Strength and deformation characteristics of steel fibrous concrete beams

Strength and deformation characteristics ofshredded rubber tire – sand mixtures

Sompote Youwai and Dennes T. Bergado

Abstract: The volume of scrap tires, an undesired urban waste, is increasing every year. One of the possible alterna-tives for this waste is to use shredded tires alone or mixed with soil as a lightweight backfill. This paper presents theresults of triaxial tests on compacted shredded rubber tire – sand mixtures. The tests were carried out with differentmixing ratios of shredded rubber tires and sand. With an increasing proportion of sand in the mixture, the density, unitweight, and shear strength of the mixture increased, but the compressibility decreased. The dilatancy characteristics ofshredded rubber tires mixed sand were relatively similar to a cohesionless material and can be explained within a criti-cal state framework. A proposed constitutive model broadly captures the strength and deformation characteristics of ashredded rubber tire – sand mixture at different mixing ratios.

Key words: shredded rubber tires, triaxial testing, constitutive model.

Résumé : Les pneus de rebut constituent un résidu urbain indésirable dont le volume s’accroît chaque année. Un desusages possibles de ce résidu est l’utilisation des pneus déchiquetés seuls ou mélangés avec le sol comme remblai lé-ger. Cet article présente les résultats d’essais triaxiaux sur les pneus de caoutchouc déchiquetés mélangés ou non avecdu sable. Les essais ont été réalisés avec différents rapports de mélange de pneus de caoutchouc déchiquetés et desable. Avec une portion croissante de sable dans le mélange, la densité et le poids volumique du mélange ont aug-menté, mais la compressibilité a diminué et la résistance au cisaillement a augmenté. Les caractéristiques de dilatancedes pneus de caoutchouc avec un mélange de sable ont été relativement similaires à un matériau pulvérulent et peuventêtre expliquées dans le cadre d’un état critique. Un modèle constitutif tel que proposé peut englober en gros les carac-téristiques de résistance et de déformation d’un mélange de pneus de caoutchouc déchiquetés et de sable à divers rap-ports de mélange.

Mots clés : pneus de caoutchouc déchiquetés, essai triaxial, modèle constitutif.

[Traduit par la Rédaction] Youwai and Bergado 264

Introduction

The growing volume of used rubber tires has prompted in-terest in developing new ways to reuse or recycle them.Shredded used tires are now being used in landfill engineer-ing as subgrade reinforcement for constructing roads oversoft soil, as well as aggregate in leach beds for septic sys-tems, and as a substitute for leachate collection stone inlandfills (Park et al. 1993; Ahmed and Lovell 1992).Crumbed or shredded used tires are also being used as anenergy producing material, an admixture in bituminous con-crete, and in low-grade rubber products, such as truck bedliners, doormats, and cushioning foams (Jones 2001). Thewhole rubber tire can also be used as reinforcement in theconstruction of retaining walls and slopes (Garga andO’Shaughnessy 2000). One possible practical applicationconsists of using shredded tires alone or mixed with soil as alightweight material for embankment fill. The product of tire

shredding is usually referred to as “tire chips”; they are gen-erally between 12 and 50 mm in size and have most of thesteel belting removed. The term “tire shreds” or “roughshreds” are used for larger sizes (Lee et al. 1999).

The average specific gravity of tire chips has been shownto be 1.22, or about 57% of that of sand. The most signifi-cant factor controlling the unit weight of the mixed materialis the relative proportion of soil to tire chips. The efficiencyof packing is controlled by other factors, such as the sizeand shape of the tire chips (Edil and Bosscher 1994).

Shear strength is a fundamental mechanical property thatgoverns the stability of embankment structures. Loose tirechips have a friction angle of repose of 37–40°, whereas theangle of repose for compacted tire chips can be as high as85° (Edil and Bosscher 1994). These values imply that tirechips exhibit a higher friction angle of repose than normalsoil. The internal friction angle of shredded rubber tire –sand mixtures obtained from direct shear tests have beenshown to range from 25 to 65° (Edil and Bosscher 1994;Foose et al. 1996). The significant factors affecting thestrength of the mixed material are normal stress, shreddedrubber tire content, sand matrix, and unit weight. The inter-nal friction angle can be predicted using the concept of soilreinforced with random fibers (Gray and Al-Refeai 1986).

