simple expression of the dynamic stiffness of grouped piles

Chapter 2SIMPLE EXPRESSION OF THE DYNAMICSTIFFNESS OF GROUPED PILES2.1 INTRODUCTIONPiles, grouped beneath a superstructure, interact with the surrounding soil during anearthquake, and the dynamic pile-soil-pile interaction often affects the motion of thesuperstructure to a considerable extent. Straightforward evaluation of the pile-soil-pileinteraction, however, is cumbersome especially in dealing with tens or hundreds of pilesgrouped together. Hence a simplified approach for the evaluation of such dynamicpile-soil-pile interaction is highly desirable for the purpose of treating the dynamicbehavior of an entire soil-foundation-structure system. Some research has been carriedout with the objective of developing such a simplified approach. Attempts include theRing-Pile method [Takemiya, 1986] and Closely-Spaced-Plates model [Ohira and Tazo,1985]. In these methods, respectively, piles with the soil caught among them arere-grouped into several concentric cylinders (piles arranged in concentric circles) andinto soil-pile-striped upright plates, allowing close evaluation of interaction effects to bemade with less time and trouble. This chapter presents a further simplified approach inwhich a group of piles is viewed as a single equivalent upright beam.Careful examination of the deflections of grouped piles reveals that most piles areindeed flexible in practice in the sense that they do not deform over their entire lengths.Instead, pile deflections become negligible below their active lengths. With the activelengths provided for different soil-pile systems, it is shown in the latter half of thischapter that pile-cap (grouped-piles-head) stiffness can be approximated in terms of the

14 SIMPLE EXPRESSIONS OF THE DYNAMIC STIFFNESS OF GROUPED PILESmass, damping and stiffness parameters; the parameters are invariant of frequency andare dependent only on the mechanical properties of soil and pile. The method presentedin this report requires real-time manipulation of soil-structure interaction parameters inaccordance with the development of non-linear features of soils and piles. The presentsimple expression of pile-cap stiffness, thus proves to be useful despite the availabilityof efficient numerical programs for analyzing pile-soil-pile interaction.2.2 EQUIVALENT SINGLE UPRIGHT BEAMIn discussing the equivalent upright beam, straightforward evaluation of pile-soil-pileinteraction is first necessary to provide rigorous solutions. Based on the numericalscheme presented by Tajimi and Shimomura [Thin-Layered Method, 1976] that allowssoil-embedded foundation interaction effects to be rigorously evaluated, a numericalprogram “TLEM”(Ver. 1.1) has been developed for soil-pile group interaction analyses[Konagai, 1998d]. The Thin-Layered Element Method is a method for describing soilstrata rather than for foundations. In this method, a soil deposit is treated as an infinitestratified medium with the inclusion of a cylindrical hollow in which the foundation isfitted. The piles are assumed to be upright Timoshenko or Beronoulli-Euler beams. Theevaluation of pile-soil-pile interaction effects in this program is based on thesuperposition method that was originally proposed by Poulos [1968, 1971]. In thisapproximation, only two piles are considered in the formulation of a global flexibilitymatrix, and other piles’ effects on these two piles are totally ignored (Figure 2.1).Kanya and Kausel [1982] have shown that the superposition scheme gives reasonableresults not only for static loads but for dynamic loads as well.active pilexθs d = 2r 0passive pileyFigure 2.1 Active and passive piles

SIMPLE EXPRESSIONS OF THE DYNAMIC STIFFNESS OF GROUPED PILES 15In contrast to the above approach, the present single upright beam is a composite ofn piles and the soil caught among them embedded in a horizontally stratified infinitepsoil deposit with material damping of the frequency-independent hysteretic type (Figure2.2). Following the TLEM assumption, the soil deposit overlying its rigid bedrock2π R 0, of thisshould include a cylindrical hollow of radius R 0. The cross-section,hollow is assumed to be identical to the beam’s cross-sectionAGenclosed with thebroken line circumscribing the outermost piles in the group (Figure 2.2a). The motionof the hollow is assumed to be compatible with that of the beam. The soil-pilecomposite together with its exterior soil is divided into n L horizontal slices as shownin Figure 2.2. The following assumptions are adopted to derive the stiffness matrix ofthe equivalent single beam:(1) Pile elements within a horizontal soil slice are all deformed at once keeping theirintervals constant, and the soil caught among the piles moves in a body with thepiles.(2) Frictional effects due to bending of piles (external moments on each individual pilefrom soil) are ignored.(3) The top ends of piles are fixed to a rigid cap.(4) All upper or lower ends of the sliced pile elements arranged on the cut-end of a soilslice remain on one plane (Note this assumption does not necessarily mean thateach pile’s cross-section remains in parallel with this plane. See Figure 2.2b).centroidA gjR 0w 1u jcentroidR 0w jn Lh jn p piles(a) soil-grouped piles systemremain on one plane(b) sliced elementsFigure 2.2 Assumptions for evaluation of equivalent single beam

