Horizontal vibration of a cylindrical rigid foundation embedded in poroelastic half-space

Computers and Geotechnics 106 (2019) 296–303

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Research Paper

Horizontal vibration of a cylindrical rigid foundation embedded in poroelastic half-space

T

⁎

Changjie Zhenga, , Rui Heb, George Kouretzisc, Xuanming Dinga a

Key Laboratory of New Technology for Construction of Cities in Mountain Area, College of Civil Engineering, Chongqing University, Chongqing 400045, China College of Harbour, Coastal and Offshore Engineering, Hohai University, Nanjing 210098, China c ARC Centre of Excellence for Geotechnical Science and Engineering, Faculty of Engineering and Built Environment, The University of Newcastle, Callaghan, NSW 2308, Australia b

A R T I C LE I N FO

A B S T R A C T

Keywords: Dynamic soil-foundation interaction Horizontal vibration Poroelastic soil

This paper presents an analytical solution that describes the response of a cylindrical rigid foundation embedded in poroelastic soil to a horizontal harmonic load. The soil underlying the foundation base is modelled as homogeneous poroelastic half-space while the soil around the shaft of the cylindrical foundation is modelled as a series of infinitesimally thin poroelastic layers. Using the proposed solution we investigate how the dynamic interaction of the foundation with its surrounding poroelastic soil is affected by the geometry and mass of the foundation, as well as the properties of the soil surrounding the shaft and underlying the foundation base.

1. Introduction Considering the dynamic interaction between an embedded rigid foundation and soil is fundamental for the analysis of structures subjected to seismic loads and machine vibrations. Kuhlemeyer [15], Kaldjian [13], Lysmer and Kuhlemeyer [16], Urlich and Kuhlemeyer [23], among others, recognized the importance of dynamic interaction phenomena, and investigated the dynamic impedance of cylindrical foundations embedded in an elastic half-space. Later, Apsel and Luco [1] proposed an integral equation technique, which they employed to obtain the impedance functions of a massless foundation embedded in layered elastic half-space. The mentioned studies consider the halfspace as single-phase and linear elastic material. However, in many cases of practical interest, it is more appropriate to consider the halfspace as a two-phase medium, consisting of its solid skeleton and pores filled with fluid (poroelastic material). Biot [2] first established the governing equations to describe propagation of elastic waves in poroelastic materials, which form the basis of a number of studies on dynamic poroelastic soil-structure interaction. For example, Halpern and Christiano [9], Bougacha et al. [3], Philippacopoulos [20] and Dargush and Chopra [6] studied the vertical dynamic response of surface foundations on a poroelastic half space. Senjuntichai et al. [21] investigated the vertical vibration of a massless foundation embedded in a poroelastic half space, using an indirect boundary integral equation method. Cai et al. [4], Cai and Hu [5] and Hu et al. [10,11] proposed a novel approach to describe the vertical and rocking vibration of a rigid ⁎

massive foundation embedded in poroelastic soil, in which the poroelastic soil is divided into two independent parts: the soil underlying the foundation base is modelled as homogeneous poroelastic half-space or a homogeneous poroelastic layer resting on a rigid base; while the soil along the shaft of the foundation is assumed to consist of a series of infinitesimally thin layers, following the Baranov-Novak concept. Horizontal vibration of foundations has attracted much less attention compared with the vertical vibrations, owing mainly to the complexity of the problem. Some important contributions in this field were made by Gladwell [8], Kassir et al. [14] and Jin and Liu [12], who investigated the rocking and horizontal vibrations of surface foundations on elastic and poroelastic half-space, while Novak and Beredugo [18] presented approximate formulas to quantify the effects of vertical, horizontal and rocking vibrations of a rigid foundation embedded in elastic soil. To cover this gap in the literature, we present in this paper a simplified analytical solution to describe the horizontal vibration of a rigid cylindrical foundation embedded in poroelastic half-space, and subjected to harmonic loads. The solution is based on the model proposed by Cai and Hu [5] for the case of vertical harmonic loads, where the soil surrounding the shaft of the foundation and the soil below its base are considered independently to obtain the reaction acting on the foundation. More specifically, the soil reaction at the base of the foundation is considered equal to that of a foundation resting on the soil surface, while the reaction from the soil surrounding the shaft is obtained while using Novak’s plane strain model. The proposed model is verified against published solutions for the special case of single-phase

corresponding author. E-mail address: [email protected] (C. Zheng).

https://doi.org/10.1016/j.compgeo.2018.11.009 Received 21 May 2018; Received in revised form 15 November 2018; Accepted 15 November 2018 0266-352X/ © 2018 Elsevier Ltd. All rights reserved.

Computers and Geotechnics 106 (2019) 296–303

C. Zheng et al.

Feiω t

H

independent poroelastic thin layers

due to the relative motion between the solid matrix and pore fluid, and is equal to the ratio of the fluid viscosity over the permeability of the two-phase medium; a 0 = ωa ρ / G is the dimensionless frequency; ω is the circular frequency. For brevity, the time factor eia0 t is suppressed from all of the equations. The constitutive relations of the poroelastic material are:

rigid cylindrical foundation a

σij = λ∗δij e + ui, j + uj, i − αδij p

λs, Gs

p=

Rseiω t

O Rbeiω t

−

(3)

M ∗wi, i

(4)

where e = ui, i is the dilatation of the solid matrix; σij are the stress components of the bulk material; p is the pore pressure. Eqs. (1) and (2) can be reduced to the following forms [12]:

r

ui, jj + (λ∗ + 1) uj, ji + a02 (1 − ρ∗ϑ) ui − (α − ϑ) pi = 0

poroelastic half-space

λb, Gb

−αM ∗e

ρ∗a02

p, ii +

M ∗ (α − ϑ)

p+

ρ∗a02 (α

− ϑ)

ϑ

(5)

e=0

(6)

From Eq. (5) we obtain:

z

(λ∗

Fig. 1. Schematic of a foundation embedded in poroelastic soil.

