conditions along the contact surface between the soil and the foundation are welded ... resting on a half-space, or on a stratum over rigid or compliant bedrock.
BOUNDARY ELEMENT METHOD APPLIED TO POROELASTODYNAMIC PROBLEMS: APPLICATION TO SOILFOUNDATION INTERACTION
HADID1 Mohamed and GHERBOUDJ2 Faouzi 1Maître de conférence, Ecole Nationale des Travaux Publics, ENTP. 2Postgraduant, Ecole Nationale des Travaux Publics, ENTP
Abstract: A boundary element model is presented for the computation of dynamic stiffness of trip foundations resting on or embedded in visco poroelastic soils according to the Biot’s theory. The boundary conditions along the contact surface between the soil and the foundation are welded or smooth for the skeleton. And either drained or undrained hydraulic conditions for the fluid (i.e. previous or impervious foundation) are used. Specific results are compared with known solutions. 1. Introduction: The analysis of the dynamic soil-structure interaction phenomena in fluid-saturated porous media is of great interest in earthquake geotechnical engineering because of its relationship with soil liquefaction problems. According to the widely accepted Biot’s theory (1956), a macroisotropic, homogeneous porous medium will sustain two compressional waves and one shear wave. The fast compressional wave (P1 wave) and the shear wave (SV wave) are similar to the corresponding ones in elastic theory. The slow compressional wave (P2 wave), however, is associated with a diffusion type process at low frequency due to fluid viscosity (Yang & Sato 1998) and is particular to porous media. Several formulation based on Biot’s theory were proposed in order to evaluate the response of porous media to wave propagation. Some of these formulations were purely analytical (Simon and al. 1984, Tabatabaie and al. 1994, Rajapakse and Senjuntichai 1995, Bo and Hua 1999, Mehiaoui and hadid 2003 and 2005) whereas others were numerical like the finite elements method (Zienkiewicz & Shiomi 1984), the thin layer elements method (Nogami & Kazama 1992) and the boundary elements method (Dominguez 1993). Based on Biot’s theory for dynamic behavior of poro-elastic media, Halpern and Christiono (1986) where the first to compute compliance coefficients of rigid plates resting on the saturated poroelastic half-space. Kassur and Xu (1988) have done a similar study for the strip footing on a poroelastic halfplane. They developed obtained a Green functions expressions in terms of integrals that are carried out numerically. More recently, Bougacha et al. (1993) obtained dynamic stiffness of a strip and circular foundations on two-phase poro-elastic stratum by a finite element approach. Using a Laplace domain boundary element formulation, Chang and Dargush (1995) and Chopra and Dargush (1995) have also obtained some impedance function for rigid foundations on poroelastic soils. Japon et al. (1997) have been presented a boundary element model for the computation of impedances for strip foundation resting on a half-space, or on a stratum over rigid or compliant bedrock. Dynamic stiffness and vibrations of rigid surface footings on poroelastic half-space or a stratum have been computed and studied by a number of investigators, mostly for vertical motions. From the knowledge of the authors, only vertical motion for embedded foundations are studied by Senjuntichai and al. (2006) and Todorovska and al. 2006. In this paper, the dynamic response of a rigid strip foundation, in visco poro-elastic soils, under horizontal, vertical and rocking loading is obtained. The technique is based on the boundary element formulation for poroelastic media obtained by Dominguez (1991, 1992) and Cheng et al. (1991). The contact condition between the soil and the foundation may be pervious or impervious smooth or welded. The results presented in this paper are for surface and embedded strip foundations in a poroelastic soils.
