soil behaviour optimisation procedure of … · in the field. two optimization techniques are...

*1 Younes ABED, Department of civil engineering, faculty of technology, university of Blida, Algeria, [email protected] 2 Djillali AMAR BOUZID, Department of civil engineering, faculty of technology, university of Blida, Algeria 3 Ilhem TOUMI, Department of civil engineering, faculty of technology, university of Blida, Algeria

SOIL BEHAVIOUR OPTIMISATION PROCEDURE OF

COHESIONLESS SOIL AROUND PRESSUREMETERE

PRESFEREMETRE ÇATLAK TOPRAKLIK TOPRAKLİĞININ TOPRAK

DAVRANIŞI OPTİMİZASYON PROSEDÜRÜ

Younes AED*1

Djillali AMAR BOUZID2 Ilhem TOUMI TOUMI

3

ABSTRACT

This paper presents a methodology for identifying soil parameters that takes into account

different constitutive equations. The procedure, applied here to identify the parameters of

generalized Prager model associated to the Drucker & Prager failure criterion from a

pressuremeter expansion curve, is based on an inverse analysis approach, which consists of

minimizing the function representing the difference between the experimental curve and the

curve obtained by integrating the model along the loading path in the in-situ testing. The

numerical process implemented here is based on a finite element procedure. Some parameters

effects on the simulated curve, examples identification and stresses path around pressuremeter

probe are also presented.

Keywords: Pressuremeter test; finite element method; inverse analysis.

ÖZET

Bu makale, farklı yapısal denklemleri hesaba katan toprak parametrelerini tanımlamak için bir

metodoloji sunmaktadır. Bir basınçölçer genişleme eğrisinden Drucker & Prager hata kriteri

ile ilişkili olan genelleştirilmiş Prager modelinin parametrelerini tanımlamak için burada

uygulanan prosedür, deney eğrisi ile deney eğrisi arasındaki farkı temsil eden bir ters analiz

yaklaşımına dayanmaktadır. In-situ testte yükleme yolu boyunca modelin bütünleştirilmesiyle

elde edilen eğri. Burada uygulanan sayısal işlem sonlu elemanlar prosedürüne dayanmaktadır.

Simüle edilmiş eğri üzerindeki bazı parametrelerin etkileri, basınç göstergesi probu

etrafındaki örnek teşhis ve gerilmeler yolu da sunulmuştur.

Anahtar Kelimeler: Basınçölçer testi; Sonlu elemanlar yöntemi; Ters analiz.

1. INTRODUCTION

During the three past decades, a tremendous progress has been made in the development of

constitutive models, which are able to accurately reproduce the soil behaviour. However, the

degree of complexity of these constitutive models (in many cases) inhibits their incorporation

into general purpose numerical codes, thus restricting their usefulness in engineering practice.

The successful application of in-situ testing of soils heavily depends on development of

interpretation methods of tests. The pressuremeter test simulates the expansion of a

cylindrical cavity and because it has well defined boundary conditions, it is more unable to

rigorous theoretical analysis (i. e. cavity expansion theory) then most other in-situ tests.

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7. Geoteknik Sempozyumu 22-23-24 Kasım 2017, İstanbul

Soil parameters can be identified from pressuremeter tests by using an inverse analysis of the

cavity pressure–cavity strain relationship measured in the field. To perform this, the

pressuremeter test is generally simulated as the expansion of an infinitely long cylindrical

cavity inside an infinite uniform medium by means of closed-form analytical solutions

(Moussavi et al., 2011) or approximate numerical models, such as the finite elements (Zanier,

1985; Carter et al.,1986; Cambou et al., 1990; Yu and Houlsby, 1991and 1995; Fahey and

Carter, 1993; Olivari and Bahar, 1995; Cudmani and Osinov, 2001; Hsieh et al., 2002;

Levasseur, 2008; Shahin et al., 2008; Javadi and Rezania, 2009; Zhang et al., 2009;

Levasseur et al, 2010; Liang and Sharo, 2010; Zhang et al., 2013; Abed et al., 2014 and Al-

Zubaidi, 2015).

Engineers prefer closed-form analytical solutions than numerical models because the recourse

to numerical methods is often time consuming and computationally non-effective. However,

the use of approximate numerical models becomes the only possible alternative, in the case

where it is difficult to obtain a closed-form solution due to the complexity of the soil

constitutive model. Furthermore, in most elasto-plastic models, a relatively large number of

parameter needs to be defined, which makes empirical curve fitting impractical so that the use

of optimization algorithms for matching simulations to experiments becomes a necessity.

