Assessing the dynamic stiffness of piled-raft foundations by means of a multiphase model

Computers and Geotechnics 71 (2016) 124–135

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Research Paper

Assessing the dynamic stiffness of piled-raft foundations by means of a multiphase model Viet Tuan Nguyen, Ghazi Hassen, Patrick de Buhan ⇑ Université Paris-Est, Laboratoire Navier (Ecole des Ponts ParisTech, IFSTTAR, CNRS UMR 8205), 6-8 avenue Blaise Pascal, Cité Descartes, Champs-sur-Marne, 77455 Marne-la-Vallée Cedex 2, France

a r t i c l e

i n f o

Article history: Received 12 February 2015 Received in revised form 13 August 2015 Accepted 31 August 2015

Keywords: Piled raft foundation Impedance Dynamic stiffness Homogenization Multiphase model Soil–pile interactions Finite element method

a b s t r a c t The problem of evaluating the dynamic impedance of a vertically loaded piled raft foundation is investigated in this paper, based on the macroscopic description of the pile-strengthened soil as a twophase linear elastic continuum. Conceived as an extension of the classical homogenization approach, this multiphase model incorporates elastic interaction laws between the soil and the reinforcing piles, which can easily be identified from the solution to a specific auxiliary problem. The equations of elastodynamics associated with this model can then be implemented into a dedicated finite element numerical code, the use of which makes it possible to produce reliable and accurate predictions for vertical impedance of large pile groups in a much easier and quicker way than with direct numerical simulations. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Evaluating the dynamic stiffness of a foundation is a key ingredient to the analysis of soil–structure interaction effects [6,25]. The specific, but more and more frequently encountered case of piled raft or pile group foundations, has received increasing attention for a few decades, first through the vibration analysis and design of individual piles in the soil mass in the context for instance of pile driving techniques, then more recently in order to assess the improvement of the foundation dynamic performance to be expected from its reinforcement by a large number of piles placed under the footing. Several engineering design methods are based on an extension to dynamics of the so called simplified Winkler model, widely employed in foundation engineering design [21]: see among others Novak [16], Novak and Sheta [17], Gazetas and Dobry [7], Mylonakis and Gazetas [14]. According to this model, the piles are schematized as one dimensional beams, interacting with the surrounding soil through continuous distributions of elementary springs and dashpots. More recently, Pacheco et al. [18] have extended the model in order to introduce inertia effects of the soil through ‘‘soil lumped masses”. This kind of method can lead to ⇑ Corresponding author. E-mail address: [email protected] (P. de Buhan). http://dx.doi.org/10.1016/j.compgeo.2015.08.014 0266-352X/Ó 2015 Elsevier Ltd. All rights reserved.

analytical or semi-analytical formulations, at least for a small number of piles, but is facing two major difficulties. The first concerns the identification of the appropriate constitutive parameters to be assigned to the mass–spring–dashpot systems, whether this identification is being made from fitting model predictions with experimental or numerical (notably finite element and/or boundary element methods) results. It should be checked in such a case, that the identified parameters are really intrinsic and do not depend on each configuration. The second, and perhaps more fundamental criticism of the Winkler model is that the static as well as dynamic behavior of the soil mass as a loaded deformable continuum is not considered in such analyses, the mass–spring–dashpot systems behaving independently from each other. A more rigorous and comprehensive approach to the design of piled raft foundation under dynamic loading would consist in resorting to full numerical methods, such as the finite element and/or boundary element techniques, both the soil and the reinforcing piles being treated as continuous media in mutual interaction with one another through contact surfaces. Some examples of such simulations may be found in Kaynia and Kausel [11], Wu and Finn [26], Maeso et al. [13], Padron et al. [19, 20], Giannakou et al. [8]. Unfortunately, since such numerical methods prove to be highly computational time consuming as the number of piles increases, their use is most often limited to designing large projects and cannot therefore form the basis of a practical engineering design tool.

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as sketched in Fig. 1. In order to evaluate the vertical dynamic stiffness (or impedance) of such a reinforced structure, a harmonic uniform vertical rigid body translation of the form:

