Stable response of axisymmetric two-phase water-saturated soil

1022Cai et al. /J Zhejiang Univ SCI 2004 509):1022-1027Jourmal of Zhejiang University SCIENCEISSN 1009-3095http://www.zju.edu.cn/jzusJzUSE-mail: jzus@zju.edu.cnStable response of axisymmetric two-phase water-saturated soilCAI Yuan-qiang (蔡袁强)，MENG Kai (孟楷), XU Chang-jie (徐长节)(Institute of Geotechnical Engineering, College of Civil Engineering and Architecture,Zhejiang University, Hangzhou 310027, China)'E-mail: caiyq@cls. zju.cdu.cnReceived Jan. 16, 2003; revision accepted Jan. 1, 2004Abstract:and displacement for axisymmetric harmonic excitations in the two-phase saturated soil with subjacent rock-stratum. Theinfluence of the coefficient of permeability and loading frequency on the soil displacement at the ground surface werestudied. The results showed that higher loading frequency led to more dynamic characteristics; and that the effect of the soilpermeability was more obvious at higher frequencies.Key words: Biot's dynamic consolidation, Lamb's problem, Saturated subsoildoi:10.1631/jzus.2004.1022Document code: ACLC number: TU91INTRODUCTIONfective and practical, in most cases they were toodifficult to solve. Only under few conditions couldIn the analysis of the dynamic response of we get the precise solutions by means of integralsaturated subsoil, sometimes, especially under transform, series, etc. Lamb (1904) studied thehigher frequency dynamic force, it may be neces- vibration of point sources and linear sources insary to consider the inertia force of both the soil andsingle phased half-space, but for natural founda-the water. Moreover, instead of studying the subsoiltions it would be more appropriate to regard themas an elastic half-space model, study of subsoil with as two-phased media. Halpern and Christiano (1986)subjacent rock-stratum has more practical meaning. and Jin (1999) studied the response of poroelasticIn this work, the vibration of saturated soil under half-space to steady-state harmonic suface trac-axisymmetric cyclic loading was studied on the tions. Chen et al.(2002) also considered the vibra-basis of the above background.tion of point sources and linear sources in half-space.Biot (1956) proposed a three-dimensionalHowever, all of the work mentioned aboveconsolidation theory of saturated soils and gave the only considered the vibration of in half- space,equations of poroelasticity foundation under dif-which could not reflct the influence of the echoesferent boundary conditions with the use of Laplacegenerated by the bottom boundary. From her studyTransform. Although inertia coupling of soil andof subsoil with subjacent rock-stratum, Zhang et al.water and other complex conditions were consid- (2002) got the integral solutions under harmonicered in the theory to make the equations more ef- force of low frequency, and Bougacha et al.(1986)got the solution of foundations with finite-elementProject (No. 81067) supported by Fok Ying Tung Educationmethod. In this paner Hanka1 transform was used toFoundation, Education Ministry of Chinaget the integra中国煤化工，and theMHCNMHG.Cai et al. /J Zhejiang Univ SCI 2004 5(9):1022-10271023boundary conditions were combined to yield thedσ_ Pjg .中+ρjii+(2a)analytic solutions of the soil with subjacentdr kynrock-stratum. The effects of the soil permeabilitydσ_ Pj8and loading frequency on the soils were also stud-dz kav。+pjw+(2b)ied in this work. Examples showed that the fre-quency of the force influences the results signifi-cantly and that the effect of the permeability ofsoil ka and g are the dynamic coefficients of perme-is influenced by the frequency.ability and acceleration of gravity, respectively.When the compressibility of soils and water is ne-glected, the compatible equations can be expressedEQUATIONSas:When inertia coupling and the compressibility(du. u, dw)(3)of soil and water are neglcted, according to Biot's .lddzdrrdzconsolidation theory the axisymmetric balanceequation of soil can be expressed as:SOLUTIONh+Gde.0z2,_. u _ 1 aσ_ ρ;_ P(1a)G drA flexible circular foundation rests on theλ+Gde1 8σ(1b)layered soils. The z-axis of the cylindrical coordi-G dzGdzGGnate system coincides with the vertical axis ofsymmetry of the foundation. The vibration of thewhere λ and G are the Lame coefficients; ul, w, Vr soils is induced by the loading Poeror, where o is theand vz are the radial displacement of soil, the ver-angular velocity of the loading and i=、-1.tical displacement of soil, the radial displacementIt is convenient to introduce the dimensionlessof water relative to soil, and the vertical displace-constants and variables:ment of water relative to soil respectively; σ, n, Psand Pf are the porewater pressure, porosity, densityof soil particles and density of water respectively;子21d. 