Dynamic vertical interaction of a foundation–soil system generated by seismic waves

Journal of Sound and Vibration 333 (2014) 2378–2389

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Dynamic vertical interaction of a foundation–soil system generated by seismic waves Peng Wang, Jun Wang n, Yuanqiang Cai, Chuan Gu College of Architecture and Civil Engineering, Wenzhou University, Wenzhou 325035, PR China

a r t i c l e in f o

abstract

Article history: Received 12 April 2013 Received in revised form 13 November 2013 Accepted 17 November 2013 Handling Editor: L.G. Tham Available online 17 February 2014

Based on Biot0 s dynamic poroelastic theory, a foundation–soil interaction model is established to investigate the vertical vibrations of a rigid circular foundation on poroelastic soil excited by incident plane waves, including the fast P waves and SV waves. Scattering waves caused by the foundation and fluid–solid coupling due to the pore water in the soil are also considered in the model. The solution of the vertical vibrations of the foundation subjected to seismic waves are obtained by solving two sets of dual integral equations derived from the mixed boundary-value conditions. The different vertical vibrations of foundation rest on elastic and saturated halfspace are compared. The influences of incident angle, permeability of soil and foundation mass on the vertical vibrations of the foundation are then discussed. The results show that resonant phenomenon of the foundation is observed at certain excitation frequencies; the effects of the pore water on the foundation vertical vibrations are significant. In addition, significant differences are found when the foundation is excited by P waves and SV waves, respectively. & 2013 Elsevier Ltd. All rights reserved.

1. Introduction The environmental vibration caused by dynamic load, such as high-speed railway, subway blast load, may lead to the dynamic interaction between the soil and the foundation of the structure at the nearly area, which imposes a negative effect on the use and safety of the structure. There are two basic mixed boundary-value problems in a substructure formulation of the soil–foundation interaction problem [1]. The first problem involves the calculation of the force–displacement relationship for the foundation under the actions of external forces and moments, and the second problem involves the calculation of the response of the foundation subjected to seismic waves. The first problem has been well investigated by many researchers. For example, Halper and Christiano [2] obtained the vertical vibration of a rigid rectangular foundation on the poroelastic half-space based on the Biot0 s theory for the first time. The influence of different boundary permeable conditions was also examined. Philippacopoulos [3] presented an analytical study of the vertical response of a rigid circular foundation on saturated layered soil. Kassir et al. [4] analyzed the coupling problem between the horizontal and rocking vibration of a circular foundation on the saturated soil. Jin [5,6] reported a minute analysis of the vertical response of a circular foundation on the poroelastic half-space by solving a set of dual integral equations. Cai and Hu [7] adopted the Novak model to study the vertical vibration of a embedded circular foundation in the soil. These studies aimed to find the impedance or compliance function of the foundation. The propagation and scattering of waves were not considered. Nevertheless, the vibrations caused by external loads actually propagate in the form of waves in soil. Furthermore, the waves are scattered by the foundation in the soil, which leads to the dynamic interaction between the

n

Corresponding author. Tel./fax: þ 86 571 8820 8774. E-mail address: [email protected] (J. Wang).

0022-460X/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jsv.2013.11.048

P. Wang et al. / Journal of Sound and Vibration 333 (2014) 2378–2389

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Fig. 1. Difference between the models for the first and second problem: (a) model for the first problem and (b) model for the second problem.

