A Xi Symmetric Fundamental Solutions

8/14/2019 A Xi Symmetric Fundamental Solutions

1/7

h t t p : / / w w w , z j u . e d u . c n / j z u s ; E - m a i l : j z u s @ z j u . e d u . c nISSN 1009 - 3095 Journal of Zhejiang U niversity SCIENC E V . 4 , N o . 4 , P . 3 9 3 - 3 9 9 , J u l y - A u g . , 2 0 0 3

Axisymmet r ic fundamenta l so lu t ionsfor a f in i te layer with impeded boundar ies"

CHENG Ze-hai (~ -~ )* , CHEN Yu n- mi n( ~; f/ ~)LING Dao -sh eng (~J ~) , TANG Xia o-w u(N ~)

( College of Civil Engineering an d Architecture, Zhejian g University, Hangzhou 31 00 27 , China )*E-mail : czhl 002 @ 163. com

Received Sept. 12,2002 ; revision accepted Dec. 2,2002A b s t r a c t : Axisymmetric fundamental solutions that are applied in the consolidation calculations of a finiteclay layer with impeded boundaries were d erive d. Laplace and Hankel integral transforms were utilized withrespect to time and radial coord inates , respectively in the analysis. The derivation of fundamental solutionsconsiders two boundary-va lue problems involving unit point loading and ring loading in the vertica l. The solut-ions are extend ed to circular distributed and strip distributed normal load. The computation and analysis of set-tlements, vertical total stress and excess pore pressure in the consolidation layer subject to circular loading arepresented.Key words : Consolidation, Integral transform, Finite layer, Impeded boundariesD o c u m e n t c o d e : A CL C nu mb er : TB121

INTRODUCTIONBlot' s consolidation theory was proposed in

1941 from continuous medium fundamentalequations. The theory taking into acco unt thecoupling between the solid strains and pore pres-sure dissipation is called "coupled consolidationtheory". The solution of governing equations of aporous solid is more complicated than those in-volving ideal elastic solids . McNamee et al.(196 0a) presen ted a solution for plane and axi-ally symmetric problems in terms of two displace-ment functions. The general solution for dis-placement functions is obtained through the ap-plication of Laplace and Fourier integral trans-forms for plane problems and Laplace and Han-kel integral transforms for axisymmetric prob-lems. Gibson et al. (1970) obtained a solutionfor plane strain and axially symmetric consolida-tion of a clay layer on a smooth impervious base.Puswewala et al. (1988) presented axisymmet-tic fundamental solutions for a completely satu-rated porous elastic solid. Gu et al. (1992) pr-esented a solution for an axisymmetric verticalloaded multi-layer base. Huang et al. (1996 )

also obtained an analytical solution for finitelayer through expanding the field of solution.

All the above investigations assumed that theconsolidation layer surface is fully permeable orimpermeable, but that the surface might be af-fected by geotechnical engineering impedimentssuch as sand cushion in preloading, replacementlayer of embankment impeding permeation of theconsolidation layer . The impeded layer can besimplified impeded boundaries of the sub-consol-idation layer. Xie (19 96) studied one dimen-sional consolidation of layered soils with impededbounda ries . However results for two-dimensionalproblem with impeded boundaries have not beenreported.This paper deals with the derivation of axi-symmetric fundamental solutions that are appliedin the consolidation calculations of a finite layerwith impeded boun darie s. The derivation of fun-damental solutions considers two boundary-valueproblems involving unit point loading and ringloading in the vertical direction. The solutionsare extended to circular loading and strip load-ing. The calculation and analysis of settlements,vertical total stress and excess pore pressure in

* Project(No. 500117) supportedby the NationaI NaturaI Science Foundation of Zhejiang Province


2/7

394 CHENG Zehai , CHEN Yunmin e t a l .

t h e c o n s o l i d a t i o n l a y e r s u b j e c t t o c i r c u l a r l o a d i n ga r e p r e s e n t e d .G O V E R N I NG E Q U A T I O N S A N D G E N E R A L SO -LUTION

F o l l o w i n g B i o t ( 1 9 4 1 ) , t h e d i s p l a c e m e n te q u a t i o n s f o r a x i s y m m e t r i c d e f o r m a t i o n s o f ac o m p l e t e l y s a t u r at e d i s o t ro p i e a n d h o m o g e n e o u sp o r o e l a s t i e s o l i d i n a c y l i n d r i c a l c o o r d i n a t e s y s -t e m a r e e x p r e s s e d i n t h e f o l l o w i n g :

