The consolidation of a stratified soil with vertical and horizontal drainage

Int J Meeh Sc~ Pergamon Press Ltd 1964 Vo] 6, pp 187-197 Pnnted in Great Britain

THE

CONSOLIDATION VERTICAL

OF A STRATIFIED

AND HORIZONTAL

SOIL WITH

DRAINAGE

M R HOR~E D e p a r t m e n t o f Civil E n g i n e e r i n g , U m v e r m t y o f M a n c h e s t e r ( R e c e w e d 7 A u g u s t 1963}

S u m m a r y - - T h e p r o b l e m c o n m d e r e d is t h a t o f t h e c o n s o l i d a t i o n o f a soil stratffied in h o m z o n t a l layers, w i t h t h e s r m u l t a n e o u s p o s s l b i h t y o f overall d r a i n a g e b o t h v e r t m a l l y t o p e r v i o u s layers a b o v e or b e l o w t h e serms o f s t r a t a , a n d h o r i z o n t a l l y t o v e r t i c a l s a n d d r a i n s A g e n e r a l s o l u t i o n is o b t a i n e d for a n y s e q u e n c e o f h o r i z o n t a l s t r a t a in w h i c h t h e coefficmnts o f c o m p r e s s i b i l i t y a n d p e r m c a b i h t y for h o m z o n t a l a n d v e r t i c a l flow are r e g a r d e d as a r b i t r a r y f u n c t m n s o f t h e v e r t m a l c o - o r d i n a t e S o l u t m n s are o b t a i n e d for b o t h t h e t w o dn~nenslonal a n d t h e r a d i a l l y s y m m e t r i c a l cases, a n d a special t r e a t m e n t is g i v e n for e q u a l a l t e r n a t e layers s u c h as clay a n d silt or clay a n d s a n d INTRODUCTION

THE provision of vertical sand drains facilitates the consohdatIon of a soil stratum by introducing horizontal drainage through the soil to the sand drain Vertical drainage to pervious layers above or below the stratum occurs simultaneously, and it has been shown (Carrillo 1) t h a t the separate solutions for homzontal and vertical drainage may be supemmposed to obtain the total rate of consolidation A review of the application of this method to various problems involving sand drams has been given by Rlchart ~ The problem becomes more involved when the soil exists in a series of horizontal strata, since drainage then tends to follow first vertical routes to the more pervious layers, whence drainage takes place preferentially towards the sand drams The present paper gives a general solution for any sequence of horizontal strata, in which the coefficients of compressibility and of permeability with respect to horizontal and vertical flow (not necessarily identical) are admitted as arbitrary functions of the vertical co-ordinate Both the twodimensional and the three-dimensional (radially symmetrical) problems are solved for conditions of free vertical strain under uniform vertical load Particular solutions are presented for the cases of alternate layers ABABA, e t c , in which layers A are identical with each other in thickness and physical properties, and so also are layers B Some interesting approximate results are obtained when layers A have low permeability, while layers B, much thinner than layers A, have high permeability This solution has relevance to the problem of laminated clays. TWO-DIMENSIONAL

FLOW

The soft m a s s m d i c a t e d in F i g 1 is a r r a n g e d m a series o f h o r i z o n t a l layers, so t h a t if OZ is t h e v e r t i c a l axis o f reference, all soil p r o p e r t i e s arc f u n c t m n s of t h e c o - o r d i n a t e d i s t a n c e z o n l y T h e soft m a s s is b o u n d e d b y a n i m p e r m e a b l e p l a n e x = L a n d t w o i m p e r m e a b l e p l a n e s parallel t o O Z X If, t h e r e f o r e , u is t h e excess p o r e w a t e r pressure, ~u/~x = 0 a t 187

188

M R

HORNE

x = L a n d ~u/~y = 0 a t t h e v e r t i c a l b o u n d a r i e s p a r a l l e l t o O Z X The fourth ~eitmal b o u n d a r y x = 0 is a s s m n e d t o b e f u l l y p e r m e a b l e , w h e n c e u = 0 w h e n x = 0 T h e h o r i z o n t a l b o u n d a r i e s z = 0 a n d z = H m a y i n d e p e n d e n t l y b e f u l l y p e r m e a b l e (u = 0) oi fiflly i m p e r m e a b l e (~u/~z = O)

!']

ooofo,

1 d_~u=0 dx

-'~

.

_

o, :o x 01. .

