HORIZONTAL RESPONSE OF PILES IN LAYERED SOILS
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By George Gazetas 1 and Ricardo Dobry, 2 Members, ASCE
ABSTRACT: An inexpensive and realistic procedure is developed for estimating
the lateral dynamic stiffness and damping of flexible piles embedded in arbitrarily layered soil deposits. Starting point is the determination of the pile deflection profile for a static force at the top using any reasonable method—beamon-Winkler foundation, finite elements, well-instrumented pile load tests in the
field, etc. Material as well as radiation damping due to waves emanating at
different depths from the pile-soil interface are rationally taken into account;
the overall equivalent damping at the top of the pile is then obtained as a function of frequency by means of a suitable energy relationship. The method is
applied to study the dynamic behavior of three different piles embedded in
two idealized and one actual layered soil deposit; the results of the method,
obtained by hand computations, compare favorably with the results of three
dimensional dynamic finite element analyses.
INTRODUCTION
Current state-of-the-art procedures for analysis and design of single
piles subjected to static lateral loads are mostly of a semiempirical nature.
They use the "beam-on-Winkler-foundation" model in which the soil
support at different depths is approximated by independent nonlinear
"springs," whose deformation characteristics are described by p-y curves
based on field load tests (22,23,32,37,38). Theoretical methods have also
been developed, which treat the soil as a continuum and utilize boundary-element type (2,29,30) or finite-element (9,31) formulations. However, most of them are still used primarily for research rather than as
design tools.
On the other hand, established methods for dynamic analysis of laterally
loaded piles are of a theoretical nature and make use of viscoelastic wavepropagation concepts to model the dynamic soil reactions against the
pile (5,8,15,16,17,18,20,24,26,27,34,35). No experimental data is yet
available in the form of "p-y versus frequency" curves obtained from
dynamic field tests. Attempts to analytically develop such dynamic p-y
curves on the basis of the nonlinear soil behavior established in the laboratory have been recently reported by Kagawa and Kraft (14,15), and
Angelides and Roesset (1). However, additional work is needed to merge
these dynamic response solutions with the results of actual (static) field
tests; this seems at present the only way to develop simple methods of
dynamic response analysis of piles for the wide range of frequencies and
load intensities encountered in practice (3).
In this paper an attempt is made to develop an inexpensive and realistic procedure for estimating the lateral response of flexible piles
embedded in a layered soil deposit, and subjected to harmonic headloading.
'Assoc. Prof, of Civ. Engrg., Rensselaer Polytechnic Inst., Troy, N.Y. 12181.
2
Prof. of Civ. Engrg., Rensselaer Polytechnic Inst., Troy, N.Y. 12181.
Note.—Discussion open until June 1,1984. To extend the closing date one month,
a written request must be filed with the ASCE Manager of Technical and Professional Publications. The manuscript for this paper was submitted for review and
possible publication on March 24, 1983. This paper is part of the Journal of Geotechnical Engineering, Vol. 110, No. 1, January, 1984. ©ASCE, ISSN 0733-9410/
84/0001-0020/$01.00. Paper No. 18496.
20
J. Geotech. Engrg., 1984, 110(1): 20-40
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The applicability of the proposed method is illustrated with three case
studies involving piles embedded in three different linearly hysteretic
soil deposits: (1) Homogeneous stratum with constant Young's modulus;
(2) an inhomogeneous stratum with modulus increasing linearly with
depth; and (3) a realistic layered soil deposit. Excellent agreement is obtained with the pertinent results of rigorous dynamic finite-element (FE)
analyses (5,8,17). Note that for cases (1) and (2) the FE results have already been independently published.
PROBLEM DEFINITION
The problem studied in this paper is that of a floating or end-bearing
fixed-head 'flexible' pile embedded in a layered soil deposit and subjected to harmonic lateral excitation at the top [Fig. 1(a)]. On such a pile,
due to the restriction imposed by the pile cap, lateral loading is applied
with no rotation of the top.
A variety of pile cross-sections [see Fig. 1(b)] can be studied with this
method. The pile is treated as an elastic flexural beam, having Young's
modulus Ep, width b = 2B measured perpendicular to the direction of
loading, and corresponding area moment of inertia Ip. The pile is assumed to be slender enough to exhibit 'flexible' behavior under horizontal loading. In practice, most laterally loaded piles are indeed 'flexible' ('long piles' in Ref. 8) in the sense that they do not deform over
their entire length L. Instead, pile deflections and stresses become negligible below an 'active length/ la [Fig. 1(a)]. This length depends on how
stiff the pile is compared to the soil, but it usually is less than 10-15
pile-diameters (17,18,31,40).
Table 1 presents simple expressions for preliminary estimates of the
active length, Z„, of various pile cross sections. These formulas were derived from the results of rigorous analyses for two idealized and rather
extreme soil profiles—a homogeneous stratum of modulus Es and a linearly inhomogeneous stratum of modulus Es = Es z/b (see Refs. 17, 31,
40 for details). The pile cross section shape factors, S, appearing in the
Poe'" 1
looding
FIG. 1.—Problem Geometry and Pile
Cross Sections Considered
FIG. 2.—Static (Y„) versus Dynamic (Y4)
Pile Displacement Shapes
21
J. Geotech. Engrg., 1984, 110(1): 20-40
TABLE 1.—Active Length Under Dynamic Loading for Preliminary Estimations
Active Length/Width, IJb
Soil profile
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0)
Homogeneous (constant modulus: Es)
Inhomogeneous (modulus proportional
to depth: £ s = Es z/b)
Expression
(2)
3.3(EPS/ES)1/5
Typical range
(3)
3.2( EPS/ES)1/6
5-15
8-20
Note: Ep = Young's modulus of pile; Es = Young's modulus of soil at a depth
z = b; S = pile cross section dimensionless shape factor given in Table 2.
formulas of Table 1, have been selected such that the bending stiffness
and radiation damping characteristics of the various pile cross sections
are consistently reproduced. Table 2 gives the corresponding expressions for S.
The steady-state horizontal displacement y(t) = yd exp (mi) at the head
of the pile is related to the harmonic horizontal exciting force P(t) = P0
exp (mi) through the complex-valued dynamic impedance function
K+mC = yd
(1)
in which i = V ( _ l ) ; w = frequency of excitation in rad/sec (w = 2ir/
where / is in Hz); P0 = amplitude of the forcing function; and yd = yd(f)
= the (complex) amplitude of the horizontal motion. The complex nature
of yA stems from the presence of damping in the system, as a result of
which forces and displacements are, generally, out of phase.
