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Dynamic response of a flexible plate on saturated soil layer
ARTICLE IN PRESS
Soil Dynamics and Earthquake Engineering 26 (2006) 637–647 www.elsevier.com/locate/soildyn
Dynamic response of a flexible plate on saturated soil layer S.L. Chena, L.Z. Chena,, J.M. Zhangb a
Department of Civil Engineering, Shanghai Jiaotong University, Shanghai, People’s Republic of China b School of Civil Engineering, Tsinghua University, Beijing, People’s Republic of China Accepted 30 December 2005
Abstract An analytical approach is developed to study the dynamic response of a flexible plate on single-layered saturated soil. The analysis is based on Biot’s two-phased theory of poroelasticity and also on the classical thin-plate theory. First, the governing differential equations for saturated soil are solved by the use of Hankel transform. The general solutions of the skeleton displacements, stresses, and pore pressures, derived in the transformed domain, are subsequently incorporated into the imposed boundary conditions, which leads to a set of dual integral equations describing the corresponding mixed boundary value problem. These governing integral equations are finally reduced to the Fredholm integral equations of the second kind and solved by standard numerical procedures. The accuracy of the present solution is validated via comparisons with existing solutions for an ideal elastic half-space. Furthermore, some numerical results are presented to show the influences of the layer depth, the plate flexibility, and the soil porosity on the dynamic compliances. r 2006 Elsevier Ltd. All rights reserved.
1. Introduction The dynamic response of a foundation was first considered by Reissner [1] by assuming a uniform contact stress distribution under the footing for mathematical simplification. Since then, this dynamic mixed boundary value problem has been the subject of extensive studies in the field of geomechanics, soil–structure interaction, and earthquake engineering. Important contributions include those of Bycroft [2], Awojobi and Grootenhuis [3], Luco and Westmann [4], Lin [5], and Todorovska et al. [6]. As a first approximation, the forced vibration of a foundation is most often investigated by modeling the soil as a single-phase linear elastic medium. However, real soils in general are two-phase materials involving a solid skeleton and pore fluids, and thus should be more realistically regarded as poroelastic materials. In the last two decades, strong interest has been expressed in reexamining the problem of foundation vibration by assuming the supporting medium to be a saturated soil model following Biot’s poroelastic theory [7–8]. For Corresponding author. Tel.: +86 21 6293 2102; fax: +86 21 6293 3082.
E-mail addresses: [email protected] (S.L. Chen), [email protected] (L.Z. Chen). 0267-7261/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.soildyn.2006.01.014
example, Halpern and Christiano [9] analyzed the timeharmonic response of a rigid plate in smooth contact with a saturated poroelastic half-space. Kassir and Xu [10] and Kassir et al. [11] examined the vibration of rigid rectangular strip and circular foundations on a poroelastic half-space. Philippacopoulos [12] studied the similar problem by considering the supporting medium as a partially saturated poroelastic half-space. Additionally, Bougachia et al. [13] and Senjuntichai and Rajapakse [14] obtained the dynamic solutions for a rigid footing on a multilayered poroelastic medium using the finite element method and the dynamic Green’s function approach, respectively. In a further paper Senjuntichai and Sapsathiarn [15] solved the forced vertical vibration of a circular plate embedded in a multilayered poroelastic medium. It should be noted that, in deriving the dynamic force–displacement relationship of a vibrating foundation, the previous studies usually treat the soil as a semi-infinite saturated medium, and most of the results are based on the assumption of a rigid foundation. In engineering applications, however, there arise many cases in which the thickness of the soil stratum is not great compared with the dimensions of the foundation. For such cases, the theory corresponding to the saturated half-space is no
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longer appropriate and may lead to substantial errors [16,17]. On the other hand, the assumption of a rigid foundation may not always be valid. In fact, recent dynamic tests of actual buildings have observed significant out-of-plane deformations of foundations [18,19], indicating the necessary to take into account the foundation flexibility in the analysis. It is thus of practical interest and necessary to examine in detail the dynamic response of a flexible plate on a single-layered saturated soil. The paper is addressed to present an analytical solution for this particular problem. First, the governing equations of dynamic poroelasticity, within the framework of Biot’s theory, are solved by means of Hankel transforms, and the required general solutions for the displacement and stress components of the saturated soil are obtained in a straightforward way. These general solutions, in combination with the boundary conditions, then yield a set of dual integral equations which correspond to the mixed boundary value problem for the vibration of a flexible plate. The dual integral equations can be further reduced to a Fredholm integral equation of the second kind and be solved using the standard numerical procedures. The dynamic compliance functions are finally derived and some numerical results are presented to explore the influences of layer thickness, plate flexibility, and soil porosity on the foundation response.
r2 u¯ r
The problem considered is shown in Fig. 1. A massless circular plate with flexible rigidity D and Poisson ratio nf is assumed to rest on a saturated soil layer and the plate center subjected to a harmonic vertical force, Feiot, with o being the circular frequency. Based on Biot’s theory of poroelasticity, the nondimensional equations governing the motion of the saturated soil in the cylindrical coordinates system
¯ þ 1Þ r2 u¯ z þ ðl¯ þ a2 M
qe ¯ qe ¼ a20 u¯ z r¯ w a20 v¯ z , þ aM q¯z q¯z
¯ 0 v¯ r , ¯ qe ¼ r¯ w a20 u¯ r ma ¯ qe þ M aM ¯ 20 v¯ r þ iba q¯r q¯r
(1b) (1c)
¯ 0 v¯ z , ¯ qe ¼ r¯ w a20 u¯ z ma ¯ qe þ M aM (1d) ¯ 20 v¯ z þ iba q¯z q¯z in which the time factor eiot has been omitted since steadystate vibrations are considered here and all the parameters and variables been normalized with respect to the radius of the plate, r0, the mass density of bulk material r, and the shear modulus of the solid skeleton, G, respectively, i.e., r¯ ¼ r=r0 ; z¯ ¼ z=r0 , u¯ r ¼ ur =r0 , u¯ z ¼ uz =r0 , v¯ r ¼ vr =r0 , v¯ z ¼ ¯ ¯ vp ¯ w ¼ rw =r; m ¯ ¼ m=r, z =r 0; r ffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffilffi ¼ l=G, M ¼ M=G, a0 ¼ ¯ r=G or0 and b ¼ r0 b= rG; ur and uz are the displacement components of the solid matrix in radial (r) and vertical (z) directions; vr and vz are the average fluid displacements relative to the solid matrix in the r and z directions; e and e are the matrix dilation and the fluid dilation relative to the solid, which are expressed as qur ur quz qvr vr qvz þ þ ; e¼ þ þ ; qr r qz qr r qz l, G are the Lame’s constants of the solid matrix; a, M are, respectively, the Biot’s compressibility parameters of the soil skeleton and water; rw is the mass density of the water and rs mass density of grains [r ¼ nrw+(1n) rs, n ¼ porosity]; m ¼ rw =n; b is a parameter accounting for the internal friction due to the relative motion between the solid matrix and pore water, and is equal to the ratio between the fluid viscosity and the intrinsic permeability of the medium; q2 1 q q2 þ þ qr2 r qr qz2 denotes the symmetric Laplacian operator. The constitutive relations can be expressed as r2 ¼
Feiωt Flexible plate r uz
∆v
1 ¯ qe ¼ a20 u¯ r r¯ w a20 v¯ r , ¯ þ 1Þ qe þ aM u¯ r þ ðl¯ þ a2 M 2 q¯r q¯r r¯ (1a)
e¼
2. Governing equations and general solutions
r0
(r, y, z), in the frequency domain, can be expressed as [7]
q¯uz ¯ þ le, q¯z
(2a)
q¯ur q¯uz þ , q¯z q¯r
(2b)
s¯ z ¼ 2
w Saturated soil layer h
t¯ zr ¼
¯ Me, ¯ s¯ f ¼ aMe Impervious rough base
z Fig. 1. Geometry of the problem.
