Foundation Depth for Bridge Piers

ORIGINAL CONTRIBUTION

Foundation Depth for Bridge Piers

G. Veerappadevaru • T. Gangadharaiah •

T. R. Jagadeesh

Received: 17 August 2012 / Accepted: 17 February 2014 / Published online: 11 March 2014

� The Institution of Engineers (India) 2014

Abstract The safety of bridge piers built in rivers hav-

ing the bed is one of the prime aspects in the study of

scouring process around bridge piers. The stability of

bridge piers depends on the depth of foundation provided

below maximum scour level. The stability analysis of

bridge piers is carried based on moment of forces acting

on the caisson pier when the pier slides and tilts slightly in

downstream from its position. The experiments are con-

ducted for three pier shapes on two sediment beds and for

different flow conditions. The curves indicating the sta-

bility limits are compared with Lacey’s recommendations

which are used in present day practice in India. The

analysis presented here indicates that the Lacey’s recom-

mendation for railway bridges is safe and for some cases

of the road bridges depends on grip length, angle of tilt

and weight of caisson.

Keywords Bridge pier � Scour depth � Foundation depth �Forces

Abbreviations

B Average width of caisson

C Position of fluid force in

terms of (h ? hs) above

the maximum scour level

CD Drag coefficient for fluid

force taken as CD = 1.0

d50 Median grain size

D Diameter of cylindrical pier

Dc Diameter of caisson pier

eb Lateral shift between FR

and W

eb� Dimensionless lateral

shift = (eb/Dc)

f Silt factor

FD� Flow intensity parameter

Ff Hydrodynamic force

Ff � Dimensionless fluid force

acting on caisson pier

FR Soil reaction force

developed at the base

of caisson

FS Lateral soil pressure acting

on caisson surface

Fs0 Soil pressure above the

deepest portion of scour

level

FVR Vertical force

h Depth of flow for upstream

of pier

g Acceleration due to

gravity

G Grip length

GF = (G - G1) Depth of caisson pier on

which soil acts on the

downstream of caisson pier

G1 Distance above the base of

caisson where maximum

soil pressure acts on

downstream face

G. Veerappadevaru (&) � T. Gangadharaiah

Department of Civil Engineering, Siddaganga Institute of

Technology, Tumkur 572103, Karnataka, India

e-mail: [email protected]

