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Limit analysis of reinforced slopes
Geotextilesand Geomembranes 7 (1988) 203-220
Limit Analysis of Reinforced Slopes
Andrzej Institute
Sawicki & Danuta LeBniewska
of Hydroengineering, (Received
IBW PAN, ul.Cysters6w
14 December
11,80-953 Gdarisk-Oliwa,
1987; accepted 27 February
Poland
1988)
ABSTRACT
This paper deals with the plastic limit analysis of reinforced soil slopes. First, the theoretical model of reinforced cohesive soil is presented, then, the formulation of boundary value problems by a method of characteristics described. The presented theory is applied to solve the problem of bearing capacity of both weightless and weighty reinforced slopes. Some results of experiments performed on small-scale slopes reinforced with fabric are presented and compared against theoretical predictions. The results presented show that the plastic model of reinforced soil and the method of characteristics serve as a useful tool in the analysis of reinforced soil structures.
1 INTRODUCTION The plastic limit analysis of reinforced soil slopes is presented in this paper. First, the theoretical model of reinforced cohesive soil, treated as a macroscopically homogeneous and anisotropic material, is described. The model proposed is based on the rigid-plastic theory of reinforced earth presented by Sawicki4 and then developed by LeSniewska.3 The idea of this generalization has appeared because such soils as clays, loams and silts are more and more being used together with reinforcing materials to make earth structures stronger. A classic reinforced earth is rather well recognized as an engineering material, because a large number of reinforced earth structures have been built since H. Vidal introduced this composite in 1966, see Ingold.’ The concept of soil reinforcement has been increasingly developed 203
Geotextiles and Geomembranes 02651144/88/$03.50 Ltd, England.
Printed
in Great Britain
@ 1988 Elsevier
Science Publishers
204
Andrzej Sawicki, Danuta LeSniewska
and used to solve a variety of geotechnical problems, including wide applications of geotextiles. Reinforced slopes are an example of the typical application of geotextiles in geotechnical engineering. In order to design properly such structures one has to know how the structure behaves under a given load, and how to predict a maximum load under which the structure will not collapse. Sawicki and LeBniewska5 presented the method of estimation of bearing capacity of reinforced soil retaining walls, that was based on the rigid-plastic model of reinforced cohesionless soil and plastic limit theorems. In this paper we present a more rigorous approach to the problem of bearing capacity of reinforced soil structures, and discuss this approach with respect to reinforced slopes. The solution to the problem of bearing capacity of reinforced slopes has
(b) Fig. 1. (a) Reinforced
soil--coordinate
system. (b) Illustration reinforced soil.
of macro- and microstresses
in
Limit analysis of reinforced slopes
205
been obtained with the help of the method of characteristics, for both weightless and weighty models of a structure. In the case of weightless soil the solution has been obtained in a closed form. For steep slopes this solution depends on the angle of slope inclination, whilst that angle does not appear in the solution for mild slopes. In the case of weighty reinforced soil the solution is possible to obtain using numerical methods. Some results of experiments performed on small scale slopes reinforced with fabric are presented and compared against theoretical predictions. The results presented show that the plastic model of reinforced soil and a method of characteristics serve as a useful tool in the analysis of reinforced soil and can also be useful in engineering design of such structures, constructions.
2 THEORETICAL
MODEL OF REINFORCED
SOIL
Reinforced soil is treated in this paper as a macroscopically homogeneous and anisotropic material, the gross behaviour of which depends on the mechanical properties and interactive contributions of the constituents, i.e. the soil and the reinforcement. We consider the soil unidirectionally reinforced, the vector n indicating the direction of reinforcement, Fig. l(a). 8 is the angle between the x axis and the direction of reinforcement. Both constituents of reinforced soil are assumed to work together, that is slippage on the interfaces between the soil and the reinforcement is neglected. Three stress tensors are defined at every point of reinforced soil; one tensor for macrostresses cr, and two tensors for microstresses, d and d, in the soil and reinforcement respectively. In indicial notation these tensors are related by
u: = ;
(crx - ~urJcos*e)
(1)
where r), and r), are volumetric fractions of the soil and reinforcement, R is a plastic locus for the reinforcement in extension, a,, = qr R, and 5 is a parameter describing the behaviour of the reinforcement (5 = -1 means
206
Andrzej Sawicki, Danuta LeSniewska
that the reinforcement works in a plastic state). For details see Sawicki,4 and Fig. l(b). The soil is assumed to obey Mohr-Coulomb yield condition: f” = (a:-~~“yz-(~~++~+2H)*sin*~+
4(fQ
= 0
(2)
where H = c/sin C#I,Cdenoting cohesion. Sawicki4 considered the case H = 0, since the French authors (Schlosser for example) excluded cohesive soils from application in reinforced earth structures. Substitution of eqns (1) into the yield condition (2) leads to the following yield condition expressed in terms of macrostresses: f = (UX- uy -
(3)
= 0
where Hk = r), H for abbreviation. The associated flow rule is assumed to be valid for the soil:
(4) where A” denotes a non-negative that the relation (4) is equivalent forced soil:
i,i =
.
A -
scalar function. Lesniewska” has shown to the associated flow rule for the rein-
af CkTij
where G; and Eijdenote strain rates in the soil and the composite respectively. The condition that the soil and the reinforcement work together can be expressed as follows: .s Eij
=
. EL,
(6)
It must be stressed that in a general case (5 # 0) the principal directions of the tensor i do not coincide with the principal directions of u, which is a characteristic feature of anisotropic materials, see Hill.’ The following relations describe the behaviour of reinforcement: E, < 0 extension E, > 0 compression
(7)
* = 0 rigid reinforcement Eli where E, = iii Q ni is a strain rate in the direction
of reinforcement.
