strain-softening behavior of waste containment system interfaces

Technical Paper by R.B. Gilbert and R.J. Byrne 

STRAIN-SOFTENING BEHAVIOR OF WASTE 

CONTAINMENT SYSTEM INTERFACES 

ABSTRACT: Laboratory test results indicate that many geosynthetic interfaces in liners 

and covers for waste containment exhibit strain-softening behavior. A peak shear 

strength is mobilized at small displacements, but post-peak strength reductions occur 

with increasing displacement. Mechanisms that contribute to strain-softening include 

clay particle reorientation, geosynthetic polishing, and geosynthetic failure. Test results 

obtained from direct shear and torsional ring shear devices are summarized to illustrate 

these mechanisms. The design implications of strain-softening behavior are 

then investigated using analytical and numerical models. Deformations in the waste under 

gravity are sufficient in many cases to limit the available shear resistance along an 

interface to less than the peak strength. It is recommended that a factor of safety greater 

than one be achieved in all containment system slope designs, assuming that residual 

strengths are mobilized along the base of the entire containment system. 

KEYWORDS: Slope stability, Geosynthetic interfaces, Waste containment, Strain- 

Softening. 

AUTHORS: R.B. Gilbert, Assistant Professor of Civil Engineering, The University 

of Texas at Austin, Department of Civil Engineering, Cockrell Hall 9.227, Austin, 

Texas 78712, USA, Telephone: 1/512-471-4929, Telefax: 1/512-471-6548, and R.J. 

Byrne, Principal, Golder Associates Inc., 4104-148 th Avenue N.E., Redmond, 

Washington 98052, USA, Telephone: 1/206-883-0777, Telefax: 1/206-882-5498. 

PUBLICATION: Geosynthetics International is published by the Industrial Fabrics 

Association International, 345 Cedar St., Suite 800, St. Paul, Minnesota 55101, USA, 

Telephone: 1/612-222-2508, Telefax: 1/612-222-8215. Geosynthetics International is 

registered under ISSN 1072-6349. 

DATES: Original manuscript received 2 October 1995, revised version received 15 

April 1996 and accepted 19 April 1996. Discussion open until 1 November 1996. 

REFERENCE: Gilbert, R.B. and Byrne, R.J., 1996, “Strain-Softening Behavior of 

Waste Containment Interfaces”, Geosynthetics International, Vol. 3, No. 2, pp. 

181-203. 

GEOSYNTHETICS INTERNATIONAL S 1996, VOL. 3, NO. 2 

181

GILBERT AND BYRNE D Strain-Softening of Waste Containment System Interfaces 

1 INTRODUCTION 

Modern waste containment facilities, such as landfills, waste piles and impoundments, 

are equipped with multiple-component lining and cover systems to contain 

waste and protect the environment. These containment systems incorporate a variety 

of soils and geosynthetics intended to prevent contaminant migration. However, the interfaces 

between the various soil and geosynthetic components may provide limited resistance 

to shear. Therefore, waste containment facilities must be designed to prevent 

slope failures resulting from slippage along containment system interfaces. 

Recent laboratory shear testing results indicate that many common interfaces in 

waste containment systems exhibit strain-softening behavior (Long et al. 1993). A peak 

interface shear strength is developed at a small shear displacement, typically less than 

approximately 15 mm. With additional shear displacement, the shear strength decreases 

and ultimately approaches a residual or minimum value. Analysis of a major 

slope failure in a waste containment system indicates that strain-softening behavior 

contributed to the failure (Byrne et al. 1992; Stark and Poeppel 1994). Peak interface 

shear strengths may not be available to resist instability in waste containment systems; 

however, designs based on residual strengths may be excessively conservative and costly. 

In this paper, the causes and design implications of strain-softening behavior for containment 

system interfaces are addressed. Laboratory test results are summarized for 

typical containment system interfaces that exhibit strain-softening behavior. Design 

implications of strain-softening behavior are then investigated using a one-dimensional 

analytical model, and a two-dimensional numerical model. 

2 REVIEW OF PREVIOUS RESEARCH ON STRAIN-SOFTENING SOILS 

The potential for reductions in shearing resistance with deformation in soils has been 

studied and well documented over the past 60 years. Hvorslev (1936) performed torsional 

ring shear tests on remolded clay specimens and measured residual shear 

strengths as small as 70% of the peak shear strength. Skempton (1964) presented direct 

shear test results for an overconsolidated clay that indicated a post-peak strength reduction. 

Skempton attributed this “strain-softening” behavior to: (i) an increase of water 

content in the failure zone due to dilation during shear; and (ii) reorientation of “flaky 

clay particles” parallel to the direction of shear. Bishop et al. (1971) presented torsional 

ring shear test results for five plastic clays that exhibit significant post-peak reductions 

in strength. In addition to dilation and clay particle reorientation, the authors included 

the loss of cementation bonds with displacement as a mechanism to explain strain-softening. 

Many investigators have also concluded that strain-softening behavior in soils can 

lead to slope instability. Skempton (1964) analyzed seven failures of natural and excavated 

slopes in overconsolidated clays, and estimated that the available shear resistance 

along the failure surface was less than the peak strength and near the residual strength 

in most cases. Bjerrum (1967) summarized data from 60 slope failures in plastic clays, 

concluding that many of the failures could not be explained with peak shear strengths 

alone. Lefebvre (1982) demonstrated that post-peak strengths provide the best estimate 

182 GEOSYNTHETICS INTERNATIONAL S 1996, VOL. 3, NO. 2


of the available shear resistance for natural clay slopes in eastern Canada. La Rochelle 

(1988, Chapter 15) contended, “... the peak strength cannot be relied upon when dealing 

with problems of stability in strain-softening soils.”. 

Skempton (1964) ascribed observed reductions in the available shear resistance to a 

progressive failure mechanism. Exceedance of the peak strength at any point along a 

failure surface will “throw additional stress” to adjacent points due to strain-softening. 

This additional stress can trigger strain-softening at the adjacent points. As a result, the 

available shear resistance along the slip surface will approach the residual strength of 

the soil. Bjerrum (1967) provided a qualitative evaluation of progressive failure mechanisms, 

while numerical models have been used to provide quantitative evaluations 

(Duncan and Dunlop 1969; Lo and Lee 1973; Law and Lumb 1978; Chowdhury and 

Grivas 1982; Wiberg et al. 1990). These studies indicated that the potential for progressive 

failure depends on the initial distribution of stress and strain in the slope and the 

shape of the shear stress versus displacement relationship for the soil. The available 

strength in strain-softening soils may also deteriorate with time because of weathering 

and creep (Casagrande and Wilson 1951; Bjerrum 1967; Singh and Mitchell 1969; Nelson 

and Thompson 1977). 