T. R. Jagadeesh

HMS Institute of Technology, Tumkur 572103, Karnataka, India

123

J. Inst. Eng. India Ser. A (August–October 2013) 94(3):169–177

DOI 10.1007/s40030-014-0055-1

H Total depth of bridge pier

from maximum flood level

to the foundation base of

caisson pier

hs0 Difference in soil levels

between front deepest

scour level to sediment

level in rear side of caisson

hs Scour depth

KA= [(1 - sin/)/(1 ? sin/)] Active co-efficient of soil

Kp= [(1 ? sin/)/(1 - sin/)] Passive co-efficient of soil

L Length of nose pier

Mf Moment of hydrodynamic

force acting on caisson pier

Ms Moment of lateral soil

force about the base

n Exponent of power law for

velocity distribution

Q Discharge intensity

r Parameter used in Eq. 22

RL Regime depth of flow

V Velocity of flow

Vc Critical velocity of sediment

u Velocity at y

U Maximum velocity at free

surface

W Weight of bridge pier and

deck

W* Dimensionless weight of

bridge and deck

x0, x1 Solutions of Eq. 22

y Vertical depth measured

above maximum scour

level

lFs Vertical component of

force due to soil friction

d Ratio G1/G

q Density of water

qs Density of sediment

rg Geometric standard devi-

ation of sediment size

distributionP

Summation sign

/ Angle of repose measured

in water under submerged

state of soil

/1, /2 Parameters used in

solution of Eq. 22

Nose pier Rounded end with

rectangular section in

the middle portion

Introduction

The bridges built across the rivers are life line to the eco-

nomic growth of the region. The multiple span bridges have

piers connected to the abutments by deck. The failure of

bridge pier or abutment causes the catastrophic effects on the

superstructure of the bridge, which in turn cutoff the com-

munication, resulting in inconvenience to road users. The

improper foundation depth below the maximum scour level

(grip length) is one of the reasons for the failure of piers. The

stability of bridge piers depends on the depth of foundation

provided below bed level to the pier. This depth consists of

the scour depth caused by the flow in front of the pier (hs) and

the grip length (G) provided below the maximum scour level

to the base level of the pier. Most of the bridge piers are

located on the caisson foundations. The caissons are usually

circular structures built at the bed level and larger in their

size in comparison to the pier dimensions. The piers are fixed

at the top of caisson. The safety of bridge piers with the

caisson depends on the equilibrium of forces and their

moments. The hydrodynamic force of incoming flow acting

on the pier and the vortex force due to the scouring vortex

acting at the deepest zone of the scour depth are causing

instability to the caisson piers. The weight of bridge deck and

the self weight of pier with caisson and the bearing capacity

of soil at the base of the caisson are considered as stabilizing

forces. The lateral soil pressure on the caisson surface acts as

major stabilizing forces on the piers with caisson.

Grip length (G) is the foundation depth for bridge pier

measured below the maximum scour level to the bottom of

the pier with caisson foundation. The researchers [1] have

advocated a method to predict the grip length based on the

regime depth of flow in rivers with cohesionless sediment

bed. His method relates to the regime depth of flow RL to

the discharge intensity q and silt factor ‘f’ as,

RL ¼ 1:35q2

f

� �1=3

ð1Þ

Silt factor ‘f’ is related with the median grain size dm = d50

in mm as,

f ¼ 1:76ffiffiffiffiffiffid50

pð2Þ

The regime scour depth RL = h is the depth of flow

measured from the free surface to the general bed level on

the upstream of the bridge pier. He proposed that the

maximum scour depth hs will be equal to the regime depth

RL. The maximum scour depth hs is the depth measured

below general bed level to the deepest portion of scour

level in front of the pier with caisson. This indicates that

the maximum scour level in front of the bridge pier from

the free surface will be equal to (h ? hs) = 2RL. It has

170 J. Inst. Eng. India Ser. A (August–October 2013) 94(3):169–177

123

been reported earlier about the total depth of bridge pier

(H) from the maximum flood level to the foundation base

of the caisson pier for road bridges as per the following

relation [2, 3],

H ¼ ðRL þ RL þ2

3RLÞ ¼ ðhþ hs þ GÞ ¼ 8

3RL or

G

2RL

� �

¼ G

hþ hs

� �

¼ 1

3ð3Þ

In addition, freeboard is to be added for final value of H,

for railway bridges

H ¼ ðRL þ RL þ RLÞ ¼ ðhþ hs þ GÞ ¼ 3RL orG

2RL

� �

¼ G

hþ hs

� �

¼ 1

2ð4Þ

At present, the Indian codes for the design of bridge pier

foundation follow the above practice (IRC: 45 and IS:

3955, 1967) [19, 20].

Prediction of Scour Depth

Prediction of scour depth in front of the piers is one of

the important investigations under taken by many inves-

tigators. The scour depth was related to gross character-

istics of flow properties, sediment properties and

geometry of the pier. The major contributions to the scour

depth based on the gross characteristics are due to the

investigations made by earlier researchers [1, 4–9]. The

scour process in front of pier is due to the vortex formed

at the junction of the pier with the sediment bed. This has

attracted the attention of many investigators. They related

the vortex characteristics with scour depth. The prediction

of scour depth based on the vortex characteristics were

investigated by the researchers [10–15]. Recently, the

prior investigators [16] have proposed a method to predict

the scour depth based on the equilibrium of power of

scouring vortex formed at the deepest scour hole. Based

on the long duration experimental data on scouring pro-

cess around bridge pier available in literature and they

related as,

hþ hs

Dc

� �

¼ 0:7FD� ð5Þ

where flow parameter FD� is related as

FD�3 ¼ q3

ðD2chÞ ðgd50Þ3=2

1

qs�qq

� �Vcffiffiffiffiffiffigd50

pð6Þ

where q is the discharge intensity in terms of m3/s/m, d50 is

the median size of sediment, qs and q are the densities of

sediment and water respectively, Vc is critical velocity

taken equal to critical shear velocity as per Shield’s crite-

rion, Dc is the caisson width and g is acceleration due to

gravity. This equation is used in the estimation of grip

length.