FIG

1

×

L

Co-ordinates and boundary conditions for two-dnnenmonal consolidation

T h e a s s u m p t i o n s u s u a l an t h e t h e o r y o f c o n s o h d a t l o n ( T e r z a g h l a) a r e a d o p t e d , a p a r t f r o m t h e a s s u m e d d e p e n d e n c e o f s o d p r o p e r t m s o n z I n e a c h e l e m e n t a r y l a y e r , t h e r e will b e a c o e f f i c i e n t o f c o m p r e s s , b l h t y av, c o e f f i c m n t s o f p e r m e a b d l t y k z a n d k z f o r h o r i z o n t a l a n d v e r t m a l f l o w r e s p e c t i v e l y , a n d a v o i d s r a t i o e. I t will b e a s s u m e d t h a t a~, kx, k~ a n d (1+%) remain, to a sufficmnt degree of approx,mation, constant with respect to the degree of consolidation, whence they are to be regarded as functions of z only. With the soil c o m p l e t e l y s a t u r a t e d , a n d w i t h t h e e x c e s s p o r e w a t e r p r e s s u r e l n t t , a l l y e v e r y w h e r e z e r o , a u n i f o r m v e r t m a l p r e s s u r e u 0 is s u d d e n l y a p p l i e d a t t i m e t = 0 t o t h e b o u n d a r y z = H As a result, the excess pore water pressure u becomes everywhere of value u 0 e x c e p t a t t h e p e r m e a b l e b o u n d a r m s , w h e r e u -- 0 T h e d l f f e r e n t , a l e q u a t i o n f o r t h e s u b s e q u e n t d i s s i p a t i o n o f p o r e w a t e r p r e s s u r e is ~u ~2 u b2 u ~ = cz ~ - ~ + c z ~z ~

(1)

w h e r e , if Yw is t h e d e n s i t y o f w a t e r , k~(1 + e v ) c ~ = - -

av Yw

and

k~(1 + e v ) c ~ = - -

av ~'w

I n a c c o r d a n c e w i t h t h e p r e v i o u s a s s u m p t i o n s , t h e c o e f f i c i e n t s o f c o n s o h d a t t o n cz, c z w i t h r e s p e c t t o h o r i z o n t a l a n d v e r t m a l flow a r e f u n c t i o n s o f z o n l y Conmder the doubly mfimte series solutmn

u =

N .....

Z

N-i

~, aN.fNn(z) s i n

.=*

(2N-- 1)~x 2L

e -bN,,t

(2)

w h e r e fiv,,(z) is a f u n c t i o n o f z a n d t h e a N . a n d bN. a r e c o n s t a n t s Thin solution satmfies t h e c o n d * t , o n s u = 0 a t x = 0 a n d ~u/~x = 0 at x = L S u b s t * t u t m g f o r u m e q u a t i o n (1), t h e s o l u t i o n is s a t m f a c t o r y p r o m d e d , f o r e a c h f u n c t i o n fiv.(z), t h e f o l l o w i n g dLfferential e q u a t i o n m s a t i s f i e d f o r all z

c.f~.(z)+{bN.

(2N--1)''2cz}f~v.(z)4-L~ = 0

(3)

The boundary conditions at z = 0 and z = H are known to be either fN.(z) = 0 (fully p e r m e a b l e ) o r f ~ . ( z ) = 0 F o r e a c h v a l u e o f N , t h e r e will b e a s e r m s o f f u n e t , o n s .f~Vl(Z),fN2(z), e t c , e a c h w i t h t h e a s s o c i a t e d v a l u e o f t h e c o n s t a n t blvl, b~.~, e t c , e a c h s a t m f y m g e q u a t i o n (3) a n d t h e b o u n d a r y c o n d i t i o n s a t z = 0 a n d z = H I t r e m a i n s t o determine the coefficients aN. which define the complete solution

The consohdatmn

o f a s t r a t f f i e d soft w i t h v e r t i c a l a n d h o r i z o n t a l d r a i n a g e

189

I t is s h o w n m A p p e n d i x I t h a t a n y t w o f u n c t m n s f2v,,(z),f~,,,(z) t h a t s a t i s f y e q u a t i o n (3) a n d t h e b o u n d a r y c o n d l t m n s e i t h e r f,v.(z),f~v. (z) = 0 o r f',v,~(z),f~,, (z) = 0 a t z = 0 a n d z = H satisfy also the orthogonahty relation

fotz fNn(z)fN'='(z) dz

= 0

(4)