3 STRAIN-SOFTENING OF CONTAINMENT SYSTEM INTERFACES 

3.1 Introduction 

Strain-softening is prevalent in many waste containment system interfaces. Laboratory 

shear test results reported for typical interfaces that exhibit varying degrees of 

strain-softening are summarized in Table 1. In order to quantify the magnitude and rate 

of strain-softening, the shear stress, τ, versus displacement, δ, relationships have been 

approximated with a linear model (Figure 1). The magnitude of strain-softening is described 

by the ratio of residual to peak strength: 

ζ = τ r 

τ p 

(1) 

In cases where the displacement limit of the test device did not allow for mobilization 

of a residual strength, the residual strength was estimated from extrapolation to the 

asymptote (Figure 1). 

The initial slope of the shear stress versus displacement curve is represented by τ p /δ p , 

while the slope of the post-peak portion of the curve is represented by (τ r - τ p )/(δ r - δ p ). 

The value of δ r was estimated such that the area under the actual stress-displacement 

curve is comparable to that under the linear model (Figure 1). A non-dimensional measure 

for the slope (or rate) of post-peak strength reduction relative to the initial, pre-peak 

slope is given as follows: 

Ω = − (τ r − τ p )∕(δ r − δ p ) 

τ p ∕δ p 

(2) 


183


(δ p , τ p ) 

Measured 

Linear model 

Shear stress, (kPa) 

τ 

(δ r , τ r ) 

Normal stress = 410 kPa 

Displacement, δ (mm) 

Figure 1. Large-scale direct shear test results on smooth geomembrane/compacted clay 

liner interface (Byrne et al. 1992). 

As Ω increases, the rate of post-peak strength reduction with displacement increases. 

3.2 Mechanisms of Strain-Softening 

Mechanisms that may contribute to strain-softening at containment system interfaces 

include the following: (i) reorientation of clay particles parallel to the direction of 

shearing; (ii) polishing of geosynthetic interface surfaces; and (iii) mechanical failure 

of a geosynthetic component at the interface. 

3.2.1 Clay Particle Reorientation 

Compacted clay liners are commonly constructed using medium to high plasticity 

clays to achieve low hydraulic conductivity. As reported by many others in the past 

(Skempton 1964; Bishop et al. 1971; Stark and Eid 1994), plastic clays are susceptible 

to strain-softening because they generally contain flexible, plate-shaped clay particles. 

With shear deformation, these particles tend to align parallel to the direction of shear, 

reducing the available shear resistance. 

Strain-softening due to clay particle reorientation has been observed at interfaces between 

geomembranes and clay liners (see appended Table 1). Striations on the clay surface 

at the completion of shear provide evidence that clay particle reorientation has contributed 

to the post-peak strength loss. Such striations have been observed both in the 

field and laboratory after shear (Byrne et al. 1992). 

The magnitude of strain-softening due to clay particle reorientation is affected by the 

compacted moisture content of the clay. Byrne et al. (1992) presented data indicating 



that the peak strength of a smooth geomembrane/compacted clay liner interface decreased 

more rapidly with increasing moisture content than did the residual strength for 

moisture contents wet of optimum. Hence, ζ values increased with increasing moisture 

content. 

The magnitude of strain-softening at the geomembrane/clay interface may be greater 

than within the clay. Stark and Poeppel (1994) used a torsional shear device to measure 

residual strengths in a plastic clay liner that were about 90% of the peak strengths (i.e. 

ζ = 0.9). At the interface between a smooth geomembrane and the same clay liner, ζ 

values were approximately 0.6. Two possible explanations for this behavior are: (i) the 

flat geomembrane surface enhances the reorientation of clay particles; and (ii) polishing 

of the geomembrane surface during shear contributes to the post-peak strength loss. 

3.2.2 Geosynthetic Polishing 

Geosynthetics are polished during shear. Interfaces with smooth and textured geomembranes 

exhibit post-peak strength reductions due to geomembrane polishing 

(Table 1). Smooth geomembranes apparently have a small-scale roughness (Vallejo and 

Zhou 1995) that is reduced or polished during shear. This polishing action is evident 

by the shiny appearance and formation of striations on smooth geomembranes during 

shear (Byrne et al. 1992). The magnitude of post-peak strength reduction due to polishing 

alone can be significant. For example, geomembrane polishing is the primary mechanism 

for strain-softening at the nonwoven geotextile/smooth geomembrane interface; 

ζ values range between 0.62 and 0.81 in Table 1 for this interface. 

Similar behavior occurs at interfaces with textured geomembranes. Direct shear tests 

were conducted on the interface between a nonwoven geotextile and a textured geomembrane 

(Li 1995). The geomembrane was made from high density polyethylene 

(HDPE), and the textured surface was developed through a coextrusion process. The 

nonwoven geotextile was a continuous filament, needle-punched geotextile made from 

polyester, and had a mass per unit area of 400 g/m 2 . The geomembrane specimens were 

circular with a diameter of 60 mm, while the geotextile specimens were square with a 

length of 240 mm. The displacement limit of the device was 25 mm. Accumulated shear 

displacements were developed by: removing the normal load after shearing a distance 

of 25 mm; removing the upper, circular shear box and returning it to its starting location; 

reapplying the normal load; and then continuing the shear displacement. After each 25 

mm increment of shear, the nonwoven geotextile specimen was replaced but the textured 

geomembrane specimen was not. 

The measured shear resistance versus displacement for each increment of continuous 

displacement is shown in Figure 2a. Post-peak strength reductions after accumulated 

displacements of 140 mm were substantial, with ζ values of approximately 0.6 (Table 

1). Also, the peak strength was not re-mobilized after replacing the geotextile at each 

25 mm increment of displacement (Figure 2a). These results indicate that geomembrane 

polishing may have a significant strain-softening effect on this nonwoven geotextile/textured 

geomembrane interface. 

For comparison purposes, a continuous displacement test was performed on the same 

interface using a double-interface shear device (Gilbert et al. 1995). The textured geomembrane 

specimens were 100 mm in length, and they were sheared over 300 mm long 

geotextile specimens. The results shown in Figure 2a indicate that similar behavior was 


185


(a) 

(b) 



τ 


τ 


Figure 2. Accumulated displacement results from direct shear tests on a nonwoven 

geotextile/textured geomembrane interface: (a) geotextile replaced after each 25 mm 

increment; (b) geomembrane replaced after each 25 mm increment. 

Note: σ = normal stress. 



obtained in the accumulated and continuous displacement tests. However, the true residual 

strength may not have been mobilized, even in the continuous displacement test, 

because continuous displacement of the geotextile material was limited by the length 

of the geomembrane specimen. 