16 SIMPLE EXPRESSIONS OF THE DYNAMIC STIFFNESS OF GROUPED PILESWith assumptions (1) and (4), there are only two degrees of freedom for each cut-end ofall slices of the soil-pile composite, namely, sway and rocking motions respectivelyu ( { u u L } Tw ( = { w w L } T) (Figuredesignated as { }= ) and { }12u N L2.2b). The rocking motions are expressed in terms of the anti-symmetric vertical motion{ w } at the outermost edge ( r = R0) of the equivalent beam with respect to the beam’scentroid. In sway motions, all nppiles are equally displaced (assumption (1)), causingthe bending stiffness, EI , of the equivalent beam to be simply nptimes as large asthe bending stiffness of an individual pile. Assumptions (3) and (4) imply that axialmotions of the piles control the overall anti-symmetric rocking motion of the equivalentbeam just as reinforcements in a concrete beam do. Therefore, another bending stiffnessparameter, EI G , is introduced to describe the rocking motion of the beam. Thisstiffness parameter EI G is evaluated following the same procedure as that used for theevaluation of bending stiffness of a reinforced concrete beam (See APPENDIX I).Lateral external forces { p x} and moments { M } are finally described in matrixnotation in terms of {} u and { w } as specified in Equation (A12) in APPENDIX I:12w NL−1−11st column of[ ][ ] [ ][ L][ D]/R ⎤0andL D L M⎥zeros for other columns ⎧ u ⎫⎥⎪⎪LLLLLLLL ⎥⎨L⎬−1−1[ D] [ L] /R and [ Q] with D added to the ⎥⎪⎪ ⎭⎡⎧ ⎫⎢⎪px ⎪⎢⎨L⎬ = ⎢ LLLLLLLL⎪ M ⎪ ⎢1st row of0⎪ ⎪ ⎢⎩R0⎭ ⎢⎣zeros for other rowswhere, [ L ], [ D ] and [ ]M1,1upper - left corner⎥⎩w⎥⎦(2.1)Q are assembled global matrices corresponding to theindividual layer parameters of 1 / hj( hj= thickness of the j-th layer), h j/ EI andG 2EI / R0hj, respectively, (See Equations (A2), (A4) and (A10) in APPENDIX IQ , respectively).defining [ L ], [ D ] and [ ]“TLEM” has been upgraded for evaluation of the behaviors of an equivalent singlebeam (Ver. 1.2). Figure 2.3 shows pile cap stiffnesses k xxfor sway motions of 2×2and 3×3 steel pile groups (Table 2.1) plotted as functions of frequency. The results forthe equivalent beams are shown as open circles. Each pile group is embedded in thesame homogeneous soil deposit (Table 2.2) equally divided into 20 slices. Downwarddips in these plots of k xxoccur at essentially the resonance frequencies of the soilstratum for vertical shear wave propagation. As a whole, however, every real part of thepile cap stiffnesses decreases slowly as the frequency increases, whereas its imaginarypart representing the damping of a soil-pile group system shows a general upward trendto the right. The curves for the equivalent single beams agree well with rigoroussolutions from “TLEM” (Ver. 1.1).

SIMPLE EXPRESSIONS OF THE DYNAMIC STIFFNESS OF GROUPED PILES 17Table 2.1 Parameters for steel pilesE p (tf/m 2 ) ρ (t/m 3 ) r 0 (m) Thickness (m) Length (m)2.1×10 7 7 0.3 0.0089 20Table 2.2 Parameters for soilρ (t/m 3 ) v (m/s)T1.5 80 0.49ν4.02 2 pilesd4.03 3 pilesImpedance k xx(10 4 tf/m)rigorous solution3.0 s3.0equivalent beamreal part2.01.0imag. part0.00 2 4 6 8 10Frequency (Hz)Impedance k xx(10 4 tf/m)real part2.0s / d = 2.51.0imag. part0.00 2 4 6 8 10Frequency (Hz)Figure 2.3 Variations of stiffness parameters for sway motions of pile groupsAssumption (1) taken in this chapter to derive the stiffness matrix of theequivalent-upright beam (Equation (2.1)) implies that the spacing between piles, s,should be within a certain limit. To investigate this constraint on the spacing betweenpiles, the results of the program “TLEM” (Ver. 1.2) were compared with the rigorousresults obtained from “TLEM” (Ver. 1.1). Here, hollow cylindrical steel piles (Table2.1) embedded in a homogeneous soil with the density ρ and the shear wave velocityvT(Table 2.2) were considered. The variations of the ratios between approximate andrigorous solutions with respect to normalized frequency ω s / vTare shown in Figure2.4 for three different values of spacing ( s/ d = 25, . s / d = 3. 33 and s/ d = 50). . Fora wide range of cases examined, “TLEM” (Ver. 1.2) is found to produce insignificanterror below a certain limit of spacing, s/ d < 3. Below this limit, however, it is noted