Terms K = K(f) and C = C(f) can be interpreted as the pile-head
equivalent "spring" and "dashpot" coefficients; they are both functions
of the frequency / = w/Zir. Physically, K reflects the stiffness and inertia
characteristics of the pile-soil system, while C expresses the energy loss
due to both, hysteretic action in the soil (material or hysteretic damping)
and geometric spreading of waves away from the pile (radiation damping). Alternatively, the equivalent damping ratio D = D(f) may be used
in place of C (1,8,17,40):
D
_ MC _ rfC
(2)
2K
K
The objective of the method developed herein is to inexpensively obTABLE 2.—Pile-Cross Section Shape Factor
Pile cross section
(1)
Circular (diameter: b)
Pipe (diameters: outside b, inside b,)
Concrete-Filled Steel Pipe Pile (diameters:
outside b, inside 6,)
Rectangular (lateral width: 2B, length: 2A)
Shape factor, S
(2)
1
1 - (bM
1 - (bM + Earn../
Es««i)(b,/6)4
1.7(A/Bf
J. Geotech. Engrg., 1984, 110(1): 20-40
tain realistic estimates of K(f) and C(/) or D(f), at the head of a laterally
loaded pile.
OUTLINE OF PROPOSED METHOD
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The proposed approximate method involves the following four steps:
1. The horizontal-displacement profile, ys(z), of the pile subjected to a
statically applied horizontal load of magnitude P0 is obtained using the
best procedure(s) available. For instance, one may utilize a beam-onWinkler-foundation type formulation along with pertinent p-y response
curves (13,22,23,32,37,38), or a boundary-element integral method
(2,29,30), or a finite-element code (9,18,31). Alternatively, one or several
well instrumented full-scale lateral pile-load tests in the field may be
used to provide ys(z). The static value, Ks, of the "spring" coefficient K
is then directly computed as Po/ys(0), in which ys(0) = static displacement of the pile head.
2. Two parallel dashpots are assumed attached to the pile at every elevation, and their characteristic coefficients, cm and cr, are determined.
(The dimensions of cm and cr are those of a "dashpot/unit length of pile.")
The first dashpot is intended to simulate the material dissipation of energy in the soil. Its coefficient, c,„, is estimated on the basis of the "effective" shear strain, ye(z), induced in the soil at each particular elevation; 7e(z) is in turn related to the static deflection profile, ys(z), obtained
in the first step. The second dashpot represents the radiation of energy
by waves spreading geometrically away from the pile-soil interface. Its
coefficient, cr, is obtained, at a particular frequency, from the solution
of appropriate plane-strain wave propagation problem(s), using soil moduli
consistent with the "effective" shear strains, 7e(z).
3. The overall "dashpot" coefficient C = C{f) at the head of the pile is
computed from the values of cr and cm distributed along the pile (step
2), in conjunction with the static pile deflection profile, ys(z) (step 1). To
this end, the following simple energy-conservation relationship is used:
C = J (c, + cm) Y2s{z)dz
(3)
Jo
in which: Ys(z) = ys(z)/ys(0) = static deflection profile normalized to a
unit top amplitude. Eq. 3 is analogous to usual expressions in classical
dynamics for replacing distributed stiffness with a generalized spring.
The main approximation in Eq. 3 involves the use of the static (/ = 0)
rather than the dynamic (/ ¥^ 0) pile displacements. Note, however, that
the main influence of / is upon the magnitude of the pile displacements
rather than on their shape. Evidence in support of the above argument
is offered in Fig. 2, which compares the static shape, Ys(z) of a fixedhead pile with its dynamic shapes, Yd(z), at two different frequencies.
All these shapes were computed using a dynamic finite-element (FE)
formulation for a circular pile of L = 25 • b, embedded in a soil stratum
with modulus proportional to depth and overlying a rigid base (39,40).
The two frequencies studied correspond, respectively, to the fundamental resonant frequency of the stratum, and to a fairly high frequency. It
23
J. Geotech. Engrg., 1984, 110(1): 20-40
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is apparent that the three curves are essentially identical at shallow depths.
The discrepancies observed at greater depths are of no major practical
consequence, since the value of C from Eq. 3 is controlled by the larger
values of Yd at shallow depths.
4. The variation with frequency of both the "spring" coefficient, K(f),
and of the damping ratio, D(f), are estimated from the static stiffness,
Ks, and from the results of steps 2 and 3. Specifically, K(f) is derived
from Ks, approximately corrected to account for possible resonance phenomena and high-frequency effects. Fig. 3 provides the basis for these
corrections. In this_ figure, the ratio K/Ks is plotted versus the frequency
factor a0 = 2n/ B/Vs for flexible piles embedded in sbc different types of
soil deposits. (B = b/2 = radius of the pile; and Vs = [Es/2ps(l + v)]1/2
is a reference value of the S-wave velocity in each stratum; Es is indicated
for each deposit in Fig. 3.) The results in Fig. 3 were obtained using
dynamic FE analyses, and include linear and nonlinear soil behavior as
well as a wide range of pile stiffnesses (1,17,18,35,40). It is evident that,
IWKENEDUS DEPOSIT OF O&WEN CLAY;
Undralned shear stregth iu"96kPa; vO.49;
strain S-wsve velocity V, w S T e / s
2*f8/V S| „,
FIG. 3.—Lateral Dynamic Stiffness versus Frequency from 3-D Finite-Element
Analyses [E, = 2p V2S (1 + v); Es = 2P V] (1 + v)]
24
J. Geotech. Engrg., 1984, 110(1): 20-40
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in many cases, K = Ks would be a realistic approximation for the frequency range examined. A correction may be necessary for the resonant
dip which appears whenever a stiff, rock-like formation is present at
some depth. This dip invariably occurs at essentially the fundamental
frequency, /„, of the soil stratum for vertical S-wave propagation. The
value of /„ can be easily obtained from published solutions, simplified
procedures, or 1-D wave propagation analyses (e.g., 7, 12, 17). The ratio
K(f„)/Ks at resonance depends mainly on the material damping ratio, p,
of the soil, approaching unity as p increases. K(f„)/Ks also approaches
unity as the stiffness of the underlying formation decreases. The results
in Fig. 3 obtained for p = 0.05 and a perfectly rigid base correspond to
cases where this effect is most pronounced.
With regard to the damping ratio, D(/), the curves obtained from Eqs.
