嵌岩桩非线性动力学特性

Journal of Shanghai University( English Edition), 2007, 11(3): 213-217Digital Object Identifier(DOi): 10.1007 /s 11741-007-0304-1Nonlinear dynamic characteristics of piles embedded in rockHU Chun-lin(胡春林)123, CHENG Chang-jun(程昌均)12, HU Sheng-gang(胡胜m2 p. R. China1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 2002. Department of Mechanics, College of Sciences, Shanghai University, Shanghai 200444, P. R. China3. School of Civil Engineering and Architecture, Wuhan University of Technology, Wuhan 430070, P. R. ChinaAbstract The nonlinear dynamic characteristics of a pile embedded in a rock were investigated. Suppose that both theaterials of the pile and the soil around the pile obey nonlinear elastic and linear viscoelastic constitutive relations. Thenonlinear partial differential equation governing the dynamic characteristics of the pile was first derived. The Galerkin methodwas used to simplify the equation and to obtain a nonlinear ordinary differential equation. The methods in nonlinear dynamicswere employed to solve the simplified dynamical system, and the time-path curves, phase-trajectory diagrams, power spectrumPoincare sections and bifurcation and chaos diagrams of the motion of the pile were obtained. The effects of parameters onthe dynamic characteristics of the system were also considered in detailKeywords lateral motion of the pile, nonlinear elastic material, linear viscoelastic material, bifurcation and chaos2000 Mathematics Subject Classification 74D10, 37L651 Introductiontype linear viscoelastic constitutive relations. Due tothe difficulty in solving the nonlinear partial differen-The pile foundations have been widely used in en- tial equation, the Galerkin method is used to simplifyineering. Due to the interaction between piles and the governing equation and to obtain a simplified dy-soil. the load transfer. deformation and motion are namic system. Methods in nonlinear dynamics yieldvery complicated, so all the nonlinear mechanical anal-the time-path curves, phase-trajectory diagrams, powersis, numerical simulation and experiment are very dif-spectrum, Poincare sections, and bifurcation and chaosficult. There have been many papers on the linear vi- diagrams of the simplified system. The influences ofbration and the dynamic response, but few papers are material and structure parameters on the nonlinear dyon nonlinear dynamic behavior and nonlinear vibrationof pilesll-3, especially, for the cases that the materialsof piles and soils are composed of nonlinear elastic and2 Mathematical model of problemviscoelastic ones. Novak 3] gave an overall overview onConsider a pile with the length l, the outer diameterdynamic analysis of piles and introduced the linear and D, the inner diameter d, the cross-sectional area A, andnonlinear dynamic theories, calculation methodsmaterial density p and the pile is embedded in a rock.qualitative conclusions. Wang, et al 4 presentedSuppose that the pile is at rest at the initial time, andsemi-analytical solution for the vibration of piles with that the origin of coordinate axes coincides with the ge-he variable section in a layered soil. The nonlinear dy- ometry center at the top of the pile, and its directionsnamical behavior of piles has been researched in 5. In are shown in Fig.1addition. Suire and Cederbaum(6)en anIt would be assumed that the material of the pilenalyzed the stability and chaotic motion in columns obeys the constitutive equation as followsChengs researched the vertical chaotic motion of vis- o= Eole sgn(-e)B+ye )+n dIn this paper, the partial differential equation gov- where a(, T)and e(c, T)are the axial stress and strainerning the nonlinear dynamic characteristics of the pile of the pile at the time T; E0, B and 7 are the elastic co-is first derived, in which, both the materials of the efficients of the material; n is the