18 SIMPLE EXPRESSIONS OF THE DYNAMIC STIFFNESS OF GROUPED PILESthat the error can yet become significant as the non-dimensional frequency increasesbeyond a certain limit (See thick lines in Figure 2.4 for large number of piles).ffAn earthquake causes the free-field ground motion { u M w } Tin which verticaldisplacement vector { w f } can be ignored in many of the practical cases encountered.The piles in this soil deposit, however, will not follow the free-field deformation pattern.This deviation of the displacements from the free-field soil displacements is denoted byss{ w } Tf sf s{ u u w + w } Tu M . Equation (2.1) is also used to evaluate effective foundation input motion+ M . The effects of soil-embedded-foundation kinematicinteraction are portrayed in the form of two kinematic displacement factors in sway androcking motionsf sf s su uT = 1+1e,swayfu, w1+ w1w1Te, rocking= ≅(2.2a), (2.2b)ff1u1u1plotted as functions of frequency. In Equation (2.2b), the vertical component offfree-field ground motion w 1is ignored.2 by 21.00.83 by 34 by 45 by 5|approx. solution||rigorous solution|0.6ω s / v T= 0.620.4s / d = 2.50.2s / d = 3.3s / d = 5.00.00.0 0.5 1.0nondimensional frequency (= ωs /v T)Figure 2.4Variation of ratios between approximate and rigoroussolutions with respect to normalized frequency ω s / vT

SIMPLE EXPRESSIONS OF THE DYNAMIC STIFFNESS OF GROUPED PILES 19Figure 2.5 shows the kinematic displacement factors of a 2×2 PC pile group plottedas functions of non-dimensional frequency ω s / vT(s/ d = 2 , See Tables 2.3 and 2.4),and they are in good agreement with rigorous solutions by Fan et al. (1982).It is again to be remembered that the piles behaving in accordance with assumption(1) are completely equal with each other not only in their deformations but also inlengthwise distributions of internal force and moment. The dynamic pile-soil-pileinteraction effects are thus excluded. Even for a static loading, any discussion based onthe assumption does not reflect the fact that outermost piles sustain heavier loads thanthose on inner piles (static pile-soil-pile interaction). Yet, the present single uprightbeam, as has been shown above, satisfactorily approximates the motions of a pile groupwith a reduced number of parameters. These parameters allow the stiffness parametersof a pile cap to be described in a further simplified manner; a discussion of lateraltranslation follows in Section 2.3.Table 2.3 Parameters for pilesEpIp (tf m2 ) ρ (t/m 3 ) r 0 (m) length (m)2.4×10 5 2.0 0.5 15Table 2.4 Parameters for surface soil depositρ (t/m 3 ) v (m/s) ν Thickness (m)T1.75 100 0.40 20kinematic displacement factorsT e, swayand T e, rocking1.51.00.50.0after Fan et al.sway (E p/E s= 1000)rocking (E p/E s= 1000)sway (E p/E s= 10000)rocking (E p/E s= 10000)rockingswayTLEM0.0 0.1 0.2 0.3 0.4 0.5normalized frequency (ω d/v T)Figure 2.5 Kinematic displacement factors of pile groups

