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 Missouri University of Science and Technology 

Scholars' Mine

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(1988) - S%#$ I%!! C&%%#% C!3%H3%3 G%%##! E%%

 J5 13

Observed Natural Frequency Pile Vijay K. PuriSouthern Illinois University, Carbondale, Illinois

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7/25/2019 Observed Natural Frequency Pile

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 roceedings: Second International Conference on Case Histories in Geotechnical Engineering, June 1-5 1988 St. Louis, Mo., Paper No. 4.41

Observed and Predicted Natural Frequency of a ·Pile Foundation

lfijay K.

Puri

1\ssistant

Professor

Southern Illinois University, Carbondale, Illinois,

JSA

SYNOPSIS: Vertical

and horizontal

vibra t ion

tes ts were conducted

on

a

45 em diameter concrete·driven pile.

The

pile

length was

17

m

The response

of the

pile was

computed following the

commonly

used approach of

Novak

1974, 1977) and

Novak

and

El-Sharnouby (1983). A

comparison

was made

of the predicted and observed response, the

resul ts

of

which

are discussed in the paper.

INTRODUCTION

Piles are used

to

support structures in a

var iety

of

situations

involving

both

s ta t ic

and

dynamic

loads.

Piles

are used

as

foundations

for machines and

other

vibrating equipment

when

operating conditions

of

the

machine limit the

vibration amplitudes

to very

small

values

and usual

block type foundations are not feasible.

A pile foundation supporting a machine may

be excited

in

vertical , horizontal,

rocking or torsional modes

of

vibration

depending upon the nature of unbalanced forces

generated by the machine.

The response

of

a

p i le sup

ported machine foundation is

generally

obtained by using

one

of the simplif ied approaches such

as

(1) using

the

concept

of elast ic subgrade

react ion (Barkan

[1962],

Maxwell

et al. [1969]), for obtaining equivalent soi l

springs, (2) treating the p i le as

a cantilever

fixed a t

the

lower

end, (3) t reat ing the p i le problem as a case of

one dimensional wave propagation in a rod

(Richart , Hall

and

Woods [1970]), and

4)

extending the solut ion of

Baranov 1967) for

embedded foundations

and

determining

the

st i ffness

and

damping

of the

so i l -p i le

system

from

the

elastic

half

space

approach Novak

[1974,

1977],

Novak and El-Sharnouby [1983], Novak

and Howell [1977]).

Well

designed

machine foundations

are

characterized

by

small

vibration

amplitudes.

Therefore,

l inear theo

r ies seem

adequate for calculat ing the

response

of such

foundations.

The

theories of

Novak

(1974,

1977),

and

Novak and

El-Sharnouby 1983) have

been commonly used

for

th is purpose. Very l i t t l e

data

comparing the predicted

and the observed

pile response

is presently avai lable.

In this paper an attempt

has

been

made

to compare the

observed

response of a

fu l l

scale p i le with i t s calculat

ed

response

using

the approach of

Novak

(1974, 1977),

Novak and

El-Sharnouby 1983)

and Prakash

and Puri

(1988). The cases

of

constant shear

modulus

with depth

homogeneous

soi l

profile)

and the parabolic

shear modu-

lus variation with

depth

(parabolic soi l prof i le) have

been

considered. The

resul ts

of th is comparison

are

discussed in

th is

paper.

FIELD

TESTS O

PILE

Forced horizontal

and vert ical

vibra t ion

tes ts were

conducted

on a

45

em. diameter pile

driven

17

m into

a

deposit of clayey s i l t (Puri e t a l . , 1977).

A

reinforc

ed

concrete cap

1.2 m

x 1.2

m

x 0.8 m. high was cast

monolithically with the pile

head for mounting the vibra

t ion generating equipment.

The

vibrat ions were monitored

1703

with

the help of

acceleration transducers mounted on the

pi le

at

mud l ine.

The

output

from

the acceleration t rans

ducers

was amplified using

universal

amplifiers

and

recorded on ink writing oscillographs (str ip chart record

er .

A

typical

amplitude

versus

frequency

plot for

one

of

these

tes ts

is

shown

in

Fig.

1.

Free

horizontal

vibrat ion

tes ts were

also conducted

on th is pi le

by pulling

and

suddenly releasing.

A

typi

cal free vibrat ion record

is shown

in

Fig.

2.

The values

of observed natural frequencies are

shown

in Table 1.

The

values of dynamic shear modulus at

the si te

were determin

ed by conducting

block

vibrat ion,

wave

propagation and

standard

penetration

tes ts .