03e 1 03pf( V 2 - ) U r + ( 2 r / -- 1 ) Y r r + G 03rr

0 3 eV 2 U r "at+ ( 2 r / - 1 ) ~z z + 1 03pf 0G 03z -

- - 01 )

0 3 ec V 2 e -- 0 3tW h e r e

032 1 03 0 2V 2 - t- - - - - - t-- 0 3 r 2 r 0 3 r 0 3 z 2

1 [ ( ) 03u~1O-~-_rUr_ + re = -7- 03r -ff-z-zlr/ = ( 1 - v ) / ( 1 - 2 v )c = 2 G r l k

I n t h e a b o v e e q u a t i o n s , U r ( r , z , t ) a n d u +( r , z , t ) d e n o t e d i s p l a e e m e n t s i n r a n d z d i r e c -t i o n s , r e s p e c t i v e l y ; p f ( r , z , t ) i s t h e e x e e s sp o r e p r e s s u r e ; G a n d v d e n o t e s h e a r m o d u l u sa n d P o i s s o n ' s r a t io o f t h e b u l k m a t e r i a l ; k i s t h ee o e f f ie i e n t o f p e r m e a b i l i t y o f t h e m e d i u m ; e i st h e d i l a t a t i o n o f t h e b u l k m a t e r i a l .

T h e s t r e s s - s t r a i n r e l a t i o n s a r e e x p r e s s e d a sCY r

2 G = e r r + ( q - - 1 ) e + -P f

2 GPs

2 GPi

2 G

O'002 G = e00 + ( r / - 1 ) e + - -~ zz

2 G = e = + ( r l - 1 ) e + - -O'zr2 G - e +, ( 7 )

W h e r e a r r ( r , z , t ) , a o o ( r , z , t ) , a = ( r , z ,t ) a n d az r ( r , z , t ) a r e n o n - z e r o s t r e s s c o m p o -n e n t s o f th e b u l k m a t e r i a l ; e r r ( r , z , t ) , e o o ( r ,

z , t ) , e = ( r , z , t ) a n d e z r ( r , z , t ) a r e b u l ks t r a i n c o m p o n e n t s .

F o l l o w i n g M c N a m e e e t a l . ( 1 9 6 0 b ) , d i s -p l a c e m e n t s U r ( r , Z , t ) a n d u , ( r , z , t ) a n d p o r ep r e s s u r e p f ( r , z , t ) a r e e x p r e s s e d i n t e r m s o ft w o f u n c t i o n s r a n d ~b i n t h e f o l l o w i n g f o r m :

a~ 034 ( 8 )Ur -- 03r + z 03r03r 034Uz - 03r + Z Orr - r ( 9 )

p f = 2 G ( 03q~ V 2 ~ b ) ( 1 0 )- a T - ~

I t i s c o n v e n i e n t t o d e d i m e n s i o n a l i z e a l l q u a n -t i t ie s w i t h r e s p e c t t o l e n g t h a n d t i m e b y s e l e c t i n ga c e r t a i n l e n g t h " R " a s u n i t y , a n d " R 2 / c ' ' a s a2 ) u n i t o f t i m e , r e s p e c t i v e l y . T h e s u b s t i tu t i o n o fE q s . ( 8 ) - ( 1 0 ) i n E q s . ( 1 ) - ( 3 ) y i e l d t h e

3 ) f o l l o w i n g g o v e r n i n g e q u a t i o n s f o r f u n c t i o n s 9 a n dso :

V 4 @ = V 2 0 3 ~ ( 1 1 )03 tV 2 ~ = 0 ( 1 2 )

L a p l a c e a n d H a n k e l t r a n s f o r m s o f E q s . ( 1 1 )a n d ( 1 2 ) r e s u l t in t h e fo l l o w i n g t w o o r d i n a r y d i f -f e r e n t i a l e q u a t i o n s . D e f i n e ~ hl a n d ~ bh l a s z e r o -o r d e r H a n k e l t r a n sf o r m a n d L a p l a c e t r a n s fo r m o ff u n e t i o n s ~ a n d @ , r e s p e c t i v e l y .

032 - - ~2 0 3 2

( 0 @ 2 - ~ 2 ) ~ b h l= 0 ( 1 4 )w h e r e p i s L a p l a c e t r a n s fo r m p a r a m e t e r ; ~ i sH a n k e l t r a n sf o r m p a r a m e t e r ; u = ~ / ~ e + P .