20SIMPLE EXPRESSIONS OF THE DYNAMIC STIFFNESS OF GROUPED PILES2.3. ACTIVE PILE LENGTH AND PILE CAP STIFFNESS2.3.1 Active pile lengthIn practice, most laterally loaded piles are ‘flexible’ in the sense that they do not deformover their entire length L. Instead, pile deflections become negligible below an activelength L (Figure 2.6). This length depends on how stiff the pile is in comparisonawith the surrounding soil. In engineering practice, Chang’s formula is widely used; inthis a pile is supported by discrete soil springs K hd , and the characteristic parameter isintroduced as β =4K hd / 4EIwith Khdesignating the coefficient of subgradereaction and d the pile diameter. The length given by 1 / β is thus directly relevant tothe active pile length La. When a soil is treated as an elastic continuum, however, it isto be recognized that Khis not an inherent constant in the soil, but rather dependenton d . In addition, the active pile length is more rationally evaluated by replacing K hdwith the shear modulus of soil µ . Some formulas for rather extreme soil profiles havebeen presented by Randolf(1981), Velez (1983) and Gazetas (1983), and in general, L ais closely related to the following parameter L 0:EIL0= 4(2.3)µwhere, EI = bending stiffness of the pile, and µ = shear modulus of soil(representative value). The active length L ais thus given as:L a= α L 0 0(2.4)with the parameter α0reflecting different soil profiles. For an nppile group, EI inEquation (2.3) will presumably be replaced with EI ( = n E I ) specified byEquation (A5) in APPENDIX I.R 0pppL aFigure 2.6Active length of pile

SIMPLE EXPRESSIONS OF THE DYNAMIC STIFFNESS OF GROUPED PILES 212.3.2 Pile cap stiffnessIt is assumed that only the soil above the active pile length, L a, is deformed asshown in Figure 2.7a. The upper soil is then divided into vertical soil columns. Giventhe prescribed vibration mode ψ z / L ) that satisfies ψ ( 0) = 1 and ψ ( 1) = 0 , these(acolumns can be replaced with simple-damped oscillators. Reducing the cross-section ofeach soil column, the soil deposit above L ais modeled by an infinite plane supportedby Winkler-type springs (Figure 2.7b). Lame’s constants λ p, µ p( µ p= shearmodulus) and mass density ρ pof the soil plane and Winkler-type spring constant k pfor the model are expressed in terms of ψ as:L a22λ = λ( z)( ψ ( z / L ))dz , µ = µ( z)( ψ ( z / L ))dz , ρ = ρ( z)( ψ ( z / L ))p∫0L aapL a2∫0⎛ dψ( z / La)and ∫ ⎟ ⎞kp= µ ( z)⎜dz(2.5a)-(2.5d)⎝ d z0⎠A frequency parameter, ω 0, is introduced herein as:ω = k p0(2.6)ρpFor a homogeneous soil, parameters λ p, µ pand ρ pin Equations (2.5a)-(2.5c) arerewritten asλp= λα 1L a, µp= µα 1Laand ρp= ρα 1La(2.7a)-(2.7c)12with α1 = ∫( ψ1( ζ ))d ζ .0Even for inhomogeneous soils too, similar expressions may be derived with λ , µ , andρ interpreted as representative values of λ( z ), µ( z ) and ρ( z ), and the parameterα1portraying different soil profiles.apL a∫0a2dzψR 0λ µ ρk pppm gpL ak gz(a) Vertically sliced soil above L a(b) Equivalent modelFigure 2.7 Soil deformation

22SIMPLE EXPRESSIONS OF THE DYNAMIC STIFFNESS OF GROUPED PILESThe expression of soil stiffness, k s, for the lateral motion of a massless bodyembedded in the soil plane in Figure 2.7b is completely identical to that given in Novaket al. [1978] regardless of the presence of Winkler-type springs kp, i.e.:2 4K1(b0) K1(a0) + a0K1(b0) K0( a0) + b0K0( b0) K1(a0)ks= πµpa0(2.8)b0K0( b0) K1(a0) + a0K1(b0) K0( a0) + b0a0K0( b0) K0( a0)where K 0and K 1are modified Bessel functions of the first and second order,respectively. Both a 0and b 0are normalized circular frequencies. As shown inAPPENDIX II, the only difference from Novak’s solution, owing to the presence ofWinkler-type springs k pappears as an inclusion of the frequency parameter ω 0ina 0and b 0as:0R0a0= ω 0R0η , b0= ω ωη with η = − ⎛ v Tv L⎝ ⎜ ⎞1 ⎟(2.9a)-(2.9c)ω0 ⎠in which ω is the circular frequency, and= µ / ρ (= transverse wave velocity in the plane) (2.9d)vT pppvL p= ( λp+ 2µp) / ρp(= longitudinal wave velocity in the plane) (2.9e).Since a0and b 0are respectively functions of vT pand vL p, Equation (2.8) is in turna function of the Poisson’s ratio ν . The expression of Equation (2.8) for a Poisson’sratio equal to 0.5 is obtained by taking a limit as ν → 0. 5 in Equation (2.8), i.e.:* 2k s = 2S + msω (2.10)where m s( = ρπ p R 2 0 ) is the soil mass of the same volume as the cylindrical hollow inthe soil plane, andaK aS* 0 1 ( 0= 2πµ )(2.11)K ( a )0 02Table 2.5 Values of ξ kand ξ mPoisson’s ratio,νξ kξ m0.50 2.000 1.00000.47 1.831 0.53360.45 1.741 0.37400.43 1.667 0.26280.40 1.580 0.14280.35 1.476 0.03520.25 1.351 00.20 1.311 00.10 1.252 00.00 1.213 0