Experimental Details

The aim of the experiments is to find the condition at which

the bridge piers with caisson destabilize. The experiments

were conducted in a flume of length 9.5 m, width 0.54 m,

and depth 0.36 m. The sediment is filled up to 0.22 m deep.

The stones packed screen provided at the inlet will calm

down the disturbances created due to the inflow at the inlet

of the flume. Wave dampeners are also provided at the

entry section of the flume to avoid the disturbance on the

free surface. The flow enters on smooth parabolic wooden

profile provided at the entrance. In order to avoid the scour

at the entrance section stone pebbles of around 30 mm size

for a length of 30 cm and 10 mm size stone pebbles for a

length of 20 cm are provided after the wooden parabolic

profile respectively.

Water is allowed to flow in the flume to achieve uniform

flow. The flow is then stopped and pier model is inserted to

the required level. The caisson portion of pier is sliced into

different thickness. The thicknesses of the slices are 15, 21,

22.5, 27.5 and 35 mm. To achieve the different depth of

caisson, slices of required sizes are arranged and held

together by bolt and nut. The bolt passes through a hole

provided at centre line of slices and pier. The depth of pier

below bed level varied in steps of 1.5, 2, 2.5, 3 and 3.5

times pier diameter (D). The required depth of flow (h) is

adjusted to allow the flow to reach previous uniform flow

condition. The scour depths are measured at regular time

intervals of 5, 10, 15, 20, 25, 30, 40, 50, and 60 and after at

every 20 min interval up to 600 min. A careful observation

of the stability of pier is made. When the pier tilts, then the

pier is considered as failed. This procedure is repeated for

different depth of caisson foundation and for different

depths of flow conditions. Some of the piers did not indi-

cate any failure even though the scour depth reached the

pier foundation. However, such cases are considered as

failure. The dead weight kept on the top of the pier rep-

resents the load of bridge deck.

Two types of sediments are used separately; coarse sedi-

ment of median size 0.65 mm and fine sediment of median

size 0.25 mm, their angle of repose inside water for sedi-

ments are 26.3� and 21� respectively. The geometric stan-

dard deviation for d50 = 0.65 mm is rg = 1.67 and for

d50 = 0.25 mm is rg = 1.20. These sizes are chosen to

represent the dominance of bed load transport for 0.65 mm

sediment bed and the dominance of suspended load transport

J. Inst. Eng. India Ser. A (August–October 2013) 94(3):169–177 171

123

for 0.25 mm sediment bed. The depth of flow is changed to

D, 2D, 3D and 3.5D where D is the pier diameter. Three types

of piers namely the cylindrical pier, the cylindrical pier with

caisson of same shape and the nose caisson pier are used as

shown in the Fig. 1. The experimental data, for each flow run

collected are; discharge, average depth of flow, scour depths

with time and scour depth at the time of pier failure and also

the weight on the pier. The size and the rotational speed of

primary vortex in front of the caisson pier in the deepest

portion of the scour hole are also measured. The results are

analyzed using equilibrium of forces acting on the pier model

for different situation of flow.

Forces Acting on Caisson Piers

Type of forces acting on bridge piers with caisson are listed

below:

• Hydrodynamic force due to water flow Ff.

• Force due to horseshoe vortex created at the deepest

portion of the scour hole FVR.

• Soil pressure developed on the lateral surface of the

foundation.

• Weight of pier and the bridge deck load along with the

vehicle on the bridge.

• Reaction due to bearing capacity of soil developed on

the base of caisson pier foundation.

• Frictional forces developed on the lateral surface of

caisson pier due to the lateral soil force acting on the

pier surface.

