Predicting scour at river bridge abutments over time - USMredac.eng.usm.my/html/publish/2015_01.pdf · conducted on compound and uniform abutments under clear water ... In the clear

Predicting scour at river bridgeabutments over time&1 Reza Mohammadpour PhDQ1

Post-doctoral researcher, River Engineering and Urban DrainageResearch Centre (REDAC), Universiti Sains Malaysia, EngineeringCampus, Seri Ampangan, Nibong Tebal, Penang, Malaysia;Department of Civil Engineering, Estahban Branch, Islamic AzadUniversity, Estahban, Iran

&2 Aminuddin Ab. GhaniProfessor and Deputy Director, REDAC, Universiti Sains Malaysia,Engineering Campus, Seri Ampangan, Nibong Tebal, Penang,Malaysia

&3 Nor Azazi ZakariaProfessor and Director, REDAC, Universiti Sains Malaysia, EngineeringCampus, Seri Ampangan, Nibong Tebal, Penang, Malaysia

&4 Thamer Ahmed Mohammed AliProfessor, Department of Civil Engineering, Faculty of Engineering,Universiti Putra Malaysia, UPM Serdang, Selangor, Malaysia

1 2 3 4

Good prediction of scour depth over time is essential for the design of river bridge foundations. In this study, two

empirical models are developed to predict the temporal variation of local scour at uniform-section bridge abutments

and those with enlarged bases, known as ‘compound’ abutments. The concepts of the principal vortex and the

volumetric rate of sediment transport theory are employed to develop the models. The critical bed-shear stress at the

scour hole is chosen as a criterion to determine the equilibrium scour depth. To validate the models, experiments are

conducted on compound and uniform abutments under clear water conditions. A comparison between the calculated

and observed data indicates that the models are able to predict the temporal variation of the scour depth with

reasonable accuracy, particularly for abutments with short lengths.

NotationQ3

Ast cross-sectional area of principal vortex withinscour hole during scour process at time t

At total area of the primary vortexA0 cross-sectional area of principal vortex before

scouringB width of abutmentBf width of foundationBu ???C coefficient??d50 median size of sedimentg acceleration of gravity;u*t ????? (= (τt/ρ)

0·5)st excess dimensionless tractive force at time tDv diameter of principal vortexdes equivalent scour depthdse equilibrium scour depthdst,I, dst,II, dst,III instantaneous scour depths for case I, case II

and case IIIe porosity of sedimentK ???????????L abutment length

Lf foundation lengthn time stepqst volumetric sediment transport rate per unit

movable widthqwt sediment transport rate in weight per unit

movable width of scour holeqvt ?????????????qvtc scour volume per unit width of scourte equilibrium timeU mean velocity upstream of obstacle, defined as

discharge divided by flow cross-sectional areaUc critical velocity of sedimentu* shear velocity of approach flowuc* critical shear velocity of bed sedimentu*t temporal variation of bed shear velocity at any

time t at nose of abutmentx, z time-dependent coordinates of scour holey flow depthZf foundation levelα correlation factorβ constantγs ????? (= ρsg)

1

Water ManagementVolume 00 Issue WM00

Predicting scour at river bridge abutmentsover timeMohammadpour, Ab. Ghani, Zakaria andMohammed Ali

Proceedings of the Institution of Civil EngineersWater Management 00 July 2015 Issue WM00Pages 1–16 http://dx.doi.org/10.1680/wama.14.00136Paper 1400136Received 09/11/2014 Accepted 21/05/2015Keywords:Q2

ICE Publishing: All rights reserved

Δ ??????? (= (ρs−ρ)/ρ)ρs mass density of sedimentρ mass density of waterσ standard deviationτc sediment critical shear stressτt bed-shear stress at abutment nose at time tØ sediment angle of repose

1. IntroductionLocal scour around a pier and abutment is the main reason forbridge failure. Local scour is caused by the development ofthree-dimensional vortices around the abutment. An accurateprediction of the scour depth at the abutment is very importantfor the design of bridges because it affects the determination ofthe foundation depth. Underestimation of the scour depth canlead to a shallow foundation depth, thus providing a chance toexpose the foundation to the scour hole, which is dangerousfor bridge safety. However, overestimation of the scour depthresults in an uneconomic bridge design (Kumar and Kothyari,2012).

Recently, different techniques using empirical, mathematicaland soft computing methods have been recommended to solveproblems in civil engineering (Mohammadpour et al., 2013a,2015). The temporal variation of local scour around abutmentsand piers is an essential aspect of hydraulic engineering (Deyand Barbhuiya, 2005). Therefore, a robust technique is necess-ary for accurate prediction of local scour. The temporal vari-ation of scour depth around uniform piers was investigated byChabert and Engeldinger (1956),Q4 Shen et al. (1969), Yanmazand Altinbilek (1991), Kothyari et al. (1992) andMohammadpour et al. (2014b). The most complete work wasconducted by Ettema (1980) with different types of sediments.Melville and Chiew (1999) developed a method for predictingthe temporal variation of scour at piers based on the data ofEttema (1980) and their new experimental data.Mohammadpour et al. (2013c) investigated the temporal vari-ation of local scour at short abutments, L/y<1 (where L and yare the length and the flow depth, respectively). They reportedthat approximately 80–90% of the maximum scour depthoccurred during the initial 20–40% of the equilibrium time.Recently, the significance of the temporal scour has beenfurther emphasised over the equilibrium scour depth (Ahmedand Rajaratnam, 2000; Ballio and Orsi, 2001; Ballio et al.,2010; Cardoso and Bettess, 1999; Chang et al., 2004; Kothyariand Ranga Raju, 2001; Mohammadpour et al., 2011; Olivetoand Hager, 2002).

Numerous studies have been conducted to estimate the scourdepth at abutments (Ballio and Orsi, 2001; Cardoso and Fael,2010; Coleman, 2005; Coleman et al., 2003; Mia and Nago,2003; Mohammadpour et al., 2013b; Oliveto and Hager, 2002;Yanmaz and Kose, 2007). These studies have mostly focusedon abutments with uniform cross-sections (or geometry).