Making an intensive use of coupled finite element–boundary element methods, Taherzadeh et al. [24] have provided numerical evaluations for the dynamic stiffness of pile group foundations under dynamic combined loadings. They established simple user-friendly analytical formulas, the coefficients of which were identified from the previously determined numerical database. The range of applicability of such formulas is however restricted to the particular configuration of a rectangular group of endbearing or floating piles with homogeneous soil characteristics. This contribution advocates the use of an alternative approach based on the general concept of macroscopic behavior of the pile-reinforced soil, which more specifically pertains to the case when a relatively large number of identically oriented and evenly spaced piles is involved in the design of the foundation. Implementing this basic idea, the paper is organized as follows. Starting from a direct fem-based simulation of a piled raft foundation subject to harmonic vertical loading (Section 3), an alternative homogenization method is first proposed in which the reinforced soil is modelled as an anisotropic linear elastic medium (Section 4), resulting in a considerable reduction of the computational effort needed for evaluating the impedance of the foundation. Pointing out the limitations of such a homogenization procedure, which tends to overestimate the foundation dynamic stiffness, a multiphase model, already successfully developed for piled raft foundations under static loading conditions [2], is presented in Section 5 in the context of a linear elastic dynamic behavior. It is characterized by the introduction of two soil–pile interaction stiffness parameters which can be easily identified from the solution of a static auxiliary problem (Section 6). The implementation of this model in a finite element code is finally applied to an illustrative example in Section 7. This application gives clear evidence of the good performance of the so obtained numerical tool based on the multiphase model, in that it is able to provide accurate and reliable estimates for the foundation vertical impedance with dramatically reduced computational times as compared with direct numerical simulations.

dðtÞex ¼ d0 expðixtÞex

where d0 is the displacement amplitude and x the angular frequency, is classically prescribed to the footing, as shown in Fig. 1. The global response of the structure is expressed through the evolution with time of the vertical resultant force exerted by the footing on the reinforced ground, equal to:

Z

FðtÞ ¼ FðtÞex

FðtÞ ¼ F 0 exp½iðxt þ /Þ

O

z

ð3Þ

K¼

F F0 expði/Þ ¼ K 0 ðxÞ expði/ðxÞÞ ¼ d d0

ð4Þ

The dynamic structural stiffness K 0 and out-of phase angle /, which both characterize the overall steady state response of the piled group foundation under vertical harmonic loading, will now be computed as functions of the angular frequency x on the basis of the following set of hypotheses and data. The soil is modelled as an isotropic linear elastic medium with the following typical characteristics: Es ¼ 45 MPa;ms ¼ 0:3, while the reinforcing piles are made of a much stiffer linear elastic material (concrete, metal): Ep ¼ 20; 000 MPa;mp ¼ 0:2. Perfect bonding (i.e. no slip condition) is assumed between the piles and the surrounding soil mass, as well as between the rigid raft and the reinforced ground, so that (1) represents a displacement-prescribed boundary condition on top of the pile group. Three different configurations will be considered as regards the distribution of piles, the length of which is always kept fixed equal to L = 16 m, whereas their spacing s and radius q are varied proportionally, so that the reinforcement volume fraction defined as:

g¼p

q2

ð5Þ

s2

B' y

x

ð2Þ

Under such conditions, the vertical impedance (or dynamic stiffness) of the foundation is classically introduced, defined as:

B B

rxx ðy; z; tÞdS

where S is the area of contact between the footing and the reinforced ground. As soon as the harmonic regime is established, this resultant force is varying with the same frequency as the prescribed footing motion:

The problem to be investigated is that of a rigid square footing of side B acting upon a homogenous elastic soil layer of depth H, which has been preliminary (that is before the installation of the raft footing) reinforced by a group of periodically distributed vertical ‘‘floating” piles of length L (
exp(i t )

with FðtÞ ¼ S

2. Problem statement

0

ð1Þ

s

s

s

L

H B' Fig. 1. Square piled raft footing under vertical dynamic loading.

s

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remains constant, equal to 1.77%, for all the three configurations reported in Table 1 and Fig. 2, where the side of the raft footing has been taken equal to B = 10 m. This means that the same amount of reinforcement is introduced in the soil for the three configurations. The different calculations have been performed on a structure with a horizontal extension of B0 ¼ 20 m and a depth equal to H = 24 m (Fig. 1).

3. Performing direct simulations with a fem-based numerical code The dynamic behavior of the piled raft foundation subject to harmonic vertical loading is first investigated through a direct numerical simulation performed for the three examined configurations, making use of a three dimensional finite element code. Due to obvious symmetry properties, only the eighth of the structure with appropriate boundary conditions (namely smooth contact with rigid planes) needs to be modelled. Fig. 3 displays the three corresponding three-dimensional meshes made of tetrahedral elements with quadratic interpolation of the displacements. A particular refinement is adopted both inside the ends of the piles and in the surrounding soil. Table 2 summarizes the corresponding numbers of elements and nodes,

Table 1 Pile radius and spacing in the three considered configurations. Configuration

Number of piles

q (m)

s (m)

g (%)

I II III

9 16 25

0.25 0.1875 0.15

3.33 2.5 2

1.77 1.77 1.77

0.25m

0.1875m

0.15m

B

B

B

(I)

(II)

B

(III)

Fig. 2. Reinforcement layout under the footing (view from above).

