the relationship of the unsaturated soil

7/29/2019 The Relationship of the Unsaturated Soil

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81 7

An equation o representgrain-sizedistributionMurray D. Fredlund, D.G. Fredlund, and G. Ward Wilson

Abstract: The grain-sizedistribution s commonly used or soil classification;however, here is also potential to usethe grain-sizedistributi on as a basis fo r estimating soil beha viour.For example,much emphasishas recently beenplacedon the estimation of the soil-water characteri stic urve. Many methods proposed n the literatureuse the grain-size distribution as a startingpoint to estimate he soil- water characteris tic urve. Two mathematical orms are pre-sented o representgrain-sizedistributi on curves, namely, a unimodal form and a bimodal form. The proposedequa-tions provide methods or accurately epr esenting niform, well-grad ed soils, and gap-gradedsoils. The five-parameterunimodal equationprovidesa closer fit than previous wo-parameter, og-normal equationsused to fit uniform and well-graded soils. The unimodal equa tion also improves representation f the silt- and clay-sizedportions of the grain-sizedistribution curve.Key words: grain-sizedistribution, sieve analysis,hydrometer analysis, soil classification,probability density function.R6sum6 : La distribution granulom6triqueest utilis6e courammentpour la classificationdes sols; cependant, l es t pos-sible d'utiliser 6galement a distributiongranulom6trique omme base d'6valuation du comportementdu sol . Parexemple,beaucoupd'emphasea 6t6 mise r6cemmentsur la d6terminationde la courbe caract6ristique ol-eau.Plusieursm6thodesproposdes ans a litt6ratureutilisent la distributiongranulom6trique omme point de d6partpour 6tablir lacourbe caract6rist ique ol-eau.Deux formes math6matiques ont pr6sent6es our reproduire es courbes de distributiongranulom6triqe:nomm6ment, une forme unimodal e et un e lbrme bimodale. Les 6quati onspropos6es ournissentdesmdthodespour repr6senter vec pr6cisiondes sols I granulom6trieuniforme, 6tal6eet discontinue.U6quationunimodale cinq parambtres ournit une meilleure concordance ue les 6quationsant6rieures og normales deuxparambtres tilis6espour reproduire es courbes des sols ir granulom6trieuniforme et 6tal6e.L'6quation unimodaleam6liore aussi a repr6sentation es portions de silt et de grosseursargileusesde la cour be de distributi ongranulom6trique.Mots clls : distribution granulom6trique, nalysepar tamisage,analyseh l'hydromEtre, classificationdes sols, fbnctionde densi td robab i l is t ique .[Traduit par la R6dactionl

IntroductionThe grain-sizedistribution s a simple,yet informative estroutinely performed n soil mechanics o classify soils. Re-cent research as madeuse of the grain-sizedistribution as abasis for the estimation of other soil propertiessuch as thesoil-water characterist ic urve through mathematicalanaly-si s (Guptaan d Larson 1979a,1979b;Arya and Paris 98l;Haverkamp and Parlange 1986). Mathematically represent-ing the grain-size distribution provides several benefits.First, the soil may be classifiedusing the best-fitparameters.Second, he mathematicalequationcan be used as the basis

for analysis elated o estimating he soil-watercharacteristiccurve. Third, a mathematicalequationcan provide a methodof representing he entire curve between measured datapoints. Representinghe soil as a mathematical unction alsoprovides ncreased lexibility in searching or similar soils ndatabases.

ReceivedNovember 20, 1998. Accepted December3, 1999.Publishedon the NRC ResearchPresswebsite on Ausust 4.2000.M.D. Fredlund, D.G. Fredlund, and G.W. Wilson.Departmentof Civil Engineering,University of Saskatchewan,Saskatoon,SK S7N 5A9, Canada.

American Society for Testing and Materials standardsDl140-54 and D422-63 (ASTM 1964a, 1964b) provide abasic testing and reporting method whereby the results of asieve and hydrometer analysis are plotted on a semilog-arithmic graph. An interpretati on method for the series ofplotted points s specified n the procedure.Manual interpre-tation methods,such as sketching n a completecurve, haveoften been used o providea completegrain-sizedistributioncurve. Gardner 1956)proposeda two-parameter,og-normaldistribution to provide representation f grain-sizedistribu-tion data.Both methodsare feasiblebut have imitations thatare discussed ater in the paper.This paperproposes wo new models o fit grain-sizedata,namely, he use of a unimodal and a bimodal mathematicalfunction. The two new equationsprovide greater lexibilityfor fitting a wide variety of soils.

