dynimac behavior of vertical cylinde rdue the wave force

8/10/2019 Dynimac Behavior of Vertical Cylinde Rdue the Wave Force

1/19

CHAPTER 4 3

DYNAMICBEHAVIOR

OF

VERTICALCYLINDER DUETO

WAVE

FORCE

Toru

Sawaragi*

andTakayukiNakamura**

ABSTRACT

This

paper

describes

the

dynamic

behavior

ofa

fixed

cylindrical

pile

due

toboththe

in-line

or

longitudinal

force

and

lift

or

transverse

forcein

regular

waves.Resonant

response

of

the

pile

duetothe

lift

force

in

the

directionnormal

to

thewavepropagationdirection

is

dis-

covered

attheperiodratios

of

T

w

/T

n

=2,3,4,5and

6

( T

w

:the

waveperiod,

T

n

:

the

naturalperiod

of

thepile).

Furthermore,

the

resonant

responses

in

the

wave

propagationdirectiondue

to

the

in-lineforcealso

appearat

the

sameperiodratios,inaddition

to

thewellknownresonancepoint

o f

T

w

/T

n

=l.Moreover,dynamicdisplacements

of

the

pile

in

thedirection

normal

to

thewavepropagation

direction

are

longer

than

those

in

the

wavepropagationdirection whentheperiodratioislonger

than1.6

and

Keulegan-Carpenternumberi slargerthan

6 .

Next,forthe

purpose

of

the

ocean

structural

design,the

methodsof

estimating

the

dynamicdisplacements

n

bothdirectionsand

ofestimating

thedynamicdisplacements

onsideringbotharederivedbyusing

Morison's

equation

and

lift

forceequationformulatedbytheauthors.Thedisplace-

ments

calculated

are

compared

exactly

with

the

experimental

results

to

investigate

thevalidityof

theproposedmethod.

INTRODUCTION

In

recent

studies

ofwave

force

on a cylindricalpile,ithasbeen

discovered

hat

a

lift

forceacts

onthe

pile

in

the

direction

normal

to

the

wavepropagationdirection,

in

additionto

a

in-line

forceacting

onthepile,

as

describedby Morison's

equation,

in

the

wave

propagation

direction.

It

was

pointedout

by

Bidde ,Sarpkaya

2

and

the

authors

3

that

thelift

forcehas

a

magnitudeas

largeas

the

in-line

force,and

thatthe

frequencyof

thelift

forceishigher

than

that

of

the

wave

and

the

in-line

force.Onthe

other

hand,

considering

the

fact

that

he

natural

frequency( f

n

)

is

generallyhigher

than

the

wavefrequency( f

w

) ,

thelift

force

may

be

important

when

the

resonance

response

ofa

fixed

off-shore

structure

n

waves

is

examined.

In

fact,

Wiegel

et

al

reported

that

2-foot

pilevibrates

largely

withthe

vibration

period

of

2. 5

seconds

inthedirectionnormaltothewave

propagation

direction

due

to

thealternatebreakingof

the

largevortices

under

the

largewave

condition

with

thewaveperiod

being

about13

seconds.

And

theyalso

reported

that

the

testpile wasbrokenby

the

latteralvibrationdescribed

above.

With

the

above-described

background,first,

in

this

paper,

the

influ-

ence

ofliftforce

onthe

dynamicresponseof

a

cylindrical

pile

of

cantilever

type

was

investigated

byexperiments,and

the

effects

of

a

periodratio( T

w

/Tn)r

a

frequency

ratio

( f

w

/f

n

)and

Keulegan-Carpenter

number

forthedynamicresponse

are

discussed.

( T

w

the

waveperiod

and

*Professor,

Departmentof

Civil

Engneering,

Suita,

Osaka,565,

Japan

**Assistant

Professor,

Department

of

Ocean

Engineering,

Ehime,Japan

2 3 7 8


2/19

BEHAVIOR OFCYLINDER

2379

equal

to

l/f

w

,T

n

:

thenaturalperiod

of

thepileand

equalto

l/f

n

).

Secondly,In

order

to

estimate

the

dynamic

response

in

the

in-line

andnormal

direction,

equationsondynamicdisplacements

intwodirections

are

derived

byusing

the

Morison's

equationonthe

in-line

forceand

the

liftforce

equation

formulated

by

the

authors. Furthermore,

the

combineddynamicdisplacement

iscalculated,

and

these

calculated

results

are

compared

with

the

experimental

results.

EXPERIMENT

The

wave

tank

used

in

this

experiment

was

a0.7mwide,0.95m

deep

and

30mlong

wavechannel

at

the

Hydraulics

Laboratoryof

Civil

Engineering,

Osaka

University.

flaptypewavegeneratorwas

located

atoneendof

thewave

tank

and

apebblebeach

was

installedat

the

other

endof

the

wave

tankto

absorb

the

waveenergy.

Model

cylinders

usedin

thisexperiment

were

two

kinds

of

cantilever

type

structure

withaconcentrated

mass

at

it s

top

asshownschematically

in

Fig.

1(A)

and

( B ) .

Each

model

pile

consisted

ofthree

parts,i.e.,

a

concentrated

mass,

a

circularcylinder

anda

spring

bar. The

mass

was

madeofsteelandhad

the

samediameter

as

that

of

the

cylinder.

Th e

springbar

was

also

made

of

steelandhada

circular

crosssection

with

diameter

of

5=5mm

for

themodel

pile

ofFig.

1(A)and

5.9mm

for

that

of

Fig.

1(B).

The

modelpile

of

Fig.l(A)wasfixed

on

theshelfin

the

squareboxmadeof

steel

with

the

same

height

as

that

of

the

horizontal

flatbed. Inthis

case,a2.5cmcylindermadeofarcylicresin

was

used

foracircularcylinder

and

the

water

depth

waskept

constantat

35cm

above

thehorizontal

bed.

On

the

other

hand,the

model

pile

of

Fig.1(B)

was

fixed

onthechannel-shaped

steel

havinga

heightof

5cm

that

was

rigidlyconnected

to

the

bottom

of

the

wave

tank.

In

this

case,

a

3cm

cylinder

was

used

andthe

depth

of

the

water

was

kept

constant

at

65cm

above

the

bottomof

the

wave

tank.

The,model

pile

ofFig.

1(A)

was

used

only

for

the

purposeofmeasuring

the

dynamicresponse

inthecomparatively

small

ranges

of

T

w

/T

n

andtheone

ofFig.

1(B)

was

used

for

thatof

T

w

/T

n

being

large.