viscosity coefficientpile and the soil obey nonlinear elastic and differential- sgn(中国煤化工 In the case of smallReceived Oct 25, 2005; Revised Mar 6, 2006Project supported by the National Natural Science Foundation of ChCNMHGthe Shanghai LeadingAcademic Discipline Project(Grant No. Y0103)CorrespondingauthorHuChun-lin,PhdCandidateProf.,E-mail:chunlinhu@163.com214Journal of shanghai Universitydeformation, the axial strain E may be giverFor given lateral displacement and rotation angle atthe top of the pile we have the following side conditioons(2) at the side I=0where y is the distance to the neutral axis, and v(a, T)isvlx=o=ym(1-cos(wT),the lateral displacement of the pile. From Eqs. (1)and(2), the bending moment M= AoydA is obtainedgivenM=-on64a2+b0-(D4-d4 a2Uwhere ym and w are constants6x2At the same time, from the above assumption, theD6-d6/82D4-d483E0Yπitial condition is512v(a, T)IT=0due, r)=0.T=0Introduce the variable transformation(1-cos(wT)),then we have the side conditions above V,Fig1 a pile embedded in a rockV|z=0=0,vl=t=0,assume that the material of the soil is aelastic and linear viscoelastic one. From theavmodel, the lateral resisting force of the soil orIs given asr,)=k+ky2+k3+23 Solutions of problemOne can see that it is difficult to solve the nonlin-where c is the damping coefficient and ki( i= 1, 2, 3)are ear Eq- (5), so the Galerkin method is used to simplifythe stiffness coefficients of the soilEq (5). The authors in 9 pointed out that the lowIt is not difficult to obtain the nonlinear differential order Galerkin system can well simulate the long-timeequation governing the lateral motion of the pile as fol- mechanical behaviors of viscoelastic structures. There-lowsfore, we consider the solution of Eq (5)as followspA+Eo兀v(a, T)=l(r)(1-cosa3D5-d5 82v a4U(=)2EoB2EoB6-82u/a3uY2The solution(11) satisfies the side conditions(10). Sub-+6E0π512ax2(ax3stituting eqs.(⑨)and(11) into eq、⑤5), multiplying twosides of Eq (5)by(1-cos(r), and then integrating6-d6/92u)20,D4-d405uon[0, l] yield the ordinary equation in w(T)+307x512(Bx2)m+m6Bm+D(k1+k2v2+k303+5)d210/Dc-16D)4-d)dm中国煤化工dTFor a pile embedded in a rock, we have the followingCNMHGa )uaside conditions at the side x =l:5k2D,24x2-495k3D3PA72 72pA m(1-cos(ur) w2Vol 11 No. 3 Jun 2007HU C L, et al.: Nonlinear dynamic characteristics of piles embedded in rockEoT'D4-d4k,D3Al464A83x2-15 k2D 64E0BoJ4 D5-d52x2-312pAD6-d6 64EorT20π2+30(1-cos(wr)512AZ84π6-42x4+210x2-31532x4-200x2+3153k3DcOS ( w14x6160pAT)F22-3 cuD2x2-3u2-Um sin(wT)x2 9[ ym cos(wT)2≈2x4-10m2+1515x42元2-3k1D46-42x4+2102-315150A=(1-m49-42+2072-315k42x64x6-42π4+210x2-315g2=B3(1-cos(m)3=0in which16Eo5 D4-d464E0B D5-dsE3B2pA464Introducing dimensionless variables and parameters64E0x5D6-dt=BT,B= DA+36A64 leads to the nonlinear E33-B2pA16512differential equation satisfied by the dimensionless dis-(),DklB2pA'B2DAd 2w dwdt2+ dt +(A10+ Au cos(2t)+A1 cos 2 t)23x'-3 DewCl(14)+(A20+A21 cos(2t)w+A3w+F10+Fil sin(S2t)The initial condition( 8)now become+F21 cos(J2t)+F22 cos(52t)+F23 cos(S2t)=0=0.(15)All the coefficients in Eq (13)are nondimensional, 4 Numerical examples and results anal-hichyA10=E1-E22U'm+ E33Vm+kil+12-2-k12VmThe Runge-Kutta method is used to solve Eq (13)with(14)under the initial condition (15)and the964-6002+945methods in nonlinear dynamics are used to analyzethe obtained numerical results which include the time.8兀2-15path curves, phase-trajectory diagrams, power specA1=E2m-2E3212πk120mtrum. Poincare sections and bifurcation and chaos di-9674-600x2+945agrams in the case of given parameters.In computation, the following parameters are given120x2-245D=10m,d=0m,l=6m,p=24×103kg/m213mEo=21×100Pa262×102,y=1.19×10A2=BE3+160+4k3n=1.0×105N.s/m3,k1=42×106N/m3,A0=kh2+120x2-25k2=-1.3×105N/m4,k=13×109N/m572xklum中国煤化工,m=69mmA3=x4E3+k13CNMHGAt the same time, we nave B=5.2880 1/s,522x4-102+15klUm59409and0.0115. Therefore, the coefficients in15x4Eq (13)can be obtained216Joumal of Shanghai UniversityThe time-path curves, phase-trajectory diagramsIn order to assess the effects of parameters on thepower spectrum and Poincare sections describing the motion characteristics, we assume that the parameterlateral motion of the viscoelastic piles are shown in vm(or ym)in Eq (16)is alterable, and thatFigs. 