However, for technical and economic reasons, actual abut-ments are constructed in the piles group with the foundation(Ataie-Ashtiani et al., 2010; Sheppard and Glasser, 2004). Theflow pattern around the compound abutment/pier (or pier) isdifferent from those around uniform abutments/piers Q5(Kumarand Kothyari, 2012). A few studies have been undertaken topredict the equilibrium depth of the local scour at compoundpiers (Coleman, 2005; Ferraro et al., 2013; Q6Jones et al., 1992;Melville and Coleman, 2000; Melville and Raudkivi, 1996;Mia and Nago, 2003; Mohammadpour et al., 2014a; Morenoet al., 2012; Parola et al., 1996; Sheppard and Glasser, 2004). Q7

Even fewer studies are available related to the temporal vari-ation of scour around compound piers (Kumar et al., 2003; Luet al., 2011; Melville and Raudkivi, 1996). Melville andRaudkivi (1996) developed a method to predict the maximumscour depth around compound piers. Mia and Nago (2003)proposed a semi-empirical model to estimate the temporalvariation of scour at uniform circular piers. This method wasmodified by Lu et al. (2011) to calculate the temporal scourdepth at compound piers with unexposed foundations. Theyused the volumetric rate of the sediment transport equation ofYalin (1977) and the concept of primary vortex (Kothyariet al., 1992) to develop the model.

Despite the extensive use of compound abutments, there islimited information about the temporal variation of localscour around compound abutments. A compound abutmentincludes a rectangular abutment placed on top of a larger rec-tangular foundation with or without piles. However, if theeffect of the foundation is integrated into the estimation of thescour depth, the conservative approach would be unnecessary.Therefore, an attempt to predict the temporal variation oflocal scour at compound abutments is necessary.

The main objective of this study is to develop two empiricalmethods for the prediction of temporal variation of local scourat uniform and compound abutments. The models were devel-oped based on the concept of the horseshoe vortex and thevolumetric rate of sediment transport theory. To verify the pro-posed models, experiments were conducted under clear waterconditions and the computed results were compared to theobserved data.

2. Experimental set-upAll of the experiments were conducted at the REDAC PhysicalModelling Laboratory of the Universiti Sains Malaysia. Theexperimental investigations were conducted in a flume of 6·0 mlong, 0·6 m wide and 0·6 m deep. The flume was equipped witha 0·25 m deep sand recess through the channel with severalfloatable screens located at the flume inlet to reduce the flowturbulence. A camera was fixed inside the transparent abut-ment to monitor the progress of the scouring process. Thescour depth was measured by a ruler that was fixed at theabutment nose, and a picture was taken by the camera every4min (Figure 1).

2



PROOFS

Three short, uniform abutments and four short, compoundabutments were used to determine the time variation of localscour (Table 1). For the short abutments, a ratio of L/y<1 wasselected for both the uniform and compound abutments(Figure 2). For this type of abutment, Melville (1992) showedthat the depth of flow has no effect on the scour depth.The discharge was controlled by a valve at the flume inlet andwas measured using a calibrated V-notch weir at the flumeoutlet.

For all of the experiments, uniform sand was used with d50=0·60 mm and a geometric standard deviation of σ=1·2. Tomaintain clear water conditions, the mean flow velocity wasset close to the critical velocity of the sediment (Uc), where Ucwas estimated using the Shields diagram and the expressionsgiven by Melville and Coleman (2000). At the end of eachexperiment, the water was drained off carefully without anydisturbance in the local scour. A point gauge with an accuracyof ±1mm was used to measure the topography of the scourhole around the abutments.

3. Test durationIn the clear water conditions, the equilibrium scour depth isachieved asymptotically with time (Melville, 1992; Melvilleand Chiew, 1999). Because the development of the scour holenever stops (Oliveto and Hager, 2002), several definitions wererecommended for equilibrium scour depth from previousstudies. In the practical tests, the duration of the experimentswas selected based on other criteria. Kumar et al. (1999)stopped their experiments when the scour depth did notchange by more than 1mm over a period of 3 h. Mia andNago (2003) stopped their tests when the variation of thescour depth was less than 1mm over 1 h. Ferraro et al. (2013)showed that the different durations are attributed to the differ-ent thicknesses and locations of the pile cap. However, in acomplex pier, a higher scouring rate and a longer time to reachequilibrium were observed when a thicker pile cap was used.

In this study, long duration experiments were conducted forthe uniform abutment of AB 2 (~67 h) and AB 3 (~84 h).Mohammadpour et al. (2013b) showed that more than 90% ofthe maximum scour depth around an abutment can beobtained after 42 h (2500 min). However, after this time, thevariation of the scour depth was observed to be less than1 mm for each time interval of 7 h; this criterion was also used

Flow

Figure 1. Transparent abutment and ruler with camera(Mohammadpour, 2013)

Experiment no. Abutment type

Foundation Abutment

Flow depth: cm Length ratio: L/LfLf: cm Bf: cm Zf: cm L: cm B: cm

AB 1 Uniform — — — 4·0 — 11·1 —

AB 2 Uniform — — — 5·5 — 11·0 —

AB 3 Uniform — — — 7·0 — 9·7 —

FA 21 Compound 5·5 11·0 1·5, 3·5, 5·0 4·0 0·73 10·5<y<13·1 0·73FA 33 Compound 9·0 18·0 1, 5, 7 7·0 0·78 10·5<y<13·1 0·78FA 42 Compound 12·0 24·0 1, 5, 7, 8 5·5 0·46 10·5<y<13·1 0·46FA 43 Compound 12·0 24·0 3, 5, 7, 8 7·0 0·58 10·5<y<13·1 0·58

Table 1. Abutment geometry characteristics for the present study

Bf

B f

B

B

Bu

L

L

Lf

Lf

Figure 2. Compound vertical wall abutment (Mohammadpour,2013)

3



PROOFS

for the complex pier. As shown in Table 2, different durationsare attributed to the different foundation locations.