The

data of these

tests was

interpreted following the approach

suggested by Prakash

and

Puri

1982,

1988). The

detai ls of the tests for dy

namic

shear modulus determination are not discussed in

this paper. The value

of

dynamic shear modulus

2t

the

level

of pi le

t ip was determined

to

be

650

kg/em •

COMPUTED RESPONSE

OF

THE

PILE

Soil and

Pile

Properties

Diameter of the pile d = 45.0 em

Embedded length 1 = 17.0 m

5

2

Young s modulus of

concrete E

= 2.2

x

10

kg/em

p

The weight

of

pile

cap and

pile

= 2686.5

kg

above

the

mud

l ine

Mass moment

of

iner t ia

of

H

= 4686.4 kg

em/ 2

pi le cap and pile (above mud line) about horizontal

axis

of vibrat ion M

m

Dynamic

Shear

Modulus of

Soil

= 650 kg/cm

2

a t

the level of

pile t ip

Gs

Natural Frequency of Vertical Vibrations

The

natura l

frequency of the pile in ver t ical

vibrations

was

calculated from

Eq. 1.

/ -.

lk :

. - 2 T J ;;

(1)

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in

which

Natural

frequency

of ver t ical

vibrat ions, Hz

kz Equivalent spring for the soi l -pi le system

for ver t ical vibrations

and

m= Mass

of the cap

and pile above

the

mud

l ine.

The value of k:r is

obtained

from Eq. 2 (Novak and

El-Sharnouby

1977).

EPA

kz

=

f...

0

2)

in

which,

EP = Young

1

s modulus of pile

material

A = Area of cross-section

of the

pile

ro

Pile

radius

and

Stiffness

parameter.

The values

of fw

1

for

the

case of

homogeneous

soil profi le

and

parabolic soi l profile

are obtained from

Fig. 3.

The

values

of the natural frequency

of vertical

vibrations so

calculated

are

shown

in Table 1.

Natural

Frequency

and

Amplitude

of Horizontal

Vibrations

The undamped natural

frequencies

of

horizontal

vibrations of the

soi l

pile system were computed

by

treat

ing

i t

as a case of coupled rocking and

sliding

and using

Eq. 3 .

w2 = ~ ± \/ ~ _ k; 3)

1,2

2 m

M . ·

4 m M . mM'

in which, w,.

1

and w,.2 are the two natural frequencies in

coupled

rocking and sl iding.

kx =

Translation stiffness

coefficient

k4>

=

Rotational stiffness

coefficient

k x ~

=

Cross-stiffness

coefficient

and, Mm =

Mass moment of

inertia

of

the

pile and pile

cap.

The values

of

kx ,

k.;

and

k..,.;

are obtained following the

approach

of

Novak and

El-Sharnouby

(1983) and

Prakash

and

Puri

1988).

4a)

4b)

4c)

in

which,

=

Moment

of iner t ia of pile cross-section

and fxt , f. and fx<PI are

st i ffness parameters that are

obtained

from

Table 2.

From

the

calculated values of

w .1;z

the two natural

frequencies/,.

t,:z are

O:btained from Eq.

5.

/ 1.2 = w

1

 z Hz

21r

5)

The

values of / ,

1.:2

for

the homogeneous soi l

profi le

and the parabolic soi l

profi le

are shown in

Table

1

in

which the natural

frequency

of free vibrations is also

shown.

17 4

The

amplitude

of

horizontal vibrations

Ax

a t

mud

l ine

was

calculated

for

different

operating

frequencies

in the range 7 to 16 Hz by using Eq. 6 (Beredugo and

Novak,

1972).

Ax =Px

6)

in which Px

=

Horizontal

exciting force.

The values

of

a

1

,

a

2

,

E

 

and E2

for use

in Eq.

6

are

obtained as follows.

in which,

and

w

=

Operating speed

in

ex= Translational damping constant

c$= Rotational damping constant

Cross

damping constant

7a)

7b)

7d)

The values of ex c$ and are

obtained

as follows

(Novak and El

Sharnouby, 1983).

EPIP

c.=V .;2

E/P

Cx.;=

7V

x4>2

0

s

in w h i ~ h V

8

Shear wave-velocity in so i l

(Sa)

(Bb)

(8c)

and

fx2 ,

/.;z and fx<P2 are damping parameters

that are

obtained from Table 2.

The values of horizontal

amplitude

Ax

were

calculated

for

the cases of homogeneous and parabolic

soi l

profi le .

The calculated values of peak resonant) horizontal

amplitudes

for

the

soi l pile system

for al l cases

are.

given

in Table 1. The calculated values of horizontal

amplitudes

at different frequencies have been

plotted in

Fig.