( 4 ) S o l u t i o n o f t h e a b o v e t w o e q u a t i o n s a r e@hl = A ( ~ , p ) e -z~ + B ( ~ , p ) e -~7 +

( 5 ) C ( ( , p ) e ~ + D ( ~ , p ) e *r ( 1 5 )9~hl = E ( ~ ) e -"~ + F ( ~ ) e ++~ ( 1 6 )

( 6 ) B a s e d o n E q s . ( 4 ) - ( 1 6 ) , g e n e r a l s o l u ti o n s f o rd i s p l a c e m e n t s , e x c e s s p o re p r e s s u r e a n d s t re s s e sf o r a x i s y m m e t r i c d e f o r m a t i o n s o f a c o m p l e t e l ys a t u r a t e d p o r o u s e l a s t i c s o l i d c a n b e w r i t t e n a sf o l l o w s :

g+ = f + ~ J o ( r ( ) E ~'A e -+'g + ?+,B e ++? _ ~'CeZ ~ _J o


3/7

Axisym me tr ic fundamental solutions for a f in i te layer with impeded boundar ies 395$ D e ~7 - ( z ~ + 1 ) E e - ~ + ( z ~ - 1 ) F e Z ~ d ~

( 1 7 ).I(7 ~ 2 J l ( r ~ ) [ A e - Z r + B e - Z 7 + C eZ ; +=

D e ~ - z E e - ~ - z F e Z ; ] d ~ ( 1 8 )p f = 2 G ~ J o ( r ~ ) [ r l p B e - Z Z + r lp D e~ 7 +0~ E e - ~ - ~ F e Z ~ ] d ~ ( 1 9 )f o= = 2 G ~ J 0 ( r ~ ) [ - ~ Ae-Z ~ _ ~ B e - * r _0~ C e z~ - ~ D e ~7 + ( z ~ + 1 ) E e - z~ + ( z ~ -1 ) F e " ; ~ d ~ ( 2 0 )

Xoo

~ z r = 2 G ~ 2 J l ( r ~ ) [ -0~A e - ~r _ y B e - z7 +

~ C e zr + y D e ~7 + z ~ E e - ~ - z ~ F e ~ ] d ~ ( 2 1 )w h e r e th e s u p e r p o s e d b a r d e n o t e s t h e L a p l a c et r a n s fo r m o f t h e r e l e v a n t q u a n t i t y .

C o m b i n i n g t h e a b o v e g e n e r a l s o l u t i o n s w i t hd i f f er e n t b o u n d a r y c o n d i t i o n s , f u n d a m e n t a l s o l u-t i o n s c a n b e o b t a i n e d .F U N D A M E N T A L S O L U T I O N S

1 . F u n d a m e n t a l s o l u t i o n s su b j e c t t o p o i n t lo a d i n g

H

F i g . 1

P0 Impeded layerk~ F0Consolidation layer

k2,c,G,v

ZA x i s y n n n e t r i c p o i n t l o a d i n g

I n F i g . 1 , H d e n o t e s th e t h i c k n e s s o f t h e i m -p e d e d l a y e r ; h i s t h e t h i c k n e s s o f t h e c o n s o l i d a -t i o n l a y e r ; k l d e n o t e s t h e c o e f f i c i e n t o f p e r m e -a b i l i ty o f t h e i m p e d e d l a y e r ; k 2 i s t h e c o e f f i c i e n to f p e r m e a b i l i t y o f t h e c o n s o l i d a t i o n . I t is a s -s u m e d t h a t p e r m e a t i o n o n l y t a k e s p l a c e v e r t i c a l lyi n t h e i m p e d e d l a y e r n o t c o n s i d e r i n g t h e v e r t i c a ld e f o r m a t i o n o f t h e i m p e d e d l a y e r a n d s h e a r i n gs t re s s o v e r t h e p l a n e ( z = 0 ) . F o l l o w i n g T e r z a -g h i o n e d i m e n s i o n a l c o n s o l i d a ti o n t h e o r y , t h eb o u n d a r y c o n d i t i o n s f o r t h e s u r f a c e o f th e c o n s o l -

i d a t i o n l a y e r a r e d e r i v e d i n t h e f o l lo w i n g f o r m s .k2 3~-z k l ~Pf ( z = O ) ( 2 2 )~ r z ---- 0 ( Z = O ) ( 2 3 )a = = p o a ( r ) / ( 2 = r ) ( z = O ) ( 2 4 )

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

( z=h )( z=h )( z = h ) ( 2 7 )

B y L a p l a c e t r a n s f o r m a n d t h e z e r o o r d e r H a n k e lt r a n s f o r m , t h e b o u n d a r y c o n d i t i o n s a r e as f o l -l o w s :