Limit analysis of reinforced slopes
207
There are two basic mechanisms of reinforced soil failure. The first mechanism depends on the simultaneous plastic flow of both the soil and the reinforcement, the last working in extension (in < 0). In this case obviously 5 = -1. In the second mechanism the reinforcement remains rigid whilst the soil becomes plastic. In this case the parameter 5 describing the behaviour of reinforcement can be determined with the help of the third part of eqn (7) and the associated flow rule (4). The second part of eqn (7) is not important in comparison with those mentioned above, especially for geotextile reinforced soil, because the reinforcement does not generally work in compression (buckling, etc.). In such cases the reinforced soil behaves like a pure soil with no reinforcement. Some graphical illustrations of the yield condition (3), for the two practically important failure mechanisms of reinforced non-cohesive soil, are presented by Sawicki.4 We deal with the plane plastic flow of reinforced soil, so the equilibrium equations are of the form:
aux au,, _+---_ ax ay
0
au,
aflxy
ay
ax
A+-=
(8) y
where y denotes the unit weight of reinforced soil (the y direction is assumed to be a vertical one). Equations (3) and (8) form the system of equations with respect to macrostresses vXx,crYand gXV.There are two differential equations (8) and one algebraic equation (3). It is possible to reduce the number of unknowns down to two using the following parametrization: u.X = ~(1+sin~#~cos2p)+&r~(cos2~-sinf#~cos2p)-H~ 5
= V( 1 - sin 4 cos 2p) - &O(cos 28 - sin 4 cos 2~) - H,
fl.XJ’=
(9)
crsin+sin2p+$&jr0(sin0-sin+sin2p)
where p denotes the angle between the x axis and the direction principal microstress in the soil, and
of a bigger
u = ;(uX+uY+2Hk) Substitution Lesniewska3
of eqns (9) into the yield condition (3) gives an identity. has shown that the relations (7) can be replaced by:
5 = -lforp-O-E>0 {f5-l,O)forp-8-E=0 5 = Oforp-f3-E<:0
(reinforcement
yields in extension)
(reinforcement
remains rigid)
(reinforcement works in compression)
(10)
Andrzej
208
wheree
= p--. 4
Equations
Sawicki,
Danuta
LeSniewska
4 2
(10) are equivalent
to the following formula:
< = -H(p-0-e)
(11)
where H( ) denotes the Heaviside function. Substitution of eqns (9) and (11) into the equilibrium equations (8) gives the following system of differential equations with respect to (T and p:
(1 +sin+cos2p)
-
~-[(2~~-&~~)sint#1sin2p+~~~~(cos2H
+~~,,(sin20 - sin+sin2p)6(p
- H-E)]
x
= 0 (12)
-sin$cos2p)fi(p-O-•)]z
dP
= y
The system of eqns (12) is more general than that presented by Sokolovsk? for an ideal soil with no reinforcement, because of the delta function S(p - 8 - E) (or Dirac’s function). We have to complete the system of equations (12) by adding the following expressions for respective differentials:
au
-dx+tdy=d~ 3X
(13)
The system of partial differential equations (12) and (13) can be analysed using standard procedures, see Sokolovski.6 This system is of a hyperbolic
209
Limit analysis of reinforced slopes
type, provided that the following relations are not satisfied simultaneously: CJ = 0 and 5 = 0 (or a0 = 0). From the characteristic polynomial of eqns equations for stress (12) and (13) we obtain the following differential characteristics: dy -= d_,~
(2~ -
+ COSC#J) +$u~coS$[coS($~ + sin+)+&cos+[sin(+
- 20) 5 cos~16(~-~-E)
- 20) +
sin+la(P - fV-E) (14)
There are two cases that have to be considered separately. corresponds top # 8 + E. In this case the stress characteristics
dy -= dx
sin 2p k cos C#I cos2p
+
sin+
(15)
= tan(p+E)
In the second case, when p is approaching
dy -= dx
The first one are given by:
8 + E(P+
8 + E), we
cos(4 - 20) t cos 4 = tan(0+E*t) sin(+ - 28) + sin 4
have:
(16)
In both cases, (15) and (16), the differential istics are of the final form:
equations
for stress character-
dy
= tan(p?E) dx which is similar to that for a pure soil. Equation (17) means that there is a continuous transition from extended and rigid reinforcement, as well as from the rigid reinforcement and that compressed. In the case of (15) the following relations are satisfied along characteristics da?(2q-
= y(dyktan4dx)
(18)
where for the reinforcement working in compression (5 = 0) we get formulae similar to those for a pure soil. For the case of (16) there is:
In a way similar to that presented equations for velocity characteristics:
dy
-
dx
= tan(p k E), dv, + dv, tan@ ~fr6) = 0
above
we obtain
the respective
(20)
Equations (20) are valid for arbitrary 5. It follows from above considerations that the stress and velocity characteristics coincide.
210
Andrzej Sawicki, Danuta LeSniewska
3 BEARING
CAPACITY OF A WEIGHTLESS REINFORCED SLOPE
Let us consider a weightless slope reinforced in the x direction as shown in Fig. 2. The angle between the slope and the horizontal is denoted by p. A structure is loaded on the upper edge by a uniformly distributed load p. The following three cases have to be considered separately:
(21)
c Y\ Fig. 2. Reinforced
slope.
The following differential equations are satisfied along characteristics in the regions of compressed, extended and rigid reinforcement respectively: du+2utan+dp
= 0
d~+(2~+u0)tan~dp dckugij(P-E)dp A solution respectively:
= 0 = 0
can be easily obtained
u = Cexp(*2ptan+) u = Cexp(k2ptan+) u =
k;u,H(p-E)+C
(22)
using an analytical
procedure.