(I)

(II) Fig. 3. Finite element meshes of the structure used in the computations.

(III)

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V.T. Nguyen et al. / Computers and Geotechnics 71 (2016) 124–135 Table 2 Mesh characteristics for the three configurations. Configuration

Nodes

Elements

I II III

28,866 39,381 61,138

19,812 28,084 43,556

the number of which is multiplied by a factor two between the 3  3 piles and the 5  5 piles configurations. The finite element formulation of the elastodynamics of such a system may be put in the following classic generic matrix form:

€ þ ½CfUg _ þ ½KfUg ¼ fFg ½MfUg

ð6Þ

where {U} is the vector of all nodal kinematic variables associated with the mesh, [M] the mass matrix, [K] the stiffness matrix and [C] is the Rayleigh damping matrix which has been conventionally taken equal to the following linear combination of the mass and stiffness matrices:

½C ¼ a½M þ b½K with a ¼ b ¼ 0:01

ð7Þ

It is to be noted that the same above relationship has been used in the forthcoming finite element simulations relating to the homogenized piled raft (Section 4) as well as to the structure modelled as a two-phase system (Section 7). Furthermore, in order to simulate horizontal radial damping, absorbing boundary conditions have been prescribed, namely those proposed by Lysmer and Kuhlemeyer [12]. They consist in applying dashpots on the lateral sides of the model, the corresponding viscous coefficients being equal to the soil mass density q times the P-wave (respectively S-wave) velocity for the normal (resp. tangential) dashpots. Again, since the primary objective of this contribution is to compare the performance of the different numerical simulations, based on the direct approach (this section), the homogenization method (Section 4) and the multiphase approach (Section 7), the same absorbing boundary conditions have been imposed on the three numerical models. Results of the different simulations performed in the frequency domain are shown in Fig. 4 in the form of curves giving the evolution of the non dimensional ratio K 0 =K s0 , where K s0 denotes the static stiffness of the un-piled raft foundation, as well as the out-of-phase angle /, as functions of the non dimensional frequency defined by:

 As could be expected, the stiffness of the foundation under quasi-static conditions (that is for a ? 0), is multiplied by a factor varying from 2.03 for a group of 3  3 piles (configuration (I)) to 2.18 for the group of 5  5 piles (intercepts of the impedance amplitude curves of Fig. 4). As an extension of this preliminary observation, its appears that the three impedance curves are relatively close to each other, the value of the impedance amplitude being always a (slightly) increasing function of the pile group density n = B/s for all the loading frequencies, except for a ffi 1.77.  The question of the amount of computational effort required for performing such calculations is essential. As an illustrative example, producing one single point on the lower impedance curve corresponding to configuration (I), i.e. n = 3, takes 10 min on a 2.1 GHz computer using the finite element code. The computational time amounts to 20 min for each point of the intermediate curve (n = 4) and up to 40 min to for upper one (n = 5). This could be easily explained by the fact that, as shown by Table 2, the size of the finite element mesh, and thus of the numerical problem, is quite significantly increasing with the density of the pile group. Therefore, one can easily imagine that for a foundation involving a large number of piles, drawing the entire impedance curve would take many hours, thus precluding the use of such a heavy numerical procedure for performing quick parametric analyses. 18 16

(I) n=3

14

(II) n=4 12

(III) n=5

10

K0 /K 0s 8 6 4 2 0 0

1

2

3

4

5

6

7

a

0.9

a¼

xB Vs

ð8Þ

0.8

where V s denotes the velocity of a shear wave propagating in the virgin soil of mass density qs ¼ 1750 kg=m3 , calculated as:

0.7

qffiffiffiffiffiffiffiffiffiffiffiffi V s ¼ Gs =qs ffi 100 m=s with

0.6

Gs ¼ Es =2ð1 þ ms Þ ¼ 17:3 MPa

0.5

ð9Þ

It is worth noting that this non-dimensional frequency could also be expressed as follows:

a ¼ 2pne

ð10Þ

where e is the ratio between the pile spacing s and the shear wavelength k ¼ 2pV s =x:

e ¼ s=k while n = B/s = 3, 4, 5 is the number of rows of the pile group. These results deserve the following comments:

(I) n=3 (II) n=4 (III) n=5

ð11Þ

(rad) 0.4 0.3 0.2 0.1 0 0

1

2

3

4

5

6

7

a

Fig. 4. Non dimensional impedance amplitude and out of phase angle as functions of the non-dimensional frequency for the three pile reinforcement configurations.

