( \ Proceedings: Second International Conference on Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamics,
March 11-15, 1991, St. Louis, Missouri, Paper No. 5.69
~
Seismic Response Analysis of Pile-Supported Structure:
Assessment of Commonly Used Approximations
Toyoaki Nogami
Reed L. Mosher
University of California at San Diego, La Jolla, California
U.S. Army Engineer Waterways Experiment Station, Vicksburg,
Mississippi
H. Wayne Jones
U.S. Army Engineer Waterways Experiment Station, Vicksburg,
Mississippi
SYNOPSIS: The seismic response of a pile-supported structure is formulated by the approach developed
by the first author.
Using this formulation, some of the crude approximations frequently used in the
seismic response analysis of a soil-pile-structure system are examined.
Those involved in the
analysis procedure are assessed under the linear elastic condition.
A commonly used nonlinear soil
model for the dynamic pile response analysis is also assessed.
It is found that those approximations
routinely used in the analysis procedure and numerical modelling can cause significant errors in the
computed response of a pile-supported structure.
INTRODUCTION
mechanical system only, whereas the free-field
nonlinearity affects both the outside and inside
mechanical systems.
When a linear elastic soil
medium is considered, the radius of t[le rigid
ring is set equal to the radius of the pile;
i.e. no inside mechanical soil model.
Details
of those mechanical systems developed in both
the frequency-domain and time-domain, can be
found in the papers published by the first
author.
According to the model shown, each of
the vertical and horizontal displacements are
expressed at the pile shafts as
A structure is frequently supported by a pile
foundation.
When this structure is analyzed for
its seismic responses, the pile foundation must
be properly taken into account in the analysis.
Seismic responses of pile
foundations
are
complex and their analyses generally require a
large amount of computations, particularly when
pile groups and nonlinear soil behavior are
considered.
Thus, various crude approximations
are
used
in
the
analysis.
This
paper
investigates errors caused by some of those
approximations frequently used in the seismic
response analysis of pile-supported structures.
fp
(1)
where u
vector containing displacements at the
piles; uo
free-field displacement; 1 = unit
vector; f
flexibility matrix of the system
(soil); and p
vector containing soil-pile
interaction forces.
The nonlinearity caused by
the soil-pile interaction affects only the
diagonal terms of the matrix f.
Novak
(1975)
has developed an approach to
analyze the dynamic response of linear elastic
single pile foundations within the frame of the
Winkler's hypothesis.
Nogami and his colleagues
have extended this approach for nonlinear pile
foundations and pile groups, including 1) linear
elastic pile foundations in the frequency-domain
analysis (Nogami, 1980; Nogami, 1983; Nogami,
198 5) , 2) nonlinear pile foundations in the
frequency-domain analysis
(Nogami and Chen,
1987a), 3) linear elastic pile foundations in the
time-domain analysis (Konagai and Nogami, 1987;
Nogami and Konagai, 1986; Nogami and Konagai,
1988a) and 4) nonlinear pile foundations in the
time-domain analysis (Nogami and Konagai, 1987b;
Nogami et al., 1988b; Nogami et al., 1991).
This approach has been verified by various
people (e.g. Sanchez-Salinero, 1983; Nogami,
1983; Roesset, 1984) and is used herein for the
assessment of the approximations frequently used
in the seismic response analysis of pilesupported structures.
Eq. 1 is coupled with the equations of motion of
pile shafts to formulate the dynamic response of
pile foundations.
Those equations are typically
formulated with a lumped mass pile model or a
continuous beam pile model.
When a continuous
lf.
f~ORIZONTl\T.
SLICE
~-
----
-.> Plt.T.F:S
SOIL MODEL AND FORMULATION OF SEISMIC RESPONSE
OF PILE-SUPPORTED STRUCTURES
Vertical pile groups are considered herein as a
general case.
A horizontal slice of a soil-pile
system with a unit thickness is idealized as
shown in Fig. 1, in which each one of pile
shafts
is
enclosed by
a
rigid
ring
and
mechanical systems are located outside and
inside of the ring.