BackgroundNumerous methodshave been developed or particle-sizeanalysis in the laboratory and field. These include theelutriation method, the test tube shaking method, theWiegner sedimentationcylinder, the photoelectric method,the pipette method, and the hydrometermethod, n additionto the sieveanalysis.Of thesemethods,only the pipetteand

Can.Geotech. .37: 817-827 2000\ @ 2000 NRC Canada


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Fredlund t al.bimodal or gap-graded s of value and the four-parameterlog-normal equationshave not been ound to be satisfactoryfor fitting these ypes of grain-sizedistribution.There are three general categoriesof grain-size distribu-tions (Holtz and Kovacs 1981): well-graded soils, uniformsoils, and gap-graded soils. This paper focuses on thesethree categories f grain-sizedistribution and providesequa-tions to fit the experimental data for each category. Thewell-graded and uniform soils are examined using aunimodal method of fitting an equation, and then a mathe-maticalmeansof representing gap-graded oil is presented.

Unimodalequationfor grain-sizedistribution dataThe selectionof an appropriate,mathematical quation n-volved a review of a variety of equations hat could be usedto frt soils data. It has been observed hat the soil-water char-acteristic curve possesses shape similar to that of thegrain-size distribution curve. This is probably to be ex-

pected,since the soil-water characteristiccurve provides arepresentation f the void distribution n a soil, whereas hegrain-sizecurve provides nformation on the distribution ofthe solid phase of the soil. Since the solids plus the voidsadd up to the total soil volume, it is to be expected hat thedistribution of the solids phase(i.e., grain-sizedistribution)would tend to bear an inverse elationship o the distributionof voids (i.e., representedby the soil-water characteristiccurve), and vice versa.A summary of severalof the equations hat have beenused o fit the soil-watercharacteristic urve is given in Ta-ble l. Brooks and Corey (1964) and Gardner (1974) pte-sented hree-parameter quationsand van Genuchten 1980)and Fredlund and Xing (1994) presented our-parameterequations. t would seem easonablehat a form of equationsimilar to thoseshown n Table I could be used o representthe grain-sizedistribution.An accurate representation of the clay fraction of thegrain-sizedistribution was considerednecessaryo completethe mathematical function. Since the Fredlund and Xing(1994) equation allows independentcontrol over the lowerend of the curve (i.e., the fine particle size range), t was se-lected as the basis or the development f a grain-sizedistri-bution equation. The reversed scale of the grain-sizedistribution and characteristics nique to the grain-sizedis-tribution required he original Fredlundand Xing equation obe modifred to the form shown as follows:tll Pe(A

8 1 9

Fig. 1. Grain-sizedata fit with unimodal equation or a clayeysilt: (a ) best-fit curve, R2 = 0.998; (b ) arithmetic probability den-sity function; (c ) logarithmic probability density function (soilnumber 10030).

so o u.E6o 4 n .of?3 2 0

(a ) to o80

(c) to o80

0 ]*0.0001

(b) oooo50004000E

: 3000E- 2000

0 .001 0 .01 0 .1 1Particle ize mm)

*M.Fredlund Unimodal - - USCS o/oClay -- - USCS % Sandr Experimental -' - USCS % Sil t

100000.0001 0. 001 0. 01 0 .1 1 10 100

Part ic lesize (mm)-* - Part ic le-size rithmetic -. - USCS % Silt . Part ic le diameter- ' - U S C S % C l a Y - - - U S C S % S a n d

Eo A nc v v'aoo , ^z + uocoo- 20

0 + -0.0001

wnerePo(d) is the percentage, y mass, of particles passingpartrcularslze;

an, s a parameterdesignating he inflection point on thecurie and is related to the initial breaking point on thecurve:rzn, s a parameter related to the steepestslope on thecurie (i.e., uniformity of the particle-sizedistribution);

Part ic lesize (mm)-----. Part ic le-size og PD F r Experimental - - USCS % Sit t' ' ' ' . M . F red lund n imoda l - -USCS % Clay - - - - - " -USCS% Sandt t (- \ ' . .11-"j tn lexn t t r . [? ll it L . l l

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Fredlundet al.