2. Governed equations and Boundary element model Porous medium is assumed to be a fluid-filled poroelastic material governed by Biot’s equations. The constitutive equations are:
τ ij = ( λ e + Qε ) δ ij + 2με ij
(1)
τ = Qe + Rε
(2)
Where: τ ij are the solid skeleton stress components; τ is the fluid equivalent stress = − np ( p= pore pressure); n are the porosity; ε ij are solid skeleton strain components = 1/ 2(uij + u j ,i ) ; δ ij is the cronecker delta function.; e = ui ,i and ε = U i ,i are the solid and fluid dilatation, respectively; u is the displacement of the solid; U is the displacement of the pore fluid; λ , u are the lamé constants for the drained solid skeleton; and Q and R are Biot’s constants. The equilibrium equations in terms of the solid and fluid displacement for a time harmonic excitation can be written in the frequency domain as
⎛ Q2 ⎞ 2 μΔu i + ⎜ λs + μ + ⎟ e ,i + Q ε ,i + X i = −ω ( ρ11u i + ρ12U i ) + i ωb (u i − U i ) R ⎠ ⎝
(Qe + R ε ),i + X i′ = −ω 2 ( ρ12u i + ρ12U 1i ) − i ωb (u i −U i )
(3) (4)
X and X’ are body forces in the solid and fluid phase, respectively; ρ11 = (1 − n) ρ s − ρ12 ; ρ 22 = nρ f − ρ12 ; ρ12 = −n(τ α − 1) ρ f are the mass coefficients, ρ s and ρ f are solid and fluid phase densities respectively; τ α is the dynamic tortuosity; b = n 2 μ k is the dissipation constant; where k (m/s) is the hydraulic conductivity of the poroelastic medium and μ is the absolute viscosity of fluid phase. The reciprocal relation can be obtained as usual starting from the equilibrium equations for timeharmonic behavior
τ ij , j + X i = −ω 2 ( ρ11ui + ρ12U i ) + iωb ( ui − U i )
(5)
τ ,i + X i′ = −ω 2 ( ρ12ui + ρ 22U i ) − iωb ( ui − U i )
(6)
Weighing the equation (5) with ui* and the equation (6) with U i , adding the two equations, integrating over the body Ω , and using integration by parts twice as usually done for derivation of reciprocal relations in static and dynamic problems. The following reciprocal relation is obtained
∫ (t u Γ
* i i
+ τ U n* ) d Γ + ∫
Ω
(X u
* i i
+ X i′U i* ) d Ω = ∫ ( ti*ui + τ *U n ) d Γ + ∫ Γ
Ω
( X u + X ′ U ) dΩ * i i
*
i
i
(7)
Where Γ is the boundary of the body Ω . t i = τ ij n j and U n = U i ni , n being the unit normal to the boundary. The integral representation of the solid displacement ui is obtained by substitution of the fundamental solution derived from the following body forces:
X i* = δ ( x − ξ ) δ ij X i′* = 0
(8)
In which δ ( x − ξ ) is the Dirac delta function. ξ indicates the point of application of the delta function and δ ij is the Kronecker delta.
