Behaviour of granular materials reinforced by continuous threads

Computers and Geotechnics 7 (1989) 83-98

BEHAVIOUR OF GRANULAR MATERIALS REINFORCED BY CONTINUOUS THREADS

P. Villard and P. Jouve

Laboratoire de Gtnie Civil Ecole Nationale Sup~rieure de Mtcanique 44072 - Nantes Cedex 03 - France

ABSTRACT The experimental research carried out on soil reinforcement has resulted in the creation of a new product composed of sand and continuous fibres. In this paper, we propose a numerical model which fully describes the behaviour of the material. The main hypothesis, based on experimental considerations, has led us to divide the material into two intimately interwoven parts : a granular matrix and a network of continuous threads. Their behaviours are respectively described by a law of elastoplasticity (proposed by Verrneer) for the matrix, and by a non-linear relation integrating all characteristics of threads. Two programs, using the finite element method, have t~en developed. The first, which deals with the axial-symmetrical problems, allows comparison between modelling and triaxial test results. The second program, adapted to plane strain problems, allows calculations for the retaining walls and banks. The results obtained open up possibilities for its development in the design of reinforced structures.

INTRODUCTION Over the last ten years in civil engineering, a great deal of research has been carried out to find new products and new techniques for building retaining structures such as walls, embankments and steep slopes, as cheaply as possible. One of these new materials, is composed of sand and continuous threads (ref. 2), and has very interesting mechanical properties, i.e. flexibility and strength. Its name is Texsol, and it is marketed by S.A.T. (Socitt6 d'Application du Texsol, France). Many experiments have been carried out at C.E.R. (Centre d'Exptrimentation Routi~re) and L.R.C.P. (Laboratoire Rtgionale des Ponts et Chausstes) in Rouen. These are simple compression tests', triaxial tests, and experiments on real walls (scale 0.5 or 1). The wealth of 83 Computers and Geotechnics 0266-352X/89/S03.50 © 1989 Elsevier Science Publishers Ltd, England. Printed in Great Britain

84 results from these tests have demonstrated the general mechanical behaviour of the material, and have made it possible to build real structures, especially steep slopes (ref. 3). The aim of this paper is to propose a numerical model for Texsol, which corresponds to the main features of test results. Two programs, based on the finite element method, have been developed, one for axial-symmetrical problems (to simulate triaxial tests) and the other for plane problems (to simulate structures).

THE HYPOTHESIS During the construction of the reinforced material, the threads and the granular material are simultaneously poured out. The threads draw a dense network of large continuous loops, which run into the spaces between the grains (fig. 1).

/

~

thread

grain of sand

FIGURE 1 - Diagram of reinforced material.

The threads, which are generally made with polyester or polypropylene, have a diameter of between 0.068 and 0.383 mm. The weight ratio (weight of threads / weight of reinforced material) is between 0.1 and 0.3 %, which gives 10 to 100 cm length of threads per cm 3 of sand. The mixture is so dense that at first, we can consider that the threads do not slip in relation to the grains. As the threads are very thin, they do not disturb the soil skeleton. So, we suppose that : - the reinforced material can be regarded as a continuum, consisting of a supcrposition of two continuums bound together : one models the granular material, and the other the network of continuous threads.

85

This hypothesis is very important for our model, and it is valid when the reinforced rnaterial is not close to failure. Near this point, the mechanical phenomenon is not yet fully understood, especially with regard to the flow of grains around the threads.

THREAD NETWORK MODEL 1 - B e h a v i o u r of a unidirectional network

a - In our method, we assume that the network of threads is like a three-dimensional continuum. As the threads are very thin, they can only bear tensile forces, and their stiffness regarding compression, torsion, bending or shearing, can be ignored. Their behaviours are studied from tensile tests, and we consider them as elastic, linear or not. First of all, let us assume that all the threads are in the same direction. At a point M of the network, this direction is characterized by two angles ¢x and I~, which are defined with regard to a local system of axis (~, ~ , z'). With cartesian coordinates, the local system is parallel to the general system (X0, ~0, Z'0) ; with cylindrical coordinates, the directions taken are (~,]Y, z'), see fig. 2. -4-

Zo m

.

q

.'~

17

Cartesian coordinates

Cylindrical coordinates

FIGURE 2 - Local axis systems.

