Use of Grain-Size Functions in Unsaturated Soil Mechanics

8/17/2019 Use of Grain-Size Functions in Unsaturated Soil Mechanics

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Use of Grain-Size Functions in Unsaturated Soil Mechanics

Murray D. Fredlund

G. Ward Wilson2

Delwyn G. Fredhmd2

A b s t r a c t

The grain-size distr ibution is commonly used for soil classification; however, there is

also potential to use the grain-size distribution as a basis for estimating unsaturated

soil behavior. Mathematically representing the grain-size distribution provides

several benefits to soil mechanics. An example is the estimation of the soil-water

characteristic curve from the grain-size distribution. Much emphasis has recently

been placed on the estimation of unsaturated soil-property functions from the soil-

water characteristic curve. Several methods have been proposed that use the grain-

size distribution as the basic information for the estimation of the soil-water

characteristic curve.

Two mathematical forms are presented to represent grain-size distribution

curves; namely, a tmimodal form and a bimodal form. The equations presented in

this paper can provide a close representation of a wide variety of grain-size

distribution for different soil types.

I In t roduc t ion

The grain-size distribution is a simple, yet informative classification test routinely

performed in soil mechanics. Valuable information regarding the amount of each

particle size can be determined in the laboratory through the use of a series of sieves

and hydrometer analysis. Recent research has made use of the grain-size distribution

as the basis for the estimation of soil properties such as the soil-water characteristic

curve Gupta and Larson, 1979; Arya and Paris, 1981; Haverkamp and Parlange,

1 9 8 6 . I t

is of value to be able to mathematically represent the grain-size distribution

curve as a continuous function that will allow further analysis to be performed. The

Graduate student Departmemof Civil Engineering,Universityof Saskatchewam Saskatoora Sask.,

STN 5A9

: Professor of CivilEngineering,Departmentof Civil Engineering,Universityof Saskatchewan,

Saskatoom Sask., S7N 5A9

69

Advances in Unsaturated Geotechnics

D o w n l o a d e d f r o m a

s c e l i b r a r y . o r g b y U n i v

e r s i t y o f T e x a s a t A r l i n g t o n o n

1 2 / 0 4 / 1 5 .

C o p y r i g h t A S C E .

F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e

s e r v e d .


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70 ADVAN CES N UNSATURATEDGEOTECHNICS

spec i fic ob jec t ive o f this paper i s to de ve lop a m athemat ica l desc r ip tion for the gra in-

size distribution o f an y given soil . This form s the foun dation step fo r the general

proced ure to determ ine re lated soil prop er ty functions an d cons t i tut ive re la t ions.

M athe m atica lly representing the grain -size distribution pro vid es several

ben efits to soil mechanics. Firstly, the soil can be classified using the best-fit

parameters . Seco ndly, the mathe m atical equation can be used as the basis for fur ther

soil analysis and descr ipt ion. A n exam ple is the est imation o f the soi l-w ater

characteristic curv e fro m the grain-size distribution (Fred iund et al, 1997). Thirdly, a

mathem at ica l equa t ion can provide a m ethod o f r epresent ing the en t ire curve

be tween m easured da ta poin ts.

Two mode ls to f i t g ra in- s ize da ta a re proposed in th is paper . These mode ls

cons is t o f a un im oda l and a b imoda l m a themat ica l func tion . The two ne w eq ua t ions

prov ide g reat f lexibi li ty for f i t t ing a w ide va r ie ty o f soils.

2 De finition o f variables

The grain -size distr ibution for a soi l is def ined as the re la t ionship betw een perce nt

passing (by mass) and the par t ic le s ize . I t has a lso been cal led the mass-based

aggreg ate s ize distribution or the AS D. T he particle size represents the size of

par t ic les that can pass a par t icular s ieve mesh. The percent passing represents the

m ass percen tage o f par t ic les passing a par t icular s ieve size .

3 Background

ASTM D422-54T (1958) presented a s tandard for de te rmining the gra in- s ize

distribution. Standard sieve sizes, repor t ing metho ds, and me thod s for perfo rm ing a

hydro m ete r ana lys is a re presented . Th e s ieve ana lys is a l lowed poin ts on the gra in-

size distribution to be determined for par t ic le s izes greater than the 200 sieve or

0 .074 ram. T he hydrom ete r ana lys is presented by A STM , s tandard izes a metho d for

determ ining the gra in-size distribution for par ticles sm aller than the 200 sieve.

Interpreta t ion o f the gra in-size distribution is typic al ly carr ied ou t manu ally.

Gard ner (1956) used a two -param eter , log-norm al distribution to f it gra in-size

dis tr ibu tion da ta . K em per and Chepi l (1965 ) fur the r conf i rmed the w ork o f Gardner

( ioc c it .) . T he lo g-norm al distribution of ten fa i led to provid e a c lose f i t o f the g ra in-

size distribution a t the extremes o f the curv e (Gardner , 1956; Hag en e t a l, 1987) .

W agner and D ing (1994) later im proved upo n the log-normal equa t ion by present ing

three and four parame ter log-n orm al equations.

Cam pbe l l (1985) presented a c lass if icat ion d iagram based on the assum pt ion

that the par t ic le-size distribution is appro xim ately log-norm al. This ass um ption led to

the par t ic le-size distribution being ap prox ima ted with a G aussian distribution

function. W ith this assumption, any com binatio n of sand, s i lt , and c lay can be

represented by a g eom etr ic (or log) m ean pa r t ic le d iamete r and a geom etr ic s tandard

devia tion. Va lues were sum m arized in a mod if ied U SD A textural c lassif ica tion char t

by Shir iz i and B oersm a (1984) .





