56TH CANADIAN GEOTECHNICAL CONFERENCE4TH JOINT IAH-CNC/CGS CONFERENCE2003 NAGS CONFERENCE

56ième CONFÉRENCE CANADIENNE DE GÉOTECHNIQUE4ième CONFÉRENCE CONJOINTE AIH-CCN/SCG

2003 NAGS CONFÉRENCE

VALIDATION OF DISCRETE FRAMEWORK FOR FIBERREINFORCEMENTChunling Li, University of Texas at Austin, USAJorge G. Zornberg, University of Texas at Austin, USA

ABSTRACT: An experimental study was conducted to validate a previously-proposed discrete methodology for thedesign of fiber-reinforced soil. The analysis of shear strength of fiber-reinforced soil using a discrete approach can beconducted by independent characterization of soil specimens and of fiber specimens since the contributions of soil andfibers are treated separately, and the non-conventional laboratory testing of the composite fiber-reinforced soilspecimens in the traditional composite approach can be avoided. A fiber-induced distributed tension can be defined foruse in stability analyses using the proposed discrete framework.

The experimental testing program involves tensile testing of fibers as well as triaxial testing of unreinforced and fiber-reinforced specimens. The triaxial tests were conducted using both sand and clay soils, reinforced with different types ofpolypropylene fibers. As predicted by the discrete framework, the fiber-induced distributed tension was observed to beproportional to the fiber content and fiber aspect ratio when failure is characterized by pullout of individual fibers.Overall, very good agreement was found between experimental and predicted results for both sand and clay soils.Specifically, the discrete framework was able to accurately predict the contribution of randomly distributed fibers for thevarious soil types, aspect ratios, and fiber contents considered in the experimental testing program.

RÉSUMÉ: Une étude expérimentale a été dirigée pour valider une méthodologie discrète précédemment proposé pourla conception de sol fibre renforcé. L'analyse de force de cisailles d'utilisation de sol fibre renforcé une approchediscrète peut être dirigée par la caractérisation indépendante de spécimens de sol et de spécimens de fibre puisque lescontributions de sol et les fibres est séparément traitée, et le laboratoire non conventionnel essayant des spécimens desol fibre renforcés composés dans l'approche composée traditionnelle peut être évité. Une tension distribuée fibrepersuadé peut être définie pour l'usage dans la stabilité analyse l'utilisation de la structure discrète proposée.

Le programme essayant expérimental implique l'essai extensible de fibres de même que l'essai triaxial de spécimens deunreinforced et fibre renforcé. Les tests triaxiaux ont été dirigés le sable d'utilisation et les sols d'argile, renforcés avecles types différents de fibres de polypropylene. Comme prédit par la structure discrète, le fibre persuadé la tensiondistribuée a été observée pour être proportionnelle au contenu de fibre et à la proportion d'aspect de fibre quand l'échecest caractérisé par individuelles retraite de fibres individuel. Général, le très bon accord a été trouvé entre les résultatsexpérimentaux et prédits pour les sols de sable et argile. En particulier, la structure discrète pouvait précisément prédirela contribution de fibres au hasard distribuées pour les divers types de divers sol, les diverses proportions d'aspect, etles contenus de fibre considérés dans le programme essayant expérimental.

1. INTRODUCTION

Fiber reinforcement has become a promising solution tothe stabilization of thin soil veneers and localized repair offailed slopes. Randomly distributed fibers can maintainstrength isotropy and avoid the existence of the potentialplanes of weakness that can develop parallel tocontinuous planar reinforcement elements. The design offiber-reinforced soil slopes has typically been performedusing composite approaches, where the fiber-reinforcedsoil is considered a single homogenized material.Accordingly, fiber-reinforced soil design has required non-conventional laboratory testing of composite fiber-reinforced soil specimens which has discouragedimplementation of fiber-reinforcement in engineeringpractice.

Several composite models have been proposed toexplain the behavior of randomly distributed fibers withina soil mass (Maher and Gray, 1990, Michalowski andZhao, 1996, Ranjan et al., 1996). The mechanisticmodels proposed by Gray and Ohashi (1983) and Maher

and Gray (1990) quantify the “equivalent shear strength”of the fiber-reinforced composite as a function of thethickness of the shear band that develops during failure.Information needed to characterize shear banddevelopment for these models is, however, difficult toquantify (Shewbridge and Sitar, 1990). Common findingsfrom the various testing programs implemented toinvestigate composite models include: (i) randomlydistributed fibers provide strength isotropy in a soilcomposite; (ii) fiber inclusions increase the “equivalent”shear strength within a reinforced soil mass; and (iii) the“equivalent” strength typically shows a bilinear behavior,which was experimentally observed by testing ofcomparatively weak fibers under a wide range ofconfining stresses.