Cz

unless n = n'

S i n c e it is a l s o k n o w n t h a t

f:

sin

(2N-1)~rXsln(2N'-l)CrXdx 2L 2L

= 0

(5)

u n l e s s N = N ' , ~t f o l l o w s t h a t F l" J0 30

c~

u n l e s s ZT = /V' a n d e q u a t m n (2) b y

n = n'

sm When

( 2 N - l) ~ x 2L

= 0tox

( 2 N ' - 1) 2L

.x

-- 0

t = 0, u = u 0 w h e n c e , m u l t i p l y i n g

fN,,(Z) s i n cz and integrating fromx

s,n

(6) both

sides of

( 2 N - - 1)~rx

2L

= Landz

= 0toz

= Hat

t = 0,

2 u 0 ~11f'v'(z) dz ,10 Cz aN" = ( 2 N - - - ~ ;"~ ) ) 'Id z ---'--'" -

2

J0

(7)

cz

T h e f u n c t t o n s / N n ( z ) a n d t h e c o n s t a n t s arch, bN. m t h e c o m p l e t e solu$1on g i v e n b y e q u a t i o n (2) h a v e t h u s b e e n d e r i v e d f o r t h e g e n e r a l c a s e o f t w o - d n n e n s l o n a l flow, a s s u m i n g t h a t s o l u t i o n s h a v e b e e n o b t a i n e d f o r e q u a t i o n (3) T h e s o l u t i o n o f e q u a t i o n (3) f o r a n y succession of sml layers of finite thmknesses with constant propertms within the layers m g i v e n m A p p e n d L x 2. T h e p a r t m t f l a r s o l u t i o n f o r e q u a l a l t e r n a t e l a y e r s will n o w b e g i v e n . SOLUTION

FOR

EQUAL ALTERNATE LAYERS UNIFORM PROPERTIES

T h e soil m a s s c o n s i s t s a l t e r n a t e l y k z = ]%p ]cz = ]czl, % = evl, c z = cxz p r o p e r t i e s av~ , ]%2, ]c~=, %2, cx= a n d cz~ the layers, see Fig 2 It m assumed

~IHI

OF

ALTERNATELY

o f l a y e r s o f t h i c k n e s s H 1 w i t h p r o p e r t m s a~ = avl, a n d c z = czl, a n d o f l a y e r s o f t h m k n e s s H 2 w i t h T h e o m g m is t a k e n a t t h e i n t e r f a c e b e t w e e n t w o o f t h a t t h e r e is a l a r g e n u m b e r o f l a y e r s a n d t h a t t h e

"~'

J

°°

H2 ! I

*.°4 *

Layer I

-:

X

Layer 2 I L

•:F= FIG 2

"-v

Two-dLmenslonal consohdatlon with equal alternate layers

o v e r a l l d r a i n a g e is h o r i z o n t a l , s o t h a t b y s y m m e t r y 19u/~z = 0 a t z = H1/2 ( m i d p l a n e o f l a y e r 1) a n d a t z = -Hi/2 ( m i d p l a n e o f l a y e r 2) I f t h e s e q u e n c e o f l a y e r s is b o u n d e d

190

M R HORNE

a b o v e or below b y a fully p e r m e a b l e layer, the m e t h o d of Carmllo m a y be applied to d e t e r m i n e the effect of overall vertmal drainage T a k i n g particular values of N and n, let fN,,(z) = f N . l ( z ) in layer 1 and f~v,~2(z) in layer 2 T h e n from e q u a t i o n (3), f ~ c n l ( z ) + ~ 2 N n l f N n l ( Z ) -~ 0

(Sa)

tt 2 fg..2(z) + [3N,~zfzv.2(z ) = 0

(Sb)

where flN.i and flN.~ are real or i m a g i n a r y constants g i v e n by bN.