The aim of the present work is to develop an effective and consistent procedure for the

identification of the engineering properties of soils by interpreting results from pressuremeter

tests. In particular, the paper presents a methodology for the identification of parameter values

in the Generalized Prager Model (GPM), (the reader is referred to the papers by Abed and

Bahar, 2010 and Abed et al., 2014) of the engineering properties of soils by inverse analysis

of the experimental cavity pressure–cavity strain curve measured during pressuremeter tests

in the field. Two optimization techniques are presented; one is based on the simplex method

of Nelder and Mead (Nelder and Mead, 1965) and the other is based on the decomposition of

the pressuremeter curve.

2. THE PROPOSED MODEL

The GP-DP (which stands for Generalized Prager associated with Drucker and Prager

Models) is an elasto-plastic constitutive model that is able to predict the nonlinear stress–

strain behaviour of soils. Based on the use of a kinematic hardening function, a bilinear stress-

strain behaviour can be simply described by means of a single internal variable (Figure 1a).

Considering the idea of kinematic hardening and multiple combined internal variables, a

piecewise linear stress-strain response can be described (Figure 1b). The stress-strain curve is

approximated by linear segments along which the tangent shear modulus is assumed to be

constant (Bahar et al., 1999).

(a) (b)

Figure 1. Stress-strain relationships: (a) bilinear; (b) piecewise linear.

222


The model, based on a set of several single Prager models is defined by the compliance of ( )

elastic elements and their associated ( ) yield surfaces. Eventually, a single linear spring

elastic element of compliance ( ) and a single yield surface of threshold stress ( ) can be

introduced and connected in series to the collection of elements to represent the initial elastic

strain and failure respectively. Thus the model is defined by ( ) parameters that can be

represented by a discrete spectrum of compliance.

The adopted Kinematic hardening model, assumes that during the plastic deformation the

loading surface translates as a rigid body in the stress space keeping the size, shape and

orientation of the initial yield surface. The expression for the yielding surface is:

(1)

Where is a constant and are the coordinates of the center of loading surface which

changes as the development of plastic strains continues. The simplest version for determining

the parameter is to assume a linear dependence of :

(2)

Where is the work hardening constant, characteristic of a given material. Eq. (2) may be

taken as the definition of the linear work hardening.

The plastic strain is given by the normality rule (standard material):

(3)

(4)

The consistency condition means that the yield surface keeps the same radius, this helps to

determine the plastic proportionality factor with:

(5)

Where the underscript denotes the th element of chain;

is the modulus of the element of the chain;

is the normal external vector to the yield surface.

3. YIELD FUNCTION AND PLASTIC FLOW RULE

The failure criterion associated with the proposed model is that of Drucker and Prager

(Drucker and Prager, 1952). This criterion is based on the assumption that the octahedral

shear stress at failure depends linearly on the octahedral normal stress through material

constants. It was established as generalisation of the Mohr-Coulomb criterion for soil:

(6)

223


Where, the constants and are material constants.

isthe second invariant of the stress deviator tensor and is the first invariant of the stress

tensor, and are defined as follows:

(7)

(8)

,

, , are the principal effective stresses.

The parameters and can be calibrated to make the Drucker-Prager criterion fit different

parts of the Mohr-Coulomb criterion (Chen and Mizuno, 1990). The Mohr-Coulomb criterion

and two different Drucker-Prager fits are shown in Figure 2.

Figure 2. Mohr-Coulomb and Drucker-Prager criteria on the octahedral plane.

The two fits are:

1. Triaxial extension fit: The triaxial extension corners of the Mohr-Coulomb criterion

coincide with the Drucker-Prager criterion.

2. Triaxial compression fit: The triaxial compression corners of the Mohr-Coulomb

criterion coincide with the Drucker-Prager criterion.

4. FINITE ELEMENT SIMULATION OF PRESSUREMETER TEST

In this work, the computed cavity pressure–cavity strain curves were obtained from the finite

element simulation of an infinitely long cylindrical cavity expanding inside a cohesionless

soil.

In order to apply the finite element method, the 1D expansion of an infinitely long cylindrical

cavity is simulated. The segment is divided into segments of unequal lengths

(Figure 3). To ensure a good accuracy of stresses and strains, the mesh is relatively fine in the

vicinity of the borehole. With growing distance to the probe, the elements increase

progressively in size. With reference to the middle point of the Pressuremeter membrane, the

224


dimension of the finite element mesh used is in width. The soil around the probe is

modelled using a linear mesh (Abed et al., 2014).

Figure 3. Finite element mesh around pressuremeter borehole.