The deformation characteristics of rubber chips have beenstudied by Wu et al. (1997) who performed triaxial tests on

Can. Geotech. J. 40: 254–264 (2003) doi: 10.1139/T02-104 © 2003 NRC Canada

254

Received 23 April 2002. Accepted 29 October 2002.Published on the NRC Research Press Web site athttp://cgj.nrc.ca on 11 March 2003.

S. Youwai and D.T. Bergado.1 School of Civil Engineering,Asian Institute of Technology, P.O. Box 4, Klong LuangPathumthani 12120, Thailand.

1Corresponding author (e-mail: [email protected]).

I:\cgj\CGJ40-02\T02-104.vpThursday, March 06, 2003 10:00:45 AM

Color profile: Generic CMYK printer profileComposite Default screen

pure tire chips with different sizes, shapes, and gradations,although the sizes of the tire chips were significantly smallerthan those used in field applications (normally 50–75 mm orlarger in the field). The internal friction angles were seen tovary in the range of 45– 57°. Lee et al. (1999) carried outtriaxial tests on both pure tire chips and tire chips mixedwith sand. In these tests, the maximum size of tire chips wasabout 30 mm and the tires represented 40% by volume ofthe mix. Lee et al. (1999) developed a hyperbolic model todescribe the deformation characteristics of the rubber chip –sand mixture.

In conclusion, as seen from previous research, shreddedrubber tires can be employed as lightweight fill for embank-ment construction because of its high strength and low unitweight. However, the disadvantages of using shredded rub-ber tires alone as fill material are: high deformation, com-paction problems, and a self-heating mechanism. Addingsoil to shredded rubber tires not only improves the deforma-tion characteristics but also increases the strength of thebackfill and reduces the self-heating problem. However, in-creasing the proportion of soil in the mixture increases theweight of the mixed material, which can cause more settle-ment in soft foundations.

Previous studies on the strength and deformation charac-teristics of shredded rubber tires mixed with sand are notsufficient to permit the evaluation of appropriate constitutivemodels. To date, there has been no implementation of theelastoplastic constitutive model for shredded rubber tireswith sand backfill. In this research, triaxial tests of shreddedrubber tire – sand mixtures mixed in different ratios, wereconducted to investigate the strength and deformation char-acteristics and to provide guidelines for the selection of ap-propriate mixing ratios for practical use. An appropriateconstitutive model is proposed to simulate the deformationand strength characteristics of shredded rubber tire – sandmixtures at different mixing ratios.

Theoretical background

In this paper, the stress and strain parameters used in theanalysis are calculated as follows:

[1] q = −′ ′σ σ1 3

[2] p′ = +′ ′σ σ1 323

and

[3] η =′

qp

where σ1′ , σ2′ , and σ3′ are the principal effective compressivestresses, q is the deviator stress, p′ is the mean effectivestress, and η is the stress ratio.

The distortional strain, εs, and the volumetric strain, εv,are defined as

[4] ε ε εs 1 323

= −( )

and

[5] εv = ε1 + 2ε3

where ε1, ε2, and ε3 are the principal compressive strains.The axial strain, ε1, is defined as

[6] ε10

0

= ∂ =

∫ L

LLL

L

L

ln (compression being positive)

where L0 is the initial height and L is the current height.The volumetric strain εv is given by

[7] εv = ∂ =

∫ V

VVV

V

V0

0ln (compression being positive)

where V0 is the initial volume and V is the current volume.The dilatancy, d, is defined as:

[8] d = ∂ε∂ε

v

s| |

According to volumetric strain, the sign conventions of adilatancy are positive in a compressive increment (contrac-tion) and negative in an expansive increment (dilation), simi-lar to the sign convention of previous studies (e.g., Wang etal. 2002; Li and Dafalias 2000; Jefferies 1993).

Generally, soil has two characteristics that are apparent tothe casual observer: plasticity and density dependence(Jefferies and Been 2000). Plasticity is apparent because thedeformations of soil are largely irrecoverable. Density de-pendence is obvious because soil can exist over a range ofdensities at constant stress, and dense soil behaves quite dif-ferently from loose soil. Unlike cohesive soil, sand does notdemonstrate a unique relationship between its void ratio, e,and mean stress, p′, for a particular stress ratio, η (Li andDafalias 2000). The state parameter, Ψ was defined as fol-lows (Been and Jefferies 1985):

[9] Ψ = e – ec

where e is the current void ratio, and ec is the critical voidratio on the critical state line.