( 2 N - 1)2~"2

~-g~

fl}.l =

The solution of e q u a t i o n z = Hi~2 is

(9a)

Czl

bNn fl~

~)

(2N -- 1)2 ~'~

yLY

=

c~)\ (9b)

Cz2

(8a) satisfying the necessary condition f~rni(z)= 0 at

[~HI fNnl(Z) = A l COSflNni[~---z \ /

(10a)

fN.2(z) = A2 c o s f l N . 2 ( ~ + z )

(10b)

Similarly f r o m e q u a t i o n (8b),

F r o m c o n t m m t y at z = 0, f~wl(0) = fN~2(0), whence

cos N ul -

cos N '2

Since t h e complete expression for u (equation (2)) contains coefficients ann t h a t are at this stage a r b i t r a r y , it is sufficient to p u t A 1 cos ~N~I H i -- A s c o s ~ N 2 H 2 - 1 2 Hence S 1 fNnl(Z) ----

COSfiN.1 H I 2

(1 la)

H 2

f~n2(z) =

flN.2H2 COS

(llb)

2

F o r c o n t i n u i t y of vertical flow at z = 0,

Hence H2

--0

(12)

~ ( 2 N - - 1)2~ "2 (Cxl_Cx2) = 0 Czl final -- Cz2 flNn$ + 4L 2

(13)

kzi fl~vnl t a n f l ~

+ kz2 fiN.2 t a n

2

E h m m a t i n g bN. b e t w e e n equations (9a) and (9b),

The c o n s o h d a t l o n of a stratified soft w , t h vert,cal and homzontal drainage

191

Corresponding to each v a l u e of N equat,ons (12) and (13) ymld a series of pa,rs of roots (fiNn. 8~¢n). (SNa~. flNaa), e t c . f r o m whmh. b y m e a n s of e q u a t m n (9a) or (9b). the q u a n t , t y bN~ m a y be derived F,nally. the values of ann. as o b t a i n e d f r o m e q u a t m n (11). become I

8u0

aN"-- (2N--1)rr HI (1 --czl + ~

.

82v~*HI-1 . z ~N~a c.2

PN..1czl 2 tan ~

fl~) + tan a

Cz2 \

8N.aHa 2

(14)

f~Nna2-12

I t should be n o t e d t h a t some of t h e roots 82v~,. 82v~2 will be l m a g m a r y , caus,ng h y p e r b o h c f u n c t m n s to replace t r , g o n o m e t r m ftmctmns W h e n the later stages of consohdatlon only are of interest (e g ff ,t is required to k n o w the t , m e at w h m h 90 per cent c o n s o h d a t m n has been achmved) the first t e r m s only (8nl. 81.a) need to be conmdered, since the o t h e r t e r m s decay m u c h more rapidly I f layer 2 m m o r e p e r m e a b l e t h a n layer 1. the lowest solutmn of e q u a t m n s (12) and (13) has 8 m (denoted b y 81) real a n d 8 ' ' 2 (denoted b y zSa) imaginary, so t h a t e q u a t m n (12) becomes

kz181

t a n ^--~!~ =

kza82t a n h 82-~ 1-12--

(15)

while equations (9a) and (9b) become ~T2 a = b = ~-i~ cz1-~-81Czl

-STa

~ C x 2 - 8 2 Caz 2

(16)

The " d e c a y pemod" (the t i m e t a k e n for pore pressure to decrease b y a factor of e -1) is g i v e n b y 1/b F o r a u m f o r m sod of coefficient of c o n s o h d a t m n c~ w i t h respect to hor,zontal drainage, b = (Tr~/4La) c~, while for the layered a r r a n g e m e n t m w h m h c~a > Czl, e q u a t , o n (16) shows t h a t 7T2

77"a

4L ~ c~1 < b < ~

c~a

Hence. as would be expected, t h e decay p e r m d m Intermediate b e t w e e n the decay permds for uniform masses of the two m a t e r m l s t a k e n separately LAYERS

OF

LOW PERMEABILITY LAYERS OF HIGH

INTERSPERSED PERMEABILITY

BY

THIN

Suppose that, in Fig 2, layers 2 are of very much hlgher permeablhty than layers I, and that they have coefficmnts of compresslblhty elther of the same order as layers l, or lower, so that cx2>~cxl , cz2>~czl Also, let H2~H I Consldermg first terms only, it follows from equat,on (llb) that the ratlo of excess pore pressure at the boundary of a layer 2 to the pore pressure m the told plane of the layer m cosh (Sa H2/2) Since the layer is thm and hlghly permeable, this m vlrtually umty, whence 82 H2/2 "~ 0, and equation (15) becomes 8~ = 2kzl 81 t a n S . H1 kza ~ 2

(17)