Inthis

experiment,

fivekindsofconcentrated massweremountedon

these

modelpiles,considering

the

efficiencyof

the

wave

generatorand

valuesofperiod

ratio(T

w

/T

n

).

Th e

values

of

thesemasses

are

tabulated

in

Table

1(A)

and

( B )

for

the

model

pile

ofFig.1(A)and( B )

respectively.

In

thistable,

the

natural

period

T

n

and

thenatural

frequency

f

n

of

thepile

measured

from

theexperiment

of

free

vibration

in

water,

andthe

logarithmic

decrement

5measured

from

the

experiment

offree

vibration

in

air

arealso

tabulated

foreach mass.

In

order

to

clarify

the

effects

of

Keulegan-Carpenter

numberand

theperiod

Fig

Structuralmodelof

or

frequency

ratio

onthe

dynamic

experimental

cylinders

responseofthemodelpile,theregion


3/19

2380

COASTAL

ENGINEERING1978

EXP.

CYL.

mass

(g)

T

(sec)

(Hz)

6

(A)

0.276

0.599

0.914

0.504

0.740

0.930

1.984

1.351

1.075

0.040

0.043

0.045

(B)

0

0.142

0.298

0.386

3.356

2.591

0.993

0.053

of

the

modelratio

( T

w

/T

n

)wave

fixed

between0.8

and

7.5,

and

the

rangeof

rmsK-C

number

( r i t i s

K-C),

whichi s

theroot

mean

square

valueof

eulegan-

Carpenter

number

at

each

verti-

cal

elevation

ofthecylinder,

wasfrom

2

to20.Therangeof

rms

Reynoldsnumber(

rmsRe

) ,

whichisthe

root

meansquare

valueof

Reynolds

number

at

each

verticalelevationofthe

cylin-

Table1

ynamic

characteristic

der,wasfrom about2000to8000.

of

tne

model

pile

The

wave

condition

used

inthis

study

was

as

follows,

the

wave

height

was

fixedbetween2cm and16cm,andthatof

thewave

period

was0.6sec

to2.3sec.

Inthisexperment,

a

16-mmcine-cameraas

located

right

above

the

pile

to

measure

the

dynamic

displacement

at

the

top

of

it .

Also

the

strain

gagesweremountednear

the

fixed

endof

the

cantilever

to

measurethe

dynamicoverturningmoment

in

bothdirections.

Thewavegageusedwasa

parallel-wireresistance

type

and

was

installesat

the

side

of

themodel

pile.

Furthermore,

in

ordertosynchronizethe16mm-movierecord

with

recordsofwatersurfaceelevationand

dynamic

moment,pulsesignalsof

10Hzwereutilized.Themovie

records

were

analyzed

withan

electronic

gragh-pen

system,and

then

locusof

thetopof

the

model

pile

was

re-

producedit hagraphicdisplaysystem.

DYNAMICBEHAVIOROF

THE

MODEL

PILE

1 )

DYNAMIC

LOCUS

OFTHE

MODEL

PILE

Typical

lociofthetopofthemodel

pile

during

onewave

cycle

of

theincidentwave

(except

(B-l))

areshownschematically

in

Fig.2with

the

frequency

ratioas

a parameter.

In

Fig.

2 ,the

X-axis

is

the

direction

of

the

wavepropagation

direction

and

Y-axis

is

the

direction

normal

to

the

wave

propagationdirection.

From

this

figure,the

following

results

areappeared.

( A ) Inthe

range

of

frequency

ratio

( f

w

/f

n

)largerthan

0.9 Fig.2

(A-l)

~

(A-3)),thedisplacementof

thetopof

thepilein

theX directionis

predominant

in

comparison

with

that

in

the

Ydirection

andthelocusshowsanealystraightline in theXdirection.Becauseof

the

well-known

resonance

atf

w

/f

n

=

ldueto

the

in-line

force,the

pile

vibrates

largely

inthe

X

direction.

Inthis

case,

the

frequencies

of

thedisplacementsinboth

directions

possess

the

wave frequency

as

shown

in

Fig.

3

( A )

Here,

Fig.3

shows

the

time

historiesof

thedisplacements

in

bothdirectionsandcorrespondstothe

locus

shown

in

Fig.

2 ,respec-

tively.

( B )

In

the

range

of

frequency

ratioranging

from

0.6

to

0.9,

the

locuslookslinealetter

of

infinitysign

(oo),

as

shown

in

Fig.2

(B-l)and(B-2)

0

In

this

case,

theY-displacement hasthesecondharmonic

frequency,asshowninFig.

3

( B ) ,but

the

X-displacementhas

only

the

wavefrequency.

( C )

Whenthefrequency

ratio

ranges

from0.4to0.6,

the

locusis

nearly

a double

ellipse

asshowninFig.

2(C-l)and

(C-2).

Inthiscase

,

theY-displacementis

much

greater

thantheX

displacement,


4/19

BEHAVIOROF

CYLINDER

2381

(A-l)l

Ycm T

w

=0.85

1:

Xcm

H=6.1

fw/fn-1.10

ms

K ~C

CI

~4.2

(A-3)

H>

(A-2)

Xcm

T

w

=0.94sec

H=3.3cm

f

w

/f

n

=0.99rmsK-C=2.6

lYcm

Xcm

T

w

=0.75sec

H=7.3cm

fw/fn=0.99rmsK-C=4.2

- f n

0

0 .

0

1.1

B-D^icm

T

w

=1.10sec

Xcm

H=11.3cm

fw/fn=0.82

rmsK-C=11.2

^

Ycm

(B-2)

T

w

=1.13sec

0.9

- H

Xcm

1 H=8.4cm

f

w

/f

n

=0.66

rmsK-C=8.5

o;

1.1

1.7

(C-]>

.Ycm

T

w

=1.50sec

H=7.7cm

Xcm

f

w

/fn=0.49

rmsK-C=ll.l

Ycm (c-2)

T

w

=1.50sec

)Xcm

H=3.7cm

0.6

f

w

/f

n

=0.49

rmsK-C=5.3

0.4

1.7

2.5

?

m

T

w

=1.60

sec

H=9.6cm

Xcm

f/f

n

=0.33

rmsK-C=15.0

Ye

i

(D-2)

Xcm

T

w

=1.61sec

H=7.4cm

f/fn=0.32

rmsK-C=ll.6

Ycit

y

D

-

3

T

w

=2.30sec

H=3.9cm

f

w

/f

n

=0.32

rmsK-C=9.1

0.4

A

0.3

2.5

A

3.3

(E-l)

Ycm

Tw

=i.55

sec

H=13.4cm

Xcm

Ycm

5

(E-2)

T

w

=1.55sec

JCcm

H=7

.

