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Loss of stability in geosynthetic-reinforced
slopes
Terrance Ma1, Sina Javankhoshdel1*
1 Rocscience Inc., 54 St Patrick St, Toronto, ON, M5T 1V1
Abstract. Geosynthetics are commonly considered to provide restoring
forces against sliding during overall slope stability analyses. Where a
slipping surface intersects a geosynthetic layer, the geosynthetic layer
produces a reaction force either in the opposite direction of sliding, in the
direction of the geosynthetic alignment, or in some intermediate direction.
The provision of geosynthetic reinforcements typically increases the factor
of safety in limit equilibrium against overall sliding in the design of a
mechanically stabilized earth (MSE) wall, and for this reason has become
popular among practitioners. However, geosynthetics are typically
installed in contiguous layers. These layers are potential interfaces for
sliding which need to be checked with respect to slope stability. In other
words, it is possible for a slope to become unstable via partial sliding along
the interface of a geosynthetic. In this paper, a comprehensive method of
analysis is demonstrated via an example which evaluates the stability of a
slope reinforced by geosynthetics. All the cases of failure with respect to
slope stability are considered via the dual treatment of the geosynthetic
elements as weak layers and supporting elements in a limit equilibrium
analysis in software.
1 Introduction
Geosynthetic materials are widely used to increase safety with regards to slope stability in
embankments, landfills, and other applications due to their relative ease of installation and
economic efficiency (Ankita et al. 2016; Zhao & Karim 2018). During analysis of slope
stability, geosynthetic sheets provide restoring forces against any sliding surfaces which
pass through them, leading to an increase to the factor of safety (FOS) against sliding.
To ensure public safety, a comprehensive analysis of slope stability involves
accounting for all possible failure modes that can reasonably be anticipated to occur during
the design period. The most common methods of failure that are considered when a sliding
surface intersects a geosynthetic layer are (a) pullout of the reinforcement along with the
sliding mass, (b) stripping of the sliding mass around the reinforcement, (c) tensile and
shear failure in the geosynthetic material (Estherhuizen et al. 2001; Forsman & Slunga
1994; Jewell 1996).
A less commonly considered mode of failure which may need to be considered when
evaluating the stability of slopes containing geosynthetics is the possibility of slipping of
* Corresponding author: sina.javankhoshdel@rocscience.com
the sliding mass along the top of the geosynthetic layer. The shear strength that is provided
by the top interface of the geosynthetic is based on the interface properties (typically,
practitioners prescribe an adhesion strength and friction angle). These interfaces are often
modelled as thin, weak layers in the slope stability analysis as they constitute zero-volume
discontinuities in the strength parameters within a slope model. This mode of failure is also
considered in the limit equilibrium analysis for other types of slope reinforcements such as
gabion walls, where sliding failure can occur through the interfaces between gabion layers
(Javankhoshdel et al. 2022).
If there are many weak layers in a slope stability model, the search for the critical slip
surface can become very complicated and time consuming, because of the need to consider
sliding along all the weak layers. In this paper, a new method is presented whereby sliding
along the various geosynthetic layers in a model can be considered heuristically during the
search for a critical slip surface in limit equilibrium. A benefit of the proposed method is
that it can complete the analysis much faster without having to consider every possibility of
weak layers for every slip surface. A numerical example is presented to demonstrate use of
the method, and the results are compared against that which was obtained using finite
element analysis.
2 Treatment of Weak Layers in Limit Equilibrium
In limit equilibrium, the search for a critical slip surface involves computing the FOS for
various shapes of sliding surfaces. The shape of the sliding surface is optimized in a way
that yields the lowest FOS, and the optimum solution is considered to the be critical slip
surface representing the overall FOS of the slope. For each sliding surface, the sliding mass
is discretized into slices above it, and each slice is solved with respect to force and/or
moment equilibrium to determine the shear strength contribution at its base. The FOS is the
ratio of available shear strength to the acting shear forces at the bases of the slices, where
values above unity suggest stability. To account for uncertainty and relative lack of
precision in this method, design standards commonly require FOS above a certain threshold
(for example, 1.5).
For this study, a weak layer in a limit equilibrium model is a zero-thickness layer
which exhibits its own material properties with respect to sliding. If during the optimization
process a slip surface contacts a weak layer, then the slip surface is truncated by the weak
layer such that the sliding occurs along the weak layer.
Geosynthetic elements are represented as lines on the model section, extending from
the slope face into the soil. When a slip surface passes through a geosynthetic element, a
restoring force from the geosynthetic is applied onto the sliding mass, based on the strength
of the geosynthetic and the location of contact.