Equilibrium Between Vortex Force and Soil Pressure

Above Maximum Scour Level

Magnitude of force created by vortex FVR and force due to

soil above the maximum scour level Fs0 are found to be

proportional to each other and act opposite in their direc-

tion. The positions of vortex force and soil pressure above

maximum scour level are fairly at the same level. Moments

created by them are considered to be fairly equal and

opposite. Hence, their contributions to the stability of the

caisson piers are not considered.

Hydrodynamic Force on Caisson Pier

Hydrodynamic force acting on the caisson pier is consid-

ered as,

Ff ¼ CD

qV2

2hDþ hsDcð Þ ð7Þ

where CD = drag coefficient, V = average velocity of flow

in front of caisson pier and the projected area (hD ? hsDc).

The moment of hydrodynamic force acting on caisson pier

above the maximum scour level is written as

Mf ¼ Ff Cðhþ hsÞ ¼CDqV2

2ðhDþ hsDcÞCðhþ hsÞ ð8Þ

where C(h ? hs) represent a position at which the

hydrodynamic force acts. The evaluation of ‘C’ is

attempted by assuming a power law type velocity

distribution in front of the caisson pier as,

u ¼ Uy

hþ hs

� �1=n

ð9Þ

where u is velocity at y, y is measured vertically above the

maximum scour level and U is a maximum velocity at free

surface and ‘n’ is a power law exponent. The average

velocity V is obtained by integrating over the depth

(h ? hs) as,

V ¼ Un

ðnþ 1Þ ð10Þ

Moment of hydrodynamic force is written as,

Mf ¼Zhþhs

0

CD

qu2

2� B � y dy ð11Þ

where B is average width of caisson; computed as,

B ¼ ðhDþ hsDcÞðhþ hsÞ

¼ Projected area=total depth ð12Þ

On integration and substituting U in terms of V, one can

express Mf as,

Mf ¼CDqV2

2Bðhþ hsÞ2

ðnþ 1Þ2n

ð13Þ

Substituting the value of B from Eq. 12 in Eq. 13 and

equating this with Eq. (8), one gets a value for C as,

Fig. 1 Details of plan and elevation of caisson piers


123

C ¼ ðnþ 1Þ2n

ð14Þ

Soil Force and Its Moments

A general case of pier tilt slightly downstream is con-

sidered as shown in Fig. 2a, b. The soil pressure acts both

on the downstream face and the upstream face of the

caisson pier. The distribution of soil pressure based on the

Terzaghi’s triangular distribution is assumed [17] as

shown in Fig. 2a. The modification of pressure is shown

in Fig. 2b. Based on the modification of pressure distri-

bution, the magnitude of soil force and the moments are

computed. The depth G1 is the position at which the

maximum soil pressure acts on the downstream side of

caisson pier and it is measured above the base and it is

indicated in Fig. 2b.

Soil force ¼ RFS ¼ ðqs � qÞgðKP � KAÞDcGG

2

� ðqs � qÞgðKp � KAÞDc

GG1

2ð15Þ

Moment of soil forces above the base of caisson

pier = RMs

RMs ¼ ðqs � qÞgðKP � KAÞDc

G3

6� 2G

G1

2

G1

3

� �

Denoting G1 = dG

RMs ¼ ðqs � qÞgðKP � KAÞDc

G3

6ð1� 2d2Þ ð16Þ

For the stability of the caisson pier, the total moments

about the rear edge of the caisson pier ‘O’ should become

zero for its critical condition.

Vertical Forces and Their Moments About the Caisson

Base

The weight W and the soil reaction force FR when they are

displaced from the centre line location of the caisson and

they are assumed to act as couple is,

RFV ¼ R Vertical forcesð Þ ¼ W � FR � lFs ¼ 0

where lFs is vertical component of force due to lateral soil

pressure. This is considered to act on caisson lateral

surface.

RFH ¼ R Horizontal forcesð Þ ¼ Ff � � FS

Magnitude of lFs in comparison to W or FR is considered

to be small and hence its contribution to moment on

position ‘O’ is neglected. Since RFV = 0 then W = FR,

when these forces act at different location one can consider

them as creation of force couple displaced at a distance eb

as shown in Fig. 2a. Hence moment due to W or FR will be,

MV ¼ Web

where eb is the lateral shift between FR and W. This is

considered as stabilizing moment.

