characteristics of the expected maximum annual flow distributions. Results are
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presented from a case study using a bridge in the UK, which revealed that a time-
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dependent increase in the mean of the expected maximum annual flow has a more
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pronounced effect on scour performance as compared to an increase of its variability
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alone. Amongst the cases examined, however, the most adverse effect on local scour
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performance is observed from the simultaneous increase in both mean and variability of
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the expected maximum annual flow. The results also highlighted the significance of the
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foundation depth and local scour model parameter in relation to the changing flow
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characteristics.
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keywords: Climate change, local scour, bridge pier, probabilistic assessment, Monte
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Carlo simulation, flow modelling.
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Nomenclature
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B is the river width (m)
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D is the pier diameter (m)
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DF is the foundation depth (m)
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F0 is the Froude number
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G(t) is the time-dependent safety margin
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H2 is used to measure the heterogeneity of the pooling group
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k is the shape parameter of the GEV distribution
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K1 is a coefficient used to consider the influence of pier shape on the predicted scour
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K2 is a coefficient used to consider the angle of attack on the predicted scour
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K3 is a coefficient used to consider the streambed conditions on the predicted scour
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K4 is a coefficient used to consider the material bed size on the predicted scour
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N is the number of simulations in each year
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n is the Manning’s coefficient
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pf is the probability of scour failure.
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Q is the flow magnitude (m3/s)
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QMED is the expected maximum annual flow with return period T = 2 years
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R is the resistance
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s is the longitudinal slope of the channel
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S(t) is the time-dependent load effect
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V is the flow velocity (m/s)
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y0 is the depth (m) of the flow upstream of the bridge pier
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ymax is the maximum local scour depth (m)
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α is the scale parameter of the GEV distribution
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Γ is the gamma function
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ε is the location parameter of the GEV distribution
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λsc is a model parameter in the HEC-18 local scour equation
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μ is the sample mean of the expected maximum annual flow
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σ is the sample standard deviation of the expected maximum annual flow
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1 Introduction
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Scour is identified as one of most widespread causes of structural failure in bridges spanning
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over rivers (Imam & Chryssanthopoulos, 2012; Imhof , 2004; Smith, 1976; Wardhana &
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Hadipriono, 2003). Scour is characterised by the erosion/removal of (underwater) river bed
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material in the vicinity of the piers and abutments (Melville & Coleman, 2000), leading to the
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development of scour holes. Scour hole depths exceeding the foundation depth of the
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piers/abutments can lead to structural instability and sudden bridge failure (Maddison, 2012;
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Van Leeuwen & Lamb, 2014). A number of factors are associated with the scour depths
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which may potentially develop at the pier and/or abutments of a bridge including, among
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others, the geometrical characteristics of the pier/abutment, the river characteristics including
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bed material and angle of attack as well as the flow magnitude at the bridge location. The
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scour phenomenon has been extensively investigated and a number of – mainly empirical –
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models are available which allow the quantification of scour depths considering the most
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significant influencing factors such as pier geometry, river and flow characteristics (Breusers,
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Nicollet & Shen, 1977; Melville, 1997; Melville & Coleman, 2000; Richardson & Davis,
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2001; Sheppard, Melville & Demir, 2014).
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The prediction of scour depths in practice involves significant uncertainty caused by
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the variability associated with the different influencing factors, including bridge, river and
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flow characteristics as well as the scour prediction models themselves, which have been
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developed empirically through small-scale laboratory experiments. A source of uncertainty
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which is becoming increasingly relevant when predicting future scour performance of bridges
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is the potential influence of climate change. The increased risk of scour of bridges due to
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climate change has been recognised worldwide, but only on a qualitative basis (DEFRA,
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2012; DoT, 2005; TRB, 2008). The potential consequences of climate change on bridge scour
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performance are currently unknown, on a quantitative basis, and need to be investigated to
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assist the development and implementation of adaptation strategies (Meyer & Weigel, 2011).
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Hence, capturing the aforementioned uncertainties in the analysis would allow a more
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reliable performance assessment of scour prone bridges and assist towards more efficient
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decision-making under conditions of uncertainty.
Some previous studies examined the probability of scour failure, without, however,
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considering the potential influence of climate change in performance assessment (Briaud,
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Brandimarte, Wang, D’Odorico, 2007; Briaud, Gardoni & Yao, 2014; Deco & Frangopol,
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2011; Johnson, 1992; Muzzammil & Siddiqui, 2009; NCHRP, 2003; Stein, Young, Trent &
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Pearson, 1999). On the other hand, risk-based frameworks considering the impact of climate
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risks on the long-term performance of civil engineering structures are becoming increasingly
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relevant within the context of infrastructure management (Stewart & Deng, 2015). Very few
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studies in the literature examine the potential effects of climate change on scour performance
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of bridge piers. For instance, Kirshen et al. (2002) examined, on a deterministic basis, the
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potential effects of climate change on scour performance by considering up to 30% assumed
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increases in the magnitude of the 100 year flow.