1.

DISCUSSION AND

CONCLUSIONS

1. The

computed

natural frequencies of ver t ical

vibrations of the pi le

for

the homogeneous and parabolic

soil profi les

are

46.0 and 38.8

Hz

respectively Table 1).

The observed natural frequency

of

ver t ical vibrations

is

32.2

Hz. The computed values of natural

frequency

for

the

homogeneous soi l profile is

43%

higher than

the

observed

natural frequency

of

vertical vibrations. For

the para

bolic soil profile, the calculated natural frequency

is

20.5% larger than

the observed

natural frequency.

2.

For the case of

coupled rocking and sl iding, the cal

culated

values of

smaller

natural frequency

fn

1

are

30.9

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1d

11.89 Hz for the homogeneous and parabolic soi l

rofiles

respectively

(Table 1). The observed natural

requency

is

10.3

Hz. The calculated natural

frequency

)r the uniform

soi l

profile

is

substantially higher than

1e observed natural frequency of horizontal vibrations.

)r the case of parabolic

soi l

profile

the

calculated

atural frequency is

about

15% higher

than

the

observed

atural

frequency.

The computed natural frequency of

horizontal free

lbrations for

the

parabolic soi l

profile

is 12.9 Hz

and

s 12% higher than the

observed

frequency of

free

lbrations

(Table 1).

The observed

values

of

the peak horizontal

vibration

nplitude is

0.44

mm. which is higher than the calculated

nplitudes

for

the

homogeneous soi l profile (0.08745 mm.)

1d

the parabolic

soi l profile (0.116 mm.). In the fre

~ e n c y range considered (Fig.

1), the

computed amplitudes

E horizontal

vibrat ions are

generally

smaller

and

near

~ s o n n c e

they are substantially smaller

than

the

)served values.

Because of

the limited nature of

the

study,

i t

is not

)Ssible

to

draw any general conclusions,

but

i t seems

~ a t in this particular case the assumption

of

a

parabol

:

soi l

profile has given reasonable

values of

natural

requencies both

for

the case of

vert ica l as well as

)rizontal vibrations.

EFERENCES

arkan, D. D.

(1962).

Dynamics

of

Bases and

Foundations. McGraw-Hill,

New

York.

aranov, V. A. (1967).

On

the calculation

of

excited

vibrations of

an embedded

foundation

(in

Russian).

Vopr.

Dyn.

Prochn. 14, 1 9 5 ~ 2 0 9

Beredugo, Y.

0 .

and Novak, M. (1972). Coupled

horizon

tal and rocking

vibration

of embedded footings. Can.

Geotech. J . 9(4), 477-497.

Maxwell,

A. A., Fry,

z. B., and

Poplin, J . K. (1969).

Vibratory loading of pile foundations.

ASTM Spec.

Tech. Publ. STP

444,

338-361.

Novak, M. (1974). Dynamic stiffness and damping

of pi les.

Can. Geotech.

J. (11(4),

574-598.

Novak, M. (1977). Vertical vibration

of

floating piles.

J. Eng. Mech. Div., Am Soc. Civ. Eng. 103(EM-1),

153-168.

Novak,·M., and

El-Sharnouby, B.

(1983).

Stiffness

and

damping constants of single piles. J. Geotech. Eng.

Div., Am. Soc. Civ. Eng. 109(GT-7), 961-974.

Novak, M.,

and Howell,

J. F. (1977). Torsional

vibrations

of

pile

foundations.

J . Geotech. Eng.

Div.,

Am Soc.

Civ. Eng. 103(GT-4), 271-285.

Prakash, S.,

and

Puri, V.

K. (1988), Foundations for

Machines: Analysis and Design. John Wiley and Sons,

New York.

Puri, V. K., Bhargava, S., Nandakumaran, P., and Arya,

A.

s.

(1977), Evaluation of

Dynamic

Soil

Pile

Con

stants

from In-Situ

Tests.

International Symposium

on Soil-Structure Interaction, India.

Richart, F. E.,

Hall, J . R.,

and Woods, R. D.

(1970).

Vibrations of Soils and

Foundations.

Prentice-Hall,

Englewood Cliffs ,

New

Jersey.

