limit analysis of geosynthetic-reinforced soil slopes - IGS ...

Technical Paper by A. Zhao
LIMIT ANALYSIS OF
GEOSYNTHETIC-REINFORCED SOIL SLOPES
ABSTRACT: A kinematic solution of the plasticity theory applied to the stability of
geosynthetic-reinforced soil slopes is presented in this paper. Translational and rotational
failure mechanisms are considered and rigorously compared. For slopes heavily
reinforced with geosynthetics, the limit loads obtained using a translational failure
mechanism are smaller than those obtained using a rotational failure mechanism, while
for slopes with a reduced amount of geosynthetic reinforcement, the latter yields better
results. A rotational failure mechanism consistently yields lower stability factor values
for load-free geosynthetic reinforced slopes. Limit analysis method solutions are
compared to solutions obtained using limit equilibrium and slip-line methods.
KEYWORDS: Geosynthetic-reinforced slopes, Limit analysis, Limit equilibrium,
Slip-line method.
AUTHOR: A. Zhao, Technical Director, Tenax Corporation, 4800 East Monument
Street, Baltimore, Maryland 21205, USA, Telephone: 1/410-522-7000, Telefax:
1/410-522-3977.
PUBLICATION: Geosynthetics International is published by the Industrial Fabrics
Association International, 345 Cedar St., Suite 800, St. Paul, Minnesota 55101-1088,
USA, Telephone: 1/612-222-2508, Telefax: 1/612-222-8215. Geosynthetics
International is registered under ISSN 1072-6349.
DATES: Original manuscript received 30 September 1996, revised version received
16 December 1996 and accepted 20 December 1996. Discussion open until 1 September
1997.
REFERENCE: Zhao, A., 1996, “Limit Analysis of Geosynthetic-Reinforced Soil
Slopes”, Geosynthetics International, Vol. 3, No. 6, pp. 721-740.
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ZHAO D Limit Analysis of Geosynthetic-Reinforced Soil Slopes
1 INTRODUCTION
Limit equilibrium methods are often used to design geosynthetic-reinforced slopes.
Various limit equilibrium methods have been used in different studies (e.g. Leshchinsky
and Volk (1985), Schmertmann et al. (1987) and Jewell (1990)). The limit analysis approach
of plasticity is used in this paper as a solution technique. The stability of geosynthetic-reinforced
slopes is assessed using translational and rotational failure mechanisms.
A translational failure mechanism consists of a number of rigid blocks separated
by internal planar rupture failure surfaces. This mechanism can be considered as a general
case to the commonly used two-wedge failure mechanism. A rotational failure
mechanism comprises a rigid block with a logarithmic rupture surface. A translational
failure mechanism is used in the paper by Michalowski and Zhao (1995) to study the
stability of reinforced slopes, and a rotational failure mechanism is used in the paper
by Michalowski and Zhao (1994) to study the effect of geosynthetic reinforcement
length and distribution on the safety of slopes. However, a rigorous comparison of these
two failure mechanisms for the analysis of geosynthetic-reinforced slopes is lacking.
Since the same kinematic admissibility requirement applies to both failure mechanisms,
it is possible to use the limit analysis method to rigorously evaluate these two
failure mechanisms for geosynthetic-reinforced slopes.
The objective of this paper is not to provide comprehensive design charts; instead,
this study will improve the understanding of the failure mechanism for geosynthetic-reinforced
slopes, by examining various limit analysis methods.
The fundamental equations of the limit analysis approach to the stability of geosynthetic-reinforced
slopes are presented in Section 2, followed by detailed formulations
of both translational and rotational failure mechanisms. The two failure mechanisms
are then rigorously compared using limit loads and rupture surfaces. Comparisons of
limit analysis method solutions to solutions obtained using limit equilibrium and slipline
methods are also presented.
