Influence of a soil-strip interface failure condition on the yield-strength of reinforced earth

Computers and Geotechnics 7 (1989) 3 18

INFLUENCE ON

OF THE

A

SOIL-STRIP

INTERFACE

YIELD-STRENGTH

OF

FAILURE

REINFORCED

CONDITION EARTH

P. de BUHAN and L. SIAD Laboratoire de M~canique des Solides (X, Mines, Ponts, U ~ , . CNRS) Ecole Polytechnique, 91128 Palaiseau, France

ABSTRACT

A comprehensive approach to the yield-strength of reinforced earth when considered

as a macroscopically homogeneous material is presented within the framework of both the yield

design and homogenization theories. In this paper, a more specific attention is paid to the case when a failure condition relating to the interfaces between the soil and the reinforcing strips has to be taken into account. A closed form expression of the corresponding macroscopic strength criterion of reinforced earth is given, along with a geometrical interpretation in the space of stresses. The analysis clearly shows a significant reduction in the overall strength of reinforced earth when compared to the case of perfect bonding. The consequences of such a reduction on the stability of some typical reinforced soil structures are then carefully examined by means of the yield design kinematic method using 4¢rigid blocks~ failure mechanisms.

1. I N T R O D U C T I O N

The idea of implementing a homogenization method for analysing reinforced soils proceeds from the intuition that, from a "macroscopic" point of view, that is insofar as the overall properties of a reinforced soil structure are investigated,

the constitutive composite soil may be perceived as a homogeneous

but anisotropic material.

Such a method has been successfully applied to the

failure design of reinforced earthworks through the previous determination of a macroscopic strength criterion (see for instance

[i] [2] [3] [4]). But up to

now this method has been mainly restricted to the assumption of perfect bonding between the soil and the reinforcing strips,

thus disregarding any slipping

phenomenon between them. The purpose of this contribution is to present an extension of the yield design homogenization method allowing to take a specific soil-strip interface failure condition into account. 3

Computers and Geotechnics 0266-352X/89/S03.50 (~ 1989 Elsevier Science Publishers Ltd, England. Printed in Great Britain

2. MACROSCOPIC STRENGTH CRITERION OF REINFORCED EARTH

Since we are looking forward to a p p l y i n g the h o m o g e n i z a t i o n method reinforced earth structures subjected to "plane strain" conditions,

~o

it seems

advisable to start from a d e s c r i p t i o n of reinforced earth as a two-~?men~3~na!

mult~layer material made up of three components

•

the n a t i v e

friction angle

~

(fig.

~)

soil w h i c h is g e n e r a l l y a dry e o h e s i o n l e s s , and a specific weight

sand with a

7 ,

Reinforcing •

. ~: ' : . O / S ! ~ ~ i n t e r f Q c e

AH I " / ' ,

0

Fig.

. /'

×

I : R e i n f o r c e d e a r t h as a m u I ~ a y 6 ~

• the reinforcing

composite material.

layers (metallic strips or geomembranes)

by their tensile strength

Rt

characterized

per unit length along the transverse direction.

• the interfaces between the soil and the reinforcements w h i c h can be regarded as extremely thin layers of a h o m o g e n e o u s material.

2.1. The perfect bonding assumption.

This a s s u m p t i o n amounts to considering

the interfaces as infinitely resistant.

In such a case it has been proved

([5] [6]) that the m a c r o s c o p i c failure condition of reinforced earth can be w r i t t e n as :

fh°m(E) =

Min _o

o

~o~o

{fs(~ _ 0 ~ x ® e ) } ~ 0

(1)

(*)

...............................................................................

(*)

Compressive stresses are counted positive.

the compressive strength of the reinforcements

Due to failure by buckling, is taken equal to zero.

where

fs

condition

denotes

the convex yield

of the soll

function

relating

to the Coulomb's

failure

:

I/2 f(s)(~) =

= [ (Oxx - Oyy) 2 + 4 02y ]x

~

in the

whereas

ao = Rt/AH

cement)

represents

verse

area,

so t h a t

(AH : distance the ~

tensile

o

has

way

between

strength

the

+ 0

yy

) sin~ ~ 0

two successive

of

dimension

the

layers

reinforcements

(2)

per

of reinforunit

trans-

of a stress.