5cm

5

f

w

/f

n

=0.25

rmsK-C=8.7

ft

0.2:

0

4

I ]

CF-1)0.5..Ycm

fw/fn

_ ,cm

0.5

yT

w

=1.49

=6.20

H=14.3cm

rmsK-C=15.9

5

IYcm

(

F

_2)

0.5

Xcm

iw

=

l-49sec

=9.3cm

f

w

/f

n

=0.20

rmsK-C= Q.4_

5

Ycm

(F-3)

,, T

w

=1.49

nlc

m

sec

'

5

H=6.8cm

f

w

/f

n

=0.20

rmsK-C=7.6

ft

D.2

0

0.20

0

5

U

(G-l)

Ycm

T

w

=l

.80se(

_ ctn

-

5

H=l

0.3cm

f

w

/fn=0.17

rmsK-C=14.0

0.5

|

Ycm

(G-2)

JCcm

Tw=

i.80sec

-

5

H=8.

0cm

fw/fn=0.17

rmsK-C=10.9

0.17

6

Fig.2oci

f

ynamic

isplacements

t

heop

of

he

ylinder


5/19

2 3 8 2

COASTALENGINEERING1978

rmsK-C=2.6

T

w

=0.94sec

H=3.3cm

X(cm)

( C )f

w

/f

n

=0.49rmsK-C=ll.l

T

w

=1.50secH=7.7cm

X(cm)

( B )f

w

/f

n

=0.823

rmsK-C=11.2

T

w

=1.10sec=11.3cm

X(cm)

2

Tw

n

f lA/lftrt

af \

rJ \

-A rA f u flMl-fMMft/lMMln/W

[VTUYITTO

T

irn

'vyirW

I

0 5 10 15 20 25

( E )

f

w

/f

n

=0.25

rmsK-C=15.6

T=1.55secH=13.4cm

( D )f

w

/f

n

=0.33

rmsK-C=15.0

Tw=1.60secH=9.6cm

X(cm)

Tw

10

1 5

20

2 5

t(sec)

(F)

f

w

/f

n

=0.20

rmsK-015.9

T

1.49sec

=14.3cm

X(cm)

11

Tw

-1

Y(cm)

10

t(sec)

F i g .3

ime

history

o f

t h e

Xa n d

Ydisplacement


6/19

BEHAVIOR OFCYLINDER 383

because

the

pile

i s

resonantedby

thelift

force

componentwith

the

frequency

two

times

as

large

as

the

wave

frequency.

Consequently,

the

Ydisplacementvibrateslargely with

the

secondharmonic

frequencyo

the

wave

as

shown

in

Fig.

3( C ) .

Onthe

other

hand,

the

X

displacement

has

oth

the

wavefrequency

and

the

secondharmonicfrequencyof

the

wave.

Since

thecharacteristics

of

the

lift

force

frequencywill

be

given

later,

readers

may

want

to

refer

t o

Fig.

6 .

( D ) Witha

frequency

ratio

rangingfrom0.3t o

0.4

the

locus

shows

a

long

ellipse

and

a

triple

ellipse

asshown

in

Fig.2(D-l),

(D-2)and

(D-3).

In

this

case,the

pile

i s

resonantedatf

w

/f

n

=l/3by

the

lift

force

component

which

corresponds

t o

thethird

harmonicfrequencyofthe

wave.

Therefore,

theYdisplace-

mentvibrates

largely

with

the

frequency

asshowninFig.3( D ) .

From

this

figure,

i tcan

be

seenthat

the

Xdisplacement

has

also

thethird

harmonic

frequencyofthe

wave

inaddition

to

the

wavefrequency,and

like

the

case

of

( C ) ,the

Ydisplacement

i s

largerthan

the

X

displace-

ment.( E ) Whenthe

frequencyratioisnearlyequal

to

0.25,

the

locus

i s

similarto

the

figure

of

a

tetra

ellipse

and

he

Y-displacement

hasalsothe

more

significant

magnitude

compared

ith

the

X

displace-

ment

(see

Fig.

3(E)).

Furthermore,

the

smaller

the

valueof

the

frequency

ratio,

asshown

inFig.2

(F-l),

(F-2),(F-3),(G-l)

and

(G-2),the

more

complicatedthedynamiclocusbecomesowing

totheappearance

ofhigher

harmonic

frequency

componentsinbothdisplacements,and

in

therangeof

frequencyrationearly

equalto1/5

and

1/6,

i t

can

be

seen

that

the

Ydisplacement

cannotbeneglected

in

comparison with

the

Xdisplacement.

Here,

the

effect

ofrmsK-Conthe

locusi snotclearlydistinguishable,

butthe

followingfeatures

may

bepointed

out:when

thefrequencyratio

nearly

equals

1 ,theYdisplacement

appears

onlyatcomparatively

small

values

of

rmsK-C,

and

in

therangeof

frequency

ratiosmallerthan

0.9,

the

Ydisplacement

decreaseswithdecreasing

values

of

rmsK-C

and

the

Y

displacement

is

equal

to

orsmallerthanthe

X

displacement

when

the

valueof

rmsK-C

i s

comparatively

small.

Thereason

for

thehigherharmonic

frequencyof

thewave

of

the

X

displacementwillbepresented

later.

2 )

RESONANT

CHARACTERISTICS

OF

THEPILE

Inorderto

examinethe

resonantcharacteristics

of

the

piledue

to

thein-lineandlift

forces,

theresonant

curves

inbothdirectionswere

obtained.

Fig.4

and

Fig.

5

show

theresonant

curves

inthe

X

and

Y .

directions

respectively

withrmsK-C

asa

parameter.

In

these

figures,

theabscissa

istheperiodratio(l/(f

w

/f

n

))and

the

ordinatei s

the

so-called

amplification

ratio,

i.e.

the

ratio

of

thedynamic

displacement

t o

the

static

displacement

due

to

thewaveforces. Here,the

static

displacements,

X

s

and

Y

s

,

are

calculated

by

means

of

thestructural

model

shown

in

Fig.

9 ,and

byusing

the

Morison's

equation

onthe

in-line

forceand

the

lift

force

equation

( E q .

7 )derived

bythe

authorsonthe

lift

force.

The

linear

wave

theory

is

also

used.

Thewave

force

i s

integrated

fromthe

bottom

of

thecircularcylinder

to

theelevating

water

surfaceas

asinusoidalwave.

In

these

figures,

X-^and

Yp^are

meausred

one-tenth

maximumdynamic

displacements

of

X

and

Y

respectively,

since

the

Ydisplacement

was

irregular

inregular

waves

asshown

in

Fig.