Moreover. because of t h e thinness of layer 2 c o m p a r e d w , t h layer 1. t h e overall decay pemod 1/b will be v e r y m u c h greater t h a n it would be if the entire soft mass were to cons*st of the m a t e r i a l m layer 2 only. whence b.~ (rra/4L 2) c~2. and so. from e q u a t i o n (16). ~ra c ~ a - rra kxa 8~ ~ 4L a cz2 4L a k.2

(18)

I t follows from (17) and (18) t h a t 8. H I t a n fll H1 7ra H1 Ha k~a 2 2 = 16 L ~ kzl

(19)

M R HORNE

192

I t IS c o n v e n i e n t to express b, using the first e q u a l i t y (16), in the form

~2 +

s

whence

-

(20)

-

k.1 Hi]

~rz r kzz H Q f~ c,i = ~ %ILF kx~-~lj

(21)

D i v i d i n g e q u a t i o n (21) b y e q u a t m n (19), it follows t h a t F =

fli Hi~2

tan (f~lH1/2)

(22)

The n u m e r m a l e l n n m a t m n of f l l H z / 2 b e t w e e n equations (19) and (22) enables F to be expressed m t e r m s of t h e nondunenmonal q u a n t i t y L = k~i H l H= k~= and t h e resulting relatmnshlp is shown by t h e lowest c u r v e m F~g 3 I t follows from e q u a t m n (20) t h a t F is a measure of t h e e x t e n t to w h m h t h e highly p e r m e a b l e b u t t h i n

10

I

I

~

I

0-9 ' "HII O0 07

I • ".r"

'J

I

I,

I

I

I

I

II

Iill I I

II

II

II

III

I

~-Loyers [

I tt

"Ht ~

o~:E~

,Leyer$ 2 / / / / / / / / / /

•

~:~

._~

06

~-ffec,IveCr•Crl FL~--~~ ///¢////////~~Z "r,''lJ / /'///////

L.yQr. '

/¢o Ov

A

L'O, yel'~t

F09 04

,,% r - - - - - -

03 OZ •

~/~/~

"

E f f e xc x,t L ~

0

H~=<::H= a

i

i

0"003 0~305

i

t I i

001

I

0 02

J

I

I

t

I

0 05

I li

0 I

1

0 2

I

I

l

05

~==n=j

kx2"~l>kxz I

I

I

I I

I

2

3

2

(rl-r Q) k_~ L;: 'HI Hz kr z or H-~Z ~'~/z F I o 3 A p p r o x u n a t e solution for effective coefficient of consolidation w i t h homzontal drainage of soil conszstmg of horizontal layers of low p e r m e a b i l i t y interspersed b y t h i n layers of high p e r m e a b i l i t y layers 2 are effective in decreasing the decay period 1/b below the v a l u e 4L2/rr 2 c:~1 which would obtain for layers 1 in t h e absence of layers 2 W h e n F approaches u m t y , i e w h e n L~ kzl H I H~ kx2 is large, t h e t h i n layers are fully effective, since the r a t e of dismpatlon of pore pressure m increased m the ratio ( k ~ l H l + k ® ~ H = ) (k~lH1), i e . m t h e s a m e ratio as t h a t of t h e Increase of rate of flow t h a t would be caused for a uniform u m t pressure g r a d i e n t along OX. W h e n L ~ kzl H1 H2 kx2

T h e c o n s o h d a t m n o f a stratffied soll w i t h v e r t i c a l a n d h o r i z o n t a l d r a i n a g e

193

is smaller, t h e d r a m p a t h a l o n g O X t h r o u g h layers 1 m sufficmntly s h o r t for t h e m o r e p e r m e a b l e layers 2 t o be less effective m i n c r e a s i n g t h e r a t e o f c o n s o h d a t m n H e n c e , p r o v x d e d H 1 a n d H a r e m a i n u n c h a n g e d , a scale effect m m t r o d u c e d , causing a n a p p a r e n t increase m t h e effective eoefficmnt o f c o n s o h d a t m n w~th increase m d l m e n s m n L T h i s effect h a s b e e n o b s e r v e d e x p e r i m e n t a l l y b y R o w O m t e s t s on l a m i n a t e d clays T h e a b o v e t r e a t m e n t m, h o w e v e r , only a p p r o x i m a t e a n d fuller n u m e r m a l s t u d m s are r e q m r e d THREE-DIMENSIONAL

(RADIALLY

SYMMETRIC)