We get
(23a) -$u,)
(23b) (234
where C denotes a constant. It follows from eqns (23a, b) that the stress characteristics are logarithmic spirals for the reinforcement being in a plastic state. The Heaviside function appearing in eqn (23~) means that in the region of rigid reinforcement there is a jump of (T, the value of which depends on the boundary conditions. The
Limit analysis of reinforced slopes
details of analysis of particular boundary value problems are presented Lesniewska.’ We restrict our attention only to final results.
211
by
3.1 ThecaseOIP<2e The mechanism of failure is presented in Fig. 3, where the plastic flow of reinforced soil occurs in the region ACDEFB. In the triangle ABC both the soil and the reinforcement become plastic. The stress characteristics are the straight lines given by y=tan
+.?e (
In the triangle characteristics equations:
+C 1 BEF the reinforcement are also the straight
y = tan@*E)
works in compression. The stress lines described by the following
(25)
In the curvilinear triangle BCE there are two families of characteristics, namely the straight lines and logarithmic spirals. The sector BD divides the region BCD in which reinforcement is extended, and the region BDE in which the reinforcement works in compression. The sector BD corresponds top = E. The maximum load carried by a structure is given by the following formula:
p =mo(l +sin4)exp
[ (:++)
tan41
Fig. 3. Mechanism of failure for the case 05/3<2E.
Fig. 4. Mechanism of failure for the case p = 2E.
212
Andrzej Sawicki, Danuta LeSniewska
It follows from eqn (26) that the solution obtained does not depend on the value of p. The same formula as eqn (26) follows from kinematical considerations, so we have obtained the exact solution to the problem of bearing capacity of a weightless reinforced slope, for a given range of angles p.
3.2 The case /3 = 2~ Figure 4 shows the mechanism of failure which differs from that presented in Fig. 3. The triangle ABC is a region of active pressure in which both components of reinforced soil are in a plastic state. The stress characteristics are here the straight lines. In the region BCD the stress characteristics are straight lines and logarithmic spirals. The soil and the reinforcement work in a plastic state in this region. The triangle BDE is a region of passive earth pressure in which the reinforcement remains rigid. The characteristics are also straight lines, the direction of the second family coinciding with the direction of reinforcement. The bearing capacity is given in this case by the formula (26). 3.3 The case 26 < p I 7r/2 The mechanism of failure is shown in Fig. 5, where in all the region ACDEB the plastic how of both constituents takes place. The reinforcement is extended. The triangle ABC forms a region of active pressure, in the triangle BDE the state of passive pressure occurs. In these two triangles the stress characteristics are straight lines. In the region BCD the characteristics are respectively straight lines and logarithmic spirals. The bearing capacity of a structure depends in this case on the value of /3. Both the static and kinematic approaches lead to the following formula:
p = flg,(l+sin$)exp
[2 (5
- pg)tan4]
where
f_rg= f_rOsinp
sin /3 + d/sin2 fi - ~0s’ 4 cos2 $I
(27)
Limit analysis of reinforced slopes
Fig. 5. Mechanism
In the case of p = m/2 for the bearing capacity Lesniewska.’ Equation (27) is the reinforced steep slopes
4 BEARING
213
of failure for the case 2~ < /3 5 IT/~.
(vertical slope) we obtain from eqn (27) the formula of the reinforced soil retaining wall, see Sawicki and exact solution to the problem of bearing capacity of (weightless) whilst eqn (26) applies to mild slopes.
CAPACITY
OF WEIGHTY
REINFORCED
SLOPES
In the case of weighty reinforced slopes it is not possible to obtain a solution to the problem of bearing capacity in a closed, analytic, form, as it was presented in Section 3. In order to get a solution to a particular problem one has to solve numerically the system of eqns (17) and (18)) or (19), depending on the mechanism of failure. We would like to present in this paper some final results of numerical computations performed for a hypothetical geotextile reinforced slope. We have assumed the following data: the height of a slope H = 10 m, the unit weight of reinforced soil y = 17 kN/m3, the angle of internal friction 4 = 34”, the cohesion c = 0. We have computed examples for various angles p and many values of the parameter oo. Figure 6 shows the net of characteristics and the computed distribution of load on the upper edge of a structure, for p = 60” and a0 = 80 kPa. The bearing capacity is 5-l MN in this case. Similar results, but for o. = 120 kPa, are shown in Fig. 7. In this case, the bearing capacity is 7-4 MN. A comparison of Figs 6 and 7 suggests that the value of u. influences the shape of slip lines. Indeed, as the reinforcement becomes stronger, the shape of the plastic region becomes similar to that for a weightless slope. A synthesis of numerical experiments regarding the bearing capacity of
214
Andrzej Sawicki, Danuta LeSniewska
p(w)Dwl.1
4 .4
Fig. 6. Stress characteristics
Fig. 7. Stress characteristics
for cm = 80 kPa.
for CT,,= 120 kPa.
Limit analysis of reinforced slopes
Fig. 8. Dependence
of bearing capacity on /3 and aa.
a 6* + 4: + .z-
(cl Fig. 9. Dependence
of bearing capacity on /3 and a~.