P f = 0 ) ( 2 8 )p f k zk 2 ~ - z = 1 H -~rr = 0 a = 0 ) ( 2 9 )~zz = p o / ( 2 7 v p ) z = 0 ) ( 3 0 )a rz = 0 z = h ) ( 3 1 )3 p f _ 0 z = h ) ( 3 2 )O zu = = 0 z = h ) ( 3 3 )

B a s e o n E q s . ( 1 7 ) - ( 3 3 ) , t h e f u n d a m e n t a l s o -l u t i o n s i n t h e L a p l a c e d o m a i n c a n b e o b t a i n e d i nt h e f o l l o w i n g f o r m .

( 1 ) D i s p l a c e m e n t in t h e r a d i a l d i r e c t i o n :- P 0u r = - 1 6 G p ~. I : ~ J l ( r ~ )e - Z ( : ~ + g ) -h ( ~ ' + 2 r ( ~ e * r 2 e 2 h r +a 1 + e4hga2) (e 2z7 + e 2hz ) _ eZT+2h~ u + ~3/~1) 4-e ZT+2 (h + z )g ( 7 a 2 - - a 3 P l ) + e Z T + 2 h ( 7 + .~) ( yOL1 4-0 1 4 / ~ 1 ) 4 " e 2 h ( 7 + ~ ' ) + z ( 7 + 2 ~ ) ( - - y o~ 2 4 " 0 ~ 4 / ~ 1 ) 4 "e 2 h T + zy +4h '~ ( _ ~OL2 4" p Z ~ 4 ) 4" e ZT+4 h ~( 70~2 - -P Z T " ~ 3 ) - - e Z ( 7 + 2 g ) ( ~ ,0 ~ 1 4 " p z ~ c ~ 3 ) 4 " e 2 h T + zT + 2zg ,( 7 a 1 4 " p z ~ t 4 ) ) / ( c o s h ( h u ( - H k2 ~3 +2 h k l p ~ ? + H k 2 ~ 3 c o s h ( 2 h ~ ) + k l ( ~2 + p ~ ? ) .s i n h ( 2 h ~ ) ) + y s i n h ( h u ( - 2 k1 ~ c o s h ( h ~ ) 2 +H k 2 ( 2 h p ~ ? + ( - ~2 + p r ] ) s i n h ( 2 h ~ ) ) ) ) d ~

( 3 4 )2 ) D i s p l a c e m e n t i n t h e v e rt i c a l d i r e c t i o n :

