DYNAMIC RESPONSE OF CIRCULAR FOOTINGS ON SATURATED POROELASTIC HALFSPACE

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1 3 th Word Conference on Earthquake Engineering Vancouver, B.C., Canada August -6, 4 Paper No. 38 DYNAMIC RESPONSE OF CIRCULAR FOOTINGS ON SATURATED POROELASTIC HALFSPACE Bo JIN SUMMARY The dynamic responses of footings on a saturated poroeastic haf space and subjected to horizonta and rocking harmonic excitation are studied. By treating the footing as a rigid disk, the probems for vibrations of disk on poroeastic haf space are formuated to Fredhom integra equations of the second kind which are soved numericay. The compiance coefficients for the horizonta and rocking vibration of disk are presented to iustrate the dynamic characteristics of footings on a poroeastic haf space. INTRODUCTION In soi-structure interaction probems, the dynamic compiance of circuar footings on a hafspace pay a key roe in determining the response of surface structures to dynamic oadings, in particuar, seismic excitation and machine vibration [-7]. Most formuations of dynamic interaction probems represented the materia of hafspace as being either an eastic or viscoeastic soid. However, in certain cases, it may be more appropriate to represent the hafspace as a two-phase medium. Recenty, much interest has been shown in reexamining this probem by considering the supporting medium to be a fuid-saturated poroeastic [8-]. This paper anayticay studies the horizonta and rocking vibrations of circuar footings on a saturated poroeastic hafspace. Firsty the pressure-soid dispacement form of the harmonic equations of motion for a poroeastic soid are deveoped from the form of the equations originay presented by Biot []. Then Fredhom integra equations for the horizonta and rocking vibrations of footings on a saturated poroeastic hafspace are estabished according to the mixed boundary vaue condition. Numerica resuts for the horizonta and rocking compiance are presented and compared with the corresponding idea eastic soutions to demonstrate the infuence of poroeastic materia parameters. GOVERNING EQUATIONS At this stage, it is convenient to non-dimensionaize a quantities with respect to ength and stress by seecting the radius of the disk a as a unit of ength and the shear moduus of the hafspace as a unit of stress. Biot s wave equations [] can be written for the present case in terms of dimensioness variabes as Department of Engineering Mechanics, Tongji University, Shanghai, China. Emai: jiniuyk@onine.sh.cn

2 i, jj ( j, ji i, i = u + λ + ) u + a ( ρ ϑ) u ( α ϑ) p () ρ a ρ a ( α ϑ) p, ii + p + e = () M ϑ ϑ in which e = u ii, is soid strain and u i (i=,,3) are the dimensioness soid dispacements; p is the dimensioness pore fuid pressure; α, M = M µ, λ = λ µ, ρ = ρ f ρ, m = m ρ and b = ab ρµ are the dimensioness materia parameters, λ and µ = Lame constants; α and M = Biot s parameters accounting for compressibiity of the two-phase materia; ρ and ρ f = mass densities of the buk materia and the pore fuid, respectivey; m = a density-ike parameter that depends on ρ f and the geometry of the pores; b = a parameter accounting for the interna friction due to the reative motion between the soid matrix and the pore fuid. The parameter b is equa to the ratio between the fuid viscosity and the intrinsic permeabiity of the medium. If interna friction is negected then b =. ϑ = ρ a ( m a ib a ) is a dimensioness parameter. a =ωa ρ µ is a dimensioness frequency ω is the frequency of the motion. For brevity, the time factor, e ia t, t is a dimensioness time has been omitted from Eqs.() and () and aso from the seque. The constitutive reations can be expressed as σij = λ δije + µ ( ui, j + u j, i ) αδij p (3) p = αm e + M ζ (4) ζ = w i, i (5) and σ ij = tota stress component of the buk materia. INTEGRAL EQUATIONS FOR HORIZONTAL AND ROCKING VIBRATIONS The footing is modeed as a rigid circuar disk resting on a poroeastic haf space. A cyindrica coordinates system r, θ, z is empoyed; r, θ pane coincides with the haf-space surface with z axis directed into the haf-space. The origin of the coordinate system is ocated at the center of the circuar disk. For horizonta vibration the reationship between the force (H) and the dispacement (D h ) is: ad ( ν) H = 8µ a h C hh he dynamic compiance coefficient can be expressed as foows: (6) c hh = (7) ν ( ) Φ ( in which ν is Poisson s ratio and Φ ( ) satisfies the Fredhom integra equations of the second kind: r t t Φ( r) + [ Φ ( Φ ( ] + K( r, Φ(