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An important parameter which directly affects the depth of the developing scour hole
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is the magnitude of the flow which may potentially be encountered in a river at the bridge
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location. The flow magnitude in a river is governed by a several factors such as catchment
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characteristics, precipitation patterns, etc. Climate change is predicted to cause changes in the
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river flow characteristics due to changes in the precipitation patterns and catchment
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characteristics (IPCC, 2012; Robson, 2002; UKCP09, 2009).
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A number of methods have been devised to predict the potential flow in a particular
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river, for instance through statistical analysis of historic flow records or alternatively using
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the rainfall-runoff method which requires knowledge of the precipitation patterns and the
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catchment descriptors (Robson & Reed, 1999). The statistical analysis of historic flow
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records through the widely-used WINFAP-FEH 3 software allows the estimation of the
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statistical properties of the expected maximum annual flow for any river in the UK.
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In this paper, a methodology is presented for the reliability assessment of scour prone
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bridges considering the potential effects of climate change. In this paper, statistical analysis
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of the expected maximum annual flow is combined with Monte Carlo simulation (MCS) to
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compute the probability of scour failure, through scenario-based modelling. The Flood
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Estimation Handbook (FEH) and the statistical software WINFAP-FEH 3 are used to
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estimate the statistical properties of the expected maximum annual flow for the river where
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the bridge is located. The potential influence of climate change on the flow characteristics is
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considered through gradual changes in the mean and variability (i.e. standard deviation) of
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the expected maximum annual flow distribution. The uncertainties associated with the
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different factors which influence scour performance are taken into account through suitable
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distributions. Results are presented from a case study using a bridge in the UK, in which
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scour reliability profiles are presented for a number of scenarios which assume time-
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dependent changes in the distribution of the expected maximum annual flow. Furthermore,
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the influence of foundation depth and local scour model parameters are also investigated in
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relation to the changing flow characteristics.
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2 Climate change and river flow
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It is broadly accepted in the literature that climate change due to anthropogenic emissions of
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greenhouse gases (i.e. CO2) is taking place (IPCC, 2007; 2013). A noticeable increase in the
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global temperature has been observed during the 20th century and it is expected to continue to
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follow an increasing trend in the coming years (IPCC, 2013). The prediction of the future
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temperature increase is associated with high uncertainty, since the future global CO2
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emissions are determined by a number of factors, which themselves are highly volatile,
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including among others, government policy, global population and economic growth and
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emergence of new technologies (Climate Change Act, 2008; Lutz & Samir, 2010; Swiss Re,
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2013a; WEF, 2012).
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The built infrastructure, including buildings, transport systems and bridges can be
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adversely affected by climate change (Kirshen, Ruth & Anderson, 2008; Meyer & Weigel,
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2011; Posey, 2012; Kumar & Imam, 2013). Natural disasters such as hurricanes, extreme
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precipitation and flooding can have major socioeconomic impacts, including among others
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loss of life and damage to the built infrastructure (e.g. loss of service in transport and other
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infrastructure systems, damage to buildings/bridges, etc.) leading to potentially significant
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macro-economic effects (Lehner, Doll, Alcamo, Henrichs & Kaspar, 2006; Swiss Re, 2013b).
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Alterations in the climatic and weather conditions due to climate change can potentially
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increase the uncertainty associated with the magnitude as well as the prediction of extreme
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weather events including extreme precipitation and river flows (IPCC, 2012). The available
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evidence suggests that these changes may prevail as temporal changes in the statistical
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properties and distribution of key climatic parameters such as temperature and precipitation
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(IPCC, 2012; Katz, 1993). To enable both the quantification of the potential effects of climate
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change and the development of future adaptation strategies, a number of potential emission
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scenarios have been devised by IPCC which cover a period up to the end of the 21st Century
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(IPCC, 2007; 2013). Increasing flood frequency and magnitude due to increasing
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precipitation and/or changes to the catchment characteristics can have a significant effect on
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the scour performance of bridges.