SIMPLE EXPRESSIONS OF THE DYNAMIC STIFFNESS OF GROUPED PILES 23It is found that the stiffness k sfor any Poisson’s ratio other than 0.5 can beapproximately expressed in the same form as Equation (2.10) but with a smallmodification [Nogami and Konagai, 1986], i.e.:* 2ks = ξk( ν) ⋅ S + ξm( ν)⋅msω(12)where, ξk( ν) and ξ m( ν) are functions dependent only on Poisson’s ratio ν . Thevalues ξk( ν) and ξ m( ν) are given in Table 2.5.Konagai et al. [1992, 1998b] have shown that assuming plane stress condition overthe entire extent of the soil plane allows k sto approximate closely the rigoroussolution of the soil stiffness, and thus, Poisson’s ratio ν in Equation (2.12) must bereplaced with ν * for a plane-stress medium, which is expressed as:*λ*pν = (2.13)*2( λ + µ )pp2λ µwhere, λ* p pp =(2.14)λ p + µ pIt is noted that ν * ranges from 0 to 1/3, and thus, ξm ( *ν ) in Equation (2.12) iscompletely equal to zero. Equation (2.12) is then rewritten as;* *ks= ξk( ν ) ⋅S(2.15)The function K1( a0) / K0( a0) is approximated by 1+ 0.4 / a0, when the absolute valueof a0is larger than 0.01 [Konagai and Nogami, 1998a]. This simplification leads to:* aK 0 1( a0)*k s = 2πµ pξk( ν ) ≅2πµ pξk( ν ) ⋅ a0( 1+04 . / a0)(2.16)K0( a0)Two limiting cases of ω → 0 and ω →∞ are addressed herein. For the static case( ω ≅ 0 ), η of Equation (2.9c) approaches 1. Replacing ω 0and v in Equation(2.9a) with those specified in Equations (2.6) and (2.9d), respectively, non-dimensionalfrequency a 0in the static case is expressed as:ak= (2.17)p0R 0µpSubstituting into Equation (2.17) Equations (2.5d) and (2.5b) which specify kpandµp, respectively, Equation (2.17) is rewritten as:α2R0a0= (2.18)α1L a12⎛ dψ( ζ ) ⎞where, α2= ∫ ⎜ ⎟ dζwith ζ = z / La0 ⎝ dζ⎠Equation (2.16) is thus simply written as:T p

24SIMPLE EXPRESSIONS OF THE DYNAMIC STIFFNESS OF GROUPED PILES⎛α⎞ ⎛⎞2R0R0k ≅ ⎜ +⎟ =⎜ +⎟s2πξkα1µLa0.4 µ La2πξkα20. 8πξkα1(2.19)⎝ α1La⎠ ⎝ La⎠For the dynamic case (ω →∞), non-dimensional frequency a 0converges on:ω R0a0= i = ia(2.20)vT pEquation (2.16) is thus approximated by:k ≅ L i ⋅ 2πξα ⋅ a + 0. πξ α(2.21)s( )µa k 18k1From Equations (2.19) and (2.21), soil stiffness will presumably be approximated as:⎧⎛R⎫0⎞k ⎨⎜⎟s≅ µ La2πξkα2+ 0.8πξkα1+ i ⋅ 2πξkα1⋅ a⎬(2.22)⎩⎝La⎠⎭Even without the soil above the active pile length, the pile group exhibits its ownstiffness, k (Figure 2.7b), which is described as:kg4EIp µ L0α3gα3= α3 3= µ L3 3 0LaLaα0122≅ (2.23)⎛ d ψ ( ζ )where, ∫ ⎟ ⎞α3=⎜ dζ20 ⎝ dζ⎠Both ksand k gsustain the massamong the piles. This massmgLa02 20ψ z)mgof the embedded pile group with soil caughtmgis approximated by:20≅ ∫ ρ πR( dz = ρ πRL α(2.24)ssaTherefore the overall stiffness k xxof the pile cap for sway motion is given as:kxx = ks + kg −mgω 2 (2.25)From Equations (2.22), (2.23) and (2.24), Equation (2.25) is rewritten as:⎡⎪⎧R ⎛⎞⎪⎫⎤0α⎢⎨⎜32k⎟xx≅ µ La2πξkα2+⎬ + ⋅ ⋅ − ⋅ ⎥⎢0.8πξkα1+3i 2πξkα1a πα1a (2.26)⎣⎪⎩La⎝ α0 ⎠⎪⎭⎥⎦Substituting Equation (2.3) into Equation (2.26), one obtains:⎡⎪⎧R ⎛⎞⎪⎫⎤0α⎢⎨⎜32k⎟xx≅ µ L0 2πξkα2+⎬ + ⋅ ⋅ − ⋅ ⎥⎢0.8πξkα0α1+2i 2πξkα0α1a πα0α1a (2.27)⎣⎪⎩L0⎝α0 ⎠⎪⎭⎥⎦It is now obvious that k xxin Equation (2.27) has the following simple form withfrequency-independent stiffness k 0, and damping and mass parameters c 0and m 0respectively:2≅ k + i ⋅ c ⋅ a − m ⋅(2.28)where,k xx0 00a1