Orz = 0 (2 5 )O p f _ 0 ( 2 6 )3 zLt z = 0


4/7

396 CHENG Zehai , CHEN Yunmin e t a l .

P O I = ~( r + ~)uz - 4G p~ Jo Jo ( r ~ ) e - ( 7 ~ e ~ ( e 2 * r -e Z h 7 ) ( 2 H k 2 ( e 2 h ( ( _ a l _ o t 2 e g h g ) + eZ7 +2h~o( - 7 " ~ + p ~ 3 - a 3 /3 ~ ~ ) + e ~ + 2 h ~ r + ~ ( ) ' ( ~ l -p r ia 4 + a 4 f l l ( ) - e zz+2(h+~)~ ( ) ' ( a2 - pr /a3 -a 3 / 9 1 ~ ) + e 2 h ( 7 + r ) + ' ( 1 + 2 ~ ) ( " ) " ~ a 2 - - p r ] a 4 - -a 4 p 1 g ) -I- e " ( r + 2 r ( ) ' g a 1 + p a 3 ( - - 1 + z g ) r ] ) +e 2 h ) ' + " ) ' + 2 z r - - u - - p a 4 ( - - 1 + Z ~ ) r ] ) +e ~ 7 +4 h ~ ( u 2 - - p a 3 ( 1 + z ~ ) r ] ) + e 2 h 7+ z T + 4 h ' ~ o( - u + p a 4 ( 1 + z ~ ) r l ) ) / ( H k 2 ( ( 1 + e 2 h T ) "( - 1 + e 2hg )2 ~3 + 4 e h ( 7 + 2 ~ ) 7 s i n h ( h u ( 2 h p ~ r ] +( _ ~2 + p r / ) s i n h ( 2 h ~ ) ) ) + k l ( ( 1 - e 2 h T ) ( 1 +e 2 h g ) 2 ) '~ + 4 e h ( 7 + 2 g) c o s h ( h ) ' ) ( 2 h p ~ r I + ( ~2 +p q ) s i n h ( 2 h ~ ) ) ) ) d ~ ( 3 5 )3 ) V e r t i c a l t o t al c o m p r e s s i v e s t r e s s :- P o I = ~(7+ ~)a = - 2 p T r Jo ~ J o ( r ~ ) e - ( ~ 2 e Z r ( e 2 Z l +e 2 h7 ) ( - - 2 H k 2 ( e 2 h ' ~ + a 1 + a 2 e 4 hg ) - e Z T + 2 h ~ .( ~ /~ 0 ~ 1 - - p r l o t 3 + O t 3 ~ 1 ~ ) - I - e ~ 7 + 2 h ( 7 + g ) ( u 1 - -P W 4 + a 4 / ~ l ~ ) + e Z T + 2 ( h + z ) g ( " )' ~~ - - P W 3 - -a 3 / ~ 1 ~ ) + e 2 h ( 7 + ~ ) + ~ (7 + 2 ~ ) ( _ y ~ z 2 + p r] o ~4 +a 4/ 31 g ) - e = ( r + i r + p a 3 ( - 1 + z ' ~ ) r ) +e 2 h T + z T + 2 z ' ~ (y~0r 1 + p a 4 ( - - 1 + z ' ~ ) : 7 ) + e " 7 + 4 h ~ .( u - p a 3 ( 1 + z ~ ) r / ) + e Zhr+ ~r +4hg( - - u +p e r 4 ( 1 + z ~ ) r ] ) ) / ( H k 2 ( ( 1 + e 2 h r ) ( - 1 +eZhr ) 2 ~ 3 + 4 e h (~ + 2 r y s i n h ( h u ( 2 h p ~ r ] +( _ ~ 2 + p 7 ) s i n h ( 2 h ~ ) ) ) + k l ( ( 1 - e Z h r ) ( 1 +e 2 h ~ )2 u + 4 e h ( Z + Z ~ ) c o s h ( h u ( 2 h p ~ r ] + ( ~ 2 +p r ]) s i n h ( 2 h g ) ) ) ) d ~ ( 3 6 )4 ) S h e a r i n g s t r e s s :

a = - 8 p ~ J o(u 2~r - e2h ~) ( - - 2H k2 ~ 'e 2 t4" + a l + a2 e4 ~ ' ) +e ~ + a h ~ ( 7 a l + a 3 f l l ) + e ~ + 2 ( ~ + " ) ~ ( ) ' a z - a 3 / ~ 1 ) -e Z 7 + 2h( 7 + [) . ( 7 0 1 -t- a 4/~1 ) -I- e TM7 + ~) + z(7 + 2~) .( - ~ + ~ 4 / ~ 1 ) + e ~ + z ~ - p ~ 4 ~ ) -e zl+4h'~ ( ") , 'a2 -- p z r i a 3 ) - e z (7+ 2g) ( y a ~ + p z r l a 3 ) +e2hT+ z7 + 2zg ( rot1 + p z T . ] ~ 4 ) ) / ( c o s h ( h u( - H k 2 ~3 + 2 h k l p ~ r ] + H k 2 ~ 3 c o s h ( 2 h ~ ) +k l ( ~2 + p r ] ) s i n h ( 2 h ~ ) ) + y s i n h ( h u( - 2 k l ~ c o s h ( h ~ ) 2 + H k 2 ( 2 h p ~ r ] +( _ ~2 + p r / ) s i n h ( 2 h ~ ) ) ) ) d ~

( 3 7 )

5 ) E x c e s s p o r e - w a t e r p r e s s u r e :P o [ = e - . ,( 7+ ~ ' )- h(} '+2~ ')P U - 8 7 rJ o r / ~ J~ ( r ~ ) ( - 1 +

e Z h ~ ) ( o t 3 e Z T ( e 2z~ + e 2 h ~ ) _ o t 4 e * Y + Z h T ( e 2 z ~ +e 2h~) _ a l e Z~( e 2Z~ + e z hT) + a 2 e Z ~ + Z h ~ ( e 2 ~z +e 2 h r ) ) / ( c o s h ( h u ( - - H k 2 ~3 + 2 h k l p ~ r ] +H k 2 ~ 3 c o s h ( 2 h ~ ) + k l ( ~ : + p r / ) s i n h ( 2 h ~ ) ) +7 s i n h ( h u ( - 2 k l ~ c o s h ( h ~ ) 2 +H k e ( 2 h p ~ r / + ( _ ~2 + p r / ) s i n h ( 2 h ~ ) ) ) ) d ~