4.1 Solution of the problem in elastic zone

In elasticity, the analytical solution of the differential equation governing the expansion of a

cylindrical cavity is given according to the elastic parameters (Clark, 1995):

(9)

(10)

The deformation occurs at a constant stress without volume changes (figure 4).

Figure 4. Yield zone around the expanding cavity

4.2 Solution of the problem in elasto-plastic zone

The behaviour law can be written by:

(11)

Where, the matrix is function of the stress state and the loading path. It describes the soil

behaviour for a loading increment (figure 4). It may be written as:

225


For (12)

For (13)

5. NUMERICAL PROGRAM

The determination of the parameters of the constitutive model from the pressure-meter test

consists in solving the following inverse problem: to find a set of parameters which minimize

the difference between the experimental data (here the pressuremeter curve defined as the

applied pressure versus the cavity wall deformation) and the simulated curve. This problem is

classically defined by an objective function which evaluates, for a given set of parameters, the

discrepancy between the model prediction and the experimental data.

To achieve this purpose, a computer program baptised "Press-Sim" is developed in order to

identify automatically the parameters of the proposed behaviour law. In order to evaluate the

objective function, we use an algorithm that allows the decomposition of the pressuremeter

curve in three parts.

The proposed program can be used in two ways:

Direct use: knowing the model parameters, the program allows the determination of the

simulated curve;

Indirect use: from the experimental curve, the program allows the identification of the model

parameters.

6. PARAMETERS EFFECT ON SIMULATED CURVE

The procedure followed in order to show some parameters effect on the simulated curve was

by considering the sensitivity of some parameters. For this, the use of the numerical program

allows to get a simulated pressuremeter curve. To carry out this analysis, the experimental

pressuremeter curve deduced from the thick cylinder tests realized on Hostun sand at ENPC

(Ecole des Ponts ParisTech) by Dupla (Dupla, 1995) has been used.

6.1 Dilatancy effect on the pressuremeter curve

We present in figure 5 the effect of dilatancy on the value of the limit pressure. We remark

that an increase of dilatancy factor leads to an increase in the limit pressure.

226


100

200

300

400

500

600

700

800

0 10 20 30 40 50 60 70 80

Axial strain (%)

Ap

plie

d p

ressu

re (

kP

a)

=0,01

=0,03

=0,05

Model parameters

=40°

E=10MPa

A=0,008

Legend:

Figure 5. Effect of parameter on limit pressure value.

6.2 Effect of the consolidation pressure on the pressuremeter curve

Figure 6 presents the consolidation pressure effect on the pressuremeter curve. The corrected

limit pressure is almost proportional to the pressure of consolidation, which is in accordance

with experimental observation (Dupla, 1995).

100

200

300

400

500

600

700

800

900

1000

1100

1200

0 10 20 30 40 50 60 70 80 90 100

Model parameters:

E=10 MPa

A=0,0025

=45° kPa

Volumetric strain (%)

Ap

plie

d p

ressu

re (

kP

a)

c=100

c=200

c=400

Figure 6. Consolidation pressure effect on the pressuremeter curve

6.3 Influence of the finite length of the pressuremeter probe

To study the influence of the finite length of the pressuremeter probe, a series of simulation of

slenderness ratio (L/D = 5, 10, 15 and 20). It appears in figure 7 that the geometry of the

227


probe has a considerable influence on the pressuremeter curve. This difference is inversely

proportional to the ratio L/D which is consistent with the experiment (Zanier, 1985)

100

200

300

400

500

600

0 10 20 30 40 50 60 70 80 90 100

Volumetric strain (%)

Ap

plie

d p

ressu

re (

kP

a)

L/D=5

L/D=10

L/D=15

L/D=20

Figure 7. Slenderness ratio effect

6.4 Stresses path around pressuremeter

The proposed method allows visualising stresses path in all the point of a chosen

discretization. Figure 8 indicates the distribution of the main stresses at the end of loading for

the self-boring pressuremeter test undertaken on the Fos sand in France. Three different areas

of soil from the borehole wall to the infinite radius are considered. Plasticity appears between

the radial stress r and circumferential stress in the horizontal plane. This first plastic area

extends between the radius a (borehole wall) and b (external radius of the plastic area). As

shown by Wood & Wroth (Wood & Wroth, 1977) and Monnet (Monnet, 2007), plasticity my

also appear in the vertical plane between the vertical stress z and circumferential stress in

an area between the radius b and radius c (external radius of both plastic areas). An elastic

area extends beyond the radius b.

