SEVENTH FRAMEWORK PROGRAMME

Capacities Specific Programme

Research Infrastructures

Project No.: 227887

SERIES SEISMIC ENGINEERING RESEARCH INFRASTRUCTURES FOR

EUROPEAN SYNERGIES

Work package [WP8 – TA4 UNIVBRIS]

SMELI Dynamic Behaviour of Soils Reinforced with Long Inclusions (Piles)

- Final Report -

User Group Leader: Prof. C. Boutin

Revision: Final

January 2013

SERIES 227887 SMELI Project

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ABSTRACT

The document reports the outcomes of an experimental test campaign for the validation

of the dynamic response modelling of group of piles. To describe these complex piles/soil

interactions under dynamic loadings, a modelling based on homogenization has been

developed in the laboratory DGCB/CNRS of the Ecole Nationale des Travaux Publics de

l’Etat (Université de Lyon). Materials are assumed linear elastic as usual in geophysics

under sufficiently small deformations. This approach accounts for the pile group effect

(soil is reinforced by a periodic distribution of linear inclusions) and includes the key

physical parameters, i.e. the stiffness soil/pile ratio and the dimensions and the density of

the piles lattice. For the normal range of problematic soils and current deep foundation,

considering transverse motions (more critical under earthquakes) it is shown that the co-

existence of the mechanisms of shear in the soil and bending in the piles leads to a non-

conventional behaviour where both deformation and curvature equally govern the stress

state (see, [Sudret, deBuhan 1988], [de Buhan, Hassen, 2008], [Soubestre, 2011],

[Boutin, Soubestre, 2011], [Soubestre, Boutin, 2012]. As a consequence, the dynamic

behaviour is qualitatively and quantitatively modified compared to un-reinforced soils.

Following these theoretical results, it has been proceed to experimental validation using

the shaking table facilities existent at University of Bristol. Within the SERIES project

(Seismic Engineering Research Infrastructures for European Synergies), a number of

careful designed experiments on analogue materials reinforced with linear inclusions

were performed, cf [Soubestre et al., 2012]. The experiments naturally include the pile

group effect and the analysis of the response in the spectral domain provided valuable

information on the actual bending effect due to the reinforcement under transverse

motions.

Keywords: Piles group foundations, Inner bending media, Experimental Analysis, Numerical Simulations, Shaking Table Tests

SERIES 227887 SMELI Project

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SERIES 227887 SMELI Project

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ACKNOWLEDGMENTS

The research leading to these results has received funding from the European Union

Seventh Framework Programme [FP7/2007-2013] for access to the University of Bristol

EQUALS laboratory under grant agreement n° 227887 [SERIES].

The team of the project would also like to thank Luiza Dihoru, Frédérique Sallet,

Edward Skuse and David Ward for their very active technical support. The gathered

outcomes result from the incomparable competency and reliability of Matt Dietz in

managing the subtleties of the shaking table and of the data process.

SERIES 227887 SMELI Project

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SERIES 227887 SMELI Project

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REPORT CONTRIBUTORS

Ecole Nationale des Travaux Publics

de l’Etat, Université de Lyon/CNRS

Claude Boutin, Stéphane Hans, Jean

Soubestre

University of Bristol Matt Dietz, Erdin Ibraim, Colin Taylor

Technical University of Lodz Marek Lefik, Marek Wojciechowski

Technical University of Civil

Engineering of Bucharest

Loretta Batali, Horatiu Popa

University of Iceland Jonas Snaebjornsson

Engineering and Technical Staff

Ecole Nationale des Travaux Publics

de l’Etat, Université de Lyon/CNRS

Frédérique Sallet

University of Bristol Luiza Dihoru, David Ward,

Edward Skuse

SERIES 227887 SMELI Project

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CONTENTS

Contents………………………………………………………………………………….vi

1 Brief Overview of the Project………………………………………………….……5

1.1 Theoretical modeling of a reinforced layer………………………………………5

1.2 Constitution of specimens………………………………………………………..6

1.3 Instrumentation and experimentations…………………………………………...6

2 Detailed description of the experimental tests campaign………………………….7

2.1 Physical model…………………………………………………………………...7

2.1.1 Constitutive materials……………………………………………………....7

2.1.2 Design of the sample of reinforced media…………………………….…...7

2.2 Experimental procedure……………………………………………………….…8

2.2.1 Implementation and tested configurations………………………………....8

2.2.2 Measurements…………………………………………………………….10

2.2.3 Shaking table tests………………………………………………………...11

3 Theoretical analysis of the experimental setup…………………………………...13

3.1 Transverse "homogeneous" modes………………………………………...........13

3.1.1 Boundary conditions……………………………………………………....13

3.1.2 Modal characteristics………………………………………………….…..14

3.2 Theoretical modal characteristics of the different tested configurations…...........15

4 Experimental results versus theoretical predictions……………………………....18

4.1 Checking of linearity and homogeneity………………………………………...18

4.2 Frequency and mode shape of the fundamental mode……………………….....18

4.2.1 Identification of fundamental frequency and mode shape…………..........18

4.2.2 Experimental versus theoretical fundamental frequencies…………..........20

4.2.3 Experimental versus theoretical fundamental mode shapes………….…...21

4.3 Analysis of inner bending……………………………………………………….22

4.3.1 White noise response……………………………………………...............22

4.3.2 Inner bending involved in the fundamental mode………………………...25

4.4 Higher modes…………………………………………………………………....27

4.4.1 Second homogeneous mode……………………………………………....27

4.4.2 Inhomogeneous mode………………………………………………….....30

5 Conclusion…………………………………………………………………..........33

6 Scientific dissemination……………………………………………………..........35

References…………………………………………………………………………..…..37

Appendix: Non-homogeneous transverse modes…………………………………….....38

1. Brief overview of the Project

The document reports the outcomes of the experimental test campaign on dynamicresponse of group of piles realized on the shaking table facilities of University of Bristol withinthe SERIES project (Seismic Engineering Research Infrastructures for European Synergies)cf. [Soubestre et al, 2011]. The main objective of the tests is to identify and quantify theactual bending effect due to reinforcement under transverse motions. The focus of thisreport is to underline the scientific learning issued from the careful designed experiments onanalogue materials reinforced with linear inclusions. The interest for earthquake engineeringlies in a deeper understanding and, in turn, a better account of Soil-Structure-Interactionfor group of piles through an improvement of the design methods. These aspects are goingto be discussed in the following. Everything reported herein aims to summarize and finalizeour research on this topic. In order to highlight the bending effect, four types of boundary

Fig. 1: Fiber reinforced material. (a) Periodic lattice of parallel identical homogeneousstraight beams embedded in a matrix. (b) Period geometry and dimensions.

conditions were experimented and compared. The actual impact of the matrix/inclusionstiffness ratio were investigated by changing the reinforcement concentration.

1.1. Theoretical modeling of a reinforced layer

Recall first the theoretical homogenized modelling to be validated experimentally, [Sou-bestre, Boutin, 2012]. In harmonic regime at the circular frequency ω, the homogeneoustransverse motion of a reinforced media of infinite lateral extension is governed by the fol-lowing equation, where U is the transverse motion, EpIp is the inertia of piles, S the periodarea of the periodic reinforcement pattern and 〈ρ〉 the mean density :

−Ep Ip

|S|d4 U

dx4+ G

d2 U

dx2+ 〈ρ〉 ω2 U = 0 (1)

The shear modulus G matches the shear soil modulus with an excellent approximation forweak reinforcement concentration as usual in practice. For a layer of reinforced material, this

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equation, together with the boundary conditions (e.g. clamped, free) enables to determine thetheoretical eigen-modes and frequencies. These latter can also be identified experimentally.The comparisons between experiments and theory, mainly based on the fundamental moderesponse of the system, have been used to assess the validity the modelling.

1.2. Constitution of specimens

The leading objective of the project is the identification of the inner bending phenome-non, in the deformation range where the soils behaves quasi-linearly. For this reason theexperiments use analogue elastic materials insuring the respect of the basic assumptionsof the theoretical modelling. This choice avoids the inherent difficulties of using sand (orother reconstituted soil) among which, possible vertical gradient of properties, lateral inho-mogeneities close to the inclusions, limited domain of linear behaviour that would imposemuch smaller deformations and consequently less accurate measurements, complex scalingof properties due to the small scale specimen,...