2 KINEMATIC APPROACH OF THE LIMIT ANALYSIS METHOD
A kinematic solution of the plasticity theory applied to the stability of geosyntheticreinforced
soil slopes is presented in this paper. This approach is based on the upperbound
theorem of plasticity. The proof of the upper-bound theorem requires that the following
assumptions be made: (i) the soil is a perfectly plastic material; (ii) the yield
function of the soil is convex in the stress space; and (iii) the soil obeys the flow rule
associated with the yield condition. The following upper-bound theorem states that the
energy dissipation rate is at least as large as the rate of work by external forces in any
kinematic admissible failure mechanism (Drucker et al. 1952):
σ * ij ε * ij dV ≥ T * i v i dS + γ i v * i dV i, j = 1, 2, 3
(1)
V
S
V
where: ε ij * = strain rate tensor in a kinematic admissible velocity field; σ ij * = stress tensor
associated with ε ij * ; γ i = unit weight vector; T i * = vector of traction on the boundary
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ZHAO D Limit Analysis of Geosynthetic-Reinforced Soil Slopes
S; v i = velocity on the loaded boundary, S; andv * i = velocity in the volume, V (Note:
v * i = v i on the loaded boundary S of the volume V).
In geosynthetic-reinforced soil structures, the total energy dissipation during the incipient
plastic failure process is equal to the sum of the energy dissipation in the soil
and in the reinforcement. The energy dissipation by the geosynthetic reinforcement can
be included as an additional energy dissipation term on the left-hand side of Equation
1. The geosynthetic reinforcement is assumed to dissipate energy during incipient collapse
only in the tensile mode (the geosynthetic reinforcement is assumed to have no
resistance to bending and compression). It is further assumed that the soil is uniform
and homogeneous, i.e. no planes of weakness along the interface between the soil and
geosynthetic reinforcement exist. It should be noted that a pullout failure mode is not
considered in this paper. For the rigid block collapse mechanism considered in this paper,
all energy dissipation takes place along the velocity discontinuities. The energy dissipation
rate per unit area of the velocity discontinuity by tensile failure of the geosynthetic
reinforcement is as follows (Figure 1):
t∕ sin ξ
d r = k t ε * x sin ξdx = k t [v] cos(ξ − Ô) sinξ
0
where: ε x ∗ = strain rate in the direction of the geosynthetic reinforcement; t = thickness
of the rupture layer; ξ = angle of inclination of the geosynthetic reinforcement to the
rupture surface; [v] = magnitude of the velocity jump across the velocity discontinuity;
Ô = internal friction angle of the soil; x = incremental horizontal length of geosynthetic
reinforcement; and k t = tensile strength of the geosynthetic reinforcement per unit cross
section of the soil-geosynthetic composite. For uniformly placed geosynthetic reinforcement,
k t can be calculated as follows:
(2)
k t = T s
(3)
[v]
Geosynthetic
reinforcement
Displaced geosynthetic
reinforcement
Figure 1.
Rupture of the geosynthetic reinforcement across a velocity discontinuity.
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ZHAO D Limit Analysis of Geosynthetic-Reinforced Soil Slopes
where: T = tensile strength of the geosynthetic reinforcement; and s = spacing of the
geosynthetic reinforcement layers.
The following Mohr-Coulomb failure criterion and an associative flow rule are used
to describe the plastic behavior of the soil:
ε ij = λ ∂ f (σ ij)
∂ σ ij
where: ε ij = strain rate tensor; σ ij = stress tensor; λ = non-negative scalar function; and
the failure criteria function, f(σ ij ), is expressed as follows:
f (σ ij ) = (σ x + σ y + 2c cot Ô)sinÔ − (σ x − σ y ) 2 + 4σ 2 xy (5)
where c is the soil cohesion. Thus, the energy dissipation rate in the soil per unit area
of the velocity discontinuity is derived as follows:
(4)
d m = c[v]cosÔ
(6)
For a rigid block collapse mechanism, Equation 1 can now be written as:
c[v]cosÔ dl + k t [v] cos(ξ − Ô)sinξdl ≥ T * i v i dS + γ i v * i dV
L * L * S
V
where: l = infinitesimal area of the discontinuity surface; and L * = total surface area
of all velocity discontinuities. By using Equation 7 it is possible to find an upper-bound
to the true limit load (or critical height) of a geosynthetic-reinforced slope. A good estimation
of the upper-bound using the kinematic approach can be obtained by considering
a failure mechanism in which geometric parameters are varied in an optimization
scheme where the smallest value of the unknown load is sought. The “Constrained Simplex”
(Complex) method developed by Box (1965) is used in the optimization scheme
in this paper.