structural foundation and the soil. Therefore, the model established for the first problem could not be applied to the dynamic interaction between the foundation and the soil excited by seismic waves (the second problem). The difference between the models for the first and second problem is shown in Fig. 1. For the second problem, Kobori et al. [8,9] investigated the vibration of a rigid circular disc on an elastic half-space subjected to obliquely incident SH waves and Rayleigh waves, respectively. The translation and torsional displacements of the disc were obtained. Byrcoft [10] and Luco [11] presented that the solutions of the basic radiation and scattering problem were relative. The relationship between the two problems is such that the foundation input motion vector could be easily calculated if the contact tractions corresponding to the solution of the radiation problem and the free-field ground motion were known. Luco [12] and Pais [13] extended this finding to more soil–foundation interaction models subsequently. The dynamic response of the circular foundation and embedded strip foundation under non-vertically incident waves, including SH, P and SV waves, were studied. Oien [14] studied the diffraction of harmonic waves with a Green0 s function approach by making use of a movable rigid strip bonded to the surface of an elastic half-space. Tham et al. [15] investigated the response of a group of flexible surface foundation subjected to harmonic incident Rayleigh and SH waves by a combined BEM and FEM procedure. Since the existence of the under-ground water affects the wave propagation in soil medium to some extent, the fully saturated poroelastic soil model, which is superior to the elastic one, is more desirable for the analysis of the dynamic response of soil–foundation interaction. Todorovska and Rjoub [16] presented a simple 2D model of soil–structure interaction for a building supported by a circular foundation embedded in a homogeneous poroelastic half-space. The wavefunction expansion method was used to represent the motion of the soil. Zhou et al. [17] studied the scattering of plane P wave by circular-arc alluvial valley in poroelastic half-space using the complex variable function method. The isolation of plane P and S waves in a homogeneous poroelastic half-space with a row of rigid or elastic piles was investigated by Cai et al. [18,19]. Although many studies have been carried out to investigate the soil–foundation interaction problem, the analysis of dynamic response of the foundation on poroelastic half-space under seismic load is limited. Furthermore, Most of the existing work did not concern three dimensional models. Therefore, it is meaningful to study the performance of the foundation on the poroelastic soil excited by incident waves. As the actual vibration of the foundation excited by seismic waves is too complex to be analyzed directly, it is usually treated as a superposition of all kinds of fundamental vibrations, such as vertical, rocking, horizontal and torsional vibration [5,6,11,12]. Therefore, it is essential to investigate to these fundamental vibrations of the foundation. In previous works [20–22], the authors studied the torsional and rocking vibrations of the surface and embedded foundation excited by plane waves. So far, the vertical vibration of the foundation on the poroelastic soil has not been investigated. In this paper, an approximate analytical approach is proposed to study the dynamic response of the foundation on the poroelastic ground under seismic load. The poroelastic soil media is governed by Biot0 s dynamic poroelastic theory, which takes into account the coupling effects between the soil particles and pore water. By imposing the continuity conditions at the foundation–soil interface, the problem is formulated into two sets of dual integral equations with the aid of Hankel transform and the scattering theory of incident waves. The dual integral equations are then reduced to Fredholm integral equations of the second kind, which can be solved by a semi-analytical method. The effects of the permeability of the soil, the boundary drainage, the incident angle, and the mass of the foundation on the vertical vibrations of the foundation are extensively discussed.

2. Governing equations of the poroelastic soil The system is shown in Fig. 2. The soil medium is modeled as a poroelastic half-space with an unsealed boundary, which is fully saturated by the viscous fluid. A rigid circular foundation, with center of curvature at O1 and radius r0, rests on the poroelastic half-space. The foundation is subjected to the action of an obliquely incident plane waves. In the followings, pffiffiffiffiffiffiffiffi the excitation and the response has harmonic time dependence of the type eiωt (ω is frequency of the motion and i ¼  1). For brevity, the factor eiωt is omitted from all expressions.

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Fig. 2. Rigid circular foundation on poroelastic half-space and the incident waves.

Taking the advantage of the axisymmetry of vertical motion, governing equations for the poroelastic soil proposed by Biot [23,24] can be expressed in the cylindrical coordinate system O(r, θ, z) as follows:   1 ∂e ∂ξ €r ¼ ρu€ r þ ρf w μ ∇2  2 ur þðλ þ α2 M þμÞ αM (1) ∂r ∂r r μ∇2 uz þðλ þ α2 M þ μÞ

∂e ∂ξ €z αM ¼ ρu€ z þ ρf w ∂z ∂z

(2)

αM

ρf ∂e ∂ξ € r þ bw _r M ¼ ρf u€ r þ w ∂r ∂r n

(3)

αM

∂e ∂ξ ρ € z þ bw _z M ¼ ρf u€ z þ f w ∂z ∂z n

(4)

where ur and uz are the radial and vertical displacement of the solid matrix, respectively; wr and wz are the average fluid displacement relative to the solid matrix in the r and z direction; e is dilatation of the solid and e¼div(u); ξ is the fluid dilatation relative to the solid and ξ¼  div(w); μ and λ are the Lamé constants of the soil skeleton; α is the constant accounting for compressibility of the solid; M is the constant accounting for compressibility of the fluid; ρ¼nρf þ(1 n)ρs, n is porosity, ρf and ρs are the densities of fluid and solid material; b is the coupling coefficient between the fluid and solid, and defined as the ratio between the fluid viscosity and the intrinsic permeability of the porous medium. The constitutive relations for a homogeneous poroelastic material can be expressed as srr ¼ λe þ2μ∂ur =∂r