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In the UK, during the years 1961-1995, an increase is observed in the extreme
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precipitation events during winter (Osborn, Hulme, Jones & Basnett, 2000). A decreasing
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annual trend of rainfall due to climate change in a particular area can include significant
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increases in precipitation associated with seasonal trends (e.g. during winter). In general, the
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climatic projections for the UK (UKCP09, 2009) indicate this situation. The impacts of
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climate change, however, are catchment specific and their magnitude can differ remarkably
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even for catchments located close to each other (Prudhomme, Kay, Crooks & Reynard,
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2013). Climate change may also affect the vegetation type within an area (Walther et al.,
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2002); this in turn can influence the evapotranspiration and runoff characteristics of a
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catchment (Peel, McMahon, Finlayson & Watson, 2002). Recently, increasing trends have
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been observed in relation to flood frequency and magnitude in the UK (Prudhomme, Jakob &
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Svensson, 2003); however, the findings of Robson (2002) suggest that these increasing trends
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are associated with natural climatic variability rather than climate change. Despite these
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observations, Robson (2002) suggested that climate change expressed as increases in rainfall
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should not be ignored as a potential source of future increase of flooding in the UK. In his
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study, the causes which limit the ability to identify trends in flow data are highlighted.
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The preceding discussion indicates that at present it is difficult to precisely quantify
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the effect of climate change in terms of precipitation and temperature changes on fluvial
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flood frequency and magnitude. However, the results of several studies suggest that in some
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areas flood frequency and magnitude will increase in the future (i.e. occurrence of extreme
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events will become more frequent). In view of these limitations, this paper aims to quantify
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the effect of potential changes of flood characteristics (i.e. changes in the frequency and
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magnitude of the expected maximum annual flow due to climate change) on the probability
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of local scour failure. Initially, statistical analysis of existing flow records is used to obtain
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the distribution of the expected maximum annual flow (see section 4.2). Thereafter, gradual
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changes are introduced to the statistical properties of the flow to account for the potential
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effect of climate change, i.e. indirectly accounting for changes in the precipitation patterns
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and/or catchment characteristics for a specific location. Results are presented from a case
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study using a bridge in the UK to investigate the potential effects of changing mean and
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variability of the expected maximum annual flow on the probability of local scour failure in
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bridge piers.
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3 Local scour prediction model
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A number of models have been proposed in literature for the estimation of local scour in
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bridge piers (Breusers, Nicollet & Shen, 1977; Melville, 1997; Melville & Coleman, 2000;
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Richardson & Davis, 2001; for a review of local sour models see Sheppard, Melville, and
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Demir (2014)). In this paper, local scour is estimated using the HEC-18 design equation
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(Arneson, Zevenbergen, Lagasse & Clopper, 2012), given by equation (1), which considers
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scour as a time-independent process, i.e. temporal effects of local scour development are not
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modelled (Chang, Lai & Yen, 2004; Melville & Chiew, 1999).
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ymax
D
 2 y0 K1 K 2 K 3 K 4  
 y0 
0.65
F0 0.43
(1)
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where, ymax is the maximum scour depth (m), y0 is the depth (m) of the flow upstream of the
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bridge pier, K1, K2, K3 and K4 are coefficients which allow for pier shape, angle of attack,
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streambed conditions and the river bed material size, D is the pier diameter and F0 is the
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Froude number given by equation (2):
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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)
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and the flow depth y0 is given by equation (4) (BD 97/12, 2012):
 nQ 
y0   1 2 
 Bs 
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(4)
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where, Q is the flow (m3/s), B is the river width (m), n is the Manning’s coefficient and s is
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the longitudinal slope of the channel.
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K1 depends on the pier nose shape and it can take the following values: 1.1 for a
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square nose, 1.0 for round nose or circular cylinder, 0.9 for a sharp nose. Coefficient K2 is a
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function of the angle of attack of the river flow with respect to the pier (Arneson,
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Zevenbergen, Lagasse & Clopper, 2012). K3 depends on whether there is clear-water scour or
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the river bed is plane (K3=1.1) as opposed to the case of having dune bed configurations with
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different dune heights (K3 ranging between 1.1 to 1.3 depending on the dune height). K4
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depends on the diameter of the river bed material and can range between 0.4 (fine material) to
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1.0 (coarse material).
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Equation (4) can be used for wide rivers with ratios B/y0 exceeding about 10, giving
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conservative predictions for cases where this is less than about 10 (BD 97/12, 2012). In this
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paper, the flow Q is estimated using the statistical analysis procedures implemented in the
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software WINFAP-FEH 3 (for more details see section 4.2 and 5.1 in this paper and (Robson
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& Reed, 1999).