+ 2) e, ii + a02 (1 − ρ∗ϑ) e − (α − ϑ) p, ii = 0

(7)

Eq. (2) can be expressed as:

e=−

soil, and accordingly is used to investigate parametrically the effect of the governing parameters of the problem on the dynamic response of the foundation.

ϑ 1 p − ∗ p M (α − ϑ) ρ∗a02 (α − ϑ) , ii

(8)

Substituting Eq. (10) into Eq. (7) yields:

p, iiii + β1 p, ii + β2 p = 0

2. Formulation of the governing equations of the problem

(9)

Eq. (8) can be expressed in cylindrical coordinates (r , θ , z ) as: Owing to complexity of the problem, several simplifying assumptions originally introduced by Cai and Hu [5] are also adopted in the following, with the main of these depicted in Fig. 1. A cylindrical rigid foundation with radius a, mass mh is considered embedded in a poroelastic half-space, at depth H . The soil underlying the base of the foundation is modelled as homogeneous poroelastic half-space with Lame constants λb and Gb . The soil along the vertical shaft of the foundation is considered as a series of infinitesimally thin independent layers with Lame constants λs and Gs . To facilitate the solution we ignore the gradient of the normal stress components in the vertical direction of these independent thin layers, and the contact surface between the foundation base and the poroelastic half-space is assumed to be impermeable or fully permeable. In addition, no slippage is allowed along the interface between the foundation shaft and its surrounding soil. The horizontal force Feiωt acting on the head of the foundation is harmonic, and we assume that the foundation translates only horizontally Uheiωt i.e. no rotation occurs. This horizontal force is transferred to soil via the base reaction Rbeiωt and the shaft reaction Rseiωt, which are considered uncoupled and are calculated independently. In order to formulate the governing equations, it is convenient to use dimensionless quantities with respect to length and stress by selecting the radius of the foundation, a as a unit of length and the shear modulus of the half-space as a unit of stress. Biot’s wave equations of the half-space can be written in terms of dimensionless variables in the frequency domain as:

e=−

ϑ 1 ∇2 p − ∗ p M (α − ϑ) ρ∗a02 (α − ϑ)

(10)

Then Biot’s wave equations in cylindrical coordinates can be obtained as [12]:

∇2 ur −

∂p 1 ⎛ ∂uθ u ∂e 2 =0 + r ⎞ + (λ∗ + 1) + a02 (1 − ρ∗ϑ) ur − (α − ϑ) ∂r r ⎝ r ∂θ r ⎠ ∂r (11)

∇2 uθ −

∂p 1 ⎛ uθ ∂u ∂e − 2 r ⎞ + (λ∗ + 1) + a02 (1 − ρ∗ϑ) uθ − (α − ϑ) r ∂θ r⎝r r ∂θ ⎠ r ∂θ (12)

=0 ∇2 uz + (λ∗ + 1)

∂p ∂e =0 + a02 (1 − ρ∗ϑ) uz − (α − ϑ) ∂z ∂z

(13)

∇4 p + β1 ∇2 p + β2 p = 0 ∇2 =

where

ϑ=

ρ∗a02 m∗a02 − ib∗a0

∂2 ∂r 2

;

(m∗a02 − iba0) a02 − (ρ∗a02)2 (λ∗ + 2) M ∗

+

(14)

1 ∂ 1 ∂2 ∂2 ∂u u 1 ∂u + 2 2 + 2; e = ∂rr + rr + r ∂θθ r ∂r r ∂θ ∂z (m∗a02 − iba0)(λ∗ + 2 + α∗M ∗) + M ∗a02 − 2αM ∗ρ∗a02

β1 =

(λ∗ + 2) M ∗

;

+

∂uz ; ∂z

β2 =

.

3. Analytical solution 3.1. General solution for the reaction at the base of the foundation

ui, jj + (λ∗ + α 2M ∗ + 1) uj, ji + αM ∗wj, ji + a02 ui + ρ∗a02 wi = 0

(1)

αM ∗uj, ji + M ∗wj, ji + ρ∗a02 ui + (m∗a02 − ib∗a 0) wi = 0

(2)

Following Cai and Hu [5], the reaction at the foundation’s base fb can be calculated by employing the solution for a surface disk resting on the top of a poroelastic half-space. The rigid foundation is assumed to be perfectly bonded to the surrounding soil (no slippage condition), and the interface between the half-space and overlying soil is assumed to be impermeable (∂p / ∂z = 0 ) or fully permeable ( p = 0 ). These assumptions lead to the following boundary conditions at z = 0 :

in which ui and wi (i = 1, 2, 3) are the dimensionless displacements of the solid phase and of the fluid phase, respectively; α , M ∗ = M / G , λ∗ = λ / G , ρ∗ = ρf / ρ , m∗ = m / ρ and b∗ = ab/ ρG are the dimensionless material parameters. α and M are the Biot’s parameters accounting for the compressibility of the two-phase material; ρ and ρf are the mass densities of the bulk material and of the pore fluid, respectively; m is a mass parameter equal to the ratio of the mass density of the fluid ρf over the porosity; b is a parameter accounting for the internal friction

ur (r , θ , 0) = U0 cos θ uθ (r , θ , 0) = −U0 sin θ 297

r⩽1 r⩽1

(15) (16)