(2c)
where s¯ z ¼ sz =G, t¯ zr ¼ tzr =G, and s¯ f ¼ sf =G are again the introduced nondimensional variables; sz is the vertical component of effective normal stress; tzr is the shear stress; and sf denotes the excess pore pressure. The above governing equations [Eqs. (1a)–(1d)] can be solved by employing the conventional Hankel transform technique, which has been described in detail by Chen et al. [17] and Chen [20]. The mth Hankel integral transform of a
ARTICLE IN PRESS S.L. Chen et al. / Soil Dynamics and Earthquake Engineering 26 (2006) 637–647
function f(r, z) with respect to r and the inverse relationship are defined as [21] Z 1 m f~ ðp; zÞ ¼ H m ½f ðp; zÞ ¼ rf ðr; zÞJ m ðrpÞ dr, (3a) 0
Z
1
f ðr; zÞ ¼
m
pf~ ðp; zÞJ m ðrpÞ dp,
(3b)
0
where p is the parameter for the Hankel transform; and Jm denotes the Bessel function of the first kind of order m. By taking appropriate Hankel transforms of Eqs. (1a)–(1d) and after some manipulations, the fundamental solutions for displacements can be derived as 0 u~¯ z ðp; z¯ Þ ¼ cðA1 sinh c¯z A2 cosh c¯zÞ þ dðB1 sinh d z¯ B2 cosh d z¯ Þ
þ p2 ðR1 cosh j¯z R2 sinh j¯zÞ,
ð4Þ
ð5Þ
r¯ w a20 d3 ¼ ; ¯ 0 ma iba ¯ 20
ð8Þ
1 t~¯ zr ðp; z¯ Þ ¼ 2pcðA1 sinh c¯z þ A2 cosh c¯zÞ þ 2pdðB1 sinh d z¯ þ B2 cosh d z¯ Þ
ð9Þ
0 s~¯ f ðp; z¯ Þ ¼ a1 ðA1 cosh c¯z A2 sinh c¯zÞ
ð10Þ
¯ 2 , k2 ¼ ðl¯ þ 2Þd lp ¯ 2; in which k1 ¼ ðl¯ þ 2Þc2 lp 2 2 ¯ ¯ a1 ¼ ða þ d1 ÞMp1 , a2 ¼ ða þ d2 ÞMp2 . 3. Formulation of mixed boundary value problem
ð6Þ
pjd3 ðR1 sinh j¯z R2 cosh j¯zÞ, ð7Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 2 in which c ¼ p p1 , d ¼ p p2 , j ¼ p s , p1, p2, and s are the complex wave numbers associated with the dilatational waves of the first and second kind and with the rotational wave, respectively, which are given by the relations qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b1 þ b21 4b2 , p21 ¼ q2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b1 b21 4b2 p22 ¼ , 2 ¯ 0 ma a2 ðiba ¯ 20 Þ þ r¯ 2w a40 s2 ¼ 0 , ¯ 0 ma iba ¯ 20 ¯ 0 Þ 2aM ¯ r¯ w a20 þ Ma ¯ 20 ¯ þ 2Þðma ðl¯ þ a2 M ¯ 20 iba b1 ¼ , ¯ ðl¯ þ 2ÞM ¯ 0 Þa2 r¯ 2 a4 ðma ¯ 20 iba 0 w 0 b2 ¼ , ¯ ðl¯ þ 2ÞM ¯ þ 2Þp21 a20 ðl¯ þ a2 M d1 ¼ , ¯ 21 r¯ w a20 aMp ¯ þ 2Þp22 a20 ðl þ a2 M , 2 ¯ 22 r¯ w a0 aMp
þ 2p2 jðR1 sinh j¯z R2 cosh j¯zÞ,
2
1 v~¯ r ðp; z¯ Þ ¼ pd1 ðA1 cosh c¯z þ A2 sinh c¯zÞ þ pd2 ðB1 cosh d z¯ þ B2 sinh d z¯ Þ
d2 ¼
0 s~¯ z ðp; z¯ Þ ¼ k1 ðA1 cosh c¯z A2 sinh c¯zÞ þ k2 ðB1 cosh d z¯ B2 sinh d z¯ Þ
þ a2 ðB1 cosh d z¯ B2 sinh d z¯ Þ,
1 u~¯ r ðp; z¯ Þ ¼ pðA1 cosh c¯z þ A2 sinh c¯zÞ þ pðB1 cosh d z¯ þ B2 sinh d z¯ Þ
pjðR1 sinh j¯z R2 cosh j¯zÞ,
and A1, A2, B1, B2, R1, R2 are arbitrary functions of p. It is noted that the radicals c, d, and j are selected in such a way that their real parts are always nonnegative. Making use of Eqs. (2a)–(2c), the general expressions for the stresses of the solid matrix and pore water are obtained as follows:
pðp2 þ j 2 ÞðR1 cosh j¯z R2 sinh j¯zÞ,
0 v~¯ z ðp; z¯ Þ ¼ cd1 ðA1 sinh c¯z A2 cosh c¯zÞ þ dd2 ðB1 sinh d z¯ B2 cosh d z¯ Þ
þ p2 d3 ðR1 cosh j¯z R2 sinh j¯zÞ,
639
Consider the vibration problem of a flexible plate which is in smooth contact with the underlying saturated soil layer of thickness h (Fig. 1). It is assumed that the ground surface is fully permeable both within and exterior to the contact area and that the base of the layer is completely rough, rigid and impervious. The boundary conditions at z ¼ 0 and z ¼ h can then be written in dimensionless quantities as t¯ zr ð¯r; 0Þ ¼ 0;
ð0p¯rp1Þ,
(11a)
s¯ z ð¯r; 0Þ ¼ 0;
ð1p¯rp1Þ,
(11b)
s¯ f ð¯r; 0Þ ¼ 0;
ð0p¯rp1Þ,
(11c)
u¯ z ð¯r; 0Þ ¼ D¯ v w即 rÞ;
ð0p¯rp1Þ,
(11d)
¯ ¼ 0; u¯ z ð¯r; hÞ
ð0p¯rp1Þ,
(11e)
¯ ¼ 0; u¯ r ð¯r; hÞ
ð0p¯rp1Þ,
(11f)
¯ ¼ 0; ð0p¯rp1Þ, v¯ z ð¯r; hÞ (11g) ¯ ¯ where h ¼ h=r0 , Dv ¼ Dv =r0 , w ¯ ¼ w=r0 with Dv denoting the vertical displacement at the center of the plate and w the plate deflection relative to its center. Introduce the following column vector: n oT 0 1 0 0 1 0 Sðp; z¯ Þ ¼ u~¯ z ðp; z¯ Þ; u~¯ r ðp; z¯ Þ; v~¯ z ðp; z¯ Þ; s~¯ z ðp; z¯ Þ; t~¯ zr ðp; z¯ Þ; s~¯ f ðp; z¯ Þ (12) and set z¯ ¼ 0 and z¯ ¼ h¯ respectively in Eqs. (4)–(6) and (8)–(10), the six functions A1, A2, B1, B2, R1, R2 can be eliminated, which leads to the following relationship in
ARTICLE IN PRESS S.L. Chen et al. / Soil Dynamics and Earthquake Engineering 26 (2006) 637–647
640
matrix form: ¯ ¼ j Sðp; 0Þ, Sðp; hÞ
(13)
where j denotes a 6 6 matrix whose elements jij ði; j ¼ 126Þ are functions of p and are explicitly given in Appendix. In view of Eqs. (11a), (11c), and (11e)–(11g), one have n oT 0 1 0 0 Sðp; 0Þ ¼ u~¯ z ðp; 0Þ; u~¯ r ðp; 0Þ; v~¯ z ðp; 0Þ; s~¯ z ðp; 0Þ; 0; 0 , (14a) n oT ¯ ¼ 0; 0; 0; s~¯ 0 ðp; hÞ; ¯ t~¯ 1 ðp; hÞ; ¯ s~¯ 0 ðp; hÞ ¯ Sðp; hÞ . z zr f
(14b)
It then follows from Eq. (13) that 0 0 u~¯ z ðp; 0Þ ¼ R11 ðpÞs~¯ z ðp; 0Þ
(15)
in which R11 ðpÞ ¼
1 j ðpÞ½j22 ðpÞj33 ðpÞ j32 ðpÞj23 ðpÞ DðpÞ 14 þ j24 ðpÞ½j32 ðpÞj13 ðpÞ j12 ðpÞj33 ðpÞ þj34 ðpÞ½j12 ðpÞj23 ðpÞ j22 ðpÞj13 ðpÞ ,
The general solution for Eq. (17) is F¯ d 1 2 2 r¯ ln r¯ þ C 1 r¯ þ C 2 w即 rÞ ¼ 2p 4 Z 1 r¯2 1 1 þd BðpÞ 4 2 4 J 0 ðp¯rÞ dp, p p 4p 0
where C1, C2 can be determined from the plate boundary conditions, 2 d w ¯ vf dw¯ ½w þ ¼0 (19) ¯ r¯¼0 ¼ 0; r¯ d¯r r¯¼1 d¯r2 and are given by Z 1 1 þ vf J 01 ðpÞ vf J 1 ðpÞ p BðpÞ þ 2 þ C1 ¼ dp p3 p 2p2 F¯ ð1 þ vf Þ 0 3 þ vf ; C 2 ¼ 0. ð20Þ 8ð1 þ vf Þ Therefore, the final expression for w即 rÞ is F¯ d¯r2 ln r¯ F¯ dð3 þ vf Þ¯r2 þ 8p 16pð1 þ vf Þ Z 1 r¯2 1 1 þd BðpÞ 4 2 4 J 0 ðp¯rÞ dp p p 4p 0 Z 1 2 1 þ vf J 0 1 ðpÞ vf J 1 ðpÞ d¯r dp. BðpÞ þ 2 þ p p3 2ð1 þ vf Þ 0 2p2
w即 rÞ ¼
DðpÞ ¼ j11 ðpÞ½j22 ðpÞj33 ðpÞ j32 ðpÞj23 ðpÞ þ j21 ðpÞ½j32 ðpÞj13 ðpÞ j12 ðpÞj33 ðpÞ þ j31 ðpÞ½j12 ðpÞj23 ðpÞ j22 ðpÞj13 ðpÞ. Substituting Eq. (15) into Eq. (11d) gives Z
1
p1 ½1 þ HðpÞBðpÞJ 0 ðp¯rÞ dp ¼
0
wð¯ D¯ v ¯ rÞ , þ 1v 1v
(16)
0 ps~¯ z ðp; 0Þ,
where BðpÞ ¼ HðpÞ ¼ ½pR11 ðpÞ=ð1 vÞ 1, and v is Poisson’s ratio of soil. It can be proved that limp!1 pR11 ðpÞ ¼ ð1 vÞ, i.e., limp!1 HðpÞ ¼ 0. The deflection of the flexible plate, w即 rÞ in Eq. (16), must satisfy
Z
1
ð21Þ Substituting Eq. (21) into Eq. (16) and expressing s¯ z ð¯r; 0Þ in terms of its Hankel transform, one can finally obtain a set of dual integral equations describing the vibration problem of a flexible plate on saturated soil layer as
¯v F¯ dð3 þ vf Þ¯r2 D F¯ d¯r2 ln r¯ þ 1 v 8pð1 vÞ 16pð1 þ vf Þð1 vÞ Z 1 p1 HðpÞBðpÞJ 0 ðp¯rÞ dp 0 Z 1 r¯2 d 1 1 þ BðpÞ 4 2 4 J 0 ðp¯rÞ dp 1v 0 p p 4p Z 1 2 1 þ vf J 0 1 ðpÞ vf J 1 ðpÞ d¯r BðpÞ þ 2 þ dp p p3 2ð1 þ vf Þð1 vÞ 0 2p2
p1 BðpÞJ 0 ðp¯rÞ dp ¼
0
ð0 r¯ 1Þ,
ð22aÞ
the following differential equation [22]: 1 q d q¯r
ð18Þ
Z
Z 1 d 1d 1 F¯ wð¯ BðpÞJ 1 ðp¯rÞ dp, þ ¯ rÞ ¼ 2 r d¯ r p 2p¯ r ¯ d¯r 0
1
BðpÞJ 0 ðp¯rÞ dp ¼ 0
ð¯r41Þ.
(22b)
0
(17) where the dimensionless flexural rigidity d is defined as d ¼ Gr30 =D and the normalized force F¯ ¼ F =Gr20 .
These integral equations with the normal contact stress being the unknown can be reduced to a Fredholm integral equation of the second kind by employing the following
ARTICLE IN PRESS S.L. Chen et al. / Soil Dynamics and Earthquake Engineering 26 (2006) 637–647
(23)
Substituting this representation for B(p) appearing in Eqs. (22a) and (22b), it is seen that Eq. (22b) is satisfied identically, while Eq. (22a) is equivalent to the following integral equation of the Fredholm type: Z 1 yðxÞ þ Kðx; yÞyðyÞ dy ¼ 1, (24) 0
4. Numerical results and discussion
where the kernel function K(x, t) can be proved to be symmetric with respect to its arguments and takes the form Z 1 Kðx; yÞ ¼ HðpÞcos px cos py dp 0 2 ð1 vf Þx2 y2 2d x y2 ln x2 þ ln y2 pð1 vÞ 4 4 2ð1 þ vf Þ 1 2 2 2 2 ½ðx yÞ lnðx yÞ þ ðx þ yÞ lnðx þ yÞ . 8 ð25Þ In deriving Eq. (24) the following relationship has been used:
By applying the trapezoidal rule to the finite integral appearing in Eq. (24), the Fredholm integral equation derived above can be discretized into a system of algebraic equations and subsequently solved numerically [26]. Once the integral equation has been solved, the dynamic compliances can be determined immediately from Eq. (26). In this section, the numerical results for dynamic response of flexible foundations on saturated soil layer are presented. The objective is to compare the feasibility and the accuracy of the present solution against previous results and also to evaluate the influence of some governing parameters on the foundation response. Fig. 2 presents a
(26)
Eq. (26) essentially presents the force-displacement relationship for the flexible plate. In this equation, ð1 vÞ=4Gr0 corresponds to the static vertical compliance for a rigid plate with radius r0 on an elastic half-space characterized by shear modulus G and Poisson’s ratio v, R1 while the factor C v ¼ 1= 0 yðxÞ dx is usually defined as the vertical dynamic compliance [4] and can be obtained by direct integration based on the numerical solution of y(x) from Eq. (24). Finally, the nondimensional displacement of the plate can be obtained by substitution of Eq. (26) into (21). The resulting expression is Z 1 uz ð¯r; 0Þ wð¯rÞ d¯r2 ln r¯ ¼1 ¼1þ yðtÞ dt Dv Dv 2pð1 vÞ 0 Z 1 dð3 þ vf Þ¯r2 yðtÞ dt 4pð1 þ vf Þð1 vÞ 0 2d þ pð1 vÞ Z 1 Z 1 1 1 yðtÞ dt 3 J 0 ðp¯rÞ 3 p p 0 0 0 2 r¯ J 1 ðpÞ vf J 01 ðpÞ þ cos pt dp. ð27Þ p p 2ð1 þ vf Þ The analysis method described above can be readily extended to study the rocking vibration of a flexible plate where the assumption of axial symmetry will not be valid. In that case, the basic solutions for the three-dimensional wave equations for saturated soil can be derived using the
1.0 v = 1/4, Present study v = 1/4, Luco and Westmann (1971)
0.8
Re[Cv]
Dv ð1 vÞ ð1 vÞ 1 ¼ Cv ¼ . R1 4Gr0 4Gr0 F 0 yðxÞ dx
Fourier expansions and Hankel integral transform with respect to the circumferential and radical coordinates, respectively. The details should be referred to [23]. In addition, the boundary conditions similar to that given by Eq. (11) can be established with minor modification [24,25]. Consequently, one may yield a set of dual integral equations describing the rocking oscillation of a flexible plate following the procedure presented by Eqs. (11)–(26), and the dynamic rocking compliance can thus be obtained.