The soil-pile interaction
produces
the
nonlinearity
in
the
inside
SYSTF'.MS
Fig. 1
931
Horizontal Slice in Soil-Pile System and
Plan View of Soil Model
beam model is used, the flexural and axial
responses of pile shafts are described by,
respectively,
a4
EI
~
-i- u
+ m
dt
2
( 6)
where U
vector containing the
lateral
displacement (U) and rotational displacement (<l>}
of the rigid cap; P
vector containing the
lateral force (P} and moment (M} applied at the
rigid cap by the super-structure motions; Kf =
stiffness matrix of the pile group attached to a
rigid cap; and ~P = force produced at the cap by
the free-field soil motion, containing a and ~.
-p
(2)
a2
-EA ~ + m
dz
i
-p
7
where: EI = diagonal matrix containing flexural
rigidities of piles; EA
diagonal matrix
containing axial rigidities of piles; and m =
diagonal matrix containing masses of piles.
Writing Eq. 1 in the either time-domain or
frequency-domain, Eq. 2 can be solved in one of
those domains.
For
simplicity,
the
super-structure
is
considered to be a mass attached to the top of
the pile cap.
Combining the structure and Eq.4,
the equation of motion of the pile-supported
super-structure
subjected
to
the
seismic
excitation is written as
With the steady state response to the harmonic
horizontal bedrock motion, the lateral soil
response is expressed as.
( 7}
where M9
mass of super-structure.
split into
(3)
where a and b
constants determined by the
boundary conditions of the free-field soil; uo(H)
= bedrock displacement; and y = W/v 3 with (J) =
circular frequency and v 3 = shear wave velocity
of soil
(8)
where U = U1 + U2.
Eq. 8 is interpreted such
that the motions, U1, are transmitted to excite
the super-structure and generate the feedback
motions, U2,
(second equation) and that the
transmitted
motions,
U1,
are the seismic
responses of pile foundation at the cap without
any super-structure according to the first
equation. This is illustrated in Fig. 2.
Substituting Eq. 3 into Eq. 1, the expressions
of the lateral and axial pile responses in the
frequency-domain can be obtained from Eqs. 1 and
2 as, respectively,
u (z)
=f..( Ane "-nz + Bne -Anz + Cne lA.nz + Dne -iAnz~n +
n~l
APPROXIMATE SEISMIC RESPONSE ANALYSIS OF PILESUPPORTED STRUCTURE
uo (H) ( ae iyz + ~e -iyz J 1
(4)
u(z}
Approximate Methods
Various approximations are
adopted in the seismic response analysis of
pile-supported structures.
Those often used are
1}
no pile-soil-pile interaction for a pile
group,
2}
frequency independent spring and
dashpot for the Winkler subgrade model and 3}
analysis by applying the free-field ground
=!~neiA"z+ Bne-iA.nz~n
n=l
where; N = number of piles; An, Bn, Cn and Dn =
constants determined by the boundary conditions
of the piles; An and ~n = n-the eigenvalue and
eigenvector obtained from (A 4 EI[k-w2m)~ = 0 for
the lateral response and (A2EA+k-w2m)~ = 0 for
axial response, with k = f-1; a = a (y4EI+k-w2m}lk; and ~ = b ('{4EI+k-w2m) -lk.
Then, all other
responses associated with the flexural and axial
pile response are expressed using Eq. 4 as,
respectively,
3
-
'C'(output)
__!_
u u
-
J/111/7111/1//
UQ (
2
-1
-1
d
d
-d
(<J>(z),EI P(z),EI M(z})= ( dzu(z},-Ju(z},--2 u ( z
dz·
dz
-]
P (z}
__Q
dz
H ){input)
II
>)
u u
Ot (output)
(5}
EA
Eq. 7 can be
u(z)
---
After determining the unknown constants in Eq. 4
for the boundary conditions of the piles, the
force and displacement responses of the piles
are completely described by Eqs. 4 and 5.
With
those
expressions,
the
force-displacement
relationship of a pile group attached to a rigid
cap can be written in the form of
/11111//11/11
UQ (H)
(input)
!
O(output.)
o-o 1
+
o,
J;·
-
0 1 (input.)