3.2.3 Geosynthetic Failure 

Mechanical failure of a geosynthetic component at the interface can lead to large and 

sudden reductions in post-peak strength. Two examples of this behavior are the interface 

between a nonwoven geotextile and a textured geomembrane and the interface between 

a textured geomembrane and a reinforced geosynthetic clay liner (GCL) (Table 

1). 

The accumulated displacement tests described above for a nonwoven geotextile/textured 

geomembrane interface (Figure 2a) were repeated by replacing the textured geomembrane, 

instead of the geotextile, after each 25 mm shear increment (Li 1995). The 

test results are shown in Figure 2b. The peak strength was reduced after accumulated 

displacements, providing evidence of strength loss due to geotextile failure (Figure 2b). 

This behavior is consistent with torsional ring shear test results presented by Stark et 

al. (1996) for nonwoven geotextile/textured geomembrane interfaces; they observed 

fiber alignment, pull-out and filament breakage in scanning electron microscope 

photographs of the nonwoven geotextiles after shear. However, a comparison of Figure 

2b (geomembrane replaced) and Figure 2a (geotextile replaced) shows that more of the 

peak strength was re-mobilized when the geomembrane was replaced. This result indicates 

that geomembrane polishing may be more significant for this particular nonwoven 

geotextile/textured geomembrane interface at a high normal stress of 690 kPa. 

Two different failure mechanisms have been identified for the interface between a 

textured geomembrane and a reinforced GCL. Gilbert et al. (1996a) tested a GCL that 

had a bentonite layer encased between a nonwoven geotextile and a woven geotextile. 

The bentonite layer was reinforced with polypropylene fibers that were needle-punched 

through the bentonite from the nonwoven geotextile into the woven geotextile. The 

GCL was hydrated, and the textured geomembrane was then sheared against the woven 

geotextile. For normal stresses up to 14 kPa, the shear surface formed at the interface 

between the textured geomembrane and woven geotextile. Post-peak strength reductions 

likely occurred due to a combination of geotextile failure at the interface and geomembrane 

polishing, with ζ values ranging from 0.65 to 0.70 (Table 1). For normal 

stresses greater than 14 kPa, the needle-punched reinforcing fibers pulled out from the 

woven geotextile, and shear occurred through the bentonite layer; strengths at large displacements 

approached the strength of unreinforced bentonite. Post-peak strength reductions 

as a result of this internal failure mechanism were more severe than at the interface, 

with ζ values ranging from 0.33 to 0.57 (Table 1). 

Byrne (1994) tested a modified version of this reinforced GCL product that had an 

adhesive applied to the woven geotextile to prevent fiber pull-out. The stronger bond 

between the reinforcing fibers and the woven geotextile prevented a shear failure 

through the bentonite, even at normal stresses exceeding 14 kPa. The magnitudes of 

post-peak strength reductions were similar to those obtained on the unmodified version 

at lower normal stresses, with ζ values ranging from 0.59 to 0.76 (Table 1). One observation 

made during this test was that the peak strength was possibly related to rough- 


187


ness on the woven geotextile surface due to the protruding needle-punched fibers that 

were covered with adhesive. These protuberances were sheared off with displacement, 

leading to post-peak reductions in strength (Byrne 1994). 

3.3 Discussion of Laboratory Test Results 

The interface test results described above and summarized in Table 1 indicate that 

strain-softening effects for containment system interfaces can be substantial. Peak 

strengths are mobilized at small displacements between 1 and 15 mm. Strain-softening 

can lead to post-peak strengths that are as small as 30% of the peak strength. The greatest 

strength reductions (i.e. the lowest ζ values) are generally associated with polishing 

and/or mechanical failure of a geosynthetic material. For a given interface, the peak and 

residual displacements and the strength reduction factor, ζ, are roughly independent of 

normal stress (Table 1). Hence, the reduction rate factor, Ω, is also approximately independent 

of normal stress (Table 1). Finally, values for the post-peak strength reduction 

rate factor, Ω, vary over several orders of magnitude, and the highest reduction rates are 

generally associated with geosynthetic failure. 

Measurement of strain-softening behavior may be difficult with some test devices 

due to displacement limitations (Figure 1). Values for δ r in Table 1 exceed the displacement 

limits of many test devices. A limited comparison between conventional direct 

shear devices (with displacement limitations) and torsional ring shear devices (with unlimited, 

continuous displacement) is possible with the data in Table 1. Results are reported 

from both devices for the following interfaces: smooth geomembrane/compacted 

clay; nonwoven geotextile/smooth geomembrane; and nonwoven 

geotextile/textured geomembrane. The measured magnitude of strain-softening, ζ, and 

rate of strain-softening, Ω, are generally smaller for the torsional ring shear device. A 

possible explanation for this result is that the true residual strength is not mobilized with 

conventional direct shear devices (even with accumulated displacements), and they 

therefore provide an unconservative estimate of strain-softening potential. However, 

it is not possible to draw convincing conclusions because different geosynthetic products 

and normal stresses were used in the different test programs. 

4 DESIGN CONSIDERATIONS FOR STRAIN-SOFTENING 

INTERFACES 

4.1 General 

Slopes in waste containment systems are commonly designed using a limit equilibrium 

approach: the available or limiting shear resistance is compared with the shear resistance 

required to maintain equilibrium. It is difficult to determine the available shear 

resistance along an interface exhibiting strain-softening behavior. It may be unsafe to 

assume that the peak strength is available, while it may be excessively conservative and 

costly to assume that only the residual strength is available. 

The available shear resistance will depend on deformations in the containment system. 

A simplified cross-section through a typical containment system slope is shown 

in Figure 3. This condition represents a below grade landfill cell during filling or a 



Side 

slope 

Buttress 

f 

Ψ 

H S 

L S 

H B 

L B 

Figure 3. 

Simplified, one-dimensional model of a waste containment slope. 

closed landfill in a valley. It could also represent a cover slope by replacing the waste 

with soil in the model. The waste on the slope is supported by a buttress of waste at the 

toe of the slope. The waste will compress while transmitting the driving load in the slope 

to the buttress, and slippage will occur along the interface to mobilize shear resistance. 

As discussed qualitatively by both Skempton (1964) and Bishop (1967), these deformations 

may progressively reduce the available shear resistance on the slip surface due to 

strain-softening behavior. A quantitative model for evaluating strain-softening effects 

on the available shear resistance in liner and cover systems is described and applied in 

the following sections. 