Such a condition

2.2. Interface failure condition.

following

xx

are the three independent components of the stress xy (x , y) coordinate system : see fig. i),

(Oxx , ayy , and tensor

- (~

can be expressed

in the

:

g(I) = g(o , T) < o

where

g

is a convex function

any facet

located

(3)

of the stress vector

at the interface

T(O

, T)

acting upon

(fig. 2).

T

g(O',T)=O

~T SOIL

>~///%TRIP Fig. 2 : So~l-strip interface strength condition.

Two such conditions

will be alternatively

considered

- a purely cohesive

condition

type)

g(!)

(c

denotes

=

(TRESCA's

IT[ - C ~ 0

the shearing strength of the interface).

in this study

:

(4)

- a purely

frictional

condition

g(k)

(6 : friction

=

(COULOMB's type)

Irl - ~ tan ~ < 0

(5)

angle)•

It has been shown within the f r a m e w o r k of the yield design h o m o g e n i z a t i o n theory [7]

that the expression of the m a c r o s c o p i c

s t r e n g t h condition of rein-

forced e a r t h must be m o d i f i e d as follows.

fh°m(z int ~

with

n

denoting

=

Max {fhom Z

(~J

,

g ( ~ . n)} < 0

the unit vector normal to the interface.

One can immediately observe that condition for instance

(6)

g(T)

(6) reduces to (i) as soon as

is taken equal to a n e g a t i v e constant

(case of perfect

bonding).

3. GEOMETRICAL

REPRESENTATIONS

It is w o r t h drawing the g e o m e t r i c a l representatioL~s of the different above. mentioned c r i t e r i a in the space of stresses w i t h coordinates

Z

xx

, Z

yy

and

/fz xy Thus,

the classical M o h r - C o u l o m b condition

space a convex domain

Gs

(2) of the soil defines in this

representing the set of allowable stress tensors.

This domain turns out to be a circular cone w i t h a v e r t e x at the origin and its axis co{nciding w i t h the line of stress

(Z

xx

Likewise, ted by

G h°m

= Z

yy

,

~xy

A

a s s o c i a t e d w i t h the h y d r o s t a t i c states

= O)

inequation (i) shows that the m a c r o s c o p i c

strength domain deno-

may be obtained as the convex envelope of the circular cone

and of the cone shifted by the distance

Then , the

domain

G h°m int

constructed as the intersection

- o

o

along the

Z - axis (see fig.3) xx

relating to the strength condition

of

Gh°m

with a domain

Gin t = {~ ; g ( ~ . n) = g(Zyy

, Zxy) < O}

Gs

Gin t

(6) may be

defined as :

IV~Exy

Exx -

~o

m

A/

Fig. 3 : S t r e n g t h domains of stresses

so that amounts

Gin t

(E

xx

, E

G

yy

xy

is a cylindrical

construction

of

and

homt Gin

int

to the

earth due to the introduction this condition

Another in drawing equation

Z

xx

accentuates

homogeneous

+ Z

yy

the different

by a deviatoric

= cst , normal to

A

of an interface

the anisotropy

,

in

failure

of reinfor-

plane,

domains

(fig. 4).

S = (Zyy - Z x x ) / ~

consists

i.e. a plane of

Let then :

P = (Zxx + Eyy)//2

This

material.

convenient way of representing

their cross-sections

- axis, which

in the case of perfect bonding.

condition.

ced earth as a macroscopically

~xx

gives clear evidence of the decrease

strength of reinforced Futhermore,

i n t h e space

Gh ° m

). Case o f a T r e s c a ' s i n t e r f a c e .

domain parallel

to the whole space of stresses

geometrical

GT M

s '

, /2 E

T = /~ Zxy

T

A

/"

S

G~

Gh°m

Fig. 4

:

Geom6t~ical representation

Combining criterion

expressions

(i),

(2),

in a d e v i a t o ~ c plane

(3) and

p

(6) the m a c r o s c o p i c

=

cst.

