Analytical approach to some practical aspects of local scour around bridge piers

Analytical approach to so

m

e pract

i

cal aspects of local scour around bridg

e

piers

Tetsur

o

Ts

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j

irn

oto

E

co

l

e

P

o

l

y

t

ec

h

n

i

que

F

ede

r

a/e de Lausanne, S

wi

tze

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la

n

d

ABSTRACT: Local scour is one of the most important causes on bridge failures during floods, and recent changes of river environments makes i t difficult to predict the scour depth and d.esign an appropriate protection works. In this study, firstly an analytical model to describe scouring process around a pier has been derived based on reasonable understanding of its essential mechanism. And then, manipulating this model, the similarity law to govern a distorted model test has been established, and furthermore, the subsidiary effects which cannot be reproduced even in a distorted model has been reasonably evaluated.

1 INTRODUCTION

Local scour is one of the most important causes of bridge failures du:ring floods particularly in rivers under severe degradation. Recently, in most rivers in Japan sufficient river improvements including erosion control and c:onstruction of high dams in their upstream region have been achieved, and reversely but in-evitably they bring about SE!vere river degradation. Therefore, appropriate pre-diction of scour depth and technology of reducing scouring are much desired in these days. Although many research works have been already done (as summarized in the literatures: Breusers, Nicollet & Shen 1977, Nakagawa & Tsujimoto 1986), proper prediction and proposal of reducing technology for local scour are s t i l l difficult because the situations of rivers surrounding local scour have become complicated and nevertheless higher accuracy of prediction of the scour depth is required accompanying unfai.r or partial development of river engineering and technology. Therefore, we have to make hydraulic model test to predi.ct scouring phenomena around bridge piers for respective cases.

However, scouring phenomena around bridges cannot necessarily be perfectly reproduced in a hydraulic model mainly because local scour around a bridge pier is often subjected to multi-scales, as shown schematically in Fig. 1. If the arguments here are limited only to the spatial scales, the governing characte -ristic scales are listed below:

(1) sand diameter: bed materials are

often composed of graded materials; and their gradation curve becomes important.

(2) flow depth and pier diameter are especially important to the local scour itself.

(3) dunes and bars have big influence on scouring process; particularly their migration degenerates its characteristics. (4) scales of supplementary structures: recently, we often observe a supplementary structures to a bridge pier for prevention of scouring or reducing scour depth. Particularly in rivers under severe degradation, such a protection work becomes too remarkable, or the foundation of the pier itself often becomes exposed to the flow severely.

In general these scales are so different each other that ordinary geometrical similarity between prototype and model cannot be achieved, and in such a case a "distorted model" is required to

Flow

1--<1

D

Fig.1 Local scour subjected to various scales.

be designed. Furthermore not only a distortion between the horizontal and the vertical scales, but also one between the

scale of flow depth and sediment size are

applied using a specially prepared

discussion of the "similaritv law".

Particularly, few has been discussed for

the latter type distorted model, and thus

it will be discussed related to the scour

depth in this paper.

In spite of such a technique of a

distorted model, still some subsidiary

effects on local scour, such as effects of

supplementary structures, bed forms,

gradation of bed materials and so on,

cannot be reproduced in the hydraulic

model. Moreover, for the sake of

simplicity or easiness in making physical

model in the laboratory, such subsidiary

effects are sometimes neglected but they

are tried to be considered separately by

another approach. That is an analytical

programme.

In order to discuss about the similitude

to govern a hydraulic model test,

particularly distorted model, and to

predict the subsidiary effects which are

neglected in a hydraulic model test, we

have to prepare a reasonable mathematical

description of the essential mechanism,

and then, we are able to use or extend the

result of the model test to evaluate the

prototype phenomenon. In other words, we

have to manipulate these two approaches

(hydraulic model test and analytical

approach) complimentarily.

2 ANALYTICAL MODELLING OF LOCAL SCOUR

AROUND A BRIDGE PIER

2.1 Two Categories of Scour Phenomena

Local scour is in general classified into

two categories: clear water scour and

scour with continuous sediment motion;

according to the different features of

time variations of scour depth as

illustrated in Fig.2 (Shen 1971). In the

case of clear water scour, the scour depth

reaches a "final scour depth" asympto-tically, at which the bed shear stress due

to the vortex decreases as weak as the

critical tractive force. Meanwhile in the

case of scour with continuous sediment

motion an "equilibrium scour depth"

appears as the result of a balance of

sediment carried away from the scour hole

by the vortex and that supplied from the

undisturbed region, and thus it somehow

fluctuates temporally affected by

irregular size dunes migration. Moreover

in Fig.3, the difference of the relations

between the final scour depth and the

equilibrium scour depth against the flow

parameter is demonstrated using the

laboratory data of Chabert & Engeldinger

(1956). The former increases with the

sediment number defined as Ns=U/ ✓ (cr/p-l)gd

.c

a.