2-3 can be easily corrected to account for the fact that no radiation
damping can be generated in a soil stratum at frequencies lower than
/„. Fig. 4 schematically illustrates the proposed modification of Eq. 2 in
the frequency range 0 < / s 2/„.
Determination of the dashpot coefficients cr and cm along the pile is a
crucial task of the proposed method and will be now discussed in some
detail.
RADIATION DASHPOT COEFFICIENTS
Several models, based on 1, 2 or 3-D wave propagation idealizations,
are available for evaluating the distributed radiation dashpot coefficients
cr = c r (/;z).
The 1-D model proposed by Berger et al. (4) and adopted by others
(14,15,27) utilizes the analogy between the dynamic response of any
1-D wave, such as one traveling along the axis of a cylinder, and a viscous dashpot (21). According to this analogy, a dashpot with coefficient
c = pAV fully absorbs the energy of a wave traveling with velocity V
along a cylinder of cross-sectional area A and mass density p [Fig. 5(a)].
Berger et al. (4) assumed that a horizontally-moving pile cross section
of effective width b = IB would solely generate 1-D P-waves traveling
in the direction of shaking and 1-D SH-waves traveling in the direction
«—Radiotion (Viscous) Damping Ratio (D r )
«—Material (Hysterfltic) Damping Ratio ( 0 m )
FREQUENCY, f
FIG. 4.-—Approximate Modification of Damping versus* Frequency Curve at Frequencies N@ar Resonance
25
J. Geotech. Engrg., 1984, 110(1): 20-40
l-D Model o 1 Betger «l 01(4)
Wlniisly Long Rod-,
I Wow Propojalion
P-Wovtl
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Rod
EndsotlhiiPeinl
dEn<ttonni»poini-^
f V-Wove Spwd } j } - f
- —
~-
!
,vo
•=>
~u
«t.,.0
|
1
Plan
5
I
1=0
Hfflitontel Saclion
FiG. §.—1-D and 2-D Radiation Damping Models
perpendicular to shaking, as sketched in Fig. 5(b). Thus, using the aforementioned analogy, their model computes
cr = 4BpsVs 1 +
(4)
vs
as the coefficient of the viscous dashpot that will fully absorb the energy
of all the waves originating at the pile-soil interface. In Eq. 4, Vp and Vs
stand for the P-wave and S-wave velocities of the soil at the depth of
interest. Vp and Vs are related through the Poisson's ratio, v, of the soil:
1/2
-"•[£2]
(5)
For all its simplicity, the model has two drawbacks. First, by assuming
that waves propagate only within two narrow zones of constant cross
section (width b = IB), the model derives a frequency-independent cr
(Eq. 4). In reality, even a square cross section would generate waves
spreading in all directions, and cr is a function of frequency, as it is shown
subsequently.
Second, the use of Vp as the appropriate wave velocity in the compression-extension zone implies that a perfect constraint is provided in the
near field by the two lateral boundaries, so that ex = ez = 0 [Fig. 5(b)].
As a result, the proposed cr (Eq. 4) exhibits a very high sensitivity to
variations in Poisson's ratio, v, and tends to infinity as v approaches 0.50
(Eq, 5). As no such jump to infinity has been found in rigorous studies
of the problem when v = 0.50 (26,35), the use of Vp in Eq. 4 is clearly
unrealistic. The reason for this is that the constraint, zx = cz = 0 is in26
J. Geotech. Engrg., 1984, 110(1): 20-40
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consistent with the assumed stress-free surrounding soil in the x-direction in Berger's model. Therefore, in a previous publication (27), it was
proposed to accept Berger's basic expression, but using a velocity, V <
Vp, instead of Vp in Eq. 4. The authors have defined three possible candidates for V. One possibility is V = Vc, in which Vc is obtained by
assuming as boundary conditions, e2 = 0 and ax = 0; the expression for
Vc = [2/(1 - v)]1/2 Vs. A second possibility is V = VL, in which VL =
rod wave velocity, defined by the boundary conditions ax = az = 0 and
by the expression VL = [2(1 + v)]1/2 Vs. The third possibility is V = Vu
in which
Vu = -
3.4 Vs
(6)
ir(l - v)"
is the Lysmer's analog "wave velocity," which has proven useful for the
understanding of surface foundations subjected to vertical oscillations
(Dobry, R. and Gazetas, G., "Stiffness and Damping of Arbitrary-Shaped
Machine Foundations"). All three velocities, Vc, VL and Vu are smaller
than Vp and have reasonable values as v = 0.5. Exactly at v = 0.5: V„ =
oo, Vc = 2VS, VL = 1.73 Vs and Vu = 2.16 Vs. The use of either of the
latter three velocities in Eq. 4 would recognize the fact that, in the soil
near the pile, compression-extension oscillations propagate with at least
some degree of normal straining in the lateral, x, direction.
A step further toward a better understanding of the radiation damping
was provided by the plane-strain model used by Novak and coworkers
(26; see also 34). They obtained a rigorous solution to the corresponding
elastodynamic boundary value problem of an infinite soil space subjected to horizontal oscillations from a rigid, vertical, infinitely long circular inclusion [Fig. 5(c)]. In this case e2 = 0 and the solution is twodimensional. The triangles in Fig. 6 depict the radiation "dashpot" coefficient, cr, obtained from Ref. 26 and normalized by 4B ps Vs, as a function of the frequency factor a0 = 2itf B/Vs for two values of the Poisson's
r
-
*
Soil Profile
,5„r
• TT-8-O—CL_Q_O
9
• S 1 I
0 ^^
»
%
O
•
bl
•
I
Symbol
>^-Free-Head Piles
6
S-
\~f—l
a2B
Fixed-Head Piles
itttta^^
Pile Cross-section
0"
0.5
o„ =27TfB/Vs
101
I04
SEp/g, or
FIG. 6.—Radiation Dashpot Coefficient
of Circular Pile Cross Section: Evaluatlon of the Approximate Plane-Strain
Model Developed by Authors
10'
10'
3E,/E,
FIG. 7.—Coefficient 8 of Eq. 12 versus
Relative Pile Stiffness
27
J. Geotech. Engrg., 1984, 110(1): 20-40
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ratio (0.25 and 0.40). B = the radius of the pile and Vs = the S-wave
velocity of the surrounding soil. Notice the monotonic decrease of cr with
frequency, in contrast with the frequency-independent cr obtained from
the 1-dimensional Eq. 4.