The angles ¢ and 13, which characterize the direction of threads, are defined in fig. 3, (x, ~ , z') being the local axis system, and (~, ~, ~ ) a system for which the thread direction is W. Let E be Young's modulus of the thread material, and p the volumetric ratio of threads (material volume / network volume).

86

o

Y

~-_"

FIGURE 3 - Diagram of the direction of a thread.

Let us consider a section S of the continuum, perpendicular to the direction w . The whole section of the corresponding threads is p S . For a tensile strain e'ww of the threads, the tensile force is :

F' = p S Ee'ww Thus, the stress in the continuum relative to e'ww is : F I

C'ww -

- E' eww'

with

E' = p E

(1)

S E' is the equivalent elastic modulus of the continuum. We can note that the real stress in the threads is equal to o'ww/P • The strain tensor of the continuum is written [ e ] for the system of axis (x, y , z ), and [ e'] for the system (LI, v , w ) . Similarly, the stress tensor is written [ c ] and [ o ' ] . Between [ e ] and [ e'] there is the relationship : [e'] in which :

= [Pl[el[P]

T

(2)

87

[P]

=

cos0teos~

sinotcos~

- sinl] ]

-sinc¢

coso~

0

sinoc sinl~

cosl3

cosct sin 13

(3)

(2) and (3) give for the direction w : i

Eww

= ~'XX C0S2{~ sin2~ +

~'yy sin2a sin2~ + Ez z cos2~

+ 2 ey z sino~ sinl3cos~ + 2 ex y sinct cosa sin2~ +

2 ex z

cosct sinl] cos~

(4)

.

For the stress tensor [ o'], with regard to the system (U, ~¢, ~¢), the component O'w w is given by (1) and (4) ; the other components are equal to zero. Thus, the components of [ a ] are obtained by : [g]

= [P]T[c¢] [P]

(5)

These components are functions of angles a and ~, and they are linear functions of strains. For the finite element method, it is more convenient to change the formulation into another matix formulation. Let us introduce the stiffness coefficients in the following form :

~x

kl i k12 k13 k14 k15 k16

~xx

Y3

k21 k22 k23 k24 k25 k26

%

zz

k31 k32 k33 k34 k35 k36

xy

k41 k42 k43 k44 k45 k46

e~z 2exy

~z

k51 k~2 k53 k54 k55 k56

2 exz

Gyz

k61 k62 k63 k64 k65 k66

2eyzJ

G

(6)

For the direction of threads, if the continuum is compressed ( ew w < 0 ), all the stiffness coefficients are zero. If the continuum is stretched ( ew w > 0 ), the stiffness matrix is symmetrical ( kij = kj i ) and the coefficients have the following values :

88 k l l = E' cos4cx sin4l~

k12 = E' cos2cx sin2¢x sin4[~

k13 = E' c0s21~ sin2[~ cos2[~

kla = E' cos3C~ s i n s sin4~

k15 = E' cos3~ sin3~ cos[~

kl6 = E' cos2~ sinct sin3[~ cos~

k22 = E' sin4cx sin4[~

k23 = E' sin2~ sin2[~ c0s213

k24 = E' coscx sin3cx sin4[~

k25 = E' c o s a sin20~ sin3[~ cos[~

k26 = E' sin30~ sin3[~ cos[~

k33 = E'cos4[~

k34 = E' coscx s i n s sin2[$ cos2[~

k35 = E' cosCx sin[~ cos3[~

k36 = E' s i n s sin[! c0s313

k44 = E' cos2u sin2a sin4[3

ka5 = E' cos2ct sincx sin3[~ cos[~

k46 = E' coscx sin2cx sin3[3 cosl3

k55 = E' c0s2c¢ sin2[~ c0s213

k56 = E' c o s a sincx sin2[3 cos213

(7)

k66 = E' sin2¢x sin2[~ cos2[~ We can note that, if all the threads are distributed in the same direction, the behaviour of the equivalent continuum is completely anisotropic. Although the stiffness matrix of the continuum is symmetrical, it cannot be considered as a matrix of Hooke's law with non linear coefficients.