Computers and Geotechnics 106 (2019) 296–303

C. Zheng et al.

(17)

order Hankel transform of σz∗ and obtain

(18)

X¯1 (ξ , z ) = 2ξγ1 a1 Ae−γ1 z + 2ξγ2 a2 B e−γ2 z + (Cξ − γ3 D)e−γ3 z

where U0 = Uh/ a is the dimensionless horizontal displacement of the foundation. According to the above conditions, it is convenient to express displacements and stresses as follows:

γ γ Y¯1 (ξ , z ) = −2 1 (χ1 + γ12 a1) Ae−γ1 z − 2 2 (χ2 + γ22 a2) B e−γ2 z ξ ξ

τrz (r , θ , 0) = τθz (r , θ , 0) = 0 σz (r , θ , 0) = 0,

r>1

∂p (r , θ , 0)/ ∂z = 0 or p (r , θ , 0) = 0

r⩾0

∗ ⎧ ur (r , θ , z ) ⎫ ⎧ur (r , z ) ⎫ ∗ ⎪ ⎪ uz (r , θ , z ) ⎪ uz (r , z ) ⎪ ⎪ ⎪ ⎪ ∗ ⎪ ⎪ τrz (r , θ , z ) ⎪ ⎪ τrz (r , z ) ⎪ cos(θ) = ⎨ σz (r , θ , z ) ⎬ ⎨ σz∗ (r , z ) ⎬ ⎪ ⎪ ∗ ⎪ ⎪ ⎪ e (r , θ , z ) ⎪ ⎪ e (r , z ) ⎪ ⎪ p (r , θ , z ) ⎪ ⎪ p∗ (r , z ) ⎪ ⎭ ⎩ ⎩ ⎭

⎧ uθ (r , θ , z ) ⎫ = ⎨ ⎩ τθz (r , θ , z ) ⎬ ⎭

∗ ⎧ uθ (r , ∗ τ ⎨ ⎩ θz (r ,

2 − ⎡ ⎜⎛ξ + γ32 ⎟⎞ C + γ3 D⎤ e−γ3 z ⎢ ⎥ ξ ⎠ ⎝ ⎣ ⎦

(19)

z) ⎫ sin(θ) z)⎬ ⎭

(20)

∂p∗ 1 1 ∇22 U + (λb∗ + 1) ⎛ − e∗⎞ + a02 (1 − ρ∗ϑ) U − (α − ϑ) ⎛ − p∗ ⎞ r ⎠ ∂ r r ⎝ ⎠ ⎝ ∂r (21) =0

∇14 p∗

+

+ β2

p∗

λb∗

f¯ (ξ , z ) =

∫0

(24)

ur∗

uθ∗; ∇n2

∂2 = ∂r 2

1 ∂ r ∂r

rf (r , z ) Jn (ξr ) dr

z) =

Ae−γ1 z

+

B e−γ2 z

n2 r2

3ξ 2a1 ⎞ −γ z 3ξ 2a2 ⎞ −γ z 2 2 U¯ (ξ , z ) = ⎛g1 − Ae 1 + ⎛g2 − Be 2 ξ⎝ 2 ⎠ ξ⎝ 2 ⎠ 2 + ⎜⎛ Cγ3 + D⎟⎞ e−γ3 z ⎝ξ ⎠ where

L22 =

γi =

⎟

ξ 2 − Li2 (i = 1, 2 );

β1 − β12 − 4β2 2

⎜

γ3 =

; S 2 = (1 − ρ∗ϑ) a02 ; χi =

r>1

(37)

Y¯ (ξ , 0) J2 (ξr ) dξ = 0

r>1

(38)

Y¯ (ξ , z ) = ξY¯1 (ξ )/2

(39)

+ 4γ1 γ2 γ3 (a1 γ12 − a2 γ22 + χ1 − χ2 ) + ξ 2 (γ2 g3 − γ1 g4 )]

ϑM ∗Li2 − ρ∗a02

; ρ∗a02 (α − ϑ) M ∗

(42)

+ (2γ32 + ξ 2)(γ2 g3 − γ1 g4 ) + 4γ1 γ2 γ3 ξ 2 (a1 − a2)] ξ γ3 (γ32 + ξ 2)(γ1 g4 − γ2 g3) + 2γ1 γ2 γ32 (g2 − g1)]

κ=

(43)

(44)

while for fully permeable interface between the half-space and overlying soil

Q11 (ξ ) = κ [2ξ 2γ32 (a1 − a2) + 4ξ 2γ3 (γ2 a2 − γ1 a1) + 4γ3 (γ1 g1 − γ2 g2) (45)

(28)

Q12 (ξ ) =

(46)

Q21 (ξ ) =

κ [4γ32 (g2

− a1) + 4γ3 (γ1 a1 − γ2 a2) + g3 − g4]

− g1) +

6ξ 2γ32 (a1

+ 4γ3 (γ1 g1 − γ2 g2) +

ξ 2 (g3

− a2) + 4ξ 2γ3 (γ2 a2 − γ1 a1) − g4 )]

(47)

Q22 (ξ ) = κ [4γ32 (g1 − g2) + 6γ32 ξ 2 (a2 − a1) + (2γ32 + ξ 2)(g3 − g4 ) + 4γ3 ξ 2 (γ1 a1 − γ2 a2)] (31)

L12 =

ai =

(41)

Q22 (ξ ) = κ [4γ32 (γ2 g1 − γ1 g2) + 6γ32 ξ 2 (γ1 a2 − γ2 a1)

⎟

ξ 2 − S2 ;

(40)

κξ 2 [2γ32 (a2

(30)

⎜

X¯ (ξ , 0) J0 (ξr ) dξ = 0

+ (2γ32 + ξ 2)(g3 − g4 )]

