simple expression of the dynamic stiffness of grouped piles

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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
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