0.6
0.4
0.2
0.0 0
1
2
(a)
3
4
5
a0 0.0
-0.2
Im[Cv]
integral representation: Z 1 2 Dv p BðpÞ ¼ yðxÞcos px dx. p 1v 0
641
-0.4
-0.6 v = 1/4, Present study v = 1/4, Luco and Westmann (1971)
-0.8
-1.0 0 (b)
1
2
3
4
5
a0
Fig. 2. Comparison of dynamic compliance functions for a flexible plate on the ideal elastic half-space: (a) real part; (b) imaginary part.
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642
comparison of the dynamic compliance functions for a vertically loaded rigid circular footing on a homogeneous elastic half-space given by Luco and Westmann [4] with those obtained from the present study. It should be mentioned here that the current solutions for this reduced problem were computed by setting rw ¼ b ¼ m ¼ d ¼ 0:01, h=r0 ¼ 50, and v ¼ 14. Excellent agreement is noted between the two results and the accuracy of the present solution scheme is therefore confirmed. Next, parametric studies will be conducted for the dynamic response of a flexible plate resting on a saturated soil layer, with particular emphasis on the influences of the soil layer depth, the plate flexibility, and the porosity. The following nondimensional material para¯ ¼ 12:2, m meters vf ¼ 0:167, v ¼ 0:25, M ¯ ¼ 1:1, a ¼ 0:97, ¯b ¼ 10, r¯ w ¼ 0:53, and different values of h¯ ¼ 1; 2; 5; 10 and d ¼ 0; 1; 10; 100; 1000, unless otherwise stated, will be adopted in the numerical study. Note that a value of d ¼ 0 corresponds to a fully rigid foundation. Fig. 3 presents the influence of the thickness of soil layer (h¯ ¼ 1; 2; 5; 10) and the non-dimensionalized frequency parameter a0 ð0pa0 p5Þ on the dynamic compliance function Cv. Also shown in this figure is the corresponding compliances for a semi-infinite saturated soil (h¯ ! 1) 3.0 2.5
h/r0 = 5
Re[Cv]
2.0
h/r0 = 2 h/r0 = 1
1.5 h/r0 = 10
1.0 0.5
h/r0 = ∞
0.0 0
1
2
3
(a)
4
5
3
5
a0 0.0
Im[Cv]
-0.4
h/r0 = 1
h/r0 = 5
-0.8 h/r0 = 10 -1.2
h/r0 = 2 h/r0 = ∞
-1.6
-2.0 0 (b)
1
2
4 a0
Fig. 3. Influence of soil layer thickness on dynamic compliance functions for d ¼ 10: (a) Real part; (b) imaginary part.
derived from [26], which is denoted by ‘‘3’’. It is found that both real and imaginary parts of the dynamic compliance functions, which correspond respectively to the in-phase and out-of-phase components of the displacement of flexible plate, exhibit significant oscillatory variations with the dimensionless frequency particularly when the soil layer thickness is small. This is the outcome of resonance phenomena: waves generated from the vibration foundation reflect at the rigid base and return back to the surface. As a result, peaks occur in the real and imaginary parts of the dynamic compliance at specific frequencies of vibration. Further, the first resonant frequencies for soil layers with different depth are observed to coincide with the natural frequencies of the purely elastic strata in vertical P-waves, which are given by [2,27] a0p;1
p r0 2ð1 vÞ 1=2 ¼ 2 h 1 2v
(28)
and equal to 2.72, 1.36, 0.54, and 0.27 for h=r0 ¼ 1; 2; 5, and 10, respectively. These results are consistent with the conclusions for the classical elastic case. Fig. 3 also shows that, as the depth of the soil layer increases, the fluctuations become less pronounced and a gradually decaying oscillatory is noted only within low frequencies. Especially for the deepest soil layer studied, h¯ ¼ 10, the variation of the compliance functions with a0 has already been in good agreement with the one corresponding to a semi-infinite soil, the major departure occurs in the frequency range of 0pa0 p0:25. This indicates that the stress and strain fields caused by the exciting force are of limit extent and increasing layer depth beyond a value corresponding to h¯ ¼ 10 has practically no effect on the foundation response. Fig. 4 shows the variation of the compliance function Cv with the normalized frequency a0 for five different relative flexibility d ¼ 0; 1; 10; 100; 1000 and for a single value of the soil layer thickness, h¯ ¼ 2. Evidently, the vibration response of the flexible plate is quite sensitive to variations in d. The magnitudes of both real and imaginary parts of the dynamic compliance functions increase with the increasing plate flexibility. However, the variations of the compliance functions with the frequency are similar for all the values of plate flexibility considered, one common feature is that the compliance functions would yield peak amplitudes at some resonant frequencies due to the presence of the standing wave generated between the surface and the rigid base. Referring to Fig. 4, it is noted that the first resonant frequencies are essentially located near a0 ¼ 1:35, which is again consistent with the one calculated from Eq. (28). The porosity dependence of the plate response has also been examined. Fig. 5 presents the numerical results with three different values of soil porosity, namely, n ¼ 0:2; 0:48; and 0:8 being taken into account. Since the nondimensional soil properties r¯ w ¼ rw =r, m ¯ ¼ m=r, pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi b¯ ¼ r0 b= rG , and a0 ¼ r=G or0 are functions of the
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643
3
8
1000
6
n= 0.48 Re[Cv]
Re[Cv]
2 4 100
n= 0.80 1 n = 0.20
10
2
1 0
δ=0
0 0
1
1
0 2
(a)
3
4
5
2
4
3
5
a0
(a)
a0 0.0 0
δ=0
Im[Cv]
10
-1
Im[Cv]
n= 0.80
−0.4
1
100 -2
−0.8
n = 0.48
−1.2
1000 −1.6
n = 0.20
-3 −2.0 1 -4 0 (b)
1
2
3
4
2
4
3
5
a0
(b) 5
a0
Fig. 5. Influence of soil porosity on dynamic compliance functions for h¯ ¼ 2: (a) real part; (b) imaginary part.
Fig. 4. Influence of plate flexibility on dynamic compliance functions for h¯ ¼ 2: (a) real part; (b) imaginary part.
bulk density r ½r ¼ n rw þ ð1 nÞ rs , which varies with changing soil porosity, three sets of material parameters should therefore be adopted, as presented in Table 1, to cover the three cases mentioned above. It is important to note that, for direct comparison, the two compliance functions for n ¼ 0:2 and 0:8 have been plotted against adjustedp nondimensional frequencies defined, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi respectively, by a00 ¼ 0:53=0:42 a0 and a00 ¼ 0:53=0:75 a0 . As seen in Fig. 5, the response curves exhibit undulations associated with the natural frequencies of the soil layer and the influence of porosity is noticeable within the frequency range of 1pa0 p3. With the soil porosity ascending from 0.2 to 0.8, the resonant peaks for both real and imaginary parts of the dynamic compliance would increase initially but then decrease and the corresponding resonant frequencies shift to larger values. Fig. 6 presents a comparison of the dynamic compliance between the saturated soil layer and the ideal elastic layer. The parameters adopted in the numerical calculation are the same as those in Figs. 2 and 3, except that the nondimesional frequency for the elastic case has been qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi modified to be a00 ¼ 1 ðr¯ 2w =mÞ ¯ a0 . From Fig. 6, it is
Table 1 Nondimensional parameters for saturated soil layer with different porosities Soil Mass porosity n density of water r¯ w
Densitylike parameter m ¯
Internal Biot Biot friction parameter parameter ¯ parameter M a b¯
0.20 0.48 0.80
2.11 1.10 0.93
21.4 10.0 7.1
0.42 0.53 0.75
12.2 12.2 12.2
0.97 0.97 0.97
clear that the effect of the pore fluid is to generally considerably decrease the real part of the compliance and accordingly contribute to the reduction of vibration response. This phenomenon is particularly obvious in case the frequency approaches the resonant value. Compared with the elastic case, the dynamic response of plate on the saturated stratum shows a relatively smooth variation with a0. Finally, the displacement profile of the exited flexible plate against the nondimensional distance r/r0 is presented in Fig. 7 for different plate flexibilities of d ¼ 1; 10; 100; 1000.
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4
0.0 0.2
2
0.4
1000
∆v
Saturated soil layer Ideal elastic layer
uz(r,0)
Re[Cv]
3
100 0.6 10
1
0.8
0
1.0 0.0
δ=1 0
1
2
(a)
3
4
5
a0
0.2
0.4
(a)
0.6
0.8
1.0
0.8
1.0
r/r0 0.0
0.0 0.2 1000
-0.8 -1.2
Saturated soil layer Ideal elastic layer
0.4 ∆v
uz(r,0)
Im[Cv]
-0.4
100
10
0.6 δ=1
0.8
-1.6 1.0 0.0
-2.0 1 (b)
2
3
4
5
a0
Fig. 6. Comparison of dynamic compliance functions between saturated soil layer and ideal elastic layer: (a) real part; (b) imaginary part.
In this figure, the displacement is normalized by uz ð¯r; 0Þ=Dv
and solutions are given for two dimensionless frequencies a0 ¼ 2 and 5. The displacement variations are very similar for these two frequencies, and as expected, the displacement tends to increase as the plate become more flexible. On the other hand, the plate displacement in general exhibits its peak value at the center and decreases along the plate radius, although under certain combinations of a0 and d, the minimum vertical displacement may be developed in the middle part of the plate.