Fig. 2 Transmitted and Feedback Motions in
Seismic Response of Pile-Supported Structure
932
R
R
Homogeneous soil profiles
(A,B and C)
and
inhomogeneous soil profiles (AX, BX, and CX) as
shown in Fig. 4 are considered.
Assuming the
linearly increasing shear modulus with depth,
inhomogeneous profiles are defined such that
their
fundamental
natural
frequencies
are
identical to those of the homogeneous profiles.
Fig.
5
shows
the
normalized acceleration
response of the free-field ground surface to
harmonic horizontal bedrock motions.
61"
0
60 6
0
R
0
0
0-
9-rTu: r:Rnur
At-: "' 0.62055 x !Ofi
klp!'l
El ,.,. 0.12726 X !Ofi
kfp!Pft
wC'fp.ht
Fig. 3
/
r.
s
2
(klps/ft )
c
A
3R.4(i
65
192. )I
G
A single mass, effective only to the lateral
translational motion, is considered for the
super-structure.
The masses supported by single
piles are Ms
5 kips. sec2 /ft for piles in
homogeneous profiles and Ms = 1.292 kips.sec2/ft2
for piles in inhomogeneous profiles.
Those M5
result in an identical Kxx ((J) = 0. 02v 5 /ro) /M 5 ratio
between profile B and profile BX.
The masses
supported by 9-pile groups are nine times of
those of the structures supported by single
piles,
and thus are respectively Ms
45
kips.sec2/ft and 11.628 kips.sec2/ft for profile
B and profile BX.
Pile Foundations Considered
AX
r.
2
!Pnp:th = 0.501 lo;f
576.92
r.
s
v
(I)
2. 5
(2)
(3)
('•)
(5)
10.2
20.)
4R. 3
(6)
61.0
)J.l
s
2
(klps/ft )
BX
ex
I 2. 7
50.9
IOI. 7
165. J
JR. I
I 52.6
.105. I
'•95. 9
2~ 1.6
305.1
9l5.t.
72''· 7
Both harmonic and random motions are considered
for input horizontal bedrock motions.
The time
history of the random bedrock motion is shown in
Fig. 6.
s
s
n
= o. 3
~
o. 05
y
= I 10 pes
JJIJI))))fi,~
12
HO!!oGENEOUS PROFJLF.S
ProfileA At B. C
JNHOIIOGENF.OIJS PROFILES
Prnfiles AX, RX, CX
R -
Fig. 4
Soil Profiles Considered
surface motion at the base of the pile ~upported
structure,
in which the third approximation
corresponds to the use of the free-field ground
surface motion for U1 in the second equation in
Eq. 8.
ll.,B,C
v
The soil model presented herein can reproduce
the dynamic
responses
very
well,
if
the
mechanical systems in the soil model are defined
from the dynamic response behavior of multiple
infinitely
long
rigid
massless
cylinder~
vertically inserted in an infinite medium
(Konagai and Nogami, 1987; Nogami, 1983: Nogami
et al., 1986, 1987a, 1987b, l988a, 1988b, 1991).
Thus, defining the soil model by this approach,
the afore-given formulation is used to examine
the effects of the
above listed approximations
and is referred herein as "relatively rigorous
method".
Fig. 5
0
0
....
__ ...
I
I
......
__ .....
Free-Field Ground Surface Response to
Harmonic Bedrock Motions
10
8
<!
.
z
8
The
above
listed
first
approximation
is
introduced by defining the model parameters for
a single cylinder in the medium rather than a
group of cylinders.
The second is introduced by
using the soil model parameters defined at (J) =
0. 02 v 5 /ro for all frequencies, in addition to
the conditions used in the first approximation:
ro = radius of pile and v 5 = shear wave velocity
of the medium.
;2
'"..,
"'
u
u
"'
"'u0
:'i
"'!-<"'
:0
0.
F.
-8
-10
0
Conditions
Considered
Pile
foundations
considered include those made of a single pile
and 3x3 piles attached to a rigid cap (Fig. 3).
4
6
8
10
Tlt!F. (~F:r..)