A discrete approach for the design of fiber-reinforced soilslopes was recently proposed to characterize thecontribution of randomly distributed fibers to stability(Zornberg, 2002). In this approach, fiber-reinforced soil ischaracterized as a two-component (soil and fibers)material. Fibers are treated as discrete elements that

Li, C., and Zornberg, J.G. (2003). “Validation of Discrete Framework for Fiber Reinforcement.” Proceedings of the 2003 North American Conference on Geosynthetics, Winnipeg, Canada, September 28 - October 1 (CD-ROM).

contribute to stability by mobilizing tensile stresses alongthe shear plane. Consequently, independent testing ofsoil specimens and of fiber specimens, but not of fiber-reinforced soil specimens, can be used to characterizefiber-reinforced soil performance.

This paper initially reviews the main concepts of thediscrete approach and subsequently validates theframework for design purposes.

2. DISCRETE FRAME WORK FOR FIBERREINFORCEMENT

2.1 Background

The volumetric fiber content, �, used in the proposeddiscrete framework is defined as:

VV = f

� [1]

where Vf is the volume of fibers and V is the controlvolume of fiber-reinforced soil.

The gravimetric fiber content, �w, typically used inconstruction specifications, is defined as:

s

fw W

W = �

[2]

where Wf is the weight of fibers and Ws is the dry weightof soil.

The dry unit weight of the fiber-reinforced soil composite,�d , is defined as:

VWW

= sfd

�

� [3]

The contribution of fibers to stability leads to an increasedshear strength of the “homogenized” compositereinforced mass. However, the reinforcing fibers actuallywork in tension and not in shear. A major objective of thediscrete framework is to explicitly quantify the fiber-induced distributed tension, t, which is the tensile forceper unit area induced in a soil mass by randomlydistributed fibers.

Specifically, the magnitude of the fiber-induceddistributed tension is defined as a function of properties ofthe individual fibers. In this way, as in analysis involvingplanar reinforcements, limit equilibrium analysis of fiber-reinforced soil can explicitly account for tensile forces.

The interface shear strength of individual fibers can beexpressed as:

avenicif c cc = f ,,, tan ��� ���� [4]

where c and � are the cohesive and frictional componentsof the soil shear strength and �n,ave is the average normalstress acting on the fibers. The interaction coefficients, ci,cand ci,�, commonly used in soil reinforcement literature forcontinuous planar reinforcement, is adopted herein torelate the interface shear strength to the shear strength ofthe soil. The interaction coefficients are defined as:

ca = c ci ,

[5]

�

�� tan

tan, = c i

[6]

where a is the adhesive component of the interface shearstrength between soil and the polymeric fiber, tan� is theskin-frictional component.

The pullout resistance of a fiber of length lf should beestimated over the shortest side of the two portions of afiber intercepted by the failure plane. The length of theshortest portion of a fiber intercepted by the failure planevaries from zero to lf /2. Statistically, the averageembedment length of randomly distributed fibers, le,ave,can be analytically defined by:

4,f

avee

l = l [7]

where lf is total length of the fibers.

The average pullout resistance can be quantified alongthe average embedment length, le,ave , of all individualfibers crossing a soil control surface A. The ratio betweenthe total cross sectional area of the fibers Af and thecontrol surface A is assumed to be defined by thevolumetric fiber content �. That is:

AA

= f�

(8)

When failure is governed by the pullout of the fibers, thefiber-induced distributed tension, tp, is defined as theaverage of the tensile forces inside the fibers over thecontrol area A. Consequently, tp can be estimated as:

� �avenicip ccc = t ,,, tan ���� � ������ [9]

where � is the aspect ratio defined as:

=ld

f

f

� [10]

where df is the equivalent diameter of the fiber.

When failure is governed by the yielding of the fibers, thedistributed tension, tt , is determined from the tensilestrength of the fiber:

ultft = t ,�� � [11]

where �f,ult is the ultimate tensile strength of the individualfibers.

