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5.2. Linear-equivalent methodologyTo account approximately for the effects of nonlinearity andslippage [7], it is assumed that an embedded cylindrical body issurrounded by a linear viscoelastic medium composed of twoparts—an outer infinite region and an inner weak layer surround-ing the cylindrical body as shown in Fig. 8. Soil nonlinearity, aswell as the weakened bond and slippage are presumed to beaccounted for by a reduced soil shear modulus and increasedsoil damping of the inner soil layer. The stiffness of the compo-site medium is calculated assuming that both media arehomogeneous, isotropic and viscoelastic with frequencyindependent material damping, the pile is rigid, infinitely long,circular and massless. The inner soft medium is considered asmassless.The complex dynamic stiffness (kwc) of the composite mediumcan be written [7] askwc ¼ GðSw1c þiSw2cÞð9Þ in which Sw1c and Sw2c are the dimensionless stiffness anddamping parameters of composite medium, respectively, and Gis the shear modulus of outer soil medium.The soil reactions of the composite medium can be substitutedinto the theory developed by Novak and Aboul-Ella [28] and thetotal complex vertical stiffness of one pile (Kww) is obtained asKww ¼ EpApRfw1c þioRVsfw2c ð10Þwhere fw1c and fw2c are the dimensionless stiffness and dampingconstants which describe the total stiffness of pile for thecomposite medium, Ep is the Young’s modulus of pile, Ap is thecross-sectional area of pile, and R is radius of the pile.By knowing the stiffness and damping constants for all thelayers in the pile–soil system under consideration, the pileresponse to vertical dynamic loading can be obtained using soil-structure numerical model DYNA 5.6. Comparison between theoretical and experimental resultsTo better understand the significance and influence ofnonlinearity, this section is pided in two parts: (1) comparisonwith linear method; (2) comparison with linear-equivalentmethod. For linear method, the stiffness and damping character-istics do not vary with the excitation level. However, theseimpedance characteristics vary with excitation intensity byintroducing weak cylindrical zone around the pile for linear-equivalent method.6.1. Comparison with linear methodTheoretical calculations were performed using the properties ofpile material and measured soil parameters (Fig. 2 and Table 1). Inthis analysis, no weak zone around the pile was considered and thevalue of damping ratio of soil was assumed constant (Ds=0.15) withdepth. The soil below the pile tip is assumed to be homogeneous.The properties of pile material used in this analysis are given inTable 3. Comparison of experimental response curves and thetheoretical predictions by linear method are presented here for threedifferent cases. Figs. 9a and b show the response curves of the singlepile and pile group, respectively, under vertical vibration. In the firstcase, it can be seen from these figures that the linear method highlyoverestimates both the stiffness and damping. Another comparisonis shown in Fig. 9c for second case. It is observed that the predictedstiffness values are higher than the observed values but the dampingvalues are very close to the test results. In third case, it can be seen inFig. 9d that predicted values of resonant amplitudes are muchhigher than the test results but the natural frequencies are close tothe observed results. The common observation from these threecases is that the linear method overestimates both the values ofstiffness and damping due to the perfect bonding between pile andsoil which is assumed in laws of linear elasticity. The theoreticalcalculated average values of stiffness and damping are almost 38%and 25% higher than the experimental results. So the linear theory isnot so very useful to predict the dynamic behaviour of pile undervertical vibration because in practice the pile and the soil bonding rarely perfect and slippage or even separation often occur at thecontact surface between the soil and pile.6.2. Comparison with linear-equivalent methodIn order to predict the nonlinear dynamic behaviour of piles,two different linear-equivalent models are used in this study: (1)using constant boundary zone parameters with depth and noseparation of pile and soil and (2) using varying boundary zoneparameters with depth and considering the separation of pile andsoil.In the first case, the nonlinear dynamic responses arecalculated by introducing a cylindrical weak zone around thepile but no pile separation was considered. Here the shearmodulus ratio (Gm/G) and inner weak zone soil damping (Dsm)are assumed constant with depth. The values of shear modulus ofouter zone (G) are considered from Fig. 2. The magnitude ofthickness of weak zone (tm) increases with increasing excitationintensity. The weak zone mass participation factor (MPF), thatrepresent the fraction of weak zone mass to be added to the pilemass at each layer, decreases with increasing excitation level. Theboundary zone parameters that are chosen for matching thetheoretical results with the observed response curve are shown inTable 4. It can be noted from Table 4 that the damping of innerzone is higher than the damping of outer zone. The pile materialsused for linear-equivalent methods are same as in the case oflinear methods (Table 3).Comparison of the experimental response curves with thetheoretical prediction by linear-equivalent method (no pileseparation) is shown in Fig. 10a for single pile (L/d=10,Ws=12 kN, Case 1) and Fig. 10b for pile group (L/d=15, s/d=2,Table 4Boundary-zone parameters for nonlinear vertical vibration.Excitationintensity(Nm)Shearmodulusratio(Gm/G)Thicknessratio(tm/R)Dampingratio ofinner zone(Dsm)Dampingratio ofouter zone(Ds)Poisson’sratioMPF0.187 0.15 0.35 0.3 0.15 0.35 0.70.278 0.15 0.55 0.3 0.15 0.35 0.60.366 0.15 0.75 0.3 0.15 0.35 0.50.450 0.15 0.95 0.3 0.15 0.35 0.4 Ws=12 kN, Case 1). It can be noted that the linear-equivalentmethod predicts the measured data quite well for verticalvibration. It can be concluded from these results that theemployed mathematical model, incorporating the weakboundary-zone, is capable of capturing the nonlinear verticalvibration of piles which is not possible in case of linear method.The results show that the resonant frequency of piles reduces andthe resonant amplitude increases as the excitation intensityincreases. For example, when the excitation intensity increasesfrom 0.187 to 0.450Nm, the resonant frequency reduces from23.71 to 21.16Hz for single pile (L/d=10, Ws=12 kN, Case 1) andfrom 34.21 to 29.44Hz for pile group (L/d=15, s/d=2, Ws=12 kN,Case 1).源'自^751;文,论`文'网]www.751com.cn