21.5
216
Andrzej Sawicki, Danuta LeSniewska
reinforced slopes is presented in Fig. 8. Figure 8(a) shows the dependence of bearing capacity P (obtained from integration of the computed load distribution p) on the angle p, for four values of the parameter (TV.Similar results, but for a weightless slope, are presented in Fig. 8(b). It follows from these results that the value of angle p does not influence strongly the bearing capacity of a structure, except for ps approaching the value of 7r/2 - 4, in the case when its own weight is taken into account. And conversely, in the case of a weightless slope the bearing capacity decreases as p approaches 7~/2 - C#J,but still the influence of p is rather small. A factor which strongly influences the bearing capacity of a structure is the value of co. Figure 9 shows the dependence of bearing capacity, for both weighty and weightless slopes, on the value of (T(,, for three values of the angle p. The influence of its own weight is visible for steep slopes (Figs 9(a) and 9(b)). In the case of p = 60” (Fig. 9(c)) there is practically no difference between the results computed for weighty and weightless slopes. It is worthy of note that the dependence between P and go is linear in all the cases analysed, except for the values of p greater than 140 kPa, for a weighty structure. It must be remembered that the results for a weightless slope have been obtained analytically, whilst those for a weighty structure numerically.
5 EXPERIMENTAL
RESULTS
In order to compare theoretical predictions with a real behaviour of reinforced slopes a set of small scale experiments has been performed. The experimental technique, materials, etc., are similar to those presented by Sawicki and Lesniewska,s so in this paper we would like to present only some final results. In above-mentioned experiments the values of a vertical load were measured, as well as mechanisms of failure photographed for various values of p and a,,. Figure 10 shows the comparison of predicted and measured results. Solid lines present the theoretical predictions for both weighty and weightless slopes, the dots correspond to experimental results. It seems that the agreement between theoretical predictions and experimental results is rather good. The influence of the weight of a slope is rather small because of small dimensions of experimental structures (24 cm high). Figure 11 shows the net of characteristics and a maximum load at failure computed for the experimental data: H = 24 cm, p = 60”, y = 17 kN/m’, C$ = 34”, c = 0, (T” = 46.9 kPa. In this case the bearing capacity was P = 4.15 kN. The dots indicate places in which the reinforcement (textile bands) got broken, in two subsequent experiments. These points lie close to the predicted slip line. In this case, as well as in other cases not reported in
217
Limit analysis of reinforced slopes
Fig. 10. Comparison
of theoretical
and experimental
results.
218
Andrzej Sawicki, Danuta LeSniewska p p(x)
LMPal
Fig. 11. Stress characteristics
against experimental
3 t
2m s a l-
I
I
35
30 @ Fig. 12. Influence
I
40
(degrees)
of I#Ion bearing capacity.
results.
Limit analysis of reinforced slopes
219
this paper, there is a good agreement between theoretical prediction and experimental results. Finally, Fig. 12 shows the influence of the angle of internal friction 4 (in the range of practically important values) on the bearing capacity of two slopes, characterized by p = 60” and 90”.
6 CONCLUSIONS The aim of the present paper was to present the plastic limit analysis of reinforced slopes. First, the theoretical model of reinforced soil, more general than that presented by Sawicki,4 was described. Then, the formulation of boundary value problems was presented and applied to solve the problem of bearing capacity of reinforced slopes. In the case of weightless structures the closed form solution was obtained, see formulae (26) and (27), for mild and steep slopes respectively. It seems that these formulae can be useful in engineering applications because of their simple form, at least to estimate the collapse load for real structures. In the case of weighty structures the solution to a particular problem can be obtained by numerical methods. Some results of such computations were presented in this paper and compared with predictions obtained from eqns (26) and (27). Finally, a comparison of experimental results with theoretical predictions was shown and briefly discussed. The conclusion is that the theory proposed gives realistic predictions, i.e. the bearing capacity and location of slip lines. The mathematical part of this paper is addressed to those readers who work on the application of plasticity theory in geomechanics. The results presented in Sections 2 and 3 give some basic material, as for example equations for characteristics, which can be easily checked and then applied by those who are familiar with the methods of limit analysis. This part of the text is very important from the view point of the theory of reinforced soil, as well as from the practical viewpoint, since it leads to some closed solutions of engineering problems, such as for example eqns (26) and (27). Numerical results presented in this paper illustrate the theory proposed, so no analysis is presented showing the influence of all the parameters, which characterize the reinforced soil, on the bearing capacity. For example, the influence of (T” is of a fundamental importance, but it is not possible to study this influence in the whole range of values, since it is not possible to obtain solutions for an unreinforced slope, the angle of inclination of which is bigger than the angle of a natural slope. The practical implications of the theory presented are in eqns (26) and (27), as well as in the method of determination of bearing capacity of
220
Andrzej Sawicki, Danuta LeSniewska
weighty reinforced slopes. In the authors belief these results are original, because no rational method for the estimation of bearing capacity of reinforced soil structures has been previously proposed.
ACKNOWLEDGEMENT The research reported in this paper was carried out in the main task No. 02.01-1.5 supervised by the Institute of Fundamental Technological Research, Warsaw, Poland.
REFERENCES 1. Hill, R., The Mathematical Theory of Plasticity. Clarendon Press, Oxford, 1956. 2. Ingold, T. S., Reinforced Earth. Thomas Telford Ltd, London, 1982. 3. Lesniewska, D., Statics and kinematics of reinforced soil-an application to bearing capacity of slopes, Ph.D. Thesis. Institute of Hydroengineering, Gdansk, Poland, 1987. 4. Sawicki, A., Plastic limit behaviour of reinforced earth, Jnl Geot. Engng, ASCE 109(7) (1983) 1000-5. 5. Sawicki, A. & Lesniewska, D., Failure modes and bearing capacity of reinforced soil retaining walls, Geotextiles and Geomembranes 5 (1987) 29-44. 6. Sokolovski, V. V., Statics of Granular Media. Pergamon Press, New York, 1965.