0 Q

...

0 <.) <I) Zse Time Scour with Continuous Sediment Motion

Time

Fig.2 Two types of local scour.

No

NsJ Ns2

I

ScourjC- W.S. Scour with Continuous

'

•

1-

Sediment Motion 2 .0 , - - - i - - - , - : . - - , - - 0 - - r - ~ - ~ Z5/D 1.5 1.0 0.5 0

0 Max

• Min 0

8

0

II

0 2 J 4 5

N5a:::UN(a/p-l)g

d

Fig.3 Final and equilibrium scour

depths.

(U=approaching flow velocity; cr=mass

density of sand; p=mass density of fluid;

g=gravity acceleration; d=sediment size);

but the latter is almost constant (rather

decreasing) against the flow parameter and

the difference between the maximum and the

minimum scour depths is appreciable, which

is caused by temporal fluctuation as seen

also in Fig.2.

2. 2 Analytical Modelling to describe

Scouring Process around Pier

Several researchers (Laursen 1963, Carsten

of the equilibrium scour depth using a somehow macroscopic control-volume as sho"'n in Fig. 4, where sediment trarniport cannot be reasonably described because flO"' and sediment behaviour around a pier show spatially wide variations.

On the other hand in order to develop a reasonable description of vortex and sediment behaviour and thus in order to improve the applicability of a scour model

to various boundary conditions, the

authors (Tsujimoto & Nakagawa 1986) have adopted more microscopic control-volume (see Fig. 5 I, at which flow and sediment behaviours are reasonably and analytically modelled and for which the continuity law of sediment is applied.

Main cause of loca 1 scour a round a bridge pier is the horse-shoe vortex which is formed as a rotating flow concentrated

at the front foot of a pier is shedding

along the sides of the pier and the

bottom. Ordinarily, the scour hole is

kept geometrically similar (almost

reverse-cone), and thus the scouring

mechanism at the front foot must symbolize the whole phenomenon. Particularly in such a "stagnant plane analysis", the hor:,e-shoe vortex can be simplified as a

rotational flow, and the bed shear stress

due to i t can be comparatively easily

evaluated (Tsujimoto & Nakagawa 1986) by

applying the conservation law of the flow

circulation proposed by Shen, Schneider &

Karaki (1969).

Sand particles are picked-up from the

vortex region S1 shown in Fig.S, and a bed

of the region (S1+S2> is uniformly

descended by scouring because o.f the

geometrical similarity almost assured by

the frictional property of sands. In the

case of scour with continuous sediment motion, sands are supplied through the arc AB (see Fig.5), and i t less changes with the scour depth.

The continuity of sediment in the scour hole can be expressed as follows:

psvS1A3d3ot A2d2 -Osin6t=(S1+s21ozs·(l-pO) - ·. - - J ,_ __ '-' J _; ( 1)

I

scour hole

qaout

Fig. 4 Macroscopic control-volume for sediment transport near the scour hole.

in which Pav-sediment pick-up rate in the

vortex region; A2 , A3=2- and 3-dimensional geometrical coefficients of sands; Po= porosity of sands; 0Bin=sediment discharge transported into the scour hole (:q80-AB:

q8a~sediment transport rate in undisturbed

region; AB=length of the arc AB); and Psv'

s

1 and

s

2 are functions of the scour depth ~D=z5 (D=pier diameter).

The difference between the

control-volume ABCD of Fig. 6 between flows with

and without pier,

t.r,

is redistributed to the vortex formed at the front foot of the

pier, and it is estimated if the potential

flow in the horizontal plane is applied,

(2)

in which us O ~surface velocity of

approaching flow, which hardly changes

with scour depth. When the local

coefficient of resistance in the vortex

Bridge Pier

2",.

h

G.

r

Hole

-ey-Fig. 5 Microscopic control-volume for

sediment transport inside of scour hole.