The fiber-induced distributed tension t to be used in thediscrete approach to account for the tensile contributionof the fibers in limit equilibrium analysis is:

� �tp tt = t ,min [12]

The critical normal stress, �n,crit , which defines thechange in the governing failure mode, is the normalstress at which failure occurs simultaneously by pulloutand tensile breakage of the fibers. That is, the followingcondition holds at the critical normal stress:

pt tt � [13]

An analytical expression for the critical normal stress canbe obtained as follows:

��

���

� tan,

,,,

��

���

�

i

ciultfcritn c

cc

[14]

2.2 Equivalent Shear Strength of Fiber-Reinforced Soil

As in analyses involving planar inclusions, the orientationof the fiber-induced distributed tension should also beidentified or assumed. Specifically, the fiber-induceddistributed tension can be assumed to act: a) along thefailure surface so that the discrete fiber-induced tensilecontribution can be directly “added” to the shear strengthcontribution of the soil in a limit equilibrium analysis; b)horizontally, which would be consistent with designassumptions for reinforced soil structures using planarreinforcements; and c) in a direction somewhere betweenthe initial fiber orientation (which is random) and theorientation of the failure plane.

This equivalent shear strength of fiber-reinforcedspecimens can be defined as a function of the fiber-induced distributed tension t, and the shear strength ofthe unreinforced soil, S:

tct S= S neq ������ ���� tan [15]

where � is an empirical coefficient that accounts for theorientation of fiber and the efficiency of the mixing offibers. � is equal to 1, if the fibers are randomlydistributed and working with 100% efficiency, otherwise �should be smaller than 1.

Depending on whether the mode of failure is fiber pulloutor yielding, the equivalent shear strength can be derivedby combining (9) or (11) with (15). It should be noted thatthe average normal stress acting on the fibers, �n,ave,does not necessarily equal the normal stress on theshear plane �n . For randomly distributed fibers, �n,avecould be represented by the octahedral stresscomponent. However, a sensitivity evaluation undertakenusing typical ranges of shear strength parameters showthat �n,ave can be approximated by �n without introducingsignificant error.

Accordingly, the following expressions can be used todefine the equivalent shear strength when failure isgoverned by fiber pullout:

� � npeqpeqpeq c = S �� �� ,,, tan [16]

� � cc = c cipeq ����� ,, 1 ��� [17]

� � � � ����� � tan1tan ,, ����� ipeq c = [18]

Equivalently, the following expressions can be obtained todefine the equivalent shear strength when failure isgoverned by tensile breakage of the fibers:

� � nteqteqteq c = S �� �� ,,, tan [19]

ultfteq c = c ,, ��� ��� [20]

� � �� tantan , = teq [21]

The above expressions yield a bilinear shear strengthenvelope, which is shown in Figure 1.

Figure 1 Representation of the equivalent shear strengthaccording to the discrete approach

Seq=S+t

1

tan�

�n

c

S

�n,crit

S

t

3. EXPERIMENTAL VALIDATION

3.1 Experimental testing program

A triaxial compression testing program on fiber-reinforcedsoil was implemented to validate the proposed discreteframework. Both cohesive and granular soils were used inthe testing program, and the soil properties weresummarized in Table 1.

Table 1 Summary of soil propertiesSoil type Soil 1 Soil 2USCS

classificationSP CL

LL % - 49PL % - 24IP % - 25% Fines 1.4 82.6

The tests were conducted using commercially availablepolypropylene fibers, and the properties of fibers weresummarized in Table 2. A series of tensile test wereperformed in general accordance with ASTM D2256-97 toevaluate the ultimate tensile strength of fibers. Theaverage tensile strength of the fibers was approximately425,000 kPa.

The triaxial testing program involved consolidated drained(CD) tests for SP soils and consolidated undrained (CU)tests for CL soils. The specimens have a diameter of 71mm and a minimum length-to-diameter ratio of 2. The CUtests were performed in general accordance with ASTMD4767, and the specimens were back pressure saturatedand the pore water pressure was measured. Theunreinforced tests of SP soil yielded an effective shearstrength envelope defined by cohesion of 6.1 kPa andfriction angle of 34.3°, while the cohesion and frictionangle of CL soil were 12.0 kPa and 31.0° respectively.