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ADVANC ES IN UNSATURATEDGEOTECHNICS 71

T he f i r st l im i t a t i on a s soc i a t e d w i t h u s i ng a l og - no r m a l type o f e qu a t i on i s t he

assumpt ion tha t the gra in-s i ze d i s t r ibu t ion i s symmet r i c . In rea l i ty , the gra in-s i ze

d i s t r i bu t i on i s o f t e n non - s ym m e t r i c a nd c a n be be t t e r f i t by a d i f f e r e n t t ype o f

e qua t ion . S e c ond l y , a m e t hod f o r f i tt i ng s o i l s t ha t a r e b i m o da l o r ga p - g r a de d i s o f t en

o f va l ue a nd t he f ou r - pa r a m e t e r l og - no r m a l e qua t i ons ha ve no t be e n f ound t o be

s a t i sf a c t o r y f o r f i t ti ng ga p - g r a de d g r a i n - s iz e d i s tr i bu t ions .

T he r e a r e t h r e e p r i m a r y type s o f g r a i n - s iz e d i s t ri bu t ions . T he s e t h r e e t ype s o f

d i s t r i bu t i ons a r e know n a s well graded soi ls , uniform s o i l s , a nd gap graded soi ls .

F i gu r e 1 i l lu s t r a te s e a c h t ype o f g r a i n - s i z e d is t ri bu t ion . T h i s pa pe r f oc us e s on t he s e

t h r e e c a t e go r i e s o f g r a i n - s i z e d i s t r i bu t i ons a nd p r ov i de s e qua t i ons t o f i t t he

e xpe r i m e n t a l da t a f o r e a c h c a t e go r y . We l l - g r a de d s o i l s a nd un i f o r m s o i l s a r e

e xa m i ne d a nd a un i m oda l m e t hod o f f i t t i ng a n e qua t i on i s de ve l ope d . T he n a

m a t he m a t ic a l m e t hod o f r e p r e s e n t ing a g a p - g r a de d s o il i s s ubs e que n t l y p r e s en t e d .

S i e v e

U . S . S t a n d a r d )

N o . 2 0 0 1 0 0 4 0 1 0 4 3 i n .

I U n i f o r m

9 , ,

o 1

0 . 0 0 1 0 . 0 1 0 .1 1 1 0 1 0 0

Grain-size diameter mm )

t

2 0 ~

m

~ o

10

F i g . 1

T hr e e p r i m a r y t ype s o f g r a in - s i z e d i s t r ibu t i on c u r ve s H o l m a n d

Kovacs , 1981)

4. Un imod al Equ ation or the Gra in size Distribution

T he s e l e c ti on o f a n a pp r op r i a t e , m a t he m a t i c a l e qua t i on i nvo l ve s a r e v i e w o f a va r i e t y

o f e qua t i ons tha t c ou l d be u s e d t o f i t s o i l s da t a. I t ha s be e n obs e r ve d t ha t t he s o i l -

w a t e r c ha r a c t e r i st i c c u r ve p os s e s s e d a s ha pe s i m i l a r t o t ha t o f the g r a i n - s iz e

d i s t ri bu t i on . T h i s i s t o b e e xpe c t e d s i nc e t he s o i l - w a t e r c ha r a c t e r i s ti c c u rve de s c r i be s





1 2 / 0 4 / 1 5 .



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72 ADVANCES1N UNSATURA TEDGEOTECHNICS

the void d is t r ibu t ion in a so il w hi le the gra in s ize curve provides inform at ion on the

dis t ribu t ion o f the so l id phase of the so i l. S ince the so l ids p lus the voids add up to

the to ta l so i l vo lum e, i t is to be expec ted tha t the d is t ribu t ion o f the so l ids phase i .e .,

g ra in- s ize d is t r ibu t ion) would tend to bea r an inverse type re la t ionship to the

distr ibution o f vo ids i .e ., soil-water c haracter is t ic curve) , and vic e versa .

Equa t ions used to f i t the so i l -wa te r charac te r i s t ic curve have been proposed

by Broo ks and Corey , 1964; Gardner , 1974; van Genuch ten , 1980; Burdine , 1953;

M ualem, 1976; F redlund and Xing , 1994. Bro oks and Co rey 1964) and Gardner

1974) presented three pa ramete r equa t ions whi le van Gen uchten 1953) and

Fredlund and X ing I99 4) presented four pa ram ete r equa tions . I t w ou ld app ear tha t

a s im i la r forms o f equa t ions could be used to represent the gra in- s ize d is tr ibu tion .

A n accu ra te representa tion o f the c lay f rac t ion o f the gra in- s ize d is tr ibu t ion

was cons ide red necessa ry in orde r to comple te the mathemat ica l func t ion . The

F r e d lund a nd X ing 1994) e qua tion al lows inde pe nde n t c on t ro l ove r the low e r e nd o f

the curv e i .e ., th e f ine par ticle s ize range) , and w as se lected as the ba sis for the

deve lo pm ent o f a gra in-s ize d is t r ibu t ion equa t ion . T he reversed sca le o f the gra in-

size distr ibution as well as character is t ics unique to the gra in-size distr ibution,

requi red tha t the or ig ina l F redlund and X ing 1994) equa t ion to be mod if ied to the

f o r m s h o w n b e l o w :

where :

a~ = pa ram ete r equa l to the inf lec t ion poin t on the curve an d re la ted to

the ini tial b reaking poin t o f the curve ,

ngr pa ramete r r e lated to the steepes t s lope of the curve ,

mg~ = pa ramete r r e la ted to the shape o f the curve ,

d~g - - param eter re la ted to th e d iam eter o f the f ines in a soi l,

d = d iamete r of any pa rt ic le s ize und er cons ide ra t ion , and

dm = diam ete r o f the m inim um a l lowable s ize par ticle.