Sep1r1 t ion Po1nt

Fig. 6 Control-volume to consider the

region cl>v=vb/u*v (vb-flow velocity at the

outer edge of the vortex core; u*v=✓tv/p;

and tv=bed shear stress due to the vortex)

is assumed to be constant, the bed shear

stress due to the vortex can be evaluated as follows (Tsujimoto & Nakagawa 1986):

(3)

( 4)

(5)

in which cl>s=us0/u*Q; to-pu*o 2 =bed shear

stress in undisturbed region; and no,

noD=radius of the vortex core and its

value at C=0 (f.lo is almost constant being

approximately 0. 2) . The value of YO

becomes about 1.5, and kro has been

reasonably estimated to be 1/7 (Tsujimoto

& Nakagawa 1986). The sediment pick-up

rate due to the vortex can be evaluated

against the above-estimated bed shear stress using the formula established by

Nakagawa & Tsujimoto (1980), which is

written as

( 6)

in which Ps*=Ps ✓d/ (O"/p-l)g; Ps=pick-up rate; t*=u* 2 /[(0'/p-l)gdJ; t•c=dimension

-less critical tractive force; and the

constants have been determined as follows:

Fo=0.03; k2=0.7; and m=3 (Nakagawa &

Tsujimoto 1980) . A dimensionless expression of Eq. (1) is in which '1'1 <Cl= (

~~)

[As (Cl +l l ·A.1 <Cl Psv0 •·'Vp (Cl 'l'2<Cl= (~) ClB0• [As<Cl+½J (7) (8) (9) (10) qg*=qg/ ✓ (cr/p-l)gd3; L1=A1D=horizontal

scale of vortex region; As=AsD=horizontal

scale of scour hole (see Fig.5);

'Vp(Cl=Psv<C)/psvo; and Psvo=Psv at C-0.

Al <Cl and As <Cl have been formulated

(Tsujimoto & Nakagawa 1986), roughly as

follows:

Submerged Weight

of Sand Pier

Fig.7 Illustration of Scour hole.

A1(CJ=Oo(l+kroC> (l+sin9cl (11)

As <C> = [CR+ (2-cRl kc,Po JC+ (2-cR) no ( 12) in which CR-cotQr; cl>r=angle of repose of

sands; and 9c=angle indicated in Fig.7,

Using the aforementioned analytical

model, both categories of scouring

processes can be described. Particularly,

the final scour depth for clear water

scour is obtained for tv ➔tc ~c=critical

tractive force), as follows:

in which 11-to /tc. Ordinarily,

scour with continuous sediment its equilibrium scour depth is of the final scour depth for 1971), and thus (13) l)>l means motion, and roughly 90% 11➔ 1 (Shen (14)

3 SIMILARITY LAW FOR LOCAL SCOUR

In this chapter, the similitude for a

distorted model of local scour, where the

scale of sediment size is distorted from

the other geometrical scales, is dis

-cussed. Here the "distortion ratio" Ed is

defined as follows:

(15) in which the subscript r means the "scale ratio", and Ed*l means a distorted model.

The final or the equilibrium scour

depths around a circular cylinder are

roughly related to the governing flow

parameter, 1)=-to/tc, as explained in the

above chapter, and thus llr=l is the first

requisite condition for the similarity.

The characteristic parameter 1) is a

function of relative flow depth Z=h/d

{ Fr2 } z-2/3

t•c<7.66) 2 (cr/p-1) (16)

in which E'r=Froude number of undisturbed

flow; and 7.66 is a constant involved in

the Manning-Strickler equation, which has

been applied as a resistance formula for

undisturbed flow over a rough bed.

Obviously, z is directly related to the

distorted ratio Ed.

If llr=l provides (zs/D) r=l by some

adjustment of other parameters, this

distorted model is quite "exactly

designed". In other words, the bed

materials to satisfy the following

equation must be used in such a model.

(17)

in which (t*clr is almost unity for larger

grain-size, Reynolds number (Xau*d/v>a70;

v=kinemati,c viscosity of fluid) . It means

that we often have to use a light

materials. Then, the scale of scour depth

is quite same to that of pier diameter,

and we can easily predict the actual scour

depth in the prototype by a model test.

However, we often use the following

not-well prepared distorted-model for the sake

of convenience in the laboratory, because

in general it is difficult to prepare a

large amount of light granular materials

though we cannot design any non-distorted

model in cJrder to keep the similarity on

the pattern of sediment motion. However,

even if one uses this kind of "not well

-prepared distorted model", we can devise

to evaluate the true value of scour depth

in the prototype.