(7)
3 TRANSLATIONAL FAILURE MECHANISM
The geosynthetic-reinforced slopes considered in this paper are restricted to those
resting on firm foundations and having a horizontal top surface. The translational failure
mechanism is shown in Figure 2a, and consists of rigid blocks separated by rupture
surfaces. Kinematic admissibility requires that all velocity jump vectors be inclined at
an angle Ô (internal friction angle of the soil) to the rupture surface. The velocity hodograph
is shown in Figure 2b. The geometric parameters chosen for the translational failure
mechanism are the angles, ξ k and ζ k , and the dimensionless parameter z k = h k /H (k
=1,2,3...N) where, N = number of rigid blocks used in the failure mechanism, H =
slope height, and h k = vertical height of rigid block k bounded by rupture surfaces.
Using trigonometric relationships, the velocity of each block, v k , and the velocity
jumps between blocks, [v] k , can be derived as a function of the vertical velocity compo-
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ZHAO D Limit Analysis of Geosynthetic-Reinforced Soil Slopes
(a)
q
[v] k
Ô
ζ k
v k
Ô
lk
ξ k
Geosynthetic reinforcement
(b)
v n
[v] n
v 1
v 0
[v] 1
Figure 2. Translational failure mechanism in a geosynthetic-reinforced slope: (a) slope
dimensions and notation; (b) velocity hodograph.
nent of the first block, v 0 . The velocity magnitude of the first block is calculated as follows:
v 1 =
v 0
sin(ξ 1 − Ô)
(8)
For k = 2 ... N, the velocities v k and [v] k are as follows:
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ZHAO D Limit Analysis of Geosynthetic-Reinforced Soil Slopes
v k = v k−1
sin(ξ k−1 + ζ k−1 − β − 2Ô)
sin(ξ k + ζ k−1 − β − 2Ô)
[v] k = v k−1
sin(ξ k−1 − ξ k )
sin(ξ k + ζ k−1 − β − 2Ô)
(9)
(10)
where β is the slope inclination angle.
The total energy dissipation by the geosynthetic reinforcement, D r , is the sum of the
dissipated energy along each velocity discontinuity. Equation 2 and the hodograph in
Figure 2b lead to the following expression:
D r = N
1
k t v k cos(ξ k − Ô) z k H
(11)
By adding the energy dissipation in the soil, the left-hand side of Equation 7 becomes:
D = N
1
c ⋅ cos Ô(v k l k + [v] k s k ) + N
1
k t v k cos(ξ k − Ô) z k H
(12)
where l k and s k are the surface areas of the velocity discontinuities as shown in Figure
2a. The following expression gives the rate of work due to the uniform load, q,ontop
of the slope:
W q = qLv 0
(13)
where L is the area of the loaded boundary. The rate of work due to the soil self-weight
is:
W γ = N
1
A k γ v k sin(ξ k − Ô)
(14)
where: A k = volume of block k (with unit thickness); and γ = unit weight of the soil.
