simple expression of the dynamic stiffness of grouped piles

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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
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