V¯ (ξ , z ) = −ξa1 Ae−γ1 z − ξa2 B e−γ2 z + De−γ3 z

+

(36)

(27)

(29)

+ γ2 a2

B e−γ2 z

∫0

∞

(35)

Q21 (ξ ) = κ [4γ32 (γ1 g2 − γ2 g1) + 6ξ 2γ32 (γ2 a1 − γ1 a2)

∂2 , ∂z 2

C e−γ3 z

z ) = γ1 a1

Ae−γ1 z

∞

(26)

e¯∗ (ξ , z ) = χ1 Ae−γ1 z + χ2 B e−γ2 z u¯ z∗ (ξ ,

∫0

r⩽1

Q12 (ξ ) = κξ 2 [2γ32 (γ1 a2 − γ2 a1) + 4γ1 γ2 γ3 (a1 − a2) + γ2 g3 − γ1 g4]

where Jn (ξr ) is the Bessel function of the first kind of the order n and ξ is the Hankel transform parameter. The general solutions for first-order Hankel transform of p∗, e∗ and uz∗ as well as second-order Hankel transform of U and zero-order Hankel transform of V can be obtained respectively:

p¯∗ (ξ ,

1 [Q21 (ξ ) X¯ (ξ , 0) + Q22 (ξ ) Y¯ (ξ , 0)] J2 (ξr ) dξ = 0 ξ

Q11 (ξ ) = κ [2ξ 2γ32 (γ2 a1 − γ1 a2) + 4γ1 γ2 γ3 (a1 γ12 − aγ22 + χ1 − χ2 )

+ − + = λb / Gb ; U = + = − where n = 0, 1, 2 . The nth-order Hankel integral transform of a function f (r , z ) with respect to r is defined as [22] ∞

∫0

∞

r⩽1

and for impermeable interface between the half-space and overlying soil

(25)

uθ∗ ; V

1 [Q11 (ξ ) X¯ (ξ , 0) + Q12 (ξ ) Y¯ (ξ , 0)] J0 (ξr ) dξ = 2U0 ξ

+ (2γ32 + ξ 2)(γ2 g3 − γ1 g4 )]

=0 ur∗

∞

X¯ (ξ , z ) = ξX¯1 (ξ )/2,

(23)

ϑ 1 ∇12 p∗ − ∗ p∗ M (α − ϑ) ρ∗a02 (α − ϑ) β1 ∇12 p∗

∫0

where

∂p∗ 1 1 ∂e∗ ∇20 V + (λb∗ + 1) ⎛ + e∗⎞ + a02 (1 − ρ∗ϑ) V − (α − ϑ) ⎛ + p∗ ⎞ r ⎠ r ⎠ ⎝ ∂r ⎝ ∂r (22) =0 ∂p∗ ∂e∗ =0 + a02 (1 − ρ∗ϑ) uz∗ − (α − ϑ) ∂z ∂z

(34)

where g3 = λb∗ χ1 − 2γ12 a1 − α ; g4 = λb∗ χ2 − 2γ22 a2 − α . Applying Hankel transforms on the above boundary conditions, we can express the mixed boundary-value problem with the following integral equations

∂e∗

e∗ = −

(33)

σ¯ z∗ (ξ , z ) = g3 Ae−γ1 z + g4 B e−γ2 z − 2γ3 C e−γ3 z

Substituting Eqs. (19) and (20) into Eqs. (11)–(14) yields after some manipulations:

∇12 uz + (λb∗ + 1)

(32)

β1 + β12 − 4β2

2 λb∗ χi + χi − α + ϑ

S2 − Li2

ξ γ3 [(γ32 + ξ 2)(g4 − g3) − 2γ3 (γ1 g1 − γ2 g2)]

κ=

;

(48)

(49)

Setting

;

lim Q11 (ξ ) = l1,

ξ →∞

gi = χi + (γi2 + ξ 2) ai . ∗ ∗ and Y1 = τrz∗ + τθz , we now apply the zero-order Letting X1 = τrz∗ − τθz Hankel transform of X1, second-order Hankel transform of Y1 and first-

lim Q12 (ξ ) = l2,

ξ →∞

lim Q21 (ξ ) = l3,

ξ →∞

lim Q22 (ξ ) = l4

ξ →∞

(50) Using Sonine’s integrals [17], The dual integral equations can be 298

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C. Zheng et al.

where E3 and E4 are undetermined coefficients. Taking into account of the definition of the solid dilatation, we obtain

reduced to a pair of Fredholm integral equations of the second kind [19,12]:

l2 ⎡ l1 ⎣

∫r

Φ1 (r ) + l + 2 l1

∫0

1

1 Φ2 (t ) dt − Φ2 (r )⎤ + t ⎦

1

∫0

1

K11 (r , t )Φ1 (t ) dt

K12 (r , t )Φ2 (t ) dt = 1

Thus

(51)

1 l4 1 Φ2 (r ) + Φ1 (t ) dt − Φ2 (r ) + l3 r r 1 l + 4 K22 (r , t )Φ2 (t ) dt = 0 l3 0

∫

∫0

1

H (2) (L1 r ) − H0(2) (L1 r ) H (2) (L2 r ) − H0(2) (L2 r ) ur = ⎡L1 a1 E1 2 + L2 a2 E2 2 ⎢ 2 2 ⎣

K21 (r , t )Φ1 (t ) dt

∫

K11 (r , t ) =

rt

K12 (r , t ) =

∫0

∫0

rt

K21 (r , t ) =

rt

K22 (r , t ) =

∫0

∞

∫0

rt

∞

∞

ξ[

(52)