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V.T. Nguyen et al. / Computers and Geotechnics 71 (2016) 124–135

4. Homogenization-based approach In order to circumvent the above mentioned numerical difficulties attached to the use of direct simulations, a homogenization approach could be envisaged in which the composite reinforced zone is replaced by a homogeneous equivalent continuum, taking advantage of the fact that the pile arrangement is periodic, while the following condition are always satisfied (see values adopted above):

Es ffi 0:22%  1; Ep

and g ffi 1:77%  1

ð12Þ

Under such conditions, it can be shown [4] that the macroscopic elastic behavior of the reinforced soil, that is the behavior of the homogeneous equivalent medium, can be expressed as:

R ¼ ks ðtrEÞ1 þ 2Gs E þ aExx ex  ex ;

with a ¼ gEp

ð13Þ

where R and E denote the macroscopic stress and strain tensors, respectively, while a represents the axial stiffness of the piles per unit transverse area to the reinforcement direction Ox. The classical linear isotropic elastic behavior of the soil is recovered in the absence of reinforcement (a ¼ 0). Besides, due to the fact that the reinforcement volume fraction is small while the reinforcing material mass density is of the same order as that of the soil, the mass density of the homogenized reinforced soil could be could be approximated by that of the soil:

qhom ¼ gqp þ ð1  gÞqs qs

ð14Þ

Following the homogenization procedure, the reinforced zone located beneath the raft footing is replaced by a homogenized reinforced zone (Fig. 5) made of an equivalent elastic continuum the constitutive elastic behavior of which is given by Eq. (13). Since the homogenized structure displays exactly the same symmetries as the initial one, only the eighth of the homogenized piled raft foundation is considered in the dynamic analysis. The corresponding finite element mesh is represented in Fig. 6, comprising 3768 10-noded tetrahedral elements (that is 5846 nodes) allowing for a quadratic approximation of the displacement field. Unlike in the direct finite element analysis, the size of the mesh elements in the reinforced zone is in no way different from that required for the analysis a homogeneous foundation. Furthermore the same mesh can be used for the three configurations, which considerably reduces the preprocessing of the computations. The results of the numerical simulations performed in the frequency domain are represented in Fig. 7. The performance of such

Fig. 6. Finite element mesh of the homogenized structure.

a homogenization method in terms of computational times is quite apparent from this figure (red curves), since the determination of any point of the curves takes less than 30 s, which is considerably lower than the 10, 20 and 40 min necessary for obtaining the same kind of results through the direct simulations of configurations (I) to (III). It should be noted that the advantage of the homogenization approach as compared with direct simulations is all the more apparent as the number of piles is increased. On the other hand, comparing the values of the impedance obtained from the direct numerical simulations with those derived from the use of the above described homogenization procedure, it turns out that the latter tends to overestimate the former in the region of low frequencies (0 < a < 1) as well as for higher frequencies (a > 2). Far from being negligible, the difference between homogenization-based and direct predictions may be significant

homogenized reinforced zone

Fig. 5. Initial and homogenized piled raft foundation.

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V.T. Nguyen et al. / Computers and Geotechnics 71 (2016) 124–135

ject to quasi static conditions. According to this model, a detailed presentation of which can be found in Sudret and de Buhan [23], de Buhan et al. [4] and quite recently Bourgeois et al. [2,3], the composite reinforced soil is schematized as a two-phase system that is as the superposition of two continuous media, called matrix and reinforcement phases, the kinematics of each phase being

20 18 (I) n=3 (II) n=4

16

(III) n=5 HOM

14

characterized by a displacement field: nm for the matrix and nr for the reinforcement (Fig. 8). The dynamic equilibrium equations, expressed for each phase separately, may be written as:

12

K 0 /K 0s 10 8

divrm þ I ¼ qm @ 2 nm =@t2

6

divðnr ex  ex Þ  I ¼ qr @ 2 nr =@t2

4

where the gravity forces are neglected. In the above equations nr denotes the density of axial force in the reinforcement per unit transverse area to the piles orientation

2

0

1

2

3

4

5

6

ex, equal to the normal effort N in the reinforcing pile divided by the cross sectional area of the unit cell:

a

0 7

nr ¼

0.9 0.8

(I) n=3

0.7

(II) n=4 (III) n=5

ð15Þ

N s2

ð16Þ

while I (resp. I) is the interaction force density exerted by the reinforcement (resp. matrix) phase on the matrix (resp. reinforcement) phase. As it will be shown later on (see Eq. (30)), the reinforcement phase mass density, and thus the inertia term of the second dynamic equilibrium equation (15), can be neglected, so that the