The chosen analogue elastic materials are a polyurethane foam for the matrix and steelfor the cylindrical inclusions. The perfect adherence between the matrix and inclusion isobtained by inserting the inclusions in the matrix after drilling holes of slightly smallersection.

1.3. Instrumentation and experimentations

Experiments were realized on the EQUALS shaking table. Tests under white noise witha sufficiently wide spectrum enable to determine the fundamental and few higher eigenfrequencies. They have been complemented by harmonic tests at the fundamental frequencyfor an accurate mode shape determination. The level of the excitation is chosen sufficientlysmall for satisfying the assumption of small elastic deformations while sufficiently large fora good accuracy of the measurements. 3D accelerometers were located on the specimen inorder to identify eigen frequency and modal deformation of the matrix/inclusions system.The inclusions were instrumented by strain gauges, and several LVDT were also installed.

These several points are presented in detail in the next section.

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2. Detailed description of the experimental tests campaign

Laboratory Experiments were carried to observe the phenomena predicted by the homo-genised model. Since the presence of fibers alter the eigen frequencies and transverse modeshapes of the soft matrix, a physical model was prepared in order to conduct shear vibrationtests and modal analysis using a shaking table. The experiments were performed within theframework of the European SERIES project at the University of Bristol’s Earthquake andLarge Structures (EQUALS) laboratory.

The performance of the EQUALS shaking table is optimised with a test specimen ofmetric dimensions that produce a frequency range of about 0.5-25 Hz. Such characteris-tics facilitates (i) measurements with sensors of usual centi-metric dimensions distributedon the surface or in the composites, (ii) the realization of mechanical devices reproducingdifferent kind of boundary conditions, (iii) records of dynamics phenomena with reasonabletime sampling.

2.1. Physical model

2.1.1. Constitutive materials

The materials match the basic assumptions of the theoretical modelling, namely : largestiffness contrast between the matrix and reinforcement, isotropic linear elastic materialsand a perfect adherence at their interface.

The matrix is a made of polyurethane foam of density ρm = 48 kg/m3. Its mechanicalproperties foam was measured using standard laboratory equipment adapted to soft mate-rials. The foam exhibited linear elastic behaviour up to 4 − 5% of axial strain. A Young’smodulus Em = 54 kPa and Poisson’s ratio νm = 0.11 were derived, which gives a shearmodulus Gm = 24.3 kPa. The experiments are conducted with a global distortion level ofabout 0.1% to ensure that the foam remained within its linear elsatic range.

The reinforcements are made of round mild steel seamless tube, the mechanical proper-ties of which are, classically, Young’s modulus Ep = 210 GPa, Poisson’s ratio νp = 0.3 anddensity ρp = 7800 kg/m3.

2.1.2. Design of the sample of reinforced media

The physical model is designed so that both bending and shear mechanisms occur whenthe reinforced layer reach its resonance. The dimensional analysis of the governing equationof shear dynamics (1) gives the following relations in order of magnitude :

Ep Ip = O(G S H2) = O(〈ρ〉 S H4 ω2

)(2)

The footprint of the test specimen (2.13 m× 1.75 m) was chosen so as to not clutter up the3m x 3m shaking table. A 25 cm spacing between fibers permits a reasonable periodicitywith 35 fibers giving the surface value S = 0.25 × 0.25 m2. The polyurethene and steelmaterials establish fix the mechanical properties Gm, ρm, Ep and ρp. The shear modulusG of the composite is approximatively equal to the that of the polyurethene matrix one :G ≈ Gm = 24.3 kPa. Setting the steel tube fibers diameter and wall thickness respectivelyto 12.7 mm and 3.25 mm, it fixes the mean density 〈ρ〉 and the fibers inertia Ip. Then, with

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Fig. 2: Experimental reinforced media fixed on the shaking table.

a model height of H = 1.25m, the left hand side of the relation (2) is satisfied and the righthand side of (2) gives a fundamental frequency less than 10 Hz, appropriate for a accuratecontrol of the shaking table.

2.2. Experimental procedure

2.2.1. Implementation and tested configurations

With this design, the 2.13 × 1.75 × 1.25 m3 tall foam matrix weighs about 225 kg whileeach steel tube fiber, of 1.3 m lengths, weighs about 1 kg. Fibers are inserted in periodicallydistributed holes drilled in the matrix with a diameter of 1 mm smaller than the fiberdiameter in order to have a good matrix/fiber adherence. Two concentrations with periodicarrangements were tested (see Figure 3) :- 35 fibers distributed on a 7 by 5 square grid of 250 cm side, giving a fiber concentration cof 0.2%,- 17 fibers distributed in staggered rows at

√2×250cm centres (obtained by removing every

second fiber), giving c = 0.1%,- 9 fibers distributed in square grid at 500mm centres (obtained by taking out three in everyfour fiber), giving a fiber concentration c = 0.05%. In the following, the experiments in thisconfiguration will not be commented since they lead to similar conclusions than 35 and 17inclusion configurations, with less marked effects of the fibers.For both fiber concentration, the four pairs of boundary conditions defined on paragraph 3were tested. For the clamped-free boundary condition, the fibers were bolted to the base-plate. For the clamped-translational boundary condition, a light (about 7.25kg) rigid lattice

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Fig. 3: Schematic views of the physical model with 35 inclusions

was fixed at the top of the fibers to avoid the rotation and just allow the translation (Figure2-b). The rotational condition at the bottom of the fibers, was realized by placing a steelball between the base-plate and the fiber to allow rotation without translation.

Fig. 4: Rigid lattice of aluminium bars suppressing the rotation at the top of the inclusions.(a) Schematic top view. (b) Photography of the system. (c) Zoom.

In all cases, the specimen is secured on the shaking table via a base-plate to which thefoam matrix is adhered. Figure 2 displays an overview of the model settled on the shakingtable. Note also that a standalone fiber and the foam matrix without fibers were also tested.

In the following a given configuration is designed by the number of fiber and the type of

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boundary condition, e.g., 35CF, or 17RF. Tests 0CF and 1CF refer respectively to the foammatrix without fiber, and to the standalone fiber.

Tab. 1: Notation of the different experimental configurations.

Number of inclusions35 17 1 0

Boundary conditions

Clamped-Free 35CF 17CF 1CF 0CFClamped-Translational 35CT 17CT

Rotational-Free 35RF 17RFRotational-Translational 35ART 17RT

Fig. 5: Schematic inclusions distributions. (a) 35. (b) 17. (c) Unreinforced block of foam.

2.2.2. Measurements

Measurands are acceleration and strain. Single-axis accelerometers (SETRA 141A) aremounted on the shaking table, on the uppermost surface of the matrix, and on the 50 mmfree length of the fiber protruding from the top of the matrix, and on one vertical face of thesample. The accelerometers were to (i) to check the uniformity of the fiber and matrix motionon the horizontal free surface, (ii) to determine the response spectrum of the composite layerand identify its fundamental (and higher ) frequency, and (iii) to obtain an assessment ofthe shape of the fundamental mode (from the three accelerometers mounted on the verticalface, at 31.2 cm, 62.5 cm from the base and 121.1 cm, i.e. at the top).

Six fibers were instrumented using Vishay strain gauges (CEA Serie, 120ohm, 3mm long)to monitor bending. The instrumented fibers are each equipped with 3 pairs of longitudinalstrain gauges located at three vertical z ordinates. To determine the bending for each pair,one gauge is faced in the +Y direction, the other in the -Y direction. The 3 pairs of gaugeare located respectively (i) at the fiber bottom (3.85 cm from the base) denoted B+/-,(ii) at the middle (62.5 cm from the base) denoted M+/- and (iii) at the top (121.15 cmfrom the base) denoted T+/-. To secure the wires, the strain gauge cabling is fed into theinterior of the fiber through small holes (≈ 1 mm diameter) drilled through the wall. In

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Fig. 6: Accelerometers. (a) On the uppermost foam block surface. (b) On the free length ofa reinforcement.

addition, the signals resulting from the strain gauges provide an accurate measurement ofeigen frequencies. Sensors positions are similar for each experiments. An example is displayedin the Figure8 for the configuration 35CF.