Fig. 6
933
Acceleration Bedrock Motion Time History
As the second equation in Eq.
Computed Results
8 indicates, seismic responses of pile-supporte d
structures are governed by the transmitted
pile
the
of
impedance
and
(U1)
mot ions
foundation {Kf), all which are affected by the
are
Kf and U1
pile foundation in general.
computed for the single piles by the relatively
rigorous method and the computed results are
Transmitted motions in
shown in Figs. 7 and 8.
the figure are normalized by multiplying the
lateral motion by 1/uo(O) and the rocking motion
free-field lateral
by L/uo (0), in which uo (0)
pile
motion at the ground surface and L
Nondimension al stiffness parameters
length.
shown in Fig. 8 are obtained by multiplying the
real part of the impedances Kj by 1/ (EsL) ,
1/ (E 5 L2), and 1/ (EsL3) respectively for j = xx,
in which E 5 = Young's modulus of soil
x~ and~~'
and
( 9)
Nondimension al damping parameters are obtained
by multiplying the imaginary parts of Kj by
and l/(E 5 L3) lao
1/(E 5 L 2 ) /ao
1/ (E 5 L) lao,
respectively for j = xx, x~ and ~~,in which ao =
The transmitted motions contain not only
wro/v 5 .
motion but also rotational motion,
lateral
the
The lateral transmitted
as seen in the figure.
motions tend to increase a little bit first and
then decrease with frequency to become smaller
The
than the free-field ground surface motion.
curves shift to the left with increasing the
soil softness, particularly for inhom0geneou s
The stiffness
than homogeneous profiles.
parameters of single-pile foundations vary very
little with frequency but the damping parameters
low
at
frequency
with
decrease
rapidly
Transmitted motions and impedance
frequencies.
functions for 9-pile groups in profiles B and BX
Similar trends as
are shown in Figs. 9 and 10.
those observed for single-pile foundations can
be seen in the transmitted motions but the
rotational ones are much smaller than those for
The pile-soil-pi le
the single-pile foundations.
interaction effects in the transmitted motions
are negligibly small at very low frequencies and
become more pronounced at higher frequencies.
The pile-soil-pi le interaction affects far more
damping
and
stiffness
the
significant ly
parameters than the transmitted motions.
Seismic responses of the structures supported by
single piles are computed for harmonic bedrock
motions by the relatively rigorous method and
The approximate method
the approximate method.
adopts crude approximation s including frequency
independent soil model and use of the free-field
The
ground surface motion as input U1 motion.
computed results for profiles B and BX are shown
Those approximatio ns appear to be
in Fig. 11.
acceptable for the piles in homogeneous soil but
overestimate the response for the piles in
seismic
Similarly,
soil.
inhomogeneo us
responses of the structures supported by 9-pile
harmonic bedrock
for
computed
are
groups
Two different approximate methods are
motions.
In the first approximate
used in this case.
method, the frequency independent soil model is
used and also no pile-soil-pi le interaction
In the second
effects are taken into account.
ground
free-field
the
approximate method,
surface motion is used for U1 in addition to
1.6
'. 6
1.2
!.2
O.R
0. A
10
---- -----------.-==..--- --------
_ _ t. _ _ t _ _ _ j
_
70
__!.._._....L____l__L_~.--1---
t\!!l'LI l!rDF.
Nll'J.J!IJI)f:
12
"8
g·~
g
~
~
8
!<
'"~
----
----
PHASE S!IIFI
10
I
-2on
'
''
'· I'
-220
I
13
w
A
<;IJIF"I
12
~
~
('c
-,-:..r--- --------=--.::.. :-_----
t'I!ASE
;::
"'
~
,.....
g
~
'"w"'
~
8
~
"':?.
\
AX
_ _ L__L_
2
),
''
'
~x __ ) , ex-/',
Transmitted Motions for Single Piles
934
1
L_'-..1_~~ -lf>O
10
p.