When Ed;'l but (cr/p-1) r=l (model test

using natural sands), it concludes that

(zs/D)ri'l, but (zs/D)r can be estimated by

the following equation since we have the

governing equation for scour depth based

on a reasonable description of its

mechanism such as Eq. (13) (for clear water

scour). in which -0

(h)

[

~

]

=Fr·

d

7. 66✓cr/p-1 1 1'-1 (18) (19)

Eq. (18) gives us the relation between

(zs/D)r and Ed. Fig.8 shows the

experimental data as for the relation

between (zs/D) r and Ed through the

"distorte,j model tests without adjustment

of the rellative density of bed materials",

obtained by Nakagawa & Suzuki (1975), in

which (h/d) p is the value of (h/d) for

prototype .. The theoretical curves and the

experimental data show fairly good

agreementa. Fr=0.8 (h/d)p=500 (h/d)p=200

0

.

1 _ ~ ~ - - ~ - - ~ 1 0

0

.

1

1.0

Uo (cm/s) D=20cm D=Scm 0 undistorted d=0.052cm d=0.07cm 29.71 0.027cm ■ 0.100cm

....

23.32 0.027cm 0. 100cm □ 14.43 0.027cm 0.027cm

....

28.28 0.063cm 0.100cm

Fig.8 Relation between (zs/D)r and

Ed-4 SUBSIDIARY EFFECTS ON LOCAL SCOUR

4.1 Effect of Supplementary Structure

Some of bridge piers with supplementary

structures are idealized or simplified as

a pier with double-scale as shown in

Fig. 9. The analytical model for single-scale pier may be applied to double-scale

pier with some modification, where the

fact that the vortex formation at the

front foot of a lower part of the pier is

different from a single pier has been

properly taken account of the analysis in

order to estimate the variation of bed shear stress due to the vortex (Tsujimoto et al. 1987).

At first, u5

o

has been replaced by the

undisturbed flow velocity at the height of

the top of the lower-part pier as

uso·fE (.0.H/h), in which fE (~) represents the undisturbed flow profile

dimension-lessed by u5

o

and h (~=y/h); and .0.H is

shown in Fig.9. Secondly, from the direct

measurement of the vortex scale using

dye-method, Eq. (3) has been modified as

tv -2 [ 0. 6kp] 2

in which kp•scale of foundation indicated in Fig.9; and H*=horizontal scale of the vortex dimensionlessed by main pier

diameter {D), and it can be expressed as

Then, the final scour depth of

double-scale pier is expressed by

_ .6H O • 6 ( .6H) ,---:-.::. (1))- -_D - -_a ·fE -_h ·'IY011 (21) such a (22) (23) (24) (25) in which a=kw/kp; and b=a(.6H/D)+l. Hence the final scour depth in this case is

subjected to .6H/D (height of protection);

kp (width of protection), h/D, and 11.

Fig.10 shows that the calculated result is

D River Bed

Pier

Foundation or Protection Work

Fig.9 Simplified model of pier and

protection works.

z

.

o

zsr ZsfO 1,, l.O

o

.

,

0 l]=tol'tc=0.7 h/D=l.2 o.z 0.4 0.6 l.A l .O llH/h

Fig.10 Effect of duble-scales of pier on

final scour depth.

consistent with some experimental data (Tsujimoto et al. 1987), in which zfo=

final scour depth of a single pier of

diameter D (without protection work).

4.2 Effect of Dune Migration

As seen in Fig. 2, in the case of scour

with continuous sediment motion, the

fluctuation of scour depth is appreciable

and thus practically important from the

view point of its prediction. Previously,

most of researchers and engineers regarded

the maximum scour depth as a sum of the

ordinary scour depth (predicted without considering the effect of dunes) and dune

height or somehow down-filtered dune

height (Shen et al. 1969). However, based

on the present analytical model for

essential mechanism of local scour around

a pier, we can describe the fluctuation of scour depth due to the migration of dunes.

In other words, we can more physically

reasonably predict the maximum scour depth

based on more statistically reasonable sense.

Tsujimoto & Nakagawa (1986) analyzed the

responding properties of scour depth for

sinusoidally fluctuating sediment

transport supplied into the scour hole.

Particularly, a linearized governing

equation of scour-depth fluctuation is

written as

(26)

in which ~•=perturbation of scour depth

dimensionlessed by pier diameter D; qsf=

perturbation of bed-load transport rate;

t*=d (CJ/p-l)g/d; As (~el · [As (~e) +1]

As <~e) +1/2 (27)

and bed-load transport rate affected by

dune migration can be expressed as dA

era

<t> =crao+craf <t> = n-po> YB <t> [

uw~]

(28)

dt in which y5=bed elevation at the upstream

edge of the scour hole measured from the

level of dune troughs; and Uw=dune

celerity.