Fig. 7: Strain gauges. (a) Lengths of reinforcement instrumented with six strain gauges. (b)Cabling.

2.2.3. Shaking table tests

The shaking table of the Bristol Laboratory for Advanced Dynamics Engineering (BLADE)at the University of Bristol was employed. This apparatus consists of a 6 tonne, 3m by3m cast-aluminium seismic platform with 21 tonne payload. Mounted within a 100 tonneconcrete block that is itself secured to bedrock, the platform is driven by eight 70kN servo-hydraulic actuators of 0.3m stroke capable of giving full control of motion in all six degreesof freedom simultaneously. Hydraulic power for the actuators is provided by five pairs ofhydraulic pumps capable of delivering 900 l/min at a working pressure of 230bar.The principal of the test is simple. The sample is firmly clamped on the table. The shakingtable imposes a horizontal uni-directional rigid body motion. The sample responds to the

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Fig. 8: Sensors positions and numbering for the configuration 35CF.

motion imposed at its base and its motion is recorded using the sensors. Two types of shakingtable signals are used to drive the table in the long direction of the specimen : white noiseand harmonic sinusoidal waveforms. Different magnitudes (0.1 g maximum) of white noiseexcitations (frequency content between 1 and 60Hz) enables identification of the first eigenfrequencies of the model. Harmonic sinusoidal waveforms are used to excite the model at itseigen frequencies for accurate determination of the mode shapes. The records are treatedusing the classic tools of signal processing, mainly the Fast Fourier Transfrom.

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3. Theoretical analysis of the experimental setup

In view of experiments designed to illustrate the consequences of the non-local transversebehavior and the influence of the complementary boundary conditions, we investigate themodes of a reinforced domain of finite thickness H in the direction of the beams (0 ≤ x1 ≤H), and of infinite or finite lateral extension 2D. This latter dimension is assumed sufficientlylarge compared to the period size (2D ≫ l) to apply the homogenized modelling. Due tothe imposed horizontal motion of the shaking table, we focus on the eigen modes associatedto such a kinematic on the bottom level, that may be split in transverse homogeneous andnon-homogeneous modes, see [Soubestre, 2011], [Soubestre, Boutin, 2012]. Motions in theform U0

α(x1)aα correspond to a uniform in-plane macroscopic displacement, and are designedas "homogeneous" modes, while "inhomogeneous" modes correspond to motions in the formU0

α(x)aα (see Appendix).

3.1. Transverse "homogeneous" modes

Transverse "homogeneous" modes involves horizontal motions in the form Uα(x1)aα.Without loss of generality, we consider transverse modes polarized in the direction a

2, i.e.

U0 = U0

2(x1)a2

. Thus, using the lightened notations U0

2= U ; x1 = z, 〈σ̃2〉21 = τ , it is shown

in Soubestre and Boutin (2012) that the homogeneous transverse modes are driven by thefollowing scalar equations (with usual notations) :

dτ

dx= −〈ρ〉 ω2 U ; τ = G

d U

dx− Ep Ip

|S|d3 U

dx3

that gives :

−Ep Ip

|S|d4 U

dx4+ G

d2 U

dx2+ 〈ρ〉 ω2 U = 0 (3)

Note that the coexistence of bending and shear mechanisms lead to define an intrinsic lengthL of the reinforced media, such that both terms are of same magnitude :

L =

√EpIp

|S|µm

3.1.1. Boundary conditions

Boundary conditions must be specified at the lower end and at the top of the layer todetermine the transverse modal characteristics of the system. Due to the bending effect, theconditions involve the motion U , the mean stress τ , and also the beam rotation ∂U/∂x1 andthe beam momentum Ep Ip∂

2U/∂x2

1. Hence, four types of simple boundary conditions and

their associated formulations expressed in terms of displacements are identified (′ stands forderivative, τ = G U ′ − (Ep Ip/ |S|) U ′′′) :

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Condition Translation Expression Rotation Expression

Free (F) : Y es τ = 0 Y es Ep IpU′′ = 0

Clamped (C) : No U = 0 No U ′ = 0

Rotational (R) : No U = 0 Y es Ep IpU′′ = 0

Translational (T) : Y es τ = 0 No U ′ = 0

Fig. 9: Scheme of the four types of boundary conditions considered in this work. From leftto right : clamped-free (CF), rotational-free (RF), clamped-translational (CT), rotational-

translational (RT).

10−2

10−1

100

101

102

103

104

0

2

4

6

8

10

12

14

16

18

20

K

f1/f

1

f2/f

1

f3/f

1

17.55

6.27

1 1

3

5

Dominating bending

Dominating shear

Fig. 10: Variation of the eigen frequencies ratios versus K for clamped-free boundary condi-tions.

3.1.2. Modal characteristics

Instead of the second degree differential equation of elastic Cauchy continua, the go-verning equation is of the fourth degree (of the same nature as the governing equation of

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sandwich beams). The general solution is in the form :

U(x) = a ch(δ2

x

H

)+ b sh

(δ2

x

H

)+ c cos

(δ1

x

H

)+ d sin

(δ1

x

H

)(4)

with

δ2

1δ2

2=

ω2 |S| 〈ρ〉 H4

Ep Ip; δ2

2− δ2

1=

G |S| H2

Ep Ip= K

Note that the dimensionless parameter K :

K =G |S| H2

Ep Ip=

H2

L2

can be regarded either as the ratio of the shear effects compared to bending effects, or asthe square of the ratio between the height of layer and the intrinsic length. Thus, bendingdominates when K is small, or equivalently H ≪ L, shear dominates when K is large orequivalently H ≫ L.

The modal analysis is performed for four pairs of boundary conditions (the first and se-cond conditions correspond respectively to the bottom (x = 0) and to the top (x = H) of thecomposite, see Figure 9) : clamped-free (CF), clamped-translational (CT), rotational-sliding(RT) and rotational-free (RF). In each case these conditions lead to a set of four linearequations. The modes correspond to the non trivial solutions which are obtained when thedeterminant vanishes. The null-determinant condition results in the modal equations givenhereafter :

– Clamped-Free :K

δ2

1δ2

2

+th(δ2) tan(δ1)

δ1 δ2

+2

K

(1 +

1

cos(δ1)ch(δ2)

)= 0

– Clamped-Sliding : tan(δ1) +(δ2

1+ K)

δ1 δ2

th(δ2) = 0

– Articulated-Sliding : cos(δ1) = 0

– Articulated-Free : (K + δ2

1) th(δ2) −

δ3

1

δ2

tan(δ1) = 0

Hence, the modal equation can be solved numerically to derive the solutions δ1i and δ2i

corresponding to the ith mode as a function of K. Eigen frequencies and mode shapes of thecomposite can then be derived. As an example, Figure 10 depicts the first and second modecharacteristics versus K for clamped-free boundary conditions. As expected, it appears inFigure 10-a that for small values of K (dominating bending) the composite has the sameratio of second frequency to fundamental frequency as a clamped-free bending beam (namely(δ12δ22)/(δ11δ21) = f2/f1 ≈ 6.27 ; f3/f1 ≈ 17.55), and for large values of K (dominatingshear) the ratio tends to that of a clamped-free shear composite (i.e. f2/f1 = 3, f3/f1 = 5).Logically, mode shapes depicted in Figure 10-b vary from bending to shear mode shapes asK increases. Similar trends are observed with the others boundary conditions.