( 3 8 )w h e r e

a 1 = - k I + H k 2 (a 2 = k l + H k 2a 3 = - k l + H k 2 ua 4 = k 1 + H k 2 uf ll = P r l ( 2 h - z )

2 . F u n d a m e n t a l s o l u t io n s u b j e ct to r in g lo a d i n gT h e s y s t e m u n d e r c o n s i d e r a t i o n i s s h o w n i n

F i g . 2 . A p p l i c a t i o n o f L a p l a c e t r a n s fo r m a n dH a n k e l t ra n s f o rm y i e l d s t h e f o ll o w i n g b o u n d a r yc o n d i t i o n s o v e r th e p l a n e ( z = 0 ) :

Hf I m p e d e d l a y e r

k,r

C o n s o l i d a t i o n l a y e rk 2 , c , G , v

F i g . 2 A x i s y m m e t r i c ring loading

k 2 O p f k l p f ( 3 9 )3 z - H~r z ---- 0 ( 4 0 )

a = = R J I ( R ~ ) ( 4 1 )O n l y c h a n g e P o / 2 7 r i n t h e fu n d a m e n t a l s o l u -

t i o n f o r p o i n t l o a d i n t o R J o ( R ~ ) , a n d t h e f u n -d a m e n t a l s o l u ti o n f o r r in g l o a d c a n b e o b t a i n e d .


5/7

Axisym metr ic fundamental solutions for a f in i te layer with impeded boundar ies 397

S O L U TI O N S F O R O T H E R D I S T R IB U T E D L O A D

1 . S o l u t i o n s f o r c i r c u l a r l o a d i n gT h e s y s t e m u n d e r c o n s i d e r a t i o n i s sh o w n i n

F i g . 3 . U s e o f L a p l a c e t r a n s f o r m a n d H a n k e lt r a n s f o r m y i e l d s t h e f o ll o w i n g b o u n d a r y c o n d i -t i o n s o v e r ( z = O ) :

Impeded layerb b y, r 1 6 2q ............. V V V V x

0 -\ \

Consolidation layer _ 9M(x,z)Vz

F i g . 4 S t r i p l o a d i n g

1 q Impeded layerH ~ ' k~V V v V v v d_ R O R Consolidation layerh i

kvc, G,v

I ZF i g . 3 A x i y m m e t r i c c i r c u l a r l o a d i n g

k 2 ~ P f k l P f3z - -Hr rz = 0

( 4 2 )( 4 3 )

a= = qR JI ( R[ ) / (p~ ) ( 4 4 )0 n l y c h a n g e Po/2~r i n t h e f u n d a m e n t a l s o l u-

t i o n f o r p o i n t l o a d i n g i n t o qRJ1 (R ~)I ~ , a n d t h ef u n d a m e n t a l s o l u t i o n fo r r i n g lo a d i n g c a n b e o b -t a i n e d .2 . S o l u t i o n s fo r s t r ip l o a d i n g

T h e s y s t e m s u b j e c t t o s t r ip l o a d i n g i s s h o w ni n F i g . 4 . B a s e d o n th e f u n d a m e n t a l s o l u t io n s f o rp o i n t l o a d i n g , t h e i n fi n i te i n t e g r a t i o n i n t h e yd i r e c t i o n a n d f i n i te i n t e g r a t i o n i n t h e x d i r e c t i o no f t h e f u n d a m e n t a l s o l u t io n s y i e l d s t h e s o l u t io n o fp o i n t M i n t h e m e d i u m f o r s t r ip l o a d i n g .

N U M E R I C A L C O M P U T A T I O N A N D A N A L Y S I S( C I R C U L A R L O A D I N G )

A l l t h e a b o v e s o l u ti o n s s h o u l d b e e v a l u a t e dn u m e r i c a l l y . I n t h e n u m e r i c a l q u a d r a t u r es c h e m e , a l l i n f in i t e i n t e g r a l s w i t h r e s p e c t t o t h eH a n k e l t r a n s f o r m p a r a m e t e r ~ a r e e v a l u a t e d u s -i n g t h e S i m p s o n r u le ; a n d t i m e d o m a i n s o l u t io n sa r e c o m p u t e d u s i n g th e a p p r o x i m a t e L a p l a c e i n -v e r s e fo r m u l a s u g g es t ed b y S c h a p e r y ( 1 9 6 2 ) .1 . T h e s e t t l e m e n t a t t h e c e n t e r o f t h e l o a d e d a r e a