Equ a t ion [ 1 i s r efer red to as the un im oda l equ a t ion and c an be used to f i t a

wid e va r ie ty of so il s a s show n in F igs . 2 , and 3 . A q uas i -New ton leas t squares

regress ion a lgo r i thm w as used to ad jus t t iuee o f the f ive pa ramete r s to f i t the

equa t ion to each so i l. T he a lgor ithm pro gress ive ly m m imiges the squared d i f fe rences

betw een the equ ation and exper imental data . T he be st- f it par t ic le s ize distr ibution

func t ion can be p lo t ted over the gra in- s ize d is t r ibu t ion da ta , typ ica l ly on a

logar i thm ic scale.

The u nim oda l equat ion provides s ign i f icant improvem ents in the f i t o f gra in-

size data ov er prev ious mathematical represen ta t ions i .e ., log-n orm al distribution)





1 2 / 0 4 / 1 5 .



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ADVAN CES IN UNSATURATEDGEOTECHNICS 73

o f pa r t i c le s p r e s e n t i n a s o i l. A no t he r f o r m c a n a l s o be u s e d t o v i s ua l i z e t he

d i s t r ibu t ion of pa r t i c l e s i zes by d i f fe ren t i a t ing the pa r t i c l e s i ze d i s t r ibu t ion curve .

T he d i f f e r e n t i a t ion p r odu c e s a pa r t i c l e - s iz e p r ob a b i l i t y de ns i t y f unc t i on P D F ) . T he

d i f f e re n t i a te d fo r m o f t he t m i m oda l g r a i n - s iz e e qua t ion c a n be s e e n i n E q . [ 2] . T he

pa r a m e t e r s p r e s e n t e d i n t he p a r t i c le - s i z e p r oba b i l i t y de ns i t y f unc t i on , P D F , a r e t he

sam e as de f in ed in Eq . [ 1 .

7 0 M F r e d l u n d / ~

c ~ U n i m o d a l ~ / ,,~

._

6 0

== 5 0 ~ , t ' s e .

=

o . 4 0 J ~ ' , \ L o g P D F

7

3 0 ~

/

o 2 0 \

| 1 0 - -- ~ f i : ~

Q" 0 ~ " ;

0 . 0 0 0 1 0 . 0 0 1 0 . 0 1 0 . 1 1

P a r t i c l e s i z e ( m m )

1 0 1 0 0

9 E x p e r i m e n t a l ._ U S C S % S i l t

- - U S C S % C la y -- - U S C S % S a n d

Fig . 2 Lo gar i thm ic pro bab i l i ty den s i ty func t ion for un i form s i l t fxom the P i lo t But t e

a r e a o f S a s ka t c he w a n op t i m um c om p r e s s ion ) be s t fi t w i t h the un i m o da l

equa t ion .

1 0 0

9 0

8 0

7 0

6 0

5 0

O

o . 4 0

E

3 0

2

9 10

0

0 . 0 0 0 1

- - ] C l a y [ : S i l t ; p a n d ~

: r ,

~, P a r t i c le - s i z e : ~ ' !

0 0 0 1 0 0 1 0 1 I 1 0

P a r t ic l e s i ze ( m m )

9 E x p e r i m e n t - - U S C S % S i lt

.... U S C S % C l a y " U S C S % S a n d

1 0 0

F i g . 3 L og a r i t hm i c p r ob a b i l i t y de ns i t y f unc t ion f o r Rub i c on S a n dy L oa m be s t f i t

w i t h t he un i m o da l e qua t ion .





1 2 / 0 4 / 1 5 .



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74 ADVAN CES N UNSATURATEDGEOTECHNICS

dd

d

1 1+ -'x n

[lfexp 1 )+ l~ -Ig r 1 ~ 1 + ~ 1 7 7 j ~ e ~ l)+ I~ ln g r ]ln [ ex ~ l)+ I~ ln g e ll

n l +

7 d ,~ [2]

Th e par t ic le s ize distributions presented in this pap er are calcula ted u sing E q. [2].

The h ighes t po in t on the PDF p lo t i s the mode or the most f requent pa r t ic le s ize .

S ince Eq. [2] is a PD F, the na tura l laws o f probab i l i ty hold a nd the a rea under the

di f fe ren t ia ted curve m ust equa l 1 as sho wn b e low.

7 t ]

~ d d

w here: x = par t ic le-size diam eter .

Equ a t ion [2] can a lso be used to ca lcu la te probabi li ties . Equ a t ion [4] show s ho w to

calcula te the probabil i ty that a soi l par t ic le diameter wil l fa l l in a cer ta in range.

Eq uation [2] can be ar ithmetical ly integrated betw een spec if ied par t ic le diameter

s izes and the probabi l i ty can be de te rm ined b y the fo l low ing rela t ionship .

x--d2

p r o b a b i l i t y d 1 < d < d 2 ) = I p x ) d x [4]

I t i s convenien t to represent the PD F fun c t ion in a d i f fe ren t man ner w hen p lo t t ing on

a logar i thmic scale. T he a ri thmetic PD F fun c t ion wi l l o f ten appear d is tor ted w hen

plo t ted on a logar i thmic scale. T he peak o f Eq . [4] wi l l no t r epresent the mo st

f requent par t ic le s ize be cause o f the logar i th mic distr ibution o f the par t ic le-size

sca le . T o ov ercom e th is l imi ta tion , the PD F func t ion i s o f ten represented as show n in

Eq. [5] . Tak ing the log of pa r tic le s ize and d i f fe ren t ia t ing the gra in- s ize equa t ion

produces a PD F tha t appears more phy s ica l ly real is tic. Th e peak o f Eq . [5] wi l l

represent the most frequent particle size.

p , ( d ) = d P p - - d P p i n 1 0 ) ~ [ 5 ]

dlog(d) dd

where: pl(d) = logar i thmic probabil i ty densi ty function.