3.2. Theoretical modal characteristics of the different tested configurations

Since the material properties, geometry and the boundary conditions of the test specimenare exactly (almost) known, the homogenized model can be employed to determine the

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0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Mode 1

x/H

−1 −0.5 0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Mode 2

K = 10−2 K = 1 K = 10 K = 102 K = 103−1 −0.5 0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Mode 3

Bending

Bending

ShearShear

Bending

Shear

Fig. 11: Mode shapes evolution versus K for clamped-free boundary conditions.

expected behavior. Table 2 details the main geometrical parameters for the 35 and 17 fiberconfigurations. A reduction in fiber concentration from 0.2 % (with 35 fibers) to 0.1% (with17 fibers) causes, the dimensionless parameter K = (G S H2)/(Ep Ip) to approximatelydouble and the intrinsic length L to reduce by around 30%. A value K ≈ 10 engenders theparticipation of both shear and bending mechanisms and gives frequency ratios intermediatebetween pure shear and pure bending ; for K ≈ 20 the effect of shear is enhanced (c.f. Figure10). L represents about the third of the height H for 35 fibers and the quarter for 17 fibers,which means that bending and shear mechanisms take place for both fiber concentrations.

Tab. 2: Geometrical parameters of the physical model for 35 and 17 inclusions.

Number of c H/l L L/H Kinclusions [m]

35 0, 20% 5, 00 0, 41 33% 9, 4317 0, 10% 3, 54 0, 29 23% 18, 82

Table 3 gives the values of the theoretical eigen frequencies of the homogeneous andnon-homogeneous (cf. Appendix) transverse modes for both configurations in the frequencyrange of interest of 0-25 Hz. It is useful to underline some trends that can be observedexperimentally.

In accordance with the intuition, for a given configuration, the fundamental frequencies(homogeneous modes) are distributed by decreasing order for the clamped-translational,clamped-free, rotational-translational, and rotational- free boundary conditions (i.e. as theboundary conditions becomes "softer").

Passing from 35 to 17 fibers leads to reduces the bending stiffness by a factor of about2, and a decrease in the density of around 7%. The shear stiffness remains constant due to

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Tab. 3: Theoretical eigen frequencies of first homogeneous and non-homogeneous modescalculated from the homogenised model.

Configuration Homogeneous modes Inhomogeneous modesf1 f2 f2/f1 f1/1 f1/2 f1/3 f2/1

[Hz] [Hz] [Hz] [Hz] [Hz] [Hz]

35CF 5, 88 23, 42 3, 98 9, 25 15, 45 22, 22 24, 4835CS 6, 47 28, 34 4, 38 9, 63 15, 68 22, 38 29, 2235AF 4, 25 18, 47 4, 35 8, 31 14, 90 21, 84 19, 8035AS 4, 53 22, 18 4, 89 8, 46 14, 99 21, 90 23, 30

17CF 5, 59 20, 34 3, 64 9, 37 16, 06 23, 26 21, 6817CS 5, 82 23, 15 3, 97 9, 52 16, 14 23, 32 24, 3417AF 4, 40 16, 78 3, 81 8, 72 15, 68 23, 01 18, 3917AS 4, 52 18, 82 4, 16 8, 78 15, 72 23, 03 20, 27

the low fiber concentration. For configurations clamped at the bottom, when the number offibers decreases the fundamental frequencies decrease ; the effect of the loss of stiffness dueto the decreasing number of fibers is more significant than the effect of the loss of mass. Forthe rotational-translational boundary condition, the fundamental frequency is equivalent forboth fiber concentrations and the effects compensate one another. For the rotational-freeboundary condition, the trend is reversed and the fundamental frequency increases as fibersare removed indicating that the loss of mass is more significant that the loss of stiffness.As the fibers are not constrained at their extremities, they do not participate to the systemstiffness but only contribute by their mass.

Note that if the bending effect were disregarded, the change from 35 to 17 reinforcementswould mainly decrease the mean density. Hence, for all type of boundary conditions anincrease of frequencies would be noticed.

Frequencies of inhomogeneous modes are distributed as those of homogeneous modes. Asfj/n corresponds to the coupling between the jthhomogeneous transverse mode and the nth

transverse compression mode, necessarily fj/n > fj (c.f. Eq. (10) in Appendix). Note thatf1/2 is inferior to f2 for each configuration and f1/3 is inferior to f2 only for the cases 35CF,35CT and 35RT. Thus, inhomogeneous modes between the two first homogeneous modesmust exist for both tested configurations.

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4. Experimental results versus theoretical predictions

4.1. Checking of linearity and homogeneity

To confirm that the physical model conforms with the assumption of linearity the 35CFtest configuration was excited by three different magnitudes of white noise : a0, 1.5 × a0

and 2 × a0. Figure 12 shows the acceleration response in terms of transfer functions bet-ween several response accelerometers distributed across the top of the sample (on foammatrix and steel fibers) and a reference accelerometer secured to the platform of the shakingtable. The linear response of the system is confirmed with the observation that the transferfunctions are identical for the three excitation levels. Moreover, the coincidence of transferfunctions derived from accelerometers placed on the foam and on the fibers indicate thatboth components follow the same horizontal translation motion, as derived by the modelunder the assumption of perfect adherence. Further, the peak of fundamental frequency iswell pronounced and the very low dispersion of records at different points of the top surfaceindicates that the system has an in-plane homogeneous kinematic within a frequency rangeof about [0-12Hz] that includes the fundamental mode.

Fig. 12: Experimental check of linear response and homogeneity. Transfer function modulusbetween accelerometers located on inclusion/table (left) and foam/table (right) for whitenoise excitations of different mean amplitudes : a0 (top), 1.5 × a0 (middle) and 2 × a0

(bottom)

4.2. Frequency and mode shape of the fundamental mode

4.2.1. Identification of fundamental frequency and mode shape

The frequency response functions of the response accelerometers located at three differentz ordinates on a vertical surface of the matrix enable the determination of (three points on)

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Fig. 13: Fundamental mode identification from white noise tests for all the configurations.Modulus of the Transfer Function between accelerometers on top foam surface and referenceaccelerometer on the table. The two top figures correspond to the standalone inclusion and

the un-reinforced foam block.

the model shape of each test configuration. Indeed, the motion at a point z of the compositeis the sum of the input motion imposed at the base, and of the motion relative to the base.

19

This relative motion is a forced motion that is, because of linearity, proportional to the inputmotion. As usual in modal analysis, the relative motion can be decomposed on the modalbasis {Φi} :

U(x, ω) = Ubase(ω)

(1 +

∑

i

Φi(x) di(ω) pi

); di(ω) =

ω2

i

ω2

i − ω2 + 2 i ξ1 ωi ω

where pi and di(ω) are respectively the modal participation coefficient (∑

i pi = 1) andFourrier Transform of the impulse response of the ith mode characterized by its angularfrequency ωi and weak damping ξi ≪ 1. For frequency close to the fundamental frequencywhere d1 attains its peak value, d1(ω) ≈ d1(ω1) = (2 i ξ1)

−1 ≫ di(ω). Consequently, forweakly damped systems (as in the present case) a peak of displacement appears at the eigenfrequency and we have :

U(x, ω1) − Ubase(ω1) ≈ Ubase(ω1)Φi(x)pi

2 i ξ1

(5)

Coming back to accelerations a, the maximum amplification in the spectrum response indi-cates the fundamental frequency. Furthermore :

a(x, ω1) − abase ≈ abaseω2

1Φ1(x)

p1

2 i ξ1

(6)

Hence, introducing the normalized fundamental mode of unit top motion, i.e. Φ1,norm(x) =Φ1(x)/Φ1(H) the mode shape can be determined experimentally from the following ratio ofrecorded acceleration spectra :

Φ1,norm(x) =at(x, ω1) − abase(ω1)

at(H, ω1) − abase(ω1)

The relevant frequency response functions for the two test configurations are presented along-side those for the unreinforced matrix and the standalone fiber in Figure 13.

4.2.2. Experimental versus theoretical fundamental frequencies

In Figure 14, the experimental fundamental frequency of the different configurationsidentified from those data are compared to the predictions given by (i) the homogenisedmodel with macroscopic shear/bending coupling, (ii) a beam model governed by pure ben-ding (with the same boundary conditions) and (iii) the usual composite model governed byshear only (with fixed-free conditions). For clamped-free boundary conditions the measuredfrequencies match those of the shear/bending homogenised model ones, with an error ofabout 1%, while they significantly depart form the predictions of the pure bending and theusual composite model. The error is slightly more important for the clamped-translationalconfiguration, respectively with 6.2% and 3.3%, and reaches 16% for the rotational-freeor -translational boundary conditions. Moreover, the evolution trends of the measured fre-quencies are consistent with those predicted by the theoretical model discussed in the sec-tion 2.1.2. The fundamental frequency decreases when passing from 35 to 17 fibers for the

20

clamped-free and clamped-translational boundary conditions and, on the contrary, increasesfor the rotational-free case. The compensation of the loss of mass by the loss of stiffnesspredicted by the homogenised model passing from the configurations 35RT to 17RT is alsoobserved experimentaly.