SIMPLE EXPRESSIONS OF THE DYNAMIC STIFFNESS OF GROUPED PILES 25k0R0c0m0= c1+ c2, = c3andµ L0L0µ L0µ L0= c4(2.29a)-(2.29c)with c1= 2πξ kα2,α3c2= 0. 8πξ kα0α1+2α0, c3= 2πξ kα0α1and c4= πα0α1.(2.30a)-(2.30d)The above equations show some important features of the pile cap stiffness. Among theparameters specified in the above equations, c1, c2, c3and c 4are dependent on theshape function ψ ( ζ ), which may not differ drastically in different soil-pile systems aslong as piles exhibit a flexible nature, and k / L 0µ0alone includes a term proportionalto 0 0. Equation (2.28) was derived with the intention of showing what could be themost important key parameters that determine . The assumption taken to derive theequation is good enough for this purpose, but certainly is an oversimplification of reality.Since soils below active pile lengths are not allowed to deform at all, the assumption isliable to lead to overestimation of the stiffness parameter k 0and underestimation ofthe damping parameter c0. Therefore, parameters c 1, c2, c3and c 4were obtainednot directly from Equations (2.30a)-(2.30d), but in such a way that the overall errorwould be minimized for the variety of soils and pile parameters examined. Theparameters that have been considered are: 1) pile parameters such as group-pile stiffness,EI ( = npEpIp), and active pile length ratio, L0 / L; and 2) soil parameters includingshear modulus µ and material damping D . In this discussion, only a homogeneoussoil profile with a square arrangement of piles is considered. The best fit of the valuesfrom Equation (2.28) to rigorous solutions of k xxwas obtained by setting c 1, c 2, c 3and c 4at 2π , π /2, 2π and π /4, respectively. Some representative cases areshown in Figures 2.8a-2.8f.The present simple expression of k xx(Equation (2.28)) allows the effects of overallsite non-linearity to be reflected by simply replacing the shear modulus of the intact soil,µ , with the complex modulus, µ'( 1 + iD ); this describes equivalent-linear features ofthe soil experiencing dynamic loading, and is obtained from shear-modulus-reductionand damping ratio curves of the soil. This manipulation, however, causes the stiffnessand damping parameters k 0and c 0in Equations (2.29a) and (2.29b) to be slightlydependent on frequency as:k0⎛ R0π ⎞= ⎜2π+ ⎟ −2πD⋅ a(2.31a)µ ' L ⎝ L 2 ⎠00c0⎛ R0π π 2⎞= 2π+ ⎜2π+ − a ⎟ D(2.31b)µ ' L0⎝ L02 4 ⎠When the effect of D cannot be ignored in Equations (2.31a) and (2.31b), appropriatekxx