2 - Behaviour

of a multidirectional

network

For a network having threads in several directions, we assume that the stiffness of the equivalent continuum is equal to the sum of all stiffness matrices weighted by coefficients of distribution : Kij =

E

A((z,~) k ij(lx,~)

(8)

et,[3

kij(cx,13) is a stiffness coefficiem, corresponding to the distribution of threads in the A((~,~)

direction ((x,[~) ; is the distribution function of threads in the different directions (ct,~), so that :

89

=

1

In (8), the coefficients Kij depend on the directions in which the threads are stretched or compressed. That gives a non-linear nature to the behaviour of the continuum, the coefficients of the stiffness matrix [ K ] depending on the principal directions of the strains { e }. If the directions of threads are continuously diswibuted in an angular sector S, the expression (8) is generalized by integration on S, as follows : I"

K..i j =

/~.(ot,13) kij(ot, ~) dS

(9)

;L(ot,15) is the distribution density, so that : I

~,(ot,~) dS = 1

For integration, we use the parameters ot and ~, the quantity dS being given by (fig. 4) : dS = sin~ dot d~

/ FIGURE 4 - Definition of the elementary surface of integration. If the distribution of directions is uniform in the sector S, the function X(ot,~) is constant and equal to :

90 Generally, analytical calculation is impossible for any deformation state of the continuum, due to the great difficulty in determining the limits of the tensile sector. However, for simple cases (like plane strains), it is possible to obtain Ki j with an analytical calculation.

MODELLING OF THE SAND Numerous studies of the behaviour of granular materials have already been made, and many formulations have been proposed. The development of numerical processings has favoured the creation of more or less sophisticated models, which are much more suited to the finite element method. From among all these models, we have chosen that of Vermeer, because it takes into account the phenomena of dilatancy and hardening with accuracy (ref. 4). This model is established from the results of triaxial tests, and it describes the behaviour of the material under loading, unloading and reloading conditions. Its last formulation, which considers a three-dimensional space of stresses, uses five parameters : [5, ee0, e¢0, (I)p and (I)¢v. It is obtained by adding to the elastic strains, the plastic strains due to shearing and those due to isotropic compression. The elastic strains use a non-linear law (a power law) and are given by a potential function. It needs two parameters : 13 the power of the law, and ee0 the elastic volumelaic strain in isotropic compression at the reference stress P0. The plastic shearing strains use a yield function F t and a non-associate potential function G 1. The equation of the yield surface is : F 1 = - 3 p II c + A III(~ = 0 , where p ,

(10)

IIo and IIIo are respectively the mean pressure, the second and the third

invariants of the stress tensor. A is a coefficient with a complex formulation, depending on the shear hardening coefficient, and on (I)p the peak angle of internal friction. In fig. 5, we represent the yield surface ( F 1 = 0 ) in the octahedral plane of the principal stress space, for different values of hardening. The formulation of G 1 has the form F 1 (see (10)), and uses the parameter (I)cv, the friction angle at critical void ratio.

91

,.,).

t r no

,

i

-

o

!

~,

It ~,

~

•

t t

s

• •

Io I

o

jo

r~

\cr I L_ iS

•

s o

FIGURE 5 - Yield surface F 1 in the octahedral plane for different values of hardening. Finally, the plastic strains due to isotropic compression, use a yield function F 2 (with hardening). This function def'mes the cap surface, and makes it possible to calculate the strains by an associate flow rule. It depends on the fifth parameter ee0, which represents the plastic volumetric strain at pressure p = P0.

CALCULATION OF STRUCTURES At present, we think that the t'mite element method is the most convenient for calculating the displacements and the stresses in real structures. So, we have created two programs using this method : one for axial symmetrical structures and the other for plane structures. The bases of these two programs are similar : - the same mesh in a vertical plane using triangular elements with three nodes per element, -

the same iterative process to solve the non-linear system of equations,

- the same method to solve the algebraic equations at each iteration. The main difference between these programs concerns the constitutive law of the reinforced material. For the sand, we have used Vermeer's model with the following stress states : ( 6r r, 6z z, t~r z, 60 o ) for one, and ( a x x, az z, 6x z ) for the other. For the threads, the integration of the stiffness coefficients on the tensile zone is more difficult for axial-symmetrical problems, because it needs numerical integration.

92 The relations between nodal forces and elemental stresses are linear, likewise those between elemental strains and nodal displacements (we assume that the displacements remain small). On the other hand, as we have seen in previous chapters, the constitutive laws of each part of the reinforced material, are significantly non-linear. Therefore, in order to solve the system of equations, it is necessary to use an iterative process. For our programs, we have found that the best process is to combine the Newton-Raphson method with a step by step method. At each iteration, we use an approximate calculation ; the computed stresses are different from the exact stresses by the quantities 8 c i j, which are called : "unbalanced stresses". These quantities are used again at the next iteration, and we stop this process when they are sufficiently small. This method is shown in diagrammatic form in fig. 6.

a A

~ °---'-' --- -

x

i

ei-i ~ei

c

FIGURE 6 - Diagram of the iterative process.