Table 2 Summary of fiber propertiesSP tests CL tests

Linear density (denier) 1000 & 360 2610Fiber content (%) 0.2 & 0.4 0.2 & 0.4Length of fibers (mm) 25 &51 25 & 51Type of fiber fibrillated &

tapefibrillated

The governing failure mode for the polymeric fibers usedin this investigation is pullout because of thecomparatively high tensile strength and short length of thefibers. Accordingly, the triaxial testing program conductedin this study focuses only on the first portion of thebilinear strength envelope shown in Figure 1.

3.2 Stress-strain behavior

SP soils reinforced with fibers Figure 2 shows the stress-strain behavior of SP soilspecimens reinforced with 360 denier fibers, and placed

at gravimetric fiber contents of 0, 0.2 and 0.4 %.Specimens were tested under confining pressure of 70kPa. The peak deviator stress increases approximatelylinearly with increasing fiber content, which is consistentwith the discrete framework (see equation (9)). The post-peak shear strength loss is smaller in the reinforcedspecimens than in the unreinforced specimens. However,the initial portions of the stress-strain curves of thereinforced and unreinforced specimens are approximatelysimilar. Accordingly, the soil appears to take most of theapplied load at small strain levels, while the load resistedby the fibers is more substantial at higher strain level. Thelarger strain corresponding to the peak deviator stressdisplayed by the fiber-reinforced specimens suggests thatfibers increase the ductility of the reinforced soilspecimen. These findings are confirmed in Figure 3,which shows the test results obtained under higherconfining stress (140 kPa). The effect of fiber length on the stress-strain behavior isshown in Figure 4. The specimens were prepared usingfibers with a different fiber type (1000 denier) than thatused in the tests shown in Figures 2 and 3. Thespecimens were prepared using the same gravimetricfiber content, but with varying fiber length. The specimensreinforced with longer (50 mm) fibers displayed highershear strength. The peak deviator stress increaseslinearly with increasing aspect ratio, which is alsoconsistent with the trend indicated by equation (9). Thestrain corresponding to the peak strength increases withincreasing fiber length. When the governing failure modeis pullout, the fiber-induced distributed tension reaches itspeak when the pullout resistance is fully mobilized. Forlonger fibers, it usually requires a larger interface sheardeformation to fully mobilize the interface strength.Consequently, the macroscopic axial strain at peak stressshould be larger for specimen reinforced with longerfibers. Figure 5 shows a similar trend for the case of testsconducted under higher confining pressures.

0

50

100

150

200

250

300

350

400

450

500

0 5 10 15 20 25Axial strain (%)

Devi

ator

stre

ss (k

Pa)

�w=0%

�w=0. 2%

�w=0. 4%

Figure 2 Stress-strain behavior of specimens prepared

using �w=0, 0.2 and 0.4% with lf =25 mm fibers (360denier), ���70 kPa, Soil 1

0

100

200

300

400

500

600

700

800

900

1000

0 5 10 15 20 25Axial strain (%)

Dev

iato

r str

ess

(kPa

)

�w=0%

�w=0. 2%

�w=0. 4%

Figure 3 Stress-strain behavior of specimens prepared

using �w=0, 0.2 and 0.4% with lf =25 mm fibers (360denier), ���140 kPa, Soil 1

0

50

100

150

200

250

300

350

0 5 10 15 20 25Axial strain (%)

Devi

ator

stre

ss (k

Pa)

unreinforced

l f=25 mm

l f=50 mm

Figure 4 Stress-strain behavior of specimen preparedusing �w=0.2 %, with lf=25 mm and 50 mm fibers (1000

denier), ��=70 kPa, Soil 1

0

100

200

300

400

500

600

0 5 10 15 20 25Axial strain (%)

Devi

ator

stre

ss (k

Pa)

unreinforced

l f=25 mm

l f=50 mm

Figure 5 Stress-strain behavior of specimen preparedusing �w=0.2 %, with lf=25 mm and 50 mm fibers (1000

