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~