A s s h o w n i n F i g . 3 , f o r kl /k2 = 1 , R = h= l m , q = 1 N / m 2 , v = 0 , H f r o m 0 m t o 5 m ,

t h e e ff e c t o f t h e i m p e d e d l a y e r t h i c k n e s s o n t h ep r o g r e s s o f s e t t l e m e n t a t t h e c e n t e r o f t h e l o a d e da r e a i s s h o w n i n F i g . 5 . T h e r e s u l ts h a v e b e e np l o t t e d i n t h e f o r m o f c u r v e s o f 2 G u = / q R a g a i n s tc t /R 2 . I t i s a p p a r e n t t h a t t h e i m p e d e d l a y e rt h i c k n e s s h a s a n i m p o r t a n t i n f l u e n c e . W h e n t h ei m p e d e d l a y e r t h i c k n e s s i s z e r o , t h e b o u n d a r y i sf u l l y p e r m e a b l e .2 . T h e t o p s u r f a c e s e t t l e m e n t o f c o n s o l i d a t i o n l a y e r

F o r k l / k 2 = 1 , R = h = l m , H = l m , q =1 N / m 2 , v = O , w h e n t h e i m p e d e d l a y e r t h i c k n e s si s l m , t h e to p s u r f a c e s e t t l e m e n t o f t h e c o n s o l i -

-0 .5-0 .6

a


6/7

3 9 8 C H E N G Z e h a i , C H E N Y u n m in e t a l .

d a t i o n l a y e r d e v e l o p s w i th t i m e a s s h o w n i n F i g .6 . T h e r e a r e s o m e s li g ht u p h e a v a l s b e y o n d t h el o a d e d a r e a .

- 0 . 2 r0

X // . :000l0 . 4 ~ , # / ~ 1 [ m T :O .O 1o . 81 _ / / , T : I1 ~ ! [ x T = 1 0] 9 T : 1 0 01.2 ~ Radial (r/R)

F i g . 6 T h e d e v e l o p m e n t w i t h t i m e o f t h e t o p s u r fa c es e t t l e m e n t o f t h e c o n s o l i d a t i o n l a y e r

3 . T o t a l c o m p r e s s i v e s t r es s at 0 . 5 m u n d e r t h e c e n te ro f t h e l o a d e d a r e a

F o r k l / k 2 = 1 , R = h = l m , q = 1 N / m 2 , v= 0 , t h e c h a n g i n g w i t h t i m e o f v e r t i c a l t o t a lc o m p r e s s i v e s t re s s a t O . 5 m u n d e r t h e c e n t e r o f

t h e l o a d e d a r e a d u r i n g c o n s o l i d a t i o n i s s h o w n i nt h e F i g . 7 . T h e r e s u l t s h a v e b e e n p l o t t e d i n t h ef o r m o f c u r v e s o f a= /q a g a i n s t ct /R 2 . T h et h i c k e r t h e i m p e d e d l a y e r , t h e s m a l l e r t h ec h a n g e r a n g e o f t h e t o t a l s t r e s s . I t i s c l e a r l ys h o w n i n F i g . 8 th a t t h e c h a n g e o f v e r t i c a l t o t a ls t r e s s i s r e l a t e d t o t h e c h a n g e o f e x c e s s p o r e p r e -s s u r e .

1.061.05

.w 1.04~"~ 1.03

1.021.01

@ H = 0| H=O.1@ f ' , , @ H = I. / _ \ @ H = 2

, m ? :0.00001 0.001 0.1 10

0

T=ct /R = z(a) (b)

F i g . 7 ( a ) E f f e c t o f t h e i m p e d e d l a y e r t h i ck n e s s a n d ti m e o n v e r t ic a l t o ta l c o m p r e s s i v e st r e ss ; ( b ) s k e t c hm a p f o r t h e p o s i t i o n o f c a l c u l a t i o n p o i n t

4 . E x c e s s p o r e p r e s s u r eF o r k l / k 2 = 1 , R = h = l m , q = 1 N / m 2 , v

= 0 , t h e c h a n g i n g w i t h t i m e o f e x c e s s p o r e p r e s -s u r e a t O . 5 m u n d e r t h e c e n t e r o f t h e l o a d e d a r e ai s s h o w n i n t h e F i g . 8 . T h e r e s u l t s h a v e b e e n

p l o t t e d i n t h e f o r m o f c u r v e s o f p f / q a g a i n s t c t /R 2 . M a n d e l - C r y e r e f f e c t is o b v i o u s fo r H = 0 .M a n d e l - C r y e r e f f e c t i s w e a k e n e d w i t h th i c k e n i n go f t h e i m p e d e d l a y e r .