子2du, u, dwPs 元=1_MV2=N== G’r Orρ=(1-n)p, +np;VC√VpGThe balance equation of water can be expressed as:_rQ元=号，Q=kGP= Poeiotwhere ro is the radius of the foundation.After introducing the dimensionless variables,Eqs.(1a)-(3) can be written as:n, ka,Pw,p,h,p, G(元+1)ge+V2i-= pa3u+ p,af,(4a)↓(元+1)9+V中国煤化工(4b)Fig.1 Description of the model and coordinate systemYHCNMH G1024Cai et al. 1J Zhejiang Univ SCI 2004 509):1022-1027=ba。r, +ρjia} +-i,a(5a)F(p,z) and F(p,z) respectively.=ba. +ρ,wa+Pv.a(5b)F(p,z)=」" rf(r,2)J(pr)dr;ndp. 印。 didi. i, diFr(p.z)=fJ"rCr.a),(or)dr. .(6)drFdzdi' FdAnd sometimes we translate F'(p,z) into j(r,z)From Eqs.(5a) and (5b) we can get:to reduce the number of the unknown variables.By means of zero order Hankel transform, Eqs.(7b), (8b) and (8c) yield示=c(i-1σ示)，可=(而一pa云denagP;dz能-qE=0, q°=ξ2+D(9a)bni-p,aod'σ。dz2~ G'(9b)Introduce the equations above into Eqs.(4a) andd证.-sw=(1+c))d_(元+1de(4b):(9c)dz2?dds2=ξ2 -o2(ρ+cρ)/G(元+1)CE+V2u_-(1+o)0-a(ρ+cρ, )u)F;江F(7a)And by means of first order Hankel transforms, Eq(元+1)de +aδ_(7a)ande= du+"+dwyield+V的-(1+c)g=-a(P+cp,)i (7b) .drrdzsi'=(i+1)ξe-(+c)ξσ(9d))r(7a)+ E (7a)+。(7b), then we can get:e=ξu' +。 dw(9e)aV°e-V°σ+βe=0.λ+2w(ρ+cρ;)(8a)Then from Eqs.(9a)-(9e), we can get:1. +.(1+c)Ge=Ae* + Be-*,(10a)Introduce Eqs.(7a) and (7b) into Eq.(6):δ=CnAex +C2Beni +Agef +B,eti (10b)V际=Ee，ε=0+c)p,0°cGu'=C;Ae +C,Ber +C24.e5(10c)+C3B,e-t +Ae* +B.e-*Then from Eqs.(8b) and (8a) we can get:;[(h+G)-(1+c)C2J?-s2 )G'V'e=De，D=5Pj +2cp, -cρ(8c)(1+c)ξ(h + 2G)c(E2-s2)GWe denote the Hankel transform of the firstw=CqAe*-CxBex +C242e*(10d)order and the zero order of a function jJ(r,z) by-C2B.中国煤化工HCNMHGCai et al.1J Zhejiang Univ SCI 2004 5(9): 1022-10271025Ci=-ECGs, Cx=-C2, A=-EA, B.=gB,respectively obtained from Eqs.(7b)-(7g).From Eqs.(8a) and (8b), we can get:du_[z° °σ°i°σ°Q°TAnd for Tgr =G(-+")，σ:=he+2G当and(12)=Taxo[u°动°σ!动σ°Q°]Q=k, we can get: .Toxo=0x(T"66)~At the ground surface, we can get: ox=P(0Srdu 0wσ, =ie+2di 0- doSro) and Trx=σ=0, where P is the load applied to thedzof|z’subsoil. And at the subjacent rock-stratum u, w, QBy means of Hankel transform, the equationsare known as u=w=Q=0. When the boundary con-ditions are combined, we can get the other un-above yieldknowns with the values deduced above.动=2qC3(Aez - Be*)+2ξC2(Age5*- B,e-i)(4,e*-B,e*)(10e)EXAMPLE AND DISCUSSIONσ:=(h-2生C;)(4ex +Bex*)With the equations listed above, we can cal-(10f) culate the dynamic response of the soils. The inte--2ξC32(4e* +Bge*)-2E(Ae* +B,e-*)grants are very oscillatory and time-consuming toQ=ξC219(Aex - Be"*)+ξ(Aei*-B,e*) (10g)obtain, and MATHEMATICA was applied to re-duce the truncation error in the calculation.The conditions of the examples are listed asfollows:BOUNDARY CONDITIONSMaximum pressure of the force: p=10 kPaRadius of the pressure: ro=1 mBoundary condition: the pore pressure of thedisplacement of soil, the vertical displacement ofsurface is equal to zero; the subjacent boundary is .soil, the vertical stress, the shear stress, the pore-unpenetrable.water pressure and the quantity of the water flowingFig.2 shows the vertical displacements of theout of the soil, respectively. At the ground surfaceground surface under different loading frequencywhere the thickness is equal to zero, Eqs.(10b)-and permeability. The solid line curves in the fig-(10g) can be expressed as: .ures are the maximum displacements of the soil,and the broken line curves are the displacements[u°w°σ°σ°Q°T(11a)when the loading force attains maximum pressure.=T%&x[4 BA,B2A B.]TThe figures obviously show that the fluctua-tion does not occur under static conditions. BecauseThen we can get the same expression at the there is a rigid subjacent boundary, the wave motionsubjacent rock- stratum as: .created by the vibration of the force may intrferewith its reversed echo in the soil. Thus, with theu”动σ!站σ°Q](11b)increase of the radius, the swing of the vertical=Tbox6[A BA2 B2Ag B]displacement at the surface ground will not at-tenuate monotononslv_ hut decreases with fluctua-The figures/elements in T 6x6 and T *6x6 can be tion. This phen中国煤化工h the freq-"YHCNMHG.1026Cai et al. /J Zhejiang Univ SCI 2004 509):1022-10272「1t三6810.1 l-1 L2i0r(m).r (m)(a)b)0.80.60.4 .0.