The parameter Ψ measures the offset of the current statefrom the critical state line at the same mean stress, p, in thee – ln(p) plane, Fig. 1. When the state of the soil is at pointA, the magnitude of the parameter Ψ is negative. This stateis defined as denser than critical. When the state of the soilis at point E, the magnitude of the parameter Ψ is positive.This state is defined as looser than critical. The tendency ofdilation at point A is higher than at point E. Moreover, whenthe state of soil is denser than critical (point A in Fig. 1), thephase transformation stress ratio, Md, peak angle of shearingresistance, and dilatancy, d, increase with increasing abso-lute magnitude of the parameter, |Ψ|, which is the distancefrom the critical state line to the current state of the soil(Been and Jeffries 1985; Li and Dafalias 2000). Thus, thisparameter strongly affects the strength and deformationcharacteristics of sand in the monotonic loading condition.

To clarify the stress, strain, and dilatancy characteristicsof sand, an example of a conventional drained triaxial com-pression test (CID) is shown in Fig. 1. The state of soil isdenser than critical at point A, and the soil is isotopicallycompressed to point A. In this case, the soil is subjected to

© 2003 NRC Canada

Youwai and Bergado 255



compression loading. The slope of the drained stress path inthe q–p plane is equal to 3. Initially, the magnitude of thedilatancy is positive (contractive). When the stress pathreaches point B at the phrase transformation state, Md, themagnitude of the dilatancy is changed to negative (dilative).Then, the stress path moves up to the peak stress ratio, Mb,and then drops down to critical state line, Mc. The criticalstate is not affected by the state parameter and soil fabric(Been et al. 1991), but the phase transformation stress ratio,Md, and the peak stress ratio, Mb, are mainly affected by thecurrent state parameter of the soil, Ψ, (Wood et al. 1994;Manzari and Dafalias 1997; Li and Dafalia 2000). With ahigher preshear mean stress, p′, the initial stress state is atpoint F, as shown in Fig. 1. The distance from the criticalstate line to point F is closer than to point A; this results in alower dilation compared to the case that has a lowerpreshear mean stress (point A).

Triaxial compression test

Isotropic consolidated drained (CID) tests were performedon samples with different mixing ratios of sand and shred-ded rubber tires. The testing program is summarized in Ta-ble 1. A total of 18 samples, consisting of shredded rubbertires and sand, mixed in the ratio of 20:80, 30:70, 40:60, and50:50 by weight, were used in the triaxial tests. Generally,the volumetric proportion of the geotechnical mixture con-trols the mechanical behavior of the mix. Thus, the mixingratios by volume of the shredded rubber tires and sand aretabulated in Table 1. The mixing ratios by volume were cal-culated based on the volume of particles for each mix as fol-lows:

[10] aGR

RG

RG

vsr

m

m

ss

m

sr

1

(100 )=

− +

where av is the mixing ratio by volume of shredded rubbertires in the mixture, Rm is the mixing ratio by dry weight ofshredded rubber tires in the mixture (%), Gsr is the specificgravity of shredded rubber tires, and Gss is the specific grav-ity of sand. An isotropic confining pressure was applied atthree different values, 50, 100, and 200 kPa, to cover thestress range in field applications of lightweight fill used inembankment construction.

Triaxial cell

A standard triaxial cell, capable of handling 100 mm di-ameter and 200 mm high samples, was used in a tempera-ture-controlled room. A schematic diagram of the triaxialtesting machine is shown in Fig. 2. The cell consisted ofthree principal components, namely: the base, the chambercylinder, and the top head of the cell. The base, which forms

© 2003 NRC Canada

256 Can. Geotech. J. Vol. 40, 2003

Fig. 1. Schematic illustration of the stress path for the drainedtriaxial compression test.

Fig. 2. Schematic diagram of the triaxial test setup.



the pedestal on which the sample rests, incorporated threedifferent connections. The cell chamber was filled withde-aired water and was pressurized to obtain the desired cellpressure. Two drainage lines were connected from the bot-tom of the sample. One of them was used in combinationwith a pressure transducer to measure the pore-water pres-sure in the undrained tests, and the other was used to flushwater to remove air bubbles during sample preparation. Thedrainage line was located in the top cap of the sample to ap-ply a back pressure in the drained test condition. Thirdly, atthe top part of the cell, a displacement transducer and a loadcell were installed to obtain the deformation and force actingon the specimens.