Stability Analysis of Caisson Pier

The stability analysis of caisson pier is carried by consid-

ering the moment of all forces on its rear edge ‘O’ at its

base level.

RFH ¼ R Horizontal forcesð Þ ¼ Ff � � FS

RM ¼ 0 ¼ CDqq2

2ðhþ hsÞðhDþ hsDcÞ Gþ hþ hsð ÞC½ �

� DcgðKp � KAÞðqs � qÞG3

6ð1� 2d2Þ �Web ð17Þ

Dividing by DcgðKP � KAÞðqs � qÞ ðhþhsÞ36

in Eq. 17, one

gets

Fig. 2 a Soil pressure distribution when the caisson pier slightly tilts.

b Assumed distribution of soil pressure


123

3CD

ðKP � KAÞq

ðqs � qÞq2

gðhþ hsÞ3ðhDþ hsDcÞðhþ hsÞDc

Gþ Cðhþ hsÞ½ � � G

hþ hs

� �3

ð1� 2d2Þ

� Web

DcgðKP � KAÞ6

ðqs � qÞðhþ hsÞ3ð18Þ

Denoting CD

ðKP�KAÞq

ðqs�qÞq2

gðhþhsÞ2ðhDþhsDcÞðhþhsÞDc

¼ Ff � and

W� ¼W

gðqs � qÞðKP � KAÞðhþ hsÞ3

eb� ¼ ðeb=DcÞ ð19Þ

Equation 17 becomes

3Ff �

ð1� 2d2ÞG

hþ hs

þ C

� �

� G

hþ hs

� �3

�6W�eb�

ð1� 2d2Þ¼ 0

ð20Þ

Denoting further

f ðdÞ ¼ ð1� 2d2Þ ð21Þ

Values of f(d) in relation to d is computed based on the

Eq. 21 and listed in Table 2.

Rearranging Eq. 20 in the form of cubic form in terms

of G/(h ? hs) as,

G

hþ hs

� �3

� G

hþ hs

� �3Ff �

f ðdÞ

� �

�3Ff �

f ðdÞ C �2W�eb�

Ff �

� �

¼ 0

ð22Þ

This is cubic equation similar to equation shown below,

x3 � qx� r ¼ 0 ð23Þ

whereG

hþ hs

� �

¼ x; q ¼3Ff �

f ðdÞ; r ¼3Ff �


Ff �

� �

Solution for the cubic equation is taken from Pipe [18]

based on condition27r2

4q3[ or\1

Solution when27r2

4q3[ 1

Substituting q and r, one gets

273Ff�

f ðdÞ

� �2

C� 2W�eb�

Ff�

� �2

43Ff �

f ðdÞ

� �3¼ 9

4

f ðdÞðC� 2W�eb� Þ2

Ff �[1 ð24Þ

Solution is given as

x0 ¼2ffiffiffi3p q1=2 cosh

/1

3cosh /1 ¼

3

q

� �3=2r

2ð25Þ

Substituting for q and r in the above equation, one gets

cosh /1 ¼1

2

3f ðdÞ3Ff �

� �3=23Ff �


f ðdÞ

� �

cosh /1 ¼3

2

f ðdÞFf �

� �1=2

C �2W�eb�

Ff �

� �

Then

/1 ¼ cosh�1 3

2

ffiffiffiffiffiffiffiffiffif ðdÞFf �

s

C �2W�eb�

Ff �

� �

ð26Þ

and first solution is,

x0 ¼2ffiffiffi3p q1=2 cosh

/1

3

Substituting q and /1, one gets

x0 ¼2ffiffiffi3p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3Ff �

f ðdÞ

� �s

cosh1

3cosh�1 3

2


s

C �2W�eb�

Ff �

� �" #( )

On simplification

x0 ¼ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiFf �

f ðdÞ

� �s

cosh1

3cosh�1 3

2


s

C �2W�eb�

Ff �

� �" #( )