TABLE 1. Comparison of Observed and Computed

Data

OBSERVED VALUES

COMPUTED VALUES

UNIFORM

PARABOLIC

VIBRATION

ITEM

SOIL

SOIL PROFILE

MODE

FORCED

FREE

PROFILE

VIBRATIONS VIBRATIONS

FORCED

FORCED FREE

VIBRATIONS

VIBRATIONS

VIBRATIONS

VERTICAL

f nz

Hz

32.2

-

46.0

38.8

-

HORIZONTAL

fnl

Hz 10.3

11.5 30.9

11.84

12.9

fn2

Hz

-

 

77.6

45.7

I

-

Ax

mm

0.44

-

0.08745

0.116

I

1705

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E. . .J

v

G• ,u

( ) (2)

0.25 10,000

2,500

1,000

500

250

0.40

10,000

2,500

1,0110

5110

250

0.25

10,000

2,500

I,IXIO

5110

250

11.40

10,000

2,5110

I,INIII

5110

250

0.44

0.40

o. 36

0.32

0.28

....

l 0.20

0.16

0.12

0.08

0.04

Table 2. Stiffness and Damping

Parameters

for

Horizontal

Vibrations of Long Piles

(Novak and

El-Sharnouby,

1983)

Stiffness Parameters

· ~ ·

/.,

g

(3) (4) (5)

(6)

(a)

1/omogelleOll

Soil Profile

0.2135

-0.11217

0.0042

0.11021

0.2998

-0.0429

O.Oii9

0.11061

0.3741

-0.0668 0.0236 0.0123

0.4411

-0.0929 0.0395

0.11210

0.5186

-0.1281 0.11659

0.11358

0.2207

-0.0232 0.1111.47 11.111124

0.3097

-0.11459 0.0132

ll.llll68

0.3860

-0.0714 0.11261

0.0136

0.4547

-0.0991

0.0436

0.11231

0.5336

-0.1365 0.11726

1).0394

(b)

Pt1rt1bolic Soil

Profile

0.18110

-11.11144

O.IMI19 O.IIIMJ8

0.2452

-11.11267

O.IMI-47

0.111120

0.3110()

-11.1141111

O.IMJ86

0.1Xl37

0.341 9

-0.11543

11.111.36 O.IM159

11.41149

-0.117.14

11.11215

0.111194

0.1857

-11.1115]

11.1111211

11.1111119

0.2529

-11.1121 -1

11.111151

II.IM122

0.309-1

-0.11426

11.11119-1

11.111141

0.3596

-11.11577

11.11149

0,11()()5

0.41711

-0.07811

0.0236

1Ullll3

g, and

are

for

pinned

end.

LEGEND

Q

Obset Ved

1:1 Computed-Para-

bolic Soil Profi le

A

Computed-Uniform

Soi l

Profi le

10 11 14

16

18

Frequency

Hz

Damping

Parameten

/,H

··l

/.<l

(7)

(8) (9)

0.1577 -0.0333 0.0107

0.2152 -0.0646

0.0297

0.2598

-().()985

0.0579

0.2953 -0.1337

0.0953

0.329 )

-0.17tl6

0.1556

0.1634

-0.0358

0.0119

0.2224

-0.(1692

0.0329

0.2677

-0.1052

0.0641

0.3034

-0.1425

0.1054

0.3377

-0.1896

0.1717

0.1450

-0.0252

0.0060

0.21125 -0.1)484

0,()159

0.249 )

-0.0737

0.0303

11.29111

-11.111111

0.0491

0.3361 -0.1370

0.0793

11.15118 -0.11271

0.11067

11.21111

-11.11519

0.0177

11.258 .1

-11.117911

0.0336

11.31MI9

-0.11179

0.0544

0.3468 -11.1461

0.0880

~ ~ ~ ~ ~ ~ ~ ~ ~ - - ~ - - _

,..;

;::

~ C 1 : 1 ? f - ~ l 2 ' - = ; : = = 2 = = = i = = . . . . ,

,..;

ell

0 . 0 4 1 . / : A ; ~ : : : : : : : : ; ; : : 6 1 : = ~ = ~

;:j O . Q £ 1 - . - . , , . C . . - ~ ~ ~ S . . - ~ ~

0

I

...2;

- st1ffnes s

{

2

a.mgtnK

I

I ' c ~ • : SO

f z

(10)

0.0054

0.0154

0.0306

0.0514

0.()864

0.0060

0.0171

0.0339

0.0570

0.0057

0.0028

0.0076

0.0147

0.0241

0.0398

().()031

0.0084

0.0163

0.0269

0.0443

]

-

 ..

a)

]

Fig. 1.

Typical

Amplitude

vs.

Frequency Plots

b)

Fig.

2.

0.1

0.2 sec.

Typical Free aorizontal)

Vibration Record

1706

Fig.

3. Stiffness and Damping For

Fixed-Tip Vertical ly Vibrating Piles

a) Homogeneous

b) Parabolic

Soil

Profiles (Novak and

El-Sharnouby,

1983)