4.2 Analytical Model 

4.2.1 Introduction 

A simplified, analytical model can provide useful insight into the available shear resistance 

along interfaces in containment systems. Gilbert et al. (1996b) developed an 

analytical model in which the buttress and side slope are treated as one-dimensional, 

elastic columns on strain-softening interfaces (Figure 3). Strain-softening at the interface 

is modeled using a linear representation of the stress-displacement relationship 

(Figure 1). The effects of strain-softening are quantified as follows: 

R f = s limit 

s p 

(3) 

where: R f = reduction in available shear resistance due to deformations; s limit = average 

available shear resistance (i.e. the limiting shear resistance); and s p = average shear resistance 

available for an incompressible waste. R f can range from 1.0 (the maximum 

strength is available) to ζ (only the residual strength is available). 

The reduction factor, R f , can be expressed as a function of ζ, Ω, and the following 

relative stiffness factor: 


189


M = τ p∕δ p 

EH 

L 2 

(4) 

where: M = non-dimensional factor; E = modulus of elasticity for the column; H = column 

thickness; and L = length of the interface along the column base. A detailed derivation 

of the solution for R f along the buttress and side slope interfaces is given by Gilbert 

et al. (1996b). The results from this derivation are provided below to study strain-softening 

behavior for typical interface properties (Table 1) and to serve as a practical design 

guide. 

4.2.2 Buttress 

The relationship between (R f ) B and ζ Β ,andΩ Β and M B for the buttress (where the subscript 

B denotes buttress) is shown in Figure 4. Since the buttress geometry is generally 

not a column with a constant height, it is approximated by a column with an equivalent 

height, H B , equal to the waste height and an equivalent length, L B , such that the volume 

of waste in the column is equal to the volume of waste in the buttress (Figure 3). These 

equivalent geometrical parameters are used in Equation 4 to find M B .AsM B increases, 

the available shear resistance along the interface decreases (Figure 4). M B will increase 

as: the interface stiffness increases (M B ∝ (τ p /δ p ) B ); the axial stiffness of the waste decreases 

(M B ∝ 1/EH B ); and the length of the buttress increases (M B ∝ L 2 B 

). For a given 

value of M B , the available shear resistance decreases as the rate of strain-softening for 

the base interface, Ω Β , increases. 

Typical values of E for waste range from 100 to 10,000 kPa depending on the type 

of waste and placement method (Sharma et al. 1989; Fassett 1993), while L B could range 

from 10 to 1000 m and H B could range from 10 to 100 m. When these values are combined 

with the range of τ p /δ p values compiled in Table 1, possible M B values range from 

∞ 

1 

0 

Figure 4. Nondimensionalized interface shear resistance reduction factors versus relative 

stiffness factor for waste buttress. 



0.1to1.0×10 8 . While, Ω values in Table 1 range from 0.006 to 0.3. Therefore, the 

available interface shear resistance for a buttress could potentially range from the peak 

to the residual shear strength depending on the waste properties, interface properties, 

and landfill geometry (Figure 4). 

The reduction in available shear resistance occurs very quickly with increasing values 

of M B beyond a limiting M B value for a given Ω Β (Figure 4). This limiting value of 

M B is insensitive to ζ B , and is approximately equal to 1/Ω Β ,forΩ Β ≤ 1. Consequently, 

a preliminary design approach is to compare M B with 1/Ω Β by calculating the product 

M B Ω Β .IfM B Ω Β > 1, then strain-softening effects are likely and the residual interface 

shear strength should be used in design. Note that M B Ω Β > 1 is equal to the following: 

(τ p − τ r ) 

> EH B 

(δ r − δ p ) L 2 

B 

B 

Hence, the effects of strain-softening in the buttress are essentially insensitive to the 

pre-peak slope of the shear resistance versus displacement curve for the base interface, 

(τ p /δ p ) B . This conclusion is useful because it may be difficult to determine the peak displacement 

accurately in laboratory testing. 

4.2.3 Side Slope 

(5) 

The available shear resistance along the side slope interface (R f ) S is related to ζ S , Ω S 

and M S (where the subscript S denotes side slope), as well as the applied shear stress and 

the stiffness provided by the buttress at the toe of the slope. The column height, normal 

stress at the interface and shear stress at the interface all increase down the slope (Figure 

3). However, since each of these variables is proportional to the waste thickness along 

the slope for a frictional interface, an equivalent column with constant height, normal 

stress and shear stress along the interface can be used to approximate the wedge of waste 

on the side slope. A detailed derivation and discussion of this approximation are given 

in Gilbert et al. (1996b). The equivalent column height, H S , for obtaining M S from Equation 

4 is the distance along a line perpendicular to the interface at the toe of the side 

slope, while the equivalent column length, L S , is the length of the side slope interface 

(Figure 3). 

The applied shear stress along the interface is represented by the following non-dimensional 

factor: 

β S = 

f toe 

(τ p ) toe 

(6) 

where: f toe = applied shear stress at the toe of the slope; and (τ p ) toe = peak, interface shear 

resistance at the toe. For a frictional interface, β S can be approximated by tanΨ/tan(Ô p ) S , 

where Ψ is the slope angle (Figure 3) and (Ô p ) S is the peak friction angle for the slope 

interface. Strain-softening will occur in this model only if β S > 1. The buttress stiffness 

is represented by a factor, α S , that relates the stiffness of the buttress to that of the side 

slope. For practical purposes, Gilbert et al. (1996b) show that α S can be approximated 

by the following: 


191


α s ≈ L S 

L B 

cos 2 θ 

(7) 

where θ is the orientation of the resultant force between the side slope and buttress. A 

conservative assumption for θ is the slope angle, Ψ. Typical values of α S are close to 1.0. 

The relationship between (R f ) S and M S ,andζ S and M S is shown in Figure 5 for given 

values of β S and α S . Gilbert et al. (1996b) show that (R f ) S is relatively insensitive to β S 

(for values greater than 1) and α S . Similar to the buttress case, the reduction in shear 

resistance from peak to residual values occurs rapidly with increasing M S once a limiting 

value of M S is exceeded. However, the limiting M S value for a given Ω S value is 

smaller than that for the buttress (Figure 4). Therefore, it is more likely that the residual 

strength will be mobilized along the side slope rather than the buttress. Further, given 

typical values for M S and Ω S (Table 1), it is unlikely that an average stress greater than 

the residual value could be mobilized along a typical side slope in a containment system. 

4.3 Design Example 

4.3.1 Introduction 

An example waste containment slope is analyzed in this section to demonstrate how 

to: apply the analytical model presented above; compare the simplified, analytical 

model results with those from a more sophisticated, numerical model; and gain further 

insight into the potential effects of strain-softening on the available shear resistance in 

1 

∞ 

Figure 5. 

Interface shear resistance versus relative stiffness factor for waste slope. 



containment systems. The geometry for the example slope is shown in Figure 6. The 

length of the base is fixed at 91.5 m, and the maximum stable slope height is determined 

for various interface and waste properties. 