2-5. One can see that the motion of the simpli- rameters are kept to be constant. The obtainfied system is chaotic.cation diagram is shown in Fig. 6. It can beas Um <0.01, the motion of the simplified system is periodic, but as the parameter Um is limited to the range0.010 Um 0.011, the motion is quasi-periodic andwill branch. With the increase of the parameter Um, themotion displays a blast of chaotic motion. This meansthat the small vibration of the pile usually is periodicbut for a big enough displacement at the top of the pile,the motion may be quasi-periodic or chaotic.Fig 2 Time-path curve-24%0050010003Fig 6 Bifurcation diagram versus Umw(x103)Assume that the parameter n2(or w)in Eq (16 )isFig 3 Phase-trajectory diagramalterable, and that other parameters are kept to be con-stant. The obtained bifurcation diagram is shown inFig. 7. It can be seen that as 2< 1. 2, the lateral vi-bration of the pile is periodic, but as the parameter 2is limited to the range 1.2 2 5.1, the motion isquasi-periodic and will branch. As S2 falls in the range5.1 32<11.0, the motion displays a blast of chaoticmotion, but as 2> 11.0, the motion is quasi-periodicnd will branchFig 4 Power spectrumrTL中国煤化工ram versus 2CNMHG(or the viscosity coefit r and aamp coemcient c) in Eq (16)is alterableand that other parameters are kept to be constant. TheFig 5 Poincare sectionbtained bifurcation diagram is shown in Fig 8. It canVol 11 No. 3 Jun 2007HU C L, et al.: Nonlinear dynamic characteristics of piles embedded in rockbe seen that as the parameter B> 22.5, the motion is 5 Conclusionsperiodic, but as B 6.5 or 10.1 0.88 m, the motion will branchcal vibration [J]. Earthquake Engineering and Struc-tural Dymamics, 1974, 5: 277-293and display a blast of chaotic motionAssume that the length l of the pile is alterable, and2 WANG Kui-hua. Vibration of inhomogeneous viscous-that other parameters are kept in Eq (16) to be con-elastic pile embedded in layered soils with general Voigtmodel J]. Joumal of Zhejiang University, 2002, 36 (3)stant. The obtained bifurcation diagram is shown in565-571 (in Chinese)Fig 10. It can be seen that as l> 6.2 m, the motion3) NovAK M. Piles under dynamic loads [C)//Proceedingsis periodic, but as l 6.2 m, the motion shapes willof Second International Conference on Recent Advancesbranch and display a blast of chaotic motion.in Geotechnical Earthquake Engineering and Soil Dy-Tokyo: AA Balkema Publ4 WANG Hong-zhi, CHEN Yun-min, CHEN Ren-pengemi-analytical solution for vibration of pile with variable section in layered soil [J]. Jourmal of vibration andShock, 2001, 20(1): 55-58(in Chines5] JIA Qi-fen, YAN Min, CHEN Yu-shu. The nonlinear com-plex dynamical behavior of foundation piles J. Journalof Tianjin University, 2000, 33(2): 545-548(in Chi-15Lnese)6 SUIRE G, CEDERBAUM G. Periodic and chaotic behav-Fig 9 Bifurcation diagram versus Dior of viscoelastic nonlinear bars under harmonic excitations J. International Journal of Mechanical Sciences,1995,37(5):753-7727 CHEN Li-qun, CHENG Chang-jun. Stability and chaoticmotion in columns of nonlinear viscoelastic material JApplied Mathematics and Mechanics, 2000, 21(6): 890-896 (in Chinese)8 HU Chun-lin, CHENG Chang-jun. Study on the vertical chaotic motion of the nonlinear viscoelasticile J. Journal of Rock and Soil Mechanics, 200中国煤化工unOn investigations of0120140160CN MH GStic structures basedChinese Jounal of Nature,199,21(1):1-4( in Chinese)Fig 10 Bifurcation diagram versus L(Editor CHEN Ai-ping)