Horizontal Response of Piles in Layered Soils

HORIZONTAL RESPONSE OF PILES IN LAYERED SOILS

By George Gazetas1 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—beam-on-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 wave-propagation 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 head-loading.

'Assoc. Prof, of Civ. Engrg., Rensselaer Polytechnic Inst., Troy, N.Y. 12181. 2Prof. 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 Geo-technical Engineering, Vol. 110, No. 1, January, 1984. ©ASCE, ISSN 0733-9410/ 84/0001-0020/$01.00. Paper No. 18496.

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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 FIG. 2.—Static (Y„) versus Dynamic (Y4) Cross Sections Considered Pile Displacement Shapes

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TABLE 1.—Active Length Under Dynamic Loading for Preliminary Estimations

Soil profile 0)

Homogeneous (constant modulus: Es) Inhomogeneous (modulus proportional

to depth: £s = Es z/b)

Active Length/Width, IJb

Expression (2)

3.3(EPS/ES)1/5

3.2( EPS/ES)1/6

Typical range (3)

8-20

5-15

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 = - (1) yd

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 2K K

(2)

The objective of the method developed herein is to inexpensively ob-

TABLE 2.—Pile-Cross Section Shape Factor

Pile cross section (1)

Shape factor, S (2)

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)

1 1 - (bM 1 - (bM + Earn../

Es««i)(b,/6)4

1.7(A/Bf

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tain realistic estimates of K(f) and C(/) or D(f), at the head of a laterally loaded pile.

OUTLINE OF PROPOSED METHOD

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-on-Winkler-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 fixed-head 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

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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/VS|„,

FIG. 3.—Lateral Dynamic Stiffness versus Frequency from 3-D Finite-Element Analyses [E, = 2p V2

S (1 + v); Es = 2P V] (1 + v)]

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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 = cr(/;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 (Dr)

«—Material (Hysterfltic) Damping Ratio ( 0 m )

FREQUENCY, f

FIG. 4.-—Approximate Modification of Damping versus* Frequency Curve at Fre­quencies N@ar Resonance

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Wlniisly Long Rod-,

Rod EndsotlhiiPeinl

I Wow Propojalion

dEn<ttonni»poini-^

f V-Wove Spwd } j } - f

l-D Model o

P-Wovtl - —

Plan

1 Betger «l

~-!

•=>

| 1 5

01(4)

,vo

~ u « -t.,.0

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 + vs (4)

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:

-"•[£2] 1/2

(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 in-

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consistent with the assumed stress-free surrounding soil in the x-direc-tion 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

i r ( l - v)" (6)

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 two-dimensional. 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

* I • TT-8-O—CL_Q_O O 6 ,5„r 9 • S 1 I % \~f—l

0 ^ ^ »a2B

Pile Cross-section

0.5 o„ =27TfB/Vs

r

-•

0"

>^-Free-Head Piles

Fixed-Head Piles

101 I04

SEp/g, or

Soil Profile

blS-

i t t t t a ^ ^

10'

3E,/E,

Symbol

•

10'

FIG. 6.—Radiation Dashpot Coefficient FIG. 7.—Coefficient 8 of Eq. 12 versus of Circular Pile Cross Section: Evalua- Relative Pile Stiffness tlon of the Approximate Plane-Strain Model Developed by Authors

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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' = 1 + 4BPsys

3.4

.ir(l - v).

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

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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

/ \ 3 / 4

w.~2\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-and-error 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.:

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Hz) = ^ \ (10) ys(z)

Notice that p(z) has units of force/length and k(z) of force/(length)2. The determination of fc(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 Ep/Es.

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 free-head 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 re-

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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/m2), constant Poisson's ratio v = 0.40 and constant hysteretic damping ratio p = 0.05. In addi­tion: soil mass density ps = 1.90 T/m3; 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 - I I ! i I 1 1 ' " T 0 tn 2'n 10 20

f in 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

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from Fig. 7 for S Ep/Es = 500), we construct the normalized profile, Y$(z) = ys(z)/y0(z), which is shown in Fig. 8(a). The static stiffness is (30,36)

/ \ 1 / 4

Ks = (4EpIp)1/4 k3/i - ( 4 x 2.5 x 108 x — x l4 x 0.344J

x (2.15 x 105)3/4 ~ 6.4 x 10s k N / m (14)