The lining system along the base and side slope has the same interface properties: a 

peak secant friction angle = 9.3_; a residual secant friction angle = 4.1_; andζ = 

tan(4.1_)/tan(9.3_) = 0.44. If the peak interface strength is available along both the side 

slope and base, then the maximum slope height that is stable (i.e. the slope height with 

a factor of safety = 1.0) is 28.3 m according to a limit equilibrium analysis based on 

Spencer’s method (Spencer 1967). The maximum, stable slope height decreases to 13.1 

m if the residual strength is all that is available along both the side slope and base interfaces. 

Therefore, the maximum stable slope height is between 13.1 and 28.3 m, depending 

on deformations within the system. 

Three different strain-softening rates and two different waste types are investigated. 

The three shear resistance versus displacement curves, which correspond to Ω values 

of 0.05, 0.005 and 0.001, are shown in Figure 7. These Ω values incorporate the range 

of typical values for containment system interfaces (Table 1). In each case, the peak 

displacement, δ p , is equal to 1.0 mm, the peak secant friction angle, Ô p , is equal to 9.3_, 

and the strength reduction factor, ζ, is equal to 0.44. The base and side slope interfaces 

are assumed to have the same stress versus displacement curves (i.e. Ω B = Ω S = Ω). Α 

municipal waste, with E = 718 kPa and γ = 11.0 kN/m 3 , and a hazardous waste, with 

E = 7,180 kPa and γ = 17.2 kN/m 3 , are used in the analysis. 

4.3.2 Analytical Model Results 

The slope configuration at the maximum possible height of 28.3 m is adopted as a 

nominal slope geometry for the analysis (i.e. to determine L S , H S , L B and H B ). This nominal 

geometry is conservative since the smallest R f values will be obtained for the highest 

and longest slope. The resulting values for L S , H S , L B and H B are given in appended Table 

2. 

First, consider the municipal waste. For the buttress, the peak shear resistance on the 

base, (τ p ) B , is equal to the normal stress on the landfill base multiplied by tanÔ p . Hence, 

(τ p ) B = (28.3 m)(11.0 kN/m 3 )tan(9.3_) = 51 kPa. Further, M B is obtained from Equation 

4 as follows: 

(51 kPa)∕(0.001 m) 

M B = (8) 

(718 kPa)(28.3 m) (49.0 m)2 ≈ 6, 000 m 2 

Vertical distance (m) 

1V:2H 

Side 

slope 

Buttress 

1V:3H 

Horizontal distance (m) 

Figure 6. 

Configuration of waste containment slope for design example. 


193


Figure 7. 

Secant friction angle versus displacement curves for interfaces in design example. 

Since the product M B Ω B exceeds 1 for all values of Ω B greater than 0.0002 (Table 2), 

there is a high potential for strength reductions due to strain-softening for all of the 

strain-softening curves in Figure 7. In fact, the values of (R f ) B for the buttress estimated 

from Figure 4 are all significantly less than 1.0, ranging from 0.49 to 0.61 (Table 2). 

As Ω B decreases, the magnitude of (R f ) B increases. For the slope, M S = 8000, β S =3.1_ 

from Equation 6, and α S = 1 from Equation 7. The resulting reduction factor, obtained 

from Figure 5, is equal to its smallest possible value, (R f ) S = ζ S = 0.44, for the three different 

Ω S values (Table 2). Therefore, it is unlikely that an average stress greater than the 

residual strength will be available along the side slope interface. 

Next, consider the hazardous waste. Since the hazardous waste is stiffer than the municipal 

waste, deformations will be smaller. For the hazardous waste, M B and M S are 

about one order of magnitude smaller than for the municipal waste. Consequently, at 

a given value of Ω, R f values for the hazardous waste are greater than or equal to those 

for the municipal waste (Table 2). For the buttress, (R f ) B is as large as 0.85 when Ω Β is 

equal to its smallest value of 0.001. Note that in this case, the product of M B Ω Β approaches 

a value of 1. However, the reduction factor along side slope, (R f ) S , is close to 

ζ S for all Ω S values, even with the order of magnitude decrease in M S . 

4.3.3 NumericalModelResults 

A two-dimensional, numerical analysis was performed using the Universal Distinct 

Element Code (ITASCA 1993) to verify the analytical results. The waste was modeled 

as an elastic, Mohr-Coulomb material using the property values given in Table 2. The 

shear resistance versus displacement relationships for the side slope and base interfaces 

were modeled using a continuously yielding model (ITASCA 1993). Figure 7 shows the 



three curves that were used corresponding to Ω values of 0.05, 0.005 and 0.001. The 

interfaces were assumed to have an infinite normal stiffness (i.e. they were treated as 

rigid boundaries). The waste was discretized into 612 quadrilateral elements with 28 

contact points along the interfaces, and waste placement was modeled as an instantaneous 

process. The significant improvements in the numerical model versus the analytical 

model are: (i) waste deformations were modeled in two dimensions versus one dimension; 

(ii) the waste was modeled as a continuum instead of individual columns on 

the base and side slope; and (iii) the interface shear resistance versus displacement relationships 

were more realistically modeled by nonlinear functions (Figure 7) rather than 

linear functions. 

The maximum, stable slope height was determined using the numerical model for 

each of the cases analyzed previously, and the results are presented in Table 2. At this 

slope height, the mobilized resistances were integrated along the base and side slope 

interfaces to determine the average resistances and the corresponding values of R f from 

Equation 3. The analytical and numerical results compare favorably. Both results show 

that R f values increase with decreasing Ω values, and increase with increasing waste 

stiffness. In addition, the magnitude of R f obtained from the analytical model is within 

15% of the value obtained from the numerical model. Finally, the estimate of R f from 

the analytical model is slightly conservative compared to the numerical model. 

This example demonstrates the potential for strain-softening along containment system 

interfaces due to deformations within the waste. A conventional design approach 

for this slope is to apply a factor of safety of 1.5 to the peak interface strength in a limit 

equilibrium analysis. The resulting slope height for this design approach is 22.0 m. In 

five of the six cases presented in Table 2, this design slope height would have be unstable. 

4.4 Design Recommendations 

The analytical and numerical results presented in this paper indicate that reductions 

in the available interface strength from peak to residual values are likely due to deformations 

in an elastic waste material under gravity. Since typical waste materials are 

not elastic and exhibit reductions in stiffness at large strains, the potential for strainsoftening 

reductions may even be greater. In addition, other factors exist that might 

cause strain-softening reductions in resistance such as: high, localized stresses imposed 

during construction; creep along the interface or within the containment system components; 

waste settlement; and foundation settlement. Therefore, it is strongly recommended 

that the potential for instability be explored in a limit equilibrium analysis using 

residual strengths along all interfaces. The factor of safety against failure for this 

analysis should be greater than one unless the consequences of failure are not significant 

or a reliance on greater strengths is clearly justified. 