Limit Analysis of Reinforced Slopes
Andrzej Institute
Sawicki & Danuta LeBniewska
of Hydroengineering, (Received
IBW PAN, ul.Cysters6w
14 December
11,80-953 Gdarisk-Oliwa,
1987; accepted 27 February
Poland
1988)
ABSTRACT
This paper deals with the plastic limit analysis of reinforced soil slopes. First, the theoretical model of reinforced cohesive soil is presented, then, the formulation of boundary value problems by a method of characteristics described. The presented theory is applied to solve the problem of bearing capacity of both weightless and weighty reinforced slopes. Some results of experiments performed on small-scale slopes reinforced with fabric are presented and compared against theoretical predictions. The results presented show that the plastic model of reinforced soil and the method of characteristics serve as a useful tool in the analysis of reinforced soil structures.
1 INTRODUCTION The plastic limit analysis of reinforced soil slopes is presented in this paper. First, the theoretical model of reinforced cohesive soil, treated as a macroscopically homogeneous and anisotropic material, is described. The model proposed is based on the rigid-plastic theory of reinforced earth presented by Sawicki4 and then developed by LeSniewska.3 The idea of this generalization has appeared because such soils as clays, loams and silts are more and more being used together with reinforcing materials to make earth structures stronger. A classic reinforced earth is rather well recognized as an engineering material, because a large number of reinforced earth structures have been built since H. Vidal introduced this composite in 1966, see Ingold.’ The concept of soil reinforcement has been increasingly developed 203
Geotextiles and Geomembranes 02651144/88/$03.50 Ltd, England.
Printed
in Great Britain
@ 1988 Elsevier
Science Publishers
204
Andrzej Sawicki, Danuta LeSniewska
and used to solve a variety of geotechnical problems, including wide applications of geotextiles. Reinforced slopes are an example of the typical application of geotextiles in geotechnical engineering. In order to design properly such structures one has to know how the structure behaves under a given load, and how to predict a maximum load under which the structure will not collapse. Sawicki and LeBniewska5 presented the method of estimation of bearing capacity of reinforced soil retaining walls, that was based on the rigid-plastic model of reinforced cohesionless soil and plastic limit theorems. In this paper we present a more rigorous approach to the problem of bearing capacity of reinforced soil structures, and discuss this approach with respect to reinforced slopes. The solution to the problem of bearing capacity of reinforced slopes has
(b) Fig. 1. (a) Reinforced
soil--coordinate
system. (b) Illustration reinforced soil.
of macro- and microstresses
in
Limit analysis of reinforced slopes
205
been obtained with the help of the method of characteristics, for both weightless and weighty models of a structure. In the case of weightless soil the solution has been obtained in a closed form. For steep slopes this solution depends on the angle of slope inclination, whilst that angle does not appear in the solution for mild slopes. In the case of weighty reinforced soil the solution is possible to obtain using numerical methods. Some results of experiments performed on small scale slopes reinforced with fabric are presented and compared against theoretical predictions. The results presented show that the plastic model of reinforced soil and a method of characteristics serve as a useful tool in the analysis of reinforced soil and can also be useful in engineering design of such structures, constructions.
2 THEORETICAL
MODEL OF REINFORCED
SOIL
Reinforced soil is treated in this paper as a macroscopically homogeneous and anisotropic material, the gross behaviour of which depends on the mechanical properties and interactive contributions of the constituents, i.e. the soil and the reinforcement. We consider the soil unidirectionally reinforced, the vector n indicating the direction of reinforcement, Fig. l(a). 8 is the angle between the x axis and the direction of reinforcement. Both constituents of reinforced soil are assumed to work together, that is slippage on the interfaces between the soil and the reinforcement is neglected. Three stress tensors are defined at every point of reinforced soil; one tensor for macrostresses cr, and two tensors for microstresses, d and d, in the soil and reinforcement respectively. In indicial notation these tensors are related by
u: = ;
(crx - ~urJcos*e)
(1)
where r), and r), are volumetric fractions of the soil and reinforcement, R is a plastic locus for the reinforcement in extension, a,, = qr R, and 5 is a parameter describing the behaviour of the reinforcement (5 = -1 means
206
Andrzej Sawicki, Danuta LeSniewska
that the reinforcement works in a plastic state). For details see Sawicki,4 and Fig. l(b). The soil is assumed to obey Mohr-Coulomb yield condition: f” = (a:-~~“yz-(~~++~+2H)*sin*~+
4(fQ
= 0
(2)
where H = c/sin C#I,Cdenoting cohesion. Sawicki4 considered the case H = 0, since the French authors (Schlosser for example) excluded cohesive soils from application in reinforced earth structures. Substitution of eqns (1) into the yield condition (2) leads to the following yield condition expressed in terms of macrostresses: f = (UX- uy -
(3)
= 0
where Hk = r), H for abbreviation. The associated flow rule is assumed to be valid for the soil:
(4) where A” denotes a non-negative that the relation (4) is equivalent forced soil:
i,i =
.
A -
scalar function. Lesniewska” has shown to the associated flow rule for the rein-
af CkTij
where G; and Eijdenote strain rates in the soil and the composite respectively. The condition that the soil and the reinforcement work together can be expressed as follows: .s Eij
=
. EL,
(6)
It must be stressed that in a general case (5 # 0) the principal directions of the tensor i do not coincide with the principal directions of u, which is a characteristic feature of anisotropic materials, see Hill.’ The following relations describe the behaviour of reinforcement: E, < 0 extension E, > 0 compression
(7)
* = 0 rigid reinforcement Eli where E, = iii Q ni is a strain rate in the direction
of reinforcement.
