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8/17/2019 Use of Grain-Size Functions in Unsaturated Soil Mechanics
1/15
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 .
8/17/2019 Use of Grain-Size Functions in Unsaturated Soil Mechanics
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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) .
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 .
8/17/2019 Use of Grain-Size Functions in Unsaturated Soil Mechanics
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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
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 .
8/17/2019 Use of Grain-Size Functions in Unsaturated Soil Mechanics
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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)
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 .
8/17/2019 Use of Grain-Size Functions in Unsaturated Soil Mechanics
5/15
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 .
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 .
8/17/2019 Use of Grain-Size Functions in Unsaturated Soil Mechanics
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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.
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 .
8/17/2019 Use of Grain-Size Functions in Unsaturated Soil Mechanics
7/15
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 .
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 .
8/17/2019 Use of Grain-Size Functions in Unsaturated Soil Mechanics
8/15
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
P a r t ic l e d i a m e t e r m m )
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
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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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
P a r t ic l e d i a m e t e r m m )
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 .
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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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.
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 .
8/17/2019 Use of Grain-Size Functions in Unsaturated Soil Mechanics
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
O
r
1 0 0
9 0
8 0
70
60
50
4 0
3 0
2 0
10
0
0 . 0001
I C l ay I
0.001
I I
S i l t i
i ;
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
P a r t i c l e s i z e ( m m )
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
Q .
~
OJ
D_
100
9 0
80
70
6 0
5 0
4 0
3 0
2 0
10
0
0.0001
JC l a y J
i
i jsi I s a o o
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
P a r t i c l e s i z e ( m m )
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 )
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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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
P a r t i c l e s i z e ( m m )
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
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 .
8/17/2019 Use of Grain-Size Functions in Unsaturated Soil Mechanics
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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
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 .
8/17/2019 Use of Grain-Size Functions in Unsaturated Soil Mechanics
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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
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