In Fig.11, the standard deviation of the

fluctuating scour depth Oz is related to

the standard deviation of duned bed

elevation Oy {the experimental data were

collected by Suzuki et al. (1983)).

Non-linear effects were partially taken

account of the above-mentioned analytical

model to evaluate the ratio of Oz to cry in

success by Tsujimoto & Nakagawa (1986).

Furthermore according to Eq. (26), the

Fourier transforms of scour depth and

are related as iWZ~Y, in which w=27tf (f=frequencyJ; and i=imaginary unit. sine•= the spectrum is an ensemble mean of product of the Fourier transform of the random process and its complex conjugate,

we obtain the following relationship.

(29)

in which Sz(W)=frequency spectrum of the fluctuating scour depth; and Sy (W) = frequency spectrum of sand waves. Since Sy (fJ is usually proportional to f-2 in

its equilibrium (.higher) frequency range as c:larified by Hino (1968) and others,

S z ( f') is proportional to c 4 _{in that}

rang•e, and it can be recognized in Fig. 12.

These statistical informations as for the

fluctuation of scour depth may become more important for maintenance of a pier and

desi,gning protection works around a pier.

Oz (cr.i) 0. S G.5 1.0 l.5 Oy (cm] 2.0

Fig .11 Standard deviations of duned-bed

elevation and fluctuating scour depth.

Fig .12 Frequency spectra of sand waves

and :fluctuating scour depth.

4.3 Effect of Sediment Sorting for Graded

Bed Materials

In a model test of graded bed materials,

the finer parts of them cannot be

reproduced with keeping the similarity of

the pattern of sediment motion between prototype and model, but we often conduct a model test using a uniform sand and then we have to evaluate the effect of

gradation of bed materials anyhow.

According to the laboratory experiments,

after scour hole is somehow established,

the sediment sorting appears remarkably.

In region A of Fig.13 (the experiments

were conducted by Nakagawa & Suzuki

(1977)), at the front foot of the pier,

.... B""·~- -::--.:.

~

j

Flow

.

C

c:;>

;;

-=

:; ,:

"'

~ I!

"

0. 100

l

RUN 6 ~ Smin I o90min ~

I

50

-

..

,..t

~

"

0 ,t 0.01 0.1 _d(cm) 100 RUN6

~--~

;;,

~ 50 r Smin

B

1760min 0 0.01 O.l _d(cm) 100 ~ ■ 5min I:: ~ ■ ,:j 30min -..; ,:j RUN6 D 90min 50 1--- --+.,..,0~ :

C

cP~

0 ~

~

1.0 1.0 0 '--4---l..=Sl:..-.-4--- - + -+-+->-' 0.01 0.1 _{d(cm) l.O}

Fig.13 Sediment sorting accompanied with local scour around a pier.

where the maximum scour depth is final~y achieved, the armour coat is formed. And

the time variation of bed materials

compositions at other typical points

observed in the experiments are also demonstrated in Fig.13. From this figure we can understand that the local scour in

the case of graded bed materials is

inevitably accompanied with local sediment sorting phenomena.

Since we have an analytical model to

describe scouring process (Chapter 2) and

also analytical models to describe

sediment motion for each grain size of

sand mixtures and the subsequent temporal variation of bed surface constitution

(Nakagawa et al. 1977), we can evaluate

the difference between the progress of

scouring and the final scour depth in the case of graded bed materials and those in

the case of uniform sand, and thus we can

predict the prototype scour behaviour from the result of model test using uniform material.

Particularly in the case of clear water

scour, armour coat formation is

appreciable. As the difference between the sediment amount picked-up and carried away from the vortex region (BCrE in

Fig.5) and that supplied by "avalanche"

from the slope region (ABCD) to the vortex

region, the process rate of scouring is

expressed by the following equation.

(30)

in which Pi =fraction of the i-th sand

class; and the subscz:ipt i indicates the

sand class of di. Moreover, the time variation of the number of the i-th class particles exposed at the bed surface in the vortex region, ni!t), can be described

by the following equation.

in which

(32)

(33)

(H)

in which Pio=initial value of Pi. Using Eqs. (30) and ( 31), the scouring process and the accompanying armour coat formation can be simultaneously described. For more

widely distributed bed materials, scouring is more suppressed and the final scour

depth is more reduced due to armouring, as

seen from calculated examples shown in Fig.14, in which the value 11 has been

calculated foz: median diameter, dso, of

original sand mixtures.