Consideration of residual strengths may significantly reduce the maximum allowable 

grade or length of slopes in a given design. In the example presented previously, the 

design slope height would be 13.1 m versus the 22.0 m height obtained using peak 

strengths with a factor of safety equal to 1.5. One approach to reduce the effect of strainsoftening 

is to isolate the strain-softening interface by including an overlying interface 

with a smaller peak strength. This approach is only useful if the shear strength of the 

upper interface is significantly greater than the residual strength of the strain-softening 


195


interface. A second design approach is to include a stiff component, such as a geogrid, 

above the interface of concern. This stiff component will increase the effective axial 

stiffness of the overlying material, decreasing the level of deformation (e.g. M B in Figure 

4 and M S in Figure 5 are reduced). 

5 CONCLUSIONS 

The laboratory test results summarized in this paper demonstrate that many geosynthetic 

interfaces in waste containment systems exhibit strain-softening behavior. The 

mechanisms leading to strain-softening include clay particle reorientation, geosynthetic 

polishing, and geosynthetic failure. Peak interface strengths are typically mobilized 

at 1 to 15 mm of displacement, and post-peak strengths can be as small as 30% of the 

peak strength. Geosynthetic polishing and failure typically cause the most severe reductions 

in strength. 

Analyses of strain-softening effects on the available shear resistance in a waste containment 

system indicate that the potential for reductions in mobilized resistance increases 

as: (i) waste stiffness decreases relative to the initial shear stiffness of the interface; 

(ii) length of the slip surface increases; and (iii) rate of strain-softening with 

displacement increases. For a buttress, the limiting shear resistance will likely be less 

than the peak strength if Equation 5 is satisfied. For a side slope, it is unlikely that the 

limiting shear resistance will be greater than the residual strength if the applied shear 

stress is greater than the peak strength. 

It is strongly recommended that a factor of safety greater than one be achieved in all 

containment system slope designs, assuming that residual strengths are mobilized along 

the entire slip surface. Future research should focus on measuring deformations and 

mobilized shear resistances in existing waste containment facilities. 

ACKNOWLEDGMENTS 

The authors acknowledge The University of Texas at Austin and Golder Associates 

Inc. for providing financial and technical support for this research. They also acknowledge 

Timothy D. Stark and James H. Long, Associate Professors of Civil Engineering 

at the University of Illinois at Urbana-Champaign, for providing raw data from test results 

and review comments. 

REFERENCES 

Bishop, A.W., Green, G.E., Garga, V.K., Andresen, A. and Brown, J.D., 1971, “A New 

Ring Shear Apparatus and Its Application to the Measurement of Residual Strength”, 

Géotechnique, Vol. 21, No. 4, pp. 273-328. 

Bjerrum, L., 1967, “Progressive Failure in Slopes of Overconsolidated Plastic Clay and 

Clay Shales”, Journal of Soil Mechanics and Foundations, Vol. 93, No. 5, pp. 3-49. 



Byrne, R.J., 1994, “Design Issues with Strain-Softening Interfaces in Landfill Liners”, 

Proceedings of Waste Tech ’94, Charleston, South Carolina, USA, January 1994, Session 

4, Paper 4. 

Byrne, R.J., Kendall, J. and Brown, S., 1992, “Cause and Mechanism of Failure, Kettleman 

Hills Landfill B-19, Unit IA”, Stability and Performance of Slopes and Embankments-II, 

Seed, R.B. and Boulanger, R.W., Editors, Geotechnical Special Publication 

No. 31, ASCE, 1992, proceedings of a specialty conference held in Berkeley, California, 

USA, Vol. 2, pp. 1188-1215. 

Casagrande, A. and Wilson, S.D., 1951, “Effect of Rate of Loading on Strength of Clays 

and Shales at Constant Water Content”, Géotechnique, Vol. 2, No. 3, pp. 251-263. 

Chowdhury, R.N. and Grivas, D.A, 1982, “Probabilistic Model of Progressive Failure 

of Slopes”, Journal of Geotechnical Engineering, Vol. 108, No. 6, pp. 803-817. 

Duncan, J.M. and Dunlop, P., 1969, “Slopes in Stiff-Fissured Clays and Shales”, Journal 

of Geotechnical Engineering, Vol. 95, No. 2, pp. 467-492. 

Fassett, J.B., 1993, “Geotechnical Properties of Municipal Solid Waste”, Special Project 

Report, School of Civil Engineering, Purdue University, West Lafayette, Indiana, 

USA, 64 p. 

Gilbert, R.B., Liu, C.N., Wright, S.G. and Trautwein, S.J., 1995, “A Double Shear Test 

Method for Measuring Interface Strength”, Proceedings of Geosynthetics ’95,IFAI, 

Vol. 3, Nashville, Tennessee, USA, February 1995, pp. 1017-1029. 

Gilbert, R.B., Fernandez, F.F. and Horsfield, D., 1996a, “Shear Strength of a Reinforced 

Geosynthetic Clay Liner”, Journal of Geotechnical Engineering, in press. 

Gilbert, R.B, Long, J.H. and Moses, B.E., 1996b, “Analysis of Progressive Slope Failure 

in Waste Containment Systems”, International Journal for Numerical and Analytical 

Methods in Geomechanics, Vol. 20, No. 1, pp. 35-56. 

Hvorslev, M.J., 1936, “A Ring Shear Apparatus for the Determination of the Shearing 

Resistance and Plastic Flow of Soils”, Proceedings of the First International Conference 

on Soil Mechanics and Foundation Engineering, Vol. II, Harvard University, 

Cambridge, Massachusettes, USA, pp. 125-129. 

ITASCA, 1993, “UDEC, Universal Distinct Element Code, Version 2.0, Volume I: 

User’s Manual”, ITASCA Consulting Group, Minneapolis, Minnesota, USA, 764 p. 

La Rochelle, P., 1988, “The Art and Science of Geotechnical Engineering at the Dawn 

of the Twenty-First Century”, Cording, E.J., Hall, W.J., Haltiwanger, J.D., Hendron, 

Jr., A.J. and Mesri, G., Editors, Prentice Hall, Englewood Cliffs, New Jersey, USA, 

620 p. 

Law, K.T. and Lumb, P., 1978, “A Limit Equilibrium Analysis of Progressive Failure 

in the Stability of Slopes”, Canadian Geotechnical Journal, Vol. 15, No. 1, pp. 

113-122. 