ξ[

Q11 − 1] J−1/2 (ξr ) J−1/2 (ξt ) dξ l1

Q12 − 1] J−1/2 (ξr ) J 3(ξt ) dξ l2

(53)

2π

∫0 ∫0

1

(54)

(55)

Q ξ [ 22 − 1] J3/2 (ξr ) J3/2 (ξt ) dξ l4

(56)

[τrz (r , θ , 0) cos θ − τθz (r , θ , 0) sin θ] rdθdr

(69)

p (1, θ) = 0

(70)

Substituting Eqs. (62), (67) and (68) into Eqs. (69) and (70) yields

E2 = δ2 E1, E3 = δ3 E1

1 [Y1 (r , 0) − X1 (r , 0)] sin θ 2

8Gb a (f + if2 ) Uh 2−ν 1

(58)

δ3 = −

(59)

H1(2) (L1) H1(2) (L2) L1 a1 H2(2) (L1) + L2 a2 δ2 H2(2) (L2) H2(2) (S )

1

0

1

(60)

f2 = Im

1

(∫ Φ (t) dt) 0

1

(61)

l

Rs = ∫0 ∫0

p (r , θ) =

(L1 r ) +

(L2 r )] cos θ

e (r , θ) =

(L1 r ) +

χ2 E2 H1(2)

(L2 r )] cos θ

(75)

T=

Rs = S1 + iS2 Gb aUh

T = 2πGs∗ l

[(λ∗ + 2) χ1 − α ] H1(2) (L1) + δ2 [(λ∗ + 2) χ2 − α ] H1(2) (L2) − δ3 SH1(2) (S ) L1 a1 [H2(2) (L1) − H0(2) (L1)] + L2 a2 δ2 [H2(2) (L2) − H0(2) (L2)]

Substituting Eqs. (62) and (63) into Eqs. (21) and (22) yields

V (r ) = −L1 a1 E1 H0(2) (L1 r ) − L2 a2 E2 H0(2) (L2 r ) + E4 H0(2) (Sr )

(65)

(76)

in which S1 and S2 are the real and imaginary parts of the shaft reaction factor, and

(63)

(64)

[σr (a, θ) cos θ − τrθ (a, θ) sin θ] adzdθ

where l = H / a is the slenderness ratio of the foundation. Accordingly, we define the shaft reaction factor as

(62)

U (r ) = L1 a1 E1 H2(2) (L1 r ) + L2 a2 E2 H2(2) (L2 r ) + E3 H2(2) (Sr )

2π

− δ3 SH1(2) (S )}

where E1 and E2 are undetermined coefficients; H1(2) ( ) is the second kind of Hankel function. Substituting Eq. (62) into Eq. (24) results in the following expression for the solid dilatation:

[χ1 E1 H1(2)

(74)

= πaGs lE1 {[(λ∗ + 2) χ1 − α ] H1(2) (L1) + δ2 [(λ∗ + 2) χ2 − α ] H1(2) (L2)

As discussed earlier, we ignore the stress gradient in the vertical direction for each infinitesimally thin poroelastic layer, therefore stresses σz , τrz and τθz of the soil layers surrounding the foundation are assumed zero. Then the Laplace operator and solid dilatation in Eqs. (11), (12), (10) and (14) for the surrounding soil become ∂u u 1 ∂u ∂2 1 ∂ 1 ∂2 ∇2 = 2 + r ∂r + 2 2 ; e = ∂rr + rr + r ∂θθ . ∂r r ∂θ The solution for the pore pressure can be obtained as

E2 H1(2)

H2(2) (Sr ) + H0(2) (Sr ) ⎤ cos θ ⎥ 2 ⎦

Therefore, the total dynamic reaction along the shaft of the foundation, Rs , can be calculated as

3.2. General solution for the shaft reaction

[E1 H1(2)

(73)

H (2) (L1 r ) − H0(2) (L1 r ) H (2) (L2 r ) − H0(2) (L2 r ) ur = E1 ⎡L1 a1 2 + L2 a2 δ2 2 ⎢ 2 2 ⎣ + δ3

(∫ Φ (t ) dt),

(72)

The radial displacement of the soil at its interface with the foundation is

where

f1 = Re

(71)

where (57)

Substituting Eqs. (58) and (59) into Eq. (57) yields:

Rb =

ur (1, θ) = U0 cos θ , uθ (1, θ) = −U0 sin θ

δ2 = −

τθz (r , θ , 0) =

(68)

According to the assumptions mentioned earlier, the boundary conditions at the shaft are:

where

1 τrz (r , θ , 0) = [X1 (r , 0) + Y1 (r , 0)] cos θ 2

H2(2) (Sr ) − H0(2) (Sr ) ⎤ sin θ ⎥ 2 ⎦

+ E3

Q ξ [ 21 − 1] J3/2 (ξr ) J−1/2 (ξt ) dξ l3

(67)

H (2) (L1 r ) + H0(2) (L1 r ) H (2) (L2 r ) + H0(2) (L2 r ) uθ = ⎡L1 a1 E1 2 + L2 a2 E2 2 ⎢ 2 2 ⎣

The force-displacement relationship for the foundation base can be obtained from the resultant contact stress:

Rb = Gb a2

H2(2) (Sr ) + H0(2) (Sr ) ⎤ cos θ ⎥ 2 ⎦

+ E3

where ∞

(66)

E4 = E3

+ δ3 [H2(2) (S ) + H0(2) (S )] (77) where 299

Gs∗

= Gs / Gb is the non-dimensional shear modulus of the soil

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C. Zheng et al.