Additional support for the validity of the 2-dimensional plane-strain
model for cr has been provided by Roesset and coworkers (34,35,5), who
used an efficient 3-D FE formulation to relate local soil reactions to corresponding pile displacements. "Spring" and "dashpot" coefficients
comparable to the plane-strain valeus were then obtained by suitably
averaging the local values. For 'flexible' piles floating in a deep soil deposit, the resulting cr values are plotted as circles in Fig. 6, for v = 0.25
and v = 0.40. There is excellent agreement between Roesset's 3-D and
Novak's 2-D values in Fig. 6.
Kagawa and Kraft (14,15) used a somewhat different averaging procedure with the results of a 3-D FE analysis to derive "spring" and
"dashpot" coefficients comparable to those of the plane-strain case. They
finally decided, however, to adopt the 1-dimensional model of Berger
et al. (4) (Eq. 4), primarily because of its simplicity and versatility in
approximately modeling nonlinear soil behavior.
An alternative simple and versatile approximate plane-strain model,
which does not have the limitations of Berger's model, has been developed by the authors. This approximate plane-strain model is based on
the intuitive assumption that compression-extension waves propagate in
the two quarter-planes along the direction of loading, while shear waves
are generated in the two quarter-planes perpendicular to the direction
of loading. Fig. 5(d) illustrates the basic elements of the model for the
case of a square pile cross section. Only horizontal soil deformations are
allowed within each quarter-plane, and all straight lines originally normal to the corresponding direction of wave-propagation remain normal
during the oscillation. Each of the four quarter-planes is assumed to vibrate independently of the three others. If the pile cross section is circular, it is replaced by a square section having the same perimeter 2ITB.
By assuming that S-waves propagate with velocity Vs in two quarter planes
and that compression-extension waves propagate with velocity Vu in the
other two, and by adding up the energies radiated away in the four
quarter planes [Fig. 5(d)], the following expression is derived for the radiation dashpot coefficient associated with a circular cross section of radius B [or for a square section of side (8/IT)B]:
C
3.4
' = 1+
.ir(l - v).
4B Ps y s
5/4-,
,
,3/4
^J «; V 4
(7a)
in which a0 = 2TT/B/VS (Gazetas, G., and Dobry, R., "Simple Radiation
Damping Models for Piles and Footings").
The expression for cr computed from Eq. 7a is plotted in Fig. 6 for v
= 0.25 and v = 0.40. It is evident that the predictions of the approximate
plane-strain model compare very favorably with the results of the more
rigorous calculations by Novak and Roesset.
At very shallow depths, however, Eq. 7a probably overpredicts the
value of c,.. The reason for this is that the presence of the (stress-free)
ground surface will facilitate the generation of surface type waves instead
28
J. Geotech. Engrg., 1984, 110(1): 20-40
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of, or in addition to, plane-strain body waves, and surface waves propagate with velocities closer to V$ than to Vu,. The authors propose, as
an approximate way of accounting for this effect, to use the velocity Vs(z)
of the soil for all four quarter planes in the model of Fig. 5(d), at depths
less than zr = 2.5b below the ground surface. Hence, at such shallow
depths Eq. 7a is replaced by
2
/
\3/4
w.~ \V
a;m z Zr=2 5b
*
-
{7b)
It is interesting to draw an analogy between the recommended zr =
2.5b and the current procedures of estimating lateral p-y design curves
for statically loaded piles (23,32,38). These procedures recognize the importance of "near-surface" effects by reducing the soil resistance in a
zone extending from the surface down to a depth zr, which is typically
also of the order of 2.5-3 pile diameters.
MATERIAL (HYSTERETIC) DASHPOT COEFFICIENTS
The first step in evaluating the distributed material dashpot coefficients, c,„ = cm (z), is the estimation of the hysteretic damping ratio, (3 =
P(z), in the soil. For a given soil, p is mainly a function of the amplitude
of the induced shear strains. A pile section at depth z oscillating with
an amplitude yd(z) induces in the surrounding soil an average shear strain
of amplitude
7e(z)
~ YiF y"(z)
(8)
This expression has been proposed by Kagawa and Kraft (14,15) and is
an extension of the Matlock (23) relationship between pile-deflection and
normal strain.
Again, as a first approximation, the static pile deflection, ys(z) may be
used in Eq. 8 in place of y,*(z). If greater accuracy is desired, a trial-anderror procedure can be readily devised, but in most cases this may not
be warranted in view of the many other uncertainties involved in defining the soil properties for specific engineering applications.
Once ye is known, the damping ratio p can be estimated from widely
available experimental data in the form of damping-versus-strain curves
(e.g., Ref. 33). Typical values for p at different levels of strain, yer are:
for 7e = lCT5, p « 0.02; for ye = 10 -4 , p « 0.05; and for ye = 10~3, p =
0.10-0.15.
The dashpot coefficient, cm, is related to p by an expression similar to
Eq. 2, in which C is replaced by cm, D by p, and K by a local soil modulus, k = k(z). Thus, Eq. 2 gives
cm~2k^
(9)
to
k = a secant modulus defined as the ratio of the static local soil reaction
against a unit length of pile, p = p(z), over the corresponding pile deflection, ys(z); i.e.:
29
J. Geotech. Engrg., 1984, 110(1): 20-40
Hz) = ^ \
(10)
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ys(z)
Notice that p(z) has units of force/length and k(z) of force/(length) 2 .
The determination offc(z)is fairly straightforward and falls within the
first step of the proposed method, i.e., the estimation of the static deflection profile, ys(z). If a beam-on-Winkler-foundation type formulation
with a linear elastic subgrade k„(z) were used to derive ys(z), then k(z)
would be
k{z) = k0(z)b
(11)
On the other hand, if a nonlinear "p-y" type analysis were done in step
1, k(z) would be obtained directly from p(z) and ys(z), by means of Eq.
10. Conversely, p(z) and ys(z) can be backfigured from available instrumented field load test results in order to estimate k(z) with Eq. 10.
Furthermore, if theory of elasticity is used to predict ys(z), using
boundary-integral or finite-element codes, k(z) can be obtained from the
local soil Young's modulus £s(z) (5,8,14,15,20,25,35):
k(z) = 8 E,(z)
(12)
The coefficient, 8, independent of z, might be selected such that the top
deflection of the pile supported by independent elastic springs of modulus 8Es(z), per unit length, is the same as the "true" deflection of the
pile embedded in an elastic continuum of Young's modulus Es(z). For
long 'flexible' piles, 8 turns out to be mainly a function of the type of
soil profile, the type of head loading and the relative stiffness of the pile
with respect to soil. Fig. 7 provides guidance for the selection of 8 in
practical applications. The hatched ranges in the figure bound results
obtained for two extreme soil profiles (a homogeneous and a linearly
inhomogeneous), two extreme types of loading (fixed-head and free-head
conditions), and a wide range of pile to soil stiffness ratios, as expressed
by S Ep/Es or S E p /E s .