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4 Probabilistic assessment of local scour
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4.1 Framework for probabilistic analysis
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Reliability analysis allows the estimation of the failure probability, pf, of a structure for
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different limit states. The performance function for the limit state of a bridge is given by
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equation (5):
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G t   R  S t 
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(5)
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where, G(t) is the time-dependent safety margin, R and S(t) are the resistance (i.e. foundation
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depth) and time-dependent load effects (i.e. maximum scour depth given as a function of flow
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magnitude, river and pier characteristics, etc.), respectively. The performance function for
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local scour in bridge piers is given by equation (6):
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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)
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where, DF is the foundation depth. G(t) ≤ 0 indicates the failure realization of the limit state.
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Using the statistical properties, including distribution type, for the random variables of
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equation (6) and assuming that the resistance DF and the load effects ymax(t), are statistically
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independent, the instantaneous (annual) probability of failure, pf(t), is given by equation (7):
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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  







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In this paper, equation (7) is evaluated using Monte Carlo simulation (MCS)
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implemented in MATLAB software. Alternatively, the availability of a closed form
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expression for the limit state function enables the use of other reliability analysis methods
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(e.g. FORM). The use of MCS, however, allows to estimate the failure probability as well as
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to obtain the distribution of the scour depth (Eq. 1) – which is not known a-priory – for the
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analysis cases examined. The cumulative (time-dependent) probability of failure, at any point
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within a time period, is given by equation (8), provided that the failures are statistically
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independent.
(7)
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k
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

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
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separated as aleatory and epistemic uncertainties (Frangopol & Liu, 2007; Kiureghian &
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Ditlevsen, 2009; Merz & Thieken, 2005). Within the context of scour assessment under
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changing flow magnitude (i.e. expected maximum annual flow) due to climate change, the
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variability of flow magnitude is associated with both natural variability and epistemic
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uncertainty. Aleatory uncertainty in relation to flow modelling is due to the lack of a specific
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emission scenario for future climate change since climate change itself is a function of
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several variables which are not always possible to be objectively quantified such as future
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global population, government policy, technological breakthroughs etc. (see also Section 2 of
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this paper). On the other hand, epistemic uncertainty is caused by the lack of understanding
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of how a specific climate change scenario (e.g. increased precipitation, etc.) will affect the
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flow conditions of a particular catchment. Furthermore, the available models for scour are
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associated with epistemic uncertainty, since the inherently complex nature of scour is
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modelled through approximations developed from laboratory experiments of small-scale pier
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models (e.g. Equation 1). This uncertainty type can be reduced by developing more accurate
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models of the phenomenon through, for instance, additional experimentation (this is
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discussed later in this paper).
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4.2 Procedure for probabilistic assessment local scour in piers
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The Monte Carlo based procedure for the probabilistic assessment of local scour considering
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the potential effects of climate change (see Figure 2) is implemented in MATLAB using a
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sample size of N = 2×106 per year. Flow events in different years are assumed to be
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independent.
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The expected maximum annual flow, which is denoted here as QMED, is modelled by
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fitting a suitable distribution to flood data using the statistical procedures implemented in
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WINFAP-FEH 3 (Robson & Reed, 1999). This approach is based on the creation and analysis
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of a pooling group of several catchments of similar hydrological characteristics, with the
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available years of flow records for each station (in the different catchments) contributing to
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the total number of station years. It should be noted that each water year in the UK starts on
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the 1st October (Robson & Reed, 1999). Initially, the location of the bridge is established
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using the maps of Flood Estimation Handbook (FEH) CD-ROM and the availability of
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nearby stations is examined. In the case that a nearby station exists it can be used by the
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WINFAP-FEH software to create the pooling group. In the case that no station is available,
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the catchment descriptors of the selected area can be used instead (Fig. 1). A pooling group is
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then created in the WINFAP-FEH software by specifying the required number of station
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years (i.e. number of flow records in terms of years) to be analysed. The pooling group is
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checked for heterogeneity using the heterogeneity measure H2. A pooling group is
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considered homogeneous when the individual sites in the group follow the same distribution
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when standardised by QMED (Robson & Reed, 1999). For H2 > 2, a revision of the pooling
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group is recommended prior to the fitting of a suitable distribution (for more details see
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Robson and Reed (1999)).
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Equations (2)-(4) are used to calculate the Froude number, flow velocity and flow
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depth, respectively, while equation (1) is used to compute the depth of local scour, for each
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set of the N randomly generated values for the variables (e.g. expected maximum annual
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flow, pier width, river slope, etc., see Table 1) in each year, equations (5)-(8) are used to
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calculate the annual (instantaneous) and time-dependent (cumulative) failure probability for a
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specific value of foundation depth. The foundation depth and the load effects y(t), i.e. the
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scour depth, are assumed to be statistically independent. In actual bridges, the foundation
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depth is governed by several factors, one of which is the maximum expected scour depth.