Materials

The materials used in the research were sand (Ayutthayasand) and shredded rubber tires. The index properties of thematerials are tabulated in Table 2. The particle size distribu-tions of both the shredded rubber tires and the sand intriaxial testing are illustrated in Fig. 3. The sample size ratiois defined as the diameter of the specimen divided by themaximum particle size. As the sample size ratio approachessix, the effects of sample size become negligible (Head1982; Indraratna et al. 1993; Marachi et al. 1972). Due tothe limitation of the maximum sample size ratio, the maxi-mum particle size of the shredded rubber tires used in thetriaxial tests was limited to 16 mm, smaller than those usedin the field application (normally 50–80 mm or larger). In

addition, the particle size and shape of the shredded rubbertire material was relatively uniform to eliminate anyanisotropic and internal reinforcing effects on conditions inthe mixed material.

Sample preparation

The sand was mixed with distilled water to obtain a watercontent of 7.5% and stored for 3 days to cure. For each

© 2003 NRC Canada


No.Mixing ratio rubber:sand(by weight, %)

Mixing ratio rubber(by volume, %)

Preshear meanstress p (kPa)

Preshear dry unitweight (kN/m3)

Preshearvoid ratio

Steadystate*

1 0:100 0 50 17.35 0.51 Yes2 0:100 0 100 17.46 0.50 Yes3 0:100 0 200 17.52 0.50 Yes4 20:80 37 50 15.23 0.36 Yes5 20:80 37 100 15.29 0.36 Mdil6 20:80 37 200 15.35 0.35 Yes7 30:70 50 50 14.43 0.30 Mdil8 30:70 50 100 14.54 0.29 Mdil9 30:70 50 200 14.77 0.27 Mdil

10 40:60 61 50 13.08 0.31 Dil11 40:60 61 100 13.14 0.30 Dil12 40:60 61 200 13.18 0.30 Dil13 50:50 70 50 11.51 0.37 Dil14 50:50 70 100 11.86 0.33 Dil15 50:50 70 200 12.42 0.27 Dil16 100:0 100 50 6.72 0.68 Mdil17 100:0 100 100 7.03 0.61 Mdil18 100:0 100 200 7.37 0.53 Mdil

*Yes, steady state apparently reached; Dil, sample still dilating at the end of the test; Mdil, small amount of dilation, sample close to steady state atend of test.

Table 1. Summary of testing program.

Material Specific gravity Effective size, D10 (mm) Classification USCS

Ayutthaya sand 2.67 0.27 SPShredded rubber tires 1.15 5.00 —

Table 2. Index property of tested materials.

Fig. 3. Particle size distribution.



designated mixing ratio, the proportion of each material wasdetermined. The material weight for each fraction wasblended thoroughly in a container prior to the compaction ofeach layer. A vacuum split mold (100 mm diameter ×200 mm high) consisting of two half-cylinders was assem-bled and placed on the bottom part of the triaxial cell. Clear-ance was provided between the bottom of the mold and theflange of the bottom cap. The rubber membrane was in-stalled in the split vacuum mold before compacting the sam-ple.

Preparation of the samples used in the research was basedon the method for preparing test specimens usingundercompaction (Ladd 1978). The hammer weight used tocompact the samples was 24.5 kN with a 304.8 mm dropheight (ASTM 1991). The sample material was divided intofive layers. To ensure uniform compaction energy in eachlayer, the blows per layer were varied from 21 blows for thefirst layer to 25 blows for the fifth. Samples were then com-pacted by a dynamic method equivalent to standard Proctorenergy. After compacting all the material into the mould, thetop cap was placed on the top surface of the sample in sucha way that the clearances between the end of the mould andthe flange of the top cap were the same as those at the bot-tom.

Testing procedure

Each specimen was saturated prior to consolidation. Afterassembling the triaxial cell, it was filled with de-aired water.A low emulsification oil was poured on top of the cell to re-duce the piston friction and to seal the chamber. The cellpressure was increased to 30 kPa for stability of the sample.This low confining pressure minimized unrecorded volumechanges during the saturation stage. Following this, the sam-ple was first flushed by allowing de-aired water to flow fromthe bottom of the sample to push any air bubbles out throughthe top drainage line. This was continued until all large airbubbles were flushed out. Both the cell pressure and backpressure were raised slowly and simultaneously to 210 and200 kPa, respectively, by using a Bishop ram. After this, thecell pressure and back pressure were raised to the desiredvalues, and both pressure lines were connected to aself-compensating mercury control device. The saturationwas checked by measuring the pore pressure parameter, B,(ratio of the change in pore-water pressure to the change incell pressure).