3 .

From

Fig.

4 ,

itisclear

that

theresonant

responsedue

to

a

in-line


7/19

2384

COASTAL ENGINEERING1978

SYM.

rmsK-C

o

0-3

o

3-6

6-10

10-15

o

15

T

w

/T

n

Fig.4

amplification

atio

n

X-direction

0

23

M

if

*^A

S

V

T

w

/T

n

force

appears

atthe

period

ratiosT

w

/T

n

=1,2,3,4,5

and6 ,

but

there

isno

response

atthe

period

ratio

T

w

/T

n

=7 . Among

these

resonances,

the

well-

knownresonance

at

T

w

/Tn=l

i s

the

most

predominant,

butthe

resonance,at

the

period

ratio

T

w

/T

n

=2and

3arealso

comparatively

large.

The

reason

for

the

appearance

of

the

response

at

T

w

/T

n

=2

and

3

maybe

due

to

the

fact

that

the

in-lineforce

( F x )

andtheover-

turningmoment( M

x

)

caused

bythein-

lineforcehave

higher

frequency

componentsthan

the

wave

basedon

the

non-linearityofthe

dragforceand

the

finite

amplitude

natureofthewater

wave. Agoodexample

illustratingthis

fact

ispresentedin

Table

2 .

Thistable

shows

the

resultof

aharmonic

analysis

of

F

x

and

M

x

acting

on

averticalcircu-

i g . 5

Amplificationr a t i o

i n Y-direction

larcylinderinwavesduringone

wave

cycle.

Here

F

x

andM

x

arecalcul-

ated

byusingtheMorison's

equation

andthe

linearwavetheory.

Table

2

( I )

is

the

result

of

the

consideration

of

the

effect

of

the

finite

amplitudenature,

the

waveforcestillbeingconsideredasasinusoidal

wave,i.e.the

integral

regionof

the

wave

force

is

from

the

bottom

of

the

circularcylinderto

the

elevatingwater

surface.n

the

other

hand

Table

2

( I I )

indicates

the

resultof

neglecting

the

above^described

effect.

Itisseen

that

F

x

and

M

x

have

higher

harmonics

than

the

wave

frequency

as

shown

in

Table

2( I )and

(II). Moreover,i t

isclear

that

these

components

with

thesecondharmonic

frequency

ofthewavediffer

significantlybetween

( I )

and

(II),

butthissignificantdifference

between( I )and

( I I )

cannotbeseen

when n=3.

The

non-linearityofthe


8/19

BEHAVIOR OF CYLINDER

2 3 8 5

nf

w

(I)

(ID

n=l

18.46x10-3

18.38xl0"

3

(Kg)

(Kg)

2

1.63

0.10

FX

3

2.13

2.08

4

0.06

0.10

5

0.31

0.31

6

0.08

0.10

n=l

353.63xl0"

3

348.25xl0"

3

(K gcm ) (K gcm )

2

56.03

1.89

M

X

3

43.96 40.98

4 3.76

1.89

5 5.84

6.15

6

1.29

1.89

T=l.5sec

H=8cm

rmsK-C=

11.53 D=2.5cm

Table

2

Harmonic

anlysisof

F

x

andM -

mostpredominant

incaseof

rmsK-C

beinglargerthan

3

dragforceand

the

effectofthefinite

amplitudenatureof

the

water

wave

on

the

dynamics

of

the

pile

willbe

described

later

on

in

detail.

FromFig.

5 ,

itis

evident

that

the

reson-

ant

response

due

tothe

lift

force

appears

at

the

same

period

ratios

as

those

in

theX

dir-

ection. Inthis

case,

however,

the

resonant

condition

depends

on

rmsK-C,

i.e.

the

reso-

nance

at

T

w

/T

n

=l

is

predominantforvalues

of

rmsK-C

smaller

than

3 ,andthe

resonances

atT

w

/T

n

=2and3arethe

I t

may

be

considered

that

thesefacts

have

a

close

relationwiththe

frequency

characteristics

of

a

lift

forceand

the

magnitudeasshowninFig.

6

and

7 o Fig.

6

shows

the

variation

of

thepredominantnon-dimensional

lift

energy

Sj

J

nf

w

)M/a

2

i,

n=l-4)

foreach

harmoniccomponent

of

the

wave

frequency

withrmsK-C.

This

figurewasobtainedby

using

the

experimen-

tal

result

of

thewave

forceonarigidly

supported

vertical

circular

cylinder

and

was

presented

in

Ref.(3),

too.

Here,

SL(nf

w

)Af

is

the

lift

energy

for

the

n-th

harmonic

of

the

wave

frequency,

and

a

2

,

is

the

var-

ianceof

the

lift

force. Fromthis

figure,

i t

canbeseenthat

the

predominant

lift

frequency

equals

the

wave

frequency

inthe

rangewhere

rmsK-Cissmallerthan

3

approximately,correspondstothesecondharmonic

frequencyof

the

waveinthe

rmsK-C

range

of6to

12,

andequals

the

thirdharmonic

frequency

of

thewaveintherangeof

rmsK-C

largerthan

1 3 ,

andthe

rest

is

the

transition

region

from

f

w

to

2f

w

and

from

2f

w

to

3f

w

. Fig.

7

showstheratiooftheone-tenth maximumlift

force

( F T J / I O )

to

the

meanvalueof

the

maximumin-lineforces( F

Tm

)withrmsK-Casa

parameter,

and

this

figure

was

obtained

by

using

the

same

experimental

results

describedabove.

Furthermore,

theexperimental

results

of

Sarpkaya

5

',

using

the

U-shapedwater-tunnel,

aregivenby

the

dottedline

inFig.

7 .

Fromthis

figure,

the

magnitude

of

the

lift

forceincreases

rapidlyascomparedwith

the

in-lineforceasrmsK-Cincreases(from5to

1 0 )

andi treaches

the

maximum valueof

1.1

timesthe

in-line

forceat

rmsK-C=10.

Therefore,

from

the

characteristics

of

the

lift

forcedescribedabove,

i tcanbe

considered

thatthe

resonance

atT

w

/T

n

=lintheY-direction

appearsonlywhenrmsK-Ci slowerthan3 ,

due

to

thepredominant

lift

force

component

having

the

waver

frequency(see

Fig.6)

However,

this

resonance

can

beneglected

as

shown

later,

because

the

magnitudeof

the

lift

force

is

comparatively

smallerthan

thatof

the

in-line

forcewhen


9/19

2386

COASTAL

ENGINEERING1978

rmsK-C

is

smaller

than

3as

showninFig.7 ,andthe

dynamic

displacement

in

the

Y-direction

isvery

small

compared

withthatintheX-direction

atthisperiodratio.

