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A typical comparison of observed and predicted responsecurve for different L/d ratios is shown in Fig. 10c. It can be seenthat both the observed and predicted stiffness and damping ofpile group increases as the L/d increases. A typical comparison ofobserved and predicted response is shown in Fig. 10d for differents/d ratios. It can be noted that both the observed and predictedstiffness and damping increases as s/d of pile group increases.In second case, the linear-equivalent model provides for thegradual expansion of the yielded zone around the pile and alsothe separation of pile and top layer of soil as the excitationlevel increase to make the nonlinear analysis more realistic.For different excitation intensities, the soil parameter in theweakened zone are adjusted so that the nonlinear theoreticalresponse curves approach the observed results. The variations ofboundary zone parameters with depth for different excitationlevels are shown in Fig. 11. It can be noted that as the excitationintensity increases, the shear modulus ratio (Gm/G) reduces,whereas the thickness ratio (tm/R) and weak zone soil damping(Dsm) increases. The values of Gm/G are increased with depth butthe tm/t and Dsm are decreased with depth for all excitation level.For all results presented by this linear-equivalent numericalmodel, 40% of weak zone mass is added to the pile for allexcitation intensity. Using the ratio Gm/G=0 in the topmost layer,the separation between the pile and soil is accounted for. Thedepth of anticipated separation (ls) ranges from 1.8d (= 0.18m)for We=0.187Nm to 2.4d(=0.24m) for We=0.450Nm. Theseparation lengths were not measured in the field and themeasurement of separation length given above were inferred byusing trial and error technique for matching the theoretical andobserved response curves.Comparison between the observed results and theoreticalsolutions using pile separation is presented here for the threecritical cases from the test results as mentioned in case of thelinear method. The comparison curves are shown in Figs. 12a and b for single pile (L/d=10, Ws=12 kN, Case 1) and pile group(L/d=15, s/d=2, Ws=12 kN, Case 1), respectively. It can be seenfromFigs. 12a and b that both the predicted stiffness and dampingvalues are decreased by introducing the weak cylindrical zonearound the pile and by providing sufficient pile separation withsoil and a very close agreement with observed results areachieved. From Fig. 12c, it is observed that the modified linear-equivalent method sufficiently reduces the stiffness values andthe predicted resonant frequencies agree with the observedvalues. Another comparison curve is shown in Fig. 12d for thirdcase. It is observed that the predicted values of damping are closeto the experimental results, particularly at higher excitationintensity. Hence, it can be said that modified linear-equivalentmethod with pile separation with soil is capable to predict theresonant frequency and amplitude values accurately at allexcitation levels.A typical comparison between theoretical and experimentalresults using both linear-equivalent methods is summarized inTable 5. Some of the salient features of the nonlinear verticalvibration of piles can be established from these results. Theresonant frequency of pile reduces and resonant amplitude ofpiles increases as the excitation intensity increases. Theembedment of pile cap has a significant effect on the dynamicresponse of piles. It is observed that the pile cap embedmenteffect considering only side reactions of cap increases the totalpile stiffness and damping. The effect of pile spacing is also animportant parameter for estimation of stiffness and damping.
The stiffness and damping of pile group increases with increasings/d ratio. More accurate prediction of resonant frequency andamplitudes of pile are achieved using the linear-equivalentmethods with pile separation as compared to the no pileseparation. It can be said that the linear-equivalent model withvarying boundary-zone parameters with depth and using theappropriate pile separation with soil, can be considered asrealistic and versatile model of nonlinear analysis of pile.7. Separation of pile and soilAn attempt is made to predict the length of separation of pilewith the soil from the experimental results. It is a difficult task tomeasure the separation between the pile and the soil at the timeof testing for such small amplitude of vibration. In this study,different separation lengths for single pile of L/d=10 were chosenfor different exciting moment by trial and error technique untilthe optimum match between observed and theoretical resultswere achieved. With the same separation length as in L/d=10 forsingle pile, it was found that the match with experimental resultswas satisfactory for other configuration of pile (both single andgroup pile of different L/d and s/d). Hence, the separation lengthsof piles are independent of L/d and s/d and it varies only withexciting moment.In order to combine the effect of static loads and embedmentconditions of pile cap on separation length, two sets of plotof maximum vibration amplitude versus separation length aregenerated—one set for embedded pile cap conditions; andanother set for no contact conditions of pile cap. Best fit curvesare drawn through the actual data points of each set. Another bestfit curve is drawn combining all sets of actual data points (bothCase 1 and Case 2). The three best fitting curves through the dataassociated with the present soil conditions can be mathematicallyexpressed by the following relationship:(i) Pile cap embedded into soil (Case 1),lsd¼ 0:6549 lnDvd þ6:7523 for 0:0004r Dvdr0:0012 ð11Þ(ii) No contact of pile cap with soil (Case 2),lsd¼ 0:7743 lnDvd þ7:3311 for 0:0007r Dvdr0:0017 ð12Þ(iii) For both embedded condition of pile cap (Case 1 and Case2),lsd¼ 1:2435Dvd 0:2575for 0:0004r Dvdr0:0017 ð13Þwhere ls is the separation length between the pile and soil, Dv isthe maximum vibration amplitudes of pile, and d is the pilediameter. The above relationships are developed based on presentsoil–pile conditions and the characteristics of boundary-zoneparameters. However, these relationships can be used as a basicguideline for similar type of soil conditions for preliminaryassessment of the separation between the pile and soil.