Fig. 14: Fundamental frequency versus the fibers concentration for the different boundaryconditions. Experimental data (red circles, 0.01% for 17 fibers, 0.02% for 35 fibers) andhomogenised model (blue line). Beam model (green dotted line) and pure shear composite

model (green dashed line) are drawn for comparison.

Regarding those experimental results, it appears that among the four tested boundaryconditions, the clamped-free condition is associated with the least discrepancy between theo-retical and experimental results. This may be because these boundary conditions are mosteasily realized within the laboratory. As for the top translational boundary condition, theframe grid system (Figure 4) seems to be not sufficiently rigid to completely avoid the toprotation of the fibers (consistently with the fact that the clamped-translational configura-tion gives slightly lower frequency than predicted). Finally, concerning the bottom rotationalboundary condition, the presence of the matrix does not seem to let the fiber rotate totallyfreely (the experimental rotational-free or -translational frequencies are higher than thetheoretical ones). Nevertheless, in any case, the values and distributions of the fundamen-tal frequencies measured experimentally are in reasonable agreement with the theoreticalpredictions.

4.2.3. Experimental versus theoretical fundamental mode shapes

The experimental fundamental mode shapes deduced from the accelerations measure-ments (Figure 13) on the configurations with 35 fibers are represented in the Figure 15

21

and compared to theoretical predictions given by (i) the homogenised model with macro-scopic shear/bending coupling, (ii) a beam model governed by pure bending (with the sameboundary conditions) and (iii) a composite model governed by shear only (with fixed-freeconditions). For the clamped-free, -translational and rotational-free boundary conditions, thetheoretical homogenised model prediction better matches the experimental data, than thebeam model and/or composite model predictions. For the rotational-translational boundarycondition, the experimental mode shape seems to be linear and departs more significantlyfrom all the theoretical predictions. Observations are similar for the configurations with 17fibers.

4.3. Analysis of inner bending

The presence of a inner bending effect is evidenced from strain gauge signals recordedunder white noise tests. In fact, in the homogenized model, the bending term is brought bythe beams. Thus, in a given beam section of position x, as the momentum M(z) is relatedto the strain extension/compression e±(z) suffered by the lateral faces by the classic relation(rp is the external radius of the circular section) :

e±(x) = ±rpM(x)

EpIp

the measurement of e± at a given level is directly proportional to the inner momentum atthis level.

4.3.1. White noise response

In Figure 16, the strain data resulting from white noise excitation are presented in theform of transfer functions between measured strain (at bottom (B), middle (M) and top (T)level in the fiber) and the reference acceleration recorded on the platform of the shakingtable..

Note first that the transfer functions spectrum of the strain gauges provide an inde-pendent way to accurately determine the fundamental frequency of each configuration. Thefact that, for a given configuration, all accelerometers and gauges data (located on differentpoints) lead to the very same values underline the fact that whole system behaves as a mo-nophasic macroscopic medium.

Figure 16 also highlights that the strain distribution in the different test configurationsclearly depends on the boundary conditions. Recall that for a cantilever beam with purebending behaviour, the moment is maximum at the embedding end and minimum at thefree end, as observed on the standalone fiber were the strain decrease from the clampedbottom to the free top. The trends are the same for configurations with 35 or 17 fibers. Theclamped-free boundary condition leads to a strain distribution comparable to the standalonefiber one, but with lower amplitudes. The strains with the clamped-translational conditionsare similar to those of the clamped-free case at the bottom and at the middle, but the topstrain is much larger, in accordance with the null rotation imposed by the top rigid grid.

22

Fig. 15: Fundamental mode shapes of the configuration with 35 (top) and 17 (bottom) fibersunder the four types of boundary conditions. Experimental data (red circles deduced fromthe record of accelerometers mounted on the vertical face, at 31.2 cm, 62.5 cm from the baseand 121.1cm, i.e. at the top) and homogenised model (blue line). Beam model (green dotted

line) and pure shear composite model (green dashed line) are drawn for comparison.

The strains of the rotational-translational conditions are inversely distributed to those ofthe clamped-free case, since the bottom rotation is free and the top rotation vanishes. Withfree rotation at both extremities (rotational-free condition), the strain at the middle (due

23

Fig. 16: Evidence of fundamental mode under white noise of all the configurations. Modulusof the Transfer Function between strain gauges on the inclusions (at bottom, middle and

top level) and reference accelerometer on the table.

solely to the inertia) is similar to the one of the rotational-translational case, while top andbottom strains are very small.

24

Fig. 17: Harmonic forcing at the fundamental frequency for all the configurations. Timeresponse of the strains gauges on inclusions.

4.3.2. Inner bending involved in the fundamental mode

A more detailed analysis can be carried out from harmonic tests performed at the funda-mental frequency (identified experimentally from white noise tests) of each test configuration.Figure 17 displays the time response of the strain gauges (at bottom, middle and top levelin the fiber) in configurations containing 35 and 17 fibers with different boundary conditions(for comparison the record on standalone fiber is also given). In all cases, the experimental

25

data clearly shows the phase opposition and the identical amplitudes of the strains recordedby each pair of gauge. Extension-compression of the same amplitude on the opposite side ofthe same section of fiber is a clear signature of the bending state. The distribution of strainis identical to the one obtained with white noise, discussed previously.

For the clamped-free and clamped-translational boundary conditions, the gauge B- (orB+) is in phase opposition with the gauges M- and T- (or M+ and T+). A sign inversion inthe momentum between the bottom and the middle of the fiber exists, which correspondsto an inflexion point in the fundamental mode shape. On the contrary, in the rotational-translational case, the gauges B-, M- and T- (or B+, M+ and T+) are in phase. There is nomomentum sign inversion : one side of the fiber is entirely in compression while the otherone is in extension. Finally, for the rotational-free boundary condition, only the gauges inthe middle (M) exhibit strain.

An additional comparison between theory and experiment can be performed using thefact that the momentum in the beam is related to the second derivative (curvature) of thetransverse motion. Hence the axial strain can be written as :

e±(z) = ±rpM(z)

EpIp= ∓rpU

′′(z)

Therefore, in harmonic regime at the fundamental frequency, using expression (5) of thedisplacement we have :

e±(z, ω1) = ∓rpU′′(z, ω1) ≈ ∓rpUbase(ω1)Φ

′′

1(z)

pi

2 i ξ1

Thus, coming back to accelerations and normalized mode :

e±(z, ω1) = ∓rpΦ′′

1,norm(z)[ω2

1abase(ω1)Φ(H)

pi

2 i ξ1

]

and, noticing from (6) that the last term in square brackets equals a(H, ω1)−abase, one finallyobtains the following relation showing that, at the fundamental frequency, the (opposite)axial strains e± measured on opposite sides of fibers are proportional to the curvature of thenormalised fundamental mode shape :

e±(z, ω1)ω2

1

rp

1

at(H, ω1) − abase(ω1)= ∓Φ′′

1,norm(z)

Since Φ1,norm, thus its curvature Φ′′

1,norm is given by the theory, this last formula enables tocompare the theoretical curvature and the curvature deduced from the strain and accelera-tion measurement (recorded conjointly). Figure 18 displays the experimental data and thetheoretical predictions from the homogenised model with macroscopic shear/bending cou-pling, in the different tested configurations. In addition, the curvature of the fundamentalmode of the pure bending beam model (with the same boundary conditions) and of compo-site model (with fixed-free condition) are also presented. For the clamped-free, -translationaland rotational-free boundary conditions, a good qualitative and quantitative agreement is

26

observed between homogenised model and experiments. It is noticeable that the beam and/orshear composite models predictions provide a poor match to the data. Discrepancies (withthe three models) are larger for the rotational-translational boundary condition, which isthe most difficult to carry out experimentally as mentioned in section 4.2.1.