5. Conclusions An analytical study is presented on the vertical vibration behavior of a flexible foundation resting in smooth contact with a saturated soil layer. By solving Biot’s equations of dynamic poroelasticity in Hankel transform domain and considering the corresponding drainage boundary conditions, the problem is formulated as integral equations in terms of the contact stresses between the foundation and the soil, and subsequently reduced to a Fredholm integral equation of the second kind whose solution is then computed. Comparisons with existing solutions
(b)
0.2
0.4
0.6 r/r0
Fig. 7. Displacement profile for flexible plate on saturated soil layer: (a) a0 ¼ 2; (b) a0 ¼ 5.
for the purely elastic case confirm the accuracy of the present formulation and also the numerical computation involved. Selected numerical results illustrate that the influence of the soil layer thickness and the flexibility of the foundation on its dynamic response is significant. For shallower soil layer, both real and imaginary parts of the dynamic compliance functions display very pronounced fluctuations with the nondimensional frequency, with resonant peaks located near the natural frequencies of the deposit. However, as the soil stratum becomes deeper, the dynamic compliance show less oscillatory variations and the effect ¯ of layer depth is almost negligible when hX10. As to the effect of plate flexibility, it is observed that an increase in its value tends to increase magnitudes of both real and imaginary parts of the dynamic compliance functions, although quite similar trends are found for different plate flexibilities considered in this paper. Numerical results also indicate that the corresponding resonant frequencies would shift to lower values as the soil porosity decrease, and the presence of the pore fluid reduces the vibration response of the plate. It is finally found that the plate displacement generally decreases along plate radius, and tends to increase as the plate become more flexible.
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Appendix
2a2 p2 cosh ch¯ x 2a1 p2 þ cosh d h¯ x a2 k 1 a1 k 2 ¯ þ cosh j h, x
j22 ¼
The elements of the matrix u are given by j11 ¼
j12 ¼
d2 ðp2 þ j 2 Þ þ 2d3 p2 cosh ch¯ ðd2 d1 Þðp2 j 2 Þ d1 ðp2 þ j 2 Þ 2d3 p2 cosh d h¯ þ ðd2 d1 Þðp2 j 2 Þ 2p2 ¯ cosh j h, þ 2 p j2 2a2 pc sinh ch¯ x 2a1 pd sinh d h¯ x pða2 k1 a1 k2 Þ ¯ sinh j h, jx 1 cosh ch¯ d2 d1 1 ¯ cosh d h, d1 d2
j13 ¼
j14 ¼
j15 ¼
a2 c sinh ch¯ x a1 d sinh d h¯ x p2 ða2 a1 Þ ¯ sinh j h, jx ðd3 d2 Þp cosh ch¯ ðj p2 Þðd1 d2 Þ ðd1 d3 Þp þ 2 cosh d h¯ ðj p2 Þðd1 d2 Þ p ¯ cosh j h, þ 2 ðp j 2 Þ
j23 ¼
j25 ¼
j26 ¼
d2 ðp2 þ j 2 Þ 2d3 p2 p sinh ch¯ cðd2 d1 Þðp2 j 2 Þ d1 ðp2 þ j 2 Þ þ 2d3 p2 p sinh d h¯ þ dðd2 d1 Þðp2 j 2 Þ 2pj ¯ sinh j h, 2 p j2
a2 p a1 p a2 a1 ¯ cosh ch¯ þ cosh d h¯ þ p cosh j h, x x x
ðd3 d2 Þp2 sinh ch¯ cðj 2 p2 Þðd2 d1 Þ ðd1 d3 Þp2 þ 2 sinh d h¯ dðj p2 Þðd2 d1 Þ j ¯ sinh j h, 2 p j2 ðk2 2p2 Þp cosh ch¯ x ðk1 2p2 Þp cosh d h¯ x pðk2 k1 Þ ¯ cosh j h, x d2 ðp2 þ j 2 Þ 2d3 p2 d1 cosh ch¯ ðd2 d1 Þðp2 j 2 Þ d1 ðp2 þ j 2 Þ 2d3 p2 þ d2 cosh d h¯ ðd2 d1 Þðp2 j 2 Þ 2d3 p2 ¯ þ 2 cosh j h, p j2
j31 ¼
j32 ¼
j16 ¼
j21 ¼
p p ¯ sinh ch¯ þ sinh d h, cðd2 d1 Þ dðd1 d2 Þ
j24 ¼
2
ðk2 2p2 Þc sinh ch¯ x ðk1 2p2 Þd þ sinh d h¯ x p2 ðk2 k1 Þ ¯ þ sinh j h, jx
645
2a2 pcd1 sinh ch¯ x 2a1 pdd2 sinh ph¯ x pða2 k1 a1 k2 Þd3 ¯ sinh j h, jx
j33 ¼
j34 ¼
d1 d2 ¯ cosh ch¯ cosh d h, d2 d1 d1 d2
a2 cd1 a1 dd2 p2 d3 ða2 a1 Þ ¯ sinh j h, sinh ch¯ sinh d h¯ jx x x d1 ðd3 d2 Þp cosh ch¯ ðj 2 p2 Þðd2 d1 Þ d2 ðd1 d3 Þp 2 cosh d h¯ ðj p2 Þðd2 d1 Þ pd3 ¯ þ 2 cosh j h, ðp j 2 Þ
j35 ¼
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646
ðk2 2p2 Þcd1 sinh ch¯ x ðk1 2p2 Þdd2 þ sinh d h¯ x ðk2 k1 Þp2 d3 ¯ þ sinh j h, jx
j36 ¼
d2 ðp2 þ j 2 Þ 2d3 p2 k1 sinh ch¯ cðd2 d1 Þðp2 j 2 Þ d1 ðp2 þ j 2 Þ 2d3 p2 þ k2 sinh d h¯ dðd2 d1 Þðp2 j 2 Þ 4p2 j ¯ þ 2 sinh j h, p j2
2a2 pc 2a1 pd sinh ch¯ þ sinh d h¯ x x ða2 a1 Þpðp2 þ j 2 Þ ¯ sinh j h, þ jx
j54 ¼
j55 ¼
j41 ¼
j42 ¼
j43
2a2 p 2a1 p k1 cosh ch¯ k2 cosh d h¯ x x 2p ¯ ða2 k1 a1 k2 Þ cosh j h, x
k1 k2 ¯ sinh ch¯ sinh d h, ¼ cðd2 d1 Þ dðd1 d2 Þ
j44 ¼
j56 ¼
ðk2 2p2 Þ 2pc sinh ch¯ x ðk1 2p2 Þ 2pd sinh d h¯ x ðk2 k1 Þpðp2 þ j 2 Þ ¯ sinh j h, jx d2 ðp2 þ j 2 Þ 2d3 p2 a1 sinh ch¯ cðd2 d1 Þðp2 j 2 Þ d1 ðp2 þ j 2 Þ 2d3 p2 ¯ þ a2 sinh d h, dðd2 d1 Þðp2 j 2 Þ
j61 ¼
a2 k 1 a1 k 2 a2 a1 2 ¯ cosh ch¯ cosh d h¯ 2p cosh j h, x x x ðd3 d2 Þpk1 sinh ch¯ cðj 2 p2 Þðd2 d1 Þ ðd1 d3 Þpk2 2 sinh d h¯ dðj p2 Þðd2 d1 Þ 2pj ¯ sinh j h, þ 2 ðp j 2 Þ
2p2 ðd3 d2 Þ cosh ch¯ ðj p2 Þðd2 d1 Þ 2p2 ðd1 d3 Þ þ 2 cosh d h¯ ðj p2 Þðd2 d1 Þ p2 þ j 2 ¯ cosh j h, 2 p j2 2
j62 ¼
2a2 pa1 2a1 pa2 ¯ cosh c¯z cosh d h, x x
j45 ¼
j46
ðk2 2p2 Þk1 ðk1 2p2 Þk2 cosh ch¯ þ cosh d h¯ ¼ x x ðk2 k1 Þ 2 ¯ þ 2p cosh j h, x
j51 ¼
j52
d2 ðp2 þ j 2 Þ 2d3 p2 2p cosh ch¯ ðd2 d1 Þðp2 j 2 Þ d1 ðp2 þ j 2 Þ 2d3 p2 2p cosh d h¯ ðd2 d1 Þðp2 j 2 Þ 2pðp2 þ j 2 Þ ¯ cosh j h, þ p2 j 2
4p2 ca2 4p2 da1 sinh ch¯ þ sinh d h¯ ¼ x x ðp2 þ j 2 Þ ¯ ða2 k1 a1 k2 Þsinh j h, þ jx
j53 ¼
2p 2p ¯ cosh ch¯ þ cosh d h, d 2 d1 d1 d2
j63 ¼ j64 ¼
a1 a2 ¯ sinh ch¯ sinh d h, cðd2 d1 Þ dðd1 d2 Þ
a1 a2 a 2 a1 ¯ cosh ch¯ cosh d h, x x a1 ðd3 d2 Þp sinh ch¯ cðj 2 p2 Þðd2 d1 Þ a2 ðd1 d3 Þp ¯ 2 sinh d h, dðj p2 Þðd2 d1 Þ
j65 ¼
a1 ðk2 2p2 Þ cosh ch¯ x a2 ðk1 2p2 Þ ¯ þ cosh d h, x
j66 ¼
where x ¼ s2 ða1 a2 Þ. References [1] Reissner E. Stationare, axialsymmetrische, durch eine schuttelnde masse erregte schwingungen elastischen halbraumes. Ingenieur Archiv 1936; 381–96. [2] Bycroft GN. Forced vibration of a rigid circular plate on a semiinfinite elastic space and on an elastic stratum. Philos Trans R Soc London (A) 1956;248:327–68. [3] Awojobi AO, Grootenhuis P. Vibration of rigid bodies on semiinfinite elastic media. Proc R Soc London (A) 1965;287:27–63. [4] Luco JE, Westmann RA. Dynamic response of circular footings. UCLA Engineering Report [no. 7113], 1971.
ARTICLE IN PRESS S.L. Chen et al. / Soil Dynamics and Earthquake Engineering 26 (2006) 637–647 [5] Lin YE. Dynamic response of circular plates resting on viscoelastic half space. J Appl Mech 1978;45:379–84. [6] Todorovska MI, Hayir A, Trifunac MD. Antiplane response of a dike on flexible embedded foundation to incident SH-waves. Soil Dynam Earth Eng 2001;21:593–601. [7] Biot MA. The theory of propagation of elastic waves in a fluidsaturated porous solid. I, Low-frequency range. J Acoust Soc Am 1956;28:168–78. [8] Biot MA. Mechanics of deformation and acoustic propagation in porous media. J Appl Phys 1962;33:1482–98. [9] Halpern MR, Christiano P. Response of poroelastic halfspace to steady-state harmonic surface tractions. Int J Numer Anal Mech Geomech 1986;10:609–32. [10] Kassir MK, Xu J. Interaction functions of a rigid strip bonded to saturated elastic half space. Int J Solids Struct 1988;24:915–36. [11] Kassir MK, Bandyopadyay KK, Xu J. Vertical vibration of a circular footing on a saturated half space. Int J Eng Sci 1989;27:353–61. [12] Philippacopoulos AJ. Axisymmetric vibration of disk resting on saturated layered half space. J Eng Mech Div ASCE 1989; 115(10):2301–22. [13] Bougacha S, Tassoulas JL, Roesset JM. Dynamic stiffness of foundations on fluid-filled poroelastic stratum. J Eng Mech Div ASCE 1993;119(8):1649–62. [14] Senjuntichai T, Rajapakse RKND. Dynamics of a rigid strip bonded to a multi-layered poroelastic half-plane. In: Selvadurai APS, editor. Mechanics of poroelastic media. Dordrecht, The Netherlands: Kluwer Academic; 1996. [15] Senjuntichai T, Sapsathiarn Y. Forced vibration of circular plate in multilayered poroelastic medium. J Eng Mech Div ASCE 2003; 129(11):1330–41.