,
FRF/)UFt!CY i'AF:Arlr.IER,
FREQUENCY PAR.J\f'!El ER,
Fig. 7
\
4~r::.W
0
0
----------
"
/
k
"
::i
~
R
~
~
~
BX
ex
~
z
~
4-
AX
~
0
0
12
l2
s
10
~
"
z
i
0.02
FRF.QtlENC.Y Pi\Rflf-1f.TER •
Fig. 8
R
~
IN T'ROFIJ.f.
30 <o
1.?:
o.of:l
O.OR
Stiffness and Damping Parameters of Single Piles
1.6
C/\PI'F.D 9-Pt!.F CROUP
o.or.
~
1.6
"
CflrrF.D 9-PILF: GROUP TN PROrJJ.f: BX
R = .10 ro
iHHl I NTERfiCT TnN
1.2
z
0
~
8
;::"
0
0.8
~
~
IHTI!Ol!T INTERACT roN
':!
~
!<
0.8
':!
~
3
0.4
j
L._L_
_t__[_
16
16
--MIPI.TT(J!lr:
12
0.4
--
-
Ar!I'L !TilDE
12
I~
\ilTHO!lT lNTERA\.TION
I r1! TNTERACTJ()N
W1Til
FREQUF:NCY PARAHnER,
Fig. 9
FRF.QUF.NCY l'fiRMIETF:R.
Transmitted Motions for 9-Pile Groups
935
0.10
I'ROFIJ.F. B
PROF"ll.l~
10
0.8
1
I
R-o1fh
-
R
PROFILF.: B
0
0.8
--k
I
I
I
I
0.6
J
I
1
\ \
r-7
s
'-
/
/
"
'>.;-~-----
' '-
0.,
a
o. 2
'-
NO TNIERACTJON
FRF:QllF.NCY PARM!ETEil,
0. I
0.1
0.2
0. I
0. 2
0. I
0.)
0.2
FRF.QUF.NCY PARAHF.H:P:,
~-~ td
0.1
-~-~ ru
PROF !I.E RX
~
II
II
II
32
I
I
~
"
0
0
0
,,
I
I
I
I
I
\
0
7
~
0
0
I
\
0
I
I
1
~- ~ ,_
~,
NO
Fig.
10
~~ (.,,)
-
~
'\\NO INfFRAr.rtON
/,.- '-.......
___ __/_....:>-..,
........ /
(
o.)
0.2
FREQUENCY PMM!HF.R,
\
--\----.......
tJO IN I i'~RACl I ON
I Nl F.RA£:'1" I ON
o.!
FREQUENCY PARM!F.TF:R,
0
\
~
"'
R=-10r
I
2'· ·I
I
0
0
PROFII,E RX
~~
(.o.J
0.1
--o. 3
o. 2
FREQUENCY ffd{N!F.TF:R,
~~LV
Stiffness and Damping Parameters of 9-Pile Groups
The real seismic motions are random and contain
Thus, the above
various frequency components.
observed errors caused by approximations at
various frequencies are all included in the
One random earthquake
response time history.
bedrock motion time history is used to see how
those errors are reflected in the time-domain
Fig. 13 shows the seismic responses
responses.
of the structures supported by single-piles,
computed by the relatively rigorous method and
As expected from the
the approximate method.
bedrock
harmonic
for
observation
previous
motions, the approximate method produces the
responses very close to those computed by the
relatively rigorous method for profile B but
amplifies the predominant frequency component
Fig. 14
responses excessively for profile BX.
shows the seismic responses of the structures
computed by the
supported by 9-pile groups,
relatively rigorous method and the approximate
The approximate method herein is the
method.
second approximate method explained previously.
This approximate method amplifies the high
frequency component responses excessively for
This is more pronounced for the
both profiles.
approximations used in the first approximate
The results computed for profiles B and
method.
In the results
12.
BX are shown in Fig.
obtained for profile B, the difference between
the peak values computed by the relatively
rigorous method and the first approximate method
appears to mostly result from the differences in
the stiffness and damping parameters between the
two cases: the first approximate method yields
The use
higher stiffnesses and lower dampings.
of the free-field ground surface motion for U1
does not produce significant errors in this
In the results obtained for
particular case.
profile BX, the response curves computed by the
first
the
and
method
rigorous
relatively
approximate method are very similar to the
transmitted motions presented in Fig. 9 and are
On the other
relatively close to each other.
hand, the response curve computed by the second
approximate method is significantly different
Therefore, the use of the
from the other two.
causes
motion
surface
ground
free-field
in the computed seismic
significant errors
structure supported by 9-pile
response of the
group in profile BX.