Kinematic admissibility requires that the following inequality be satisfied:
ξ k + ζ k−1 − β − 2Ô > 0
k = 2, , N
(15)
Having derived the energy dissipation rates and the rate of work due to the external
forces, Equation 7 can be used to compute the limit load, or critical height of the geosynthetic-reinforced
slope. The following expression represents the limit load, q/k t ,ontop
of a reinforced slope:
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ZHAO D Limit Analysis of Geosynthetic-Reinforced Soil Slopes
q
k t
= 1
Lv 0 N
1
v k cos(ξ k − Ô)z k H + N
1
c
k t
cos Ô(v k l k + [v] k s k ) − N
1
A k
γ
k t
v k sin(ξ k − Ô)
The critical height of a geosynthetic-reinforced slope is represented by a dimensionless
“stability factor”, γH/k t , that is calculated as follows:
γH
=
k t
N
1
1
(A k ∕H)v k sin(ξ k − Ô) N
1
v k cos(ξ k − Ô)z k H + N
1
c
cos Ô(v
k k l k + [v] k s k ) − q Lv
t k t 0
The effect of the number of rigid blocks used in the translational failure mechanism
on the limit loads of two geosynthetic-reinforced slopes is shown in Figure 3. The limit
load obtained using a six-block failure mechanism is 18.4% less than the limit load obtained
using a two-block failure mechanism for the 45_ slope. These values are within
the expected range of results: the greater number of blocks used, the more accurate the
upper-bound solution. The effect of the number of blocks used in the analyses on the
computed limit load decreases as the slope increases. For a 60_ slope, the limit load
obtained using a six-block failure mechanism is 8.2% lower than the limit load obtained
using a two-block failure mechanism. The failure mechanisms for the 45_ slope using
two- and six-block translational failure mechanisms are presented in Figures 4 and 5,
(16)
(17)
Figure 3. Effect of the number of rigid blocks used in a translational failure mechanism
analysis of a geosynthetic-reinforced slope (Ô =30_, γH/k t =4).
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ZHAO D Limit Analysis of Geosynthetic-Reinforced Soil Slopes
respectively. The greater the number of rigid blocks used in the translational failure
mechanism, the smoother the failure surface.
The influence of the internal friction angle of the soil, Ô, on the failure surface geometry
is shown in Figure 6. Figure 6 indicates that the greater the internal friction angle
of the soil, the greater the limit loads, and the narrower the failure surfaces.
4 ROTATIONAL FAILURE MECHANISM
The rotational failure mechanism is also examined in order to compute the limit loads
(or critical heights) for geosynthetic-reinforced slopes. The failure surface (velocity
discontinuity) is assumed to pass through the toe of the slope. The normality rule in plasticity
theory requires that the velocity discontinuity vector be inclined to the rupture
surface at the internal friction angle of the soil, Ô, and the shape of the rupture surface
in the rigid rotation mechanism must be a log-spiral:
r = r o e (θ−θ o tan Ô)
(18)
Figure 4. Two-block translationalfailure mechanism in a geosynthetic-reinforced slope(β
=45_; Ô =30_, γH/k t =4).
Figure 5. Six-block translational failure mechanism in a geosynthetic-reinforced slope (β =
45_; Ô =30_, γH/k t =4).
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ZHAO D Limit Analysis of Geosynthetic-Reinforced Soil Slopes
Ô= 25_, q/k t =2.43
Ô= 30_, q/k t =4.08
Ô= 35_, q/k t =6.44
Ô= 40_, q/k t =9.98
Figure 6. Failure surfaces in a geosynthetic-reinforced slope using the translational failure
mechanism and a range of soil friction angles (β =60_, γH/k t =4).
where r o is the radius at the initial angle, θ o , as shown in Figure 7. The magnitude of
the velocity jump along the failure surface propagates according to the following expression:
[v] = [v] o e (θ−θ o tan Ô)
(19)
where: [v] o =r o ω; and ω = velocity of rotation about the log-spiral center, O.
The geosynthetic reinforcement energy dissipation rate along the entire log-spiral
failure surface is calculated by integrating the unit energy dissipation given in Equation
2 as follows:
D r =
(20)
where:
L * k t [v] cos(ξ − Ô) sinξdl
(21)
ξ = π 2 − θ + Ô (22)
dl = rdθ
cos Ô
By substituting Equations 18, 19, 21 and 22 into Equation 20, the following expression
is obtained:
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ZHAO D Limit Analysis of Geosynthetic-Reinforced Soil Slopes
O
θ h
θ
θ o
r o
L
q
H
[v]
Ô
ξ
Log-spiral failure surface
Geosynthetic reinforcement
Figure 7.