An isotropic consolidation stress was then applied to thesample by increasing the cell pressure while maintaining theback pressure constant at 200 kPa. A time span of at least2 h was allowed for any creep deformation to be completed.At the end of consolidation, the axial load was increased at aconstant rate of axial strain until an axial strain of 25% wasreached or failure occurred. A strain-controlled rate of shear-ing of 0.19% strain/min was used during testing. The choiceof the strain rate was made on the basis of 95% minimumpore-water pressure dissipation during loading (Head 1982).During shear, data of axial deformation and axial load wererecorded from the dial gauges and load cell, respectively.

Testing results

The average preshear dry unit weight of the mixed mate-rial increased linearly with increasing amounts of sand in themixture, as shown in Fig. 4. The unit weight of the shreddedrubber tire – sand mixture was found to be less than that ofcompacted sand by about 13–60%, depending on the mixingratio. Unlike sand, the deformation characteristics of theshredded rubber tires consisted of two components, namely:the particle compression and the particle rearrangement.Generally, the void ratio is defined as the ratio between thevolume of void and the volume of solid. For shredded rubbertires, the apparent volume change is caused not only by thevolume change of the pore spaces but also by the high defor-mation of tire chip particles. However, it is difficult to sepa-rate these two deformation components. Thus, in this paper,these two deformation components were employed in thecalculation of the void ratio, and the calculated void ratio ofthe mixture is apparently lower than that of the normal soil.The preshear void ratios of each mix ratio were calculatedfrom the preshear dry unit weight, as tabulated in Table 1.

Strength characteristics

The relationships between the maximum major principalstress, σ1, and the minor principal stress, σ3, are illustrated inFig. 5. The strength of the shredded rubber tire – sand mix-ture was increased linearly with increasing confining pres-sure (σ3). The strength of the shredded rubber tire – sandmixture increased linearly with increasing confining pres-sure (σ3). The strength of the mixed material increased withincreasing amounts of sand in the mix because the shearstrength of sand is higher than that of the shredded rubbertires. The peak internal friction angle was found to varyfrom 30 to 34° with increasing proportions of sand in themix. There was also an apparent cohesion intercept resultingfrom the compaction effort.

The strength characteristics obtained in the research wereless than those recorded by previous investigators (Lee et al.1999; Wu et al. 1997) because of the different sizes and

© 2003 NRC Canada


Fig. 4. The dry unit weight and modified compression index(λ*) with different mixing ratios of the shredded rubber tire –sand mixture (by weight).



shapes of shredded rubber tire chips and the different samplepreparation procedures. The size of shredded rubber tirechips used in this research was relatively uniform and theaspect ratio, defined as the ratio of length to width of the tirechips, was approximately equal to unity. In this situation, theshredded rubber tire chips could not act as reinforcement asin the concept of sand reinforced with random fibers (Fooseet al. 1996). The influence of different compaction proce-dures is commonly attributed to the different forms of soilfabric (Sivakumar and Wheeler 2000). The peak strength ofthe granular material depends on the void ratio and the soilfabric. Accordingly, the different compaction procedures canproduce various magnitudes of strength.

Deformation characteristics

The relationships between the increment of volumetricstrain, ∆ε, and the effective mean stress, ∆p′, of the mixedmaterial can be expressed as a logarithmic regression as fol-lows:

[11] ∆εv = λ* ∆[ln(p′)]

The values of the modified compression index λ*(Vermeer and Brinkgreve 1995), which is the compressionratio divided by 2.303 (CR/2.303), of the shredded rubbertire – sand mixture are presented in Fig. 4. With an increas-ing proportion of shredded rubber tires in the mixed mate-rial, the modified compression index λ* increased. Eventhough rubber chip particle has low compressibility becausethe Poisson’s ratio is very close to 0.5 causing high bulkmodulus, the shear modulus of rubber chip particle is lowand, consequently, deforms (distorts) easily. When subjectedto stress, a rubber chip particle in the mix is deformedthereby reducing the void between the rubber tire chip parti-cles. Thus, the rubber shredded tire – sand mixtures have

lower compressibility compared to pure sand. Themagnitude of modified compression index increased with in-creasing mixing ratios from 30:70 to 40:60 by dry weight ofshredded rubber tires and sand because the volume of shred-ded rubber tires in the mixture increased as tabulated inTable 1. The behavior of mixed material was changed froma sand matrix with reinforced shredded rubber tires to shred-ded rubber tires with sand filling the voids. These resultsprovide potential guidelines for selecting the optimum mixratio between sand and shredded rubber tire chips. However,the weight of the backfill should be considered in selectingthe mixing ratio because increasing the proportion of sandincreases the weight of the mixed material.