4. Scour mechanism at uniform andcompound abutments

The flow patterns around abutments are very complex, and theflow complexity increases with the development of the scourhole (Ghodsian and Vaghefi, 2009). The flow pattern aroundan abutment can be categorised into four components: a prin-cipal vortex, wake vortices, a down-flow and a secondaryvortex (Figure 3).

The vortices around the abutment develop the local scourin three parts: in front of, to the side of and downstream ofthe abutment (Dey and Barbhuiya, 2005). The down-flow at theupstream face of the abutment and the principal vortex at thebase of the abutment are the main causes for the development

of the scour hole. The shape of the principal vortex is ellipticalwith inner and outer core regions (Kwan and Melville, 1994).The wake vortices are created downstream of the abutment dueto the separation of flow around the abutment. The secondaryvortices rotate in the opposite direction of the primary vorticesand develop a small hole on the outside of the initial scourhole. Before the presence of the foundation in the scour hole,the trend of scouring at a compound abutment is similar to thatof a uniform abutment. However, if the foundation appears intoscour hole Q8at a compound abutment, it weakens the vortices infront of the abutment and postpones the scour process. Duringthis time, the flow develops the scour hole parallel to and infront of the abutment.

Over time, the scour hole in front of the abutment is graduallyenlarged, especially at the foundation nose, and a shallowgroove develops around the foundation. Subsequently, thescour depth increases due to the formation of another vortexupstream from the foundation.

5. Scour zonesTo investigate the effects of the foundation level (Zf ) on a com-pound abutment, three locations (cases) were considered for thefoundation level. In case I, the foundation level was locatedbelow the equilibrium scour depth (Figure 4). In case II, thescour depth reached the top of the foundation. In thiscase, the principal vortices were weakened by the foundation.In case III, the top of the foundation was located within thescour hole. In Figure 4, the instantaneous scour depths for

Case Experiment U: cm/s U/Uc Duration: min L: cm Zf: cm L/y Zf/L dse: cm dse/L

I AB 1 27·6 0·96 3052 4 — 0·36 — 6·7 1·68I AB 2 28·6 0·98 4004 5·5 — 0·5 — 9·5 1·73I AB 3 26·5 0·95 5080 7 — 0·72 — 11·8 1·68II FA 21 27·6 0·96 3000 4 1·5 0·36 0·38 6 1·50III FA 21 27·7 0·96 2500 4 3·5 0·36 0·88 3·5 0·88III FA 21 29·7 0·98 2133 4 5 0·38 1·25 5 1·25II FA 33 27·2 0·95 2136 7 1 0·64 0·14 12·1 1·73II FA 33 28·5 0·98 2316 7 5 0·66 0·71 10·3 1·47II FA 33 28·3 0·97 2822 7 7 0·61 1·00 7·5 1·07II FA 42 28·4 0·97 2262 5·5 1 0·46 0·20 8·7 1·58II FA 42 29·4 0·99 2905 5·5 5 0·54 0·91 7·3 1·33III FA 42 29·4 0·99 3033 5·5 7 0·46 1·27 7 1·27III FA 42 28·1 0·98 3206 5·5 8 0·46 1·45 8 1·45II FA 43 28·9 0·95 2248 7 3 0·62 0·43 14·5 2·07II FA 43 28 0·97 2641 7 5 0·65 0·71 14·1 2·01III FA 43 28 0·97 3430 7 7 0·64 1·00 7 1·00III FA 43 28·3 0·96 3498 7 8 0·63 1·14 8 1·14III FA 43 27·8 0·95 3512 7 9 0·62 1·29 9 1·29

Table 2. Summary of the experimental results for the presentstudy

Secondary vortex

Principal vortex

Down-flowWake vortex

Flow

Figure 3. Flow pattern around an abutment

4



PROOFS

case I, case II and case III are denoted by dst,I, dst,II and dst,III,respectively.

Case I will occur when the abutment has a small dimension orthe foundation is located at a high level (under the equilibriumscour depth, dse). In this case, the compound abutment issimilar to the uniform abutment with a maximum scour depthof 2L (Melville, 1992). In case II, the foundation limits thedevelopment of scouring and the instantaneous scour depth(dst,II) remains constant at the foundation level (Zf ).

Mohammadpour (2013) indicated that this case can beobserved in the range of 1<Zf/L<2 and the scour depth isapproximately equal to the foundation level (ds~Zf ).

In case III, the foundation exposes into Q9the scour hole, and thefoundation size (Lf ) influences the scour hole. This conditionwas observed for 0≤Zf/L<1.

6. Proposed model for uniform abutmentThe empirical model was developed based on the concept ofthe horseshoe vortex at the abutment. Referring to the methodproposed by Mia and Nago (2003), the bed-load sedimenttransport equation of Yalin (1977) can be employed tocompute the time variation of local scour at uniform and com-pound abutments. The sediment transport rate at any time isdefined by the following equation

1:qwt

Δγsd50u�t¼ β st 1� 1

a stln 1þ a stð Þ

� �

where qwt= the sediment transport rate in weight per unitmovable width of the scour hole; Δ=(ρs−ρ)/ρ; ρs= the massdensity of the sediment; ρ=the mass density of water; γs=ρs g;d50= the median size of the sediment; g= the acceleration ofgravity; u*t=(τt/ρ)

0·5; τt=the bed-shear stress at the abutmentnose at time t; β=a constant to be determined; and st=anexcess dimensionless tractive force at time t that can beexpressed by (Lu et al., 2011)

2: st ¼ u�tu�c

� �2

� 1

where u*c=(τc/ρ)0·5 and τc= the sediment critical shear stress.