ð27Þ

when

27r2

4q3¼ 9

4

f ðdÞFf �

C �2W�eb�

f ðdÞ

� �

\1 ð28Þ

The second solution is given as, x1 ¼ 2ffiffi3p q1=2 cos

/2

3where

cos /2 ¼3

q

� �3=2r

2¼ 3

2


s

C �2W�eb�

Ff �

ð29Þ

/2 ¼ cos�1 3

2


s

C �2W�eb�

Ff �

Then solution for second condition becomes

x1 ¼2ffiffiffi3p

ffiffiffiffiffiffiffiffiffi3Ff �

f ðdÞ

s

cos1

3cos�1 3

2


s

C �2W�eb�

Ff �

� �( )

On further simplification

x1 ¼ 2

ffiffiffiffiffiffiffiffiffiFf �

f ðdÞ

s

cos1

3cos�1 3

2


s

C �2W�eb�

Ff �

� �( )

ð30Þ

Equations (27) and (30) represents limiting state of stability

to the caisson piers when it slightly tilts.

Experimental Verification

The computation of flow parameter Ff � is carried out using

the experimental data for both cases of unstable and stable

state of piers that are based on the Eqs. 27 and 30. The plot


123

of Ff � against the grip length in terms of G/(h ? hs) is

shown in Fig. 3. The data indicating unstable condition are

marked with dark. The data indicating stable and unstable

state of piers are mixed in region G/(h ? hs) \ 0.2 and

Ff � \ 0.004 Lacey’s limiting state for stability of bridge

piers built for road and railways are also indicated in the

Fig. 3. The experimental data for unstable state falls below

these limiting values.

Limiting State of Stability for Different Values of c

and d

The stability analysis is carried for the case when d = 0,

i.e. the pier slides downstream does not tilt. In such case f

(d) = 1.0. The vertical force and the weight of the caisson

pier acts along the centre line of the caisson pier and they

are equal and opposite, hence eb� = 0. The stability

depends on the position of fluid force acting on the caisson

pier. This is represented by the value of ‘C’. The limiting

state of stability of caisson pier for d = 0, f (d) = 1.0, for

various values of ‘C ‘is carried out using the Eq. 27 or 30

and are plotted in Fig. 4. One can observe that as the value

of ‘C’ decreases, the value of G/(h ? hs) also decreases for

the given fluid force Ff � . All the stability limiting curves

fall below the Lacey’s recommendation for both road and

railway bridge piers. This indicates that the position of

fluid force acting nearer to bed level will result in less grip

length.

The effect of tilt of the caisson pier is indicated in the

Fig. 2a, b as d and its function f (d) = (1 - 2d2). Here, tilt

considered such that at eb� = 0.0. For various values of

dimensionless positions of tilt in terms of (1 - 2d 2) are

plotted for C = 1.0, 0.75, 0.667, and 0.55, in the Figs. 5, 6,

7, and 8 respectively. The experimental values for unstable

positions of caisson piers are also marked in these figures.

One can observe that the experimental values fall below the

grip length except for two points which are very near to the

limiting state of stability curves. As the d value decreases,

the limiting value of grip length also decreases. The sta-

bility limits recommended by Lacey are also marked in

these figures. In all the cases considered, the recommen-

dations advocated for the railway bridge design are safe.

However, for road bridges, the stability limit curves cross

the Lacey recommendation indicating, the magnitude of

grip length needed is more for the given flow conditions

represented by Ff � .

Limiting State of Stability

The presence of tilt such that the force couple created

between the weight force of the caisson and the load it

carries with soil reaction force FR in the form of eb� is

considered. The magnitude of eb� and W* are to be known

for this analysis. In order to see this effect on stability

curves magnitude of (eb�W�=Ff � ) is assumed as 0.1. The

stability limiting curves are computed for various values of

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.001 0.002 0.003 0.004 0.005

Ff*

G/(

h+h s

)

Caisson piers-USPNose piers-USPCylindrical piers-USPCaisson piers-SPNose piers-SPCylindrical piers-SP

- For Railway bridge - - For Road bridge

Fig. 3 Experimental data showing stable and unstable state for

different piers. (USP Unstable piers, SP stable piers)