From such a fact, one can expect a

reducing of scour depth by artificial

intrusion of coarse materials to the

"d 1: 1.61 2:2.09 50 3:2.97· --0.1 1.0 d(cm)

I

o0

- - - ~

-

'-~

Zs/D 10-1 10-2

D/dso=100

✓-

'~

«,,

/ 2 ,"' I'\"'\ .0 3 11=1.0 4 10-31...i...t=.Ji...1.--'-.J...LJ-...L-'--.I..U.--'-...L.1...J I 03 104 105 106 107 i✓ (a/p-1 )g/d50

Fig.14 Effect of gradation of bed

materials on scouring process.

102 10 I

zs

(cm) toO 10-2 1.0 Zsf ZsfO 0.5 l (sec) 103 104 105 106 1.0

0 L-1..__._.,____.::::::::;

•

====="=

.,.

0 20 40 60 80 I 00

P=PERCENT AGE OF COARSER MATER La\.LS

zso=Scour depth for P=0%

Fig.15 Eftect of intrusion of coarse materials as scour-reducing work.

original bed materials around a bridge

pier. The calculated results and the

experimental data on the scour-depth

reducing effect by intrusion of coarse

materials are shown in Fig .15, in which

crd=✓d₈~ /d16 ; df=diameter for which f% of

sediment mixture is finer; P-volumetric

percentage of intruded coarser materials;

and zfo=final scour depth for original bed

materials. The applicability of the

present analytical model is confirmed by

this figure.

5 CONCLUSIONS

Local scour around bridge piers is often

subjected to several different scales

besides the flow depth, sediment diameter

and the pier diameter; such as the scales

of bed forms, supplementary structures and

prediction formulae of scour depth are not

necessarily reasonably applied to such

cases. Hence, we often have to conduct

hydraulic model tests. However, such

mulLi-scale S"bjected phenomena are

difficult to be reproduced perfectly in

hydraulic models. A distorted model is

sometimes a powerful means but s t i l l

subsidiary effects on scouring cannot be

reproduced.

An analytical model to describe the

essential mechanism of local scour around

a cylindrical pier has been derived based

on a kind of "stagnant plane analysis of

local flow" and reasonable bed-load

transport model. Based on this model, the

similitude for a distorted model test,

particularly in which the scale of

sediment size is distorted from the other

geometrical scales, has been established.

Furthermore the subsidiary effects on

local scour have been reasonably

evaluated, which are in general not

reproduced in a model test: effects of

supplementary structures, dune migrations,

and gradation of bed materials. Most of

the present analytical treatments have

been well verified by experimental data. The obtained results provide a powerful

help in manipulating hydraulic model tests

and analytical approaches complimentarily

in order to predict the scour depth under

various conditions and furthermore to

devise effective disaster prevention works

against local scour.

REFERENCES

Breusers, H.N.C., G. Nicollet & H.W. Shen

1977. Local scour around cylindrical

piers. J.Hyd.Res. 15,3:211-252.

Carstens, M.R. 1966. Similarity laws for

localized scour. J.Hyd.Div. ASCE, 92HY3:

13-36.

Chabert, J. & P. Engeldinger lll56. Etude

des affouillements auteur des piles des

ponts. Laboratoire National

d'Hydraulique, Chatou, France:6.

Hine, M. 1968. Equilibrium range spectra

of sand waves. J.Fluid Mech. 34:565-573.

Laursen, E.M. 1963. An analysis of relief bridge scour. J.Hyd.Div. ASCE, 89HY3:

93-118.

Nakagawa, H. & K. Suzuki 1975a. Armoring

effect on local scour a.round bridge piers. Annuals, D.P.R.I. Kyoto Univ. 188:689-700 (in Japanese).

Na.kagawa, H. & K. Suzuki 1975b.

Characteristics of local scour around a

bridge piers by tidal currents. Proc.

22nd Japanese Conf. on Coastal Engrg. JSCE:21-27 (in Japanese).

Nakagawa, H. and T. Tsujimoto 1980.

Sand bed instability due to bed load

motion. J.Hyd.Div. ASCE, 106HY12: 2029-2051.

Nakagawa, H. & T. Tsujimoto 1986. Local scour around structures in rivers. In

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