The stress–strain relationships of shredded rubber tires areillustrated in Fig. 6. The relationships between the deviatoricstress, q, and the distortional strain, εs, were relatively linearwith the distortional strains being less than 10%. At highconfining pressures, there was no failure even though thedistortional strain reach 20%. For all of the confiningstresses, the relationship between the volumetric strain andthe distortional strain was linear with the distortional strainsless than 10%. From the critical state framework, the stressstate of the compacted specimen is at point A, which isdenser than the critical state as shown in Fig. 1. The distancebetween the critical state line and the current soil state is re-duced with increasing effective mean stress, p′ (at point E).Due to the reduction of the absolute value of the state pa-rameter, |Ψ|, the dilatancy decreased with increasing confin-ing stress. Due to the high deformation of the rubber tirechips, the initial dilatancy, d (which is the slope of the rela-tionship between the volumetric strain and the distortionalstrain), increased with increasing confining stress.

The stress–strain relationships of the sand with differentconfining stresses are plotted in Fig. 7. The peak strengthdeveloped at distortional strains, εs, of 1.7, 2.5, and 3.5% for

© 2003 NRC Canada


Fig. 5. The relationship between the maximum major principal stress and the minor principal stress of the shredded rubber tire – sandmixture at different mixing ratios.



confining stresses at 50, 100, and 200 kPa, respectively. Thevolumetric strain was initially compressive and then expan-sive. The tendency to maximum dilation (expansion) de-creased with increasing confining stress. Compared toshredded rubber tires alone, the distortional strain at thepeak strength of the sand was less than that of the shreddedrubber tires due to the high deformation of the rubber tirechips. In addition, the magnitude of the maximum positivevolumetric strain (compression) was significantly less thanthe shredded rubber tire samples. Therefore, combining sandwith shredded rubber tires can improve the deformationcharacteristics of the shredded rubber tires.

The stress–strain relationships of the shredded rubbertires – sand mixtures at different mix ratios are plotted inFigs. 8–11. Due to the high deformation of the rubber chipparticles, the magnitude of the distortional strain, εs, at thepeak deviator stress, q, and at the phase transformation stateincreased with an increasing proportion of rubber tire chipsin the mixture. Consequently, the contraction increased andthe dilation after the peak strength decreased. The dilatancyvalue, which is the increment of the volumetric strain (εv) di-vided by the increment of distortional strain (εs), consistedof both an initial positive value (compression) and then anegative value (expansion). The dilatancy characteristicswere similar to the behavior of cohesionless material, having

both negative and positive magnitudes of dilatancy. Withincreasing confining pressure, the tendency of dilatancy de-creased with increasing confining stress due to the reductionin the absolute value of the state parameter, |Ψ|. The phasetransformation stress ratio, Md, of all of the mixes was lessthan the critical state, Mc, and peak stress ratio, Mb.

Overall, the deformation characteristics of the shreddedrubber tire – sand mixtures were relatively similar to sandand can be explained by using the critical state frameworkwith the state parameter, Ψ. The deformation portion of therubber chips can be classified into two portions, namely: de-formation of the tire chip particles and rearrangement of theparticles. This is different from the common characteristicsof soil wherein deformation mainly occurs from the reloca-tion of particles, with negligible deformation of the soil par-ticle. Thus, the critical state for the mixed material isdifficult to determine because of the deformation of the tirechip particles at the steady state. Even for the sand, the truesteady state is difficult to determine and exists in the labora-tory only for uniform clean sand. It is difficult to verify itscorrectness (Atkinson and Bransby 1982; Poulos et al. 1985;Been et al. 1991). From the test results, a mild tendency todilate at the end of the test is shown in Table 1. In thesecases, the condition at the end of the test was postulated tobe represented by a steady state condition.

© 2003 NRC Canada


Fig. 6. Comparison between the drained triaxial compression testand the model simulation of shredded rubber tires.