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The reciprocal relation gives
(
)
c ij u i + ∫ t ij*u i d Γ + ∫ τ *jU n d Γ = ∫ u ij* t i d Γ + ∫ τU nj* d Γ + ∫ X i u ij* + X i′U ij* d Ω (9) Γ
Γ
Γ
Γ
Ω
To obtain the integral representation of the fluid stress, the fundamental solution corresponding to the following body forces is used:
⎡ 1 ⎤ X i′* = ⎢ ln r ⎥ ⎣ 2π ⎦ ,i
for 2D for 3D
⎡ −1⎤ X i′* = ⎢ ⎥ ⎣ 4πr ⎦ ,i X i* =
iωb + ω 2 ρ X i′* 2 − iωb + ω ρ f
(10)
Where r is the distance to the point ξ at which the fluid stress is going to be represented. The reciprocal relation gives
⎡ * ⎛ * X i′* ni cτ * * ⎜ τ τ + + Γ = + + t u U d j u U ⎢ ij i j n ∫Γ ⎢ i ij ⎜ nj − iωb + ω 2 ρ f − iωb + ω 2 ρ f ∫Γ ⎝ ⎣
(
)
⎡ ⎛ X i′ * + ∫ ⎢ X i u ij* + X i′⎜U ij* + Ω ⎜ − iωb + ω 2 ρ f ⎢⎣ ⎝ Where
J =
⎞⎤ ⎟⎥ dΓ ⎟⎥ ⎠⎦ ⎞⎤ ⎟⎥ dΩ ⎟⎥ ⎠⎦ (11)
1 and Z = J (iwb + w 2 ρ12 ) iwb − w 2 ρ 22
Using vector notation, the representation for a point “i” can be written for two dimensional domain under zero body forces conditions as
c i u i + ∫ p *ud Γ = ∫ u * p d Γ Γ
(12)
Γ
u and p are boundary variable vectors for displacements and stresses respectively:
⎡ u1 ⎤ ⎡ t1 ⎤ ⎢ ⎥ u = ⎢u2 ⎥ and , p = ⎢⎢ t 2 ⎥⎥ ⎢⎣ τ ⎥⎦ ⎢⎣U n ⎥⎦
u * et p* are fundamental solution tensors * * ⎡ u11 u21 ⎢ * u * = ⎢u12* u22 * ⎢u13* u23 ⎣
−τ 1* ⎤ ⎥ −τ 2* ⎥ ; −τ 3* ⎥⎦
Where Uˆ n*3 = U n*3 − JX 'α* nα = (J τ 3,* α + Zu )nα and
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* ⎡ t11* t21 ⎢ * p* = ⎢t12* t22 ⎢t * t * ⎣ 13 23
−U n*1 ⎤ ⎥ −U n*2 ⎥ −Uˆ n*3 ⎥⎦
⎡1 c = ⎢⎢ 0 ⎢⎣ 0 i
0 1 0
0 ⎤ 0 ⎥⎥ − J ⎥⎦
for a smooth boundary point. A boundary element discretization of Eq. (8) leads to a system of 3N equations:
Hu = Gp
(13)
where N is the number of nodes on the boundary; u is a vector containing solid displacement and fluid stress at boundary nodes; p is a vector containing solid traction and fluid normal displacement at boundary nodes; and H, G are 3N · 3N system matrices obtained by integration of the 2-D poroelastic fundamental solution over the boundary elements. Along the contact surface between the soil and the foundation, the displacements of the skeleton are constrained by the displacements of the foundation, and the motion of the fluid is constrained by the drainage condition. The equilibrium of the normal traction on the solid p ns and the total normal traction on the poroelastic medium T ns
p ns + T ns = 0
(14)
Equilibrium between the tangential traction on the solid pts and on the porous medium T t p
pts + T t s = 0
(15)
WithT np = t np + τ p , whereby t np = normal traction on the skeleton; τ p = pore pressure. And T t p = t tp = tangential traction on skeleton. Compatibility of displacements along the interface
u ns = u np = U np
(16)
u st = u tp
(17)
Whereby u ns and u ts denote displacement components of solid zone; u np and u tp denote displacement components of skeleton of porous zone; and U nP denotes normal displacement of pore fluid. If the solid material is pervious, the prior compatibility and equilibrium conditions along the interface become as follows: (14) remains the same with T np = t np , ( (τ p = p p = 0); (15) and (17) de not change; and (16) becomes u ns = u np . Note that the condition on the normal displacement of the pore fluid has been substituted by a condition on the pore pressure. 3. Dynamic stiffness functions of strip foundations After application of the conditions at the contact elements, H and G matrices will be of 2N x 2N . Considering the matrix T that transforms the displacement vector u f of a rigid foundation into the translational displacement vector u, the relationship between the force P and the displacement uf with respect to a rigid foundation is given by
P = T t AG −1 H T u f
(18)
Where A is a diagonal matrix, the term of which represents the area governed by the i-th element. The impedance matrix K is expressed as follows K = T t A G −1 H T
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(19)
The dynamic stiffness coefficients will be expressed as
K d (ω ) = K s (k (ω ) + iω c(ω ))
(20)
K = K S (k + ia0c )(1 + 2i β )
(21)
or
The compliance can be written in adimensionalized form by multiplying it with the shear modulus of the soil. The values of the compliance are obtained for dimensionless frequencies from 0 to 3. The dimensionless frequency is defined as: a0 = ω B / C s . Where B is the half-width of the strip foundation and Cs is the shear wave velocity. The shear modulus for nonzero hysteretic damping β is substituted by a complex equivalent term G* as: G*=G (1+2i β ). Where β is the material damping of hysteretic type of solid skeleton. Constant boundary elements are used to discretize the soil surface. First, the length of the free surface discretized must be grater to three times the half-width of the strip foundation for the surfaced foundation and seven times for embedded foundations with embedment ratio E/B=0.5 and E/B=1 (fig . 1). This value could be found by a simple numerical test, this test consist to compute dynamic stiffness for different values of the distance D. The appropriate distance is obtained when no significant variation of impedance function are denoted for a greater discretisation length.