(5)

szz ¼ λeþ 2μ∂uz =∂z

(6)

sθθ ¼ λe þ2μur =r

(7)

τrz ¼ μð∂ur =∂z þ ∂uz =∂rÞ

(8)

p_ f ¼ αM e_ þM ξ_

(9)

where pf is the pore fluid pressure; srr, sθθ, szz and τrz denote the effective stress components of the solid skeleton. 3. The total waves in saturated half-space 3.1. Incident P wave The seismic excitation, which is represented by an incident plane fast P waves as shown in Fig. 2, can be expressed by the potential  ikp1 ðx φin 1 ¼ a0 e

sin θin  z cos θin Þ

(10)

For an incident plane fast P wave, the reflection from the free-surface of the poroelastic half-space generates three reflected plane waves: fast and slow P waves, and SV waves. A general representation for the reflected wave is  ikp1 ðx φre 1 ¼ a1 e

sin θp1 þ z cos θp1 Þ

(11)

 ikp2 ðx φre 2 ¼ a2 e

sin θp2 þ z cos θp2 Þ

(12)

P. Wang et al. / Journal of Sound and Vibration 333 (2014) 2378–2389

ψ re ¼ b1 e  iks ðx

sin θs þ z cos θs Þ

2381

(13)

where a0 is the amplitude of the incident wave; θin is the angle of incidence; a1, a2 and b1 are the amplitudes of the reflected fast P, slow P and SV waves, respectively; θp1, θp2 and θs are the angles of the reflected fast P, slow P and SV waves, moreover, θin ¼θp1; kp1, kp2 and ks are the wavenumbers of the fast P, slow P and SV waves in the medium. Expressions for the amplitudes, angles and wavenumbers of the reflected waves can be found in Lin [25]. The potentials can be represented in the coordinate system O(r, θ, z) ikp1z z  ike r φin 1 ¼ a0 e

cos θ

(14)

 ikp1z z  ike r φre 1 ¼ a1 e

cos θ

(15)

 ikp2z z  ike r φre 2 ¼ a2 e

cos θ

(16)

ψ re ¼ b1 e  iksz z  ike r

cos θ

(17)

where kp1z ¼kp1 cos θp1, kp2z ¼ kp2 cos θp2, ksz ¼ks cos θs, and ke ¼kp1 sin θp1 ¼kp2 sin θp2 ¼ks sin θs. re re re In the above equations, φin are considered as free-field waves describing the wave propagation in 1 , φ1 , φ2 and ψ poroelastic half-space in the absence of foundation. The presence of the rigid foundation, which can be regarded as a secondary wave source, modifies the free-field motion and gives rise to the scattering waves. According to the scattering theory of elastic waves suggested by Pao [26], the total scattering wave field in soil can be divided into two parts. One corresponds to rigid-body scattering waves generated when the free-field waves are scattered by the fixed rigid foundation on the poroelastic half-space, which is denoted as uS. And the other one is regarded as radiation scattering waves generated by the vibration of the rigid foundation, which is denoted as uR. Therefore, the total wave fields consist of free-field waves, rigid-body scattering waves and radiation scattering waves. 3.2. Incident SV wave When the seismic excitation is presented by SV wave, there are two possible critical angles for SV wave incidence because two P waves co-exist in the poroelastic medium. As the incident angle reaches the critical angle, the reflected P wave becomes a surface wave. When the incident angle exceeds the critical angle, the reflected P wave will decrease exponentially with depth. The critical angles can be determined from θcr1 ¼ sin  1 ðkp1 =ks Þ

(18)

θcr2 ¼ sin  1 ðkp2 =ks Þ

(19)

Considering the critical angels, the free-field waves can be expressed as ψ E ¼ b0 eksz z  ike r

cos θ

(20)

 kp1z z  ike r φre 1 ¼ a1 e

cos θ

(21)

 kp2z z  ike r φre 2 ¼ a2 e

cos θ

(22)