FLOW

T h e soft m a s s m a g a i n c o n t a i n e d m h o r i z o n t a l layers, so t h a t t h e p r o p e r t m s a~, k,, k=, e~, c, a n d c= (where r d e n o t e s t h e r a d m l d l r e c t m n a n d z t h e v e r t m a l , Fig. 4) are f u n c t m n s

u=O or d~-=O Oo°

H

I

~'--u=O j

@

O

°~ °o

b

4

° •

° °°

L

Fie

•

OB

PI_

- I - r0

C o - o r d i n a t e s a n d b o u n d a r y c o n d ] t m n s for radially s y m m e t m c a l consohdatmn

o f z o n l y H o r i z o n t a l d r a i n a g e t a k e s place t o w a r d s t h e t u n e r c y h n d r m a l surface r = r 0 a t w h m h u = 0, while a t t h e o u t e r e y l m d r m a l surface r = rl, ~u/~r = 0 A t t h e h o r i z o n t a l surfaces z = 0 a n d z = H , e i t h e r u = 0 o r ~u/~z -- 0 T h e differential e q u a t m n for t h e d m s l p a t m n of p o r e p r e s s u r e ]s (TerzaghP)

/~Su

8u

1 ~u\

~u

~-/= ~ ' [ - ~ + r ~r) +~' ~

(23)

Conmder t h e solution

..... U---~

~,

N~I

( Z ann fNn(z) Uo a2¢

nffil

(24)

w h e r e Uo(aN r/ro) is a f u n c t i o n o f Bessel f u n c t i o n s o f zero o r d e r a n d o f t h e first k m d (J0) a n d o f zero o r d e r a n d o f t h e s e c o n d k i n d (Y0) g w e n b y

(25) T h e q u a n t l t m s ~2v are t h e r o o t s o f t h e e q u a t i o n /

r~\

\

%I

u,l~,~-'j = o

(26)

w h e r e t h e f u n c t i o n Ul(a N r/r0) m t r o d u c e s Bessel f t m c t l o n s o f o r d e r u m t y a n d o f t h e first (J1) a n d s e c o n d (Y1) k i n d s m t h e f o r m

=

,27)

194

M R HOR:NE

I t follows from e q u a t i o n (25) t h a t the solution (24) satisfies the condition u = 0 at r = r 0. Smce

and

Y~(aN~o ) = --a'v Yl(aN--rtr0 \ ro] where the prime denotes differentiation w i t h respect to r, it follows from equations (24)-(27) t h a t , as required, &u/&r = 0 at r = r 1 N o w t h e function U0(a N r/ro) satisfies t h e Bessel e q u a t i o n of zero order

U~+

Uo+~ro]

0= 0

(28)

Substituting e q u a t i o n (24) in e q u a t i o n (23) and using (28), it follows t h a t e q u a t i o n (24) is a solution of (23) p r o v i d e d

[ (U]er fNn(Z)

ezfNn(z )+ bNn--

= 0

(29)

This e q u a t i o n corresponds to (3) for t h e two-dimensional problem, a n d the functions The eoeffiemnts ann are derived b y m u l t i p l y i n g b o t h sides of e q u a t i o n (24) b y

f,v,~(z) and t h e constants bNn m a y be o b t a i n e d in a skrmlar m a n n e r fNn(z) rUo aN Cz

setting u = u 0 and t = O, a n d m t e g r a t m g b e t w e e n limits r = r 0 to r~, z = 0 to H m a d e of t h e orthogonal relations

rU o aN

Uo aN"

dr = 0

for

Use is

N#N'

J "go

and of the orthogonal relations

f

ttf'vn(z)f~v~(Z)dz=O 0

n#n'

for

Cz

established in A p p e n d i x 1 W h e n N = N ' we h a v e , since U0(aN ) = 0 and Ux(a N rl/ro) = 0, o

r0!

',

2

- ~ U~(aN)

Finally,

r~rUo(a:~ r--1 dr = -- r2~-°Ul(agv ) o

~-~ence

\

to?