HOM

0.6

interaction force density I is oriented along the reinforcement direction: I ¼ Iex . Within the framework of linear isotropic elasticity, the constitutive laws of the phases are expressed as:

0.5

(rad) 0.4 0.3

rm ¼ km trðem Þ1 þ 2Gm em

0.2

for the matrix phase, and

0.1

nr ¼ ar er

0 0

1

2

3

4

5

6

7

a

Fig. 7. Homogenization-based calculations (red curves) vs. direct evaluations of impedance amplitudes and phase differences as functions of the non-dimensional loading frequency. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

as illustrated in Table 3 which gives the different estimates for two particular values of the non-dimensional frequency. The homogenization procedure may overestimate the piled raft foundation dynamic impedance by more than 30% for a = 1.01 (f = 1.6 Hz). 5. A generalization of the homogenization approach: the multiphase model An extension and correlative improvement of the homogenization approach can be obtained through the adoption of a so-called multiphase model, which has been first developed for the design of reinforced soil structures, and notably piled raft foundations, sub-

Table 3 Estimates of the non dimensional foundation dynamic impedance K 0 =K s0 obtained from direct and homogenization-based simulations. a ¼ xB=V s

(I) n = 3

(II) n = 4

(III) n = 5

Homogenization

1.01 (f = 1.6 Hz) 3.28 (f = 5.2 Hz)

0.69 4.54

0.75 4.91

0.78 5.04

0.91 5.51

ð17Þ

ð18Þ

for the reinforcement phase, where km and Gm are the Lamé constants of the matrix phase and ar is the axial stiffness of the reinforcement phase, the strain variables em and er being defined from the displacement fields as:

em ¼ 1=2ðgradnm þ T gradnm Þ and er ¼

@nrx @x

ð19Þ

As regards the interaction forces, the constitutive law is expressed as a linear relationship between the interaction force density I and the difference between the displacements of the two phases along the reinforcement orientation:

I ¼ cI ðnrx  nm x Þ

ð20Þ

Moreover, a second kind of interaction forces corresponds to those developed at the pile lower tips located at the depth (x = L), modelled as a surface density of forces applied on the reinforcement phase by the matrix phase (Fig. 9):

nr ðx ¼ LÞ ¼

Nðx ¼ LÞ ¼ p s2

ð21Þ

This surface density of interaction forces generates a discontinuity of the vertical stress component across the surface in the matrix phase:

rmxx ðx ¼ Lþ Þ  rmxx ðx ¼ L Þ ¼ p

ð22Þ

so that the elastic constitutive law relative to such a pile tip interaction may be expressed as:

p ¼ cp ðnrx  nm x Þðx ¼ LÞ

ð23Þ

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B

B

y

O

y

O

L

L

x

x

soft clay

pile

matrix

m

s

s

reinforcem ent

r

Fig. 8. Principle of the multiphase model [2].

y

L

m

m

ex

interaction force density

V

I ex

I ex

ex

r

ex

0

pe x

pe x x

nr ex

x

tip interaction

Fig. 9. Interaction forces between matrix and reinforcement phases [2].

Remarks.  It is worth noting that the interaction constitutive laws (20) related to the pile shaft interaction and (23) concerning the pile tip interaction, are strongly reminiscent of the classical load– transfer curves employed in the piled raft foundation design methods. The main difference lies in the fact that the latter refer to individual piles, whereas the former are associated with a multiphase macroscopic description of the pile-reinforced soil.  It is worth noting that, applying the asymptotic homogenization technique [1] to the derivation of the effective axial dynamic behavior of linear elastic fiber-reinforced materials, Soubestre and Boutin [22] have obtained macroscopic equilibrium equations which appear to be quite similar to those of the above described multiphase model in the particular situation of a sufficiently high stiffness contrast between the soil and the pile. 6. Identification of the multiphase model constitutive parameters First of all, it is important to observe that the above described multiphase model can be considered as an extension of the homogenization approach (de Buhan and Hassen [4] for a comprehensive analysis in the case of static or quasi static loading conditions). Indeed, assuming that both phases are given the same kinematics, which can be achieved when the interaction stiffness parameter tends to infinity (perfect bonding condition1):

cI !1

r m m r I ¼ cI ðnrx  nm x Þ ! nx ¼ nx ! n ¼ n ¼ n

and summing up Eqs. (15), while using the constitutive relationships (17) and (18), allows to recover a single global dynamic equilibrium equation:

divR ¼ ðqm þ qr Þ@ 2 n=@t 2

ð25Þ

along with a macroscopic elastic constitutive law:

R ¼ rm þ nr ex  ex ¼ km ðtrEÞ1 þ 2Gm E þ aExx ex  ex

ð26Þ

since, due to the perfect bonding condition (24):

em ¼ 1=2ðgradn þ T gradnÞ ¼ E and er ¼

@nx ¼ Exx @x

ð27Þ

As a consequence, the different elastic parameters appearing in the constitutive equations of the matrix and reinforcement phases may be determined in a straightforward way from the homologous characteristics of the soil and the individual reinforcing inclusions. Thus, as a result of identifying (13) and (26), the axial stiffness of the reinforcement phase is simply calculated as the stiffness of one individual inclusion per unit area transverse to the pile orientation:

ar ¼ a ¼ gEp

ð28Þ

while the elastic constants of the matrix phase may be identified with those of the soil:

km ks ; 1 It should be emphasized that such a perfect bonding condition is expressed at the macroscopic scale of the two-phase model, and not at the soil–pile interface as such [4].

ð24Þ

Gm Gs

ð29Þ

Meanwhile, the mass densities of the matrix and reinforcement phases involved in the dynamic equilibrium equations (15) are simply evaluated as:

V.T. Nguyen et al. / Computers and Geotechnics 71 (2016) 124–135

qm ¼ ð1  gÞqs ffi qs ; qr ¼ gqp ffi 0 since q  1

ð30Þ

so that:

qm þ qr ¼ qhom qs

ð31Þ

On the other hand, identifying the interaction stiffness parameters cI and cp, appearing in the constitutive equations (20) and (23), proves to be a bit more complex task. The procedure adopted under quasi-static conditions by Bourgeois et al. [2] or Hassen et al. [10] for piled embankments, will be directly applied here, assuming that the stiffness interaction parameters cI and cp will remain independent of the solicitation frequency, in exactly the same way as the elastic constants of the matrix and reinforcement phases. As explained with full details in Bourgeois et al. [2], the numerical determination of the matrix-reinforcement interaction coefficients is based on a comparison between the direct numerical solution and the multiphase-based analytical simulation of an auxiliary problem, defined as follows (Fig. 10(a)). A soil layer of thickness H ¼ L þ h and infinite extension in the horizontal directions, has been reinforced by a periodic distribution of vertical piles of length L, each individual pile being subjected on its head to the same vertical loading Q. The soil and piles are given the same mechanical and geometrical properties as those selected at the beginning of the paper. The evaluation of the settlements profiles as well as the determination of the axial force distributions along the piles can be achieved either through a finite element simulation performed on a representative unit cell (Fig. 10(c)) or by considering the equivalent two-phase system represented in Fig. 10(b), for which an analytical solution can be developed [2]. Again, the solution to the unit cell problem of Fig. 10(c) is obtained numerically by means of the finite element code using

131

the mesh of Fig. 10(d). On account of the symmetry of this problem, only the eighth of the unit cell is considered with the following boundary conditions: smooth contact with a rigid fixed plane of the lateral and bottom surfaces of the unit cell; upper end pile surface subjected to a vertical uniform pressure equal to Q =ðpq2 Þ, while the soil surface remains stress free. In all three configurations, a uniform pressure of Q =ðpq2 Þ ¼ 1 MPa was applied on the upper section of the pile, so that the corresponding stress in the reinforcement phase is equal to:

nr ðx ¼ 0Þ ¼ q ¼ Q =s2 ¼ ðQ=pq2 Þg ¼ 0:0177 MPa

ð32Þ

The stress and settlement values are calculated at different depths, in the pile as well as the soil, either analytically by the multiphase method or numerically by the finite element method. The interactions parameters are then determined by applying a least square method aimed at determining the couple (cI, cp) for which the analytical solution best fits the numerical points. This procedure leads to the following results summarized in Figs. 11 and 12, as well as in Table 4. Fig. 11 displays the axial force distributions along the pile length for the three configurations, the symbols corresponding to the results of the finite element simulations, while the solid curves represent the best fitted analytical simulations using the multiphase model. As can be clearly seen from this figure, the axial compressive force in the piles is decreasing from its maximum prescribed head value (32), to a non-zero value approximately equal to 0.0035 MPa at the piles lower tip. The corresponding settlement profiles in the reinforcement phase evaluated from the numerical (points) and analytical (solid curves) simulations are shown in Fig. 12. It clearly demonstrates that, for a given amount of reinforcement placed into the soil

Fig. 10. (a) The auxiliary problem with its (b) multiphase model, and (c) numerical model with its (d) mesh discretization.