The value of a can be obtained from the following equation(Lu et al., 2011):

3: a ¼ 2�45 u�cðρs=ρÞ0�4

ffiffiffiffiffiffiffiffiffiffiffiffiΔgd50

p

The volumetric sediment transport rate per unit movable width(qst) can be determined as

4: qst ¼ qwt ð1� eÞΔγs

where e=the porosity of the sediment and is a fraction of thevoid volume over the total volume. The qst can be expressedusing Equations 1 and 4 as

5: qst ¼ β ð1� eÞ d50u�tst 1� 1a st

ln 1þ a stð Þ� �

Flow

B

B

LLf

L

Bf

Lf

Bf

A A

B

B

(a)

(b)

(c)

Abutment

Foundation

Foundation

Section B–B

Section A–A

Flow

Plan

Initial bed level

Initial bed level

Zf

Case ICase IICase III

Case I

Case II

Case IIIZf

dst,III dst,IIdst,I

∅

∅

dst,IIIdst,IIdst,I

Abutment

y

Figure 4. Three cases for compound abutments below the initialbed: (a) plan view; (b) section A-A; (c) section B-B

5



PROOFS

In this study, the scour hole around the abutment was approxi-mated by half of a frustum of the inverted cone, where theangle of the frustum is equal to the sediment angle of repose(Ø). The shape of the scour hole can remain almost constantand just become larger with respect to time with a diameter atthe base equal to the abutment length (L). However, the volu-metric expressions of the scour hole can be derived with regardto this approximation (Mia and Nago, 2003; Yanmaz andAltinbilek, 1991). As shown in Figure 5, the volume of thescour hole around a uniform abutment was calculated as

6: Vvt ¼ π

2

ðdst0

x2dz� BLdst

where B=the width of the abutment; L=the abutment length;and x and z are time-dependent coordinates of the scour hole.Writing x in terms of z, as shown in Figure 5, and integratingleads to

7: Vvt ¼ π L2

2dst þ π L

2 tan Ød2st þ

π

6 tan2 Ød3st � BLdst

The surface width of the scour hole at time t is equal todst/tan Ø (Figure 5). Therefore, the volume of the scour holeper unit movable width of the scour hole at time t can bedetermined as

8: qvt ¼ Vvt

dst= tan Øð Þ

¼ π

6 tan Ød2st þ

π L2

dst þ π L2

2tan Ø � BL tanØ

The solution of this equation for dst is expressed as:

9: dst ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6 tanØ

πqvt þ 6BL

π� 3L2

4

� �tan2Ø

s� 3L

2tan Ø

where qvt should be defined. As shown in Figure 6, A0 is thecross-sectional area of the principal vortex before scouring,and Ast is the cross-sectional area of the principal vortexwithin the scour hole during the scour process at time t. Then,the total area of the primary vortex (At) can be expressed as:

10: At ¼ A0 þ Ast

It is assumed that the shape of the primary vortex is circularbefore scour (Dey and Barbhuiya, 2005; Kothyari et al., 1992);therefore

11: A0 ¼ π

4D2

v

Barbhuiya and Dey (2003) recommended the followingequation to calculate Dv at vertical wall abutments

12: Dv ¼ 0�295L yL

� �0�205

and

13: Ast ¼ 12

d2st

tan Øþ 0�087 dst

tan Ø

where Dv=the diameter of the principal vortex; y=flow depth;and L=the abutment length.

The presence of obstacles is well known to increase the sensi-tivity of the motion of sediments in their vicinity. Chabert andEngeldinger (1956) suggested that no scour would occur at apier for U/Uc<0·5, where U is the mean velocity upstreamfrom the obstacle, defined as the discharge divided by the flowcross-sectional area, and Uc is the critical velocity of the sedi-ment. Chiew (1995) suggested that the critical value of U/Uc is0·3. Hager and Oliveto (2002) showed that this value dependson the flow contraction. Fael et al. (2006) indicated that thecritical flow for scour inception at the abutment decreases withincreasing L/y, and for a long abutment with L/y>5·56, thisvalue is smaller than 0·5. Cardoso et al. (2002) corroboratedthe conclusion of Chabert and Engeldinger (1956) for shortabutments. In the present study, all of the experiments wereconducted for short abutments; therefore, the value of U/Uc≥

0·5 was chosen for scour inception.

Z

X

X

L

X = L + Z/tan∅

dst /tan ∅

dst

dse

dse /tan ∅

∅

∅

Figure 5. Geometric definition of scour hole around a uniformabutment

6



PROOFS

The scour initiates at a vertical wall abutment when u*≥0·5uc*(Dey and Barbhuiya, 2005), where u*= the shear velocity ofthe approach flow and uc*=the critical shear velocity of thebed sediment. To estimate the flow shear velocity, the logarith-mic mean flow velocity for a rough bed can be expressed as

14:Uu�

¼ 5�75 logy

2d50þ 6

where U=the mean velocity and y=the flow depth. The flowvelocity and shear stress around short abutments (L/y<1) wereinvestigated byQ10 Ahmed and Rajaratnam (2000). They foundthat the bed shear stress increases substantially near the abut-ment and reaches a maximum value at the abutment nose. Inaddition, before the scour begins (or when At=A0), the shearstress at the nose of the abutment (τ*t) was estimated to beequal to 3·63 times the upstream bed shear stress (τ*).Therefore, with respect to u�t=u� ¼

ffiffiffiffiffiffiffiffiffiffiffiffiτ�t=τ�

p ¼ ffiffiffiffiffiffiffiffiffi3�63p ¼ 1�91,

the equation from Kothyari et al. (1992) can be modified foran abutment to describe the temporal variation of the bedshear stress at the nose of the abutment as

15: u�t ¼ 1�91 u� A0

At

� �C

where u*t=the temporal variation of the bed shear velocity atany time t at the nose of the abutment. At the equilibrium con-dition (dst=dse), the area of the scour depth (Ase) is equal to0·5dse

2 /tan Ø (Figure 5). However, in this state, the scour depthat the nose of the abutment is fixed and equal to dse. Then, u*tcan be assumed to equal to u*c and another form of Equation15 for the equilibrium time is

16:u�c

1�91 u� ¼A0

A0 þ Ase

� �C

In Figure 7, the experimental data from the present and pre-vious studies were used to estimate the coefficient. The valueof C was found to be 0·16. In addition, the experimental datawere also employed to determine the constant β in Equation 5.As shown in Figure 8, the value of aqst/d50u*t was plottedagainst 1+ast, and a value of 5·0 was obtained for β. Then,Equation 5 can be rewritten as