Limit analysis of reinforced slopes
207
There are two basic mechanisms of reinforced soil failure. The first mechanism depends on the simultaneous plastic flow of both the soil and the reinforcement, the last working in extension (in < 0). In this case obviously 5 = -1. In the second mechanism the reinforcement remains rigid whilst the soil becomes plastic. In this case the parameter 5 describing the behaviour of reinforcement can be determined with the help of the third part of eqn (7) and the associated flow rule (4). The second part of eqn (7) is not important in comparison with those mentioned above, especially for geotextile reinforced soil, because the reinforcement does not generally work in compression (buckling, etc.). In such cases the reinforced soil behaves like a pure soil with no reinforcement. Some graphical illustrations of the yield condition (3), for the two practically important failure mechanisms of reinforced non-cohesive soil, are presented by Sawicki.4 We deal with the plane plastic flow of reinforced soil, so the equilibrium equations are of the form:
aux au,, _+---_ ax ay
0
au,
aflxy
ay
ax
A+-=
(8) y
where y denotes the unit weight of reinforced soil (the y direction is assumed to be a vertical one). Equations (3) and (8) form the system of equations with respect to macrostresses vXx,crYand gXV.There are two differential equations (8) and one algebraic equation (3). It is possible to reduce the number of unknowns down to two using the following parametrization: u.X = ~(1+sin~#~cos2p)+&r~(cos2~-sinf#~cos2p)-H~ 5
= V( 1 - sin 4 cos 2p) - &O(cos 28 - sin 4 cos 2~) - H,
fl.XJ’=
(9)
crsin+sin2p+$&jr0(sin0-sin+sin2p)
where p denotes the angle between the x axis and the direction principal microstress in the soil, and
of a bigger
u = ;(uX+uY+2Hk) Substitution Lesniewska3
of eqns (9) into the yield condition (3) gives an identity. has shown that the relations (7) can be replaced by:
5 = -lforp-O-E>0 {f5-l,O)forp-8-E=0 5 = Oforp-f3-E<:0
(reinforcement
yields in extension)
(reinforcement
remains rigid)
(reinforcement works in compression)
(10)
Andrzej
208
wheree
= p--. 4
Equations
Sawicki,
Danuta
LeSniewska
4 2
(10) are equivalent
to the following formula:
< = -H(p-0-e)
(11)
where H( ) denotes the Heaviside function. Substitution of eqns (9) and (11) into the equilibrium equations (8) gives the following system of differential equations with respect to (T and p:
(1 +sin+cos2p)
-
~-[(2~~-&~~)sint#1sin2p+~~~~(cos2H
+~~,,(sin20 - sin+sin2p)6(p
- H-E)]
x
= 0 (12)
-sin$cos2p)fi(p-O-•)]z
dP
= y
The system of eqns (12) is more general than that presented by Sokolovsk? for an ideal soil with no reinforcement, because of the delta function S(p - 8 - E) (or Dirac’s function). We have to complete the system of equations (12) by adding the following expressions for respective differentials:
au
-dx+tdy=d~ 3X
(13)
The system of partial differential equations (12) and (13) can be analysed using standard procedures, see Sokolovski.6 This system is of a hyperbolic
209
Limit analysis of reinforced slopes
type, provided that the following relations are not satisfied simultaneously: CJ = 0 and 5 = 0 (or a0 = 0). From the characteristic polynomial of eqns equations for stress (12) and (13) we obtain the following differential characteristics: dy -= d_,~
(2~ -
+ COSC#J) +$u~coS$[coS($~ + sin+)+&cos+[sin(+
- 20) 5 cos~16(~-~-E)
- 20) +
sin+la(P - fV-E) (14)
There are two cases that have to be considered separately. corresponds top # 8 + E. In this case the stress characteristics
dy -= dx
sin 2p k cos C#I cos2p
+
sin+
(15)
= tan(p+E)
In the second case, when p is approaching
dy -= dx
The first one are given by:
8 + E(P+
8 + E), we
cos(4 - 20) t cos 4 = tan(0+E*t) sin(+ - 28) + sin 4
have:
(16)
In both cases, (15) and (16), the differential istics are of the final form:
equations
for stress character-
dy
= tan(p?E) dx which is similar to that for a pure soil. Equation (17) means that there is a continuous transition from extended and rigid reinforcement, as well as from the rigid reinforcement and that compressed. In the case of (15) the following relations are satisfied along characteristics da?(2q-
= y(dyktan4dx)
(18)
where for the reinforcement working in compression (5 = 0) we get formulae similar to those for a pure soil. For the case of (16) there is:
In a way similar to that presented equations for velocity characteristics:
dy
-
dx
= tan(p k E), dv, + dv, tan@ ~fr6) = 0
above
we obtain
the respective
(20)
Equations (20) are valid for arbitrary 5. It follows from above considerations that the stress and velocity characteristics coincide.
210
Andrzej Sawicki, Danuta LeSniewska
3 BEARING
CAPACITY OF A WEIGHTLESS REINFORCED SLOPE
Let us consider a weightless slope reinforced in the x direction as shown in Fig. 2. The angle between the slope and the horizontal is denoted by p. A structure is loaded on the upper edge by a uniformly distributed load p. The following three cases have to be considered separately:
(21)
c Y\ Fig. 2. Reinforced
slope.
The following differential equations are satisfied along characteristics in the regions of compressed, extended and rigid reinforcement respectively: du+2utan+dp
= 0
d~+(2~+u0)tan~dp dckugij(P-E)dp A solution respectively:
= 0 = 0
can be easily obtained
u = Cexp(*2ptan+) u = Cexp(k2ptan+) u =
k;u,H(p-E)+C
(22)
using an analytical
procedure.