The presence of the inflexion point (sign inversion of curvature) in the fundamental modeshape is obvious on Figure 18 for the clamped-free and -translational boundary conditions.Note that for the clamped-free boundary conditions, the sign inversion traduces the cou-pling between bending and shear effects within the composite. Indeed, the measured pointscoincide with the theoretical homogenised modal prediction with a shear/bending coupling.In contrast, for clamped-translational case, the inflexion point is due to the blocked rotationat the top of the fibers.

4.4. Higher modes

Figure 19 to 22 shows the typical response in the frequency range [0-35 Hz] recordedfrom accelerometers and strain gauges under horizontal white noise test (illustrated by theconfigurations with 35 fibers, as 17 fibers gives similar results). Four frequency bands canbe defined in all the tested configurations, with the same qualitative observations but withsome differences in the frequency intervals.

0 ; ≃12 Hz : the behaviour is dominated by the fundamental homogeneous mode studiedin the previous section. Both accelerometers and strain gauges capture the resonance.

≃12 ; ≃17Hz : the accelerometers transfer functions significantly decrease (below 1) whilethe strain gauges vary smoothly. In this frequency range, the acceleration at the topof the system is inferior to the acceleration of the table. This "antiresonance" pheno-mena, is likely due to a gyration mode of the specimen that induces a rotation of thetop surface according to the horizontal axis perpendicular to the table motion. Thisinterpretation is supported by the records of opposite vertical accelerations at oppositeextremity of the specimen. This kinematics does not produce bending in the fiber. Aclose look of this complex mode (that may involves sliding between matrix and fibers)is beyond of the scope of this project.

≃17 ; ≃22 Hz : The strain gauges show an amplification corresponding to a resonance.The accelerometers transfer functions are not significantly amplified and takes distinctvalues at different locations. This indicates an inhomogeneous kinematic of the systemcorresponding to an inhomogeneous mode.

≃22 ; ≃27Hz : One notes an amplification of both accelerometers and strain gauges andthe modulus of the transfer functions of the accelerometers located on different pointsof the top surface are identical. This corresponds to the second homogeneous mode ofthe system.

In the sequel we consider the second homogeneous mode, then the non-homogeneous mode.

4.4.1. Second homogeneous mode

Let us first compare the theoretical and experimental frequencies f2,th and f2,exp.Theexperimentally measured frequencies f2,exp are close to those predicted theoretically with a

27

Fig. 18: Curvature of the fundamental mode shapes of the different configurations. Experi-mental data (red circles deduced from the record of accelerometers mounted on the verticalface, at 31.2cm, 62.5cm from the base and 121.1cm, i.e. at the top) and homogenised model(bue line). For comparison beam model and composite model are presented respectively in

green dashed-dotted line and green dashed line.

maximum error of 11% for the case 35CS as shown in Figure 23. Moreover, the evolutiontrends of the measured frequencies follow the homogenised model that predicts, for all theboundary conditions, that f2 decreases when passing from 35 to 17 inclusions (the loss of

28

0 5 10 15 20 25 30 35

100

101

|TF|

35CF - Accelerometers

Foam7/tableFoam20/tableFoam17/tableFoam14/tableFoam4/table

0 5 10 15 20 25 30 3510

−1

100

101

102

35CF - Gauges

f (Hz)

|TF|

5B−/table5B+/table5M+/table5T−/table5T+/table

(× 10 µe/g)

Fig. 19: Response of the configuration 35CF under white noise. Modulus of the transferfunction between foam accelerometers/table (top) and inclusions gauges/table (bottom).

0 5 10 15 20 25 30 35

100

101

35CS - Accelerometers

|TF|

Foam7/tableFoam3/tableFoam4/table

0 5 10 15 20 25 30 3510

−1

100

101

102

35CS - Gauges

f (Hz)

|TF|

5B−/table5B+/table5M+/table5T−/table5T+/table

(× 10 µe/g)

Fig. 20: Response of the configuration 35CT under white noise. Modulus of the transferfunction between foam accelerometers/table (top) and inclusions gauges/table (bottom).

mass dominates the loss of bending stiffness).

The different configurations have also been submitted to an harmonic forcing regime atthe frequency corresponding to the second homogeneous mode. According to same procedureas for the fundamental mode, the Figure 24 presents the second mode curvature determi-ned experimentally from measured strains and theoretically by the homogenised model. Onenotes a qualitative agreement between experiments and theory but quantitatively, the dis-crepancies are more pronounced than for the fundamental mode.

29

0 5 10 15 20 25 30 35

100

101

35AF - Accelerometers

|TF|

Foam7/tableFoam20/tableFoam17/tableFoam14/tableFoam4/table

0 5 10 15 20 25 30 3510

−1

100

101

102

35AF - Gauges

f (Hz)

|TF|

6B−/table6B+/table6M−/table6M+/table6T−/table6T+/table(× 10 µe/g)

Fig. 21: Response of the configuration 35RF under white noise. Modulus of the transferfunction between foam accelerometers/table (top) and inclusions gauges/table (bottom).

0 5 10 15 20 25 30 35

100

101

35AS - Accelerometers

|TF|

Foam7/tableFoam20/tableFoam17/tableFoam14/tableFoam4/table

0 5 10 15 20 25 30 3510

−1

100

101

102

35AS - Gauges

f (Hz)

|TF|

3B−/table3M−/table3T+/table

(× 10 µe/g)

Fig. 22: Response of the configuration 35RT under white noise. Modulus of the transferfunction between foam accelerometers/table (top) and inclusions gauges/table (bottom).

4.4.2. Inhomogeneous mode

As mentioned in section 3, inhomogeneous modes are due to the finite size of the physicalmodel submitted to a free stress state on its vertical surfaces.

The strain-gauges frequency response functions permit to identify the inhomogeneousmodes more easily than the accelerometers frequency response functions. An intermediateresonance clearly appears between the two first homogeneous modes of the configurations

30

Fig. 23: Second homogeneous mode frequency versus the fibers concentration for the dif-ferent boundary conditions. Experimental data (red circles, 0.01% for 17 fibers, 0.02% for35 fibers) and theoretical homogenised model (blue line). Beam model (green dotted line)

and usual composite model (green dashed line) are drawn for comparison.

35CF and 35CT (c.f. Figures 19 and 20). The inhomogeneous modes peaks are less markedand separated for the cases 35RF and 35RT (c.f. Figures 21 and 22). The configurations withbottom rotational boundary seem to be submitted to disturbing effects may be related to theproximity of the gyration mode. In the following, the results concerning the rotational-freeand -translational boundary conditions are given for information purpose only, but must beconsidered less reliable than those concerning the clamped-free and -translational conditions.

The measured frequencies seem to correspond to the theoretical frequency of the modecoupling the first transverse mode to the second compression mode. One may ascribe theabsence of the inhomogeneous mode coupling the first transverse mode to the first com-pression mode to the fact that it could be masked by the gyration mode. The theoreticaland experimental frequencies f1/2 of the different configurations are compiled in Table 4.The error between the theoretical prediction and the measure of f1/2 is less than 15% forthe cases 35CF and 35CT (12.6%) and significantly decreases for 17 fibers (about 1%). Ho-wever, the evolution trends when passing from 35 to 17 fibers are opposite to those of thehomogenised model. Nevertheless, this discrepancy has to be moderated due to the weakvariations of frequencies considered here. Thus, the theoretical mode that couples the firsttransverse mode to the second compression mode seems to be a good explanation for thisinhomogeneous mode experimentally observed.

The differences between f1/2,th and f1/2,exp and their evolutions may come from the factthat length of the physical model (D = 2, 13m) is of the order of the compression wavelengths

31

Fig. 24: Experimental (red circles) and theoretical (homogenised model in bue line - beammodel in green dashed-dotted line - composite model in green dashed line) curvature ofthe second homogeneous mode shapes of the configurations containing 35 inclusions with

different boundary conditions.

that develop within the experimental system. Indeed, for a composite of finite dimension,the theoretical homogenised description is valid in the whole system except on the edge, ona distance of about one period. Hence, in the tested configurations, the boundary layer mayhave a less negligible effect upon inhomogeneous modes that directly depend on the size Dof the physical model, than on homogeneous modes.