strength

writes i

[(z

- 0 - Z xx

fh°m(E~ ~ 0 int "----"

with

g(Zyy

-

~

d

o

,z

xy

)2 + 4 Z 2 ] yy xy

0

/2 -- (Z

+ Z xx

- O) sin,# ~< 0 yy

,

)~o

that is :

I/2 [(s + ~/£f) 2 + m 2]

Z E Gh°m = int

with

- d

o

g((P + S)I~"

~< (P - d//f) s i n e

~ o ~< O

,

Tl£f) ~

(7)

o

It follows any deviatoric truncated

that the cross-section plane

of the boundary surface of G h°m int is the envelope of a family of circles,

(P = cst)

by the curve of equation

trical representation,

the boundary

types of regions

and which might be qualitatively

• Regions

A

correspond

which both components

• Regions

B

surface of

respectively described

(~ = - O

the macroscopic

o

for

the tensile

: case of the figure)

states of stress

C

correspond

inducing a

to reach its ultimate

to the case when the strength capacities

the soil and the reinforcing

leading

to a failure mode by slipping

strips.

Figures 5-a and 5-b display the representations G h°m int

• a Tresca's

• a Coulomb's

of the macroscopic

in a deviatoric plane in the particular

cases of :

interface

g((P + S ) / ~

, T/~)

= IT/~I

- C ~< 0

interface

g((P + S ) / ~

, TI/~) = I T / E l

- (p + s) tan 6 1 ~

In the latter case it can be observed criterion

~

the strips remain within failure.

• Finally region~

strength domain

stresses

(~ = 0 : failure by buckling).

of the interfaces are fully mobilized, between

from this geome-

earth come up to failure,

strips being m a x i m u m

represent

whereas

(fig. 4).

as follows.

failure mode where the soil is the only constituent strength,

~ 0

G h°m being divided into int denoted by A , B , C in Fig. 4,

to the set of macroscopic

of the reinforced

load in the reinforcing or equal to nought

, T/~)

"failure modes" could then be predicted

Three possible

three different

g((P + S ) / ~

by

remains unaltered

as far as

~ > ~

~< 0 .

that the macroscopic .

strength

10

T

Gh°m

l

Gh°m

............../ 4=i=--~~ ' , ' ~ I / I I I i ¢ I

,'

/i/i,'//,,

/0 h 0 r'tq'

G int

int

b) Cou~omb's interface

a) Tr~sca's interface Fig. 5 4. STABILITY OF A REINFORCED EARTH EMBANKMENT

As a first example of a p p l i c a t i o n of the theory p r e s e n t e d a reinforced earth wall such as that shown in Fig. to evaluate the critical height H

H

6 will be considered. of the wall,

beyond w h i c h the structure will collapse.

dimensional analysis that

H

The p r o b l e m is

i.e. the m a x i m u m value of

It can be easily shown from

may be w r i t t e n in the following form :

H* =

~

K'W

m)

X where and

K m

is a non d i m e n s i o n a l factor, the "stability" factor of the wall~ a non d i m e n s i o n a l parameter c h a r a c t e r i z i n g the strength of the

interface, namely

m = C/0

and

m =

tan6 tan

o

in the case of a Tresca's

interface,

in the case of a frictional interface.

Our objective is to obtain an upperbound estimate for through

the yield design kinematic approach

K

(or

H )

([8]) using "rigid block" fai-

lure mechanisms.

From now on, the two types of interfaces will be considered separately.

11 4.1. Tresca's intemfaoe. The mechanisms blocks city

involved

consist

of two translating

(fig. 6) : the upper block is moving downwards with a vertical U

, while the lower triangular block

V , making an angle

jump across

AB

~

with respect

is tangential,

OAB

to line

velo-

is moving with a velocity

OA

and such that the velocity

that is

u = l~I = I~I cos(0 + ~) .

The virtual work done by the external velocity

field

v

writes

then

forces

(~)

in the corresponding

:

i P(y , v) = ~ y U ( H

2 - h z) t a n @

(8)

.