3.0

3.3. Dynamic response of the foundation

2.5

The governing equation of the rigid foundation can be now established using Newton’s second law

2.0

ρa2Bh ¨ 2−ν Uh + (k + ic ) Uh = F Gb 8Ga

1.5

Kh

along the shaft of the foundation.

(78)

k 0.5

(79)

0.0

and k, c are the real and imaginary components of the impedance

2−ν k = f1 + S1 8 c = f2 +

2−ν S2 8

(81)

(82)

3.5

The amplitude of the horizontal displacement of the rigid foundation is denoted asUh max

3.0

(83)

where Mh is the horizontal dynamic response factor, which is used to quantify dynamic interaction effects on the horizontal vibration of the foundation, and is defined as:

Mh =

3

4

2.0 1.5

Present solution l=1 Present solution l=2 Apsel and Luco (1987) l=1 Apsel and Luco (1987) l=2

1.0

1 (k − Bh a02)2 + c 2

2

a0

2.5

k

F 2−ν Mh Gb a 8

1

Fig. 2. Comparison of the horizontal dynamic impedance obtained with the proposed solution for the special case of single-phase soil against the solution of Gazetas [7] for a surface foundation.

4.0

Uh max =

0

(80)

The dimensionless equivalent horizontal dynamic impedance is defined as Kh

Kh = k + ic

c

1.0

where Bh is the mass ratio of the foundation

2 − ν mh Bh = 8 ρa3

Present solution Gazetas (1983)

0.5

(84)

0.0

0

1

2

4. Verification and discussion

3

a0

4

5

6

4

5

6

30

The solution derived above can be rather straightforwardly programmed, and employed to quickly calculate the horizontal response factor of a rigid foundation embedded in poroelastic soil, for varying problem parameters. Following the verification of the derived solution against the solutions of Gazetas [7] and Apsel and Luco [1] for singlephase soil, we study parametrically the influence of the slenderness ratio; the mass ratio of the foundation; the shear modulus of the backfill; the soil porosity; and the base permeability boundary conditions on the dynamic response of the foundation. Unless otherwise specified, impermeable boundary condition at the foundation base is adopted in the numerical results, and the following dimensionless parameters are considered: α = 0.97 , M ∗ = 10 , λb∗ = 1.5, b∗ = 0.1, ρ∗ = 0.53, m∗ = 1.1, Bh = 4 , l = 0.5, Gs∗ = 1.

Present solution l=1 Present solution l=2 Apsel and Luco (1987) l=1 Apsel and Luco (1987) l=2

25

c

20 15 10 5 0

4.1. Comparison with existing solutions for the special case of single-phase soil

0

1

2

3

a0

Fig. 3. Comparison of the components of the horizontal dynamic impedance obtained with the proposed solution for the special case of single-phase soil against the boundary integral solution of Apsel and Luco [1] for an embedded foundation.

To verify the accuracy of the present solution, first we obtain the reduced solution for a surface foundation on an elastic single-phase half-space by setting l = 0 , α = b∗ = M ∗ = ρ∗ = 10−5 . Results plotted in Fig. 2 suggest that the horizontal dynamic impedance calculated with the present solution for the special case of single-phase soil matches the solution of Gazetas [7] for a surface foundation, across a wide range of dimensionless frequency α0 values. Fig. 3 presents a comparison of the components of the horizontal dynamic impedance obtained with the reduced solution, against the solution derived by Apsel and Luco [1] for an embedded foundation. Two values of the foundation’s slenderness ratio (l = 1, 2) are considered. The imaginary component of the horizontal dynamic impedance (damping) matches Apsel and Luco’s [1]

solution. However, some discrepancies are observed in values of the real component (stiffness) in the low frequency range (a0 < 1), for both l values. Similar observations are also reported by Cai and Hu [5], and are attributed to ignoring the vertical stress gradient in the vertical direction in the present solution, which results in underestimating stiffness in the very low frequency range.

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2.0

1.2

l=0.25 l=0.5 l=1 l=2

1.8 1.6 1.4

b=0.01 b=0.1 b=1 b=10 b=100

1.0 0.8

1.0

Mh

Mh

1.2

0.8

0.6 0.4

0.6 0.4

0.2

0.2 0.0 0.0

0.2

0.4

0.6

0.8

1.0

a0

1.2

1.4

1.6

1.8

0.0 0.0

2.0

0.2

0.4

0.6

0.8

1.0

a0

1.2

1.4

1.6

1.8

2.0

Fig. 6. Effect of porosity parameter b on the horizontal dynamic response factor.

Fig. 4. Effect of slenderness ratio on the horizontal dynamic response factor.

4.2. Parametric evaluation of the horizontal dynamic response factor faster, and the response is predominantly drained. The present solution allows considering different soil properties around the shaft of the foundation (e.g. backfill material) compared to the natural subsoil, via assigning different moduli. Taking advantage of that, in Fig. 7 we present the effect of the shear modulus of the backfill soil on the response factor, for different α0 values. As expected, as the stiffness of the material around the shaft increases (increasing Gs∗ values) the horizontal reaction provided by the soil along the shaft of the foundation increases, resulting in reduced response factor amplitudes. However, this effect becomes trivial for α0 values above the resonance frequency. In order to shed some light on the mechanisms of interaction between the foundation and its surrounding soil, we present in Fig. 8 the contribution of shaft stiffness and damping to the real and imaginary components of total impedance. The plots correspond to resonance frequency a0 = 0.54, and the contribution of the shaft is depicted as function of the slenderness ratio of the foundation. As expected, the contribution of shaft resistance tends to zero as l approaches zero, and will asymptotically tend to 1.0 as l approaches infinity. Notice also that the contribution of the shaft to damping is considerably higher than its contribution to total stiffness. This suggests that radiation damping mainly takes place in the soil surrounding the shaft, however the contribution of the poroelastic base soil on stiffness is considerable, and will dominate the response for shallow foundations. Next we attempt to quantify the effect of considering soil as twophase medium in the analysis of the horizontal dynamic response of the foundation, which can be captured explicitly with the present solution.