936
PROFIJ,F. R
l''HII' f I.E RX
RE:J.Tl.TIVF:LY
R I GORO!JS H8THOD
---FRF:Q. JNDEl'F.tm. SO!l.ST!I"F.
TRANSMIT1Ell Hfll"TON lDF.NTI(",/'11.
TO FRF:E-Flf.LD f'lOTlfiN
''
'
'~
~--
FRF.QUF.NCY FARAHF.TF.R •
Fig.
11
FRF:QUF.NCY PARAMETER •
Responses of Structures Supported by Single Piles to Harmonic Bedrock Motions
rROFJ!.F. R
R " 10 r
,.....
I
I
I
/j
I,
..---
/
i
NO INTF:RACTION
FRF.Q. 1NtlEPF.Nn. SOJT, STIFF.
"ITRASN!'ITE!l N()'l ION lDENTJr.AL
10 FREF.-flf.LO Mn1 ION
I
'"
1\
I.
\
\
\
\
i
'"'
§
' ' ' .......... ~--
:~
I
I
I
I
~
\\
\
I
I
C<
\\
I
I
I
I
I
~
\'
FRF.QUF.NCY PARJ\HETF.R.
inhomogeneous
profile.
I\
/
I
I
I
·'w
. /~
I
............
12
r\
0
Q
'
Fig.
R • 10 r
~
~
I
//
I'
FROFJJ,E HX
w
I
I
I
'
~
RF:LJI.TIVELY
RIGOROUS HF.THOD
NO 1 NTERACTJON
FREQ. DF.PENI). SOli. STIFF.
0
I
I
/
/
/
/
;::-..._
:-....
/
·...._/
2 H
it~-!'! W
FREQIJF.NCY rARMIETER.
Response of Structures Supported by 9-Pile-Groups to Harmonic Bedrock Motions
profile
than
the
homogeneous
the ring.
Details ot those systems can be tound
in the papers by the first author contained in
the references.
CONVENTIONAL WINKLER MODEL FOR NONLINEAR SOILPILE INTERACTION
Under the steady state harmonic response, the
complex force-displaceme nt relationship of the
conventional model is schematically illustrated
in Fig. 15: the real and imaginary parts are
respectively the backbone curve and the curve
related to the damping (area of the hysteresis
loop).
The damping in this case is simply the
summation of the nonlinear damping and radiation
damping, and thus the nonlinearity induced in
the vicinity of the pile always increases the
damping since the dashpot is not affected by the
nonlinear behavior.
Fig. 16 shows the complex
A conventional Winkler model is made of the
nonlinear
spring
and dashpot
placed
in
a
mutually parallel position, in which the latter
is to reproduce the radiation damping and its
property is independent of the displacement
(Matlock
et
al.,
1978).
The
soil
model
presented herein can account for the nonlinear
soil-pile interaction by using the nonlinear
mechanical system inside the rigid ring and the
frequency independent mechanical system outside
937
SINGLE rll.E IN PROFILE BX
0.020-
0.020
0.010
~
0.005
0.005
ffi
i
~
0.000
::!
"~ -0.005
~
0.010
"'~
j
REI.ATIVF.I.Y
R I GDROUS METHOD
"'"
!;;
e!
0.015.
;:;
0.015
-0.005
;':
~
-0.010-
-0.010-
-0.015
-0.015-0.020 _ _ _ ___,______..J.__ _ _...J _ _ _ _L . . . _
10
8
4
2
0
-0.020
-----L- ----'--- --L--
0.020
"'~
0.010
~
~
0.005-
~
i
::!
"'~ -0.005
-0.005
~
~ -0.010
~ -0.010
-0.015
-0.015.