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However, this correlation is not considered in the present analysis due to the lack of sufficient
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information. The procedure (Figure 2) is demonstrated through a case study; the different
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analysis cases examined are discussed in subsequent sections of this paper.
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5 Bridge case study
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5.1 Modelling of expected maximum annual flow
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The bridge considered in this case study is assumed to be located on the river Earn in
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Scotland, UK, assuming alluvial riverbed conditions (Gilvear & Black, 1999). Initially, the
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location of the examined bridge is established on the maps of FEH CD-ROM 3 software and
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the available nearby stations are identified. This station is then used in WINFAP-FEH 3
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software to create the pooling group using stations from similar catchments with a total of
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1000 station years (annual maxima series). During the revision of the pooling group the total
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number of station years reduced to 827 due to removal of stations with unreliable records.
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The value of the heterogeneity measure H2 indicates that the revised pooling group is
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sufficiently homogeneous and no review is needed (i.e. standardised test value H2=0.0901).
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WINFAP-FEH 3 provides a number of options for estimating QMED, which is the
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maximum annual flow with a return period T = 2 years, for instance using the catchment
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descriptors or annual maxima (AM) series (for more details see Robson and Reed (1999)). In
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this paper, a QMED of 250.2 m3/s is estimated from AM series of the station.
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Analysis of pooling group flood data (i.e. annual maxima series) using the WINFAP-
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FEH 3 software indicates that the generalized extreme value (GEV) distribution, given by
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equation (9) (Kottegoda & Rosso,1997), is the most suitable distribution for modelling the
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magnitude of the expected maximum annual flow; the cumulative distribution function is
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expressed as follows:
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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
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GEV distribution parameters obtained from the statistical analysis of stations in similar
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catchments (i.e. pooling group) in WINFAP are α = 0.222, ε = 0.919 and k = 0.002. The scale
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and location parameters of the GEV distribution are given by equations (10) and (11),
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respectively (Kottegoda & Rosso,1997).

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k 2 2
 1  2k    2 1  k 
 
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
1   1  k  
k
(10)
(11)
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where, μ is the sample mean, σ is the sample standard deviation and k, ε and α are the shape,
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location and scale parameters of the generalized extreme value distribution, respectively
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(Kottegoda & Rosso,1997), and Γ is the gamma function which is defined by the following
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integral:
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
  x    t x1et dt
0
for x > 0, otherwise   x   0
(12)
The potential effects of climate change on scour are examined through a parametric
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study in which the scale and location parameters are gradually changed for increasing values
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in the mean μ and variability (i.e. standard deviation σ) of the flood magnitude in equations
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(10) and (11). The random variables and analysis scenarios considered are described next.
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5.2 Random variables and analysis cases
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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
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of the random variables considered in this case study. In this study, the bed material is
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assumed to be deterministic and time invariant; however, over time, changing flow properties
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may change the size of bed material. The river and pier dimensions (i.e. widths of the river
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and pier, respectively) are also treated as random. Examination of historic drawings from
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existing bridges have assisted the selection of the mean values for these variables, with the
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selected bridge being considered as a representative example for this area. Actual
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measurements taken during a site visit could allow for deterministic values to be used for
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these variables. To this end, actual measurements on several (similar) piers within the same
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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   2sc 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. To this
615
end, further research would facilitate the estimation of statistical properties including
616
distribution type for this variable.
617
618
8 Acknowledgements
619
The authors would like to acknowledge the financial support provided by EPSRC (UK) through Grant
620
EP/I00744X/1. We would also like to thank Professor Marios Chryssanthopoulos for constructive
621
discussions.
622
623
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776
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784
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786
787
788
789
790
791
792
793
794
795
796
797
798
32
799
Tables
800
Table 1: Statistical properties of variables.
Bridge piers
River
Variables
Mean
COV
Distribution
Reference
Width (m)
65
0.05
Normal
Assumed
Streambed conditions (Coef. K3)
1.1
0.05
Uniform
NCHRP (2013)
Bed material size (K4)
1.0
-
-
-
Slope
0.0032
0.05
Lognormal
Assumed
Manning’s coefficient
0.035
0.28
Lognormal
NCHRP (2013)
Foundation depth (m)
4.5
-
-
Assumed
Pier nose shape (Coef. K1)
1.0
-
-
-
Angle of attack (Coef. 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