Rigid rotation failure mechanism in a geosynthetic-reinforced soil slope.
h
D r = k t r 2 o ω 1
cos Ô θ e 2(θ−θ o)tanÔ
sin θ cos(θ − Ô)dθ
θo
(23)
Integration leads to:
D r = 1 2 k t r 2 o [sin 2 θ h e 2(θ h −θ o)tanÔ
− sin 2 θ o ]ω
(24)
where: θ = log-spiral angle; θ o = initial log-spiral angle (top of slope); θ h = final log-spiral
angle (toe of slope) (Figure 7).
The rate of energy dissipation in the soil, D m , and the rate of work due to the soil selfweight,
W γ , were derived by Chen et al. (1969), and are given as follows for completeness:
D m = 1 1
2 cr2 o
tan θ [e2 (θ h −θ o)tanÔ
− 1]ω
W γ = r 3 o γ (f 1 − f 2 − f 3 ) ω
(25)
(26)
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ZHAO D Limit Analysis of Geosynthetic-Reinforced Soil Slopes
where function f 1 , f 2 and f 3 are dependent on the geometry of the slope, the geometry
of the failure surface, and the internal friction angle of the soil as follows:
f 1 (θ o , θ h ) =
1
3(1 + 9tan 2 Ô) (3 tan Ô cos θ h + sin θ h )e 3(θ h −θo)tanÔ (3 tan Ô cos θ o + sin θ o )
f 2 (θ o , θ h ) = 1 6
L
r o
2cosθ o − L r o
sin θ o
f 3 (θ o , θ h ) = 1 6 e(θ h −θ o)tanÔsin(θ h − θ o ) − L r o
sin θ h
cos θ o − L r o
+ cos θ h e (θ h −θ o)tanÔ
(27)
(28)
(29)
where:
L
r o
= sin(θ h − θ o )
− sin(θ h + β) H
sin θ h sin θ h sin β
(30)
H
r o
= e (θ h −θ o)tanÔ
sin θ h − sin θ o
r o
(31)
The rate of work due to a uniform load q on top of the slope is:
W q = qL(r o cos θ o − L∕2) ω
(32)
For a stable slope, the energy dissipation rate must be greater than the rate of work
for any kinematic admissible mechanism. Again, the energy balance in Equation 7 is
used to formulate the solution for one unknown quantity. The limit load, q/k t , and the
stability factor, γH/k t , for a geosynthetic-reinforced slope can be calculated using the
following expressions:
q
=
1
k t L
r
2cosθ o − L
o r o
γH
k t
=
⎡
⎪
⎣
2 γH (f
k 1 − f 2 − f 3 )
tan Ô (e2(θ h −θo)tanÔ − 1) + sin 2 θ h e 2(θ h −θo)tanÔ − sin 2 θ o − t
c 1
k t
H
r o
2(f 1 − f 2 − f 3 )
c k t
1
tan Ô (e2(θ h −θ o)tanÔ − 1) + sin 2 θ h e 2(θ h −θ o)tanÔ − sin 2 θ o − q k t
H
r o
⎪ ⎤ ⎦
L
ro
2cosθ o − L r o
(33)
(34)
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ZHAO D Limit Analysis of Geosynthetic-Reinforced Soil Slopes
5 COMPARISON OF THE TWO FAILURE MECHANISMS
The failure surfaces obtained using the translational and rotational failure mechanisms
are presented in Figure 8 assuming a slope inclination angle, β =70_, γH/k t =
6, and Ô =35_. Five blocks were used in the translational failure mechanism. The
normalized limit loads, q/k t , from the translational and rotational failure mechanisms
are 3.66 and 3.52, respectively.