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ADVANCES IN UNSATURATED GEOTECHNICS 7

T h e p r o b a b i l i t y o f t h e l o g a r it h m i c P D F c a n b e c a l c u l a te d a s f o l lo w s .

x=log d 2 )

prob ab i l iO : d 1 < d < d 2 ) = I p / x ) d r

x = l og d l )

[6]

T h e p r o b a b i l i t y d e n s i t y f u n c t i o n f o r v a r i o u s g r a i n - s iz e c u r v e s c a n b e s e e n i n F i g s . 2 ,

a n d 3 .

5. Pa ram etric Stud y o f the Pr op ose d Grain-size Distribution Equ ation

A p a r a m e t r i c s t u d y o f th e p r o p o s e d u n i m o d a l e q u a t i o n s h o w s b e h a v i o r s i m i l a r t o t h a t

o f t h e o r ig i n a l F r e d l u n d a n d X i n g 1 9 9 4 ) e q u a t i o n . T h e a g , p a r a m e t e r i s r e la t e d t o t h e

i n i ti a l b r e a k o f t h e e q u a t i o n a n d i ts e f fe c t o n t h e g r a i n - s iz e d i s t r ib u t i o n c u r v e c a n b e

s e e n i n F i g . 4 w h e r e a g~ i s v a r i e d a n d t h e o t h e r e q u a t i o n p a r a m e t e r s a r e h e l d c o n s t a n t .

T he a~,,p a r a m e t e r p r o v i d e s a n i n d i c a t i o n o f t h e l a r g e s t p a r t i c l e s i z es .

t

E

o

r

100

9

8

70

60

50

40

30

2 0

1 0

0

I I I I

0 0 0 1 0 0 1 0 1 1 1 0 1 0 0

P a r t i c l e d i a m e t e r m m )

Fig . 4 Ef fec t o f va ry ing the ag~ pa ra m e te r w h i l e ng~ = 4 .0 , mgr = 0 .5 , drg~= 1000,

a n d d m = 0 . 0 0 1 .

F i g u r e 5 s h o w s h o w t h e p a r a m e t e r n g , i n f l u e n c e s t h e s l o p e o f th e g r a i n - s iz e

d i s tr i b u ti o n . T h e p o i n t o f m a x i m u m s l o p e o f t h e g r a i n - si z e d i s tr i b u t i o n in d i c a te s t h e

g r a d a t i o n o f t h e p a n i c l e s i z e s i .e . , o n a l o g a r i t h m s c a l e ) i n t h e s o i l a s s e e n i n F i g . 5 .

T h e p a r a m e t e r

mg

c o n t r o l s th e b r e a k o n t o t h e f i n e r p a r ti c l e s iz e o f t h e s a m p l e . T h e

e f f e c t o f t h e

m g

p a r a m e t e r c a n b e s e e n i n F i g . 6 . T h e p a r a m e t e r , d rg ,, a f f e c ts t h e s h a p e

o f t h is f i n e p a r t ic l e s i ze o f th e c u r v e . H o w e v e r , t h e a m o u n t o f v a r i a t i o n p r o d u c e d o n

t h e c u r v e i s q u i t e m i n i m a l a s s h o w n i n F i g . 7 . I n s o m e c a s e s t h e d rg ~ c a n b e m o d i f i e d

t o i m p r o v e th e f i t o f t h e o v e r a l l e q u a t i o n . I t w a s f o u n d t h a t a v a l u e o f 0 .0 0 1 f o r d rg ,

p r o v i d e d a r e a s o n a b l e f i t i n m o s t c a s es .





1 2 / 0 4 / 1 5 .



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76 ADVANCES IN UNSATURATED GEOTECHNICS

t -

tO

t ~

rt

100

9 0

8

7 0

6 0

5 0

4 0

3 0

2 0

10

0

I I I I

0.001 0.01 0.1 1 10 100

P a r t ic l e d i a m e t e r m m )

F i g . 5 E f f e c t o f v a r y i n g t h e n ~ p a r a m e t e r w h i l e a g = 1 .0 m g~ = 0 .5

drg

= 1 0 0 0 a n d

dm= 0 . 0 0 1

100

9O

o i i i i i

0

g _ o

2 0

10

0

0.001 0.01 0.1 1 10 100


F i g . 6 E f f e c t o f v a r y i n g t h e m g~ p a r a m e t e r w h i l e a g = 1 .0 ng~ = 4 . 0 dr~ = 1 0 0 0 a n d

d = 0 . 0 0 1





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ADVANCES IN UNSATUR ATEDGEOTECHNICS 77

t ~

t /

t /

t~

r

G

13.

