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The inverse problem of nonlinear vibration is encounteredhere from the measured nonlinear response to identify theparameters of the pile–soil system. The nonlinear restoring force,the damping and the effective mass of the system are determinedfrom the measured nonlinear response curves. The methodologyof the inverse problem formulated by Novak [26] has been used inthe present study. This study is confined to the inverse problem ofsteady state oscillation excited by a harmonic force whoseamplitude increases with the square of excitation frequency. Itis also assumed that the stiffness of pile–soil system isindependent of frequency. The reduction of natural frequency isrecognized using backbone curve which can be established on themeasured response curve. The backbone curve describes thevariation of the undamped natural frequencies O(A) withamplitude. A simple relation is proposed [26] assuming that therestoring force is nonlinear and the damping force is linear with acharacter of viscous damping.O¼ ffiffiffiffiffiffiffiffiffiffiffiffiffio1o2p ð3Þwhere o1 and o2 are the frequencies corresponding to the pointsof interaction between the response curve and a line passingthrough the origin of coordinates. Two sets of response curve (Set1: Single pile, L/d=20, Ws=10 kN, Pile cap embedded into soil;Set 2: Pile group, 2 2, L/d=15, s/d=4, Ws=12 kN, No contact ofpile cap with soil) were chosen amongst the all observed datawhere maximum nonlinearity was found. The backbone curveO(A) was constructed to each response curve by intersecting theresponse curves by a trace of lines shown in Fig. 4 for single pileand in Fig. 5 for pile group. It can be observed that each responsecurve has its own backbone curve but the nature of thesebackbone curves is different for single pile and pile group. Fromthe nature of backbone curve it can be seen that the stiffnesscharacteristic of the pile–soil system varies with the differentlevel of excitation intensity. The stiffness characteristics of systemcan be established corresponding to the inpidual backbonecurves of both single pile and pile group.The stiffness characteristic can be expressed assuming therestoring force F(A) as nonlinear for every steady state amplitudeA which linearizes to give the equivalent linear stiffness depend-ing on amplitude A, that is,KeðAÞ¼ FðAÞAð4ÞThe restoring force can be expressed by a power series asFðAÞ¼ k1Aþk2A3þk5A5þ ... þknAnð5Þwhere ki is constant.The amplitude dependent natural frequency is given byO2ðAÞ¼ffiffiffiffiffiffiffiffiffiffiffiKeðAÞmeffs¼ 1meff ðk1 þk3A2þk5A4þ ... þknAn 1Þð6Þwhere meff is the effective mass of the system.The damping and effective mass can be determined in severalways [26] using the geometric properties of the nonlinearresponse curves. For the nonlinear response, the effective mass(meff) is much greater than the mass of total static load (ms=Ws/g;Ws is the total static load on pile) including the mass of pile cap,steel ingots and exciter. The apparent additional mass can beexpressed in terms of the coefficientx¼ meff msmsð7ÞThe values of effective mass and damping are given in Table 2for both single pile and pile group. It can be seen that the effectivemass decreases and damping increases with increasing excitationintensities. The lower values of additional effective mass at higher excitation indicate that partial separation might have occurredwith higher excitation intensity. The characteristics ofthe restoring force can be determined from the backbone curve,O, and the calculated values of effective mass to each responsecurve asFðAÞ¼ Ameff O2ð8ÞFrom the response curves the restoring force versus displace-ment characteristics are plotted in Fig. 6 for single pile and inFig. 7 for pile group. The tangent at the origin determines thestiffness of the system and it is observed that the overall stiffnessdecreases with increasing excitation intensity. The values of thestiffness at different excitation intensities are summarized inTable 2. The nonlinear response curves (Figs. 4 and 5) were backcalculated using the nonlinear theory with the calculated pile–soilsystem mass, damping, and the characteristic of restoring force. Itcan be seen that theoretical nonlinear response curves agree withthe measured data quite well for vertical vibrations. Thus, acomprehensive and straight forward theoretical analysis is possible with the system having a nonlinear restoring force andlinear damping.5. Novak’s continuum approach—dynamic interaction factorapproach5.1. Linear methodologyIn this study, the continuum approach proposed by Novak andAboul-Ella [28,29] are used to obtain the impedance of pileembedded in layered medium. The stiffness and dampingparameters of single piles were determined from complex andfrequency dependent impedance functions derived from analy-tical solutions of the foundation vibration problems. In thisapproach, the dynamic soil reactions to the displacements of apile element were calculated assuming that the soil consists ofinfinitely thin layers extending horizontally to infinity.The pile group interaction analysis incorporated in thisanalysis is based on the practical concept of ‘‘interaction factors’’proposed by Kaynia and Kausel [30] for dynamic loading. Theinteraction factor approach assumes that the pile group effect canbe assessed by simply superimposing the interaction effects oftwo piles at a time. This approach allows important factors suchas frequency variation, pile spacing, pile length to diameter ratio,relative pile–soil rigidity, and pile tip condition to be consideredin the interaction factors. The pile group impedances arecalculated from the single pile impedances and the dynamicinteraction factors in a rigorous manner, using closed-formformulae derived by Novak and Mitwally [31].To account for the embedment effect of pile cap, Novak andBeredugo [33] introduced some expressions for stiffness anddamping of embedded footing under vertical vibrations usingplane strain soil model. In this analysis, the soil reactions at thebase area of the cap are normally not included and only the soilreactions on the vertical sides of pile cap are considered. The sidereactions due to pile cap embedment results in increased pilestiffness and damping. The theoretical calculations of single piles and pile groups under vertical vibration were performed using thecomputer program DYNA 5, which employ the previous assump-tions and procedure.