Notice the sensitivity of 8 to the type of loading conditions ("fixed"
versus "free"). On the other hand, the stiffness ratio appears to be much
more important for free-head than for fixed-head loaded piles. For typical soils and piles, 8 « 1.0-1.2 for fixed-head and 8 « 1.5-2.5 for freehead conditions. Kagawa and Kraft (14,15) derived a similar 8 factor by
equating the work done by the soil reactions along the pile; their results
are generally consistent with those of Fig. 7.
Three numerical examples are now presented illustrating the detailed
application of the method and demonstrating its technical and economic
advantages. Note that all computations in these examples, beyond the
determination of the static response, can be performed by hand, without
the use of a computer.
FIRST APPLICATION: PIPE PILE IN STIFF HOMOGENEOUS SOIL STRATUM
An end-bearing steel pipe-pile having outside and inside diameters b
= 1.0 m and b, = 0.9 m, is embedded in a 25-m deep homogeneous
overconsolidated soil stratum underlain by rigid bedrock. The pile is
subjected to fixed-head type harmonic oscillations, to which the soil re30
J. Geotech. Engrg., 1984, 110(1): 20-40
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sponds as a linear hysteretic solid of constant Young's modulus Es =
172.0 MPa (1 MPa = 1,000 kPa; 1 kPa = 1 kN/m 2 ), constant Poisson's
ratio v = 0.40 and constant hysteretic damping ratio p = 0.05. In addition: soil mass density ps = 1.90 T/m 3 ; shear modulus Gs = Es/[2(1 + v)]
= 172.0/2.8 « 61.5 MPa; and S-wave velocity Vs = (61,500/1.90)1/2 «
180.0 m/s. These properties are typical of stiff overconsolidated clay
deposits.
The question is to determine the dynamic values of K(f) and D(/) at
the head of the pile for loading frequencies, /, ranging from 0-20 cycles/
sec (Hz), using the proposed method.
Some preliminary computations must be done to ensure that the method
is indeed applicable in this case. Tables 1 and 2 are consulted for estimating the dynamic active pile length, l„. The cross section shape factor
S = 1 - (0.9/1)4 = 0.344 leading to a pile-soil stiffness ratio, S Ep/E$ =
0.344 X 2.5 x 108/172,000 = 500, in which Ep = 2.5 X 108 kPa is the
modulus of steel. Therefore,
/SE\V5
I, = 3.3 I — l J
b = 3.3 x (500)1/5 x 1.0 = 11.5 m
(13)
which is clearly less than the actual pile length, L = 25 m. Hence, the
pile is 'flexible' and our simplified method can be utilized.
Step 1.—-The static pile deflection profile, ys{z), due to a unit horizontal force, P„ = 1, and without any rotation at the top is readily available
in closed-form from beam-on-Winkler-foundation analysis (30,36). After
estimating k = 8ES « 1.25 x 172.0 - 215.0 MPa (where 8 is obtained
Lorqe-Diameter Steel Pipe-Pile in
Homogenous Soil Stratum
0-0.05
4'
A"T
I
0
I
tn
! i
2'n
I
10
1
1
20
f i n Hz
(o)
(b)
FIG. 8.—First Application of the Proposed Method: (a) Problem Geometry and Static
Pile Deflection Profile; (h) Comparison of Predictions by Method with Independently Published Dynamic FE Results
31
J. Geotech. Engrg., 1984, 110(1): 20-40
from Fig. 7 for S Ep/Es = 500), w e construct the normalized profile, Y$(z)
= ys(z)/y0(z), which is s h o w n in Fig. 8(a). The static stiffness is (30,36)
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/
\1/4
Ks = (4E p I p ) 1/4 k3/i - ( 4 x 2.5 x 108 x — x l 4 x 0.344J
x (2.15 x 10 5 ) 3/4 ~ 6.4 x 10 s k N / m
(14)
Step 2.—We start with material damping. Since d a m p i n g ratio a n d soil
modulus are both i n d e p e n d e n t of z, Eqs. 9 a n d 12 yield a depth-independent coefficient
= 2 X 2.15 x 105 x - = 4.3 x 1 0 5 -
cm = 2k(0
CO
(15)
(0
The distributed radiation d a s h p o t coefficients for shallow a n d greater
depths are computed from Eqs. 7. For z s zr = 2.5 x 1 = 2 . 5 m
cr = cri = 4B P s V s (1.67a; 1 / 4 )
= 4 x 0.5 x 1.9 x 180 X 1.67a0~1/4 = 1,142a; 1 '' 4
(16)
while for z > 2.5 m, given that v = 0.40,
cr» cn « 4B P s y s (2.578« 0 - 1 / 4 ) « l,763a 0 ~ 1/4
(17)
Step 3.—The pile-head " d a s h p o t " coefficient, C, is estimated from the
energy relationship, Eq. 3:
/-25
C = cm\
,-2.5
Y\dz + cn
Jo
Jo
[25
Y]dz + cr2
Y*dz
J2.5
(18)
The above three integrals can be evaluated analytically in this particular
case. They are found to be approximately equal to 2.23, 1.84 a n d 0.39,
respectively. Consequently, Eqs. 15-18 yield:
C - 2.23 x 4.3 x 1 0 5 - + (1.84 x 1,142 + 0.39 x l,763)fl 0 _1/4
do
» 9.59 x 105 - + 2,788fl0_1/4
(19)
CO
Alternatively, the effective pile-head d a m p i n g ratio is
coC
9.59X105S
tab 180
2,788fl0_1/4
D« — =
-. + — x — x —
•
2KS
2 x 6.4 x 105
V3
0.5
2 x 6.4 x 105
« 0.75(3 + 0.78a; 3 / 4
(20a)
D « 0.038 + 0.037/ 3 / 4
(20b)
Step 4.—The fundamental natural frequency, / „ , of the soil stratum
due to vertically propagating (horizontally polarized) S waves is
Vs
180
/„ = 0.25 — = 0.25 x — = 1.80 H z
(21)
H
25
At frequencies lower t h a n 1.80 H z the d a m p i n g ratio is taken as conor
32
J. Geotech. Engrg., 1984, 110(1): 20-40
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stant, equal to 0.038; at frequencies higher than 2/„ = 3.60 Hz, D is given
by Eq. 20b; and a linear interpolation is assumed for the intermediate
frequency range, 1.80 < / < 3.60. The resulting D = D(f) is shown in
Fig. 8(b) and compared to the "actual" curve, obtained from a FE analysis performed by the authors using the formulation of Blaney et al. (5).