Fig. 7. Comparison between the drained triaxial compression testand the model simulation of sand.



From the triaxial test results, mixing sand into the voidsof the shredded rubber tires not only improved the deforma-tion characteristics but also slightly increased the strength ofthe mixed material. However, as lightweight backfill for em-bankment on soft ground foundation, an increasing propor-tion of sand in the mixture increases the dry unit weight andinduces settlement in the soft foundation. Thus, theappropriate mixing ratio of shredded rubber tires and sandshould be well selected based upon the optimum unit weightof the backfill and the desirable deformation characteristicsof the foundation.

Constitutive model

As part of the research, a constitutive model was devel-oped to simulate the stress–strain characteristics of shreddedrubber tire – sand mixtures at different mixing ratios. In pre-vious research, the deformation characteristics of shreddedrubber tires were simulated by using a hyperbolic model(Lee et al. 1999). However, the hyperbolic model cannotcapture the overall dilatancy characteristics, which are bothpositive and negative.

The concept of critical state (Roscoe et al. 1958) has beensuccessfully applied to simulate the stress–strain characteris-tics of geomaterials in both cohesive and granular materials.

Constitutive models based on the critical state frameworkfor sand have been proposed by many investigators (e.g.,Jefferies 1993; Wood et al. 1994). For shredded rubber tiresmixed with sand, the deformation characteristics of themixed material increase the tendency to elastic behavior.Therefore, an appropriate model should contain both elasticand plastic characteristics. The hypoplasticity model, basedon the critical state framework, was employed to simulatethe stress–strain characteristics of the shredded rubber tire –sand mixtures. The model used was proposed by Li andDafalias (2000). The yield criteria based on the concept thatplastic deformation occurs whenever the stress ratio, η, ex-ceeds its historic maximum and a constant stress ratio pathinduces no plastic deformations. The yield locus, f, conicalshape in stress space, can be expressed as

[12] f = q – ηp′ = 0

From the theory of plasticity (Dafalias 1986), the generalequations are illustrated as follows:

© 2003 NRC Canada


Fig. 8. Comparison between the drained triaxial compression testand the model simulation of the shredded rubber tire – sandmixture at a mixing ratio of 20:80 (by weight).

Fig. 9. Comparison between the drained triaxial compression testand the model simulation of the shredded rubber tire – sandmixture at a mixing ratio of 30:70 (by weight).



[13]∂∂

qp

G

K′

=

3 0

0

−+ −

−−

1

3

9 3

3

2

2K G K d

G KG

KGd K dp

s

vηη

η∂ε∂ε

where ∂q is the increment of deviatoric stress; ∂p′ is the in-crement of mean stress; G is the elastic shear modulus; K isthe elastic bulk modulus; Kp is the plastic modulus; η is thestress ratio (q/p); d is the dilatancy; ∂εs is the increment ofdistortional strain, and ∂εv is the increment of volumetricstrain.

The elastic part is defined as follows:

[14] G Ge

ep p= −

+′0

2

1( )2.97

a

[15] K Gvv

= +−

2 13 1 2

( )( )

where Go is the initial elastic shear modulus; e is the voidratio; pa is the atmospheric pressure; and ν is Poisson’s ratio.

The plastic part is defined as follows:

[16] K hGM n

p e= −

ηΨ

[17] d dM

m= −

0 e Ψ η

where Kp is the plastic modulus; h, m, and n are the modelparameters; M is the critical stress ratio; Ψ is the state pa-rameter; d0 is the initial dilatancy, and η is the stress ratio.

The state parameter, Ψ, of the soil was based on the dif-ferent void ratios between the critical state and the currentstate of the soil, as shown in Fig. 1. The critical state linewas postulated to be a linear relationship between the voidratio, e, and the logarithmic of the mean stress, ln(p), Thus,based on eq. [9], the state parameter, Ψ, can be expressed asfollows:

[18] Ψ = e – [e τ – λc ln(p)]

where λc is the slope of the critical state line and eτ is thecritical void ratio when the mean stress is equal to 1.

For shredded rubber tires, the initial dilatancy increasedwith increasing mean stress. Thus, eq. [15] was modified asfollows:

© 2003 NRC Canada


Fig. 10. Comparison between the drained triaxial compressiontest and the model simulation of the shredded rubber tire – sandmixture at a mixing ratio of 40:60 (by weight).