1 0 0

9 0

8 0

70

6 0

5 0

4 0

30

2 0

10

0

r l J l l l

0.001

I I I

t t t l

7 7 0

0.01

III ] l llll llfl ELll lff IIIIIIII

~

IIII I I I I I ~ I L I } I J l I I I

1 1 1 11 1 1 /1 1 1 1 L l ] _ l ~ ~ i ~

111 I I ~ I I U [ I I ] ] ] B I I J J l ~

III _ ~ l J J l [ l I 1 I IIH II

~ [ J J J J J J J L l J I I I I I I

' , ~ I i i i i i i i H - l t , l l l l l l l l

0.1 1 10

I


0 0

F i g . 7 E f f e c t o f v a r y i n g t h e

drg~

pa ram ete r w hi l e as , , = 1 .0 , ng~ = 4 .0 , mg~ = 0 .5 , and

d m = 0 . 0 0 1

T h e u n i m o d a l e q u a t i o n ( E q . [ 1 ] ) a p p e a r s t o h a v e v e r s a t i l i t y i n h a n d l i n g a w i d e

v a r i e t y o f s o i l ty p e s .

6 . B i m o d a l E q u a t i o n f o r t h e G r a i n si z e D i s t r ib u t i o n C u r v e

T h e r e i s a l i m i t a t i o n i n u s i n g t h e u n i m o d a l e q u a t i o n ( i . e . , E q . [ 1 ] ) w h e n t h e s o i l s a r e

g a p - g r a d e d . I n t h i s c a s e , i t i s n e c e s s a r y t o c o n s i d e r t h e u s e o f a b i m o d a l , b e s t -f i t.

S o i l s f r e q u e n t l y h a v e p a r t i c l e s i z e d i s t r i b u t i o n s t h a t a r e n o t c o n s i s t e n t w i t h a

u n i m o d a l d i s t ri b u t i o n a n d a s a r e s u lt , a t te m p t s t o f i t t h e u n i m o d a l e q u a t i o n t o c e r t a i n

da t a s e t s can o f t en l ead t o unsa t i s f ac t o r y r e su l t s .

T h e c h a r a c t e r i s t ic s h a p e o f a b i m o d a l o r g a p g r a d e d s o i l i s t h e d o u b l e ' h u m p

s e e n i n t h e e x p e r i m e n t a l d a t a . T h e s e a n o m o l i e s i n d i c a t e t h a t t h e p a r t i c l e s a r e

c o n c e n t r a t e d a ro u n d t w o s e p a r a t e p a r t i c l e s i z e r a n g e s . F r o m a m a t h e m a t i c a l

s ta n d p o in t , a g a p - g r a d e d s o il c an b e v i e w e d a s a c o m b i n a t i o n o f t w o o r m o r e

s e p a r a te s o il s ( D u m e r , 1 9 9 4 ) . T h i s a l l o w s f o r t h e s t a c k i n g o f m o r e t h a n o n e

u n i m o d a l e q u a t i o n .





1 2 / 0 4 / 1 5 .



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78 ADV ANCES IN UNSAT URAT EDGEOTECHNICS

.ca =

l l I j 7 ]

lnlexp(1)+l~_lng~ll 'g~- 1 ~ 1 + ~

[7]

w h e r e :

k = t h e n u m b e r o f s u b s y s t e m s f o r t h e t o ta l p a r t ic l e - s iz e

d i s t r i bu t i on ,

w i = t h e w e i g h t i n g f a c t o r s f o r t h e s u b c u r v e s , s u b j e c t t o 0 < w i < 1 a n d

Ew i = 1 .

F o r a b im o d a l c u r v e , k w o u l d b e e q u a l t o 2 a n d th e n u m b e r o f p a r a m e t e r s t o b e

d e t e r m i n e d w o u l d b e 4 t i m e s [ k + ( k - 1 ) ] . T h e u n i m o d a l e q u a t i o n i s u s e d a s t h e b a s i s

f o r t h e b i m o d a l e q u a t io n . T h e f i n a l e q u a t i o n f o r a b i m o d a l c u r v e i s s h o w n b e l o w i n

i t s e x t e n d e d f o r m .

w h e r e :

ab i = p a r a m e t e r r e l a te d t o t h e i n i ti a l b r e a k i n g p o i n t s o n t h e c u r v e ,

n bi = p a r a m e t e r r e l a t e d t o t h e s t e e p e s t s l o p e o n a p o r t i o n o f t h e c u r v e ,

m b i = p a r a m e t e r r e l a t e d t o t h e s h a p e o f t h e c u r v e ,

jb i = p a r a m e t e r r e l a t e d to t h e s e c o n d b r e a k i n g p o i n t a l o n g t h e c u r v e ,

k bi = p a r a m e t e r r e l a t e d t o th e s e c o n d s t e e p s l o p e a l o n g t h e c u r v e ,

lbi = p a r a m e t e r r e l a t e d t o t h e s e c o n d s h a p e o f t h e c u r v e ,

d rb i = p a r a m e t e r r e l a t e d t o t h e a m o u n t o f f r e e s i n a s o i l ,

d = d i a m e t e r o f a n y p a r t i c l e s i z e u n d e r c o n s i d e r a t i o n , a n d

dm = d i a m e t e r o f t h e m i n i m u m a l l o w a b l e s i z e p a r ti c le .

T h e b i m o d a l d a t a s e t s c a n b e c l o s e l y f it u s i n g t h e b i m o d a l b e s t - f it e q u a t i o n ( F i g . 8 ).

H o w e v e r , t h e b i m o d a l f i t p r o v i d e d o n l y a n adequatei t o f u n i m o d a l d a t a s e ts . I n

o t h e r w o r d s , u n i m o d a l d a t a s e ts w e r e b e t t e r f it u s i n g t h e u n i m o d a l e q u a t i o n . T h e

r e s u l ts o f f i tt i n g t h e b i m o d a l c u r v e t o s e v e r a l d i f f e r e n t s o i ls c a n b e s e e n i n F i g s . 8 to

10.





1 2 / 0 4 / 1 5 .



s e r v e d .