17: qst ¼ 5�0 ð1� eÞd50u�tst 1� 1a st

ln 1þ a stð Þ� �

The sediment transport rate (qst ¼ ðqvt n � qvtn�1Þ=ðtn � tn�1Þ)was used to estimate qst at any time step of n. However, thescour depth at any time step of n can be determined in termsof qvt as

18: dstn ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6 tanØ

πqvtn þ

6BLπ

� 3L2

4

� �tan2Ø

s� 3L

2tan Ø

Flow FlowPrimary vortex (At = A0)

Primary vortex at time t (At = A0 + Ast)

A0 A0

Ast

Initial bed level Initial bed level

∅

Abutment Abutment

B B

ss yy

(a) (b)

Figure 6. Primary vortex in front of a uniform abutment: (a)before scouring, At =A0; (b) during scouring, At =A0+Ast

log[A0 /(A0 + Ast)]

log(

u *c/1

·91u

*)

u*t

A0 + Ase

0·16A0

00

–0·2

–0·2

–0·1

–0·4

–0·3

Dey and Barbhuiya (2005)

Yanmaz and Kose (2007) Ballio and Orsi (2001)

Fael et al. (2006) Lim (1997) Dongol (1994)

Present study

–0·6 –0·8 –1·0 –1·2 –1·4 –1·6 –1·8 –2·0

1·91u*

Figure 7. Experimental data used to find coefficient C

7



PROOFS

and

19: qvtn ¼Xni¼0

qsti

The following steps that were modified from Mia and Nago(2003) were recommended for the computation of the scourdepth at an abutment under clear water conditions. The inputdata include the flow depth (y), the approach mean velocity(U ), the critical velocity of the sediment (Uc), the abutmentdimensions (L and B), the sediment angle of repose (Ø) andthe median size of the sediment (d50).

(a) Calculate the fixed value of A0, u* and u*c usingEquations 11, 12 and 14.

(b) At the first time step (t0=0 and n=0), no scour hasoccurred (dst0=0 and qst0=0), so Ast=0, and At=A0.Using Equation 15 gives that u*t0=1·91 u*. Then, thevalues of st0 and a can be estimated using Equations 2and 3, respectively.

(c) For the next time step (n=1), calculate the value of qst1based on Equation 17.

(d) Compute qvt1=qst0+qst1 using Equation 19 and calculatedst1 using Equation 18.

(e) Determine the total cross-section (At) using Equations11–13 based on the last value of dst.

(f) Compute new values of st and u*t using Equations 2 and15, respectively.

(g) Estimate the new value of qstn for the next step usingEquation 17 and calculate qvtn using Equation 19.

(h) Calculate dst using Equation 18 for the new time step oftn=tn−1+1.

(i) Steps (d ) through (i) can be repeated to obtain the vari-ation of the scour depth until u*t becomes approximatelyequal to u*c.

7. Proposed model for compound abutmentThis method is presented for the cases where the foundation ofa compound abutment is located below the initial bed level(Figure 4). As mentioned previously, three conditions arise forthe calculation of local scour. In case I, the foundation levelis located below the scour hole and the foundation has noeffect on the local scour. In this case, the proposed method-ology for a uniform abutment can be used to predict the scourdepth.

In case II, the principal vortices in front of the abutment areweakened by the foundation, and the scour depth is confinedby the top of the foundation. In such cases, the foundationis similar to an obstacle for the vortices, and the principalvortex rests on the top surface of the foundation. In theseconditions, no further scour occurs at the upstream portion ofthe footing and the principal vortices just expand the volumeof the scour hole to the sides and downstream of theabutment.

aqst/d

50u *t

(1 + ast)

Equation 17 for β = 5·0

Present study0·8

0·7

0·6

0·5

0·4

0·3

0·2

0·1

01 1·05 1·10 1·15 1·20 1·25 1·30 1·35 1·40


Cardoso and Bettess (1999)Yanmaz and Kose (2007)

Best fit function

Figure 8. Determination of β for an abutment

Flow

Initial bed level

dsct∅

∅

Abutment

Foundation

Bf

Zf

y(Bf–B)/2

B

Ws

Zf /tan∅

Figure 9. Definition sketch for scour at a compound abutment

Flow

Primary vortex at time t ( At = A0 + KAstI+ AstII)

Initial bed level

dsct

∅

Abutment

Foundation

Bf

Bu

Zf

Scour hole

y

B

AstI

A0

AstII

Figure 10. Principal vortices in front of a compound abutment

8



PROOFS

In case III (0<Zf/L≤1), at first, the principal vortex increasesthe scour depth to the top of foundation. As shown inFigure 9, owing to the effect of the foundation, the scour depthis considered to be fixed at the Zf level for a specific time (timelag). During the time lag, the principal vortices expand thescour hole horizontally over the top surface of the foundation.

This time (or time lag) is equal to the duration of the scourprocess at a uniform abutment for a depth equal to (Bf−B)/2(Figure 9). After this step, the foundation is exposed to thescour hole and another principal vortex is generated by thefoundation. Therefore, the total time lag needed to create aprincipal vortex in front of the foundation is equal to the timelag mentioned above, plus the required time for the scour to

reach the top of the foundation, Zf (Figure 9). Therefore, forcalculating the total required time lag, the equivalent scourdepth (des) is recommended to be

20: des ¼ αBf � B

2

� �þ Zf ¼ αBu þ Zf

The correlation factor α was used to consider the effect of thevortex produced between the top of the foundation and theinitial bed level. The value of α is between 0·00 and 1·00.