Furthermore,

atT

w

/T

n

=2,theresonantresponse

appears

only

whenrmsK-C

islarger

than

3 ,due

tothepredominantsecond

harmonicfrequency

shown

inFig.

6 .

From

the

investigationdescribed

above,it

can

beconcluded

thatthe

resonant

characteristics

ofthepile

due

toboth

the

in-lineand

the

liftforcehavea

closerelationto

the

charcteristlcs

of

the

wave

forces,includingthefrequency

and

magnitude

of

the

wave

force.

5

0

F i g .

6

Predominant

l i f t energy

v e r s u s

rmsK-C

1 5

20

rmsK-C


10/19

BEHAVIOR OF

CYLINDER

2 3 8 7

FL

1 0

r

* T , r

1.0

0.5

0. 0

F i g ,

r s

^5

v

Sarpkaya

3 )

MAGNITUDEOFTHE

Y-BISPLACEMENT

From

the

practical

point

of

view,

i t

may

be

important

toknow

the

magnitude

of

the

Y-displacementinrela-

tion

t o

the

X-displace-

ment.

Fig.

8

showsthe

variationof

the

ratio

of

the

Y-displacement

totheX-displacement

in

terms

of

T

w

/T

n

.

Here,

the

one-tenth

maximum

displacements

in

bothdirectionsare

used.

From

thisfigure,

itisclear

that

the

X-displacement

ispredominantwhenthe

period

ratio

issmaller

than

1.5

approximately. Onthe

other

hand,the

Y-displacement

i s

predominantin

the

range

where

the

period

ratio

islargerthan

1.5

andespeciallythe

predominance

of

the

Y-displacement

i s

conspicuousnear

theresonance

points

described

above

except

atTw/T

n

=l,

when

rmsK-C

i slarger

than

6 .

This

reasoncan

begiven

by

thecharacteristics

of

the

lift

force

as

shown

inFig.

6

and

Fig.7 .

Therefore,

from

the

above-mentionedexperimental

results,i tcanbe

pointedout

thatrather

thana

in-line

force,

aliftforce

is

the

more

significant

force

when

the

naturalperiodof

thestructure

i s

lower

than

the

wave

period

andrmsK-C

is

higher

than

6 .

5 10

1 5

r

msK-c20

7

R a t i o

of

maximum

one-tenth

l i f t

f o r c e

t o

i n - l i n ef o r c ev e r s u s

r m s K - C

ESTIMATION

OF

DYNAMIC

RESPONSE

3. 0

' 1 0

X

P l V

2.0

1.0

0.0

/

o

S Y M .

rmsK-C

o

0-3

0

3-6

f f i

6-10

10-5

O

15

--

V

- -

F i g .

8 R a t i o

of

Y-displacementt o

ment v e r s u s

period r a t i o

6

7

Tw/Tn

X-displace-

1 )FORMULATIONOF

THE

LIFT

FORCE

EQUATION

Asmentioned

above,

the

computationofa

lift

forceisnecessary

in

order

to

estimate

thedynamic

response

of

a

structure

due

to

i t .

However,

i t

i s

difficult

to

formulate

thelift

forceequation

which

can

express

the

time

variation

of

the

lift

force,because

the

lift

force

is

generatedby

the

alternatebreaking

ofthe

eddiesand

i t

is

irregular

even

in

regu-

lar

waves. Therefore,


11/19

2388

COASTAL

ENGINEERING1978

120

(El)

90

60

60

(e

2

2)

30

0

(3/3)

-30

lu

- s -

e s f

- W -

-60

30

KM)

0

30

F i g .

10

< 5 - e -

* 4 >

o o

15

rmsK-C

W

_

45

rmsK-C

jLi-

>L-L

o 1 [ 5

rmsK-C

t h eformulation

of

t helift

force

is

performedempirically

based

on

t h eexperimentalresultofwave

forcesonarigidlysupported

vertical

circularcylinder.

It

maybeassumedthathefo r -

mula

is

expressed

byt h esuper-

position

of

eachpredominantfr e-

quencycomponent

of

t h e

li ftfo r c e

asshown

in

Fig.6,givenby

Eq.(l).

< L .

P h a s e

angle

o f t h e n - t h

harmoniclift f o r c e

f L ( t ) = 0

L

tf

a T

X

os(2nTTf

w

t-e

n

(1)

Here,

f

L

t):the

lift

force

per

unit

J

e ng t h;SL ( n f

w

:the

variance

of

t h e

li ft

force;

and

n

:the

phase

- i

angle

betweent h e

n-th

harmonic

lift

force

componentandt h e

inci-

dent

wave.

The

spectral

energyof

t hen-th

harmonicliftfo r c e

may

be

given

by

t he

experimental

resultof

Fig.

6.

In

t h i s

s t u dy,t h e

non-

dimensional

n-thharmoniclift

force

energy

is

given

byt heempirical

formula

which

is

specifiedatt h e

right

side

ofFig.6,and

it

is

shownby

t h e

solidline

int his

figure.Further,t h ephase

angle

wasobtainedby

usingt heresult

of

harmonicanalysis

of

botht h e

measured

li ft

force

and

wave

records.

Th e

change

of

phase

angle

with

rms

K-CisshowninFi g .9,in

which

t h epredminant

regionof

the

n-th

harmonic

lift

force

is

also

shown by

anarrow

mark.Th escattering

of

t h eexperimentalresults

is

relatively

large,but

if

attention

is

focussedon

each

predominantregione

n

may

be

consideredasaconstant

value,i.e.e

2

/2

=25,E3/3=-15an d

Ei,/4=0.

However,as

j

isscattered

from

6 0t o

120in

t hepredominant

region

of

f

w

fr equ enc y,

its eemst obe

quite

all

rightt o

considert ha t

t h e

average

value

of

eiis9 0 ,becauset h e

magnitude

oft h eli ft

force

is

quite

small

in

comparison

with

t h e

in-line

fo r c e

in

t h e

region

where

rmsK-C

is

smaller

than

5,as

shown

in

Fig.7.

Ont h eotherhand,Chakrabaltietal

s)

havepresentedt heliftforce

asshowninEq

2.

fL t)=4p

Du

mJcLnsin 2niTf

w

t-a

n

)

(2)

:

t h e

ere,U :t h e

maximum

horizontalwater

particle

velocity;

C^n

li ftcoefficientf ort h en-th

harmonicliftforce;D t h ediamterofa

circular

cylinder;

an dp:t h edencity

of

water.