(23) ψ re ¼ b1 e  ksz z  ike r cos θ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 where ke ¼kp1 sin θp1 ¼kp2 sin θp2 ¼ks sin θs; kp1z ¼ i kp1  ke when θs oθcr1 and kp1z ¼  ke  kp1 when θs Zθcr1; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 kp2z ¼ i kp2  ke when θs oθcr2 and kp2z ¼  ke kp2 when θs Zθcr2; ksz ¼ks cos θs. When the incident waves are SV waves, the total wave fields also consist of free-field waves, rigid-body scattering waves and radiation scattering waves here. 4. Boundary conditions and the solution The problem can be expressed entirely in terms of dimensionless parameters. It is defined that all length quantities are nondimensionalized by r0, and stress and with respect to μ. In addition, other pffiffiffiffiffiffiffi ffi pore pressure are nondimensionalized pffiffiffiffiffi n parameters are introduced: ω0 ¼ r 0 ω ρ=μ, λn ¼ λ=μ, ρn ¼ ρf =ρ, M n ¼ M=μ, b ¼ br 0 = ρμ. There and in the follows, asterisk indicates nondimensional parameters. The Hankel transform with respect to the radial coordinate and its inverse Hankel transform are employed to solve the problem, which are defined as [27]: Z 1 v f~ ðεÞ ¼ rf ðrÞJ v ðεrÞdr (24) 0

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v f~ ðrÞ ¼

Z

1 0

εf ðεÞJ v ðεrÞdε

(25)

where Jv(εr) is the vth-order Bessel function of the first kind. Substituting the dimensionless parameters into Eqs. (1)–(9) and applying the Hankel transforms to them, the following general solutions can be obtained based on the procedure proposed by Cai [6]: n

e~ 0 ðε; zn Þ ¼ A1 e  q1 z þA2 e  q2 z n

n

p~ 0f ðε; zn Þ ¼ c21 A1 e  q1 z þ c22 A2 e  q2 z n

(26) n

(27)

n

u~ 1r ðε; zn Þ ¼ c31 A1 e  q1 z þ c32 A2 e  q2 z þA3 e  q3 z n

n

n

(28) n

u~ 0z ðε; zn Þ ¼ c41 A1 e  q1 z þ c42 A2 e  q2 z þ c43 A3 e  q3 z n

n

(29) n

s~ 0zz ðε; zn Þ ¼ ðλ  2q1 c41 ÞA1 e  q1 z þ ðλ 2q2 c42 ÞA2 e  q2 z  2q3 c43 e  q3 z n

n

s~ 1rz ðε; zn Þ ¼ ðq1 c31 þεc41 ÞA1 e  q1 z  ðq2 c32 þεc42 ÞA2 e  q2 z  ðq3 þ εc43 Þe  q3 z

(30) n

(31)

where A1, A2, A3 are arbitrary functions to be determined from the boundary conditions. Expressions for q1, q2, q3, c21, c22, c31, c32, c41, c41 and c43 are given in the appendix. There, the assumption that the vertical vibration is independent of the rocking vibration is adopted as Philippacopoulos [3], Jin [5,6], Cai [7], etc. So, the perfect bond between the soil and the foundation at z ¼0 can be stated as follows: srz ðr n ; 0Þ ¼ 0 uz ðr n ; 0Þ ¼ AT0

0 rr n o1

(32)

0 r rn r 1

(33)

where AT0 denotes the nondimensional vertical displacement of the rigid foundation. Two types of drainage conditions are considered at the surface of the poroelastic half-space. An unsealed half-space boundary implies pf ðr n ; 0Þ ¼ 0

0 r rn o 1

(34)

szz ðr n ; 0Þ ¼ 0

1 or n o1

(35)

and for a sealed boundary, it is szz ðr n ; 0Þ þ pf ðr n ; 0Þ ¼ 0 pf ðr n ; zn Þ ¼0 zn

1 o rn o 1

0 r r n o1

(36) (37)

where pf (rn, 0) is the nondimensional pore fluid pressure at the surface of the poroelastic space, srz (rn, 0) and szz (rn, 0) denote the nondimensional effective stress components of the solid skeleton at the surface of the poroelastic space. According to the definition of the wave field in above, the displacements corresponding to rigid-body scattering waves are of the same magnitudes and the adverse directions at the contact surface between the foundation and the soil comparing to these caused by the free-field waves. And because the radiation scattering waves are owing to the vibrations of the rigid foundation, the displacements generated by the radiation scattering waves are equal to the vibration amplitudes of the foundation at the contact surface. So for the unsealed boundary, following the definition of rigid-body scattering waves and radiation scattering waves, Eqs. (33) and (35) can be rewritten as uRz ðr n ; 0Þ ¼ AT0 sRzz ðr n ; 0Þ ¼ 0

0 r rn r 1

(38)

1 or n o1

(39)

þ re n ðr ; 0Þ uSz ðr n ; 0Þ ¼ uin z

sSzz ðr n ; 0Þ ¼ 0

0 r rn r 1

(40)