As long as the principal vortex reaches the outer top edge ofthe foundation, part of the foundation will also be exposed tothe flow. In such a scenario, the methodology for computingthe temporal variation of the scour depth is based on the effectof the principal vortices in front of both the abutment and thefoundation. As indicated in Figure 10, the principal vorticesare classified in three groups: the vortex in front of the abut-ment and on the initial bed level (A0), the vortex between thetop of the foundation and the initial bed (Ast,I) and the vortexin front of the foundation (Ast,II). The principal vortex betweenthe initial bed and the top of the foundation (Ast,I) is a func-tion of the foundation level (Zf) and the dimensions of thefoundation in front of the abutment (Bu=(Bf−B)/2). Then, acorrection factor of K should be used to consider the effect ofboth Z and Bu on Ast,I. The total area of the principal vortices(At) can be expressed as

21: At ¼ A0 þ K Ast;I þ Ast;II

For a compound abutment, the area of the vortex above theinitial bed (A0) is similar to that of a uniform abutment andcan be estimated using Equation 11. However, Equation 13was employed to calculate Ast,II in the following equation,

00

0·2

0·2

0·3

0·4

0·5

0·6

0·7

0·8

0·1

0·4 0·6 0·8

Z/L

1·0 1·2 1·4 1·6

α

Figure 12. Variation of α in terms of Z/L

L

Z

dsctX = Lf + z/ tan∅

dsct/ tan∅

∅

Lf

Lf–L

X

X

Z

Figure 11. Geometric definition of a scour hole at a compoundabutment

0

Equation 26 for (Bu/L)(Z/L) < 0·3Equation 27 for (Bu/L)(Z/L) ≥ 0·3

0·2 0·4 0·6 0·8Z/L

1·0 1·2 1·40

1

2

3

4

5

6

7

K v

alue

Figure 13. Variation of K-value in terms of Z/L and Bu/L

9



PROOFS

Start

Read:L, Lf, B, Bf, d50, y , u , uc and ∅

t0=0; n=0; Ast=0; then At = A0; qst=0; dsct=0;

Dv from Equation 2; u* and u*c from Equation 14

Calculation of A0 from Equation 1;

Calculation of u*t from Equation 15; a from Equation 3;

Calculation of qst from Equation 17; qvtn from

Equation 19;

Calculation of dsctn from Equation 24; Ast,I from Equation 23;

Calculation of Ast from Equation 13; At from Equation 10;

Equation 25 – 27Calculation of α and k using

u*t from Equation 15; st from Equation 2

u*t from Equation 15; qvtn from Equation 19

Ast,II from Equation 22; At from Equation 21

dstn< Z

dsctn= Z

u*t≈u*c

u*t ≈ u*c

dsctn=dstn

dstn≤des

dstnfrom Equation 18

st from Equation 2 and dse from Equation 20

n=n + 1; tn=tn + 1;

n=n + 1;tn=tn + 1;

Yes

Yes

Yes

Yes

End

No

No

No

No

Figure 14. Algorithm for computing the temporal variation of thescour depth at a compound abutment (Modified from Kothyariet al., 1992)

10



PROOFS

2·01·8

1·61·41·21·00·80·6

0·40·2

0·2 0·3 0·4 0·5 0·6 0·7 0·8 0·9 1·00·0

0·0 0·1

y = 11·1 cmL = 4·0 cmd50 = 0·6 mm

U/Uc = 0·96

AB I (Present study)Computed (Present study)

d s/L

d s/L

d s/L

d s/L

d s/L

d s/L

d s/L

t/te t/te

t/te t/te

t/te

t/te

t/te

2·62·82·22·01·81·61·41·2

0·81·0

0·6

0·20·4

0·2 0·3 0·4 0·5 0·6 0·7 0·8 0·9 1·00·0

0·0 0·1

y = 18·3 cmL = 5·0 cmd50 = 5·6 mm

U/UC = 1·0

Computed (Present study)Ballio and Orsi 2001, L = 5 cm

2·01·8

1·61·41·21·00·80·60·40·2

0·2 0·3 0·4 0·5 0·6 0·7 0·8 0·9 1·00·0

0·0 0·1

y = 11·0 cmL = 5·5 cmd50 = 0·6 mm

U/Uc = 0·98

AB II (Present study)Computed (Present study)

2·0

1·8

1·61·41·21·00·80·6

0·40·2

0·2 0·3 0·4 0·5 0·6 0·7 0·8 0·9 1·00·0

0·0 0·1

y = 20·0 cmL = 8·0 cmd50 = 0·26 mm

U/Uc = 0·92

Dey and Barbhuiya 2005Computed (Present study)

Dey and Barbhuiya 2005Computed (Present study)

2·01·8

1·61·41·21·00·80·6

0·40·2

0·2 0·3 0·4 0·5 0·6 0·7 0·8 0·9 1·00·0

0·0 0·1

y = 9·7 cmL = 7·0 cmd50 = 0·6 mm

U/Uc = 0·95

AB III (Present study)Computed (Present study)

2·0

1·81·61·41·21·00·8

0·6

0·40·2

0·2 0·3 0·4 0·5 0·6 0·7 0·8 0·9 1·00·0

0·0 0·1

y = 20·0 cmL = 10·0 cmd50 = 0·26 mm

U/Uc = 0·92

0·2 0·3 0·4 0·5 0·6 0·7 0·8 0·9 1·00·0 0·1

y = 9·2 cmL = 10 cmd50 = 5·0 mm

U/Uc = 1·0

Computed (Present study)Ballio and Orsi 2001, L = 10 cm2·4

2·22·01·81·61·41·2

0·81·0

0·6

0·20·4

0·0

Figure 15. Comparison of the observed and computed temporalvariations of scour depth at a uniform abutment

11



PROOFS

where the scour depth in this area is equal to dst−Z

22: Ast;II ¼ 12

dsct � Zfð Þ2tan Ø

þ 0�087 dsct � Zfð Þtan Ø

A trapezoidal shape was assumed for calculating the vortexarea in the region of Ast,I as

23: Ast;I ¼ Bf � B2

þ dscttan Ø

� �Zf � Z2

f

tan Ø

From Figure 11 and Equation 6, the following equations weredeveloped for compound abutments to estimate the scourvolume per unit width of scour (qvtc)

In each step, the numerical solution can be used to calculatethe dst in this equation.

8. Results and discussionTable 2 shows the results of all of the experiments for bothuniform and compound abutments. AB and FA were used foruniform and compound abutments, respectively. For the com-pound abutments, Zf is located in the range of 0 to 2L (casesII and III).