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.002 0.004 0.006 0.008 0.01 0.012

Ff*

G/(

h+h s

)

C=1C=0.75C=0.667C=0.55

123,4

1234

Caisson piersNose piers

Cylindrical piers


Unstable

Stable

Lacey's Recommendations

Experimental points for failure of:

Fig. 4 Stability limits for grip length for different positions of fluid

force action when pier slides downstream

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.002 0.004 0.006 0.008 0.01 0.012

Ff*

G/(

h+h s

)

C=1δ=0.5δ=0.4δ=0.3δ=0.2δ=0.1δ=0.0

123, 45, 6

123456


Cylindrical piers



Experimental pointsfor failure of:-

Unstable

Stable

Fig. 5 Grip length limiting curves for C = 1.0 and for different dvalues


123

f(d) and for C = 1.0. Figure 9 shows both solid lines for

different values of d and eb� = 0.0 and dotted lines for

corresponding values of d when eb� = 0.0. One can

observe that the dotted lines fall below the corresponding

solid lines indicating that the lower limiting case for sta-

bility. It may be observed that the stability limiting curves

fall below the corresponding values of C = 1.0 and for

eb� = 0.0. This indicates the presence of weight force

which gives more stability as long as it acts as stabilizing

moment direction.

Discussions and Conclusions

The stability analysis of the caisson piers considering fluid

forces causing instability and lateral soil force causing

stability are performed for the case when pier slightly tilts

from its position. The analysis indicates that the recom-

mendations for grip length of Lacey for railway bridges

that are safe for magnitude of fluid forces considered here.

However, Lacey’s recommendation for grip length for road

bridges is in unsafe zone depending on the position of fluid

force acting and the angle of tilt.

The experimental data fall within the limiting curve for

stability analyzed.

The value of r in cubic Eq. 23 has to be positive for the

solution given in Eqs. 27 and 30. The value of r will be

positive when (eb�W�=Ff � ) \ C. The value of C is con-

sidered to vary from 0.55 to 1.0 as indicated in Table 1. To

represent the effect of (eb�W�=Ff � ), it is taken as 0.1 for the

computation and its effect is shown in Fig. 9.

The stability of the caisson piers are studied both

experimentally and by doing analysis of forces acting on it.

These limiting values are compared with Lacey’s

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.002 0.004 0.006 0.008 0.01 0.012

Ff*

G/(

h+h s

)

C=0.75δ=0.5δ=0.4δ=0.3δ=0.2δ=0.1δ=0.0

123, 45,6

123456


Cylindrical piers

- For Railway bridge - -For Road bridge

Experimental points for failure of:-


Unstable

Stable


-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.002 0.004 0.006 0.008 0.01 0.012

Ff*

G/(

h+h s

)

C=0.667δ=0.5δ=0.4δ=0.3δ=0.2δ=0.1δ=0.0

123,45,61

23456


Cylindrical piers




Unstable

Stable


-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.002 0.004 0.006 0.008 0.01 0.012

Ff*

G/(

h+h s

)

C=0.55δ=0.5δ=0.4δ=0.3δ=0.2δ=0.1δ=0.0

123, 45, 6 1

23456


Cylindrical piers


Experimental pointsfor failure of


Unstable

Stable


-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.002 0.004 0.006 0.008 0.01 0.012

Ff*

G/(

h+h s

)

C=1δ=0.5δ=0.4δ=0.3δ=0.2δ=0.1δ=0.0

1 2 3, 45, 6

123456


Cylindrical piers




Unstable

Stable 1'2'3', 4'5', 6'

Fig. 9 Grip length limiting curves for C = 1.0 and for different

values of d and (eb�W�=Ff � ) = 1.0

Table 1 Computation of C for different values of power law expo-

nent ‘n’

n 1 2 3 4 7 10

C 1 0.75 0.667 0.625 0.571 0.55

The velocity distribution for various values of n are represented as

n = 1 linear, n = 2 parabolic and n = 10 towards more uniform type

of flow


123

recommendation for grip length. The analysis of forces

indicates that the Lacey’s recommendation for railway

bridge is in the safe limits. However, the recommendation

of grip length for road bridges depends on an angle of tilt,

weight of caisson pier and load it carries (Table 2).