O~N

2u2o Ul(aN) aN

aNn

rl 2 2( r o )T1 Uo a N -

{(r0 -- )

--

"| n fN.(Z) dz Jo

cz

(30)

U~,aN,}(Hf~in(z! dz Jo

ez

Thus t h e complete solution represented b y e q u a t i o n (24) has been derived The ease of a l t e r n a t e layers of the same materials is solved as m t h e two-dunenslonal ease E q u a t i o n s (ga) and (gb), ( l l a ) and ( l l b ) , (12) and (13) a p p l y w i t h c~i, c~2 replaced b y C,l, cr2 respeetwely, and ( 2 N - 1) 7r/2L replaced b y aN/r o The coefficients ann m e q u a t i o n (24) are given b y 1

BaNU°UI(aN) ann=

{(r~)2 U , / a oi

r~,

N,. o)

f~N"l Czl U~(aN)} H1 (1

~- + ~

2

tan finn1 H1 + 1 2

tan ~

~ N " ' CZ2

t a n fllv~ H2 2

+ t a n , f~N~ H ~ )

+ H2(l+^ 2__ t a n f l N n 2 H ~ + t a n , / 3 N . a H , ] c~2 \ /~N.2H2 2 2 !

(31)

The consolidation of a stratified soil w i t h v e r t i c a l a n d horizontal drainage

195

The solution w h e n layers 1 are of low p e r m e a b i l i t y and layers 2 of high p e r m e a b i l i t y , b u t t h i n c o m p a r e d w i t h layers 1, IS o b t a i n e d as m the two-du~aensmnal case E q u a t i o n (19) is replaced b y /~1 H1 tan/~, H I ~ H1H~ Icr2 (32) 2 2 4 re~ k~i while e q u a t i o n (20) becomes b

=

0tl2Crl[l

-kr2H2]

(33)

a n d e q u a t i o n (22) remains applicable. The q u a n t i t y al is t h e first root of the equatxon

\

rI] = o %/

E l i m l n a t m g flzHi/2 b e t w e e n equations (22) a n d (32), F m a y be expressed as a functzon of

r~ k~l H, H 2 kr2 To facilitate comparison w i t h the two-dunensional case it IS more c o n v e n i e n t to express F m t e r m s of (r 1 -- r0) ~ k~I H, H 2 kr2 since (r 1 --r0) corresponds to L, t h e greatest horizontal drainage path. Results are g i v e n for various ratios of o u t e r to tuner r a d m s (ri/ro) in F i g 3 I t m a y be n o t e d t h a t t h e twodimensional case m a y be i n t e r p r e t e d as corresponchng to mfmlte radii, i e to the case rl/r o = 1 I t is of interest t h a t , for a given horizontal drainage p a t h , , e for g i v e n

(ri -

r0) 2

k~i

H I H~ k~2 the t h i n highly p e r m e a b l e layers h a v e t h e greater effect on drainage the greater the ratio rl/r o

Acknowledgement---The a u t h o r ' s interest was d r a w n to this p r o b l e m b y Professor P W Rowe, and t h e a u t h o r gratefully acknowledges his a d w c e and cmticism REFERENCES 1 2 3 4

N F K P

CARRII~O, J Math & P h y s 21, 1 (1942) E RICItART, Trans A m e r Soc Cwl Engrs 124, 709 (1959) TERZAGHI, Theoretscal Sozl Mechanics, Chap 13 Wiley, N e w Y o r k (1943) W R o w E , Geotechnulue 9, 107 (1959) APPENDIX PROOF

1

OF ORTttOGONAL RELATIONS EQUATIONS (3) A N D (29)

(4) F O R

I n e q u a t i o n (3), let {blvn (2N--1)~lr2c~}4L 2 Cz

= EN.(z )

where, since c~ and c~ are functions of z, so also is ENn(z ). F o r t h e two functions fNn(z) a n d fN~ (z), e q u a t i o n (3) becomes /fvn(z) + E N n ( z ) / ~ ( z ) = 0

(A1)

f~,, (z)+ E N . (z)fN, , (z) = 0

(A2)

196

M R HOI~NE

M u l t i p l y e q u a t i o n (A1) b y f.¥n (z) a n d i n t e g r a t e w i t h r e s p e c t t o z b e t w e e n limltu = = q) andz = H Then

f EiV,,(Z)yN.(Z)fNn (z) dz

= =

by

_ [f~n(z)fNn ' f/f

p Nn

zffitl + (z)]~_o

f

A ,w ( z ) f N, . (z) dz

p

(A3)