The agreement is very good in Fig. 8(b), throughout the wide frequency
range examined.
Particularly remarkable is the successful prediction that the effective
hysteretic damping ratio at the top of the pile-soil system is only about
75% of the hysteretic damping ratio in the soil. Such a value may seem
strange, but in fact it is a natural consequence of the assumed perfectly
elastic behavior for one of the two components of the system, the pile.
The variation with frequency of the pile-head "spring" coefficient, K,
is readily constructed with the help of Fig. 3(a) for S Ep/Es = 500, since
K, (6.4 X 106 kN/m) and /„ (1.80 Hz) are already known. Fig. 8(b) portrays K = K(f). In this case the "actual" FE curve essentially coincides
with the constructed curve, as the example in Fig. 8 is the same as the
case used to construct Fig. 3(a). The agreement between predicted and
"actual" K(f) curves may not necessarily be as good in more general
cases.
SECOND APPLICATION: CONCRETE PILE IN SOFT NORMALLY
CONSOLIDATED CLAY STRATUM
A b = 0.35 m, L = 14 m circular concrete pile of modulus Ep = 2.5 x
107 kPa is embedded in a deposit of very soft, normally consolidated
saturated clay underlain by rigid bedrock (Fig. 9(a)). The soil responds
Smojl-Diomater Concrete Pile in Inhomoqenous Soil Strotom
In)
lb)
FIG. 9.—Second Application of the Proposed Method: (a) Problem Geometry and
Static Pile Deflection Profile; (b) Comparison of Predictions by the Method with
Independently Published Dynamic FE Results
33
J. Geotech. Engrg., 1984, 110(1): 20-40
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as a linear hysteretic solid having undrained Young's modulus Es =
380ff„, constant Poisson's ratio v = 0.49 and constant damping ratio (3
= 0.05. [Note that the assumption of constant p = 0.05 is made only in
order to compare the results with those of a previously published FE
study (40).] Term av is the effective vertical stress and at a particular
depth z is equal to av = (ys - yw) z » (16.5 - 10)z = 6.52, in which -ys
= 16.5 kN/m 3 is the saturated unit weight of the soil and ya » 10 kN/
m3 is the unit weight of water. Thus, Es increases linearly with depth:
£s « 380(6.52) » 2,470z; the characteristic modulus at a one-diameter
depth is Es = 2,470 x 0.35 » 864 kPa and the corresponding S-wave
velocity Vs * [864/(1.65 X 2.98)]1/2 « 13.3 m/s. The wave velocity at the
middle of the deposit is about 60 m/s. [Note that these soil properties
are somewhat similar to those of the Drammen clay used in the pile
study of Angelides and Roesset (1).]
In this case the crosssection shape factor S = 1 and, thus, S Ep/Es =
2.5 X 107/864 « 29,000, and, from Table 1, the dynamic active length is
/SE\1/6
/„ = 3.2 I —v-J b « 3.2 X (29,000)1/6 X 0.35 = 6.2 m < L = 14 m . . . . (22)
indicating a clearly 'flexible' pile for which the proposed method is
applicable.
Step 1.—The normalized static pile deflection profile, Ys(z), obtained
from a FE analysis (5), is shown in Fig. 9(a). The static stiffness is found
to be about 6,600 kN/m, a value which checks with the expression proposed in Refs. 17 and 40.
(S E x °'35
Ks - 0 . 6 0 b E , l - ^
= 0.60 x 0.35 x 864 x (29,000)035 - 6,615 kN/m
(23)
Step 2.—From Fig. 7 for S Ep/Es = 29,000 obtain 8 = 1.13; hence
cm = cjz) = 28ES - = 2 x 1.13 x 2,4702 x - = (5,582 - J z
W
(0
\
(24)
(0/
which, in this case, is proportional to depth. From Eqs. 7 we obtain for
2 < zr = 2.5 x 0.35 = 0.88 m:
cri = cr{z) = 4BpsFs(2){1.669[a0(z)]-1/4} - 93.5/" 1/4 z 5/8
{25a)
and for z > 0.88 m:
cn = cri(z) « 4BPsVs(2){2.97[fl0(2)r1/4} - 166/- 1/4 z 5/8
Step 3.—Eq. 3 takes the form
,•5.20
w
+166
r-
zY2sdz+
C - 5,582Jo
(25b)
/.0.88
3.5
L Jo
z5/8Y2sdz
z5/eY2sdz f~m
(26)
The three integrals are numerically computed to be 1.35, 0.42 and 0.70,
34
J. Geotech. Engrg., 1984, 110(1): 20-40
respectively; hence
C « 7,536 - + 156f-1H
(27)
to
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The effective damping ratio at the pile-head becomes
a>C 7,536 -B 2 T T / X 1 5 6 ,.,
~^r^5+Tk^5>^-0.57P
D
+
0.074^
(28)
Step 4.—The fundamental natural frequency, /„, of the soil stratum in
vertical S-waves is given by (7,40)
1/2
/14^1/2
x
= 1.15 Hz
(29)
v ;
IT
14 \0.35/
At frequencies, /, lower than 1.15 Hz the damping ratio, D, is taken
constant, equal to 0.57 B = 0.57 X 0.05 - 0.0285; in the range / > 2/„ =
2.30 Hz, D is given by Eq. 28; and in the intermediate frequency range
D is estimated by a linear interpolation. The resulting D = D(/) compares very favorably with the "actual" FE curve, as is evidenced in Fig.
9(b). Notice that for this particular soil profile the effective hysteretic
damping ratio of the system is a smaller fraction (57%) of the hysteretic
damping in the soil, compared to the damping in the homogeneous profile of the previous example. Finally, the variation with frequency of the
pile-head "spring" coefficient, K, is constructed with the help of Fig.
3(d) for S Ep/Es = 29,000, once Ks (6,615 kN/m) and/„ (1.15 Hz) have been
determined.