We get
(23a) -$u,)
(23b) (234
where C denotes a constant. It follows from eqns (23a, b) that the stress characteristics are logarithmic spirals for the reinforcement being in a plastic state. The Heaviside function appearing in eqn (23~) means that in the region of rigid reinforcement there is a jump of (T, the value of which depends on the boundary conditions. The
Limit analysis of reinforced slopes
details of analysis of particular boundary value problems are presented Lesniewska.’ We restrict our attention only to final results.
211
by
3.1 ThecaseOIP<2e The mechanism of failure is presented in Fig. 3, where the plastic flow of reinforced soil occurs in the region ACDEFB. In the triangle ABC both the soil and the reinforcement become plastic. The stress characteristics are the straight lines given by y=tan
+.?e (
In the triangle characteristics equations:
+C 1 BEF the reinforcement are also the straight
y = tan@*E)
works in compression. The stress lines described by the following
(25)
In the curvilinear triangle BCE there are two families of characteristics, namely the straight lines and logarithmic spirals. The sector BD divides the region BCD in which reinforcement is extended, and the region BDE in which the reinforcement works in compression. The sector BD corresponds top = E. The maximum load carried by a structure is given by the following formula:
p =mo(l +sin4)exp
[ (:++)
tan41
Fig. 3. Mechanism of failure for the case 05/3<2E.
Fig. 4. Mechanism of failure for the case p = 2E.
212
Andrzej Sawicki, Danuta LeSniewska
It follows from eqn (26) that the solution obtained does not depend on the value of p. The same formula as eqn (26) follows from kinematical considerations, so we have obtained the exact solution to the problem of bearing capacity of a weightless reinforced slope, for a given range of angles p.
3.2 The case /3 = 2~ Figure 4 shows the mechanism of failure which differs from that presented in Fig. 3. The triangle ABC is a region of active pressure in which both components of reinforced soil are in a plastic state. The stress characteristics are here the straight lines. In the region BCD the stress characteristics are straight lines and logarithmic spirals. The soil and the reinforcement work in a plastic state in this region. The triangle BDE is a region of passive earth pressure in which the reinforcement remains rigid. The characteristics are also straight lines, the direction of the second family coinciding with the direction of reinforcement. The bearing capacity is given in this case by the formula (26). 3.3 The case 26 < p I 7r/2 The mechanism of failure is shown in Fig. 5, where in all the region ACDEB the plastic how of both constituents takes place. The reinforcement is extended. The triangle ABC forms a region of active pressure, in the triangle BDE the state of passive pressure occurs. In these two triangles the stress characteristics are straight lines. In the region BCD the characteristics are respectively straight lines and logarithmic spirals. The bearing capacity of a structure depends in this case on the value of /3. Both the static and kinematic approaches lead to the following formula:
p = flg,(l+sin$)exp
[2 (5
- pg)tan4]
where
f_rg= f_rOsinp
sin /3 + d/sin2 fi - ~0s’ 4 cos2 $I
(27)
Limit analysis of reinforced slopes
Fig. 5. Mechanism
In the case of p = m/2 for the bearing capacity Lesniewska.’ Equation (27) is the reinforced steep slopes
4 BEARING
213
of failure for the case 2~ < /3 5 IT/~.
(vertical slope) we obtain from eqn (27) the formula of the reinforced soil retaining wall, see Sawicki and exact solution to the problem of bearing capacity of (weightless) whilst eqn (26) applies to mild slopes.
CAPACITY
OF WEIGHTY
REINFORCED
SLOPES
In the case of weighty reinforced slopes it is not possible to obtain a solution to the problem of bearing capacity in a closed, analytic, form, as it was presented in Section 3. In order to get a solution to a particular problem one has to solve numerically the system of eqns (17) and (18)) or (19), depending on the mechanism of failure. We would like to present in this paper some final results of numerical computations performed for a hypothetical geotextile reinforced slope. We have assumed the following data: the height of a slope H = 10 m, the unit weight of reinforced soil y = 17 kN/m3, the angle of internal friction 4 = 34”, the cohesion c = 0. We have computed examples for various angles p and many values of the parameter oo. Figure 6 shows the net of characteristics and the computed distribution of load on the upper edge of a structure, for p = 60” and a0 = 80 kPa. The bearing capacity is 5-l MN in this case. Similar results, but for o. = 120 kPa, are shown in Fig. 7. In this case, the bearing capacity is 7-4 MN. A comparison of Figs 6 and 7 suggests that the value of u. influences the shape of slip lines. Indeed, as the reinforcement becomes stronger, the shape of the plastic region becomes similar to that for a weightless slope. A synthesis of numerical experiments regarding the bearing capacity of
214
Andrzej Sawicki, Danuta LeSniewska
p(w)Dwl.1
4 .4
Fig. 6. Stress characteristics
Fig. 7. Stress characteristics
for cm = 80 kPa.
for CT,,= 120 kPa.
Limit analysis of reinforced slopes
Fig. 8. Dependence
of bearing capacity on /3 and aa.
a 6* + 4: + .z-
(cl Fig. 9. Dependence
of bearing capacity on /3 and a~.