Y

C

D

H

B V

A

Fig. 6 : "Two blocks" f a i l u r e mechanism of a reinforced e a c h wall (case of a Tresca's i n t e r f a c e ) . the maximum resisting work developed

On the other hand, nized structure by

along the discontinuity

lines

OA , AB

and

by the homogeAC , denoted

S , is defined as :

P

hot

(v) =

I

hom (n ; ~v]) d E ~int --S

where

~XH

is the velocity jump across

S

following

its normal

n , and

12

~hOmint (n_ ; ~v~)_ = sup {(~._ n) . ~v~_ ; fhOmint(L)~E < 0}

so that taking

(I),

(4) and (6) into account,

hom (n ; ~v]) = ~int ---

C U

along

AC

C U tan(0 + ~)

along

AB

along

OA

o Thence

U cos e tan (8 + ~)

finally

hom Pint

I q°U[mh

h > 0

(v) =

and thus comparing with

=

+ ~)

@

h = 0

for

K

:

H

X ~-o

~ 2 Min

{ h2~ II tan ~

A numerical m i n i m i z a t i o n parameters

if

(8), the following upper bound estimate

H

*

and

h/H

Figure 7 displays

of this analytical

'

-tan @

expression with respect

which define the goemetry

K

of the mechanism

to

yields

:

~ K+(~ , m = C / O ) o

some typical curves representing

as a function

of the friction

angle

from infinity

(perfect bonding)

~

, for different

to nought

the variations values of

(perfectly smooth

must be pointed out that in the latter case K

if

+ (H - h)(l + m tanO) tan(e + ~)]

(7 U H t a n ( O o

K

one gets after some calculation~

+ K (~ , m = O)

= tan2(~/4 + ~/2) thus coinciding with the P from the yield design static approach performed

lower bou~zd in [i].

m

interface). reduces

of

K+

ranging It

to

estimate derived

13 10 m:C/O" o

YHI(Io:K

OO

0.3 0.2 0.1 0

f

0 10

20

30

(p(o)

Fig. 7 : Variations of

4.2. Coulomb's interface. ding case is considered,

AB

make the same angle

Following

, m = C/Oo).

The same class of failure mechanisms as in the prece except that segment

to the vertical direction, while across

K+(~

37 ~0

V ~

AC

is inclined at an angle

is such that the velocity discontinuity with the horizontal direction

the same computational

procedure

as in section

(fig. 8).

4.1, one obtains

Y D

C

H

~x Fig. 8 : Failure mechan~m c o n s i d ~ e d i n t h e case of a f r i c t i o n a l

interface.

14 K

< K+(~ , m = tan 6 /tan ~)

with the p a r t i c u l a r values

of

K+(~ , O) = K

= K

and

K+(~ , m) = cst

P for

m > I.

The c o r r e s p o n d i n g t h e o r e t i c a l curves are shown in Fig. 9 b e l o w :

10

?H/~o: K

m _-tan~/tan ~

30

25

Fig.

5. BEARING

35

9 : Vania2io~

of

37

40

K+(~ , m = tan 6 /tan ~).

CAPACITY PROBLEM

A n o t h e r typical p r o b l e m w h i c h deserves to be investigated is that of the load carrying capacity of a surface footing earth half space and subjected to a load

Q

the symmetry axis of the footing. D e n o t i n g by

resting upon a reinforced

u n i f o r m l y d i s t r i b u t e d along B

the w i d t h of the footing,

and n e g l e c t i n g the contribution due to the weight of the soil, the u l t i m a t e load

Q

writes then

Q

= B O

o

N (~ , m = tan 6 /tan ~)

15

a

b

Fig. 10 : Beating capacity problem. where

N

is a non dimensional

the only one considered

for

N

interface will be

, and thus for the bearing

can be derived from the kinematic

of failure mechanisms meter

(the frictional

here).

An upper bound estimate the structure,

factor

sketched

in Fig.

capacity

10-b, depending

on the angular para-

8 :

N

whence minimizing with respect

~ N +(~ , m

to

N*

; e)

e :

<

N +(~

,

m)

•

The corresponding results are shown in Fig. ii, where it should be + that N stops increasing as soon as 6 becomes greater than or

noticed

equal to

~ .

of

approach using the family

16

30

N = Q/B (;o

m = to n&/ton @

/

20

i 10

I

15

5

Fig. 6. A N U M E R I C A L

11

of

: Variations

N+(~ , m = tan ~ /tan ~).