26SIMPLE EXPRESSIONS OF THE DYNAMIC STIFFNESS OF GROUPED PILESvalues of k 0and c 0must be determined taking into account the most probablepredominant frequency a in the soil-structure interaction reality. Figures 2.8a-2.8bshow that introducing the complex shear modulus of soil, the effect of material dampinghas properly been taken into account.The downward dips in these non-dimensional plots of rigorous variations of k xxvs.frequency occur at essentially the resonance frequencies of the soil stratum for verticalshear wave propagation. Thus the results from ‘TLEM’ analyses with a perfectly rigidbase laid under the soil stratum correspond to cases where this effect is mostpronounced. It is therefore more likely that the solutions adhere along the ridges ofthese plots as the bases become more flexible. As can be seen from Figure 2.8c,Equation (2.28) underestimates slightly the real part of stiffness, and overestimates itsimaginary part for lower values of shear modulus of soil.From the study of a wide range of pile parameters (viz. number of piles, diameter ofindividual piles and length of piles), it was found that Equation (2.28) is validirrespective of pile and soil parameters as long as the active-pile-length ratio, L0 / L , iswithin a certain limit. Beyond this limit, the behavior of piles deviates from the‘flexible’ nature. Figures 2.8d-2.8f show this trend of the deviation of Equation (2.28)from the results of “TLEM” (Ver. 1.2) as the ratio L0 / L increases. In these figures,L0 / L are changed by arbitrarily changing the number of piles and/or diameter ofindividual piles. These figures show that the allowable limit of L0 / L is 0.3 or less.A similar expression must also be derived for the dynamic stiffness of grouped pilesin rocking motion and for the coupled stiffness between lateral sway and rockingmotions. In this extension also, active pile length, if rationally estimated, would allowthe pile-cap stiffness to be approximately described in a similar manner. Further detailedstudy on this point will be addressed in a later publication.

SIMPLE EXPRESSIONS OF THE DYNAMIC STIFFNESS OF GROUPED PILES 27non-dimensional stiffness, k xx/(µL 0)n p=4, d=0.5m, L=10m, µ = 30000 tf/m 2 (v T= 414 m/s)2015105equivalent beam (TLEM)equation (28)real partimaginary partL 0/L= 0.10s/d= 4.1900 1 2 3non-dimensional frequency, ω R /v 0 TFigure 2.8a Variation of stifness for sway motion of pile cap(D = 0.05)non-dimensional stiffness, k xx/(µL 0)n p=4, d=0.5m, L=10m, µ = 30000 tf/m 2 ((v T)= 414 m/s)2015105equivalent beam (TLEM)equation (28)real partimaginary partL 0/L = 0.10s/d = 4.1900 1 2 3non-dimensional frequency, ω R 0 /v TFigure 2.8b Variation of stiffness for sway motion of pile cap(D = 0.20)

28SIMPLE EXPRESSIONS OF THE DYNAMIC STIFFNESS OF GROUPED PILESnon-dimensional stiffness, k xx/(µL 0)15105real partn p=4, d=0.5m, L=20mequivalent beam (TLEM)equation (28)imaginary partL 0/L = 0.10s/d = 4.19D = 0.0500.0 0.5 1.0 1.5non-dimensional frequency, ω R 0 /v TFigure 2.8c Variation of stiffness for sway motion of pile cap(µ = 1875 tf/m 2 , v T= 103.5 m/s)n p=4, d=0.5m, L=10m, µ = 30000 tf/m 2 (v T= 414 m/s)non-dimensional stiffness, k xx/µ L 02015105equivalent beam (TLEM)equation (28)real partimaginary partD= 0.10s/d= 4.1900 1 2 3non-dimensional frequency, ω R 0/v TFigure 2.8d Variation of stiffness for sway motion of pile cap(L 0/L = 0.10)

SIMPLE EXPRESSIONS OF THE DYNAMIC STIFFNESS OF GROUPED PILES 29n p=25, d=1m, L=10m, µ = 30000 tf/m 2 (v T= 414 m/s)non-dimensional frequency, k xx/(µL 0)40200-20imaginary partequivalent beam (TLEM)equation (28)D = 0.10s/d = 1.48137real part-400 2 4 6 8non-dimensional frequency, ω R 0 /v TFigure 2.8e Variation of stiffness for sway motion of pile cap(L 0/L = 0.31)n p=4, d=2m, L=10m, µ = 30000 tf/m 2 (v T= 414 m/s)non-dimensional frequency, k xx/(µL 0)80400-40imaginary partequivalent beam (TLEM)equation (28)D = 0.10s/d = 4.19real part-800 2 4 6 8 10 12non-dimensional frequency, ω R /v 0 TFigure 2.8f Variation of stiffness for sway motion of pile cap(L 0/L = 0.40)