Finally, the measured higher modes (homogeneous and non-homogeneous) are consistentwith the homogenised model predictions. Globally, the concordance of theoretical (homogeni-sed model) and experimental modal characteristics (frequencies and mode shapes curvature)is correct but less accurate than for the fundamental mode. Note that the interpretation ofinhomogeneous higher modes, inherent to the finite dimension of the physical model, is moredelicate.

32

Tab. 4: Theoretical and experimental frequencies f1/2 of the different configurations.

Configurations35CF 35CT 35RF 35RT 17CF 17CT 17RF 17RT

f1/2,th 15.45 15.68 14.90 14.99 16.06 16.14 15.68 15.72(Hz)

f1/2,exp 17.40 17.65 14.65 15.21 15.84 16.34 14.59 15.02(Hz)

error 12.6% 12.6% 1.7% 1.5% 1.4% 1.2% 7.0% 4.5%

5. Conclusion

The main learnings of this project can be summarized as follows.It is proven experimentally that the concept of generalized continua at the leading order,

which may appear as a mathematical curiosity, applies to highly contrasted composites in thecontext of elastic linear behavior and small deformations. Indeed, the tested samples weredesigned according to rules identified from the theoretical approach. The good agreementbetween experiments and theory provides evidence for the actual coupling between thebeam bending behaviour of the fibers and the elastic shear behaviour of the matrix. Suchbehavior significantly departs from the behavior of usual modeling of composites. Because ofinner bending, the modes shapes and frequency distributions of transverse homogeneous andinhomogeneous modes are modified. Nevertheless, the classic general framework of modalanalysis remains valid. The global consistency of the results obtained with the differenttested configurations (two fiber concentrations, four types of boundary conditions) givesa reasonable confidence in the theory. In particular the determining role of the "specific"boundary conditions of the fibers is caught by both modeling and experiments.

We have taken advantage of the metric dimensions of the reinforced material to installinstrumentation on the surface but also inside the media. This enables an easy check of theactual role of the reinforcements. The observed unconventional properties are of interest,either to improve the understanding of the actual behavior and the design of reinforcedsoils. The experimental data gathered through this project are presently used by membersof the project team to develop new numerical modeling of reinforced soils.

As for practical applications in earthquake engineering, the homogenised quasi-analyticalmodel with explicit boundary conditions, enables to overcome the cumulated difficultiesinduced by the weak amount of reinforcement, the mechanical contrasts and the dynamicregime. Further, the model can be simply modified to account, either for damping of thematerials by considering visco-elastic constituents by means of complex modulus (in Fourierdomain) or for imperfect contact condition between constituents by introducing elastic orvisco-elastic interface laws.

Finally, it is worth mentioning that the shear/bending behaviour differs from the usualformulation based on the Wrinkler approach, as proposed by European norms. The keydifference lies in the description of the soil effect on the pile. With the Wrinkler springs, the

33

soil is assumed to be compressed by the piles (more precisely the horizontal motion of thebeam is taken as the driving variable for the soil response), conversely to the present analysiswhere the soil is sheared together with the piles. The experiments tend to prove that thedesign rules suggested by the norm do not account for the actual mechanism governing thesoil/piles system (at least in the elastic domain).

34

6. Scientific dissemination

The theory related to this study and the results of this project have been reported in thePhD of J. Soubestre, in scientific publications, and were presented in several national andinternational conferences.

– PhD report

J. SoubestreHomogénéisation et expérimentation de milieux renforcés par inclusion linéaires. Ap-

plication aux fondations profondes. PhD- Université de Lyon/ ENTPE - INSA - July2011

– Publications

– Boutin C., Soubestre J.,Generalized inner bending continua for linear fiber reinforced materials. Internatio-nal Journal of Solids and Structures. 48 , pp 517-534, (2011)

– J. Soubestre, C. BoutinNon-local dynamic behavior of linear fiber reinforced materials Mechanics of Mate-rials (55), 16-32, (2012)

– J. Soubestre, C. Boutin, M.S. Dietz, L. Dihoru, S. Hans, E. Ibraim, C. A. TaylorDynamic Behaviour of Reinforced Soils ; Theoretical Modelling and Shaking Table

Experiments. Geotechnical, Geological, and Earthquake Engineering, Volume 22,247-263, DOI : 10.1007/978-94-007-1977-4-13 (2012)

– C. Boutin, J. Soubestre, M.S. Dietz,Experimental evidence of the high-gradient behaviour of fiber reinforced materials.

Submitted to European Journal of Mechanics A/ Solids - (2013)

– A paper is in preparation for an earthquake engineering journal

– Conferences

– J. Soubestre, C. Boutin, M.S. Dietz, L. Dihoru, S. Hans, E. Ibraim, C. A. TaylorDynamic Behaviour of Reinforced Soils ; Theoretical Modelling and Shaking Table

Experiments. Series Workshop - Ohrid - Macedonia - 2 sept 2010

– C. Boutin, J.SoubestreSecond gradient behaviour of reinforced soils and deep foundations Theoretical analy-

sis and experimental evidence. 15ème Congr. Franco-Polonais - Gdansk 7-9 Oct. 2010

35

– C. Boutin, J. SoubestreGeneralized continua for reinforced geomaterials SIAM Conf. Mathematical Com-putational Issues in the Geosciences (GS11) Long beach, California, 21-24/03, 2011.

– J. Soubestre, C. Boutin, M. DietzDynamique des systemes sol-pieux. 8ème Col. National AFPS 2011 Paris 6-9/09 2011

– C. Boutin, J. Soubestre, M. DietzModèle à double gradient pour matériaux renforcés. Modélisation théorique et expé-

rimentale. 20ème Congrès Français de Mécanique Besançon, 29/08-2/09 2011

– C. Boutin, S. Hans, J. Soubestre, C. Taylor, E. Ibraim, M. Dietz, L. DihoruDynamic behaviour of reinforced soils and deep foundations. Theoretical analysis and

experiments. 15ème World Conf. Earthquake Engng. - Lisboa 24-28/09 2012

36

Références

Auriault J.L., Boutin C., Geindreau C., 2009, "Homogénéisation de phénomènes couplés en milieux hétéro-gènes 1." Mécanique et Ingénierie des Matériaux, Hermes, Lavoisier.

Bellieud M., Bouchitté G., 2002 ,"Homogenization of soft elastic material reinforced by fibers" Asymp. Anal.32 (2), pp153-183.

Boutin C. , 1996, "Microstructural effects in elastic composites" Int. J. Solids Structures 33 (7), pp1023-1051.

Boutin C., Soubestre J., 2011, "Generalized inner bending continua for linear fiber reinforced materials."International Journal of Solids and Structures 48, pp 517-534.

de Buhan P., Hassen G., 2008, "Multiphase approach as a generalized homogenization procedure for model-ling the macroscopic behaviour of soils reinforced by linear inclusions" European Journal of Mechanics,A/Solids 27 (4), pp. 662-679

Caillerie D., 1980, "The effect of a thin inclusion of high rigidity in an elastic body" Math. Meths.Appl.Sci.2, pp251-270.

Christensen R.M., K.H. Lo., 1979, "Solutions for effective shear properties in three phase sphere and cylindermodel". J. Mech. Phys. Solids, 27, pp 315-330.

Eringen A.C.,1968, "Mechanics of micromorphic continua", IUTAM Symposium, Ed. Kröner E.Springer-Verlag (Berlin) pp18-35.

Forest S., 2006, "Milieux continus généralisés et matériaux hétérogènes", Presses de l’Ecole des Mines,ISBN : 978-2911762673

Gambin B., Kröner E., 1989, "High order terms in homogenized stress- strain relation of periodic elasticmedia " Pys. Stat. Solids (b) 151, pp513-519.

Hans S., Boutin C., 2008, "Dynamics of discrete framed structures - Unified homogenized description" J.of. Mech. Materials Strutures 33 (7), pp1023-1.