As mentioned previously, a time lag occurs in the scourprocess due to the effect of the foundation in case III. To esti-mate the total required time lag, the equivalent scour depth(des) and its coefficient (α) should be determined. As shown inFigure 12, the collected data were used to estimate this coeffi-cient. The following equation was suggested for α

25: α ¼ �0�5 Zf

L

� �þ 1�0 for 0 � Zf , 2L

The result indicates that for Zf=0, α is equal to 1·0 and thevalue of α decreases with increasing foundation level (Zf ). Forcases II and III (0≤Zf<2L), the time lag depends on both Bu

and Zf. According to Equation 20, for a large foundation(high value of Bu) at bed level (α=1), the time lag is large andlocal scour may never occur upstream from the abutment.Hjorth (1975) reported that if the upstream extension of thefoundation becomes larger than two times the pier width, it isexpected to completely protect the portion of the bed upstreamfrom the pier from the horseshoe vortex created by the pier.For case I (Zf>2L), the effect of the foundation is eliminated

and the value of α is equal to 0·0. In this case, the temporalvariation of local sour at the compound abutment is similar tothat of a uniform abutment.

Values of K were proposed to consider the effect of the foun-dation level (Zf ) and its dimension (Bu, Figure 9) on the scourdepth due to the influence of the principal vortex betweenthe top of the foundation and the initial bed. As shown inFigure 13, the data were categorised in two groups, and thefollowing equations were recommended for the variation ofK based on Zf/L and Bu/L:

26:K ¼ 5�0 Zf

L

� �2

� 5�0 ZL

� �þ 1�61 for

Bu

L

� �Zf

L

� �, 0�3

27:

K ¼ 8�20 Zf

L

� �2

� 7�9 0 ZL

� �þ 2�70 for

Bu

L

� �Zf

L

� �� 0�3

Finally, the computation of the temporal variation of scourdepth can be carried out as per the flowchart modified fromKothyari et al. (1992), as shown in Figure 14.

24: qvtc ¼ Vvt

Lf � Lþ ðdsct= tan ØÞ½ � ¼1

Lf � Lþ ðdsct= tan ØÞ½ �π

6tan2 Ød3sct þ

πLf

2 tan Ød2sct þ

π L2f

2� BfLf

� �dsct þ BfLf � BLð ÞZf

� �

60

50

40

30

20

10

00 10 20 30 40

Observed scour depth: cm

Com

pute

d sc

our

dept

h: c

m

50 60

Ballio and Orsi (2001)


Dongol (1994)

Present study

+15%

–15%

Figure 16. Comparison between the observed and computedscour depths at a uniform abutment using the proposed method

12



PROOFS

2·0FA21, Z=5·0 cmComputed

1·81·61·41·21·00·80·60·40·2

0·2 0·3 0·4 0·5 0·6 0·7 0·8 0·9 1·00·0

0·0 0·1

d s/L

d s/L

d s/L

d s/L

d s/L d s/

L

d s/L d s/

L

d s/L d s/

L

t/te t/te

t/te t/te

t/te t/te

t/te t/te

t/te t/te

2·0FA 42, Z=1·0 cmComputed

1·81·61·41·21·00·80·60·40·2

0·2 0·3 0·4 0·5 0·6 0·7 0·8 0·9 1·00·0

0·0 0·1

2·0 FA33, Z=1·0 cmComputed1·8

1·61·41·21·00·80·60·40·2

0·2 0·3 0·4 0·5 0·6 0·7 0·8 0·9 1·00·0

0·0 0·1

2·0FA 42, Z=5·0 cm

Computed1·81·61·41·21·00·80·60·40·2

0·2 0·3 0·4 0·5 0·6 0·7 0·8 0·9 1·00·0

0·0 0·1

2·0 FA33, Z=3·0 cmComputed1·8

1·61·41·21·00·80·60·40·2

0·2 0·3 0·4 0·5 0·6 0·7 0·8 0·9 1·00·0

0·0 0·1

2·0 FA 42, Z=7·0 cmComputed1·8

1·61·41·21·00·80·60·40·2

0·2 0·3 0·4 0·5 0·6 0·7 0·8 0·9 1·00·0

0·0 0·1


1·81·61·41·21·00·80·60·40·2

0·2 0·3 0·4 0·5 0·6 0·7 0·8 0·9 1·00·0

0·0 0·1


1·61·41·21·00·80·60·4

0·00·2

0·2 0·3 0·4 0·5 0·6 0·7 0·8 0·9 1·00·0 0·1


1·81·61·41·21·00·80·60·40·2

0·2 0·3 0·4 0·5 0·6 0·7 0·8 0·9 1·00·0

0·0 0·1


1·6

1·21·4

0·81·0

0·60·4

0·00·2

0·2 0·3 0·4 0·5 0·6 0·7 0·8 0·9 1·00·0 0·1

Figure 17. Comparison between the observed and computedscour depths at a compound abutment using the proposedmethod

13



PROOFS

8.1 Time variation of local scour at a uniformabutment

Figure 15 shows a comparison between the observed andcalculated temporal variations of scour depth at a uniformshort abutment using the data fromQ11Q12 Cardoso and Bettess(1999), Ballio and Orsi (2001), Dey and Barbhuiya (2005) andthe present study. Most of the runs in this figure correspondto long-duration experiments. The scour depth and timewere normalised using the abutment length (L) and the equili-brium time (te), respectively. Good agreement was foundbetween the computed scour depth and the observed data.The best result can be obtained if the bed-shear stress at thenose of the abutment is approximately equal to or a little lessthan the critical shear stress. Then, the calculations continueduntil the difference between u*t and u*c was between 0·07and 0·1.