Sincet h eliftforceinregularwavesisirregular,fx,(t)isconsidered

asarandom

function

of

t im e .With

t h eexception

of

C^,

t he

t e rm s

on

t h e

right

hand

side

of

Eq.

(2)aretheregularfunctionsor

constants.


12/19

BEHAVIOR OFCYLINDER

2 3 8 9

L

10

L

3.0

2.0

1.0

O

w

c

0

05

0

rmsK-C

Fig.

1

atio

of

significantli t

orce

to e a n

li t

orceersus

msK C

Therefore,

Ci

n

must

be

a

random

variable.

Onthe

other

hand,

thedistribu-

tion

of

thepeaklift

forceis

simlor

to

the

Rayleighdistribution

from

theauthors'experiments.

Fig,10

shows

an

example

of

the

relation

between

the

ratio

of

the

signifi-

cant

valueofthe

lift

forceanditsmeanvalue

and

rmsK-C.Thetheoreti-

calvalue

of

this

ratio

based

onthe

Rayleigh

distribution

is

1.637,

and

i t

i sshown

by

a

straitline

in

Fig.10.

As

seen

in

thisfigure,

the

experimental

values

are

scattered

aroundthe

theoretical

valueindependent

ofrmsK-C.

Moreover,the

liftforce

spectra

in

thepredominant

region

ofeach

harmonicliftforce

component

can

be

considered

tobe

a

narrow-band

spectra

) .

6)

From

the

aboveinvestigations,

thelift

force

canbe

assumed

to

be

a

random

variable

of

thenarrow-band

aussian

randomprocess.

Thus,

thevarianceofthe

liftforcecanbe

givenbyEq.( 3 )

o

=E[f(t)]

{-pDU

m

}

2

4E[C]

3

Here,QListhe

lift

coefficient

fthepeakliftforce

andi s

a

random

variable

of

Rayleigh

distribution.

Therefor,

usingthe

following

relation,

(

C

L

)rms=vi[c]=

C

L

,/io/1.8

a

L

i sgivenby

Eq.

( 4 )

from

Eq.( 3 ) .

1

Thevalidityof

Eq.(4)i s

examinedby

investigating

Eq.(5)

deducedfrom

Eq.(4).

-^

DU


13/19

2390

COASTALENGINEERING1978

IfEq.(4)isvalid,

?

and5'

ave

to

agreewith

each

other.

This

agreementof

and

5

>

sshown

in

Fig.

11,from

whichi tcanbeseen

that

and

5'agree

well

regardless

ofrmsK-C.Therefore,

the

lift

force

equation

canbe

expressed

as

Eq.(7)

fromEq.(2)andEq.(4),

fL(t)

1

r

T

n

nn*

?

S

L

(nfw)Af

J-gCLi/io

U

m

oi

cos(2nTTf

w

-e

n

)

( 7 )

15

rmsK-C

20

2 )EQUATIONOFMOTION

OFTHE

PILE

In

order

to

estimate

the

dynamic

response

of

a

single

pile

structure

(see

Fig.1),

the

pilewasidealizedbya

two-degree

of

freedomeqiva-

lent

spring-masssystem

with

a

viscousdamper

as

shown

in

Fig.

1 2 .

Th e

idealization

is

based

on

the

assumption

that

the

rigidityof

the

cylinder

section

on

Fig.1

is

muchlarger

than

hatof

the

spring

bar

section,

allowing

to

assume

the

circularcylinder

and

the

con-

centrated

mass

to

be

a

rigid

body.Inthis

case,

thedis-

F i g . l lComparison g a n d

C

v e r s u s

rmsK-C

placements

of

the

top

ofpile,

XandY

( X

andYare

for

the

X

and

Y

directionsrespectively),

are

givenby

the

horizontal

displacementsat

thebottomof

the

circular

cylinder,xand

y ,

and

the

rotation

angles

of

the

circular

cylinder,

6

X

and

8y

respectively.

Furtheremore,

assuming

sin8~6,

X

and

Y

are

givenby

Eq.( 8 ) .

X =x

L

Y

=

y+

L

( 8 )

Here,

L

the

distancefrom

the

bottom

of

the

cylinder

to

the

top

of

thepile.

Inthis

study,the

mass

of

the

spring

bar

andthe

wave

force

onthis

bar

are

assumed

to

be

egleglble,

becausethesevaluesare

very

small.Onthese

assump-

tions,the

equation

ofmotionof

the

pile

inthe

X

direction

due

to

the

in-line

force

may

be

given

by

Eq.( 9 )

and

Eq.(10).

On

the

other

hand,

that

in

the

Y

direction

m ay

be

given

by

Eq

( 1 1 )

and

Eq.(12).

In

theseequations,

i twasassumed

that

the

mutualinf-

luencebetweenthe

vibrations

of

X

and

Y

canbe

neglected.

(m+mv)TT2+(m+m

v

)G -

'dt' dt-2

x

t

A

C_

dt

nmiiiiiiiu

6EI,

-?T

X


14/19

BEHAVIOR OF

CYLINDER

391

-

|

c D P D

f

V.

-

,.&

|u -

-

^U

+

Vf&.

(9)

ZL

( m

+ m

v

) G^f + [ I

G

+ I

Gv

+ ( m +

dl

+

clegs

+

(z

-

3 x )

r^

+l

, dx

d0^ } 6

y

Z

L

1 2 )

Here,

G:

distance

fromthelower

end

of

the

cylinderto

the

center

of

gravity

including

theaddedmass

of

the

cylinder

;

G

A

:

distance

from

thelower

end

of

the

cylinderto

the

centerofgravityminus

the

added

mass

of

the

cylinder

;

El

:

flexible

rigidity

of

thespring

bar;C:

structural

damping

coefficient

I

length

of

the

spring

bar

;h:

still

water

depth

;

n

watersurface

elevation

;

IQ

moment

of

inertia

about

the

center

of

gravityduetohetotalmassminus

the

addedmass

of

the

cylinder

;

IQ

V

moment

of

inertia

aboutthecenter

o f f

gravity

dueto

the

added mass

f

cylinder;

CD

dragcoefficient

;

CM

mass

coefficient

u

horizontal

water

particle

velocity

;

m

:total

effective

massminusthe

added

mass;

m

v

:

addedmass

of

thecylindergivenby

Eq.(13).


15/19

2392 OASTAL

ENGINEERING1978

m

v

C

v

irpD

(h+-

z

L

)M 13 )

In

Eq.(13),

C

v

i s

the

coefficientofadded

mass

( C

v

= C > i

-1);

z^

:

distance

from

thebottomof

the

water

to

thelower

endof

thecylinder;z*:

z-z^;

F j )andM j )are

the

fluid

damping

force

and

moment in

the

Ydirectionres-

pectivly,

andthesearegivenby

Eq.