References

1. G. Lacey, Stable channels in alluvium, vol. 229, Paper 4736, in

Minutes of the Proceedings of the Institution of Civil Engineers,

(William Clowes and Sons Ltd., London, Great Britain, 1930),

pp. 259–292

2. R. Gopal, A.S.R. Rao, Basic and Applied Soil Mechanics. (New

Age International (P) Limited, Publisher, New Delhi, 2000),

pp. 604–638

3. B.C. Punmia, A.K. Jain, A.K. Jain, Soil Mechanics and Foun-

dations. (Laxmi Publications (P) LTD., New Delhi, 2004),

pp. 803–826

4. E.M. Laursen, A. Toch, Scour Around Bridge Piers and Abut-

ments. Iowa Highways Research Board, Bulletin. No. 4, 60

(1956)

5. C.R Neill, River-Bed Scour. Technical Publication No. 623.

(Canadian Good Roads Association, Ottawa, Canada, 1964)

6. H.W. Shen, V.R. Schneider, S. Karaki, Local scour around bridge

piers. J. Hydraul. Div. Proc. ASCE, 95, HY-6 (1969)

7. H.N.C. Breusers, Local Scour Near Offshore Structures. Delft

Hydraulics Laboratory. Publication No. 105 (1972)

8. S.C. Jain, Maximum scour around bridge piers. J. Hydraul. Eng.

ASCE 106(11), 1827–1842 (1981)

9. B.W. Melville, Scour at bridge sites Chap. 15, in Civil Engi-

neering Practice, Hydraulics/Mechanics, ed. by Cheremisinoff

et al. (1988)

10. C.J. Baker, Theoretical approach to prediction of local scour

around bridge piers. J. Hydraul. Res. IAHR 18(1), 1–12 (1980)

11. K. Kubo, M. Takezawa, in Modelling Soil–Water–Structure

Interactions, ed. by Kolkman et al (Balkema, Rotterdam, 1988)

12. W.J. Devenport, R.L. Simpson, Time-dependent and time-aver-

aged turbulence structure near the nose of a wing-body junction.

J. Fluid Mech. 210, 23–55 (1990)

13. U.C. Kothyari, R.J. Garde, K.G. Ranga Raju, Live bed scour

around cylindrical bridges. J. Hydraul. Res. IAHR, 30(5) (1992)

14. M. Muzzammil, Characteristics of horseshoe vortex at cylindrical

bridge pier models. Ph D thesis. Department of Civil Engineer-

ing, Indian Institute of Technology, Kanpur, India (1992)

15. S. Dey, Time-variation of scour in the vicinity of circular piers

vol. 136(2), in Proceedings of the Institution of Civil Engineers,

Water, Maritime and Energy, pp. 67–75 (1999)

16. G. Veerappadevaru, T. Gangadharaiah, T.R. Jagadeesh, Vortex

scouring process around bridge pier with caisson. J. Hydraul. Res.

IAHR 49(3), 378–383 (2011)

17. S. Swami, Analysis and Design of Substructures Limit State

Design, 2nd edn. (Oxford & IBH Publishing Co. Pvt. Ltd., New

Delhi, 2006), pp. 465–467

18. L.A. Pipes. Applied Mathematics for Engineers and Physicists,

2nd edn. (McGraw-Hill Book Company Inc., Tokyo, 1958),

pp. 118–119

19. IRC:45, Recommendations for estimating the resistance of soil

below the maximum scour level in the design of well foundations

of bridges. (Indian Roads Congress, New Delhi, 1972)

20. IS: 3955, Indian Standard Code of Practice of Design and Con-

struction of well foundation, I.S.I., New Delhi (1967)

Table 2 Value of f (d) for different d

d 0.0 0.1 0.2 0.3 0.4 0.5

f (d) 1.0 0.98 0.92 0.82 0.68 0.5


123

Documents

Foundation Depth for Bridge Piers