Fig. 4 depicts the variation of the horizontal dynamic response factor Mh , with the dimensionless frequency a0, for different values of the slenderness ratio l. It is reminded here that Mh provides a measure of the amplification of foundation displacements due to poroelastic soilstructure interaction effects. The amplification due to resonance increases as the slenderness ratio decreases, while for l ≥ 2 (i.e. when the embedment depth becomes equal or greater to the diameter of the foundation) resonance effects become trivial. However, notice that in the high-frequency range (a0 > 1) the response factor is not particularly sensitive to l. Fig. 5 presents the variation of the horizontal dynamic response factor Mh with a0, for different values of mass ratio of the foundation Bh . Observe that the mass of the foundation has significant effect on both the resonant frequency, and the amplitude of the horizontal dynamic response factor at the resonance frequency. Interestingly, at low frequencies (less that the resonance frequency), the dynamic response factor increases as the mass ratio increases, yet the opposite trend is observed at high frequencies. Fig. 6 depicts the variation of the horizontal dynamic response factor Mh with a0, for different values of the porosity parameter b, which accounts for soil permeability and pore fluid viscosity. Generally, coarse-grained materials such as sands feature low values of b, while higher b values are associated with low-permeability clays. Results presented in Fig. 6 suggest that dynamic interaction effects will be more prominent for foundations in sand, where higher response factor values are observed. Notice that as b decreases to b = 0.01 and 0.1 its effect on the dynamic response becomes trivial, as excess pore pressures dissipate

2.0

1.6

1.2 1.0

Mh

Gs =0.5

Bh=1

1.6

Gs =0.75

Bh=2

1.4

Gs =1

Bh=4

1.2

Bh=8

1.0

M

1.4

0.8

* *

0.8

0.6

0.6

0.4

0.4 0.2

0.2 0.0 0.0

*

1.8

Bh=0.5

0.0 0.0

0.2

0.4

0.6

0.8

1.0

a0

1.2

1.4

1.6

1.8

2.0

0.2

0.4

0.6

0.8

1.0

a0

1.2

1.4

1.6

1.8

2.0

Fig. 7. Effect of shear modulus of the soil material around the shaft on the horizontal dynamic response factor.

Fig. 5. Effect of mass ratio on the horizontal dynamic response factor. 301

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(a)

Ashaft/Abase 0.4

0.8

1.2

1.6

2.0

2.4

2.8

3.2

3.6

2.0

4.0

*

0.8

Gs =0.5

1.6

*

Gs =0.75

l=0.25

1.4

*

0.6

poroelastic (b=0.01) elastic

1.8

Gs =1

1.2

Mh

shaft stiffness/total stiffness

0.0 1.0

0.4

1.0

l=0.5

0.8 0.6

0.2

l=1

0.4 0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.0

2.0

l

Ashaft/Abase

0.8 0.6

0.4

0.8

1.2

1.6

2.0

2.4

2.8

3.2

3.6

0.4

0.6

0.8

1.0 a0

1.2

2.0

4.0

*

1.6

*

1.4

*

1.2

Gs =0.5 Gs =0.75 Gs =1

0.4

1.4

1.6

1.8

2.0

poroelastic (b=100) elastic

1.8

Mh

shaft damping/total damping

0.0 1.0

0.2

(b)

l=0.25

1.0 l=0.5

0.8 0.6

l=1

0.4

0.2

0.2 0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.0

2.0

0.2

0.4

0.6

0.8

l

1.0

a0

1.2

1.4

1.6

1.8

2.0

Fig. 9. Comparison of horizontal dynamic response factor values for elastic soil against high (a) and low (b) permeability poroelastic soil.

Fig. 8. Variation of the contribution of the soil surrounding the shaft to stiffness and damping with the slenderness ratio, for different values of its shear modulus (results for a0 = 0.54).

1.2 0.1

In Fig. 9(a) and (b) we compare the variation of the horizontal dynamic response factor Mh with a0 for elastic and for poroelastic soil. Results cover different foundation slenderness ratios (l = 0(25, 0.5, 1.0) and porosity parameters (b = 0(01, 100). Notice first that considering soil as two-phase medium has an effect on the amplitude of the horizontal dynamic response factor, but not on the resonance frequency. This effect is not straightforward. Not considering stress-flow coupling in freedraining soils (b = 0(01), results in underestimating the response factor and thus foundation displacements at resonance, as shown in Fig. 9(a). On the contrary, for low-permeability soils (b = 100), not considering stress-flow coupling will result in overestimating the amplitude of the response factor at the resonant frequency. These differences become more prominent as the slenderness of the foundation decreases, therefore stress-flow coupling must be considered for the dynamic analysis of massive shallow foundations in saturated soil. Finally, we investigate the effect of the hydraulic boundary conditions at the foundation base by plotting in Fig. 10 the variation of the horizontal dynamic response factor with the dimensionless frequency for different porosity parameter values and impermeable/full permeable boundary conditions between the half-space and overlying soil. Notice that the assumed hydraulic boundary conditions have a trivial effect on the response, at least for high-permeability soils. However the dynamic response factor attains consistently lower values when the contact between the half-space and overlying soil is assumed impermeable, a trend which is more obvious in the case of low-permeability soil (b = 10).

b=

1.0

Impermeable Permeable

1 10

Mh

0.8 0.6 0.4 0.2 0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

a0 Fig. 10. Comparison of horizontal dynamic response factor values for impermeable and permeable hydraulic boundary conditions at the foundation base.