( 1 4 )

and

(15).

;

t

c

pD(

z

^ i

+

idz

u)

M

D

- i-C

n

pdz*(-

+

*&.)

|&

+

zAd (15)

In

this

analysis,iti s

assumed

that

thelift

forcecan

becalculated

by

Eq.(7),

and

the

valuesof

dragand

mass

coefficients,

Cp

and

C j ^ ,

assumes

the

following

values,

i.e.

CD=1.5and

C j ^ = 2 . 2 ,

basedontheexperimental

results

oftheauthors

3

.

Sincetheequationsofmotiondescribedabovearenonlineardifferential

equations,

noexact

solution

can

be

obtained.

Hence,

only

approximate

solutions

canbeobtained

by

usingthenumerical

techniques.

Inthis

calculation,

Newmark

B-method

8

isused

to

solve

the

equationofmotion.

The

valueof

Bis

selected

as1/6,

which

isequivalenttoalinearacce-

leration method.

Thetime

interval,

At ,i s

taken

as

0.005

sec,

because

the

naturalfrequencyofthesecondmodeof

thevibration

modelranged

from

31

to35.5Hzfor

the

five

kinds

ofmassesshowninTable

1 .

Taking

astationaryresponseconditioninto

account,

thecalculation

time

w as

as

1 5

secondsfor

each

case.

3 )

CALCULATION

RESULT

Atfirst,

the

dynamic

displacement

intheX

direction was

computed

to

investigate

the

estimationdescribed

above(refer

to

Table

2).

Fig.

13

shows

afewexampleso f

computationresults

in

the

Xdirection

due

to

the

in-lineforce

for

valuesofperiod

ratio

about2

or3 .

In

this

figure,

the

solid

line

indicates

the

calculatedresultbythe

method

( I ) ,

which

considers

theeffectof

the

finiteamplitudenature

ofthewave,

the

latter

being

considered

as

asinusoidal wave,onthe

in-line

force,

and

the

dottedline

indicates

thecalculatedresultby

the

method

(II),which

neglects

theabove-mentioned

effect

onthe

in-line

force.

Inother

words

the

integralregionofthewaveforce

on

the

pile

isfrom

the

lower

end

of

the

cylinder

to

the

still

water

level;n

in

Eqs.(9)

and

( 1 0 )is

assumedtobe0 . Themeasuredresultsare

also

showninthis

figure

by

smallcircles.

From

thisfigure,

it

can

beseenthatthe

calculated

results

by

means

ofmethod

( I )

agree

well

with

themeasuredresults.

On

theother

hand,

there

i smuchdiscrepancy

in

the

frequencyand

magnitudeof

the

displace-

mentbetween

the

calculated

results

by method

( I I )

and

the

measured

results

at

the

periodratio

T

w

/T

n

=2,

asshown

in

Fig.

13

( A )and

( B ) .

However,

there

islittledifferencebetween

the

resultsof

methods

( I )

and

( I I )

at

T

w

/T

n

=3,as

shown

inFig.

13

( C ) .

Fromthis

fact,

it

canbe

considered

that

the

resonance

in

theX

direction

at

T

w

/T

n

=2

i s

caused

by

the

finite

amplitudenatureofwavesandthatat

T

w

/T

n

=3iscaused

bythe

non-linearityofthedrag

force.oreover,the

dynamic

displacement

in


16/19

BEHAVIOR O F

CYLINDER

2393

( A )f

w

/f

n

=0.49msK-C=9.4

T

w

=1.06sec

=10.1cm

X(cm)

( T

w

/Tn=2.0)

( B )f

w

f

n

=0.49

msK-C=ll.l

T

w

=1.50sec

=7.7cm

X(cm)

( T

w

/Tn=2.0)

( C )

f

w

/f

n

=0.32

rmsK~C=11.6

T

w

=1.61sec H=7.4cm

X(cm) ( T

w

/Tn=3.1)

I t

0

- 1

-

t(sec)

F i g .1 3Calculation

r e s u l t s

o f

cylinder

displacement

i n

t h e Xdirection

( A )

f

w

/f

n

=0.48

rmsK-C=7.2

T

w

=l.lsec

=7.6cm

v

, > ( T

w

/T

n

=2.1)

Y(cm)

( B )

f

w

/f

n

=0.33rmsK-C=15.0

Tw=1.60sec=9.6cm

Y(cm)

(

Tw

/

Tn=

3.o)

( C )

f

w

/fn=0.25

rmsK-C=15.6

Tw=1.55secH=13.4cm

fL

Cm)

(Tw/Tn=4.0)

t(sec)

F i g .

1 4

Calculationr e s u l t sof

cylinder

displacement

i n

t h e

Y

direction


17/19

2394 OASTAL ENGINEERING1978

the

Xdirection

for

other

ranges

of

the

period

ratio

werecomputedby

method( I ) . As

aresult,

i t

was

confirmed

that

the

dynamicresponse

in

the

Xdirection

can

be

calculatedbyEqs.( 9 )

and

( 1 0 )

based

on

method

( I )inthe

range

where

the

periodratioi ssmaller

than

6.5.

Next,thedynamicdisplacements

intheYdirectiondue

to

the

lift

force

werealsocomputed

by

Eqs.

( 1 1 )

and

( 1 2 )

based

onmethod

( I )

describedabove. Some

examples

at

theresonancepoints

intheYdirection

ar e

shown

inFig.

1 4 .

The

solid

line

shows

thecalculated

result

and

smallcircles

denote

the

measured

result.

In

thiscase,

asthe

Y

displacement

is

not

regular,

the

displacement

nearlyequal

tothemaximum

value

is

plottedfor

both

theexperimentaland

calculatedresults.

Thisfigureindicates

that

thecalculated

resultsagree

well

with

the

measuredresults

for

eachresonancepointincluding

the

properties

of

the

frequency

and

magnitudeof

theY

displacements.

Therefore,iti s

concluded

that

the

dynamicresponseintheYdirectiondueto

thelift

force

canbe

calculated

byEqs.(7),

( 1 1 )

and

(12).

Finally,

the

combineddynamicdisplacements

of

the

pilewere

computed

by

composing

the

calculated

displacements

in

twodirections,

because

the

maximum

dynamicdisplacement

consideringbothdisplacements

is

desired

for

an

engineeringdesign.