-0.020 - - - - - ' - - - - - - ' - - - - - . l 6
4
2
0
-0.020 - - - - - - - " - - - - - - ' - - -
0
-10
8
Fig. 13
0.020
0.020
TN rROFI!.F. "
0.015
RF:1.1\TIVF:l.Y
0.010-
~
0.005
"'ffi
6
8
10
Responses of Structure s Supported by Single Piles to Random Bedrock Motions
q-rli.F: GROtlr
,.,"'
2
TH1f:(SF.C.)
TJHF.(SF.C.)
"
HF:THOD
0.015 -
~
0.010
ffi
7\rPROX.
..
"'
MF.THOD
APPROX.
0.015
"
10
8
f!HF.(Sf.C.)
0.020
"'"
6
2
0
rTMF.(SF.C,)
R I GO ROllS
t-1F.TI!Qf)
..
0.015
~~
"'
0.010
E
ffi
0.005
9-PLLE GROUP IN PROTILE BX
RELATIVEJ;'i
R tGOROUS Mf:THOD
j
i
~
::! -0.005
~
~ -0.005 .
~ -0.010
~ -0.010
-0.015
-·0.015
-0.020'-- -----'----- - - - - - ' - - - - - ' - - - - - '
10
8
4
0
----'
_ _ _ _ _ _ _ _ _ L _ _ _ _ _ _ _ ____t.. _ _ _ _ _ _ _ l.~---'--
-0.020
8
4
0
10
'! !tlF.(SH".)
lJHE(SF.C.)
0.020
0.020
""'
0.015
~
0.010
~
~
0.005
1\PF'ROX.
HETHOD
j
~
-0.015
-0.020 L__ _ _. __ _ _, __ _ _. __ _ _, __ _-...J
10
8
6
2
0
·----L--~---L---___J
2
4
8
8
10
Tntf,(SF.C,)
TlHF.(SFC.)
Fig. 14
Motions
Responses of Structure s Supported by 9-Pile Groups to Random Bedrock
938
OJ(B)
OJ(A)
Pr (A)
Pi'(A)
Pi"(A)
u
<
OJ (A)
Pi.
ci
Pi
Pi": ;Ji due to
radiation damp
o 0 =0.2
o 0 ~0.02
30
LI~-·C "s=o.3
20
20
Linear Syslem
<t
~
Pi'(B)
Pi"(B)
Real Port
,'
CD
Linear
_,
Syslem<•
,.,.,.,.-
-CD--~
1
Real Port
E
tr
<
Complex Force-Disp lacement Relationsh ip of Convention al Soil Model
30 -
--"''
.,
Pr (9)
Pi': Pi due to
nonlinear damp
OJ (B)
P = Pr + ipi
Fig. 15
=
\.-··
10
.
--
.
,"r.--
,·.~
0
ci
-- ---
E
<t
0
15
E
<t
<t
CD.®
:/
...-~
u
~
_-::::-::.::-~cD®@ - 0.02
Gop ({3=0.7)
/@Degradation ( 8= n-0.3)
dl
_._·L__ ___ ,__. _,_·_·_·1___1
~
~~~-CD.@.@
------- ------
•_ __.__ _...._____J _ _ L_ _ _J
0.02
0.04
0.04
Disploceme~l Amp. (in. l
Displacement hmp. (in.)
Fig. 16
(f)
Imag. Part
10-
E
0
CD Jnelaslic;
:
ci
5 -
~--L--~ ____J____
_
j_~
Gap ( /3=0.7)
@ Degradation (8= n -o.3 )
10
ci
~·---
CD Inelastic
®
,\!0/
------·----------------- (])' ® .@
..,..,.,..//
~/
10
CD.®.@
_ . . . t _ _ L _ _ _ ,___ __,__ _ __j
15 ·
lmog. Part
,'
Complex Force-Disp lacement Relationsh ip of Nogami Soil Model
force-displ acement relationsh ip of the presented
soil model.