5. Concluding remarks We presented a solution for the analysis of embedded shallow foundations subjected to dynamic horizontal loads. Use of Biot’s theory for stress wave propagation in poroelastic media allows us to consider the effects of stress-flow coupling on the horizontal displacements of the foundation. The obtained solution allows shedding some light on the mechanisms governing the dynamic interaction of the foundation with its surrounding two-phase medium. We have thus shown that 302

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filled poroelastic stratum. J Eng Mech 1993;119(8):1649–62. [4] Cai YQ, Hu XQ, Xu CJ, et al. Vertical dynamic response of a rigid foundation embedded in a poroelastic soil layer. Int J Numer Anal Meth Geomech 2009;33(11):1363–88. [5] Cai YQ, Hu XQ. Vertical vibrations of a rigid foundation embedded in a poroelastic half space. J Eng Mech ASCE 2010;136(3):390–8. [6] Dargush GF, Chopra MB. Dynamic analysis of axisymmetric foundations on poroelastic media. J Eng Mech 1996;122(7):623–32. [7] Gazetas G. Analysis of machine foundation vibrations: state of the art. Int J Soil Dyn Earthquake Eng 1983;2(1):2–42. [8] Gladwell GML. Forced tangential and rotatory vibration of a rigid circular disc on a semi-infinite solid. Int J Eng Sci 1968;6(10):591–607. [9] Halpern MR, Christiano P. Steady-state harmonic response of a rigid plate bearing on a liquid-saturated poroelastic halfspace. Earthquake Eng Struct Dyn 1986;14(3):439–54. [10] Hu XQ, Cai YQ, Ding GY, Wang J. Rocking vibrations of a rigid embedded foundation in a poroelastic half-space. Int J Numer Anal Meth Geomech 2010;34(13):1409–30. [11] Hu XQ, Cai YQ, Wang J, Ding GY. Rocking vibrations of a rigid embedded foundation in a poroelastic soil layer. Soil Dyn Earthquake Eng 2010;30(4):280–4. [12] Jin B, Liu H. Horizontal vibrations of a disk on a poroelastic half-space. Soil Dyn Earthquake Eng 2000;19(4):269–75. [13] Kaldjian MJ. Trosional stiffness of embedded footings. J Soil Mech Foundat Div ASCE 1969;95(7):969–80. [14] Kassir MK, Xu J, Bandyopadyay KK. Rotatory and horizontal vibrations of a circular surface footing on a saturated elastic half-space. Int J Solids Struct 1996;33(2):265–81. [15] Kuhlemeyer RL. Vertical vibrations of footings embedded in layered media PhD. dissertation Berkeley, Calif.: Univ. of California; 1969. [16] Lysmer John, Kuhlemeyer Roger L. Finite dynamic model for infinite media. J Eng Mech Div 1969;95(4):859–78. [17] Noble B. The solution of Bessel-function dual integral equations by a multiplyingfactor method. Math Proc Cambridge Philos Soc 1963;59:351–62. [18] Novak M, Beredugo YO. The effect of embedment on footing vibrations. In: Proceedings of the First Canadian Conference on Earthquake Engineering Research, Vancouver, BC; 1971. [19] Pak RYS, Saphores JDM. Lateral translation of a rigid disc in a semi-infinite solid. Quart J Mech Appl Math 1992;45(3):435–49. [20] Philippacopoulos AJ. Axisymmetric vibration of disk resting on saturated layered half-space. J Eng Mech 1989;115(10):2301–22. [21] Senjuntichai T, Mani S, Rajapakse R. Vertical vibration of an embedded rigid foundation in a poroelastic soil. Soil Dyn Earthquake Eng 2006;26(6–7):626–36. [22] Sneddon I. The use of integral transforms. New York: McGraw-Hill; 1970. [23] Urlich CM, Kuhlemeyer RL. Coupled rocking and lateral vibrations of embedded footings. Can Geotech J 1973;10(2):145–60.

interaction effects are more prominent for heavy, shallow foundations on saturated sand, while resonance phenomena are rather trivial for light, slender foundations embedded in low-permeability clays. We have also proven that simpler total stress analysis solutions considering foundation soil as single-phase medium may result in under-estimating or over-estimating horizontal foundation displacements, depending on the permeability of soil. The presented algorithm can be rather straightforwardly programmed and used to estimate the real and imaginary components of impendence, required as input for the dynamic analysis of structures subjected to horizontal wind, wave and seismic loads; when their amplitude is sufficiently low to justify ignoring effects of soil non-linearity and slippage at the soil-foundation interface. We should reiterate here that ignoring the stress gradient along the vertical direction in the soil surrounding the shaft of the foundation results in underestimating stiffness in the low frequency range, a fact that was confirmed while comparing our results against the solution of Apsel and Luco [1] for single-phase soil. Finally, ignoring the rotation of the foundation is perhaps not realistic for slender foundations, and in such cases coupling of horizontal and rocking vibrations needs to be considered in the analysis model. Acknowledgements This work was supported by the National Natural Science Foundation of China (No. 51708064) and Fundamental Research Funds for the Central Universities (No. 106112017CDJXY200002 and No. 106112016CDJXZ208821). References [1] Apsel RJ, Luco JE. Impedance functions for foundations embedded in a layered medium: an integral equation approach. Earthquake Eng Struct Dyn 1987;15(2):213–31. [2] Biot MA. Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range. J Acoust Soc Am 1956;28(2):168–78. [3] Bougacha S, Roësset JM, Tassoulas JL. Dynamic stiffness of foundations on fluid-

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