11/15

A D V A N C E S 1 N U N S A T U R A T E D G E O T E C H N I C S 7 9

t

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r

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60

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S i l t i

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i

i F , ? o , - ,

~ 7 ; ~ ~

i ,,~.

0.01 0.1

I s a n d I ~ ~

M . F r e d l u n d

B i m o d a l

,.

Par t i c l e -s i ze

, ~

10


100

9

Ex per im enta l . ... US CS % S i l t

.... U S C S % C l ay - U S C S % S an d

F i g . 8 L o g a r i t h m i c p r o b a b i l i ty d e n s i t y f u n c t i o n fi tt e d w i t h a b i m o d a l e q u a t i o n , fo r a

g a p - g r a d e d S a p r o l it ic S o i l te s te d a t t h e U n i v e r s i t y o f S a s k a t c h e w a n

O )

t -

u )

r

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~

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D_

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i

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i M . F re d l und i

i U n i m oda l / / I =

, = / / I i

i i /=w M Fredlund

/ d ~ , / B im o d a l ,

I

i 1

0 001 0 01 0 1 I 10 100


9 E x p e r i m e n t a l - - U S C S % Si l t

.... U S C S % C l ay - - - U S C S % S an d

F i g . 9 E x a m p l e o f a b i m o d a l f i t o n a g a p - g r a d e d S a p r o l i t i c S o i l t e s t e d a t t h e

U n i v e r s i t y o f S a s k a t c h e w a n R 2 = 0 . 9 9 9 )





1 2 / 0 4 / 1 5 .



s e r v e d .


12/15

80 ADVANCES IN UNSATURATEDGEOTECHNICS

t

3

u

r

o .

O

0-

1 0 0

9 0

8 0

7 0

6 0

5 0

4 0

3 0

2 0

10

0

0.0001

c l a y l i s i l i

M . F r e d l u n d - -

U n im od al :;~11V i

. ~ . ~ M . F red lu n d

i

0.001 0.01 0.1


10 100

9 E x p e r im e n t - - U S C S %

" " U S C S % C l a y - - U S C S % S a n d

F i g 1 0 E xa m p l e o f a b i m oda l f i t o f a ga p - g r a d e d S a p r o l i t ic S o i l te s t e d a t the

U n i v e r s i ty o f S a s k a tc h e w a n ( R 2 = 0 .999)

7. Appl ica t ion o f the Mathema t ica l Func t ion fo r the Grain s i ze Dis tr ibu tion

T he g r a i n s i z e d i s t r i bu t ion ha s be e n u s e d e x t e ns i ve l y f o r t he c l a s s i f ic a t i on o f s o i l s .

T he a p p l i c a t i on o f the m a t he m a t i c a l e qua t i ons i n t h is pa pe r c a n be a pp l i e d to

ge o t e c hn i c a l e ng i ne e r i ng p r a c t ic e . T he u s e o f e qua t ions t o f it t he g r a i n - s i z e

d i s t r ibu t ion pro vide s severa l advan tages . F i r s t ly , the equa t ions presen ted in th i s

pa pe r p r ov i de a m e t hod f o r e s t i m a t i ng a c on t i nuous f unct ion . S e c ond l y ,

qua n t i f i c a t i on o f s o i l s ba s e d on t he i r g r a i n s i z e d i s t r i bu t i on i s pos s i b l e w he n

equa t ions a re f it t o da tase t s o f so i l s in form at ion . Th i rd ly , equa t ions pro vide a

c ons i s t e n t m e t hod f o r de t e r m i n i ng phys i c a l i nd i c e s s uc h a s pe r c e n t c l a y , pe r c e n t

sand , pe rce nt s i l t , and pa r t i c l e d iam ete r va r i ab les such as d /~ d.,c~ dz~ ds~ and d6o.

I t has a l so been found tha t the gra in s i ze d i s t r ibu t ion i s cen t ra l to mos t

m e t hods o f e s t i m a t i ng t he s o i l -w a t e r c ha ra c t e r is t ic c u r ve ( G up t a a nd L a r s on , 1979 ;

Ar ya and Par is , 1981; Haverk am p and Par lange , 1986 , Ranj i tka r and S under , 1989).

A n a c c u r a t e r e p r e s e n t a ti on o f t he s o i l pa r t i c l e si z e s i s e ss e n t i al w he n t he g r a i n - s i z e

d i s t ri bu t i on c u r ve i s u s e d a s t he b a s i s f o r the e s t i m a t i on o f t he s o i l - w a t e r

c ha r a c t e r is t ic c u r ve . T he e qua t i ons p r e s e n t e d i n th i s pa pe r a ppe a r t o p r ov i de an .

exce l l en t bas i s for the es t ima t ion o f the so i l -w a te r charac te r i s t i c curve (F red lun d e t .

al, 1997).

7.1 Pa ram eters o f the grain size distribution equation s

T he un i m od a l f i t o f t he g r a i n - s i z e d i s t ri bu t i on ha s be e n f i t t o m a ny e xpe r i m e n t a l l y





1 2 / 0 4 / 1 5 .



