Post on 02-Jun-2018
transcript
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
8/10/2019 Dynimac Behavior of Vertical Cylinde Rdue the Wave Force
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
8/10/2019 Dynimac Behavior of Vertical Cylinde Rdue the Wave Force
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,
8/10/2019 Dynimac Behavior of Vertical Cylinde Rdue the Wave Force
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
8/10/2019 Dynimac Behavior of Vertical Cylinde Rdue the Wave Force
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
8/10/2019 Dynimac Behavior of Vertical Cylinde Rdue the Wave Force
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
8/10/2019 Dynimac Behavior of Vertical Cylinde Rdue the Wave Force
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
8/10/2019 Dynimac Behavior of Vertical Cylinde Rdue the Wave Force
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
8/10/2019 Dynimac Behavior of Vertical Cylinde Rdue the Wave Force
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
8/10/2019 Dynimac Behavior of Vertical Cylinde Rdue the Wave Force
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,
8/10/2019 Dynimac Behavior of Vertical Cylinde Rdue the Wave Force
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.
8/10/2019 Dynimac Behavior of Vertical Cylinde Rdue the Wave Force
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
8/10/2019 Dynimac Behavior of Vertical Cylinde Rdue the Wave Force
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
8/10/2019 Dynimac Behavior of Vertical Cylinde Rdue the Wave Force
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).
8/10/2019 Dynimac Behavior of Vertical Cylinde Rdue the Wave Force
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
8/10/2019 Dynimac Behavior of Vertical Cylinde Rdue the Wave Force
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
8/10/2019 Dynimac Behavior of Vertical Cylinde Rdue the Wave Force
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
8/10/2019 Dynimac Behavior of Vertical Cylinde Rdue the Wave Force
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
8/10/2019 Dynimac Behavior of Vertical Cylinde Rdue the Wave Force
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.