647
[16] Gibson RE, Schiffman RL, Pu SL. Plane strain and axially symmetric consolidation of a clay layer on a smooth impervious base. Q J Mech Appl Math 1970;23(4):505–20. [17] Chen SL, Zhang LM, Chen LZ. Consolidation of a finite transversely isotropic soil layer on a rough impervious base. J Eng Mech Div ASCE 2005;131(12):1279–80. [18] Hudson DE. Dynamic tests of full-scale structures. J Eng Mech Div ASCE 1977;103(6):1141–57. [19] Trifunac MD, Ivanovic SS, Todorovska MI, Novikova EI, Gladkov AA. Experimental evidence for flexibility of a building foundation supported by concrete friction piles. Soil Dynam Earth Eng 1999;18:169–87. [20] Chen SL. Vertical vibrations of foundations on saturated grounds. PhD thesis, Zhejiang University, China, 2000 (in Chinese). [21] Sneddon I. The use of integral transforms. New York: McGraw-Hill; 1970. [22] Szilard R. Theory and analysis of plates: classical and numerical methods. Englewood Cliffs, NJ: Prentice-Hall, Inc.; 1974. [23] Chen SL, Chen LZ, Pan E. Three-dimensional time-harmonic Green’s functions of saturated soil under buried loading. Unpublished, 2005. [24] Iguichi M, Luco JE. Vibration of flexible plate on viscoelastic medium. J Eng Mech Div ASCE 1982;108(6):1103–20. [25] Fu B. Study on the rocking vibrations of foundations resting on saturated grounds. PhD thesis, Zhejiang University, China, 2005 (in Chinese). [26] Chen SL, Chen LZ, Pan E. Vertical vibration of a flexible plate with rigid core on saturated ground. Unpublished, 2005. [27] Gazetas G. Analysis of machine foundation vibrations: state of the art. Dynam Earth Eng 1983;2(1):2–42.
Soil Dynamics and Earthquake Engineering 26 (2006) 637–647 www.elsevier.com/locate/soildyn
Dynamic response of a flexible plate on saturated soil layer S.L. Chena, L.Z. Chena,, J.M. Zhangb a
Department of Civil Engineering, Shanghai Jiaotong University, Shanghai, People’s Republic of China b School of Civil Engineering, Tsinghua University, Beijing, People’s Republic of China Accepted 30 December 2005
Abstract An analytical approach is developed to study the dynamic response of a flexible plate on single-layered saturated soil. The analysis is based on Biot’s two-phased theory of poroelasticity and also on the classical thin-plate theory. First, the governing differential equations for saturated soil are solved by the use of Hankel transform. The general solutions of the skeleton displacements, stresses, and pore pressures, derived in the transformed domain, are subsequently incorporated into the imposed boundary conditions, which leads to a set of dual integral equations describing the corresponding mixed boundary value problem. These governing integral equations are finally reduced to the Fredholm integral equations of the second kind and solved by standard numerical procedures. The accuracy of the present solution is validated via comparisons with existing solutions for an ideal elastic half-space. Furthermore, some numerical results are presented to show the influences of the layer depth, the plate flexibility, and the soil porosity on the dynamic compliances. r 2006 Elsevier Ltd. All rights reserved.
1. Introduction The dynamic response of a foundation was first considered by Reissner [1] by assuming a uniform contact stress distribution under the footing for mathematical simplification. Since then, this dynamic mixed boundary value problem has been the subject of extensive studies in the field of geomechanics, soil–structure interaction, and earthquake engineering. Important contributions include those of Bycroft [2], Awojobi and Grootenhuis [3], Luco and Westmann [4], Lin [5], and Todorovska et al. [6]. As a first approximation, the forced vibration of a foundation is most often investigated by modeling the soil as a single-phase linear elastic medium. However, real soils in general are two-phase materials involving a solid skeleton and pore fluids, and thus should be more realistically regarded as poroelastic materials. In the last two decades, strong interest has been expressed in reexamining the problem of foundation vibration by assuming the supporting medium to be a saturated soil model following Biot’s poroelastic theory [7–8]. For Corresponding author. Tel.: +86 21 6293 2102; fax: +86 21 6293 3082.
E-mail addresses: [email protected] (S.L. Chen), [email protected] (L.Z. Chen). 0267-7261/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.soildyn.2006.01.014
example, Halpern and Christiano [9] analyzed the timeharmonic response of a rigid plate in smooth contact with a saturated poroelastic half-space. Kassir and Xu [10] and Kassir et al. [11] examined the vibration of rigid rectangular strip and circular foundations on a poroelastic half-space. Philippacopoulos [12] studied the similar problem by considering the supporting medium as a partially saturated poroelastic half-space. Additionally, Bougachia et al. [13] and Senjuntichai and Rajapakse [14] obtained the dynamic solutions for a rigid footing on a multilayered poroelastic medium using the finite element method and the dynamic Green’s function approach, respectively. In a further paper Senjuntichai and Sapsathiarn [15] solved the forced vertical vibration of a circular plate embedded in a multilayered poroelastic medium. It should be noted that, in deriving the dynamic force–displacement relationship of a vibrating foundation, the previous studies usually treat the soil as a semi-infinite saturated medium, and most of the results are based on the assumption of a rigid foundation. In engineering applications, however, there arise many cases in which the thickness of the soil stratum is not great compared with the dimensions of the foundation. For such cases, the theory corresponding to the saturated half-space is no
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638
longer appropriate and may lead to substantial errors [16,17]. On the other hand, the assumption of a rigid foundation may not always be valid. In fact, recent dynamic tests of actual buildings have observed significant out-of-plane deformations of foundations [18,19], indicating the necessary to take into account the foundation flexibility in the analysis. It is thus of practical interest and necessary to examine in detail the dynamic response of a flexible plate on a single-layered saturated soil. The paper is addressed to present an analytical solution for this particular problem. First, the governing equations of dynamic poroelasticity, within the framework of Biot’s theory, are solved by means of Hankel transforms, and the required general solutions for the displacement and stress components of the saturated soil are obtained in a straightforward way. These general solutions, in combination with the boundary conditions, then yield a set of dual integral equations which correspond to the mixed boundary value problem for the vibration of a flexible plate. The dual integral equations can be further reduced to a Fredholm integral equation of the second kind and be solved using the standard numerical procedures. The dynamic compliance functions are finally derived and some numerical results are presented to explore the influences of layer thickness, plate flexibility, and soil porosity on the foundation response.
r2 u¯ r
The problem considered is shown in Fig. 1. A massless circular plate with flexible rigidity D and Poisson ratio nf is assumed to rest on a saturated soil layer and the plate center subjected to a harmonic vertical force, Feiot, with o being the circular frequency. Based on Biot’s theory of poroelasticity, the nondimensional equations governing the motion of the saturated soil in the cylindrical coordinates system
¯ þ 1Þ r2 u¯ z þ ðl¯ þ a2 M
qe ¯ qe ¼ a20 u¯ z r¯ w a20 v¯ z , þ aM q¯z q¯z
¯ 0 v¯ r , ¯ qe ¼ r¯ w a20 u¯ r ma ¯ qe þ M aM ¯ 20 v¯ r þ iba q¯r q¯r
(1b) (1c)
¯ 0 v¯ z , ¯ qe ¼ r¯ w a20 u¯ z ma ¯ qe þ M aM (1d) ¯ 20 v¯ z þ iba q¯z q¯z in which the time factor eiot has been omitted since steadystate vibrations are considered here and all the parameters and variables been normalized with respect to the radius of the plate, r0, the mass density of bulk material r, and the shear modulus of the solid skeleton, G, respectively, i.e., r¯ ¼ r=r0 ; z¯ ¼ z=r0 , u¯ r ¼ ur =r0 , u¯ z ¼ uz =r0 , v¯ r ¼ vr =r0 , v¯ z ¼ ¯ ¯ vp ¯ w ¼ rw =r; m ¯ ¼ m=r, z =r 0; r ffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffilffi ¼ l=G, M ¼ M=G, a0 ¼ ¯ r=G or0 and b ¼ r0 b= rG; ur and uz are the displacement components of the solid matrix in radial (r) and vertical (z) directions; vr and vz are the average fluid displacements relative to the solid matrix in the r and z directions; e and e are the matrix dilation and the fluid dilation relative to the solid, which are expressed as qur ur quz qvr vr qvz þ þ ; e¼ þ þ ; qr r qz qr r qz l, G are the Lame’s constants of the solid matrix; a, M are, respectively, the Biot’s compressibility parameters of the soil skeleton and water; rw is the mass density of the water and rs mass density of grains [r ¼ nrw+(1n) rs, n ¼ porosity]; m ¼ rw =n; b is a parameter accounting for the internal friction due to the relative motion between the solid matrix and pore water, and is equal to the ratio between the fluid viscosity and the intrinsic permeability of the medium; q2 1 q q2 þ þ qr2 r qr qz2 denotes the symmetric Laplacian operator. The constitutive relations can be expressed as r2 ¼
Feiωt Flexible plate r uz
∆v
1 ¯ qe ¼ a20 u¯ r r¯ w a20 v¯ r , ¯ þ 1Þ qe þ aM u¯ r þ ðl¯ þ a2 M 2 q¯r q¯r r¯ (1a)
e¼
2. Governing equations and general solutions
r0
(r, y, z), in the frequency domain, can be expressed as [7]
q¯uz ¯ þ le, q¯z
(2a)
q¯ur q¯uz þ , q¯z q¯r
(2b)
s¯ z ¼ 2
w Saturated soil layer h
t¯ zr ¼
¯ Me, ¯ s¯ f ¼ aMe Impervious rough base
z Fig. 1. Geometry of the problem.