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ADVAN CES IN UNSATURATEDGEOTECHNICS 81

m e a s u r e d g r a i n - s i z e da t a s e t s e x t r a c t e d f r om r e s e a r c h pa pe r s . T he un i m oda l f i t

pe r f o r m e d w e l l w i t h t he e xc e p t ion o f s o i ls e xh i b i t i ng b i m oda l be ha v i o r. T he

pa r a m e t e r s o f t he un i r noda l e qua t ion va r y i n a m a nne r s i m i l a r t o t he pa r a m e t e r s i n

t he F r e d l und a nd X i ng ( 1994 ) s o i l - w a t e r c ha r a c t e r is t ic c u r ve e qua t i on . T h i s s t udy

a l s o i nve s t iga t e d w he t he r e qua t ion pa r a m e t e r s c ou l d be g r oupe d a c c o r d i ng t o s o i l

t e x t u r a l c la s s i f ic a t i ons . F o r e xa m pl e , i s t he r e a r a nge o f t he ns~ pa r a m e t e r t yp i c a l f o r

s i l ty sands? The resu l t s o f th i s re sea rch ind ica te tha t general p a r a m e t e r g r o u p s c a n b e

ident i f i ed bu t specific pa r a m e t e r g r oup i ngs c a nno t be i de n t i f i e d . T he i n f l ue nc e o f

e qua t i on pa r a m e t e r s on e a c h o t he r doe s no t a l l ow f o r specific group ings . I t wa s

f ound t ha t g r oup i ng s o i l s i s m or e s uc c e s s f u l w he n pa r a m e t e r s w i t h phys i c a l

s i gn i fi c a nc e a r e s e l ec t e d . S uc c e s s f u l g r oup i ngs o f s o i l p r ope r t i e s ha s be e n a c h i e ve d

by c om bi n i ng s o i l s a c c o r d i ng t o phys i c a l pa r a m e t e r s s uc h a s pe r c e n t c l a y , pe r c e n t

s i lt , and pe rcent s and and through the use o f va r i ab les such as d t~ ~ d3~ ds~ and

d6o.

7.2 Determining physical parameters fro m the grain-size dis tribution equat ion

O ne o f t he be ne f i t s o f t he t w o g r a i n - s iz e e qua ti ons p r e s e n t e d i n t h i s pa pe r i s t ha t

c onve n t i ona l phy s i c a l va r i a b l e s c a n be c om pu t e d f r om the c u r ve s . T he m o s t

c om m onl y u s e d va r i a b l e s a re pe r c e n t c l a y , pe r c e n t s a nd , a nd pe r c e n t s il t. A l s o u s e d

are d iam ete r va r i a b les such d lc~ d . ,~ d36 ds~ and d6o. The e qua t i ons p r e s e n t e d a re o f

t he f o r m , Pp d) w he r e d i s pa r t ic l e d i a m e t e r ( ra m ) . T h e p e r c e n t c l a y , pe r c e n t s i l t, a nd

pe r c e n t s a nd c a n t he r e f o r e be a c om pu t e d by s ubs t i t u t i ng i n t he a pp r op r i a t e

d i a m e t e r s. T he d i a m e t e r s u s e d de pe nd upon t he c r i t e r ia a s soc i a t e d w i t h t he va r i ous

c l a s s if i c a ti on m e t hods . T he d i v i s i ons c a n be de t e rm i ne d f o r a ny c l a s s i f ic a t i on

m e t hod by s ubs t it u t ing i n t o t he e qua t i ons t he a pp r op r i a t e d i a m e t e r s a s s how n i n F i g .

l l .

F i g 1 1

r

r

o

r

n

100

90

80

70

60

50

30

20

10

0 00001 0 0001

Sand

Coarse

o ool O Ol o 1 1 lO lOO

Par ticle D ia m ete r m m )

De te rmin a t ion o f the so i l f rac t ions ( i .e , , c l ay , s i lt , and sand) when

us i ng the u n i m o da i e qua t i on





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82 ADVANCES IN UNSATURATEDGEOTECHNICS

T he d i a m e t e r va r i a b l e s m u s t be c om pu t e d i n a n i nve r s e m a nne r . T he pa r t i c l e

s i z e d i a m e t e r a ns w e r s t o t he qu e s t i on , ' W ha t pa r t i c l e d i a m e t e r ha s 10 pe r c e n t o f t he

t o t a l m a s s s m a l l e r t ha n t h i s s i z e ? T a k i ng t he i nve r s e o f e i t he r t he un i m od a l o r

b i m oda l e qua t i on i s d i f fi c u lt . A h a l f - le ng t h a lgo r i t hm w a s t he r e f o r e u s e d t o c om pu t e

d i a m e t e r s f r om t he g r a i n s i ze c u r ve . A n i n i t ia l gue s s d i a m e t e r w a s s e l e c t e d a nd t he

c o r r e c t i on d i s ta nc e w a s p r og r e s s i ve l y ha l ve d un t i l the i t e ra t i on p r oc e s s y i e l de d a

m i n i m a l e r ro r .

8 . Conc lus ions

U n i m od a l a nd b i m oda l e qua t i ons a r e p r e s e n te d t o f i t e s s e n t ia l l y a ny g r a i n - s i z e

d i s t r i bu t i on da t a s e t . T he un i m oda l e qua t i on w a s f ound t o p r ov i de a good f i t o f a

va r i e t y o f s o i l s . T he e x t r e m e s o f the g r a i n - s iz e d i st r i bu t ion w e r e a l s o w e l l - f i t by t he

equa t ion .

G a p - g r a de d s o i l s c a n be f i t u s ing a b i m o da l e qua ti on . T he b i m o da l e qua t i on

a l l ow s f o r a m a t he m a t i c a l r e p r e s e n t a t i on o f a ny g r a i n - s i z e d i s t r i bu t i on w he r e t he

s a m p l e c on t a i n s t w o d i s t i nc t l y d i f f e re n t , bu t dom i na n t pa r t i c l e s i z e g r oups .