21.5
216
Andrzej Sawicki, Danuta LeSniewska
reinforced slopes is presented in Fig. 8. Figure 8(a) shows the dependence of bearing capacity P (obtained from integration of the computed load distribution p) on the angle p, for four values of the parameter (TV.Similar results, but for a weightless slope, are presented in Fig. 8(b). It follows from these results that the value of angle p does not influence strongly the bearing capacity of a structure, except for ps approaching the value of 7r/2 - 4, in the case when its own weight is taken into account. And conversely, in the case of a weightless slope the bearing capacity decreases as p approaches 7~/2 - C#J,but still the influence of p is rather small. A factor which strongly influences the bearing capacity of a structure is the value of co. Figure 9 shows the dependence of bearing capacity, for both weighty and weightless slopes, on the value of (T(,, for three values of the angle p. The influence of its own weight is visible for steep slopes (Figs 9(a) and 9(b)). In the case of p = 60” (Fig. 9(c)) there is practically no difference between the results computed for weighty and weightless slopes. It is worthy of note that the dependence between P and go is linear in all the cases analysed, except for the values of p greater than 140 kPa, for a weighty structure. It must be remembered that the results for a weightless slope have been obtained analytically, whilst those for a weighty structure numerically.
5 EXPERIMENTAL
RESULTS
In order to compare theoretical predictions with a real behaviour of reinforced slopes a set of small scale experiments has been performed. The experimental technique, materials, etc., are similar to those presented by Sawicki and Lesniewska,s so in this paper we would like to present only some final results. In above-mentioned experiments the values of a vertical load were measured, as well as mechanisms of failure photographed for various values of p and a,,. Figure 10 shows the comparison of predicted and measured results. Solid lines present the theoretical predictions for both weighty and weightless slopes, the dots correspond to experimental results. It seems that the agreement between theoretical predictions and experimental results is rather good. The influence of the weight of a slope is rather small because of small dimensions of experimental structures (24 cm high). Figure 11 shows the net of characteristics and a maximum load at failure computed for the experimental data: H = 24 cm, p = 60”, y = 17 kN/m’, C$ = 34”, c = 0, (T” = 46.9 kPa. In this case the bearing capacity was P = 4.15 kN. The dots indicate places in which the reinforcement (textile bands) got broken, in two subsequent experiments. These points lie close to the predicted slip line. In this case, as well as in other cases not reported in
217
Limit analysis of reinforced slopes
Fig. 10. Comparison
of theoretical
and experimental
results.
218
Andrzej Sawicki, Danuta LeSniewska p p(x)
LMPal
Fig. 11. Stress characteristics
against experimental
3 t
2m s a l-
I
I
35
30 @ Fig. 12. Influence
I
40
(degrees)
of I#Ion bearing capacity.
results.
Limit analysis of reinforced slopes
219
this paper, there is a good agreement between theoretical prediction and experimental results. Finally, Fig. 12 shows the influence of the angle of internal friction 4 (in the range of practically important values) on the bearing capacity of two slopes, characterized by p = 60” and 90”.
6 CONCLUSIONS The aim of the present paper was to present the plastic limit analysis of reinforced slopes. First, the theoretical model of reinforced soil, more general than that presented by Sawicki,4 was described. Then, the formulation of boundary value problems was presented and applied to solve the problem of bearing capacity of reinforced slopes. In the case of weightless structures the closed form solution was obtained, see formulae (26) and (27), for mild and steep slopes respectively. It seems that these formulae can be useful in engineering applications because of their simple form, at least to estimate the collapse load for real structures. In the case of weighty structures the solution to a particular problem can be obtained by numerical methods. Some results of such computations were presented in this paper and compared with predictions obtained from eqns (26) and (27). Finally, a comparison of experimental results with theoretical predictions was shown and briefly discussed. The conclusion is that the theory proposed gives realistic predictions, i.e. the bearing capacity and location of slip lines. The mathematical part of this paper is addressed to those readers who work on the application of plasticity theory in geomechanics. The results presented in Sections 2 and 3 give some basic material, as for example equations for characteristics, which can be easily checked and then applied by those who are familiar with the methods of limit analysis. This part of the text is very important from the view point of the theory of reinforced soil, as well as from the practical viewpoint, since it leads to some closed solutions of engineering problems, such as for example eqns (26) and (27). Numerical results presented in this paper illustrate the theory proposed, so no analysis is presented showing the influence of all the parameters, which characterize the reinforced soil, on the bearing capacity. For example, the influence of (T” is of a fundamental importance, but it is not possible to study this influence in the whole range of values, since it is not possible to obtain solutions for an unreinforced slope, the angle of inclination of which is bigger than the angle of a natural slope. The practical implications of the theory presented are in eqns (26) and (27), as well as in the method of determination of bearing capacity of
220
Andrzej Sawicki, Danuta LeSniewska
weighty reinforced slopes. In the authors belief these results are original, because no rational method for the estimation of bearing capacity of reinforced soil structures has been previously proposed.
ACKNOWLEDGEMENT The research reported in this paper was carried out in the main task No. 02.01-1.5 supervised by the Institute of Fundamental Technological Research, Warsaw, Poland.
REFERENCES 1. Hill, R., The Mathematical Theory of Plasticity. Clarendon Press, Oxford, 1956. 2. Ingold, T. S., Reinforced Earth. Thomas Telford Ltd, London, 1982. 3. Lesniewska, D., Statics and kinematics of reinforced soil-an application to bearing capacity of slopes, Ph.D. Thesis. Institute of Hydroengineering, Gdansk, Poland, 1987. 4. Sawicki, A., Plastic limit behaviour of reinforced earth, Jnl Geot. Engng, ASCE 109(7) (1983) 1000-5. 5. Sawicki, A. & Lesniewska, D., Failure modes and bearing capacity of reinforced soil retaining walls, Geotextiles and Geomembranes 5 (1987) 29-44. 6. Sokolovski, V. V., Statics of Granular Media. Pergamon Press, New York, 1965.