(2c)
where s¯ z ¼ sz =G, t¯ zr ¼ tzr =G, and s¯ f ¼ sf =G are again the introduced nondimensional variables; sz is the vertical component of effective normal stress; tzr is the shear stress; and sf denotes the excess pore pressure. The above governing equations [Eqs. (1a)–(1d)] can be solved by employing the conventional Hankel transform technique, which has been described in detail by Chen et al. [17] and Chen [20]. The mth Hankel integral transform of a
ARTICLE IN PRESS S.L. Chen et al. / Soil Dynamics and Earthquake Engineering 26 (2006) 637–647
function f(r, z) with respect to r and the inverse relationship are defined as [21] Z 1 m f~ ðp; zÞ ¼ H m ½f ðp; zÞ ¼ rf ðr; zÞJ m ðrpÞ dr, (3a) 0
Z
1
f ðr; zÞ ¼
m
pf~ ðp; zÞJ m ðrpÞ dp,
(3b)
0
where p is the parameter for the Hankel transform; and Jm denotes the Bessel function of the first kind of order m. By taking appropriate Hankel transforms of Eqs. (1a)–(1d) and after some manipulations, the fundamental solutions for displacements can be derived as 0 u~¯ z ðp; z¯ Þ ¼ cðA1 sinh c¯z A2 cosh c¯zÞ þ dðB1 sinh d z¯ B2 cosh d z¯ Þ
þ p2 ðR1 cosh j¯z R2 sinh j¯zÞ,
ð4Þ
ð5Þ
r¯ w a20 d3 ¼ ; ¯ 0 ma iba ¯ 20
ð8Þ
1 t~¯ zr ðp; z¯ Þ ¼ 2pcðA1 sinh c¯z þ A2 cosh c¯zÞ þ 2pdðB1 sinh d z¯ þ B2 cosh d z¯ Þ
ð9Þ
0 s~¯ f ðp; z¯ Þ ¼ a1 ðA1 cosh c¯z A2 sinh c¯zÞ
ð10Þ
¯ 2 , k2 ¼ ðl¯ þ 2Þd lp ¯ 2; in which k1 ¼ ðl¯ þ 2Þc2 lp 2 2 ¯ ¯ a1 ¼ ða þ d1 ÞMp1 , a2 ¼ ða þ d2 ÞMp2 . 3. Formulation of mixed boundary value problem
ð6Þ
pjd3 ðR1 sinh j¯z R2 cosh j¯zÞ, ð7Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 2 in which c ¼ p p1 , d ¼ p p2 , j ¼ p s , p1, p2, and s are the complex wave numbers associated with the dilatational waves of the first and second kind and with the rotational wave, respectively, which are given by the relations qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b1 þ b21 4b2 , p21 ¼ q2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b1 b21 4b2 p22 ¼ , 2 ¯ 0 ma a2 ðiba ¯ 20 Þ þ r¯ 2w a40 s2 ¼ 0 , ¯ 0 ma iba ¯ 20 ¯ 0 Þ 2aM ¯ r¯ w a20 þ Ma ¯ 20 ¯ þ 2Þðma ðl¯ þ a2 M ¯ 20 iba b1 ¼ , ¯ ðl¯ þ 2ÞM ¯ 0 Þa2 r¯ 2 a4 ðma ¯ 20 iba 0 w 0 b2 ¼ , ¯ ðl¯ þ 2ÞM ¯ þ 2Þp21 a20 ðl¯ þ a2 M d1 ¼ , ¯ 21 r¯ w a20 aMp ¯ þ 2Þp22 a20 ðl þ a2 M , 2 ¯ 22 r¯ w a0 aMp
þ 2p2 jðR1 sinh j¯z R2 cosh j¯zÞ,
2
1 v~¯ r ðp; z¯ Þ ¼ pd1 ðA1 cosh c¯z þ A2 sinh c¯zÞ þ pd2 ðB1 cosh d z¯ þ B2 sinh d z¯ Þ
d2 ¼
0 s~¯ z ðp; z¯ Þ ¼ k1 ðA1 cosh c¯z A2 sinh c¯zÞ þ k2 ðB1 cosh d z¯ B2 sinh d z¯ Þ
þ a2 ðB1 cosh d z¯ B2 sinh d z¯ Þ,
1 u~¯ r ðp; z¯ Þ ¼ pðA1 cosh c¯z þ A2 sinh c¯zÞ þ pðB1 cosh d z¯ þ B2 sinh d z¯ Þ
pjðR1 sinh j¯z R2 cosh j¯zÞ,
and A1, A2, B1, B2, R1, R2 are arbitrary functions of p. It is noted that the radicals c, d, and j are selected in such a way that their real parts are always nonnegative. Making use of Eqs. (2a)–(2c), the general expressions for the stresses of the solid matrix and pore water are obtained as follows:
pðp2 þ j 2 ÞðR1 cosh j¯z R2 sinh j¯zÞ,
0 v~¯ z ðp; z¯ Þ ¼ cd1 ðA1 sinh c¯z A2 cosh c¯zÞ þ dd2 ðB1 sinh d z¯ B2 cosh d z¯ Þ
þ p2 d3 ðR1 cosh j¯z R2 sinh j¯zÞ,
639
Consider the vibration problem of a flexible plate which is in smooth contact with the underlying saturated soil layer of thickness h (Fig. 1). It is assumed that the ground surface is fully permeable both within and exterior to the contact area and that the base of the layer is completely rough, rigid and impervious. The boundary conditions at z ¼ 0 and z ¼ h can then be written in dimensionless quantities as t¯ zr ð¯r; 0Þ ¼ 0;
ð0p¯rp1Þ,
(11a)
s¯ z ð¯r; 0Þ ¼ 0;
ð1p¯rp1Þ,
(11b)
s¯ f ð¯r; 0Þ ¼ 0;
ð0p¯rp1Þ,
(11c)
u¯ z ð¯r; 0Þ ¼ D¯ v w即 rÞ;
ð0p¯rp1Þ,
(11d)
¯ ¼ 0; u¯ z ð¯r; hÞ
ð0p¯rp1Þ,
(11e)
¯ ¼ 0; u¯ r ð¯r; hÞ
ð0p¯rp1Þ,
(11f)
¯ ¼ 0; ð0p¯rp1Þ, v¯ z ð¯r; hÞ (11g) ¯ ¯ where h ¼ h=r0 , Dv ¼ Dv =r0 , w ¯ ¼ w=r0 with Dv denoting the vertical displacement at the center of the plate and w the plate deflection relative to its center. Introduce the following column vector: n oT 0 1 0 0 1 0 Sðp; z¯ Þ ¼ u~¯ z ðp; z¯ Þ; u~¯ r ðp; z¯ Þ; v~¯ z ðp; z¯ Þ; s~¯ z ðp; z¯ Þ; t~¯ zr ðp; z¯ Þ; s~¯ f ðp; z¯ Þ (12) and set z¯ ¼ 0 and z¯ ¼ h¯ respectively in Eqs. (4)–(6) and (8)–(10), the six functions A1, A2, B1, B2, R1, R2 can be eliminated, which leads to the following relationship in
ARTICLE IN PRESS S.L. Chen et al. / Soil Dynamics and Earthquake Engineering 26 (2006) 637–647
640
matrix form: ¯ ¼ j Sðp; 0Þ, Sðp; hÞ
(13)
where j denotes a 6 6 matrix whose elements jij ði; j ¼ 126Þ are functions of p and are explicitly given in Appendix. In view of Eqs. (11a), (11c), and (11e)–(11g), one have n oT 0 1 0 0 Sðp; 0Þ ¼ u~¯ z ðp; 0Þ; u~¯ r ðp; 0Þ; v~¯ z ðp; 0Þ; s~¯ z ðp; 0Þ; 0; 0 , (14a) n oT ¯ ¼ 0; 0; 0; s~¯ 0 ðp; hÞ; ¯ t~¯ 1 ðp; hÞ; ¯ s~¯ 0 ðp; hÞ ¯ Sðp; hÞ . z zr f
(14b)
It then follows from Eq. (13) that 0 0 u~¯ z ðp; 0Þ ¼ R11 ðpÞs~¯ z ðp; 0Þ
(15)
in which R11 ðpÞ ¼
1 j ðpÞ½j22 ðpÞj33 ðpÞ j32 ðpÞj23 ðpÞ DðpÞ 14 þ j24 ðpÞ½j32 ðpÞj13 ðpÞ j12 ðpÞj33 ðpÞ þj34 ðpÞ½j12 ðpÞj23 ðpÞ j22 ðpÞj13 ðpÞ ,
The general solution for Eq. (17) is F¯ d 1 2 2 r¯ ln r¯ þ C 1 r¯ þ C 2 w即 rÞ ¼ 2p 4 Z 1 r¯2 1 1 þd BðpÞ 4 2 4 J 0 ðp¯rÞ dp, p p 4p 0
where C1, C2 can be determined from the plate boundary conditions, 2 d w ¯ vf dw¯ ½w þ ¼0 (19) ¯ r¯¼0 ¼ 0; r¯ d¯r r¯¼1 d¯r2 and are given by Z 1 1 þ vf J 01 ðpÞ vf J 1 ðpÞ p BðpÞ þ 2 þ C1 ¼ dp p3 p 2p2 F¯ ð1 þ vf Þ 0 3 þ vf ; C 2 ¼ 0. ð20Þ 8ð1 þ vf Þ Therefore, the final expression for w即 rÞ is F¯ d¯r2 ln r¯ F¯ dð3 þ vf Þ¯r2 þ 8p 16pð1 þ vf Þ Z 1 r¯2 1 1 þd BðpÞ 4 2 4 J 0 ðp¯rÞ dp p p 4p 0 Z 1 2 1 þ vf J 0 1 ðpÞ vf J 1 ðpÞ d¯r dp. BðpÞ þ 2 þ p p3 2ð1 þ vf Þ 0 2p2
w即 rÞ ¼
DðpÞ ¼ j11 ðpÞ½j22 ðpÞj33 ðpÞ j32 ðpÞj23 ðpÞ þ j21 ðpÞ½j32 ðpÞj13 ðpÞ j12 ðpÞj33 ðpÞ þ j31 ðpÞ½j12 ðpÞj23 ðpÞ j22 ðpÞj13 ðpÞ. Substituting Eq. (15) into Eq. (11d) gives Z
1
p1 ½1 þ HðpÞBðpÞJ 0 ðp¯rÞ dp ¼
0
wð¯ D¯ v ¯ rÞ , þ 1v 1v
(16)
0 ps~¯ z ðp; 0Þ,
where BðpÞ ¼ HðpÞ ¼ ½pR11 ðpÞ=ð1 vÞ 1, and v is Poisson’s ratio of soil. It can be proved that limp!1 pR11 ðpÞ ¼ ð1 vÞ, i.e., limp!1 HðpÞ ¼ 0. The deflection of the flexible plate, w即 rÞ in Eq. (16), must satisfy
Z
1
ð21Þ Substituting Eq. (21) into Eq. (16) and expressing s¯ z ð¯r; 0Þ in terms of its Hankel transform, one can finally obtain a set of dual integral equations describing the vibration problem of a flexible plate on saturated soil layer as
¯v F¯ dð3 þ vf Þ¯r2 D F¯ d¯r2 ln r¯ þ 1 v 8pð1 vÞ 16pð1 þ vf Þð1 vÞ Z 1 p1 HðpÞBðpÞJ 0 ðp¯rÞ dp 0 Z 1 r¯2 d 1 1 þ BðpÞ 4 2 4 J 0 ðp¯rÞ dp 1v 0 p p 4p Z 1 2 1 þ vf J 0 1 ðpÞ vf J 1 ðpÞ d¯r BðpÞ þ 2 þ dp p p3 2ð1 þ vf Þð1 vÞ 0 2p2
p1 BðpÞJ 0 ðp¯rÞ dp ¼
0
ð0 r¯ 1Þ,
ð22aÞ
the following differential equation [22]: 1 q d q¯r
ð18Þ
Z
Z 1 d 1d 1 F¯ wð¯ BðpÞJ 1 ðp¯rÞ dp, þ ¯ rÞ ¼ 2 r d¯ r p 2p¯ r ¯ d¯r 0
1
BðpÞJ 0 ðp¯rÞ dp ¼ 0
ð¯r41Þ.
(22b)
0
(17) where the dimensionless flexural rigidity d is defined as d ¼ Gr30 =D and the normalized force F¯ ¼ F =Gr20 .