30SIMPLE EXPRESSIONS OF THE DYNAMIC STIFFNESS OF GROUPED PILES2.4. SUMMARYPiles grouped beneath a superstructure can be viewed as a single equivalent uprightbeam when the piles are closely spaced. The stiffness matrix presented herein (Equation(2.1)) yields close approximations of both dynamic pile-cap stiffness and kinematicdisplacement factors. This idealization of grouped piles as a single equivalent uprightbeam and the concept of the active pile length have facilitated the derivation of a simpleexpression of pile-cap stiffness in terms of frequency-independent mass, damping andstiffness parameters (Equations (2.29a)-(2.29c)). This expression is valid irrespective ofpile and soil parameters as long as the pile group exhibits a “flexible” nature with itsactive-pile-length ratio, L0 / L, kept less than 0.3. The present simple expression ofpile-cap stiffness also allows the effects of overall site non-linearity to be reflected bysimply replacing the shear modulus of the intact soil, µ , with the complex modulus,µ'( 1+ iD ) , which describes equivalent-linear features of the soil experiencing theseismic motion.A similar expression must also be derived for the dynamic stiffness of grouped piles inrocking motion and for the coupled stiffness between lateral sway and rocking motions.Moreover, there is further scope to extend this study for inhomogeneous soil-profile andthe local non-linearity of soil that develops in the vicinity of piles. In these extensionsalso, active pile length, if rationally estimated, would allow pile-cap stiffness to beapproximately described in a similar manner. This will be discussed in futurepublications.REFERENCESFan, K., Gazetas, G., Kanya, A., Kausel, E. and Ahmad, S. [1991] “Kinematic SeismicResponse of Single Piles and Pile Groups,” Jour., Geotechnical Engineering, ASCE,117(12), 1860-1879.Gazetas, G. and Dobry, R. [1984] “Horizontal Response of Piles in Layered Soils,” Journal ofGeotechnical Engineering, ASCE, 110(1), 20-40.Kanya, A. and E. Kausel [1982] “Dynamic Stiffness and Seismic Response of Pile Groups,”NSF report, NSF/CEE-82023.Konagai, K. and Maehara, M. [1992] “Study on Hypotheses for Simple Numerical Evaluationof Soil-Embedded Structure Interaction,” Bull., Earthquake Resistant Structure ResearchCenter, 25, 39-60.Konagai, K. and T. Nogami [1998a] “Analog circuit to simulate dynamic soil-structureinteraction in shake table test,” International Journal of Soil Dynamics and EarthquakeEngineering, 17(5), 279-287.Konagai, K., A. Mikami and T. Nogami [1998b] “Simulation of Soil-Structure InteractionEffects in Shaking Table Tests,” Geotechnical Earthquake Engineering and Soil Dynamics

SIMPLE EXPRESSIONS OF THE DYNAMIC STIFFNESS OF GROUPED PILES 311998, Seattle, Geotechnical Special Technical Publication, ASCE, 75(1), 482-493.Konagai, K., Nogami, T., Katsukawa, T., Suzuki, T. and Mikami, A. [1998c] “Real TimeControl of Shaking Table for Soil-Structure Interaction Simulation,” Jour. of StructuralMechanics and Earthquake Engineering, JSCE, 598/I-44, 203-210.Konagai, K. [1998d] “Guide to TLEM,” program manual No. 5, Konagai Lab., IIS, Univ. ofTokyo.Nogami, T. and Konagai, K. [1988] “Time Domain Flexual Response of Dynamically LoadedSingle Piles,” Journal of Engineering Mechanics, ASCE, 114(9), 1512-1525.Novak, M., Nogami, T. and Aboul-Ella, F. [1978] “Dynamic Soil Reactions for Plane StrainCase,” Proc., ASCE, 104(EM4), 953-959.Ohira, A., Tazo, T, Nakahi, S. and Shimizu, K. [1985] “Study of Dynamic Behavior of Piles inSoft Soils,” Journal of Structural Engineering / Earthquake Engineering, 362/I-4, 417-426.Randolph, M. F. [1981] “Response of Flexible Piles to Lateral Loading,” Geotecnique, 31(2),247-259.Tajimi, H. and Y. Shimomura [1976] “Dynamic Analysis of Soil-Structure Interaction by theThin Layered Element Method,” Transactions of the Architechtual Institute of Japan, 243,41-51.Takemiya, H. [1986] “Ring-Pile Analysis for a Grouped Pile Foundation Subjected to BaseMotion,” Structural Engineering/Earthquake Engineering, 3(1), 195s-202s.Velez, A., Gazetas, G., and Krishnan, R. [1983] “Lateral Dynamic Response of ConstrainedHead Piles,” Journal of Geotechnical Engineering, ASCE, 109(8).Poulos, H. G. [1968] “Analysis of the Settlement of Pile Groups,” Geotechnique, 18, 449-471Poulos, H. G. [1971] “Behavior of Laterally Loaded Piles,” Jour., Soil Mechanics andFoundation Division, ASCE, 97(SM5), 733-751.

32SIMPLE EXPRESSIONS OF THE DYNAMIC STIFFNESS OF GROUPED PILES