Figure 16 shows a comparison between observed and pre-dicted scour depth using the proposed method. The data givenby Dongol (1994), Ballio and Orsi (2001), Dey and Barbhuiya(2005) and the present study were employed for this compari-son. The collected experimental data cover a wide range ofvariables, such as the abutment length ratio (L/y) ranging from0·16 to 1·00, the sediment coarseness (L/d50) ranging from 10to 462 and the flow intensity (U/Uc) ranging from 0·95 to 1·00.As shown in Figure 16, the presented model predicts thetemporal variation of scour depth with high accuracy. Theaccuracy of the model for predicting local scour at abutmentswith short lengths and low scour depths (with ds<25 cm) isbetter than those for long abutments with large scour depths(ds>25 cm).

8.2 Time variation of local scour at a compoundabutment

Figure 17 shows a comparison between the computed tem-poral variation of scour depth at a compound abutment andthe corresponding observed data in the present study. A perfectagreement was obtained between the computed and observeddata for most tests. The presented method mostly predicts thescour depth with high accuracy. The results show that thefoundation decreases the scour depth when it is placed at asuitable depth. For example, in FA 33, increasing the foun-dation level from 1·0 cm (Zf/L=0·14) to 7·0 cm (Zf/L=1·00)decreased the scour depth ratio (ds/L) from 1·73 to 1·07 (Table2). Compared to the uniform abutment, the best result at thecompound abutment was also observed when the difference ofu*t and u*c was between 0·07 and 0·1. Thus, the proposedmethod in this paper can be practically employed to design aproper foundation level for a compound abutment.

9. ConclusionsIn this study, the temporal variation of local scour at shortuniform and compound abutments (L/y<1) was investigatedexperimentally under clear water conditions. An empirical

method has been presented for predicting the temporal vari-ation of local scour at uniform abutments based on the tech-nique proposed by Mia and Nago (2003). The proposedmethod has produced an accurate prediction of the temporalvariation of scour depth at uniform abutments for the datacollected from the present study and previous studies.

Another empirical method has been developed for computingthe temporal variation of scour depth at compound abutmentswhile the top of the foundation is located below the initial bedlevel. The concept of the principal vortex (Barbhuiya and Dey,2003; Kothyari et al., 1992) was used to estimate the effect ofthe foundation at a compound abutment. The experimentaldata were used to validate the proposed method. A reasonablygood agreement was obtained between the observed and com-puted values of scour depth around compound abutments.

AcknowledgementsThe authors would like to thank Universiti Sains Malaysia forthe financial support under the RU GA grant (1001/REDAC/8022009).

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Q7 Please confirm the change of ‘Sheppard et al. 2004’ to ‘Sheppard

and Glasser, 2004’ in text citation as per reference list.

Q8 ‘if the foundation appears into scour hole’ – intended meaning is

not clear here. Please consider rewording.

Q9 ‘foundation exposes into’. Sense not clear here. Possibly the ‘foun-

dation is exposed into’?

Q10 Please confirm the change of ‘Ahmad and Rajaratnam (2000)’ to

‘Ahmed and Rajaratnam (2000)’ in text citation as per reference

list.

Q11 Please confirm the change of ‘Cardoso and Battess (1999)’ to

‘Cardoso and Bettess (1999)’ in text citation as per reference list.

Q12 Fig 15 does not mention Cardoso and Bettess data, OK?

Q13 Please provide the issue number in Ahmed and Rajaratnam 2000.

Q14 Please provide the issue number in Ataie-Ashtiani et al. 2010.

Q15 Please provide the issue number in Ballio et al. 2010.

Q16 Please provide the issue number in Barbhuiya and Dey 2003.

Q17 Please provide the issue number in Cardoso and Bettess 1999.

Q18 Please check the issue number and DOI added in Cardoso and Fael

2010.

Q19 Please provide the issue number in Chang et al. 2004.

Q20 Please provide the issue number in Chiew 1995.

Q21 Please provide the issue number in Coleman 2005.

Q22 Please provide the issue number in Coleman et al. 2003.

Q23 Please provide the issue number in Dey and Barbhuiya 2005.

Q24 Please provide the issue number and page range in Fael et al. 2006.

Q25 Please provide the issue number in Ferraro et al. 2013.

Q26 Please provide the issue number in Ghodsian and Vaghefi 2009.

Q27 Please provide the issue number in Jones et al. 1992.

Q28 Please provide the issue number in Kothyari and Kumar 2012.

Q29 Please provide the issue number in Kothyari and Ranga Raju 2001.

Q30 Please provide the issue number in Kothyari et al. 1992.

Q31 Please provide the issue number in Kumar and Kothyari 2012.

Q32 Please provide the issue number in Kumar et al. 1999.

Q33 Please provide the issue number in Kwan and Melville 1994.

Q34 Please check author names and issue no. added by copy editor in

Lu et al. 2011.

Q35 Please provide the issue number in Melville 1992.

Q36 Please provide the issue number in Melville and Chiew 1999.

Q37 Please provide the issue number in Melville and Raudkivi 1996.

Q38 Please provide the issue number in Mia and Nago 2003.

Q39 Please provide page range in Mohammadpour et al. 2011

Q40 Please provide the issue number in Mohammadpour et al. 2013a.

Q41 Please check the issue number and DOI added in Mohammadpour

et al. 2013b.

Q42 Please provide the issue number in Mohammadpour et al. 2013c.

Q43 Please provide location of publisher in Mohammadpour et al.2014a, and confirm whether 455 is the page no. referenced

Q44 Please provide the issue number in Mohammadpour et al. 2014b.

Q45 Please provide the issue number in Mohammadpour et al. 2015.

Q46 Please provide publisher name, city/town, country, state name if

USA in Moreno et al. 2012.

Q47 Please provide the issue number in Oliveto and Hager 2002.

Q48 Please provide the issue number in Parola et al. 1996.

Q49 Please provide page range in Sheppard and Glasser 2004

Q50 Please provide the issue number in Yanmaz and Altinbilek 1991.

Q51 Please provide the issue number in Yanmaz and Kose 2007.

Q52 Lim 1997 is cited on artwork for Figure 7. Please provide details

for reference list.

Documents

Predicting scour at river bridge abutments over time - USMredac.eng.usm.my/html/publish/2015_01.pdf · conducted on compound and uniform abutments under clear water ... In the clear