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V.T. Nguyen et al. / Computers and Geotechnics 71 (2016) 124–135

stiffness coefficients leading to the best fitting between the numerical points and the multiphase analytical curves parametrized by these coefficients, are reported in Table 4. It turns out from looking at the last two columns that cI/s2 and cp/s are almost constant. It follows that, for the selected set of characteristics, the interaction stiffness coefficients may be written in the form:

axial force (MPa) -0.000 0

-0.004

-0.012

-0.008

-0.016

2

depth (m)

4 6

cI ffi 75:8=s2 ;

8

12

n 5

14 16 Fig. 11. Axial force distributions along the reinforcing piles.

settlement (mm) 2.7

3.3

3.1

2.9

3.5

3.7

0 2

depth (m)

4 6

n 5

8

n 3

10

n 4

12 14 16

Fig. 12. Settlement curves for the three pile reinforcement configurations.

Table 4 Identification of the pile shaft and pile tip interaction coefficients as functions of the pile spacing. Configuration

s = B/n (m)

cI (MN/m4)

cp (MN/m3)

cIs2 (MPa)

cps (MPa)

I II III

3.33 2.5 2

6.68 12.50 18.82

5.26 7.40 9.69

74.1 78.1 75.3

17.5 18.5 19.4

(characterized by the same volume fraction for all three configurations), the average pile settlement is decreased by almost 10% when using a 5  5 piles configuration instead of the 3  3 piles configuration. From a general point of view, the matrix-reinforcement interaction stiffness parameters may be expressed as functions of the pile and soil elastic properties, as well as geometrical characteristics such as pile radius and spacing. As a result of simple dimensional analysis [2], these parameters may be written in nondimensional form as:

cI ¼

Es I g s2



ms ;

 Ep p s ;m ;g ; E

cp ¼

Es p g s



ms ;

 Ep p s ;m ;g E

ð34Þ

It follows that in the present situation one single finite element calculation is sufficient for evaluating both interaction stiffness coefficients for any value of the pile spacing. Furthermore, Eq. (33) show that, in the most general case, the determination of these coefficients comes down to establishing analytical formulas giving the non-dimensional coefficients gI and gp as functions of the reinforcement volume fraction, pile to soil stiffness ratio and Poisson’s ratios, as it has been done for instance for piled embankments [5,10].

n 3

10

cI ffi 18:5=s

7. Application to the dynamic analysis of the piled raft foundation A three dimensional finite element formulation of the piled raft foundation, modelled as a two-phase system, has been obtained in the case of static [2] as well as dynamic [15] loading conditions. The same finite element mesh as that employed in the homogenization method (see Fig. 6) comprising quadratic tetrahedral elements, can be used, with an important difference in the two-phase reinforced zone. Indeed, four instead of three, degrees of freedom are attached to each node of the mesh, namely the two horizontal displacements, which are the same for the matrix and the reinforcement phases, along with their two different vertical displacements (Fig. 13). Referring to the general matrix form (6) of the discretized dynamics problem, it should be noted in particular that the stiffness matrix [K] includes terms relating to both individual phases along with terms associated to the interactions between phases. A specifically dedicated computer code has been developed based on the previous fem formulation, and used for simulating the steady state dynamic response of the above described piledraft foundation in the three configurations of Table 1. The corresponding interaction stiffness parameters are those summarized in Table 4. The results of all simulations are reported in Fig. 14 in the form of non-dimensional impedance amplitude (and out-of-phase angle) vs. frequency curves relative to the three configurations, obtained from the direct simulation, the homogenization method and the multiphase model. It can be immediately seen that the results obtained through the multiphase approach are much closer to those of the direct simulations than those produced by the homogenization method which almost always tends to significantly overestimate the foundation impedance.

y

y

z

ð33Þ

where gI and gp are non-dimensional functions of four dimensionless variables. This implies that, all the other parameters being kept constant (and notably the reinforcement volume fraction g), the pile shaft interaction coefficient cI (resp. pile tip interaction coefficient cp) is directly proportional to 1/s2 or n2 (resp. to 1/s or n). This kind of variation is pretty well confirmed by the previously performed numerical simulations. The values of the interaction

r xy

mm yx

z

x Fig. 13. Nodal degrees of freedom of the two-phase system.

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0.9

20 18 16

Mult.

HOM

0.7

(II) n=4

Mult.

14

0.6

12

0.5

K0 /K 0s 10

(rad)

8

0.4 0.3

6

0.2

4

0.1

2 0 0

1

2

3

4

5

6

0

a

7

0

1

2

3

4

5

6

7

a

20

0.9 0.8

18

(I) n=3 HOM Mult.

0.7

HOM

16

Mult.

14

0.6

(rad)

HOM

0.8

(I) n=3

(II) n=5

12

0.5

K0 /K0s

0.4

10 8

0.3

6

0.2

4

0.1

2

0 0

1

2

3

4

5

6

7

0

a

0

20

1

2

3

4

5

6

a

7

0.9

18

HOM

0.8

Mult.