0.60
x
13.3
THIRD APPLICATION: CONCRETE FILLED STEEL PILE FLOATING
IN A REALISTIC LAYERED PROFILE
The proposed method is now tested with a realistic layered soil profile, typical of those encountered in the San Francisco Bay Area. As pictured in Fig. 10(d), the profile consists of seven main layers and is underlain by bedrock located at about 60 m below the ground surface. A
4.2 m thick sandy fill covers the site, below which lies a 7.8 m thick
layer of normally consolidated San Francisco Bay Mud. The remaining
layers consist of stiff and very stiff clays interbedded with a 2.0 m layer
of sand at a depth of 28.0 m. The properties of each layer, described in
Fig. 10(a) through their Young's modulus, E s , Poisson's ratio, v, and
hysteretic damping ratio, B, were selected to be generally consistent with
measured values in these types of soil. Note in Fig. 10 that the values
of B are different for the different layers.
A very large diameter (1.4 m) concrete-filled steel-pipe pile, of 0.085m pipe thickness, was selected for this example. This pile diameter is
somewhat larger than the largest piles usually driven on-shore for buildings and bridges. The pile was selected as an extreme example of high
stiffness, to test the proposed method and to show that even some of
the stiffest piles can be treated as "flexible" from the viewpoint of their
lateral response. In the example, the pile is embedded in the upper 34
m of the deposit, i.e., with its tip 4.0 meters within the lower (very stiff)
clay layer.
35
J. Geotech. Engrg., 1984, 110(1): 20-40
20Q
400
6QQ
<=*
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Son Fmnclieo
Gay Mud
p,«l.70t/m 3
ym?
FIG. 10.-—Third Application of the Proposed Method: (a) Realistic Layered Soil
Profile; (b) Static Pile Deflection Profile
The pile must be designed against lateral dynamic loads with frequencies, /, ranging from about 0-22 Hz.
From Table 2, the cross section shape factor is
S.l-(^V + 0. 1 o(H?y. 0 .46.Vl.40/
Vl-40/
(30)
leading to an "effective" pile modulus S Ep = 0.46 x 2.5 x 108 = 1.15
x 108 kPa. For a crude estimate of the dynamic active length, /„, we
observe [Fig. 10(1;)] that a representative soil modulus, ESfiV, can be anywhere from, say 80.0 MPa to 200.0 MPa. These values lead to a stiffness
ratio S EP/ES/OT ranging from about 580-1,440. Table 1, then, indicates
that the effective dynamic length, Z„, will most likely be on the order of
20.0 meters—certainly less than the 34.0 m actual pile length.
A static FE analysis yields the normalized deflection profile, Ys, portrayed in Fig. 10(b), and a stiffness Ks » 0.9 x 106 kN/m. (As a crude
order-of-magnitude check: using the expression suggested in Ref. 8 with
the aforementioned range of Esav values one gets the range 0.50 x 106
<KS< 1.10 x 106 kN/m.)
Application of the method is now straightforward. The soil is discretized into 18 sublayers along the length of the pile. For each sublayer,
cm, cmY2Az, cr and cfY2Az are computed (Eqs. 7, 9, 12). Table 3 depicts
these computations. The choice of 8 « 1.35 for Eq. 12 is based on Fig.
7 and on the fact that this soil profile is somewhere in-between a homogeneous and a linearly-inhomogeneous one; clearly some engineering judgment is needed for this choice.
The integral of Eq. 3 is easily evaluated from this table by adding up
Cols. 10 and 11. The effective pile-head damping ratio is
D
53,858
2K.
6
2 x 0.9 x 10
15,061 x 2irf , ...
L
Vi
3/4
—
2 x 0.9 x 106f~ « 0.33 + 0.053/ .
36
J. Geotech. Engrg., 1984, 110(1): 20-40
(31)
TABLE 3.—Computation of Damping for Pile in San Francisco Bay Area Profile
Layer
number
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0)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Az
(2)
0.42
0.42
0.43
0.43
0.64
0.64
0.64
0.64
1.71
1.71
1.93
2.18
3.21
3.21
4.30
5.00
2.00
4.00
2 = 34
P
(3)
0.04
0.04
0.04
0.04
0.04
0.04
0.04
0.04
0.07
0.07
0.07
0.07
0.03
0.03
0.03
0.02
0.02
0.02
Es
(4)
58
87
116
132
132
132
132
132
86
86
86
86
200
200
200
357
273
480
Vs
(5)
112
135
159
165
165
165
165
165
130
130
130
130
184
184
184
257
232
300
cocm
(6)
6,264
9,396
12,528
12,920
12,920
12,920
12,920
12,920
16,254
16,254
16,254
16,254
16,200
16,200
16,200
19,278
14,742
25,920
Crf'4
(7)
2,085
2,662
3,303
3,536
3,536
3,536
3,589
4,563
4^272
4,272
4,272
4,272
6,888
6,888
6,888
8,746
7,404
10,734
ys
(8)
0.996
0.978
0.962
0.920
0.88
0.81
0.75
0.68
0.58
0.41
0.27
0.16
0.07
0.03
0.015
0.010
0.002
0.0
w 2
Ys2 u>c„,Y*&z crf Y , Az
(10)
(9)
(11)
0.992
2,610
869
0.956
3,773
1,069
0.925
1,314
4,983
0.846
4,700
1,286
0.774
1,751
6,400
0.656
5,424
1,485
0.562
4,647
1,290
0.462
3,820
1,348
0.336
9,340
2,454
0.166
4,613
1,212
0.073
2,290
601
0.026
922
241
0.005
260
109
0.001
22
53
0.0002
14
6
0.0001
9
4
0.0
0
0
0.0
0
0
53,858
15,061
Note: Az: m; E s : MPa; Vs: m/s; cm and c r : kN • s/m.
07
0.6
05
0.4
D
0.3
02
0
10
20
IXIO 6
Proposed Method
3 - D Finite Element Analysis
* 0.5XI0 6
0
10
f i n Hz
20
FIG. 11.—Third Application: Comparison of Predictions by Method with Dynamic
FE Results
37
J. Geotech. Engrg., 1984, 110(1): 20-40
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Eq. 31 is plotted in Fig. 11, after being modified at frequencies lower
than 2/„, in which /„ is estimated to be 1.30 Hz, by using the simplified
Rayleigh Procedure described in Ref. 7. The very favorable agreement
with the corresponding FE curve is evident and needs no further
elaboration.
The comparison of the K(f) curves is also portrayed in Fig. 11. The
agreement here is absolutely no surprise, since the largest and the smallest values of the FE computed K(f) are within 10% of Ks.