The limit loads on geosynthetic-reinforced slopes calculated using a rotational failure
mechanism along with the limit loads obtained using a translational mechanism are presented
in Figure 9. For slopes that are heavily reinforced with geosynthetics (small γH/k t
value), the limit loads obtained using a translational failure mechanism are smaller than
the limit loads obtained using a rotational mechanism. While for slopes that have a reduced
amount of geosynthetic reinforcement (large γH/k t value), the limit loads obtained
using a translational failure mechanism are greater than the limit loads obtained
using a rotational mechanism. For a geosynthetic-reinforced slope with β =90_, the two
failure mechanisms produce almost identical solutions.
Figure 10 shows a comparison of the stability factor for load-free geosynthetic-reinforced
slopes using translational and rotational failure mechanisms. As shown in Figure
10, a rotational failure mechanism always yields lower stability factor values for loadfree
geosynthetic-reinforced slopes.
Both the limit loads and the rupture surfaces obtained using the two failure mechanisms
are practically identical. This is due to the fact that a large number of blocks were
used for the translational failure mechanism and all of the results were optimized. The
Translational failure mechanism
Rotational failure mechanism
Figure 8. Translational versus rotational failure mechanisms in a geosynthetic-reinforced
slope (β =70_, Ô =35_, γH/k t =6).
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ZHAO D Limit Analysis of Geosynthetic-Reinforced Soil Slopes
Translational failure mechanism
Rotational failure mechanism
Figure 9.
Limit loads on geosynthetic-reinforced slopes (Ô =35_).
Stability factor, γH/k t
Translational failure mechanism
Rotational failure mechanism
Figure 10.
Stability factors for load-free geosynthetic-reinforced slopes.
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ZHAO D Limit Analysis of Geosynthetic-Reinforced Soil Slopes
difference would be greater if less blocks are used, and the resulting failure surface
would not be smooth (Figure 4).
6 COMPARISON WITH LIMIT EQUILIBRIUM METHODS
In Figure 11, the limit analysis method is compared to existing limit equilibrium
methods for geosynthetic-reinforced slopes with β =60_. The analysis results are
presented using the stability factor, γH/k t , as a function of Ô. A rotational failure mechanism
was used in the limit analysis method. In a paper by Michalowski and Zhao (1995),
a translational failure mechanism was utilized. Figure 11 indicates that the limit analysis
method solution compares reasonably well to the solutions obtained using the limit
equilibrium methods. The solution obtained using the kinematic approach of the limit
analysis method is a rigorous upper-bound limit load (or critical height) solution while
the limit equilibrium methods produce approximate solutions. This is due to the fact
that, in the limit equilibrium methods, various assumptions are made regarding the
stress distribution along the failure surface, and the forces in the geosynthetic reinforcement
are usually not assumed to be fully mobilized in all layers. It has been demonstrated
by Drescher and Detournay (1993) and Michalowski (1989) that the forces in
translational failure mechanisms satisfy equilibrium equations. Also, moment equilibrium
is satisfied for a rotational failure mechanism.
and
Stability factor, γH/k t
Figure 11. Stability factors for geosynthetic-reinforced slopes using limit analysis and
limit equilibrium methods (β =65_).
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ZHAO D Limit Analysis of Geosynthetic-Reinforced Soil Slopes
7 COMPARISON WITH THE SLIP-LINE METHOD
The slip-line method is based on the derived failure criterion describing the failure
of a homogenized geosynthetic-reinforced soil composite and the application of the
method of stress characteristics. The derivation of the failure criterion for a geosynthetic-reinforced
soil composite is presented by Michalowski and Zhao (1995). The limit
loads on geosynthetic-reinforced soil slopes, walls and foundations are calculated using
the slip-line method described by Zhao (1996). The stress characteristic fields for inclined
geosynthetic-reinforced slopes at β =65_ and with soil internal friction angles
of 30_, 35_ and 40_ are shown in Figures 12 to 14, respectively.