1 or n o1

(41)

0 r rn r 1

(42)

and for the sealed boundary uRz ðr n ; 0Þ ¼ AT0

sRzz ðr n ; 0Þ þ pRf ðr n ; 0Þ ¼ 0 þ re n ðr ; 0Þ uSz ðr n ; 0Þ ¼ uin z

1 o rn o 1

(43)

0 r rn r 1

(44)

P. Wang et al. / Journal of Sound and Vibration 333 (2014) 2378–2389

sszz ðr n ; 0Þ þ psf ðr n ; 0Þ ¼ 0

1 or n o1

2383

(45)

þ re where uin , uRz and uSz are nondimensional vertical displacement components of the free-field wave, radiation scattering z wave and rigid-body scattering wave, respectively. When the incident waves are P waves, Huang [28] have given the vertical displacement of the free-field wave at the saturated half-space surface þ re n uin ðr ; 0Þ ¼ ikp1z ða0  a1 ÞJ 0 ðke r n Þ  ikp2z a2 J 0 ðke r n Þ þ z

2

b1 ke J ðke r n Þ ks 0

(46)

Applying the inverse Hankel transform to Eqs. (26)–(31), together with Eqs. (38)–(45), the following two sets of standard dual integral equations can be derived: Z 1 ε  1 ð1 þ HðεÞÞBR ðεÞJ 0 ðεr n Þdε ¼ AT0 = η 0 r r n r 1 0 Z 1 BR ðεÞJ 0 ðεr n Þdε ¼ 0 1 or n o1 (47) 0

Z

1 0

Z

þ re ε  1 ð1 þHðεÞÞBs ðεÞJ 0 ðεr n Þdε ¼  uin =η z

0

1

Bs ðεÞJ 0 ðεr n Þdε ¼ 0

0 rr n r1 1 or n o1

(48)

where BR ðεÞ ¼ εs~ 0zzR ðε; 0Þ, Bs ðεÞ ¼ εs~ 0zzs ðε; 0Þ, HðεÞ ¼ ðεf ðεÞ=ηÞ 1, η ¼ limε-1 εf ðεÞ, expressions for f (ε) is given in the appendix. According to the methods suggested by Nobel [29], Eqs. (47) and (48) can be reduced to Fredholm integral equations of the second kind Z 2εAT0 1 BR ðεÞ ¼ θR ðxÞ cos ðεxÞdx (49) πη 0 Substituting Eq. (49) into Eq. (47), the following integral equation of the Fredholm type can be obtained: Z 1 1 Mðx; yÞθR ðyÞdy ¼ 1 θR ðxÞ þ π 0 where the kernel function M(x, y) takes the form

Z

Mðx; yÞ ¼ 2

1 0

HðεÞ cos ðεxÞ cos ðεyÞdε

(50)

(51)

Employing the following integral representation: Bs ðεÞ ¼

2ε π

Z

1 0

θs ðxÞ cos ðεxÞdx

(52)

Substituting Eqs. (46) and (52) into Eq. (48), the following integral equation of the Fredholm type can be obtained: ! Z 2 1 1 b1 ke Mðx; yÞθs ðyÞdy ¼  iða0 a1 Þkp1z þ ia2 kp2z  (53) cos ðke r n Þ θs ðxÞ þ π 0 ks The kernel function M(x,y) is given by Eq. (51). Eqs. (50) and (53) can be discretized into a series of algebraic equations and be solved numerically. Combining with Eqs. (49) and (52), the total force acting at the foundation base T can be obtained Z Z T 4AT0 1 4 1 ¼ θR ðxÞdx þ θS ðxÞdx (54) 2 η 0 η μr 0 0 where T n ¼ T=μr 20 is the nondimensional form of T. Eq. (54) indicates that the force exerted on the foundation may be thought of as being composed of two parts: the first part corresponds to the force required to move the foundation by displacement U0 when there is no free-field motion, the second part corresponds to the force to hold the foundation in place when it is subjected to the action of the freefield waves. Introducing the definitions Z 4 1 K TT ¼ θR ðtÞdt (55) η 0 and Un ¼ 

ð4=ηÞ

R1

0 θS ðtÞdt K TT

(56)

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Eq. (54) can be rewritten in the form T n ¼ K TT ðAT0  U n Þ