These integral equations with the normal contact stress being the unknown can be reduced to a Fredholm integral equation of the second kind by employing the following
ARTICLE IN PRESS S.L. Chen et al. / Soil Dynamics and Earthquake Engineering 26 (2006) 637–647
(23)
Substituting this representation for B(p) appearing in Eqs. (22a) and (22b), it is seen that Eq. (22b) is satisfied identically, while Eq. (22a) is equivalent to the following integral equation of the Fredholm type: Z 1 yðxÞ þ Kðx; yÞyðyÞ dy ¼ 1, (24) 0
4. Numerical results and discussion
where the kernel function K(x, t) can be proved to be symmetric with respect to its arguments and takes the form Z 1 Kðx; yÞ ¼ HðpÞcos px cos py dp 0 2 ð1 vf Þx2 y2 2d x y2 ln x2 þ ln y2 pð1 vÞ 4 4 2ð1 þ vf Þ 1 2 2 2 2 ½ðx yÞ lnðx yÞ þ ðx þ yÞ lnðx þ yÞ . 8 ð25Þ In deriving Eq. (24) the following relationship has been used:
By applying the trapezoidal rule to the finite integral appearing in Eq. (24), the Fredholm integral equation derived above can be discretized into a system of algebraic equations and subsequently solved numerically [26]. Once the integral equation has been solved, the dynamic compliances can be determined immediately from Eq. (26). In this section, the numerical results for dynamic response of flexible foundations on saturated soil layer are presented. The objective is to compare the feasibility and the accuracy of the present solution against previous results and also to evaluate the influence of some governing parameters on the foundation response. Fig. 2 presents a
(26)
Eq. (26) essentially presents the force-displacement relationship for the flexible plate. In this equation, ð1 vÞ=4Gr0 corresponds to the static vertical compliance for a rigid plate with radius r0 on an elastic half-space characterized by shear modulus G and Poisson’s ratio v, R1 while the factor C v ¼ 1= 0 yðxÞ dx is usually defined as the vertical dynamic compliance [4] and can be obtained by direct integration based on the numerical solution of y(x) from Eq. (24). Finally, the nondimensional displacement of the plate can be obtained by substitution of Eq. (26) into (21). The resulting expression is Z 1 uz ð¯r; 0Þ wð¯rÞ d¯r2 ln r¯ ¼1 ¼1þ yðtÞ dt Dv Dv 2pð1 vÞ 0 Z 1 dð3 þ vf Þ¯r2 yðtÞ dt 4pð1 þ vf Þð1 vÞ 0 2d þ pð1 vÞ Z 1 Z 1 1 1 yðtÞ dt 3 J 0 ðp¯rÞ 3 p p 0 0 0 2 r¯ J 1 ðpÞ vf J 01 ðpÞ þ cos pt dp. ð27Þ p p 2ð1 þ vf Þ The analysis method described above can be readily extended to study the rocking vibration of a flexible plate where the assumption of axial symmetry will not be valid. In that case, the basic solutions for the three-dimensional wave equations for saturated soil can be derived using the
1.0 v = 1/4, Present study v = 1/4, Luco and Westmann (1971)
0.8
Re[Cv]
Dv ð1 vÞ ð1 vÞ 1 ¼ Cv ¼ . R1 4Gr0 4Gr0 F 0 yðxÞ dx
Fourier expansions and Hankel integral transform with respect to the circumferential and radical coordinates, respectively. The details should be referred to [23]. In addition, the boundary conditions similar to that given by Eq. (11) can be established with minor modification [24,25]. Consequently, one may yield a set of dual integral equations describing the rocking oscillation of a flexible plate following the procedure presented by Eqs. (11)–(26), and the dynamic rocking compliance can thus be obtained.
0.6
0.4
0.2
0.0 0
1
2
(a)
3
4
5
a0 0.0
-0.2
Im[Cv]
integral representation: Z 1 2 Dv p BðpÞ ¼ yðxÞcos px dx. p 1v 0
641
-0.4
-0.6 v = 1/4, Present study v = 1/4, Luco and Westmann (1971)
-0.8
-1.0 0 (b)
1
2
3
4
5
a0
Fig. 2. Comparison of dynamic compliance functions for a flexible plate on the ideal elastic half-space: (a) real part; (b) imaginary part.
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642
comparison of the dynamic compliance functions for a vertically loaded rigid circular footing on a homogeneous elastic half-space given by Luco and Westmann [4] with those obtained from the present study. It should be mentioned here that the current solutions for this reduced problem were computed by setting rw ¼ b ¼ m ¼ d ¼ 0:01, h=r0 ¼ 50, and v ¼ 14. Excellent agreement is noted between the two results and the accuracy of the present solution scheme is therefore confirmed. Next, parametric studies will be conducted for the dynamic response of a flexible plate resting on a saturated soil layer, with particular emphasis on the influences of the soil layer depth, the plate flexibility, and the porosity. The following nondimensional material para¯ ¼ 12:2, m meters vf ¼ 0:167, v ¼ 0:25, M ¯ ¼ 1:1, a ¼ 0:97, ¯b ¼ 10, r¯ w ¼ 0:53, and different values of h¯ ¼ 1; 2; 5; 10 and d ¼ 0; 1; 10; 100; 1000, unless otherwise stated, will be adopted in the numerical study. Note that a value of d ¼ 0 corresponds to a fully rigid foundation. Fig. 3 presents the influence of the thickness of soil layer (h¯ ¼ 1; 2; 5; 10) and the non-dimensionalized frequency parameter a0 ð0pa0 p5Þ on the dynamic compliance function Cv. Also shown in this figure is the corresponding compliances for a semi-infinite saturated soil (h¯ ! 1) 3.0 2.5
h/r0 = 5
Re[Cv]
2.0
h/r0 = 2 h/r0 = 1
1.5 h/r0 = 10
1.0 0.5
h/r0 = ∞
0.0 0
1
2
3
(a)
4
5
3
5
a0 0.0
Im[Cv]
-0.4
h/r0 = 1
h/r0 = 5
-0.8 h/r0 = 10 -1.2
h/r0 = 2 h/r0 = ∞
-1.6
-2.0 0 (b)
1
2
4 a0
Fig. 3. Influence of soil layer thickness on dynamic compliance functions for d ¼ 10: (a) Real part; (b) imaginary part.
derived from [26], which is denoted by ‘‘3’’. It is found that both real and imaginary parts of the dynamic compliance functions, which correspond respectively to the in-phase and out-of-phase components of the displacement of flexible plate, exhibit significant oscillatory variations with the dimensionless frequency particularly when the soil layer thickness is small. This is the outcome of resonance phenomena: waves generated from the vibration foundation reflect at the rigid base and return back to the surface. As a result, peaks occur in the real and imaginary parts of the dynamic compliance at specific frequencies of vibration. Further, the first resonant frequencies for soil layers with different depth are observed to coincide with the natural frequencies of the purely elastic strata in vertical P-waves, which are given by [2,27] a0p;1
p r0 2ð1 vÞ 1=2 ¼ 2 h 1 2v
(28)
and equal to 2.72, 1.36, 0.54, and 0.27 for h=r0 ¼ 1; 2; 5, and 10, respectively. These results are consistent with the conclusions for the classical elastic case. Fig. 3 also shows that, as the depth of the soil layer increases, the fluctuations become less pronounced and a gradually decaying oscillatory is noted only within low frequencies. Especially for the deepest soil layer studied, h¯ ¼ 10, the variation of the compliance functions with a0 has already been in good agreement with the one corresponding to a semi-infinite soil, the major departure occurs in the frequency range of 0pa0 p0:25. This indicates that the stress and strain fields caused by the exciting force are of limit extent and increasing layer depth beyond a value corresponding to h¯ ¼ 10 has practically no effect on the foundation response. Fig. 4 shows the variation of the compliance function Cv with the normalized frequency a0 for five different relative flexibility d ¼ 0; 1; 10; 100; 1000 and for a single value of the soil layer thickness, h¯ ¼ 2. Evidently, the vibration response of the flexible plate is quite sensitive to variations in d. The magnitudes of both real and imaginary parts of the dynamic compliance functions increase with the increasing plate flexibility. However, the variations of the compliance functions with the frequency are similar for all the values of plate flexibility considered, one common feature is that the compliance functions would yield peak amplitudes at some resonant frequencies due to the presence of the standing wave generated between the surface and the rigid base. Referring to Fig. 4, it is noted that the first resonant frequencies are essentially located near a0 ¼ 1:35, which is again consistent with the one calculated from Eq. (28). The porosity dependence of the plate response has also been examined. Fig. 5 presents the numerical results with three different values of soil porosity, namely, n ¼ 0:2; 0:48; and 0:8 being taken into account. Since the nondimensional soil properties r¯ w ¼ rw =r, m ¯ ¼ m=r, pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi b¯ ¼ r0 b= rG , and a0 ¼ r=G or0 are functions of the
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643
3
8
1000
6
n= 0.48 Re[Cv]
Re[Cv]
2 4 100
n= 0.80 1 n = 0.20
10
2
1 0
δ=0
0 0
1
1
0 2
(a)
3
4
5
2
4
3
5
a0
(a)
a0 0.0 0
δ=0
Im[Cv]
10
-1
Im[Cv]
n= 0.80
−0.4
1
100 -2
−0.8
n = 0.48
−1.2
1000 −1.6
n = 0.20
-3 −2.0 1 -4 0 (b)
1
2
3
4
2
4
3
5
a0
(b) 5
a0
Fig. 5. Influence of soil porosity on dynamic compliance functions for h¯ ¼ 2: (a) real part; (b) imaginary part.
Fig. 4. Influence of plate flexibility on dynamic compliance functions for h¯ ¼ 2: (a) real part; (b) imaginary part.
bulk density r ½r ¼ n rw þ ð1 nÞ rs , which varies with changing soil porosity, three sets of material parameters should therefore be adopted, as presented in Table 1, to cover the three cases mentioned above. It is important to note that, for direct comparison, the two compliance functions for n ¼ 0:2 and 0:8 have been plotted against adjustedp nondimensional frequencies defined, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi respectively, by a00 ¼ 0:53=0:42 a0 and a00 ¼ 0:53=0:75 a0 . As seen in Fig. 5, the response curves exhibit undulations associated with the natural frequencies of the soil layer and the influence of porosity is noticeable within the frequency range of 1pa0 p3. With the soil porosity ascending from 0.2 to 0.8, the resonant peaks for both real and imaginary parts of the dynamic compliance would increase initially but then decrease and the corresponding resonant frequencies shift to larger values. Fig. 6 presents a comparison of the dynamic compliance between the saturated soil layer and the ideal elastic layer. The parameters adopted in the numerical calculation are the same as those in Figs. 2 and 3, except that the nondimesional frequency for the elastic case has been qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi modified to be a00 ¼ 1 ðr¯ 2w =mÞ ¯ a0 . From Fig. 6, it is
Table 1 Nondimensional parameters for saturated soil layer with different porosities Soil Mass porosity n density of water r¯ w
Densitylike parameter m ¯
Internal Biot Biot friction parameter parameter ¯ parameter M a b¯
0.20 0.48 0.80
2.11 1.10 0.93
21.4 10.0 7.1
0.42 0.53 0.75
12.2 12.2 12.2
0.97 0.97 0.97
clear that the effect of the pore fluid is to generally considerably decrease the real part of the compliance and accordingly contribute to the reduction of vibration response. This phenomenon is particularly obvious in case the frequency approaches the resonant value. Compared with the elastic case, the dynamic response of plate on the saturated stratum shows a relatively smooth variation with a0. Finally, the displacement profile of the exited flexible plate against the nondimensional distance r/r0 is presented in Fig. 7 for different plate flexibilities of d ¼ 1; 10; 100; 1000.