228


Plasticity area between radial

and circunferantial stress

0,00 0,25 0,50 0,75 1,00

10

20

30

40

50

60

70

80

90

100

cb

Elasticity area

Plasticity area between vertical

and circunferantial stress

Str

ess (

kP

a)

Radius (m)

Radial stress

Circunferantial stress

Vertical stress

a

Figure 8. Distribution of main stresses at end of loading

7. SIMULATION EXAMPLES

7.1 Mini-pressuremeter tests carried out at IMG, Grenoble (France)

Simulations were performed on experimental curves obtained from the pressuremeter tests

carried out in the calibration chamber on the Hostun sand at IMG (Institut Mécanique de

Grenoble, France). We note that the gap between the experimental curve and the theoretical

curve for the different simulations does not exceed 5% (figures 9a-9e)

Tables 3 and 4 respectively show the comparison between model parameters and friction

angle comparison with other methods and tests.

0 100 200 300 400 500 600 700 800

0

50

100

150

200

250

300

350

400

Model parameters for H161P27 test

E=35 MPa

A=0,0008

P0=27 kPa

d=16,1 kN/m3

Vo

lum

e c

ha

ng

e (

cm

3)

Applied pressure (kPa)

Simulated curve

Experimental curve

Figure 9a. H161P27 test (Boubanga, 1990)

229


0 100 200 300 400 500 600 700

0

50

100

150

200

250

300

350

400

450

500


E=45 MPa

A=0,001

P0=18 kPa

d=16,1 kN/m3

Vo

lum

e c

ha

ng

e (

cm3

)


Simulated curve

Experimental curve

Figure 9b. H161P18 test (Boubanga, 1990)

0 50 100 150 200 250 300

0

100

200

300

400

500


E=15 MPa

A=0,01

P0=27 kPa

d=15,7 kN/m3

Simulated curve

Experimental curve

Vo

lum

e c

ha

ng

e (

cm3

)


Figure 9c. H157P27 test (Boubanga, 1990)

0 25 50 75 100 125 150 175 200 225 250

0

200

400

600

800

1000

Vo

lum

e c

ha

ng

e (

cm3

)


Simulated curve

Experimental curve


E=13 MPa

A=0,015

P0=18 kPa

d=15,5 kN/m3

Figure 9d. H155P18 test (Boubanga, 1990)

230


0 50 100 150 200 250 300 350 400 450 500

0

100

200

300

400

500


E=16 MPa

A=0,004

P0=34 kPa

d=15,5kN/m3

Vo

lum

e c

ha

ng

e (

cm

3)


Simulated curve

experimental curve

Figure 9e. H155P34 test (Boubanga, 1990) Figure 9. Comparison between simulated and experimental curve for mini-pressuremeter test

realized on Hostun sand in IMG, France

Table 1. Model parameters

Model parameters

Test d(kN/m3) P0 (kPa) E (MPa) A ’ (°)

H152P21 15.2 21 35 0.005 36 0

H155P18 15.5 18 13 0.015 36 0

H155P34 15.5 34 16 0.004 39 0

H157P47 15.7 47 15 0.01 35 0.001

H161P34 16.1 34 45 0.001 40 0.003

H161P57 16.1 57 35 0.0008 40 0.003

Table 2. Results comparison

Friction angle ’ (°)

Test Duncan Lade ECL Proposed method Triaxial test

H152P21 33 37 36 36 35

H161P27 38 39 38 40 37.5

H152P18 - - - 40 35

H157P27 36 - 34 35 35

H155P18 - 37 37 36 35

H155P34 39 40 40 39 37.5

In Table 1, we show the model parameters deduced from the different simulations.

Comparison of the friction angles in Table 2 shows that all methods including triaxial test

provide friction angles included in a thin range.

231


8. CONCLUSION

The objective of this study was to develop an inverse method of identifying the

behavior parameters of generalized Prager model, associated with the failure criterion

of Drucker and Prager.

The resolution of the inverse problem of the cylindrical cavity expansion in an

elastoplastic medium showed the ability of the method to determine soil parameters

wherever the generalized Prager model was used.

Taking into account the dilatancy effect in this study has shown its effect on the

pressuremeter curve, the limit pressure increases with the dilatancy of the material.

The corrected limit pressure is almost proportional to the pressure of consolidation,

which is consistent with experimental observation.

This study showed that the assumption of plane strain is verified with the increase of

slenderness of the probe.

The distribution of main stresses at the end of loading shows that the plasticity may

occur between the radial stress r and the circumferential stress and between the

vertical stress z and the circumferential stress.

The difference between simulated curves and experimental curves for the different

simulations is mostly tolerable.

The values of the friction angle are quite close to those deduced from the triaxial tests

and other methods.

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soil behaviour optimisation procedure of … · in the field. two optimization techniques are...

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