(57)

n

where KTT is vertical dynamic impedance for the foundation. U is called input vertical motion here and corresponds to the vertical vibrations of the rigid foundation under the action of the seismic excitation when no external forces are acting on the foundation ðT n ¼ 0Þ. Eq. (57) establishes the force and vertical vibration relationship for the rigid foundation subjected to the action force and the seismic excitation. 5. Dynamic equilibrium of the foundation By now, only the foundation vertical displacement U0 remains unknown. It can be determined from the dynamic equilibrium condition for the foundation. Combining with Eq. (54), the dynamic equilibrium implies Z Z 4AT0 1 4 1 ω20 mn AT0 ¼ θR ðxÞdx þ θS ðxÞdx (58) η 0 η 0 where mn is the dimensionless mass of the foundation, mn ¼ m=ρr 30 . According to the vibration analysis method given by Richartr [30], the vertical displacement amplitude of the foundation excited by incident P wave can be written as R1 jð4=ηÞ 0 θS ðtÞdtj ffi AT0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (59) ðkt  mn ω20 Þ2 þðct ω0 Þ2 where kt ¼  Re(KTT) and ct ¼ Im(KTT)/ω0. Likewise, the vertical vibrations of the foundation excited by incident SV wave can be obtained by employing the same procedure. The only difference is that Eqs. (46) and (53) have the following forms: þ re n uin ðr ; 0Þ ¼  kp1z a1 J 0 ðke r n Þ kp2z a2 J 0 ðke r n Þ þ z

1 θs ðxÞ þ π

Z

1

0

2

ðb0 þ b1 Þke J 0 ðke r n Þ ks 2

Mðx; yÞθs ðyÞdy ¼

ðb0 þb1 Þke a1 kp1z þa2 kp2z  ks

(60)

! cos ðke r n Þ

(61)

For the purpose of brevity, while without loss of generality, the gravity force of the foundation is neglected in the above derivations. 6. Numerical results and discussions 6.1. Comparisons with existing solutions In order to verify the accuracy of the proposed procedure for calculation of the dynamic response of the foundation excited by incident plane waves, it is applied to the analysis of the dynamic response of a circular foundation on a uniform half-space subjected to elastic waves, which has been studied by Luco [1]. Fig. 3 compares the response of the massless foundation obtained from present model and that given by Luco [1]. To be comparable, the elastic half-space soil medium is simulated by setting the poroelastic parameters negligibly small (a, Mn, ρn and bn are set to 10  5), while the other parameters are the  same with those of Luco [1]. In Fig. 3, β is the shear wave velocity, c is the apparent velocity of propagation along the x-axis, and U n =U 0  is the ratio of the input vertical motion and the vertical displacement of the soil excited by incident waves in the absence of the foundation. It can be seen that the two solutions are in close agreement with each other.

  Fig. 3. Comparison of U n =U 0  between present work and those of Luco [1]

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6.2. Numerical results for poroelastic soil In this section, the influences of the key parameters on the vertical vibrations of the foundation are discussed. The poroelastic parameters are set to α¼ 1, λn ¼1.004, Mn ¼246.78, ρn ¼0.45, and n ¼0.37, which are the same with those given in Cai [18]. The effect of excitation frequency on the vertical displacement is shown in Fig. 4 for different foundation masses, in which incident angle θ¼301, and internal friction bn ¼ 10. Regardless of the excitations of P waves or SV waves, for a given value of foundation mass mn, the vertical displacement increases with increasing excitation frequency ω0 until it reaches a peak value, and decays quickly after that. This phenomenon is referred to as resonance, and the frequency corresponding to the peak value of the displacement is defined as resonant frequency. In the range of ω0 exceeding the resonant frequency, the displacement of the foundation excited by SV waves decay more quickly than that by P waves. It is also observed in both Fig. 4a and b that the curve is sensitive to the foundation mass. When the excitation frequency is less than the resonant frequency, the increases of foundation mass leads to the increases of the displacement amplitude. But when the excitation frequency is larger than the resonant frequency, the larger the foundation mass is, the less significant the vibration of the foundation is. In addition, the resonance frequency decreases slightly as the foundation mass increases. The dynamic responses of elastic and poroelastic soil excited by P waves and SV waves are presented in Figs. 5 and 6, respectively. The incident angle θin ¼301, 451, and the dimensionless mass of the foundation mn ¼10. In the case of P wave exactions as shown in Fig. 5, the vertical displacements of poroelastic half-space are smaller than those of elastic half-space due to the existence of the pore water in soil medium. Especially, the resonant amplitude of poroelastic half-space is approximately a half of that of elastic half-space. In the case of SV wave excitations as shown in Fig. 6, similar phenomenon can also be observed when θin ¼451 (Fig. 6(b)), but the differences between the elastic and poroelastic soil are quite small. The reason is that the propagation of SV waves is independent of pore water which depends on the solid skeleton only. However, when θin ¼301, the displacement of the poroelastic soil is much larger than that of the elastic soil medium. As will be evidenced in the following, this phenomenon arises from the effect of critical angles. These results indicate that the elastic solutions can hardly be used to evaluate the performance of the foundation subjected to incident waves. Fig. 7 shows the dependence of vertical displacements on boundary drainage. The internal friction bn ¼10, dimensionless mass of the foundation mn ¼10 and θin ¼301. Comparing with a sealed boundary, an unsealed boundary contributes to