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644
4
0.0 0.2
2
0.4
1000
∆v
Saturated soil layer Ideal elastic layer
uz(r,0)
Re[Cv]
3
100 0.6 10
1
0.8
0
1.0 0.0
δ=1 0
1
2
(a)
3
4
5
a0
0.2
0.4
(a)
0.6
0.8
1.0
0.8
1.0
r/r0 0.0
0.0 0.2 1000
-0.8 -1.2
Saturated soil layer Ideal elastic layer
0.4 ∆v
uz(r,0)
Im[Cv]
-0.4
100
10
0.6 δ=1
0.8
-1.6 1.0 0.0
-2.0 1 (b)
2
3
4
5
a0
Fig. 6. Comparison of dynamic compliance functions between saturated soil layer and ideal elastic layer: (a) real part; (b) imaginary part.
In this figure, the displacement is normalized by uz ð¯r; 0Þ=Dv
and solutions are given for two dimensionless frequencies a0 ¼ 2 and 5. The displacement variations are very similar for these two frequencies, and as expected, the displacement tends to increase as the plate become more flexible. On the other hand, the plate displacement in general exhibits its peak value at the center and decreases along the plate radius, although under certain combinations of a0 and d, the minimum vertical displacement may be developed in the middle part of the plate.
5. Conclusions An analytical study is presented on the vertical vibration behavior of a flexible foundation resting in smooth contact with a saturated soil layer. By solving Biot’s equations of dynamic poroelasticity in Hankel transform domain and considering the corresponding drainage boundary conditions, the problem is formulated as integral equations in terms of the contact stresses between the foundation and the soil, and subsequently reduced to a Fredholm integral equation of the second kind whose solution is then computed. Comparisons with existing solutions
(b)
0.2
0.4
0.6 r/r0
Fig. 7. Displacement profile for flexible plate on saturated soil layer: (a) a0 ¼ 2; (b) a0 ¼ 5.
for the purely elastic case confirm the accuracy of the present formulation and also the numerical computation involved. Selected numerical results illustrate that the influence of the soil layer thickness and the flexibility of the foundation on its dynamic response is significant. For shallower soil layer, both real and imaginary parts of the dynamic compliance functions display very pronounced fluctuations with the nondimensional frequency, with resonant peaks located near the natural frequencies of the deposit. However, as the soil stratum becomes deeper, the dynamic compliance show less oscillatory variations and the effect ¯ of layer depth is almost negligible when hX10. As to the effect of plate flexibility, it is observed that an increase in its value tends to increase magnitudes of both real and imaginary parts of the dynamic compliance functions, although quite similar trends are found for different plate flexibilities considered in this paper. Numerical results also indicate that the corresponding resonant frequencies would shift to lower values as the soil porosity decrease, and the presence of the pore fluid reduces the vibration response of the plate. It is finally found that the plate displacement generally decreases along plate radius, and tends to increase as the plate become more flexible.
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Appendix
2a2 p2 cosh ch¯ x 2a1 p2 þ cosh d h¯ x a2 k 1 a1 k 2 ¯ þ cosh j h, x
j22 ¼
The elements of the matrix u are given by j11 ¼
j12 ¼
d2 ðp2 þ j 2 Þ þ 2d3 p2 cosh ch¯ ðd2 d1 Þðp2 j 2 Þ d1 ðp2 þ j 2 Þ 2d3 p2 cosh d h¯ þ ðd2 d1 Þðp2 j 2 Þ 2p2 ¯ cosh j h, þ 2 p j2 2a2 pc sinh ch¯ x 2a1 pd sinh d h¯ x pða2 k1 a1 k2 Þ ¯ sinh j h, jx 1 cosh ch¯ d2 d1 1 ¯ cosh d h, d1 d2
j13 ¼
j14 ¼
j15 ¼
a2 c sinh ch¯ x a1 d sinh d h¯ x p2 ða2 a1 Þ ¯ sinh j h, jx ðd3 d2 Þp cosh ch¯ ðj p2 Þðd1 d2 Þ ðd1 d3 Þp þ 2 cosh d h¯ ðj p2 Þðd1 d2 Þ p ¯ cosh j h, þ 2 ðp j 2 Þ
j23 ¼
j25 ¼
j26 ¼
d2 ðp2 þ j 2 Þ 2d3 p2 p sinh ch¯ cðd2 d1 Þðp2 j 2 Þ d1 ðp2 þ j 2 Þ þ 2d3 p2 p sinh d h¯ þ dðd2 d1 Þðp2 j 2 Þ 2pj ¯ sinh j h, 2 p j2
a2 p a1 p a2 a1 ¯ cosh ch¯ þ cosh d h¯ þ p cosh j h, x x x
ðd3 d2 Þp2 sinh ch¯ cðj 2 p2 Þðd2 d1 Þ ðd1 d3 Þp2 þ 2 sinh d h¯ dðj p2 Þðd2 d1 Þ j ¯ sinh j h, 2 p j2 ðk2 2p2 Þp cosh ch¯ x ðk1 2p2 Þp cosh d h¯ x pðk2 k1 Þ ¯ cosh j h, x d2 ðp2 þ j 2 Þ 2d3 p2 d1 cosh ch¯ ðd2 d1 Þðp2 j 2 Þ d1 ðp2 þ j 2 Þ 2d3 p2 þ d2 cosh d h¯ ðd2 d1 Þðp2 j 2 Þ 2d3 p2 ¯ þ 2 cosh j h, p j2
j31 ¼
j32 ¼
j16 ¼
j21 ¼
p p ¯ sinh ch¯ þ sinh d h, cðd2 d1 Þ dðd1 d2 Þ
j24 ¼
2
ðk2 2p2 Þc sinh ch¯ x ðk1 2p2 Þd þ sinh d h¯ x p2 ðk2 k1 Þ ¯ þ sinh j h, jx
645
2a2 pcd1 sinh ch¯ x 2a1 pdd2 sinh ph¯ x pða2 k1 a1 k2 Þd3 ¯ sinh j h, jx
j33 ¼
j34 ¼
d1 d2 ¯ cosh ch¯ cosh d h, d2 d1 d1 d2
a2 cd1 a1 dd2 p2 d3 ða2 a1 Þ ¯ sinh j h, sinh ch¯ sinh d h¯ jx x x d1 ðd3 d2 Þp cosh ch¯ ðj 2 p2 Þðd2 d1 Þ d2 ðd1 d3 Þp 2 cosh d h¯ ðj p2 Þðd2 d1 Þ pd3 ¯ þ 2 cosh j h, ðp j 2 Þ
j35 ¼
ARTICLE IN PRESS S.L. Chen et al. / Soil Dynamics and Earthquake Engineering 26 (2006) 637–647
646
ðk2 2p2 Þcd1 sinh ch¯ x ðk1 2p2 Þdd2 þ sinh d h¯ x ðk2 k1 Þp2 d3 ¯ þ sinh j h, jx
j36 ¼
d2 ðp2 þ j 2 Þ 2d3 p2 k1 sinh ch¯ cðd2 d1 Þðp2 j 2 Þ d1 ðp2 þ j 2 Þ 2d3 p2 þ k2 sinh d h¯ dðd2 d1 Þðp2 j 2 Þ 4p2 j ¯ þ 2 sinh j h, p j2
2a2 pc 2a1 pd sinh ch¯ þ sinh d h¯ x x ða2 a1 Þpðp2 þ j 2 Þ ¯ sinh j h, þ jx
j54 ¼
j55 ¼
j41 ¼
j42 ¼
j43
2a2 p 2a1 p k1 cosh ch¯ k2 cosh d h¯ x x 2p ¯ ða2 k1 a1 k2 Þ cosh j h, x
k1 k2 ¯ sinh ch¯ sinh d h, ¼ cðd2 d1 Þ dðd1 d2 Þ
j44 ¼
j56 ¼
ðk2 2p2 Þ 2pc sinh ch¯ x ðk1 2p2 Þ 2pd sinh d h¯ x ðk2 k1 Þpðp2 þ j 2 Þ ¯ sinh j h, jx d2 ðp2 þ j 2 Þ 2d3 p2 a1 sinh ch¯ cðd2 d1 Þðp2 j 2 Þ d1 ðp2 þ j 2 Þ 2d3 p2 ¯ þ a2 sinh d h, dðd2 d1 Þðp2 j 2 Þ
j61 ¼
a2 k 1 a1 k 2 a2 a1 2 ¯ cosh ch¯ cosh d h¯ 2p cosh j h, x x x ðd3 d2 Þpk1 sinh ch¯ cðj 2 p2 Þðd2 d1 Þ ðd1 d3 Þpk2 2 sinh d h¯ dðj p2 Þðd2 d1 Þ 2pj ¯ sinh j h, þ 2 ðp j 2 Þ
2p2 ðd3 d2 Þ cosh ch¯ ðj p2 Þðd2 d1 Þ 2p2 ðd1 d3 Þ þ 2 cosh d h¯ ðj p2 Þðd2 d1 Þ p2 þ j 2 ¯ cosh j h, 2 p j2 2
j62 ¼
2a2 pa1 2a1 pa2 ¯ cosh c¯z cosh d h, x x
j45 ¼
j46
ðk2 2p2 Þk1 ðk1 2p2 Þk2 cosh ch¯ þ cosh d h¯ ¼ x x ðk2 k1 Þ 2 ¯ þ 2p cosh j h, x
j51 ¼
j52
d2 ðp2 þ j 2 Þ 2d3 p2 2p cosh ch¯ ðd2 d1 Þðp2 j 2 Þ d1 ðp2 þ j 2 Þ 2d3 p2 2p cosh d h¯ ðd2 d1 Þðp2 j 2 Þ 2pðp2 þ j 2 Þ ¯ cosh j h, þ p2 j 2
4p2 ca2 4p2 da1 sinh ch¯ þ sinh d h¯ ¼ x x ðp2 þ j 2 Þ ¯ ða2 k1 a1 k2 Þsinh j h, þ jx
j53 ¼
2p 2p ¯ cosh ch¯ þ cosh d h, d 2 d1 d1 d2
j63 ¼ j64 ¼
a1 a2 ¯ sinh ch¯ sinh d h, cðd2 d1 Þ dðd1 d2 Þ
a1 a2 a 2 a1 ¯ cosh ch¯ cosh d h, x x a1 ðd3 d2 Þp sinh ch¯ cðj 2 p2 Þðd2 d1 Þ a2 ðd1 d3 Þp ¯ 2 sinh d h, dðj p2 Þðd2 d1 Þ
j65 ¼
a1 ðk2 2p2 Þ cosh ch¯ x a2 ðk1 2p2 Þ ¯ þ cosh d h, x
j66 ¼
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