(Z )f~ n " (Z )d Z

v i r t u e o f t h e b o u n d a r y c o n & t m n s , n a m e l y e i t h e r ]Nn(Z) ( a n d f.vn(z))= 0 or (andf~cn (z)) = 0 a t z = 0 a n d z = H S n m l a r l y f r o m e q u a t m n (A2)

f~,(z)

f:Eiv. (z)fN.(z)f~v~ (z) dz = fo f~,,(z)f~-~ (z) dz S u b t r a c t i n g (A4) f r o m (A3) a n d s u b s t i t u t i n g t h e v a l u e s of

EN,~(z) a n d

(A4) E ~ . (z),

(bN.--bs.) (H f~.(~)f~. (~) d~ = 0

Jo

cz

H e n c e , e x c e p t w h e n bNn = b~vn', I e n = ~b',

foH fNÈ(z)f~vn

(z)

dz

= 0

(A5)

Cz

Similar a r g u m e n t s a p p l y t o e q u a t m n (29)

APPENDIX SOLUTION

FOR

f~.n(z)

2

A N D b~vn F O R A R B I T R A R Y UNIFORM LAYERS

SEQUENCE

OF

L e t succesmve l a y e r s 1, 2, ,7, , s h a v e t h m k n e s s e s H j a n d p r o p e r t m s ave , kxj , kz~, eve, cx~, cz~ I n l a y e r 3, t h e f t m c t l o n f N n ( z ) [ e q u a t i o n (3)] will h a v e t h e f o r m fj(z) w h e r e

/;'+[3~fj {bNn ~

= 0

(A6)

(2N-1)2~r2cz'}4L'

=

(AT) Cz~

Hence

f~ = A s stuff, z + B,

(AS)

cosfl, z

T a k i n g a local omgln for t h e v e r t i c a l o r d i n a t e z m t h e m i d p l a n e of l a y e r 3, a n d a mmflar local omgm m t h e m i d p l a n e of l a y e r j + 1 for t h e f u n c t m n f~+l r e l a t i n g t o t h a t layer, t h e c o n d i t i o n s of c o n t m m t y a t t h e i n t e r f a c e r e q m r e (A9)

Hence

A ~ s m ~ - ~ + B j c o s fl'H' ' -2t-,+ - 1 mn f l ' + '~H ' + l " - - ~ - -¢A'k"'~'c°s~!-Bjk~'sm~

B , + , c o s f l ~ + ~" ~ + 1 -

0

(All)

--B~+I ]czO+ll ~t+1 s m B,+I H~+I _ 0 2

(A12)

-A'+lk"''+x'~'+'c°s~'+lH'+'2

T h e c o n s o h d a t l o n of a s t r a t i f i e d sod w i t h v e r t m a l a n d h o m z o n t a l d r a i n a g e

197

T h e r e w d l ex,st a l t o g e t h e r ( s - 1) e q u a t , o n s of t y p e ( A l l ) a n d ( s - 1) of t y p e (A12) I n adchtlon, e , t h e r f l = 0 o r f ~ -- 0 a t z = - H 1 / 2 m l a y e r 1, a n d e , t h e r / , -- 0 o r / , ' = 0 a t z = H 8 / 2 m l a y e r s H e n c e t h e r e w d l b e a t o t a l of 2s e q u a t , o n s m t h e 2s u n k n o w n s A , a n d B, E q u a t i n g t h e d e t e r m i n a n t of t h e coefficmnts of t h e s e u n k n o w n s t o zero, t h e r e s u l t i n g e*genvalues of t h e q u a n t i t y blvn a r e t h e r e q m r e d v a l u e s bN1, blv2, , w h e n c e t h e f~ are k n o w n T h e u n k n o w n s A j, B~ m a y all b e e x p r e s s e d in t e r m s of one of t h e s e q u a n t l t m s , w h e n c e t h e c o m p l e t e f u n c t i o n s flv~ a r e k n o w n a p a r t f r o m one u n k n o w n coefficmnt for e a c h f u n c t i o n T h e s e u n k n o w n s are t h e n d e r , v e d f r o m e q u a t , o n (7) T h e r a d i a l p r o b l e m m s o l v e d m a precisely s u n d a r m a n n e r , s t a r t i n g f r o m e q u a t i o n (29) m place of e q u a t i o n (3) T h e a b o v e t r e a t m e n t a p p h e s , p r o w d e d t h e / c ~ , c~ are r e p l a c e d b y kr~, c~ r e s p e c t i v e l y , a n d ( 2 N - 1) 7 r / 2 L is e v e r y w h e r e r e p l a c e d b y c~v/r o