Fig. 4. AT0 varied with ω0 under different mn: (a) excited by P waves and (b) excited by SV waves.

Fig. 5. Comparison between elastic and poroelastic soil for P waves: (a) θin ¼30o and (b) θin ¼45o

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Fig. 6. Comparison between elastic and poroelastic soil for SV waves: (a) θin ¼ 30 and (b) θin ¼ 45o

Fig. 7. AT0 varied with ω0 under different drainage conditions: (a) excited by P waves and (b) excited by SV waves.

a more significant vertical response. It can be explained by the fact that there is more significant soil–water interaction for the sealed boundary, which gives rise to the energy dissipation. In addition, it is also observed that the resonance frequency of the foundation decreases slightly for the sealed poroelastic half-space. The internal friction resulting from the relative motion between the solid skeleton and pore fluid is inversely proportional to the intrinsic permeability of the soil medium. The vertical displacement of the foundation subjected to P waves and SV waves under different internal friction are shown in Figs. 8 and 9, respectively. The incident angle θin ¼301, 451, and dimensionless mass of the foundation mn ¼10. An inspection of Fig. 8 shows that the resonant amplitude decreases as the internal friction bn increases under the excitations of P waves, but it is not apparent. When the excitation frequency is larger than the resonant frequency, the soil with higher intrinsic permeability has larger vibration amplitude. Under the excitations of SV waves, the vertical displacement also decreases as the internal friction increases when θin ¼451, as shown in Fig. 9b. When θin ¼301 (Fig. 9a), however, the trend turns out to be quite different. The vertical displacement is larger for the soil with lower permeability. It can be seen that the incident angle θin not only influences the amplitude of the vertical displacement, but also remodels the trend of the displacement with the variation of internal friction when the foundation is excited by SV waves. The incident angle is another key factor that directly affects the vertical displacement of the foundation excited by incident waves. Fig. 10 shows the vertical displacement of the foundation excited by incident waves of various incident angles. The internal friction bn ¼1. Fig. 11 displays the variation of vertical displacement with incident angle under different internal frictions. The excitation frequency ω0 ¼0.5, and dimensionless mass of the foundation mn ¼10. An angle of θin ¼ 01 corresponds to vertical incidence, while θin ¼901 corresponds to horizontal incidence. For incident P waves, both Figs. 10(a) and 11(a) indicate that the vertical displacement of the foundation decreases as the incident angle increases. The maximum vertical displacement occurs at vertical incidence, while the displacement corresponding to horizontal incidence is zero. It is also seen from Fig. 11(a) that the displacement of poroelastic soil is less than that of elastic medium. Furthermore, the displacement of poroelastic soil decreases as the intrinsic permeability increases. For incident SV waves (Figs. 10b and 11b), the vertical displacement generally increases with increasing incident angle until it reaches a peak value, and decays after that. Both the displacement under vertical incidence and that under horizontal incidence are exactly zero. Recalling Fig. 9, the variation of incident angle does not cause the variation of resonant frequency. In addition, oscillations of soil vertical displacement are observed when the incident angels are in the range of 151 oθin o451, as shown in Fig. 11(b). According to

P. Wang et al. / Journal of Sound and Vibration 333 (2014) 2378–2389

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Fig. 8. AT0 varied with ω0 under different bn for incident P waves: (a) θin ¼30o and (b) θin ¼45o.

Fig. 9. AT0 varied with ω0 under different bn for incident SV waves: (a) θin ¼30o and (b) θin ¼45o.

Fig. 10. AT0 varied with ω0 under different θin: (a) incident P waves and (b) incident SV waves.