Fig. 11. Comparison between the drained triaxial compressiontest and the model simulation of the shredded rubber tire – sandmixture at a mixing ratio of 50:50 (by weight).



[19] d kpp M

m=

−

d

a

e Ψ η

where kd is the model parameter and other variables are asdefined previously. The selected parameters for the constitu-tive model were independently determined for each mixingratio based on the procedure proposed by Li and Dafalias(2000), as tabulated in Table 3. The critical state parameters,consisting of M, eτ, and λc were determined by directly fit-ting the test data for the critical stress ratio and the criticalstate line in the e – ln(p) plane. The parameter m can be de-termined by the phrase transformation state at which d = 0.The equation was solved based on eq. [15] as follows:

[20] mMM

d=

1

Ψd

ln

The parameter n was determined at the drained peak stressstate, at which Kp = 0. The equation was solved based oneq. [14] as follows:

[21] nMMb b

=

1Ψ

ln

The parameter, d0, was calibrated based on the εs–εv curvesolved from eq. [15] as follows:

[22]dd

ev

q0

εε

η= −

dM

mΨ

For the shredded rubber tires, the parameter k was cali-brated based on the εs–εv curve solved from eq. [16] as fol-lows:

[23]dd

ev

sd

a

εε

η=

−

kpp M

mΨ

The parameters, h and Go, were calibrated independentlybased on experimental q–εs curves as follows (Li andDafalias 2000):

[24]∂∂ε

η

ηq

hG

e ppM

e

n

s0

a2.91 e

=

−

−

+ −

( )

( )

2

1 13

Ψ

Finally, Poisson’s ratio, ν, was back-calculated from thetest results.

The differential equations in eqs. [11]–[17] were solvednumerically in a forward difference manner with updatingvoid ratio dependence variables (e.g., Ψ, d, Kp, G, and K) ateach step. The comparison between the experimental resultsand the model simulation are illustrated in Figs. 6–11. It canbe seen that the model simulations broadly matched the ex-perimental results at different confining pressures and mixratios. The effectiveness of the critical state framework inconjunction with state-dependent dilatancy can simulate thedeformation and strength characteristics of shredded rubbertire – sand mixtures. Compared to others models (e.g., Leeet al. 1999), the proposed model can better capture overallthe dilatancy characteristics, including both negative andpositive values. The hyperbolic model can only simulate thepositive value of the dilatancy (compression) and can onlybe employed for a particular state of material (Duncan andChang 1970).

Conclusion

Drained triaxial compression tests were conducted onshredded rubber tire – sand mixtures mixed at different ra-tios. With an increasing proportion of sand in the mix, thestrength and unit weight increased and deformation due toisotropic compression decreased. The deformation was sig-nificantly reduced when the sand in the mixture was morethan 30%. When shear, the mixed material displayed bothcompression and expansion dilatancy values depending onthe ratios of the mixtures. The strength and deformationcharacteristics of shredded rubber tire – sand mixtures canbe described by the critical state framework with the statedparameter. The proposed hypoplasticity model can model thestrength and deformation characteristics of shredded rubbertire – sand mixtures.

Acknowledgements

The research behind this paper was supported by the Na-tional Metal and Material Technology Center (MTEC), Thai-land. The authors would like to express their gratitude to Dr.Suksan Horpibulsuk of the Suranaree University of Technol-ogy, Thailand, for his valuable comments and inspiring dis-cussions. The thoughtful comments from anonymousreviewers are also gratefully appreciated.

References

Ahmed, I., and Lovell, C.W. 1992. Use of waste materials in high-way construction: State of the practice and evaluation of the se-lected waste products. Transportation Research Record

© 2003 NRC Canada


Mixing ratiosand:rubber Go v h n M eτ λc D0 kd m

0:100 1.9 0.33 3.5 3.0 1.40 1.13 0.1102 — 0.5 12.0050:50 4.0 0.33 1.3 3.0 1.37 0.60 0.0460 0.80 — 1.2060:40 6.0 0.33 1.3 0.6 1.35 0.66 0.0302 0.80 — 0.4570:30 10.0 0.33 1.3 1.8 1.35 0.53 0.0302 0.65 — 2.0020:80 18.0 0.33 1.3 4.0 1.35 0.44 0.0102 1.00 — 4.00100:0 80.0 0.33 1.3 4.0 1.35 0.61 0.0132 1.70 — 0.50

Table 3. Model parameters.



© 2003 NRC Canada


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