Fig. 2. Grain-sizedata fit with unimodal equation or a siltysand: (a ) best-fit curve, R2 = 0.985; 1b; arithmetic probabilitydensity function; (c) logarithmic probability density function (soilnumber 3).( u ) t o o i

BO604020

;0 ;0.0001 0.001 0 .0 1 0 .1Particlesize (mm)-M.Fredlund Unimodal - - USCSo/ oClay -- - USCS % Sand

r Experimental - - USCS%Sil t

(b; +oo350300

821

Fig. 3. Grain-size data fit with unimodal equation for a sandyclay: (a ) best-fit curve, R2 = 0.999; (b ) arithmetic probabilitydensity function; (c) logarithmic probability density function(soi l number 11648).(a) 1oo

80

00.0001 0.001 0.01 0.1 '1 10 100Particle ize mm)

*M.FredlundUnimodal . - USCS% Clay -- - USGS% Sandr ExDerimental - - 'USCS % Silt(b; sooo4500400035003000

2500200015001000500

00.0001 0 .001 0 .01 0.1 1 10Particle-sizemm)

- Part ic le-size ri thmeticPD F _ . _ .USCS % Silt- . . - U S C S % C l a y - - U S C S % S a n d

(c ) 1o o80

0.0001 0.001 0.01 0. 1 '1 10 10 0Particle ize mm)

------.Particle-sizeogPDF r Experimental - . - .USCS% Sitt' ' '" [/.Fredlund nimodal - - USCS% Clay -'-"' USCS%Sand

curve. The parameter ' is related o the initial break of theequation and is more precisely the inflection point on thecurve. Its effect on the grain-sizedistribution curve can beseen n Fig.5a, where dn. s varied from 0. to 10 while theother equationparameteisare held constant.The parameterar . provides an indication of the largestparticle sizes.

EcEE 250200

1501 00500.0001 0 .001 0 .01 0.1 1 10

Particle ize mm)- Part ic le-size ri thmeticPD F _ . _ .USCS % Silt- . . U S C S % C l a y - - ' U S C S % S a n d

(c ; t oo

0

so o u.E8 + ozoP8 2 0

;


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82 2Figure 5b shows how the parametern* influences he slopeof the grain-sizedistribution.The point of maximum slopealong he grain-sizedistributionprovidesan ndication of t hedominant particle size (i.e., on a logarithmic scale) n thesoil. In the parametric epresentation, ' is varied from I to 4.The parameterzr. influences the break onto the finer par-ticle size of the sample. The effect of varying the parame-term% rom 0.3 to 0.9 can be seen n Fig.5c. Th e parameterd.n,alfects the shape along the finer particle size portion ofthe curve.However, he influenceon the curve s quite mini-mal as shown n Fig. 5d. In somecasesdrn,can be modifiedto improve the fit of the overall equation.With the best-fitanalysisshown,drn.was adjustedmanually o improve he fitof the curve to the data. t was found that a value of 0.001for dr* provided a reasonable it in most cases.

Bimodal equation for the grain-sizedistribution curveThere s a limitation in using the unimodal equation i.e.,eq. [1]) when the soils are gap-graded s shown n Fig. 6. Inthis case, t is necessary o consider the use of a bimodalequation when performing the best-fit analysis. Soils fre-quently have particle-sizedistributions hat are not consis-tent with a unimodaldistribution and, as a result, attempts ofit the unimodal equation to certain data sets can often leadto a misrepresentation f the characterof the particle-sizedistribution.This is particularly mportantwhen the equation

Can. Geotech . . Vo l .37 ,2OOO

Fig. 4. Variation of R2 enor as the amount of fines representedin a soil increases.

10.990.980.97

N ^ ^ ^E u.Yo0.950.940.930.92

!F6mF;mn.% o o o'.xw a' oo ooe o

5 o ^ o o, a o t8'

o

0.1 1Particle ize mm)

20 40 60 80 10 0%ClaY

is differentiatedand used for further analyses e.g.,estima-tion of the soil-water characteristic urve).The characterist ic hapeof a bimodal or gap-graded oi l isthe double "hump" often observed from experimental data.These humps indicate that the particle-size distribution isconcentrated around two separate particle sizes. From amathematical tandpoint,a gap-graded oi l can be viewed asa combinationof two or more separate oils (Durner 1994).This allows for the "stackins" of more than one unimodalequation:

Fig. 5. Parameter ariation: (a ) effect of varying the parametera* . while nr,= 4.0,mr,= 0.5, d,er= 1000, andd^= 0.001; (b ) effect ofvarying the parameter r. while a*. = 1.0,mr , = 0.5, d,g.= 1000, and d. = 0.001; (c ) effect of varying the parameterm* while ar, =1.0, ?sr 4.0, dry,= 1000, andd, - 0.001; (d ) effect of varying the parameter .r , while a*. = 1.0,nr,= 4.0,mr,=0.5, andd- = 0.001.

E.B oo6oE a , no -o

20

100(a)80

'100(c )80

10 0(b)80

;


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i :;' : ^--^ r--ti i /. r f ,

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the relationship of the unsaturated soil

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