Lin [25], the coefficients of reflected P waves oscillate dramatically when the incident angle approaches to the critical angle. It therefore results in the fluctuations of soil vertical displacement.

7. Conclusions The dynamic response of a circular rigid foundation on poroelastic medium excited by incident plane waves is investigated based on Biot0 s dynamic poroelastic theory. The Hankel transform is employed to solve the governing equations

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Fig. 11. AT0 varied with θin under different bn: (a) excited by P waves and (b) excited by SV waves.

analytically. The vertical displacements of the foundation are evaluated by solving two sets of Fredholm integral equations describing the boundary conditions. Based on the results of parametric study, the following conclusions can be drawn: (1) The resonant phenomenon arises regardless of P wave excitations or SV wave excitations. Both the resonant displacement and resonant frequency decrease as the foundation mass increases. (2) Pore water imposes significant effect on the vertical vibrations when the foundation is excited by incident P waves. The peak value of the vertical displacement of the foundation on poroelastic half-space is smaller than that of elastic medium. The vertical displacements of an unsealed boundary are larger than those of a sealed boundary for P and SV waves. (3) The vertical displacements decrease slightly as internal friction bn increases when subjected to incident P waves. However, the influence of bn on the vertical displacements varies with incident angle when the foundation is subjected to SV waves. In most cases, the vertical displacements also decrease as bn increases. But when the angle is less than the value at which the total reflection occurs, the influence of bn varies due to the sharp peaks of the reflection coefficients of P waves. (4) The vertical displacements of the foundation subjected to P waves decreases as incident angle increases. The maximum vertical displacements occur at vertical incidence, while the displacements under horizontal incidence are zero. When the foundation is excited by SV waves, the vertical displacements generally increase from zero at the beginning and decay to zero as incident angle increases. The maximum displacements occur when θin ¼ 651. The displacement curves show slight fluctuations when the incident angle is equal to the critical angle or the total reflection occurs.

Acknowledgments The work presented in this paper is supported by the National Science Fund for Distinguished Young Scholars (Grant no. 51025827), the National Natural Science Foundation of China (Grant no. 51208383) and the Natural Science Foundation of Zhejiang Province (Grant no. LQ12E08008). Appendix Expressions for q1, q2, q3, c21, c22, c31, c32, c41, c41, c43 and f (ε) appearing in above are given by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 q1 ¼ ε2  ðβ4 þ β24 4β5 Þ; q2 ¼ ε2  ðβ4  β24 4β5 Þ; q3 ¼ ε2  ω20 ρn ω20 D1 2 2 β2 þ β1 ðq21  ε2 Þ ; β3

c22 ¼

β2 þ β1 ðq22  ε2 Þ β3

εðλn þ1Þ  c21 εðα þD1 Þ ; q21 þω20 þ ρn ω20 D1 ε2

c32 ¼

εðλn þ 1Þ  c22 εðα þ D1 Þ q22 þ ω20 þ ρn ω20 D1  ε2

c21 ¼

c31 ¼

c41 ¼

εc31  1 ; q1

c42 ¼

εc32  1 ; q1

c43 ¼

ε q3

P. Wang et al. / Journal of Sound and Vibration 333 (2014) 2378–2389

f ðεÞ ¼

2389

c21 c42 c22 c41 þc43 D2 ðλ  2q2 c42 Þc21  ðλ 2q1 c41 Þc22  2q3 c43 D2

for the unsealed boundary, f ðεÞ ¼

 q1 c42 þ q2 c41 þ q2 c43 D3 ½ðλ þ2εÞq2 c21  2εq2 c31 þ q2   ½ðλ þ 2εÞq1 c22 2εq1 c32 þq1   2εq2 D3 n

for the sealed boundary, where D1 ¼ ðnρn ω0 =inb  ρn ω0 Þ, β1 ¼ D1 ðλn þ 2Þ, β2 ¼ D1 ω20 ð1 þD1 ρn Þ ρn ω20 ðαþ D1 Þ2 , β3 ¼ D2 ¼

ρn ω20 ðα þD1 Þ ρn ω2 ρn ω2 β ðα þ D1 Þρn ω20 β3 β ; β4 ¼ 2  n 0 ; β5 ¼  n 0 2  ; n β1 M D1 D1 β 1 M D1 β 1 M

ðq1 c31 þεc41 Þc22  ðq2 c32 þ εc42 Þc21 q q ðc31  c32 Þ þεq2 c41  εq1 c42 D3 ¼ 1 2 : q3 þ εc43 q1 q3 þεq1 c43

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