CONCLUSIONS AND ADDITIONAL CAPABILITIES OF THE MODEL
Determining the dynamic response of laterally loaded piles embedded
in a layered soil deposit is not a routine operation. At present, it requires
the use of rather complicated and not widely available computer programs—a time and money consuming process.
The writers have presented a practical alternative; a simplified procedure whose starting point is the estimation of the static pile deflection
profile. From there on, the method is based on simple physical approximations which have been to a large extent verified, both directly and
indirectly. These approximations refer to the nature of radiation and material damping at different depths along the pile-soil interface; the way
these individual effects combine; and the influence of the natural frequency of the whole deposit on the response of the pile.
Not only are the computations of the method very simple and straightforward such that a computer is not required, but, also, the engineer
has a clear picture of the assumptions and uncertainties involved at every
step of the analysis. Thus, he is encouraged to use his judgment
throughout the process. Moreover, noncircular pile geometries can be
easily analyzed with the help of Table 2 and use of the simple radiation
damping model.
Equally significant is the potential of the proposed method for considering other effects. Phenomena such as the lateral variation of soil
modulus due to the influence of pile installation, and the nonlinear soil
behavior during large-amplitude vibration can in principle be handled
by the method, although further research is still needed.
In conclusion, the proposed model is simple, has so far compared very
favorably with dynamic finite-element analyses, and seems very promising in areas where the current state-of-the-art of pile analysis is not
well developed.
ACKNOWLEDGMENT
The authors are grateful to William J. Gardner for his useful suggestions.
APPENDIX.—REFERENCES
1. Angelides, D. C , and Roesset, J. M., "Nonlinear Lateral Dynamic Stiffness
of Piles," Journal of the Geotechnical Engineering Division, ASCE, Vol. 107, No.
GT11, Nov., 1981, pp. 1443-1460.
2. Banerjee, P. K., and Davies, T. G., "The Linear Behavior of Axially and Laterally Loaded Single Piles Embedded in Nonhomogeneous Soils," Geotechnique, Vol. 28, No. 3, 1978, pp. 309-326.
38
J. Geotech. Engrg., 1984, 110(1): 20-40
Downloaded from ascelibrary.org by ZHANGYUAN NI on 02/25/19. Copyright ASCE. For personal use only; all rights reserved.
3. Bea, R. G., "Dynamic Response of Piles in Offshore Platforms," Special Technical Publication on Dynamic Response of Pile Foundations: Analytical Aspects, ASCE,
O'Neill and Dobry, eds., Oct., 1980, pp. 80-109.
4. Berger, E., Mahin, S. A., and Pyke, R., "Simplified Method for Evaluating
Soil-Pile Structure Interaction Effects," Proceedings of the 9th Offshore Technology Conference, OTC Paper 2954, Houston, Tex., 1977, pp. 589-598.
5. Blaney, G. W., Kausel, E., and Roesset, J. M., "Dynamic Stiffness of Piles,"
Second International Conference on Numerical Methods in Geomechanics, Vol. II,
Virginia Polytechnic Institute and State University, Blacksburg, Va., June,
1976, pp. 1001-1012.
6. Desai, C. S., and Kuppusamy, T., "Applications of a Numerical Procedure
for Laterally Loaded Structures," Numerical Methods in Offshore Piling, Institution of Civil Engineers, London, England, 1980, pp. 93-99.
7. Dobry, R., Oweis, I., and Urzua, A., "Simplified Procedures for Estimating
the Fundamental Period of a Soil Profile," Bulletin of the Seismological Society
of America, Vol. 66, No. 4, 1976, pp. 1293-1321.
8. Dobry, R., Vicente, E., O'Rourke, M. J., and Roesset, J. M., "Horizontal
Stiffness and Damping of Single Piles," Journal of the Geotechnical Engineering
Division, ASCE, Vol. 108, No. GT3, Mar., 1982, pp. 439-459.
9. Faruque, M. O., and Desai, C. S., "3-D Material and Geometric Nonlinear
Analysis of Piles," Proceedings of the Second International Conference on Numerical Methods in Offshore Piling, University of Texas at Austin, Tex., 1982, pp.
553-575.
10. Franklin, J. N., and Scott, R. F., "Beam Equation with Variable Foundation
Coefficient," Journal of the Engineering Mechanics Division, ASCE, Vol. 105,
No. EM5, 1979, pp. 811-827.
11. Gazetas, G., "Analysis of Machine Foundation Vibrations: State-of-the-Art,"
International Journal of Soil Dynamics and Earthquake Engineering, Vol. 2, No. 1,
1983 (presented in the International Conference of Soil Dynamics, held at
Southampton University, July, 1982), pp. 2-43.
12. Gazetas, G., "Vibrational Characteristics of Soil Deposits with Variable Wave
Velocity," International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 6, No. 1, 1982, pp. 1-20.
13. Georgiadis, M., and Butterfield, R., "Laterally Loaded Pile Behavior," Journal
of the Geotechnical Engineering Division, ASCE, Vol. 108, No. GT1, 1982, pp.
155-165.
14. Kagawa, T., and Kraft, L., "Seismic p-y Response of Flexible Piles," Journal
of the Geotechnical Engineering Division, ASCE, Vol. 106, No. GT8, Aug., 1980,
pp. 899-918.
15. Kagawa, T., and Kraft, L. M., "Lateral Load-Deflection Relationship of Piles
Subjected to Dynamic Loadings," Soils and Foundations, Vol. 20, No. 4, 1980,
pp. 19-36.
16. Kaynia, A. M., and Kausel, E., "Dynamic Behavior of Pile Groups," Proceedings of the 2nd International Conference on Numerical Methods in Offshore Piling, University of Texas at Austin, Tex., 1982, pp. 509-531.
17. Krishnan, R., "Static and Dynamic Response of Piles," thesis submitted to
the Rensselaer Polytechnic Institute, in Troy, New York, in 1982, in partial
fulfillment of the requirements for the Degree of Master of Science.
18. Kuhlemeyer, R. L., "Static and Dynamic Laterally Loaded Floating Piles,"
Journal of the Geotechnical Engineering Division, ASCE, Vol. 105, No. GT2, Proc.
Paper 14394, Feb., 1979, pp. 289-304.
19. Kuhlemeyer, R. L., and Lysmer, J., "Finite Element Method Accuracy for'
Wave Propagation Problems," Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 99, No. SM5, Proc. Paper 9703, May, 1973, pp. 421-427.
20. Liou, D. D., and Penzien, J., "Mathematical Modeling of Piled Foundations,"
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