The failure surface calculated using the slip-line method is compared to the failure
surface from the limit analysis method (Figure 15). The two failure surfaces are similar.
The limit load comparison using the slip-line and limit analysis methods is shown in
Figure 16. The slip-line method yields lower limit load values at lesser γH/k t values, and
β = 65_
Ô = 30_
γH/k t = 2
q/k t = 4.33
Figure 12. Stress characteristics for a geosynthetic-reinforced slope obtained using the
slip-line method.
β = 65_
Ô = 35_
γH/k t = 2
q/k t = 6.16
Figure 13. Stress characteristics for a geosynthetic-reinforced slope obtained using the
slip-line method.
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ZHAO D Limit Analysis of Geosynthetic-Reinforced Soil Slopes
β = 65_
Ô = 40_
γH/k t = 2
q/k t = 8.91
Figure 14. Stress characteristics for a geosynthetic-reinforced slope obtained using the
slip-line method.
Slip-line method
Limit analysis method
Figure 15. Comparison of failure surfaces in a geosynthetic-reinforced slope obtained
using the slip-line and limit analysis methods (β =65_, Ô =35_, γH/k t =2).
higher limit load values at greater γH/k t values. Even though the slip-line method does
not necessarily yield exact solutions for the limit loads, the solutions are typically good
approximations, and are typically lower than the results using the kinematic approach.
However, the limit loads presented in Figure 16 are the result of a problem that is not
well-posed in that the length of the loaded boundary is not defined “a priori”, but is defined
as a part of the solution. The calculated length of the loaded boundary using the
slip-line and limit analysis methods are not the same. The slip-line solution presented
in this paper is based on a “partial stress field”; when this stress field is extended to the
entire body, a lower bound solution to the true load is obtained.
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ZHAO D Limit Analysis of Geosynthetic-Reinforced Soil Slopes
Figure 16. Comparison of limit loads on geosynthetic-reinforced slopes using the slip-line
and limit analysis methods (β =70_, Ô =35_).
8 CONCLUSIONS
The kinematic approach of the plasticity theory to the stability of geosynthetic-reinforced
soil slopes is presented in this paper. For slopes heavily reinforced with geosynthetics,
the limit loads obtained using a translational failure mechanism are smaller than
the limit loads obtained using a rotational failure mechanism, while for slopes with a
reduced amount of reinforcement, the latter yields better results. A rotational failure
mechanism always yields lower stability factor values for load-free geosynthetic-reinforced
slopes. For translational failure mechanisms, a more accurate solution is
achieved when a greater number of rigid blocks are used, particularily for slopes with
smaller inclination angles.
A kinematic approach to the limit analysis method of geosynthetic-reinforced slopes
is a rigorous upper-bound limit load (or critical height) solution. The application of
three design methods (i.e. limit analysis, limit equilibrium and slip-line methods) to the
stability analysis of geosynthetic-reinforced slopes, are assessed. Comparisons of the
limit analysis method to the limit equilibrium and the slip-line methods indicate that
the limit analysis method produces reasonably similar results. The kinematic approach
may prove to be useful in the design of geosynthetic-reinforced slopes.
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ZHAO D Limit Analysis of Geosynthetic-Reinforced Soil Slopes
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Slopes”, Soils and Foundations, Vol. 9, No. 4, pp. 23-32.
Drescher, A. and Detournay, E., 1993, “Limit Load in Translational Failure Mechanisms
for Associative and Non-Associative Materials”, Geotechnique, Vol. 43, No.
3, pp. 443-456.
Drucker, D.C., Prager, W. and Greenberg, H.T., 1952, “Extended Limit Design Theorems
for Continuous Media”, Quarterly of Applied Mathematics, Vol. 9, No. 4, pp.
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Jewell, R.A., 1990, “Revised Design Charts for Steep Reinforced Slopes”, Reinforced
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Walls”, Transportation Research Record 1131, pp. 5-16.