Furthermore,

comparison

between

the

computed

andmeasured

combined

dynamic

responses

gives

the

whole

judgement

forthe

validityof

the

estimation method

ofthe

dynamic

responses

in

both

direc-

tions.

Fig.15

showsthis

comparison,

andtheright-hand

side

of

this

figurei s

the

calculated

resultwhile

theleft-hand

side

indicates

the

measuredresult. AsinFig.2 ,

the

X-axis

i s

the

direction

of

the

wave

propagationdirection

and

the

Y-axisis

the

direction

normal

to

the

wave

propagationdirection.

Because

of

the

irregularityof

the

Ydisplacement,

the

combined

dynamicdisplacementsduring

one

wavecyclein

which

the

maximum

combined

displacement

appears

areplotted

inFig.

1 5 . I t

i s

apparent

thatthe

period

raito

gradually

increases

from

( A )

of

T

w

/T

n

l

( f

w

/f

n

l)to( G )

of

T

w

/T

n

*5

( f

w

f

n

*l/5).

A

little

difference

between

the

measuredand

calculated

locus

i s

observed

in

thecaseof

( G ) ,

inFig.

1 5 .

However,

taking

intoconsiderationtheirregularityof

theYdisplacement,

the

calculated

results

cansafely

be

said

to

have

goodagreements

with

the

experimentalresults.

I tis

concluded

that

the

combined

dynamic

displacementcanb ecalculatedbyEqs.(7),( 9 ) ,

(10),( 1 1 )

and

(12).

CONCLUSION

The

dynamicbehavior

of

a

fixed

circular

pile

due

to

the

in-line

and

the

lift

forcesisinvestigated

rom

the

theoreticaland

experimental

standpoint

ofview.It

enabledusto

arrive

at

thefollowing

conclusions.

First,the

resonantresponsesof

a

singlecircular

pile

due

to

the

lift

force

inthe

directionnormal

to

thewavepropagationdirection

are

found

to

take

place

at

theperiod

ratios

of

T

w

/T

n

=2,3,4,5

and6,when

rmsK-Ci slarger

than

3

Furthermore,

the

resonant

responses

inthe

wave

propagationdirectiondue

to

the

in-line

force

also

appearat

the

same.period

ratiosas

the

former

case,

inadditiontothewellknownresonanceat

Tw /

T

n

=

l-Moreover,

dynamic

displacements

ofthe

cylinder

due

tothe

lift

forceinthedirection normal

to

thewavepropagationdirectionare

larger

thanthose

in

thewavepropagationdirectiondue

tothein-line

forceat

theabove-mentionedresonance

points

except

at

T

w

/T

n

=l,

when

rmsK-C

is


18/19

BEHAVIOR OF

CYLINDER

2395

(M EASDR MENT)

(CALCULATION)

(M EASUR MENT)

(CALCULATION)

(A)

f

w

/f

n

=0.82 (T

w

/T

n

=1.22)

rmsK-C=5.1

T

w

=0.91sec

H=6.9cm

jY(clm)

|| |

n

. JY^m)

(E)

f

w

/f

n

=0.33

(T

w

/T

n

=3.03)

rmsK-C=13.0

T=1.60sec H=9.6cm

(B')

f

w

/f

n

=0.66 (T

w

/T

n

=1.52)

rmsK-

T

c=a

iiii

X(em)

3sec

(C) t

w

/tn=0 'i9 (T

w

VT

n

=2.64)

rmsK-C=ll.l

,

Y(dm)

-3- Y(:m)

|

?

H

(cm)

.4cm

(I) f

w

/f

n

=0.249 (f

w

/T

n

=4:02)

rmsK-C=15.6

T=l,55sec

H=13.4cm

T l.fosjsc

]H=7|.7a|n

(D) f

w

/f

n

=0.44

(T

w

/T

n

=2.27)

rmsK-C=9.2

Y(cm)

|Y(cm)

(G)

f

w

/f

n

=0.20

(T

w

/T

n

=5.0)

rmsK-C=15.9

T=1.49sec H=14.3cm

(cm)

X(cm)

Fig.

5

Calculation

esultsfynam icoci


19/19

2396 OASTAL

ENGINEERING1978

larger

than

6 .

herefore,

thelift

force

is

moresignificant

thanthe

in-line

force

whenthenaturalperiod

of

the

structure

is

smaller

than

the

waveperiodandrmsK-C

i s

comparatively

large.

Secondly,thedynamicdisplacementsin

bothdirectionsand

the

combined

dynamic

displacementcan

be

calculatedbyapplying

theMorison's

formulaand

thelift

force

equation

to

theequationofmotionin

each

direction.

REFERENCES

1)

Bidde,

D.D :

Laboratory

study

of

liftforcesoncircularpiles,

Journal

oft he

Waterways,

Harborsan dCoastal

Engineering

Division,

ASCE,

Vol.9 7,

No.WW4,

1 9 7 1 ,

pp.595-614.

2)Sarpkaya,

T.:

Forceson

rough-walled

circularcylinders

inharmonic

flcv,

Proc.15t h

Conf.

onCoastalEngineering,

1 9 7 6 ,

pp.2301-2330.

3)

Sawaragi,

T ,Nakamura,T.andKita,

H.

Characteristics

of

lift

forcesonacircular

pile

in

waves,Coastal

Engineering

in

Japan,

vol.19 ,

1 9 7 6 ,pp.59-71.

4)Wiegel,

R.L.,

Beebe,K.E.

an d

Mo o n,

Ji

Ocean

wave

forcesoncircular

cylindrical

piles,

Journal

of

t heHydraulicsDivision,

ASCE,

Vol.

83 ,

No.HY2,

1957,

pp.1199-1-36.

5)

Sarpkaya,

T. Forcesoncylindersandspheres

in

a

sinusoidally

oscillating

fluid,

Journal

ofApplied

Mechanics,

ASME,

Vol.

42,

No.1,

1 9 7 5 ,

pp.32-37.

6)Chakrabalti,S.K,Wolbelt,

A.L.

an dTom,

W.A.

Wave

forceson

verticalcircularcylinder,

JournalofWaterways,

Harbors

an d

Coastal

Engineering

Division,

ASCE,

Vol.102,No.

WW2,

1 9 7 6 ,

pp.203-221.

7)Davenport,

W.B.

Jr

a n d

Root,W.L.:

Anintroduction

to

t h e

theory

ofrandomsignals

an d

noise,

McGraw-HillInc.,1958,

pp.145-175.

8)

Biggs,

J.M.:

Introduction

to

structural

dynamics,

McGraw-Hill,Inc.,

1964,

pp.1-33.

dynimac behavior of vertical cylinde rdue the wave force

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