宣2-0.2ψr/三1t-0.4 t /-0.6 t-0.8.1 φ8 ior(m)c)d)3F0.40.2昌(-0.24681(-0.40t-1电e)f)Fig.2 Curve of one-layered soils(a) ao=3.46, b=42.5; (b) ao=0.346, b=42.5;(c)a, = 3.46, b=425;(d) an =0.346, b=425; (e)a,=3.46, b=4250; (f) ao =0.346, b= 4250;uency of the force. By comparing (a), (c) and (e) the seepage velocity but also the deformation of therespectively with (b), (d) and (f) in Fig.2, we can soil skeleton. The figures show that with the de-see that the fluctuation of the soil is more obvious crease of the coefficient of permeability, the swingswhen 0 equals 100 rad/s. We can also see from the of the surface displacement also attenuate. Thisbroken line in the figures that the phase difference conclusion agrees with the results of Yang and Songof the force and the displacement increases with the (1999), who studied the dynamic problems of soilfrequency of the force. These phenomena accord foundations by finite-element method. But what iswith the results of similar low-frequency situation remarkable here is the combined effect of the soilpresented by Zhang Yu-hong.permeability中国煤化工》，At lowThe permeability of the soil influences not only frequency the: ao equalsTHCNMHG.Cai et al.1J Zhejiang Univ SCI 2004 5(9): 1022-102710270.346, the swing of the soils in which b equalsCONCLUSION4250 differs only by about 10% from that of thesoils in which b equals 42.5. But at a higher fre-The dynamic interaction problems of harmonicquency the difference becomes more obvious.vibration of a circular flexible foundation wereWhen ao equals 3.46, the swing of the soils in whichstudied for the first time by analysis method, withb equals 4250 is two-fifth of that of the soils inthe displacement of water relative to soil and thewhich b equals 42.5. We can also see from theinfluence of a subjacent rock stratum considered.From the numerical results, conclusions can bfigures that if b goes beyond certain ranges, itsdrawn that the response of the soils strongly de-change will have lttle influence on the result. Forpends on both the properties of the saturatedexample, the swing of (e) differs only by about 20%soil-foundation system and the load acting on theof that of (), and their curves are quite similar. foundation, and other factors such as the frequencyThese phenomena accord with the result of soils of of the load and Darcy's permeability coefficient ofthe half-space model studied by Chen et al.(2002). the medium. These should not be neglected in de-Fig.3 shows the curve of the vertical swing at termining the response of structures to dynamicethe center point of the load under different loading loadings. Because some integrants for computingfrequencies. Generally speaking, when the swing ofproblems cannot be calculated at the present time,the dynamic load is invariable, the work done by in this paper we only presented the vertical dis-the dynamic load in a half cycle decreases with the placements of the soils at the ground surface, andincrease of the frequency. Thus the vertical swing, other results will come out with further studies.which is mainly determined by the work, may alsogenerally attenuate. For example, the swing of the Referencescenter point shown in Fig.2a is obviously less than Biot, M.A.. 1956. Theory of propagation of elastic waves in athat shown in Fig.2b. But from the curve we can seefluid-saturated porous solid. The Journal of the Acous-tical Society of America, 28(2):168- 191.that it does not decrease monotonously. Within therange of the frequency shown in the figure, the peakBougacha, S., Tassoulas, JL., Roesset, J.M., 1986. Analysis offoundations on fluid-filled poroelastic stratum. J. Eng.value of the swing occurs at about 18.5 Hz.Mech, ASCE, 119:1632- 1648.Halpern, M.R., Christiano, P, 1986. Response of poroelastichalf- space to steady state harmonic surface tractions. Int.1.2J. Numer. Anal. Mech. Geomech, 10:609-632.Jin, B., 1999. The vertical vibrations of an elastic circular plate、 0.on a fluid-saturated porous half-space. Int. J. Eng. Sci,0.637:379- 393.0.Lamb, H., 1904. On the propagation of tremors over the sur-face of an elastic solid. Phil Trans Roy Soc, 203: 1-42.Yang, J,, Song, E.X., 1999. Dynamic analysis of infinite satu-rated soil foundation by finite elements. J. of Tsinghua152025303540)4550University, 39(12):82- 85 (in Chinese).0 (Hz)Zhang, Y.H, Huang, Y, Wang, Z.J., 2002. Analysis proceduresof dynamic response of 3-D non-axisymmetric layeredFig.3 Curve of vertical displacement of the loadingsaturated soil. Earthquake Engineering and Engineeringcenter for different frequencyVibration, 22(4):47-52.中国煤化工MHCNMH G.