In the proposed method, the geosynthetic elements are also treated as weak layers in
the model. Where a slip surface contacts a geosynthetic weak layer and slides along it
rather than passing through it, the effective adhesion and friction angle of the interface
between the geosynthetic and the layer of soil directly above it are used to determine the
shear strength at the bases of the slices. This method was coded in Slide2 and used to
produce the results shown in this paper.
3 Heuristic Particle Swarm Method
The optimization problem which minimizes the FOS while varying the slip surface is not
trivial, and requires use of advanced global optimization techniques, such as the Particle
Swarm Method (Kennedy & Eberhart 1995). A variation of this method based on Kenney
& Eberhart (1997), which allows for the addition of binary optimization variables, called
the Heuristic Particle Swarm method (Chen et al. 2024), is adopted for this study.
In the proposed optimization method, the FOS is minimized while changing the shape
of the slip surface polyline (a vector of x and y coordinates) and a binary state condition for
each weak layer in the model as being used (state 1) or ignored (state 0) to clip the slip
surface if contact occurs. In other words, when the slip surface encounters a weak layer, it
will either be forced to slide along the weak layer (state 1, “on”) or intersect through the
weak layer (state 0, “off”). In the latter case, a force is exerted from the geosynthetic onto
the sliding mass based on its governing strength in pullout, stripping, and tensile failure.
Formally, the optimization problem takes the following form:
Minimize FOS, by varying: (1a)
X = {x1, y1, x2, y2, xi, yi, …, xn, yn} (1b)
B = {b1, b2, bj, …, bm} (1c)
where (xi,yi) are the coordinates of the points on the slip surface polyline constrained to the
slope geometry, and bi is the state of weak layer i. For this study, the default of n = 8
surface vertices is adopted, and m is the number of weak layers in the model.
The modified Particle Swarm method (Kennedy and Eberhart 1997) is an efficient,
heuristically-based algorithm used to solve the above optimization problem, and can be
summarized as follows:
1. Simulate a random solution by assigning random values to X and B and compute
its associated FOS, storing this information into an entity called a particle. Repeat
for Np particles.
2. Each particle keeps track of its individual best (i.e. lowest FOS) solution so far,
and the best solution found globally so far.
3. Over NI iterations, mutate every particle by randomly interpolating the values of X
and B between its current position, its individual best position, and the global best
position.
4. At the end of each iteration, update the particle’s individual best and global best
positions.
For binary variables, the state of each weak layer j is determined randomly based on a
“stickiness” variable, vj, stored uniquely for each particle. For each weak layer, a random
number between zero and unity is generated and if it is less than the quantity Cw in Eq. (2)
then bj = 1; otherwise bj = 0.
Cw = (1 + exp(-vjt+1))-1 (2)
The stickiness value affects the likelihood of a weak layer being “on” or “off” – as vj
becomes increasingly positive, Cw approaches 1; whereas for increasingly negative vj, Cw
approaches 0. During the mutation step in each iteration, vj is increased if the attracting
particle has bj = 1 and decreased if the attracting particle has bj = 0. Over time, particles
with probabilities associated with the activation of the critical weak layers will dominate
the swarm population.
Now, owing to the nature of the specific problem of slope stability with weak layers, a
couple of minor modifications were made to the Kennedy and Eberhart (1997) method to
improve the performance of the algorithm for this study:
If a slip surface does not touch a certain weak layer j, then the stickiness vj for the
particle that generated the slip surface is not updated for that weak layer (Chen et
al. 2024).
If there are multiple overlapping weak layers within the sliding mass over a given
slip surface, then there will be an increase chance for the upper layer(s) to be
deactivated for that particle, to provide equal opportunity for the underlying layers
to be considered (Chen et al. 2024).
To encourage continual global searching, at the end of each iteration, the 40% of
particles with highest individual best FOS are discarded and re-generated with
random solutions.
Further details of the Heuristic Particle Swarm method can be found in the Chen et al.
(2024) study, where it was adopted for 3D limit equilibrium.
4 EXAMPLE
A typical 2:5-sloped embankment with the dimensions and material properties shown in
Figure 1 is used to demonstrate the use of the proposed method. The model was produced
and analyzed using the Slide2 software. The General Limit Equilibrium (GLE) method of
limit equilibrium was used to calculate FOS.
Figure 1. Analysis of a typical embankment reinforced with geosynthetic sheets
In the absence of reinforcement, this slope is unstable because the slope angle exceeds
the angle of repose for the fill material. It is assumed that the soil is retained, and Huesker
Fortrac 55T Synthetic Geogrids are used to stabilize the slope, with the following
properties:
Allowable tensile strength = 27.0893 kN/m
Strip coverage = 100%
Anchorage: Both ends
Anchorage connection strength = 27.0893 kN/m
Interface shear strength method = Friction Angle & Adhesion
Interface friction angle = 15°
Interface adhesion = 0 kPa
Where a reinforcement is intersected by the slip surface without sliding along it, the
restoring force onto the sliding mass is assumed to act parallel to the reinforcement and is
considered as an active force in the equilibrium equation.