Fig .1. : Boundary discretization of strip foundation. Fig. 2 shows a comparison of the normalised stiffness and damping coefficients obtained by the present approach with the results of Kassir and Xu (1988) and japon et al (1997), the vertical, horizontal and rocking compliances of a rigid strip footing resting on a poro-viscoelastic halfspace was analyzed. A smooth pervious contact conditions was considered between soil and foundation first. In witch the soil have the following properties corresponding to a dense sand: shear modulus of the skeleton G= 3.2175 x 107 N/m²; Poisson’s ratio ν=0.25; porosity n=0.35; density of the solid material ρs =1425 Kg/m3; density of the fluid ρf=1000 Kg/m3; permeability k=10-4m.s-1, correspond to a seepage force constant b=1.1986 x 107 N .s/m4); and Biot’s constants Q=4.61 x 108 N/m² and R=2.4823 x108 N/m². The results show a very good agreement can be observed between the two sets of results presented in fig. 2 which validate the model used. To show the importance of the assumed contact conditions, the forgoing analysis for k=10-3m/s was assuming completely different contact conditions. The boundary conditions along the contact surface between the soil and the foundation are welded or smooth for the skeleton. And either drained or undrained hydraulic condition for the fluid (i.e. previous or impervious foundation). Fig.3 shows the results for horizontal, vertical and rocking motion, the effect of the contact conditions is only important for the real part of the vertical stiffness. In the case of horizontal as well as rocking motion, the effect of the contact conditions is negligible. In the case of vertical motion, the foundation stiffness is increases when the foundation is welded to the soil and impervious. This effect increases with frequency. Next, an embedded foundation is considered. Perfect contact between the foundation and soil exists in this case and the adjacent soil moves with the foundation. Modelling of the soil surface surrounding the soil foundation is necessary. The variation of the vertical, horizontal and rocking motion amplitudes of the embedded rigid foundation with embedment ratio E/B=0 E/B=0.5 and E/B=1 respectively (fig.4) are done.
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4. Conclusion The frequency domain boundary element formulation presented in this paper can be successfully employed to solve linear poro-elastodynamic problems under plane strain/plane stress conditions and appears to be well suited for poro-elastodynamic foundation interaction problems. Thus, it requires discretization along the boundary only and it automatically accounts for the radiation condition. A small amount of free-field around the foundation has been discretized. Our results have been compared with available data in the literature and attest of the accuracy and computational efficiency of the formulation. Although our results are given here only for rigid, flat or embedded strip foundations in homogeneous visco poro-elastic soil, the approach can be extended to foundations on layered visco poro-elastic soil without many difficulties.
Fig .2. Comparison of the results
Fig .3. Type of the contact effect
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Fig .4 Embedement effect
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