0 .50 .4

- ~ 0 . 30 .20 .1

0

@ a = 5| H : 2|H=O.1

0.00001 0.001 0.1 1 10T = ct /R 2

(a)F i g . 8

R

r

0

(b)( a ) E f f e c t o f t h e i m p e d e d l a y e r t h i c k n e ss a n d t i m e o n e x c e ss p o r e p r e s s u r e ; ( b ) s k e t c h m a p f o r th e

p o s i t i o n o f c a l c u l a t i o n p o i n t


7/7

Ax is y m me tr ic fu n d a m e n ta l s o lu tio n s fo r a f in i te la y e r w i th imp e d e d b o u n d a r ie s 3 9 9

CONCLUSIONS

Explici t solut ions for Laplace t ransforms ofd i s p l acem en ts , t r ac t ions , po re p res s u re ware de-r ived for a f in i te layer with impeded boundar iesand s ub jec ted to ax i s ym m et r ic po in t loads andr ing loads . The Lap lace and Han ke l t r ans fo rm sut i l ized in th is paper are ef f ic ient . The solut ionsare ex ten ded to c i r cu la r d i s t r ibu ted and s t rip d i s -t r ibu ted norm al load ings . The r es u l t i s conver -gent and sat is factory .

As a resul t of the calcul at ion and anal ys is , i ti s apparen t tha t the th icknes s o f the im ped edlayer has an im por tan t in f luenc e on s e t t l em en t ,ver t ical to tal s t ress and pore pressure of the con-sol idat ion layer . The ver t i cal to tal s t ress is t ime-depen dan t du r ing cons o l ida t ion . The th icker theim peded layer i s , the s m al le r the chang in g r angeof the ver t ical to tal s t ress and the we aker theMande l -C ryer ef f ec t . The im peded layer de laysthe set t leme nt progress of the co nsol ida t ionlayer .

R e f e r e n c e sBiot, M . A . , 1941. General theory of three-dimensional

consolidation. J. Appl . Phys , 12,155.Gibson, R. E ., Schiffman, R. L. and Pu, S .L ., 1970.

Plane strain and axially symmetric consolidation of aclay layer on a smooth impermeable Base. Q.J. Mech.Appl. Math, 23:505 - 519.

Gu, Y. Z. and Jin , B. , 1992. Bolt consolidation analyticalsolutions for multi-layer base subject to axisymmetricloading. J. Geotechnical Engineering, 20 : 17 - 21.

Huang, C. Z. and Xiao, Y ., 1996. Analytical solutions fortwo dimensional consolidation problems. J. Geotechni-cal Engineering, 18:47 - 54.

McNamee, J. and Gibson, R. E ., 1960a. Plane strain andaxially symmetric problems of the Consolidation of aSemi-infinity Clay Stratum. Q . J. Mech . Appl . Ma th ,13:210 - 227.

McNamee, J. and Gibson, R. E. ~ 1960b. Displacementfunction and linear transforms applied to diffusionThrough Porous Elastic Media. Q. J. Mech. Appl.M a t h . , 13: 89- 111.

Puswewala, U. G. A.a nd Rajapakse, R. K. N. D. ,1988 .Axisymmetric fundamental solutions for a completelysaturated porous elastic solid. Int . J . Engng. Sci ,26(5) :419 - 436.

Schapery, R . A . , 1962. Approximate methods of transforminversion for viscoelastic stress analys is. Proc. 4th U.S.Nat. Cong.on Appl. Mech, p. 1075 .

l i e , K . H . , 1996. One dimensional consolidation analysisof layered soils with impeded boundaries. J. ZhejiangUniversity, 30( 5 ) : 567 - 575 ( in Chinese).

H e l e n Z h a n g , m a n a g i n g e d it o r o f J Z U Sj z u s @ z j u . e d u . c n T e l / F a x 8 6 - 5 71 - 8 7 9 5 2 2 7 6 ~

W e l c o m e v is it in g o u r j o u rn a l w e b s i t e .h t t p : / / w w w . z j u . e d u . c n / j z u s

W e l c o m e c o n t ri b u ti o n s & s u b s c r i p ti o n f r o m a l l o v e r th e w o r ldT h e e d i to r w o u l d w e l c o m e y o u r v i e w o r c o m m e n t s o n a n y i te m in

t h e j o u r n a l , o r r e l a te d m a t t e r sP l e a s e w r i t e t o ;

Documents

A Xi Symmetric Fundamental Solutions