EXAMPLE

Let us consider soil

35 37

25

for i l l u s t r a t i v e

is a c o h e s i o n l e s s

purpose

a numerical

sand w i t h a typical value of

example.

~ = 37 °

The b a c k f i l l

for the friction

angle.

Under cements, and ii)

the a s s u m p t i o n

the k i n e m a t i c

of perfect

bonding

a p p r o a c h w i l l yield

between

:

K+(~

for the s t a b i l i t y

factor

K

= 37 ° , m = 1) ~ 8

of the w a l l

and N+(@

for the b e a r i n g

capacity

= 37 ° , m = I) ~ 20

factor

N

the soil and the r e i n f o r -

the following

of the footing.

estimates

(see fig.

9

17

Assuming now that the strength of the interfaces between the soil and the reinforcements is governed b y a C o u l o m b ' s c r i t e r i o n (i.e.

with tan~= 0.8 tan

~ ~ 31°), the corresponding estimates become :

K+(~ = 37 ° , m = 0.8) ~ 8

and

N+(~ = 37 ° , m = 0.8) ~ 13.5 .

This numerical example clearly shows that the influence of a strength condition at the interface is quite negligible in the case of the reinforced wall (unless

m

becomes smaller than 0.5), whereas it proves very important

in the case of the footing, since the predicted bearing capacity is cut ~own by nearly 33 %.

7. C O N C L U S I O N

The homogenization method provides a convenient framework for designing reinforced earth structures,

taking an interface failure condition between the

soil and the reinforcements into account. However,

it should be noted that,

unlike the other components of reinforced earth (soil and reinforcements)

the

strength characteristics of which can be measured through classical experimental procedures

(e.g.

: tensile tests for the reinforcements,

and "triaxial"

tests or direct shear tests for the soil), the question of how to get a definite estimate for the strength of the interfaces remains largely unanswered up to now. In particular,

it would be advisable to take great care when deri-

ving such characteristics to be used within that framework (e.g. a friction angle

6)

from "pull out" tests as often suggested [9], sinch such tests

involve not only the interfaces but also the surrounding soil and are therefore difficult to interpret.

18 REFERENCES

I.

de Buhan, P. Mangiavacchi, R., Nova, R., Pellegrini, G. and SalenGon, 7. Yield design of reinforced earth walls through a homogenization method (Accepted for publication in Geotechnique).

2.

de Buhan, P. and Salen~on, J. Analyse de stabilit~ d'ouvrages en sols renforc~s par une m~thode d'homog~n~isation. Revue FranGaise de Ggotechnique, n ° 41, (1987), 29-43.

3.

Sawicki, A. Plastic limit behaviour of reinforced earth. Jl. of Geotech. Eng. Div., A.S.C.E., vol. 109, n ° 7, (1983),

4.

Sawicki, A. and Lesniewska, D. Failure modes and bearing capacity of reinforced Geotextiles and Geomembranes, 5, (1987), 29-44.

soil retaining walls.

5.

de Buhan, P., SalenGon, J. and Siad, L. Crit~re de r~sistance pour le mat~riau "terre arm~e". C.R. Ac. Sc., 302, II, Paris, (1986), 377-381.

6.

Siad, L. Analyse de stabilit~ des ouvrages g~n~isation. Thesis, E.N.P.C., Paris, (1987).

7.

1000-1005.

en terre arm~e par une m~thode d'homo-

de Buhan, P. Approche fondamentale du calcul ~ la rupture des ouvrages Thesis, Univ. P. et M. Curie, Paris, (1986).

8.

Salen~on, J. Calcul ~ la rupture et analyse limite. Presses de I'E.N.P.C., Paris, (1983).

9.

Schlosser, F. and Guilloux, A. Le frottement dans le renforcement des sols. Revue Franqaise de G~otechnique, n ° 16, (1981), 65-77.

en sols renforc~