Lefebvre, G., 1982, “Use of Post-Peak Strength in Slope Stability Analysis”, Symposium 

on Slopes on Soft Clays, Swedish Geotechnical Institution Report No. 7, Linkoping, 

pp. 263-292. 

Li, M.H., 1995, “Strength of Textured Geomembrane and Nonwoven Geotextile Interfaces” 

M.Sc. Thesis, The University of Texas at Austin, Austin, Texas, USA, 124 p. 


197


Lo, K.Y. and Lee, C.F., 1973, “Stress Analysis and Slope Stability in Strain-Softening 

Materials”, Geotechnique, Vol. 23, No. 1, pp. 1-11. 

Long, J.H., Daly, J.J. and Gilbert, R.B., 1993, “Structural Integrity of Geosynthetic Liner 

and Cover Systems for Solid Waste Landfills”, Report, Project No. OSWR 05-005, 

Office of Solid Waste Research, University of Illinois Center for Solid Waste Management 

and Research, Urbana, Illinois, USA, 284 p. 

Nelson, J.D. and Thompson, E.G., 1977, “A Theory of Creep Failure in Overconsolidated 

Clay”, Journal of Geotechnical Engineering, Vol. 103, No. 11, pp. 1281-1294. 

Sharma, H.D., Dukes, M.T. and Olsen, D.M., 1989, “Field Measurements of Dynamic 

Moduli and Poisson’s Ratios of Refuse and Underlying Soils at a Landfill Site”, Geotechnics 

of Waste Fills - Theory and Practice, Landva, A. and Knowles, G.D., Editors, 

ASTM Special Publication 1070, proceedings of a symposium held in Pittsburgh, 

Pennsylvania, USA, pp. 57-70. 

Singh, A. and Mitchell, J.K., 1969, “Creep Potential and Creep Rupture of Soils” Proceedings 

of the Seventh International Conference on Soil Mechanics and Foundation 

Engineering, Vol. 1, Mexico City, Mexico, pp. 379-384. 

Skempton, A.W., 1964, “Long-Term Stability of Clay Slopes”, Géotechnique, Vol. 14, 

No. 2, pp. 75-101. 

Spencer, E., 1967, “A Method of Analysis of the Stability of Embankments Assuming 

Parallel Inter-Slice Forces”, Géotechnique, Vol. 17, No. 1, pp. 11-26. 

Stark, T.D. and Eid, H.T., 1994, “Drained Residual Strength of Cohesive Soils”, Journal 

of Geotechnical Engineering,Vol. 120, No. 5, pp. 856-871. 

Stark, T.D. and Poeppel, A.R., 1994, “Landfill Liner Interface Strengths from Torsional 

Ring Shear Tests”, Journal of Geotechnical Engineering, Vol. 120, No. 3, pp. 

286-293. 

Stark, T.D., Williamson, T.A. and Eid, H.T., 1996, “HDPE Geomembrane/Geotextile 

Interface Shear Strength”, Journal of Geotechnical Engineering, Vol. 122, No. 3, pp. 

197-203. 

Vallejo, L.E. and Zhou, Y., 1995, “Fractal Approach to Measuring the Roughness of 

Geomembranes”, Journal of Geotechnical Engineering, Vol. 121, No. 5, pp. 

442-446. 

Wiberg, N.E., Koponen, M. and Runesson, K., 1990, “Finite Element Analysis of Progressive 

Failure in Long Slopes”, International Journal for Numerical and Analytical 

Methods in Geomechanics, Vol. 14, No. 9, pp. 599-612. 

NOTATIONS 

Basic SI units are given in parentheses. 

E = modulus of elasticity (of a column of waste or landfill cover) (Pa) 

f toe = applied shear stress at the toe of the slope (Pa) 

H = column thickness (m) 



H Β = column thickness for a buttress (of a landfill cover or waste) (m) 

H S = column thickness for a slope (of a landfill cover or waste) (m) 

L = length of the interface along the column base (m) 

L Β = length of the interface along the column base for a buttress (m) 

L S = length of the interface along the column base for a slope (m) 

M = stiffness factor (dimensionless) 

M Β = stiffness factor for a buttress (dimensionless) 

M S = stiffness factor for a slope (dimensionless) 

R f = reduction in available shear resistance due to deformation 

(dimensionless) 

(R f ) Β = reduction in available shear resistance due to deformation of a buttress 


(R f ) S = reduction in available shear resistance due to deformation of a slope 


s limit = average available shear resistance (Pa) 

s p = average available shear resistance for an incompressible waste (Pa) 

α S = buttress stiffness (dimensionless) 

β S = non-dimensional factor representing the applied shear stress along a 

slope interface (dimensionless) 

γ = unit weight of soil or waste (N/m 3 ) 

γ d = dry unit weight of compacted clay liner (N/m 3 ) 

ζ = strain-softening coefficient (dimensionless) 

ζ Β = strain-softening coefficient for a buttress (dimensionless) 

ζ S = strain-softening coefficient for a slope (dimensionless) 

δ = displacement (m) 

δ p = peak displacement (m) 

δ r = residual displacement (m) 

θ = orientation of the resultant force between the side slope and buttress (_) 

ν = Poisson’s ratio (dimensionless) 

σ = normal stress (Pa) 

τ = shear stress (Pa) 

τ p = peak shear stress (Pa) 

(τ p ) Β = peak interface shear resistance for a buttress (Pa) 

(τ p ) toe = peak interface shear resistance at the toe of a slope (Pa) 

τ r = residual shear stress (Pa) 

Ô = interface friction angle (_) 

(Ô p ) S = peak interface friction angle for a slope (_) 

ψ = slope angle (Figure 3) (_) 


199


Ω = rate of post-peak strength reduction (dimensionless) 

Ω Β = rate of post-peak strength reduction for a buttress (dimensionless) 

Ω S = rate of post-peak strength reduction for a slope (dimensionless) 


GILBERT AND BYRNE 

D 

Strain-Softening of Waste Containment System Interfaces 

Table 1. Summary of strain-softening characteristics for selected containment system interfaces. 