M a t he m a t i c a l r e p r e s e n t a ti on o f the g r a i n - s iz e d i s t r ibu t i on p r ov i de s num e r ous

be ne f i ts . Cu r ve s c a n b e i de n t i f ie d a nd c a t e go r i z e d . L i ke w i s e , the g r a i n - s i z e c u rve s

c a n be l oc a t e d i n a da t a ba s e u s i ng s e a r c h i ng t e c hn i qu i e s. G r a i n - s i z e va r i a b l e s (i .e . ,

clay, dlo~ d6g

e t c .) c a n be m a t he m a t i c a l l y de t e r m i ne d f r om t he e qua t ion . T he

un i m oda l a nd b i m o da l e qua t ions p r ov i de a m e t hod f o r f i t ti ng t he t h r e e p r i m a r y type s

o f w e l l - g r a de d s o i l s , un i f o r m s o i ls , a nd ga p - g r a d e d s o i ls .

T he p r opos e d c on t i nuous m a t he m a t i c a l f unc t i on f o r t he g r a i n - s i z e c u r ve s e t s

the s t age for fur the r ana ly s i s to e s t im a te the so i l -wa te r charac te r i s t i c curve o f a so i l .

9 . Re f e rences

A r y a , L . M . , a nd P a r i s J. F . , 1981, A phys i c o e m pi r i c a l m ode l t o p r e d i c t t he s o i l

m o i s t u r e c ha r a c t e r is t ic f r om pa r t i c l e - s i z e d i s t r ibu t i on a nd bu l k de ns i t y da t a , S o i l

Sc ience S oc ie ty o f A m er ica Journa l , Vol . 45 , pp . 1023-1030.

Br ooks R . H . a nd C or e y A . T . , 1964 , H ydr a u l i c P r ope r t i e s o f P o r ous M e d i a ,

Co lorad o S ta te Un iv . Hyd ro l . Paper , No . 3 , 27 , pp . M arch 1964.

Burd ine , N . T . , 1953, R e la t ive pe r m eab i l i ty ca lcu la t ion s f rom p ore s i ze

d i s t r ibu t ion da ta , Journa l o f Pe t ro leu m T echno logy , Vol . 5 , No . 3 , pp . 71-78 .

Ca m pbe l l , G . S . , 1985 , Soi l Phy s ics wi th Bas ic , E l sev ie r , N ew York .

Fred lun d , D . G . , and X ing , A . , 1994 , Equa t ion s for the so i l -wa te r charac te r i s t i c

curve , Can adian G eotech nica l Journa l , Vol . 31 , No . 3 , pp . 521-532 .

Fred lun d , M. D . , F red lun d , D . G . , and W i l son , G . W. , 1997, P red ic t ion o f the

S o i l - W a t e r Cha r a c t e r is t ic Cur ve f r om G r a i n - S i z e D i s t r i bu t ion a nd V o l um e - M a s s





1 2 / 0 4 / 1 5 .



s e r v e d .


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ADVANCES N UNSATURATEDGEOTECHNICS 83

Proper ties, 3rd Brazi l ian Sym po sium on Unsaturated Soils , R io de Janeiro, Apri l 22-

25.

Gardner , W . R. , 1974, The perm eabil i ty problem, So il Science, 117, 243-249.

Gupta , S . C. , and W. E. I_arson, 1979, Est imating soi l-water re tention

characteristics fro m particle size distribution, orga nic m atter perce nt, and bulk

density, W ater Resou rces Research Journal , Vo l. 15, N o. 6, pp. 1633-1635.

Hagen, L . J . , Sk idmore , E . L . and Fryrea r , D.W. , 1987, Using two s ieves to

characterize dry soi l aggreg ate s ize distribution, Transact ion s o f the A SA E, 30 1) ,

162-165.

Ha verkam p, R. , and Parlange, J . Y. , 1986) , Predic t ing the water- re tention curve

fro m a particle -size distribution: 1. Sa nd y soils w itho ut organ ic matter, S oil Science,

Vo l. 142, No . 6, pp. 325-339.

Hol tz , R . D. and Kovacs , W. D. , 1981, An in t roduc t ion to geotechnica l

engineering, Prentice-Hall, Inc., En gle w oo d Cliffs, N ew Jersey .

Kem per , W. D. and Chepil , W . S., 1956, Size distribution o f aggregates. In

M ethods o f Soil Ana lys is, pa r t 1 . ed . C .A. B lack Ag ron om y 9 :499-510.

Koh nke , H., 1968, Soil Phys ics , M cGraw -Hi l l Book Com pany , N ew Y ork .

M ualem, Y. , 1976, A new mo de l for pred ic ting the hydraul ic condu c t iv i ty o f

unsaturated po rou s m edia , W ater Resource s Res. , 12, 513-522.

Ran ji tkar S ., and Sun der B., 1989, Predic t ion o f hyd raulic proper t ies of

unsaturated granular soils based on g rain size data , Ph.D T hesis , Un iversi ty of

Massachusetts, 75-131.

Shirizi M . A, and B oersm a L., 1984, A u nifyin g quanti ta t ive analysis o f soil

texture , Soil Science So cie ty o f A m erica Journal, 48, 142-147.

van Genuch ten , M. T , 1980, A c losed form equa t ion for pred ic t ing the hydraul ic

cond uc t iv i ty o f unsa tura ted so il s, Soi l Sc ience Soc ie ty Am er ica Journa l, pp . 892-890.

W agner , L. E. , and Ding, D. , 1994, Rep resenting aggre gate s ize distributions as

m odif ied lognormal d is tr ibu tions , Am er ican Soc ie ty of Agr icu l tura l Eng ineers, Vo l .

37, No. 3, pp. 815-821.




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s e r v e d .

Use of Grain-Size Functions in Unsaturated Soil Mechanics

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