16

(II) n=4

HOM Mult.

0.7

14

(III) n=5

0.6

12

0.5

K0 /K 0s 10

(rad) 0.4

8

0.3

6 4

0.2

2

0.1

0 0

1

2

3

4

5

6

7

a

0 0

1

2

3

4

5

6

7

a

Fig. 14. Non dimensional impedance amplitude and out-phase angle as functions of dimensionless loading frequency evaluated using the direct method, the homogenization method and the multiphase model.

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Table 5 Comparison between direct, homogenization and multiphase evaluations of the piled raft foundation impedance amplitude. a

n=3

n=4

direct/mult./hom.

direct/mult./hom.

1.01

0.69

0.71 (2.8%)

0.91 (32%)

0.75

0.755 (0.6%)

0.91 (21%)

0.78

0.79 (1.3%)

0.91 (17%)

3.28

4.54

4.90 (8%)

5.51 (21%)

4.91

5.1 (3.8%)

5.51 (12%)

5.04

5.22 (3.6%)

5.51 (9.3%)

This point is illustrated by Table 5 where all the calculated impedances have been reported for the already considered two particular frequencies. It thus appears that in the domain of relatively low frequencies (a = 1.01), the dynamic impedances calculated through the multiphase model are less than 3% higher than the evaluations given by the direct approach taken as reference values, while the homogenization-based estimates are 17–32% higher. For higher frequencies (a = 3.28), the relative gap varying from 9.3% to 21% observed for the evaluations of the homogenization approach with respect to those of the direct one, is still significantly reduced to 3–8% when making use of the fem-implemented multiphase model. As regards the performance of the multiphase approach in terms of computational time reduction, the computational time required for calculating the piled foundation impedance for a particular loading frequency remains lower than 30 s which is almost the same as for the homogenization-based finite element method (see Section 4). This means that drawing the entire impedance curve of a given piled raft foundation takes no more than a few minutes whatever the number of reinforcing piles, instead of several hours for a group of 5  5 piles when using a direct numerical simulation.

n=5 direct/mult./hom.

formulas such as (33) or (34). Since it has been clearly demonstrated in this contribution that the stiffness coefficients relating to the soil–pile interactions do not significantly depend on the loading frequency, the latter formulas can be established from the numerical solution of a linear elastic static auxiliary problem, such as that displayed in Fig. 10. Resulting from the finite element implementation of the multiphase model described in this paper, a numerical code has been set up making it possible to undertake parametric analyses concerning the dynamic behavior of piled raft foundations in a rapid, efficient and reliable way, which remained so far out of range of direct numerical simulations. While such a code, along with the multiphase model of pile reinforced soil upon which it is based, has been validated in the case of vertical loading from the comparison with direct simulations, its ability to assess the impedance of this kind of foundations subject to more complex combined loadings, such as horizontal or rocking-type solicitations, remains to be further investigated. As already established in the particular case of static loading conditions [9], an enriched multiphase model accounting for the pile shear and flexural behavior, in addition to the axial one, has to be developed [15]. References

8. Concluding remarks Perceived as an extension of the classical homogenization procedure, a multiphase model, previously developed in the context of static loading conditions, has been successfully applied to the design of piled raft foundations subject to vertical dynamic loading. Confirming what has been already observed in static or quasi-static conditions, the decisive advantage of this model, incorporated into a finite element formulation, over a classical direct simulation where the soil and the reinforcements are treated as geometrically distinct elements, is twofold. In the first place, contrary to the direct numerical approach, which requires a highly refined mesh discretization of the reinforced soil region, thus leading in some cases to an oversized numerical problem, the multiphase model allows for a treatment of the reinforced soil structure as if it were homogeneous, with no more need of a highly refined mesh as in the direct simulation. As it has been clearly shown on the illustrative examples presented in this paper, the computational effort corresponding for instance to the determination of the vertical dynamic impedance of a piled raft foundation, is dramatically reduced by a factor almost equal to one hundred in the case of a square raft reinforced by a 5  5 pile group, with a probably further increase of this factor with the number of piles. Secondly, but still importantly, the meshing procedure associated with the use of the multiphase simulation tool, is considerably alleviated, since no tedious re-meshing of the structure is necessary when changing some key parameters of the reinforcement layout such as the pile diameter or spacing. Indeed, only two constitutive stiffness parameters of the multiphase model ought to be modified in such a case: the axial stiffness of the reinforcement phase on the one hand, through a simple modification of the reinforcement volume fraction (see Eq. (28) of the main text), the interaction stiffness parameters on the other hand, derived from

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