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of Reinforced Soil Structures”, Journal of Geotechnical Engineering,ASCE,
Vol. 121, No. 2, pp.152-162.
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on Safety of Slopes”, Proceedings of the Fifth International Conference on
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1994, pp. 495-498.
Michalowski, R.L., 1989, “Three Dimensional Analysis of Locally Loaded Slopes”,
Geotechnique, Vol. 39, No. 1, pp. 27-38.
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NOTATIONS
Basic SI units are given in parentheses.
c = soil cohesion (N/m 2 )
d m = energy dissipation rate in soil per unit area of velocity discontinuity
(Nm -1 s -1 )
d r = energy dissipation rate in geosynthetic reinforcement per unit area of
velocity discontinuity (Nm -1 s -1 )
D = total energy dissipation rate in soil and geosynthetic reinforcement
(Nm -1 s -1 )
D m = total energy dissipation rate in soil (Nm -1 s -1 )
D r = total energy dissipation rate in geosynthetic reinforcement (Nm -1 s -1 )
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ZHAO D Limit Analysis of Geosynthetic-Reinforced Soil Slopes
f 1 , f 2 , f 3 = functions dependent on geometry of slope and failure surface, and Ô
(Equations 27, 28 and 29, respectively) (dimensionless)
H = slope height (m)
h k = vertical height of rigid block bounded by rupture surfaces (Figure 2) (m)
k t = tensile strength of geosynthetic reinforcement per unit area of composite
(N/m 2 )
L = area of loaded boundary (m 2 )
L * = total surface area of all velocity discontinuities (m 2 )
l = infinitesimal area of discontinuity surface (m 2 )
l k = surface area of velocity discontinuity (m 2 )
N = number of rigid blocks in translational failure mechanism
(dimensionless)
q = uniform load on top of the slope (N/m 2 )
r = log-spiral radius (m)
r o = log-spiral radius at initial angle, θ o (m)
S = loaded boundary area (m 2 )
s = spacing of geosynthetic reinforcement layers (m)
s k = surface area of velocity discontinuity (m 2 )
t = rupture layer thickness (m)
T = tensile strength of geosynthetic reinforcement per unit width (N/m)
T * i = vector of traction on the boundary S (N/m 2 )
V = volume (m 3 )
V k = volume of block k with unit thickness (Equation 14) (m 3 )
v i = velocity vector on the loaded boundary S (m/s)
v * i = velocity in the volume V (m/s)
v k = velocity of rigid blocks in translational failure mechanism (m/s)
[v] = magnitude of velocity jump vector (m/s)
x = incremental length of geosynthetic reinforcement (m)
z k = dimensionless parameter, z k = h k /H
W q = rate of work due to uniform load q (Nms -1 )
W γ = rate of work due to self-weight of soil (Nms -1 )
α = inclination angle of geosynthetic reinforcement with respect to x-axis (_)
β = slope inclination angle from horizontal (_)
γ = unit weight of soil (N/m 3 )
γ i = unit weight of soil vector (N/m 3 )
ε ij = strain rate tensor (s -1 )
ε * ij = strain rate tensor in kinematic admissible velocity field (s -1 )
ε * x = strain rate in direction of reinforcement (s -1 )
GEOSYNTHETICS INTERNATIONAL S 1996, VOL. 3, NO. 6
739

ZHAO D Limit Analysis of Geosynthetic-Reinforced Soil Slopes
ζ k = angular variable used in translational failure mechanism (Figure 2) (_)
θ = log-spiral angle (_)
θ h = final log-spiral angle (top of slope) (_)
θ o = inital log-spiral angle (toe of slope) (_)
λ = non-negative scalar function (dimensionless)
ξ , ξ k = angle of inclination of reinforcement to failure surface (_)
σ ij = stress tensor associated with ε ij (N/m 2 )
σ * ij = stress tensor associated with ε * ij (N/m 2 )
Ô = internal friction angle of soil (_)
ω = velocity of rotation (_/s)
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