In this model, the nonlinear
behavior in the vicinity of the pile foundation
(mechanica l system inside the ring) generates
the damping but at the same time reduces the
energy transmitte d to the infinity (radiation
damping)
The net effect is to increase the
damping at the frequencie s where the radiation
damping is small
(low frequencie s),
but to
reduce the damping at the frequencie s where the
radiation damping is large (high frequencie s).
approxima tions can cause significan t errors in
the computed responses.
This is generally
pronounced for pile groups and soil profiles
containing soft soils at shallow depth.
A
commonly used nonlinear Winkler soil mode fails
to reproduce the coupling between the nonlinear
and radiation dampings, and thus overestima tes
the damping in the high frequency component
responses.
All those warrant us that the
seismic response of pile-suppo rted structures
must be analyzed by using the methods and
numerical models based on a careful and rational
considerat ion.
CONCLUSIONS
Seismic responses of pile-suppo rted structures
are formulated with the approach developed by
the first author.
Approxima tions frequently
used in the seismic response analysis of pilesupported structures are assessed by using this
formulatio n.
Some of the routinely used
ACKNOWLEDGEMENT
The work presented in this paper was sponsored
by the U.S. Army Corps of Engineers, Waterways
Experiment Station.
939
REFERENCES
Konagai, K. and T. Nogami (1987), "Time-Domain
Axial Response of Dynamically Loaded Pile
Groups", J. Enrg. Mechanics, ASCE, Vol. 113,
No. EM3: 417-430.
Matlock, H., S.H. Foo and L.-L. Bryant (1978),
"Simulation of Lateral Pile Behavior", Proc.
Earthquake Engineering and Soil Dynamics, ASCE,
Pasadena, July: 600-619.
Nogami, T. (1980), "Dynamic Stiffness and
Damping of Pile Groups in Inhomogeneous Soil",
ASCE Special Technical Publication on Dynamic
Response of Pile Foundation: Analytical Aspect,
October: 31-52.
Nogami, T. (1983), "Dynamic Group Effect in
Vertical Response of Piles in Group", J.
Geotech Engrg, ASCE, Vol. 109, No. GT2: 228243.
Nogami, T. (1985), "Flexural Response of Grouped
Piles Under Dynamic Loading", Int. J.
Earthquake Engrg. and Structural Dynamics, Vol.
13, No.3: 321-336.
Nogami, T. and K. Konagai (1986), "Time-Domain
Axial Response of Dynamically Loaded Single
Piles", J. Engrg. Mechanics, Vol. 112, No. EMl:
1241-1252.
Nogami, T. and H. L. Chen (1987), "Prediction of
Dynamic Lateral Response of Nonlinear Single
Pile Foundation", ASCE Special Technical
Publication on Experiment, Analysis and
Observation, April: 39-52.
Nogami, T. and K. Konagai (1987b), "Dynamic
Response of Vertically Loaded Nonlinear Pile
Foundations", J. Engrg. Mechanics, Vol. 113,
No. EM2: 147-160.
Nogami, T. and K. Konagai (1988a), "Time-Domain
Flexural Response of Dynamically Loaded Single
Piles", J. Engrg. Mechanics, ASCE, Vol. 114,
EM9: 1512-1513.
Nogami, T., K. Konagai and J. Otani (1988b),
"Nonlinear Pile Foundation Model for TimeDomain Dynamic Response Analysis", Proc. 9 th
World Conf. Earthq. Engrg., Vol. 3, TokyoKyoto, Aug. : 593-598.
Nogami, T., J. Otani K. Konagai and H. J. Chen
(1991), "Nonlinear Soil-Pile Interaction Model
for Dynamic Lateral Motion", J. Geotech.
Engrg., ASCE, to appear.
Novak, M. (1975), "Dynamic Stiffness and Damping
of Piles", J. Can. Geotech. Engrg., NRC of
Canada, Vol. 11, No. 4: 574-698.
Roesset, J. M. (1984), "Dynamic Stiffness of
Pile Group", ASCE Special Technical Publication
on Analysis and Design of Pile Foundations,
October: 263-286.
Sanchez-Salinero , I. (1983), "Dynamic Stiffness
of Pile Group", Geotechnical Engineering Report
GR83-5, Civil Engineering Dept., Univ. Texas at
Austin, July.
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