denier), ��=140 kPa, Soil 1

CL soils reinforced with fibers

Figure 6 compares the stress-strain behavior of bothunreinforced and fiber-reinforced specimen using soil 2.The reinforced specimen were prepared at �w=0.2%,using 2-inch long 2610 denier fibers. Both specimenswere compacted at optimum moisture content to 90% ofthe maximum dry density achieved in the standardProctor test as specified in ASTM D 698, and testedunder confining pressure ��=98 kPa. Due to theundrained test condition, the effective confining stresschanges with the excess pore water pressure induced inthe process of shearing. The peak shear strength wasselected in terms of the maximum value of (��’/��’). Theincrement of deviator stress due to fiber addition is not asobvious as in the case of SP sand. However, the porewater pressure generated during shearing is larger forreinforced specimen than for unreinforced specimen (seeFigure 7). Consequently the effective confining stressinside the reinforced specimen is smaller than that insidethe unreinforced specimen. The fiber-reinforced specimenachieved equal or higher peak deviator stress than theunreinforced specimen under a lower effective confiningstress. This shows that the addition of fibers increasesthe shear strength of the reinforced specimen. Sincepositive water pressure is associated with the tendency ofvolume shrinkage, this observation shows that fiberreinforcement restrains the dilatancy of the reinforcedsoil. Other researchers (Michalowski and Cermak, 2003,Consoli et al., 1998) reported that fiber-reinforcedspecimens displayed smaller volume dilatation thanunreinforced specimen in consolidated drained (CD) test.This observation confirms their findings in a different testcondition.

Similar observation can be made from Figures 8 and 9,which shows the stress-strain behavior and the porewater pressure evolution obtained using 25-mm fibersplaced at 0.2% and 0.4% gravimetric fiber contents. Asthe fiber content increases, the pore water pressuregenerated during undrained shearing also increases.

0

20

40

60

80

100

0 5 10 15 20Axial strain (%)

Dev

iato

r str

ess

(kPa

)

�w=0.2%

�w=0%

Figure 6 Stress-strain behavior of specimens preparedusing �w=0, 0.2 %, with lf=50 mm fibers (2610 denier),

��=98 kPa, Soil 2

- 20

0

20

40

60

80

100

120

0 5 10 15 20

Axial strain (%)

Pore

wat

er p

ress

ure

(kPa

)�w=0.2%

�w=0%

Figure 7 Excess pore water pressure of specimensprepared using �w=0, 0.2 %, with lf=50 mm fibers (2610

denier), ��=98�kPa, Soil 2

0

20

40

60

80

100

120

140

0 5 10 15 20Axial strain (%)

Dev

iato

r str

ess

(kPa

)

�w=0.2%

�w=0.4%

Figure 8 Stress-strain behavior of specimens preparedusing �w=0.2, 0.4%, with lf=25 mm fibers (2610 denier),

��=116 kPa, Soil 2

- 20

0

20

40

60

80

100

120

0 5 10 15 20

Axial strain (%)

Pore

wat

er p

ress

ure

(kPa

) �w=0.2%

�w=0.4%

Figure 9 Excess pore water pressure of specimensprepared using �w =0.2, 0.4 %, with lf=25 mm fibers (2610

denier), ��=116kPa, Soil 2

3.3 Shear strength behavior

Equations (16) through (18) were used to predict theequivalent shear strength for fiber-reinforced specimens.Interaction coefficients (ci,c and ci,�) of 0.8 are assumed inthe analyses conducted in this study. The interface shearstrength obtained from pullout test results conducted onwoven geotextiles was considered representative of theinterface shear strength on individual fibers. For practicalpurposes, interaction coefficients can be selected fromvalues reported in the literature for continuous planarreinforcements. This is because pullout tests conductedusing a variety of soils and planar geosynthetics havebeen reported to render interaction coefficient valuesfalling within a narrow range (Koutsourais et al., 1998,Michalowski and Cermak, 2003). � is assumed to be 1.0for randomly distributed fibers. Table 3 summarized thevalues of parameters used in the analyses.

Table 3 Summary of parameters used in the prediction

� ���°� c (kPa) ci,c ci,�Soil1

1.0 34.3 6.1 0.8 0.8

Soil2

1.0 31.0 12.0 0.8 0.8

The effect of fiber content on shear strength is shown inFigure 10, which compares the experimental data andpredicted shear strength envelopes obtained from Soil 1using 25 mm fibers with linear density of 360 denierplaced at fiber contents of 0.0%, 0.2%, and 0.4%. Theexperimental results show a clear increase in equivalentshear strength with increasing fiber content. No majorinfluence of fibrillation is perceived in the results of thetesting program. The shear strength envelope for theunreinforced specimens was defined by fitting theexperimental data. However, the shear strengthenvelopes shown in the figure for the reinforcedspecimens were predicted analytically using the proposeddiscrete framework. A very good agreement is observedbetween experimental data points and predicted shearstrength envelopes. As predicted by the discreteframework, the distributed fiber-induced tension increaseslinearly with the volumetric fiber content. Similarobservation can be made in Figure 11, which shows theresults obtained from Soil 2 using 50- mm-long fibers withlinear density of 2610 denier.