Type of interface 

(Reference) 

Test device 

(displacement 

limit) 

Normal 

stress 

(kPa) 

τp 

(kPa) 

τr 

(kPa) ζ = τ r 

δp 

τ p 

(mm) 

δr 

(mm) Ω = 

Smooth geomembrane/Compacted clay 1 

(Byrne et al. 1992) 

Large-scale 

direct shear 

(50 mm) 

150 

280 

410 

38 

45 

45 

23 

26 

30 

0.60 

0.58 

0.66 

3 

3 

3 

25 

20 

27 

Smooth geomembrane/Compacted clay 1 

(Stark and Poeppel 1994) 

Torsional 

ring shear 

(unlimited) 

48 

96 

280 

13 

22 

29 

7 

12 

22 

0.54 

0.55 

0.76 

3 

3 

3 

45 

20 

20 

Nonwoven geotextile/Smooth 

geomembrane 2 

(Stark and Poeppel 1994) 

Torsional 

ring shear 

(unlimited) 

79 

158 

234 

400 

13 

23 

37 

58 

8.4 

15 

23 

42 

0.65 

0.65 

0.62 

0.72 

1 

1 

2 

1 

64 

37 

44 

50 

Nonwoven geotextile/Smooth 

geomembrane 3 

(Gilbert et al. 1995) 

Double interface 

direct shear 

(200 mm) 

70 

140 

210 

280 

350 

410 

14 

25 

45 

55 

67 

87 

11 

19 

33 

40 

54 

67 

0.79 

0.76 

0.73 

0.73 

0.81 

0.77 

1 

2 

2 

3 

1 

1 

30 

18 

15 

18 

26 

25 

Nonwoven geotextile/Textured 

geomembrane 4 

(Li 1995) 

Direct shear 

(accumulated) 

16 

340 

690 

15 

180 

320 

10 

110 

190 

0.67 

0.61 

0.59 

7 

8 

12 

65 

50 

75 

τ r−τp 

δr−δp 

τp∕δp 

0.05 

0.07 

0.04 

0.03 

0.08 

0.04 

0.006 

0.010 

0.018 

0.006 

0.007 

0.030 

0.041 

0.055 

0.008 

0.010 

0.04 

0.07 

0.08 

Likely mechanisms of 

strain-softening 

Clay particle orientation 

Geomembrane polishing 

Clay particle orientation 




Geotextile failure 


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D 


Nonwoven geotextile/Textured 

geomembrane 5 

(Stark et al. 1996) 

Torsional 

ring shear 

(unlimited) 

48 

96 

190 

290 

30 

60 

120 

180 

11 

22 

41 

67 

0.37 

0.37 

0.34 

0.37 

6 

6 

6 

6 

220 

140 

240 

130 

0.02 

0.03 

0.02 

0.03 



Textured geomembrane/Woven 

geotextile component of reinforced 

GCL 6 

(Gilbert et al. 1996a) 

Large-scale 

direct shear 

(50 mm) 

3.5 

6.9 

14 

28 

69 

350 

2.5 

4.6 

10 

16 

39 

100 

1.7 

9 

13 

42 

3 

7 

0.68 

0.65 

0.70 

0.57 

0.33 

0.42 

7 

11 

14 

11 

15 

12 

41 

46 

41 

41 

48 

50 

0.1 

0.1 

0.2 

0.2 

0.3 

0.2 



(normal stress≤14 kPa) 

Needle-punched fiber failure 

(normal stress>14 kPa) 

Textured geomembrane/Woven 

geotextile component of reinforced 

GCL *7 

(Byrne 1994) 

Large-scale 

direct shear 

(50 mm) 

96 

290 

480 

60 

110 

190 

46 

69 

113 

0.76 

0.63 

0.59 

8 

13 

13 

45 

56 

70 

0.05 

0.11 

0.09 



Notes: 1 Smooth HDPE geomembrane, 1.5 mm thick; compacted clay is a soil-bentonite admixture compacted at nominal w = 30%, and γ d =15kN/m 3 . 2 Nonwoven, 

needle-punched, polypropylene geotextile, 205 g/m 2 ; HDPE geomembrane, 1.5 mm thick. 3 Nonwoven, needle-punched, polyester geotextile, 440 g/m 2 ; HDPE 

geomembrane, 1.5 mm thick. 4 Nonwoven, needle-punched, polyester geotextile, 400 g/m 2 ; HDPE geomembrane, 1.5 mm thick, coextruded texturing. 5 Nonwoven, 

needle-punched, polypropylene geotextile, 270 g/m 2 ; HDPEgeomembrane, 1.5mm thick,coextruded texturing. 6 HDPE geomembrane,1.5 mm thick, laminated texturing; 

GCL comprises bentonite between nonwoven and woven geotextiles, needle-punched reinforcement, hydrated. 7 HDPE geomembrane, 1.5 mm thick, laminated texturing; 

GCL * comprises bentonite between nonwoven and woven geotextiles, needle-punched reinforcement with adhesive, hydrated. 

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Table 2. Results of analysis of example waste containment slope. 

Nominal 

Case buttress 

geometry 

Nominal 

slope 

geometry 

Waste 

properties 

Waste Interface 

Interface 

properties 1 softening 

Strain 

softening Strain 

curve 2 Strain 

1-D analytical results 2-D numerical results 

Buttress Slope 

softening 

rate 3 , Ω MB (Rf)B MS 

βS 

(_) (Rf)S 

Buttress Slope 

Buttress 

(Rf)B 

Slope 

(Rf)S 

Maximum 

stable 

height 

(m) 

A 0.05 6000 0.49 8000 3.1 0.44 0.56 0.44 18.9 

Municipal 

waste 

LB = 49.0 m 

HB = 28.3 m 

LS = 63.2 m 

HS = 31.6 m 

E = 718 kPa Ôp =9.3_ 

ν =0.3,Ô =35_ 

γ =11.0 kN/m 3 

δp =1mm 

ζ = 0.44 

αS =1 

γ = 11.0 kN/m 3 ζ = 0.44 

B 0.005 6000 0.51 8000 3.1 0.44 0.60 0.44 19.5 

C 0.001 6000 0.61 8000 3.1 0.44 0.69 0.44 20.4 

A 0.05 950 0.57 1300 3.1 0.44 0.61 0.44 19.8 

Hazardous 

waste 

LB = 49.0 m 

HB = 28.3 m 

LS = 63.2 m 

HS = 31.6 m 

E =7180kPa Ôp =9.3_ 

ν =0.3,Ô =35_ 

γ =17.2 kN/m 3 

δp =1mm 

ζ = 0.44 

αS =1 

γ = 17.2 kN/m 3 ζ = 0.44 

B 0.005 950 0.63 1300 3.1 0.44 0.72 0.44 22.0 

C 0.001 950 0.85 1300 3.1 0.45 0.90 0.53 25.9 

Notes: 1 Buttress and slope have same interface properties: (Ô p)B = (Ôp)S = Ôp ; (δp)B = (δp)S = δp ; ζB = ζS = ζ. 2 Interface secant friction angle versus displacement curves 

are shown in Figure 7; curve applies to both buttress and slope interfaces. 3 Buttress and slope have same interface properties: ΩB = ΩS = Ω. E= Young’s modulus; ν = 

Poisson’s ratio; Ô = friction angle; and γ = unit weight. 

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strain-softening behavior of waste containment system interfaces

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