EXAMPLES OF APPLICATION 1 - C o m p a r i s o n o f o u r m o d e l w i t h l a b o r a t o r y test results

The cylindrical samples, used for laboratory tests, are made in moulds. Their diameters are 100 m m and 150 ram, and their slenderness ratio is 2. The continuous threads are thrown out by ejectors, and are mixed with the sand which flows out of a convoyor by gravitaty. This way, the sample is made by successive layers in the mould, the thread distribution being greater in horizontal planes.

93 For the tests, the threads were polyester, polyamide and kevlar. Their behaviour under tension, was mesured at C.E.R. in Rouen, by tensile tests. For the simulation, we used the thread behaviours presented in fig. 7. The cN/tex is a unity used in textile manufacture ; we have the equivalence : 1 cN/tex = (Yt/Yw) 10 MPa, in which 7t and Yware the unit weights of threads and water. For normal threads, the value of the ratio 7t/7w is approximately 1.4.

cN/tex (3)

80 60 40

/, I •

(i) polyester T163 (2) polyamide 235/34 Dtex (3) kevlar TI90

experimentation

(1)

/,

•

simulation

20

5

10

15

FIGURE 7 - Behaviour of threads under tension.

As there were no experimental results concerning the real thread distribution in the samples, for the presented simulations we considered that the reinforced material was homogeneous and isotropic. The results of these simulations are compared with those of triaxial tests on different kinds of sand and threads, carried out at C.E.R. in Rouen. TABLE 1 : Characteristics of reinforced materials

sand

threads

Z/'Yw weightratio

example

Op

~.

unit weight

nature

(I)

38,7 °

31 °

1.63

polyamide

1.14

0,287 %

(2)

41,6 °

32 °

1.86

kevlar

1.42

0,148 %

(3)

46,5 °

33 °

1.85

polyester

1.38

0,105 %

94 We assumed Vermeer's model corresponds to sands used for these tests. Table 1 summarizes the main characteristics of the reinforced materials which were employed. The other constitutive parameters of the sands were close, and they were relatively insignificant in our simulation, We used the following values :

=

015

ee0 = 4 10 -3

kPa

,

Be0 = 3.5 10.3

° 1 - ~3

1600

800

I

.I

I

5

10

15

--

~i %

experimentation simulation

FIGURE 8 - Example h

01

kPa

-

kPa

o 3

(~i - 03

1000 2000

500

100C

E1% I

0

5I

10

=-

FIGURE 9 - Example 2.

5

I

10

L

FIGURE 10 - Example 3.

96 The confining pressure was 75 kPa. The stress-strain curves of the tests are compared with those obtained with our model, by using the mechanical properties of the components (see figs 8, 9 and 10). For all the examples, we notice that before failure, computed curves were rather similar to experimental curves. In ref. 1, we show that it is possible to have a closer correspondence between these two kinds of curves, by adjusting the weight ratio of threads with a coefficient. The nature of this coefficient depends on the constitutive coefficients, the shape of the grains, the efficiency of threads, etc... It is a purely empirical coefficient, and we do not yet know how to estimate its value before testing. For the results, presented in figs 8, 9 and 10, we did not use the axial-symmetrical program, because we assumed that the stress field was uniform in the sample. In ref. 5, we completed this study by simulating different boundary conditions for the samples with this program, and we show that the axial stress clearly varies with these conditions. We have thus shown that it is possible to model the flexibility of the reinforced material, from knowledge of the different components. Close to failure, we think that the hypothesis of non-slip, is no longer true. This explains that our model does not yet predict failure.

2 - Simulation of an experimental wall We now consider a slender wall, subjected to hydrostatic pressure. This example is inspired by an experimental wall built at C.E.R. in Rouen, the loading being imposed by water power. The dimensions of the wall are : 4 m high and 0.8 m thick ; the mesh has 85 nodes and 128 elements (fig. 11). At the base, the boundary conditions are : no displacements (horizonal or vertical) for simulating an embedment. The fibres are made with high tensile (H.T.) polyester (presented in fig. 7, Tt/Tw = 1.38). They are uniformely distributed in the (X,Y) plane, the weight ratio being 0.2 %. The unit weight of sand is equal to 16.5 kNm -3 , and the values of the other parameters are : ~p = 40° ,

Oev = 32° ,

I~ = 0.25 ,

eeo = 4 10-3 ,

~e0 = 3.5 10-3 .