The effect of fiber aspect ratio on shear strength is shownin Figure 12, which compares the experimental andpredicted shear strength envelopes of specimens of Soil1 placed at �w=0.2%, with 25 and 50 mm-long fibers. Aspredicted by the discrete framework, increasing the fiberlength increases the pullout resistance of individual fibers,and results in a higher fiber-induced distributed tension.Consequently, for the same fiber content, specimensreinforced with longer fibers will have higher equivalentshear strength. This trend agrees well with theexperimental data. Similar observation can be made fromFigure 13, which shows the results obtained from Soil 2using 25 mm and 50 mm-long fibers and placed at �w=0.4%.

0

100

200

300

400

0 100 200 300 400�' (kPa)

��(k

Pa)

Exp. data, 0% fibersStren. envelope, 0% fibers (best fit)Exp. data, 0.2% fibers (fibr.)Exp. data, 0.2% fibers (tape)Stren. envelope, 0.2% fibers (predicted)Exp. data, 0.4% fibers (fibr.)Exp. data, 0.4% fibers (tape)Stren. envelope, 0.4% fibers (predicted)

Figure 10 Comparison between predicted andexperimental shear strength results for specimens

reinforced at ��w =0, 0.2%, 0.4% with 25 mm-long fibers(360 denier), Soil 1

0

20

40

60

80

100

0 20 40 60 80 100�' (kPa)

��(k

Pa)

Exp. data, 0% fibersStren. envelope, 0% fibers(best fit)Exp. data, 0.2% fibersStren. envelope, 0.2% fibers(predicted)Exp. data, 0.4% fibersStren. envelope, 0.4% fibers(predicted)

Figure 11 Comparison between predicted andexperimental shear strength results for specimens

reinforced at �w=0, 0.2%, 0.4% with 50 mm-long fibers(2610 denier), Soil 2

3.4 Combined effect of aspect ratio and fiber content

Additional insight into the validity of the proposed discreteapproach can be obtained by comparing the resultsobtained for specimens reinforced with 50 mm-long fibersplaced at a fiber content of 0.2% with those obtained forspecimens reinforced with 25 mm-long fibers placed at afiber content of 0.4%. That is specimens with a constantvalue of (�w·��� As inferred from inspection of equation (9)the fiber-induced distributed tension is directlyproportional to both the fiber content and the fiber aspectratio. Consequently, the predicted equivalent shearstrength parameters for the above combinations of fiberlength and fiber content are the same. Figures 14 and 15combine these experimental results.

0

100

200

300

400

0 100 200 300 400

�' (kPa)

��(k

Pa)

Exp. data 0% fibersStren. envelope, 0% fibers (best fit)Exp. data, 25mm (tape)Exp. data, 25mm (fibr)Stren. envelope, 25mm (predicted)Exp. data 50mm (tape)Exp. data 50mm (fibr)Stren. envelope, 50mm (predicted)

Figure 12 Comparison between predicted andexperimental shear strength results for specimens

reinforced at �w =0.2%, with 25 mm-long and 50 mm-longfibers (1000 denier), Soil 1

0

20

40

60

80

100

0 20 40 60 80 100�' (kPa)

��(k

Pa)

Exp. data, 0% fibersStren. envelope, 0% fibers (best fit)Exp. data, 25mm fibersStren. envelope, 25mm fibers(predicted)Exp. data, 50mm fibersStren. envelope,25mm fibers(predicted)

Figure 13 Comparison between predicted andexperimental shear strength results for specimens

reinforced at �w =0.4%, with 25 mm-long and 50 mm-longfibers (2610 denier), Soil 2

The good agreement between experimental results andpredicted values provides additional evidence of thesuitability of the proposed discrete approach. From thepractical standpoint, it should be noted that using 50 mm-long fibers placed at a fiber content of 0.2% correspondsto half the reinforcement material than using 25 mm-longfibers placed at a fiber content of 0.4%. That is, for thesame target equivalent shear strength the firstcombination leads to half the material costs than thesecond one. It is anticipated, though, that difficulty inachieving good fiber mixing may compromise the validityof the relationships developed herein for comparativelyhigh aspect ratios (i.e. comparatively long fibers) and forcomparatively high fiber contents. The fiber content orfiber length at which the validity of these relationships iscompromised should be further evaluated. Nonetheless,good mixing was achieved for the fiber contents and fiberlengths considered in this investigation, which were

selected based on values typically used in geotechnicalprojects.