3 + (, ) ( ) K r t Φ t = r 4 Φ ( r [ ( ( ] K ( r, ) + 3 r Φ Φ + Φ 4 + (, ) ( ) 3 K r t Φ t = r (8) ( r (9) Q (, = K ( r ] J ( ξ () Q ( Q(, = rt ξ[ ] J 3 3 = K ( r, ] J ( ξr) J 3 ( ξ () K ( r ( ξ (3) Q ( ), = ξ K ( r ] J ( ξ (4) In the interest of brevity the functions Q ij ( ( i =, ; j =, ) and i ( i =, 4 ) have been omitted. For rocking vibration the reationship between the rocking moment (M) and the rocking ange of the disk (φ ) is: 3( ν) M φ = C 3 mm (5) 8µ a The dynamic compiance coefficient are given as foows: C mm = (5) ν 3( ) t Φ ( in which Φ ( satisfies the Fredhom integra equation of the second kind: Φ( r ) + K( r, Φ( = r r (6) ξ ( = K( r, ] J ( ξ ( The functions (, ( ) and have been omitted. ξ (7) NUMERICAL RESULTS After obtaining Φ ( t ) from Eqs.(8) and (9), and Φ (t ) from Eq.(6) we can cacuate the horizonta and rocking compiance coefficient C hh and C mm respectivey. The seected poroeastic materias are considered in the numerica study. The dimensioness parameters of the poroeastic materias are λ =.5, M =, ρ = 53., m =. and α =. 97. In addition, b =. or. The arger the

4 permeabiity, the smaer the b is. Figs.-4 show the horizonta and rocking compiance coefficients varying with dimensioness frequency a. In the figures the soid ine and dotted ine represent b =. and b = respectivey, and Ο represents the soution for an eastic haf-space obtained by Luco and Westmann [6]. It is found from Figs.-4 that rea part and imaginary part of the horizonta or rocking compiance increase with increasing the permeabiity of the fuid in poroeastic medium (or the decrease of b ). The rea part of the horizonta compiance is increased about %, whie the rea part rocking compiance is increased about 4%. The imaginary part is increased about 7% for horizonta vibration and 8% for rocking vibration when b is decreased from. to.. It shoud be noted that the findings in Figs.- are vaid for eastic response ony. If the structure of the soi changes during vibration excess pore pressures are deveoped and the materia properties wi change. Re[ C hh ] Im[ C hh ] a a FIG. Rea part of horizonta compiance FIG. Imaginary part of horizonta compiance Re[ C mm ] Im[ C mm ] a a FIG.3 Rea part of rocking compiance FIG.4 Imaginary part of rocking compiance

5 CONCLUSIONS This paper anayticay examines the horizonta and rocking vibration of a rigid disk on a poroeastic haf space. With aid of integra transforms, the horizonta and rocking vibrations of a disk on a poroeastic haf space is formuated as Fredhom integra equations of the second kind. The numerica resuts presented in the paper indicated that 8~% differences exist for the horzonta and rocking compiance when permeabiity varies from b =. to b =. The differences between the rocking compiance for poroeastic medium and for eastic medium are smaer than 8% which can be negected in the practica engineering, whie differences for the horizonta compiance are sma than 4%. REFERENCES. Sung, T. Y., Vibrations in semi-infinite soids due to periodic surface oadings. ASTM Spec. Techn. Pub., Symposium on Soi Dynamics, 953; 56: Bycroft, G. N., Forced vibrations of a rigid circuar footing on a semi-infinite eastic space and on a eastic stratum. Phi. Trans. Roy. Soc. Lond. A, 956; 48; Awajobi, A. O., and Grootenhuis, P., Vibration of rigid bodies on semi-infinite eastic media. Proc. Roy. Soc. A, 965; 87: Robertson, I. A., Forced vertica vibration of a rigid circuar disc on a semi-infinite eastic soid. Proc. Camb. Phi. Soc. A, 966; 6: Gadwe, G. M. L., Forced tangentia and rotatory vibration of a rigid circuar disc on a semiinfinite soid. Int. J. Engrg. Sci., 968; 6: Luco, J. E., and Westmann, R. A., Dynamic response of circuar footings. J. Engrg. Mech., ASCE, 97; 97(5): Veetsos, A. S., and Wei, Y. T., Latera and rocking vibrations of footings. Journa of Soi Mechanics and Foundations Division, ASCE, 97; 97(9): Phiippacopouos, A. J., Axisymmetric vibration of disk resting on saturated ayered haf space. J. Engrg. Mech., ASCE, 989;5(): Jin, B., and Liu, H., Vertica dynamic response of a disk on a saturated poroeastic haf space. Soi Dynamics and Earthquake Engineering, 999; 8(6): Jin, B., The vertica vibration of an eastic circuar pate on a fuid-saturated porous haf space, Internationa Journa of Engineering Science, 999; 37(3): Zeng, X., Rajapakse, R. K. N. D., Vertica vibrations of a rigid disk embedded in a poroeastic medium. Int. J. Numer. Ana. Meth. Geomech., 999; 3(5): Biot, M. A., Mechanics of deformation and acoustic propagation in porous media, J. App. Phys., 96; 33(4):

Displacement Solution for a Static Vertical Rigid Movement of an Interior Circular Disc in a Transversely Isotropic Tri-Material Full-Space

Displacement Solution for a Static Vertical Rigid Movement of an Interior Circular Disc in a Transversely Isotropic Tri-Material Full-Space Vo:8, No:, 4 Dispacement Soution for a Static Vertica Rigid Movement of an Interior Circuar Disc in a Transversey Isotropic Tri-Materia Fu-Space D. Mehdizadeh, M. Rahimian, M. Eskandari-Ghadi Internationa