The initial stress state in the wall is complicated, because it assumes an initial tension of threads createdby the weight loading, which needs a preliminary calculation. Then, it is possible

96 to simulate the hydrostatic loading of the wall by a gradual increase in the water level. The results are computed for the heights given in table 2.

TABLE

2 -

Different heights of water level.

loading number

1

2

3

4

5

6

7

8

water height, m

1.08

1.88

2.42

2.80

3.12

3.37

3.62

4.00

For the different loadings, the shapes of the displacements are similar, likewise the stresses. For example, the loading n ° 8 leads to fig. 12, and n ° 5 to fig. 13. The results show that the lower part of the wall is highly stressed. More particularly, we notice that there is a tension zone where the threads are highly stretched, and a strong compression zone where only the sand is stressed. It resembles the bending of a reinforced concrete wall.

\t\f

i\ 1\ i \i \I i 4 ~NIX \t\1 I \IN \i\1 I Nt\ \t\1 I \1\ \ I\i i Ni\

FIGURE

11 -

m

Diagram of the wall.

FIGURE 12 - Displacements of the walt.

97

hd

m

xx unity kPa

-i0 -16

~

-23

-25

-5o

-2o

-100 0

20

-200

100 d~ 0.4

FIGURE 13 - Isostress curves.

0.8

m

FIGURE 14 - Displacement of the top.

In fig. 14, the curve represents the horizontal displacement of the top dp, relative to the height hd of the water level. It shows that there is a critical level (about 2.5 m), beyond which the displacement is pronounced. As we have already stated, the phenomenon of failure cannot yet be simulated by our model. However, our study has demonstrated that it is possible to predict the flexibility of a structure. When the test results on the real wall are published, we will be able to compare them with those obtained by simulation with the precise dimensions and real characteristics of materials.

CONCLUSIONS In this paper, we have shown how to model the reinforcement of sand by continuous threads. Our model has made it possible to study the flexibility of structures built with this material. We have begun to simulate the influence of different parameters, such as the weight ratio of threads, their tensile strength, the nature of soil, etc... These results are presented in ref. 5. The comparison with tests is still partial, because there is always some scattering for experimental results, and it is difficult to get the expected value for each parameter. Therefore, in

98 order to be able to validate future models, a databank of tests must be available where test conditions are all precisely defined. Our present research work is concentrated on the study of failure mechanisms of reinforced materials. We think that failure is due to the thread slipping between the grains of sand when the mean stress in the sand is not sufficient, or when the tension in the threads is too strong. However, this research is difficult because the reinforced material can no longer be considered as a continuum. Nevertheless, by combining our flexibility method with a limit state method, it now becomes possible to calculate the stresses and to design structures.

ACKNOWLEDGEMENTS Great appreciation is expressed to Mr Blivet of the "Laboratoire Rdgional des Ponts et Chaussdes" in Rouen, and to Mr Khay of the "Centre d'Expdrimentation Routi~re" in Rouen, for their help, especially for the experimental results. The authors are also grateful to Mr Leflaive of the S.A.T. (Socidtd d'Application du Texsol) for having initiated and encouraged this research.

REFERENCES 1.

Jouve P. and Villard P. Moddlisation du renforcement des mat&iaux granulaircs par des •s continus. 8th France-Poland svmm)sium of Auulicd Soils Mech~lni¢~,Grenoble (1987), 549-565.

2. Leflaive E. Le renforcement des mat6riaux granulaires avec des ills continus. Proe. ~l-Int. Conf. on Geotextiles, Las Vegas (1982), 721-729. 3.

Leflaive E., Khay M. and Blivet J.C. Un nouveau mat&iau : le Texsol. .Bull. Liai. des Lab. des Pont~ et Chauss6es. Paris, n° 125, may-june 1983, 105-114.

4. Verrneer P.A. A five-constant model unifying well established concepts. Proc. In~, workshou on Constitutive Behaviour of Soil. Grenoble (1982), vol.2, 134-166 5. ViUard P. Etude du mnforcement des sables par des ills continus. Mod61isation et applications. Thesis of doct., Ecole Nationale SupErietrre de M6eanique, Nantes (1988).