The Heuristic Particle Swarm search method was used with 50 particles of n = 8
vertices over 500 iterations to search for the critical non-circular slip surface. Surface
altering was also enabled with 20 iterations. Sliding surfaces shallower than 1.0 m in depth
were excluded from the analysis using a depth filter. The result of the search, which was
completed in 26 seconds on a laptop computer, shows a critical slip along the bottom-most
geogrid, with FOS = 1.3, as shown in Figure 2.
Figure 2. Result of heuristic search with geosynthetic layers considered as weak layers
An alternative version of the search was performed using the original particle swarm
method and using the option of considering all the weak layers independently for every slip
surface produced during the search. Such an analysis produced the same result (FS = 1.3)
but took 142 seconds (5.5 times longer) to complete. Although for a simple model such as
this example the difference in computational time is not highly consequential, one can
imagine the amount of time saved computing a more complicated model if the proposed
method computes the same result over five times faster.
Upon closer examination of the slip surfaces in Figure 2, the minimum FOS surface
associated with sliding along each geosynthetic is given in Table 1, and the corresponding
surfaces are shown in Figure 3. Note that sliding along the uppermost geosynthetic layer
was not considered due to the minimum depth filter of 1.0 m.
Table 1. Minimum FOS associated with Sliding Along Each Layer
Layer (from bottom) FOS
1 1.30
2 1.37
3 1.45
4 1.53
5 1.65
6 1.79
7 1.97
8 2.26
9 2.76
10 N/A*
* Mode excluded due to depth filter
Figure 3. Critical slip surface for each layer of sliding
5 Verification
The critical slip surface with FS = 1.3 in this example was verified using finite element
analysis in RS2. The weak layers of the geosynthetic were considered using the Joints
feature, and a similar Shear Strength Reduction (SSR) factor of 1.26 was obtained, shown
in Figure 4.
Figure 4. Critical slip surface in shear reduction factor analysis, with shear plastic strain distribution
There is a slight difference between the finite element result and the limit equilibrium
analysis result because the assumptions of the analyses are different. For instance, the
angles of the interslice resultant forces are assumed in limit equilibrium whereas
equilibrium among the internal elements is solved in finite elements. The distribution of
applied force is also only applied within the region above the sliding mass for limit
equilibrium, whereas it is distributed into the elements during finite elements.
Note that to ignore the local failure mode of shallow sliding along the inclined surface
of the slope, a thin layer of elastic material was added at that location.
6 Sensitivity Analysis
A brief sensitivity analysis was also conducted, varying the friction angle of the interface
between 5°, 25° and 35° (the original case was 15°). Upon increasing the interface friction
angle sufficiently, the failure mode changes to overall slope failure through the underlying
layer of soil. The results of the sensitivity analysis are shown in the following figures
(Figure 4 for 5°, Figure 5 for 25° and Figure 6 for 35°).
Figure 4. Critical slip surface for interface friction angle of 5°
Figure 5. Critical slip surface for interface friction angle of 25°
Figure 6. Critical slip surface for interface friction angle of 35°
It can be seen from Figure 6 that the FOS corresponding to overall slope failure in the
underlying soil layer is 1.72. A transition of failure mode occurs between friction angles
25° and 35°, with FOS = 1.71 occurring at the bottom-most geosynthetic layer for a friction
angle of 25°. As the friction angle increases, the FOS for sliding along the bottom-most
geosynthetic layer increases, until the overall FOS is governed by failure in the underlying
soil layer.
In this example, it was observed that if interface failure occurs, the bottom-most
geosynthetic governs the sliding. This phenomenon was also observed for some examples
of gabion walls (Javankhoshdel et al. 2022). However, it is always important to consider all
the possible modes in the analysis. In cases where different types geosynthetics are used
along the height of the slope, the strength of the interface may vary between the types of
materials used, and failure can occur via sliding on any one of the geosynthetics (not
necessarily the bottom-most layer).
7 Conclusion
In this study, a new heuristic method of considering sliding along weak layers is used to
model slope failure via sliding along geosynthetic layers. It is important to consider this
failure mode since the interface between a geosynthetic and its surrounding materials may
not always be strong enough to force the governing failure mode below the embankment or
retaining wall. For instance, in many cases such as the example presented in this study, the
critical slip surface can occur along the bottom-most interface. The proposed method offers
fast and efficient computations for considering multiple weak layers in a model.
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