0

100

200

300

400

0 100 200 300 400

�' (kPa)

��(k

Pa)

Experimental data, 0% fibersStren. envelope, 0% fibers (best fit)Exp. data, 0.2% fibers (fibr.), 50 mmExp. data, 0.2% fibers (tape), 50 mmExp. data, 0.4% fibers (fibr.), 25 mmExp. data, 0.4% fibers (tape), 25 mmStren. envelope, (predicted)

Figure 14 Consolidated shear strength results forspecimen reinforced with 50 mm-long fibers (1000 denier)placed at �w =0.2% and 25 mm fibers placed at �w =0.4%,

Soil 1

0

20

40

60

80

100

0 20 40 60 80 100

�'(kPa)

��(k

Pai)

Exp. data, 0% fibers

Stren. envelope, 0% fibers (best fit)

Exp. data, 0.2% fibers, 50 mm

Exp. data, 0.4% fibers, 25mm

Stren. envelope (predicted)

Figure 15 Consolidated shear strength results forspecimen reinforced with 50 mm-long fibers (2610 denier)placed at �w=0.2% and 25 mm fibers placed at �w =0.4%,

Soil 2

Figure 16 shows the stress-strain behavior of specimenreinforced with 50 mm fibers placed at �w =0.2% and 25mm fibers placed at �w =0.4%. While the discreteapproach was developed only to predict the shearstrength response, the results in the figure show thatfiber-reinforced specimens prepared using a constantvalue of (�w·���display similar stress-strain behavior. Thissimilar response is observed for both fibrillated and tapefibers, suggesting that the fibrillation procedure does nothave a significant impact on the mechanical response offiber-reinforced soil. The experimental results suggestthat the proportionality of shear strength with the fibercontent and fiber aspect ratio predicted by the discreteframework can be extrapolated to the entire stress-strainresponse of fiber-reinforced specimens.

0

50

100

150

200

250

300

350

0 5 10 15 20 25Axial strain (%)

Dev

iato

r str

ess

(kPa

)

0% f i ber s0. 4% f i ber s( t ape) , 25mm0. 2% f i ber s( t ape) , 50mm0. 4% f i ber s( f i br . ) , 25mm0. 2% f i ber s( f i br . ) , 50mm

Figure 16 Comparison between stress-strain behavior forspecimen reinforced with 50mm fibers (1000 denier)

placed at �w=0.2% and 25 mm fibers placed at �w =0.4%,��=70 kPa

4. CONCLUSIONS

The discrete approach for fiber-reinforced soil wasvalidated in this investigation using experimental datafrom a triaxial testing of both sand and clay. The effect offiber reinforcement on stress-strain behavior and shearstrength was investigated and compared with theanalytical results of the discrete approach. The mainconclusions drawn from this investigation are:

(a) The addition of fibers can significantly increase thepeak shear strength and limit the post peak strengthloss of both cohesive and granular soil. An increasein fiber content leads to increasing strain at failureand, consequently, to a more ductile behavior.

(b) The fiber reinforcement tends to restrain the volumedilation of the soil in drained condition, orequivalently, increase the positive water pressure inundrained condition.

(c) The peak shear strength increases with increasingaspect ratio. The strain at peak deviator stressincreases with increasing fiber aspect ratio.

(d) As predicted by the discrete framework, theexperimental results confirmed that the fiber-induced distributed tension increases linearly withfiber content and fiber aspect ratio when failure ischaracterized by pullout of individual fibers.

(e) Experimental results conducted using specimenswith a constant (�w·�) value show not only the sameshear strength but also display a similar stress-strain behavior.

(f) If good mixing can be achieved, fibers withcomparatively high aspect ratio can lead to lowerfiber contents while reaching the same targetequivalent shear strength, resulting in savings ofreinforcement material.

(g) Overall, for both sand and clay specimens, thediscrete approach was shown to predict accuratelythe shear strength obtained experimentally usingspecimens reinforced with polymeric fibers testedunder confining stresses typical of slopestabilization projects.

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