Hashin Z.,1983, "Analysis of composite materials",J. Appl. Mech. 50, pp481-505.Koo K.K., Chau K.T., Yang X., Lam S.S., Wong Y.L., 2003, "Soil-pile-structure interaction under SH wave

excitation", Earthquake Engng. Struct. Dyn, vol. 32, p. 395-415.Makris N., Gazetas G.,1992 "Dynamic pile-soil-pile interaction. Part II. "Earth. Engng. Struct. Dyn. 21,

pp145-162.Mylonakis G., Gazetas G.,1999, "Lateral vibration and internal forces of group piles in layered soil", Journal

of geotechnical and geoenvironnemental engineering, p. 16-25.Pideri C., Seppecher P., 1997, "A second gradient material resulting from the homogenization of an hetero-

geneous linear elastic medium" Continuum Mech. and Thermodyn. 9, pp 241-257.Postel M., 1985, "Réponse sismique de fondations sur pieux". PhD thesis, Ecole Centrale de Paris, Paris.Sanchez-Palencia E.,1980, Non Homogeneous Media and Vibration Theory, Springer-Verlag, Berlin.Soubestre J. "Homogénéisation et expérimentation de milieux renforcés par inclusions linéaires" PhD Thesis

- ENTPE-Insa, 261p.Soubestre J., Boutin C., 2012, "Non-local dynamics of linear fiber reinforced materials" Mechanics of Ma-

terials, 55, 16-32.Soubestre J., Boutin C., Dietz M.S., Dihoru L., Hans S., Ibraim E., Taylor C. A. : 2012, "Dynamic Behaviour

of Reinforced Soils -Theoretical Modelling and Shaking Table Experiments"Geotechnical, Geological, andEarthquake Engineering 22, 247-263.

Sudret B., De Buhan P., 1999 "Modélisation multiphasique de matériaux renforcés par inclusions linéaires"C.R. Acad. Sci. IIb 327, pp7-12.

37

7. Appendix : Non-homogeneous transverse modes

To study non-homogeneous transverse modes (of motion polarized for example in thedirection a

2) we consider a transverse motion of the form U(x) a

2. The in-plane macro-

equilibrium gives the set :

−Ep Ip

|S|∂4 U

∂x4

1

+ G∂2 U

∂x2

1

+ (λ + 2G)∂2 U

∂x2

2

+ G∂2 U

∂x2

3

+ 〈ρ〉ω2 U = 0 (7)

(λ + G)∂2 U

∂x2∂x3

= 0 (8)

Equation (8) imposes an affine dependance according either to x2 or to x3, and the generalsolution is split in two fields (aα and bα are constant coefficients) :

U = (A(x1, x2)(a3 x3 + b3) + B(x1, x3)(a2 x2 + b2)) a2

Both kind of fields (related to A and B) can be studied independently, and in each case the li-near dependence simplifies. It leads to focusing on generic solutions in the form (disregardingthe linear dependence on the "third" variable) :

U = U(x1, x2) a2

or U = U(x1, x3) a2

Fields U(x1, x2) are investigated as variables separated functions, i.e. U(x1, x2) = h(x1) ×g(x2). Introducing this expression in (7) and dividing by h(x1) × g(x2), leads to the twofollowing equations governing h(x1) and g(x2) :

−Ep Ip

|S| h′′′′ + Gh′′ + 〈ρ〉ω2(1 − β)h = 0 and (λ + 2G)g′′ + 〈ρ〉ω2βg = 0 (9)

where β is a constant to be determined according to the boundary conditions. Note thatfields U(x1, x3) = h(x1)× g(x3) lead to same equations (9) except that (λ + 2G) is replacedby G.

Consider a slot of reinforced medium of height H , of infinite lateral extension along x3

(hence there is no x3 dependence in the modes) and of finite lateral extension D along x2

(−D/2 ≤ x2 ≤ D/2). By assuming that the lateral surfaces x2 = ±D/2 are free of stress,we have :

σ22(x1,±D/2) = (λ + 2G)∂ U

∂x2

(x1,±D/2) = 0 therefored g

dx2

(±D/2) = 0

For β < 0, g takes the form g(x2) = a ch(δ x2) + b sh(δ x2), with δ =√

〈ρ〉ω2|β|/C22

22, and

the boundary conditions on lateral faces cannot be satisfied. Thus β must be positive, andg is in the form g(x2) = a cos(δ x2) + b sin(δ x2). Consequently, the boundary conditions canbe fulfilled provided that β takes positive discrete values βn, each of these values associatedwith a function gn(x2) characterizing compression modes in the direction a

2:

〈ρ〉ω2βn = C22

22

(n π

D

)2

n ∈ N and gn(x2) = ancos

(2n πx2

D

)+bnsin

((2n + 1) πx2

D

)

38

Equation (9) which governs h and equation (3) which defines homogeneous modes simplydiffer by the inertial coefficient (1 − βn) which may either be positive (e. g. homogeneousmodes reached when n = 0, i.e. βn = 0) or negative (e.g. for large transverse compres-sion modes orders n). By introducing the index n to specify the solution, the roots of thecharacteristic equation of the hn function (associated with βn) deduced from (9) read :

δ2

1n δ2

2n =ω2〈ρ〉(1 − βn) |S|H4

Ep Ip

=|S|H4

Ep Ip

(ω2〈ρ〉 − C22

22

(n π

D

)2)

δ2

2n − δ2

1n =C12

12|S| H2

2 Ep Ip

= K

Thus, if 1 − βn > 0, then δ2

1n δ2

2n > 0 and we have, similarly to homogeneous modes (4), anoscillating solution of the form :

hn(x1) = a ch(δ2n

x1

H

)+ b sh

(δ2n

x1

H

)+ c cos

(δ1n

x1

H

)+ d sin

(δ1n

x1

H

)

Whereas, if 1−βn < 0, then δ2

1nδ2

2n < 0 and we obtain a combination of exponential solutionsthat correspond to modes confined in the vicinity of the top and bottom boundaries :

hn(x1) = a ch(δ2n

x1

H

)+ b sh

(δ2n

x1

H

)+ c ch

(δ1n

x1

H

)+ d sh

(δ1n

x1

H

)

−1 0 10

0.5

1

−1 0 10

0.5

1

−1 0 10

0.5

1

−1 0 10

0.5

1

−1 0 10

0.5

1

−1 0 10

0.5

1

−1 0 10

0.5

1

−1 0 10

0.5

1

−1 0 10

0.5

1

−1 0 10

0.5

1

−1 0 10

0.5

1

x2=-D/2

−1 0 10

0.5

1

x2=-D/4

−1 0 10

0.5

1

x2=0−1 0 10

0.5

1

x2=D/4

−1 0 10

0.5

1

x2=D/2

n=3

n=2

n=1

Fig. 25: Inhomogeneous mode shapes (j = 1, n = 1; 2; 3) of a reinforced medium of para-meter K = 10 with clamped-free boundary conditions.

Nevertheless, in both cases, top and bottom boundary conditions expressed through U(x)for homogeneous modes can be directly transposed to hn(x1). Therefore, modal equations

39

of inhomogeneous modes are identical to those of homogeneous modes derived in section 3except that δ1 and δ2 have to be replaced by δ1n and δ2n for 0 < βn < 1 or by i δ1n (withi2 = −1) and δ2n for βn > 1. The modal frequencies fn/j are now characterized by the ordern of the compression modes (along x2) and the order j of the associated transverse modes(along x1) :

fn/j =1

2 π H2

√Ep Ip

〈ρ〉 |S|

√

δ2

1n/j δ2

2n/j +(n π

D

)2 C22

22|S| H4

Ep Ip(10)

and the corresponding mode shapes read :

inhomΦn/j(x1, x2) = hn/j(x1) × gn(x2)

In Figure 25, an illustration of mode shapes inhomΦn/1(x1, x2) coupling the first transversemode (j = 1) for each of the first three lateral compression modes (n = 1, 2, 3), is presentedfor a reinforced medium of parameter K = 10 with clamped-free boundary conditions.

40