Step 2.—We start with material damping. Since damping ratio and soil modulus are both independent of z, Eqs. 9 and 12 yield a depth-inde­pendent coefficient

cm = 2k- = 2 X 2.15 x 105 x - = 4.3 x 1 0 5 - (15) (0 CO (0

The distributed radiation dashpot coefficients for shallow and greater depths are computed from Eqs. 7. For z s zr = 2.5 x 1 = 2 . 5 m

cr = cri = 4BP sV 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 « 4BPsy s(2.578«0-1 /4) « l,763a0~1/4 (17)

Step 3.—The pile-head "dashpo t" coefficient, C, is estimated from the energy relationship, Eq. 3:

/-25 ,-2.5 [25

C = cm\ Y\dz + cn Y]dz + cr2 Y*dz (18) Jo Jo J2.5

The above three integrals can be evaluated analytically in this particular case. They are found to be approximately equal to 2.23, 1.84 and 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)fl0_1/4

do

» 9.59 x 105 - + 2,788fl0_1/4 (19)

CO

Alternatively, the effective pile-head damping ratio is

coC 9 .59X10 5 S 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)

or D « 0.038 + 0.037/3 / 4 (20b) Step 4.—The fundamental natural frequency, / „ , of the soil s tratum

due to vertically propagating (horizontally polarized) S waves is Vs 180

/„ = 0.25 — = 0.25 x — = 1.80 Hz (21) H 25

At frequencies lower than 1.80 Hz the damping ratio is taken as con-

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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

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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/m3 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

/ S E \ 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 (24) W (0 \ (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/"1/4z5/8 {25a)

and for z > 0.88 m:

cn = cri(z) « 4BPsVs(2){2.97[fl0(2)r1/4} - 166/-1/4z5/8 (25b)

Step 3.—Eq. 3 takes the form ,•5.20 r- /.0.88

C - 5,582- zY2sdz+ 3.5 z5/8Y2

sdz w Jo L Jo

+166 z5/eY2sdz f~m (26)

The three integrals are numerically computed to be 1.35, 0.42 and 0.70,

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respectively; hence

C « 7,536 - + 156f-1H (27) to

The effective damping ratio at the pile-head becomes

a>C 7,536 -B 2 T T / X 1 5 6 ,., D~^r^5+Tk^5>^-0.57P + 0 . 0 7 4 ^ (28)

Step 4.—The fundamental natural frequency, /„ , of the soil stratum in vertical S-waves is given by (7,40)

1/2 0.60 13.3 / 1 4 ^ 1 / 2

x x = 1.15 Hz (29) IT 14 \0.35/ v ;

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.

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, Es, 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 0.085-m 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.

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ym?

Son Fmnclieo Gay Mud

p,«l.70t/m3

20Q 400 6QQ <=*

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 .1o(H?y.0 .46.- (30) Vl.40/ Vl-40/

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 discre-

tized into 18 sublayers along the length of the pile. For each sublayer, cm, cmY2Az, cr and cfY

2Az 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 2K.

53,858

2 x 0.9 x 106 2 x 0.9 x 106

15,061 x 2irf , ... — Lf~Vi « 0.33 + 0.053/3/4. (31)

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TABLE 3.—Computation of Damping for Pile in San Francisco Bay Area Profile

Layer number

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

Ys2

(9) 0.992 0.956 0.925 0.846 0.774 0.656 0.562 0.462 0.336 0.166 0.073 0.026 0.005 0.001 0.0002 0.0001 0.0 0.0

u>c„,Y*&z (10)

2,610 3,773 4,983 4,700 6,400 5,424 4,647 3,820 9,340 4,613 2,290

922 260

53 14 9 0 0

53,858

crfwY2, Az (11)

869 1,069 1,314 1,286 1,751 1,485 1,290 1,348 2,454 1,212

601 241 109

22 6 4 0 0

15,061

Note: Az: m; Es: MPa; Vs: m/s; cm and cr: kN • s/m.

07

0.6

0 5

0.4

D

0.3

0 2

0 10 20

IXIO6

* 0.5XI06

0 10 20

f in Hz

FIG. 11.—Third Application: Comparison of Predictions by Method with Dynamic FE Results

37

Proposed Method

3 -D Finite Element Analysis

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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," Geotech-nique, Vol. 28, No. 3, 1978, pp. 309-326.

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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.

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